text:
\section{Introduction}

	We consider a system of $N$ non-interacting fermions in a harmonic trap and study the semiclassical structure of its one-particle density matrix associated with the corresponding Gibbs (equilibrium) state of the trapped gas at temperature $T>0$. More precisely, given a family of one-particle normalized nonnegative trace class operators $\op_{\beta}$ acting on $L^2(\R^d)$ and indexed by the parameter $\beta$, proportional to $T^{-1}$, we consider the following quantities
	\begin{equation}\label{eq:quantum-grad}
		\Dhx\op_{\beta} :=\com{\nabla,\op_{\beta}} \quad\text{ and }\quad \Dhv\op_{\beta} :=\com{\tfrac{x}{i\hbar},\op_{\beta}},
	\end{equation}
	which we refer to as the quantum gradients of $\op_{\beta}$,
	and show that for any $p\in [2,\infty]$,
	\begin{equation*}
		\op_\beta,\, \sqrt{\op_\beta} \in \cW^{1,p}(\sfm)
	\end{equation*}
	uniformly in $\hbar$, where $\cW^{1,p}(\sfm)$ are the semiclassical analogue of weighted Sobolev spaces equipped with the norm
	\begin{equation*}
		\Nrm{\op}{\cW^{1,p}(\sfm)}^p := \Nrm{\op\,\sfm}{\L^p}^p+\Nrm{\Dhx\op\,\sfm}{\L^p}^p + \Nrm{\Dhv\op\,\sfm}{\L^p}^p
	\end{equation*}
	 and $\sfm:=1+\n{\opp}^n$. Here $\opp=-i\hbar\nabla$ is the momentum operator and $\L^p$ are the semiclassical versions of the Lebesgue spaces endowed with the rescaled Schatten norms
	 \begin{equation*}
	 	\Nrm{\op}{\L^p} := h^{\frac{3}{p}} \Tr{\n{\op}^p}^{\frac{1}{p}}
	 \end{equation*}
	 where $h = 2\pi\hbar$.

	The quantities~\eqref{eq:quantum-grad} play key roles in obtaining semiclassical estimates, such as in the problems of the derivation of the Vlasov equation from quantum mechanics and of the Hartree and the Hartree--Fock equations from many-body quantum mechanics. To the author's knowledge, they first appeared in~\cite{benedikter_mean-field_2014}, where the Hartree equation was derived in the mean-field regime for arbitrary long times from a system of $N$ fermions interacting through a smooth potential, under the assumption that the initial state $\op^\init$ of the Hartree dynamics is a pure state, i.e. $(\op^\init)^2=\op^\init$, and satisfies the following semiclassical structures
	\begin{equation}\label{eq:semiclassical-bounds}
		\Nrm{\Dhx\op^\init}{\L^1} \leq C \quad\text{ and }\quad \Nrm{\Dhv\op^\init}{\L^1}\leq C.
	\end{equation}
	Similarly, bounds on semiclassical Schatten norms of commutators are used in~\cite{golse_empirical_2019} to obtain uniform in $\hbar$ mean-field estimates, in~\cite{lafleche_strong_2021} in the context of the semiclassical limit, in \cite{golse_convergence_2021} in the context of the convergence of numerical schemes, in~\cite{golse_semiclassical_2020} to get estimates on the quantum Wasserstein distance.

	When the interaction is the Coulomb potential only partial results are available on a time scale of order one (see \cite{porta_mean_2017}) and they rely on a semiclassical structure expressed in terms of $L^p$ norms of the diagonal of the operator $|\Dhv\op|$, with $p>3$ (see also \cite{lafleche_strong_2021}).

	In~\cite{fournais_optimal_2019} and~\cite{benedikter_effective_2022}, the authors obtained semiclassical bounds of the form~\eqref{eq:semiclassical-bounds} for pure states in the non-interacting case, which in turn provided explicit examples of initial pure states that satisfy the assumptions in~\cite{benedikter_mean-field_2014} on the semiclassical structures. In this paper, we extend this analysis to the finite positive temperature case, where the one-particle reduced density matrix $\op$ is no longer a projection. When dealing with mixed states the analogue relevant quantities considered in~\cite{benedikter_mean-field_2016-1, chong_many-body_2021} are
	\begin{equation}\label{eq:mixed-semiclassical-bounds}
		\Nrm{\Dhx\sqrt{\op^\init}}{\L^p} \leq C \quad\text{ and }\quad\Nrm{\Dhv\sqrt{\op^\init}}{\L^p} \leq C.
	\end{equation}
	In~\cite{benedikter_mean-field_2016-1}, the interaction is smooth and Equation~\eqref{eq:mixed-semiclassical-bounds} is required to hold true for the the initial state for $p=1$. In~\cite{chong_many-body_2021}, singular interactions of the form $\n{x}^{-a}$ with $a\leq 1$ are considered and Equation~\eqref{eq:mixed-semiclassical-bounds} is required to hold for $p\in [2,\infty]$.

	In this work, we focus on quasi-free Gibbs states, which is an important class of examples as they model the equilibria of an ideal gas at finite positive temperature. It is well-known that the one-particle reduced density matrix associated to a Gibbs state can be computed explicitly (see Proposition~\ref{prop:quasi-free}). Using the explicit expression \eqref{eq:explicit-form}, we prove bounds on the weighted $\L^p$ norms of the quantum gradients \eqref{eq:quantum-grad} with $\op_\beta$ given by the one-particle reduced density matrix associated to the Gibbs state of a system of non-interacting fermions confined by a harmonic potential, and on its partition function. Our main result is the following.

	\begin{theorem}\label{thm:main-thm}
		Let $H = \frac{\n{\opp}^2+\n{x}^2}{2}$ be the $d$-dimensional harmonic oscillator Hamiltonian and
		define the family of density matrix operators given by
		\begin{equation}\label{eq:explicit-form}
			\op_\beta : =\dfrac{1}{N\,h^d}\(\id+e^{\beta(H-\mu)}\)^{-1}
		\end{equation}
		where $\mu$ depends on $\beta$, $N$ and $h$ and is chosen so that $h^d\tr(\op_\beta)=1$ holds. Then for any $p\geq 2$,
		$\op_\beta\in\mathcal{W}^{1,p}(\sfm)$ and $\sqrt{\op_\beta}\in\mathcal{W}^{1,p}(\sfm)$ uniformly with respect to $\hbar\in(0,1)$. More explicitly, let
		\begin{equation}\label{eq:def_Z}
			Z_\beta:= h^d\tr(e^{-\beta H}), \quad \text{ and } \quad Z_\mu:= \lambda\, e^{-\beta\mu}
		\end{equation}
		with $\lambda := Nh^d$, then the following holds. For any fixed $\beta \in \R_+$, $\hbar \in (0, 1)$
		and for $p \in [1, \infty]$, we have the bound
		\begin{equation}\label{thm:main_result_III}
			\Nrm{\Dh\op_\beta}{\L^p} \leq C_{d,p}\, \frac{\beta^{\frac{1}{2}-\frac{d}{p}}}{Z_\mu}  \frac{\max\(2\sqrt{2},\beta\hbar\)^{\frac{1}{2}-\frac{1}{p}}}{\(\theta(\beta\hbar)\)^\frac{1}{p}}
		\end{equation}
		where $\theta(x) = \th(x)/x$ with $\th(x) = \tfrac{e^x-e^{-x}}{e^x+e^{-x}}$, and
		\begin{align}\label{thm:main_result_I}
			C_{\lambda,\beta}^{-1}\,Z_{\beta} \leq Z_\mu \leq Z_{\beta}
		\end{align}
		with $C_{\lambda,\beta} = 2$ if $\mu \le d\hbar/2$ and $C_{\lambda, \beta}= 1 + e^{\beta\lambda^{1/d}/\pi}$ if $\mu \ge d\hbar/2$.
	\end{theorem}

	\begin{remark}
	As a corollary of our result, we exhibit a class of initial states satisfying the regularity assumptions of \cite[Theorem~3.1]{chong_many-body_2021}.
	\end{remark}

	\begin{remark}
		Our proof in the case $p=\infty$ is rather general and in particular can be adapted without difficulty to other external trapping potentials $V(x)$, that is to the case of an Hamiltonian of the form $H = \frac{\n{\opp}^2}{2} + V(x)$.
	\end{remark}

	\begin{remark}
		As indicated in \cite[Figure 1]{chong_many-body_2021}, there are three scaling regimes to keep in mind when studying mean-field dynamics of quantum systems (both bosonic and fermionic), namely, $N h^d\rightarrow 0, N h^d\rightarrow \infty$, and $N h^d$ converges to some constant (in fact, $N h^d=1$ which we called the critical scaling).

		Let $\beta>0$ be fixed. In the {lower density} scaling regime $N h^d\rightarrow 0$,  we have that $e^{\beta\mu}\rightarrow 0$ or equivalently $\mu\rightarrow -\infty$, which is an immediate consequence of  \eqref{thm:main_result_I} in the above theorem and the fact that $Z_\beta$ is bounded (cf. Section \ref{sect:bounds_on_inverse_fugacity}). It is not difficult to see that in this case $\op_\beta=\(Nh^d + Z_\mu\,e^{\beta H}\)^{-1}$ will converge to $Z_\beta^{-1} \,e^{-\beta H}$. In other words, in the semiclassical limit $\hbar\rightarrow 0$, $\op_\beta$ will converge to the Maxwell--Boltzmann distribution. This result should not be surprising to the reader since in the lower density scaling regime $Nh^d\to 0$, the Fermi gas behaves like a classical ideal gas.

		In the case of the higher density scaling regime $N h^d\rightarrow \infty$, we recall that as in \cite[Equation~(16)]{chong_many-body_2021}, $\lambda = Nh^d \leq \frac{1}{\Nrm{\op}{\L^\infty}} =: \cC_\infty^{-1}$. This shows that $\Nrm{\op}{\L^\infty}\rightarrow 0$ as $\lambda\rightarrow \infty$. In short, the higher density scaling regime is not accessible to Fermi gases. From a more physical perspective, a high density Fermi gas exhibits quantum ``degeneracy'' behavior, consequence of the Pauli exclusion principle, which limits the gas from getting arbitrarily dense.

		Finally, in the critical scaling case, we see that $\op_\beta$ does not converge to the Maxwell--Boltzmann distribution but instead to the Fermi--Dirac distribution. The divergence from classical behavior is attributed to the fact that at the critical scaling we are in fact modeling the degeneracy behavior of the Fermi gas. See e.g.~\cite[Chapter 17.5]{chapman_mathematical_1990}.
	\end{remark}

	\begin{remark}
		In the semiclassical limit $\hbar\to 0$, then our main Inequality~\eqref{thm:main_result_III} implies
		\begin{equation*}
			\Nrm{\Dh\op_\beta}{\L^p} \leq C_{d,p,\lambda}\, \beta^{\frac{1}{2}+\frac{d}{p'}} + o(\hbar)
		\end{equation*}
		which is optimal in the sense that it is consistent with the behavior of the classical norm $\Nrm{\nabla\(Z_\beta^{-1}e^{-\beta\n{z}^2}\)}{L^p(\RRd)}$ in terms of $\beta$. It is however interesting to see that this scaling seems to change at low temperature and when $\hbar$ is not negligible (i.e. when $\beta\hbar$ is large).
	\end{remark}

\subsection{Second quantization formalism}
	To introduce quasi-free Gibbs states, we will need the second
	quantized description of a many-body system of non-interacting fermions in an external trap.
	Here we give a brief review of the Fock space formalism. For a more comprehensive treatment, the reader could consult \cite{bratteli_operator_1981, solovej_many_2014}.

	The fermionic (antisymmetric) Fock space over the one-particle Hilbert space $\h = L^2(\R^d)$ is defined to be the closure of
	\begin{equation*}
		\cF :=\bigoplus^\infty_{n=0} \h^{\wedge n} = \CC \oplus \h \oplus (\h\wedge \h)\oplus \cdots
	\end{equation*}
	with respect to the norm induced by the standard associated inner product. Here the $n$-th sector of $\cF$ given by $\h^{\wedge n}$ denotes the subspace of $\h^n = L^2(\R^{dn})$ containing functions that satisfy the antisymmetry property
	\begin{equation*}
		\psi(x_1, \ldots, x_n) = \operatorname{sgn}(\sigma)\, \psi(x_{\sigma(1)}, \ldots, x_{\sigma(n)})
	\end{equation*}
	for any permutation $\sigma$. An important vector to note is the vacuum vector $\Omega = (1, 0, 0, \ldots)\in \cF$ which describes a pure state with no particles.

	The operator-valued distribution $a_x$ and $a^\ast_x$ on $\cF$ are defined by their actions on the $n^\text{th}$ sector of $\cF$ as follows. Let $\psi \in \h^{\wedge n}$, then
	\begin{align*}
		(a_x\psi)(x_1, \ldots, x_{n-1}) &:= \sqrt{n}\,\psi(x, x_1, \ldots, x_{n-1}),
		\\
		(a_x^\ast\psi)(x_1,\ldots, x_{n+1}) &:= \sum^{n+1}_{j=1}\frac{(-1)^{j-1}}{\sqrt{n+1}}\, \delta_x(x_j)\, \psi(x_1,\ldots,\cancel{x_j},\ldots, x_{n+1}).
	\end{align*}
	Furthermore, for each $\varphi \in \h$, we associate to it the annihilation operator and its adjoint the creation operator, defined by
	\begin{equation*}
		a(\varphi)= \intd \conj{\varphi(x)}\,a_x\dd x \quad\text{ and } \quad a^\ast(\varphi)= \intd \varphi(x)\,a^\ast_x\dd x.
	\end{equation*}
	It can be shown that creation and annihilation operators are bounded on $\cF$ with operator norm $\Nrm{a(\varphi)}{\infty} = \Nrm{a^\ast(\varphi)}{\infty} = \Nrm{\varphi}{\h}$ and that they satisfy the canonical anticommutation relations, i.e.
	\begin{equation*}
		\{a(\varphi), a(\phi)\} = \{a^\ast(\varphi), a^\ast(\phi)\} = 0 \quad \text{ and } \quad \{a^\ast(\varphi), a(\phi)\}  = \inprod{\phi}{\varphi}_\h\id
	\end{equation*}
	hold for all $(\varphi, \phi) \in \h^2$ where $\{\sfA, \sfB\}:=\sfA\sfB+\sfB\sfA$ is the standard anticommutator of the operators $\sfA$ and $\sfB$.

	In this work, we are interested in trapped non-interacting Fermionic systems modeled by the one-particle Hamilton operator
	\begin{equation}\label{eq:trap-hamiltonian}
		H = \tfrac{1}{2} \n{\opp}^2 + V(x),
	\end{equation}
	where $\opp = -i \hbar \grad_x$ and $V$ is an external trapping potential. The second quantization of the one-particle operator $H$ on $\cF$  is defined to be the operator $\sfH=\dG(H):=\intd a^\ast_x H a_x\dd x$ whose action on the $n^\text{th}$ sector is given by
	\begin{equation*}
		\(\dG(H)\Psi\)(x_1,\ldots, x_n)= \sum^n_{j=1} (-\tfrac{\hbar^2}{2}\lapl_{x_j}+V(x_j))\Psi(x_1,\ldots, x_n).
	\end{equation*}
	Another useful operator on $\cF$ is the number operator
	\begin{equation*}
		\cN = \bigoplus^\infty_{n=0} n\,\id_{\h^{\wedge n}}.
	\end{equation*}

\subsection{Harmonic oscillators}\label{sec:harmonic_oscillator}

	To simplify the presentation, we choose the external potential $V$ to be the harmonic trapping potential, that is,
	\begin{equation}\label{eq:harmonic-hamiltonian}
		H = \tfrac12 \n{\opp}^2+ \tfrac12 \n{x}^2 = \sum^d_{i=1} \tfrac12 \,\opp_i^2+\tfrac12\, x_i^2 = \sum^d_{i=1}H_i.
	\end{equation}
	Since $H$ is the $d$-dimensional quantum harmonic oscillator, it is well-known that it emits a discrete spectrum where the eigenvalues and eigenvectors are given by
	\begin{align*}
		E_{n} &= \(\n{n}_1 + \tfrac{d}{2}\) \hbar &&\text{for } n = (n_1, \ldots, n_d) \in \N_{0}^d,
		\\
		u_{E_{n}} &= \phi_{n_1}\otimes\cdots \otimes\phi_{n_d} &&\text{with } H_i\phi_{n_i} = \(n_i+\tfrac12\)\hbar\, \phi_{n_i}.
	\end{align*}
	where $\n{n}_1 = n_1+\ldots+n_d$ and the functions $\phi_{n_i}$ are the standard one-variable Hermite functions
	\begin{equation*}
		\phi_{n_i}(x_i)= c_{n_i} \,e^{-x^2/2}P_{n_i}(x_i) \quad \text{ with } \quad c_{n_i}^{-1} = 2^{\frac{n_i}{2}} (n_i!)^{\frac12} \(\pi\hbar\)^\frac14,
	\end{equation*}
	with $P_{n_i}(x)$ the Hermite polynomials. Moreover, for each eigenvalue $E_{n}$ the corresponding multiplicity is given by $g_{\n{n}_1, d} = \binom{\n{n}_1+d-1}{d-1}$. In particular, by the spectral decomposition of $H$, we could now rewrite $\sfH$ as
	\begin{equation}
		\sfH = \sum_{n\in \Z_0^d} E_{n} \, a^\ast(u_{E_n})\,a(u_{E_n}).
	\end{equation}

\subsection{Thermal states}

	We consider quantum states that are given by density operators $\opom$ on $\cF$, that is, the expectation of an observable $\sfA$ is given by $\bangle{\sfA}_{\opom} = \tr_{\cF}\(\sfA\, \opom\)$ for every $\sfA \in \cB(\cF)$, where $\cB(\cF)$ denotes the space of bounded operators on the Fock space. The Gibbs equilibrium of a trapped non-interacting Fermi gas associated to some positive temperature $T>0$ is defined to be the unique minimizer of the Gibbs free energy functional
	\begin{equation*}
		\ecF(\opom) = \tr_{\cF}\(\sfH\,\opom\) - \kb\, T\,\ecS(\opom) \quad \text{ with } \quad \ecS(\opom)=-\tr_{\cF}\(\opom\, \log\opom\)
	\end{equation*}
	in $\cX_N=\{\opom \in \cB(\cF) \mid \opom \ge 0,\ \tr_{\cF}\(\opom\)=1,\ \tr_{\cF}\(\cN\,\opom\)=N\}$ where $\ecS$ is the von Neumann entropy. It can be checked that the Gibbs state associated to the temperature $T$ is given by the normalized positive trace class operator
	\begin{equation}\label{def:gibbs_states}
		\gibbs = \frac{1}{\cZ_N}\,e^{-\beta \(\sfH - \mu\cN\)}
	\end{equation}
	where $\beta = 1/(\kb\,T)$ and the chemical potential $\mu$ is chosen so that $\Tr{\cN \gibbs}=N$.
	Here $\cZ_N$ is the grand canonical partition function
	\begin{equation*}
		\cZ_N = 1+\sum_{n=1}^\infty e^{n\beta\mu} \tr_{\h^{\wedge n}} \exp\bigg(-\beta \sum^{dn}_{j=1} H_j\bigg)
		= \prod_{n \in \Z_0^d} (1+e^{-\beta (E_{n}-\mu)}).
	\end{equation*}

\subsection{Quasi-free states}

	Here we will only give a rudimentary introduction to the tools necessary for the subsequent sections. A more comprehensive exposition of the following content can be found in \cite{bratteli_operator_1981, solovej_many_2014}. The state $\opom$ is said to be quasi-free if for all $n\in \N$, we have that
	\begin{align*}
		&\small \tr_{\cF}\(a^{\sharp_1}(f_1)\cdots a^{\sharp_{2n+1}}(f_{2n+1})\opom\) = 0,
		\\
		&\small \tr_{\cF}\(a^{\sharp_1}(f_1)\cdots a^{\sharp_{2n}}(f_{2n})\opom\) = \sum_{\sigma} (-1)^{\sigma} \prod^n_{j=1} \Tr{a^{\sharp_{\sigma(j)}}(f_{\sigma(j)})a^{\sharp_{\sigma(j+n)}}(f_{\sigma(j+n)})\opom},
	\end{align*}
	where $a^{\sharp_j}$ stands for either $a$ or $a^*$, and the sum is taken over all permutations $\sigma$ satisfying $\sigma(1) < \sigma(2)<\ldots <\sigma(n)$, and $\sigma(j) < \sigma(j+n)$, for all $j\in\set{1,\ldots, n}$. The definition indicates that quasi-free states are determined by the one-particle reduced density matrix operator defined via its integral kernel
	\begin{align}
		\op(x, y)= \frac{1}{N h^d}\tr_{\cF}\(a^\ast_y\, a_x\, \opom\),
	\end{align}
	and the antisymmetric pairing function defined by
	\begin{align}
		\opal(x, y) = \frac{1}{N h^d}\tr_{\cF}\(a_y \, a_x\, \opom\).
	\end{align}
	More compactly, we introduce the generalized one-particle density matrix $\opG:\h\oplus\h^\ast\rightarrow \h\oplus\h^\ast$ defined by
	\begin{align}\label{def:generalized_opdm}
		\opG =
		\begin{bmatrix}
			\op & \opal\\
			-\mathds{J}\opal\mathds{J} & \id -\mathds{J}\op \mathds{J}^\ast
		\end{bmatrix}
	\end{align}
	where $\mathds{J}:\h\rightarrow \h^\ast$ is  the map $\mathds{J}(\phi)=\inprod{\phi}{\cdot}$ and $\mathds{J}^\ast$ is its adjoint operator. This observation is summarized by the following proposition (see e.g. \cite[Appendix G]{solovej_many_2014}).

	\begin{prop}
		Let $\opG:\h\oplus\h^\ast\rightarrow \h\oplus\h^\ast$ be an operator of the form \eqref{def:generalized_opdm}. Then $\opG$ is the generalized one-particle density matrix of some quasi-free state with finite particle number if and only if $\opG\ge 0$ and $\tr\op<\infty$.
	\end{prop}

	In the case of $\gibbs$, it can be readily shown that its generalized one-particle density matrix is given by
	\begin{equation*}
		\opG_{\beta}
		=
		\begin{bmatrix}
			\op_\beta & 0
			\\
			0 & \id-\mathds{J}\op_\beta \mathds{J}^\ast
		\end{bmatrix}
	\end{equation*}
	where $\opal = 0$, which we refer to as the gauge-invariant condition. The following proposition, whose proof can be found for example in \cite[Proposition~5.2.23]{bratteli_operator_1981}, makes the link between the Fock space Gibbs state~\eqref{def:gibbs_states} and the one body Gibbs state considered in our main theorem.

	\begin{prop}\label{prop:quasi-free}
		Let $\gibbs$ denotes the Gibbs state \eqref{def:gibbs_states} with $\Tr{e^{-\beta\sfH}} <\infty$. Then $\gibbs$ is quasi-free and its one-particle reduced density matrix operator is given by Equation~\eqref{eq:explicit-form}.
	\end{prop}

\section{Semiclassical thermal regularity}

In this section, we give the proof of the main theorem.

\subsection{Preliminaries}
	For a trapped non-interacting Fermi gas with Hamiltonian~\eqref{eq:trap-hamiltonian}, let us write $G = e^{-\beta H}$ and $\lambda = N h^d$. Consider the quantities
	\begin{subequations}
		\begin{align}
			\opg_\beta &= Z_\beta^{-1} \,e^{-\beta H},
			\\
			\op_\beta &= \lambda^{-1}\(\id + e^{\beta (H-\mu)}\)^{-1} = G_\mu \(\id+\lambda\, G_\mu\)^{-1},
		\end{align}
	\end{subequations}
	where $G_\mu = \lambda^{-1}\,e^{-\beta\(H-\mu\)} = Z_\mu^{-1} \,G$ and $Z_\beta$ and $Z_\mu$ are given in Equation~\eqref{eq:def_Z}. The quantum gradients of these operators are given by the following lemma.
	\begin{lem}
		Let $\Dh$ be a quantum gradient. Then we have
		\begin{subequations}
			\begin{align}\label{eq:gradient_gaussian}
				\Dh\opg_\beta &= -\frac{\beta}{Z_\beta} \int_0^1 G^{1-s} \(\Dh H\) G^s \dd s,
				\\\label{eq:gradient_fermi_dirac}
				\Dh\op_\beta &= -\beta \int_0^1 G_\mu^{1-s}(\id + \lambda G_\mu)^{-1} \(\Dh H\) (\id + \lambda G_\mu)^{-1} G_\mu^s\dd s.
			\end{align}
		\end{subequations}
	\end{lem}

	\begin{proof}
		Notice that for any operators $A$ and $B$, we have that
		\begin{equation*}
			\dpt \com{A,e^{tB}} = \com{A,B\,e^{tB}} = \com{A,B}e^{tB} + B\, \com{A,e^{tB}}.
		\end{equation*}
		Therefore, $\dpt\(e^{-tB}\com{A,e^{tB}}\) = e^{-tB}\com{A,B}e^{tB}$ and we obtain the Duhamel-like formula
		\begin{equation}
			\com{A,e^{tB}} = \int_0^t e^{\(t-s\)B}\com{A,B}e^{sB}\dd s.
		\end{equation}
		In particular, taking $t=1$ and $B=-\beta H$, we obtain the identity
		\begin{equation*}
			\Dh G = -\beta \int_0^1 G^{1-s} \(\Dh H\) G^s\dd s
		\end{equation*}
		from which we deduce Identity~\eqref{eq:gradient_gaussian}. Then observe that if $A$ is an invertible operator, by the Leibniz rule for commutators, $0 = \Dh\!\(AA^{-1}\) = \(\Dh A\)A^{-1} + A\Dh\!\(A^{-1}\)$ and so
		\begin{equation}\label{eq:derivative_A_inverse}
			\Dh\!\(A^{-1}\) = - A^{-1} \(\Dh A\)A^{-1}.
		\end{equation}
		In particular, since $\Dh\(\id+\lambda G_\mu\) = \lambda \Dh G_\mu$, we deduce that
		\begin{align*}
			\Dh\op_\beta &= \(\Dh G_\mu\) \(\id + \lambda G_\mu\)^{-1} - G_\mu \(\id+\lambda G_\mu\)^{-1} \lambda \(\Dh G_\mu\) \(\id + \lambda G_\mu\)^{-1}
			\\
			&= \(\id + \lambda G_\mu\)^{-1} \(\Dh G_\mu\) \(\id + \lambda G_\mu\)^{-1}
		\end{align*}
		which leads to Formula~\eqref{eq:gradient_fermi_dirac}.
	\end{proof}
	Taking the square root of $G$ changes $\beta$ by $\beta/2$. Hence it follows that
	\begin{equation*}
		\Dh\sqrt{\opg_\beta} = -\frac{\beta}{2\sqrt{Z_\beta}} \int_0^1 G^{(1-s)/2} \(\Dh H\) G^{s/2} \dd s.
	\end{equation*}
	On the other hand, it is more difficult to compute explicitly $\Dh\sqrt{\op_\beta}$. The following lemma allows us to bound $\sqrt{\op_{\beta}}$ by reducing the problem to estimating $G$.

	\begin{lem}
		Let $\sfm$ be a self-adjoint operator. Then
		\begin{equation*}
			\Nrm{\Dh\sqrt{\op_\beta}}{\L^p(\sfm)} \leq \(\Nrm{\Dh \sqrt{G_\mu}}{\L^p(\sfm)} + \frac{\lambda}{2} \Nrm{\sqrt{\op_\beta}}{\L^q(\sfm)} \Nrm{\Dh G_\mu}{\L^r}\)
		\end{equation*}
		for any $1\le p, q, r\le \infty$ such that $\frac{1}{p}= \frac{1}{q} + \frac{1}{r}$.
	\end{lem}

	\begin{proof}
		By Identity \eqref{eq:derivative_A_inverse}, we have that
		\begin{equation}\label{eq:gradient_sqrt}
		\begin{aligned}
			\Dh\sqrt{\op_\beta}
			= \(\Dh \sqrt{G_\mu} - \sqrt{\op_\beta} \(\Dh \sqrt{\id+\lambda G_\mu}\)\) \(\id+\lambda G_\mu\)^{-\frac{1}{2}}.
		\end{aligned}
		\end{equation}
		To bound $\Dh \sqrt{\id+\lambda G_\mu}$, we proceed as in \cite[Lemma~7.1]{chong_many-body_2021}. Let $A := \lambda G_\mu$.
		Since $0\leq A \leq 2\,c := \lambda Z_\mu^{-1}$, we deduce that $\Nrm{A-c}{\L^\infty} \leq c$ and so the following series is absolutely convergent
		\begin{equation*}
			\sqrt{\id + A} = \sqrt{1+c}\,\sqrt{\id+\tfrac{1}{1+c}\(A-c\)} = \sum_{n=0}^\infty \binom{1/2}{n} \(\tfrac{1}{1+c}\)^{n-\frac{1}{2}} \(A-c\)^n.
		\end{equation*}
		Since it follows from the Jacobi identity that
		\begin{equation*}
			\Dh\(A-c\)^n = \sum_{k=1}^n \(A-c\)^{k-1} \(\Dh A\) \(A-c\)^{n-k},
		\end{equation*}
		then we deduce the estimate
		\begin{align*}
			\Nrm{\Dh\sqrt{\id+A}}{\L^p} &\leq \sum_{n=1}^\infty \n{\binom{1/2}{n}} \(\tfrac{1}{c+1}\)^{n-\frac{1}{2}} n \Nrm{\Dh A}{\L^p} \Nrm{A-c}{\L^\infty}^{n-1}
			\\
			&\leq \frac{1}{2\sqrt{c+1}} \,\sum_{n=1}^\infty \binom{-1/2}{n-1} \(\tfrac{-c}{c+1}\)^{n-1} \Nrm{\Dh A}{\L^p} = \frac{1}{2} \Nrm{\Dh A}{\L^p}.
		\end{align*}
		We conclude by applying the fact that $\Nrm{\Dh\sqrt{\op_\beta} \,\sfm}{\L^p}= \Nrm{\sfm\,\Dh\sqrt{\op_\beta}}{\L^p}$, which follows from taking the adjoint, and the triangle inequality to Formula~\eqref{eq:gradient_sqrt}. This yields the desired result.
	\end{proof}

\subsection{Bounds on the inverse fugacity}\label{sect:bounds_on_inverse_fugacity}

	In the rest of this paper, we assume $H$ is the harmonic oscillator Hamiltonian $H = \frac{\n{\opp}^2 + \n{x}^2}{2}$. In this case of the partition function in $\opg_\beta$ has the closed form
	\begin{equation}
		Z_\beta =  \(\frac{2\pi}{\beta}\)^d \frac{1}{\operatorname{shc}(\frac{\beta\hbar}{2})^d}
	\end{equation}
	where $\operatorname{shc}(x) = \sh(x)/x$ denotes the hyperbolic sinc function. In particular $Z_\beta \leq (2\pi/\beta)^d$ and $Z_\beta \sim (2\pi/\beta)^d$ as $\beta\hbar\rightarrow 0$. The function $Z_\mu$, that one could call the inverse fugacity, can be compared to the partition function $Z_\beta$ as proved in the following proposition.

    \begin{prop}\label{prop:bounds_on_chemical_potential}
		Let $Z_\mu= \lambda\, e^{-\beta\mu}$ with $\lambda = Nh^d$. Then the following inequality holds
		\begin{equation}\label{eq:bound_cN}
			C_{\lambda,\beta}^{-1}\,Z_{\beta} \leq Z_\mu \leq Z_{\beta}
		\end{equation}
		with $C_{\lambda,\beta} = 2$ if $\mu \leq d\hbar/2$ and $C_{\lambda,\beta} = 1 + e^{\beta\lambda^{1/d}/2\pi}$ if $\mu \ge d\hbar/2$.
	\end{prop}

	\begin{proof}[Proof of Proposition \ref{prop:bounds_on_chemical_potential}]
		Let $c_\mu = e^{\beta\mu} = N h^d/Z_\mu$. Notice that
		\begin{equation*}
			1 = h^d \Tr{\op_\beta} = \frac{1}{N} \Tr{\big(\id + c_\mu^{-1}e^{\beta H}\big)^{-1}} \leq \frac{c_\mu}{N} \Tr{e^{-\beta H}},
		\end{equation*}
		which implies the first inequality in Formula~\eqref{eq:bound_cN}.

		Next, define the function $g:\R_+\to\R_+$ by $g(r) = \big(1+c_\mu^{-1}\,e^{\beta r}\big)^{-1}$, then for any $R>0$, it holds $g(r) \geq \big(1+c_\mu^{-1}\,e^{\beta r}\big)^{-1} \,\indic_{r\leq R}$, which leads to
		\begin{equation*}
			1 = \frac{1}{N} \tr\(g(H)\) \geq \frac{1}{N\(1+c_\mu^{-1}\,e^{\beta R}\)}\, \tr\(\indic_{H\leq R}\).
		\end{equation*}
		Recalling the property of the harmonic oscillator given in Section~\ref{sec:harmonic_oscillator}, one sees that the trace of the characteristic function is nothing but
		\begin{equation*}
			\Tr{\indic_{H\leq R}} = \n{\Set{n\in\N^d_0 : \(\n{n}_1+\tfrac{d}{2}\)\hbar \leq R}}.
		\end{equation*}
		Since $\n{n}_1 \geq \sup_{j} n_j =: \n{n}_\infty$, then this can be crudely estimated by
		\begin{equation}
			\Tr{\indic_{H\leq R}} \geq \n{\Set{n\in\N^d_0 : \n{n}_\infty \leq \tfrac{R}{\hbar}-\tfrac{d}{2}}} = {\(\floor{\tfrac{R}{\hbar}-\tfrac{d}{2}}+1\)^d}.
		\end{equation}
		In particular, since $\mu\ge d\hbar/2>0$, then taking $R = \mu = \frac{\ln c_\mu}{\beta}$ yields $\tfrac{\mu}{\hbar}-\frac{d}{2} \leq \big(N^{1/d}+1\big)$ and so since $N \geq 1$,
		\begin{equation}\label{eq:bound_mu}
			\mu \leq 2\,N^{1/d}\hbar + \frac{d\hbar}{2}.
		\end{equation}

		Now, let us obtain upper bounds for $c_\mu$. First, observe that we have the lower bounds
		\begin{equation}\label{ineq:lower_for_c_N}
			1 = \frac{c_\mu}{N} \Tr{\big(\id + c_\mu e^{-\beta H}\big)^{-1}e^{-\beta H}} \geq \frac{Z_\beta\,c_\mu}{Nh^d\(1+c_\mu e^{-d\beta\hbar/2}\)}.
		\end{equation}
		If $c_\mu \le e^{d\beta\hbar/2}$ (i.e. $\mu\le d\hbar/2$) then it follows from Inequality \eqref{ineq:lower_for_c_N} that
		\begin{align*}
			c_\mu \le 2\, N h^d Z_\beta^{-1}.
		\end{align*}
		On the other hand, if $c_\mu \ge e^{d\beta\hbar/2}>1$ (i.e. $\mu \ge d\hbar/2$), then it follows from Inequality \eqref{eq:bound_mu} and Inequality \eqref{ineq:lower_for_c_N} that we have the bound
		\begin{align}
				c_\mu \leq \(1+c_\mu\, e^{-d\beta\hbar/2}\) N h^d Z_\beta^{-1} \leq \(1+e^{2\beta\hbar N^{1/d}}\) N h^d Z_\beta^{-1}
		\end{align}
		This completes our proof of the proposition.
	\end{proof}

	\subsection{\texorpdfstring{$\L^\infty$}{L-infty} bounds} This section is devoted to the proof of the following proposition.
	\begin{prop}\label{prop:regu_thermal_harmonic}
		Let $\beta >0$ and $H = \frac{\n{\opp}^2 + \n{x}^2}{2}$, then we have the estimates
		\begin{align*}
			\Nrm{\Dh\opg_\beta}{\L^\infty} &\leq \frac{2}{Z_\beta}\max\(\sqrt{\beta}, \beta\sqrt{\hbar}\),
			\\
			\Nrm{\Dh\op_\beta}{\L^\infty} &\leq \frac{2}{Z_\mu} \max\(\sqrt{\beta}, \beta\sqrt{\hbar}\).
		\end{align*}
	\end{prop}

	Prior to giving the proof of the above proposition, let us make the following observation. Since $\Dhx{H} = x$ and $\Dhv{H} = \opp$, we deduce that $\Dh_{z} H =\opz := (x,\opp)$ where $z=(x, \xi)$. By Identity \eqref{eq:gradient_gaussian}, we have that
	\begin{equation*}
		\Dh\op_\beta = -\beta \int_0^1 (\id + \lambda G_\mu)^{-1}G_\mu^{1-s} \,\opz\, G_\mu^s(\id + \lambda G_\mu)^{-1}\dd s
	\end{equation*}
	which implies the estimate
	\begin{equation*}
		\Nrm{\Dh\op_\beta}{\L^p} \leq \beta\,\int_0^1 \Nrm{G_\mu^{1-s} \opz\, G_\mu^s}{\L^p} \d s = \beta\, Z_{\mu}^{-1}\,\int_0^1 \Nrm{G^{1-s} \opz\, G^s}{\L^p} \d s.
	\end{equation*}
	Since $\Nrm{G}{\L^\infty}\leq 1$, then we have the estimate
	\begin{equation}\label{eq:split_integral}
		\begin{aligned}
			\Nrm{\Dh\op_\beta}{\L^p} &\leq \beta\, Z_\mu^{-1}\int_0^{1/2}  \Nrm{G^{1-s} \opz}{\L^p} \d s + \beta\, Z_\mu^{-1}\int_{1/2}^{1}  \Nrm{\opz \,G^{s}}{\L^p} \d s
			\\
			&\leq 2\,\beta\, Z_\mu^{-1} \int_{1/2}^1  \Nrm{\opz\,G^{s}}{\L^p} \d s.
		\end{aligned}
	\end{equation}
    Hence to estimate $\Dh\op_\beta$ and $\Dh\opg_\beta$, it remains to estimate the value of $\Nrm{x\,e^{-\beta s H}}{\L^p}$ and $\Nrm{\opp\,e^{-\beta s H}}{\L^p}$ for $s\in[\frac12,1]$. Let start with the case $p=\infty$.
	\begin{lem}\label{lem:weights_vs_gaussian}
		Let $\beta>0$, then we have the estimate
		\begin{align*}
			\Nrm{\n{x}^n e^{-\beta H}}{\L^\infty}^{2/n} = \Nrm{\n{\opp}^n e^{-\beta H}}{\L^\infty}^{2/n} \leq n\max\(\tfrac{2}{\beta}, \sqrt{2}\,\hbar\).
		\end{align*}
	\end{lem}

	\begin{remark}
		In the classical case, it is not difficult to prove that the maximum of the function $x\mapsto \n{x}^n e^{-\beta \n{x}^2}$ is $\(\frac{n}{2e\beta}\)^{n/2}$, and more generally,
		\begin{equation*}
			\Nrm{\n{x}^n e^{-\beta \n{x}^2}}{L^p}^p = \omega_d\,\Gamma(\tfrac{d+np}{2}) \(\beta p\)^{-\frac{d+np}{2}}
		\end{equation*}
		where $\omega_d$ is the volume of the $d$-dimensional unit ball and $\Gamma$ is the gamma function.
	\end{remark}

	\begin{proof}[Proof of Lemma~\ref{lem:weights_vs_gaussian}]
		It is sufficient to prove a bound on the first quantity because $H$ is symmetric in $x$ and $\opp$. Let $\varphi\in L^2$ and $\psi = e^{-t H} \varphi$. Notice that for any $t\geq 0$, $\Nrm{\psi}{L^2} \leq \Nrm{\varphi}{L^2}$. Let $y := \Nrm{\n{x}^{n} \psi}{L^2}^{2/n}$. Since $2\,\dpt\psi = - \big(\n{x}^2-\hbar^2\Delta\big)\psi$, then integrating by parts yields
		\begin{multline*}
			\dt y^n = - \intd\n{\psi}^2 \n{x}^{2\(n+1\)} \d x - \hbar^2 \Re{\intd \nabla\(\conj{\psi} \n{x}^{2n}\) \cdot\nabla\psi\dd x}
			\\
			\leq - \intd \n{\psi}^2 \n{x}^{2\(n+1\)} + \hbar^2 \n{\nabla\psi}^2 \n{x}^{2n} \d x + 2\,n\,\hbar^2  \intd \n{\psi} \n{x}^{2n-1} \n{\nabla\psi}\dd x.
		\end{multline*}
		Applying Young's inequality for the product, we get
		\begin{align*}
			\dt y^n &\leq - \intd \n{\psi}^2 \n{x}^{2(n+1)} \d x + \(n\hbar\)^2 \intd\n{\psi}^2 \n{x}^{2\(n-1\)}\d x
			\\
			&\leq - c^{-1}\,y^{n+1} + \(n\hbar\)^2  c\, y^{n-1}
		\end{align*}
		where $c = \Nrm{\varphi}{L^2}^{2/n}$. This yields the differential inequality
		\begin{equation}
			 y' \leq  -\frac{1}{n\,c}\(y^{2} - \(n\hbar\,c\)^2\).
		\end{equation}
		This ordinary differential equation has a fixed point at $y = n\hbar\,c$. If initially, $y \leq \hbar\,c$, then $y'\geq 0$ but $y$ remains smaller than $n\hbar\, c$. If not, then at any time $y'< 0$ and $\hbar\,c < y(t) < y(0)$. If initially $y > \sqrt{2}\,n\,\hbar \,c$, then as long as it remains true, it holds
		\begin{equation*}
			y' \leq -\frac{1}{2\,n\,c}\, y^2
		\end{equation*}
		which implies
		\begin{equation*}
			y(t) \leq \max\(\frac{2}{t}, \sqrt{2}\,\hbar\) n\, c
		\end{equation*}
		and proves the result by taking $t=\beta$.
	\end{proof}

	To complete the proof of Proposition~\ref{prop:regu_thermal_harmonic}, we use the above lemma to get that for any $s\in[\tfrac12,1]$,
	\begin{equation*}
		\Nrm{x\,e^{-\beta s H}}{\L^\infty} \leq 2\max\(\tfrac{1}{\sqrt{\beta}}, \sqrt{\hbar}\).
	\end{equation*}
	Finally, we conclude using Inequality~\eqref{eq:split_integral}.

	\subsection{\texorpdfstring{$\L^p$}{Lp} bounds for \texorpdfstring{$2\le p<\infty$}{2<= p<infty}}
	\begin{prop}\label{prop:Lp_regu_thermal_harmonic}
		Let $\beta> 0$ and $\hbar \in (0,1)$. Suppose $p \in [2, \infty]$ then there exists $C_{d,p} > 0$ such that
		\begin{align*}
			\Nrm{\Dh\opg_\beta}{\L^p} & C_{d,p}\, \frac{\beta^{\frac{1}{2}-\frac{d}{p}}}{Z_\beta}  \frac{\max\(2\sqrt{2},\beta\hbar\)^{\frac{1}{2}-\frac{1}{p}}}{\theta(\beta\hbar)^\frac{1}{p}},
			\\
			\Nrm{\Dh\op_\beta}{\L^p} &\leq C_{d,p}\, \frac{\beta^{\frac{1}{2}-\frac{d}{p}}}{Z_\mu}  \frac{\max\(2\sqrt{2},\beta\hbar\)^{\frac{1}{2}-\frac{1}{p}}}{\theta(\beta\hbar)^\frac{1}{p}}.
		\end{align*}
		where where $\theta(x) = \th(x)/x$ with $\th(x) = \tfrac{e^x-e^{-x}}{e^x+e^{-x}}$.
	\end{prop}

	\begin{lem}[Moments of the thermal state]\label{lem:moments_of_thermal_state}
		Let $\beta>0$ and $n>-d$. Then
		\begin{align*}
			h^d\Tr{\n{x}^n\opg_\beta} = h^d\Tr{\n{\opp}^n\opg_\beta} = \frac{C_{d,n}}{\(\beta\,\theta(\beta\hbar/2)/2\)^{n/2}}
		\end{align*}
		where $C_{d,n} = \Gamma(\tfrac{d+n}{2})/\Gamma(\tfrac{d}{2})$.
	\end{lem}

	\begin{proof}
		Let $H_\circ = \frac{{x}^2+\n{\opp}^2}{2}$ be the one-dimensional Hamiltonian of the harmonic oscillator, and $\psi_n$ be its eigenvalues verifying
		\begin{equation*}
			H_\circ\psi_n = \(n+\tfrac{1}{2}\)\hbar\,\psi_n.
		\end{equation*}
		The Wigner transform of the corresponding density operator $\ket{\psi_n}\bra{\psi_n}$, is classically given by (see e.g. \cite[Section~5.04]{groenewold_principles_1946} or \cite[Theorem~1.105]{folland_harmonic_1989})
		\begin{equation*}
			f_n(z) = 2\(-1\)^n e^{-\n{z}^2/\hbar}\, L_n\!\(\frac{2\n{z}^2}{\hbar}\)
		\end{equation*}
		where $z = (x,\xi)$ and $L_n$ is the Laguerre polynomial of order $n$ defined by
		\begin{equation*}
			L_n(z) = \frac{e^{x}}{n!} \,\partial_x^n\!\(x^ne^{-x}\) = \sum_{k=0}^{n} \binom{n}{k} \frac{(-x)^k}{k!}
		\end{equation*}
		By the formula of the generating function of the Laguerre polynomials, we deduce that
		\begin{equation*}
			\sum_{n=0}^\infty t^n f_n(z) = \frac{2}{1+t}\, e^{-\frac{\n{z}^2}{\hbar}\frac{1-t}{1+t}}
		\end{equation*}
		Taking $t = e^{-\beta\hbar}$, we obtain the Wigner transform of the thermal state. Since in dimension $d$ it is factorized, it yields to the following formula for the Wigner transform of $Z_\beta^{-1}\,e^{-\beta H}$
		\begin{equation*}
			f_\beta(z) = \(\frac{\beta\,\theta(\beta\hbar/2)}{2\pi}\)^d  e^{-\beta \n{z}^2\theta(\beta\hbar/2)}
		\end{equation*}
		Its spatial moments are given by
		\begin{align*}
			h^d\Tr{\n{x}^n\opg_\beta} &= \iintd f_\beta(z)\n{x}^n\d z = \(\frac{\beta\,\theta(\beta\hbar/2)}{2\pi}\)^{d/2} \intd \n{x}^n e^{-\beta \n{x}^2\theta(\beta\hbar/2)}\dd x
		\end{align*}
		which yields the result.
	\end{proof}

	\begin{proof}[Proof of Proposition \ref{prop:Lp_regu_thermal_harmonic}]
		Since $h^d\Tr{\n{\n{x}^n G}^2} = Z_{2\beta}\,h^d\Tr{\n{x}^{2n} \opg_{2\beta}}$, then, by Lemma \ref{lem:moments_of_thermal_state}, we have the identity
		\begin{equation*}
			\Nrm{\n{x}^n G}{\L^2}^2 = \Nrm{\n{\opp}^n G}{\L^2}^2 = \frac{C_{d,2n}\,Z_{2\beta}}{\(\beta\,\theta(\beta\hbar)\)^{n}}
		\end{equation*}
		whenever $n>-d/2$. Now applying linear interpolation of Schatten norms and Lemma~\ref{lem:weights_vs_gaussian}, we obtain the intermediate Schatten norm bounds
		\begin{equation*}
			\Nrm{\n{x}^n G}{\L^p} = \Nrm{\n{\opp}^n G}{\L^p} \leq C_{d,n,p} \, Z_{2\beta}^\frac{1}{p} \, \frac{\max\(\tfrac{2}{\beta}, \sqrt{2}\,\hbar\)^{n\(\frac{1}{2}-\frac{1}{p}\)}}{\(\beta\,\theta(\beta\hbar)\)^\frac{n}{p}}
		\end{equation*}
		where $C_{d,n,p} = \, C_{d,2n}^\frac{1}{p} \, n^{n\(\frac{1}{2}-\frac{1}{p}\)}$. Finally, by Identity \eqref{eq:gradient_gaussian}, we arrive at
		\begin{multline*}
			\Nrm{\Dh \opg_\beta}{\L^p}
			\leq \frac{2\beta}{Z_\beta}\int_{1/2}^1  \Nrm{\opz\,G^{s}}{\L^p} \d s
			\\
			\leq C_{d,1,p} \frac{2\beta}{Z_\beta} \sup_{s\in[\frac12,1]} \, Z_{2s\beta}^\frac{1}{p} \, \frac{\max\(\tfrac{2}{s\beta}, \sqrt{2}\,\hbar\)^{\frac{1}{2}-\frac{1}{p}}}{\(s\beta\,\theta(s\beta\hbar)\)^\frac{1}{p}}
			\\
			\leq C_{d,p}\, \frac{\beta^{\frac{1}{2}-\frac{d}{p}}}{Z_\beta}  \frac{\max\(2\sqrt{2},\beta\hbar\)^{\frac{1}{2}-\frac{1}{p}}}{\(\theta(\beta\hbar)\)^\frac{1}{p}}
		\end{multline*}
		where we used the fact that $Z_\beta \leq (2\pi/\beta)^d$ and $C_{d,p} = 2^{\frac{5}{4}+\frac{2d+1}{p}}\,C_{d,1,p}\,\pi^\frac{d}{p}$. Similarly, Inequality~\eqref{eq:split_integral} implies the same bound for $\op_\beta$ with $Z_\beta$ is replaced by $Z_\mu$.
	\end{proof}

{\bf Acknowledgements.} J.C. was supported by the NSF through the RTG grant DMS- RTG 184031. C.S. acknowledges the support of the Swiss National Science Foundation through the Eccellenza project PCEFP2\_181153 and of the NCCR SwissMAP.

\bibliographystyle{abbrv}

redpajama_set_name: RedPajamaArXiv


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Rede Evangelizar, under the guidance of Father Reginaldo Manzotti, was launched in 2011, based in the city of Curitiba, south-west of São Paulo. As its success continued to grow, it extended its in-house production and post facilities, adding its first EditShare servers in 2017.
This major expansion, designed and implemented by EditShare's local system integration partner CIS Group, includes a FLOW Ingest server and EFS online storage, as well as FLOW asset management, metadata controller and additional production nodes. The integral FLEX Cloud Sync software applications ensure that all media is securely backed up, with archives and restore functionality in the Amazon Web Services (AWS) cloud.
"We had found our original EditShare installation to be a very powerful tool for us," said Renan Borges, post production coordinator at Rede Evangelizar é Preciso. "As our productions grew in size and complexity, we needed to boost our operational performance and allow more users access to our content. EditShare and CIS Group readily understood our requirements, and proposed an excellent technical solution, support for the migration, and the ability to move our archive to the cloud which gives us lifecycle cost savings as well as easier operations."
Said Bacho, Chief Revenue Officer at EditShare added, "CIS Group was able to put together a very attractive package, including long-term licensing and continuing support, it meant we could deliver all the functionality Rede Evangelizar needed, at the right price point, and help them achieve the operational streamlining they sought."
"It's important to see that the partnership between CIS and EditShare has brought a level of trust to our customers. With the upgrade of Rede Evangelizar's system to the new EditShare FLOW tools, we will optimize the end-to-end workflow, offer greater productivity and efficiency, and also guarantee greater production quality," said Alberto Santana, Territory Account Manager at CIS Group. The extensions to the EditShare architecture at Rede Evangelizar are being installed at the end of 2022, and will go live in the first quarter of 2023.
For more information on EditShare solutions, please click here to get in touch.
EditShare's cloud-enabled remote editing and project management technology was recently recognized by the National Academy of Television Arts and Sciences (NATAS) with a prestigious 2021 Emmy® Award for Technology and Engineering.
About CIS Group
Since 1988, CIS Group has been a Media Technology solutions provider & systems integrator with core competencies across the professional services spectrum. Over the years our scope has evolved to include: Consulting & Workflow Design, Integration, Full System Commissioning & Installation, Technical & Operational Training, Maintenance & Support, DevOps, and Managed Services. Focused exclusively on the MediaTech space, CIS has delivered countless solutions aimed at a variety of use cases throughout the Content Supply Chain – including live production, post-production, archiving, and distribution. In relying on CIS's expertise, our clients have been able to outsource a lot of their technical challenges, and focus their energy and resources on how the technology we implement ultimately delivers business value and creative value to their organizations. Enabling our customers to achieve their desired business goals and operational goals by effectively implementing our solutions is what we mean when we say that customer success is in our DNA.
Learn more at www.cisgroup.tv
Meriam Khan
MKM Marketing Communications
meriam@mkm-marcomms.com

redpajama_set_name: RedPajamaCommonCrawl


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Ukrop News 24
Breaking News Ukraine on political and economic developments, the geopolitical processes, crime statistics
APU has built a new training complex
The construction costs amounted to almost 40 million hryvnia On the basis of the complex can simultaneously engage up to three companies of soldiers by day and one Rota during the occupation by night. Yavarouskaya on a military firing range in Lviv region completed the construction of a training complex called the Fight in the village. This was reported by Ukrainian military portal on Monday, December 31
Mayweather had the two minutes to defeat Nasukawa
Floyd Mayweather Jr. Eminent American triumphantly returned to the ring after a long pause. The former world champion in five weight categories Floyd Mayweather defeated Tensina Nasukawa in the main event of the show RIZIN 14, which was held in Japan
In the Kharkov area man died after crossing the border
Male December 28, crossed the border through the checkpoint Strelice The man crossed the border through the checkpoint Strilecha and apparently accidentally went in the wrong direction, said the guards. In the village of hoptivka Dergachi district of Kharkiv region on December 29 was found unconscious man, who soon died. Said this yesterday, the Center of emergency medical aid and disaster medicine
Ukraine's Central election Commission closed polling stations in Russia
CEC abolished the polling stations on the territory of Russia The Central election Commission has considered the submission of the MFA of Ukraine on amendments to the list of permanent foreign polling stations. The Central election Commission of Ukraine to eliminate five polling places for voting in the Russian Federation. It is stated in the decree No. 274, which was published on the CEC website on Monday, December 31
Natsukawa: I will destroy Mayweather
Japanese fighter on the upcoming confrontation with Floyd Mayweather Jr. Japanese MMA fighter Tensin Natsukawa shared expectations of the match with Floyd Mayweather Jr. The fight will take place December 31 in Saitama (Japan) on organization tournament Rizin and will be held according to the rules of professional Boxing within the weight category up to 66.7 kg. the Meeting will last three rounds for three minutes, the result will not…
For the year NBU bought $1.4 billion on the interbank market
The national Bank actively joined the reserves in December The regulator did not oppose the trends that prevailed in the foreign exchange market of Ukraine. At the end of December a positive balance of operations of the National Bank of Ukraine on the interbank currency market amounted to $339 million as a whole for 2018 – almost $1.4 billion. This is stated in the Macroeconomic and monetary review that is…
Mayweather outweighed Nasukawa
Between American and Japanese fighter, the difference was more than four kilograms. In the Japanese city of Saitama on Monday, December 31, the tournament will be held Rizin 14, the main event will be the fight between former world Boxing champion Floyd Mayweather and Japanese kickboxer Tensina Nasukawa. On the eve of the meeting, the athletes held an official ceremony of weighing, the results of which found that weight Floyd…
Lomachenko congratulated Mustache with obtaining the title of boxer of the year
World champion in the lightweight happy for his compatriot and friend. The absolute world champion on Boxing in the first heavyweight Oleksandr Usyk became the best boxer of the year by The Ring. Another Ukrainian world champion, but the lightweight Vasyl Lomachenko congratulated his friend for receiving this award. The Ring also gave the Mustache a personal belt
Mayweather for the fight with Nasukawa will receive $ 9 million in 9 minutes
The former world champion in 9 minutes will receive a lump sum. The former holder of the title of world champion in five weight categories American Floyd Mayweather told how much will a friendly battle with Japanese Tensina Nasukawa. "What if I told you that I will get $ 9 million for the 9 minutes of battle in Tokyo, you would do the same? I like to call this nine-minute…
Galatasaray will sell ex-player of Dinamo, not to pay bonuses to Kiev
Moroccan too costly for Turkish Granby The ex-Dynamo Kiev midfielder Younes Belhanda now playing for Galatasaray, could leave the Turkish top club, says Fanatik. As reports the edition, the reason for the transfer is money. Moroccan in the summer of 2017 moved from the Kyiv club in Turkey for 8 million euros, while the contract also specified a number of bonuses
REX on SBU expelled the poles, who burned the coat of arms of Ukraine in the fireplace
mike Sid on Ukraine will help Canada to restore the population of bees
Mohammed Ali Muthana on Egypt extended the duties on rebar from Ukraine
esa on The attack in Kabul: there are photos of killed Ukrainians
Donnie Trump on In Odessa, a man blew up a car with his ex-wife
Production of Boeing 737 Max discontinued
Durov called the iCloud tool for surveillance
The number of Netflix subscribers in the world exceeded 167 million
Toyota has recalled 3.4 million cars worldwide
Yarmolenko will miss three weeks
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This website has been created by the CCSD Department of School Social Work, in partnership with local child-serving agencies, to provide you with critical material regarding your role as a mandated reporter and insight to common indicators of child abuse and neglect.
This website will also provide information related to the district's mandated reporter protocol and applicable resources within both the district and community.
View ALL Mandatory Reporting Videos below or view segments on other pages.
Instruction and assessment for CCSD employees.


The Dynamic Vitality Method brings to Rancho La Puerta, a blend of sports science, body mindfulness and qigong practices to provide foundational skills for vibrant aging, self-healing, and injury prevention. In every class of Dynamic Walking, Swimming and Tai Chi & Qigong, participants discover their energetic patterns, train to transform them, and learn practices to take home. Each session presents basic principles plus builds on prior learnings. Classes at can be taken separately or paired for greater synergy. Stay tuned for our possible 2019 dates at Rancho La Puerta.
Our Dynamic Vitality Method was the featured presentation April 7-14, 2018 at beautiful Rancho La Puerta in Tecate, Mexico 90 minutes from San Diego. The weather was excellent for the 18 classes we taught in Dynamic Walking, Dynamic Swimming and Dynamic Tai Chi and Qigong. Guests described feeling lighter, taller, younger and stronger using the Dynamic Walking Method. Dynamic Swimmers were amazed at the distances they could swim effortlessly gliding like marine mammals. "Energized yet relaxed," reported the Dynamic Tai Chi & Qigong participants - just like our students in Marin.


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Diageo unveils 'weird and wonderful' whisky arm
18th November, 2015 by Lauren Eads
Diageo has launched an experimental whisky arm tasked with producing "unorthodox, weird and wonderful whiskies".
Smoky Goat is described as a "smoky sweet" blended Scotch.
Called Whisky Union, the project was unveiled during Diageo's investors conference held in New York last week, and is described as an "experimental craft whiskey off-shoot within Diageo".
"Our whiskey makers' creations are about exploration of flavour and pushing the 55 boundaries of blending", said David Gates, head of premium core spirits at Diageo during the conference. "They are unorthodox, weird and wonderful and possibly like nothing you've ever tasted before"
The project is in line with Diageo's recently unveiled strategy to target growth in the Scotch whisky sector, and compete with the rise of innovative American and New World whiskies.
The first two whiskies from Whisky Union will be launched on a trial basis in Berlin, Hamburg, Munich and Vienna, with a global release to follow. They include Smoky Goat (€24), a "smoky sweet" blended Scotch whisky, and Boxing Hares (€22), a 35% abv spirit drink made from Scotch whisky infused with hops. A third whisky – a blend of Scotch, Canadian and American whiskies – called Huxley, will launch shortly after.
Boxing Hares is a 35% abv spirit drink made from Scotch whisky infused with hops.
"We know that these Whiskey Union products may well be quite polarising – and that's just fine", said Gates. "Some of them will fail – and that's just fine. This concept is about experimenting, testing and giving freedom to our 'makers' imaginations and giving our consumers a range of completely different styles and personalities of whiskey."
Some of its creations will be labeled as Scotch whisky, when they adhere to industry regulations, while others will not be.
"It's worth stating here that we think the SWA regulations are a good thing," said Gates. "They give consumers complete reassurance that when they buy a product labelled Scotch whisky, they know exactly what they are getting and that it is of the very highest standard. But there is room to play. When our innovations comply completely to the SWA regulations we shall call then Scotch whiskies and when they don't we won't."
All whiskies produced by Whisky Union will be released under a "new business model", which Diageo hopes will attract a new group of whisky drinkers.
"We believe these brands and a new way of bringing them to market will allow us to reach a new group of people who will not have engaged with whiskey before", explained Gates. "And we are going to do it in a different way."
One Response to "Diageo unveils 'weird and wonderful' whisky arm"
Chairface42 says:
I love when I can buy stuff produced by family owned small businesses in a craft way. Oh, sorry, I commented on the wrong article.


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American Textile Company Continues to Grow
American Textile Company (ATC) continues to pioneer the bedding industry by moving west to open a new production and shipping facility in Salt Lake City, Utah.
DUQUESNE, Pa. – This 55,000-square foot facility comes as a result of ATC's continued growth in the bed pillow business and officially opened its doors in May 2007. "American Textile Company is excited about the opening of this new facility, which reinforces our commitment to growing our bed pillow business," said Reid Ruttenberg, ATC chairman. American Textile Company will also open its new showroom in the home textile building on Fifth Avenue in New York City at this year's summer market held in August. The showroom will move to its new sixth-floor space, which will be double the size of the previous fifth-floor location.
The opening of this new showroom will allow us to better showcase our cutting-edge, performance-based bedding products to our national retail customers," said Lance Ruttenberg, executive vice president and vice president of operations. In addition to the opening of the Salt Lake City facility and new showroom, the company has experienced tremendous growth in the past year, including an expansion of its headquarters in Duquesne, PA, which doubled the size of the bed pillow operation.


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MegaCortex variant redesigned a self-executing, incorporates features of previous version
News by Teri Robinson
A new variant of MegaCortex ransomware is targeting enterprises across the US and Europe
A new variant of MegaCortex ransomware making its way across the US and Europe has been recast as a self-executing menace that doesn't require a password and is aimed at enterprises, according to a technical analysis released by researchers at Accenture iDefense.
"The disadvantage of the first version was that actors had to run the ransomware manually or risk of leaking the password. This prevented global distribution of the ransomware," Accenture said. "The MegaCortex Version 2 author has updated the ransomware to remove these disadvantages and redesigned the ransomware to self-execute."
"It seems this threat actor has done its homework regarding which business model works best," said Mounie Hahad, who heads Juniper Networks Threat Labs. "It has learned from the infamous SamSam group that also delivers ransomware manually after infiltrating an organisation."
As a result attackers can "precision-deliver highly potent malware while keeping it somewhat difficult to obtain by security researchers," he said.
The new version of MegaCortex integrates the first iteration's script features. It also "decrypts the main payload and executes in memory; detects and terminates security tools; [and] detects and stops various types of software such as backup software, database software and Web server software so there is no update to files related to that software," the analysis showed, as well as "hardcodes the password into the ransomware to allow the ransomware to decrypt the main payload automatically; and integrates the loader, main module and worker into a single executable."
Ransomware incidents have ramifications beyond a particular targeted company, "affecting the entire ecosystem," including business partners, suppliers and vendors, Matan Or-El, co-founder and CEO of Panorays, stressed. "This ransomware interrupts corporate operations and causes a Denial-of-Service to the supply chain."
On the plus side, the MegaCortex variant "is fairly easy to detect, should the threat actor decide to use it more widely or put it up on a ransomware-as-a-service offering," said Hahad.
Noting "a variety of actions" companies have at their disposal to mitigate supply chain risks," Matan recommends they "evaluate the cyber-posture of their third parties and demand that they adhere to a certain security standard."
They should also have a set policy for securely dealing with third parties, like severing those "connections with a high-risk vendor" that don't meet a set security threshold or requiring a password change for those vendors that represent a medium risk. Finally, organisations "should continuously monitor the security posture of their third parties, receive notification of any change in their security and act according to the policy they put in place," said Matan.
This article was originally published on SC Media US.
Attacks & Hacking Malware Crime & Threats Cyber-criminals & threat actors
MyDoom is alive, mailing, and evading AV that assumes you've upgraded
Sophisticated Android spyware toolset 'Monokle' linked to sanctioned Russian defence contractor
Cyber-crimes cost £2.3 million per minute in 2018
Emotet back from Christmas break to wreak havoc on networks
Security and breach concerns a major pretext for phishing scams
Conversation-hijack threat gets personal: security professionals speak out


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JRS/USA Updates
Empowering Refugees
Checkout Resources
How To Get Involved With JRS/USA
Want to get involved with JRS/USA? Use this resource guide.
Host a Screening of Not Just Football
Host a screening of Not Just Football, a film by Paolo Casalis that documents the origin story of Darfur United, a soccer team created by Darfuri Refugees. Partner with JRS/USA and iAct in bringing this documentary to your local community.
40 Prayers for 40 Days, A Lenten Prayer Guide
Whether you have a regular daily prayer you'd like to dedicate during the time of Lent, or you'd like to use Lent as an opportunity to focus your energy towards migrants and refugees, we have a guide for prayer for each day of Lent.
Use these Stations of the Cross to accompany Jesus on his journey, as we watch and pray over displaced people around the world.
USA: A Prayer of Gratitude
This prayer of gratitude is from a detainee at a U.S. Federal Detention Center whom JRS/USA serves as part of our Detention Chaplaincy Project, a program that provides pastoral and religious assistance to meet the needs of migrants and refugees detained in five US federal detention centers located in Florida, Texas, Arizona and New York.
Implementing The Global Compact on Refugees from a Faith Perspective
This policy note is a response to the Global Compact on Refugees agreed to at the UN General Assembly in New York.
JRS/USA Refugee Simulation: Walk a Mile in My Shoes
Gather your family, friends, classmates, or fellow parishioners and see what it might be like to Walk a Mile in a refugee's shoes.
EXPLORE MORE WITH JRS
Jesuit Refugee Service/USA
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JRS Great Lakes Africa
JRS Latin America and Caribbean
JRS/USA is a 501(c)(3) not-for-profit organization. Federal Tax ID 52-1355257.
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knowing the behaviour of debris suspended in a fluid has many very important functions throughout various fields, together with engineering and geophysics. Comprising major elements, this publication starts with the well-developed concept of debris in viscous fluids, i.e. microhydrodynamics, really for unmarried- and pair-body dynamics. half II considers many-body dynamics, protecting shear flows and sedimentation, bulk stream houses and collective phenomena. An interlude among the 2 elements presents the elemental statistical concepts had to hire the result of the 1st (microscopic) within the moment (macroscopic). The authors introduce theoretical, mathematical techniques via concrete examples, making the cloth available to non-mathematicians. additionally they contain many of the many open questions within the box to inspire additional learn. as a result, this is often a fantastic advent for college students and researchers from different disciplines who're imminent suspension dynamics for the 1st time.
Breakwaters and closure dams belong to the main fantastic hydraulic constructions. layout and building of those buildings are so interrelated right figuring out isn't attainable and not using a thorough wisdom of concept and a formal figuring out of sensible concerns. This ebook deals this mixture to the graduate scholar.
With out right hydraulic fill and compatible specialized gear, many significant infrastructure initiatives corresponding to ports, airports, roads, business or housing tasks couldn't be realised. but finished information regarding hydraulic fill is hard to discover. This completely researched ebook, written by means of famous specialists, takes the reader step by step in the course of the complicated improvement of a hydraulic fill undertaking.
With thirty in step with cent of the world's oil and fuel construction coming from offshore parts, the development of expert vessels to accomplish offshore operations is a vital a part of the undefined. notwithstanding, with exploration and creation being played in more and more exacting destinations, the scope for disputes bobbing up from price overruns, scheduling delays and technical problems is great.


You are receiving this from a student who is applying to be a part of Kids Prosper Kids (KPK). KPK is a student-led non-profit organization made up of 4th - 12th graders who demonstrate the drive, integrity, commitment and servant heart to create positive change in their environment and throughout the world.
Please answer the below questions and hit "send" when you are finished. Your responses will be sent to the KPK Student Director and placed in the student's portfolio for future use. Thank you so much for your recommendation. To learn more about Kids Prosper Kids please click here.
Please share strengths that you see in him/her and why you are recommending student for Kids Prosper Kids.


Geoff Shepard, shown here visiting the Old Orange County Courthouse, is former White House Domestic Council Associate Director under Richard Nixon. He has been uninvited to participate in a Chapman Law School Watergate symposium. He said,"My presentation was entitled, 'Due Process and the Watergate Trials: Did Politics Trump the Rule of Law?' How could that possibly not be appropriate for a law school's symposium on Watergate?"
Geoff Shepard is former White House Domestic Council Associate Director under Richard Nixon. Shepard who grew up in Irvine and now lives in Philadelphia, has been uninvited to participate in a Chapman Law School Watergate symposium. "All I wanted to do was to raise questions for consideration about the fairness of the Watergate trials," he said.
Donald P. Kennedy Hall is the centerpiece of the Chapman University School of Law in Orange. Chapman University's Law Review is presenting "The 40th Anniversary of Watergate: A Commemoration of the Rule of Law."
Geoff Shepard, shown here visiting the Old Orange County Courthouse, is former White House Domestic Council Associate Director under Richard Nixon. Shepard who grew up in Irvine and now lives in Philadelphia has been uninvited to participate in a Chapman Law School Watergate symposium. "My goal was to raise the disturbing possibility that the political hysteria that accompanied the Watergate scandal overwhelmed our Constitution's guarantee of the most basic of rights: the right to a fair trial," he said.
Watergate's like a hornet's nest. Poke it, and someone's going to get stung.
It's time to put on your protective netting.
Behind the anniversary's impressive title are some impressive names. To name one: Former counsel to the president and Watergate plea bargainer, John Dean.
Perhaps nearly as interesting – and Dean's been plumbing Watergate through seven administrations – is one man not on the agenda.
Well he wasn't, then he was, then he wasn't.
Welcome to Watergate, a sometimes strange world in which an 18-and-half-minute blank tape can speak volumes.
To understand the people involved with Watergate in the 21st Century it helps to understand that those who earned their bones under Watergate is a small club, two deeply divided clubs really.
There are the Richard Nixon loyalists and there those who helped prosecute the president.
Others sift through Watergate on a quest for justice.
With the chief prosecutors as well as most of Nixon's top staff dead, it falls to the young bucks to fight it out.
At Chapman's two-day symposium, Geoff Shepard, a senior member of the Nixon's Domestic Council staff, was hoping to raise questions about due process.
Shepard wrote to the law school dean and was told they were full up. But the student-run Law Review put Shepard on the working agenda and agreed to pay expenses. For the final agenda, all names remained. Except Shepard's.
I meet Shepard at the Nixon library where he has just emerged from a meeting with the Nixon Foundation.
With his political antennae intact, Shepard immediately makes it clear that while he does some work for the foundation, he speaks only for himself.
But don't mistake Shepard for a man with blind ambition.
In neutral territory at a nearby coffee shop, I learn that Shepard has a playful sense of humor, went to elementary school in Irvine, first heard Nixon speak in high school in Long Beach and coincidentally, like Nixon, went to Whittier College.
While in law school at Harvard, Shepard was named a White House fellow and found himself inside the Nixon Administration at age 24. He spent the next five years working for the president.
Like others I've talked to who served under the 37th president, Shepard saw a different Nixon from the one many saw on television. In person, the president often was personable, kind. He even had a touch of vulnerability.
During Watergate, Shepard served as principal deputy to Nixon's defense counsel. During the Plumbers trial, Shepard testified as a government witness and, later, was subpoenaed in the Watergate cover-up trial.
Six months into Gerald Ford's presidency, Shepard became a corporate lawyer in Philadelphia. But his loyalty to Nixon never waned.
For three decades, Shepard has hosted annual reunions for the Nixon-Ford policy planning staffs. He also has produced numerous Nixon Legacy Forums, which are sponsored by the private Nixon Foundation and feature discussions with former officials.
"Secret Plot" is one of the targets that Chapman officials take aim at when they dismiss Shepard as a serious scholar.
They explain that the symposium isn't a free-wheeling debate. Rather, it is more like a class geared to help students qualify for the school's Law Review.
To be sure, Shepard's book comes out of right field arguing that Democrats leveraged Watergate to elect, well, Democrats. But of Watergate figures, it focuses on one Republican in particular, the seminar's keynote speaker, John Dean.
While Ronald Rotunda, law school professor, would phrase it differently, he might agree with Shepard that Dean is unique.
Rotunda, a Republican like Chapman's law school dean, is a highly regarded Constitutional scholar, has a fondness for lawyer jokes and – like Shepard – has first-hand Watergate experience.
An advisor to Ken Starr during the prosecution of President Clinton, Rotunda served as assistant majority counsel for the Watergate Committee. It was then that he got to know Dean.
During a wide-ranging interview, Rotunda allows that he reviewed Shepard's 18-page draft for the symposium.
"Look, I don't want to hurt people's feelings," Rotunda says of Shepard's draft, but there wasn't much intellectual content and it was marked with innuendoes.
Rotunda discounts Watergate doubters, saying that once the Watergate tapes surfaced, the questions about what people knew melted away. The tapes made it clear who knew what and when.
So if the mysteries are solved, why should we care about Watergate four decades later?
Laws were passed to curtail presidential powers and there were political reforms. Rotunda also points out that history repeats itself, especially when we ignore it.
I ask Shepard what drives his effort to question to Dean in a public forum.
In the end, the buzzing is about truth. But whose truth?
David Whiting's column appear four days a week; dwhiting@ocregister.com.


Four people were robbed and one man was shot early Sunday morning near the Capital City Club.
ATLANTA -- Police have released a sketch of a suspect wanted in connection with a violent armed robbery this past weekend in an exclusive Brookhaven community.
The video shows a suspect who was allegedly involved in a Forest Park carjacking.
Police are not sure if the same suspect was involved in both crimes, but did confirm the carjacked vehicle is the same one in the Brookhaven robbery and shooting.
On Monday, police identified the victims of the violent armed robbery and shooting early Sunday morning outside the Capital City Club in Brookhaven.
Four people told police they were leaving a wedding reception and waiting on an Uber at the corner of West Brookhaven Drive and Capitol City Lane when they were robbed. One mam was shot in the abdomen when he tried to follow the suspect.
Crime Stoppers has increased the reward in in this case to $10,000 for information leading to the arrest and indictment of the suspects.


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Peaceful Schools International works with schools around the world to help them become more peaceful places. We do this by working directly with our member schools, and by organizing events and taking on international projects that promote peace education. We also provide a range of different services available to any school.
What is peace education?
PSI defines peace education simply as "learning to live well together". We also use the definition of peace education from UNICEF:
"the process of promoting the knowledge, skills, attitudes and values needed to bring about the behavior changes that will enable children, youth and adults to prevent conflict and violence, both overt and structural; to resolve conflict peacefully; and to create conditions conducive to peace, whether at a personal, interpersonal, inter-group, national, or international level."
Peace in Action
The responsibility for creating a global culture of peace is far too lofty a goal for educators alone. We can, however, strive to create and sustain a culture of peace within our schools. To do so, we must teach peace. It cannot be achieved by means of a simple recipe or by wishing for it anymore than we can achieve it by writing tough policies or merely mandating it.
There are basic principles inherent in the belief that it is necessary to equip children and youth with the attitudes, knowledge and skills essential to the prevention and resolution of conflict. Every student has the right to feel safe in school. The prevention of conflict and the peaceful resolution of conflict can be taught and should be taught.
There are currently many books and other resource materials available that contain innovative, practical lessons in cooperation, respecting differences, communicating effectively, expressing emotions in a positive manner and resolving conflict creatively and non-violently. No single book or program will meet everyone's needs. When there is a variety of resources made available in the school, teachers will be free to use those that are relevant to the needs of their students, being aware that their needs will vary from year to year.
In some schools and school districts, peace education has been incorporated into curricular activities. Social studies, health and language arts are some of the subject areas that are conducive to such integration. In other schools and school districts, peace education activities are introduced in an extracurricular manner. For example, many schools host Peace Festivals, invite guest speakers and hold assemblies with a focus on peace. How peace education occurs is not as important as that it occurs.
What is a peaceful school?
A peaceful school is one in which students, staff and parents work together to ensure that everyone feels safe, valued, and respected. Based on responses from educators, students and parents, PSI sees six essential ingredients in creating a peaceful school:
Schools use a collaborative approach to decision making and develop a climate of cooperation, support and understanding. For example, students, parents and teachers are all involved in developing a school's discipline policy.
Schools provide curricular and/or extracurricular peace education initiatives. For example, schools might host a peace festival where they share their experiences of peace with parents and the community.
Teaching methods stress participation, cooperation, problem solving and respect for differences. Students are encouraged to be open minded and accepting of others who may look different, have different customs or hold beliefs that do not correspond with their own.
Student and community centered conflict resolution strategies such as peer mediation are available.
The school is involved in community service projects. For example, students may pick up litter in their neighbourhood, fundraise for a specific cause or group, or collect food and clothing to donate.
Opportunities for professional development on creating a positive school climate are available to all staff. This may include training in areas such as crisis response, dealing with bullying, peer mediation, anti-racism or anti-sexism programs, or cooperative learning strategies.
An example: Kingslake Public School, Toronto, Ontario
A very multicultural Toronto school, Kingslake has been a member of Peaceful Schools International since 2004. Originally, the school's focus was on building a culture of peace within the Kingslake community. Drawing on PSI resources, the school created peace books and poems and organized activities to promote positive interaction. One member of the staff reported that, "We knew our efforts were producing results when we overheard a student reminding a new Kingslake student that, 'We don't do that at Kingslake. We're a peaceful school.'"
Having entrenched peaceful practices at Kingslake, the school has recently expanded its focus by twinning with Mayagba Primary School in Sierra Leone, through PSI. The schools share emails, photos and books and learn about each other's cultures. At each step of the way, Kingslake staff has felt strongly supported by PSI. Representatives from the organization have visited staff, made presentations to students, and provided resource materials, leading to close and meaningful connections between the school and PSI.


Hot Diamonds Forever Heart Stud Earrings. Beautifully cut stud earrings set with a real diamond.
Hot Diamonds earrings are beautifully designed in sterling silver, 18ct yellow and rose gold and are always adorned with at least one real diamond. The perfect accessory to any outfit and can be matched with the extending range of Hot Diamonds necklaces, rings and bracelets.


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MuseumSights and activitiesBritish Motor Museum
The British Motor Museum is home to the world's largest collection of historic British cars. It includes hundreds of historic cars including several cars that have played a pivotal role in the evolution of British motoring as well as famous cars from film and television.
What to see at the British Motor Museum
The museum is a semi-circular building with the Time Road display arranged around the outside wall. This exhibit chronicles the history of British motoring from 1896 to 1996. It not only shows how cars have evolved but it also depicts the history of the motoring experience with exhibits showing pivotal moments such as the introduction of cat's eyes, the first driving test and the UK's first motorway.
The Design & Concepts display features prototype vehicles that never made it into production including the MG EX-E and the Triumph Lynx.
The sports cars exhibit features leading British sports cars from Aston Martin, Lotus, McLaren, Morgan and Triumph.
The museum's Jaguar exhibit features the highlights of the Jaguar Heritage Trust's collection of over 150 cars, which includes the only surviving D-type prototype and the only XJ13 ever produced. The Jaguar exhibit also includes an interactive display that charts Jaguar's history.
The Land Rover display includes a tracked Land Rover 'Cuthbertson', the very first Land Rover, the very first Range Rover and the millionth Land Rover Discovery 4. This display also includes an interactive exhibit that chronicles the evolution of the Land Rover brand.
The first production Land Rover (1948) is one of more than 300 classic British cars on display at the British Motor Museum. (Photo: DeFacto [CC BY-SA 4.0])
Making British Cars is an exhibit that tells the story of the British motor industry, showing pivotal moments in the industry's history and charting the rise and fall of various companies in the industry.
Under the Skin is an interactive display that explains how cars work. It includes steering simulators plus plenty of buttons to push and levers to pull.
Highlights of the museum's exhibits include famous cars from film and television including the hot pink FAB1 from the Thunderbirds film, the Land Rover Defender from the James Bond Skyfall film and the Land Rover 'Judge Dredd' City Cab. The museum is also home to several royal vehicles including the Rover P5 used by the Royal family.
The museum is home to several famous cars from film and television including the FAB1 car from the 2004 Thunderbirds film. (Photo: Maccer [CC BY-SA 4.0])
Other notable vehicles from the museum's collection of over 300 classic cars include both the very first and the very last production models of the Rover 75, the very first Mini produced and the very last Rover Mini Cooper, the Sinclair C5, the last Aston Martin DB7 and the last Austin Montego.
The collections centre
The vehicles that are not on display inside the main museum building are kept in the museum's collections centre. Unlike most other museums, visitors can also visit the collection centre meaning that you are able to see all the cars in the museum's collection.
As this is a storage facility, as opposed to a museum, there is not a lot of information provided for the cars at the collection centre. However, there are volunteers on hand who can answer any questions and there are also two daily guided tours of the centre.
Visiting the British Motor Museum
The British Motor Museum is located in Gaydon, 14.5km (9 miles) southeast of Warwick. Gaydon is a small village that also happens to be home to Aston Martin's headquarters, one of two Jaguar Land Rover engineering centres and a test track on the site of a former RAF base.
Ample free parking and the museum's location in an industrial area near a small village means that it is easiest to access if you're driving. You can visit by public transport, however, with bus routes 77, 77A, 78, X77 stopping around a 10-minute walk from the museum. These buses connect the museum with Leamington Spa with services running every hour or so and a journey time of around 40 minutes.
The museum is open daily. Although admission isn't cheap, it is reasonably priced considering how much there is to see here.
Your entry fee includes free tours of the museum that depart daily at 11am and 2pm and tours of the museum's collections centre run at noon and 3pm every day.
The museum and the collections centre are both fully wheelchair accessible.
The museum's Junction 12 Cafe serves tea and coffee, cakes and light meals.
It is a surprisingly large museum for a small village and you can easily spend between two and three hours here.
Free entry to the British Motor Museum with the Explorer Pass
The Shakespeare's England Explorer Pass gives you free entry to the British Motor Museum and 17 other attractions in Warwickshire.
We may earn a small commission if you purchase the Shakespeare's England Explorer Pass after clicking this link.
Learn more about the Explorer Pass
Free guided tours
Compton Verney Art Gallery and Park
Leamington Spa Art Gallery & Museum
Banbury Road, Gaydon CV35 0BJ
Bus 77, 77A, 78, X77 (stop: British Motor Museum)
enquiries@britishmotormuseum.co.uk
https://www.britishmotormuseum.co.uk/
Admission £14.50, students £12.50, seniors (60 and over) £12.50, children (5–16) £9, children (under 5) free; free with the Shakespeare's England Explorer Pass
Now closed Opens at 10.00 am
Open 10am–5pm daily


Yes I am a recovering addict. It is not easy to write about addiction especially when the addiction is not understood. It certainly wasn't understood by me for a very long time. I attempt to write my story because my hope is it may help others. This starts years ago – way too young. I was mature and smart for my age thirteen going on thirty. My addiction started then and lasted well into my thirties. In fact, I have only been "clean" for the past two years and even those two years have not been without setbacks and multiple falls off the proverbial wagon. One thrill after another for years, I would easily grow weary of one and push on to the next dealer of choice. None of them were ever enough. I would leave one as soon as the thrill dulled and go to the next higher, greater thrill. I lost family time, money and eventually ended up close to homeless, broke and struggling to put food on the table. I wrecked my car just trying to find the next fix; I did serious harm to myself esteem, my emotional and physical health.
What was this addiction that took so much from me and ruled my life for twenty-five years? Love. Yes my addiction to love – specifically the love of a man ruled my life for over two decades. Now as I look back it's hard to believe that I spent twenty-five years of my life hell bent on finding "the one." Where does my addiction start? For me, like most of us something was missing from my childhood. I suppose for me it was to be "adored." From the time I was quite little I thought when I find "the one" I will be adored! I finally will be that precious object protected, cherished. I found "the one" off and on for the next twenty years, getting high falling in love, much like I imagine the lighting of the crack pipe. Soon enough the love didn't feel right anymore, and I was out on the street looking for the next fix, never took me long to find another dealer. I was a serial monogamist for twenty years. Marriages did not work out, I was an expert at moving on, but it was after divorcing my children's father and hitting the street again, I found the dealer that would end all dealers. I prayed for someone to come along and adore me and suppose the devil supplied that very thing, or maybe my God knew the lesson I needed. A man came along that said everything I had waited my whole life to hear he even literally said over and over "I adore you Cameron." I gave everything I had emotionally, spiritually, mentally and physically. I could not lose this time! This was what I had waited for all my life! The only problem was it wasn't real; he wasn't what he said – it was all a fantasy and a fantasy that I failed miserably to see the obvious reality. See that's what addiction does to you, everyone around you can see you flailing, they can see the obvious answers but you have convinced yourself you are ok! You can handle this! Even though it goes against everything you believe even though you hate the situation you 'think' you have it under control. No one can tell you different, They don't understand how much joy you have from this "love." Yet your business is failing as you wait for him to make you honest. You cry until you have massive migraines, you add to the chemicals in your body multiple times in a year to even out the depression that has become too much for you to handle.
It took me three years of therapy, loss of a career and hitting rock bottom to find the courage to grow. The journey that I call my desert was excruciating, so many times I wanted to just lay down and die, but I with God's help kept going – sometimes it was three steps forward and two steps back, sometimes it was three steps forward four steps back, but I kept moving forward inch by inch. The pain and loss seemed to never end. I would ask "How long does this last??" The withdrawals, the constant pain, the dreams, the death of the fantasy. At times it was minute by minute to walk on my own, sometimes day by day some days I didn't make it. Some days I just stayed home and cried. I cried a lot. I thought a lot about the desert times in the bible, I thought "How did the Israelites manage to wander forty years in the desert before making it to the promise land?!?" I can tell you it is a long and arduous journey you think it will never end, but oh Lord how beautiful the milk and honey flow when you reach the other side. I am here to tell you that you can make it!There is light at the end of the tunnel, not only that but it's more beautiful because of the journey and what you learned along the way. I am now clean and sober. I can now say I no longer rely on the love of a man to get my high. I am high on me. Funny thing I learned after all those months the thing that I was missing was right here all along. I could tell you how bad the men were that I met and loved along the way, and some definitely were not saints by any means, but the truth is the problem was always with me. I allowed them to abuse me – I wasn't in the place to say no to the things that were not in my best interest.
So if you are struggling with the addiction to love, if you base your happiness on the next man or woman in your life. If you are in an abusive relationship, if you are the other woman, if you are with someone that hurts you emotionally or you can't seem to let go. Take the first step by getting help real therapy. Reach out to other women, they have probably been there is one way or another.
Today will be hard, tomorrow as well; in fact there may be many days ahead that are tough. I can tell you the work is so worth the end product. I am now living a life the young me could not even imagine. I have found me, I know what I like! I am reading and studying and meditating on the thousands of things I want to do with my life, and not one of them requires the love of a man. Don't get me wrong I am not against marriage or relationships, but I am against being addicted to those relationships having to have a mate to be completed, to be whole, to be happy.
My faith, my family & my true self is plenty.
Much like this lotus flower I feel that I have come to bloom out of the mud of life and it's a beautiful thing!
This blog is not to say I am never down or never tempted or don't have those battles. Like any kind of addict it rears it's ugly head off and on, but I have done the hard work and I usually am able to bring myself back to self love. Life's a journey and so is spiritual and emotional wellness. God bless you and keep you as you make your journeys!
A lot of people can't handle the real life love when the newness wears off. Good for you for realizing this and "coming clean." Love ya! And so very proud of you.
This entry was posted on January 26, 2015 by Cameron.


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The Jones Eye Institute is a 9-story facility built to provide hands-on care by physicians with years of specialty training in all areas of the eye. The physicians at J.E.I. represent every ophthalmic sub-specialty, and the patients enjoy a multitude of services ranging from an optical shop to a LASIK surgery program.
Architectural Firm: Polk Stanley Roland Curzon Porter Architects, Ltd.
Baptist Health Medical Center – Heber Springs has been committed to serving Heber Springs and surrounding communities with a complete range of medical services and support groups since opening in 1968. The hospital became part of Baptist Health in 1996 and was relocated to a new location in 2007. The facility includes an expanded emergency department and operating suite, centralized outpatient clinic, in-house MRI services, two covered entrances, a large cafeteria, and updated inpatient rooms.
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The first phase on the Baptist Medical Springhill Hospital facility was completed in 1998. This 2001 expansion was for the addition of the third floor Patient-Bed Wing facilities.


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FTIR-ATR spectroscopic analyses of changes in wood properties during particle- and fibreboard production of hard- and softwood trees
Müller, G., Schöpper, C., Vos, H., Kharazipour, A., and Polle, A. (2009). "FTIR-ATR spectroscopic analyses of changes in wood properties during particle- and fibreboard production of hard- and softwood trees," BioRes. 4(1), 49-71.
Fourier transform infrared attenuated total reflectance (FTIR-ATR) spectroscopy was combined with multivariate data analysis to investigate the chemical changes in wood during particle- and medium density fibreboard (MDF) production of grand fir (Abies grandis [Douglas ex D. Don] Lindl.) and European beech (Fagus sylvatica L.). The mechanical and technological properties of the novel particle- and fibreboards from beech or grand fir wood were similar to those of conventional panels from pine and spruce. This indicates that these timbers can be used as resources for wood-based panel production. Principal component analysis of FTIR spectra differentiated wood, fibres, particles, MDF, and particleboards of both species in the whole production process. Modifications in the spectra of fibres and particles suggested that cellulose properties of wood were changed during mechanical pulping. Different binders and hydrophobic additives were clearly traceable and discernable in wood composites. Samples from the same production step were clustered together, indicating high homogeneity of the raw materials, and intermediate and final products, respectively. This suggests that FTIR spectroscopy in combination with cluster analysis is a useful tool to assess product quality and can be further developed to control and optimize production processes for innovative wood-based panels.