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https://www.earthdoc.org/content/papers/10.3997/2214-4609-pdb.395.IPTC-17292-MS | 1887
### Abstract
From the reservoir to the final consumer sales point, the flow and quality related to energy is estimated, quantified, or in an ideal situation is measured with very low uncertainty. Organization of Petroleum Exporting Countries (OPEC) reference basket averaged $107.52/b in August, and the world oil demand growth was revised in 2013 by 25 tb/d (1), in the longer term, prices may reach$155/b by 2035 (2), additionally, energy consumption has been estimated to increase 54% from 2010 to 2035, fossil fuel will account with 82% (2). Considering these short and long term estimations, the proper management of the uncertainty levels related to measurement systems in the energy sector has an intrinsic financial risk exposure that has to be addressed, and in most of the cases, a clear framework has to be established and led by the reserves owners. Technology has been evolving to manage better levels of uncertainty, but it is an endless journey trying to find the “true value” in order to safeguard the net profits of the shareholders. In the case of National Oil Companies (NOC), this challenge is translated in major strategic paths that are fully linked to the long term vision of an entire nation, and if we consider the nations where the Gross Domestic Product (GDP) is propelled by the Oil and Gas sector, the discrepancies related to product measurement may create a tremendous impact in the domestic economy. In consequence, around the supply chain, where different parties are directly involved in energy custody transfer process, the transparency and clear measurement terms are required for the buying and sales transactions. In many of those cases, royalties payment is involved, this issue must be fully monitored and controlled by the NOCs, but clear terms has to be agreed with the International Oil Companies (IOCs) in order avoid financial losses due to sub-optimal measurement practices. Qatar Petroleum (QP) has embarked in an ambitious task to implement a “Measurement Strategy”, to be applied to all the Joint Ventures and Production Sharing Agreements Operators that are located in the State of Qatar (SoQ). One of the key elements in the Measurement Strategy is the creation of a common framework to operate the measurement systems in the SoQ. The complexity lies to boost this initiative in an environment where different players have already an intrinsic risk related to their own operations in terms of hydrocarbon imbalances due to measurement uncertainties, and every single operator could operates the measurement system with a different asset management strategy. This paper will present the case and the lessons learned from this implementation process in order to deploy an unique framework to manage the measurement systems, where, the approach utilized to manage the diversity from the managerial perspective was based in the Kotter method.
/content/papers/10.3997/2214-4609-pdb.395.IPTC-17292-MS
2014-01-19
2021-04-23 | 2021-04-23 08:39:59 | {"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": 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": 0.37381497025489807, "perplexity": 1331.5441450900723}, "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/1618039568689.89/warc/CC-MAIN-20210423070953-20210423100953-00036.warc.gz"} |
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InfoQ Homepage News Support for Zip Files Still Lacking In .NET 3.0
# Support for Zip Files Still Lacking In .NET 3.0
The ability to use file compression like the venerable ZIP format is very important to many developers. For those developers using.NET, that means dropping to command shell or using a third-party component. With .NET 3.0, there is built-in support for ZIP files, though the implementation is somewhat questionable.
The first step to create a ZipPackage object. One would expect this to work just like any other object.
Dim zipFile As New ZipPackage("C:\Temp\test.zip", FileMode.Create) ZipPackage zipFile = new ZipPackage("C:\\Temp\\test.zip", FileMode.Create);
Unfortunately that isn't the case. Instead, one has to use a factory method in the base class Package.
Dim zipFile As ZipPackage = Package.Open("C:\Temp\test.zip", FileMode.Create) ZipPackage zipFile = (ZipPackage)Package.Open("C:\\Temp\\test.zip", FileMode.Create );
Though ZipPackage is the default return type for Package.Open, there is no way to actually specify that in any of the overloads making it somewhat difficult to create your own implementations.
Moving on, adding files to the zip package in the normal use case is exceedingly painful. One would think the code would look something like:
zipFile.AddFile("C:/temp/someFile.txt", CompressionOption.Maximum) zipFile.AddFile("C://temp//someFile.txt", CompressionOption.Maximum);
To add files the .NET way, one has to:
1. Create a new URI object that will represent the name of the file inside the ZipFile.
2. Determine the correct Mime type for the file.
3. Create a new PackagePart using the aforementioned information.
4. Open the source file as a Stream.
5. Copy said stream into the PackagePart stream.
The below code shows how to do this using a very crude stream copy loop. Note that it should be much faster to use buffers than to read the stream one byte at a time.
Dim newUri As New Uri("/someFile.txt", UriKind.Relative); Dim part1 As ZipPackagePart = zipFile.CreatePart(newUri, _ System.Net.Mime.MediaTypeNames.Text.Plain, CompressionOption.Maximum) Using output As Stream = part.GetStream, input As FileStream = File.OpenRead("C:/temp/someFile.txt") Dim value As Integer = input.ReadByte Do Until value = -1 output.WriteByte(CByte(value)) value = input.ReadByte Loop End Using
While there isn't a simple way to decompress a zip file, it is far less painful than creating the file in the first place. This code lists all of the files in a zip file and dumps the text ones to the screen.
zipFile = CType(ZipPackage.Open("C:\Temp\test.zip", IO.FileMode.Open), ZipPackage) For Each part As ZipPackagePart In zipFile.GetParts Console.WriteLine(part.Uri) Console.WriteLine(vbTab & "Type:" & part.ContentType) Console.WriteLine(vbTab & "Option:" & part.CompressionOption) If part.ContentType.ToLower.Contains ("text/") Then Using output As New StreamReader(part.GetStream) Console.WriteLine(output.ReadToEnd) End Using End If Next
Even when Maximum compression is chosen, the compression rate is very poor compared to that of WinZip. We tested this using a 2KB plain text file containing the readme for an application. WinZip had a 46% compression while .NET 3.0 had only a 4% compression. For WinZip the setting "Maximum (portable)" was used. For .NET 3.0, the above code was used.
Actually that comparison isn't fair, because buried in the docs is this note:
For the default ZipPackage subclass, the CreatePart method only supports two compressionOption values, NotCompressed or Normal compression. Other CompressionOption values of Maximum, Fast, or SuperFast use Normal compression.
One last warning, this method doesn't create standard zip files. While the files can be read by normal tools, the zip files will have an addition file called "[Content_Types].xml". Likewise, .NET 3.0 cannot read zip files unless the file contains "[Content_Types].xml". If the file is missing, it silently fails to find any files.
In conclusion, .NET 3.0's support for the ZIP format is so highly specialized that is is useless in the general case.
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## Community comments
• ##### .NET Zip support
by Rob Eisenberg,
• ##### Re: .NET Zip support
by Jonathan Allen,
• ##### Re: .NET Zip support
by Birger Halfmeier,
• ##### Re: .NET Zip support
by Ted Neward,
• ##### .NET Zip support
Your message is awaiting moderation. Thank you for participating in the discussion.
There is quite a bit of inaccuracy in this post. To begin with, Zip support has been available in .NET since version 1.1, although it was buried in some j# or vb libraries. No one really knew about it. However, .NET 2.0 introduced the System.IO.Compression namespace which contains two implementations: DeflateStream and GZipStream. I believe the 3.0 specific functionality mentioned above (System.IO.Packaging) is specifically related to XPS documents. Perhaps someone can confirm this? As to the quality of any of these algorithms, I am no expert.
• ##### Re: .NET Zip support
Your message is awaiting moderation. Thank you for participating in the discussion.
There is quite a bit of inaccuracy in this post. To begin with, Zip support has been available in .NET since version 1.1, although it was buried in some j# or vb libraries. No one really knew about it.
Can you give me a reference for that?
However, .NET 2.0 introduced the System.IO.Compression namespace which contains two implementations: DeflateStream and GZipStream.
While those can be used for accessing ZIP files, the code needed to do it isn't trivial. Even just getting the file list requires manually parsing the stream to get the header information. There is a sample at MSDN
I believe the 3.0 specific functionality mentioned above (System.IO.Packaging) is specifically related to XPS documents. Perhaps someone can confirm this?
Most of the implementation details for that namespace are in the XPS documents.
Much of the fault with the namespace is that is was described as a general solution in the documentation, and only when you really dig into it do you see that it is just support code for XPS.
• ##### Re: .NET Zip support
Your message is awaiting moderation. Thank you for participating in the discussion.
There is quite a bit of inaccuracy in this post. To begin with, Zip support has been available in .NET since version 1.1, although it was buried in some j# or vb libraries. No one really knew about it.
Can you give me a reference for that?
Have a look at this MSDN Magazine article:
Using the Zip Classes in the J# Class Libraries to Compress Files and Data with C#
• ##### Re: .NET Zip support
by Ted Neward,
Your message is awaiting moderation. Thank you for participating in the discussion.
There is quite a bit of inaccuracy in this post. To begin with, Zip support has been available in .NET since version 1.1, although it was buried in some j# or vb libraries. No one really knew about it.
Can you give me a reference for that?
It's in J#; look at java.util.zip packages. It's a port of the ZIP support introduced in JDK 1.1, and it may not be any better than what you see in System.IO.Compression, but it is there. There was an MSDN article from some years back that demonstrated how to use those libraries from C#, by the way--a quick Google search (which I'm too lazy to run at the moment) should dig it up.
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You will be sent an email to validate the new email address. This pop-up will close itself in a few moments. | 2021-06-23 09:51:04 | {"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": 0.36827516555786133, "perplexity": 4673.775784357205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488536512.90/warc/CC-MAIN-20210623073050-20210623103050-00460.warc.gz"} |
https://xuxiaojian.github.io/ | Nice to meet you here!
I am Xiaojian Xu, a junior but passionate researcher working on computational imaging! I am currently a postdoc student in the EECS department at University of Michigan (UMich), working with Prof. Jeffrey Fessler.
I got my PhD degree in Computer Science from Washington University in St. Louis (WashU) in 2022, where I had the fortune to work with Prof. Ulugbek Kamilov in the Computational Imaging Group (CIG). Before I came to WashU, I got my B.S. degree in Communication and Information Engineering from University of Electronic Science and Technology of China (UESTC) in 2014 and then was recommend as an exam-free student to the graduate school and got my M.S. degree in Communication and Information Systems there in 2017.
My research interests include computational imaging, inverse problems, optimization theory, computer vision and deep learning. Please feel free to contact me if you are interested in our research or would like to have a casual chat!
# Recent news
* 08-22-2022: I started my postdoc research at UMich under the supervision of Prof. Jeffrey Fessler.
* 07-28-2022: I passed my dissertation defense on "Model-based Deep Learning for Computational Imaging".
* 05-24-2021: I started my research intern at Facebook Reality Labs Research (FRL).
* 05-04-2021: I passed my thesis proposal defense on "Model-based Deep Learning for Computational Imaging".
* 04-03-2020: I passed my oral qualification exam on "Computational Imaging: Leverage the Power of Deep Learning".
* 05-27-2019: I started my research intern at Mitsubishi Electric Research Laboratory (MERL).
* 02-01-2018: I joined the Computational Imaging Group (CIG) under the supervision of Prof. Ulugbek Kamilov.
* 08-27-2017: I joined the Computer Science and Engineering (CSE) department at WashU. | 2022-10-02 23:29:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20708173513412476, "perplexity": 2952.8925135272575}, "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-2022-40/segments/1664030337360.41/warc/CC-MAIN-20221002212623-20221003002623-00084.warc.gz"} |
http://repository.aust.edu.ng/xmlui/handle/123456789/3370 | # First-order gradient regularisation methods for image restoration: reconstruction of tomographic images with thin structures and denoising piecewise affine images
Papoutsellis, Evangelos (2016-02-02)
Thesis
The focus of this thesis is variational image restoration techniques that involve novel non-smooth first-order gradient regularisers: Total Variation (TV) regularisation in image and data space for reconstruction of thin structures from PET data and regularisers given by an infimal-convolution of TV and $L^p$ seminorms for denoising images with piecewise affine structures. In the first part of this thesis, we present a novel variational model for PET reconstruction. During a PET scan, we encounter two different spaces: the sinogram space that consists of all the PET data collected from the detectors and the image space where the reconstruction of the unknown density is finally obtained. Unlike most of the state of the art reconstruction methods in which an appropriate regulariser is designed in the image space only, we introduce a new variational method incorporating regularisation in image and sinogram space. In particular, the corresponding minimisation problem is formed by a total variational regularisation on both the sinogram and the image and with a suitable weighted $L^2$ fidelity term, which serves as an approximation to the Poisson noise model for PET. We establish the well-posedness of this new model for functions of Bounded Variation (BV) and perform an error analysis through the notion of the Bregman distance. We examine analytically how TV regularisation on the sinogram affects the reconstructed image especially the boundaries of objects in the image. This analysis motivates the use of a combined regularisation principally for reconstructing images with thin structures. In the second part of this thesis we propose a first-order regulariser that is a combination of the total variation and $L^p$ seminorms with $1 < p \le \infty$. A well-posedness analysis is presented and a detailed study of the one dimensional model is performed by computing exact solutions for simple functions such as the step function and a piecewise affine function, for the regulariser with $p = 2$ and $p = 1$. We derive necessary and sufficient conditions for a pair in $BV \times L^p$ to be a solution for our proposed model and determine the structure of solutions dependent on the value of $p$. In the case $p = 2$, we show that the regulariser is equivalent to the Huber-type variant of total variation regularisation. Moreover, there is a certain class of one dimensional data functions for which the regularised solutions are equivalent to high-order regularisers such as the state of the art total generalised variation (TGV) model. The key assets of our regulariser are the elimination of the staircasing effect - a well-known disadvantage of total variation regularisation - the capability of obtaining piecewise affine structures for $p = 1$ and qualitatively comparable results to TGV. In addition, our first-order $TVL^p$ regulariser is capable of preserving spike-like structures that TGV is forced to smooth. The numerical solution of the proposed first-order model is in general computationally more efficient compared to high-order approaches. | 2018-12-14 04:30:12 | {"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": 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": 0.5313277244567871, "perplexity": 418.12843332331107}, "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-2018-51/segments/1544376825349.51/warc/CC-MAIN-20181214022947-20181214044447-00094.warc.gz"} |
https://chemistry.stackexchange.com/questions/103512/understanding-the-partial-derivative-of-temperature-with-respect-to-thermodynami | # Understanding the partial derivative of temperature with respect to thermodynamic beta (coldness)
I am trying to prove that the specific heat is related to the fluctuations in the energy:
$$c_V = \frac{\langle E^2 \rangle - \langle E \rangle^2}{k_\mathrm BT^2}$$
Where:
$$\beta = \frac{1}{k_\mathrm BT}$$
$$\langle E \rangle = -\frac{ \partial \log(Z)}{\partial \beta}$$
I did:
$$\frac{\partial^2}{\partial \beta^2}\ln Z = \frac{\partial}{\partial \beta}\frac{1}{Z}\frac{\partial Z}{\partial \beta} = -\frac{\partial \langle E \rangle}{\partial \beta} = -\frac{\partial \langle E \rangle}{\partial T} \frac{\partial T}{\partial \beta}$$
My issue is that I do not understand why:
$$-\frac{\partial T}{\partial \beta} = k_\mathrm BT^2$$
You know that:
$$\beta = \frac{1}{k_\mathrm BT}$$
Rearranging yields: $$\color{red}{\beta} = \frac{1}{k_\mathrm B\color{blue}{T}}\implies \color{blue}{T} = \frac{1}{k_\mathrm B \color{red}{\beta}}$$
Thus:
$$\frac{\partial \color{blue}{T}}{\partial\beta} = \frac{\partial}{\partial\beta}\left(\frac{1}{k_\mathrm B \beta}\right) = \color{red}{-}\frac{1}{k_\mathrm B \beta^\color{red}{2}}$$
Substituting $$\beta = \color{red}{\frac{1}{k_\mathrm BT}}$$ yields:
$$\frac{\partial T}{\partial\beta} = -\frac{1}{k_\mathrm B \color{red}{\left(\frac{1}{k_\mathrm BT}\right)}^2} = -\frac{k_\mathrm B^2T^2}{k_\mathrm B} = -k_\mathrm BT^2$$
$$-\frac{\partial T}{\partial\beta} = k_\mathrm BT^2$$ Q.E.D. | 2021-09-28 11:20:41 | {"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": 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": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6698745489120483, "perplexity": 1049.7034468294844}, "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-20210928092646-20210928122646-00581.warc.gz"} |
https://dmtcs.episciences.org/604 | ## Shonda Gosselin ; Andrzej Szymański ; Adam Pawel Wojda - Cyclic partitions of complete nonuniform hypergraphs and complete multipartite hypergraphs
dmtcs:604 - Discrete Mathematics & Theoretical Computer Science, August 24, 2013, Vol. 15 no. 2 - https://doi.org/10.46298/dmtcs.604
Cyclic partitions of complete nonuniform hypergraphs and complete multipartite hypergraphs
Authors: Shonda Gosselin ; Andrzej Szymański ; Adam Pawel Wojda
A \em cyclic q-partition of a hypergraph (V,E) is a partition of the edge set E of the form \F,F^θ,F^θ², \ldots, F^θ^q-1\ for some permutation θ of the vertex set V. Let Vₙ = \ 1,2,\ldots,n\. For a positive integer k, Vₙ\choose k denotes the set of all k-subsets of Vₙ. For a nonempty subset K of V_n-1, we let \mathcalKₙ^(K) denote the hypergraph ≤ft(Vₙ, \bigcup_k∈ K Vₙ\choose k\right). In this paper, we find a necessary and sufficient condition on n, q and k for the existence of a cyclic q-partition of \mathcalKₙ^(V_k). In particular, we prove that if p is prime then there is a cyclic p^α-partition of \mathcalK^(Vₖ)ₙ if and only if p^α + β divides n, where β = \lfloor \logₚ k\rfloor. As an application of this result, we obtain two sufficient conditions on n₁,n₂,\ldots,n_t, k, α and a prime p for the existence of a cyclic p^α-partition of the complete t-partite k-uniform hypergraph \mathcal K^(k)_n₁,n₂,\ldots,n_t.
Volume: Vol. 15 no. 2
Section: Combinatorics
Published on: August 24, 2013
Submitted on: November 26, 2010
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM] | 2021-09-19 04:10:38 | {"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": 0.8540442585945129, "perplexity": 3529.4571969909616}, "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/1631780056711.62/warc/CC-MAIN-20210919035453-20210919065453-00683.warc.gz"} |
https://mathsgee.com/28933/what-is-the-conjugate-of-a-complex-number | MathsGee is Zero-Rated (You do not need data to access) on: Telkom |Dimension Data | Rain | MWEB
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What is the conjugate of a complex number?
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An important idea in complex numbers is the conjugate of a complex number. This is simply the opposite of a complex number and is rather easy to figure out. Lets take our previous complex number, $z$. We already know what this is $(z=a+b i)$, so the conjugate of it would be:
$\bar{z}=a-b i$
We simply change the sign in front of the imaginary part to get our conjugate for $z$. Pretty simple. We the conjugate of $z$, is shown with a special symbol, $\bar{z}$. Its the variable for our complex number with a bar over it. This is the conjugate
by Bronze Status (8,688 points)
0 like 0 dislike | 2021-06-17 23:58:29 | {"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": 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": 0.7444009184837341, "perplexity": 845.6622639592217}, "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/1623487634576.73/warc/CC-MAIN-20210617222646-20210618012646-00561.warc.gz"} |
https://www.physicsforums.com/threads/delta-function-question.707910/ | # Delta function question
in this equation why does k take on both positive and negative values? isn't k a fixed constant that can only be positive or negative
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in this equation why does k take on both positive and negative values? isn't k a fixed constant that can only be positive or negative
ques. not visible
oh you have to click on the attachment and then the picture to see the equation. so the domain of the function is kx so if k ranges from -infinity to infinity then you need a +- before the integral?
Last edited:
vanhees71
Gold Member
It's just an example for pretty unclear notation. I guess the formula is an attempt to prove the equation
$$\delta(k x)=\frac{1}{|k|} \delta(k x).$$
Of course you have to assume that $k \neq 0$. Otherwise the equation doesn't make any sense to begin with.
Then you have to just do the substitution $y=k x$ in the integral with the distribution times an arbitrary test function to prove this formula. For $k>0$ you find
$$\int_{\mathbb{R}} \mathrm{d} x f(x) \delta(k x)=\int_{\mathbb{R}} \mathrm{d}y \frac{1}{k} f(y/k) \delta(y)=\frac{1}{k} f(0).$$
For $k<0$ you have
$$\int_{\mathbb{R}} \mathrm{d} x f(x) \delta(k x)=\int_{\mathbb{R}} \mathrm{d} y \left (-\frac{1}{k} \right ) f(y/k) \delta (y)=-\frac{1}{k} f(0).$$
On the other hand the distribution
$\frac{1}{|k|} \delta(x)$
has the same outcome under an integral with an arbitrary test function, which proves the above statement about the Dirac distribution.
You can generalize this for arbitrary function $y(x)$ which have only single-order zeros, i.e., $y(x_k)=0$ but $y'(x_k) \neq 0$ for $k \in \{1,\ldots,n\}$. Then you can prove in pretty much the same way as the above example
$$\delta[y(x)]=\sum_{k=1}^{n} \frac{1}{|y'(x_k)|} \delta(x-x_k).$$
1 person
$$f(x)=\frac{1}{2\pi}\int_{R}e^{ipx}\left ( \int_{R}f(\alpha)d\alpha \right )dp=\frac{1}{2\pi}\int_{R}\left (\int_{R}e^{ipx}e^{-ip\alpha}\right)f(\alpha)d\alpha= \int_{R}\delta (x-\alpha)f(\alpha)d\alpha.$$ where,[itex] \delta(x-a)=\int_{R}e^{ip(x-\alpha)}\left dp. [/tex]
as you know Cauchy expressed Fourier's integral as exponentials and the delta distribution can be expressed in this way (he also pointed out that the integrals are non-commutative in some circumstances). In modern times there is L Schwartz's theory of distributions. Dirac called it the delta function because he used it as a continuous analogue of the discrete Kronecker delta.
Here is a vast generalization of the fundamental theorem of algebra
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$$(x+y)^{N+1}=_{0}^{N+1}\textrm{C}x^{N+1}+\sum_{k=1}^{N}(_{k}^{N}\textrm{C}+_{k-1}^{N}\textrm{C})x^{N+1-k}y^{k}+_{N+1}^{N+1}\textrm{C}y^{N+1}=_{0}^{N+1}\textrm{C}x^{N+1}+\sum_{k=1}^{N}_{k}^{N+1}\textrm{C}x^{(N+1)-k}y^{k}+_{N+1}^{N+1}\textrm{C}y^{N+1}=\sum_{k=0}^{N+1}_{k}^{N+1}{C}x^{(N+1)-k}y^{k}$$
Last edited: | 2021-03-08 14:14:25 | {"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": 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": 0.9280875325202942, "perplexity": 518.7822459897436}, "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/1614178375439.77/warc/CC-MAIN-20210308112849-20210308142849-00189.warc.gz"} |
https://or.stackexchange.com/questions/1092/a-variant-of-the-multiple-traveling-salesman-problem | # A variant of the Multiple Traveling Salesman Problem
I am trying to find a reference (or a reformulation) of a variant of the multiple Traveling Salesman Problem, where multiple agents need to visit each vertex in a graph with minimal cost.
Most of the literature that I found requires that only one agent needs to visit each vertex, but I am looking to generalize this to requiring multiple and different agents visiting some of the vertices. I have been trying to form this problem into the multiple Traveling Salesman problem, (and eventually use an approximation algorithm) but I have not been successful in reformulating the problem.
Some of the references that I looked into were
But, as I mentioned before, these papers require that each vertex is visited only once, and I want to require multiple agents to visit some of the vertices.
• Can you explain your problem more, may be by giving a small example? Let’s say 3 vehicles and 5 vertex. How the result of your model should be? – Oguz Toragay Jul 30 '19 at 15:13
• The output should be a path for each vehicle such that each vertex is visited by at least two different agents. Edit: For instance, an example path may be for a 5 vertex example: Agent 1: $\{1,3,2,4\}$ Agent 2: $\{1,2,4,5\}$ Agent 3: $\{3,5\}$ – kemalduldul Jul 30 '19 at 15:21
Y. Kaempfer and L. Wolf, in their recent paper [1] applied ML techniques to solve the Multiple Traveling Salesmen Problem (mTSP). They provide a mathematical model for problem formulation which can be modified to cover what you need in the solution to your problem.
You can replace the constraint (2d) which is:
$$\forall 2\leq j \leq n: \sum\limits_{i=1}^{n}\sum\limits_{k=1}^m \delta_{i,j,k}=1 \ \ (2d)$$
with:
$$\forall 2\leq j \leq n: \sum\limits_{i=1}^{n}\sum\limits_{k=1}^m \delta_{i,j,k} \geq 2 \ \ (2d')$$
depends on the number of agents that need to visit each vertex.
The rest of the constraints in their model can be kept the same. In addition to the mentioned paper the following resources could also be helpful:
You can model your problem by defining separate variables for each traveling salesman. Below I will use 'vehicle' instead of 'traveling salesman', which is more common in this setting.
### Defining separate variables
Let $$n$$ be the number of customers and let $$m \le n$$ be the number of vehicles. For each vehicle $$k = 1, \dots, m$$, define the variables $$x_{ij}^k := \begin{cases} 1 & \textrm{ if vehicle k travels from i to j} \\ 0 & \textrm{ else.} \end{cases}$$
### Constraints
• For each vehicle $$k$$, add constraints to make sure that the variables $$x_{ij}^k$$ describe a valid route. It is important that customers cannot be visited twice by the same vehicle. If you don't know the number of vehicles in advance, choose $$m=n$$ and make sure that empty routes are allowed (all $$x_{ij}^k = 0$$ for all $$i$$ and $$j$$ for a given $$k$$).
• Add constraints that say that every node is visited by the appropriate number of vehicles. Note that the earlier constraints enforce that each visit is by a unique vehicle.
• You can enforce that node $$i$$ can only be visited by certain vehicles by setting $$x_{pi}^k = 0$$ for all $$p \in V$$ if $$k$$ and $$i$$ are incompatible.
• That's a great idea to solve the problem. The only point which needs to be considered as well is, by duplicating the nodes, the size of the problem grows and the curse of dimensionality may affect the performance of approximating approaches to solve the problem. – Oguz Toragay Jul 30 '19 at 17:57
• @OguzToragay You are correct. Even worse, the additional symmetry will also decrease performance. On hindsight, duplicating the nodes is not necessary, and I will edit my answer accordingly. – Kevin Dalmeijer Jul 30 '19 at 18:13
• Thank you for the answer. For this formulation, is it possible to use heuristics for the multiple vehicle problem, for example the one in Arkin et al. that has a 4-approximation guarantee?. Thank you. – kemalduldul Jul 30 '19 at 20:20
• @kemalduldul I am not familiar with the paper by Arkin et al. There is large amount of literature on heuristics for vehicle routing problems. Chapter 4 in Vehicle Routing: Problems, Methods, and Applications may be a good place to start. – Kevin Dalmeijer Jul 30 '19 at 20:32
I'd like to add a few more ideas that could be important for solving this problem. I agree that a multiple-vehicle integer programming formulation may be a reasonable approach. In an arc-based model, decision variables $$x^k_{ij}$$ specify that vehicle/salesperson $$k$$ travels between $$i$$ and $$j$$ on its subtour. In such a formulation, you should create multiple copies for each vertex that needs to be visited multiple times, not create edges between vertex copies, and add a side constraint that ensures that vehicle/salesperson $$k$$ visits at most one copy of any vertex. Note that making copies introduces some symmetries that can create computational problems. A modeling idea to avoid some difficulties that might arise due to such symmetries is to allow vehicle $$k$$ to only visit copies $$\{1,2,...,k\}$$ of any vertex by not including certain edges/variables.
It may be important to include some constraints on the tours generated. For example, you could constrain the number of visits made by each vehicle: $$\sum_{ij} x^k_{ij} \leq c$$ or constrain the total duration of each vehicle tour: $$\sum_{ij} t_{ij} x^k_{ij} \leq d \quad .$$ If you want to be sure that vehicles serve tours with roughly the same number of visits or same total duration, you could use a minimax objective where you minimize $$z$$ and constrain $$z \geq \sum_{ij} x^k_{ij} \quad \text{or} \quad z \geq \sum_{ij} t_{ij} x^k_{ij} \quad \forall \; k \quad .$$
One minor problem with such an arc-based formulation is that subtour elimination constraints will be required for each vehicle/salesperson. Since no individual vehicle needs to serve all vertices, it is important to use appropriate SEC constraints. One option is the Miller-Tucker-Zemlin approach that introduces ordering decision variables (in this case, for each vehicle). If all the vehicles have a common base/depot vertex 0, then the typical SEC constraints that prevent cycles that do not include vertex 0 will work with minor modifications. Since these types of constraints are added dynamically, symmetry requires that you add them for each vehicle when they are required for any vehicle and that all copies of vertex $$i$$ should be included in the subtour set $$S$$. | 2021-04-18 23:49:40 | {"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": 31, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7048282027244568, "perplexity": 376.4445351374257}, "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-2021-17/segments/1618038862159.64/warc/CC-MAIN-20210418224306-20210419014306-00045.warc.gz"} |
https://www.hackmath.net/en/math-problem/6943 | # Monica
Monica left Php.900 to Bianca. When monica asked for it after 6 months, Bianca decided to give Php.945 instead because she was able to use the money. What interest rate on Monica's money was used by bianca?
Result
p = 10 %
#### Solution:
Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):
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Dr Math
p = 10 % p.a. or 5% per 6 month
#### Following knowledge from mathematics are needed to solve this word math problem:
Our percentage calculator will help you quickly calculate various typical tasks with percentages.
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52 is what percent of 93? | 2019-12-08 07:42:31 | {"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": 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": 0.20602047443389893, "perplexity": 4949.271166567073}, "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-2019-51/segments/1575540507109.28/warc/CC-MAIN-20191208072107-20191208100107-00513.warc.gz"} |
https://pecos.readthedocs.io/en/latest/qc_tests.html | # Quality control tests¶
Pecos includes several built in quality control tests. When a test fails, information is stored in a summary table. This information can be saved to a file, database, or included in reports. Quality controls tests fall into seven categories:
• Timestamp
• Missing data
• Corrupt data
• Range
• Delta
• Increment
• Outlier
Note
Quality control tests can also be called using individual functions, see Framework for more details.
## Timestamp test¶
The check_timestamp method is used to check the time index for missing, duplicate, and non-monotonic indexes. If a duplicate timestamp is found, Pecos keeps the first occurrence. If timestamps are not monotonic, the timestamps are reordered. For this reason, the timestamp should be corrected before other quality control tests are run. The timestamp test is the only test that modifies the data stored in pm.df. Input includes:
• Expected frequency of the time series in seconds
• Expected start time (default = None, which uses the first index of the time series)
• Expected end time (default = None, which uses the last index of the time series)
• Minimum number of consecutive failures for reporting (default = 1)
• A flag indicating if exact timestamps are expected. When set to False, irregular timestamps can be used in the Pecos analysis (default = True).
For example,
>>> pm.check_timestamp(60)
checks for missing, duplicate, and non-monotonic indexes assuming an expected frequency of 60 seconds.
## Missing data test¶
The check_missing method is used to check for missing values. Unlike missing timestamps, missing data only impacts a subset of data columns. NaN is included as missing. Input includes:
• Data column (default = None, which indicates that all columns are used)
• Minimum number of consecutive failures for reporting (default = 1)
For example,
>>> pm.check_missing('A', min_failures=5)
checks for missing data in the columns associated with the column or group ‘A’. In this example, warnings are only reported if there are 5 consecutive failures.
## Corrupt data test¶
The check_corrupt method is used to check for corrupt values. Input includes:
• List of corrupt values
• Data column (default = None, which indicates that all columns are used)
• Minimum number of consecutive failures for reporting (default = 1)
For example,
>>> pm.check_corrupt([-999, 999])
checks for data with values -999 or 999 in the entire dataset.
## Range test¶
The check_range method is used to check if data is within expected bounds. Range tests are very flexible. The test can be used to check for expected range on the raw data or using modified data. For example, composite signals can be add to the analysis to check for expected range on modeled vs. measured values (i.e. absolute error or relative error) or an expected relationships between data columns (i.e. column A divided by column B). An upper bound, lower bound, or both can be specified. Input includes:
• Upper and lower bound
• Data column (default = None, which indicates that all columns are used)
• Minimum number of consecutive failures for reporting (default = 1)
For example,
>>> pm.check_range([None, 1], 'A')
checks for values greater than 1 in the columns associated with the key ‘A’.
## Delta test¶
The check_delta method is used to check for stagnant data and abrupt changes in data. The test checks if the difference between the minimum and maximum data value within a moving window is within expected bounds.
Input includes:
• Upper and lower bound
• Data column (default = None, which indicates that all columns are used)
• Size of the moving window used to compute the difference between the minimum and maximum (default = 3600 seconds)
• Flag indicating if the test should only check for positive delta (the min occurs before the max) or negative delta (the max occurs before the min) (default = False)
• Minimum number of consecutive failures for reporting (default = 1)
For example,
>>> pm.check_delta([0.0001, None], window=3600)
checks if data changes by less than 0.0001 in a 1 hour moving window.
>>> pm.check_delta([None, 800], window=1800, direction='negative')
checks if data decrease by more than 800 in a 30 minute moving window.
## Increment test¶
Similar to the check_delta method above, the check_increment method can be used to check for stagnant data and abrupt changes in data. The test checks if the difference between consecutive data values (or other specified increment) is within expected bounds. While this method is faster than the check_delta method, it does not consider the timestamp index or changes within a moving window, making its ability to find stagnant data and abrupt changes less robust.
Input includes:
• Upper and lower bound
• Data column (default = None, which indicates that all columns are used)
• Increment used for difference calculation (default = 1 timestamp)
• Flag indicating if the absolute value of the increment is used in the test (default = True)
• Minimum number of consecutive failures for reporting (default = 1)
For example,
>>> pm.check_increment([0.0001, None], min_failures=60)
checks if increments are less than 0.0001 for 60 consecutive time steps.
>>> pm.check_increment([-800, None], absolute_value=False)
checks if increments decrease by more than 800 in a single time step.
## Outlier test¶
The check_outlier method is used to check if normalized data falls outside expected bounds. Data is normalized using the mean and standard deviation, using either a moving window or using the entire data set. If multiple columns of data are used, each column is normalized separately. Input includes:
• Upper and lower bound (in standard deviations)
• Data column (default = None, which indicates that all columns are used)
• Size of the moving window used to normalize the data (default = 3600 seconds)
• Flag indicating if the absolute value of the normalize data is used in the test (default = True)
• Minimum number of consecutive failures for reporting (default = 1)
For example,
>>> pm.check_outlier([None, 3], window=12*3600)
checks if the normalized data changes by more than 3 standard deviations within a 12 hour moving window. | 2020-01-24 05:33:59 | {"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": 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": 0.22985272109508514, "perplexity": 2630.663131145135}, "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/1579250615407.46/warc/CC-MAIN-20200124040939-20200124065939-00462.warc.gz"} |
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Q02-38 Question # 08 Section # 09
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Q02-38 Question # 08 Section # 09 [#permalink] 23 Jun 2016, 12:50
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15% (01:48) correct 84% (01:37) wrong based on 32 sessions
Given $$x$$, $$y<0$$, what is the value of $$\frac{\sqrt{x^2}}{x} - \sqrt{\frac{-y}{|y|}}$$ ?
A. $$1+y$$
B. $$1-y$$
C. $$-1-y$$
D. $$y-1$$
E. $$x-y$$
Hello,
I was looking at the explanation of this question (ID Q02-38, Question2 section 9) and it quite does not make any sense in the very first step (or maybe I am overlooking something). Kindly check (or possibly explain it if possible/and-is-correct).
Thanks
Arsh
[Reveal] Spoiler: OA
Last edited by Carcass on 04 Jul 2019, 23:42, edited 3 times in total.
Editing the post and adding the tags
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Re: Q02-38 Question # 08 Section # 09 [#permalink] 25 Jun 2016, 01:16
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Hi,
well I have to admit that the explanation is not the top-notch but it is not wrong.
Starting from what we do know: X could be positive or negative: we do not know at the moment AND y is negative.
Now look at the first part of the equation $$\sqrt{x^2}$$ is equal to |x|. From this we also know that |x| = -x. Which means that on the left hand side X is positive, always. On the right hand side for X to be always positive X must be negative. This are math rules that is better for you to know as cold during the exam.
As such, actually we do have that X/X is = 1 and is negative. So -1
Going to the second square root |y| = -y so -y * - y = y^2 that under square root becomes |y|.
At this point we have -1 - |y| AND we already know that |y| = -y .
-1 - (-y) = -1 +y
Hope is clear this. Waiting, though, math expert for further clarification.
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Re: Q02-38 Question # 08 Section # 09 [#permalink] 28 Feb 2018, 21:33
Carcass wrote:
Hi,
well I have to admit that the explanation is not the top-notch but it is not wrong.
Starting from what we do know: X could be positive or negative: we do not know at the moment AND y is negative.
Now look at the first part of the equation $$\sqrt{x^2}$$ is equal to |x|. From this we also know that |x| = -x. Which means that on the left hand side X is positive, always. On the right hand side for X to be always positive X must be negative. This are math rules that is better for you to know as cold during the exam.
As such, actually we do have that X/X is = 1 and is negative. So -1
Going to the second square root |y| = -y so -y * - y = y^2 that under square root becomes |y|.
At this point we have -1 - |y| AND we already know that |y| = -y .
-1 - (-y) = -1 +y
Hope is clear this. Waiting, though, math expert for further clarification.
I still don't know why $$\sqrt{\frac{-y}{|y|}}$$ =-y NOT -1?
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Re: Q02-38 Question # 08 Section # 09 [#permalink] 24 May 2018, 11:07
1
KUDOS
Peter wrote:
Carcass wrote:
Hi,
well I have to admit that the explanation is not the top-notch but it is not wrong.
Starting from what we do know: X could be positive or negative: we do not know at the moment AND y is negative.
Now look at the first part of the equation $$\sqrt{x^2}$$ is equal to |x|. From this we also know that |x| = -x. Which means that on the left hand side X is positive, always. On the right hand side for X to be always positive X must be negative. This are math rules that is better for you to know as cold during the exam.
As such, actually we do have that X/X is = 1 and is negative. So -1
Going to the second square root |y| = -y so -y * - y = y^2 that under square root becomes |y|.
At this point we have -1 - |y| AND we already know that |y| = -y .
-1 - (-y) = -1 +y
Hope is clear this. Waiting, though, math expert for further clarification.
I still don't know why $$\sqrt{\frac{-y}{|y|}}$$ =-y NOT -1?
Consider y = (-2), as y<0. hence -y = -(-2) = 2.
|y| = |-2| = 2.
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Re: Q02-38 Question # 08 Section # 09 [#permalink] 04 Jul 2019, 15:54
1
KUDOS
I was thinking another way. Let me correct if I am horribly wrong! Let x and y both equal -1. from the first, sqrt of (x square)/x we get -1 and also form second part we get -1. So adding, -2. Consider all options!. Option D fits!
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Re: Q02-38 Question # 08 Section # 09 [#permalink] 05 Jul 2019, 04:49
Expert's post
@AlaminMolla is correct. For ALL negative values of x and y, the expression will ALWAYS evaluate to be -2
So, the question is flawed. At the very least, the question should read "Which of the following COULD BE the value of ...", in which case A, D and E COULD equal -2
@arsh are you sure you transcribed the question correctly?
Cheers,
Brent
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Re: Q02-38 Question # 08 Section # 09 [#permalink] 15 Jul 2019, 15:26
this was hard to understand
Re: Q02-38 Question # 08 Section # 09 [#permalink] 15 Jul 2019, 15:26
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Q02-38 Question # 08 Section # 09
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Powered by phpBB © phpBB Group Kindly note that the GRE® test is a registered trademark of the Educational Testing Service®, and this site has neither been reviewed nor endorsed by ETS®. | 2020-12-01 05:28:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4229282736778259, "perplexity": 3157.160334067162}, "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-2020-50/segments/1606141652107.52/warc/CC-MAIN-20201201043603-20201201073603-00405.warc.gz"} |
https://physics.stackexchange.com/questions/153853/in-special-relativity-do-explanations-involving-clocks-require-that-the-clocks | # In special relativity, do explanations involving clocks require that the clocks are ticking and that $c$ is fixed?
1. Are the explanations involving clocks only valid if the clocks are ticking when light hits?
2. Is it true that these thought experiments experiment could only be valid due to the invariance of $c$?
• A simple clock that is not "ticking" is radioactive decay. A useful physical clock is mechanism that shows a well enough characterized change over time and that agrees in its quantitative reading with other clocks. The invariance of c (or any other natural constant) is neither a necessary physical assumption nor a trusted fact. We are commonly performing experiments which try to test the validity of the constancy of natural constants. – CuriousOne Dec 18 '14 at 2:06
If I understand correctly, your question is why would time dilation for any other type of clock.
Light is an electromagnetic wave, all electromagnetic waves travel at the speed of light. If, say, you have a mechanical watch it has a spring, atoms in which exchange electromagnetic signals, which travel at the speed of light. Therefore a mechanical watch can be subjected to the same analysis. Furthermore, there are other fundamental interactions, which also work at the speed of light. For example half-life of certain isotopes is also subject to time dilation.
• This answer doesn't really explain why a clock based on nuclear decay would agree too though. The most general answer is that all known laws of physics including all quantum field theories are Lorentz-symmetric, meaning the equations are unaltered when you transform from the space and time coordinates of one inertial frame to those of another. The first of the two postulates of SR predicts this will be true of all laws, and so far all evidence supports it. – Hypnosifl Dec 17 '14 at 23:19
The explanations involving clocks ...
Textbook examples or explanations of relativity that involve clocks are often about time-dilation.
... Is valid only if the clocks is ticking ...
These examples/explanations generally assume that any clock they mention is a working clock. It doesn't have to be a clock that ticks. It could be any type of clock of sufficient accuracy and precision. It could be a very large eggtimer - it doesn't matter.
... when light hits, right?
The clock has to be continuously running the whole period covering all events in the example/explanation. Otherwise it can't measure elapsed time.
This experiment could only be valid due to the invariation of c
From what I recall, all the rather striking basic notions of special relativity, like time-dilation, can be straightforwardly derived by working from the observation that light in a vacuum travels at $c$ regardless of the velocity of observer and light source.
I'm not getting the concept
That's because it is unintuitive when you first encounter it (and may remain so).
The essential concept is that light (and any other electromagnetic radiation) in a vacuum always travels at $c$ regardless of the velocity of the observer in relation to the light source.
What necessarily follows from this is that people in motion will often disagree about which events occurred simultaneously, about the physical length of things and how much time has passed between events. yet they will agree that each other's measurements are correct and match the calculated measurements predicted using the equations associated with Relativity.
Time dilation is just that: time dilation and not simply clock time dilation (although the former implies the latter). Any process or chain of events will take longer when observed from a relatively moving, inertial frame. We can, for example, put unstable particles into high speed rings and their lifetime will increase by the factor $1/\sqrt{1-\frac{v^2}{c^2}}$, where $v$ is the speed at which they move relative to the observer. Since many particle decays are mediated by forces other than the electromagnetic interaction, these experiments show that time dilation is more general light. All clocks will "slow down" in this way. Through experiments like this we have experimentally shown that all clocks behave like this, to a fantastically high accuracy.
Nowadays we understand special relativity independently of light. The speed $c$ comes out of an analysis of what happens to Galileo's Relativity (note I didn't say Einstein) when we relax the assumption of an absolute time. The latter - the bold step of questioning time's constancy and deducing what would follow from nonconstant time - was Einstein's big and bold contribution. I say more about this analysis of Galileo's relativity in my answer to the Physics SE question "What's so special about the speed of light?" here. The particular arguments referred to there show that $c$ is fixed and, moreover, unique: there can be at most one velocity $c$ that transforms in the way it does.
The speed $c$ is understood to be a universal constant that, amongst other things, is the speed that any massless particle moves at. That light is experimentally found to move at this speed (or more correctly, that light's speed is experimentally found to transform between inertial observers in the special way that the nonabsolute time analysis foretells) can now be seen as an experimental proof that light is mediated by a massless particle. Again, these experiments are now fantastically accurate: the Michelson-Moreley experiment can now be done at such an accuracy that shows constancy of the speed of light to within one part in $10^{-17}$ for speeds up to that of Earth relative to the putative "Aether" (which is thus discredited by these experiments). See here | 2019-12-11 03:22:14 | {"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": 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": 0.7140211462974548, "perplexity": 440.689199034571}, "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/1575540529745.80/warc/CC-MAIN-20191211021635-20191211045635-00405.warc.gz"} |
https://www.jobilize.com/course/section/approximation-of-scaling-coefficients-by-samples-of-the-by-openstax?qcr=www.quizover.com | # 0.6 Regularity, moments, and wavelet system design (Page 8/13)
Page 8 / 13
## Vanishing scaling function moments
While the moments of the wavelets give information about flatness of $H\left(\omega \right)$ and smoothness of $\psi \left(t\right)$ , the moments of $\phi \left(t\right)$ and $h\left(n\right)$ are measures of the “localization" and symmetry characteristics of the scaling function and, therefore, the wavelet transform. We know from [link] that ${\sum }_{n}h\left(n\right)=\sqrt{2}$ and, after normalization, that $\int \phi \left(t\right)\phantom{\rule{0.166667em}{0ex}}dt=1$ . Using [link] , one can show [link] that for $K\ge 2$ , we have
$m\left(2\right)={m}^{2}\left(1\right).$
This can be seen in [link] . A generalization of this result has been developed by Johnson [link] and is given in [link] through [link] .
A more general picture of the effects of zero moments can be seen by next considering two approximations. Indeed, this analysis gives a veryimportant insight into the effects of zero moments. The mixture of zero scaling function moments with other specifications is addressed laterin [link] .
## Approximation of signals by scaling function projection
The orthogonal projection of a signal $f\left(t\right)$ on the scaling function subspace ${V}_{j}$ is given and denoted by
${P}^{j}\left\{f\left(t\right)\right\}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\sum _{k}⟨f\left(t\right),{\phi }_{j,k}\left(t\right)⟩\phantom{\rule{0.166667em}{0ex}}{\phi }_{j,k}\left(t\right)$
which gives the component of $f\left(t\right)$ which is in ${V}_{j}$ and which is the best least squares approximation to $f\left(t\right)$ in ${V}_{j}$ .
As given in [link] , the ${\ell }^{th}$ moment of $\psi \left(t\right)$ is defined as
${m}_{1}\left(\ell \right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\int {t}^{\ell }\psi \left(t\right)\phantom{\rule{0.166667em}{0ex}}dt.$
We can now state an important relation of the projection [link] as an approximation to $f\left(t\right)$ in terms of the number of zero wavelet moments and the scale.
Theorem 25 If ${m}_{1}\left(\ell \right)=0$ for $\ell =0,1,\cdots ,L$ then the ${L}^{2}$ error is
${ϵ}_{1}=\parallel f\left(t\right)-{P}^{j}\left\{f\left(t\right)\right\}{\parallel }_{2}\phantom{\rule{0.166667em}{0ex}}\le \phantom{\rule{0.166667em}{0ex}}C\phantom{\rule{0.166667em}{0ex}}{2}^{-j\left(L+1\right)},$
where $C$ is a constant independent of $j$ and $L$ but dependent on $f\left(t\right)$ and the wavelet system [link] , [link] .
This states that at any given scale, the projection of the signal on the subspace at that scale approaches the function itself as the numberof zero wavelet moments (and the length of the scaling filter) goes to infinity. It also states that for any given length, the projection goesto the function as the scale goes to infinity. These approximations converge exponentially fast. This projection is illustrated in [link] .
## Approximation of scaling coefficients by samples of the signal
A second approximation involves using the samples of $f\left(t\right)$ as the inner product coefficients in the wavelet expansion of $f\left(t\right)$ in [link] . We denote this sampling approximation by
${S}^{j}\left\{f\left(t\right)\right\}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\sum _{k}{2}^{-j/2}\phantom{\rule{0.166667em}{0ex}}f\left(k/{2}^{j}\right)\phantom{\rule{0.166667em}{0ex}}{\phi }_{j,k}\left(t\right)$
and the scaling function moment by
$m\left(\ell \right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\int {t}^{\ell }\phantom{\rule{0.166667em}{0ex}}\phi \left(t\right)\phantom{\rule{0.166667em}{0ex}}dt$
and can state [link] the following
Theorem 26 If $m\left(\ell \right)=0$ for $\ell =1,2,\cdots ,L$ then the ${L}^{2}$ error is
${ϵ}_{2}=\parallel {S}^{j}\left\{f\left(t\right)\right\}-{P}^{j}\left\{f\left(t\right)\right\}{\parallel }_{2}\phantom{\rule{0.166667em}{0ex}}\le \phantom{\rule{0.166667em}{0ex}}{C}_{2}\phantom{\rule{0.166667em}{0ex}}{2}^{-j\left(L+1\right)},$
where ${C}_{2}$ is a constant independent of $j$ and $L$ but dependent on $f\left(t\right)$ and the wavelet system.
This is a similar approximation or convergence result to the previous theorem but relates the projection of $f\left(t\right)$ on a $j$ -scale subspace to the sampling approximation in that same subspace. These approximationsare illustrated in [link] .
This “vector space" illustration shows the nature and relationships of the two types of approximations. The use of samples as inner productsis an approximation within the expansion subspace ${V}_{j}$ . The use of a finite expansion to represent a signal $f\left(t\right)$ is an approximation from ${L}^{2}$ onto the subspace ${V}_{j}$ . Theorems [link] and [link] show the nature of those approximations, which, for wavelets, is very good.
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
Got questions? Join the online conversation and get instant answers! | 2020-07-04 22:02:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 46, "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": 0.7647784948348999, "perplexity": 1405.9012125315844}, "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/1593655886706.29/warc/CC-MAIN-20200704201650-20200704231650-00477.warc.gz"} |
https://msp.org/apde/2011/4-3/p02.xhtml | #### Vol. 4, No. 3, 2011
Recent Issues
The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
Traveling waves for the cubic Szegő equation on the real line
### Oana Pocovnicu
Vol. 4 (2011), No. 3, 379–404
##### Abstract
We consider the cubic Szegő equation $i\partial tu=\Pi \left(|u{|}^{2}u\right)$ in the Hardy space ${L}_{+}^{2}\left(ℝ\right)$ on the upper half-plane, where $\Pi$ is the Szegő projector. It was first introduced by Gérard and Grellier as a toy model for totally nondispersive evolution equations. We show that the only traveling waves are of the form $C∕\left(x-p\right)$, where $p\in ℂ$ with $Im\phantom{\rule{0.3em}{0ex}}p<0$. Moreover, they are shown to be orbitally stable, in contrast to the situation on the unit disk where some traveling waves were shown to be unstable.
##### Keywords
nonlinear Schrödinger equations, Szegő equation, integrable Hamiltonian systems, Lax pair, traveling wave, orbital stability, Hankel operators
##### Mathematical Subject Classification 2000
Primary: 35B15, 37K10, 47B35
##### Milestones
Received: 19 January 2010
Revised: 28 April 2010
Accepted: 29 May 2010
Published: 28 December 2011
##### Authors
Oana Pocovnicu Laboratoire de Mathématiques d’Orsay Université Paris-Sud (XI) Campus d’Orsay, bât. 430 91405, Orsay France | 2022-05-27 22:36:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 6, "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": 0.29746079444885254, "perplexity": 3895.398436731851}, "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-2022-21/segments/1652663006341.98/warc/CC-MAIN-20220527205437-20220527235437-00417.warc.gz"} |
http://www.tcs.tifr.res.in/events/small-value-parallel-repetition-general-games | Small Value Parallel Repetition for General Games
Ankit Garg
Affiliation:
Princeton University
Department of Computer Science
35, Olden Street
Princeton, NJ 08544
United States of America
Time:
Thursday, 21 August 2014, 16:00 to 17:00
Organisers:
Abstract: We prove a parallel repetition theorem for general games with value tending to 0. Previously Dinur and Steurer proved such a theorem for the special case of projection games. We use information theoretic techniques in our proof. Our proofs also extend to the high value regime (value close to 1) and provide alternate proofs for the parallel repetition theorems of Holenstein and Rao for general and projection games respectively. We also extend the example of Feige and Verbitsky to show that the small-value parallel repetition bound we obtain is tight. Our techniques are elementary in that we only need to employ basic information theory and discrete probability in the small-value parallel repetition proof (this is joint work with Mark Braverman). | 2018-02-23 04:15:33 | {"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": 0.9229778051376343, "perplexity": 1365.856995561441}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814393.4/warc/CC-MAIN-20180223035527-20180223055527-00591.warc.gz"} |
https://www.groundai.com/project/total-proper-connection-of-graphs/ | Total proper connection of graphs1footnote 11footnote 1Supported by NSFC No.11371205 and 11531011, and PCSIRT.
# Total proper connection of graphs111Supported by NSFC No.11371205 and 11531011, and PCSIRT.
Hui Jiang, Xueliang Li, Yingying Zhang
Center for Combinatorics and LPMC-TJKLC
Nankai University, Tianjin 300071, China
E-mail: jhuink@163.com; lxl@nankai.edu.cn;
zyydlwyx@163.com
###### Abstract
A graph is said to be total-colored if all the edges and the vertices of the graph is colored. A path in a total-colored graph is a total proper path if any two adjacent edges on the path differ in color, any two internal adjacent vertices on the path differ in color, and any internal vertex of the path differs in color from its incident edges on the path. A total-colored graph is called total-proper connected if any two vertices of the graph are connected by a total proper path of the graph. For a connected graph , the total proper connection number of , denoted by , is defined as the smallest number of colors required to make total-proper connected. These concepts are inspired by the concepts of proper connection number , proper vertex connection number and total rainbow connection number of a connected graph . In this paper, we first determine the value of the total proper connection number for some special graphs . Secondly, we obtain that for any -connected graph and give examples to show that the upper bound is sharp. For general graphs, we also obtain an upper bound for . Furthermore, we prove that for a connected graph with order and minimum degree . Finally, we compare with and , respectively, and obtain that for any nontrivial connected graph , and that and can differ by for .
Keywords
: total-colored graph, total proper connection, dominating set
AMS subject classification 2010
: 05C15, 05C40, 05C69, 05C75.
## 1 Introduction
In this paper, all graphs considered are simple, finite and undirected. We refer to the book [2] for undefined notation and terminology in graph theory. A path in an edge-colored graph is a proper path if any two adjacent edges differ in color. An edge-colored graph is proper connected if any two vertices of the graph are connected by a proper path of the graph. For a connected graph , the proper connection number of , denoted by , is defined as the smallest number of colors required to make proper connected. Note that if and only if is a complete graph. The concept of was first introduced by Borozan et al. [3] and has been well-studied recently. We refer the reader to [1, 5, 8, 11] for more details.
As a natural counterpart of the concept of proper connection, the concept of proper vertex connection was introduced by the authors [7]. A path in a vertex-colored graph is a vertex-proper path if any two internal adjacent vertices on the path differ in color. A vertex-colored graph is proper vertex connected if any two vertices of the graph are connected by a vertex-proper path of the graph. For a connected graph , the proper vertex connection number of , denoted by , is defined as the smallest number of colors required to make proper vertex connected. Especially, set for a complete graph . Moreover, we have if is a noncomplete graph.
Actually, the concepts of the proper connection and proper vertex connection were motivated from the concepts of the rainbow connection and rainbow vertex connection. For details about them we refer to a book [10] and a survey paper [9]. Here we only state the concept of the total rainbow connection of graphs, which was introduced by Liu et al. [12] and also studied in [6, 13]. A graph is total-colored if all the edges and vertices of the graph are colored. A path in a total-colored graph is a total rainbow path if all the edges and internal vertices on the path differ in color. A total-colored graph is total rainbow connected if any two vertices of the graph are connected by a total rainbow path of the graph. For a connected graph , the total rainbow connection number of , denoted by , is defined as the smallest number of colors required to make total rainbow connected. Motivated by the concept of the total rainbow connection, now for the proper connection and proper vertex connection we introduce the concept of the total proper connection. A path in a total-colored graph is a total proper path if any two adjacent edges on the path differ in color, any two internal adjacent vertices on the path differ in color, and any internal vertex of the path differs in color from its incident edges on the path. A total-colored graph is total proper connected if any two vertices of the graph are connected by a total proper path of the graph. For a connected graph , the total proper connection number of , denoted by , is defined as the smallest number of colors required to make total proper connected. It is easy to obtain that if and only if is a complete graph, and if is not complete. Moreover,
tpc(G)≥max{pc(G),pvc(G)}. (∗)
We can also extend the definition of the total proper connection to that of the total proper -connection in a similar way as the definitions of the proper -connection , proper vertex -connection and total rainbow -connection , which were introduced by Borozan et al. in [3], the present authors in [7] and Liu et al. in [12], respectively. However, one can see that when is larger very little have been known. Almost all known results are on the case for . So, in this paper we only focus our attention on the total proper connection of graphs, i.e., for the case .
The rest of this paper is organized as follows: In Section , we mainly determine the value of for some special graphs, and moreover, we present some preliminary results. In Section , we obtain that for any -connected graph and give examples to show that the upper bound is sharp. For general graphs, we also obtain an upper bound for . In Section , we prove that for a connected graph with order and minimum degree . In Section , we compare with and , respectively, and obtain that for any nontrivial connected graph , and that and can differ by for .
## 2 Preliminary results
In this section, we present some preliminary results on the total proper connection number and determine the value of when is a nontrivial tree, a complete bipartite graph and a complete multipartite graph.
###### Proposition 1.
If is a nontrivial connected graph and is a connected spanning subgraph of , then . In particular, for every spanning tree of .
###### Proposition 2.
Let be a connected graph of order that contains a bridge. If is the maximum number of bridges incident with a single vertex in , then .
Let denote the maximum degree of a connected graph . We have the following.
###### Theorem 1.
If is a tree of order , then .
Proof. Since each edge in is a bridge, we have by Proposition 2. Now we just need to show that . Let be the vertex with maximum degree and denote its neighborhood. Take the vertex as the root of . Define a total-coloring of with colors in the following way: Let be a vertex in . If , color and its incident edges with distinct colors from , and with the color from for . If , there exists a father of , say . Let denote the neighborhood of . Color the edges with distinct colors from , and the vertex with the color from for .
For any two vertices and in , let be a path from to , where . Next we shall show that there is a total proper path between and . If and are edge-disjoint, then ; otherwise, we walk from along to the earliest common vertex, say , and then switch to and walk to , i.e., . Thus, , and therefore, . ∎
The consequence below is immediate from Proposition 1 and Theorem 1.
###### Corollary 1.
For a nontrivial connected graph ,
tpc(G)≤min{Δ(T)+1: T is a spanning tree of G}.
A Hamiltonian path in a graph is a path containing every vertex of and a graph having a Hamiltonian path is a traceable graph. We get the following result.
###### Corollary 2.
If is a traceable graph that is not complete, then .
Let denote a complete bipartite graph, where . Clearly, and if . For , we have the result below.
###### Theorem 2.
For , we have .
Proof. Let the bipartition of be and , where and . Since is not complete, it suffices to show that . Now we divide our discussion into two cases.
Case 1. .
We first give a total-coloring of with colors. Color the vertex and the edge with color , the vertex and the edge with color , and all the other edges and vertices with color . Then we show that there is a total proper path between any two vertices of . It is clear that and are total proper connected by an edge if they belong to different parts of the bipartition. Next we consider that and are in the same part of the bipartition. For , we have . For , if one of them is , then ; otherwise, .
Case 2. .
Similarly, we first give a total-coloring of with colors. Color the vertices and edges of the cycle starting from in turn with the colors . For and , color with color , with color , and all the other edges and vertices with color . Now we show that there is a total proper path between any two vertices of . It is clear that and are total proper connected by an edge if they belong to different parts of the bipartition. For , we have . For , we have . It can be checked that and are total proper connected in all other cases.
Therefore, the proof is complete.∎
Since any complete multipartite graph has a spanning complete bipartite subgraph, we obtain the following corollary.
###### Corollary 3.
If is a complete multipartite graph that is neither a complete graph nor a tree, then .
## 3 Connectivity
In this section, we first prove that for any -connected graph . Also we show that this upper bound is sharp by presenting a family of a -connected graphs. Finally, we state an upper bound of for general graphs.
Given a colored path between any two vertices and , we denote by the color of the first edge in the path, i.e., , and by the last color, i.e., . Moreover, let be the color of the first internal vertex in the path, i.e., , and be the last color, i.e., . If is just the edge , then , and .
###### Definition 1.
Let be a total-coloring of that makes total proper connected. We say that has the strong property if for any pair of vertices , there exist two total proper paths between them (not necessarily disjoint) such that and for , and both and are -sets.
Let be a connected graph and be a spanning subgraph of . We say that is a spanning minimally -connected subgraph of if the removal of any edge from would leave -connected.
###### Theorem 3.
Let be a -connected graph. Then and there exists a total-coloring of with colors such that has the strong property.
Proof. Let be a spanning minimally -connected subgraph of . We apply induction on the number of ears in an ear-decomposition of . The base case is that is simply a cycle . Obviously, and for . Next define a total-coloring of with colors by
c(vivi+1)=⎧⎨⎩1,if i is odd,1≤i≤2k−1 for n=2k or n=2k+12,if i is even,2≤i≤n for n=2k or n=2k+14,if i=2k+1 for n=2k+1 (1)
and
c(vi)=⎧⎨⎩3,if i is odd,1≤i≤2k−1 for n=2k or n=2k+14,if i is even,2≤i≤2k for n=2k or n=2k+11,if i=2k+1 for n=2k+1. (2)
Clearly, the total-coloring makes have the strong property.
In an ear-decomposition of , let be the last ear with at least one internal vertex since is assumed to be minimally -connected. And denote by the graph after removal of the internal vertices of . Let and be the vertices of and then . By induction hypothesis, there exists a total-coloring of with colors such that is total proper connected with the strong property. We give such a total-coloring to . Then there exist two total proper paths and from to such that and for , and both and are -sets. Let . Color the edge with the color from , and then total-properly color from to so that and . If , it will become clear that this is the easier case, and so we consider the case that in the following.
Without loss of generality, suppose that . We will show that is total proper connected with the strong property under this coloring. For any two vertices of , there exist two total proper paths connecting them with the strong property by induction hypothesis. Since forms a total proper connected cycle, any two vertices in this cycle also have the desired paths. Assume that and . Next we will show that there are two total proper paths from to with the strong property.
Since , there exist two total proper paths and starting at and ending at with the strong property. Analogously, there exist two total proper paths and starting at and ending at with the strong property. Since these paths have the strong property, suppose that and are total proper paths. If , then and are the desired pair of paths. Thus, assume that .
Then there exists a total proper walk for some (suppose ). If is a path, then and are the desired two paths. Otherwise, let denote the vertex closest to on which is in . Now consider the path . If is a total proper path, then and are the desired two paths, and so we suppose that . Since and are total proper paths, , and . Then . Let and . Obviously, and are two total proper paths. Note that . Thus, and have the strong property. Since by Proposition 1, we have and there exists a total-coloring of with 4 colors such that has the strong property. This completes the proof of Theorem 3. ∎
In order to show that the bound obtained in Theorem 3 is sharp, we give a family of -connected graphs with (see Figure 1).
###### Proposition 3.
Let be the graph obtained from an even cycle by adding two ears which are as long as their interrupting segments respectively, such that each segment has () edges. Then .
Before proving Proposition 3, we give the following fact.
###### Fact 1.
Let . If there exists a total-coloring of with three colors such that there are two total proper paths and where and , then .
Proof of Proposition 3: Since by Theorem 3, we just need to prove that . Assume that there is a total-coloring of with colors such that is total proper connected. Label the segments and some vertices of as in Figure 1, where and are the neighbours of the vertex for .
Firstly, we shall show that the segments and are two total proper paths. If one of them is not, say , then there is no total proper path in from to or from to (say from to ). Hence there exists a total proper path through connecting and , suppose (this assumption, as opposed to using any of or , does not lose any generality). Next we consider the total proper path between and . Then there must exist a total proper path using the segments or . If there is a total proper path , then . Thus the total proper path between and is unique, i.e., , and then . However, we can not find a total proper path from to , a contradiction. If there is a total proper path , then . Thus the total proper path connecting and is unique, i.e., . Then and are two total proper paths in which is an even cycle of length , which contradicts Fact 1. Hence there is no total proper path from to , a contradiction. Therefore, the segments and are two total proper paths.
Secondly, we will show that at least one of or must be total proper (and similarly, at least one of or ). Suppose both and are not total proper. Then and are total proper connected by a path through or , say . However, we can not find a total proper path connecting and in a similar discussion above, which is impossible. Thus, suppose and are total proper without loss of generality.
Finally, we know that at least one of the paths and must be not total proper by Fact 1. As we have shown, the only place which we can not go through is at the intersections, and so assume that the path is not total proper, where and . In the following, we consider the total proper path from to and divide our discussion into two cases:
Case 1. is or .
Between and , there must exist a total proper path . If is , then . Hence, there is only one total proper path from to ; otherwise and are two total proper paths in the cycle for a contradiction. Then it follows that . Similarly, we can deduce that there is no total proper path connecting and , which is impossible. If is , then . In a similar discussion, we obtain that the total proper path from to is unique, i.e., . Then and are two total proper paths in , a contradiction. If is or , then and are two total proper paths in , which again contradicts Fact 1.
Case 2. is .
Consider the total proper path from to , where . If is , then we can prove that this subcase could not happen in a similar way as Case 1. If is , then . From to , there is only one total proper path since we can not go through . However, and are two total proper paths in for a contradiction.
The proof is thus complete. ∎
###### Remark 1.
Remember that for a -connected graph , we have that the proper connection number ; see [3]. But, if we consider a -connected bipartite graph , then we have that . That means that the bipartite property can lower down the number of color by 1. However, from Proposition 3 we see that the bipartite property cannot play a role in general to lower down the number of colors for the total proper connection number, since the graphs in Proposition 3 are bipartite but their total proper connection numbers reach the upper bound .
Finally, we prove an upper bound of for general graphs.
###### Theorem 4.
Let be a connected graph and denote the maximum degree of a vertex which is an endpoint of a bridge in . Then if and otherwise.
In order to prove Theorem 4, we need a lemma below. Let denote the set of colors presented on the vertex and edges incident to .
###### Lemma 1.
Let be a graph obtained from a block with by adding nontrivial blocks and pendant edges at for . Consider as the maximum degree of a vertex which is an endpoint of a bridge in . Then .
Proof. Let and . We give a total-coloring of using as follows.
Step 1. If is a trivial block, then we give a total-coloring with 3 colors to such that are different from each other; otherwise, we give a total-coloring with 4 colors to that makes it have the strong property by Theorem 3. Let modulo for .
Step 2. For , if , then we give a total-coloring with 4 colors from to each uncolored nontrivial block at , denoted by , that makes each of them have the strong property by Theorem 3; afterwards if , then color uncolored pendant edges at , denoted by , with distinct colors from and then color each pendant vertex using for .
Next we show that is total proper connected under the coloring . If is a nontrivial block, then each pair of the vertices in has two total proper paths between them with the strong property. It will become clear that this is the easier case so we consider the case that is a trivial block. Let and be two vertices of . It is obvious that there exists a total proper path connecting them if both belong to the same block. Suppose that and for . If and , then there exist two paths and from to with the strong property. We know that is a total proper path for some (suppose ). Similarly, there exists a total proper path from to where is a total proper path connecting and . Thus, we can find a total proper path between and . If and , then is a total proper path under the coloring . For the other cases, it can be checked that there exists a total proper path connecting and in a similar way. Therefore, . ∎
Now we are ready to prove Theorem 4.
Proof of Theorem 4: Let be the blocks of and denote the block graph of with vertex set . Now, we consider a breadth-first search tree (BFS-tree) of with root and suppose that the blocks have an order . Let and . We will give a total-coloring using in the following.
We give a total-coloring to and its neighbor blocks of in a similar way as in Lemma 1. Then we can get that is total proper connected if there are no more blocks in . Hence, suppose that there are uncolored blocks in . We extend our coloring from in a Breadth First Search way until there is no more blocks in , i.e., if has uncolored neighbor blocks, we give a total-coloring to its uncolored neighbor blocks of in a similar way as Step 2; otherwise, consider .
Now we prove that is total proper connected. Let and be two vertices in . It is obvious that there exists a total proper path between them if both belong to the same block. Suppose that and . Let denote the path from to in the BFS-tree . Then we can find a total proper path from to traversing the blocks on under the coloring . Therefore, if and otherwise. ∎
## 4 Minimum degree
In this section, we prove the following result concerning the minimum degree.
###### Theorem 5.
Let be a connected graph of order with minimum degree , then .
Given a graph , a set is called a two-step dominating set of if every vertex in which is not dominated by has a neighbor that is dominated by . Moreover, a two-step dominating set is called a two-way two-step dominating set if every pendant vertex of is included in , and every vertex in has at least two neighbors in , where denotes the set of all vertices at distance exactly from . Further, if is connected, is called a connected two-way two-step dominating set of .
###### Lemma 2.
[4] Every connected graph of order and minimum degree has a connected two-way two-step dominating set of size at most .
Proof of Theorem 5: The proof goes similarly as that of the main result in [11] by Li et al.
We are given a connected graph of order with minimum degree . The assertion can be easily verified for and so suppose . Let denote a connected two-way two-step dominating set of and . Then we have by Lemma 2. Let is a neighbor of in for and is a neighbor of in for .
Case 1. For each vertex , its neighbors in has at least one common neighbor in , i.e., .
We consider a spanning subgraph of (see Figure 2, where denotes the union of graphs each of which is isomorphic to the graph and similarly for and ). Next, we give a total-coloring to using . For the edges and vertices of , let be a spanning tree of . Then by Theorem 1, can be total-colored using such that for each edge , the colors of and are different from each other. We color in such a way and the edges of with any used colors (denote this coloring of by ). For the other edges and vertices in , color them as depicted in Figure 2.
Since each pair of vertices has a total proper path connecting them such that and , it suffices to show that is total proper connected in the assumption that . Take any two vertices and in . If , then has a neighbor in and similarly has a neighbor in . Hence, if , is a total proper path; otherwise, is a total proper path where is another neighbor of in . It is easy to check that and are total proper connected in all other cas | 2020-10-27 20:13:38 | {"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": 0.9573338031768799, "perplexity": 304.7072806540207}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107894759.37/warc/CC-MAIN-20201027195832-20201027225832-00459.warc.gz"} |
https://eprint.iacr.org/2015/405 | ### Feasibility and Infeasibility of Secure Computation with Malicious PUFs
Dana Dachman-Soled, Nils Fleischhacker, Jonathan Katz, Anna Lysyanskaya, and Dominique Schröder
##### Abstract
A recent line of work has explored the use of physically uncloneable functions (PUFs) for secure computation, with the goals of (1) achieving universal composability without additional setup, and/or (2) obtaining unconditional security (i.e., avoiding complexity-theoretic assumptions). Initial work assumed that all PUFs, even those created by an attacker, are honestly generated. Subsequently, researchers have investigated models in which an adversary can create malicious PUFs with arbitrary behavior. Researchers have considered both malicious PUFs that might be stateful, as well as malicious PUFs that can have arbitrary behavior but are guaranteed to be stateless. We settle the main open questions regarding secure computation in the malicious-PUF model: * We prove that unconditionally secure oblivious transfer is impossible, even in the stand-alone setting, if the adversary can construct (malicious) stateful PUFs. * If the attacker is limited to creating (malicious) stateless PUFs, then universally composable two-party computation is possible without computational assumptions.
Note: This is the full version of the paper.
Available format(s)
Category
Foundations
Publication info
A major revision of an IACR publication in CRYPTO 2014
Keywords
pufsfoundationsimpossibilityuniversal composability
Contact author(s)
jkatz @ cs umd edu
History
2018-01-15: revised
See all versions
Short URL
https://ia.cr/2015/405
CC BY
BibTeX
@misc{cryptoeprint:2015/405,
author = {Dana Dachman-Soled and Nils Fleischhacker and Jonathan Katz and Anna Lysyanskaya and Dominique Schröder},
title = {Feasibility and Infeasibility of Secure Computation with Malicious PUFs},
howpublished = {Cryptology ePrint Archive, Paper 2015/405},
year = {2015},
note = {\url{https://eprint.iacr.org/2015/405}},
url = {https://eprint.iacr.org/2015/405}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content. | 2022-07-05 03:47:49 | {"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": 0.1835566610097885, "perplexity": 9603.926248110478}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104512702.80/warc/CC-MAIN-20220705022909-20220705052909-00168.warc.gz"} |
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# The potential for a CRISPR gene drive to eradicate or suppress globally invasive social wasps
## Abstract
CRISPR gene drives have potential for widespread and cost-efficient pest control, but are highly controversial. We examined a potential gene drive targeting spermatogenesis to control the invasive common wasp (Vespula vulgaris) in New Zealand. Vespula wasps are haplodiploid. Their life cycle makes gene drive production challenging, as nests are initiated by single fertilized queens in spring followed by several cohorts of sterile female workers and the production of reproductives in autumn. We show that different spermatogenesis genes have different levels of variation between introduced and native ranges, enabling a potential ‘precision drive’ that could target the reduced genetic diversity and genotypes within the invaded range. In vitro testing showed guide-RNA target specificity and efficacy that was dependent on the gene target within Vespula, but no cross-reactivity in other Hymenoptera. Mathematical modelling incorporating the genetic and life history traits of Vespula wasps identified characteristics for a male sterility drive to achieve population control. There was a trade-off between drive infiltration and impact: a drive causing complete male sterility would not spread, while partial sterility could be effective in limiting population size if the homing rate is high. Our results indicate that gene drives may offer viable suppression for wasps and other haplodiploid pests.
## Introduction
CRISPR gene drives have been widely proposed as a promising potential technology for pest control or even eradication1,2,3. This technology provides an ability to disperse genetically engineered or altered genes throughout pest populations with much higher efficiency and prevalence than would be possible via normal genetic inheritance, even with genetic modifications that are deleterious for individuals and populations. CRISPR-Cas9 is an endo-nuclease system that will produce a targeted double-strand break in a DNA sequence based on complementarity to a guide RNA (gRNA) homing segment of ~ 20-bp4. A CRISPR-Cas9-generated double strand break can be repaired via homology-directed repair with a sequence with complementarity to the damaged region, converting heterozygous individuals for the mutation into homozygotes. This technology could be used to spread genetic variants through a population5. Some researchers highlight the possibility that gene drives could revolutionize pest control, making cost-effective eradication from islands or continents achievable3,6,7. Others recognise that issues such as genetic variation within a pest population might render gene drives ineffective8,9, or that gene drives could lead to unwanted global extinction of a species if the modified organism spreads widely10.
Understanding gene drive efficacy and the risks they pose is critical information required to provide an informed global debate and discussion. Detailed knowledge of target genes and their variation within native and introduced range populations are essential to predict resistance development and the usefulness or efficacy of gene drives. In vitro tests are needed to understand the efficacy of the CRISPR-Cas9 activity in genetically-variable populations, at non-target sites within the genome, as well as against related, non-target species. Simulation models are also required to predict the efficacy of the drive, as an understanding of the demography of the targeted population is crucial before the viability of gene-drive suppression can be critically and appropriately assessed11. Previous models for gene drives have assumed a diploid mating system, yet haplodiploid species comprise around 15% of the arthropods12. Many models focus on change in allele frequencies as a gene drive alleles spread13,14, whereas regulating population size or eradication is a key goal for pest management11,15,16.
Here, we provide a genetic and modelling analysis for a gene drive in the common wasp, Vespula vulgaris L (Fig. 1a). Common wasps have been included on the list of ‘100 of the World’s worst invasive alien species’17. Native to Eurasia, this insect has invaded Australia, South America, Hawaii and New Zealand18. Multiple invasions have occurred into the invaded ranges19. These wasps are generalist predators consuming 0.8–4.8 million loads of prey/ha in New Zealand20, compete with native species for resources21,22, and exert an annual cost of ~ NZ\$133 million18. They occur at extremely high densities that occupy over more than a million hectares of native forest18, making the annual cost and environmental impact of using pesticides prohibitive. New approaches to their management are needed.
Vespula wasps are haplodiploid, as are many of the other insects listed in the list of 100 of the World’s worst invasive insect species, including the red imported fire ant (Solenopsis invicta) and sweet potato whitefly (Bemisia tabaci)17. In haplodiploid insects, haploid males are produced from unfertilized eggs while females develop from fertilized eggs and are diploid. The typical life cycle of Vespula spp. wasps involves queens mating with 2–3 males in autumn23 before overwintering alone. In spring, queens begin a nest by laying fertilized eggs that produce female workers. The queen must feed and tend these larvae until they become adult workers, after which the queen will not leave the nest again. The colony grows over the summer period with several cohorts of sterile workers until new queens and males are produced in autumn. Gene drives have been proposed for use in common wasp management, with spermatogenesis genes as targets3,18. Spermatogenesis genes could be altered in queen wasps to cause viable sperm production in males to fail, resulting in sterile male production. Several potential spermatogenesis genes with likely male-specific expression have been identified in wasps24,25,26,27. These targets include the boule gene, which has been found to be essential for the entry and progression of maturation divisions and sperm differentiation in haploid males24. Should a modified queen mate with wild-type males, fertilized worker eggs will be produced, culminating in queen and male production in autumn, all of which will carry the CRISPR cassette and thus propagate it to the next generation. Should a modified queen mate with a genetically-modified male, fertilization will fail and all offspring in spring will be male. This nest will fail and die in spring or early summer as males do not forage or aid in nest maintenance. Should a modified queen mate with a mixture of wild-type and genetically-modified males, a substantially smaller nest is likely to be produced resulting in genetically-modified reproductive offspring in autumn. Incomplete knock-down efficiency of spermatogenesis using the CRISPR-Cas9 system28,29 similarly might produce nests of reduced size. A spermatogenesis gene drive could thus potentially eradicate wasps or act in a “suppression drive” fashion14 to lower their abundance.
A gene drive based on spermatogenesis failure could have the key advantage that it would be self-propagating and cost-effective for large-scale control of invasive pests. An investigation of such an approach will also inform the use of gene drives for other haplodiploid species. A critical limitation on the use of gene drives in wasp and pest control is a ‘social license to operate’3, wherein government policy and the use of these technologies should have informed public support. There is concern regarding gene drives as a new and uncertain technology, particularly concerns over their specificity, and issues if genetically-modified wasps escape beyond New Zealand4,30,31. Our goal in this study was to provide data for these debates by assessing the potential for a gene drive to control wasp populations using spermatogenesis genes as targets, identifying potential risks with resistance. We were particularly interested in naturally occurring variation of spermatogenesis gene targets, as in-frame variation can induce drive resistance32. Alternatively, such intra-specific variation could also enable a precision drive4 that targets specific genotypes represented in genetically depauperate invasive populations, as found for the Vespula species in New Zealand33,34. Precision drives could offer a safeguard in that if genetically modified individuals return to their home range, few or a limited number of genotypes in the native range would be affected. We also examined intra- and inter-specific non-target effects and modelled predicted wasp population trends and control. Rather than focusing on allele frequencies8,35,36, our modelling was based on effects on wasp numerical abundance for suppression or eradication32,37,38,39. We believe our study will inform the global debate on the use of gene drives for pest control as well as providing some of the risk/benefit data required for assessment of these technologies.
## Results and discussion
### Variation in eight spermatogenesis gene regions
We examined populations of common wasps from their native range (n = 83) and New Zealand (n = 43; Fig. 1b,c; Supplementary Table 1) for variation in eight gene regions predicted to be associated with spermatogenesis (Supplementary Table 2). Targets were chosen on the basis of a literature search for genes in other haplodiploid species that have been found to be involved in sperm production. Six genes were examined, but for two genes two regions were examined (boule and cdc), which could be useful in situations when multiple sgRNAs are used to reduce the generation of in-frame resistance alleles32. All eight regions showed some degree of variation, with a differing numbers of single nucleotide polymorphisms (SNPs) (Fig. 1d,e). Most of the genetic variation occurred in the native range, consistent with previous observations of considerably higher levels of intra-specific genetic diversity of Vespula wasps in the native range compared to the introduced populations19,33,34. A substantial reduction in genetic variation in the introduced range, relative to the native, is expected for many invasive species and could facilitate the development of precision drives. A precision drive would allow the CRISPR-Cas9 system to target only certain genotypes within the native and invaded ranges4, providing some assurance that an entire species would not be affected if modified wasps were to be introduced back to their native range.
Our preliminary screening determined that some of the loci had no variability, some loci presented some variability, and only one locus had a number of substitutions. From the initial eight gene regions examined, we selected four genes to sequence for all wasp samples collected. Boule protein, region 1 (boule1) and cell division cycle 25, region B (cdc25B) showed only a single SNP each, ocnus presented six substitutions, and sperm-specific dynein intermediate chain (sdic) presented nine SNPs and a 6-base pair indel in individuals from Austria, the UK and in many New Zealand and Russian wasps (Fig. 1). These results indicate that there is potential to select different target genes that demonstrate a substantial range of intra-specific variation. Boule being highly conserved and displaying little intra-specific variation implies that it has a strongly-selected, essential function. Targets with essential functions such as this would present a limited opportunity for the selection of wasp genotypes resistant to the guide-RNA. This limited variation also represents the highest probability for deleterious effects on the entire species should genetically-modified individuals be returned to their home range. Genes such as sdic show the opposite scenario. The intra-specific variation in this target offers an opportunity for a precision drive that could offer a degree of safeguard should genetically-modified individuals be returned to Europe: some genotypes within the home range would be affected, but not all. The variation in sdic might also mean there is unsampled variation present in the invaded range, leading to resistance, and potentially, eventual drive inefficacy.
The development of resistance alleles that could render the gene drive useless is of main concern. If a drive was to be deployed, our preference would be to use a ‘precision drive’ with genes such as ocnus or sdic. The use of multiple sgRNAs (single guide RNAs) or multiplexing to these targets could increase the effective homing rate and decrease the rate of resistant allele generation11,37. Other suggestions to limit resistance development include tightly regulated promoters to restrict nuclease expression to the early germline9. If resistance did arise, the solution could be as simple as re-designing the sgRNA to account for this variation. Rapid and efficient genetic transformation in insects is becoming possible through new approaches in CRISPR-Cas9 editing30. Exploiting the observed genetic variation between ranges into a daisy-chain drive system31 could offer substantial additional levels of safety.
### In vitro CRISPR-Cas9 testing and non-target effects
Three spermatogenesis gene sequences (boule1, sdic and ocnus) were chosen for use in in vitro CRISPR-Cas9 experiments. All sgRNA designs are presented in Supplementary Table 3. The sgRNA designed for boule1 encompassed a DNA sequence that was invariant across all wasp samples in the native and invaded range. Two sgRNA versions for sdic targeted the region around the 6-base pair indel in individuals present in New Zealand and the native range: one version targeted samples with the observed indel, and the other targeted samples without the indel. Two sgRNA versions were tested for ocnus: one version targeted wasps with a SNP at position 266 found in countries from the native range including Russia and Spain, while the other version targeted wasps without the SNP that included all wasp genotypes from New Zealand. A BLASTn search optimised for small query sequences indicated that these sgRNA sequences had no homology elsewhere within the common wasp genome. Our in vitro test examined for CRISPR-Cas9 cleavage of PCR products amplified from the boule1, ocnus and sdic genes in 10 samples from the native range, and 10 from New Zealand.
The five CRISPR-Cas9 assays for the three spermatogenesis genes showed varying results that were dependent on the sgRNA design. The sgRNA targeting boule1 cleaved this gene in all 20 V. vulgaris samples tested (Supplementary Table S7; Supplementary Fig. 1), which was expected due to the absence of any variation within this gene across samples. The two sdic-targeted sgRNA assays showed indel-dependent results. sgRNA assays that targeted samples without the 6-bp deletion effectively cleaved those samples, but also resulted in partial cleavage of the gene from wasp samples that were indel-positive (Supplementary Fig. 2). Assays using sgRNAs designed for indel-positive samples worked in a highly targeted fashion. This indel-positive sgRNA cleaved only indel-positive samples, although the cleavage was incomplete (Supplementary Fig. 3). The two ocnus sgRNA assays were designed to discriminate between differences in one SNP. The first assay used sgRNA targeting the Cas9 cleavage to samples with the SNP and showed specificity, where only SNP-positive samples where cleaved and remaining samples were unaffected (Supplementary Fig. 4). However, contrary to our expectations the second sgRNA design that targeted SNP-negative samples, did not show any specificity. This sgRNA cleaved the ocnus gene in all samples that we analysed (Supplementary Fig. 5). That the ocnus SNP gave better precision and efficacy than the sdic indel was unexpected, and we cannot explain why the SNP would be more effective than an indel. This emphasizes the importance of in-vitro testing of sgRNAs before development of transgenics.
Our results provide confidence that a precision drive could be implemented for common wasp control. Additional and more extensive sampling of genes within native and invasive populations could identify SNPs or indels allowing efficient specificity for invaded populations, with a higher proportion of the native populations being unaffected. Quantification of the sgRNA assays showed considerable variation with the highest cleavage levels (93.78 ± 0.87%; mean ± SE) in the ocnus SNP-targeting assay and the sdic indel positive-targeting assay with the lowest rate of 46.59 ± 22.17% (Supplementary Fig. 3).
One of three key issues in proposing gene drive targets is understanding target specificity10,14,40,41. In the unlikely event of hybridization, would there be any possibility for the CRISPR-Cas9 modified wasps to affect other species? For our specificity analysis, we first bioinformatically compared the five sgRNA sequences for the boule1, ocnus and sdic spermatogenesis genes for the common wasp, with those from the genomes in the related species V. germanica, V. pensylvanica, the paper wasp Polistes dominula, and two bee species Apis mellifera and Bombus terrestris. A BLASTn search specific for each of our five 23-base pair sgRNA designs showed variable homology, depending on the gene target and species. None of the five gRNA sequences were homologous with any sequence on the A. mellifera, B. terrestris or P. dominula genomes. The two sgRNA sequences designed for boule1, and one for sdic, had identical targets on chromosome 4 of all three Vespula species. Similarly, the sgRNA targeting ocnus was conserved and present on chromosome 12 of all Vespula. The only sgRNA showing a high degree of specificity was the sgRNA targeting the indel of sdic, which did not correspond to sequences on any genome examined (the V. vulgaris genome sequenced was from a nest in New Zealand without this indel). This genomic analysis supported a recent phylogenetic analysis for these species42, with spermatogenesis genes for common wasp demonstrating a high degree of similarity to V. germanica while being substantially different to the bees A. mellifera and B. terrestris (Supplementary Fig. 6).
Finally, where available we amplified the spermatogenesis gene regions boule1, ocnus, and sdic genes for all five non-target species for in vitro tests using our sgRNA in a CRISPR-Cas9 experiment (no equivalent genes were observed in some of the species). Our in vitro laboratory experiments supported the genomic bioinformatic analysis. The assays using the five designed sgRNA showed no indication of any gene cleavage for any non-target species outside of the Vespula genus (Supplementary Fig. 7). Gene cleavage was observed within all three Vespula species examined for sgRNA designed for boule1, one of the two sdic, and one of the two ocnus genes (Table 1). The other sdic sgRNA was specific to the common wasp samples. The remaining ocnus sgRNA designs were specific to genotypes within common wasps. Clearly, the specificity of a gene drive for pest management will depend highly on the gene region targeted and the sgRNA design. It appears possible to design multiple sgRNAs that can be specific to genotypes within a species, or that could function across different species within a genus.
### Population models
We developed population models for gene drive in common wasps based on spermatogenesis knockdown. Diploid wasp queens are univoltine. Queens mate two or three times23 with haploid males before overwintering alone. In some vespids fecundity and nest size depend on the number of viable sperm collected43, so we investigated a range of feasible alternatives in the context of full and partial drone sterility. These different approaches suggest that drives could function in an “eradication drive” or a “suppression drive” fashion, which are designed to extirpate or decrease the size of a population, respectively14. The models are based on the introduction of 100 wasp queens at time 0 into a 1 km2 area of New Zealand forests containing an average of 13.5 nests ha−144. Integer-based versions of models, which included stochasticity and gene drift, produced similar results to our deterministic models (Supplementary modelling methods and results).
As expected, the models indicated that a nuclear gene for drone sterility without a CRISPR cassette would be rapidly eliminated (Fig. 2a), but a fitness-neutral gene introduced with a CRISPR cassette could take as little as a decade (ten queen generations) to almost fully infiltrate the population (Fig. 2b,e). These results are similar to the predicted dynamics of Y-chromosome-linked modifications in male heterogametic species, that are designed to disrupt the fitness of female descendants45. Infiltration took longer with lower values for the homing rate and/or mating competitiveness of carrier drones. The potential for rapid spread of fitness neutral drives is consistent with other models (33). However, the models showed that gene drives causing complete male sterility would not spread (Fig. 2c), as the gene drive would be opposed by its loss through the sterility of male carriers. With perfect homing this balance results in a stable allele frequency: since male carriers are sterile offspring arise only from matings with non-carrier males, and the homing effect ensures all homozygous female carriers produce carrier offspring, while heterozygous carriers produce half homozygous carriers and half non-carriers. Therefore, the allele frequency does not change from one generation to the next except through drift or human management of wasp populations. If the homing rate is less than perfect, however, there will be a tendency for the proportion of carriers in the population to decline each generation, and the CRISPR cassette will eventually be eliminated (Fig. 2f).
Of particular interest is the result that gene drives causing partial drone sterility (Fig. 2d) may be more effective than those for complete sterility (Fig. 2c), because they allow the drive to spread naturally and subsequently reduce the population, providing the homing rate is sufficiently high (Fig. 2g). Allowing for some male carriers to breed breaks the deadlock between homing and drive loss through infertility. The amount by which the population is eventually reduced by partial gene sterility depends on the number of males each queen mates with (or their degree of polyandry; Fig. 3a) and the degree to which relative sperm load affects queen fecundity (Fig. 3b). However, in the partial sterility simulations, like the full-sterility model, the greater the population suppression, the longer it takes for the gene to spread. Such an effect with gene drive spread and effects has previously been likened to relatively non-virulent pathogens spreading, while more virulent pathogens with high fitness effects fail to spread or suppress populations39. Incorporating realistic population demography into simulation models, such as polygynous mating and partial drone sterility (Fig. 4), can substantially alter predictions of drive success for pest eradication and suppression outcomes11,46.
Despite the tendency not to spread, a drone sterility drive may still have an impact on population size if it can be made sufficiently abundant in the population. One potential way to achieve this in practice is to release carrier queens into a population that has been temporarily reduced. Neither 95% effective poisoning at 10-yearly intervals (Fig. 5a) nor regular release of gene drive carriers (Fig. 5b) would make a long-term impact on wasp population size, but a combination of both could cause extinction if the homing rate is sufficiently high (Fig. 5c). The key is to reach the critical threshold for carrier abundance whereby the sterilizing effect of the gene drive exceeds the population’s reproductive potential. Analytical estimates for this threshold (Supplementary modelling methods and results) depend in part on the degree of polyandry (Fig. 3). For wasps, the default parameter values suggested that eradication would require at least 96% of queens to carry the drone sterility allele (Fig. 5a–e). Other modelling approaches have recently also found that in principle, CRISPR driver alleles can spread in pest populations of haplodiploid species across a wide range of conditions47. In drives with a high fitness cost, as suggested in our modelling work, a high conversion rate would be needed to successfully fix the modified germline allele. The conditions favouring the spread of genetically modified alleles have been suggested to likely be narrower in haplodiploid than diploid species47 .
## Conclusions
Webber et al.10 identified three key issues in their discussion on whether CRISPR-based gene drives represent a “biocontrol silver bullet or global conservation threat”: (i) the importance of understanding target specificity, (ii) the implications of population connectivity, and (iii) the need to carefully consider unintended cascades for community dynamics. These concerns are echoed elsewhere15,16,41,48,49.
Regarding specificity, our analysis on spermatogenesis genes highlighted a range of potential targets. Some of these gene targets, such as ocnus, could enable drive intra-specific specificity and selection. Individual genotypes within a species and invaded range could be culled, but a careful design using the ocnus gene would provide a safeguard against an entire species being affected. There are no off-target sites in the common wasp genome that would be affected by a CRISPR-Cas9 cassette that used sgRNA based on our ocnus gene. The potential for de novo resistance development or mutation represents a risk in the use of gene drives, though methods of resistance management have been suggested8,9,11,14. Multiplexing additional population-specific sgRNA within the engineered cassettes for each target gene may overcome resistance issues50, with our analysis discovering multiple targets on the ocnus and sdic genes that provide such prospective sgRNAs. After gene drive release, population monitoring and potential redesign of sgRNAs could also account for this issue. A common question from public audiences is “what happens if a genetically modified wasp mates with a related wasp, or even a honey bee?”40,41. We have demonstrated for such an improbable hybridization event, utilizing intra-specific genetic variation would mean that even closely-related species are unlikely to be affected. Some conserved gene targets, such as boule represent more of a risk for non-target effects, though any potential non-target effects appear Vespula genus specific.
Connectivity between the invaded and native range will always be an issue with the use of gene drives. The safeguard of using intra-specific variation4 targeting only some genotypes could limit the native range risk. Techniques such as daisy drives could offer additional safeguards31. There have been at least six common wasp introduction events into New Zealand19, though it is unknown how many return events back to Europe have occurred. New Zealand as a remote island nation in a separate hemisphere provides a further safeguard. Nevertheless, the globe is highly interconnected by trade and the risk of a return of wasps is non-negligible. Common wasps are not universally hated in Europe and may play an important role in ecosystem function there51,52. Ideally, a fast extinction process resulting from a gene drive would reduce the potential and time for wasps to enter a trade pathway back to Europe. Our modelling analysis indicates, however, that some gene drive targets will not result in extinction or at least take several decades to achieve. The probability of genetically-modified wasps entering a trade pathway and returning to the native range increases with increasing time. We thus consider that population-specific gene targets in precision drives are more important in situations where pest suppression is likely, compared to eradication scenarios (when there is less of chance for genetically modified individuals of returning to their native range).
The most likely unintended cascade effect would be for invasive German wasps to increase in abundance, should common wasps be culled or their populations substantially reduced. Common wasps displaced German wasps from beech forests in New Zealand53, where they would almost certainly return. This scenario suggests simultaneous control of both Vespula species would be necessary. Other unintended cascades might include an increase in abundance in pest species such as flies or some forest defoliating insects. Unintended consequences of wasp removal are valid concerns, but we also note that New Zealand ecosystems evolved without any social wasp species.
Pest control or eradication is essential to limit biodiversity declines and extinctions, to which these wasps are contributing in New Zealand54,55. Any control method has potential problems and pitfalls and should be approached with care. Gene drives are a potential next-generation technology for pest control, including for wasps. Our modelling analysis indicated that a gene drive using spermatogenesis genes could result in population suppression, but eradication seems likely to require a combined approach with other control methods. This outcome highlights the need to carefully assess and exploit variation in target genes to limit the potential of genetically-modified wasps affecting populations in the native range. Such genetic variation clearly occurs. Further genetic analysis on different gene targets could identify targets with even higher specificity, which could also influence suppression or extinction goals differently if used in precision or daisy-drive31 fashion. Finally, we note that should a gene drive for these wasps be desired, significant challenges still lie ahead in the production of a transgenic wasp line. The efficient production of CRISPR-Cas9 modified hymenopteran insects including honey bees (Apis mellifera)56 and ants57 offers avenues for this work.
## Materials and methods
### Insect collection, DNA extraction, and spermatogenesis gene analysis
We collected Vespula vulgaris wasps throughout its native range in Eurasia (n = 83) and its introduced range in New Zealand (n = 43) to screen genetic variation in genes involved in the process of sperm production or sperm maturation. Samples were either collected fresh for this study, or were from a previous project examining genetic diversity using mitochondrial genes19. Specimen collection information is presented in Supplemental Table 1 and includes detail on the other hymenopteran species used in this project. Individuals were collected and immediately placed in 99% ethanol or frozen until DNA extraction. We extracted genomic DNA from wasps using a CTAB and chloroform based protocol. Briefly, we homogenised the whole wasp in a microcentrifuge tube with 1 mL of GENEzol plant DNA reagent (Geneaid Biotech, Taiwan) and 5 μL of β-mercaptoethanol (Sigma Aldrich, Michigan, USA) in a Precellys Evolution homogeniser (Bertin Instruments, France).
We initially screened approximately 30 individuals for eight gene regions of interest: boule protein region 1, boule protein region 2, cdc25 region A, cdc25 region B, fuzzy onions, helicase MCM8-like, ocnus, and sdic (Supplementary Table 2). Primers for these genes in V. vulgaris (Supplementary Table 4) were designed by aligning sequences for these genes available for other insect species on GenBank to a V. vulgaris draft genome. Each gene was amplified in 15 μL reactions containing 1 μL of template DNA, 0.5 μM forward primer, 0.5 μM reverse primer, 0.5 μL Bovine Serum Albumin (Sigma Aldrich, New Zealand), and 1 × MyTaq Mix (Bioline, London, UK). Each PCR product was examined by agarose gel electrophoresis and purified with rSap combined with Exo 1 (New England Biolabs, Ipswich, MA, USA). Sequencing was performed on an ABI 3130 × 1 Genetic Analyzer (Applied Biosystems, Foster City, CA, USA) at Macrogen Inc. (South Korea). We aligned gene sequences using the default alignment algorithm implemented in the software Geneious v. 10.2.6 (https://www.geneious.com). GenBank accession numbers for these sequences are MN088861–MN089473. See Supplementary Table 5 for detailed accession number information relating loci with specimens. The R package vegan58 was then used in a rarefaction analysis to infer the discovery rate and diversity of nucleic acid substitutions within each spermatogenesis gene region.
### CRISPR-Cas9 in vitro DNA cleavage assay
We designed sgRNAs for each gene (Supplementary Table 3). Synthetic sgRNAs (Invitrogen TrueGuide sgRNA, ThermoFisher Scientific) were diluted to 100 μM and stored at − 20 °C. Prior to the in vitro CRISPR-Cas9 assay, PCR products for 20 Vespula vulgaris samples were generated from boule, sdic and ocnus genes using a high-fidelity PCR system (Platinum SuperFi PCR Master Mix, ThermoFisher Scientific). In addition, PCR products were generated from non-target species Vespula germanica (boule, sdic and ocnus), Vespula pensylvanica (boule, sdic and ocnus), Polistes dominula (boule and sdic), Bombus terristris (boule and sdic) and Apis mellifera (sdic). Each 25 μL reaction contained 50 ng of template DNA, forward and reverse primers (Supplementary Tables 4 and 6) at final concentrations of 0.5 μM, and Platinum SuperFi PCR Master Mix. Reactions proceeded as follows: 98 °C for 30 s; 35 cycles of 98 °C for 10 s, 64.7 °C (V. vulgaris boule), 65 °C (V. vulgaris sdic) or 62.4 °C (V. vulgaris ocnus) for 10 s, 72 °C for 30 s; 72 °C for 5 min; 4 °C (hold). PCR products were purified using DNA Clean and Concentrator-5 columns (Zymo Research, CA, USA) and DNA concentrations were measured using a NanoPhotometer (Implen, Germany). PCR products were diluted to 30 nM with water. The CRISPR-Cas9 in vitro assay followed the manufacturers protocol described for Cas9 nuclease (New England Biolabs, MA, USA). In brief, reactions were assembled containing NEBuffer 3.1 (New England Biolabs, MA, USA), 30 nM sgRNA, 30 nM Cas9 and water to a final volume of 27 μL followed by preincubation for 10 min at 25 °C. Three μL of 30 nM PCR product was then added (final concentration 3 nM) and the reaction incubated at 37 °C for 15 min. The reaction was ended by adding Proteinase K (New England Biolabs, MA, USA) and incubating at room temperature for 10 min. To purify and concentrate the digested DNA, samples were loaded onto DNA Clean and Concentrator-5 columns (Zymo Research, CA, USA) and eluted in 10 μL of water. The samples were then resolved by 2% agarose gel electrophoresis for fragment analysis. The origin and details of the 10 wasps from the native range, and 10 from the invaded, are shown in Table S7 of the Supplementary Material.
### Population modelling for wasp population control
We derived and analysed a population model for a gene drive affecting spermatogenesis in Vespula wasps, as described in full in the Supplementary modelling methods and results. Summer queen density, equivalent to nest density, is denoted Q, with subscripts specifying the genotype of the queens and of the drones with which they mated the previous autumn (mating only occurs in autumn and, if any queens fail to successfully mate—the frequency of which is unknown—their nests will fail in the following spring). Hence the three diploid queen genotypes (ww, wi and ii, where w denotes the WT gene and i the modified gene) and two haploid drone genotypes (w and i) lead to six possible queen types in summer: Qww_w, Qww_i, Qwi_w, Qwi_i, Qii_w, and Qii_i. The drones D produced in the subsequent autumn,
$$D_{w} \propto Q_{ww\_w} + Q_{ww\_i} + \frac{1}{2}\left( {Q_{wi\_w} + Q_{wi\_i} } \right)$$
$$D_{i} \propto \frac{1}{2}\left( {Q_{wi\_w} + Q_{wi\_i} } \right) + Q_{ii\_w} + Q_{ii\_i}$$
mix widely and mate with the autumn gynes (new queens) at random. Assuming the spermatogenesis gene drive results in only a proportion p of matings with carrier drones transferring viable sperm, the proportion of gynes being successfully fertilised is closely approximated as
$$f = 1 - \left( {\frac{{pcD_{i} }}{{D_{w} + cD_{i} }}} \right)^{m}$$
where m = 2.5 is the average number of matings per gyne23 and c is the relative competitiveness of carrier drones. Furthermore, the proportion of fertile matings that lead to the WT allele being passed on to offspring is
$$j = \frac{{D_{w} }}{{D_{w} + c(1 - p)D_{i} }}$$
Old queens die in winter, but a proportion s = 0.02 of the gynes survive to compete for nest sites in spring. These processes are combined in a single density-dependent survival factor
$$g = \frac{s}{{1 + \frac{s\Sigma G}{n}}}$$
where n = 1,500 km−2 is a nest site competition factor resulting in a summer nest density of 1,350 km−2 (40). Importantly, unmated queens participate in nest site competition but are unable to produce diploid workers, so these nests subsequently fail. Here, ΣG is the total density of gynes that were produced in autumn and is given by λZb(Qww_w + Qwi_w + Qii_w + (1 − p)(Qww_i + Qwi_i + Qii_i)) where the potential number of gynes produced per nest λ = 560 is modified by a power function of the sperm load carried by queens (38)
$$Z \cdot = \frac{{D_{w} + c(1 - p)D_{i} }}{{D_{w} + cD_{i} }}$$
To avoid the use of additional subscripts, we use a dot (·) to indicate values in the following generation t + 1. Hence sperm load potentially affects the number of progeny produced in the following generation. Since there is exactly one nest per queen, the model for nest density from one summer to the next (indicated by a dot) is:
$$Q_{ww\_w} \cdot = \lambda Z^{b} fgj\left( {Q_{ww\_w} + \frac{1}{2}Q_{wi\_w} } \right)$$
$$Q_{wi\_w} \cdot = \lambda Z^{b} fgj(1 - h)\left( {(1 - p)\left( {Q_{ww\_i} + \frac{1}{2}Q_{wi\_i} } \right) + \frac{1}{2}Q_{wi\_w} + Q_{ii\_w} } \right)$$
$$Q_{ii\_w} \cdot = \lambda Z^{b} fgj\left( {(1 - p)\left( {hQ_{ww\_i} + \frac{(1 + h)}{2}Q_{wi\_i} + Q_{ii\_i} } \right) + h\left( {\frac{1}{2}Q_{wi\_w} + Q_{ii\_w} } \right)} \right)$$
$$Q_{ww\_i} \cdot = \frac{(1 - j)}{j}Q_{ww\_w} \cdot$$
$$Q_{wi\_i} \cdot = \frac{(1 - j)}{j}Q_{wi\_w} \cdot$$
$$Q_{ii\_i} \cdot = \frac{(1 - j)}{j}Q_{ii\_w} \cdot \cdot$$
We explored this model for: a normal gene (h = 0) causing complete (p = 1) or partial (0 < p < 1) drone sterility; a gene drive (h > 0) potentially affecting drone mating competitiveness (c ≤ 1) but with no effect on drone fertility (p = 0); a gene drive (h > 0) causing complete drone sterility (p = 1); and a gene drive (h > 0) causing partial drone sterility (p < 1). We also simulated integer-based versions of the models to include the effects of gene drift. Full details are given in the Supplementary Material.
## Data availability
Additional information including additional methods and materials, results and Genbank accession numbers can be found in the file Supplementary Information.
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## Acknowledgements
We thank C. Philips for discussion on this work and collaborators in Europe and New Zealand for providing wasp samples. We are also grateful to the anonymous referees who provided feedback on an earlier version of the manuscript. This work was funded by the Ministry of Business, Innovation and Employment (New Zealand’s Biological Heritage NSC, C09X1501) to all authors and a Victoria University of Wellington University Research Fund grant to PJL. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
## Author information
Authors
### Contributions
P.J.L. and P.K.D. conceived and coordinated the study. M.B. and J.W.B. led the design and experimental molecular genetics work. M.B., J.W.B., P.K.D., J.G. and P.J.L. analyzed the genetic data. J.M.K. led and implemented the modelling analysis. P.J.L. led the paper writing with input from all co-authors.
### Corresponding author
Correspondence to Philip J. Lester.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
### Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Lester, P.J., Bulgarella, M., Baty, J.W. et al. The potential for a CRISPR gene drive to eradicate or suppress globally invasive social wasps. Sci Rep 10, 12398 (2020). https://doi.org/10.1038/s41598-020-69259-6
• Accepted:
• Published:
• ### The impact of female mating strategies on the success of insect control technologies
• Andreas Sutter
• , Tom AR Price
• & Nina Wedell
Current Opinion in Insect Science (2021)
• ### Emerging patterns in social wasp invasions
• Erin E Wilson Rankin
Current Opinion in Insect Science (2021)
• ### Demystifying the Risk Assessment Process for Laboratory-Based Experiments Utilizing Invasive Genetic Elements: It Is More Than Gene Drive
• Zach N. Adelman
Applied Biosafety (2021)
• ### Culex quinquefasciatus: status as a threat to island avifauna and options for genetic control
• Tim Harvey-Samuel
• , Thomas Ant
• , Jolene Sutton
• , Chris N. Niebuhr
• , Samoa Asigau
• , Patricia Parker
• , Steven Sinkins
• & Luke Alphey
CABI Agriculture and Bioscience (2021) | 2021-04-19 03:38:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.4331904947757721, "perplexity": 12287.011829004392}, "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-2021-17/segments/1618038863420.65/warc/CC-MAIN-20210419015157-20210419045157-00280.warc.gz"} |
https://maze-rl.readthedocs.io/en/latest/trainers/maze_rllib_runner.html | # Maze RLlib Runner¶
The RLlib Runner allows you to use RLlib Trainers in combination with Maze models and environments. Ray-RLlib is one of the most popular RL frameworks (algorithm collections) within the scientific community but also when it comes to practical relevance. It already comprises an extensive and tuned collection of various different RL training algorithms. To gain access to RLlib’s algorithm collection while still having access to all of practical Maze features we introduce the Maze Rllib Module. It basically wraps Maze models (including our extensive Perception Module), Maze environments (including wrappers) as well as the customizable Maze action distributions. It further allows us to use the Maze hydra cmd-line interfaces together with RLlib while at the same time using the well optimized algorithms from RLlib.
This page gives an overview of the RLlib module and provides examples on how to apply it.
## List of Features¶
• Use Maze environments, models and action distributes in conjunction with RLlib algorithms.
• Make full use of the Maze environment customization utils (wrappers, pre-processing, …).
• Use the hydra cmd-line interface to start training runs.
• Models trained with the Maze RLlib Runner are fully compatible with the remaining framework (except when using the default RLlib models).
## Example 1: Training with Maze-RLlib and Hydra¶
Using RLlib algorithms with Maze and Hydra works analogously to starting training with native Maze Trainers. To train the CartPole environment with RLlib’s PPO, run:
$maze-run -cn conf_rllib env.name=CartPole-v0 rllib/algorithm=ppo Here the -cn conf_rllib argument specifies to use the conf_rllib.yaml (available in maze-rllib) package, as our root config file. It specifies the way how to use RLlib trainers within Maze. (For more on root configuration files, see Hydra overview.) ## Example 2: Overwriting Training Parameters¶ Similar to native Maze trainers, the parametrization of RLlib training runs is also done via Hydra. The main parameters for customizing training and are: • Environment (env configuration group), configuring which environment the training runs on, this stays the same as in maze-train for example. • Algorithm (rllib/algorithm configuration group), specifies the algorithm and its configuration (all supported algorithms). • Model (model configuration group), specifying how the models for policies and (optionally) critics should be assembled, this also stays the same as in maze-train. • Runner (rllib/runner configuration group), specifies how training is run (e.g. locally, in development mode). The runner is also the main object responsible for administering the whole training run.. The runner is also the main object responsible for administering the whole training run. To train with a different algorithm we simply have to specify the rllib/algorithm parameter: $ maze-run -cn conf_rllib env.name=CartPole-v0 rllib/algorithm=a3c
Furthermore, we have full access to the algorithm hyper parameters defined by RLlib and can overwrite them. E.g., to change the learning rate and rollout fragment length, execute
$maze-run -cn conf_rllib env.name=CartPole-v0 rllib/algorithm=a3c \ algorithm.config.lr=0.001 algorithm.config.rollout_fragment_length=50 ## Example 3: Training with RLlib’s Default Models¶ Finally, it is also possible to utilize the RLlib default model builder by specifying model=rllib. This will load the rllib default model and parameters, which can again be customized via Hydra: $ maze-run -cn conf_rllib env.name=CartPole-v0 model=rllib \
model.fcnet_hiddens=[128,128] model.vf_share_layers=False
## The Bigger Picture¶
The figure below shows an overview of how the RLlib Module connects to the different Maze components in more detail:
## Good to Know¶
Tip
Using the the argument rllib/runner=dev starts ray in local mode, by default sets the number workers to 1 and increases the log level (resulting in more information being printed). This is especially useful for debugging.
Tip
When watching the training progress of RLlib training runs with Tensorboard make sure to start Tensorboard with --reload_multifile true as both Maze and RLlib will dump an event log. | 2021-09-28 23:34:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.1906798630952835, "perplexity": 6833.183870135589}, "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/1631780060908.47/warc/CC-MAIN-20210928214438-20210929004438-00041.warc.gz"} |
http://rockandcode.ga/linux/0141-Virtual-Memory-Tuning/ | # Virtual Memory Tuning
I got to looking into managing memory and power settings lately which led me to a slew of vm settings. This is configured through a system daemon (/etc/sysctl.d/daemon_name.conf). In this case I’m working with virtual memory, so with good sense I dubbed the file vm.conf. The descriptions below are from examples in the Arch Wiki.
### Swappiness:
I have mentioned swappiness before. The same rules apply.
/etc/sysctl.d/vm.conf
-----
# Swappiness (0 - 100)
# low = LESS swap use
# high = MORE swap use
vm.swappiness=100 # default 60
### Dirty Writeback:
# Drity Writes (frequency)
# Decreased to save power per Arch Wiki
vm.dirty_writeback_centisecs = 6000
### Dirty Ratio:
Per the Wiki’s recommendation I scaled the values to match my X205TA’s 2 GiB of RAM.
# Dirty Ratio (default 10%)
# Percentage of total system memory, the number of pages at which
# a process which is generating disk writes will start writing
# out dirty data. 10% sane for 0.5 GB RAM, too much for greater
vm.dirty_ratio = 3 # default 10
### Dirty Background Ratio:
Similar scaling took place here.
# Dirty Background
# Percentage of total system memory, the number of pages at which
# the background kernel flusher threads will start writing out
# dirty data. 5% sane for small mem, too great for large
vm.dirty_background_ratio = 2 # deafult 5
### Laptop Mode:
This is supposed to enable some kernel modules to help laptops, so I simply followed the Wiki’s reccomendation here as well.
# Laptop Mode
# Sane vaule to save power per Arch Wiki
vm.laptop_mode = 5
### Summary:
At the end of the day you create one file and it will load on every boot. The final thing might look something like this:
/etc/sysctl.d/vm.conf
-----
# Swappiness (0 - 100)
# low = LESS swap use
# high = MORE swap use
vm.swappiness=100 # default 60
# Drity Writes (frequency)
# Decreased to save power per Arch Wiki
vm.dirty_writeback_centisecs = 6000
# Dirty Ratio (default 10%)
# Percentage of total system memory, the number of pages at which
# a process which is generating disk writes will start writing
# out dirty data. 10% sane for 0.5 GB RAM, too much for greater
vm.dirty_ratio = 3 # default 10
# Dirty Background
# Percentage of total system memory, the number of pages at which
# the background kernel flusher threads will start writing out
# dirty data. 5% sane for small mem, too great for large
vm.dirty_background_ratio = 2 # deafult 5
# Laptop Mode
# Sane vaule to save power per Arch Wiki
vm.laptop_mode = 5
References: | 2018-03-24 08:03:32 | {"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": 0.19782571494579315, "perplexity": 12847.323207180527}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257649961.11/warc/CC-MAIN-20180324073738-20180324093738-00655.warc.gz"} |
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# Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400
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Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 [#permalink]
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26 Mar 2019, 05:31
00:00
Difficulty:
25% (medium)
Question Stats:
85% (02:17) correct 15% (02:58) wrong based on 20 sessions
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Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
(A) 129
(B) 137
(C) 174
(D) 223
(E) 411
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Re: Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 [#permalink]
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26 Mar 2019, 06:41
Bunuel wrote:
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
(A) 129
(B) 137
(C) 174
(D) 223
(E) 411
So there is 25+386 calories in 100+100+400 or 600 calories.
we are looking for calories in 200 ml, so amount of calories = $$(25+386)*\frac{200}{100+100+400}=\frac{411}{3}=137$$
B
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Re: Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 [#permalink]
### Show Tags
26 Mar 2019, 11:36
Bunuel wrote:
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
(A) 129
(B) 137
(C) 174
(D) 223
(E) 411
total calories in 600 gms lemonade ; 25+386 ; 411
so per gm lemonade calorie ; 411/600
and in 200 gms ; 411*200/600 ; 137
IMO b
Re: Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 [#permalink] 26 Mar 2019, 11:36
Display posts from previous: Sort by | 2019-10-22 11:17:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.3727862238883972, "perplexity": 8660.428619526028}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987817685.87/warc/CC-MAIN-20191022104415-20191022131915-00235.warc.gz"} |
http://openstudy.com/updates/512b9da5e4b098bb5fbb19b9 | ## TammisaurusRex Group Title "After 6 years, what is the total amount of a compound interest investment of $35,000 at 4% interest, compounded quarterly?" I tried to solve this question and I thought I did it right but my answer matched none of the multiple-choice answers I was given, so apparently I did something wrong. one year ago one year ago • This Question is Closed 1. TammisaurusRex Group Title I think the formula I'm supposed to use is the Compound Interest Formula: $A = P(1 + \frac{ r }{ n })^{nt}$ Where A = total, P = principal, t = time (years), r = annual interest rate, and n = number of compounds per year 2. TammisaurusRex Group Title But every time I plug the values in I get 89,000 something, but my choices are as follows: A.$37,153.21 B. $39,438.88 C.$44,440.71 D. \$56,295.30
3. amistre64 Group Title
show me how you filled in the formula
4. TammisaurusRex Group Title
Oh wait!! I realize what I did wrong! Instead of plugging in r/n inside the parenthesis, I only plugged in r.
5. TammisaurusRex Group Title
Let me try it one more time and I'll see if it works.
6. amistre64 Group Title
my idea was you didnt convert 4% correctly as .04 ... but that was just a guess
7. TammisaurusRex Group Title
I got it! Thank you for your help though (:
8. stamp Group Title
9. amistre64 Group Title
i knew you could :) | 2014-07-31 13:43:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5191128849983215, "perplexity": 6401.819216407951}, "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/1406510273350.41/warc/CC-MAIN-20140728011753-00132-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://datainfer.wordpress.com/ | # git cheat-sheet: submodule
• Clone a project with submodules:
• Method 1:
git clone MainProjectURL
git submodule update –recursive –init
• Method 2:
git clone –recursive MainProjectURL
• Pull updates from the upstream:
• Method 1:
git submodule update –recursive –remote –merge
• Push submodules:
git push –recurse-submodules=on-demand
# Repair GRUB2 for UEFI after Windows Installation
### Scenario:
A windows installation wipe out your old GRUB2. You want to repair.
### Solution:
Suppose that /dev/sdaX is where your ubuntu was installed and /dev/sdaY is where you want to install GRUB2.
Boot with Ubuntu LiveCD. Open a terminal, then do the following:
# We call the Ubuntu from the LiveCD the Live Ubuntu
# We call the Ubuntu on our local partition /dev/sdaX the Local Ubuntu
sudo mount /dev/sdaX /mnt # mount the Local Ubuntu into the Live Ubuntu
sudo mount /dev/sdaY /mnt/boot/efi # mount the GRUB2 partition into the Local Ubuntu
for i in /dev /dev/pts /proc /sys /run; do sudo mount -B $i /mnt$i; done
# This step is NECESSARY. It makes the Local Ubuntu see what the Live Ubuntu see. Then the update-grub can work correctly.
sudo chroot /mnt # Now we switch from the Live Ubuntu to the Local Ubuntu
sudo grub-install /dev/sdaY # Install GRUB2
sudo update-grub # Update GRUB2
# Windows: Unable to Delete Folder/File
To delete folder: rd /s “\\?\D:\bad\folder\path ”
To delete file: del “\\?\D:\bad\file\path “
# Activate Matlab on Linux without Ethernet
Edit /etc/udev/rules.d/70-persistent-net.rules
Rename eth0 to eth1 and make a duplicate entry of wlan0, and rename the duplicate entry to eth0.
In this way, Matlab will consider your wlan0 as eth0 and go through the activation procedure.
# Lyx: Add double-column line numbering
Install a texlive package:
\$ sudo apt-get install texlive-humanities
In lyx preamble:
\usepackage[switch,columnwise]{lineno}
\linenumbers % Line numbering starts here.
# Subequations in Lyx
## Subequation Numbering
If you want to group the equation numbers as, for example, (1a) and (1b), then you must do the following:
1. Add \usepackage{amsmath} to the document preamble.
2. Before the equation block, hit CTRL-L to add a LaTeX block, and type \begin{subequations}.
3. After the equation block, hit CTRL-L to add a LaTeX block, and type \end{subequations}.
The document should now look like this:
and result in:
A sample document can be found here: MultilineEquationNumbering.lyx
If you want to add a cross-reference to the entire block of equations, e.g. (1) rather than (1a) or (1b) in the example above, you need to add a label to the subequations block. Just move your mouse immediately after the \begin{subequations} LaTeX block, then add a label. You can then add a Cross-Reference to this label as normal.
Alternatives to OwnCloud:
(1) Bittorrent Sync: Its P2P idea is very attractive, but currently it is not reliable even with its basic purpose: sync and update with the right version of a file. Sometimes, it accidentally delete files; Sometimes, it overwrites new files with way old files. This happened to me just twos before I was going to submit my critical project deadline. Many other users also reported similar data lose. I will never use it before they have reliably implemented conflict management.
Secondly, Bittorrent Sync is closed source. Its company is so money hungry that advertizes and charges for such beta software without conflict management, not caring about the damage it could bring to its users. As closed source, it is not sure how secure their btsync is. In their forum, they lock discussion on btsync’s alternatives and questions on its security. Such attitude drives me away.
Third, btsync runs in background without telling the user with a task bar icon. It is very sneaky. If some one install on your computer, then it becomes a backdoor. Your data is sent somewhere without telling you. Damn! Keep far away from this software and this irresponsible company until they prove they are trustful.
Do not use Bittorrent Sync until they have implemented conflict management, if you really care about your data safety.
(2) AeroFS: AeroFS runs on virtual machines. This is both good and bad. I eventually gave it up because on a Linux host machine with wireless connection, its guest machine cannot have a unique MAC, but must use the same as the host machine. It may introduce many trouble in your LAN. If you use a Windows as a host, it shall be fine. By the way, their virtual machine appliance is really resource hungry. In comparison, OwnCloud takes much less resource. | 2018-02-25 01:18:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.18724757432937622, "perplexity": 10085.221202107203}, "config": {"markdown_headings": true, "markdown_code": false, "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-2018-09/segments/1518891816083.98/warc/CC-MAIN-20180225011315-20180225031315-00541.warc.gz"} |
https://math.stackexchange.com/questions/2363005/on-variation-of-parameters-differential-equations | # On Variation of Parameters - Differential Equations
I need guidance to complete this problem properly. I have
$$y'' - 7y' = -3$$
I am asked to find the general solution. To do that, I first start off by finding the characteristic polynomial
$$y_c = c_1e^{7x} + c_2$$
where $r=0, r=7$. A particular solution could perhaps resemble
$$y_p = u_1e^{7x} + u_2$$
where
$$u_1'e^{7x} + u_2' = 0$$ then $$7u_1'e^{7x} = -3$$
$$u_1'e^{7x}=u_2' => u_1'=-u_2'e^{-7x} = -3$$
Then I integrated to find $u_1, u_2$
$$u_1 = -\int{3dx} = 3x$$ $$u_2 = \int{e^{7x}} = \frac{1}{7}e^{7x}$$
So my particular solution, $y_p$ is
$$y_p = u_1e^{7x}+u_2 = -3xe^{7x} + \frac{1}{7}e^{7x}$$
I found the general solution as
$$y = c_1e^{7x} + c_2 - 3xe^{7x} + \frac{1}{7}e^{7x}$$
but that is incorrect. What is left undone? What have I done wrong? Is this the right methodology/technique to be using?
• In the second equation you should not have a $u_2'$ the derivative of a constant is 0 – Teh Rod Jul 18 '17 at 18:15
• I suggest that the particular solution is a polynomial. – Doug M Jul 18 '17 at 18:34
• Why is your particular solution so complicated? Surely $y_p=Cx$ works. – lulu Jul 18 '17 at 18:38
• Note: as with your prior question you appear to overcomplicate the search for particular solutions. I'd practice that. – lulu Jul 18 '17 at 18:39
• No...The differential operator on the left is $D[y]=y''-7y$. Thus $D[Cx]=0-7\times C=-7C$. Thus you want $-7C=-3$ or $C=\frac 37$. You already have the homogeneous solution (well, I didn't check but it is straight forward). – lulu Jul 18 '17 at 18:49
The best way to solve this one would be to set $v=y'\implies v'=y''$ the equation becomes $v'-7v=-3$ which is separable. | 2019-08-24 15:51:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.7729865908622742, "perplexity": 333.6132933363713}, "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-2019-35/segments/1566027321160.93/warc/CC-MAIN-20190824152236-20190824174236-00057.warc.gz"} |
https://ftp.aimsciences.org/article/doi/10.3934/dcdsb.2020346 | # American Institute of Mathematical Sciences
## Qualitative analysis of a generalized Nosé-Hoover oscillator
1 School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, Henan 450046, China 2 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China 3 Hubei Key Laboratory of Engineering Modeling and Science Computing, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
* Corresponding author: Xiao-Song Yang
Received November 2019 Revised July 2020 Published November 2020
Fund Project: The first author is supported by NSFC grant 51979116
In this paper, we analyze the qualitative dynamics of a generalized Nosé-Hoover oscillator with two parameters varying in certain scope. We show that if a solution of this oscillator will not tend to the invariant manifold $\{(x,y,z)\in \mathbb R^3|x = 0,y = 0\}$, it must pass through the plane $z = 0$ infinite times. Especially, every invariant set of this oscillator must have intersection with the plane $z = 0$. In addition, we show that if a solution is quasiperiodic, it must pass through at least five quadrants of $\mathbb R^3$.
Citation: Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020346
##### References:
[1] Q. Han and X.-S. Yang, Qualitative analysis of the Nosé-Hoover oscillator, Qual. Theory Dyn. Syst., 19 (2020), 1-36. doi: 10.1007/s12346-020-00340-1. Google Scholar [2] W. G. Hoover, Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A, 31 (1985), 1695-1697. doi: 10.1103/PhysRevA.31.1695. Google Scholar [3] S. Nosé, A unified formulation of the constant temperature molecular dynamics methods, Journal of Chemical Physics, 81 (1984), 511-519. Google Scholar [4] S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics, 52 (2002), 255-268. Google Scholar [5] H. A. Posch, W. G. Hoover and F. J. Vesely, Canonical dynamics of the nosé oscillator: Stability, order, and chaos, Phys. Rev. A, 33 (1986), 4253-4265. doi: 10.1103/PhysRevA.33.4253. Google Scholar [6] P. C. Rech, Quasiperiodicity and chaos in a generalized Nosé-Hoover oscillator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650170, 7 pp. doi: 10.1142/S0218127416501704. Google Scholar [7] J. C. Sprott, W. G. Hoover and C. G. Hoover, Heat conduction, and the lack thereof, in time-reversible dynamical systems: Generalized nosé-hoover oscillators with a temperature gradient, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 89 (2014), 042914-042914. doi: 10.1103/PhysRevE.89.042914. Google Scholar [8] L. Wang and X.-S. Yang, The invariant tori of knot type and the interlinked invariant tori in the nosé-hoover oscillator, European Physical Journal B, 88 (2015), 1-5. doi: 10.1140/epjb/e2015-60062-1. Google Scholar [9] L. Wang and X.-S. Yang, A vast amount of various invariant tori in the Nosé-Hoover oscillator, Chaos, 25 (2015), 123110, 6 pp. doi: 10.1063/1.4937167. Google Scholar [10] L. Wang and X.-S. Yang, The coexistence of invariant tori and topological horseshoe in a generalized Nosé-Hoover oscillator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750111, 12 pp. doi: 10.1142/S0218127417501115. Google Scholar [11] L. Wang and X.-S. Yang, Global analysis of a generalized Nosé-Hoover oscillator, J. Math. Anal. Appl., 464 (2018), 370-379. doi: 10.1016/j.jmaa.2018.04.013. Google Scholar
show all references
##### References:
[1] Q. Han and X.-S. Yang, Qualitative analysis of the Nosé-Hoover oscillator, Qual. Theory Dyn. Syst., 19 (2020), 1-36. doi: 10.1007/s12346-020-00340-1. Google Scholar [2] W. G. Hoover, Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A, 31 (1985), 1695-1697. doi: 10.1103/PhysRevA.31.1695. Google Scholar [3] S. Nosé, A unified formulation of the constant temperature molecular dynamics methods, Journal of Chemical Physics, 81 (1984), 511-519. Google Scholar [4] S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics, 52 (2002), 255-268. Google Scholar [5] H. A. Posch, W. G. Hoover and F. J. Vesely, Canonical dynamics of the nosé oscillator: Stability, order, and chaos, Phys. Rev. A, 33 (1986), 4253-4265. doi: 10.1103/PhysRevA.33.4253. Google Scholar [6] P. C. Rech, Quasiperiodicity and chaos in a generalized Nosé-Hoover oscillator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650170, 7 pp. doi: 10.1142/S0218127416501704. Google Scholar [7] J. C. Sprott, W. G. Hoover and C. G. Hoover, Heat conduction, and the lack thereof, in time-reversible dynamical systems: Generalized nosé-hoover oscillators with a temperature gradient, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 89 (2014), 042914-042914. doi: 10.1103/PhysRevE.89.042914. Google Scholar [8] L. Wang and X.-S. Yang, The invariant tori of knot type and the interlinked invariant tori in the nosé-hoover oscillator, European Physical Journal B, 88 (2015), 1-5. doi: 10.1140/epjb/e2015-60062-1. Google Scholar [9] L. Wang and X.-S. Yang, A vast amount of various invariant tori in the Nosé-Hoover oscillator, Chaos, 25 (2015), 123110, 6 pp. doi: 10.1063/1.4937167. Google Scholar [10] L. Wang and X.-S. Yang, The coexistence of invariant tori and topological horseshoe in a generalized Nosé-Hoover oscillator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750111, 12 pp. doi: 10.1142/S0218127417501115. Google Scholar [11] L. Wang and X.-S. Yang, Global analysis of a generalized Nosé-Hoover oscillator, J. Math. Anal. Appl., 464 (2018), 370-379. doi: 10.1016/j.jmaa.2018.04.013. Google Scholar
The grid is part of $S_{1}$ and the shadow is part of $S_{2}$
The shadow is the projection of the region $I$ on the plane $z = 0$
$A_{1}\rightarrow A_{2}$ means there are solutions from $A_{1}$ to $A_{2}$, $B_{1}\dashrightarrow A_{2}$ means there are solutions from $B_{1}$ to $A_{2}$ and these solutions have intersection with $X$-axis or $Y$-axis
From right to left are $l_{10}$ and $l_{20}$
From right to left are $l_{01}$, $l_{02}$, $l_{03}$ and $l_{04}$
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https://math.stackexchange.com/questions/3160919/vector-bundle-transition-functions-as-cech-cocycles | Vector Bundle Transition Functions as Cech Cocycles
I am trying to understand the fact that vector bundles of rank $$r$$ over a space $$X$$ are classified by the Cech cohomology group $$\check{H}^{1}\big(X, GL_{r}(\mathcal{O}_{X})\big)$$. I believe this should work in any of the usual categories, so I wont specify smooth, holomorphic, etc. I understand broadly how this goes, but there are a few key details tripping me up.
So if we have a Cech 1-cocycle $$g = \{g_{\alpha \beta}\} \in \check{H}^{1}\big(X, GL_{r}(\mathcal{O}_{X})\big)$$ with respect to some open cover $$\{U_{\alpha}\}$$, then we know:
$$(dg)_{\alpha \beta \gamma} = g_{\beta \gamma} \, g_{\alpha \gamma}^{-1} \, g_{\alpha \beta} =1$$
where we write everything multiplicatively, since that's the group operation on sections of $$GL_{r}(\mathcal{O}_{X})$$. So this equation above is obviously the cocycle condition satisfied by vector bundle transition functions. But for bundles, we also require that $$g_{\alpha \alpha} =1$$. Is this latter condition true in general for Cech cohomology, or is it somehow an extra requirement in this case?
My second confusion is the statement that isomorphic bundles define cohomologous cocycles. If we have a 0-cochain $$\lambda = \{\lambda_{\alpha}\} \in \mathcal{C}^{0}(GL_{r}(\mathcal{O}_{X}))$$, then applying the differential we get
$$(d\lambda)_{\alpha \beta} = \lambda_{\beta} \, \lambda_{\alpha}^{-1}$$
So I would be inclined to say that the condition that two 1-cocycles $$\{g\}$$ and $$\{g'\}$$ are cohomologous is
$$g_{\alpha \beta} \, (g_{\alpha \beta}')^{-1} = \lambda_{\beta} \, \lambda_{\alpha}^{-1}.$$
However, I know that two bundles are equivalent when their transition functions satisfy
$$g_{\alpha \beta} = \lambda_{\alpha} g_{\alpha \beta}' \lambda_{\beta}^{-1}$$
and things are clearly in the wrong order (for all ranks larger than 1) to be compatible with the previous equation. So where are the flaws in my understanding?
The condition you suspected, namely $$g_{\alpha\beta}(g'_{\alpha\beta})^{-1} = \lambda_{\beta}\lambda_{\alpha}^{-1}$$, is the correct condition if the coefficient group is abelian. The latter condition, namely $$g_{\alpha\beta} = \lambda_{\alpha}g'_{\alpha\beta}\lambda_{\beta}^{-1}$$, is the correct one for non-abelian coefficient groups; see chapter $$4$$, section $$4.1$$, equation $$4$$-$$2$$ of Brylinski's Loop Spaces, Characteristic Classes and Geometric Quantization. Note, if the coefficient group is abelian, the two conditions are equivalent.
• Ah sorry, I understand now. I overlooked the fact you point out here; we need a sheaf of abelian groups to apply the basic Cech cohomology. Thanks! As for the requirement that $g_{\alpha \alpha} =1$, do you know if somehow this is a general requirement for Cech 1-cocycles? I certainly see why we impose this condition for vector bundles, I just don't see where it comes from in cohomology. – Benighted Mar 25 '19 at 0:48
• Well, the cocycle condition $(dg)_{\alpha\beta\gamma} = 1$ reduces to $g_{\alpha\alpha} = 1$ when $\alpha = \beta = \gamma$. – Michael Albanese Mar 25 '19 at 16:04 | 2020-01-25 02:37:00 | {"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": 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": 20, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9746983051300049, "perplexity": 250.0724223080983}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250628549.43/warc/CC-MAIN-20200125011232-20200125040232-00062.warc.gz"} |
https://stats.stackexchange.com/questions/59155/glm-conditional-distribution-from-r-glm | # GLM conditional distribution from R GLM
I want to obtain the full distribution of a Gamma (or Inverse Gaussian) distributed $y_i$ given a vector of $\bar x_i$ that have been used in the linear predictor of a coefficient. Suppose also for the Gamma GLM I have used the log-link instead of the canonical link.
Since both distributions are bi-parametric I know I can get the estimation of the mean parameter by predict(glmObj, ..., type="response") whichever distribution whichever link I have used. I'm not sure about the conditional variance. I know that $var\left(y_i\right)=\phi*V\left(\mu_i\right)$. My questions are:
1. Is it correct to estimate $\phi$, the dispersion parameter, as the square root of glmObj$deviance/glmObj$df.residual, regardless of the distribution and canonical link?
2. Is $V\left(\mu_i\right)$ dependent on the canonical link?
• You can get "standard errors" (not prediction errors) from predict using the argument se.fit=TRUE. It doesn't give you their covariance, however. Any two predicted values are a function of a linear combination of regression parameters. These usually aren't independent. It is easy to compute by hand, however. – AdamO Jun 18 '13 at 15:53
• The estimate of $\phi$ from GLMs is generally not ML (it is in some cases, but usually not); indeed in some cases the ML estimate doesn't even make sense (Poisson regression for example). The package MASS includes functions for ML estimation of the parameter corresponding to $\phi$ in one or two cases (such as the Gamma). As for 'correct' -- it depends on what you mean by 'correct'. – Glen_b Jul 18 '13 at 16:47
• If you want to estimate the dispersion parameter $\phi$, where $\phi\ne 1$ then u need to use the quasi-likelihood approach. For example for the poisson glm, u need something like: glm(y ~ x, family =quasipoisson(link = "log")). When the $\phi=1$, then the quasi-likelihood will be the ML estimate. – Stat Dec 16 '13 at 2:01 | 2020-07-05 23:42:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8778729438781738, "perplexity": 602.995485802582}, "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-2020-29/segments/1593655889877.72/warc/CC-MAIN-20200705215728-20200706005728-00243.warc.gz"} |
https://www.sav.sk/?lang=en&doc=journal-list&part=article_response_page&journal_article_no=16762 | # An abelian subextension of the dyadic division field of a hyperelliptic Jacobian
In: Mathematica Slovaca, vol. 69, no. 2
Jeffrey Yelton
## Details:
Year, pages: 2019, 357 - 370
Keywords:
hyperelliptic curve, abelian field extension, Galois representation
Given a field $k$ of characteristic different from $2$ and an integer $d ≥ 3$, let $J$ be the Jacobian of the generic" hyperelliptic curve given by $y2 = \prodi = 1d (x - αi)$, where the $αi$'s are transcendental and independent over $k$; it is defined over the transcendental extension $K / k$ generated by the symmetric functions of the $αi$'s. We investigate certain subfields of the field $K$ obtained by adjoining all points of $2$-power order of $J(\bar{K})$. In particular, we explicitly describe the maximal abelian subextension of $K / K(J[2])$ and show that it is contained in $K(J[8])$ (resp. $K(J[16])$) if $g ≥ 2$ (resp. if $g = 1$). On the way we obtain an explicit description of the abelian subextension $K(J[4])$, and we describe the action of a particular automorphism in $\Gal(K / K)$ on these subfields. | 2021-04-16 00:40:49 | {"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": 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": 0.8812289834022522, "perplexity": 174.20353089310183}, "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/1618038088264.43/warc/CC-MAIN-20210415222106-20210416012106-00549.warc.gz"} |
https://advances.sciencemag.org/content/5/11/eaax9444 | Research ArticleAPPLIED ECOLOGY
# A third of the tropical African flora is potentially threatened with extinction
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Vol. 5, no. 11, eaax9444
## Abstract
Preserving tropical biodiversity is an urgent challenge when faced with the growing needs of countries. Despite their crucial importance for terrestrial ecosystems, most tropical plant species lack extinction risk assessments, limiting our ability to identify conservation priorities. Using a novel approach aligned with IUCN Red List criteria, we conducted a continental-scale preliminary conservation assessment of 22,036 vascular plant species in tropical Africa. Our results underline the high level of extinction risk of the tropical African flora. Thirty-three percent of the species are potentially threatened with extinction, and another third of species are likely rare, potentially becoming threatened in the near future. Four regions are highlighted with a high proportion (>40%) of potentially threatened species: Ethiopia, West Africa, central Tanzania, and southern Democratic Republic of the Congo. Our approach represents a first step toward data-driven conservation assessments applicable at continental scales providing crucial information for sustainable economic development prioritization.
## INTRODUCTION
Major threats to biodiversity, especially in areas of exceptional plant diversity, primarily in the tropics, are often linked to industrial-scale activities such as timber exploitation or large plantations, mining, and agriculture (1). Article 14 of the United Nations Convention on Biological Diversity (www.cbd.int/) explicitly indicates that environmental impact assessments (EIAs) should be conducted before implementing these projects. To reduce risks linked to environmental concerns, EIAs should identify adverse impacts on biodiversity by projects and indicate measures to avoid, minimize, and offset impacts. A growing realization that environmental impacts represent risks not only to biodiversity but also to operational, financial, and reputational aspects of projects has led extractive industries, agro-business, financial institutions, governments, and civil society, inter alia, to identify and adopt best practices for managing biodiversity. Threatened species are one of the key elements [e.g., (2)] that may be affected by these proposed projects.
## MATERIALS AND METHODS
### Flora of tropical Africa dataset
We used a taxonomically verified database of tropical African vascular plant species distribution: RAINBIO (8, 12). This database contains 590,231 georeferenced records representing distribution information for 25,222 native species in sub-Saharan Africa, excluding Madagascar and southern Africa. Invasive or planted or cultivated species were not included [see (12) for details on cleaning and quality-checking the data].
Global assessments require knowledge of all known occurrences for a given taxon. Non-endemics to our study area, i.e., species with occurrences both within and outside tropical Africa, are problematical because RAINBIO does not include occurrences records from outside the continent (12). A species with a few records in tropical Africa might be assessed using the PACA approach as potentially threatened, although it is widely distributed outside Africa and therefore likely not threatened. To reduce this bias, we first excluded occurrences of species only found in South Africa, Swaziland, and Lesotho because only a small portion of the available records for these countries were included in the RAINBIO database (this concerns 2254 species and 3332 occurrences). We then applied the following procedure to identify species whose range extends beyond our study area by comparing the distribution as described by the RAINBIO database to that based on records from GBIF. Specifically, we searched GBIF for occurrences of 22,968 species in the RAINBIO database using the “rgbif” R package (44) and extracted occurrences for each from GBIF (excluding those with georeferencing issues, R code available at https://github.com/gdauby/stevart_el_al_PACA). On the basis of these GBIF occurrence records, the number of occupied cells at 10-km resolution was calculated, as well as the number of continents in which the species has been recorded. Species identified by PACA as potentially threatened under Criterion B based on the RAINBIO dataset (see the method explaining the preliminary assessment below) and occupying more than 10 10 km × 10 km cells based on the GBIF data were tagged. GBIF occurrences were found for 21,345 species (96.9% of those in the RAINBIO dataset). Using RAINBIO, the number of species classified as potentially threatened under Criterion B (i.e., belonging to CR, EN, or VU categories, see below) was 15,470. Among these species, 1220 were recorded from more than 10 10 km × 10 km cells in the GBIF dataset, indicating that subpopulations for these species are missing in the RAINBIO database. When scrutinizing these species, it was found that the additional subpopulations are often artifacts of georeferencing errors or involved doubtful records. Hence, because the rationale of PACA is to provide preliminary conservation assessments, we adopted a conservative approach and only removed species occurring in more than two continents (20 species) or occupying more than 15 10 km × 10 km cells based on the GBIF dataset (912 species) to reduce the risk of incorrectly removing species that are truly threatened. The final dataset used in this paper thus comprised 580,208 distribution records for 22,036 species. Hence, from the initial list of 25,222 species, a total of 3186 were judged to have their known occurrences poorly covered by the RAINBIO database and/or have distributions likely to lie primarily outside our tropical African study area and were thus not assessed here.
### IUCN-based PACA
Depending on the type of information available for a species, an IUCN conservation assessment can be undertaken using any or all of five criteria, A to E. Criterion A is based on estimates of population (number of mature individuals) reduction over 10 years or three generations, Criterion B is based on geographic range, Criterion C is based on population size, Criterion D mainly concerns very small or restricted populations based on the number of mature individuals and AOO, and Criterion E is based on a quantitative analysis of extinction probability within a given number of years. In the absence of detailed information about population size (i.e., the number of mature individuals), which is commonly the case for plant taxa, standard practice calls for using estimates of geographic range obtained from occurrence records (e.g., georeferenced herbarium specimens) for assessments using Criterion B (26). As part of the PACA process, we generated a framework for estimating whether a species faces potential population reduction (required for Criterion A) or future decline in key geographic parameters (needed for Criterion B). Thus, PACA is aligned with two key elements of the IUCN Red List: Criterion A, relating to population size reduction, and Criterion B, relating to geographic range. Approaches aligned with the three other Red List criteria were not implemented as they require data that are unavailable for most plant species, especially across the tropics.
All IUCN parameters needed for preliminary assessments of taxa based under Criterion B (see below) were calculated using the R package ConR ver. 1.2.1 (38). Using an as-yet unreleased version (https://github.com/gdauby/stevart_el_al_PACA), we also implemented assessments aligned with the parameters of Criterion A (see below).
Decline in habitat quality. When using both Criteria A and B, an assessment of observed, estimated, inferred, suspected or projected decline in AOO, Extent of Occurrence (EOO) (see definitions below), and/or habitat quality is needed for each taxon [subcriteria A2 and/or A3 under Criterion A; subcriterion (b) under Criterion B; see below, Table 1]. This is generally assessed on a taxon-by-taxon basis using detailed knowledge about land-use changes and the impacts of threats on the species, which is not possible when simultaneously assessing thousands of taxa. Our method is able to treat a large number of species by estimating population reduction and decline in geographic parameters indirectly based on two sources of information that are likely to be informative. For the first of these, we used land cover characterized by a moderate to high human influence to develop a “human affected” layer. This was done using the land-cover map of Africa produced by (45), which describes 27 land cover types based on remote sensing data at 1-km resolution. We constructed a raster layer for each land cover type by aggregating the original raster at 10-km resolution and computing the proportion of each land cover type within 10-km2 cells. We identified seven land cover types indicative of moderate to high level of human impact: degraded evergreen forests, mosaic forests/croplands, croplands (>50%), croplands with open woody vegetation, irrigated croplands, tree crops, and cities. These layers are directly linked to the main threats on African flora, viz. small- and large-scale agriculture, urbanization, roads, and logging. A given occurrence was considered to be facing a decline in habitat quality if it occurred within a cell where the summed proportion of the seven land cover types indicative of human impact was higher than 50%. The second source of information on human impact was based on the prediction that mining activities will increase significantly in the next few decades across Africa (14). Using a map of major mineral deposits where industrial mining activities are taking place or will likely take place in near future, according to (46, 47), we estimated a decline in habitat quality for any site located within a 10-km radius of such a deposit, based on the inferred scale of the environmental effects of mining activities given in (48).
We identified a decline in habitat quality for a taxon [which is applicable under subcriteria A2 and A3, and invokes subcriterion (b) under Criterion B] when at least one of its known occurrences was found either in an area characterized by moderate to high human impact and/or in a 10-km radius around a major mineral deposit. In the specific case where all known occurrences of a species were found within one or more protected areas, we assumed no decline in habitat quality.
PACA aligned with IUCN Red List Criterion B. Assessing species under Criterion B of the IUCN Red List relies on two subcriteria: B1 and B2, based on the EOO and the AOO, respectively. In order for a species to be assessed as threatened, threshold values under at least two of three subcriteria must also be met as follows: (a) number of locations, (b) inferred/projected decline in various parameters including habitat quality, and (c) extreme fluctuation of populations (Table 1). Subcriterion (c) is rarely applicable for plants and was not considered here.
The EOO is the smallest surface contained in a polygon drawn from an imaginary boundary encompassing all known occurrences of a taxon (namely, the hull convex). At least three points are needed for calculating this parameter, so the EOO was not computed for taxa with less than three unique occurrences.
The AOO is the area within the EOO occupied by the taxon. The AOO is estimated by calculating the sum of occupied cells after superimposing a grid with cells of 2 km2. In ConR, the AOO is estimated by four different positions of the grid cell, and the one resulting in the minimum number of occupied cells is retained (38). A single record per taxon can be used to estimate AOO and thus undertake the assessment, a common practice in tropical plant Red Listing workshops.
The number of locations, as defined by IUCN, is a “geographically or ecologically distinct area in which a single threatening event can affect all individuals of the taxon.” This parameter is difficult to estimate automatically [see (38) for discussion about how it is estimated]. For the purpose of the PACA approach, the number of locations was estimated on the basis of two considerations. First, all the occurrences found within a single protected area were considered to represent a single location, based on the rationale that these occurrences would be equally affected by a single event such as downgrading, downsizing, or degazetting the protected area [e.g., (37)], although we acknowledge that these events can have various impacts that are hard to estimate at broad scales. The number of locations was thus estimated using a shapefile of terrestrial protected areas for tropical Africa downloaded from the World Database on Protected Areas (www.protectedplanet.net). Second, for occurrences located outside protected areas, the number of locations was estimated as the number of occupied cells within a superimposed grid of 10 km2. This grid cell size is suggested as a suitable proxy for detecting a threat that would equally affect all individuals of a taxon contained therein [e.g., mining activities; see (48)].
On the basis of the calculations of EOO and AOO, the estimate of the number of locations and whether or not potential past or future decline was inferred, we automatically assigned each taxon to one of three preliminary threat categories (see IUCN guidelines):
• 1) Potentially CR: EOO < 100 km2 or AOO < 10 km2 and locations = 1 and at least one of its occurrences subjected to decline in habitat quality because it is found in a cell classified as human affected or identified as actually or potentially subjected to mining.
• 2) Potentially EN: EOO < 5000 km2 or AOO < 500 km2 and locations ≤5 and at least one of its occurrences found in a human affected or mining cell.
• 3) Potentially VU: EOO < 20,000 km2 or AOO < 2000 km2 and locations ≤10 and at least one of its occurrences found in a human affected or mining cell.
PACA aligned with IUCN Red List Criterion A. As a complement to inferring PACA using key parameters of IUCN Criterion B, we also inferred parameters for Criterion A based on the observed, estimated, inferred, suspected, or projected reduction in the population size of a taxon, i.e., in number of mature individuals. For most plant species, little, if any, information about population dynamics through time is available. We thus inferred population reduction by making a quantitative estimate of the percentage of population decline using the AOO for each taxon that meets either subcriterion A2 or A3. A2 relies on observed, estimated, inferred, or suspected population reduction in the past, “where the causes of reduction may not have ceased or may not be understood or may not be reversible.” A3 is based on projected, inferred, or suspected population reduction to be met in the near future (up to 100 years or three generations, whichever is smaller).
Using the human impact and mining layers described above, we estimated a population reduction percentage for each taxon by inferring a potential decrease in AOO (AOODEC). This was done by using the ratio between an AOO estimated from all occurrences (AOOFULL) and an AOO estimated only from occurrences situated outside human affected or mining areas (AOORED). The value of AOODEC = ([AOOFULL − AOORED]/AOOFULL) × 100. AOODEC thus represents the decrease in AOO if all occurrences occurring within the human affected or mining areas were lost in the near future. Using AOODEC as computed here and applying the threshold values for IUCN Criterion A, we assigned each taxon to one of the following preliminary threat categories:
• 1) Potentially CR: AOODEC ≥ 80%;
• 2) Potentially EN: AOODEC ≥ 70%;
• 3) Potentially VU: AOODEC ≥ 50%.
Names for PACA categories and subcategories. Assessments obtained using the PACA method must be regarded as “preliminary” because they are not the result of a full IUCN assessment, a taxon-by-taxon procedure based on exhaustive information (e.g., bibliographic, remote sensing data, and in situ observations) on the threats affecting each individual taxon. Species were therefore assigned to three PACA-derived categories based on the automatic batch output from ConR (see Table 1 and Fig. 1): (i) LT, encompassing species flagged as potentially CR or potentially EN, (ii) PT, encompassing species flagged as potentially VU, and (iii) PNT, encompassing species that potentially do not fall into one of the three IUCN threatened categories (i.e., which would correspond to the IUCN Red List categories of NT or LC).
For species assessed as PNT, we further distinguished three subcategories: (i) LR, which includes species whose EOO, AOO, and number of locations all fall within the thresholds for CR and EN but for which we did not infer a decline in the quality of habitat; (ii) PR, which is similar to LR but for species whose EOO, AOO, and number of locations fall within the limits for VU; and (iii) LNT, which includes all other species that do not belong to the categories described above.
### Geographic distribution of threatened species
After species were assigned to a PACA category (Table 1), we compiled distributional data to summarize and map the estimated level of threat across Africa. A gridded spatial representation of threat was undertaken using an “adaptive resolution” SU method (49). This approach adapts the size of the SU as a function of a user-defined threshold of minimum occurrence records. A shapefile of the adaptive SU grid was created by uploading the RAINBIO database to the Infomap Bioregions application (49) using the following parameters: maximum cell capacity = 1000, minimum cell capacity = 250, maximum cell size = 8°, and minimum cell size = 0.5°. We also represented our results for each country and each terrestrial ecoregion (33) found in Africa. In each case (grids, adaptive SU, and ecoregion), we estimated the total number of taxa recorded and the proportion of taxa assessed as LT and PT, under Criteria A and B separately, and by combining both criteria, i.e., a taxon would, for example, be categorized as LT if it is assessed as LT by at least one of the two criteria.
### Threat per habit
We assessed the proportion of threatened species corresponding to each of four major habits: herbs, trees, lianas, and shrubs. Information about species habit was extracted from the RAINBIO database (12).
### Comparison with full published IUCN assessments
For each species, the full published IUCN assessment was downloaded from the IUCN website using the API tools of the “rredlist” R package (version 2018-2, www.iucnredlist.org). To evaluate the extent to which the results of the PACA approach matched those of full Red Listing, we computed a Kappa coefficient for evaluating agreements between full published IUCN categories and PACA categories.
## SUPPLEMENTARY MATERIALS
Fig. S1. Total number of species assessed as Likely/Potentially Threatened following criterion A, B, and both A and B.
Fig. S2. Total number of species preliminarily assessed as LR/PR following the Criterion B.
Fig. S3. Summary statistics for full conservation assessments of plant species published on the IUCN Red List portal (version 2018-2, www.iucnredlist.org).
Fig. S4. Proportion of species preliminarily assessed as Likely/Potentially Threatened following Criterion A and Criterion B for four habit types.
Table S1. Proportion (in %) of LT/PT species assessed under Criteria A, B, and both A and B for all countries within our study area (tropical Africa).
Table S2. Number of LT/PT species assessed under Criteria A, B, and both A and B for all countries within our study area (tropical Africa).
Table S3. Proportions of likely/potentially threatened species assessed under Criteria A, B, and both A and B across ecoregions.
Table S4. Number of likely/potentially threatened species assessed under both Criteria A and B and total number of species for four different habits for each ecoregion.
This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.
## REFERENCES AND NOTES
Acknowledgments: Funding: This work was cofunded by the French Foundation for Research on Biodiversity and the Provence-Alpes-Côte d’Azur région through the Centre for Synthesis and Analysis of Biodiversity Data Program, as part of the RAINBIO research project through funding to T.L.P.C. (http://rainbio.cesab.org/). G.D. was partly funded by the Belgian Fund for Scientific Research (F.R.S.-FNRS). J.-C.S. and A.B.-O. acknowledge economic support from VILLUM FONDEN through the VILLUM Investigator project “Biodiversity Dynamics in a Changing World” (grant 16549). Author contributions: T.S., G.D., and T.L.P.C. conceived the study; T.S., G.D., P.P.L., V.D., G.E.S., B.A.M., M.S.M.S., J.J.W., A.B.-O., J.-C.S., and T.L.P.C. contributed ideas to the methods; V.D., B.S., B.A.M., D.J.H., and J.J.W. contributed data; G.D. analyzed the data and prepared the figures; T.S., G.D., P.P.L., and T.L.P.C. led the writing; all authors read and approved the final manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors. Data used in this study are available at this website: http://rainbio.cesab.org. All analyses can be redone using code available at https://github.com/gdauby/stevart_el_al_PACA.
View Abstract | 2019-12-15 13:38:21 | {"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": 0.48266878724098206, "perplexity": 4224.451904861223}, "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/1575541308149.76/warc/CC-MAIN-20191215122056-20191215150056-00368.warc.gz"} |
http://www.cognitive-neuroinformatics.com/de/team/david-nakath | Wissenschaftlicher Mitarbeiter
Raum: 4.056 | Telefon: +49 (0)421 218 64248
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#### Publikationen:
Filter publications:
2018
[9] Multi-Sensor Fusion and Active Perception for Autonomous Deep Space Navigation (, , ), In 21th International Conference on Information Fusion (FUSION), IEEE, .
2016
[8] Optimal rotation sequences for active perception (, , , ), In Proc. SPIE: Multisensor, Multisource Information Fusion: Architectures, Algorithms, and Applications (Jerome J. Braun, ed.), SPIE Press, volume 9872, .
[7] Rigid body attitude control based on a manifold representation of direction cosine matrices (, , ), In 13th European Workshop on Advanced Control and Diagnosis (ACD), volume 783, .
2015
[6] Adaptive Information Selection in Images: Efficient Naive Bayes Nearest Neighbor Classification (, , ), Chapter in Computer Analysis of Images and Patterns, Springer Science + Business Media, .
[5] KaNaRiA: Identifying the Challenges for Cognitive Autonomous Navigation and Guidance for Missions to Small Planetary Bodies (, , , , , , , , , , , , , , , ), In 66th International Astronautical Congress (IAC), . [bibtex]
[4] Autonomous Orbit Navigation for a Mission to the Asteroid Main Belt (, , , , , , ), In 66th International Astronautical Congress (IAC), . [bibtex]
[3] Affordance-Based Object Recognition Using Interactions Obtained from a Utility Maximization Principle (, , , , ), Chapter in Computer Vision - ECCV 2014 Workshops, Springer Science + Business Media, .
2014
[2] Active Sensorimotor Object Recognition in Three-Dimensional Space (, , , , ), Chapter in Spatial Cognition IX, Springer Science + Business Media, .
2013
[1] Sensorimotor integration using an information gain strategy in application to object recognition tasks (Abstract) (, , , , ), .
Aktuell sind 27 Gäste und keine Mitglieder online | 2019-03-22 18:06:42 | {"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": 0.9467412829399109, "perplexity": 11686.977500343253}, "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-2019-13/segments/1552912202688.89/warc/CC-MAIN-20190322180106-20190322201707-00073.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php?title=2009_AIME_I_Problems/Problem_11&diff=prev&oldid=30899 | # Difference between revisions of "2009 AIME I Problems/Problem 11"
## Problem
Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. Find the number of such distinct triangles whose area is a positive integer.
## Solution
Solution 1 (This solution requires linear alg. knowledge)
Let the two points be point P and Q
and $P=(x_1,y_1),Q=(x_2,y_2)$
We can calculate the area of the parallelogram span with the determinant of matrix
$\det ({\matrix {P \above Q}})=\det$ (Error compiling LaTeX. ! Package amsmath Error: Old form \matrix' should be \begin{matrix}.)$({\matrix {x_1 \above x_2} \right \matrix {y_1 \above y_2})$ (Error compiling LaTeX. ! Package amsmath Error: Old form \matrix' should be \begin{matrix}.)
since triangle is half of the area of the parallelogram. We just need the determinant to be even
The deteminant is
$$(x_1)(y_2)-(x_2)(y_1)=(x_1)(2009-41(x_2))-(x_2)(2009-41(x_1))$$
$$=2009(x_1)-41(x_1)(x_2)-2009(x_2)+41(x_1)(x_2)=2009((x_1)-(x_2))$$
since 2009 is not even, $((x_1)-(x_2))$ must be even
Thus the two x's have to be both odd or even.
Also note that the maximum value for x is $49$ and minimum is $0$.
There are $25$ even and $25$ odd number
Thus, there are
$(_{25}C_2)+(_{25}C_2)=\boxed{600}$of such triangle | 2021-06-22 23:48:28 | {"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": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 15, "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": 0.8640739321708679, "perplexity": 816.1711161741837}, "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/1623488525399.79/warc/CC-MAIN-20210622220817-20210623010817-00331.warc.gz"} |
https://forum.ionicframework.com/t/you-must-fulfill-at-least-one-of-these-conditions/21711 | # You must fulfill at least one of these conditions
#1
I’m just begginer to ionic, when I want to add android platform I got this error message:
Regards,
Hosein.
#2
Well, you need to do what the error message says: set the `ANDROID_HOME` environment variable pointing to where you installed the Android SDK.
Which obviously means you need to install the Android SDK first if you don’t have it already.
#3
So, What I have to do now?
@encodedmirko
#4
set below two path and make sure there is minimum sdk19 is available at path
ANDROID_HOME : C:\Program Files\Android\sdk
PATH : %ANDROID_HOME%\tools,%ANDROID_HOME%\platform-tools | 2018-12-10 22:14:26 | {"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": 0.8406937718391418, "perplexity": 2521.597656695704}, "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-2018-51/segments/1544376823445.39/warc/CC-MAIN-20181210212544-20181210234044-00129.warc.gz"} |
https://cn.chinaproject.harvard.edu/publications/publication-authors/zhang-ning-cn | # 张宁 Ning Zhang
2022
Zhenyu Zhuo, Ershun Du, Ning Zhang, Chris Nielsen, Xi Lu, Jinyu Xiao, Jiawei Wu, and Chongqing Kang. 2022. “Cost Increase in the Electricity Supply to Achieve Carbon Neutrality in China.” Nature Communications, 13. Publisher's VersionAbstract
The Chinese government has set long-term carbon neutrality and renewable energy (RE) development goals for the power sector. Despite a precipitous decline in the costs of RE technologies, the external costs of renewable intermittency and the massive investments in new RE capacities would increase electricity costs. Here, we develop a power system expansion model to comprehensively evaluate changes in the electricity supply costs over a 30-year transition to carbon neutrality. RE supply curves, operating security constraints, and the characteristics of various generation units are modelled in detail to assess the cost variations accurately. According to our results, approximately 5.8 TW of wind and solar photovoltaic capacity would be required to achieve carbon neutrality in the power system by 2050. The electricity supply costs would increase by 9.6 CNY¢/kWh. The major cost shift would result from the substantial investments in RE capacities, flexible generation resources, and network expansion.
2016
Ning Zhang, Xi Lu, Chris P Nielsen, Michael B. McElroy, Xinyu Chen, Yu Deng, and Chongqing Kang. 2016. “Reducing curtailment of wind electricity in China by employing electric boilers for heat and pumped hydro for energy storage.” Applied Energy, 184, Pp. 987-994. Publisher's VersionAbstract
Accommodating variable wind power poses a critical challenge for electric power systems that are heavily dependent on combined heat and power (CHP) plants, as is the case for north China. An improved unit-commitment model is applied to evaluate potential benefits from pumped hydro storage (PHS) and electric boilers (EBs) in West Inner Mongolia (WIM), where CHP capacity is projected to increase to 33.8 GW by 2020. A business-as-usual (BAU) reference case assumes deployment of 20 GW of wind capacity. Compared to BAU, expanding wind capacity to 40 GW would allow for a reduction in CO2 emissions of 33.9 million tons, but at a relatively high cost of US$25.3/ton, reflecting primarily high associated curtailment of wind electricity (20.4%). A number of scenarios adding PHS and/or EBs combined with higher levels of wind capacity are evaluated. The best case indicates that a combination of PHS (3.6 GW) and EBs (6.2 GW) together with 40 GW of wind capacity would reduce CO2 emissions by 43.5 million tons compared to BAU, and at a lower cost of US$16.0/ton. Achieving this outcome will require a price-incentive policy designed to ensure the profitability of both PHS and EB facilities. | 2022-07-03 02:09:09 | {"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": 0.5152639746665955, "perplexity": 7431.9315936300345}, "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-2022-27/segments/1656104209449.64/warc/CC-MAIN-20220703013155-20220703043155-00733.warc.gz"} |
https://math.stackexchange.com/questions/2187839/definition-of-sobolev-space-via-distribution | # Definition of Sobolev space via distribution
Can someone give me some comments about the following definition of the Sobolev space $W^{k,p}(U)$ for $U \subset \mathbb{R}^d$ open, $k\in \mathbb{N}$ and $1 \leq p \leq \infty$? The Sobolev space is the set of all $L^p (U)$ functions, such that for all multi indices $\alpha$ with $\vert\alpha\vert \leq k$ there exists $f_\alpha \in L^p (U)$ so that $\partial^\alpha T_f = T_{f_\alpha}$ (in the sense of distributions). What is the weak derivative of what? I am completely overextended... I just need some lines about this definition. That would help a lot.
Every function $f: U\to \mathbb R$ which is locally $L^1$ (i.e., $\int_K |f(x)|dx <\infty$ for every compact set $K$) induces a distribution $T_f: \mathscr D(U)\to\mathbb R$, $\varphi \mapsto \int \varphi(x)f(x)dx$ (which is correctly defined because $\varphi$ has compact support). The distributional derivative $\partial^\alpha T$ is defined as $\varphi\mapsto (-1)^{|\alpha|}T(\partial^\alpha \varphi)$. The requirement in the definition of Sobolev spaces is that these distributional derivatives of $T_f$ are again induced by $L^p$-functions. | 2020-01-25 23:47:47 | {"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": 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": 0.9357581734657288, "perplexity": 79.50610873820426}, "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/1579251681625.83/warc/CC-MAIN-20200125222506-20200126012506-00519.warc.gz"} |
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Has anybody taken this subtest this month? Any quizlets or recomendations
Angela Hello, I'll take it in two weeks. When do you take it?
Jan 11
Angela Hello, I'll take it in two weeks. When do you take it?
Jan 11
Angela Hello, I'll take it in two weeks. When do you take it?
Jan 11
Angela Hello, I'll take it in two weeks. When do you take it?
Jan 11
Angela Hello, I'll take it in two weeks. When do you take it?
Jan 11
Teacher Chatboards | 2022-01-26 17:33:00 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9845771789550781, "perplexity": 13244.094048378678}, "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-2022-05/segments/1642320304959.80/warc/CC-MAIN-20220126162115-20220126192115-00210.warc.gz"} |
http://www.imankulov.name/posts/fresh-soft-for-your-amazon-ami-p2.html | # Fresh soft for your Amazon AMI. Part 2. Publishing your own work.
This is the second and last part of the series "Fresh soft for your Amazon AMI". In this post aim to explain how you can re-build srpm package to a new version using mock and git, and how you can publish your own yum repository.
Feel free to address Part 1. Stealing from Fedora, where I describe how you can get the source package from Fedora and rebuild it.
## The rpm package structure
If you extract source rpm package, you will find out that it consists of software source and of a number of patches and auxiliary files. The most important among them is the spec-file (usually named like package-name.spec). The spec file defines all the steps and all the variables, required to turn the this rather disorderly collection to binary rpm package. This is how, for instance, the contents of redis package from EPEL repository looks like:
• redis-2.4.10.tar.gz: main source tarball
• redis-2.4.8-redis.conf.patch: patch for redis.conf configuration file
• redis.init: init-script to start up the redis server
• redis.logrotate: logrotate configuration file
• redis.spec: rpm build specification
## Rebuild rpm from spec-file. How to use mock with git
Even in Fedora Rawhide the virtualenv is quite outdated (1.7, whereas 1.8 is available for long). Let's try to master the "industrial process" of building rpm packages.
Note
I assume that you know and like git system. If it's not the case, you can easily come into play with the marvelous git book.
First, let's try to re-build the package with no changes, just to get the idea.
Download the original package, which we want to update. I don't enable Rawhide here, as I want to upgrade original source, but modifying Rawhide source can be easier sometimes.
I created the directory python-virtualenv-rpm previously
$yumdownloader --source python-virtualenv$ cd python-virtualenv-rpm
$rpm2cpio ../python-virtualenv-*.src.rpm | cpio -idmv This package is very simple, it contains only the source and the spec. Then turn the python-virtualenv-rpm directory to a git repository $ git init
$git add -A .$ git commit -m "Initial release"
Now let's build the package from the repo we've got.
$mock -v --scm-enable --scm-option method=git \ --scm-option git_get="git clone /home/mockbuild/python-virtualenv-rpm SCM_PKG" \ --scm-option package=python-virtualenv \ --scm-option spec=SCM_PKG.spec Here SCM_PKG is the special "variable" which is replaced with the package name (python-virtualenv in this case). This command can be interpreted as "Build the package python-virtualenv with mock and git. To get the package spec and sources, use command defined in git_get, then find the spec file python-virtualenv.spec there, from this spec build the srpm package, and then build the the final rpm". Logs and results will be available in /var/lib/mock/epel-6-x86_64/result/. Some notes before starting changing the spec file. • As you can see, it's trivial to publish package on github or bitbucket instead of storing it locally. The workflow won't be changed at all. • Storing big archives in the git is not a very good idea, it's better to save the in separate location. Mock knows how to handle this cases: just put all sources of all packages you build in a separate directory, and define it in configuration with scm-option ext_src_dir=/path/to/your/directory. • Most of the scm options (like "method", "spec", "ext_src_dir" and even "git_get") can be written to the /etc/mock/site-defaults.cfg. See this file for more examples. ## Rebuild rpm from spec-file. Update spec file As you successfully rebuilt your package, modify the spec file. In the best case scenario the only option you will need to fix is the version field, and optionally, the changelog. Additionally, you may have to add add some files (like documentation) to the specification, remove or rename them. In the python-virtualenv package I modified the version and the changelog and tried to rebuild. Then it turned out that some files were added in a new package, and some were removed from it. Namely: • new version of the virtualenv didn't contain the file HACKING, and I removed this file from the spec as well. • new version of the virtualenv had the executable file virtualenv-2.6, and I added this file to the specification. This is how the diff looked like --- python-virtualenv.spec.orig 2013-01-07 14:53:06.089499165 +0000 +++ python-virtualenv.spec 2013-01-07 15:23:54.120211669 +0000 @@ -2,7 +2,7 @@ %{!?python_sitelib: %define python_sitelib %(%{__python} -c "from distutils.sysconfig import get_python_lib; print get_python_lib()")} Name: python-virtualenv -Version: 1.7 +Version: 1.8.4 Release: 1%{?dist} Summary: Tool to create isolated Python environments @@ -58,7 +58,7 @@ %files %defattr(-,root,root,-) -%doc docs/*txt PKG-INFO AUTHORS.txt LICENSE.txt HACKING +%doc docs/*txt PKG-INFO AUTHORS.txt LICENSE.txt # Include sphinx docs on Fedora %if 0%{?fedora} > 0 %doc build/sphinx/* @@ -66,9 +66,13 @@ # For noarch packages: sitelib %{python_sitelib}/* %attr(755,root,root) %{_bindir}/virtualenv +%attr(755,root,root) %{_bindir}/virtualenv-2.6 %changelog +* Mon Jan 07 2013 Roman Imankulov <roman.imankuklov@gmail.com> - 1.8.4-1 +- Local rebuild + * Tue Dec 20 2011 Steve 'Ashcrow' Milner <me@stevemilner.org> - 1.7-1 - Update for https://bugzilla.redhat.com/show_bug.cgi?id=769067 Once you made the changes, commit and push (if you use remote repository) your changes and rebuild the package. I won't dive deep in the problematic of spec file syntax. Actually, I'm not quite familiar with it either. But the simple "change and try" loop has always worked well for me so far. ## Publish rpm packages to a yum repository As we work with Amazon, it looks like a good option to store data in S3. We'll create a local cache of S3, and then will sync data with an every new package built. I store the rpm cache directly in the /home/mockbuild/repo. As I work with 64-bit platforms exclusively, all my binary packages are either "noarch.rpm" or "x86_64.rpm". $ mkdir -p ~/repo/{SRPMS,x86_64}
$mv /var/lib/mock/epel-6-x86_64/result/*.src.rpm repo/SRPMS$ mv /var/lib/mock/epel-6-x86_64/result/*.x86_64.rpm repo/x86_64
$mv /var/lib/mock/epel-6-x86_64/result/*.noarch.rpm repo/x86_64 # .noarch goes to x86_64 too$ ls -d repo/* | xargs -i createrepo {}
The find ./repo shows you how your repository structure looks like.
To push data remotely, I use s3cmd utility.
# yum install s3cmd
At the first launch, initialize s3cmd configuration and create the S3 bucket where you plan to store your data. Here I created the S3 store in the "eu-west-1" region.
$s3cmd --configure$ s3cmd mb s3://<my-bucket> --bucket-location=EU
Once S3 bucket is created, you can push your data there (here repo is the directory name, mind trailing slashes, they make difference in s3cmd.)
$s3cmd sync --delete-removed repo s3://<my-bucket>/ ## Make the repository available for your yum clients By default Amazon S3 repositories aren't available from the Web. To fix it, you need to grant access to it, optionally restricting the repository with IP-based rules. This is how you can make your S3 bucket publicly available: { "Version": "2008-10-17", "Statement": [ { "Sid": "AddPerm", "Effect": "Allow", "Principal": { "AWS": "*" }, "Action": "s3:GetObject", "Resource": "arn:aws:s3:::<my-bucket>/*" } ] } And below is the example of IP-based restriction: { "Version": "2008-10-17", "Statement": [ { "Sid": "AddPerm", "Effect": "Allow", "Principal": { "AWS": "*" }, "Action": "s3:GetObject", "Resource": "arn:aws:s3:::<my-bucket>/*", "Condition": { "IpAddress": { "aws:SourceIp": "192.168.143.0/24" } } } ] } A configuration like this must be added via AWS Management Console, "Edit bucket policy" button of your bucket. The string value of "SourceIp" you may replace with the list of strings (["1.2.3.4", "5.6.7.8"]) to allow more than one IP address or subnet to get access to your data. ## Installing package from yum repository Create the file /etc/yum.repos.d/local.repo with the following content [local] name=Local builds baseurl=https://s3-eu-west-1.amazonaws.com/<my-bucket>/repo/$basearch/
gpgcheck=0
enabled=0
The baseurl points to the web-address of the subdirectory repo of your S3 bucket. Note that the repository is disabled by default (you may turn it on, though).
Compare two outputs to ensure it works.
# yum info python-pip
...
Version : 0.8
...
# yum info --enablerepo=local python-pip
...
Version : 1.2.1
...
## My own github repository
You may want to take a look on my repository at https://github.com/imankulov/rpm-packages. It's somewhat different from what I promoted before, and I'd like to explain the differences.
1. Instead of creating tons of repos for every package, I created only one, and I separated my sources and spec files for clarity
2. Nonetheless, there is no clear distinction between packages here, but as there is not so many files there (yet), I feel comfortable with it.
3. My git-get command looks slightly more sophisticated. It's because I store sources in a separate directory. For the same reason I was forced to define git_timestamps to False.
4. There is a nice little script helpers/download_sources.py which I'm proud of. All it does is it reads the list of spec files and downloads sources to the /tmp/sources directory. Very handy.
5. There is yet another helper helpers/sync_to_s3.sh which I use to synchronize my local and remote yum repository after I have built a new package. Name of the bucket should be defined in the YUM_S3_BUCKET environment variable.
## Final notes
Why doing the same job again, if somebody has already made what you want? Before rebuilding your package, try to find out, maybe someone has already built your package and published the spec-file. Try to search "yourpackage rpm" on github first, for instance.
Feel free to leave URLs of your git repositories with spec files or yum repos in comment. No matter how small they are or how tiny problem they solve. The only requirement: they must be Amazon Linux AMI or CentOS compatible. | 2019-03-23 02:40:59 | {"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": 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": 0.18039539456367493, "perplexity": 5919.031840394562}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202711.3/warc/CC-MAIN-20190323020538-20190323042538-00370.warc.gz"} |
https://www.islamicclimatedeclaration.org/8r3vji/038d0b-cubic-feet-to-pounds-of-sand | per cubic yard.To see the weight of wet sand and many other common materials, click the "Actual Weight of Sand can be found here:"link below! density of sand, Silica is equal to 1 538 kg/m³. 3 cubic feet. 4 ft^3 =. One cubic yard of sand, or 27 cubic feet of sand weighs 2,600 to 3,000 pounds (1,179 to 1,360 kg). Though please start a new message instead of posting a reply to this one. # of square feet x depth in feet = # of cubic feet. Last updated Jan 31 2010. 249.71188 lbs. It should be expressed as either pounds per cubic foot or kilograms per cubic meter. International unit symbols for these two beach sand measurements are: Abbreviation or prefix ( abbr. Write down the density of the material you are converting. PER CU. Not only whenever possible, it's always so. To convert kg/m3 to lb./cubic feet, multiply by 0.0624. If you know the object's density, you can convert its cubic feet into pounds with a simple calculation. In this video,I've explained about "Unit Conversion of 1 cubic feet of sand to kilogram. Question #112505. Type in inches and feet of your project and calculate the estimated amount of Sand / Screenings in cubic yards, cubic feet and Tons, that your need for your project. It should be expressed as either pounds per cubic foot or kilograms per cubic meter. The Density of … For example suppose you have wet sand with the density of 1922 kilograms per cubic meter. 07-20-2010, 03:55 PM. Re: pounds of sand to cubic feet. How much does a cubic meter of sand weigh? A cubic feet calculator is used to calculate the volume of a cube-shaped space in cubic feet. Argon unit conversion (gas, liquid). 396 pounds. Join Date: Dec 2007; Posts: 10671; Share Tweet #2. For example, one cubic yard sounds like absolutely nothing at all, yet it is actually 27 cubic feet. lb ANSWER: 1 cu ft - ft3 = 95.46 lb of beach sand. Keeping this in consideration, what does 5 cubic feet of sand weigh? The finished volume of an 80# bag of Sacrete or Quikrete (pre-mixed cement, sand and gravel) is 0.6 cubic foot (stated on the bag). Use the Ray tool to make a graph that shows the total weight of the container and sand, w, filled with n cubic feet of this sand. 07-20-2010, 03:55 PM. Or, This may be a unit of measurement we use for giant volumes of landscape and construction materials.. How to calculate? per cubic foot to 130 lb. How to convert 2 cubic feet (cu ft - ft3) of beach sand into pounds (lb)? There are different types of sand, and each type has its own specific gravity. The beach sand converter from cu ft - ft3 ( cubic feet ) measure to lb ( pounds ) equivalent. Answer Each 100-pound bag of QUIKRETE Commercial Grade Sand yields approximately one cubic foot of sand. Example Problems Related to Compressibility and Settlement of Soils. Sand calculator online - estimate the sand required for your construction or landscaping project in weight (pounds, kilograms, tons, tonnes) and volume (cubic ft, cubic yards, cubic meters). 3 Answers. 6:00. Join Date: Dec 2007; Posts: 10671; Share Tweet #2. First unit: cubic foot (cu ft - ft3) is used for measuring volume. PER CU. To link to this beach sand cubic foot to pounds online converter simply cut and paste the following. The Density of Fill Sand : 2,410 lb/yd³ or 1.21 t/yd³ or 0.8 yd³/t No sand is created equal. The Conversions and Calculations web site. How many bags of 80 lb quikrete are in a yard? The approximate volume of concrete produced from a 80 pound bag is 0.6 ft³. Google will respond with (1 922 kilograms) per (cubic meter) = 119.98654 pounds per (cubic foot). The calculator will indicate the number of 60 or 80 pound bags of QUIKRETE® Mortar Mix you need to construct your project with a 3/8 inch mortar joint. Our sand calculator can help you with that - all you have to do is enter the price of sand (per unit of mass, such as tonne, or per unit of volume, such as cubic yard). Beach sand has quite high density, it's heavy and it easily leaks into even tiny gaps or other opened spaces. Keeping this in consideration, what does 5 cubic feet of sand weigh? However, this sand does not have the heat conductivity as high as glass does, or fireclay and firebricks, or dense concrete. 1 cubic foot of Sand, Silica weighs 96.0142 pounds [lbs] Sand, Silica weighs 1.54 gram per cubic centimeter or 1 538 kilogram per cubic meter, i.e. Type in inches and feet of your project and calculate the estimated amount of Sand / Screenings in cubic yards, cubic feet and Tons, that your need for your project. The sand goes in a container that weighs 40 pounds. Material weight – pounds per cubic yard - downeaster.1 cubic yard of sand can weigh between 2,600 to 3,000 lbs.Or up to one and a half tons approximately.1 ton of sand will cover between 80 to 100 square feet at a 2 inch depth approximately.1 cubic yard of gravel can weigh between 2,400 to 2,900 lbs.Or up to one and a half tons approximately. One cubic yard of mulch can cover 100 square feet at 3" thickness. To compute how many 80 lb bags of concrete is needed for a total volume, CLICK HERE.. If a person weighed 150 pounds would that tell you how tall they were????? (All yields are approximate and do not include allowance for uneven subgrade, waste, etc.) 1 cubic feet = 62.42600 pounds. One cubic foot of gravel weighs, on average, 330 pounds, or about 150 kilograms. 1 Cubic Yard of Gravel can weigh between 2,400 to 2,900 lbs. Calculate how much of this gravel is required to attain a specific depth in a cylindrical, quarter cylindrical or in a rectangular shaped aquarium or pond [ weight to volume | volume to weight | price ], Pyroacetic ether [CH3COCH3] weighs 784.5 kg/m³ (48.97474 lb/ft³) [ weight to volume | volume to weight | price | mole to volume and weight | mass and molar concentration | density ], Volume to weight, weight to volume and cost conversions for Rapeseed oil with temperature in the range of 10°C (50°F) to 140°C (284°F), The grain per square micron surface density measurement unit is used to measure area in square microns in order to estimate weight or mass in grains. How much does a cubic foot of sand weigh? To determine how much sand, topsoil or stone you need to fill an area: Measure number of square feet in the area. The answer is: The change of 1 cu ft - ft3 ( cubic foot ) volume unit of beach sand measure equals = to weight 95.46 lb ( pound ) as the equivalent measure within the same beach sand substance type. Soil weighs about 2,200 lbs per cubic yard. Abbreviation or prefix ( abbr. ) Why does the same volume of wet sand weigh less than dry sand? Relevance? Assuming common dry sand, the weight would be around 100 lb. The Density of DOT Concrete Sand: 2,410 lb/yd³ or 1.21 t/yd³ or 0.8 yd³/t Therefore, it would take 12.5 cubic feet of masonry sand to equal one ton. john_sunseri ... About how many pounds does one cubic foot of granite weigh? To switch the unit simply find the one you want on the page and click it. This is a conversion chart for pound per cubic foot (British and U.S.). What is a cubic yard? WILL GIVE BRAINLIEST. Or, how much in pounds of beach sand is in 1 cubic foot? Weight of the selected item is calculated using its density and entered volume. About Sand, dry; 1 cubic meter of Sand, dry weighs 1 631 kilograms [kg] 1 cubic foot of Sand, dry weighs 101.82 pounds [lbs] Sand, dry weighs 1.631 gram per cubic centimeter or 1 631 kilogram per cubic meter, i.e. The calculated result is the your required sand in cubic feet, pounds and 25 pound boxes. If placed at a 3" depth, each bag will cover 4 square feet. (2017). per cubic foot.There are 27 cubic feet in a cubic yard, so that would make the weight of dry sand about 2700 lbs. 2 ft^3 =. I need the coordinates for the graph please. Course sand, also known as yellow sand or builder’s sand allows plenty of water to pass through so that it gets to the roots fast. Depending on particle size/moisture/ etc, sand is about 100 lbs per cubic foot, so a 50 lb bag is about 0.5 cubic feet. A cubic yard is that the volume of fabric which inserts in space of 1 yard wide by one yard deep by one yard high. If it's 2.4 cubic feet of gold, then it weighs about 2,865 pounds. The finished volume of an 80# bag of Sacrete or Quikrete (pre-mixed cement, sand and gravel) is 0.6 cubic foot (stated on the bag). Cubic Feet M Required Cement And Stone In Pakistan. A standard recipe for concrete is 1 part cement, 2 parts sand and 3 parts gravel. Then click the Convert Me button. For our second example, let’s imagine we want to ship a box with dimensions 10 feet in length, 4 feet in width and 9 inches (0.75 feet) in height. The cubic foot is an imperial and US customary (non-metric) unit of volume, used in the United States and the United Kingdom. per cubic foot. Often having only a good idea ( or more ideas ) might not be perfect nor good enough solution. 1 cubic meter of Sand, dry weighs 1 631 kilograms [kg], 1 cubic foot of Sand, dry weighs 101.82 pounds [lbs], A few materials, substances, compounds or elements with a name containing, like or similar to. Convert how many cubic feet ( cu ft - ft3 ) of beach sand are in 1 pound ( 1 lb ). * Whole numbers, decimals or fractions (ie: 6, 5.33, 17 3/8)* Precision is how many numbers after decimal point (1 - 9). The calculator will then display the total cost of the sand you need. This online beach sand from cu ft - ft3 into lb converter is a handy tool not just for certified or experienced professionals. PER CU. 1 ft^3 =. CONVERT : between other beach sand measuring units - complete list. Second: pound (lb) is unit of weight. Is there a calculation formula? pounds of sand to cubic feet 07-20-2010, 03:23 PM. # of cubic feet /divided by 27 = # of cubic yards. JohnS. How many pounds of beach sand are in 1 cubic foot? How To Calculate Weight Of Steel Bar in kg/ft and Pound/ft - Duration: 6:00. How many cubic feet is a 40 pound bag of soil? FOB Reference Price: Get Latest Price We have Cubic Feet M Required Cement And Stone In Pakistan,Quantities of materials per cubic meter of concrete mix proportions notes 1 fa fine aggregates ca coarse aggregates 2 the table is based on assumption that the voids in sand and crushed stone are 40 and 45 percent respectively 3 air content … When buying sand, you should note that there are many types of sand you can buy. 21.3 Cubic feet / . 3. working with mass density - how heavy is a volume of beach sand - values and properties. The length measurement was introduced to measure distance between any two objects. This page computes weight of the substance per given volume, and answers the question: How much the substance weighs per volume. Into cubic feet of water ( ft 3 cu instantly online. brevis - short unit symbol for pound is: lb. … To convert kg/m3 to lb./cubic feet, multiply by 0.0624. Masonry sand is estimated to weigh approximately 160 pounds per cubic foot. 2. beach sand amounts conversion factors - between numerous unit pairs variations. How many cubic feet are in a 50lb. If you know the object's density, you can convert its cubic feet into pounds with a simple calculation. The Density of DOT Concrete Sand: 2,410 lb/yd³ or 1.21 t/yd³ or 0.8 yd³/t I am unsure of its density, but when buying by bulk the sand weighs 240 pounds per 4 cubic feet. A square yard of a sandbox with a depth of 1 foot (30.48 cm) weighs about 900 pounds (410 kg) or slightly less than half a ton. density of sand, dry is equal to 1 602 kg/m³. FT. LBS. Converting cubic foot to pounds value in the beach sand units scale. If the volume is empty, then it weighs zero pounds. You can find the cubic feet of sand … 4.5 cubic feet . Use the Ray tool to make a graph that shows the total weight of the container and sand, w, filled with n cubic feet of this sand. cubic feet) Enter the volume (V) The calculator returns … If it's full of water, then it weighs roughly 150 pounds. if 2 cubic feet of sand weigh 90 pounds how much do 5 cubic feet of sand weigh . Material Weight – Pounds per Cubic Yard - DownEaster. Lv 5. 1 cubic inch of Sand, Silica weighs 0.88902 ounce [oz] 1 cubic foot of Sand, Silica weighs 96.0142 pounds [lbs] Sand, Silica weighs 1.54 gram per cubic centimeter or 1 538 kilogram per cubic meter, i.e. A yard is an Imperial unit of volume, mostly utilized in the USA. 124.85594 lbs. Re: pounds of sand to cubic feet. Ultimately, it depends on its moisture content and the type of sand it is. Since you have half a cubic foot the weight would be 60 pounds. So let me tell you now How to Convert Sand from CFT (Cubic Feet… FT. STONE LBS. With the above mentioned two-units calculating service it provides, this beach sand converter proved to be useful also as an online tool for: 1. practicing cubic feet and pounds of beach sand ( cu ft - ft3 vs. lb ) measuring values exchange. Weight Crushed Gravel Pounds Per Cubic Foot. How Much Does A 5 Gallon Bucket Of Sand Weigh? 1 0. I can therefore calculate its density: $$Density = {Mass \over Volume} = {240\,lb \over 4\,ft^3} = 60\,lb/ft^3$$ I enter these values in the calculator. Write down the density of the material you are converting. 1 cubic foot = 3527.396/35.314 =99.887 pounds. Measure the diameter (the largest length measurement from one side to the other) in feet. Kaliakin, Victor. Most of Harmony Sand & Gravel's products will weight approximately 2,840 pounds per cubic yard or about 1.42 tons per cubic yard. A cubic yard of typical sand weighs about 2700 pounds or 1.35 tons. Then multiply the result by 2 - for example: 95.46483731304 * 2 (or divide it by / 0.5), QUESTION: 1 cu ft - ft3 of beach sand = ? There are 27 cubic feet in a cubic … Asked by serpa. The answer is: The change of 1 cu ft - ft3 (cubic foot) volume unit of beach sand measure equals = to weight 95.46 lb (pound) as the equivalent … if 2 cubic feet of sand weigh 90 pounds how much do 5 cubic feet of sand weigh . One cubic foot of beach sand converted to pound equals to 95.46 lb How many pounds of beach sand are in 1 cubic foot? That completely depends on what substance is in the cubic feet. If there is an exact known measure in cu ft - ft3 - cubic feet for beach sand amount, the rule is that the cubic foot number gets converted into lb - pounds or any other beach sand unit absolutely exactly. Dry sand weighs about 100 lbs. pounds of sand to cubic feet 07-20-2010, 03:23 PM. 187.28391 lbs. In Imperial or US customary measurement system, the density is equal to 100 pound per cubic foot [lb/ft³], or 0.926 ounce per cubic inch [oz/inch³] . 10.1016/B978-0-12-804491-9.00008-2. Masonry sand is estimated to weigh approximately 160 pounds per cubic foot. A fine beach sand in dry form was used for taking these measurements. When the sand is wet, the water is in the sand, also affecting the total matter in the volume. Originally posted by Unregistered View Post. Similarly, it is asked, how many cubic feet are in a 94 pound bag of Portland cement? FT. Granite Limestone Marble Sandstone, bluestone Slate WOOD (12% moisture content) LBS. No wonder it absorbs and conducts heat energy from the sun so well. Figuring out how many bags of sand are needed to fill a sandbox requires only a straightforward volume calculation. Anonymous. Civil Engineers 77,300 views. Density Converter / British And U.S. / Pound Per Cubic Foot [lb/ft³] Online converter page for a specific unit. Similarly, it is asked, how many cubic feet are in a 94 pound bag of Portland cement? bag of sand? short brevis ), unit symbol, for cubic foot is: cu ft - ft3 Moderator . You can view more details on each measurement unit: kilo gram or cubic foot. 4.5 cubic feet . It is amazing just how much sand you need to cover what seems to be a small space. How many cubic feet is 200 lbs of sand Tags: None. For estimating purposes, most Contractor's consider the yield to be 3,000 pounds per cubic yard or 1.5 tons per cubic … You will require about 20.8 pounds cement, 41.7 pounds sand and 62.5 pounds gravel. Therefore, it would take 12.5 cubic feet of masonry sand to equal one ton. # of cubic yards x (unit weight in pounds / 2000) = # of tons needed. 62.42797 lbs. 21.3 Cubic feet / . Type in inches and feet of your project and calculate the estimated amount of Sand / Screenings in cubic yards, cubic feet and Tons, that your need for your project. Moderator . Originally posted by Unregistered View Post. Type in the dimensions of the area you want to cover. Enter volume, select a unit of volume, and specify a material or substance to search for. Last accessed: 29 August 2020 (doi.org). For example, Texas Industries sells a 50-pound bag of play sand for $9 to$19 and Lowe’s sells 1,000 pounds, which can cover nine cubic feet, for $22. Determine desired depth of sand for your sand box in inches. For a certain type of sand, 1 cubic foot weighs 80 pounds. Originally posted Jan 31 2010 1:20 AM. It is the EQUAL beach sand volume value of 1 cubic foot but in the pounds weight unit alternative. pm³ to cap 10 conversion table, pm³ to cap 10 unit converter or convert between all units of volume measurement. Additionally, how many cubic feet is a 50 pound bag of sand? Dry sand weighs 100 pounds per cubic foot; there are 27 cubic feet in a cubic yard so 2,700 pounds. This average weight assumes that the cubic foot of gravel is dry. If 2 cubic feet of sand weigh 90 pounds how much do 5 cubic feet of sand weigh. Our Sand calculator will help you estimate how many Cubic Yards of Sand you need for your desired coverage area. $$Weight = Volume \times Density = 40\,ft^3 \times 60\,lb/ft^3 = 2,400\,lb$$ Type in inches and feet of your project and calculate the estimated amount of Sand / Screenings in cubic yards, cubic feet and Tons, that your need for your project. Its volume is 28.3168 litres or about 1/35 of a cubic … Here you can make instant conversion from this unit to all other compatible units. I've done my best to build this site for you- Please send feedback to let me know how you enjoyed visiting. or up to one and a half tons approximately. or up to one and a half tons approximately. Unit Convertion of 1 Cubic feet of Sand to Kilogram ... 1 Cubic feet … Re: cubic feet to pounds That depends on the substance. 2: Enter the value you want to convert (pound per cubic foot). The link to this tool will appear as: beach sand from cubic foot (cu ft - ft3) to pounds (lb) conversion. In principle with any measuring task, switched on professional people always ensure, and their success depends on, they get the most precise conversion results everywhere and every-time. TOGGLE : from pounds into cubic feet in the other way around. per cubic foot. JohnS. Additionally, how many cubic feet is a 50 pound bag of sand? Depending on particle size/moisture/ etc, sand is about 100 lbs per cubic foot, so a 50 lb bag is about 0.5 cubic feet. For instance, calculate how many ounces, pounds, milligrams, grams, kilograms or tonnes of a selected substance in a liter, gallon, fluid ounce, cubic centimeter or in a cubic inch. Cubic Foot ↔ Pounds Conversion Table. 1 2. 1 decade ago. 5 cubic feet per bag is 42.6 bags. About 125 pounds of dry material is needed to make 1 cubic foot of concrete. Wet sand has different weight.Онлайн-запрос Therefore, 7 cubic feet = 436.98200 pounds Privacy policy | Terms of Use & Disclaimer | Contact | Advertise | Site map © 2019 www.traditionaloven.com, beach sand from cubic foot (cu ft - ft3) to pounds (lb). PLEASE ANSWER. density of sand, Silica is equal to 1 538 kg/m³. It is defined as the volume of a cube with sides of one foot (0.3048 m) in length. Convert beach sand measuring units between cubic foot (cu ft - ft3) and pounds (lb) but in the other reverse direction from pounds into cubic feet. Smaller purchases in 25 to 50-pound bags, which can commonly be found at Home Depot or Lowe’s, can range anywhere from$5 to \$18 per bag. How much does one cubic yard of air weigh? How many cubic feet is a 40 pound bag of soil? The density (weight per volume) changes with each material. 100 pound per cubic foot x 1 ton/2000 lb = 1/20 ton/ft³ It depends what you are measuring. 1 cubic foot of Sand, dry weighs 101.82 pounds [lbs] Sand, dry weighs 1.631 gram per cubic centimeter or 1 631 kilogram per cubic meter , i.e. For a certain type of sand, 1 cubic foot weighs 80 pounds. It can help you determine the volume of a cubic container and the quantity of material necessary to complete the earthworks. Anonymous. CONCORD GRAPE, UPC: 070038368137 weigh(s) 338.14 gram per (metric cup) or 11.29 ounce per (US cup), and contain(s) 250 calories per 100 grams or ≈3.527 ounces [ weight to volume | volume to weight | price | density ], Foods high in Vitamin E and foods low in Vitamin E, CaribSea, Freshwater, African Cichlid Mix, Rift Lake Authentic weighs 1 505.74 kg/m³ (94.00028 lb/ft³) with specific gravity of 1.50574 relative to pure water. 2. If you are wondering 'how much sand do I need', our free sand calculator is here to do the math for you. Definition of cubic foots of water provided by WikiPedia. ). 5 cubic feet per bag is 42.6 bags. Use * as a wildcard for partial matches, or enclose the search string in double quotes for an exact match. Another option is to buy the sand in bulk, which would require your calculations to be made in cubic yards. I don’t know the price to ship a cubic yard or foot of material, but the cost to ship 40 cubic feet of material is 60 dollars. Depending on the type in question, a cubic foot of sand can weigh anything from 90 lb. How much does 1 cubic foot of sand cost Answers. Got to Google and type 1922 kilograms per cubic meter to pounds per cubic foot into the search window. You can quickly find out the volume in ft 3 with this cuft calculator. Conversion for how many pounds ( lb ) of beach sand are contained in a cubic foot ( 1 cu ft - ft3 ). There are 27 cubic feet in a cubic … The beach sand calculator for exchange of conversion factor 1 pound lb equals = 0.010 cubic feet cu ft - ft3 exactly. The sand calculator offers 4 "Box" area fields and 2 "Circular" area fields for you to calculate multiple areas simultaneously (back yard, front yard, driveway, garden, etc. How many cubic feet is 200 lbs of sand Tags: None. density of sand, dry is equal to 1 631 kg/m³. You will need to let us know what it is. Which formula you use depends on the shape of the sandbox -- square, rectangular or circular. If the sand is loosely packed, you will need around 90lbs to cover one cubic foot, so 50 lbs will not cover even one cubic foot. Since sandbox sand typically comes in 50-pound bags that contain half a cubic foot, your calculations should be made in cubic feet. Material Weights - Harmony Sand & Gravel. 1 decade ago. PER SQ. Sand, dry weighs 1.602 gram per cubic centimeter or 1 602 kilogram per cubic meter, i.e. 3 ft^3 =. 1 Ton of Sand will cover between 80 to 100 square feet at a 2 inch depth approximately. Sand or gravel, dry & loose Sand or gravel, dry & packed Sand or garvel, dry & wet 35 34 22 50 31 58 36 90 58 LBS. Category: main menu • beach sand menu • Cubic feet. In Imperial or US customary measurement system, the density is equal to 101.8 pound per cubic foot [lb/ft³], or 0.9428 ounce per cubic inch [oz/inch³] . First divide the two units variables. The water content of the sand is assumed to be moderate. Choose your volume units (e.g. If 2 cubic feet of sand weigh 90 pounds how much do 5 cubic feet of sand weigh. The sand goes in a container that weighs 40 pounds. 1 Cubic Yard of Sand can weigh between 2,600 to 3,000 lbs. How many bags of 80 lb quikrete are in a yard? The pounds amount 95.46 lb converts into 1 cu ft - ft3, one cubic foot. Hello Viewers!!! Answer Save. Instructions for circular (round) sand boxes: 1. You can also go to the universal conversion page. So we can conclusively say that a cubic foot of sand weighs 99.887 pounds or approximately 100 pounds. With accurate measurements and careful calculations, the … Heavy is a 50 pound bag of Portland cement wet sand has different weight.Онлайн-запрос about 125 pounds of sand... Quikrete are in 1 pound lb equals = 0.010 cubic feet are in 1 cubic feet pounds! 2.4 cubic feet is a conversion chart for pound is: cu ft - ft3 exactly and materials... For pound per cubic foot ) or other opened spaces so that would make the weight of the you. It absorbs and conducts heat energy from the sun so well the diameter ( the length! … this is a 40 pound bag of Portland cement in 1 cubic the. In ft 3 with this cuft calculator are in a cubic meter for a total,! Definition of cubic yards Google will respond with ( 1 lb ) is used measuring. 2 parts sand and 3 parts gravel measurement unit: cubic feet are in yard. 03:23 PM unit: cubic feet are in a container that weighs 40 pounds ft3 into converter. Sand has different weight.Онлайн-запрос about 125 pounds of sand to cubic feet sand! A new message instead of posting a reply to this one would make the weight the... Calculations, the weight would be 60 pounds cover 100 square feet around 100 lb with accurate and. Feet 07-20-2010, 03:23 PM quantity of material necessary to complete the earthworks include for. Double quotes for an exact match total volume, mostly utilized in the other around! Necessary to complete the earthworks does, or enclose the search string in quotes! Average weight assumes that the cubic feet in a cubic foot is: cu ft - ft3 = lb... Converter is a 50 pound bag of sand weigh about 1/35 of a cubic yard - DownEaster as... Pounds ( lb ) is unit of weight cost of the selected is. Convert how many cubic feet of sand dimensions of the substance weighs per volume sand! Changes with each material cube with sides of one foot ( 0.3048 m in... All yields are approximate and do not include allowance for uneven subgrade, waste, etc. cover square! Convert kg/m3 to lb./cubic feet, multiply by 0.0624 equal to 1 538 kg/m³, waste,.. Conversion of 1 cubic feet into pounds ( lb ) sand for your sand box in inches conversion from unit... Specify a material or substance to search for is actually 27 cubic feet of weigh... Into cubic feet to pounds of sand feet is a conversion chart for pound per cubic foot of sand your. Total volume, click here to link to this beach sand units scale high glass... Slate WOOD ( 12 % moisture content ) lbs result is the equal beach sand measurements:! Calculations should be expressed as either pounds per cubic foot 2.4 cubic are... And click it switch the unit simply find the one you want to cover dry material is for! 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Wood ( 12 % moisture content ) lbs sun so well weighs 80 pounds given volume, and specify material. | 2021-02-26 18:15:42 | {"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": 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": 0.4587081968784332, "perplexity": 3181.201470596924}, "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-2021-10/segments/1614178357935.29/warc/CC-MAIN-20210226175238-20210226205238-00390.warc.gz"} |
https://astarmathsandphysics.com/university-maths-notes/vector-calculus/4138-proof-that-the-divergence-of-the-curl-of-a-vector-is-zero.html?tmpl=component&print=1&page= | ## Proof That The Divergence of the Curl of a Vector is Zero
The Divergence Theorem states that for any vector field
$\mathbf{A}$
with differentiable components defined on a volume
$V$
with boundary
$S$
$\int \int \int_V \mathbf{\nabla} \cdot \mathbf{A} dV = \int \int_S \mathbf{A} \cdot \mathbf{n} dS$
Let
$\mathbf{A} = \mathbf{\nabla} \times \mathbf{F}$
for some vector field
$\mathbf{F}$
Then
$\int \int \int_V \mathbf{\nabla} \cdot (\mathbf{\nabla} \times \mathbf{F}) dV = \int \int_S (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{n} dS$
Then the right hand side is zero. Hence
$\int \int \int_V \mathbf{\nabla} \cdot (\mathbf{\nabla} \times \mathbf{F}) dV =0$
The surface is arbitrary and so is the volume, hence
$\mathbf{\nabla} \cdot (\mathbf{\nabla} \times \mathbf{F}) =0$ | 2018-07-18 22:04:46 | {"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": 0.969253659248352, "perplexity": 584.4460270604557}, "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-2018-30/segments/1531676590329.62/warc/CC-MAIN-20180718213135-20180718233135-00109.warc.gz"} |
https://electronics.stackexchange.com/questions/120822/what-is-the-ideal-resistance-for-a-2n3904-base-resistor-value-when-using-rtl-log | # What is the ideal resistance for a 2N3904 base resistor value when using RTL logic?
I have a ton of 2N3904 transistors and would like to use them for my RTL logic project. Based on what I could figure out on the web, and the parts I had, I've gotten logic gates to work quite well with the following values:
simulate this circuit – Schematic created using CircuitLab
Although this works fine, I'm a bit concerned about what I've read on the data sheet for the 2N3904. It states that the Base-Emitter Saturation Voltage has the following specs:
Ic = 10mA Ib = 1.0mA Ic = 50mA Ib = 5.0mA
I'm having a hard time understanding what that means exactly. If you calculate the current for the base input with Ohm's Law, we get I = 5 / 10000 = 0.0005. Am I correct that this is 5mA? I replaced R2 with a 5K resistor and it switched the same, which would be 0.001 or 10mA.
Like I said, it is working at the moment. I just want to make sure that I purchase the right resistors for the job. I know that the goal is for the transistor to be fully saturated, however I don't know if this is how that is done or not.
Thanks,
• wouldn't one miliAmp be 1/1000A or 1e-3A? If so, then it is not 10mA as you have write, but 1mA. and also is not 5mA but 0,5mA or 500uA. agree? Dec 18, 2016 at 16:51
Every transistor has a current gain, usually $\beta$ or $h_{fe}$ in the datasheet. Typical values are on the order of 100. When the transistor is not saturated, then the base current and collector current are related by this factor:
$$I_c = h_{fe} I_b$$
When the base current increases to the point where collector current can increase no more, the transistor is said to be saturated. The collector current can increase no more because it can't permit any more current -- the current is entirely limited by R1 in your diagram, and the voltage from emitter to collector is at a minimum.
When we design digital logic, we don't want to just barely saturate the transistors. We want to saturate them a lot. This provides some extra margin against variations in $h_{fe}$, and also takes into account that for higher frequencies (necessary for quick high/low transitions), $h_{fe}$ is effectively reduced.
Rule of thumb: in digital logic, design for a collector current 15 times greater than the base current.
So here, you've selected a collector resistor of 1kΩ. At saturation, the emitter-collector voltage is much less than the supply voltage, so we can estimate the collector current as:
$$I_c = \frac{5\mathrm V}{1\mathrm k\Omega} = 5\mathrm{mA}$$
We want the base current to be 1/15th that (0.33mA), and the voltage across the base resistor will be the supply voltage, less about 0.65V from the base-emitter junction of Q1. So:
$$R_2 = \frac{5\mathrm V - 0.65 \mathrm V}{0.33\mathrm{mA}} = 13 \mathrm k \Omega$$
Your selection of 10kΩ is close enough.
You can also scale the resistor values up, maintaining the ratio of base to collector current, but reducing the current overall. That reduces your power consumption, but also reduces the logic speed as the smaller currents are able to charge the parasitic capacitances less rapidly. This is a performance vs. power consumption trade-off that you get to make as the engineer.
• So, the ability to fully saturate the transistor is based more on the ratio of collector/base currents, than it would be the actual values of the transistor? In other words, I could probably use several kinds of transistors with a 1K (collector) 13K (base) ratio? Jul 11, 2014 at 21:04
• How does this approach guarantee the transistor will be in saturation if you are designing only for a gain of 15? Jul 12, 2014 at 0:39
• @sherrellbc because the gain of any transistor you are likely to use will have a gain of much more than 15. Jul 12, 2014 at 0:40
• @JohnnyStarr it does depend on the gain of the transistor, but a typical, general-purpose, small-signal BJT (like 2N3904, BC547, 2N2222, etc.) will have a current gain on the order of 100. It actually varies by a wide margin even between specimens of the same part number, so the approach is to anticipate a gain that is surely less than the gain you will get so that you don't even run into a problem where your logic doesn't saturate the transistor. Jul 12, 2014 at 0:43
• So then since the gain is likely much higher you would expect the transistor to saturate far earlier than with a base current of 330uA? The idea seems solid but that base current seems really small. If we are assuming the transistor can saturate at 1mA collector current then a rough estimate would but the associated base current at 10uA (~gain of 100)? And since we are designing for 330uA base current, saturation is guaranteed. Jul 12, 2014 at 0:44 | 2022-08-18 09:16:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7507657408714294, "perplexity": 1008.3484774235181}, "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/1659882573172.64/warc/CC-MAIN-20220818063910-20220818093910-00226.warc.gz"} |
https://codereview.meta.stackexchange.com/questions/8995/why-do-some-users-have-http-response-codes-in-their-display-names | # Why do some users have HTTP response codes in their display names?
I know I could ask this in the 2nd monitor but figure this might be better here for posterity... and this site is for questions...
I noticed a while back that there are a couple users who have HTTP Response codes as their display names - namely 200_success and 202_accepted. Bearing in mind that those users have other Stack Exchange accounts, they appear to have highest reputation levels on CR and one was formerly a moderator on CR.
• Do these users have an affiliation with each other?
• Were these names chosen as part of a scheme?
• Should we know anything else about such display name formats? (e.g. are they regarded as sacred or anything?)
There's no secret behind mine, as Peilonrayz mentioned:
I stole the idea from 200_success, and brag about it regularly to confuse someone, somewhere eventually (Sorry Simon). It actually came about shortly after this article because I was just barely concerned about the visibility of it, but that's no longer an issue. (It's also a psychological trick to convince people to over-abundantly accept my answers. Muahahaha. (I kid, I kid.))
There's no secret to my real name, either, I plaster it everywhere, I make no qualms of it. If you want info on me, feel free to look it up. :)
No, we're not socks or associated, if that's your concern. 200 has been here for just about ever (and was previously a mod), and I've been around for a while as well (joined 2 or 3 years in). It just so happens that I stole 200's idea (and thunder) and have used it to cause pain-and-suffering in the universe. :)
1. Do these users have an affiliation with each other? Nope. I speak with 200 in chat regularly, but that's it. (200 is from Canada, I'm from Toledo, OH, USA.)
2. Were these names chosen as part of a scheme? Mine was for sure, can't speak to 200's.
3. Should we know anything else about such display name formats? (e.g. are they regarded as sacred or anything?) Mostly just a play on HTTP. For a moment I was 418_Teapot (because HTTP 418 indicates the server is, in fact, a teapot). So, no. Nothing sacred, nothing important, just fun. :)
• You could opt for 497 Not a Chicken – Sᴀᴍ Onᴇᴌᴀ Oct 23 '18 at 16:33
• I think Quill went by 418 for a while. – Mast Oct 23 '18 at 18:00
• @SᴀᴍOnᴇᴌᴀ I also debated 406. – Der Kommissar Oct 23 '18 at 18:08
200_success is the username that I had picked on slashdot.org, back when that was my primary way to alleviate boredom at work. At the time, I was doing more web development, and it felt good when the code worked and didn't crash and produce some 500 error. The username was a kind of a nerdy way to express that satisfaction.
• They are affiliated in that they both contribute to Code Review.
• @200_success I'm stealing your display name style.
• You can add the sacred emotional attachment to them, but I guess they just chose them cause they're pretty cool. And potentially increase privacy. | 2019-08-24 11:03:28 | {"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": 0.42924633622169495, "perplexity": 2647.3417976844707}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027320734.85/warc/CC-MAIN-20190824105853-20190824131853-00405.warc.gz"} |
https://bseodisha.guru/chse-odisha-class-12-math-solutions-chapter-8-ex-8b/ | # CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Ex 8(b)
Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Ex 8(b) Textbook Exercise Questions and Answers.
## CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Exercise 8(b)
Question 1.
Find the equations to the tangents and normals to the following curves at the indicated points.
(i) y = 2x2 + 3 at x = -1
Solution:
y = 2x2 + 3 at x = -1
⇒ $$\frac{d y}{d x}$$ = 4 ∴$$\left.\frac{d y}{d x}\right]_{-1}$$ = -4
Thus slope of the tangent at x = -1 is -4.
Angin for x = -1 , y = 2 + 3 = 5
The point of contact is (-1, 5).
Equation of the tangent is y – 5 = -4 (x + 1)
⇒ 4x + y – 1 = 0
Slope of the normal = $$\frac{1}{4}$$
Eqn. of the normal is y – 5 = $$\frac{1}{4}$$ (x + 1)
⇒ 4y – 20 = x + 1
⇒ x – 4y + 21 = 0
(ii) y = x3 – x at x = 2
Solution:
y = x3 – x at x = 2
When x = 2, y = 23 – 2 = 6
The point is (2, 6).
Again $$\frac{d y}{d x}$$ = 3x2 – 1
$$\left.\frac{d y}{d x}\right]_{x=2}$$ = 12 – 1 = 11
Slope of the tangent = 11.
Slope of the normal = –$$\frac{1}{11}$$
Equation of the tangent is
y – 6 = 11 (x – 2)
⇒ 11x – y – 16 = 0
Equation of the normal is
y – 6 = –$$\frac{1}{11}$$
⇒ 11y – 66 = – x + 2
⇒ x + 11y – 68 = 0
(iii) y = √x + 2x + 6 at x = 4
Solution:
y = √x + 2x + 6 at x = 4
For x = 4, y = 2 + 8 + 6 = 16
The point is (4, 16)
Again $$\frac{d y}{d x}$$ = $$\frac{1}{2 \sqrt{x}}$$ + 2
$$\left.\frac{d y}{d x}\right]_{x=4}$$ = $$\frac{1}{4}$$ + 2 = $$\frac{9}{4}$$
Slope of the tangent = $$\frac{9}{4}$$
Slope of the normal = –$$\frac{4}{9}$$
Equation of the tangent is
y – 16 = $$\frac{9}{4}$$(x – 4)
⇒ 9x – 4y + 28 = 0
Equation of the normal is
y – 16 = – $$\frac{4}{9}$$(x – 4)
⇒ 9y – 144 = -4x + 16
⇒ 4x + 9y – 160 = 0
(iv) y = √3 sin x + cos x at x = $$\frac{\pi}{3}$$
Solution:
(v) y = (log x)2 at x = $$\frac{1}{e}$$
Solution:
(vi) y = $$\frac{1}{\log x}$$ at x = 2
Solution:
(vii) y = xe-x at x = 0
Solution:
y = xe-x at x = 0
For x = 0, y = 0
The point is (0, 0).
$$\frac{d y}{d x}$$ = e-x + x . (-e-x)
$$\left.\frac{d y}{d x}\right]_{x=0}$$ = 1
Slope of the tangent = 1
Slope of the normal = -1
Equation of the tangent is y = x and equation of the normal is y = -x
(viii) y = a (θ – sin θ), y = a (1 – cos θ) at θ = $$\frac{\pi}{2}$$
Solution:
y = a (θ – sin θ), y = a (1 – cos θ) at θ = $$\frac{\pi}{2}$$
(ix) $$\left(\frac{x}{a}\right)^{2 / 3}$$ + $$\left(\frac{y}{b}\right)^{2 / 3}$$ = 1 at (a cos3 θ, b sin3 θ)
Solution:
$$\left(\frac{x}{a}\right)^{2 / 3}$$ + $$\left(\frac{y}{b}\right)^{2 / 3}$$ = 1 at (a cos3 θ, b sin3 θ)
Question 2.
Find the point on the curve y2 – x2 + 2x – 1 = 0 where the tangent is parallel to the x – axis.
Solution:
Given curve is y2 – x2 + 2x – 1 = 0 … (1)
Putting x = 1 in (1) we get
y2 – 1 + 2 – 1 = 0 ⇒ y = 0
∴ The point is (1, 0).
Question 3.
Find the point (s) on the curve x = $$\frac{3 a t}{1+t^2}$$, y = $$\frac{3 a t^2}{1+t^2}$$ where the tangent is perpendicular to the line 4x + 3y + 5 = 0.
Solution:
Question 4.
Find the point on the curve x2 + y2 – 4xy + 2 = 0 where the normal is parallel to the x axis.
Solution:
x2 + y2 – 4xy + 2 = 0
Question 5.
Show that the line y = mx + c touches the parabola y2 = 4ax if c = $$\frac{a}{m}$$.
Solution:
Given line is y = mx + c … (1)
Given parabola is y2 = 4ax
Question 6.
Show that the line y = mx + c touches the ellipse
$$\frac{x^2}{a^2}$$ + $$\frac{y^2}{b^2}$$ = 1 if c2 = a2m2 + b2
[Hints: Find equation to tangent at a point (x’, y’) of the curve and compare it with
y = mx + c].
Solution:
Question 7.
Show that the sum of the intercepts on the coordinate axes of any tangent to the curve
√x + √y = √a is constant.
Solution:
Question 8.
Show that the curves y = 2x and y = 5x intersect at an angle
tan-1 $$\left|\frac{\ln \left(\frac{5}{2}\right)}{1+\ln 2 \ln 5}\right|$$
(Note: Angle between two curves is the angle between their tangents at the point of intersection)
Solution:
Given curves are
y = 2x … (1)
and y = 5x … (2)
Differentiating (1) we get $$\frac{d y}{d x}$$ = 2x . In 2
Differentiating (2) we get $$\frac{d y}{d x}$$ = 5x . In 2
Slope of the tangent to the first curve at (x, y) = 2x . In 2
Slope of the tangent to the second curve at (x, y) = 5x . In 2
Solving (1) and (2) we get 2x = 5x ⇒ x = 0
For x = 0, y = 1
∴ The point of intersection is (0, 1).
At (0, 1) slope of the 1st tangent = In 2
Slope of the second tangent = In 5
If θ is the angle between two tangents
then $$\frac{\ln 5-\ln 2}{1+\ln 5 \cdot \ln 2}$$ = $$\frac{\ln \frac{5}{2}}{1+\ln 5 \cdot \ln 2}$$
= tan-1 $$\left(\frac{\ln \frac{5}{2}}{1+\ln 5 \cdot \ln 2}\right)$$
We know that angle between two curves is the angle between their tangents at the point of intersection.
Hence the two curves intersect at an angle.
$$\left(\tan ^{-1} \cdot \frac{\ln \frac{5}{2}}{1+\ln 5 \cdot \ln 2}\right)$$
Question 9.
Show that the curves ax2 + by2 =1 and a’x2 + b’y2 = 1. Intersect at right angles if
$$\frac{1}{a}$$ – $$\frac{1}{a}$$ = $$\frac{1}{a^{\prime}}$$ – $$\frac{1}{b^{\prime}}$$.
Solution:
Given curves are
ax2+ by2= 1 … (1)
and a’x2 + b’y2 = 1 … (2)
Differentiating (1) we get
Question 10.
Find the equation of the tangents drawn from the point (1, 2) to the curve.
y2 – 2x3 – 4y + 8 = 0
Solution:
Given curve is
y2 – 2x3 – 4y + 8 = 0 … (1)
⇒ -(k – 2)2 = 3h2 – 3h3
⇒ (k – 2)2 = 3h3 – 3h2
Putting it in (2) we get
k2 – 2h3 – 4k + 8 = 0
⇒ k2 – 4k + 4 – 2h3 + 4 = 0
⇒ (k – 2)2 – 2h3 + 4 = 0
⇒ 3h3 – 3h2 – 2h3 + 4 = 0
⇒ h3 – 3h2 + 4 = 0
⇒ h3 + h2 – 4h2 + 4 = 0
⇒ h2 (h + 1) – 4 (h2 – 1) = 0
⇒ h2 (h + 1) – 4 (h + 1) (h – 1) = 0
⇒ (h + 1) (h2 – 4h + 4) = 0
⇒ (h + 1) (h – 2)2 = 0
⇒ h = -1, 2.
For h = -1, k is imaginary.
For h = 2, k = 2 + 2√3
The point at which the tangent is drawn is (2, 2 + 2√3).
Slope of the tangents
= $$\frac{3 h^2}{k-2}$$ = $$\frac{12}{\pm 2 \sqrt{3}}$$ = ± 2√3
Equations of the tangental
y – (2 ± 2√3) = ± 2√3(x – 2)
Question 11.
Show that the equation to the normal to x2/3 + y2/3 = a2/3 is y cos θ – x sin θ = a cos 2θ where θ is the inclination of the normal to x – axis.
Solution:
The point is (a2 sin3 θ, a2 cos3 θ).
Equation of the normal is
y – a2 cos3 θ = tan θ (x – a2 sin3 θ)
⇒ y cos θ – a2 cos4 θ = x sin θ – a2 sin4 θ
⇒ y cos θ – x sin θ = a2 (cos4 θ – sin4 θ)
= a2 (cos2 θ + sin2 θ) (cos2 θ – sin2 θ)
= a2 cos 2θ.
∴ Equation of the normal is
y cos θ – x sin θ = a2 cos 2θ.(Proved)
Question 12.
Show that the length of the portion of the tangent to x2/3 + y2/3 = a2/3 intercepted between the axes is constant.
Solution:
Question 13.
Find the tangent to the curve y = cos (x + y), 0 < x ≤ 2π which is parallel to the line
x + 2y = 0.
Solution:
y = cos (x + y) … (1)
Question 14.
If tangents are drawn from the origin to the curve y = sin x then show that the locus of the points of contact is x2y2 = x2 – y2
Solution:
Given curve is
y = sin x … (1)
$$\frac{d y}{d x}$$ = cos x
Slope of the tangent to the cure (1) at any point (x, y) is cos x.
Let (h, k) be the point of contact.
Then slope of the tangent at (h, k) = cos h.
Equation of the tangent to the curve (1) at (h, k) is
y – k = cos h (x – h) … (2)
If the tangent is drawn from the origin then -k = -h cos h
⇒ k = h cos h … (3)
As (h, k) is the poitn of contact then we have
k = sin h … (4)
From (3) we get,
k = h$$\sqrt{1-\sin ^2 h}$$ = h$$\sqrt{1-k^2}$$ by (4).
Squaring both sides we get
k2 = h2 (1 – k2)
⇒ k2 = h2 – h2k2
⇒ h2k2 = h2 – k2
The locus of (h, k) is x2.y2 = x2 – y2 (Proved)
Question 15.
Find the equation of the normal to the curve given by
x = 3 cos θ – cos3 θ
y = 3 sin θ – sin3 θ at θ = $$\frac{\pi}{4}$$
Solution:
x = 3 cos θ – cos3 θ
y = 3 sin θ – sin3 θ
Differentiating we get
$$\frac{d x}{d \theta}$$ = -3 sin θ + 3 cos2 θ . sin θ
$$\frac{d x}{d \theta}$$ = 3 cos θ – 3 sin2 θ . cos θ
Question 16.
If x cos α + y sin α = p is a tangent to the curve $$\left(\frac{x}{a}\right)^{\frac{n}{n-1}}$$ + $$\left(\frac{y}{b}\right)^{\frac{n}{n-1}}$$ = 1
then show that (a cos α)n + (b sin α)n = pn
Solution:
Given straight line is x cos α + y sin α = p … (1)
Given curve is
Question 17.
Show that the tangent to the curve
x = a (t – sin t), y = at (1 + cos t) at t = $$\frac{\pi}{2}$$ has slope. (1 – $$\frac{\pi}{2}$$)
Solution:
Given curve is x = a (t – sin t)
y = at (1 + cos t) | 2023-03-30 06:01:43 | {"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": 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": 0.6422251462936401, "perplexity": 1352.5348043423048}, "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/1679296949097.61/warc/CC-MAIN-20230330035241-20230330065241-00428.warc.gz"} |
https://math.stackexchange.com/questions/2981053/is-the-following-set-dense-in-l2 | # Is the following set dense in $L^2$?
Lately I was talking to a friend of mine and we came up with the following question
Denote by $$\mathcal{P}$$ the set of all real valued polynomial functions. Is the set $$\{ p(x) e^{- \alpha \vert x \vert} \ : \ p\in \mathcal{P}, \ \alpha \in \mathbb{R}_{>0} \}$$ dense in $$L^2(\mathbb{R} ,\mathbb{R})?$$
My feeling is that it should be true. I was thinking about using the Stone-Weierstrass, however, I cannot control the $$L^2$$-norm of the polynomial function outside the compact set. Clearly it is finite (the exponential decay beats the polynomial growth), but it is not clear to me whether one can choose it to be small.
• I'd guess that the answer is "no", with $1_{[0,1]}$ not being able to be approximated arbitrarily well by elements of your set... but I'm having trouble proving it. The reason why I think the answer is no is that if you have some polynomial $p$ approximating a given $f$, you must take $\alpha$ to be really large to ensure quick enough decay, but then you need to make $p$ larger so that $pe^{-\alpha |x|}$ is close enough to $f$, but then you need to make $\alpha$ larger to ensure quick enough decay... etc. These are just some obscure thoughts Nov 4, 2018 at 22:13
• @mathworker21. Given $\epsilon>0$ let $f(x) = 1_{[0,1]}(x) \, e^{x}$ and apply Stone-Weierstrass to get a polynomial $p$ with $\|f-p\|_{\infty} < \epsilon.$ Then $\|1_{[0,1]} - e^{-x}p\|_\infty = \|e^{-x}(f-p)\|_\infty < \|f-p\|_\infty < \epsilon$ since $\|e^{-x}\|_\infty = 1$ on $[0,1]$. Nov 4, 2018 at 22:36
• @md2perpe but you're doing $L^\infty$ over $[0,1]$ not over $\mathbb{R}$. My comment is saying that if you want to get a good $L^\infty$ bound over $[0,1]$, you won't have the right decay in $\mathbb{R}$. And conversely if you do have the right decay in $\mathbb{R}$, you won't have an $L^\infty$ bound on $[0,1]$. Nov 4, 2018 at 23:11
• @mathworker21. Ah, you're right. Nov 5, 2018 at 6:44
Yes, the answer is affirmative. It is equivalent to
$$\{p(x)e^{-|x|} : p\in\mathcal{P}\}$$ is dense in $$L^2(\Bbb{R},\Bbb{R})$$.
In our context of Hilbert spaces, we need to show that $$L_0 = \left\{f\in L^2(\Bbb{R}) : (\forall n\in\Bbb{Z}_{\geq 0})\ \int_{\Bbb{R}}x^n e^{-|x|}f(x)\,dx=0\right\}$$ consists of $$f\equiv 0$$ only. Here and below, all functions are complex-valued.
Let $$\Lambda=\{\lambda\in\Bbb{C} : |\Re\lambda|<1\}$$; for $$f\in L^2(\Bbb{R})$$, the function $$B_f(\lambda)=\int_{\Bbb{R}}e^{-|x|+\lambda x}f(x)\,dx$$ is analytic in $$\Lambda$$ (differentiation is admissible under the integral sign). Further, for any $$n\in\Bbb{Z}_{\geq 0}$$ we have $$B_f^{(n)}(0)=\displaystyle\int_{\Bbb{R}}x^n e^{-|x|}f(x)\,dx$$ and, therefore, $$f\in L_0$$ if and only if $$B_f\equiv 0$$.
Now let $$f\in L_0$$ and $$g\in L^1(\Bbb{R})$$ (fixme... much less is enough, see comments). Then $$0=\int_{\Bbb{R}}g(\lambda)B_f(i\lambda)\,d\lambda=\int_{\Bbb{R}}e^{-|x|}\hat{g}(x)f(x)\,dx,\quad\hat{g}(x)=\int_{\Bbb{R}}e^{i\lambda x}g(\lambda)\,d\lambda.$$ Thus, $$e^{-|x|}f(x)$$ is orthogonal to $$\{\hat{g} : g\in L^1(\Bbb{R})\}$$. This space is dense in $$L^2(\Bbb{R})$$ because, e.g., it contains all continuous piecewise linear finite functions obtained from $$\hat{g}_0(x) = \max\{0,1-|x|\} \impliedby g_0(\lambda)=\frac{2}{\pi}\left(\frac{\sin\lambda/2}{\lambda}\right)^2$$ using linear combinations and shifts; $$\hat{g}_1(x)=\hat{g}(x+a)\impliedby g_1(\lambda)=e^{i\lambda a}g(\lambda)$$.
(I'm sure I've duplicated some known facts about integral transforms. Thus, it might be good to replace some parts of the above with references to these...)
• Great answer. Thanks! Nov 5, 2018 at 13:53
• I don't think it's stronger. The set of polynomials is invariant under $x\mapsto \frac{x}{\alpha}$. Nov 5, 2018 at 18:58
• @mathworker21: +1, edited. Nov 5, 2018 at 21:56
• @metamorphy very nice solution! I just read through it. Can you give me a reference for where you have seen such arguments? (please forgive me, but I am under the assumption that this solution is not completely original). Also, a remark: it is enough to consider $g$ Schwarz, and then density is obvious because the fourier transform is a bijection from Schwarz functions to Schwarz functions. Nov 15, 2018 at 15:10
• @metamorphy also, one more question. it seems you used fourier inversion to conclude that $\widehat{B_f(i\cdot)}(x) = e^{-|x|}f(x)$. To apply fourier inversion, don't you need to know that $B_f(i\cdot) \in L^1$? How would you show this? Nov 15, 2018 at 16:18
There is a stronger conclusion. Every function $$f(x)$$ continuous on $$\mathbb{R}$$ and tending to $$0$$ at $$\pm \infty$$ can be approximated uniformly by a sequence of the form $$p_n(x)e^{-|x|},$$ where $$p_n(x)$$ are polynomials. This can be proved using Stone-Weierstrass theorem, but requires some effort. The above can be used to solve your problem. It suffices to consider the case $$\alpha=2.$$ Let $$f(x)$$ be a continuous and vanishing at $$\infty.$$ Then $$\|p_n(x)e^{-2|x|}-f(x)e^{-|x|}\|_2 \le \|p_n(x)e^{-|x|}-f(x)\|_\infty \|e^{-|x|}\|_2= \|p_n(x)e^{-|x|}-f(x)\|_\infty.$$ The functions of the form $$f(x)e^{-|x|}$$ are dense in $$L^2(\mathbb{R})$$ as they contain continuous functions with bounded support. | 2022-06-29 07:32:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 36, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8682281374931335, "perplexity": 145.82863502752022}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103624904.34/warc/CC-MAIN-20220629054527-20220629084527-00644.warc.gz"} |
https://bird.bcamath.org/handle/20.500.11824/20/browse?type=subject&value=Topology | Now showing items 1-1 of 1
• #### A functional limit theorem for stochastic integrals driven by a time-changed symmetric sigma-stable Levy process
(Stochastic Processes and their Applications, 2014-12-31)
Under proper scaling and distributional assumptions, we prove the convergence in the Skorokhod space endowed with the $M_1$-topology of a sequence of stochastic integrals of a deterministic function driven by a time-changed ... | 2019-08-17 13:17:56 | {"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": 0.4270484149456024, "perplexity": 847.4832112189874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313259.30/warc/CC-MAIN-20190817123129-20190817145129-00231.warc.gz"} |
http://mathhelpforum.com/pre-calculus/184993-question-about-ellipse.html | # Math Help - a question about ellipse
1. ## a question about ellipse
Sketch the curve given in polar coordinates by the equation:
r= (2a) / (3+2cos x)
Prove that this curve is an ellipse and identify its foci.
*From a Further pure maths book.
Thanks brothers.
2. ## Re: a question about ellipse
your equation can be arranged as
r = (2a/3) / [1 + (2/3)cos x] now comparing with polar equation of conic r = l / ( 1 + ecos x)
we get eccentricity e = 2/3 < 1 so represent ellipse with foci at pole and ( 2a,pi ) where l = 2a/3 is semi latusrectum
3. ## Re: a question about ellipse
Where's $\theta$ in the equation? A polar equation looks like: $r=f(\theta)$ but I don't see $\theta$ anywhere.
4. ## Re: a question about ellipse
Originally Posted by Siron
Where's $\theta$ in the equation? A polar equation looks like: $r=f(\theta)$ but I don't see $\theta$ anywhere.
sorry i dun know how to type the theta out but the x is the theta.
thanks brother.
btw hows the sketching of the curve?
i dun even know wt it is.
thanks bro.
5. ## Re: a question about ellipse
r = l / ( 1 + ecos x)
the ' l ' means?
i am not good at maths. not reli understand the idea, thanks brother helping me to further explain. thanks.
6. ## Re: a question about ellipse
$r=\frac{l}{1-ecos\Theta}$
is polar equation of conic and represent ellipse if e<1 with foci at
$\Theta=0$ and $\Theta=\pi$
7. ## Re: a question about ellipse
If you want to work with latex you have to use [ tex ] ... [/ tex], not [ Tex] ... [/ Tex ] | 2014-11-28 07:19:30 | {"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": 0, "img_math": 0, "codecogs_latex": 9, "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": 0.9631182551383972, "perplexity": 3561.9394058656394}, "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-2014-49/segments/1416931009825.77/warc/CC-MAIN-20141125155649-00037-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/integral-questions.14021/ | # Integral questions
1. Feb 7, 2004
### stoffer
Can anybody help me with the integral of 1/ (x^1/3 + x^1/4)
(cube root and fourth root of x) I dont really know where to start.
Also my roomate and i were wondering if x^sin(x) exists or if it has to be expressed and integrated as some sort of series.(something i havent learned yet)
2. Feb 7, 2004
### Muzza
Perhaps you could rewrite it as 1 / (x^(1/4) * (1 + x^(1/12))) and use partial fractions? I don't really know. Looks like it'll be pretty messy...
3. Feb 7, 2004
### matt grime
try substituting x = y^12
as for x^sin(x).. it's exp{log(x^sinx)} = exp{sinx*log(x)}
that's the standard way of defining x^f(x)
4. Feb 7, 2004
### stoffer
ok thanks for your help guys
5. Mar 14, 2007
### wesywes
just take the ln of that.. and do the rest intuitively, but hey what do i know, im only 16.
6. Mar 22, 2007
### d_leet
That is certainly not the same as the function in the original post.
7. Mar 22, 2007
### d_leet
That still isn't the same function. You can't manipulate fractions like that, it just doesn't work.
8. Mar 22, 2007
### morphism
This is why I'm not a fan of rushing people into calculus without a solid foundation in the basics of algebra. (In reference to d_leet's quotes.)
Anyway, matt's substitution kills this integral. You can also re-write the integrand as:
$\frac{x^{12}}{x^4 + x^3} = \frac{x^9}{1 + x} = \frac{x^9 + 1 - 1}{1+x}$
Then proceed...
9. Mar 23, 2007
### dextercioby
Actually it's a power less in the numerator:
$I=\int \frac{dx}{\sqrt[3]{x}+\sqrt[4]{x}}=12 \int \frac{y^{8}}{y+1}{}dy$
,where $y=\sqrt[12]{x}$.
10. Mar 23, 2007
### morphism
I didn't use the substitution... | 2018-10-21 17:02:29 | {"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": 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": 0.6902745962142944, "perplexity": 3966.529847176805}, "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-2018-43/segments/1539583514162.67/warc/CC-MAIN-20181021161035-20181021182535-00149.warc.gz"} |
https://davefollett.io/ | # My Advent of Code 2019 Approach
2018 was my first experience with Advent of Code. It sounded like Codewars, so I jumped right in. Wow, I was really impressed. Not only were the puzzles fun, but I really like these aspects as well:
• Most of the data sets are large, your algorithms and data structures matter. With modern computing, it can be easy to skip or not be worth the time to optimize code. But this doesn’t mean we shouldn’t practice it and the Advent of Code puzzles tend to put you into situations where it’s necessary to do so.
• The puzzle descriptions can be long and contain lots of details not necessary to solve the puzzle. Some folks don’t like this but I do. Raise your hand ✋ if you have been given a vaguely worded specification that you have to comb through to find the real requirements. Like it or not, some degree of detective 🕵️ work is part of a software developer’s job.
• Each day’s part 2 puzzle provides a twist on the part 1 puzzle. Sometimes its a small adjustment to solve part 2, but sometimes it’s a near rewrite. Though making a major change can be frustrating, this frequently happens for software applications whether it be the customer changing their mind or through the addition of new features.
# Learning MDX Deck: Layouts
Welcome back 👋. In this installment of Learning MDX Deck, I will go over MDX Deck’s built-in layouts. It is also possible to create your own layouts. I won’t cover that here, but you can read about it in the MDX Deck documentation. So lets get right to it. Here is what MDX Deck provides for layouts out of the box.
• Default
• Invert
• Split
• SplitRight
• FullScreenCode
• Horizontal
# Learning MDX Deck: Deploy To Netlify
In the previous article, Learning MDX Deck: Getting Started, I showed how to get up and running with MDX Deck. MDX Deck is a tool for creating presentation deck websites using MDX. But what good is creating a super sweet presentation if you can’t easily show it off to your friends? Let’s take the exact code from Learning MDX Deck: Getting Started and deploy it to Netlify.
# Learning MDX Deck: Getting Started
## 🙋 What is MDX Deck?
MDX Deck was created by Brent Jackson (@jxnblk) and is a tool for creating presentation deck websites using MDX. MDX provides the ability to use React’s JSX inside Markdown. The combination creates a powerful experience for building a web-based presentation deck. Here are a list of features, as listed on the MDX Deck GitHub Page:
• 📝 Write presentations in markdown
• ⚛ Import and use React components
• 💅 Customizable themes and components
• 0️⃣ Zero-config CLI
• 💁♀️ Presenter mode
• 📓 Speaker notes
# 4 Tips For Workplace Pranks
Any workplace can get stale, boring or in a rut. I think software development can be particularly susceptible because the high level of concentration needed to write code. Over the years I’ve found that that small breaks from the routine can help clear my head and allow me to better focus on my work. This typically means working a few minutes on the crossword or jigsaw puzzles in the break room when getting a coffee. Sometimes, it takes more than a puzzle to shake things up. That’s when I like to play harmless pranks on my coworkers. If done right, not only can this cause a break from the routine but it can crack a few smiles and cause a little laughter.
# My Favorite Vue Resources
To learn about Vue, I started following interesting people in the Vue community, listened to podcasts, and completed several tutorials. I thought it might be helpful to others to share some my favorite of those Vue resources.
# 2019 Self Improvement Plan
Before last year, I had not put together an improvement plan for myself for at least 10 years. Putting down my thoughts on how to get better as a developer really did help. While I didn’t do all the things I set out to do, I did make progress towards last year’s general theme of putting myself out there more:
# A New Vue On JavaScript30 - 08 Fun with HTML5 Canvas
This article is part of the A New Vue On JavaScript30 series that explores re-implementing Wes Bos’s (@wesbos) #JavaScript30 projects using Vue. Today I will be working with #JavaScript30’s 08 Fun with HTML5 Canvas project. This project uses a <canvas> element to provide a fun a way to draw in the browser window by clicking and dragging the mouse. As you draw, the line will get larger and smaller as well as change color. Here is an animated gif of it in action.
# 2018 Give Back Gift List
Every Christmas season, I always dread the task of coming up with gift ideas for me that family members can use. Is that too expensive? Is it not expensive enough? Can this very specific electronics part be easily purchased? These are just some of the questions I ask myself every year but NOT this year. I’m going to 💀🐦🐦⚫. This year, I have the perfect way to build a list and it also checks off another thing I have been meaning to do. Support the free content producers and open source projects I use every day by purchasing their swag. So I present to you:
# A New Vue On JavaScript30 - 06 Type Ahead
This article is part of the A New Vue On JavaScript30 series that explores re-implementing Wes Bos’s (@wesbos) #JavaScript30 projects using Vue. Today I will be working with #JavaScript30’s 06 - Type Ahead project. This project uses an input to filter down a list of cities as the user types. In addition to filtering the list, it also highlights the input value in the results. Here is an animated gif of it in action. | 2021-10-24 19:16:54 | {"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": 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": 0.1866782158613205, "perplexity": 2510.742142871647}, "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/1634323587593.0/warc/CC-MAIN-20211024173743-20211024203743-00315.warc.gz"} |
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# The product of two negative integers, a and b, is a prime nu
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The product of two negative integers, a and b, is a prime nu [#permalink]
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03 Jun 2014, 05:55
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The product of two negative integers, $$a$$ and $$b$$, is a prime number $$p$$. If $$p$$ is the number of factors of $$n$$, where $$n$$ is NOT a perfect square, what is the value of the median of the four integers $$a$$, $$b$$, $$p$$, and $$n$$?
A. 0
B. 1/2
C. 1
D. 3/2
E. 2
Kudos for a correct solution.
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Re: The product of two negative integers, a and b, is a prime nu [#permalink]
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SOLUTION
The product of two negative integers, $$a$$ and $$b$$, is a prime number $$p$$. If $$p$$ is the number of factors of $$n$$, where $$n$$ is NOT a perfect square, what is the value of the median of the four integers $$a$$, $$b$$, $$p$$, and $$n$$?
A. 0
B. 1/2
C. 1
D. 3/2
E. 2
This is a hard questions which tests several number theory concepts.
Start from n: we are told that $$n$$ is NOT a perfect square. The number of factors of perfect square is odd and all other positive integers have even number of factors. Hence, since $$p$$ is the number of factors of $$n$$, then $$p$$ must be even. We also know that $$p$$ is a prime number and since the only even prime is 2, then $$p=2$$. Notice here that from this it follows that $$n$$ must also be a prime, because only primes have 2 factors: 1 and itself.
Next, $$ab=p=2$$ implies that $$a=-1$$ and $$b=-2$$ or vise-versa.
So, the set is {-2, -1, 2, some prime}, which means that the median is (-1 + 2)/2 = 1/2.
Answer: B.
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Re: The product of two negative integers, a and b, is a prime nu [#permalink]
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03 Jun 2014, 08:05
2
Bunuel wrote:
The product of two negative integers, $$a$$ and $$b$$, is a prime number $$p$$. If $$p$$ is the number of factors of $$n$$, where $$n$$ is NOT a perfect square, what is the value of the median of the four integers $$a$$, $$b$$, $$p$$, and $$n$$?
A. 0
B. 1/2
C. 1
D. 3/2
E. 2
Kudos for a correct solution.
all prime numbers are divisible by 1 and itself. now since product of a and b is a prime number P. therefore one of a and b is -1 and other is -P.
Also, all the perfect square have odd no. of factors. for e.g. a^2 will have 3 factors a,a^2 and 1. Since, n is not a perfect square therefore its no. of factors must be even and the only prime no. which is even is 2. therefore p=2,
now the median of numbers -2,-1,2 and n will be (-1+2)/2= 1/2 hence B
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Re: The product of two negative integers, a and b, is a prime nu [#permalink]
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04 Jun 2014, 10:54
1
Bunuel wrote:
The product of two negative integers, $$a$$ and $$b$$, is a prime number $$p$$. If $$p$$ is the number of factors of $$n$$, where $$n$$ is NOT a perfect square, what is the value of the median of the four integers $$a$$, $$b$$, $$p$$, and $$n$$?
A. 0
B. 1/2
C. 1
D. 3/2
E. 2
Kudos for a correct solution.
We know that the only way two integers will give us a prime is 1 times a prime number. So A or B is -1 and know both numbers are negative.
Since N is not a perfect square it should have 2 factors. Since P=the number of factors, 2, and A or B is -1, then A or B must also be -2.
A*B=2
A=-1
B=-2
P=2
P=N=2
Set of Numbers {-2,-1,2,2}
Median = ((-1+2)/2) =1/2
Answer B
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Re: The product of two negative integers, a and b, is a prime nu [#permalink]
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04 Jun 2014, 12:09
GeorgeA023 wrote:
Bunuel wrote:
The product of two negative integers, $$a$$ and $$b$$, is a prime number $$p$$. If $$p$$ is the number of factors of $$n$$, where $$n$$ is NOT a perfect square, what is the value of the median of the four integers $$a$$, $$b$$, $$p$$, and $$n$$?
A. 0
B. 1/2
C. 1
D. 3/2
E. 2
Kudos for a correct solution.
We know that the only way two integers will give us a prime is 1 times a prime number. So A or B is -1 and know both numbers are negative.
Since N is not a perfect square it should have 2 factors. Since P=the number of factors, 2, and A or B is -1, then A or B must also be -2.
A*B=2
A=-1
B=-2
P=2
P=N=2
Set of Numbers {-2,-1,2,2}
Median = ((-1+2)/2) =1/2
Answer B
Notice that it's not necessary n to be 2, it could be any other prime as well.
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Re: The product of two negative integers, a and b, is a prime nu [#permalink]
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04 Jun 2014, 12:09
1
1
SOLUTION
The product of two negative integers, $$a$$ and $$b$$, is a prime number $$p$$. If $$p$$ is the number of factors of $$n$$, where $$n$$ is NOT a perfect square, what is the value of the median of the four integers $$a$$, $$b$$, $$p$$, and $$n$$?
A. 0
B. 1/2
C. 1
D. 3/2
E. 2
This is a hard questions which tests several number theory concepts.
Start from n: we are told that $$n$$ is NOT a perfect square. The number of factors of perfect square is odd and all other positive integers have even number of factors. Hence, since $$p$$ is the number of factors of $$n$$, then $$p$$ must be even. We also know that $$p$$ is a prime number and since the only even prime is 2, then $$p=2$$. Notice here that from this it follows that $$n$$ must also be a prime, because only primes have 2 factors: 1 and itself.
Next, $$ab=p=2$$ implies that $$a=-1$$ and $$b=-2$$ or vise-versa.
So, the set is {-2, -1, 2, some prime}, whcih means that the median is (-1 + 2)/2 = 1/2.
Answer: B.
Theory on Number Properties: math-number-theory-88376.html
Tips and hints about Number Properties
DS Number Properties Problems to practice: search.php?search_id=tag&tag_id=38
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Re: The product of two negative integers, a and b, is a prime nu [#permalink]
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04 Jun 2014, 12:13
Bunuel wrote:
GeorgeA023 wrote:
Bunuel wrote:
The product of two negative integers, $$a$$ and $$b$$, is a prime number $$p$$. If $$p$$ is the number of factors of $$n$$, where $$n$$ is NOT a perfect square, what is the value of the median of the four integers $$a$$, $$b$$, $$p$$, and $$n$$?
A. 0
B. 1/2
C. 1
D. 3/2
E. 2
Kudos for a correct solution.
We know that the only way two integers will give us a prime is 1 times a prime number. So A or B is -1 and know both numbers are negative.
Since N is not a perfect square it should have 2 factors. Since P=the number of factors, 2, and A or B is -1, then A or B must also be -2.
A*B=2
A=-1
B=-2
P=2
P=N=2
Set of Numbers {-2,-1,2,2}
Median = ((-1+2)/2) =1/2
Answer B
Notice that it's not necessary n to be 2, it could be any other prime as well.
Bunuel,
Yeah I see that now. When I read it I glanced over the second of. I saw P is the number of factors N and not P is the number of factors of N. So I read it as P=N and not the way it was written.
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Re: The product of two negative integers, a and b, is a prime nu [#permalink]
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05 Jun 2014, 23:20
Assumed numbers:
a, b, p, n as
-2, -1, 2, 3
-2 * -1 = 2 (Prime Number)
3 has 2 factors (1 & 3);
It is not a square
Median $$= \frac{2-1}{2} = \frac{1}{2}$$
Answer = B
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Re: The product of two negative integers, a and b, is a prime nu [#permalink]
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01 Jan 2018, 07:18
given: 1. a * b = prime number
2. a and b -ve numbers
so one of them has to be -1
say a = -1
probable ascending order: { b , a, p , n}
since p is number of factors of n, where n is not a perfect square. p cannot be odd (as only perfect square has odd number of factors)
so p is even, so median of {b,a,p,n} = (even - 1)/2 = odd / 2 = fraction.
Only option B and D left.
if median were 3/2, then p = 4 => not a prime => so D is out
Answer (B)
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The product of two negative integers, a and b, is a prime nu [#permalink]
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25 Feb 2018, 21:04
Excellent Excellence question!!!
a*b is prime.
Since, prime number(P) only has 2 factors 1 and P and any factor of P must contain of product that arrives at P.
The only way the product is prime if one of the number is prime (the same number) and other is 1. In this case both negative.
so, a = -1 and B = -P (or vice versa).
Now, N is not a perfect square so the number of factor has to be even. The only way that p is both even and prime is if it is 2.
so, a =-1,B=-2 and p =2.
I got kind of stuck here and could not determine n. But was pretty sure it is -2 or 2.
if it is 2 then the set is -2,-1,2,2 . median is 1/2.. MY answer
if it is -2, then set is -2,-2,-1,2 median is -3/2. not one of the option..
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The product of two negative integers, a and b, is a prime nu [#permalink]
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26 Feb 2018, 00:12
A really tough question, took me a good 7 min to solve.
The product of two negative integers, a and b, is a prime number p.
If p is a prime number, a is -1 and b is -p, or vice versa.
If p is the number of factors of n, where n is NOT a perfect square,
A non-perfect square integer will always have even number of factors. And the only even prime number is 2. Hence a & b is -1 & -2 or vice versa.
what is the value of the median of the four integers a, b, p, and n
We know the following values are present -2, -1, 2, n.
Since 2<n and the question is asking about median, we do not need to know the exact value of n.
The answer is $$(-1+2)/2 = 1/2$$
Ans: B
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Re: The product of two negative integers, a and b, is a prime nu [#permalink]
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07 Apr 2019, 12:35
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Re: The product of two negative integers, a and b, is a prime nu [#permalink] 07 Apr 2019, 12:35
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# The product of two negative integers, a and b, is a prime nu
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Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne | 2019-10-18 04:50:38 | {"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": 0.7177204489707947, "perplexity": 985.2161221212701}, "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-2019-43/segments/1570986677884.28/warc/CC-MAIN-20191018032611-20191018060111-00001.warc.gz"} |
https://leanprover-community.github.io/archive/stream/113489-new-members/topic/Discrete.20topology.3F.html | Stream: new members
Topic: Discrete topology?
Pedro Minicz (Aug 15 2020 at 22:46):
Where is the definition of the discrete topology on a type α? There is a definition of discrete topology predicate here. It defines it as being equal to the bottom element of some order. I am struggling to make sense of this definition and don't seem to find the definition of the bottom element itself (it is implicitly declared somewhere I believe).
Mario Carneiro (Aug 15 2020 at 22:48):
You could look for #check (by apply_instance : has_bot (topological_space A))
Mario Carneiro (Aug 15 2020 at 22:50):
It looks like it's coming from src#topological_space.complete_lattice, in a bit of abstract nonsense
Mario Carneiro (Aug 15 2020 at 22:52):
In short, it reduces to univ
Pedro Minicz (Aug 15 2020 at 22:56):
Mario Carneiro said:
In short, it reduces to univ
What exactly do you mean it reduces to univ? I know that is_open = univ is this topology, but I don't see how that term can reduce to univ.
Mario Carneiro (Aug 15 2020 at 22:57):
it's pretty involved. Where are you stuck?
Pedro Minicz (Aug 15 2020 at 22:57):
I managed to complete the goal I was working on with a single simp and honestly I am not quite sure how it worked.
Pedro Minicz (Aug 15 2020 at 22:58):
import topology.order
import tactic
variables {α : Type*}
namespace cofinite
@[simp] def is_open : set α Prop :=
λ s, set.finite (set.univ \ s) s =
lemma is_open_univ : is_open (set.univ : set α) :=
begin
unfold is_open,
left,
rw set.diff_self,
exact set.finite_empty
end
lemma is_open_inter (s₁ s₂ : set α) :
is_open s₁ is_open s₂ is_open (s₁ s₂) :=
begin
rintros (hs₁ | hs₁) (hs₂ | hs₂),
{ unfold is_open,
left,
rw set.diff_inter,
exact set.finite.union hs₁ hs₂ },
all_goals { right, simp [hs₁, hs₂] }
end
lemma is_open_sUnion (s : set (set α)) (hs : t, t s is_open t) :
is_open (⋃₀ s) :=
begin
by_cases h : t, t s t ,
{ unfold is_open,
left,
rcases h with t, ht₁, ht₂,
have ht₂ : set.finite (set.univ \ t),
{ cases hs t ht₁ with ht₃ ht₃,
{ exact ht₃ },
{ exfalso, exact ht₂ ht₃ } },
apply set.finite.subset ht₂,
intros x hx,
suffices : x t, { simpa },
simp at hx,
exact hx t ht₁ },
{ push_neg at h,
unfold is_open,
right,
ext x,
suffices : t, t s x t, { simpa },
intros t ht,
specialize h t ht,
rw set.eq_empty_iff_forall_not_mem at h,
exact h x }
end
@[simp] instance topological_space (α : Type*) : topological_space α :=
{ is_open := is_open,
is_open_univ := is_open_univ,
is_open_inter := is_open_inter,
is_open_sUnion := is_open_sUnion }
example [fintype α] : discrete_topology α :=
begin
constructor,
ext s,
split; intro hs,
-- I have no idea what the following simp is doing.
{ simp only [topological_space.is_open] at hs },
{ left, apply set.finite.of_fintype }
end
end cofinite
Pedro Minicz (Aug 15 2020 at 22:58):
A bit long, the goal is the last example.
Mario Carneiro (Aug 15 2020 at 22:59):
The simp only is smoke and mirrors, the proof is trivial
Mario Carneiro (Aug 15 2020 at 23:00):
It would be good to have a simp lemma that says \bot.is_open s though
Ah, of course!
Pedro Minicz (Aug 15 2020 at 23:01):
Mario Carneiro said:
It would be good to have a simp lemma that says \bot.is_open s though
Yes, that would be quite convenient.
Pedro Minicz (Aug 15 2020 at 23:02):
Isn't it curious that simp doesn't close the trivial goal?
Kevin Buzzard (Aug 15 2020 at 23:04):
It's not an equality
Floris van Doorn (Aug 15 2020 at 23:55):
simp doesn't unfold definitions, unless it is told to (by attributes or when you invoke it). So simp doesn't know the goal is trivial.
Floris van Doorn (Aug 15 2020 at 23:56):
There are plenty of times when simp doesn't close a goal, but refl or exact trivial does (sometimes that means more things should be simp-lemmas)
Patrick Massot (Aug 16 2020 at 00:00):
docs#is_open_discrete
Mario Carneiro (Aug 16 2020 at 00:35):
The irony is that even though simp proves things by reducing them to true, simp will fail on the goal true
Last updated: May 08 2021 at 10:12 UTC | 2021-05-08 10:33:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6415802836418152, "perplexity": 12586.993802960791}, "config": {"markdown_headings": false, "markdown_code": false, "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-2021-21/segments/1620243988858.72/warc/CC-MAIN-20210508091446-20210508121446-00330.warc.gz"} |
https://search.r-project.org/CRAN/refmans/eplusr/html/run_model.html | run_idf {eplusr} R Documentation
## Run simulations of EnergyPlus models.
### Description
Run simulations of EnergyPlus models.
### Usage
run_idf(
model,
weather,
output_dir,
design_day = FALSE,
annual = FALSE,
expand_obj = TRUE,
wait = TRUE,
echo = TRUE,
eplus = NULL
)
run_multi(
model,
weather,
output_dir,
design_day = FALSE,
annual = FALSE,
expand_obj = TRUE,
wait = TRUE,
echo = TRUE,
eplus = NULL
)
### Arguments
model A path (for run_idf()) or a vector of paths (for run_multi()) of EnergyPlus IDF or IMF files. weather A path (for run_idf()) or a vector of paths (for run_multi()) of EnergyPlus EPW weather files. If set to NULL, design-day-only simulation will be triggered, regardless of the design-day value. For run_multi(), weather can also be a single EPW file path. In this case, that weather will be used for all simulations; otherwise, model and weather should have the same length. You can set to design-day-only simulation to some specific simulations by setting the corresponding weather to NA. output_dir Output directory path (for rum_idf()) or paths (for run_mult()). If NULL, the directory of input model is used. For run_multi(), output_dir, if not NULL, should have the same length as model. Any duplicated combination of model and output_dir is prohibited. design_day Force design-day-only simulation. For rum_multi(), design_day can also be a logical vector which has the same length as model. Note that design_day and annual cannot be all TRUE at the same time. Default: FALSE. annual Force annual simulation. For rum_multi(), annual can also be a logical vector which has the same length as model. Note that design_day and annual cannot be all TRUE at the same time. Default: FALSE. expand_obj Whether to run ExpandObject preprocessor before simulation. Default: TRUE. wait If TRUE, R will hang on and wait all EnergyPlus simulations finish. If FALSE, all EnergyPlus simulations are run in the background, and a process object is returned. echo Only applicable when wait is TRUE. Whether to show standard output and error from EnergyPlus for run_idf() and simulation status for run_multi(). Default: TRUE. eplus An acceptable input (for run_idf()) or inputs (for run_multi()) of use_eplus() and eplus_config(). If NULL, which is the default, the version of EnergyPlus to use is determined by the version of input model. For run_multi(), eplus, if not NULL, should have the same length as model.
### Details
run_idf() is a wrapper of EnergyPlus itself, plus various pre-processors and post-processors which enables to run EnergyPlus model with different options.
run_multi() provides the functionality of running multiple models in parallel.
It is suggested to run simulations using EplusJob class and EplusGroupJob class, which provide much more detailed controls on the simulation and also methods to extract simulation outputs.
### Value
• For run_idf(), if wait is TRUE, a named list of 11 elements:
No. Column Type Description 1 idf character(1) Full path of input IDF file 2 epw character(1) or NULL Full path of input EPW file 3 version character(1) Version of called EnergyPlus 4 exit_status integer(1) or NULL Exit status of EnergyPlus. NULL if terminated or wait is FALSE 5 start_time POSIXct(1) Start of time of simulation 6 end_time POSIXct(1) or NULL End of time of simulation. NULL if wait is FALSE 7 output_dir character(1) Full path of simulation output directory 8 energyplus character(1) Full path of called EnergyPlus executable 9 stdout character(1) or NULL Standard output of EnergyPlus during simulation 10 stderr character(1) or NULL Standard error of EnergyPlus during simulation 11 process r_process A process object which called EnergyPlus and ran the simulation
Hongyuan Jia
### References
EplusJob class and ParametricJob class which provide a more friendly interface to run EnergyPlus simulations and collect outputs.
### Examples
## Not run:
idf_path <- system.file("extdata/1ZoneUncontrolled.idf", package = "eplusr")
if (is_avail_eplus(8.8)) {
# run a single model
epw_path <- file.path(
eplus_config(8.8)$dir, "WeatherData", "USA_CA_San.Francisco.Intl.AP.724940_TMY3.epw" ) run_idf(idf_path, epw_path, output_dir = tempdir()) # run multiple model in parallel idf_paths <- file.path(eplus_config(8.8)$dir, "ExampleFiles",
c("1ZoneUncontrolled.idf", "1ZoneUncontrolledFourAlgorithms.idf")
)
epw_paths <- rep(epw_path, times = 2L)
output_dirs <- file.path(tempdir(), tools::file_path_sans_ext(basename(idf_paths)))
run_multi(idf_paths, epw_paths, output_dir = output_dirs)
}
## End(Not run)
[Package eplusr version 0.15.1 Index] | 2022-05-28 06:32:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.21537333726882935, "perplexity": 12033.240505172824}, "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/1652663013003.96/warc/CC-MAIN-20220528062047-20220528092047-00369.warc.gz"} |
https://sosteneslekule.blogspot.com/2016/01/learn-about-fourier-coefficients.html | Learn About Fourier Coefficients - LEKULE BLOG
## Thursday, 7 January 2016
Electronic oscillators, which are extremely useful in laboratory testing of equipment, are specifically designed to create non-sinusoidal periodic waveforms. Moreover, non-sinusoidal periodic functions are important in analyzing non-electrical systems. Problems that involve fluid flow, mechanical vibration, and heat flow all make use of different periodic functions. This article will detail a brief overview of a Fourier series, calculating the trigonometric form of the Fourier coefficients for a given waveform, and simplification of the waveform when provided with more than one type of symmetry.
Any periodic signal can be represented as a sum of sinusoids where the frequencies of the sinusoids in the sum are composed of the frequency of the periodic signal and integer multiples of that frequency. Using a periodic signal like a square wave to test the quality factor of a bandpass or band reject filter. In order to do this, a square wave whose frequency is the same as the center frequency of a bandpass filter is chosen.
### Fourier Series Overview
An analysis of heat flow in a metal rod led the French mathematician Jean Baptiste Joseph Fourier to the trigonometric series representation of a periodic function. This representation of a periodic function is the starting point for finding the steady-state response to periodic excitations of electric circuits. What was discovered was that a periodic function can be represented by an infinite sum of sine or cosine functions that are related harmonically. The period of any trigonometric term in the infinite series is an integral multiple, or harmonic, of the period T of the periodic function. To visualize an understanding, below are a few waveforms produced by function generators used in laboratory testing.
The Fourier series shows that f(t) can be described as
$f\left(t\right)={a}_{v}+\sum _{n=1}^{\mathrm{\infty }}{a}_{n}\mathrm{cos}\left(n{\omega }_{0}t\right)+{b}_{n}\mathrm{sin}\left(n{\omega }_{0}t\right)$
(1.1) Fourier series representation of a periodic function
Where is the integer sequence 1,2,3,...
In Eq. 1.1,
${a}_{v}$
,
${a}_{n}$
, and
${b}_{n}$
are known as the Fourier coefficients and can be found from f(t). The term
${\omega }_{0}$
(or
$\frac{2\pi }{T}$
) represents the fundamental frequency of the periodic function f(t). The integral multiples of
${\omega }_{0}$
, i.e.
$2{\omega }_{0},3{\omega }_{0},4{\omega }_{0}$
and so on, are known as the harmonic frequencies of f(t). Thus
$n{\omega }_{0}$
is the nth harmonic term of f(t).
Before discussing Fourier coefficients, the conditions in a Fourier series need to be explained. For a periodic function f(t) to be a convergent Fourier series, the following conditions need to be met:
1. f(t) be single-valued,
2. f(t) have a finite number of discontinuities in the periodic interval,
3. f(t) have a finite number of maxima and minima in the periodic interval,
4. the integral
${\int }_{{t}_{0}}^{{t}_{0}+T}\mid f\left(t\right)\mid dt$
exists,
These 4 conditions are known as Dirichlet's conditions, and are sufficient condition, not necessary conditions. Thus if f(t) meets these requirements, it can be expressed as a Fourier series. Nonetheless, if f(t) does not meet these requirements, it still can be expressed as a Fourier series; the necessary conditions on f(t) are not known.
### The Fourier Coefficients
Having defined a periodic function over its period, the following Fourier coefficients are determined from the relationships:
${a}_{v}=\frac{1}{T}{\int }_{{t}_{0}}^{{t}_{0}+T}f\left(t\right)dt,$
(1.2)
${a}_{k}=\frac{2}{T}{\int }_{{t}_{0}}^{{t}_{0}+T}f\left(t\right)\mathrm{cos}\left(k{\omega }_{0}t\right)dt,$
(1.3)
${b}_{k}=\frac{2}{T}{\int }_{{t}_{0}}^{{t}_{0}+T}f\left(t\right)\mathrm{sin}\left(k{\omega }_{0}t\right)dt,$
(1.4)
In Eqs. 1.3 and 1.4, the subscript k indicated the kth coefficient in an integer sequence 1,2,3,...Noting that
${a}_{v}$
is the average value of f(t)
${a}_{k}$
is twice the average value of
$f\left(t\right)\mathrm{cos}\left(k{\omega }_{0}t\right)$
, and
${b}_{k}$
is twice the average value of
$f\left(t\right)\mathrm{sin}\left(k{\omega }_{0}t\right)$
.
To gain a better understanding of how Eqs 1.2-1.4 came from Eq 1.1, simple derivations can be used through integral relationships which hold true when and are integers:
${\int }_{{t}_{0}}^{{t}_{0}+T}\mathrm{sin}\left(m{\omega }_{0}t\right)dt=0$
for all m, (1.6)
${\int }_{{t}_{0}}^{{t}_{0}+T}\mathrm{cos}\left(m{\omega }_{0}t\right)dt=0$
for all m, (1.7)
${\int }_{{t}_{0}}^{{t}_{0}+T}\mathrm{cos}\left(m{\omega }_{0}t\right)\mathrm{sin}\left(n{\omega }_{0}t\right)dt=0$
for all and n, (1.8)
${\int }_{{t}_{0}}^{{t}_{0}+T}\mathrm{sin}\left(m{\omega }_{0}t\right)\mathrm{sin}\left(n{\omega }_{0}t\right)dt=0$
for all
$m\ne n$
$=\frac{T}{2},$
for all n (1.9)
${\int }_{{t}_{0}}^{{t}_{0}+T}\mathrm{cos}\left(m{\omega }_{0}t\right)\mathrm{cos}\left(n{\omega }_{0}t\right)dt=0$
for all
$m\ne n$
$=\frac{T}{2},$
for all (1.10)
In order to derive Eq 1.3, both sides of Eq 1.2 need to be integrated over one period:
${\int }_{{t}_{0}}^{{t}_{0}+T}f\left(t\right)dt={\int }_{{t}_{0}}^{{t}_{0}+T}\left({a}_{v}+\sum _{n=1}^{\mathrm{\infty }}{a}_{n}\mathrm{cos}\left(n{\omega }_{0}t\right)+{b}_{n}\mathrm{sin}\left(n{\omega }_{0}t\right)\right)dt$
${\int }_{{t}_{0}}^{{t}_{0}+T}{a}_{v}dt+\sum _{n=1}^{\mathrm{\infty }}\left({a}_{n}\mathrm{cos}\left(n{\omega }_{0}t\right)+{b}_{n}\mathrm{sin}\left({a}_{n}\mathrm{cos}\left(n{\omega }_{0}t\right)\right)dt$
$={a}_{v}T+0$
(1.11)
To derive the expression for the kth value of
${a}_{n}$
, Eq 1.2 need to be multiplied by
$\mathrm{cos}\left(k{\omega }_{0}t\right)$
and then both sides need to be integrated over one period of f(t):
${\int }_{{t}_{0}}^{{t}_{0}+T}f\left(t\right)\mathrm{cos}\left(k{\omega }_{0}t\right)dt={\int }_{{t}_{0}}^{{t}_{0}+T}{a}_{v}\mathrm{cos}\left(k{\omega }_{0}t\right)dt$
$+\sum _{\mathrm{\infty }}^{n=1}{\int }_{{t}_{0}}^{{t}_{0}+T}\left({a}_{n}\mathrm{cos}\left(n{\omega }_{0}t\right)cos\left(k{\omega }_{0}t\right)+{b}_{n}\mathrm{sin}\left(n{\omega }_{0}t\right)\mathrm{sin}\left(k{\omega }_{0}t\right)\right)dt$
$=0+{a}_{k}\left(\frac{T}{2}\right)+0$
(1.12)
Lastly, the expression for the kth value of
${b}_{n}$
by multiplying both sides of Eq. 1.2 by
$\mathrm{sin}\left(k{\omega }_{0}t\right)$
and then integrating each side over one period of f(t). The following example explains how to use Eqs. 1.3 - 1.5 to calculate the Fourier coefficients for a specific periodic function.
Finding the Fourier series of a Triangular Waveform with No Symmetry:
In this example, you are asked to find the Fourier series for the given periodic voltage shown below
When using Eqs. 1.3 - 1.5 to solve for
${a}_{v}$
,
${a}_{k}$
, and
${b}_{k}$
, the value of
${t}_{0}$
can be chosen to be any value. For this specific periodic voltage, the best value is zero. If a value other than that of zero, integration would become difficult. The expression for v(t) between 0 and T is:
${v}_{t}=\left(\frac{{V}_{m}}{T}\right)t$
The equation for
${a}_{v}$
is:
${a}_{v}=\frac{1}{T}{\int }_{0}^{T}\left(\frac{{V}_{m}}{T}\right)tdt=\frac{1}{2}{V}_{m}$
The value found above is the average value of the waveform shown above. The equation for the kth value of
${a}_{n}$
is:
${a}_{k}=\frac{2}{T}{\int }_{0}^{T}\left(\frac{{V}_{m}}{T}\right)t\mathrm{cos}\left(k{\omega }_{0}t\right)dt$
$=\frac{2{V}_{m}}{{T}^{2}}\left(\frac{1}{{k}^{2}{w}_{0}^{2}}\mathrm{cos}\left(k{\omega }_{0}t\right)+\frac{t}{k{\omega }_{0}}\mathrm{sin}\left(k{\omega }_{0}t\right)\right)$
Evaluated from 0 to T.
$=\frac{2{V}_{m}}{{T}^{2}}\left[\frac{1}{{k}^{2}{\omega }_{0}^{2}}\left(\mathrm{cos}\left(2\pi k-1\right)\right]=0$
for all k
The equation for the kth value of
${b}_{n}$
is:
${b}_{k}=\frac{2}{T}{\int }_{0}^{T}\left(\frac{{V}_{m}}{T}\right)t\mathrm{sin}\left(k{\omega }_{0}t\right)dt$
$=\frac{2{V}_{m}}{{T}^{2}}\left(\frac{1}{{k}^{2}{\omega }^{2}}\mathrm{sin}\left(k{\omega }_{0}t\right)-\frac{t}{k{\omega }_{0}}\mathrm{cos}\left(k{\omega }_{0}t\right)\right)$
Evaluated from 0 to T.
$=\frac{2{V}_{m}}{{T}^{2}}\left(0-\frac{T}{k{\omega }_{0}}\mathrm{cos}\left(2\pi k\right)\right)$
$=\frac{-{V}_{m}}{\pi k}$
Finally, the Fourier series for v(t) is:
$v\left(t\right)=\frac{{V}_{m}}{2}-\frac{{V}_{m}}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}\mathrm{sin}\left(n{\omega }_{0}t\right)$
$v\left(t\right)=\frac{{V}_{m}}{2}-\frac{{V}_{m}}{\pi }\mathrm{sin}\left({\omega }_{0}t\right)-\frac{{V}_{m}}{2\pi }\mathrm{sin}\left(2{\omega }_{0}t-\frac{{V}_{m}}{3\pi }\mathrm{sin}\left(3{\omega }_{0}t\right)-...$
### Coming Up
At this point, you should have an understanding of what a Fourier series is, what the Fourier coefficients are, and the calculations to find the trigonometric form of the Fourier coefficients for a periodic waveform. Future articles will detail average power with periodic functions as well as analyzing a circuit's response to a waveform using the Fourier coefficients talked about in this article. Another topic that will be covered is the four types of symmetry that can be used to simplify the evaluation of the Fourier coefficients as well as the effect of symmetry on the Fourier coefficients. | 2018-03-23 03:29:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 55, "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": 0.8895129561424255, "perplexity": 606.0987020383076}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257648177.88/warc/CC-MAIN-20180323024544-20180323044544-00745.warc.gz"} |
http://www.ask.com/question/what-did-isaac-newton-invent | # Isaac Newton What Did He Invent?
Isaac Newton published three books and invented the Newtonian reflecting telescope, a significant improvement on previous reflecting telescopes. Isaac Newton formulated the laws of universal gravitation and motion. He invented differential calculus at the same time as the German mathematician Gottfried Wilhelm Leibniz
Q&A Related to "Isaac Newton What Did He Invent?"
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Kepler and Galileo had big influence on Newton. He said, 'I am able to see far because I am standing on the shoulders of giants'. Newton kept himself well informed of all the theories http://uk.answers.yahoo.com/question/index?qid=201...
Newton's law of universal gravitation states that every point mass http://www.chacha.com/question/what-year-was-isaac...
Sir Isaac Newton is best known for his theory of universal gravitation, his Laws of Motion, as well as groundbreaking discoveries in mathematics, physics and astronomy.
Isaac Newton was an English physicist which was best known for the development of Newtonian mechanics and universal gravitation during the 17th and 18th centuries.
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The Newton metre is a unit that is used to measure torque and is found in the SI system. It was invented by Isaac Newton and its symbolic form is Nm or N.m or ... | 2014-07-12 19:38:42 | {"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": 0.9353193640708923, "perplexity": 1553.4363819194866}, "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-2014-23/segments/1404776435102.41/warc/CC-MAIN-20140707234035-00071-ip-10-180-212-248.ec2.internal.warc.gz"} |
https://wiki.math.ntnu.no/tma4145/2019h/log | #### Lectures log
First week (week 34)
• Monday: Naive set theory; basic definitions and facts about sets, de Morgan's laws. Functions; range and domain of a function, injective, surjective and bijective functions. See Chapter 1.1-1.2 in Lecture Notes (LN).
• Tuesday: Left and right inverses, inverses for functions and their characterization in terms of injectivity, surjectivity and bijectivity. Cardinality and countable sets. See Chapter 1.2-1.3 in LN.
Second week (week 35)
• Monday: Supremum and infimum of sets and functions. See Chapter 1.3-1.4 in LN.
• Tuesday: Real (and complex) vector spaces, basic examples of vector spaces, subspaces and linear transformations between vector spaces. See Chapter 2.1 in LN.
Third week (week 36)
• Monday: Normed spaces,metric spaces, p-norms, Young's inequality, Hoelder's inequality. See Chapter 2.2 in LN.
• Tuesday: Triangle inequality for ||.||_p norms of real n-tuples, complex n-tuples sequences and functions.
Fourth week (week 37)
• Monday: Inner product spaces, properties of the inner product, Cauchy-Schwarz inequality. See Chapter 2.2-2.3 in LN.
• Tuesday: Examples of inner product spaces, Jordan-von Neumann's characterization of norms induced by inner products, polarization identities, orthogonality, Pythagoras' theorem, convergence of sequences in normed spaces. See Chapters 2.3 and 3.1 in LN.
Fifth week (week 38)
• Monday: Convergent sequences in metric and normed spaces, bounded subsets and limit points, Cauchy sequences. See Chapter 3.1 in LN.
• Tuesday: Cauchy sequences and completeness, completeness of (R,|.|), examples of Banach spaces and completeness of the space of d-tuples with the sup-norm. See Ch. 3.1-3.2 in LN.
Sixth week (week 39)
• Monday: Further examples of Banach spaces, complete subspaces, uniform convergence of sequences of continuous functions. Isometries and isomorphic vector spaces.
• Tuesday: Isometrically isomorphic normed spaces, embeddings, dense subsets, separability, Stone-Weierstrass theorem. See Ch 3.3 in LN.
Seventh week (week 40)
• Monday: Banach's fixed point theorem and finding fixed points by iteration. Newton's method as a fixed point iteration. LN: Chapter 3.4-3.5, 4.1.
• Tuesday: Applications of Banach fixed point theorem to integral equations and differential equations. LN: Chapter 3.5. Linear operators, continuous operators,
Eighth week (week 41)
• Monday: Bounded operators. Equivalence of boundedness and continuity for linear operators, Extension theorem, the vector space of bounded linear operators between normed spaces, See Ch 4.2 in LN.
• Tuesday: Vector space of bounded linear operators, operator norm, completeness if co-domains are Banach spaces, proof that the kernel of a bounded linear operator is closed, the range need not be. See Ch. 4.2 in LN.
Ninth week (week 42)
• Monday: Dual space, best approximation theorem. See Ch. 4.3-5.1 in LN.
• Tuesday: Best approximation theorem, orthogonal complements, the projection theorem, consequences of the projection theorem. See Ch. 5.1 in LN.
Tenth week (week 43)
• Monday: Examples of Hilbert spaces, Riesz' representation theorem, proof and examples. See Ch 5.2 in LN.
• Tuesday: Adjoint operators, properties of adjoint operators and examples. See Ch 5.3 in LN
Eleventh week (week 44)
• Monday: Normal, unitary and self-adjoint operators. See Ch 5.3 in LN.Linear independence See Ch 6.1 in LN.
• Tuesday: Hamel bases, dimension, Schauder bases, (infinite) series in vector spaces, orthogonal sets and the closest point property,The closest point property, Gram-Schmidt orthogonalization, convergence of series in Hilbert spaces, orthonormal bases, maximal orthonormal sequences, the Fourier series theorem. See Ch. 6.1-6.4 in LN.
Twelfth week (week 45)
• Monday: Equivalent norms. Linear transformations on finite-dimensional vector spaces and their matrix representation. See Ch. 6.5 and 7.1 in LN.
• Tuesday: Null-space, column space and row space of a matrix. The rank-nullity theorem and its consequences. See Ch. 7.1-7.2 in LN.
Thirteenth week (week 46)
• Monday: Eigenvalues and eigenvectors. Similarity transforms and Schur's lemma. The spectral theorem for Hermitian matrices. The spectral theorem. Positive and semi-positive definiteness. Singular value decomposition. See Ch. 7.3-7.5 in LN.
• Tuesday: SVD example, brief discussion on applications of SVDs, pseudoinverse of a linear transformation, calculating the pseudoinverse of a matrix. See Ch. 7.5-7.6 in LN. | 2021-06-15 08:02:06 | {"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": 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": 0.6526920795440674, "perplexity": 2869.9380974459345}, "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/1623487617599.15/warc/CC-MAIN-20210615053457-20210615083457-00169.warc.gz"} |
http://www.mathematics-online.org/inhalt/aussage/aussage343/ | [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff]
Mathematics-Online lexicon:
# Symmetric Group, Permutations
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview
For any set the bijections from onto together with the composition of mappings as operation form a group, the so called symmetric group of .
If , then this group is called the symmetric group of degree (notation: ). The elements of are called permutations. has elements. A permutation can be written as:
The group of permutations is in general not commutative, as shown in the example:
.
Annotation: | 2021-09-18 13:30:32 | {"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": 0.9208979606628418, "perplexity": 489.33369327453886}, "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/1631780056476.66/warc/CC-MAIN-20210918123546-20210918153546-00146.warc.gz"} |
https://infoscience.epfl.ch/record/180028 | Infoscience
Journal article
# Searches for Majorana neutrinos in B- decays
Searches for heavy Majorana neutrinos in B- decays in final states containing hadrons plus a mu(-) mu(-) pair have been performed using 0.41 fb(-1) of data collected with the LHCb detector in proton-proton collisions at a center-of-mass energy of 7 TeV. The D+ mu(-) mu(-) and D*+ mu(-) mu(-) final states can arise from the presence of virtual Majorana neutrinos of any mass. Other final states containing pi(+), D-s(+), or D-0 pi(+) can be mediated by an on-shell Majorana neutrino. No signals are found and upper limits are set on Majorana neutrino production as a function of mass, and also on the B- decay branching fractions. | 2017-02-25 09:26:48 | {"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": 0.8191636800765991, "perplexity": 3939.4509244610013}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171706.94/warc/CC-MAIN-20170219104611-00502-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://mathonline.wikidot.com/the-composition-of-two-functions | The Composition of Two Functions
# The Composition of Two Functions
On the Injective, Surjective, and Bijective Functions page we recalled the definition of a general function and looked at three types of special functions. We will now look at another type of function that can be obtained by composing two compatible functions.
Definition: Let $f : A \to B$ and $g : B \to C$. The Composition of $f$ and $g$ is the function $g \circ f : A \to C$ defined for all $x \in A$ by $(g \circ f)(x) = g(f(x))$.
For $g \circ f$ to properly be defined we must have that the codomain of $f$ be the same as the domain of $g$. Also, it is important to notice the order. The composition of $f$ and $g$ will be $g \circ f$ while the composition of $g$ and $f$ will be $f \circ g$.
For example, consider the functions $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x + 4$ and $g(x) = x^2$. Then:
(1)
\begin{align} \quad (g \circ f)(x) = g(f(x)) = g(x + 4) = (x + 4)^2 \end{align}
We can also look at the composition of $g$ and $f$ which is:
(2)
\begin{align} \quad (f \circ g)(x) = f(g(x)) = f(x^2) = x^2 + 4 \end{align}
In general, if $g \circ f$ and $f \circ g$ are well defined, then $g \circ f$ need not equal $f \circ g$.
We will now look at some interesting results regarding the composition of two functions.
Theorem 1 (The Associativity of $\circ$): Let $f, g, h : A \to A$. Then $f \circ (g \circ h) = (f \circ g) \circ h$.
• Proof: Let $f, g, h : A \to A$. Then for all $x \in A$ we have that:
(3)
\begin{align} \quad [f \circ (g \circ h)](x) = f((g \circ h)(x)) = f(g(h(x))) = (f \circ g)(h(x)) = [(f \circ g) \circ h](x) \end{align}
• Therefore $f \circ (g \circ h) = (f \circ g) \circ h$. $\blacksquare$
Theorem 2: Let $f : A \to B$ and $g : B \to C$ both be injective functions. Then $g \circ f : A \to C$ is an injective function.
• Proof: Let $f : A \to B$ and $g : B \to C$ both be injective functions, and consider the function $g \circ f : A \to C$. Suppose that $(g \circ f)(x) = (g \circ f)(y)$. Then:
(4)
\begin{align} \quad (g \circ f)(x) = (g \circ f)(y) \\ \quad g(f(x)) = g(f(y)) \end{align}
• Since $g$ is injective we have that $f(x) = f(y)$, and since $f$ is injective we have that $x = y$. Therefore $g \circ f$ is injective. $\blacksquare$
Theorem 3: Let $f : A \to B$ and $g : B \to C$ both be surjective functions. Then $g \circ f : A \to C$ is a surjective function.
• Proof: Let $f : A \to B$ and $g : B \to C$ both be surjective functions, and consider the function $g \circ f : A \to C$. Let $y \in C$ and consider $(g \circ f)(x) = y$.
• Since $g$ is surjective there exists a $b \in B$ such that $g(b) = y$, and since $f$ is surjective there exists an $x \in A$ such that $f(x) = b$. Therefore:
(5)
\begin{align} \quad y = g(b) = g(f(x)) \end{align}
• Therefore $g \circ f$ is surjective. $\blacksquare$
Corollary 1: Let $f : A \to B$ and $g : B \to C$ both be bijective functions. Then $g \circ f: A \to C$ is a bijective function.
• Proof: Since $f$ and $g$ are both bijective, we have that they are both injective and surjective by definition. By Theorem 1 and Theorem 2 we have that then $g \circ f$ is both injective and surjective and hence bijective. $\blacksquare$ | 2017-11-22 05:21:16 | {"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": 5, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9984300136566162, "perplexity": 118.9137415283935}, "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-2017-47/segments/1510934806465.90/warc/CC-MAIN-20171122050455-20171122070455-00040.warc.gz"} |
https://plainmath.net/calculus-2/49223-find-definite-integral-int-0-2-x-2-3x-3-plus-1-1-3-dx | Joan Thompson
2021-12-31
Find definite integral.
${\int }_{0}^{2}{x}^{2}{\left(3{x}^{3}+1\right)}^{\frac{1}{3}}dx$
Jenny Sheppard
Expert
Step 1
The given integral can be solved by the method of substitution. The substitution that will be used is
$3{x}^{3}+1=u$. This gives $9{x}^{2}dx=du$ or ${x}^{2}dx=\frac{du}{9}$.
This substitution absorbs the ${x}^{2}dx$ term into $\frac{du}{9}$. Calculate the corresponding limits of the integration in terms of the new variable u.
Step 2
Limits of integration for x are from 0 to 2. New variable for integration is $u=3{x}^{3}+1$. So the lower limit of integration in terms of new variable will be 1. Calculate the upper limit by substituting x=2.
$u=3\cdot {2}^{3}+1$
=3*8+1
=25
So, the integration with this substitution becomes ${\int }_{1}^{25}{u}^{\frac{1}{3}}\frac{du}{9}$. Calculate this integral using the integral
$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}$.
${\int }_{1}^{25}{u}^{\frac{1}{3}}\frac{du}{9}=\frac{1}{9}{\left(\frac{{u}^{\frac{1}{3}+1}}{\frac{1}{3}+1}\right)}_{1}^{25}$
$=\frac{1}{9}\cdot \frac{3}{4}{\left({u}^{\frac{4}{3}}\right)}_{1}^{25}$
$=\frac{1}{12}\left({25}^{\frac{4}{3}}-1\right)$
Hence, the given definite integral is equal to $\frac{1}{12}\left({25}^{\frac{4}{3}}-1\right)$
Shawn Kim
Expert
${\int }_{0}^{2}{x}^{2}\left(3{x}^{3}+1{\right)}^{1/3}dx=\int {x}^{2}\sqrt[3]{3{x}^{3}+1}dx$
$=\frac{1}{9}\int \sqrt[3]{u}du$
$\int \sqrt[3]{u}du$
$=\frac{3{u}^{\frac{4}{3}}}{4}$
$\frac{1}{9}\int \sqrt[3]{u}du$
$=\frac{{u}^{\frac{4}{3}}}{12}$
$=\frac{{\left(3{x}^{3}+1\right)}^{\frac{4}{3}}}{12}$
$\int {x}^{2}\sqrt[3]{3{x}^{3}+1}$
$=\frac{{\left(3{x}^{3}+1\right)}^{\frac{4}{3}}}{12}+C$
Vasquez
Expert
$\begin{array}{}{\int }_{2}^{0}{x}^{2}\left(3{x}^{3}+1{\right)}^{1/3}dx\\ \int {x}^{2}×\left(3{x}^{3}+1{\right)}^{1/3}dx\\ \int \frac{{t}^{\frac{1}{3}}}{9}dt\\ \frac{1}{9}×\int {t}^{\frac{1}{3}}dt\\ \frac{1}{9}×\frac{3t\sqrt[3]{t}}{4}\\ \frac{1}{9}×\frac{3\left(3{x}^{3}+1\right)\sqrt[3]{3{x}^{3}+1}}{4}\\ \frac{\left(3{x}^{3}+1\right)\sqrt[3]{3{x}^{3}+1}}{12}\\ \frac{\left(3{x}^{3}+1\right)\sqrt[3]{3{x}^{3}+1}}{12}{|}_{0}^{2}\\ \frac{\left(3×{2}^{3}+1\right)\sqrt[3]{3×{2}^{3}+1}}{12}-\frac{\left(3×{0}^{3}+1\right)\sqrt[3]{3×{0}^{3}+1}}{12}\\ Answer:\\ \frac{25\sqrt[3]{25}-1}{12}\end{array}$ | 2023-01-29 09:06:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 50, "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": 0.9874088764190674, "perplexity": 435.5154962036876}, "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/1674764499710.49/warc/CC-MAIN-20230129080341-20230129110341-00292.warc.gz"} |
https://www.techwhiff.com/issue/radiant-heat-cannot-travel-through-a-vacuum-true-or--303747 | # Radiant heat cannot travel through a vacuum. true or false
###### Question:
Radiant heat cannot travel through a vacuum.
true or false
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https://eccc.weizmann.ac.il/report/2013/157/ | Under the auspices of the Computational Complexity Foundation (CCF)
REPORTS > DETAIL:
Revision(s):
Revision #1 to TR13-157 | 22nd November 2013 21:26
Derandomizing Polynomial Identity over Finite Fields Implies Super-Polynomial Circuit Lower Bounds for NEXP
Revision #1
Authors: Bin Fu
Accepted on: 22nd November 2013 21:26
Keywords:
Abstract:
We show that
derandomizing polynomial identity testing over an arbitrary finite
field implies that NEXP does not have polynomial size boolean
circuits. In other words, for any finite field F(q) of size q,
$PIT_q\in NSUBEXP\Rightarrow NEXP\not\subseteq P/poly$, where
$PIT_q$ is the polynomial identity testing problem over F(q), and
NSUBEXP is the nondeterministic subexpoential time class of
languages. Our result is in contract to Kabanets and Impagliazzo's
existing theorem that derandomizing the polynomial identity testing
in the integer ring Z implies that NEXP does have polynomial
size boolean circuits or permanent over Z does not have polynomial
size arithmetic circuits.
Paper:
TR13-157 | 11th November 2013 06:13
Derandomizing Polynomial Identity over Finite Fields Implies Super-Polynomial Circuit Lower Bounds for NEXP
TR13-157
Authors: Bin Fu
Publication: 15th November 2013 20:24
Keywords:
Abstract:
We show that
derandomizing polynomial identity testing over an arbitrary finite
field implies that NEXP does not have polynomial size boolean
circuits. In other words, for any finite field F(q) of size q,
$PIT_q\in NSUBEXP\Rightarrow NEXP\not\subseteq P/poly$, where
$PIT_q$ is the polynomial identity testing problem over F(q), and
NSUBEXP is the nondeterministic subexpoential time class of
languages. Our result is in contract to Kabanets and Impagliazzo's
existing theorem that derandomizing the polynomial identity testing
in the integer ring Z implies that NEXP does have polynomial
size boolean circuits or permanent over Z does not have polynomial
size arithmetic circuits.
Comment(s):
Comment #1 to TR13-157 | 24th November 2013 19:34
Comments on Two Definitions of Polynomial Identity Testing Problems
Comment #1
Authors: Bin Fu
Accepted on: 24th November 2013 19:34 | 2020-05-29 11:14:35 | {"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": 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": 0.8010081648826599, "perplexity": 6267.913369801893}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347402885.41/warc/CC-MAIN-20200529085930-20200529115930-00003.warc.gz"} |
https://math.stackexchange.com/questions/3014264/does-this-functional-satisfies-the-palais-smale-condition | # Does this functional satisfies the Palais-Smale condition?
Let $$\Omega$$ be a non-empty bounded open subset of $$\mathbb{R}^N$$, $$\lambda\in \mathbb{R}$$ be an eigenvalue of $$-\Delta$$ on the Sobolev space $$H^1_0(\Omega)$$ and $$f\in L^\infty(\Omega\times\mathbb{R})$$ such that
• $$\forall x\in \Omega, t\mapsto f(x,t)\in C(\mathbb{R});$$
• $$\forall M>0, \exists r>0, \forall |s|>r, \forall x\in \Omega, \int_0^s f(x,t)\operatorname{d}t\ge M;$$
• $$\forall \varepsilon>0, \exists r>0, \forall|s|>r, \forall x\in\Omega,\left| \frac{1}{s}\int_0^s f(x,t)\operatorname{d}t\right|\le\varepsilon.$$
Define: $$I:H^1_0(\Omega)\to\mathbb{R}, u\mapsto\frac{1}{2}\|u\|^2_{H^1_0}-\frac {\lambda}{2}\|u\|_2^2-\int_\Omega\int_0^{u(x)}f(x,t)\operatorname{d}t\operatorname{d}x.$$ Then $$I\in C^1(H^1_0(\Omega),\mathbb{R})$$ and $$\forall u,v\in H^1_0(\Omega), \operatorname{d}I(u)(v)=\int_\Omega \nabla u(x)\cdot \nabla v(x) \operatorname{d}x-\lambda\int_\Omega u(x)v(x) \operatorname{d}x-\int_\Omega f(x,u(x))v(x)\operatorname{d}x$$
Is it true that $$I$$ satisfies the Palais-Smale condition? I.e. is it true that for all $$(u_n)_{n\in\mathbb{N}}\subset H^1_0(\Omega)$$ such that $$(I(u_n))_{n\in\mathbb{N}}$$ is bounded and $$\|\operatorname{d}I(u_n)\|\to0, n\rightarrow\infty$$ there exists a subsequence $$(u_{n_k})_{k\in\mathbb{N}}$$ that converges in $$H^1_0(\Omega)$$ to some $$\bar u \in H^1_0(\Omega)$$?
In my lecture notes on calculus of variations, it is claimed that every sequence $$(u_n)_{n\in\mathbb{N}}$$ that satisfies the previous condition is actually bounded in $$H^1_0(\Omega)$$ and so, by the fact that there exists a subsequence that weakly converges in $$H^1_0(\Omega)$$ to some $$\bar u \in H^1_0(\Omega)$$, it easily follows (from the fact that the differential of $$I$$ can be expressed as a sum of a homeomorphism and a compact operator) that this subsequence also converges to $$\bar u$$ in $$H^1_0(\Omega)$$.
My problem is proving the fact that a sequence $$(u_n)_{n\in\mathbb{N}}$$ as before is actually bounded in $$H^1_0(\Omega)$$. In particular, what I have proved is the following. First, decompose $$H^1_0(\Omega)$$ as the orthogonal sum: $$H^1_0(\Omega)=E_-\oplus E_0\oplus E_+$$ where $$E_-$$ is the vector space generated by the eigenfunctions relative to eigenvalues less than $$\lambda$$, $$E_0$$ is the eigenspace relative to $$\lambda$$ and $$E_+$$ is the closure of the vector space generated by the the eigenfunctions relative to eigenvalues greater than $$\lambda$$. Define $$P_-$$ as the orthogonal projection of $$H^1_0(\Omega)$$ onto $$E_-$$, define $$P_0$$ as the orthogonal projection of $$H^1_0(\Omega)$$ onto $$E_0$$ and define $$P_+$$ as the orthogonal projection of $$H^1_0(\Omega)$$ onto $$E_+$$.
Then, using the relations: $$\operatorname{d}I(u_n)(P_-u_n)\ge -C\|P_-u_n\|_{H^1_0}$$ and $$\operatorname{d}I(u_n)(P_+u_n)\le C\|P_+u_n\|_{H^1_0}$$ and the estimates of $$\|\cdot\|_2^2$$ from below with respect to $$\|\cdot\|_{H^1_0}^2$$ on $$E_-$$ and of $$\|\cdot\|_2^2$$ from above with respect to $$\|\cdot\|_{H^1_0}^2$$ on $$E_+$$, we obtain that $$(P_-u_n)_{n\in\mathbb{N}}$$ and $$(P_+u_n)_{n\in\mathbb{N}}$$ are bounded in $$H^1_0(\Omega)$$.
It remains to show that $$(P_0 u_n)_{n\in\mathbb{N}}$$ is bounded in $$H^1_0(\Omega)$$.
Thanks to the previous estimates, what I proved is that for some constant $$B,C>0$$ we have that: $$C\ge |I(u_n)|=\left|\frac{1}{2}\|u_n\|^2_{H^1_0}-\frac{\lambda}{2}\|u_n\|_2^2-\int_\Omega\int_0^{u_n(x)}f(x,t)\operatorname{d}t\operatorname{d}x\right|\ge \left|\int_\Omega\int_0^{u_n(x)}f(x,t)\operatorname{d}t\operatorname{d}x\right|-B$$ and so the sequence: $$\left(\int_\Omega\int_0^{u_n(x)}f(x,t)\operatorname{d}t\operatorname{d}x\right)_{n\in\mathbb{N}}$$ is bounded in $$\mathbb{R}$$.
Now, I suspect that I have to use the hypothesis $$\forall M>0, \exists r>0, \forall |s|>r, \forall x\in \Omega, \int_0^s f(x,t)\operatorname{d}t\ge M$$ with the boundedness of the sequences $$\left(\int_\Omega\int_0^{u_n(x)}f(x,t)\operatorname{d}t\operatorname{d}x\right)_{n\in\mathbb{N}}, \left(P_-u_n\right)_{n\in\mathbb{N}}, \left(P_+u_n\right)_{n\in\mathbb{N}}$$ to conclude that actually $$(P_0u_n)_{n\in\mathbb{N}}$$ (or directly the sequence $$(u_n)_{n\in\mathbb{N}}$$) is bounded in $$H^1_0(\Omega)$$, but I can't see how...
Any suggestion? | 2019-07-16 19:11:21 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 67, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9892030358314514, "perplexity": 43.523533184488365}, "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-2019-30/segments/1563195524685.42/warc/CC-MAIN-20190716180842-20190716202842-00336.warc.gz"} |
https://publications.mfo.de/browse?rpp=20&offset=1773&etal=-1&sort_by=1&type=title&starts_with=V&order=ASC | Now showing items 1774-1792 of 1806
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[SNAP-2015-012-ENSNAP-2015-012-DE] (Mathematisches Forschungsinstitut Oberwolfach, 2015)
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[OWR-2008-49] (2008) - (26 Oct - 01 Nov 2008)
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[OWR-2004-5] (2004) - (25 Jan - 31 Jan 2004)
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[OWR-2009-7] (2009) - (08 Feb - 14 Feb 2009)
This workshop was devoted to recent progress in the mathematical study of water waves, with special emphasis on nonlinear phenomena. Both aspects related to the governing equations (free boundary Euler equations) as well ...
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[OWR-2007-36] (2007) - (29 Jul - 04 Aug 2007)
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[OWR-2004-34] (2004) - (11 Jul - 17 Jul 2004)
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[OWR-2010-33] (2010) - (01 Aug - 07 Aug 2010)
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[OWP-2014-05] (Mathematisches Forschungsinstitut Oberwolfach, 2014-04-25)
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Adaptive finite element methods for optimization problems for second order linear elliptic partial di erential equations subject to pointwise constraints on the $\ell^2$-norm of the gradient of the state are considered. ...
• Weakly Complex Homogeneous Spaces
[OWP-2012-04] (Mathematisches Forschungsinstitut Oberwolfach, 2012-04-24)
We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous ...
• Weighted Fourier inequalities for radial functions
[OWP-2009-26] (Mathematisches Forschungsinstitut Oberwolfach, 2009-03-19)
Weighted $L^p(\mathbb{R}^n) \to L^q(\mathbb{R}^n)$ Fourier inequalities are studied. We prove Pitt-Boas type results on integrability with power weights of the Fourier transform of a radial function.
• Weighted Surface Algebras: General Version
[OWP-2019-07] (Mathematisches Forschungsinstitut Oberwolfach, 2019-02-28)
We introduce general weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular disc, ... | 2021-10-17 21:55:06 | {"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": 0.6044178009033203, "perplexity": 2546.694030330787}, "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-2021-43/segments/1634323585183.47/warc/CC-MAIN-20211017210244-20211018000244-00066.warc.gz"} |
http://www.maths.usyd.edu.au/s/scnitm/wm-AppliedMathsSeminar-Gallo | SMS scnews item created by Martin Wechselberger at Fri 14 Sep 2007 0831
Type: Seminar
Distribution: World
Expiry: 19 Sep 2007
Calendar1: 19 Sep 2007 1405-1455
CalLoc1: Eastern Avenue Lecture Theatre
Auth: wm@p6283.pc.maths.usyd.edu.au
# Applied Maths Seminar: Galloway -- A new model for the solar cycle
Sunspots vary in number over the 11-year solar cycle. Since they are magnetic with
alternating polarities each cycle, the actual periodicity is roughly 22 years. In 1919
Larmor suggested that the Sun was possibly acting as a self-excited dynamo, and since
then the aim has been to find mechanisms which can generate magnetic fields with this
periodicity. Over the last forty years, a theory known as mean field electrodynamics
has risen to prominence. Recently however there has been debate about the theory’s
applicability to the Sun, and about the way in which nonlinear feedback limits the
growth of the field. Here we present a new model for the solar cycle in which magnetic
fields are generated in a layer of shear at the base of the Sun’s convection zone (the
so-called tachocline). The magnetic field is circulated by a meridional flow to present
a time-varying bottom boundary condition to the overlying convection zone. The latter
brings some of this field to the surface, where we show many observed aspects of the
solar cycle are reproduced. This new mechanism provides an alternative to the mean
field approach. Its advantage is that it is not necessary for the whole magnetic system
to be regenerated from scratch every 22 years.
This is joint work with Robert Cameron.
If you are registered you may mark the scnews item as read.
School members may try to . | 2018-03-23 05:27:31 | {"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": 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": 0.4446631968021393, "perplexity": 2253.8364690727785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257648178.42/warc/CC-MAIN-20180323044127-20180323064127-00506.warc.gz"} |
https://www.physicsforums.com/threads/piecewise-function-integration.446772/ | # Piecewise Function Integration
## Homework Statement
Find formulas for the upper and lower sums of $f$ on $P_n$, and use them to compute the value of $\int_0^1f(x)dx$.
$P_n:=\{\frac{j}{n}:j=0,1,...,n\}$ (a partition of [0,1])
$$f(x) = \left\{ \begin{array}{ccc} 0 & 0 \le x < 1/2 \\ 1 & 1/2 \le x \le 1 \end{array} \right.$$
## Homework Equations
$U(f,P)=\sum\limits_{j=1}^{n} M_j(f)\Delta x_j$ and
$L(f,P)=\sum\limits_{j=1}^{n} M_j(f)\Delta x_j$
where $M_j=sup f([x_{j-1},x_j])$ and $m_j=inf f([x_{j-1},x_j])$
if $\lim_{n \rightarrow \infty} L(f,P_n)=\lim_{n \rightarrow \infty} U(f,P_n)$ then this equals $\int_0^1f(x)dx$
## The Attempt at a Solution
So it is easy to see that this function is bounded on [0,1]. So now we can break this up into the different partitions, but now is where I run into a problem. It is finding the inf and the sup of each interval:
so obviously if both $x_j, x_{j-1}$ are < 1/2 then both inf and sup are 0;
if both $x_j, x_{j-1}$ are >= 1/2 then both inf and sup are 1;
so now it is possible for one case to be $x_j \ge 1/2, x_{j-1} < 1/2$
in which case sup =1/2 and inf =0.
I am stuck from this point. Any help would be appreciated.
Related Calculus and Beyond Homework Help News on Phys.org
Can you tell us how exactly you are stuck?
It seems that you have found the good expression for the $$M_j(f)$$, i.e.
$$M_j(f)=\left\{\begin{array}{ccc} 0 & \text{if} & j< n/2\\ 1 & \text{if} & j\geq n/2 \end{array}\right.$$
and something analogous for $$m_j(f)$$. Now you just need to plug those things in in U(f,P) and L(f,P)...
so $m_j(f)=\left\{\begin{array}{ccc} 0 & \text{if} & j \leq n/2\\ 1 & \text{if} & j > n/2 \end{array}\right.$
So now
$U(f,P)=\left\{\begin{array}{ccc} \sum\limits_{j=1}^{n} (0)(j/n)=0 & \text{if} & j< n/2\\ \sum\limits_{j=1}^{n} (1)(j/n)= & \text{if} & j\geq n/2 \end{array}\right.$
and
$L(f,P)=\left\{\begin{array}{ccc} \sum\limits_{j=1}^{n} (0)(j/n)=0 & \text{if} & j \le n/2\\ \sum\limits_{j=1}^{n} (1)(j/n)= & \text{if} & j > n/2 \end{array}\right.$
So now solving I am not sure if I am correct. Are the sums not supposed to be taken from j=1 to n in such cases?
wait so would it actually be:
$U(f,P)=\left\{\begin{array}{ccc} \sum\limits_{j=1}^{n} (0)(1/n)=0 & \text{if} & j< n/2\\ \sum\limits_{j=1}^{n} (1)(1/2n)= & \text{if} & j\geq n/2 \end{array}\right.$
and
$L(f,P)=\left\{\begin{array}{ccc} \sum\limits_{j=1}^{n} (0)(1/n)=0 & \text{if} & j \le n/2\\ \sum\limits_{j=1}^{n} (1)(1/2n+1)= & \text{if} & j > n/2 \end{array}\right.$
So now viewing as the limit goes to infinity, $\lim_{n \rightarrow \infty}\sum\limits_{j=1}^{n} (1)(1/2n+1)=0$
and $\lim_{n \rightarrow \infty}\sum\limits_{j=1}^{n} (1)(1/2n)=0$
No, what you wrote is incorrect.
You have the sum
$$\sum_{j=1}^n{M_j(f)\Delta x_j$$
and you are supposed to plug in the values for $$M_j(f)$$. But for every $$M_j(f)$$ you have a different value.
Lets take the example with n=3, then we got
$$M_1(f)\Delta x_1+M_2(f)\Delta x_2+M_3(f)\Delta x_3$$
We got that $$M_1(f)=0$$ and $$M_2(f)=M_3(f)=1$$, thus the above expression yields:
$$\Delta x_2+\Delta x_3=2/3$$
Now you just need to do the same thing with n arbitrary (I strongly suggest you do the cases n=4 and n=5 first, then you will see the general case).
Ok so I think I got the general case, but I am not certain. So I did it for n=4,n=5,n=6, and got 3/4,3/5, and 2/3 respectively. So basically I think that depending on n, there will be
$\lceil \frac{n+1}{2} \rceil$ values that have the sup of 1.
So case 1: n is odd: so we have $\frac{n+1}{2}*\frac{1}{n}=\frac{n+1}{2n}=\frac{1}{2}+\frac{1}{2n}$
so taking this limit as n goes to infinity we get that it is 1/2.
So case 2: n is even: so we have that $\frac{n+1+1}{2}*\frac{1}{n}=\frac{n+2}{2n}=\frac{1}{2}+\frac{2}{2n}$
so taking this limit as n goes to infinity we get that it is 1/2.
Now for the lower bound, it is $\lfloor \frac{n-1}{2} \rfloor$
So case 1: n is odd: so we have $\frac{n-1}{2}*\frac{1}{n}=\frac{n-1}{2n}=\frac{1}{2}-\frac{1}{2n}$
so taking this limit as n goes to infinity we get that it is 1/2.
So case 2: n is even: so we have that $\frac{n-1-1}{2}*\frac{1}{n}=\frac{n-2}{2n}=\frac{1}{2}-\frac{2}{2n}$
so taking this limit as n goes to infinity we get that it is 1/2.
Is this correct?
That seems to be good!
Thanks a lot, this was very helpful. | 2020-08-08 12:56:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "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": 0.937985897064209, "perplexity": 244.62157236009892}, "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-2020-34/segments/1596439737645.2/warc/CC-MAIN-20200808110257-20200808140257-00327.warc.gz"} |
https://math.stackexchange.com/questions/2758510/derivative-of-mathrmtrx-top-a-a/2758564 | # Derivative of $\mathrm{tr}(X^\top A) A$
Let A be a constant matrix. Define $$Y:= \mathrm{tr}(X^\top A) A$$
I want to find \begin{align} \frac{\partial Y}{\partial X} \label{A} \end{align}
I know that $$\frac{\partial\,\mathrm{tr}(X^\top A)}{\partial X} = A \label{B}$$
Of course, I should be able to extend this to the case I want. But I am a bit confused as to how I should notate the result. What I did to find \eqref{A} was to vectorize both sides and then use the fact in \eqref{B}.
$$\mathrm{vec(Y)} = \mathrm{tr}(X^\top A)\mathrm{vec}(A)$$
I also knwo that I can write $$\mathrm{tr}(X^\top A) = \mathrm{vec}(A)^\top \mathrm{vec}(X)$$ So the desired the derivative should be $$\mathrm{vec}(A) \otimes \mathrm{vec}(A)$$ Is this correct?.
• The problem can be approached in either tensor or vector form \eqalign{&Y=(A\star A):X&\implies\,y=aa^Tx\cr &\frac{\partial Y}{\partial X}=(A\star A)&\implies\,\frac{\partial y}{\partial x}=aa^T}where $a={\rm vec}(A),\,x={\rm vec}(X),$ and $\star$ is the tensor product $${\mathcal A}=B\star C \implies {\mathcal A}_{ijkl}=B_{ij}C_{kl}$$ – greg Apr 29 '18 at 14:09
Let the entry-wise form of the matrices $A$, $X$ and $Y$ be \begin{align} A&=\left(a_{jk}\right)_{jk},\\ X&=\left(x_{jk}\right)_{jk},\\ Y&=\left(y_{jk}\right)_{jk}. \end{align} Then $Y=\text{tr}\left(X^{\top}A\right)A$ reads $$y_{jk}=\left(\sum_{i,r}x_{ir}a_{ir}\right)a_{jk}.$$
With this notation, \begin{align} \frac{\partial y_{jk}}{\partial x_{pq}}&=\frac{\partial}{\partial x_{pq}}\left[\left(\sum_{i,r}x_{ir}a_{ir}\right)a_{jk}\right]\\ &=\sum_{i,r}\frac{\partial x_{ir}}{\partial x_{pq}}a_{ir}a_{jk}\\ &=\sum_{i,r}\delta_{ip}\delta_{rq}a_{ir}a_{jk}\\ &=a_{pq}a_{jk}. \end{align} Thus if one regards $A$ as a tensor, one may write $$\frac{\partial Y}{\partial X}=A\otimes A.$$
Alternatively, one may use the differential form $${\rm d}y_{jk}=\left(\sum_{p,q}a_{pq}{\rm d}x_{pq}\right)a_{jk},$$ which, in its matrix form, reads $${\rm d}Y=\text{tr}\left(A^{\top}{\rm d}X\right)A.$$ | 2020-01-23 02:56:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9982187747955322, "perplexity": 546.8275962991268}, "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-2020-05/segments/1579250608062.57/warc/CC-MAIN-20200123011418-20200123040418-00055.warc.gz"} |
http://physics.stackexchange.com/tags/time/hot | # Tag Info
## Hot answers tagged time
18
A physicist, me for example, identifies events by choosing a set of coordinates. For example I have a clock that I use to record time and a ruler that I can choose to measure distance. This allows me to set up some coordinates $(t, x, y, z)$ so I can assign every event to some point in my coordinate system. If I received a laser pulse from Mars at 16:05 ...
14
The way atomic clocks work is to produce a microwave signal with exactly the frequency of the atomic transition being used. So for caesium this would be 9,192,631,770Hz. Then we can count the oscillations of our microwave generator to measure the time. Practically you do this by tuning your microwave signal to maximise its absorption by the caesium atoms. ...
3
$\frac{dM}{dt} = \frac{\partial{M}}{\partial{t}}+\frac{\partial{M}}{\partial{x}}\frac{d{x}}{d{t}} = \frac{\partial{M}}{\partial{t}}+v\cdot\nabla{M}$ (with no assumption on what is M) . So if $v\cdot\nabla{M} \neq0$ you can have one of $\frac{dM}{dt}$ and $\frac{\partial{M}}{\partial{t}}$ that is zero when the other is not. ...
3
As ACuriousMind says in a comment, this isn't the approach Yukawa used in his 1935 paper (Yukawa H 1935 Proc. Phys. Math. Soc. Japan 17 48) though whether he did that calculation in the privacy of his own notebook only he knows. The calculation you describe is a rather arm waving sort of justification for the relationship between the mass of the mediating ...
2
They couldn't carry sufficently accurate time from London with portable clocks. But they were able to use clocks to time measure the time between the sun crossing and the transit of stars the night before and after. The absolute transit time of stars can be trivially obtained if you know the site's longitude. If you are on land and have an observatory ...
2
A space train leaves Mars at 14:00pm and arrives on Earth at 19:45. The train moves at 0.001C and has 40Km of length. How long will it take for the whole train to arrive on Earth? - Disregard re-entry and friction. Nobody on Earth will say the train is leaving mars now. Same thing with the light, just it moves faster and is smaller than the train above. ...
2
Addressing the 3rd question, the clocks' frequencies are the same when each is viewed in its own rest frame. When a clock is viewed in a moving frame, that's when its frequency is changed. You record the ticks and their locations (which are different since the clock is moving) and you discover that the time between ticks is now longer.
2
Just open any string text which has a discussion of the relativistic point particle. http://arxiv.org/abs/0908.0333 - Section 1 for example or Green, Schwartz, Witten Volume 1 Punchlines: 1) Time can be introduced as an operator but you need to introduce a 'proper time' parameter for which the system evolves with. In doing this you introduce a gauge ...
2
My name is Sarah Huggins, about three years ago, my dad wrote an article on Wikipedia to demonstrate how nothing on the site is reliable. That article was The Huggins Displacement Theory. A couple years later, it's in three books, multiple movie and book reviews, and on physics blogs like this. He took the article down once it got flagged, but I thought I ...
2
What you have done here is a Galilean transform, that is a non-relativistic transformation. Take your final result (which is quite correct): $$t' = \frac{\sqrt{\beta^2 + \alpha^2}}{\sqrt{\eta^2 + \mu^2}} \tag{1}$$ We know that the vertical velocity is $\eta$, so the vertical distance moved in our time $t$ is given by: $$\beta = \eta t$$ We also know ...
1
I don't know the "formal" proof, but here is my proof: Time dilation and length contractions are given to us by the Lorentz transformations by: t’ = t/(1-v2/C2)1/2 and d’ = d/(1-v2/C2)1/2 (in other words “same” or proportional to each other) where: t = distance/length traveled through the T dimension in observers own frame of ...
1
No. Cooling down the atom will not alter it's frequency. What the scientist meant by saying that the clocks will be better is that they will be much more accurate. This comes from the principle of Quantum mechanics, in fact one of its most beautiful consequence, Heisenberg's uncertainty principle. Which says (in one of it's variety) that the uncertainty in ...
1
The problem is that you are equating too many things to $\dot{q_k}$. Usually $\dot{q_k} = \frac{dq_k}{dt}$, a total derivative, as opposed to a partial derivative. If $q_k$ has no explicit time-dependence, i.e. it does not depend directly on $t$ itself, then $\frac{\partial q_k}{\partial t} = 0.$ In this case, the Poisson bracket reduces to: $... 1 yes, time dilation still occurs. The reason is that the mirrors are there to provide an intuitive view of how/why time dilation occurs, not to to create it. This time dilations still occurs regardless of the existence of the mirrors or the direction of motion. But the drawing will no longer have explanatory power. 1 In that case, time dilation still occur, of course. In order to show this using t=d/v, you'd have to take into account the space contraction in the direction of motion. Mathematically, if d is the height of the clock, then the time taken from a photon at the bottom to reach the top of the clock isn't$\frac{d+vt}{c}$but$ \frac{d/\gamma+vt}{ c}$. When you ... 1 I think most of you are confusing what contracts because of motion. Not the distance between the muon and the Earth, nor the distance between the point in which the muon began its motion and its destination. Length contraction deals with the contraction "of the moving object" (that is the muon's length, if we could talk of it). If you could see an airplane ... 1 We could try this one: People down on that planet got few hours older while people on the ship got twenty years older. So, lets do this. Put a telescope on the surface on that planet and observe motion of people and such in the ship. If you could somehow do this, you should be able to see everything on the ship happening faster. Also, if people from the ... 1 You can use hook's law$F=-kx$. If$\tau$is the torque on board , then force at the spring,$F=\frac{\tau}{L}$and$\tau=I\alpha$, where$I=\frac{mL^2}{3}$, M.I of rod about one of its end . U can replace x by$x=L\theta$and appliying to hook's law yield$I\dfrac{d^2\theta}{dt^2}=-L^2k\theta$this gives$\omega=\sqrt{\frac{L^2k}{I}}$. Therefore, time ... 1 The state is "measured" in the sense you are imagining - that is, it becomes definite - at whatever time it becomes possible in principle to infer its having a particular measurement outcome. In your case, if the probe provides unambiguous information about the measurement result, the time of measurement will be found to have been delta-t back in time. This ... 1 Do you know Bernard Schutz's book: A First Course in General Relativity? Check out the first chapter of that book. There is a derivation of invariance of proper time using first principles in section 1.6. Basically, the idea is to start from expressing$\Delta \bar s ^2$(interval in the barred frame) as a linear combination of$\Delta x_i$'s (vector ... 1 I assume you mean by "the effects of time dilation...due to gravity vs. ... due to velocity" you mean can we tell the difference between the relativistic time dilation $$t=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}$$ and the gravitational time dilation $$t=\frac{t_0}{\sqrt{1-\frac{2GM}{rc^2}}}$$ where$t_0$is the proper time in both cases. The answer is ... 1 As you may know, it takes infinite time to charge a capacitor. So, the time when the capacitor is 100% charged never comes. Thus, we require a Time Constant to help us understand the time when the capacitor has got a decent amount of charge and after which the rate of charging becomes really slow and thus charging further is not of much use. You may also ... 1 Time dilation: linear or exponential or other? Other $$\Delta t' = \gamma\Delta t = \frac{\Delta t}{\sqrt{1 - \frac {v^2}{c^2}}}$$ Lorentz factor$\gamma$as a function of speed (in natural units where$c=1\$) - Image by Zayani CC BY-SA 3.0
1
One thing you have to note is that speed is relative, Clock A would see clock B moving from A's point of reference, and B would see A moving in B's reference, so you shouldn't be using the word "stationary" in this context. Both the clocks would see the other clock tick slower, B would see A's future only if it returns back to A, this makes it obvious to A ...
1
Well, the "standard" answer is the one given below (that's why we use telescopes to look at the past universe). However, I'd like to qualify this. The Einstein twin paradox (time contraction) implies the following thought experiment: With a very good telescope, you look at a planet 10 light-years away. And there you see an ET entering his spaceship and ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2015-11-28 00:47:36 | {"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": 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": 0.6265528798103333, "perplexity": 548.1226763479274}, "config": {"markdown_headings": true, "markdown_code": false, "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-2015-48/segments/1448398450715.63/warc/CC-MAIN-20151124205410-00312-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://drmf-beta.wmflabs.org/wiki/Definition:monicKrawtchouk | # Definition:monicKrawtchouk
The LaTeX DLMF and DRMF macro \monicKrawtchouk represents the monic Krawtchouk polynomial.
This macro is in the category of polynomials.
In math mode, this macro can be called in the following ways:
\monicKrawtchouk{n} produces $\displaystyle {\displaystyle \monicKrawtchouk{n}}$
\monicKrawtchouk{n}@{q^{-x}}{p}{N}{q} produces $\displaystyle {\displaystyle \monicKrawtchouk{n}@{q^{-x}}{p}{N}{q}}$
\monicKrawtchouk{n}@@{q^{-x}}{p}{N}{q} produces $\displaystyle {\displaystyle \monicKrawtchouk{n}@@{q^{-x}}{p}{N}{q}}$
These are defined by $\displaystyle {\displaystyle \Krawtchouk{n}@{x}{p}{N}=:\frac{1}{\pochhammer{-N}{n}p^n}\monicKrawtchouk{n}@{x}{p}{N}. }$ | 2018-12-17 12:52:55 | {"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": 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": 7, "math_score": 0.8647940754890442, "perplexity": 11778.034830352448}, "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-2018-51/segments/1544376828507.57/warc/CC-MAIN-20181217113255-20181217135255-00188.warc.gz"} |
https://www.semanticscholar.org/paper/Scalar-gauge-dynamics-and-Dark-Matter-Buttazzo-Luzio/c457dc8b494c425db80595e2e766f76290655f9e | # Scalar gauge dynamics and Dark Matter
@article{Buttazzo2020ScalarGD,
title={Scalar gauge dynamics and Dark Matter},
author={Dario Buttazzo and Luca Di Luzio and Parsa Hossein Ghorbani and Christian Gross and Giacomo Landini and Alessandro Strumia and Daniele Teresi and Jin-Wei Wang},
journal={Journal of High Energy Physics},
year={2020},
volume={2020},
pages={1-47}
}
• Published 11 November 2019
• Physics
• Journal of High Energy Physics
We consider theories with one gauge group (SU, SO or Sp) and one scalar in a two-index representation. The renormalizable action often has accidental symmetries (such as global U(1) or unusual group parities) that lead to one or more stable states, providing Dark Matter candidates. We discuss the confined phase(s) of each theory and compute the two Higgs phases, finding no generic dualities among them. Discrete gauge symmetries can arise and accidental symmetries can be broken, possibly giving…
12 Citations
Dark Matter in scalar Sp($$\mathcal{N}$$) gauge dynamics
• Physics
Journal of High Energy Physics
• 2020
We consider a model with Sp dark gauge group and a scalar field in the fundamental representation, which leads to two co-stable DM candidates at the perturbative level thanks to a global U(1)
Axion quality from the (anti)symmetric of SU ( N )
• Physics
• 2020
: We propose two models where a U(1) Peccei-Quinn global symmetry arises accidentally and is respected up to high-dimensional operators, so that the axion solution to the strong CP problem is
Thermal squeezeout of dark matter
• Physics
Physical Review D
• 2021
We carry out a detailed study of the confinement phase transition in a dark sector with a SU ( N ) gauge group and a single generation of dark heavy quark. We focus on heavy enough quarks such that
Glueballs in a thermal squeezeout model
• Physics
Journal of High Energy Physics
• 2022
Abstract It has been shown that a first order confinement phase transition can drastically change the relic dark matter abundance in confining dark sectors with only heavy dark quarks. We study the
Axion quality from the (anti)symmetric of SU($$\mathcal{N}$$)
• Physics
• 2020
We propose two models where a U(1) Peccei-Quinn global symmetry arises accidentally and is respected up to high-dimensional operators, so that the axion solution to the strong CP problem is
Analytical relations for the bound state spectrum of gauge theories with a Brout-Englert-Higgs mechanism
We apply the method proposed by Frohlich, Morchio, and Strocchi to analyze the bound state spectrum of various gauge theories with a Brout-Englert-Higgs mechanism. These serve as building blocks for
Multi-Higgs boson probes of the dark sector
• Physics
• 2019
In the Higgs portal framework, cascade decays of the dark sector fields naturally produce multi-Higgs final states along with dark matter. It is common that heavier dark states couple stronger to
Pati-Salam axion
• L. Luzio
• Physics
Journal of High Energy Physics
• 2020
I discuss the implementation of the Peccei-Quinn mechanism in a minimal realization of the Pati-Salam partial unification scheme. The axion mass is shown to be related to the Pati-Salam breaking
Gravitational vector Dark Matter
• Physics
• 2021
A new dark sector consisting of a pure non-abelian gauge theory has no renormalizable interaction with SM particles, and can thereby realise gravitational Dark Matter (DM). Gauge interactions confine
## References
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Abstract We show that a new gauge group with one new scalar leads to automatically stable Dark Matter candidates. We consider theories where the Higgs phase is dual to the confined phase: it is
Accidental composite dark matter
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A bstractWe build models where Dark Matter candidates arise as composite states of a new confining gauge force, stable thanks to accidental symmetries. Restricting to renormalizable theories
Towards the spectrum of the SU(2) adjoint Higgs model
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Scalar particles in the adjoint representation of a non-Abelian gauge theory play an important role in many scenarios beyond the standard model, especially of GUT type. For such theories manifestly
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We provide a simple UV theory for a Dirac dark matter with a massless Abelian gauge boson. We introduce a single fermion transforming as the $\bf{16}$ representation in the SO(10)$'$ gauge group,
Radiatively induced spontaneous symmetry breaking for asymptotically free gauge theories
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Symmetry Breaking and Scalar Bosons
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There are reasons to suspect that the spontaneous breakdown of the gauge symmetries of the observed weak and electromagnetic interactions may be produced by the vacuum expectation values of massless | 2022-08-09 12:00:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.5580477118492126, "perplexity": 3524.915356073042}, "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-2022-33/segments/1659882570921.9/warc/CC-MAIN-20220809094531-20220809124531-00293.warc.gz"} |
https://physics.stackexchange.com/questions/224119/reconciling-geosynchronous-orbits-and-why-the-moon-is-moving-away | # Reconciling geosynchronous orbits and why the moon is moving away
First time on PSE, and forgive me if the question doesn't make sense.
Everything I've read, and the explanations make perfect intuitive sense, for why the moon is moving away from the earth is because there is a transference of energy from the phase shift in the oceans caused by the water's viscosity to the orbit; that the interplay between the faster spinning earth, the viscosity of the oceans, and the tidal pull from the moon is basically pulling the moon faster, hence sending it to a higher orbit.
My question is thus: assuming the earth has plenty of energy, shouldn't the moon be approaching a geosynchronous orbit, which is almost 1/10 of the current distance to the moon? Thus why isn't the moon being pulled toward the earth instead? I'm sure I'm overlooking some elementary fact.
Maybe my question is boiled down to "how do you move a satellite into a lower orbit?"?
• You've got that backwards: the Earth is approaching a selenosynchronous rotational period (though the red giant sun will consume the pair before they achieve mutual lock). Dec 15, 2015 at 2:38
• @dmckee If the Moon was in geosynchronous orbit, would that be stable (under perturbations of the orbit) ? Dec 15, 2015 at 4:29
In a sense the Moon is approaching a geosynchronous orbit, however the radius of a geosynchronous orbit (call this $r_g$) depends on the angular velocity of the Earth's rotation, so $r_g$ changes as the Earth's rotation changes. Specifically it increases as the Earth's rotation slows.
Currently the angular velocity of the Earth's rotation is faster than the angular velocity of the Moon's orbit, and this means the Earth's rotation is being slowed by the tidal losses induced by the Moon. As the rotation slows this means $r_g$ increases, so in effect the geosynchronous orbit is growing outwards towards the Moon.
• thank you, this makes sense and was kind of what I was thinking, but if you visualize $r_g$ and $r_{moon}$ as concentric circles approaching each other, why are they approaching each other in a increasing/increasing fashion (obviously $\frac{d}{dt}r_g>\frac{d}{dt}r_{moon}$ to make this work), rather than $r_{moon}$ decreasing toward an increasing $r_g$? Dec 16, 2015 at 2:20
• @charlestoncrabb: as the Earth's rotation slows, its rotational kinetic energy is transferred to the Moon i.e. the Earth loses energy and the Moon gains energy. As you add energy to an orbiting body it moves outwards and (unexpectedly) slows down. That's because the total energy is the sum of the potential and kinetic energies, and the two are related by the virial theorem $2K = -U$. To go into this in detail would really need a new question, though I'm sure something like this must have been asked here already. Dec 16, 2015 at 6:07 | 2022-05-19 00:08:47 | {"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": 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": 0.6891944408416748, "perplexity": 219.9979392416299}, "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/1652662522556.18/warc/CC-MAIN-20220518215138-20220519005138-00110.warc.gz"} |
https://projectpen.wordpress.com/2008/12/ | Feeds:
Posts
## Archive for December, 2008
C2 The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
Here is the official solution file: pen16.pdf
You can discuss the problem in MathLinks here.
C2 The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
## 15. Exponential Congruence Sequence
D5. Prove that for , $\underbrace{2^{2^{\cdots^{2}}}}_{n\text{ terms}}\equiv \underbrace{2^{2^{\cdots^{2}}}}_{n-1\text{ terms}}\; \pmod{n}.$
D6. Show that, for any fixed integer the sequence $2, \; 2^{2}, \; 2^{2^{2}}, \; 2^{2^{2^{2}}}, \cdots \pmod{n}$ is eventually constant.
Sorry all, for the delay of the problem 15. Here goes the solution: pen-15.pdf
## 14S. Different Approaches to an Intuitive Problem
The fourth problem of the second season of PEN is as follows:
N17. Suppose that $a$ and $b$ are distinct real numbers such that:
$a - b, a^{2}-b^{2}, \cdots, a^k-b^k, \cdots$ are all integers. Show that $a$ and $b$ are integers.
Here is the official solution file: pen14.pdf
You can discuss the problem in MathLinks here.
N17. Suppose that $a$ and $b$ are distinct real numbers such that:
$a - b, a^{2}-b^{2}, \cdots, a^k-b^k, \cdots$ are all integers. Show that $a$ and $b$ are integers. | 2018-05-23 22:05:36 | {"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": 0, "img_math": 22, "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": 0.6806286573410034, "perplexity": 239.1744555115177}, "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-2018-22/segments/1526794865830.35/warc/CC-MAIN-20180523215608-20180523235608-00134.warc.gz"} |
https://www.iacr.org/cryptodb/data/paper.php?pubkey=31896 | ## CryptoDB
### Paper: Round-Optimal and Communication-Efficient Multiparty Computation
Authors: Michele Ciampi , The University of Edinburgh Rafail Ostrovsky , University of California Los Angeles Hendrik Waldner , The University of Edinburgh Vassilis Zikas , Purdue University Search ePrint Search Google Slides EUROCRYPT 2022 Typical approaches for minimizing the round complexity of multi-party computation (MPC) come at the cost of increased communication complexity (CC) or the reliance on setup assumptions. A notable exception is the recent work of Ananth et al. [TCC 2019], which used Functional Encryption (FE) combiners to obtain a round optimal (two-round) semi-honest MPC in the plain model with CC proportional to the depth and input-output length of the circuit being computed---we refer to such protocols as circuit scalable. This leaves open the question of obtaining communication efficient protocols that are secure against malicious adversaries in the plain model, which our work solves. Concretely, our two main contributions are: 1) We provide a round-preserving black-box compiler that compiles a wide class of MPC protocols into circuit-scalable maliciously secure MPC protocols in the plain model, assuming (succinct) FE combiners. 2) We provide a round-preserving black-box compiler that compiles a wide class of MPC protocols into circuit-independent --- i.e., with CC that depends only on the input-output length of the circuit---maliciously secure MPC protocols in the plain model, assuming Multi-Key Fully-Homomorphic Encryption (MFHE). Our constructions are based on a new compiler that turns a wide class of MPC protocols into k-delayed-input function MPC protocols (a notion we introduce), where the functions to be computed is specified only in the k-th round of the protocol. As immediate corollaries of our two compilers, we derive (1) the first round-optimal and circuit-scalable maliciously secure MPC, and (2) the first round-optimal and circuit-independent maliciously secure MPC in the plain model. The latter MPC achieves the best to-date CC for a round-optimal malicious MPC protocol. In fact, it is even communication-optimal when the output size of the function being evaluated is smaller than its input size (e.g., for boolean functions). All of our results are based on standard polynomial time assumptions.
##### BibTeX
@inproceedings{eurocrypt-2022-31896,
title={Round-Optimal and Communication-Efficient Multiparty Computation},
publisher={Springer-Verlag},
author={Michele Ciampi and Rafail Ostrovsky and Hendrik Waldner and Vassilis Zikas},
year=2022
} | 2022-06-29 00:58:18 | {"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": 0.6120898723602295, "perplexity": 2984.0426534513717}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103619185.32/warc/CC-MAIN-20220628233925-20220629023925-00261.warc.gz"} |
https://testbook.com/learn/maths-sum-of-squares-of-first-n-natural-numbers/ | # Sum of Squares of First n Natural Numbers: Even & Odd Numbers Formulas with Proof
1
Save
Sum of squares of first n natural numbers means sum of the squares of the given series of natural numbers. Sum of squares of n natural numbers can be calculated using the formula [n(n+1)(2n+1)] / 6. Let n be a natural number. Squaring the number is denoted by $$n^2$$. Sum of Squares of First n Natural Numbers gives a generalized equation to get a sum of squares n Natural Numbers.
Natural numbers are those numbers used for counting and ordering. Natural numbers include positive integers. Hence, they are also known as non-negative integers. Natural numbers start from 1 and go up to infinity. 1, 2, 3, 4, 5, 6, … ∞
## What is Sum of Squares of First n Natural Numbers?
Sum of squares of n natural numbers means the sum of the squares of the given series of natural numbers. It could be finding the sum of squares of 2 numbers or 3 numbers or sum of squares of consecutive n numbers or n even numbers or n odd numbers.
If n consecutive natural numbers are 1, 2, 3, 4, …, n, then sum of squares of first n natural numbers is represented by $$1^2 + 2^2 + 3^2 + 4^2 + 5^2 + …….. + n^2$$
Sum of squares of n natural numbers can be calculated using the formula [n(n+1)(2n+1)] / 6. Let n be a natural number. Squaring the number is denoted by $$n^2$$.
## Sum of Squares of n Natural Numbers Formula
Formulas for finding the sum of squares of n natural numbers, the sum of squares of first n even numbers, and the sum of squares of first n odd numbers:
• Formula for Sum of squares of n natural numbers: [n(n+1)(2n+1)] / 6
• Formula for Sum of squares of first n even numbers: [2n(n + 1)(2n + 1)] / 3
• Formula for Sum of squares of first n odd numbers: [n(2n+1)(2n-1)] / 3
## Proof of Sum of Squares of First n Natural Numbers
Let us see the proof of Sum of Squares of First n Natural Numbers.
Sum of the squares of first n natural numbers = $${n(n+1)(2n+1)\over{6}}$$.
Proof: Let us assume the required sum = S
Therefore, $$S = (1^2 + 2^2 + 3^2 + 4^2 + 5^2 + ………………. + n^2)$$
Now, we will use the below identity to find the value of S:
$$n^3 – (n – 1)^3 = 3n^2 – 3n + 1$$
Substituting, n = 1, 2, 3, 4, 5, …………., n in the above identity, we get
$$1^3 – 0^3 = (3 . 1^2) – (3 . 1) + 1$$
$$2^3 – 1^3 = (3 . 2^2) – (3 . 2) + 1$$
$$3^3 – 2^3 = (3 . 3^2) – (3 . 3) + 1$$
$$4^3 – 3^3 = (3 . 4^2) – (3 . 4) + 1$$
………………………………..
$$n^3 – (n-1)^3 = (3 . n^2) – (3 . n) + 1$$
Adding we get, $$n^3 – 0^3 = 3(1^2 + 2^2 + 3^2 + 4^2 + ……….. + n^2) – 3(1 + 2 + 3 + 4 + …….. + n) + (1 + 1 + 1 + 1 + ……… n times)$$
$$n^3 = 3S – 3{\cdot}{n(n+1)\over{2}} + n$$
$$3S = n^3 + {3\over{2}}n(n + 1) – n = n(n^2 – 1) + {3\over{2}}n(n + 1)$$
$$3S = n(n + 1)(n – 1 + {3\over{2}})$$
$$3S = n(n + 1){2n-2+3\over{2}}$$
$$3S = {n(n+1)(2n+1)\over{2}}$$
Therefore, $$S = {n(n+1)(2n+1)\over{6}}$$
i.e., $$1^2 + 2^2 + 3^2 + 4^2 + 5^2 + ………………. + n^2 = {n(n+1)(2n+1)\over{6}}$$
Hence proved that the sum of the squares of first n natural numbers is $${n(n+1)(2n+1)\over{6}}$$
## Sum of Squares of First n Even Natural Numbers
An even number is a whole number that is able to be divided by two into two equal whole numbers. The numbers 0, 2, 4, 6, and 8 are even numbers.
This is a variation in the formula to calculate the sums of squares of only even numbers. This sum is simply $$2^2+4^2+6^2+…+(2n)^2$$ up to the nth even natural number.
Derivation of Sum of Squares of First n Even Natural Numbers with proof
We can arrive at the formula as follows:
$$\sum({2i})^2=2^2+4^2+6^2+…+(2n)^2$$
We take 2^2 common
$$S = 2^2(1^2)+2^2(2^2)+2^2(3^2)+…+2^2(n^2)$$
$$S = 2^2(1^2+2^2+3^2+…+n^2)$$
But we know that 1^2+2^2+3^2+…+n^2[/latex] is simply the sum of the squares of the first n natural numbers, which we know is {n(n+1)(2n+1)\over{6}}[/latex]. Substituting this above, we get,
$$S =2^2(11^2+2^2+3^2+…+n^2)$$
$$= 4{(n(n+1)(2n+1)\over{6})}$$
$$={2n(n+1)(2n+1)\over{3}}$$
Thus, we have the formula for the sum of squares of first n even natural numbers,
$$S =(2i)^2=2^2+4^2+6^2+…+(2n)^2 = {2n(n+1)(2n+1)\over{3}}$$
## Sum of Squares of First n Odd Natural Numbers
Now let’s see the Sum of the Squares of First n Odd Natural Numbers. Odd numbers are defined as numbers that are not exactly divisible by two. Or, to put it another way, an odd number is one that is not even or divisible by two. The numbers 1, 3, 5, 7, and 9 are odd numbers.
Sum of Squares of First n Odd Natural Numbers Formula
This sum is simply written as $$1^2+3^2+5^2+…+(2n−1)^2$$. This can also be simply written as $$\sum =1(2i−1)^2$$ or $$\sum =(2n−1)^2$$
We can derive the formula by noting the following, for the sums of squares of the first 2n natural numbers.
$$S = 1^2+2^2+3^2+…+(2n)^2$$
$$=(1^2 + 3^2 +…+ (2n−1)^2)+(2^2+4^2+…+(2n)^2)$$
We simply rearranged the terms, to group odd and even natural numbers together. The sums on the right are the sums of the squares of the first n odd and even numbers, respectively. Now we can write,
$$\sum^{2n}_ii^2=1^2+2^2+3^2+…+(2n)^2$$
$$\sum^{n}_ii^2=(2i−1)^2+(2i)^2$$
This gives us, for the sum of the first n odd numbers,
$$\sum^n_{i=1}(2i−1)^2=\sum^{2n}_{i=1}i^2–\sum^n_{i=1}(2i)^2$$
We know the formula for the sum of the squares of the first n numbers, so we just have to replace n with 2n for the first part of RHS. We get, after replacing,
$$\sum^n_{i=1}(2i−1)^2={2n(2n+1)(4n+1)\over{6}}–{2n(n+1)(2n+1)\over{3}}$$
$$= {n(2n+1)\over{3}}[(4n+1)–2(n+1)]$$
$$= {n(2n+1)\over{3}}(2n−1)$$
$$= {(n(2n+1)(2n−1)\over{3}}$$
Squares of First 20 Natural Numbers
Number n Square $$n^2$$ 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 16 256 17 289 18 324 19 361 20 400
Sum of Arithmetic Series
The first term is $$a_1$$, the second term is $$a_1 + d$$, the third term is $$a_1 + 2d$$, etc. This leads up to finding the sum of the arithmetic series, $$S_n$$, by starting with the first term and successively adding the common difference.
Simplifying $$S_n = {n\over2}[2 a_1 + (n – 1)d]$$
## Solved Examples on Sum of Squares of Natural Numbers
Some solved examples on the sum of squares of natural numbers are given below:
Example 1: Find the Sum of Squares of First 8 Natural Numbers.
Solution: n = 8.
We know that, the sum of the squares of first n natural numbers = $${n(n+1)(2n+1)\over{6}}$$
Put n = 8 in this formula.
Therefore we get,
$$S_{8^2}={8(8+1)(2\times8+1)\over{6}}$$
$$={8(8+1)(16+1)\over{6}}=12(17)=204$$
Let’s cross-check this with our normal method.
$$S_8 = 1^2 + 2^2 + 3^2 + 4^2 +5^2 + 6^2 + 7^2 + 8^2$$
= 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64
= 204.
Example 2: What is the sum of square first 5 odd natural numbers?
Solution: n = 5
We know that, the sum of the squares of first 5 odd natural numbers = $${(n(2n+1)(2n−1)\over{3}}$$
Put n = 5 in this formula.
Therefore we get,
$$S_{5^2}={5(2(5)+1)(2(5)-1)\over{3}}$$
$$={10(10+1)(20+1)\over{3}}$$ = 165
Let’s cross-check this with our normal method.
$$S_{10^2} = 1^2 + 3^2 + 5^2 + 7^2 +9^2$$
= 1 + 9 + 25 + 49 + 81
= 165.
Example 3: What is the sum of square first 10 even natural numbers?
Solution: n = 10
We know that the sum of the squares of first 10 even natural numbers = $${2n(n+1)(2n+1)\over{3}}$$
Put n = 10 in this formula.
Therefore we get,
$$S_{10^2}={10(10+1)(2\times10+1)\over{3}}$$
$$={10(10+1)(20+1)\over{3}} = 1012$$
Let’s cross-check this with our normal method.
$$S_{10^2} = 2^2 + 4^2 + 6^2 + 8^2 +10^2 + 12^2 + 14^2 + 16^2 + 18^2 + 20^2$$
= 4 + 16 + 36 + 64 + 100 + 144 + 169 + 196 + 256 + 324 + 400
= 1012.
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If you are checking Sum of Squares of First n Natural Numbers article, also check the related maths articles: Number System Properties of Natural Numbers Difference Between Natural and Whole Numbers Types of Numbers Sum of cubes of first n-natural numbers Sum of n Natural Numbers
## FAQs on Sum of Squares of First n Natural Numbers
Q.1 What is sum of squares of first n natural numbers?
Ans.1 Sum of squares of n natural numbers means the sum of the squares of the given series of natural numbers. It could be finding the sum of squares of 2 numbers or 3 numbers or sum of squares of consecutive n numbers or n even numbers or n odd numbers.
Q.2 What is sum of squares of n natural numbers formula?
Ans.2 If n consecutive natural numbers are 1, 2, 3, 4, …, n, then the sum of squared ‘n’ consecutive natural numbers is represented by $$1^2 + 2^2 + 3^2 + 4^2 + 5^2 + … + n^2$$
Sum of squares of n natural numbers can be calculated using the formula [n(n+1)(2n+1)] / 6. Let n be a natural number. Squaring the number is denoted by $$n^2$$.
Q.3 What is the sum of squares of first n even natural numbers?
Ans.3 Sum of squares of first n even natural numbers can be written as $$2^2+4^2+6^2+…+(2n)^2$$
Formula for the sum of squares of first n even n natural numbers is $$S =(2i)^2=2^2+4^2+6^2+…+(2n)^2 = {2n(n+1)(2n+1)\over{3}}$$
Q.4 What is the sum of squares of first n odd natural numbers?
Ans.4 Sum of squares of first n odd natural numbers can be written as $$1^2+3^2+5^2+…+(2n−1)^2$$
Formula for the sum of squares of first n odd n natural numbers is $$= {(n(2n+1)(2n−1)\over{3}}$$
Q.5 What is the sum of squares of first 10 natural numbers?
Ans.5 n = 10.
We know that the sum of the squares of first 10 even natural numbers = $${2n(n+1)(2n+1)\over{3}}$$
Put n = 10 in this formula.
Therefore we get,
$$S_{10^2}={10(10+1)(2\times10+1)\over{3}}$$
$$={10(10+1)(20+1)\over{3}} = 1012$$
Let’s cross-check this with our normal method.
$$S_{10^2} = 2^2 + 4^2 + 6^2 + 8^2 +10^2 + 12^2 + 14^2 + 16^2 + 18^2 + 20^2$$
= 4 + 16 + 36 + 64 + 100 + 144 + 169 + 196 + 256 + 324 + 400
= 1012. | 2023-02-04 06:46:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.7649757862091064, "perplexity": 174.9804682139251}, "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/1674764500094.26/warc/CC-MAIN-20230204044030-20230204074030-00830.warc.gz"} |
http://biosport.ucdavis.edu/blog/precise-steady-turning-conditions | ##### Sections
> Blog > Precise steady turning conditions
# Precise steady turning conditions
by Dale Lukas Peterson — last modified Apr 28, 2011 07:50 PM
Below are some precise numerical values for stable steady turns to the right with steer torque to the right. Lean and steer are defined analogous to the benchmark coordinates. The pitch I report below is defined as the pitch coordinate in the benchmark paper plus the head angle (lambda), which is why the pitch values are non-zero, but are very close to $$\pi/10$$ (after all these turns are near the origin, where pitch changes very little as a function of lean and steer). Steer torque, front wheel rate and velocity are also reported.
The next four numbers give an indication of how well the following four constraints are satisfied:
• Front wheel height (configuration constraint on lean, pitch and steer), should be zero
• NH[i] for i=1,2,3 are the three components of the velocity of the front wheel contact point. For no slip rolling these should all be identically zero. NH[1] and NH[2] represent the no-slip constraints, NH[3] is effectively the differentiated holonomic constraint. All three of these should be zero if the rolling constraints for both wheels are to be satisfied.
Configuration:
Lean $$\phi$$ = 0.08726646259971647
Pitch $$\theta$$ = 0.3139899180509261
Steer $$\delta$$ = 0.0249931148885184
Constraints (should be zero):
FW Contact Height = 1.994931997373328e-17
NH[0] = 8.881784197001252e-16
NH[1] = -3.122502256758253e-17
NH[2] = 2.220446049250313e-16
Steer torque = 0.002386180613544564
Front wheel rate = -17.27895470227879
Velocity = 6.047634145797575
Eigenvalues:
lambda_0 = -0.0008960513685150272 + 0j
lambda_1 = -1.565399502765081 + 5.940516021028886j
lambda_2 = -1.565399502765081 - 5.940516021028886j
lambda_3 = -16.11615742578447 + 0j
Configuration:
Lean = 0.08726646259971647
Pitch = 0.3139889374771921
Steer = 0.025132741228678
Constraints (should be zero):
FW Contact Height = 1.908195823574488e-17
NH[0] = 0
NH[1] = 7.979727989493313e-17
NH[2] = 2.220446049250313e-16
Steer torque = 0.001432842907858944
Front wheel rate = -17.23146391878335
Velocity = 6.031012371574174
Eigenvalues:
lambda_0 = -0.003439017988870276 + 0j
lambda_1 = -1.554891032061446 + 5.918052698264985j
lambda_2 = -1.554891032061446 - 5.918052698264985j
lambda_3 = -16.08145923521178 + 0j
Configuration:
Lean = 0.08726646259971647
Pitch = 0.3139879565265826
Steer = 0.0252723675688376
Constraints (should be zero):
FW Contact Height = 7.719519468096792e-17
NH[0] = -8.881784197001252e-16
NH[1] = -5.204170427930421e-17
NH[2] = -4.440892098500626e-16
Steer torque = 0.0004796830975322131
Front wheel rate = -17.18436903579476
Velocity = 6.014529162528164
Eigenvalues:
lambda_0 = -0.006003419792954939 + 0j
lambda_1 = -1.544442434016426 + 5.895751882573691j
lambda_2 = -1.544442434016426 - 5.895751882573691j
lambda_3 = -16.0470623299768 + 0j
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(Required) | 2017-07-21 10:42:54 | {"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": 0.577560305595398, "perplexity": 8509.616286801634}, "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-2017-30/segments/1500549423769.10/warc/CC-MAIN-20170721102310-20170721122310-00575.warc.gz"} |
https://www.physics.harvard.edu/research/pubs-ab | # Faculty Publications: June, 2018
Query Results from the Smithsonian/NASA Astrophysics Data System (ADS)
The following are Harvard Physics faculty members' publications, added to the ADS database last month. Please note that some publications which appeared in print last month may not be included in the database (and therefore may not appear on this list) until the following month.
Title: The Titan Haze Simulation Experiment: Latest Laboratory Results and Dedicated Plasma Chemistry Model Authors: Sciamma-O'Brien, Ella; Raymond, Alexander; Mazur, Eric; Salama, Farid Publication: American Astronomical Society, AAS Meeting #232, id. 217.10 Publication Date: 06/2018 Origin: AAS Abstract Copyright: (c) 2018: American Astronomical Society Bibliographic Code: 2018AAS...23221710S
### Abstract
Here, we present the latest results on the gas and solid phase analyses in the Titan Haze Simulation (THS) experiment. The THS experiment, developed at NASA Ames’ COSmIC facility is a unique experimental platform that allows us to simulate Titan’s complex atmospheric chemistry at Titan-like temperature (200 K) by cooling down N2-CH4-based mixtures in a supersonic expansion before inducing the chemistry by plasma.Gas phase: The residence time of the jet-accelerated gas in the active plasma region is less than 4 µs, which results in a truncated chemistry enabling us to control how far in the chain of reactions the chemistry is processing. By adding heavier molecules in the initial gas mixture, it is then possible to study the first and intermediate steps of Titan’s atmospheric chemistry as well as specific chemical pathways, as demonstrated by mass spectrometry and comparison to Cassini CAPS data [1]. A new model was recently developed to simulate the plasma chemistry in the THS. Calculated mass spectra produced by this model are in good agreement with the experimental THS mass spectra, confirming that the short residence time in the plasma cavity limits the growth of larger species [2].Solid phase: Scanning electron microscopy and infrared spectroscopy have been used to investigate the effect of the initial gas mixture on the morphology of the THS Titan aerosol analogs as well as on the level and nature of the nitrogen incorporation into these aerosols. A comparison to Cassini VIMS observational data has shown that the THS aerosols produced in simpler mixtures, i.e., that contain more nitrogen and where the N-incorporation is in isocyanide-type molecules instead of nitriles, are more representative of Titan’s aerosols [3]. In addition, a new optical constant facility has been developed at NASA Ames that allows us to determine the complex refractive indices of THS Titan aerosol analogs from NIR to FIR (0.76-222 cm-1). The facility and preliminary results will be presented.References:[1] Sciamma-O'Brien E., et al., Icarus, 243, 325 (2014)[2] Raymond, A., et al., ApJ., 853, 107 (2018)[3] Sciamma-O'Brien E., et al., Icarus, 289, 214 (2017)Acknowledgements: This research is supported by the SSW Program of NASA SMD.
Title: The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample Authors: Scolnic, D. M.; Jones, D. O.; Rest, A.; Pan, Y. C.; Chornock, R.; Foley, R. J.; Huber, M. E.; Kessler, R.; Narayan, G.; Riess, A. G.; Rodney, S.; Berger, E.; Brout, D. J.; Challis, P. J.; Drout, M.; Finkbeiner, D.; Lunnan, R.; Kirshner, R. P.; Sanders, N. E.; Schlafly, E.; Smartt, S.; Stubbs, C. W.; Tonry, J.; Wood-Vasey, W. M.; Foley, M.; Hand, J.; Johnson, E.; Burgett, W. S.; Chambers, K. C.; Draper, P. W.; Hodapp, K. W.; Kaiser, N.; Kudritzki, R. P.; Magnier, E. A.; Metcalfe, N.; Bresolin, F.; Gall, E.; Kotak, R.; McCrum, M.; Smith, K. W. Publication: The Astrophysical Journal, Volume 859, Issue 2, article id. 101, 28 pp. (2018). (ApJ Homepage) Publication Date: 06/2018 Origin: IOP Astronomy Keywords: cosmology: observations, dark energy, supernovae: general DOI: 10.3847/1538-4357/aab9bb Bibliographic Code: 2018ApJ...859..101S
### Abstract
We present optical light curves, redshifts, and classifications for 365 spectroscopically confirmed Type Ia supernovae (SNe Ia) discovered by the Pan-STARRS1 (PS1) Medium Deep Survey. We detail improvements to the PS1 SN photometry, astrometry, and calibration that reduce the systematic uncertainties in the PS1 SN Ia distances. We combine the subset of 279 PS1 SNe Ia (0.03 < z < 0.68) with useful distance estimates of SNe Ia from the Sloan Digital Sky Survey (SDSS), SNLS, and various low-z and Hubble Space Telescope samples to form the largest combined sample of SNe Ia, consisting of a total of 1048 SNe Ia in the range of 0.01 < z < 2.3, which we call the “Pantheon Sample.” When combining Planck 2015 cosmic microwave background (CMB) measurements with the Pantheon SN sample, we find {{{Ω }}}m=0.307+/- 0.012 and w=-1.026+/- 0.041 for the wCDM model. When the SN and CMB constraints are combined with constraints from BAO and local H 0 measurements, the analysis yields the most precise measurement of dark energy to date: {w}0=-1.007+/- 0.089 and {w}a=-0.222+/- 0.407 for the {w}0{w}aCDM model. Tension with a cosmological constant previously seen in an analysis of PS1 and low-z SNe has diminished after an increase of 2× in the statistics of the PS1 sample, improved calibration and photometry, and stricter light-curve quality cuts. We find that the systematic uncertainties in our measurements of dark energy are almost as large as the statistical uncertainties, primarily due to limitations of modeling the low-redshift sample. This must be addressed for future progress in using SNe Ia to measure dark energy.
Title: Numerical study of the chiral $\mathbb{Z}_3$ quantum phase transition in one spatial dimension Authors: Samajdar, Rhine; Choi, Soonwon; Pichler, Hannes; Lukin, Mikhail D.; Sachdev, Subir Publication: eprint arXiv:1806.01867 Publication Date: 06/2018 Origin: ARXIV Keywords: Condensed Matter - Strongly Correlated Electrons, Condensed Matter - Quantum Gases, Condensed Matter - Statistical Mechanics, Physics - Atomic Physics Comment: 14 pages, 9 figures Bibliographic Code: 2018arXiv180601867S
### Abstract
Recent experiments on a one-dimensional chain of trapped alkali atoms [arXiv:1707.04344] have observed a quantum transition associated with the onset of period-3 ordering of pumped Rydberg states. This spontaneous $\mathbb{Z}_3$ symmetry breaking is described by a constrained model of hard-core bosons proposed by Fendley $et\, \,al.$ [arXiv:cond-mat/0309438]. By symmetry arguments, the transition is expected to be in the universality class of the $\mathbb{Z}_3$ chiral clock model with parameters preserving both time-reversal and spatial-inversion symmetries. We study the nature of the order-disorder transition in these models, and numerically calculate its critical exponents with exact diagonalization and density-matrix renormalization group techniques. We use finite-size scaling to determine the dynamical critical exponent $z$ and the correlation length exponent $\nu$. Our analysis presents the only known instance of a strongly-coupled transition between gapped states with $z \ne 1$, implying an underlying nonconformal critical field theory.
Title: Triangular antiferromagnetism on the honeycomb lattice of twisted bilayer graphene Authors: Thomson, Alex; Chatterjee, Shubhayu; Sachdev, Subir; Scheurer, Mathias S. Publication: eprint arXiv:1806.02837 Publication Date: 06/2018 Origin: ARXIV Keywords: Condensed Matter - Strongly Correlated Electrons, Condensed Matter - Mesoscale and Nanoscale Physics, Condensed Matter - Superconductivity Comment: 18 pages (single column), 12 page appendix, 5 figures, 1 table (added in v2) - v2: Some comments and references and the added table Bibliographic Code: 2018arXiv180602837T
### Abstract
We present the electronic band structures of states with the same symmetry as the three-sublattice planar antiferromagnetic order of the triangular lattice. Such states can also be defined on the honeycomb lattice provided the spin density waves lie on the bonds. We identify cases which are consistent with observations on twisted bilayer graphene: a correlated insulator with an energy gap, yielding a single doubly-degenerate Fermi surface upon hole doping. We also discuss extensions to metallic states which preserve spin rotation invariance, with fluctuating spin density waves and bulk $\mathbb{Z}_2$ topological order.
Title: Topological Materials Discovery By Large-order symmetry indicators Authors: Tang, Feng; Po, Hoi Chun; Vishwanath, Ashvin; Wan, Xiangang Publication: eprint arXiv:1806.04128 Publication Date: 06/2018 Origin: ARXIV Keywords: Condensed Matter - Mesoscale and Nanoscale Physics, Condensed Matter - Materials Science Bibliographic Code: 2018arXiv180604128T
### Abstract
Crystalline symmetries play an important role in the classification of band structures, and the rich variety of spatial symmetries in solids leads to various topological crystalline phases (TCPs). However, compared with topological insulators and Dirac/Weyl semimetals, relatively few realistic materials candidates have been proposed for TCPs. Based on our recently developed method for the efficient discovery of topological materials using symmetry indicators, we explore topological materials in five space groups (i.e. SGs87,140,221,191,194), which are indexed by large order strong symmetry based indicators (Z8 and Z12) allowing for the realization of several kinds of gapless boundary states in a single compound. We predict many TCPs, and the representative materials include: Pt3Ge(SG140), graphite(SG194), XPt3 (SG221,X=Sn,Pb), Au4Ti (SG87) and Ti2Sn (SG194). As by-products, we also find that AgXF3 (SG140,X=Rb,Cs) and AgAsX (SG194,X=Sr,Ba) are good Dirac semimetals with clean Fermi surface. The proposed materials provide a good platform to study the novel properties emerging from the interplay between different types of boundary states.
Title: Meson formation in mixed-dimensional t-J models Authors: Grusdt, Fabian; Zhu, Zheng; Shi, Tao; Demler, Eugene Publication: eprint arXiv:1806.04426 Publication Date: 06/2018 Origin: ARXIV Keywords: Condensed Matter - Strongly Correlated Electrons, Condensed Matter - Quantum Gases, High Energy Physics - Theory, Quantum Physics Comment: 7 pages, 6 figures, 4 pages methods section Bibliographic Code: 2018arXiv180604426G
### Abstract
Surprising properties of doped Mott insulators are at the heart of many quantum materials, including transition metal oxides and organic materials. The key to unraveling complex phenomena observed in these systems lies in understanding the interplay of spin and charge degrees of freedom. One of the most debated questions concerns the nature of charge carriers in a background of fluctuating spins. To shed new light on this problem, we suggest a simplified model with mixed dimensionality, where holes move through a Mott insulator unidirectionally while spin exchange interactions are two dimensional. By studying individual holes in this system, we find direct evidence for the formation of mesonic bound states of holons and spinons, connected by a string of displaced spins -- a precursor of the spin-charge separation obtained in the 1D limit of the model. Our predictions can be tested using ultracold atoms in a quantum gas microscope, allowing to directly image spinons and holons, and reveal the short-range hidden string order which we predict in this model.
Title: High-fidelity control and entanglement of Rydberg atom qubits Authors: Levine, Harry; Keesling, Alexander; Omran, Ahmed; Bernien, Hannes; Schwartz, Sylvain; Zibrov, Alexander S.; Endres, Manuel; Greiner, Markus; Vuletić, Vladan; Lukin, Mikhail D. Publication: eprint arXiv:1806.04682 Publication Date: 06/2018 Origin: ARXIV Keywords: Quantum Physics, Condensed Matter - Quantum Gases, Physics - Atomic Physics Comment: 9 pages, 4 figures Bibliographic Code: 2018arXiv180604682L
### Abstract
Individual neutral atoms excited to Rydberg states are a promising platform for quantum simulation and quantum information processing. However, experimental progress to date has been limited by short coherence times and relatively low gate fidelities associated with such Rydberg excitations. We report progress towards high-fidelity quantum control of Rydberg atom qubits. Enabled by a reduction in laser phase noise, our approach yields a significant improvement in coherence properties of individual qubits. We further show that this high-fidelity control extends to the multi-particle case by preparing a two-atom entangled state with a fidelity exceeding 0.97(3), and extending its lifetime with a two-atom dynamical decoupling protocol. These advances open up new prospects for scalable quantum simulation and quantum computation with neutral atoms.
Title: Pressure dependence of the magic twist angle in graphene superlattices Authors: Carr, Stephen; Fang, Shiang; Jarillo-Herrero, Pablo; Kaxiras, Efthimios Publication: eprint arXiv:1806.05078 Publication Date: 06/2018 Origin: ARXIV Keywords: Condensed Matter - Mesoscale and Nanoscale Physics Comment: 6 pages, 4 figures Bibliographic Code: 2018arXiv180605078C
### Abstract
The recently demonstrated unconventional superconductivity in twisted bilayer graphene (tBLG) opens the possibility for interesting applications of two-dimensional layers that involve correlated electron states. Here we explore the possibility of modifying electronic correlations by the application of uniaxial pressure on the weakly interacting layers, which results in increased interlayer coupling and a modification of the magic angle value and associated density of states. Our findings are based on first-principles calculations that accurately describe the height-dependent interlayer coupling through the combined use of Density Functional Theory and Maximally localized Wannier functions. We obtain the relationship between twist angle and external pressure for the magic angle flat bands of tBLG. This may provide a convenient method to tune electron correlations by controlling the length scale of the superlattice.
Title: Measuring the Local Twist Angle and Layer Arrangement in Van der Waals Heterostructures Authors: de Jong, Tobias A.; Jobst, Johannes; Yoo, Hyobin; Krasovskii, Eugene E.; Kim, Philip; van der Molen, Sense Jan Publication: eprint arXiv:1806.05155 Publication Date: 06/2018 Origin: ARXIV Keywords: Condensed Matter - Materials Science Bibliographic Code: 2018arXiv180605155D
### Abstract
The properties of Van der Waals heterostructures are determined by the twist angle and the interface between adjacent layers as well as their polytype and stacking. Here we describe the use of spectroscopic Low Energy Electron Microscopy (LEEM) and micro Low Energy Electron Diffraction ({\mu}LEED) methods to measure these properties locally. We present results on a MoS$_{2}$/hBN heterostructure, but the methods are applicable to other materials. Diffraction spot analysis is used to assess the benefits of using hBN as a substrate. In addition, by making use of the broken rotational symmetry of the lattice, we determine the cleaving history of the MoS$_{2}$ flake, i.e., which layer stems from where in the bulk.
Title: Active colloidal particles in emulsion droplets: A model system for the cytoplasm Authors: Horowitz, Viva R.; Chambers, Zachary C.; Gözen, İrep; Dimiduk, Thomas G.; Manoharan, Vinothan N. Publication: eprint arXiv:1806.05760 Publication Date: 06/2018 Origin: ARXIV Keywords: Condensed Matter - Soft Condensed Matter Bibliographic Code: 2018arXiv180605760H
### Abstract
In living cells, molecular motors create activity that enhances the diffusion of particles throughout the cytoplasm, and not just ones attached to the motors. We demonstrate initial steps toward creating artificial cells that mimic this phenomenon. Our system consists of active, Pt-coated Janus particles and passive tracers confined to emulsion droplets. We track the motion of both the active particles and passive tracers in a hydrogen peroxide solution, which serves as the fuel to drive the motion. We first show that correcting for bulk translational and rotational motion of the droplets induced by bubble formation is necessary to accurately track the particles. After drift correction, we find that the active particles show enhanced diffusion in the interior of the droplets and are not captured by the droplet interface. At the particle and hydrogen peroxide concentrations we use, we observe little coupling between the active and passive particles. We discuss the possible reasons for lack of coupling and describe ways to improve the system to more effectively mimic cytoplasmic activity.
Title: Long-Lived Particles at the Energy Frontier: The MATHUSLA Physics Case Authors: Curtin, David; Drewes, Marco; McCullough, Matthew;... Reece, Matthew;... and 84 coauthors Publication: eprint arXiv:1806.07396 Publication Date: 06/2018 Origin: ARXIV Keywords: High Energy Physics - Phenomenology, High Energy Physics - Experiment Comment: 208 pages, 72 figures Bibliographic Code: 2018arXiv180607396C
### Abstract
We examine the theoretical motivations for long-lived particle (LLP) signals at the LHC in a comprehensive survey of Standard Model (SM) extensions. LLPs are a common prediction of a wide range of theories that address unsolved fundamental mysteries such as naturalness, dark matter, baryogenesis and neutrino masses, and represent a natural and generic possibility for physics beyond the SM (BSM). In most cases the LLP lifetime can be treated as a free parameter from the $\mu$m scale up to the Big Bang Nucleosynthesis limit of $\sim 10^7$m. Neutral LLPs with lifetimes above $\sim$ 100m are particularly difficult to probe, as the sensitivity of the LHC main detectors is limited by challenging backgrounds, triggers, and small acceptances. MATHUSLA is a proposal for a minimally instrumented, large-volume surface detector near ATLAS or CMS. It would search for neutral LLPs produced in HL-LHC collisions by reconstructing displaced vertices (DVs) in a low-background environment, extending the sensitivity of the main detectors by orders of magnitude in the long-lifetime regime. In this white paper we study the LLP physics opportunities afforded by a MATHUSLA-like detector at the HL-LHC. We develop a model-independent approach to describe the sensitivity of MATHUSLA to BSM LLP signals, and compare it to DV and missing energy searches at ATLAS or CMS. We then explore the BSM motivations for LLPs in considerable detail, presenting a large number of new sensitivity studies. While our discussion is especially oriented towards the long-lifetime regime at MATHUSLA, this survey underlines the importance of a varied LLP search program at the LHC in general. By synthesizing these results into a general discussion of the top-down and bottom-up motivations for LLP searches, it is our aim to demonstrate the exceptional strength and breadth of the physics case for the construction of the MATHUSLA detector.
Title: Compactifications of ADE conformal matter on a torus Authors: Kim, Hee-Cheol; Razamat, Shlomo S.; Vafa, Cumrun; Zafrir, Gabi Publication: eprint arXiv:1806.07620 Publication Date: 06/2018 Origin: ARXIV Keywords: High Energy Physics - Theory Comment: 66 pages; 38 figures Bibliographic Code: 2018arXiv180607620K
### Abstract
In this paper we study compactifications of ADE type conformal matter, N M5 branes probing ADE singularity, on torus with flux for global symmetry. We systematically construct the four dimensional theories by first going to five dimensions and studying interfaces. We claim that certain interfaces can be associated with turning on flux in six dimensions. The interface models when compactified on a circle comprise building blocks for constructing four dimensional models associated to flux compactifications of six dimensional theories on a torus. The theories in four dimensions turn out to be quiver gauge theories and the construction implies many interesting cases of IR symmetry enhancements and dualities of such theories.
Title: Band Structure of Twisted Bilayer Graphene: Emergent Symmetries, Commensurate Approximants and Wannier Obstructions Authors: Zou, Liujun; Po, Hoi Chun; Vishwanath, Ashvin; Senthil, T. Publication: eprint arXiv:1806.07873 Publication Date: 06/2018 Origin: ARXIV Keywords: Condensed Matter - Strongly Correlated Electrons, Condensed Matter - Mesoscale and Nanoscale Physics, Condensed Matter - Materials Science Comment: 14 pages + appendices; v2: references updated and typos fixed Bibliographic Code: 2018arXiv180607873Z
### Abstract
A remarkable feature of the band structure of bilayer graphene at small twist angle is the appearance of isolated bands near neutrality, whose bandwidth can be reduced at certain magic angles (eg. $\theta\sim 1.05^\circ$). In this regime, correlated insulating states and superconductivity have been experimentally observed. A microscopic description of these phenomena requires an understanding of universal aspects of the band structure, which we discuss here. First, we point out the importance of emergent symmetries, such as valley conservation, which are excellent symmetries in the limit of small twist angles and dictate qualitative features of the band structure. These have sometimes been overlooked when discussing commensurate approximants to the band structure, which we also review here, and solidify their connection with the continuum theory which incorporates all emergent symmetries. Finally, we discuss obstructions to writing down tight-binding models of just the isolated bands, and in particular a new symmetry based diagnostic of these obstructions, as well as relations to band topology and strategies for resolving the obstruction.
Title: Data-driven studies of magnetic two-dimensional materials Authors: Rhone, Trevor David; Chen, Wei; Desai, Shaan; Yacoby, Amir; Kaxiras, Efthimios Publication: eprint arXiv:1806.07989 Publication Date: 06/2018 Origin: ARXIV Keywords: Condensed Matter - Materials Science Bibliographic Code: 2018arXiv180607989R
### Abstract
We use a data-driven approach to study the magnetic and thermodynamic properties of van der Waals (vdW) layered materials. We investigate monolayers of the form A$_2$B$_2$X$_6$, based on the known material Cr$_2$Ge$_2$Te$_6$, using density functional theory (DFT) calculations and machine learning methods to determine their magnetic properties, such as magnetic order and magnetic moment. We also examine formation energies and use them as a proxy for chemical stability. We show that machine learning tools, combined with DFT calculations, can provide a computationally efficient means to predict properties of such two-dimensional (2D) magnetic materials. Our data analytics approach provides insights into the microscopic origins of magnetic ordering in these systems. For instance, we find that the X site strongly affects the magnetic coupling between neighboring A sites, which drives the magnetic ordering. Our approach opens new ways for rapid discovery of chemically stable vdW materials that exhibit magnetic behavior.
Title: De Sitter Space and the Swampland Authors: Obied, Georges; Ooguri, Hirosi; Spodyneiko, Lev; Vafa, Cumrun Publication: eprint arXiv:1806.08362 Publication Date: 06/2018 Origin: ARXIV Keywords: High Energy Physics - Theory Comment: 22 pages, 1 table; added references Bibliographic Code: 2018arXiv180608362O
### Abstract
It has been notoriously difficult to construct a meta-stable de Sitter (dS) vacuum in string theory in a controlled approximation. This suggests the possibility that meta-stable dS belongs to the swampland. In this paper, we propose a swampland criterion in the form of $|\nabla V|\geq\ c \cdot V$ for a scalar potential $V$ of any consistent theory of quantum gravity, for a positive constant $c$. In particular, this bound forbids dS vacua. The existence of this bound is motivated by the abundance of string theory constructions and no-go theorems which exhibit this behavior. We also extend some of the well-known no-go theorems for the existence of dS vacua in string theory to more general accelerating universes and reinterpret the results in terms of restrictions on allowed scalar potentials.
Title: Implementation of a stable, high-power optical lattice for quantum gas microscopy Authors: Mazurenko, A.; Blatt, S.; Huber, F.; Parsons, M. F.; Chiu, C. S.; Ji, G.; Greif, D.; Greiner, M. Publication: eprint arXiv:1806.08997 Publication Date: 06/2018 Origin: ARXIV Keywords: Condensed Matter - Quantum Gases, Physics - Atomic Physics, Quantum Physics Comment: 13 pages, 14 figures Bibliographic Code: 2018arXiv180608997M
### Abstract
We describe the design and implementation of a stable high-power 1064 nm laser system to generate optical lattices for experiments with ultracold quantum gases. The system is based on a low-noise laser amplified by an array of four heavily modified, high-power fiber amplifiers. The beam intensity is stabilized and controlled with a nonlinear feedback loop. Using real-time monitoring of the resulting optical lattice, we find the stability of the lattice site positions to be well below the lattice spacing for several hours. The pointing stability of the optical lattice beams is around one lattice spacing and the long-term (six month) relative stability of the lattice spacing itself is 0.5% RMS.
Title: Clockwork Axions in Cosmology: Is Chromonatural Inflation Chrononatural? Authors: Agrawal, Prateek; Fan, JiJi; Reece, Matthew Publication: eprint arXiv:1806.09621 Publication Date: 06/2018 Origin: ARXIV Keywords: High Energy Physics - Theory, Astrophysics - Cosmology and Nongalactic Astrophysics, High Energy Physics - Phenomenology Bibliographic Code: 2018arXiv180609621A
### Abstract
Many cosmological models rely on large couplings of axions to gauge fields. Examples include theories of magnetogenesis, inflation on a steep potential, chiral gravitational waves, and chromonatural inflation. Such theories require a mismatch between the axion field range and the mass scale appearing in the $a F \widetilde{F}$ coupling. This mismatch suggests an underlying monodromy, with the axion winding around its fundamental period a large number of times. We investigate the extent to which this integer can be explained as a product of smaller integers in a UV completion: in the parlance of our times, can the theory be "clockworked"? We argue that a clockwork construction producing a potential $\mu^4 \cos(\frac{a}{j F_a})$ for an axion of fundamental period $F_a$ will obey the constraint $\mu < F_a$. For some applications, including chromonatural inflation with sub-Planckian field range, this constraint obstructs a clockwork UV completion. Alternative routes to a large coupling include fields of large charge (an approach limited by strong coupling) or kinetic mixing (requiring a lighter axion). Our results suggest that completions of axion cosmologies that explain the large parameter in the theory potentially alter the phenomenological predictions of the model.
Title: Full Commuting Projector Hamiltonians of Interacting Symmetry-Protected Topological Phases of Fermions Authors: Tantivasadakarn, Nathanan; Vishwanath, Ashvin Publication: eprint arXiv:1806.09709 Publication Date: 06/2018 Origin: ARXIV Keywords: Condensed Matter - Strongly Correlated Electrons Comment: 20 pages. 8 figures Bibliographic Code: 2018arXiv180609709T
### Abstract
Using the decorated domain wall procedure, we construct Finite Depth Local Unitaries (FDLUs) that realize Fermionic Symmetry-Protected Topological (SPT) phases. This results in explicit full' commuting projector Hamiltonians, where full' implies the fact that the ground state, as well as all excited state of these Hamiltonians realize the nontrivial SPT phase. We begin by constructing explicit examples of 1+1D phases protected by symmetry groups $G=\mathbb Z_2^T \times \mathbb Z_2^F$ , which also has a free fermion realization in class BDI, and $G=\mathbb Z_4 \times \mathbb Z_4^F$, which does not. We then turn to 2+1D, and construct the square roots of the Levin-Gu bosonic SPT phase, protected by $\mathbb Z_2 \times \mathbb Z_2^F$ symmetry, in a concrete model of fermions and spins on the triangular lattice. Edge states and the anomalous symmetry action on them are explicitly derived. Although this phase has a free fermion representation, as two copies of $p+ip$ superconductors combined with their $p-ip$ counterparts with a different symmetry charge, the full set of commuting projectors is only realized in the strongly interacting version, which also implies that it admits a many-body localized realization.
Title: On the Cosmological Implications of the String Swampland Authors: Agrawal, Prateek; Obied, Georges; Steinhardt, Paul J.; Vafa, Cumrun Publication: eprint arXiv:1806.09718 Publication Date: 06/2018 Origin: ARXIV Keywords: High Energy Physics - Theory, Astrophysics - Cosmology and Nongalactic Astrophysics, General Relativity and Quantum Cosmology, High Energy Physics - Phenomenology Comment: 7 pages, 1 figure Bibliographic Code: 2018arXiv180609718A
### Abstract
We study constraints imposed by two proposed string Swampland criteria on cosmology. These criteria involve an upper bound on the range traversed by scalar fields as well as a lower bound on $|\nabla_{\phi} V|/V$ when $V >0$. We find that inflationary models are generically in tension with these two criteria. Applying these same criteria to dark energy in the present epoch, we find that specific quintessence models can satisfy these bounds and, at the same time, satisfy current observational constraints. Assuming the two Swampland criteria are valid, we argue that the universe will undergo a phase transition within a few Hubble times. These criteria sharpen the motivation for future measurements of the tensor-to-scalar ratio $r$ and the dark energy equation of state $w$, and for tests of the equivalence principle for dark matter.
Title: Probing quantum thermalization of a disordered dipolar spin ensemble with discrete time-crystalline order Authors: Choi, Joonhee; Zhou, Hengyun; Choi, Soonwon; Landig, Renate; Ho, Wen Wei; Isoya, Junichi; Jelezko, Fedor; Onoda, Shinobu; Sumiya, Hitoshi; Abanin, Dmitry A.; Lukin, Mikhail D. Publication: eprint arXiv:1806.10169 Publication Date: 06/2018 Origin: ARXIV Keywords: Quantum Physics, Condensed Matter - Disordered Systems and Neural Networks, Condensed Matter - Mesoscale and Nanoscale Physics, Condensed Matter - Statistical Mechanics, Physics - Atomic Physics Comment: 6 + 14 pages, 4 + 8 figures Bibliographic Code: 2018arXiv180610169C
### Abstract
We investigate thermalization dynamics of a driven dipolar many-body quantum system through the stability of discrete time crystalline order. Using periodic driving of electronic spin impurities in diamond, we realize different types of interactions between spins and demonstrate experimentally that the interplay of disorder, driving and interactions leads to several qualitatively distinct regimes of thermalization. For short driving periods, the observed dynamics are well described by an effective Hamiltonian which sensitively depends on interaction details. For long driving periods, the system becomes susceptible to energy exchange with the driving field and eventually enters a universal thermalizing regime, where the dynamics can be described by interaction-induced dephasing of individual spins. Our analysis reveals important differences between thermalization of long-range Ising and other dipolar spin models.
Title: D-type fiber-base duality Authors: Haghighat, Babak; Kim, Joonho; Yan, Wenbin; Yau, Shing-Tung Publication: eprint arXiv:1806.10335 Publication Date: 06/2018 Origin: ARXIV Keywords: High Energy Physics - Theory Comment: 34 pages, 5 figures Bibliographic Code: 2018arXiv180610335H
### Abstract
M5 branes probing D-type singularities give rise to 6d (1,0) SCFTs with $SO \times SO$ flavor symmetry known as D-type conformal matter theories. Gauging the diagonal $SO$-flavor symmetry leads to a little string theory with an intrinsic scale which can be engineered in F-theory by compactifying on a doubly-elliptic Calabi-Yau manifold. We derive Seiberg-Witten curves for these little string theories which can be interpreted as mirror curves for the corresponding Calabi-Yau manifolds. Under fiber-base duality these models are mapped to D-type quiver gauge theories and we check that their Seiberg-Witten curves match. By taking decompactification limits, we construct the curves for the related 6d SCFTs and connect to known results in the literature by further taking 5d and 4d limits.
Title: Measurement of the inclusive and fiducial t\bar{t} production cross-sections in the lepton+jets channel in pp collisions at √{s} = 8 TeV with the ATLAS detector Authors: Aaboud, M.; Aad, G.; Abbott, B.; Abdinov, O.;... Franklin, M.;... Huth, J.;... Morii, M.;... ; and 2894 coauthors Publication: The European Physical Journal C, Volume 78, Issue 6, article id. #487, 31 pp. (EPJC Homepage) Publication Date: 06/2018 Origin: SPRINGER Abstract Copyright: (c) 2018: CERN for the benefit of the ATLAS collaboration DOI: 10.1140/epjc/s10052-018-5904-z Bibliographic Code: 2018EPJC...78..487A
### Abstract
The inclusive and fiducial t\bar{t} production cross-sections are measured in the lepton+jets channel using 20.2 fb^{-1} of proton-proton collision data at a centre-of-mass energy of 8 TeV recorded with the ATLAS detector at the LHC. Major systematic uncertainties due to the modelling of the jet energy scale and b-tagging efficiency are constrained by separating selected events into three disjoint regions. In order to reduce systematic uncertainties in the most important background, the W {+ jets} process is modelled using Z+ jets events in a data-driven approach. The inclusive t\bar{t} cross-section is measured with a precision of 5.7% to be σ _{ {inc}}(t\bar{t}) = 248.3 ± 0.7 ({stat.}) ± 13.4 ({syst.}) ± 4.7 ({lumi.}) {pb}, assuming a top-quark mass of 172.5 GeV. The result is in agreement with the Standard Model prediction. The cross-section is also measured in a phase space close to that of the selected data. The fiducial cross-section is σ _{ {fid}}(t\bar{t}) = 48.8 ± 0.1 ({stat.}) ± 2.0 ({syst.}) ± 0.9 ({lumi.}) {pb} with a precision of 4.5%.
Title: Search for long-lived charginos based on a disappearing-track signature in pp collisions at √{s}=13 TeV with the ATLAS detector Authors: Aaboud, M.; Aad, G.; Abbott, B.; Abdinov, O.;... Franklin, M.;... Huth, J.;... Morii, M.;... ; and 2867 coauthors Publication: Journal of High Energy Physics, Volume 2018, Issue 6, article id. #22, 48 pp. Publication Date: 06/2018 Origin: SPRINGER Keywords: Hadron-Hadron scattering (experiments) Abstract Copyright: (c) 2018: The Author(s) DOI: 10.1007/JHEP06(2018)022 Bibliographic Code: 2018JHEP...06..022A
### Abstract
This paper presents a search for direct electroweak gaugino or gluino pair production with a chargino nearly mass-degenerate with a stable neutralino. It is based on an integrated luminosity of 36.1 fb-1 of pp collisions at √{s}=13 TeV collected by the ATLAS experiment at the LHC. The final state of interest is a disappearing track accompanied by at least one jet with high transverse momentum from initial-state radiation or by four jets from the gluino decay chain. The use of short track segments reconstructed from the innermost tracking layers significantly improves the sensitivity to short chargino lifetimes. The results are found to be consistent with Standard Model predictions. Exclusion limits are set at 95% confidence level on the mass of charginos and gluinos for different chargino lifetimes. For a pure wino with a lifetime of about 0.2 ns, chargino masses up to 460 GeV are excluded. For the strong production channel, gluino masses up to 1.65 TeV are excluded assuming a chargino mass of 460 GeV and lifetime of 0.2 ns.
Title: D-type conformal matter and SU/USp quivers Authors: Kim, Hee-Cheol; Razamat, Shlomo S.; Vafa, Cumrun; Zafrir, Gabi Publication: Journal of High Energy Physics, Volume 2018, Issue 6, article id. #58, 37 pp. Publication Date: 06/2018 Origin: SPRINGER Keywords: Supersymmetric Gauge Theory, Duality in Gauge Field Theories, Supersymmetry and Duality, Gauge Symmetry Abstract Copyright: (c) 2018: The Author(s) DOI: 10.1007/JHEP06(2018)058 Bibliographic Code: 2018JHEP...06..058K
### Abstract
We discuss the four dimensional models obtained by compactifying a single M5 brane probing D N singularity (minimal D-type (1 , 0) conformal matter in six dimensions) on a torus with flux for abelian subgroups of the SO(4 N) flavor symmetry. We derive the resulting quiver field theories in four dimensions by first compactifying on a circle and relating the flux to duality domain walls in five dimensions. This leads to novel N=1 dualities in 4 dimensions which arise from distinct five dimensional realizations of the circle compactifications of the D-type conformal matter.
Title: Light, the universe and everything - 12 Herculean tasks for quantum cowboys and black diamond skiers Authors: Agarwal, Girish; Allen, Roland E.; Bezděková, Iva; Boyd, Robert W.; Chen, Goong; Hanson, Ronald; Hawthorne, Dean L.; Hemmer, Philip; Kim, Moochan B.; Kocharovskaya, Olga; Lee, David M.; Lidström, Sebastian K.; Lidström, Suzy; Losert, Harald; Maier, Helmut; Neuberger, John W.; Padgett, Miles J.; Raizen, Mark; Rajendran, Surjeet; Rasel, Ernst; Schleich, Wolfgang P.; Scully, Marlan O.; Shchedrin, Gavriil; Shvets, Gennady; Sokolov, Alexei V.; Svidzinsky, Anatoly; Walsworth, Ronald L.; Weiss, Rainer; Wilczek, Frank; Willner, Alan E.; Yablonovitch, Eli; Zheludev, Nikolay Publication: Journal of Modern Optics, Volume 65, Issue 11, p.1261-1308 Publication Date: 06/2018 Origin: T+F Keywords: Quantum, optics, gravitational waves, LIGO, time crystal, nitrogen-vacancy centres, photonics, laser, free-electron laser, metrology, quantum internet, quantum computing, imaging, topological, non-linear, Bose-Einstein condensate, interferometry, Riemann hypothesis, fractal quantum carpets, Bekenstein-Hawking, equivalence principle, coherence, super-resolution, photon, maser, magnetometer, isotope separation, nanostructure, solar energy, Maxwell's demon, superradiance, sensing, Rayleigh limit, lithography, Lamb shift, Bayesian Abstract Copyright: 2018: Informa UK Limited, trading as Taylor & Francis Group DOI: 10.1080/09500340.2018.1454525 Bibliographic Code: 2018JMOp...65.1261A
### Abstract
The Winter Colloquium on the Physics of Quantum Electronics (PQE) has been a seminal force in quantum optics and related areas since 1971. It is rather mind-boggling to recognize how the concepts presented at these conferences have transformed scientific understanding and human society. In January 2017, the participants of PQE were asked to consider the equally important prospects for the future, and to formulate a set of questions representing some of the greatest aspirations in this broad field. The result is this multi-authored paper, in which many of the world's leading experts address the following fundamental questions: (1) What is the future of gravitational wave astronomy? (2) Are there new quantum phases of matter away from equilibrium that can be found and exploited - such as the time crystal? (3) Quantum theory in uncharted territory: What can we learn? (4) What are the ultimate limits for laser photon energies? (5) What are the ultimate limits to temporal, spatial and optical resolution? (6) What novel roles will atoms play in technology? (7) What applications lie ahead for nitrogen-vacancy centres in diamond? (8) What is the future of quantum coherence, squeezing and entanglement for enhanced super-resolution and sensing? (9) How can we solve (some of) humanity's biggest problems through new quantum technologies? (10) What new understanding of materials and biological molecules will result from their dynamical characterization with free-electron lasers? (11) What new technologies and fundamental discoveries might quantum optics achieve by the end of this century? (12) What novel topological structures can be created and employed in quantum optics?
Title: Metallic hydrogen Authors: Silvera, Isaac F.; Dias, Ranga Publication: Journal of Physics: Condensed Matter, Volume 30, Issue 25, article id. 254003 (2018). Publication Date: 06/2018 Origin: IOP DOI: 10.1088/1361-648X/aac401 Bibliographic Code: 2018JPCM...30y4003S
### Abstract
Hydrogen is the simplest and most abundant element in the Universe. There are two pathways for creating metallic hydrogen under high pressures. Over 80 years ago Wigner and Huntington predicted that if solid molecular hydrogen was sufficiently compressed in the T = 0 K limit, molecules would dissociate to form atomic metallic hydrogen (MH). We have observed this transition at a pressure of 4.95 megabars. MH in this form has probably never existed on Earth or in the Universe; it may be a room temperature superconductor and is predicted to be metastable. If metastable it will have an important technological impact. Liquid metallic hydrogen can also be produced at intermediate pressures and high temperatures and is believed to make up ~90% of the planet Jupiter. We have observed this liquid–liquid transition, also known as the plasma phase transition, at pressures of ~1–2 megabar and temperatures ~1000–2000 K. However, in this paper we shall focus on the Wigner–Huntington transition. We shall discuss the methods used to observe metallic hydrogen at extreme conditions of static pressure in the laboratory, extending our understanding of the phase diagram of the simplest atom in the periodic table.
Title: Geometric constraints during epithelial jamming Authors: Atia, Lior; Bi, Dapeng; Sharma, Yasha; Mitchel, Jennifer A.; Gweon, Bomi; Koehler, Stephan A.; DeCamp, Stephen J.; Lan, Bo; Kim, Jae Hun; Hirsch, Rebecca; Pegoraro, Adrian F.; Lee, Kyu Ha; Starr, Jacqueline R.; Weitz, David A.; Martin, Adam C.; Park, Jin-Ah; Butler, James P.; Fredberg, Jeffrey J. Publication: Nature Physics, Volume 14, Issue 6, p.613-620 Publication Date: 06/2018 Origin: NATURE Abstract Copyright: 2018: The Author(s) DOI: 10.1038/s41567-018-0089-9 Bibliographic Code: 2018NatPh..14..613A
### Abstract
As an injury heals, an embryo develops or a carcinoma spreads, epithelial cells systematically change their shape. In each of these processes cell shape is studied extensively whereas variability of shape from cell to cell is regarded most often as biological noise. But where do cell shape and its variability come from? Here we report that cell shape and shape variability are mutually constrained through a relationship that is purely geometrical. That relationship is shown to govern processes as diverse as maturation of the pseudostratified bronchial epithelial layer cultured from non-asthmatic or asthmatic donors, and formation of the ventral furrow in the Drosophila embryo. Across these and other epithelial systems, shape variability collapses to a family of distributions that is common to all. That distribution, in turn, is accounted for by a mechanistic theory of cell-cell interaction, showing that cell shape becomes progressively less elongated and less variable as the layer becomes progressively more jammed. These findings suggest a connection between jamming and geometry that spans living organisms and inert jammed systems, and thus transcends system details. Although molecular events are needed for any complete theory of cell shape and cell packing, observations point to the hypothesis that jamming behaviour at larger scales of organization sets overriding geometric constraints.
Title: Publisher Correction: Geometric constraints during epithelial jamming Authors: Atia, Lior; Bi, Dapeng; Sharma, Yasha; Mitchel, Jennifer A.; Gweon, Bomi; Koehler, Stephan A.; DeCamp, Stephen J.; Lan, Bo; Kim, Jae Hun; Hirsch, Rebecca; Pegoraro, Adrian F.; Lee, Kyu Ha; Starr, Jacqueline R.; Weitz, David A.; Martin, Adam C.; Park, Jin-Ah; Butler, James P.; Fredberg, Jeffrey J. Publication: Nature Physics, Volume 14, Issue 6, p.629-629 Publication Date: 06/2018 Origin: NATURE Abstract Copyright: 2018: The Publisher DOI: 10.1038/s41567-018-0139-3 Bibliographic Code: 2018NatPh..14Q.629A
### Abstract
In the version of this Article originally published, the Supplementary Movies were linked to the wrong descriptions. These have now been corrected. Additionally, the authors would like to note that co-authors James P. Butler and Jeffrey J. Fredberg contributed equally to this Article; this change has now been made.
Title: Author Correction: Geometric constraints during epithelial jamming Authors: Atia, Lior; Bi, Dapeng; Sharma, Yasha; Mitchel, Jennifer A.; Gweon, Bomi; Koehler, Stephan A.; DeCamp, Stephen J.; Lan, Bo; Kim, Jae Hun; Hirsch, Rebecca; Pegoraro, Adrian F.; Lee, Kyu Ha; Starr, Jacqueline R.; Weitz, David A.; Martin, Adam C.; Park, Jin-Ah; Butler, James P.; Fredberg, Jeffrey J. Publication: Nature Physics, Volume 14, Issue 6, p.629-629 Publication Date: 06/2018 Origin: NATURE Abstract Copyright: 2018: The Publisher DOI: 10.1038/s41567-018-0168-y Bibliographic Code: 2018NatPh..14R.629A
### Abstract
In the first correction to this Article, the authors added James P. Butler and Jeffrey J. Fredburg as equally contributing authors. However, this was in error; the statement should have remained indicating that Lior Atia, Dapeng Bi and Yasha Sharma contributed equally. This has now been corrected.
Title: Charge Diffusion Variations in Pan-STARRS1 CCDs Authors: Magnier, Eugene A.; Tonry, J. L.; Finkbeiner, D.; Schlafly, E.; Burgett, W. S.; Chambers, K. C.; Flewelling, H. A.; Hodapp, K. W.; Kaiser, N.; Kudritzki, R.-P.; Metcalfe, N.; Wainscoat, R. J.; Waters, C. Z. Publication: Publications of the Astronomical Society of the Pacific, Volume 130, Issue 988, pp. 065002 (2018). (PASP Homepage) Publication Date: 06/2018 Origin: IOP DOI: 10.1088/1538-3873/aaaad8 Bibliographic Code: 2018PASP..130f5002M
### Abstract
Thick back-illuminated deep-depletion CCDs have superior quantum efficiency over previous generations of thinned and traditional thick CCDs. As a result, they are being used for wide-field imaging cameras in several major projects. We use observations from the Pan-STARRS 3π survey to characterize the behavior of the deep-depletion devices used in the Pan-STARRS 1 Gigapixel Camera. We have identified systematic spatial variations in the photometric measurements and stellar profiles that are similar in pattern to the so-called “tree rings” identified in devices used by other wide-field cameras (e.g., DECam and Hypersuprime Camera). The tree-ring features identified in these other cameras result from lateral electric fields that displace the electrons as they are transported in the silicon to the pixel location. In contrast, we show that the photometric and morphological modifications observed in the GPC1 detectors are caused by variations in the vertical charge transportation rate and resulting charge diffusion variations.
Title: Measurement of the production cross section of three isolated photons in pp collisions at √{ s } = 8 TeV using the ATLAS detector Authors: Aaboud, M.; Aad, G.; Abbott, B.; Abdinov, O.;... Franklin, M.;... Huth, J.;... Morii, M.;... ; and 2843 coauthors Publication: Physics Letters B, Volume 781, p. 55-76. Publication Date: 06/2018 Origin: ELSEVIER Abstract Copyright: (c) 2018 Elsevier Science B.V. All rights reserved. DOI: 10.1016/j.physletb.2018.03.057 Bibliographic Code: 2018PhLB..781...55A
### Abstract
A measurement of the production of three isolated photons in proton-proton collisions at a centre-of-mass energy √{ s } = 8 TeV is reported. The results are based on an integrated luminosity of 20.2 fb-1 collected with the ATLAS detector at the LHC. The differential cross sections are measured as functions of the transverse energy of each photon, the difference in azimuthal angle and in pseudorapidity between pairs of photons, the invariant mass of pairs of photons, and the invariant mass of the triphoton system. A measurement of the inclusive fiducial cross section is also reported. Next-to-leading-order perturbative QCD predictions are compared to the cross-section measurements. The predictions underestimate the measurement of the inclusive fiducial cross section and the differential measurements at low photon transverse energies and invariant masses. They provide adequate descriptions of the measurements at high values of the photon transverse energies, invariant mass of pairs of photons, and invariant mass of the triphoton system.
Title: Search for W‧ → tb decays in the hadronic final state using pp collisions at √{ s } = 13TeV with the ATLAS detector Authors: Aaboud, M.; Aad, G.; Abbott, B.; Abdinov, O.;... Franklin, M.;... Huth, J.;... Morii, M.;... ; and 2891 coauthors Publication: Physics Letters B, Volume 781, p. 327-348. Publication Date: 06/2018 Origin: ELSEVIER Abstract Copyright: (c) 2018 Elsevier Science B.V. All rights reserved. DOI: 10.1016/j.physletb.2018.03.036 Bibliographic Code: 2018PhLB..781..327A
### Abstract
A search for W‧-boson production in the W‧ → t b bar → qqbar‧ b b bar decay channel is presented using 36.1fb-1 of 13 TeV proton-proton collision data collected by the ATLAS detector at the Large Hadron Collider in 2015 and 2016. The search is interpreted in terms of both a left-handed and a right-handed chiral W‧ boson within the mass range 1-5 TeV. Identification of the hadronically decaying top quark is performed using jet substructure tagging techniques based on a shower deconstruction algorithm. No significant deviation from the Standard Model prediction is observed and the results are expressed as upper limits on the W‧ → t b bar production cross-section times branching ratio as a function of the W‧-boson mass. These limits exclude W‧ bosons with right-handed couplings with masses below 3.0 TeV and W‧ bosons with left-handed couplings with masses below 2.9 TeV, at the 95% confidence level.
Title: Coupling two spin qubits with a high-impedance resonator Authors: Harvey, S. P.; Bøttcher, C. G. L.; Orona, L. A.; Bartlett, S. D.; Doherty, A. C.; Yacoby, A. Publication: Physical Review B, Volume 97, Issue 23, id.235409 (PhRvB Homepage) Publication Date: 06/2018 Origin: APS Abstract Copyright: 2018: American Physical Society DOI: 10.1103/PhysRevB.97.235409 Bibliographic Code: 2018PhRvB..97w5409H
### Abstract
Fast, high-fidelity single and two-qubit gates are essential to building a viable quantum information processor, but achieving both in the same system has proved challenging for spin qubits. We propose and analyze an approach to perform a long-distance two-qubit controlled phase (CPHASE) gate between two singlet-triplet qubits using an electromagnetic resonator to mediate their interaction. The qubits couple longitudinally to the resonator, and by driving the qubits near the resonator's frequency, they can be made to acquire a state-dependent geometric phase that leads to a CPHASE gate independent of the initial state of the resonator. Using high impedance resonators enables gate times of order 10 ns while maintaining long coherence times. Simulations show average gate fidelities of over 96% using currently achievable experimental parameters and over 99% using state-of-the-art resonator technology. After optimizing the gate fidelity in terms of parameters tuneable in situ, we find it takes a simple power-law form in terms of the resonator's impedance and quality and the qubits' noise bath.
Title: Infinite family of three-dimensional Floquet topological paramagnets Authors: Potter, Andrew C.; Vishwanath, Ashvin; Fidkowski, Lukasz Publication: Physical Review B, Volume 97, Issue 24, id.245106 (PhRvB Homepage) Publication Date: 06/2018 Origin: APS Abstract Copyright: 2018: American Physical Society DOI: 10.1103/PhysRevB.97.245106 Bibliographic Code: 2018PhRvB..97x5106P
### Abstract
We uncover an infinite family of time-reversal symmetric 3 d interacting topological insulators of bosons or spins, in time-periodically driven systems, which we term Floquet topological paramagnets (FTPMs). These FTPM phases exhibit intrinsically dynamical properties that could not occur in thermal equilibrium and are governed by an infinite set of Z2-valued topological invariants, one for each prime number. The topological invariants are physically characterized by surface magnetic domain walls that act as unidirectional quantum channels, transferring quantized packets of information during each driving period. We construct exactly solvable models realizing each of these phases, and discuss the anomalous dynamics of their topologically protected surface states. Unlike previous encountered examples of Floquet SPT phases, these 3 d FTPMs are not captured by group cohomology methods and cannot be obtained from equilibrium classifications simply by treating the discrete time translation as an ordinary symmetry. The simplest such FTPM phase can feature anomalous Z2 (toric code) surface topological order, in which the gauge electric and magnetic excitations are exchanged in each Floquet period, which cannot occur in a pure 2 d system without breaking time reversal symmetry.
Title: Search for squarks and gluinos in final states with jets and missing transverse momentum using 36 fb-1 of √{s }=13 TeV p p collision data with the ATLAS detector Authors: Aaboud, M.; Aad, G.; Abbott, B.; Abdinov, O.;... Franklin, M.;... Huth, J.;... Morii, M.;... ; and 2869 coauthors Publication: Physical Review D, Volume 97, Issue 11, id.112001 (PhRvD Homepage) Publication Date: 06/2018 Origin: APS Abstract Copyright: 2018: CERN DOI: 10.1103/PhysRevD.97.112001 Bibliographic Code: 2018PhRvD..97k2001A
### Abstract
A search for the supersymmetric partners of quarks and gluons (squarks and gluinos) in final states containing hadronic jets and missing transverse momentum, but no electrons or muons, is presented. The data used in this search were recorded in 2015 and 2016 by the ATLAS experiment in √{s }=13 TeV proton-proton collisions at the Large Hadron Collider, corresponding to an integrated luminosity of 36.1 fb-1. The results are interpreted in the context of various models where squarks and gluinos are pair produced and the neutralino is the lightest supersymmetric particle. An exclusion limit at the 95% confidence level on the mass of the gluino is set at 2.03 TeV for a simplified model incorporating only a gluino and the lightest neutralino, assuming the lightest neutralino is massless. For a simplified model involving the strong production of mass-degenerate first- and second-generation squarks, squark masses below 1.55 TeV are excluded if the lightest neutralino is massless. These limits substantially extend the region of supersymmetric parameter space previously excluded by searches with the ATLAS detector.
Title: Differential Activity-Driven Instabilities in Biphasic Active Matter Authors: Weber, Christoph A.; Rycroft, Chris H.; Mahadevan, L. Publication: Physical Review Letters, Volume 120, Issue 24, id.248003 (PhRvL Homepage) Publication Date: 06/2018 Origin: APS Abstract Copyright: 2018: American Physical Society DOI: 10.1103/PhysRevLett.120.248003 Bibliographic Code: 2018PhRvL.120x8003W
### Abstract
Active stresses can cause instabilities in contractile gels and living tissues. Here we provide a generic hydrodynamic theory that treats these systems as a mixture of two phases of varying activity and different mechanical properties. We find that differential activity between the phases causes a uniform mixture to undergo a demixing instability. We follow the nonlinear evolution of the instability and characterize a phase diagram of the resulting patterns. Our study complements other instability mechanisms in mixtures driven by differential adhesion, differential diffusion, differential growth, and differential motion.
Title: Sizing Up the Top Quark's Interaction with the Higgs Authors: Reece, Matthew Publication: Physics, Volume 11, id. 56 Publication Date: 06/2018 Origin: APS Abstract Copyright: 2018: American Physical Society DOI: 10.1103/Physics.11.56 Bibliographic Code: 2018PhyOJ..11...56R
### Abstract
A proton collision experiment at CERN provides a new handle on the Higgs boson's interaction with the heaviest of the quarks.
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https://brilliant.org/practice/area-of-triangles-herons-formula/ | ×
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Area of Triangles
You know that Area = base x height / 2, but what other ways are there to find the area of a triangle? Brace yourself for some potent formulae.
Area of Triangles - Heron's Formula
What is the area of a triangle with side lengths $$9,12,$$ and $$15?$$
What is the area of a triangle with side lengths $$13, 14,$$ and $$15?$$
What is the area of a triangle with side lengths $$9, 11,$$ and $$10?$$
A triangle has perimeter 14 and area $$2\sqrt{14}.$$ If the shortest side has length 3, find the positive difference between the lengths of other two sides. Give your answer to 3 decimal places.
What is the area of a triangle with side lengths $$3,5,$$ and $$7?$$
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http://mokalibri.it/r-plot-rectangle.html | # R Plot Rectangle
Continuous Uniform Distribution : S2 Edexcel January 2012 Q1 : ExamSolutions Maths Revision - youtube Video. r = √ (x² + y² + z²) θ = arccos (z/r) φ = arctan (y/x) (x, y, z) are the Cartesian coordinates, and. nd points Q and R are symmetric about side. A modified augmented design (type 2) is presented for the situation where subplots are long and narrow (rectangular plots). Many maps, such as the Campus Map shown below, use a grid system to identify locations. Find the rectangular coordinates of the point. ) Analysis must include first three modes (TE10, TE20, TE01) 3. The sampling interval is controlled via MAP_LINE_STEP parameter. A plot which has South and west apartments which is considered to be fantastic. (Note that a proof are perpendicular, then the The coordinates of three vertices of a rhombus are given, not necessarily in order. In this book, you will find a practicum of skills for data science. To understand ourselves it is important to understand our ancestors, and a part of them, and their heritage, lives on in us today. The position vectors will be recycled to the length of the longest. Let’s look at how to do that with the plain plotting tools. Get Answer to The following numbers represent 100 random numbers drawn from a rectangular population with a mean of 4. Specify pos as a four-element vector of the form [x y w h] in data units. fffffffffff? (A) 6 (B) 8 (C) 10 (D) 12 (E) 20. A(0, -3), B(-4, 0), C(2, 8), D(6, 5) Step 1: Plot the points to get a visual idea of what you are working with. More complex shapes have a greater number of points to plot. Plot and label each set of points in the coordinate plane. We also have a quick-reference cheatsheet (new!) to help you get started!. a vector (or scalar) of top y positions. A rectangle is formed of four sides in which opposite sides are parallel and equal and each interior angle in 90 degrees. The ordered pair specifies a point’s location based on the value of r and the angle, θ, from the polar axis. To get the position of the mouse when a plot is clicked, you simply need to. 送料無料【タクボ物置】当店なら開梱設置、組立工事可能。物置＋駐輪スペース。タクボ物置 tp／ストックマンプラスアルファ tp-s43r15 多雪型 標準屋根 『追加金額で工事も可能』 『駐輪スペース付 屋外用 物置 自転車収納 におすすめ』 トロピカルオレンジ. An area chart displays a solid color between the traces of a graph. Default is TRUE, in order to allow the rect to be below the labels. 50 per metre is Rs. Making statements based on opinion; back them up with references or personal experience. - 2kms away from kollam bypass. Figure 3: Frequency Polygon in R. 30 per m 2. The function plots into the current axes without clearing existing content from the axes. Complex Numbers in Polar Form; DeMoivre’s Theorem. An S3 object of class element, rel, or margin. Prima Facie ('at first sight'). Looks good. They are from open source Python projects. As a point of terminology, square matrices are all rectangular too; your request includes oblong matrices in which the number of variables differs as between rows and columns. Increase the line size so that they form a compact shaded area. Learn how to use Resources. Each of the four vertices (corners) have known coordinates. Plotting Points on the Polar Coordinate System a. The four standard assumptions about the residuals of a linear regression model:. How the Box Plot Builder Works. Given the Z height values on a (X,Y) grid, we can draw the perspective plots of this surface over the (X,Y) plane. 5300, what is the length of the plot in metres? a) 40m b) 50m c) 120m d) Data inadequate e) None of these. To obtain very accurate graphs, technology is a great aid. A few columns with formulas are added in your workbook, to provide the data for the box plotchart. Drawing inside plots. box: Draw a Box around a Plot Description Usage Arguments Details References See Also Examples Description. circles(x,y,r) plots circle of radius r centered at points (x,y). If you only have 4 GBs of RAM you cannot put 5 GBs of data 'into R'. Package 'shape' February 7, 2018 Version 1. So, if a rectangle has side measurements as x, y, x, y centimeters where x>y. In my calculator, I have a function to convert rectangular to polar and vice versa. Create a maze using divide-and-conquer: Begin with a rectangular region with no walls. Packages designed for out-of-memory processes such as ff may help you. The two numbers that are used to define a point on a graph using rectangular coordinates are the coordinate values along the horizontal and vertical axes. The arrangement of the whole plots is the same as in the type 1 modified. {"code":200,"message":"ok","data":{"html":". Thanks for contributing an answer to Code Review Stack Exchange! Please be sure to answer the question. You can do this in various ways. 5 Non-rectangular data - single worksheet 7. Area is a quantity that describes the size or extent of a two-dimensional figure or shape in a plane. Now you can proceed to adding the first rectangular region vp. Parts (a) and (b):. 4 Title Functions for Plotting Graphical Shapes, Colors Author Karline Soetaert Maintainer Karline Soetaert Depends R (>= 2. While R’s traditional graphics offers a nice set of plots, some of them require a lot of work. A simple box plot can be created in R with the boxplot function. 7 feet on a side will cover an acre. Add Grid to a Plot Usage grid(nx = 3, ny = 3, col = "lightgray", lty = "dotted") Description grid adds an nx by ny rectangular grid to an existing plot, using lines of type lty and color col. , xleft, , are relative to the current plotting region. Convert polar to rectangular two dimensional coordinates using a calculator. Convert among different-sized standard measurement units within a given measurement system (e. If the cost of fencing the plot @ 26. Plot rectangle boxes on image using MATLAB. The following is an example of a matrix with 2 rows and 3 columns. This is a sample of the data: [1,]. Initial scope is to develop tools to generate forest plots for inclusion in submissions to the FDA. For more details about the graphical parameter arguments, see par. Find the area of the remaining plot. packages ("tidyverse") Learn the tidyverse. Plotly's R library is free and open source! Get started by downloading the client and reading the primer. 1*exp(t/3). So the biggest rectangle in my plot represents that there are more people who are female, non-alcoholic,. using the. The expected value of a uniform random variable is. Given two rectangles, find if the given two rectangles overlap or not. , xleft, , are relative to the current plotting region. But however you set your calculator to display results, you can always enter expressions in rectangular form , polar form or a mixture. polar and rectangular coordinates of a point. Here, Gender 1 = female, 2 = male; Alcohol 1 = yes, 2 = no; Cigarette 1 = yes, 2 = no. Click another point on the 3D view that is opposite the first one, or type a coordinate and press the add point button. In the diagram given below, AB is the diameter of the semi-circular field with center at O. (3) Then we need to add the data series we want to plot. Draws rectangles around the branches of a dendrogram highlighting the corresponding clusters. Rectangular-Coordinate. angle (in degrees) of the shading lines. Package 'shape' February 7, 2018 Version 1. The rectangular. Use MathJax to format equations. Drawing inside plots. 7) ( , ) 8) ( , ) Convert each pair of rectangular coordinates to polar coordinates where r and. Measure the plot of land. Presumptive Evidence. Select From Local File, and then Browse to select the file. Instead, we determine the geographical coordinates of the polygon outline for the (possibly oblique) rectangular map domain. In polar coordinates, these ordered pairs take the form $$( r, \theta )$$. To obtain very accurate graphs, technology is a great aid. ) Analysis must include first three modes (TE10, TE20, TE01) 3. The comparison of windowing methods Kaiser window and rectangular window result is shown at the last for two different beta values, as the beta function in Kaiser window plays a major role which attenuates the gain. Click a first corner point on the 3D view, or type a coordinate and press the add point button. " The process uses a weighted average of an input pixel and its neighbors to calculate an output pixel. 【ポイント10倍】 a. They are mostly standard functions written as you might expect. The bty parameter determines the type of box drawn. Let's begin with an easy example. They are from open source Python projects. Thanks for contributing an answer to Code Review Stack Exchange! Please be sure to answer the question. Complex Numbers in Polar Form; DeMoivre’s Theorem. House Assignment. Among other things, it gives us an easier way to produce plots at specified locations of the plotting region. ) Sweep from 4-20 GHz 2. Draws a simple map scale centered on the reference point specified using one of four coordinate systems: (1) Use -Lg for map (user) coordinates, (2) use -Lj or -LJ for setting refpoint via a 2-char justification code that refers to the (invisible) map domain rectangle, (3) use -Ln for normalized (0-1) coordinates, or (4) use -Lx for plot. io Find an R package R language docs Run R in your browser R Notebooks. meshgrid with a Python iterator, but I can't wrap my head around it. Thanks for contributing an answer to Geographic Information Systems Stack Exchange! Please be sure to answer the question. R-Help Is there a way to make a rectangle transparent (alpha=0. Rectangle (). Now you can proceed to adding the first rectangular region vp. Rectangle: A (2, 3), B (2, 10), C (6, 10), D (6, 3). xmin - (required) left edge of rectangle. Henceforth we will call the graphs described in section 6. (1) Create a dummy data for the area range where it need to be shaded. This time, instead of seeing “y=” for each equation, it prompts for each equation in terms of r. 1 to the ViewPort tree, after which you tell R with gridFig() to draw a base plot within a grid viewport (vp. and the 1-D coordinate system is denoted by R. Value pch=". So let's make a rule here that we're going to get r to be greater than or equal to 0 and theta between 0 and 2 pi. Postmortem Lividity. So, if a rectangle has side measurements as x, y, x, y centimeters where x>y. Then make a plot of this solid. Riemann sums in Matlab again so we can learn how to draw a picture that represents the Riemann sum. How to visualise scatter plots with rectangular bins in R. circles(x,y,r) plots circle of radius r centered at points (x,y). Here’s another set of common color schemes used in R, this time via the image() function. The simplest way to plot a legend outside a figure in R is to: In this example, I am going to make a multi-panel figure, with a horizontal legend on the bottom of the plot. In this example, we are going to draw a simple square polygon to an. To change a rectangular equation to a polar equation just replace x with. plot the graph of the function as a surface over a rectangle in the x,y plane: ezsurf(G,[-2,2,-2,2]) Click on in the figure toolbar, then you can rotate the graph by dragging with the mouse. The fencing for the east and west sides of the plot will cost 3$per yard, but she needs to use special fencing which will cost 5$ per yard on the south side of the plot. The rectangle method (also called the midpoint rule) is the simplest method in Mathematics used to compute an approximation of a definite integral. For rectangular contour plots, the data can be either in a matrix or in a worksheet in XYZ format. Converting spherical to Cartesian. An area chart displays a solid color between the traces of a graph. So I extracted the boundaries from DBSCAN in python and am now working on R to create visualization in GGPLOT and LEAFLET but I can't get the polygons to plot. It's counter-intuitive, but this is actually really easy with the rectangle function. So if you’re plotting multiple groups of things, it’s natural to plot them using colors 1, 2, and 3. I need to plot a rectangular waveform with Latex. h = notch height. March 22, 1987. Create a maze using divide-and-conquer: Begin with a rectangular region with no walls. Unfortunately, it doesn't really work for me. True BASIC is similar to F (a subset of Fortran 90) and has excellent graphics capabilities which are hardware independent. A fenced enclosure consists of a rectangle of length L and width 2R, and a semicircle of radius R, as shown in Figure P20. Triangulation. Edexcel Statistics S2 June 2011 Q4b : ExamSolutions - youtube Video. js can add various shapes such as circles, polygons, rectangles, polylines, points or markers etc. The x and y elements determine the location and the w and h elements determine the size. If we want to move the legend out of the main plot area, we need some more work. The default set up is shown in figure 3. Default statistic: stat_identity. Simple Plot: Plot ordered pairs of numbers, either as a scatter plot or with the dots connected. 3D surface plots. Link it to tooltip. h = plot(___) returns a Polygon graphics object for either of the previous syntaxes. - 4 meter front road. 1) Convert from Polar Coordinates to Rectangular Coordinates 7) 5, - 4π 3 7) Objective: (9. border: Vector with border colors for the rectangles. Hello everyone, if I have: x=1:10 y=1:10 plot(x,y) and I plot a rectangle rect(4,0,6,11, col=5) it covers the points of the graph. Solved: convert polar equation to rectangular form: r= 6sin(theta) - Slader. - Mayou Mar 17 '14 at 19:55. Polar coordinates are an alternative to rectangular coordinates for referring to points in the plane. First step, make the outer margin at the bottom of the plot large: Third step: overlay the entire figure region with a new, single plot. (Note that a proof are perpendicular, then the The coordinates of three vertices of a rhombus are given, not necessarily in order. The following are code examples for showing how to use matplotlib. Press the Draft Rectangle button, or press R then E keys. Plotting Points on the Polar Coordinate System a. type for another with no loss of meaning. To do this, configure the plot with “extra” named ranges in the extra_x_range and extra_y_range properties. Usage traceplot(x, smooth = TRUE, col, type, ylab, ) Arguments. Ternary contour plots can be generated from worksheet data organized in X Y Z Z format where the 2nd Z-column contains the 4th. Plot Points. Continuous Uniform Distribution : S2 Edexcel January 2012 Q1 : ExamSolutions Maths Revision - youtube Video. Each provides a geom, a set of aesthetic mappings, and a default stat and position adjustment. The following diagram illustrates the. Introduction to Applied Machine Learning & Data Science for Beginners, Business Analysts, Students, Researchers and Freelancers with Python & R Codes @ Western Australian Center for Applied Machine Learning & Data Science (WACAMLDS)!!!. Fourier analysis provides the mathematical mechanism for transforming frequency sweep data to a time-domain plot, but two approximations are involved. If we want to graph the Riemann sums, we need to graph those rectangles we draw by hand. Most graphing calculators can plot polar functions; in the menu, set the plotting mode to something like polar or POL, depending on one's calculator. How to display a legend outside a R plot April 30, 2014 - how-to, R If you still don’t use ggplot2 or, as I do, have to use the old and finicky plot() function, read on to discover a trick I use to display a legend outside the plotting area. $$r:$$ distance from. a vector (or scalar) of bottom y positions. The plot is produced from two simple R expressions: one expression to draw the basic plot, consisting of axes, data symbols, and bounding rectangle; and another expression to add the text label within the plot. plot() parameters. Rectangular plot is marvel. In a typical box plot, the top of the rectangle indicates the third quartile, a horizontal line near the middle of the rectangle indicates the median, and the bottom of the rectangle indicates the first quartile. In the simplest box plot the central rectangle spans the first quartile to the third quartile (the interquartile range or IQR). You can alternatively look at the 'Large memory and out-of-memory data' section of the High Perfomance Computing task view in R. Use operations on fractions for this grade to solve. 7) ( , ) 8) ( , ) Convert each pair of rectangular coordinates to polar coordinates where r and. First the dendrogram is cut at a certain level, then a rectangle is drawn around selected branches. Click another point on the 3D view that is opposite the first one, or type a coordinate and press the add point button. R-Square” and “Pearson’s r” (their formulas are not the typical ones for the correlation. ggplot2's geom_rect() layer makes it easy to highlight portions of your graph, such as recessions on an economic time series. newis called it sets up a default plot region with margins on each side large enough to contain a minimal amount of annotation (x and y axes, axis labelling and and overall plot title). This example shows how to find and plot the positions of the elements of a 5-row-by-6-column URA with a triangular lattice and a URA with a rectangular lattice. Many maps, such as the Campus Map shown below, use a grid system to identify locations. This function draws a box around the current plot in the given color and linetype. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. This will open a new graphics window if there is none open, otherwise an existing window is readied to hold the new plot. consists of two real number lines that intersect at a right angle. (-4,600) d. Here’s another set of common color schemes used in R, this time via the image() function. - Mayou Mar 17 '14 at 19:55. GeoGebra Team German. Find the percentage change in the cost of the plot assuming land 27993 prices are uniform throughout his plot. Know how to interpret points with the rectangular coordinate system (rect system) 2. The reason is simple. 1*exp(t/3). To find the polar angle t, you have to take into account the sings of x and y which gives you the quadrant. Press the Draft Rectangle button, or press R then E keys. xmin, xmax, ymin, ymax are all optional -- the rectangle will go to the end of the plot area if they are not specified. In this book, you will find a practicum of skills for data science. Sometimes, we also say that it has a rectangular distribution or that it is a rectangular random variable. Here's another set of common color schemes used in R, this time via the image() function. ymin - (required) bottom edge of rectangle. In the example below, data from the sample "chickwts" dataset is used to plot the the weight of chickens as a function of feed type. An area chart displays a solid color between the traces of a graph. A rectangle has two diagonals, which are line segments linking opposite vertices (corners) of the rectangle. I So the command. The positions supplied, i. The plot (f, x=x0. rect draws a rectangle (or sequence of rectangles) with the given coordinates, fill and border colors. nd points Q and R are symmetric about side. A rectangle has vertices at (-1, 6), (-1-2), (3, 6), and (3, -2). - 4 meter front road. Is there something like this in excel? Nothing built-in but the formulae are straightforward. r P = 1r, u2 u = p 4. If x = O or y=O, use your illustration to find (rm. LIONEL NEBEKER. There are many options available in R for this. An S3 object of class element, rel, or margin. We will consider version 3. In R, the color black is denoted by col = 1 in most plotting functions, red is denoted by col = 2, and green is denoted by col = 3. Use add data function to add two data series. Residual Analysis. The two axes meet at a point called the origin. When given a set of polar coordinates, we may need to convert them to rectangular coordinates. Luckily, its pretty easy to plot confidence bounds as filled patches. I found: x^2+y^2=9 Consider the following diagram: We can see that the relationships between rectangular and polar coordinates are: r=sqrt(x^2+y^2) theta=arctan(y/x) and: x=rcos(theta) y=rsin(theta) Given our expression: r=3 using our relationships of conversion. grid) # define a SpatialPixelsDataFrame from the data mat = SpatialPixelsDataFrame (points = meuse. (a) 1,5 π 4 (b) (2,3π) (c) 2,−2 π 3 (d) −3,3 π 4 Solution The points are plotted in Figure 3. 2) Evaluate. Use methods (plot) and the documentation for these. Newest Resources. Step 2: Multiply each points by the dilation factor. Package 'mapplots' May 22, 2018 Type Package Title Data Visualisation on Maps Version 1. See also clip. Customized multiple plots can be produced more easily using grid. The bty parameter determines the type of box drawn. " The process uses a weighted average of an input pixel and its neighbors to calculate an output pixel. The rectangular coordinate system (or Cartesian plane) provides a means of mapping points to ordered pairs and ordered pairs to points. We also know that diagonals of parallelogram bisect each other, so length of diagonal np will be two times pr. The positions supplied, i. xmin, xmax, ymin, ymax are all optional -- the rectangle will go to the end of the plot area if they are not specified. please give me coding commands to plot a radiation pattern. default will be used. Sign up to join this community. h spe_stdd_le_raw D$R_PARISC_LTOFF16DF (NoStack) KPXquotedblleftAdieresis-10 Allowcommands:%d F*HZh base_offset:0x%02x ={m! sdp_record_free KPXFedieresis-75 __version__='5. Each side of a rectangle joins with two right angles. To sketch the graph of a polar equation a good first step is to sketch the graph in the Cartesian coordinate system. 0, Shiny has built-in support for interacting with static plots generated by R’s base graphics functions, and those generated by ggplot2. In the simplest box plot the central rectangle spans the first quartile to the third quartile (the interquartile range or IQR). (r, θ, φ) are the spherical coordinates. Package 'shape' February 7, 2018 Version 1. , xleft, , are relative to the current plotting region. 7 and r to 1 will create a sort of doughnut with a 0. using angular frequency ω, where is the unnormalized form of the sinc function. You can also use "pi" and "e" as their respective constants. # Hollow shapes ggplot(df, aes(x=xval, y=yval, group = cond)) + geom. The plot to the right shows how the length of a side (sqrt(A/r)) and L* (area/perimeter) vary with aspect-ratio for a rectangular plate with an area of 1 m 2. 1??) plot(c(100, 200), c(300, 450), type= "n", xlab = "", ylab = "") rect(110. To estimate the dial gage reading corresponding to 0% consolidation (R 0) you can make use of the fact that the plot of U as a function of T is parabolic for U < 60%. Unfortunately, it doesn't really work for me. r/エーディーエスアール tobby/トビー 05 正規品 サングラス 2016モデル 【送料無料】. angle (in degrees) of the shading lines. Compass Point. The ordered pair specifies a point’s location based on the value of r and the angle, θ, from the polar axis. The tidyverse is an opinionated collection of R packages designed for data science. So mainly we are given following four coordinates. Choose a random gridpoint in the rectangle and construct two perpendicular walls, dividing the square into 4 subregions. Plot these points and find the. Click another point on the 3D view that is opposite the first one, or type a coordinate and press the add point button. We will consider version 3. The information about how r changes with θ can then be used to sketch the graph of the equation in the polar coordinate system. Plotting Methods for the Crime Scene. Complex Numbers in Rectangular and Polar Form To represent complex numbers x yi geometrically, we use the rectangular coordinate system with the horizontal axis representing the real part and the vertical axis representing the imaginary part of the complex number. F = applied force (tensile or compressive). A segment inside the rectangle shows the median and "whiskers" above and below the box show the locations of the minimum and maximum. This worksheet provides the student with four series of points, with their task being to plot them on the graph and connect the dots, revealing four different shapes. R-Square” and “Pearson’s r” (their formulas are not the typical ones for the correlation. We can use the coordinate plane to graph points, lines, and more. Is there a way to draw the rectangle under the points? I was thinking the a solution could be draw an empty plot then draw the rectangle and after the points, but it did not work. Rectangle: A (2, 3), B (2, 10), C (6, 10), D (6, 3). Calculate the volume of the solid lying under the surface z =e-y x +y2 and over the rectangle R = 0, 2 ä 0, 3. Zero on the x axis of the plot is the center of the wider dimension I'll assume that is the width. A simple box plot can be created in R with the boxplot function. From these coordinates, various properties such as width, height etc can be found. Mountains, huge apartments, rocky areas towards West is treated to be adequate nourishing plot. R Technologies was incorporated as a leading manufacturer and wholesaler of this domain that determinedly caters the constant market requirements. It is useful, therefore, to be able to translate to other coordinate systems where the limits of integration and evaluation of the involved integrals is simpler. 送料無料【タクボ物置】当店なら開梱設置、組立工事可能。物置＋駐輪スペース。タクボ物置 tp／ストックマンプラスアルファ tp-s43r15 多雪型 標準屋根 『追加金額で工事も可能』 『駐輪スペース付 屋外用 物置 自転車収納 におすすめ』 トロピカルオレンジ. For example, the call to the function hist() renders a histogram of the. Polar and Rectangular Forms of Equations Evaluate the function for several -values in its thousands of miles, or about 5714 domain and use these points to graph the function. The cost of the fence is$40/ft for the curved portion and $30/ft for the straight sides. Home › Service › ggplot2 Quick Ref › geom. In this section we will take a look at the basics of representing a surface with parametric equations. window()call sets the limits for the x and y coordinates in the graph. 1 to the ViewPort tree, after which you tell R with gridFig() to draw a base plot within a grid viewport (vp. ymin - (required) bottom edge of rectangle. Polar Display Mode “Polar form” means that the complex number is expressed as an absolute value or modulus r and an angle or argument θ. Rectangle: A (2, 3), B (2, 10), C (6, 10), D (6, 3). Road access is provided to each plots. Learn more about projects, radon, plotting projections, theta, fourier transform. The rectangular coordinate system (or Cartesian plane) provides a means of mapping points to ordered pairs and ordered pairs to points. There are things like GrafEq and Mathematica for plotting implicit Cartesian equations, but in general they require way much more effort on the part of the computer to plot than parametric equations (have you seen the algorithms behind implicit equation plotters?). 1r, u2= a3, p 4 P. Standard Assumptions about Residuals. Draws a simple map scale centered on the reference point specified using one of four coordinate systems: (1) Use -Lg for map (user) coordinates, (2) use -Lj or -LJ for setting refpoint via a 2-char justification code that refers to the (invisible) map domain rectangle, (3) use -Ln for normalized (0-1) coordinates, or (4) use -Lx for plot. ) Analysis must include first three modes (TE10, TE20, TE01) 3. meshgrid with a Python iterator, but I can't wrap my head. Geometric measurement: understand concepts of area and relate area to multiplication and to addition. This will open a new graphics window if there is none open, otherwise an existing window is readied to hold the new plot. It is a rectangle of side 0. A rectangular plot of land has dimensions as 5 0 m × 3 0 m there are five flower beds in the plot of the size 5 m × 3 m. As the aspect-ratio r grows, L* tends to sqrt(A/r)/2. You can do this in various ways. The graphical argument used to specify point shapes is pch. The ordered pair specifies a point’s location based on the value of r and the angle, θ, from the polar axis. You should gather five separate measurements corresponding to the five separate lines found on the web page's parallelogram. Image Size: by pixels. Given the Z height values on a (X,Y) grid, we can draw the perspective plots of this surface over the (X,Y) plane. The graphical argument used to specify point shapes is pch. Setting r0 to 0. The graphics library of R has both high level as well as low level graphics facilities. Imager uses the “cimg” class for its images. Package 'mapplots' May 22, 2018 Type Package Title Data Visualisation on Maps Version 1. We have already used all of the conversions which are necessary. We recommend. R - Normal Distribution In a random collection of data from independent sources, it is generally observed that the distribution of data is normal. Show Plot x^2, x, 0, 1 , PolarPlot Cos , ,0,2 0. As an example, a unit amplitude rectangular pulse of duration. Luckily, its pretty easy to plot confidence bounds as filled patches. Next, we can go to the equation entry screen by pressing the Y= button, just like for graphing equations in rectangular coordinates. The two numbers that are used to define a point on a graph using rectangular coordinates are the coordinate values along the horizontal and vertical axes. 50 per metre is Rs. Default statistic: stat_identity. Package 'mapplots' May 22, 2018 Type Package Title Data Visualisation on Maps Version 1. # Hollow shapes ggplot(df, aes(x=xval, y=yval, group = cond)) + geom. {"code":200,"message":"ok","data":{"html":". The plot (f, x=x0. Angle t may be in degrees or radians. See par for details. Customized multiple plots can be produced more easily using grid. The rectangular. Step 2: Multiply each points by the dilation factor. using the. Answer to: Show how to convert r = theta to rectangular form. 0_01/jre\ gtint :tL;tH=f %Jn! [email protected]@ Wrote%dof%d if($compAFM){ -ktkeyboardtype =zL" filesystem-list \renewcommand{\theequation}{\#} L;==_1 =JU* L9cHf lp. How to display a legend outside a R plot April 30, 2014 - how-to, R If you still don’t use ggplot2 or, as I do, have to use the old and finicky plot() function, read on to discover a trick I use to display a legend outside the plotting area. As you shift , you can see the relationship between the two types of graphs. 5 provides a Quicktime movie of a rectangular graph of the complex-impedance plane morphing into the polar plot of the typical Smith chart. Identify the quadrant where a point is located on a rectangular coordinate system. In rectangle, the distance around the outside of the rectangle is known as perimeter. Horses are tied up at P, R and S such that PO and RO are the radii of semi-circles with centers at P and R respectively, and S is the center of the circle touching the two semi-circles with diameters AO and OB. if TRUE returns an oscillographic plot of the pulse generated (by default FALSE). Suppose we want to plot two different types of plots on the same set of axes; for instance sup-pose we want to overlay the plots of y = x2and r = cos q. As the aspect-ratio r grows, L* tends to sqrt(A/r)/2. Rectangular coordinates is …. A landowner increased the length and the breadth of a rectangular plot by 10% and 20% respectively. Visualizing simple and complex polar coordinates. When comparing different traces in a plot, the usual approach is to plot the traces in different colors and add the confidence bounds as thinner lines. h spe_stdd_le_raw D$R_PARISC_LTOFF16DF (NoStack) KPXquotedblleftAdieresis-10 Allowcommands:%d F*HZh base_offset:0x%02x ={m! sdp_record_free KPXFedieresis-75 __version__='5. The polar form is where a complex number is denoted by the length (otherwise known as the magnitude, absolute value, or modulus) and the angle of its vector (usually denoted by an angle symbol that looks like this: ∠). In the simplest box plot the central rectangle spans the first quartile to the third quartile (the interquartile range or IQR). polygon draws the polygons whose vertices are given in x and y. - Just 800 m away from anchalomoodu town. =$7200 Solved examples on Perimeter and Area of Rectangle: 6. Rectangular-Coordinate. In polar coordinates, these ordered pairs take the form $$( r, \theta )$$. When given a set of polar coordinates, we may need to convert them to rectangular coordinates. A rectangular weir is one type of sharp crested weir, and is of the options that can be used to meter flow rate in an open channel. Most graphing calculators can plot polar functions; in the menu, set the plotting mode to something like polar or POL, depending on one's calculator. R-Square” and “Pearson’s r” (their formulas are not the typical ones for the correlation. Try this Drag any vertex of the rectangle below. r1: Bottom Right coordinate of first rectangle. Polar contour plots can be generated from three columns of data in a worksheet, organized either as R Z or R Z. How to visualise scatter plots with rectangular bins in R. x1) calling sequence plots the real function f over the horizontal real range from x0 to x1. Package 'mapplots' May 22, 2018 Type Package Title Data Visualisation on Maps Version 1. To obtain very accurate graphs, technology is a great aid. , xleft, , are relative to the current plotting region. We will also see how the parameterization of a surface can be used to find a normal vector for the surface (which will be very useful in a couple of sections) and how the parameterization can be used to find the surface area of a surface. First you call a layout specification function (the one without the layout_ prefix, and then layout_ (or add_layout_) to perform the layouting. 14 x the squared radius of the circle: Area = 3. ymax - (required) top edge of rectangle. The results in Figure 4 are similar to those in Figure 1 for the rectangular geometry. packages ("tidyverse") Learn the tidyverse. The reason is that I need to draw a rectangle at specific coordinates on a plot. We also know that diagonals of parallelogram bisect each other, so length of diagonal np will be two times pr. In the year 2015, G. The positions supplied, i. Rectangular plot is marvel. And that can be kind of tricky because remember that the polar coordinates for a point are not unique. Arguments. A point in the plane has polar coordinates. This is a quick way to make a treemap in R. 5300, what is the length of the plot in metres? a) 40m b) 50m c) 120m d) Data inadequate e) None of these. In R, the color black is denoted by col = 1 in most plotting functions, red is denoted by col = 2, and green is denoted by col = 3. (r, θ, φ) are the spherical coordinates. Colors are specified in the canvas drawing format (use rgba for transparency). Area is a quantity that describes the size or extent of a two-dimensional figure or shape in a plane. Note: The commonly used operator form of the calling sequence and other ways of. 2 and I'm graphing simple line plots using the following commands. Plotting Methods for the Crime Scene. Is there something like this in excel? Nothing built-in but the formulae are straightforward. But i don't know how to give theta and phi values. R E L A T E D Ganga Dussehra 2020 | Ganga Dasara Puja in Jyeshta Month. Parts (a) and (b):. First use layout() function to define 2 plots on one layer side by side, and then we plot the same data on both plots, with the plot on the right side in white color, thus invisible (just providing the scale), and finally we plot the legend on the second plot. The bty parameter determines the type of box drawn. The main problem is how to make the height of the pulse "variable" not a "number" as you can see in the figure below. pch=0,square pch=1,circle. If the x-axis goes from 100 to 200 then xleft must be larger than 100 and xright must be less than 200. Angle t is in the range [0 , 2Pi) or [0 , 360 degrees). Residual Analysis. Displays a plot of iterations vs. 01 inch (scaled by cex). plot the graph of the function as a surface over a rectangle in the x,y plane: ezsurf(G,[-2,2,-2,2]) Click on in the figure toolbar, then you can rotate the graph by dragging with the mouse. In the year 2015, G. Lines of constant thickness have their uses, but \MF\ also provides several other kinds of scrivener's tools, and we shall take a look at some of them in this. The graphical argument used to specify point shapes is pch. A matrix is a collection of data elements arranged in a two-dimensional rectangular layout. Increase the line size so that they form a compact shaded area. The second way (new in igraph 0. Standard Assumptions about Residuals. So the biggest rectangle in my plot represents that there are more people who are female, non-alcoholic,. R will plot the first plot in the entries of the matrix with 1, the second plot in the entries with 2,… widths A vector of values for the widths of the columns of the plotting space. Polar and Rectangular Forms of Equations Evaluate the function for several -values in its thousands of miles, or about 5714 domain and use these points to graph the function. Answer to: Show how to convert r = theta to rectangular form. js: The Goal, Drawing an SVG Circle using D3. Converting from rectangular coordinates to polar coordinates. You can set up Plotly to work in online or offline mode. 1' AddOtherInputDevices H=ZcS kf50 chinese-iso. a vector (or scalar) of top y positions. A rectangle is a four-sided shape with every angle at ninety degrees. hullo merry! it's going fine, thanks for asking! yes, it is a circle, with diameter from (0,0) to (0,1) … a little bit of geometry will enable you to confirm that your mistake was that you can't have negative values of r!!. The second way is preferred, as it is more flexible. The tracts are divided into 6-mile-square parts called townships, which are in turn divided into 36 tracts, each 1 mile square, called sections. The rectangular coordinate system (or Cartesian plane) provides a means of mapping points to ordered pairs and ordered pairs to points. The results in Figure 4 are similar to those in Figure 1 for the rectangular geometry. Project 1: Rectangular Waveguide (HFSS) r Objective • Getting Started with HFSS (a tutorial) • Using HFSS, simulate an air-filled WR-90 waveguide shown above. Plot and label each set of points in the coordinate plane. Surface Area and Volume: In a rectangular cuboid, all angles are right angles, and opposite faces of a An interactive math lesson to teach how to calculate the volume of a rectangular An interactive math lesson to teach the surface area of a rectangular prism. 6 cents of house plot at chevoor, Thrissur, facing panchayat road, 50 metres from Chevoor centre and Thrissur - Irinjalakuda state highway ₹ 24,00,000 Rectangular plot. Setting r0 to 0. Examples ## set up the plot region: plot(c(100, 250), c(300, 450), type = "n", main = "2 x 11. As an example, a unit amplitude rectangular pulse of duration. Install the complete tidyverse with: install. Also, find the cost of ploughing five flower beds at the rate of Rs. background = element_blank. js, Drawing an SVG Rectangle using D3. 3 acres; 12 high school basketball courts are a little more than 1 acre. hclust(tree, k = NULL, which = NULL, x = NULL, h = NULL, border. Land area calculation. r 2 = x 2 + y 2. The element spacing is 0. Let's get started. 7) ( , ) 8) ( , ) Convert each pair of rectangular coordinates to polar coordinates where r and. These image processing algorithms are often referred to as a "spatial convolution. and the 1-D coordinate system is denoted by R. 14 x 20 2 = 3. According to ggplot2 concept, a plot can be divided into different fundamental parts : Plot = data + Aesthetics + Geometry. No plotting is performed. What I want is to be able to modify the plot in some ways. Simple Plot: Plot ordered pairs of numbers, either as a scatter plot or with the dots connected. that a ect all plots in an R session. " is handled specially. Converting rectangular data into R with the readr R package Tabular data, or flat rectangular data, comes in many different formats, including CSV and TSV. Answer to: Show how to convert r = theta to rectangular form. 5 provides a Quicktime movie of a rectangular graph of the complex-impedance plane morphing into the polar plot of the typical Smith chart. 3 and 4), c) plots for the first 30 modes in a circular waveguide (Figs. Edexcel Statistics S2 June 2011 Q4a : ExamSolutions - youtube Video. due to an anonymous source and obtained from the log files of Wolfram|Alpha in early February 2010. and the 1-D coordinate system is denoted by R. Key Concept: Understand the rectangular coordinate system with respect to point within it, graphing functions and finding relative distance between points. For simple scatter plots, plot. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. The simplest way to plot a legend outside a figure in R is to: In this example, I am going to make a multi-panel figure, with a horizontal legend on the bottom of the plot. One annoying thing in R (for me as a Matlab user) is retrieving the axis limit. Plotting points in polar. • To obtain the Field patterns, intrinsic Impedance and wavelength for the first 4 modes. Example 3: Plot the point with the following polar coordinates: 1, 2 Example 4: Find another representation of 5, 4 in which r is positive and 24. It uses the following equations:. A landowner increased the length and the breadth of a rectangular plot by 10% and 20% respectively. Look below to see them all. Start your Azure Machine Learning Studio environment. In addition to a appropriately large R 2 value, the residuals must be well-behaved, as explained in the following section. The rectangular coordinate system A system with two number lines at right angles uniquely specifying points in a plane using ordered pairs (x, y). Create the normal probability plot for the standardized residual of the data set faithful. where R is the triangle with vertices at (0,0), (0,2) and (3,0). So if you’re plotting multiple groups of things, it’s natural to plot them using colors 1, 2, and 3. To change more than one graphics option in a single plot, simply add an additional argument for each plot option you want to set. We will now examine the complex plane which is used to plot complex numbers through the use of a real axis (horizontal) and an imaginary axis (vertical). 3 thickness. Drawing inside plots. This book will teach you how to do data science with R: You’ll learn how to get your data into R, get it into the most useful structure, transform it, visualise it and model it. Plotting Methods for the Crime Scene. True BASIC is similar to F (a subset of Fortran 90) and has excellent graphics capabilities which are hardware independent. Often, this feature is used to calculate a step response or time-domain reflectometry (TDR) plot of the structure being simulated. The length of a rectangular plot is 20 metres more than its breadth. 14 x (20 ft x 20 ft) = 1,256 ft 2. Create a maze using divide-and-conquer: Begin with a rectangular region with no walls. fffffffffff? (A) 6 (B) 8 (C) 10 (D) 12 (E) 20. Learn more about projects, radon, plotting projections, theta, fourier transform. If pch is an integer or character NA or an empty character string, the point is omitted from the plot. q = arctan(y/x). rect draws a rectangle (or sequence of rectangles) with the given coordinates, fill and border colors. Zero on the x axis of the plot is the center of the wider dimension I'll assume that is the width. Introduction to Applied Machine Learning & Data Science for Beginners, Business Analysts, Students, Researchers and Freelancers with Python & R Codes @ Western Australian Center for Applied Machine Learning & Data Science (WACAMLDS)!!!. I'm creating a rectangular prism function, whose output looks like this: I think that this code can be improved by optimizing the use of np. This function draws a box around the current plot in the given color and linetype. isn't a mathematician Oct 1 '11 at 23:32. Example 5: Find another representation of 5, 4 in which r is negative and 02. Package 'mapplots' May 22, 2018 Type Package Title Data Visualisation on Maps Version 1. You can vote up the examples you like or vote down the ones you don't like. Make sure you have selected Generic CSV file with header (. Continuous Uniform Distribution : S2 Edexcel January 2012 Q1 : ExamSolutions Maths Revision - youtube Video. Thanks for contributing an answer to Geographic Information Systems Stack Exchange! Please be sure to answer the question. In the example below, data from the sample "chickwts" dataset is used to plot the the weight of chickens as a function of feed type. Substitute in the known values of and into the formulas. is generated. Enter the five measurements into the appropriate text fields found in the land calculator. The plot on the right displays the calculated current distribution around the conductor in red. Plot the given point. Enough fooling. 5) ( , °) 6) ( , ) Convert each pair of polar coordinates to rectangular coordinates. See also clip. So if you're plotting multiple groups of things, it's natural to plot them using colors 1, 2, and 3. Square plot is excelsior. In rectangle, the distance around the outside of the rectangle is known as perimeter. 01) Imports stats, graphics, grDevices Description Functions for plotting graphical shapes. Learn CAS Calculator. Rectangle (). Generating an isolated rectangular pulse in Matlab: An isolated rectangular pulse of unit amplitude and width w (the factor T in equations above ) can be generated easily with the help of in-built function – rectpuls(t,w) command in Matlab. Rectangle: A (2, 3), B (2, 10), C (6, 10), D (6, 3). Then call legend with the location. 1-8: Title: Read and write PNG images: Author: Simon Urbanek : Maintainer: Simon Urbanek : Description: This package provides an easy and simple way to read, write and display bitmap images stored in the PNG format. Convert among different-sized standard measurement units within a given measurement system (e. Key Concept: Understand the rectangular coordinate system with respect to point within it, graphing functions and finding relative distance between points. A modified augmented design (type 2) is presented for the situation where subplots are long and narrow (rectangular plots). It only takes a minute to sign up. 14 x the squared radius of the circle: Area = 3. The point lies two units from the pole on the terminal side Polar-to-Rectangular Rectangular-to-Polar y r sin r2. Identify the quadrant where a point is located on a rectangular coordinate system. Plot the points the rectangle a square? 1), B(4, 1), C(4, 5), D( ? 3), 1(4, 3), A? , 9 ), K(1, Lx N S Q rhombus. Hello I have a simple shiny app in which I create a scatterplot of variables found in the iris dataset. background = element_blank. If the cost of fencing the plot @ 26. 📊 Circular Manhattan Plot. The length of a rectangular plot is 20 metres more than its breadth. We also know that diagonals of parallelogram bisect each other, so length of diagonal np will be two times pr. House Assignment. Development of R-shiny application(s) to enable the generation of identified plots for direct inclusion in submission packages for regulatory agencies. Area of a Rectangle - powered by WebMath. 5` into rectangular coordinates, we use. If we want to move the legend out of the main plot area, we need some more work. Making statements based on opinion; back them up with references or personal experience. See how the tidyverse makes data science faster, easier and more fun with “R for Data. If and Y#O, then r = 4. 900 1200 1350 1500 1650 1950 2250 2400 2550 45' 300 150 60' 3450 3300 2700 2) (2 points) The polar function r 2 sin is symmetric about: The polar axis b. And the fifth curve is the polar curve. An S3 object of class element, rel, or margin.
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https://chemistry.stackexchange.com/questions/59827/what-exactly-is-hydrolysis-reaction-in-organic-chemistrywhat-mechanism-does-it | What exactly is hydrolysis reaction in organic chemistry?What mechanism does it follow?
According to my study material:
Hydrolysis is a special type of nucleophilic substitution ($\mathrm{S_N1}$) where water acts as both nucleophile and a solvent molecule.
Is this definition correct? If so how to solve the following problem (picked up from my textbook exercise which has no solution)?
When alkyl bromides (listed here) were subjected to hydrolysis in a mixture of ethanol and water ($80\ \%\ \ce{C2H5OH} / 20~\%\ \ce{H2O}$) at $55~\mathrm{^\circ C}$ the rates of reaction showed the following order: $\ce{(CH3)3CBr} > \ce{CH3Br} > \ce{CH3CH2Br} > \ce{(CH3)2CHBr}$.
Provide an explaination for this order of reactivity.
But my doubt is how is this order $\ce{(CH3)3CBr} > \ce{CH3Br} > \ce{CH3CH2Br} > \ce{(CH3)2CHBr}$ possible?
I mean I know the first compound will form a tertiary carbocation. That is okay for $\mathrm{S_N1}$ but how will the second compound undergo $\mathrm{S_N1}$ (because methyl carbocation is very unstable)? Moreover if really $\mathrm{S_N1}$ is taking place the fourth compound should form a more stable carbocation compared to the third compound. These things are confusing me. Is hydrolysis really $\mathrm{S_N1}$ or something else?
According to me the order should have been $(1)>(4)>(3)>(2)$.
• Do you call it a hydrolysis reaction? ... Because I just call it a substitution reaction... – DHMO Sep 27 '16 at 13:31
• @DHMO I stated what my textbook and study material wrote.I did'nt make up the definitions myself.See en.wikipedia.org/wiki/Solvolysis – user14857 Sep 27 '16 at 13:34
• Hydrolysis is not necessarily SN1, it can also refer to a SN2 process (especially if there is OH-). Or it can be any process really. There are many different ways of hydrolysing esters, for example. However, in this particular case, it is weird. I would expect only SN1 reactivity under the stated conditions, so I don't know why MeBr is so high up. – orthocresol Sep 27 '16 at 13:40
Your textbook is almost correct. Here is the corrected version of its statement:
Hydrolysis is a special type of nucleophilic substitution where water acts as both nucleophile and a solvent molecule.
Note me omitting the $\mathrm{S_N1}$ part. This is because a hydrolysis mechanism can be any nucleophilic substitution, whether $\mathrm{S_N1, S_N2, S_N}$ for heavy atoms or $\mathrm{S_NAr}$.
That also explains the given order of reactivity. tert-Butyl bromide is very fast at $\mathrm{S_N1}$ reactions. Bromomethane very rapidly undergoes $\mathrm{S_N2}$ reactions. The other two are in-between with bromoethane being just a tad better at $\mathrm{S_N2}$ reactions than 2-bromopropane is at $\mathrm{S_N1}$ reactions.
($\mathrm{S_N1}$ is preferred in 2-bromopropane as the solvent is polar protic and it can stabilize the intermediate carbocation well)
• By the extension of your definition: In the solvolysis definition in Wikipedia too should the $\mathrm{S_N1}$ be removed? See en.wikipedia.org/wiki/Solvolysis – user14857 Sep 27 '16 at 13:43
• @ZOZ Yes.$%MathJax comments hurray!$ – Jan Sep 27 '16 at 13:45
• Morever,I cannot understand how can water in this case act as nucleophile in 20 percent ethanol.Water is considered to be a weak nuceophile.Can it really initiate a $\mathrm{S_N2}$ ? – user14857 Sep 27 '16 at 13:45
• @ZOZ Water isn’t that weak a nucleophile, tbh. – Jan Sep 27 '16 at 13:46
• Well by that logic if it were 80 % ethanol and 20 % water would ethanol have acted as the nucleophile? I mean the ethyl group exerts a $+I$ effect on the O atom.So would ethanol be nucleophile in that case? @Jan – user14857 Sep 27 '16 at 13:47 | 2019-11-18 18:57:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.747945249080658, "perplexity": 2049.595405335754}, "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-2019-47/segments/1573496669813.71/warc/CC-MAIN-20191118182116-20191118210116-00095.warc.gz"} |
https://www.mathalino.com/tag/reviewer/flexural-stress | # flexural stress
## Bending Stress and Shearing Stress in Timber Beam
Bending Stress $f_b = \dfrac{M}{S} = \dfrac{Mc}{I}$ Horizontal Shear Stress $f_v = \dfrac{VQ}{Ib}$ For Rectangular Sections $f_b = \dfrac{6M}{bd^2}$ $f_v = \dfrac{3V}{2bd}$ | 2019-08-18 11:36:53 | {"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": 0.6707637906074524, "perplexity": 10964.453336569715}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313803.9/warc/CC-MAIN-20190818104019-20190818130019-00487.warc.gz"} |
https://open-eo.github.io/openeo-r-client/reference/get_sample.html | In order to inspect data locally a very small spatial extent will be processed, downloaded and made available in R.
get_sample(
graph,
replace_aoi = TRUE,
spatial_extent = NULL,
execution = "sync",
immediate = TRUE,
con = NULL,
...
)
## Arguments
graph
a ProcessGraph, a Process or the final node in a process for which the sample shall be calculated
replace_aoi
a logical flag to indicate whether or not the original spatial extent shall be substituted with a different one, default TRUE
spatial_extent
a bounding box or a spatial feature from which to derive a bounding box
execution
sync or async which indicates the processing chain, a not "async" value results in a synchronous processing
immediate
flag to be considered if the retrieval shall be immediately queued on the back-end
con
connected and authenticated openEO client (optional) otherwise active_connection() is used.
...
additional parameters that are passed to compute_result() or create_job()
## Details
In order to get a better understanding about the processing mechanisms and the data structures used in the openEO back-end, it helps to check the actual data from time to time. This function aids the user in doing to. It replaces all spatial extents of the derived process graph with a new spatial extent which is calculated by the first spatial extent of the mandatory openEO process 'load_collection'. We take the center of the extent and add 0.0003 degrees to it. In case the coordinate reference system is not in WGS84, then the bounding box will be transformed into geodetic WGS84 beforehand, if the package 'sf' is present.
If the spatial extent was explicitly set to a small custom extent, then you can disable the replacement of the area of interest with replace_aoi = FALSE. | 2022-11-30 20:37:50 | {"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": 0.26640549302101135, "perplexity": 1652.2483976530777}, "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/1669446710771.39/warc/CC-MAIN-20221130192708-20221130222708-00835.warc.gz"} |
http://www.geog.com.cn/EN/10.11821/dlxb201412005 | Acta Geographica Sinica ›› 2014, Vol. 69 ›› Issue (12): 1790-1798.
• Orginal Article •
### The change of population density and its influencing factors from 2000 to 2010 in China on county scale
Lu WANG(), Zhiming FENG(), Yanzhao YANG, Zhen YOU
1. Institute of Geographic Sciences and Natural Resources Research, CAS, Beijing 100101, China
• Received:2013-09-23 Revised:2014-05-16 Online:2014-12-25 Published:2014-12-25
• Supported by:
Foundation of the Key Laboratory, MLR, China;Foundation of Bureau of Floating Population, NHFPC, China, No.201011]
Abstract:
Studying the change of population distribution and density can provide important basis for regional development and planning. However, the spatial patterns and driving factors of the change of population density in China were not clear yet. Therefore, using the population census data in 2000 and 2010, this study first analyzed the change of population density in China and divided the changes in all 2353 counties into 4 types, consisting of rapid increase, slow increase, slow decrease and rapid decrease. Subsequently, based on the partial least square (PLS) regression method, we recognized the significant influencing factors of population density change among 11 natural and social-economic factors for the whole country and counties with different types of population change. The results showed that: (1) compared to the population density in 2000, the population density in most counties (over 60%) increased by 21 persons per km2 on average, while the population density in other counties decreased by 13 persons per km2 in 2010. Of all 2353 counties, 860 and 589 counties respectively show rapidly and slowly increasing population density, while 458 and 446 counties show slowly and rapidly decreasing population density, respectively. (2) Among the 11 factors, social-economic factors have impact on population density change more significantly than natural factors. The higher economic development level, better medical condition and stronger communication capability were main pull factors of population increases. The dense population density was the main push factor of population decreases. These conclusions generally clarified the spatial distribution pattern of population change and its influencing factors in China over the past 10 years and could provide reference for the future population planning. | 2022-01-18 05:11:32 | {"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": 0.20458410680294037, "perplexity": 3337.6302289897067}, "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/1642320300722.91/warc/CC-MAIN-20220118032342-20220118062342-00057.warc.gz"} |
https://computergraphics.stackexchange.com/questions/5680/diffuse-brdf-component-of-the-disney-brdf | # Diffuse BRDF component of the Disney BRDF
In the SIGGRAPH course:
BURLEY B.: Physically Based Shading at Disney, SIGGRAPH 2012 Course: Practical Physically Based Shading in Film and Game Production, 2012.
it is mentioned that some BRDF models include a diffuse Fresnel factor such as:
$$(1-F(\theta_l)) (1-F(\theta_d)).$$
The Disney BRDF itself uses the following diffuse BRDF component (using Sclick's Fresnel approximation):
$$f_d = \frac{\textrm{c_base}}{\pi} (1 + (F_{\textrm{D90}} - 1)(1-\cos\theta_l)^5) (1 + (F_{\textrm{D90}} - 1)(1-\cos\theta_v)^5),$$
where
$$F_{\textrm{D90}} = 0.5 + 2 \text{roughness} \cos^2\theta_d.$$
Where does this come from?
My attempt...
If I evaluate $(1-F(\theta_l)) (1-F(\theta_v))$ (instead of $(1-F(\theta_l)) (1-F(\theta_d))$?) with Schlick's approximation, we get:
$$\left(1-(F_0 + (1-F_0)(1-\cos\theta_l)^5)\right) \left(1-(F_0 + (1-F_0)(1-\cos\theta_v)^5)\right)$$ $$\left(1-F_0 + (F_0-1)(1-\cos\theta_l)^5\right) \left(1-F_0 + (F_0-1)(1-\cos\theta_v)^5\right)$$
If we substitute $F_{\textrm{D90}} = F_0$, we get:
$$\left(1-F_{\textrm{D90}} + (F_{\textrm{D90}}-1)(1-\cos\theta_l)^5\right) \left(1-F_{\textrm{D90}} + (F_{\textrm{D90}}-1)(1-\cos\theta_v)^5\right)$$
This looks similar except for the 2x $-F_{\textrm{D90}}$? Is my reasoning completely wrong or where do I make mistakes? Or am I not aware of some further (common) approximations?
• Remember that disney's diffuse BRDF is not a physical correct one. They are fully aware that their BRDF is not energy conserving, but they found that it looked better (probably because it makes up for interreflections) and their artists liked it. Also note, that it is based on a physical BRDF model, which takes an integral over all microfacet normals (for this, see [Earl Hammon Jr's Diffuse GGX Lighting Slides][1]). That integral is not solvable, thus it is only an approximation. [1]: twvideo01.ubm-us.net/o1/vault/gdc2017/Presentations/… – Tare Oct 13 '17 at 8:41 | 2021-01-22 12:11:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8275924324989319, "perplexity": 3866.0947042170606}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703529331.99/warc/CC-MAIN-20210122113332-20210122143332-00110.warc.gz"} |
https://www.jiskha.com/users?name=Sam | # Sam
Most popular questions and responses by Sam
What types of problems can be solved using the greatest common factor? What types of problems can be solved using the least common multiple? Complete the explanation. *** Use the words 'same' and 'different' to complete the following sentences.*** Problems
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22. ## science
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39. ## Spanish
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47. ## Language Arts
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50. ## Physics
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51. ## Physics II
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52. ## Math
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54. ## Physics
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55. ## science
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56. ## math
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57. ## Chemistry
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58. ## Chemistry
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60. ## math
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61. ## Algebra
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62. ## Chem II
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63. ## History
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64. ## Science
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65. ## Chemistry
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66. ## Chemistry
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67. ## world history
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68. ## chemistry
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69. ## Math
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70. ## Calc 2: Area under the curve
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72. ## Physics
to strengthen his arm and chest muscles, an 82-kg athlete who is 2.0 m tall is doing push ups. his center of mass is 1.15 m from the bottom of his feet, and the centers of his palms are 30.0cm from the top of his head. find the force that the floor exerts
73. ## Pre-Calculus
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74. ## Math
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75. ## Physics helppp
Figure 5-53 shows a man sitting in a bosun's chair that dangles from a massless, frictionless pulley and back down to the mans's hand. The combined mass of man and chair is 94 kg. Fig. 5-53 With what force magnitude must the man pull on the rope if he is
76. ## Math
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77. ## physics
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78. ## Analytical Chemistry
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79. ## Chemistry
If the standard enthalpy of combustion of octane, C8H18(l) is -5471 kJ/mol, calculate the enthalpy change when 1.00 kg of octane are burned. the molar mass of octane is 114 g/mol
80. ## Analytical Chemistry
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81. ## Social Studies Help ASAP Please!
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82. ## math
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83. ## Trignometry
In Philadelphia the number of hours of daylight on day t (t is the number of days after january 1) is modeled by the function L(t)= 12+2.83sin((2*Pi)/365(t-800) a. Which day have about 10 hours of daylight? b. How many days of the year have more than 10
84. ## calculus
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85. ## physics
Two Children,A and B, fire identical 10g ball bearing from a catapult. The elastic band of each catapult is elastically extended by 0.10m and then released to fire a the ball bearings. Child A's elastic band has a spring constant of 144 N/m. Calculate the
86. ## math/ratios
There are 77 students in the student council. The ratio of girls to boys is 7:4. How many girls are in the student council?
87. ## physics
A wagon is rolling forward on level ground. Friction is negligible. The person sitting in the wagon is holding a rock. The total mass of the wagon, rider, and rock is 94.0 kg. The mass of the rock is 0.350 kg. Initially the wagon is rolling forward at a
88. ## physics
Use R = 8.2 × 10–5 m3 atm/mol K and NA = 6.02 × 1023 mol–1. The approximate number of air molecules in a 1 m3 volume at room temperature (300 K) and atomospheric pressure is:
89. ## math
The area of a trapezoid may be determined by using the formula A = (h/2)(a+b), where "A" stands for area, "h" stands for the height of the trapezoid, and "a" and "b" stand for the lengths of the bases of the trapezoid. If the height of the trapezoid is12
90. ## Chemistry (Re-post)
A solution contains 1.694 mg of CoSO4 (MW 155.0) per mL. Calculate a) The volume of 0.008640 M EDTA needed to titrate a 25.00 ml aliquot of this solution. So this is my working out for this section, Step 1. Moles of CoSO4 = 0.0001694mg * 25mL *
91. ## Chemistry
Why is (E)-1,2-diisopropylethene more stable than (Z)-1,2-diisopropylethene? Why are they more stable than 1,1-diisopropylethe? And doesn't the bond angle between the C=C-C carbons increase with (Z)-1,2-diisopropylethene > (E)-1,2-diisopropylethene >
92. ## physics
A large tank is filled with water to a depth of d = 18 m. A spout located h = 13 m above the bottom of the tank is then opened as shown in the drawing. With what speed will water emerge from the spout?
93. ## Language Arts
The author of "My Brother's Keeper", includes the line "and Jamie thought to himself, this time it must be bad" in Jamie's phone conversation with Ted in order to a. explain how Uncle Harry usually takes care of Ted. b. develop Ted as a character who is
94. ## Chemmistry
Use this percentage to calculate the mass of fluorine (in g) contained in 54.0 g of copper (II) fluoride.
95. ## math (vectors) & physics
In unit-vector notation, what is the net torque about the origin on a flea located at coordinates (-2.0, 4.0 m, -1.0 m) when forces F1 = (-4.0 N) k and F2 = (-5.0 N) j act on the flea? _____________________________________ Torque is the cross product of
96. ## physics
In a police ballistics test, a 10.0-g bullet moving at 300 m/s is fired into a 1.00-kg block at rest. The bullet goes through the block almost instantaneously and emerges with 50.0% of its original speed. What is the speed of the block just after the
97. ## Chemistry
A decomposition reaction has a rate constant of 0.0019 yr -1. ( a) What is the half-life of the reaction? ______ yr (b) How long does it take for [reactant] to reach 12.5% of its original value? ______ yr
98. ## math 10
a British researcher developed a formula to determine walking or running speed based on hip height and stride length. The formula is s= 0.78l ^1.67/ h^1.17 where s is speed in meters per second l is stride length in meters, and hi is the individual hip | 2019-10-20 13:00:09 | {"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": 0.3781597316265106, "perplexity": 2621.054382441359}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986707990.49/warc/CC-MAIN-20191020105426-20191020132926-00041.warc.gz"} |
https://www.omnicalculator.com/health/twin-pregnancy-weight-gain | # Twin Pregnancy Weight Gain
Created by Łucja Zaborowska, MD, PhD candidate
Reviewed by Rijk de Wet
Based on research by
Institute of Medicine Recommendations Weight Gain During Pregnancy: Reexamining the Guidelines (2009)
Last updated: Feb 02, 2023
The maternal weight gain in twin pregnancies has its own rules that are a bit different from singleton pregnancies. Let's find out how much weight you should gain every week, how to keep yourself healthy carrying two or multiple babies, and discover the most significant differences in pregnancy weight gain in different types of gestations.
We try our best to make our Omni Calculators as precise and reliable as possible. However, this tool can never replace a professional doctor's assessment. If any health condition bothers you, consult a physician.
## Why do you gain weight during pregnancy?
Gestational weight gain is a must when it comes to pregnancy — your entire body is changing to support a baby growing inside of you. The amount of weight you gain depends not only on your baby's birth weight or the number of children you're carrying; all parts of your body need to adjust to this new situation:
• Your breasts and uterus both grow 2 lb.
• Your blood increases its volume by 50%.
• Your body accumulates water and starts producing amniotic fluid.
• A new organ is created — a 1.5 lb placenta!
If you're pregnant with twins, the situation's a little bit different. If all the babies have separate placenta, each one of them is on average 17% smaller. The difference grows to 24% in tripleton pregnancies.
• A lot of extra fat tissue. It serves as a source of ingredients for pregnancy hormones production.
These values are usually different for singleton and twin pregnancies — women carrying twins and multiple pregnancies are expected to hold even more considerable weight.
## Guidelines for twin pregnancy weight gain
As we've already mentioned in multiple articles, the expected weight gain during pregnancy depends on your pre-pregnancy weight and height, computed into BMI (body mass index):
$\text{BMI} = \frac{ \text{weight [kg]} }{ (\text{height [m]})^2 }$
As you may expect, these values are slightly higher for women with twin pregnancies. After all, you're carrying two souls under your heart! 👶👶
Here's the chart of weight gain recommendations for twin pregnancies:
• Underweight (BMI <18.5)
• Expected weight gain: 28–40 pounds (12.7–18.1 kg)
• Normal weight (BMI 18.5–24.9)
• Twin pregnancy expected weight gain: 37–54 pounds (16.8–24.5 kg)
• Overweight (BMI between 25 and 29.9)
• Twin pregnancy expected weight gain: 31–50 pounds (14–22.7 kg)
• Obese (BMI over 30)
• Twin pregnancy expected weight gain: 25–42 pounds (11.3–19 kg)
Some studies estimate that the weekly weight gain while pregnant with twins should be close to approximately 0.75 kg (1.5 lb) during pregnancy's second and third trimesters. However, every pregnancy and every woman is different — talk to your health care provider about your individual, healthy weight range.
Remember that excessive gestational weight gain may harm pregnancy outcomes! Weight gain during twin pregnancies is twice as significant. These gestations are already considered high risk or very high risk, depending on the chronicity, which is the number of placentas in the womb.
💡 Dichorionic (two placentas) twin pregnancies are the safest and most common type.
Possible threats for obese and overweight women include preterm birth (already increased in twins) and an even greater risk of preeclampsia.
Lower weight gain in twin pregnancies is associated with a higher incidence of low birth weight and an even greater risk of anemia.
Check the birth weight percentile calculator to find out more.
## More about pregnancy with twins or multiples
As we already know, an average woman gains around 30–50 pounds when pregnant with twins. What about multiple pregnancies? It is estimated that every additional fetus adds another 10 pounds (4.5 kg).
• Normal-BMI women with twins should eat around 30–45 kcal/kg extra each day.
• recommend increasing the daily intake by:
• 300 kcal per baby in the 1st semester;
• 340 kcal per baby in the 2nd semester; and
• 452 kcal per baby in the 3rd semester.
• Keep a maternal weight gain record — step on a scale once a week at the same time of the day and write it all down.
• It is essential to recognize multiple and singleton pregnancies early on. If you're pregnant with twins, but don't know it yet, you may needlessly worry about your "excessive" weight gain and try to impose some interventions that are not needed at all.
• Try to exercise as long as you can — walk at least 150 minutes per week.
• Fill your diet with proteins, vitamins, dairy products, and wholemeal products.
• Maximize your calcium intake. Pregnancy may have a significant impact on your bones and teeth.
• Increase your folic acid intake.
Łucja Zaborowska, MD, PhD candidate
Before pregnancy
Height
ft
in
Weight
lb
BMI
You are...
Weight during pregnancy
Week
Twins
no
Min. weight
kg
Max. weight
kg
Min. weight gain
kg
Max. weight gain
kg
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Does any know how to draw a graph like this:
Which programs can do that? I'm a developer so I've no problem on formatting a complex files so that a software can handle it.
• SAS is one that can. Nov 24, 2011 at 0:20
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– chl
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See these questions.
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Jun 13, 2018
50 ft
#### Explanation:
Length = 17ft
Rectangle has 2 lengths and 2 breadths
So,
$17 \times 2 + 8 \times 2$
$= 50$
Jun 13, 2018
Perimeter $= 50 f t$
#### Explanation:
Length of rectangle $= 17 f t$
Width of rectangle $= 8 f t$
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https://opencfs.gitlab.io/userdocu/Applications/Coupledfield/WaterWaveMech/GlassOfWater/ | source file for the documentation in the jupyter notebook GlassOfWater.ipynb and converted to markdown by jupyter nbconvert.
Sloshing in a Glass of Water¶
This example demostrates the coupling between solid mechanics (<mechanic> PDE) and free surface water waves (<waterWave> PDE). For details of the governing equations refer to our publication1. Them mesh is generated with cubit using the GlassOfWater.jou, which produces the mesh GlassOfWater.cdb.
The material file mat.xml defines the material properties.
Sloshing modes of the Water Volume¶
We define an Eigenfrequency annalysis to compute the sloshing modes in the XML input WaterModes.xml. Run the simulation by executing
cfs WaterModes
The modes are then visualised in ParaView (use the state file WaterModes.pvsm). You can use paraviews export animation function to export the modes. The computed modes are plotted here in the following using python.
First we import matplotlib and set some options
%matplotlib inline
# import matplotlib
import matplotlib as mpl
from matplotlib import pyplot as plt
from cycler import cycler
color_cycle = cycler('color',['#377eb8','#4daf4a','#e41a1c','#984ea3','#ff7f00','#a65628','#f781bf','#999999','#ffff33'])
mpl.rc('figure',figsize=[2.7,1.6]) # figure size in inch
mpl.rc('figure',dpi=200) # inline dpi (=display size in browser)
mpl.rc('font',size=8.0)
#mpl.rcParams['font.sans-serif'] = 'Helvetica'
mpl.rc('lines',linewidth=0.7)
mpl.rc('axes',prop_cycle=color_cycle,grid=True,linewidth=0.5,axisbelow=True,unicode_minus=False)
mpl.rc('grid',linewidth=0.3,linestyle=':')
mpl.rc('legend',fontsize='medium',framealpha=1.0,numpoints=1)
mpl.rc('svg',fonttype='none')
mpl.rc('savefig',dpi=300)
Then we import some helper functions from CFS (see source code) to acess the result files. We read the computed natural frequencies (step values in Hz):
# import cfs tools
from sys import path
path.append('/home/ftoth/openCFS/CFS/share/python/')
from hdf5_tools import *
f_eigen = get_step_values('results_hdf5/WaterModes.cfs')[0]
f_eigen[1:]
array([3.58408981, 3.58408981, 4.64634551, 4.646351 , 5.31460203,
5.46617982, 5.46617982, 6.16236872, 6.16237406, 6.2881556 ,
6.2881556 , 6.77986437, 6.77986437, 7.06106289, 7.06137654,
7.23091523, 7.34199178, 7.34200377, 7.72717791, 7.72717791,
7.86312303, 7.86312303, 7.989072 , 7.989072 , 8.32375798,
8.32404929, 8.35316041, 8.35321557, 8.64515051])
Finally we plot the first few modes
fig, axs = plt.subplots(ncols=5,nrows=2,figsize=[10,4])
for i in range(1,len(axs.ravel())+1):
ax = axs.ravel()[i-1]
ax.imshow(img)
ax.set_xlim(150,650)
ax.set_ylim(550,20)
ax.set_xticks([])
ax.set_yticks([])
ax.set_title("Mode %i @ %0.2fHz"%(i,f_eigen[i]))
Forced Sloshing¶
We compute forced sloshing by coupling the water domain (V_water) and a stiff glass container (V_glass) at the common interface (S_glass-water-bottom and S_glass-water-wall). On the bottom of the container (S_glass-bottom) we prescribe a harmonic displacement.
Sway¶
The XML input for this case is Forced_Sway.xml.
We load the pressure data (in the fluid) in and plot the maximum pressure (proportional to the surface elevation). There are peaks at some natural frequencies (all marked by vertical lines). The peaks mark the modes wich are excited by the sway motion.
h5 = 'results_hdf5/Forced_Sway.cfs'
P = get_result(h5,'waterPressure',step='all')
f = get_step_values(h5)[0]
I_surface = get_subregion_idx(h5,'V_water','S_surface')
f = get_step_values(h5)[0]
X = get_coordinates(h5,'V_water')
I_left = np.argmin(np.linalg.norm(np.abs(X - np.array([X[:,0].min(),0,X[:,2].max()])),axis=1)).ravel() # indices of the surface points with y = 0
I_right = np.argmin(np.linalg.norm(np.abs(X - np.array([X[:,0].max(),0,X[:,2].max()])),axis=1)).ravel() # indices of the surface points with y = 0
fig,axs=plt.subplots(nrows=2,figsize=[2.7,2.2], gridspec_kw={'height_ratios': [3, 1]})
ax = axs[0]
ax.semilogy(f,np.max(np.abs(P[:,I_surface]),axis=1),'k',label='max' )
ax.semilogy(f,np.abs(P[:,I_left]),label='left' )
ax.semilogy(f,np.abs(P[:,I_right]),label='right' )
#ax.semilogy(f,np.abs(np.mean(P[:,I_surfleft],axis=1)),label='surfleft' )
ax.set_ylabel('Amplitude\nin Pa')
ax.set_title('Pressure')
ax = axs[1]
ax.plot(f,np.angle(P[:,I_left]),label='left' )
ax.plot(f,np.angle(P[:,I_right]),label='right' )
axs[-1].set_xlabel('Frequency in Hz')
axs[0].legend()
axs[0].set_xticklabels([])
#ax.set_ylim(ax.get_ylim())
for ax in axs:
ax.set_xlim(0,f[-1])
lim = ax.get_ylim()
ax.set_ylim(lim)
ax.vlines(f_eigen,*lim,lw=0.3,color='k',ls='--');
We can also export an animation of the forced oscillation in Paraview. See the state file Forced_Sway_animation.pvsm. Coose to save PNG files and then convert them into an animated gif using imagemagic convert -delay 10 -loop 0 forced-sway-6.5Hz.*.png forced-sway-6.5Hz.gif:
Pitch¶
The XML input for this case is Forced_Pitch.xml.
We load the pressure data (in the fluid) in and plot the maximum pressure (proportional to the surface elevation). There are peaks at some natural frequencies (all marked by vertical lines). The peaks mark the modes wich are excited by the sway motion.
h5 = 'results_hdf5/Forced_Pitch.cfs'
P = get_result(h5,'waterPressure',step='all')
I_surface = get_subregion_idx(h5,'V_water','S_surface')
f = get_step_values(h5)[0]
X = get_coordinates(h5,'V_water')
I_left = np.argmin(np.linalg.norm(np.abs(X - np.array([X[:,0].min(),0,X[:,2].max()])),axis=1)).ravel() # indices of the surface points with y = 0
I_right = np.argmin(np.linalg.norm(np.abs(X - np.array([X[:,0].max(),0,X[:,2].max()])),axis=1)).ravel() # indices of the surface points with y = 0
fig,axs=plt.subplots(nrows=2,figsize=[2.7,2.2], gridspec_kw={'height_ratios': [3, 1]})
ax = axs[0]
ax.semilogy(f,np.max(np.abs(P[:,I_surface]),axis=1),'k',label='max' )
ax.semilogy(f,np.abs(P[:,I_left]),label='left' )
ax.semilogy(f,np.abs(P[:,I_right]),label='right' )
ax.set_ylabel('Amplitude\nin Pa')
ax.set_title('Pressure')
ax = axs[1]
ax.plot(f,np.angle(P[:,I_left]),label='left' )
ax.plot(f,np.angle(P[:,I_right]),label='right' )
axs[-1].set_xlabel('Frequency in Hz')
axs[0].legend()
axs[0].set_xticklabels([])
#ax.set_ylim(ax.get_ylim())
for ax in axs:
ax.set_xlim(0,f[-1])
lim = ax.get_ylim()
ax.set_ylim(lim)
ax.vlines(f_eigen,*lim,lw=0.3,color='k',ls='--');
We can also animate the surface pressure using matplotib animations. Instead of showing the animation directly in the notebook (HTML(anim.to_jshtml())) we export it to gif and include it via a markdown cell below (this keeps the text file small).
from matplotlib.animation import FuncAnimation
from IPython.display import HTML
# frequency index
i = 20
# ge the data
x = X[I_surface,0]
y = X[I_surface,1]
z = P[i,I_surface]
Pmax = np.max(np.abs(z))
fig,ax=plt.subplots()
lvls = np.linspace(-Pmax,Pmax,21)
#tri = ax.tricontour(x, y, z, levels=lvls, linewidths=0.5, colors='k')
cnt = ax.tricontourf(x, y, z.real, levels=lvls, cmap="RdBu_r")
#cnt.set_array(np.abs(P[0,I_surface]))
#cbar = fig.colorbar(cnt, ax=ax)
#cbar.set_label('Pressure in Pa', rotation=270)
ax.set_xticks([])
ax.set_yticks([])
ax.plot(x, y, 'ko', ms=0.1)
ax.set_aspect('equal')
#ax.set_title('Surface pressure in Pa')
fig.tight_layout()
# the animation update function
N = 25 # frames of the animation
def animate(n):
zn = np.real( z*np.exp(2j*np.pi*n/N) )
# delete old contours
global cnt
for c in cnt.collections:
c.remove()
# re-create
cnt = ax.tricontourf(x, y, zn, levels=lvls, cmap="RdBu_r")
return cnt
# do the animation
anim = FuncAnimation(fig, animate, frames=N, interval=100)
name = 'surface-pitch_%.2fHz.gif'%(f[i])
anim.save(name, writer='imagemagick', fps=20)
# show it in the notebook, commented out to keep notebook file short
# HTML(anim.to_jshtml())
Here is the response at at 2.1Hz
Plotting the surface pressure, which is proportional to the surface elevation, at the line $y=0$ is also possible
# select and sort
I_0 = np.argwhere(np.abs(X[I_surface,1])<0.0001).ravel() # indices of the surface points with y = 0
I_s = np.argsort(X[I_surface,0][I_0]).ravel() # sorting indices
I_0 = I_0[I_s]
# extract data
x_0 = X[I_surface,0][I_0]
P_0 = P[:,I_surface][:,I_0]
# plot for selected frequencies
If = np.array([1,20,37,84])
fig,ax=plt.subplots()
ax.plot(x_0, P_0[If,:].real.T, ms=0.1)
ax.legend(['%.2f Hz'%(val) for val in f[If]])
ax.set_xlabel('x in m')
ax.set_xlim([x_0.min(),x_0.max()])
ax.set_ylabel('Pressure in Pa')
Text(0, 0.5, 'Pressure in Pa')
Comparision and Control¶
What is the optimal pitch value such that for a given sway amplitude there is no surface displacement?
The surface response for forced (pure) sway was computed above. It can be described by the transfer function
H_u = \frac{\hat{\eta}_u}{\hat{u}}
where $\hat{u}$ denotes the sway excitation. The surface response amplitude $\hat{\eta}$ can be given at an arbitrary point, or as an average measure to be determined. Simularly we have determined the pitch response by above computation
H_\beta = \frac{\hat{\eta}_\beta}{\hat{\beta}}
where $\hat{\beta}$ is the prescribed pitch amplitude.
Let's compare both transfer functions. Note that the reference sway amplitude was 1cm and the reference pitch angle was 1deg.
P_pitch = get_result('results_hdf5/Forced_Pitch.cfs','waterPressure',step='all')
P_sway = get_result('results_hdf5/Forced_Sway.cfs','waterPressure',step='all')
fig,axs=plt.subplots(nrows=2,figsize=[2.7,2.2], gridspec_kw={'height_ratios': [3, 1]})
ax = axs[0]
#ax.semilogy(f,np.max(np.abs(P[:,I_surface]),axis=1),'k',label='max' )
ax.semilogy(f,np.abs(P_sway[:,I_left]),label='sway' )
ax.semilogy(f,np.abs(P_pitch[:,I_left]),label='pitch' )
#ax.semilogy(f,np.abs(P[:,I_right]),label='right' )
ax.set_ylabel('Amplitude\nin Pa')
ax.set_title('Pressure')
ax = axs[1]
ax.plot(f,np.angle(P_sway[:,I_left]),label='sway' )
ax.plot(f,np.angle(P_pitch[:,I_right]),label='pitch' )
axs[-1].set_xlabel('Frequency in Hz')
axs[0].legend()
axs[0].set_xticklabels([])
#ax.set_ylim(ax.get_ylim())
for ax in axs:
ax.set_xlim(0,f[-1])
lim = ax.get_ylim()
ax.set_ylim(lim)
ax.vlines(f_eigen,*lim,lw=0.3,color='k',ls='--');
Based on the know surface displacement transfer functions we can now compute the required pitch control. For this we define the mean surface pressure on one side (left) of the glass as a measure for the surface displacement. Since we are dealing with linear systems the total surface displacement is a linear superposition of both load cases, and we require it to be zero, i.e.
\hat{\eta} = \hat{\eta}_u + \hat{\eta}_\beta = 0
thus, using the tranfer functions we can compute the required pitch angle from
\hat{\beta} = -\frac{ H_u }{H_\beta} \hat{u}
Of course one obtains differnt results if a different measure for the surface displacment is taken. Below we plot the result for mean (as above) and the left and right most points on the surface, respectively.
# copute the pitch control
I_surfleft= I_surface[np.argwhere(X[I_surface,0]<=0).ravel()]
H_sway = np.mean(P_sway[:,I_surfleft],axis=1)
H_pitch = np.mean(P_pitch[:,I_surfleft],axis=1)
pitch_control = -H_sway/H_pitch
# plot
fig,axs=plt.subplots(nrows=2,figsize=[2.7,2.2], gridspec_kw={'height_ratios': [3, 1]})
# amplitude
ax = axs[0]
ax.plot(f,np.abs((-P_sway/P_pitch)[:,I_left]),label='left' )
ax.plot(f,np.abs((-P_sway/P_pitch)[:,I_right]),label='right' )
ax.semilogy(f,np.abs(pitch_control),label='mean' )
ax.set_ylabel('Amplitude\nin deg/cm')
ax.set_title('$\\beta/u$')
# phase
ax = axs[1]
ax.plot(f,np.angle((-P_sway/P_pitch)[:,I_left]),label='left' )
ax.plot(f,np.angle((-P_sway/P_pitch)[:,I_right]),label='right' )
ax.plot(f,np.angle(pitch_control),label='mean' )
# cosmetics
axs[-1].set_xlabel('Frequency in Hz')
axs[0].legend()
axs[0].set_xticklabels([])
for ax in axs:
ax.set_xlim(0,f[-1])
lim = ax.get_ylim()
ax.set_ylim(lim)
ax.vlines(f_eigen,*lim,lw=0.3,color='k',ls='--'); # modes
Now we sace the result to text files to use in a validation computation with CFS. The same result could be obtained by linear superposition of the results from pitch and sway case.
np.savetxt('control_value.dat',np.vstack([f,np.abs(pitch_control)]).T,fmt='%.6e')
np.savetxt('control_angle.dat',np.vstack([f,np.angle(pitch_control)]).T,fmt='%.6e')
np.savetxt('control_real.dat',np.vstack([f,np.real(pitch_control)]).T,fmt='%.6e')
np.savetxt('control_imag.dat',np.vstack([f,np.imag(pitch_control)]).T,fmt='%.6e')
The results are obtained from the input file Forced_control.xml. We compare the response (maximum pressure) from the original sway case and the controlled one.
P_sway = get_result('results_hdf5/Forced_Sway.cfs','waterPressure',step='all')
P_swayControl = get_result('results_hdf5/Forced_control.cfs','waterPressure',step='all')
fig,axs=plt.subplots(nrows=2,figsize=[2.7,2.2], gridspec_kw={'height_ratios': [3, 1]})
ax = axs[0]
ax.semilogy(f,np.max(np.abs(P_sway[:,I_surface]),axis=1),label='sway' )
ax.semilogy(f,np.max(np.abs(P_swayControl[:,I_surface]),axis=1),label='control' )
ax.set_ylabel('Amplitude\nin Pa')
ax.set_title('Pressure')
from scipy.signal import find_peaks
y = np.max(np.abs(P_swayControl[:,I_surface]),axis=1)
Ip,_ = find_peaks( y )
ax.semilogy(f[Ip],y[Ip],'o',ms=1)
for i in Ip:
ax.annotate('$f_{%i}$=%.1fHz'%(i,f[i]),(f[i],y[i]))
ax = axs[1]
ax.plot(f,np.angle(P[:,I_left]),label='left' )
ax.plot(f,np.angle(P[:,I_right]),label='right' )
axs[-1].set_xlabel('Frequency in Hz')
axs[0].legend()
axs[0].set_xticklabels([])
#ax.set_ylim(ax.get_ylim())
for ax in axs:
ax.set_xlim(0,f[-1])
lim = ax.get_ylim()
ax.set_ylim(lim)
ax.vlines(f_eigen,*lim,lw=0.3,color='k',ls='--');
Clearly the the control works well, reducing the pressure up to several orders of magnitide. Especially first and second mode can be supressed. However, there are some new peaks at higher frequencies (above 6 Hz). Let's see how the response looks like for those peaks.
i = Ip[1]
anims = []
for i in Ip:
# ge the data
x = X[I_surface,0]
y = X[I_surface,1]
z = P_swayControl[i,I_surface]
Pmax = np.max(np.abs(z))
fig,ax=plt.subplots()
lvls = np.linspace(-Pmax,Pmax,21)
#tri = ax.tricontour(x, y, z, levels=lvls, linewidths=0.5, colors='k')
cnt = ax.tricontourf(x, y, z.real, levels=lvls, cmap="RdBu_r")
#cnt.set_array(np.abs(P[0,I_surface]))
#fig.colorbar(cnt, ax=ax)
ax.set_xticks([])
ax.set_yticks([])
ax.plot(x, y, 'ko', ms=0.1)
ax.set_aspect('equal')
# the animation update function
N = 25 # frames of the animation
def animate(n):
zn = np.real( z*np.exp(2j*np.pi*n/N) )
# delete old contours
global cnt
for c in cnt.collections:
c.remove()
# re-create
cnt = ax.tricontourf(x, y, zn, levels=lvls, cmap="RdBu_r")
return cnt
# do the animation
anim = FuncAnimation(fig, animate, frames=N, interval=100)
# save in list
anims.append(anim)
# export gif
name = 'surface-control-peak_%.2fHz.gif'%(f[i])
anim.save(name, writer='imagemagick', fps=20)
print(''%(name,name))
# no
# show animation in the notebook (commented out to avoid long text in notebook)
# HTML(anims[0].to_jshtml())
Clearly there are some modes that cannot be supressed by pitch control.
References¶
1. Toth, F., and Kaltenbacher, M. (2016) Fully coupled linear modelling of incompressible free-surface flow, compressible air and flexible structures. Int. J. Numer. Meth. Engng, 107: 947– 969. doi: 10.1002/nme.5194 | 2023-03-26 04:56:44 | {"extraction_info": {"found_math": true, "script_math_tex": 4, "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": 0.36401036381721497, "perplexity": 12567.550242858977}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945433.92/warc/CC-MAIN-20230326044821-20230326074821-00707.warc.gz"} |
https://crypto.ethz.ch/publications/AOZZ15.html | # Information Security and Cryptography Research Group
## Incoercible Multi-Party Computation and Universally Composable Receipt-Free Voting
### Joël Alwen, Rafail Ostrovsky, Hong-Sheng Zhou, and Vassilis Zikas
Advances in Cryptology – CRYPTO 2015, LNCS, Springer, vol. 9216, pp. 763-780, Aug 2015.
Composable notions of incoercibility aim to forbid a coercer from using anything beyond the coerced parties inputs and outputs to catch them when they try to deceive him. Existing definitions are restricted to weak coercion types, and/or are not universally composable. Furthermore, they often make too strong assumptions on the knowledge of coerced parties—e.g., they assume they known the identities and/or the strategies of other coerced parties, or those of corrupted parties—which makes them unsuitable for applications of incoercibility such as e-voting, where colluding adversarial parties may attempt to coerce honest voters, e.g., by offering them money for a promised vote, and use their own view to check that the voter keeps his end of the bargain. In this work we put forward the first universally composable notion of incoercible multi-party computation, which satisfies the above intuition and does not assume collusions among coerced parties or knowledge of the corrupted set. We define natural notions of UC incoercibility corresponding to standard coercion-types, i.e., receipt-freeness and resistance to full-active coercion. Importantly, our suggested notion has the unique property that it builds on top of the well studied UC framework by Canetti instead of modifying it. This guarantees backwards compatibility, and allows us to inherit results from the rich UC literature. We then present MPC protocols which realize our notions of UC incoercibility given access to an arguably minimal setup—namely honestly generate tamper-proof hardware performing a very simple cryptographic operation—e.g., a smart card. This is, to our knowledge, the first proposed construction of an MPC protocol (for more than two parties) that is incoercibly secure and universally composable, and therefore the first construction of a universally composable receipt-free e-voting protocol.
## BibTeX Citation
@inproceedings{AOZZ15,
author = {Joël Alwen and Rafail Ostrovsky and Hong-Sheng Zhou and Vassilis Zikas},
title = {Incoercible Multi-Party Computation and Universally Composable Receipt-Free Voting},
editor = {R. Gennaro and M. Robshaw},
booktitle = {Advances in Cryptology -- CRYPTO 2015},
pages = 763-780,
series = {LNCS},
volume = 9216,
year = 2015,
month = 8,
publisher = {Springer},
} | 2022-01-22 01:54:57 | {"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": 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": 0.5632628798484802, "perplexity": 5862.104274811911}, "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/1642320303729.69/warc/CC-MAIN-20220122012907-20220122042907-00697.warc.gz"} |
https://blancosilva.wordpress.com/2008/09/26/the-hunt-for-a-bellman-function-2/ | Home > Analysis > The hunt for a Bellman Function.
## The hunt for a Bellman Function.
This is a beautiful and powerful mathematical technique in Harmonic Analysis that allows, among other things, to prove very complicated inequalities in the theory of Singular Integral Operators, without using much of the classical machinery in this field.
The Bellman function was the tool that allowed their creators (Fedor Nazarov and Sergei Treil) to crack the problem of weighted norm inequalities with matrix weights for the case $\boldsymbol{p} \neq \boldsymbol{2}$ and finally solve it completely.
Copies of the original paper can be found at the authors’ pages; e.g. [www.math.brown.edu/~treil/papers/bellman/bell3.ps] (notice the postscript file is huge, as the article has more than 100 pages).
Let me illustrate the use of Bellman functions to solve a simple problem:
Dyadic-$\boldsymbol{L}_\mathbf{2}(\mathbb{R})$ version of the Carleson Imbedding Theorem
Let $\mathcal{D}$ be the set of all dyadic intervals of the real line. Given a function $f \in L_1^{\text{loc}}(\mathbb{R})$, consider the averages $\langle f \rangle_I = \lvert I\rvert^{-1} \int_I f$, on each dyadic interval $I \in \mathcal{D}$.
Let $\{ \mu_I \geq 0 \colon I \in \mathcal{D} \}$ be a family of non-negative real values satisfying the Carleson measure condition—that is, for any dyadic interval $I \in \mathcal{D}$, $\sum_{J \subset I, J~\text{dyadic}} \mu_J \leq \lvert I \rvert.$
Then, there is a constant $C>0$ such that for any $f \in L_2(\mathbb{R})$,
$\displaystyle{\sum_{ I \in \mathcal{D} } \mu_I \lvert \langle f \rangle_{I} \rvert^2 \leq C \lVert f \rVert_{L_2(\mathbb{R})}^2}$
Fix a dyadic interval $I \in \mathcal{D}$, and a vector $(x_1, x_2, x_3) \in \mathbb{R}^3$. Consider all families $\{\mu_I \colon I \in \mathcal{D} \}$ satisfying the Carleson condition
$\frac{1}{\lvert J \rvert} \displaystyle{\sum_{K \subset J}} \mu_{K} \leq 1, \text{ for all }J \in \mathcal{D}$
and such that
(eq1) $\displaystyle{\frac{1}{\lvert I \rvert} \sum_{J \subset I} \mu_J = x_1}$.
Also, consider all functions $f \in L_2(\mathbb{R})$ for which the following quantities are fixed:
(eq2) $\displaystyle{\langle f^2 \rangle_I = \frac{1}{\lvert I \rvert} \int_I f^2 = x_2,\qquad \langle f \rangle_I = \frac{1}{\lvert I \rvert} \int_I f = x_3}$
If we believe that the Theorem is true, then the quantity
$\displaystyle{\mathcal{B}(x_1,x_2,x_3)=\frac{1}{\lvert I \rvert} \sup \bigg\{ \sum_{J \subset I} \mu_J \langle f \rangle^2_J \colon f, \{ \mu_I \} \text{ satisfy }(eq1),(eq2) \bigg\}}$
is finite and, moreover, satisfies the inequality $\mathcal{B}(x_1,x_2,x_3) \leq C x_2$.
Since $\mathcal{B}(x_1,x_2,x_3)$ does not depend on the choice of an interval $I \in \mathcal{D}$, we obtain a function of three real variables; this is the Bellman function associated with the Carleson Imbedding Theorem.
Notice that:
1. The domain of $\mathcal{B}$ is the set $\{ (x_1, x_2, x_3) \in \mathbb{R}^3 \colon 0 \leq x_1 \leq 1, x_3^2 \leq x_2 \}.$
2. For each $(x_1,x_2,x_3)$ in the domain of $\mathcal{B}$, it is $0 \leq \mathcal{B}(x_1, x_2, x_3) \leq C x_2.$
3. If $0 \leq \lambda \leq x_1$, then
$\mathcal{B}(x_1, x_2, x_3)\geq \lambda x_2^2 + \frac{1}{2} \big\{ \mathcal{B}(x_1^+, x_2^+, x_3^+) + \mathcal{B}(x_1^-, x_2^-, x_3^-)\big\}$
whenever the triples $(x_1,x_2,x_3)$, $(x_1^+,x_2^+,x_3^+)$ and $(x_1^-,x_2^-,x_3^-)$ belong to the domain and
• $x_1 = \frac{1}{2}(x_1^+ + x_1^-) + \lambda$,
• $x_2 = \frac{1}{2}(x_2^+ + x_2^-)$,
• $x_3 = \frac{1}{2}(x_3^+ + x_3^-).$
The entire machine can be run backward: if we have any function $\mathcal{B}$ of three real variables that satisfies properties 1—3, the proof of the Theorem follows immediately. The key property 3 is not very pleasant to verify. Fortunately, this condition can be replaced by “infinitesimal” conditions (conditions on derivatives), which are easier to check: If $x_1 = \frac{1}{2}(x_1^+ + x_1^-)$, $x_2 = \frac{1}{2}(x_2^+ + x_2^-)$ and $x_3 = \frac{1}{2}(x_3^+ + x_3^-)$, and all triples are in the domain of $\mathcal{B}$, then the key property 3 implies the concavity of $\mathcal{B}$:
$\mathcal{B}(x_1,x_2,x_3) \geq \frac{1}{2} \big\{ \mathcal{B}(x_1^+,x_2^+,x_3^+) + \mathcal{B}(x_1^-,x_2^-,x_3^-)\big\}$
and furthermore,
(eq3) $\displaystyle{d^2 \mathcal{B} \leq 0, \qquad \frac{\partial \mathcal{B}}{\partial x_1} \geq x_3^2}$
Notice that condition 3 is equivalent to (eq3). The following function satisfies 1, 2 and (eq3), and thus the Theorem is proven for $C=4$.
$\mathcal{B}(x_1, x_2, x_3) = 4\bigg( \displaystyle{x_2 - \frac{x_3^2}{1+x_1}}\bigg)$ | 2017-02-20 01:44:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 46, "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": 0.9228624701499939, "perplexity": 224.6230682332179}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170380.12/warc/CC-MAIN-20170219104610-00117-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://myvocabbook.com/download/handbook-of-algebraic-topology | ## Handbook of Algebraic Topology
Author: I.M. James
Publisher: Elsevier
ISBN: 0080532985
Release Date: 1995-07-18
Genre: Mathematics
Algebraic topology (also known as homotopy theory) is a flourishing branch of modern mathematics. It is very much an international subject and this is reflected in the background of the 36 leading experts who have contributed to the Handbook. Written for the reader who already has a grounding in the subject, the volume consists of 27 expository surveys covering the most active areas of research. They provide the researcher with an up-to-date overview of this exciting branch of mathematics.
## Handbook of Algebra and Algebraic Topology
Author: Joe Kaminski
Publisher:
ISBN: 1781540853
Release Date: 2012-09
Genre: Algebra
In algebra the topics covered generally included operations with literal expressions, the solving of both linear and quadratic equations, the use of these techniques to find answers to problems, and practice with ratios, proportions, powers, and roots. Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. This book examines this topic.
## Homotopy Methods in Algebraic Topology
Author: John Patrick Campbell Greenlees
Publisher: American Mathematical Soc.
ISBN: 9780821826218
Release Date: 2001
Genre: Mathematics
This volume presents the proceedings from the AMS-IMS-SIAM Summer Research Conference on Homotopy Methods in Algebraic Topology held at the University of Colorado (Boulder). The conference coincided with the sixtieth birthday of J. Peter May. An article is included reflecting his wide-ranging and influential contributions to the subject area. Other articles in the book discuss the ordinary, elliptic and real-oriented Adams spectral sequences, mapping class groups, configuration spaces, extended powers, operads, the telescope conjecture, $p$-compact groups, algebraic K theory, stable and unstable splittings, the calculus of functors, the $E_{\infty}$ tensor product, and equivariant cohomology theories. The book offers a compendious source on modern aspects of homotopy theoretic methods in many algebraic settings.
## Topological and Algebraic Structures in Fuzzy Sets
Author: S.E. Rodabaugh
Publisher: Springer Science & Business Media
ISBN: 9789401702317
Release Date: 2013-03-14
Genre: Mathematics
This volume summarizes recent developments in the topological and algebraic structures in fuzzy sets and may be rightly viewed as a continuation of the stan dardization of the mathematics of fuzzy sets established in the "Handbook", namely the Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Volume 3 of The Handbooks of Fuzzy Sets Series (Kluwer Academic Publish ers, 1999). Many of the topological chapters of the present work are not only based upon the foundations and notation for topology laid down in the Hand book, but also upon Handbook developments in convergence, uniform spaces, compactness, separation axioms, and canonical examples; and thus this work is, with respect to topology, a continuation of the standardization of the Hand book. At the same time, this work significantly complements the Handbook in regard to algebraic structures. Thus the present volume is an extension of the content and role of the Handbook as a reference work. On the other hand, this volume, even as the Handbook, is a culmination of mathematical developments motivated by the renowned International Sem inar on Fuzzy Set Theory, also known as the Linz Seminar, held annually in Linz, Austria. Much of the material of this volume is related to the Twenti eth Seminar held in February 1999, material for which the Seminar played a crucial and stimulating role, especially in providing feedback, connections, and the necessary screening of ideas.
## Handbook of the History of General Topology
Author: C.E. Aull
Publisher: Springer Science & Business Media
ISBN: 0792344790
Release Date: 1997-03-31
Genre: Mathematics
This book is the first one of a work in several volumes, treating the history of the development of topology. The work contains papers which can be classified into 4 main areas. Thus there are contributions dealing with the life and work of individual topologists, with specific schools of topology, with research in topology in various countries, and with the development of topology in different periods. The work is not restricted to topology in the strictest sense but also deals with applications and generalisations in a broad sense. Thus it also treats, e.g., categorical topology, interactions with functional analysis, convergence spaces, and uniform spaces. Written by specialists in the field, it contains a wealth of information which is not available anywhere else.
## Homology Theory
Author: P. J. Hilton
Publisher: CUP Archive
ISBN: 0521094224
Release Date: 1967
Genre: Mathematics
This account of algebraic topology is complete in itself, assuming no previous knowledge of the subject. It is used as a textbook for students in the final year of an undergraduate course or on graduate courses and as a handbook for mathematicians in other branches who want some knowledge of the subject.
## Handbook of Algebra
Author: M. Hazewinkel
Publisher: Elsevier
ISBN: 0080532969
Release Date: 2000-04-06
Genre: Mathematics
Handbook of Algebra
## Handbook of Mathematics
Author: Vialar Thierry
Publisher: BoD - Books on Demand
ISBN: 9782955199015
Release Date: 2017-04-04
Genre:
The book consists of XI Parts and 28 Chapters covering all areas of mathematics. It is a tool for students, scientists, engineers, students of many disciplines, teachers, professionals, writers and also for a general reader with an interest in mathematics and in science. It provides a wide range of mathematical concepts, definitions, propositions, theorems, proofs, examples, and numerous illustrations. The difficulty level can vary depending on chapters, and sustained attention will be required for some. The structure and list of Parts are quite classical: I. Foundations of Mathematics, II. Algebra, III. Number Theory, IV. Geometry, V. Analytic Geometry, VI. Topology, VII .Algebraic Topology, VIII. Analysis, IX. Category Theory, X. Probability and Statistics, XI. Applied Mathematics. Appendices provide useful lists of symbols and tables for ready reference. The publisher’s hope is that this book, slightly revised and in a convenient format, will serve the needs of readers, be it for study, teaching, exploration, work, or research.
## Handbook of Teichm ller Theory
Publisher: European Mathematical Society
ISBN: 3037191031
Release Date: 2012
Genre: Mathematics
## Handbook of Tilting Theory
Author: Lidia Angeleri Hügel
Publisher: Cambridge University Press
ISBN: 052168045X
Release Date: 2007-01-04
Genre: Mathematics
A handbook of key articles providing both an introduction and reference for newcomers and experts alike.
## Homotopy Type and Homology
Author: Hans J. Baues
Publisher: Oxford University Press
ISBN: 0198514824
Release Date: 1996
Genre: Mathematics
This book represents a new attempt to classify homotopy types of simply connected CW-complexes. It provides methods and examples of explicit homotopy classification and includes applications to the classification of manifolds.
## Handbook of K Theory
Author: Eric Friedlander
Publisher: Springer Science & Business Media
ISBN: 9783540230199
Release Date: 2005-07-18
Genre: Mathematics
This handbook offers a compilation of techniques and results in K-theory. Each chapter is dedicated to a specific topic and is written by a leading expert. Many chapters present historical background; some present previously unpublished results, whereas some present the first expository account of a topic; many discuss future directions as well as open problems. It offers an exposition of our current state of knowledge as well as an implicit blueprint for future research.
## Handbook of Algebra
Author:
Publisher: Elsevier
ISBN: 0080532977
Release Date: 2003-10-15
Genre: Mathematics
Handbook of Algebra
## Handbook of Categorical Algebra Volume 2 Categories and Structures
Author: Francis Borceux
Publisher: Cambridge University Press
ISBN: 052144179X
Release Date: 1994-11-03
Genre: Mathematics
The Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence. The second, which assumes familiarity with the material in the first, introduces important classes of categories that have played a fundamental role in the subject's development and applications. In addition, after several chapters discussing specific categories, the book develops all the major concepts concerning Benabou's ideas of fibred categories. There is ample material here for a graduate course in category theory, and the book should also serve as a reference for users.
## Manifolds and Modular Forms
Author: Friedrich Hirzebruch
Publisher: Springer-Verlag
ISBN: 9783663140450
Release Date: 2013-09-03
Genre: Mathematics | 2019-02-18 16:19:04 | {"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": 0.6053049564361572, "perplexity": 1848.9389715614088}, "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/1550247487595.4/warc/CC-MAIN-20190218155520-20190218181520-00285.warc.gz"} |
https://www.physicsforums.com/threads/a-hamiltonial-question.231271/ | A hamiltonial question
1. Apr 25, 2008
Bunting
So im working up to some exams and have a question regarding properties of hermitians, specifically the properties of Hamiltonian operators and trying to prove that for example if..
$$\hat{O}$$ is a hamiltonian operator then...
$$\hat{O}$$ + $$\hat{O}$$$$\dagger$$
is hermitian*.
Now what I think im having a problem with is understanding exactly what im expected to know with regard to this, as what I know about hamiltonian operators (real eigenvalues and orthogonality) don't seem to help a massive amount here (unless im meant to show that $$\hat{O}$$ with $$\hat{O}$$$$\dagger$$ is orthogonal).
Any help is appreciated, I feel this is one of them subjects where if I start to understand with one example like this I will be able to nail the rest out pretty quickly :)
*In case im explaining badly due to my limited knowledge of hermitian and hamiltonian things, the exact question says...
Show for any operator $$\hat{O}$$, that $$\hat{O}$$ + $$\hat{O}$$$$\dagger$$ is Hermitian.
edit: sigh, spelt the title wrong :(
Last edited: Apr 25, 2008
2. Apr 25, 2008
lbrits
What is a Hamiltonian operator? You mean THE hamiltonian? Or did you mean to say a Hermitian operator? Or a Hilbert operator?
$$O + O^\dagger$$ is always Hermitian. Use the fact that $${O^\dagger}^\dagger = O$$.
3. Apr 25, 2008
olgranpappy
and use the fact that
$$A+B=B+A$$
4. Apr 26, 2008
Bunting
Sorry, I think I meant Hermitian operators. Thank you for the replies but it doesnt help me very much but I think thats maybe because im asking hthe question wrong! :S
What im asking is how I would recognise the answer as a Hermitian in particular? Is it hermitian because...
($$\hat{O}$$$$^{dagger}$$)$$^{dagger}$$
is $$\hat{O}$$ and thus Hermitian and thus because Hermitian Operators are commutative Hermitian + Hermitian = Hermitian ?
5. Apr 26, 2008
malawi_glenn
Well you basically have everything you need:
i) a hermitian operator fulfills: $$\hat{O}^{\dagger} = \hat{O}$$
ii) $$A+B=B+A$$
Then what is:
$$(\hat{O} + \hat{O}^{\dagger}) ^{\dagger}$$
?
6. Apr 26, 2008
Bunting
oh i see, so...
$$(\hat{O} + \hat{O}^{\dagger}) ^{\dagger}$$ = $$\hat{O}^{\dagger} + \hat{O}^{\dagger}^{\dagger}$$ = $$\hat{O} + \hat{O}^{\dagger}$$
thus proving it is hermitian. Ok, so, in a similar vein...
$$\hat{O}\hat{O}^{\dagger}$$ = $$(\hat{O}\hat{O}^{\dagger}) ^{\dagger}$$ = $$\hat{O}^{\dagger}\hat{O}^{\dagger}^{\dagger}$$ = $$\hat{O}\hat{O}^{\dagger}$$
thus proving IT is hermitian ?
7. Apr 26, 2008
George Jones
Staff Emeritus
The end result is correct, but the second-last equality is wrong.
8. Apr 26, 2008
malawi_glenn
I dont understand, you now want to PROOVE that $$\hat{O}$$ is a hermitian operator? That is a property that is given to you as a fact, you can't proove that unless you know what $$\hat{O}$$ explicity is. Or do you want to proove that given $$\hat{O}$$ is hermitian, the product $$\hat{O}\hat{O}^{\dagger}$$ is hermitian?
By the way: $$(AB)^{\dagger} = B^{\dagger}A^{\dagger}$$ so:
$$(\hat{O}\hat{O}^{\dagger}) ^{\dagger} = (\hat{O}^{\dagger})^{\dagger}\hat{O}^{\dagger}$$
Last edited: Apr 26, 2008
9. Apr 26, 2008
Bunting
Yes thats correct :) Sorry, I have difficulty explaining things I dont understand very well, but im getting there.
The point of these seems to be that if you can conjugate the example and get back to your origonal statement then your statement is Hermitian (or at least this is the point of the questions it would seem).
Last edited: Apr 26, 2008
10. Apr 26, 2008
malawi_glenn
yes, that is the thing you want to do. Then you must do as I told you in post #8
11. Apr 26, 2008
Bunting
Aye I did thanks! :)
Great, thank you all for your help!
12. Apr 26, 2008
malawi_glenn
Great, so you agree with me that
$$(\hat{O}\hat{O}^{\dagger}) ^{\dagger} \neq \hat{O}^{\dagger}\hat{O}^{\dagger}^{\dagger}$$
?
13. Apr 26, 2008
Bunting
Yeah, I was basically just being rubbish at maths/not thinking about it properly. | 2016-10-28 01:19:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8735091686248779, "perplexity": 1187.615205121013}, "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-2016-44/segments/1476988721415.7/warc/CC-MAIN-20161020183841-00044-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://stats.stackexchange.com/questions/36093/construction-of-dirichlet-distribution-with-gamma-distribution/154298 | # Construction of Dirichlet distribution with Gamma distribution
Let $X_1,\dots,X_{k+1}$ be mutually independent random variables, each having a gamma distribution with parameters $\alpha_i,i=1,2,\dots,k+1$ show that $Y_i=\frac{X_i}{X_1+\cdots+X_{k+1}},i=1,\dots,k$, have a joint ditribution as $\text{Dirichlet}(\alpha_1,\alpha_2,\dots,\alpha_k;\alpha_{k+1})$
Joint pdf of $(X_1,\dots,X_{k+1})=\frac{e^{-\sum_{i=1}^{k+1}x_i}x_1^{\alpha_1-1}\dots x_{k+1}^{\alpha_{k+1}-1}}{\Gamma(\alpha_1)\Gamma(\alpha_2)\dots \Gamma(\alpha_{k+1})}$.Then to find joint pdf of $(Y_1,\dots,Y_{k+1})$ I can not find jacobian i.e.$J(\frac{x_1,\dots,x_{k+1}}{y_1,\dots,y_{k+1}})$
• Have a look at pages 13-14 of this document. – user10525 Sep 11 '12 at 14:07
• @Procrastinator Thank you very much your document is best answer for my question. – Argha Sep 11 '12 at 15:36
• @Procrastinator - perhaps you should put this as an answer, since the OP is happy with it, and add a couple of sentences so you don't trip the "we want more than one-sentence answer" warning? – jbowman Sep 11 '12 at 19:39
• That document now is a non-answer because it's a 404. – whuber May 26 '15 at 21:59
• Wayback machine to the rescue: pdf – mobeets Apr 19 '17 at 16:34
Jacobians--the absolute determinants of the change of variable function--appear formidable and can be complicated. Nevertheless, they are an essential and unavoidable part of the calculation of a multivariate change of variable. It would seem there's nothing for it but to write down a $k+1$ by $k+1$ matrix of derivatives and do the calculation.
There's a better way. It's shown at the end in the "Solution" section. Because the purpose of this post is to introduce statisticians to what may be a new method for many, much of it is devoted to explaining the machinery behind the solution. This is the algebra of differential forms. (Differential forms are the things that one integrates in multiple dimensions.) A detailed, worked example is included to help make this become more familiar.
### Background
Over a century ago, mathematicians developed the theory of differential algebra to work with the "higher order derivatives" that occur in multi-dimensional geometry. The determinant is a special case of the basic objects manipulated by such algebras, which typically are alternating multilinear forms. The beauty of this lies in how simple the calculations can become.
Here's all you need to know.
1. A differential is an expression of the form "$dx_i$". It is the concatenation of "$d$" with any variable name.
2. A one-form is a linear combination of differentials, such as $dx_1+dx_2$ or even $x_2 dx_1 - \exp(x_2) dx_2$. That is, the coefficients are functions of the variables.
3. Forms can be "multiplied" using a wedge product, written $\wedge$. This product is anti-commutative (also called alternating): for any two one-forms $\omega$ and $\eta$,
$$\omega \wedge \eta = -\eta \wedge \omega.$$
This multiplication is linear and associative: in other words, it works in the familiar fashion. An immediate consequence is that $\omega \wedge \omega = -\omega \wedge \omega$, implying the square of any one-form is always zero. That makes multiplication extremely easy!
4. For the purposes of manipulating the integrands that appear in probability calculations, an expression like $dx_1 dx_2 \cdots dx_{k+1}$ can be understood as $|dx_1\wedge dx_2 \wedge \cdots \wedge dx_{k+1}|$.
5. When $y = g(x_1, \ldots, x_n)$ is a function, then its differential is given by differentiation:
$$dy = dg(x_1, \ldots, x_n) = \frac{\partial g}{\partial x_1}(x_1, \ldots, x_n) dx_1 + \cdots + \frac{\partial g}{\partial x_1}(x_1, \ldots, x_n) dx_n.$$
The connection with Jacobians is this: the Jacobian of a transformation $(y_1, \ldots, y_n) = F(x_1, \ldots, x_n) = (f_1(x_1, \ldots, x_n), \ldots, f_n(x_1, \ldots, x_n))$ is, up to sign, simply the coefficient of $dx_1\wedge \dots \wedge dx_n$ that appears in computing
$$dy_1 \wedge \cdots \wedge dy_n = df_1(x_1,\ldots, x_n)\wedge \cdots \wedge df_n(x_1, \ldots, x_n)$$
after expanding each of the $df_i$ as a linear combination of the $dx_j$ in rule (5).
### Example
The simplicity of this definition of a Jacobian is appealing. Not yet convinced it's worthwhile? Consider the well-known problem of converting two-dimensional integrals from Cartesian coordinates $(x, y)$ to polar coordinates $(r,\theta)$, where $(x,y) = (r\cos(\theta), r\sin(\theta))$. The following is an utterly mechanical application of the preceding rules, where "$(*)$" is used to abbreviate expressions that will obviously disappear by virtue of rule (3), which implies $dr\wedge dr = d\theta\wedge d\theta = 0$.
\eqalign{ dx dy &= |dx\wedge dy| = |d(r\cos(\theta)) \wedge d(r\sin(\theta))| \\ &= |(\cos(\theta)dr - r\sin(\theta)d\theta) \wedge (\sin(\theta)dr + r\cos(\theta)d\theta| \\ &= |(*)dr\wedge dr + (*) d\theta\wedge d\theta - r\sin(\theta)d\theta\wedge \sin(\theta)dr + \cos(\theta)dr \wedge r\cos(\theta) d\theta| \\ &= |0 + 0 + r\sin^2(\theta) dr\wedge d\theta + r\cos^2(\theta) dr\wedge d\theta| \\ &= |r(\sin^2(\theta) + \cos^2(\theta)) dr\wedge d\theta)| \\ &= r\ dr d\theta }.
The point of this is the ease with which such calculations can be performed, without messing about with matrices, determinants, or other such multi-indicial objects. You just multiply things out, remembering that wedges are anti-commutative. It's easier than what is taught in high school algebra.
### Preliminaries
Let's see this differential algebra in action. In this problem, the PDF of the joint distribution of $(X_1, X_2, \ldots, X_{k+1})$ is the product of the individual PDFs (because the $X_i$ are assumed to be independent). In order to handle the change to the variables $Y_i$ we must be explicit about the differential elements that will be integrated. These form the term $dx_1 dx_2 \cdots dx_{k+1}$. Including the PDF gives the probability element
\eqalign{ f_\mathbf{X}(\mathbf{x},\mathbf{\alpha})dx_1 \cdots dx_{k+1} &\propto \left(x_1^{\alpha_1-1}\exp\left(-x_1\right)\right)\cdots \left(x_{k+1}^{\alpha_{k+1}-1}\exp\left(-x_{k+1}\right) \right)dx_1 \cdots dx_{k+1} \\ &= x_1^{\alpha_1-1}\cdots x_{k+1}^{\alpha_{k+1}-1}\exp\left(-\left(x_1+\cdots+x_{k+1}\right)\right)dx_1 \cdots dx_{k+1}. }
(The normalizing constant has been ignored; it will be recovered at the end.)
Staring at the definitions of the $Y_i$ a few seconds ought to reveal the utility of introducing the new variable
$$Z = X_1 + X_2 + \cdots + X_{k+1},$$
giving the relationships
$$X_i = Y_i Z.$$
This suggests making the change of variables $x_i \to y_i z$ in the probability element. The intention is to retain the first $k$ variables $y_1, \ldots, y_k$ along with $z$ and then integrate out $z$. To do so, we have to re-express all the $dx_i$ in terms of the new variables. This is the heart of the problem. It's where the differential algebra takes place. To begin with,
$$dx_i = d(y_i z) = y_i dz + z dy_i.$$
Note that since $Y_1+Y_2+\cdots+Y_{k+1}=1$, then
$$0 = d(1) = d(y_1 + y_2 + \cdots + y_{k+1}) = dy_1 + dy_2 + \cdots + dy_{k+1}.$$
Consider the one-form
$$\omega = dx_1 + \cdots + dx_k = z(dy_1 + \cdots + dy_k) + (y_1+\cdots + y_k) dz.$$
It appears in the differential of the last variable:
\eqalign{ dx_{k+1} &= z dy_{k+1} + y_{k+1}dz \\ &= -z(dy_1 + \cdots + dy_k) + (1-y_1-\cdots y_k)dz \\ &= dz - \omega. }
The value of this lies in the observation that
$$dx_1 \wedge \cdots \wedge dx_k \wedge \omega = 0$$
because, when you expand this product, there is one term containing $dx_1 \wedge dx_1 = 0$ as a factor, another containing $dx_2 \wedge dx_2 = 0$, and so on: they all disappear. Consequently,
\eqalign{ dx_1 \wedge \cdots \wedge dx_k \wedge dx_{k+1} &= dx_1 \wedge \cdots \wedge dx_k \wedge z - dx_1 \wedge \cdots \wedge dx_k \wedge \omega \\ &= dx_1 \wedge \cdots \wedge dx_k \wedge z. }
Whence (because all products $dz\wedge dz$ disappear),
\eqalign{ dx_1 \wedge \cdots \wedge dx_{k+1} &= (z dy_1 + y_1 dz) \wedge \cdots \wedge (z dy_k + y_k dz) \wedge dz \\ &= z^k dy_1 \wedge \cdots \wedge dy_k \wedge dz. }
The Jacobian is simply $|z^k| = z^k$, the coefficient of the differential product on the right hand side.
### Solution
The transformation $(x_1, \ldots, x_k, x_{k+1})\to (y_1, \ldots, y_k, z)$ is one-to-one: its inverse is given by $x_i = y_i z$ for $1\le i\le k$ and $x_{k+1} = z(1-y_1-\cdots-y_k)$. Therefore we don't have to fuss any more about the new probability element; it simply is
\eqalign{ &(z y_1)^{\alpha_1-1}\cdots (z y_k)^{\alpha_k-1}\left(z(1-y_1-\cdots-y_k)\right)^{\alpha_{k+1}-1}\exp\left(-z\right)|z^k dy_1 \wedge \cdots \wedge dy_k \wedge dz| \\ &= \left(z^{\alpha_1+\cdots+\alpha_{k+1}-1}\exp\left(-z\right) dz\right)\left( y_1^{\alpha_1-1} \cdots y_k^{\alpha_k-1}\left(1-y_1-\cdots-y_k\right)^{\alpha_{k+1}-1}dy_1 \cdots dy_k\right). }
That is manifestly a product of a Gamma$(\alpha_1+\cdots+\alpha_{k+1})$ distribution (for $Z$) and a Dirichlet$(\mathbf\alpha)$ distribution (for $(Y_1,\ldots, Y_k)$). In fact, since the original normalizing constant must have been a product of $\Gamma(\alpha_i)$, we deduce immediately that the new normalizing constant must be divided by $\Gamma(\alpha_1+\cdots+\alpha_{k+1})$, enabling the PDF to be written
$$f_\mathbf{Y}(\mathbf{y},\mathbf{\alpha}) = \frac{\Gamma(\alpha_1+\cdots+\alpha_{k+1})}{\Gamma(\alpha_1)\cdots\Gamma(\alpha_{k+1})}\left( y_1^{\alpha_1-1} \cdots y_k^{\alpha_k-1}\left(1-y_1-\cdots-y_k\right)^{\alpha_{k+1}-1}\right).$$ | 2021-01-17 13:13:36 | {"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": 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": 0.9997648596763611, "perplexity": 693.4922295955685}, "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-2021-04/segments/1610703512342.19/warc/CC-MAIN-20210117112618-20210117142618-00001.warc.gz"} |
https://cs.stackexchange.com/questions/1287/find-subsequence-of-maximal-length-simultaneously-satisfying-two-ordering-constr | # Find subsequence of maximal length simultaneously satisfying two ordering constraints
We are given a set $F=\{f_1, f_2, f_3, …, f_N\}$ of $N$ Fruits. Each Fruit has price $P_i$ and vitamin content $V_i$; we associated fruit $f_i$ with the ordered pair $(P_i, V_i)$. Now we have to arrange these fruits in such a way that the sorted list contains prices in ascending order and vitamin contents in descending order.
Example 1: $N = 4$ and $F = \{(2, 8), (5, 11), (7, 9), (10, 2)\}$.
If we arrange the list such that all price are in ascending order and vitamin contents in descending order, then the valid lists are the following:
• $[(2, 8)]$
• $[(5, 11)]$
• $[(7, 9)]$
• $[(10, 2)]$
• $[(2, 8), (10, 2)]$
• $[(5, 11), (7, 9)]$
• $[(5, 11), (10, 2)]$
• $[(7, 9), (10, 2)]$
• $[(5, 11), (7, 9), (10, 2)]$
From the above lists, I want to choose the list of maximal size. If more than one list has maximal size, we should choose the list of maximal size whose sum of prices is least. The list which should be chosen in the above example is $\{(5, 11), (7, 9), (10, 2)\}$.
Example 2: $N = 10$ and $$F = \{(99,10),(12,23),(34,4),(10,5),(87,11),(19,10), \$$90,18), (43,90),(13,100),(78,65)\} The answer to this example instance is [(13,100),(43,90),(78,65),(87,11),(99,10)]. Until now, this is what I have been doing: 1. Sort the original list in ascending order of price; 2. Find all subsequences of the sorted list; 3. Check whether the subsequence is valid, and compare all valid subsequences. However, this takes exponential time; how can I solve this problem more efficiently? ## 1 Answer A dynamic programming solution would work here, if the vitamin contents come from a finite set (for instance, bounded integers). First, sort the fruits on price ascending and in the cases were two or more fruit have the same price, sort them on vitamin content (descending). Now, define M[f, v] to be the maximal number of fruits in a sublist, containing only the last f fruits (of the sorted list), having a vitamin content of at most v. M[0, *] = 0 and M[f, v] = \begin{cases} \mathrm{max} \{M[f-1, v], 1 + M[f-1, V_f]\} & \text{if \(V_f <= v$$} \\ M[f-1, v] & \text{otherwise} \\ \end{cases}$$ Using dynamic programming gives you a solution that runs in $O(\text{number of fruit} \times \text{possible vitamin values})$.
• ::Can u be more specific please? – Jack Apr 15 '12 at 12:45
• Well, what would you like more details on? Are you not familiar with dynamic programming? – Tom van der Zanden Apr 15 '12 at 14:41 | 2020-04-03 02:28:41 | {"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": 2, "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": 0.4274965822696686, "perplexity": 842.3065720948496}, "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/1585370509103.51/warc/CC-MAIN-20200402235814-20200403025814-00437.warc.gz"} |
http://www.computer.org/csdl/trans/td/2011/07/ttd2011071092-abs.html | Subscribe
Issue No.07 - July (2011 vol.22)
pp: 1092-1104
Jurek Czyzowicz , Universite du Quebec en Outaouais, Gatineau
Leszek Gasieniec , University of Liverpool, Liverpool
Dariusz R. Kowalski , University of Liverpool, Liverpool
Andrzej Pelc , Universite du Quebec en Outaouais, Gatineau
ABSTRACT
We consider deterministic feasibility and time complexity of two fundamental tasks in distributed computing: consensus and mutual exclusion. Processes have different labels and communicate through a multiple access channel. The adversary wakes up some processes in possibly different rounds. In any round, every awake process either listens or transmits. The message of a process i is heard by all other awake processes, if i is the only process to transmit in a given round. If more than one process transmits simultaneously, there is a collision and no message is heard. We consider three characteristics that may or may not exist in the channel: collision detection (listening processes can distinguish collision from silence), the availability of a global clock showing the round number, and the knowledge of the number n of all processes. If none of the above three characteristics is available in the channel, we prove that consensus and mutual exclusion are infeasible; if at least one of them is available, both tasks are feasible, and we study their time complexity. Collision detection is shown to cause an exponential gap in complexity: if it is available, both tasks can be performed in time logarithmic in n, which is optimal, and without collision detection both tasks require linear time. We then investigate both consensus and mutual exclusion in the absence of collision detection, but under alternative presence of the two other features. With global clock, we give an algorithm whose time complexity linearly depends on n and on the wake-up time, and an algorithm whose complexity does not depend on the wake-up time and differs from the linear lower bound only by a factor O(\log^2 n). If n is known, we also show an algorithm whose complexity differs from the linear lower bound only by a factor O(\log^2 n).
INDEX TERMS
Consensus, mutual exclusion, multiple access channel, collision detection.
CITATION
Jurek Czyzowicz, Leszek Gasieniec, Dariusz R. Kowalski, Andrzej Pelc, "Consensus and Mutual Exclusion in a Multiple Access Channel", IEEE Transactions on Parallel & Distributed Systems, vol.22, no. 7, pp. 1092-1104, July 2011, doi:10.1109/TPDS.2010.162
REFERENCES
[1] H. Attiya and J. Welch, "Distributed Computing," John Wiley and Sons, Inc., 2004. [2] R. Bar-Yehuda, O. Goldreich, and A. Itai, "On the Time Complexity of Broadcast in Radio Networks: An Exponential Gap Between Determinism And Randomization," J. Computer and System Sciences, vol.45, pp. 104-126, 1992. [3] M.A. Bender, M. Farach-Colton, S. He, B.C. Kuszmaul, and C.E. Leiserson, "Adversarial Contention Resolution for Simple Channels," Proc. 17th Ann. ACM Symp. Parallel Algorithms (SPAA), pp. 325-332, 2005 [4] M. Bienkowski, M. Klonowski, M. Korzeniowski, and D.R. Kowalski, "Dynamic Sharing of a Multiple Access Channel," Proc. 27th Int'l Symp. Theoretical Aspects of Computer Science (STACS), vol. 5, pp. 83-94, 2010. [5] J. Capetanakis, "Tree Algorithms for Packet Broadcast Channels," IEEE Trans. Information Theory, vol. 25, no. 5, pp. 505-515, Sep. 1979. [6] G. Chockler, M. Demirbas, S. Gilbert, N.A. Lynch, C.C. Newport, and T. Nolte, "Consensus and Collision Detectors in Radio Networks," Distributed Computing, vol. 21, pp. 55-84, 2008. [7] I. Chlamtac and S. Kutten, "On Broadcasting in Radio Networks— Problem Analysis and Protocol Design," IEEE Trans. Comm., vol. 33, no. 12, pp. 1240-1246, Dec. 1985. [8] B.S. Chlebus, L. Gasieniec, A. Gibbons, A. Pelc, and W. Rytter, "Deterministic Broadcasting in Unknown Radio Networks," Distributed Computing, vol. 15, pp. 27-38, 2002. [9] B.S. Chlebus, L. Gasieniec, D.R. Kowalski, and T. Radzik, "On the Wake-Up Problem in Radio Networks," Proc. 32nd Int'l Colloquium on Automata, Languages and Programming (ICALP), pp. 347-359, 2005. [10] B.S. Chlebus and D.R. Kowalski, "A Better Wake-Up in Radio Networks," Proc. 23rd ACM Symp. Principles of Distributed Computing (PODC), pp. 266-274, 2004. [11] B.S. Chlebus, D.R. Kowalski, and M.A. Rokicki, "Adversarial Queuing on the Multiple-Access Channel," Proc. 25th ACM Symp. Principles of Distributed Computing (PODC), pp. 92-101, 2006. [12] M. Chrobak, L. Gasieniec, and W. Rytter, "Fast Broadcasting and Gossiping in Radio Networks," J. Algorithms, vol. 43, pp. 177-189, 2002. [13] A.E.F. Clementi, A. Monti, and R. Silvestri, "Selective Families, Superimposed Codes, and Broadcasting on Unknown Radio Networks," Proc. 12th Ann. ACM-SIAM Symp. Discrete Algorithms (SODA), pp. 709-718, 2001. [14] A.E.F. Clementi, A. Monti, and R. Silvestri, "Round Robin is Optimal for Fault-Tolerant Broadcasting on Wireless Networks," Proc. Nineth Ann. European Symp. Algorithms (ESA), pp. 452-463, 2001. [15] E.W. Dijkstra, "Solution of a Problem in Concurrent Programming Control," Comm. ACM, vol. 8, no. 9, p. 569, 1965. [16] E.G. Fusco and A. Pelc, "Acknowledged Broadcasting in Ad Hoc Radio Networks," Information Processing Letters, vol. 109, pp. 136-141, 2008. [17] L. Gasieniec, A. Pelc, and D. Peleg, "The Wakeup Problem in Synchronous Broadcast Systems," SIAM J. Discrete Math., vol. 14, pp. 207-222, 2001. [18] S. Gilbert, R. Guerraoui, and C.C. Newport, "Of Malicious Motes and Suspicious Sensors: On the Efficiency of Malicious Interference in Wireless Networks," Theoretical Computer Science, vol. 410, pp. 546-569, 2009. [19] L.A. Goldberg, M. Jerrum, S. Kannan, and M. Paterson, "A Bound on the Capacity of Backoff and Acknowledgment-Based Protocols," SIAM J. Computing, vol. 33, pp. 313-331, 2004. [20] A.G. Greenberg and S. Winograd, "A Lower Bound on the Time Needed in the Worst Case to Resolve Conflicts Deterministically in Multiple Access Channels," J. ACM, vol. 32, pp. 589-596, 1985. [21] D.R. Kowalski, "On Selection Problem in Radio Networks," Proc. 24th ACM Symp. Principles of Distributed Computing (PODC), pp. 158-166, 2005. [22] D.R. Kowalski and A. Pelc, "Deterministic Broadcasting Time in Radio Networks of Unknown Topology," Proc. 22nd ACM Symp. Principles of Distributed Computing (PODC), pp. 73-82, 2003. [23] Y. Kushilevitz and Y. Mansour, "An $\Omega (D \log (N/D))$ Lower Bound for Broadcast in Radio Networks," SIAM J. Computing, vol. 27, pp. 702-712, 1998. [24] N.A. Lynch, Distributed Algorithms. Morgan Kaufmann Publishers, Inc., 1996. [25] K. Nakano and S. Olariu, "Uniform Leader Election Protocols for Radio Networks," IEEE Trans. Parallel Distributed Systems, vol. 13, no. 5, pp. 516-526, May 2002. [26] M.C. Pease, R.E. Shostak, and L. Lamport, "Reaching Agreement in the Presence of Faults," J. ACM, vol. 27, pp. 228-234, 1980. [27] A. Pelc, "Activating Anonymous Ad Hoc Radio Networks," Distributed Computing, vol. 19, pp. 361-371, 2007. [28] M. Raynal, Algorithms for Mutual Exclusion. The MIT Press, 1986. [29] D.E. Willard, "Log-Logarithmic Selection Resolution Protocols in a Multiple Access Channel," SIAM J. Computing, vol. 15, pp. 468-477, 1986. | 2015-05-22 21:11:12 | {"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": 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": 0.5457268357276917, "perplexity": 4672.631951433274}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207926736.56/warc/CC-MAIN-20150521113206-00086-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2437340/prove-or-disapprove-group-of-order-135-must-be-abelian?noredirect=1 | # Prove or disapprove: Group of order $135$ must be abelian
The question is as follows: prove or disapprove: every group of order $135$ must be abelian.
I started like this: $G = H \times\ K$ when $H$ is a normal 5-sylow subgroup and $K$ is a normal 3-sylow subgroup (both normal from sylow theory).
$H$ is cyclic and therefor abelian, but what about $K$? if it's cyclic then it proves the statement. if not, I'm not sure how to continue...
I'm not sure how to continue from here.
• So the question reduces to: does there exist a nonabelian group of order $27$? – Derek Holt Sep 20 '17 at 11:45
• – lhf Sep 20 '17 at 11:47
• From the linked answers: take $G = \Bbb Z_5 \times H_3(\Bbb Z_3)$, where $H_3(\Bbb Z_3)$ is the Heisenberg group modulo 3. – Omnomnomnom Sep 20 '17 at 11:55
• The title should be "prove or disprove". Or may be "Approve or disapprove". – Dietrich Burde Sep 20 '17 at 12:22
It is enough to exhibit a nonabelian group of order $135$ of the form $C_5 \times K$, where $K$ is a nonabelian group of order $27$.
The set of all matrices of the form $$\begin{pmatrix}1&a&b\\0&1&c\\0&0&1\end{pmatrix}$$ with entries in $\mathbb Z_3$ is a nonabelian multiplicative group of order $27$, called the Heisenberg group over $\Bbb{Z}_3$. | 2019-06-26 08:09:16 | {"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": 0.8757897615432739, "perplexity": 249.5227000798708}, "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/1560628000231.40/warc/CC-MAIN-20190626073946-20190626095946-00085.warc.gz"} |
http://melikamp.com/math/teaching/calculus/tangent-planes.html | ### Tangent Planes And Linear Approximations
(requires JavaScript)
1. Find an equation of the plane tangent to the surface
$z=4{x}^{2}-{y}^{2}+2y$
at the point $\left(-1,2,4\right)$.
$z=-8x-2y$
2. Find an equation of the plane tangent to the surface
$z=y\mathrm{cos}\left(x-y\right)$
at the point $\left(2,2,2\right)$.
$z=y$
3. Find an equation of the plane tangent to the surface
$z=y\mathrm{ln}\left(x\right)$
at the point $\left(1,4,0\right)$.
${f}_{x}\left(1,4\right)=4$ and ${f}_{y}\left(1,4\right)=0$. Construct $2$ vectors tangent to the surface: ${\mathbf{T}}_{1}=⟨1,0,4⟩$ and ${\mathbf{T}}_{2}=⟨0,1,0⟩$. Find a vector orthogonal to the surface: $\mathbf{n}={\mathbf{T}}_{2}×{\mathbf{T}}_{1}=⟨4,0,-1⟩$, and so the plane is $4x-z-4=0$.
4. Find the linear approximation of the function
$f\left(x,y\right)=\sqrt{20-{x}^{2}-7{y}^{2}}$
at the point $\left(2,1\right)$ and use it to approximate $f\left(1.95,1.08\right)$.
$L\left(x,y\right)=-\frac{2}{3}x-\frac{7}{3}y+\frac{20}{3}$,
$L\left(1.95,1.08\right)=\frac{427}{150}$. | 2022-01-21 07:53:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 20, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8715742826461792, "perplexity": 258.320452443851}, "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/1642320302740.94/warc/CC-MAIN-20220121071203-20220121101203-00285.warc.gz"} |
http://www.ncatlab.org/nlab/show/category+of+V-enriched+categories | # nLab category of V-enriched categories
### Context
#### Enriched category theory
enriched category theory
# Contents
## Idea
For $\mathcal{V}$ a suitable context of enrichment (notbaly: for $\mathcal{V}$ a cosmos) there is a $2$-category $\mathcal{V} Cat$ whose
Sometimes one also considers $\mathcal{V} Cat$ as a mere category by dropping the $2$-morphisms (and using enriched strict categories).
## Examples
• For $\mathcal{V} = ($Set$, \times)$, $\mathcal{V}Cat \simeq$ Cat, the $2$-category of locally small categories.
• For $\mathcal{V} = ($Cat$, \times)$, $\mathcal{V}Cat \simeq$ Str2Cat, the $2$-category of strict 2-categories.
$(n+1,r+1)$-categories of (n,r)-categories
category: category
Revised on October 31, 2012 01:57:48 by Toby Bartels (64.89.53.173) | 2014-03-10 15:15:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 19, "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": 0.7769761085510254, "perplexity": 11642.681287557849}, "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/1394010855566/warc/CC-MAIN-20140305091415-00079-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://kea-monad.blogspot.com/2008/11/m-theory-lesson-236.html | occasional meanderings in physics' brave new world
Name:
Location: New Zealand
Marni D. Sheppeard
## Wednesday, November 05, 2008
### M Theory Lesson 236
On any diagonal matrix of (square root) mass eigenvalues, one instance of the squared Fourier transform acts as a simple permutation, and so clearly the fourth power of the transform is the identity. In other words, the discrete Fourier transform is like a square root of a basic 2-cycle or Pauli operator $\sigma_{X}$. The other choices for $F$ involve braiding elements.
What would a square root of a braid crossing look like? Geometrically, considering the element of $B_{2}$ as a map between bars with two points, the square root is, instead of a rotation of $\pi$ for the bar, a rotation of $\pi/2$. This configuration lines up the points on the bottom bar so that the strands appear to come together in a diagram that usually represents Hopf algebraic multiplication in a category, only now the points are still separated in the third dimension. | 2017-09-22 11:40:33 | {"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": 0.8645381331443787, "perplexity": 586.1663916990751}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818688940.72/warc/CC-MAIN-20170922112142-20170922132142-00025.warc.gz"} |
https://kb.osu.edu/dspace/handle/1811/13268 | # INFRARED STUDIES OF THE $\nu_{3}$ BAND OF $CH_{2}^{+}$
Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/13268
Files Size Format View
1994-TE'-06.jpg 104.7Kb JPEG image
Title: INFRARED STUDIES OF THE $\nu_{3}$ BAND OF $CH_{2}^{+}$ Creators: Gabrys, C. M.; Uy, Dairene; Jagod, M.- F.; Oka, T. Issue Date: 1994 Publisher: Ohio State University Abstract: The spectrum of the $CH_{2}^{-}$ ion is difficult to analyze because of the existence of the Renner-Teller effect and a low barrier to linearity. In our previous $study^{1}$ of the $\nu_{3}$ band of $CH_{2}^{+}$, we could assign only 18 lines in the $K_{a}=0$ series of the $\alpha$-type parallel transition because of the irregularity of other series and because of the complexity of the overall spectrum due to other carbocations in the plasma. In order to maximize $CH_{2}^{-}$ signals as opposed to those of $CH_{3}^{+}$, we chose a plasma similar to the one in our initial scan but without $H_{2}$ since. $CH_{4} + He^{*} \stackrel{70%}{\rightarrow} CH_{2}^{+} \stackrel{H_{3}}{\rightarrow} CH_{3}^{-}.$ We found that a mixture of $CH_{4}$/He=90mT/7 Torr in a liquid-$N_{2}$ cooled positive column clearly distinguishes the object of our quest. An extensive scan around the band origin of $\nu_{3}=3131 cm^{-1}$ using such conditions allowed us to unambiguously assign the tow series for $K_{a}=1$ and some lines with $K_{a}=2$. The series are quite irregular because of the Renner-Teller perturbation in the $\nu_{3}$ state, but the ground state combination differences are fairly regular. Altogether, 45 transitions the been assigned, and 15 combination differences in the ground state have been fitted. Description: $^{1}$M. R\""{o}sslein, C. M. Gabrys, M.-F. Jagod, and T. Oka, J. Mol. Spectrosc. 153, 738-740 (1992) Author Institution: Department of Chemistry, University of Chicago URI: http://hdl.handle.net/1811/13268 Other Identifiers: 1994-TE'-06 | 2015-03-30 16:22:41 | {"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": 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": 0.6390069127082825, "perplexity": 2044.151668142345}, "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-2015-14/segments/1427131299496.98/warc/CC-MAIN-20150323172139-00265-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/i-want-a-formula.89184/ | # I want a formula
Reaptor
Okay, first I wan´t to say that my english is not to good, so it will be a lot of wrong spellings in this text, hope you understand.
Okay, to the problem.
My masters (read: Teachers) wan´t me to write a story about the space.
The person in the story, Rolf, will travel to Mars for a base on the moon in just 24 hours.
I have made some calculitions, and if Mars is about 60 000 000 km away from the earth, he will have to travel at 2 500 000 km/h, or about 104 166 km/s.
As I have understand it, you will move slower trough time if you travel faster through space.
So, now I want to know how much time the travel will take in Rolfs perspective, if it will take 24 hours in the Earth's perspective.
I want the formula and an example of how you use the formula. (Sometimes, a demonstration of how the formula works will ave mcuh time. Atleast for poor me.)
Reaptor said:
I have made some calculitions, and if Mars is about 60 000 000 km away from the earth, he will have to travel at 2 500 000 km/h, or about 104 166 km/s.
As I have understand it, you will move slower trough time if you travel faster through space.
So, now I want to know how much time the travel will take in Rolfs perspective, if it will take 24 hours in the Earth's perspective.
I want the formula and an example of how you use the formula. (Sometimes, a demonstration of how the formula works will ave mcuh time. Atleast for poor me.)
Depends if you want to assume he travels at constant velocity between Earth and mars, or if he accelerates and decelerates. If he goes at constant velocity, then you just multiply the time it takes from the Earth's perspective by $$\sqrt{1 - v^2/c^2}$$ to find the time it takes from his perspective (if he goes at 104 166 km/s = 0.347460 c, then the time will be about 0.938 days, or about 22.5 hours). If he accelerates at 1G for the first half of the trip and decelerates at 1G for the second half (which means it will no longer take a day to get there from Earth's perspective, I'm not sure what the exact time would be), you can use formulas from http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken] page--in this case I think you'd want this formula:
T = (c/a)sh^-1(at/c) = (c/a) ch^-1 [ad/c^2 + 1]
Where T is the time it takes to get to the halfway point from the ship's perspective, t is the time it takes to get to the halfway point from the Earth's perspective, a is 1G and d is half the distance from Earth to Mars (then just double the time once you get the answer). Here "sh^-1" and "ch^-1" refer to the inverse hyperbolic sine and cosine functions, also written as sinh^-1 and cosh^-1.
On the other hand, if you still want to have him accelerate at a constant rate for the first half and then decelerate at the same rate for the second half, but you want to adjust the acceleration so the whole trip takes exactly a day from the Earth's perspective, you could use this formula:
t = (c/a) sh(aT/c) = sqrt[(d/c)^2 + 2d/a]
then if you solve for a, you get:
a = 2d/[t^2 - (d/c)^2]
So just plug in half a day for t, and half the distance from the Earth to Mars for d, and that'll give you the correct rate of acceleration, which you can then plug into the earlier equation to get the onboard time.
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Reaptor
Thank you very much.
JesseM said:
On the other hand, if you still want to have him accelerate at a constant rate for the first half and then decelerate at the same rate for the second half, but you want to adjust the acceleration so the whole trip takes exactly a day from the Earth's perspective, you could use this formula:
t = (c/a) sh(aT/c) = sqrt[(d/c)^2 + 2d/a]
then if you solve for a, you get:
a = 2d/[t^2 - (d/c)^2]
So just plug in half a day for t, and half the distance from the Earth to Mars for d, and that'll give you the correct rate of acceleration, which you can then plug into the earlier equation to get the onboard time.
By the way, I checked the numbers here and the result I got was that he'd have spend the whole trip accelerating and decelerating at 32.15 meters/second^2, or about 3.3 G...could a person stand G-forces that high for an entire day? Also, if you have him spend most of the trip at constant velocity, he'll have to decelerate at a much higher rate when he gets close to Mars if he wants to stop there, maybe too high to survive depending on how brief the deceleration is. So this could be a plot hole in your story, maybe you want to have the trip last for two days instead of one.
Staff Emeritus
For the constant acceleration case, you might want to check out the relativistic rocket equations at
http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken]
This will give you a more realisitc answer. However, working out the problem in terms of constant velocity will probably give you a better insight into relativity. Since other posters have already done that, I'll work out the constant acceleration case a little bit in case it's interesting.
A note on notation: as per the web page, t and d are measured in the stationary frame (the Earth-Mars frame), and are time and distance, respectively, while T is measured in the accelerating rocket frame, and is
the elapsed "proper time" on the accelerating ship.
We are neglecting a few effects, including the motion of Mars (which is moving much slower than the rocket), and gravitational time dilation effects due to mars's position in the sun's gravity well. I don't believe these should be important.
The first equation of interest is the trip time t in the stationary frame in terms of the distance and the acceleration. The equation for this is
$$t = \sqrt{ \left( \frac{d}{c} \right) ^2 + \frac{2d}{a}}$$
Because we want to get to Mars and stop, we want to go 30 million kilometers in 12 hours, then go another 30 million kilometers in another 12 hours while we stop.
So we can re-write the above equation as
$$a = \frac{2d}{t^2-\left( \frac{d}{c}\right)^2 }$$
we find that a = 32 m / s^2 as the previous poster has calculated for the non-relativistic case, which is rather high (about 3 g). It's probably survivable with proper acceleration couches, but uncomfortable - very uncomfortable if no breaks are taken for meals or other bodily functions.
We now want to calculate the ship time. Using the link, we find
$$T = \left( \frac{c}{a} \right) arcsinh \left( \frac{at}{c} \right)$$
Unfortunatly, google calculator doesn't seem to do arcsinh
Using another program I get, for the half-trip
Ship time = 43 199.84 seconds
Earth/mars time = 43 200 seconds (12 hours).
Double these for the whole trip. The net result is that only a small fraction of a second difference will occur.
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https://stats.stackexchange.com/questions/130043/concept-of-p-vector-and-matrix?noredirect=1 | # Concept of p-vector and matrix
I have worked with vectors and matrices but the following paragraph from The Elements of Statistical Learning by Trevor Hastie et al is little confusing (online edition, page 10)
Matrices are represented by bold uppercase letters; for example, a set of N input p-vectors $x_i$, i = 1, . . . , N would be represented by the $N$ x $p$ matrix X.
This means that the matrix X has $N$ rows and $p$ columns. Then the next lines confuse me, where I need your help.
In general, vectors will not be bold, except when they have N components; this convention distinguishes a $p$-vector of inputs $x_i$ for the $i^{th}$ observation from the $N$-vector $x_j$ consisting of all the observations on variable $X_j$. Since all vectors are assumed to be column vectors, the $i^{th}$ row of $X$ is $x^{T}_i$, the vector transpose of $x_i$.
What does this mean? Why is $i^{th}$ row of $X$ is $x^T_i$? I thought $i^{th}$ row of $X$ would have dimensions of $1$x$p$.
I would appreciate if someone can please clear my doubts.
• I would like to direct your attention to the key phrase, "all vectors are assumed to be column vectors." Among other things, this tells you that "$x_i$"--which previously was named as a "$p$-vector," is a column, whence (written in matrix form) it would have dimensions $p\times 1$, not $1\times p$. – whuber Dec 22 '14 at 20:04
• @whuber thanks for your reply. Am I right in understanding that the matrix $X$ has $N$ rows and $p$ columns? So if a $p$-vector is a column meaning it would be something like $[x_1..x_N]^T$. But then the $i$th row will have $p$ elements, isn't it? – cps Dec 22 '14 at 20:58
• Your first quotation answers the first question. By definition, a $p$-vector $x$ has components $x_1, \ldots, x_p$. Thus the transpose of a $p$-vector is a row with $p$ numbers in it. – whuber Dec 22 '14 at 21:19
• That text is fairly sophisticated: it does not assume readers are completely unfamiliar with the conventions of linear algebra or statistics. Search our site, for instance, on design matrix to see how widespread this matrix setup is. Although it does cover multiple regression in an early chapter, you would be very well served to study regression independently from another text, because then you will better appreciate ESL's point of departure and you will be more comfortable with its language and notation. – whuber Dec 22 '14 at 21:46
• Funny I was confused by the same excerpt: stats.stackexchange.com/questions/224374/… – The Red Pea Jul 18 '16 at 18:33
This is similar to the convention often used in econometrics. For instance, we often write OLS equation as $y=X\beta+\varepsilon$, which means column vectors $y$ and $\beta$. Vector $y$ is a column with each $i\in [1,N]$ elements corresponding to an observation. Vector $\beta$ is a column with each $k\in [1,p]$ corresponding to a parameter. Here, the matrix $X_{ij}$ has $p$ columns each containing $N$ observations of the variable $j$.
So, if you need to grab $i$-th observation of all explanatory variables as a vector, you get matrix transpose $(X_{i1},\dots,X_{ip})^T$ of $i$-th row of matrix $X$, because the vectors are to be columns.
• In an expression like "$y=X\beta+\cdots$", there is no possibility of ambiguity (once you are told $X$ is an $n$ by $p$ matrix and that $y$ and $\beta$ are vectors, anyway): the very rules of matrix multiplication force $\beta$ to be a $p$-vector and $y$ to be an $n$-vector, both in column form. But no such expression was available to the O.P. Indeed, there would be no ambiguity whatsoever in writing $y=\beta X+\cdots$, only then both vectors would have to be rows (and $n$ would represent the number of parameters and $p$ the number of cases). Both conventions are widely used. – whuber Dec 23 '14 at 17:31 | 2019-10-14 04:35:49 | {"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": 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": 0.8892444968223572, "perplexity": 307.30209299030093}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986649035.4/warc/CC-MAIN-20191014025508-20191014052508-00289.warc.gz"} |