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A generalized Lyapunov-type inequality in the frame of conformable derivatives
Thabet Abdeljawad1,
Jehad Alzabut1 &
Fahd Jarad2
We prove a generalized Lyapunov-type inequality for a conformable boundary value problem (BVP) of order \(\alpha \in (1,2]\). Indeed, it is shown that if the boundary value problem
$$ \bigl(\textbf{T}_{\alpha }^{c} x\bigr) (t)+r(t)x(t)=0,\quad t \in (c,d), x(c)=x(d)=0 $$
has a nontrivial solution, where r is a real-valued continuous function on \([c,d]\), then
$$ \int_{c}^{d} \bigl\vert r(t) \bigr\vert \,dt> \frac{\alpha^{\alpha }}{(\alpha -1)^{\alpha -1}(d-c)^{ \alpha -1}}. $$
Moreover, a Lyapunov type inequality of the form
$$ \int_{c}^{d} \bigl\vert r(t) \bigr\vert \,dt> \frac{3\alpha -1}{(d-c)^{2\alpha -1}} \biggl( \frac{3 \alpha -1}{2\alpha -1} \biggr) ^{\frac{2\alpha -1}{\alpha }},\quad \frac{1}{2}< \alpha \leq 1, $$
is obtained for a sequential conformable BVP. Some examples are given and an application to conformable Sturm-Liouville eigenvalue problem is analyzed.
In [1], it was proved that if the boundary value problem (BVP)
$$ x^{\prime \prime }(t)+r(t) x(t)=0,\quad t \in (c,d), x(c)=x(d)=0, $$
has a nontrivial solution, where r is a real-valued continuous function, then
$$ \int_{c}^{d} \bigl\vert r(s) \bigr\vert \,ds> \frac{4}{d-c}. $$
Undoubtedly, the Lyapunov inequality (3) proves cooperative and supportive in differential equations. Indeed, it is frequently used to dominate certain quantities for the sake of proving qualitative properties of solutions for differential equations. The Lyapunov inequality has been extensively under consideration for integer-order differential equations. As fractional differential equations become of high interest due to their demonstrated applications [2], nevertheless, the last few years have witnessed the appearance of many papers which systematically studied the fractional analog of this inequality and other types of inequalities; we refer the reader to [3–11] for Lypanouv type and to [12] for Gronwall type. The newly defined conformable fractional calculus was initiated in [13] and studied later on by the current author in [14, 15], where many properties of conformable operators were introduced. However, the progress in this direction is still at its earliest stage [16–18]. In this paper, we prove a Lyapunov-type inequality for a conformable fractional BVP. Moreover, a Lyapunov type inequality is obtained for a sequential conformable BVP. Some examples are given and an application to conformable Sturm-Liouville eigenvalue problem is analyzed. The new obtained inequalities generalize the existing ones.
Preliminaries on conformable derivatives
This section is devoted to the presentation of some preliminaries about higher order fractional conformable derivatives developed in [14].
([13, 14])
The (left) conformable fractional derivative starting from c of a function \(f:[c,\infty )\rightarrow \mathbb{R}\) of order \(0< \alpha \leq 1\) is defined by
$$ \bigl(T_{\alpha }^{c} g\bigr) (t)= \lim _{\epsilon \rightarrow 0} \frac{g(t+ \epsilon (t-c)^{1-\alpha } )-g(t)}{\epsilon }. $$
In the case when \(c=0\), we write \(T_{\alpha }\). If \((T_{\alpha }^{c} g)(t)\) exists on \((c,d)\) then \((T_{\alpha }^{c} g)(c) = \lim_{t \rightarrow c^{+}}(T_{\alpha }^{c} g)(t)\).
Note that if g is differentiable then
$$ \bigl(T_{\alpha }^{c} g\bigr) (t)= (t-c)^{1-\alpha } g^{\prime }(t). $$
Moreover, the conformable fractional integral of order \(0<\alpha \leq 1\) starting at \(c\geq 0\) is defined by \((I_{\alpha }^{c} g)(t)= \int_{c}^{t}g(x)(x-c)^{\alpha -1}\,dx\) and following [13] by \((I_{\alpha }^{c} g)(t)=\int_{c}^{t}g(x)x^{\alpha -1}\,dx\).
In the case of higher order, the following definition becomes true.
([14])
Let \(n<\alpha \leq n+1 \) and set \(\gamma = \alpha -n\). Then the conformable fractional derivative starting from c of a function \(g:[c,\infty )\rightarrow \mathbb{R}\) of order α, where \(g^{(n)}(t)\) exists, is defined by
$$ \bigl(\textbf{T}_{\alpha }^{c} g\bigr) (t)= \bigl(T_{\gamma }^{c} g^{(n)}\bigr) (t). $$
In the case \(c=0\), we write \(\textbf{T}_{\alpha }\).
Note that if \(\alpha =n+1\) then \(\gamma =1\) and the fractional derivative of g becomes \(g^{(n+1)}(t)\). Also when \(n=0\) (or \(\alpha \in (0,1)\)) then \(\gamma =\alpha \) and the definition coincides with that in Definition 1. From (6), it is immediate that if \(n< \alpha \leq n+1\) then \(\gamma =\alpha -n\) and if moreover, the \((n+1)\)th derivative (or the derivative of \(g^{(n)}\)) exists then we have
$$ \bigl( \textbf{T}_{\alpha }^{c} g\bigr) (t)= \bigl(T_{c}^{\gamma }g^{(n)}\bigr) (t)=(t-c)^{1- \gamma } g^{(n+1)}(t)=(t-c)^{1-\alpha +n}g^{(n+1)}(t). $$
Lemma 1
Assume that \(g:[c,\infty )\rightarrow \mathbb{R}\) is continuous and \(0< \alpha \leq 1\). Then, for all \(t>c\), we have
$$ T_{\alpha }^{c} I_{\alpha }^{c} g(t)=g(t). $$
In the case of higher order, the following definition is valid.
Let \(\alpha \in (n,n+1]\) and set \(\gamma = \alpha -n\). Then the left conformable fractional integral starting at c of order α is defined by
$$ \bigl(I_{\alpha }^{c} g \bigr) (t)= \textbf{I}_{n+1}^{c} \bigl((t-c)^{\gamma -1}g\bigr)= \frac{1}{n!} \int_{c}^{t} (t-x)^{n}(x-c)^{\gamma -1}g(x)\,dx. $$
Notice that if \(\alpha =n+1\) then \(\gamma =1\) and hence \((I_{\alpha } ^{c} g)(t)=(\textbf{I}_{n+1}^{c} g)(t)=\frac{1}{n!}\int_{c}^{t} (t-x)^{n} g(x)\,dx\), which is the iterative integral of g, \(n+1\) times over \((c,t]\).
Recall that the left Riemann-Liouville fractional integral of order \(\alpha >0\) starting from c is defined by
$$ \bigl(_{c}\textbf{I}^{\alpha }g\bigr) (t)= \frac{1}{\Gamma (\alpha )} \int_{c} ^{t} (t-s)^{\alpha -1} g(s)\,ds. $$
We see that \((I_{\alpha }^{c} g)(t)= ({}_{c}\textbf{I}^{\alpha }g)(t)\) for \(\alpha = n+1\), \(n=0,1,2,\ldots \) .
In virtue of [14], we recall that \(({}_{c}\textbf{I}^{\alpha }(t-c)^{\mu -1})(x) =\frac{\Gamma (\mu )}{\Gamma (\mu +\alpha)} (x-c)^{\alpha +\mu -1}\), \(\alpha , \mu >0\). Indeed, if \(\mu \in \mathbb{R}\) such that \(\alpha +\mu -n>0\) then the conformable fractional integral of \((t-c)^{\mu }\) of order \(\alpha \in (n,n+1]\) is
$$ \bigl(I_{\alpha }^{c} (t-c)^{\mu }\bigr) (x)=\bigl(\textbf{I}_{n+1}^{c} (t-c)^{\mu + \alpha -n-1}\bigr) (x)= \frac{\Gamma (\alpha +\mu -n)}{\Gamma (\alpha + \mu +1)} (x-c)^{\alpha +\mu }. $$
The following is a generalization of Lemma 1.
Assume that \(f:[c,\infty )\rightarrow \mathbb{R}\) such that \(g^{(n)}(t)\) is continuous and \(n< \alpha \leq n+1\). Then, for all \(t>c\), we have
$$ \textbf{T}_{\alpha }^{c} I_{\alpha }^{c} g(t)=g(t). $$
Let \(\alpha \in (n,n+1] \) and \(g:[c,\infty ) \rightarrow \mathbb{R}\) be \((n+1)\) times differentiable for \(t>c\). Then, for all \(t>c\), we have
$$ I_{\alpha }^{c} \textbf{T}_{\alpha }^{c}(g) (t)= g(t)-\sum_{k=0}^{n} \frac{g ^{(k)}(c)(t-c)^{k}}{k!}. $$
From [14], we recall that the solution of the following conformable fractional initial value problem:
$$ \bigl(T_{\alpha }^{c} x\bigr) (t)= \lambda x(t),\quad 0< \alpha \leq 1, x(c)=x_{0}, t>c, $$
is \(x(t)=x_{0} e^{\lambda \frac{(t-c)^{\alpha }}{\alpha }}\).
The following example shows why it is useful to work in conformable differential systems.
Consider the following second order ordinary differential equation:
$$ t^{2-2\alpha } y^{\prime \prime }+t^{1-\alpha }\bigl((1-\alpha )t^{-\alpha }-2\bigr)y^{\prime }-3y=0, \quad 0< \alpha \leq 1, $$
where \(y(t)\) is assumed to have second order continuous derivative. In view of (13), one can figure out that it is not easy to solve it using standard methods. However, equation (13) is rewritten in the form of the sequential (local) conformable differential equation as
$$ y^{(2\alpha )}-2y^{(\alpha )}-3y=0, $$
where \(y^{(2\alpha )}=t^{1-\alpha }(t^{1-\alpha }y^{\prime })^{\prime }= t^{1-\alpha }[t^{1-\alpha }y^{\prime \prime }+(1-\alpha )t^{- \alpha }y^{\prime }]\) such that \(y^{(2\alpha )}\) stands for \((T_{\alpha }^{0} T_{\alpha }^{0} y)(t)\) and \(T_{\alpha }^{0} \) is defined in (5). By using the operator method, we may write (14) in the form
$$ \bigl(D^{\alpha }-3\bigr) \bigl(D^{\alpha }+1\bigr)y=0. $$
The solution of the above equation has the form
$$ y(t)=c_{1}e^{3 \frac{t^{\alpha }}{\alpha }}+c_{2} e^{- \frac{t^{ \alpha }}{\alpha }}. $$
For \(\alpha = \frac{1}{2}\), (13) reduces to \(ty^{\prime \prime }+ (\frac{1}{2}-2\sqrt{t})y^{\prime }-3y=0\), which by the above arguments has a solution \(y(t)=c_{1}e^{6 \sqrt{t}}+c_{2} e^{- 2 \sqrt{t}}\). It is worth mentioning that, for \(\alpha =1\), the solution \(y(t)=c_{1}e^{3t}+c_{2} e^{-t}\) of the ordinary differential equation \(y^{\prime \prime }-2y^{\prime }-3y=0\) is verified.
A Lyapunov-type inequality for a conformable BVP
Consider the following (local) conformable BVP:
$$ \bigl(\textbf{T}_{\alpha }^{c} x\bigr) (t)+r(t)x(t)=0, \quad 1< \alpha \leq 2, c< t< d, x(c)=x(d)=0. $$
\(x(t)\) is a solution of the BVP (15) if and only if it satisfies the integral equation
$$ x(t)= \int_{c}^{d} H(t,s) r(s) x(s)\,ds, $$
where H is the Green function for (15) defined by
$$ H(t,s)= \textstyle\begin{cases} \frac{(t-c) (d-s)}{d-c}\cdot (s-c)^{ \alpha -2}, & c \leq t \leq s\leq d, \\ ( \frac{(t-c) (d-s)}{d-c}-(t-s) )\cdot (s-c)^{ \alpha -2}, & c \leq s \leq t\leq d. \end{cases} $$
Applying the integral \(I_{\alpha }^{c}\) to (15) and making use of Definition 3 and Theorem 1 with \(n=1\) and \(\gamma =\alpha -1\), we obtain
$$ x(t)=c_{1}+c_{2} (t-c)- \int_{c}^{t} (t-s)r(s) x(s) (s-c)^{\alpha -2}\,ds. $$
The condition \(x(c)=0\) implies that \(c_{1}=0\) and the condition \(x(d)=0\) implies that \(c_{2}=\frac{1}{d-c} \int_{c}^{d} (d-s)r(s)x(s) (s-c)^{\alpha -2}\,ds\) and hence
$$ x(t)=\frac{t-c}{d-c} \int_{c}^{d} (d-s)r(s)x(s) (s-c)^{\alpha -2}\,ds- \int_{c}^{t} (t-s)r(s) x(s) (s-c)^{\alpha -2}\,ds. $$
Then, using
$$\begin{aligned}& \int_{c}^{d} (d-s)r(s) x(s) (s-c)^{\alpha -2}\,ds \\& \quad = \int_{c}^{t} (d-s)r(s) x(s) (s-c)^{\alpha -2}\,ds+ \int_{t}^{d} (d-s)r(s) x(s) (s-c)^{\alpha -2}\,ds, \end{aligned}$$
the proof is concluded. □
The Green function H defined above has the following properties:
\(H(t,s)\geq 0\) for all \(c\leq t,s \leq d\).
\(\max_{t \in [c,d]} H(t,s)=H(s,s)\) for \(s \in [c,d]\).
\(H(s,s)\) has a unique maximum, given by
$$ \max_{s \in [c,d]} H(s,s)= H \biggl(\frac{c+(\alpha -1)d}{\alpha }, \frac{c+( \alpha -1)d}{\alpha } \biggr)=\frac{(d-c)^{\alpha -1}(\alpha -1)^{\alpha -1}}{\alpha^{\alpha }}. $$
Define the functions \(h_{1}(t,s)=\frac{(t-c) (d-s)}{d-c}\cdot (s-c)^{ \alpha -2}\) and \(h_{2}(t,s)= ( \frac{(t-c) (d-s)}{d-c}-(t-s) )\cdot (s-c)^{ \alpha -2}\).
It is clear that \(h_{1}\geq 0\). To determine the sign of \(h_{2}\), we observe that \((t-s)=\frac{t-c}{d-c} (d- (c+ \frac{(s-c)(d-c)}{(t-c)}) )\) and that \(c+\frac{(s-c)(d-c)}{(t-c)} \geq s\) if and only if \(s\geq c\). Together with \((s-c)^{ \alpha -2} \geq 0 \) we conclude that \(h_{2}\geq 0\) as well. Hence, the proof of the first part is complete.
It is clear that \(h_{1}(t,s)\) is an increasing function in t. Differentiating \(h_{2}\) with respect to t for every fixed s and following similar analysis as in first part we conclude that \(h_{2}\) is a decreasing function.
Let \(g(s)=H(s,s)=\frac{(s-c)^{\alpha -1}(d-s)}{d-c}\). Then it is sufficient to show that \(g^{\prime }(s)=0\) if \(s=\frac{c+( \alpha -1)d}{\alpha }\) and hence the proof is completed.
If the BVP (15) has a nontrivial solution, where r is a real-valued continuous function on \([c,d]\), then
Let \(x \in Y=C[c,d]\) be a nontrivial solution of the BVP (15), where \(\Vert x \Vert = \sup_{t \in [c,d]}\vert x(t) \vert \). By Lemma 3, x must satisfy
$$ x(t)= \int_{c}^{d} H(t,s) r(s)x(s)\,ds. $$
Therefore
$$ \Vert x \Vert \leq \max_{t \in [c,d]} \int_{c}^{d} \bigl\vert H(t,s)r(s) \bigr\vert \,ds \Vert x \Vert $$
$$ \max_{t \in [c,d]} \int_{c}^{d} \bigl\vert H(t,s)r(s) \bigr\vert \,ds \ge 1. $$
By using the properties of the Green function H proved in Lemma 4, we end up with
$$ \frac{(d-c)^{\alpha -1}(\alpha -1)^{\alpha -1}}{\alpha^{\alpha }} \int _{c}^{d} \bigl\vert r(s) \bigr\vert \,ds >1. $$
Inequality (17) is an immediate conclusion of (18). □
If \(\alpha =2\), then (17) reduces to the classical Lyapunov inequality (3). We also invite the reader to compare the obtained generalized Lyapunov inequality in Theorem 2 and the one obtained recently and independently in [19]. The approach in [19] is different and the authors there proved the existence of solution in the space \(AC^{2}[c,d]=\{u \in C^{1}[c,d]: u^{\prime }\in AC[c,d]\}\). Further, we see that our obtained inequality provides, for example when applied to the Sturm-Liouville eigenvalue problem, sharper lower estimate for the eigenvalues. Indeed, in Section 5 we can see that the lower estimate \(\frac{\alpha^{\alpha }}{(\alpha -1)^{\alpha -1}}\) is bigger than \(4 (\alpha -1)\) for \(1<\alpha \leq 2\). This is due to the observation that \(( \frac{\alpha }{\alpha -1} ) ^{\alpha }\geq 4\), for \(1<\alpha \leq 2\). Further, for convenience, in the next section we prove a sequential type Lyapunov inequality version as well.
A Lyapunov-type inequality for a sequential conformable BVP
Consider the following sequential conformable BVP:
$$ x^{(2\alpha )}(t)+r(t)x(t)=0,\quad c< t< d,\frac{1}{2}< \alpha \leq 1, x(c)=x(d)=0. $$
We shall prove a Lyapunov inequality for (19).
$$ x(t)= \int_{c}^{d} G(t,s) r(s) x(s)\,ds, $$
where G is the Green function of (19) defined by
$$ G(t,s)= \textstyle\begin{cases} g_{1}(t,s), & c \leq s \leq t\leq d, \\ g_{2}(t,s), & c \leq t \leq s\leq d, \end{cases} $$
such that
$$ g_{1}(t,s)=(s-c)^{\alpha -1} \biggl[ \frac{(s-c)^{\alpha }}{\alpha }- \frac{(t-c)^{ \alpha }(s-c)^{\alpha }}{\alpha (d-c)^{\alpha }} \biggr] $$
$$ g_{2}(t,s)=(s-c)^{\alpha -1} \biggl[ \frac{(t-c)^{\alpha }}{\alpha }- \frac{(t-c)^{ \alpha }(s-c)^{\alpha }}{\alpha (d-c)^{\alpha }} \biggr] . $$
Applying the integral \(I_{\alpha }^{c}\) twice to (19) and using Definition 3, we get
$$ y(t)=c_{1}+c_{2} \frac{(t-c)^{\alpha }}{\alpha }- \int_{c}^{d} r(s) x(s) (s-c)^{\alpha -1} \biggl[ \frac{(t-c)^{\alpha }}{\alpha }-\frac{(s-c)^{ \alpha }}{\alpha } \biggr]\,ds. $$
The condition \(x(c)=0\) implies that \(c_{1}=0\) and the condition \(x(d)=0\) implies that \(c_{2}=\frac{\alpha }{(d-c)^{\alpha }} \int_{c} ^{d} r(s)x(s) (s-c)^{\alpha -1} [ \frac{(d-c)^{\alpha }}{\alpha }-\frac{(s-c)^{ \alpha }}{\alpha } ]\,ds\) and hence
$$\begin{aligned} x(t) &= \frac{(t-c)^{\alpha }}{(d-c)^{\alpha }} \int_{c}^{d} r(s)x(s) (s-c)^{\alpha -1} \biggl[ \frac{(d-c)^{\alpha }}{\alpha }-\frac{(s-c)^{ \alpha }}{\alpha } \biggr]\,ds \end{aligned}$$
$$\begin{aligned} &\quad {}- \int_{c}^{t} r(s) x(s) (s-c)^{\alpha -1} \biggl[ \frac{(t-c)^{\alpha }}{\alpha }-\frac{(s-c)^{\alpha }}{\alpha } \biggr]\,ds. \end{aligned}$$
Finally, the proof is concluded by splitting the first integral in (21) to
$$\begin{aligned} &\int_{c}^{t} r(s)x(s) (s-c)^{\alpha -1} \biggl[ \frac{(d-c)^{\alpha }}{ \alpha }-\frac{(s-c)^{\alpha }}{\alpha } \biggr]\,ds \\ &\quad {}+ \int_{t}^{d} r(s)x(s) (s-c)^{\alpha -1} \biggl[ \frac{(d-c)^{\alpha }}{\alpha }-\frac{(s-c)^{\alpha }}{\alpha } \biggr]\,ds. \end{aligned}$$
The Green function G defined above has the following properties:
\(G(t,s)\geq 0\) for all \(c\leq t,s \leq d\).
\(\max_{t \in [c,d]} G(t,s)=G(s,s)\) for \(s \in [c,d]\).
\(f(s)=G(s,s)\) has a unique maximum, given by
$$\begin{aligned} \max_{s \in [c,d]} G(s,s) &= G\bigl(\Lambda (c,d,\alpha ),\Lambda (c,d, \alpha )\bigr) \\ & = \frac{(d-c)^{2\alpha -1}}{3\alpha -1} \biggl( \frac{2\alpha -1}{3 \alpha -1} \biggr) ^{\frac{2\alpha -1}{\alpha }}, \end{aligned}$$
$$ \Lambda (c,d,\alpha )=c+ (d-c) \biggl[ \frac{(2\alpha -1)}{(3\alpha -1)} \biggr] ^{\frac{1}{\alpha }}. $$
Define the two functions \(g_{1}\) and \(g_{2}\) as in Lemma 5.
The proof follows by noting that the function \(g_{1}\geq 0\) since \(g_{1}(t,s)\) is decreasing in t for any s and \(g_{1}(d,s)=0\) for any s. Also, \(g_{2}\geq 0\) since \(g_{2}(t,s)\) is increasing in t for any s and \(g_{2}(c,s)= 0\) for any s.
The proof of this part follows by noting that the function \(g_{1}(t,s)\) is decreasing in t for any s and that \(g_{2}(t,s)\) is increasing in t for any s by realizing that \(( 1-\frac{(s-c)^{ \alpha }}{(d-c)^{\alpha }} ) \geq 0\) for all s.
Let \(f(s)=G(s,s)=(s-c)^{\alpha -1} [ \frac{(s-c)^{ \alpha }}{\alpha }-\frac{(t-c)^{\alpha }(s-c)^{\alpha }}{\alpha (d-c)^{ \alpha }} ] \). Then one can show that \(f^{\prime }(s)=0\) if \(s=\Lambda (c,d,\alpha )\) and hence the proof is concluded.
$$ \int_{c}^{d} \bigl\vert r(t) \bigr\vert \,dt> \frac{1}{ G(\Lambda (c,d,\alpha ),\Lambda (c,d, \alpha ))}=\frac{3\alpha -1}{(d-c)^{2\alpha -1}} \biggl( \frac{3\alpha -1}{2\alpha -1} \biggr) ^{\frac{2\alpha -1}{\alpha }}. $$
Let \(x \in X=C[c,d]\) be a nontrivial solution of the BVP (19) where \(\Vert x \Vert = \sup_{t \in [c,d]}\vert x(t) \vert \). By Lemma 5, x must satisfy
$$ y(t)= \int_{c}^{d} G(t,s) r(s)x(s)\,ds. $$
Taking the norm leads to
$$ \Vert x \Vert \leq \max_{t \in [c,d]} \int_{c}^{d} \bigl\vert G(t,s)r(s) \bigr\vert \,ds \Vert x \Vert $$
or equivalently
$$ \max_{t \in [c,d]} \int_{c}^{d} \bigl\vert G(t,s)r(s) \bigr\vert \,ds \ge 1. $$
By using the properties of the Green function \(G(t,s)\) given in Lemma 6, we come to the conclusion that
$$ G\bigl(\Lambda (c,d,\alpha ),\Lambda (c,d,\alpha )\bigr) \int_{c}^{d} \bigl\vert r(s) \bigr\vert \,ds >1, $$
from which (23) follows. □
Since \(G(\Lambda (c,d,\alpha ),\Lambda (c,d,\alpha ))\) tends to \(\frac{d-c}{4}\) as \(\alpha \rightarrow 1\) then the classical Lyapunov inequality (3) is obtained again: \(\alpha \rightarrow 1\). In this case, one may also deduce that \(x^{(2\alpha )}(t) \rightarrow x^{\prime \prime }(t)\) as \(\alpha \rightarrow 1\).
Consider the following conformable Sturm-Liouville eigenvalue problem of order \(\alpha \in (1,2]\).
$$ \bigl(\textbf{T}_{0}^{\alpha }x\bigr) (t)+\lambda x(t)=0, \quad 0< t< 1, x(0)=x(1)=0. $$
If λ is an eigenvalue of (24), then by Theorem 2, we must have \(\vert \lambda \vert > \frac{\alpha^{\alpha }}{(\alpha -1)^{\alpha -1}}\). If x is twice differentiable, then by means of (5), equation (24) is equivalent to
$$ t^{2-\alpha }x^{\prime \prime }(t)+\lambda x(t)=0, \quad t \in (0,1), x(0)=x(1)=0, $$
which can be considered as a type of generalized Sturm-Liouville eigenvalue problem. It is not easy to find the eigenvalues and eigenfunctions of this problem. However, if we consider the sequential conformable Sturm-Liouville eigenvalue problem
$$ x^{(2\alpha )}(t)+\lambda^{2} x(t)=0,\quad 0< t< 1, x(0)=x(1)=0, \frac{1}{2}< \alpha \leq 1, $$
then by using the operator method its solution is given by
$$ x(t)=c_{1}\cos \lambda \frac{t^{\alpha }}{\alpha }+c_{2}\sin \lambda \frac{t^{\alpha }}{\alpha }. $$
The boundary conditions imply that the eigenvalues are \(\lambda = n \alpha \pi \), \(n \in \mathbb{Z}\) and the correspondent eigenfunctions are \(\sin \lambda \frac{t^{\alpha }}{\alpha }\). Notice that, for any \(n \in \mathbb{Z}\), we have
$$ \lambda^{2}=n^{2}\alpha^{2}\pi^{2} > G\bigl(\Lambda (0,1,\alpha ),\Lambda (0,1,\alpha )\bigr)^{-1}= (3\alpha -1) \biggl( \frac{3\alpha -1}{2\alpha -1} \biggr) ^{\frac{2\alpha -1}{\alpha }}. $$
This is a direct verification of Theorem 3.
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The first and the second author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.
Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh, 11586, Saudi Arabia
Thabet Abdeljawad & Jehad Alzabut
Department of Mathematics and Computer Science, Çankaya University, Ankara, 06790, Turkey
Fahd Jarad
Thabet Abdeljawad
Jehad Alzabut
Correspondence to Thabet Abdeljawad.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Abdeljawad, T., Alzabut, J. & Jarad, F. A generalized Lyapunov-type inequality in the frame of conformable derivatives. Adv Differ Equ 2017, 321 (2017). https://doi.org/10.1186/s13662-017-1383-z
Received: 15 June 2017
DOI: https://doi.org/10.1186/s13662-017-1383-z
26D15
Lyapunov inequality
conformable derivative
Green's function
boundary value problem
Sturm-Liouville eigenvalue problem
Advances in Fractional Differential Equations and Their Real World Applications


Abbreviation for Sixteen Personality Factor Questionnaire.
Using ANOVA to analyze scores of members who have been there for different times in different dimensions and the result shows that there are no significant differences in the scores of 16PF or 15FQ in each dimension.
In our previous research, variables such as treatment, HB score on day 1, and Vigilance and Tension scores in 16PF may influence the value of HB score at month 6.
overall, positive item totals, and negative item totals) and 16PF Sten scores can be found in Table 1.
In one of his studies, he administered the 16PF to 630 full-time music students in Britain to determine if there were any personality differences between players of different instruments.
Paralelamente, se han confirmado ciertas relaciones con otros constructos como los valores y modelos de personalidad como el del 16PF, que es preciso matizar.
Der 16-Personlichkeits-Faktoren-Test 16PF [The 16-Personality-Factor-Test.
Table Showing Low, Average and High scores on different Personality Factors measured through 16PF of Men (n=50) and Women (n=30) prison inmates.
Inteligencia emocional e desempenho no trabalho: um estudo com MSCEIT, BPR-5 e 16PF.
Personalidade, interesses e habilidades: um estudo correlational da BPR-5, LIP e do 16PF.
Teacher scores on each of the 16 primary factors measured on the 16PF (personal factor) personality scale were regressed on their principals' effectiveness ratings.
For example, the insights gained from such assessments as the 16PF Assessment[R], Myers-Briggs Type Indicator[R] the DISC[R] and many other similar instruments will give you, in broad strokes, insights about yourself that you can explore in more depth by asking others about their perceptions of you.
Este hallazgo se asemeja a los de otro estudio realizado en Colombia con mujeres maltratadas de la ciudad de Bucaramanga, por Fajardo y otros (27), en el que se aplico la prueba de personalidad 16PF para obtener medidas independientes de varios factores psicologicos.


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Q: How to guarantee space for an ImageView, keep it from shrinking? I need to create an XML with the following content:
* Two TextViews with varying text (1- 3 rows), one TextView below the other.
* One ImageView to the right of the 2 TextViews, centered vertically, 30x30 px.
One major limitation is that it can't take the whole screen when inflated into a PopupWindow, which means that I cannot use the attributes fill_parent in many places.
I tried a lot of different layouts, but the ImageView keeps getting pushed away by the TextViews when the text is long. When the text is "half long" the image gets tiny but is still visible. I want it to always be 30x30 px, the text can wrap and use another line instead.
I tried to specify values for width and minWidth. Also tried layout_weight="1" for the ImageView. Also tried wrapping the ImageView into a LinearLayout and give that the layout_weight="1" parameter. Nothing works.
Here's an example of what is not working:
 <LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
    android:layout_width="wrap_content" android:layout_height="wrap_content">

    <LinearLayout android:layout_width="wrap_content"
       android:layout_height="wrap_content" android:orientation="vertical">

       <TextView android:id="@+id/popupTitle" android:layout_width="wrap_content"
      android:layout_height="wrap_content"/>

       <TextView android:id="@+id/popupContent"
      android:layout_width="wrap_content" android:layout_height="wrap_content"/>
    </LinearLayout>

    <ImageView android:layout_width="wrap_content"
       android:layout_height="wrap_content" android:src="@drawable/img_30x30_px" />
 </LinearLayout>

A: I've had a similar problem and i found that the TableView layout worked for me. It took a while but you can play with the stretch and strink columns proeprties to change the behaviour of how the columns expand to match there content.
Note that you should be able to set the linearLayout of your text to fill_parent (to fill the column space).
Read this on how TableRow works http://developer.android.com/resources/tutorials/views/hello-tablelayout.html

<TableRow>
    <!-- Column 1 -->
    <LinearLayout android:layout_width="fill_parent"
   android:layout_height="wrap_content" android:orientation="vertical">

            <TextView
                android:layout_column="1"
                android:text="Open..."
                android:padding="3dip" />
            <TextView
                android:text="Ctrl-O"
                android:gravity="right"
                android:padding="3dip" />
    </LinearLayout>
    <!-- Column 2 -->
    <ImageView android:layout_width="wrap_content"
        android:layout_height="wrap_content" android:src="@drawable/img_30x30_px" />
</TableRow>

(Afraid i'm not on a machine with Android so i couldn't test the above code).

A: Thank you so much Emile! It works! You made my weekend! :)
I had to try it right away (althought it's friday evening), and here's what my xml now looks like:
<TableLayout xmlns:android="http://schemas.android.com/apk/res/android"
android:layout_width="wrap_content" android:layout_height="wrap_content"
android:shrinkColumns="0">

<TableRow>
<!-- Column 1 -->
<LinearLayout android:layout_width="fill_parent"
android:layout_height="wrap_content" android:orientation="vertical">

  <TextView android:id="@+id/popupTitle"
  android:layout_width="wrap_content" android:layout_height="wrap_content"/>

  <TextView android:id="@+id/popupContent"
  android:layout_width="wrap_content" android:layout_height="wrap_content" />
  </LinearLayout>

  <!-- Column 2 -->
  <ImageView android:layout_width="wrap_content" android:layout_height="wrap_content"
  android:layout_gravity="center_vertical" android:src="@drawable/img_30x30_px" />

</TableRow>
</TableLayout>

redpajama_set_name: RedPajamaStackExchange


text:
Terrorism and Tourism in the Asia Pacific Region: Is Travel and Tourism in a New World After 9/11? | East-West Center | www.eastwestcenter.org
Terrorism and Tourism in the Asia Pacific Region: Is Travel and Tourism in a New World After 9/11?
Christopher Edmonds and James Mak
East-West Center Working Papers, Economics Series, No. 86
The paper reviews trends in travel and tourism in selected Asia Pacific countries before and after the terrorist attacks of September 11, 2001 (9/11) to consider the question of whether or not global tourism has fundamentally changed since 9/11. Tourism is an important economic sector in several Asia Pacific countries and is a "fragile" industry in that it is highly susceptible to external shocks such as wars, outbreaks of deadly contagious diseases, incidents of terrorism, and so on. The first part of the paper presents a stylized picture of industry response following terrorist incidents and other major negative shocks to tourism, and reviews international tourist arrivals at selected Asia Pacific destinations. A richer body of data available for Japan and the U.S. allows examination of the extent of substitution between domestic and international travel, and the impact of changed travel behavior on tourist spending. The paper finds that there has been significant substitution of domestic travel for overseas travel by nationals of both countries after 9/11, and that this has had a dramatic impact on the Hawaii tourism market. The paper explores some of the reasons for the differences observed in post-9/11 travel recoveries across Asia Pacific countries.
Additional titles in the East-West Center Working Papers series
The Impact of 9/11 and Other Terrible Global Events on Tourism in the U.S. and Hawaii
ASEAN Matters for America / America Matters for ASEAN
Australia Matters for America / America Matters for Australia
India Matters for America / America Matters for India
Korea Matters for America / America Matters for Korea
Korea Matters for America/America Matters for Korea

redpajama_set_name: RedPajamaCommonCrawl


Heroes, in the simplest term, are a group of people. To answer questions about any group, we first must try to understand the meanings of the title used to describe them.
It is immediately apparent that Webster understood the power of the wisdom of three. In other words, to be hero / heroine one must possess and reveal three character traits at the same moment in time.
The Law of the Hero!
First, they must act in a selfless manner. Being a warrior requires putting the cause before the safety of one's self.
Second, the person must be noble of character. To be noble is to be of high moral character; having no one above you.
Third, one must demonstrate great courage when the time dictates the need.
Who was one of your childhood heroes / heroines?
As a young man the Lone Ranger was one of my earliest heroes. I grew up around relatives who were cowboys; I had an interest in horses and looked up to people who wanted to help others. Every week my Dad and I would watch this ordinary man who wore a mask overcome all odds to help people battle against the criminal elements of the times. He never asked for money, sought praise or backed away from doing what was right. His examples set the standards for millions of people all around the world.
Who are the heroes / heroines of today's children?
Do they meet the standards of selflessness, noble of character and the courage to act appropriately when times are tough?
The conclusion of the group I met with was that as a society and in our personal, professional and spiritual lives, we have unconsciously allowed ourselves to be moved away from the definitions that matter most. We no longer pay attention to the truth. Instead, we are easily swayed toward a relativistic mindset.
The perceived heroes / heroines of today are not viewed as selfless individuals living morally honorable lives nor are they people courageous enough to stand up for the truth. Just the opposite seems to be present in most cases. Today's examples, for the most part, are self-centered and close minded. In some cases they brag of living an immoral lifestyle. When courage is talked about or demonstrated it is to support causes that go against family values.
We did all agree that heroes and heroines are very important to any society. These individuals bring communities together in tough times. They offer examples of a life to be followed and in spite of the fear that might be present; they challenge each of us to do what is right for the good of someone else.
We each left the meeting wanting to live in accordance with the Law of the Hero.


text:
A Medalha de Ouro Kurchatov () é uma condecoração concedida desde 1962 pela Academia de Ciências da Rússia em física nuclear e tecnologia nuclear. É denominada em memória de Igor Vasilyevich Kurchatov. Na época da União Soviética foi concedida entre 1962 e 1989 a cada três anos e na época da Rússia foi concedida novamente a partir de 1998, em intervalos irregulares entre dois e cinco anos.

Recipientes

1962: Pyotr Spivak e Yuri Prokoviev
1965: Yuriy Prokoshkin, Vladimir Rykalin, Valentin Petruhin e Anatoly Danubians
1968: Anatoly Alexandrov
1971: Isaak Kikoin
1974: Juliï Borisovich Khariton e Savely Moiseevich Feinberg
1977: Jakov Seldovich e Fyodor Shapiro
1980: Isai Izrailevich Gurevich e Boris Nikolsky
1981: William d'Haeseleer
1983: Vladimir Mostovoy
1986: Venedikt Dzhelepov e Leonid Ponomarev
1989: Georgy Flyorov e Yuri Oganessian
1998: Aleksey Ogloblin
2000: Nikolay Dollezhal
2003: Yuri Trutnev
2008: Oleg Filatov
2013: Eugene Avrorin

Prêmios de física
Prêmios de ciência e engenharia da Rússia
Prêmios da Academia de Ciências da Rússia
Prêmios estabelecidos em 1962

redpajama_set_name: RedPajamaWikipedia


RALEIGH - The North Carolina Division of Motor Vehicles will open its Henderson license plate agency as a temporary site today at 8:30 a.m. The office will be located at 946-D Andrews Avenue and will replace the previous agency that closed on October 1.
Office hours for the agency will be from 8:30 a.m. -4:30 p.m. on Monday - Friday. The agency can be contacted by phone at (252) 438-3528 and its fax number is (252) 436-2489.


Newspapers, TV anchors, celebrities, politicians – they all talk about climate change and global warming interchangeably, as if these two were one and the same thing.
Are they? Do Climate Change and Global Warming mean the same?
While climate change and global warming are thus different, there is however a clear relationship: Just as temperature is one component of climate, global warming is a component of climate change.
In summary: Global warming is not the same as climate change; rather, global warming is a component of climate change.
This section is from the Climate Change for Idiots series. See answers to all questions from here.
« Why is a small change in average global temperature a concern?


Lady Oh Janey is a design and fashion brand which is all about the creation of Parisian inspired clothing pieces, lovely patterns and intricate fashion design illustrations. The artist aims to have these patterns produced in a wide variety of accessories and clothing for women and girls. Lady Oh Janey mixes her love of florals, fashion and pastel tones. All the clothes are made in limited quantities and can be sent back to us for recycling or donation. Lady Oh Janey would also like to share her love for food, travel, diy and motivational stories through her blog. Discover more about the brand, the newest clothing collection and her art pieces by subscribing to our mailing list.


Web Hosting > VodaHost Web Hosting Support > Email Problems (POP, SMTP & Webmail) > Email Problems, What Did I Do Wrong?
View Full Version : Email Problems, What Did I Do Wrong?
Have you ticked the "Server needs authentication" box in your email client?
I want to make my email addresses on my web page go to send message. and connect with my msn email account. Is this a pop3 function? Which video do I need to study?
you a email when they click on it.
I guess I didn't understand. I thought I did as you said. but with the URL set to circuspet.com it just goes back to my index http://www.circuspet.com/aboutus.html If there is nothing wrong with what you see,what could it be? URL? Control panel? Not enough fish in my diet?
Yes, I did that and it made no difference. Could it be that the username you assigned me was epguacom yet my e-mail account is set up for info@epgusa.com. Still looking for the solution to my problem.
The link should be "mailto" type, not "http" type. In the URL field type in, after the "mailto:" your email address.
You must use the password and username that you have set for your email account.
Sorry I'm not more computer savvy.
Works great. piano piano se va lontano. Now to battle other demons.
Spoke too soon maybe, what is Authentacation and how do I get it?
In your Outlook Express settings, Tools, Accounts, select the account, Properties, Server, last box at the bottom: Server requires authentication. Tick it.


At Gerald Jones Company, we pride ourselves on providing top-quality services and helping our customers create the home they've always wanted. That's why we offer custom windows. Select the size, shape, type of glass and function that's a perfect fit for your home, and then leave the rest to our professionals. We'll install your new awning window with precision and attention to detail.
Contact Gerald Jones Company for your custom awning window in Raleigh, Durham, Chapel Hill and beyond, or ask us about our other specialty windows, including glass block windows, accent windows, geometric windows, and more. We look forwarding to working with you for all your home improvement needs!
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What started as a group of friends sharing a love of fine craft beer and home brewing, led to us joining forces to create Something Wicked Brewing Company!
We brew with a desire to bring our custom creations to you and to help grow the local Craft Beer Scene in Hanover, PA. With a passion and pride put into all our beers we work to push the limits of what traditional beers can be and bring to you a constant rotation of newly crafted choices.
Our beers will both tempt and tease your pallet. Four flagship beers cover a broad spectrum of styles: IPA, Kolsch, Blonde Ale and a Stout. Our rotating tap handles will offer a variety of styles and flavors sure to please the novice beer drinker and the most hardened Hophead.
A small fare menu includes Hanover's original Revonah pretzel logs, handmade seasonal flatbread pizzas and snack items.
Something Wicked is always brewing - stop in and see what!


text:
By January 27, 2021 0 comments
vivo Brand Launch - Great New Beautiful Smartphones Have Arrived on our Online Store
With our business continuing to grow, we are delighted to announce our brand new partnership with vivo! As long-term fans of the brand, we cannot wait for our customers to get their hands on these truly stunning smartphones.
The vivo Brand Story
Established 11 years ago, vivo has gone from strength to strength, manufacturing beautiful and highly impressive smartphones in this ever-competitive market. Having begun operations in China, the most competitive consumer electronics market in the world, vivo has managed to succeed and expand globally. With international expansion beginning back in 2014, vivo entered the large Indian and South East Asian markets and has since gained an immense reputation in the industry.
By Q1 2015, despite not even being in the European and North American markets, vivo was ranked as one of the top 10 smartphone manufacturers in the world, with a 2.7% market share. Following this, vivo continued to grow, entering many others markets in 2017, such as Russia, Sri Lanka, Taiwan, Hong Kong, & Bangladesh, and then North America in 2018. Selling smartphones in the US under the BLU brand name, vivo, again, managed to maintain their forward momentum and gain even more success.
Now we come to 2020. With over 10,000 employees and operations in over 100 countries, vivo has finally announced its entry into Europe. With this, based in the UK, eFones.com are incredibly happy to announce that we will be selling vivo's brand new smartphones, going directly up against the smartphone behemoths of Apple, Samsung, and Huawei.
With their ambitious goals, vivo reached a sponsorship deal with FIFA become the official smartphone brand of the 2018 and 2022 FIFA World Cups and signed a deal with UEFA as an official partner of the Euro 2020 and 2024 finals.
The Stunning Smartphones
So, what does vivo need to do to succeed in the UK market? Simply put, vivo needs to create strong performing and beautifully designed smartphones at great value for money. Luckily, vivo has done exactly that, bringing UK consumers four brand new stylish smartphones that rival its already well-established and highly rated competitors.
The Flagship – vivo X51 5G
As the flagship device in vivo's line-up, the X51 5G offers everything you could ever need from a smartphone. That means a picture-perfect and ultra-smooth display, powerful processor, super-fast 5G network capability, and incredible cameras (which come with their very own splash of industry-first innovation).
To start, the standout feature of the vivo X51 5G is definitely its Gimbal camera system, a feature not seen on any other smartphone currently available. If you don't already know, a Gimbal is essentially a pivot mechanism that allows an object to rotate about a single axis. This allows the X51 5G's super-sharp 48MP primary camera to keep stable while the phone itself moves freely. Because of this, the main camera of the X51 5G has incredible image stabilization, minimising blur and reducing shake of video further than any other smartphone. Its rivals typically have what is called OIS (optical image stabilization). Overall, the Gimbal double-ball suspension mechanism allows the camera to cover angles 300% larger than OIS. As well as providing ultra-stable video and sharper images, the Gimbal mechanism also allows the camera to track objects within its view, maintaining perfect focus.
It is clear to see the X51 5G's stunning design. Being super slim and lightweight, coupled with its gorgeous Full HD 6.56-inch AMOLED display, the vivo X51 5G is truly elegant and easy on the eye.
Elsewhere, with its 90Hz refresh rate, scrolling with the X51 5G is a great experience, being super smooth and a joy to use. Powering the phone is Qualcomm's second-most powerful processor from 2020, the Snapdragon 765G. Bringing fast performance, great image processing, and (most importantly) 5G, the 765G chip makes slowdowns while using your device a thing of the past.
As a flagship Android smartphone, the vivo X51 5G would not be complete without features such as an under-display fingerprint scanner and some form of fast charging. With this, the vivo X51 5G's under-display fingerprint sensor allows you to unlock your phone extremely quickly and easily, and more importantly, securely. Additionally, the vivo X50 5G features 33W FlashCharge, enabling you to charge the device from zero to 57% in only 30 minutes – very convenient if you ever find yourself low on charge during the day.
vivo's Mid-Range Competitor – The Y70
Bringing many features from vivo's very own flagship smartphone, the Y70 (shown above) provides great value for consumers during these tough times. For just £279, the Y70 offers a great display that utilises AMOLED technology. Because of this, you get a truly immersive experience, with its vibrant colours and deep blacks produced from each pixel. At Full HD resolution and large 6.44-inch size, the Y70 is up there with the best smartphones for entertainment, whether you're watching high-definition video, looking at photos, or playing mobile games.
The Y70 comes with impressive cameras as well, with a sharp 48MP primary sensor, capable of shooting 4K video, as well as secondary macro and depth cameras for great close-up images and portrait photos.
Just like the premium X51 5G, the vivo Y70 utilises an under-display fingerprint sensor for fast and secure unlock, as well as the 33W FlashCharge for convenient power whenever you need. One addition that the Y70 retains is the 3.5mm headphones jack, meaning if you don't have wireless headphones yet, you'll still be able to use any wired ones you may have lying around.
The Budget Smartphones – vivo Y20s & Y11s
Much like their more advanced counterparts, the vivo Y20s and Y11s feature very stylish and slim designs, but at impressively low price points. Being very similar, both the Y20s (£149 – shown on left) and Y11s (£119 – shown on right) include the largest batteries of vivo's new line-up, being 5,000mAh in size. This means that these phones will easily last a full day of fairly intensive use, and even multiple days if used lightly.
These budget smartphones also feature identical screens, with expansive 6.51-inch Halo FullView displays, great for all kinds of viewing experiences. With their thin bezels, these displays really come to life, being very immersive, providing no visual distractions.
The two main differences between these two devices are the cameras, and the storage/memory options. The entry-level vivo Y11s features dual-rear cameras, with a 13MP wide and a 2MP depth sensor. The Y20s on the other hand adds a 2MP macro camera with its triple camera system, perfect for taking images of close-up objects, such as tiny wildlife.
The entry-level Y11s comes with 32GB storage (that is expandable via microSD) and 3GB RAM, whereas the Y20s includes 128GB storage (also expandable) and an improved 4GB RAM.
An Exciting Partnership
At eFones.com, we cannot wait to start our long-term relationship with vivo and look forward to an exciting future ahead. We know that so many of our customers will love vivo's new smartphones, all bringing sleek designs, plenty of features, all at great value for money.
Get the flagship vivo X51 5G for just £749 here https://www.efones.com/products/vivo-x51-5g
Get the impressive vivo Y70 for just £279 here https://www.efones.com/products/vivo-y70
Get the capable vivo Y20s for just £149 here https://www.efones.com/products/vivo-y20s
Get the entry-level vivo Y11s for just £119 here https://www.efones.com/products/vivo-y11s
Want to pay in smaller payments? By purchasing direct from eFones.com you can take advantage of the Klarna service, allowing you to split the cost of the device over your choice of 3, 6, 12 months interest free or 24 months at just 18.9% APR.


Over the past five years or so, there has been an emerging awareness of a body of work that arose during the Second World War. Fragments of it have been known for some years to disparate groups of academics and their students, but it is as if now, in the 21st century, these writings are forming themselves into a body, steadily enjoining us to give them our attention.
The group has become known as the 'Death Cell Philosophers' and includes, among others, such writers as Simone Weil, Etty Hillesum, Dietrich Bonhoeffer, Edith Stein, Maria Skobstova and Corrie ten Boom, whose sister Betsie is quoted above. These people did not know each other; it is possible that Edith Stein and Etty Hillesum crossed paths at Westerbork transport camp, but if so no record of their meeting has been found. But all were concerned, in diverse ways, to document transformation as it happened to them in the crucible of mid-20th-century Europe, in the years 1932–45, and to pass on what they had learnt. The form these writings took was neither memoir in the usual sense, nor history, nor fiction; possibly the word 'chronicles' could help to describe these records of passage, of change from one state to another, in the context of the extreme pressure that came with the war.
People respond to pressure in many ways. What distinguishes this group of people is the resolution to be changed for the better in the context of absolute mortal threat – leading them in some cases even to forfeit the chance to escape it. As Rowan Williams has put it, these people were concerned with "the discovery of a real and completely, powerfully transforming divine faithfulness, present even in the depths of the nightmare…".
To discuss properly all of the writers I have mentioned above would be impossible in the scope of this short article, so I propose here to concentrate on two of them – Simone Weil and Etty Hillesum. Of the six philosophers above, these two participated most explicitly in what Dom Sylvester Houédard of Prinknash Abbey would later call 'the wider ecumenism' – a phrase coined to describe a contemplative stance that, while respecting the metaphors and lived experiences of particular faiths, is not constrained by their boundaries.
The writings of Simone Weil are in some ways the least related to the circumstances of the war, as she did not experience the camps herself – in fact, she may have died before their existence was fully known. Nevertheless, paradoxically, a thread of mortal pressure runs right through her work, and gains strength as the war progresses. This pressure was created not by Weil's proximity to the Nazis, but, outwardly, by her distance from the fight against them, and inwardly, her distance, as she felt it, from what she considered to be grace.
Simone Weil was born in Paris in 1909 to a Jewish agnostic family; in secondary school she studied with the philosopher Emile Chartier, aka Alain, and was one of his most brilliant students. After university she embarked upon a teaching career, and also spent time as a manual worker in a Renault factory to gain first hand experience of the privations of working people at the time. Her notebooks also contain wonderful accounts of two trips to Italy, where her focus begins to change towards her own kind of Christianity. This was deeply influenced, as it continued to be in the coming years, by her engagement with the Bhagavad Gita, the Zen Teachings of Daisetsu Teitaro Suzuki and the English mystical poets, particularly George Herbert.
But eventually, of course, the Weil family's lack of engagement with formal Judaism did not prevent them from being forced to leave Paris in 1940. After a period in Marseille, in which Weil developed important friendships with the philosopher Gustave Thibon and Fr J.M. Perrin, she accompanied her parents to New York. Immediately upon arrival she began petitioning to be allowed to return to England, which she did in 1942, working with the Free French in London until her death in 1943 at a sanatorium in Kent. The cause of death was tuberculosis exacerbated by her lifelong habit of eating little; this practice had intensified with the war, and her decision to eat no more than the French troops were allowed.
Prayer consists of attention. It is the orientation of all the attention of which the soul is capable towards God. The quality of the attention counts for much in the quality of prayer. Warmth of heart cannot make up for it.
Never is a genuine effort of the attention wasted. It always has its effect on the spiritual plane and in consequence on the lower one of the intelligence… Without our knowing or feeling it, this apparently barren effort has brought more light into the soul. The result will one day be discovered in prayer.
She brought her own intensely cultivated power of attention to bear upon particular texts in a way that for her was absolutely transformational. The most notable instances of this were her contemplation of Herbert's poem 'Love', and her daily focused recitation of the 'Our Father'. Her efforts and experiences are recorded in essays and 'pensées sans ordres' collected from her notebooks and published articles.
Etty Hillesum (1914–43), by contrast to Weil, was directly in the line of fire during the war. As a young woman coming of age in Amsterdam, she pursued studies in Russian language and literature, living a rather chaotic and emotionally turbulent life, until that path was cut off from her by the occupation and the strictures imposed upon Jews by the Nazis in the period 1940–43. However, as the situation worsened she found herself among a group of friends and colleagues who valued education, books, art, and – for want of better words – 'the inner life'. She began a transformative friendship with the figure central to the group, Julius Spier, who had been a colleague of Jung. Her interior resources were strengthened and deepened, recalling, when she resolved "to regulate my life", the diary entries of Claudine Moine, hundreds of years before, in the midst of another war.
It's happened to me a few times recently – in the street or wherever I chance to be, I stop with bated breath for a moment and have to ask myself, is this really my life? So full, so rich, so intense and so beautiful?
She was not forcibly detained at Westerbork but had a certain amount of freedom of movement through a job she had taken in Amsterdam with the Jewish Council. Her friends urged her to make her escape, but she did not. She stayed where she was, moving between the camp and her flat in Amsterdam until she and her family were sent to Auschwitz in 1943, finishing her diaries with the words: "We should be willing to act as a balm for all wounds." Astonishingly, a postcard that she threw from the transport train as it departed, was picked up by a farmer and sent to its intended. "We left the camp singing", she wrote.
Her letters and diaries are, in my view, required reading for anyone wishing to learn about – and, more aptly, from – those years.
The pressures of the 21st century are of course markedly different from those of the 20th but it is becoming clear that the Death Cell Philosophers' strategies for dealing with adversity, including the most extreme, can be of enormous help to us now. Their individual resolutions to be positively transformed in the face of oppression, and their efforts to rally all of their interior faculties to the cause of loving service, may indeed serve as models for us, as we enter the next phase of the digital age.
Main image: Westerbork Transit Camp. Photograph from BeeldBankWO2.
Article Thumbnail: Jewish prisoners boarding the train to Auschwitz from Westerbork transit camp. Photograph from BeeldBankWO2.
Our thanks to Beedlebankwo2 for images of Westerbork Transit Camp; to the Jewish Historical Museum of Amsterdam for material on Etty Hillesum. For the Simone Weil material we have not been able to trace copyright holders, but if we have inadvertently overlooked them, we will be pleased to make the necessary arrangements at the first opportunity.
Etty Hillesums's diaries have been published in English in full, combining in one volume An Interrupted Life (Diaries 1941–43) and Letters from Westerbork, translated from the Dutch by Arnold J. Pomerans (Henry Holt & Co., New York, 1996). These two can often be found published separately.
The quote from Rowan Williams comes from his Foreword to Patrick Woodhouse's Etty Hillesum: A Life Transformed (Bloomsbury, 2009).
For Claudine Moine, see You Looked at Me: The Spiritual Testimony of Claudine Moine. Trans. Gerald Carroll (Clarke & Co, Cambridge, 1989).
Some of Simone Weil's most essential writings can be found in Waiting for God, translated by Emma Craufurd (Putnam & Sons, 1951); Gateway to God (Fontana, 1971); Intimations of Christianity Among the Ancient Greeks (Routledge & Kegan Paul, 1957); Gravity and Grace (Routledge Classics, 2002); War and the Iliad (NYRB Classics, 2001); Letter to a Priest (Routledge Classics, 2002).
Her political writings include: The Need for Roots (Routledge Classics, 2001); Oppression and Liberty (Routledge Classics, 2002). Her Oeuvres Complètes has been been published by Gallimard, and she is widely anthologised.
Dietrich Bonhoeffer is best known for his Letters and Papers from Prison (SCM, 1953). His other works include Life Together, The Cost of Discipleship, and Ethics, all of which have recently been reprinted.
Maria Skobstova and Edith Stein are both profiled in the Modern Spiritual Masters series, published by Orbis.
Corrie ten Boom's experiences were chronicled in The Hiding Place (Hodder & Stoughton, 2004).
For more on Dom Sylvester Houédard's 'wider ecumenism', see Charles Verey, 'Dom Sylvester's 'Wider Ecumenism': Considered Words, Explosive Meaning', in Notes from the Cosmic Typewriter (Ed. Nicola Simpson, Occasional Paper, 2012).


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Australian Reinsurance Pool Corporation (ARPC) is a corporate Commonwealth entity established under the Terrorism Insurance Act 2003 (TI Act). ARPC is also subject to requirements under the Public Governance, Performance and Accountability Act 2013 (prior to 1 July 2014, the Commonwealth Authorities and Companies Act 1997) and is within the Treasury portfolio. ARPC is responsible for administering the terrorism reinsurance scheme, providing primary insurers with reinsurance for commercial property and associated business interruption losses arising from a declared terrorist incident.
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text:
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Peter Berkowitz On The John Batchelor Show (19:22)
Hoover Institution fellow Peter Berkowitz discusses his Real Clear Politics article "Double Jeopardy At The University Of California."
Berkowitz discusses liberal bias on college campuses on KQED's Forum
Peter Berkowitz, the Tad and Dianne Taube Senior Fellow at the Hoover Institution, discusses whether or not students at the University of California are receiving a biased and compromised education from activist professors.
Berkowitz discusses his op-ed "Professors Proselytizing Liberalism"
Peter Berkowitz, the Tad and Dianne Taube Senior Fellow at the Hoover Institution, chair of the Koret-Taube Task Force on National Security and Law, and cochair of the the Boyd and Jill Smith Task Force on Virtues of a Free Society, notes, on Wall Street Journal TV, that public colleges are legally obligated to keep the classrooms free of politics and that classrooms should be places where students are free to explore ideas.
Do Campuses Tilt Left?
Research | Articles | by Mary Eberstadt
Every once in a while, something you read is so otherwise inexplicable that satire seems the safest bet...
Peggy Noonan's Words To Live By
In a June 4, 2010, Wall Street Journal column, republished in her new collection, "The Time of Our Lives," Peggy Noonan tells the heartbreaking story of 28-year-old Detroit Tigers' pitcher Armando Galarraga.
Affirmative Action and the Demotion of Truth
For several decades, the number of students attending college in the United States has been growing rapidly: Over the last 20 years or so, enrollments have risen by about 50 percent, and over the last 50 years they have more than quadrupled. During this time, especially the last two decades, the polarization of our politics has markedly intensified.
Duke Erodes Liberal Education
On May 8, the Duke University student newspaper published a stirring letter addressed to the school community that was co-signed by 101 students and former students. The letter protested the decision of the university's Sanford School of Public Policy to decline to renew the contract of Evan Charney, associate professor of the practice of public policy and political science, and called on the provost to reverse the decision.
The Golden State's Me Generation
In the midst of the Great Recession California students protest in favor of themselves. . . .
Conservatism and the University Curriculum
The political science departments at elite private universities such as Harvard and Yale, at leading small liberal arts colleges like Swarthmore and Williams, and at distinguished large public universities like the University of Maryland and the University of California, Berkeley, offer undergraduates a variety of courses on a range of topics...
Research | Articles | by Shelby Steele
Why Abraham Lincoln matters—even now. By Shelby Steele.
This is a story about using American politics to promote the highest of ideals and to realize the worthiest of accomplishments...
Hoover's Uncommon Knowledge Featured Nov. 29 On New Fox Nation
News | News/Press
The Hoover Institution's Uncommon Knowledge with Peter Robinson will appear on the new Fox Nation streaming service on Nov. 29 with an exclusive interview of economist and Hoover senior fellow Thomas Sowell.
Hoover Announces 2009–10 National Fellows
The Hoover Institution's annual postdoctoral W. Glenn Campbell and Rita Ricardo-Campbell National Fellows have been named for the 2009–10 academic year.
Why Here, Why Now? Why Did The United States Enjoy Dramatic Improvements In The Standard Of Living During The Last Century?
Hoover Institution economists John Cogan, Lee Ohanian, Terry Anderson, and George Shultz examine the causes for and the reasons behind so many improvements being made to the quality of life in the United States over the past century. They analyze the role that free markets, property rights, innovation, regulation, taxes, and national security played in these remarkable achievements.
Rebels With a Cause: Themselves
Students turn protest into another form of narcissism. By Peter Robinson.
A Most Ingenious Trick
Research | Articles | by Matt Ridley
Matt Ridley, author of The Rational Optimist, insists that we humans must face the truth about ourselves—no matter how good it might be. An interview with Peter Robinson.
Overseers, Directors, and Scholars Meet at Hoover Headquarters to Discuss Strategy and New Developments in US Policy
The Summer Board of Overseers meeting commenced on July 10th, 2018 with a dinner featuring remarks by Victor Davis Hanson, the Martin and Illie Anderson Senior Fellow at Hoover. Here Hanson talks with Joel C. Peterson, chairman of Hoover's Board of ...


text:
/*
 * @author max
 */
package com.intellij.util.ui;

import com.intellij.openapi.util.text.StringUtil;
import org.jetbrains.annotations.NotNull;
import org.jetbrains.annotations.Nullable;

import java.awt.*;
import java.lang.annotation.Annotation;

/**
 * @author max
 * @author Konstantin Bulenkov
 */
public class ColorUtil {
  private ColorUtil() {
  }

  public static Color softer(@NotNull Color color) {
    if (color.getBlue() > 220 && color.getRed() > 220 && color.getGreen() > 220) return color;
    final float[] hsb = Color.RGBtoHSB(color.getRed(), color.getGreen(), color.getBlue(), null);
    return Color.getHSBColor(hsb[0], 0.6f * hsb[1], hsb[2]);
  }

  public static Color darker(@NotNull Color color, int tones) {
    final float[] hsb = Color.RGBtoHSB(color.getRed(), color.getGreen(), color.getBlue(), null);
    float brightness = hsb[2];
    for (int i = 0; i < tones; i++) {
      brightness = Math.max(0, brightness / 1.1F);
      if (brightness == 0) break;
    }
    return Color.getHSBColor(hsb[0], hsb[1], brightness);
  }

  public static Color brighter(@NotNull Color color, int tones) {
    final float[] hsb = Color.RGBtoHSB(color.getRed(), color.getGreen(), color.getBlue(), null);
    float brightness = hsb[2];
    for (int i = 0; i < tones; i++) {
      brightness = Math.min(1, brightness * 1.1F);
      if (brightness == 1) break;
    }
    return Color.getHSBColor(hsb[0], hsb[1], brightness);
  }

  public static Color dimmer(@NotNull Color color) {
    float[] rgb = color.getRGBColorComponents(null);

    float alpha = 0.80f;
    float rem = 1 - alpha;
    return new Color(rgb[0] * alpha + rem, rgb[1] * alpha + rem, rgb[2] * alpha + rem);
  }

  private static int shift(int colorComponent, double d) {
    final int n = (int)(colorComponent * d);
    return n > 255 ? 255 : n < 0 ? 0 : n;
  }

  public static Color shift(Color c, double d) {
    return new Color(shift(c.getRed(), d), shift(c.getGreen(), d), shift(c.getBlue(), d), c.getAlpha());
  }

  public static Color withAlpha(Color c, double a) {
    return toAlpha(c, (int)(255 * a));
  }

  public static Color toAlpha(Color color, int a) {
    Color c = color != null ? color : Color.black;
    return new Color(c.getRed(), c.getGreen(), c.getBlue(), a);
  }

  public static Color withAlphaAdjustingDarkness(Color c, double d) {
    return shift(withAlpha(c, d), d);
  }

  public static String toHex(@NotNull final Color c) {
    final String R = Integer.toHexString(c.getRed());
    final String G = Integer.toHexString(c.getGreen());
    final String B = Integer.toHexString(c.getBlue());
    return (R.length() < 2 ? "0" : "") + R + (G.length() < 2 ? "0" : "") + G + (B.length() < 2 ? "0" : "") + B;
  }

  /**
   * Return Color object from string. The following formats are allowed:
   * #abc123,
   * ABC123,
   * ab5,
   * #FFF.
   *
   * @param str hex string
   * @return Color object
   */
  public static Color fromHex(String str) {
    str = StringUtil.trimStart(str, "#");
    if (str.length() == 3) {
      return new Color(
        17 * Integer.valueOf(String.valueOf(str.charAt(0)), 16).intValue(),
        17 * Integer.valueOf(String.valueOf(str.charAt(1)), 16).intValue(),
        17 * Integer.valueOf(String.valueOf(str.charAt(2)), 16).intValue());
    }
    else if (str.length() == 6) {
      return Color.decode("0x" + str);
    }
    else {
      throw new IllegalArgumentException("Should be String of 3 or 6 chars length.");
    }
  }

  @Nullable
  public static Color fromHex(String str, @Nullable Color defaultValue) {
    try {
      return fromHex(str);
    }
    catch (Exception e) {
      return defaultValue;
    }
  }

  /**
   * Checks whether color is dark or not based on perceptional luminosity
   * http://stackoverflow.com/questions/596216/formula-to-determine-brightness-of-rgb-color
   *
   * @param c color to check
   * @return dark or not
   */
  public static boolean isDark(@NotNull final Color c) {
    // based on perceptional luminosity, see
    return (1 - (0.299 * c.getRed() + 0.587 * c.getGreen() + 0.114 * c.getBlue()) / 255) >= 0.5;
  }

  @NotNull
  public static Color mix(@NotNull Color c1, @NotNull Color c2, double balance) {
    balance = Math.min(1, Math.max(0, balance));
    return new Color((int)((1 - balance) * c1.getRed() + c2.getRed() * balance + .5),
                     (int)((1 - balance) * c1.getGreen() + c2.getGreen() * balance + .5),
                     (int)((1 - balance) * c1.getBlue() + c2.getBlue() * balance + .5),
                     (int)((1 - balance) * c1.getAlpha() + c2.getAlpha() * balance + .5));
  }
}

redpajama_set_name: RedPajamaGithub


There is almost no country without an anthem and Nigeria isn't an exception. In this post, we'll be telling you all you need to know about the national anthem. So sit back, relax and let's take a brief but important journey back in time to when Nigeria had no National Anthem.
This anthem was adopted in the year 1960, and was going to be the one and only generally adopted anthem, except for constant criticism from different people. Quite a number of notable personalities in the country had reservations about a European, which was the same tribe as our colonial masters, gifting us our national anthem.
The main voice of discontentment was from the popular Nigerian newspaper – Daily Service, which was run by Egbé Omo Òduduwá a Yoruba group, known for "saying it as it is".
They began a rebellious campaign against the national anthem, and their campaign led to the establishment of a committee which voiced it for notable personalities and any interested persons of the general public to sign a petition for or against the continued use of the national anthem.
In 1977, the petition was against the continued use of the national anthem. 1978 made it eighteen years that the old national anthem was in use and in 1978, a national contest to create a new anthem was organized by the National Publicity Committee and many entries were sent in, but only five was shortlisted.
The "Arise, O Compatriots" anthem has since been the official national anthem of Nigeria since 1978 till today. Benedict Elide Odiase recently passed away on June 12, 2013 but his name (and the five composers') would forever be attached and recalled, anytime the history of the national anthem is talked about.


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text:
Laurence Leherte
Unit of theoretical and structural physico-chemistry
NAmur MEdicine & Drug Innovation Center
Namur Institute of Structured Matter
2 Meeting abstract
Applications of Leveling Methods to Properties of Small Molecules and Protein Systems
Leherte, L., 2018, Innovations in Computational Chemistry: Theoretical and Quantum Chemistry at the Dawn of the 21st Century. Carbó-Dorca, R. & Chakraborty, T. (eds.). p. 197-248 51 p.
Research output: Contribution in Book/Catalog/Report/Conference proceeding › Chapter
Multiresolution analysis
Charge density
Understanding structure and dynamics of small peptides and proteins through the lens of network science
Fossepre, M., Leherte, L., Laaksonen, A. & Vercauteren, D., 2018, Methods and Principles in Medicinal Chemistry series : Biomolecular Simulations in Structure-based Drug Discovery. Gervasio, F. L., Spiwok, V., Mannhold, R., Buschmann, H. & Holenz, J. (eds.). Wiley-VCH
Biological systems
Determination of Protein Coarse-Grain Charges from Smoothed Electron Density Distribution Functions and Molecular Electrostatic Potentials
Leherte, L. & Vercauteren, D., 2010, Handbook of Computational Chemistry Research. Collett, C. T. & Robson, C. D. (eds.). Hauppauge (NY): Nova Science Publishers, Inc., p. 153-191 39 p.
Protein-Protein Docking Using Three-Dimensional Reduced Representations and Based on a Genetic Algorithm
Becue, A., Meurice, N., Leherte, L. & Vercauteren, D., 2008, Models, Mysteries, and Magic of Molecules. Boeyens, J. C. A. & Ogilvie, J. F. (eds.). Guildford: Springer, p. 301-323 23 p.
Topological Analysis of Proteins as Derived from Medium and High Resolution Electron Density: Applications to Electrostatic Properties
Leherte, L., Guillot, B., Vercauteren, D., Pichon-Pesme, V., Jelsch, C., Lagoutte, A. & Lecomte, C., 2007, The Quantum Theory of Atoms in Molecules - From Solid State to DNA and Drug Design. Matta, C. F. & Boyd, R. J. (eds.). Weinheim, Germany: Wiley-VCH, p. 285-316 32 p.
Interpretation Techniques for Promolecular Electron Density Distribution Functions
Leherte, L. & Vercauteren, D., 2005, Abstracts of Papers of the American Chemical Society . Vol. 230. p. COMP-273
Multiresolution Analysis of Topological Representations of Structural and Physico-Chemical Properties of Pharmacological Molecules
Biname, J., Leherte, L. & Vercauteren, D., 2002, Abstracts of Papers of the American Chemical Society. Vol. 223. p. CINF-005
Theoretical Studies of External Donors' Properties Used in Propylene Polymerisation
Hansenne, C., Pattou, D., Standaert, D., Debras, G., Leherte, L. & Vercauteren, D., 2002, Abstracts of Papers of the American Chemical Society . Vol. 223. p. POLY-325
Theoretical Analysis of Hydrolysis of Sulfur Fluorides SFn (n = 3 - 6) in the Gas Phase
Larin, A., Meurice, N., Leherte, L., Rajzmann, M., Vercauteren, D. & Trubnikov, D., 2001, Gaseous Dielectrics IX. Christophorou, L. G. & Olthoff, J. K. (eds.). p. 425-430 6 p.
Determination of Screening Descriptors for Chemical Reaction Databases
Dury, L., Leherte, L. & Vercauteren, D., 2000, Principles of Data Mining and Knowledge Discovery: 4th European Conference, PKDD 2000. Zighed, D. A., Komorowski, J. & Zytkow, J. (eds.). Vol. 1910. p. 388-394 7 p.
Development of a Genetic Algorithm Method Especially Designed for the Comparison of Molecular Models: Application to the Elucidation of the Benzodiazepine Receptor Pharmacophore
Meurice, N., Leherte, L., Vercauteren, D., Bourguignon, J-J. & Wermuth, C-G., 1997, Computer-Assisted Lead Finding and Optimization. Eds., H., de, V., Waterbeemd, B. & Testa, G. (eds.). Bâle: Verlag Helvetica Chimica Acta, p. 497-509 13 p.
Elaboration of an Original Genetic Algorithm Approach Dedicated to the Study of Molecular Similarity
Meurice, N., Leherte, L. & Vercauteren, D., 1997, Abstracts of Papers of the American Chemical Society. Vol. 213. p. COMP-393
Development of a Genetic Algorithm Method Applied to the Study of the Similarity between Ligands for the Benzodiazepine Receptors
Topological Analysis Approach for the Comparison of Structurally Different Molecules - Applications to Benzodiazepine-Related Compounds
Leherte, L., Meurice, N. & Vercauteren, D., 1996, Abstracts of Papers of the American Chemical Society. Vol. 212. p. COMP-185
Knowledge Representation Tools for Molecular Scene Analysis
Glasgow, J., Fortier, S., Conklin, D., Allen, F. & Leherte, L., 1995, Computer Tools for Molecular Modeling in the Biotechnology : Proceedings of the 28th Annual Hawaii International Conference . Hunter, L. & Shriver, B. D. (eds.). Vol. V. p. 217-226 10 p.
Shape information from a critical point analysis of calculated electron density maps: Application to DNA-drug systems
Leherte, L., Allen, F. & Vercauteren, D., 1995, E.C.C.C.1 Computational Chemistry. Vol. 330. p. 801-801 1 p.
Topological Analysis of Electron Density Maps of Cyclodextrin Complexes with Chiral Guests
Latour, T., Leherte, L. & Vercauteren, D., 1995, E.C.C.C.1 Computational Chemistry. Vol. 330. p. 802-802 1 p.
Topological Analysis of Protein Electron-Density Maps - Rules for the Recognition of Secondary Structure Patterns
Leherte, L. & Vercauteren, D., 1995, Abstracts of Papers of the American Chemical Society. Vol. 209. p. COMP-166
Utility of Orthographic Projections in Molecular Graphics Applications
Baudoux, G., Leherte, L., Vanderveken, D., Durant, F., Paddeu, G. & Vercauteren, D., 1995, Scientific Visualization 95: proceedings of the international symposium, Caligari, Italy, 27-29 September 1995. Scateni, R. (ed.). Singapore: World Scientific, p. 112-122 11 p.
Computerized Structural-Analysis of Zeolitic Networks - Conceptualization of a Zeolite Secne through Graphs Comparison
Latour, T., Leherte, L. & Vercauteren, D., 1994, Abstracts of Papers of the American Chemical Society. Vol. 208. p. COMP-101
Molecular Graphics Applied to the Investigations of Sorbates in Zeolites
Vercauteren, D., Leherte, L., Vanderveken, D. & Derouane, E., 1993, Computer Aided Innovation of New Materials II. Doyama, M., Kihara, J., Tanaka, M. & Yamamoto, R. (eds.). p. 997-1002 6 p.
Molecular Modelling and Molecular Graphics of Sorbates in Molecular Sieves
Vercauteren, D., Leherte, L., Vanderveken, D., Horsley, J., Freeman, C. & Derouane, E., 1993, Elementary Reaction Steps in Heterogeneous Catalysis. Joyner, R. W. & van, R. A. (eds.). Vol. 398. p. 389-401 13 p.
A Theoretical View and Approach to the Physics and Chemistry of Zeolites and Molecular Sieves
Derouane, E., André, J-M., Leherte, L., Galet, P., Vanderveken, D., Vercauteren, D. & Fripiat, J., 1990, Theoretical Aspects of Heterogeneous Catalysis. Moffat, J. B. (ed.). p. 1-51 51 p.


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Home 1 › Silver dirham of the famous Toqtamysh Khan (782-801 AH/1380-1398 AD), Qirim mint , Juchid Jochid Mongols (Sagdeyeva #390) 2
Silver dirham of the famous Toqtamysh Khan (782-801 AH/1380-1398 AD), Qirim mint , Juchid Jochid Mongols (Sagdeyeva #390)
Al-Sultan al-adil Tokhtamish Khan ("The Just Sultan Tokhtamish Khan") within a scallopped cirle / Zarb fi balad (Qirim), no date ("Minted in the city (of Qirim)") within a complex double multifoil. 15mm, 0.94 grams. Mint of Qirim . Sagdeeva (2005) #390.
Tokhtamysh was the prominent khan of the White Horde, who briefly unified the White Horde and Blue Horde subdivisions of the Golden Horde into a single state. He was a descendant of Genghis Khan's eldest grandson, Orda Khan or his brother Tuqa-Timur. Tokhtamysh appears in history in 1376, trying to overthrow his uncle Urus Khan, ruler of the White Horde, and fleeing to the great Timur. Tokhtamysh dreamed of emulating his ancestors and made plans to reunite the Ulus Jochi. In 1380, he invaded the Blue Horde by fording across the Volga. The ruler of the Blue Horde, Mamai, was killed shortly after the Battle of Kulikovo, making Tokhtamysh's victory over the horde all the easier. In just six years, Tokhtamysh had reunified the Mongol lands from to Lake Balkhash. Believing he could emulate the successes of Genghis Khan he suddenly invaded Transoxiana, the heart of Timur's realm. This proved to be his downfall, as Timur attacked the Golden Horde and defeated Tokhtamysh at the Terek. Timur sacked the capital, Sarai Berke, vassalized the Golden Horde, and placed a puppet ruler, Koirichak, on the throne of Orda's Ulus and appointed Temur Qutlugh khan of the Horde. Tokhtamysh escaped to the Ukrainian steppes and asked for help from the Grand Duke Vytautas of Lithuania. In the great Battle of the Vorskla River (1399) the combined forces of Tokhtamysh and Vytautas were defeated by two of Timur's generals, khan Temur Qutlugh and emir (murza, visir) Edigu. The defeated Tokhtamysh was killed in Tyumen by Edigu's men in 1406.
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