text:
\begin{document}

\def\a{a}
\def\ak{a^\dagger}
\def\bra#1{\langle #1|}
\def\ket#1{| #1\rangle}
\def\skal#1#2{\langle #1| #2\rangle}

\def\bran#1{\langle\widetilde{ #1|}}
\def\ketn#1{\widetilde{| #1\rangle}}

\def\norma#1{e^{-{{{\vert #1 \vert} \over 2}}^2}}
\def\normaa#1{e^{{{{\vert #1 \vert} \over 2}}^2}}
\def\tay#1#2#3{{#1^{\!^{(\!#2\!)}}\!\!(#3)}}
\def\tayf#1#2#3{{{{#1^{\!^{(\!#2\!)}}\!\!(#3)}}\over{#2!}}}
\def\pocjed#1{\begin{equation}\label{eq:#1}}
\def\jed#1{(\ref{eq:#1})}
\def\op#1{{#1}}

\title{Construction of the phase operator using logarithm of the annihilation operator}
\author{Aleksandar Petrovi\'c}
\address{University of Belgrade, Serbia.}
\ead{a.petrovic.phys@gmail.com}

\begin{abstract}
We investigate a lema that excludes existence of the phase operator and present a condition to avoid the lema. A
method for construction of an analytic function $f$ of the annihilation operator $\a$ is given. $f(z)$ is analytic
on some compact domain that does not separate the complex plane. Using these results we obtain $\ln\op a$. Since
$\lbrack\op{a}^\dagger \op{a}\ ,-i\ln\op{a}\rbrack=i$, we can use $\ln\op a$ to construct an operator $\op{\Phi}$,
which satisfies the definition of the phase operator.
\end{abstract}
\pacs{03.65.Ca, 03.65.Ta}

\section{Introduction}
The annihilation operator in Hilbert space is operator $\a$ which satisfies relation:

\pocjed{komutator}
\lbrack\a,\ak\rbrack=1.
\end{equation}
A self-adjoint number operator $N$ and the corresponding Fock states are constructed using $\a$:
\pocjed{operator_br_cestica} N=\ak\a, \ \ \ \ \  N\ket n =n\ket n ,\ \ \ n=0,1,2,3,...
\end{equation}
The phase operator is an operator conjugated to number operator:

\pocjed{def4}
\lbrack\op{N},\op{\Phi}\rbrack=i I.
\end{equation}
It is an operator which corresponds to the phase of the harmonic oscillator. Its eigenvalues are phases of the
quantum harmonic oscillator. An explicit form of this operator has not been given, and some authors have
questioned its existence \cite{carruthers},\cite{leonhardt}.
Best approach to construction of the phase operator is given by Pegg and Barnett \cite{Pegg}.
They constructed an operator and corresponding eigenfunctions which give a good approximation in
finite dimensional space. However, when dimension of space tends to $\infty$, this operator does not exists as an operator on a Hilbert space.

\noindent
Although form of the phase operator has not been given, basis of its
eigenvalues is well known \cite{levi}:

\pocjed{vektor_faze_def}
\ket{\varphi}_f = {1\over{\sqrt{2\pi}}}\sum\limits_{k=0}^\infty e^{ik\varphi} \ket{n}\ .
\end{equation}
The vectors $\ket\varphi_f$ form a non-orthogonal basis:

\pocjed{scal_proiz}
_f\skal{\varphi^\prime} \varphi _f={1\over{2}}\delta(\varphi^\prime - \varphi) +
{1\over{4\pi}}(1+i\cot{{1\over{2}}(\varphi^\prime - \varphi)}).
\end{equation}

\noindent
The $\ket\varphi_f$ vectors have a useful
property, which in fact recommends them as eigenvectors of the phase operator. Their time evolution is natural:

\pocjed{evolucija}
\e^{iNt}\ket\varphi_f=\ket{\varphi+t}_f .
\end{equation}
A recent result, giving an expression for analytic functions of the annihilation operator \cite{rad1_arxiv}, is
used here as a starting point for construction of the phase operator. Hence, for the first time we can construct a
correct expression for $\ln\op{a}$, which will then be used as a base for the further construction of the phase
operator.
\noindent Namely, an explicit construction of an operator which satisfies commutation relation
\jed{def4} and has vectors \jed{vektor_faze_def} as eigenvectors, is given.
\noindent
In section 2, we give a set
of conditions for operator $\op{\Phi}$ which circumvent the non-existence arguments. In section 3, we construct an
expression for analytic functions of the annihilation operator and discuss some of its properties. In section 4 we
construct $\ln\op{a}$. In section 5, starting from $\ln\op{a}$, we construct the phase operator $\op{\Phi}$.

\section{Existence of the phase operator}

The usual proof of non existence of the phase operator is based on reduction to contradiction \cite{leonhardt}.
Namely, assuming this operator exists, using \jed{operator_br_cestica} and \jed{def4} we obtain:

\pocjed{faza_n}
\bra{n}\lbrack\op{N},\op{\Phi}\rbrack \ket{m}=(n-m)\bra{n} \op{\Phi} \ket{m} \not = i I.
\end{equation}
\noindent However, if it is assumed that the vectors  $\ket{n}$ do
not belong to the domain of the operator $\op\Phi$, then argument
\jed{faza_n} breaks down. We see that a necessary condition for
existence of the phase operator $\op{\Phi}$ is:

\pocjed{kontra4}
\ket{n}\not\in D(\op{\Phi}),\ \ \ n=0,1,2,3,...
\end{equation}
leading to that $\op{\Phi}$ in $\ket n$ representation cannot be expressed as a matrix. If $\op{\Phi}$ exists, it
can be expressed as a product of two or more matrices. Indeed, in section 5, we will construct an operator
$\op{\Phi}$ that satisfies these conditions.

\section{Analytic functions of the annihilation operator}

In this section, we briefly describe a construction of function $f$ of the annihilation operator $\a$. $f(z)$ is
analytic on some compact domain that does not separate the complex plane. A more detailed analysis of this topic
is given in \cite{rad1_arxiv}. First, we give note about Runge's approximation theorem and a sequence of
polynomials which approximate $f(z)$ on the whole domain. Then we construct a new form of identity which is well
suited for construction of $f(\a)$. We proceed constructing $f(\a)$ and finish with an analysis of some of its
properties.

\subsection{Runge's approximation theorem}

{\bf Theorem:}  If $f$ is an analytic function on a compact domain $\Omega$ that does not separate the complex
plane, then there exists a sequence $P_l(z)$ of polynomials such that  converges uniformly to $f(z)$ on $\Omega$
\cite{saff},\cite{markushevic},\cite{alfors}:

\pocjed{runge1}
f(z)=\sum\limits_{l=0}^\infty
P_l(z-z_0)=\sum\limits_{l=0}^\infty\sum\limits_{k=0}^{d_l} c_k^{(l)}  (z-z_0)^k , c_k^{(l)}\in\mathbb{C}, z,z_0
\in \Omega.
\end{equation}
\noindent Runge's theorem  is an existence theorem, i.e. it does not give values of $c_k^{(l)}$. Functions which
can be approximated using polynomials \jed{runge1} are also $\ln z$ and $z^\lambda, \lambda\in\mathbb{R}$. In this
paper Mittag-Leffler expansion \cite{markushevic} is used to approximate function $\ln z$.

\subsection{New identity resolution}

Eigenstates $\ket\alpha$ of annihilation operator $\a$ are called coherent states:

\pocjed{koherent1}
\a\ket{\alpha} =\alpha \ket{ \alpha }  , \alpha\in\mathbb{C}.
\end{equation}
$\ket\alpha$ can be expressed in terms of Fock states $\ket{n}$:

\pocjed{koherent_preko_n}
\ket{\alpha}=\norma
\alpha\sum\limits_{n=0}^\infty {{\alpha^n}\over{\sqrt{n!}}} \ket n , \alpha\in\mathbb{C}.
\end{equation}
Non-normalized coherent states are:

\pocjed{nenormirana_stanja}
\ketn \alpha=\rme^{\alpha\ak} \ket 0 = \normaa \alpha \ket \alpha.
\end{equation}
Coherent states form an overcomplete and non-orthogonal set which spans the resolution of identity \cite{arsa}:

\pocjed{stara_jedinica}
I={1\over\pi}\int\rmd^2\alpha\ket\alpha\bra\alpha .
\end{equation}
In formal analogy with the spectral theorem one can write an entire function of the annihilation operator:

\pocjed{entire_function}
f(\a)={1\over\pi}\int\rmd^2\alpha f(\alpha)\ket\alpha\bra\alpha .
\end{equation}
However, expression \jed{entire_function} is not valid for non-entire functions \cite{arsa}, so we construct a new
identity resolution:

\pocjed{nova_jedinica1} I=-i\oint\displaylimits_{\vert\gamma\vert=R}
{{\rmd\gamma}\over{\gamma}}\ketn{\gamma+z_0}\bran\gamma\; J\;\rme^{-z_0 \ak}\ , R>0, z_0\in\mathbb{C},
\end{equation}
where

\pocjed{nova_jedinica2} J={1\over{2\pi}}\sum\limits_{n=0}^\infty{{n!}}\ket n\bra n  .
\end{equation}

\subsection{Construction of analytic functions of the annihilation operator}

Using sum \jed{runge1} and resolution of identity \jed{nova_jedinica1}, an analytic function of annihilation
operator can be constructed:

\begin{eqnarray}\label{eq:Mittag2}
f(\a) &=& \sum\limits_{l=0}^\infty\sum\limits_{k=0}^{d_l}
c_k^{(l)}
(\a-z_0)^k \cdot I\nonumber\\
&=& -i\sum\limits_{l=0}^\infty\sum\limits_{k=0}^{d_l} c_k^{(l)} \oint\displaylimits_{\vert \gamma\vert=R}\!\!\!
\rmd\gamma \gamma^{k-1} \ketn{\gamma+z_0} \bran\gamma \; J\; \rme^{-z_0\ak}.
\end{eqnarray}

\noindent
Using definition \jed {nenormirana_stanja}  of $\ketn\gamma$ and performing integration, we obtain

\pocjed{Runge3}
f(\a)=\sum\limits_{l=0}^\infty\sum\limits_{k=0}^{d_l} c_k^{(l)} \sum\limits_{n=0}^\infty
\sum\limits_{m=k}^{n+k} {n\choose m-k} \sqrt{{m!}\over{n!}} \;\; z_0^{n-m+k}   \;\;\ket n \bra m\;\rme^{-z_0 \ak},
\end{equation}

\noindent
which for $\alpha\in\Omega$ gives:

\pocjed{Mittag3a}
f(\a)\ket\alpha=f(\alpha)\ket\alpha .
\end{equation}

\noindent
We can rewrite \jed {Runge3}, collecting coefficients at dyads, as

\begin{eqnarray}\label{eq:Runge5}
f(\a) &=& \hat\chi\rme^{-z_0 \ak},\\
\ \ \ \hat\chi &=& \sum\limits_{l=0}^\infty\sum\limits_{n=0}^\infty\sum\limits_{m=0}^\infty\chi_{nm}^{(l)} \ket n
\bra m\ ,\nonumber\\
\chi_{nm}^{(l)} &=& \sqrt{{m!}\over{n!}} \sum\limits_{k=p}^{s_l} c_k^{(l)}{n\choose m-k} \; z_0^{n-m+k}\ ,\nonumber \\
\ \ \ p&=&\max\{0,m-n\},\;\;s_l=\min\{m,d_l\} .\nonumber
\end{eqnarray}

\subsection{Some properties of analytic functions of annihilation operator}

\noindent
Considering the basic relation between the annihilation and creation operator
$\lbrack\op{a},\op{a}^\dagger\rbrack=1$, one would expect

\begin{eqnarray}
\lbrack f(\op{a}),\op{a}\rbrack&=& 0,\label{eq:rel1}\\
\lbrack f(\op{a}),\op{a}^\dagger\rbrack&=&f^{'}(\op{a})\label{eq:rel2}
\end{eqnarray}

\noindent Since $f(\a)$ is not an entire function of $\a$, relations \jed{rel1} and \jed{rel2} need to be proven
directly. \noindent Relation \jed{rel1} can be calculated explicitly using \jed{Runge5}.
\noindent
To prove
relation \jed{rel2} we first need to construct $f^{\prime}(z)$ . It is easy to see that

\begin{eqnarray}\label{eq:izvod}
f^{\prime}(z)&=&\sum\limits_{l=0}^\infty\sum\limits_{k=0}^{d_l -1}\bar{c}_k^{(l)}(z-z_0)^k,\ z,z_0\in\Omega,\\
\ \ \bar{c}_k^{(l)}&=&(k+1)c_{k+1}^{(l)}.\nonumber
\end{eqnarray}

\noindent
Using \jed{izvod} and \jed{Runge5} we can prove \jed{rel2} directly.

\section{Logarithm of the annihilation operator}

In this section we construct an operator $\ln \a$, which is a good starting point for construction of the phase
operator \cite{arsa1},\cite{arsa}. A complex function $f(z)=\ln z$ is analytic on a simply connected subdomain
$\Omega$ in $\mathbb{C}$, which is obtained by making a cut in the complex plain along a ray originating at zero.
For simplicity, we chose a cut along the negative part of the $x$ axis. We also set $z_0=1$ in \jed{runge1}. Any
other choice of a cut and $z_0$ leads to an equivalent construction of $\ln \a$. \noindent As already noted, the
Runge theorem does not explicitly give coefficients $c_{k}^{(l)}$ in \jed{runge1}. A convenient method for
computing $c_k^{(l)}$ for the function $f(z)=\ln z$ is Mittag-Leffler expansion in the star \cite{markushevic}.
This method gives explicit constants in the expansion of $\ln z$:

\begin{eqnarray}\label{eq:ln0}
\ln z&=&\lim_{p \rightarrow \infty}\sum\limits_{l=1}^p\sum\limits_{k=1}^p\ c_k^{(l)}\ (z-1)^k\ ,\ z\in\Omega,\\
\ c_{k}^{(l)}&=&d_{k}^{(l)}\ {{(-1)^{(k+1)}}\over{k}}, k>0,\nonumber\\
\ d_{k}^{(l)} &=& {k!\over {l!}}\ \Gamma^k_p\ \Theta^l_p\ E^{(k)}_l \ ,\nonumber
\end{eqnarray}

\noindent
where

\begin{eqnarray}\label{eq:ln3}
\Gamma_p&=&2 H^{-2}_p,\\
\Theta_p &=& 1- e^{-{1\over{2}} H^2_p},\nonumber\\
E^{(k)}_l &=& {1\over{k!}}{{\partial^k}\over{\partial \rho ^k}} \prod_{s=0}^{l-1} (\rho + s)
\biggm\vert_{\rho=0}=c(l,k) .\nonumber
\end{eqnarray}
c(l,k) is unsigned Stirling number of the first kind. $H_p$ is any sequence satisfying the following condition:

\pocjed{ln6}
2 H_p^2\ e^{{1\over{2}} H^2_p} < p ,\ \ \ \ \  p>0.
\end{equation}

\noindent
This sequence of polynomials converges locally uniformly to $\ln z$ on $\Omega$. \noindent Using the
obtained coefficients in the expansion \jed{ln0} and relation \jed{Runge5} we can represent $\ln \op a$ as the
following limit:

\pocjed{ln7}
\ln\op{a} = \lim_{p \rightarrow
\infty}\sum\limits_{l=0}^p \sum\limits_{n=1}^p\sum\limits_{m=1}^p
\chi_{nm}^{(l)} \ket n \bra m \;\; \rme^{- \ak} ,
\end{equation}

\begin{displaymath}
\chi_{nm}^{(l)} = \sqrt{{m!}\over{n!}} \sum\limits_{k=k1}^m
c_k^{(l)} {n\choose m-k}        \;\;\; ,k1=\max\{0,m-n\} .
\end{displaymath}

\noindent
$\ln\op{a}$ has nice properties needed for construction of the phase operator \cite{arsa}. Using
\jed{rel1} and \jed{rel2} it is obvious that: \pocjed{ln9} \lbrack\op{a}^\dagger \op{a},-i \ln\op{a}\rbrack=
\op{a}^\dagger \lbrack\op{a},-i \ln\op{a}\rbrack + \lbrack\op{a}^\dagger,-i \ln\op{a}\rbrack\op{a}
={i\over{\op{a}}}\ \op{a} = i.
\end{equation}
We can conclude that $\ln \op{a}$ is conjugate to number operator,
and therefore is a good base for construction of the phase
operator.

\section{$\ln \op{a}$ and the phase operator}

\noindent
Let

\pocjed{faza1}
\op{\Phi}=-i \op{Y}^{-1} \ln\op{a} \op{Y} ,
\end{equation}
where

\pocjed{faza2} \op{Y}=\sum\limits_{k=0}^\infty{1\over{\sqrt{k!}}} \ket k \bra k .
\end{equation}
$\op{Y}$ is a diagonal operator so it is obvious

\pocjed{faza3} \lbrack\op{a}^\dagger \op{a}\ ,\op{\Phi}\rbrack=i.
\end{equation}
On the other hand, vectors \jed{vektor_faze_def} and coherent
states \jed{koherent1} can be combined:

\pocjed{faza4} \ket{\varphi}_f={e^{1\over{2}}\over{\sqrt{2\pi}}}
\op{Y}^{-1} \ket{e^{i\varphi}}
\end{equation}

\noindent Vector $\ket{\varphi}_f$ is an eigenvector of the
operator $\op{\Phi}$ in equation \jed{faza1}:

\pocjed{faza5} \op{\Phi} \ket{\varphi}_f = -i \op{Y}^{-1}
\ln\op{a} \op{Y} {e^{1\over{2}}\over{\sqrt{2\pi}}} \op{Y}^{-1}
\ket{e^{i\varphi}} = -i {e^{1\over{2}}\over{\sqrt{2\pi}}}
\op{Y}^{-1}  (i\varphi) \ket{e^{i\varphi}}=\varphi
\ket{\varphi}_f,
\end{equation}

\begin{displaymath}
\varphi\in(-\pi,\pi).
\end{displaymath}
Hence, the operator $\op{\Phi}$ is a phase operator.

\section{Appendix}

\noindent
To determine action of the operator $f(a)$ defined in \jed{Runge3}, on the vector $\ket\alpha$, we first
compute the following matrix element:

\pocjed{app1}
\bra{m} e^{-z_0 \op{a}^\dagger} \ket\alpha =
\norma\alpha \bra{m} \ketn {\alpha-z_0} =\norma\alpha
{{{(\alpha-z_0)^m}}\over{\sqrt{m!}}}
\end{equation}
Using \jed{Runge3} we can see

\begin{eqnarray}\label{eq:app2}
f(\a)\ket\alpha &=& \sum\limits_{l=0}^\infty\sum\limits_{k=0}^{d_l} c_k^{(l)} \sum\limits_{n=0}^\infty
\sum\limits_{m=k}^{n+k}{n\choose m-k}{1\over{\sqrt{n!}}}\;\; z_0^{n-m+k} \norma\alpha (\alpha-z_0)^m\;\;\ket n\nonumber\\
&=&\norma\alpha\sum\limits_{l=0}^\infty\sum\limits_{k=0}^{d_l} c_k^{(l)} (\alpha - z_0)^k \sum\limits_{n=0}^\infty
{1\over{\sqrt{n!}}} \sum\limits_{m=0}^{n} {n\choose m} z_0^{n-m} (\alpha - z_0)^m \;\;\ket n \nonumber
\end{eqnarray}
Last sum in previous equation is $\alpha^n$, and finally

\pocjed{app3}
f(\a)\ket\alpha=\norma\alpha\sum\limits_{l=0}^\infty\sum\limits_{k=0}^{d_l} c_k^{(l)} (\alpha - z_0)^k
\sum\limits_{n=0}^\infty {{(\alpha - z_0)^n}\over{\sqrt{n!}}} \;\;\ket n = f(\alpha)\ket\alpha
\end{equation}

\ack
The autor would like to thank Drs M. Arsenovi\'c, D. Arsenovi\'c, D. Davidovi\'c and J.Ajti\'c for their
suggestions and help in preparation of this manuscript.

\section*{References}

config: arxiv


TITLE: When moving something by entrainment in a fluid stream, where does the energy come from?
QUESTION [0 upvotes]: The particular Something I had in mind here would be air bubbles that are pulled downwards against their buoyancy by a stream of water falling down a shaft. The work required to push those bubbles down must decrease energy somewhere else, so where does it come from? Is the stream's velocity reduced?

REPLY [2 votes]: Entrainment works by the entraining medium giving momentum and energy to the entrained medium. The Wiki page on the injector pump should get you started.


TITLE: Goodstein's sequences and theorem.
QUESTION [4 upvotes]: I was wondering if anyone would be able to provide me with any textbooks that discuss the Goodstein sequence and the Goodstein theorem and which provide a proof also?
Thanks.

REPLY [5 votes]: A very nice introduction to this area is

MR0891258(88g:03084). Simpson, Stephen G. Unprovable theorems and fast-growing functions. In Logic and combinatorics (Arcata, Calif., 1985), 359–394, Contemp. Math., 65, Amer. Math. Soc., Providence, RI, 1987.

Simpson describes the paper as inspired by the question of whether there could be "a comprehensive, self-contained discipline of finite combinatorial mathematics". He argues that the incompleteness results he describes (the unprovability of Goodstein's theorem in Peano arithmetic being one example) indicate that the answer is no. The paper is expository and assumes almost no background from logic. For instance, it introduces ordinals and provides many intuitions for several of the results it describes. I think it is a joy to read. I actually really like the collection it appears in. Many of the papers in it (although in general much more technical) deal with similar results (unprovability of Hydra-Hercules games, variants of Kruskal's tree theorem, etc).
One of the results Simpson's presents is Goodstein's, and the paper contains a decent sketch (together with references to papers proving its unprovability in $\mathsf{PA}$).
Specifically on Goodstein's theorem, you may also enjoy reading the paper The termite and the tower by Will Sladek. Will wrote it while an undergraduate student at Caltech, as part of their Mathematical writing course; I served as a mentor through the write-up process. I think the metaphor in the title is a great way of describing Goodstein's theorem.
While Will was working on the paper, I thought it would be nice to include some numbers illustrating how long some Goodstein sequences take to terminate. This ballooned into a short paper of my own,

MR2585906(2011c:03139). Caicedo, Andrés Eduardo. Goodstein's function. Rev. Colombiana Mat. 41 (2007), no. 2, 381–391.

In the paper I present a formula for computing Goodstein's function, the map that to each $n$ assigns the number of steps that it takes the Goodstein sequence starting at $n$ to terminate.
Recall that the sequence is defined as follows: Let $n\ge2$. The base-$n$ expansion of $m\in\mathbb N$ is the unique way of writing $m$ as a sum $\sum_{j=0}^k a_j n^k$ where each $a_j$ is a base-$n$ "digit" (a number in $\{0,1,\dots,n-1\}$. The complete base-$n$ expansion is the result of iterating this representation, by also writing in base $n$ all the exponents $k$ of powers of $n$ in the expansion of $m$, and all the exponents in the expansion of these exponents, and so on.
An example I give in the paper is $m=266$. Its base-$2$ expansion is $266=2^8 + 2^3 + 2^1$. To obtain the complete base-$2$ expansion, we write $8$ as $2^3=2^{2^1+1}$, and $3$ as $2^1+1$, thus
 $$ 266= 2^{2^{2+1}} + 2^{2+1} + 2 $$ where, for clarity, I am just writing $2$ instead of $2^1$ throughout.
The change of base function $R_n$ assigns to each $m\in\mathbb N$ the result of replacing with $n+1$ each $n$ in the complete base-$n$ expansion of $m$.
For instance, $$ R_2(266)=3^{3^{3+1}}+3^{3+1}+3=443426488243037769948249630619149892887. $$
The Goodstein sequence starting at $m$ is the sequence $(m)_k$ where $(m)_1=m$ and $$(m)_{k+1}=\begin{cases} R_{k+1}((m)_k)-1&\text{if }(m)_k>0,\\0&\text{if }(m)_k=0.\end{cases}$$
That the sequence terminates means that $(m)_k=0$ for some $k$. Goodstein's theorem is the statement that the sequence terminates for any $m$, and Goodstein's function $\mathcal G$ assigns to $m$ the least $k$ such that $(m)_k=0$.
The "tower" in Will's paper refers to the stack of exponentials in the complete base-$n$ representation of a number. The "termite" is the $-1$, the small bit we chip away at each step. The usual proof of Goodstein's theorem
replaces at each step $k$ each $k$ in the complete base-$k$ representation of the number with an $\omega$, thus associating to each step in the sequence an ordinal number written in Cantor normal form. What one checks is that these ordinals are decreasing. For example, the sequence for $m=4$ is
$(4)_1=4=2^2$, $(4)_2=26=3^3-1=2\cdot 3^2+2\cdot 3+2$, $(4)_3=41=2\cdot 4^2+2\cdot 4+1$, $(4)_4=60=2\cdot 5^2+2\cdot 5$, $(4)_5=83=2\cdot 6^2+6+5$, etc. The associated sequence of ordinals is $$\omega^\omega>2\cdot \omega^2+2\cdot \omega+2>2\cdot\omega^2+2\cdot \omega+1>2\cdot\omega^2+2\cdot \omega>2\cdot\omega^2+\omega+5>\dots$$
The sequence keeps going for a long time indeed, but it eventually reaches zero, with
 $$ \mathcal G(4)=3\cdot 2^{402653211}-2.$$
The formula I obtain verifies that Goodstein's function $\mathcal G$ is fast growing, in this case meaning that it grows so quickly that given any function $f\!:\mathbb N\to\mathbb N$ provably total in Peano arithmetic, there is a $k$ such that for all $n\ge k$, $\mathcal G(n)>f(n)$.


TITLE: Use calculus to find minimum value of $S$
QUESTION [0 upvotes]: $S = x^2/2 + (16000√2)/x$
I know I have to find the derivative and get the stationary points, then use the second derivative to find the one which gives a minimum value and then put that x value back into S to find the minimum value.
However, it seems like there are some issues with how I am differentiating because I keep getting absurd answers which I won't waste time outlining..
any advice?
Cheers

REPLY [1 votes]: Here is an alternative answer using AM-GM:
When $x < 0$, $f'(x) = x - \frac{16000 \sqrt2}{x^2}$ is always negative, so there is no local minimum there.
Therefore if the local minimum exists, it must exist when $x > 0$. Using AM-GM we have:
$$\frac{x^2}{2} + \frac{8000 \sqrt{2}}{x} + \frac{8000 \sqrt{2}}{x} ≥ 3 \cdot \sqrt[3]{\frac{x^2}{2} \cdot \frac{8000 \sqrt{2}}{x} \cdot \frac{8000 \sqrt{2}}{x}}$$
$$≥ 3 \sqrt[3] {64 \ 000 \ 000}$$
$$≥ 1200$$


TITLE: Analog of a discrete fibration for sheafs?
QUESTION [0 upvotes]: Given a discrete fibration $ P : \text{Dis}(C^{\text{op}}) $ you can map to a copresheaf.
$$ [[P]](x) = \Sigma s, C(P(s), x)$$
What is the equivalent if you want to map to a cosheaf on a site $(C, J)$?
I'm not so good with sheafs so I find this confusing.
I have a feeling you want a subset of discrete fibrations such that
$$ \forall s, \text{id}_{P(s)} \in J(P(s)) $$
You're thinking of discrete fibrations as mapping identity maps to identity maps.
I am not quite sure what you want for mapping to a sheaf either. I think you want something like:
$$ [[P]](x) = \Sigma s, \{ f : C(P(s), x) | f \in J(s) \}$$
But this seems off to me.
There should also be an analogous operation for "sheafification" mapping discrete fibrations to the subset of fibrations.
I think mostly I'm looking for a name I can google to lookup more information.

REPLY [1 votes]: $\def\C{\mathcal{C}}\def\DFib{\text{DFib}}$I will try to sketch a possible course of action for addressing this problem. The story can be told from different perspectives at different levels of generality (I forgot most of them, so I was hoping for some other user to answer before me...).
The moral of this story is that

Under the equivalence
$$ \Gamma : [\C^\text{op}, {\sf Set}] \cong \DFib(\C) : \Delta$$
sheaves correspond to those discrete fibrations that are stacks, i.e. such that two sets (the set of "descent data" and the set of "effective descent data") are in bijection.

In a sense, the conditions that define discrete fibration "satisfying descent" (i.e. those for which the above bijection exists) are just the sheaf conditions passed under the mirror of the equivalence above: a presheaf $F$ has an associated discrete fibration $\Gamma F$ (its category of elements, where an object is a pair $(U, s\in FU)$) over $\C$, and $F$ is a sheaf if and only if for every $U$ and every cover of $U$, $F$ fits into an equaliser diagram
$$\textstyle
FU \to \prod_i FU_i \rightrightarrows \prod_{i,j} FU_{ij}
$$
so $\Gamma F$ will "satisfy descent" if it satisfies a similar property. In order to find which property is this, you have to understand what $FU, FU_i$ and $FU_{ij}$ are in terms of $\Gamma F$ (and the parallel maps induced by restriction of sections, of course).


TITLE: A convergence in sobolev spaces involving time
QUESTION [1 upvotes]: Suppose $\Omega $ is bounded and
$$u^m\xrightarrow[]{w^*} u \quad \text{in} \,\, L^{\infty} (\mathbb{R}^+,H^2(\Omega)\cap H_0^1(\Omega))$$
and
$$u^m_t\xrightarrow[]{w^*} u_t \quad \text{in} \,\,L^\infty (\mathbb{R}^+,L^2(\Omega))\cap L^2 (\mathbb{R}^+,H_0^1(\Omega))$$
I want to prove
$$u^m\xrightarrow[]{strongly} u \quad \text{in} \,\,L^2 ((0,T),H_0^1(\Omega))$$
It is used in some papers and non of them have a proof. I know it comes from Aubin-Lions Lemma but the most general lemma I found doesn't work. The lemma works If we show:
$$u^m\xrightarrow[]{w^*} u \quad \text{in} \,\,L^2 ((0,T),H^2(\Omega)\cap H_0^1(\Omega))$$
But how? I will be thankful for any hint.

REPLY [2 votes]: You already have $u^m \to u$ weak-* in $L^\infty(\mathbb{R}^+; H_0^1(\Omega))$. Hence, $u^m \to u$ weak-* in $L^\infty(0,T; H_0^1(\Omega))$. And this gives $u^m \to u$ weak in $L^2(0,T; H_0^1(\Omega))$. No need for Aubin-Lions here.


TITLE: Why is the electric field highest in regions of highest resistance?
QUESTION [4 upvotes]: I know Kirchhoff's law and that current must be constant. But what exactly is the phenomenon, from first principles, that causes the electric field to be stronger in a region with lower carrier mobility/concentration.
I thought it might be repulsion from electrons  that pushes the electrons in front ahead (as charge instantaneously accumulates) but I can't really wrap my head around how exactly more work is done to move a charge through a higher resistance.

REPLY [2 votes]: $E = - \frac{\mathrm dV}{\mathrm dx}$
means that when voltage changes rapidly with distance inside the resistor, the electric field will be high.  You could have two, cylindrical, 1000 ohm resistors, one of which is 5 mm long and the other is 50 mm long.  If they are in the same circuit (carrying the same current), the 5 mm resistor will contain the higher electric field because the voltage changes more rapidly across it.


TITLE: Exercise, measure theory
QUESTION [3 upvotes]: I need help with this exercise:

This exercise is in a chapter where I learn the monotone convergence theorem and Fatous lemma, so I assume I shall use them.
Since $\textbf{1}_Ef_n\rightarrow \textbf{1}_Ef$, this exercise would have been very easy if the sequences were increasing, sadly they are not.
Using Fatou's lemma I get:
$\liminf_{n \rightarrow \infty} \int_Ef_nd\mu \ge \int\liminf_{n \rightarrow \infty}\textbf{1}_Ef_n d\mu=\int_Ef d\mu$.
If I could get another with lim sup, I would maybe be able to be done.
Another problem is that since the $f_n$'s may not be increasing, I do not really know that:
$lim_{n \rightarrow \infty}\int_Ef_nd\mu$ even exist?, because this sequence may not even be monotone?
I guess I should use the part where the entire integral is less than infinity. But I do not really see how to use it. I guess this tells me that the function is finite a.e., but how does that help?

REPLY [3 votes]: Hint:
Replace $f_n$ by a seuqence converging to $f$, that is also dominated by $f$. For example take $g_n= \min (f_n,f)= \frac{f_n+f-|f_n-f|}{2}$.
Then use dominated convergence, and the fact that $f,f_n\geq 0$.
Your goal is to show that $\|f_n-f\|_1\to 0$, and then the claim will follow.

REPLY [1 votes]: You use the part about infinity to say that
$$
\int_E fd\mu =\int f d\mu - \int_{E^C} f d\mu
$$
(the same holds for the $f_n$ similarly, at least from a certain index)
and this, repeating the same argument with fatou as you did in the question with $E^C$, gives you
$$
\liminf_{n\to\infty} \int_{E^C} f_n d\mu \geq \int_{E^C} f d\mu
$$
then you get
$$
\liminf_{n\to\infty} \int_{E} f_n d\mu= \int f d\mu - \liminf_{n\to\infty} \int_{E^C} f_n d\mu \leq \int f d\mu - \int_{E^C} f d\mu = \int_{E} f d\mu.
$$
I'm a bit rusty on the properties of liminf, does this convince you?