text:
# American Institute of Mathematical Sciences

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doi: 10.3934/jimo.2018140

## Necessary optimality condition for trilevel optimization problem

 1 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China 2 School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

* Corresponding author

Received  January 2017 Revised  September 2017 Published  September 2018

Fund Project: This work was supported by the Natural Science Foundation of China (11871383, 11401487), and the Basic and Advanced Research Project of Chongqing(cstc2016jcyjA0239)

This paper mainly studies the optimality conditions for a class of trilevel optimization problem, of which all levels are nonlinear programs. We firstly transform this problem into an auxiliary bilevel optimization problem by applying KKT approach to the lower-level problem. Then we obtain a necessary optimality condition via the differential calculus of Mordukhovich. Finally, a theorem for existence of optimal solution is derived via Weierstrass Theorem.

Citation: Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018140
##### References:

show all references

##### References:
 [1] Paul B. Hermanns, Nguyen Van Thoai. Global optimization algorithm for solving bilevel programming problems with quadratic lower levels. Journal of Industrial & Management Optimization, 2010, 6 (1) : 177-196. doi: 10.3934/jimo.2010.6.177 [2] Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483 [3] Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1213-1223. doi: 10.3934/jimo.2018092 [4] Yong Xia. New sufficient global optimality conditions for linearly constrained bivalent quadratic optimization problems. Journal of Industrial & Management Optimization, 2009, 5 (4) : 881-892. doi: 10.3934/jimo.2009.5.881 [5] Liwei Zhang, Jihong Zhang, Yule Zhang. Second-order optimality conditions for cone constrained multi-objective optimization. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1041-1054. doi: 10.3934/jimo.2017089 [6] Tadeusz Antczak, Najeeb Abdulaleem. Optimality conditions for $E$-differentiable vector optimization problems with the multiple interval-valued objective function. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019089 [7] Mansoureh Alavi Hejazi, Soghra Nobakhtian. Optimality conditions for multiobjective fractional programming, via convexificators. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-9. doi: 10.3934/jimo.2018170 [8] Sebastian Albrecht, Marion Leibold, Michael Ulbrich. A bilevel optimization approach to obtain optimal cost functions for human arm movements. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 105-127. doi: 10.3934/naco.2012.2.105 [9] Dan Li, Li-Ping Pang, Fang-Fang Guo, Zun-Quan Xia. An alternating linearization method with inexact data for bilevel nonsmooth convex optimization. Journal of Industrial & Management Optimization, 2014, 10 (3) : 859-869. doi: 10.3934/jimo.2014.10.859 [10] Michael Hintermüller, Tao Wu. Bilevel optimization for calibrating point spread functions in blind deconvolution. Inverse Problems & Imaging, 2015, 9 (4) : 1139-1169. doi: 10.3934/ipi.2015.9.1139 [11] Henri Bonnel, Ngoc Sang Pham. Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions. Journal of Industrial & Management Optimization, 2011, 7 (4) : 789-809. doi: 10.3934/jimo.2011.7.789 [12] Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101 [13] Jing Quan, Zhiyou Wu, Guoquan Li. Global optimality conditions for some classes of polynomial integer programming problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 67-78. doi: 10.3934/jimo.2011.7.67 [14] Yuhua Sun, Laisheng Wang. Optimality conditions and duality in nondifferentiable interval-valued programming. Journal of Industrial & Management Optimization, 2013, 9 (1) : 131-142. doi: 10.3934/jimo.2013.9.131 [15] Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361 [16] Xiao-Bing Li, Qi-Lin Wang, Zhi Lin. Optimality conditions and duality for minimax fractional programming problems with data uncertainty. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1133-1151. doi: 10.3934/jimo.2018089 [17] Xiaoni Chi, Zhongping Wan, Zijun Hao. Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1111-1125. doi: 10.3934/jimo.2015.11.1111 [18] Roya Soltani, Seyed Jafar Sadjadi, Mona Rahnama. Artificial intelligence combined with nonlinear optimization techniques and their application for yield curve optimization. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1701-1721. doi: 10.3934/jimo.2017014 [19] Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 [20] Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

2018 Impact Factor: 1.025

format: plain


text:
Parallel Plate Capacitors Video Lessons

Concept

# Problem: Two 2.10 cm × 2.10 cm plates that form a parallel-plate capacitor are charged to ± 0.708 nC.Part AWhat is the electric field strength inside the capacitor if the spacing between the plates is 2.60 mm?Part BWhat is the potential difference across the capacitor if the spacing between the plates is 2.60 mm?Show your work.

###### FREE Expert Solution

Part A

Electric field strength inside the capacitor is:

$\overline{){\mathbf{E}}{\mathbf{=}}\frac{\mathbf{V}}{\mathbf{d}}}$

The potential difference between the plates is:

$\overline{){\mathbf{V}}{\mathbf{=}}\frac{\mathbf{Q}}{\mathbf{C}}}$

Capacitance C is:

$\overline{){\mathbf{C}}{\mathbf{=}}{{\mathbf{\epsilon }}}_{{\mathbf{0}}}\frac{\mathbf{A}}{\mathbf{d}}}$

96% (426 ratings)
###### Problem Details

Two 2.10 cm × 2.10 cm plates that form a parallel-plate capacitor are charged to ± 0.708 nC.

Part A

What is the electric field strength inside the capacitor if the spacing between the plates is 2.60 mm?

Part B

What is the potential difference across the capacitor if the spacing between the plates is 2.60 mm?


text:
# [OS X TeX] O no, not this again - fonts

Arno Kruse arnokruse at macnews.de
Sun Nov 14 05:51:26 EST 2004

Dear badly affected TeX-friends,

after googling unsuccessfully for a while I dare to ask some questions.
I know I am an idiot as I ask questions that have been answered hundred
of times (million of times?); and very probably everybody knows the
So it will do that ONE reader has time, pleasure and patience to answer
out of the list.

1. I have bought BBEdit. Now I try to learn using it (I have never
imagined before that an apparently simple program like an editor could
be complicated in this way.) But I don´t succeed in telling BBEdit to
work with TeX.

2. A very kind member of this list (thanks to J.) has sent me  the
Lucida-fonts - I have tried to buy them in vain. Another very kind
member  (thanks to Herb!) has helped me to install them properly, but
now I have the problem that a simple expression like
$\sin(x)$
looks nastyly: "sin" appears much smaller than "x".

3. I like the font myriad very much. I own it. I want to use it in
LaTeX. I do not know what to do to do so.

arno
--------------------- Info ---------------------
Mac-TeX Website: http://www.esm.psu.edu/mac-tex/
& FAQ: http://latex.yauh.de/faq/
TeX FAQ: http://www.tex.ac.uk/faq
List Post: <mailto:MacOSX-TeX at email.esm.psu.edu>


text:
Solved

# Batch conversion of XML files to PDF

Posted on 2016-07-14
59 Views
I receive a number of XML or docxml file to be converted to pdf format. I did research and found this script:
'See http://msdn2.microsoft.com/en-us/library/bb238158.aspx
Const wdFormatPDF = 17  ' PDF format.
Const wdFormatXPS = 18  ' XPS format.

Const WdDoNotSaveChanges = 0

Dim arguments
Set arguments = WScript.arguments

' Make sure that there are one or two arguments
Function CheckUserArguments()
If arguments.Unnamed.Count < 1 Or arguments.Unnamed.Count > 2 Then
WScript.Echo "Use:"
WScript.Echo "<script> input.doc"
WScript.Echo "<script> input.doc output.pdf"
WScript.Quit 1
End If
End Function

' Transforms a doc to a pdf
Function DocToPdf(docInputFile, pdfOutputFile)

Dim fileSystemObject
Dim wordApplication
Dim wordDocument
Dim wordDocuments
Dim baseFolder

Set fileSystemObject = CreateObject("Scripting.FileSystemObject")
Set wordApplication = CreateObject("Word.Application")
Set wordDocuments = wordApplication.Documents

docInputFile = fileSystemObject.GetAbsolutePathName(docInputFile)
baseFolder = fileSystemObject.GetParentFolderName(docInputFile)

If Len(pdfOutputFile) = 0 Then
pdfOutputFile = fileSystemObject.GetBaseName(docInputFile) + ".pdf"
End If

If Len(fileSystemObject.GetParentFolderName(pdfOutputFile)) = 0 Then
pdfOutputFile = baseFolder + "\" + pdfOutputFile
End If

' Disable any potential macros of the word document.
wordApplication.WordBasic.DisableAutoMacros

Set wordDocument = wordDocuments.Open(docInputFile)

' See http://msdn2.microsoft.com/en-us/library/bb221597.aspx
wordDocument.SaveAs pdfOutputFile, wdFormatPDF

wordDocument.Close WdDoNotSaveChanges
wordApplication.Quit WdDoNotSaveChanges

Set wordApplication = Nothing
Set fileSystemObject = Nothing

End Function

' Execute script
Call CheckUserArguments
If arguments.Unnamed.Count = 2 Then
Call DocToPdf(arguments.Unnamed.Item(0), arguments.Unnamed.Item(1))
Else
Call DocToPdf(arguments.Unnamed.Item(0), "")
End If
'end
Set arguments = Nothing

which does fine, except that I can only do one file at a time.

Is there a way to change this so that it can covert a bunch of files in a folder that have the .xml or .docxml extension?

Thanks!
0
Question by:wasmithpfs
• 4
• 3
• 2

LVL 32

Accepted Solution

Robberbaron (robr) earned 450 total points (awarded by participants)
ID: 41712614
sure.... do you want to do all .xml files ?  how do you want the output ?

add this to the top of your current script...

'-------------multiple XML files ----
Set objFSO = CreateObject("Scripting.FileSystemObject")
sStartFolder = "C:\ee"

Set objFolder = objFSO.GetFolder(sStartFolder)

Set colFiles = objFolder.Files

For Each objFile in colFiles
If LCase(objFSO.GetExtensionName(objFile.Name)) = "xml" Then
Wscript.Echo objFile.Name
DocToPdf objfile.Name,""
End If
Next
Wscript.Echo
wscript.quit

'----------------------------------------------
'See http://msdn2.microsoft.com/en-us/library/bb238158.aspx
Const wdFormatPDF = 17  ' PDF format.
Const wdFormatXPS = 18  ' XPS format.

0

Author Comment

ID: 41712803
Hi Robberbaron (robr),

Thanks for responding. Yes I wan to do all xml or docxml files; the outout needs to be pdf.

wasmithpfs
0

Author Comment

ID: 41713192
Hi Robberbaron (robr),

When I run the updated code, I get  an
ActiveX component can't create object: 'Word.Application'
error. I   have  the Office 365' version of Word  on my Windows 10 machine--might that be causing this issue?

wasmithpfs
0

LVL 32

Expert Comment

ID: 41713733
could be... I cant test as i have Office2010 installed. (so it works for me)

what happens if you start Word before running the vbs command ?
1

Author Comment

ID: 41719318
Hi--
where are you running the vbs from? the cmd line or from run.exe?  if I run it from  cmd line and reference the target directory like this:
C:\Users\WDMobile2\Desktop\batchconvert.vbs  C:\Users\WDMobile2\Desktop\filestoconvert

I get an "expect end of statement" error that references this line of code:
Set objFolder = objFSO.GetFolder(sStartFolder)

If I run it from the run.exe and enter: C:\Users\WDMobile2\Desktop\batchconvert.vbs,  i get a different error: "object ' required wordDocument'
...which references this line of code (from my original code):
wordDocument.SaveAs pdfOutputFile, wdFormatPDF.

0

LVL 32

Assisted Solution

Robberbaron (robr) earned 450 total points (awarded by participants)
ID: 41719985
1/ i run from commandline.   use cscript nameofscript.vbs
2/ the target folder is currently hardcoded into script.
sStartFolder = "C:\ee"
2.1/  so try changing that first, then we an add commandline parameter read.
3/ the second error is within the previous code so its the same as your first problem where the Word.Appliciation object could not be created.
1

LVL 52

Assisted Solution

Joe Winograd, EE MVE earned 50 total points (awarded by participants)
ID: 41719994
Hi wasmithpfs,

To test the theory that the problem may be Office 365, you may download and install Office 2016, which comes with most (although not all) Office 365 subscriptions, as discussed in this 5-minute EE video Micro Tutorial:
How to install the Office 2016 desktop applications that come with the free trial of Office 365 Home

Since robr says it works with Office 2010, it will likely work with Office 2016. Worth a try, assuming your Office 365 subscription includes the desktop apps. And if it doesn't, you could use a trial version that does, and for free (to at least test the theory), as explained in this other 5-minute EE video Micro Tutorial:
How to get a free trial of Office 365 with the Office 2016 desktop applications

Regards, Joe
1

Author Comment

ID: 41744274
Hi Robberbaron,

I followed your suggestion of using cscript in the command line and it worked perfectly!

Thanks so much! :)

--wasmithpfs
0

LVL 52

Expert Comment

ID: 41768431
Nearly all the credit (450 points) to Robberbaron for the right answer, as stated by the asker: "it worked perfectly!" A small credit (50 points) to Joe Winograd for a good way to test the theory that the problem may be Office 365.
0

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text:
Enter the distance between the inside radius of the bend and the neutral bend line, and the sheet metal thickness to calculate the K Factor.

## K Factor Formula

The following formula is used to calculate the K Factor of a bend.

K = D / T

• Where K is the K factor
• D is the distance from the inside radius of a bend and the neutral bend line
• T is the thickness of the material

## K Factor Definition

What is a k factor? A k factor is a term used in manufacturing used to describe the ratio of the distance between the inner radius and the neutral bend line. This k factor is used to determine the length of a flat sheet that is required to make a bend of a specific radius.

## Example Problem

How to calculate a k factor?

1. First, determine the distance between the inner radius and neutral bend line.

For this example, this distance is found to be 2.25″.

2. Next, determine the thickness of the material.

In this material, the being used is 1.5″

3. Finally, calculate the k factor.

Using the formula, the k factor is found to be 2.25/1.5 = 1.5


text:
Nothing Pleases Me

# npm

## markdown-pp-js

1.0.0 • Public • Published

# Markdown Preprocessor (MarkdownPP)

The Markdown Preprocessor is a JavaScript rewrite of John Reese's Python module, designed to add extended features on top of the excellent Markdown syntax defined by John Gruber. These additions are mainly focused on creating larger technical documents without needing to use something as heavy and syntactically complex as Docbook.

MarkdownPP uses a set of selectable modules to apply a series of transforms to the original document, with the end goal of generating a new Markdown document that contains sections or features that would be laborious to generate or maintain by hand.

Documents designed to be preprocessed by MarkdownPP should try to follow the convention of naming files with a .mdpp extension, so that MarkdownPP can generate a document with the same name, but with the standard .md extension. As an example, this document in raw format is named "readme.mdpp", and the generated document from MarkdownPP is named "readme.md" so that GitHub can find and process that document when viewing the repository.

1. Installation and Usage
2. Modules
2.1. Includes
2.2. IncludeURLs
2.3. Table of Contents
2.4. Reference
2.5. LaTeX Rendering
2.6. YouTube Embeds
3. Examples
4. Support
5. References

## 1. Installation and Usage

You can download or clone the source code from GitHub, or you can install from the npm repository with:

npm install markdown-pp-js --save

or

yarn install markdown-pp-js --saved

There are two components to the project: a JavaScript module, index.js, and a a simple command line interface to the module, markdownpp.

Assuming you have a file named foo.mdpp, you can generate the preprocessed file foo.md by running the following command:

$node markdownpp foo.mdpp -o foo.md  If you do not specify an output file name, the results will be printed to stdout, enabling them to be piped to another command. By default, all available modules are enabled. You can specify a list of modules to exclude: $ node markdownpp foo.mdpp -o foo.md -e latexrender,youtubembed

To see usage instructions, including a list of enabled modules, supply the -h or --help arguments:

$node markdownpp --help  ## 2. Modules ### 2.1. Includes In order to facilitate large documentation projects, MarkdownPP has an Include module that will replace a line of the form !INCLUDE "path/to/filename" with the contents of that file, recursively including other files as needed. File foo.mdpp: Hello  File bar.mdpp: World!  File index.mdpp: !INCLUDE "foo.mdpp" !INCLUDE "bar.mdpp"  Compiling index.mdpp with the Include module will produce the following: Hello World!  Furthermore, the Include module supports the shifting of headers in the file to be included. For example, File foo.mdpp: # Foo ## Bar  File index.mdpp: # Title ## Subtitle !INCLUDE "foo.mdpp", 2  Compiling index.mdpp with the Include module and using 2 as shift parameter will yield: # Title ## Subtitle ### Foo #### Bar  ### 2.2. IncludeURLs Facilitates the inclusion of remote files, such as files kept in a subversion or GitHub repository. Like Include, the IncludeURL module can replace a line of the form !INCLUDEURL "http://your.domain/path/to/filename" with the contents returned from that url, recursively including additional remote urls as needed. IncludeURL runs immediately after the Include module finishes executing. This means that is it possible to include local files that then require remote files, but impossible parse !INCLUDE statements found in remote files. This is prevent ambiguity as to where the file would be located. Remote file http://your.domain/foo.mdpp: Hello  Remote file http://your.domain/bar.mdpp: Remote World!  Local file index.mdpp: !INCLUDEURL "http://your.domain/foo.mdpp" !INCLUDEURL "http://your.domain/bar.mdpp"  Compiling index.mdpp with the IncludeURL module will produce the following: Hello Remote World!  ### 2.3. Table of Contents The biggest feature provided by MarkdownPP is the generation of a table of contents for a document, with each item linked to the appropriate section of the markup. The table is inserted into the document wherever the preprocessor finds !TOC at the beginning of a line. Named  tags are inserted above each Markdown header, and the headings are numbered hierarchically based on the heading tag that Markdown would generate. ### 2.4. Reference Similarly, MarkdownPP can generate a list of references that follow Markdown's alternate link syntax, eg [name]: <url> "Title". A list of links will be inserted wherever the preprocessor finds a line beginning with !REF. The generated reference list follows the same alternate linking method to ensure consistency in your document, but the link need not be referenced anywhere in the document to be included in the list. ### 2.5. LaTeX Rendering Lines and blocks of lines beginning and ending with$ are rendered as LaTeX, using Google's charting engine.

For example,

$\displaystyle \int x^2 = \frac{x^3}{3} + C$

becomes

### 2.6. YouTube Embeds

As GitHub-flavored Markdown does not allow embed tags, each line of the form !VIDEO "[youtube url]" is converted into a screenshot that links to the video, roughly simulating the look of an embedded video player.

For example,

!VIDEO "http://www.youtube.com/embed/7aEYoP5-duY"

becomes

## 3. Examples

Example file.mdpp:

# Document Title

!TOC

## Header 1
### Header 1.a
## Header 2

!REF

[github]: http://github.com "GitHub"

The preprocessor would generate the following Markdown-ready document file.md:

# Document Title

1\. [Header 1](#header1)
1.1\. [Header 1.a](#header1a)
2\. [Header 2](#header2)

## Header 1

### Header 1.a

## Header 2

*	[GitHub][github]

[github]: http://github.com "GitHub"

## 4. Support

If you find any problems with MarkdownPP, or have any feature requests, please report them to GitHub, and I will respond when possible. Code contributions are always welcome, and ideas for new modules, or additions to existing modules, are also appreciated.

## Keywords

### Install

npm i markdown-pp-js

### Repository

github.com/MikeRalphson/markdown-pp-js

### Homepage

github.com/MikeRalphson/markdown-pp-js#readme

3

1.0.0

MIT


text:
# Thread: Find the Rate of Change

1. ## Find the Rate of Change

Find the rate of change of the distance between the origin and a moving point on the graph of
y = x^2 + 1 It dx/Dr = 2 cm/second

My work:

I proceeded taking the implicit derivative of the given function to find dy/dt.

dy/dt = 2x* dx/dt

dy/dt = 2x* (2) and the I plugged x = 0 from the origin value for x.

dy/dt = 2(0)* (0)

2(2x^3+3x)/sqrt(x^4+3x^2+1)

My Answer is zero and the book's answer is a crazy rational function.

2. ## Re: Find the Rate of Change

Originally Posted by nycmath
Find the rate of change of the distance between the origin and a moving point on the graph of
y = x^2 + 1 It dx/Dr = 2 cm/second

My work:

I proceeded taking the implicit derivative of the given function to find dy/dt.

dy/dt = 2x* dx/dt

dy/dt = 2x* (2) and the I plugged x = 0 from the origin value for x.

dy/dt = 2(0)* (0)

2(2x^3+3x)/sqrt(x^4+3x^2+1)

My Answer is zero and the book's answer is a crazy rational function.

-Dan

3. ## Re: Find the Rate of Change

Originally Posted by nycmath
Find the rate of change of the distance between the origin and a moving point on the graph of
y = x^2 + 1 It dx/Dr = 2 cm/second

My work:

I proceeded taking the implicit derivative of the given function to find dy/dt.

dy/dt = 2x* dx/dt

dy/dt = 2x* (2) and the I plugged x = 0 from the origin value for x.

dy/dt = 2(0)* (0)

2(2x^3+3x)/sqrt(x^4+3x^2+1)

My Answer is zero and the book's answer is a crazy rational function.

Sorry, just a little joke.

The problem is asking about how fast the distance is changing between a point on the graph and the origin. So say we have a point on the graph (x, y). What is the distance between that point and the origin? Once you get that you can take your derivative. (And recall that y = x^2 - 1 before you differentiate.)

-Dan

4. ## Re: Find the Rate of Change

This is what I did:

(1) I used the distance formula for points to find d.

d = sqrt{(x-0)^2 + (x^2-1-0)}

d = sqrt{x^4-2x^2+1}

(2) I then differentiated implicitly and found dr/dt to be

[2x(2x^2-1)]/sqrt{x^4-2x^2+1}

It is not correct.

5. ## Re: Find the Rate of Change

Originally Posted by nycmath
d = sqrt{(x-0)^2 + (x^2-1-0)}
Your parabola is $\displaystyle y = x^2 + 1$. You have a minus sign there. And don't forget the (x - 0)^2 term.

-Dan

6. ## Re: Find the Rate of Change

Almost there...take a look.

dr/dt = 2(2x^3+ 3)/sqrt(x^4+3x^2+1)

The numerator should be
2(2x^3+3x). I am missing an "x" for the term 3x. See it?

7. ## Re: Find the Rate of Change

Originally Posted by nycmath
Almost there...take a look.

dr/dt = 2(2x^3+ 3)/sqrt(x^4+3x^2+1)

The numerator should be
2(2x^3+3x). I am missing an "x" for the term 3x. See it?
$\displaystyle r = \sqrt{x^4 + 3x^2 + 1}$

$\displaystyle \frac{dr}{dt} = \frac{1}{2} (x^4 + 3x^2 + 1)^{-1/2} \cdot (4x^3 + 6x) \frac{dx}{dt}$

-Dan

8. ## Re: Find the Rate of Change

I clearly see my error. When taking the derivative of the inside using the chain rule, I mistakenly multiplied 4x^3 and 6x by the value given for dx/dt or 2. Thank you for your help. Math is fun.


text:
# Repeated measures analysis

I was reading a book about repeated measures analysis, but I really couldn't drill enough into the concept behind the math equations. In the book, it says like this:

What I can't understand is the distribution for y. You know when we talked about normal distribution for a random variable X, we defined it X ~N ( µ, σ2). But here I really don't know how to interpret vector V in normal distribution. I mean if the shape for normal distribution is as follows then what will the shape be for normal distribution with a vector V within it.

And at the end if you have come up with a better explanation for concept behind GEE, I would be really thankful to share it here.

Thanx

This distribution is an extension of the univariate normal distribution called multivariate normal distribution with parameters $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$. I guess everything you need to know about it can be found on the Wikipedia page (or at least this is a good starting point).


text:
# Roger Dashen

Roger Dashen
BornMay 5, 1938
DiedMay 25, 1995 (aged 57)
NationalityAmerican
Alma materCaltech
Known forQuantum field theory

Roger Frederick Dashen (May 5, 1938 in Grand Junction, Colorado – May 25, 1995 in La Jolla) was an American theoretical physicist who studied particle physics and quantum field theory.

Dashen studied physics at the Harvard University (where he also played in the football team), where he graduated "summa cum laude" in 1960. Then he went to Caltech, where he earned his PhD in 1964. After that, he was a professor at Caltech in 1967 and the Institute for Advanced Study in 1969. In 1986, he became a professor at the University of California, San Diego (UCSD) in 1988 was Head of Faculty. He played a leading role in the establishment of a supercomputer center at UCSD and in the establishment of the Institute for Theoretical Physics of the National Science Foundation at the University of California, Santa Barbara.

In the 1960s he worked partly in collaboration with Murray Gell-Mann with current algebras of particle physics and models with chiral symmetry. In 1964, he worked on S-matrix methods to calculate electromagnetic corrections to strong interactions, including the mass difference between protons and neutrons with Steven Frautschi.[1] In the 1970s, he studied with Brosl Hasslacher and André Neveu quantum field theoretical model theories (such as the Sine-Gordon model and the Gross-Neveu model) for extended particles and semiclassical approximations ([WKB approximation] []) for their analysis.[2] They developed the Dashen-Hasslacher-Neveu method (DHN) for quantization of solitons using path integrals. After the discovery of instantons in the quantum (QCD) by Polyakov, he examined it with David Gross and Curtis Callan.[3] In the 1980s, he worked with lattice gauge theory s and found with Neuberger (regardless of Giorgio Parisi) is an upper bound for the Higgs Boson - mass.[4]

Dashen was also involved in sound propagation in the ocean (with military applications in mind), where he applied quantum field theoretical methods (such as path integrals and the renormalization group) on the problem of random scattering of sound in the ocean. He worked with his JASON colleagues Kenneth Watson and Frederik Zachariason.[5] Dashen was senior scientific adviser to the US Navy, including the committee on the safety of SSBNs, the rocket-carrying submarines, and matters relating to submarine warfare. He was a member of the JASON Defense Advisory Group.

Dashen was a member of the National Academy of Sciences since 1984.

He was married and had two daughters.

## Books

• Roger Dashen and Steve Adler: Current Algebras. Applications to Particle Physics . Benjamin 1968

## References

1. ^ Dashen and Frautschi. S-Matrix Method for Calculating electromagnetic corrections to strong interactions . In: Physical Review B . Band 135, 1964, 1190
2. ^ Dashen, Hasslacher and Neveu. non perturbative Models and extended hadron models in field theory Part 1 Semi Classical Functional Methods . In: Physical Review D . Volume 10, 1974, p 4114, Part 2 Two dimensional models and extended hadrons . In: Physical Review D . Volume 10, 1974, p.4 130; The Particle Spectrum in Model Field Theories from semiclassical functional integral techniques . In: Physical Review D . Volume 11, 1975, p 3424; Semi Classical bound states in asymptotically free theory on . In: Physical Review D . Volume 12, 1975, p 2443
3. ^ Callan, Dashen and Gross: The structure of the vacuum gauge theory . In: Physics Letters B . Volume 63, 1976, p 334; Toward a theory of the strong interactions . In: Physical Review D . Volume 17, 1979, p 2717
4. ^ From the assumed triviality of ${\displaystyle {\phi ^{4}}}$ theories.. Et al According to later work of Kuti, Lee, Shen is the consequent upper limit in the Glashow-Weinberg-Salam theory (standard model of electroweak interactions) at 700 GeV.
5. ^ Dashen, Watson, Zachariason, Walter Munk and Stanley Flatte. Propagating sound in a fluctuating ocean . Cambridge University Press, 1979


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## Posts Tagged ‘math.CO’

### The extremal utility belt: Cauchy-Schwarz

March 10, 2010

I found this little gem in an old post on Terry Tao’s blog. There’s not really enough content in it to merit an entire Extremal Toolbox post, but it’s too cool not to point out.

Theorem. Graphs of order $n$ and girth at least 5 have $o(n^2)$ edges.

Proof. Suppose that $G$ has $1/2*c*n^2$ edges. Define the function $A: V^2 \rightarrow \{0, 1\}$ to be the “adjacency characteristic function.” Now, by Cauchy-Schwarz:

$n^4 \Sigma A(x_1, y_1)A(x_1, y_2)A(x_2, y_1)A(x_2, y_2)$

$\geq (n \Sigma A(x, y_1) A(x, y_2))^2$

$\geq (\Sigma A(x, y))^4 = \frac{1}{16} c^4 n^8$.

Clearly for $c$ fixed, $\frac{1}{16} c^4 n^8$ is unbounded as $n \rightarrow \infty$. But the first expression is $n^4$ times the number of (possibly degenerate) 4-cycles in the graph; it’s easy to check that there are $O(n^3)$ degenerate 4-cycles, so as n is unbounded our graph must contain a 4-cycle. QED

Now, it’s possible to get better bounds by cleverly doing “surgery” on the graph and just using pigeonhole. (See here for details.) But it’s tricky and far less beautiful than the argument with Cauchy-Schwarz, which really demonstrates one way in which C-S can be thought of as “strengthening” pigeonhole.

### 2^n \geq n: The graph theory proof?

February 15, 2010

Theorem. For every positive integer n, $2^n \geq n$.

Proof. Consider a tree on n vertices $T = (V, E)$ with root $v_0$. Assign to each edge $\{v, w\}$ an element of the vector space $GF(2)^V$, obtained by setting 1s in the the coordinates corresponding to v and w and 0s elsewhere. I claim that these vectors are linearly independent; for suppose otherwise, and letthe vectors corresponding to $S \subset E$ sum to 0. There is a natural “distance” function on E w/r/t $v_0$; let $e_0 = \{s, t\}$ have maximal distance in S, and suppose WLOG that t is farther than s from $v_0$. Then the coordinate corresponding to t is nonzero for exactly one element of S, and the sum over all elements of S must therefore be nonzero. This is a contradiction. So in particular these |E| = n-1 elements are distinct and nonzero, which means (by Pigeonhole) that there are at most $2^n-1$ of them.

### Full HAP tables for the second 1124 sequence

January 12, 2010

Since it’s not yet on Thomas’ blog, and I need to use the data, I thought I’d make it publicly available. All credit goes to Thomas Sauvaget for the sequence-to-HTML code, and to Polymath for the sequence itself. (Probably someone specific discovered it, but I don’t know who.) Tables below the fold.

### The extremal toolbox: A matrix problem

January 10, 2010

I’m starting a new series of posts this semester where I get “back to basics.” One of the few areas of mathematics in which I can claim anything even in the same connected component as “expertise” is extremal combinatorics. Unfortunately for me and my lazy, big-picture brain, though, extremal combinatorics is very much a “problem-solving” subject, with a relatively small number of tools that are used to solve all sorts of different problems. So without some practice solving these problems, or expositing the solutions, it’s easy to get rusty.

Hence, “The Extremal Toolbox.” In each post, I’ll take a (solved!) problem in extremal combinatorics — anything from Sperner’s theorem to Kakeya over finite fields, as long as there’s an extremal flavor — and try to break down a proof into its component parts.

Today I’m going to examine a problem which appeared on MathOverflow some time ago, which I didn’t quite solve (but came within epsilon of!) The relevant post is here; if you don’t care to click through, here’s the problem.

Let $M$ be an $n \times n$ matrix with non-negative integer entries. Suppose further that if $m_{ij}$ is 0, then the sum of all the entries in the ith row or the jth column is at least $n$. Then the sum of all the entries in $M$ is at least $n^2/2$.

## (more…)

### Bleg: What’s the most recent day no one alive was born

January 5, 2010

Inspired by Michael Lugo’s post on reconstructing a person from their DOB, zipcode, and gender.

If you, for whatever reason, ever watch the Today show, you’ll notice that one of the recurring features is the hosts listing the names of some men and women who are turning 100. Becoming a centenarian is a reasonably big accomplishment — in the U.S., it nets you a congratulatory letter from the President, for example. But if you look into it, you’ll notice that you can find someone turning 100 on pretty much any given day. Usually not someone particularly well-known, but certainly someone. (I tried to find someone famous and vaguely math-related who just turned or is turning 100 for this post, but couldn’t; however, the fascinating economist Ronald Coase turned 99 last week.) It’s almost certainly true that on any given day, someone somewhere in the world is in fact celebrating their 100th birthday. But go ten years further, and you find almost no one who lives to 110. Actually, I know of only one supercentenarian, living or not, who is interesting for reasons apart from his longevity — the late Vietoris, the topologist, probably best known as half of the Vietoris-Rips complex and the Mayer-Vietoris sequence. Odds are pretty good that no one alive is turning 110 today, or tomorrow, or (sadly) New Years’ Day.

So… a question is starting to take shape. On every day between December 29, 1909, and today, someone was born who is still living today. But much earlier than that, and the above statement begins to be false. So what’s the most recent day that no one living was born on?

### What’s a “locally determined graph property?”

January 1, 2010

This has nothing to do with the rest of the post, but I’ll put it here so you read it before you get bored. I’d like to thank my readers (all seven of you) for supporting this blog in the first six months or so of its existence, and hope that you’ll stick around (and be joined by hundreds of new readers…) to hear my sporadic ramblings and wild ravings in the next year. Here’s to a happy and successful 2010!

Over at MathOverflow, Gjergji Zaimi asks (in a criminally under-voted-for question): How can we obtain global information from local data in graph theory?  This is something that perhaps everyone working in or around graph theory has asked themselves, in some form, at some point — I know I have. So it’s not surprising that Gjergji’s question has received many different answers with many different interesting things to say.

I originally wanted to write a post trying to “answer” Gjergji’s question as best I could, but quickly realized the futility of that goal — it’s such a broad and deep question that I doubt if anyone could answer it concisely, and I know I couldn’t! So instead I’ll just talk about an $\epsilon$ of the question — what does it even mean, “local data?”

### The coupon collectors’ problem

December 28, 2009

This is a considerably lower-level post than usual, which I’ll (following Terry Tao) also blame on the holidays; there’s another, even less mathematical post in the works which I hope to finish sometime tomorrow.

How many times do you need to flip a coin before you expect to see both heads and tails? How many times do you need to roll a die before you expect to see all the numbers 1-6? These are two instances of the coupon collectors’ problem. Wikipedia gives not one, but two nice solutions to the problem, but there’s an even nicer “back-of-the-envelope” calculation which gives you the correct asymptotics for virtually nothing, and (I like to think) shows the power of thinking “categorically” at even a very low level.

So let’s give a statement of the problem. A company — say Coca-Cola, for concreteness — is holding a contest where everyone who collects one each of n different “coupons” wins some prize. You get a coupon with each purchase of a Coke, and each coupon is equally likely. What’s the expected number of Cokes you have to buy in order to collect all the coupons?

If you do some experimentation (or calculation) with small instances, you’ll see that this number seems to be growing somewhat faster than n. For n = 2, for example, the expected number is 3, and for n = 3 it’s 7. But how much faster? Like $n^2 \text{?}~n \sqrt{n}$? Or just a constant times n?

None of the above, as it happens, and you might have already guessed (or known) that the correct order of growth is $O(n~log~n)$. Here’s how you can figure this out for yourself.

Think of the collection as a function from Coke bottles to (equivalence classes of) coupons. If we’ve collected all the coupons, the function is surjective. So we can rephrase “What is the probability that, after I buy m Coke bottles, I have collected all n coupons” as “What is the probability that a random function from a set with m elements to a set with n elements is surjective?” Actually, we’ll estimate the probability that it’s not surjective.

If the function isn’t surjective, then its image contains at most n-1 elements. Fixing n-1 elements, the probability that our random function takes some element of the domain to this subset is $\frac{n-1}{n}$. Since the appropriate events are all independent, the probability that the random function takes every element to the subset is therefore $(\frac{n-1}{n})^m$.

Now there are n possibilities for the subset of size n-1, so we apply the union bound and say that the probability that our random function is not surjective is at most $n (\frac{n-1}{n})^m$. (Of course, this is an upper bound, and there is an error term; but we’ll return to that in a bit.)

So we want this expression to be smaller than, say, 1/10, which means that $(\frac{n-1}{n})^m = 1/10n$. But when n is large, we have that $(1 - \frac{1}{n})^n$ is about 1/e, so m has to be on the order of $ln~n$!

Now we’ll backtrack a bit. How do we know that the union bound was reasonably tight? After all, we counted functions whose image had size n-2 twice! Well, if you go back through the analysis and do inclusion-exclusion, you’ll see that the probability winds up being close to 1 when $m << n log n$ — but I don’t know of a computation-free way to argue that $O(n log n)$ is asymptotically right! Does anyone else?

So how is this “categorical thinking?” Well, it’s not, really. Category theory only really starts to get mildly interesting when you talk about functors, and doesn’t come into its own right until natural transformations are introduced. But if you’ve learned to think categorically, you see morphisms where other people see objects — in this case, a function where others might see a set — and while this is rarely enough to apply abstract-nonsense tools, it is enough to broaden your intuition and see paths you might have otherwise missed. And this is at least as useful.

### The importance of choosing the right model

December 13, 2009

I’ve had the idea for this post bouncing around in my head for several months now, but now that I don’t have classes to worry about I can finally get around to writing it. I want to talk about some “pathological examples” in computer science, in particular in complexity and computability theory.

I wanted to post a picture of William H. Mills with this post, in the spirit of Dick Lipton’s blog. Unfortunately I can’t find one! Mills was a student of Emil Artin in the ’40s; he finished his thesis in 1949 and promptly (as far as I can tell) disappeared until the ’70s, when we find some work by a William H. Mills on combinatorial designs. After this, there’s nothing of note until Dr. Mills passed away several years ago at the age of 85.

While his work (assuming it’s his) on combinatorial designs looks interesting, W.H. Mills’ small place in history is assured by a short and unassuming paper from 1947, published when he was still a student. In this paper he showed that there’s a constant, which he called A, such that $[A^{3^n}]$ is prime for all integers $n \geq 1$. Nowadays it’s called Mills’ constant.

### Where do graphs live?

December 5, 2009

This post came out of some thoughts I posted (anonymously, but mostly because I didn’t feel like registering) over at nLab. I don’t think it’s a secret that I’m heavily interested in the relationships between category theory and combinatorics, and more generally the ways in which we can use “structured” algebraic objects and “continuous” topological objects to gain information about the unstructured discrete objects in combinatorics. That said, the folks over at the nLab work on some crazy abstract stuff, which seems about as far away as possible from the day-to-day realities of graph theory or set systems. And maybe it is — but I hope it’s not, and as far as I’m concerned, this is a windmill that deserves to be tilted at. (After all, it might be a giant.)

So as my jumping-off point, I’ll take my observation from last time that the relationship between graphs and digraphs is analogous to the one between groupoids and categories. I briefly mentioned something called a quiver, which can be thought of as any of the following:

• Another name for a digraph, which categorical people use when they don’t want us combinatorialists stomping in and getting the floor all muddy;
• A “free category,” i.e., one in which there are no nontrivial relations between composition of morphisms;
• An algebraic object whose representations we want to consider; it’s worth thinking of this way mostly because of the “freeness,” although if you try to define it more formally you’ll probably end up with the previous definition;
• What you get when you take (part of) a category and forget all the rules for how morphisms compose.

This last point is the most interesting one for our purposes, since it’s clearly an algebraic object but isn’t as restrictive as “free category,” and thus has a chance of capturing the unstructured behavior of the combinatorial zoo. But it’s tricky to turn this into a rigorous definition that actually includes everything we want to be a quiver… so we’ll just use “quiver” as a fancy name for “digraph.” However, there’s an important philosophical lesson to be learned from the final point, so I’ll set it off:

Philosophical lesson. The edges of a quiver shouldn’t carry any information except for the vertices they are incident to; more generally, paths in a quiver shouldn’t carry any information except for their sequence of vertices.

### More on graphs and digraphs

November 16, 2009

So this question’s been bugging me ever since I first thought it up, and I figured (in the spirit of MaBloWriMo, which by now is pretty much dead on this blog) that I’d ask about it here — I need to give Math Overflow a break.

The question concerns adjoint functors, which I don’t understand half as well as I’d like, but enjoy thinking about anyway. One of the (many!) motivating examples that adjoint functors generalize is the common “free/forgetful” dichotomy. For instance, there’s a functor from the category of groups (say) to the category of sets, which is defined by simply “forgetting” the group structure and giving back the underlying set. This functor doesn’t have an inverse, of course; that would make the two categories isomorphic, which is way too much to expect. Nor does it have an “inverse up to natural transformation.” That would make the categories equivalent, which is almost as good as isomorphism. But it does have the next-best thing after that: a functor in the opposite direction which comes with a natural isomorphism on some hom-sets. This is the free functor, that assigns to each set the free group on that set. These functors are called adjoint functors.


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# NCERT Solutions | Class 11 Maths Chapter 8 | Binomial Theorem

## CBSE Solutions | Maths Class 11

Check the below NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem Pdf free download. NCERT Solutions Class 11 Maths  were prepared based on the latest exam pattern. We have Provided Binomial Theorem Class 11 Maths NCERT Solutions to help students understand the concept very well.

### NCERT | Class 11 Maths

Book: National Council of Educational Research and Training (NCERT) Central Board of Secondary Education (CBSE) 11th Maths 8 Binomial Theorem English

#### Binomial Theorem | Class 11 Maths | NCERT Books Solutions

You can refer to MCQ Questions for Class 11 Maths Chapter 8 Binomial Theorem to revise the concepts in the syllabus effectively and improve your chances of securing high marks in your board exams.

## NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem Ex 8.1

Expand each of the expressions in Exercises 1 to 5.

Ex 8.1 Class 11 Maths Question 1.
$${ \left( 1-2x \right) }^{ 5 }$$
Solution.

Ex 8.1 Class 11 Maths Question 2.
$${ \left( \frac { 2 }{ x } -\frac { x }{ 2 } \right) }^{ 5 }$$

Solution.

Ex 8.1 Class 11 Maths Question 3.
$${ \left( 2x-3 \right) }^{ 6 }$$
Solution.

Ex 8.1 Class 11 Maths Question 4.
$${ \left( \frac { x }{ 3 } +\frac { 1 }{ x } \right) }^{ 5 }$$
Solution.

Ex 8.1 Class 11 Maths Question 5.
$${ \left( x+\frac { 1 }{ x } \right) }^{ 6 }$$
Solution.

Using binomial theorem, evaluate each of the following

Ex 8.1 Class 11 Maths Question 6.
$${ \left( 96 \right) }^{ 3 }$$
Solution.

Ex 8.1 Class 11 Maths Question 7.
$${ \left( 102 \right) }^{ 5 }$$
Solution.

Ex 8.1 Class 11 Maths Question 8.
$${ \left( 101 \right) }^{ 4 }$$
Solution.

Ex 8.1 Class 11 Maths Question 9.
$${ \left( 99 \right) }^{ 5 }$$
Solution.

Ex 8.1 Class 11 Maths Question 10.
Using Binomial Theorem, indicate which number is larger$${ \left( 1.1\right) }^{ 10000 }$$ or 1000.
Solution.
Splitting 1.1 and using binomial theorem to write the first few terms we have

Ex 8.1 Class 11 Maths Question 11.
Find $${ \left( a+b \right) }^{ 4 }-{ \left( a-b \right) }^{ 4 }$$. Hence, evaluate $${ \left( \sqrt { 3 } +\sqrt { 2 } \right) }^{ 4 }-{ \left( \sqrt { 3 } -\sqrt { 2 } \right) }^{ 4 }$$.
Solution.
By binomial theorem, we have

Ex 8.1 Class 11 Maths Question 12.
Find $${ \left( x+1 \right) }^{ 6 }+{ \left( x-1 \right) }^{ 6 }$$. Hence or otherwise evaluate $${ \left( \sqrt { 2 } +1 \right) }^{ 6 }+{ \left( \sqrt { 2 } -1 \right) }^{ 6 }$$.
Solution.
By using binomial theorem, we have

Ex 8.1 Class 11 Maths Question 13.
Show that $${ 9 }^{ n+1 }-8n-9$$ is divisible by 64, whenever n is a positive integer.
Solution.
We have to prove that $${ 9 }^{ n+1 }-8n-9=64k$$

Ex 8.1 Class 11 Maths Question 14.
Prove that $$\sum _{ r=0 }^{ n }{ { 3 }^{ r } }$$ 8Cr = 4n
Solution.
We have,

## NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem Ex 8.2

Expand binomials using the binomial expansion method step-by-step.

Ex 8.2 Class 11 Maths Question 1.
Find the coefficient of x5 in (x + 3)8
Solution.
Suppose x5 occurs in the (r + 1)th term of the expansion (x + 3)8

Ex 8.2 Class 11 Maths Question 2.
a5 b7in (a-2b)12
Solution.
Suppose a5 b7 occurs in the (r + 1)th term of the expansion (a – 2b)12.

Write the general term in the expansion of

Ex 8.2 Class 11 Maths Question 3.
(x2 – y)6
Solution.

Ex 8.2 Class 11 Maths Question 4.
(x2 – yx)12, x ≠ 0
Solution.
We have given, (x2 – yx)12 = (x2 + (- yx))12, x ≠ 0

Ex 8.2 Class 11 Maths Question 5.
Find the 4th term in the expansion of (x – 2y)12.
Solution.

Ex 8.2 Class 11 Maths Question 6.
Find the 13th term in the expansion of $${ \left( 9x-\frac { 1 }{ 3\sqrt { x } } \right) }^{ 18 }$$, x ≠ 0
Solution.

Find the middle terms in the expansions of

Ex 8.2 Class 11 Maths Question 7.
$${ \left( 3-\frac { { x }^{ 3 } }{ 6 } \right) }^{ 7 }$$
Solution.
As the exponent 7 is odd, so there will be two middle terms in the expansion

Ex 8.2 Class 11 Maths Question 8.
$${ \left( \frac { x }{ 3 } +9y \right) }^{ 10 }$$
Solution.

Ex 8.2 Class 11 Maths Question 9.
In the expansion of (1 + a)m+n, prove that coefficients of am and an are equal.
Solution.

Ex 8.2 Class 11 Maths Question 10.
The coefficients of the (r – 1)th, rth and (r + 1)th terms in the expansion of (x + 1 )n are in the ratio 1: 3: 5. Find n and r.
Solution.

Ex 8.2 Class 11 Maths Question 11.
Prove that the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion of (1 + x)2n-1.
Solution.

Ex 8.2 Class 11 Maths Question 12.
Find a positive value of m for which the coefficient of x2 in the expansion (1 + x)m is 6.
Solution.


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# Integration with square roots HELP!

(8x+1)^0.5

## The Attempt at a Solution

I tried using substitution but it clearly doesn't work because nothing in the brackets equals when derived.

Anyone help me with the beginning steps?

Dick


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## Amplitude and Phase Modulation

Amplitude and phase modulation is a linear baseband modulation technique in which the message modulates the amplitude and phase of a constant frequency signal. Two carrier signals of the same frequency are out of phase with each other by 90°, a condition known as orthogonality or quadrature. The transmitted signal is created by adding the two carrier signals together. At the receiver, the two signals can be coherently separated (demodulated) because of their orthogonality property.

Communications Toolbox™ software includes these modulation and demodulation functions and blocks to model amplitude phase shift keying (APSK), digital video broadcast system APSK (DVBS-APSK), and MIL-188-QAM.

### APSK

A pure M-APSK constellation is composed of NC concentric rings or contours, each with uniformly spaced PSK points. The M-APSK constellation set is

`$\chi =\left\{\begin{array}{cc}{R}_{1}\mathrm{exp}\left(j\left(\frac{2\pi }{{M}_{1}}i+{\varphi }_{1}\right)\right),& i=0,\dots ,{M}_{1}-1,\\ {R}_{2}\mathrm{exp}\left(j\left(\frac{2\pi }{{M}_{2}}i+{\varphi }_{2}\right)\right),& i=0,\dots ,{M}_{2}-1,\\ ⋮& ⋮\\ {R}_{{N}_{\text{C}}}\mathrm{exp}\left(j\left(\frac{2\pi }{{M}_{{N}_{\text{C}}}}i+{\varphi }_{\text{Nc}}\right)\right),& i=0,\dots ,{M}_{{N}_{\text{C}}}-1,\end{array}$`

where:

• The modulation order is equal to the sum of all Ml for l = 1, 2, ... , NC.

• NC is the number of concentric rings. NC ≥ 2.

• Ml is the number of constellation points in the lth ring.

• Rl is the radius of the lth ring.

• ϕl is the phase offset of the lth ring.

• $j=\sqrt{-1}$

### DVBS APSK

Digital video broadcasting (DVB) standards specify S2 [1], S2X [2], and SH [3] standard-specific amplitude phase shift keying (APSK) modulation.

• DVB-S2 standard specifies QPSK modulation and concatenated convolutional and Reed-Solomon channel coding, and is now used by most satellite operators worldwide for television and data broadcasting services.

• DVB-S2X standard based systems offer the ability to operate with very low carrier-to-noise and carrier-to-interference ratios (SNR down to -10 dB), to serve markets such as airborne (business jets), maritime, civil aviation internet access, VSAT terminals at higher frequency ranges or in tropical zones, small portable terminals for journalists and other professionals. Furthermore, the S2X system provides transmission modes offering significantly higher capacity and efficiency to serve professional links characterized by very high carrier-to-noise and carrier-to-interference ratios conditions.

• DVB-SH standard provides an efficient transmission system using frequencies below 3 GHz suitable for satellite services to handheld devices, in terms of reception threshold and resistance to mobile satellite channel impairments.

### MIL-STD-188-110

MIL-STD-188-110 is a US Department of Defense standard for HF communications using serial PSK mode of both data and voice signals.

The standard specifies physical layer modulation schemes for tactical and long-haul communications. The modulation scheme specified by the standard is a mix of QAM and APSK. For a detailed description of the modulation scheme, see MIL-STD-188-110B & C [5].

## References

[1] ETSI Standard EN 302 307 V1.4.1: Digital Video Broadcasting (DVB); Second generation framing structure, channel coding and modulation systems for Broadcasting, Interactive Services, News Gathering and other broadband satellite applications (DVB-S2), European Telecommunications Standards Institute, Valbonne, France, 2005-03.

[2] ETSI Standard EN 302 307-2 V1.1.1: Digital Video Broadcasting (DVB); Second generation framing structure, channel coding and modulation systems for Broadcasting, Interactive Services, News Gathering and other broadband satellite applications (DVB-S2X), European Telecommunications Standards Institute, Valbonne, France, 2015-02.

[3] ETSI Standard EN 302 583 V1.1.1: Digital Video Broadcasting (DVB); Framing structure, channel coding and modulation for Satellite Services to Handheld devices (SH), European Telecommunications Standards Institute, Valbonne, France, 2008-03.

[4] Sebesta, J. “Efficient Method for APSK Demodulation.” Selected Topics on Applied Mathematics, Circuits, Systems, and Signals (P. Pardalos, N. Mastorakis, V. Mladenov, and Z. Bojkovic, eds.). Vouliagmeni, Athens, Greece: WSEAS Press, 2009.

[5] MIL-STD-188-110B & C: "Interoperability and Performance Standards for Data Modems." Department of Defense Interface Standard, USA.