text:
Truckstop Review – Red Lion, Northampton
The Red Lion truckstop at Northampton was established over 30 years and is just 200 yards from junction 16 of the M1 making it very convenient place for truckers to park to get a meal.
It's a big establishment that offers plenty of facilities. The restaurant offers varied cuisines and there's a bar, convenience store, laundry facility, clean toilets and showers.
It certainly was popular – especially with foreign drivers – on the day TSN visited to check it out. The Red Lion is open Monday to Friday from 0500-2300, on Saturdays from 0630 to 2300 and on Sundays from 1000-2300.
The standard breakfast is a sausage, two rashers of bacon, hash brown, beans, egg and a fried slice. The all-day breakfast has two bacon, two sausages, two eggs, beans, tomatoes and chips for £5.95. The full breakfast is the same but has three hash browns and mushrooms, but no chips. The vegetarian breakfast has two fried eggs, two Quorn sausages, two hash browns, mushrooms, beans and tomatoes. You get a free tea or coffee with any breakfast
The mega belly buster at £6.95 has four bacon, three each of sausages, hash browns and eggs, plus mushrooms, black pudding, beans and tomatoes.
Also available are beans, cheese, eggs (fried, scrambled or poached) on toast. You can also build your own breakfasts with all the various 'components' to suit your exact requirements.
Baguettes and sandwiches
The Red Lion offers a house special of cheese, steak and onion baguette at £4.30. Then there is the range of other baguettes priced between £3.35 and £3.95 and includes ham salad, ham & eggs, bacon & cheese, tuna mayo, sausage & onions, bacon & sausage, corn beef, bacon, lettuce & tomato and cheese salad.
Also available are sandwiches priced between £2.20 through to £3.95 and include bacon, sausage, ham salad, tuna, corn beef, sausage & egg, steak & onion, bacon & mushrooms, sausage & bacon, bacon & egg, cheese & onion and cheese salad.
The panini range – from £3.50 to £4.20 – includes steak & onions, ham & cheese, tuna cheese melt, and cheese with either mushrooms, peppers or tomatoes.
Jacket potatoes come as plain with butter at £2.99 or with one topping at £4.25 or two toppings at £4.95 – with a choice of cheese, beans, coleslaw, bacon, ham, chilli con carne and tuna mayo.
Salads are chicken and bacon Caesar, green, sweet chilli chicken all at £6.75 while ham & pineapple or tuna and egg are both £5.95 a go.
The Red Lion has an extensive menu of main meals with dishes for everybody's palette.
Tortilla chips with cheese, jalapenos, salsa and chilli is £5 – a vegetarian option is £4.
Cheesy chips are £2.50 and cheesy garlic bread is £3.50. Pizzas are £5 for margherita or £6.50 for either pepperoni or chicken. A lamb kofta roll in naan bread is £5.95
From the grill, there is the sticky platter at £9.95, the Full Monty mixed grill at £11.95, minted lamb steaks at £9.45, 10oz Gammon steak at £7.95, rump steak at £9.95, ribeye steak at £10.45 and barbeque spare ribs at £9.95. You can upgrade any of these to 'surf and turf' for £3.
There is a varied selection of other main meals. They include Hunter's chicken at £6.45, Black pepper chicken nibblets at £6.25 and Jerk chicken at £7.45.
The popular dishes include cod and chips (£7.95), scampi (£6.95), spaghetti bolognaise (£6.95), lasagne (£7.25), penne arrabbiata (£6.35), shepherd's pie (£7.25) and chicken, ham & leek, steak & ale or steak & kidney pies (all £6.45). Sausage, mash & Yorkshire pudding is £6.75.
For those liking something a bit hotter, then there is chicken tikka masala and chicken jalfrezi (both £7.95), beef madras (£8.45), Tandoori chicken (£7.75) and lamb kofta (£6.25). Chilli con carne is £6.95.
There is a wide range of burger options to suit all tastes. A standard 6oz beef burger is £5.95 while a cheeseburger or chicken burger are both £6.25. A cheese and bacon burger is £6.40 while a sticky barbeque beef burger with cheese is just five pence more.
The 'Big daddy' or 'Express lift tower' burgers are both £7.95 while the 'Mamma Mia' burger is priced at £6.95. there are also veggie or spicy bean burgers both at £5.95.
All burgers come with chips and coleslaw and you can upgrade to sweet potato fries for an extra quid.
The traditional favourites, apple pie, spotted dick or syrup sponge are all £3.99 while the chocolate sponge and sticky toffee puddings are both priced at £4.50. They come with custard, or vanilla ice cream
The Red Lion has showers for both sexes. There is also an outdoor washroom and toilet for those leaving early, which are also open 24 hours.
Each shower is in its own cubical, with its own separate dry area and sink area to provide privacy. After each use, a manned attendant will clean it and bring it back to standard for the next use.
Its overnight security includes infrared CCTV cameras that monitor the site 24 hours a day. There is parking for over 150 HGVs with manned parking attendants.
High mounting banks stop intruders entering the site. The Red Lion is also a member of the Park Mark Scheme, which helps it to constantly improve the security of the site. A new security building just been built, offering more lights around the premises.
There is also a large, well-stocked, shop for drivers specialising in cooking equipment, CB radios and accessories, vehicle hardware and truck accessories, in-cab decoration and entertainment, cab care products and clothing. It also stocks every day essentials such as toiletries and medication.
The site also has a bar and shows all live football games on big screen TVs. The bar sells draught beers, bottles, cans, ciders, spirits, wines and soft drinks. There are 'beer nights' twice a week as well as extended Footy Hour on football nights, giving drivers discounted pints during the game. In addition there are special one-off promotion nights.
There is a free-to-use ATM and the Red Lion offers a laundry service, with coin-operated machines that have built-in detergent and softener.
It's clear this is a truckstop that is geared up to serve drivers. There is plenty of room for trucks, although several were parked in the approach road on the day of our visit. However, it was very busy, so you need to be on your best to park in some of the tight spots that were vacant!
The food was delivered promptly. It wasn't outstanding, but it certainly wasn't poor, and the choice was excellent. The staff were helpful and polite, and the place was clean and pleasant to eat at.
Red Lion Truckstop A45,Weedon Road, Upper Heyford, Northampton, NN7 4DE
E-mail: info@redliontruckstop.com
Website: www.redliontruckstop.com
Truckstop Review – Redbeck Motel, Wakefield
Whitwood Café, Castleford


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text:
\section{Introduction And Main Motivations}
We have proposed an application to the area of optimization on manifolds,
written as a digest of \cite{TO} enhanced with references to the most recent literature. We have the most interesting result on optimization on Riemannian manifold as follow.
\begin{theorem}\label{oth}
	\cite{TO} Let $(N, g)$ be a Riemannian submanifold of a Riemannian manifold $(\overline{M}, \overline{g})$ and $f : N \rightarrow \mathbb{R}$ be a differentiable function. If $x_{0} \in N$ is the solution of the optimum problem $\min\limits_{x_{0} \in N} f(x_{0})$, then
	\begin{enumerate}
		\item[(a)] $(grad \hspace{0.1 cm} f)(x) \in T_{x}^{\perp}N$,
		\item[(b)] the bilinear form $\Theta : T_{x}N \times T_{x}N \rightarrow \mathbb{R}$,
		$$\Theta(E, F) = Hess_{f}(E, F) + \overline{g}(h^{'}(E, F), (grad \hspace{0.1 cm} f)(x))$$
	\end{enumerate}
	is positive semi-definite, where $h^{'}$ is the second fundamental form of $N$ in $\overline{M}$ and $grad \hspace{0.1 cm} f$ denotes the gradient of $f$.
\end{theorem}
In summary, optimization on manifolds is about exploiting tools of differential geometry to build optimization schemes on abstract manifolds, then
turning these abstract geometric algorithms into practical numerical methods
for specific manifolds, with applications to problems that can be rephrased
as optimizing a differentiable function over a manifold. This research program has shed new light on existing algorithms and produced novel
methods backed by a strong convergence analysis.
We close by pointing out that optimization of real-valued functions on
manifolds, as formulated in Theorem 1.1, is not the only place where
optimization and differential geometry meet and also is the Riemannian
geometry of the central path in linear programming. Using the terminology optimization method, F. Casorati \cite{c} defined an extrinsic invariant, called Casorati curvature of a submanifold of a Riemannian manifold, as the normalized square length of the second fundamental form. This concept extended the study of the principal direction of a hypersurface of a Riemannian manifold. The geometrical aspects and the significance of the Casorati curvatures discussed by some eminent geometers \cite{1decu1,1kowalczyk,3verstraelen1}. It showed an increasing development in pure Riemannian geometry. Therefore, geometers were curious to obtain the optimal inequalities for the Casorati curvatures of a submanifold of different ambient spaces. Such inequalities with a pair of conjugate affine connections involving the normalized scalar curvature of statistical submanifolds in different ambient spaces were derived (for example \cite{vilcu,lee3}).

On the other hand, the concept of warped product manifolds has many applications in physics as well as in differential geometry. For instance, different models of space-time in general relativity are expressed in terms of warped geometry, and the Einstein field equations and modified field equations have many exact solutions as the warped products. In 1969, the idea of warped product manifolds has been initiated by R.L. Bishop and B. O'Neil \cite{article.8} with manifolds of negative curvature. These manifolds are the most fruitful and natural generalization of Riemannian product manifolds.

A statistical structure can be considered as a fruitful generalization of a Riemannian structure (a pair of a Riemannian metric and its Levi-Civita connection). By following this idea in complex geometry, H. Furuhata gave the definition of a holomorphic statistical manifold as a nice generalization of a pair of a Kaehler structure and Levi-Civita connection (see \cite{1f,00furu}). It could be treated as the statistical counterpart of the notion of complex manifold. Later on, Furuhata et al. \cite{9furu1} introduced the statistical counterpart of a Sasakian manifold and defined the notion of a Sasakian statistical manifold in the contact geometry. Vilcu et al. \cite{4vilcu} investigated the existence of the almost quaternionic structures on the statistical manifolds. They defined the concept of quaternionic Kaehler-like statistical manifolds. By extending some results derived by K. Takano concerning statistical manifolds endowed with almost complex \cite{k1} and almost contact structures \cite{k}, they obtained the main curvature properties of quaternionic Kaehler-like statistical submersions. L. Todjihounde established a dualistic structure on the warped product manifold in \cite{k2}. In the light of his work, Furuhata et al. \cite{f2} studied the statistical counterpart of a Kenmotsu manifold and introduced the notion of Kenmotsu statistical manifolds. In addition, it can be locally treated as the warped product of a holomorphic statistical manifold and a line. They proved that a Kenmotsu statistical manifold of constant $\phi-$sectional curvature is constructed from a holomorphic statistical manifold. They equipped a Kenmotsu manifold with an affine connection and gave a method of how to construct a Kenmotsu statistical manifold of constant $\phi-$sectional curvature as the warped product of a holomorphic statistical manifold and a line. Recently, in the light of Todjihounde's work, C. Murathan and B. Sahin \cite{murthan} obtained Wintgen-like inequality for statistical submanifolds of statistical warped product manifolds. They also studied how to construct Kenmotsu-like statistical manifold and cosymplectic-like statistical manifold based on the existence of Kaehler-like statistical manifold
 \begin{theorem}\label{0t}
 	Let $N$ be an $m-$dimensional statistical submanifold of a statistical warped product manifold of type $\mathbb{R} \times_{\mathfrak{f}} \overline{M}(\overline{c})$. Then the normalized Casorati curvatures $\delta_{\mathbb{\mathcal{C}}}(m-1)$ and $\delta_{\mathbb{\mathcal{C}}}^{\ast}(m-1)$ satisfy
 	\begin{align}\label{118}
 	\rho^{\nabla,\nabla^{\ast}}\leq&2\delta_{\mathcal{C}}^{0} + \frac{1}{m-1}\mathcal{C}^{0} +\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}\bigg) +\frac{3\overline{c}}{4m(m-1)\mathfrak{f}^{2}}||\phi||^{2} \nonumber\\
 	&-\frac{2}{m}\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}}- \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}+ \frac{\mathfrak{f}^{''}}{\mathfrak{f}}\bigg)||\mathrm{T} ||^{2}- \frac{m}{2(m-1)}\Big(||\mathcal{H}||^{2} + ||\mathcal{H}^{\ast}||^{2}\Big),
 	\end{align}
 	where $2\mathcal{C}^{0} = \mathcal{C} + \mathcal{C}^{\ast}$ and $2\delta_{\mathcal{C}}^{0}(m-1) = \delta_{\mathcal{C}}(m-1) + \delta_{\mathcal{C}}^{\ast}(m-1)$.\\
 \end{theorem}
One can prove that the normalized scalar curvature is bounded above by the normalized Casorati curvatures $\widehat{\delta}_{\mathbb{\mathcal{C}}}(m-1)$ and $\widehat{\delta}_{\mathbb{\mathcal{C}}}^{\ast}(m-1)$, that is,
\begin{corollary}\label{c31}
	Let $N$ be an $m-$dimensional statistical submanifold of a statistical warped product manifold of type $\mathbb{R} \times_{\mathfrak{f}} \overline{M}(\overline{c})$. Then the normalized Casorati curvatures $\delta_{\mathbb{\mathcal{C}}}(m-1)$ and $\delta_{\mathbb{\mathcal{C}}}^{\ast}(m-1)$ are given by
	\begin{align}\label{p8}
	\rho^{\nabla,\nabla^{\ast}}\leq& 2\widehat{\delta}_{\mathcal{C}}^{0} + \frac{1}{m-1}\mathcal{C}^{0} +\Big(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}\Big) +\frac{3\overline{c}}{4m(m-1)\mathfrak{f}^{2}}||\phi||^{2} \nonumber\\
	&-\frac{2}{m}\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}}\bigg)||\mathrm{T}||^{2}- \frac{m}{2(m-1)}\Big(||\mathcal{H}||^{2} + ||\mathcal{H}^{\ast}||^{2}\Big).
	\end{align}
\end{corollary}
In the sequel, we deduce some properties when equality case hold in Theorem \ref{0t} and Corollary \ref{c31}. Therefore  we can announce the following result to analogous of Theorem \ref{0t} and Corollary \ref{c31}.
\begin{theorem}\label{t31}
	Let $(\mathbb{R}, dz, \overline{\nabla}^{\mathbb{R}})$ be a trivial statistical manifold and $(\overline{M}, \overline{g}, \overline{\nabla}, \mathcal{J})$ be a holomorphic statistical manifold of constant holomorphic sectional curvature $\overline{c}$. If $N$ be an $m-$dimensional statistical submanifold of a statistical warped product manifold of type $\mathbb{R} \times_{\mathfrak{f}} \overline{M}(\overline{c})$. Then the equality holds in the inequalities (\ref{118}) and (\ref{p8}) if and only if the symmetric and bilinear imbedding curvature tensors of $N$ with respect to $\overline{\nabla}$ and $\overline{\nabla}^{\ast}$ satisfy $h = - h^{\ast}$.
\end{theorem}
If $\phi(TN)= T^{\perp}N$, then $N$ is called a Legendrian submanifold. In particular case, $n=m$. As an application, we can directly give the following result based on Theorem \ref{0t}.

\begin{corollary}
	Let $(\mathbb{R}, dz, \overline{\nabla}^{\mathbb{R}})$ be a trivial statistical manifold and $(\overline{M}, \overline{g}, \overline{\nabla}, \mathcal{J})$ be a holomorphic statistical manifold of constant holomorphic sectional curvature $\overline{c}$. If $N$ be an $m-$dimensional Legendrian submanifold of a statistical warped product manifold of type $M^{2m+1} = \mathbb{R} \times_{\mathfrak{f}} \overline{M}^{2m}(\overline{c})$, then the normalized Casorati curvatures $\delta_{\mathbb{\mathcal{C}}}(m-1)$ and $\delta_{\mathbb{\mathcal{C}}}^{\ast}(m-1)$ satisfy
	\begin{align*}
		\rho^{\nabla,\nabla^{\ast}} \leq 2\delta_{\mathcal{C}}^{0} + \frac{1}{m-1}\mathcal{C}^{0} +\Big(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}\Big) - \frac{m}{2(m-1)} \Big(||\mathcal{H}||^{2} + ||\mathcal{H}^{\ast}||^{2} \Big),
	\end{align*}
	where $2\mathcal{C}^{0} = \mathcal{C} + \mathcal{C}^{\ast}$ and $2\delta_{\mathcal{C}}^{0}(m-1) = \delta_{\mathcal{C}}(m-1) + \delta_{\mathcal{C}}^{\ast}(m-1)$. Moreover, the equality holds in the inequality if and only if the symmetric and bilinear imbedding curvature tensors of $N$ with respect to $\overline{\nabla}$ and $\overline{\nabla}^{\ast}$ satisfy $h = - h^{\ast}$.
\end{corollary}

In the initial paper, Chen established inequalities between the scalar curvature, the sectional curvature and the squared norm of the mean curvature of a submanifold in a real space form. He also obtained the inequalities between $k$-Ricci curvature, the squared mean curvature and the shape operator for the submanifolds in the real space form with arbitrary codimension \cite{chen}. Since then different geometers obtained similar inequalities for different submanifolds and ambient spaces. T. Oprea \cite{toprea} derived Chen-Ricci inequality by means of optimization techniques which applied in the setup of Riemannian geometry. Aydin et al. \cite{a} derived a Chen-Ricci inequality for statistical submanifolds in a statistical manifold of constant curvature. A. Mihai et al. \cite{a2} established a similar inequality with respect to a sectional curvature of the ambient Hessian manifold. Therefore, an important application of Theorem \ref{0t} is the following..

\begin{theorem}\label{t14}
	Let $N$ be an $m-$dimensional statistical submanifold of a statistical warped product manifold of type $\mathbb{R} \times_{\mathfrak{f}} \overline{M}(\overline{c})$. Then for each unit vector $E \in T_{\wp}N$, $\wp \in N$, we have
	\begin{align}\label{r11}
	Ric^{\nabla, \nabla^{\ast}}(E)\leq& \bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}}- \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}\bigg)(m-1)+ \frac{3\overline{c}}{4\mathfrak{f}^{2}}||\phi E||^{2}\notag\\
	+&\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}}\bigg)\Big((2-m)g^{2}(E, \mathrm{T}) - ||\mathrm{T}||^{2}\Big)\notag\\
	+&\frac{m^{2}}{2} ||\mathcal{H}^{0}||^{2} - 2(m-1) \max \overline{\mathcal{K}}^{\overline{\nabla}, \overline{\nabla}^{\ast}} (E \wedge \cdot),
	\end{align}
	where $\max \overline{\mathcal{K}}^{\overline{\nabla}, \overline{\nabla}^{\ast}} (E \wedge \cdot)$ denotes the maximum of the sectional curvature function of $\overline{M}(\overline{c})$ with respect to $\overline{\nabla}$ and $\overline{\nabla}^{\ast}$ restricted to $2-$plane sections of the tangent space $T_{\wp}N$ which are tangent to the unit vector $E$. Moreover, the equality holds in the inequality (\ref{r11}) if and only if
	\begin{align}
	h^{0}(E, E) = \frac{m}{2} \mathcal{H}^{0}(\wp)\,\,\,,\,\text{and}\,\,\,\,\, h(E, F) = -h^{*}(E, F),
	\end{align} for all $F \in T_{\wp}N^{m}$ orthogonal to $E$.
\end{theorem}
Ricci-flat manifolds are Riemannian manifolds whose Ricci curvature vanishes. Ricci-flat manifolds are special cases of Einstein manifolds, where the cosmological constant need not vanish.

Since Ricci curvature measures the amount by which the volume of a small geodesic ball deviates from the volume of a ball in Euclidean space, small geodesic balls will have no volume deviation, but their "shape" may vary from the shape of the standard ball in Euclidean space. For example, in a Ricci-flat manifold, a circle in Euclidean space may be deformed into an ellipse with equal area. In physics, Ricci-flat manifolds represent vacuum solutions to the analogs of Einstein's equations for Riemannian manifolds of any dimension, with vanishing cosmological constant. Therefore we consider statistical submanifolds are Ricci-flat then we give the following result as.
\begin{corollary}\label{c13}
	Let $N$ be an $m-$dimensional statistical submanifold of a statistical warped product manifold of type $\mathbb{R} \times_{\mathfrak{f}} \overline{M}(\overline{c})$, If $N$ is Ricci-flat then we have
	\begin{align}
	2(m-1) \max \overline{\mathcal{K}}^{\overline{\nabla}, \overline{\nabla}^{\ast}} (E \wedge \cdot)&\leq\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}\bigg)(m-1) + \frac{3\overline{c}}{4\mathfrak{f}^{2}} \nonumber \\
	&+\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}}\bigg)\Big((2-m)g^{2}(E, \mathrm{T}) - ||\mathrm{T}||^{2}\Big)\nonumber \\
	&+\frac{m^{2}}{2} ||\mathcal{H}^{0}||^{2}.
	\end{align}
\end{corollary}

\section{Preliminaries And Notations}
A Riemannian manifold $(\overline{M},\overline{g})$ with an affine connection $\overline{\nabla}$ is said to be a statistical manifold \cite{1f} $(\overline{M},\overline{g},\overline{\nabla})$ if $\overline{\nabla}$ is a torsion free connection on $\overline{M}$ and the covariant derivative $\overline{\nabla}\overline{g}$ is symmetric. A statistical manifold is a Riemannian manifold $(\overline{M},\overline{g})$ endowed with a pair of torsion-free affine connections $\overline{\nabla}$ and $\overline{\nabla}^{\ast}$ satisfying
\begin{eqnarray*}
E\overline{g}(F, G) = \overline{g}(\overline{\nabla}_{E}F, G) + \overline{g}(F, \overline{\nabla}^{\ast}_{E}G),
\end{eqnarray*}
for any $E, F, G \in \Gamma(T\overline{M})$. The connections $\overline{\nabla}$ and $\overline{\nabla}^{\ast}$ are called dual connections. The notion of conjugate connection was introduced by Amari \cite{l} into statistics. Later, his studies were developed by Lauritzen \cite{LS}. Clearly, $\big(\overline{\nabla}^{\ast}\big)^{\ast} = \overline{\nabla}$. Also, a dual connection of any torsion free affine connection $\overline{\nabla}$ is given by $2\overline{\nabla}^{0} = \overline{\nabla} + \overline{\nabla}^{\ast}$, where $\overline{\nabla}^{0}$ is the Levi-Civita connection on $\overline{M}$. Moreover, if $(\overline{\nabla}, \overline{g})$ is a statistical structure on $\overline{M}$, then $(\overline{\nabla}^{\ast}, \overline{g})$ is also a statistical structure.

\begin{definition}
\cite{cm} Let $(\overline{M}, \overline{\nabla}, \overline{\nabla}^{\ast}, \overline{g})$ be a statistical manifold. If $(\overline{M}, \overline{g}, \mathcal{J})$ is an almost Kaehler manifold then $(\overline{M}, \overline{\nabla}, \overline{\nabla}^{\ast}, \overline{g}, \mathcal{J})$ is called an almost Kaehler statistical manifold, that is, a $2-$form $\omega$ on $\overline{M}$ is closed, where $\omega$ is defined by
\begin{eqnarray*}
\omega(E, F) = \overline{g}(E, \mathcal{J}F),
\end{eqnarray*}
for any $E, F \in \Gamma(T\overline{M})$.
\end{definition}

\begin{definition}
\cite{00furu} Let $(\overline{M}, \overline{\nabla}, \overline{g}, \mathcal{J})$ be a Kaehler manifold with an affine connection $\overline{\nabla}$ on $\overline{M}$. Then $(\overline{M}, \overline{\nabla}, \overline{g}, \mathcal{J})$ is said to be a holomorphic statistical manifold if
\begin{enumerate}
\item $(\overline{M}, \overline{\nabla}, \overline{g})$ is a statistical manifold, and
\item a $2-$form $\omega$ on $\overline{M}$ is $\overline{\nabla}-$parallel (means $\overline{\nabla}\omega = 0$).
\end{enumerate}
\end{definition}

\begin{definition}
\cite{00furu} A holomorphic statistical manifold $(\overline{M}, \overline{\nabla}, \overline{g}, \mathcal{J})$ is of constant holomorphic curvature $\overline{c} \in \mathbb{R}$ if and only if
\begin{eqnarray}\label{d}
\overline{\mathcal{S}}(E, F)G &=& \frac{\overline{c}}{4} \big\{\overline{g}(F, G)E - \overline{g}(E, G)F + \overline{g}(\mathcal{J}F, G)\mathcal{J}E \nonumber \\
&&- \overline{g}(\mathcal{J}E, G)\mathcal{J}F + 2\overline{g}(E, \mathcal{J}F)\mathcal{J}G \big\}.
\end{eqnarray}
for any $E, F, G \in \Gamma(T\overline{M})$. It is denoted by $\overline{M}(c)$.
\end{definition}

Let $(\overline{M}, \overline{\nabla}, \overline{g})$ be a statistical manifold and $N$ be a submanifold of $\overline{M}$. Then $\big(N, \nabla, g\big)$ is also a statistical manifold with the induced statistical structure $(\nabla, g)$ on $N$ from $(\overline{\nabla}, \overline{g})$ and we call $\big(N, \nabla, g\big)$ as a statistical submanifold in $\big(\overline{M}, \overline{\nabla}, \overline{g}\big)$. The fundamental equations in the geometry of Riemannian submanifolds (see \cite{11}) are the Gauss and Weingarten formulae and the equations of Gauss, Codazzi and Ricci. In the statistical setting, Gauss and Weingarten formulae are respectively defined by \cite{vos}
\begin{eqnarray}\label{e}
\left .
\begin{tabular}{ccc}
$\begin{aligned}
\overline{\nabla}_{E}F &= \nabla_{E}F + h(E, F)V, \hspace{0.6 cm} \overline{\nabla}_{E}^{\ast}F = \nabla_{E}^{\ast}F + h^{\ast}(E, F)V,
\vspace{0.2 cm}\\
\overline{\nabla}_{E}V &= - A_{V}(E) + \nabla_{E}^{\perp}V, \hspace{0.9 cm} \overline{\nabla}_{E}^{\ast}V = - A_{V}^{\ast}(E) + \nabla_{E}^{\perp \ast}V,
\end{aligned}$
\end{tabular}
\right \}
\end{eqnarray}
for any $E, F \in \Gamma(TN)$ and $V \in \Gamma(T^{\perp}N)$, where $\overline{\nabla}$ and $\overline{\nabla}^{\ast}$ (resp., $\nabla$ and $\nabla^{\ast}$) are the dual connections on $\overline{M}$ (resp., on $N$). The symmetric and bilinear imbedding curvature tensor of $N$ in $\overline{M}$ with respect to $\overline{\nabla}$ and $\overline{\nabla}^{\ast}$ are denoted by $h$ and $h^{\ast}$, respectively. The relation between $h$ (resp. $h^{\ast}$) and $A_{V}$ (resp. $A_{V}^{\ast}$) is defined by \cite{vos}
\begin{eqnarray}\label{f1}
\left .
\begin{tabular}{ccc}
$\begin{aligned}
\overline{g}(h(E, F), V) &= g(A_{V}^{\ast}E, F),
\vspace{0.2 cm}\\
\overline{g}(h^{\ast}(E, F), V) &= g(A_{V} E, F),
\end{aligned}$
\end{tabular}
\right \}
\end{eqnarray}
for any $E, F \in \Gamma(TN)$ and $V \in \Gamma(T^{\perp}N)$.

Let $\overline{R}$ and $R$ be the curvature tensor fields of $\overline{\nabla}$ and $\nabla$, respectively. The corresponding Gauss, Codazzi and Ricci equations are respectively given by \cite{vos}
\begin{eqnarray}\label{i111}
\overline{g}(\overline{R}(E, F) G, H) &=& g(R(E, F) G, H) + \overline{g}(h(E, G), h^{\ast}(F, H)) \nonumber \\
&&- \overline{g}(h^{\ast}(E, H), h(F, G)),
\end{eqnarray}
\begin{eqnarray}\label{i1}
(\overline{R}(E, F) G)^{\perp} &=& \nabla_{E}^{\perp}h(F, G) - h(\nabla_{E} F, G) - h(F, \nabla_{E}G) \nonumber \\
&&-  [\nabla_{F}^{\perp}h(E, G) - h(\nabla_{F}E, G) - h(E, \nabla_{F}G) ],
\end{eqnarray}
\begin{eqnarray}\label{i11}
\overline{g}(\overline{R}^{\perp}(E, F) V, U) &=& g(R(E, F) V, U) + g([A_{V}^{\ast}, A_{U}]E, F),
\end{eqnarray}
for any $E, F, G, H \in \Gamma(TN)$ and $V, U \in \Gamma(T^{\perp}N)$, where $R^{\perp}$ is the Riemannian curvature tensor on $T^{\perp}N$. Similarly, $\overline{R}^{\ast}$ and $R^{\ast}$ are respectively the curvature tensor fields with respect to $\overline{\nabla}^{\ast}$ and $\nabla^{\ast}$. We can obtain the duals of all equations (\ref{i111})-(\ref{i11}) with respect to $\overline{\nabla}^{\ast}$ and $\nabla^{\ast}$. Then the curvature tensor fields of $\overline{M}$ and $N$ are respectively given by \cite{article.82,article.822}
\begin{eqnarray}\label{1}
\overline{\mathcal{S}} = \frac{1}{2}(\overline{R} + \overline{R}^{\ast}),  \hspace{0.2 cm} \textrm{and} \hspace{0.2 cm}  \mathcal{S} = \frac{1}{2}(R + R^{\ast}).
\end{eqnarray}
Thus, the sectional curvature $\mathbb{K}^{\nabla,\nabla^{\ast}}$ on $N$ of $\overline{M}$ is given by \cite{article.82,article.822}
\begin{eqnarray}\label{2}
\mathbb{K}^{\nabla,\nabla^{\ast}}(E \wedge F) &=& g(\mathcal{S}(E, F)F, E) \nonumber\\
&=& \frac{1}{2}(g(R(E, F) F, E) + g(R^{\ast}(E, F)F, E)),
\end{eqnarray}
for any orthonormal vectors $E, F \in T_{\wp}N$, $\wp \in N$.

The notion of warped product manifolds came up as a nice generalization of Riemannian product manifolds. According to R.L. Bishop and B. O'Neil, these manifolds were defined as follows \cite{article.8}:

\begin{definition}
Let $(N_{1}, g_{1})$ and $(N_{2}, g_{2})$ be two (pseudo)-Riemannian manifolds and $\mathfrak{f} > 0$ be a differentiable function on $N_{1}$. Consider the product $\rho: N_{1} \times N_{2} \longrightarrow N_{1}$ and $\delta: N_{1} \times N_{2} \longrightarrow N_{2}$. Then the warped product $N = N_{1} \times_{\mathfrak{f}} N_{2}$ is the product manifold $N_{1} \times N_{2}$ equipped with the Riemannian structure such that
\begin{eqnarray}\label{15}
\overline{g}(E, F ) = g_{1}(\rho_{\ast} E,  \rho_{\ast} F) + \mathfrak{f}^{2}(u) g_{2}(\delta_{\ast} E, \delta_{\ast} F)
\end{eqnarray}
for any $E, F \in \Gamma(T_{(u, v)}N)$, $u \in N_{1}$ and $v \in N_{2}$, where $\ast$ is the symbol for the tangent maps. The function $\mathfrak{f}$ is called the warping function of the warped product.
\end{definition}

Let $\chi(N_{1})$ and $\chi(N_{2})$ be the set of all vector fields on $N_{1} \times N_{2}$ which is the horizontal lift of a vector field on $N_{1}$ and the vector lift of a vector field on $N_{2}$, respectively. Thus, it can be seen that $\rho_{*}(\chi(N_{1})) = \Gamma(TN_{1})$ and $\delta_{*}(\chi(N_{2})) = \Gamma(TN_{2})$. So, $\rho_{*}(X) = E \in \Gamma(TN_{1})$, $\rho_{*}(Y) = F \in \Gamma(TN_{1})$, $\delta_{*}(U) = G \in \Gamma(TN_{2})$ and $\delta_{*}(V) = H \in \Gamma(TN_{2})$. We recall a general result for a dualistic structure on the warped product manifold $N_{1} \times_{\mathfrak{f}} N_{2}$ given by L. Todjihounde in \cite{k2}.

\begin{proposition}\label{prop}
Let $(g_{1}, \nabla^{N_{1}}, \nabla^{N_{1}\ast})$ and $(g_{2}, \nabla^{N_{2}}, \nabla^{N_{2}\ast})$ be dualistic
structures on $N_{1}$ and $N_{2}$. For $X, Y \in \chi(N_{1})$ and $U, V \in \chi(N_{2})$, $\overline{\nabla}$, $\overline{\nabla}^{\ast}$ on $N_{1} \times N_{2}$ satisfy
\begin{itemize}
\item[(a)] $\overline{\nabla}_{X} Y = \nabla_{E}^{N_{1}}F$, \hspace{2.56 cm} $\overline{\nabla}_{X}^{\ast}Y = \nabla_{E}^{N_{1}\ast}F$,
\item[(b)] $\overline{\nabla}_{X}U = \overline{\nabla}_{U}X = \frac{E(\mathfrak{f})}{\mathfrak{f}}G$, \hspace{1.3 cm} $\overline{\nabla}_{X}^{\ast}U = \overline{\nabla}_{U}^{\ast}X = \frac{E(\mathfrak{f})}{\mathfrak{f}}G$,
\item[(c)] $\overline{\nabla}_{U}V = \nabla^{N_{1}}_{G}H - \frac{\overline{g}(U, V)}{\mathfrak{f}}grad \textrm{   } \mathfrak{f}$, \hspace{0.3 cm} $\overline{\nabla}_{U}^{\ast}V = \nabla^{N_{1}\ast}_{G}H - \frac{\overline{g}(U, V)}{\mathfrak{f}}grad \textrm{   } \mathfrak{f}$.
\end{itemize}
Then $(\overline{g}, \overline{\nabla}, \overline{\nabla}^{\ast})$ is a dualistic structure on $N_{1} \times N_{2}$.
\end{proposition}

We consider a statistical warped product manifold of type $M = \mathbb{R} \times_{\mathfrak{f}} \overline{M}$ with metric $\overline{g} = g_{1} + \mathfrak{f}^{2}(z)g_{2}$, where $\mathbb{R}$ is a trivial statistical manifold with metric $g_{1} = dz^{2}$ and $\overline{M}$ is an almost Kaehler statistical manifold \cite{cm} with metric $g_{2}$ and dual affine connections $\nabla^{\overline{M}}$ and $\nabla^{\overline{M}\ast}$. The structure vector field on $M$ is denoted by $\xi = \partial z$. Any arbitary vector field on $M$ is defined by $Z = \eta(Z) + E$, where $E$ is any vector field on $\overline{M}$ and $dz = \eta$. Moreover, a new tensor field $\phi$ of type $(1, 1)$ on $M$ can be defined by uing tensor field $\mathcal{J}$, that is, $ \phi Z = \mathcal{J}E$. R. Gorunus et al. named this type of statistical warped product as an almost Kenmotsu statistical manifold \cite{kenmotsu}. The statistical curvature tensor $\overline{\mathcal{S}}$ of $M = \mathbb{R} \times_{\mathfrak{f}} \overline{M}(\overline{c})$, where $\overline{M}(\overline{c})$ is a holomorphic statistical manifold of constant holomorphic sectional curvature $\overline{c} \in \mathbb{R}$, is given by \cite{kenmotsu}
\begin{eqnarray}\label{r}
\overline{\mathcal{S}}(E, F, G, H) &=& \frac{1}{2}\big[\overline{R}(E, F, G, H) + \overline{R}^{\ast}(E, F, G, H)\big]\nonumber \\
&=& \alpha\big[\overline{g}(E, H)\overline{g}(F, G) - \overline{g}(E, G)\overline{g}(F, H)\big]\nonumber \\
&&+ \beta\big[\overline{g}(E, G)\overline{g}(F, \partial z)\overline{g}(H, \partial z) \nonumber \\
&&- \overline{g}(F, G)\overline{g}(E, \partial z)\overline{g}(H, \partial z) \nonumber \\
&&+ \overline{g}(F, H)\overline{g}(E, \partial z)\overline{g}(G, \partial z) \nonumber \\
&&- \overline{g}(E, H)\overline{g}(F, \partial z)\overline{g}(G, \partial z)\big] \nonumber \\
&&+ \gamma\big[\overline{g}(E, \phi G)\overline{g}(\phi F, H) - \overline{g}(F, \phi G) \overline{g}(\phi E, H) \nonumber \\
&&+ 2 \overline{g}(E, \phi F)\overline{g}(\phi G, H)\big],
\end{eqnarray}
where $\partial z = \frac{\partial}{\partial z}$ denotes the unit tangent vector field on $\mathbb{R}$ and $\alpha$, $\beta$ and $\gamma$ are given by
\begin{eqnarray*}
\alpha &=& \frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}, \\
\beta &=& \frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}}, \\
\gamma &=&  \frac{\overline{c}}{4\mathfrak{f}^{2}}.
\end{eqnarray*}

\section{A Family of Optimal Casorati Inequalities}

Let $N$ be any $m-$dimensional submanifold of a $(2n+1)-$dimensional almost Kenmotsu statistical manifold $M^{2n+1} = \mathbb{R} \times_{\mathfrak{f}} \overline{M}^{2n}$. If $\{e_{1}, \dots, e_{m}\}$ and $\{\xi_{1}, \dots, \xi_{p}\}$ are respectively orthonormal basis of $T_{\wp}N^{m}$ and $T_{\wp}^{\perp}N^{m}$ for $\wp \in \overline{M}$. Then the mean curvature vectors $\mathcal{H}$ and $\mathcal{H}^{\ast}$ of $N$ are
$$\mathcal{H} = \frac{1}{m}\sum_{i = 1}^{m}h(e_{i}, e_{i}),$$
and
$$\mathcal{H}^{\ast} = \frac{1}{m}\sum_{i = 1}^{m}h^{\ast}(e_{i}, e_{i}).$$
Also, we set
$$h_{ij}^{k} = g(h(e_{i}, e_{j}), e_{k}),$$
and
$$h_{ij}^{\ast k} = g(h^{\ast}(e_{i}, e_{j}), e_{k}),$$
for $i, j \in \{1, \dots, m\}$, $k \in \{1, \dots, p\}$.\\

The squared norm of second fundamental forms $h$ and $h^{\ast}$ are denoted by $\mathcal{C}$ and $\mathcal{C}^{\ast}$, respectively, called the {\it Casorati curvatures} of $N$ in $\overline{M}$. Therefore, we have
\begin{eqnarray}\label{13}
\mathcal{C} = \frac{1}{m}||h||^{2}, \hspace{0.4 cm} \textrm{            }
 (\textrm{resp.} \textrm{            } \mathcal{C}^{\ast} = \frac{1}{m1}||h^{\ast}||^{2}),
\end{eqnarray}
where
\begin{eqnarray*}
||h||^{2} = \sum_{k = 1}^{p} \sum_{i, j = 1}^{m}\big(h_{ij}^{k}\big)^{2}, \hspace{0.4 cm} \textrm{            }
(\textrm{resp.} \textrm{            } ||h^{\ast}||^{2} = \sum_{k = 1}^{p} \sum_{i, j = 1}^{m}\big(h_{ij}^{\ast k}\big)^{2}).
\end{eqnarray*}

If we consider a $s-$dimensional subspace $W$ of $TN$, $s \geq 2$, and an orthonormal basis $\{e_{1}, \dots, e_{s}\}$ of $W$. Then the scalar curvature of the $s-$plane section $W$ is defined as
\begin{eqnarray*}
\tau(W) = \sum_{1 \leq i < j \leq s}\mathcal{S}(e_{i}, e_{j}, e_{j}, e_{i}),
\end{eqnarray*}
and the Casorati curvatures of the subspace $W$ are as follows:
\begin{eqnarray*}
\mathcal{C}(W)= \frac{1}{s}\sum_{k = 1}^{p}\sum_{i, j = 1}^{s}\big(h_{ij}^{k}\big)^{2}, \hspace{0.4 cm} \textrm{            }
 (\textrm{resp.} \textrm{            } \mathcal{C}^{\ast}(W)= \frac{1}{s}\sum_{k = 1}^{p}\sum_{i, j = 1}^{s}\big(h_{ij}^{\ast k}\big)^{2}).
\end{eqnarray*}

\begin{enumerate}
\item[(1)] The normalized Casorati curvatures $\delta_{\mathcal{C}}(m - 1)$ and $\delta_{\mathcal{C}}^{\ast}(m - 1)$ are defined as
\begin{eqnarray*}
[\delta_{\mathcal{C}}(m - 1)]_{\wp} &=& \frac{1}{2}\mathcal{C}_{\wp} +  (\frac{m + 1}{2m} )\textrm{inf} \{\mathcal{C}(W)| W: \text{a hyperplane of} \textrm{  }  T_{\wp}N\}, \\
\textrm{and} \hspace{0.2 cm} [\delta_{\mathcal{C}}^{\ast}(m - 1)]_{\wp} &=& \frac{1}{2}\mathcal{C}_{\wp}^{\ast} +  (\frac{m + 1}{2m} )\textrm{inf} \{\mathcal{C}^{\ast}(W)|W: \text{a hyperplane of} \textrm{  }  T_{\wp}N\}.
\end{eqnarray*}
\item[(2)] The normalized Casorati curvatures $\widehat{\delta}_{\mathcal{C}}(m - 1)$ and $\widehat{\delta}_{\mathcal{C}}^{\ast}(m - 1)$ are defined as
\begin{eqnarray*}
[\widehat{\delta}_{\mathcal{C}}(m - 1)]_{\wp} &=& 2\mathcal{C}_{\wp} -  (\frac{2m - 1}{2m} )\textrm{sup} \{\mathcal{C}(W)| W: \text{a hyperplane of} \textrm{  } T_{\wp}N\}, \\
\textrm{and} \hspace{0.2 cm} [\widehat{\delta}_{\mathcal{C}}^{\ast}(m - 1)]_{\wp} &=& 2\mathcal{C}_{\wp}^{\ast} -  (\frac{2m - 1}{2m} )\textrm{sup} \{\mathcal{C}^{\ast}(W)| W: \text{a hyperplane of} \textrm{  } T_{\wp}N\}.
\end{eqnarray*}
\end{enumerate}
\subsection{Proof of Theorem \ref{0t}}
The scalar curvature $\tau^{\nabla, \nabla^{\ast}}$ of $N$ is given by
\begin{eqnarray}\label{p1}
2 \tau^{\nabla, \nabla^{\ast}} &=& \sum_{1 \leq i < j \leq m} \mathcal{S}(e_{i}, e_{j}, e_{j}, e_{i}) \nonumber \\
&=& \frac{1}{2}\sum_{1 \leq i < j \leq m} \{R(e_{i}, e_{j}, e_{j}, e_{i}) + R^{\ast}(e_{i}, e_{j}, e_{j}, e_{i})\}\nonumber \\
&=& \sum_{1 \leq i < j \leq m} R(e_{i}, e_{j}, e_{j}, e_{i})\nonumber \\
&=& m(m-1) (\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}) - 2(m-1) (\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}})||\mathrm{T} ||^{2} + \frac{3\overline{c}}{4\mathfrak{f}^{2}}||\phi||^{2} \nonumber \\
&&+ \frac{1}{2}\sum_{i, j}\{\overline{g}(\hbar^{\ast}(e_{i}, e_{i}), \hbar(e_{j}, e_{j}))  + \overline{g}(\hbar(e_{i}, e_{i}), \hbar^{\ast}(e_{j}, e_{j}))\} \nonumber \\
&&- 2 \sum_{i, j} \overline{g}(\hbar(e_{i}, e_{j}), \hbar^{\ast}(e_{i}, e_{j})),
\end{eqnarray}
where $\mathrm{T} = \partial z - \sum_{k=1}^{p}\lambda_{k}\xi_{p}$ is a vector field tangent to $N$ and $\lambda_{1}, \dots, \lambda_{p}$ are smooth functions over $N$. Now we define a polynomial $\mathbb{Q}$ in terms of the components of the second fundamental form $h^{0}$ (with respect to $\overline{\nabla}^{0}$) of $N$.
\begin{eqnarray}\label{p6}
\mathbb{Q} &=& \frac{1}{2} m(m - 1)\mathcal{C}^{0} + \frac{1}{2}(m-1)(m + 1)\mathcal{C}^{0}(W) + \frac{m}{2}(\mathcal{C} + \mathcal{C}^{\ast})+ \frac{3\overline{c}}{4\mathfrak{f}^{2}}||\phi||^{2}\nonumber\\
&&+ m(m-1) (\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}) - 2(m-1) (\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}})||\mathrm{T} ||^{2} \nonumber \\
&&- \frac{m^{2}}{2}(||\mathcal{H}||^{2} + ||\mathcal{H}^{\ast}||^{2})- 2 \tau^{\nabla, \nabla^{\ast}},
\end{eqnarray}
where $W$ is a hyperplane of $T_{\wp}N$.

Without loss of generality, we assume that $W$ is spanned by $\{e_{1}, \dots, e_{m}\}$. Then, by (\ref{p1}) and (\ref{p6}), we derive
\begin{eqnarray*}
\mathbb{Q} &=& \sum_{k = 1}^{p}  [\sum_{i,j = 1}^{m} \frac{m+3}{2} (h_{ij}^{0 k})^{2} + \frac{m+1}{2} \sum_{i,j=1}^{m-1} (h_{ij}^{0 k})^{2} - 2 (\sum_{i = 1} h_{ii}^{0 k})^{2}] \nonumber \\
&=& \sum_{k = 1}^{p}  [ 2(m+2)\sum_{1 \leq i < j \leq m - 1} (h_{ij}^{0 k})^{2} + (m+3)\sum_{i=1}^{m-1} (h_{im}^{0 k})^{2}\nonumber \\
&&+ m \sum_{i=1}^{m-1} (h_{ii}^{0 k})^{2} - 4 \sum_{1 \leq i < j \leq m} (h_{ii}^{0 k}h_{jj}^{0 k}) + \frac{m-1}{2}(h_{mm}^{0 k})^{2}] \nonumber \\
&\geq& \sum_{k = 1}^{p}  [\sum_{i = 1}^{m-1} m (h_{ii}^{0 k})^{2} + \frac{m-1}{2} (h_{mm}^{0 k})^{2} - 4 \sum_{1 \leq i < j \leq m} h_{ii}^{0 k}h_{jj}^{0 k} ].
\end{eqnarray*}
For any $k \in \{1, \dots, p\}$, we define a quadratic form $\mathcal{P}_{k} : \mathbb{R}^{m} \rightarrow \mathbb{R}$ by
\begin{align}\label{p7}
\mathcal{P}_{k}(h_{11}^{0k}, h_{22}^{0k}, \dots, h_{m-1m-1}^{0k}, h_{mm}^{0k}) &=\sum_{i = 1}^{m-1} m (h_{ii}^{0 k})^{2} + \frac{m-1}{2} (h_{mm}^{0 k})^{2} \nonumber \\
 &-4 \sum_{1 \leq i < j \leq m} h_{ii}^{0 k}h_{jj}^{0 k}.
\end{align}
Further, we consider the constrained extremum problem $\min P_{k}$ subject to
$$\overline{\mathcal{N}} : \sum_{i=1}^{m} h_{ii}^{0k} = \alpha^{k},$$
where $\alpha^{k}$ is a real constant.

From (\ref{p7}), we find that the critical points $$h^{0 c} = (h_{11}^{0k}, h_{22}^{0k}, \dots, h_{m-1m-1}^{0k}, h_{mm}^{0k})$$ of $\overline{\mathcal{N}}$ are the solutions of the following system of linear homogeneous equations.
\begin{eqnarray}\label{16}
\left .
\begin{tabular}{ccc}
    $\frac{\partial \mathcal{P}_{k}}{\partial h_{ii}^{0 k}} = 2(m+2)(h_{ii}^{0k}) - 4\sum_{r = 1}^{m} h_{rr}^{0k} = 0$ \vspace{0.2 cm} \\

    $\frac{\partial \mathcal{P}_{k}}{\partial h_{mm}^{0 k}} = (m-1)h_{mm}^{0k} - 4\sum_{r = 1}^{m-1} h_{rr}^{0k}  = 0$,
\end{tabular}
\right \}
\end{eqnarray}
for $i \in \{1, 2, \dots, m-1\}$ and $k \in \{1, \dots, p\}$. Hence, every solution $h^{0c}$ has
\begin{eqnarray*}
h_{ii}^{0k} = \frac{1}{m+1}\alpha^{k}, \hspace{1 cm}  h_{mm}^{0k} = \frac{4}{m+3}\alpha^{k},
\end{eqnarray*}
for $i \in \{1, 2, \dots, m-1\}$ and $k \in \{1, \dots, p\}$.

Now, we fix $x \in \overline{\mathcal{N}}$. The bilinear form $\Theta : T_{x}\overline{\mathcal{N}} \times T_{x}\overline{\mathcal{N}} \rightarrow \mathbb{R}$ has the following expression:
\begin{eqnarray*}
\Theta(E, F) = Hess_{P_{k}}(E, F) + <h^{'}(E, F), (grad \hspace{0.1 cm} P_{k})(x)>,
\end{eqnarray*}
where $h^{'}$ denotes the second fundamental form of $\overline{\mathcal{N}}$ in $\mathbb{R}^{m}$ and $<\cdot,\cdot>$ denotes the standard inner product on $\mathbb{R}^{m}$. The Hessian matrix of $\mathcal{P}_{k}$ is given by
\begin{eqnarray*}
Hess_{P_{k}} = \left(
 \begin{array}{ccccc}
    2(m+2) & -4 & \dots & -4 & -4\\
    -4 & 2(m+2) & \dots& -4 & -4\\
    \vdots & \vdots \ & \ddots & \vdots & \vdots\\
    -4 & -4  & \dots &  2(m+2) & -4 \\
    -4 & -4  & \dots &  -4 & (m-1)
  \end{array}
\right).
\end{eqnarray*}
Take a vector $E \in T_{x}\overline{\mathcal{N}}$, which satisfies a relation $\sum_{i=1}^{m} E_{i} = 0$. As the hyperplane is totally geodesic, that is, $h^{'} = 0$ in $\mathbb{R}^{m}$, we get
\begin{eqnarray*}
\Theta(E, E) &=& Hess_{P_{k}}(E, E) \\
&=& 2(m+2) \sum_{i=1}^{m-1} E_{i}^{2} + (m-1)E_{m}^{2} - 8 \sum_{i \neq j =1}^{m} E_{i}E_{j} \\
&=& 2(m+2) \sum_{i=1}^{m-1} E_{i}^{2} + (m-1)E_{m}^{2} - 4[ (\sum_{i = 1}^{m} E_{i})^{2} - \sum_{i = 1}^{m}E_{i}^{2}] \\
&=& 2(m+4) \sum_{i=1}^{m-1} E_{i}^{2}+(m+3) E_{m}^{2} \\
&\geq& 0.
\end{eqnarray*}
However, the point $h^{0c}$ is the only optimal solution, i.e., the global minimum point of problem and reaches a minimum $\mathbb{Q}(h^{0c}) = 0$ for the solution $h^{0c}$ of the system (\ref{16}). It follows that $\mathbb{Q} \geq 0$. Thus, we have
\begin{eqnarray*}
2 \tau^{\nabla, \nabla^{\ast}} &\leq& \frac{1}{2} m(m - 1)\mathcal{C}^{0} + \frac{1}{2}(m-1)(m + 1)\mathcal{C}^{0}(W) + \frac{m}{2}(\mathcal{C} + \mathcal{C}^{\ast}) + \frac{3\overline{c}}{4\mathfrak{f}^{2}}||\phi||^{2}\nonumber\\
&&+ m(m-1) (\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}) - 2(m-1) (\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}})||\mathrm{T} ||^{2} \nonumber \\
&&- \frac{m^{2}}{2}(||\mathcal{H}||^{2} + ||\mathcal{H}^{\ast}||^{2}).
\end{eqnarray*}
The normalized scalar curvature $\rho$ of $N$ is defined as
\begin{eqnarray}\label{4}
\rho^{\nabla, \nabla^{\ast}} = \frac{2\tau^{\nabla, \nabla^{\ast}} }{m(m - 1)}.
\end{eqnarray}
Hence, we get
\begin{eqnarray*}
\rho^{\nabla, \nabla^{\ast}} &\leq& \frac{1}{2}\mathcal{C}^{0} + \frac{m+1}{2m}\mathcal{C}^{0}(W) + \frac{1}{2(m-1)}(\mathcal{C} + \mathcal{C}^{\ast}) + \frac{3\overline{c}}{4m(m-1)\mathfrak{f}^{2}}||\phi||^{2}\nonumber\\
&&+ (\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}) - \frac{2}{m} (\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}})||\mathrm{T} ||^{2} \nonumber \\
&&- \frac{m}{2(m-1)}(||\mathcal{H}||^{2} + ||\mathcal{H}^{\ast}||^{2})
\end{eqnarray*}
for every tangent hyperplane $W$ of $N$. If we take the infimum over all tangent hyperplanes $W$. Therefore our assertion (\ref{118}) follows. This completes the proof of Theorem \ref{0t}.

\subsection{Proof of Corollary \ref{c31}}
For this, we consider the following polynomial
\begin{eqnarray}\label{p66}
Q &=& 2m(m - 1)\mathcal{C}^{0} - \frac{1}{2}(m-1)(m + 1)\mathcal{C}^{0}(W) + \frac{m}{2}(\mathcal{C} + \mathcal{C}^{\ast})+ \frac{3\overline{c}}{4\mathfrak{f}^{2}}||\phi||^{2}\nonumber\\
&&+ m(m-1) (\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}) - 2(m-1) (\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}})||\mathrm{T} ||^{2} \nonumber \\
&&- \frac{m^{2}}{2}(||\mathcal{H}||^{2} + ||\mathcal{H}^{\ast}||^{2}) - 2 \tau^{\nabla, \nabla^{\ast}},
\end{eqnarray}
where $W$ is a hyperplane of $T_{\wp}N$. The rest proof is similar to the proof of Theorem \ref{0t}.

\subsection{Proof of Theorem \ref{t31}}
The equality holds in the inequalities (\ref{118}) and (\ref{p8}) if and only if the second fundamental form with respect to $\overline{\nabla}^{0}$ is zero, that is, $h^{0} = 0$. This further gives $h = - h^{\ast}$. Partially, we say that $h$ and $h^{\ast}$ are linearly dependent. This completes the proof of Theorem \ref{t31}.

\section{Chen-Ricci inequality for statistical submanifolds}
The curvatures invariants are widely used in the field of differential geometry and in physics also. The Ricci curvature is the essential term in the Einstein field equations, which plays a key role in general relativity. It is immensely studied in differential geometry which gives a way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from the ordinary Euclidean $n-$space. A Riemannian manifold is said to be an Einstein manifold if the Ricci tensor satisfies the vacuum Einstein equation. The lower bounds on the Ricci tensor on a Riemannian manifold enable one to find global geometric and topological information by comparison with the geometry of a constant curvature space form. Now, we give the proof of second main theorem as follow.

\subsection{Proof of Theorem \ref{t14}}
Let $N$ be an $m-$dimensional statistical submanifold of a statistical warped product manifold of type $\mathbb{R} \times_{\mathfrak{f}} \overline{M}(\overline{c})$. We choose $\{e_{1}, \dots, e_{m}\}$ as the orthonormal frame of $T_{\wp}N$ such that $e_{1} = E$ and $||E|| = 1$, and $\{e_{1}, \dots, e_{p}\}$ as the the orthonormal frame of $T_{\wp}N$. Then by (\ref{r}) and (\ref{1}), we have
\begin{align}\label{r1}
\sum_{i = 2}^{m}\overline{\mathcal{S}}(e_{1}, e_{i}, e_{1}, e_{i})=& \bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}\bigg)(m-1) - \bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} \frac{\mathfrak{f}^{''}}{\mathfrak{f}}\bigg)\nonumber \\
&+\bigg(\sum_{i=2}^{m} g(e_{i}, e_{i}) g^{2}(E, \mathrm{T}) + \sum_{i= 1}^{m} g^{2}(\mathrm{T}, e_{i}) - g^{2}(E, \mathrm{T})\bigg)\nonumber \\
&+\frac{3\overline{c}}{4\mathfrak{f}^{2}}||\phi E||^{2}\nonumber \\
&=\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}\bigg)(m-1)+ \frac{3\overline{c}}{4\mathfrak{f}^{2}}||\phi E||^{2} \nonumber \\
&+\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}}\bigg)\bigg((2-m)g^{2}(E, \mathrm{T}) - ||\mathrm{T}||^{2}\bigg).
\end{align}\label{}
From (\ref{i111}), dual of (\ref{i111}) and (\ref{1}), we get
\begin{align*}
2\overline{\mathcal{S}}(e_{1}, e_{i}, e_{1}, e_{i})=& 2\mathcal{S}(e_{1}, e_{i}, e_{1}, e_{i}) - g(h(e_{1}, e_{1}), h^{\ast}(e_{i}, e_{i}))\\
&-g(h^{\ast}(e_{1}, e_{1}), h(e_{i}, e_{i}))
+2g(h(e_{1}, e_{i}), h^{\ast}(e_{1}, e_{i}))\\
&=2\mathcal{S}(e_{1}, e_{i}, e_{1}, e_{i}) - \{4g(h^{0}(e_{1}, e_{1}), h^{0}(e_{i}, e_{i}))\\
&- g(h(e_{1}, e_{1}), h(e_{i}, e_{i})) - g(h^{\ast}(e_{1}, e_{1}), h^{\ast}(e_{i}, e_{i}))\\
&- 4g(h^{0}(e_{1}, e_{i}), h^{0}(e_{1}, e_{i})) + g(h(e_{1}, e_{i}), h(e_{1}, e_{i}))\\
&+ g(h^{\ast}(e_{1}, e_{i}), h^{\ast}(e_{1}, e_{i}))\}\\
&=2\mathcal{S}(e_{1}, e_{i}, e_{1}, e_{i}) - 4\sum_{k=1}^{p}(h^{0k}_{11}h^{0k}_{ii} - (h^{0k}_{1i})^{2})\\
&+ \sum_{k=1}^{p}(h^{k}_{11}h^{k}_{ii} - (h^{k}_{1i})^{2}) + \sum_{k=1}^{p}(h^{\ast k}_{11}h^{\ast k}_{ii} - (h^{\ast k}_{1i})^{2}).
\end{align*}
Summing over $2 \leq i \leq m$ and using (\ref{r1}), we derive
\begin{align*}
2\bigg[&\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}}- \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}\bigg)(m-1)+ \frac{3\overline{c}}{4\mathfrak{f}^{2}}||\phi E||^{2} \\
&+\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}}\bigg)((2-m)g^{2}(E, \mathrm{T}) - ||\mathrm{T}||^{2})\bigg]\\
=&2Ric^{\nabla, \nabla^{\ast}}(E) - 4\sum_{k=1}^{p}\sum_{i=2}^{m}(h^{0k}_{11}h^{0k}_{ii} - (h^{0k}_{1i})^{2}) \\
&+ \sum_{k=1}^{p}\sum_{i=2}^{m}(h^{k}_{11}h^{k}_{ii} - (h^{k}_{1i})^{2}) + \sum_{k=1}^{p}\sum_{i=2}^{m}(h^{\ast k}_{11}h^{\ast k}_{ii} - (h^{\ast k}_{1i})^{2}),
\end{align*}
where $Ric^{\nabla, \nabla^{\ast}}(E)$ denotes the Ricci curvature of $N$ with respect to $\nabla$ and $\nabla^{\ast}$ at $\wp$. Further, we derive
\begin{align}\label{g2}
2Ric^{\nabla, \nabla^{\ast}}(E) - 2\bigg[&\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}\bigg)(m-1)+ \frac{3\overline{c}}{4\mathfrak{f}^{2}}||\phi E||^{2}\nonumber \\
&+\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}}\bigg)((2-m)g^{2}(E, \mathrm{T}) - ||\mathrm{T}||^{2})\bigg]\nonumber \\
=&4\sum_{k=1}^{p}\sum_{i=2}^{m}(h^{0k}_{11}h^{0k}_{ii} - (h^{0k}_{1i})^{2})- \sum_{k=1}^{p}\sum_{i=2}^{m}(h^{k}_{11}h^{k}_{ii} - (h^{k}_{1i})^{2})\nonumber \\
&-\sum_{k=1}^{p}\sum_{i=2}^{m}(h^{\ast k}_{11}h^{\ast k}_{ii} - (h^{\ast k}_{1i})^{2}).
\end{align}
Using the following relations
\begin{align*}
\sum_{i=2}^{m}\overline{R}(e_{1}, e_{i}, e_{1}, e_{i}) = \sum_{i=2}^{m}R(e_{1}, e_{i}, e_{1}, e_{i}) - \sum_{k=1}^{p}\sum_{i=2}^{m}(h^{k}_{11}h^{k}_{ii} - (h^{k}_{1i})^{2})
\end{align*}
and
\begin{align*}
\sum_{i=2}^{m}\overline{R}^{\ast}(e_{1}, e_{i}, e_{1}, e_{i})= \sum_{i=2}^{m}R^{\ast}(e_{1}, e_{i}, e_{1}, e_{i}) - \sum_{k=1}^{p}\sum_{i=2}^{m}(h^{\ast k}_{11}h^{\ast k}_{ii} - (h^{\ast k}_{1i})^{2}).
\end{align*}
Now we subsitute into (\ref{g2}), we arrive at
\begin{align*}
2Ric^{\nabla, \nabla^{\ast}}(E)-& 2\bigg[\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}\bigg)(m-1)+ \frac{3\overline{c}}{4\mathfrak{f}^{2}}||\phi E||^{2} \\
&+\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}}\bigg)((2-m)g^{2}(E, \mathrm{T}) - ||\mathrm{T}||^{2})\bigg]\\
=&4\sum_{k=1}^{p}\sum_{i=2}^{m}(h^{0k}_{11}h^{0k}_{ii} - (h^{0k}_{1i})^{2}) + \sum_{i=2}^{m}\overline{R}(e_{1}, e_{i}, e_{1}, e_{i})\nonumber \\
&-\sum_{i=2}^{m}R(e_{1}, e_{i}, e_{1}, e_{i}) + \sum_{i=2}^{m}\overline{R}^{\ast}(e_{1}, e_{i}, e_{1}, e_{i})\nonumber \\
&-\sum_{i=2}^{m}R^{\ast}(e_{1}, e_{i}, e_{1}, e_{i})\\
=&4\sum_{k=1}^{p}\sum_{i=2}^{m}(h^{0k}_{11}h^{0k}_{ii} - (h^{0k}_{1i})^{2}) - 2 \sum_{i=2}^{m}\overline{\mathbb{K}}(E \wedge e_{i})\nonumber \\
&-2 \sum_{i=2}^{m}\overline{\mathbb{K}}^{\ast}(E \wedge e_{i}).
\end{align*}
Taking into account of the following equation
$$2\sum_{i=2}^{m}\overline{\mathbb{K}}^{\overline{\nabla}, \overline{\nabla}^{\ast}} (E \wedge e_{i}) = \sum_{i=2}^{m}\overline{\mathbb{K}}(E \wedge e_{i}) + \sum_{i=2}^{m}\overline{\mathbb{K}}^{\ast}(E \wedge e_{i}),$$
we obtain
\begin{align}\label{g3}
Ric^{\nabla, \nabla^{\ast}}(E) &-\bigg[\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}\bigg)(m-1) + \frac{3\overline{c}}{4\mathfrak{f}^{2}}||\phi E||^{2}  \nonumber \\
&+\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}}\bigg)((2-m)g^{2}(E, \mathrm{T})- ||\mathrm{T}||^{2})\bigg] \nonumber \\
&+2(m-1) \max \overline{\mathbb{K}}^{\overline{\nabla}, \overline{\nabla}^{\ast}} (E \wedge \cdot) \nonumber \\
&=2\sum_{k=1}^{p}\sum_{i=2}^{m}(h^{0k}_{11}h^{0k}_{ii} - (h^{0k}_{1i})^{2}) \leq 2\sum_{k=1}^{p}\sum_{i=2}^{m}h^{0k}_{11}h^{0k}_{ii}.
\end{align}
Let us define the quadratic form $\mathcal{P}_{k} : \mathbb{R}^{m} \rightarrow \mathbb{R}$ by
\begin{align*}
\mathcal{P}_{k}(h_{11}^{0k}, h_{22}^{0k}, \dots, h_{mm}^{0k}) = \sum_{k=1}^{p}\sum_{i=2}^{m}h^{0k}_{11}h^{0k}_{ii}.
\end{align*}
We consider the constrained extremum problem $\max \mathcal{P}_{k}$ subject to
\begin{align*}
\overline{\mathcal{N}}:\sum_{i=1}^{m}h^{0k}_{ii} = \alpha^{k},
\end{align*}
where $\alpha^{k}$ is a real constant. The gradient vector field of the function $\mathcal{P}_{k}$ is given by $$grad\hspace{0.1 cm} \mathcal{P}_{k} = (\sum_{i=2}^{m}h^{0k}_{ii}, h^{0k}_{11}, h^{0k}_{11}, \dots, h^{0k}_{11}).$$ For an optimal solution $h^{0c} = (h_{11}^{0k}, h_{22}^{0k}, \dots h_{mm}^{0k})$ of the problem in question, the vector $grad \hspace{0.1 cm} \mathcal{P}_{k}$ is normal to $\overline{\mathcal{N}}$ at the point $h^{0c}$. It follows that
\begin{align*}
h^{0k}_{11} = \sum_{i=2}^{m}h^{0k}_{ii} = \frac{\alpha^{k}}{2}.
\end{align*}
Now, we fix $x \in \overline{\mathcal{N}}$. The bilinear form $\Theta : T_{x}\overline{\mathcal{N}} \times T_{x}\overline{\mathcal{N}} \rightarrow \mathbb{R}$ has the following expression:
\begin{align*}
\Theta(E, F) = Hess_{P_{k}}(E, F) + <h^{'}(E, F), (grad \hspace{0.1 cm} P_{k})(x)>,
\end{align*}
where $h^{'}$ denotes the second fundamental form of $\overline{\mathcal{N}}$ in $\mathbb{R}^{m}$ and $<\cdot,\cdot>$ denotes the standard inner product on $\mathbb{R}^{m}$. The Hessian matrix of $P_{k}$ is given by
\begin{eqnarray*}
Hess_{P_{k}} = \left(
 \begin{array}{ccccc}
    0 & 1 & \dots & 1 \\
    1 & 0 & \dots& 0 \\
    \vdots & \vdots \ & \ddots & \vdots \\
    1 & 0  & \dots &  0 \\
    1 & 0  & \dots &  0
  \end{array}
\right).
\end{eqnarray*}
We consider a vector $E \in T_{x}\overline{\mathcal{N}}$, which satisfies a relation $\sum_{i=2}^{m} E_{i} = -E_{1}$. As $h^{'} = 0$ in $\mathbb{R}^{m}$, we get
\begin{align*}
\Theta(E, E) =&Hess_{P_{k}}(E, E)\\
=& 2 \sum_{i=2}^{m} E_{1}E_{i} \\
=&(E_{1} + \sum_{i=2}^{m} E_{i})^{2} - (E_{1})^{2} - (\sum_{i=2}^{m} E_{i})^{2}\\
=&-2(E_{1})^{2} \leq 0.
\end{align*}
However, the point $h^{0c}$ is the only optimal solution, i.e., the global maximum point of problem. Thus, we obtain
\begin{eqnarray}\label{g4}
\mathcal{P}_{k} \leq \frac{1}{4}(\sum_{i=1}^{m} h_{ii}^{0k})^{2} = \frac{m^{2}}{4}(\mathcal{H}^{0k})^{2},
\end{eqnarray}
where $\mathcal{H}^{0}$ denotes mean curvature vector of $N$ with respect to Levi-Civita connection $\overline{\nabla}^{0}$. On combining (\ref{g3}) and (\ref{g4}), we get our desired inequality (\ref{r11}).

Moreover, the vector field $E$ satisfies the equality case if and only if
\begin{align*}
h_{1i}^{0r} = 0 \Rightarrow h_{1i}^{r} = - h_{1i}^{*r},
\end{align*}
and
\begin{align*}
h^{0r}_{11} = \sum_{i=2}^{m}h^{0r}_{ii}, \hspace{1 cm} i \in \{2, \dots, m\}, \textrm{  } r \in \{1, \dots, p\}.
\end{align*}
It can be rewritten as
\begin{align*}
h^{0r}_{11} = \frac{m}{2}\mathcal{H}^{0}.
\end{align*}
This completes the proof of Theorem \ref{t14}.
\subsection{Proof of Corollary \ref{c13}}
For Ricci flat submanifold then the Ricci curvature should be vanish, that is, $Ric^{\nabla, \nabla^{\ast}}(E)=0$. Then from \eqref{r11}, we get required proof of Corollary.\\

Some immediate consequences of Theorem \ref{t14}:
\begin{corollary}
Let $N$ be an $m-$dimensional anti-invariant submanifold of a statistical warped product manifold of type $\mathbb{R} \times_{\mathfrak{f}} \overline{M}(\overline{c})$. Then for each unit vector $E \in T_{\wp}N$, $\wp \in N$, we have
\begin{align*}
Ric^{\nabla, \nabla^{\ast}}(E)&\leq\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}\bigg)(m-1) \\
&+\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}}\bigg)\Big((2-m)g^{2}(E, \mathrm{T}) -||\mathrm{T}||^{2}\Big)\nonumber \\
&+\frac{m^{2}}{2} ||\mathcal{H}^{0}||^{2} - 2(m-1) \max \overline{\mathcal{K}}^{\overline{\nabla}, \overline{\nabla}^{\ast}} (E \wedge \cdot).
\end{align*}
Furthermore, the equality holds in the inequality if and only if $h^{0}(E, E) = \frac{m}{2} \mathcal{H}^{0}(\wp)$ and $h(E, F)=-h^{*}(E, F)$, for all $F \in T_{\wp}N^{m}$ orthogonal to $E$.
\end{corollary}

\begin{corollary}
Let $N$ be an $m-$dimensional invariant submanifold of a statistical warped product manifold of type $\mathbb{R} \times_{\mathfrak{f}} \overline{M}(\overline{c})$. Then for each unit vector $E \in T_{\wp}N$, $\wp \in N$, we have
\begin{align*}
Ric^{\nabla, \nabla^{\ast}}(E) &\leq\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}}\bigg)(m-1) + \frac{3\overline{c}}{4\mathfrak{f}^{2}} \nonumber \\
&+\bigg(\frac{\overline{c}}{4\mathfrak{f}^{2}} - \frac{(\mathfrak{f^{'}})^{2}}{\mathfrak{f}^{2}} + \frac{\mathfrak{f}^{''}}{\mathfrak{f}}\bigg)\Big((2-m)g^{2}(E, \mathrm{T}) - ||\mathrm{T}||^{2}\Big)\nonumber \\
&+\frac{m^{2}}{2} ||\mathcal{H}^{0}||^{2}-2(m-1) \max \overline{\mathcal{K}}^{\overline{\nabla}, \overline{\nabla}^{\ast}} (E \wedge \cdot).
\end{align*}
In addition, the equality holds in the inequality if and only if $h^{0}(E, E) = \frac{m}{2} \mathcal{H}^{0}(\wp)$ and $h(E, F) = -h^{*}(E, F)$, for all $F \in T_{\wp}N^{m}$ orthogonal to $E$.
\end{corollary}

\section*{Acknowledgement}
The authors would like to thank to anonymous referees for their comments to improve the paper.

redpajama_set_name: RedPajamaArXiv


Time management is difficult if you do not have the right techniques. This article will help you learn about managing your time. Use these tips and start managing your time.
Take the time to properly plan for the upcoming day. Establish tomorrow's agenda at the end of each work day so that you will be fully prepared. A great way to end your work day is to prepare your to-do list for the next day. With your activities planned, you'll feel that it is easy to dive right in to your tasks.
You can manage time very well by using a calendar. Many people like the convenience of writing on paper calendars. Some people like electronic calendars on phones and computers for their flexibility. Whatever type you prefer, a calendar will help you manage your time more effectively by keeping your tasks straight.
If you struggle with effective time management, take time to analyze what's awry with the current workflow. You should be focusing on the important tasks at hand. If not, then get to the root of the reason why. If you're serious about managing time wisely, you must identify what is keeping you from doing that.
If it's tough for you to manage your time, plan the day the night before. This might be due to a to-do list that you create at the finish of your days, or it might be something more in-depth. If you do this, you can relax a little knowing what you will be prepared for the next day.
Try saying no. A lot of people get stressed because they can't say no. Check your schedule if you're overbooked. Can you pass a task to someone else? Ask your family and friends for help.
Every day when you awake, plan how your day will go. Write down everything you plan to do and the amount of time you will need. Keeping a daily plan keeps you on an efficient path.
If you wish to work in privacy, simply close the door to your office. An open door is seen as a sign that you are available to them for any questions or problems. A closed door gives you total privacy. People realize that you need to concentrate on your work in order to get your tasks done on time.
Stay focused on what you are doing until it's done. Avoid distraction by tasks that come up while working on one already. You may find that people try to layer on additional "important" tasks while you still are working on completing the first. Do not let them do this. Complete one thing prior to moving on to another one.
Look for time management classes in your area. You will discover many techniques to help you. Some companies offer their employees time management classes, as they feel it will help them succeed. If that is not the case at your place of business, you can probably find one offered at the community college in your area.
To get things done, make a priority list with all tasks placed in the order of importance. That will help you get a clearer picture of what you are up against. Consider the important things you need to finish each day. Put these first on your list. Then, you can work your way down to what is least important.
Try to prepare yourself mentally for the tasks you must accomplish. Having the wrong mindset at the onset of a task will be a hindrance. Convince yourself to focus for a given time and then follow through.
To save money on transportation and some time, bundle your errands together. Do more than just make a stop at the supermarket by also combining your post office stop and your haircut as well. If picking your kids up at school is on the agenda, head out early and get another errand done too.
The Pomodoro method is a great approach to try. It says to work for 25 minutes, then take a 5 minute break. With that method, you'll be able to stop feeling that you have to work even harder. You can work when you want and move on for the day.
Think about what you want to accomplish in life. It usually is the case that if you would like to accomplish something, then time is not a problem. Find those activities you can cut from your list and think of what you desire. If you schedule time for the things you really want to do, you will be much happier.
Give yourself some time for wiggle room for any large project. Items like those can have unexpected issues pop up. These problems may delay the process and change the completion time you had scheduled. Be prepared for them by giving yourself a buffer.
Deadlines are a powerful incentive for managing your time better. If you have a firm deadline for a task, you will be more likely to get it done on time. So, you probably are going to find it useful to assign solid finish lines to any tasks you have that are currently devoid of them. When you set a firm completion date, you will become more productive with your tasks.
Tasks that can be handled quickly should be dealt with when they arise. Other than that, do it later. If something keeps happening repeatedly, make it a habit to take care of it immediately.
Give yourself rewards during the day when you finish a task. This will give you motivation and incentive. A little treat can go a long way in getting you to repeat your behavior.
Make yourself lists of what you need done when it's time for the day to be over. Include everything you need to accomplish and when you'd like to be done by. This helps you feel less anxious as you proceed through your day.
You can probably see that some people use effective time management plans. All you need is to get your feet wet and work towards positive results. Use the tips you went over here and things should easily fall into place for you!


After being so into nail art, sometimes it's hard for me to just go with a plain color. But we all need something really simple and understated once in awhile.
This is two coats of Essence Iced Strawberry (the name just makes me happy). This cream pink has a beige shimmer and is more of a frost finish. I'm often not a fan of the frost but liked how this turned out.
It chipped after two days but I was washing my hands a ton. A nice polish for an interview or for those non nail art friendly work places.
Its a great color on you!


The Waywayseecappo Wolverines have acquired 20-year-old Forward Trygve Many Guns from the OCN Blizzard in a three-way trade involving Peyton Malcolm heading to the SJHL's La Ronge Ice Wolves.
In 39 games last season with OCN, the Siksika Alberta native scored 11 goals and 15 assists while adding 145 penalty minutes. According to Wolverines Head Coach and General Manager, Taylor Harnett, Many Guns will fit it nicely with the group in Waywayseecappo.
There's no question Many Guns is off to a great start to the year. The 6-foot-3, 220-pound winger has scored 3 goals and 3 assists in only 4 games with OCN.
The Wolverines would like to thank Peyton Malcolm for his hard work and dedication during his time in Waywayseecappo and wish him the best with his new team moving forward.


Each class is 2 hours making a total of 28 hours.
Price:You can book modules as follows. Classes 13 and 14 are only available to people who have attended Classes 1- 12. All prices in Hong Kong dollars.
Fashion Show: Students who take both fashion figure drawing and collection building courses will be eligible to join ITS fashion show at the end of the year.
An advanced course to be completed after previous courses have been completed by the student, or they show proficiency to be demonstrated by portfolio of work. This course gives a detailed practical analysis of the fashion design process in building a collection to the student's desire (and links to sewing and patternmaking classes where students sew/execute their own collection). The course runs on a single project basis which takes the student from concept to collection presentation boards showing the fully designed collection. Final class will culminate in an in-depth critical analysis of design given by the facilitator in the presence of all students so that they may also understand critical feedback.
Designed for high school students who will be submitting a body of work for tertiary education, working professionals who wish to sharpen their existing portfolios or who are looking to change professions into fashion. This course will look at and develop skills for presenting design projects including technical flat drawings, theme boards, customer profile, and presentation boards.
Monday, Wednesday and Friday 10 am – 12 noon or 1-3 pm.
Or by appointment – please call to arrange other timings (conditions apply).
Students can choose to attend more than one session a week if they wish.
Please contact us if you are interetsed in doing a fashion portfolio development course or a fashion portfolio preparation course.


At CHK our goal is to help provide the best surgical recovery, arthritis management & sports conditioning possible for local pets. Swimming is an ideal form of rehabilitation for dogs recovering from surgery or injury. It's also beneficial for dogs with joint issues or arthritis. It's a low impact, non-weight bearing workout ideal for helping with weight loss.
Swimming incorporates stretching, aerobics and strength training. Our HydroWorx pool also has a warm water massage feature which can aid in increasing blood flow and decreasing tension & pain for your dog. Swimming in our warm water pool doesn't only help your dog physically, it's relaxing & fun and you will find improvement in your dog's spirit as well!
Emily Lewis is our swim coach. Prior to working here at CHK she was a surgical technician at several local veterinary hospitals. Through that experience she is familiar with post operative care for most surgical patients. Linette Poquette, the owner of Comfort Hill Kennel, is also an animal health technician. They work in partnership with a patient's veterinary team to provide the best care possible. They are able to follow prescriptions for desired length of swim, resistance, water massage and more.
Please call us at 802-877-8308 for more information. Pool time is available by appointment only.


text:
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Consumers warm to Celebrating Australian Made
@AuManufacturing continues our editorial series – Celebrating Australian Made – with some good news about public sentiment to making locally. New consumer research from Roy Morgan shows the preference for Australian-made goods continued to increase in 2020, with the huge majority of 93% of Australians stating they are more likely to buy products made in…
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Building products group CSR has boosted sales and profits across diverse construction markets, announcing full year ended March 31 net profit after tax up 17 per cent to $146.1 million. A slowdown in residential construction was more than made up through strong cost control and operational efficiency increases, with the company's building products segment reporting…
Crumbs for industry, funds for dead-end fossil fuels in 2021-22 budget
Analysis and Commentary, Manufacturing NewsBy Peter Roberts May 12, 2021 Leave a comment
By Peter Roberts Last year's $1.4 billion Manufacturing Modernisation fund – that is a paltry few hundred million a year – is essentially the sum total of the federal government's response to the question marks raised over the role of manufacturing by Covid-19 supply chain disruptions. There was almost nothing not previously announced, and only…
Budget's patent box scheme and an RDTI review
By Denham Sadler A limited patent box scheme will be launched by the federal government, offering tax breaks for medical and biotech innovations in an effort to boost commercialisation and retain IP in Australia. The federal budget, handed down by Treasurer Josh Frydenberg on Tuesday night, included the establishment of a patent box scheme targeted…
Budget to fund local mRNA vaccine manufacturing
By Denham Sadler The federal government has said it will develop an onshore mRNA vaccine manufacturing capability in Australia, but the actual amount of funding allocated to the significant task has been kept secret. In the 2021-22 federal budget, unveiled by Treasurer Josh Frydenberg on Tuesday night, the government confirmed that the industry department would…
Australian Good Design Awards nominations close on Friday
There are only a few days left to nominate for the 2021 Australian Good Design Awards, with entries closing on Friday May 14. The awards have been running since 1958, making them one of the longest-running international design awards, and are recognised by the World Design Organization (WDO) as Australia's peak international design endorsement and…
Quickstep announces alliance targeting post-pandemic airline bounce
ASX-listed composites business Quickstep has signed a non-binding MoU with Triumph Aviation Services Asia (TASA) covering Australasian aftermarket airline and other MRO customers. TASA and Quickstep's Aerospace Services division (QAS) intend to form a "strategic alliance" with a broad scope including "services such as MRO, spares pooling and repair development for the aero structures market…
Ellume establishes US diagnostics plant
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redpajama_set_name: RedPajamaCommonCrawl


The Judge's entry, "Debts in Wrong Places ," is a humorous take-off of Garth Brooks' country music hit song, "Friends in Low Places" — certainly a popular song in Texas, where Judge Trust practiced before ascending to the bench here in the Eastern District of New York.
Steven Horowitz and Gideon Kendall, creators of the Bankruptcy Bill  cartoon strip, sponsored the contest and came up with a great new cartoon featuring the comic likeness of Judge Trust that you absolutely must see!!! Click this link to see this new comic strip: Bankruptcy Bill Song Contest Cartoon .
When I had told my wife, Arlene, of the contest last fall, she said she had to enter — just to poke fun at me. Her song, "A Bankruptcy Wife's Lament ," is about a bankruptcy attorney who is so busy that he ignores his wife and family (me???).
My song, "Debt Free Girl ," a take-off on Billy Joel's hit, "Uptown Girl,"didn't win a prize, but I still think it's great fun in any event.


Click on any image below to find out more details and prices.
"Your festoon lights were invaluable in our dark field and added a magical touch to the evening, and the tree uplighters created a real wow factor. Thanks for all your help."
"Thank you for all your help and advice, the fairy lights were perfect for inside the marquee and as you said, they provided just the right amount of light for the evening. Thank you again."


Early on into our move, I found a fixer-upper campaign piece for sale on Facebook. We hadn't unpacked everything and my workshop was nothing more than a junk drawer for empty boxes and miscellaneous curtain rods. But the price was right (and it was an easy pick up on my way home) so I brought it home without a plan.
Since the paint color from our inspiration piece was a custom mix, I got the paint color-matched to finally get a name: True Taupewood by Behr.
It's a warm shade of greige that reminds me of wet potters clay.
This color is super moody when it wants to be.
Like these brass recessed pulls.
Despite what it looks like a first glance, those pulls aren't on individual drawers - they are actually on a pair of doors that open up into a cabinet.
This little guy was shipped off to it's new home in Richmond. But, if you are interested in a custom order like it, email me at cate@stylemutthome.com.


text:
Q: Instaling ThingsBoard on Docker with PostreSql So I'm trying to install ThingsBoard using Docker on a MacBook Pro.
I'm following the instrutions on (https://hub.docker.com/r/thingsboard/tb-postgres/)
After the instalation I need to configure the Docker image of thingsboard to use an external database from PostgreSql.
Anyone who can help me how to do it? I read some documentation and they say to change the .env and tb.env file...but can't find these files on my disk.

A: The image you reference has a preinstalled Postgres database inside, so no need for external one.
If you want separate container with database, you can follow instructions on the tool's git repository. The .env file is located there and this is what they mean by the .env file to edit, to choose which database should be deployed.
Other than that, to connect this container to a standalone database, you'd have to tweak it's internal configuration.

redpajama_set_name: RedPajamaStackExchange


Join us for a tour of all things EBSCOhost. Be prepared to cover a lot of ground! There is a wealth of resources available here.
Go to https://ebscotraining.webex.com/ebscotraining/k2/j.php?MTID=t14422e81c3d322792ea30b2fc9c44a9d and register.


Button Moon is a multi shot cake by Sky Crafter.
Cake / Barrage from Sky Crafter Fireworks called Button Moon. Blue pearls rising to pastel colour bursts with silver glitter. Beautiful. Contact the Online Fireworks team if you have any questions on Button Moon by Sky Crafter Fireworks.


text:
Q: How can I process a tarfile with a Python multiprocessing pool? I'm trying to process the contents of a tarfile using multiprocessing.Pool.  I'm able to successfully use the ThreadPool implementation within the multiprocessing module, but would like to be able to use processes instead of threads as it would possibly be faster and eliminate some changes made for Matplotlib to handle the multithreaded environment.  I'm getting an error that I suspect is related to processes not sharing address space, but I'm not sure how to fix it:
Traceback (most recent call last):
  File "test_tarfile.py", line 32, in <module>
    test_multiproc()
  File "test_tarfile.py", line 24, in test_multiproc
    pool.map(read_file, files)
  File "/ldata/whitcomb/epd-7.1-2-rh5-x86_64/lib/python2.7/multiprocessing/pool.py", line 225, in map
    return self.map_async(func, iterable, chunksize).get()
  File "/ldata/whitcomb/epd-7.1-2-rh5-x86_64/lib/python2.7/multiprocessing/pool.py", line 522, in get
    raise self._value
ValueError: I/O operation on closed file

The actual program is more complicated, but this is an example of what I'm doing that reproduces the error:
from multiprocessing.pool import ThreadPool, Pool
import StringIO
import tarfile

def write_tar():
    tar = tarfile.open('test.tar', 'w')
    contents = 'line1'
    info = tarfile.TarInfo('file1.txt')
    info.size = len(contents)
    tar.addfile(info, StringIO.StringIO(contents))
    tar.close()

def test_multithread():
    tar   = tarfile.open('test.tar')
    files = [tar.extractfile(member) for member in tar.getmembers()]
    pool  = ThreadPool(processes=1)
    pool.map(read_file, files)
    tar.close()

def test_multiproc():
    tar   = tarfile.open('test.tar')
    files = [tar.extractfile(member) for member in tar.getmembers()]
    pool  = Pool(processes=1)
    pool.map(read_file, files)
    tar.close()

def read_file(f):
    print f.read()

write_tar()
test_multithread()
test_multiproc()

I suspect that the something's wrong when the TarInfo object is passed into the other process but the parent TarFile is not, but I'm not sure how to fix it in the multiprocess case.  Can I do this without having to extract files from the tarball and write them to disk?

A: You're not passing a TarInfo object into the other process, you're passing the result of tar.extractfile(member) into the other process where member is a TarInfo object. The extractfile(...) method returns a file-like object which has, among other things, a read() method which operates upon the original tar file you opened with tar = tarfile.open('test.tar').
However, you can't use an open file from one process in another process, you have to re-open the file. I replaced your test_multiproc() with this:
def test_multiproc():
    tar   = tarfile.open('test.tar')
    files = [name for name in tar.getnames()]
    pool  = Pool(processes=1)
    result = pool.map(read_file2, files)
    tar.close()

And added this:
def read_file2(name):
    t2 = tarfile.open('test.tar')
    print t2.extractfile(name).read()
    t2.close()

and was able to get your code working.


text:
President Ho Chi Minh's birthday marked in Algeria, Russia
The Vietnamese Embassy in Algeria on May 19 held a ceremony to mark the 127th birth anniversary of the late President Ho Chi Minh (May 19).
Activities in Singapore, Mexico mark late President's birthday
Argentine media spotlights late President Ho Chi Minh
Exhibition spotlights untold stories about President Ho Chi Minh
Vietnamese Ambassador Pham Quoc Tru speaks at the ceremony (Source: VNA)
Hanoi (VNA) - The Vietnamese Embassy in Algeria on May 19 held a ceremony to mark the 127th birth anniversary of the late President Ho Chi Minh (May 19).
Addressing the event, Vietnamese Ambassador to Algeria Pham Quoc Tru briefed participants on the life and career of the late leader as well as his role and influence in the struggle for national liberation and in international workers' movements.
President Ho Chi Minh has continuously been an example for all Vietnamese people to follow, the ambassador said.
He called on the Vietnamese community in Algeria to learn and follow the moral example of President Ho Chi Minh, upholding traditions to build Vietnam into a country with civilisation, prosperity, peace and stability in the new context in the international and regional arena.
Earlier, Ambassador Tru and the overseas Vietnamese offered incense to the President at his alter at the Embassy.
The same day, staff of the Vietnamese Embassy in Russia staff laid flowers at the President Ho Chi Minh statue Moscow to mark the late leader's 127th birthday.
The event drew representatives from Vietnamese representative offices, the Vietnamese community, the Vietnamese business association in Russia, the Vietnam – Russia Friendship Association, and the Russia Veterans Association, among others.-VNA
President Ho Chi Minh 127th birthday Algeria Russia Ambassador Pham Quoc Tru moral example Moscow Related stories Russia Related stories Algeria
Ho Chi Minh City has been ranked fifth as Asia Pacific investors' most preferred market for investments by CBRE Group, Inc.
Da Nang treasures contributions of foreigners: city leader
Da Nang has appreciated the contributions and accompany of foreigners in the city in a tough year of 2020, and hoped to continue to receive more assistance from foreign individuals, organisations and businesses during the recovery process in the city and the central region as a whole, said Chairman of the municipal People's Committee Le Trung Chinh.


text:
Q: Crystal Reports: Use Sub-Report parameter values in Main-Report I have a Sub-Report that takes 2 date parameters (Start and End date)
A Main-Report that has Sub-report nested in it. When I refresh the report, I get a prompt to enter the dates.
All I'm trying to do is display those dates in (start date - end date) format on Main-Report.
Here's what I tried.
Created a formula (SharedStartDate) in Sub-Report as
Shared StringVar startDate :=  ToText({?@start_DT},"MM/dd/yyyy");
startDate;

Created another formula in Main-Report with the same name (SharedStartDate)
Shared StringVar startDate;
startDate

When I use this formula in the main-report, I get empty text.
I'm not sure what Am i doing wrong, I'm pretty new working with CR
Thank you guys in advance for any suggestions.

A: Take care of few things while passing the value to main report.

*

*Formula need to be placed on the sub report before passing the value.

*When value is used in main report then the resulting formula should be kept in coming sections in the design.. not in the preceeding sections then you won't get the value.

*one more option would be (not suggested)since you are manuplating parameters then link the parameters from mainreport to subreport and use that parameter at any section irrespective of sub report location in report.


You Don't Need a Bailout -- You Need Social Media!
FACT: 25% of Ford's marketing spending is on Digital/Social Media.
FACT: Ford is the ONLY U.S. auto company that didn't take a government loan.
Are you new to blogging? New to social media? Welcome aboard. In planning for 2010 and beyond, you're going to want to increase your use of these tools.
If you've been around the internet in the last 6 months, you've watched the video Social Media Revolution.
What you might not have seen is the equally impressive sequel: Social Media ROI.
This video below is a must watch. No matter what business you are in, for 2010, say NO to bailouts! Say YES to Social Media!
you can visit my main Northern Virginia & D.C. real estate website at www.brianblock.com.
ExpressCopy. com | Design. Print. Mail. All Online!
Should I Have My House Appraised Prior To The Listing?
Add a splash of GOOGLE JUICE to your Blog!


text:
UHS student on a mission to end hunger
Katie Strawbridge, Reporter
Despite living in a modern era, poverty and hunger plague the world. 1 in 9 people do not have enough to eat, causing 3 million young lives to be lost every year. This means that every 10 seconds, a child dies of hunger-related disease. While these numbers are daunting, Carolyn Pascal, a senior at Urbana High School, believes that something can be done to begin to solve this situation.
For a very long time, Pascal has been aware and active about these issues, stating that she had been concerned about the hungry since she was in middle school. "When I was 12 years old, I began holding community outreach events at local grocery stores. Over the next three years, I hosted 20 outreach events and collected more than 3,600 items," Pascal said. Even after all of these efforts, she knew that there was still more to be done to help.
"I founded Educate Girls to Eradicate Hunger to inform the public of the power of providing secondary education to girls in developing nations and to raise funds to educate girls," Pascal said about her organization, which has, to date, provided 71 girls in Malawi with tuition, room and board, uniforms and more for an entire year. Its message has also spread to thousands of people after only a year after its founding, and continues to do so.
Pascal chose to help Malawi, a country in Africa, since it is one of the poorest nations in the world. Only 7% of girls attend secondary school there. Since families must pay for their children to be sent to schools, Pascal explained, "families opt to educate their sons, leaving daughters trapped in the cycle of generational poverty." She chose the first school that she has helped through the UNICEF KIND project, which has a subcomponent focusing exclusively on educating girls.
Keeping teen girls in school in impoverished countries such as Malawi may end the cycle of poverty in a generation, making girls' education a better method for eradicating hunger rather than simply increasing food production. Educating girls greatly reduces their risk of contracting HIV, dying during childbirth, and suffering a host of hunger-related diseases, many of which are fatal to children, as well as becoming child brides and having more children than they can afford. Girls who receive a secondary education also tend to marry later, have fewer children, earn higher wages, and support healthier families. In Pascal's words, "an educated girl uplifts her entire family from poverty."
Knowing this, Pascal became determined to inform the public about this topic and raise funds to educate girls, all while still being in high school herself. She has held many fundraisers, including a mailing campaign, an email campaign, a book fair, and most notably, the annual Educate Girls Superhero 5K with 1K Fun Run. The inaugural race in 2018 was held at Monocacy Village Park, and the 2019 race was held at Urbana District Park. Both races engaged over 100 runners and volunteers.
Pascal has also started the Indigenous Peoples Literature Society, after learning how imperialism and colonization has stripped many people of their indigenous cultures and identities. Through this society, literature by indigenous authors is discussed at many book clubs and events. Some places where these discussions are held include the Urbana Regional Library, Barenes and Noble, the Frederick Distance Festival the SheLEADS Youth Summit, and the Educate Girls Superhero 5k race. With this platform, 1,964 people have been exposed to the voices of indigenous authors.
As for the future, Pascal will be going to Bowdoin College, where she plans on studying economics. She hopes to be able to eventually study abroad, where she can learn more about different cultures. Inspired through her experience of learning about the impoverished and hungry, Pascal plans on going to law school to fight for the underrepresented.
Learn more about Pascal's organization at: EducateGirlstoEradicateHunger.org


text:
Imprint: Puffin
Format: Board Book
Flip Flap Spot
Hill, Eric
An exciting new Spot format featuring 100 fancydress outfits for little hands to mix and match.
Who does Spot want to be today? A cowboy? A spaceman? A pirate? A tiger?
Or maybe a mismatch of them all . . .
Join Spot as he dresses up in lots of different fancy-dress outfits, from a goal-scoring wizard to a magical cowboy! This flip-flap book allows for a crazy array of silly and mismatched outfits, and is perfect for little hands to play with.
This fantastically fun little book flips and flaps to create 100 different Spot outfits!
Pub date: 30 October 2017
Eric Hill (1927-2014) was born in London and attended Pooles Park Elementary and Tollington Park Central schools until the outbreak of World War Two. Prior to the outbreak of hostilities, 12 year-old Eric was evacuated to the small village of Bluntisham in Huntingdonshire. But he returned to London for Christmas and stayed there for the rest of the war. Eric left school at fifteen to start his first job with a shipping office in the City. The following year he joined an art studio as a messenger. This was a bigger break than he had imagined, because at the studio was a well-known Austrian cartoonist, named Tim, who encouraged him to start drawing cartoons. Gradually Eric became more of an art assistant than a messenger and spent more time at the drawing board under Tim's guidance. Tim was a superb artist and cartoonist and later had his own BBC TV show for children. After a stint with the Royal Air Force in Germany, Eric Hill returned to the studio again. His cartooning work began to be published in national magazines and newspapers. He started a weekly strip for the London magazine Illustrated and did small story sketches for Lilliput. After a year of drawing cartoons, Eric Hill left the studio and joined the art department of an advertising agency as a visualiser. The next important step was an appointment with F.H.K Henrion at Erwin Wasey, a large advertising agency in Park Lane. In 1976, Eric's son Christopher was born. When Christopher was two, Eric made up a story to read to him at bedtime about a small puppy. The extraordinary success of Where's Spot?, published in 1980, gave Eric Hill the confidence to work full-time as an author. The second title, Spot's first Walk, was published a year later. The publishing phenomenon that is Spot was born! There are over 40 million books in print read by children in more than 100 countries. The adventures of Spot have been translated into 65 languages, including minority languages such as Inupiaq, Occitan, Frisian and Farosese. Spot has different names around the world including Spotty (Italy), Dribbel (the Netherlands) and Fleck (Germany)! The animated TV series has aired in over 80 countries and the films have been included in the permanent collection of the prestigious Museum of Television and Radio History in New York as an outstanding example of British children s programming. A study by UK-based Millard Brown reports that Spot has 99% awareness amongst UK parents and 83% amongst children.
In 2008 Eric was awarded an OBE for services to children's literature.
The Very Hungry Caterpillar: Easter Colours
I Just Couldn't Wait to Meet You
Kate Ritchie, Hannah Sommerville
Bluey: Time to Play!
Where's Spot?
Mem Fox, Freya Blackwood
The Wonderful Things You Will Be
Emily Winfield Martin
No One Likes a Fart
Zoe Foster Blake, Adam Nickel
Peppa Pig: Peppa Visits the Great Barrier Reef
Anna Walker
LGB The Poky Little Puppy
Janette Sebring Lowrey
LGB The Little Red Caboose
Marian Potter
LGB The Fire Engine Book
Tibor Gergely
LGB The Tale Of Peter Rabbit
All About Spot
LGB The Saggy Baggy Elephant
Byron Jackson
Roo Knows Blue
Renee Treml
LGB The Little Red Hen
J.P. Miller
LGB I'm A T. Rex!
Dennis Shealy


text:
Aerospace, Archives, Energy and Power
OBJECTS OF ENGINEERING-Sabre model
The Sabre model can be seen at IMechE, made by the company it is a detailed example of an engine model.
Napier Sabre V (series VII) aero engine, 1946, scale model. From Napier's Archive at IMechE.
D Napier & Son had been working on large aero engines since the end of World War I. The Lion, in modified form, had powered several of the Supermarine Schneider Trophy competitors and several land speed record cars. However, by the late 1920s, work had started on replacements.
Gloster VI seaplane with Napier Lion VIID engine, winner of the 1929 Schneider Trophy. From Napier's Archive at IMechE.
Two H-block designs, the H-16 Rapier and the H-24 Dagger were then produced. The H-block consisted of two horizontally opposed engines, lying one atop or beside another. In 1927, Harry Ricardo published a study on the concept of the sleeve valve engine. He argued that traditional poppet valve engines would be unlikely to produce much more than 1,500 hp; to pass this limit a sleeve valve would have to be used, to increase volumetric efficiency. Ricardo started work with Bristol Engines on sleeve-valve designs and Major Frank Halford started work with Napier.
Halford and Napier used the opposed H layout of the Dagger but the new design positioned the cylinder blocks horizontally so to use sleeve valves. All of the accessories were grouped above and below the cylinder blocks, rather than being at the front and rear of the engine as in most contemporary designs.
Sabre crankshaft. Pt No 19800/1. From Napier's Archive at IMechE.
The first Sabre engines were ready for testing in January 1938, limited to 1,350 hp. By March, they were passing tests at 2,050 hp and by June 1940, when the Sabre passed the Air Ministry's 100-hour test, the first production versions were delivering 2,200 hp. By the end of the year, they were producing 2,400 hp – the 1940 Rolls-Royce Merlin II was generating just over 1,000 hp.
The first operational aircraft to be powered by Sabre were the Hawker Typhoon and Hawker Tempest. Other aircraft using the Sabre were, early prototype and production variants of the Blackburn Firebrand, the Martin-Baker MB 3 and a Hawker Fury. However, the introduction of jet engines after World War II led to the quick demise of the Sabre – there was less need for high power military piston aero engines. Napier turned to turboprop engines instead.
Napier's Archive is at IMechE, email us for information.
Archives, Institution of Mechanical Engineers
imechearchive January 21, 2019 February 14, 2019 Aeronautical, Aviation, Napier
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Dimensions: 16.5" x 14" x 7.75"
You don't need to be a doctor to carry this well designed Jack Georges Voyager classic leather doctors bag! It has a steel frame opening for easy access to everything inside. The Jack Georges doctors bag has a heavy duty leather carry handle and a removable adjustable shoulder strap and you can also slip it over the trolley handle of your rolling luggage securely. There are metal feet on the bottom of the Jack Georges leather doctors bag! We include FREE shipping and a FREE leather bag tag with your initials monogrammed!