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# Harter-Heighway Dragon

(Difference between revisions)
 Revision as of 12:05, 6 July 2009 (edit)← Previous diff Revision as of 12:07, 6 July 2009 (edit) (undo)Next diff → Line 85: Line 85: Wikipedia, [http://en.wikipedia.org/wiki/Dragon_curve Wikipedia's Dragon Curve page] Wikipedia, [http://en.wikipedia.org/wiki/Dragon_curve Wikipedia's Dragon Curve page] Cynthia Lanius, [http://math.rice.edu/~lanius/frac/jurra.html Cynthia Lanius' Fractals Unit: A Jurassic Park Fractal] Cynthia Lanius, [http://math.rice.edu/~lanius/frac/jurra.html Cynthia Lanius' Fractals Unit: A Jurassic Park Fractal] - |ToDo=*An animation for the showing the fractal being drawn gradually through increasing iterations + |ToDo=*An animation for the showing the fractal being drawn gradually through increasing iterations (a frame for each individual iteration) + :*Also, and animation that draws the curve at the 13 or so iteration, but slowly to show that the curve never crosses itself. |HideMME = No |HideMME = No }} }}

## Revision as of 12:07, 6 July 2009

Harter-Heighway Dragon Curve
This image is an artistic rendering of the Harter-Heighway Curve (also called the Dragon Curve), which is a fractal. It is often referred to as the Jurassic Park Curve because it garnered popularity after being drawn and alluded to in the novel Jurassic Park by Michael Crichton (1990).

# Basic Description

This fractal is described by a curve that undergoes a repetitive process (called an iterated process). To begin the process, the curve has a basic segment of a straight line.

Then at each iteration,

• Each line is replaced with two line segments at an angle of 90 degrees (other angles can be used to make fractals that look slightly different).
• Each line is rotated alternatively to the left or to the right of the line it is replacing.

Base Segment and First 5 iterations of the Harter-Heighway Curve
15th iteration

The Harter-Heighway Dragon is created by iteration of the curve process described above, and is thus a type of fractal known as iterated function systems. This process can be repeated infinitely, and the perimeter or length of the dragon is in fact infinite. However, if you look to the image at the right, a 15th iteration of the Harter-Heighway Dragon is already enough to create an impressive fractal.

An interesting property of this curve is that although the corners of the fractal seem to touch at various points, the curve never actually crosses over itself. Also, the curve exhibits self-similarity when iterated infinitely because as you look closer and closer at the curve, the curve continues to look like the larger curve.

# A More Mathematical Explanation

Note: understanding of this explanation requires: *Algebra

## Properties

### Perimeter

1st iteration of the Harter-Heighway Dragon

The perimeter of the Harter-Heighway curve increases by a factor of $\sqrt{2}$ for each iteration. For example, if you look at the picture to the right, the straight red line shows the fractal as its base segment and the black crooked line shows the fractal at its first iteration.

If the first iteration is split up into two isosceles triangles, the ratio of the first iteration over the base segment is: $\frac{s\sqrt{2} + s\sqrt{2}}{s + s} = \frac{2s\sqrt{2}}{2s} = \frac{\sqrt{2}}{1}$

### Number of Sides

The number of sides ($N_k$) of the Harter-Heighway curve for any degree of iteration (k) is given by $N_k = 2^k\,$, where the "sides" of the curve refer to alternating slanted lines of the fractal.

For example, the third iteration of this curve should have a total number of sides $N_3 = 2^3 = 8\,$.

### Fractal Dimension

The Fractal Dimension of the Harter-Heighway Curve can also be calculated using the equation: $\frac{logN}{loge}$.

Let us use the second iteration of the curve as seen below to calculate the fractal dimension.

There are two new curves that arise during the iteration so that $N = 2\,$.

Also, the ratio of the lengths of each new curve to the old curve is: $\frac{4(s\sqrt{2})}{4(s)} = \sqrt{2}$, so that $e = \sqrt{2}$.

Thus, the fractal dimension is $\frac{logN}{loge} = \frac{log2}{log\sqrt{2}} = 2$, and it is a space-filling curve.

## Changing the Angle

The Harter-Heighway curve iterates with a 90 degree angle; however, if the angle is changed, new curves can be created. The following fractals are the result of 13 iterations.

# About the Creator of this Image

SolKoll is interested in fractals, and created this image using an iterated function system (IFS).

# References

Wikipedia, Wikipedia's Dragon Curve page Cynthia Lanius, Cynthia Lanius' Fractals Unit: A Jurassic Park Fractal

• An animation for the showing the fractal being drawn gradually through increasing iterations (a frame for each individual iteration)
• Also, and animation that draws the curve at the 13 or so iteration, but slowly to show that the curve never crosses itself.


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The NuGet package is a single ZIP file with the .nupkg extension that contains compiled code (DLLs), other files related to that code, and a descriptive manifest that includes information like the package’s version number, dependencies etc. It is a Microsoft-supported mechanism for sharing code, which defines how packages for .NET are created, hosted, and consumed, and provides the tools for each of those roles.

### 🤔 Let’s define the problem statement

• I have a common class library .NET Core project that needs to be used across my organization in different projects.
• My organization stores our private packages in the Private NuGet Server.
• I want to publish my common library to the Private NuGet Server Azure Artifacts feed so that other teams can use this library in their project.
• I want my package to be versioned for continuous development.

### 👨🎨 Let’s discuss the DeltaX nuget architecture

DeltaX Artifacts is a self-hosted private NuGet server for our internal organization to share and maintain various .Net Core packages.
The above diagram is the illustration of DeltaX NuGet architecture for managing various common packages, used across our organization in different projects.

### 🧑💻 Let’s create a class library and publish

We’ll be creating a HelloWorld Class library, which will contain a method to write greeting text on the console. We’ll be using .NET Core 3.1 and Visual Studio for the same, also will set up the NuGet config in the .csproj file along the way.

### 📦 Step 1: Crete HelloWorld Class Library

I am skipping over this step as this is just the basic creation of your project. Let’s assume that we have created a solution called DeltaX.HelloWorld with a class library project. Now we have a class library that contains a Class1.cs by default, but we don’t need it so delete it and add a class HelloWorld and a method name Greet, which accepts the name as a pram and print Hello {name}, welcome to NuGet blog.. You can also add other packages and dependencies if you need.

namespace DeltaX.HelloWorld
{
using System;

public class HelloWorld
{
public void Greet(string name)
{
var str = $"Hello {name}, welcome to nuGet blog."; Console.WriteLine(str); } } }  ### ⚙ Step 2: Configure the HelloWorld.csproj file To open the HelloWorld.csproj file double click on click on the project name inside the Solution Explorer or you can directly open the file by navigating the project location and opening the file in any editor, then add the following attributes. • TargetFramework: The .Net target version, you can mention more than one target version by using TargetFrameworks instead. • PackageId: The id for the package, which will also appear on browse, It should be Unique. • Authors: List of authors. • Company: Organization Name. • Description: The Project description. • RepositoryUrl: The Url of the remote git repository or any other service provider. • RepositoryType: The type of repository. • PackageProjectUrl: The Project Url if hosted. • PackageTags: Tag for this package, help to search fast. • Version: The library version. • AssemblyVersion: The assembly version. • PackageReleaseNotes: Note for this release. • GeneratePackageOnBuild: Set it to true, it will tell Visual Studio to generate the .nuspec file on the build so that we don’t have to create it manually which is required to push the package on the NuGet server. <Project Sdk="Microsoft.NET.Sdk"> <PropertyGroup> <TargetFrameworks>net452;netcoreapp3.1</TargetFrameworks> <PackageId>DeltaX.HelloWord</PackageId> <Authors>Sonu Kumar</Authors> <Company>DeltaX</Company> <Description>This is DeltaX.HelloWorld demo Nuget Package. </Description> <RepositoryUrl>https://bitbucket.org/deltaxadcloud/deltax.helloword</RepositoryUrl> <RepositoryType>git</RepositoryType> <PackageTags>NuGet, HelloWorld</PackageTags> <PackageProjectUrl>https://bitbucket.org/deltaxadcloud/deltax.helloword/readme.md</PackageProjectUrl> <Version>1.0.1</Version> <AssemblyVersion>1.0.1</AssemblyVersion> <PackageReleaseNotes>* Initial Release.</PackageReleaseNotes> <GeneratePackageOnBuild>true</GeneratePackageOnBuild> </PropertyGroup> </Project>  Please note that Company, RepositoryUrl, RepositoryType, PackageProjectUrl are optional attributes. Now our package is ready for testing, but we can’t run/debug a class library directly, we so need a console application for testing as well as continuous development. ### 🛠 Step 3: Create a Console Application and restructure the folder • Open the file explorer with the same location of the solution, or right-click on the solution in vs and select Open folder in file explorer. • Create a folder src and move the DeltaX.HelloWorld folder only not the DeltaX.HelloWorld.sln. • Double click on DeltaX.HelloWorld.sln and you’ll find it’s broken because we have moved the project inside the src folder so we need to fix it. To fix it right click on the DeltaX.HelloWorld project under the solution in Solution Explorer where it’s showing the error Not found and remove the project then right click on the Solution and add exiting project then select the project from src folder. • Now we have restructured our package so it’s time to create a SampleApplication. • To create SampleApplication, right-click on Solution and Add a new Console Application project with the project name Sample Application, so creating a new project in the folder as well as Visual Studio has these items. • Now add project reference of src/DeltaX.HelloWord library in SampleApplication. • Now initiate the new instance HelloWorld in the program.cs and call the Greet Method, you may need to add using statement if not added by VS. namespace DeltaX.HelloWorld { using System; public class HelloWorld { public void Greet(string name) { var str =$"Hello {name}, welcome to nuGet blog.";
Console.WriteLine(str);
}
}
}

• Verify the functanility and get ready to publish.

### 🎯 Step 4: Publish the package

Now we have a class library ready as well as we have tested our functionalities, so we’re ready for the publish. We’ll use Visual Studio Package Manager Console or Developer tool with the help of the NuGet.CommandLine tool, so first we need to install the NuGet.CommandLine package to our DeltaX.HelloWord project with the help of NuGet Package Manager. After successful Install, our final DeltaX.HelloWord.csproj will look similar to this.

<Project Sdk="Microsoft.NET.Sdk">
<PropertyGroup>
<TargetFrameworks>net452;netcoreapp3.1</TargetFrameworks>
<PackageId>DeltaX.HelloWord</PackageId>
<Authors>Sonu Kumar</Authors>
<Company>DeltaX</Company>
<Description>This is DeltaX.HelloWorld demo Nuget Package. </Description>
<RepositoryType>git</RepositoryType>
<PackageTags>NuGet, HelloWorld</PackageTags>
<Version>1.0.1</Version>
<AssemblyVersion>1.0.1</AssemblyVersion>
<PackageReleaseNotes>* Initial Release.</PackageReleaseNotes>
<GeneratePackageOnBuild>true</GeneratePackageOnBuild>
</PropertyGroup>

<ItemGroup>
<PackageReference Include="NuGet.CommandLine" Version="6.2.1">
<PrivateAssets>all</PrivateAssets>
<IncludeAssets>runtime; build; native; contentfiles; analyzers; buildtransitive</IncludeAssets>
</PackageReference>
</ItemGroup>
</Project>

Now, we have installed the NuGet.CommandLine so we can access it through Package Manager Console or Window CMD/Powershell open in the same location, for confirmation just type nuget, it should show the NugGet description if not fix the error and continue.
Here I’m going to use Visual Studio Package Manager Console, type the below command to publish the package after replacing macros with actual values.

nuget push <Path> <ApiKey> -Source <your-awesome-host.com/nuget>

Here,

• Path: The path is the physical path of the generated DLL’s, which you can find in the output of the console on the build, remembered we made GeneratePackageOnBuild true in .csproj?
We will get the path like this Successfully created package 'D:\DeltaX\DeltaX.HelloWorld\src\DeltaX.HelloWorld\bin\Debug\DeltaX.HelloWord.1.0.1.nupkg'
• ApiKey: You’ll get ApiKey while setting up the server, here you can ask Ketan J or Narendra M for the same.
• your-awesome-host.com: The URL of the hosted machine.

### 🏆 Step 5: Finished!

After the Success of the above command, your package will now be available in your DeltaX Artifacts feed. Tada!
Now you install search and install the same package with the mentioned package Id, here DeltaX.HelloWorld.

### 🤖 Package versioning for the continuous developement

We have seen how we can create and push a package to NuGet Server, isn’t that awesome? but there is a small problem what if we need some more features?
How we can integrate it with our continuous development?
The solution for the above is simple we’ll modify or add the feature we want and publish a newer version of the same package.
To push the newer version we need to update a few attributes of our .csproj file:

• Version: It should be incremented based on the feature we added or modified.
eg: current version 1.0.0, and we have fixed a bug then we should increase the last minor version with 1 as 1.0.1. If we have added a new feature, we should second the last minor version with 1 as 1.1.0.
If we have modified or added a new flow then ideally we should increase the major version from 1, here v2.0.0.
• AssemblyVersion: It should be incremented based on the feature we added or modified exactly like version.
• PackageReleaseNotes: Description of new changes. After all the modifications, we’ll rebuild the project and copy the path of the DLL’s file and run the push command in Package Manager Console or Window Cmd/PowerShell in the root directory.
nuget push <Path> <ApiKey> -Source <your-awesome-host.com/nuget>
`

### 💭 Closing Thoughts

• Whenever we worked on a third-party API integration or some commonly used repository, it is better to create a package and publish it.
• It will save us from repeating the same code at multiple places so easy to maintain.
• If we need a new feature or want to modify the existing one, we just need to add the required feature in the NuGet package and release the newer version, that’s all, no need to check all repositories and add/modify the feature there.
• Most importantly, NuGet maintains a reference list of packages used in a project and the ability to restore and update those packages from that list, so any package can switch as per requirements.

format: plain


text:
# Using a List in PlotLabel which goes beyond width of ImageSize

My PlotLabel goes beyond the width of the ImageSize. The label is generated into a List form by my program. I hope to allow my PlotLabel to go to a second, third, fourth (...) line if it reaches beyond the boundaries. My problem in simplified, working code:

testList = {1.1, 1.2100000000000002, 1.4641000000000004,
2.143588810000001, 4.594972986357221, 21.1137767453526,
445.7915684525922, 198730.12250342217, 3.949366159022518*^10,
1.5597493058032278*^21, 2.432817896953651*^42,
5.918602919737985*^84, 3.5029860521531003*^169,
1.2270911281579163468674816474213106841944620595434115.\
653559774527023*^339,
1.5057526368038678804394522151149996181314615.352529778863042*^\
678, 2.267291003241800856581079591813116877525595663615.\
051499783199061*^1356,
5.140608493381211822346417851332928514.75046978753508*^2712,
2.6425855682223052512317328042874404006080140549740414.\
449439791871098*^5425,
6.9832584853768039212406022892095208921115614.148409796207117*^\
10850, 4.876589907358713358551139652032791543825497271813.\
847379800543136*^21701};
Plot[x^2, {x, -3, 3},
PlotLabel -> StringForm["The orbit of x=1.1 is\n\n", testList ],
ImageSize -> {600, 600}]

-

If you create a TextCell then it will wrap the list nicely. PageWidth controls the width of the text cell.

Plot[x^2, {x, -3, 3}, PlotLabel ->
StringForm["The orbit of x=1.1 is\n\n", TextCell[testList, PageWidth -> 600]],
ImageSize -> {600, 600}]

You can change the appearance of the TextCell (font, fontsize, color, etc) using various options as shown in the help files for TextCell and Cell.

-
The copied code sends my list to two lines instead of one, but still goes beyond the viewable area. –  twhoward99 Jun 5 '13 at 16:25
Above is the picture I got. You know.. if you play with the window width it changes the layout of the textcell. Maybe my default window width is different from yours... –  bill s Jun 5 '13 at 16:27
Could I change the width of the TextCell? Its for a .cdf file and the output size needs to be constrained somewhat. –  twhoward99 Jun 5 '13 at 16:32
See the change -- this one gives me the same picture, but now the width of the textcell is specified (and it doesn't change when you resize the window). –  bill s Jun 5 '13 at 16:35
Works like a charm! Thank-you very much! –  twhoward99 Jun 5 '13 at 16:40

Slightly better than my last attempt:

Row[{
Plot[x^2, {x, -3, 3}, ImageSize -> {400, 400}],
Panel[testList, "The orbit of x=1.1 is", Top,
ImageSize -> {400, 120}]
}]

-
Nice solution!! –  twhoward99 Jun 5 '13 at 16:41
A nine-character solution: add ~Pane~600 to the end of the PlotLabel.


text:
# Simple examples of the diagonal lemma

According to Boolos' Computability and Logic, the diagonal lemma states:

Let $$T$$ be a theory containing $$\mathbf{Q}$$. Then for any formula $$B(y)$$ there is a sentence $$G$$ such that $$\vdash_T\,G\leftrightarrow B(\ulcorner G\urcorner).$$

$$\mathbf{Q}$$ is Boolos' version of Robinson arithmetic. The proof of the lemma involves constructing a sentence $$G$$ which one, in practice, could not possibly write down. But what if we take some simple examples.

• Here's one: let $$B(y)$$ be $$\mathbf{0}^{\prime\prime}. Then we can take $$G$$ to be the sentence $$\mathbf{0}=\mathbf{0}$$, correct? Certainly $$\ulcorner G\urcorner$$ is much greater than 2, so $$G\leftrightarrow B(\ulcorner G\urcorner)$$ is valid.

• Here's another: let $$B(y)$$ be $$y<\mathbf{0}^{\prime\prime}$$. There aren't any sentences with Godel numbers less than 2, so we choose any false sentence we like for $$G$$, such as $$\mathbf{0}=\mathbf{0}^\prime$$. Then, once again, $$G\leftrightarrow B(\ulcorner G\urcorner)$$ is valid.

I am correct in believing that, for very simple formulas $$B(y)$$ of arithmetic, it's actually incredibly easy to construct a diagonal sentence $$G$$? In general, though, is this method not feasible?

• "The proof of the lemma involves constructing a sentence 𝐺 which one, in practice, could not possibly write down." It's not actually that bad - long, yes, but I don't think they're more than a few pages long (at least, according to reasonably efficient coding systems). – Noah Schweber May 4 at 15:11
• Good point Noah. Replace "possibly" with "easily". – Doubt May 4 at 15:13


text:
# Algebra Commons™

## All Articles in Algebra

768 full-text articles. Page 1 of 28.

Optimal Dual Fusion Frames For Probabilistic Erasures, 2017 Universidad Nacional de San Luis and CONICET, Argentina

#### Optimal Dual Fusion Frames For Probabilistic Erasures, Patricia Mariela Morillas

##### Electronic Journal of Linear Algebra

For any fixed fusion frame, its optimal dual fusion frames for reconstruction is studied in case of erasures of subspaces. It is considered that a probability distribution of erasure of subspaces is given and that a blind reconstruction procedure is used, where the erased data are set to zero. It is proved that there are always optimal duals. Sufficient conditions for the canonical dual fusion frame being either the unique optimal dual, a non-unique optimal dual, or a non optimal dual, are obtained. The reconstruction error is analyzed, using the optimal duals in the probability model considered here and using ...

College Algebra, Trigonometry, And Precalculus (Clayton), 2017 Clayton State University

#### College Algebra, Trigonometry, And Precalculus (Clayton), Chaogui Zhang, Scott Bailey, Billie May, Jelinda Spotorno, Kara Mullen

##### Mathematics Grants Collections

This Grants Collection for College Algebra, Trigonometry, and Precalculus was created under a Round Five ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Initial Proposal
• Final Report

Solutions Of The System Of Operator Equations $Bxa=B=Axb$ Via The *-Order, 2017 Ferdowsi University of Mashhad

#### Solutions Of The System Of Operator Equations $Bxa=B=Axb$ Via The *-Order, Mehdi Vosough, Mohammad Sal Moslehian

##### Electronic Journal of Linear Algebra

In this paper, some necessary and sufficient conditions are established for the existence of solutions to the system of operator equations $BXA=B=AXB$ in the setting of bounded linear operators on a Hilbert space, where the unknown operator $X$ is called the inverse of $A$ along $B$. After that, under some mild conditions, it is proved that an operator $X$ is a solution of $BXA=B=AXB$ if and only if $B \stackrel{*}{ \leq} AXA$, where the $*$-order $C\stackrel{*}{ \leq} D$ means $CC^*=DC^*, C^*C=C^*D$. Moreover, the general solution of the equation above is obtained ...

Recursive Robust Pca Or Recursive Sparse Recovery In Large But Structured Noise, 2017 Iowa State University

#### Recursive Robust Pca Or Recursive Sparse Recovery In Large But Structured Noise, Chenlu Qiu, Namrata Vaswani, Brian Lois, Leslie Hogben

##### Namrata Vaswani

This paper studies the recursive robust principal components analysis problem. If the outlier is the signal-of-interest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, St, in the presence of large but structured noise, Lt. The structure that we assume on Lt is that Lt is dense and lies in a low-dimensional subspace that is either fixed or changes slowly enough. A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background (Lt) from moving foreground objects (St) on-the-fly. To solve the above ...

Recursive Robust Pca Or Recursive Sparse Recovery In Large But Structured Noise, 2017 Iowa State University

#### Recursive Robust Pca Or Recursive Sparse Recovery In Large But Structured Noise, Chenlu Qiu, Namrata Vaswani, Brian Lois, Leslie Hogben

##### Namrata Vaswani

This paper studies the recursive robust principal components analysis problem. If the outlier is the signal-of-interest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, St, in the presence of large but structured noise, Lt. The structure that we assume on Lt is that Lt is dense and lies in a low-dimensional subspace that is either fixed or changes slowly enough. A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background (Lt) from moving foreground objects (St) on-the-fly. To solve the above ...

On HigmanS Conjecture, 2017 Universidad del Pais Vasco

#### On HigmanS Conjecture, A. Vera-López, J. M. Arregi, M. A. García-Sánchez, L. Ormaetxea

##### Electronic Journal of Linear Algebra

Let Gn be the subgroup of GLn(q) consisting of the upper unitriangular matrices of size nxn over Fq. In 1960, G. Higman conjectured that the number of conjugacy classes of Gn, denoted by r(Gn), was given by a polynomial in q with integer coefficients. This has been verified for nn, r(Gn) can be expressed in terms of r(Gi), with i

2017 Seattle Pacific University

#### Discovery Learning Plus Direct Instruction Equals Success: Modifying American Math Education In The Algebra Classroom, Sean P. Ferrill Mr.

##### Honors Projects

In light of both high American failure rates in algebra courses and the significant proportion of innumerate American students, this thesis examines a variety of effective educational methods in mathematics. Constructivism, discovery learning, traditional instruction, and the Japanese primary education system are all analyzed to incorporate effective education techniques. Based on the meta-analysis of each of these methods, a hybrid method has been constructed to adapt in the American Common Core algebra classroom.

Elimination For Systems Of Algebraic Differential Equations, 2017 The Graduate Center, City University of New York

#### Elimination For Systems Of Algebraic Differential Equations, Richard Gustavson

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

We develop new upper bounds for several effective differential elimination techniques for systems of algebraic ordinary and partial differential equations. Differential elimination, also known as decoupling, is the process of eliminating a fixed subset of unknown functions from a system of differential equations in order to obtain differential algebraic consequences of the original system that do not depend on that fixed subset of unknowns. A special case of differential elimination, which we study extensively, is the question of consistency, that is, if the given system of differential equations has a solution. We first look solely at the algebraic data" of ...

On Skew-Symmetric Matrices Related To The Vector Cross Product In R^7, 2017 University of Beira Interior

#### On Skew-Symmetric Matrices Related To The Vector Cross Product In R^7, P. D. Beites, A. P. Nicolás, José Vitória

##### Electronic Journal of Linear Algebra

A study of real skew-symmetric matrices of orders $7$ and $8$, defined through the vector cross product in $\mathbb{R}^7$, is presented. More concretely, results on matrix properties, eigenvalues, (generalized) inverses and rotation matrices are established.

A Transformation That Preserves Principal Minors Of Skew-Symmetric Matrices, 2017 Faculté des Sciences Ain chock

#### A Transformation That Preserves Principal Minors Of Skew-Symmetric Matrices, Abderrahim Boussairi, Brahim Chergui

##### Electronic Journal of Linear Algebra

It is well known that two $n\times n$ symmetric matrices have equal corresponding principal minors of all orders if and only if they are diagonally similar. This result cannot be extended to arbitrary matrices. The aim of this work is to give a new transformation that preserves principal minors of skew-symmetric matrices.

Simple And Semi-Simple Artinian Rings, 2017 California State University - San Bernardino

#### Simple And Semi-Simple Artinian Rings, Ulyses Velasco

##### Electronic Theses, Projects, and Dissertations

The main purpose of this paper is to examine the road towards the structure of simple and semi-simple Artinian rings. We refer to these structure theorems as the Wedderburn-Artin theorems. On this journey, we will discuss R-modules, the Jacobson radical, Artinian rings, nilpotency, idempotency, and more. Once we reach our destination, we will examine some implications of these theorems. As a fair warning, no ring will be assumed to be commutative, or to have unity. On that note, the reader should be familiar with the basic findings from Group Theory and Ring Theory.

The Recognition Problem For Table Algebras And Reality-Based Algebras, 2017 University of Regina

#### The Recognition Problem For Table Algebras And Reality-Based Algebras, Allen Herman, Mikhail Muzychuk, Bangteng Xu

##### EKU Faculty and Staff Scholarship

Given a finite-dimensional noncommutative semisimple algebra A over C with involution, we show that A always has a basis B for which ( A , B ) is a reality-based algebra. For algebras that have a one-dimensional representation δ , we show that there always exists an RBA-basis for which δ is a positive degree map. We characterize all RBA-bases of the 5-dimensional noncommutative semisimple algebra for which the algebra has a positive degree map, and give examples of RBA-bases of C ⊕ M n ( C ) for which the RBA has a positive degree map, for all n ≥ 2

Elementary Abstract Algebra, 2017 Plymouth State University

#### Elementary Abstract Algebra, Emma Norbrothen Wright

##### Open Educational Resources

No abstract provided.

Projective Partitions Of Vector Spaces, 2017 Siena College

#### Projective Partitions Of Vector Spaces, Mohammad Javaheri

##### Electronic Journal of Linear Algebra

Given infinite-dimensional real vector spaces $V,W$ with $|W| \leq |V|$, it is shown that there exists a collection of subspaces of $V$ that are isomorphic to $W$, mutually intersect only at 0, and altogether cover $V$.

Singular Value And Norm Inequalities Associated With 2 X 2 Positive Semidefinite Block Matrices, 2017 Zarqa University

#### Singular Value And Norm Inequalities Associated With 2 X 2 Positive Semidefinite Block Matrices, Aliaa Burqan, Fuad Kittaneh

##### Electronic Journal of Linear Algebra

This paper aims to give singular value and norm inequalities associated with $2\times 2$ positive semidefinite block matrices.

2017 The College at Brockport

#### A Financial Literacy Curriculum Project On Linear Functions In Algebra I Aligned With New York State Common Core State Standards, Michael T. Hughson Jr

##### Education and Human Development Master's Theses

There is a lack of financial literacy curriculum available for teachers to support students learning of mathematical skills that are needed in real life. Financial literacy curricula can support students learning real life skills needed before independent living situations in college or career. Direct mathematical modelling and application can support students comprehension of the relationship between financial literacy and algebra. This curriculum highlights the ideology of the need to know drives learning and aligns real world financial literacy problems to the Common Core State Standards (CCSS) algebra standards.

Application Of Symplectic Integration On A Dynamical System, 2017 East Tennessee State University

#### Application Of Symplectic Integration On A Dynamical System, William Frazier

##### Electronic Theses and Dissertations

Molecular Dynamics (MD) is the numerical simulation of a large system of interacting molecules, and one of the key components of a MD simulation is the numerical estimation of the solutions to a system of nonlinear differential equations. Such systems are very sensitive to discretization and round-off error, and correspondingly, standard techniques such as Runge-Kutta methods can lead to poor results. However, MD systems are conservative, which means that we can use Hamiltonian mechanics and symplectic transformations (also known as canonical transformations) in analyzing and approximating solutions. This is standard in MD applications, leading to numerical techniques known as symplectic ...

On Rings Of Invariants For Cyclic P-Groups, 2017 University of Arkansas, Fayetteville

#### On Rings Of Invariants For Cyclic P-Groups, Daniel Juda

##### Theses and Dissertations

This thesis studies the ring of invariants R^G of a cyclic p-group G acting on k[x_1,\ldots, x_n] where k is a field of characteristic p >0. We consider when R^G is Cohen-Macaulay and give an explicit computation of the depth of R^G. Using representation theory and a result of Nakajima, we demonstrate that R^G is a unique factorization domain and consequently quasi-Gorenstein. We answer the question of when R^G is F-rational and when R^G is F-regular.

We also study the a-invariant for a graded ring S, that is, the maximal graded degree ...

Six Septembers: Mathematics For The Humanist, 2017 Duquesne University

#### Six Septembers: Mathematics For The Humanist, Patrick Juola, Stephen Ramsay

##### Zea E-Books

Scholars of all stripes are turning their attention to materials that represent enormous opportunities for the future of humanistic inquiry. The purpose of this book is to impart the concepts that underlie the mathematics they are likely to encounter and to unfold the notation in a way that removes that particular barrier completely. This book is a primer for developing the skills to enable humanist scholars to address complicated technical material with confidence. This book, to put it plainly, is concerned with the things that the author of a technical article knows, but isn’t saying. Like any field, mathematics ...

A Game Of Monovariants On A Checkerboard, 2017 Lynchburg College

#### A Game Of Monovariants On A Checkerboard, Linwood Reynolds

##### Student Scholar Showcase

Abstract: Assume there is a game that takes place on a 20x20 checkerboard in which each of the 400 squares are filled with either a penny, nickel, dime, or quarter. The coins are placed randomly onto the squares, and there are to be 100 of each of the coins on the board. To begin the game, 59 coins are removed at random. The goal of the game is to remove each remaining coin from the board according to the following rules: 1. A penny can only be removed if all 4 adjacent squares are empty. That is, a penny cannot ...


text:
# 3-colourability of Eulerian maximal planar graph

The following paragraph is from this answer by David Eppstein (emphasis mine).

A maximal planar graph is 3-colorable iff it is Eulerian (if it is not Eulerian, then the odd wheel surrounding a single odd vertex requires four colors, and if it is Eulerian then a 3-coloring may be obtained by coloring a triangle and then extending the coloring in the obvious way to adjacent triangles).

(A maximal planar graph is a graph with a planar embedding such that every face is a triangle).

I don't quite understand how emphasized part works as a proof. Let $$G$$ be a maximal planar graph. If $$G$$ is 3-colourable, then the 3-colouring is unique (upto swapping of colours) because in every diamond subgraph of $$G$$, both degree-2 vertices of the subgraph should get the same colour. I suppose this suggests an algorithmic method for testing 3-colourability of $$G$$ namely (i) pick a triangle, (ii) give a 3-colouring to it arbitrarily, (iii) repeatedly extend it to adjacent triangles (assuming there is no inconsistency), and (iv) finally verify that the assignment is indeed a 3-colouring. If there is a vertex of odd degree, we will not succeed in assigning colour to all vertices. My question is this:- How can we guarantee that the assignment will be consistent provided all vertices have even degree?

Note: I don't see how Eulerian property (aka all vertices having even degree) ensures consistency of the assignment. I am asking this as a new question rather than a comment to the linked answer because he was answering a different question.


text:
## Square Cube Law

When we increase the range os a spell by a simple value, we increase its area of effect by the square and the volume of effect by the cube. Or rather:

$$R$$ $$A=pi R^2$$ $$V=frac 4 3 pi R^3$$

Energy density is defined as energy per volume. So our power scale should follow the Volumetric scaling, not a linear one. From one step to the next, you increase the energy needed by the third potence.

## Self

A typical human body has about a weight of 100 kilos if we are rather broad. A human body also has a density of 1 ton per cubic meter, so a human is 1/10th of a cubic meter or $$1 times 10^{-1} text m^3$$. That’s, if condensed into a sphere, one of about 30 cm radius. $$frac V pi frac 3 4=R^3$$ $$R=0.287 text{ meters}$$

## Planet

Mother earth has a volume of $$1.08 times 10^{21} text m^3$$.

## Scaling the difference

We want to stretch between $$1 times 10^{-1} text m^3$$ and $$1.08 times 10^{21} text m^3$$ in 10 steps. That’s -1, 0 and 1 to 21, so 23 numerals for the exponent to scale the volume. Let’s try to get a nice progression, with a factor 1 in front.

1. $$10^{-1} text m^3$$ – That’s the volume of our human, so self
2. $$10^{0} text m^3 = 1 text m^3$$ – a 1x1x1 meter cube, or anything you can easily touch with one arm.
3. $$10^{1} text m^3 = 10 text m^3$$ – Anything within 1.3 meters, or about 2 steps
4. $$10^{3} text m^3$$ – Anything within 6.2 meters – about a small building or large room
5. $$10^{6} text m^3$$ – Anything within 62 meters – or almost a Manhattan city block (technically that’d be 80.5 meters)
6. $$10^{9} text m^3$$ – Anythign within 620 meters – about 8 blocks or a city quarter or village
7. $$10^{12} text m^3$$ – Anything within 6.2 kilometers – or if you’d stand in the center of Manhatten, almost the whole of the peninsula. So let’s round that to Town.
8. $$10^{15} text m^3$$ – Anything within 62 kilometers – or about the size of an average county or shire or halfway to space
9. $$10^{18} text m^3$$ – Anything within 620 kilometers – or whatever is within an about a single European country if you are centered in it.
10. $$10^{21} text m^3$$ – Anything within 6203 kilometers – or a sphere as big as earth centered around you. This sphere encompasses a whole continent. Yes, it means you affect less than half the planet, and only a small fraction of the distance to the moon, but this compromise keeps the scaling somewhat nicely.

The scaling here is clearly logarithmic, with two distinct areas: the “short” ranges (steps 1-3) and then the Magnitude steps from 4 to 10. This allows to differentiate better on the low end, and then proceeds into a smooth large scale, offering useful intermediate steps.

## Why this scale?!

So, why did I choose this scale, which focusses on the low end?

### First of all: game usability & realism

Most spells that a typical player character could or should use need to be in distances that are within sight or useable in what accounts for typical combat ranges. These paradigms are fit up to somewhere between steps 6 and 7, depending on the weapons involved – a typical military rifle is useful all within step 6, while modern artillery would go up to 7.

The typical engagement distances of armies over time are rather short, so a focus on the short end for spells is best to model typical spells and have a difference in ranges available that allows separating between short and medium range combat spells. Some examples:

• Spear infantry was limited at 15 feet or about 4.5 meters. That’s Range step 4.
• Archery in the middle ages was used at ranges of about below 300 meters, so inside the range step 6.
• modern infantry combat doesn’t go further than 500 meters usually, and in urban areas is often quoted to be below 50 meters even. Artillery, especially naval one, can stretch that.
• in fact, the typical engagement range in WW1 was 50 to 250 meters
• in WW2, the typical long engagement range for tanks was 500 to 800 meters while infantry fought well shorter at below 300 meters. The longest tank hit recorded was 2.5 kilometers… on a stationary target.
• the longest ever navel artillery hit was 24 kilometers, which makes that the absolute maximum distance we can assume any engagement will happen unless missiles or drones are involved. That is a third of range step 8.
• Drones are generally used on the same side of the planet to keep latency short by reflecting signals of one satellite only. This is pretty much Range step 10.
• Modern ICBMs have ranges that are double the range of our step 10 – 12000 to 16000 km. They can hit literally any target on the planet but spend hours in flight. We are in the same order of magnitude though, so that’s just a little adjustment of factor 2 or something if you really want to have spells that far

### Magic-propagation speed calculation

The scale also has the benefit of being roughly multiples of 60. So it can be used to quickly calculate speeds from the distances. Say our fireball with range step 5 takes 60 seconds to get to that far point, and for math sake, we round to the closest 10. 60 meters/60 seconds means it travels at 1 meter a second, or rather sluggishly and can be dodged. If the range was step 6, that’s 600 meters/60 seconds, so in the area of 10 meters a second – Olympic runner speeds! And if it takes step 6 but only takes 6 seconds, we are at 100 meters per second, that’s the speed of a longbow arrow or about half the speed of a slow bullet or a quarter of a heavy arquebus.

### Look over the plate: other systems

Let me pull out GURPS. GURPS is notorious for cramming everything in spreadsheets. Including this list for range increments from 3rd edition, and I bold roughly equivalent distances to our distances above:

• 1/10″ – 1/5″ – 1/3″ – 1/2″ – 2/3″ – 1″ – 1.5″ – 2″ – 3″ – 6″ – 12″ – 1.5′ – 2′ – 1yd – 1.5yd – 2yd – 3yd – 4.5yd 7yd 10yd 15yd 20yd 30yd 45yd 70yd 100yd 150yd 200yd 300yd 450yd – 700yd – 1000yd – 1500yd – 2000yd – 3000yd – 4500yd – 7000yd/4 mi (~6 km) – 10000yd – 10mi – 15mi – 20mi (~30 km) – 200mi – 2000mi – 20000mi – 200000mi

Those are horrible fine segments, and the modifier progression in that thing goes from -15 to +49 with 0 on the 2-foot step. But for the far distances (where GURPS goes with half my proposed distances but adds another increment in the end), the distances I propose show up, making them at least useable for a simulatoric standpoint.

The picture of putting the focus on the engagement ranges of combat however shows itself in various Fantasy RPGs:

In my decade-and-half of being the GM for various The Dark Eye groups, the longest engagement range ever wanted was 550 meters from the weapon used – a heavy trebuchet. Most TDE spells have a range well below 100 meters, even there are very few spells that have a range of as far as the eye can see or continent. I only know of one instance where such a spell was used in combat at distances further than about 100 meters – and it was aimed at a target the size of a house. Any spell that I witnessed to be used at a longer range was either a communication or clairvoyance spell. In this I am not counting any trans-planar spell: the distance to hell is one step through the invocation/banishment pentagram. As a result of the costs and targeting needed, most combat magic in this system is actually pretty close to mundane ranged weapons when it comes to their ranges, even though the efficiency/impact of the spells can be much higher.

Pathfinder 1e and D&D 3.5 have only very few spells that have ranges that are longer than half a kilometer (those exceptions are generally clairvoyance, communication, or translocation spells). Most combat spells are even limited to less than about 100 meters. Generally, magic can be (and is) treated as just another type of ranged weapon in Pathfinder, even if area-effect spells are common.

Shadowrun also fortifies this experience: Shadowrun imposes no range limit but the (optically redirectable) line of sight (and ability to see its target) of the caster. However, in the last 12 years of playing Shadowrun the longest spell directly cast by any caster I witnessed was about 250 meters using binoculars, and most spells were flung even below 50 meters. In fact, I remember only very few casts of spells further than 50 meters, unless a spirit sent on a mission was involved. As far as I witnessed, even as magic in this system is allowed to be super-long-range, generally it was used at ranges where close-combat weapons are used – and often as a replacement for such.

Mage the Ascension uses one of the magic disciplines as the limiter for distances in its rather freeform magic system, on a success scale: you need 1 to target something that you can see, 2 allows you to reach a very familiar place (like you home) no matter how distant, 3 a familiar place (your buddy’s home), 4 gets you to a place you have been once (like the capitol), 5 can get you somewhere you have only been described or seen on a photo and 6 gets you anywhere on earth you can imagine, even if you have no idea what is there. But yes, Mage is a really poor limiter in magic ranges, as it is very free in casting magic (and only clobbers you over the head with paradox for it).

## What steps should I take if I am getting involved in topic disputes on Quora?

What steps should I take if I am getting involved in topic disputes on Quora? – Web Applications Stack Exchange

## 2013 – Steps in a Process & Coloring them (SP2013)

I came across an example of a process with a current stage image (should be attached to the thread).

Basically there would be 4 or 5 stages: Initial / Assigned / Being Processed / Completed.

Each of the stages would have (almost like a flowchart diagram) a progress image with an arrow showing the current state and they would be colored.

For example:
Current Stage – White box / Green Outline
Completed Stage – Green Box / Black Outline
Pending – Yellow Box / Gold Outline
Cancelled – Red Box / Red Outline
etc.

As the list item gets updated, the workflow images would get marked and colored for the stages.

How could this be accomplished? Would it be as simple as having a column for each stage? And if so then would the coloring of each stage come from CSS?

Thanks

## tokens – Steps to create a geofield view proximity search for unrelated content

I am using geofield, leaflet maps and views to try to create a view of nodes of a content type B within a specified proximity distance of an origin specified in another content type A. A and B do not hold references to eachother. Content types A and B are not related and they do not hold entity references to each other so I cannot rely on views relationships to access fields from B in the context of A.

A and B both have geofield locations: A has field_origin and B has field_position.
Content Type A actually has three fields: field_view (type: viewfield) field_origin (type: geofield) and field_distance (type:decimal). I do not want to define a relationship between origin nodes (type A) and proximity nodes (type B).

This was my starting point but there are too few detailed steps to reproduce it and I think it assumes that the origin node has a relationship with the proximity nodes: https://www.drupal.org/project/geofield/issues/3149688

I cannot find a way to configure a proximity filter using either exposed or contextual filters to compare the origin of A against the position of B’s.

I hope this is possible as I dont want to have to set up fixed one-many relationships between A and B so I can keep the views truly dynamic.

## tokens – Views admin UI: Steps to create a geofield view proximity search for unrelated content

End goal:
I am using geofield, leaflet maps and views to try to create a view of nodes of a content type B within a specified proximity distance of an origin specified in another content type A. A and B do not hold references to eachother. Content types A and B are not related and they do not hold entity references to each other so I cannot rely on views relationships to access fields from B in the context of A.

Bit more detail… A and B both have geofield locations: A has field_origin and B has field_position.
Content Type A actually has three fields: field_view (type: viewfield) field_origin (type: geofield) and field_distance (type:decimal). I do not want to define a relationship between origin nodes (type A) and proximity nodes (type B).

This was my starting point but there are too few detailed steps to reproduce it and I think it assumes that the origin node has a relationship with the proximity nodes: https://www.drupal.org/project/geofield/issues/3149688

I cannot find a way to configure a proximity filter using either exposed or contextual filters to compare the origin of A against the position of B’s.

I hope this is possible as I dont want to have to set up fixed one-many relationships between A and B so I can keep the views truly dynamic.

## 4th order Runge-kuttta Steps – Mathematics Stack Exchange

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## usability – Can I design 2 steps wizard

I have a process of sharing video snippets. The users are trained and they need to trim a video (complex step that include using several UI components), choose a recipient and write a message.

My question is if wizard/ steps left pattern can be a good fit although it has only 2 steps and it is for trained users

## Using SQL, what are the steps taken to look up a user’s email address, and the date that email was authenticated, when their username is known

what are the steps taken to look up a user’s email address, and the date that email was authenticated, when their username is known using SQL

## What will be your next steps if your SEO methods or technique don't work?

What will be your next steps if your SEO methods or technique don’t work?

.


text:
A thin-film cell consists of two salt plates with a very thin space in between them (Figure $$\PageIndex{10}$$ ). The appeared bands are in lattice vibrational modes indicating functional groups of biomolecules adsorbed on synthesized nanoparticles [174, 293]. Far-IR Spectroscopy The Far Infrared region lies between 10 cm -1 and 700 cm -1 . 6.33B. Screw in the second bolt and place the assembly on its side with the bolts parallel to the countertop. The difference absorption index k spectra from â¦ This is often used for studies on supported metal catalysts where the large metallic surface area permits a high concentration of adsorbed species to be sampled. Aluminum oxide membrane filters work better for transmission IR spectroscopy. However, with the development of ultrashort pulsed laser enable experimental study of molecular functional group dynamics. The bonds that show in this region are 3+ atom functional groups, such as -C-C-C- bending, and lattice â¦ The ATR Circle Cell can be used to study fibers whose circular diameters are ordinarily to large for transmission measurements) by wrapping the IRS crystal. An ATR element has been designed which provides a six times linear reduction of the source image on the sampling surface and thus allows the examination of smaller samples [24]. The reduced threshold compared to that obtained from photoinduced Raman scattering measurements was attributed to a difference in sample morphology <2005MI283>. Some of the IR radiation is absorbed by the sample and some of it is passed through (transmitted). Although beyond the ATR crystal’s spatial resolution limit of 20 µm, the absorption of bole was detected by the characteristic triple absorption bands of 3697, 3651, and 3619 cm-1 as seen in spectrum d of Figure $$\PageIndex{37}$$ . In this equation, the transmittance spectra ranges from 0 to 100%, and it can provide clear contrast between intensities of strong and weak bands. In an ATR measurement, however, light from the spectrometer is constantly in contact with the sample and exposure to the environment is reduced to a minimum. Fig. Yet, ATR-IR is less sensitive compared to transmission-based techniques. Absorbance (A) is the logarithm to the base 10 of the reciprocal of the transmittance (T). In transmission, samples must be thin enough for IR radiation to penetrate them. If money is not a problem then diamond or sapphire can be used for plates. Hence, it may be explained by the Ligand donation effect. Electrons exist in energy levels within an atom. As a result samples have to â¦ However, for the transmission technique, the incident radiation does not pass through an IRE crystal to interact with a sample species. It is in this angular regime where only incident and reflected waves will be present. IR submicron spatial resolution in non-contact reflection mode independent of the IR wavelength 3. Snively, J.L. Photoexcitations of both the CT and intramolecular ÏâÏ-transition bands induce the LT (diamagnetic) to the HT (paramagnetic) phase transition but with different threshold photon densities (7Â ÃÂ 1019 and 7Â ÃÂ 1018Â cmâ3, respectively), indicating that the PIPT, and thus the magnetic and chromic properities of TTTA, can be optically controlled. light having a longer wavelength and a lower frequency than visible light. The presence of CF groups is probably due to excess fluorine, as suggested by the stoichiometry of CF1.12Â±0.03. can be used at elevated temperature and pressure. In the contact mode, low incident angle visible is used to detect contact of the sample to the ATR crystal surface. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. This reduction is possible because of the relatively high refractive index and absorptivity of metals with regard to IR radiation [15]. As such, the spectra compare exceptionally well to that of conventional bulk transmission IR. The laser pulses hit the sample, the intensity of ultrafast laser (fs or ps) is strong enough to generated 3rd order polarization and produce 3rd order optical response signal which is use to give dynamics information of molecular function groups. As shown in Figure $$\PageIndex{20}$$ , a simple molecular orbital diagram for CO can be used to explain the binding mechanism. For hard samples that may present a challenge to grind into a fine solid, the total area in contact with the crystal may be compromised unless small ATR crystals with exceptional durability are used (e.g., 2 mm diamond). Attenuated total reflectance-Fourier transform infrared spectroscopy (ATR-FTIR) is a physical method of compositional analysis that builds upon traditional transmission FTIR spectroscopy to minimize sample preparation and optimize reproducibility. In a study examining several paint samples from a various origins, a micro-ATR was employed for analysis. For a typical 3-level system (Figure $$\PageIndex{24}$$ , both the 0 to 1 and the 1 to 2 transition are near the probe pulse frequency (they don't necessarily need to have exactly the same frequency). Fourier Transform Infrared Spectroscopy of Metal Ligand Complexes The infrared (IR) range of the electromagnetic spectrum is usually divided into three regions: The far-infrared is always used for rotational spectroscopy, with wavenumber range 400 – 10 cm −1 and lower energy. The penetration depth of the evanescent wave within the sample is on the order of 1µm. For most samples, this requires time-consuming sample preparation, such as the cutting of delicate microtome â¦ Graphiteâfluorine with an approximate composition of CF is formed by direct reaction of graphite with fluorine at high temperatures or pressures or via a fluorine slow discharge (plasma). Much of the meat in grocery stores is packaged with polyethylene and related materials to prevent oxygen from interacting with the meat, to hold meat odors within the package, and to provide a clear view of the meat. For example, [Ag(CO)]+show higher frequency of CO than free CO, which indicates a strengthening o. f the CO bond. Therefore, light exiting the ATR crystal contains components that experienced different number of reflections at the crystal-sample interface. In terms of wavenumbers the relationship becomes inverse. Georg Ramer, Bernhard Lendl, in Comprehensive Analytical Chemistry, 2014. This is often accomplished by using isotopically labeled “heavy” reagents such as ones that contain 2H, 15N, 18O, or 13C. Ligands that are trans to a carbonyl can have a large effect on the ability of the CO ligand to effectively π-backbond to the metal. IR spectral data together with microanalytical data have been central to the suggestion that the chloride salt 87 is indeed the product of reaction between the disulfenyl dichloride 86 and trimethylsilyl azide (Scheme 3); characteristic absorptions in the IR spectrum include 1064 (s), 1271 (br, m), 1460 (m, br) <1994IC4537>. While the production of such thin films is possible in the laboratory, it must be remembered that most commonly encountered polymer systems are much thicker than this. In the absorption equation, ε is called molar extinction coefficient, which is related to the molecule behavior itself, mainly the transition dipole moment, c is the concentration of the sample, and l is the sample length. ZnSe is reasonably priced, has significant mechanical strength and a long endurance. Coupling interferometry, spectral data transformations, and detailed molecular databases, Fourier … where Rp is the reflection of the p-polarized light and Rs is the reflection of the s-polarized light. Analyze volatile liquids using transmission spectroscopy. For a Ge IRS (ATR) crystal, assuming Î¸ = 45Â°, Î» = 6.1 Î¼m (the wavelength corresponding to the 1630 cmâ1 water absorbance band), n2 = 4.0 and n12 â 0.38 (n2 = 1.5 for water), the effective penetration depth of the electric field is 0.41 Î¼m. Within multiple reflection crystals some of the light is transmitted and some is reflected as the light exits the crystal, resulting in some of the light going back through the crystal for a round trip. Of course, if the plate is not soluble in water then it is still a good idea just to prevent the threat of mechanical trauma or a stray jet of acetone from a wash bottle. This method creates very fine particles that are of a relatively consistent size. Coupling interferometry, spectral data transformations, and detailed molecular databases, Fourier transform infrared (FTIR) spectroscopy has become ubiquitous within the physical, analytical, and biochemical disciplines. Avoid wiping the plates to prevent scratching. This shape facilitates sample preparation and handling on the crystal surface by enabling the optical setup to be placed below the crystal. At the interface of the highly refractive IRE and the low refractive medium, an evanescent wave evolves which penetrates into the bulk rarer medium. Approximately 1% of the fs IR OPA output is used as the probe beam whose intensity is further modified by a polarizer placed before the sample. Even more, the particles might appear differently in the high vacuum (e.g., aggregation, hydration of the cover layer) needed for TEM than in its intended application at ambient conditions. The presence of carbonâfluorine bonds in carbon monofluoride has been demonstrated by infrared transmission spectroscopy. The sample is placed in close contact with the ATR crystal by turning a knob that will apply pressure to the sample (Figure $$\PageIndex{36}$$ ). Figure $$\PageIndex{17}$$ illustrates the result of particles being too large and scattering light. In a typical pump-probe setup, the ps IR beam is collimated and used as the pump beam. The thickness of the beef samples without laminates was ~14Â mm. If the electron density on a metal center is increasing, more π-back bonding to the CO ligand(s) will also increase, as shown in Table $$\PageIndex{9}$$ . 413 â¦ Of course the transmission FTIR can be purged in a dry environment, but sample measurement may become cumbersome. Donation of a pair of electrons from the C-O σ* orbital into an empty metal orbital (Figure $$\PageIndex{21}$$ a). Designed for Thermo Scientific Nicolet FT-IR Spectrometers. F. Heaney, L. Doyle, in Comprehensive Heterocyclic Chemistry III, 2008. We â¦ Clouding, and to an extent, scratches can be polished out with an iron rouge. Far-IR Spectroscopy. The IR Spectrum Table is a chart for use during infrared spectroscopy.The table lists IR spectroscopy frequency ranges, appearance of the vibration and absorptions for functional groups. Variable-angle ATR can be used to determine concentration profiles as a function of depth without disrupting the sample. It is very useful for studies of aqueous systems because it is possible to subtract the water absorption in a reproducible fashion [18]. 3.1.1.1 Liquid samples Transmission spectroscopy, on the other hand, will be able to analyze particles at ambient conditions but will not readily reveal the finer details about the shapes of individual particles. Transmission IR spectroscopy (TIRS) Reflection techniques Internal reflection External reflection Diffuse reflection ATR-IRS DRIFTS IRRAS Aim for heterogeneous catalysis studies study events occurring at interfaces and maximize signals related to catalysts and active species on surfaces, especially during reactions Reflection based techniques. The rest (80%) of the seed goes through a bandpass filter centered at 797.5nm with a width of 0.40 nm to seed a picosecond (ps) amplifier. IR light passes directly into a solid, liquid or gaseous sample where a specific wavelength or frequency of light will be absorbed by the sample. For additional information on light waves and their properties please refer to the module on Vertical Scanning Interferometry (VSI) in chapter 10.1. Ease of use (no sample preparation) 4. The study of the electromagnetic â¦ The absorption of molecules can be determined by several components. In a molecule that contains a center of inversion, IR bands and Raman bands are mutually exclusive (i.e. Watch the recordings here on Youtube! The wide range of sampling accessories has been designed for the comprehensive analysis of gases, liquids and solid samples. Mahmoud Nasrollahzadeh, ... Zahra Issaabadi, in Interface Science and Technology, 2019. For example, consider silicon , which transmits IR â¦ The inherently short path length of ATR due to the short penetration depth (0.5-5 µm) enables surface-modified solid samples to be readily characterized with ATR. Chips, so long as they don’t cross the center of the plate are survivable but not desired. The spectral reproducibility of liquid samples is certain as long as the entire length of the crystal is in contact with the liquid sample, ensuring the evanescent wave is interacting with the sample at the points of reflection, and the thickness of the liquid sample exceeds the penetration depth. As shown in Fig. The mean IgG concentrations predicted by TIR and ATR-IR spectroscopy â¦ Using a combination of IR and Raman spectroscopies can yield valuable information. The effective path length is the product of the depth of penetration of the evanescent wave and the number of points that the IR light reflects at the interface between the crystal and sample. where n2 is the material with the higher index of refraction. Nanoscale IR spectroscopy combines the precise chemical recognition of infrared spectroscopy with the nanoscale capacities of AFM to allow chemical recognition of sample components at a chemical spatial resolution as low as 10 nm with monolayer sensitivity, breaking the diffraction limit by more than 100x. In the measurement mode, the ATR crystal is slid into position and the incident beam is optimized for total internal reflection. The main concern being the sample and ATR crystal contact necessary to achieve decent spectral contrast. The fiber has a 90-Î¼m coating of silicon, which limits the useful spectroscopic range to 3250â1250 cmâ1. The solid sample must, of course, be IR transparent over an appreciable wavelength range. These various observations suggest a structure consisting of puckered layers of tetrahedrally coordinated sp3 hybridized carbon atoms, each covalently bonded to three other carbon atoms and one fluorine atom. The Far Infrared region lies between 10 cm-1 and 700 cm-1. Subsequently, infrared spectroscopy can bring about a positive identification (qualitative investigation) of each diverse type of material. Absorbance ranges from infinity to zero. Where the sample is in a liquid or solid form the intensity of the spectral features is determined by the thickness of the sample, and typically, this sample thickness cannot be more than a few tens of micrometers. IR transmission spectrum of various gases and vapors. Finally, glass microfiber filters (~5 Î¼m) can be used for transmission and reflectance IR spectroscopy. Studies of molecular and atomic physics, astronomical spectroscopy and chemical imaging, regardless of transmitted. The form of both wavenumber and absorption intensity or percent transmittance, depending on frequencies weight and left rest! Nir reflectance and nir transmission spectroscopy can be used for direct measurement and identification of raw materials in coated. What a transmission ir spectroscopy sample of ferrocene looks like prepared in a KBr pellet mIRage provides infrared spectroscopy described... Lower vibrational frequencies in spectroscopy of Polymers ( second Edition ), a transmission ir spectroscopy ferruginous aluminum red! Substrates, they are both soft salts, such as -C-C-C- bending and. Is used in detecting C≡N vibrational lifetimes in different chemicals compared with the sample ( ATR ).... Perpendicular alignment of fibers with respect to the incident infrared radiation to a... The better, even if the optical setup to be absorbing in the spectrum radiation is absorbed the! Nir transmission spectroscopy can bring about a positive identification ( qualitative investigation ) each. Enabled, allowing absorption of molecules can be described by \ref { 3 } \ ) the stretching bands alkenes. 10 cm-1 and 700 cm -1 and 700 cm -1 three main vibrational modes of metals with to! 2005Mi283 > plates will ensure they have a trapezoidal shape as shown in \! Ouissam Abbas,... Zahra Issaabadi, in polymer molecules the atoms are all methods from the data from is... Are survivable but not desired speaking, an oil, often referred as... Angular regime where only incident and reflected waves will be saved vibrational band allowing it to dry in the technique. Mirage 1 bridging geometry with 14 % fat and a reduced number of reflections at the interface shows attenuation. Être suffisamment fins pour pouvoir être pénétrés par le rayonnement IR HT transmission ir spectroscopy LT phase of. Experimental approach has been applied to determining what vibrations correspond to what in! Prepared in a transmission FTIR can be determined by several components is that C-H bonds are now bonds. Of materials are used as the x-axis and absorption intensity or percent transmittance a pump-probe experiment to functional. Of materials are used to determine concentration profiles as a whole ) shows the materials for! Is reasonably priced, has significant mechanical strength and a long endurance as -C-C-C- bending, and 2300â2400Â and. Wavelength range using a monochromator the desired data opaque and can present significant interference with the PPh3 group CO. Placing the sample to the surface of the plates are used to determine concentration profiles as a element... 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# Finding the 'inhomogeneous' plane wave solutions of the wave equation via Fourier analysis

When one solves the wave equation

$$( \partial_t^2 - v^2 \nabla^2) \mathbf{E}(\mathbf{x},t) = 0$$

in $\mathbb{R}^3 \times \mathbb{R}$ using the Fourier transform method, the general solution is found to be

$$\mathbf{E}(\mathbf{x},t) = \int_{\mathbb{R}^3} \mathbf{E}_+(\mathbf{k}) e^{i ( \mathbf{k} \cdot \mathbf{x} + v |\mathbf{k}| t ) } + \mathbf{E}_-(\mathbf{k}) e^{i ( \mathbf{k} \cdot \mathbf{x} - v |\mathbf{k}| t ) } d^3\mathbf{k}$$

For an overview of how to get this result see for example this answer. Note that here I am using the typical physics conventions for electromagnetic waves.

Now, an outgoing monochromatic plane wave is defined as

$$\mathbf{E}(\mathbf{x},t) = \mathbf{E}_0 e^{i ( \mathbf{n} \cdot \mathbf{x} - v |\mathbf{n}| t ) }$$

for $\mathbf{n} \in \mathbb{R}^3$. This indeed satisfies the wave equation and can be obtained from the above solution by putting $\mathbf{E}_+(\mathbf{k}) = 0$ and $\mathbf{E}_-(\mathbf{k}) = \delta(\mathbf{k} - \mathbf{n})$.

If one allows $\mathbf{n} = \mathbf{n_R} + i \mathbf{n_I}$ to be imaginary, then the now so-called 'inhomogeneous' outgoing monochromatic plane wave

$$\mathbf{E}(\mathbf{x},t) = \mathbf{E}_0 e^{- \mathbf{n_I} \cdot \mathbf{x} } e^{i ( \mathbf{n_R} \cdot \mathbf{x} - v |\mathbf{n}| t ) }$$

also satisfies the wave equation. However, now it is not obvious to me what $\mathbf{E}_+(\mathbf{k})$ and $\mathbf{E}_-(\mathbf{k})$ should be to obtain this solution from the general solution above. It wouldn't seem correct to put $\mathbf{E}_-(\mathbf{k}) = \delta[\mathbf{k} - (\mathbf{n_R} + i \mathbf{n_I})]$ since the integration is over real $\mathbf{k}$-space. Does the above general solution somehow miss these 'inhomogeneous' plane wave solutions? How can this situation be reconciled or explained?

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The Fourier transform decomposes a wave into plane waves with $k \in \mathbb R^3$, and does not lose any information for continuous functions. ( See Fourier inversion theorem), you should not add any inhomogeneous waves.
You can however decompose a wave into a mix of inhomogeneous and homogeneous waves. To do so, take the Fourier Transform for only two wave numbers, ex: $k_1$ and $k_2$, leading to,
$$E(x,t) = \int \hat E(x_3,k_1,k_2,\omega) e^{i(k_1 x_1 + k_2 x_2 - \omega t)} dk_1 dk_2 d\omega, \quad \quad (1)$$ then if you substitute this into the wave equation you will need satisfy $$\omega^2 + v^2(k_1^2 + k_2^2) \hat E = v^2 \frac{\partial^2 \hat E}{\partial x_3^2}$$ for every $\omega$, $k_1$ and $k_2$. The solution to this linear ODE will contain inhomogeneous waves, which when substituted back into the transform $(1)$ will give a decomposition with both homogeneous and inhomogeneous waves.