TITLE: partial fractions when the fraction cannot be decomposed
QUESTION [5 upvotes]: I am trying to find partial fractions of $\frac {1}{(x^2+1)^2}$. All the coefficients I get are zeros except the coefficient for the constant term which is 1, leaving me with the fraction I started with, so it seems like the fraction cannot be decomposed. How can I then go about writing this fraction as the sum $\frac {1}{2}\left [\frac {1}{x^2+1}-\frac{x^2-1}{(x^2+1)^2}\right]$? Is it just manipulation and trial and error?

REPLY [2 votes]: Just to clarify:
In cases of irreducible denomiantors over the real numbers such as your case, we treat it like a regular fraction with a repeated factor in the denominator but instead of letting the numerators of each fraction produced by the partial fraction decomposition be constant, instead we let the numerator be a polynomial of degree one lower than the factor in question.
That was rather a mouthful! To clarify, in your case:
$$\frac{1}{(x^2+1)^2}\equiv\frac{Ax+B}{x^2+1}+\frac{Cx+D}{(x^2+1)^2}$$
Finding those coefficients, however, yields $D=1$ and this is the only non zero coefficient. This is why your fraction is already in  partial fraction form.


TITLE: Some questions about the arithmetic function $\,g(n)=\sum_{(k,n)=1}\tau(k)$
QUESTION [2 upvotes]: Given the arithmetic function $\tau(n)=\sum_{d|n}1$, let's define the function
$$g(n)=\sum_{1\le k\le n\;(k,n)=1}\tau(k)$$
For $n\ge1$ the first values of $g(n)$ are $1,1,3,3,8,3,14,7,14,8,...$
Analyzing the sequence of the first $10^4$ values of the function, I could see that:

$g(n)=n\,$ only for $\,n=3,40,576,606$;
if the argument of the function is a prime $(p)$, we have $\,g(p)\sim p\cdot \log(p)$.

I ask if it's possible to prove that
$$\lim_{k\rightarrow\infty}\frac{g(p_k)}{p_k\cdot \log(p_k)}=\lim_{k\rightarrow\infty}\frac{\sum_{i=1}^{p_k-1}\tau(i)}{k\cdot \log\,k\cdot(\log\,k+\log\,\log\,k)}=1$$
Many thanks.
[same question here]

REPLY [3 votes]: In general, the summatory function
$$
S_q(x) = \sum_{\substack{n\leq x\\ (n,q)=1}} \tau(n)
$$
can be evaluated asymptotically via a standard computation using Perron's Formula. This sum comes up when studying the divisor function in coprime residue classes (see this paper, for instance). I won't write out all the details, but one can show that
$$
S_q(x) = x \left(\frac{\phi(q)}{q}\right)^2\left(\log x + 2\gamma-1+2\sum_{p\mid q} \frac{\log p}{p-1}\right) + O\left(\tau(q)x^{\delta}(\log x)^2\right)
$$
for some $\delta < 1$. Taking $x=q=n$, we find that in general
$$
\sum_{\substack{k\leq n\\ (k,n)=1}} \tau(k) \sim \frac{\phi(n)^2}{n}\log n.
$$
When $n$ is prime, this gives the desired limit.
Edit: It should be noted that
$$
\sum_{p\mid n} \frac{\log p}{p-1} \ll \log\log n,
$$
so this doesn't contribute to the leading order term.


TITLE: What does it mean for a data set to have Gaussian-distributed noise?
QUESTION [1 upvotes]: I need to find an answer for this question.
What does it mean for a data set to have Gaussian-distributed noise?
Can anyone help?

REPLY [1 votes]: I think that's normal assumption, because Gaussian distribution is easy to treat.
And also assume there are many small independent noise source in your data, then by Central Limit Theorem, your noise will be Gaussian.


text:
\section{Introduction}

The health emergency is the first challenge each country has faced in the fight to SARS-CoV-2.

The next challenge, even more critical, is the safe reactivation of the production processes to prevent economic and social crises.

Even if the vaccine should be successful, its distribution to the whole population will require more than one year, thus forcing a long coexistence with the virus.

During this time, it will be crucial to understand how the infection is spreading across the population and take the best countermeasures to contain it without further lockdowns.

The epidemic models currently used provide a macro-scale description of the spread, which allows forecasting the hospitals' load during the lockdown, but are partially ineffective when preventing the lockdown.

An effective tool to analyze the current virus spread situations and identify the presence of infected individuals is the use of swabs, which are used extensively.

However, the limited availability and the time required to test a large number of subjects make them only a partial answer to the reconstruction of the disease transmission among the population.

The scientific community agrees that contact-tracing applications could provide micro-scale data useful to contain the infection by driving the choice of the daily swab selection process.

These applications can generate millions of punctual data that can be combined with the clinical data available to the Local Health Units (LHUs) to reconstruct infection chains and networks and, thus, identify potentially infected people.\footnote{

For instance, the contact-tracing app adopted in Italy is \url{https://www.immuni.italia.it/}.}

Currently, only direct contacts with the positive subjects are tested for SARS-CoV-2, while a more accurate analysis can suggest testing those subjects which have a high probability of being infected even if they had not direct contact with the positive one.

However, dealing with millions of data is unfeasible for humans and Artificial Intelligence (AI) techniques are commonly considered the most promising technology for such tasks.

Remarkably, \citet{tsai2012security} have already adopted AI techniques for the contagion containment problem in a setting different from ours (\emph{i.e.}, fake news containment in social networks).

In the SARS-CoV-2 setting, AI can be adopted to develop predictive models from data and reasoning on them to suggest where and when to use swabs to test for infected people.

However, the potential of these techniques is not fully understood yet.

More importantly, their applicability is limited by the severe European privacy regulation that requires the information on the individuals to only reside on their personal devices in an anonymized fashion.

A natural choice to address this problem is the adoption of distributed algorithms capable of managing contacts information in a decentralized fashion~\cite{DBLP:conf/atal/AliKT05,DBLP:conf/ijcai/WuZJ13}.

Distributed algorithms are extensively studied in the multi-agent systems literature, where every agent locally controls a subset of variables and exchanges messages with the other agents to achieve a solution (\citet{shoham2008multiagent}).

\paragraph{Original Contributions} In this work, we present a distributed algorithm, namely \alg, capable, in a decentralized fashion as to avoid the violation of the privacy regulation, of identifying two classes of people whose roles are prominent in the infection from contact-tracing data: \emph{asymptomatic} subjects, \emph{i.e.}, infected subjects not showing any symptom, and \emph{presymptomatic} subjects, \emph{i.e.}, individuals in the initial phase of the disease.

The former ones are an important fraction of the infected people, and their identification is difficult, while the latter can transmit the disease with high probability.

We develop a micro-scale epidemic model based on the recent virus transmission studies by~\citet{ferretti2020quantifying}.

The \alg{} algorithm exploits this model to reconstruct infection chains present in the population using tracing app data and ranks the individuals according to the likelihood of being infected, so that the limited amount of daily swab tests available to the healthcare authorities can be used effectively.

Finally, we experimentally compare on a synthetic setting the \alg{} algorithm with currently adopted policies to select swabs.

config: arxiv


TITLE: Random variable transformation with floor function
QUESTION [0 upvotes]: Can someone help me with the following question:
For each $n\geq 1$, let $X_{n}$ be a random variable following an exponential distribution with mean $n$. Determine $F_{n}$: the distribution function of $X_{n} - 10 [\frac{X_{n}}{10}]$, where $[u]$ denotes the greatest integer lesser or equal than $u$. What is the limit of $F_{n}$ when $n \rightarrow + \infty$.
I've tried to calculate that by using Probability Total Law, but since I have the same variable both in $[X_{n}/10]$ and out the floor function, I was not able to make it.
Using computational simulation I've found that the answer should be a Uniform distribution (0, 10) but I can't prove it mathematically.

REPLY [0 votes]: $F_n$ can be computed readily. Let $Y_n=X_n-10[\frac{X_n}{10}]$.  Then for $0\le y \le 10$,
$P(Y_n \le y)=\sum_{k=0}^{\infty}P(10k\le X_n\le 10k+y)=\sum_{k=0}^{\infty}(e^{\frac{-10k}{n}}-e^{\frac{-(10k+y)}{n}})=(1-e^{\frac{-y}{n}})\frac{1}{1-e^{\frac{-10}{n}}}$  For large n, $F_n(y)\approx \frac{y}{10}$, which is uniform for $0\le y\le 10$.
.


TITLE: If X is complete, and if $p_n$ is a sequence in X. Assume that $\Sigma_{n=1}^{\infty} d(p_{n}, p_{n+1})$ converges. Show that $p_n$ converges.
QUESTION [0 upvotes]: I want to show that $p_n$ a sequence in X converges if $\Sigma_{n=1}^{\infty} d(p_{n}, p_{n+1})$ converges. Here's the outline of what I've done so far:

If X is a complete metric space, then every Cauchy sequence converges. WTS that $p_n$ is a Cauchy sequence. By triangle inequality and letting m=n+k, we have that:
$d(p_n, p_m) = d(p_n, p_{n+k}) \leq d(p_n, p_{n+1}) + d(p_{n+1}, p_{n+2}) .... d(p_{n+k-1}, p_{n+k})$.

And I'm stuck. I know I by definition of Cauchy that I need $\epsilon$ such that $\forall \epsilon >0, \exists N \text{ such that } d(p_n, p_m)< \epsilon$.
Help would be much appreciated!

REPLY [1 votes]: If $\Sigma_{n=1}^{\infty} d(p_{n}, p_{n+1})$ converges then the sequence of partial sums, $\{S_n\}$ is Cauchy where $$S_k=\Sigma_{n=1}^{k} d(p_{n}, p_{n+1})$$. Then for any given $\epsilon>0$,$\exists m_0\leq n_0\in\mathbb{N}$ such that $S_n-S_{m-1}<\epsilon$ for all $m\geq m_0, n\geq n_0\in\mathbb{N}$ with $m\leq n$. The rest you have done already!


TITLE: Existence of Lipschitz reparametrization
QUESTION [3 upvotes]: Suppose we are given a continuous path,
$$\gamma:[0,1]\rightarrow (X,d)\text{,}$$
in a metric space $(X,d)$. When we deal with differentiable enough paths in Riemann manifolds we can give a parametrization that is $l$-Lipschitz, where $l$ is the length of the path. If have that $\gamma$ has finite length $l$ in the sense that
$$l=\sup\left\{\sum_{i=0}^{n-1}d(\gamma(t_i),\gamma(t_{i+1}))\,|\, n\in \mathbb{N}; \forall i< n,t_i\in[0,1]\text{ and }t_i<t_{i+1};t_0=0;t_n=1\right\}$$
is finite, can we guarantee the existence of a parametrization that makes the path $l$-Lipschitz?
Note: The necessity for this result comes that I need this fact to prove that certain sequences of path in metric spaces have nice properties in order to obtain a limit path. And in this way proving that compact path metric spaces are geodesic metric spaces.

REPLY [4 votes]: Yes, every rectifiable path can be parametrized by arclength (and a locally rectifiable path too, if you allow the parameter domain to be an unbounded interval).
Let $L(x)$ be the length of $\gamma_{|[0,x]}$. This is a nondecreasing function of $x$. It is constant precisely on the intervals where $\gamma$ is constant. Thus, $\gamma\circ L^{-1}$ is well defined. This is the desired parametrization.


text:
\begin{document}

\title{A Fast Stochastic Plug-and-Play ADMM for Imaging Inverse Problems}

\author{Junqi Tang,

        Mike~Davies,~\IEEEmembership{Fellow,~IEEE}
\thanks{J.Tang and M.Davies are with the School of Engineering, University of Edinburgh. Correspondence to J.Tang@ed.ac.uk}

}

\markboth{June, 2020}
{Shell \MakeLowercase{\textit{et al.}}: Bare Demo of IEEEtran.cls for IEEE Journals}

\maketitle

\begin{abstract}
 In this work we propose an efficient stochastic plug-and-play (PnP) algorithm for imaging inverse problems. The PnP stochastic gradient descent methods have been recently proposed and shown improved performance in some imaging applications over standard deterministic PnP methods. However, current stochastic PnP methods need to frequently compute the image denoisers which can be computationally expensive. To overcome this limitation, we propose a new stochastic PnP-ADMM method which is based on introducing stochastic gradient descent inner-loops within an inexact ADMM framework. We provide the theoretical guarantee on the fixed-point convergence for our algorithm under standard assumptions. Our numerical results demonstrate the effectiveness of our approach compared with state-of-the-art PnP methods.
\end{abstract}

\begin{IEEEkeywords}
  Stochastic ADMM, Plug-and-Play Priors.
\end{IEEEkeywords}

\IEEEpeerreviewmaketitle

\section{Introduction}

Recent trends in the research of computational imaging have been focusing on developing algorithms which are able to jointly utilize the power of classical physical models and advanced image priors \cite{venkatakrishnan2013plug, egiazarian2007compressed,kamilov2017plug, romano2017little,reehorst2018regularization}. These methods typically take the form of well-known optimization algorithms, and plug in a pretrained deep neural network \cite{zhang2017beyond} or a patch-based denoiser with non-local denoising properties \cite{buades2005non,dabov2008image,chen2017trainable,talebi2013global}. In this work we propose a novel stochastic plug-and-play method for imaging inverse problems. Consider the following observation model for a linear inverse problem:
\begin{equation}\label{eq:1}
b = A x^\dagger + w, \ \ \ A \in \mb{R}^{n \times d}
\end{equation}
where $x^\dagger$ denotes the vectorized (raster) ground truth image, $A \in \mb{R}^{n \times d}$ represents the forward measurement model, $b \in \mb{R}^n$ denotes the observation, while $w \in \mathbb{R}^n$ represents the random additive noise. Traditionally, in order to get a good estimate of the ground truth $x^\dagger$, we typically seek to find the minimizer of a composite objective function:
\begin{equation}\label{reg_ls}
    x^\star \in \arg \min_{x \in \mathbb{R}^d} \left\{ f(x) + g(x)\right\},
\end{equation}
where $f(x) = \frac{1}{n}\sum_{i = 1}^n f_i(x) = \frac{1}{n}\sum_{i = 1}^n f(a_i, b_i, x)$ is the data fidelity term which is assumed to be convex and smooth, such as the least-square loss $ f(x) = \frac{1}{2n}\|Ax - b\|_2^2$, and here we denote $a_i$ as the i-th row of $A$ and $b_i$ as the i-th element of $b$. Meanwhile $g(x)$ in (\ref{reg_ls}) denotes a regularization term which encodes image priors, with classical examples including the sparsity-inducing regularization in wavelet domain and the total-variation regularization \cite{chambolle2016introduction}, etc.  The composite loss function (\ref{reg_ls}) can be effectively minimized via a class of iterative algorithms which are known as the proximal splitting methods \cite{combettes2011proximal}, including the forward-backward splitting \cite{lions1979splitting,2009_Beck_Fast,beck2009fast}, primal-dual splitting\cite{chambolle2011first,chambolle2015convergence,pesquet2014class}  and the Douglas-Rachfort splitting/alternating direction method of multipliers (ADMM)\cite{douglas1956numerical, boct2015inertial, boyd2011distributed}, etc.

Considering the link between proximal operators and denoising, researchers \cite{venkatakrishnan2013plug,egiazarian2007compressed,kamilov2017plug} have discovered that if they simply replace the proximal operator on $g$ with a direct call to an off-the-shelf denoising algorithm, such as NLM \cite{buades2005non}, TNRD \cite{chen2017trainable}, BM3D \cite{dabov2008image}, or the DnCNN \cite{zhang2017beyond}, excellent image recovery/reconstruction results can often be attained. Although the imaging community has very limited theoretical understanding and convergence analysis for such algorithms so far, these ad-hoc approaches have shown state-of-the-art performances in various imaging applications.

 Recently, inspired by the success of stochastic gradient descent (SGD) methods in solving large-scale optimization tasks in machine learning \cite{bottou2010large,shalev2011pegasos,kingma2014adam} and some imaging applications \cite{chambolle2018stochastic,chouzenoux2017stochastic,ehrhardt2017faster}, Sun et al \cite{sun2019online} have extended the deterministic plug-and-play ISTA/FISTA method \cite{kamilov2017plug} and proposed PnP-SGD in order to improve the computational efficiency. In each iteration of PnP-SGD, a minibatch stochastic gradient is computed as an unbiased estimator of the full gradient, which yields a computational benefit in each iteration. However, as discussed in \cite{tang2019practicality}, current stochastic gradient methods in general need to compute the proximal operator/denoisers more frequently than the deterministic gradient methods within the same amount of gradient evaluations. When the denoiser is computationally expensive, the actual performance benefit of using stochastic gradient techniques may be compromised due to this computational overhead. In this work, we seek to address this issue for stochastic PnP methods and propose a more practical approach for utilizing the power of stochastic gradient techniques to accelerate the deterministic plug-and-play algorithms.

 \subsection{Main contributions}
 This paper's contribution is two-fold:

 \begin{itemize}
     \item  We propose an efficient stochastic PnP method which empirically improves upon previous stochastic approach in \cite{sun2019online} by reducing the number of calls on the modern denoisers which are usually a bottleneck for computation. We demonstrate the effectiveness of our approach in X-ray computed tomography (CT) imaging problems.
     \item We provide a theoretical fixed-point convergence analysis for our stochastic PnP algorithm under standard assumptions.
 \end{itemize}

 \section{Stochastic PnP-ADMM}

In a recent work of Sun et al \cite{sun2019online}, a stochastic PnP algorithm is proposed for image restoration and reconstruction, which can be written as the following:
  \begin{eqnarray*}
 && \mathrm{\textbf{PnP-SGD}} - \mathrm{Initialize}\ x^0, z^0 = x^0 \\
 &&\mathrm{For} \ \ \ k = 1, 2,...,  t\\
&&\left\lfloor
\begin{array}{l}
x^{k}=  \mathcal{D} [z^{k-1}-\eta \cdot \triangledown f_{S_k}(z^{k-1})]\\
z^{k} = x^{k} + \alpha^k(x^k - x^{k-1})
\end{array}
\right.
 \end{eqnarray*}
 where $\triangledown f_{S_k}(x^{k-1})$ denotes a minibatch stochastic gradient with a randomly subsampled index $S_k$, chosen uniformly at random from a partitioned index $\hat{\I} = \{\I_1, \I_2, ..., \I_K\}$, where $\I_1\cup\I_2\cup...\cup\I_K = [n]$ and $\I_i \cap \I_j = \emptyset$, $\forall i \neq j \in [K]$. The PnP-SGD algorithm is essentially a plug-and-play variant of the stochastic proximal gradient descent \cite{rosasco2014convergence} which is based on the forward-backward splitting \cite{lions1979splitting}. In each iteration of PnP-SGD, a minibatch stochastic gradient estimate is computed:
\begin{equation}\label{sg}
    \triangledown f_{S_k}(z^k) = \frac{1}{m}\sum_{i \in S_k}  \triangledown f_i(z^k),
\end{equation}
where $m = \frac{n}{K}$. It first performs a stochastic gradient descent step with a step-size $\eta$, then a denoising step is computed using an off-the-shelf denoiser \cite{buades2005non,dabov2008image,chen2017trainable,zhang2017beyond} denoted as $\D(\cdot)$. Finally a momentum step is performed for empirical convergence acceleration with a momentum parameter $\alpha_k$ as in the FISTA algorithm \cite{2009_Beck_Fast}. The computational benefit of PnP-SGD over its deterministic counterparts (PnP-ISTA/PnP-FISTA \cite{kamilov2017plug}) comes from using an approximation of the full gradient by the minibatch gradient (\ref{sg}) which can be efficiently computed. However, the PnP-SGD does not have the capability to reduce the cost of computing the denoising step -- the denoiser has to be called at each iteration. To overcome this computational bottleneck, one plausible approach is to decouple the gradient step and the denoising step via Douglas-Rachford splitting/ADMM instead of the forward-backward splitting. In this work we study and propose a stochastic gradient extension of the PnP-ADMM algorithm\footnote{We write the update rule of PnP-ADMM in this paper using the equivalent Douglas-Rachford splitting reformulation \cite[Section 9.1]{ryu2019plug} for the simplicity of notation in analysis.} \cite{venkatakrishnan2013plug}:
 \begin{eqnarray*}
 && \mathrm{\textbf{PnP-ADMM}} - \mathrm{Initialize}\ x^0=z^0 \in \mb{R}^d; \\
 &&\mathrm{For} \ \ \ k = 0, 2,...,  t\\
&&\left\lfloor
\begin{array}{l}
y^{k}=  \mathrm{prox}_{\tau f} [z^k]\\
x^{k+1} = \mathcal{D}[2 y^k - z^{k}]\\
z^{k+1} = z^{k} + x^{k+1} - y^k.
\end{array}
\right.
 \end{eqnarray*}
where in each iteration an exact proximal step on the data-fidelity term $f(x)$ is computed with a constant step-size $\tau$:
  \begin{equation}\label{prox_step}
    \mathrm{prox}_{\tau f} (\cdot) = \arg \min_{x} \frac{1}{2} \|x - \cdot \|_2^2 + \tau f(x).
\end{equation}
In classical ADMM the step-size $\tau$ can be any positive constant to ensure convergence. The update rule of the PnP-ADMM can be written as $z^{k+1} = T(z^k)$, where $T(\cdot)$ is an operator defined as \cite{ryu2019plug}:
\begin{equation}
    T = \frac{1}{2}I + \frac{1}{2}(2\mc{D} - I)(2\mr{prox}_{\tau f} - I).
\end{equation}
Our proposed solution, presented in algorithm 1, is to use SGD with momentum to \emph{approximately} solve the prox step (\ref{prox_step}) within PnP-ADMM framework. We denote the number of inner-iterations at the $k$-th outerloop as $N_k$. In each inner-iteration a stochastic gradient descent step is performed with a step-size $\eta_k$, and then followed by a momentum step for empirical acceleration. Unlike the PnP-SGD which needs to call the denoiser in every stochastic gradient descent iteration, the proposed method only needs to compute the denoiser once every $N_k$ iterations. Our theoretical analysis is restricted to the case where we set $\alpha_j = 0$ and the parameters $N_k$ and $\eta_k$ are chosen adaptively in each outer-iteration. However, in practice, a constant step size $\eta$ which is inversely proportional to the Lipschitz constant $\eta_k = O(\frac{1}{L})$, $\tau = O(1)$, a constant number of inner-iterations $N_k = O(K)$, and a FISTA-like momentum parameter $\alpha_j = \frac{j-1}{j+3}$ \cite{chambolle2015convergence} are suggested for good empirical performance.

 \begin{algorithm}[t]\label{AA}
   \caption{--- Stochastic PnP-ADMM}
\begin{algorithmic}

   \State  {\bfseries Initialization:} number of inner-iterations: $[N_1, N_2, ...,N_K]$, momentum parameter sequence: $[\alpha_1, \alpha_2,...,\alpha_{\max_{j\in[K]} N_j}]$, partition index $\hat{\I} = \{\I_1,\I_2,...,\I_K\}$, $z^0 \in \mb{R}^d$, $y_0^0 \in \mb{R}^d$.
   \For{$k = 1$ {\bfseries to} $K$}
   \For{$j =1$ {\bfseries to} $N_k$}
    \State Randomly sample $S_j \in \hat{\I}$ with replacement.
    \State Compute a stochastic gradient estimator $\triangledown f_{S_j}(y^k_{j-1})$
   \State        $v^k_{j}=  y^k_{j-1}-\eta_k \cdot [\tau \triangledown f_{S_j}(y^k_{j-1}) + y^k_{j-1} - z^k]$

    \State {\bfseries Momentum:} $y^k_{j} = v^k_{j} + \alpha_j (v^k_{j} - v^k_{j-1})$

   \EndFor
   \State $x^{k+1} = \mathcal{D}( 2 y^k_{N_k} - z^k)$
   \State $z^k = z^{k-1} + x^{k+1} - y^{k}_{N_k}$
   \State $y^{k+1}_0 = x^{k+1}$
   \EndFor
\State Output $x^{K}$
\end{algorithmic}
\end{algorithm}

\section{Convergence Analysis}

In this section we provide theoretical analysis for our stochastic PnP-ADMM. When we run a stochastic gradient-based innerloop, we are effectively making an approximation of the proximal step $y^{k}=  \mathrm{prox}_{\tau f} [z^{k}]$, hence we can write our stochastic PnP-ADMM algorithm as the inexact recursion:
\begin{equation}\label{DR}
    z^{k+1} = T(z^k) + \ve^k.
\end{equation}
where $\ve^k$ denotes the approximation error. Now the desired fixed-point convergence can be established for (\ref{DR}), if we make the following standard assumptions as in \cite{ryu2019plug} on the denoiser and the data-fidelity term.

\subsection{Generic Assumptions}

\begin{myh}
 The denoiser satisfies:
 \begin{equation}
     \|(\D - I)(x) - (\D - I)(y)\|_2 \leq \be \|x - y\|_2, \ \forall x, y \in \mb{R}^d,
 \end{equation}
 with $\be > 0$.
\end{myh}
It is easy to show that A.1 implies a relaxed non-expansiveness condition on $\D(\cdot)$ which reads $\|\D(x) - \D(y)\|_2 \leq (1+ \be) \|x - y\|_2$ and is satisfied for a wide class of modern denoisers such as NLM and properly trained DnCNNs \cite{ryu2019plug}.

\begin{myh}
 $f(\cdot)$ is $\mu$-strongly-convex:
 \begin{equation}
     f(x) - f(y) - \langle \triangledown f(y), x-y \rangle \geq \mu \|x - y\|_2^2,\ \forall x, y \in \mb{R}^d,
 \end{equation}
 with $\mu > 0$. Meanwhile for a given minibatch partition index $\hat{\I} = \{\I_1, \I_2, ..., \I_K\}$ such that $f(\cdot) = \frac{1}{K}\sum_{k=1}^K f_{\I_k}(\cdot)$, each $f_{\I_k}$ is $L$-smooth, such that $\forall x, y \in \mb{R}^d$:
  \begin{equation}
     f_{\I_k}(x) - f_{\I_k}(y) - \langle \triangledown f_{\I_k}(y), x-y \rangle \leq L \|x - y\|_2^2.
 \end{equation}
\end{myh}

The strong-convexity assumption is necessary for our analysis. It seems pessimistic since a number of imaging inverse problems do not have strong-convexity. We believe that the assumption on strong-convexity could be relaxed, e.g. following the ideas from \cite{oymak2017sharp,bolte2007lojasiewicz,liang2017local,pmlr-v70-tang17a,tang2018rest}. On the other hand, one can instead run Algorithm 1 on a regularized objective $\hat{f}(x) = f(x) + \frac{\e}{2}\|x\|_2^2$ to manually enforce strong-convexity, which is a classical trick in convex optimization \cite{nesterov2013introductory}. Nevertheless, we believe that relaxing this assumption is an important future direction for the analysis.

\subsection{Analysis}

We first apply an existing convergence result for SGD for establishing the approximation accuracy of the inner-loop:
\begin{mylem}\label{lem_sg}
Under Assumption A.2, denote that for $k$-th outer-loop of Stochastic PnP-ADMM $y_\star^{k}=  \mathrm{prox}_{\tau f} [z^k]$, and define the following quantities for each outer-iteration $k$:
\begin{equation}
\begin{aligned}
   & \sigma_k^2 := \E_q[\|\tau\triangledown f_{\I_q}(y^k_\star) + y^k_\star - z^k\|_2^2], \\& \xi_k := \|y_\star^{k} - x^{k}\|_2^2,
    \end{aligned}
\end{equation}
then if the step size $\eta_k = \frac{\mu_0 \varepsilon}{2 \varepsilon \mu_0 L_0 + 2\sigma_k^2}$, $\alpha_j = 0$ for all $j$, $N_k = 2\log(\frac{\xi_k}{\ve})(\frac{L_0}{\mu_0} + \frac{\sigma_k^2}{\mu_0^2 \ve})$ with $\mu_0 = \tau \mu + 1$, $L_0 = \tau L + 1$, then we have the approximation error of the proximal step bounded as:
\begin{equation}
    \E \|y_{N_k}^k - \mathrm{prox}_{\tau f} [z^k]\|_2^2 \leq \varepsilon,
\end{equation}
where the expectation is taken over the random sampling of the indices within the inner-loop.
\end{mylem}

{\bf Proof.}
We first observe that the proximal step $y_\star^{k}=  \mathrm{prox}_{\tau f} [z^k]$ can be written precisely as a finite-sum optimization problem of the follow form:
\begin{equation}
\begin{aligned}
    \mr{prox}_{\tau f}(z^k) &= \arg \min_x \frac{1}{2}\|x - z^k\|_2^2 + \tau f(x)\\
    &= \arg \min_x \frac{1}{K}\sum_{q=1}^K [\tau f_{\I_q}(x) + \frac{1}{2}\|x - z^k\|_2^2],
\end{aligned}
\end{equation}
which is a $(\tau \mu + 1)$-strongly-convex objective and each of the element in the sum is $(\tau L + 1)$-smooth.

According to \cite[Theorem 2.1]{needell2014stochastic}, if we run SGD (starting at $x^k \in \mb{R}^d$) with uniform random sampling and a  step size $\frac{\mu_0\ve}{2\ve\mu_0 L_0 + 2\sigma_k^2}$, then after $N = 2\log(\frac{\|y_\star^k - x^k\|_2}{\ve})(\frac{L_0}{\mu_0} + \frac{\sigma_k^2}{\mu_0^2 \ve})$ iterations, we have $\E \|y_{N_k}^k - y_\star^k\|_2^2 \leq \varepsilon$. $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Box$

Now, we are able to prove the fixed-point convergence for the inexact recursion (\ref{DR}), and hence for our proposed method.

\begin{mythm}\label{main_thm2}
Assume A.1 and A.2 with $\be < 1$, denote positive values $\sigma$ and $\xi$ such that $\forall k$,  $\sigma_k \leq \sigma$ and $\xi_k \leq \xi$, and the quantities $\mu_0$, $L_0$, $\sigma_k^2$, $\xi_k$ are defined as in Lemma \ref{lem_sg}. If we choose the step-size parameters as $\tau > 1/(1+\be-2\be^2)$, $\alpha_j = 0$, $\eta_k = \frac{\mu_0}{2 \mu_0 L_0 + 2k\sigma^2}$, $N_k = 2\log(k\xi)(\frac{L_0}{\mu_0} + \frac{k\sigma^2}{\mu_0^2})$, we have the following fixed-point convergence for Algorithm 1:
\begin{equation}
    \E\|z^{k+1} - z^k\|_2 \rightarrow 0,
\end{equation}
when $k \rightarrow +\infty$.
\end{mythm}

Our main theorem suggests that for the basic form of Algorithm 1 where we choose the momentum $\alpha_j = 0$, the outerloop step-size $\tau = O(1)$, the inner-loop step-size decreasing in $k$ and the number of inner-iterations increasing in $k$, Algorithm 1 is guaranteed to converge to a fix point. However our numerical results in section IV suggest that we may set the number of inner-loop $N_k$ and step size $\eta_k$ to be constant and use FISTA-type of momentum \cite{2009_Beck_Fast,beck2009fast,chambolle2015convergence} for good empirical performance in practice.

\subsection{Proof for Theorem \ref{main_thm2}}
Firstly, due to assumption A.1 we have:
\begin{equation}
    \begin{aligned}
     \be \|x - y\|_2 &\geq \|(\D - I)(x) - (\D - I)(y)\|_2 \\&\geq \|\D(x) - \D(y)\|_2 - \|x - y\|_2,\ \forall x, y \in \mb{R}^d,
     \end{aligned}
\end{equation}
hence $\|\D(x) - \D(y)\|_2 \leq (1+ \be) \|x - y\|_2 ,\ \forall x, y \in \mb{R}^d$. Denote $u_k = \mr{prox}_{\tau f} (z^k) - \yk = y_\star^k - \yk$, we have:
\begin{equation}
    \begin{aligned}
        \|\ve^k\|_2 &= \|z^{k+1} - T(z^k)\|_2\\
                    &\leq \|u_k\|_2 + \|\D(2\yk - z^k) - \D(2\yk - z^k + 2 u_k) \|_2\\
                    &\leq (3+2\be)\|u_k\|_2.
    \end{aligned}
\end{equation}
Applying Lemma \ref{lem_sg} gives $ \E \|\ve^k\|_2 \leq \frac{3+2\be}{k}$. Now according to \cite[Theorem 2]{ryu2019plug}, under assumption A.1 and A.2, we can ensure that:
\begin{equation}
    \|T(x) - T(y)\|_2 \leq \delta \|x - y\|_2,\ \forall x, y \in \mb{R}^d,
\end{equation}
where $\delta = \frac{1 + \be + \be\tau\mu + 2 \be^2\tau\mu}{1 + \tau\mu +2\be\tau\mu}$. Moreover, if $\tau > 1/(1+\be-2\be^2)$ and $\be < 1$, then $\delta < 1$. Hence we have:
\begin{equation}
\begin{aligned}
     \E\|z^{k+1} - z^k\|_2 &= \E\|T(z^{k}) - T(z^{k-1}) + \ve^k - \ve^{k-1}\|_2 \\
     &\leq \E\|T(z^{k}) - T(z^{k-1})\|_2 \\
      &\ \ +\E\|\ve^k\|_2 +\E\| \ve^{k-1}\|_2\\
           &\leq \delta \E\|z^k - z^{k-1}\|_2 + \frac{3 + 2\beta}{k} + \frac{3+2\beta}{k-1}\\
     &\leq \delta \E\|z^k - z^{k-1}\|_2 + \frac{15}{k}
\end{aligned}
\end{equation}
If we recursively apply the same argument we will get:
\begin{equation}
    \E\|z^{k+1} - z^k\|_2 \leq \delta^k \E\|z^1 - z^0\|_2 + \frac{15}{k}\sum_{i = 0}^{k-1} \frac{k\delta^i}{k - i}.
\end{equation}
Then we use a classic criterion to show the boundedness of series $\sum_{i=0}^{+\infty} v_i$ where $v_i = \frac{k\delta^i}{k - i}$. For any finite $p > 1$, we have:
\begin{equation}
    \lim_{i \rightarrow + \infty} i^pv_i = \lim_{i \rightarrow + \infty} \frac{\delta^i i^p k}{k - i} = 0,
\end{equation}
where we take $k \rightarrow + \infty$ and $i \leq k-1$, and then:
\begin{equation}
    \sum_{i = 0}^{k-1} \frac{k\delta^i}{k - i} < + \infty, \ \ \ \ \frac{15}{k}\sum_{i = 0}^{k-1} \frac{k\delta^i}{k - i} \rightarrow 0.
\end{equation}
Hence by taking $k \rightarrow +\infty$, we have $\E \|z^{k+1} - z^k\|_2 \rightarrow 0$. Thus finishes the proof for Theorem \ref{main_thm2}.

\begin{figure}[t]
	\centering
    \includegraphics[width=.469\textwidth]{pnp_f1.pdf}
	\caption{The estimation error plot of the compared algorithms on low-dose CT example}
	\label{fig:1}
\end{figure}

\section{Numerical Experiments}

 For our numerical experiments, we choose the X-ray CT imaging as an example since it is know to favor stochastic gradient methods \cite{tang2019practicality}. We compare our algorithm with the state-of-the-art stochastic PnP method with momentum acceleration proposed by Sun et al \cite{sun2019online}, as well as the PnP FISTA algorithm \cite{kamilov2017plug}. We use MATLAB R2018a in a machine with 1.6 GB RAM, 1.80 GHz Intel Core i7-8550 CPU.

 We first test the compared methods on low-dose CT imaging problems, where low-energy noisy X-ray measurements with $I_0 = 10^3$ are used, which demands strong image priors are used in order to achieve good-quality reconstructions. Meanwhile we also compare these algorithms in sparse-view CT imaging with $I_0 = 10^{4}$, where fewer X-ray measurements are taken compared to the number of pixels to be inferenced\footnote{The numerical result in this example suggests that empirically the strong-convexity is not needed for the stochastic PnP-ADMM to converge.}. For low-dose CT example, we choose the penalized weighted least-squares objective as the data-fidelity term, which is tailored for low-dose CT \cite{wang2006penalized}. For sparse-view CT example, we choose the standard least-squares loss as the data fidelity term. For the randomized methods we partition the data into 10 minibatches. The noisy CT observations are obtained via $y \sim \mathrm{Poisson}(I_0 e^{- Ax})$ where the forward operator $A$ is implemented using the {\it AIRtools} package \cite{hansen2012air}. For our algorithm, we set $N_j = 10$ for all $j$ such that in each inner-loop we make exactly one pass of the data, outer-loop step-size $\tau = 1$, inner-loop step-size $\eta_k = \frac{1}{L}$, and the momentum parameter $\alpha_j = \frac{j-1}{j+3}$ as suggested in \cite{chambolle2015convergence}.

 \begin{figure}[t]
	\centering
    \includegraphics[width=.469\textwidth]{pnp_f2.pdf}
	\caption{The estimation error plot of the compared algorithms on sparse-view CT example}
	\label{fig:3}
\end{figure}

We choose the BM3D \cite{dabov2008image} with the denoiser-scaling \cite{xu2020boosting} as the denoiser:
\begin{equation}
    \D_\gamma (x) = \frac{1}{\gamma} \mr{BM3D}(\gamma x),
\end{equation}
and maximize the reconstruction performance for each of the compared algorithms via grid-searching the parameter $\gamma$.

We present the numerical results of the algorithms in Figure 1 for low-dose CT inverse problem of size $A \in \mb{R}^{88256 \times 65536}$, and in Figure 2 for sparse-view CT imaging task of size $A \in \mb{R}^{92160 \times 262144}$. We plot the estimation error $\log_{10}\|x - x^\dagger\|_2$ to the ground-truth image $x^\dagger$, against the actual run time as well as the number of datapasses. We can observe that both PnP-SGD and our method are much faster than PnP-FISTA in terms of datapass. The PnP-SGD appears to be faster than our method in terms of number of datapasses. However, in terms of actual run time, the PnP-SGD is slower than our method, due to the need to compute the costly BM3D at each stochastic gradient iteration.

\section{Conclusion}

In this work we propose a stochastic PnP-ADMM algorithm which is able to provide practical acceleration with stochastic gradient techniques, for efficiently solving imaging inverse problems. This is an effective approach to make the stochastic PnP schemes truly practical, by reducing the computational overhead of the modern denoisers. We provide a fixed-point convergence analysis, and demonstrate the effectiveness of our method in numerical experiments.

\section*{Acknowledgment}
This work is supported by ERC Advanced grant 694888, C-SENSE.

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\end{document}


TITLE: Let $f(x) =x+\tan^3x $ and $g(x) =f^{-1}(x)$ then find the value of ...
QUESTION [1 upvotes]: Problem :
Let $f(x) = x + \tan^3 x $ and $g(x) =f^{-1}(x)$ then find the value of
$56g'(\frac{\pi}{4}+1)$
My approach :
$f'(x) = 1+3\tan^2x \sec^2x$
$f'(\frac{\pi}{4})=7$
now how to find $g'(\frac{\pi}{4}+1)$
Please guide not getting any clue on this.

REPLY [4 votes]: Notice that, by definition of an inverse function:
$f(g(x))=x$
Then:
$\frac{d}{dx} f(g(x))=1$
$f'(g(x)) \cdot g'(x)=1$ (by the Chain Rule)
And, as @Alex G. pointed out:
$g'(x)=\frac{1}{f'(g(x))}$
So, then it should follow immediately that:
$g'(\frac{π}{4}+1)=\frac{1}{f'(g(1+\frac{π}{4}))}$
And then just do the calculations...
EDIT: Sure enough, trying some stereotypical tangent values works. In this case we find that:
$g(\frac{π}{4}+1)=\frac{π}{4}$
So, therefore:
$g'(\frac{π}{4}+1)=\frac{1}{f'(\frac{π}{4})}$
$f'(\frac{π}{4})=1+3\tan(\frac{π}{4})\sec^2(\frac{π}{4})=1+3\sec^2(\frac{π}{4})=7$
So your answer is then, $g'(\frac{π}{4}+1)=\frac{1}{7}$


text:
\label{sec:the-su3-theory}

\subsection{Planar algebras and canopoleis}
\label{ssec:planar-algebras}

We'll give a brief recap of the construction of Morrison and Nieh here, and refer the reader to \cite{BN05_MR2174270}, \cite{MorrN:sl(3)}, and \cite{Web07_MR2308960} for more technical details regarding planar algebras and canopoleis.

Recall that an \emph{oriented planar arc diagram} is, colloquially, just an oriented crossingless tangle in a disk with (possibly) some smaller disks removed, and with the remaining holes given some ordering. Two such diagrams can be composed whenever the outer boundary of one diagram matches one of the inner boundaries of the other: we just shrink the first diagram and paste it into the second, giving a new planar arc diagram. More generally, let $P$ be a planar arc diagram with $n$ holes; we'll label each from 1 up to $n$, and think of the outer boundary of $P$ as the ``$0$th hole." If $\mathcal{Q}_i$ is the set of planar arc diagrams that match the boundary of the $i$th hole of $P$, then $P$ defines an operation $P: \mathcal{Q}_1 \times ... \times \mathcal{Q}_n \rightarrow \mathcal{Q}_0$. See Figure \ref{fig:operad} for an example.

\begin{figure}[ht]
\begin{align*}
\mathfig{0.2}{webs/operad1}: \left( \mathfig{0.2}{webs/operad3}, \mathfig{0.2}{webs/operad2} \right) \longmapsto \mathfig{0.2}{webs/operad5}
\end{align*}
\caption[Composition in the oriented planar arc diagram operad]{Composition in the oriented planar arc diagram operad.}
\label{fig:operad}
\end{figure}

This operation on oriented planar arc diagrams gives them the structure of a \emph{colored operad}, where the coloring just refers to the labels (incoming and outgoing strands) on the disk boundaries. Such an operad can act on a collection of objects in some monoidal category $\mathcal{C}$: we associate to each color $s_i$ an object $\mathcal{P}(s_i)$, and to each collection of composable colors $s_1,...,s_n, s_0$ we associate the space of maps $\Hom{}{\mathcal{P}(s_1) \times ... \times \mathcal{P}(s_n)}{\mathcal{P}(s_0)}$. Of course, a properly colored planar arc diagram $P$ specifies a map $\mathcal{P}(s_1) \times ... \times \mathcal{P}(s_n) \rightarrow \mathcal{P}(s_0)$.

\begin{defn}
A \emph{planar algebra in} $\mathcal{C}$ is a collection $(\mathcal{P}(s_i)) \in \Ob{\mathcal{C}}$ that admits the above action of the operad of oriented planar arc diagrams.
\end{defn}

\begin{figure}[ht]
\begin{align*}
\mathfig{0.2}{webs/operad1} : \left( \mathfig{0.2}{webs/tangle_disk1}, \mathfig{0.2}{webs/tangle_disk2} \right) \longmapsto \mathfig{0.2}{webs/planar_tangle}
\end{align*}
\caption[A planar algebra]{A planar algebra: the operad action on oriented tangle diagrams.}
\label{fig:planar-tangle}
\end{figure}

Practically speaking, this structure gives us an associative way of ``multiplying" elements of our collection, in a planar fashion. As an easy example, the set of oriented tangle diagrams forms a planar algebra in the category of sets (see Figure \ref{fig:planar-tangle}), with generating set $\left\{ \mathfig{0.06}{webs/positive_disk},\mathfig{0.06}{webs/positive_disk}\right\}$; we shall, of course, consider these diagrams up to Reidemeister equivalence. Similarly, Kuperberg's $\su{3}$ spider forms a planar algebra in the category of $\Z[q,q^{-1}]$-modules, where we quotient by the $\su{3}$ spider relations (Equations \ref{eq:spider}); the spider is generated, as a planar algebra, by $\left\{ \mathfig{0.06}{webs/sink_disk},\mathfig{0.06}{webs/source_disk}\right\}$.   We can thus view the $\su{3}$ quantum link invariant as a map of planar algebras, which is convenient for both computational efficiency and organizing philosophy.

The goal, then, is to categorify this local picture of a quantum invariant: to do this, we invoke the notion of a canopolis, first appearing in \cite{BN05_MR2174270}.

\begin{defn}
A \emph{canopolis} is a planar algebra in some (monoidal\footnote{Here the monoidal structure is just given by cartesian product.}) category of categories $(\mathcal{C}(s_i))$. In particular, both the collection of objects and the collection of morphisms form planar algebras.
\end{defn}

Now a planar arc diagram $P$ will define a \emph{functor} $P:\mathcal{C}(s_1) \times ... \times \mathcal{C}(s_n) \rightarrow \mathcal{C}(s_0)$. We can view each category $\mathcal{C}(s)$ as a ``can" (rather than just a disc) with a specified label $s$ that can be plugged into a cylinder with a matching label in $P \times [0,1]$: objects will live on the tops and bottoms of cans, and morphisms will live inside cans. Further, the fact that $P$ defines a functor guarantees that planar operations commute with the usual composition of morphisms within their categories. Thus, we can build a ``city of cans" by composing vertically or horizontally in any order. It will also be useful to talk about maps between canopoleis.

\begin{defn}
A \emph{canopolis morphism} $\mathbf{C} \rightarrow \mathbf{C'}$ is a collection of functors $$(\mathcal{C}(s_i)) \rightarrow (\mathcal{C}'(s_i))$$ that commute with all planar algebra operations.
\end{defn}

Our first example of a canopolis will be the categorification of the set of oriented tangles. Let $\Ortang(s)$ be the category of tangle cobordisms with fixed boundary denoted by $s \in \mathcal{S}$, where $\mathcal{S}$ indexes the set of strand intersections with the boundary circle and their orientations, up to cyclic permutation. Then we define $\Ortang$ to be the canopolis in the category $\bigcup_{s \in \mathcal{S}} \Ortang(s)$.
(Note that we need more than just the cobordisms between two individual crossings to generate all possible tangle cobordisms.)
Here we want all morphisms considered up to four-dimensional isotopy; when viewing a generic morphism as a movie of tangle diagrams, this means we mod out by the movie moves.

\subsection{Categorifying the $\su{3}$ spider}
\label{ssec:categorifying}

A more interesting example, because it involves relations, is the categorification of the $\su{3}$ spider. Let $\Cob{3}_s$ be the category of cans with fixed boundary as above: the objects are $\su{3}$ webs and the morphisms are seamed cobordisms (or \emph{foams}), which are just CW-complexes modeled on $``Y" \times [0,1]$, plus some additional data.

\begin{defn}[from \cite{MorrN:sl(3)}]
Given two webs $D_1$ and $D_2$ drawn in a disc, both with boundary
$\partial$, a \emph{seamed cobordism} from $D_1$ to $D_2$ is a
2-dimensional CW-complex $F$ with
\begin{itemize}
\item exactly three oriented $2$-cells meeting along each oriented singular $1$-cell, such that the orientations on the $2$-cells all induce the same orientation on the seam;
\item a cyclic ordering on those three $2$-cells;
\item and an identification of the boundary of $F$ with $D_1 \cup D_2
\cup (\partial \times [0,1])$ such that
\begin{itemize}
\item the orientations on the sheets induce the orientations on the
edges of $D_1$, and the opposite orientations on the edges of $D_2$,
\item and the cyclic orderings around the singular seams agree with the
cyclic orderings around a vertex in $D_1$ or $D_2$ given by its
embedding in the disc; the anticlockwise ordering for ``inwards" vertices, the clockwise ordering for ``outwards" vertices.
\end{itemize}
\end{itemize}
Note that a foam is an abstract space; while its boundary is identified with lines on the surface of the can $D^2 \times [0,1]$ (and thus it is picturesque to view foams as in Figure \ref{fig:canopolis-action}), the foam doesn't literally live in the can.
\end{defn}

\begin{figure}[ht]
\begin{align*}
\mathfig{0.3}{foams/canopolis1b} : \left( \mathfig{0.1}{foams/canopolis2b}, \mathfig{0.1}{foams/canopolis3b} \right) \longmapsto \mathfig{0.3}{foams/canopolis4b}
\end{align*}
\caption[Planar composition of foams in $\Cob{3}$.]{Planar composition of foams in $\Cob{3}$. It is convenient to view these foams in cans, though really they are not embedded there.}
\label{fig:canopolis-action}
\end{figure}

We define $\Cob{3}$ to be the canopolis in the category $\bigcup_{s \in \mathcal{S}} \Cob{3}_s$ (see Figure \ref{fig:canopolis-action}), where, for $R$ a ring in which 2 and 3 are invertible\footnote{In this paper, we'll assume $R=\Z[\frac{1}{2}, \frac{1}{3}]$.}, we allow formal $R$-linear combinations of morphisms, and where we impose the following local relations on foams:

\begin{itemize}
\item``Closed foam" relations:
\begin{align}
\label{eq:closed_foams}
\mathfig{0.075}{cobordisms/sphere} & = 0 &
\mathfig{0.075}{cobordisms/torus} & = 3 \\
\notag
\mathfig{0.15}{cobordisms/double_torus} & = 0 &
\mathfig{0.16}{cobordisms/triple_torus_disc} & = 0
\end{align}

\item The ``neck cutting" relation:
\begin{equation}
\label{eq:neck_cutting}
 \rotatemathfig{0.04}{90}{cobordisms/cylinder} = \frac{1}{3}   \rotatemathfig{0.04}{90}{cobordisms/neck_cutting_left}
      - \frac{1}{9} \rotatemathfig{0.04}{90}{cobordisms/neck_cutting_middle}
      + \frac{1}{3} \rotatemathfig{0.04}{90}{cobordisms/neck_cutting_right}

\end{equation}

\item The ``airlock" relation:
\begin{equation}
\label{eq:airlock}
    \rotatemathfig{0.05}{90}{cobordisms/airlock} = -
    \rotatemathfig{0.05}{90}{cobordisms/cup_cap}
\end{equation}

\item The ``tube" relation
\begin{equation}
\label{eq:tube_relation}
    \mathfig{0.1}{cobordisms/tube_relation/tube} =
    \frac{1}{2}
    \mathfig{0.1}{cobordisms/tube_relation/two_bubbles_lower_kiss} +
    \frac{1}{2}
    \mathfig{0.1}{cobordisms/tube_relation/two_bubbles_upper_kiss}
\end{equation}

\item The ``three rocket" relation:
\begin{equation}
\label{eq:rocket_relation}
  \mathfig{0.1}{cobordisms/rocket_relation/rocket_z}
 + \mathfig{0.1}{cobordisms/rocket_relation/rocket_x}
 + \mathfig{0.1}{cobordisms/rocket_relation/rocket_y} = 0
\end{equation}

\item The ``seam-swap" relation: reversing the cyclic order of the three 2-cells
attached to a closed singular seam is equivalent to multiplication by $-1$.

\item The ``sheet relations" (which can be derived from the relations above):
\begin{align}
  \mathfig{0.05}{cobordisms/sheet_algebra/blister} & = 0 &
 \label{eq:choking-torus-multiplication}
   \mathfig{0.065}{cobordisms/sheet_algebra/two_handle_discs} & =
   -3 \mathfig{0.06}{cobordisms/sheet_algebra/handle} \\
 \label{eq:torus-choking-torus}
 \mathfig{0.065}{cobordisms/sheet_algebra/handle_handle_disc} & = 0 &
  \mathfig{0.065}{cobordisms/sheet_algebra/two_handles} & = 0
\end{align}
The first of these four is the extremely useful ``blister relation."
\end{itemize}

\begin{remark}
\begin{enumerate}
    \item $\Cob{3}$ is generated, as a canopolis, by the cup, cap, saddle, zip, and unzip morphisms below.
        \begin{equation*}
            \mathfig{0.08}{foams/birth} \quad
            \mathfig{0.08}{foams/death} \quad
            \mathfig{0.08}{cobordisms/saddle2} \quad
            \mathfig{0.08}{cobordisms/zip} \quad \mathfig{0.08}{cobordisms/unzip}
        \end{equation*}
    \item As a consequence of the local relations, all closed foams in $\Cob{3}$ can be evaluated to scalars. (Lemma 3.3 in \cite{MorrN:sl(3)})
\end{enumerate}
\end{remark}

As it turns out, $\Cob{3}$ will benefit from slightly more structure. First we'll make it into a graded canopolis by endowing web diagrams with formal grading shifts given by powers of $q$. Further, define the grading of a morphism $C$ from $q^{m_1}D_1$ to $q^{m_2}D_2$ by
\begin{align}
\label{eq:grading}
\deg C = 2\chi(C) - B + \frac{V}{2} + m_2 - m_1
\end{align}
where $B$ is the number of boundary points on $D_i$ and $V$ is the total number of trivalent vertices in the webs $D_1$ and $D_2$. Note that the local relations above are degree homogeneous, and that degree is additive under canopolis composition.

Second, we form $\Mat{\Cob{3}}$ by introducing formal direct sums of objects, and allowing matrices of morphisms between these direct sums. Morrison and Nieh prove that the graded decategorification of $\Mat{\Cob{3}}$  is, in fact, Kuperberg's $\su{3}$ spider.

Finally, for the \emph{coup de grace}, we arrive at the canopolis $\Kom{\Mat{\Cob{3}}}$ by considering chain complexes (up to chain homotopy equivalence) with objects and morphisms in $\Mat{\Cob{3}}$. We'll have to be slightly more explicit about the action of planar arc diagrams now that cans will be associated with complexes and chain maps, rather than just objects and morphisms in $\Cob{3}$. However, the rule is simple: apply the usual construction for tensor product of complexes, but use the planar arc diagram to ``multiply" objects and morphisms instead of $\otimes$. (See Appendix \ref{ssec:planar-comps} for details.) For convenience, let's make the abbreviation $\Kob{3} := \Kom{\Mat{\Cob{3}}}$.

\subsection{A link homology}
\label{ssec:link-homology}

Having defined the relevant canopoleis, Morrison and Nieh proceed to construct a link homology that categorifies the $\su{3}$ quantum link invariant, i.e., a map $Kh(\su{3}): \Ob{\Ortang} \rightarrow \Ob{\Kob{3}}$. Such a map is easily defined on objects in $\Ortang$ by the following categorified skein relations:
\begin{equation*}
\xymatrix@R-1mm{
 \mathfig{0.06}{webs/positive_crossing} \ar@{|->}[r] & \quad \; & \Bigg( \bullet \ar[r]
    & q^2\phantom{-} \mathfig{0.06}{webs/two_strand_identity} \ar[r]^{\mathfig{0.06}{cobordisms/zip}} & q^3 \mathfig{0.06}{webs/upwards_H_diagram} \ar[r] & \bullet \Bigg) \\
 \mathfig{0.06}{webs/negative_crossing} \ar@{|->}[r] & \Bigg( \bullet \ar[r] &
    q^{-3} \mathfig{0.06}{webs/upwards_H_diagram} \ar[r]^{\mathfig{0.06}{cobordisms/unzip}} & q^{-2} \mathfig{0.06}{webs/two_strand_identity} \ar[r] & \bullet \Bigg) &
}
\end{equation*}
The homological heights here are $-2$, $-1$, $0$, $1$, and $2$; the webs with $q^{\pm2}$ shifts lie at height 0 in each case. (Let's also establish the following nomenclature for the webs in this picture: we'll call the ones with $q^{\pm2}$ shifts the \emph{smoothly-resolved} webs for these crossings, and the ones with $q^{\pm3}$ shifts the \emph{I-resolved} webs for these crossings.)

These crossings will compose under planar operations to make larger tangles, as will the associated complexes. One important subtlety is that planar composition of complexes is independent of the order of composition, up to chain isomorphism, but this isomorphism is not the obvious permutation: one needs to sprinkle some minus signs into the permutation to make it a chain map. The upshot is that (1) for well-definition of complexes, all crossings in a tangle diagram must be equipped with an ordering (of course, this is equivalent to the ordering of holes in a planar arc diagram), and (2) there are (slightly) nontrivial chain maps that will reorder the crossings. See Appendix \ref{ssec:sign-conventions} for details.

To complete our map on objects, we need only check that the map $Kh({\su{3}})$ is invariant under isotopy of tangles, i.e., that Reidemeister moves applied to the source tangle do not change the homotopy type of the resulting complex. This is essentially done in \cite{MorrN:sl(3)} by constructing a chain homotopy equivalence for each version of the oriented Reidemeister moves, though we will provide additional details in Section \ref{sec:reidemeister_maps}.

\subsection{A canopolis morphism?}

There is a natural way to define $Kh(\su{3})$ on morphisms (tangle cobordisms) as well. Since a surface in 4-space can be presented by a movie, we can view a cobordism as a sequence of tangles diagrams that, at each stage, differ by a Reidemeister or Morse move. Thus we need only define chain maps for these six generating moves. This is easy: Morse moves induce the obvious gluing of \mbox{0-,} \mbox{1-,} or 2-handles into a foam, and Reidmeister moves already have chain maps defined for the link homology. Again, the heart of the issue here is whether these induced maps are well-defined, i.e., invariant under the movie moves. If they are, then $Kh(\su{3})$ is a canopolis morphism.


TITLE: Why is the acceleration of a rotating body less than if it wasn't rotating?
QUESTION [0 upvotes]: Suppose an ideal string is passed from a pulley of any mass, (placed on a smooth horizontal surface), with sufficient friction so that the rope  doesn't slide on it! Now if the pulley is attached to a block over an ideal pulley to another block hanging! Now the block hanging can translate downwards with any acceleration! So my question is why does the pulley has less acceleration in comparison to the hanging block?

REPLY [1 votes]: Because the acceleration of the mass is the same as the acceleration of a dot on the string. As the pulley moves, the string unwinds due to the rotation. You have $dx=dx'+Rd\theta$, so $a=a'+R\alpha$

The black square is a mark on the string just to show how much it moved: dx, which is the total displacement of the string, which is the same as that of the hanging block. dx' is how much the center of the pulley moved, and $d\theta$ how much it rotated as he moved.


TITLE: $x+ix=i$ solution?
QUESTION [1 upvotes]: 3 years without math exercises , today I found a question in Quora about the solution of $x+ix =i$.
So I reasoned this way :
\begin{align} x+ix=i &\implies x=i−ix &\\
& \implies x=i(1−x) &\\
& \implies x/(1−x)=i\\
&\implies x^2/(1−x)^2=−1 \\
&\implies x^2=−[(1−x)^2]\\
& \implies x^2=−[1−2x+x^2] \\
&\implies x^2=−x^2+2x−1\\
&\implies 2x^2−2x+1=0\\
&\implies x^2−x+(1/2)=0
\end{align}
$$
a=1\quad b=−1 \quad c=1/2\\
\delta=b^2−4ac =−1=i^2
$$
so
$$x_1=(−b−i)/2∗a=(1−i)/2 \\
x_2=(−b+i)/2∗a=(1+i)/2$$
However the solution, as in site, is only $(1+i)/2$. What am I doing wrong?

REPLY [1 votes]: $[1].$ Suppose $A\implies B\implies C \implies x\in \{0,1\}.$ You cannot conclude, from this, that $x\in \{0,1\}\implies A.$ This does not mean that it isn't useful to use one-way implications. It's very useful but it's not the whole story. You have $A\implies x\in \{0,1\}$ and then  you may find
(i).  $x=0\implies \neg A$ and $x=1\implies \neg A,$ and conclude that $A$ is false for all $x,$ OR
(ii). $x=0 \implies A$ and $x=1\implies  \neg A,$ and conclude that $A\iff x=0,$ OR
(iii). $x=0\implies \neg A$ and $x=1\implies A,$ and conclude that $A\iff x=1,$ OR
(iv) $x=0\implies A$ and $x=1\implies A,$ and conclude that $A\iff (x=0 \lor x=1).$
$[2].$ Let $a,b \in \Bbb R.$ Then $a+ib \ne 0$ iff $a,b$ are not both $0.$ And if  $a+ib\ne 0 $ then $a+ib$ has a multiplicative inverse $(a/r-ib/r)=(a-ib)/r$ where $r=a^2+b^2.$ Because $$(a+ib)(a-ib)/r=(a^2-i^2b^2)/r=(a^2+b^2)/r=1.$$ So if $x,y,z \in \Bbb C$ and $z\ne 0$ then $x=y\iff xz=yz.$ (Just as when $x,y,z\in \Bbb R$ and $z\ne 0$).
If $a=b=1$ then $r= a^2+b^2=2$ so $1/(a+ib) =(a-ib)/r=(1-i)/2.$
Therefore $$x+ix= i\iff  x(1+i)=i\iff$$ $$\iff x\cdot (1+i)[1/(1+i)]=i\cdot[1/(1+i)] \iff$$ $$\iff x\cdot 1=i\cdot [(1-i)/2]\iff$$ $$\iff x=[i-i^2]/2=[i-(-1)]/2=(i+1)/2. $$
Notice the implications are two-way so there is nothing more to do.
In brief, solving a linear equation in one variable $(x)$ in $\Bbb C$ is done just the way you do it in $\Bbb R.$


text:
\begin{document}

\title{Complex geometry of quantum cones}

\author{Tomasz Brzezi\'nski}
 \address{ Department of Mathematics, Swansea University,
  Swansea SA2 8PP, U.K.}
  \email{T.Brzezinski@swansea.ac.uk}
  \keywords{Quantum cone, differential calculus, smooth algebra, holomorphic structure.}

\begin{abstract}
The algebras obtained as fixed points of the action of the cyclic group $\ZZ_N$ on the coordinate algebra of the quantum disc are studied. These can be understood as coordinate algebras of quantum or non-commutative cones. The following observations are made. First, contrary to the classical situation, the actions of $\ZZ_N$ are free and the resulting algebras are homologically smooth. Second, the quantum  cone algebras  admit differential calculi that have all the characteristics of calculi on {\em smooth} complex curves. Third, the corresponding volume forms are exact, indicating that the constructed algebras describe manifolds with boundaries.
\end{abstract}

\maketitle

\section{Introduction}
Recent studies of  deformed orbifolds or manifolds with singularities revealed  an intriguing phenomenon: many classical surfaces $\Sigma$ with singular points admit non-commutative or $q$-versions $\Sigma_q$ that are smooth. Smoothness can be understood on several levels. Most naively, one observes that the multiple roots of the defining polynomial relations become separated upon the quantization  \cite{BrzFai:tea}, \cite{Brz:Sei}. Furthermore, the classically non-free action of a finite group on a smooth surface that gives rise to the singularity becomes free once the surface and the action are quantized  \cite{Brz:smo}. On a more sophisticated level, the deformed coordinate algebras $\cO(\Sigma_q)$ are {\em homologically smooth}, i.e.\ they fit into a finite exact sequence of finitely generated and projective $\cO(\Sigma_q)$-bimodules \cite{Brz:smo}. Often, $\cO(\Sigma_q)$ is  a (twisted) {\em Calabi-Yau algebra} of dimension two, i.e\  $\cO(\Sigma_q)$ is not only homologically smooth, but also its Hochschild cohomology with values in the enveloping algebra of $\cO(\Sigma_q)$ is trivial except in degree two, where it is isomorphic to $\cO(\Sigma_q)$ with the bimodule structure twisted by an automorphism \cite{Liu:hom}. This last observation can be interpreted as an existence of the volume form, and is tantamount  to the Poincar\'e duality \cite{Van:rel}.

Every two-dimensional orbifold is obtained by glueing cones, hence quantum cones seem to be a natural point to start. In this note we carry out parts of the programme of {\em differential smoothing} initiated in \cite{BrzSit:pil} and discuss differential structures on quantum cones that allow one to view them as complex curves.   We define coordinate algebras of quantum cones $\cO(C^N_{q,\gamma})$  as fixed points of the action of cyclic groups $\ZZ_N$ on the non-commutative disc algebra $\cO(D_{q,\gamma})$ \cite{KliLes:two}. This action turns out to be free and  $\cO(C^N_{q,\gamma})$  are (twisted) Calabi-Yau algebras. We derive differential calculi on $\cO(C^N_{q,\gamma})$ from that on $\cO(D_{q,\gamma})$ and show that they define holomorphic structures on $\cO(C^N_{q,\gamma})$ in the sense of \cite{KhaLan:hol}. The resulting volume forms on $\cO(C^N_{q,\gamma})$ are exact, which might indicate that quantum cones have boundaries.

\section{Quantum cones  are smooth}\label{sec.smooth}
The coordinate algebra $\cO(D_{q,\gamma})$  of the quantum disc is defined as the complex $*$-algebra generated by $z,z^*$ subject to relation
\begin{equation}\label{disc}
z^*z - qzz^* = \gamma,
\end{equation}
where $q,\gamma \in \RR$, $q\neq 0$ are parameters. This is obviously a deformation of the plane (or the unit disc), which corresponds to $q,\gamma =0$.  $\cO(D_{q,0})$ is the quantum or Manin's plane, and the case $\cO(D_{0,1})$ is the quantum oscillator algebra. In fact, if $\gamma\neq 0$, then the generators can be rescaled, so that $\gamma$ can be eliminated from \eqref{disc}. If $q\in (0,1)$,  the algebra   $\cO(D_{q,1-q})$ can be completed to the $C^*$-algebra isomorphic to the  Toeplitz  algebra \cite{KliLes:two}. For an accessible introduction to geometry of the quantum disc we refer to \cite[Chapter~1]{Vak:qua}.

$\cO(D_{q,\gamma})$  admits an action of the cyclic group $\ZZ_N$, which can be described in terms of the $\ZZ_N$-grading on generators as $\deg(z) =1$, $\deg(z^*) = N-1 = -1\  {\rm mod}\ N$. The fixed point subalgebra of this action or the degree-zero part of  $\cO(D_{q,\gamma})$ is generated by $a=zz^*$, $b=z^N$, $b^* = z^{*N}$. These satisfy the  relations
\begin{equation}\label{cone}
ab = q^N ba + \gamma [N]_q b, \qquad bb^* = \prod_{l=0}^{N-1} (q^{-l}a + \gamma [-l]_q), \qquad  b^*b = \prod_{l=1}^{N} (q^{l}a + \gamma [l]_q),
\end{equation}
where $[n]_q = (1-q^{n})/(1-q)$ are $q$-integers. The complex $*$-algebra generated by a selfadjoint  $a$ and $b,b^*$ subject to relations \eqref{cone} is called the {\em coordinate algebra of the quantum $N$-cone} and is denoted by $\cO(C^N_{q,\gamma})$. In case $q=\gamma =0$, $a$, $b$, $b^*$ can be thought of as coordinates and then relations \eqref{cone} describe a cone in $\CC\times \RR$, hence the name. The polynomials in $a$ on the right-hand sides of the last two equations in \eqref{cone} have no repeated roots as long as $\gamma\neq 0$. One might therefore expect that in this case the algebras  $\cO(C^N_{q,\gamma})$  describe {\em smooth} non-commutative surfaces.

An action of a (finite) group $G$   on a quantum space $\Sigma_q$ is free if and only if the corresponding coordinate algebra $\cO(\Sigma_q)$ is {\em strongly $G$-graded}. This means that for all $g\in G$, one can find finite number of elements $X_i $ of $G$-degree $g$ and $Y_i$ of degree $g^{-1}$ such that $\sum_i X_iY_i =1$. In the case of the cyclic group $\ZZ_N$ suffices it to find such $X_i$ of degree $1$ and $Y_i$ of degree $N-1$; see \cite[Section~AI.3.2]{NasVan:gra}.

\begin{theorem}\label{thm.free}
For all $\gamma\neq 0$, the $\ZZ_N$-action on $D_{q,\gamma}$ is free. Consequently, $\cO(C^N_{q,\gamma})$ are homologically smooth algebras for all values of $N>1$, $q$ and $\gamma \neq 0$.
\end{theorem}
\begin{proof}
Set $X_0 = \alpha_0 z^{*N-1}$, $X_i = \alpha_i a^{i-1} z$, $Y_0 =  z^{N-1}$ and $Y_i = z^*$, $i=1,\ldots, N-1$, where all the $\alpha_r\in \CC$ are to be determined. By equations \eqref{disc},
$$
z^{*N-1} z^{N-1} = \prod_{l=1}^{N-1} (q^{l}a + \gamma [l]_q)  =: \sum_{r=0}^{N-1} \beta_r a^r.
$$
 The comparison of the coefficients of the powers of $a$ in the condition $\sum_{i=0}^{N-1} X_iY_i =1$, yields the system of $N$ equations for the $\alpha_r$: $\alpha_0 \beta_0 = 1$ and $\alpha_0 \beta_i +\alpha _i =0$, for $i=1,2,\ldots, N-1$, whose
determinant is $\beta_0 = \gamma^{N-1} \prod_{l=1}^{N-1}[l]_q\neq 0$. Hence the required $X_i$ and $Y_i$ can be found.

Since the enveloping algebra of $\cO(D_{q,\gamma})$ is left Noetherian of finite global dimension and since the $\ZZ_N$-actions defining the cones are free if $\gamma\neq 0$, the algebras $\cO(C^N_{q,\gamma})$ are homologically smooth by \cite[Corollary~6]{Kra:Hoc} or \cite[Criterion~1]{Brz:smo}.
\end{proof}

The freeness of the action of $\ZZ_N$ on $D_{q,\gamma}$ is equivalent to the statement that the quantum disc is the total space of a $\ZZ_N$-principal quantum bundle over the quantum cone $C^N_{q,\gamma}$ \cite{BrzMaj:gau}, \cite{BrzHaj:Che}. This bundle is non-trivial, i.e.\ $\cO(D_{q,\gamma})$ is not isomorphic to $\cO(C^N_{q,\gamma})\ot \CC\ZZ_N$ as a $\ZZ_N$-graded left $\cO(C^N_{q,\gamma})$-module. The triviality would require the existence of  invertible elements in $\cO(D_{q,\gamma})$ of degree $n$, for  all $n\in \ZZ_N$. However, only non-zero scalar multiples of the identity are invertible in $\cO(D_{q,\gamma})$, and these have degree zero.

\begin{theorem}\label{thm.C-Y}
For all $\gamma\neq 0$,  $\cO(C^N_{q,\gamma})$ are (twisted) Calabi-Yau algebras.
\end{theorem}
\begin{proof}
Define $x:= 1 + \gamma^{-1}(q-1) a$. The relations \eqref{cone} yield
\begin{equation}\label{cone.z}
xb = q^Nbx, \qquad bb^* = \left(\frac{\gamma}{1-q}\right)^N\prod_{l=0}^{N-1} \left(1-q^{-l}x\right), \qquad  b^*b = \left(\frac{\gamma}{1-q}\right)^N\prod_{l=1}^{N} \left(1-q^{l}x\right).
\end{equation}
Since the change of variables $a\mapsto x$ is reversible, equations \eqref{cone.z} simply give a different presentation of $\cO(C^N_{q,\gamma})$. This presentation allows one to view $\cO(C^N_{q,\gamma})$ as a {\em generalized Weyl algebra} over the polynomial ring in the variable $x$ \cite{Bav:fin}. Since the polynomials on the right hand side of \eqref{cone.z} have no repeated roots, $\cO(C^N_{q,\gamma})$ are twisted Calabi-Yau algebras by \cite[Theorem~4.5]{Liu:hom}.
\end{proof}

The inspection of relations \eqref{cone.z} makes it clear that, for $\gamma\neq 0$ and $N$ odd, the algebras $\cO(C^N_{q,\gamma})$ are isomorphic to coordinate algebras of quantum real weighted projective spaces $\cO(\RR\PP^2_q(N;+))$ \cite{Brz:Sei}.

\section{Differential and complex geometry of quantum cones}\label{sec.complex}
By a {\em differential calculus} over an algebra $A$ we mean a pair $(\Omega (A), d)$, where $\Omega (A) = \oplus_{n\in \NN}  \Omega^n (A)$ is an $\NN$-graded algebra  such that $\Omega^0 (A) = A$, and  $d: \Omega^ n(A) \to \Omega^{n+1}(A)$  is the degree-one linear map that satisfies the graded Leibniz rule, is such that $d\circ d=0$ and, for all $n\in \NN$,  $\Omega^n(A)=Ad(A)d(A)\cdots d(A)$ ($d(A)$ appears $n$-times).  If $A$ is a complex $*$-algebra, then it is often requested that $\Omega (A)$ be a $*$-algebra and that $*\circ d = d\circ *$. In this situation one refers to $\Omega (A)$ as to a {\em $*$-differential calculus}. In this case, following \cite{KhaLan:hol}, a {\em complex structure on $A$}  is  the bi-grading decomposition of $\Omega (A)$,
$$
\Omega^n(A) = \bigoplus_{k+l =n} \Omega^{(k,l)}(A),
$$
such that $*:\Omega^{(k,l)}(A)\to \Omega^{(l,k)}(A)$, and the decomposition $d= \delta +\bdelta$ into differentials $\delta: \Omega^{(k,l)}(A)\to \Omega^{(k+1,l)}(A)$, $\bdelta: \Omega^{(k,l)}(A)\to \Omega^{(k,l+1)}(A)$, such that
$\delta(a)^* = \bdelta(a^*)$, for all $a\in \Omega (A)$.

To construct a complex structure on $\cO(C^N_{q,\gamma})$
start with the  differential $*$-calculus $\Omega (D_{q,\gamma})$ on $\cO(D_{q,\gamma})$ generated by one-forms $dz$ and $dz^*$ subject to relations
\begin{equation}\label{calculus.d}
zdz = q^{-1}dz z, \qquad z^*dz = qdz z^*, \qquad dz\wedge dz^* = -q^{-1}dz^*\wedge dz, \qquad dz\wedge dz =0;
\end{equation}
see e.g.\ \cite{SinVak:ana}. Note that $\Omega (D_{q,\gamma}) = \cO(D_{q,\gamma}) \oplus \Omega^1 (D_{q,\gamma}) \oplus \Omega^2 (D_{q,\gamma})$, where $\Omega^1 (D_{q,\gamma})$ is a a free left and right  $\cO(D_{q,\gamma})$-module with free generators $dz$ and $dz^*$, and   $\Omega^2 (D_{q,\gamma})$ is a free  $\cO(D_{q,\gamma})$-module generated by $dz\wedge dz^*$. We define the differential $*$-calculus $\Omega (C^N_{q,\gamma})$ on $\cO(C^N_{q,\gamma})$ as the differential $*$-subalgebra of $\Omega (D_{q,\gamma})$  generated by $a=zz^*$, $b= z^N$ and their differentials $da$, $db$. Set
$$
\omega_0 := dz z^*, \qquad \omega_1 := d b = [N]_qz^{N-1} dz, \qquad \omega:= dz\wedge dz^*.
$$

\begin{theorem}\label{thm.complex}
Assume that $\gamma\neq 0$. Let $\Omega^{(1,0)} (C^N_{q,\gamma})$ be the submodule of $\Omega^{1} (C^N_{q,\gamma})$ generated by $\omega_0$, $\omega_1$, let $\Omega^{(0,1)} (C^N_{q,\gamma})$ be the submodule of $\Omega^{1} (C^N_{q,\gamma})$ generated by $\omega_0^*$, $\omega_1^*$, and let $\Omega^{(1,1)} (C^N_{q,\gamma}) = \Omega^{2} (C^N_{q,\gamma})$. Set $\Omega^{(k,l)} (C^N_{q,\gamma}) =0$ if $k>1$ or $l>1$. Let $\delta$ be the projection of $d$ onto $\Omega^{(1,0)} (C^N_{q,\gamma})$ and let $\bdelta$ be the projection of $d$ onto $\Omega^{(0,1)} (C^N_{q,\gamma})$. Then the above bi-grading decomposition of $\Omega (C^N_{q,\gamma})$ defines a complex structure on $\cO(C^N_{q,\gamma})$. Furthermore, $\Omega^{2} (C^N_{q,\gamma}) = \cO(C^N_{q,\gamma})\, \omega = \omega\, \cO(C^N_{q,\gamma})$.
\end{theorem}
\begin{proof}
The only non-trivial statement to be proven here is that both $\omega_0$ and $\omega_0^*$ belong to $\Omega^{1} (C^N_{q,\gamma})$ (there sum is in $\Omega^{1} (C^N_{q,\gamma})$, since it is equal to $da$). If this is established, then the bi-graded decomposition of $\Omega^{1} (C^N_{q,\gamma})$ and maps $\delta$, $\bdelta$ are well-defined. Also, since $\omega = -d\omega_0$, the last assertion will follow. We start by proving the following
\begin{lemma}\label{lem.calculus}
For all $q\in (0,1)$ and positive $n\in \NN$,  if an ideal in the polynomial ring $\CC[w]$ contains both $X_n(w)= \prod_{l=1}^{n-1} (1+q^{-l}w)$ and $ Y_n(w)= \prod_{l=1}^{n-1} (1+q^{l}w)$, then it also contains the identity.
\end{lemma}
\begin{proof}
By induction in $n$. Clearly, the statement is true for $n=1$. Assume it is true for $k$, i.e.\ there exist  polynomials $f(w)$ and $g(w)$ such that $f(w)X_k(w) +g(w)Y_k(w) =1$. Let  $J$ be an ideal containing $X_{k+1}(w)$ and $Y_{k+1}(w)$. Then
\begin{eqnarray*}
(1+q^{-k}w)(1+q^k w) &=&  (f(w)X_k(w) +g(w)Y_k(w))(1+q^{-k}w)(1+q^{k}w) \\
&=&  f(w) X_{k+1}(w)(1+q^{k}w) + g(w)Y_{k+1}(w)(1+q^{-k}w) \in J.
\end{eqnarray*}
Similarly, by considering $ (f(qw)X_k(qw) +g(qw)Y_k(qw))(1+qw)(1+q^{-k+1}w)(1+q^{-k}w)$ one finds that $(1+qw)(1+q^{-k}w)(1+q^{-k+1}w)  \in J$, and hence also $(1+q^{-1}w)(1+q^{k}w)(1+q^{k-1}w)  \in J$, by symmetry. By comparing powers of $w$, one easily finds that there exist complex numbers $\alpha_0, \alpha_1, \alpha_2$ such that $(\alpha_0 +\alpha_1w)(1+q^{k}w) +\alpha_2(1+qw)(1+q^{-k+1}w) =1$. Multiplying this by $1+q^{-k}w$ we find that  $1+q^{-k}w\in J$. Again, by symmetry $1+q^{k}w\in J$, and clearly there exists a linear combination of these that makes up 1.
\end{proof}

Define $w:= {\gamma}^{-1}({1-q})qa - q $. Using relations \eqref{disc} and \eqref{calculus.d} one finds
\begin{equation}\label{omegas}
b^*db = q[N]_q \left(\frac{\gamma}{1-q}\right)^{N-1} \prod_{l=1}^{N-1} (1+q^{l}w)\, \omega_0, \quad bdb^*= [N]_q\left(\frac{ q^{-1}\gamma}{1-q}\right)^{N-1} \prod_{l=1}^{N-1} (1+q^{-l}w)\, \omega_0.
\end{equation}
Hence $\prod_{l=1}^{N-1} (1+q^{l}w)\omega_0,  \prod_{l=1}^{N-1} (1+q^{-l}w)\omega_0 \in \Omega^{1} (C^N_{q,\gamma})$ and, by Lemma~\ref{lem.calculus}, $\omega_0 \in \Omega^{1} (C^N_{q,\gamma})$. Since the calculus is closed under the $*$-operation also $\omega_0^* \in \Omega^{1} (C^N_{q,\gamma})$. Now the theorem follows.
\end{proof}

The existence of relations such as \eqref{omegas} means that the generators $\omega_0$ and $\omega_1$ of  the  left (or right) $\cO(C^N_{q,\gamma})$-module $\Omega^{(1,0)} (C^N_{q,\gamma})$ are not independent.  That $\Omega^{(1,0)} (C^N_{q,\gamma})$  is a finitely generated and projective $\cO(C^N_{q,\gamma})$-module may be argued as follows. $\Omega^{(1,0)} (C^N_{q,\gamma})$ is a submodule of the free $\cO(D_{q,\gamma})$-module  $\cO(D_{q,\gamma})\, dz$, generated by $z^{N-1}dz$ and $z^*dz$, hence it can be identified with the $\cO(C^N_{q,\gamma})$-submodule of $\cO(D_{q,\gamma})$ generated by $z^{N-1}$ and $z^*$. This is exactly the $\cO(C^N_{q,\gamma})$-module consisting of all elements in $\cO(D_{q,\gamma})$ of degree $N-1$. By Theorem~\ref{thm.free}, $\cO(D_{q,\gamma})$ is strongly graded, hence any submodule of elements of a fixed degree is  projective by \cite[A~I.3.3~Corollary]{NasVan:gra}.  Therefore  $\Omega^{(1,0)} (C^N_{q,\gamma})$ can be seen as module of sections of a non-commutative vector bundle over the quantum cone. Furthermore, since $\cO(D_{q,\gamma})$ has no invertible elements of degree $1$ or $N-1$, by the same arguments as those following the proof of Theorem~\ref{thm.free}, the module $\Omega^{(1,0)} (C^N_{q,\gamma})$ is  not free. This is in full agreement with the identification of  $\cO(C^2_{q,1-q})$ as the quantum equatorial sphere; holomorphic cotangent bundle over the sphere is not trivial. The employment of the $*$-conjugation yields similar statements for $\Omega^{(0,1)} (C^N_{q,\gamma})$.

Perhaps it is no too surprising that the quantum cones $C^N_{q,\gamma}$ admit natural complex structures, for the considerations of Section~\ref{sec.smooth} substantiate the claim that they are smooth quantum manifolds. The exactness of the volume form might indicate that quantum cones correspond to non-commutative manifolds with boundaries.  The non-triviality  of the cotangent bundle indicates that  $C^N_{q,\gamma}$ correspond to non-parallelizable manifolds. For these reasons one might think about the non-commutatrve smoothing of the cone as of blowing up the singular point into a balloon with a neck rather than as of rounding it.

\section{Outlook and speculations}
My motivation for writing this note was to draw the reader's attention to unexpected results of actions of finite groups on quantum spaces. The classically non-free actions are freed upon quantization, singularities are removed, and the geometric and topological natures of the spaces of fixed points are transformed. This opens up an exciting possibility of deploying the full power of differential geometry in areas in which it could not be utilized previously. Already the study of one of the simplest examples leads to surprises, poses new questions and opens up avenues for further research. An immediate direction of studies would be to try and understand quantum cones as Riemannian surfaces, whether from the spectral point of view \cite{Con:non} or in a more algebraic setup \cite{Maj:Rie}, \cite{BegMaj:sta}. Once these aspects are understood in sufficient depth, one can attempt to build physical models based on quantum cones. In another direction the {\em differential smoothness} of quantum cones in the sense of \cite{BrzSit:pil}, i.e.\ the existence of an isomorphism between complexes of differential and integral forms \cite{BrzElK:int}, should be investigated. The lessons thus learnt should be applied to other deformations of orbifolds including those that can be obtained by  glueing of quantum cones.

Hitherto we discussed only algebraic aspects of quantum cones, leaving aside topological issues.  In view of relations \eqref{cone.z}, the $C^*$-algebra completions of the $\cO(C^N_{q,\gamma})$ to algebras of continuous functions $\cC(C^N_{q,\gamma})$ can be carried out in the same way as for the quantum real weighted projective spaces \cite{Brz:Sei}. Consequently,  $\cC(C^N_{q,\gamma})$ is a pullback of $N$-copies of the Toeplitz algebra along the symbol maps or the algebra of continuous functions on the $N$-copies of  quantum discs glued along their circular boundaries; see \cite{CalMat:cov}. In particular  $\cC(C^2_{q,\gamma})$ is isomorphic to the algebra of continuous functions on the quantum equatorial or generic Podle\'s sphere  \cite{Pod:sph}. This identifies $C^2_{q,\gamma}$ as a deformation of a compact manifold without a boundary. Classically, volume forms for such manifolds are never exact, yet the volume forms described in Theorem~\ref{thm.complex} are exact! The exactness of the volume form is compatible with the interpretation of $C^2_{q,\gamma}$ as the cone but not as the sphere. On the other hand, the interpretation of $C^2_{q,\gamma}$ as a sphere is compatible with the non-triviality of cotangent bundles.

The identification of $C^2_{q,\gamma}$ leads to another question: how is it possible for the sphere to arise from an action of a finite group on the disc? A possible heuristic answer might be to  think about the quantum disc as having an internal structure: the quantum disc is composed of layers of quantum discs. The $\ZZ_2$-action does not bulge  the disc into a cone, but rather it gently separates two internal layers, which are still joint at the common circle, thus forming a sphere or a balloon. Speculatively, one might interpret the classical circle that the equatorial Podle\'s sphere contains as the rim of the neck of the balloon (rather than as the equator), and thus reconcile this point of view with the interpretation of the differential smoothing offered at the end of Section~\ref{sec.complex}. For the  action of  $\ZZ_N$, $N$ internal layers are separated.

\section*{Acknowledgements.}
It is my pleasure to thank the organizers of the workshop on Noncommutative Field Theory and Gravity, Corfu 2013 for creating a friendly and inspiring research environment.


TITLE: Are the following functions convex/concave? $f(x, y) = x^{3}-\ln(x^2y)$ and $f(x,y)=x^\alpha y^\beta-x-y$, where both $\alpha, \beta\in(0,1)$
QUESTION [1 upvotes]: The definition of concave/convex function in a open,convex set A is:

Let $f : A ⊆ R^n −→ R$ with $A$ open and convex and let $f ∈ C^2(A)$;
  then $f$ is convex in $A$ if and only if, for all $x ∈ A$, $D2f(x)$
  (Hessian) is Positive SD;  $f$ is concave in A if and only if, for all
  $x ∈ A, D2f(x)$ is Negative SD.

I would like to know whether the following functions are convex/concave in their maximal domain of existence. If not, is there a 'maximal' open set A where they are convex/concave?
1) $f(x, y) = x^{3}-\ln(x^2y)$.
2) $f(x,y)=x^\alpha y^\beta-x-y$, where both $\alpha, \beta \in(0,1)$.
**

Here is my attempt:

**
1)
The maximal interval is:
$D=\{(x, y) ∈ R^2 : x \neq 0, y>0\}$
Clearly this set is not convex, so that we can not have convexity/concavity of f. Looking at the Hessian:
$H=\pmatrix{6x+\frac{2}{x^2}&0 \cr 0&\frac{1}{x^2}\cr}$
We get that the function is convex in $A=\{(x, y) ∈ R^2 : x > 0, y>0\}$
2)
The maximal interval is:
$D=\{(x, y) ∈ R^2 : x \geq 0, y\geq 0\}$
Clearly this set is not open, so that we can not have convexity/concavity of f. But I think we can have convexity/concavity in $A=\{(x, y) ∈ R^2 : x > 0, y>0\}$  Looking at the Hessian:
$H=\pmatrix{\alpha(\alpha-1)x^{\alpha-2}y^\beta&\beta(\alpha)x^{\alpha-1}y^{\beta-1}\cr \beta(\alpha)x^{\alpha-1}y^{\beta-1}&\beta(\beta-1)x^\alpha y^{\beta-2}\cr}$
We get then that the determinant is: $\beta (\beta-1)\alpha (\alpha-1) x^{2(\alpha-1)}y^{2(\beta -1)}-\beta^2 \alpha ^2 x^{2(\alpha-1)}y^{2(\beta -1)}$, where the term $a_{11}$ of the hessian matrix is $\alpha(\alpha-1)x^{\alpha-2}y^\beta\leq0$ $\forall x,y$. So the hessian is NSD if the determinant is $\geq 0$. But it looks like the sign of the determinant depends on $\alpha,\beta$. But I can not get for which values of $\alpha, \beta$ the determinant is positive or zero.
Can you help me? Is my first attempt correct? And also what about the second one??

REPLY [1 votes]: Your answer is correct. The first function is convex in the region you found and there is no way for the first function to be concave because the determinant and the first principal minor in the Hessian of the first function always have the same sign. For the second part though, we need the determinant and the first principal minor($a_{11}$) to be positive  thus $\beta(\beta-1)\alpha(\alpha-1)x^{2(\alpha-1)}y^{2(\beta -1)}-\beta^2\alpha^2x^{2(\alpha-1)}y^{2(\beta -1)}\ge0\Rightarrow\beta(\beta-1)\alpha(\alpha-1)-\beta^2\alpha^2\ge0\Rightarrow\alpha\beta(1-\alpha-\beta)\ge0$.
Also $\alpha(\alpha-1)x^{\alpha-2}y^\beta \ge 0 \Rightarrow \alpha(\alpha-1)\ge 0$ because the domain is $x\ge0,y\ge0$. Thus we need to find where these two expression $f(\alpha ,\beta)=\alpha\beta(1-\alpha-\beta),g(\alpha ,\beta)=\alpha(\alpha-1)$ are positive. In the following figure we see where $f(\alpha,\beta)$ is positive. Four regions exist $\alpha,\beta \in [0,1]$ or $\alpha\ge1,\beta\le0$ or $\alpha\le0,\beta\ge1$ or $\alpha\le0,\beta\le0$. We can find these regions from the fact that the multiplications of three factor $\alpha$, $\beta$ and $(1-\alpha-\beta) $ must be positive and since each factor is linear, there are $2^3-1$ regions total (rach one being positive or negative)(please consider the horizontal axis as $\alpha$ and the vertical axis as $\beta$)

In the second figure we see where the second one $g(\alpha,\beta)$ is positive. This one is easier, since is only a function of $\alpha$ and we have two linear factors and $2^2-1$ regions total.

and by intersection we arrive at the three regions that satisfies convexity $\alpha\ge1,\beta\le0$ or $\alpha\le0,\beta\ge1$ or $\alpha\le0,\beta\le0$. There is also the assumption that $\alpha,\beta \in (0,1)$. If we take this into consideration, then there is no way this function can be convex. But hopefully there is one region that this function can be concave. For concavity the Hessian must be negative definite or equivalently, $a_{11}$ must be negative and the determinant must be positive, Therefore $f(\alpha,\beta)$ must be positive and $g(\alpha,\beta)$ must be negative. The only region satisfying this according to above explanations is the intersection of the region $\alpha,\beta \in (0,1)$ and under the line $\alpha+\beta=1$, the triangular region $ABC$ in the first figure.


TITLE: Compute the Cayley table of $\mathbb{Z}^*_{10}$
QUESTION [0 upvotes]: Could someone please help me with this question: Compute the Cayley table of the group $\mathbb{Z}^*_{10}$ and give an explicit definition for the different homomorphisms $\psi_1 ,$$\psi_2$$\quad$ $\mathbb{Z}^*_{10}$$\to$$\mathbb{Z}^*_{10}$.

REPLY [1 votes]: Here are simple steps you may follow.
1.List the non zero element of modular 10 which are 1,2,3,4,5,6,7,8,9 .
2.Determine the value of gdc(k,n)=1
i.e gcd(1,10)=1 , gcd(2,10)=2,gcd(3,10)=1, gcd(4,10)=2,gcd(5,10)=5,gcd(6,10)=2,gcd(7,10)=1,gcd(8,10)=2,gcd(9,10)=1.
3.The elements you need to form the Cayley table are 1,3,7,9


TITLE: Number of solutions in linear equation with 3 variables
QUESTION [1 upvotes]: Is there a way how to determine number of solutions in linear equation like this: $ax + by + cz = d$, where $a,b,c,x,y,z,d$ are non-negative integers and $a,b,c,d$ are known?

REPLY [0 votes]: Above equation shown below:
$ax + by + cz = d$  -------$(1)$
For (a, b, c, d)= (3, 2, 5, 32), equation $(1)$ has parametric solution given below:
$x=(1-v)$
$y=(-5u+9v+87)$
$z=(2u-3v-29)$
For $(u,v)$ = $(15, 0)$ we get,
$(x, y, z)$ = $(1, 12, 1)$