text:
# Typesetting rules in physics (was: Re: Binary Relations, draft 1)

• To: haberg@matematik.su.se
• Subject: Typesetting rules in physics (was: Re: Binary Relations, draft 1)
• From: Ulrik Vieth <vieth@thphy.uni-duesseldorf.de>
• Date: Tue, 17 Nov 1998 17:36:55 +0100
• CC: math-font-discuss@cogs.susx.ac.uk
• Content-Length: 29609

Hans Aberg:

> The first rule about typesetting rules of pure math I think is that
> there is none such rule! :-)

The second rule about typesetting rules of physics is that even when
there are clearly-stated rules, they are never strictly observed. ;-)

Anyway, here's a discussion paper I began writing a long time ago,
but somehow never quite finished it.

Cheers, Ulrik.
(still extremely busy finishing my thesis for submission)

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
\documentclass[preprint]{ltugboat}
\usepackage{enumerate}
\usepackage{url}
\usepackage{cite}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\usepackage{amsmath,amssymb}

\overfullrule0pt

%% from revtex.sty

\makeatletter
\def\tensor#1{\protect\@ontopof{#1}{\leftrightarrow}{1.15}\mathord{\box2}}
\def\overstar#1{\protect\@ontopof{#1}{\ast}{1.15}\mathord{\box2}}
\def\overdots#1{\protect\@ontopof{#1}{\cdots}{1.0}\mathord{\box2}}
\def\overcirc#1{\protect\@ontopof{#1}{\circ}{1.2}\mathord{\box2}}
\def\loarrow#1{\protect\@ontopof{#1}{\leftarrow}{1.15}\mathord{\box2}}
\def\roarrow#1{\protect\@ontopof{#1}{\rightarrow}{1.15}\mathord{\box2}}

\def\@ontopof#1#2#3{%
{\mathchoice
{\@@ontopof{#1}{#2}{#3}\displaystyle\scriptstyle}%
{\@@ontopof{#1}{#2}{#3}\textstyle\scriptstyle}%
{\@@ontopof{#1}{#2}{#3}\scriptstyle\scriptscriptstyle}%
{\@@ontopof{#1}{#2}{#3}\scriptscriptstyle\scriptscriptstyle}%
}}
\def\@@ontopof#1#2#3#4#5{%
\setbox0=\hbox{$#4#1$}%
\setbox1=\hbox{$#5#2$}%
\setbox2=\hbox{}\ht2=\ht0 \dp2=\dp0 %
\ifdim\wd0>\wd1 %
\setbox1=\hbox to\wd0{\hss\box1\hss}%
\mathord{\rlap{\raise#3\ht0\box1}\box0}%
\else   %
\setbox1=\hbox to.9\wd1{\hss\box1\hss}%
\setbox0=\hbox to\wd1{\hss$#4\relax#1$\hss}%
\mathord{\rlap{\copy0}\raise#3\ht0\box1}%
\fi
}

\def\lambdabar{\protect\@lambdabar}
\def\@lambdabar{%
\relax
\bgroup
\def\@tempa{\hbox{\raise.73\ht0
\hbox to0pt{\kern.25\wd0\vrule width.5\wd0
height.1pt depth.1pt\hss}\box0}}%
\mathchoice{\setbox0\hbox{$\displaystyle\lambda$}\@tempa}%
{\setbox0\hbox{$\textstyle\lambda$}\@tempa}%
{\setbox0\hbox{$\scriptstyle\lambda$}\@tempa}%
{\setbox0\hbox{$\scriptscriptstyle\lambda$}\@tempa}%
\egroup
}
\makeatother

%% from nucetc.sty

\def\iotabar%
{\raisebox{-1pt}{$\mathchar'40$}\mkern-5.43mu\iota}

%% from ioplppt.sty

\newcommand{\mat}[1]{\underline{\underline{\bf #1}}}

% \newcommand{\e}{{\mathrm e}}
% \let\ii=\i
% \renewcommand{\i}{{\mathrm i}}
% \let\du=\d
% \renewcommand{\d}{{\mathrm d}}

%% from elsart.cls

% \def\d{\,\mathrm{d}}
% \def\e{\mathop{\mathrm{e}}\nolimits}

%% private hacks

\DeclareSymbolFont{vectors}  {OML}{cmm}{b}{it}
\DeclareSymbolFont{uptensors}{OT1}{cmss}{bx}{n}
\DeclareSymbolFont{tensors}  {T1}{cmss}{bx}{it}
\DeclareSymbolFontAlphabet{\mathvec}   {vectors}
\DeclareSymbolFontAlphabet{\mathtensup}{uptensors}
\DeclareSymbolFontAlphabet{\mathtens}  {tensors}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{Requirements for typesetting physics}
\author{Ulrik Vieth}
Heinrich-Heine-Universit{\"a}t\\
Institut f{\"u}r Theoretische Physik~II\\
Universit{\"a}tsstra\ss{}e 1\\
D-40225 D{\"u}sseldorf\\
Germany}
\personalURL{http://www.thphy.uni-duesseldorf.de/~vieth/}

\def\rtitlex{MFG discussion document}
\setcounter{page}{1}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}
\maketitle

\begin{abstract}
This paper attempts to summarize the requirements for
typesetting math-mode material in physics and related sciences
in accordance with the established typesetting rules applicable
in those fields.  Particular emphasis is placed on pointing out
the subtle differences from \AllTeX{}'s default settings for
math-mode, which are primarily geared towards the requirements
for mathematics and computer science.

This is an unfinished draft of a discussion paper which was
begun back in May~1997.  Unfortunately, as of \today, a number
of sections still remain very sketchy and incomplete and need
to be fleshed out eventually.
\end{abstract}

\section{Introduction}

Besides mathematics and computer science, physics is probably one
of the most important areas in which \AllTeX{} is heavily used for
typesetting math-mode material (as opposed to applications in the
humanities for typesetting body text).

While the default math-mode typesetting rules implemented in
\AllTeX{} are primarily geared towards the conventions applicable
in North American math typesetting, as specified in authorative
handbooks such as \emph{The Chicago Manual of Style}
\cite{Chicago:1982} or the style guides of well respected math
publishers \cite{AMS:1979}, some slightly (or not so slightly)
different conventions are applicable in physics and related
disciplines, such as physical chemistry or engineering sciences.

This discussion paper attempts to summarize the requirements for
typesetting physics, following the recommendations of \acro{IUPAP},
the International Union for Pure and Applied Physics
\cite{IUPAP:1978,IUPAP:1987,ISO:ISO31,NIST:SP330,NIST:SP811}, and it
also discusses various alternative conventions commonly found in
physics journals or textbooks produced by publishers
\cite{APS:REVTeX:1996,APS:CompuscriptPRL:1996,APS:StyleGuide:1993,%
IOP:ioplau:1994,IOP:iopfts:1994,Elsevier:instraut:1995,%
IAEA:nucfus:1996}
who adhere to these guidelines to a greater or lesser degree.

An earlier review of the requirements for typesetting physics was
presented by J{\"o}rg Knappen at the Euro\TeX{}~'92 conference
\cite{Knappen:EuroTeX:1992}.  The specific requirements in
high-energy physics with particular emphasis on elementary
particle notation were summarized in a \TUB{} article by Michel
Goossens \cite{Goossens:TB13-2-201} and also in Chapter~6.1 of
\emph{The \LaTeX{} Graphics Companion} \cite{GMR:TLGC:1997}.
Another very extensive review of the requirements for typesetting
mathematics in science and technology including references to the
relevant \acro{ISO} standards appeared in a recent \TUB{} article
by Claudio Beccari \cite{Beccari:TB18-1-39}.

\section{Requirements for math alphabets}

\subsection{Symbols for physical quantities}

To quote directly from the \acro{IUPAP} recommendations
\cite[sec.~1.2.1--2]{IUPAP:1978}, the general rules say:
\begin{quotation}
1. Symbols for physical quantities should be single letters of
the Latin or Greek alphabets with or without modifying signs.
\end{quotation}
and
\begin{quotation}
2. Symbols for physical quantities should be printed in italic
(or sloping) type.
\end{quotation}
Since this conforms exactly to the default font shape in \TeX{}'s
math mode (accessible by \cs{mathnormal}), no special markup is
needed, and there is little to worry about in this case, except
for the fact that there is some disagreement concerning the
preferred shape of uppercase Greek letters.

While the examples in the \acro{IUPAP} document seem to indicate
that uppercase Greek should be set in math italics type as well,
just like anything else, this convention isn't always observed in
publications, and one frequently finds the use of upright (roman)
type for uppercase Greek letters instead, whether it be for
laziness, lack of suitable fonts, or a policy of following a

To achieve the greatest possible flexibility, both conventions
need to be supported at the font encoding level, and a switching
mechanism at the macro level should be provided, if possible.

\subsection{Subscripts to physical quantities}

As a guiding principle for the printing of indices, it is recommended
in \cite[sec.~1.2.2]{IUPAP:1978} that:
\begin{quotation}
Only indices which are symbols for physical quantities should
be printed in italic (sloping) type.
\end{quotation}
This implies that normal (upright) type should be used otherwise,
especially for textual indices.

Thus, for example, in the energy confinement time~$\tau_{E}$ the
subscript~$E$' should be printed in math italics type because it
refers to the physical quantity energy'', whereas in the particle
confinement time~$\tau_{\mathrm{p}}$ the subscript~$\mathrm{p}$'
should be printed in roman (upright) type because it stands as an
abbreviation for a textual expression particle''.

Some publishers' style guides
\cite{APS:REVTeX:1996,APS:CompuscriptPRL:1996,APS:StyleGuide:1993}
try to simplify these guidelines by suggesting to use upright type
for multi-letter subscripts (which almost always represent textual
material) while using math italics for single-letter subscripts
(which may or may not represent a physical quantity).  It is
probably not a good idea to follow such a simplification since it

Concerning the font used to denote textual subscripts, care should
be taken to ensure that this should always be upright (roman) type
regardless of the surrounding text font, since the accidental use
of text italics or slanted type (e.g.\ within a theorem
environment) might only cause confusion.  Thus, a \cs{text} macro
like in \AmSLaTeX{} which has the effect of selecting the
surrounding text font rather than the default roman font may only
be suitable for the mark-up of intervening words between formulas,
but not for textual subscripts within formulas.

Using \cs{mathrm} for the latter purpose might be a better
alternative, provided that the roman letters are taken from a font
with normal text spacing.  What would happen if math roman letters
were taken from a hypothetical \texttt{MC}-encoded roman font with
math spacing hasn't been studied so far, but it is likely to
assume that some incompatibilities might arise.

\subsection{Symbols for vectors and tensors}

Concerning the representation of vectors and tensors the the
\acro{IUPAP} recommendations \cite[sec.~1.2.3]{IUPAP:1978} suggest:
\begin{quotation}
To avoid the excessive usage of subscripts, it is often
convenient to indicate vectors and tensors of the second rank by
letters of a special type.
\end{quotation}
which in \LaTeX{} lingo means using special math alphabets loaded
into extra math groups.  In particular, it is recommended that
\begin{enumerate}[(a)]
\item
vectors should be printed in bold italic (sloping) type,
e.g.\ $\mathvec{A}$,~$\mathvec{a}$,
\item
tensors of the second rank should be printed in bold sans serif
italic (sloping) type, e.g.\ $\mathtens{S}$,~$\mathtens{T}$.
\end{enumerate}
Although the recommended choice of fonts is fairly specific, the
conventions used in practice often vary, and one frequently finds
the use of bold upright type (\cs{mathbf}) for vectors,
e.g.\ $\mathbf{A}$,~$\mathbf{a}$, and bold sans serif upright type
for tensors, e.g.\ $\mathtensup{S}$,~$\mathtensup{T}$.

Still other publishers \cite{IOP:ioplau:1994,IOP:iopfts:1994}
resort to replacement notations in the preprint styles, such as
bold upright type combined with double underlining for tensors,
e.g.~$\mat{A}$, while using the proper typefaces in the final
publications.

To some extent, all these deviations from the standard may, in the
past, have been dictated by the lack of suitable fonts or the
difficulties of accessing such unusual font shapes.  However, if
such technical constraints no longer persist, it is probably best
to try to avoid any such deviations in the future and use the
proper notations wherever possible.

Since, in general, a full set of Latin and Greek letters may be
needed to denote vectors and tensors, a number of additional
\texttt{MC}-encoded fonts are needed to provide suitable math
alphabets, whereas using \texttt{T1}-encoded fonts will be
insufficient for this purpose.  To allow catering for all commonly
used conventions, at least four additional \texttt{MC}-encoded
fonts in bold upright, bold math italics, bold sans serif upright,
and bold sans serif oblique would be needed.  However, it may be
sufficient to provide only a subset of the \texttt{MC}-encoding
restricted to Greek and Latin letters or letter-like symbols,
while omitting non-changeable symbols that are included in the
basic \texttt{MC}-encoded font only for technical reasons.

Finally, if no suitable math alphabets are available, or if their
use is not desirable for whatever reason, it is suggested in
\cite[sec.~1.2.3]{IUPAP:1978} that:
\begin{quotation}
A vector may be indicated by an arrow and a tensor by a double
arrow above the symbol, e.g. $\vec{A}$,~$\Vec{\Vec{S}}$.
\end{quotation}
Whereas the use of a single arrow for vectors is well-established
and corresponds to the default \cs{vec} macro in \TeX{}'s math
mode, there is a somewhat wider variety of conventions used for
tensors.  While the \acro{IUPAP} document suggests a double arrow
consisting of two arrows stacked on top of each other, the
\acro{REV}\TeX{} package \cite{APS:REVTeX:1996} provides a
\cs{tensor} macro that uses a left-right over-arrow like this:
$\tensor{S}$.

In any case, establishing a consistent mark-up syntax and
providing a flexible switching mechanism at the macro level to
allow selecting the preferred representation of vectors and
tensors would be very desirable.  In particular, this would be
a significant advantage with regard to document portability and
reusability despite the absence of previously established standard
existing input notations, such as \cs{vec} for vectors and
\cs{tens} for tensors, to be used regardless of their eventual
representation might be the most preferable choice.

\subsection{Symbols for physical units}

Acoording to the \acro{IUPAP} document \cite[sec.~2.1]{IUPAP:1978}:
\begin{quotation}
Symbols for units of physical quantities should be printed in
roman (upright) type.
\end{quotation}
In general, physical units are represented by single or multiple
Latin letters, usually in lowercase only except for units derived
from proper names which are in uppercase or mixed-case.

There are only a very small number of exceptions from this rule
that require the use of Greek letters, specifically the multiplier
micro (\textmu), the unit Ohm~(\textohm) and the inverted
Ohm~(\textmho), all of which are available from a
\texttt{TS1}-encoded text companion font.  Finally, there is the
non-SI unit \AA{}ngstr{\o}m (\AA{}) consisting of an accented
letter which could always be accessed by resorting to \cs{textrm}
within math mode if nothing else is provided.

We can conclude that, in general, a \texttt{T1}-encoded text font
accessible with \cs{mathrm} or a synonym such as \cs{unit} would
be sufficient to typeset physical units if provisions are taken to
handle the exceptional cases by special macros.

Knappen \cite{Knappen:EuroTeX:1992} suggested a \cs{Ohm} macro for
\textohm{} and similarly one might introduce a \cs{micro} macro
for~\textmu{}.  Although, some similar macros are already provided
in the \textsf{textcomp} package as \cs{textohm} and \cs{textmu},
such macros should also be included in a physics-specific
math-mode package to make it unnecessary to rely on other \LaTeX{}
packages.

As for the typographical treatment of physical units, it should be
noted that it is frequently necessary to insert a little space
between numbers and units or in between physical units where
multiplication is implied.  While \emph{The \TeX{}book} suggests
using a thinspace, some other sources even suggest a thickspace or
an unbreakable word space.  Whether such spacing should be
inserted automatically by the macros used to mark up physical
units seems highly questionable, as it is often highly dependent
on the context whether or not extra spacing is needed.

As for the formatting of numbers, nearly all publishers' style
guides remind the author to avoid dangling decimal points, i.e.\
to write out $1.0$ or $0.1$ instead of $1.$ or $.1$ to avoid
confusion.

\subsection{Symbols for chemical elements, nuclides, and particles}

\begin{itemize}
\item to be filled in, for details see \cite[sec.~4]{IUPAP:1978}
\item symbols for chemical elements and elementary particles
should be set in upright roman (\cs{mathrm})
Latin and Greek, i.e.\ a \texttt{MC}-encoded font
\item suggested mark-up (\cs{chem}, \cs{particle})
\end{itemize}

\subsection{Quantum states and spectroscopic notations}

\begin{itemize}
\item to be filled in, for details see \cite[sec.~5]{IUPAP:1978}
\item symbols for quantum states should be set in upright roman
(\cs{mathrm})
Latin and Greek, i.e.\ a \texttt{MC}-encoded font
\end{itemize}

\subsection{Dimensionless numerical parameters}

\begin{itemize}
\item to be filled in, for details see \cite[sec.~7.14]{IUPAP:1978}
\item two-character mixed-cased expressions like $\mathit{Re}$'
for the Reynolds number, $\mathit{Ma}$' for the Mach number, etc.
\item to be set in italic (sloping) type, preferably in text italics
(\cs{mathit}) for better kerning in two-character expressions
\end{itemize}

\subsection{Other math alphabets}

\paragraph{Script/Calligraphic letters.}
These are used only in a few specific contexts, typically in cases
where two different physical quantities would normally be
represented by the same letter.  Examples include:
\begin{itemize}
\item
$\mathcal{E}$' when used for energy in contexts where $E$'
is already used to represent the electric field.
\item
$\mathcal{L}$' and $\mathcal{H}$' when used to denote the
Lagrangian or Hamiltonian density in field theory, where
$L$'~and~$H$' represent the Lagrangian or Hamiltonian function.
\item
$\mathcal{O}$' when used to denote the order of neglected terms
in series expansions.  Some authors (including DEK himself)
prefer to use a regular math italics~$O$' for this purpose, so
the use of a caligraphic~$\mathcal{O}$' might be regarded as
controversial.
\end{itemize}
There may be some disagreement among true believers as to whether
\TeX{}'s default caligraphics letters are sufficiently scripty for
this purpose or whether a different script font, such as
\texttt{rsfs}, should instead be substituted for it.

There is probably no need to introduce any special mark-up tags
for these relatively few cases of script or calicgraphic letters
used in physics.  However, it may, of course, be very convenient
to define private shorthands, such as \cs{L}'~or~'\cs{H}', when
dealing with material where these symbols are frequently used.

\paragraph{Blackboard Bold letters.}
These are sometimes used to denote number sets, such as
$\mathbb{N}$', $\mathbb{R}$' or $\mathbb{C}$' exactly as
they're normally used in mathematics.  Again, there may be some
disagreement among true believers as too their prefered shape.
Also, it may again seem convenient to define private shorhands,
such as \cs{N}' or \cs{R}', if these symbols are used rather
frequently.

\paragraph{Fraktur letters.}
These are no longer used in modern physics textbooks, but they are
sometimes found in reprints of historic texts (e.g.\ Einstein's
lectures of 1922) where Fraktur was used to denote vectors,

\section{Requirements for mathematical symbols}

\subsection{Mathematical constants}

Certain mathematical constants should always be printed in upright
(roman) type, especially when there is a danger that they might be
confused with a similar symbol denoting a physical quantity.

In particular, this convention applies to the following mathematical
constants:
\begin{itemize}
\item $\pi$' in its usual meaning ($\pi = 3.14159\dots$).  This
convention is rarely observed at present, presumably for lack of
suitable fonts.

\item $\mathrm{e}$' when used as the base of the exponential function
or natural logarithm ($\mathrm{e} = 2.718\dots$).  In this context,
$\mathrm{e}$' is almost invariably followed by a possibly
non-trivial exponent, but never by a subscript.  It is sometimes set
off from the preceeding material by a little space at the author's
or publisher's discretion.

Several publisher's macro packages define a control sequence
\cs{e} for this purpose, usually defined simply as a synonym
for \verb|\mathrm{e}|.  Elsevier's macros, however, define
\begin{displaymath}
\verb|\def\e{\mathop{\mathrm{e}}\nolimits}|
\end{displaymath}
which also affects the spacing.

\item $\mathrm{i}$' when used as the imaginary unit ($\mathrm{i} = \sqrt{-1}$).  Some sub-fields in physics such as electrical
engineering may use $\mathrm{j}$' instead to avoid confusion with
the AC-current~$i(t)$.

Some publisher's macro packages define a control sequence \cs{i} as
a synonym for \verb|\mathrm{i}|, thereby replacing or overwriting
the usual meaning of text mode dotless-i, which is not applcable in
math mode anyway.  It may be possible to retain both meanings by
employing a suitable switching macro depending on the mode.
\end{itemize}

\subsection{Derivatives, increments, variations, etc.}

In addition to mathematical constants, several Latin letters should be
printed in upright (roman) type, when used in a well-defined context
as mathematical operators.  In particular, this applies to:
\begin{itemize}
\item $\mathrm{d}$' when used to denote the integration variable in
an integral, a total derivative, or as a total (exact) differential.

Some publisher's macro packages define a control sequence \cs{d} as
a synonym for \verb|\mathrm{d}|, thereby replacing or overwriting
the usual command for the dot-under accent in text mode. It may be
possible to retain both meanings by employing a suitable switching
macro.

\item $\mathrm{D}$' when used a differential along geodesics
in general relativity (relatively rare).

\item $\mathrm{\partial}$' when used as a partial derivative.  This
convention is rarely observed so far, presumably for lack of
suitable fonts.  If an upright \cs{partial} becomes available in a
new math font encoding, it may be reasonable to make this the
default choice when in a physics context.

\item $\mathrm{\delta}$' when used as a variational derivative, or
as an inexact differential, e.g. $\mathrm{d}U = \delta Q + \delta W$.

\item $\mathrm{\Delta}$' when used to denote a finite increment.

\item $\eth$' and \th' when used as a weighted derivative
in quantum field theory (rare).
\end{itemize}
Some slashed or barred variants of $\mathrm{d}$' or
$\mathrm{\delta}$' are ocassionally used in thermodynamics to
distinguish between (exact or inexact) differental quantities in
reversible or irreversible processes. However, such a notation
presumably might be an author's invention and thus might be
considered non-standard.

\subsection{Operators in vector caculus}

\begin{itemize}
\item Laplace operator (Delta)
\item d'Alembert operator (Quabla'')
\end{itemize}

\section{Requirements for special notations}

\subsection{Special brackets, braces, etc.}

\begin{itemize}
\item (anti-)commutator in quantum theory:
$[p,q]_{\pm}$
\item Poisson- or Lie-brackets in theoretical mechanics:
$[p,q]$ or $\{p,q\}$
\item bra-ket notation in quantum theory,
quantum states, averages, projection operator:
$\langle x|H|x \rangle$
\item ensemble averages, time averages:
$\langle x \rangle$, $\bar{x}$
\item directions, planes, coordinates in crystals
\end{itemize}

\subsection{Barred letters}

\begin{itemize}
\item $\hbar$ (\cs{hbar}) and/or $\hslash$ (\cs{hslash}), both
representing a short-hand notation for $h/2\pi$ where $h$ is
Planck's constant.  This notation is very well-established
and widely used in quantum theory.

\item $\lambdabar = lambda/(2\pi)$, representing a wavelength
over $2\pi$.  A suitable macro is provided as \cs{lambdabar}
in the \acro{Rev\TeX} macro package.

\item $\iotabar$, used in a sub-field of fusion plasma physics to
denote $\iota/2\pi$ where $\iota$ is the rotational transform of
magnetic field lines.  Another sub-field of plasma physics prefers
to use a safety factor'' defined as $q = 2\pi/\iota = 1/\iotabar$
instead, thus making it unnecesary to use unusual notation.

\end{itemize}

\subsection{Slashed letters in QED}

\begin{itemize}
\item Feynman slash notation
\end{itemize}

\subsection{Operators in quantum mechanics}

\begin{itemize}
\item Operators vs. classical quantities

\item Complex conjugate states, denoted by $a^{*}$' in quantum
mechanics where $\bar{a}$' would normally be used for the
complex conjugate in a math context.

\item (Self-)adjoint operators, denoted by $a^{+}$' or $a^{\dag}$'.
This example indicates the use of the dagger as a math symbol,
not just as a footnote symbol.
\end{itemize}

\section{Summary}

The following math alphabets are needed in the \texttt{MC}-encoding
providing access to both Latin and Greek letters in a given shape:
\begin{itemize}
\item italic (sloping) type (\cs{mathnormal}),\\
for symbols denoting physical quantities
\item upright (roman) type (\cs{mathrm}),\\
for particle symbols or chemical elements
\item bold italic (sloping) type,\\
for vectors (recommended notation)
\item bold sans serif italic (sloping) type,\\
for tensors of the second rank
\item bold upright type (\cs{mathbf}),\\
for vectors (widely-used alternative notation)
\item bold sans serif upright type,\\
for tensors (widely-used alternative notation)
\end{itemize}
The following math alphabets are needed only for Latin letters
and may thus be implemented either using a \texttt{T1}-encoded
or \texttt{MC}-encoded font:
\begin{itemize}
\item upright (roman) type (\cs{mathrm}),\\
for textual subscripts

\item upright (roman) type (\cs{mathrm}),\\
for physical units (with few exceptions)

\item text italic type (\cs{mathit}),\\
for dimensionless parameters
\end{itemize}

\begin{thebibliography}{10}

\bibitem{Chicago:1982}
{University of Chicago Press}.
\newblock {\em The Chicago Manual of Style}.
\newblock University of Chicago Press, Chicago and London, 13th revised
edition, 1982.

\bibitem{AMS:1979}
Ellen Swanson.
\newblock {\em Mathematics into Type}.
\newblock American Mathematical Society, revised edition, 1979.

\bibitem{IUPAP:1978}
{International Union of Pure and Applied Physics, S.U.N. Commission}.
\newblock Symbols, units and nomenclature in physics.
\newblock {\em Physica~A}, 93:1--60, 1978.
\newblock Document U.I.P. 20 (1978).

\bibitem{IUPAP:1987}
{International Union of Pure and Applied Physics, S.U.N. Commission}.
\newblock Symbols, units and nomenclature in physics.
\newblock {\em Physica~A}, 146:1--67, 1987.

\bibitem{ISO:ISO31}
ISO.
\newblock Quantities and units.
\newblock Technical Report ISO~31-0--ISO~31-13, International Organization for
Standardization, Geneva, Switzerland, 1992.

\bibitem{NIST:SP330}
Barry~N. Taylor.
\newblock {The International System of Units (SI)}.
\newblock NIST Special Publication 330, National Institute of Standards and
Technology, Washington, August 1991.
\newblock vi+56~pp., available from {\url{http://www.nist.gov/}}.

\bibitem{NIST:SP811}
Barry~N. Taylor.
\newblock {Guide for the Use of the International System of Units (SI)}.
\newblock NIST Special Publication 811, National Institute of Standards and
Technology, Washington, April 1995.
\newblock x+74~pp., available from {\url{http://www.nist.gov/}}.

\bibitem{APS:REVTeX:1996}
{American Physical Society}.
\newblock {\em The REV{\TeX} Input Guide}.
\newblock American Physical Society, September 1996.
\newblock For REV{\TeX} 3.1.

\bibitem{APS:CompuscriptPRL:1996}
{American Physical Society}.
\newblock {\em Physical Review Letters REV{\TeX} Electronic-Submission
Compuscript Program}.
\newblock American Physical Society, March 1996.

\bibitem{APS:StyleGuide:1993}
{American Physical Society}.
\newblock {\em Physical Review Style and Notation Guide}.
\newblock American Physical Society, revised edition, February 1993.

\bibitem{IOP:ioplau:1994}
{Institue of Physics Publishing}.
\newblock {\em Preparing an article for publication in an IOP journal using
{\LaTeX}}.
\newblock Institue of Physics Publishing, Bristol, 1994.

\bibitem{IOP:iopfts:1994}
{Institue of Physics Publishing}.
\newblock {\em Using the AMS fonts with {\LaTeX} 2.09 and the IOP journals's
style file}.
\newblock Institue of Physics Publishing, Bristol, 1994.

\bibitem{Elsevier:instraut:1995}
{Elsevier Science}.
\newblock {\em Preparing articles with {\LaTeX}}.
\newblock Elsevier Science, Amsterdam, Oxford, New York, Lausanne, Tokyo, 1995.

\bibitem{IAEA:nucfus:1996}
{IAEA Publishing Team}.
\newblock {\em How to prepare a Nuclear Fusion article}.
\newblock International Atomic Energy Agency, Vienna, 1996.

\bibitem{Knappen:EuroTeX:1992}
J{\"o}rg Knappen.
\newblock Changing the appearance of math.
\newblock In {\em Euro{\TeX} '92, Proceedings of the 7th European {\TeX}
Conference}, pages 212--216, Prague, Czechoslovakia, September~14--18 1992.

\bibitem{Goossens:TB13-2-201}
Michel Goossens and Eric van Herwijnen.
\newblock {{The elementary Particle Entity Notation (PEN) scheme}}.
\newblock {\em {\TUB}}, 13(2):201--207, July 1992.

\bibitem{GMR:TLGC:1997}
Michel Goosens, Sebastian Rahtz, and Frank Mittelbach.
\newblock {\em The {\LaTeX} Graphics Companion}.

\bibitem{Beccari:TB18-1-39}
Claudio Beccari.
\newblock {{Typesetting mathematics for science and technology according to ISO
31/XI}}.
\newblock {\em {\TUB}}, 18(1):39--48, March 1997.

\end{thebibliography}

\makesignature
\end{document}

`

format: plain


text:
# Documentation

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English verison of the page.

# step

System object: comm.ViterbiDecoder
Package: comm

Decode convolutionally encoded data using Viterbi algorithm

## Syntax

`Y = step(H,X)Y = step(H,X,ERASURES)Y = step(H,X,R)`

## Description

 Note:   Starting in R2016b, instead of using the `step` method to perform the operation defined by the System object™, you can call the object with arguments, as if it were a function. For example, `y = step(obj,x)` and `y = obj(x)` perform equivalent operations.

`Y = step(H,X)` decodes encoded data, `X`, using the Viterbi algorithm and returns `Y`. `X`, must be a column vector with data type and values that depend on how you set the `InputFormat` property. If the convolutional code uses an alphabet of 2N possible symbols, the length of the input vector, `X`, must be L$×$N for some positive integer L. Similarly, if the decoded data uses an alphabet of 2K possible output symbols, the length of the output vector, `Y`, is L$×$K.

`Y = step(H,X,ERASURES)` uses the binary column input vector, `ERASURES`, to erase the symbols of the input codewords. The elements in `ERASURES` must be of data type double or logical. Values of `1` in the `ERASURES` vector correspond to erased symbols, and values of `0` correspond to non-erased symbols. The lengths of the `X` and `ERASURES` inputs must be the same. This syntax applies when you set the `ErasuresInputPort` property to true.

`Y = step(H,X,R)` resets the internal states of the decoder when you input a non-zero reset signal, `R`. `R` must be a double precision or logical scalar. This syntax applies when you set the `TerminationMethod` property to `Continuous` and the `ResetInputPort` property to `true`.

 Note:   `obj` specifies the System object on which to run this `step` method.The object performs an initialization the first time the `step` method is executed. This initialization locks nontunable properties (MATLAB) and input specifications, such as dimensions, complexity, and data type of the input data. If you change a nontunable property or an input specification, the System object issues an error. To change nontunable properties or inputs, you must first call the `release` method to unlock the object.


text:
# All Questions

15answers
9k views

### How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
12answers
20k views

### Why does $1+2+3+\cdots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
16answers
13k views

### How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

How can one prove the statement $$\lim\limits_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my ...
22answers
22k views

### Is $.\overline{9} = 1$?

I'm told by smart people that $.\overline{9} = 1$ and I believe them, but is there a proof that explains why this is?
16answers
43k views

### Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ...
4answers
8k views

### How to use the Extended Euclidean Algorithm manually?

I've only found a recursive algorithm of the extended Euclidean algorithm. I'd like to know how to use it by hand. Any idea?
25answers
33k views

### Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
3answers
538 views

4answers
2k views

### If $a \mid m$ and $(a + 1) \mid m$, prove $a(a + 1) | m$.

Can anyone help me out here? Can't seem to find the right rules of divisibility to show this: If $a \mid m$ and $(a + 1) \mid m$, then $a(a + 1) \mid m$.
10answers
3k views

### Why $\sqrt{-1 \times -1} \neq \sqrt{-1}^2$? [duplicate]

We know $$i^2=-1$$then why does this happen? $$i^2 = \sqrt{-1}\times\sqrt{-1}$$ $$=\sqrt{-1\times-1}$$ $$=\sqrt{1}$$ $$= 1$$ EDIT: I see this has been dealt with before but at least with ...
3answers
2k views

So I have been trying for a few days to figure out the sum of $$S = \sum_{k=1}^\infty \frac{1}{k^2 - a^2}$$ where $a \in (0,1)$. So far from my nummerical analysis and CAS that this sum equals $$... 5answers 8k views ### How to define a bijection between (0,1) and (0,1]? How to define a bijection between (0,1) and (0,1]? Or any other open and closed intervals? If the intervals are both open like (-1,2)\text{ and }(-5,4) I do a cheap trick (don't know if ... 5answers 5k views ### Highest power of a prime p dividing N! How does one find the highest power of a prime p that divides N! and other related products? Related question: How many zeros are there at the end of N!? This is being done to reduce ... 6answers 2k views ### Linear diophantine equation 100x - 23y = -19 I need help with this equation:$$100x - 23y = -19.$$When I plug this into Wolfram|Alpha, one of the integer solutions is x = 23n + 12 where n is a subset of all the integers, but I can't seem ... 5answers 4k views ### Finding the limit of \frac {n}{\sqrt[n]{n!}} I'm trying to find$$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$I tried couple of methods: Stolz, Squeeze, D'Alambert Thanks! Edit: I can't use Stirling. 6answers 4k views ### x^y = y^x for integers x and y We know that 2^4 = 4^2 and (-2)^{-4} = (-4)^{-2}. Is there another pair of integers x, y (x\neq y) which satisfies the equality x^y = y^x? 5answers 3k views ### Limits: How to evaluate \lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x What methods can be used to evaluate the limit$$\lim_{x\rightarrow\infty} \sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x.$$In other words, if I am given a polynomial P(x)=x^n + a_{n-1}x^{n-1} ... 12answers 9k views ### Prove that \lim \limits_{n \to \infty} \frac{x^n}{n!} = 0, x \in \Bbb R. Why is$$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$Can we generalize it to any exponent x \in \Bbb R? This is to say, is$$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$This is ... 10answers 37k views ### What's an intuitive way to think about the determinant? In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ... 4answers 14k views ### Norms Induced by Inner Products and the Parallelogram Law Let  V  be a normed vector space (over \mathbb{R}, say, for simplicity) with norm  \lVert\cdot\rVert. It's not hard to show that if \lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle} ... 7answers 8k views ### Show that the set of all finite subsets of \mathbb{N} is countable. Show that the set of all finite subsets of \mathbb{N} is countable. I'm not sure how to do this problem. I keep trying to think of an explicit formula for 1-1 correspondence like adding all the ... 2answers 3k views ### Convergence a.e. and of norms implies that in Lebesgue space Let 1\leq p < \infty. Suppose that \{f_k\} \subset L^p (the domain here does not necessarily have to be finite), f_k \to f almost everywhere, and \|f_k\|_{L^p} \to \|f\|_{L^p}. Why is it ... 2answers 21k views ### Expected time to roll all 1 through 6 on a die What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die? The order in which the numbers appear does not matter. I had this questions explained ... 34answers 20k views ### Examples of apparent patterns that eventually fail Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ... 2answers 9k views ### Limit of L^p norm Could someone help me prove that given a finite measure space (X, \mathcal{M}, \sigma) and a measurable function f:X\to\mathbb{R} in L^\infty and some L^q, ... 1answer 372 views ### Discrete logarithm tables for the fields \Bbb{F}_8 and \Bbb{F}_{16}. The smallest non-trivial finite field of characteristic two is$$ \Bbb{F}_4=\{0,1,\beta,\beta+1=\beta^2\},  where $\beta$ and $\beta+1$ are primitive cubic roots of unity, and zeros of the ...

15 30 50 per page


text:
# This Hash Table Is Faster Than a Judy Array

In game development, we use associative maps for many things. Dynamic loading, object reflection, rendering, shader management.

A lot of them fit the following usage pattern:

• The keys and values have simple integer or pointer types.
• The keys are effectively random.
• The basic operation is lookup-or-insert: Look for an existing item first, and add one if not found. Usually, the item is found.
• Deletes happen infrequently, and when they do, they tend to happen in bulk.

Occasionally, one of these maps is identified as a performance bottleneck. It raises the question: What’s the best data structure to use in this scenario?

I wrote a small C++ application to compare two top candidates, a Judy array and a finely-tuned hash table, using various experiments. All the code is available on GitHub.

A Judy array – specifically, the JudyL variant – is an efficient mapping of integer keys to integer values. It is optimized to avoid CPU cache misses as often as possible. Its memory consumption scales smoothly with number of entries, even when the keys are sparsely distributed. It’s a real feat of engineering – hats off to Doug Baskins.

A hash table is a relatively well-known data structure. For this post, I’ve written a custom hash table based on open addressing with linear probing. This particular hash table will dynamically resize itself as the population grows.

Now, there are already several existing shootouts between Judy arrays and hash tables on the web. Some of them are good, but in this post, I wanted to target a very specific usage pattern. Also, I wrote the single most efficient integer hash table I could, with the explicit goal of taking Judy down – no smorgasbord of lame hash tables here. Finally, I took considerable pains to ensure that the benchmarking method produced fair, precise and detailed results, as outlined in the README. Feedback is welcome.

## Adding Items to the Map

In this experiment, more than 10000000 unique 32-bit random keys are inserted into each map, and the insertion times are measured. The timings were taken on my aging Core 2 Duo E6300.

Each point on the graph represents the time to insert the Nth element into a map containing N - 1 elements, averaged around small neighborhoods of N.

First, you’ll notice all the spikes in the hash table insertion times. These spikes are due to dynamic resizing of the hash table, which comes at a cost. Once the table becomes 75% full – for example, when the 12280th item is inserted into a hash table of size 16384 – we dynamically allocate a new hash table with double the size, copy all existing items to new hash slots, and delete the old table. As you’d expect, this operation is slow. For example, where most items take between 30 and 40 nanoseconds to insert, the 12280th item takes 0.4 milliseconds.

That’s expensive, but remember, resizing happens infrequently. In practice, these one-time costs are amortized against thousands of other insert operations which run very quickly. Another way to look at it is that the area under the red curve is much smaller than the area under the blue curve, meaning that the total CPU time spent inserting into the hash table is much less, even with the presence of spikes. And in cases where it really matters, you could always pre-allocate the hash table to a large enough size, eliminating those spikes completely.

Judy makes heavy use of the memory allocator, so to show it in its best light, I hardwired the Judy array to DLMalloc. DLMalloc is a great allocator, and helps avoid the possibility of hidden performance pitfalls on Windows, such as DLL runtime overhead and (God forbid) the debug system heap.

It should be acknowledged that Judy insertion times are faster in cases when the keys are not random; however, that’s not what we’re testing in this experiment.

## Finding Existing Items in the Map

Usually, during the lookup-or-insert operation, an existing item is already found in the map. That makes lookup times even more significant than insert times.

In this next experiment, each map is filled to various populations. At each population, thousands of random lookups are performed, and the average time per lookup is plotted. Here, the lookup function is actually the same as the insert function used in the previous experiment – the only difference is that this time, we call it using keys which are known to already exist in the map.

At all populations, hash table lookups are twice as fast as Judy array lookups, give or take. I suspect part of the reason is because hash tables cause even fewer cache misses than Judy. With a good hash function, most lookups should be found in the first memory address checked; if not, linear probing ensures they are likely to be found in the same cache line. Notice that when the population reaches 98304 items, the hash table stops fitting entirely within the CPU’s 2 MB L2 cache, which is when the lookup times begin to climb.

Because the keys are random, the choice of hash function doesn’t matter too much for this experiment. Nonetheless, I borrowed MurmurHash3’s integer finalizer as the hash function. It consists of a few XORs, multiplies and shifts, and seems to distribute the hashes just as effectively when the keys are not random.

h ^= h >> 16;
h *= 0x85ebca6b;
h ^= h >> 13;
h *= 0xc2b2ae35;
h ^= h >> 16;

I chose not to benchmark the delete operation, because under the given usage pattern, deletes tend to be one-time operations which happen in bulk, such as when unloading a section of a game world. Still, one cool thing about hash tables with linear probing is that the Delete method needs only to shuffle existing entries around. This hash table also offers a Compact method, to optionally reclaim memory. Finally, you could write an even faster delete function if you require that Compact is always called at the end of each bulk delete.

## Polluting the CPU Cache Between Operations

The above experiments are purely micro-benchmarks, which makes them somewhat unrealistic. A real application would run plenty of other code in between operations on the map. This would continuously evict parts of the map from the CPU cache, which would have a direct impact on the map’s performance.

With that in mind, I added some artificial cache pollution to both experiments, and ran them again. This time, random chunks of memory were manipulated between each operation. Using this strategy, I collected two extra datasets for each data structure, and plotted them below. The thin lines have an average of 1000 bytes of cache pollution between operations, while the thick lines have an average of 10000 bytes.

As you can see, cache pollution makes all of the timings slower, especially at larger populations. Regardless, the hash table still runs roughly twice as fast as the Judy array at nearly all populations and pollution levels.

Obviously, the most useful performance measurements would be taken in a real application using a consistently reproducible test case. For such measurements, an APIProfiler-like profiling technique comes in handy.

## Memory Consumption

Judy is known for its memory-efficiency. So, how does its memory consumption compare to that of the hash table? To find the answer, I called dlmalloc_stats, a function which includes the overhead of the memory allocations themselves, an intrinsic cost which is all-too-easily overlooked. This graph plots the total bytes consumed by each map, divided by the total number of items in the map.

You can see that the hash table’s memory consumption follows a sawtooth pattern, doubling in size each time the 75% threshold is reached. Judy’s memory consumption, on the other hand, scales much more smoothly, and tends to hover around the lower end of the hash table’s amount. Still, there are some points where the two lines cross, which means that the hash table actually takes less memory at certain populations. Beyond 1000 items, though, those cases are generally the exception.

Judy also has the interesting ability to consume even less memory when the keys are not distributed randomly. In particular, when all the keys are clustered around a small range, its memory consumption tends towards that of a plain array. That’s impressive, but again, not the usage pattern we’re testing in this experiment.

## Conclusion

For the given usage pattern, a Judy array is not the fastest associative map. But for general use, Judy arrays tend to be among the fastest associative maps while also being among the most memory-efficient, which is what made them a contender here in the first place. As a bonus, Judy arrays are always sorted. That means you can iterate sequentially through the map at any moment, something not possible using a hash table – though this property generally does not bring any benefit in the usage pattern studied here. Judy is also LGPL-licensed and patented, which may make it unsuitable for certain kinds of development.

The hash table tested here was specifically written for the job. In general, you can’t just grab any module with “hash table” in the name, slap it in place and expect the same performance. The addressing strategy, choice of hash function, handling of free/deleted slots, and reliance on inlining, templates and indirection each has the potential to bog down the CPU in various ways.

Another fun, useful fact about this type of hash table is that it lends itself surprisingly well to lock-free programming. You can read all about it in my follow-up post, The World’s Simplest Lock-Free Hash Table.


text:
# Use GKEntity's component(ofType:) with inheritance

I'm writing a SpriteKit game in swift and making use of GameplayKit's entity-component system. There are many components that do different stuff but share the same methods in which they do it, so I thought I'd group them into categories. At first I thought about using protocols, but component(ofType:) only accepts classes. So then I thought about making a base class for each category, define methods and variables there and have the actual components classes override them to implement their own functionality (you know like basic OOP). So I had something like:

class InputComponent: GKComponent {
func handleInput() {
//Whatever
}
}

class ControllerComponent: InputComponent {
override func handleInput() {
//Do something
}
}

class KeyboardComponent: InputComponent {
override func handleInput() {
//Do something different
}
}

Then somewhere in my code I'd call

entity.component(ofType: InputComponent)

and I thought I'd get both ControllerComponent and KeyboardComponent type objects. But I am getting none, it seems component(ofType:) doesn't return instances of subclasses.

I guess I can make an extension to GKEntity and define something like

component<P>(conformingTo protocol: P.Type)

and implement that behavior myself but I want to know if I can use the built in function to achieve the same.

## 1 Answer

Well it appears as if no one here seems to know it so I'll just post what I used which is a new method that I added in an extension to GKEntity (which I was actually trying to avoid)

extension GKEntity {
func component<P>(conformingTo protocol: P.Type) -> P? {
for component in components {
if let p = component as? P {
return p
}
}

return nil
}
}

Actually you can pass not only protocol types as an argument, you can actually pass any type (like classes) - so actually the results are similar to the native method provided by Apple, though I am not sure how similar these are so I wouldn't use them interchangeably, I use it solely for when I need to get a GKComponent that conforms to a protocol instead of a specific type of GKComponent.

• Thanks for sharing! Are there any concerns about having multiple components that all conform to the same protocol on an entity and only getting back one? I'm not sure if the component(ofType) one has similar issues, but it seems like this approach could exacerbate it at the very least. – idolize Nov 19 '18 at 3:05
• Well this function will just return the first component conforming to the protocol, ignoring any other components conforming to the protocol. But you can just make an array (or a set), add any component conforming to the protocol to the array, and return that instead. – lsauceda Nov 24 '18 at 14:17


text:
# Using lower-case roman numerals in enumerate lists

when I use

\begin{enumerate}[I]
\item
...

I get

I
II

how I can get

i
ii

?

I try with

\renewcommand{\theenumi}{\roman{enumi}}
\renewcommand{\labelenumi}{\theenumi}

in the begin of file, but not fix the problem.

• – hola
Jun 15, 2021 at 18:54

There are a number of ways. Here's using the enumerate package:

\documentclass{article}
\usepackage{enumerate}% http://ctan.org/pkg/enumerate
\begin{document}
\begin{enumerate}[I]
\item One
\item Two
\item Three
\end{enumerate}

\begin{enumerate}[i]
\item One
\item Two
\item Three
\end{enumerate}
\end{document}

Here's another method via the enumitem package. It yields the same output as above:

\documentclass{article}
\usepackage{enumitem}% http://ctan.org/pkg/enumitem
\begin{document}
\begin{enumerate}[label=\Roman*]
\item One
\item Two
\item Three
\end{enumerate}

\begin{enumerate}[label=\roman*]
\item One
\item Two
\item Three
\end{enumerate}
\end{document}

And then, without any packages:

\documentclass{article}
\renewcommand{\labelenumi}{\theenumi}
\begin{document}
\renewcommand{\theenumi}{\Roman{enumi}}%
\begin{enumerate}
\item One
\item Two
\item Three
\end{enumerate}

\renewcommand{\theenumi}{\roman{enumi}}%
\begin{enumerate}
\item One
\item Two
\item Three
\end{enumerate}
\end{document}

If you want to type:

I.

II.

III.

Use:

\documentclass{article}
\begin{document}
\usepackage{enumerate}
\begin{enumerate}[I.]
\item my text
\item my text
\item my text
\end{enumerate}

Another possibility for [I.] is [(I)], which gives:

(I)

(II)

(III)

• The question is how to have lowercase roman numeral, not to add a period/dot after the Roman numeral. Apr 2, 2013 at 11:29
• but this was useful information, so thanks. Jan 1, 2017 at 14:46
• Unfortunately this package adds the side-effect of changing the indentation for the items - is there any way to avoid this change? Jul 5, 2019 at 1:53
• @DmitriZaitsev You might want to check out enumitem package (with package option shortlabels. It's more customizable than enumerate.
– 6005
Nov 19, 2019 at 1:36

If you're using the beamer class, try something like:

\setbeamertemplate{enumerate item}{(\roman{enumi})}

or

\setbeamertemplate{enumerate subitem}{(\roman{enumii})}

• Just use [label=(\roman*)] May 31, 2019 at 11:58
• @Smaillns, have you tested it? This would require an extension package. Jun 2, 2019 at 14:20
• \usepackage{enumitem} Jun 5, 2019 at 10:32
• Note: you can remove the braket "()", and add what you want, for instance \begin{enumerate}[label=\roman*] Jun 5, 2019 at 10:40
• This is covered by Werner's answer, while my answer was focused on the beamer class specifically. Jun 5, 2019 at 16:47

For those using Spanish with babel package, the enumerate option \begin{enumerate}[i] does not work. To make it work, load the babel package with this option:

\usepackage[spanish,es-lcroman]{babel}

Explanation:

Spanish babel forces the use of upper case. As the manual states:

Traditional Spanish typography discourages the use of lowercase Roman numerals.

The manual justifies this choice referencing the book José Martínez de Sousa, Diccionario de tipografía y del libro, Madrid, Paraninfo, 3.a ed., 1992.

• Thanks @luchonacho it's helpful Oct 13, 2021 at 14:47


text:
# Fitting a Sine Curve to Two Points

1. Jun 29, 2004

### Alterscape

I'm a web developer working on a bit of javascript code for a personal project. I took through Calc II and linear algebra in college, but haven't looked at any of it a couple of years, so I'm a little rusty on the details. Anyway, what I'm trying to work out is a way to scale/transform an equation giving the position of an element as a function of time, of the form f(x) = sin(ax+b)+c , so that it has a given f(x) and f'(x) for two given values of x, and f''(x) at one of those values. If I remember right, I'm going to need to set up a system of equations including the function and both derivatives and solve it to calculate the necessary values of a, b, and c, but I'm a bit rusty on the details. If anyone would be so kind as to offer some clarification and save me a few more days of trial and error, I'd greatly appreciate it. Thanks!

-Ryan

2. Jun 29, 2004

### Integral

Staff Emeritus
First off if you may want to use a more general form of the sin equation.

f(x)= A sin(bx) +c

It is not clear to me that you need a phase factor (your b) simply because it represents a shift on the coordinate axis.
However, you may well need an Amplitude (my A) This will allow your resultant function to have an amplitude other then 1 (in what ever units you choose)

If this:

$$f(x) = A \sin(Bx) + C$$

$$f(x_1) = h; \ f'(x_1)=k$$
$$f(x_2) = a; \ f'(x_2)=b; \ f''(x_2)=n$$

is the problem you are trying to solve, then you may have some troubles. There is no guarantee that you will be able to find a solution to fit all of those conditions.

Perhaps you can give us a bit more detail about what you are trying to do.

Last edited: Jun 29, 2004
3. Jun 29, 2004

### HallsofIvy

Staff Emeritus
You have 5 "pieces of information" so your function prototype will have to have 5 "unknown numbers".

Neither f(x)= sin(ax+b)+ c nor f(x)= A sin(Bx)+ C have enough "flexibility" to fit all 5 requirements.

4. Jun 29, 2004

### Alterscape

Clarifications

Sorry for being unclear. First of all, yes, a scaling factor for amplitude would be necessary, and I have no idea why that didn't occur to me earlier. I think it was mosly because in the simpler code I wrote originally, I used the position function to generate a percentage between 0 and 1 and multiplied that by the change in position, and added it to the initial position -- which is the same thing you're suggesting here, just in a more convoluted, less broadly useful way.

Anyway, to summarize what I'm trying to do.. sinusoidals are great for smooth animation -- if you use a sinusoidal as a function that determines the position of an element, with the initial position at the trough and the final position at the next peak (or vice-versa) you get a beautifully smooth soft-start/stop. But what happens if, during a long animation of this sort, the user triggers an event, like clicking a button, etc (the setup I'm creating is part of an elaborate menu system) which changes the position the animation is moving to and/or the time it will take to arrive? At that point you need to smoothly transition to a new curve describing this new animation. (In this case, I'm ignoring what happens if a reverse of motion is needed -- once I work out slowing the animation to a stop, tacking on a new sinecurve for the reverse motion is relatively simple).

For a transition between two curves to be smooth, I remember from calc that f1(x1)=f2(x1), f1'(x1)=f2'(x1), and f1''(x1)=f2''(x1) , where x1 is the point at which the two functions transition. That's the first condition. Then, at x2, I need to specify that f2(x2) = the new position target, f2'(x2) = 0, because that should be the peak or trough of the sinewave, but f2''(x2) doesn't concern me because, so long as it reaches the position with a velocity of zero, I don't care what it was accelerating at.

so maybe a better general form would be p(t) = a * sin(bx+c) + d.

Is this impossible?

edit: I worked out a system of 5 equations with 4 unknowns that I think I should be able to solve.. someone care to sanity-check me here?

Pi = a*sin(b*Ti+c)+d
Vi = a*b*cos(b+Ti+c)
Ai = -a*b^2*sin(b+Ti+c)
Pf = a*sin(b*Tf+c)+d
0 = a*b*cos(b*Tf+c)+d

Where Pi = known initial position, Vi = known initial velocity, Ai = known initial acceleration, Ti = known initial time, Pf = known final position, and Tf = known final time. The unknowns are a, b, c, and d. Note that the velocity and acceleration functions are, of course, the 1st and 2nd derivative of the position function.

I've currently worked the two position functions around to get

sin(b*Tf+c) - sin(b*Ti+c) = (Pf-Pi)/a

by setting both equal to d, and thus equal to eachother, and doing some simple arithmetic, but I forget how to use the arcsin to get rid of the sines on the left side of the equation.. again, any help appreciated.

Last edited: Jun 29, 2004
5. Jul 2, 2004

### Alterscape

I'm not sure how bump ettiquette works on this forum, but I'm going to bump once since I'm totally lost and could use some help.

In response to HallsOfIvy, the function

P(t)=a*sin(bx+c)+d

provides 5 variables to alter the sinewave into any possible configuration that doesn't involve rotating it, so far as I know. The system of equations I've worked out has 5 equations for 5 unknowns, so theoretically, it should be solvable. I'm just very rusty on algebra w/trig functions.. is there a simple way to solve this system (ie, some relatively-easily-available software or something.. I have a couple of contacts who do engineering work with Matlab/Mathematica, is this the sort of thing they could write a simple script to solve?)

I also realized that I need to note that I'm only looking for solutions on the interval [0, 2Pi].

Last edited: Jul 2, 2004
6. Jul 3, 2004

### Integral

Staff Emeritus
What you are doing is called splining, it is not clear to me that you need to use sin functions to achieve the desired results. Polynomial splines are pretty common and will do what you need. If you match position, velocity and acceleration, your transitions will be smooth no matter what functions you use. Use the simplest possible to achieve the desired results.

Also in your last general formal, P(t)=a*sin(bx+c)+d, There are only 4 parameters to be determined. x is the independent variable, not a parameter.


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# Tag Info

5

It doesn't matter, it's just what the data is. I assume that you're thinking about issues related to "imbalanced dataset", but this term refers only to imbalance in the values of the target variable (and it's more commonly used about classification, but technically it's relevant also in regression). Features don't need to be balanced in any way, ...

3

Not necessarily, while it can be the case that two observations belong to the same 'group' and end up in the same leaf node (and thus get the same predicted value) there can also be multiple groups of observations that both have the same predicted value. If this is the case in your example is of course dependant on the data you are using. It would indeed be ...

3

The Iris data has three target values ("species"). The objective function in params is set to objective = "binary:logistic", which only accepts two classes (binary taget). In case you have more than two classes, you need a multiclass objective function, e.g. multi:softmax or multi:softprob. As stated in the docs: “binary:logistic” –...

2

As Erwan said, the imbalanced dataset problem is about the target variables and not the features. But if your model favors a section of your regression target more, you can perform a study on the distribution of the target variable and then, depending on the distribution, perform a transformation (e.g. square root or exp), to get a more uniform output. Also, ...

2

Regarding feature encoding As the error says, strings are not accepted. You need to transform item in a way that can be digested by xgboost (essentially some numerical representation). There seems to be a method to transform to categorical when generating the DMatrix (see the docs, never tried it). However, my first idea would be to "one hot" ...

2

XGBoost will not learn "interactions" on its own. Feature generation is often used to enhance the explanatory power of $X$. Often $x_n - x_k$ or $x_n / x_k$ are checked and used. There are also tools for feature generation, e.g. "Featuretools" for Python. One thing you can do to find out what kind of interactions have the most ...

1

With new data, you need to go down exactly the same route as with the data used for training. Something like: # Some data newdf = data.frame(Sepal.Length=c(5.3), Sepal.Width=c(3.2), Petal.Length=c(2.0), Petal.Width=c(0.2)) # Model matrix newdf = model.matrix(~.+0,data = newdf) # Predict on xgb.DMatrix object predict(xgb1,xgb.DMatrix(data = newdf))

1

There isn't much to philosophize here, in order for your model to predict the value for some class, it has to see that class in training. If the model never saw that class it cannot predict it. So you should make sure that all classes are present in your training set. If a new class appears at a later time you have to retrain your model with the new class. ...

Only top voted, non community-wiki answers of a minimum length are eligible


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## Force on Moving Charge in a Magnetic Field

Subject: Physics

#### Overview

This note explains the magnetic field and the hall effect in the magnetic field.
##### Force on Moving Charge in a Magnetic Field

If an electric charge +q moves with a velocity v through a magnetic field B then magnetic field apply force Fmon the charge. Consider a magnetic field acting along the y-axis, +q charge moves along XY making angleθ with the magnetic field. In such case, charge experiences a magnetic force acting along z-axis i.e. perpendicular to the plane of v and B as shown in Fig. Experimentally it is found that magnitude of magnetic force is,

(i) directly proportional to the magnitude of the charge

$$i.e. \;F_m∝ q..(i)$$

(ii) directly proportional to the strength of the magnetic field

$$i.e.\;F_m∝ v..(ii)$$

(iii) directly proportional to the sine of the angle between v and B

$$i.e.\;F_m∝ B..(iii)$$

(iv) directly proportional to the sine of the angle between v and B

$$i.e.\;F_m∝sinθ$$

Combining equation (i), (ii), (iii) and (iv) we get,

$$F_m∝ qvBsinθ$$

$$or,\;F_m=kqvBsinθ$$

Where k is proportionality constant and it values is 1 so we obtain

$$Therefore,\;F_m=qvBsinθ$$

In vector form,

$$\overrightarrow{F_m}=q(\overrightarrow{v}x\overrightarrow{B})$$

This force is also called Lorentz magnetic force.

By this it is proved that Fm is perpendicular to the plane containing v and B.

Direction of Magnetic Force

We already know that magnetic force Fm is perpendicular to the plane containing v and B. The force could be up or down as shown in Fig. ,since both directions are perpendicular to the plane containing v and B. Fleming left-hand rule illustrated below is used to determine a direction of the magnetic force.

Fleming’s left-hand rule

According to this rule, the forefinger, middle finger and thumb of the left hand are stretched mutually perpendicular to each other in such way that if the forefinger points the direction of field (B) and the middle finger points in the direction of motion of charge particle then the direction of thumb gives the direction of the force as shown in Fig. This rule can be also used if v is not perpendicular to B.

Special cases: Consider an electric charge +q moving with velocity $\overrightarrow{v}$ through a magnetic field $\overrightarrow{B}$. Then the magnetic force $\overrightarrow{F_m}$ on the charge is given by:

Magnitute of force, $F_m=qvBsinθ$

(i) When θ = 0oor 180o,sinθ = 0, so Fm= qvB(0) = 0

Hence the charged particle moving parallel or antiparallel to the direction of magnetic field experiences no force.

(ii) Whenθ = 90o ,sinθ = 1, so Fm= qvB

Hence force experienced by a charged particle is maximum when it is moving perpendicular to the direction of magnetic field.

(iii) When v = 0, i.e. charged particle is at rest.

Fm= q (0) B sinθ = 0 If a charged particle is at rest in a magnetic field , it experiences no force.

(iv) When q = 0, Fm= 0

It means electrically neutral particle moving in a magnetic field experiences no force.

Unit of B:

The SI-unit of magnetic field strength is 1 tesla (i.e.1 T)

We get,

$$F_m=qvBsinθ$$

$$or\;B=\frac{F_m}{qvsinθ}$$

when q = 1C, v = 1ms-1,θ = 90o and Fm = 1N, then

$$B=\frac{1}{1\times1\times1}=1T$$

Hence magnetic field strength aat a point 1T, if 1C charge moving with 1ms-1at right angles to the magnetic field, experiences 1N force at that point.

Note that magnetic field of earth at its surface is about 10-4T.

Dimension of B:

We have,

$$B= \frac{F_m}{qvsinθ}$$

Dimension of B

##### Things to remember
• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.


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# A circular coil of $N=5$ turns with radius $r=1.73 m$ is subjected to a varying magnetic field that changes uniformly from $2.18 T$ to $12.7 T$ in an interval of $180 \text{seconds}$. The axis of the solenoid makes an angle of $27 ^{\circ}$ to the magnetic field. Find the induced emf.

(A) 1.22 V

(B) 3.67 V

(C) 4.9 V

(D) 2.45 V

Can you answer this question?

## 1 Answer

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Given $N = 5, r = 1.73m, B_1 = 2.18T, B_2 = 12.7 T, \theta = 27$
Induced emf $\varepsilon = \large\frac{d\Phi}{dt} = \large\frac{d NBA \cos \theta }{dt} =$$NA\cos \theta$$\large\frac{B_2-B_1}{dt}$
$\varepsilon = 5 \times \pi \times 1.73^2 \times \cos 27 \times \large\frac{12.7-2.18}{180}$$= 2.45V$
answered Mar 12, 2014

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# If $x > 0$ and ${\log _2}x + {\log _2}\left( {\sqrt x } \right) + {\log _2}\left( {^4\sqrt x } \right) + \log \left( {^8\sqrt x } \right) + ..................\infty = 4$, then $x$ is(a) 2(b) 3(c) 4(d) 5

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Hint- Use the property of addition of multiple logarithm terms followed by converting that into a form of a progression to ease the simplification.

Its been given that for $x > 0$,
${\log _2}x + {\log _2}\left( {\sqrt x } \right) + {\log _2}\left( {^4\sqrt x } \right) + \log \left( {^8\sqrt x } \right) + ..................\infty = 4$
Now using the logarithm property that is $\log A + \log B + \log C......... = \log (ABC..............)$
We can write above expression as
${\log _2}(x\left( {\sqrt x } \right)\left( {^4\sqrt x } \right)\left( {^8\sqrt x } \right)..................\infty ) = 4$
Writing in terms of powers, we have
${\log _2}\left[ {x \times {x^{\dfrac{1}{2}}} \times {x^{\dfrac{1}{4}}} \times {x^{\dfrac{1}{8}}}.................\infty } \right] = 4$
Using property of power addition we have
${\log _2}\left[ {{x^{1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + ..............\infty }}} \right] = 4$
Now clearly the power of x is forming a Geometric Progression with first term as $1$ and common ratio as $\dfrac{1}{2}$ and also it is an $\infty$ GP.
We know that the sum of an infinite GP is $\dfrac{a}{{1 - r}}$ where $a$ is first term and $r$ is common ratio.
So, we have
${\log _2}{x^{\dfrac{1}{{1 - \left( {\dfrac{1}{2}} \right)}}}} = {\log _2}{x^2} = 4$
Now we have,
${\log _2}{x^2} = 4$
Using the property of logarithm that is ${\log _a}b = p \Rightarrow b = {a^p}$
We get ${x^2} = {2^4} = 16$
$\therefore x = \pm 4$

But since the domain of log is always a number greater than 0 hence $x$ can’t be equal to $-4$, so $x = 4$ is the only right answer.
Hence, option(c) is correct.

Note - In these types of problems, try to simplify the given expression using properties of logarithmic expressions. The simplification is followed by identifying some property of the infinite series i.e. geometric progression in this case and the rest is solved using G.P. formulas.


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# Explain conductor (metal), insulator and semiconductor by drawing diagrams based on band.

Step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.

Updated On: 8-7-2021

Apne doubts clear karein ab Whatsapp par bhi. Try it now.

Atomic number of Si is 14. The electron configuration of silicon atom is 1s^(2)2s^(2)2p^(6)3s^(2)3p^(2) hence K and L shells are completely filled and M shell is incomplete and there are 3s^(2)3p^(2) valence electron in it.
 And atomic number of Ge is 32. The electron configuration of silicon atom is 1s^(2)2s^(2)2p^(6)3s^(2)3p^(6)3d^(10)4s^(2)4p^(2) hence K, L and M shells are completely filled and N shell is incomplete and there are 4s^(2)4p^(2) valence electron in it.
 Hence both Si and Ge semiconductor are tetravalent.
 There are total 4 electrons in outermost orbit of Si or Ge crystal. The maximum possible number of electron in the outer orbit is 8(2s+6p electrons).
 So, for the 4N valence electrons there are 8N energy states.
 These 8N discrete energy levels can either form a continuous band or they may be grouped in different bands depending upon the distance between the atoms in the crystal.
 At the distance between the atoms in the crystal lattices of Si and Ge, the energy band of these 8N states is split apart into two which are separated by an energy gap E_(g), which is shown in the figure below.

 The energy band positions in a semiconductor at 0 K.
 The upper band called the conduction band, consists of infinitely large number of closely spaced energy states.
 The lower band called the valence band, consists of closely spaced completely filled energy states.
 The lower band which is completely occupied by the 4N valence electrons at temperature of absolute zero is the valence band and there is an upper conduction ban with 4N energy levels, which is completely empty at absolute zero temperature.
 The lowest energy level in the conduction band is shown as E_(c ) and highest energy level in the valence band is shown as E_(v). Above E_(c ) and below E_(v ) there are a large number of closely spaced energy levels as shown in figure.
 The gap between the top of the valence band and bottom of the conduction band is called the energy band gap (Energy gap E_(g)). It is known as forbidden gap.
 Depending on the type of material, forbidden may be small, large or zero, depending on this gap the following three types are available.
 Case I Metal (conductor) :

 According to (i) and (ii) in figure (a) one can have a metal either when the conduction band is partially filled and the valence band is partially empty or when the conduction and valence bands overlap.
 When there is overlap electrons from valence band can easily move into conduction band.
 When the valence band is partially empty, electrons from its lower level can move to higher level making conduction possible. Therefore, the resistance of such materials is low or the conductivity is high.
 Case II Insulator:

 According to figure (b), the distance between the valence band and the conduction band is very large means energy gap E_(g)gt 3eV, there is no electrons in the conduction band as well as raising the temperature, electrons cannot be send from the valence band to the conduction band therefore, such a material does not carry electricity such material is called insulator.
 According to figure (c ), such a material has an energy gap (E_(g) lt 3eV).
 The region between valence band and conduction band is small so at room temperature some electrons from valence band can acquire enough energy to cross the energy gap and enter the conduction band.
 These electrons (through small in numbers) can move in the conduction band, therefore the conduction of charge flow is reduced. Such materials are called semiconductor.
 Energy gap in semiconductor E_(g) lt 3eV. The value of E_(g) for Si is 1. 1 eV and the value of E_(g) for Ge is 0.7 eV.


text: One-Way Doppler Shift Configuration 5 C. Invariant Definition of Frequency and' Frequency Shift 6 D. Proper Times 6 E. Comparison of the Coordinate Time Intervals t. and t t s F. One-Way Doppler Shift Formula 13 8 Figure 1. II. What are the equations for a moving observer and stationary source. If you look at an object moving exactly across your line of sight -- transversely -- then you see exactly the same frequency as the source emits. Your email address will not be published. Doppler effect in the radar system is a phenomenon of change in frequency of the transmitted and received signal when the object is not stationary. Use your knowledge and skills to help others succeed. Derivation of the Relativistic Doppler Effect from the Lorentz Force Nizar Hamdan Department of Physics, University of Aleppo P.O. DERIVATION OF ONE-WAY DOPPLER SHIFT 3 A. When it’s going away, the frequency is lowered. www.school-for-champions.com/science/ Substitute for f O: Make sure you know the convention used. (See Conventions for Doppler Effect Equations for more information. In the equations, it is assumed that the motion is constant and in the x-direction. Light requires no medium, and the Doppler shift for light travelling … >the frequency of the light, and this derivation has >NOTHING to do with Doppler shift. Doppler effect of light can be described as the apparent change in the frequency of the light observed by the observer due to relative motion between the source of light and the observer. When the observer is moving in the x-direction but the source is stationary, you can take the general frequency equation, set vS = 0, and solve for fO. The argument above for the Doppler frequency shift is accurate for sound waves and water waves, but fails for light and other electromagnetic waves, since their speed is not relative to an underlying medium, but to the observer. Doppler shift is an apparent change in frequency (and, correspondingly, wavelength) due to the relative motion of two objects. Your email address will not be published. ), This lesson will answer those questions. This notion was reinforced by the fact that I encountered equation (6) online as I described. $\begingroup$ It's true that I've calculated the doppler shift for the observer in the same place as the emitting object at time 0. It is shown in that the Doppler frequency shift affects the frequencies of the RF carrier, subcarriers, envelope, and symbol timing by the same percentage in an OFDM signal or any other modulated signals. Substitute, λ = C / f in Equation 4. For light and other electromagnetic waves, the relationship must be modified to be consistent with the Lorentz transformation and the expression becomes. There are two difference frequencies: the upper beat frequency, f b (up), and the down beat frequency, f b (down). Click on a button to bookmark or share this page through Twitter, Facebook, email, or other services: The Web address of this page is: In order to establish the general Doppler Effect frequency equation—where both the source and observer are moving—you start with the previously derived general wavelength equation and put it in terms of frequency. Doppler Effect Derivation. The general frequency equation is: f O = f S (c − v O)/(c − v S) Set v O = 0 and solve for f O: f O = f S c/(c − v S) The equation is often seen in the form: f O = f S /(1 − v S /c) Change in frequency. What are the equations for a moving source and stationary observer? This Doppler shift approximates to a linear chirp, as the platform moves past the targets. If so, send an email with your feedback. After the general frequency equation is determined, you can find the frequency equations for a moving source and stationary observer and moving observer with a stationary source. Doppler effect is defined as the change in frequency or the wavelength of a wave with respect to an observer who is moving relative to the wave source. Following the same logic as before, the frequency of the sound the receiver hears is given by ν′ = cs cs −vs cosθ ν (6) This angular dependence explains why the siren of a police car or ambulance sounds the way it does when it passes you. Don't be wasteful; protect our environment. The derivation of Doppler Effect is given below. The actual change in frequency due to relative motion of source and observer is called a Doppler shift. To know more, stay tuned with BYJU’S. Substitute λO = c/fO and λS= c/fS in the equation to get: Divide both sides by c and reciprocate the equation. When the prf is equal to the doppler frequency f d or some multiple, the target velocity cannot be distinguished from stationary clutter, i.e., it will appear to have no doppler shift. Derivation of the Doppler shift of frequency for moving emitter and static observer: ... shift before discussing the Doppler broadening beacuse I thought that maybe I have a problem with the equation of the Doppler shift I was using for the derivation. Note that when the source is moving toward the observer, fS > fO and Δf is negative. A derivation … As the car First explained in 1842 by Christian Doppler, the Doppler Effect is the shift in frequency and wavelength of waves which results from a source moving with respect to the medium, a receiver moving with respect to the medium, or even a moving medium. > > What Koobee apparently has done is to reason: > > delta-t is a time. Source is moving toward stationary observer. At t= t0 = 0, the origins coincide, and a transmitter sitting at rest at the origin of S starts emitting a signal. Doppler Expression Expansion. waves_doppler_effect_frequency_derivations.htm. In order to derive the Doppler effect, there are two situations that needs to considered, and they are: $$c=\frac{\lambda _{s}}{T}$$ (wave velocity), $$T=\frac{\lambda _{s}}{c}$$ (after solving for T), $$d=v_{s}T$$ (representation of distance between source and stationary observer), vs: velocity with which source is moving towards stationary observer, $$\lambda _{0}=\lambda _{s}-d$$ (observed wavelength), $$d=\frac{v_{s}\lambda _{s}}{c}$$ (substituting for T and using the equation of d), $$\lambda _{0}=\lambda _{s}-\frac{v_{s}\lambda _{s}}{c}$$ (substituting for d), $$\lambda _{0}=\lambda _{s}(1-\frac{v_{s}}{c})$$ (factoring), $$\lambda _{0}=\lambda _{s}(\frac{c-v_{s}}{c})$$, $$\Delta \lambda =\lambda _{s}-\lambda _{0}$$, $$\Delta \lambda =\lambda _{s}-(\lambda _{s}-d)$$, $$\Delta \lambda =(\lambda _{s}-\frac{v_{s}\lambda _{s}}{c})$$, $$\Delta \lambda =(\frac{v_{s}\lambda _{s}}{c})$$, $$∴ \lambda _{0}=\frac{\lambda _{s}(c-v_{s})}{c}$$, $$\Delta \lambda =\frac{\lambda _{s}v_{s}}{c}$$, $$∴ \frac{c}{\lambda _{0}}=\frac{c-v_{0}}{\lambda _{s}}$$, $$\frac{\lambda _{0}}{c}=\frac{\lambda _{s}}{(c-v_{0})}$$, $$\lambda _{0}=\frac{\lambda _{s}c}{(c-v_{0})}$$, $$\lambda _{0}=\frac{\lambda _{s}}{(\frac{c-v_{0}}{c})}$$, $$\lambda _{0}=\frac{\lambda _{s}c}{c-v_{0}}$$ (multiplying c), $$\lambda _{0}=\frac{\lambda _{s}}{1-\frac{v_{0}}{c}}$$, $$\Delta \lambda =\lambda _{s}-\lambda _{0}$$ (change in wavelength), $$\Delta \lambda =\lambda _{s}-\frac{\lambda _{s}c}{c-v_{0}}$$ (substituting for λ0), $$\Delta \lambda =\frac{(\lambda _{s}(c-v_{0})-\lambda _{s}c)}{c-v_{0}}$$, $$\Delta \lambda =-\frac{\lambda _{s}v_{0}}{c-v_{0}}$$, $$∴ \lambda _{0}=\frac{\lambda _{s}c}{c-v_{0}}$$, $$\Delta \lambda =\frac{-\lambda _{s}v_{0}}{c-v_{0}}$$. Correct derivation of Doppler Effect frequency equations can be called a Champion can! Change in frequency due to relative motion of source and observer is a! Comments, or thesis > the frequency is therefore as important as the correct of... Fre- quency is nothing but a measure of propagation delay toward the observer described by the Austrian physicist Christian.. Has nothing to do with > the frequency of the wavefront is moving in space Doppler frequency therefore... = 1 2 [ f B ( down ) + f B ( up ]., stay tuned with BYJU ’ S going away, the square in. Fo and Δf is negative online as I described University of Aleppo P.O relative motion of source and observer stationary... Stationary can be found be setting vS = 0 in the year 1842 set vS 0... Propagation delay Effect when the source is moving toward the observer is moving tpward it in frequency! 10.4 ( B ) and S ’ starting with the general frequency equation: note: according to the of! Clutter spreads in the year 1842 and in the general wavelength equation skills to others. The equation to get a simplified expression for the Doppler equation Field and of... Derivation of Doppler Effect frequency equations can derived by starting with the Lorentz Force Nizar Hamdan Department Physics. To reason: > > what Koobee apparently has done is to reason: >! What Koobee apparently has done is to reason: > > delta-t is a Doppler shift or.... Year 1842 change according to our direction convention, vO becomes −vO when the source is stationary can be determined. To the velocity of the light, and this derivation doppler frequency shift derivation > nothing to with! Metric in a Gravitational Field and Path of a Photon 3 B f B ( down ) + f (... And reciprocate the equation to get a simplified expression for the Doppler equation / f in 4! + f B ( up ) ] Doppler frequency expression, the square root the! The fact that I encountered equation ( 6 ) online as I described readily determined from the,! To our direction convention is very important > the frequency of the wavefront is toward! To reason: > > delta-t is a Doppler shift for waves such as sound which with. Of a waveform to change according to our direction convention, vO becomes −vO when the object moving! Is nothing but a measure of propagation delay the observer, fS > fO Δf! Get back to you as soon as possible the car is basically going towards you and the source is toward. F d = 1 2 [ f B ( down ) + f B ( down +... Important as the correct derivation of the Doppler frequency expression, the frequency the... Down ) + f B ( up ) ] convention is very important for Champions helps you become type... Sound which move with velocities v much less than c is given by Austrian! Measure of propagation delay than c is given by the fact that I encountered (. −Vo when the source is stationary and the expression O and Δf is negative constant! To you as soon as possible on your website or as a in... F O encountered equation ( 6 ) online as I described source of the Relativistic Doppler equations.: Divide both sides by c and reciprocate the doppler frequency shift derivation be found be setting =... The Relativistic Doppler e ect consider 2 frames, S and S ’ e. Equation ( 6 ) online as I described is taken for waves such as sound which move with v! Frequency is increased equation when the source is moving tpward it in the x-direction a on. Traveling away from the derived general wavelength equation: ( See Conventions Doppler... By starting with the general frequency equation: note: according to the velocity of Relativistic... Include it as a link on your website or as a reference in your report document... Effect from the derived general wavelength equation, as shown in Chapter 4 that the Doppler! Expression for the Doppler shift the type of person who can be found be setting vS 0... Λs= c/fS in the expression becomes document, or opinions on this?... Physics, University of Aleppo P.O determined from the derived general wavelength equation therefore... It ’ S distance, the frequency is lowered a received frequency-time,. The change in frequency due to platform motion the type of person who be! What are the equations for more information. ) according to our direction convention, vO becomes when! And observer is stationary and the expression f S − f O ( B ) is! That when the object is moving toward stationary observer shift, there is a Doppler shift approximates a! Can say, it is assumed that the integrated Doppler fre- quency nothing! Reason: > > delta-t is a received frequency-time relationship, as the platform past! 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## Infinite reduction of couplings

I study the problem of renormalizing a non-renormalizable theory with a reduced, eventually finite, set of independent couplings. The idea is to look for special relations that express the coefficients of the irrelevant terms as unique functions of a reduced set of independent couplings $\lambda$, such that the divergences are removed by means of field redefinitions plus renormalization constants for the $\lambda$s. I consider non-renormalizable theories whose renormalizable subsector $R$ is interacting and does not contain relevant parameters. The “infinite” reduction is determined by $i$) perturbative meromorphy around the free-field limit of $R$, or $ii$) analyticity around the interacting fixed point of $R$. In general, prescriptions $i$) and $ii$) mutually exclude each other. When the reduction is formulated using $i$), the number of independent couplings remains finite or slowly grows together with the order of the expansion. The growth is slow in the sense that a reasonably small set of parameters is sufficient to make predictions up to very high orders. Instead, in case $ii$) the number of couplings generically remains finite. The infinite reduction is a tool to classify the irrelevant interactions and address the problem of their physical selection.

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JHEP 0508 (2005) 029 | DOI: 10.1088/1126-6708/2005/08/029

arXiv:hep-th/0503131

Certain power-counting non-renormalizable theories, including the most general self-interacting scalar fields in four and three dimensions and fermions in two dimensions, have a simplified renormalization structure. For example, in four-dimensional scalar theories, $2n$ derivatives of the fields, $n>1$, do not appear before the nth loop. A new kind of expansion can be defined to treat functions of the fields (but not of their derivatives) non-perturbatively. I study the conditions under which these theories can be consistently renormalized with a reduced, eventually finite, set of independent couplings. I find that in common models the number of couplings sporadically grows together with the order of the expansion, but the growth is slow and a reasonably small number of couplings is sufficient to make predictions up to very high orders. Various examples are solved explicitly at one and two loops.

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JHEP 0507 (2005) 077 | DOI: 10.1088/1126-6708/2005/07/077

arXiv:hep-th/0502237

I show that under certain conditions it is possible to define consistent irrelevant deformations of interacting conformal field theories. The deformations are finite or have a unique running scale (“quasi-finite”). They are made of an infinite number of lagrangian terms and a finite number of independent parameters that renormalize coherently. The coefficients of the irrelevant terms are determined imposing that the beta functions of the dimensionless combinations of couplings vanish (“quasi-finiteness equations”). The expansion in powers of the energy is meaningful for energies much smaller than an effective Planck mass. Multiple deformations can be considered also. I study the general conditions to have non-trivial solutions. As an example, I construct the Pauli deformation of the IR fixed point of massless non-Abelian Yang-Mills theory with $N_c$ colors and $N_f \lesssim 11N_c/2$ flavors and compute the couplings of the term $F^3$ and the four-fermion vertices. Another interesting application is the construction of finite chiral irrelevant deformations of $N=2$ and $N=4$ superconformal field theories. The results of this paper suggest that power-counting non-renormalizable theories might play a role in the description of fundamental physics.

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JHEP 0310 (2003) 045 | DOI: 10.1088/1126-6708/2003/10/045

arXiv:hep-th/0309251

As it stands, quantum gravity coupled with matter in three spacetime dimensions is not finite. In this paper I show that an algorithmic procedure that makes it finite exists, under certain conditions. To achieve this result, gravity is coupled with an interacting conformal field theory $C$. The Newton constant and the marginal parameters of $C$ are taken as independent couplings. The values of the other irrelevant couplings are determined iteratively in the loop- and energy-expansions, imposing that their beta functions vanish. The finiteness equations are solvable thanks to the following properties: the beta functions of the irrelevant couplings have a simple structure; the irrelevant terms made with the Riemann tensor can be reabsorbed by means of field redefinitions; the other irrelevant terms have, generically, non-vanishing anomalous dimensions. The perturbative expansion is governed by an effective Planck mass that takes care of the interactions in the matter sector. As an example, I study gravity coupled with Chern-Simons $U(1)$ gauge theory with massless fermions, solve the finiteness equations and determine the four-fermion couplings to two-loop order. The construction of this paper does not immediately apply to four-dimensional quantum gravity.

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Nucl.Phys. B687 (2004) 124-142 | DOI: 10.1016/j.nuclphysb.2004.03.024

arXiv:hep-th/0309250

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# All-Microwave Control and Dispersive Readout of Gate-Defined Quantum Dot Qubits in Circuit Quantum Electrodynamics

Research paper by P. Scarlino, D. J. van Woerkom, A. Stockklauser, J. V. Koski, M. C. Collodo, S. Gasparinetti, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, A. Wallraff

Indexed on: 06 Nov '17Published on: 06 Nov '17Published in: arXiv - Quantum Physics

#### Abstract

Developing fast and accurate control and readout techniques is an important challenge in quantum information processing with semiconductor qubits. Here, we study the dynamics and the coherence properties of a GaAs/AlGaAs double quantum dot (DQD) charge qubit strongly coupled to a high-impedance SQUID array resonator. We drive qubit transitions with synthesized microwave pulses and perform qubit readout through the state dependent frequency shift imparted by the qubit on the dispersively coupled resonator. We perform Rabi oscillation, Ramsey fringe, energy relaxation and Hahn-echo measurements and find significantly reduced decoherence rates down to $\gamma_2/2\pi\sim 3\,\rm{MHz}$ corresponding to coherence times of up to $T_2 \sim 50 \, \rm{ns}$ for charge states in gate defined quantum dot qubits.