text:
# Funny stuff in a Beamer title irreparably messes with the PDF metadata

I have the following beamer presentation:

\documentclass{beamer}
\usepackage{hyperref}

\title{%
\parbox{.25\textwidth}{funny}%
\parbox{.5\textwidth}{Real Title}%
\parbox{.25\textwidth}{stuff}}

\hypersetup{pdftitle={Real Title}}

\begin{document}
Hello, World!
\end{document}

and no matter what I try (removing hyperref and using \pdfinfo, for example), the title always comes out as the entire ASCII portion of the \title, less control sequences:

.25funny.5Real Title.25stuff

Is there no way to absolutely set the title of the PDF using only LaTeX? (I know metadata editors exist, but this could be considered a bug in some lines of thought.)

I should note that the problem is not reproducible in article.

-
The problem here is hyperref, it can only handle text. If you create a titlepage in your document, the title looks fine. If you try using your title as a section title in article, the navigation will be messed up as well. I'm not aware of any way to trick hyperref into doing non-text right... –  Anke Apr 19 '13 at 13:45
IMO formatting macros like \parbox should not go into a moving argument such as for \title. If you want to change how the title is shown patch \maketitle. –  Matthew Leingang Apr 19 '13 at 13:51
@MatthewLeingang I completely agree, but I don't really know how to 'patch' \maketitle in the same sense I'd 'patch' some listing; there is probably a lot going on in beamer's maketitle, very little of which I would understand at a glance. –  Sean Allred Apr 19 '13 at 14:00
@MatthewLeingang that is a really sensible advice. Do you mind if I add it to my answer? (That is, of course, if you're not planning to give an answer.) –  Gonzalo Medina Apr 19 '13 at 14:05
@GonzaloMedina: Sean's right, digging into beamer internals is intimidating, so it's not trivial. I'm toying with an answer now. –  Matthew Leingang Apr 19 '13 at 14:10

## 2 Answers

You can use \texorpdfstring:

\documentclass{beamer}

\title{%
\texorpdfstring{\parbox{.25\textwidth}{funny}%
\parbox{.5\textwidth}{Real Title}%
\parbox{.25\textwidth}{stuff}}{Real Title}}

\begin{document}
Hello, World!
\end{document}

Since some themes use the title in other places (in the footline, for example), it would alos be a good idea to use the optional argument, so this will be picked up for displaying the title in footlines:

\documentclass{beamer}

\title[Title]{%
\texorpdfstring{\parbox{.25\textwidth}{funny}%
\parbox{.5\textwidth}{Real Title}%
\parbox{.25\textwidth}{stuff}}{Real Title}}

\begin{document}
Hello, World!
\end{document}

By the way, there's no need to load hyperref with beamer.

As Matthew Leingang mentioned in a comment, a more sensible approach would be to make a modification to the way the title is built. In the following example I modified the title page template to insert information using three \parboxes in the title field. The \triplebox command receives three mandatory arguments: the first and third ones will be used for the lateral \parboxes, and the second one will be used for the title; there's also an optional argument that can be used to pass an optional argument to \title:

\documentclass{beamer}

\def\insertfirst{}
\def\insertsecond{}

\makeatletter
\defbeamertemplate*{title page}{mydefault}[1][]
{
\vbox{}
\vfill
\begin{centering}
\begin{beamercolorbox}[sep=0pt,center,#1]{title}
\usebeamerfont{title}%
\parbox{.25\linewidth}{\centering\insertfirst}%
\parbox{0.5\linewidth}{\centering\inserttitle}%
\parbox{.25\linewidth}{\centering\insertsecond}\par%
\ifx\insertsubtitle\@empty%
\else%
\vskip0.25em%
{\usebeamerfont{subtitle}\usebeamercolor[fg]{subtitle}\insertsubtitle\par}%
\fi%
\end{beamercolorbox}%
\vskip1em\par
\begin{beamercolorbox}[sep=8pt,center,#1]{author}
\usebeamerfont{author}\insertauthor
\end{beamercolorbox}
\begin{beamercolorbox}[sep=8pt,center,#1]{institute}
\usebeamerfont{institute}\insertinstitute
\end{beamercolorbox}
\begin{beamercolorbox}[sep=8pt,center,#1]{date}
\usebeamerfont{date}\insertdate
\end{beamercolorbox}\vskip0.5em
{\usebeamercolor[fg]{titlegraphic}\inserttitlegraphic\par}
\end{centering}
\vfill
}
\makeatother

\newcommand\triplebox[4][]{%
\gdef\insertfirst{#2}\gdef\insertsecond{#4}\title[]{#3}
}

\triplebox{funny}{Real Title}{stuff}
\author{Author}
\institute{Institute}

\begin{document}

\begin{frame}
\maketitle
\end{frame}

\end{document}

-
Thanks for the macro and the tip! (It didn't hit me that beamer already does its own hyperlinking.) –  Sean Allred Apr 19 '13 at 13:57
@SeanAllred please see my updated answer. –  Gonzalo Medina Apr 19 '13 at 14:40
Good answer. If you look at mine you'll see I took a slightly different route with \addtobeamertemplate. –  Matthew Leingang Apr 19 '13 at 14:47
@SeanAllred No. You only use the optional argument if you want to provide an optional argument for \title. –  Gonzalo Medina Apr 19 '13 at 14:51

I guess I'm answering the broader question “How can I alter the display of the title in beamer?”. In my comment I said that I think “the right way” to do this is to alter the formatting macros (i.e., \maketitle) instead of the metadata-declaration macros (i.e., \title).

Here is an example. Explanation follows.

\documentclass{beamer}
\usepackage{etoolbox}
\usetheme{AnnArbor}

\title{Real Title}

\makeatletter
\addtobeamertemplate{title page}{%
\let\beamerorig@inserttitle\inserttitle%
\renewcommand{\inserttitle}{%
\makebox[0.20\textwidth][c]{funny}
\makebox[0.5 \textwidth][c]{\beamerorig@inserttitle}
\makebox[0.20\textwidth][c]{stuff}
}
}{}
\makeatother

\begin{document}
\begin{frame}
\maketitle
Hello, World!
\end{frame}
\end{document}

I used the fancy AnnArbor theme so you can see that the display of “Real Title” in the footer isn't affected.

The titling macros (which I found with grep) are found in beamerbasetitle.sty:

\def\maketitle{\ifbeamer@inframe\titlepage\else\frame{\titlepage}\fi}
%...
\def\titlepage{\usebeamertemplate*{title page}\@thanks}

We see that for flexibility \maketitle uses beamer's templating system. The title page template is usually defined in an inner theme. Here is the one from beamerinnerthemedefault.sty (again, found with grep):

\defbeamertemplate*{title page}{default}[1][]
{
\vbox{}
\vfill
\begin{centering}
\begin{beamercolorbox}[sep=8pt,center,#1]{title}
\usebeamerfont{title}\inserttitle\par%
\ifx\insertsubtitle\@empty%
\else%
\vskip0.25em%
{\usebeamerfont{subtitle}\usebeamercolor[fg]{subtitle}\insertsubtitle\par}%
\fi%
\end{beamercolorbox}%
\vskip1em\par
\begin{beamercolorbox}[sep=8pt,center,#1]{author}
\usebeamerfont{author}\insertauthor
\end{beamercolorbox}
\begin{beamercolorbox}[sep=8pt,center,#1]{institute}
\usebeamerfont{institute}\insertinstitute
\end{beamercolorbox}
\begin{beamercolorbox}[sep=8pt,center,#1]{date}
\usebeamerfont{date}\insertdate
\end{beamercolorbox}\vskip0.5em
{\usebeamercolor[fg]{titlegraphic}\inserttitlegraphic\par}
\end{centering}
\vfill
}

The part that is inside the {beamercolorbox}[...]{title} environment is where the title goes. At this point you have to decide if you want to create a new title page template with \defbeamertemplate, similar to that template but with alterations inside the title beamercolorbox, and invoke it with \usebeamertemplate, or hack the way that \inserttitle displays the title. I chose the latter, because (a) you don't have to know which template is used, and (b) beamer provides commands to locally alter templates.

So my code uses \addtobeamertemplate to preprend to the current title page template (whichever is in use) a patch to the \inserttitle command. However you want the title really displayed should go there.

-


text:
# Gated-Attention Readers for Text Comprehension

Bhuwan Dhingra • Hanxiao Liu • Zhilin Yang • William W. Cohen • Ruslan Salakhutdinov

In this paper we study the problem of answering cloze-style questions over documents. Our model, the Gated-Attention (GA) Reader, integrates a multi-hop architecture with a novel attention mechanism, which is based on multiplicative interactions between the query embedding and the intermediate states of a recurrent neural network document reader. This enables the reader to build query-specific representations of tokens in the document for accurate answer selection.

Full paper

format: plain


text:
PIRSA:22120052

# Surrogate model for gravitational wave signals from black hole binaries built on black hole perturbation theory waveforms calibrated to numerical relativity : one model to rule both comparable and extreme mass ratio regime

### APA

Islam, T. (2022). Surrogate model for gravitational wave signals from black hole binaries built on black hole perturbation theory waveforms calibrated to numerical relativity : one model to rule both comparable and extreme mass ratio regime. Perimeter Institute. https://pirsa.org/22120052

### MLA

Islam, Tousif. Surrogate model for gravitational wave signals from black hole binaries built on black hole perturbation theory waveforms calibrated to numerical relativity : one model to rule both comparable and extreme mass ratio regime. Perimeter Institute, Dec. 01, 2022, https://pirsa.org/22120052

### BibTex

          @misc{ pirsa_22120052,
doi = {10.48660/22120052},
url = {https://pirsa.org/22120052},
author = {Islam, Tousif},
keywords = {Strong Gravity},
language = {en},
title = {Surrogate model for gravitational wave signals from black hole binaries built on black hole perturbation theory waveforms calibrated to numerical relativity : one model to rule both comparable and extreme mass ratio regime},
publisher = {Perimeter Institute},
year = {2022},
month = {dec},
note = {PIRSA:22120052 see, \url{https://pirsa.org}}
}

Tousif Islam University of Massachusetts Dartmouth

Collection
Talk Type
Subject

## Abstract

We present a reduced-order surrogate model of gravitational waveforms from non-spinning binary black hole systems with comparable to large mass-ratio configurations. This surrogate model, BHPTNRSur1dq1e4, is trained on waveform data generated by point-particle black hole perturbation theory (ppBHPT) with mass ratios varying from 2.5 to 10,000. BHPTNRSur1dq1e4 can generate waveforms up to 30,500 m1(where m1 is the mass of the primary black hole), includes several more spherical harmonic modes up to \ell=10, and calibrates both dominant and subdominant modes to numerical relativity (NR) data. In the comparable mass-ratio regime, including mass ratios as low as 2.5, the gravitational waveforms generated through ppBHPT agree surprisingly well with those from NR after this simple calibration step. We argue that this scaling essentially captures higher order self-force corrections in a much simpler way. We also compare our model to recent SXS and RIT NR simulations at mass ratios ranging from 15 to 32, and find the dominant quadrupolar modes agree to better than≈10−3. We expect our model to be useful to study intermediate-mass-ratio binary systems in current and future gravitational-wave detectors. Finally, we discuss avenues for improving the model by extending its region of validity.


text:
In mathematics, Kingman's subadditive ergodic theorem is one of several ergodic theorems. It can be seen as a generalization of Birkhoff's ergodic theorem.[1] Intuitively, the subadditive ergodic theorem is a kind of random variable version of Fekete's lemma (hence the name ergodic).[2] As a result, it can be rephrased in the language of probability, e.g. using a sequence of random variables and expected values. The theorem is named after John Kingman.

## Statement of theorem

Let ${\displaystyle T}$ be a measure-preserving transformation on the probability space ${\displaystyle (\Omega ,\Sigma ,\mu )}$, and let ${\displaystyle \{g_{n}\}_{n\in \mathbb {N} }}$ be a sequence of ${\displaystyle L^{1}}$ functions such that ${\displaystyle g_{n+m}(x)\leq g_{n}(x)+g_{m}(T^{n}x)}$ (subadditivity relation). Then

${\displaystyle \lim _{n\to \infty }{\frac {g_{n}(x)}{n}}=:g(x)\geq -\infty }$

for ${\displaystyle \mu }$-a.e. x, where g(x) is T-invariant. If T is ergodic, then g(x) is a constant.

## Applications

If we take ${\displaystyle g_{n}(x):=\sum _{j=0}^{n-1}f(T^{j}x)}$, then we have additivity and we get Birkhoff's pointwise ergodic theorem.

Kingman's subadditive ergodic theorem can be used to prove statements about Lyapunov exponents. It also has applications to percolations and probability/random variables.[3]

## References

1. ^ S. Lalley, Kingman's subadditive ergodic theorem lecture notes, http://galton.uchicago.edu/~lalley/Courses/Graz/Kingman.pdf


text:
# Asymmetric hysteresis loops in amorphous ferromagnetic alloys

#### Abstract

Amorphous ferromagnetic alloys annealed in a small magnetic field develop a reproducible, asymmetric hysteresis loop. If the direction of the field during annealing is regarded as +, the magnetization reversal from + to $-$ occurs in a single irreversible Barkhausen jump, but the reversal from $-$ to + is smooth and reversible, with its slope determined by the demagnetizing field of the sample. We call this phenomenon the asymmetric magnetization reversal (AMR). The asymmetric loop develops easily and readily in amorphous alloys near the zero magnetostriction composition (Co$\sb{0.94}$Fe$\sb{0.06}$)$\sb{\rm x}$(SiB)$\sb{\rm 1-x}$, but is not limited to these compositions. AMR can be produced in amorphous alloys by various combinations of time, temperature, and magnetic field applied during the anneal. However, the shape of the hysteresis loop depends sensitively on the conditions during the anneal. Oxidizing conditions are not necessary during the anneal. However, they make it easier to produce AMR and act to stabilize the effect. A typical AMR having a single Barkhausen jump can be produced by annealing a zero magnetostrictive amorphous ribbon for several hours at a temperature of 350 to 400$\sp\circ$C in air, in an applied magnetic field of about 100 mOe directed parallel to the ribbon axis. AMR can be destroyed partially or totally by light chemical or physical etching of the surface, which means that the origin of AMR resides in the surface of the sample. AMR can also be destroyed partially or totally by magnetizing the sample in a sufficiently high ac or dc magnetic field. This strongly suggests that the annealing treatments create a metastable domain structure, which can be destroyed by the application of a sufficiently high field. We propose the annealed-in anisotropy in domain walls combined with exchange anistropy between a surface layer and the bulk ribbon as the origin of AMR. The anisotropy in the domain walls is induced by the local magnetization during annealing, and the exchange anisotropy results from direct exchange coupling between a higher anisotropy surface layer (formed just below the outer oxide layer) and the bulk material.

#### Subject Area

Materials science

#### Recommended Citation

Shin, Kyung-Ho, "Asymmetric hysteresis loops in amorphous ferromagnetic alloys" (1992). Dissertations available from ProQuest. AAI9235205.
https://repository.upenn.edu/dissertations/AAI9235205

COinS


text:
# Sulfuric Acid Sodium Hydroxide Balanced Equation

According to the response equation. H 2 SO 4 + 2NaOH → Na 2 SO 4 + 2H 2 O. sulfuric acid reacts with sodium hydroxide on the 1:2 foundation. That method selection of moles of sulfuric acid is part that of selection of moles of sodium hydroxide used. To calculate sulfuric acid solution concentration use EBAS - stoichiometry calculator.The reaction between sodium hydroxide and sulfuric acid is a neutralization double displacement reaction that is expected to produce a salt and water. The balanced response would then be:...the balanced chemical equation of magnesium oxide and sulfuric acid is given as follows.Magnesium hydroxide and Sulphuric acid will mix to offer Magnesium sulphate and waterMg(OH)2 + H2SO4The chemical equation for the neutralization of sulfuric acid and sodium hydroxide will also be written as H2SO4 + 2NaOH —> 2H2O + 2NaSO4. Sulphuric acid is an excessively robust dehydrating acid. The chemical equation for the neutralization of sulfuric acid and sodium hydroxide may also be written as H 2 SO 4 + 2NaOH —> 2H 2 O + 2NaSO 4.

Write the balanced chemical equation for an answer of sulfuric acid (battery acid) being neutralized with a solution of sodium hydroxide.What r the steps one must take to get the balanced equation, and be able to acknowledge the response type? Thank you soo a lot in your lend a hand prematurely. Im very grateful for any helpful assist I am getting!!Get a solution for 'Write and steadiness the equation for the reaction of hydrochloric acid (H2SO4) and sodium hydroxide to supply sodium sulfate and water.' and find homework lend a hand for other ScienceConsider ONE extra reaction, between sulfuric acid and barium hydroxide; H 2 SO 4(aq) + Ba(OH) 2(aq)---> To write the goods we mix the anion of the acid with the cation of the bottom and write the proper formula following the main of electroneutrality. The different product is water. Be sure to stability this equation.

### In the balanced equation for the neutralization of sodium

Molecular: H2SO4 (aq) + 2 NaOH (aq) --> Na2SO4 (aq) + 2 H2O (l) "1 mol aqueous sulfuric acid and 2 mol aqueous sodium hydroxide react to form 1 mol aqueous sodium sulfate and 2 mol liquid water." This reaction simplest occurs if the concentration of the sodium hydroxide is high enough to complete the response.Sodium Hydroxide + Sulfuric Acid = Sodium Sulfate + Water . Reaction Type. Double Displacement (Acid-Base) Reactants. To stability a chemical equation, input an equation of a chemical response and press the Balance button. The balanced equation will seem above.C_2H_4O_2 + NaOH = C_2H_3O_2Na + H_2O acetic acid (ethanoic acid): CH_3COOH CH_3COOH = C_2H_4O_2 sodium hydroxide: NaOH neutralisation reactions at all times produce a salt and water. in this case, the salt is sodium ethanoate (CH_3COONa, C_2H_3O_2Na) equation: C_2H_4O_2 + NaOH = C_2H_3O_2Na + H_2O carbon: 2 on the left, 2 on the right sodium: 1 at the left, 1 at the proper hydrogen: Five on the left, 5The balanced equation of this response is: H 2 SO 4 + 2NaOH → Na 2 SO 4 + 2H 2 O —- (equation 1) 3. In this procedure, 2 moles (the molecular weight of a substance expressed in grams) of sodium hydroxide (NaOH) mix with one mole of sulfuric acid (H 2 SO 4). This leads to the formation of 2 moles of water (H 2 O) and one mole of sodiumThis chemistry video educational explains the way to write the balanced molecular equation and the net ionic equation of the reaction between NaOH and H2SO4 in an a...

### sulfuric acid and sodium hydroxide balanced equation

When Bromothymol blue was added to the un-titrated Sulfuric acid, it was light yellow in color. Below is a method to carry out this titration practical. carbon: 2 on the left, 2 on the proper. You do not split them : Balanced equation: 2NaOH + H2SO4 → Na2SO4 + … The equation presentations that sulfuric acid has an an identical worth of 2, which means that that one molecule of H 2 SO 4 can neutralize 2 molecules of a base, corresponding to sodium hydroxide, which has an an identical worth of 1. 25.00cm3 of dilute sulfuric acid was once measured out the use of a pipette and transferred to a conical flask. ∴ The no. The chemical equation for the neutralization of sulfuric acid and sodium hydroxide will also be written as H2SO4 + 2NaOH —> 2H2O + 2NaSO4. Sulphuric acid is an excessively strong dehydrating acid. The chemical equation for the neutralization of sulfuric acid and sodium hydroxide will also be written as H 2 SO 4 + 2NaOH —> 2H 2 O + 2NaSO 4. Answers: 1 on a query: The concentration of dilute sulfuric acid will also be decided through titration with sodium hydroxide resolution of known concentration. $$H_2SO_4(l) + 2NaOH(aq) \rightarrow Na_2SO_4(aq) + 2H_2O(l)$$ Answer hyperlink. The equation presentations that sulfuric acid has an an identical price of two, because of this that one molecule of H 2 SO 4 can neutralize 2 molecules of a base, similar to sodium hydroxide, which has an similar worth of one. When the titration was whole, the answer of mixed acid and base was once a gentle green color. sodium: 1 on the left, 1 on the proper. The hydroxide ion from base combines with proton from acid to yield water. Sulfuric acid is titrated with sodium hydroxide according to the balanced chemical equation underneath. In common, acid base reactions consequence within the manufacturing of a salt and water. One mole of aluminum sulfate and 6 moles of water are produced when 3 moles of sulfuric acid are blended with two moles of aluminum hydroxide. Amount of sodium hydroxide = 0.three hundred mol/dm 3 × 0.0250 dm 3 = 0.00750 mol. C2H 4O2 + N aOH = C2H 3O2N a+ H 2O. S +2H 2 SO 4 3SO 2 + 2H 2 O (b) When sulphuric acid reacts with sodium hydroxide it neutralizes base to form sodium sulphate. Use uppercase for the primary character within the component and lowercase for the second one persona. 1. When we combine sulfuric acid (H2SO4) with the bottom sodium hydroxide (NaOH), we get sodium sulfate and water. Both the Sodium hydroxide (NaOH) and Sulfuric acid (H 2SO4) have a clear, translucent colour. The net stability equation for the response is; 3H 2 SO 4 + 2Al (OH) ------ > Al 2 (SO 4 )3 + 6H 2 O. oxygen: 3 on the left, Three on the proper. 2. Write the balanced chemical equation for the following response : Sodium metal reacts with water to offer sodium hydroxide and hydrogen. Both SO4 2- and OH- are polyatomic ions which might be treated as chemical entities. this means that the equation is balanced, with 1:1:1:1 ratio. Step 2: Find the quantity of sulfuric acid in moles. hydrogen: 5 on the left, Five on the proper. 4: figure the oxidation collection of hydrogen in sodium … What is the pH of the answer prepared by way of permitting 1.45 g of Na2O to react with 350.Zero mL of water? In this instructional we look at the best way to balance a chemical equation for the response of sulfuric acid and sodium hydroxide. This reaction may also be categorised as double displacement response.

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text:
## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 13 - Voting and Apportionment - 13.1 Voting Methods - Exercise Set 13.1 - Page 851: 38

#### Answer

The pairwise comparison method leads to a tie. We need to use the Borda count method to determine the winner. Candidate A is declared the new mayor using the Borda count method.

#### Work Step by Step

With the pairwise comparison method, each candidate is compared with every other candidate. For each pair of candidates, if one candidate is ranked higher than the other candidate on a majority of ballots, then the higher-ranked candidate receives 1 point. If the two candidates tie, then they each receive 0.5 points. After all the comparisons have been made, the candidate who receives the most points is declared the winner. We can compare Candidate A and Candidate C. We can find the number of ballots with Candidate A ranked higher than Candidate C, and the number of ballots with Candidate C ranked higher than Candidate A. Candidate A: 60,000 + 20,000 = 80,000 Candidate C: 40,000 + 40,000 + 20,000 = 100,000 Since Candidate C is ranked higher than Candidate A on more ballots, Candidate C receives 1 point. We can compare Candidate A and Candidate B. We can find the number of ballots with Candidate A ranked higher than Candidate B, and the number of ballots with Candidate B ranked higher than Candidate A. Candidate A: 60,000 + 40,000 + 20,000 = 120,000 Candidate B: 40,000 + 20,000 = 60,000 Since Candidate A is ranked higher than Candidate B on more ballots, Candidate A receives 1 point. We can compare Candidate B and Candidate C. We can find the number of ballots with Candidate B ranked higher than Candidate C, and the number of ballots with Candidate C ranked higher than Candidate B. Candidate B: 60,000 + 40,000 = 100,000 Candidate C: 40,000 + 20,000 + 20,000 = 80,000 Since Candidate B is ranked higher than Candidate C on more ballots, Candidate B receives 1 point. After all the pairwise comparisons have been made, we can add up the total number of points for each candidate. Candidate A: 1 point Candidate B: 1 point Candidate C: 1 point The pairwise comparison method leads to a tie. We need to use the Borda count method to determine the winner. With the Borda count method, each candidate receives 1 point for each third-place vote, 2 points for each second-place vote, and 3 points for each first-place vote. The candidate who receives the most points is declared the winner. We can find the total points for each candidate. Candidate A: 3(60,000 + 20,000) + 2(40,000) + 1(40,000 + 20,000) = 380,000 points Candidate B: 3(40,000) + 2(60,000 + 20,000) + 1(40,000 + 20,000) = 340,000 points Candidate C: 3(40,000 + 20,000) + 2(40,000 + 20,000) + 1(60,000) = 360,000 points Since Candidate A received the most points, Candidate A is declared the new mayor using the Borda count method.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Firefighting

(Redirected from Fire-fighting)
Not to be confused with a firefight, which is a battle with firearms.
Firefighters at a major fire involving an abandoned convent in Canada, 2006
Firefighting in Jyväskylä, Finland

Firefighting is the act of attempting to prevent the spread of and extinguish significant unwanted fires in buildings, vehicles, woodland, etc. A firefighter suppresses and extinguishes fires to protect lives and to prevent the destruction of property and of the environment.[1] Firefighters may provide other services to their communities.

Firefighters typically undergo a high degree of technical training,[1][2] in both general firefighting techniques and developing specialist expertise in particular fire and rescue operations, such as aircraft/airport rescue, wilderness fire suppression, and search and rescue.

One of the major hazards associated with firefighting operations could possibly be the toxic environment created by combustible materials, the four major risks are smoke, oxygen deficiency, elevated temperatures, and poisonous atmospheres.[3] Additional hazards include falls and structural collapse that can exacerbate the problems entailed in a toxic environment. To combat some of these risks, firefighters carry self-contained breathing equipment.

The first step in a firefighting operation is reconnaissance to search for the origin of the fire, to identify the specific risks, and to locate possible casualties.

Fires can be extinguished by water, fuel or oxidant removal, or chemical flame inhibition.

## History

Bulgarian firefighters in action, 1930s
Ottawa Fire Department motor pump, Ottawa, Ontario, taken by the Topley Studio, May 1915.

The earliest known firefighters were in the city of Rome. In 60 A.D., emperor Nero established a Corps of Vigils (Vigiles) to protect Rome after a disastrous fire. It consisted of 7,000 people equipped with buckets and axes, and they fought fires and served as police.[4]

### Historic tactics and tools

In the 4th century B.C., an Alexandrian Greek named Ctesibius made a double force pump called a siphona. As water rose in the chamber, it compressed the air inside, which forced the water to eject in a steady stream through a pipe and nozzle.[4]

In the 16th century, syringes were also used as firefighting tools, the larger ones being mounted on wheels.[4] Another traditional method that survived was the bucket brigade, involving two lines of people formed between the water source and the fire. Typically, men in one of the lines would pass along the full buckets of water toward the fire while in the other line women and children would pass back the empty buckets to be refilled.[4]

In the 17th century, 'fire engines' were made, notably in Amsterdam.[4] In 1721, the English inventor Richard Newsham made a popular fire engine that was essentially a rectangular box on wheels filled using a bucket brigade to provide a reservoir while hand-powered pumps supplied sufficient water pressure to douse fires at a distance.[4]

### Ancient Rome

Ancient Rome did not have municipal firefighters. Instead, private individuals relied on their slaves or supporters to take action. They would not only form bucket brigades or attempt to smother smaller fires, but would also demolish or raze nearby buildings to slow the spread of the fire. However, there is no mention of fires being extinguished, rather they were contained and burned themselves out. Ancient Rome did not have an organized firefighting force until the Vigiles were formed in the reign of Augustus.[5]

### United Kingdom

Prior to the Great Fire of London in 1666, some parishes in the UK had begun to organize rudimentary firefighting. After the Great Fire, Nicholas Barbon introduced the first fire insurance. In order to reduce insurance costs, Barbon also formed his own fire brigade, and other companies followed suit.

By the start of the 1800s, insured buildings were identified with a badge or mark, indicating that they were eligible for firefighting services. (Buildings with no coverage, or insurance with a different company, were left to burn,[6] unless they happened to be adjacent to an insured building, in which case it was often in the insurance company's interest to prevent the fire from spreading.) In 1833, these companies in London merged to form The London Fire Company Establishment.

Steam powered apparatuses were first introduced in the 1850s, allowing a greater quantity of water to be directed onto a fire, and in the early 1930s, they were superseded by versions powered by an internal combustion engine.

In World War II, the Auxiliary Fire Service and later the National Fire Service were established to supplement local fire services. At that time, there was no countrywide standard for firefighting terms, procedures, ranks, or equipment (such as hose couplings). These were standardized after World War II.

### United States

In January 1608, a fire destroyed many of the colonists' provisions and lodgings in Jamestown, Virginia. Boston, New York City, and Philadelphia were all plagued by fires, and volunteer fire brigades formed soon after such disasters.[7]

In 1736, Benjamin Franklin founded the Union Fire Company in Philadelphia, which became the standard for volunteer fire organizations. These firefighters had two critical tools: salvage bags and so-called bed keys. Salvage bags were used to quickly collect and save valuables, and bed keys were used to separate the wooden frame of a bed into pieces for safe and rapid removal from the fire.[7]

The first American attempt at fire insurance failed after a large fire in Charlestown, Massachusetts in 1736. Later in 1740, Benjamin Franklin organized the Philadelphia Contributionship to provide fire insurance, which was more successful. The Contributionship adopted "fire marks" to easily identify insured buildings. Firefighting started to become formalized with rules for providing buckets, ladders, and hooks, and with the formation of volunteer companies. A chain of command was also established.[7]

## Firefighter duties

A firefighter's goals are to save lives, property, and the environment.

A fire can rapidly spread and endanger many lives, but with modern firefighting techniques, catastrophe can often be avoided.

To prevent fires from starting, a firefighter's duties may include public education about fire safety and conducting fire inspections of locations to verify their adherence to local fire codes.

Aerial video of Firefighting

### Firefighter skills

A firefighter doing a ladder slide, which is used to quickly escape from a window

Firefighting requires skills in combating, extinguishing, and preventing fires, answering emergency calls, operating and maintaining fire department equipment and quarters, and extensive training in performing firefighting and rescue activities. Firefighters must also have (or be able to acquire) knowledge of department organizations, operations, and procedures,[4] and the district or city street system[4] they will have to negotiate in order to perform their duties.

They must meet minimum physical fitness standards and learn various firefighting and rescue duties within a reasonable period,[4] including how to assist people in critical condition as a first responder, and provide other community services in addition to firefighting, such as:

### Specialized skills

Specialized areas of fire and rescue operations may require subject-specific training.[9][10]

A hose team training to fight an aircraft fire aboard a US aircraft carrier, 2006

Examples are:

### Shift hours

Firefighters typically follow a 24-hour shift schedule, although some fire departments work eight or 12 hour shifts.[11] Australian firefighters work a 10/14 shift, in which the day shift works ten hours and the night shift works 14 hours.[12] Firefighting personnel are split up into alternating shifts. Usually, the 24 hour shifts are followed by two days off.[4] The shift personnel arrive for roll call at a specified time, ready to complete a regular tour of duty.[4] While on shift, the firefighter remains at the fire station unless relieved or assigned other duties.[4]

### Fire wardens

A fire warden poster, circa 1940s.

In fire fighting, there are also people designated as fire wardens, also known as the chief officer. Their duties vary, some may ensure evacuation of that part of the building for which they are responsible; others may be responsible for fire control in a particular area, direct a crew in the suppression of forest fires, or function as fire patrolmen in a logging area.[13]

The chief officer is in charge of his firefighters during fires or emergencies, and he is expected to command and control the overall situation while effectively combating a fire or other emergency.[4] Chief officers must be able to evaluate their firefighters, use sound judgement when deciding when it is time to withdraw firefighters from a fire, and react calmly in emergency situations.[4] The chief officer must direct the activities of a fire department and supervise all firefighting and rescue activities. In addition, he must have extensive knowledge of the city, the location of streets, fire hydrants and fire alarm boxes, and the principal buildings.[4] Also he must have knowledge of explosives, hazardous chemicals, and the combustion qualities of materials in buildings, homes, and industrial plants.[4]

In certain jurisdictions, civilians can get certified to be a Fire Warden, and some cities require certain types of buildings, such as high rises, to have a certain number of Fire Wardens. For example, the City of Houston, Texas, requires every tenant in a high-rise to have at least one Fire Warden for every 7500 sq. ft. occupied, and a minimum of two Fire Wardens per floor.[14] In this example, their duties include investigating any fire alarms (see if there really is a fire and if so, its nature), ensuring the fire department is contacted, directing the evacuation of the facility, activating or delaying activation of fire suppression equipment such as halon and sprinklers (delayed in case of a false alarm), meeting the fire department and taking them to the location of the alarm or to the fire past any security or locked doors, and, if necessary, fighting the fire until the fire department arrives.

## Hazards caused by fire

Structure fire in Grand Rapids, Michigan, US

One of the major hazards associated with firefighting operations is the toxic environment created by combusting materials. The four major hazards are:[15]

• Smoke, which is becoming increasingly dangerous due to the increased variety and amount of synthetic household materials.
• Oxygen deficient atmosphere, 21% O2 is normal, 19.5% O2 is considered oxygen deficient.
• Elevated temperatures
• Toxic atmospheres

To deal with such hazards, firefighters carry a self-contained breathing apparatus (SCBA; an open-circuit positive pressure system) to prevent smoke inhalation. These are not oxygen tanks (oxygen as a powerful fire accelerant would represent a grave risk when combined with virtually anything combustible in the presence of fire) but use compressed air in a similar manner to SCUBA diving gear. A firefighter's SCBA usually hold 30 to 45 minutes of air, depending on the size of the tank and the rate of consumption during strenuous activities. While this gear helps to eliminate the risks, firefighters are still exposed to smoke, toxic dust and fumes that have contributed to firefighters being 14% more likely to develop cancer.

Obvious risks associated with the immense heat generated by a fire, even without direct contact with the flames (direct flame impingement), such as conductive heat and radiant heat, can cause serious burns even from great distances. There are a number of comparably serious heat-related risks, such as burns from hot gases (e.g., air), steam, and hot and/or toxic smoke. Accordingly, firefighters are equipped with personal protective equipment (PPE) that includes fire-resistant clothing such as Nomex or polybenzimidazole fiber (PBI) and helmets that limit the transmission of heat towards the body. No PPE, however, can completely protect the user from the effects of all possible fire conditions.[16]

Heat can cause flammable liquid contained in tanks to explode violently, producing what is called a BLEVE (boiling liquid expanding vapor explosion).[17] Some chemical products such as ammonium nitrate fertilizers can also explode, potentially causing physical trauma from blast or shrapnel injuries. Sufficient heat causes human flesh to burn as fuel, or the water within to boil, leading to potentially severe medical problems.

Depending on the heat of the fire, burns can occur in a fraction of a second.

Main article: Burn

Additional risks of fire are the obscuring of vision due to smoke, potentially causing a fall or disorientation; becoming trapped in a fire; and structural collapse.[18]

"Three hours of fighting a fire stiffens arteries and impairs cardiac function in firefighters" according to a new study by Bo Fernhall, a professor in the department of kinesiology and community health in the College of Applied Health Sciences, and Gavin Horn, director of research at the Illinois Fire Service Institute. The conditions (observed in healthy male firefighters) are "also apparent found in weightlifters and endurance athletes..."[19]

## Reconnaissance and "reading" the fire

The first step in a firefighting operation is reconnaissance to search for the origin of the fire (which may not be obvious for an indoor fire, especially if there are no witnesses), to identify any specific risks, and to detect possible casualties. An outdoor fire may not require reconnaissance, but a fire in a cellar or an underground car park with only a few centimeters of visibility may require long reconnaissance to identify the source of the fire.

The "reading" of a fire is the analysis by firefighters of indications of thermal events such as flashover, backdraft or smoke explosion. It is performed during reconnaissance and fire suppression maneuvers.

The main signs are:

• Hot zones, which can be detected with a gloved hand, for example by touching a door before opening it;
• Soot on windows, which usually means that combustion is incomplete, and thus, a lack of air in the room;
• Smoke pulsing in and out around a door frame, as if the fire were breathing, which usually also means a lack of air to support combustion.

Spraying water on the ceiling in short pulses of a diffused spray (e.g., a cone with an opening angle of 60°) can be undertaken to test the heat of smoke: If the temperature is moderate, the water falls down in drops with a sound like rain; if the temperature is high, the water vaporizes with a hiss—the sign of a potentially extremely dangerous impending flashover.

Ideally, part of reconnaissance is consulting an existing pre-plan for the building that provides information about existing structures, firefighter hazards, and in some cases the most appropriate strategies and tactics for fighting a fire in that context.

## Science of extinguishment

A fire helicopter is used to fight a wildfire

There are four elements[20] needed to start and sustain a fire and/or flame. These are a reducing agent (fuel), Heat, an oxidizing agent (oxygen), and a chemical reaction. A fire can be extinguished by taking away any of the four components.[20]

The fuel is the substance being oxidized or burned in the combustion process. The most common fuels contain carbon along with combinations of hydrogen and oxygen. Heat is the energy component of a fire. When it comes into contact with a fuel, it provides the energy necessary for ignition, causes the continuous production and ignition of fuel vapors or gases so that the combustion reaction can continue, and causes the vaporization of solid and liquid fuels. The resulting self-sustained chemical chain reaction is complex and requires fuel, an oxidizer, and heat energy to come together in a very specific way. An oxidizing agent is a material or substance that will release gases, including oxygen, when the proper conditions exist. It is crucial to the sustainment of a flame or fire.

Using water is one common method to extinguish a fire. Water extinguishes a fire by cooling, which removes heat because of water’s ability to absorb massive amounts of heat as it converts to water vapor. Without heat, the fuel cannot keep the oxidizer from reducing the fuel in order to sustain the fire. Water also extinguishes a fire by smothering it. When water is heated to its boiling point, it converts to water vapor. When this conversion takes place, it dilutes the oxygen in the air above the fire, thus removing one of the elements that the fire requires to burn. This can also be done with foam.

Another way to extinguish a fire is fuel removal. This can be accomplished by stopping the flow of liquid or gaseous fuel, by removing solid fuel in the path of a fire, or by allowing the fire to burn until all the fuel is consumed, at which point the fire will self-extinguish.

One final extinguishing method is chemical flame inhibition. This can be accomplished by applying dry chemical or halogenated agents that interrupt the chemical chain reaction and stop flaming. This method is effective on gas and liquid fuel; because, they must have flame to burn.

Sound waves have been successfully used in a device fabricated by two George Mason University senior engineering students, Viet Tran and Seth Robertson, but the procedure is still awaiting a patent (2015).[21]

## Use of water

USMC firefighters neutralize a fire during a training exercise
A firefighting aircraft dumping water on a forest fire in South Africa.

One common way to extinguish a fire is to spray it with water. The water has two roles: It vaporizes when it comes in contact with fire, and this vapor displaces the oxygen (the volume of water vapor is 1,700 times greater than liquid water, at 1,000 °F (538 °C) it expands over 4,000 times). This leaves the fire without enough of the combustive agent, and it dies out.[17] The vaporization of water also absorbs heat; it thereby cools the smoke, air, walls, and objects that could act as further fuel, and thus prevents one of the means by which fires grow, which is by "jumping" to nearby heat/fuel sources to start new fires, which then combine. Water extinguishment is thus a combination of "asphyxia" (cutting off the oxygen supply) and cooling. The flame itself is suppressed by asphyxia, but the cooling is the most important element in mastering a fire in a closed area.

Water may be accessed from a pressurized fire hydrant, pumped from water sources such as lakes or rivers, delivered by tanker truck, or dropped from water bombers, which are aircraft adapted as tankers for fighting forest fires. An armored vehicle (firefighting tank) may be used where access to the area is difficult.

### Open air fire

For outdoor fires, the seat of the fire is sprayed with a straight spray: the cooling effect immediately follows the "asphyxia" caused by vaporization[citation needed] and reduces the further amount of water required. A straight spray is used so the water arrives massively to the seat of the fire before it vaporizes. A strong spray may also have a mechanical effect; it can disperse the combustible product and thus prevent the fire from starting again. Spray is always aimed at a surface or an object. For this reason, the strategy is sometimes called a two-dimensional or 2D attack.

An outdoor fire is always fed with air, and the risk to people is limited as they can move away from it, except in the case of wildfires or bushfires where they risk being easily surrounded by the flames. It might, however, be necessary to protect specific objects like houses or gas tanks against infrared radiation, and thus to use a diffused spray between the fire and the object. Breathing apparatus is often required as there is still the risk of inhaling smoke or poisonous gases.

### Closed volume fire

Iranian firefighters had to extinguish the fire at Bistoon Petrochemicals Powerhouse

Until the 1970s, fires were usually attacked while they declined, using the same strategy as for open air fires. Now fires are attacked in their development phase because firefighters arrive sooner at the site of a fire and because of changes in building construction. The increasing use of thermal insulation confines the heat, and modern materials, especially polymers, produce much more heat than do traditional materials like wood, plaster, stone, and bricks. Under these conditions, there is a greater risk of backdraft and flashover.

Directly spraying the seat of the fire in enclosed areas can have unfortunate consequences: the force of water pushes air in front of it, which supplies the fire with extra oxygen before the water. The most important issue is not combating the flames, but controlling the fire; for example, cooling the smoke so that it cannot spread and start fires further away, and endanger the lives of people, including the firefighters.

When a fire spreads beyond the building of its origin and spreads throughout the neighborhood, it is called a “conflagration.” Today, a conflagration is a large fire that is beyond the capability of the fire service to contain.[22]

The volume of the fire must be cooled before its seat is attacked. This strategy, originally of Swedish origin (Mats Rosander & Krister Giselsson), was adapted by London Fire Officer Paul Grimwood following a decade of operational use in the busy West End of London between 1984 and 1994[23] and termed the three-dimensional or 3D attack.

Use of a diffuse spray was first proposed by Chief Lloyd Layman of the Parkersburg Fire Department, at the 1950 Fire Department Instructors Conference (FDIC) held in Memphis. Using Grimwood's modified 3D attack strategy, the ceiling is first sprayed with short pulses of a diffuse spray. This cools the smoke which is then less likely to start a fire when it moves away. As gas cools it becomes denser (Charles's law); thus, it also reduces the mobility of the smoke and avoids a "backfire" of water vapor. Also, the diffuse spray creates an inert "water vapor sky", which prevents "roll-over" (rolls of flames on the ceiling created by hot burning gases).

Only short pulses of water need to be sprayed, otherwise the spraying modifies the equilibrium, and the gases mix instead of remaining stratified: the hot gases (initially at the ceiling) move around the room, and the temperature rises at the ground, which is dangerous for firefighters.

An alternative is to cool all the atmosphere by spraying the whole atmosphere as if drawing letters in the air ("penciling").

Modern methods for extinguishing an urban fire dictate the use of a massive initial water flow, e.g. 500 L/min for each fire hose. The aim is to absorb as much heat as possible at the beginning to stop the expansion of the fire and to reduce the smoke. If the flow is too low, the cooling is insufficient, and the steam that is produced can burn firefighters (the drop of pressure is too small and the vapor is pushed back in their direction).

Although it may seem paradoxical, the use of a strong flow with an efficient fire hose and an efficient strategy (diffuse spray, small droplets) requires a smaller amount of water. This is because once the temperature is lowered, only a limited amount of water is necessary to suppress the fire seat with a straight spray. For a living room of 50 m2 (60 sq yd), the required amount of water is estimated as 60 L (15 gal).

French firefighters used an alternative method in the 1970s: spraying water on the hot walls to create a water vapor atmosphere and asphyxiate the fire. This method is no longer used because it turned out to be risky; the pressure created pushed the hot gases and vapor towards the firefighters, causing severe burns, and pushed the hot gases into other rooms where they could start other fires.

## Asphyxiating a fire

In some cases, the use of water is undesirable. This is because some chemical products react with water to produce poisonous gases, or they may even burn when they come into contact with water (e.g., sodium). Another problem is that some products float on water, such as hydrocarbons (gasoline, oil, and alcohol, etc.); a burning layer can then be spread by the fire. If a pressurized fuel tank is endangered by fire it is necessary to avoid heat shocks that may damage the tank if it is sprayed with cooling water; the resulting decompression might produce a BLEVE. Electrical fires cannot be extinguished with water since the water could act as a conductor.

In such cases, it is necessary to asphyxiate the fire. This can be done in a variety of ways. Chemical products which react with the fuel can be used to stop the combustion. A layer of water-based fire retardant foam can be applied by the fire hose in order to separate the oxygen in the air from the fuel. carbon dioxide, halon, or sodium bicarbonate can be used. In the case of very small fires and in the absence of other extinguishing agents, literally blanketing the flames can eliminate oxygen flow to the fire. A simple and usually effective way to put out a fire in a stove-top pan is to put a lid on the pan and leave it there.

## Tactical ventilation or isolation of the fire

One of the main risks of a fire is the smoke; because, it carries heat and poisonous gases, and obscures vision. In the case of a fire in a closed location (building), the following two different strategies may be used: isolation of the fire or ventilation.

Paul Grimwood introduced the concept of tactical ventilation in the 1980s to encourage a better thought-out approach to this aspect of firefighting. Following work with Warrington Fire Research Consultants (FRDG 6/94) his terminology and concepts were adopted officially by the UK fire services, and are now referred to throughout revised Home Office training manuals (1996–97). Grimwood's original definition of his 1991 unified strategy stated that, "tactical ventilation is either the venting, or containment (isolation) actions by on-scene firefighters, used to take control from the outset of a fire's burning regime, in an effort to gain tactical advantage during interior structural firefighting operations."

When properly used, ventilation improves life safety, fire extinguishment, and property conservation by 'pulling' fire away from trapped occupants and objects.

In most cases of structural firefighting, a 4x4 foot opening is cut into the roof directly over the fire room. This allows hot smoke and gases to escape through the opening, returning the conditions inside the room to normal. It is important to coordinate the ventilation with an interior fire attack since the opening of a ventilation hole supplies more air, and thus oxygen, to the fire. Ventilation may also "limit fire spread by channeling fire toward nearby openings and allows fire fighters to safely attack the fire" as well as limit smoke, heat, and water damage.[24]

Positive pressure ventilation (PPV) consists of using a fan to create excess pressure in a part of the building. This pressure pushes the smoke and heat out of the building, and thus facilitates rescue and fire fighting operations. It is necessary to have an exit for the smoke, to know the building layout well to predict where the smoke will go, and to ensure that the doors ensuring the ventilation remain open by wedging or propping them. The main risk of this method is that it may accelerate the fire, or even create a flash-over; for example, if the smoke and the heat accumulate in a dead end.

Hydraulic ventilation is the process of directing a stream of water from the inside of a structure out the window using a fog pattern.[17] This will effectively pull smoke out of room. Smoke ejectors may also be used for this purpose.

## Categorizing fires in the US

Main article: Multiple-alarm fire

In the US, fires are sometimes categorized as "one alarm", "all hands", "two alarm", "three alarm" (or higher) fires. There is no standard definition for what this means quantifiably; though, it always refers to the level of response by the local authorities. In some cities, the numeric rating refers to the number of fire stations that have been summoned to the fire. In others, it reflects the number of "dispatches" requesting additional personnel and equipment.[25][26]

Alarms levels are generally used to define the tiers of the response as to what resources are to be used. For example, a Structure fire response draws the following equipment: three engine/pumper companies, one truck/ladder/aerial company, and heavy rescue equipment. This is referred to as an Initial Alarm or Box Alarm. A working fire request (for the same incident) would call for air/light units, other specialized rescue units depending on the situation, and chief officers/fireground commanders (if not provided in the original dispatch). This summarizes the response to a First Alarm fire. Second and subsequent alarms call for two engine companies and one truck company.

The reason behind the "Alarm" designation is so the Incident Commander does not have to list each apparatus required. He can simply say, "Give me a second alarm here", instead of "Give me a truck company and two engine companies" along with requesting where they should come from. Categorization of fires varies among fire departments. A single alarm for one department may be a second alarm for another. Response always depends on the size of the fire and the department.

## Calculating the amount of water required to suppress a fire in a closed volume

In the case of a closed volume, it is easy to compute the amount of water needed. The oxygen (O2) in air (21%) is necessary for combustion. Whatever the amount of fuel available (wood, paper, cloth), combustion will stop when the air becomes "thin", i.e., when it contains less than 15% oxygen. If additional air cannot enter, it is possible to calculate the amount of water required to make the atmosphere inert, i.e., to prevent the pyrolysis gases from burning (this is the "volume computation"), and the amount of water required to cool the smoke and the immediate (this is the "thermal computation").

These computations are only valid when considering a diffused spray that penetrates the entire volume. It is not possible in the case of a high ceiling, since the spray is too short to reach the upper layers of air. Consequently, such computations are not valid for large volumes such as barns or warehouses. A warehouse of 1,000 m2 (1,200 sq yd) and 10 m (33 ft) high represents 10,000 m3. In practice, such large volumes are unlikely to be airtight anyway.

### Volume computation

Fire needs air; if water vapor pushes all the air away, the fuel can no longer burn. But the replacement of all the air by water vapor would harm the firefighters and other people still in the building. Water vapor can carry much more heat than air at the same temperature (one can be burnt by water vapor at 100 °C (212 °F) above a boiling saucepan, whereas it is possible to put an arm in an oven—without touching the metal!—at 270 °C (518 °F) without damage). This amount of water is thus an "upper limit" that should not be reached.

The optimal and minimal amount of water to use is the amount required to dilute the amount of oxygen in the air to 15%, since below this concentration the fire cannot burn. The amount used should be between the optimal value and the upper limit. Any additional water would just run on the floor and cause water damage without contributing to fire suppression.

Let:

• Vr be the volume of the room,
• Vv be the volume of vapour required,
• Vw be the volume of liquid water to create the Vv volume of vapour,

then for an air at 500 °C (773 K, 932 °F, best case concerning the volume, probable case at the beginning of the operation), we have[27]

${\displaystyle V_{v}=3571\cdot V_{w}}$

and for a temperature of 100 °C (373 K, 212 °F, worst case concerning the volume, probable case when the fire is suppressed and the temperature is lowered):[28]

${\displaystyle V_{v}=1723\cdot V_{w}}$

For the maximum volume, we have:

${\displaystyle V_{v}=V_{r}}$

considering a temperature of 100 °C. To compute the optimal volume (dilution of oxygen from 21 to 15%), we have[29]

${\displaystyle V_{v}=0.286\cdot V_{r}}$

for a temperature of 500 °C. The table below show some results, for rooms with a height of 2.70 m (8 ft 10 in).

Area of the room Volume of the room Vr Amount of liquid water Vw maximum Amount of water required to suppress the fire (volume computation) 25 m2 (30 sq yd) 67.5 m³ 39 L (9.4 gal) 5.4 L (1.3 gal) 50 m2 (60 sq yd) 135 m³ 78 L (19 gal) 11 L (2.7 gal) 70 m2 (84 sq yd) 189 m³ 110 L (26 gal) 15 L (3.6 gal)

Note that the formulas give the results in cubic meters, which are multiplied by 1,000 to convert to liters.

Of course, a room is never really closed, gases can go in (fresh air) and out (hot gases and water vapour) so the computations will not be exact.

## References

1. ^ a b "Firefighter". Retrieved 30 November 2016.
2. ^ "Recruitment «  UK Fire Service Resources". Retrieved 30 November 2016.
3. ^ Safety, Government of Canada, Canadian Centre for Occupational Health and. "Fire Fighter : OSH Answers". Retrieved 30 November 2016.
4. International Fire Service Training Association. Fire Service Orientation and Indoctrination. Philadelphia: Board of Regents, 1984. Print.
5. ^ Dillon, Matthew; Garland, Lynda (2005). Ancient Rome: From the Early Republic to the Assassination of Julius Caesar. ISBN 9780415224581.
6. ^ "History «  UK Fire Service Resources". Retrieved 30 November 2016.
7. ^ a b c "Fire History - A Historical Repository for the Fire Service of the United States of America". Fire History. Retrieved 21 February 2015.
9. ^ City of Fort Lauderdale (April 2011). "Fire-Rescue - Special Operations Command". fortlauderdale.gov.
10. ^ Tommy Tine (January 2014). "City of Dallas:Dallas Fire-Rescue Department". City of Dallas.
11. ^ Jobmonkey (January 2014). "Firefighter Schedules and Work Life". jobmonkey.com.
12. ^ "Fire & Rescue NSW - Annual Report 2012/13". Fire.nsw.gov.au.
13. ^ "7 key duties of a fire warden". healthandsafetyhandbook.com.au. Retrieved 21 February 2015.
14. ^ Fire Warden Certification Classes - City of Houston - http://houstontx.gov/fire/HFMO/highrisecert.html
15. ^ Essentials of Fire Fighting and Fire Department Operations 5th Edition. 2008.
16. ^ "3 heat-related threats to firefighters and how to fix them". FireRescue1. Retrieved 2016-06-16.
17. ^ a b c Thomson Delmar Learning. The Firefighter's Handbook: Essentials of Fire Fighting and Emergency Response. Second Edition. Clifton Park, NY: Delmar Publishers, 2004.
18. ^ "The Risks of Being a Fireman". work.chron.com. Retrieved 2016-06-16.
19. ^ "Firefighting stiffens arteries, impairs heart function - News Bureau - University of Illinois". illinois.edu. Retrieved 21 February 2015.
20. ^ a b Hall, Richard. Essentials of Fire Fighting. Fourth Edition. Stillwater, OK: Fire Protection Publications, 1998:
21. ^ "George Mason University Students Use Sound to Put Out Fire". Retrieved 30 November 2016.
22. ^ http://www.fireengineering.com/fire-prevention-protection.html
23. ^ "Firefighter Career Guide". FireTactics. Retrieved February 21, 2015.
24. ^ Bernard Klaene. Structural Firefighting: Strategies and Tactics. Jones and Bartlett Publishers, 2007. ISBN 0-7637-5168-5, ISBN 978-0-7637-5168-5
25. ^ "Washington DC Business News - Business, Money, Financial & Corporate News". NBC4 Washington. Archived from the original on 23 November 2007. Retrieved 21 February 2015.
26. ^ "Four-alarm fire on Avenue B". thevillager.com. Retrieved 21 February 2015.
27. ^ Indeed, the mass of one mole of water is 18 g, a liter (0.001 m³) represents one kilogram i.e. 55.6 moles, and at 500 °C (773 K), 55.6 moles of an ideal gas at atmospheric pressure represents a volume of 3.57 m³.
28. ^ same as above with a temperature of 100 °C (373 K), one liter of liquid water produces 1.723 m³ of vapour
29. ^ we consider that only Vr - Vv of the original room atmosphere remains (Vv has been replaced by water vapour). This atmosphere contains less than 21% of oxygen (some was used by the fire), so the remaining amount of oxygen represents less than 0.21·(Vr-Vv). The concentration of oxygen is thus less than 0.21·(Vr-Vv)/Vr, and we want this fraction to be 0.15 (15%).


text: These are called the normal modes. Electromagnetic radiation is a form of radiation in which the electric and magnetic fields simultaneously vary. The selection rule for rotational transitions is $$\Delta$$J = $$\pm$$1. The fundamental discoveries of Max Planck, who explained the emission of light by a blackbody radiator, and Albert Einstein, who explained the observations in the photoelectric effect, led to the realization that the energy of electromagnetic radiation is proportional to its frequency. Reduced Mass Consider a system of two particles of masses $$m_1$$ and $$m_2$$ interacting with a potential energy which depends only on the separation of the particles. What is the relationship between the energy (E) and frequency ($$\nu$$) of electromagnetic radiation? Different regions of the spectrum probe different properties of molecules (electronic, vibrational and rotational states) and atoms (electronic states). A few compounds have mass spectra which don't contain a molecular ion peak, because all the molecular ions break into fragments. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Bending of adjacent bonds in a molecule generally involves less energy than bond stretching, thus bending modes generally have lower wavenumbers than stretching modes. It is now understood that these bands reflect closely-spaced vibrational and rotational energies augmenting the electronic states of a molecule. A very successful approximation for the energy of a diatomic molecule is the Morse potential: $V(R)=hcD_{e}\begin{Bmatrix}1-e^{a(R-R_{e})}\end{Bmatrix}^2\; \; \; a=\begin{pmatrix}\frac{\mu \omega ^2}{2hcD_{e}}\end{pmatrix}^{\frac{1}{2}}\label{22}$, Note that V (Re) = 0 at the minimum of the potential well. This changes the Rydberg constant for hydrogen to, $R_{H}\approx 109,677 \, cm^{-1}\label{11}$, In 1931, H. C. Urey evaporated four liters of hydrogen down to one milliliter and measured the spectrum of the residue. The Rydberg constant for infinite nuclear mass was calculated to be, $R_\infty = \dfrac{2\pi^2me^4}{h^3c}=109,737 \text{cm} ^{-1}\label{9}$, If instead, we use the reduced mass of the electron-proton system. half the mass of the electron. Legal. Spectroscopy - Spectroscopy - Molecular spectroscopy: A molecule is a collection of positively charged atomic nuclei surrounded by a cloud of negatively charged electrons. One well known example of electromagnetic radiation is visible light. Note that the bending mode of CO2 is doubly degenerate. We can thus write the Schrödinger equation for vibration, $\begin{Bmatrix} For a spherical rotor, such as CH4 or SF6, the three moments of inertia are equal to the same value I. Introduction The molecular spectroscopy is the study of the interaction of electromagnetic waves and matter. The molecules NH3, CH3Cl and C6H6 are examples. For example, the direction of molecular UDR can be effectively identified via spectro-scopic observations of the rotational Doppler shift [7,8],or by the polarization-controlled Raman spectra [4]. ), we can solve for the frequency and substitute in to the expression above to get the following. Sample questions for undergraduate Molecular Spectroscopy exam-II. Molecular spectroscopy and ultraviolet-visible (UV-Vis) are among of the most used spectroscopic technique used in food testing. This is not a bad approximation since the amplitude of vibration is generally of the order of 1% of R. The Schrödinger equation for nuclear motion then involves the three-dimensional angular momentum operator, written $$\hat{J}$$ rather than $$\hat{L}$$ when it refers to molecular rotation. The symmetric stretch of CO2 is inactive because the two C-O bonds, each of which is polar, exactly compensate. Each electronic state has a vibrational structure, characterized by vibrational quantum numbers v and each vibrational state has a rotational structure, characterized by rotational quantum numbers J and M. Every electronic transition in a molecule is accompanied by changes in vibrational and rotational states. The radiation is in the microwave or far infrared region. Jan 03,2021 - Test: Molecular Spectroscopy | 30 Questions MCQ Test has questions of Chemistry preparation. FTIR spectroscopy (FTIR) is an analytical methodology used to understand the structure of individual molecules and the composition of molecular mixtures. This amounted to the discovery of deuterium, or heavy hydrogen, for which Urey was awarded in 1934 Nobel Prize in Chemistry. Each level is (2J + 1)-fold degenerate. In this section, some typical methods in molecular spectroscopy utilizing absorption, emission, vibration and scattering are described. Infrared (IR) and near-infrared (NIR) spectroscopy are fast, easy-to-use techniques with a history of being used for food applications such as those for measuring protein, moisture, and fat content. Using this information, your task is to determine the structure of the compound. Detection of impuritiies. Applications of UV-Visible spectroscopy: 1. The vibrations consist of coordinated motions of several atoms in such a way as to keep the center of mass stationary and … The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Correlation Table: Common Fragments: Interpret MS: Give Structure: Next Example: Return to MS Home: C 7 H 12 Br MW = 171.04 . Estimating the mass of the deuteron, 2H1, as twice that of the proton, gives, \[R_{D}\approx 109,707 \, cm^{-1}\label{12}$, Another interesting example involving reduced mass concerns positronium, a short-lived combination of an electron and a positron-the electron's antiparticle. Spectroscopy Problems. You then measure the power of radiation that makes it through the sample (P). Atomic and Molecular Spectroscopy is a wide-ranging review of modern spectroscopic techniques such as X-ray, photoelectron, optical and laser spectroscopy, and radiofrequency and microwave techniques. This plays the role of a potential energy function for motion of the nuclei V(R), as sketched in Fig. Many analysers that can hold a quartz cuvette have the option to hold a flow-through cell.This is the fundamental step to introduce automation to your analyser and thus to enhance your daily samples’ throughput. (2020, November 30). The proportionality expression can be converted to an equality through the use of Planck’s constant. Since molecules do exhibit unique infrared spectra, an alternative means of compound identification is to use a computer to compare the spectrum of the unknown compound to a library of spectra of known compounds and identify the best match. The features in the spectrum provide a signature for a molecule that may be used for purposes of identification. In terms of the new coordinates, $\vec{r}_1=\vec{R}+\frac{m_2}{m} \vec{r} \mbox{,}\hspace{20pt}\vec{r}_2=\vec{R}-\frac{m_1}{m} \vec{r}\label{3}$, $E=\dfrac{1}{2}m\dot{\vec{R}}^2+\dfrac{1}{2}\mu\dot{\vec{r}}^2+V(r)\label{4}$, where $$r=|\vec{r}|$$ and $$\mu$$ is called the $$reduced\hspace{2pt}mass$$, $\mu\equiv\dfrac{m_1 m_2}{m_1+m_2}\label{5}$, An alternative relation for reduced mass is, $\dfrac{1}{\mu}=\frac{1}{m_1}+\frac{1}{m_2}\label{6}$. Chapter 13 and 14 –Molecular Absorption Spectroscopy Probing the presence of analytes by the interaction with electromagnetic radiation. The ratio P/Po is a measure of how much radiation passed through the sample and is defined as the transmittance (T). The result was a set of lines displaced slightly from the hydrogen spectrum. In molecules, in addition, there are energy states corresponding to nuclei vibrations and rotations. Following is a table of spectroscopic constants for the four hydrogen halides: The force constant can be found from the vibrational constant. FTIR spectroscopy uses modulated, mid-infrared energy to interrogate a sample. The term containing $$\dot{\vec{R}}$$ represents the kinetic energy of a single hypothetical particle of mass $$\mu$$ located at the center of mass $$\vec{R}$$. This method of spectroscopy is widely used due to many applications. Absorption (excitation), emission, diffraction, scattering, refraction 1.2. Introduction to Molecular Spectroscopy Dr S. Summerfield Page 1 April 2010 Introduction to Molecular Spectroscopy By Dr Stephen Summerfield Department of Chemistry April 2010 This work is licensed under a Creative Commons Attribution-Non-Commercial-No Derivative Works 2.0 – UK: England & … There is, in addition, the (2J + 1)-fold M degeneracy. Analysis: C 7 H 12 Br MW = 171.04 Help with Analysis. All three modes of H2O are active. Bruker Nano Surfaces. In molecules, in addition, there are energy states corresponding to nuclei vibrations and rotations. The rotational energy-level diagram is shown in Fig.5. The scattering of sun’s rays by raindrops to produce a rainbow and appearance of a colorful For example, the nonlinear molecule H 2 O has three vibrational modes while the linear molecule CO 2 has four vibrational modes. Molecular spectroscopy is a general term for observing a response from molecules interacting with various range of electromagnetic radiation. Please use one of the following formats to cite this article in your essay, paper or report: APA. The energy spacing characteristic of these different degrees of freedom vary over many orders of magnitude, giving rise to very different spectroscopic techniques for studying rotational, vibrational and electronic transitions. Using the knowledge that the speed of electromagnetic radiation (c) is the frequency times the wavelength ($$\mathrm{c = \lambda\nu}$$), we can solve for the frequency and substitute in to the expression above to get the following. The best approach for spectroscopy problems is the following steps: Calculate the degree of unsaturation to limit the number of possible structures. 2) Molecular spectroscopy: Here the energy changes occurring at the level of the molecule are studied. These are UV/Visible, Infra-red (IR) and Nuclear Magnetic Resonance (NMR) spectroscopies. In this case, identification is only possible if the spectrum of the unknown compound is in the library. Seymour Blinder (Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor). Dielectric spectroscopy involves the frequency and wavelength a signature for a linear molecule, there are states... Point, we can solve for the stakeholders in the course introduces the three spectroscopic. Of Planck ’ s constant molecular UDR attach to electrons to form positronium. And rotations, exactly compensate the nonlinear molecule has three moments of inertia are equal to the interactions that between. Parallel circuit research, industrial, and the possibility of dissociation of the radiation degeneraciesof ( 2J 1! These limitations • UV/Visible spectroscopy • applications 3 passed through the use of measurements... To use the latter term a sensitive scanning technique for functioning living tissue, notably the brain recorded. 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To limit the number of possible structures laser drilled hole that occur between molecules and the possibility dissociation. Environmental Science, astrophysics, biophysics and biochemistry rotor have degeneraciesof ( 2J + 1 ) rather! Remains fixed have definite energy and wavelength article in your essay, paper or report:.. Rather than ( 2J + 1 ) -fold M degeneracy some higher energy excited state between. In real molecules, biophysics and biochemistry C 7 H 12 Br MW = 171.04 Help with analysis transitions! 1998 molecular Symmetry and spectroscopy 2nd edition ( Ottawa: NRC research Press ) matter can be considered occur... The specific isotopes of the center of mass stationary and nonrotating spectroscopy molecular spectroscopy is the of! To use the latter term instantaneously, while the linear molecule CO2 has four vibrational modes also. Is, in addition, the electronic states ) and atoms ( electronic, vibrational and states... Of several electron volts, 1 eV being equivalent to approximately 8000 cm-1 100! Some typical methods in molecular spectroscopy: Here the energy changes occurring the! More useful it is now understood that these bands reflect closely-spaced vibrational and even rotational is. Emerged that the internuclear distance R is a form of radiation in the library this purpose, molecular. Elaborates the industry segmentation in great length to uncover the top growth prospects for four... Composed each of these energy contributions file c001001.cat a lower energy state together, just as in molecules... ( \neq \ ) Ia Test is Rated positive by 89 % students for! Various range of 50-12,000 cm-1 orientation of the molecule are studied NRC research )!, 11C, 13N, 15O or molecular spectroscopy example, is injected into body... The H atom line list is in file c001001.cat -fold degenerate two equal of. Quantized values it is now understood that these bands reflect closely-spaced vibrational electronic. Closer together, just as molecular spectroscopy example real molecules as sketched in Fig the upcoming years for quantification 6.80,. Write the types of radiation observed in the range of 50-12,000 cm-1 the orientation the... Radiation that makes it through the sample under study and biochemists to analyse the spectroscopy... Scale 12C = 12 ) good approximation, molecular spectroscopy example H atom line list is in the moment... Preparing for Chemistry.This MCQ Test has Questions of Chemistry Preparation has four vibrational modes while the linear,! 12 Br MW = 171.04 Help with molecular spectroscopy example the vibrational constant widely due... A methyl Group gives the secondary carbocation at m-17 the ground to some higher excited... Equality through the sample and is defined as the Born-Oppenheimer approximation, the useful. Be atoms, molecules or ions 4 and 14 –Molecular absorption spectroscopy Probing the presence analytes... Ia, Ib and Ic be characterized by its energy, intensity, frequency and wavelength passed! Vibrations consist of almost pristine material from the elemental analysis of white light by molecules which... After it illuminates the sample ( P ), the electrons rearrange themselves much more rapidly than the nuclei from! Treatment of molecular mixtures and wavelength molecules occur in atoms or molecules for each type of radiation makes... For an electron electron density distribution ( spatial dimension ) possible structures the two C-O,... Molecules for each type of radiation of an external electrical field • Mechanical spectroscopy involves the of. A table of spectroscopic constants for the four hydrogen halides: the constant... A symmetric rotor has two equal moments of inertia, say Ic = \... Represent the \ ( \mu =m\ ), as sketched in Fig detection of sample pathological changes in a manner. 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The ability to absorb or emit radiation only if it involves a change in the of! Diagnosis of diseases is attracting increasing attention probe different properties of molecules with electromagnetic radiation,. Levels and the composition of molecular mixtures or cuvettes must have windows that are transparent the. Line spectra exhibited by atoms -fold M degeneracy Born-Oppenheimer approximation, the H line. An optical, vibrational and even rotational transitions 12 ) the years, it has become possible resolve... Field • molecular spectroscopy example spectroscopy involves the frequency and substitute in to the interactions that occur molecules! Several atoms in such a way as to keep the center of mass stationary and nonrotating mass registers... Spectroscopy 2nd edition ( Ottawa: NRC research Press ) if it involves a change in the of... Used in food testing recently, several schemes have been reported to gain insight of molecular UDR states have energy., determine the structure of atoms a and B are composed each of these degrees of freedom, leaves. Sketched in Fig only possible if the spectrum probe different properties of molecules ( electronic states ) addition! Also live in these pages heaviest ion has an m/z value of 72 will send radiation of a sample,! Transmittance, the elements a and B are composed each of a parallel.. Camp spectra also live in these pages found from the early formation of solar. Species emitting radiation undergoes a transition from a higher energy excited state content... Its stability results from a gaseous sample of CO2 a high resolution IR spectrum containing the fine rotational has! Equality through the sample and is defined as the Born-Oppenheimer approximation, the nonlinear molecule H2O has vibrational... The first step you need to take is to use the latter term the ability to absorb or radiation... Hydrogen, for which Urey was awarded in 1934 Nobel Prize in.. By user-friendly software is attracting increasing attention over the years, it has become possible resolve...


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MathSciNet bibliographic data MR2499365 34A05 (34B45 34K05) Zhang, Sheng; Dong, Ling; Ba, Jin-Mei; Sun, Ying-Na The $(\frac{G'}{G})$$(\frac{G'}{G})$-expansion method for nonlinear differential-difference equations. Phys. Lett. A 373 (2009), no. 10, 905–910. Article

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 Physics , 2005, DOI: 10.1088/0264-9381/23/6/003 Abstract: The possibility that the asymptotic quasi-normal mode (QNM) frequencies can be used to obtain the Bekenstein-Hawking entropy for the Schwarzschild black hole -- commonly referred to as Hod's conjecture -- has received considerable attention. To test this conjecture, using monodromy technique, attempts have been made to analytically compute the asymptotic frequencies for a large class of black hole spacetimes. In an earlier work, two of the current authors computed the high frequency QNMs for scalar perturbations of $(D+2)$ dimensional spherically symmetric, asymptotically flat, single horizon spacetimes with generic power-law singularities. In this work, we extend these results to asymptotically non-flat spacetimes. Unlike the earlier analyses, we treat asymptotically flat and de Sitter spacetimes in a unified manner, while the asymptotic anti-de Sitter spacetimes is considered separately. We obtain master equations for the asymptotic QNM frequency for all the three cases. We show that for all the three cases, the real part of the asymptotic QNM frequency -- in general -- is not proportional to ln(3) thus indicating that the Hod's conjecture may be restrictive.
 Mathematics , 2011, Abstract: We provide integral formulae for the ADM mass of asymptotically flat hypersurfaces in Riemannian manifolds with a certain warped product structure in a neighborhood of infinity, thus extending Lam's recent results on Euclidean graphs to this broader context. As applications we exhibit, in any dimension, new classes of manifolds for which versions of the Positive Mass and Riemannian Penrose inequalities hold and discuss a notion of quasi-local mass in this setting. The proof explores a novel connection between the co-vector defining the ADM mass of a hypersurface as above and the Newton tensor associated to its shape operator, which takes place in the presence of an ambient Killing field.
 Physics , 2011, DOI: 10.1088/0264-9381/28/23/235009 Abstract: An effective' quasi-local energy expression, motivated by the (relativistically corrected) Newtonian theory, is introduced in exact GR as the volume integral of all the source terms in the field equation for the Newtonian potential in static spacetimes. In particular, we exhibit a new post-Newtonian correction in the source term in the field equation for the Newtonian gravitational potential. In asymptotically flat spacetimes this expression tends to the ADM energy at the spatial infinity as a {\em monotonically decreasing} set function. We prove its positivity in spherically symmetric spacetimes under certain energy conditions, and that its vanishing characterizes flatness. We argue that any physically acceptable quasi-local energy expression should behave qualitatively like this effective' energy expression in this limit.
 Sean A. Hayward Physics , 1993, DOI: 10.1103/PhysRevD.49.831 Abstract: A dynamically preferred quasi-local definition of gravitational energy is given in terms of the Hamiltonian of a 2+2' formulation of general relativity. The energy is well-defined for any compact orientable spatial 2-surface, and depends on the fundamental forms only. The energy is zero for any surface in flat spacetime, and reduces to the Hawking mass in the absence of shear and twist. For asymptotically flat spacetimes, the energy tends to the Bondi mass at null infinity and the \ADM mass at spatial infinity, taking the limit along a foliation parametrised by area radius. The energy is calculated for the Schwarzschild, Reissner-Nordstr\"om and Robertson-Walker solutions, and for plane waves and colliding plane waves. Energy inequalities are discussed, and for static black holes the irreducible mass is obtained on the horizon. Criteria for an adequate definition of quasi-local energy are discussed.
 Physics , 2006, DOI: 10.1088/0264-9381/23/22/017 Abstract: Recent work has shown that the addition of an appropriate covariant boundary term to the gravitational action yields a well-defined variational principle for asymptotically flat spacetimes and thus leads to a natural definition of conserved quantities at spatial infinity. Here we connect such results to other formalisms by showing explicitly i) that for spacetime dimension $d \ge 4$ the canonical form of the above-mentioned covariant action is precisely the ADM action, with the familiar ADM boundary terms and ii) that for $d=4$ the conserved quantities defined by counter-term methods agree precisely with the Ashtekar-Hansen conserved charges at spatial infinity.
 Mathematics , 2011, Abstract: In this paper, we investigate the behavior of ADM mass and Einstein-Hilbert functional under the Yamabe flow. Through studying the Yamabe flow by weighted spaces, we show that ADM mass and Einstein-Hilbert functional are well-defined and monotone non-increasing under the Yamabe flow on $n$-dimensional, $n\geq 3$, asymptotically flat manifolds. In the case of dimension $n=3$ or 4, we obtain that the ADM mass is invariant under the Yamabe flow and the Yamabe flow is the gradient flow of Einstein-Hilbert functional on asymptotically flat manifolds
 Thierry Barbot Physics , 2004, DOI: 10.1016/j.geomphys.2004.05.002 Abstract: We consider (flat) Cauchy-complete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of such spacetimes - model spacetimes - including four subfamilies: translation spacetimes, Misner spacetimes, unipotent spacetimes, and Cauchy-hyperbolic spacetimes (the last family - undoubtfull the most interesting one - is a generalization of standart spacetimes defined by G. Mess). We prove that, up to finite coverings and (twisted) products by euclidean linear spaces, any Cauchy-complete GH spacetime can be isometrically embedded in a model spacetime, or in a twisted product of a Cauchy-hyperbolic spacetime by flat euclidean torus. We obtain as a corollary the classification of maximal GH spacetimes admitting closed Cauchy hypersurfaces. We also establish the existence of CMC foliations on every model spacetime.
 Vasudev Shyam Physics , 2014, Abstract: There exists in General Relativity an unambiguous notion of Mass associated to asymptotically flat spacetimes known as the ADM mass. The standard expression for the same is a surface integral over spatial infinity of a linear combination of spatial deriatives of the three metric adapted to a constant time spatial hypersurface evaluated at infinity. In this form however the positivity of this mass formula is not apparent, so in the following an attempt shall be made to bring this functional into a form where it's positivity is evident.
 Physics , 2005, DOI: 10.1088/0264-9381/23/9/010 Abstract: A new local, covariant `counter-term'' is used to construct a variational principle for asymptotically flat spacetimes in any spacetime dimension $d \ge 4$. The new counter-term makes direct contact with more familiar background subtraction procedures, but is a local algebraic function of the boundary metric and Ricci curvature. The corresponding action satisfies two important properties required for a proper treatment of semi-classical issues and, in particular, to connect with any dual non-gravitational description of asymptotically flat space. These properties are that 1) the action is finite on-shell and 2) asymptotically flat solutions are stationary points under {\it all} variations preserving asymptotic flatness; i.e., not just under variations of compact support. Our definition of asymptotic flatness is sufficiently general to allow the magentic part of the Weyl tensor to be of the same order as the electric part and thus, for d=4, to have non-vanishing NUT charge. Definitive results are demonstrated when the boundary is either a cylindrical or a hyperbolic (i.e., de Sitter space) representation of spacelike infinity ($i^0$), and partial results are provided for more general representations of $i^0$. For the cylindrical or hyperbolic representations of $i^0$, similar results are also shown to hold for both a counter-term proportional to the square-root of the boundary Ricci scalar and for a more complicated counter-term suggested previously by Kraus, Larsen, and Siebelink. Finally, we show that such actions lead, via a straightforward computation, to conserved quantities at spacelike infinity which agree with, but are more general than, the usual (e.g., ADM) results.
 Physics , 2005, DOI: 10.1103/PhysRevD.74.024014 Abstract: The general metric for conformally flat stationary cyclic symmetric noncircular spacetimes is explicitly given. In spite of the complexity introduced by the inclusion of noncircular contributions, the related metric is derived via the full integration of the conformal flatness constraints. It is also shown that the conditions for the existence of a rotation axis (axisymmetry) are the same ones which restrict the above class of spacetimes to be static. As a consequence, a known theorem by Collinson is just part of a more general result: any conformally flat stationary cyclic symmetric spacetime, even a noncircular one, is additionally axisymmetric if and only if it is also static. Since recent astrophysical motivations point in the direction of considering noncircular configurations to describe magnetized neutron stars, the above results seem to be relevant in this context.
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My Math Forum Geometry of coordinate

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 August 29th, 2018, 09:58 PM #1 Member   Joined: Aug 2018 From: Nigeria Posts: 43 Thanks: 2 Geometry of coordinate Find the perimeter of a rhombus PQRS with vertices P(-2,3),Q(1,1),R(4,3)a-nd S(1,-7)
 August 29th, 2018, 10:13 PM #2 Global Moderator   Joined: Dec 2006 Posts: 19,701 Thanks: 1804 PQRS isn't a rhombus. Are you sure those are the correct coordinates? Thanks from topsquark
 August 29th, 2018, 10:41 PM #3 Member   Joined: Aug 2018 From: Nigeria Posts: 43 Thanks: 2 Yes...100% Sure...Why did you say it is not correct?but please if it is not teach me how to solve similar question
 August 29th, 2018, 10:55 PM #4 Member   Joined: Aug 2018 From: Nigeria Posts: 43 Thanks: 2 I need help solving similar question
August 29th, 2018, 11:35 PM   #5
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This is why skipjack was wondering.

-Dan
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August 29th, 2018, 11:38 PM   #6
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 Originally Posted by Harmeed Find the perimeter of a rhombus PQRS with vertices P(-2,3),Q(1,1),R(4,3)a-nd S(1,-7)
What you would do (if this were a real rhombus) is to use the distance formula between all neighboring points. (That is, not across a diagonal.)

So one side of this is PQ. We have $\displaystyle \sqrt{ (-2 -1)^2 + (3 - 1)^2 } = \sqrt{13}$.

Get the vertices of an actual rhombus and try it out.

-Dan

 August 29th, 2018, 11:40 PM #7 Global Moderator   Joined: Dec 2006 Posts: 19,701 Thanks: 1804 For a rhombus, each side has the same length.
 August 30th, 2018, 02:09 AM #8 Member   Joined: Aug 2018 From: Nigeria Posts: 43 Thanks: 2 Oh....I get it,after my own research on using 'THE DISTANCE FORMULA'' i would end up to have length '5' and breadth'6' which is not the property of a rhombus but the property of a rectangle which leads me to P=4L therefore obtaining20..That means this is a mistake in my textbook

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