text:
\section{Introduction}

Broadly speaking, Korn's inequalities are an important part of the mathematical analysis of many partial differential equations and their associated finite element methods. In particular, the \emph{standard} Korn's inequalities provide an upper bound for the $L_2$ norm of the gradient of a sufficiently-smooth vector field in terms of the $L_2$ norm of the symmetric gradient and (sometimes) an auxiliary norm/seminorm. These types of Korn's inequalities can be classified into two general categories: i) continuous inequalities, and ii) discrete inequalities. The continuous inequalities are usually formulated for sufficiently smooth vector fields which are well-defined (with regard to smoothness) over the entirety of an open, bounded, Lipschitz domain. In contrast, discrete inequalities are usually formulated for sufficiently smooth vector fields which are well-defined in a piecewise fashion on Lipschitz subdomains of a larger domain. Naturally, the associated subdomains are usually convex, polyhedral elements which belong to a triangulation. The continuous class of inequalities can be used to analyze the non-discretized behavior of partial differential equations, or the behavior of numerical methods in which the entire computational domain is covered by a single element. One may consult~\cite{horgan1995korn} for a brief review of continuous Korn inequalities and their applications in the field of continuum mechanics. Conversely, the discrete class of inequalities can be used to analyze finite element methods which operate on the aforementioned partial differential equations. In particular, the discrete standard Korn's inequalities are useful for analyzing finite element methods for applications in linear elasticity~\cite{brenner2008mathematical,riviere2008discontinuous,di2013locking,di2015hybrid,hong2016robust}, and incompressible fluid dynamics~\cite{di2011mathematical,evans2013isogeometricA,evans2013isogeometricB,john2016finite,hong2016uniformly,fu2018strongly}.

Recently, an interest in \emph{generalized} Korn's inequalities has emerged. The generalized Korn's inequalities are very similar to their standard counterparts, with the following distinction: they provide an upper bound for the gradient of a vector field in terms of the trace-free symmetric gradient instead of the standard symmetric gradient. The trace-free symmetric gradient naturally arises in the context of compressible fluid dynamics, where the trace of the velocity gradient tensor (i.e.~the divergence of the velocity field) is non-zero~\cite{anderson1990modern}. In addition, there are applications in the areas of general relativity, Cosserat elasticity, and geometry~\cite{fuchs2009application,schirra2009regularity,faraco2005geometric}. Several classes of continuous generalized Korn's inequalities have been previously developed by Dain~\cite{dain2006generalized}, Schirra~\cite{schirra2012new}, Feireisl and Novotn{\`y}~\cite{feireisl2009singular}, and Breit et al.~\cite{breit2017trace}. However, to the author's knowledge, there have been no rigorous efforts to develop discrete generalized Korn's inequalities. In principle, the development of such discrete inequalities will facilitate the construction of new, robust finite element methods in the aforementioned application areas. In addition, we note that there is an existing class of `versatile mixed finite element methods' for simulating incompressible/compressible fluid dynamics problems that will immediately benefit from the introduction of such inequalities~\cite{chen2020versatile,miller2020versatile}. In particular,  discrete generalized Korn's inequalities are required in order to construct rigorous error estimates for these methods. For all of the reasons mentioned above, this paper will attempt to derive the required class of generalized inequalities.

In order to obtain the desired generalized inequalities, we will follow the approach of Brenner and coworkers~\cite{Brenner04,brenner2004poincare,brenner2015piecewise}. We note that other researchers have performed extensive work on constructing discrete standard Korn's inequalities, including Knobloch and Tobiska~\cite{knobloch2003korn}, Attia and Starke~\cite{attia2005discrete}, Mardal and Winther~\cite{mardal2006observation}, Zhang~\cite{zhang2017discrete}, and Lee~\cite{lee2018robust}; however, we choose to primarily focus on the work of Brenner in this paper. The interested reader is encouraged to consult~\cite{hong2016presentation} for a compact review of the other work on this topic.

In~\cite{Brenner04}, Brenner constructs a class of discrete standard Korn's inequalities for piecewise $H^1$ vector fields in 2D and 3D. In~\cite{brenner2004poincare}, similar techniques are used to construct Poincar{\'e}--Friedrichs discrete inequalities for piecewise $H^2$ vector fields in 2D. Finally, in~\cite{brenner2015piecewise}, standard Korn and Poincar{\'e}--Friedrichs discrete inequalities are obtained for classes of meshes generated by independent refinements, for piecewise $H^1$ vector fields in 2D and 3D.

We intend for the present work to be a natural extension of~\cite{Brenner04} to the case of generalized Korn's inequalities, and piecewise $H^2$ vector fields in 3D. The organization of this paper is as follows. In section 2, we provide an overview of generalized Korn's inequalities at the continuous level. In section 3, we present several new generalized Korn's inequalities at the discrete level. In section 4, we provide proofs for the discrete inequalities. Finally, in section 5, we conclude with a brief summary of the present work.

\section{The Continuous Case}

Let us begin by introducing some important notation and concepts which will be beneficial during the remainder of our presentation.

One may consider a bounded, connected, open, polyhedral domain $\Omega$ in $\mathbb{R}^3$, with boundary $\partial \Omega$. By construction, the boundary of this domain is Lipschitz. On this domain, we define a generic vector field $\bm{u} \in \left[H^2 (\Omega)\right]^3$. In a natural fashion, we define the following norms and seminorms which produce finite results when acting upon $\bm{u}$
\begin{align*}
    \left\| \bm{u} \right\|_{L_2 \left(\Omega\right)} &= \left( \int_{\Omega} | \bm{u} |^2 \, dx \right)^{1/2}, \\[1.5ex]
    \left|\bm{u}\right|_{H^{1}(\Omega)}^{2} &= \sum_{j=1}^{3} \left| u_j \right|^{2}_{H^{1} (\Omega)}, \qquad
    \left\| \bm{u} \right\|_{H^1(\Omega)}^{2} = \left\| \bm{u} \right\|_{L_2 \left(\Omega\right)}^2 + \left| \bm{u} \right|_{H^{1}(\Omega)}^{2},
    \\[1.5ex]
    \left|\bm{u}\right|_{H^{2}(\Omega)}^{2} &= \sum_{j=1}^{3} \left| u_j \right|^{2}_{H^{2} (\Omega)}, \qquad
    \left\| \bm{u} \right\|_{H^2(\Omega)}^{2} = \left\| \bm{u} \right\|_{L_2 \left(\Omega\right)}^2 + \left| \bm{u} \right|_{H^{1}(\Omega)}^{2} + \left| \bm{u} \right|_{H^{2}(\Omega)}^{2},
\end{align*}
where $|\bm{u}|$ is the standard Euclidean norm of $\bm{u}$, and
\begin{align*}
    \left| u_j \right|_{H^{k}(\Omega)} = \left( \sum_{|\bm{\alpha}| = k} \int_{\Omega} \left| D^{\bm{\alpha}} u_j \right|^2 \, dx \right)^{1/2},
\end{align*}
is the Hilbert seminorm of $u_j$ for $H^k (\Omega)$. In the equation above, $\bm{\alpha} = \left(\alpha_1, \alpha_2, \alpha_3\right)$ is the multi-index, and $D^{\bm{\alpha}}$ is the generalized, multi-dimensional derivative operator.

Next, we introduce the following, more specialized, derivative operators
\begin{align*}
    \epsilon_{ij} \left(\bm{u}\right) &= \frac{1}{2} \left(\text{grad}_{ij}(\bm{u}) + \text{grad}_{ji}(\bm{u}) \right) = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right), \\[1.5ex]
    \text{div} \left(\bm{u}\right) &= \sum_{i=1}^{3} \frac{\partial u_i}{\partial x_i}, \\[1.5ex]
    \text{curl} \left(\bm{u} \right) &= \left(\frac{\partial u_3}{\partial x_2} - \frac{\partial u_2}{\partial x_3}, \frac{\partial u_1}{\partial x_3} - \frac{\partial u_3}{\partial x_1}, \frac{\partial u_2}{\partial x_1} - \frac{\partial u_1}{\partial x_2} \right)^t,
\end{align*}
where $\bm{\epsilon} \left( \cdot \right)$ is the symmetric gradient operator, $\text{grad}(\cdot)$ is the standard gradient operator, $\text{div}\left(\cdot \right)$ is the divergence operator, and $\text{curl} \left(\cdot \right)$ is the curl operator. In addition, we introduce the trace-free symmetric gradient operator, which takes the following form
\begin{align*}
    \bm{\epsilon} \left(\bm{u}\right) - \frac{1}{3} \text{div} \left(\bm{u} \right) \mathbb{I},
\end{align*}
where $\mathbb{I}$ is the $3\times3$ identity matrix. Now, we can define the following norms which remain finite for all $\bm{u} \in \left[H^{1}(\Omega) \right]^3$
\begin{align}
    &\left\| \bm{\epsilon} \left(\bm{u}\right) \right\|_{L_2(\Omega) \times L_2(\Omega)} = \left( \int_{\Omega} \left| \bm{\epsilon} \left(\bm{u}\right) \right|^{2} \, dx \right)^{1/2}, \label{norm_one} \\[1.5ex]
    &\left\| \bm{\epsilon} \left(\bm{u}\right) - \frac{1}{3} \text{div} \left(\bm{u} \right) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)} = \left( \int_{\Omega} \left| \bm{\epsilon} \left(\bm{u}\right) - \frac{1}{3} \text{div} \left(\bm{u} \right) \mathbb{I} \right|^{2} \, dx \right)^{1/2}, \label{norm_two}
\end{align}
where the quantity $\left| \bm{\epsilon} \left(\bm{u}\right) \right|$ is the standard Frobenius inner-product applied to $\bm{\epsilon} (\bm{u})$.

We note that if $\bm{u}$ is pointwise divergence-free, then the norm definitions in Eqs.~\eqref{norm_one} and \eqref{norm_two} are identical, and the necessary analysis has already been previously established in other work, (see the Introduction). However, for the case of a generic $\bm{u}$ which is not divergence-free, there are some important results which remain to be investigated.

Towards this end, one may introduce the kernel space of the trace-free symmetric gradient
\begin{align*}
    \mathbf{CK} \left(\Omega\right) = \left\{2 \left(\bm{a} \cdot \bm{x} \right) \bm{x} - \left| \bm{x} \right|^2 \bm{a} + \rho \bm{x} + \bm{Q} \bm{x} + \bm{b} : \bm{a} \in \mathbb{R}^{3}, \bm{b} \in \mathbb{R}^{3}, \rho \in \mathbb{R}, \; \text{and} \; \bm{Q} \in \mathfrak{so}\left(3\right)   \right\},
\end{align*}
where $\bm{x} = \left(x_1, x_2, x_3\right)$ is a coordinate vector, and $\mathfrak{so}(3)$ is the space of $3\times 3$ skew-symmetric matrices. The kernel space (above) is identical to the space of conformal Killing vectors, which was originally identified in~\cite{dain2006generalized} and further studied in~\cite{schirra2012new}. This space contains infinitesimal rigid body motions, Killing vectors, and dilatations. By construction, we have that
\begin{align}
    \bm{\epsilon} \left(\bm{u}\right) - \frac{1}{3} \text{div} \left(\bm{u}\right) \mathbb{I} = \bm{0} \qquad \Longleftrightarrow \qquad \bm{u} \in \mathbf{CK} \left(\Omega\right). \label{kernel_condition}
\end{align}
It turns out that proving Korn-type inequalities for the trace-free symmetric gradient is less straightforward than proving the equivalent inequalities for the standard symmetric gradient. The main challenges arise from the fact that the kernel space for the symmetric gradient is simpler than $\mathbf{CK} (\Omega)$, as it is merely the space of infinitesimal rigid body motions
\begin{align*}
    \mathbf{RM}(\Omega) = \left\{\bm{Q} \bm{x} + \bm{b} : \bm{b} \in \mathbb{R}^{3} \; \text{and} \; \bm{Q} \in \mathfrak{so}\left(3\right)  \right\}.
\end{align*}
The key complication is that $\mathbf{RM}(\Omega)$ is linear in $\bm{x}$, whereas the space $\mathbf{CK} (\Omega)$ is quadratic.

With the above discussion in mind, it is somewhat surprising to find that Korn's inequalities for the trace-free symmetric gradient exist. In particular, in accordance with~\cite{feireisl2009singular} (Theorem 11.22, pg. 466), we have that
\begin{align*}
    \left\| \bm{u} \right\|_{H^{1} (\Omega)} \leq C \left( \left\| \bm{\epsilon} \left(\bm{u}\right) - \frac{1}{3} \text{div} \left(\bm{u} \right) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)} + \int_{\Omega} \left| \bm{u} \right| \, dx \right), \qquad \forall \bm{u} \in \left[ H^{1} (\Omega) \right]^{3}.
\end{align*}
Now, after some simple manipulations with H{\"o}lder's inequality, we obtain
\begin{align}
    \left\| \bm{u} \right\|_{H^{1} (\Omega)} \leq C_{\Omega} \left( \left\| \bm{\epsilon} \left(\bm{u}\right) - \frac{1}{3} \text{div} \left(\bm{u} \right) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)} + \left\| \bm{u} \right\|_{L_2(\Omega)} \right), \label{korn_basic_cont_full}
\end{align}
and
\begin{align}
    \left| \bm{u} \right|_{H^{1} (\Omega)} \leq C_{\Omega} \left( \left\| \bm{\epsilon} \left(\bm{u}\right) - \frac{1}{3} \text{div} \left(\bm{u} \right) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)} + \left\| \bm{u} \right\|_{L_2(\Omega)} \right). \label{korn_basic_cont}
\end{align}
Here, the Korn's inequality has been rewritten in the standard fashion: namely, it provides a bound for the gradient norm of $\bm{u}$ in terms of the trace-free gradient norm and the $L_2$ norm. It remains for us to specialize this inequality, and replace the $L_2$ norm with a broader class of more useful seminorms. Towards this end, we define a class of seminorms on $\left[ H^1(\Omega) \right]^{3}$ that satisfy the following constraints
\begin{align}
    \bm{\Phi} \left(\bm{v}\right) \leq C_{\bm{\Phi}} \left\| \bm{v} \right\|_{H^1(\Omega)}, \qquad \forall \bm{v} \in \left[ H^1(\Omega) \right]^{3}, \label{phi_cond_one}
\end{align}
and
\begin{align}
    \bm{\Phi} \left(\bm{m}\right) = 0 \qquad \text{and} \qquad \bm{m} \in \mathbf{CK}(\Omega) \qquad \Longleftrightarrow \qquad \bm{m} = \text{a constant vector}, \label{phi_cond_two}
\end{align}
where $C_{\bm{\Phi}}$ is a constant that depends on our particular choice of seminorm. It turns out that all seminorms that satisfy the second criterion, will also be invariant under the addition of a constant vector, such that
\begin{align}
    \bm{\Phi} \left(\bm{v} + \bm{c} \right) = \bm{\Phi} \left(\bm{v} \right), \label{phi_cond_three}
\end{align}
where $\bm{c} \in \mathbb{R}^{3}$.

Now, based on the seminorm properties given above, and Eq.~\eqref{korn_basic_cont}, it can be shown that
\begin{align}
    \left| \bm{v} \right|_{H^{1} (\Omega)} \leq C_{\Omega} \left( \left\| \bm{\epsilon} \left(\bm{v}\right) - \frac{1}{3} \text{div} \left(\bm{v} \right) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)} + \bm{\Phi} \left(\bm{v}\right) \right), \qquad \forall \bm{v} \in \left[ H^1(\Omega) \right]^{3}.
    \label{korn_seminorm_cont}
\end{align}
The proof of Eq.~\eqref{korn_seminorm_cont} uses the fact that $H^{1}$ is compactly embedded into $L_2$. Some details of the proof are given in Appendix A, Lemma~\ref{seminorm_lemma}.

There are an infinite number of seminorms which satisfy the required constraints (Eqs.~\eqref{phi_cond_one} and \eqref{phi_cond_two}). However, we choose to focus on the following seminorms due to their utility for practical applications
\begin{align}
    \label{seminorm_def_one} \bm{\Phi}_{1} \left(\bm{v} \right) &= \left\| G \bm{v} \right\|_{L_2(\Omega)}, \\[1.5ex]
    \bm{\Phi}_{2} \left(\bm{v} \right) &= \sup_{ \substack{\bm{m} \in \mathbf{CK}(\Omega) \\ \label{seminorm_def_two}
    \left\| \bm{m} \right\|_{L_2(\partial \Omega)} = 1, \int_{\partial \Omega} \bm{m} \, ds = \bm{0}}} \int_{\partial \Omega} \bm{v} \cdot \bm{m} \, ds,

\end{align}
where
\begin{align*}
    G \bm{v} = \bm{v} - \frac{1}{|\Omega|} \int_{\Omega} \bm{v} \, dx,
\end{align*}
is the projection from $\left[L_{2}(\Omega)\right]^{3}$ on to the orthogonal complement of constant vector fields. Generally speaking, we have that
\begin{align}
    \left\| G\bm{v} \right\|_{L_2(\Omega)} \leq C \left\| \bm{v} \right\|_{L_2(\Omega)}, \qquad \forall \bm{v} \in \left[L_2(\Omega)\right]^{3}. \label{ortho_bound}
\end{align}

\section{The Discrete Case}

Our domain $\Omega \subset \mathbb{R}^3$ can tessellated by non-overlapping tetrahedral elements. We denote the individual elements by $T$, and the overall partition of the domain into these elements by $\mathcal{T}$. The resulting partition has the following properties:
\begin{align*}
    \overline{\Omega} = \bigcup_{T \in \mathcal{T}} \overline{T},
\end{align*}
and
\begin{align*}
    T \cap T' = \emptyset \qquad \text{unless} \qquad T = T'.
\end{align*}
We assume that the minimum interior angle of any tetrahedron $T \in \mathcal{T}$ is given by $\theta_{\mathcal{T}}$. In addition, we assume that the partition is conforming, such that all interior faces are shared by two tetrahedra, and all boundary faces are associated with only one tetrahedron. We denote the individual faces in the partition by $\sigma$. The total set of interior faces is denoted by $S\left(\mathcal{T}, \Omega\right)$, and for each $\sigma \in S\left(\mathcal{T}, \Omega\right)$, we can define a ``$-$" side and a ``$+$" side. Next, we define a jump operator
\begin{align*}
    \left[\!\left[ \bm{u} \right] \! \right]_{\sigma} = \bm{u}_{+} - \bm{u}_{-},
\end{align*}
where $\bm{u}$ is a generic vector that has well-defined values on the interior faces. It is also convenient to define a quadratic projection operator $\pi_{\sigma}$ on each face, such that $\pi_{\sigma}: \left[L_2(\sigma)\right]^3 \rightarrow \left[P_2(\sigma)\right]^3$. This operator satisfies the following standard projection inequality
\begin{align}
    \left\| \pi_{\sigma} \left[\!\left[ \bm{u} \right] \! \right]_{\sigma} \right\|_{L_2(\sigma)} \leq C \left\| \left[\!\left[ \bm{u} \right] \! \right]_{\sigma} \right\|_{L_2(\sigma)}, \qquad \forall \bm{u} \in \left[L_2(\sigma)\right]^{3}. \label{proj_ineq}
\end{align}
The definitions (above) allow us to effectively characterize vector fields which possess discontinuities along the interior faces. We are particularly interested in the class of (possibly discontinuous) piecewise $H^2$ vector fields $\bm{u} \in \left[H^2(\Omega, \mathcal{T})\right]^{3}$, where
\begin{align*}
    \left[H^2(\Omega, \mathcal{T})\right]^{3} = \left\{ \bm{u} \in \left[ L_{2}(\Omega) \right]^{3} : \bm{v}_{T} = \bm{v}|_{T} \in \left[H^{2}(T) \right]^{3} \quad  \forall T \in \mathcal{T} \right\}.
\end{align*}
The associated broken seminorms can be defined as follows
\begin{align*}
    \left| \bm{u} \right|_{H^{1}(\Omega, \mathcal{T})} = \left( \sum_{T \in \mathcal{T}} \left| \bm{u} \right|^{2}_{H^1(T)} \right)^{1/2}, \qquad  \left| \bm{u} \right|_{H^{2}(\Omega, \mathcal{T})} = \left( \sum_{T \in \mathcal{T}} \left| \bm{u} \right|^{2}_{H^2(T)} \right)^{1/2}.
\end{align*}
In addition, we can define piecewise derivative operators $\forall T \in \mathcal{T}$
\begin{align*}
    \bm{\epsilon}_{\mathcal{T}} (\bm{u}) |_{T} = \bm{\epsilon} \left( \bm{u}_{T} \right), \qquad \text{div}_{\mathcal{T}} (\bm{u}) |_{T} = \text{div} \left(\bm{u}_{T} \right).
\end{align*}

With the above discussion in mind, we can formally state a new class of generalized, discrete Korn's inequalities for all $\bm{v} \in \left[H^2(\Omega, \mathcal{T})\right]^{3}$
\begin{align}
    \left| \bm{v} \right|_{H^{1} (\Omega, \mathcal{T})}^{2} \leq C \Bigg(& \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2} + \left(\bm{\Phi} (\bm{v}) \right)^{2} \label{korn_seminorm_disc_proj} \\[1.5ex]
    \nonumber &+ \sum_{\sigma \in S(\mathcal{T}, \Omega)} \left(\text{diam} \, \sigma\right)^{-1} \left\| \pi_{\sigma} \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} \right\|^{2}_{L_2(\sigma)} \Bigg).
\end{align}
In accordance with Eq.~\eqref{proj_ineq}, we can immediately eliminate the dependence of the inequality on the projection operator $\pi_{\sigma}$, and obtain
\begin{align}
        \left| \bm{v} \right|_{H^{1} (\Omega, \mathcal{T})}^{2} \leq C \Bigg(& \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2} + \left(\bm{\Phi} (\bm{v}) \right)^{2} \label{korn_seminorm_disc} \\[1.5ex]
    \nonumber &+ \sum_{\sigma \in S(\mathcal{T}, \Omega)} \left(\text{diam} \, \sigma\right)^{-1} \left\| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} \right\|^{2}_{L_2(\sigma)} \Bigg).
\end{align}
The Korn's inequalities above can be expressed explicitly for each seminorm, as follows
\begin{align}
        \left| \bm{v} \right|_{H^{1} (\Omega, \mathcal{T})}^{2} \leq C \Bigg(& \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2} + \left\| G \bm{v} \right\|_{L_2(\Omega)}^{2} \\[1.5ex]
    \nonumber &+ \sum_{\sigma \in S(\mathcal{T}, \Omega)} \left(\text{diam} \, \sigma\right)^{-1} \left\| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} \right\|^{2}_{L_2(\sigma)} \Bigg), \\[1.5ex]
           \left| \bm{v} \right|_{H^{1} (\Omega, \mathcal{T})}^{2} \leq C \Bigg(& \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2} + \sup_{ \substack{\bm{m} \in \mathbf{CK}(\Omega) \\
    \left\| \bm{m} \right\|_{L_2(\partial \Omega)} = 1,  \int_{\partial \Omega} \bm{m} \, ds = \bm{0}}} \left( \int_{\partial \Omega} \bm{v} \cdot \bm{m} \, ds \right)^{2}, \\[1.5ex]
     \nonumber &+ \sum_{\sigma \in S(\mathcal{T}, \Omega)} \left(\text{diam} \, \sigma\right)^{-1} \left\| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} \right\|^{2}_{L_2(\sigma)} \Bigg),
\end{align}
or equivalently,
\begin{align}
        \left| \bm{v} \right|_{H^{1} (\Omega, \mathcal{T})}^{2} \leq C \Bigg(& \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2} + \left\| G \bm{v} \right\|_{L_2(\Omega)}^{2} \label{useful_one} \\[1.5ex]
    \nonumber &+ \sum_{\sigma \in S(\mathcal{T}, \Omega)} \left(\text{diam} \, \sigma\right)^{-1} \left\| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} \right\|^{2}_{L_2(\sigma)} \Bigg), \\[1.5ex]
           \left| \bm{v} \right|_{H^{1} (\Omega, \mathcal{T})}^{2} \leq C \Bigg(& \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2} + \left\| \bm{v} \right\|^{2}_{L_2(\partial \Omega)} \label{useful_two} \\[1.5ex]
    \nonumber &+ \sum_{\sigma \in S(\mathcal{T}, \Omega)} \left(\text{diam} \, \sigma\right)^{-1} \left\| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} \right\|^{2}_{L_2(\sigma)} \Bigg).
\end{align}
The inequalities in Eqs.~\eqref{useful_one} and \eqref{useful_two} are the most useful in practice. In particular, they can be used for mixed finite element methods and discontinuous Galerkin methods. They can also be used for continuous Galerkin methods. In this latter case, all of the jump terms vanish, and one obtains the following simplified inequalities
\begin{align}
        \left| \bm{v} \right|_{H^{1} (\Omega, \mathcal{T})}^{2} \leq C \Bigg(& \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2} + \left\| G \bm{v} \right\|_{L_2(\Omega)}^{2} \Bigg), \\[1.5ex]
           \left| \bm{v} \right|_{H^{1} (\Omega, \mathcal{T})}^{2} \leq C \Bigg(& \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2} + \left\| \bm{v} \right\|^{2}_{L_2(\partial \Omega)} \Bigg).

\end{align}
It remains for us to prove the key discrete result (Eq.~\eqref{korn_seminorm_disc_proj}), from whence all other discrete results follow. This will be the focus of the next section.

\section{Proofs of Discrete Results}
In this section, we will construct a rigorous proof for Eq.~\eqref{korn_seminorm_disc_proj}. This will require the introduction of two lemmas, prior to the final result. The final result itself will be stated in a theorem at the end of the section.

Prior to the first lemma, it is necessary to introduce a mapping operator, along with some additional notation. Let us denote the set of all vertices in $\mathcal{T}$ as $\mathcal{V} (\mathcal{T})$, and the set of all edge midpoints as $\mathcal{M} (\mathcal{T})$. Thereafter, one may define a mapping operator $E$ which maps piecewise quadratic vector fields to continuous piecewise quadratic vector fields. More precisely, we define $E: V_{\mathcal{T}} \rightarrow W_{\mathcal{T}}$, where
\begin{align*}
    V_{\mathcal{T}} &= \left\{\bm{v} \in \left[L_2(\Omega)\right]^{3}: \bm{v}_{T} = \bm{v}|_{T} \in \left[P_2(T) \right]^{3} \quad \forall T \in \mathcal{T} \right\}, \\[1.5ex]
    W_{\mathcal{T}} &= \left\{\bm{w} \in \left[H^1(\Omega)\right]^{3}: \bm{w}_{T} = \bm{w}|_{T} \in \left[P_2(T) \right]^{3} \quad \forall T \in \mathcal{T} \right\},
\end{align*}
and where
\begin{align}
    \left(E \bm{v}\right)(p) &= \frac{1}{|\mathcal{T}_p|} \sum_{T \in \mathcal{T}_{p}} \bm{v}_{T} (p), \qquad \forall p \in \mathcal{V}(\mathcal{T}), \label{edef_one} \\[1.5ex]
    \left(E \bm{v} \right)(m) &= \frac{1}{|\mathcal{T}_m|} \sum_{T \in \mathcal{T}_{m}} \bm{v}_{T} (m), \qquad \forall m \in \mathcal{M}(\mathcal{T}). \label{edef_two}
\end{align}
Here, we define $\mathcal{T}_p$ as the set of tetrahedra sharing a common vertex $p$
\begin{align*}
    \mathcal{T}_p = \left\{ T \in \mathcal{T} : p \in \partial T \right\},
\end{align*}
and $\mathcal{T}_m$ as the set of tetrahedra sharing a common edge with midpoint $m$
\begin{align*}
    \mathcal{T}_m = \left\{ T \in \mathcal{T} : m \in \partial T \right\}.
\end{align*}
Figure~\ref{tet_chain_fig} provides an example of a shared vertex and a shared edge midpoint.
\begin{figure}[h!]
\includegraphics[width=12cm]{tet_chain.png}
\caption{A collection of tetrahedra: $T_1, T_2, \ldots, T_5$. The tetrahedra share two common vertices, one of which ($p$) is shown on the left. The tetrahedra also share a single edge midpoint ($m$) which is shown on the right.}
\label{tet_chain_fig}
\end{figure}
In general, we assume that the number of tetrahedra sharing a vertex or midpoint is relatively small, i.e.
\begin{align*}
    | \mathcal{T}_{p} | &\lesssim 1, \qquad \forall p \in \mathcal{V}(\mathcal{T}), \\[1.5ex]
    | \mathcal{T}_{m} | &\lesssim 1, \qquad \forall m \in \mathcal{M}(\mathcal{T}).
\end{align*}
Here, and throughout the remainder of this section, we say that $A \lesssim B$ when $A \leq c_0 B$, and $c_0$ is a constant do not depend on the mesh. In a natural fashion, we say that $A \lesssim B$ and $B \lesssim A$ implies $A \approx B$.

In addition,
the continuous piecewise quadratic vector fields (which reside in $W_{\mathcal{T}})$ can be represented using piecewise quadratic, continuous Galerkin finite elements. These elements can be visualized as shown in figure~\ref{p2_fig}.
\begin{figure}[h!]
\includegraphics[width=5cm]{p2_interp.png}
\caption{A collection of continuous, piecewise quadratic, finite elements. The degrees of freedom for each element are represented using solid dots.}
\label{p2_fig}
\end{figure}

We are now ready to introduce the first lemma of this section.

\begin{lemma}
The pointwise differences between the piecewise vector fields $\bm{v} \in V_{\mathcal{T}}$ and the continuous mapping $E \bm{v} \in W_{\mathcal{T}}$ are given by the following inequalities
\begin{align}
    \left| \bm{v}_{T} (p) - E \bm{v} (p) \right|^2 &\lesssim \sum_{\sigma \in \Xi_p} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (p) \right|^2, \qquad \forall T \in \mathcal{T}, \; \forall p \in \mathcal{V} (\mathcal{T}), \label{jump_one} \\[1.5ex]
    \left| \bm{v}_{T} (m) - E \bm{v} (m) \right|^2 &\lesssim \sum_{\sigma \in \Xi_m} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (m) \right|^2, \qquad \forall T \in \mathcal{T}, \; \forall m \in \mathcal{M} (\mathcal{T}), \label{jump_two}
\end{align}
where $\Xi_p$ is the set of interior faces sharing $p$ as a common vertex
\begin{align*}
    \Xi_{p} = \left\{ \sigma \in S(\mathcal{T},\Omega) : p \in \partial \sigma \right\},
\end{align*}
and $\Xi_m$ is the set of interior faces sharing $m$ as a common edge midpoint
\begin{align*}
    \Xi_{m} = \left\{ \sigma \in S(\mathcal{T},\Omega) : m \in \partial \sigma \right\}.
\end{align*}
\end{lemma}

\begin{proof}
The proof of Eq.~\eqref{jump_one} appears in Lemma 2.1 of~\cite{Brenner04}. It remains for us to prove Eq.~\eqref{jump_two}. We begin by using the definition of $E$ from Eq.~\eqref{edef_two} in order to obtain the following expression
\begin{align}
    \bm{v}_{T} (m) - E\bm{v} (m) = \frac{1}{|\mathcal{T}_m|} \sum_{T' \in \mathcal{T}_m} \left(\bm{v}_{T}(m) - \bm{v}_{T'}(m) \right).
    \label{lemma_1a}
\end{align}
Next, we introduce a sequence of tetrahedra which share the edge midpoint $m$, where $T = T_1$ and $T' = T_k$, and $k > 1$. An example of such a sequence is $T_1, T_2, \ldots, T_{j}, T_{j+1}, \ldots, T_{k-1}, T_k$, where $k \leq |\mathcal{T}_{m}|$. Furthermore, suppose that $T_j$ and $T_{j+1}$ share a common face $\sigma_j \in \Xi_m$. Under these circumstances, the following identity holds
\begin{align}
    \left| \bm{v}_{T}(m) - \bm{v}_{T'}(m) \right|^{2} &= \left| \sum_{j=1}^{k-1}\left(\bm{v}_{T_j}(m) - \bm{v}_{T_{j+1}}(m) \right) \right|^2 \label{lemma_1b} \\[1.5ex]
    \nonumber & \lesssim \sum_{j=1}^{k-1} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma_j}(m) \right|^2,
\end{align}
where we have used the root-mean-square-arithmetic-mean inequality on the last line. Upon combining Eqs.~\eqref{lemma_1a} and \eqref{lemma_1b}, and using the root-mean-square-arithmetic-mean inequality a second time, we obtain the desired result (Eq.~\eqref{jump_two}).
\end{proof}

\begin{lemma}
Suppose that a seminorm $\bm{\Phi}$ satisfies the following inequalities
\begin{align}
    \bm{\Phi} (\bm{w}) &\leq C_{\bm{\Phi}} \left\| \bm{w} \right\|_{H^1(\Omega, \mathcal{T})}, \qquad \bm{w} \in \left[ H^{1}(\Omega, \mathcal{T}) \right]^{3}, \\[1.5ex]
    \left(\bm{\Phi}(\bm{v} - E \bm{v} )\right)^{2} & \lesssim \sum_{\sigma \in S(\mathcal{T},\Omega)} (\text{diam} \, \sigma) \left( \sum_{p \in \mathcal{V}(\sigma)} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (p) \right|^{2} + \sum_{m \in \mathcal{M}(\sigma)} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (m) \right|^{2} \right), \quad \forall \bm{v} \in V_{\mathcal{T}}, \label{seminorm_bound}
\end{align}
in conjunction with
\begin{align*}
    \bm{\Phi} (\bm{m}) = 0 \qquad \text{and} \qquad \bm{m} \in \mathbf{CK}(\Omega) \qquad \Longleftrightarrow \qquad \bm{m} = \text{const},
\end{align*}
where $\mathcal{V}(\sigma)$ is the set of all vertices of $\sigma$, and $\mathcal{M}(\sigma)$ is the set of all edge midpoints of $\sigma$. Under these circumstances, the following inequality holds for all $\bm{v} \in V_{\mathcal{T}}$
\begin{align}
    \left| \bm{v} \right|^{2}_{H^{1}(\Omega, \mathcal{T})} & \lesssim \left\| \epsilon_{\mathcal{T}}(\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|^{2}_{L_2(\Omega) \times L_2(\Omega)} + \left(\bm{\Phi}(\bm{v})\right)^{2} \label{v_estimate} \\[1.5ex]
    \nonumber &+\sum_{\sigma \in S(\mathcal{T},\Omega)} (\text{diam} \, \sigma) \left( \sum_{p \in \mathcal{V}(\sigma)} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (p) \right|^{2} + \sum_{m \in \mathcal{M}(\sigma)} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (m) \right|^{2} \right).
\end{align}
\end{lemma}

\begin{proof}
    We begin by assuming that the diameter of each triangle is comparable in size to the diameters of the nearby faces
    %
    \begin{align}
        \text{diam} \, T \approx \text{diam} \, \sigma, \qquad &\forall T \in \mathcal{T}_{p}, \; \sigma \in \Xi_{p}, \; p \in \mathcal{V}(\mathcal{T}), \label{diam_criterion} \\[1.5ex]
        \nonumber & \forall  T \in \mathcal{T}_{m}, \; \sigma \in \Xi_{m}, \; m \in \mathcal{M}(\mathcal{T}).
    \end{align}
    %
    Next, we develop the following piecewise $H^1$ estimate by using standard $L_{\infty}$ error estimates (cf.~\cite{brenner2008mathematical}, Chapter 4), in conjunction with Eqs.~\eqref{jump_one}, \eqref{jump_two}, and \eqref{diam_criterion}
    %
    \begin{align}
        \left| \bm{v} - E \bm{v} \right|^{2}_{H^{1}(\Omega, \mathcal{T})} &\lesssim \sum_{T \in \mathcal{T}} \left(\text{diam} \, T\right) \left( \sum_{p \in \mathcal{V}(T)} \left| \bm{v}_{T} (p) - E\bm{v}(p) \right|^{2} + \sum_{m \in \mathcal{M}(T)} \left| \bm{v}_{T} (m) - E\bm{v}(m) \right|^{2} \right) \label{h1_estimate} \\[1.5ex]
        \nonumber & \lesssim \sum_{\sigma \in S(\mathcal{T},\Omega)} (\text{diam} \, \sigma) \left( \sum_{p \in \mathcal{V}(\sigma)} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (p) \right|^{2} + \sum_{m \in \mathcal{M}(\sigma)} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (m) \right|^{2} \right).
    \end{align}
    %
    In addition, we develop the following $H^1$ estimate based on the triangle inequality, root-mean-square-arithmetic-mean inequality, Eqs.~\eqref{korn_seminorm_cont}, \eqref{seminorm_bound}, and \eqref{h1_estimate}, and Eq.~\eqref{grad_bound} from Lemma~\ref{grad_lemma} in Appendix A
    %
    \begin{align*}
        \left| \bm{v} \right|^{2}_{H^{1}(\Omega,\mathcal{T})} &\leq \left| E \bm{v} \right|^{2}_{H^{1}(\Omega)} + \left| \bm{v} - E \bm{v} \right|^{2}_{H^{1}(\Omega, \mathcal{T})} \\[1.5ex]
        & \lesssim \left\| \bm{\epsilon}(E\bm{v}) - \frac{1}{3} \text{div} (E\bm{v}) \mathbb{I} \right\|^{2}_{L_2(\Omega) \times L_2(\Omega)} + \left( \bm{\Phi}(E\bm{v}) \right)^{2} + \left| \bm{v} - E \bm{v} \right|^{2}_{H^{1}(\Omega, \mathcal{T})} \\[1.5ex]
        &\lesssim \left\| \bm{\epsilon}_{\mathcal{T}}(\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|^{2}_{L_2(\Omega) \times L_2(\Omega)} + \left\| \bm{\epsilon}_{\mathcal{T}}(\bm{v} - E\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v} - E\bm{v}) \mathbb{I} \right\|^{2}_{L_2(\Omega) \times L_2(\Omega)} \\[1.5ex]
        &+ \left( \bm{\Phi}(\bm{v}) \right)^{2} + \left( \bm{\Phi}(\bm{v} - E\bm{v}) \right)^{2} + \left| \bm{v} - E \bm{v} \right|^{2}_{H^{1}(\Omega, \mathcal{T})} \\[1.5ex]
        &\lesssim \left\| \bm{\epsilon}_{\mathcal{T}}(\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|^{2}_{L_2(\Omega) \times L_2(\Omega)} + \left( \bm{\Phi}(\bm{v}) \right)^{2} + \left( \bm{\Phi}(\bm{v} - E\bm{v}) \right)^{2} + \left| \bm{v} - E \bm{v} \right|^{2}_{H^{1}(\Omega, \mathcal{T})} \\[1.5ex]
        & \lesssim \left\| \bm{\epsilon}_{\mathcal{T}}(\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|^{2}_{L_2(\Omega) \times L_2(\Omega)} + \left( \bm{\Phi}(\bm{v}) \right)^{2} \\[1.5ex]
        & + \sum_{\sigma \in S(\mathcal{T},\Omega)} (\text{diam} \, \sigma) \left( \sum_{p \in \mathcal{V}(\sigma)} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (p) \right|^{2} + \sum_{m \in \mathcal{M}(\sigma)} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (m) \right|^{2} \right).
    \end{align*}
    %
    The last line contains the desired result (Eq.~\eqref{v_estimate}).
\end{proof}

In the following examples, we will demonstrate that the inequality in Eq.~\eqref{seminorm_bound} holds for the seminorms in Eqs.~\eqref{seminorm_def_one} and \eqref{seminorm_def_two}.

\begin{example}
    For the seminorm in Eq.~\eqref{seminorm_def_one}, we have that
    %
    \begin{align*}
        \left(\bm{\Phi}_{1}(\bm{v}-E\bm{v})\right)^{2} & = \left\| G(\bm{v} - E \bm{v}) \right\|_{L_2(\Omega)}^{2} \\[1.5ex]
        &\leq C \left\| \bm{v} - E \bm{v} \right\|_{L_2(\Omega)}^{2} \\[1.5ex]
        & \lesssim \sum_{T \in \mathcal{T}} \left(\text{diam} \, T \right)^{3} \left( \sum_{p\in\mathcal{V}(T)} \left| \bm{v}_{T}(p) - E \bm{v}(p) \right|^{2} + \sum_{m\in\mathcal{M}(T)} \left| \bm{v}_{T}(m) - E \bm{v}(m) \right|^{2} \right) \\[1.5ex]
        &\lesssim \sum_{\sigma \in S(\mathcal{T},\Omega)} (\text{diam} \, \sigma)^{3} \left( \sum_{p \in \mathcal{V}(\sigma)} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (p) \right|^{2} + \sum_{m \in \mathcal{M}(\sigma)} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (m) \right|^{2} \right),
    \end{align*}
    %
    where we have used the boundedness of the projection operator (Eq.~\eqref{ortho_bound}), a standard $L_{\infty}$ estimate~\cite{brenner2008mathematical}, and Eqs.~\eqref{jump_one}, \eqref{jump_two}, and \eqref{diam_criterion}.
\end{example}

\begin{example}
    For the seminorm in Eq.~\eqref{seminorm_def_two}, we have that
    %
    \begin{align*}
        \left(\bm{\Phi}_{2}(\bm{v}-E\bm{v})\right)^{2} &= \sup_{ \substack{\bm{m} \in \mathbf{CK}(\Omega) \\
        \left\| \bm{m} \right\|_{L_2(\partial \Omega)} = 1, \int_{\partial \Omega} \bm{m} \, ds = \bm{0}}} \left( \int_{\partial \Omega} \left( \bm{v} - E \bm{v} \right) \cdot \bm{m} \, ds \right)^{2} \\[1.5ex]
        &\leq C \left\| \bm{v} - E \bm{v} \right\|_{L_2(\partial\Omega)}^{2}
        \\[1.5ex]
        & \lesssim \sum_{\substack{T \in \mathcal{T} \\ \partial T \cap \partial \Omega \neq \emptyset}} \left(\text{diam} \, T \right)^{2} \left( \sum_{p\in\mathcal{V}(T)} \left| \bm{v}_{T}(p) - E \bm{v}(p) \right|^{2} + \sum_{m\in\mathcal{M}(T)} \left| \bm{v}_{T}(m) - E \bm{v}(m) \right|^{2} \right) \\[1.5ex]
        &\lesssim \sum_{\sigma \in S(\mathcal{T},\Omega)} (\text{diam} \, \sigma)^{2} \left( \sum_{p \in \mathcal{V}(\sigma)} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (p) \right|^{2} + \sum_{m \in \mathcal{M}(\sigma)} \left| \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} (m) \right|^{2} \right),
    \end{align*}
    %
    where we have used H{\"o}lder's inequality, a standard $L_{\infty}$ estimate~\cite{brenner2008mathematical}, a discrete trace inequality, a root-mean-square-arithmetic-mean inequality, and Eqs.~\eqref{jump_one}, \eqref{jump_two}, and \eqref{diam_criterion}.
\end{example}

In order to provide an appropriate context for our discussion of the final theorem, we will introduce the following interpolation operator for all $\bm{v} \in \left[H^{2}(T)\right]^{3}$
\begin{align}
    &\left| \int_{T} \left(\bm{v} - \Pi_{T} \bm{v} \right) dx \right| = 0, \qquad \left| \int_{T} \text{div} \left(\bm{v} - \Pi_{T} \bm{v} \right) dx \right| = 0, \label{interp_def} \\[1.5ex]
    \nonumber &\left| \int_{T} \text{curl} \left( \bm{v} - \Pi_{T} \bm{v} \right) dx \right| = 0, \qquad \left| \int_{T} \text{curl} \left( \text{curl} \left( \bm{v} - \Pi_{T} \bm{v} \right) \right) dx \right| = 0.
\end{align}
This interpolation operator is noteworthy because it projects $\bm{v}$ on to the kernel space $\mathbf{CK} (T)$. In fact, it can be shown that for all $\bm{m} \in \mathbf{CK}(T)$
\begin{align*}
    & \left| \int_{T} \bm{m} \, dx \right| = 0, \qquad \left| \int_{T} \text{div} (\bm{m}) \, dx \right| = 0, \\[1.5ex]
    &\left| \int_{T} \text{curl}(\bm{m}) \, dx \right| = 0, \qquad \left| \int_{T} \text{curl}(\text{curl} (\bm{m})) \, dx \right| = 0, \qquad \Longleftrightarrow \qquad \bm{m} = \bm{0}.
\end{align*}
The proof of the above statement appears in Appendix B, Lemma~\ref{projection_lemma}. Using the element-wise definition of the interpolation operator above, we can construct a global interpolation operator
\begin{align*}
    &\Pi : \left[H^{2}(\Omega,\mathcal{T})\right]^{3} \rightarrow V_{\mathcal{T}}, \\[1.5ex]
    & \left(\Pi \bm{v}\right) |_{T} = \Pi_{T} \bm{v}_{T}, \qquad \forall T \in \mathcal{T}.
\end{align*}
In addition, upon combining Eq.~\eqref{interp_def} with Eq.~\eqref{local_korn}, (the latter of which is derived in Appendix C, Theorem~\ref{local_korn_lemma}), we obtain
\begin{align}
    \left| \bm{v} - \Pi_{T} \bm{v} \right|_{H^{1}(T)} &\lesssim \left\| \bm{\epsilon} (\bm{v} - \Pi_{T} \bm{v}) - \frac{1}{3} \text{div}(\bm{v} - \Pi_{T} \bm{v}) \mathbb{I} \right\|_{L_2(T) \times L_2(T)} \label{project_error_one} \\[1.5ex]
    \nonumber & = \left\| \bm{\epsilon} (\bm{v}) - \frac{1}{3} \text{div}(\bm{v}) \mathbb{I} \right\|_{L_2(T) \times L_2(T)},
\end{align}
for all $T \in \mathcal{T}$ and $\bm{v} \in \left[H^{2}(\Omega,\mathcal{T})\right]^{3}$. Finally, by a standard Poincar{\'e}-type inequality, we have that
\begin{align}
    \left\| \bm{v} - \Pi_{T} \bm{v} \right\|_{L_2(T)} \lesssim \left(\text{diam} \, T \right) \left| \bm{v} - \Pi_{T} \bm{v} \right|_{H^{1}(T)}, \label{project_error_two}
\end{align}
for all $T \in \mathcal{T}$ and $\bm{v} \in \left[H^{1}(\Omega,\mathcal{T})\right]^{3}$. We are now ready to state the final result.

\begin{theorem}
    Suppose that a seminorm $\bm{\Phi}: \left[H^{1}(\Omega, \mathcal{T})\right]^{3} \rightarrow \mathbb{R}$ satisfies the following condition
    %
    \begin{align}
        \bm{\Phi} \left(\bm{v} - \Pi \bm{v} \right) \lesssim \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}}(\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}, \qquad \forall \bm{v} \in \left[ H^{2}(\Omega, \mathcal{T}) \right]^{3}, \label{seminorm_proj_bound}
    \end{align}
    %
    where $\Pi : \left[ H^{2}(\Omega, \mathcal{T}) \right]^{3} \rightarrow V_{\mathcal{T}}$. Under these circumstances, the following inequality holds
    %
    \begin{align}
        \left| \bm{v} \right|^{2}_{H^{1}(\Omega,\mathcal{T})} \leq \kappa\left(\theta_{\mathcal{T}} \right) \Bigg( &\left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}}(\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2} + \left(\bm{\Phi}(\bm{v})\right)^{2} \label{final_result} \\[1.5ex]
        \nonumber &+ \sum_{\sigma \in S(\mathcal{T}, \Omega)} \left(\text{diam} \, \sigma \right)^{-1} \left\| \pi_{\sigma} \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} \right\|^{2}_{L_2(\sigma)} \Bigg), \qquad \forall \bm{v} \in \left[H^{2}(\Omega, \mathcal{T})\right]^{3},
    \end{align}
    %
    where $\kappa: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ is a continuous function, which does not depend on $\mathcal{T}$.
\end{theorem}

\begin{proof}
    We start by constructing the following $H^1$ estimate using the triangle inequality, the root-mean-square-arithmetic-mean inequality, Eq.~\eqref{kernel_condition}, Eq.~\eqref{v_estimate}, and Eq.~\eqref{project_error_one}
    %
    \begin{align}
        \left| \bm{v} \right|_{H^{1}(\Omega, \mathcal{T})}^{2} &\leq \left| \bm{v} - \Pi \bm{v} \right|^{2}_{H^{1}(\Omega,\mathcal{T})} + \left| \Pi \bm{v} \right|^{2}_{H^{1}(\Omega, \mathcal{T})} \label{theorem_a} \\[1.5ex]
        \nonumber &\lesssim \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}}(\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2} + \left(\bm{\Phi}(\Pi \bm{v}) \right)^{2} \\[1.5ex]
        \nonumber &+\sum_{\sigma \in S(\mathcal{T},\Omega)} (\text{diam} \, \sigma) \left( \sum_{p \in \mathcal{V}(\sigma)} \left| \left[\!\left[ \Pi \bm{v} \right] \! \right]_{\sigma} (p) \right|^{2} + \sum_{m \in \mathcal{M}(\sigma)} \left| \left[\!\left[ \Pi \bm{v} \right] \! \right]_{\sigma} (m) \right|^{2} \right).
    \end{align}
    %
    In addition, we can use the triangle inequality and Eq.~\eqref{seminorm_proj_bound} to obtain the following estimate
    %
    \begin{align}
        \bm{\Phi}(\Pi \bm{v}) & \leq \bm{\Phi}(\bm{v} - \Pi \bm{v}) + \bm{\Phi}(\bm{v}) \label{theorem_b} \\[1.5ex]
        \nonumber &\lesssim \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}}(\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)} + \bm{\Phi}(\bm{v}).
    \end{align}
    %
    It remains for us to construct an upper bound for the jump terms on the RHS of Eq.~\eqref{theorem_a}. Towards this end, we re-introduce the quadratic projection operator
    %
    \begin{align}
        \nonumber &\pi_{\sigma} : \left[L_{2}(\sigma)\right]^{3} \rightarrow \left[P_2(\sigma)\right]^{3}, \qquad \forall \sigma \in S(\mathcal{T}, \Omega).

    \end{align}
    %
    Then, by the triangle inequality, the root-mean-square-arithmetic-mean inequality, the discrete trace inequality, and standard inverse estimates (see~\cite{brenner2008mathematical}, Chapter 4), we obtain
    %
    \begin{align}
        \left| \left[\!\left[ \Pi \bm{v} \right] \! \right]_{\sigma} (p) \right|^{2} &= \left| \left[\!\left[ \Pi \bm{v} - \bm{v} + \bm{v} \right] \! \right]_{\sigma} (p) \right|^{2} \label{project_p} \\[1.5ex]
        \nonumber &= \left| \pi_{\sigma} \left[\!\left[ \Pi \bm{v} - \bm{v} + \bm{v} \right] \! \right]_{\sigma}(p) \right|^{2} \\[1.5ex]
        \nonumber &\lesssim \left| \pi_{\sigma} \left[\!\left[ \bm{v} - \Pi \bm{v} \right] \! \right]_{\sigma} (p) \right|^{2} + \left| \pi_{\sigma} \left[\!\left[ \bm{v} \right] \! \right]_{\sigma}(p) \right|^{2} \\[1.5ex]
        \nonumber &\lesssim \left(\text{diam} \, \sigma\right)^{-2} \left( \left\| \pi_{\sigma} \left[\!\left[ \bm{v} - \Pi \bm{v} \right] \! \right]_{\sigma} \right\|_{L_2(\sigma)}^{2} + \left\| \pi_{\sigma} \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} \right\|_{L_2(\sigma)}^{2} \right).
    \end{align}
    %
    Following a similar argument, we have that
    %
    \begin{align}
        \left| \left[\!\left[ \Pi \bm{v} \right] \! \right]_{\sigma} (m) \right|^{2} \lesssim \left(\text{diam} \, \sigma\right)^{-2} \left( \left\| \pi_{\sigma} \left[\!\left[ \bm{v} - \Pi \bm{v} \right] \! \right]_{\sigma} \right\|_{L_2(\sigma)}^{2} + \left\| \pi_{\sigma} \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} \right\|_{L_2(\sigma)}^{2} \right). \label{project_m}
    \end{align}
    %
    Next, we can construct the following $L_2$ estimate by using the discrete trace inequality in conjunction with Eqs.~\eqref{project_error_one}, \eqref{project_error_two}, and \eqref{proj_ineq}
    %
    \begin{align}
        \left\| \pi_{\sigma} \left[\!\left[ \bm{v} - \Pi \bm{v} \right] \! \right]_{\sigma} \right\|_{L_2(\sigma)}^{2} &\leq C \left\| \left[\!\left[ \bm{v} - \Pi \bm{v} \right] \! \right]_{\sigma} \right\|_{L_2(\sigma)}^{2} \label{theorem_c} \\[1.5ex]
        \nonumber & \lesssim \sum_{T \in \mathcal{T}_{\sigma}} \Bigg( \left(\text{diam} \, T \right) \left| \bm{v}_{T} - \Pi_{T} \bm{v}_{T} \right|_{H^{1}(T)}^{2} \\[1.5ex]
        \nonumber &+ \left(\text{diam} \, T\right)^{-1} \left\| \bm{v}_{T} - \Pi_{T} \bm{v}_{T} \right\|_{L_{2}(T)}^{2} \Bigg) \\[1.5ex]
        \nonumber &\lesssim \sum_{T \in \mathcal{T}_{\sigma}} \left(\text{diam} \, T \right) \left\| \bm{\epsilon}(\bm{v}_{T}) -\frac{1}{3} \text{div} (\bm{v}_{T}) \mathbb{I} \right\|_{L_2(T) \times L_2(T)}^{2},
    \end{align}
    %
    where $\mathcal{T}_{\sigma}$ is the set of tetrahedra sharing the common side $\sigma$. Now, upon combining Eqs.~\eqref{diam_criterion}, \eqref{project_p}, \eqref{project_m}, and \eqref{theorem_c}, we obtain
    %
    \begin{align}
        &\sum_{\sigma \in S(\mathcal{T},\Omega)} (\text{diam} \, \sigma) \left( \sum_{p \in \mathcal{V}(\sigma)} \left| \left[\!\left[ \Pi \bm{v} \right] \! \right]_{\sigma} (p) \right|^{2} + \sum_{m \in \mathcal{M}(\sigma)} \left| \left[\!\left[ \Pi \bm{v} \right] \! \right]_{\sigma} (m) \right|^{2} \right) \label{theorem_d} \\[1.5ex]
        \nonumber & \lesssim \sum_{\sigma \in S(\mathcal{T},\Omega)} (\text{diam} \, \sigma)^{-1} \left\|\pi_{\sigma} \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} \right\|_{L_2(\sigma)}^{2} + \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}}(\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2}.
    \end{align}
    %
    Lastly, on combining Eqs.~\eqref{theorem_a}, \eqref{theorem_b}, and \eqref{theorem_d} we obtain
    %
    \begin{align*}
        \left| \bm{v} \right|_{H^{1}(\Omega, \mathcal{T})}^{2} &\lesssim \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}}(\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2} + \left( \bm{\Phi} (\bm{v}) \right)^{2} \\[1.5ex]
        &+ \sum_{\sigma \in S(\mathcal{T},\Omega)} (\text{diam} \, \sigma)^{-1} \left\|\pi_{\sigma} \left[\!\left[ \bm{v} \right] \! \right]_{\sigma} \right\|_{L_2(\sigma)}^{2}.
    \end{align*}
    %
    This equation is identical to the desired result (Eq.~\eqref{final_result}).
\end{proof}

In the following examples, we will demonstrate that the inequality in Eq.~\eqref{seminorm_proj_bound} holds for the seminorms in Eqs.~\eqref{seminorm_def_one} and \eqref{seminorm_def_two}.

\begin{example}
    For the seminorm in Eq.~\eqref{seminorm_def_one}, we have that
    %
    \begin{align*}
    \left(\bm{\Phi}_{1}(\bm{v} - \Pi \bm{v} )\right)^{2} & = \sum_{T \in \mathcal{T}} \left\|G(\bm{v}_{T} - \Pi_{T} \bm{v}_{T}) \right\|_{L_{2}(T)}^{2} \\[1.5ex]
    &\leq C \sum_{T \in \mathcal{T}} \left\|\bm{v}_{T} - \Pi_{T} \bm{v}_{T} \right\|_{L_{2}(T)}^{2} \\[1.5ex]
    &\lesssim \sum_{T \in \mathcal{T}} \left(\text{diam} \, T\right)^{2} \left\| \bm{\epsilon} (\bm{v}_{T}) - \frac{1}{3} \text{div}(\bm{v}_{T}) \mathbb{I} \right\|_{L_2(T) \times L_2(T)}^{2} \\[1.5ex]
    &\lesssim \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2},
    \end{align*}
    %
    where we have used Eqs.~\eqref{ortho_bound}, \eqref{project_error_one}, and \eqref{project_error_two}.
\end{example}

\begin{example}
    For the seminorm in Eq.~\eqref{seminorm_def_two}, we have that
    %
    \begin{align*}
    \left(\bm{\Phi}_{2}(\bm{v} - \Pi \bm{v} )\right)^{2} &= \sup_{ \substack{\bm{m} \in \mathbf{CK}(\Omega) \\
        \left\| \bm{m} \right\|_{L_2(\partial \Omega)} = 1, \int_{\partial \Omega} \bm{m} \, ds = \bm{0}}} \left( \int_{\partial \Omega} \left( \bm{v} - \Pi \bm{v} \right) \cdot \bm{m} \, ds \right)^{2} \\[1.5ex]
        &\leq C \left\| \bm{v} - \Pi \bm{v} \right\|_{L_2(\partial\Omega)}^{2} \\[1.5ex]
        &\lesssim \sum_{\substack{T \in \mathcal{T} \\ \partial T \cap \partial \Omega \neq \emptyset}} \left(\text{diam} \, T\right) \left\| \bm{\epsilon} (\bm{v}_{T}) - \frac{1}{3} \text{div}(\bm{v}_{T}) \mathbb{I} \right\|_{L_2(T) \times L_2(T)}^{2} \\[1.5ex]
        &\lesssim \left\| \bm{\epsilon}_{\mathcal{T}} (\bm{v}) - \frac{1}{3} \text{div}_{\mathcal{T}} (\bm{v}) \mathbb{I} \right\|_{L_2(\Omega) \times L_2(\Omega)}^{2},
    \end{align*}
    %
    where we have used a discrete trace inequality, and Eqs.~\eqref{project_error_one} and \eqref{project_error_two}.
\end{example}

\section{Conclusion}

In this work, we introduced a class of discrete generalized Korn's inequalities. These inequalities are similar to standard Korn's inequalities, with the key caveat that they involve the trace-free symmetric gradient operator instead of the symmetric gradient operator. We have shown that the trace-free symmetric gradient operator requires a careful analysis, as it possesses a more complicated kernel space than the standard symmetric gradient operator. As a result, it is necessary for us to introduce a new quadratic mapping procedure, and a new projection operator in order to construct the corresponding discrete Korn's inequalities. To the best of our knowledge, this endeavor has been successful in generating a new class of discrete inequalities for piecewise $H^2$ vector fields in 3D.

There are several interesting possibilities for future research. In particular, the extension of this work to 2D remains an open research question. In particular, the kernel space becomes infinite dimensional in this case, unless the vector field vanishes on the boundary~\cite{dain2006generalized}. In principle, such an extension is possible, although it is beyond the scope of the current work. In addition, we note that there are still opportunities to extend our analysis to higher dimensions, where more complicated derivative operators are present. This extension may be of particular interest for space-time finite element methods in 4D.

redpajama_set_name: RedPajamaArXiv


text:
Before long, Shovels & Rope will be on everyone's radar. An interview with the roots-rock duo, who performs at Infinity Hall in Hartford on Oct. 14.
'Super Talented' Foresters Debut Full Album 'House Stories' On CTNOW
Evan Nork, a teenager from Bethany, started writing songs when he was four. They stayed in his head until he could figure out what to do with them.
Indie Singer-Songwriter Conor Oberst To Play College Street
With Selena Gomez, Justin Timberlake and the rest of the pop stars who started out their singing careers as teenagers, it's easy to forget that there are
A TRAGEDY'S LONG SHADOW
By LYNNE TUOHY; Courant Staff Writer

redpajama_set_name: RedPajamaCommonCrawl


text:
SHERIFF'S OFFICE STRESSES THE IMPORTANCE OF EMERGENCY PREPAREDNESS DURING NATIONAL PREPAREDNESS MONTH
August 28, 2014 /in Press Releases /by Monmouth County Sheriffs Office
Monmouth County: Sheriff Shaun Golden is reinforcing the message to prepare, plan, stay informed and get involved as National Preparedness Month gets underway Sept. 1. National Preparedness Month, sponsored by the Federal Emergency Management Agency and the Department of Homeland Security is an effort to educate the public on how to prepare for emergencies.
"The Monmouth County Sheriff's Office, in partnership with numerous agencies, prepares for emergencies such as natural disasters, mass casualties and chemical threats throughout the year, by conducting drills, training, tabletops exercises and educational programs," said Sheriff Shaun Golden whose agency runs the Office of Emergency Management. "These efforts are critical in ensuring that we can effectively and efficiently respond to crises situations, but, it's vital the pubic plays a role as well, by preparing, planning and staying informed."
One of the many ways the sheriff's office assists the public in disaster preparedness is through its STORM Program. Seniors Taking on Readiness Measures was created as a result of Hurricane Irene and Superstorm Sandy. It equips those who are among the most vulnerable populations with emergency preparedness information, assists them in providing a family disaster plan and provides them with an emergency kit.
The program will be presented in Shrewsbury, Keansburg, Manasquan and Howell during National Preparedeness Month.
"Many residents are well informed and involved, but they fall short in creating a plan and building the kit," said Mike Oppegaard, OEM Coordinator. "STORM gives them that plan and equips them with the basic necessities in the event they need to go to a shelter."
The sheriff's office is also reinforcing to residents to have emergency notifications and public safety information at their fingertips, through the Monmouth County Sheriff's Office Application (app) for smartphones. The application is designed for Android and iOS platforms and can be downloaded for free.
The app provides instant access to sheriff's office related information, which includes real-time emergency notifications through its integrated push notification system that automatically displays the message on the phone. Users are able to be instantly alerted to weather, or other emergency related situations, from the Monmouth County Sheriff's Office and the agency's Office of Emergency Management.
"It can be challenging during emergency situations to provide consistent, official messages to a large group of residents. In addition to social media and our website, this app is another tool that we have to keep the public informed and encourage all residents to utilize," said Sheriff Golden.
In addition to the ongoing efforts of emergency preparedness, the sheriff's office, in partnership with Neptune Township OEM, will be participating in the September 30th America's PrepareAthon which provides individuals, organizations, and communities with an opportunity to prepare for specific hazards through drills, discussions and exercises. The event will take place in Neptune Township from 5 p.m. to 8 p.m. in the parking lot of the Headliner, 1401 Route 35.
News Release-National Preparedness Month
/wp-content/uploads/2015/12/Monmouth-County-Sheriffs-Office.png 0 0 Monmouth County Sheriffs Office /wp-content/uploads/2015/12/Monmouth-County-Sheriffs-Office.png Monmouth County Sheriffs Office2014-08-28 12:31:192014-10-24 10:56:10SHERIFF'S OFFICE STRESSES THE IMPORTANCE OF EMERGENCY PREPAREDNESS DURING NATIONAL PREPAREDNESS MONTH
SHERIFF'S OFFICE IS ASKING DRIVERS TO STAY SAFE AND SOBER THIS HOLIDAY... SHERIFF SHAUN GOLDEN HELPS SENIORS IN DISASTER PREPAREDNESS


The activites of the department are variable including teaching, research, collaboration researches with related department in the Medical Research Institute, The registration of postgraduate students with different backgrounds either in biological science, Medicine, Veterinary, pharmacy and faculty of education (biology department) for M.Sc and Ph.d degrees. Curricula are designed for obligatory and elective courses to fulfill the requirements needed.
The reaearch activities : include the participation of staff in members. Scientific seminars either in the department or sharing others in the conferences and congresses either local or international with scientific papers.
The future planes will include the proposal of projects either by stuff members of the department or by sharing with other departments in the Medical Research Institute either local or international.
The services include preparation of samples either paraffin or frozen , Sectioning , Aplication of different stains, Photography and image analysis. This is offered for researchers in faculties of Science, Medicine, Veterinary, Agriculture, and related department in the Medical Research Institute. This activists are paid and constitute an income to the department to be used for other research activities.


text: Q: Textview cursor not visible on becomeFirstResponder : Swift So I have a viewController on top of which I add another viewController as a popUp. This popUp has a textview in it, wherein the cursor won't be visible even though I can type and it shows it in the textview. I have already tried changing the tintColor as well as checked all the constraints but it just won't show up. A strange thing does happen when I select all the text the cursor randomly appears. I have tried changing the textview but with no luck, any help is appreciated!

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Jacqui Verellen is a singer-songwriter residing in Toronto, Canada. Known for her gentle vocals and her soul-stirring lyrics, Jacqui's music is all about story-telling and connectedness. Influenced by storytellers such as Joni Mitchell, Ingrid Michaelson, Lauren Daigle, Sleeping at Last and Taylor Swift - it's no wonder she has had success as a songwriter, as well; placing in many esteemed songwriting competitions, including top 15 in the International Songwriting competition. Jacqui's sound is a blend of warm instrumentals, honest lyricism and cool, electronic vibes. Her goal is to help her listener's through tough times, and to inspire them to feel connected to the source. Jacqui's upcoming EP, "Honest," was created from a place of suffering and then healing. Jacqui is more than excited to share what is a spiritual journey with her listeners, dropping her debut EP "Honest," March 2019, single by single. Jacqui plays four instruments, and has an interesting background, having completed an Honours Opera degree, psychology studies and a Music Business Diploma. Jacqui began as a teacher and now works within the Canadian music industry as well.


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text:
Kentucky Town Mourns Its First Female Mayor, Lucy Lou, A Dog
RABBIT HASH, Ky. (AP) — The town of Rabbit Hash, Kentucky, is known for a string of highly esteemed mayors, one of whom died this week. She was 12, and a dog. The Enquirer reports Rabbit Hash's first female mayor, Lucy Lou, died Sept. 10. The colorful political figure was a fixture in the media,...
Scientist Ester Gaya examines the fungus Isaria sinclairii on an insect also known as a zombie fungus at Kew Gardens' fungarium in London, Tuesday, Sept. 11, 2018. The release Wednesday of the scientists at the renowned Royal Botanic Gardens at Kew "State of the World's Fungi" report, is touted as the first ever global look at the way fungi help provide food, medicine, plant nutrition, lifesaving drugs _ and can also spread death and destruction at an alarming pace. (AP Photo/Frank Augstein)
September 12, 2018 - 12:38 am
Royal Botanic Garden Seeks Respect For World's Fungus
LONDON (AP) — The scientists at the renowned Royal Botanic Gardens at Kew are trying to correct an injustice: They don't believe fungus gets the respect it deserves. That's one reason behind the release Wednesday of their "State of the World's Fungi" report, touted as the first ever global look at...
FILE - In this July 25, 2011, file photo, a group of wild horses cools off in the ocean breeze on the beach in Corolla, N.C. As North Carolina braces for Hurricane Florence, some tourists and residents are worried about the famous wild horses that roam the Outer Banks. But Sue Stuska, a wildlife biologist based at Cape Lookout National Seashore, said the horses instinctively know what to do in a storm. (AP Photo/Virginian-Pilot, Steve Earley, File)
North Carolina's Wild Horses Expected To Survive Hurricane
For many vacationers on North Carolina's Outer Banks, a trip there is not complete without at least catching a glimpse of the majestic wild horses that roam the islands. As Hurricane Florence approaches, many are expressing concern about how the horses will fare during the powerful Category 4 storm...
This photo provided by the Oregon Justice Resource Center shows Josh Horner with his wife Kelli Horner after a hearing in Bend, Ore., Monday, Sept. 10, 2018. A 50-year sentence in a sex abuse case against Horner was dismissed Monday by Deschutes County District Attorney John Hummel after the Oregon Innocence Project found holes in the 2017 conviction that undermined the credibility of the complainant, including that Horner had shot the dog in front of her. (Jenny Coleman/Oregon Justice Resource Center via AP)
Discovery Of Dog Saves Oregon Man From Sex-Crime Conviction
SALEM, Ore. (AP) — The discovery of a black Lab named Lucy led to the unravelling of a criminal case Monday against an Oregon man who had begun serving a 50-year prison sentence. Joshua Horner, a plumber from the central Oregon town of Redmond, was convicted on April 12, 2017, of sexual abuse of a...
Official Defends Trump Plan To Revamp Endangered Species Act
WASHINGTON (AP) — A top Trump administration official on Monday defended a plan to revamp the Endangered Species Act, saying the proposed changes would result in more effective, quicker decisions on species protection. Deputy Interior Secretary David Bernhardt dismissed criticism by environmental...
Side Of Seagrass Please: Scientists Find Omnivorous Shark
IRVINE, Calif. (AP) — Ruining the reputation of sharks as bloodthirsty predators, California researchers said they have found a shark that enjoys a side of seagrass with its prey. Bonnethead sharks not only eat grass while chomping fish and squid — they also digest the plant and gain nutrition from...
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Man Kills Dog Washington Family Got After Vegas Shooting
BELLINGHAM, Wash. (AP) — A Washington state family is mourning after a neighbor shot and killed their dog — a Labradoodle they got to help deal with stress and anxiety following last year's mass shooting in Las Vegas. The Bellingham Herald reports Lona and Joseph Johnson were at the Route 91...
Man Fined $124 For Kicking Seagull That Ate Cheeseburger
HAMPTON, N.H. (AP) — A man accused of kicking a seagull that tried to eat his cheeseburger at a New Hampshire beach has been fined $124. Police investigated the report from a bystander at Hampton Beach earlier this summer. NH1 reports the man Nate Rancloes, said he had just returned from getting a...
This photo of three images taken from October 2017 video produced by Sierra Research Laboratories shows mice congregated around and climbing on a device that was supposed to scare them away. The images were included in a civil complaint filed with the U.S. Attorney's Office for the Southern District of New York against BHH, the New York based company that produces the device. On Wednesday, Sept. 5, 2018, a New York judge said pictures of mice lounging around the anti-rodent device are reason enough to let a lawsuit proceed against the company that sells them. (Sierra Research Laboratories via AP)
Judge Cites Pictures Of Mice, Lets Lawsuit Proceed
NEW YORK (AP) — Pictures of mice lounging around an anti-rodent device designed to make them flee were cited by a judge who let a class-action lawsuit proceed Wednesday against a company that sells and markets them. "It is often said that a picture is worth a thousand words," U.S. District Judge...
FILE - In this March 3, 2013 file photo elephants drink water in the Chobe National Park in Botswana. A conservation group says elephant poaching has increased in Botswana, which has long been viewed as a rare refuge for elephants in Africa. (AP Photo/Charmaine Noronha, File)
Botswana, Long An Elephant Refuge, Faces Poaching Threat
JOHANNESBURG (AP) — Botswana, long viewed as a rare refuge for African elephants, is coming under increasing threat from poachers. Poachers are killing elephants in the southern African country in growing numbers after wiping out large numbers of elephants in nearby Zambia and Angola, a...


text:
The Time Machine (1960) | Sci-Fi Saturdays
by Jovial Jay June 15, 2019
by Jovial Jay
George Pal returns to Sci-Fi Saturdays with his classic adaptation of HG Wells' The Time Machine.
While The Time Machine may not technically be the first Hollywood film to deal with time travel, it is certainly the most notable and the one that kick-started the sub-genre. Without this film, time travel films may have never achieved the notoriety that they have today.
The trailer makes sure to mention some very key elements that audiences may be interested in. It name checks HG Wells, Jules Verne and George Pal, to ensue audiences know that they are getting quality entertainment. Surprisingly the trailer describes the people of the future; the Eloi being the slaves (and food) of the Morlocks. Granted, in 1960 audiences were likely to see a trailer once or twice prior to seeing the film, so it's hard to say how this reveal would affect audiences of the time. It's definitely a "spoiler" by today's usage! The trailer is narrated by Paul Frees (who has a small role in the film as the voice of the "rings"), who poses a lot of questions to the audience, such as "what happens when boy meets girl thousands of years hence." Upselling the love story, the action, and the Morlocks seemed to be the biggest points that the studio wanted to hit with this advertisement. All in all, a pretty good representation of the finished film.
Presented below is the trailer for the film.
The Fiction of The Film
The Time Machine presents the bulk of the film in an extended flashback, with the beginning and end of the films taking place in early 1900. The film opens on a Friday evening, the 5th of January, 1900, when four men arrive at the home of their friend George. David Filby (Alan Young), Dr. Philip Hillyer (Sebastian Cabot), Anthony Bridewell (Tom Helmore), and Walter Kemp (Whit Bissell) are escorted into the dining room by George's housekeeper Mrs. Watchett (Doris Lloyd). As the quartet sits down to dine, George (Rod Taylor) bursts in, disheveled and bedraggled, asking what day it is. He proceeds to explain to the others just what he has encountered.
The Time Machine title card.
Five days previous, on December 31, 1899, George shows his four guests a miniature model of his time machine. Explaining the nature of the four dimensions (left/right, forward/back, up/down, and time), George flicks a switch and sends the model forward in time. Several of his guests are unimpressed believing it a parlor trick. Only Filby stays behind and begs George to destroy the real time machine, as it will only cause George's own destruction. George promises not to walk out of the house and bids Filby a good evening.
George retires to his workshop where he has the full-scale model of the machine. He sits down in the device and tests the controls ever so gently. He realizes the candle and his clock have changed their states, so he decides to press on further. George's journey is portrayed by the flickering of the sun, the opening and closing of his plants, and the changing fashions on the mannequin in Filby's Department Store across the street (all portrayed through the use of stop-motion animation). His first stop is on September 13, 1917 during World War I, where he exits the machine to meet Filby's son James (also played by Alan Young) who explains that his father died during the war.
Distressed by the thought of a continued conflict, George presses further on into the future. This time landing on June 19, 1940 during the blitz of London in World War II. As he continues, his house disappears, having been struck by a bomb and he lands a third time in August 18, 1966 where the air raid sirens beckon civilians underground for safety. An aged James Filby again meets George, who hasn't changed a day since their previous meeting, just before an Atomic Satellite attacks overhead. George is able to once again flee into the future.
George burst into the parlor during dinner with an unbelievable story.
His final destination is Oct 12 in the year 802,701. There he finds a peaceful Eden with many young men and women swimming, lazing around and eating fruit. He believes he has truly found the perfect future. But he soon realizes that these Eloi, as they call themselves, have no knowledge of technology. They are ignorant of the histories of the planet and their people, save the stories told to them by the "talking rings," memory tapes left by a race long past. George discovers Weena (Yvette Mimieux), an attractive Eloi girl, who tells him of the Morlocks, who in the meantime have dragged his time machine into their temple.
The Morlocks enslave the Eloi, using them for slave labor as well as food. George is disgusted by the state mankind has left the world in. When Weena is kidnapped by the Morlocks, George (and no one else) goes to rescue her. He manages to fight off a number of Morlocks, freeing many other Eloi, and awakening the spark of violence (or perhaps self-preservation) in the Eloi. However he becomes trapped in the Morlock temple, with Weena on the outside. After defeating the Morlocks around him, George activates the machine and returns to 1900, here he drags the machine back into his workshop. He enters the parlor, as seen at the beginning of film. He thanks his guests for coming and thanks Filby for being a friend. However, Filby is suspicious and follows George to his workshop, but he has already returned to the future, having taken three unknown books with him. The film ends as Filby ponders what they could possibly be.
"You've all the time in the world." – David Filby
George shows Walter kemp, Dr. Hillyer, Anthony Bridewell, and David Filby his time machine model.
By the end of the 1950s, sci-fi films had become more geared towards horror and less towards space, science, rockets and futurism. Therefore, it's heartening when the 1960s kicked off with one of the most science-fictiony films of the decade! A true harbinger that some of the best science-fiction films ever were coming!
Last week's film, Journey to the Center of the Earth was a Victorian-style adventure film from 19th Century writer Jules Verne that ended the decade of the 50s. This week, the new decade begins with a Victorian-style adventure film from 19th Century writer H.G. Wells. This was not the first work by Wells that had been adapted to film. The Island of Dr Moreau (as both a French silent film (1913), and a German silent film (1921), but more popularly as Island of Lost Souls; 1932) and The Invisible Man (1933) had been made previously. These two movies are considered more in the horror genre, even though they have sci-fi roots. Although these films are outside the purview of this series, having been created prior to 1950, Sci-Fi Saturdays did cover the 1953 adaptation of Wells' The War of the Worlds, his most famous adaptation to date.
The full size machine, which birthed a thousand cinematic journeys in time.
While Verne and Wells were both 19th century writers, their writing occurred in two different parts of the century, and stemmed from different philosophies. Verne's work is set in his present using fantastic technology to achieve his stories' ends, while Wells appears to be more of a futurist, delving into the future of society, sometimes with horrific outcomes. The Time Machine was the third film to become a heavy influence into the steampunk movement. Verne's 20,000 Leagues and Journey to the Center of the Earth, were the first two influences, whose elements were covered in the Sci-Fi Saturdays articles on those films. Suffice to say that prop designer William Ferrari influenced many other people with his distinctive design.
Enter again George Pal. Pal produced both War of the Worlds and The Time Machine, but interestingly enough, while he decide to update Worlds to a modern setting (from the Victorian London to 1950s California), he kept The Time Machine in its original setting. The Time Machine indicated the pinnacle of Pal's science-fiction work. He had always wanted to adapt this story, and by 1960 the time was right. His future films would focus more on Fantasy, rather than Sci-Fi. As with War of the Worlds, this adaptation earned Pal and his team an Academy Award for Best Special Effects, which seems odd by today's standards. The prop work is extraordinary, but the matte paintings and optical effects leave much to be desired.
Even though the character is called George, this Easter Egg reveals he is really HG Wells (also the date is exactly 1,019 years prior to this article!)
Genre-fication
The Time Machine may not have been the first piece of fiction on time travel or the first time travel film, but it is certainly the most popular at this time. As an iconic film it defined the subgenre for at least 25 years until Back to the Future came out. Other authors tackled the idea of traveling to another time, but did it in a variety of ways. Six years before Wells released his publication of The Time Machine, Mark Twain broached the subject of time travel in A Connecticut Yankee in King Arthur's Court. While time travel purists may reject stories like Connecticut Yankee as fantastical rather than scientific, it qualifies, at least for the period it was released. At least three films adaptations were released prior to 1960, the most popular being the 1949 version with Bing Crosby, but after that Pal's Time Machine became the cornerstone for future time travel ideas. Time travel would continue to gain popularity throughout the genre until the 1980s when films like The Terminator, Bill & Ted's Excellent Adventure and Back to the Future would catapult the genre into new heights.
This particular adaptation of the story addressed traveling through time in a linear fashion as one would a river; hopping off the boat onto the shore in the various times. Issues such as causality, predestination paradoxes and other troubling problems with time travel are not addressed. In fact, the narrator, George (who is only ever referred to by this name, and could be Wells, or Pal), moves extremely linearly in his travels. At one point Dr. Hillyer blurts out that the future is irrevocable, so what would the point of time travel be. George counters that his inventions are for scientific curiosity, additionally he adds to escape the horrors of war and chaos that his must live through in the present time.
Man down! The time traveller is thrown from his vehicle when he stops too quickly!
The film also moves linearly in its character arc. George returns to the past in a panic that he has lost Weena forever. His desperation to get back to her is misplaced, as he has a machine that could literally travel to any moment. The film chooses to use this momentum and potential plot hole to build suspense. But also, and most likely, to keep the complexity of the film to a minimum, Pal's hero must return to the moment he left. As Filby states, "he has all the time in the world."
The Time Machine also returns the narrative of the genre to a more introspective mode. Time travel in essence is a thematic device used to allow the author to discuss regret or longing for the past, and hope or fear of the future; things that mankind has troubled with throughout its existence. Pal's version poses some very serious philosophical questions that allow the audience to take away more from the film than it posits.
George in the garden of Eden circa 802,701 AD.
Societal Commentary
As mentioned above, the film version of The Time Machine asks many questions about mankind's existence, which is something that not many sci-fi films from the 1950s had attempted to date. George is decidedly against war, specifically the Boer War in South Africa. He hates it so much so that he is affronted by his friends when they suggest his brain power would be better off served creating weapons to help England win the war. The fact that he ends up stopping during multiple battles only serves to propel him further into the future, until he finds a world free from the horrors of killing.
Unfortunately the future that George seeks comes with its own problems, in this case, the collapse of society. The portrayal of the future society is somewhat simplistic, yet allegorical. Separating the races into two distinct castes, the worker (Morlock) and the management (Eloi). But also the cattle (Eloi) and the farmer (Morlock). Wells had written the two races from the standpoint of a member of Victorian society. By the time the film was made, it became allegorical for different things, such as the Cold War – or the eventual fate for mankind after a nuclear tragedy. The "talking rings" present a little information that war raged on between "east and west" for 326 years, presumably leading to the irradiation of the surface. A ring describes the choices made by the survivors to burrow beneath the ground, or "to take our chances in the sunlight. Small as those chances might be." It is strange that 800 Centuries hence, the same triggers exist for the populace. Morlocks play the "sirens" and the Eloi stream below ground like sheep. Something that audiences of the 1960s would understand all too well, with their experience with air raid drills.
Perhaps unintended, but the future looks like an Aryan's dream, complete with little blonde boys and girls living without a care. The film gives very little information about the future, mostly due to the ignorance of the Eloi. In a holistic view, it's hard to see this small representation of society as equivalent to every place in the world. After all this is only intended to be England in the future, which could very well become so homogeneously white that there are no racial boundaries, save the division between Eloi and Morlock. More likely, it's the narrow view of Hollywood during the 1960s. The story they set to tell was about one man's travels through time, and not sociological statement of the future.
The ruins of a once great society, now a home for the Eloi to live in apathy.
The film also has much to say about history, or at least the study of the past. George is aghast that the Eloi have no government or work, and seem to have forgotten any lessons learned by their ancestors. While their culture initially appears to be ideal, it quickly becomes apparent that trade-offs were made in their ascent to this place. The Eloi are mostly content, having their needs met by the Morlocks. It's unclear how the harvests work, except maybe at night, for the Morlocks appear extremely sensitive to light. But the Eloi are kept fed and happy without lifting much of a finger for themselves. George also sees the lack of apathy the Eloi have for fellow kind, when Weena is drowning in a river and none of the Eloi pay her any mind. The film seeks to remind the audience that a purposeful life, one with introspection, work, and effort is better than living a sheltered and pampered one.
A final interesting question that the film poses, that would be answered differently for each viewer, is what knowledge makes up society. Filby realizes that in George's final trip, he has taken three volumes from his library, but neither he nor Mrs. Watchett know which ones. What titles from 1900 would be useful in rebuilding the minds of the Eloi? Perhaps some science text, but also some philosophy text might be useful. This is a great hook to keep audiences discussing the film long after the movie ends.
George confronts the Eloi on their idyllic lifestyle and lack of knowledge.
The Science in The Fiction
One thing that good science fiction films do is to create a plausible reality in which the technology that is central to the film, works. It's a fine line between delving into the science too much and not addressing it at all. The Time Machine does a good job of explaining a number of basic tenets about time travel. When Doctor Hillyer becomes dismissive of George's "introduction" to the 4th dimension, George puts him on the spot to explain the first three. Explaining to the audience that traveling through time is akin to moving left/right, or up/down provides a clear vision in ones head. It's a very basic setup, but enough to give the audience of basic understanding, and a visual cue, for the types of travel being discussed.
Another important scientific element that is demonstrated, quite clearly, is the theory of relativity. George lights a candle next to his clock prior to stepping into his time machine. After moving forward a couple of hours, as demonstrated by the advancement of the clock and the shortening of the candle, his watch reads that only a few moments have passed. From his perception, relative to the room, he has moved forward hours in a matter of seconds. No need to get into a debate about Einstein and mathematics. The film use "show don't tell" brilliantly in this case.
Finally, the film attempts to address the fact that space is the same throughout time. As George tries to explain to Dr. Hillyer the fact that his scale model is still in the same space (ie location) but just at a different time from them. Again, another clear setup for the end of the film where George needed to drag his time machine back into his workshop so that he would be outside the Morlock temple when he returned to the future. Unfortunately, what most science-fiction time travel stories ignore is that the Earth moves in space, so that one point "now" is not the same point in "an hour." Technically one could find themselves floating in space if they actually were to only travel in time.
The menacing Morlocks! Bloodthirsty underground carnivores stoking the fires of progress.
As more and more science-fiction films were created, a larger catalog of props were available to be used. Nowadays companies mass produce props for use in all types of films, but in 1960 a film was limited by its budget and what it could rent (or use from a previous film). The Time Machine utilizes several things from previous MGM films. The first, and maybe most distinctive, are the steps up to the Eloi dining hall. These were built for Kismet (1944), and reused in lots of MGM productions including several episodes of The Twilight Zone. Additional two props from Forbidden Planet can be seen. The first is a costume from the crew, used here as a "futuristic" uniform for the Air Raid Marshall in 1966. The second is the astrolabe used on board the ship for navigation, found as a discarded piece of tech in the talking rings chamber. Not to be outdone, the time machine prop would also appear in future films and television shows such as Gremlins and The Big Bang Theory.
Many sci-fi films reuse props from previous film. Here MGM makes use of a prop and a costume from Forbidden Planet (1956).
Some of the actors would go on to other sci-fi roles. Whit Bissell is the most prominent sci-fi actor, having been in films such as The Invasion of the Body Snatchers (1958) and Target Earth (1954), plus Soylent Green (1973). The female lead, Yvette Mimieux, would return to sci-fi in the 1980 Disney film The Black Hole. And Alan Young, who is more known for his television work as Wilbur on Mr. Ed, or his voice work as Scrooge McDuck, would appear in The Cat From Outer Space (1978) and the 2002 remake of The Time Machine. Additionally, that remake, which is a very similar version of the story, enhanced a bit with stops in other times, was directed by Simon Wells, HG Wells' grandson.
I last wrote about this film over 4 years ago days after the death of Rod Taylor, as part of a Time Travel blog, celebrating 2015 – "the future" as seen from the Back to the Future franchise. My prose may have improved, but this film still stands as a stunning piece of fiction. For its scientific flaws and simplicity, The Time Machine is still an amazing film that deals with the reality of living in the time one is born. George did not feel part of his time and yearned for a better life, but his travel into the unknown almost cost him his life, as Filby had predicted. The grass is always greener as they say, but in time travel films the protagonist always has a chance to explore "the other side" and decide if the journey is worthwhile.
Coming Next Week
Jovial Jay
Having grown up on comics, television and film, "Jovial" Jay feels destined to host podcasts and write blogs related to the union of these nerdy pursuits. Among his other pursuits he administrates and edits stories at the two largest Star Wars fan sites on the 'net (Rebelscum.com, TheForce.net), and co-hosts the Jedi Journals podcast over at the ForceCast network.
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AbstractThe NASA P8 hypersonic inlet has been redesigned using an automated methodology incorporating a 3-D Reynolds-averaged Navier-Stokes code, a gradient-based optimizer and artificial intelligence methods. The objective of the redesign is to cancel the cowl shock and the additional cowl-generated compression at the centerbody by appropriate contouring of the centerbody boundary. The original P8 inlet, designed in the early 1970s, was intended to achieve this same objective, but detailed experimental measurements indicated that a substantial reflected shock system was present. We have obtained several different redesigns which achieve approximate cancelation of the cowl shock and additional cowl-generated compression. The choice of the objective function, which is used to drive the optimization, has a significant impact on the final design. Several different formulations for the objective function have been employed, and improvements of 60% to 80% have been achieved.


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Sunrise and sunset times in Bakers Creek, Queensland
Check out today's and tomorrow's sunrise and sunset times in Bakers Creek, Queensland, Australia, as well as the whole calendar for January 2023.
January 31, 2023. Current time: 3:20 am (Australia/Brisbane timezone)
Day length: 13 hours, 5 minutes
15 hours, 29 minutes left for today's sunset in Bakers Creek
Tomorrow will be 1 minute shorter than today in Bakers Creek
Bakers Creek,
Latitude: -21.216669
Longitude: 149.199997
Time zone: Australia/Brisbane
February 2023 - Bakers Creek, Queensland - Sunrise and sunset calendar
Sunrise and sunset times, civil twilight start and end times as well as solar noon, and day length for every day of February in Bakers Creek.
In Bakers Creek, Queensland, the first day of February is 13 hours, 5 minutes long. The last day of the month is 12 hours, 34 minutes, so the length of the days gets 30 minutes shorter in February 2023.
Year distribution of sunrise and sunset times in Bakers Creek, Queensland - 2023
The following graph shows sunrise and sunset times in Bakers Creek for every day of the year. There are two jumps in the graph that represent the hour change for Daylight Saving Time (DST) in Bakers Creek, Queensland.
Nabowla
Nangwarry
Pegarah
Longford, New South Wales
Kogan
Koorboora


Before taking them to home, you have to make certain you quantified your chair's measurement. Thus, it is going to match your seat completely. Assessing any chairs before shifting the pads will likely probably be really important. Thus, you're going to be able to avoid useless pads to get plumbing chairs. You can see right now that the pads that you have obtained may not fit the chair on your plumbing.
It can function as the optimal/optimally selection for comfortable seats that are easy to clean. The very best thing concerning micro fiber is that there are several sort of style and style and layout available on the market.
The design has developed in every portion of dwelling includes how plumb. People like to have their home to be comfortable, but this day comfy isn't enough because the look of the look also very significant. In addition, the professional also take significant of its own specialty. Even tub area is not area where it's seen the most unlike living area, still it's essential to get great design for this.
how plumb will soon be very beneficial to individuals whose smaller plumbing appears stuffy. A plumbing is supposed to give us a excellent feeling in it, since this is exactly the area where we get ready foods for our loved ones. If our plumbing appears stuffy, we will acquire anxious readily. Still we have to face the undeniable fact which our plumbing is not small. What if we all do about that? We've to redesign or update it. It is not that simple for sure, but nevertheless, there are ideas to create it look more spacious.


In today's NHL rumors, there are updates on the status of Jeff Skinner's contract in Buffalo and questions about what to do if he doesn't sign before the trade deadline. Meanwhile, the New York Islanders are looking to upgrade their scoring and one insider thinks they might be looking to an interesting and surprising name. What's happening with the potential trade pieces coming out of New York and is there a team interested in Andre Burakovksy?
Elliotte Friedman was on Calgary's Sportsnet 960 and spoke about Jeff Skinner's contract negotiations with the Buffalo Sabres. The two sides still haven't reached a deal on an extension and in a lot of cases, that would be cause to trade the scorer knowing you could lose him for nothing in free agency as a pending UFA. The Sabres might feel differently.
Islanders Interested In Ilya Kovalchuk?
Darren Dreger had a Tuesday morning radio hit on Montreal's TSN 690 and wondered what the plan would be for the New York Islanders heading into the trade deadline. "I do feel like Lamoriello is definitely interested in doing something, and maybe adding a little bit more scoring punch. He doesn't have a whole lot of areas of concern," Dreger said.
That something could be Ilya Kovalchuk out of Los Angeles.
Dreger went on to say, "I had somebody just kind of whisper in my ear yesterday – and let's not call it a source or anything firm, just wild speculation is what it is – wondering whether Lamoriello might have interest in Ilya Kovalchuk with the Los Angeles Kings." He mentioned stranger things have happened and that if placed in the right role with the club Kovalchuk could flourish.
The question becomes the long-term nature of Kovalchuk's contract. After signing a three-year $18.75 million contract with the Kings, it might be a reach to think the Islanders would want to take on all that term.
Darren Dreger had a Tuesday morning radio hit on Toronto's TSN 1050 and discussed the options for the New York Rangers at the trade deadline. Dreger suggests there is a laundry list of players other teams might be interested in.
Dreger also mentioned Kevin Hayes late last week during another interview and wondered about the possibility of Hayes being moved to help another team's playoff run but then circling back to sign with the Rangers in the summer.
LeBrun described Burakovksy as a young player who was great during the Capitals' Stanley Cup run and could be acquired for not much while giving the Avs control over another contract. As an RFA, GM Joe Sakic can then decide what he wants to do during the summer.


Rift Valley Fever is an acute mosquito born viral disease caused by a virus belonging to the family Bunyaviridae and the genus Phlebovirus. It is a fever causing virus which most commonly affects domesticated animals, such as cattle, buffalo, sheep, goats and camel. The virus also has the ability to infect and cause illnesses in human beings. Rift valley fever was first reported in livestock by veterinary officers in Kenya's Rift Valley in the early 1910's. Rift valley fever exists in most of the sub-saharan Africa, Western Africa and Madagascar, but it is also found in the Eastern and Southern regions of Africa where sheep and cattle are reared. Rift valley fever outbreak was also reported in Saudi Arabia in September, 2000. Outbreaks of rift valley fever increase the likelihood of contact between humans and deceased animals, which can lead to an epidemic of rift valley fever in humans. In 1977, the virus was detected in Egypt and caused a large outbreak in both humans and animals, resulting in over 600 human deaths in the country.
Although rift valley fever is only confined to Middle-East and Africa, it can easily spread to other countries. The incidences of rift valley fever peak in the summer, which is essentially the season when the virus gets optimum conditions to grow. Increasing incidence rates and prevalence of rift valley fever are the major factors expected to drive the rift valley testing market during the forecast period. However, costs associated with the diagnostic testing in Middle-East and Africa is a leading restraining factor that might hinder the growth of the rift valley fever testing market.
The global market for rift valley fever testing is highly competitive with various players operating in the global space. Both the test types — reverse transcriptase PCR and ELISA — are some of the most common techniques used in research, clinical diagnosis and related markets. Some of the established brands of rift valley fever testing are Thermo Fisher Scientific, Inc. (U.S), Merck KGaA (Germany), New England Biolabs, Inc., (U.S), Promega (U.S), Illumina, Inc, Agilent Technologies, Inc. (U.S.) and F.Hoffmann-la Roche Ltd. (Switzerland).
Geographically, the global rift valley fever testing market is classified into 8 regions: North America, Latin America, Western Europe, Eastern Europe, Asia-Pacific, China, Japan, Middle East and Africa. North America is expected to be a leader in rift valley fever testing market owing to concentration of key market players in the region. The market in Middle East & Africa and Asia Pacific is expected to grow at significant CAGR due to expansion of rift valley fever product offerings by key players. Europe is expected to hold the second largest share in the global rift valley fever testing market throughout the forecast period.
Examples of some of the major players in the rift valley fever testing market are Thermo Fisher Scientific, Inc. (U.S), Merck KGaA (Germany), New England Biolabs, Inc., (U.S), Promega (U.S), Illumina, Inc, Agilent Technologies, Inc. (U.S.), F.Hoffmann-la Roche Ltd. (Switzerland), Takara Bio, Inc. (Japan), QIAGEN N.V. (The Netherlands), Becton, Dickson and Company (U.S), Biocartis NV (Belgium), Bioline Reagents Limited (United Kingdom), Dnavision SA (Belgium), Eppendorf AG (Germany) and Exiqon A/S (Denmark).


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The Océ PlotWave 750 printing system makes it easy for you to get great results. Everything from media roll placement and toner loading to print job management has been carefully designed to be ergonomic and intuitive. The Green Button approach to large format scanning offers great quality output at the touch of a button. Get results right the first time with no need for troubleshooting. Thanks to low noise and heat emission, the Océ PlotWave 750 wide format printing system can be comfortably placed in a departmental environment.