text:
\begin{document}

\title{The Hochschild cohomology ring of the standard
Podle\'s quantum sphere}

\author{Ulrich Kr\"ahmer}

\address{University Gardens, Glasgow G12 8QW, UK}

\email{ukraehmer@maths.gla.ac.uk}

\begin{abstract}
The cup and cap product in twisted
Hochschild
(co)\-ho\-mo\-lo\-gy is computed
for the standard quantum
2-sphere and used
to construct a cyclic
2-cocycle that represents the
fundamental Hochschild class.
\end{abstract}
\maketitle

\section{Introduction}
The aim of this
article is to compute for the quantised
coordinate ring $A=\mathbb{C}_q[S^2]$
(the standard Podle\'s quantum 2-sphere) the
cup and cap product
\begin{eqnarray}
		  \smallsmile
&:& H^m(A,{}_\sigma A) \otimes
		  H^n(A,{}_\tau A) \rightarrow
		  H^{m+n}(A,{}_{\tau \circ \sigma} A),\nonumber\\
		  \smallfrown
&:& H_n(A,{}_\sigma A) \otimes
		  H^m(A,{}_\tau A) \rightarrow
		  H_{n-m} (A,{}_{\tau \circ \sigma}
		  A)\nonumber
\end{eqnarray}
in the Hochschild (co)homology
of $A$ with coefficients in the bimodules
${}_\sigma A$ that arise
by twisting the canonical bimodule
structure of $A$
by $ \sigma \in \mathrm{Aut} (A)$.
In \cite{tomuli2} we carried out similar
computations for $\mathbb{C}_q[SU(2)]$.
For $\mathbb{C}_q[S^2]$
they become much simpler
and their conceptual content that was
somewhat hidden in
\cite{tomuli2} between lengthy
computations becomes more transparent.

For
the coordinate ring of a smooth variety
$X$, the
Hochschild-Kostant-Rosenberg
theorem identifies
$\Lambda(A):=
(\bigoplus_{n \ge 0} H^n(A,A),\smallsmile)$
with the algebra of multivector fields on $X$,
and $\bigoplus_{n \ge 0} H_n(A,A)$
as a $\Lambda(A)$-module (via $\smallfrown$)
with the differential forms $ \Omega (X)$ on $X$.
For noncommutative algebras,
$\Lambda(A)$ tends to be
fairly degenerate. However, twisting by
$ \sigma $
allows one to consider richer cohomology
rings that encode more
information about $A$.

For example, the
generators of the Drinfeld-Jimbo quantisation
of the Lie algebra of $SU(2)$
act via twisted rather than usual derivations
on $A=\mathbb{C}_q[S^2]$.
These give rise to
two cohomology classes
$[\partial_{\pm 1}] \in
H^1(A,{}_{\sigma^{-1}_\mathrm{mod}} A)$,
where $\sm$ is Woronowicz's
modular automorphism determined
for example by (\ref{podushka}) below.
We will see that
they behave under the cup product
similar to the corresponding
classical $SU(2)$-invariant
vector fields on
$S^2=SU(2)/U(1)$,
$$
		  [\partial_1] \smallsmile [\partial_1]=
		  [\partial_1] \smallsmile [\partial_{-1}]+
		  q^2 [\partial_{-1}] \smallsmile [\partial_1]=
		  [\partial_{-1}] \smallsmile [\partial_{-1}]=0,
$$
and use them to define
a functional
on $H_2(A,{}_\sm A) \simeq \mathbb{C}$
of the form
\begin{equation}\label{blinde}
		  \varphi ([\omega]):=
		  q^{-1} \int [\omega] \smallfrown
  		  ([\partial_1] \smallsmile [\partial_{-1}])
		  \in \mathbb{C}.
\end{equation}
Here $[\omega] \in H_2(A,{}_\sm A)$
is acted on by
$[\partial_1] \smallsmile [\partial_{-1}]
\in H^2(A,{}_{\sigma^{-2}_\mathrm{mod}} A)$
to produce a class in
$H_0(A,{}_{\sigma^{-1}_\mathrm{mod}} A)$,
and then one applies a certain twisted
trace $\int \in (H_0(A,{}_\sigma A))^*$
in order to obtain a numerical invariant of
$[\omega]$.

The functional $\varphi$
provides a dual
description of the fundamental class
$[\d A] \in H_2(A,{}_\sm A)$ that corresponds
under the Poincar\'e-type duality \cite{ulineu}
\begin{equation}\label{podushka}
		  H_n(A,{}_\sm A) \simeq H^{2-n}(A,A)
\end{equation}
to
$1 \in H^0(A,A)$.
From the practical point of view, one can
use $\varphi$ to determine the homology class
of a given Hochschild cycle, and
for $\mathbb{C}_q[SU(2)]$ this tool
allowed us
to compute the
cyclic homology \cite{kmt}
built upon $H_n(A,{}_\sigma A)$
as a special case of Connes-Moscovici's
Hopf-cyclic homology \cite{cm}.

The trace $\int$ in (\ref{blinde}) is for
$\mathbb{C}_q[S^2]$ actually a character
(namely the restriction of the counit
$\varepsilon$ of $\mathbb{C}_q[SU(2)]$
to $\mathbb{C}_q[S^2]$), so
on the level of
chains $a_0 \otimes a_1 \otimes a_2$
in the standard Hochschild complex,
$\varphi$ acts as
$$
		  \varphi (a_0,a_1,a_2)=
		  q^{-1}\varepsilon(a_0)F(a_1)E(a_2),
$$
where $E,F : A \rightarrow k$ are
the (untwisted) derivations given by
$$
		  E(a):=\varepsilon(\partial_{-1}(a)),
		  \quad
		  F(a):=\varepsilon(\partial_1(a)).
$$
There seems to be a general
principle behind this that we observed already
in \cite{tomuli2}. Therein, the trace $\int$
needed was the integral over
the unquantised maximal torus in quantum
$SU(2)$. For the quantum 2-sphere
the corresponding submanifold of $S^2$
is simply a point, namely one of the two
leaves of the symplectic foliation of the
Poisson manifold $S^2$ quantised by $A$.

Finally we discuss how to add a
counter term $\eta$ to $\varphi$ in order
to obtain a functional on cyclic
homology without changing
the functional on $H_2(A,{}_\sm A)$.
Schm\"udgen and Wagner have defined a
cyclic 2-cocycle in \cite{sw} that looks
like $ \varphi $, only the trace
$\int$ is the Haar functional of
$\mathbb{C}_q[SU(2)]$, and this makes
their functional trivial on
Hochschild homology \cite{tom}.

The structure of the paper is as follows.
In Sections~2-5 we recall the definition
of Hochschild (co)homology, of the cup and
cap product, give some more
background about the above mentioned
Poincar\'e duality and introduce then
the algebra $A=\mathbb{C}_q[S^2]$
we want to study.
In Section~6 we recall from \cite{tom}
an explicit formula for the fundamental
class $\d A$ of $A$. In Section~7
we determine the twisted
centre of $A$ and its cap
product action on the second
Hochschild homology: in \cite{tom}
it was shown that
$H_2(A,{}_\sigma A)=0$ except when
$\sigma=\sigma^n_\mathrm{mod}$
for some $n \ge 1$, and then
$H_2(A,{}_{\sigma^n_\mathrm{mod}} A) \simeq
\mathbb{C}$. Here
we identify the sum of all these
nontrivial homology groups with the free
module of rank 1 over a polynomial ring
$k[x_0]$ that constitutes the twisted centre of
$A$. We continue in Section~8 with
recalling from \cite{tom}
(but in a slightly simplified form)
the computation of
the zeroth Hochschild homology groups of
$A$ and of the twisted
traces on $A$, describing in addition
the cap product action of the twisted centre.
Section~9 is the first really interesting one,
here we compute the cup product between
twisted derivations of $A$ that arise from the
action of the quantised Lie algebra of
$SU(2)$ on $A$. We observe
similar as in \cite{tomuli2} that these
twisted derivations generate a quantised
exterior algebra. These computations
are then used in Section~10 to
define and discuss $ \varphi $.
Section~11 recalls the definition and some key
properties of cyclic homology, and finally
we show in Section~12 that $ \varphi $ can be
altered by a Hochschild
coboundary to obtain a cyclic cocycle.\\

I acknowledge support by the EPSRC
fellowship EP/E/043267/1.

\section{Hochschild (co)homology with coefficients in
 ${}_\sigma A$}\label{twihoco}
Let $A$ be a unital associative algebra over a field $\k$
and $ \sigma \in \mathrm{Aut} (A)$ be an
automorphism.
We denote by ${}_\sigma A$ the $A$-bimodule
which is $A$ as vector space with left and right
$A$-actions given by
$a \triangleright b \triangleleft c:=\sigma(a)bc$,
$a,b,c \in A$, and by $H_n(A,{}_\sigma A)$
and $H^n(A,{}_\sigma A)$ the Hochschild
(co)homology groups of $A$ with coefficients in
${}_\sigma A$. Explicitly,
$H_n(A,{}_\sigma A)$ is the homology
of the chain complex
$C^\sigma_n:=A^{\otimes n +1}$ with boundary map
$\b_n : C^\sigma_n \rightarrow C^\sigma_{n-1}$
given by
\begin{eqnarray}
		  \b_n(a_0 \otimes \ldots \otimes a_n)
&=& \sum_{i=0}^{n-1} (-1)^i
		  a_0 \otimes \ldots \otimes a_ia_{i+1} \otimes
		  \ldots \otimes a_n\nonumber\\
&& +(-1)^n \sigma (a_n) a_0 \otimes \ldots \otimes a_{n-1}.\nonumber
\end{eqnarray}
Usually we will write
$\b(a_0,\ldots,a_n)$ instead of
$\b_n(a_0 \otimes \ldots \otimes a_n)$
and similarly for
other multilinear maps.
Dually, $H^n(A,{}_\sigma A)$
is the cohomology of the cochain complex
$C^n_\sigma:=\mathrm{Hom}_k(A^{\otimes n},A)$
with coboundary map given by
\begin{eqnarray}
		  (\b^n \psi)(a_0,\ldots,a_n)
&=& \sigma (a_0) \psi (a_1,\ldots,a_n)
		  \nonumber\\
&& +  \sum_{i=0}^{n-1} (-1)^{i+1}
		  \psi(a_0,\ldots,a_ia_{i+1},\ldots,a_n)
		  \nonumber\\
&& +(-1)^{n+1} \psi
		  (a_0,\ldots,a_{n-1})a_n.\nonumber
\end{eqnarray}

In degree 0, we identify
$ \psi : A^{\otimes 0}:=k \rightarrow A$ with
$ a:=\psi (1) \in A$.
This is a cocycle precisely when
$ab=\sigma(b)a$ for all $b \in A$. Thus
$H^0(A,{}_\sigma A)$ consists of the
$ \sigma $-central elements of $A$. In
degree 1, a cocycle is a
$ \sigma $-twisted derivation
$ \psi : A \rightarrow A$, $ \psi (ab)=\sigma (a)
\psi (b) + \psi (a)b$,
and $H^1(A,{}_\sigma A)$
is the space of all such
derivations modulo those of
the form $ \psi (a)=ba-\sigma(a)b$
for some $b \in A$.
For more information and details, see
e.g.~\cite{ce,tomuli,kmt,loday,weibel}.

\section{The cup and cap product}
The cup product is the map
$$
		  \smallsmile \,\,: H^m(A,{}_\sigma A) \otimes
		  H^n(A,{}_\tau A) \rightarrow
		  H^{m+n}(A,{}_{\tau \circ \sigma} A),\quad
		  \sigma , \tau \in \mathrm{Aut} (A)
$$
given on the level of cochains by
\begin{equation}\label{verz}
		  (\varphi \smallsmile \psi)
		  (a_1,\ldots,a_{m+n})=
		  \tau (\varphi (a_1,\ldots,a_m))
		  \psi (a_{m+1},\ldots,a_{m+n}).
\end{equation}
For any monoid
$G \subset \mathrm{Aut} (A)$, it
turns
$$
		  \Lambda_G(A):=
		  \bigoplus_{n \in \mathbb{N},\sigma \in G}
		  H^n(A,{}_\sigma A)
$$
into an $\mathbb{N} \times G$-graded
algebra that we would like to view as some analogue of
an algebra of multivector fields on a classical space.
Twisted derivations play here the role of vector
fields, and the following easily checked
(see \cite{tomuli2}) relations demonstrate
their behaviour under $\smallsmile\,$:
\begin{lemma}\label{relands}
In degree 0, $\smallsmile$
reduces to the opposite product of $A$,
\begin{equation}\label{oppo}
		  a \smallsmile b=ba,\quad
		  a \in H^0(A,{}_\sigma A),
		  b \in H^0(A,{}_\tau A),
\end{equation}
and for
$c \in H^0(A,{}_\sigma A)$ and twisted
derivations $\varphi \in C^1_\sigma,
\psi \in C^1_\tau$
we have
$$
		  \psi \smallsmile c= \tau(c) \smallsmile
			\psi,\quad
		  [\varphi] \smallsmile [\psi]=
		  -[\sigma ^{-1} \circ \psi \circ \sigma]
		  \smallsmile [\varphi] \in
		  H^2(A,{}_{\tau \circ \sigma} A).
$$
\end{lemma}

Dually,
$$
		  \Omega^G(A):=
		  \bigoplus_{n \in \mathbb{N},\sigma \in G}
		  H_n(A,{}_{\sigma^{-1}} A)
$$
becomes an $ \mathbb{N} \times G$-graded
(right) module over $ \Lambda_G(A)$
via the cap product
$$
		  \smallfrown \,\,: H_n(A,{}_\sigma A) \otimes
		  H^m(A,{}_\tau A) \rightarrow
		  H_{n-m} (A,{}_{\tau \circ \sigma} A),\quad m \le n.
$$
Explicitly, this is given between
$ \varphi \in C^m_\tau$ and
$a_0 \otimes \ldots \otimes a_n \in C_n^\sigma$ by
$$
		  (a_0 \otimes \ldots \otimes a_n) \smallfrown
		  \varphi =
		  \tau (a_0) \varphi (a_1,\ldots,a_m) \otimes
		  a_{m+1} \otimes \ldots \otimes a_n \in
		  C_{n-m}^{\tau \circ \sigma}.
$$
In particular, the cap product
with a twisted central
element
$c \in H^0(A,{}_\sigma A)$ is simply given by
multiplication from the \emph{left},
\begin{equation}\label{spaetzle}
		  (a_0 \otimes \ldots \otimes a_n) \smallfrown c=
		  \sigma (a_0)c \otimes \ldots \otimes a_n=
		  ca_0 \otimes \ldots \otimes a_n.
\end{equation}
 For more information,
see e.g.~\cite{ce,tomuli2,nesttsygan}.

\section{Poincar\'e duality}
The cup and cap
product structures are intimately
related to Poincar\'e-type dualities
between homology and
cohomology. As became clear in
recent years, there is for many algebras $A$ a
distinguished automorphism
$ \sm $ and for all
$ \tau \in \mathrm{Aut} (A)$ a canonical
$k$-linear isomorphism
\begin{equation}\label{podu}
		  H^n(A,{}_\tau A) \simeq
		  H_{\mathrm{dim}(A)-n}(A,{}_{\tau \circ \sm} A),
\end{equation}
where
$ \mathrm{dim} (A)$ is the dimension of $A$ in
the sense of \cite{ce}, see
e.g.~\cite{bz,farinati,tomuli,uli,ulineu}
but first of all \cite{vdb} for this story. Under the above isomorphism,
the canonical element $1 \in H^0(A,A)$ corresponds
to a class $[\d A] \in H_{\mathrm{dim}(A)}(A,{}_\sm A)$,
and then the isomorphism is given by the cap
product with this fundamental class. In \cite{tomuli2} we carried
this out explicitly for the standard quantised coordinate ring
$\mathbb{C}_q[SU(2)]$ (see e.g.~\cite{chef} for
background on quantum groups), and the aim here is to
do the same for the standard
quantum 2-sphere of
Podle\'s that we introduce in the next section.
For the coordinate ring of a smooth affine variety
such a duality will hold if and only if the variety is
Calabi-Yau, that is, if the line bundle on $X$
whose sections are
the top degree K\"ahler
differentials $\Omega^{\mathrm{dim}(X)}(X)$
is trivial (in general one has to twist
not by an automorphism but by this module, see
e.g.~\cite{uli}). This happens
if and only if there is a nowhere (i.e.~in no
localisation at prime ideals) vanishing element
in $\Omega^{\mathrm{dim}(X)}(X)$,
and under the
Hochschild-Kostant-Rosenberg isomorphism
$\Omega^{\mathrm{dim}(X)}(X) \simeq
H_{\mathrm{dim}(X)}(k[X],k[X])$ such an element will be
identified with the fundamental class $[\d k[X]]$.

\section{The Podle\'s sphere}\label{aldlos}
From now on we fix $\k=\mathbb{C}$, an element
$q \in \k \!\setminus\!\!\{0\}$ assumed to be not a root
of unity, and $A$ is the standard Podle\'s quantum 2-sphere \cite{Po},
that is, the universal $\k$-algebra
generated by $x_{-1},x_0,x_1$
satisfying the relations
$$
		  x_{\pm 1}x_0=q^{\mp 2}x_0x_{\pm 1},\quad
		  x_{\pm 1}x_{\mp 1}=
		  q^{\mp 2}x_0^2+q^{\mp 1}x_0.
$$
It follows easily from these relations that
the elements
$$
		  e_{ij}:=\left\{
\begin{array}{ll}
x_0^ix_1^j\quad & j \ge 0,\\
x_0^ix_{-1}^{-j}\quad &j <0,
\end{array}\right.\quad i \in \mathbb{N},j \in
\mathbb{Z}
$$
form a vector space basis of $A$.

We denote by $G$ the
automorphism group of $A$.
The defining relations imply that
for any $\lambda \in \k \!\setminus\!\!\{0\}$
there is a unique
$\sigma_\lambda \in G$
with $\sigma_\lambda (x_n)=\lambda^n x_n$.
\begin{lemma}
Any $\sigma \in G$ is of the form
$\sigma_\lambda$ for some $ \lambda $,
that is, $G \simeq \k \!\setminus\!\!\{0\}$.
\end{lemma}
\begin{pf}
It is straightforward to classify the characters
of $A$ and to see that the intersection of their
kernels is the ideal generated by
$x_0$. It follows that any automorphism
$ \sigma $ maps $x_0$ to a nonzero
scalar multiple of $x_0$. Hence
$x_0 \sigma(x_{\pm 1})=
q^{\pm 2}\sigma (x_{\pm 1})x_0$ which implies
$\sigma (x_{\pm 1})=f_\pm x_{\pm 1}$
for some $f_\pm \in k[x_0]$.
Inserting into the defining relations
gives the claim.
\end{pf}

See
also \cite{dijkhuizen,chef} for more information
about this algebra.

\section{The fundamental class}
As shown in \cite{ulineu}, $A$
satisfies (\ref{podu}) with $ \mathrm{dim}(A)=2$
(as it probably should be for a quantum 2-sphere),
and with $\sm$ determined uniquely by
$$
		  \sm(x_n)=q^{2n} x_n.
$$
So $H_2(A,{}_\sm A) \simeq H^0(A,A)$,
the centre of $A$. This consists only of the
scalars, hence $[\d A]$ is unique
up to normalisation.
Hadfield has computed explicit vector space
bases of all $H_n(A,{}_\sigma A)$ for general
automorphisms $ \sigma $ \cite{tom} and
has given
in particular an explicit cycle representing
$[\d A] \in H_2(A,{}_\sm A)$:
\begin{eqnarray}
		  \d A
&:=& 2  x_1 \otimes
		  (x_{-1} \otimes x_0-q^2x_0 \otimes x_{-1}) \nonumber\\
&& + 2 x_{-1} \otimes (q^{-2}
		  x_0 \otimes x_1-x_1 \otimes x_0)\nonumber\\
&&+1 \otimes (qx_1 \otimes x_{-1}-q^{-1}x_{-1} \otimes x_1
		  +(q-q^{-1})x_0 \otimes x_0) \nonumber\\
&&+2x_0 \otimes (x_1 \otimes x_{-1}-x_{-1} \otimes x_1
		  +(q^2-q^{-2})x_0 \otimes x_0).
		  \nonumber
\end{eqnarray}

\section{The twisted centre and
$H_2(A,{}_\sigma A)$}
As we pointed out above,
the fundamental class is
classically (meaning for the
coordinate ring of a
smooth variety) represented by a
nowhere vanishing
algebraic differential form
of top degree, and this
form can be multiplied by any
regular function to give a new
top degree form. Analogously
we can act via the
cap product by any
twisted central element
$a \in H^0(A,{}_\tau A)$ on the
fundamental class $[\d A] \in H_2(A,{}_\sm A)$
and obtain another homology
class in
$H_2(A,{}_{\tau \circ \sm} A)$. This operation
clarifies completely the
structure of all the other
nonvanishing
$H_2(A,{}_\sigma A)$ that were computed by
Hadfield, since they become altogether
identified with a free $k[x_0]$-module of rank 1:
\begin{lemma}
The twisted centre $\Lambda^0_G(A)$
of $A$ is the subalgebra
generated by
$x_0 \in H^0(A,{}_\sm A)$, and
for every $[\omega] \in H_2(A,{}_\sigma A)$,
$\sigma$ any automorphism, there exists
exactly one polynomial $f \in k[x_0]$ such that
$[\omega]=[\d A] \smallfrown f$.
\end{lemma}
\begin{pf}
Since any automorphism fixes $x_0$,
any twisted central
element must commute with $x_0$ and is hence a
polynomial in $x_0$ (use the vector space basis
$e_{ij}$). Conversely, it is clear that
$x_0 \in H^0(A,{}_\sm A)$ and hence
$x_0^i=x_0 \smallsmile \ldots \smallsmile x_0
\in H^0(A,{}_{\sigma_{q^{2i}}} A)$ (recall Lemma~\ref{relands}).
The second part follows by Poincar\'e duality (\ref{podu}),
but of course also from Hadfield's explicit
 computations of all the nontrivial
$H_2(A,{}_\sigma A)$.
\end{pf}

\section{Twisted traces and
$H_0(A,{}_\sigma A)$}\label{traces}
As a vector space,
$H_0(A,{}_\sigma A)$ can be described
as follows \cite{tom}:
\begin{lemma}\label{hzero}
For $ \sigma = \sigma_\lambda$,
the following is a vector space basis of
$H_0(A,{}_\sigma A)$:
$$
		  \{[1]\} \cup
		  \{[x_{\pm 1}^j]\,|\,j \neq 0,\lambda = 1\} \cup
		  \{[x_0]\,|\,\lambda \neq q^{2i},i >0\} \cup
		  \{[x_0^i]\,|\,\lambda = q^{2i},i >0\}.
$$
\end{lemma}
\begin{pf}
By definition, $H_0(A,{}_\sigma A)$ is as a vector
space the quotient of $A$ by the subspace spanned by
elements of the form $\b(a,b)=ab-\sigma(b)a$.
Since
$$
		  a \otimes bc=ab \otimes c+\sigma(c)a \otimes b
		  -\b(a,b,c),
$$
one has $\b(a,bc)=\b(ab,c)+\b(\sigma (c)a,b)$,
so $ \mathrm{im}\, \b$ is spanned by the elements
$b^k_{ij}:=\b(e_{ij},x_k)$, $i \in \mathbb{N}$, $j \in \mathbb{Z}$,
$k=-1,0,1$ which are for $ \sigma=\sigma_\lambda$ given by
\begin{eqnarray}
&& b^{-1}_{ij} =
		  (1-\lambda^{-1}q^{2i})e_{ij-1},\quad j \le 0\nonumber\\
&& b^{-1}_{ij} =
		  (q^{-4j+2}-\lambda^{-1}q^{2i+2})e_{i+2j-1}
		  +(q^{-2j+1}-\lambda^{-1}q^{2i+1})e_{i+1j-1},\quad j>0,\nonumber\\
&& b^0_{ij} = (q^{-2j}-1)e_{i+1j},\nonumber\\
&& b_{ij}^1 = (1-\lambda q^{-2i})e_{ij+1},\quad j
 \ge 0\nonumber\\
&& b_{ij}^1 = (q^{-4j-2}-\lambda q^{-2i-2})e_{i+2j+1}
+(q^{-2j-1}-\lambda q^{-2i-1})e_{i+1j+1},\quad j<0.\nonumber
\end{eqnarray}
Reducing this by sheer inspection gives that $ \mathrm{im}\, \b$ is
spanned by the elements
\begin{eqnarray}
&& e_{i+1j},\quad (\lambda-1)e_{0j},\quad j \neq 0,\nonumber\\
&& (\lambda-q^{2i+4})q^{-2i-2}e_{i+20}
+(\lambda-q^{2i+2})q^{-2i-1}e_{i+10},\quad i \ge 0.\nonumber
\end{eqnarray}
The claim follows easily.
\end{pf}

Dually, $H_0(A,{}_\sigma A)$ can be described
in terms of
$\sigma$-twisted traces, that is,
linear functionals
$\int : A \rightarrow k$ satisfying
$$
		  \int a b = \int \sigma (b)a,\quad
		  a,b \in A.
$$
Such traces obviously descend to
well-defined functionals on
$H_0(A,{}_\sigma A)$ which we denote for
simplicity by the same symbol.
For each of the basis elements in
Lemma~\ref{hzero}, we can (and do)
define one such trace
\begin{eqnarray}
&& \int_{[x_{\pm 1}^j]} e_{kl} :=
		  \left\{
			\begin{array}{ll}
			 1 \quad & k=0,\pm j=l,\\
			 0 \quad & \mbox{otherwise},
			\end{array}\right.\quad j \ge 0,\nonumber\\
&& \int_{[x_0]} e_{kl} := \left\{
									\begin{array}{ll}
			1 \quad & k=1,l=0,\\
		  (-1)^{k+1}q^{1-k}
		  \frac{1-\lambda q^{-2}}{1-\lambda
		  q^{-2k}}\quad & k>1,l=0,\\
									0 \quad & \mbox{otherwise},
									\end{array}\right.
		  \nonumber\\
&& \int_{[x_0^i]} e_{kl} := \left\{
									  \begin{array}{ll}
										1 \quad & k=i,l=0,\\
										0 \quad & \mbox{otherwise},
									  \end{array}\right.\quad i>1.\nonumber
\end{eqnarray}
Note that $\int_{[x_0]}$ is defined in such
a way that the case $\lambda=q^2$ is
included. Note also that $\int_{[1]}$
is in fact the character $\varepsilon$
determined by $ \varepsilon (x_n)=0$.
Any automorphism of $A$ leaves
$ \mathrm{ker}\,\varepsilon$ invariant, so this is
a twisted trace with respect to all
automorphisms of $A$.
Since we have
for all elements $[\omega],[\eta]$ of the basis of
$H_0(A,{}_\sigma A)$ from Lemma~\ref{hzero}
$$
		  \int_{[\omega]} [\eta]=\left\{
\begin{array}{ll}
1 \quad & [\omega]=[\eta],\\
0 \quad & \mbox{otherwise},
\end{array}\right.
$$
we can (and will)
use the $\int_{[\omega]}$ to determine
the homology class of a given 0-cycle.
For example, it helps describing the cap product action of the
twisted centre on
$H_0(A,{}_\sigma A)$:
\begin{lemma}
The action of
$\Lambda^0_G(A)$ on
$H_0(A,{}_{\sigma_\lambda} A)$, is determined by
\begin{eqnarray}
&&  [x_{\pm 1}^j] \smallfrown x_0=0,\quad j
 >0,\nonumber\\
&& [x_0^i] \smallfrown x_0=[x_0^{i+1}]=\left\{
		  \begin{array}{ll}
			0 \quad & i=0,\lambda=q^{2k},k>0,\\
			-q
		  \frac{1-\lambda}{q^2-\lambda}[x_0]\quad & i=1,\lambda \neq q^2.
		  \end{array}\right.\nonumber
\end{eqnarray}
\end{lemma}
\begin{pf}
As we remarked in (\ref{spaetzle}), $x_0$ acts on
cycles by multiplication from the left. The claim
follows by applying all the above
constructed twisted
traces to the resulting cycles.
\end{pf}

In particular, the span of the classes
$[x_0^i] \in H_0(A,{}_{\sigma^i_\mathrm{mod}}A)$ is the
orbit of $[1] \in H_0(A,A)$ under
the action of the twisted centre. In a sense, the
span of the $[x_{\pm 1}^j]$ can be viewed as the orbit
of $[1]$ under the cap product with $x_{\pm 1}$, although
the latter do not belong to the twisted centre of
$A$: for any subalgebra $B \subset A$ with
$ \sigma (B) \subset B$, there is a map
$H_n(B,{}_\sigma B) \rightarrow H_n(A,{}_\sigma A)$
given on the level of cycles by the embedding of $B$
into $A$ in each tensor component. If
a class is in the image of this map, then taking the
cap product with a twisted central element of $B$ is
well-defined, and this applies here to the case
$B$ is the subalgebra generated by $x_1$ or $x_{-1}$, respectively.

\section{Three twisted derivations}
The Podle\'s sphere is a module algebra over
the Hopf dual $\mathbb{C}_q[SU(2)]^\circ$ of
the quantised coordinate ring of
$SU(2)$, hence twisted primitive
elements therein act as twisted derivations on
$A$. We will not need Hopf algebra
theory later, so we rather state the following
lemma that the reader can
verify directly by checking
compatibility with the
defining relations of $A$:
\begin{lemma}
The assignments
\begin{eqnarray}
&& \partial_1 : x_{-1},x_0,x_1 \mapsto
		  0,qx_{-1},1+(q+q^{-1})x_0,\nonumber\\
&& \partial_0 : x_{-1},x_0,x_1 \mapsto
		  -x_{-1},0,x_1,\nonumber\\
&& \partial_{-1} : x_{-1},x_0,x_1 \mapsto
		  1+(q+q^{-1})x_0,q^{-1}x_1,0\nonumber
\end{eqnarray}
can be extended uniquely to
1-cocycles $ \partial_i \in C^1_{\sigma^{-|i|}_\mathrm{mod}}$,
$$
		  \partial_i(ab)=\sigma^{-|i|}_\mathrm{mod}(a) \partial_i(b)+
		\partial_i (a) b,\quad a,b \in A,\quad i=-1,0,1.
$$
\end{lemma}

The following
lemma describes the cup product action of
$\Lambda^0_G(A)$ on these derivations.
Admittedly, the result is slightly weird:
\begin{lemma}
The $\Lambda^0_G(A)$-module generated by
$[\partial_0]$ is free, but
$[\partial_{\pm 1}] \smallsmile x_0=0$.
\end{lemma}
\begin{pf}
One checks directly that
one has for all $a \in A$
$$
		  (\partial_{\pm 1} \smallsmile x_0)(a)=
		  x_0 \partial_{\pm 1}(a)=
		  \pm \frac{1}{q-q^{-1}} (x_{\mp 1}a
		  -ax_{\mp 1}),
$$
so the derivations
$\partial_{\pm 1} \smallsmile x_0$ are
inner. On the other hand, the
computation of $e_{jk} x_1-
\sigma^i_\mathrm{mod}(x_1)e_{jk}=
b^1_{jk}$  in the proof of Lemma~\ref{hzero}
shows that no inner derivations
in $C^1_{\sigma^i_\mathrm{mod}}$ can map
$x_1$ to
$x_0^ix_1=(\partial_0 \smallsmile x_0^i)(x_1)$.
\end{pf}

For the reason explained at the end of the
previous section, it does make sense to take the cup
product between $ \partial_{\pm 1}$ and
$x_{\mp 1}^j$, although these are not twisted central,
and this produces new twisted
derivations. Acting with them on the fundamental class and
comparing the result with the generators of
$H_1(A,{}_\sigma A)$ computed in \cite{tom}
allows one to describe all twisted derivations
of $A$, see \cite{tomuli2} where we carried this out
for $ \mathbb{C}_q[SU(2)]$. However, there is little
gain in this for the main purpose of the
present paper which is to obtain
a functional describing $\d A$ in a dual fashion, so I
leave out these calculations.

Let us compute instead the algebra generated by the
$[\partial_i]$.
Classically, a differential
form can be contracted with a
vector field to reduce its degree, and in the
quantum case this is generalised by
the cap product action of (the
cohomology class of) a
twisted derivation on a homology
class.
Here is the full orbit of
$\d A$ under this action of
the $ \partial_i$:
\begin{eqnarray}
 \d A \smallfrown \partial_0
&=& 2q^{-2} x_{-1}x_0 \otimes x_1
		  +2q^2x_1x_0 \otimes x_{-1}
		  -2(q^2+q^{-2})x_0^2 \otimes x_0
		  \nonumber\\
&&+	  q^{-1}x_{-1}  \otimes x_1
		  +qx_1  \otimes x_{-1}
		  -2(q+q^{-1})x_0  \otimes x_0 ,\nonumber\\
		  (\d A \smallfrown \partial_0)
		  \smallfrown \partial_0
&=& 2(q^2-q^{-2})x_0^3+3(q-q^{-1})x_0^2,
		  \nonumber\\
		  (\d A \smallfrown \partial_0)
			\smallfrown \partial_{-1}
&=& 2(-q^5+q^{-1})x_1x_0^2
		  +(-2q^2+1+q^{-2}1)x_1x_0 +
		  q^{-1}x_1,\nonumber\\
		  (\d A \smallfrown \partial_0)
			\smallfrown \partial_1
&=& 2(q-q^{-5})x_{-1}x_0^2+
		  (q^2+1-2q^{-2})x_{-1}x_0
		  +qx_{-1},\nonumber
\end{eqnarray}
\begin{eqnarray}
		  \d A \smallfrown \partial_{-1}
&=& -2q^{-1}x_1^2 \otimes x_{-1}
		  -2q^{-1}x_0^2 \otimes x_1
		  +2(q^3+q^{-3}) x_1 x_0 \otimes x_0
		  \nonumber\\
&&-(1+q^{-2})x_0 \otimes x_1
		  +(1+q^{-2})x_1 \otimes x_0 -
		  q^{-1} \otimes x_1,\nonumber\\
		  (\d A \smallfrown \partial_{-1})
			\smallfrown \partial_0
&=& 2(q^{-3}-q^3) x_1x_0^2
		  +(-q^2-1+2q^{-2})x_1x_0
		  -q^{-1}x_1,\nonumber\\
		  (\d A \smallfrown \partial_{-1})
			\smallfrown \partial_{-1}
&=&2(q^2-q^{-6})x_1^2x_0
		  +(q^{-3}-q^{-5})x_1^2,\nonumber\\
		  (\d A \smallfrown \partial_{-1})
			\smallfrown \partial_1
&=& 2(q^{-8}-1)x_0^3
		  +(-q-2q^{-1}+q^{-5}+
		  2q^{-7})x_0^2 \nonumber\\
&&+(-2-q^{-2}+q^{-4})x_0 -q^{-1},\nonumber
\end{eqnarray}
\begin{eqnarray}
		  \d A \smallfrown \partial_1
&=& 2q x_{-1}^2 \otimes x_1
		  +2qx_0^2 \otimes x_{-1}
		  -2(q^3+q^{-3})x_{-1}x_0 \otimes x_0 \nonumber\\
&&+(q^2+1)x_0  \otimes x_{-1}
		  +(-q^2-1)x_{-1}  \otimes x_0 +q \otimes x_{-1},\nonumber\\
 (\d A \smallfrown \partial_1) \smallfrown \partial_0
&=& 2(q^3-q^{-3})x_{-1}x_0^2
		  +(2q^2-1-q^{-2})x_{-1}x_0-qx_{-1},\nonumber\\
(\d A \smallfrown \partial_1) \smallfrown \partial_{-1}
&=& 2(1-q^8)x_0^3
		  +(-2q^7-q^5+2q+q^{-1}) x_0^2 \nonumber\\
&& +(-q^4+q^2+2)x_0+q,\nonumber\\
(\d A \smallfrown \partial_1) \smallfrown \partial_1
&=& 2(q^6-q^{-2})x_{-1}^2x_0
		  +(q^5-q^3)x_{-1}^2.\nonumber
\end{eqnarray}
In homology, this reduces to:
\begin{eqnarray}
&& ([\d A] \smallfrown [\partial_i])
			\smallfrown [\partial_i] = 0,\nonumber\\
&& ([\d A] \smallfrown [\partial_0]) \smallfrown
		  [\partial_{-1}] =
		  -([\d A] \smallfrown [\partial_{-1}])
			\smallfrown [\partial_0] =
			q^{-1}[x_1],\nonumber\\
&& ([\d A] \smallfrown [\partial_0])
		  \smallfrown [\partial_1] =
		  -([\d A] \smallfrown [\partial_1])
			\smallfrown [\partial_0] =
			q[x_{-1}],\nonumber\\
&& ([\d A] \smallfrown [\partial_1])
			\smallfrown [\partial_{-1}] =
		  -q^2([\d A] \smallfrown [\partial_{-1}])
		  \smallfrown [\partial_1] =
		  (q^2+1)[x_0]+q.\nonumber
\end{eqnarray}
From this, we see:
\begin{lemma}
The $\partial_i$ satisfy no other
relations than those dictated by
Lemma~\ref{relands},
$$
		  [\partial_i] \smallsmile [\partial_j] =
		  -q^{2ij}[\partial_j] \smallsmile
		  [\partial_i],\quad i \le j.
$$
\end{lemma}
\begin{pf}
Poincar\'e duality tells that the
$\Lambda_G(A)$-module
$ \Omega^G(A)$ is free and
generated by $\d A$, so
the $ [\partial_i]$ satisfy under $\smallsmile$ all
the relations
they satisfy as linear maps on
$\Omega^G(A)$. The result follows
from the above computations.
\end{pf}

\section{The volume form}
Finally, we now put
$$
		  \varphi : \Omega^G(A) \rightarrow k,
		  \quad
		  [\omega] \mapsto
		  q^{-1} \int_{[1]} [\omega] \smallfrown
 		  ([\partial_1] \smallsmile [\partial_{-1}]),
$$
or explicitly on chain level
$$
		  \varphi (a_0,a_1,a_2)=
		  q^{-1} \int_{[1]}
		  \sigma^{-2}_\mathrm{mod}(a_0)
		  \sigma^{-1}_\mathrm{mod}(\partial_1(a_1))
		  \partial_{-1}(a_2).
$$
Our above computation of
$([\d A] \smallfrown [\partial_1])
\smallfrown [\partial_{-1}]$ implies
$$
		  \varphi (\d A)=1,
$$
so the functional $\varphi$
provides a dual description of
the fundamental class.
It is a very useful
tool (and probably
the only really applicable one) for
checking whether or not a given
$\sm$-twisted 2-cycle
has trivial class in $H_2(A,{}_\sm A)$ or
not, which as a result of the above
computations (recall that
$H_2(A,{}_\sm A) \simeq \mathbb{C}$)
is the fact if and only if
$ \varphi$ vanishes on the
cycle.

Note that we remarked in Section~\ref{traces}
that $\int_{[1]}$ is actually a character that
we also denote by $\varepsilon$ (when
embedding $A$ as usual into the quantised
coordinate ring of $SU(2)$,
this character becomes the restriction of
the counit). Hence $ \varphi $ can also be
written as simple as
$$
		  \varphi (a_0,a_1,a_2)=
		  q^{-1}\varepsilon(a_0)F(a_1)E(a_2),
$$
where $E,F : A \rightarrow k$ are
the (untwisted) derivations given by
$$
		  E(a):=\varepsilon(\partial_{-1}(a)),
		  \quad
		  F(a):=\varepsilon(\partial_1(a)).
$$
A moment's reflection now gives
the explicit formula
\begin{equation}\label{shock}
		  \varphi (e_{ij},e_{kl},e_{mn})=
		  q^{-1} \delta_{i0}\delta_{j0}
		  \delta_{k0}\delta_{l1}\delta_{m0}
		  \delta_{n-1}.
\end{equation}
This looks surprisingly simple (not to say
banal), usually one
expects $\int$ to be
some sort of integral. However, we
observed already in \cite{tomuli2}
for the case of quantum
$SU(2)$ that the functional appearing when
expressing the volume form dual to
$\d A$ as above is given by something like
the integral of the restriction of functions to
a maximal torus which is a Poisson subgroup
of the Poisson group quantised by
$\mathbb{C}_q[SU(2)]$. Here we have the
same phenomenon,
but it appears much sharper since the functional $\int_{[1]}$ is
really integration over (meaning evaluation in)
a single point. It somehow
seems that the homological
information about a quantum space can be
supported in
a classical subspace of smaller dimension.

Note also that one can alternatively define
$$
		  \varphi_\pm : \omega \mapsto
		  \pm q^{\mp 1} \int_{[x_{\mp 1}]}
		  \omega \smallfrown (\partial_0
		  \smallsmile \partial_{\pm 1}),
$$
where $\int_{[\pm 1]}$ is the trace (no twist)
dual to $[x_{\pm 1}] \in H_0(A,A)$. Again,
the above computations give
$\varphi_\pm(\d A)=1$,
hence $\varphi=\varphi_+=\varphi_-$
as functionals on
$H_2(A,{}_\sm A) \simeq \mathbb{C}$
(but not as
functionals on $C_2^\sm$),
and for some purposes
this representation of the functional
might be better suited that $\varphi $.

\section{Cyclic homology}
In the classical case $A=k[X]$,
$X$ a smooth variety, exterior derivation
$\d$ turns the algebraic differential forms
$\Omega(X)$ into
a cochain complex that computes
the algebraic de
Rham cohomology $H^n(X)$ of $X$.
In the noncommutative case,
Connes' cyclic homology provides a subtle
analogue of $H^n(X)$. The extension
of cyclic homology to
the twisted coefficients ${}_\sigma A$
arose in the work of Kustermans, Murphy
and Tuset on covariant differential
calculi over quantum groups \cite{kmt},
but can also be viewed as a special case of
Connes-Moscovici's Hopf-cyclic
homology \cite{cm}, see e.g.~our article
\cite{tomuli2} and the references therein
for more background.\footnote{As Jack
Shapiro informed me, he will
discuss in a forthcoming article
the corresponding version of
noncommutative de Rham
theory \cite{jack}.}

The precise
relation between
$HC_n(k[X])$ ($ \sigma = \mathrm{id}$)
and $H^n(X)$ is
\begin{equation}\label{derham}
		  HC_n(k[X]) \simeq
		  \Omega^n(X)/\mathrm{im}\, \d \oplus
		  H^{n-2}(X) \oplus H^{n-4}(X) \oplus \ldots,
\end{equation}
so all differential $n$-forms (not only the
closed ones in $ \mathrm{ker}\, \d$!) have
classes in $HC_n(k[X])$, and this gives
a map
$$
		  \mathsf{I} : \Omega^n(X) \simeq
		  H_n(k[X],k[X]) \rightarrow
 		  HC_n(k[X]).
$$
Furthermore, $\d$ (applied to
$\Omega^n(X)/\mathrm{im}\, \d$)
gives a well-defined map
$$
		  \mathsf{B} :
 		  HC_n(k[X]) \rightarrow
		  \Omega^{n+1}(X) \simeq
		  H_{n+1}(k[X],k[X])
$$
which kills all the $H^{n-2i}(X)$
summands in (\ref{derham}), and finally
there is
$$
		  \mathsf{S} : HC_n(k[X]) \rightarrow
		  HC_{n-2}(k[X])
$$
that cuts off the first term
$\Omega^n(X)/\mathrm{im}\, \d$
and leaves the rest
untouched (using the obvious embedding
$H^{n-2}(X) \rightarrow
\Omega^{n-2}(X)/\mathrm{im}\, \d$
in the next summand).

This whole picture carries over to the general
noncommutative case and becomes condensed
into Connes' $\mathsf{SBI}$-sequence,
see \cite{loday,weibel} for the details.
The upshot of this is that
$HC_n(A)$ contains a whole lot of
ballast, the really interesting part is only
the image of the natural map
$\mathsf{I}$ coming from Hochschild
homology, which equals the kernel of the
periodicity map $\mathbf{S}$.

One way to define the cyclic
theory is in terms of the operator
$$
		  \t : C_n^\sigma \rightarrow
		  C^\sigma_n :
		  a_0 \otimes \ldots \otimes a_n \mapsto
		  (-1)^n \sigma(a_n) \otimes a_0 \otimes
		  \ldots \otimes a_{n-1}.
$$
Its coinvariants
$C^\sigma_n/\mathrm{im}\,(\mathrm{id}-\t)$
form a quotient complex of $(C^\sigma_n,\b)$
whose homology
is $HC_n^\sigma(A)$ ($k$ should contain
$ \mathbb{Q} $ and $\sigma$ should
be diagonalisable, otherwise the result might
be not what one wants).
As a consequence,
a linear functional
$\psi : C_n^\sigma \rightarrow k$
with $ \psi (\mathrm{im}\, \b)=0$ descends
to a functional on
$HC^\sigma_n(A)$ if it is invariant under $\t$,
$\psi = \psi \circ \t$. Clearly, $\psi$
also induces a functional on
$H_n(A,{}_\sigma A)$, but this might
vanish even when the one on
$HC^\sigma_n(A)$ doesn't
(namely when there exists
$\chi : C_{n-1}^\sigma \rightarrow k$
with $\psi = \chi \circ \b$, but no such
$\chi$ which is cyclic,
$\chi=\chi \circ \t$). Such a functional
on $HC_n^\sigma(A)$ that vanishes on
$H_n(A,{}_\sigma A)$
corresponds to the above described
ballast in cyclic homology,
it vanishes in the classical case
on the leading term
$\Omega^n(X)/\mathrm{im}\, \d$ of
$HC_n(k[X])$ and is rather
a functional on some $HC^\sigma_{n-2k}(A)$,
$k>0$, that is promoted to a functional on
$HC^\sigma_n(A)$ using the periodicity
operation $\mathsf{S}$.

\section{The case of the Podls\'s sphere}
Let now $A$ be again the Podle\'s sphere.
Schm\"udgen and Wagner have constructed
in \cite{sw} a nontrivial cyclic 2-cocycle
on $A$ which later was shown by Hadfield
to be trivial when viewed on
Hochschild homology. It is now natural to ask
whether our volume form $ \varphi $
constructed above does also
give rise to a nontrivial functional
on cyclic homology.

It is easily checked
that a functional $\psi : C^\sigma_n
\rightarrow k$ vanishing on $ \mathrm{im}\, \b$
is cyclic if and
only if $\varphi (1,a_1,\ldots,a_n)=0$ for
all $a_1,\ldots,a_n \in A$. Using this one sees that
our $ \varphi $ itself is not cyclic since by
(\ref{shock}) we have
$$
		  \varphi (1,x_1,x_{-1})=q^{-1} \neq 0.
$$
However, we can alter $\varphi$ by a
coboundary to make it cyclic,
so the problem is only a matter of
representing
the functional on $H_2(A,{}_\sm A)$
properly:
\begin{lemma}
Define $ \eta := \phi \circ \b :
C_2^\sm \rightarrow k$, where
the linear functional
$\phi : A \otimes A \rightarrow k$
vanishes on all $e_{kl} \otimes e_{mn}$
except on the following ones:
$$
		  \phi (1,x_0):=\frac{1}{q^{-2}-1},\quad
		  \phi (1,x_0^2):=\frac{1}{q-q^{-1}},\quad
		  \phi(x_0,x_0):=\frac{1}{2(q-q^{-1})}.
$$
Then $\eta$ induces the trivial functional
on $H_2(A,{}_\sm A)$ and
$\varphi + \eta$ is cyclic.
\end{lemma}
\begin{pf}
By its definition, $ \eta $ vanishes on
$ \mathrm{im}\, \b$ and defines the trivial
functional on $H_2(A,{}_\sm A)$.
By (\ref{shock}), the only chain of the form
$1 \otimes e_{kl} \otimes e_{mn}$
on which $ \varphi $ does not
vanish is
$1 \otimes x_1 \otimes x_{-1}$, where
$ \varphi $ has the value $q^{-1}$,
and one easily checks using
$$
		  \eta (1,a_1,a_2)=
		  \phi (a_1,a_2)
		  -\phi (1,a_1a_2)
		  +\phi (\sm(a_2),a_1)
$$
that similarly $ \eta (1,e_{kl},e_{mn})$
vanishes except when
$e_{kl} \otimes e_{mn}=x_1 \otimes x_{-1}$
and then it equals      $-q^{-1}$. The result
follows.
\end{pf}

config: arxiv


TITLE: Complex number calculations given values for z1 and z2
QUESTION [0 upvotes]: I have the question:

Given that $z_1 = 4 - 3i$  and $z_2 = -3 + 4i$, determine:
  a) $2z_1 - z_2$
  b) $z_1z_2$
  c) $z_1\overline{z_1}$

Here is my attempt. Is it correct?
a)
$$\begin{align}2z_1 - z_2&=2(4-3i)-(-3+4i)\\&=8-6i+3-4i\\&=8+3-6i-4i\\&=11-10i\end{align}$$
b)
$$\begin{align}z_1z_2&=(4-3i)(-3+4i)\\&=-12+16i+9i-12i^2\\&=-12+16i+9i-12(-1)\\&=-12+16i+9i+12\\&=16+9i\\&=25i\end{align}$$
c)
$$\begin{align}z_1\overline{z_1}&=(4-3i)(4+3i)\\&=16+12i-12i-9i^2\\&=16-9(-1)\\&=16-(-9)\\&=25\end{align}$$

REPLY [1 votes]: After 1 year on MSE why not start typing in Latex?
It is not that much complicated or different from normal writing for basic stuff like indices z_1=$z_1$, exponents i^2=$i^2$,z^*=$z^*$ and \\=line break.
You can go for advanced stuff as you go along with the site.
Look for instance at a transcription of $(a)$.
$2z_1-z_2\\
=2(4-3i)-(-3+4i)\\
=8-6i+3-4i\\
=8+3-6i-4i\\
=11-10i$
$2z_1-z_2\\
=2(4-3i)-(-3+4i)\\
=8-6i+3-4i\\
=8+3-6i-4i\\
=11-10i$

Anyway, notice that for $z=x+iy$ then $$zz^*=x^2+y^2$$
You can use this shortcut instead of doing $\require{cancel}(x+iy)(x-iy)=x^2-\cancel{ixy}+\cancel{ixy}-i^2y^2=x^2-(-1)y^2=x^2+y^2$.
So $z_1z_1^*=4^2+3^2=16+9=25$.
And remark that $z_2=4i-3=i(4+3i)=i(4-3i)^*=iz_1^*$
So $z_1z_2=i(z_1z_1^*)=i(25)=25i$


TITLE: Wasserstein distance for two normal cdfs
QUESTION [2 upvotes]: Just came to know about a fact that the Wasserstein distance between two cumulative distribution functions, $F_1$ and $F_2$, can be expressed as: $$\int_{-\infty}^{\infty} |F_1 (x)-F_2(x)| dx$$.
Is it possible to compute this distance in case $F_1$ and $F_2$ are two normal distributions, with constant variance $1$ and means $\mu_1 \neq \mu_2$ by hand?
I have no clue how to work with the absolute value along with the integral which does not seem to have a closed form!

REPLY [3 votes]: Suppose that $\mu_2 > \mu_1$.
The graph of density of $N(\mu_1, 1)$ lies to the left of the graph of density of $N(\mu_2, 1)$. As it was already mentioned by Jean Marie, it follows that $F_1 > F_2$.
It's known that if $\xi$ has distribution function $F(x)$ then $\mathbf{E}\xi = \int_{-\infty}^{0} (- F(x))dx + \int_{0}^{\infty} (1  - F(x))dx$.
Suppose that $\xi_i \sim N(\mu_i, 1)$. We have
$$\int_{-\infty}^{\infty} |F_1(x) - F_2(x)|dx = \int_{-\infty}^{\infty} (F_1(x) - F_2(x))dx =$$
$$= \int_{-\infty}^{0} (- F_2(x))dx -  \int_{-\infty}^{0} (- F_1(x))dx + \int_{0}^{\infty} (F_1(x) - F_2(x))dx =$$
$$= \int_{-\infty}^{0} (- F_2(x))dx -  \int_{-\infty}^{0} (- F_1(x))dx + \int_{0}^{\infty} ((1  - F_2(x)) - (1-F_1(x)))dx =$$
$$=\int_{-\infty}^{0} (- F_2(x))dx + \int_{0}^{\infty} (1  - F_2(x))dx  -$$
$$ -  \bigg( \int_{-\infty}^{0} (- F_1(x))dx + \int_{0}^{\infty} (1  - F_1(x))dx
 \bigg)= $$
$$= \mathbf{E}\xi_2 - \mathbf{E}\xi_1 = \mu_2 - \mu_1.$$
It follows that the answer is $|\mu_2 - \mu_1|.$


TITLE: Two articulations points connected by an edge. Must this edge be a bridge?
QUESTION [2 upvotes]: Assume wie have an undirected Graph $G$ (not a multigraph). Assume the vertex $u$ and $v$ are  articulation points and there exists an edge $e$ between $u$ and $v$. I believe that the edge $e$  does not have to be a
bridge. However, I simply cannot find a counterexample

REPLY [1 votes]: @Especially Lime has already given the simplest example while I was trying to draw the above graphs in Paint but I still find it insightful to show how to find a counter-example.
The graph $G$ is the first example I could think of where $u$ and $v$ are clearly cut vertices and $e$ is a bridge. Then we can try to add an edge to this graph to make $e$ not a bridge. In order to do that, we can first try to add an edge to the graph (shown in red) where this new edge $f$ connects the components of $G-e$ so that $e$ is not a bridge anymore. Then we can check whether $u$ and $v$ are still cut vertices or not in $G+f$. In my example, they are still cut vertices so we found a counter-example. And if that wasn't the case, we should have added the new edge to somewhere else. In the worst case, we might have to change our first graph $G$.


TITLE: Irreducible algebraic sets with intersecting parts
QUESTION [2 upvotes]: Let $V = V(F)$ be an irreducible hypersurface in $A^n(k)$.
To show:
If $W$ is an irreducible algebraic set in $A^n(k)$ with $V \subset W$, then $V = W$.
The ideas I got so far:
Since $V, W$ are irreducible, the generated ideals, $I(V)$ and $I(W)$, are prime.
Since $V, W$ are algebraic sets with $V \subset W$, $I(V) \supset I(W)$.
Is there a possibility to proof $I(V) \subset I(W)$ with the help of the information that those ideals are prime / $V, W$ are irreducible?

REPLY [0 votes]: Well, let's think: the most annoying thing to say is you probably want $W$ not to be all of $\mathbb{A}^n_k$.
Now, the inclusion $V(f) \subseteq V(I)$ is equivalent to $(f) \supseteq I$. Suppose $(f) \supsetneq I$, i.e. $f \notin I$. Well, take any nonzero $i \in I$, and write it as $f \cdot p$. By the fact that $I$ is prime, we deduce $p \in I$. Writing $p = f \cdot p'$, we deduce $p'$ is in $I$. Continuing in this fashion, we deduce $p$ is divisible by arbitrarily high powers of $f$, a contradiction.


TITLE: Cantor space with prefix metric
QUESTION [0 upvotes]: Consider the metric space $\{$ $a_0a_1a_2....$ $:$ $a_i=0,1$ $\}$
with the metric $d(x,y)=min(I: a_i\neq b_i)$ and 0 otherwise.
Edit: I meant to say $d(x,y)=\frac{1}{2^{min(I:a_I\neq b_I)}}$ when $x\neq y$.
Show that the space is complete.
I'm not really sure how to construct a limit, perhaps there's a theorem I must consider? May I have some hints?

REPLY [0 votes]: Let $(x_n)_n$ be a Cauchy sequence in $2^w$. So there exists $N$ such that whenever  $m,n\geq N$, we have $d(x_n,x_m)<\frac{1}{2^K}$.
This is equivalent to saying that  $x_N|i=x_m|i$ are equal for all $i\leq K$ for all $m\geq N$. Define $x\in 2^w$ to be such that $x|i=x_N|i$ for all $i\leq K$.
So choosing $K$ to be such that $\frac{1}{2^K}<\epsilon$. whenever $n\geq N$, we have $x_n|i=x|i$ for all $i\leq K$. Thus, $d(x,x_n)\leq\frac{1}{2^K}<\epsilon$.
Note, the following lemma has been used:
Lemma 1: For $x,y \in 2^w$, $d(x,y)<\frac{1}{2^N}$ if and only if $x_i=y_i$ for all $i\leq N$.
Proof: Suppose there exists $i\leq N$ such that $x_i\neq y_i$. Put $K=min(j : x_j\neq y_j)$. Therefore, $d(x,y)=\frac{1}{2^K} \geq \frac{1}{2^i}$ $\geq$ $\frac{1}{2^N}$. Contradiction. The converse holds similarly.


TITLE: Prove that $n^k < 2^n$ for all large enough $n$
QUESTION [7 upvotes]: If $k\ge 2$ is an integer,
prove by elementary means
(no calculus or limits)
that
there is a $N(k)$
such that
$n^k < 2^n$
for all integers $n \ge N(k)$.
Give an explicit form for $N(k)$.

REPLY [2 votes]: Pick $n > 2k$. Then,
$$
2^n > \binom{n}{k + 1} = \frac{n (n - 1) \cdot \ldots \cdot (n - k)}{(k+1)!}
\geq \frac{n^{k + 1}}{2^{k+1} (k+1)!}
$$
by a trivial combinatorial argument for the first inequality, and using $n - k > n/2$ in the second inequality.
Therefore $2^n > n^k$ as soon as
$$
n > 2^{k + 1} (k + 1)!
$$


text:
\begin{document}

\title[Order-to-chaos transition]{Order-to-chaos transition in the model of quantum pendulum subjected to noisy perturbation}

\author{D.~V.~Makarov, L.~E.~Kon'kov}

\address{
Laboratory of Nonlinear Dynamical Systems,\\
Pacific Oceanological Institute of the Russian Academy of Sciences,\\
43 Baltiiskaya st., 690041 Vladivostok, Russia, URL: http://dynalab.poi.dvo.ru}

\begin{abstract}

Motion of randomly-driven quantum nonlinear pendulum is considered.
Utilizing one-step Poincar\'e map, we demonstrate that classical phase space
corresponding to a single realization of the random perturbation
can involve domains of finite-time stability.
Statistical analysis of the finite-time evolution operator (FTEO) is carried out
in order to study influence of finite-time stability on quantum dynamics.
It is shown that domains of finite-time stability give rise to ordered
patterns in distributions of FTEO eigenfunctions.
Transition to global chaos is accompanied by smearing of these patterns;
however, some of their traces survive on relatively long timescales.

\end{abstract}

\vspace{2pc}
\noindent{\it Keywords}: finite-time stability, random perturbation, quantum chaos, one-step Poincar\'e map, finite-time evolution operator

\section{Introduction}

It is well-known that deterministic classical systems with few
degrees of freedom can exhibit complicated chaotic behaviour that is
similar to behaviour of systems with intrinsic random fluctuations.
However,
the methods used for studying deterministic chaotic systems
and systems under stochastic driving differ substantially.
Representing the external perturbation as some random process,
we add uncertainty into the equations of motion.
Each realization of the perturbation creates an unique
trajectory of the system, therefore, efficient description of the
system's response implies the usage of statistical analysis.

Nevertheless, single realizations of the random perturbation can be considered
as deterministic functions. As long as the temporal Fourier spectrum of the perturbation
is broad, there is no way for survival of impenetrable stable domains in phase space \cite{WW-TB,Abdullaev11}.
However, statistical analysis of the finite-time Lyapunov exponents in randomly-driven
systems shows that remarkable fraction of phase space area maintains stability on timescales
which exceed significantly the so-called Lyapunov time

\begin{equation}
 t_\mathrm{L}=\frac{1}{\lambda_L},
\end{equation}

where $\lambda_L$ is the global Lyapunov exponent calculated with $t\to\infty$ \cite{WT01,Laffargue2013}.
Such trajectories form bundles like branched flows in quantum point contacts \cite{Topinka2000,Topinka2001,Kaplan-PRL02,BoLiu13},
or coherent ray clusters in ocean acoustics \cite{Chaos}.
One shouldn't confuse this kind of coherent phenomena with the domains of particle clusterization
in random fields \cite{Klyatskin_Gurarie}.
The main difference is the location of initial conditions for particles.
In the case of the coherent clusters, the initial conditions belong to continuous
manifolds in phase space.
Location of that manifolds depends on particular realization of the perturbation.
Such manifolds can be found out in various ways. For example, one can calculate
the map of finite-time Lyapunov exponents in phase space \cite{OM,Finn_Apte-Chaos},
or compute eigenfunctions of the transfer operator \cite{Froyland-PRL}.
It turns out that system's behaviour under a single realization can significantly
differ from the picture obtained via statistical averaging.
There can occur phenomena which are typical for deterministic systems, for example,
intermittency and capturing into dynamical traps \cite{Zas,Traps,Tomc_Laksh}.
Thus, one needs some general approach which can be used
for both deterministic and noisy cases.

The problem of interrelation between deterministic and statistical approaches
also arises in quantum systems whose classical counterparts exhibit chaotic behaviour.
Periodic orbit theory \cite{Gutzwiller,Stockman} provides classical interpretation of
quantum spectra and, in addition, reveals some non-classical features,
like scars of wavefunctions \cite{Heller}.
One can suggest that some peculiarities of deterministic quantum systems should
manifest themselves in quantum systems involving classical noise.
The issue of particular interest is how deterministic phenomena associated with periodic orbits
are revealed under stochastic driving, when there is no periodic orbits in the strict sense.

In the present paper, we demonstrate approach that allows one to analyze quantum systems under weak random perturbation
in the framework of deterministic theory.
In that approach, it is implied that system's behaviour possesses some features which are common for all typical
realizations of the perturbation.
In this way, any realization is treated as a deterministic process with known spectral properties.
On the classical level, our approach is based on the one-step Poincar\'e map originally introduced in \cite{JPA,PRE73}.
It serves as a tool for finding out phase space patterns repeating themselves after some time interval.
Some of these patterns have regular form and can be referred to as domains of finite-time stability.
We use the quantum counterpart of the one-step Poincar\'e map, the so-called finite-time evolution operator,
for exploring manifestations of finite-time stability in quantum motion.
Mathematically equivalent approach was used for studying sound propagation in
a randomly-inhomogeneous oceanic waveguide \cite{UFN,PRE87}.
In the present paper, we use this approach for a purely quantum problem, namely for the quantum nonlinear pendulum
subjected to broadband perturbation.
Our main goal is to study how domains of finite-time stability influence quantum dynamics.

It is worth mentioning that the condition of invariance under translation over finite time interval
is too restrictive, and there can be phase space patterns which
don't satisfy it but correspond to non-chaotic dynamics.
A striking example is the so-called branched flows in quantum point contacts \cite{Kaplan-PRL02,BoLiu13}.
Our approach is rather designed for oscillatory motion subjected to external noise.
In particular, it can be implemented in the problems of noise-driven dissociation or ionization \cite{Singh_Kenfack_Rost-PRA08,Kenfack08-NJP,Feng_Chu},
where regular phase space domains prevent transitions into unbounded states, therefore,
efficient destruction of these domains is of great importance.
The same problem arises in randomly-driven quantum ratchets with cold atoms \cite{PLA}.

The paper is organized as follows.
The next section describes the model under consideration.
In section \ref{Onestep}, we study classical motion of a randomly-driven pendulum by means of
one-step Poincar\'e map.
Signatures of classical finite-time stability in quantum dynamics are explored
in section \ref{FTEO}. In Conclusion, we summarize the results obtained.

\section{Model}
\label{Model}

Consider the quantum Hamiltonian

\begin{equation}
\eqalign{
 &\hat H = -\frac{\hbar^2}{2}\frac{\partial}{\partial x} + U(x) + \varepsilon V(x,t),\\
 U(x) = -&\cos{x}, \quad V(x,t) = f(t)\sin{x} - f(t+\Delta)\cos{x}},
 \label{qH}
\end{equation}

where $\varepsilon\ll 1$, and $f(t)$ is so-called harmonic noise \cite{HN,Anischenko}.
The corresponding Schr\"odinger equation reads

\begin{equation}
 i\hbar\frac{\partial\Psi}{\partial t} = \hat H\Psi.
 \label{shrod}
\end{equation}

In the present paper we simplify the analysis by considering only quantum states with zero quasimomentum.
It corresponds to the periodic boundary conditions $\Psi(-\pi,t)=\Psi(\pi,t)$.
This simplification is quite reasonable if we deal with semiclassical regime when energy bands are flat, and tunneling between
neighbouring potential wells is fairly weak. It demands the Planck constant to be small, therefore,
we set $\hbar=0.1$.
The model (\ref{qH}) was used in \cite{PLA} in the context of quantum ratchet phenomena.

Harmonic noise is described by coupled stochastic differential equations

\begin{equation}
 \dot f=y,\quad
 \dot y=-\Gamma y-\omega_0^2f + \sqrt{2\beta\Gamma}\xi(t),
 \label{ou2d}
\end{equation}

where $\Gamma$ is a positive constant, and
$\xi(t)$ is Gaussian white noise.
The terms $f(t)$ and $f(t+\Delta)$ in (\ref{qH})
correspond to identical realizations of harmonic noise and
differ only by the temporal shift $\Delta$.
The first two moments of harmonic noise are given by

\begin{equation}
 \left<f\right>=0,\quad
 \left<f^2\right>=\frac{\beta}{\omega_0^2}.
\end{equation}

We set $\beta=1$, that is, the perturbation strength is solely determined
by the parameter $\varepsilon$.
In the case of low values of $\Gamma$,
the power spectrum of harmonic noise has the peak at the frequency

\begin{equation}
 \omega_{\mathrm{p}}=\sqrt{\omega_0^2-\frac{\Gamma^2}{2}}.
\label{w_p}
\end{equation}

Width of the peak is given by the formula

\begin{equation}
\Delta\omega = \sqrt{\omega_{\mathrm{p}}+\Gamma\omega'}-
\sqrt{\omega_{\mathrm{p}}-\Gamma\omega'},
 \label{width}
\end{equation}

where $\omega'=\sqrt{\omega_0^2-\Gamma^2/4}$.
As $\Gamma\to 0$, $f(t)\to \sin(\omega_0t+\phi_0)$, where $\phi_0$ is determined
by initial conditions in (\ref{ou2d}).
Setting $f(0)=1$, $y(0)=0$, and

$ \Delta=\pi/(2\omega_0)$,

one can easily find that
$V(x,t)=\sin(x+\omega_0t)$ in the case of $\Gamma=0$.
Hence, it turns out that, for $\Gamma>0$, $V(x,t)$
behaves like a plane wave whose amplitude and
phase velocity fluctuate with time. The plane-wave acts as a dragging
force for particles and leads to the onset of directed current, i.~e. the ratchet effect.
This kind of ratchets is known as travelling potential ratchets \cite{JETPL,PRE75,JTPL08,EPJB}.
In the semiclassical regime, direction of the current coincides with the direction of
the perturbation phase velocity, provided dynamical barriers preventing the transition of particles
into infinite regime are destroyed. More intricate behaviour is observed in the deep quantum regime \cite{Fromhold-PRA13}.
In the present work we use the following values
of parameters: $\omega_0=1$, $\Gamma=0.1$ and $\varepsilon=0.05$.

\section{One-step Poincar\'e map}
\label{Onestep}

Let's begin with the classical level and consider the classical counterpart of the Hamiltonian (\ref{qH})

\begin{equation}
H=\frac{p^2}{2}+U(x)+\varepsilon V(x,\,t).
\label{ham-st}
\end{equation}

Corresponding equations of motion read

\begin{equation}
\frac{dx}{dt}=p,\quad
\frac{dp}{dt}=-\frac{dU}{dx}-\varepsilon\frac{dV}{dx}.
\label{sys-xp}
\end{equation}

Let's consider some arbitrarily chosen realization of $V(x,t)$.
Then we can treat $V(x,t)$ as a deterministic function and
refer to (\ref{sys-xp}) as the system of ordinary differential equations.
As $V(x,t)$ is an oscillating function of time, the domains of finite-time stability
may involve components which transform to themselves without mixing in course of evolution from $t=0$
to $t=\tau$.
These components can be found out by means of the one-step Poincar\'e map \cite{JPA,PRE73,Gan1}

\begin{equation}
p_{i+1}=p(t=\tau;\,p_i,x_i),\quad
x_{i+1}=x(t=\tau;\,p_i,x_i),
\label{map}
\end{equation}

where $p(t=\tau;\,p_i,x_i)$ and $x(t=\tau;\,p_i,x_i)$ are
solutions of  (\ref{sys-xp}) with initial conditions
 $p(t=0)=p_i$, $x(t=0)=x_i$.

 \begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.3\textwidth]{p04a.eps}
\includegraphics[width=0.3\textwidth]{p04b.eps}\\
\includegraphics[width=0.3\textwidth]{p04c.eps}
\includegraphics[width=0.3\textwidth]{p04d.eps}
\caption{Phase space portraits constructed via one-step Poincar\'e map with $\tau=4\pi$.
Figures (a)-(d) correspond to different realizations of harmonic noise.
}
\label{fig-p04}
\end{center}
\end{figure}

 One-step Poincar\'e map is equivalent to the usual Poincar\'e map
with the Hamiltonian

\begin{equation}
\bar H=\frac{p^2}{2}+U(x)+\varepsilon\tilde V(x,\,t),
\label{ham-period}
\end{equation}

\begin{equation}
\tilde V(x,\,\bar t+n\tau)=V(x,\,\bar t),\quad
0\leqslant \bar t\leqslant\tau,
\label{xi-sr}
\end{equation}

$n$ is an integer.
Thus, we replace the original system by the equivalent
time-periodic one. Validity of this replacement is provided by the time restriction
to the interval $[0:\tau]$. An alternative approach for generalization of Poincar\'e map
onto stochastic dynamical systems was offered in \cite{Hitczenko,Berglund}.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.3\textwidth]{p20a.eps}
\includegraphics[width=0.3\textwidth]{p20b.eps}\\
\includegraphics[width=0.3\textwidth]{p20c.eps}
\includegraphics[width=0.3\textwidth]{p20d.eps}
\caption{Phase space portraits constructed via one-step Poincar\'e map with $\tau=20\pi$.
Figures (a)-(d) correspond to different realizations of harmonic noise.
}
\label{fig-p20}
\end{center}
\end{figure}

Following the analogy with the usual Poincar\'e map, one can deduce the main property of one-step Poincar\'e map:
{\em if a trajectory of (\ref{map}) forms a closed continuous curve in phase space,
any point belonging to it is an initial condition for a trajectory of (\ref{sys-xp}) that
remains stable at $t=\tau$.}
The inverse statement is not generally true, therefore, one-step Poincar\'e map
yields a sufficient but not necessary criterion of stability.
This means that one-step Poincar\'e map basically underestimates area of regular domains.

As we artificially reduce the problem to a time-periodic one,
theory of time-periodic Hamiltonian systems can be invoked \cite{JPA,PRE73}.
Phase space structure of one-step Poincar\'e map is determined
by resonances

\begin{equation}
 m_1\tau = m_2T,
 \label{rescond}
\end{equation}

where $T$ is the period of unperturbed motion. Period is function of the action variable  defined as \cite{Zas}

\begin{equation}
 I = \frac{1}{2\pi}\oint p\,dx.
\end{equation}

Each resonance has certain width in the space of the action.
Width can be calculated using the theory of nonlinear resonance. It can be shown that the distance between neighbouring dominant nonlinear
resonances decreases with increasing $\tau$ as $\tau^{-1}$ \cite{JPA}. This results in resonance overlapping
and gradual transition to global chaos according to the Chirikov criterion \cite{Chirikov79}.

Figures \ref{fig-p04} and \ref{fig-p20} demonstrate phase portraits constructed by means of map (\ref{map})
with $\tau=4\pi$ and $\tau=20\pi$, respectively. Notably, figures corresponding to the same value of $\tau$
but different realizations of harmonic noise represent very similar patterns with nearly the same fraction of regular phase space area.
In the case of $\tau=4\pi$ (see Fig.~\ref{fig-p04}), the central part of phase space
maintains stability for all realizations of harmonic noise.
This domain corresponds to the vicinity of the stable equilibrium point of the unperturbed system.
Chaos emerges in the neighbourhood of the unperturbed separatrix.
Increasing of $\tau$ to $20\pi$ results in remarkable shrinking of the regular phase space area,
as it is illustrated in Fig.~\ref{fig-p20}.
Phase space region corresponding to finite motion becomes submerged into the
chaotic sea. Stable domains inside the unperturbed separatrix may survive only as small islands.
For some realizations, they disappear completely. For example, the internal part of the chaotic sea presented
in Fig.~\ref{fig-p20}(d) is almost uniform, without any apparent islands.
Further increasing of  $\tau$ results in complete disappearance of stable islands in
the phase space region enclosed by the unperturbed separatrix.

\section{Finite-time evolution operator}
\label{FTEO}

Phase space portraits presented in the preceding section indicate fast destruction of stable domains.
Let's consider how this process is revealed in quantum dynamics.
Quantum counterpart of one-step Poincar\'e map is the  operator $\hat G$ defined as

\begin{equation}
 \hat G(\tau)\bar\Psi(x)=\exp\left(-\frac{i}{\hbar}\hat{H}\tau\right)\bar\Psi(x)
 =\left.\Psi(x,\,t)\right\vert_{t=\tau},
\label{evolution}
\end{equation}

where $\bar{\Psi}(x)=\Psi(x,\,t=0)$.
Hereafter we shall refer to  $\hat G$ as the finite-time evolution operator (FTEO).
Each realization of harmonic noise creates its own realization of the FTEO.
FTEO was firstly utilized in \cite{Kol97} for the problem of noise-driven quantum diffusion.
Wave analogue of the FTEO was used in  \cite{Levelspacing,Tomc11,UFN,PRE87,Hege13}.

Peculiarities of classical phase space should be reflected in spectral properties
of the FTEO.
Eigenvalues and eigenfunctions of the FTEO satisfy the equation

\begin{equation}
\hat G\Psi_m(x)=g_m\Psi_m(x)\equiv e^{-i\epsilon_m/\hbar}\Psi_m(x).
\label{eigen}
\end{equation}

Quantity $\epsilon_m$ is the analogue of quasienergy in time-periodic quantum systems.
As increasing of $\tau$ results in destruction of regular domains, one should expect
transition in statistics of level spacings $s=\epsilon_{m+1}-\epsilon_m$ from
Poissonian to Wigner-like regime \cite{Kol97}. This expectation can fail in the presence
of periodic-orbit bifurcations \cite{Levelspacing,UFN}.
Moreover, analysis of level spacing statistics doesn't provide
accurate estimate of the regular phase space area \cite{PRE87}.
In this way, analysis of FTEO eigenfunctions seems to be more robust way.
To facilitate the analysis,
eigenfunctions $\Phi_m$  can be expanded over eigenstates of the unperturbed potential

\begin{equation}
 \Phi_m(x) = \sum\limits_n c_{mn}\phi_n(x).
 \label{fl_expand}
\end{equation}

Chaos implies extensive  transitions between energy levels \cite{BerKol}, therefore,
a chaos-assisted eigenfunction of FTEO should be compound of many unperturbed eigenstates.
Thus, one can use the participation ratio

\begin{equation}
\nu = \left(
\sum\limits_{m}\vert c_{mn}\vert^4
\right)^{-1},
 \label{npc}
\end{equation}

as  measure of ``chaoticity''.
Phase space region associated with an eigenfunction can be found by means of the parameter \cite{Viro05}

\begin{equation}
\mu=\sum\limits_{m=1}^M\vert c_{mn}\vert^2m.
 \label{muu}
\end{equation}

Indeed, the formula $\left<I\right>=\hbar(\mu+1/2)$ yields the expectation value of the classical action corresponding to the eigenfunction.
Parameters $\nu$ and $\mu$ provide suitable classification of eigenfunctions and can be used for tracking the transition from order
to chaos by means of numerical simulation.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.34\textwidth]{mu_nu1.eps}
\includegraphics[width=0.34\textwidth]{mu_nu2.eps}\\
\includegraphics[width=0.34\textwidth]{mu_nu3.eps}
\includegraphics[width=0.34\textwidth]{mu_nu4.eps}
\caption{Distribution of FTEO eigenfunctions in the  $\mu$-$\nu$ plane.
Values of $\tau$:
(a) $4\pi$, (b) $10\pi$, (c) $20\pi$, (d) $100\pi$.
}
\label{fig-munu}
\end{center}
\end{figure}

Numerical simulation was conducted with the ensemble of 100 realizations of the FTEO.
100 eigenfunctions with the lowest values of $\mu$ are taken into account
for each realization.
An informative view is provided by distributions of FTEO eigenfunctions
in the $\mu$-$\nu$ space.
These distributions corresponding to different values of $\tau$
are presented in Fig.~\ref{fig-munu}.
For relatively small values of $\tau$, the dots corresponding to eigenfunctions form ordered patterns
consisted of distinct slightly biased lines. Such patterns were earlier observed
in \cite{PRE87}, where they were called ``stalagmites''. Each ``stalagmite'' is formed by eigenfunctions
localized near periodic orbits of the one-step Poincar\'e with the same location in the action space.
Transition to chaos is accompanied by delocalization of eigenfunctions and smearing of
``stalagmites''. It should be noted that smearing is partially suppressed by dynamical localization \cite{Stockman},
that is, weakly unstable periodic orbits retain the ability to trap eigenfunctions \cite{PRE76}.
For $\tau=4\pi$, stalagmites occur in the whole range of $\mu$ values, except for the vicinity of $\mu=25$.
This value of $\mu$ corresponds to the phase space region near the unperturbed separatrix of the pendulum,
where the classical chaotic sea is originated initially.
As $\tau$ increases, the smeared domain grows indicating gradual transition to chaos due to overlapping
of classical resonances (\ref{rescond}).
Notably, chaos-assisted delocalization firstly emerges in the range of small values of $\mu$, corresponding to finite motion.
This infers efficient destruction of invariant curves impeding transition between finite
and infinite regimes. Results of Ref.~\cite{PLA} show that the destruction of invariant curves leads to the onset of directed current.

Nevertheless, traces of eigenfunctions with good persistence to chaos are visible even for $\tau=100\pi$.
For instance, there is a small slightly smeared horizontal stripe near $\mu \simeq 80$ (see Fig.~\ref{fig-munu}(d))
corresponding to infinite motion with relatively high velocities.
That phase space region is characterized by inequality $T\ll\tau$ that anticipates
weak influence of resonances (\ref{rescond}), and, hence, weakness of chaos induced by their overlapping.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.4\textwidth]{npc.eps}
\caption{
Fractions of eigenfunction ensemble, corresponding to different regimes of localization, vs time.
}
\label{fig-npc}
\end{center}
\end{figure}

As domains of finite-time stability in phase space give rise to FTEO eigenfunctions
with small $\nu$, one can estimate their contribution using the cumulative distribution

\begin{equation}
F(\nu)=\int\limits_{1}^{\nu}\rho(\nu')\,d\nu',
\label{cumul1}
\end{equation}

where $\rho(\nu')$ is the corresponding probability density function.
We refer to the case $\nu\le 2$ as the regime of strong localization.
Indeed, inequality $\nu\le 2$ implies that a FTEO eigenfunction is mainly contributed from
one or two unperturbed eigenstates.
In addition, let's consider two regimes of moderate localization: $\nu\le 5$ and $\nu\le 10$.
The sense of the latter two criteria becomes apparent if we take into account that
finite motion corresponds to the 25 lowest unperturbed eigenstates.
Thus, the inequality $\nu\le 5$ ($\nu\le 10$) picks out FTEO eigenfunctions
occupying less than 20 percents (40 percents) of the phase space area enclosed by the unperturbed separatrix.
As it follows from Fig.~\ref{fig-npc}, fraction of strongly localized eigenfunctions rapidly decays
down to nearly zero. It indicates the absence of significant long-living regular domains in classical phase space.
It should be mentioned that the opposite situation was observed in Ref.~\cite{PRE87}, where slow decay of $F(2)$
was linked to the presence of degenerate tori in phase space. In the present case there is no degenerate tori,
therefore, there is no such route to the persistence of regular domains.
Fractions of moderately localized functions decay much slower, and their impact remains significant
even at $\tau=100\pi$. It can be understood as manifestation of dynamical localization, i.~e.
partial suppression of chaos-assisted diffusion due to the interference.

\section{Conclusion}

In the present paper, we demonstrate approach designed for studying quantum manifestations
of classical finite-time stability under weak external random perturbation.
The approach is based on construction of the finite-time evolution operator (FTEO).
It is emphasized that statistical properties of
FTEO eigenfunctions are closely linked to classical phase space structure revealed by one-step
Poincar\'e map that can be regarded as the classical counterpart of the FTEO.
In particular, the stalagmite-like patterns on distributions of eigenfunctions in the space of parameters
$\mu$ and $\nu$ are related to wavefunction concentrations near the periodic orbits of the one-step Poincar\'e map.
Increasing of time results in emergence of chaos, therefore,
periodic orbits loss their stability. This leads to smearing of the  ``stalagmites''.
Chaos-assisted destruction of ``stalagmites'' represents an alternative view onto the order-to-chaos transition
in randomly-driven quantum systems.

In the present work, we consider stable domains satisfying the condition of invariance
under translation over finite time interval.
They occur owing to the same mechanism as stable islands in time-periodic systems.
Namely, they are remnants of nonlinear resonances of one-step Poincare map and arise in somewhat regular and predictable way.
To underline the latter circumstance, we present phase space portraits corresponding to different realizations,
and one can see that regular domains appear in the same ranges of the action (or energy) values.

This work is supported by the Russian Foundation of Basic Research under project 13-01-12404,
and by the Siberian and Far-Eastern Branches of the Russian Academy of Sciences under joint project 12-II-SO-07-022.
Authors are grateful to Michael Uleysky for the assistance in preparation of the manuscript.

\section*{References}

\providecommand{\newblock}{}


TITLE: Giving an asymptotically tight bound on sum $\sum_{k=1}^n (\log_2 k)^2$
QUESTION [2 upvotes]: I am trying to give an asymptotically tight bound for the sum $\displaystyle\sum_{k=1}^{n} (\text{lg}k)^{2}$, where lg denotes the base 2 logarithm.
I really have no idea where to begin, and I've tried looking at consecutive ratio terms, I've tried changing the summand to $2^{2\text{lg}\text{lg}k}$, etc.
I know that $\Theta$ is linear, so really I just need an asymptotically tight bound on $(\text{lg}k)^{2}$, and I should be fine from there. But still, no luck...
Any help, hints, etc. would be greatly appreciated.

REPLY [7 votes]: I will give two answers below -- the first one is simple, and applies if you do not care about getting the exact constant in the asymptotic (i.e., if a $\Theta(\cdot)$ result is enough). The second is a tad longer, but gives the exact leading term; and actually more (i.e., it will also give the two or three next low-order terms, in this particular case).

Method 1: Upper and lower bounds.
You have
$$
\sum_{k=\lfloor n/2\rfloor }^n (\log k)^2 \leq \sum_{k=1}^n (\log k)^2 \leq n (\log n)^2
$$
and
$$
\sum_{k=\lfloor n/2\rfloor }^n (\log k)^2 \geq \frac{n}{2}\left(\log \left\lfloor\frac{n}{2}\right\rfloor\right)^2 \geq \frac{n}{2}\left(\log \frac{n}{2} - 1\right)^2 = \frac{n}{2}\left(\log n - 2\right)^2 \operatorname*{\sim}_{n\to\infty}  \frac{1}{2} n(\log n)^2.
$$
Putting it together,
$$
\sum_{k=1}^n (\log k)^2 = \Theta\left( n \log^2 n \right).
$$

Method 2: Using integrals, often much easier to handle than sums.
Intuitively, a comparison series-integral would most likely do the trick. Indeed, $f\colon x > 1\mapsto (\ln x)^2$ is increasing, which is a good hint it'd work. (I am dropping the base 2 of the logarithm, since it is only a constant factor in the end result.)
Now, for every $k \geq 1$, for all $x\in[k,k+1)$
$$
(\ln k)^2 \leq (\ln x)^2\leq (\ln(k+1))^2
$$
so that
$$
\sum_{k=1}^{n-1} (\ln k)^2 \leq \sum_{k=1}^{n-1} \int_{k}^{k+1}(\ln x)^2 dx
= \int_{1}^{n}(\ln x)^2 dx \leq \sum_{k=1}^{n-1} (\ln(k+1))^2=\sum_{k=2}^{n} (\ln k)^2.
$$
This implies, rearranging terms (and since $\ln 1 = 0$), that
$$
\int_{1}^{n}(\ln x)^2 dx \leq \sum_{k=1}^{n} (\ln k)^2 \leq \int_{1}^{n}(\ln x)^2 dx+ (\ln n)^2
$$
and now, it only remains to compute (e.g., by integration by parts):
$$
\int_{1}^{n}(\ln x)^2 dx = \big[x(\ln^2 x - 2\ln x +2)\big]^n_1 = (1+o(1) n \ln^2 n
$$
in order to conclude:
$$
\sum_{k=1}^{n} (\ln k)^2 \operatorname*{\sim}_{n\to\infty}n \ln^2 n
$$
and, actually,
$$
\sum_{k=1}^{n} (\ln k)^2
= n \ln^2 n -2n \ln n + 2n +o(n)
$$
since the difference between the upper and lower bounding integrals is $\ln^2 n = o(n)$.

To sum up:
$$
\sum_{k=1}^{n} (\log_2 k)^2
= n \log_2^2 n -({2}{\ln 2})n\log_2 n + {2}{(\ln 2)^2}n +o(n)
$$


TITLE: Cylindrical triple-integral bounds
QUESTION [1 upvotes]: The integral to solve is as follows:

$$\iiint(x^2-y^2)^2 dV$$
Bounded in the first octan by $z=\frac{1}{x^2+y^2},x+y=2,x+y=1$.

In order to solve it and practice transition to cylindrical bounds, I tried to turn it into a cylindrical triple-integral. The bounds, as I saw it, turn into:
$z=\frac{1}{r^2}, r=\frac{1}{\cos\theta+\sin\theta},r=\frac{2}{\cos\theta+\sin\theta}$ while $0\le \theta \le \frac{\pi}{2}$ as we are in the first octant. So I get the following integral:

$$\int_0^\frac{\pi}{2}\int_\frac{1}{cos\theta+sin\theta}^\frac{2}{{cos\theta+sin\theta}}\int_0^\frac{1}{r^2}r^5 (cos^2(2\theta))dzdrd\theta$$

Solving it until the last integration leads to:

$$\frac{15}{4}\int_0^\frac{\pi}{2}\frac{\cos^2(2\theta)}{(\cos\theta+\sin\theta)^4}d\theta$$

Which leads me to believe the boundaries I have for $r$ are wrong, yet I can't figure out why - it seems logical to convert the boundaries to that. Any help in spotting the mistake will be extremely appreciated!

REPLY [1 votes]: HINT: Write the denominator as $(1+\sin2\theta)^2$. You should get an integral of something like $(\sec 2\theta-\tan 2\theta)^2$, which is quite doable.


TITLE: Error analysis and how values in references are determined
QUESTION [1 upvotes]: Question 1:Most science textbooks have appendixes that have a value for some physical property of some object. This includes diameter of electrons, viscosity of fluids, boiling points, etc. My question is, are the values presented in such appendixes (or other data bases) averages?
Question 2: Also, suppose I try to calculate the boiling point of water using equations. I then compare my result with data obtained from the CRC Handbook. How would I determine how accurate my result is? Would I use percentages or is there another way to calculate the error? Note: by percentages I mean $$\vert\frac{result-reference\,\,value}{reference\,\,value}\vert\cdot100$$

REPLY [1 votes]: 1) This varies by textbook.  A common format you'll see is h=6.62606957(29)×10−34 (from Wikipedia: Planck constant).  The digits in parentheses indicate they are uncertain.  Hence, you'd expect that h is known to at least 0.00000001/6.6260957 (pretty well known.)  Other references will explicitly state what the error bars are, or may simply cite the sources.  You'd expect the error bars to be in the references in the latter case.
2) This is a fine way to determine your accuracy.  I'm presuming, of course, that your error bars are much larger than the uncertainty in the boiling point of water.  Don't be so sure that's the case!  Water boiling point changes with air pressure, humidity, saline content...


TITLE: Limits on linear acceleration at constant torque
QUESTION [1 upvotes]: Suppose I have wheels driven by a motor. The wheels make contact with a horizontal surface, so that the wheels accelerate in translational and rotational motion.  If I want to know the linear acceleration of the wheels I use: $a=\alpha r$.
The torque on the wheel is given by:
$\tau=I \alpha \rightarrow \alpha=\frac{\tau}{I}$
The moment of inertia for a solid wheel rotating on its axis is given by $I=\frac{1}{2}mr^2$.
We get: $\alpha=\frac{\tau}{\frac{1}{2}mr^2}$ and so $a=\frac{2r\tau}{mr^2}=\frac{2\tau}{mr}$.
From this I conclude that the linear acceleration gets bigger when I take a smaller radius. But what are the limits of the linear acceleration at a constant torque when the radius gets smaller?
Lets say the torque of the motor is 1Nm, the mass is 0.1kg and the radius is 0.01m. In this case the linear acceleration is: $a=\frac{2}{0.001}=2000 m/s^2$. This seems to be way too high. Is the limit the maximum angular velocity of the motor?

REPLY [1 votes]: Naturally infinite in newtonian mechanics? If you have constant force and if mass goes to zero acceleration clearly goes to infinite(of course, in the realm of classical newtonian mech.)


TITLE: Evaluate this complex line integral
QUESTION [0 upvotes]: I think I'm close but there's a part I'm not sure of. I have to find the value of $$\int|z|^2(\bar z + i)dz$$ and $\gamma$ is the circle given by $|z - i| = 1$ with counterclockwise orientation. I already figured out that $z(t) = cos(t) + i(1 + sin(t))$ so $|z|^2 = 2 + 2sin(t)$ and $\bar z = cos(t) - i(1 + sin(t))$. However when I multiply by $z'(t)$, I get $-sin(t) + icos(t)$ but my teacher said the integrand is not supposed to be analytic. Can someone see if I made a mistake somewhere?

REPLY [1 votes]: In your notation, $\overline z+i=\cos t-i\sin t$ and $dz=i(\cos t+i\sin t)\,dt$.
Then
$$(\overline z+i)\,dz=i(\cos t-i\sin t)(\cos t+i\sin t)\,dt=i\,dt$$
so your integral is
$$\int_0^{2\pi}i(2+2\sin t)\,dt$$
etc.


TITLE: Find the radius of convergence, $a_n=f^{n}(0)/n!$ and $f(z)=z+f(z^2)$
QUESTION [0 upvotes]: Let $f$ be an analytic function such that $$f(z)=z+f(z^2)$$
Let $a_n=f^{n}(0)/n!$
Then find the radius of convergence of $\sum_{n=1}^{\infty}{a_n}z^{n}$
$f'(z)=1+f'(z^2)2z$ so $a_1=1$
$f''(z)=f'(z^2)2+f''(z^2)4z^2$ so $a_2=2/2!$
But this process become lengthy as we go to higher derivatives. Is there a more cleaver and interesting way to solve this problem?
Thanks

REPLY [2 votes]: Comparing coefficients we get $a_n=a_{2n}, n \geq 1$ and $a_1=1$. Hence $a_{2^{n}}=1$ for all $n$ and $a_n=0$ for $n$ odd, $n \neq 1$. . The radius of convergence $R$is given by $\frac 1 R=\lim \sup |a_n|^{1/n}$ and hence $R=1$.


TITLE: If $f(z)=z+a_2 z^2 + a_3 z^3 + \cdots$ and $\sum_{n\geq 2} n|a_n| \leq 1$, then $f$ is univalent
QUESTION [0 upvotes]: I am trying to understand the proof from this book of the following theorem:
$\textbf{Theorem}:$ If $f(z)= z+ \sum_{n\geq 2} a_n z^n$ is such that $\sum_{n\geq 2} n|a_n| \leq 1$, then $f$ is univalent in the unit disk $\mathbb{D}$.
In the proof, the authors first proved that the series converges and hence $f$ is an analytic function in $\mathbb{D}$. Till now I have no problem in understanding. The remaining part of the proof given in the book is as follows:
Let $|z_0|<1$, then
\begin{equation} (f(z)-f(z_0))-(z-z_0) = \sum_{n\geq 2}a_n (z^n -z_{0}^{n})  = (z-z_0)\sum_{n\geq 2} a_n (z^{n-1} +z^{n-2}z_0 + \cdots + z_{0}^{n-1}). \end{equation}
As $|z^{n-1} +z^{n-2}z_0 + \cdots + z_{0}^{n-1}|<n$ for $|z|<1$, we have $$|(f(z)-f(z_0))-(z-z_0)| < |z-z_0|\sum_{n\geq 2} n|a_n| \leq |z-z_0|.$$
According the Rouche's theorem, $f(z)-f(z_0)$ and $z-z_0$ have the same number of zeros in $\mathbb{D}$, that is $f(z)=f(z_0)$ has exactly one solution.
$\textbf{My doubt}:$ In Rouche's theorem, the inequality must hold on the boundary of the region, but in this proof $|z|<1$. So how can we apply Rouche's theorem here?
$\textbf{My explanation}: $ From the proof we can say that the inequality holds on any circle inside $\mathbb{D}$, as we can fix $|z|=\delta <1$, and $|z_0|<|z|$. We get the conclusion by allowing $\delta \rightarrow 1^-$.
$\textbf{My question}:$ Is my explpanation correct? Or I am missing out some basic fact?

REPLY [6 votes]: This is not an explanation of the steps in the book but I would like to point out that the proof in the book looks a bit silly to me. You don't require Rouche's Theorem. There is a very elementary 'High School' proof: suppose $f(z)=f(w)$. Then $z-w=a+a_2(w^{2}-z^{2})+a_3(w^{3}-z^{3})+\cdots$. Note that $|w^{n}-z^{n}| =|z-w|(|w^{n-1}+w^{n-2}z+\cdots+wz^{n-2}+z^{n-2}|< n|z-w|$ if $z \neq w$.  Using the hypothesis we  arrive at the contradiction $|z-w| <|z-w|$ if $z \neq w$.


TITLE: $\|x\|_X \leq C \|Ax\|_Y,\quad C\gt 0$ Why is $A$ injective?
QUESTION [0 upvotes]: Why does $A:X\to Y$ where $X$ is a banach space and $Y$ is a normed space, where $A$ is a surjective bounded linear operator, where:
$$\|x\|_X \leq C \|Ax\|_Y,\quad C\gt 0$$
Mean that $A$ is also an injective operator?
I can't see it. Not sure where to start, it doesn't seem like it immediately jumps out at me.

REPLY [1 votes]: If $Ax = 0$, then $0\le \|x\|_X\le C\|Ax\|_Y = 0$, so .....


TITLE: Property of a closed path
QUESTION [8 upvotes]: I have the feeling this property holds true but I cannot find a way to prove it:

Let $\gamma: [0,1] \to \mathbb{R}^n$ be a differentiable closed path, i.e. $\gamma(0) = \gamma(1)$. Then there exist two values $\lambda_1 \neq \lambda_2 \in [0,1[$ such that
$$ \left<\gamma'(\lambda_1),  \gamma(\lambda_2) - \gamma(\lambda_1) \right> \geq 0, $$
and
$$ \left<\gamma'(\lambda_2),  \gamma(\lambda_1) - \gamma(\lambda_2) \right> \geq 0. $$

Graphically this means that I can find two points on my path such that the projection of the tangent vector at these points on the line linking these two points face each other.
I would be happy to discuss how this problem could be approached :)
EDIT: I did not want to influence the reflexion in a way that would lead to a dead-end but here is the way I tried to approach the problem.
Let define the function $f: [0,1] \to \mathbb{R}$ as follows:
$$f(\lambda)
=
\left<\gamma'(\lambda),  \gamma(\lambda^{+}) - \gamma(\lambda) \right>
+
\left<\gamma'(\lambda^{+}),  \gamma(\lambda^{+}) - \gamma(\lambda) \right>,
$$
with $\lambda^{+}(\lambda) = mod(\lambda + 0.5, 1)$. Hence, for instance $\lambda^{+}(0.25) = 0.75$ and $\lambda^{+}(0.9) = 0.4$. Obviously $\lambda^{+}(\lambda^{+}(\lambda)) = \lambda$.
We have that $f$ is continuous and $f(\lambda^{+}) = - f(\lambda)$.
Therefore, one can find with the Intermediate Value Theorem a value $\lambda_1$ such that $f(\lambda_1) = 0$. If I define and $\lambda_2 = \lambda^+(\lambda_1)$, this implies that
$$
\left<\gamma'(\lambda_1),  \gamma(\lambda_2) - \gamma(\lambda_1) \right>
+
\left<\gamma'(\lambda_2),  \gamma(\lambda_2) - \gamma(\lambda_1) \right>
= 0
$$
$$
\Leftrightarrow
\left<\gamma'(\lambda_1),  \gamma(\lambda_2) - \gamma(\lambda_1) \right>
=
\left<\gamma'(\lambda_2),  \gamma(\lambda_1) - \gamma(\lambda_2) \right>.
$$
This means that I can find two points on my curve $\gamma(\lambda_1) $ and $\gamma(\lambda_2)$ such that the projection of the tangent vector at these points on the line between these points either point towards each other, either away from each other.
I am however stuck on how to prove that the former should hold (I believe it should based on intuition and a lot of examples I tried).

REPLY [3 votes]: What if you want to prove they are orthogonal? Let me rewrite the problem with $\gamma:S^1\to\mathbb{R}^n$, where $S^1$ is the circle $\mathbb{R}/\mathbb{Z}$.
Take $d:S^1\times S^1\to\mathbb{R}$ as $d(x,y)=\|\gamma(x)-\gamma(y)\|^2$, using here the Euclidean norm. This function is continuous from a compact space to $\mathbb{R}$, hence it has a maximum $(\lambda_1,\lambda_2)$. We assume $d(\lambda_1,\lambda_2)>0$ of course.
Then, since this point is a maximum, it is true that $\frac{d}{dx}d(x,\lambda_2)|_{x=\lambda_1}=0$ and $\frac{d}{dy}d(\lambda_1,y)|_{y=\lambda_2}=0$ by differentiability of $\gamma$ (I assume there is no problem at $\lambda=0$ concerning the differentiability).
Then, computing these two derivatives, you obtain exactly
$$\left<\gamma'(\lambda_1),  \gamma(\lambda_2) - \gamma(\lambda_1) \right> = 0,$$
$$\left<\gamma'(\lambda_2),  \gamma(\lambda_1) - \gamma(\lambda_2) \right> = 0.$$
This works perfectly even if if the curve is differentiable but not regular.
In general, you could have equality but not strict positivity, I am thinking about a circular path.


TITLE: Why is axiom of dependent choice necesary here? (noetherian space implies quasicompact)
QUESTION [1 upvotes]: In the proof of noetherian space implies quasicompact I am reading, it goes as follows: Let $X$ be a noetherian space.
Let $U$ be the collection of open subsets of $X$ that can be expressed as a finite union of opens sets in the covering. If $U$ does not contain $X$, then there exists an infinite ascending chain of sets in $U$ (axiom of dependent choice), hence contradiction.
I know it is similar to Why is axiom of choice needed? (Equivalent conditions for Noetherian) , but
I am just not seeing why the axiom of dependent choice is necessary here...
I would greatly appreciate some explanation. Thank you!
Ps by Noethrian space, I mean Noetheiran topological space: Every descending chain of closed subsets of $X$ eventually becomes constant.

REPLY [4 votes]: The issue is that without Dependent Choice stating that every chain is finite does not imply the existence of a maximal element.
It is a weird situation, because it means that every finite chain can be extended. However if you have an infinite sequence given, then you do know it is eventually stable.
The proof relies on being able to choose some arbitrary closed sets which is smaller, and that the induction must go through to define an infinite sequence, rather than just arbitrarily long finite sequences. And this is where the axiom of choice (or rather, dependent choice) is used.


TITLE: How to prove the unique linear map that extends a function is only injective iff the vectors of the function are linearly independent
QUESTION [1 upvotes]: Let $F$ be a field, let $V_1,V_2$ be vector spaces over F, and suppose $B = ({u_1,u_2,...,u_k})$is a basis of $V_1$ (where $k=dimV_1$).Consider functions $φ,ψ:B→V,$
and write $f$ and $g$ for the linear extensions from $V_1$ to $V_2$ of $φ$ and $ψ$ respectively.
Prove f is injective iff the vectors  $φ(u_1).... φ(u_k)$ are linearly independent.
Attempt:
Suppose the set $\{φ(v_{1}),φ(v_{2}),\ldots,φ(v_{n})\}\subset V_{1}$ is linear independent and the linear mapping $f:V_{1}\rightarrow V_{2}$ in injective. Now
\begin{align*}
& \alpha_{1}f(φ(v_{1})) + \alpha_{2}f(φ(v_{2})) + \ldots + \alpha_{n}f(φ(v_{n})) = 0 \Longrightarrow f(\alpha_{1}φ(v_{1}) + \alpha_{2}φ(v_{2}) + \ldots + \alpha_{n}φ(v_{n})) = 0\Longrightarrow\\\\
& \alpha_{1}φ(v_{1}) + \alpha_{2}φ(v_{2}) + \ldots + \alpha_{n}φ(v_{n}) = 0 \Longrightarrow \alpha_{1} = \alpha_{2} = \ldots = \alpha_{n} = 0
\end{align*}
Thus the set $\{f(φ(v_{1})),f(φ(v_{2})),\ldots,f(φ(v_{n}))\}$ is linear independent.
Now I'm stuck on this second part.
$g$ is surjective if and only if the set ${ψ(u) : u ∈ B}$ is a spanning set of $V_2$.
So I know for this one we know that since $B$ is a basis, then B is a spanning set of $V_1$ but I'm unsure how to continue? Can anyone check my proof and assist with this one?

REPLY [1 votes]: $\textbf{For the first part:}$
$\phi(u_1),...,\phi(u_k)$ independent $\implies f$ injective:
if $f(a) = f(b)$, then
$f(\sum_{i=1}^k c_i u_i) = f(\sum_{i=1}^k c_i' u_i)$
$\iff \sum_{i=1}^k c_i f(u_i) = \sum_{i=1}^k c_i' f(u_i)$
$\iff \sum_{i=1}^k (c_i - c_i') f(u_i) = 0$
$\iff \sum_{i=1}^k (c_i - c_i') \phi(u_i) = 0$
$\implies c_i-c_i'=0$ $\forall i$, since $\phi(u_1),...,\phi(u_k)$ independent
For the converse, if $\phi(u_1),...,\phi(u_k)$ dependent, then $\exists c_1,...c_k$ not all $0$ such that
$\sum_{i=1}^k c_i \phi(u_i) = 0$
$\iff f(\sum_{i=1}^k c_i u_i) = 0$
but $f(0) = 0$ since $f$ is linear
and $\sum_{i=1}^k c_i u_i \ne 0$ since not all $c_i$ are $0$ and $u_i$ are a basis
so $f$ is not injective
$\textbf{For the second part:}$
$\psi(u_1),...,\psi(u_k)$ span $\implies g$ surjective:
Let $v \in V_2$, then since $\psi(u_1),...,\psi(u_k)$ span, $\exists c_1,...,c_k$ such that
$v = \sum_{i=1}^k c_i \psi(u_i)$
$= \sum_{i=1}^k c_i g(u_i)$
$= g(\sum_{i=1}^k c_i u_i)$
For the converse, just note for $v \in V_2$,
$v = g(a) = g(\sum_{i=1}^k c_i u_i) = \sum_{i=1}^k c_i g(u_i) = \sum_{i=1}^k c_i \psi(u_i)$


text:
\chapter{Introduction}
The goal of inner model theory is to construct and analyze canonical models of set theory. The simplest example of such a model is G\"odel's constructible universe \(L\), the smallest model of set theory that contains every ordinal number. One sense in which \(L\) is canonical is that seemingly every question about the internal structure of \(L\) can be answered. For example, G\"odel proved that \(L\) satisfies the Generalized Continuum Hypothesis. This stands in stark contrast with the universe of sets \(V\), many of whose most basic properties (for example, whether the Continuum Hypothesis holds) cannot be determined in any commonly accepted axiomatic system.

To what extent does \(L\) provide a good approximation to the universe of sets? On the one hand, the principle that every set belongs to \(L\) (or in other words, \(V= L\)) cannot be refuted using the ZFC axioms, since \(L\) itself is a model of the theory \(\text{ZFC}+ V = L\). If \(V = L\), then \(L\) approximates \(V\) {\it very }well.
On the other hand, the model \(L\) fails to satisfy relatively weak large cardinal axioms. If one takes the stance that these large cardinal axioms are true in the universe of sets, one must conclude that \(V \neq L\). Moreover, it follows from large cardinal axioms that \(L\) constitutes only a tiny fragment of the universe of sets. For example, assuming large cardinal axioms, the set of real numbers that lie in \(L\) is countable.

Are there canonical models generalizing \(L\) that yield better approximations to \(V\)?  It turns out that there is a hierarchy of canonical models beyond \(L\), satisfying stronger and stronger large cardinal axioms. The program of building such models has met striking success, reaching large cardinal axioms as strong as a Woodin limit of Woodin cardinals. Based on the pattern that has emerged so far, it is plausible every large cardinal axiom has a canonical model.

At present, however, a vast expanse of large cardinal axioms are not yet known to admit canonical models. A key target problem for inner model theory is the construction of a canonical model with a supercompact cardinal. Work of Woodin suggests that the solution to this problem alone will yield an \emph{ultimate canonical model} that inherits essentially all large cardinals present in the universe. There is therefore hope that the goal of constructing inner models for all large cardinal axioms could might be achieved in a single stroke. If this is possible, the resulting model would be of enormous set theoretic interest, since it would closely approximate the universe of sets and yet admit an analysis that is as detailed as that of G\"odel's \(L\).

This dissertation investigates whether there can be a canonical model with a supercompact cardinal. To do this, we develop an abstract approach to inner model theory. This is accomplished by introducing a combinatorial principle called the Ultrapower Axiom, which is expected to hold in all canonical models. If one could show that the Ultrapower Axiom is inconsistent with a supercompact cardinal, one would arguably have to conclude that there can be no canonical model with a supercompact cardinal.

Supplemented with large cardinal axioms, the Ultrapower Axiom turns out to have surprisingly strong and coherent consequences for the structure of the upper reaches of the universe of sets, particularly above the first supercompact cardinal. These consequences are entirely consistent with what one would expect to hold in a canonical model, yet are proved by methods that are completely different from the usual techniques of inner model theory. The coherence of this theory provides compelling evidence that the Ultrapower Axiom is consistent with a supercompact cardinal. If this is the case, it seems that the only possible explanation is that the canonical model for a supercompact cardinal does indeed exist. Optimistically, studying the consequences of the Ultrapower Axiom will shed light on how this model should be constructed.
\subsection{Outline}
We now describe the main results of this dissertation. \\

\noindent {\bf \sc \cref{MOChapter}.} In this introductory chapter, we introduce UA in the context of the problem of the linearity of the Mitchell order on normal ultrafilters. We  show first that UA holds in all canonical inner models, a result that is philosophically central to this dissertation. More precisely, we prove that UA is a consequence of Woodin's Weak Comparison principle:

\begin{repthm}{WCtoUA}
	Assume that \(V = \textnormal{HOD}\) and there is a \(\Sigma_2\)-correct worldly cardinal. If Weak Comparison holds, then the Ultrapower Axiom holds.
\end{repthm}

We then show that UA implies the linearity of the Mitchell order:
\begin{repthm}{UAMO}[UA]
	The Mitchell order is linear.
\end{repthm}

Two applications of this result to longstanding problems of Solovay-Reinhardt-Kanamori \cite{Kanamori} are explained in the introduction to \cref{MOChapter}.\\

\noindent  {\bf \sc \cref{KetonenChapter}.} This chapter introduces the Ketonen order, a generalization of the Mitchell order to all countably complete ultrafilters on ordinals. The restriction of this order to weakly normal ultrafilters was originally introduced by Ketonen. The first proof of the wellfoundedness of the generalization of this order to countably complete ultrafilters is due to the author:
\begin{repthm}{KOWellfounded}
	The Ketonen order is wellfounded.
\end{repthm}

The main theorem of this chapter explains the fundamental role of the Ketonen order in applications of the Ultrapower Axiom:
\begin{repthm}{LinKetThm}
	The Ultrapower Axiom is equivalent to the linearity of the Ketonen order.
\end{repthm}
In addition, we analyze the relationship between the Ketonen order and various well-known orders like the Rudin-Keisler order and the Mitchell order. \\

\noindent {\bf \sc \cref{GMOChapter}.} The topic of this chapter is the generalized Mitchell order, which is defined in exactly the same way as the usual Mitchell order on normal ultrafilters but removing the requirement that the ultrafilters involved be normal. This order is not linear (assuming there is a measurable cardinal), and in fact it is quite pathological when considered on ultrafilters in general. The two main results of this chapter generalize the linearity of the Mitchell order to nice classes of ultrafilters:
\begin{repthm}{DoddMO}[UA]
	The generalized Mitchell order is linear on Dodd sound ultrafilters.
\end{repthm}
Dodd soundness is a generalization of normality that was first isolated in the context of inner model theory by Steel \cite{Schimmerling}. A uniform ultrafilter \(U\) on a cardinal \(\lambda\) is Dodd sound if the map \(h : P(\lambda)\to M_U\) defined by \(h(X) = j_U(X)\cap [\text{id}]_U\) belongs to \(M_U\). The concept is discussed at great length in \cref{DoddSection}.

A better-known generalization of normality is the concept of a normal fine ultrafilter (\cref{NormalFineDef}), introduced by Solovay, and underpinning the theory of supercompact cardinals. The second result of this chapter generalizes the linearity of the Mitchell order to this class of ultrafilters:
\begin{repthm}{GCHLinear}[UA]
	Suppose \(\lambda\) is a cardinal such that \(2^{<\lambda} = \lambda\). Then the generalized Mitchell order is linear on normal fine ultrafilters on \(P_{\textnormal{bd}}(\lambda)\).
\end{repthm}
Here \(P_{\textnormal{bd}}(\lambda)\) denotes the set of bounded subsets of \(\lambda\).\\

\noindent {\bf \sc \cref{RFChapter}.} We turn to another fundamental order on ultrafilters, the Rudin-Frol\'ik order. The structure of the Rudin-Frol\'ik order on countably complete ultrafilters is intimately related to the Ultrapower Axiom. For example, we point out the following simple connection:
\begin{repcor}{RFDirected}\index{Rudin-Frol\'ik order!directedness}
	The Ultrapower Axiom holds if and only if the Rudin-Frol\'ik order is directed on countably complete ultrafilters.
\end{repcor}
On the other hand, it is well-known that the Rudin-Frol\'ik order is not directed on ultrafilters on \(\omega\).

The chapter is devoted to deriving deeper structural features of the Rudin-Frol\'ik order from UA. The most interesting one is that it is locally finite:
\begin{repthm}{RFFinite}[UA]
	A countably complete ultrafilter has at most finitely many predecessors in the Rudin-Frol\'ik order up to isomorphism.
\end{repthm}
Given the finiteness of the Rudin-Frol\'ik order, it turns out to be possible to represent every ultrafilter as a finite iterated ultrapower consisting of \emph{ irreducible ultrafilters}, ultrafilters whose ultrapowers cannot be factored as an iterated ultrapower (\cref{FactorizationThm}). We apply this to analyze ultrafilters on the least measurable cardinal under UA:
\begin{repthm}{KunenUA}[UA]
	Every countably complete ultrafilter on the least measurable cardinal \(\kappa\) is isomorphic to \(U^n\) where \(U\) is the unique normal ultrafilter on \(\kappa\) and \(n\) is a natural number.
\end{repthm}
This generalizes a classic theorem of Kunen \cite{KunenLU}.\\

\noindent {\bf \sc \cref{GCHChapter}.} This chapter exposits two inner model principles that follow abstractly from UA in the presence of a supercompact cardinal:
\begin{repthm}{HODGen}[UA]
	Assume there is a supercompact cardinal. Then \(V\) is a generic extension of \(\textnormal{HOD}\).
\end{repthm}
Thus UA almost implies \(V = \text{HOD}\). This is best possible in the sense that it is consistent that UA holds and \(V\) is a \emph{ nontrivial} generic extension of HOD.

The main result of the chapter is that UA implies the Generalized Continuum Hypothesis:
\begin{repthm}{MainTheorem}[UA]
	Suppose \(\kappa\) is supercompact. Then for all \(\lambda\geq \kappa\), \(2^\lambda = \lambda^+\).
\end{repthm}
Thus UA almost implies the GCH. This is best possible in the sense that it is consistent that UA holds but CH fails.\\

\noindent {\bf \sc \cref{SCChapter1}.} This chapter initiates an analysis of strongly compact and supercompact cardinals under UA. In this chapter, we investigate the structure of the least strongly compact cardinal, introducing the ``least ultrafilters" \(\mathscr K_\lambda\), and proving that they witness its supercompactness:
\begin{repthm}{StrongSuper}[UA]
	The least strongly compact cardinal is supercompact.\\
\end{repthm}

\noindent {\bf \sc \cref{SCChapter2}.} The main result of this chapter is the \emph{ Irreducibility Theorem} (\cref{SuccessorIrredThm}, \cref{LimitIrredThm}) that relates supercompactness and irreducibility (that is, Rudin-Frol\'ik minimality) under UA. The original impetus for proving this theorem was to analyze all larger strongly compact cardinals:
\begin{repthm}{MenasUA}[UA]
	A cardinal \(\kappa\) is strongly compact if and only if it is supercompact or a measurable limit of supercompact cardinals.
\end{repthm}
We also analyze various other large cardinals using UA. For example, we consider huge cardinals (\cref{HugeThm}) and rank-into-rank cardinals (\cref{I3Thm}).


TITLE: Evaluating $\int_1^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}(x)\:\mathrm{d}y\:\mathrm{d}x$ using polar coordinates?
QUESTION [1 upvotes]: How is the following integral found using polar coordinates.
$$\int_1^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}(x)\:\mathrm{d}y\:\mathrm{d}x$$
I know the the part of the domain the circle being asked in r is from 1 to 2.
Then for theta it seems to be from $\frac{7\pi}{4}$ to $\frac{\pi}{4}$
So then I do
$\int_1^2 r\cos(\theta)(r) dr$
I get $\int r^2 \cos(\theta)$
then
$\frac{r^3}{3}\cos(\theta)$ plug in my my 2 and 1 and get.
$\int_{7\pi/4}^{\pi/4}\frac{7}{3}\cos(\theta)d\theta$
take my integration
I get
$\frac{7}{3}\sin(\theta)$ plug my numbers in and I get
$\frac{16\sqrt{2}}{6}$
but the correct answer is $2\sqrt{3}$

REPLY [1 votes]: Hint:
Note that $-\frac{\pi}{3} \leq \theta \leq \frac{\pi}{3}$ and that for a given $\theta$ the range of $r$ is from $\frac{1}{\cos(\theta)}$ to $2$.
Therefore:
$$I = \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \int_{\frac{1}{\cos (\theta)}}^2 r \cos (\theta)r \:\mathrm{d}r\:\mathrm{d}\theta \Rightarrow $$
$$I = \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \int_{\frac{1}{\cos (\theta)}}^2 r^2  \:\mathrm{d}r \cos (\theta) \:\mathrm{d}\theta$$