text:
\section{Introduction}
This paper focuses on the semilinear systems of the form
\begin{equation}\label{stiff equation}
\dot{y}(t)=Ay+f(y),\quad y(t_0)=y_{0},
\end{equation}
where $A$ is a linear unbounded differential operator or a matrix which has  eigenvalues with large negative real part or with purely imaginary eigenvalues of large modulus, and the non-linear term $f$ is supposed to be nonstiff satisfying Lipschitz condition. The semilinear system \eqref{stiff equation} arises in many applications, such as the charged-particle dynamic \cite{Cary}, the  rapidly rotating shallow water equations for semi-geostrophic particle motion \cite{Cotter}, and the semi-discretization of semilinear PDEs.
Equation \eqref{stiff equation} is usually stiff and
one popular class of  numerical integrators that are suitable for such problems is  the exponential integrators \cite{Hochbruck2010}. These methods  normally permit larger step sizes and provide higher accuracy than the non-exponential integrators. The basic idea behind such methods is to solve the stiff part with an exact solver.
Based on the variation of constants formula,  the exact solution of \eqref{stiff equation} is given by
\begin{equation}\label{exact-stiff equation}
y(t_0+h)=\text{exp}(hA)y_0+h\int_0^1\text{exp}((1-\tau)hA)f(y(t_0+\tau h))d\tau,
\end{equation}
where the integration interval is $[t_0,t_0+h]$.
Most exponential integrators can be obtained from an appropriate approximation of the integral in exact solution \eqref{exact-stiff equation}. For example,
the exponential Euler method is obtained by interpolating the
nonlinear term at $y_0$ with the form
\begin{equation*}\label{EEM}
y_{1}=\text{exp}(hA)y_0+h\phi(hA)f(y_0),
\end{equation*}
and the implicit exponential Euler method is given by interpolating the
nonlinear term at  $y_{1}$ with the form
\begin{equation*}\label{IEEM}
y_{1}=\text{exp}(hA)y_0+h\phi(hA)f(y_{1}),
\end{equation*}
where $\phi(z):=\frac{e^z-1}{z}$ \cite{Hochbruck2010}.
More examples of  exponential integrators can be found in, e.g., \cite{Hochbruck2010, Hochbruck2008}.

Equation \eqref{stiff equation} might possess  important geometric structures. In particular, the canonical Hamiltonian structure corresponds to  equations of the form
\begin{equation}\label{Hamilonian stiff equation}
\dot{y}(t)=J\nabla H(y),\quad y(t_0)=y_{0},
\end{equation}
where
$$H(y)=\frac{1}{2}y^TMy+ U(y),$$
and
\begin{equation*}\label{Canonical skew-symmetric matrix}
J = \left[ \begin{matrix}
0 & I_{m}\\
-I_{m} &  0
\end{matrix} \right].
\end{equation*}
with  $I_{m}$ an identity matrix, $M$ a symmetric matrix and $U(y)$ a scalar function. Two prominent features of equation \eqref{Hamilonian stiff equation} are the conservation of the energy function $H(y)$ and the preservation of the symplecticity.
In this work,  we intend to focus on more general equations than the canonical Hamiltonian systems, where the matrix  $J$ in \eqref{Hamilonian stiff equation} is allowed to be a constant skew-symmetric matrix or a negative semidefinite matrix. A skew-symmetric matrix $J$ grantees the conservation of  the energy, while a negative semidefinite matrix $J$ will lead to a dissipative system with Lyapunov function $H(y)$  monotonically decreasing.  In view of these special structures in equation \eqref{Hamilonian stiff equation}, a type of candidate  methods will be the structure-preserving exponential integrators which
have superior qualitative behavior over long-time integration compared with the general-purpose designed higher-order methods \cite{Hairer2006}. Examples include symmetric methods \cite{Celledoni2008}, symplectic methods \cite{Wuxinyuan2018} and energy-preserving methods \cite{Wuxinyuan2016}.  Here we would like to consider particularly energy-preserving exponential integrators. Such methods in previous work are fully implicit, e.g., \cite{Wuxinyuan2016,miyatake2014energy,cui2021mass,shen2019geometric}, except for the recent work \cite{jiang2020linearly}, where a linearly implicit method for the nonlinear Klein--Gordon equation was considered using the scalar auxiliary variable (SAV) \cite{shen2019new} approach. For linearly implicit methods, only one linear system is solved in each iteration and thus less computational cost is needed.
For this consideration, we expect to construct linearly implicit methods in this work.

There are two mostly used techniques in creating linearly implicit energy-preserving methods for general conservative/dissipative systems with gradient flow, according to the authors' best knowledge. The first one follows from Furihata and co-authors, where multiple-point methods are used so that the nonlinearity can be portioned out \cite{furihata2011discrete}.  Further studies of this method were presented in  \cite{dahlby2011general} and \cite{eidnes2020linearly} using the concept of polarized energy and polarized discrete gradient.
The second technique is to combine the linearly implicit Crank-Nicolson method and the invariant energy quadratization (IEQ) \cite{zhao2017numerical} or the scalar auxiliary variable (SAV) \cite{shen2019new} approach.
Both the IEQ-based and the SAV-based methods are applicable for nonlinear systems, including nonpolynomials; however, these methods need bounded free energy regarding the nonlinear terms. Besides, the linearly implicit methods constructed based on the second approach have no symmetric property. This paper will focus on the technique using polarized energy for problems with polynomial energy functions. There are several reasons for choosing this technique. First, there are huge amounts of PDEs and ODEs with polynomial energy functions, e.g., the nonlinear Schrödinger equations, the nonlinear wave equations, the KdV equations, the Camassa-Holm equations, the wind-induced oscillator,
the polynomial pendulum oscillator, and so on. Second, there are no other restrictions for the nonlinear terms except for being polynomials. Third, the general scheme of the methods based on the first technique looks much more straightforward than the second technique. Last but not least, the linearly implicit methods constructed based on the first technique can be symmetric, and it has been shown that symmetric methods applied to (near-)integrable reversible systems share similar properties to symplectic methods: linear error growth, long-time near-conservation of first integrals and existence of invariant tori \cite{hairer2006geometric}. Thus methods with symmetric property usually provide prominent long-time behavior.

The paper is organized as follows. First, we construct the symmetric linearly implicit energy-preserving exponential integrators and discuss their properties in Section 2. In Section 3, numerical examples are presented to illustrate the performance of the proposed method. In the last section,  we conclude the paper with a summary of the properties and advantages shared by the  method.

\section{Symmetric linearly implicit energy-preserving exponential scheme}
In this section, we combine the idea of constructing linearly implicit methods using polarized energy \cite{dahlby2011general}  and the idea of constructing energy-preserving exponential methods using discrete gradient \cite{Wuxinyuan2016} to build the symmetric linearly implicit energy-preserving exponential integrators.  To present the method more intuitively, we restrict the nonlinear term $U(y)$ (potential energy) in equation \eqref{Hamilonian stiff equation} to be a cubic polynomial. However, the method is also applicable to problems with any higher-order polynomials, for which the results will be introduced briefly in this paper too.

The critical point of using polarized energy to construct a linearly implicit method is to portion out the nonlinearity over consecutive time steps. This can be carried out by constructing quadratic polarized energy and then performing the polarized discrete gradient method, similarly as shown in \cite{eidnes2019linearly} for a cubic polynomial. A systematical way of constructing a quadratic polarization for higher-order polynomial functions is presented in  \cite{dahlby2011general}, for example
\begin{itemize}
    \item  $U(x)=x^2$  can be polarized by  $\bar{U}(x,y)=\theta\frac{x^2+y^2}{2}+(1-\theta)xy$, $\theta\in [0,1]$,
    \item   $U(x)=x^3$  can be polarized by  $\bar{U}(x,y)=x\frac{x+y}{2}y,$
      \item   $U(x)=x^4$ can be polarized by  $\bar{U}(x,y)=x^2y^2,$
       \item  $U(x)=x^5$ can be polarized by  $\bar{U}(x,y,z,w)=xyzw\frac{x+y+z+w}{4}$,
    \item $U(x)=x^6$ can be polarized by  $\bar{U}(x,y,z)=x^2y^2z^2$.
\end{itemize}
Following from \cite{eidnes2019linearly},  $\overline{\nabla}\bar{U}$ is said to be a polarized discrete gradient for a polarized  energy $\bar{U}$ if  the following conditions hold
\begin{equation}\label{discrete eq}
\begin{split}
\bar{U}(y,z) - \bar{U}(x,y) &=
\frac{1}{2}(z-x)^T \overline{\nabla}\bar{U}(x,y,z), \\
\overline{\nabla}\bar{U}(x,x,x)&= \nabla U(x).
\end{split}
\end{equation}
We will use the polarized discrete gradient to construct the linearly implicit energy-preserving exponential integrators. Consider the variation of constants formula on interval $[t_0,t_0+2h]$ for problem \eqref{stiff equation}  in the form
\begin{equation}\label{Variation of constants 2step}
y(t_0+2h)=\text{exp}(2hJM)y_0+2h\int_{0}^{1}\text{exp}(2(1-\xi)hJM)J\nabla U(y(t_0+2h\xi))d\xi.
\end{equation}
Substituting $\nabla U(y(t_0+2h\xi))$ with the polarized discrete gradient $\overline{\nabla}\bar{U}(y_0,y_1,y_2)$ in \eqref{Variation of constants 2step}, we obtain the energy-preserving exponential integrator for problem \eqref{Hamilonian stiff equation} as follows
\begin{equation}\label{approx EP exponential integrator}
y_{n+2}=\text{exp}(2hJM)y_n+2h\phi(2hJM)J \overline{\nabla}\bar{U}(y_n,y_{n+1},y_{n+2}) \quad\quad \text{(LIEEP)}.
\end{equation}

\begin{remark}
Suppose that a quadratic polarization of a higher-order polynomial function $U(y_n)$  has the form  $\bar{U}(y_{n},\cdots,y_{n+i},\cdots y_{n+p-1})$. Then the generalization of the polarized discrete gradient in \eqref{discrete eq} can be given by
\begin{equation}\label{discrete eq_higherorder}
\begin{split}
\bar{U}(y_{n+1},\cdots y_{n+p}) -\bar{U}(y_{n},\cdots y_{n+p-1})&=
\frac{1}{p}( y_{n+p}-y_{n})^T \overline{\nabla}\bar{U}(y_{n},\cdots, y_{n+p}), \\
\overline{\nabla}\bar{U}(y_{n},\cdots, y_{n+p})&= \nabla U(y_n).
\end{split}
\end{equation}
\end{remark}

\begin{remark}\label{orderp-scheme}
Suppose that the polarized discrete gradient for a higher-order polynomial energy function $U(y_n)$ is given by   $\overline{\nabla}\bar{U}(y_{n},y_{n+1},\cdots, y_{n+p})$ following from equation \eqref{discrete eq_higherorder}. Then the linearly implicit energy-preserving exponential integrator for problem  \eqref{Hamilonian stiff equation}  can be  given by
\begin{equation}\label{approx EP exponential integrator_higherorder}
y_{n+p}=\text{exp}(phJM)y_n+ph\phi(phJM)J \overline{\nabla}\bar{U}(y_{n},y_{n+1},\cdots, y_{n+p}).
\end{equation}
\end{remark}
\begin{remark}
Scheme \eqref{approx EP exponential integrator} is  a special case of scheme \eqref{approx EP exponential integrator_higherorder} with $p=2$.
\end{remark}

Before presenting the main theorems about the proposed method's conservative properties, we begin with a lemma.

\begin{lemma}\label{lemma}
For any symmetric matrix $M$, positive integer $p$ and  scalar $h>0$, the matrix
\begin{equation*}\label{lemmaEq}
B=\text{exp}(phJM)^TM\text{exp}(phJM)-M
\end{equation*}
is zero  when $J$ is skew-symmetric and  negative semidefinite  when $J$ is negative semidefinite.

\end{lemma}
The proof of this lemma follows directly from Lemma 2.2 in \cite{Wuxinyuan2016} since the result does not change when  $h$ is replaced by $ph$.

\begin{theorem}\label{enerPre-cubic}
Scheme \eqref{approx EP exponential integrator} preserves the following polarized energy
\begin{equation}\label{Penergy}
\bar{H}(y_n,y_{n+1})=\frac{1}{4}(y_n^TMy_n+y_{n+1}^TMy_{n+1})+\bar{U}(y_n,y_{n+1}).
\end{equation}
\end{theorem}

\begin{proof}\label{proof_PEP}
We first assume that the matrix $M$ is not singular and denote by  $V=2hJM$, $M^{-1}\overline{\nabla}\bar{U}=\overline{\nabla}\tilde{U}$. The energy error has the form
\begin{equation}\label{energyer}
\begin{split}
&\bar{H}(y_{n+1},y_{n+2})-\bar{H}(y_n,y_{n+1})\\
&=\frac{1}{4}(y_{n+2}^TMy_{n+2}+y_{n+1}^TMy_{n+1})-\frac{1}{4}(y_n^TMy_n+y_{n+1}^TMy_{n+1})\\
&+\bar{U}(y_{n+1},y_{n+2})-\bar{U}(y_n,y_{n+1}).\\
\end{split}
\end{equation}
By replacing $y_{n+2}=\text{exp}(V)y_n+2h\phi(V)J \overline{\nabla}\bar{U}(y_n,y_{n+1},y_{n+2})$ and using $\phi(V)V=\text{exp}(V)-I$, we get the following equations
\begin{equation}\label{linearEER}
\begin{split}
&\frac{1}{4}(y_{n+2}^TMy_{n+2}+y_{n+1}^TMy_{n+1})-\frac{1}{4}(y_n^TMy_n+y_{n+1}^TMy_{n+1})\\
&=\frac{1}{4}y_n^T(\text{exp}(V)^TM\text{exp}(V)-M)y_n+hy_n^T\text{exp}(V)^TM\phi(V)J\overline{\nabla}\bar{U}\\
&+h^2(\overline{\nabla}\bar{U})^TJ^T\phi(V)^TM\phi(V)J\overline{\nabla}\bar{U}\\
&=\frac{1}{4}y_n^T(\text{exp}(V)^TM\text{exp}(V)-M)y_n+\frac{1}{2}y_n^T\text{exp}(V)^TM(\text{exp}(V)-I)\overline{\nabla}\tilde{U}\\
&+\frac{1}{4}(\overline{\nabla}\tilde{U})^T(\text{exp}(V)-I)^TM(\text{exp}(V)-I)\overline{\nabla}\tilde{U},
\end{split}
\end{equation}
and
\begin{equation}\label{nonlinearEER}
\begin{split}
&\bar{U}(y_{n+1},y_{n+2})-\bar{U}(y_n,y_{n+1})\\
&=\frac{(y_{n+2}-y_n)^T}{2}\overline{\nabla}\bar{U}(y_n,y_{n+1},y_{n+2})\\
&=\frac{1}{2}y_n^T(\text{exp}(V)^T-I)\overline{\nabla}\bar{U}+h(\overline{\nabla}\bar{U})^TJ^T\phi(V)^T\overline{\nabla}\bar{U}\\
&=\frac{1}{2}y_n^T(\text{exp}(V)^TM-M)\overline{\nabla}\tilde{U}+\frac{1}{2}\overline{\nabla}\tilde{U}^TV^T\phi(V)^TM\overline{\nabla}\tilde{U}\\
&=\frac{1}{2}y_n^T(\text{exp}(V)^TM-M)\overline{\nabla}\tilde{U}+\frac{1}{2}\overline{\nabla}\tilde{U}^T(\text{exp}(V)^TM-M)\overline{\nabla}\tilde{U}.
\end{split}
\end{equation}
Inserting equations \eqref{linearEER} and \eqref{nonlinearEER} to equation \eqref{energyer}, we obtain the following results
\begin{equation}\label{energyer final}
\begin{split}
&\bar{H}(y_{n+1},y_{n+2})-\bar{H}(y_n,y_{n+1})\\
&=\frac{1}{4}y_n^T(\text{exp}(V)^TM\text{exp}(V)-M)y_n+\frac{1}{2}y_n^T(\text{exp}(V)^TM\text{exp}(V)-M)\overline{\nabla}\tilde{U}\\
&+\frac{1}{4}\overline{\nabla}\tilde{U}^T(\text{exp}(V)^TM\text{exp}(V)-M)\overline{\nabla}\tilde{U}+\frac{1}{4}\overline{\nabla}\tilde{U}^T(\text{exp}(V)^TM-M\text{exp}(V))\overline{\nabla}\tilde{U}\\
&=\frac{1}{4}(y_n+\overline{\nabla}\tilde{U})^T(\text{exp}(V)^TM\text{exp}(V)-M)(y_n+\overline{\nabla}\tilde{U})\\
&+\frac{1}{4}\overline{\nabla}\tilde{U}^T(\text{exp}(V)^TM-M\text{exp}(V))\overline{\nabla}\tilde{U}\\
&=0,
\end{split}
\end{equation}
where the last step follows from the fact that $\text{exp}(V)^TM-M\text{exp}(V)$ is skew-symmetric, and $\text{exp}(V)^TM\text{exp}(V)-M$ is also skew-symmetric from Lemma \ref{lemma}.

For a singular $M$, one can find a series of non-singular and symmetric matrices $M_\epsilon$ such that $M_\epsilon= M$ when $\epsilon\rightarrow 0$. For any $M_\epsilon$, we can follow the proof above and show that the polarized energy function in the form
\begin{equation*}\label{PH with epsilon}
\bar{H_\epsilon}(y^\epsilon_n,y^\epsilon_{n+1})=\frac{1}{4}({y^\epsilon_n}^TM_\epsilon y^\epsilon_n+{y^\epsilon_{n+1}}^TM_\epsilon y^\epsilon_{n+1})+\bar{U}(y^\epsilon_n,y^\epsilon_{n+1})
\end{equation*}
is preserved by the approximation given by
$$y^\epsilon_{n+2}=\text{exp}(2hJM_\epsilon)y_n^\epsilon+2h\phi(2hJM_\epsilon)J \overline{\nabla}\bar{U}(y_{n}^\epsilon,y_{n+1}^\epsilon,y_{n+2}^\epsilon)$$
for the following problem
\begin{equation*}\label{}
\dot{y}(t)=JM_\epsilon y+J\nabla U(y(t)),\quad y(t_0)=y_{0}.
\end{equation*}
Therefore, $\bar{H_\epsilon}(y^\epsilon_n,y^\epsilon_{n+1})=\bar{H}(y_n,y_{n+1})$ is preserved by method \eqref{approx EP exponential integrator} when $\epsilon\rightarrow 0$.

\end{proof}
For problems \eqref{Hamilonian stiff equation} with higher-order polynomial energy functions, similar conservation property can be obtained by scheme \eqref{approx EP exponential integrator_higherorder}, as shown in the following corollary.
\begin{corollary}\label{orerp-energy}
Scheme \eqref{approx EP exponential integrator_higherorder} preserves the following polarized energy
\begin{equation}\label{polaEn higherorder}
\bar{H}(y_{n},\cdots y_{n+p-1})=\frac{1}{2p}\sum_{i=0}^{p-1}y_{n+i}^TMy_{n+i}+\bar{U}(y_{n},\cdots, y_{n+p-1}).
\end{equation}
\end{corollary}
The proof is similar to Theorem \ref{enerPre-cubic} and thus is omitted here.
\begin{theorem}\label{enerdissip-cubic}
If $J$ is a constant negative semidefinite matrix, scheme \eqref{approx EP exponential integrator} preserves the polarized Lyapunov function $\bar{H}$ for problem \eqref{Hamilonian stiff equation}:
$$\bar{H}(y_{n+1},y_{n+2})\le \bar{H}(y_{n+1},y_{n}),$$
where $\bar{H}(y_n,y_{n+1})$ is defined  by equation \eqref{Penergy}.
\end{theorem}

\begin{proof}
Let us suppose $M$ to be non-singular; otherwise we follow the similar procedure in the proof for Theorem \ref{enerPre-cubic}, i.e., constructing a series of $M_\epsilon$ convergent to $M$ to achieve the result.

For a constant negative semidefinite matrix $J$, the error of Lyapnov function has the same form as the energy error in \eqref{energyer final}:
\begin{equation*}\label{Lyapunov final}
\begin{split}
&\bar{H}(y_{n+1},y_{n+2})-\bar{H}(y_n,y_{n+1})\\
&=\frac{1}{4}(y_{n+2}^TMy_{n+2}+y_{n+1}^TMy_{n+1})-\frac{1}{4}(y_n^TMy_n+y_{n+1}^TMy_{n+1})\\
&+\bar{U}(y_{n+1},y_{n+2})-\bar{U}(y_n,y_{n+1})\\
&=\frac{1}{4}(y_n+\overline{\nabla}\tilde{U})^T(\text{exp}(V)^TM\text{exp}(V)-M)(y_n+\overline{\nabla}\tilde{U})\\
&+\frac{1}{4}\overline{\nabla}\tilde{U}^T(\text{exp}(V)^TM-M\text{exp}(V))\overline{\nabla}\tilde{U}\\
&\le 0,
\end{split}
\end{equation*}
where the last step follows from the fact that $\text{exp}(V)^TM-M\text{exp}(V)$ is skew-symmetric, and  $\text{exp}(V)^TM\text{exp}(V)-M$ is negative semidefinite according to Lemma \ref{lemma}.

\end{proof}
Similarly, for problems \eqref{Hamilonian stiff equation} with a higher-order polynomial  energy, we have the following corollary.
\begin{corollary}
If $J$ is a constant negative semidefinite matrix, scheme \eqref{approx EP exponential integrator_higherorder} preserves the polarized Lyapunov function:
$$\bar{H}(y_{n+1},\cdots, y_{n+p})\le \bar{H}(y_{n},\cdots, y_{n+p-1}),$$
where $\bar{H}(y_{n},\cdots y_{n+p-1})$ has the same form as equation \eqref{polaEn higherorder}.
\end{corollary}
The proof is similar to Theorem \ref{enerdissip-cubic} and thus is omitted here.

\begin{theorem}\label{LIEEP_symmetric}
Scheme \eqref{approx EP exponential integrator} is symmetric.
\end{theorem}\label{proof-LIEEP_symmetric}
\begin{proof}
Exchanging $y_{n}$, $y_{n+1}$ $\leftrightarrow$ $y_{n+2} $, $ y_{n+1}$ and replacing $2h$ by $-2h$ in \eqref{approx EP exponential integrator}, we obtain
 \begin{equation*}\label{symmetric}
y_n=\text{exp}(-V)y_{n+2}-2h\phi(-V)J\overline{\nabla}\bar{U}(y_{n+2},y_{n+1},y_{n}).
\end{equation*}
Following from the definition in \eqref{discrete eq} and the  cyclic permutation free property of the polarized energy \cite{dahlby2011general},

 we get
 \begin{equation}\label{symmetric_PDG-}
\overline{\nabla}\bar{U}(y_{n+2},y_{n+1},y_{n})=\overline{\nabla}\bar{U}(y_{n},y_{n+1},y_{n+2}).
 \end{equation}
Using $\text{exp}(V)\phi(-V)=\phi(V)$ and the relation in \eqref{symmetric_PDG-}, we then obtain
 \begin{equation*}\label{symmetric}
y_{n+2}=\text{exp}(V)y_{n}+2h\phi(V)J\overline{\nabla}\bar{U}(y_{n},y_{n+1},y_{n+2}).
\end{equation*}
\end{proof}

Scheme \eqref{approx EP exponential integrator_higherorder} for problems with higher-order polynomial $U(y)$ also holds the symmetric property if the polarization of the function $U(y)$ is invariant when the order of its arguments is reversed. This can be obtained  by symmetrizing over dihedral group \cite{dahlby2011general}. Although only cyclic permutation free is required in the definition of the polarized energy,  we can actually always get a permutation free quadratic polarization for any higher order polynomial $\bar{U}(y)$. In fact,  the polarization examples shown above are all permutation free. In this paper, we always consider permutation free polarization, i.e., quadratic polarization satisfying the following  condition
\begin{equation*}\label{permutation free}
\bar{U}(y_{n},\cdots y_{n+p-1})=\bar{U}(y_{n+i_1-1},\cdots y_{n+i_p-1}),\quad (i_1,\cdots,i_p)\in S_p,
\end{equation*}
where $S_p$ is a symmetric group.

\begin{corollary}
Scheme \eqref{approx EP exponential integrator_higherorder}
is symmetric if the polarization of function $U(y)$ is permutation free.
\end{corollary}
The proof is similarly to the proof of Theorem \ref{LIEEP_symmetric} except that the cyclic permutation free property should be replaced by the permutation free property.

\section{Numerical experiment}
The proposed method is suitable for conservative or dissipative differential equations of the form \eqref{Hamilonian stiff equation} with $J$ a constant skew-symmetric or negative semidefinite matrix and $U(y)$ a scalar polynomial function of any order. These equations include the highly oscillatory conservative or dissipative  ODEs and also the semi-discrete systems arising from PDEs.  In this section, we test our method by three differential equations. The first two examples are used to demonstrate the efficient behavior of the method compared to the fully implicit method, e.g., the energy-preserving exponential integrators based on the averaged vector field method, denoted by EAVF. The third example is chosen to show the application of the method for problems with higher-order polynomial energy functions.

EAVF method was put forward in  \cite{Wuxinyuan2016}, which has the form
 \begin{equation}\label{EAVF}
y_{n+1}=\text{exp}(hJM)y_n+h\phi(hJM)J \overline{\nabla}U(y_n,y_{n+1}), \quad\quad
\end{equation}
where $\overline{\nabla}U(y_n,y_{n+1})=\int_{0}^1 \nabla U((1-\tau)y_n+\tau y_{n+1})d\tau$.
Besides, scheme \eqref{EAVF} has been shown to preserve the discrete energy of the form
\begin{equation}\label{EAVF-energy}
H(y_n)=\frac{1}{2}y_n^TMy_n+U(y_n).
\end{equation}
The integral in EAVF method is evaluated by the 2-point GL quadrature formula, which gives an exact approximation of the integration. In most cases, the terms $\text{exp}(phJM)$ and $\phi(phJM)$ ($p$ is a positive integer number) can not be calculated explicitly, and  we use the
MATLAB package proposed in \cite{berland2007expint} to compute them, where Pade approximations are used.

In the experiments, the global error is defined by
\begin{equation*}\label{gler}
\underset{n\ge 0}{\max}\|y_n-y(t_n)\|,
\end{equation*}
where $t_n=t_0+nh$ with $h$ the time step size, and $y(t_n)$ is the reference exact solution.
In this work, we compute the reference solution by  the 6-order continuous Runge–Kutta (CRK) method \cite{hairer2010energy} with the form
\begin{equation*}\label{CRK6GL5}
\begin{cases}
      y_{n+1/3}=y_n+hJ\int_0^1\big(\frac{37}{27}-\frac{32}{9}\sigma+\frac{20}{9}\sigma^2\big)\nabla H(Y_\sigma)d\sigma\\
      \vspace{5pt}
     y_{n+2/3}=y_n+hJ\int_0^1\big(\frac{26}{27}+\frac{8}{9}\sigma-\frac{20}{9}\sigma^2\big)\nabla H(Y_\sigma)d\sigma\\
     y_{n+1}=y_n+hJ\int_0^1\nabla H(Y_\sigma)d\sigma
    \end{cases},
\end{equation*}
where
 \begin{equation*}
 \begin{split}
Y_\sigma=&-\frac{(3\sigma-1)(3\sigma-2)(\sigma-1)}{2}y_n+\frac{3\sigma(3\sigma-2)(3\sigma-3)}{2}y_{n+1/3}\\
&-\frac{3\sigma(3\sigma-1)(3\sigma-3)}{2}y_{n+2/3}+\frac{\sigma(3\sigma-1)(3\sigma-2)}{2}y_{n+1},
 \end{split}
\end{equation*}
and the integrals are evaluated exactly by the 5-point GL quadrature.
For all fully implicit schemes, we solve the nonlinear system   by  the fixed point iteration with tolerance as $10^{-14}$. All the numerical results presented are obtained from schemes implemented in MATLAB (2020a release), running on a MacBook Pro
with a dual-core 2.6 GHz Intel 6-Core i7 processor and 16 GB of 2667 MHz DDR4
RAM.

\vspace{2pt}

\noindent\textbf{Test problem one}.  We consider an averaged system in wind-induced oscillation \cite{mclachlan1998unified}
\begin{equation}\label{problem1}
\begin{split}
\dot{x}_1&=-\zeta x_1-\lambda x_2+x_1x_2,\\
\dot{x}_2&=\lambda x_1-\zeta x_2+\frac{1}{2}(x_1^2-x_2^2),
\end{split}
\end{equation}
where $\zeta\ge 0$ is a damping factor and $\lambda$ is a detuning parameter with $\zeta=r\text{cos}(\theta)$, $\lambda=r\text{sin}(\theta)$, $r\ge 0$, $0\le\theta\le\pi/2$.
Equation \eqref{problem1} can be rewritten into the form \eqref{Hamilonian stiff equation} with
\begin{equation*}\label{problem1 J}
J=\begin{bmatrix}
    -cos(\theta)      & -sin(\theta)   \\
    sin(\theta)      & -cos(\theta)
\end{bmatrix}
\quad
M=\begin{bmatrix}
    r      & 0   \\
    0      & r
\end{bmatrix},
\end{equation*}
\begin{equation*}
U=-\frac{1}{2}sin(\theta)(x_1x_2^2-\frac{1}{3}x_1^3)+\frac{1}{2}cos(\theta)(\frac{1}{3}x_2^3-x_1^2x_2).
\end{equation*}

Its energy function (when $\theta=\pi/2$) or Lyapunov function (dissipative case, when $\theta\le\pi/2$) is
\begin{equation*}\label{problem1 H}
H=\frac{1}{2}r(x_1^2+x_2^2)-\frac{1}{2}sin(\theta)(x_1x_2^2-\frac{1}{3}x_1^3)+\frac{1}{2}cos(\theta)(\frac{1}{3}x_2^3-x_1^2x_2).
\end{equation*}
The matrix exponential in scheme \eqref{approx EP exponential integrator} for problem \eqref{problem1} can be calculated explicitly as follows
\begin{equation*}\label{exp(V)}
exp(V)=\begin{bmatrix}
    exp(-2hcr)cos(2hsr)      & -exp(-2hcr)sin(2hsr)   \\
    exp(-2hcr)sin(2hsr)      & exp(-2hcr)cos(2hsr)
\end{bmatrix},
\end{equation*}
with $c=cos(\theta)$ and $s=sin(\theta)$.
We can obtain a polarized discrete gradient $\nabla\bar{U}(x^n,x^{n+1},x^{n+2})$ based on a polarization of $U$ given by
 \begin{equation}\label{polarisation of U}
 \begin{split}
\bar{U}(x^n,x^{n+1})=&-\frac{1}{2}sin(\theta)\big(a\frac{x_1^n+x_1^{n+1}}{2}x_2^nx_2^{n+1}+(1-a)\frac{x_1^n(x_2^{n+1})^2+x_1^{n+1}(x_2^{n})^2}{2}\\
&-\frac{1}{3}x_1^n\frac{x_1^n+x_1^{n+1}}{2}x_1^{n+1}\big)+\frac{1}{2}cos(\theta)\big(\frac{1}{3}x_2^n\frac{x_2^n+x_2^{n+1}}{2}x_2^{n+1}\\
&-ax_1^nx_1^{n+1}\frac{x_2^n+x_2^{n+1}}{2}
-(1-a)\frac{x_2^n(x_1^{n+1})^2+x_2^{n+1}(x_1^{n})^2}{2}\big).
 \end{split}
\end{equation}
Then we get the linearly implicit energy-preserving scheme in the form of \eqref{approx EP exponential integrator} and the polarized energy in the form of \eqref{Penergy}.

Consider the initial vector $x_1(0)=0, \quad x_2(0)=1$, $r=20$, step size $h=1/20$, and the parameters $\theta=\pi/2$ or $\theta=\pi/2-10^{-4}$. For  LIEEP method, the starting point $[x_1^1, x_2^1]$ is computed by the Matlab function \emph{ode45}. $\theta=\pi/2$ provides a conservative system, and Figure  \ref{energy upwind-conserve} confirms that EAVF method preserves the discrete energy \eqref{EAVF-energy} and LIEEP method preserves the polarized energy \eqref{Penergy}. While  $\theta=\pi/2-10^{-4}$ leads to a dissipative system, and Figure  \ref{energy upwind-dissipative} shows that EAVF method and  LIEEP method preserves the dissipation of the Lyapunov function in \eqref{EAVF-energy} and \eqref{Penergy}, respectively.

\begin{figure}[H]
\centering
      \begin{subfigure}[b]{0.45\textwidth}
      \centering
                \includegraphics[width=0.99\textwidth]{energyplot_conservative_T1000.png}
                \caption{$\theta=\pi/2$}
      \label{energy upwind-conserve}
        \end{subfigure}\hspace{18pt}
          \begin{subfigure}[b]{0.45\textwidth}
        \centering
                \includegraphics[width=0.99\textwidth]{energy_dissipative_T1000.png} \caption{$\theta=\pi/2-10^{-4}$}
      \label{energy upwind-dissipative}
        \end{subfigure}
      \caption{ The energy behaviour of both  EAVF and  LIEEP method with time step size $h=1/20$. }
      \label{energy upwind}
\end{figure}

In Figure \ref{energy upwind cons efficiency order conserv} and  \ref{energy upwind cons efficiency order dissip}, we consider the global errors and the computational cost using step sizes  $h=1/10/2^{i}$, with $i=0,1,\cdots, 5$. Surprisingly, Figure  \ref{orderplot upwind-conserve} shows that LIEEP method is superconvergent for the conservative system ($\theta=\pi/2$). We find that this behavior is closely related to  the parameter $a$ in the polarized potential energy in \eqref{polarisation of U}. We have tried $a=0, 1/2,1/4, 1$, but only $a=1/2$ gives a three-order behaviour. Figure  \ref{efficiencyplot upwind-conserve} shows that the proposed method is more efficient than the fully implicit EAVF method.  When $\theta=\pi/2-10^{-4}$, i.e., the system is dissipative,  the superconvergent behavior disappears for LIEEP method, see  Figure \ref{orderplot upwind-dissipative}. From this figure, we observe that LIEEP method has a convergent issue when the  step size is $h=1/10$, but with the decrease of the time step size, LIEEP method gets convergent and behaves even better than  EAVF method.  Figure \ref{efficiencyplot upwind-dissipative} indicates that the proposed method is much more efficient than EAVF method for the dissipative system.

\begin{figure}[H]
\centering
      \begin{subfigure}[b]{0.45\textwidth}
      \centering
                \includegraphics[width=0.99\textwidth]{orderplot_T1000_conservative.png}
                 \caption{order plot}
      \label{orderplot upwind-conserve}
        \end{subfigure}\hspace{18pt}
          \begin{subfigure}[b]{0.45\textwidth}
        \centering
                \includegraphics[width=0.99\textwidth]{T1000_timecost_conservative.png}
                  \caption{efficiency plot}
      \label{efficiencyplot upwind-conserve}
        \end{subfigure}
      \caption{$T=1000$, $\theta=\pi/2$, the time step sizes are  $h=1/10/2^{i}$, for $i=0,1,\cdots, 5$.}
      \label{energy upwind cons efficiency order conserv}
\end{figure}

\begin{figure}[H]
\centering
      \begin{subfigure}[b]{0.45\textwidth}
      \centering
                \includegraphics[width=0.99\textwidth]{orderplot_T1000_dissipative.png}
                  \caption{order plot}
      \label{orderplot upwind-dissipative}
        \end{subfigure}\hspace{18pt}
          \begin{subfigure}[b]{0.45\textwidth}
        \centering
                \includegraphics[width=0.99\textwidth]{T1000_timecost_dissipative.png}
                  \caption{efficiency plot}
      \label{efficiencyplot upwind-dissipative}
        \end{subfigure}
      \caption{$T=1000$, $\theta=\pi/2-10^{-4}$, the time step sizes are  $h=1/10/2^{i}$, for $i=0,1,\cdots, 5$.}
      \label{energy upwind cons efficiency order dissip}
\end{figure}

\noindent\textbf{Test problem two.}
 We consider  a continuous generalization of an $\alpha$-FPU (Fermi-Pasta-Ulam) system \cite{macias2009implicit}:
\begin{equation}\label{problem2}
\frac{\partial^2 u}{\partial t^2}=\beta\frac{\partial^3 u}{\partial t\partial x^2}+\frac{\partial^2 u}{\partial x^2}(1+\epsilon(\frac{\partial u}{\partial x})^p)-\gamma\frac{\partial u}{\partial t}-m^2u,
\end{equation}
where $\epsilon>0$, $\beta\ge 0$ is the coefficient of the internal damping,  $ \gamma\ge 0$ is the coefficient of the external damping, and $(x,t)\in[0, L]\times[0, T]$. Taking $\frac{\partial u}{\partial t} =v$, equation \eqref{problem2} can be rewritten as
\begin{equation}\label{problem2-rewritten}
\begin{split}
\partial_t u&=v\\
\partial_t v&=\beta\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 u}{\partial x^2}(1+\epsilon(\frac{\partial u}{\partial x})^p)-\gamma v-m^2u.
\end{split}
\end{equation}
Denoting by $y=[u,v]^T$, equation \eqref{problem2-rewritten} can be reformulated as the following Hamiltonian form
\begin{equation*}\label{problem1-rewritten-Hamiltonian pde}
\frac{\partial y}{\partial t}=\mathcal{Q}\frac{\delta \mathcal{H}}{\delta y},
\end{equation*}
where
\begin{equation*}\label{Hamiltonian}
\mathcal{Q}=\begin{bmatrix}
    0& 1 \\
    -1 & \beta\partial^2_x-\gamma
  \end{bmatrix},\quad
\mathcal{H}=\int_0^L E(t, u, v, u_x)dx,
\end{equation*}
with
\begin{equation*}\label{local energy}
E(t, u, v, u_x)=\frac{1}{2}u_x^2+\frac{m^2}{2}u^2+\frac{1}{2}v^2+\epsilon\frac{u_x^{p+2}}{(p+2)(p+1)}.
\end{equation*}
The function $E(t, u, v, u_x)$ physically represents the local energy density of  system \eqref{problem2} at any time $t$.

Consider $p=1$ and the homogeneous Dirichlet boundary conditions $u(0,t)=u(L,t)=0$. Discretizing $\partial_x^2$ with the central difference operator and $\partial_x$ with the forward difference operator,   we obtain the following semi-discrete  ODE system

\begin{equation*}\label{problem1 semi-ODE}
\dot{y}=Q(My+\nabla U(y)),
\end{equation*}
where
\begin{equation*}\label{problem1 J}
Q=\begin{bmatrix}
    0      & I  \\
    -I      & \beta D- \gamma I
\end{bmatrix},
\quad
M=\begin{bmatrix}
    m^2I-D      & 0   \\
    0      & I
\end{bmatrix},
\quad
U(y)=\sum_{j=0}^{N-1}\frac{\epsilon}{6}(\frac{u_{j+1}-u_j}{\Delta x})^3.
\end{equation*}
Setting $w_j^n=\frac{u_{j+1}^n-u_j^n}{\Delta x}$,
and defining the polarized energy
\begin{equation*}\label{problem PFU polarised energy}
\bar{U}(w_j^n, w_j^{n+1})=\sum_{j=0}^{N-1}\frac{\epsilon}{6}w_j^n\frac{w_j^n+w_j^{n+1}}{2}w_j^{n+1},
\end{equation*}
we can obtain the polarized discrete gradient
\begin{equation*}\label{PFU polarised discrete gradient}
\begin{split}
\bar{\nabla}\bar{U}(w_j^n, w_j^{n+1}, w_j^{n+2} )
&=\frac{\epsilon}{6\Delta x}w_{j-1}^{n+1}
(w_{j-1}^{n}+w_{j-1}^{n+1}+w_{j-1}^{n+2})\\
&-\frac{\epsilon}{6\Delta x}w_{j}^{n+1}(w_{j}^{n}+w_{j}^{n+1}+w_{j}^{n+2}),
\end{split}
\end{equation*}
and the discrete gradient
\begin{equation*}\label{PFU  discrete gradient}
\bar{\nabla}U(w_j^n )
=\frac{\epsilon}{2\Delta x}(w_{j-1}^{n})^2-
\frac{\epsilon}{2\Delta x}(w_{j}^{n})^2.
\end{equation*}

We consider $m=0$, $\epsilon=\frac{3}{4}$,  $L=128, \quad T=100$ and spatial step size $\Delta x=1$.  The initial conditions are set to be $u_j(0)=q_j(0)$, $v_j(0)=\dot{q}_j(0)$ and
{\small
\begin{equation*}\label{PFU  initial condition}
q_j(t)=
5\text{ln}\frac{1+\text{exp}\big(2(\alpha(j-97)+t\text{sinh}(\alpha))\big)}{1+\text{exp}\big(2(\alpha(j-96)+t\text{sinh}(\alpha))\big)}+5\text{ln}\frac{1+\text{exp}\big(2(\alpha(j-32)+t\text{sinh}(\alpha))\big)}{1+\text{exp}\big(2(\alpha(j-33)+t\text{sinh}(\alpha))\big)},
\end{equation*}
}
where $\alpha=0.1$.
For LIEEP method, the starting point $y_1$ is computed by the 6-order CRK method.

In Figure \ref{FPU  energy betas gamma0}, we fix the external damping coefficient to be zero ($\gamma=0$) and present the energy behavior of LIEEP method for systems with different internal damping coefficients and a long simulation time $T=500$.  We observe that the numerical method preserves the energy when there is no damping ($\beta=0$) and also preserves the dissipation property when the internal damping coefficient is greater than zero, consistent with what is observed in \cite{macias2009implicit},  where a fully implicit four-step method is considered. Similar behavior is observed in  Figure \ref{FPU  energy beta0 gammas}, where the internal damping coefficient is set to be zero ($\beta=0$). Figure  \ref{FPU beta 0 order efficiency} and \ref{FPU beta 2 order efficiency} confirm that both EAVF  and  LIEEP method are of order 2 in time, and the comparison of the computational cost between these two methods gives a clear evidence that the proposed method is more efficient than  EAVF method. In this experiment, we also present the numerical solutions given by LIEEP method for systems with different settings of $\beta$ and $\gamma$, see Figure  \ref{FPU solution}. These figures clearly demonstrate the dissipative nature of the
external damping coefficient, see the change of the colors between Figure  \ref{FPU gamma 0 solution} and  Figure \ref{FPU gamma 0005 solution}, and the internal damping coefficient, see the change of the shapes between Figure \ref{FPU gamma 0 solution} and Figure \ref{FPU beta2 solution}. These observations in the numerical solutions are in accordance with the results shown by the fully implicit four-step method in \cite{macias2009implicit}.

\begin{figure}[H]
\centering
      \begin{subfigure}[b]{0.45\textwidth}
      \centering
                \includegraphics[width=0.99\textwidth]{FPU_energy_beta_T500.png}
                \caption{$\gamma=0$}
      \label{FPU  energy betas gamma0}
        \end{subfigure}\hspace{18pt}
          \begin{subfigure}[b]{0.45\textwidth}
        \centering
                \includegraphics[width=0.99\textwidth]{FPU_energy_gamma_T500.png}
                \caption{$\beta=0$. }
      \label{FPU  energy beta0 gammas}
        \end{subfigure}
      \caption{Polarized energy by LIEEP method for $\alpha$-FPU system with different settings of internal and external damping coefficients. $T=500$ and time step size $h=0.025$.}
      \label{FPU  energy beta gamma}
\end{figure}

\begin{figure}[H]
\centering
      \begin{subfigure}[b]{0.45\textwidth}
      \centering
                \includegraphics[width=0.99\textwidth]{FPU_efficientcy_beta0.png}
        \end{subfigure}\hspace{18pt}
          \begin{subfigure}[b]{0.45\textwidth}
        \centering
                \includegraphics[width=0.99\textwidth]{FPU_order_beta0.png}
        \end{subfigure}
      \caption{ $T=100$, $\gamma=0.005$, $\beta=0$ and time step size $h=\frac{1}{2^i}$, $i=1,\cdots,5$, \textit{Left:} efficiency comparison; \textit{right:} order plot. }
      \label{FPU beta 0 order efficiency}
\end{figure}

\begin{figure}[H]
\centering
      \begin{subfigure}[b]{0.45\textwidth}
      \centering
                \includegraphics[width=0.99\textwidth]{FPU_efficientcy_beta2.png}
        \end{subfigure}\hspace{18pt}
          \begin{subfigure}[b]{0.45\textwidth}
        \centering
                \includegraphics[width=0.99\textwidth]{FPU_order_beta2.png}
        \end{subfigure}
      \caption{$T=100$, $\gamma=0$, $\beta=2$ and time step size $h=\frac{1}{2^i}$, $i=1,\cdots,5$. \textit{Left:} efficiency comparison; \textit{right:} order plot. }
      \label{FPU beta 2 order efficiency}
\end{figure}

\begin{figure}[H]
\centering
      \begin{subfigure}[b]{0.45\textwidth}
      \centering
                \includegraphics[width=0.99\textwidth]{beta0gamma0_solu.png}
                \caption{$\gamma=0$, $\beta=0$.}
      \label{FPU gamma 0 solution}
        \end{subfigure}

          \begin{subfigure}[b]{0.45\textwidth}
        \centering
                \includegraphics[width=0.99\textwidth]{beta0gamma005_solu.png}
                  \caption{$\gamma=0.005$, $\beta=0$.}
      \label{FPU gamma 0005 solution}
        \end{subfigure}
          \begin{subfigure}[b]{0.45\textwidth}
        \centering
                \includegraphics[width=0.99\textwidth]{FPU_beta2_gamma0.png}
                 \caption{$\gamma=0$, $\beta=2$.}
      \label{FPU beta2 solution}
        \end{subfigure}
      \caption{ The numerical solution of the  $\alpha$-FPU system with different settings of internal and external damping coefficients on time interval $[0, 500]$ and with time step size $h=0.025$.   }
      \label{FPU solution}
\end{figure}

\vspace{2pt}

\noindent\textbf{Test problem three.}
We consider the polynomial pendulum oscillator, and the main focus of this example is to illustrate the energy conservation property shown in Corollary \ref{orerp-energy} for the proposed method in Remark \ref{orderp-scheme}. Consider the nonlinear pendulum problem with the Hamiltonian
\begin{equation*}\label{pendulum energy}
H(p,q)=\frac{1}{2}p^2+1-\text{cos}q,
\end{equation*}
and a truncated Taylor expansion of the cosine function:
\begin{equation}\label{pendulum Taylor}
H(p,q)=\frac{1}{2}p^2+\frac{1}{2}q^2-\frac{1}{24}q^4+\frac{1}{720}q^6.
\end{equation}
The approximation in \eqref{pendulum Taylor} to the original problem will be more accurate if even higher-order polynomial is used and   $\lvert q\rvert$ is sufficiently small, e.g., $\lvert q\rvert<\frac{1}{2}$ \cite{iavernaro2009high}.
Denoting by $y=[q;p]$, the polynomial pendulum oscillator with energy function \eqref{pendulum Taylor} can be rewritten into form \eqref{Hamilonian stiff equation} with $m=2$, $J$ the canonical skew-symmetric matrix, M the identity matrix and
\begin{equation}\label{pendulum poly U}
U(y)=-\frac{1}{24}q^4+\frac{1}{720}q^6.
\end{equation}
Consider a polarization of \eqref{pendulum poly U} as follows
\begin{equation}\label{polaried pendulum poly U}
\bar{U}(y_n,y_{n+1},y_{n+2})=-\frac{1}{24}q_nq_{n+1}q_{n+2}\frac{q_n+q_{n+1}+q_{n+2}}{3}+\frac{1}{720}{q_n}^2{q_{n+1}}^2{q_{n+2}}^2.
\end{equation}
We can obtain a polarized discrete gradient of the form
\begin{equation*}\label{polarised gradient U}
\begin{split}
\bar{\nabla}\bar{U}(y_n,y_{n+1},y_{n+2},y_{n+3} )=&\frac{1}{240}{q_{n+1}}^2{q_{n+2}}^2(q_{n}+q_{n+3})\\
&-\frac{1}{24}q_{n+1}q_{n+2}(q_n+q_{n+1}+q_{n+2}+q_{n+3}).
\end{split}
\end{equation*}
Take the initial value as  $q_0=0.5$, $p_0=1$ and the integration interval as $[0,1000]$. We compute the first two starting points $y_1$ and $y_2$ by  Matlab function \emph{ode15s}. The polarized energy is reported in Figure \ref{pendulum energy fig}, and we observe that it is exactly preserved by LIEEP method defined by equation \eqref{approx EP exponential integrator_higherorder}. In this figure, we also present the original discrete energy by LIEEP method , i.e.,
\begin{equation}\label{discrete pendulum energy}
H(p_n,q_n)=\frac{1}{2}p_n^2+1-\text{cos}q_n.
\end{equation}
Although LIEEP method does not preserve the exact original energy, Figure \ref{pendulum energy fig} shows that the discrete energy given by  LIEEP method in the form of \eqref{discrete pendulum energy}  stays oscillated and bounded over a long-time integration. Besides, we observe  that the numerical solution by LIEEP method applied to the truncated equation provides an approximation with a similar behaviour as the exact solution of the nonlinear pendulum oscillator if  a small time step size is considered, e.g., h=0.3, i.e.,  the phase space is a cylinder,  as illustrated in Figure \ref{pendulum solution}.

\begin{figure}[H]
\centering
      \begin{subfigure}[b]{0.45\textwidth}
      \centering
                \includegraphics[width=0.99\textwidth]{pendulum_energy.png}
                 \caption{Energy} \label{pendulum energy fig}
        \end{subfigure}\hspace{18pt}
          \begin{subfigure}[b]{0.45\textwidth}
        \centering
                \includegraphics[width=0.99\textwidth]{solution_pendulum.png}
                 \caption{solution}\label{pendulum solution}
        \end{subfigure}
      \caption{ In \ref{pendulum energy fig}, the time step size is $h=1$, the polarized energy is defined by equation  \eqref{polaEn higherorder} with $p=3$ and $\bar{U}(y_n,y_{n+1},y_{n+2})$ defined by \eqref{polaried pendulum poly U}; the discrete energy is defined by equation \eqref{discrete pendulum energy}. In \ref{pendulum solution}, the time step size is $h=0.3$.}
      \label{pendulum}
\end{figure}

\section{Conclusion}
This paper constructs a novel symmetric linearly implicit  exponential integrator that holds the conservative properties for semi-linear problems  with polynomial energy functions. The method is developed based on combining the idea of using polarized discrete gradient to build linearly implicit methods and the idea of using discrete gradient to create energy-preserving exponential integrators.
Besides conservative properties, the method is shown to be symmetric, which guarantees excellent long-time behavior.

We test our methods on three types of differential equations, including an oscillated ODE, i.e., the averaged wind-induced oscillator, an oscillated PDE, i.e.,  the damped FPU problem, and also an ODE with higher-order polynomial energy fucntion, i.e., the polynomial pendulum oscillator. The numerical experiments confirm that the proposed method preserves the polarized energy or the Lyapunov function, and the method is of order two. Moreover, it has been shown that the proposed method has superconvergent behavior for some particular systems when a proper polarized energy is considered.
Compared with the fully implicit method (EVAF), our method shows a significantly lower computational cost. In view of the nice properties and the good behavior, we recommend the proposed method for problems with polynomial energy function.

\section*{Acknowledgement}
The author would like to thank  Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme \emph{Geometry, compatibility and structure preservation in computational differential equations (2019)} under Grant number EP/R014604/1,  where work on this paper was partly carried out.

The author would also like to thank the
European Union Horizon 2020 research and innovation programme under the Marie Sk\l{}odowska-Curie grant agreement  No. 691070 CHiPS.

\bibliographystyle{elsarticle-num-names}

redpajama_set_name: RedPajamaArXiv


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It's interesting that such places seem to be categorized as whatever is most expedient for any given proposal.
"If its a public place, then people there shouldn't have an expectation of privacy, right?"
"Well, no. You see, it may be open to the public, but it's privately owned, so people there still have an expectation of privacy."
"Oh, okay. Well if it's private property, they can allow smoking in it, right?"
"Well, no. You see, it may be private property, but it's open to the public. Therefore the public has free reign to ban things it doesn't like."
I wish our country would make up its goddamn mind once in a while.


Ondřej Filip, the Chairman of the Board of NIX.CZ, the Czech neutral peering node, was re-elected last week into the Board of the European Internet Exchange Association (EURO-IX). His mandate in this important position will last for two years.
The mission of EURO-IX is to support, develop, strengthen and improve the Internet Exchange Point (IXP) community. This involves the coordination of technical standards, development of common procedures, sharing and publishing of statistics and other useful information.


text:
Abortion bans invite Roe v. Wade Supreme Court challenge
Kristan Hawkins
Rep. Terri Collins, R-Alabama, gets a standing ovation after her near-total-ban-on-abortion bill passed the House at the Alabama Statehouse in Montgomery, Ala., on Tuesday, April 30, 2019. (Mickey Welsh/The Montgomery Advertiser via AP) more >
By Valerie Richardson - The Washington Times - Wednesday, May 15, 2019
When and if the Supreme Court does agree to hear a challenge to the 1973 Roe v. Wade decision, Republican state Rep. Terri Collins wants to see it come from Alabama.
Her bill effectively banning abortion in nearly all circumstances was signed Wednesday by Gov. Kay Ivey, making Alabama one of almost a dozen states this year to approve sweeping pro-life legislation as Republicans jump on the overturn-Roe bandwagon barreling toward the Supreme Court.
Such bills have almost no chance of surviving the inevitable lower-court challenges, but that's the point. Republican lawmakers are spoiling for a legal fight, hoping that their state's pro-life bill will become the vehicle for the high court's 5-4 conservative majority to put the brakes on Roe.
"States are actually rushing to become that first state to become abortion-free," said Kristan Hawkins, presidents of Students for Life of America. "They're trying to out-compete each other. And this is actually a good thing for us, that we've got so many cases."
The Alabama bill, which won final passage Tuesday in the state House, comes as the most restrictive in the nation, prohibits abortion except in cases when the life of the mother is in danger or pregnancy poses a "serious health risk." Its sponsor, Ms. Collins, beat back attempts to add rape and incest exceptions, arguing that the bill needed to be as "clean" as possible for the courts.
"My goal with this bill, and I think all of our goal, is to have Roe v. Wade turned over, and that decision be sent back to the states so that we can come up with our laws that address and include amendments and things that address those issues," said Ms. Collins at a Wednesday press conference on AL.com.
Meanwhile, House Majority Leader Steny H. Hoyer joined other Democrats in denouncing the abortion measures, accusing Republican state legislatures of trying to one-up each other with ever-more restrictive bills.
"These laws are part of a systematic effort by Republican lawmakers around the country to outdo each other with how restrictive they can be in taking away reproductive choice for women with the goal of undermining Roe v. Wade," said Mr. Hoyer in a statement.
Hot on Alabama's heels are Louisiana and Missouri. Both state legislatures took up Wednesday bills to prohibit abortion after the fetal heartbeat can be detected, or between six to eight weeks' gestation, which have already been signed this year in Georgia, Kentucky, Mississippi and Ohio.
None of those bills has taken effect. The Kentucky bill was blocked by a judge on March 15, shortly after it was signed by Gov. Matt Bevin. Another Kentucky measure banning second-trimester "dilation and evacuation" procedures was halted by a federal judge last week.
"We profoundly disagree with the court's decision and will take this case all the way to the Supreme Court, if necessary, to protect unborn children from being dismembered limb by limb while still alive," said Bevin spokeswoman Elizabeth Goss Kuhn in a statement.
Missouri Gov. Mike Parson cheered pro-life state legislators Wednesday as the Senate considered the measure, which would also ban abortions based on race, sex selection and potential for Down Syndrome.
"It's time to make Missouri the most Pro-Life state in the country!" tweeted the Republican governor. "Thanks to leaders in the House and Senate, we are one vote away from passing one of the strongest #ProLife bills in the country - standing for life, protecting women's health, and advocating for the unborn."
The legislative surge behind bills sharply curtailing abortion access comes despite enormous opposition from powerful pro-choice groups like Planned Parenthood and the ACLU, which have vowed to defeat the bills in court, and Democrats stunned by the momentum.
"It's kind of chilling what's happening nationally, including Wisconsin," said Democratic Assembly Minority Leader Gordon Hintz at a news conference.
Democrats have made it clear they plan to leverage the issue in the 2020 campaign. Sen. Kirsten Gillibrand, New York Democrat, announced Wednesday she was going to Georgia to "fight back, hard, on the front lines" as part of her presidential primary bid.
The Democratic Senatorial Campaign Committee said the bills show the urgency of taking back the Senate "to protect the Supreme Court and defend Roe v. Wade" from President Trump, who has appointed two conservative justices in his first term.
"This isn't a scene from The Handmaid's Tale. This is happening in Alabama — in our country — in the year 2019," said Sen. Kamala Harris, California Democrat, in a fundraising plea on behalf of pro-choice groups.
Whether this year's round of bills will pass Supreme Court muster is another question. While some Republicans argue that the heartbeat bills are the ideal test cases, Americans United for Life senior counsel Clarke D. Forsythe disagreed.
He said less-restrictive measures such as the pain-capable bills, which prohibit abortions after the fetus is able to feel pain at about 20-weeks' gestation, may be more advantageous. Such bills have passed in 21 states.
"The heartbeat bill or an earlier prohibition such as Alabama's might be the least attractive to the court," said Mr. Forsythe.
He argued that the court under Chief Justice John Roberts has shown that it prefers incremental action in previous abortion cases. In addition, the court may want to take up measures with more public support, such as the pain-capable bills.
"Certainly legal notions about legitimacy are going to be present in any case reexamining Roe," said Mr. Forsythe.
Ms. Hawkins said such differences of opinion are inevitable as red states chart their own courses.
"There's a lot of different legal strategies here, and there's not one group at the helm directing all these," said Ms. Hawkins. "I'm not worried. I'm watchful to see which case the Supreme Court does pick up."
• Valerie Richardson can be reached at vrichardson@washingtontimes.com.
Copyright © 2022 The Washington Times, LLC. Click here for reprint permission.

redpajama_set_name: RedPajamaCommonCrawl


YOU HAVE LIMITED TIME. YOU NEED RESULTS NOW. YOU'RE IN THE RIGHT PLACE.
Every executive, business leader and business owner has a unique story. Maybe you've managed all the details of your business on your own for decades but as your business has grown you need to optimize your energy and your time. Maybe your business is disorganized or inefficient and you need help developing systems so help your team work more efficiently. Maybe you've spent years building your business from the ground up, but now it's time to create a better work/life balance so you can focus on the people who matter most to you. Maybe you want to improve your response time to customer inquiries, but aren't ready to bring someone on a full-time.
Whatever it is, that vision that drives you and keeps you moving forward... that's your story… and it creates momentum for your business.
But it's not always that easy, is it? And that's why you're here. Because no matter what you try the results simply aren't there. There aren't enough hours in the day to leave you with the feeling that you are truly getting it all done while keeping everyone happy.
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READY TO GET REAL RESULTS SO YOU CAN GET BACK TO THE WORK YOU LOVE? WE REALLY SHOULD TALK.


Address to the people of Nova Scotia by the Directors of the Nova Scotia Society for Developing and Encouraging Home Manufactures with the constitution of the Society.
By: Nova Scotia Society for Developing and Encouraging Home Manufactures.
Nova Scotia tour book; an official guide to the highways of Nova Scotia.
By: Nova Scotia Travel Bureau.
Nova Scotia tour book; an official guide to the highways of Nova Scotia. 1967-68.
By: Presbyterian Church of Nova Scotia. Synod.
An address to the people of Nova Scotia [electronic resource] / [J. McCully].
Chemins de fer > Nouvelle-Écosse > Finances.
Railroads > Nova Scotia > Finance.
Dated: Brunswick Place, Halifax, 8th May, 1863.


text:
Is the future of pharmacy under threat from technology?
The future of pharmacy has been an ongoing discussion for some years. The advances in technology of all kinds, and their impact on the profession, are a serious concern. With advances in artificial intelligence and 'fuzzy' logic (an approach to computing based on 'degrees of truth'), and in logistics and supply (e.g. advances in the design, and use of drones), we must be concerned for the future of pharmacy.
Imagine the following scenario: the patient is interviewed by the GP (via the internet, possibly by a GP avatar) and is prescribed a medicine. The prescription is sent electronically to a central dispensary where it is picked, labelled and checked against the patient's records for drug interactions, etc. The dispensed medicine is then packed into a shipper, loaded on to a drone and delivered to the patient's address. Before the medicine can be taken from the shipper, the patient would complete a 'medicines awareness' test, via the internet on how to use their medicine properly. (This scenario is not perfect; I simply use it as an illustration.)
This may seem far-fetched, and I do not believe it is going to happen immediately, but it could happen someday. Much of the technology already exists, some may need further refinement, other parts are still to be developed. We may not like it, but we have to be aware of the potential.
Where is the pharmacist in all this? Would pharmacists be needed with the use of universal barcodes, etc? I do not see independent community pharmacies constructing a fully automated central dispensary. I am not sure the major chains could finance such a programme on a national basis (but, in collaboration with Amazon?). Perhaps community pharmacy still has a future, but for how long?
We must also consider the future of pharmacy education. We currently have 31 schools of pharmacy in the UK. If my 'doomsday' scenario were to become reality, how many pharmacists would we need for the community, including GP practices, and hospital services, and how many would industry require? More importantly, how many pharmacy graduates would we need each year?
R. Christian Moreton
Waltham, Massachusetts, United States
Disclosure: Dr Moreton is vice-president of pharmaceutical sciences at Finnbrit Consulting, which provides advice on drug development and excipients.
The Pharmaceutical Journal, PJ, September 2017, Vol 299, 7905;299(7905):DOI:10.1211/PJ.2017.20203135
Appropriate management of urinary tract infections in older patients
Transitional change is required to reposition pharmacy as we enter 2023
Public and pharmacist views on the appearance of and messaging on antibiotics to address inappropriate use


text:
Q: Optimizing Kernel Density Bandwidth using Python I am attempting to build a class that automatically determines the optimal bandwidth for a kernel density estimate. I am using the FFTKDE method of KDEpy for my purposes, since I am required to calculate this quantity very quickly. I am aware that there are the options of Scott's Rule, Silverman's Rule and Improved Sheather-Jones in KDEpy, but I am keen to directly optimise mysomething bandwidth.
I would like to calculate the Maximum likelihood cross-validation (MCLV) in order that I optimize it. However, my code is terribly slow, since I need to estimate the KDE for each data point in 20,000 data points, each iteration of the minimisation.
The equation I am attempting to optimize looks like this (see page 8, here):

My code for calculating this loss is as follows:
import numpy as np
from KDEpy import FFTKDE
from tqdm import tqdm
import time

def MLCV(data, bw):
   N = len(data)
   idx = np.ones(N, bool)
   logs = np.empty(N)
   for i in tqdm(range(N)):
     idx[i]=False
     x_kde, y_kde = FFTKDE(bw=bw).fit(data[idx]).evaluate(2**13)
     idx[i]=True
     logs[i] = np.sum(np.log(y_kde))

   MLCV = np.sum(logs)/N - np.log((N-1)*bw)
   return MLCV

data = np.random.normal(size=20000)
bw = 0.01
t0 = time.process_time()
mlcv = MLCV(data, bw)
t1 = time.process_time()
print("MLCV = {:3.3E}, Elapsed Time = {:3.3}s".format(mlcv, t1-t0))

Output:
MLCV = -1.077E+05, Elapsed Time = 46.3s

Can anyone suggest a means of making this faster/an alternative, quicker algorithm?
I have also considered simply minimising the negative log of the output, which I have seen elsewhere:
def L(data, bw):
    x_kde, y_kde = FFTKDE(bw=bw).fit(data).evaluate(2**13)
    return -np.sum(np.log(y_kde))

However, my intuition tells me neither method is the correct solution, since I cannot directly calculate the values for the actual data, only interpolate them, due to the FFT method requiring points to be on a grid.
Is there a loss that suits my needs? Can anyone suggest a better solution than I have come up with?

redpajama_set_name: RedPajamaStackExchange


Alas, the bus that picks us up from CPH everyday is not the new one. It's another one which I really thought was the one going to be replaced, but it's still running (barely!) The first day we were coming back from school the driver yelled to another guy hanging on to the outside and he 'spider-man-ish' crawled inside the bus, yelled at 3 of the kids to get off the engine cover and then whisked it open. The engine is on the inside of the bus to the left of the driver (they drive on the opposite side of the road). He started pumping something and water, I hope it was water, was squirting everywhere. The bus revved into action and we continued on down the road. Yesterday, while on the roasting bus waiting for the driver to show up something occurred to me. This little voice in my head said "Melinda, stop being on time". Somewhere in my 30's I became a time natzi. I was "one of those people who are always on time; or even early!" I gave myself plenty of time to get from point A to point B and rarely had that rushed 'Oh crap, I'm late" feeling.
As I sat jammed into a seat –and Anja li's sleeping head in my lap, with multiple pairs of eyes watching my every move. I thought "why am I on this bus, when the driver is still in the school and ½ the kids are running around outside the bus; outside the bus in the cool breeze?" I thought multiple times, in that sweltering situation, "just get off!" but then I would have to crawl over a couple of kids after waking up little Anjali. I also may not get this seat back (seats are a luxury on Nepali buses). So I sat there and reveled in the occasional breeze that made its way past all the little dark headed children surrounding me. I started making eye contact with whoever looked at me until that became somewhat of a game. FINALLY, 30 minutes later, the bus driver moseyed on over to the bus and the kids started pilling on. The driver whipped open the engine cover and did the pump thing himself while gunning the engine and soon like The Little Engine that Could, we were chug chug chugging on down the road.
Today as I was showing the photo's of the new bus to one of the teachers, 'my' bus driver came over and sat down. The teacher then translated what 'my' driver said (while looking at the photo's). He said that I should take photos of his bus (the yellow bus) since I had only taken them of the 'new' bus. The teacher then went on to say that I should take MORE photos of the yellow bus since that is the bus that takes me to and from school. That bus should be more important to me! I felt really bad as I didn't realize that these guys take such pride in their work/bus. I said I would promptly take a photo of the bus and motioned to get up. They said, "NO, not yet the sun is too high. Take the photo around 4pm, then it will be a better shot". Wow! That's some love!! As their instruction; I waited and took photos of the yellow bus. And here in this blog, I give respect to the most important bus in Nepal; here is the beauty in all its glory!
In this photo taken from my seat, there are 7 actual seats in front of me (2 rows of 2 seats on the left and 1 row of 2 seats and then one front seat on the right). In all there were 17 or more children (not counting the ones standing up! There is no such saying as "we're full" in Nepal.


I gave this presentation on E.P.I.C. content 4 times to 4 different groups in a recent 2-week period. Whew!
In this episode, you'll hear the presentation itself as well as why I decided to include certain things, like video and music clips.
Business storytelling has been all the rage for the past few years: we've been told to share stories in our presentations, on our websites, and on social media about our companies, our customers, and our team members.
But, for the most part, these stories have fallen flat and no longer have the novelty they once did.
A bunch of random stories included here and there or sharing your personal bio or your business origin story isn't enough.
Too much content that's created, whether it's presentations, videos, podcasts, articles, or social media, is uninspired, undifferentiated, and unmemorable.
How can you attract and retain your audience? How can you avoid getting into – or get out of – a rut of routine, unoriginal, and tired content?
Your content and your presentations need to be E.P.I.C.
In this episode, you'll learn what E.P.I.C. stands for, how to use it for your content and presentations, and the transformation you'll see for yourself and those you serve.
→ Get the free PDF to Make Your Content E.P.I.C.
→ Need help creating your E.P.I.C. presentation? Schedule a free consultation call to learn how we can work together.
Feel we are going to work together some time…soon.
What you are doing in your moments are EPIC with your podcasts and the group I am in on Facebook!
Love your content. Even though, I am on year 2 of my business and 1st year in events, I have gaines knowlege on how to best serve my clients.
Thanks so much, Betty! So glad to hear the podcast has been helpful.


Bargetto Winery was founded as South Montebello Vineyard and Wine Company by John & Phillip Bargetto in 1910 and is based in Corralitos, CA. It produces 40,000 cases. The company is led by its President Martin Bargetto. Its web site can be found at www.bargetto.com.


text:
300 - Natural Resources/Environ (1125)
(-) Senate Committee on Energy and Natural Resources (1)
(-) Senate Committee on the Budget (9)
Amendment SA 2669 to H.J.Res. 61 (Continuing Appropriations Resolution, 2016)
Amendment SA 2669 to H.J.Res. 61, Proposed by Mr. McConnell for Mr. Cochran on September 22, 2015
S. 1134, St. Croix River Crossing Project Authorization Act
Cost estimate for the bill as reported by the Senate Committee on Energy and Natural Resources on December 16, 2011
S. 1769, Rebuild America Jobs Act
Cost estimate for the bill as introduced on October 31, 2011
S. 1660, American Jobs Act of 2011
Cost estimate for the bill as introduced in the Senate on October 5, 2011
Cost estimate for the bill as introduced on September 13, 2011
S. 223, FAA Air Transportation Modernization and Safety Improvement Act
Cost Estimate February 2, 2011
Direct spending and revenues effects estimate for the bill as introduced on January 27, 2011
H.R. 3619, Coast Guard Authorization Act of 2010
Pay-as-you-go table for the bill as amended, and further amended by a draft Senate amendment as provided to CBO by the Senate Budget Committee on September 29, 2010
H.R. 2847, Hiring Incentives to Restore Employment Act
Cost estimate for Senate Amendment 3310, as introduced in the nature of a substitute for H.R. 2847 on February 11, 2010.
Aviation Investment and Modernization Act of 2008
Direct spending and revenue effects of draft legislation provided to the CBO on April 28, 2008.
H.R. 1442, An act to complete the codification of title 46, United States Code, "Shipping", as positive law
Cost estimate for the bill as ordered reported by the Senate Committee on the Judiciary on September 7, 2006


text:
The 2014 United States House of Representatives elections in Mississippi were held on Tuesday, November 4, 2014 to elect the four members of the United States House of Representatives from the state of Mississippi, one from each of the state's four congressional districts. The elections coincided with other elections to the United States Senate and House of Representatives and various state and local elections, including a Senate election in Mississippi.

Overview

District
Results of the 2014 United States House of Representatives elections in Mississippi by district:

District 1

The incumbent was Republican Alan Nunnelee, who had represented the district since 2011. He was re-elected with 60% of the vote in 2012 and the district has a PVI of R+16.

Nunnelee was the only Congressman in Mississippi who did not face a primary opponent in 2014. Democrats Ron Dickey and Rex Weathers, who was a candidate for the seat in 1992 and 1996 and the nominee for the seat in 1998 and 2002, ran.  Nunnelee died on February 6, 2015, shortly after starting his third term in office.

Danny Bedwell, Chairman of the Mississippi Libertarian Party, was the Libertarian nominee.

Primary results

General results

District 2

The incumbent was Democrat Bennie Thompson, who had represented the district since 1993. He was re-elected with 67% of the vote in 2012 and the district had a PVI of D+13.

Damien Fairconetue was running against Thompson in the Democratic primary. No Republican filed to run for the seat.

Primary results

General results

District 3

The incumbent was Republican Gregg Harper, who had represented the district since 2009. He was re-elected with 80% of the vote in 2012 and the district had a PVI of R+14.

Hardy Caraway, an Independent candidate for the 2nd district in 1984 and the Republican nominee for the 2nd district in 2000 was running against Harper in the Republican primary. Democrats Jim Liljeberg; Doug Magee, who was a Republican candidate for the 4th district in 1988; and Dennis Quinn also ran.

Primary results

Primary runoff results

General results

District 4

The incumbent was Republican Steven Palazzo, who had represented the district since 2011. He was re-elected with 64% of the vote in 2012 and the district had a PVI of R+21.

Palazzo was first elected in 2010, defeating Democratic incumbent Gene Taylor. He was targeted by the Club for Growth.

Chris McDaniel, a Republican member of the Mississippi State Senate, considered challenging Palazzo in the primary, but chose instead to run for the United States Senate. Taylor, who served in the U.S. House from 1989 to 2011, has switched parties, and ran for the seat as a Republican.
Tavish Kelly,  Tom Carter, and Ron Vincent, who was a candidate for the seat in 2012, also ran against Palazzo.

Democrats Trish Causey and Matthew Moore, who was the nominee for the seat in 2012, also ran.

Primary results

General results

References

External links
U.S. House elections in Mississippi, 2014 at Ballotpedia
Campaign contributions at OpenSecrets

United States House
2014
Mississippi

redpajama_set_name: RedPajamaWikipedia


Eat to Die or Eat to Live?
Sounds horrible doesn't it? Who wants to eat to hasten death? But the sad fact is that many (most?) Americans have a lousy diet and exit on the Standard American Deadly Diet. Every day, the fast food joints are full, drive-throughs bumper to bumper while people await their burgers, fries and sodas. In the evenings pizza delivery cars/trucks course the byways delivering death in a box.
Tragically, most people also understand that this diet is truly deadly but continue consuming these harmful substances, seemingly happily, as their arteries clog, their blood sugar spikes and cancers grow in their bodies; too small to be detected but they are there. Even when faced with facts from thousands of studies, papers, documented evidence and in the light of places like the rare 'Blue Zones' around the planet, people seemingly just don't care and happily consume these 'foods' that they KNOW are not healthy.
They mistakenly think it won't be THEM who drops dead from a heart attack/stroke. No, not me who will be diagnosed with breast, prostate, pancreatic cancer. No, not me facing foot/leg amputation because of ravaging diabetes. Never me; it's always the poor 'other guy/gal'.
Well you are WRONG. It WILL BE YOU next time. It is inevitable, if you continue to eat crap your body will turn into crap. Your body is constantly struggling to rebuild itself 24 X 7 and the tools & building blocks are what you eat. Want to live in a house built of fat, salt, sugar and toxins? OR, would you rather rebuild yourself with strong fibrous materials full of nutrients that will build a strong, healthy energized body? These are not silly questions, they are the FACTS that YOU must and DO face every time you open your pie hole and put something down your gullet.
Allopathic Medicine & Drugs are NOT the answer!
These same people often say, my doctor will save me. I can take statins for my high cholesterol. I can take metformin for my diabetes, chemo will kill the cancer. Bypass operations will replace my clogged arteries and stents will keep them open. Go ahead and believe the commercials; but they are FALSE FLAGS luring you into an early, often horrible, death. And to put this in perspective, 614,000 patients die from heart disease and 591,000 from cancer, making medical errors the third leading cause of death. More people die from medical errors than from lung diseases (147,000 deaths), from stroke (133,000 deaths) or from accidents (136,000 deaths). Make you feel better? (Reference: https://www.huffingtonpost.com/cary-a-presant-md/medical-errors-are-the-th_b_11117808.html).
Most regular doctors receive little or no training in simple dietary nutrition. Poor nutrition is the leading cause of the major lifestyle disease like cardio/pulmonary disease, cancer and diabetes; while diabetes contributes to both. These leading causes of death and disability, not to mention horrendous expen$e are, for the most part, entirely PREVENTABLE and in many cased fully reversible. But you will never hear this from the medical industry that pumps billion$ of drugs into the population, preforms thousands of, often unnecessary, surgeries.
For centuries, early humans would consult the local 'Witch Doctor' who would prescribe herbs, tonics and root medicines to cure many ailments. Later cannabis became a popular remedy for a plethora of ailments. Now these very effective remedies are considered 'voodoo' and 'old wives tales' and those practitioners and your great grandmother, are now gone. Corrupt governments with the aid of major drug manufacturers have long warred against natural medicines that, in many cases, are MORE effective than the patented drugs and have none of the deadly side effects of many of the prescription (often addictive) drugs prescribed by your local physician.
So where do you find the magic cures to prevent your early death? How about the produce section in your local grocery store? Who to consult for guidance? How about a local Plant Based Chef specializing and certified in plant-based cuisine? You may find that the 'medicine' is not only effective but it is DELICIOUS and the side effects are not deadly, but energizing and healthy. What a concept, Health Through FOOD, not drugs, surgery and chemicals!
The choice is entirely yours. Your life and, more often than not, the lives of your family are in your hands; or better put, in your kitchen pantry. Will you continue to pollute your body and face an often-horrible painful death or will you live, like those in the Blue Zones, a long and energetically happy life?
Eat to Live or Eat to Die?


win-win cooperation". We hope we can have a friendly relationship with businessman from all over the world. for Pet Bottle Crusher , pet bottle crusher , metso mobile crusher , Our company has abundant strength and possesses a steady and perfect sales network system. We wish we could establish sound business relationships with all customers from at home and abroad on the basis of mutual benefits.
"While in the past few years, our organization absorbed and digested innovative technologies both equally at home and abroad. Meanwhile, our organization staffs a group of experts devoted for the advancement of Pet Bottle Crusher , pet bottle crusher , metso mobile crusher , We aspire to meet the demands of our customers globally. Our range of products and services is continuously expanding to meet customers' requirements. We welcome new and old customers from all walks of life to contact us for future business relationships and achieving mutual success!


We worked with a transportation company in Saudi Arabia seeking a transportation management solution that would enable them to manage orders and shipments around the country. Our job was to identify the software solution, customize it, and train users on how to use it.
We worked with the client to create a requirements document that allowed us to understand the processes the client had in place and develop any processes where gaps existed. This enabled us to identify 10 different potential software solutions.
We identified several of these which would work with their desired accounting solution without being more complicated that what the company would need while fitting into the client's budget. We narrowed down the selections to two that we felt would best fit the requirement and the client chose the one they preferred.
Once we identified the preferred solution, we used the requirements document to customize the solution for the client. We found that upgrades of some of their current solutions provided deeper integration and cost savings. So, with the client's approval we made the necessary upgrades.
We also developed several web interfaces to allow certain team members to interface with the software without needing to access the core components. Furthermore, we customized the software dashboard.
Our solution helped the client tremendously, streamlining its operations. We also helped the client save additional licensing fees thanks to the web interfaces we developed. Our customization of the dashboard provided the client full visibility of what was going on in the company.
We also developed instruction materials for the client that walked them through how to use the software. Each role had clear instructions on how to interact with the software and what would be expected as a result. We walked the client through the training materials and demonstrated use so that the client was able to train the other team members on the software. Recommendations were also made on implementing the solution.


All The Sea Ranch homes currently listed for sale in The Sea Ranch as of 04/25/2019 are shown below. You can change the search criteria at any time by pressing the 'Change Search' button below.
If you have any questions or you would like more information about any The Sea Ranch homes for sale or other The Sea Ranch real estate, please contact us and one of our knowledgeable The Sea Ranch REALTORS® would be pleased to assist you.
"Stunning three bedroom and three bathroom Sea Ranch home. Enjoy this 4,104 square foot home in one of the greatest parts of California. Carpet and wood flooring. Attached two car garage. Lots of windows throughout. Tons of natural light throughout the home. High ceilings throughout. Large open kitchen with an island. Washer and dryer area. Wrap around balcony. Beautiful community. Built in 1992. Walk to the beach..."