text:
\section{}

\makeatother

\begin{abstract}
Noether's celebrated theorem associating symmetry and conservation laws in classical field theory is adapted to allow for broken symmetry in geometric mechanics and is shown to play a central role in deriving and understanding the generation of fluid circulation via the Kelvin-Noether theorem for ideal fluids with stochastic advection by Lie transport (SALT).
\end{abstract}

\section{Noether's theorem in geometric mechanics}
\subsection{Euler-Poincar\'e reduction}
Geometric mechanics deals with group-invariant variational principles. In this setting, Noether's theorem \cite{noether1918invariante, noether1971invariant} plays a key role. Given the tangent lift action $G\times TM\to TM$ of a Lie group $G$ on the tangent bundle $TM$ of a manifold $M$\footnote{$M$ is called the \emph{configuration manifold} in classical mechanics.} on which $G$ acts transitively, Noether's theorem states that each Lie symmetry of a Lagrangian $L:TM\to \mathbb{R}$ defined  in the action integral $S=\int L(q,v)dt$ for Hamilton's variational principle $\delta S = 0$ with $(q,v)\in TM$ implies a conserved quantity for the corresponding Euler-Lagrange equations defined on the cotangent bundle $T^*M$.
The conserved quantities arising from Noether's theorem in the case where the configuration manifold $M$ is a Lie group $G$ were studied by Smale, in \cite{smale1970topologya, smale1970topologyb}, where it was shown that the reduction procedure $TG\to TG\setminus G\simeq\mathfrak{g}$ leads to dynamics which take place on the dual $\mathfrak{g}^*$ of the Lie algebra $\mathfrak{g}$. The dynamical variable $m\in\mathfrak{g}^*$ in the dual Lie algebra is now called the momentum map (Smale called it angular momentum). In general, the configuration manifold $M$ is not a Lie group. However, when a Lie group $G$ acts transitively on a configuration manifold $M$ the proof of Noether's theorem induces a cotangent-lift momentum map $J: T^*M\to\mathfrak{g}^*$. The momentum map induced this way is an infinitesimally equivariant Poisson map taking functions on the cotangent bundle $T^*M$ of $M$ to the dual Lie algebra $\mathfrak{g}^*$ of the Lie group $G$. The momentum map $J: T^*M\to\mathfrak{g}^*$ is equivariant and Poisson, even if $G$ is not a Lie symmetry of the Lagrangian in Hamilton's principle.  Momentum maps naturally lead from the Lagrangian side to the Hamiltonian side. The Hamiltonian dynamics on $T^*M$ involves symplectic transformations.  However, as we shall discuss below, for the class of Hamiltonians which can be defined as $H\circ J: \mathfrak{g}^*\to \mathbb{R}$, the momentum map induces Euler-Poincar\'e motion on the Lagrangian side and Lie-Poisson motion on the Hamiltonian side. To illustrate these remarks, we return to the situation in which the configuration manifold, $M$, is a Lie group, $G$.

For hyperregular Lagrangians, the Legendre transform to the Hamiltonian side is invertible and one may reconstruct the solution on $G$ from its representation on $T^*G\setminus G\simeq\mathfrak{g}^*$. In that case, solving the equations describing the evolution of the momentum map on the dual Lie algebra $\mathfrak{g}^*$ is equivalent to solving the equations on the  cotangent bundle $T^*G$ when the configuration manifold is $G$. When the Lie group $G$ acts transitively, freely and properly on the configuration manifold $M$, then one may reconstruct the solution on $M$ from its representation on $T^*G\setminus G\simeq\mathfrak{g}^*$. The last statement is proven for finite-dimensional Lie groups $G$ in, e.g., \cite{abraham1978foundations}.

The Lie-group reduced equations defined on the dual Lie algebra $\mathfrak{g}^*$ via Smale's  procedure of reduction by symmetry $T^*G\setminus G\simeq\mathfrak{g}^*$ are called Euler-Poincar\'e equations after \cite{poincare1901forme}. Provided the Lagrangian is hyperregular, the Euler-Poincar\'e reduction procedure can be expressed in terms of the cube of linked commutative diagrams shown in figure \ref{fig:cube}.
\begin{figure}[H]
\small
\centering
\begin{tikzcd}[row sep=3em, column sep=small]
&
L:TG\to\mathbb{R} \arrow[dl] \arrow[rr,  "\text{Legendre transform}", leftrightarrow] \arrow[dd]
&
&
H:T^*G\to\mathbb{R} \arrow[dl] \arrow[dd]
\\
\text{Euler-Lagrange eqns} \arrow[rr, crossing over, Leftrightarrow]
&
& \text{Hamilton's eqns}
\\
& \ell:\mathfrak{g}\to\mathbb{R} \arrow[dl] \arrow[rr, "\text{Legendre \hspace{0.25cm} transform}", leftrightarrow]
&
& \hslash:\mathfrak{g}^*\to\mathbb{R} \arrow[dl]
\\
\text{Euler-Poincar\'e eqns} \arrow[rr, Leftrightarrow] \arrow[from=uu, crossing over]
&
& \text{Lie-Poisson eqns} \arrow[from=uu, crossing over]
\end{tikzcd}
\caption{The cube of commutative diagrams for geometric mechanics on Lie groups. Euler-Poincar\'e reduction (on the left side) and Lie-Poisson reduction (on the right side) are both indicated by the arrows pointing down. The diagrams are all commutative, provided the Legendre transformation and reduced Legendre transformation are both invertible.}
\label{fig:cube}
\end{figure}
To summarise the notation in figure \ref{fig:cube}, $G$ denotes the  configuration manifold which is assumed to be isomorphic to a Lie group, $TG$ is the tangent bundle, $T^*G$ is the cotangent bundle, $TG\setminus G \simeq \mathfrak{g}$ is the Lie algebra and $T^*G\setminus G\simeq \mathfrak{g}^*$ is the dual of the Lie algebra. The Lagrangian is a functional $L:TG\to\mathbb{R}$ and the Hamiltonian is a functional $H:T^*G\to\mathbb{R}$. Euler-Poincar\'e reduction takes advantage of Lie group symmetries to transform the Lagrangian and Hamiltonian into group-invariant variables, which leads to a reduced Lagrangian $\ell:\mathfrak{g}\to\mathbb{R}$ and a reduced Hamiltonian $\hslash:\mathfrak{g}^*\to\mathbb{R}$. The diagram comprising the face of the cube involving these functionals in figure \ref{fig:cube} commutes if the Legendre transform is a diffeomorphism. This is guaranteed if the Lagrangian or Hamiltonian is hyperregular. The Euler-Lagrange equations and Hamilton's equations are related via a change of variables, which also holds for the Euler-Poincar\'e equations and the Lie-Poisson equations. Many finite dimensional mechanical systems may be described naturally in this framework. The classic example is the rotating rigid body, discussed from the viewpoint of symmetry reduction by Poincar\'e in \cite{poincare1901forme}. In his 1901 paper, Poincar\'e also raised the issue of \emph{symmetry breaking}, by introducing the vertical acceleration of gravity, which breaks the  $SO(3)$  symmetry for free rotation and restricts it to  $SO(2)$ for rotations about the vertical axis.

Stochasticity may also be included in the framework of Euler-Poincar\'e reduction by symmetry. The first attempt to include noise consistently in finite-dimensional symplectic Hamiltonian  mechanics was by \cite{bismut1982mecanique} and reduction by symmetry of stochastic systems was studied by \cite{lazaro2008stochastic}.

\paragraph{Plan of the paper.}
In the present work, we will review Euler-Poincar\'e reduction of stochastic infinite dimensional variational systems with symmetry breaking.  The infinite dimensional case is interesting because it is the natural setting for fluid dynamics, quantum mechanics and elasticity. The foundations of the finite dimensional stochastic geometric mechanics are established in \citet{cruzeiro2018momentum}. We will explore the infinite dimensional case in context of fluid dynamics, where symmetry under the smooth invertible maps of the flow domain is broken by the spatial dependence of the initial mass density.

\subsection{Sobolev class diffeomorphisms}
Consider an $n$-dimensional compact and oriented smooth manifold $M$, equipped with a Riemannian metric $\langle\,\cdot\,,\,\cdot\,\rangle$. This will be the spatial domain of flow and $X\in M$ will denote the initial position of any given fluid particle. The manifold $M$ is acted upon by a group of Sobolev class diffeomorphisms. In \cite{ebin1970groups} it is shown that the space of $C^\infty$ diffeomorphisms, defined by $\mathfrak{D}=\{g\in C^\infty(M,M)|\,g\text{ is bijective and } g^{-1}\in C^\infty(M,M)\}$, is not the convenient setting to study fluid dynamics, but that one should use $\mathfrak{D}^s = \{g\in H^s(M,M)|\,g\text{ is bijective and } g^{-1}\in H^s(M,M)\}$, the space of Sobolev class diffeomorphisms with $s$ weak derivatives. The reason for this choice is that the smooth diffeomorphisms constitute a Fr\'echet manifold for which there is no inverse or implicit function theorem and no general solution theorem for ordinary differential equations. Each of these latter features would prohibit the study of geodesics.

The space of Sobolev class diffeomorphisms is both a Hilbert manifold and a topological group if $s>n/2+1$, as was shown by \cite{ebin1967space}. The Hilbert manifold structure implies the existence of function inverses and the implicit function theorem, as well as the existence of a general solution theorem for ordinary differential equations. This additional structure also implies that one can construct the tangent space of $\mathfrak{D}^s$ in the usual way and study geodesics. The space $\mathfrak{D}^s$ is the configuration space for continuum mechanics and each $g\in\mathfrak{D}^s$ is called a configuration. A fluid trajectory starting from $X\in M$ at time $t=0$ is given by $x(t)=g_t(X)=g(X,t)$, with $\mathfrak{D}^s\ni g:M\times\mathbb{R}^+\to M$ being a continuous one-parameter subgroup of $\mathfrak{D}^s$. In the deterministic case, computing the time derivative of this one-parameter subgroup gives rise to the \emph{reconstruction equation}, given by
\begin{equation}
\frac{\partial}{\partial t}g_t(X) = u(g_t(X),t),
\label{eq:reconstructiondeterministic}
\end{equation}
where $u_t(\,\cdot\,)=u(\,\cdot\,,t)\in \mathfrak{X}^s$ is a time dependent vector field with flow $g_t(\,\cdot\,)=g(\,\cdot\,,t)$. The initial data is given by $g(X,0)=X$. Here $\mathfrak{X}^s=H^s(TM)$ denotes the space of Sobolev class vector fields on $M$, which is also the Lie algebra associated to the Sobolev class diffeomorphisms.

\subsection{Stochastic advection by Lie transport (SALT)}
In the setting of stochastic advection by Lie transport (SALT), which was introduced by \cite{holm2015variational}, the deterministic reconstruction equation in \eqref{eq:reconstructiondeterministic} is replaced by the semimartingale
\begin{equation}
{\sf d}g(X,t) = u(g_t(X),t)dt + \sum_{i=1}^M \xi_i(g_t(X))\circ dW_t^i,
\label{eq:reconstructionstochastic}
\end{equation}
where the symbol $\circ$ means that the stochastic integral is taken in the Stratonovich sense. The initial data is given by $g(X,0)=X$. The $W_t^i$ are independent, identically distributed Brownian motions, defined with respect to the standard stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$. Such a noise was shown to arise from a multi-time homogenisation argument in \cite{cotter2017stochastic}. The $\xi_i(\,\cdot\,)\in\mathfrak{X}^s$ are called data vector fields and are prescribed. These data vector fields represent the effects of unresolved degrees of freedom on the resolved scales of fluid motion and account for unrepresented processes. They are determined by applying empirical orthogonal function analysis to appropriate numerical and/or observational data. For instance, for an application to the two dimensional Euler equations for an ideal fluid, see \cite{cotter2019numerically} and for an application to a two-layer quasi-geostrophic model, see \cite{cotter2018modelling}. Stochastic models enable the use of a variety of methods in data assimilation, which are discussed in \cite{cotter2019particle}. It is not difficult to make sense of \eqref{eq:reconstructiondeterministic}, but understanding \eqref{eq:reconstructionstochastic} is more complicated. In \cite{de2020implications}, a \emph{stochastic chain rule} is shown to exist. This stochastic chain rule is called the \emph{Kunita-It\^o-Wentzell (KIW) formula} and helps interpret the semimartingale in \eqref{eq:reconstructionstochastic}. The KIW formula will also be used later to prove the stochastic Kelvin circulation theorem. First, however, the space $\mathfrak{D}^s$ needs to be given more structure.

The space $\mathfrak{D}^s$ inherits a \emph{weak Riemannian structure} from the underlying manifold $M$ in a natural way. For $g\in \mathfrak{D}^s$ and $V,W\in T_g\mathfrak{D}^s$, one can define the following bilinear form
\begin{equation}
(V,W)=\int_M \langle(V(X),W(X)\rangle_{g(X)}\mu(dX),
\label{eq:weakriemannian}
\end{equation}
where $\mu$ is the volume form on $M$ induced by the metric. The Riemannian structure induced by \eqref{eq:weakriemannian} is weak because the topology is of type $L^2$, which is strictly weaker than the $H^s$ topology. This bilinear form is a linear functional on the Hilbert space $T_g\mathfrak{D}^s$ and can be used to define the dual space $T_g^*\mathfrak{D}^s$. The pairing between $V\in T_g\mathfrak{D}^s$ and $\alpha\in T_g^*\mathfrak{D}^s$ is given by
\begin{equation}
\langle \alpha,V \rangle = \int_M \alpha(X)\cdot V(X).
\label{eq:dualitypairing}
\end{equation}
Hence the metric on $M$ and the volume form $\mu(dX)$ can be used to construct the isomorphism between $T\mathfrak{D}^s$ and $T^*\mathfrak{D}^s$ as $V(X)\mapsto\alpha(X)=V^\flat(X)\mu(dX)$, where $\flat:TM\to T^*M$ is one of the musical isomorphisms that are induced by the metric on $M$. The group $\mathfrak{D}^s$ is not a Lie group; since right multiplication is smooth, but left multiplication is only continuous. Hence $\mathfrak{D}^s$ is a topological group with a weak Riemannian structure. In general, these properties are not sufficient to guarantee the existence of an exponential map. However, \cite{ebin1970groups} showed that an exponential map can exist in many important cases. In particular, they showed that the geodesic spray associated to \eqref{eq:weakriemannian} (with and without forcing) is smooth.\footnote{The \emph{geodesic spray} is the vector field whose integral curves are the geodesics.} The smoothness of the geodesic spray persists even though $H^s$ diffeomorphisms are considered rather than smooth diffeomorphisms. Combined with the existence of an exponential map, the smoothness property implies a regular interpretation of the Euler-Poincar\'e equations on $\mathfrak{D}^s$, provided that one uses right translations and right representations of the group on itself and its Lie algebra, as shown in \cite{holm1998euler}. However, due to the presence of the volume form $\mu(dm)$, the bilinear form \eqref{eq:weakriemannian} is not right-invariant under the action of the entire $H^s$ diffeomorphism group, although there is right-invariance under the action of the isotropy subgroup $\mathfrak{D}^s_\mu = \{g\in\mathfrak{D}^s | \, g_*\mu = \mu\}$. Since this subgroup is a proper subgroup, as it is smaller than $\mathfrak{D}^s$ itself. Thus, one speaks of \emph{symmetry breaking}.

In deriving the equations of ideal deterministic fluid dynamics, one needs to keep track of the volume form as well. The appropriate mathematical setting for this is an \emph{outer semidirect product group}. This means that one constructs a new group from two given groups with a particular type of group operation. For continuum mechanics, the ingredients are $\mathfrak{D}^s$ and $V^*$, where $V^*$ is a vector space of tensor fields. This vector space is the space of \emph{advected quantities} and it will always contain at least the volume form $\mu$.  \smallskip

\begin{definition}[Advected quantity]
A fluid variable is said to be \emph{advected}, if it keeps its value along Lagrangian particle trajectories. Advected quantities are sometimes called \emph{tracers}, because the evolution histories of scalar advected quantities with different initial values (labels) trace out the Lagrangian particle trajectories of each label, or initial value, via the \emph{push-forward} of the full diffeomorphism group, i.e., $a_t=g_{t\,*}a_0= a_0g_t^{-1}$, where $g_t$ is a time-dependent curve on the manifold of diffeomorphisms that represents the fluid flow.
\end{definition}\smallskip

\begin{remark}[Advected quantities as order parameters]
When several advected quantities are involved, the space $V^*$ is the direct sum of several vector spaces, where each summand space hosts a different advected quantity. In general, each additional advected quantity decreases the dimension of the isotropy subgroup. For example, consider an ideal deterministic fluid with a buoyancy variable $b$, then the Lagrangian corresponding to the model will depend on $\mu$ and $b$ in a parametric manner. This Lagrangian will be right invariant under the action of the isotropy subgroup $\mathfrak{D}^s_{\mu,b} = \{g\in\mathfrak{D}^s|\, g_*\mu=\mu \text{ and } g_*b=b\}$. Hence, advected quantities are \emph{order parameters} and each additional order parameter breaks more symmetry. For the sake of notation, one usually writes $\mathfrak{D}^s_{a_0}$ for the isotropy subgroup, no matter how many advected quantities there are. One then uses $a$ to represent all advected quantities and $a_0$ to denote the initial value of the advected quantities.
\end{remark}

\subsection{Semidirect product group adjoint \& coadjoint actions}
The semidirect product group action is constructed in the following way. The representation of $\mathfrak{D}^s$ on a vector space $V$ is by push-forward, which is a left representation, as shown by \cite{marsden1984semidirect}. The representation of the group on itself and on its Lie algebra is a right representation. In terms of analysis, this means that all representations are smooth and no derivatives need to be counted. The group action of the semidirect product group is given by
\begin{equation}
\begin{aligned}
\bullet:(\mathfrak{D}^s\times V)\times(\mathfrak{D}^s\times V)\to(\mathfrak{D}^s\times V)\\
\quad (g_1,v_1)\bullet(g_2,v_2):= (g_1\circ g_2,v_2+(g_2)_*v_1)
\end{aligned}
\label{eq:semidirectproduct}
\end{equation}
with $g_1,g_2\in\mathfrak{D}^s$ and $v_1,v_2\in V$. The semidirect product group is often denoted as $\mathfrak{D}^s\circledS V = (\mathfrak{D}^s\times V,\bullet)$. In the group action above, $(g_2)_*v_1$ denotes the \emph{push-forward} of $v_1$ by $g_2$ and $\circ$ denotes composition. Note that the group affects both slots in \eqref{eq:semidirectproduct}, but the vector space only appears in the second slot. The identity element of the semidirect product group is $(e,0)$ where $e\in\mathfrak{D}^s$ is the identity diffeomorphism and $0\in V$ is the zero vector. An inverse element is given by
\begin{equation}
(g,v)^{-1} = (g^{-1},-(g^{-1})_*v) = (g^{-1}, -g^*v),
\end{equation}
where $g^*v$ denotes the pull-back of $v$ by $g$. To understand how reduction works for semidirect products, it is helpful to know how the group acts on its Lie algebra and on the dual of its Lie algebra. Duality will be defined with respect to the sum of the pairing \eqref{eq:dualitypairing} and the dual linear transformation $[\,\cdot\,]^*$ on $V$. This pairing induces another pairing in a natural way on $\mathfrak{X}^s\times V$. Consider two at least $C^1$ one parameter subgroups $(g_t,v_t),(\widetilde{g}_\epsilon,\widetilde{v}_\epsilon)\in \mathfrak{D}^s\times V$. Using these one parameter subgroups, one can compute the inner automorphism, or adjoint action of the group on itself. This adjoint action is defined by conjugation
\begin{equation}
\begin{aligned}
{\rm AD}:(\mathfrak{D}^s\times V)\times(\mathfrak{D}^s &\times V)\to(\mathfrak{D}^s\times V),\\
{\rm AD}_{(g_t,v_t)}(\widetilde{g}_\epsilon,\widetilde{v}_\epsilon) &:= (g_t,v_t)\bullet (\widetilde{g}_\epsilon,\widetilde{v}_\epsilon)\bullet (g_t,v_t)^{-1}\\
&= \big(g_t\circ\widetilde{g}_\epsilon\circ g_t^{-1}, g_t^*(\widetilde{v}_\epsilon - v_t + \widetilde{g}_{\epsilon*}v_t)\big).
\end{aligned}
\label{eq:AD}
\end{equation}
To see how the group acts on its Lie algebra, one can compute the derivative with respect to $\epsilon$ and evaluate at $\epsilon=0$ in the adjoint action of the group on itself. Let $\mathfrak{X}^s\ni \widetilde{u}=\frac{d}{d\epsilon}|_{\epsilon=0}\widetilde{g}_\epsilon$ and $V\ni\widetilde{b}=\frac{d}{d\epsilon}|_{\epsilon=0}\widetilde{v}_\epsilon$. This choice for a vector field is guided by the deterministic reconstruction equation in \eqref{eq:reconstructiondeterministic}. For any tensor $S_\epsilon\in T_s^r(M)$ whose dependence on $\epsilon$ is at least $C^1$ it holds that
\begin{equation}
\frac{d}{d\epsilon}\widetilde{g}_{\epsilon*}S_\epsilon = \widetilde{g}_{\epsilon*}\left(\frac{d}{d\epsilon}S_\epsilon-\mathcal{L}_{\widetilde{u}} S_\epsilon\right).
\label{eq:liechainrule}
\end{equation}
Important here is that the Lie derivative does not commute with pull-backs and push-forwards that depend on parameters, see \cite{abraham1978foundations}. The adjoint action of the group on its Lie algebra can be computed as
\begin{equation}
\begin{aligned}
{\rm Ad}:(\mathfrak{D}^s\times V)\times(\mathfrak{X}^s &\times V)\to (\mathfrak{X}^s\times V),\\
{\rm Ad}_{(g_t,v_t)}(\widetilde{u},\widetilde{b})&:= \frac{d}{d\epsilon}\Big|_{\epsilon=0}{\rm AD}_{(g_t,v_t)}(\widetilde{g}_\epsilon,\widetilde{v}_\epsilon)\\
&= (g_{t*}\widetilde{u},g^*_t\widetilde{b}-g_t^*\mathcal{L}_{\widetilde{u}}v_t).
\end{aligned}
\label{eq:Ad}
\end{equation}
By means of the pairing on $\mathfrak{X}^s\times V$, one can compute the dual action to the adjoint action \eqref{eq:Ad}. This is called the coadjoint action of the group on the dual of its Lie algebra. Let $(\widetilde{m},\widetilde{a})\in(\mathfrak{X}^s\times V)^*$, then the coadjoint action is given by
\begin{equation}
\begin{aligned}
{\rm Ad}^*:(\mathfrak{D}^s\times V)\times(\mathfrak{X}^s &\times V)^*\to(\mathfrak{X}^s\times V)^*,\\
\langle{\rm Ad}^*_{(g_t^{-1},-g_t^{-1}v_t)}(\widetilde{m},\widetilde{a}),(\widetilde{u},\widetilde{b})\rangle &:= \langle(\widetilde{m},\widetilde{a}),{\rm Ad}_{(g_t,v_t)}(\widetilde{u},\widetilde{b})\rangle,\\
{\rm Ad}^*_{(g_t^{-1},-g_t^{-1}v_t)}(\widetilde{m},\widetilde{a}) &= (g_t^*\widetilde{m}+v_t\diamond g_{t*}\widetilde{a},g_{t*}\widetilde{a}).
\end{aligned}
\label{eq:Ad*}
\end{equation}
\begin{definition}[The diamond operator]
The coadjoint action \eqref{eq:Ad*} features the \emph{diamond operator}, which is defined for $a\in V^*$, $u\in\mathfrak{X}^s$ and fixed $v\in V$ as
\begin{equation}
\langle v\diamond a, u\rangle_{\mathfrak{X}^{s*}\times\mathfrak{X}^s} := -\langle a,\mathcal{L}_u v\rangle_{V^*\times V}.
\end{equation}
Note that the diamond operator is the dual of the Lie derivative regarded as a map $\mathcal{L}_{(\,\cdot\,)}v:\mathfrak{X}^s\to V$, hence $v\diamond(\,\cdot\,):V^*\to\mathfrak{X}^{s*}$. The diamond operator shows how an element from the dual of the vector space acts on the dual of the Lie algebra.
\end{definition}

When evaluated at $t=0$, the $t$-derivatives of ${\rm Ad}$ in \eqref{eq:Ad} and ${\rm Ad}^*$ in \eqref{eq:Ad*} define, respectively, the adjoint and coadjoint actions of the Lie algebra on itself and on its dual. Denote by $\mathfrak{X}^s\ni u = \frac{d}{dt}|_{t=0}g_t$ and $V\ni b=\frac{d}{dt}|_{t=0}v_t$. The adjoint action of the Lie algebra on itself is
\begin{equation}
\begin{aligned}
{\rm ad}:(\mathfrak{X}^s\times V)\times(\mathfrak{X}^s &\times V)\to (\mathfrak{X}^s\times V),\\
{\rm ad}_{(u,b)}(\widetilde{u},\widetilde{b})&:=\frac{d}{dt}\Big|_{t=0}{\rm Ad}_{(g_t,v_t)}(\widetilde{u},\widetilde{b}),\\
{\rm ad}_{(u,b)}(\widetilde{u},\widetilde{b})&=(-\mathcal{L}_u\widetilde{u},\mathcal{L}_u\widetilde{b}-\mathcal{L}_{\widetilde{u}}b)\\
&= (-[u,\widetilde{u}],\mathcal{L}_u\widetilde{b}-\mathcal{L}_{\widetilde{u}}b),
\end{aligned}
\label{eq:ad}
\end{equation}
where the bracket $[\,\cdot\,,\,\cdot\,]$ in \eqref{eq:ad} is the commutator of vector fields. The minus sign is due to fact that group acts on itself from the right. The coadjoint action of the Lie algebra on its dual can be obtained by computing the dual to \eqref{eq:ad} or by taking the derivative with respect to $t$ and evaluate at $t=0$ in \eqref{eq:Ad*}. Either way, one arrives at
\begin{equation}
\begin{aligned}
{\rm ad}^*:(\mathfrak{X}^s\times V)\times(\mathfrak{X}^s &\times V)^*\to(\mathfrak{X}^s\times V)^*,\\
\langle{\rm ad}^*_{(u,b)}(\widetilde{m},\widetilde{a}),(\widetilde{u},\widetilde{b})\rangle &:= \langle(\widetilde{m},\widetilde{a}),{\rm ad}_{(u,b)}(\widetilde{u},\widetilde{b})\rangle,\\
{\rm ad}^*_{(u,b)}(\widetilde{m},\widetilde{a}) &= (\mathcal{L}_u\widetilde{m} + b\diamond\widetilde{a},-\mathcal{L}_u\widetilde{a}),
\end{aligned}
\label{eq:ad*}
\end{equation}
in which \eqref{eq:ad} implies the last line in \eqref{eq:ad*}. Alternatively, one can also obtain \eqref{eq:ad*} by taking the derivative with respecto to $t$ in \eqref{eq:Ad*} and evaluate at $t=0$. \smallskip

\begin{remark}[Coadjoint action and the diamond operator]
The coadjoint action is an important operator in geometric mechanics and representation theory. It was shown by \cite{kirillov1962unitary} and in further work by \cite{kostant1970quantization} and \cite{souriau1970structure} that the coadjoint orbits of a Lie group $G$ have the structure of symplectic manifolds and are connected with Hamiltonian mechanics. See \cite{kirillov1999merits} for a review. The computations of the adjoint and coadjoint actions for the semidirect product group is valuable for fluid mechanics, as they introduce the two fundamental operators that appear in the equations of motion. The Lie derivative is responsible for transport of tensors along vector fields and its dual action given by the diamond operator encodes the symmetry breaking. In particular, the diamond operator introduces the effect of symmetry breaking into the Euler-Poincar\'e equations of motion.
\end{remark}

\section{Deterministic geometric fluid dynamics}
With the adjoint and coadjoint actions defined, one can derive continuum mechanics equations with advected quantities by using symmetry reduction. Euler-Poincar\'e reduction for a semidirect product group $\mathfrak{D}^s\times V$ as developed in \cite{holm1998euler} is sketched below in figure \ref{fig:cubesdp}.
\begin{figure}[H]
\small
\centering
\begin{tikzcd}[row sep=3em, column sep=small]
&
L:T\mathfrak{D}^s\times V^*\to\mathbb{R} \arrow[dl] \arrow[rr,  "\text{Legendre transform}", leftrightarrow] \arrow[dd]
&
&
H:T^*(\mathfrak{D}^s\times V)\to\mathbb{R} \arrow[dl] \arrow[dd]
\\
\text{Euler-Lagrange eqns} \arrow[rr, crossing over, Leftrightarrow]
&
& \text{Hamilton's eqns}
\\
&
\ell:\mathfrak{X}^s\times V^*\to\mathbb{R} \arrow[dl] \arrow[rr, "\text{Legendre \hspace{0.25cm} transform}", leftrightarrow]
&
&
\hslash:(\mathfrak{X}^s\times V)^*\to\mathbb{R} \arrow[dl]
\\
\text{Euler-Poincar\'e eqns} \arrow[rr, Leftrightarrow] \arrow[from=uu, crossing over]
&
&
\text{Lie-Poisson eqns} \arrow[from=uu, crossing over]
\end{tikzcd}
\caption{The cube of continuum mechanics in the semidirect product group setting. Reduction is indicated by the arrows pointing down.}
\label{fig:cubesdp}
\end{figure}
 As shown by comparison of  figure \ref{fig:cubesdp} with figure \ref{fig:cube}, several new features arise in semidirect product Lie group reduction which differ from Euler-Poincar\'e reduction by symmetry when the configuration space itself is a Lie group. These differences can be conveniently explained by introducing the physical concept of an order parameter. As discussed earlier, the order parameters in continuum mechanics are the elements of $V^*$ which are advected by the action of the diffeomorphism group $\mathfrak{D}^s$. The advection is defined simply as the semidirect product action on the elements of $V^*$. The introduction of each additional advected state variable (or, order parameter) into the physical problem further breaks the original symmetry $\mathfrak{D}^s$. The remaining symmetry of the Lagrangian in Hamilton's principle is the isotropy subgroup $\mathfrak{D}^s_{a_0}$ of the initial conditions, $a_0$, for the entire set of advected quantities, $a$. The action of the diffeomorphism group $\mathfrak{D}^s$ on these initial conditions then describes their advection as the action of $\mathfrak{D}^s$ on its coset space $\mathfrak{D}^s\setminus\mathfrak{D}^s_{a_0}=V^*$. Once the inital values of the order parameters, $a_0$,  have been set, one must still define a Legendre transform from the Lagrangian formulation into the Hamiltonian formulation and vice versa. The Legendre transform in the setting of semidirect products is a partial Legendre transform, since it transforms between $T\mathfrak{D}^s$ and $T^*\mathfrak{D}^s$ or $T\mathfrak{D}^s\setminus\mathfrak{D}^s_{a_0} \simeq \mathfrak{X}^s$ and $T^*\mathfrak{D}^s\setminus\mathfrak{D}^s_{a_0} \simeq\mathfrak{X}^{s*}$ only after having fixed the value $a_0$ of the order parameters, which live in $V^*$. This coset reduction is what figure \ref{fig:cubesdp} shows. The remaining invariance of a functional under the action of the isotropy subgroup is called its \emph{particle relabelling symmetry}.

Our exploration continues on the Lagrangian side in figure \ref{fig:cubesdp}. Consider a fluid Lagrangian $L:T\mathfrak{D}^s\times V^*\to\mathbb{R}$. By fixing the value of $a_0\in V^*$, one can construct $L_{a_0}:T\mathfrak{D}^s\to\mathbb{R}$. If this Lagrangian is right invariant under the action of the isotropy subgroup $\mathfrak{D}_{a_0}^s$, then one can construct
\begin{equation}
\begin{aligned}
L\left(\frac{d}{dt}g\circ g^{-1},e,a_0\right) &= L_{a_0}\left(\frac{d}{dt}g\circ g^{-1},e\right)\\
&= \ell_{a_0}\left(\frac{d}{dt}g\circ g^{-1}\right) = \ell\left(\frac{d}{dt}g\circ g^{-1}, g_*a_0\right).
\end{aligned}
\label{eq:lagrangians}
\end{equation}
Here $\circ$ means composition of functions. The same procedure applies to the Hamiltonian. Since the coadjoint action is known, it is straightforward to formulate the Lie-Poisson equations. The details of Hamiltonian semidirect product reduction and also more information on the Lagrangian semidirect product reduction can be found in \cite{holm1998euler}.

The coadjoint action of the Lie algebra on its dual is also required for the Lagrangian semidirect product reduction. One can use the deterministic reconstruction equation to see that the argument of the Lagrangians in \eqref{eq:lagrangians} is
\begin{equation}
\frac{d}{dt}g\circ g^{-1} = u.
\end{equation}
Using this information, one can integrate the Lagrangian in time to construct the action functional. By requiring the variational derivative of the action functional to vanish, one can compute the equations of motion. However, due to the removal of symmetries, the variations are no longer free.

\section{Stochastic geometric fluid dynamics}
In the situation where noise is present, that is, when the reconstruction equation is  \eqref{eq:reconstructionstochastic}, the Euler-Poincar\'e variations become stochastic. Consider $g:\mathbb{R}^2\to\mathfrak{D}^s$ with $g_{t,\epsilon}=g(t,\epsilon)$ to be a two parameter subgroup with smooth dependence on $\epsilon$, but only continuous dependence on $t$. Let us denote
\[
{\sf d}\chi_{t,\epsilon}(X) = ({\sf d}g_{t,\epsilon}\circ g_{t,\epsilon})(X) = u_{t,\epsilon}(X)dt + \sum_{i=1}^N \xi_i(X)\circ dW_t^i
\]
 and
 \[
 v_{t,\epsilon}(X) = (\frac{\partial}{\partial \epsilon}g_{t,\epsilon}\circ g_{t,\epsilon})(X)\,.
 \]
 When a $\circ$ symbol is followed by $dW_t$ it means Stratonovich integration and in every other context the $\circ$ symbol is used to denote composition. Note that the data vector fields $\xi_i$ are prescribed and hence will not have a dependence on $\epsilon$.

In order to compute with these stochastic subgroups and their associated vector fields, one needs a stochastic Lie chain rule. The Kunita-It\^o-Wentzell (KIW) formula is the stochastic generalisation of the Lie chain rule \eqref{eq:liechainrule}. A proof of the KIW formula is given  in \cite{de2020implications} for differential $k$-forms and vector fields. That proof includes the technical details on regularity that will be omitted here. In the KIW formula, the $k$-form is allowed to be a semimartingale itself. Let $K$ be a continuous adapted semimartingale that takes values in the $k$-forms and satisfies
\begin{equation}
K_t = K_0 + \int_0^t G_s ds + \sum_{i=1}^N\int_0^t H_{i\,s}\circ dB_s^i,
\label{eq:kformsemimartingale}
\end{equation}
where the $B_t^i$ are independent, identically distributed Brownian motions. The drift of the semimartingale $K$ is determined by $G$ and the diffusion by $H_i$, both of which are $k$-form valued continuous adapted semimartingales with suitable regularity. Let $g_t$ satisfy \eqref{eq:reconstructionstochastic}, then \cite{de2020implications} shows that the following holds
\begin{equation}
{\sf d}(g_t^*K_t) = g_t^*\big({\sf d}K_t + \mathcal{L}_{u_t} K_t\,dt + \mathcal{L}_{\xi_i}K_t \circ dW_t^i\big).
\label{eq:kiwformula}
\end{equation}
Equation \eqref{eq:kformsemimartingale} helps to interpret the ${\sf d}K_t$ term in the KIW formula \eqref{eq:kiwformula}. This formula will be particularly useful in computing the variations of the variables in the Lagrangian. To compute these variations, one needs the variational derivative.

\paragraph{The variational derivative.} The variational derivative of a functional $F:\mathcal{B}\to\mathbb{R}$, where $\mathcal{B}$ is a Banach space, is denoted $\delta F/\delta \rho$ with $\rho\in\mathcal{B}$. The variational derivative can be defined by the first variation of the functional
\begin{equation}
\delta F[\rho]:= \frac{d}{d\epsilon}\Big|_{\epsilon=0} F[\rho+\epsilon \delta\rho] = \int \frac{\delta F}{\delta \rho}(x)\delta\rho(x)\,dx = \left\langle\frac{\delta F}{\delta \rho},\delta \rho\right\rangle.
\end{equation}
In the definition above, $\epsilon\in\mathbb{R}$ is a parameter, $\delta\rho\in\mathcal{B}$ is an arbitrary function and the first variation can be understood as a Fr\'echet derivative. A precise and rigorous definition can be found in \cite{gelfand2000calculus}. With the definition of the functional derivative in place, the following lemma can be formulated.
\medskip

\begin{lemma}
With the notation as above, the variations of $u$ and any advected quantity $a$ are given by
\begin{equation}
\delta u(t) = {\sf d}v(t) + [{\sf d}\chi_t,v(t)],\quad \delta a(t) = -\mathcal{L}_{v(t)}a(t),
\label{def:delta-var}
\end{equation}
where $v(t)\in\mathfrak{X}^s$ is arbitrary.
\end{lemma}
\begin{proof}
The proof of the variation of $a(t)$ is a direct application of the Kunita-It\^o-Wentzell formula to $a(t,\epsilon)=g_{t,\epsilon*}a_0$. Note that the data vector fields $\xi_i$ are prescribed and do not depend on $\epsilon$. Denote by $x_{t,\epsilon} = g_{t,\epsilon}(X)$. Then one has
\begin{equation}
{\sf d}g_{t,\epsilon}(X) = {\sf d}x_{t,\epsilon} = u_{t,\epsilon}(x_{t,\epsilon})\,dt + \sum_{i=1}^N \xi_i(x_{t,\epsilon})\circ dW_t^i =: {\sf d}\chi_{t,\epsilon}(x_{t,\epsilon}).
\label{eq:twoparameterstochu}
\end{equation}
The vector field associated to the $\epsilon$-dependence of the two parameter subgroup is given by
\begin{equation}
\frac{\partial}{\partial \epsilon}g_{t,\epsilon} = \frac{\partial}{\partial \epsilon}x_{t,\epsilon} = v_{t,\epsilon}(x_{t,\epsilon}).
\label{eq:twoparameterstochv}
\end{equation}
Computing the derivative with respect to $\epsilon$ of \eqref{eq:twoparameterstochu} gives
\begin{equation}
\begin{aligned}
\frac{\partial}{\partial \epsilon}{\sf d}x_{t,\epsilon} &= \frac{\partial}{\partial \epsilon}\big({\sf d}\chi_{t,\epsilon}(x_{t,\epsilon})\big)\\
&= \left(\frac{\partial}{\partial \epsilon}u_{t,\epsilon} + v_{t,\epsilon}\cdot\frac{\partial}{\partial x_{t,\epsilon}}{\sf d}\chi_{t,\epsilon}\right)(x_{t,\epsilon}),
\end{aligned}
\end{equation}
where the independence of the data vector fields $\xi_i$ on $\epsilon$ was used. Taking the differential with respect to time of \eqref{eq:twoparameterstochv} gives
\begin{equation}
\begin{aligned}
{\sf d}\left(\frac{\partial}{\partial \epsilon} x_{t,\epsilon}\right) &= {\sf d}\big(v_{t,\epsilon}(x_{t,\epsilon})\big)\\
&=  \left( {\sf d}v_{t,\epsilon}(x_{t,\epsilon}) + {\sf d}\chi_{t,\epsilon}\cdot\frac{\partial}{\partial x_{t,\epsilon}}v_{t,\epsilon}\right)(x_{t,\epsilon}).
\end{aligned}
\end{equation}
One can then evaluate at $\epsilon=0$ and call upon equality of cross derivative-differential to obtain the result by subtracting. Since $g_{t,\epsilon}$ depends on $t$ in a $C^0$ manner, the integral representation is required. The particle relabelling symmetry permits one to stop writing the explicit dependence on space,
\begin{equation}
\delta u(t)\,dt = {\sf d}v(t) + [{\sf d}\chi_t,v(t)].
\end{equation}
This completes the proof of formula \eqref{def:delta-var} for the variation of $u(t)$.
\end{proof}
The notation in \eqref{eq:twoparameterstochu} needs careful explanation, because it comprises both a stochastic differential equation and a definition. The symbol ${\sf d}\chi_{t,\epsilon}$ is used to define a vector field, whereas ${\sf d}x_{t,\epsilon}$ denotes a stochastic differential equation. This lemma makes the presentation of the stochastic Euler-Poincar\'e theorem particularly simple.
\medskip

\begin{theorem}[Stochastic Euler-Poincar\'e]\label{thm:SEP}
With the notation as above, the following are equivalent.
\begin{enumerate}[i)]
\item The constrained variational principle
\begin{equation}
\delta\int_{t_1}^{t_2}\ell(u,a)\,dt = 0
\end{equation}
holds on $\mathfrak{X}^s\times V^*$, using variations $\delta u$ and $\delta a$ of the form
\begin{equation}
\delta u = {\sf d}v + [{\sf d}\chi_t,v], \qquad \delta a = -\mathcal{L}_v a,
\end{equation}
where $v(t)\in \mathfrak{X}^s$ is arbitrary and vanishes at the endpoints in time for arbitrary times $t_1,t_2$.
\item The stochastic Euler-Poincar\'e equations hold on $\mathfrak{X}^s\times V^*$
\begin{equation}
{\sf d}\frac{\delta \ell}{\delta u} + \mathcal{L}_{{\sf d}\chi_t}\frac{\delta \ell}{\delta u} = \frac{\delta \ell}{\delta a}\diamond a\,dt,
\label{eq:stochep}
\end{equation}
and the advection equation
\begin{equation}
{\sf d}a + \mathcal{L}_{{\sf d}\chi_t}a = 0.
\label{eq:stochadv}
\end{equation}
\end{enumerate}
\end{theorem}

\begin{proof}
Using integration by parts and the endpoint conditions $v(t_1)=0=v(t_2)$, the variation can be computed to be
\begin{equation}
\begin{aligned}
\delta\int_{t_1}^{t_2}\ell(u,a)\,dt
&=
\int_{t_1}^{t_2}\left\langle\frac{\delta\ell}{\delta u},\delta u\right\rangle + \left\langle\frac{\delta\ell}{\delta a},\delta a\right\rangle\,dt\\
&= \int_{t_1}^{t_2}\left\langle\frac{\delta\ell}{\delta u},{\sf d}v + [{\sf d}\chi_t,v]\right\rangle + \left\langle\frac{\delta\ell}{\delta a}\,dt,-\mathcal{L}_v a\right\rangle\\
&= \int_{t_1}^{t_2}\left\langle -{\sf d}\frac{\delta\ell}{\delta u} - \mathcal{L}_{{\sf d}\chi_t}\frac{\delta\ell}{\delta u} + \frac{\delta\ell}{\delta a}\diamond a\,dt,v\right\rangle\\
&= 0\,.
\end{aligned}
\end{equation}
Since the vector field $v$ is arbitrary, one obtains the stochastic Euler-Poincar\'e equations. Finally, the advection equation \eqref{eq:stochadv} follows by applying the KIW formula to $a(t)=g_{t*}a_0$.
\end{proof}

\begin{remark}
The stochastic Euler-Poincar\'e theorem is equivalent to the version presented in \citet{holm2015variational}, which uses stochastic Clebsch constraints. In \cite{holm2015variational} one can also find an investigation the It\^o formulation of the stochastic Euler-Poincar\'e equation.
\end{remark}

\paragraph{Stochastic Lie-Poisson formulation.}
The stochastic Euler-Poincar\'e equations have an equivalent stochastic Lie-Poisson formulation. To obtain the Lie-Poisson formulation, one must Legendre transform the reduced Lagrangian. The Legendre transformation in the presence of stochasticity becomes itself stochastic in the following way
\begin{equation}
m := \frac{\delta\ell}{\delta u}, \qquad \hslash(m,a)\,dt + \sum_{i=1}^N\langle m,\xi_i\rangle \circ dW_t^i = \langle m,{\sf d}\chi_t\rangle - \ell(u,a)\,dt.
\label{eq:reducedstochlegendre}
\end{equation}
The stochasticity enters the Legendre transformation because the momentum map $m$ is coupled to the stochastic vector field ${\sf d}\chi_t$. The left hand side of the transformation determines the Hamiltonian, which is a semimartingale. The underlying semidirect product group structure has not changed, it is still the $H^s$ diffeomorphisms with a vector space, but the Hamiltonian has become a semimartingale. This implies that in the stochastic case the energy is not conserved, because Hamiltonian depends on time explicitly. Note that \eqref{eq:reducedstochlegendre} emphasises that the Lagrangian does not feature stochasticity in this framework. Instead, the Lagrangian represents the physics in the problem, which does not change. The stochasticity is supposed to account for the difference between observed data and deterministic modelling. The stochastic Lie-Poisson equations are given by
\begin{equation}
{\sf d}(m,a) = -{\rm ad}^*_{(\frac{\delta\hslash}{\delta m},\frac{\delta\hslash}{\delta a})}(m,a)\,dt - \sum_{i=1}^N{\rm ad}^*_{(\xi_i,0)}(m,a)\circ dW_t^i,
\label{eq:stochliepoisson}
\end{equation}
where ${\rm ad}^*$ is given in \eqref{eq:ad*}. Since both the drift and the diffusion part use the same operator in \eqref{eq:stochliepoisson}, the stochastic Lie-Poisson equations preserve the same family of Casimirs (or integral conserved quantities) as the deterministic Lie-Poisson equations. The stochastic Euler-Poincar\'e theorem has a stochastic Kelvin-Noether circulation theorem as a corollary.

Let the manifold $M$ be a submanifold of $\mathbb{R}^n$ with coordinates $X$. Then the volume form can be expressed with respect to a density. That is, $\mu(d^n X) = \rho_0(X)d^n X$. By pushing forward $\rho_0$ along the stochastic flow $g_t$, one obtains $\rho$. Let $\mathfrak{C}^s$ be the space of loops $\gamma:S^1\to\mathfrak{D}^s$, which is acted upon from the left by $\mathfrak{D}^s$. Given an element $m\in\mathfrak{X}^s$, one can obtain a 1-form by formally dividing $m$ by the density $\rho$.

The circulation map $\mathcal{K}:\mathfrak{C}^s\times V^*\to\mathfrak{X}^{s**}$ is defined by
\begin{equation}
\langle \mathcal{K}(\gamma,a),m\rangle = \oint_\gamma\frac{m}{\rho}\,.
\end{equation}
Given a Lagrangian $\ell:\mathfrak{X}^s\times V^*\to \mathbb{R}$,  the \emph{Kelvin-Noether quantity} is defined by
\begin{equation}
I(\gamma,u,a) := \oint_\gamma\frac{1}{\rho}\frac{\delta\ell}{\delta u}\,.
\end{equation}
One can now formulate the following stochastic Kelvin-Noether circulation theorem.
\medskip

\begin{theorem}[Stochastic Kelvin-Noether]\label{Thm:KelThm}
Let $u_t=u(t)$ satisfy the stochastic Euler-Poincar\'e equation \eqref{eq:stochep} and $a_t=a(t)$ the stochastic advection equation \eqref{eq:stochadv}. Let $g_t$ be the flow associated to the vector field ${\sf d}\chi_t$. That is, ${\sf d}\chi_t = {\sf d}g_t\circ g_t^{-1} = u_t\,dt + \sum_{i=1}^N \xi_i\circ dW_t^i$. Let $\gamma_0\in \mathfrak{C}^s$ be a loop. Denote by $\gamma_t = g_t\circ \gamma_0$ and define the Kelvin-Noether quantity $I(t):= I(\gamma_t,u_t,a_t)$. Then
\begin{equation}
{\sf d}I(t) = \oint_{\gamma_t}\frac{1}{\rho}\frac{\delta\ell}{\delta a}\diamond a\,dt\,.
\label{eqn:KelThm}
\end{equation}
\end{theorem}
\begin{proof}
The statement of the stochastic Kelvin-Noether circulation theorem involves a loop that is moving with the stochastic flow. One can transform to stationary coordinates by pulling back the flow to the initial condition. This pull-back yields
\begin{equation}
I(t) = \oint_{\gamma_t}\frac{1}{\rho}\frac{\delta\ell}{\delta u} = \oint_{\gamma_0}g_t^*\left(\frac{1}{\rho}\frac{\delta\ell}{\delta u}\right) = \oint_{\gamma_0}\frac{1}{\rho_0}g_t^*\left(\frac{\delta\ell}{\delta u}\right).
\end{equation}
An application of the Kunita-It\^o-Wentzell formula \eqref{eq:kiwformula} leads to
\begin{equation}
{\sf d}I(t) = \oint_{\gamma_0}\frac{1}{\rho_0}g_t^*\left({\sf d}\frac{\delta\ell}{\delta u} + \mathcal{L}_{{\sf d}\chi_t}\frac{\delta \ell}{\delta u}\right) = \oint_{\gamma_0}\frac{1}{\rho_0}g_t^*\left(\frac{\delta\ell}{\delta a}\diamond a\right)\,dt,
\end{equation}
since $u$ satisfies the stochastic Euler-Poincar\'e theorem. Transforming back to the moving coordinates by pushing forward with $g_t$ yields the final result.
\end{proof}

Thus, Theorem \ref{Thm:KelThm} explains how particle relabelling symmetry gives rise to the Kelvin-Noether circulation theorem via Noether's theorem. When the only advected quantity present is the mass density, the loop integral of the diamond terms vanishes. This means that circulation is conserved according to Noether's theorem for an incompressible fluid, or for a barotropically compressible fluid. The presence of other advected quantities breaks the symmetry further and introduces the  \emph{diamond terms} which generate circulation, as one can see in the Kelvin-Noether circulation theorem in equation \eqref{eqn:KelThm}. Consequently, the symmetry breaking due to additional order parameters can provide additional mechanisms for the generation of Kelvin-Noether circulation in ideal fluid dynamics.

\paragraph*{Outlook.}
Stochastic geometric mechanics is an active field of mathematics which has recently established its utility for a broad range of applications in science. Basically, everything that can be done with Hamilton's principle for deterministic geometric mechanics can also be made stochastic in the sense of Stratonovich. This is possible because the variational calculus in Hamilton's principle requires only the product rule and chain rule from ordinary calculus. The happy emergence of the new science of stochastic geometric mechanics was celebrated with the publication of the book \citet{albeverio2017stochastic}. This book showcases some of the recent developments in stochastic geometric mechanics. Another collection of recent developments can be found in \citet{castrillon2019journal}. An ongoing development is in the direction of \emph{rough geometric mechanics}, initiated with a rough version of the Euler-Poincar\'e theorem in \citet{crisan2020variational}. Remarkably, variational principles which are driven by geometric rough paths again only require the product rule and the chain rule. Other directions involve the inclusion of jump processes, fractional derivatives and non-Markovian processes in geometric mechanics. For example, recent work by \citet{albeverio2020weak} shows that SDEs driven by semimartingales with jumps have weak symmetries and a corresponding extension of the reduction and reconstruction technique is discussed.

\section*{Acknowledgments}
We are enormously grateful for many encouraging discussions over the years with T.S. Ratiu, F. Gay-Balmaz, C. Tronci, S. Albeverio, A.B. Cruzeiro, F. Flandoli, and also with our friends in project STUOD (stochastic transport in upper ocean dynamics) and in the geometric mechanics research group at Imperial College London.
The work of DDH was partially supported by European Research Council (ERC) Synergy
grant STUOD - DLV-856408. EL was supported by EPSRC grant [grant number EP/L016613/1] and is grateful for the warm hospitality at the Imperial College London EPSRC Centre for Doctoral Training in the Mathematics of Planet Earth during the course of this work.

\renewcommand\bibname{\sc References}
\bibliographystyle{plainnat}

redpajama_set_name: RedPajamaArXiv


New York Jets quarterback Mark Sanchez is aiming toward getting back in time for the season opener against the Tampa Bay Buccaneers.
He's got a bruised shoulder and is unlikely to practice this week.
Jets coach Rex Ryan and Sanchez decline to elaborate on the nature of the injury, which was sustained when Ryan made an ill-advised move to play the starter after Geno Smith left the game following three interceptions.
Why was Sanchez even in the game in garbage time playing with backups?
Ryan said he's not being evasive about the injury to Sanchez.


text:
Telecommuting's curse: the 24-hour workday
Portable devices allow Jamie Reid of A.P. Reid Insurance to conduct business anywhere. But such flexibility comes with a danger. 'My workday can become pretty much 24 hours if I let it,' he says.
PAUL DARROW/The Globe and Mail
Wallace Immen
Published March 8, 2013 Updated May 11, 2018
Jamie Reid is getting so used to blending his work and personal life using technology that when he booked a spa treatment as a Valentine's Day gift for his wife, he told her about it by posting the appointment on her online business calendar.
The chief executive officer of Halifax-based A.P. Reid Insurance Stores oversees a business with 12 locations across Nova Scotia and has customers throughout the country. He often keeps in touch with them by e-mail and text rather than travelling for face-to-face meetings.
But it keeps him busy. "My workday can become pretty much 24 hours if I let it," he says.
Handling much of it on portable devices gives him the flexibility to get out of the office to take his 13-year-old son to soccer practice four days a week, but it also lets the work follow him. "Sitting in the stands I'll often be dealing with proposals on a smartphone," he says.
The proliferation of powerful mobile devices means that the future will include more blurring between personal and work time, the experts say. That's giving people more flexibility in scheduling work around their personal commitments, but it also carries a risk of never being able to shut off from work mode.
"It's becoming so possible to integrate our work with everything we do that in the future, we'll need a new way to describe it," says Nora Spinks, CEO of the Vanier Institute for the Family in Ottawa. "Where once we talked about work-life balance, now we talk about blending work and life using portable devices wherever we are."
And most Canadians under 25 have never known a time when they didn't have a device they could use for both work and personal messages wherever they are, she adds. "They just talk about life."
According to an Ipsos Reid study done in January, working Canadians believe their personal and work lives are growing increasingly intertwined. As a result, 70 per cent of Canadian parents say they depend on technology such as smartphones and cloud services to connect with family and office to stay on top of their hectic schedules. It also found 78 per cent believe that technology allows their families to stay better organized.
"I think it's just a natural evolution based on what we now understand about work, and what's available to people to get things done in multiple places, so I think in general that's all for the best," Ms. Spinks says.
But this raises the spectre of a future in which people could stay plugged in to their work all the time, Ms. Spinks says. "If you didn't set limits, no one will tell you when work's done."
Norma Tombari, the director of global diversity for Royal Bank of Canada in Toronto, is seeing both sides of the issue. Mobile devices have allowed her to work when she feels most productive.
"I'm a morning person, so I can come in early to the office and leave earlier, or I can start the day at home if the weather is bad," she says.
Her responsibilities include regular commutes between the Toronto and Montreal offices, and she has found it's convenient to work in the airport waiting for planes or taking a train using WiFi connections.
But there's a downside to the flexibility: "I've found I have to have the discipline to schedule in breaks even in the middle of the day. If I don't take personal time that's a break from work, I find I'm less creative," she says.
That's the message of David Posen, a stress specialist in Oakville, Ont., in his new new book Is Work Killing You?
"I've taken to asking the question in my seminars, 'Where are you when you get your best ideas?' Almost no hands go up when I ask, 'Is it at work?' They're more likely to say they get the best ideas when they're taking a shower, or walking the dog or doing yoga and not distracted by their technology," he says.
"The problem with mobile technology is there can be so many messages and it's so easy to reply to them, that for some people their main goal in the day becomes getting through them all. If they don't set some personal boundaries they're going to end up overwhelmed in the future," Dr. Posen warns.
"My advice is to use anything that works for you, but do a reality check on whether it is really working for you."
It's warning sign, for example, if the first thing you do after waking is check your cellphone and e-mail, he says. Other signals are the need to check messages at the dinner table or a restaurant or while walking. "All along Bay Street [in Toronto] you see people who aren't watching where they're going because they're watching their smartphones."
A simple check it to ask the people you live with if they think you spend too much time on your smartphone, he says. "Get a sense from other people because they may see you more clearly than you do."
Then you need to take action to regain control of your personal time. "We're getting to the point where having cut-and-dried rules aren't going to work. So you do have to set your own rules for disengaging."
"Even though we talk about blending work and life, we have to learn to integrate technology into our lives in a way that still does leave enough personal time."
If the demand is coming from your organization, a push-back can be effective. "People need to gather up their courage and say to the boss and their colleagues that their need for instant response to every message has become disruptive. That tends to make people back off," Dr. Posen says.
Mr. Reid says he has taken steps to set limits. For instance, he turns off his devices before bedtime, sets them to vibrate in meetings and ignores them during conversations.
At the same time, he has found that being online whenever he wants is actually a way to make both his work and life simpler and easier.
"I find it less stressful to get issues that come up out of the way on the spot rather than let the work build up."
How to tame the work-life blur
Is Yahoo's decision to recall remote workers a blow to teleworking?
Solo workers come together to share desks – and a common purpose
A working world without borders
Mobile workers are the 'new norm'
As government telework expands, things to consider

redpajama_set_name: RedPajamaCommonCrawl


A recent column in Time featured the following quote from a job seeker: "I've sent out maybe 4,000 to 5,000 resumes, all over the world. This is my fulltime job. I do it seven days a week. I've got 2,300 rejection letters sitting in my computer; the rest didn't even both to respond. " It's a stark summary of a failed job search, but one that offers an important lesson: the job market has never been tougher … and the old ways of looking for a job never more inadequate. In today's fickle economy, you have to do more than put a check mark in the Done box.
This person has the right outlook. Searching for a job is "a fulltime job. " But, that phrase now has an entirely new meaning. Historically, doing a job was simply a matter of meeting the requirements specified for satisfactory performance. These standards were usually detailed in a position description or union contract. They defined both the minimum and maximum effort an employer expected of an individual worker.
Today, in contrast, those standards define only the minimum. In effect, employers want more. They are now looking for superlative performance. They expect you to fulfill the minimum requirements plus make the maximum contribution possible. Accomplish the first, and you're getting the job done; accomplish the first and the second and you're doing superlative work. The former may make you feel good, but only the latter will make an employer feel good about you.
The same is true of a job search. Meet the minimum requirements – sit at your computer for eight hours and shoot out hundreds or even thousands of resumes – and you can tell yourself that you've put in a full day's work in your hunt for a job. That may make you feel as if you've done your part, but it's unlikely to make a significant impact on employers.
What should you be doing? I think you need to double down on your job search.
In poker, to double down means to double your bet. It doesn't increase your odds of winning, but it does increase the pot if you do win. In a job search, to double down means to double your bet, as well. The goal, however, is not only to increase the size of the pot, but also to double your odds of winning the job you want.
There are two equally important actions involved in doubling down in a job search.
First, you have to increase the potential value you offer to an employer.
Then, you have to increase the odds that an employer will recognize that potential.
Action 1. Enroll in an academic course or training program at the same time you launch your job search. Figure out what skills and knowledge will most enhance your perceived value as an employee and go to work on acquiring them.
That information, of course, has to be true. But assuming it is, your entry tells an employer two things about you which will help you stand out from the crowd. It signals that you recognize the importance of keeping your skill set up-to-date and that you take personal responsibility for doing so. In effect, you've added "conscientiousness" and "initiative" to your list of competencies, and those traits double down your worth to an employer.
Action 2. Practice the Application Two-Step. Don't apply for every opening for which you're even remotely qualified, but instead, focus on a limited number where you are a perfect fit.
For each of those optimum opportunities, execute Step 1 and apply for the opening exactly as the employer specifies. Demonstrate your interest in the position, but do so by signaling that you can follow directions and won't, therefore, complicate matters for the recruiter.
After that, execute Step 2. Start networking online and off. Reach out through social media sites, your professional association, the alumni organization at your college and anyplace else you can think of where you might find a contact in the organization with the opening. Making such connections is hard work and making them work effectively for you is even harder, but done well, you will find someone who is willing to walk your resume into the employer's HR Department and put it on the recruiter's desk. That will help your resume stand out from the crowd and the resulting visibility will double your odds of being considered for the opening.
As with poker, doubling down won't guarantee that you win the job of your dreams. It will, however, double up both your potential value and your visibility to employers. Those outcomes raise the ante for you and make your job search a much safer bet.


as soon as now, if I can be believed.
The past is past. The future is to come.
will nonetheless . . . where was I? Let me pause.
and might never be done, I say, I say.
Thank you; I missed it last night and prefer synopses in sonnet form.


Just 20km west of bustling Amsterdam lies the city of Haarlem, a hidden gem of old Dutch charm that's a favourite with the locals. The city that leant its name to its vibrant Manhattan sister has a whole history of stories that has shaped each cobbled street, with tales of beer brewing, tulip trading and great fires producing the city's striking architecture. As well as providing quintessential Dutch experiences such as browsing markets in the Grote Markt or taking in the views from the Molen de Adriaan windmill, Haarlem is also home to many unique ones. Visit the oldest museum in the Netherlands, attend a beer tasting in former church-turned-microbrewery De Jopenkerk, or wander through the boutique shopping district – affectionately nicknamed De Gouden Straatjes (The Golden Streets), Haarlem has regularly been voted the best shopping destination in the Netherlands.


Who doesn't like Super Bowl commercials? Every Monday morning quarterback will be evaluating their favorite ads.
Being the only country on earth not using the metric system to measure stuff can make things confusing.
City Montessori School in Lucknow, India boasts the largest school population with an enrollment of 32,000 students. I wonder how full detention is?
The air quality in Grants Pass, OR, caused by two fires in 2013 produced air quality so bad it was too high for the air quality meter to read.
Let the shopping frenzy begin. Thanksgiving Day. What are you thankful for this year? Was it the year you thought it would be?
"Will Work For Food." This was the original marketing message of those who had fallen on hard times.
Football players make big bucks but within 2 years of retirement 78 percent will file for bankruptcy. I guess that's not really a fun fact — just a fact.
Your Customers Are Electronic, Are You?


Joondalup United can today confirm the appointment of Terry Maton as the State League Reserves coach for the 2019 season.
The experienced Maton, who holds a UEFA B Licence, joins the club following spells with Stirling Lions and Joondalup City.
At City, Maton oversaw their 18s team who finished second in consecutive seasons.
Prior to moving to Australia, Maton was based in the United Kingdom where he coached at Hayes and Yeading, Fulham Ladies National Premier League, and Middlesex Ladies Representative, among others.
Maton will support first team head coach Nick Jennings as JUFC look to bounce back from last season's disappointing campaign.
Blaine Merredew and Adam Buckley will work with Maton at Reserves level and expressions of interest are now being sought from players keen to be part of the squad.
If you wish to to get involved, simply head to this link, fill out your details, and hit 'Send'.


Top 10 Filipino Christmas Recipes Panlasang PinoyA family Cooking Food Blog that features tried and tested Filipino and Asian Recipes for the home cook. These Christmas baby food ideas and tips for the holiday season will help you make baby's first Christmas extra special! Remember: Your baby should be at least 6 months of age before you introduce him to solid foods, unless your doctor has recommended starting earlier. Another option is to combine arrowroot with homemade chicken stock (1 tbsp arrowroot per 8 fl oz (1 cup) of stock).
If your baby has already been introduced to a wide range of solid foods, then it should be fine to add a little cranberry sauce to his holiday meal.
Depending on where you are in the world, the way in which potatoes are cooked for the Christmas dinner can vary. Bread sauce cooked to a traditional recipe will contain milk, so it would not be suitable if your baby is sensitive to cow's milk or if a family history of dairy allergy exists. One note before going to the kitchen: it is important to understand that the preparation of Panettone requires a great deal of time and effort, as is the case for recipes made with sourdough bread. To begin, wrap the sourdough bread in a towel lightly dusted with flour and set it aside in a warm place for at least 2 hours, allowing the dough to rise to twice its original volume. In order to obtain perfectly leavened dough, heat the oven to 302 F, switch it off, and place the mixing bowl with the dough on the opened door. In the meantime, cut the candied cedar and orange pieces into small cubes and soak the sultana raisins in warm water. Mix 2.2 pounds of flour with 2 tablespoons of salt and place on a pastry board, making a well in the middle.
Knead the dough by hand for 20 minutes, until the dough is tough enough that it can be beat onto the board.
Using a strip of thin cardboard, surround each Panettone with the parchment paper, creating a mould. Place the Panettone on a baking sheet covered in parchment paper, and score the top of the dough with a cross. This entry was posted on Tuesday, December 23rd, 2008 at 06:17 and is filed under academia barilla, . Mince Pies, a sweet pastry crust filled with mincemeat are offered at tea time, carol concerts and Christmas parties all the way through the Christmas season. Dainty canapes and the best of English cheeses are served at Christmas parties and family gatherings alike. While Christmas cake and the world-famous Christmas pudding - flamed with cognac and served with cream and brandy butter - usually round off the festivities - supported by more mince pies and maybe a traditional sherry trifle, syllabub or chocolate yule log. Many of the traditional Christmas foods, such as mincemeat, Christmas cake and Christmas pudding are better for being made weeks in advance and being left to mature. For most families, recipes for Christmas dinner include roast turkey with all the trimmings - that's sprouts, roast potatoes, carrots, chipolata sausages wrapped in bacon, chestnuts, and bread sauce and cranberry sauce. On Boxing Day, a baked ham could take centre stage, served cold with salads and pickles and Cumberland sauce.
And all around the house there will be nibbles, and mince pies, and sweets and nuts and - of course - chocolate!
Or check out my little survival guide of Christmas Food Essentials to make sure you're well prepared for the festivities. If you want to prepare unique foods for Christmas, why don't you try German Christmas food. Indeed, preparing German christmas food will be able to make you feel calm and happy during the Christmas. This may sound fairly straight forward, however finding a nice material for the table cloth is quite difficult. Then add the potato (which will thicken the mixture) and the remaining veggies and puree to the desired consistency. In the UK, roasted potatoes are common – whereas in the US, most families serve mashed potatoes.
But they are not suitable for young babies, as these processed meats contain too much salt. Otherwise, a little creamy bread sauce with baby's Christmas dinner would be fine and would make a lovely dip for veggies if he is finger-feeding.
To reduce wrinkles that you at the moment have or to help stop even more from showing up, consume a diet that's rich in antioxidants.
We began purchasing whole cows from the local farm last year, and it's working out really well for us. We found the recipe browsing the Academia Barilla Gastronomic Library, from the Italian cookbook "Il panettone milanese" (publisher: La Spiga). Don't be discouraged if your home-made Panettone does not look as beautiful as the ones you can buy in the supermarket, because yours will surely be genuine and comparable to the original recipe. Once the dough has fully risen, place the dough on a pastry board and gradually mix in 4.5 oz of flour and some warm water, kneading the dough by hand until smooth. In a separate saucepan, melt 10.5 oz of butter on low heat, making sure that the butter does not burn. Gradually incorporate the leavened dough, the melted butter, and the sugar and egg mixture. Add the candied cedar and orange bits, as well as the raisins, into the dough, kneading the dough for 2-3 minutes until well combined. Place the ball of Panettone onto a piece of parchment paper, buttered and dusted with flour. Let the Panettone leaven in a warm place for at least 6 hours, until it has doubled in volume. This is followed by either Christmas pudding or a Christmas trifle and later tea, coffee and mince pies. Of course you can imagine what taste that you will feel later on after making this cheese sandwich. Of course you can also feel the Christmas day fully by preparing the best food through the recipes of German foods. Try and steer clear of tacky plastic table cloths or paper table cloths (unless the paper is nicely presented). During this special meal, families and friends gather together to share a delightful meal and wish everyone a Joyous Christmas.
Older babies, already comfortable with finger foods, can be given this meal cut up into bite-sized pieces. Puree them together and serve – you can adjust these quantities to create a thinner or thicker gravy if you prefer.
I suggest we all start cooking some healthy recipe like these at home if we're going to fix this horrible situation. Once the syrup is lukewarm, remove from direct heat and place the saucepan on warm water bath (Bain-Marie). For additional information, German food that becomes special in the Christmas day is bread, but what kind of food that will be introduced in here? Since it is a Christmas day, pasta will become the best option since the weather in this day will be cold because of winter. If you want to know other type of Germany foods that you can prepare for Christmas, you can just read the magazine or browse into the internet about any types of German foods that can be prepared for Christmas.
The dishes served usually comes from the Family's Christmas Recipe collection, or they can be something special and only prepared once a year. The ones on that site seem pretty easy, I tried last night and it's the first time my family has eaten well in ages, I just feel great.
This stuff will lead to a nutritious diet that provides many advantages to your total effectively-being. Depending on where you are holding the reception – this is likely to be taken care of for you.
Focus on things that can be served as finger foods like salmon pate with crackers or shrimp cocktails. When you give your physique the tools that it needs, it will be higher in a position to cut back wrinkles and give you glowing stunning skin. You can add the pasta food that you create with some additional ingredients that can make you feel satiny such as vegetables, chicken, and fruit.
Have a variety of ham, pork, salads and finger foods served in an easy to access fashion and complimented with decorations such as flowers and nicely folded napkins. If you have a specific way that you want the food prepared and served it is a good idea to let the wait staff know ahead of time so that they can cater for your desires. Many vegetarians will eat fish (if they aren't strictly vegan) so it is worth keeping this in mind.
Some Christmas plants have poisonous berries (such as mistletoe) and you don't want to guests accidentally picking at those.
Children are likely to be in attendance too – so it is a good idea to help safeguard the food area by not leaving anything on the table that is poisonous. This might be your uncle, your aunt, your dad, or your mom – it might even be yourself.However, there are situations wherein some of us have to travel away from the place that we call home to work or to permanently settle – and if you are not that designated cook, it is hard to replicate those Christmas dishes that you enjoyed.
Varieties to have strawberries run across the slope.
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Rewarded retroactively wholegrain cereals, too, to increase can be an important facet of garden design. Radiation used to kill.
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A London-based translation agency Docsbase, offers professional certified translations of divorce decree from German into English. We are fast growing translation agency specialised in certified translations of personal and legal documents. We offer a fast turnaround time, most competitive prices and precise certified translations into English. Our certified translations are accepted by the Home Office, courts, consulates and other main organisations and institutions in the UK, EU and beyond. We follow strict confidentiality procedures and can guarantee that your divorce decree will stay with us and our trusted translators at all times.
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This short book might just be life changing, it certainly contains some very deep insights. But they are oxymorons in a sense that they are so simple to put into practice despite being profound. I really enjoyed this and would recommend.
Alot of interesting ideas but it is 9 hours of listening to something like a scientific lecture and to be honest I got bored with it.
Would you try another book written by Craig Beck or narrated by Craig Beck?


text:
Q: Saving Model to multiple tables maybe on of you can help my with a little problem im dealing with right now.
I have a simple form with: Loginname, Loginpassword, Name, Firstname, Age. Now i want to save these information in two different tables. Users and UserInfo.
I have a model "user" that holds the login information. Now i want to save the additional data in the other database-table.
The user information is always linked to the normal user.
My thinking was, to store all the information in one big "user model" and upon $model->save() storing the data in the relating tables.
Especially when using the model in different modules it would be helpful to do all the magic in the model and not the controller.
Thanks in advance!

A: I think what you want todo is create a model mapper for handling your users data that will reference both your table models.
There is a good example here:
Multiple Tables for a single Model using Zend_Model_Mappers and Zend_Db_Table_Abstract

redpajama_set_name: RedPajamaStackExchange


text:
Home Articles Making Her Own Way
Making Her Own Way
Jan Randall
A career of challenges fueled Professor Emerita Jan Randall's mission to aid women in science
Professor Emerita of Biology Jan Randall's most important biological discovery came while she was she was sitting on a lawn chair, at night, by herself in the desert. Armed with a flashlight and lots of patience, she was observing an endangered kangaroo rat in southern Arizona, an animal usually identified as asocial. After many nights of study, however, Randall realized that the small rats had a secret. "I discovered that they had a unique form of communication, which is drumming their feet, and that they had a rudimentary social organization," she explained. "Even though they live alone, they know who their neighbors are."
Randall was no stranger to solitary work. When she began studying zoology in the early 1960s, she was one of only a few women pursuing the sciences. While working on her doctorate degree at Washing-ton State University, her animal behaviorist professor often went hunting and fishing with male students, but only visited her study site once. Years later at a desert research station in Arizona, Randall observed that her male colleagues were often accompanied by their wives, who worked as unpaid research assistants — giving them a career boost that she and the other female researchers did not have. "I became a pretty strident feminist, and that kind of carried me through in many ways," she said. "I was determined that I as a woman wasn't going to be thwarted. I persevered."
At San Francisco State University, where Randall began researching and teaching in 1987, she felt more at home. However, she found that there was little opportunity to bond with her female colleagues. In response, she revived SF State's Women in Science and Engineering (WISE) program, which provides opportunities for women in STEM fields to learn from and support one another through presentations, workshops and the establishment of mentoring communities.
After 17 years at SF State, Randall continues her work as an activist as a member of the board of the Endangered Species Coalition and is the author of a new book, "Endangered Species: A Reference Handbook." She also created and funded the WISE scholarship, which is awarded to an outstanding female graduate student in the College of Science & Engineering each year. "I just hope that young women now get mentors," Randall said. "I didn't really have a mentor — that was the hard part. I didn't have anyone saying I should do this or that. I had to make my own way."
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Apr222020 by adminNo Comments
The Intelligent Investor Ebook Download Free
Business, Economics, Nonfiction
The Intelligent Investor Ebook Download Free The Classic Text Annotated to Update Graham's Timeless Wisdom for Today's Market Conditions The greatest investment advisor of the twentieth century, Benjamin Graham taught and inspired people worldwide. Graham's philosophy of "value investing" -- which shields investors from substantial error and teaches them to develop long-term strategies -- has made The Intelligent Investor the stock market bible ever since its original publication in 1949. Over the years, market developments have proven the wisdom of Graham's strategies. While preserving the integrity of Graham's original text, this revised edition includes updated commentary by noted financial journalist Jason Zweig, whose perspective incorporates the realities of today's market...
Rich Dad Poor Dad PDF Download
Business, Economics, Nonfiction, Self Help
Rich Dad Poor Dad PDF Download Rich Dad Poor Dad is Robert T. Kiyosaki's story of growing up with two dads — his real father and the father of his best friend, his "rich dad" — and the ways in which both men shaped his thoughts about money and investing. The book explodes the myth that you need to earn a high income to be rich and explains the difference between working for money and having your money work for you.
Britannia Unchained PDF Download
Economics, Politics
Britannia Unchained PDF Download Britannia Unchained: Global Lessons for Growth and Prosperity At a time of profound economic uncertainty, five rising stars of the Conservative party provide a powerful vision for a resurgent Britain Britannia Unchained argues that Britain needs to abandon pride to learn from other countries to achieve success. It puts Britain firmly within the context of international competition, particularly with emerging economies. Britain is often cited as a classic example of a declining power and yet Britain in the 21st century still possesses immense advantages: the English language, an excellent higher education, the most excellent financial sector in the world, and a convenient time zone. While many commentators forecast the decline of the West, while...
Apr192020 by Free eBookNo Comments
Janesville: An American Story by Amy Goldstein
Business, Economics, History, Nonfiction, Politics, Sociology
Janesville: An American Story by Amy Goldstein eBook Download Financial Times and McKinsey Business Book of the Year Award Winner "Moving and magnificently well-researched...Janesville joins a growing family of books about the evisceration of the working class in the United States. What sets it apart is the sophistication of its storytelling and analysis." —The New York Times A Washington Post reporter's intimate account of the fallout from the closing of a General Motors' assembly plant in Janesville, Wisconsin—Paul Ryan's hometown—and a larger story of the hollowing of the American middle class. This is the story of what happens to an industrial town in the American heartland when its factory stills—but it's not the familiar tale. Most observers record the immediate shock o...
This Fight Is Our Fight by Elizabeth Warren
Autobiography, Biography, Economics, History, Memoir, Nonfiction, Politics
This Fight Is Our Fight by Elizabeth Warren eBook Download The fiery U.S. Senator from Massachusetts and bestselling author offers a passionate, inspiring book about why our middle class is under siege and how we can win the fight to save it Senator Elizabeth Warren has long been an outspoken champion of America's middle class, and by the time the people of Massachusetts elected her in 2012, she had become one of the country's leading progressive voices. Now, at a perilous moment for our nation, she has written a book that is at once an illuminating account of how we built the strongest middle class in history, a scathing indictment of those who have spent the past thirty-five years undermining working families, and a rousing call to action. Warren grew up in Oklahoma, and she's...
Everybody Lies by Seth Stephens-Davidowitz
Business, Economics, Nonfiction, Psychology, Science, Sociology, Technology
Everybody Lies by Seth Stephens-Davidowitz eBook Download Blending the informed analysis of The Signal and the Noise with the instructive iconoclasm of Think Like a Freak, a fascinating, illuminating, and witty look at what the vast amounts of information now instantly available to us reveals about ourselves and our world—provided we ask the right questions. By the end of an average day in the early twenty-first century, human beings searching the internet will amass eight trillion gigabytes of data. This staggering amount of information—unprecedented in history—can tell us a great deal about who we are—the fears, desires, and behaviors that drive us, and the conscious and unconscious decisions we make. From the profound to the mundane, we can gain astonishing knowledge about the h...
No Is Not Enough by Naomi Klein
Economics, History, Nonfiction, Politics
No Is Not Enough by Naomi Klein eBook Download "This is one attempt to uncover how we got to this surreal political moment. It is also an attempt to predict how, under cover of shocks and crises, it could get a lot worse. And it's a plan for how, if we keep our heads, we might just be able to flip the script and arrive at a radically better future." –From the Introduction Donald Trump's takeover of the White House is a dangerous escalation in a world of cascading crises. His reckless agenda—including a corporate coup in government, aggressive scapegoating and warmongering, and sweeping aside climate science to set off a fossil fuel frenzy—will generate waves of disasters and shocks to the economy, national security, and the environment. Acclaimed journalist, activist, and bes...
Mar252020 by Free eBookNo Comments
Utopia for Realists by Rutger Bregman
Economics, History, Nonfiction, Philosophy, Politics, Sociology
Utopia for Realists by Rutger Bregman eBook Download Universal basic income. A 15-hour workweek. Open borders. Does it sound too good to be true? One of Europe's leading young thinkers shows how we can build an ideal world today. After working all day at jobs we often dislike, we buy things we don't need. Rutger Bregman, a Dutch historian, reminds us it needn't be this way-and in some places it isn't. Rutger Bregman's TED Talk about universal basic income seemed impossibly radical when he delivered it in 2014. His April 2017 talk continues to make the provocative case, detailing the idea's 500-year history and even its brief success in a forgotten trial in Manitoba in the 1970s. Nearly a million views later, guaranteed basic income is being seriously considered by leading econom...
Mar202020 by adminNo Comments
Bullshit Jobs by David Graeber
Business, Economics, Nonfiction, Philosophy, Politics, Sociology
Bullshit Jobs by David Graeber eBook Download From bestselling writer David Graeber, a powerful argument against the rise of meaningless, unfulfilling jobs, and their consequences. Does your job make a meaningful contribution to the world? In the spring of 2013, David Graeber asked this question in a playful, provocative essay titled "On the Phenomenon of Bullshit Jobs." It went viral. After a million online views in seventeen different languages, people all over the world are still debating the answer. There are millions of people—HR consultants, communication coordinators, telemarketing researchers, corporate lawyers—whose jobs are useless, and, tragically, they know it. These people are caught in bullshit jobs. Graeber explores one of society's most vexing and deeply felt ...
The Fatal Conceit by Friedrich A. Hayek
Economics, History, Nonfiction, Philosophy, Politics
The Fatal Conceit by Friedrich A. Hayek eBook Download Hayek gives the main arguments for the free-market case and presents his manifesto on the "errors of socialism." Hayek argues that socialism has, from its origins, been mistaken on factual, and even on logical, grounds and that its repeated failures in the many different practical applications of socialist ideas that this century has witnessed were the direct outcome of these errors. He labels as the "fatal conceit" the idea that "man is able to shape the world around him according to his wishes." "The achievement of The Fatal Conceit is that it freshly shows why socialism must be refuted rather than merely dismissed—then refutes it again."—David R. Henderson, Fortune. "Fascinating. . . . The energy and precision with which ...


WHO ARE WE? Established in 1999, Hunter's Friend LLC is one of the nation's largest stand-alone Archery Pro-Shops. We serve bowhunters and archery enthusiasts in every state and in most countries around the world. We are an accredited member of the Better Business Bureau with an A+ rating, we are a Dunn & Bradstreet listed business establishment, and we are a proud member of the Archery Trade Association (ATA).
WHAT WE DO: To make a long story short, we make and sell archery equipment, mostly to bowhunters, and we do it a little differently than other archery retailers. Our philosophy is based on a model of excellent technical service along with great values (not in lieu of). We believe archery consumers shouldn't have to choose between getting good technical service and getting a competitive price. Archery consumers have traditionally had two choices: pay top-dollar to buy from a local pro-shop with competent technical help, or try to save money buying from a discount mail-order store and be "on your own" regarding setup and tuning. It's our belief that bowhunters and archery enthusiasts shouldn't have to choose one or the other. You should be able to get both. Over sixteen years later, that's why we're still here.
TECHNICAL SERVICE IS KEY: We've worked hard to bridge the gap between the local and mail-order concepts. We don't sell bows like most internet and mail-order archery suppliers. Less than 2% of the bows we sell are bare bows. The other 98% are pro-shop compound bow systems (package deals) which we first setup and service in our pro-shop. Today we are one of the busiest independent Archery Pro-Shops in the United States, and we're honored to be the #1 standalone dealership for many of the brands we represent. Our pro shop sets up thousands of compound bows each year, and every single bow and custom arrow set we prepare - from the cheapest to most expensive - gets our very best service and support.
WHAT WE'RE NOT: Of course, no matter how many bows we sell, you won't ever catch us acting like a "big-box" store. We're not a distribution center, a mass-merchant, a warehouse operation, or some kind of "Power Seller." We don't sell trolling-motors, kayaks, or golf bags - and our showroom doesn't even have a fish tank. When you call us, you GET A PERSON - not a prompt. In this store, we have customers, not account numbers. We know that's not how big retail normally goes, but we still prefer an old-fashioned approach. We focus on providing the smartest values and the best services in the archery industry. Archery is what we know; archery is what we love. Perhaps that's why we enjoy one of the best customer satisfaction rates of any Archery Pro-Shop in the United States. If you've haven't bought from our Pro-Shop before, we would love a chance to work for you and show you what we can do. Our crew starts building bows and fletching arrows at 6:30 every morning. If you need something, we'll be here and ready to help.
HOW IT ALL STARTED: We haven't always offered pro-shop compound bow packages to our internet customers. Since the traditional wisdom was that the archery buyer must physically be in the store to ensure a proper fit and setup on a new bow, we simply hadn't considered doing things any other way. At one time, the pro-shop was where we served our walk-in customers, and our website was just the place where we advertised our walk-in services and sold some miscellaneous parts. But as we got to know our new internet customers, we began to discover how many bowhunters abroad had no reasonable access to a fair and reputable Archery Pro-Shop. Time and time again our internet customers remarked that their "local" archery stores were too far away, too expensive, closed in the off season, only willing to offer certain brands, difficult to deal with, etc. We began to wonder if it was possible to bridge the gap between the physical Archery Pro-Shop and internet store. We had the equipment, the technical staff, and the facilities to get the job done. But was it possible to build, fit and tune-up a new bow for a customer 1,000 miles away? Could we offer our internet customers ALL the same services as our walk-in customers and make it work? In spite of what the critics would say, we decided to give it a try.
OUR FIRST ATTEMPT: In the fall of 2001, we launched the program. We began by writing and posting a buyer's help-guide on the web, called simply the Help with the Details page (sorry, we couldn't think of a more creative title). This guide gave our customers the basics on what they needed to know before placing an order: how to measure draw length, how to choose an appropriate draw weight, selecting fletchings, nock set options, peep choices, let-off percentage choices, etc. Then we picked just one compound bow for the experiment, an inexpensive single-cam rig called the Champion Buck Bow. We rigged it up really nice. Everything was included: bow, sight, rest, quiver, arrows, broadheads, sling, stabilizer, release, silencers, even a hard-shell case. We advertised that the bow would be delivered fully setup, adjusted, tuned, pre-sighted and ready-to-shoot. We nervously awaited customer response. To our complete surprise, we had orders the very next day. People seemed to like the idea and they were willing to give us a chance - what a blessing indeed. We filled orders for the Champion Buck Bow from literally all over the US & Canada, and even a few to Europe and South Africa. It certainly wasn't easy. In fact, it was a lot of work, as we wanted to get it perfect and prove our critics wrong. We spent many hours working with every bow until they were all driving tacks. We adjusted every bow to their exact draw weights, draw lengths, and let-offs. We pre-cycled, paper-tuned, group-tuned, chronographed, powder-tested, pre-sighted, and documented results on every single unit meticulously. We test-shot those bows until our arms were too tired to lift up - but we had fun doing it.
HOLDING OUR BREATH: Of course, just selling something doesn't make it successful. We knew the real proof would be the response from our customers. And we were thrilled that of the 125+ "test" bow systems built that season – zero units were returned for the money-back guarantee. Only a small handful of customers needed any adjustments whatsoever and we handled those issues largely by phone. Our pro-shop had found a way to consistently hit the nail on the head, providing internet customers with a professionally setup bow, built to fit, and shooting great right out of the case. In fact, we got a number of thank you letters and photographs from our pro shop package customers, many of whom had bowhunting success that first season. The hard work had paid-off and we were really proud of the results. Incidentally, since the inception of the program, our pro shop compound bow packages continue to have the highest customer satisfaction rate of anything we sell in our store.
THANK YOU HF CUSTOMERS: As we continued to improve our program, offering more services and premium bow lines each subsequent year, the popularity of our pro-shop compound bow packages continued to grow and mature. Today we are one of the busiest independent archery pro-shops in the United States. It would seem we have the Champion Buck Bow and a LOT of great customers to thank. While we never dreamed our pro-shop could be so successful, we are truly blessed and sincerely thankful to all of the bowhunters who took a chance on us - and to the many loyal repeat customers who buy their new compound bows and custom arrows from Hunter's Friend. We truly appreciate your business.


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Comm Daily(R) Notebook
Professor Barbara van Schewick is quoted in this article on net neutrality and broadband regulation:
FCC panelists: Net neutrality protects innovation and freedom
The Microsoft Blog
Professor Barbara van Schewick is quoted on net neutrality from a FCC workshop, where she served as an expert panelist:
Court Decision Could Lead FCC To Redefine Internet
Miller-McCune
Professor Barbara van Schewick talks to reporter Emily Badger about FCC Internet policy and net neutrality:
A federal appeals court in D.C. earlier this week threw up a roadblock to the Federal Communications Commission's plans for the future of the Internet in America. The details of the case were relatively straight-forward: Comcast was caught interfering with traffic by customers using the cumbersome file-sharing application BitTorrent, flouting a 2005 FCC Internet policy stating that Web users are entitled to access the content and applications of their choice.
2015 Parliament of the World's Religions (Past Event)
CIS Residential Fellow Valarie Kaur featured speaker!
For more information and to register visit:
http://www.parliamentofreligions.org/index.cfm?n=35&sn=14
Security of Things Forum (Past Event)
This year's Security of Things Forum will feature two tracks: Leaders and Hackers that are intended to balance high-level talks and panel discussions focused on the operational and policy impacts of securing the Internet of Things with a variety of hands-on demonstrations, tutorials and granular "shop talks" on everything to IoT device hacking to protocols and platform as a service options, to securing IoT devices in enterprises and critical infrastructure settings.
100th Annual Commencement Ceremony (Past Event)
For more information visit the College of Saint Benedict website.
Valarie Kaur, a civil rights lawyer, documentary filmmaker and interfaith leader, will deliver the commencement address during the 100th annual commencement ceremony at the College of Saint Benedict at 2 p.m. Saturday, May 9, in Clemens Field House on the CSB campus, St. Joseph, Minnesota.
DOJ Lawsuit Against California Raises Legal Questions Over Net Neutrality
The U.S. Justice Department has sued California over its net neutrality law.
California Gov. Jerry Brown recently signed the measure, which was in response to the Federal Communications Commission's decision to repeal net neutrality in 2017, which took effect this past June.
To learn more about this lawsuit, The Show spoke with Barbara van Schewick, a law professor and director of the Center for Internet and Society at Stanford Law School.
Expert: Smart Tech Is Making Us Dumb
We know that smart phones and other information technology are changing the way we live and the way we relate to other people, but could they actually be making us dumber?
Brett Frischmann says they are, and that we should question the use of digital technology and surveillance.
Can states regulate the internet?
The days are numbered for federal net neutrality regulations. In response, some states are working on their own versions to prevent internet service providers (ISP) from blocking, slowing or charging more for some web traffic. Oregon, Washington and several other states have made new rules, but a bill working its way through the California legislature would go the furthest. Marketplace Tech host Molly Wood spoke with Ryan Singel, a media and strategy fellow at Stanford Law School, about how a state can regulate a business that crosses state lines.
Stanford Legal with Pam Karlan & Joe Bankman: "Web extra: Net neutrality with guest Barbara van Schewick"
Law School professor Barbara van Schewick discusses net neutrality as the FCC plans to vote on changing those rules.