text:
\section{Three-phase Power Flow Modeling}

Whereas the use of convex relaxations is well-applicable in single-phase networks, these are not yet robust enough for the more complex OPF formulations for unbalanced three-phase systems. Its development for three-phase networks~\cite{dallanese_distributed_2013} is hampered by issues of nondegeneracy and inexactness of the semidefinite relaxations~\cite{louca_nondegeneracy_2014}. In~\cite{sankur_optimal_2018,bolognani_fast_2016}, linear approximations of the unbalanced power flow are proposed to overcome these challenges and enable three-phase OPF for a broad range of conditions.
Here we adopt the modeling developed in~\cite{arnold_optimal_2016, sankur_linearized_2016,sankur_optimal_2018}, which can be seen as an adaptation of the \emph{DistFlow} model~\cite{baran_optimal_1989} to unbalanced circuits, coined the \emph{Dist3Flow} model. We restate part of the model's derivation here to provide sufficient intuition for its use in experiments.

In this setting, each node and line segment can have up to three phases, labeled $a$, $b$, and $c$. Phases are referred to by the variables $\phi$ and $\psi$, where $\phi \in \{ a, b, c \}$, $\psi \in \{ a, b, c \}$.

If line $(m,n)$ exists, its phases must be a subset of the phases present at both node $m$ and node $n$.

The current/voltage relationship for a three phase line $(m,n)$ between adjacent nodes $m$ and $n$ is captured by Kirchhoff's Voltage Laws (KVL) in its full \eqref{eq:KVLmn1}, and compact form \eqref{eq:KVLmn2}:

\begin{gather}
	\begin{bmatrix}
		V_{m}^{a} \\ V_{m}^{b} \\ V_{m}^{c}
	\end{bmatrix}
	=
	\begin{bmatrix}
		V_{n}^{a} \\ V_{n}^{b} \\ V_{n}^{c}
	\end{bmatrix}
	+
	\begin{bmatrix}
		Z^{aa}_{mn} & Z^{ab}_{mn} & Z^{ac}_{mn} \\
		Z^{ba}_{mn} & Z^{bb}_{mn} & Z^{bc}_{mn} \\
		Z^{ca}_{mn} & Z^{cb}_{mn} & Z^{cc}_{mn}
	\end{bmatrix}
	\begin{bmatrix}
		I_{mn}^{a} \\ I_{mn}^{b} \\ I_{mn}^{c}
	\end{bmatrix}
    \label{eq:KVLmn1}
	\\
    \mathbf{V}_{m} = \mathbf{V}_{n} + \mathbf{Z}_{mn} \mathbf{I}_{mn}
    \label{eq:KVLmn2}
\end{gather}

Here, $Z^{\phi \psi}_{mn} = r^{\phi \psi}_{mn} + jx^{\phi \psi}_{mn}$ denotes the complex impedance of line $(m,n)$ across phases $\phi$ and $\psi$.

\noindent Next, we define the per phase complex power as $\mathbf{S}_{mn}^{\phi} = \mathbf{V}_{n}^{\phi} \left( \mathbf{I}_{mn}^{\phi} \right)^{*}$

, and the $3 \times 1$ vector of complex power phasors $\mathbf{S}_{mn} = \mathbf{V}_{n} \circ \mathbf{I}_{mn}^{*}$ where $\mathbf{S}_{mn}$ is the power from node $m$ to node $n$ at node $n$.
\begin{equation}
    \sum_{l:(l,m) \in \mathcal{E}} \mathbf{S}_{lm} = \mathbf{s}_{m} + \sum_{n:(m,n) \in \mathcal{E}} \mathbf{S}_{mn} + \mathbf{L}_{mn}
    \label{eq:power03}
\end{equation}

The term $\mathbf{L}_{mn} \in \mathbf{C}^{3 \times 1}$ is a nonlinear and non-convex loss term. As in \cite{gan_convex_2014} and \cite{baran_optimal_1989}, we assume that losses are negligible compared to line flows, so that $\left| L_{mn}^{\phi} \right| \ll \left| S_{mn}^{\phi} \right| \ \forall (m,n) \in \mathcal{E}$. Thus, we neglect line losses, linearizing \eqref{eq:power03} into \eqref{eq:power04}.
\begin{equation}
    \sum_{l:(l,m) \in \mathcal{E}} \mathbf{S}_{lm} \approx \mathbf{s}_{m} + \sum_{n:(m,n) \in \mathcal{E}} \mathbf{S}_{mn}
    \label{eq:power04}
\end{equation}

\noindent Now, we define the real scalar $y_{m}^{\phi} = \left| V_{m}^{\phi} \right|^{2} = V_{m}^{\phi} (V_{m}^{\phi})^{*}$, the $3 \times 1$  real vector $\mathbf{y}_{m} = \left[ y_{m}^{a} , \ y_{m}^{b} , \ y_{m}^{c} \right]^{T} = \mathbf{V}_{m} \circ \mathbf{V}_{m}^{*}$.
With these definitions, \cite{sankur_optimal_2018} derives the following equations that govern the relationship between squared voltage magnitudes and complex power flow across line $(m,n)$:

\begin{equation}
\begin{gathered}
	\mathbf{y}_{m} = \mathbf{y}_{n} + 2 \mathbf{M}_{mn} \mathbf{P}_{mn} - 2 \mathbf{N}_{mn} \mathbf{Q}_{mn} + \mathbf{H}_{mn} \\
    \mathbf{M}_{mn} = \Re \left\{ \Gamma_{n} \circ \mathbf{Z}_{mn}^{*} \right\},
    \mathbf{N}_{mn} = \Im \left\{ \Gamma_{n} \circ \mathbf{Z}_{mn}^{*} \right\} \,,
\end{gathered}
\label{eq:mag09}
\end{equation}
where $\Gamma_{n} = \mathbf{V}_{n} \left( 1 / \mathbf{V}_{n} \right)^{T} \in \mathbb{C}^{3 \times 3}$ represents a matrix with voltage balance ratios across all phases at node $n$. Hence, we have that $\Gamma_{n}\left( \phi, \phi \right) = 1$ and  $\Gamma_{n}\left( \phi, \psi \right) = V_n^{\phi}/V_n^{\psi} \triangleq \gamma_{n}^{\phi \psi}$. Furthermore, $\mathbf{H}_{mn} = \left( \mathbf{Z}_{mn} \mathbf{I}_{mn} \right) \circ \left( \mathbf{Z}_{mn} \mathbf{I}_{mn} \right)^{*} = \left( \mathbf{V}_{m} - \mathbf{V}_{n} \right) \circ \left( \mathbf{V}_{m} - \mathbf{V}_{n} \right)^{*}$ is a $3 \times 1$ real-valued vector representing higher-order terms.
Notice that we have separated the complex power vector into its active and reactive components, $\mathbf{S}_{mn} = \mathbf{P}_{mn} + j \mathbf{Q}_{mn}$.

This nonlinear and nonconvex system is difficult to incorporate into a state estimation or optimization formulation without the use of convex relaxations. Following the analysis in \cite{gan_convex_2014}, we apply two approximations. The first approximation is legitimized by the fact that the high order term $\mathbf{H}_{mn}$ is negligible, such that $\mathbf{H}_{mn} = \left[ 0 , \ 0 , \ 0 \right]^{T}$. The second approximation assumes that node voltages are ``nearly balanced" such that:

\begin{equation}
	\Gamma
    =
    \begin{bmatrix}
    	1 & \gamma_{n}^{ab} & \gamma_{n}^{ab} \\
        \gamma_{n}^{ba} & 1 & \gamma_{n}^{bc} \\
        \gamma_{n}^{ca} & \gamma_{n}^{cb} & 1
    \end{bmatrix}
    =
    \begin{bmatrix}
    	1 & \alpha & \alpha^{2} \\
        \alpha^{2} & 1 & \alpha \\
        \alpha & \alpha^{2} & 1
    \end{bmatrix}
	\forall n \in \mathcal{N} \,,
    \label{eq:gammaalpha}
\end{equation}
\noindent where $\alpha = 1 \angle 120 \degree = \frac{1}{2} ( -1 + j \sqrt{3} )$ and $\alpha^{-1} = \alpha^{2} = \alpha^{*} = 1 \angle 240 \degree = \frac{1}{2} ( -1 - j \sqrt{3} )$.
Applying the approximations for $\mathbf{H}_{mn}$ and $\Gamma_{n}$ to \eqref{eq:mag09}, we arrive at a linear system of equations:
\begin{equation}
	\mathbf{y}_{m} \approx \mathbf{y}_{n} + 2 \mathbf{M}_{mn} \mathbf{P}_{mn} - 2 \mathbf{N}_{mn} \mathbf{Q}_{mn} \,, \text{ with}
    \label{eq:mag10}
\end{equation}
\begin{equation}
\begin{aligned}
	2 & \mathbf{M}_{mn}
    = \ldots \\
	& \begin{bmatrix}
		2 r_{mn}^{aa} & -r_{mn}^{ab} + \sqrt{3} x_{mn}^{ab} & -r_{mn}^{ac} - \sqrt{3} x_{mn}^{ac} \\
		-r_{mn}^{ba} - \sqrt{3} x_{mn}^{ba} & 2 r_{mn}^{bb} & -r_{mn}^{bc} + \sqrt{3} x_{mn}^{bc} \\
		-r_{mn}^{ca} + \sqrt{3} x_{mn}^{ca} & -r_{mn}^{cb} - \sqrt{3} x_{mn}^{cb} & 2 r_{mn}^{cc}
	\end{bmatrix} \,,
\end{aligned}
\label{eq:Mmn}
\end{equation}

\begin{equation}
\begin{aligned}
	2 & \mathbf{N}_{mn}
    = \ldots \\
	& \begin{bmatrix}
		-2 x_{mn}^{aa} & x_{mn}^{ab} + \sqrt{3} r_{mn}^{ab} & x_{mn}^{ac} - \sqrt{3} r_{mn}^{ac} \\
		x_{mn}^{ba} -\sqrt{3} r_{mn}^{ba} & -2 x_{mn}^{bb} & x_{bc} + \sqrt{3} r_{mn}^{bc}\\
		x_{mn}^{ca} + \sqrt{3} r_{mn}^{ca} & x_{mn}^{cb} -\sqrt{3} r_{mn}^{cb} & -2 x_{mn}^{cc}
	\end{bmatrix} \,.
\end{aligned}
\label{eq:Nmn}
\end{equation}

We present the full set of equations that comprise a linearized model for unbalanced power flow.
We named these the \emph{LinDist3Flow} equations, as developed in~\cite{arnold_optimal_2016,sankur_linearized_2016,sankur_optimal_2018}.

\begin{tcolorbox}[ams gather]
\textbf{LinDist3Flow Equations for Three-Phase Networks} \nonumber \\
\text{Per phase node complex load:} \nonumber
\\
s_{n}^{\phi} \left( y_{n}^{\phi} \right) = \left( \beta_{S,n}^{\phi} + \beta_{Z,n}^{\phi} y_{n}^{\phi} \right) d_{n}^{\phi} + u_{n}^{\phi} - j c_{n}^{\phi} \,, \label{eqn:nodallin} \\
\forall \phi \in \mathcal{P}_{n}, \, \forall n \in \mathcal{N}
\nonumber
\\

\text{Branch power flow:} \nonumber
\\
\sum_{l:(l,m) \in \mathcal{E}} \mathbf{S}_{lm} \approx \mathbf{s}_{m} + \sum_{n:(m,n) \in \mathcal{E}} \mathbf{S}_{mn} \,, \forall m \in \mathcal{N}
\label{eqn:powerlin}
\\
\text{Magnitude and angle equations:} \nonumber
\\
\left[ \mathbf{y}_{m} \approx \mathbf{y}_{n} + 2 \mathbf{M}_{mn} \mathbf{P}_{mn} - 2 \mathbf{N}_{mn} \mathbf{Q}_{mn} \right]_{\mathcal{P}_{mn}} \,,
\label{eqn:maglin}
\\
\left[ \Theta_{m} \approx \Theta_{n} - \mathbf{N}_{mn} \mathbf{P}_{mn} - \mathbf{M}_{mn} \mathbf{Q}_{mn} \right]_{\mathcal{P}_{mn}} \,,
\label{eqn:anglelin}
\\
\text{with }\left[ \mathbf{M}_{mn}= \Re \left\{ \Gamma \mathbf{Z}_{mn}^{*} \right\} , \
\mathbf{N}_{mn}= \Im \left\{ \Gamma \mathbf{Z}_{mn}^{*} \right\} \right]_{\mathcal{P}_{mn}} \nonumber \\
\forall (m,n) \in \mathcal{E} \,. \label{eqn:MmnNmnlin}
\end{tcolorbox}

Here, we have embraced a load model with a fixed voltage sensitivity (as parameterized with $\beta_{S,n}^{\phi}, \beta_{Z,n}^{\phi}$), with $d_{n}^{\phi}$ denoting demand, $u_{n}^{\phi} := u_n^{p,\phi} + j u_n^{q,\phi}$ the controllable power injection and $c_{n}^{\phi}$ a possible capacitor bank.

The accuracy of the approximations in the power and voltage magnitude equations has been investigated in \cite{gan_convex_2014} and \cite{robbins_optimal_2016-1}. In~\cite{sankur_optimal_2018}, we perform a Monte Carlo analysis to explore the level of error introduced by the voltage angle equation assumptions.

\section{Three-phase Optimal Power Flow}

\subsection*{Voltage Balancing}
\label{sec:sim_voltage_balance}

Although the \textit{LinDist3Flow} equations allow for a flexible OPF framework comparable to the one presented for single-phase networks above, here we focus on one instance to demonstrate the generalization of decentralized OPF to three-phase unbalanced systems.
We remodel an instance first presented in~\cite{sankur_linearized_2016} that demonstrated OPF for balancing voltage magnitudes on an unbalanced radial distribution feeder.
This is an important objective as many three phase loads (induction motors for instance) are sensitive to high levels of voltage imbalance.
Furthermore, 3-phase voltage regulation equipment may actuate based on single phase measurements, and significant imbalances can lead to improper operation of these devices.

The problem is defined as
\begin{equation}
\begin{array}{rl}

  \underset{y_{n}^{\phi},P_{mn}^{\phi},Q_{mn}^{\phi},u_{n}^{\phi}} {\text{min}}

  & \sum_{n \in \N} \left[ \sum_{\phi \neq \psi} ( y_{n}^{\phi} - y_{n}^{\psi} )^{2} + \rho \sum_{\phi} \left| u_{n}^{\phi} \right|^{2} \right] \nonumber \\
  \text{s.t.}
  & \eqref{eqn:nodallin} \,, \eqref{eqn:powerlin} \,, \eqref{eqn:maglin} \,, \underline{y} \le y_{n}^{\phi} \le \overline{y}, \nonumber \\
  &  \left| u_{n}^{\phi} \right| \le \overline{u}, \quad \forall n \in \N \,,

\label{eq:OP3F}
\end{array}
\end{equation}

\noindent where $\phi,\psi \in \{ a,b,c\}$.
The parameter $\rho$ is a penalty on control action set to $0.01$.
\cite{sankur_linearized_2016}~showed the effectiveness of the \textit{LinDist3Flow} model in drastically reducing voltage imbalance across phases, and we will aim to reproduce these results in Section~\ref{sec:results_3ph}.

\section{Results for Three-Phase Optimal Power Flow}
\label{sec:results_3ph}
We also show the efficacy of Decentralized OPF for three-phase unbalanced networks, by learning and reconstructing the solutions~$u_{n}^{p,\phi},u_{n}^{q,\phi}$, with $\phi \in \{ a,b,c\}$, of the centralized OPF problem~\eqref{eq:OP3F}, as formulated in Section~2 of the main paper.
The feeder model used here is the IEEE 13 node test feeder~\cite{_ieee_2017}.
Figure~\ref{fig:Result3Phase} shows the results of applying Decentralized OPF to balance voltage magnitudes across phases for a scenario with a significant gap between phase~$b$ and phase~$c$ of $\approx 0.1$p.u..
We see that the voltage magnitudes are tightly balanced across the network. Notice that in order to balance the large gap in the lower half of the feeder (nodes 671, 692, 675, 680, 684), a small increase in gap is incurred in the left upper radial (nodes 645, 646).
Similar to the results for single-phase networks, the Decentralized OPF method also performs well at locally reconstructing OPF solutions in three-phase unbalanced networks, further validating its promise to serve as a new paradigm for regulating voltage and power flow in distribution networks.

\begin{figure}[h!]
\centering
\includegraphics[trim=0.5cm 0 1.5cm 0,width=.48\textwidth]{rbic_figs/20180325_Voltage_Balancing.eps}

\caption{Voltage magnitude balancing. In grey, voltage magnitudes across three phases without control. In black, voltage magnitudes across three phases applying decentralized OPF.}
\label{fig:Result3Phase}
\end{figure}


TITLE: Class divided into 5, probability of 2 people in the same team
QUESTION [2 upvotes]: This might be a very simple question, but: a class of 25 students is divided into 5 teams of 5 each. What is the probability of student X and Y being in the same team?
is it just 4/25?

REPLY [0 votes]: Another way to look at this is to think of how many ways you can arrange 25 kids into five five-member teams.  This is $\frac{25!}{5!5!5!5!5!}$.
Now, how many ways can you arrange it so that $X$ and $Y$ are on the same team?  In whatever team they end up (so there are 5 such equivalent scenarios) the other 23 kids will be arranged around them, 3 kids on the team our buddies ended up in, while the others will be spread out in teams of five (i.e., 3/5/5/5/5 or 5/3/5/5/5, or 5/5/3/5/5, or ... , you get the idea).
Each of these scenarios has probability $\frac{23!}{3!5!5!5!5!5!}$. Thus, the final probability will be
$$
P(X, Y on\,\,same\,\,team)=\frac{\frac{23!}{3!5!5!5!5!5!}}{\frac{25!}{5!5!5!5!5!}}\times 5=\frac{23!5}{25!3!}\times 5=\left (\frac{5}{25}\right )\left (\frac{4}{24}\right )5=\left (\frac{4}{24}\right )=\frac{1}{6}
$$


TITLE: Weak *-topology of $X^*$ is metrizable if and only if ...
QUESTION [2 upvotes]: Let $X$ be a topological vector space on which $X^*$ separates points. Prove that "the weak *-topology of $X^*$ is metrizable if and only if $X$ has a finite or countable Hamel basis"?
(A set $\beta$ is a Hamel basis for a vector space $X$ if $\beta$ is a maximal linearly independent subset of $X$. In other words $\beta$ is a Hamel basis if every $x\in X$ has a unique representation as a finite linear combination of element of $\beta$.)

REPLY [2 votes]: This should be obvious from the characterization of the weak-$^*$ topology as the topology of pointwise convergence and it's easy to show that $\mathbb R^A$ is only metrizable when the cardinality of $A$ is countable.  But by the Hahn-Banach theorem $X^*$ has a Hamel basis that's at least the same size as that of $X$ (assuming that $X$ is locally convex. $L^p$ for $0<p<1$ has no non-zero continuous linear functionals) so $X^*$ cannot be metrizable.
I'll leave the details for you to fill out.


TITLE: $f$ is convex function iff Hessian matrix is nonnegative-definite.
QUESTION [9 upvotes]: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $f \in C^2$. Show that $f$ is convex function iff Hessian matrix is nonnegative-definite.
$f(x,y)$ is convex if $f( \lambda x + (1-\lambda )y) \le \lambda f(x) + (1- \lambda)f(y)$ for any $x,y \in \mathbb{R}^2$.
Hessian matrix is nonnegative-definite if $f_{xx}'' x^2 + f_{x,y}(x+y) + f_{yy}''y^2 \ge 0$
I know the definition but I have no idea how prove the If and only if condition or first and second implication?

REPLY [5 votes]: I would use restrictions to lines: $\phi_{a,b}(t) = f(a+tb)$ where $t\in\mathbb R$ and $a,b\in\mathbb R^2$ and $b\ne 0$. The key points are:

$f$ is  convex if and only if $\phi_{a,b}$ is convex for every $a,b$. This follows from the fact that definition of convexity involves points on the same line.
The Hessian of $f$ is nonnegative definite (aka positive  semidefinite) if and only if $\phi_{a,b}''\ge 0$ for all $a,b$. Indeed, Hessian is a symmetric matrix and for such matrices being nonnegative is equivalent to $b^THb\ge 0$ for all $b\in\mathbb R^2$.  By the multivariable chain rule, $b^THb$ gives $\phi_{a,b}''$.


text:
\section{The Lie algebra \(\cgplus\)} \label{The lie algebra cgplus}
We assume \(g \geq 4\).
Let \(H \defeq \QRational^{2g}\) be
the fundamental representation of the symplectic group \(\SymplTwogQ\), and
\(\mu \colon H \tensor H \to \QRational \) be
the symplectic form on \(H\).
We fix a symplectic basis \(a_1, \dots, a_g, b_1, \dots, b_g\) with respect to \(\mu\)
, i.e. for any \(1 \leq i, j \leq g\), \(\mu(a_i, a_j) = \mu(b_i, b_j) = 0\) and
\(\mu(a_i, b_j) = - \mu(b_j, a_i) = \delta_{ij}\) where \(\delta_{ij}\) is the Kronecker delta.

\par For \(w \geq 0\), define
\begin{equation}
\cg(w) \defeq S^{w + 2} H
\qquad \text{and} \qquad
\cgplus \defeq \bigoplus_{w \geq 1} \cg(w)
\subset \bigoplus_{w \geq 0} \cg(w) \eqdef \cg.
\end{equation}
An element \(\xi \in \cg(w) \subset \cg\) is said to be
of \textit{weight} \(w\).

\par We define a Lie bracket \(\left[, \right]\) on \(\cg\) by
\begin{equation*}

\includegraphics{cgplus_defmap_liebracket-crop.pdf}
\end{equation*}
for \(k, l \geq 0\) and monomials
\(x_1 \cdots x_{k+2} \in \cg(k)\), \(y_1 \cdots y_{l+2} \in \cg(l)\).
Here \(\widehat{\phantom{x_i}}\) means the absence of that component.
This bracket is interpreted as the classical Poisson bracket on \(C^\infty (\Real^{2g})\)
restricted to polynomial functions whose coefficients in \(\QRational\).
Note that this bracket preserves weights on \(\cg\) and is \(\Symp\)-equivariant.
Then the subspace \(\cgplus \subset \cg\) becomes a Lie subalgebra.

\par The differential \(\partial_\bullet\) on the Chevalley-Eilenberg chain complex \(\wedge^\bullet \cg\)
with respect to this Lie bracket is described as
\begin{equation*}
    \partial_n (f_1 \wedge \cdots \wedge f_n)
    = \sum_{1 \leq i < j \leq n} (-1)^{i+j+1} \left[ f_i, f_j \right]
    \wedge f_1 \wedge \cdots \wedge \widehat{f_i} \wedge \cdots
    \wedge \widehat{f_j} \wedge \cdots \wedge f_n
\end{equation*}
for any \(f_1 ,\dots, f_n \in \cg\).
We also introduce weights on \(\wedge^\bullet \cg\) by declaring
that \(f_1 \wedge \cdots \wedge f_n\) is of \textit{weight} \(k_1 + \cdots + k_n\)
if \(f_1 \in \cg(k_1), \dots, f_n \in \cg(k_n)\).
We denote by \(\left( \wedge^n \cg \right)_w\)
the subspace of \(\wedge^n \cg\) generated by elements of weight \(w\).
Similarly, we introduce a notation \(\left( \wedge^n \cgplus \right)_w \subset \wedge^n \cgplus\).

\par Since the bracket map \(\left[, \right]\) is \(\Symp\)-equivariant,
so is the differential \(\partial_\bullet\).
The weight on \(\wedge^\bullet \cg\) defined above
is also preserved by \(\partial_\bullet\).
Therefore for \(n \geq 1\),
we can decompose the chain space \(\wedge^n \cgplus\) by weight
and moreover into \(\Symp\)-irreducible components
because each \(\left( \wedge^n \cgplus \right)_w\) is
a finite dimensional \(\Symp\)-module.

\par Hereafter we concentrate on \(\cgplus\) rather than \(\cg\).

\begin{remark}
We regard the vector space \(\wedge^\bullet \cgplus\) as
the quotient space of \(\bigoplus_{n \geq 1} \left( \cgplus \right)^{\tensor n}\).
Let
\(\pi \colon \bigoplus_{n \geq 1} \left( \cgplus \right)^{\tensor n} \twoheadrightarrow \wedge^\bullet \cgplus\)
be the quotient map.
We write
\begin{equation*}
    \cg(k_1) \wedge \cdots \wedge \cg(k_n) \defeq
    \pi (\cg(k_1) \tensor \cdots \tensor \cg(k_n))
\end{equation*}
for \(n \geq 2\).
We easily see that
\(\cg(k) \wedge \cg(l) = \cg(l) \wedge \cg(k) \isom \cg(k) \tensor \cg(l)\)
if \(k,l \geq 1\) and \(k \neq l\).
This notation might be unusual, however we use it for simplicity.
For example,
\begin{equation}
    \begin{aligned}
        \cg(2) \wedge \cg(1) \wedge \cg(1) &{}=
        \left(
            \begin{array}{c}
                \text{The vector subspace of } \wedge^3 \cgplus \text{ generated by } \\
                \text{the elements of the form } f_1 \wedge f_2 \wedge f_3 \\
                \text{for } f_1 \in \cg(2) ,\ f_2, f_3 \in \cg(1)
            \end{array}
        \right) \\
        &{}\isom \cg(2) \tensor \left( \wedge^2 \cg(1) \right).
    \end{aligned}
\end{equation}
\end{remark}

\par We use the following natural way
to regard \(\cg(k)\) and \(\cg(k) \wedge \cg(l)\)
as \(\Symp\)-submodules of \(H^{\tensor n}\)s.
Consider an injective map
\begin{equation}

\begin{aligned}
    \includegraphics{cgplus_defmap_injectivemap-crop.pdf}
\end{aligned}
\end{equation}
for \(k \geq 1\).
We always see degree \(n\) polynomials of \(H\)
as elements in \(H^{\tensor n}\) by this map.
Then for \(k > l \geq 1\), we see
\(\cg(k) \wedge \cg(l) \subset \wedge^2 \cgplus\)
as a submodule of \(H^{\tensor (k+l+4)}\) by an injective map
\begin{equation}
    \begin{aligned}

        \includegraphics{cgplus_defmap_iotatensor-crop.pdf}
    \end{aligned}
\end{equation}
and for \(k \geq 1\),
\begin{equation}
    \begin{aligned}

        \includegraphics{cgplus_defmap_iotawedge-crop.pdf}
    \end{aligned}
\end{equation}
using the same notation \(\iota\).
For \(m,n \geq 1\),
\(\alpha \in H^{\tensor m}\) and \(\beta \in H^{\tensor n}\),
define
\begin{equation}
    \alpha \wedge \beta \defeq
        \alpha \tensor \beta - \beta \tensor \alpha
        \in H^{\tensor (m+n)},
    \qquad
    \alpha \symmprod \beta \defeq
        \alpha \tensor \beta + \beta \tensor \alpha
        \in H^{\tensor (m+n)}.
\end{equation}

\ \\
\par Let us refer to \(H_1 (\cgplus)\) before we discuss \(H_2 (\cgplus)\).
\begin{proposition}\label{prop H1}
    \begin{itemize}
        \item[(1)] The map
            \(\partial_2 = [,] \colon (\wedge^2 \cgplus)_w \to (\wedge^1 \cgplus)_w = \cg(w)\)
            is surjective if \(g, w\geq 2\).
        \item[(2)] \(H_1 (\cgplus) = S^3 H\) if \(g \geq 2\).
    \end{itemize}
\end{proposition}
\begin{proof}
    For (1), we see that this map is zero or surjective
    because it is \(\Symp\)-equivariant
    and \(\cg(w)=S^{w+2} H\) is \(\Symp\)-irreducible.
    Now, we have
    \(\partial_2 (a_1^w a_g \wedge a_1^2 b_g) = [a_1^w a_g, a_1^2 b_g] = a_1^{w+2}\).
    This leads to surjectivity of this map.
    \par (2) follows from (1) and \(\wedge^2 \cgplus = \bigoplus_{w \geq 2} (\wedge^2 \cgplus)_w\).
\end{proof}
Contrary to this case, the chain space \((\wedge^2 \cgplus)_w\)
is not \(\Symp\)-irreducible for general \(w\).
In order to apply the similar method of confirming all the cycles are boundaries,
we must know how the chain space \((\wedge^2 \cgplus)_w\)
decomposes into \(\Symp\)-irreducible components.

config: arxiv


text:
\begin{document}

\title[Averaging with a time-dependent perturbation]{Averaging with a time-dependent perturbation parameter}
\author{\textsc{David Fajman}}
\author{\textsc{Gernot Hei}\ss\textsc{el}}
\address[\textsc{David Fajman} and \textsc{Gernot Hei}\ss\textsc{el}]{University of Vienna, Department of Physics, Strudlhofgasse 4, 1090 Vienna, Austria}
\email{david.fajman@univie.ac.at}
\email{gernot.heissel@univie.ac.at}
\author{\textsc{Jin Woo Jang}}
\address[\textsc{Jin Woo Jang}]{University of Bonn, Institute for Applied Mathematics, Endenicher Allee 60, Bonn, Germany}
\email{jangjinw@iam.uni-bonn.de}

\begin{abstract}
Motivated by recent problems in mathematical cosmology, in which temporal averaging methods are applied in order to analyse the future asymptotics of models which exhibit oscillatory behaviour, we provide a theorem concerning the large-time behaviour for solutions of a general class of systems. We thus propose our result to be applicable to a wide range of problems in spatially homogenous cosmology with oscillatory behaviour. Mathematically the theorem builds up on the standard theory of averaging in non-linear dynamical systems.
\end{abstract}

\let\thefootnote\relax\footnotetext{2010 \textit{Mathematics Subject Classification.} Primary 	34C15, 83F05,  34C29,
81Q05.

	\textit{Key words and phrases.} Spatially homogeneous cosmology, Nonlinear oscillations, Klein-Gordon equation, Averaging method, Time-dependent perturbation parameter.}

\thanks{The authors would like to thank Piotr Bizo\'n for discussions. The first and second authors acknowledge support of the Austrian Science Fund (FWF) through the Project \emph{Geometric transport equations and the non-vacuum Einstein flow} (P 29900-N27). The second and third authors gratefully acknowledge the support of the Hausdorff Research Institute for Mathematics
(Bonn), through the Junior Trimester Program on Kinetic Theory. The third author was supported by CRC 1060 \textit{The mathematics of emergent effects} at the
University of Bonn funded through the German Science Foundation (DFG)}

\newcommand{\eqdef }{\overset{\mbox{\tiny{def}}}{=}}

\addtocounter{footnote}{-1}\let\thefootnote\svthefootnote

\thispagestyle{empty}

\maketitle

\section{Introduction}\label{S: intro}

\subsection{Averaging}

The theory of averaging in nonlinear dynamical systems has its origins in the 18th century, when perturbative methods for differential equations became important to test the predictions of Newtons theory of gravitation for celestial mechanics against observational data. The motion of a planet is described by a Keplerian ellipse only in good approximation, since it represents the solution to the two-body problem planet-sun. The actual trajectory of a planet is however also influenced by all other celestial bodies such as their own satelites. The influence of these bodies results in (generally periodic) perturbations from Keplerian motion. The necessity to take into account these effects due to the advances in accuracy of astronomical observations lead to the development of perturbative methods for differential equations, and eventually to the birth of averaging methods in nonlinear dynamical systems; cf.~\cite[\textsc{App}~A]{SandersEtAl2007}.

Nowadays these methods represent widely used tools in science and engineering, backed by an increasingly well developed mathematical theory; cf. e.g.~\cite{SandersEtAl2007}. The core idea is to approximate a perturbed full system by an unperturbed averaged system. The latter is thereby obtained from the former by an integration over a scale associated with the perturbation; e.g. a period in the case of a periodic perturbation. The basic theorems then quantify the accuracy of the averaged approximation in terms of error estimates; cf. e.g.~\cite[p.~31, \textsc{Thm}~2.8.1 or p.~101, \textsc{Thm}~5.5.1]{SandersEtAl2007}. Typically the perturbation is controlled by a constant parameter, and the error estimates together with their range of validity are expressed in terms of orders of it.

With respect to modern gravitational physics, averaging often refers to spatial averaging in inhomogenous cosmology, a subject studied in particular with regards to the backreaction source terms appearing in the averaged equations, which potentially provide a natural explanation for dark energy; cf. e.g. \cite{BuchertEhlers1997, Buchert2008}. Here we however consider \emph{spatially homoegenous} cosmology and \emph{temporal} averaging over oscillatory perturbations. The latter occur for instance when the spacetime is populated by matter which satisfies a wave equation, such as scalar fields (Klein-Gordon equation) or Yang-Mills fields; cf. e.g. \cite{FaimanWyatt2021, FajmanEtAl2020, AlhoUggla2015, AlhoEtAl2015, AlhoEtAl2020}. Yet such oscillatory behaviour can even be present in perfect-fluid cosmologies, as for instance demonstrated by the cases which exhibit a phenomenon called Weyl curvature dominance or asymptotic self-similarity breaking; cf. \cite{Wainwright1999, Wainwright2000, Horwood2003}.

\subsection{Motivation}

A focus point of spatially homogenous cosmology is the investigation of the past and future asymptotics of these systems by analytical means, or, expressed in the language of the analysis of PDEs, their stability; cf. \cite{WainwrightEllis1997, Rendall2008, Coley2003, MR3186493, RyanShepley1975,PhysRevD.85.083511, MR3549249, MR3907044,MR2425407} for general reference and also \cite{AnderssonFajman, MR868737} for the stability of some prominent spacetimes.  In other words, one is interested in how the cosmological models evolved in a distant past (e.g. towards a big-bang singularity) or distant future (e.g. towards a big-crunch singularity or in a forever expanding state).  The present work is motivated by~\cite{FajmanEtAl2020} which offers an analysis of the future asymptotics of LRS Bianchi~III cosmologies with a massive scalar field. Type III thereby refers to the Bianchi type of the spatial homogeneity while LRS (locally rotationally symmetric) refers to an additional spatial rotation symmetry; cf.~\cite{RyanShepley1975, WainwrightEllis1997}. The scalar field $\phi=\phi(t)$ satisfies the Klein-Gordon equation
\begin{align}\label{E: KG}
\ddot\phi+\phi=H(-3\dot\phi) ,
\end{align}
in which the mass is set to $1$ and where a dot denotes differentiation with respect to coordinate time $t$. $H=H(t)$ is the Hubble scalar---a measure of the overall isotropic rate of spatial expansion; cf.~\cite{Rendall2008, WainwrightEllis1997}. Evidently, this cosmology is an example for one of the systems mentioned above, in which oscillations enter via a matter wave equation.

From the point of view of~\eqref{E: KG}, $H$ represents a time-dependent damping of the harmonically oscillating matter field $\phi$. Its time dependence is governed by the coupling to the Einstein equations, which can be expressed as a set of constraints and first-order evolution equations for $H$ and other geometric quantities.\footnote{In the present example of LRS Bianchi type~III the only constraint is the Hamiltonian constraint and the only evolution equation in addition to the Raychaudhuri equation is one for the (Hubble normalised) shear; cf.~\cite{FajmanEtAl2020}. The latter has only one independent component in this case.} From the explicit form of the Raychaudhuri equation (i.e. the evolution equation for $H$), it is easy to show that (for an initially expanding universe) $H$ is strictly decreasing and furthermore
\begin{align}\label{E: H to zero}
\lim_{t\to\infty} H(t) = 0;
\end{align}
cf. \cite[\textsc{Lemma}~3]{FajmanEtAl2020}.\footnote{Another important feature of the problem is, that the Raychaudhuri equation is proportional to $H^2$, while the other evolution equations are proportional to $H$.} This guaranteed smallness of $H$ for large times was motivation in~\cite{FajmanEtAl2020} to associate~\eqref{E: KG} with the simpler problem
\begin{align}\label{E: VdP1}
\ddot\phi+\phi=\epsilon\,(-3\dot\phi)
\end{align}
in which $H(t)$ is replaced by a small perturbation constant $\epsilon>0$. For problems of this kind the theory of (periodic) averaging in non-linear dynamical systems offers powerful theorems; cf.~\cite{SandersEtAl2007}. For~\eqref{E: VdP1} these provide error estimates between $\phi$ and the solution of an associated averaged equation in terms of orders of the perturbation constant~$\epsilon$. In the light of~\eqref{E: H to zero} and the association of~\eqref{E: KG} with~\eqref{E: VdP1} it then seems natural to conjecture that the error between the solution of~\eqref{E: KG} and its associated time averaged equation decays and goes to zero. If this holds, then one can obtain the future asymptotics of the full oscillatory system by analysing the much simpler averaged system, which is free of oscillations.

The above paragraph summarises the core idea entertained in~\cite{FajmanEtAl2020}, in which a respective conjecture is formulated and supported. A closely related approach has also been taken independently in~\cite{AlhoUggla2015, AlhoEtAl2015, AlhoEtAl2020}, where \cite{AlhoEtAl2015, AlhoEtAl2020} even provide theorems concerning the future asymptotics. However, the applicability of these theorems is limited to the concrete cosmological models and matter models discussed in these sources, as is also the conjecture of~\cite{FajmanEtAl2020}. Yet oscillatory problems of this kind are a reoccurring theme in mathematical cosmology, at least (but not only) w.~r.~t. matter fields which satisfy a wave equation. Thus, instead of having to prove a separate theorem for each symmetry class and matter model, one would ideally like to have only a short number of theorems available which are however applicable to a wider range of models. With the present work we aim to make a contribution in this spirit. With \textsc{Theorem}~\ref{theorem local} we provide a rigorous result concerning the large-time behaviour of a wide class of cosmological models abstracted from the system of~\cite{FajmanEtAl2020}, i.e. from the LRS Bianchi~III Einstein-Klein-Gordon system. The statement of the theorem is stronger than that of the theorems of \cite{AlhoEtAl2015, AlhoEtAl2020} and the conjecture of~\cite{FajmanEtAl2020} w. r. t. its generality and range of applicability. It is weaker in that it does not provide the solutions in the limit for $t$ to infinity, but rather gives error estimates for finite time intervals. However these time intervals increase, and the estimated errors decrease, with the time of truncation from the full to the averaged system. Since the latter can be chosen freely, our \textsc{Theorem}~\ref{theorem local} should be a versatile and useful tool to investigate the future asymptotics of cosmologies with oscillatory behaviour by analytical means.

Lastly, we note that while we do not explicitly exclude a cosmological constant for the class of models which satisfy the premises of our theorem, it is likely that the requirement that $H\rightarrow 0$ does not hold in these cases. The importance of this decay of $H$ for controlling the error between the full and the averaged solutions is also why we restrict to the future asymptotics here, since towards the past, $H$ and thus the error is increasing in the motivating model of~\cite{FajmanEtAl2020}.

\subsection{Outline}

The paper is structured as follows. In \textsc{Section}~\ref{S: background} we give some background on periodic averaging in non-linear dynamical systems, and we relate our case to the standard problems of this theory. We then formulate our result in \textsc{Section}~\ref{S: Results}: a theorem on the large-time behaviour of a general class of systems, tailored to problems in spatially homogenous cosmology with oscillatory behaviour. We provide the proof in \textsc{Section}~\ref{S: Proofs} and conclude with a summary in~\textsc{Section}~\ref{S: Summary}.

\section{Background and preparation}\label{S: background}

After a word on notation in \textsc{Section}~\ref{SS: Notation} we give some background on averaging theory. In \textsc{Section}~\ref{SS: VdP} we briefly discuss the Van der Pol equations as prototypes for periodic averaging which are closely associated to the Klein-Gordon equation~\eqref{E: KG} of our motivating problem. We then give the standard form of systems in periodic averaging. After a brief discussion of averaging methods and theorems for these systems in \textsc{Section}~\ref{SS: Methods}, we then make the association with our problems of interest in mathematical cosmology, in which we deal with a strictly decreasing perturbation function rather than a perturbation constant. For these we conjecture the existence of respective averaging statements. To this end, we define the cornerstones for our considered class of systems in \textsc{Section}~\ref{SS: class of systems}. In particular we define a required standard form and asymptotics of the perturbation function for our wide class of systems of consideration, but also mild restrictions on the right hand side functions of the systems.

\subsection{Notation}\label{SS: Notation}
Throughout the paper bold case quantities denote vectors whereas quantities in italics denote scalars. We denote the $0^{th}$ to $i^{th}$-order coefficients and the coefficient of the $i^{th}$-order remainder term in the Taylor expansion of $f(\textbf{x},t,\epsilon)$ near $\epsilon=0$ as $f^0(\textbf{x},t),...,f^i(\textbf{x},t)$ and $f^{[i+1]}(\textbf{x},t,\epsilon)$, respectively, such that
$$
f(\textbf{x},t,\epsilon)=\sum_{j=0}^i f^j(\textbf{x},t)\epsilon^j+f^{[i+1]}(\textbf{x},t,\epsilon)\epsilon^{i+1};
$$
cf. also~\cite[\textsc{p}~13, \textsc{Notation}~1.5.2]{SandersEtAl2007}. The norm $\|\cdot\|$ denotes the standard discrete $\ell^1$ norm $$\|\mathbf{u}\|\eqdef \sum_{i=1}^n|u_i|$$ for $\mathbf{u}\in \mathbb{R}^n$. We also denote as $L^\infty_{\textbf{x},t}$ (or simply $L^\infty$) the standard $L^\infty$ space in both $\textbf{x}$ and $t$ variables with the corresponding norm defined as
$$
\|f\|_{L^\infty_{\textbf{x},t}} \eqdef \sup_{\textbf{x},t}|f(\textbf{x},t)|.
$$

\subsection{Van der Pol equations as prototypes of periodic averaging}\label{SS: VdP}

For our purpose it is instructive to compare~\eqref{E: KG} to a general class of Van der Pol equations
\begin{align}\label{E: VdP2}
\ddot\phi+\phi=\epsilon\,g(\phi,\dot\phi) ,
\end{align}
with a sufficiently smooth function $g$ and where $\epsilon$ is a constant, in contrast to $H=H(t)$ in~\eqref{E: KG}; cf.~\cite[\textsc{Sec}~2.2]{SandersEtAl2007}. For small $\epsilon$, \eqref{E: VdP2} represents a perturbed harmonic oscillator which suggests a parametrisation of the solution by variation of arbitrary constants in terms of an amplitude $r=r(t)$ and phase $\varphi=\varphi(t)$:
\begin{align}\label{E: trafo}
\phi = r \sin(t-\varphi) \qquad\text{and}\qquad
\dot\phi = r \cos(t-\varphi) .
\end{align}
This transforms~\eqref{E: VdP2} into two first-order equations of the form $[\dot r,\dot\varphi]^\mathrm T = \epsilon\,\mathbf f(r,\varphi,t)$, with $\mathbf f$ $2\pi$-periodic in $t$. From~\cite[\textsc{Chap}~2]{SandersEtAl2007} this is a periodic averaging problem of the more general standard form
\begin{align}\label{E: standard form}
\dot{\mathbf x} = \epsilon\,\mathbf f^1(\mathbf x, t) + \epsilon^2\, \mathbf f^{[2]}(\mathbf x, t,\epsilon) , \quad
\mathbf x(0)=\mathbf a ,
\end{align}
with $\mathbf f$ $T$-periodic in $t$.

\subsection{Averaging methods and conjecture}\label{SS: Methods}

For this kind of problem the theory of averaging provides theorems which estimate the error between the solution $\mathbf x(t)$ of~\eqref{E: standard form} and a $\mathbf z(t)$ which solves the corresponding averaged equation
\begin{align}\notag
\dot{\mathbf z} = \epsilon\,\overline{\mathbf f}^1(\mathbf z), \quad
\mathbf z(0)=\mathbf a, \quad\text{where}\quad
\overline{\mathbf f}^1(\mathbf z) := \frac{1}{T}\int_0^T \mathbf f^1(\mathbf z, s)\,\mathbf ds .
\end{align}
For instance \cite[\textsc{p}~31, \textsc{Thm}~2.8.1]{SandersEtAl2007} states that the error between $\mathbf x(t)$ and $\mathbf z(t)$ is $O(\epsilon)$ on timescales of $O(\epsilon^{-1})$; cf. also \textsc{Lemma}~\ref{yzmain} below. Note that both the error estimate and its domain of validity are given in terms of orders of powers of the perturbation constant $\epsilon$.
\begin{remark}
The averaging can be interpreted as a time dependent coordinate
transform, a nonlinear transform for nonlinear problems, and many of the results apply to
deterministic time-dependent perturbations; cf. \cite[Chapter 3]{MR3244318} and \cite[Section 5]{MR2585314}.
\end{remark}

The similarity between~\eqref{E: KG} and~\eqref{E: VdP2} suggests to treat the former as a perturbed harmonic oscillator as well, and to apply averaging in an analogous way. Care has to be taken however since in contrast to $\epsilon$, $H$ is time-dependent and itself subject to an evolution equation of the coupled system. If valid, then a striking feature of such an approach is the possibility to exploit the fact~\eqref{E: H to zero} that $H$ is strictly decreasing and going to zero. In~\eqref{E: KG} $H$ takes the role of the perturbation parameter $\epsilon$ in~\eqref{E: VdP2}, which controls the error and the domain of validity of the error estimate. Hence, with strictly decreasing $H$ the error should decrease as well. One can obtain the information about the large-time behaviour of the more complicated full system via an analysis of the simpler averaged system.

\subsection{Our general class of systems}\label{SS: class of systems}

This ideas just summarised were already entertained in~\cite{FajmanEtAl2020} at the example of LRS Bianchi~III. Likewise~\cite{AlhoEtAl2015, AlhoUggla2015, AlhoEtAl2020} took a closely related approach for the concrete models of these sources. \cite{AlhoEtAl2015, AlhoEtAl2020} thereby give theorems concerning the respective future asymptotics.

The purpose of the present work is to formulate a theorem concerning the large-time behaviour which is applicable to a wide range of problems in spatially homogeneous cosmology. We therefore define our problem as general as possible while also as restrictive as necessary to exploit certain key features.

The systems that we consider in the following have the general form
\begin{equation}\label{ODEsystem}
\begin{bmatrix}
\dot{H}\\ \dot{\bfx}
\end{bmatrix}
=H\bfF^1(\bfx,t)+H^2\bfF^{[2]}(\bfx, t)=H
\begin{bmatrix}
0\\ \bff^1(\bfx,t)
\end{bmatrix}+H^2
\begin{bmatrix}
f^{[2]}(\bfx,t)\\ 0
\end{bmatrix},
\end{equation}
where $H=H(t)$ is positive, is strictly decreasing in $t$, and
\begin{equation}\label{E: H to zero 2}
\lim_{t\rightarrow \infty}H(t)=0.
\end{equation}
\begin{example}\label{Ex: 1}
Our prime example for a system which fits the general form \eqref{ODEsystem} is the LRS Bianchi type~III Einstein-Klein-Gordon system introduced in \cite{FajmanEtAl2020}, which is given by
\begin{align}
\dot{H}&=H^2[-(1+q)], \label{E: BIII H}\\
\dot{\Sigma}_+&=H[-(2-q)\Sigma_++1-\Sigma^2_+-\Omega], \label{E: BIII S}\\
\ddot{\phi}+\phi&=H[-3\dot{\phi}], \label{E: BIII KG}
\end{align}
with the \textit{deceleration parameter} $$q=2\Sigma^2_++\frac{1}{6H^2}(2\dot{\phi}^2-\phi^2).$$
Here $H$ is the \textit{Hubble scalar}, and we see that in this example the first component of~\eqref{ODEsystem} is the the Raychaudhuri equation~\eqref{E: BIII H}. $\phi$ is the scalar field satisfying the Klein-Gordon equation~\eqref{E: BIII KG}. $\Sigma_+$ is the only independent component of the Hubble normalized shear tensor, and $\Omega$ is the Hubble normalized energy density. The \textit{Hamiltonian constraint} $1-\Sigma_+^2-\Omega>0$ and the \textit{non-vacuum} condition $\Omega>0$ give the uniform bounds of $\Sigma_+\in (-1,1)$ and $\Omega \in (0,1)$. The Klein-Gordon equation~\eqref{E: BIII KG} has the form of a Van der Pol equation. Via an amplitude-phase transformation~\eqref{E: trafo} it can be formulated as the following two first-order equations \cite[Section 2.2]{FajmanEtAl2020}:
\begin{equation}\label{KG eq}
\begin{split}
\dot{\Omega}&=H[2\Omega(1+q-3\cos(t-\varphi)^2],\\
\dot{\varphi}&=H[-3\sin(t-\varphi)\cos(t-\varphi)],
\end{split}
\end{equation}which also implies $$q=2\Sigma_+^2+\Omega(3\cos(t-\varphi)^2-1).$$  Then $H$ and $\bfx \eqdef (\Sigma_+, \Omega, \varphi)^\top$ satisfy the system \eqref{ODEsystem}; i.e., the components of $\dot{\mathbf x}$ stand for both the remaining Einstein evolution equation \eqref{E: BIII S} and the two first-order Klein-Gordon equations \eqref{KG eq}. Note that $\bff^1$ and $f^{[2]}$ are smooth with respect to $\bfx$. Also, both $\bff^1$ and $f^{[2]}$ are in $L^\infty_{\bfx, t}.$
\end{example}
\begin{example}\label{Ex: 2}
In addition to LRS Bianchi~III also the Einstein Klein-Gordon systems of the spatially homogenous, LRS cosmologies of Bianchi types~I, II, VIII and IX as well as Kantowski-Sachs can be brought into the form~\eqref{ODEsystem}. The reader can verify that by starting from the formulation of~\cite{RendallUggla2000}, however using a Klein-Gordon field as matter model, and then performing the usual amplitude-phase transformation~\eqref{E: trafo}. For type~IX it is however expected that the condition~\eqref{E: H to zero 2} fails, since it is a closed cosmology, and as such it tends to re-collapse; cf.~\cite{WainwrightEllis1997}.
\end{example}
\begin{example}\label{Ex: 3}
The systems treated in~\cite{AlhoUggla2015, AlhoEtAl2015, AlhoEtAl2020} are also examples of the form~\eqref{ODEsystem}. In \cite{AlhoEtAl2020} the matter source is a Yang-Mills field, which demonstrates that averaging techniques in mathematical cosmology are by no means limited to Einstein-Klein-Gordon cosmology.
\end{example}

\section{Results}\label{S: Results}

We are now ready to formulate our theorem.

\subsection{Large-time behaviour}\label{SS: local-in-time}

Consider the system~\eqref{ODEsystem} with positive  and strictly decreasing $H=H(t)$, and with $\lim_{t\rightarrow \infty}H(t)=0$. We then obtain the following large-time behaviour on the system \eqref{ODEsystem}:
\begin{theorem}[Large-time behaviour]\label{theorem local}Suppose that $H=H(t)>0$ is strictly decreasing in $t$, and $$\lim_{t\rightarrow \infty}H(t)=0.$$ Fix any $\varepsilon>0$ with $\varepsilon<H(0)$ and define $t_*>0$ such that $\varepsilon=H(t_*)$.   Suppose that
$$\|\bff^1\|_{L^\infty_{\bfx,t}},\ \|f^{[2]}\|_{L^\infty_{\bfx,t}} <\infty,$$ and that $\bff^1(\bfx,t)$ is Lipschitz continuous and $f^{[2]}$ is continuous with respect to $\bfx$ for all $t\ge t_*$. Also, assume that $\bff^1$ and $f^{[2]}$ are $T$-periodic for some $T>0$. Then for all $t>t_*$ with $t=t_*+O(H(t_*)^{-\gamma})$ for any given $\gamma\in(0,1)$, we have
\begin{equation}\label{local statement}\bfx(t)-\bfz(t)=O(H(t_*)^{\min\{1,2-2\gamma\}}),\end{equation} where $\bfx$ is the solution of the system \eqref{ODEsystem} with the initial condition $\bfx(0)=\bfx_0$ and
$\bfz(t)$ is the solution of the averaged equation $$\dot{\bfz}=H(t_*)\bar{\bff}^1(\bfz), \text{ for } t>t_*$$ with the initial condition $\bfz(t_*)=\bfx(t_*)$ where the average $\bar{\bff}^1$ is defined as
$$\bar{\bff}^1(\bfz )=\frac{1}{T}\int_{t_*}^{t_*+T} \bff^1(\bfz,s)ds.$$
\end{theorem}
\begin{proof}
Cf. sections~\ref{S: firstorder} and~\ref{S: periodic averaging}.
\end{proof}
\begin{remark}

Example~\ref{Ex: 1} of \textsc{Section}~\ref{SS: class of systems} satisfies the assumptions of \textsc{Theorem}~\ref{theorem local}. We expect the same to hold for examples~\ref{Ex: 2} and~\ref{Ex: 3}, with the exception of LRS Bianchi~IX which as a closed cosmology is expected to re-collapse and thus to fail the assumption~\eqref{E: H to zero 2}; cf.~\cite{WainwrightEllis1997}.
\end{remark}
\begin{remark}We emphasise the following key properties of~\textsc{Theorem}~\ref{theorem local}, which are connected to the choice of truncation time $t_*$ to the first order system:
\begin{itemize}
\item The longer one waits with truncation, the better the estimate gets.
\item Furthermore, the longer one waits with truncation, for the longer is the estimate valid.
\end{itemize}
These are attractive properties for the analysis of the large-time behaviour of such systems. To the best of our knowledge, \textsc{Theorem}~\ref{theorem local} is the first of its kind, which gives the large-time behaviour for a general class of systems, with a time dependent perturbation parameter, and which naturally appear in spatially homogenous cosmology with oscillating matter models.
\end{remark}

\section{Proofs}\label{S: Proofs}
The proof of \textsc{Theorem}~\ref{theorem local} involves the first-order approximation of the system, the error estimates, and the use of periodic averaging. In \textsc{Section}~\ref{S: firstorder}, we introduce the first-order approximation of our system and prove the main ingredient of our theorem---\textsc{Proposition}~\ref{xymain}. In \textsc{Section}~\ref{S: periodic averaging} we apply a standard theorem of periodic averaging which closes the proof.

\subsection{First-order approximation and error estimates}\label{S: firstorder}
In order to approximate the solution $\bfx(t)$ of the system \eqref{ODEsystem}, we consider the following truncated first-order system; for $t>t_*$ with some given time $t_*$, we consider
\begin{equation}\label{firstordersystem}
\begin{bmatrix}
\dot{\mathcal{H}}\\ \dot{\bfy}
\end{bmatrix}
=\mathcal{H}(t)
\begin{bmatrix}
0\\ \bff^1(\bfy,t)
\end{bmatrix},
\end{equation}
where $\mathcal{H}(t_*)=H(t_*)$ and $\bfy(t_*)=\bfx(t_*)$; i.e., the initial data coincide. Then since $\dot{\mathcal{H}}=0,$ we have
$\mathcal{H}(t)\equiv H(t_*).$ We hereby denote $H_*\eqdef H(t_*).$
Then we introduce the following proposition on the error estimates for the first-order approximation:
\begin{proposition}\label{xymain}Fix any $t_*>0$.
Suppose that $\bfx$ and $\bfy$ are the solutions of \eqref{ODEsystem} and \eqref{firstordersystem}, respectively, with the initial conditions $\bfx(0)=\bfx_0$ and $\bfy(t_*)=\bfx(t_*)$.
Suppose further that $H=H(t)>0$ is strictly decreasing in $t$, and $$\lim_{t\rightarrow \infty}H(t)=0.$$ In addition, suppose that
$$\|\bff^1\|_{L^\infty_{\bfx,t}},\ \|f^{[2]}\|_{L^\infty_{\bfx,t}} <\infty,$$ and that $\bff^1(\bfx,t)$ is Lipschitz continuous in $\bfx$ for all $t\ge t_*$. Then for all $t>t_*$ with $t=t_*+O(H(t_*)^{-\gamma})$ for some $\gamma\in(0,1)$, we have
$$\bfx(t)-\bfy(t)=O(H(t_*)^{2-2\gamma}).$$
\end{proposition}
\begin{proof}
Since $\dot{\mathcal{H}}=0$ in \eqref{firstordersystem}, we have $\mathcal{H}(t)=H_*,$ for all $t\ge t_*.$ Now, we subtract the equation for $\bfy$ in \eqref{firstordersystem} from that for $\bfx$ in \eqref{ODEsystem} to obtain that
$$\dot{\bfx}-\dot{\bfy} = H(t)\bff^1(\bfx,t)-H_* \bff^1(\bfy,t).$$ By integrating this identity, we have
\begin{equation}\label{eq 1} \bfx(t)-\bfy(t) = \int_{t_*}^t \left(H(\tau)\bff^1(\bfx,\tau)-H_* \bff^1(\bfy,\tau)\right)d\tau,\end{equation} by the initial condition $\bfy(t_*)=\bfx(t_*).$
Now we consider the mean-value theorem for $H(\tau)$ and observe that, for some $t_c=t_c(\tau) \in (t_*,\tau)$, we have
$$H(\tau)-H_*=\dot{H}(t_c) (\tau-t_*),\ \text{ for } \tau \in (t_*,t).$$
Then \eqref{eq 1} implies that
\begin{multline*}\bfx(t)-\bfy(t) = \int_{t_*}^t \left((H_*+\dot{H}(t_c) (\tau-t_*))\bff^1(\bfx,\tau)-H_* \bff^1(\bfy,\tau)\right)d\tau\\
= H_*\int_{t_*}^t \left(\bff^1(\bfx,\tau)-\bff^1(\bfy,\tau)\right)d\tau
+\int_{t_*}^t \dot{H}(t_c) (\tau-t_*)\bff^1(\bfx,\tau)d\tau\\
=H_*\int_{t_*}^t \left(\bff^1(\bfx,\tau)-\bff^1(\bfy,\tau)\right)d\tau
+\int_{t_*}^t H^2(t_c) f^{[2]}(\bfx,t_c) (\tau-t_*)\bff^1(\bfx,\tau)d\tau,
\end{multline*}
where the last line is by the first equation on $\dot{H}$ in the system \eqref{ODEsystem}.
Therefore, we have
\begin{multline*}
\|\bfx(t)-\bfy(t)\| \\
\le H_*\int_{t_*}^t \left\|\bff^1(\bfx,\tau)-\bff^1(\bfy,\tau)\right\|d\tau
+\int_{t_*}^t H^2(t_c) |f^{[2]}(\bfx,t_c) | (\tau-t_*) \|\bff^1(\bfx,\tau)\|d\tau\\
\le H_*\int_{t_*}^t \left\|\bff^1(\bfx,\tau)-\bff^1(\bfy,\tau)\right\|d\tau
+\frac{(t-t_*)^2}{2}H_*^2\|f^{[2]}\|_{L^\infty}\|\bff^1\|_{L^\infty}\\
\le H_* c_L\int_{t_*}^t \left\|\bfx(\tau)-\bfy(\tau)\right\|d\tau
+\frac{(t-t_*)^2}{2}H_*^2\|f^{[2]}\|_{L^\infty}\|\bff^1\|_{L^\infty},
\end{multline*} because $H(\cdot)$ is decreasing,
where $c_L$ is the Lipschitz constant and the third inequality is by the Lipschitz continuity of $\bff^1.$ Now we apply the Gr\"onwall inequality and obtain
$$\|\bfx(t)-\bfy(t)\| \le \frac{(t-t_*)^2}{2}H_*^2\|f^{[2]}\|_{L^\infty}\|\bff^1\|_{L^\infty}\exp\left(H_* c_L(t-t_*)\right).$$
Therefore, for the time-scale $t=t_*+O(H_*^{-\gamma})$ for any $\gamma \in(0,1)$ we have
$$\|\bfx(t)-\bfy(t)\| \le O(H_*^{2-2\gamma})\exp(O(H_*^{1-\gamma}))=O(H_*^{2-2\gamma}),$$ as long as $H_*$ is finite. This completes the proof.
\end{proof}

\subsection{Periodic averaging}\label{S: periodic averaging}

Equipped with the first-order approximation and the error estimates, the next ingredient is a result from periodic averaging theory.

First we remind ourselves of the periodic averaging problem in standard form~\eqref{E: standard form} which we discussed in \textsc{Section}~\ref{SS: VdP}. For convenience we rewrite it here for the special case $\mathbf f^{[2]}=0$ as
\begin{equation}\label{yeq}
\dot{\bfy}=\varepsilon\bff^1(\bfy,t),\ \bfy(0)=\bfy_0,\quad \text{for } t>0
\end{equation}
with $\bfy, \ \bff^1(\bfy,t)\in \mathbb{R}^n,$ where $\bff^1$ is $T$-periodic and $\varepsilon$ is a small parameter.
Now we note that the equation for $\bfy$ of our first order system~\eqref{firstordersystem} is of this form with $\varepsilon=H_*$.
Then we fix a sufficiently large time $t_*>0$, and consider the first-order associated averaged system
\begin{equation}\label{zeq}\dot{\bfz}=\varepsilon\bar{\bff}^1(\bfz), \text{ for } t>t_*\end{equation} with the initial condition $\bfz(t_*)=\bfy(t_*)$ where $\varepsilon=H(t_*)$ and the average $\bar{\bff}^1$ is defined as
$$\bar{\bff}^1(\bfz )=\frac{1}{T}\int_{t_*}^{t_*+T} \bff^1(\bfz,s)ds.$$
Let $D\subset \mathbb{R}^n$ be a connected, bounded open set (with compact closure) containing the initial value $\bfy(t_*)=\bfz(t_*)$. Fix $L>0$ and $\varepsilon_0>0$ such that the solutions $\bfy(t,\varepsilon)$ and $\bfz(t,\varepsilon)$ with $0\le \varepsilon\le \varepsilon_0$ remain in $D$ for $t_*\le t\le t_*+L/\varepsilon.$ Then there is a basic result on the asymptotic error estimates:
\begin{lemma}[Theorem 2.8.1 of \cite{SandersEtAl2007}]\label{yzmain}
Let $\bff^1$ be Lipschitz continuous.  Also, assume that $\bff^1$ is $T$-periodic for some $T>0$. Let $\varepsilon_0$, $D$, $L$ be as above. Then there exists a constant $c>0$ such that the solutions $\bfy$ and $\bfz$ of \eqref{yeq} and \eqref{zeq} satisfy
$$\|\bfy(t,\varepsilon)-\bfz(t,\varepsilon)\|<c\varepsilon,$$for $0\le \varepsilon\le \varepsilon_0$ and $t_*\le t \le t_*+\frac{L}{\varepsilon},$ where $\|\cdot\|$ denotes the norm $$\|\mathbf{u}\|\eqdef \sum_{i=1}^n|u_i|$$ for $\mathbf{u}\in \mathbb{R}^n.$
\end{lemma}

Then Proposition \ref{xymain} and Lemma \ref{yzmain} together imply \textsc{Theorem}~\ref{theorem local} by letting $\varepsilon=H(t_*)$, using the triangle inequality and noting that the time scale $O(H(t_*)^{-\gamma})$ is smaller than $O(H(t_*)^{-1})$ for a sufficiently large time $t_*$ such that $H(t_*)=\varepsilon\ll1.$
This completes the proof of \textsc{Theorem}~\ref{theorem local}.

\section{Summary}\label{S: Summary}

Motivated by problems in spatially homogenous cosmology in \cite{FajmanEtAl2020, AlhoEtAl2015, AlhoUggla2015, AlhoEtAl2020} in which oscillations enter the system via a matter wave equation, we present a theorem regarding the large-time behaviour of such systems. The key idea is thereby to view these as perturbation problems, in which the Hubble scalar plays the role of the perturbation function---in analogy to the perturbation constant in standard problems of averaging in non-linear dynamical systems.

In standard averaging the perturbation constant estimates the error between the solutions of the full and a simpler time-averaged system. We show that for large times this is also the case for our class of systems and the Hubble scalar. Furthermore, we show that the estimated error decreases, and the time of validity of the estimate increases, with the time of truncation from the full to the averaged system, which can be chosen freely.

While related theorems have been given for the concrete examples discussed in~\cite{AlhoEtAl2015, AlhoEtAl2020}, our theorem is formulated in a very general fashion, with only mild restrictions on the systems of consideration. Since systems of this kind appear to be re-occurring in spatially homogenous cosmology with oscillatory behaviour (cf. \cite{FajmanEtAl2020, AlhoEtAl2015, AlhoUggla2015, AlhoEtAl2020}) it is desirable to craft a toolkit of theorems which is applicable to a wider class of models. To this end we offer our theorem, which to our best knowledge represents the first step in this direction.

\bibliographystyle{hplain}

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\end{document}


TITLE: Independence of Roots
QUESTION [5 upvotes]: Let $f \in F_p[x]$ be an irreducible polynomial of degree $n$, where $p$ is prime. Prove that its roots are independent over $F_p$.
EDIT: It was pointed in the answers that this claim is true only for some irreducible polynomials. My question can be changed to: prove that for any $n$, such irreducible polynomial of degree $n$ exists.
Remarks:

This is false in characteristic zero: take $x^2+1 \in \mathbb{R}[x]$.
I think it's true for any finite field (i.e., $p$ can be a prime power)

Let $\theta$ be some root of $f$.

$F_p[\theta] = F_{p^n}$, and $\{\theta^i\}_{i=0}^{n-1}$ is a basis.
By Frobenius Automorphism and by the identity $x^{p^i}-x = \prod_{a \in F_{p^i}} (x-a)$, one can show that $\{\theta^{p^i}\}_{i=0}^{n-1}$ are all the roots of $f$. It also shows that all of $f$'s roots are in the extension - they are powers of $\theta$.
Since $F_p[\theta]$ contains all $f$'s roots, the claim is equivalent to: $Span \{\theta^{p^i}\} = Span \{ \theta^i\}$

The motivation is that I saw this fact used in a cool combinatorics proof, where the bijection
$$\vec{a} \to \sum a_i \theta^{p^i}$$
from $(F_p)^n$ to $F_{p^n}$ was used, and I didn't understand why it is a bijection.
(For the curious: if we restrict this bijection to Lyndon words of length $n$ with alphabet $F_p$, and send $\sum a_i \theta^{p^i}$ to $\text{minimal polynomial of} \sum a_i \theta^{p^i}$, it becomes a bijection of Lyndon words of length $n$ to irreudcible polynomials of degree $n$)

REPLY [5 votes]: I'm afraid this is false. A simple case of a linearly dependent set of roots occurs, when they have trace zero.
For example the roots of the irreducible polynomial $p(x)=x^3+x+1\in\mathbf{Z}_2[x]$ sum up to zero:
$$
p(x)=(x-\alpha)(x-\alpha^2)(x-\alpha^4)=x^3+(\alpha+\alpha^2+\alpha^4)x^2+x+1,
$$
so $\alpha+\alpha^2+\alpha^4=0$ is a non-trivial linear dependency relation.
======================
Edit: OTOH it is not too difficult to show that $\theta$ can be chosen in such a way that we get linear independence. Look up normal basis. An argument that I happen to remember goes as follows. Let us view $F_{p^n}$ as a module $M$ over the polynomial ring $R=F_p[T]$, where we let the indeterminate $T$ act via the Frobenius automorphism: $T\cdot x=x^p$ for all $x\in M$. As $M$ is finite dimensional over $F_p$, it is obviously a torsion module, and as $R$ is a PID, we can write it as a direct sum cyclic $R$-modules in terms of the invariant factors:
$$
M=\bigoplus\sum_{i=1}^rR/p_i(T)R,
$$
where always $p_i(T)|p_{i+1}(T)$. I claim that here actually $r=1$ and $p_r(T)=q(T)=T^n-1.$
Anyway $p_r(T)=a_0+a_1T+\cdots+a_kT^k$ annihilates
all of $M$ (here $a_i\in F_P$ for $i=0,1,\ldots,k,$ at least one non-zero). It is well known that $q(T)$ annihilates $M$, so $k\le n$, because $p_r(T)$ is the lowest degree polynomial in $ann(M)$. But for all $x\in F_{p^n}$
$$
a_0x+a_1x^p+\cdots+a_kx^{p^k}=0,
$$
and this polynomial cannot have $p^n$ zeros unless $k=n$. Therefore $p_r(T)=q(T)$. But $\dim_{F_p}(R/p_i(T)R)=\deg p_i(T)$, so $n=\sum_i\deg p_i(T)$, and there is no room for any other invariant factor. Therefore
$$
M\cong_R R/q(T)R,
$$
and $M$ is a cyclic $R$-module. Now we can let $\theta$ be any generator of $M$ as an $R$-module. The elements $T^i\cdot\theta=\theta^{p^i},i=0,1,\ldots,n-1$ are then a basis. QED


TITLE: Simultaneous triangularization of two diagonal matrices and a symmetric matrix
QUESTION [2 upvotes]: I have the following quadratic eigenvalue problem:
$\det(\lambda^2M + \lambda D + J)=0$
where, M and D are both $n \times n$ real diagonal matrix with positive diagonal entries; $J$ is
1) a nearly symmetric n-by-n real matrix.
2) One of the eigenvalues of $J$ is guaranteed to be zero because each row sum of $J$ is guaranteed to be zero.($J$ looks similar to the so-called "Laplacian matrix" in Graph theory).
3) $J$ is positive semi definite,i.e. the other eigenvalues are guaranteed to be larger than zero.
I want to find a decoupled analytic solution for the expression of each eigenvalue. I thought about Schur decomposition (or, "Schur Unitary Triangularization"), i.e. whether $\exists \text{ a unitary } U $, such that $U^HJU = T$, where $T$ is a upper triangular matrix with diagonal elements as the same as the eigenvalues of $J$.
Previously, I took for granted that  $U^HMU$ and $U^HDU$ are still diagonal; however this can be wrong; thus, now I am thinking about whether there is any condition such that   $U^HMU$ and $U^HDU$ are also upper triangular.
If this can be achieved, then the determinant can be reduced to upper triangular structure and its value is:
    $\Pi (\lambda^2M_i + \lambda D_i + \mu_i) = 0$, where $\mu_i$ is the eigenvalue of $J$, and each $\lambda_i$ can be solved for from the well-known root formula of the quadratic equation.
I spent a whole day reading the literature and found so far some of the "best" "Simultaneously triangularization" sufficient conditions are from
"Simultaneous Triangularization"(Book) By Heydar Radjavi, Peter Rosenthal, pp. 19-21 Theorem 1.6.4 and 1.6.6. (Google provides a preview of this part.)
Although the theorem itself is a constructive one, it is not very straightforward to use.  And it looks like the book's content is hard for me, trying to use  "representation theory" and/or "Lie algebras" to prove things, which is far beyond my current level.
So, is there any other straightforward method to reach my goal?
What if I loosen the requirement for J, say J is strictly symmetric?
Note: Another approach I am trying to look at now is "the quadratic eigenvalue problem".

REPLY [1 votes]: The paper Triangularizing Quadratic Matrix Polynomials, Tisseur and Zaballa, 2014 shows how to transform a quadratic matrix polynomial into upper triangular form.
In general, though, what you are asking seems too strong; I don't think one can solve a QEP easily in that way. The usual routes (linearize the problem using the companion form, see e.g. http://dx.doi.org/10.1137/S0036144500381988) should work better. There are several canonical forms for pairs of matrices (QZ factorization, simultaneous diagonalization by congruence), but working with three matrices at the same time is way more difficult, and I don't think that the fact that $M$ and $D$ commute helps here.
Shameless advertising: with a co-author, we have recently studied a problem arising from Markov chain modelling in which the matrices have that exact same structure (diagonal-diagonal-generator of a cont.time Markov chain): see https://arxiv.org/abs/1605.01482. The only difference with respect to your setup is that $D$ has mixed signs (but maybe a change of variables in $\lambda$ can take care of that). I don't know what is the origin of your quadratic eigenvalue problem and what your computing goal is, but it might be related. What we do there is, essentially, finding the subspace spanned by the eigenvectors with $\lambda_i \leq 0$, or finding the general stable solution of the differential equation $M\ddot x + D \dot x + Jx=0$.


TITLE: Does ZF $\vdash$ Con(ZF) $\rightarrow$ Con(ZF+$\lnot$AC)?
QUESTION [2 upvotes]: Certainly ZFC $\vdash$ Con(ZF) $\rightarrow$ Con(ZF+$\lnot$AC), but the usual forcing argument to construct a model of ZF+$\lnot$AC seems to require choice to find a generic filter.

REPLY [3 votes]: This has a simple, almost cop-out, solution: absoluteness.
Note that the statement is in fact a first-order statement in the language of arithmetic. And such statement is provable from $\sf ZF$ if and only if it is provable from $\sf ZFC$.
Since you know the latter, you actually know the former.
Although in reality, you can just verify the following in $\sf ZF$:

The completeness theorem holds for countable languages without choice. So if $\sf ZFC$ is consistent, it has a model.
Forcing and symmetric extensions work without using the axiom of choice.

So if $\sf ZFC$ is consistent, it has a countable model, where forcing works as usual, symmetric extensions work as usual, and therefore if there is a model of $\sf ZF$ there is one where the axiom of choice fails.
It should also be added that the axiom of choice is not really needed in order to find a generic filter over a countable transitive model. The reason is that the model is in fact countable, so the elements from the forcing poset inside the model make a countable set externally. Using this fact, as well the fact that there are only countably many dense set in the model we can define a generic filter recursively without appealing to the axiom of choice.


TITLE: Faster-Than-Light Communication using Entangled Photons
QUESTION [4 upvotes]: Based on my understanding of the “Double-Slit Quantum Eraser Experiment”, documented here: http://grad.physics.sunysb.edu/~amarch/, it seems that Faster-Than-Light communication is possible.  Of course, this violates the “no-communication theorem”, so I assume it’s due to a misunderstanding on my part.
In the experiment, changing whether the polarization of the ‘p’ photon is erased or not changes whether or not interference patterns appear for the ‘s’ photon.  Suppose Alice and Bob generate many entangled photons and then travel to opposite sides of the galaxy, Alice taking the ‘p’ photons and Bob taking the ‘s’ photons.  At a predetermined time Bob releases his photons one by one into a double slit setup as in the experiment.  If he sees an interference pattern, he knows that Alice must have “erased” the polarization information on her corresponding ‘p’ photons, and if not that she has instead measured it.
Of course, even stranger situations are possible.  The fact that the ‘s’ photon in the experiment showed interference patterns when the ‘p’ photon was going to be erased, but hadn’t been yet, makes me think that photons can retrieve information from the future.  For instance, measuring the ‘s’ photon and detecting interference patterns or not, while the ‘p’ photon is bouncing around in some system, waiting to have its polarization be either erased or measured depending on the result of some future event, like an election result.  If the ‘s’ photon shows an interference pattern, then you would know the result of the electron (or know that forces beyond your control erase the information for some other reason).
I’d be very appreciative if you could clear this up for me.  My search online has not been fruitful, since the only similar questions I’ve seen just say something like “When you have entangled photons, measurements on one member of the pair cannot tell you anything about whether the other member has been measured or not.”, which seems to be contradicted by the above experiment.
Thanks!

REPLY [3 votes]: Something that confuse many people on this experiments is that you need to understand that just measuring the s photons will not show an interference pattern. The interference pattern appears only after you have selected the specific photons in s that correspond to the photons in p whose polarization has been erased. So you need this after-measurement in p information in order to see an interference pattern in s. And this information is classical, and cannot be sent faster than light.


TITLE: Determine when does a spherical curve closes.
QUESTION [0 upvotes]: So I'm plotting curves in three dimensions on my computer. I'm currently using equations in spherical coordinates to define them. These are some examples, using ISO convention ( r , θ , φ ):
$\big(a.sin(b.\theta), \theta, c\big)$
$\big(a.sin(b.\theta), \theta, \theta\big)$
$\big(a.abs(sin(b.\theta)), \theta, c.sin(d.\theta)\big)$
How could I find the closing angle for these variations?
Maybe a good starting point should be looking at 2D rhodoneas. According to wolfram, a two-dimensional rhodonea curve ($r = a.cos(\frac{r}{s}.\theta)$) closes at a polar angle of $\theta=\pi .s.p$, where $p=1$ if $r.s$ is odd and $p=2$ if $r.s$ is even. But I can't understand why that happens.
Ref:
https://mathworld.wolfram.com/Rose.html
I'm probably missing something or maybe I got something wrong, so if you have any thoughts on this I would love to know and discuss it further.

REPLY [1 votes]: If i understand correctly you are asking for the angle at which the curves are 'back at their starting point'. Let's start with the rhodoneas. You might know that the $\sin$ and $\cos$ functions are periodic with a period of $2\pi$, meaning that they are back at their starting point at an angle of $2\pi$ radians.
A curve of the form $r=a\cos(\frac{n}{s} \theta)$ is back at its starting point when $\frac{n}{s} \theta$ is a multiple of $2\pi$ for the first time. If $n$ is even (a multiple of 2) this happens at $\theta=s \cdot\pi$, because $s \cdot \pi \cdot \frac{n}{s} = n\pi$ and $n$ is a multiple of two.
If $n$ is not even this happens at $\theta=2s \cdot\pi$.
Back to your curves:
$(a \sin(b\theta), \theta, c)$ closes when $b\theta$ and $\theta$ are both a multiple of $2\pi$ for the first time. If $b$ is an integer this happens at $\theta=2\pi$, if $b$ is a rational number given by $b=\frac{p}{q}$ with $p,q$ integers, this happens at $\theta=q\cdot 2\pi$.
If $b$ is irrational, the curve never closes because an integer multiple of $2\pi$ cant also be an integer multiple of $2\pi \cdot b$.
Similar logic applies to the other two curves.
EDIT:
I forgot to mention that $\theta$ needs to be an integer multiple of $2\pi$, otherwise the curve won't end at a point equivalent to $(0,0)$.


TITLE: When dealing with an improper antiderivative, why do you take the limit from the left?
QUESTION [0 upvotes]: The example I have is $$\left[\tan x+\frac{1}{\cos x}\right]_0^{\frac{\pi }{2}}$$
In my mind, you treat it as a limit as $x \rightarrow \frac{\pi}{2}$
However, apparently it its $$\lim _{x\to \left(\frac{\pi }{2}\right)^-}\left(\tan x+\frac{1}{\cos x}\right)$$ because "you approach $\frac{π}{2}$ from the left".
Why is that so? The source is here: https://www.youtube.com/watch?v=TJzAp_NSsbM&t=0s (timestamp 4:45)

REPLY [0 votes]: Imagine that you decide to compute the integral as the Reimann sum
$$\lim_{n \to \infty} \sum_{k=1}^n g(k) \times \frac{\pi/2}{n},$$
where $g(k)$ is the value of the function $f(x)$ at $x= (k−1) \frac{π/2}{n}$.
Then in this Reimann sum, the value of $f(x)$ at any value $x \geq (\pi/2)$ is totally irrelevant.
Consequently, when examining the limit as $x$ approaches $\pi/2$ of the anti-derivative $F(x)$, you dismiss the limit as $x$ approaches $\pi/2$ from above as totally irrelevant.


TITLE: Are the Yang-Mills equation and its generalization gauge invariant?
QUESTION [4 upvotes]: I have derived the Yang-Mills equation and its generalization coupled to a current of a scalar field $\phi$ by extremalizing the action describing a $\mathrm{SU}(2)$ scalar field gauge theory:
$$\partial_\mu F^{\mu\nu} +ig\left[ A_{\mu},F^{\mu\nu}\right] = j^\nu$$
where, $\phi$ is a two components scalar field,
$$j^\nu = -ig\left[ (D_\nu \phi)^\star T^a\phi -\phi^\star T^a(D_\nu \phi)\right]T^a$$
where $D_\nu = \partial_\nu + ig A_\nu$. But when I take a gauge transformation:
$$\phi' = e^{-i\omega_a T_a}\phi = U\phi, \quad A'_\mu = UA_\mu U^{-1} -\frac{i}{g}U\partial_\mu U^{-1}$$
I find I can not take same formalism from $j'^{\nu}$ as the original $j^{\nu}$. I think there must be something wrong with my calculation, because the current should be gauge invariant. My question is therefore whether the Yang-Mills equation and its generalization is gauge invariant and how one would show this invariance.
More about my calculation, please comment
@ACuriousMind, Thanks you help and analysis.Now I will write down more of my calculation:
When I calculate the $j^{\nu}$', I find that the term :
$${\phi}^{*\alpha}{\partial _{\upsilon}\phi}^{\alpha }$$
always contains the quality:
$${ U }^{ -1 }{ T }^{ a }U$$
as
$${\phi}^{*\alpha}{\partial _{\upsilon}\phi}^{\alpha }{ U }^{ -1 }{ T }^{ a }U$$
the  ${ U }^{ -1 }{ T }^{ a }U$can not be cancelled by the commutator calculation. Base on  your above answer, can I think this phenomena is correct? I'm never work for QFT, and I learning the classical gauge field theory by my-self, please point out my mistake,Please, theanks.

REPLY [3 votes]: No, the gauge current need not be gauge invariant, since it carries a group index in non-abelian theories. You should recall that both sides of the Yang-Mills equation (and therefore the current itself) are Lie-algebra valued and therefore transform in the adjoint representation. Not even the field strength $F^a_{\mu\nu}$ is gauge invariant, but transforms in the adjoint representation of the gauge group, which is why your action should (hopefully) contain only its trace as $\mathrm{Tr}(F_{\mu\nu}F^{\mu\nu})$.
It is to be noted that, since the current is not invariant, it is not observable.


TITLE: Yoga on coherent flat sheaves $\mathcal{F}$ over projective space $\mathbb{P}^n$
QUESTION [4 upvotes]: I'm reading Mumfords's Lectures on Curves on an Algebraic Surface (jstor-link: https://www.jstor.org/stable/j.ctt1b9x2g3)
and I found in Lecture 7 (RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON
$\mathbb{P}^n$; p 47) dealing with yoga on coherent
sheaves $F$ over pojective space $\mathbb{P}^n$ I found on page 52
a proof I not understand:
Corollary 3: Given a coherent sheaf $\mathcal{F}$ on $\mathbb{P}^n \times S$,
$\mathcal{F}$ is flat over $S$
if and only if there exists an $m_0$ such that if $m \ge m_0$,
$p_* \mathcal{F}(m)$ is
locally free. Hence, in this case, the Hilbert polynomial of on
$\mathbb{P}^n _s$ is locally constant.
Proof: If $F$ is flat over $S$, then let $m_0$ be large enough so
that derived image $R^i p_*(\mathcal{F}(m))=(0)$, if $i>0, m \ge m_0$.
Using Corollary $1$ and $1 \frac{1}{2}$ one deduces that
$p_*(\mathcal{F}(m)) \otimes k(s)$ maps onto $H^0(\mathbb{P}^n _s, \mathcal{F}_s(m))$
for all $s\in S, m \ge m_0$. Then by (iii), $p_* \mathcal{F}(m)$ is locally free.
As for converse [...]
In original:

Problem: The "...Then by (iii), $p_* \mathcal{F}(m)$ is locally free..." part I not understand.
(iii) (on page 51) states:

By above we know $p_*(\mathcal{F}(m)) \otimes k(s) \to H^0(\mathbb{P}^n _s, \mathcal{F}_s(m))$
is surjective, that is we can apply (iii) to $i=1$ and deduce
$R^1p_*(\mathcal{F}(m))$ is locally free sheaf. But Mumford claims this for
$p_* \mathcal{F}(m)= R^0 p_* \mathcal{F}(m)$.
Is this an error in the proof or do I miss something?

REPLY [1 votes]: I think you can apply $(iii)$ with $i = 0$ to obtain that $p_*\mathcal{F}(m)$ is locally free. Since the surjectivity of the base change map in degree $i-1 = -1$ is trivial, you only need the surjectivity in degree $i=0$ required by the condition in $(ii)$.
In summary, you use the base change theorem several times. From Serre's vanishing theorem, you have $H^1(\mathbb{P}_s^n,\mathcal{F}(m)) = 0$ for $m$ large enough and all $s\in S$. From $(ii)$ with $i=1$, you get that $R^1p_*\mathcal{F}(m) = 0$. From $(iii)$ with $i=1$, you get that $R^0p_*\mathcal{F}(m) \otimes k(s) \rightarrow H^0(\mathbb{P}_s^n,\mathcal{F}(m))$ is surjective for every $s$ in $S$ ($R^1p_*\mathcal{F}(m)$ is zero so locally free). Then, you use $(iii)$ with $i=0$  as I explained above to deduce that $p_*\mathcal{F}(m)$ is locally free.
I hope everything is correct.