text:
Shielding effect

In chemistry, the shielding effect sometimes referred to as atomic shielding or electron shielding describes the attraction between an electron and the nucleus in any atom with more than one electron. The shielding effect can be defined as a reduction in the effective nuclear charge on the electron cloud, due to a difference in the attraction forces on the electrons in the atom. It is a special case of electric-field screening. This effect also has some significance in many projects in material sciences.

Strength per electron shell
The wider the electron shells are in space, the weaker is the electric interaction between the electrons and the nucleus due to screening. In general we can order the electron shells (s,p,d,f) as such $$S(\mathrm{s}) > S(\mathrm{p}) > S(\mathrm{d}) > S(\mathrm{f}) ,$$ where S is the screening strength that a given orbital provides to the rest of the electrons.

Description
In hydrogen, or any other atom in group 1A of the periodic table (those with only one valence electron), the force on the electron is just as large as the electromagnetic attraction from the nucleus of the atom. However, when more electrons are involved, each electron (in the nth-shell) experiences not only the electromagnetic attraction from the positive nucleus, but also repulsion forces from other electrons in shells from 1 to n. This causes the net force on electrons in outer shells to be significantly smaller in magnitude; therefore, these electrons are not as strongly bonded to the nucleus as electrons closer to the nucleus. This phenomenon is often referred to as the orbital penetration effect. The shielding theory also contributes to the explanation of why valence-shell electrons are more easily removed from the atom.

Additionally, there is also a shielding effect that occurs between sublevels within the same principal energy level. An electron in the s-sublevel is capable of shielding electrons in the  p-sublevel of the same principal energy level.

The size of the shielding effect is difficult to calculate precisely due to effects from quantum mechanics. As an approximation, we can estimate the effective nuclear charge on each electron by the following:

 * $$Z_\mathrm{eff}=Z- \sigma \,$$

Where Z is the number of protons in the nucleus and $$\sigma\,$$ is the average number of electrons between the nucleus and the electron in question. $$\sigma\,$$can be found by using quantum chemistry and the Schrödinger equation, or by using Slater's empirical formulas.

In Rutherford backscattering spectroscopy, the correction due to electron screening modifies the Coulomb repulsion between the incident ion and the target nucleus at large distances. It is the repulsion effect caused by the inner electron on the outer electron.

title: Shielding effect


text:
Polylogarithm

In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function $Lis(z)$ of order $s$ and argument $z$. Only for special values of $s$ does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams.

The polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special cases of the Lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions, nor with the offset logarithmic integral $Li(z)$, which has the same notation without the subscript.

The polylogarithm function is defined by a power series in $z$, which is also a Dirichlet series in $s$: $$\operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s} = z + {z^2 \over 2^s} + {z^3 \over 3^s} + \cdots$$

This definition is valid for arbitrary complex order $s$ and for all complex arguments $z$ with $Li–3(z)$; it can be extended to $Li–2(z)$ by the process of analytic continuation. (Here the denominator $ks$ is understood as $Li–1(z)$). The special case $Li0(z)$ involves the ordinary natural logarithm, $Li1(z)$, while the special cases $Li2(z)$ and $Li3(z)$ are called the dilogarithm (also referred to as Spence's function) and trilogarithm respectively. The name of the function comes from the fact that it may also be defined as the repeated integral of itself: $$\operatorname{Li}_{s+1}(z) = \int_0^z \frac{\operatorname{Li}_s(t)}{t} dt$$ thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orders $s$, the polylogarithm is a rational function.

Properties
In the case where the order $$s$$ is an integer, it will be represented by $$s=n$$ (or $$s=-n$$ when negative). It is often convenient to define $$\mu=\ln(z)$$ where $$\ln(z)$$ is the principal branch of the complex logarithm $$\operatorname{Ln}(z)$$ so that $$-\pi< \operatorname{Im}(\mu) \le \pi.$$ Also, all exponentiation will be assumed to be single-valued: $$z^s = \exp(s\ln(z)).$$

Depending on the order $$s$$, the polylogarithm may be multi-valued. The principal branch of $$\operatorname{Li}_s(z)$$ is taken to be given for $$| z | < 1$$ by the above series definition and taken to be continuous except on the positive real axis, where a cut is made from $$z = 1$$ to $$\infty$$ such that the axis is placed on the lower half plane of $z$. In terms of $\mu$, this amounts to $$-\pi < \arg(-\mu)\le \pi $$. The discontinuity of the polylogarithm in dependence on $$\mu$$ can sometimes be confusing.

For real argument $$z$$, the polylogarithm of real order $$s$$ is real if $z < 1$, and its imaginary part for $$z \ge 1$$ is :

$$\operatorname{Im}\left( \operatorname{Li}_s(z) \right) = -{{\pi \mu^{s-1}}\over{\Gamma(s)}}.$$

Going across the cut, if ε is an infinitesimally small positive real number, then:

$$\operatorname{Im}\left( \operatorname{Li}_s(z+i\epsilon) \right) = {{\pi \mu^{s-1}}\over{\Gamma(s)}}.$$

Both can be concluded from the series expansion (see below) of Lis(e) about μ = 0.

The derivatives of the polylogarithm follow from the defining power series:

$$z \frac{\partial \operatorname{Li}_s(z) }{ \partial z} = \operatorname{Li}_{s-1}(z)$$ $$\frac{\partial \operatorname{Li}_s(e^\mu) }{ \partial \mu} = \operatorname{Li}_{s-1}(e^\mu).$$

The square relationship is seen from the series definition, and is related to the duplication formula (see also, ):

$$\operatorname{Li}_s(-z) + \operatorname{Li}_s(z) = 2^{1-s} \operatorname{Li}_s(z^2).$$

Kummer's function obeys a very similar duplication formula. This is a special case of the multiplication formula, for any positive integer p:

$$\sum_{m=0}^{p-1} \operatorname{Li}_s(z e^{2\pi i m/p}) = p^{1-s} \operatorname{Li}_s(z^p),$$

which can be proved using the series definition of the polylogarithm and the orthogonality of the exponential terms (see e.g. discrete Fourier transform).

Another important property, the inversion formula, involves the Hurwitz zeta function or the Bernoulli polynomials and is found under relationship to other functions below.

Particular values

For particular cases, the polylogarithm may be expressed in terms of other functions (see below). Particular values for the polylogarithm may thus also be found as particular values of these other functions.

  For integer values of the polylogarithm order, the following explicit expressions are obtained by repeated application of z·∂/∂z to Li1(z): $$\operatorname{Li}_{1}(z) = -\ln(1-z)$$ $$\operatorname{Li}_{0}(z) = {z \over 1-z}$$ $$\operatorname{Li}_{-1}(z) = {z \over (1-z)^2}$$ $$\operatorname{Li}_{-2}(z) = {z (1+z) \over (1-z)^3}$$ $$\operatorname{Li}_{-3}(z) = {z (1+4z+z^2) \over (1-z)^4}$$ $$\operatorname{Li}_{-4}(z) = {z (1+z) (1+10z+z^2) \over (1-z)^5} .$$

Accordingly the polylogarithm reduces to a ratio of polynomials in z, and is therefore a rational function of z, for all nonpositive integer orders. The general case may be expressed as a finite sum: $$\operatorname{Li}_{-n}(z) = \left(z {\partial \over \partial z} \right)^n {z \over {1-z}} = \sum_{k=0}^n k! S(n+1, k+1) \left({z \over {1-z}} \right)^{k+1} \qquad (n=0,1,2,\ldots),$$ where S(n,k) are the Stirling numbers of the second kind. Equivalent formulae applicable to negative integer orders are : $$\operatorname{Li}_{-n}(z) = (-1)^{n+1} \sum_{k=0}^n k! S(n+1, k+1) \left({{-1} \over {1-z}} \right)^{k+1} \qquad (n=1,2,3,\ldots),$$

and: $$\operatorname{Li}_{-n}(z) = {1 \over (1-z)^{n+1}} \sum_{k=0}^{n-1} \left\langle {n \atop k} \right\rangle z^{n-k} \qquad (n=1,2,3,\ldots),$$

where $$\scriptstyle \left\langle {n \atop k} \right\rangle$$ are the Eulerian numbers. All roots of Li−n(z) are distinct and real; they include z = 0, while the remainder is negative and centered about z = −1 on a logarithmic scale. As n becomes large, the numerical evaluation of these rational expressions increasingly suffers from cancellation ; full accuracy can be obtained, however, by computing Li−n(z) via the general relation with the Hurwitz zeta function (see below).   Some particular expressions for half-integer values of the argument z are: $$\operatorname{Li}_1(\tfrac12) = \ln 2$$ $$\operatorname{Li}_2(\tfrac12) = \tfrac1{12} \pi^2 - \tfrac12 (\ln 2)^2$$ $$\operatorname{Li}_3(\tfrac12) = \tfrac16 (\ln 2)^3 - \tfrac1{12} \pi^2 \ln 2 + \tfrac78 \zeta(3) ,$$

where ζ is the Riemann zeta function. No formulae of this type are known for higher integer orders, but one has for instance : $$\operatorname{Li}_4(\tfrac12) = \tfrac 1{360} \pi^4 - \tfrac 1{24}(\ln 2)^4 + \tfrac1{24} \pi^2 (\ln 2)^2 - \tfrac 1 2 \zeta(\bar3, \bar1),$$

which involves the alternating double sum $$\zeta(\bar3, \bar1)=\sum_{m>n>0} (-1)^{m+n} m^{-3} n^{-1}.$$

In general one has for integer orders n ≥ 2 : $$\operatorname{Li}_n(\tfrac12) = -\zeta(\bar1, \bar1, \left\{ 1 \right\}^{n-2}),$$

where ζ(s1, …, sk) is the multiple zeta function; for example: $$\operatorname{Li}_5(\tfrac12) = -\zeta(\bar1, \bar1, 1,1,1).$$   As a straightforward consequence of the series definition, values of the polylogarithm at the pth complex roots of unity are given by the Fourier sum: $$ \operatorname{Li}_s(e^{2 \pi i m/p}) = p^{-s} \sum_{k=1}^p e^{2 \pi i m k/p} \zeta(s, \tfrac {k}{p}) \qquad (m = 1, 2, \dots, p-1),$$

where ζ is the Hurwitz zeta function. For Re(s) > 1, where Lis(1) is finite, the relation also holds with m = 0 or m = p. While this formula is not as simple as that implied by the more general relation with the Hurwitz zeta function listed under relationship to other functions below, it has the advantage of applying to non-negative integer values of s as well. As usual, the relation may be inverted to express ζ(s, m⁄p) for any m = 1, …, p as a Fourier sum of Lis(exp(2πi k⁄p)) over k = 1, …, p.

Relationship to other functions
i^{-s} \operatorname{Li}_s(e^{2\pi i x}) + i^s \operatorname{Li}_s(e^{-2\pi i x}) = \frac{(2\pi)^s}{\Gamma(s)} \zeta(1 - s, x), $$ which relation holds for $|z| < 1$ if $|z| ≥ 1$, and for $exp(s ln k)$ if $s = 1$. Equivalently, for all complex s and for complex $Li1(z) = −ln(1−z)$, the inversion formula reads $$ \operatorname{Li}_s(z) + (-1)^s \operatorname{Li}_s(1/z) = \frac{(2\pi i)^s}{\Gamma(s)} ~\zeta \left(1 - s, ~\frac{1}{2} + {\ln(-z) \over {2\pi i}} \right), $$ and for all complex s and for complex $s = 2$ $$ \operatorname{Li}_s(z) + (-1)^s \operatorname{Li}_s(1/z) = {(2\pi i)^s \over \Gamma(s)} ~\zeta \left(1 - s, ~\frac{1}{2} - {\ln(-1/z) \over {2\pi i}} \right). $$ For $s = 3$, one has $z = 1$, and both expressions agree. These relations furnish the analytic continuation of the polylogarithm beyond the circle of convergence |z| = 1 of the defining power series. (The corresponding equation of and  is not correct if one assumes that the principal branches of the polylogarithm and the logarithm are used simultaneously.) See the next item for a simplified formula when s is an integer. $$ More generally, one has for $η(s)$: $$\begin{align} \operatorname{Li}_{n}(z) + (-1)^n \operatorname{Li}_{n}(1/z) &= -\frac{(2\pi i)^n}{n!} B_n \left( \frac{1}{2} + {\ln(-z) \over {2\pi i}} \right) & (z \not\in ]0;1]), \\ \operatorname{Li}_{n}(z) + (-1)^n \operatorname{Li}_{n}(1/z) &= -\frac{(2\pi i)^n}{n!} B_n \left( \frac{1}{2} - {\ln(-1/z) \over {2\pi i}} \right) & (z \not\in ~]1;\infty[), \end{align}$$ where both expressions agree for $β(s)$. (The corresponding equation of and  is again not correct.) \chi_s(z) = \tfrac {1}{2} \left[ \operatorname{Li}_s(z) - \operatorname{Li}_s(-z) \right].$$ \operatorname{Li}_n(z) &= z _{n+1}F_{n} (1,1,\dots,1; 2,2,\dots,2; z) & (n = 0,1,2,\ldots), \\ \operatorname{Li}_{-n}(z) &= z _{n}F_{n-1} (2,2,\dots,2; 1,1,\dots,1; z) & (n = 1,2,3,\ldots) ~. \end{align}$$ Z_n(z) = {1 \over (n - 1)!} \int_z^\infty {t^{n-1} \over e^t-1} dt \qquad (n = 1,2,3,\ldots) , $$ the polylogarithm Lin(z) for positive integer n may be expressed as the finite sum : $$ \operatorname{Li}_{n}(e^\mu) = \sum_{k=0}^{n-1} Z_{n-k}(-\mu) {\mu^k \over k!} \qquad (n = 1,2,3,\ldots). $$ A remarkably similar expression relates the "Debye functions" Zn(z) to the polylogarithm: $$Z_n(z) = \sum_{k=0}^{n-1} \operatorname{Li}_{n-k}(e^{-z}) {z^k \over k!} \qquad (n = 1,2,3,\ldots) .$$ \sum_{n=1}^\infty\frac{z^nJ_{-s}(n)}{1-z^n}=\operatorname{Li}_{s}(z).$$
 * For $Γ(1 − s)$, the polylogarithm reduces to the Riemann zeta function $$\operatorname{Li}_s(1) = \zeta(s) \qquad (\operatorname{Re}(s)>1).$$
 * The polylogarithm is related to Dirichlet eta function and the Dirichlet beta function: $$ \operatorname{Li}_s(-1) = -\eta(s),$$ where $s = 0$ is the Dirichlet eta function. For pure imaginary arguments, we have: $$\operatorname{Li}_s(\pm i) = -2^{-s} \eta(s) \pm i\beta(s),$$ where $0 ≤ Re(x) < 1$ is the Dirichlet beta function.
 * The polylogarithm is related to the complete Fermi–Dirac integral as: $$F_s(\mu) = -\operatorname{Li}_{s+1}(-e^\mu).$$
 * The polylogarithm is related to the complete Bose–Einstein integral as: $$G_s(\mu) = \operatorname{Li}_{s+1}(e^\mu).$$
 * The polylogarithm is a special case of the incomplete polylogarithm function $$ \operatorname{Li}_s(z) = \operatorname{Li}_s(0,z) .$$
 * The polylogarithm is a special case of the Lerch transcendent $$\operatorname{Li}_s(z) = z\Phi(z,s,1).$$
 * The polylogarithm is related to the Hurwitz zeta function by: $$\operatorname{Li}_s(z) = {\Gamma(1 - s) \over (2\pi)^{1-s}} \left[i^{1-s} \zeta \left(1 - s, \frac{1}{2} + {\ln(-z) \over {2\pi i}} \right) + i^{s-1} ~\zeta \left(1 - s, \frac{1}{2} - {\ln(-z) \over {2\pi i}} \right) \right],$$ which relation, however, is invalidated at positive integer s by poles of the gamma function $Im(x) ≥ 0$, and at $0 < Re(x) ≤ 1$ by a pole of both zeta functions; a derivation of this formula is given under series representations below. With a little help from a functional equation for the Hurwitz zeta function, the polylogarithm is consequently also related to that function via : $$
 * For positive integer polylogarithm orders s, the Hurwitz zeta function ζ(1−s, x) reduces to Bernoulli polynomials, $Im(x) < 0$, and Jonquière's inversion formula for n = 1, 2, 3, … becomes: $$\operatorname{Li}_{n}(e^{2\pi i x}) + (-1)^n \operatorname{Li}_{n}(e^{-2\pi i x}) = -{(2\pi i)^n \over n!} B_n(x),$$ where again 0 ≤ Re(x) < 1 if Im(x) ≥ 0, and 0 < Re(x) ≤ 1 if Im(x) < 0. Upon restriction of the polylogarithm argument to the unit circle, Im(x) = 0, the left hand side of this formula simplifies to 2 Re(Lin(e2πix)) if n is even, and to 2i Im(Lin(e2πix)) if n is odd. For negative integer orders, on the other hand, the divergence of Γ(s) implies for all z that : $$\operatorname{Li}_{-n}(z) + (-1)^n \operatorname{Li}_{-n}(1/z) = 0 \qquad (n = 1,2,3,\ldots).
 * The polylogarithm with pure imaginary μ may be expressed in terms of the Clausen functions Cis(θ) and Sis(θ), and vice versa :$$\operatorname{Li}_s(e^{\pm i \theta}) = Ci_s(\theta) \pm i Si_s(\theta).$$
 * The inverse tangent integral $z ∉ (0, 1]$ can be expressed in terms of polylogarithms: $$\operatorname{Ti}_s(z) = {1 \over 2i} \left[ \operatorname{Li}_s(i z) - \operatorname{Li}_s(-i z) \right].$$ The relation in particular implies: $$\operatorname{Ti}_0(z) = {z \over 1+z^2}, \quad \operatorname{Ti}_1(z) = \arctan z, \quad \operatorname{Ti}_2(z) = \int_0^z {\arctan t \over t} dt, \quad \ldots~ \quad \operatorname{Ti}_{n+1}(z) = \int_0^z \frac{\operatorname{Ti}_n(t)}{t} dt,$$ which explains the function name.
 * The Legendre chi function χs(z) can be expressed in terms of polylogarithms: $$
 * The polylogarithm of integer order can be expressed as a generalized hypergeometric function: $$\begin{align}
 * In terms of the incomplete zeta functions or "Debye functions" : $$
 * Using Lambert series, if $$J_s(n)$$ is Jordan's totient function, then $$

Integral representations
Any of the following integral representations furnishes the analytic continuation of the polylogarithm beyond the circle of convergence |z| = 1 of the defining power series.   The polylogarithm can be expressed in terms of the integral of the Bose–Einstein distribution: $$\operatorname{Li}_{s}(z) = {1 \over \Gamma(s)} \int_0^\infty {t^{s-1} \over e^t/z-1} dt .$$

This converges for Re(s) > 0 and all z except for z real and ≥ 1. The polylogarithm in this context is sometimes referred to as a Bose integral but more commonly as a Bose–Einstein integral. Similarly, the polylogarithm can be expressed in terms of the integral of the Fermi–Dirac distribution: $$-\operatorname{Li}_{s}(-z) = \frac{1}{\Gamma(s)} \int_0^\infty {t^{s-1} \over e^t/z+1} dt .$$

This converges for $z ∉ (1, ∞)$ and all $z$ except for z real and ≤ −1. The polylogarithm in this context is sometimes referred to as a Fermi integral or a Fermi–Dirac integral. These representations are readily verified by Taylor expansion of the integrand with respect to z and termwise integration. The papers of Dingle contain detailed investigations of both types of integrals.

The polylogarithm is also related to the integral of the Maxwell–Boltzmann distribution: $$ \lim_{z \to 0} \frac{\operatorname{Li}_{s}(z)}{z} = {1 \over \Gamma(s)} \int_0^\infty {t^{s-1} e^{-t}} dt = 1. $$

This also gives the asymptotic behavior of polylogarithm at the vicinity of origin.   A complementary integral representation applies to Re(s) < 0 and to all z except to z real and ≥ 0: $$\operatorname{Li}_{s}(z) =\int_0^\infty {t^{-s} \sin[s \pi /2 - t \ln(-z)] \over \sinh(\pi t)} dt .$$

This integral follows from the general relation of the polylogarithm with the Hurwitz zeta function (see above) and a familiar integral representation of the latter.   The polylogarithm may be quite generally represented by a Hankel contour integral, which extends the Bose–Einstein representation to negative orders s. As long as the t = μ pole of the integrand does not lie on the non-negative real axis, and s ≠ 1, 2, 3, …, we have: $$\operatorname{Li}_s(e^\mu) = -{{\Gamma(1 - s)} \over {2\pi i}} \oint_H {{(-t)^{s-1}} \over {e^{t-\mu}-1}} dt$$

where H represents the Hankel contour. The integrand has a cut along the real axis from zero to infinity, with the axis belonging to the lower half plane of t. The integration starts at +∞ on the upper half plane (Im(t) > 0), circles the origin without enclosing any of the poles t = μ + 2kπi, and terminates at +∞ on the lower half plane (Im(t) < 0). For the case where μ is real and non-negative, we can simply subtract the contribution of the enclosed t = μ pole: $$\operatorname{Li}_s(e^\mu) = -{{\Gamma(1 - s)} \over {2\pi i}} \oint_H {{(-t)^{s-1}} \over {e^{t-\mu}}-1} dt - 2\pi i R$$

where R is the residue of the pole: $$R = {i \over 2\pi} \Gamma(1 - s) (-\mu)^{s-1} .$$   When the Abel–Plana formula is applied to the defining series of the polylogarithm, a Hermite-type integral representation results that is valid for all complex z and for all complex s: $$\operatorname{Li}_s(z) = \tfrac{1}{2}z + {\Gamma(1 - s, -\ln z) \over (-\ln z)^{1-s}} + 2z \int_0^\infty \frac{\sin(s\arctan t - t\ln z)} {(1+t^2)^{s/2} (e^{2\pi t}-1)} dt$$

where Γ is the upper incomplete gamma-function. All (but not part) of the ln(z) in this expression can be replaced by −ln(1⁄z). A related representation which also holds for all complex s, $$\operatorname{Li}_s(z) = \tfrac{1}{2}z + z \int_0^\infty \frac{\sin[s \arctan t - t \ln(-z)]} {(1+t^2)^{s/2} \sinh(\pi t)} dt ,$$

avoids the use of the incomplete gamma function, but this integral fails for z on the positive real axis if Re(s) ≤ 0. This expression is found by writing 2s Lis(−z) / (−z) = Φ(z2, s, 1⁄2) − z Φ(z2, s, 1), where Φ is the Lerch transcendent, and applying the Abel–Plana formula to the first Φ series and a complementary formula that involves 1 / (e2πt + 1) in place of 1 / (e2πt − 1) to the second Φ series. </li> <li> We can express an integral for the polylogarithm by integrating the ordinary geometric series termwise for $$s \in \mathbb{N}$$ as $$\operatorname{Li}_{s+1}(z) = \frac{z \cdot (-1)^s}{s!} \int_0^1 \frac{\log^s(t)}{1-tz} dt. $$ </li> </ol>

Series representations
<ol> <li> As noted under integral representations above, the Bose–Einstein integral representation of the polylogarithm may be extended to negative orders s by means of Hankel contour integration: $$\operatorname{Li}_s(e^\mu) = -{\Gamma(1 - s) \over 2\pi i} \oint_H {(-t)^{s-1} \over e^{t-\mu}-1} dt,$$

where H is the Hankel contour, s ≠ 1, 2, 3, …, and the t = μ pole of the integrand does not lie on the non-negative real axis. The contour can be modified so that it encloses the poles of the integrand at t − μ = 2kπi, and the integral can be evaluated as the sum of the residues : $$\operatorname{Li}_s(e^\mu) = \Gamma(1 - s) \sum_{k=-\infty}^\infty (2k \pi i - \mu)^{s-1}.$$

This will hold for Re(s) < 0 and all μ except where eμ = 1. For 0 < Im(μ) ≤ 2π the sum can be split as: $$\operatorname{Li}_s(e^\mu) = \Gamma(1-s) \left[ (-2\pi i)^{s-1} \sum_{k=0}^\infty \left(k + {\mu \over {2\pi i}} \right)^{s-1} + (2\pi i)^{s-1} \sum_{k=0}^\infty \left(k+1- {\mu \over {2\pi i}} \right)^{s-1} \right],$$

where the two series can now be identified with the Hurwitz zeta function: $$\operatorname{Li}_s(e^\mu) = {\Gamma(1 - s) \over (2\pi)^{1-s}} \left[i^{1-s} ~\zeta \left(1 - s, ~{\mu \over {2\pi i}} \right) + i^{s-1} ~\zeta \left(1 - s, ~1 - {\mu \over {2\pi i}} \right) \right] \qquad (0 < \operatorname{Im}(\mu) \leq 2\pi) .$$

This relation, which has already been given under relationship to other functions above, holds for all complex s ≠ 0, 1, 2, 3, … and was first derived in. </li> <li> In order to represent the polylogarithm as a power series about μ = 0, we write the series derived from the Hankel contour integral as: $$\operatorname{Li}_s(e^\mu) = \Gamma(1 - s) (-\mu)^{s-1} + \Gamma(1 - s) \sum_{h=1}^\infty \left[(-2 h \pi i - \mu)^{s-1} + (2 h \pi i - \mu)^{s-1} \right] .$$

When the binomial powers in the sum are expanded about μ = 0 and the order of summation is reversed, the sum over h can be expressed in closed form: $$\operatorname{Li}_s(e^\mu) = \Gamma(1 - s) (-\mu)^{s-1} + \sum_{k=0}^\infty {\zeta(s-k) \over k!} \mu^k .$$

This result holds for |μ| < 2π and, thanks to the analytic continuation provided by the zeta functions, for all s ≠ 1, 2, 3, …. If the order is a positive integer, s = n, both the term with k = n − 1 and the gamma function become infinite, although their sum does not. One obtains : $$\lim_{s \to k+1} \left[ {\zeta(s-k) \over k!} \mu^k + \Gamma(1 - s) (-\mu)^{s-1} \right] = {\mu^k \over k!} \left[\sum_{h=1}^k {1 \over h} - \ln(-\mu) \right],$$

where the sum over h vanishes if k = 0. So, for positive integer orders and for |μ| < 2π we have the series: $$\operatorname{Li}_{n}(e^\mu) = {\mu^{n-1} \over (n-1)!} \left[ H_{n-1} - \ln(-\mu) \right] + \sum_{k=0,k\ne n-1}^\infty {\zeta(n-k) \over k!} \mu^k ,$$

where Hn denotes the nth harmonic number: $$H_n = \sum_{h=1}^n {1 \over h}, \qquad H_0 = 0.$$

The problem terms now contain −ln(−μ) which, when multiplied by μn−1, will tend to zero as μ → 0, except for n = 1. This reflects the fact that Lis(z) exhibits a true logarithmic singularity at s = 1 and z = 1 since: $$\lim_{\mu \to 0} \Gamma(1-s)(-\mu)^{s-1} = 0 \qquad (\operatorname{Re}(s) > 1).$$

For s close, but not equal, to a positive integer, the divergent terms in the expansion about μ = 0 can be expected to cause computational difficulties. Erdélyi's corresponding expansion in powers of ln(z) is not correct if one assumes that the principal branches of the polylogarithm and the logarithm are used simultaneously, since ln(1⁄z) is not uniformly equal to −ln(z).

For nonpositive integer values of s, the zeta function ζ(s − k) in the expansion about μ = 0 reduces to Bernoulli numbers: ζ(−n − k) = −B1+n+k / (1 + n + k). Numerical evaluation of Li−n(z) by this series does not suffer from the cancellation effects that the finite rational expressions given under particular values above exhibit for large n. </li> <li> By use of the identity $$1 = {1 \over \Gamma(s)} \int_0^\infty e^{-t} t^{s-1} dt \qquad (\operatorname{Re}(s) > 0) ,$$

the Bose–Einstein integral representation of the polylogarithm (see above) may be cast in the form: $$\operatorname{Li}_s(z) = \tfrac{1}{2}z + {z \over 2 \Gamma(s)} \int_0^\infty e^{-t} t^{s-1} \coth{t - \ln z \over 2} dt \qquad (\operatorname{Re}(s) > 0).$$

Replacing the hyperbolic cotangent with a bilateral series, $$\coth{t-\ln z \over 2} = 2 \sum_{k = -\infty}^\infty {1 \over 2 k \pi i + t - \ln z} ,$$

then reversing the order of integral and sum, and finally identifying the summands with an integral representation of the upper incomplete gamma function, one obtains: $$\operatorname{Li}_s(z) = \tfrac{1}{2}z + \sum_{k = -\infty}^\infty {\Gamma(1-s, 2k \pi i - \ln z) \over (2k \pi i - \ln z)^{1-s}}.$$

For both the bilateral series of this result and that for the hyperbolic cotangent, symmetric partial sums from −kmax to kmax converge unconditionally as kmax → ∞. Provided the summation is performed symmetrically, this series for Lis(z) thus holds for all complex s as well as all complex z. </li> <li> Introducing an explicit expression for the Stirling numbers of the second kind into the finite sum for the polylogarithm of nonpositive integer order (see above) one may write: $$\operatorname{Li}_{-n}(z) = \sum_{k=0}^n \left( {-z \over 1-z} \right)^{k+1} \sum_{j=0}^k (-1)^{j+1} {k \choose j} (j+1)^n \qquad (n=0,1,2,\ldots).$$

The infinite series obtained by simply extending the outer summation to ∞ : $$\operatorname{Li}_s(z) = \sum_{k=0}^\infty \left( {-z \over 1-z} \right)^{k+1} ~\sum_{j=0}^k (-1)^{j+1} {k \choose j} (j+1)^{-s} ,$$

turns out to converge to the polylogarithm for all complex s and for complex z with Re(z) < 1⁄2, as can be verified for |−z⁄(1−z)| < 1⁄2 by reversing the order of summation and using: $$\sum_{k=j}^\infty {k \choose j} \left( {-z \over 1-z} \right)^{k+1} = \left[ \left( {-z \over 1-z} \right)^{-1} -1 \right]^{-j-1} = (-z)^{j+1}.$$

The inner coefficients of these series can be expressed by Stirling-number-related formulas involving the generalized harmonic numbers. For example, see generating function transformations to find proofs (references to proofs) of the following identities: $$\begin{align} \operatorname{Li}_2(z) &= \sum_{j \geq 1} \frac{(-1)^{j-1}}{2} \left(H_j^2+H_j^{(2)}\right) \frac{z^j}{(1-z)^{j+1}} \\ \operatorname{Li}_3(z) &= \sum_{j \geq 1} \frac{(-1)^{j-1}}{6} \left(H_j^3+3H_j H_j^{(2)} + 2 H_j^{(3)}\right) \frac{z^j}{(1-z)^{j+1}}. \end{align}$$

For the other arguments with Re(z) < 1⁄2 the result follows by analytic continuation. This procedure is equivalent to applying Euler's transformation to the series in z that defines the polylogarithm. </li> </ol>

Asymptotic expansions
For |z| ≫ 1, the polylogarithm can be expanded into asymptotic series in terms of ln(−z):

$$\operatorname{Li}_s(z) = {\pm i\pi \over \Gamma(s)} [\ln(-z) \pm i\pi]^{s-1} - \sum_{k = 0}^\infty (-1)^k (2\pi)^{2k} {B_{2k} \over (2k)!} {[\ln(-z) \pm i\pi]^{s-2 k} \over \Gamma(s+1-2k)},$$ $$\operatorname{Li}_s(z) = \sum_{k=0}^\infty (-1)^k (1-2^{1-2k}) (2\pi)^{2k} {B_{2k} \over (2k)!} {[\ln(-z)]^{s-2 k} \over \Gamma(s+1-2k)},$$

where B2k are the Bernoulli numbers. Both versions hold for all s and for any arg(z). As usual, the summation should be terminated when the terms start growing in magnitude. For negative integer s, the expansions vanish entirely; for non-negative integer s, they break off after a finite number of terms. describes a method for obtaining these series from the Bose–Einstein integral representation (his equation 11.2 for Lis(eμ) requires −2π < Im(μ) ≤ 0).

Limiting behavior
The following limits result from the various representations of the polylogarithm :

$$\operatorname{Li}_s(z) \sim_{|z|\to 0} z$$ $$\operatorname{Li}_s(e^\mu) \sim_{|\mu|\to 0} \Gamma(1 - s) (-\mu)^{s-1} \qquad (\operatorname{Re}(s) < 1)$$ $$\operatorname{Li}_s(\pm e^\mu) \sim_{\operatorname{Re}(\mu) \to \infty} -{\mu^s \over \Gamma(s+1)}\qquad (s \ne -1, -2, -3, \ldots)$$ $$\operatorname{Li}_{-n}(e^\mu) \sim_{\operatorname{Re}(\mu) \to \infty} -(-1)^n e^{-\mu} \qquad (n = 1, 2, 3, \ldots)$$ $$\operatorname{Li}_s(z) \sim_{\operatorname{Re}(s) \to \infty} z$$ $$\operatorname{Li}_s(e^\mu) \sim_{\operatorname{Re}(s) \to -\infty}  \Gamma(1-s) (-\mu)^{s-1} \qquad (-\pi < \operatorname{Im}(\mu) < \pi)$$ $$\operatorname{Li}_s(-e^\mu) \sim_{\operatorname{Re}(s) \to -\infty} \Gamma(1 - s) \left[ (-\mu - i\pi)^{s-1} + (-\mu + i\pi)^{s-1} \right] \qquad (\operatorname{Im}(\mu) = 0)$$

Wood's first limit for $z ∉ (0, ∞)$ has been corrected in accordance with his equation 11.3. The limit for $ln(−z) = −ln(−^{1}⁄_{z})$ follows from the general relation of the polylogarithm with the Hurwitz zeta function (see above).

Dilogarithm
The dilogarithm is the polylogarithm of order s = 2. An alternate integral expression of the dilogarithm for arbitrary complex argument z is : $$\operatorname{Li}_2 (z) = -\int_0^z{\ln (1-t) \over t} dt = -\int_0^1{\ln (1-zt) \over t} dt.$$

A source of confusion is that some computer algebra systems define the dilogarithm as dilog(z) = Li2(1−z).

In the case of real z ≥ 1 the first integral expression for the dilogarithm can be written as $$\operatorname{Li}_2(z) = \frac{\pi^2}{6} - \int_1^z{\ln(t-1) \over t} dt - i\pi \ln z$$

from which expanding ln(t−1) and integrating term by term we obtain

$$\operatorname{Li}_2(z) = \frac{\pi^2}{3} - \frac{1}{2}(\ln z)^2 - \sum_{k=1}^\infty {1 \over k^2 z^k} - i\pi \ln z \qquad (z \ge 1).$$

The Abel identity for the dilogarithm is given by

$$\operatorname{Li}_2 \left( \frac{x}{1-y} \right) + \operatorname{Li}_2 \left( \frac{y}{1-x} \right) - \operatorname{Li}_2 \left(\frac{xy}{(1-x)(1-y)} \right) = \operatorname{Li}_2(x) + \operatorname{Li}_2(y) + \ln(1-x) \ln(1-y)$$

$$(\operatorname{Re}(x) \le \tfrac{1}{2} \wedge \operatorname{Re}(y) \le \tfrac{1}{2} \vee \operatorname{Im}(x) > 0 \wedge \operatorname{Im}(y) > 0 \vee \operatorname{Im}(x) < 0 \wedge \operatorname{Im}(y) < 0 \vee \ldots).$$

This is immediately seen to hold for either x = 0 or y = 0, and for general arguments is then easily verified by differentiation ∂/∂x ∂/∂y. For y = 1−x the identity reduces to Euler's reflection formula $$\operatorname{Li}_2 \left(x \right) + \operatorname{Li}_2 \left(1-x\right) = \frac{1}{6} \pi^2 - \ln(x)\ln(1-x) ,$$ where Li2(1) = ζ(2) = 1⁄6 π2 has been used and x may take any complex value.

In terms of the new variables u = x/(1−y), v = y/(1−x) the Abel identity reads $$\operatorname{Li}_2(u) + \operatorname{Li}_2(v) - \operatorname{Li}_2(uv) = \operatorname{Li}_2 \left( \frac{u-uv}{1-uv} \right) + \operatorname{Li}_2 \left( \frac{v-uv}{1-uv} \right) + \ln \left( \frac{1-u}{1-uv} \right) \ln\left( \frac{1-v}{1-uv} \right),$$ which corresponds to the pentagon identity given in.

From the Abel identity for x = y = 1−z and the square relationship we have Landen's identity $$\operatorname{Li}_2(1-z) + \operatorname{Li}_2 \left( 1-\frac{1}{z} \right) = - \frac{1}{2} (\ln z)^2 \qquad (z \not \in ~]-\infty; 0]) ,$$ and applying the reflection formula to each dilogarithm we find the inversion formula $$\operatorname{Li}_2(z) + \operatorname{Li}_2(1/z) = -\tfrac{1}{6} \pi^2 - \tfrac{1}{2} [\ln(-z)]^2 \qquad (z \not \in [0; 1[) ,$$

and for real z ≥ 1 also $$\operatorname{Li}_2(z) + \operatorname{Li}_2(1/z) = \tfrac{1}{3} \pi^2 - \tfrac{1}{2} (\ln z)^2 - i\pi \ln z .$$

Known closed-form evaluations of the dilogarithm at special arguments are collected in the table below. Arguments in the first column are related by reflection x ↔ 1−x or inversion x ↔ 1⁄x to either x = 0 or x = −1; arguments in the third column are all interrelated by these operations.

discusses the 17th to 19th century references. The reflection formula was already published by Landen in 1760, prior to its appearance in a 1768 book by Euler ; an equivalent to Abel's identity was already published by Spence in 1809, before Abel wrote his manuscript in 1826. The designation bilogarithmische Function was introduced by Carl Johan Danielsson Hill (professor in Lund, Sweden) in 1828. has remarked that the dilogarithm is the only mathematical function possessing a sense of humor.

 * {| class="wikitable" style="text-align: center;"

! $$x $$ ! $$\operatorname{Li}_2(x) $$ ! $$x $$ ! $$\operatorname{Li}_2(x) $$
 * + Special values of the dilogarithm
 * $$-1 $$
 * $$-\tfrac {1}{12} \pi^2 $$
 * $$-\phi $$
 * $$-\tfrac {1}{10} \pi^2 - \ln^2 \phi $$
 * $$0 $$
 * $$0 $$
 * $$-1 / \phi $$
 * $$-\tfrac {1}{15} \pi^2 + \tfrac {1}{2} \ln^2 \phi $$
 * $$\tfrac {1}{2} $$
 * $$\tfrac {1}{12} \pi^2 - \tfrac {1}{2} \ln^2 2 $$
 * $$1 / \phi^2 $$
 * $$\tfrac {1}{15} \pi^2 - \ln^2 \phi $$
 * $$1 $$
 * $$\tfrac {1}{6} \pi^2 $$
 * $$1 / \phi $$
 * $$\tfrac {1}{10} \pi^2 - \ln^2 \phi $$
 * $$2 $$
 * $$\tfrac {1}{4} \pi^2 - \pi i \ln 2 $$
 * $$\phi $$
 * $$\tfrac {11}{15} \pi^2 + \tfrac {1}{2} \ln^2(-1 / \phi) $$
 * $$\phi^2 $$
 * $$-\tfrac {11}{15} \pi^2 - \ln^2(-\phi) $$
 * }
 * Here $$\phi = \tfrac{1}{2} (\sqrt{5}+1)$$ denotes the golden ratio.
 * $$\tfrac {11}{15} \pi^2 + \tfrac {1}{2} \ln^2(-1 / \phi) $$
 * $$\phi^2 $$
 * $$-\tfrac {11}{15} \pi^2 - \ln^2(-\phi) $$
 * }
 * Here $$\phi = \tfrac{1}{2} (\sqrt{5}+1)$$ denotes the golden ratio.
 * $$-\tfrac {11}{15} \pi^2 - \ln^2(-\phi) $$
 * }
 * Here $$\phi = \tfrac{1}{2} (\sqrt{5}+1)$$ denotes the golden ratio.
 * Here $$\phi = \tfrac{1}{2} (\sqrt{5}+1)$$ denotes the golden ratio.

Polylogarithm ladders
Leonard Lewin discovered a remarkable and broad generalization of a number of classical relationships on the polylogarithm for special values. These are now called polylogarithm ladders. Define $$\rho = \tfrac{1}{2} (\sqrt{5}-1)$$ as the reciprocal of the golden ratio. Then two simple examples of dilogarithm ladders are

$$\operatorname{Li}_2(\rho^6) = 4 \operatorname{Li}_2(\rho^3) + 3 \operatorname{Li}_2(\rho^2) - 6 \operatorname{Li}_2(\rho) + \tfrac {7}{30} \pi^2$$

given by and

$$\operatorname{Li}_2(\rho) = \tfrac{1}{10} \pi^2 - \ln^2\rho$$

given by Landen. Polylogarithm ladders occur naturally and deeply in K-theory and algebraic geometry. Polylogarithm ladders provide the basis for the rapid computations of various mathematical constants by means of the BBP algorithm.

Monodromy
The polylogarithm has two branch points; one at z = 1 and another at z = 0. The second branch point, at z = 0, is not visible on the main sheet of the polylogarithm; it becomes visible only when the function is analytically continued to its other sheets. The monodromy group for the polylogarithm consists of the homotopy classes of loops that wind around the two branch points. Denoting these two by m0 and m1, the monodromy group has the group presentation

$$\langle m_0, m_1 \vert w = m_0 m_1 m^{-1}_0 m^{-1}_1, w m_1 = m_1 w \rangle.$$

For the special case of the dilogarithm, one also has that wm0 = m0w, and the monodromy group becomes the Heisenberg group (identifying m0, m1 and w with x, y, z).

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Polylogarithm


text:
User talk:95chung chi ho

Lets talk about Tuen Mun 's unique and funny side.

Hi - could you say where your images comes from please. Secretlondon 18:50, Mar 1, 2004 (UTC)

from http://www.kcrc.com/annual/anreport/eng/new_projects/west_rail_02.htm

title: User talk:95chung chi ho


text:
Institute for Policy Studies

The Institute for Policy Studies (IPS) is an American progressive  think tank started in 1963 that is based in Washington, D.C. It was directed by John Cavanagh from 1998 to 2021. In 2021 Tope Folarin was announced as new Executive Director. It focuses on U.S. foreign policy, domestic policy, human rights, international economics, and national security.

IPS has been described as one of the five major independent think tanks in Washington. Members of the IPS played key roles in the civil rights and anti-war movements of the 1960s, in the women's and environmental movements of the 1970s, and in the peace, anti-apartheid, and anti-intervention movements of the 1980s.

1960s
The Institute for Policy Studies was founded in 1963 by Marcus Raskin and Richard Barnet as the think tank for "the most powerful of the powerless," according to a 2009 Carnegie Report. The founders were officials in the John F. Kennedy administration —Raskin, then in his twenties, was working as a White House aide for McGeorge Bundy, and Barnet served in a similar role to John J. McCloy. They had become disillusioned by priorities based on politics rather than moral issues.

Against the backdrop of the counterculture of the 1960s, the opposition to United States involvement in the Vietnam War, and the civil rights movement of the 1960s, the Institute for Policy Studies "became a brand name for its unabashedly left-wing tone" in contrast with RAND and the largely conservative think tanks. Members of these movements came to IPS headquarters in Washington, D.C.'s Dupont Circle. In a 2009 interview, Raskin said, "Very quickly, with the Vietnam war, the civil rights movement, the women’s movement, the institute became a place where different people from the movements came. People came in from demonstrations" and "camped out in the offices. Early on [the IPS] had predicted that Vietnam would be a disaster." During the presidency of Lyndon Johnson, Raskin was indicted by the federal government for the 1965 publication of "tens of thousands of copies of an IPS anti-war Vietnam Reader"—a kind of textbook for anti-war teach-ins. He was charged with encouraging people to resist the draft. In 1967, Raskin and IPS Fellow Arthur Waskow penned "A Call to Resist Illegitimate Authority", a document signed by dozens of scholars and religious leaders which helped to launch the draft resistance movement.

In 1964, several leading African-American activists joined the institute's staff and turned IPS into a base for supporting for the Civil Rights Movement. Fellow Bob Moses organized trainings for field organizers of the Student Nonviolent Coordinating Committee on the links between civil rights theory and practice, while Ivanhoe Donaldson initiated an assembly of African-American government officials. Port Huron SDS co-writer and civil rights veteran, IPS Fellow Robb Burlage launched the critical health care justice movement in 1967 with his "Burlage Report". Later Burlage founded the Health Policy Advisory Center, which published the initially monthly bulletin, Health/Pac Bulletin, first in 1968 and thereafter semi-annually and eventually quarterly for nearly 3 decades.

The IPS was also at the forefront of the feminist movement. Fellow Charlotte Bunch organized a significant women's liberation conference in 1966 and later launched two feminist periodicals, Quest and Off Our Backs. Rita Mae Brown wrote and published her notable lesbian coming-of-age novel Rubyfruit Jungle while on the staff in the 1970s.

Raskin's 2018 obituary in The Nation said that for him, "ideas were the seedlings for effective action."

IPS also organized congressional seminars and published numerous books that challenged the national security state, including Gar Alperovitz’s Atomic Diplomacy and Barnet's Intervention and Revolution. IPS was the object of repeated FBI and Internal Revenue Service probes. The Nixon administration placed Barnet and Raskin on its Enemies List.

1970s
In 1971, Raskin received "a mountain of paper" from a source that was later identified as Daniel Ellsberg. These became known as the Pentagon Papers. Raskin played his "customary catalytic role" by putting Ellsberg in touch with New York Times reporter Neil Sheehan.

In 1974, the institute created an Organizing Committee for the Fifth Estate as part of its Center for National Security Studies which published the magazine CounterSpy until 1984.

In 1976, agents of Chilean dictator Augusto Pinochet assassinated two IPS members of staff on Washington's Embassy Row. The target of the car bomb attack was Orlando Letelier, a former Chilean government minister and ambassador to the United States, one of Pinochet's most outspoken critics and the head of IPS's sister organization, the Transnational Institute (TNI). Ronni Karpen Moffitt, a 25-year-old IPS development associate, was also killed.

The Institute for Policy Studies hosts an annual human rights award in the names of Letelier and Moffitt to honor them while celebrating new heroes of the human rights movement from the United States and elsewhere in the Americas. The award recipients receive the Letelier-Moffitt Human Rights Award.

The Transnational Institute, an international progressive think tank based in Amsterdam, was originally established as the IPS's international program, although it became independent in 1973.

In its attention to the role of multinational corporations, it was also an early critic of what has come to be called globalization. Richard Barnet's 1974 examination of the power of multinational corporations, Global Reach, was one of the first books on the subject.

1980s
In the 1980s, Raskin served as chair of the SANE/Freeze campaign.

In the 1980s, IPS became heavily involved in supporting the movement against U.S. intervention in Central America. IPS Director Robert Borosage and other staff helped draft Changing Course: Blueprint for Peace in Central America and the Caribbean, which was used by hundreds of schools, labor unions, churches, and citizen organizations as a challenge to U.S. policy in the region.

In 1985, Fellow Roger Wilkins helped found the Free South Africa Movement, which organized a year-long series of demonstrations that led to the imposition of U.S. sanctions. In 1987, S. Steven Powell published his non-fiction Covert Cadre: Inside the Institute for Policy Studies in which he "providing by far the single most compendious collection of facts about IPS that anyone has yet compiled" according to a lengthy critical review by Joshua Muravchik.

In 1986, after six years of the Reagan administration, Sidney Blumenthal said that "Ironically, as IPS has declined in Washington influence, its stature has grown in conservative demonology. In the Reagan era, the institute has loomed as a right-wing obsession and received most of its publicity by serving as a target."

Conservative think tanks American Enterprise Institute and The Heritage Foundation described the IPS as the "far left" or "radical left" of the late 1980s, another conservative think tank, who engaged in what the author Joshua Muravchik coined as "communophilism".

In his 1988 book Far Left of Center: The American Radical Left Today, Emory University professor Harvey Klehr said that IPS "serves as an intellectual nerve center for the radical movement, ranging from nuclear and anti-intervention issues to support for Marxist insurgencies".

1990s
In the early 1990s, IPS began monitoring the environmental impacts of U.S. trade, investment, and drug policies.

Fellows

 * Sarah Anderson
 * Ajamu Baraka
 * Phyllis Bennis
 * John Cavanagh
 * Karen Dolan
 * Robb K. Burlage
 * John Kiriakou
 * Saul Landau
 * Marcus Raskin
 * Sanho Tree
 * Daphne Wysham
 * Frank Smith Jr.

Senior scholars

 * Maude Barlow
 * Norman Birnbaum
 * Noam Chomsky
 * Steve Cobble
 * Chuck Collins
 * Barbara Ehrenreich
 * Paul Epstein
 * Richard Falk
 * Bill Fletcher
 * Andy Levine
 * Jerry Mander
 * Jack O'Dell
 * Vandana Shiva

Funding
Start-up funding was secured from the Sears heir, Philip M. Stern, and banker, James Warburg. Most of the money came from a foundation of Samuel Rubin.

title: Institute for Policy Studies


text:
Wikipedia talk:Historical archive/Imported pictures/Pictures from southwarkphotolibrary.co.uk details

Hmm, I'm confused by CopyrightedFreeUse, which seems to be yours. The URL doesn't have any info about licensing, either way. Is CopyrightedFairUse just the same as a dedication to the public domain? Or is it something else? Have you already answered this? :) Martin 17:50, 22 Feb 2004 (UTC)

 * Martin - tbh I have no ideas what license they have released them under, and I don't think they have either. All I have is what they said to me on the email. I don't think they care who uses them. I don't know how we deal with organisations that don't know or care about the detail of licensing. Secretlondon 17:54, Feb 22, 2004 (UTC)

Yes, that is a puzzler. Hassle them is one option, but the problem is always that the person being hassled might not be the person who has the authorisation to grant licenses. Perhaps have a seperate MediaWiki message for this photo library (linking to this email exchange) until someone can get clarification. "CopyrightedFreeUse" is a really bad tag in my opinion, because it's not really clear what it might mean. Better to let people know about the uncertainty? Martin 18:34, 22 Feb 2004 (UTC)

 * Yes - fair enough. I didn't know what to call it. I'm worried about hassling them as I'm not sure anyone knows, they would probably have to ask the legal department which would cost them and it would be easier and cheaper to say no. I think the intention may be public domain, but I don't know. Secretlondon 18:37, Feb 22, 2004 (UTC)

How about MediaWiki:SouthwarkPhotoLibrary? Martin 18:45, 22 Feb 2004 (UTC)


text:
Stronsay

Stronsay is an island in Orkney, Scotland. It is known as Orkney's 'Island of Bays', owing to an irregular shape with miles of coastline, with three large bays separated by two isthmuses: St Catherine's Bay to the west, the Bay of Holland to the south and Mill Bay to the east. Stronsay is 3,275 ha in area, and 44 m in altitude at its highest point. It has a usually resident population of 349. The main village is Whitehall, home to a heritage centre.

Sights on the island include the Vat of Kirbister, a natural arch described as the "finest in Orkney", white sand beaches in the three bays, and various seabirds amongst which are Arctic terns.

Geography and geology
As with most of Orkney, Stronsay is made up of Old Red Sandstone which has produced a fine soil in many places. It is generally low-lying.

On the eastern coast, spectacular rock formations include the Vat of Kirbister – Stronsay's famous natural rock arch, often described as the finest in Orkney. The coast around Odiness Bay features numerous dramatic geos, caves, flagstone rock platforms and sea stacks, including Tam's Castle, reputed to have once housed a hermitage on its flat summit.

Stronsay is the seventh largest of the Orkney Islands. Surrounding small islands include Linga Holm, Papa Stronsay, the Holm of Huip (named after a district in north west Stronsay) and the Holms of Spurness. To the north is Sanday, to the north west Eday, to the south west Shapinsay and Mainland Orkney, and the islet of Auskerry due south.

Prehistory
Two flint arrowheads discovered at a site investigated by Orkney College in April 2007 are believed to date from the late Paleolithic or early Mesolithic periods some 10–12,000 years ago. They were found amongst a scattered collection of other tools on a farm. If confirmed, they could be the earliest human artifacts found in Scotland.

18th and 19th centuries
The island is now agricultural, but during the 18th and 19th centuries, kelp collection and herring curing employed up to 5000 people. The population was over 1000 for the entire 19th century and until the mid-20th century, with the 1891 census recording a population of 1275, excluding seasonal itinerants involved in the herring industry.

The kelp burning industry was started by James Fea of Whitehall in 1727, and thrived during the remainder of the 18th century, lasting into the early 19th century; some of the kiln ruins can still be seen.

Present day
The usual resident population of the island in 2011 was 349, compared to 342 in 2001. The island is mainly agricultural with farming and fishing the main industries. Efforts are being made by the community to develop the economy by improving tourism infrastructure.

Much of the settlement is strung out along the island's main road, but the main population centre is Whitehall village, in the north of the island, with a row of seafront houses and two piers, a link to the village's foundation as a herring fishing station in the 19th century. There is a variety of accommodation in Whitehall, including the Stronsay Hotel, Storehouse B&B and FishMart hostel.

Orkney Ferries sail from Whitehall to Kirkwall on the Orkney Mainland, and planes fly there from the island's airstrip (Stronsay Airport), also in the north of the island.

Whitehall overlooks the separate island of Papa Stronsay, across the blue waters of Papa Sound. A prominent feature of the smaller island is the monastery complex of the Transalpine Redemptorist monks who live there. Visitors are welcome and the monks can help arrange boat trips.

The island's school is well-resourced and caters for nursery, primary and secondary pupils.

Moncur Memorial Church is the Church of Scotland centre of worship on Stronsay and there is a Catholic chapel in Whitehall, which is part of the Diocese of Aberdeen.

There is a thriving craft community, with several specialised craft shops.

Wildlife

While the landscape has very few trees, the fertile soil supports a wide variety of wild flowers including oysterplant, frog orchids, adder's tongue, and also a naturalised population of Patagonian ragwort.

There is a wide variety of birds on the island, and birdwatchers have recorded many rarities. Common species include whooper swan, various ducks and geese including the greater white-fronted goose, as well as common redshank, common snipe, common quail, great skua, corn bunting and corncrake, the last of which is rare in Scotland.

Many seabirds can be seen from Lamb Ness and Lamb Head. The Stronsay Bird Reserve is said to be one of the best sites in Europe to spot rare migrants such as the American Golden Plover, Arctic Redpoll, Scarlet Rosefinch and Tawny Pipit.

Seals can also be spotted at many locations, including the Ayre of the Myers, a popular picnic spot near Whitehall, at the twin south-facing bays at Houseby and Sand of Crook or from the seal-hide near Holland Bay.

The more unusual occasional visitors which have been recorded include:
 * American golden plover
 * Arctic redpoll
 * Pied wheatear
 * Rustic bunting
 * Scarlet rosefinch
 * Tawny pipit
 * Various warblers including: Radde's warbler, marsh warbler and western subalpine warbler.

title: Stronsay


text:
TNI

TNI or Tni may refer to:

 * Satna Airport, IATA code TNI
 * Tahitian Noni International, Inc.
 * Taqramiut Nipingat Inc., an Inuit broadcasting organization in Quebec, Canada
 * Telephone Network Interface; see Network interface device (NID)
 * Indonesian National Armed Forces (Tentara Nasional Indonesia)
 * Texas Neurosciences Institute, a research and neurological clinical center
 * Thai-Nichi Institute of Technology a private college in Thailand
 * Transnationality Index
 * Transnational Institute
 * The National Interest
 * The New Inquiry
 * Trichoplusia ni, the cabbage looper moth
 * Trusted News Initiative

title: TNI


text:
Michael Fay

Michael Fay may refer to:
 * Sir Michael Fay (banker) (born 1949), New Zealand merchant banker in the America Cup's Hall of Fame
 * J. Michael Fay (born 1956), American ecologist and conservationist
 * Michael D. Fay, American war artist
 * Michael Francis Fay (born 1960), British botanist
 * Michael Fay, American caned in Singapore in 1994 for vandalism

title: Michael Fay


text:
Pie and mash

Pie and mash is a traditional working-class food, originating in the Docks of London. Often accompanied with jellied eels, the dish has been popularised as "a Cockney classic". It typically consists of a minced beef pie, mashed potato, and a parsley sauce known as liquor. Pie, mash and eel shops have been in London since the 19th century, and are still common in East and South London, and in many parts of Kent and Essex.

History
During the Victorian era, industrial air pollution tended to be worse in the east and southeast of London because of the prevailing westerly wind, with the result that the East End was settled more by the working classes, while the western part of the city was home to higher social classes.

The savoury pie had long been a traditional food, and its small handsized form also made it a transportable meal, protected from dirt by its cold pastry crust, and filled with cheap minced meat, usually mutton.

Jellied eels are often associated with pie and mash, as European eels cooked in gelatine also became a common worker's meal, since eels were one of the few forms of fish that could survive in the heavily polluted River Thames and London's other rivers at that time. Supply was plentiful through the late 19th century, particularly from the Dutch fishing boats landing catches at Billingsgate Fish Market.

During World War II, government food rationing threatened pie and mash shops with closure. However, due to public demand the Ministry of Food was persuaded to allow these traditional eateries to remain open. As a result, pie and mash supplemented family rations amid the widespread national food shortages and along with fish and chips, that other traditional British working-class dish, pie and mash increased in popularity among middle-class families too.

Since 2010, as revealed in a joint study by the Zoological Society of London and the Environment Agency, the number of eels captured in research traps in the River Thames fell from 1,500 in 2005 to 50 in 2010, meaning most eels used in pie and mash shops are now from the Netherlands and Northern Ireland.

While eel consumption continues to go down in the 21st century (in 2000, there was only one stall selling live eels in Billingsgate Market), the number of eel and pie shops has not gone down very much: there were 87 eel and pie shops in Greater London in 1995, compared to around 110 at the end of 1800.

Composition
The main dish sold is pie and mash, a minced-beef and cold-water-pastry pie served with mashed potato. There should be two types of pastry used; the bottom or base should be suet pastry and the top can be rough puff or short. It is common for the mashed potato to be spread around one side of the plate and for a type of parsley sauce to be present. This is commonly called "liquor sauce" or simply "liquor" (liquor as in a liquid in which something has been steeped or cooked), traditionally made using the water kept from the preparation of the stewed eels. However, many shops no longer use stewed eel water in their parsley liquor. The sauce traditionally has a green colour, from the parsley.

Shops
Before shops became common, trading took place from braziers or carts. It was not until late Victorian times that shops began to appear. The first recorded shop was Henry Blanchard’s at 101 Union Street in Southwark in 1844 which was described as an “Eel Pie House”. The shops have become part of the local community and heritage of their area; for example, L. Manze in Walthamstow became Grade II listed by English Heritage in 2013 due to its architectural and cultural significance.

Traditionally, pie and mash shops have white tile walls with mirrors, and marble floors, tables and work tops, all of which are easy to clean. They give the shops, never called restaurants or café, a late Victorian or Art Deco appearance.

Because of the large number of pleasure boat steamer companies offering Sunday trips on the River Thames, many Eastenders used them to explore the more gentrified west of London. The result was that many also wanted their traditional foods of ale and pie and mash, resulting in the renaming of both a hotel that they frequently visited and the island on which it sat in Twickenham to Eel Pie Island in the early 1900s.

21st-century decline and current prospect
With the gentrification of South and East London as well as the docklands area, there have been significant socio-economic and demographic changes in the London neighbourhoods traditionally most associated with pie and mash. Through the 2020s, industry experts continue to note that affluent new locals are now more interested in "lattes and paninis" than a dish typically associated with working-class Londoners. As a result of the declining customer base, historic and famous pie and mash shops including L. Manze in Walthamstow have been forced to close and their premises taken over by new incumbents. In May 2023, for example, the iconic Grade II listed art-deco building at 9 Broadway Market in Hackney, which had been home to F. Cooke's pie and mash shop for the previous 120 years, was taken over by Cubitts, a company that designs and handcrafts bespoke eyewear. W.J. Arments Eel & Pie House at Walworth, founded in 1914, with its glazed tiled walls, marble table tops, opening sash windows and a floor covered with sawdust, is one of the few shops to have thrived in the changing socio-economic climate.

title: Pie and mash


text:
Ecgberht II of Kent

Ecgberht II was king of Kent jointly with Heaberht.

Ecgberht II is known from his coins and charters, ranging from 765 to 779,   two of which were witnessed or confirmed by Heaberht.

Ecgberht II acceded by 765, when he issued his earliest surviving charter. However, around this time Offa, king of Mercia, appears to have been attempting to rule Kent directly, as he seems to have issued or confirmed a number of charters relating to Kent. According to the Anglo-Saxon Chronicle, a battle was fought at Otford in 776, and, although the outcome was not recorded, the fact that Kent seems to have remained independent for several years afterward suggests that Ecgberht was victorious. It is known that he remained king until at least 779, the date of his latest charter.

title: Ecgberht II of Kent


text: Template:Jesus

title: Template:Jesus


text:
Eday

Eday (, Aidee) is one of the islands of Orkney, which are located to the north of the Scottish mainland. One of the North Isles, Eday is about 24 km from the Orkney Mainland. With an area of 27 km2, it is the ninth-largest island of the archipelago. The bedrock of the island is Old Red Sandstone, which is exposed along the sea-cliffs.

There are various well-preserved Neolithic tombs, as well as evidence of Bronze Age settlement and the remains of a Norse-era castle. During the period of Scottish rule the substantial property of Carrick House was developed at Calfsound, which became a burgh for a short period. During the British era many agricultural improvements were introduced, although there has been a substantial decline in the population since the mid-nineteenth century. In the twenty-first century the Eday Partnership has had success in promoting the island's economy. Local placenames reflect the diverse linguistic heritage and the landscapes of the island and its surrounding seas attract abundant wildlife.

Etymology
"Eday" is a name derived from the Old Norse eið and means "isthmus island". This is a name specifically associated with economic activity used only where the isthmus has been a "route for the movement of goods and/or boats from one coast to another".

There are numerous other eið names in the islands of the North Atlantic and those in Orkney include Hoxa (Haugeið) on South Ronaldsay, Aith (found on Walls, Stronsay and the west Mainland) and Scapa in St Ola which is derived from the Norse Skálpeið. Bay of Doomy, near the central isthmus on Eday, may also have a name derived from dómr-eið, meaning "isthmus of the courthouse", indicating it could have been an important meeting place during the Norse period of Scottish history. In the 17th century Eday was also known as "Heth Øy".

In common with elsewhere in the Orkney islands, place names are generally a mixture of Norse, Scots and English influences. Any Pictish names that existed before the arrival of Scandinavian settlers on Eday appear to have been obliterated. The common suffix -quoy is from the Old Norse kví-ló and signifies an enclosure in a marshy area. Skaill on the east coast is from the Norse skáli and suggests an important farm on good fertile land that was associated with several smaller tunships. The Bay of London also has Norse origins, lund-inn meaning "woodland", although this is no longer an apt description for this largely treeless landscape. Old Norse lundi means "puffin", which creatures may once have nested in sandy land at the back of the bay.

Orkney was Christianised before the arrival of Viking settlers, and there are various local "Papa" names that reflect the activities of the pre-Norse papar monks there. The farm of Papleyhouse near Linkataing may indicate such a link to the past, although the connection is by no means certain. The name "geo", which occurs frequently around the rocky coast, is from the Norse gjá and means a narrow and deep cleft in the face of a cliff.

Geography and geology
Eday is 14 km long from north to south but only just over 500 metres wide at the narrow neck of land between the Sands of Doomy and Bay of London and has been described as being "nipped at the waist". The centre of the island is largely moorland covered with heather, and cultivation is confined to the coasts.

The highest points are Flaughton Hill at the island's centre, Fersness Hill at West Side, Vinquoy Hill to the north and Ward Hill to the south, which reaches 101 m. In Orkney this last name, which derives from the Norse varði, is a common one for the highest point on an island as in the past they were used for lighting warning beacons.

The largest body of water is the 10 ha sea southeast of Vinquoy Hill. Loch of Doomy lies on the western side of the narrow "waist" and the smaller Loch Carrick on the north coast.

The population is dispersed along the coastal farmsteads and nowhere on the island has the status of a village. Calfsound is the most populous of the settled areas, with other concentrations at Millbounds on the east coast, which has a post office and a community facility in a converted chapel, and Backaland in the south where the ferry from the Mainland docks. Eday is surrounded by other small islands that make up the "seemingly impossible green and russet jigsaw of Orkney's North Isles". Calf of Eday lies 350 m north of the settlement of Calfsound. Further east is Sanday across the Eday Sound. Stronsay and Linga Holm are to the south east and Muckle Green Holm to the south west beyond the straits known as the Fall of Warness. Egilsay lies some 5 km due west. Rusk Holm, Faray and Holm of Faray lie beyond the Sound of Faray to the northwest, and beyond them is the larger island of Westray.

In common with its neighbouring isles, Eday is largely formed from Middle Devonian Old Red Sandstone deposited in the Orcadian Basin. The Eday Group is the name for a substantial sequence of sandstones that is found at many locations in Orkney, for which Eday and the area around Eday Sound are the type area. In places it is up to 800 m thick, and is largely composed of yellow and red sandstones with intervening grey flagstones and marls. The rock is easily quarried and some of the yellow sandstones from Fersness were used in the construction of St Magnus Cathedral in Kirkwall. The Devonian sequence is deformed into a major fold, the north–south trending Eday Syncline, with the youngest part of the sequence, the Upper Eday Sandstone outcropping in the north of the island from Bay of Cusby to Red Head. The oldest part of the sequence, the Rousay Flagstones are found on the eastern side of the island at Bight of Milldale and from Kirk Taing to War Ness, and to the west from Sealskerry Bay to Fersness. Veness is formed of Upper Eday Sandstone downfaulted against the flagstones.

Prehistory
The very limited archaeological record provides scant evidence of Mesolithic life in Orkney, but the later assemblage of houses and monumental Neolithic structures in the archipelago is without parallel in the United Kingdom.

Vinquoy chambered cairn, located in a commanding position overlooking the Calf Sound, is 17 m in diameter and 2.5 m high. The narrow entrance passage of this Maeshowe-type tomb leads to a central chamber with four side-cells. Other sites of interest on Eday include the Stone of Setter standing stone that dominates the col north of Mill Loch, and which at 4.5 m high is one of the tallest monoliths in Orkney. There are two more chambered cairns at Braeside and Huntersquoy and another on the Calf of Eday. Rectangular in shape, it was excavated in 1936–37 and contains a small chamber with two compartments and a larger one with four stalls that has a separate entrance and was probably added at a later date. Although there are several Bronze Age sites on the island, they provide less dramatic remains. At Warness in the south west there is a burnt mound from this period and there are the ruins of two houses of a similar age on Holm of Faray near the Point of Dogs Bones. The Fold of Setter is an 85 m Bronze Age enclosure located to the north of Mill Loch. There is the site of a large Iron Age roundhouse containing a saddle quern at Linkataing in north west Eday. Latterly, Orkney was settled by the Picts although the archaeological evidence is sparse.

Norse colonisation
It is not known "when and how the Vikings conquered and occupied the Isles", and although Norse contacts with Scotland certainly predate the first written records in the 8th century,  their nature and frequency are unknown. The place name evidence of a Norse presence on Eday is conclusive and very little is known about the specifics of life on the island at this time. The Norse-era ruins of the Castle of Stackel Brae, which dates from the 12th or 13th century, lie under a green mound to the east of the Bay of Greentoft. The castle may have been the most important building on Eday at this time.

Scottish rule
In 1468 Orkney became part of the Kingdom of Scotland and an influx of Scottish entrepreneurs helped to create a diverse and independent community that included farmers, fishermen and merchants that called themselves comunitatis Orcadie and who proved themselves increasingly able to defend their rights against their feudal overlords. Nonetheless, the actions of the aristocracy continue to provide much of the information known about affairs on Eday at the time. In 1561, during the Reformation, Edward Sinclair was granted the feu of Eday by Adam Bothwell, Bishop of Orkney. These were turbulent times—Sinclair's duties included to defend the reforming Bishop "against whatsoever invaders" —and later that year he was one of the ringleaders of an anti-Catholic riot in Kirkwall. His son William took over the running of the Eday estate in due course, but it became burdened with debt. In 1601 when Edward was "an auld decrepit man ... aged 100 or thereby" William attempted to sell the family interest to George Sinclair the Earl of Caithness. The new proprietor sent half a dozen boatloads of "vagabondis, broken Highland men of Caithness" to Eday, much to the alarm of the notorious Earl Patrick of Orkney. Earl Patrick was able to use the poor relationship between the elderly father Edward and his son, (the former claiming William fired muskets at him and grabbed him by the neck like a dog) to take action. Acting, so he alleged, on behalf of Edward, Earl Patrick evicted William, took the Eday the rents for himself and profited from the extraction of building stone from Towback quarry.

John Stewart, Earl of Carrick, the brother of Earl Patrick, was granted Eday in 1632  and he constructed Carrick House at Calfsound shortly thereafter. He used peat to manufacture salt from salt pans at both Carrick and on the Calf of Eday. The product was described as "quite fine" in the 17th century when it was undertaken on a substantial scale although of "indifferent quality" in the early 19th century when it was being conducted as a cottage industry. Peat extraction was also an important industry in the past as Sanday and North Ronaldsay obtained most of their fuel from Eday and this material was also exported to whisky distilleries on mainland Scotland.

Stewart's ambitions for Calfsound were considerable. Described as the "town and port" of Carrick it became a burgh (the only other one in Orkney being Kirkwall) with the right to appoint baillies and hold markets but it was never likely to flourish in such a location.

British era
From the first decade of the 18th century Orkney became part of the new Kingdom of Great Britain. This was a time of great interest in agricultural improvement although the changes this brought about were not of significance in Orkney until the mid-nineteenth century. For example, no potatoes were grown on Eday until around 1780. By comparison to these gradual changes, Carrick House saw drama in 1725. The property was now owned by James Fea who had been a school friend of a "Mr Smith", a trader of Stromness. When Smith was unmasked as the notorious pirate John Gow he sought to escape the attentions of the authorities by making for Eday via a raid on Hall of Clestrain, in Orphir. When Gow's ship Revenge ran aground on the Calf of Eday, Fea's men took him prisoner and held him at Carrick House, for which Fea was given a £1,700 reward. The bell from the Revenge is still in Carrick House.

In the early nineteenth century the kelp industry provided significant employment on some of the Orkney islands, but when the market collapsed between 1830 and 1832 it caused considerable hardship. North Ronaldsay was especially hard-hit and several families were allowed to resettle from there to develop land at Westside on Eday. Rising populations meant increasing land values, especially for small tenancies. In 1843 crofts were valued at 1 £/acre on Eday, nearly three times the price for larger farms. However, the 20th century saw decline. Immigration from mainland Scotland was essentially unknown even in the late 1950s and the population in 2001 was about an eighth of the total 160 years earlier.

Transport and economy
Eday can be reached by both sea and air from the Orkney Mainland. Orkney Ferries provide daily ferry crossings to Backaland on Eday from Kirkwall. The Orkney inter-island air service, operated by Loganair, connects Kirkwall Airport with Eday London Airport. In 2014 the Orkney Islands Council began consultation to build a number of fixed crossings between seven of the Orkney Islands. This includes the possibility of a 2.6 mile bridge between Eday and Papa Westray.

Farming and crofting are mainstays of the local economy, especially livestock husbandry. Flocks of sheep are also kept on Faray.

The European Marine Energy Centre (EMEC) based at Stromness is a Scottish Government-backed research facility. They have installed a wave testing system at Billia Croo on the Orkney mainland and a tidal power testing station, overlooking the Fall of Warness, on Eday. The test site was chosen because of the marine currents that reach almost 4 m/s at spring tides. There are seven offshore testing berths connected to the 33KV North Isles section of the national grid, via an underground cable.

Eday Partnership, the local development trust is active in promoting the island's economy and has instigated numerous projects, including Eday Heritage Centre, and the purchase of a new diesel tank for the island. Eday's various community projects contributed £380,000 to the island's economy from 2005 to 2007 and a 900 kW community-owned wind turbine is planned. The income that this asset will generate is expected to reduce fuel poverty on the island, support new community enterprises and create affordable housing.

In July 2008, the island celebrated the opening of the Eday Heritage and Visitor Centre in the restored former Baptist Church. There is a heritage display area, a permanent archive, a café and a tourist information point. The Eday Oral History Project records life on the island in the past and is also housed within the centre, which has its own 6 kW wind turbine.

The island's population was 160 as recorded by the 2011 census an increase of over 30% since 2001 when there were 121 usual residents. During the same period Scottish island populations as a whole grew by 4% to 103,702.

Natural history
In the early 19th century Patrick Neill wrote of the local flora that "Eda is a mossy island; a great part of it consisting of barren marshy heaths. Juncus uliginosus here covers whole acres; and the pretty little plant Radiola millegran, or all-seed, is everywhere strewed." Over 120 species of wild plants have been recorded on the island, including bog myrtle found nowhere else in Orkney.

In the mid-17th century, Eday was described as being "absolutely full of moorland birds", and today there are red-throated divers on Mill Loch, Arctic skuas and bonxies on the moors and black guillemot offshore. Shore birds include meadow pipit, rock pipit and ringed plover. The woodland at Carrick House attracts a variety of migrants, and otters can be seen around the coasts. There are colonies of harbour and grey seals on Muckle Green Holm, Little Green Holm, Faray and Holm of Faray, and white-beaked dolphin, minke whale and killer whale are occasional visitors to the area. A ranger provides guided walks throughout the year.

Prominent natives

 * Australian pioneer union leader and Australian Labor Party politician William Spence was born on Eday in 1846.
 * John Flaws Reid, who represented Mackenzie in the House of Commons of Canada, was born on Eday in 1860.
 * Rev Norman Miller Johnson FRSE (1887–1949) was minister of Eday and died there.

title: Eday


text:
Grand Marnier

Grand Marnier is a French brand of liqueurs. The brand's best-known product is Grand Marnier Cordon Rouge, an orange-flavored liqueur created in 1880 by Alexandre Marnier-Lapostolle. It is made from a blend of Cognac brandy, distilled essence of bitter orange, and sugar, containing 40% alcohol (70 proof in the UK, 80 proof in the US). It is commonly consumed "neat" as a cordial or a digestif, and can be used in mixed drinks and desserts. Popular examples of the latter include crêpes Suzette and crêpes au Grand Marnier.

Aside from Cordon Rouge, the Grand Marnier line includes other liqueurs, most of which can be used similarly.

History
According to its official website, Grand Marnier's first distillery was built in 1827 by Jean Baptiste Lapostolle in the village of Neauphle-le-Château. His granddaughter Julia married Louis-Alexandre Marnier in 1876, and four years later, the Marniers released a signature cognac with Haitian bitter orange.

Swiss hotelier César Ritz (1850–1918) reportedly came up with the name "Grand Marnier" for Marnier-Lapostolle, who in return helped him purchase and establish the Hotel Ritz Paris. During the extravagant Belle Époque, Grand Marnier was served in the Ritz Hotel as well as the Savoy Hotel. The brand has released anniversary editions of its well-known liqueur lines and became popular in the United States in the 1980s.

On 15 March 2016, the Campari Group announced a friendly takeover offer for Société des Produits Marnier Lapostolle S.A., the owner of Grand Marnier.

Cordon Rouge
Cordon Rouge ("Red Ribbon") is an orange-flavored cognac liqueur and the original Grand Marnier liqueur, created in 1880 by Alexandre Marnier-Lapostolle. It is consumed neat and is also used in mixed drinks and desserts.

Signature Series

 * Natural Cherry – A blend of Grand Marnier Cordon Rouge with wild tropical Haitian and Dominican oranges and European Griotte cherries.
 * Raspberry Peach – A blend of European raspberries and rare red peaches from Ardèche, in the South of France, combined with Grand Marnier Cordon Rouge.

Cordon Jaune
Cordon Jaune (Yellow Ribbon) was a triple-sec Curaçao-like liqueur that was made with neutral grain spirit instead of Cognac. The production stopped on 1 January 2017, after the company was taken over by Campari.

Cuvée du Centenaire
Cuvée du Centenaire ("Centennial Edition") was first released in limited quantities in 1927 to commemorate the 100th anniversary. It is made with up to 25-year-old fine cognacs and is consumed neat. On average, Cuvée du Centenaire is $145 per bottle.

Cuvée Spéciale Cent Cinquantenaire
The Cuvée Spéciale Cent Cinquantenaire ("Special Sesquicentennial Edition") was created in 1977 to honor the 150th anniversary of the brand and is currently the most expensive variety of Grand Marnier. It is made with up to 50-year-old cognacs in frosted glass bottles featuring hand-painted Art Nouveau decorations. In 2005, it was marketed with the slogan "Hard to find, impossible to pronounce, and prohibitively expensive."

Cuvée Louis-Alexandre Marnier-Lapostolle
Cuvée Louis-Alexandre Marnier-Lapostolle is a special tribute to the founder of Grand Marnier, Louis-Alexandre Marnier Lapostolle. Made from 82% Cognac and 18% orange liqueur, it has a rich intensity of macerated citrus combined with the warming flavors of Cognac, pine, and Earl Grey tea on the finish.

As a cordial
Grand Marnier is commonly consumed "neat" as a cordial or a digestif.

In mixed drinks
Grand Marnier can be used in mixed drinks. It can be used as an alternative to Cointreau or triple sec, orange-flavored liqueurs made from neutral spirits. Some examples of these include the Cosmopolitan, Margarita, Sangria, Sidecar, and the B-52.

In food
Grand Marnier is used in the preparation of flambé dishes, such as crêpes Suzette, Grand Marnier soufflé and crème brûlée. It can also be used in the sauce of the "Canard à l'orange" roasted duck dish. It is also an ingredient in several kinds of pastries, such as liquor cream buns and the French Christmas dessert known as Bûche de Noël (Yule log). It is frequently used in recipes for cranberry sauce, as sweetness and citrus can be a contrast to the bitterness of cranberries. It can be used in some fruitcake recipes instead of brandy. It can also be drizzled over vanilla ice cream.

Awards and recognition
In 2001, Grand Marnier's Cordon Rouge won the Gold Medal at the San Francisco World Spirits Competition. It was also awarded the 4 Star recommendation from F. Paul Pacult's Kindred Spirits in the Spirit Journal Guide.

Cuvée du Centenaire:
 * 5-star recommendation from F. Paul Pacult's Kindred Spirits, the Spirit Journal Guide
 * Double Gold medal at the San Francisco World Spirits Competition 2001
 * Gold Medal at the San Francisco World Spirits Competition 2007

Cuvée Spéciale Cent Cinquantenaire:
 * "Best of the Best" Spirits in the Robb Report
 * 5-star recommendation from F. Paul Pacult's Kindred Spirits, the Spirit Journal Guide
 * Gold Medal at the Salon des Arts Ménagers 1983 – Brussels
 * "Best Liqueur” at the San Francisco World Spirits Competition 2001
 * Double Gold Medal at the San Francisco World Spirits Competition 2007

title: Grand Marnier


text:
Bleu celeste

Bleu celeste (, "sky blue") is a rarely occurring and non-standard tincture in heraldry (not being one of the seven main colours or metals or the three staynard colours). This tincture is sometimes also called ciel or simply celeste. It is depicted in a lighter shade than the range of shades of the more traditional tincture azure, which is the standard blue used in heraldry.

Generally considered to be European rather than English or Scottish, after the First World War it started to appear in England in arms and badges relating to the Royal Air Force, though it is still doubtful if any more than a dozen examples could be found in British heraldry, and most British heralds consider it unheraldic. While in the post-World War I period bleu-celeste is depicted as a darker shade, in prior times it was depicted as very light, and has even been treated as a metal, as azure charges have been placed on a bleu celeste field, and vice versa.

Regardless, bleu celeste is still used by heraldic authorities, such as Britain's College of Arms in London, with one example of the arms of Jacob Flamberg being granted in 2016 using bleu celeste in the blazon for the field, mantling and charge, in each case per pale/party with sable.

"Per pale Bleu Celeste and Sable on a Fess wavy between three Plum Trees eradicated Argent a Wolf courant per pale Sable and Bleu Celeste langued and armed Gules" and "Party Bleu Celeste and Sable doubled Argent"

Bleu celeste can also be seen in the arms of Israel and Peru and also in the arms of former Canadian Governor General Ray Hnatyshyn. In the arms of the University of Natal Athletic Union the azure is defined as "sky blue".

Light blue

In addition to bleu celeste, there is also an apparently unique example in British heraldry of the use of "light blue" in the Municipal Borough of Barnes, through which the Oxford versus Cambridge boat race passes on the Thames. The arms show the respective blades of the teams' oars, coloured dark Oxford blue and light Cambridge blue, and may be blazoned thus:

 * Azure, on a saltire Or between four ostrich feathers argent, two oars in saltire proper, the blade of that to the dexter dark blue and that to the sinister light blue.

When in 1965 that borough merged with its neighbours to form the Borough of Richmond upon Thames, the coloured oars were transferred to the supporters in the arms of the new borough.

title: Bleu celeste


text:
Amy Dillwyn

Elizabeth Amy Dillwyn (16 May 1845 – 13 December 1935) was a Welsh novelist, businesswoman, and social benefactor. She was one of the first female industrialists in Britain.

Biography
Born in Sketty, Swansea, Dillwyn was a member of a prominent family. Her father was the industrialist and politician Lewis Llewelyn Dillwyn, her older sister was the lepidopterist Mary De la Beche Nicholl, her paternal uncle was the botanist and photographer John Dillwyn Llewelyn, her paternal aunt was the photographer Mary Dillwyn, and her paternal first cousin was the astronomer and photographer Thereza Dillwyn Llewelyn. Dillwyn's paternal grandfather was the businessman, naturalist, and politician Lewis Weston Dillwyn, while her maternal grandfather was the geologist and palaeontologist Henry De la Beche. Dillwyn's paternal great-grandfather was the American abolitionist William Dillwyn, who had migrated to Great Britain.

From the 1870s to the early 1890s, Dillwyn, wrote six novels. Her themes included feminism, social reform, and a favourable view of the Rebecca Riots (1839–1843) in response to unfair taxation. Following her father's death in 1892, Dillwyn inherited his spelter works at Llansamlet, and his debts. She personally managed the industry, and by 1899 had fully repaid her father's debts. In 1902, she turned her business into a registered company. She joined the National Union of Women's Suffrage Societies and campaigned for women's suffrage.

Dillwyn never married, but considered herself married to "her wife" Olive Talbot. She was nicknamed "The Pioneer" by friends.

Family

Elizabeth Amy Dillwyn was born on 16 May 1845, in Sketty in Swansea, Wales. She was the daughter of Lewis Llewelyn Dillwyn and Elizabeth (née De la Beche). She had an older sister and brother, Mary (born 1839) and Henry (born 1843) and a younger sister, Sarah (born 1852).

Her uncle was John Dillwyn Llewelyn of Penllergare who, along with his wife Emma Thomasina Talbot, his sister (Amy's aunt) Mary Dillwyn and his daughter (Amy's cousin) Theresa Story Maskelyne (née Dillwyn-Llewellyn) were pioneers of early photography. Her paternal grandfather was the naturalist Lewis Weston Dillwyn and her maternal grandfather was the geologist Henry De la Beche. The Dillwyn family were originally Quakers and her great-grandfather was William Dillwyn, the anti-slavery campaigner from Pennsylvania who returned to campaign in Britain.

Her father became a Liberal MP (1855–1892), and was the owner of the Dillwyn Spelter Works at Swansea.

Biography
In 1864 her fiancé, Llewelyn Thomas of Llwynmadog, died shortly before their planned wedding. Research into Dillwyn's life has also shown a close relationship with Olive Talbot through letters, who she called her "wife" in diaries. From this, some theorise that the unrequited love in her novels was inspired by this real relationship. In 1866 her mother died. Between 1880 and her father's death in 1892 she had six novels published.

Following the deaths of her brother in 1890 and her father in 1892 Amy Dillwyn lost the family home at Hendrefoilan due to its being entailed to the male line, but inherited her father's debts of over £100,000 (£8 million or more today). She wore a bright purple skirt, with a yellow rose in her belt and flowers in her hat, to her father's funeral, in a statement against the Victorian conventions of elaborate funerals, which could cause families to get into debt from buying mourning clothing in haste.

She was able to rescue her father's spelter works at Llansamlet, which she managed herself, and thus saved 300 jobs. Dillwyn lived in lodgings until the business was saved, eventually moving into Tŷ Glyn, her home for the rest of her life. It took until 1899 for "Dillwyn & Co". to pay off the last of its creditors and begin returning a profit. In 1902, Dillwyn registered the business as a registered company.

She was a strong supporter of social justice and gave her support to striking seamstresses. Her unorthodox appearance, her habit of smoking cigars and her lifestyle made her a well-known figure in the local community. When the National Union of Women's Suffrage Societies was formed at the turn of the century, Dillwyn joined as one of the earliest supporters in Wales. Although rejecting the militant actions of some members, she was still a staunch member of the movement.

Death
Dillwyn died in Swansea on 13 December 1935, at the age of ninety. She was cremated and her ashes buried in the churchyard of St Paul's Church, Sketty. Probate was granted to Rice Mansel Dillwyn and her estate was valued at £114,513 7s 9d.

Legacy
Her house, Tŷ Glyn (now Mumbles Nursing Home), still stands at West Cross, Swansea and a Blue plaque has been installed on its boundary wall.

In 2018, Dillwyn (representing women in business) was chosen as one of the top 100 Welsh Women by Women's Equality Network Wales, in their project to mark the centenary of the Representation of the People Act 1918, granting some women the vote.

Amy Dillwyn's life has inspired works of art and drama.

Novels
Amy Dillwyn started writing in the 1870s, stating "I've an idea I will try and write one chapter and see how I like it". The Rebecca Rioter (as The Rebecca Rioter: A Story of Killay Life) was published by Macmillan in 1880, by "E. A. Dillwyn". Telling the fictionalised account of a Rebecca rioter, loyal to the cause even when transported to Australia, the novel shows Dillwyn's political views, liberal towards the Rebecca rioters and against English rule, despite her father's part in squashing the riots. Both The Rebecca Rioter and Chloe Arguelle were translated as into Russian by members of the liberal intelligentsia. Her novel Jill was the most commercially successful, with the original July 1884 print selling out, causing it to be reprinted in September of the same year. She eventually satisfied the reviewers' calls for a sequel with the 1887 Jill and Jack. Bith David Painting and Kirsti Bohata identify recurring themes in her novels of crusading social reform, unrequited love and criticism of the upper classes. Feminist concerns predominate, however, and many of her stories had tomboyish women as protagonists. Dillwyn also anonymously contributed to the Spectator regularly in the 1880s.

Works

 * The Rebecca Rioter (1880) (reprinted 2004 by Honno)
 * Chloe Arguelle (1881)
 * A Burglary; or Unconscious Influence (1883) (reprinted 2009 by Honno)
 * Jill (1884) (reprinted 2013 by Honno)
 * Nant Olchfa (serialised in The Red Dragon: The National Magazine of Wales, Vols X-XI, 1886–7)
 * Jill and Jack (1887)
 * Maggie Steele's Diary (1892)

title: Amy Dillwyn


text:
Guido Molinari

Guido Molinari L.L. D. (October 12, 1933 – February 21, 2004) was a Canadian artist, known nationally and internationally for his serial abstract paintings and their dynamic interplay of colours and focus on modular shapes. His Stripe series is especially celebrated. Molinari himself described their effect - and the effect of all his paintings - as creating a new kind of fictional space "because it happens in the mind and yet also involves the totality of perception".

Biography
Molinari was born in Montreal, Quebec into a family originally from the Abruzzo region in Italy. He was the son of Charles Molinari (1879–1948), a musician with the Orchestre des concerts symphoniques de Montréal and first president of the Quebec Musicians' Association; and of Evelyne Dini (1889–1966), the daughter of a sculptor so his childhood was culturally rich. He began painting at age 13 and later enrolled at the School of the Art Association of Montreal, studying with Marian Dale Scott and Gordon Webber (1948-1951). A year later he contracted tuberculosis. While he was convalescing, he studied existentialism, reading authors such as Sartre, Camus, Piaget and Nietzsche. He did not complete his formal training in art but found his own path.

Early in his career, Molinari read a 1955 article about Jackson Pollock dripping paint onto canvas and went to New York to paint abstractly. He then returned to Montreal where he held his first solo exhibition at L'Échourie, founded the Galerie L'Actuelle with Fernande Saint-Martin, his future wife (also in 1955) and was one of the founding members of the Non-Figurative Artists Association in 1956. Between 1963 and 1969, he created hard edge paintings consisting of colour in vertical bands of equal width placed on a flat picture plane called the Stripe series. The National Gallery of Canada, acquired a canvas from the series.

Molinari was selected by Lawrence Alloway for the Guggenheim International Award 1964 exhibition. His paintings were seen in New York, Honolulu, Berlin, Ottawa and Buenos Aires. In 1965, his works appeared in The Responsive Eye at the Museum of Modern Art in New York along with works by artists such as Frank Stella. They were also seen in The Deceived Eye in Fort Worth, Texas; and in a fourth show, a New York solo exhibition at East Hampton Gallery. His work was described by reviewers in glowing terms as "Pop spelled 'Pow!'— in these handsome paintings the message comes across visually".

In 1967, he received a Guggenheim Fellowship and in 1968, the prestigious David Bright Prize at the 34th Biennale di Venezia for which he and Ulysse Comtois had been selected by Brydon Smith. From 1969 to 1970, Molinari created "checkerboard" paintings (he titled them "Structures") dividing the verticals by the horizontal and in 1971, he began to bisect each stripe, creating a new format of triangles. In the late 1970s, he created the Quantificateur series and, in the years before his death, the Checkerboard paintings (he called them the Continuum series).

Molinari taught for 27 years at Sir George Williams University and Concordia University, retiring in 1997. In 2004, Concordia recognized him with a posthumous honorary doctorate.

An avid art collector, his extensive private collection included the work of Mondrian and the manuscript pages of Mondrian's original defition of Neo-Plasticism (1926), Matisse, John Cage, Jasper Johns, and Quebec artists Denis Juneau, John Lyman, and Ozias Leduc. His obituary in the National Post quoted the then director of the Art Gallery of Ontario, Matthew Teitelbaum, as saying he owned Barnett Newman, Richard Serra, Francis Bacon, Piet Mondrian and Ellsworth Kelly.

In 1997, he established the 'Molinari Quartet' through the Molinari Foundation, a group that has been active now for twenty-five plus years.

In 2004, Guido Molinari died of pneumonia after having bone cancer which migrated from his lungs. His East Montreal studio at 3290 St. Catherine Street East remains intact and open to the public.

Selected public exhibitions
From 1953, Molinari exhibited his work, primarily in America and Europe. His first solo museum survey was organised by the MacKenzie Art Gallery in Regina in 1964 and toured to the Vancouver Art Gallery. He had exhibitions or was included in group shows in a host of other institutions, including in Guido Molinari: 1951-1961, Peintures en noir et blanc/ The Black and White Paintings (1961) curated and catalogue by Gary Dufour which toured Canada; the Guggenheim Museum (Guggenheim International Award show, 1964, and in 1967); the Museum of Modern Art in New York (The Responsive Eye, 1965); in 49th Parallels: New Canadian Art (1971), Ringling Museum of Art curated and catalogue by Dennis Young; at the National Gallery of Canada (retrospective curated and catalogue by Pierre Théberge, 1976); and the Musée d'art contemporain de Montréal (retrospective, 1995).

Selected works in public collections
Molinari's work is in national collections and in international collections such as the Museum of Modern Art in New York, and the Kunstmuseum Reutlingen concrete.

Honours and awards
Molinari won a Guggenheim Fellowship in 1967, the David Bright Prize at the 34th Venice Biennale (1968), was made an Officer of the Order of Canada in 1971, received the Victor Martyn Lynch-Staunton Award (1973) and won the Prix Paul-Émile-Borduas in 1980.

Memberships
He was a member of the Royal Canadian Academy of Arts (1969).

Record sale prices
At a sale at Heffel Auction House, Vancouver, May 25, 2016, Guido Molinari's Sans titre, 84 x 96 in 213.4 x 243.8 cm, acrylic on canvas, Estimate: $100,000 - $150,000 CDN, sold for: $354,000 CDN (premium included). In a sale of 15 June 2022 at the Cowley Abbott Auction, Toronto, Molinari's Série noir/blanc, acrylic on canvas, signed and dated "11/67" on the reverse, 81 x 68 ins ( 205.7 x 172.7 cms ), realized a price of $264,000.00.

Documentaries

 * Guido Molinari: The Colour of Memory

title: Guido Molinari