text:
\begin{document}
\title{Smart Inverter Grid Probing for Learning Loads:\\
Part II -- Probing Injection Design}

\author{
	Siddharth Bhela,~\IEEEmembership{Student Member,~IEEE,}
	Vassilis Kekatos,~\IEEEmembership{Senior Member,~IEEE,} and
	Sriharsha Veeramachaneni

}

\maketitle

\begin{abstract}
This two-part work puts forth the idea of engaging power electronics to probe an electric grid to infer non-metered loads. Probing can be accomplished by commanding inverters to perturb their power injections and record the induced voltage response. Once a probing setup is deemed topologically observable by the tests of Part I, Part II provides a methodology for designing probing injections abiding by inverter and network constraints to improve load estimates. The task is challenging since system estimates depend on both probing injections and unknown loads in an implicit nonlinear fashion. The methodology first constructs a library of candidate probing vectors by sampling over the feasible set of inverter injections. Leveraging a linearized grid model and a robust approach, the candidate probing vectors violating voltage constraints for any anticipated load value are subsequently rejected. Among the qualified candidates, the design finally identifies the probing vectors yielding the most diverse system states. The probing task under noisy phasor and non-phasor data is tackled using a semidefinite-program (SDP) relaxation. Numerical tests using synthetic and real-world data on a benchmark feeder validate the conditions of Part I; the SDP-based solver; the importance of probing design; and the effects of probing duration and noise.
\end{abstract}

\begin{IEEEkeywords}
Smart inverters, power system state estimation, Farka's lemma, max-sum diversity, semi-definite relaxation.
\end{IEEEkeywords}

\section{Introduction}\label{sec:introd}
Part I of this work put forth the novel data acquisition scheme of probing-to-learn (P2L). The P2L scheme leverages smart inverters to probe an electric grid with the purpose of finding the values of non-metered loads. It also provided conditions under which a particular probing setup is successful. In particular, it was shown that given the feeder graph $\mcG$, the locations of non-metered buses $\mcO$ and probing buses $\mcM$, and the number of probing actions $T$, a simple linear program could tell whether non-metered loads could be recovered or not. Assuming noiseless data, this test relied on the generic rank of the Jacobian matrix $\bJ\left(\{\bv_t\}\right)$ related to the P2L equations. It is thus a \emph{topological} rather than a \emph{numerical observability} guarantee~\cite[Ch.~4.6]{ExpConCanBook}.

Even for the standard power flow (PF) and power system state estimation (PSSE) setups, topological observability relates to the sparsity structure of the associated Jacobian matrix. This structure alone however cannot adequately capture the numerical column rank of the Jacobian matrix. There exist specification or measurement sets whose Jacobian is full column-rank in general, but becomes ill-conditioned or even singular under specific state values (including the boundaries for voltage collapse); see e.g.,~\cite[Ch.~10]{ExpConCanBook}, \cite{Yang17}. In addition, once a probing setup is deemed topologically observable, the power injections of probing inverters could be judiciously selected to improve load or state estimates. This is challenging since P2L is an implicit nonlinear identification task, and probing injections should be comply to network constraints without knowing the non-metered loads.

The contribution of Part II is on two practical aspects of grid probing. First, a systematic approach to design probing setpoints that conform to grid safety and improve numerical accuracy is developed in Section~\ref{sec:probing}. Second, the proposed P2L task is tackled through semidefinite program (SDP)-based solvers presented in Section~\ref{sec:solvers}. The conditions of Part I along with the probing setpoint design and the solver of Part II, are numerically validated using actual residential load data from the Pecan Street project on the IEEE 34-bus benchmark feeder in Section~\ref{sec:sim}. Conclusions and current research efforts are outlined in Section~\ref{sec:conclusion}.

Adding to the notational conventions of Part I, here the symbol $\bone$ denotes the all-one vector and $\be_k$ is the $k$-th canonical vector; their dimensions would be clear from the context. The notation $\bV \succeq \bzero$ means that $\bV$ is a Hermitian (complex and conjugate symmetric) positive semidefinite matrix; the matrix trace is denoted by $\trace(\cdot)$; and $\|\ba\|_2$ is the $\ell_2$-norm of vector $\ba$. The notation $k=1:K$ is a shorthand to $k=1,\dots, K$.

\section{Designing Probing Injections}\label{sec:probing}
Suppose probing has been deemed successful for a particular $(\mcM,\mcO)$ placement of probed and non-metered buses, i.e., the setup $(\mcM,\mcO)$ has passed the test of Algorithm~1 or 2 of Part I. The next question is how to select probing setpoints that are implementable by inverters; compliant to feeder constraints; and at the same time, improve estimation accuracy. This section deals with the design of inverter setpoints during probing interval $\mcT$ with slots $t=1,\ldots,T$, for a given $T$.

To facilitate the exposition, let us stack the power injections at all probing buses $\{(p_{n},q_{n})\}_{n\in\mcM}$ in vectors $\bp_\mcM$, $\bq_\mcM$, and $\bs_\mcM:=[\bp_\mcM^\top~\bq_\mcM^\top]^\top$. Likewise, the injections at all non-metered buses $\{(p_{n},q_{n})\}_{n\in\mcO}$ are collected in $\bp_\mcO$, $\bq_\mcO$, and $\bs_\mcO:=[\bp_\mcO^\top~\bq_\mcO^\top]^\top$. The injections at slot $t$ will be denoted by a superscript $t$.

In search of a meaningful metric to design the probing injections $\{\bs_\mcM^t\}_{t=1}^T$, one could consider the minimum mean square estimation error for non-metered loads $\bs_\mcO$ or states $\{\bv_t\}_{t=1}^T$. The former is hard to derive given the implicit estimation task involved. The latter exhibits the Cramer-Rao lower bound (CRLB) of $[\bJ^\top\left(\{\bv_t\}\right)\bJ\left(\{\bv_t\}\right)]^{-1}$; a proof for this CRLB can be obtained by adopting the result in~\cite{Gang18}. Since $\bJ\left(\{\bv_t\}\right)$ depends linearly on $\{\bv_t\}_{t=1}^T$, the CRLB depends inverse quadratically on the unknown states.

To arrive at a practical solution, we resort to selecting probing setpoints so that the electric grid is driven to the most diverse states $\{\bv_t\}_{t=1}^T$ while abiding by inverter and feeder operational constraints. We conjecture that probing the grid to effect larger state variations across $\mcT$ would yield smaller condition numbers for $\bJ\left(\{\bv_t\}\right)$ and $\bJ^\top\left(\{\bv_t\}\right)\bJ\left(\{\bv_t\}\right)$.

Hence, the goal is to design $\{\bs_\mcM^t\}_{t=1}^T$ that yield the most diverse system states $\{\bv_t\}_{t=1}^T$. Since the system states depend on both $\{\bs_\mcM^t\}_{t=1}^T$ and the unknown $\bs_\mcO$ in a non-linear fashion, our design adopts a linearized power flow model. The latter can be obtained by taking the first-order Taylor's series approximation of the PF equations with respect to nodal voltages expressed in polar coordinates~\cite{BoDo15}, \cite{Deka17}. Unless a reference system state is available, the linearization occurs at the flat voltage profile of $\tbv=u_0\bone+j\bzero$, and yields the so termed \emph{linearized distribution flow} (LDF) model~\cite{BW2}, \cite{Deka17}, which can be rearranged for our analysis as
\begin{equation}\label{eq:LDF}
\by:=\begin{bmatrix}
\bu-u_0\bone\\
\boldsymbol{\theta}
\end{bmatrix}
=
\begin{bmatrix}
\bK & \bL \\
\bM & \bN
\end{bmatrix}
\begin{bmatrix}
\bs_\mcM \\
\bs_\mcO
\end{bmatrix}.
\end{equation}
The vectors $\bu$ and $\btheta$ collect the voltage magnitudes and angles at all buses excluding the substation; and matrices $(\bK,\bL,\bM,\bN)$ depend on the bus admittance matrix $\bY$; see~\cite{Deka17}, \cite{GVV17}. Armed with a linear mapping between power injections and voltages, the design of setpoints $\{\bs_\mcM^t\}_{t=1}^T$ is accomplished next in three steps.

\subsection{Build Library of Implementable Probing Setpoints}\label{subsec:S1}
The first step of the setpoint design builds a library $\mcS$ of $K\gg T$ \emph{candidate} injection vectors indexed by $k$
\begin{equation}\label{eq:S}
\mcS:=\{\bs_{\mcM}^k\}_{k=1}^K.
\end{equation}
The entries of each $\bs_{\mcM}^k$ should be \emph{implementable}, in the sense that each probing inverter should be able to inject the requested value of complex power.

To characterize the allowable range of inverter injections $(p_{n},q_{n})$ with $n\in\mcM$, two inverter classes are identified. The first class consists of inverters interfacing solar panels. When inverter $n$ interfaces a solar panel, its complex injection is limited by its apparent power capacity $\bar{s}_n$ as
\begin{equation}\label{eq:appc1}
p_{n}^2+q_{n}^2 \leq \bar{s}_n^2.
\end{equation}
Moreover, if the maximum active power that can be generated given the solar irradiance at the current probing period is $\bar{p}_{n}$, then its active power injection is limited by
\begin{equation}\label{eq:appc2}
0\leq p_{n}\leq \bar{p}_{n}.
\end{equation}

The second class consists of inverters interfacing energy storage units. The apparent power constraint of \eqref{eq:appc1} should still be enforced. If the power rate of energy storage unit $n$ is $\bar{p}_{n}$, the active injection from inverter $n$ should lie within
\begin{equation}\label{eq:appc3}
-\bar{p}_{n}\leq p_{n}\leq \bar{p}_{n}
\end{equation}
since the battery can be charged or discharged. Given the short duration of probing, limits on the state of charge have been ignored for simplicity.

Given the limitations for each inverter class, a candidate probing injection $\bs_\mcM^k\in\mcS$ can be constructed by sampling uniformly at random $p_n^k$ within \eqref{eq:appc2}--\eqref{eq:appc3} for all $n\in\mcM$. Upon fixing active injections, the reactive injections can be sampled again uniformly at random within $|q_n^k|\leq \sqrt{\bar{s}_n^2-(p_n^k)^2}$ to comply with \eqref{eq:appc1}. Scenarios where a single bus hosts multiple inverters belonging to the previous two or additional classes can be incorporated in the sampling process.

As explained in Remark~1 of Part I, a probing bus $n\in\mcM$ may be hosting controllable inverters and non-controllable assets (non-probing inverters and non-controllable loads). The process of sampling implementable injections through \eqref{eq:appc1}--\eqref{eq:appc3} can be repeated for all controllable inverters. The net injection from non-controllable assets is assumed to be metered; that is the case for the Pecan Street dataset~\cite{pecandata}. The complex powers injected into bus $n$ are summed up and used in the P2L. To keep the notation uncluttered, we will slightly abuse notation and denote this net injection at bus $n$ as $p_n+jq_n$.

The sampling process is repeated $K$ times to construct library $\mcS$. Although each candidate probing vector $\bs_\mcM^k\in\mcS$ can be implemented by inverters, the aggregate effect of probing injections may be violating feeder constraints. To handle this concern, we next reduce library $\mcS$ to only those probing injections abiding by feeder constraints.

\subsection{Maintaining only Network-Compliant Probing Setpoints}\label{subsec:S2}
Even though a probing action lasts for one second or two, the operator may still want to guarantee that it does not violate any feeder constraints. For example, voltage regulation standards dictate voltage magnitudes to remain within a pre-specified range as $\underline{u}\leq u_n\leq \overline{u}$ for all $n\in\mcN^+$. A probing injection vector $\bs_\mcM^k\in\mcS$ is deemed \emph{network-compliant} if the incurred voltage deviations are maintained within the allowable range $\underline{u}\bone\leq\bu -u_0\bone\leq \overline{u}\bone$ with the inequalities applied entry-wise. Thanks to \eqref{eq:LDF}, these voltage constraints can be expressed as linear inequality constraints on $\bs_\mcM^k$
\begin{equation}\label{eq:volt}
\underline{u}\bone\leq \bK\bs_\mcM^k +\bL \bs_\mcO \leq \overline{u}\bone.
\end{equation}

One cannot directly check whether $\bs_\mcM^k$ is network-compliant, since $\bs_\mcO$ is unknown. To bypass this complication, non-metered loads are assumed to lie within a known range
\begin{equation} \label{eq:load}
\underline{\bs}_{\mcO} \leq \bs_{{\mcO}} \leq \overline{\bs}_{\mcO}.
\end{equation}
The bounds $(\underline{\bs}_{\mcO},\overline{\bs}_{\mcO})$ can be derived from historical data, the confidence intervals of load forecasts, or the load estimates obtained during the previous probing period.

Adopting a robust design, we would like to comply with the voltage constraints in \eqref{eq:volt} for all possible values of non-metered loads in \eqref{eq:load}. To do so, we leverage the next version of Farka's lemma on the containment of polytopes.

\begin{lemma}[\cite{Eaves1982}, \cite{Mangasarian2002}, \cite{Zhao17}]\label{le:farka}
The non-empty polytope $\mcP_1:=\{\bx:\bA\bx\leq \bb\}$ with $\bA\in\mathbb{R}^{M\times N}$ is contained within the polytope $\mcP_2:=\{\bx:\bC\bx\leq \bd\}$ with $\bC\in\mathbb{R}^{K\times N}$ if and only if there exists matrix $\bE\geq \bzero$ satisfying $\bE\bA=\bC$ and $\bE\bb\leq \bd$.
\end{lemma}

Based on Lemma~\ref{le:farka}, to ensure that the polytope over $\bs_\mcO$ defined in \eqref{eq:load} is contained within the polytope of \eqref{eq:volt}, we need to solve the feasibility problem
\begin{align}\label{eq:farkas}
\mathrm{find}~&~\bE\\
\mathrm{s.to}~&~\bE\geq \bzero\nonumber\\
~&~\bE\left[\begin{array}{c}
-\mathbf{I}_{2O}\\
\mathbf{I}_{2O}
\end{array}\right]=
\left[\begin{array}{c}
-\mathbf{L}\\
\mathbf{L}
\end{array}\right]\nonumber\\
~&~\bE\left[\begin{array}{c}
-\underline{\bs}_\mcO\\
\overline{\bs}_\mcO
\end{array}\right]\leq
\left[\begin{array}{c}
\bK\bs_\mcM^k+(u_0-\underline{u})\bone\\
-\bK\bs_\mcM^k-(u_0-\overline{u})\bone
\end{array}\right].\nonumber
\end{align}
Given $\bs_\mcM^k$, if the linear program in \eqref{eq:farkas} is feasible, the candidate vector $\bs_\mcM^k$ is deemed network-compliant and is copied to the \emph{reduced library} $\mcS_r$. Otherwise, the candidate vector is not copied to $\mcS_r$ since there exist load values within $[\underline{\bs}_{\mcO},\overline{\bs}_{\mcO}]$ that violate the voltage constraints in \eqref{eq:volt}. The test of \eqref{eq:farkas} is repeated for all $\bs_\mcM^k\in\mcS$ to get the reduced library $\mcS_r:=\{\bs_\mcM^\ell\}_{\ell=1}^L$ of $L$ candidate injection vectors with $L\leq K$.

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.4]{clustT4.pdf}
	\caption{Percentage of candidate vectors $\bs_\mcM^k\in\mcS$ that violate \eqref{eq:volt} for varying voltage bounds $(\underline{u},\overline{u})$. The percentage of non-admissible vectors decreases with smaller load uncertainty and/or looser voltage regulation bounds.}
	\label{fig:clustT4}
\end{figure}

To demonstrate the importance of this library reduction step, we ran a numerical test on the IEEE 34-bus feeder for $T=4$ and $O=10$; see Fig.~\ref{fig:clustT4}. Load uncertainty in \eqref{eq:load} was confined within $\underline{\bs}_{\mcO}=(1-\frac{1}{\gamma})\bs_{\mcO}$ and $\overline{\bs}_{\mcO}=(1+\frac{1}{\gamma})\bs_{\mcO}$ for $\gamma>0$. The candidate inverter injections in $\mcS$ were randomly drawn from $\pm 0.2$ pu and tested against \eqref{eq:farkas}. For increasing $\gamma$, the uncertainty bounds in \eqref{eq:load} became tighter and progressively more candidate vectors were rendered admissible. Even for loose voltage regulation limits of $\pm 10\%$ and tight load uncertainty, more than $20\%$ of the candidates in $\mcS$ violated \eqref{eq:volt}.

The reduction from $\mcS$ to $\mcS_r$ via \eqref{eq:farkas} can be generalized. For example, limits on line and transformer flows can be expressed as linear functions of power injections and appended to \eqref{eq:volt}. Moreover, correlations in load forecasts across buses, or power factor limitations applied on a per-bus basis, both can be directly captured as linear inequalities and appended to \eqref{eq:load}. Finally, if the library has been reduced significantly so that $L<T$, the operator could broaden the voltage interval $[\underline{u},\overline{u}]$ and/or tighten the load uncertainty range in \eqref{eq:volt} if grid probing is still needed to recover non-metered loads.

\subsection{Finding Probing Setpoints with Most Diverse States}\label{subsec:S3}
Given the reduced library $\mcS_r=\{\bs_\mcM^\ell\}_{\ell=1}^L$ of implementable and network-compliant candidates, the last step is to select the $T$ candidates yielding the most diverse states. Recall that the system state $\bv^\ell$ related to probing injection $\bs_\mcM^\ell$ depends also on the unknown loads $\bs_\mcO$. Moreover, the dependence on both $\bs_\mcM^\ell$ and $\bs_\mcO$ is non-linear and implicit. The approximate LDF model of \eqref{eq:LDF} can help us circumvent these technical challenges.

The Euclidean distance between the system states induced by injections $\bs_\mcM^\ell,\bs_\mcM^{\ell'}\in\mcS_r$ will be surrogated by the Euclidean distance between the approximate states of \eqref{eq:LDF} as
\begin{equation*}
\|\bv^\ell - \bv^{\ell'}\|_2\simeq \|\by_\ell - \by_{\ell'}\|_2
\end{equation*}
for all $\ell,\ell'=1,\ldots,L$. The latter simplifies as
\begin{align*}
\|\by_\ell - \by_{\ell'}\|_2&=\left\|\begin{bmatrix}
\bK & \bL \\
\bM & \bN
\end{bmatrix}\left(
\begin{bmatrix}
\bs_\mcM^\ell \\
\bs_\mcO
\end{bmatrix}-
\begin{bmatrix}
\bs_\mcM^{\ell'} \\
\bs_\mcO
\end{bmatrix}
\right)\right\|_2\nonumber\\
&=\left\|\begin{bmatrix}
\bK\\
\bM
\end{bmatrix}
\left(\bs_\mcM^\ell-\bs_\mcM^{\ell'}\right)\right\|_2
\end{align*}
where we have exploited the linearity in \eqref{eq:LDF} together with the fact that non-metered loads remain roughly invariant during probing. We define the distance between $\bs_\mcM^\ell,\bs_\mcM^{\ell'}\in\mcS_r$ as
\begin{align}\label{eq:dist}
d(\ell,\ell')&:=\|\by_\ell - \by_{\ell'}\|_2^2\nonumber\\
&=(\bs_\mcM^\ell-\bs_\mcM^{\ell'})^\top
(\bK^\top\bK+\bM^\top\bM)
(\bs_\mcM^\ell-\bs_\mcM^{\ell'}).
\end{align}

Based on this metric, we would like to select a subset $\mcA$ of $T$ out of the $L$ candidate vectors in $\mcS_r$ so that the sum of their pairwise distances is maximized
\begin{align}\label{eq:MSD}
\max_{\mcA\subset\mcS_r}~&~\sum_{\ell\in\mcA}\sum_{\ell'\in\mcA} d(\ell,\ell')\\
\mathrm{s.to}~&~|\mcA|=T.\nonumber
\end{align}
The task in \eqref{eq:MSD} is known as the \emph{max-sum diversity} (MSD) problem, and appears frequently in information retrieval, computational geometry, and operations research~\cite{Cevallos16}. In fact, MSD can be reformulated as a binary quadratic program (QP) after introducing the $L\times L$ distance matrix $\bD$ with entries $D_{\ell,\ell'}:=d(\ell,\ell')$ as
\begin{subequations}\label{eq:MSD2}
\begin{align}
f^\star:=\max_{\bx\in\{0,1\}^L}~&~\bx^\top\bD\bx\label{eq:MSD2:cost}\\
\mathrm{s.to}~&~\bx^\top\bone=T.\label{eq:MSD2:c1}
\end{align}
\end{subequations}
Despite its simple form, the MSD task is NP-hard~\cite{Cevallos16}. However, thanks to the properties of $\bD$, the problem in \eqref{eq:MSD2} enjoys a polynomial-time approximate scheme (PTAS)~\cite{Cevallos16}.

Although $\bD$ is indefinite, the objective in \eqref{eq:MSD2:cost} can be shown to be concave under constraint \eqref{eq:MSD2:c1}. To see this, define the $2N\times L$ matrix $\tbY:=\left[\by_1~\cdots~\by_L\right]$ and use the definition of $d(\ell,\ell')$ to rewrite the objective of \eqref{eq:MSD2} as
\begin{align*}
f(\bx)&:=\bx^\top\bD\bx=\sum_{\ell=1}^L\sum_{\ell'=1}^L x_\ell x_{\ell'} D_{\ell,\ell'}\\
&=\sum_{\ell=1}^L\sum_{\ell'=1}^L x_\ell x_{\ell'} \left(\|\by_\ell\|_2^2+\|\by_{\ell'}\|_2^2-2\by_\ell^\top\by_{\ell'}\right)\\
&=\sum_{\ell'=1}^L  x_{\ell'} \|\by_{\ell'}\|_2^2\left(\sum_{\ell=1}^L x_\ell \right) + \sum_{\ell=1}^L  x_{\ell} \|\by_{\ell}\|_2^2\left(\sum_{\ell'=1}^L x_{\ell'} \right)\\
&-2\sum_{\ell=1}^L\sum_{\ell'=1}^L x_\ell x_{\ell'}\by_\ell^\top\by_{\ell'}\\
&=2T\bc^\top\bx -2\bx^\top \tbY^\top\tbY \bx.
\end{align*}
where $\bc:=\left[\|\by_1\|_2^2~\cdots~\|\by_L\|_2^2\right]^\top$. Since $\tbY^\top\tbY\succeq \bzero$, the objective $f(\bx)$ equals a concave quadratic function.

For moderate $L$ (a few hundreds), the task in \eqref{eq:MSD2} can be handled by a mixed-integer QP solver. For $T=2$, the MSD solution can be found by an exhaustive search. For larger $T$, we will use a randomized rounding approach, as adopted from \cite{Rag1987} in \cite[Remark~2]{Cevallos16}. The approach is briefly reviewed here for completeness. Its first step solves the relaxed problem
\begin{subequations}\label{eq:MSDr}
\begin{align}
\hbx:=\arg\min_{\bzero\leq \bx\leq \bone}~&~2\bx^\top \tbY^\top\tbY \bx- 2T\bc^\top\bx\label{eq:MSDr:cost}\\
\mathrm{s.to}~&~\bx^\top\bone=T.\label{eq:MSDr:c1}
\end{align}
\end{subequations}
Since the binary constraints of \eqref{eq:MSD2} are related to box constraints in \eqref{eq:MSDr}, it holds that $f(\hbx)\geq f^\star$. To construct a point $\tbx$ that is feasible for \eqref{eq:MSD2}, draw $L$-dimensional vectors $\{\tbx_i\}$ whose entries are independent Bernoulli random variables with mean $(1-\beta)\hbx$ for some $\beta>0$, say $\beta=0.1$. The so constructed binary vectors $\tbx_i$'s satisfy $\mathbb{E}[\tbx_i^\top\bone]=(1-\beta)T$ and $\mathbb{E}[\tbx_i^\top\bD\tbx_i]=(1-\beta)^2\hbx^\top\bD\hbx$. The purpose of scaling $\hbx$ by $(1-\beta)$ is to ensure $\tbx_i$'s are both feasible for \eqref{eq:MSD2} and yield relatively high cost with significant probability~\cite{Cevallos16}.

Let us now comment on the complexity for designing probing setpoints. The first step described in Section~\ref{subsec:S1} is computationally inexpensive. The second step of Section~\ref{subsec:S2} involves solving the linear program in \eqref{eq:farkas} $K$ times, once for each candidate setpoint vector. The third step of Section~\ref{subsec:S3} entails solving the linearly-constrained quadratic program of \eqref{eq:MSDr}, whose complexity is cubic in the number of variables $L$. As detailed later in Section~\ref{sec:sim}, running this design process for the IEEE 34-bus feeder and $K=100$ candidate setpoints took $1-1.5$ min depending on $(M,O)$. The tests were run on a laptop computer using generic off-the-shelf solvers.

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.4]{barplot.pdf}
	\caption{Histogram of condition numbers for the Jacobian matrix $\mathbf{J}(\{\mathbf{v}_t\}_{t=1}^4)$ obtained by randomly sampling quadruplets of $\bs_\mcM$'s from $\mcS_r$.}
	\label{fig:barplot}
\end{figure}

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.32]{clustT2.pdf}
	\caption{Probing setpoint design on the IEEE 34-bus grid for $T=2$ and $O=6$. The blue (red) lines correspond to the system states induced by the first (second) probing setpoints $\{\bs_\mcM^t\}_{t=1}^2$. Solid lines depict voltage magnitudes, and dashed lines voltage angles.}
	\label{fig:clustT2}
\end{figure}

To justify the need for this third step in probing design, we conducted a test on the IEEE 34-bus feeder for $T=4$ and $O=6$. For this test, load uncertainty was confined within a factor of $\pm 1$ times the nominal loads. Given library $\mcS$ of randomized injections drawn from $\pm 0.2$~pu and obeying \eqref{eq:appc1}--\eqref{eq:appc3}, we constructed the reduced library $\mcS_r$ based on \eqref{eq:farkas} for $[\underline{u},\overline{u}]=[0.90,1.10]$. We then solved \eqref{eq:MSDr} and followed the randomized rounding process to construct 100 binary $\tbx_i$'s. We evaluated the cost $f(\tbx_i)$ for those $\tbx_i$'s satisfying $\tbx_i^\top\bone=T$, and returned the $\tbx_i$ yielding the largest cost. The condition number of the Jacobian matrix evaluated at the so obtained $\tbx_i$ was $3\cdot 10^6$. We also calculated the condition number of the Jacobian matrix evaluated at random candidate quadruplets in $\mcS_r$. The latter condition numbers ranged within $10^7-10^9$; see Fig.~\ref{fig:barplot}. Hence, albeit MSD adds computational complexity, it is an important part of the probing design process.

To show that this MSD step provides diverse system states, we performed another test on the IEEE 34-bus feeder under the same setup, but for $T=2$. The states induced by the designed probing and nominal loads are shown in Figure~\ref{fig:clustT2}.

Upon solving \eqref{eq:MSD} near optimally, we have obtained $T$ probing injection vectors $\{\bs_\mcM^t\}_{t=1}^T$ that are: \emph{i)} implementable by inverters; \emph{ii)} network-compliant; and \emph{iii)} yield diverse system states. In the process of probing design, the first step (Section~\ref{subsec:S1}) operates on the entries of $\bs_\mcM^t$'s; the second step (Section~\ref{subsec:S2}) considers each vector $\bs_\mcM^t$ as a whole; and the third step (Section~\ref{subsec:S3}) accounts for the joint effect of probing injections $\{\bs_\mcM^t\}_{t=1}^T$.

\section{Solving the P2L tasks}\label{sec:solvers}
Recall that the P2L task with phasor data involves solving the set of non-linear equations
\begin{subequations}\label{eq:PF12}
	\begin{align}
	u_n(\bv_t)&=u_n^t &\forall n\in\mcM, t \in \mcT \label{eq:PF12u}\\
	\theta_n(\bv_t)&=\theta_n^t &\forall n\in\mcM, t \in \mcT\label{eq:PF12t}\\
	p_{n}(\bv_t)&=p_{n}^t &\forall n\in\mcM, t \in \mcT \label{eq:PF12p}\\
	q_{n}(\bv_t)&=q_{n}^t &\forall n\in\mcM, t \in \mcT \label{eq:PF12q}\\
    p_{n}(\bv_t)&=p_{n}(\bv_{t+1}) &\forall n\in\mcO,t \in \mcT' \label{eq:PFC2p}\\
	q_{n}(\bv_t)&=q_{n}(\bv_{t+1}) &\forall n\in\mcO,t \in \mcT' \label{eq:PFC2q}
	\end{align}
\end{subequations}
where $\bv_t$'s are the system states across $\mcT=\{1,\ldots,T\}$; $\{(u_n^t,\theta_n^t,p_n^t,q_n^t)\}_{n\in\mcM}$ are the probing data collected at time $t$; \eqref{eq:PF12u}--\eqref{eq:PF12q} are the $4MT$ metering equations; and \eqref{eq:PFC2p}--\eqref{eq:PFC2q} are the $2O(T-1)$ coupling equations with $\mcT':=\{1,\ldots,T-1\}$. For the P2L task with non-phasor data, the angle information in \eqref{eq:PF12t} is unavailable.

Having characterized the local identifiability for the P2L tasks in Part I, this section presents solvers for tackling P2L. If grid specifications are noiseless and the power injections in $\mathcal{O}$ remain unaltered during probing, the P2L tasks boil down to solving the equations in \eqref{eq:PF12}. The latter can be tackled by adopting the semidefinite program (SDP)-based solvers developed in \cite{ZhuGia12}, \cite{KlauberZhu15}, \cite{MLB15}, \cite{MALB16}. Here we will skip the details, which can be found in \cite{BKV17} for $T=2$, and outline the P2L solver for noisy data.

Probing data are inexact due to measurement noise and modeling inaccuracies in the metering equations of \eqref{eq:PF12u}--\eqref{eq:PF12q}. To account for small fluctuations in non-metered loads during probing, a noise term is added to the RHS of the coupling equations in \eqref{eq:PFC2p}--\eqref{eq:PFC2q}. To cope with noisy data, we extend the penalized SDP-based state estimator of~\cite{MALB16} to the P2L setting as follows
\begin{subequations}\label{eq:CPSSE}
	\begin{align}
	\min~ &\alpha \sum_{t=1}^{T}{\trace(\mathbf{M}\bV_t)} + \sum_{t=1}^{T}\sum_{k=1}^{3M} f_{k}({\epsilon_{k}^t}) + \sum_{t=1}^{T-1}\sum_{l=1}^{2O} f_{l}({\xi_{l}^t}) \label{eq:CPSSE:cost}\\
\mathrm{over}~&~		\bV_t \succeq \bzero, \{\epsilon_k^t\}_{k=1}^{3M},\quad t\in\mcT\\
~&~\{\xi_l^t\}_{l=1}^{2O},\quad t\in\mcT'\\
	\mathrm{s.to} ~&~\trace(\mathbf{M}_k\bV_t)+{\epsilon_{k}^t}=\hat{s}^t_k, \quad k=1:3M, t\in\mathcal{T} \label{eq:CPSSE:1}\\	~&~\trace(\mathbf{M}_l\bV_t)=\trace(\mathbf{M}_l\bV_{t+1})+{\xi_{l}^t}, \quad l=1:2O, t\in\mathcal{T}' \label{eq:CPSSE:O}
	\end{align}
\end{subequations}
where the given matrices $\mathbf{M}_k$ depend on $\mathbf{Y}$; see \cite{KlauberZhu15}, \cite{MLB15}.

The matrix variables $\bV_t\succeq\bzero$ have been obtained upon relaxing the rank-one constraint $\bV_t=\tilde{\bv}_t\tilde{\bv}_t^H$ on the original system states for $t\in\mcT$. The measurements $\hat{s}_k^t$ relate to state $\mathbf{v}_t$ in \eqref{eq:CPSSE:1}; and the constraints in \eqref{eq:CPSSE:O} couple the $T$ states. The auxiliary variables $\epsilon_k^t$ can be substituted from \eqref{eq:CPSSE:1}--\eqref{eq:CPSSE:O} into the objective of \eqref{eq:CPSSE}; they are introduced here only to simplify notation. The data fitting penalties $f_k$ can be either a weighted squared or absolute value, that is
\begin{subequations}
	\begin{align*}
	f_k(\epsilon_k^t) &=\left(\frac{\epsilon_k^t}{\sigma_k^2}\right)^2= \frac{\left(\hat{s}^t_k-\trace(\mathbf{M}_k\bV_t)\right)^2}{\sigma_k^2} \quad \text{or} \\
	f_k(\epsilon_k^t) &=\frac{|\epsilon_k^t|}{\sigma_k}=\frac{|\hat{s}^t_k-\trace(\mathbf{M}_k\bV_t)|}{\sigma_k}
	\end{align*}
\end{subequations}
with different $\sigma_k$'s depending on the uncertainty of the $k$-th datum. Likewise, the auxiliary variables $\xi_l^t$'s capture variations of non-metered loads and are penalized through $f_l$'s, which are defined as $f_k$'s.

The first summand in \eqref{eq:CPSSE:cost} corresponds to a regularizer promoting rank-one minimizers for $\bV_t$; a practical choice sets $\bM=\bG$ as suggested in \cite{MLB15}. The second and third summands in \eqref{eq:CPSSE:cost} are data-fitting terms. The tuning parameter $\alpha>0$ governs the balance between the regularizer and the data-fitting terms: For $\alpha=0$, the P2L cost involves only the data-fitting terms; whereas for increasing $\alpha$, more emphasis is placed on the regularizer~\cite{MALB16}. If one or more of the minimizers $\bV_t^\star$ of \eqref{eq:CPSSE} is not rank-one, the heuristic for constructing a system state $\bv_t^\star$ proposed in~\cite{MALB16} is used.

Additional constraints can be added to strengthen the SDP relaxation. For example, if non-metered buses are known to host exclusively loads, the constraints  $\trace(\mathbf{M}_l\bV_t)\leq0$ for $l=1:2O$, and $t \in \mathcal{T}$ can be appended to \eqref{eq:CPSSE}. Additional information on loads, such as the uncertainty range of \eqref{eq:load}, can be readily incorporated. As in \cite{MALB16}, if bus $n$ is known to be a zero-injection bus, then $\tilde{i}_n=\be_n^\top\bY\tbv$ has to be zero. Therefore, the constraint $\tbv\tilde{i}_n^\star=\bV \bY^\star\mathbf{e}_n=\bzero$ can be added.

Given phasor data, the metering equations corresponding to voltage magnitudes can be dropped. If the vectors of voltage phasors $\{\tbv_t\}_{t=1}^T$ are included as optimization variables, the direct measurements on the voltage phasors of $\mcM$ can be simply expressed as
\begin{equation}\label{eq:PMU1}
\tilde{v}_{t,k}+{\epsilon_{k}^t}=\hat{s}^t_k, \quad k=1:M,~t\in\mcT.
\end{equation}
To capture the dependence between $\tbv_t$ and $\tbV_t$, the non-convex constraint
\[\rank\left(	\begin{bmatrix}
	\bV_t & \tilde{\bv}_t \label{eq:PMUb} \\
	\tilde{\bv}_t^H & 1
	\end{bmatrix}\right)=1\]
can be surrogated by the next SDP constraint as in \cite{ZhuGia12}
\begin{equation} \label{eq:PMU2}
	\begin{bmatrix}
	\bV_t & \tilde{\bv}_t\\
	\tilde{\bv}_t^H & 1
	\end{bmatrix}
	\succeq \bzero, \quad t\in\mcT.
\end{equation}
Since the $\tbv_t$'s are optimization variables now, there is no need to use the heuristic of \cite{MALB16} to recover the system states.

\section{Numerical Tests}\label{sec:sim}
The topological observability criteria for the P2L task and the SDP-based solvers were numerically tested using the IEEE 34-bus feeder. The original multi-phase grid was converted to an equivalent single-phase grid~\cite{GLTL12}. The numerical tests were run on a 2.7 GHz Intel Core i5 laptop computer with 8 GB RAM using the Sedumi solver on YALMIP and MATLAB~\cite{sedumi}, \cite{YALMIP}.

\subsection{Numerical Observability}\label{subsec:Jacobian}
Since Theorems~1 and 2 of Part I rely on the sparsity pattern rather than the exact values of $\bJ\left(\{\bv_t\}\right)$, we evaluated $\bJ\left(\{\bv_t\}\right)$ for 1,000 random state sequences $\{\bv_t\}_{t=1}^T$. The scenarios of phasor and non-phasor data were tested under four probing setups. For each setup, the placement of non-metered $\mcO$ and probing buses $\mcM$ were fixed. We generated 1,000 random state sequences by randomly drawing voltage magnitudes in the range $[0.90,1.10]$ per unit and voltage angles in the range $[-1.5, 1.5]\degree$. Assuming non-phasor data first, the following four setups were constructed according to the condition of Th.~2 of Part I:
\begin{itemize}
\item Setup A meets the condition for $O=16$ and $T=2$.
\item Setup B meets the condition for $O=6$ and $T=2$.
\item Setup C does not meet the condition for $O=16$ and $T=2$, but it does for $T=4$.
\item Setup D does not meet the condition for $O=6$ and $T=2$, but it does for $T=4$.
\end{itemize}
The same setups were considered for phasor data. As discussed in Part I, setups A and B meet also the condition of Theorem~1. Additionally, setups C and D were constructed such that they meet the condition of Theorem~1 for $T=2$.

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.4]{Jacob1.pdf}\\ \vspace*{1em}
	\includegraphics[scale=0.4]{Jacob2.pdf}
	\caption{Histograms of the condition numbers for the P2L Jacobian matrices with non-phasor data for $T=2$ (top) and $T=4$ (bottom) probing actions.}
	\label{fig:Jacob}
\end{figure}

\emph{Non-phasor data:} Figure~\ref{fig:Jacob} depicts the condition number histograms obtained under the four setups for $T=2$ and $4$. Under setups A and B, although the dimensions of $\bJ\left(\{\bv_t\}\right)$ increase with $T$, the condition numbers did not. In fact, the condition number was sometimes reduced, especially in networks with large $\mcO$. For setups C and D, there was a significant shift in the histograms from $T=2$ to $T=4$, which validates Theorem~2. By and large, the condition number improves for decreasing $O$ and increasing $T$. Hence, when more loads are to be recovered, longer probing periods should be used. Of course, longer probing periods may violate the stationarity assumption on loads.

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.4]{pmuJacobT2.pdf}\\ \vspace*{1em}
 ~\includegraphics[scale=0.4]{pmuJacobT2F.pdf}
	\caption{Histograms of the condition numbers for the P2L Jacobian matrices with phasor data for $T=2$ (top) and $T=4$ (bottom) probing actions.}
	\label{fig:Jacobpmu}
\end{figure}

\emph{Phasor data:} Figure~\ref{fig:Jacobpmu} displays the condition number histograms of $\bJ\left(\{\bv_t\}\right)$ again for $T=2$ and $4$. As expected, due to the value added of phasor data, the condition numbers decrease significantly. In addition, setups C and D that failed for $T=2$ with non-phasor data, become successful with phasor probing data. The tests corroborate the criteria of Th.~1. The bottom panel of Figure~\ref{fig:Jacobpmu} displays the condition number histograms under the following two setups that did not satisfy the condition of Th.~1: \emph{i)} for $T=4$ and $O=6)$ (yellow histogram); and \emph{ii)} for $T=4$ and $O=16$ (green histogram).

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.4]{T1.pdf}
	\caption{Histograms of the condition numbers for the Jacobian matrix $\bJ\left(\bv_1\right)$ with phasor and non-phasor data for single-slot probing $(T=1)$.}
	\label{fig:T1P}
\end{figure}

\emph{Single-slot probing scenario:} We also tested the special case of $T=1$, where one fixes voltages and injections on a subset of buses $\mcM$ and tries to find the loads at the remaining buses $\mcO$. This setup is pertinent to learning ZIP loads as discussed in Part I. We tested two fixed placements of non-metered buses that met the conditions of Theorems~3 and 4, respectively. We then evaluated $\bJ\left(\bv_1\right)$ at $1,000$ random system states. Figure~\ref{fig:T1P} shows the histograms for the condition numbers of $\bJ\left(\bv_1\right)$. Bus placements that did not meet the criteria of Th.~3 and 4 exhibited condition numbers similar to those at the bottom panel of Figure~\ref{fig:Jacobpmu}.

The condition number of the Jacobian matrices in PSSE tasks for transmission systems is known to depend heavily on the specification set \cite{Clements83}, \cite{Baldick01}: A larger number of voltage magnitude and line flow measurements tends to yield a lower condition number. It is thus expected that adding line flow measurements would improve load and state estimation.

\subsection{SDP-based P2L} \label{sec:SDP}
Given noisy specifications, the P2L tasks were tackled using actual data and the SDP-based solver of \eqref{eq:CPSSE}--\eqref{eq:PMU2}. The loads on the IEEE 34-bus grid were taken from the Pecan Street dataset~\cite{pecandata}, between 10:00 a.m. and 01:40 p.m. on January 1, 2013, and in 10-minute intervals. Load sequences were scaled so that the peak active load over the tested period was $0.5$ pu. Lacking values for reactive loads, a lagging power factor of 0.9 was simulated for all loads.

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.4]{BoxT6PNoNmsd.pdf}\vspace*{1em}
	\includegraphics[scale=0.4]{BoxT6P.pdf}
	\caption{Percentage error in active power injection estimates with phasor data for $T=4$ and $O=8$ without MSD (top) and with MSD (bottom).}

	\label{fig:MSD}
\end{figure}

To simulate probing injections at buses in $\mcM$, we first created a data library $\mcS$ of $K=100$ randomized injection vectors as described in Section~\ref{subsec:S1} for $\bar{p}_n=0.2$~pu. The library $\mcS$ was then reduced to $\mcS_r$ to ensure that voltage magnitudes lie within $[0.90,1.10]$~pu for non-metered loads within $[\mathbf{0}, 2\bs_{\mcO}]$ as described in Section~\ref{subsec:S2}. For all tests, the regularization parameter was set to $\alpha=20,000$, and the functions $f_k$ and $f_l$ in \eqref{eq:CPSSE} were selected as the WLS costs. To simulate measurement noise, the probing data recorded for an actual quantity $x$ (e.g., voltage magnitude or power injection) was modeled as $\hat{x}=x(1+\epsilon)$, where $\epsilon$ is a zero-mean Gaussian random variable. The variance $\sigma^2$ of $\epsilon$ was selected to yield the desired value of signal-to-noise ratio (SNR)
\begin{equation}\label{eq:SNR}
10\cdot \log_{10}\frac{x^2}{\mathbb{E}[x^2\epsilon^2]}=-20\cdot \log_{10}\sigma.
\end{equation}
This variance is the same variance appearing in \eqref{eq:CPSSE} as $\sigma_k^2$. Likewise, to capture small load variations, non-metered loads were simulated by perturbing their nominal value $p_n$ as $\hat{p}_n^t=(1+\epsilon_n)p_n$ for $t\in\mcT$, and similarly for $q_n$'s.

To check whether the MSD step of Section~\ref{subsec:S3} improves estimation, we tested P2L with and without this step. The test considered $100$ Monte Carlo realizations for the loads at 10:00 a.m. The P2L task was run for $T=4$, $O=8$, and using phasor data. The SNR values were set respectively to $80$ and $60$~dB for metered and non-metered buses. PMUs are expected to have such high accuracy~\cite{Frigo16}. The range of percentage errors was reduced from $[-50,+50]\%$ to $[-30, +40]\%$ by selecting the $T$ most diversifying setpoints.

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.4]{BoxT4NP.pdf}
	\caption{Percentage error in active power injection estimates with non-phasor data for $T=4$ and $O=8$.}

	\label{fig:T6P}
\end{figure}

To verify the improvement of using phasor over non-phasor probing data, we repeated the previous MSD setup but now for non-phasor data. The obtained percentage errors are depicted in Figure~\ref{fig:T6P} and are of worse accuracy compared to those in the bottom panel of Figure~\ref{fig:MSD}. We also tested the single-slot probing scenario of $T=1$ under slightly different probing setups for (non)-phasor data. Figure~\ref{fig:T1} illustrates the statistics of the obtained percentage errors.

\begin{remark}[]\label{re:}
Based on the numerical tests, we have observed that load estimates generally improve when: a) the MSD step is implemented; b) phasor data are utilized; c) the duration $T$ is increased; and d) $O$ is decreased.
\end{remark}

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.4]{BoxT1NP.pdf}\\ \vspace*{1em}
	\includegraphics[scale=0.4]{BoxT1P.pdf}
	\caption{Percentage error in active power injection estimates for $T=1$ with non-phasor data and $O=4$ (top); and with phasor data and $O=6$ (bottom).}
	\label{fig:T1}
\end{figure}

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.39]{vrmseT2sm.pdf}

	\caption{RMSE for the system state with non-phasor data for $T=2$.}
	\label{fig:rmse}
\end{figure}

We next evaluated how the estimation accuracy of system states depends on the SNR. For this test, the SNRs for non-metered loads and probing data were identical. The state estimation accuracy was evaluated in terms of the root mean square error (RMSE) defined as
$\sqrt{\sum_{t=1}^T\|\bv_t-\hbv_t\|_2^2/(NT)}$
averaged over 20 Monte Carlo tests. Figure~\ref{fig:rmse} shows how the RMSE decreases for increasing SNR.

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.4]{PMUn-T4O6P.pdf}\\ \vspace*{1em}
	\includegraphics[scale=0.4]{PMUn-T6O6P.pdf}
	\caption{Active power injection estimates using probing with phasor data for $T=4$ (top) and $T=6$ (bottom).}

	\label{fig:PQ4}
\end{figure}

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.4]{PMUn-T6O6P.pdf}\\ \vspace*{1em}
	\includegraphics[scale=0.4]{PMUn-T6O6Q.pdf}
	\caption{Active (top) and reactive (bottom) power injection estimates using probing with phasor data for $T=6$ and $O=6$.}
	\label{fig:PQ5}

\end{figure}

To validate P2L over different loading conditions, we ran numerical tests for the period of 10:00 a.m. and 01:40 p.m. and every 10 min using phasor data. The SNRs for probing data and non-metered loads were again fixed to $80$ and $60$dB, respectively. Figures~\ref{fig:PQ4} and \ref{fig:PQ5} present the actual and estimated non-metered (re)active loads on buses $\{4,6,15,20,27,31\}$ for $T=4$ and $T=6$, accordingly. The plots show the load estimation improvement by increasing $T$.

Regarding the runtime of our algorithms, each P2L task took between $95-180$~sec, which were allocated as follows:
\begin{itemize}
	\item The linear programs of \eqref{eq:farkas} took $70$~sec overall, to check the feeder compliance of $K=100$ candidate probing setpoints.
	\item The quadratic program of \eqref{eq:MSDr} needed to select the $T$ most diversifying setpoints was solved in less than $10$~sec.
	\item The SDP formulation of \eqref{eq:CPSSE} together with the heuristic of \cite{MALB16} to obtain a rank-one solution took $25-100$~sec.
\end{itemize}
The load learning task for the single-slot probing setup $(T=1)$ was solved in less than $15$~sec.

\section{Conclusions}\label{sec:conclusion}
The novel data acquisition scheme of probing an electric grid via smart inverters to infer non-metered loads has been presented. Part I studied the topological observability of grid probing using (non)-phasor data in potentially meshed networks. If a probing setup is deemed topologically observable, Part II has presented a systematic methodology for designing probing injections. The goal is improved estimation accuracy and adherence to inverter and feeder constraints even without knowing non-metered loads. The computational tasks involved in grid probing have been cast as penalized SDP-based solvers and account for noisy measurements and non-stationary loads.

Numerical tests using synthetic and real-world data on benchmark feeders demonstrate the ensuing take-away simulation findings: \emph{i)} High-accuracy phasor data are better for load recovery than non-phasor data; \emph{ii)} Having the most diverse system states during probing yields better load estimates; \emph{iii)} Probing seemed to yield better estimates under broad voltage regulation range and tight load uncertainty \emph{iv)} Although increasing $T$ improved the system state accuracy, the obtained load estimates were not always better, especially for larger $O$. Nevertheless, we were able to recover a reasonable number of loads; and \emph{v)} Including the extra constraints to strengthen the SDP relaxation provided better numerical accuracy.

Several questions remain open. Developing scalable solvers perhaps along the lines of \cite{Gang18}; incorporating measurement from distribution lines and transformers~\cite{SevlianR15}; and applying our topological observability framework to detect data attacks in distribution grids; all constitute pertinent research directions.

\balance
\bibliographystyle{IEEEtran}
\bibliography{myabrv,power}

\end{document}

config: arxiv


TITLE: Singing: Resonance body open-closed or closed-closed?
QUESTION [2 upvotes]: Googling yields contradictory results, so here my question:
When I sing, my vocal chords vibrate, but my whole body is the resonance body, right? So I would say that when I think about standing waves etc. I must think of a resonantor that's closed (i.e. hard boundary conditions) at both ends, because my bottom end is certainly closed, and the top end has the vocal chords which are vibrating similar to woodwind instruments, so it's closed on that end as well.
If I am mistaken, could someone point out the physics behind this?

REPLY [3 votes]: Both "perfectly open" (zero acoustic impedance) and "perfectly closed" (infinite acoustic impedance) boundary conditions are only idealizations that never occur in practice. For the case of the human vocal tract, they aren't even very good approximations.
The "bottom end" of the resonating cavity is not, in fact, the lungs, but the vocal folds (as Georg pointed out). This end has some acoustic impedance that's neither extremely low nor extremely high. I'm sure the impedance also changes somewhat with the pitch and volume of the phonation.
The "top end" of the cavity is of course the mouth, and its impedance changes with the vowel sound you're pronouncing. For aptly-named "open" vowels such as "ah", "oh", and so on, the impedance is actually low enough that it might be a good approximation to say it's perfectly open. But for closed vowels ("ee", "oo", ...) and especially for humming with closed lips, the acoustic impedance is much higher (but still far from infinite).


TITLE: Localisation is extenstion by scalars
QUESTION [0 upvotes]: Let $R$ be a ring, $S \subset R$ a multiplicatively closed subset and $M$ and $R$-module.
There is a bilinear map $S^{-1}R \times M \to S^{-1}M$ given by $(\frac{r}{s}, m) \mapsto \frac{rm}{s}$ giving rise to an $R$-linear map
$$f: S^{-1}R \otimes_R M \to S^{-1}M$$
defined by $\frac{r}{s} \otimes m \mapsto \frac{rm}{s}$. To show that localisation is an example of an extension of scalars, I would like to show that $f$ is an isomorphism.
For surjectivity just take $r=1$.

I don't know how to show that this map is injective. Let $\frac{r}{s} \otimes m \in \ker f$. Then $\frac{rm}{s} = \frac{0}{1}$ and there exists $s' \in S$ such that $rms'=0$, but now what?

Doesn't $f$ actually just tell us about the structure of both sets as $R$-modules? I believe that to show they are isomorphic as $S^{-1}R$-modules we would need to check that the action of $S^{-1}R$ is the same on both sides? Perhaps this is trivial and omitted.

REPLY [2 votes]: For 1) You observed that $r/s \otimes m \in \ker{f}$ implies $s'rm = 0$ for some $s' \in S$.  Then use the defining laws of tensor products to manipulate the expression $r/s \otimes m$. We have $r/s \otimes m = 1/s \otimes rm = \frac{s'}{ss'} \otimes rm = \frac{1}{ss'} \otimes s'rm = \frac{1}{ss'} \otimes 0 = 0$, as desired.
For 2)  Suppose that two $S^{-1}R$ modules $M, N$ are isomorphic as $R$-modules.  So we have an $R$-linear map $f: M \rightarrow N$ that is an isomorphism.  In fact, $f$ is necessarily also $S^{-1}R$ linear.  To see this, consider the expression $r/s f(m) - f(rm/s)$.  Multiply by $s$ and use $R$-linearity of $f$ to see that this expression is $0$ in $N$.   So actually $f: M \rightarrow N$ is an isomorphism of $S^{-1}R$ modules! In conclusion, if two $S^{-1}R$ modules $M, N$ are isomorphic as $R$-modules, then they are isomorphic as $S^{-1}R$-modules.
PS)  Perhaps it's worth noting that the argument in 2) generalizes from localizations to epimorphisms of rings, i.e. if $R \rightarrow T$ is an epimorphism of commutative rings, and $M, N$ are two $T$-modules, then an $R$-linear map $f: M \rightarrow N$ is already $T$-linear, hence $M, N$ are isomorphic as $T$ modules if they are isomorphic as $R$-modules.


TITLE: Relative rate ratio versus derivative with respect to time
QUESTION [1 upvotes]: Given:

In the United States between 1850 and 1880, the number of farmers
  continued to increase, but at a rate lower than that of the general
  population.

Prove that following statement directly contradicts the
 information presented above.

The proportion of farmers in the general population increased from 68
  percent in 1850 to 72 percent in 1880.

There was the answer:

Let $x_0$ be the number of farmers in 1850 Let $x_1$ be the number of
  farmers in 1850
Let $y_0$ be the number total population in 1850 Let $y_1$ be the
  number total population in 1850
The first statement says
$x_1>x_0$ and $\frac {x_1}{x_0} < \frac{y_1}{y_0}$
The contradictory statement says:
$\frac {x_0}{y_0} = 0.68$ and $\frac {x_1}{y_1} = 0.72$ or
  $\frac{x_0}{y_0} < \frac{x_1}{y_1}$
So the two are not compatible.

This was the objection to the answer:

How do you know the rate of farmers increasing isn't the derivative
  with respect to time? and therefore
  $\frac{x_1-x_0}{30}<\frac{y_1-y_0}{30}$?

The answer isn't solid because it can't explain why the problem statement didn't refer to "rate" as the change in farmers with respect to time, and the growth of the population with respect to time.
If the problem did refer to the derivative in physics, then the "following statement" doesn't contradict the statement and therefore the answer is false?

REPLY [0 votes]: Correct. We can provide a very simple counterexample which is consistent with both statements:
$$x_0=34$$
$$y_0=50$$
$$x_1=72$$
$$y_1=100$$


TITLE: $\ell_{\infty}$ and the bounded approximation property
QUESTION [6 upvotes]: I know from several references that the Banach space of bounded sequences, namely $\ell_{\infty}$, has the bounded approximation property. But I do not have a reference where this assertion is proven.
So I would be very grateful if you can give me a reference for the result asserting that $\ell_{\infty}$ has the bounded approximation property ?
If you do not have a reference but you know how to do it, please provide me a sketch of your proof.
I just recall here the definition of the approximation property.
We say that a Banach space $X$ has the approximation property (AP) if for every $\varepsilon>0$, for every compact set $K \subset X$, there exists a finite rank operator $T \in \mathcal B(X)$ such that $\|Tx-x\| \leq \varepsilon$ for every $x \in K$. Let $\lambda \geq 1$, if in the above definition T can always be chosen so that $\|T\|\leq \lambda$, then we say that $X$ has the $\lambda$-bounded approximation property ($\lambda$-(BAP)).
Thank you in advance.

REPLY [1 votes]: Not only that I think every subspace of $L_\infty$- space also has A.P.


TITLE: Spontaneous breaking of supersymmetry
QUESTION [4 upvotes]: I am a beginner in SUSY Quantum mechanics. I had read that the symmetry is spontaneously broken if $A \left|\psi \right>_n^{(1)}\neq 0 $ and symmetric if $A \left|\psi \right>_n^{(1)}= 0. $ But analyzing the spectrum of broken symmetry I can see a symmetry in the energy values, and in case of unbroken symmetry ground state (zero energy) $H_1$ does not have corresponding energy state in $H_2$. This shows an asymmetry in spectrum of partner potentials in unbroken case.
I think I am wrong somewhere.
What do we mean by symmetry and the spontaneous breaking of symmetry here? Which symmetry is broken here?
Edit:
$A$  is the Annihilation operator. $\left|\psi \right>_n^{(1)}$ is the eigen vector for $H_1$. $H_1$ and $H_2$ are the Hamiltonian corresponding to the partner potentials $V_1$ and $V_2$.

REPLY [4 votes]: This is a perfect storm of notational dissonance between QM and QFT. Your statement

I had read that the symmetry is spontaneously broken if $A \left|\psi \right>_n^{(1)}\neq 0 $ and symmetric if $A \left|\psi \right>_n^{(1)}= 0. $

is inapposite and misconstrued—justly paradoxical. I suspect labels on symmetric phase versus SSB might not be productive here.

Review of SSQM
$$
Q=\begin{pmatrix} 0& 0\\ A& 0 \end{pmatrix}, \qquad Q^\dagger=\begin{pmatrix} 0& A^\dagger\\ 0& 0 \end{pmatrix},
$$
so
$$
H= Q Q^\dagger+ Q^\dagger Q=  \begin{pmatrix} A^\dagger A& 0\\ 0& A A^\dagger \end{pmatrix}\equiv \begin{pmatrix} H_1& 0\\ 0& H_2 \end{pmatrix}.
$$
Now
$$
A|\psi_n^{(1)}\rangle = |\psi_{n-1}^{(2)}\rangle, \qquad A^\dagger |\psi_n^{(2)}\rangle = |\psi_{n+1}^{(1)}\rangle, \qquad E^{(1)}_{n+1}= E_n^{(2)}  \qquad E_0^{(1)}=0.
$$
So $\psi^{(2)}_n$ and $\psi^{(1)}_{n+1}$ have the same eigenvalue under H, and they are degenerate pairs-- (unbroken symmetry),
$$
Q \begin{pmatrix}  \psi^{(1)}_{n+1}\\ 0 \end{pmatrix} = \begin{pmatrix}  0\\ \psi^{(2)}_{n} \end{pmatrix} .
$$

But the ground state is unique, an unpaired susy-singlet,  and non-degenerate:
$$
Q \begin{pmatrix}  \psi^{(1)}_{0}\\ 0 \end{pmatrix} = 0, \qquad A|\psi^{(1)}_0\rangle = 0 ,
$$
so, if you chose, you might call it SSBroken (but Witten calls it unbroken). A Susy rotation does not couple it with other states. So the bottom of the spectrum, the ground state, lacks the symmetry of the rest of the spectrum.
This is a narrow answer to your question, but does not alleviate the perfect storm of dissonance.

The QFT paradigm The above stands in sharp contrast to QFT with its infinite d.o.f.

To avoid confusion, I'll call the infinitesimal Susy charge here ${\cal Q}$, so that the real correspondent to the finite Q above is, instead, the full group
super-transformation, $\exp i\bar\theta {\cal Q}$, for a Grassmann angle θ. In QFT, the symmetric (unbroken) phase is characterized by
$$
{\cal Q}|0\rangle= 0, \Longrightarrow \exp (i\bar\theta {\cal Q})|0\rangle=|0\rangle .
$$
For $[H,{\cal Q}]=0$, eigenstates of the hamiltonian,
$$
H \phi |0\rangle= E \phi |0\rangle
$$
are susy-rotated to degenerate ones,
$$
H(e^{i\bar\theta {\cal Q}} \phi e^{-i\bar \theta {\cal Q}} )|0\rangle =
e^{i\bar\theta {\cal Q}} H\phi  |0\rangle = E (e^{i\bar\theta {\cal Q}} \phi e^{-i\bar \theta {\cal Q}} )|0\rangle,
$$
evocative of "what you had read". So, it is here that this is the hallmark of degeneracy of the spectrum, while the vacuum is unique, similarly to above.
In the SSB phase, we have the opposite: most states are non-degenerate, by failure of the above argument, but the vacuum is now degenerate. It is not unique:
$$
{\cal Q}|0\rangle\neq 0  \Longrightarrow \qquad |\Omega\rangle= \exp (i\bar\theta {\cal Q})|0\rangle\neq |0\rangle .
$$
This state is degenerate with the vacuum and roils and bubbles with goldstinos.
So, at the end of the day, your question dramatizes the stark difference between QM and QFT. The takeaway may well be that sticking to facts rather than labels might be the sanest option.


TITLE: Is $x+y\alpha + z\alpha^2$ a subfield?
QUESTION [2 upvotes]: Let $\alpha = \sqrt[3]{2}$. I want to Prove that the set of all numbers $x+y\alpha+z\alpha^2$, for $x,y,z \in \mathbb{Q}$, is a subfield of $\mathbb{C}$.
I have showed that this set is a subring, but I'm not sure how to show for any $a\neq 0$ in our set has a multiplicative inverse.
I have tried picking two arbitrary elements, $m,n$ in our set. Let $m= x+y\alpha+z\alpha^2$ and let $n=a+b\alpha+c\alpha^2$. Then setting our product $m\cdot n$ equal to 1 gets us:
$$(ax+2zb+2yc) + (bx+ay+2zc)\alpha + (cx + yb +za)\alpha^2 = 1.$$
I was wondering if this could lead me to my proof?

REPLY [1 votes]: A demonstration which is relies less on explicit calculation and more on abstract properties of fields, vector spaces, and linear maps:
With
$\alpha = 2^{1/3}, \tag 0$
let
$S = \{1, \alpha, \alpha^2 \}; \tag 1$
$S$ is a linearly independent set over $\Bbb Q$.
For
$\sigma = x + y\alpha + z\alpha^2, \; x^2 + y^2 + z^2 \ne 0, \tag 3$
i.e., for
$\sigma \ne 0, \tag 4$
multiplication by $\sigma$ acts as a $\Bbb Q$-linear map
$\sigma:\text{span}(S) \to \text{span}(S) \tag 5$
via
$v \mapsto \sigma v \tag 6$
for
$v \in \text{span}(S); \tag 7$
as such, $\sigma$ is injective:
$\sigma v_1 = \sigma v_2 \Longrightarrow \sigma(v_1 - v_2) = 0 \Longrightarrow v_1 - v_2 = 0 \Longrightarrow v_1 = v_2; \tag 8$
note that here we have only relied upon the fact that the ring
$\text{span}(S) \subset \Bbb C \tag{9}$
is an integral domain, being a sub-ring of the field $\Bbb C$; we have not yet explicitly invoked invertability of $\sigma$, which is fortunate since that is what we are (still engaged in) trying to prove; now since
$\dim_{\Bbb Q} \text{span}(S) = 3 < \infty, \tag{10}$
it follows that any injective linear mapping $\text{span}(S) \to \text{span}(S)$ is also surjective; in the present case this means that there is some
$\tau \in \text{span}(S) \tag{11}$
with
$\sigma \tau = 1_{\Bbb Q}; \tag{12}$
since the binary operation on the left of this equation is ordinary multiplication in $\Bbb C$, we also have
$\tau \sigma = 1_{\Bbb Q}; \tag{13}$
(12) and (13) together by definition mean
$\tau = \sigma^{-1}, \tag{14}$
and we have shown that $\text{span}(S)$ is a field as per request.


TITLE: Uniform convergence of sequence on interval $[-b,0]$
QUESTION [1 upvotes]: I am aiming to prove that there exists a continous function $\exp:\mathbb{R}\rightarrow\mathbb{R}$  using the sequence $$e_n^x=\sum_{k=0}^{n}\frac{x^k}{k!}$$
So far: defining the sequence $e_n^x=\sum_{k=0}^{n}\frac{x^k}{k!}$ given that this is a polynomial, I have shown that this is cauchy and therefore uniformly convergent on the reals, and taking $b\in\mathbb{Q}$ the sequence is also cauchy and thus uniformly convergent. Then $e_n^x$ is uniformly convergent for $x\in[0,b]$, I then showed that the limit function is continuous, then it must be that $\exp\colon[0,b] \rightarrow\mathbb{R}$. Now assuming that $e_n^x$ is uniformly convergent in $[0,x]$.
Now I am aiming to show that $\{e_n^{-x}\}$ is uniformly convergent on $[-b,0]$, here I am aiming to use the triangle inequality somehow so that if $exp$ is converging in $[0,b]$ then $\exp$ is converging on the union set of $\mathbb{Q}$ and $[0,b]$. After doing this it has to be that if $\exp:[0,b]\rightarrow\mathbb{R}$ then $\exp:[0,\infty]\rightarrow\mathbb{R}$. From here I can take a real number $r\in\mathbb{R}$ so that for every element $q\in\mathbb{Q}$ where $r<q$ then $\exp:[0,q]\rightarrow\mathbb{R}$ and we have that exp is continous at $r$ and this must hold for all reals.So then it must be that there exists $\exp:\mathbb{R}\rightarrow\mathbb{R}$.
I would like to know if me general idea is correct? Further I am having some trouble with showing that $\{e_n^{-x}\}$ is uniformly convergent on $[-b,0]$ how would we go about showing this?

REPLY [1 votes]: For $x \in [-b,b]$ and $m \ge n$ we have
$$\left|e_m(x) - e_n(x)\right| =\left|\sum_{k=n+1}^m \frac{x^k}{k!}\right| \le \sum_{k=n+1}^m \frac{|x|^k}{k!} \le \sum_{k=n+1}^\infty \frac{b^k}{k!} \xrightarrow{m,n \to \infty} 0$$
uniformly in $x$ so the sequence $(e_n)_n$ is uniformly Cauchy on $[-b,b]$ and hence converges uniformly to a continuous function $\exp : [b,b] \to \Bbb{R}$.
You only need to know that the series $\sum_{k=1}^\infty \frac{b^k}{k!}$ converges, which you can prove using the ratio test:
$$\frac{\frac{b^{k+1}}{(k+1)!}}{\frac{b^k}{k!}} = \frac{b}{k+1} \xrightarrow{k\to\infty} 0.$$


text:
\begin{document}

\begin{titlepage}
\title{
\vskip -70pt
\begin{flushright}
{\normalsize \ ITFA-2008-21}\\
\end{flushright}
\vskip 45pt
{\bf The quantum equivariant cohomology of toric manifolds through mirror symmetry}
}
\vspace{3cm}

\author{{J. M. Baptista} \thanks{ e-mail address:
    j.m.baptista@uva.nl}  \\
{\normalsize {\sl Institute for Theoretical Physics} \thanks{ address: Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands}
} \\
{\normalsize {\sl University of Amsterdam}}
}

\date{June 2008}

\maketitle

\thispagestyle{empty}
\vspace{2cm}
\vskip 20pt
{\centerline{{\large \bf{Abstract}}}}
Using mirror symmetry as described by Hori and Vafa, we compute the quantum equivariant cohomology ring of toric
manifolds. This ring arises naturally in topological gauged sigma-models and is related to the Hamiltonian
Gromov-Witten invariants of the target manifold.

\vspace{.35cm}

\end{titlepage}

\section{Introduction}

Mirror symmetry was described by Hori and Vafa in \cite{H-V} as a duality between sigma-models and Landau-Ginzburg
models. In the special case where the target of the sigma-model is Calabi-Yau, this symmetry is a duality of the
full $N=2$ supersymmetric theories, but for certain other targets, namely toric varieties and complete intersections
of hypersurfaces in toric varieties, the symmetry still exists as a duality of the associated topological theories.
The aim of this note is to illustrate in a simple setting how Hori and Vafa's description of mirror symmetry can be
used to study not only sigma-models but also gauged sigma-models, in our case with toric targets. Since these particular gauged
non-linear models can be obtained as quotients of gauged linear sigma-models (with a bigger gauge group), the
applicability of mirror symmetry is not really a surprise, for \cite{H-V} is in fact all about dualities of gauged
linear models. Nevertheless, explicit mirror predictions for gauged sigma-models do not seem to exist in the
literature, so we thought that producing one such example may be interesting.

The mathematical objects that we study here through the use of mirror symmetry are the quantum equivariant cohomology
rings of toric manifolds. In the usual non-equivariant case, the quantum cohomology rings are deformations of the de
Rham cohomology rings and encode certain genus zero Gromov-Witten invariants of the target manifold $X$. In our
equivariant (or gauged) case, the quantum rings will be deformations of the classical equivariant cohomology rings
and will presumably encode certain genus zero Hamiltonian Gromov-Witten invariants of $X$. The latter invariants study symplectic
manifolds equipped with hamiltonian actions and have been rigorously defined and studied in the mathematical
literature \cite{C-G-M-S, C-S}. Loosely speaking, they are given by integrals of natural cohomology classes over the moduli space of solutions of the vortex equations with target $X$. Their ring structure, however, does not seem to have been considered so far. The Hamiltonian Gromov-Witten
invariants can be interpreted in the language of quantum field theory as expectation values of observables in
gauged non-linear sigma-models, in a theory very much analogous to the usual sigma-models/Gromov-Witten one \cite{Bap2}.
The BPS states are then the classical vortex solutions, and the quantum equivariant cohomology rings are identified with the rings of local observables of the gauged topological theory.

Another version of equivariant Gromov-Witten theory has been described by Givental in \cite{Giv2}. That is a different construction from the one we are considering here. In particular the invariants in \cite{Giv2} are defined by equivariant integration over the moduli space of holomorphic curves, and so the vortex moduli spaces do not enter the picture. Recently, however, it has been proposed in \cite{Gon-Wood} that Givental's equivariant invariants are related to the weak coupling limit of the Hamiltonian (or gauged) Gromov-Witten theory, i.e. to the limit $e^2 \rightarrow 0$ of the gauged sigma-model. How this limit works precisely doesn't seem to be completely clear yet. Recall that, on the other extreme, the limit $e^2 \rightarrow \infty$ has already been studied for quite some time \cite{Ga-Sa, Hori-al}: it leads to a relation between the Hamiltonian Gromov-Witten invariants of $X$ and the usual Gromov-Witten invariants of the K\"ahler quotient of $X$ by the gauge group.

$\ $

Coming back to our study, the main result of this note is roughly the following. Let $X$ be a $k$-dimensional toric manifold obtained as the
K\"ahler quotient of $\CC^n$ by a linear action of the torus $T^{n-k}$ with charges $Q^a_j$, where the indices run as
$1\leq j \leq n$ and $1\leq a \leq n-k$. We assume that X is Fano, i.e. that it has positive definite first Chern class.
Then for any of the natural $T^k$-actions on $X$, the quantum equivariant
cohomology ring of this manifold is the polynomial ring
\begin{equation}
QH_T^\ast (X) \ =\ \CC [ w^1 , \ldots , w^n ]\ / \ Q(X) \ ,
\label{1.1}
\end{equation}
where $Q(X)$ is the ideal generated by the binomials
\begin{equation}
 \Big( \prod_{ \{ j: Q^a_j \geq 0 \} }  (w^j)^{Q_j^a}  \Big) - e^{-t'^a} \Big( \prod_{\{ j: Q^a_j < 0 \} }
(w^j)^{- Q_j^a} \Big)   \label{1.2}
\end{equation}
for all $a=1, \ldots , n-k$.
Recall that the usual (non-equivariant) quantum cohomology rings of toric manifolds have been rigorously computed
\cite{Bat, Giv, C-S}, and are
\begin{equation}
QH^\ast (X) \ =\ \CC [ w^1 , \ldots , w^n ]\ / \ ( I + Q(X)) \ ,  \nonumber
\end{equation}
where $I$ is a certain ideal of linear relations. Thus the result presented here shows a clear analogy with the
classical case, where the cohomology rings of toric manifolds are
\begin{equation}
H^\ast (X) \ =\ \CC [ w^1 , \ldots , w^n ]\ / \ ( I + J) \qquad {\rm and} \qquad
H_{T}^\ast (X) \ =\ \CC [ w^1 , \ldots , w^n ]\ / \ J \ ,
\label{1.3}
\end{equation}
with $J$ being the classical Stanley-Reisner ideal \cite{Bat, M-P}.

$\ $

Three questions that arise directly from the work in this note are the following.
Firstly, to check the prediction of mirror symmetry by computing the quantum equivariant ring directly on the
sigma-model side. This would presumably involve computing a few genus zero Hamiltonian Gromov-Witten invariants
of $X$. For toric $X$ we note that quite a few methods and results are already available
\cite{C-S, Bap1, Mel-Ple}.
The second question is to go on to see whether mirror symmetry can be used to predict the Hamiltonian Gromov-Witten themselves,
not just the resulting quantum ring. This will involve computing the correlation functions in the dual
Landau-Ginzburg model and then mapping them to the invariants of the gauged sigma-model. Finally, the third question
would be to generalize our computations to gauged non-abelian models with Grassmannian targets. The best shot
here is probably to use the non-abelian mirror conjecture of \cite{H-V}.

$\ $

The outline of the note is as follows. In section 2 we describe with more care the toric manifolds and actions that will
be considered here. In section 3 we use the results of \cite{H-V} to derive the Landau-Ginzburg model that is dual to the
gauged sigma-model with target $X$. In section 4 we look at the associated gauged topological theory to recognize
the observables and the role of the gauge fields. In section 5 we finally compute the quantum equivariant
cohomology ring, illustrating the result with the example of projective spaces.

\section{Toric manifolds and actions}

  Let $X$ be any $k$-dimensional K\"ahler toric manifold that can be realized as the K\"ahler quotient of $\CC^n$ by a
linear action of a torus $T^{n-k}$. We will denote by $Q^a_j$ the integral weights, or charges, of this torus action,
where the indices run as $1\leq j \leq n$ and $1\leq a \leq n-k$. Then $X$ is the space of $T^{n-k}$-orbits in a subset
of $\CC^n$ defined by equations of the form
\begin{equation}
\mu (z) \ :=\ \sum_j Q^a_j \: |z^j|^2 \  =\  r'^a \ , \qquad z \in \CC^n \ .
\label{2.1}
\end{equation}
Observe that the quotient space $X(r') := \mu^{-1} (r') / T^{n-k}$ depends on the chosen value of the parameter $r' \in {\mathbb R}^{n-k}$, and only for a certain range of $r'$ will $X(r')$ be isomorphic to the initial $X$ as a complex manifold. This range can be shown to be a convex cone in the parameter space ${\mathbb R}^{n-k}$, and is called the K\"ahler cone of $X$. For $r'$ outside the K\"ahler cone the quotient $X(r')$ can be very different from $X$, and can in particular be singular or have smaller dimension. In general the parameter space ${\mathbb R}^{n-k}$ will be subdivided into a number of non-intersecting cones of different dimensions, each cone corresponding to an isomorphism class of quotients. Varying $r'$ inside each of these cones will only affect the K\"ahler metric induced on the quotient $X(r')$, not the complex manifold itself. It can then be shown that for such a variation the K\"ahler class of the induced metric depends linearly on $r'$. More precisely, for $r'$ inside the K\"ahler cone of $X$ there exists a basis $\{\eta_a \}$ of the cohomology $H^2 (X; {\mathbb Z} )$ such that the K\"ahler form $\omega_{X(r')}$ can be written as
\[
\omega_{X(r')} \ = \ \sum_a  r'^a \eta_a \ .
\]
This shows how, inside each cone, the $r'^a$'s are effectively a parametrization of the K\"ahler class.

Another standard
fact in toric geometry is that there exists a set of primitive vectors $v^1 , \ldots , v^n  \in \mathbb{Z}^{k}$ that span
$\mathbb{Z}^{k}$ and satisfy the condition
\begin{equation}
\sum_j Q_j^a \: v^j \ =\  0
 \label{2.2}
\end{equation}
for all values of $a$. In the algebraic picture of toric manifolds these $v$'s are generators of the regular fan associated to $X$. In fact,
given the $v$'s corresponding to $X$, we can take the $Q^a$'s to be a basis of the $(n-k)$-dimensional lattice that
solves (\ref{2.2}). As can be easily checked, after a redefinition of the $r'^a$'s if necessary, these new $Q$'s define the
same quotient $X$ as the old ones. The linear transformation that takes the new $Q$'s to the old ones is invertible over
the reals but in general not over the integers. These vectors $v^1 , \ldots , v^n \in {\mathbb Z}^k$ that generate the fan encode much of the information about $X$. For example the ideal $I$ that appears in the (non-equivariant) cohomology ring of $X$ is simply defined by the linear relations
\[
\sum_{i=1}^n  \langle v^i , t \rangle \; w^i \ = 0 \ , \qquad t \in {\mathbb R}^{k}\ .
\]

$\ $

In this paper we will be considering abelian gauged sigma-models with $X$ as target manifold. Since the models are gauged,
to define them we need an abelian action on $X$. This action can be constructed through any abelian action on $\CC^n$ that
commutes with the $T^{n-k}$-action used in the definition of $X$, and so descends to $X$. For definiteness we will take the
$T^k$-action on $X$ induced by the linear action of this group on $\CC^n$ with integral weights $u_j^b$, where the indices run
as $1\leq j \leq n$ and $1\leq b \leq k$. Taking the $u$'s such that the vectors $u^1 , \ldots , u^{k}  \in \mathbb{Z}^n$
complete the set of $Q^a$'s to form a basis of $\RR^n$, we get at the end a hamiltonian $T^k$-action on $X$ that operates through
holomorphic transformations and whose generic orbits have real dimension $k$.

Gauged sigma-models are theories of maps $\phi$ from a Riemann surface to the target manifold $X$ coupled to gauge fields $A$.
These gauge fields should be regarded as connections on a principal bundle over the Riemann surface, or worldsheet. If
the classical models admit a supersymmetric extension, then in addition to $\phi$ and $A$ there will be several other bosonic
and fermionic fields. The case that concerns us are the $N=2$ gauged, supersymmetric and abelian sigma-models with target
$\CC^n$ or the toric manifold $X$. In this case both the lagrangian and the several other fields of the theory can be
conveniently rewritten in term of superfields using the Grassmann variables $\theta^{\pm}$ and $\thetab^{\pm}$. Following
the conventions of \cite{H-V} we will need the chiral fields
\begin{equation}
\Phi   \ = \ \phi (w^\pm) + \sqrt{2}\, \theta^\alpha \, \psi_\alpha (w^\pm) + 2\, \theta^+ \theta^- \, F(w^\pm)
\label{2.3}
\end{equation}
and the twisted chiral fields
\begin{align}
Y \ &= \ y(\tilde{w}^\pm) +  \sqrt{2}\, \theta^+ \,\bar{\chi}_+ (\tilde{w}^\pm) +  \sqrt{2}\, \thetab^- \,\chi_- (\tilde{w}^\pm) +
2\, \theta^+ \thetab^- \, G(\tilde{w}^\pm)   \label{2.4}  \\
\Sigma \ &= \ \sigma (\tilde{w}^\pm)  +  i \sqrt{2}\, \theta^+ \,\lamb_+ (\tilde{w}^\pm) - i \sqrt{2} \,\thetab^- \,
\lambda_- (\tilde{w}^\pm) +  2 \,\theta^+ \thetab^- \, [D - i (F_A)_{01}] (\tilde{w}^\pm)   \ , \nonumber
\end{align}
where one should expand the various fields as functions of the variables $w^{\pm} = x^{\pm} - i \theta^{\pm} \thetab^\pm$ and
$\tilde{w}^{\pm} = x^{\pm} \mp i \theta^{\pm} \thetab^\pm$, with $x^\pm = x^0 \pm x^1$ being the light-cone coordinates on the
Minkowski worldsheet. The last superfield, the superfield $\Sigma$, is not a fundamental field; it is instead the
superfield-strength of the vector multiplet $V$ associated to the abelian connection $A$, written in the Wess-Zumino gauge.
In terms of these
superfields the various supersymmetric lagrangians that we will use can all be written in a more compact and convenient
form.

\section{The dual gauged lagrangian}

Recall that the stated purpose of this note is to use mirror symmetry to study some basic properties of the gauged
sigma-model with toric target $X$ and group $T^k$. Our first task is therefore to write down the lagrangian of this model and,
more importantly, the lagrangian of the mirror theory. This task is in fact quite simple given the results of \cite{H-V}.
One starts by considering the theory before the quotients, i.e. the gauged sigma-model with target $\CC^n$ and group
$T^k \times T^{n-k}$. Its lagrangian is
\begin{align}
L \ = \ &\int \dd^4 \theta \ \bigg\{  \sum_{j=1}^{n} \bar{\Phi}_j  \Big( \exp{(2\, Q_j^a \,V'_a)}  +  \exp{(2\, u^b_j\, V_b)}\Big)
\Phi_j \ - \ \frac{1}{2e^2} \sum_{b=1}^{k} \Big(\bar{\Sigma}_b \, \Sigma_b \Big)   \label{3.1}\\
&- \frac{1}{2(e')^2} \sum_{a=1}^{n-k} \Big( \bar{\Sigma}'_a \, \Sigma'_a \Big)  \bigg\}\
 -\ \frac{1}{2} \int \dd^2 \tilde{\theta}\ \Big( t'^a\, \Sigma'_a  + t^b\, \Sigma_b  + {\rm c.c.} \Big)  \ ,  \nonumber
\end{align}
where $e$ and $e'$ are the gauge coupling constants corresponding to the two tori of the gauge theory, while $t$ and $t'$
are the usual complex parameters of gauged sigma-models, i.e. the complex combination $r - i \theta$ of the Fayet-Iliopoulos parameter
and the theta angle. For those less familiar with the jargon, the terms in the first and second integrals are respectively
called the D-terms and the F-terms of the lagrangian. Now, according to Hori and Vafa, the mirror of this theory is another
gauged sigma-model with twisted chiral fields $Y^1 , \ldots , Y^n $ and lagrangian
\begin{align}
L_{\rm dual} \ = \ &\int \dd^4 \theta\  \bigg\{   {\rm kinetic\ term}\ (Y^j , \bar{Y}^j)\  -\  \frac{1}{2e^2}
\sum_{b=1}^{k} \Big(\bar{\Sigma}_b\, \Sigma_b \Big)\ -\ \frac{1}{2(e')^2} \sum_{a=1}^{n-k} \Big(\bar{\Sigma}'_a \, \Sigma'_a \Big)
\bigg\}    \label{3.2} \nonumber \\
&+\frac{1}{2} \int \dd^2 \tilde{\theta}\ \bigg\{  \Sigma'_a\: \Big(Q_j^a Y^j - t'^a \Big)\ +\  \Sigma_b \Big(u^b_j Y^j - t^b \Big)
\ +\ \sum_{j=1}^{n} \exp{(-Y^j)} \ +\ {\rm c.c.} \bigg\}  .  \nonumber
\end{align}
In the derivation of \cite{H-V} the first two F-terms of $L_{\rm dual}$ come from the application of T-duality to $L$,
whereas the exponential F-term is a non-perturbative contribution from instantons. A very important property of $L_{\rm dual}$
is that at a D-term level the matter and gauge fields become uncoupled; furthermore, in $L_{\rm dual}$ the gauge fields
appear only through the gauge field strengths $\Sigma$, with no explicit mention of the vector multiplet $V$.

The point now is to go to the limit $e' \rightarrow \infty$ on both theories, the original and the dual. For the original
$L$ it is well known that this limit takes us to the sigma-model with quotient target $\CC^n /\!/ T^{n-k} = X$ \cite{Hori-al}.
It is here that the Fano assumption on $X$ enters, as is also explained in \cite{Hori-al}. The resulting quotient model
is still gauged, with group $T^k$,  because we have kept the constant $e$ finite. On the dual side, on the other
hand, the limit $e' \rightarrow \infty$ just imposes the constraints
\begin{equation}
Q^a_j\,  Y^j - t'^a \ = \ 0  \qquad {\rm for\ all}\quad a=1, \ldots , n-k.   \nonumber
\label{3.3}
\end{equation}
As in \cite{H-V}, these constraints are solved by
\begin{equation}
Y^j \ = \ s^j \ +\ \sum^{k}_{b=1}  v^j_b \, \Theta^b    \ ,
\label{3.4}
\end{equation}
where the $\Theta^1 , \ldots , \Theta^k$ are new twisted chiral fields associated to new complex coordinates, and the
constants $s^1 , \ldots , s^n \in \CC$ are any particular solution of the algebraic equation $Q_j^a s^j = t'^a$.
Finally, inserting (\ref{3.4}) into the limit of $L_{\rm dual}$ and denoting by $\langle \cdot , \cdot \rangle$ the
canonical inner product on $\RR^k$, we get that as $e' \rightarrow \infty$ the lagrangian $L_{\rm dual}$ reduces to
\begin{align}
\hat{L}_{\rm dual} \ = &\ \int \dd^4 \theta\  \Big[ {\rm kinetic\ term}\ (\Theta^b , \bar{\Theta}^b)\  -\  \frac{1}{2 e^2}\:
\langle \bar{\Sigma},  \Sigma \rangle \Big]    \label{3.5}  \\
&+\, \frac{1}{2} \int \dd^2 \tilde{\theta}\ \bigg\{  \langle \Sigma , u_j  \rangle \Big(\langle v^j , \Theta \rangle
+  s^j \Big)\; -\; \langle \Sigma  ,  t \rangle\;  +\;  \sum_{j=1}^{n} \exp{\big(- \langle v^j , \Theta \rangle - s^j\big)}  \
+\  {\rm c.c.} \bigg\} . \nonumber
\end{align}
This lagrangian is expected to be dual to the $e' \rightarrow \infty$ limit of $L$, i.e. dual to the lagrangian
of the gauged sigma-model with target $X$ and group $T^k$. The superpotential of this dual Landau-Ginzburg theory
is thus
\begin{equation}
W \ = \ \langle \Sigma , u_j  \rangle \Big(\langle v^j , \Theta \rangle  +  s^j \Big)
\ -\ \langle \Sigma  ,  t \rangle  \ +\  \sum_{j=1}^{n} \exp{\Big(- \langle v^j , \Theta \rangle - s^j\Big)} \ .
\label{3.6}
\end{equation}
We stress that these calculations are basically the same as in \cite{H-V}, the only difference being that here we have
kept the gauge terms in the quotient theory. In \cite{H-V} all the gauge terms were gotten rid of, because there one is only
interested in the non-gauged sigma-model with target $X$.

\section{Observables of the dual topological theory}

Our work up to this point has been to identify the Landau-Ginzburg theory dual to the original gauged sigma-model with
target $X$ and group $T^k$. We concluded that the dual theory has lagrangian (\ref{3.5}) and superpotential (\ref{3.6}).
We can now start to use this dual to extract properties of the gauged sigma-model.

The main object that concerns us in this note is the (small) quantum equivariant cohomology ring of the target $X$.
As is well known in the non-equivariant case, this ring coincides with, or can be defined as, the ring of local
observables of the A-twisted topological sigma-model with target $X$. This topological theory is
obtained by twisting the original supersymmetric theory along its vectorial R-symmetry. Now the vectorial R-symmetry is
also a symmetry of the dual model, the gauged Landau-Ginzburg model, and in particular one can also twist the latter
model to obtain a topological theory, which is then expected to be the dual of the gauged A-theory on $X$. The ring of
local observables of that topological Landau-Ginzburg model will then coincide with the ring of local observables of
the A-twisted gauged sigma-model, i.e. with the quantum equivariant cohomology ring of $X$. Thus, given this rationale,
we at present have the two following tasks: firstly to twist the dual theory (\ref{3.5}) along its vectorial R-symmetry,
and, secondly, to compute the ring of observables of the resulting topological theory.

  The whole of this recipe is very familiar from the non-gauged case. In that case the final result is that the ring of
local observables, or chiral ring, is a ring of polynomials quotiented by the ideal generated by the first partial
derivatives of the superpotential $W$. Here in the gauged case, however, we must pay a little extra attention to the
gauge fields. In fact, although the twisted chiral field strengths $\Sigma$ appear in the superpotential (\ref{3.6})
side by side with the twisted chiral matter fields $\Theta$, they are not exactly in the same footing as the latter. As we will see, at the end it turns out that the $\Sigma$'s do contribute to the ring of polynomials that corresponds to the observables but,
on the other hand, they do not contribute to the ideal by which one should mod-out, i.e. the partial derivatives of $W$
in the directions of the $\Sigma$'s do not appear in the result. The reason is that the observables determined by these partial derivatives are not trivial in the topological theory, and so cannot be moded-out. In contrast, the derivatives of $W$ with respect to the matter fields are still trivial (or $Q$-exact) observables, and hence as usual must be moded-out. This difference is explicitly revealed in the expressions for the action of the topological $Q$-operator obtained below in (\ref{4.2}).

$\ $

So we now want to determine the observables of the topological theory derived from the Landau-Ginzburg model
(\ref{3.5}). This model has the general form
\begin{equation}
\hat{L}\ = \ \int \dd^4 \theta\  \Big[ K (Y ,\bar{Y})\  -\  \frac{1}{2 e^2}\: \langle \bar{\Sigma},\Sigma \rangle \Big]\
+\ \frac{1}{2} \int \dd^2 \tilde{\theta}\ \Big\{ W( \Sigma , Y)\ +\ {\rm c.c.} \Big\} \ ,
\label{4.1}
\end{equation}

where the $\Sigma$'s and $Y$'s are the twisted chiral superfields of expressions (\ref{2.4}). As usual, after twisting,
the space of fields of the topological theory will be acted by a natural fermionic operator, $Q$, and the observables of the
theory will be the corresponding $Q$-cohomology classes. At the end of the section, after writing down the explicit form
of $Q$, we will conclude that the local observables of the topological theory are the holomorphic functions
$f (\sigma^b , y^j)$ modulo the ideal generated by the $Q$-exact functions $\partial_{y^j} W $. As mentioned before, the
derivatives $\partial_{\sigma^b} W$ do not appear. This is, in fact, all that we will subsequently need, so the reader may
wish at this point to just note the result and smoothly fly over to the next section. If not, then here is a brief
justification. Starting with the lagrangian (\ref{4.1}) and superfields (\ref{2.4}), the first task is to twist the theory
along the vectorial R-symmetry. This symmetry is defined by
\begin{align}
Y\ &\longrightarrow \  Y (e^{-i\alpha} \theta^{\pm} , e^{i\alpha} \thetab^{\pm})  \nonumber \\
\Sigma \ &\longrightarrow \  \Sigma (e^{-i\alpha} \theta^{\pm} , e^{i\alpha} \thetab^{\pm})  \ ,\nonumber
\end{align}
and in components reads
\begin{align}
\lamb_+ &\longrightarrow  e^{-i\alpha} \lamb_+   \qquad \qquad  \chib_+ \longrightarrow e^{-i\alpha} \chib_+    \nonumber \\
\lam_- &\longrightarrow  e^{i\alpha} \lam_-      \qquad \qquad  \chi_- \longrightarrow e^{i\alpha} \chi_-  \ , \nonumber
\end{align}
with all the other fields remaining invariant. Applying the usual rules for twisting a theory along an R-symmetry
\cite{Wit1, Hori-al},
we can reinterpret the various component fields as sections of different bundles and subsequently combine them into a new
set of fields, the fields of the topological theory. These new fields will all be scalars or one-forms on the worldsheet
with values either on the Lie algebra or on the pull-back bundle $y^\ast TX$. The lagrangian of the topological theory
is then obtained by writing down (\ref{4.1}) in components and substituting in the new set of fields. Now, because the fields
involved are many and we do not really need the explicit topological lagrangian for our purposes, we will skip writing it
down here. Instead we proceed directly to compute the form of the topological operator $Q$, which is the only
knowledge required to identify the ring of local observables. As usual, the action of $Q$ on the different fields can be
read out of the $N=2$ supersymmetry transformations. These are written down for instance in \cite{H-V} for the gauge fields and in
\cite{Hori-al} for the matter fields of the twisted chiral multiplet.
Since the definition of the topological operator is $Q = Q_- + \bar{Q}_+$,
to obtain the explicit $Q$-action over the fields one only needs to put the fermionic parameters $\epsilon_+ =
\bar{\epsilon}_- = 1$ and $\epsilon_- = \bar{\epsilon}_+ = 0$ in these transformations (see the conventions of \cite{H-V}).
After integrating out the auxiliary fields $D^a$ and $G^j$ and rotating to euclidean worldsheet (because topological
theories are euclidean), this yields the result:
\begin{align}
Q\,& A^a_z \ =\ - i \lam_-^a   &   Q\,& \lam_-^a \ =\ -2\sqrt{2}\, \partial_z \sigma^a     \label{4.2}\\
Q\,& A^a_{\zb} \ =\  i \lamb_+^a   &   Q\,& \lamb_+^a \ =\  2\sqrt{2}\, \partial_{\zb} \sigma^a \nonumber  \\
Q\,& \sigma^a \ = \ 0  &  Q\,& (\lamb_-^a + \lam_+^a) \ = \ 0  \nonumber \\
Q\,& \sib^a \ = \ - i \sqrt{2} (\lamb_-^a + \lam_+^a)   &  Q\,&  (\lamb_-^a - \lam_+^a)  \ = \ -2\sqrt{2} i [ \ast F^a_A +
 e^2  {\rm Re}\ ( \partial_{\sigma^{a}}W )]  \nonumber \\
Q\,& y^j \ = \ 0    &   Q\,& \chi_-^j \ =\ - 2 i \sqrt{2}\, \partial_z y^j  \nonumber  \\
Q\,& \bar{y}^j \ = \ \sqrt{2} (\chi^j_+ - \chib^j_- )    &   Q\,& \chib_+^j \ =\  2i \sqrt{2}\, \partial_{\zb} y^j \nonumber \\
Q\,& (\chi^j_+ - \chib^j_- )\ =\ 0  &  Q\,& [2 \sqrt{2}\, h_{j\bar{l}}(\chi^l_+ + \chib^l_- )] \ =\ -\, \partial_{y^j} W \ .\nonumber
\end{align}
Here $z$ is the complex coordinate on the euclidean worldsheet and $h_{j\bar{l}} = - \partial_j \partial_{\bar{l}}K$
is the hermitian metric on the geometry $T^k$-dual to $X$.
It is apparent from these expressions that the only natural $Q$-closed operators that may not be $Q$-exact are the
holomorphic combinations of the fields $y$ and $\sigma$. Furthermore, one such holomorphic combination will be
$Q$-exact if and only if it has any of the derivatives $\partial_{y^j} W$ as a factor. Observe also that the partial
derivatives $\partial_{\sigma^b}W$ appear in (\ref{4.2}) only through their real part, which is not holomorphic and therefore
not even $Q$-closed.

\section{The chiral ring of the gauged models}

Applying the results of the previous section to the lagrangian (\ref{3.5}), we see that the local observables of the dual
topological theory are the holomorphic functions $f(\sigma^b , \theta^b)$ modulo the ideal of functions generated by the
partial derivatives $\partial_{\theta^b}W$. Now, in analogy to what is done in the non-gauged case, instead of considering
all the holomorphic functions in the definition of chiral ring, we restrict ourselves to finite degree polynomials in the
variables $\sigma^b$ and $(x^b)^{\pm 1} := \exp{(\mp \theta^b)}$. This is related to the fact that in the definition of
the equivariant de Rham complex we only consider finite degree forms and polynomials in the Lie algebra.
Then the chiral ring of the dual theory is
\begin{equation}
\CC [ \sigma^1, \ldots , \sigma^k , (x^1)^{\pm 1} , \ldots , (x^k)^{\pm 1} ] \ / \ D(W) \ ,
\label{5.1}
\end{equation}
where $D(W)$ is the ideal generated by the derivatives
\begin{equation}
\partial_{\theta^b}W \ = \ -x^b\, \partial_{x^b} W \ = \ \sum_{j=1}^{n}\; u^b_j\: \Big[\langle \sigma , v^j \rangle \  -\ e^{-s^j}
\prod_{c=1}^{k}  (x^c)^{v^j_c} \Big]\ .
\label{5.2}
\end{equation}
This result can be cast in a different and perhaps simpler form. Consider the ring homomorphism
\begin{equation}
\Lambda :  \CC [\sigma^1 , \ldots , \sigma^k , w^1 , \ldots , w^n ]\ \longrightarrow \ \CC [\sigma^1 , \ldots , \sigma^k ,
(x^1)^{\pm 1} , \ldots , (x^{k})^{\pm 1} ]
\end{equation}
determined by
\begin{equation}
\sigma^b\  \longmapsto \ \sigma^c\, v^j_c \, u^b_j  \qquad {\rm and} \qquad w^j \ \longmapsto \ e^{-s^j}\: \prod_{c=1}^{k}\: (x^c)^{v^j_c} \ .
\end{equation}
It is not difficult to recognize that the matrix $B^b_c := u_j^b v_c^j$ is invertible over $\RR$. Basically this follows from
the fact that both $\{ Q^a , u^b  \}$ and $\{ Q^a , v_b \}$ are basis of $\RR^n$. It is then clear that, restricted to the
subring of polynomials in the $\sigma$'s, the map $\Lambda$ is an isomorphism to its image, which is exactly the same subring.
Furthermore, because of the assumption that the vectors $v^1 , \ldots , v^n$ span the full $\mathbb{Z}^k$, it is also
clear that the image by $\Lambda$ of the subring $\CC [ w^1 , \ldots , w^n ]$ is the full subring of Laurent polynomials
$\CC [ (x^1)^{\pm 1} , \ldots , (x^{k})^{\pm 1} ]$. Hence, we conclude, $\Lambda$ is surjective. (Observe in passing that this
was the first step that required the stated property of set of $v$'s, a property which comes from the regularity of the fan
corresponding to $X$; in particular the derivation of (\ref{5.1}) did not require the regularity assumption on $X$.)

Now, by construction, the ideal $D(W)$ of (\ref{5.2}) is the image by $\Lambda$ of the ideal $\tilde{D}$ in the domain
generated by the polynomials
\begin{equation}
\sigma^b - u_j^b \, w^j \qquad \qquad {\rm for} \qquad b = 1, \ldots , k \ .  \nonumber
\end{equation}
Furthermore, just as in \cite{Bat}, it is apparent that the kernel of $\Lambda$ is the ideal $Q(X)$ generated by the binomials (\ref{1.2})
described in the introduction. Hence we finally conclude that, up to isomorphism of rings,
\begin{align}
QH_T^\ast (X) \ &= \ \CC [\sigma^1 , \ldots , \sigma^k , w^1 , \ldots , w^n ]\ / \ (\tilde{D} + Q(X))  \label{5.4} \\
&=\ \CC [ w^1 , \ldots , w^n ]\ / \ Q(X) \ .  \nonumber
\end{align}

Having arrived at this result, a number of conclusions can be drawn in analogy with the non-equivariant case. Firstly observe that, through the definition (\ref{1.2}) of the ideal $Q(X)$, the chiral ring seems to depend on the complex parameter $t' = r' - i \theta'$ in $\CC^{n-k}$. It is easy to recognize, however, that this dependence is only apparent, and that by redefining the variables $w^j$ one can absorb any finite variation of $t'$. This means that the chiral ring of our gauged sigma-model does not depend on the value of $t'$, at least as long as the assumption that the target $X(r')$ is a smooth Fano manifold of dimension $k$ remains valid. This is analogous to the non-gauged case \cite{Bat, M-P}, and leads to the conclusion that toric targets that are isomorphic in codimension 1 have the same quantum equivariant ring. The usual example here is toric targets related by flop-type birational transformations, for in this case they can be realized as quotients $X(r')$ with the same charges $Q_j^a$ and parameters $r'$ belonging to adjacent cones in the parameter space ${\mathbb R}^{n-k}$.

Another conclusion follows from the result in \cite{Bat} that says that if $t'$ belongs to the K\"ahler cone of $X$ and $\lambda$ is a real scalar, then the formal limit $\lambda \rightarrow +\infty$ of the relations (\ref{1.2}) that define the ideal $Q_{\lambda t'}(X)$ is actually the set of relations that define the classical Stanley-Reisner ideal $J(X)$. Comparing $(\ref{1.1})$ and $(\ref{1.3})$ this shows that, as expected, the classical equivariant cohomology ring of $X$ can be obtained as a formal limit of the quantum equivariant cohomology ring. This formal limit, however, depends on the choice of the cone in $\CC^{n-k}$ to which $t'$ belongs. This corresponds to the fact that the classical equivariant ring of the quotient $X(r')$ also depends on the cone, in contrast with the quantum equivariant ring.

We thus conclude that much of the story of mirror symmetry and quantum cohomology of toric manifolds extends to the equivariant case.

$\ $

\noindent {\bf Example: projective spaces.}
The compact toric manifolds $X = \CC {\mathbb P}^k$ correspond to the assignments $n=k+1$ and $Q= (1, \ldots , 1)
\in {\mathbb Z}^{k+1}$ in the general picture above. The result (\ref{5.4}) then states that the quantum equivariant
cohomology ring of these manifolds is the polynomial ring
\begin{equation*}
Q H_{T^k}^\ast (\CC {\mathbb P}^k) \ = \ \CC [w^1 ,  \ldots , w^{k+1}] /  (w^1 \cdots w^{k+1} - e^{-t'}) \ .
\end{equation*}
The parameter space is here one-dimensional, and in the limit $t' \rightarrow +\infty$ this ring formally reduces to the classical $T^k$-equivariant cohomology
of $\CC {\mathbb P}^k$, which is
\begin{equation*}
 H_{T^k}^\ast (\CC {\mathbb P}^k) \ = \ \CC [w^1 ,  \ldots , w^{k+1}] /  (w^1 \cdots w^{k+1}) \ .
\end{equation*}
To put this in context, recall that the usual (non-equivariant) cohomology rings of $\CC {\mathbb P}^k$ are
\begin{equation*}
Q H^\ast (\CC {\mathbb P}^k) \ = \ \CC [w] /  (w^{k+1} - e^{-t'}) \qquad \ {\rm and} \qquad \
H^\ast (\CC {\mathbb P}^k) \ = \ \CC [w] /  (w^{k+1}) \ . \
\end{equation*}

\vskip 45pt
\noindent
{\bf Acknowledgements.}
I would like to thank Marcos Mari\~no and Jan de Boer for helpful conversations and the referee for suggesting several improvements.  I am partially supported by the
Netherlands Organisation for Scientific Research (NWO), Veni grant 639.031.616.


TITLE: Possible value of $x$ so that fractions are in simplest form.
QUESTION [0 upvotes]: Which of the following could be the possible value of $x$ for
  which each of the fractions is in its simplest form, where
  $\lfloor{x\rfloor}$ stands for greatest integer less than or equal to $x$ ?
$\dfrac{\lfloor{x\rfloor}+7}{10},\ \dfrac{\lfloor{x\rfloor}+18}{11},\ \dfrac{\lfloor{x\rfloor}+31}{12},\ \dfrac{\lfloor{x\rfloor}+46}{13},\cdots\cdots\cdots\cdots,\ \dfrac{\lfloor{x\rfloor}+1489}{39}$
and
$\dfrac{\lfloor{x\rfloor}+1567}{40}$

$a.)\ 95.71\quad \quad \quad \quad \quad b.)\ 93.71 \\
\color{green}{c.)\ 94.71}\quad \quad \quad \quad \quad d.)\ 92.71 \\$
I simplified this into
$\dfrac{\lfloor{x\rfloor}+7}{10},\ 1+\dfrac{\lfloor{x\rfloor}+7}{11},\ 2+\dfrac{\lfloor{x\rfloor}+7}{12},\ 3+\dfrac{\lfloor{x\rfloor}+7}{13},\cdots\cdots\cdots\cdots,\ 38+\dfrac{\lfloor{x\rfloor}+7}{39}$
and
$39+\dfrac{\lfloor{x\rfloor}+7}{40}$
I am also struggling to interpret the question.
This question if from chapter quadratic equations though it doesn't look like.
I look for a short and simple way.
I have studied maths up to $12$th grade.

REPLY [1 votes]: Interpreting the question:
The greatest integer function and the non-integral answers are a red herring, meant to distract you. Everything in the question except $x$ is an integer, so the fractional part of $x$ is immediately thrown away and not used. Your simplification is also important, since the integral parts of the expression are irrelevant as to whether the fraction is in lowest terms. Therefore, the question could be more simply phrased:
Which of these are all in simplest terms:
$$\frac{x+7}{10},\ \frac{x+7}{11},\ \frac{x+7}{12},\ \ldots,\ \frac{x+7}{40}$$
with the answer choices (a) $95$; (b) $93$; (c) $94$; (d) $92$
A fraction is not in lowest terms if the greatest common divisor for the numerator and denominator is not one: i.e. if there is any common divisor greater than one.
Solving the problem:
For each possible value of $x$, let's look at the factors of $x+7$.
(a) $95+7=102=2\cdot 3\cdot 17$. Many denominators in the list of $10$ through $40$ have a factor of $2$ or $3$, and $17$ is itself in the list. So this is not the answer.
(b) $93+7=100=2\cdot 2\cdot 5\cdot 5$. Many denominators in the list of $10$ through $40$ have a factor of $2$ or $5$. So this is not the answer.
(c) $94+7=101$ which is prime. None of those denominators can have any factor in common, so this is the answer.
(d) $92+7=99=9\cdot 11$. Many denominators in the list of $10$ through $40$ have a factor of $9$ or $11$, and $11$ is itself in the list. So this is not the answer.
Therefore the only answer is (c), $94.71$ which becomes $94$.

Quadratic equations:
Why is this in a chapter about quadratic equations? The fractions given are all in the form
$$\frac{\lfloor x\rfloor + (n+4)^2-18}{n+9}$$
for $n=1,2,\ldots, 31$. You can see the quadratic expression in $n$ in the numerator.


TITLE: How can I deploy a CG-Steihaug algorithm for trust region sub-problem solving?
QUESTION [0 upvotes]: I'm studying various optimization methods and on this occasion, I'm trying to tackle the Trust Region Problem by solving the sub-region problem with the Steihaug-CG algorithm in Python. I'm using the pseudo-code from the book (Numerical Optimization, Nocedal), which is the same Pseudo Code from this source.  I'm more than confused on the both find $\tau$ steps.

I tried several ways to decompose this, mainly by using a solver for the $m_k{(p_k)}$ function and a line search for the second possible case, but both proved completely unsuccessful. I'm also adding my Python code for further clarification:
def steihaugcg(B, gradf, delta, tol=1e-9, max_it=1000):
r=[gradf]
if norm(r[-1]) < tol: return np.zeros(B.shape[0])

def LineSearch(z, d, DELTA):
    t=.5
    for _ in range(500):
        if np.allclose(norm(z+t*d),DELTA):return t
        if norm(z+t*d) < DELTA: t = t*1.9
        if norm(z+t*d) > DELTA: t = t*0.1
    return None

d=-r[-1]
t=0
Size=B.shape[0]
z=np.zeros(Size)
for _ in range(max_it):
    if d.T@B@d <= 0:
        t = minimize(lambda t: gradf.T@(z+t*d) + 0.5*(z+t*d).T@B@(z+t*d), 1).x
        p=z+t*d
        if np.allclose(norm(p),delta):
            return p
    alpha=(r[-1].T@r[-1])/(d.T@B@d)
    z=z+alpha*d

    if norm(z) >= delta:
        t = LineSearch(z,d,delta)
        if t is not None: return z+t*d

    r.append(r[-1]+alpha*B@d)

    if norm(r[-1])<tol:print("third");return z
    beta = (r[-1].T@r[-1])/(r[-2].T@r[-2])
    d = -r[-1] + beta*d

Can somebody provide insight on how to solve the find $\tau$ subproblem?

REPLY [0 votes]: Note that $\|p_k\|^2=\|z_j+\tau d_j\|^2 = z_j^Tz_j+2\tau d_j^Tz_j+\tau^2d_j^Td_j=\Delta_k^2$. So you can solve $\tau$ using the quadratic formula.


TITLE: "infinitely oscillating" manifold
QUESTION [2 upvotes]: Find out whether $M:=\{\left(x\cdot\cos\left(\frac{1}{x}\right),\,x\cdot\sin\left(\frac{1}{x}\right)\right)|\,x>0\}$ is a submanifold of $\mathbb{R}^2$ or not.
My guess is that it's not since I've tried to construct a homeomorphism from $\mathbb{R}$ to $M$ $\left(x\mapsto \left(x\cdot\cos\left(\frac{1}{x}\right),\,x\cdot\sin\left(\frac{1}{x}\right)\right)\right)$ and failed in proving bijectivity. I then tried to construct multiple charts from $\mathbb{R}$ and $\mathbb{R}^2$ but it hasn't worked out so far because of the way $M$ looks near the origin.
Thank you very much in advance.

REPLY [3 votes]: It is a submanifold.
Consider the map $f : (0,\infty) \to \mathbb R^2, f(x) = (x\cos(\frac{1}{x}),x\sin(\frac{1}{x}))$. We have $M = f((0,\infty))$. The map $f$ is injective since $\lVert f(x) \rVert = \sqrt{x^2\cos^2(\frac{1}{x}) + x^2\sin^2(\frac{1}{x})} = \sqrt{x^2} = x$ which implies $f(x_1) \ne f(x_2)$ for $x_1 \ne x_2$. Thus $\tilde f : (0,\infty) \stackrel{f}{\to} M$ is a bijection. Moreover, $g = \lVert - \rVert_M : M \to (0,\infty)$ is continuous and we have $g \circ \tilde f = id$, therefore $\tilde f$ is homeomorphism.
We have $f'(x) = (\cos(\frac{1}{x}) + \frac{1}{x}\sin(\frac{1}{x}), \sin(\frac{1}{x}) - \frac{1}{x}\cos(\frac{1}{x}))$ and we get
$$\lVert f'(x) \rVert = \sqrt{\cos^2(\frac{1}{x}) + \frac{2}{x}\sin(\frac{1}{x})\cos(\frac{1}{x}) +  \frac{1}{x^2}\sin^2(\frac{1}{x}) + \sin^2(\frac{1}{x}) - \frac{2}{x}\sin(\frac{1}{x})\cos(\frac{1}{x}) +  \frac{1}{x^2}\cos^2(\frac{1}{x})} \\ = \sqrt{1 + \frac{1}{x^2}} \ne 0 .$$
Thus $f'(x) \ne 0$ so that $f$ is an immersion. Hence $f$ is a smooth embedding and $M$ is a smooth submanifold of $\mathbb R^2$.