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# Lambert W function

The graph of W(x) for W > −4 and x < 6. The upper branch with W ≥ −1 is the function W0 (principal branch), the lower branch with W ≤ −1 is the function W−1.

In mathematics, the Lambert W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function f(z) = zez, where ez is the exponential function, and z is any complex number. In other words

${\displaystyle z=f^{-1}\left(ze^{z}\right)=W\left(ze^{z}\right).}$

By substituting z0 = zez into the equation above, we get the defining equation for the W function (and for the W relation in general):

${\displaystyle z_{0}=W(z_{0})e^{W(z_{0})}}$

for any complex number z0.

Since the function f is not injective, the relation W is multivalued (except at 0). If we restrict attention to real-valued W, the complex variable z is then replaced by the real variable x, and the relation is defined only for x ≥ −1/e and is double-valued on (−1/e, 0). The additional constraint W ≥ −1 defines a single-valued function W0(x). We have W0(0) = 0 and W0(−1/e) = −1. Meanwhile, the lower branch has W ≤ −1 and is denoted W−1(x). It decreases from W−1(−1/e) = −1 to W−1(0) = −∞.

It can be extended to the function z = xax using the identity

${\displaystyle x={\frac {W(z\ln {a})}{\ln {a}}}.}$

The Lambert W relation cannot be expressed in terms of elementary functions.[1] It is useful in combinatorics, for instance, in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. the maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions) and also occurs in the solution of delay differential equations, such as y′(t) = a y(t − 1). In biochemistry, and in particular enzyme kinetics, a closed-form solution for the time-course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.

Main branch of the Lambert W function in the complex plane. Note the branch cut along the negative real axis, ending at −1/e. In this picture, the hue of a point z is determined by the argument of W(z), and the brightness by the absolute value of W(z).
The modulus of the principal branch of the Lambert W function, colored according to arg W(z)

## Terminology

The two main branches W0 and W−1.

The Lambert W function is named after Johann Heinrich Lambert. The main branch W0 is denoted Wp in the Digital Library of Mathematical Functions, and the branch W−1 is denoted Wm there.

The notation convention chosen here (with W0 and W−1) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth.[2]

## History

Lambert first considered the related Lambert's Transcendental Equation in 1758,[3] which led to an article by Leonhard Euler in 1783[4] that discussed the special case of wew.

The function Lambert considered was

${\displaystyle x=x^{m}+q.}$

Euler transformed this equation into the form

${\displaystyle x^{a}-x^{b}=(a-b)cx^{a+b}.}$

Both authors derived a series solution for their equations.

Once Euler had solved this equation he considered the case a = b. Taking limits he derived the equation

${\displaystyle \ln x=cx^{a}.}$

He then put a = 1 and obtained a convergent series solution.

After taking derivatives with respect to x and some manipulation, the standard form of the Lambert function is obtained.

In 1993, when it was reported that the Lambert W function provides an exact solution to the quantum-mechanical double-well Dirac delta function model for equal charges—a fundamental problem in physics—Corless and developers of the Maple computer algebra system made a library search and found that this function was ubiquitous in nature.[2][5]

Another example where this function is found is in Michaelis–Menten kinetics.

## Calculus

### Derivative

By implicit differentiation, one can show that all branches of W satisfy the differential equation

${\displaystyle z(1+W){\frac {dW}{dz}}=W\quad {\text{for }}z\neq -{\frac {1}{e}}.}$

(W is not differentiable for z = −1/e.) As a consequence, we get the following formula for the derivative of W:

${\displaystyle {\frac {dW}{dz}}={\frac {W(z)}{z(1+W(z))}}\quad {\text{for }}z\not \in \left\{0,-{\frac {1}{e}}\right\}.}$

Using the identity eW(z) = z / W(z), we get the following equivalent formula:

${\displaystyle {\frac {dW}{dz}}={\frac {1}{z+e^{W(z)}}}\quad {\text{for }}z\neq -{\frac {1}{e}}.}$

### Antiderivative

The function W(x), and many expressions involving W(x), can be integrated using the substitution w = W(x), i.e. x = wew:

{\displaystyle {\begin{aligned}\int W(x)\,dx&=xW(x)-x+e^{W(x)}+C\\&=x\left(W(x)-1+{\frac {1}{W(x)}}\right)+C.\end{aligned}}}

(The last equation is more common in the literature but does not hold at x = 0). One consequence of this (using the fact that W(e) = 1) is the identity

${\displaystyle \int _{0}^{e}W(x)\,dx=e-1.}$

## Asymptotic expansions

The Taylor series of W0 around 0 can be found using the Lagrange inversion theorem and is given by

${\displaystyle W_{0}(x)=\sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}x^{n}=x-x^{2}+{\tfrac {3}{2}}x^{3}-{\tfrac {8}{3}}x^{4}+{\tfrac {125}{24}}x^{5}-\cdots .}$

The radius of convergence is 1/e, as may be seen by the ratio test. The function defined by this series can be extended to a holomorphic function defined on all complex numbers with a branch cut along the interval (−∞, −1/e]; this holomorphic function defines the principal branch of the Lambert W function.

For large values of x, W0 is asymptotic to

{\displaystyle {\begin{aligned}W_{0}(x)&=L_{1}-L_{2}+{\frac {L_{2}}{L_{1}}}+{\frac {L_{2}\left(-2+L_{2}\right)}{2L_{1}^{2}}}+{\frac {L_{2}\left(6-9L_{2}+2L_{2}^{2}\right)}{6L_{1}^{3}}}+{\frac {L_{2}\left(-12+36L_{2}-22L_{2}^{2}+3L_{2}^{3}\right)}{12L_{1}^{4}}}+\cdots \\[5pt]&=L_{1}-L_{2}+\sum _{l=0}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{l}\left[{\begin{smallmatrix}l+m\\l+1\end{smallmatrix}}\right]}{m!}}L_{1}^{-l-m}L_{2}^{m},\end{aligned}}}

where L1 = ln x, L2 = ln ln x, and [l + m
l + 1
]
is a non-negative Stirling number of the first kind.[2] Keeping only the first two terms of the expansion,

${\displaystyle W_{0}(x)=\ln x-\ln \ln x+o(1).}$

The other real branch, W−1, defined in the interval [−1/e, 0), has an approximation of the same form as x approaches zero, with in this case L1 = ln(−x) and L2 = ln(−ln(−x)).[2]

It is shown[6] that the following bound holds for xe:

${\displaystyle \ln x-\ln \ln x+{\frac {\ln \ln x}{2\ln x}}\leq W_{0}(x)\leq \ln x-\ln \ln x+{\frac {e}{e-1}}{\frac {\ln \ln x}{\ln x}}.}$

In 2016 it was proven[7] that the branch W−1 can be bounded as follows:

${\displaystyle -1-{\sqrt {2u}}-u0.}$

### Integer and complex powers

Integer powers of W0 also admit simple Taylor (or Laurent) series expansions at zero:

${\displaystyle W_{0}(x)^{2}=\sum _{n=2}^{\infty }{\frac {-2\left(-n\right)^{n-3}}{(n-2)!}}x^{n}=x^{2}-2x^{3}+4x^{4}-{\tfrac {25}{3}}x^{5}+18x^{6}-\cdots .}$

More generally, for r, the Lagrange inversion formula gives

${\displaystyle W_{0}(x)^{r}=\sum _{n=r}^{\infty }{\frac {-r\left(-n\right)^{n-r-1}}{(n-r)!}}x^{n},}$

which is, in general, a Laurent series of order r. Equivalently, the latter can be written in the form of a Taylor expansion of powers of W0(x) / x:

${\displaystyle \left({\frac {W_{0}(x)}{x}}\right)^{r}=e^{-rW_{0}(x)}=\sum _{n=0}^{\infty }{\frac {r\left(n+r\right)^{n-1}}{n!}}\left(-x\right)^{n},}$

which holds for any r and |x| < 1/e.

## Identities

A few identities follow from the definition:

{\displaystyle {\begin{aligned}W(xe^{x})&=x&{\text{for }}x&\geq 0,\\W_{0}(xe^{x})&=x&{\text{for }}x&\geq -1,\\W_{-1}(xe^{x})&=x&{\text{for }}x&\leq -1.\end{aligned}}}

Note that, since f(x) = xex is not injective, it does not always hold that W(f(x)) = x, much like with the inverse trigonometric functions. For fixed x < 0 and x ≠ −1, the equation xex = yey has two solutions in y, one of which is of course y = x. Then, for i = 0 and x < −1, as well as for i = −1 and x ∈ (−1, 0), y = Wi(xex) is the other solution.

Some other identities:[8]

{\displaystyle {\begin{aligned}&W(x)\cdot e^{W(x)}=x,\quad {\text{therefore:}}\\[5pt]&e^{W(x)}={\frac {x}{W(x)}},\qquad e^{-W(x)}={\frac {W(x)}{x}},\qquad e^{nW(x)}=\left({\frac {x}{W(x)}}\right)^{n}.\end{aligned}}}
${\displaystyle \ln W(x)=\ln x-W(x)\quad {\text{for }}x>0.}$ [9]
${\displaystyle W\left(a\ln a\right)=\ln a\quad {\text{for }}a\in \mathbb {R} .}$
{\displaystyle {\begin{aligned}&W(x)=\ln {\frac {x}{W(x)}}&&{\text{for }}x\geq -{\frac {1}{e}},\\[5pt]&W\left({\frac {nx^{n}}{W\left(x\right)^{n-1}}}\right)=nW(x)&&{\text{for }}n,x>0\end{aligned}}}
(which can be extended to other n and x if the right branch is chosen).
${\displaystyle W(x)+W(y)=W\left(xy\left({\frac {1}{W(x)}}+{\frac {1}{W(y)}}\right)\right)\quad {\text{for }}x,y>0.}$

From inverting f(ln x):

{\displaystyle {\begin{aligned}W(x\ln x)&=\ln x\\&=W(x)+\ln W(x)\quad {\text{for }}x>0.\end{aligned}}}

Substituting −ln x in the definition:

{\displaystyle {\begin{aligned}W_{0}\left(-{\frac {\ln x}{x}}\right)&=-\ln x&{\text{for }}0&e.\end{aligned}}}

With Euler's iterated exponential h(x):

{\displaystyle {\begin{aligned}h(x)&=e^{-W(-\ln x)}\\&={\frac {W(-\ln x)}{-\ln x}}\quad {\text{for }}x\neq 1.\end{aligned}}}

## Special values

For any nonzero algebraic number x, W(x) is a transcendental number. Indeed, if W(x) is zero, then x must be zero as well, and if W(x) is nonzero and algebraic, then by the Lindemann–Weierstrass theorem, eW(x) must be transcendental, implying that x = W(x)eW(x) must also be transcendental.

${\displaystyle W\left(-{\frac {\pi }{2}}\right)={\frac {i\pi }{2}}.}$
${\displaystyle W\left(-{\frac {1}{e}}\right)=-1.}$
${\displaystyle W\left(-{\frac {\ln a}{a}}\right)=-\ln a\quad \left({\frac {1}{e}}\leq a\leq e\right).}$
${\displaystyle W\left(2\ln 2\right)=\ln 2.}$
${\displaystyle W(0)=0.}$
${\displaystyle W(1)=\Omega =\left(\int _{-\infty }^{\infty }{\frac {dt}{\left(e^{t}-t\right)^{2}+\pi ^{2}}}\right)^{-1}-1\approx 0.56714329\ldots }$  (the Omega constant).
${\displaystyle W(1)=e^{-W(1)}=\ln \left({\frac {1}{W(1)}}\right)=-\ln W(1).}$
${\displaystyle W(e)=1.}$
${\displaystyle W(e^{1+e})=e.}$
${\displaystyle W(-1)\approx -0.31813+1.33723i.}$
${\displaystyle W'(0)=1.}$

## Representations

The principal branch of the Lambert function can be represented by a proper integral, due to Poisson:[10]

${\displaystyle -{\frac {\pi }{2}}W(-x)=\int _{0}^{\pi }{\frac {\sin \left({\tfrac {3}{2}}t\right)-xe^{\cos t}\sin \left({\tfrac {5}{2}}t-\sin t\right)}{1-2xe^{\cos t}\cos(t-\sin t)+x^{2}e^{2\cos t}}}\sin \left({\tfrac {1}{2}}t\right)\,dt\quad {\text{for }}|x|<{\frac {1}{e}}.}$

On the wider domain −1/exe, the considerably simpler representation is found by Mező:[11]

${\displaystyle W(x)={\frac {1}{\pi }}\operatorname {Re} \int _{0}^{\pi }\ln \left({\frac {e^{e^{it}}-xe^{-it}}{e^{e^{it}}-xe^{it}}}\right)\,dt.}$

The following continued fraction representation also holds for the principal branch:[12]

${\displaystyle W(x)={\cfrac {x}{1+{\cfrac {x}{1+{\cfrac {x}{2+{\cfrac {5x}{3+{\cfrac {17x}{10+{\cfrac {133x}{17+{\cfrac {1927x}{190+{\cfrac {13582711x}{94423+\ddots }}}}}}}}}}}}}}}}.}$

Also, if |W(z)| < 1:[13]

${\displaystyle W(x)={\cfrac {x}{\exp {\cfrac {x}{\exp {\cfrac {x}{\ddots }}}}}}.}$

In turn, if |W(z)| > 1, then

${\displaystyle W(x)=\ln {\cfrac {x}{\ln {\cfrac {x}{\ln {\cfrac {x}{\ddots }}}}}}.}$

## Branches and range

The range of the W function, showing all branches. The orange curves are the images of the imaginary axis. The black are the images of the real axis (the quadratrix of Hippias). The purple curves is the image of a small circle around the branch point z = 0; the red curves are the images of a small circle around the point z = −1/e.

There are countably many branches of the W function, denoted by Wk(z), for integer k; W0(z) being the principal branch. The branch point for the principal branch is at z = −1/e, with a branch cut that extends to −∞ along the negative real axis. This branch cut separates the principal branch from the two branches W−1 and W1. In all other branches, there is a branch point at z = 0 and a branch cut along the whole negative real axis.

The range of the entire function is the complex plane. The image of the real axis coincides with the quadratrix of Hippias, the parametric curve w = −t cot t + it.

## Other formulas

### Definite integrals

There are several useful definite integral formulas involving the W function, including the following:

{\displaystyle {\begin{aligned}&\int _{0}^{\pi }W\left(2\cot ^{2}x\right)\sec ^{2}x\,dx=4{\sqrt {\pi }}.\\[5pt]&\int _{0}^{\infty }{\frac {W(x)}{x{\sqrt {x}}}}\,dx=2{\sqrt {2\pi }}.\\[5pt]&\int _{0}^{\infty }W\left({\frac {1}{x^{2}}}\right)\,dx={\sqrt {2\pi }}.\end{aligned}}}

The first identity can be found by writing the Gaussian integral in polar coordinates.

The second identity can be derived by making the substitution u = W(x), which gives

{\displaystyle {\begin{aligned}x&=ue^{u},\\[5pt]{\frac {dx}{du}}&=(u+1)e^{u}.\end{aligned}}}

Thus

{\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {W(x)}{x{\sqrt {x}}}}\,dx&=\int _{0}^{\infty }{\frac {u}{ue^{u}{\sqrt {ue^{u}}}}}(u+1)e^{u}\,du\\[5pt]&=\int _{0}^{\infty }{\frac {u+1}{\sqrt {ue^{u}}}}du\\[5pt]&=\int _{0}^{\infty }{\frac {u+1}{\sqrt {u}}}{\frac {1}{\sqrt {e^{u}}}}du\\[5pt]&=\int _{0}^{\infty }u^{\tfrac {1}{2}}e^{-{\frac {u}{2}}}du+\int _{0}^{\infty }u^{-{\tfrac {1}{2}}}e^{-{\frac {u}{2}}}du\\[5pt]&=2\int _{0}^{\infty }(2w)^{\tfrac {1}{2}}e^{-w}\,dw+2\int _{0}^{\infty }(2w)^{-{\tfrac {1}{2}}}e^{-w}\,dw&&\quad (u=2w)\\[5pt]&=2{\sqrt {2}}\int _{0}^{\infty }w^{\tfrac {1}{2}}e^{-w}\,dw+{\sqrt {2}}\int _{0}^{\infty }w^{-{\tfrac {1}{2}}}e^{-w}\,dw\\[5pt]&=2{\sqrt {2}}\cdot \Gamma \left({\tfrac {3}{2}}\right)+{\sqrt {2}}\cdot \Gamma \left({\tfrac {1}{2}}\right)\\[5pt]&=2{\sqrt {2}}\left({\tfrac {1}{2}}{\sqrt {\pi }}\right)+{\sqrt {2}}\left({\sqrt {\pi }}\right)\\[5pt]&=2{\sqrt {2\pi }}.\end{aligned}}}

The third identity may be derived from the second by making the substitution u = x−2 and the first can also be derived from the third by the substitution z = (tan x) / 2.

Except for z along the branch cut (−∞, −1/e] (where the integral does not converge), the principal branch of the Lambert W function can be computed by the following integral:[14]

{\displaystyle {\begin{aligned}W(z)&={\frac {z}{2\pi }}\int _{-\pi }^{\pi }{\frac {\left(1-\nu \cot \nu \right)^{2}+\nu ^{2}}{z+\nu \csc \nu e^{-\nu \cot \nu }}}\,d\nu \\[5pt]&={\frac {z}{\pi }}\int _{0}^{\pi }{\frac {\left(1-\nu \cot \nu \right)^{2}+\nu ^{2}}{z+\nu \csc \nu e^{-\nu \cot \nu }}}\,d\nu ,\end{aligned}}}

where the two integral expressions are equivalent due to the symmetry of the integrand.

### Indefinite integrals

{\displaystyle {\begin{aligned}&\int {\frac {W(x)}{x}}dx={\tfrac {1}{2}}\left(1+W(x)\right)^{2}+C.\\[5pt]&\int W\left(Ae^{Bx}\right)dx={\frac {1}{2B}}\left(1+W\left(Ae^{Bx}\right)\right)^{2}+C.\end{aligned}}}

## Solutions of equations

Many equations involving exponentials can be solved using the W function. The general strategy is to move all instances of the unknown to one side of the equation and make it look like Y = XeX at which point the W function provides the value of the variable in X. In other words:

${\displaystyle Y=Xe^{X}\;\Longleftrightarrow \;X=W(Y).}$

### Example 1

{\displaystyle {\begin{aligned}2^{t}&=5t\\[5pt]1&={\frac {5t}{2^{t}}}\\[5pt]1&=5te^{-t\ln 2}\\[5pt]{\frac {1}{5}}&=te^{-t\ln 2}\\[5pt]-{\frac {\ln 2}{5}}&=(-t\ln 2)e^{-t\ln 2}\\[5pt]W\left(-{\frac {\ln 2}{5}}\right)&=-t\ln 2\\[5pt]t&=-{\frac {W\left(-{\frac {\ln 2}{5}}\right)}{\ln 2}}.\end{aligned}}}

More generally, the equation

${\displaystyle p^{x}=ax+b}$

where

${\displaystyle p>0{\text{ and }}p\neq 1{\text{ and }}a\neq 0}$

can be transformed via the substitution of variables

${\displaystyle -t=x+{\frac {b}{a}}}$

into

${\displaystyle tp^{t}=R=-{\frac {1}{a}}p^{-{\frac {b}{a}}},}$

giving

${\displaystyle t={\frac {W(R\ln p)}{\ln p}},}$

which yields the final solution

${\displaystyle x=-{\frac {W\left(-{\frac {\ln p}{a}}\,p^{-{\frac {b}{a}}}\right)}{\ln p}}-{\frac {b}{a}}.}$

### Example 2

{\displaystyle {\begin{aligned}x^{x}&=z\\x\ln x&=\ln z\\e^{\ln x}\cdot \ln x&=\ln z\\\ln x&=W(\ln z)\\x&=e^{W(\ln z)}.\end{aligned}}}

For z ≠ 1, this can be equivalently written as

${\displaystyle x={\frac {\ln z}{W(\ln z)}},}$

since

${\displaystyle \ln z=W(\ln z)e^{W(\ln z)}}$

by definition.

### Example 3

{\displaystyle {\begin{aligned}x^{n}&=e^{-ax},\quad \ a,n\neq 0,\\x^{n}e^{ax}&=1.\end{aligned}}}

If n is rational, let ω be any pth root of unity, where p/q =|n| is a reduced fraction. Otherwise, ω=1.

Taking the nth root:

{\displaystyle {\begin{aligned}xe^{\frac {ax}{n}}&=\omega \\{\frac {ax}{n}}e^{\frac {ax}{n}}&={\frac {a\omega }{n}}\\{\frac {ax}{n}}&=W\left({\frac {a\omega }{n}}\right)\\x&={\frac {nW\left({\frac {a\omega }{n}}\right)}{a}}.\end{aligned}}}

If we want the real solutions, and a/n is real, we must have ω = ±1 and /n > -1/e. This gives at most three solutions:

${\displaystyle x={\begin{cases}{\frac {n}{a}}W_{0}\left({\frac {a}{n}}\right),&{\text{if }}{\frac {a}{n}}\geq -{\frac {1}{e}},\\{\frac {n}{a}}W_{-1}\left({\frac {a}{n}}\right),&{\text{if }}{-{\frac {1}{e}}}<{\frac {a}{n}}<0.\end{cases}}}$

And, if n is rational and p is even:

${\displaystyle x={\begin{cases}{\frac {n}{a}}W_{0}\left({\frac {-a}{n}}\right),&{\text{if }}{\frac {a}{n}}\leq {\frac {1}{e}},\\{\frac {n}{a}}W_{-1}\left({\frac {-a}{n}}\right),&{\text{if }}{\frac {1}{e}}>{\frac {a}{n}}>0.\end{cases}}}$

### Example 4

Whenever the complex infinite exponential tetration

${\displaystyle z^{z^{z^{\cdot ^{\cdot ^{\cdot }}}}}}$

converges, the Lambert W function provides the actual limit value as

${\displaystyle c=\exp(-W(-\ln z)),}$

where ln z denotes the principal branch of the complex logarithm. This can be shown by observing that zc = c if c exists, so

{\displaystyle {\begin{aligned}z&=c^{\frac {1}{c}}\\z^{-1}&=c^{-{\frac {1}{c}}}\\{\frac {1}{z}}&=\left({\frac {1}{c}}\right)^{\frac {1}{c}}\\-\ln(z)&={\frac {1}{c}}\ln {\frac {1}{c}}\\-\ln(z)&=e^{\ln {\frac {1}{c}}}\ln {\frac {1}{c}}\\\ln {\frac {1}{c}}&=W(-\ln z)\\{\frac {1}{c}}&=e^{W(-\ln z)}\\c&=e^{-W(-\ln z)},\\\end{aligned}}}

which is the result which was to be found.

### Example 5

Solutions for

${\displaystyle x\log _{b}x=a}$

have the form[5]

${\displaystyle x=e^{W(a\ln b)}.}$

### Example 6

The solution for the current in a series diode/resistor circuit can also be written in terms of the Lambert W function. See diode modeling.

### Example 7

${\displaystyle {\dot {y}}(t)=ay(t-1)}$
${\displaystyle \lambda =ae^{-\lambda },}$

${\displaystyle \lambda =W_{k}(a)\quad {\text{and}}\quad y(t)=e^{W_{k}(a)t},}$

where k is the branch index. If a ≥ 1/e, only W0(a) need be considered.

## Applications

### Instrument design

The Lambert W function was shown in 2013 to be the optimal solution for the required magnetic field of a Zeeman slower.[15]

### Viscous flows

Granular and debris flow fronts and deposits, and the fronts of viscous fluids in natural events and in the laboratory experiments can be described by using the Lambert–Euler omega function as follows:

${\displaystyle H(x)=1+W\left((H(0)-1)e^{(H(0)-1)-{\frac {x}{L}}}\right),}$

where H(x) is the debris flow height, x is the channel downstream position, L is the unified model parameter consisting of several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient.

In pipe flow, the Lambert W function is part of the explicit formulation of the Colebrook equation for finding the Darcy friction factor. This factor is used to determine the pressure drop through a straight run of pipe when the flow is turbulent.[16]

### Neuroimaging

The Lambert W function was employed in the field of neuroimaging for linking cerebral blood flow and oxygen consumption changes within a brain voxel, to the corresponding blood oxygenation level dependent (BOLD) signal.[17]

### Chemical engineering

The Lambert W function was employed in the field of chemical engineering for modelling the porous electrode film thickness in a glassy carbon based supercapacitor for electrochemical energy storage. The Lambert W function turned out to be the exact solution for a gas phase thermal activation process where growth of carbon film and combustion of the same film compete with each other.[18][19]

### Materials science

The Lambert W function was employed in the field of epitaxial film growth for the determination of the critical dislocation onset film thickness. This is the calculated thickness of an epitaxial film, where due to thermodynamic principles the film will develop crystallographic dislocations in order to minimise the elastic energy stored in the films. Prior to application of Lambert W for this problem, the critical thickness had to be determined via solving an implicit equation. Lambert W turns it in an explicit equation for analytical handling with ease.[20]

### Porous media

The Lambert W function has been employed in the field of fluid flow in porous media to model the tilt of an interface separating two gravitationally segregated fluids in a homogeneous tilted porous bed of constant dip and thickness where the heavier fluid, injected at the bottom end, displaces the lighter fluid that is produced at the same rate from the top end. The principal branch of the solution corresponds to stable displacements while the −1 branch applies if the displacement is unstable with the heavier fluid running underneath the ligther fluid.[21]

### Bernoulli numbers and Todd genus

The equation (linked with the generating functions of Bernoulli numbers and Todd genus):

${\displaystyle Y={\frac {X}{1-e^{X}}}}$

can be solved by means of the two real branches W0 and W−1:

${\displaystyle X(Y)={\begin{cases}W_{-1}\left(Ye^{Y}\right)-W_{0}\left(Ye^{Y}\right)=Y-W_{0}\left(Ye^{Y}\right)&{\text{for }}Y<-1,\\W_{0}\left(Ye^{Y}\right)-W_{-1}\left(Ye^{Y}\right)=Y-W_{-1}\left(Ye^{Y}\right)&{\text{for }}-1

This application shows that the branch difference of the W function can be employed in order to solve other transcendental equations.[22]

### Statistics

The centroid of a set of histograms defined with respect to the symmetrized Kullback–Leibler divergence (also called the Jeffreys divergence) has a closed form using the Lambert W function.[23]

### Exact solutions of the Schrödinger equation

The Lambert W function appears in a quantum-mechanical potential (see Lambert W step potential) which affords the fifth – next to those of the harmonic oscillator plus centrifugal, the Coulomb plus inverse square, the Morse, and the inverse square root potential – exact solution to the stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions. The potential is given as

${\displaystyle V={\frac {V_{0}}{1+W\left(e^{-{\frac {x}{\sigma }}}\right)}}.}$

A peculiarity of the solution is that each of the two fundamental solutions that compose the general solution of the Schrödinger equation is given by a combination of two confluent hypergeometric functions of an argument proportional to[24]

${\displaystyle z=W\left(e^{-{\frac {x}{\sigma }}}\right).}$

### Exact solutions of the Einstein vacuum equations

In the Schwarzschild metric solution of the Einstein vacuum equations, the W function is needed to go from the Eddington-Finkelstein coordinates to the Schwarzschild coordinates. For this reason, it also appears in the construction of the Kruskal–Szekeres coordinates.

### Resonances of the delta-shell potential

The s-wave resonances of the delta-shell potential can be written exactly in terms of the Lambert W function.[25]

### Thermodynamic equilibrium

If a reaction involves reactants and products having heat capacities that are constant with temperature then the equilibrium constant K obeys

${\displaystyle \ln K={\frac {a}{T}}+b+c\ln T}$

for some constants a, b, and c. When c (equal to ΔCp / R) is not zero we can find the value or values of T where K equals a given value as follows, where we use L for ln T.

{\displaystyle {\begin{aligned}-a&=(b-\ln K)T+cT\ln T\\&=(b-\ln K)e^{L}+cLe^{L}\\[5pt]-{\frac {a}{c}}&=\left({\frac {b-\ln K}{c}}+L\right)e^{L}\\[5pt]-{\frac {a}{c}}e^{\frac {b-\ln K}{c}}&=\left(L+{\frac {b-\ln K}{c}}\right)e^{L+{\frac {b-\ln K}{c}}}\\[5pt]L&=W\left(-{\frac {a}{c}}e^{\frac {b-\ln K}{c}}\right)+{\frac {\ln K-b}{c}}\\[5pt]T&=\exp \left(W\left(-{\frac {a}{c}}e^{\frac {b-\ln K}{c}}\right)+{\frac {\ln K-b}{c}}\right).\end{aligned}}}

If a and c have the same sign there will be either two solutions or none (or one if the argument of W is exactly −1/e). (The upper solution may not be relevant.) If they have opposite signs, there will be one solution.

The classical finite-size corrections to the dispersion relations of giant magnons, single spikes and GKP strings can be expressed in terms of the Lambert W function.[26][27]

## Generalizations

The standard Lambert W function expresses exact solutions to transcendental algebraic equations (in x) of the form:

${\displaystyle e^{-cx}=a_{0}(x-r)}$

(1)

where a0, c and r are real constants. The solution is

${\displaystyle x=r+{\frac {1}{c}}W\left({\frac {c\,e^{-cr}}{a_{0}}}\right).}$

Generalizations of the Lambert W function[28][29][30] include:

${\displaystyle e^{-cx}=a_{0}\left(x-r_{1}\right)\left(x-r_{2}\right),}$

(2)

where r1 and r2 are real distinct constants, the roots of the quadratic polynomial. Here, the solution is a function has a single argument x but the terms like ri and a0 are parameters of that function. In this respect, the generalization resembles the hypergeometric function and the Meijer G function but it belongs to a different class of functions. When r1 = r2, both sides of (2) can be factored and reduced to (1) and thus the solution reduces to that of the standard W function. Equation (2) expresses the equation governing the dilaton field, from which is derived the metric of the R = T or lineal two-body gravity problem in 1 + 1 dimensions (one spatial dimension and one time dimension) for the case of unequal rest masses, as well as the eigenenergies of the quantum-mechanical double-well Dirac delta function model for unequal charges in one dimension.
• Analytical solutions of the eigenenergies of a special case of the quantum mechanical three-body problem, namely the (three-dimensional) hydrogen molecule-ion.[32] Here the right-hand side of (1) (or (2)) is now a ratio of infinite order polynomials in x:

${\displaystyle e^{-cx}=a_{0}{\frac {\displaystyle \prod _{i=1}^{\infty }(x-r_{i})}{\displaystyle \prod _{i=1}^{\infty }(x-s_{i})}}}$

(3)

where ri and si are distinct real constants and x is a function of the eigenenergy and the internuclear distance R. Equation (3) with its specialized cases expressed in (1) and (2) is related to a large class of delay differential equations. G. H. Hardy's notion of a "false derivative" provides exact multiple roots to special cases of (3).[33]

Applications of the Lambert W function in fundamental physical problems are not exhausted even for the standard case expressed in (1) as seen recently in the area of atomic, molecular, and optical physics.[34]

## Numerical evaluation

The W function may be approximated using Newton's method, with successive approximations to w = W(z) (so z = wew) being

${\displaystyle w_{j+1}=w_{j}-{\frac {w_{j}e^{w_{j}}-z}{e^{w_{j}}+w_{j}e^{w_{j}}}}.}$

The W function may also be approximated using Halley's method,

${\displaystyle w_{j+1}=w_{j}-{\frac {w_{j}e^{w_{j}}-z}{e^{w_{j}}\left(w_{j}+1\right)-{\dfrac {\left(w_{j}+2\right)\left(w_{j}e^{w_{j}}-z\right)}{2w_{j}+2}}}}}$

given in Corless et al.[2] to compute W.

## Software

The Lambert W function is implemented as LambertW in Maple, lambertw in GP (and glambertW in PARI), lambertw in Matlab,[35] also lambertw in octave with the specfun package, as lambert_w in Maxima,[36] as ProductLog (with a silent alias LambertW) in Mathematica,[37] as lambertw in Python scipy's special function package,[38] as LambertW in Perl's ntheory module,[39] and as gsl_sf_lambert_W0, gsl_sf_lambert_Wm1 functions in the special functions section of the GNU Scientific Library (GSL). In the Boost C++ libraries, the calls are lambert_w0, lambert_wm1, lambert_w0_prime, and lambert_wm1_prime. In R, the Lambert W function is implemented as the lambertW0 and lambertWm1 functions in the lamW package.[40]

A C++ code for all the branches of the complex Lambert W function is available on the homepage of István Mező.[41]

## Notes

1. ^ Chow, Timothy Y. (1999), "What is a closed-form number?", American Mathematical Monthly, 106 (5): 440–448, arXiv:math/9805045, doi:10.2307/2589148, JSTOR 2589148, MR 1699262.
2. Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E. (1996). "On the Lambert W function" (PostScript). Advances in Computational Mathematics. 5: 329–359. doi:10.1007/BF02124750.
3. ^ Lambert J. H., "Observationes variae in mathesin puram", Acta Helveticae physico-mathematico-anatomico-botanico-medica, Band III, 128–168, 1758.
4. ^ Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus". Acta Acad. Scient. Petropol. 2, 29–51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350–369, 1921.
5. ^ a b Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J. (1993). "Lambert's W function in Maple". The Maple Technical Newsletter. 9: 12–22. CiteSeerX 10.1.1.33.2556.
6. ^ A. Hoorfar, M. Hassani, Inequalities on the Lambert W Function and Hyperpower Function, JIPAM, volume 9, issue 2, article 51. 2008.
7. ^ Chatzigeorgiou, I. (2013). "Bounds on the Lambert function and their Application to the Outage Analysis of User Cooperation". IEEE Communications Letters. 17 (8): 1505–1508. arXiv:1601.04895. doi:10.1109/LCOMM.2013.070113.130972.
8. ^
9. ^
10. ^ Finch, S. R. (2003). Mathematical constants. Cambridge University Press. p. 450.
11. ^ István, Mező. "An integral representation for the principal branch of Lambert the W function". Retrieved 7 November 2017.
12. ^ Dubinov, A. E.; Dubinova, I. D.; Saǐkov, S. K. (2006). The Lambert W Function and Its Applications to Mathematical Problems of Physics (in Russian). RFNC-VNIIEF. p. 53.
13. ^ Robert M., Corless; David J., Jeffrey; Donald E., Knuth (1997). A sequence of series for the Lambert W function. Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation. pp. 197–204. doi:10.1145/258726.258783. ISBN 978-0897918756.
14. ^ "The Lambert W Function". Ontario Research Centre.
15. ^ B Ohayon., G Ron. (2013). "New approaches in designing a Zeeman Slower". Journal of Instrumentation. 8 (2): P02016. arXiv:1212.2109. Bibcode:2013JInst...8P2016O. doi:10.1088/1748-0221/8/02/P02016.
16. ^ More, A. A. (2006). "Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes". Chemical Engineering Science. 61 (16): 5515–5519. doi:10.1016/j.ces.2006.04.003.
17. ^ Sotero, Roberto C.; Iturria-Medina, Yasser (2011). "From Blood oxygenation level dependent (BOLD) signals to brain temperature maps". Bull Math Biol (Submitted manuscript). 73 (11): 2731–47. doi:10.1007/s11538-011-9645-5. PMID 21409512.
18. ^ Braun, Artur; Wokaun, Alexander; Hermanns, Heinz-Guenter (2003). "Analytical Solution to a Growth Problem with Two Moving Boundaries". Appl Math Model. 27 (1): 47–52. doi:10.1016/S0307-904X(02)00085-9.
19. ^ Braun, Artur; Baertsch, Martin; Schnyder, Bernhard; Koetz, Ruediger (2000). "A Model for the film growth in samples with two moving boundaries – An Application and Extension of the Unreacted-Core Model". Chem Eng Sci. 55 (22): 5273–5282. doi:10.1016/S0009-2509(00)00143-3.
20. ^ Braun, Artur; Briggs, Keith M.; Boeni, Peter (2003). "Analytical solution to Matthews' and Blakeslee's critical dislocation formation thickness of epitaxially grown thin films". J Cryst Growth. 241 (1–2): 231–234. Bibcode:2002JCrGr.241..231B. doi:10.1016/S0022-0248(02)00941-7.
21. ^ Colla, Pietro (2014). "A New Analytical Method for the Motion of a Two-Phase Interface in a Tilted Porous Medium". PROCEEDINGS,Thirty-Eighth Workshop on Geothermal Reservoir Engineering,Stanford University. SGP-TR-202.([1])
22. ^ D. J. Jeffrey and J. E. Jankowski, "Branch differences and Lambert W"
23. ^ F. Nielsen, "Jeffreys Centroids: A Closed-Form Expression for Positive Histograms and a Guaranteed Tight Approximation for Frequency Histograms"
24. ^
25. ^ de la Madrid, R. (2017). "Numerical calculation of the decay widths, the decay constants, and the decay energy spectra of the resonances of the delta-shell potential". Nucl. Phys. A. 962: 24–45. arXiv:1704.00047. Bibcode:2017NuPhA.962...24D. doi:10.1016/j.nuclphysa.2017.03.006.
26. ^ Floratos, Emmanuel; Georgiou, George; Linardopoulos, Georgios (2014). "Large-Spin Expansions of GKP Strings". JHEP. 03 (3): 0180. arXiv:1311.5800. doi:10.1007/JHEP03(2014)018.
27. ^ Floratos, Emmanuel; Linardopoulos, Georgios (2015). "Large-Spin and Large-Winding Expansions of Giant Magnons and Single Spikes". Nucl. Phys. B. 897: 229–275. arXiv:1406.0796. doi:10.1016/j.nuclphysb.2015.05.021.
28. ^ Scott, T. C.; Mann, R. B.; Martinez Ii, Roberto E. (2006). "General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function". AAECC (Applicable Algebra in Engineering, Communication and Computing). 17 (1): 41–47. arXiv:math-ph/0607011. doi:10.1007/s00200-006-0196-1.
29. ^ Scott, T. C.; Fee, G.; Grotendorst, J. (2013). "Asymptotic series of Generalized Lambert W Function". SIGSAM (ACM Special Interest Group in Symbolic and Algebraic Manipulation). 47 (185): 75–83. doi:10.1145/2576802.2576804.
30. ^ Scott, T. C.; Fee, G.; Grotendorst, J.; Zhang, W.Z. (2014). "Numerics of the Generalized Lambert W Function". SIGSAM. 48 (1/2): 42–56. doi:10.1145/2644288.2644298.
31. ^ Farrugia, P. S.; Mann, R. B.; Scott, T. C. (2007). "N-body Gravity and the Schrödinger Equation". Class. Quantum Grav. 24 (18): 4647–4659. arXiv:gr-qc/0611144. Bibcode:2007CQGra..24.4647F. doi:10.1088/0264-9381/24/18/006.
32. ^ Scott, T. C.; Aubert-Frécon, M.; Grotendorst, J. (2006). "New Approach for the Electronic Energies of the Hydrogen Molecular Ion". Chem. Phys. 324 (2–3): 323–338. arXiv:physics/0607081. Bibcode:2006CP....324..323S. CiteSeerX 10.1.1.261.9067. doi:10.1016/j.chemphys.2005.10.031.
33. ^ Maignan, Aude; Scott, T. C. (2016). "Fleshing out the Generalized Lambert W Function". SIGSAM. 50 (2): 45–60. doi:10.1145/2992274.2992275.
34. ^ Scott, T. C.; Lüchow, A.; Bressanini, D.; Morgan, J. D. III (2007). "The Nodal Surfaces of Helium Atom Eigenfunctions". Phys. Rev. A. 75 (6): 060101. Bibcode:2007PhRvA..75f0101S. doi:10.1103/PhysRevA.75.060101.
35. ^ lambertw – MATLAB
36. ^ Maxima, a Computer Algebra System
37. ^ ProductLog at WolframAlpha
38. ^
39. ^ ntheory at MetaCPAN
40. ^ Adler, Avraham (2017-04-24), lamW: Lambert W Function, retrieved 2017-12-19
41. ^ The webpage of István Mező


text:
# Why the temperature of a gas drops in adiabatic expansion ?

Q: Why the temperature of a gas drops in adiabatic expansion ?

Sol: In adiabatic expansion , there is no exchange of heat between system and surroundings i.e. ΔQ = 0 .

By the first law of thermodynamics $\large \Delta Q = \Delta U + \Delta W$

Therefore in Adiabatic expansion ; $\large \Delta W = – \Delta U$

Thus , work done by a gas in adiabatic expansion is associated with a decrease of internal energy , so its temperature decreases .

format: plain


text:
# Is the essential derivative of a convex function continuous?

Suppose that $f:\mathbb{R}\rightarrow\mathbb{R}$ is convex on $\mathbb{R}$, but not necessarily differentiable everywhere. We know, though, that such a function will be differentiable almost everywhere, and that the set of points where $f$ is nondifferentiable, call this $\cal{A}$, is at most countable, and of Lebesgue measure zero.

Then, the derivative of $f$, call it $f'$, exists on $\mathbb{R}\setminus\cal{A}$.

Question #1: Is $f'$ continuous on $\mathbb{R}\setminus\cal{A}$?

Question #2: Does the result change if $f$ is nondecreasing?

I ask these questions because we know that a convex, differentiable function on $\mathbb{R}$ has a continuous derivative; I am just wondering if this proposition can be extended as described above.

Thanks!

• The statements and references at this 14 June 2001 ap-calculus post archived at Math Forum probably answer your questions, but I don't have time now to sort through this and be more specific. – Dave L. Renfro Nov 10 '17 at 12:34

Yes to #1, no to #2. For any a.e. defined increasing function, left limits and right limits must exist, and they match except at jumps. $f'$ is an increasing function where defined, so is continuous except for jump discontinuities, which is where $f$ is not differentiable.


text:
Normalizing axes prior to PCA

For a given centered configuration of points $X\in\mathbb{R}^{n\times 3}$, the covariance matrix is denoted by $S=\frac{1}{n}X^TX$. Recall that the 2D PCA solution is obtained by $Y=X\cdot U$, where $U\in\mathbb{R}^{3\times 2}$ is a matrix of orthonormal eigenvectors of $S$ associated with its 2 dominant eigenvalues $\lambda_1, \lambda_2$.

Now, suppose that $Y_1\in\mathbb{R}^{n\times 2}$ is obtained by normalizing each axis of $Y$. The normalization factor would be $1/\sqrt{\lambda_i}$, $i\in\{1,2\}$ (please correct me if I'm wrong here).

Also, suppose that $Z$ is obtained by normalizing each axis of $X$ to unit length. A 2D PCA solution on $Z$ is denoted by $Y_2$. Are $Y_1$ and $Y_2$ the same? I obtain that the results are different, but it hold that the axes of both $Y_1$ and $Y_2$ are orthonormal. If so, what is the distinguishing characteristic of $Y_2$ relative to $Y_1$?

Normalizing the axes of $X$ by the singular values of $Y$ is arbitrary and hence meaningless. ($X$ might just be $Y$ with the ordering of the three reversed.) No sensible relation can be expected, neither is the use of the second recipe sound.
• Please note the edit I made (configuration $Z$ is $X$ with all axes having unit lenght (1)). What property would have a 2D PCA solution from $Z$ relative to PCA solution from $X$. (they will not be the same, and not just differ by the aspect ratio, but I'm interested in the actual properties and possibly a relation to PCA on intact $X$) – usero Jul 27 '12 at 19:56


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# For which $(a,b) \in \mathbb{R}$ is $x^2+2axy+by^2$ the real part of a holomorphic function?

I am working on the following exercise:

For which $$a,b \in \mathbb{R}$$ is $$u := x^2+2axy+by^2$$ the real part of a holomorphic function in $$\mathbb{C}$$? For each of these $$(a,b)$$ find all holomorphic functions.

I think we should use the Cauchy-Riemann Differential Equations here, so for the imaginary part $$v$$ of a holomorphic function has to hold:

$$\frac{\partial u}{\partial_x} = 2x+2ay = \frac{\partial v}{\partial_y}$$

$$\frac{\partial u}{\partial_y} = 2ax+2by = -\frac{\partial v}{\partial_x}$$

To get $$v$$ from the partial derivatives I would say that we need integration. Integrating $$2x+2ay$$ over $$y$$ yields $$2xy+ay^2+C_1$$ and integrating $$(-1) \cdot (2ax+2by)$$ over $$x$$ yields $$-ax^2-2bxy+C_2$$ I do not know how to continue from here. Could you help me?

Assume that $$u$$ is the real part of a holomorphic function. By equality of mixed partial derivatives and the CR equations, $$\frac{\partial ^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{\partial}{\partial x} \frac{\partial v}{\partial y} - \frac{\partial}{\partial y} \frac{\partial v}{\partial x} = 0$$

Holds for all $$(x,y) \in \mathbb C^2$$. This gives you $$b = -1$$.

We have : $$\frac{\partial v}{\partial x} = 2y - 2ax \implies v = f(y) + 2yx - ax^2$$ for some $$C^1$$ function of $$y$$. Similarly, $$\frac{\partial v}{\partial y} = 2x + 2ay$$ implies $$v = f(x) + 2xy + ay^2$$ for some $$C^1$$ function of $$x$$. Comparing the expression by isolating $$x$$ and $$y$$ terms, $$f(x) = -ax^2$$ and $$f(y) = ay^2$$. Thus, we get $$v(x,y) = -ax^2 + 2xy + ay^2$$.

Thus, $$x^2 + 2axy - y^2 + i(-ax^2+ 2xy + ay^2)$$ is the family of holomorphic functions in question.

Dig a little deeper, and you see that : $$x^2(1-ai) +2(a+i)(xy) - y^2(1-ai) = (1-ai)(x^2 + 2ixy - y^2) = (1-ai)(x+iy)^2$$ is a scalar multiple of the square of a complex number, so of course it is holomorphic.

If $$u$$ is the real part of a holomorphic function, $$u$$ is harmonic (and this follows from Cauchy-Riemann equations).

Thus, a necessary condition for $$y$$ to be the real part of a holomorphic function is that $$\Delta u=0\\ 2+2b=0\\ b=-1$$

Conversely, a harmonic function in all of $$\mathbb{R}^2$$ is the real part of an holomorphic function (which in our case you can determine explicitly), and we are done

The constants $$C_1=f(x)$$ and $$C_2=g(y)$$. Hence $$v_1=2xy+ay^2+f(x)$$ Signe we know $$v_{1x}=2y+f’(x)=-2ax-2by$$ From Which we get $$b=-1$$ and $$f’(x)=-2ax$$. We also have $$-ax^2-2byx+g(y)=v_2$$ We know that $$-2bx+g’(y)=2x+2ay$$ From comparison we get $$g’(y)=2ay$$. The two functions are identical so it doesn’t matter which one we choose. $$v_1=2xy+ay^2-ax^2$$ $$v_2=2xy+ay^2-ax^2$$ From these calculation I conclude that $$b=-1$$ and $$a$$ can be whatever.


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## College Algebra (10th Edition)

Interval Notation: $(-1 , 1]$ Inequality Form: $-1 \lt x \le 1$
The graph involves an open parenthesis and a close bracket. This means that the number where the bracket is located (1) is included in the interval while the number where the parenthesis is located (-1) is not included in the interval. The graph covers $x$ from $-1$ to $1$. This means that the interval is $(-1 , 1]$. The inequality equivalent of this is $-1 \lt x \le 1$.


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)}}{\*\pnseclvl6\pnlcltr\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}} {\*\pnseclvl7\pnlcrm\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl8\pnlcltr\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl9\pnlcrm\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}\pard\plain \ql \li0\ri0\sl480\slmult1 \widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs20\lang1033\langfe1033\cgrid\langnp1033\langfenp1033 {\b\fs40 \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par }\pard \ql \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 {\b\fs40 \par \par \par \par }\pard \ql \li0\ri0\sl480\slmult1\widctlpar\faauto\adjustright\rin0\lin0\itap0 {\b\fs40 \par \par \par \par \par }\pard \ql \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 {\b\fs40 \par }\pard \ql \li0\ri0\sl480\slmult1\widctlpar\faauto\adjustright\rin0\lin0\itap0 {\b\fs40 \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par }\pard \ql \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 {\b\fs40 \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par }\pard\plain \s2\ql \li0\ri0\sl480\slmult1\keepn\widctlpar\faauto\outlinelevel1\adjustright\rin0\lin0\itap0 \b\fs40\lang1033\langfe1033\cgrid\langnp1033\langfenp1033 { \par \par \par \par \par \par \par \par }\pard\plain \ql \li0\ri0\sl480\slmult1\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs20\lang1033\langfe1033\cgrid\langnp1033\langfenp1033 {\b\fs40 \par \par \par \par \par \par \par \par \par \par \par \par }\pard \ql \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 {\b\fs40 \par \par }{ \par \par \par \par \par \par \par \par \par \par \par \par \par }{\b\i\fs40 \par \par }{ \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par }{\b\fs40 \par C H R O N I C L E O F T H E \par \par }{ \par \par }{\b\fs40 }{ \par \par \par Volume IV \par Appendix \par \par Table A \par The Four Angelic Guardians of the Holy People of Israel \par \par Machsi \'93Thy Refuge\'94 \par Umtsudath \'93And my Fortress\'94 \par Tsinnah \'93Thy Shield\'94 \par Usocherah \'93And Buckler\'94 \par \par Table B \par The Fifth Guardian (Honorary): Angel of the Contretemps, \par With Distinguishing Virtues \par Sir Ernest Shackleton Endurance/Faithfulness \'93Shepherd\'94 \par Kiwi Dave Wyndham Forgiveness \par Skip Cavendish Unconditional Love \par \par \par }{\b \par \par \par \par }{ \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par }{\b\fs52 Volume VII \par \par C H R O N I C L E S O F THE \par \par H U N T E R S T A R S \par \par }{ \par \par \par \par \par \par \par \par \par }{\b\fs52 \par \par \par \par \par \par }{ \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par Volume IV \par Appendix \par \par Table One \par The Homeric Trojan Cycle (Classical), \par With Brave Scout\rquote s Recovered Epics and Fragments of Epics \par }{\i \par Iliad }{ An epic work of Homer the poet, about the Wrath of Achilles during the Siege of Troy (Ilios, the City of the Serpent-Armed Goddess); Brave Scout\rquote s accounts deal with the major event of the seduction and fall of the city. \par }{\i Odyssey}{ Another work of Homer (forming part of the lost \'93Nostoi\'94 epic), following the Homeward Journey of Odysseus and his men after the Sack of Troy; this journey is rep licated in part by Homer in Atlantis, who seeks his lost homeland and his own true identity. \par }{\i *Cypria }{ A lost epic, until recovered by Brave Scout: From the \'93Judgment of Paris\'94 to the War of the Achaeans and the Trojans How Paris abducted the royal wif e of King Menelaus of Sparta according to the plan of Agamemnon, king of Mycenae, igniting the war between the Achaeans and the Ilians. Cypria, the name of the ship, was also the name of an island the Ilians controlled before it was seized by Achaean forces in the war. \par }{\i *Aethiopis}{ A lost epic, until recovered by Brave Scout: About the Achaean champion warrior Akillos, (or Achilles), and his slayings of Penthesilea the Amazon and the Ethiopian champion, Memnon, followed by his own death; the epic appears in Chronicle of the Undeleted, A.S. 2024, Part II, A Question of Any. This epic is restored in part with Brave Scout\rquote s accounts of the epic struggles of Elijah against Jezebel. Since he had struggled bitterly (with his own spirit, primarily, that unmanfully sought to destroy what he knew was the right, manful thing to do), he was eminently qualified to be the teller of the restoration. \par }{\i *The Little Iliad II \par }{Brave Scout\rquote s \'93The Horse of Tenedos\'94 and other portions dealing with this Horse of Stumbling as it continued to appear in the course of civilization and how, ultimately, the Horse became a stumbling-block to the whole, anti-Yeshua Roman Empire. \par }{\i *Sack of Troy \par }{ Also a lost epic until recovered by Brave Scout: About the capture and burning of the city by the Achaeans led by Odysseus and Agamemnon; how the power-stone was found and what happened to it, along with the destroying, flying \'93Eye of the Gods. \'94 \par }{\i Returns }{(Nostoi) of the Heroes \par Lost in antiquity (except for the Odyssey, which formed one part of it), but restored in the Returns of the Joseph Cycle by Gabriel Tall Scout; \par Also, Brave Scout\rquote s accounts of how various champions of the Alpha and Omega, the First and the Last, achieved victory after much trouble and defeat. \par }{\i Telegony}{ \par Originally about Telegonus, son of Odysseus by Circe the witch, who takes a bad turn in life with an ill-conceived marriage. \par In the Homer of Atlantis chronicles, Homer is ill-starred (\'93bewitched\'94 by a mother who vanishes and a grandmother who deceives him; who becomes both Odysseus and his son Telegonus, for he seeks his lost identity, his son, and in seeking the fathers a new identity, a new and terrible Telegonus. \par ________________ \par *Fragments of these appear in the Retro Star series \par ________________ \par \par Table Two \par The Tall Chief/Brave Scout Trojan Cycle (Pre-Classical and Classical to Modern Era) \par The Troika Horses of the Mu-Nu Point--the Mediterranean World \par A.A.S. 1230, }{\i Chronicle of the Horse of Tenedos \par }{A.A.S. 30}{\i , Chronicle of the Horse of Antirrhodus}{ \par \par A.S. 50, }{\i Chronicle of the Horse of Troas}{ \par \par The Horses of the Kaph-Lamed Point--the Eurasian World \par A.S. 1947, }{\i Chronicle of the Horse of Nanking}{ \par A.S. 1970, }{\i Chronicle of the Horse of Warsaw \par }{ \par The Troika Horses of the M-N Point--The Western World \par A.S. 1922, }{\i Another Troy \par \par }{A.S. 1938, }{\i Chronicle of the Mountain Tomb; \par }{A.S. 1938-41, }{\i Chronicle of the Oracle of Meno}{ \par \par \par \par Table Three \par Yeshua\rquote s Mount: One Horse in Two Locations \par \par The Two Horses of Z-Point--Universes I and II \par Z-Point, }{\i The Gloria Dei Procession}{ \par _________________ \par *At His investiture in 30 A.D., the Lion-Lamb Yeshua (Divine Meekness) came riding a colt of an ass into Jerusalem; As the King of Kings, Lord of the Universes (Divine Majesty), He rides a white stallion at Z-Point) \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par Volume IV \par Appendix \par \par Angel of the Contretemps Cycle \par A.S. 1922 Chronicle of the Ice Bird--Part I \par A.S. 1973 Chronicle of the Ice Bird--Part II \par A.S. 1978 Chronicle of the Ice Bird--Part III \par A.S. 1973-78 Chronicle of the Blue Bridge Salient \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par }{\b\fs24 \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par }{Volume IV \par Appendix \par \par The Royal Line of Ptolemies in Mizraim \par Years of Reign \par Ptolemy I, Soter \'93The Savior\'94 323-285 B.C. (or d. 283 B.C.) \par Ptolemy II, Philadelphus 285-247 B.C. \par Ptolemy III, Euergetes \'93Benefactor\'94 247-222 B.C. \par Ptolemy IV, Philopater 222-205 B.C. \par Ptolemy V, Epiphanes 205-181 B.C. \par Ptolemy VI, Philometor 181-146 B.C. \par Ptolemy VII, Evergetes II 146-117 B.C. \par Ptolemy VIII, Soter II, or Lathryrus 146-81 B.C. (ruled Cyprus for 18 yrs. \par while his brother Alexander I held the throne in Alexandria) \par Ptolemy IX, Ptolemy Alexander I d. 88 B.C. (ruled Mizraim while Ptolemy VIII was \par expelled to Cyprus) \par Ptolemy X, Ptolemy Alexander II 81-80 B.C. (assassinated by a mob shortly after \par accession) \par Ptolemy XI, Ptolemy Philopator Neos Dionysios, familiarly Auletes, the Flute Player \par 80-51 B.C. (held the throne after killing his daughter \par Berenice and her husband who had been made rulers) \par Ptolemy XII 51-47 B.C. \par Ptolemy XIII 47-44 (married to Cleopatra, who had been made \par queen in 47 B.C. by Julius Caesar, disappeared, probably murdered by Cleopatra to make \par room for Caesarion her son by Julius Caesar) \par Ptolemy XIV 47-30 B.C. (joint ruler with his mother Cleopatra \par until the defeat at Actium; put to death by Octavianus of Rome) \par \par }{\b\fs24 \par \par \par \par \par }{Volume IV \par Appendix \par \par Sons of Jesse of the Bethehem Weaver-Clan* \par Eldest \par 1. Eliab (or Elihu)--Firstborn \par 2. Abinadab \par 3. Shammah (or Shimma) \par 4. Nethaneel \par 5. Raddai \par 6. Ozem \par 7. Elhanan (later known as David) \par 8. One unnamed brother \par \par __________ \par *the town being small and possessing no great resources, each head of a household found it necessary to make one living from several trades or occupations. Jesse was a man of substance, being not rich, but not poor, but he found it necessar y to employ his womenfolk in weaving and rug-making, and \par his sons in pottery; one son, Elhanan, was employed in keeping the family\rquote s herd of sheep. The fineness of the weaving and rugs produced by Jesse\rquote s family from the fine wool gathered from Elhanan\rquote s flock came to overshadow the other productions, and so they became known as the Weaver-Clan though most of the family made simple, practical earthenware pots, large and small, that were sold in the much larger \par neighboring Jebusite market town of Jebus (later known as Jerusalem, after it was conquered by the army of Jesse\rquote s son, the former shepherd, Elhanan). \par \par \par \par \par \par }{\b \par }{\b\fs24 \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par }{ \par \par \par \par \par \par Volume IV \par Appendix \par \par Table A \par Names for the Families or Races of Giants \par \par Emims--Deut. 2:10--in Moab \par Zamzummins--Deut. 2:20--in the land of the Ammonites \par Horims--Deut. 2:12--in Seir, where the Edomites later settled after destroying them \par Anakim--Numbers 13:22--in Philistia and surrounding countries \par \par Table B \par \par Biblical Names of Giants \par \par Anak--father of a family or race of giants in Philistia and Surrounding Areas;Numbers 13:22 lists his sons \par as Ahiman, Sheshai, and Talmai; \par Sheshai, Ahinan, Talmai were killed by Caleb the Kenite who followed Moses and entered the Promised Land and gained possession of his allottment of land \par Sippai (I Chron. 20:4) \par Og, king of Bashan \par Isbi-benob, killed by Abishai \par Goliath (or Golyat), one of five brothers serving as champions of the five Philistine cities \par Lahmi, brother of Goliath \par Unnamed Philistine giant killed by David\rquote s nephew \par Saph (II Samuel 21:18) \par Arba (champion of Hebron, or Kiriath--gave his name to the city Kiriah-arba) \par \par Table C \par Names of Dwarves (Species or Ethnic Groups) \par \par Gammadims--Ezekiel 27:11--Tyre, serving as watchmen in the towers, along with Arvadites upon \par the walls guarding them. \par ________________________ \par Note: What were the Gammadims, a species or mutants? Evidently, they were numerous enough to \par be used to guard the towers and walls of a queen city, Tyre, so it is hard to see that they were \par mutations, which would be rare and hard to find. Anything so numerous had to be a species. The giants, too, were numerous enough to be called \'93families,\'94 which really mean they were a species, for giantism occurred not in rare indivi duals but ran in entire families. The family in Gath that claimed Goliath was one such. In the close confines of narrow towers, the military value of this strange species can be appreciated. Rather than using fewer men of normal height and size, ma n y more Gammadims with bows and also pots of boiling oil or heaving big rocks could be accommodated, thus unleashing greater havoc on the enemy soldiery below the walls. Where did they originate? Quite possibly, they were natives of the Middle East. In Africa, a species of dwarves lives to this modern day, which were called \'93Deng\'94 in ancient times but are now called \'93pygmies.\'94 It is evident, also, that the Gammadims died out, unless they were imported pygmies from Africa, which is not unlikely, since the Tyrians operated far-flung trade lines that encompassed Africa, the Middle East, Europe and even the British Isles. \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par Volume IV \par \par \par \par \par \par \par \par \par \par Volume IV \par Appendix \par \par \par Timeline--The Question of the Crucifixion of Christ \par on the Twin Earths \par \par As the chief defining Event (after the Fall of Man in Garden of Eden), the Crucifixion \par must never be treated lightly, even in fiction. It remains forever, exclusively non-fiction and supremely sacred text. Although the author and chronicler begs revelation from God on this Greatest Event \par in heaven and earth, the Holy Scriptures describe it adequately. Reverence is only needed \par to treat it properly. For that the author and chronicler prays: reverence and awe that \par will illumine the mind on what the Scriptures describe in Matthew, Mark, Luke, and John, \par and elsewhere in both Old and New Testaments. The author confesses that he took too light a view of crucifixion in first treating of the Event in this series, but he has since \par asked God for forgiveness, God who--in order to save humanity in the only way possible--paid \par such a great Price as giving up the life of His own Son on the Cross. Christ said that every \par sin committed against Him would be forgiven, and the author is assured that his superficial \par attitude is forgiven and cleansed by the Blood of Christ. \par First, there can be no double death of Christ through crucifixion. Christ cannot have died on two separate worlds and at two times, since Scripture plainly states that Christ died once for all. We can \par crucify Him again by sinning against him or denying belief in Him--but not in the sense \par that Christ is put back upon the Cross, for that was God\rquote s Supreme Act of Dealing With \par Humanity\rquote s Sin through a Substitute taking Humanity\'94s Death Sentence! The Crucifix is an emblem that is a continual reminder of the Lord\rquote s agony on the Cross, but everyone should understand that crucifixion is not taking place forever but has been completed. \par The crucifixion was completed, and it is done and over with. But what was completed in that event? On the Cross humanity was pardoned by Christ\rquote s substitution. He, who was innocent of all sin, was \par \'93made sin\'94 for our sake. God the Father looked upon Christ\rquote s substitution, his taking \par sin\rquote s guilt on Himself, as fully and forever sufficient to pay for justice\rquote s demand for punishment. \par Jesus was the Lamb provided by God to be slain for the setting free and pardon of sinful \par and condemned humanity. No other man was available or sinless to stand in humanity\rquote s stead but this man, who was God\rquote s only begotten Son. No other man was sinless and perfect, as Jesus \par was from beginning to end of his tenure on earth. He was condemned by Roman and Jewish \par authorities for claiming he was the Son of God, but he was never convicted of any wrongdoing \par or crime and sedition against the Roman rule. Sedition was the lesser false charge, which was \par entirely unsupported, since he came and went freely, speaking everywhere, and at no time did the Roman authorities arrest him for opposing their rule, since He always said that His kingd om was not of this world. His crime, which was falsely laid as the condeming charge against him, was supposedly blasphemy. But it was no blasphemy, for He truly was the Son of God! He told the truth, and they condemned Him for the truth. Even this, on the Cross, the Lord forgave them!--\'94Father,\'94 he prayed in his last dying \par moments, \'93forgive them, for they know not what they do.\'94 \par Since, then, there can be no double death of Christ, this series cannot present such a thing. \par Christ is portrayed as being born twice, it is true, for this wonderful Incarnation of the Divine into Human Flesh is still wonderful when doubled, but when it comes to Christ\rquote s death this series cannot present him as being slain twice. Christ suffered too much for tha t, and scripture clearly forbids a repeat of His death on the Cross. The crucifixion, in its entirety, is a historical fact--amply attested by the generation of Jews and Romans and many other peoples contemporary with Yeshua the Christ. His Resurrect ion also is amply supplied with historical evidence, despite the continual carping and hedging of some scholars to whom nothing so momentous could ever be acceptable (though even some of these, who have been \par most notable scholars and authorities, have b een converted to a belief in Christ or at least acknowledgement of the fact of the Resurrection, through honest research and judicious handling of the manuscripts). In this series a second crucifixion occurs, but there is no second death of Christ. W hatever can be learned by repeating the crucifixion stops short of violating His once-for-all-time death on the Cross. Science fiction must bow the knee somewhere before the Truth, and I have chosen to bow my writer\rquote s knee here, regardless of the difficul ties and inconsistencies in the plot this creates. By this point the reader should have gotten the point of the Crucifixion anyway, and to proceed to a second death would be not only blasphemous, in my opinion, but totally unnecessary. Christ\rquote s Incarnation and Nativity in Bethlehem is quintessentially reenacted (not just \'93repeated\'94 ) every time a human soul--the spiritual equivalent of the Bethlehem stable--welcomes in the Christ Spirit. The transfiguring spiritual birth that occurs then in an indiv idual heart is the same that occured twenty centuries ago in the natural setting of a Bethlehem stable-cave, and it is repeated in countless human hearts. For that reason I make no apology for staging a second Nativity on Earth II. \par \par Volume IV \par Appendix \par \par }{\b\fs52 \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par }{ \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par }{\fs40 Book Three \par \par }{\b\fs40 \par \par U N C H R O N I C L E OF T H E S E C R E T \par S H A R E R S, P A R T III \par \par A N N O S T E L L A E }{\b\i\fs40 3 0 - \par }{\b\fs40 \par }{\b\i\fs40 \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \'93Joseph\rquote s Letter\'94 \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par }{ \par \par \par \par Lost Chronicle \par \par \par \par \par \par \par \par \par \par }{\b\fs40 \par \par \par }{\b\fs56 \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par }{ \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par Not used \par \par \'93Dear Van,\'94 she had written, \'93don\rquote t ever give up on your dreams. I won\rquote t if you won\rquote t! I\rquote m gonna be an artist someday! What are you dreaming to be? I think I can guess, but I will let you tell me yourself when you can. My Mom doesn\rquote t know anything about the world like she thinks she does. She says there\rquote s nothing you need to paint, since there are enough pictures already painted. All she knows is her house and yard--and doesn\rquote t care about the rest. Daddy says artists are sponges and no good scabs, living off honest, hard-working people when they should be working at real jobs. But my art teacher says I have talent, and not to give up, and I won\rquote t. So, Van, just know that whatever happens I\rquote m your friend, and I\rquote ll stand by you, despite what others say. Follow your lucky star, and I\rquote ll follow mine! \par Your pen pal, Cecily (Cissy) Helen O\rquote Calahan. \par \par \'93P.S. I\rquote m French, Irish, and Dutch, and the Dutch is best, because I love my grandma who \par is Dutch. She says I can be an artist if that is what I want. Why can\rquote t Mom and Daddy be like her?\'94 \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par Volume VII \par }{\b\fs52 \par }{ \par Volume IV \par Appendix \par \par \par \par Champions of the Blue Bridge by Continent \par Europe \par Ilse Former Auschwitz guard \par \par No. America/Asia \par Jacob Mitsuo \par John Plummer Kim Phuc \par \par Oceania \par Te Hapuku Karaitiana \par \par No. America \par Pearl Mr. Shoop \par A Lakota chief David Guth \par \par Middle East \par Joseph II Abdullah \par \par So. America \par The Native Peoples The Latinos \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par Vol. IV \par Appendix \par \par Some Fives in World Development \par \par The Eton Five \par The Russian Five (composers) \par Five Good Emperors (Chinese) \par Five Great Kings \par Five Pecks of Rice Movement \par Five-Power Constitution \par Five-Tone Scale \par Fivers (British game) \par The Pentocracy (A secret cabal of five influential (financially, politically, and socially) persons in Britain and the U.S. organized in the early 20}{\super th}{ century for world domination) \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par }}


text:
Successive Refinement with Decoder Cooperation and its Channel Coding Duals

# Successive Refinement with Decoder Cooperation and its Channel Coding Duals

Himanshu Asnani, Haim Permuter and Tsachy Weissman 111Department of Electrical Engineering, Stanford University 222Department of Electrical Engineering, Ben Gurion University
###### Abstract

We study cooperation in multi terminal source coding models involving successive refinement. Specifically, we study the case of a single encoder and two decoders, where the encoder provides a common description to both the decoders and a private description to only one of the decoders. The decoders cooperate via cribbing, i.e., the decoder with access only to the common description is allowed to observe, in addition, a deterministic function of the reconstruction symbols produced by the other. We characterize the fundamental performance limits in the respective settings of non-causal, strictly-causal and causal cribbing. We use a new coding scheme, referred to as Forward Encoding and Block Markov Decoding, which is a variant of one recently used by Cuff and Zhao for coordination via implicit communication. Finally, we use the insight gained to introduce and solve some dual channel coding scenarios involving Multiple Access Channels with cribbing.

{keywords}

Block Markov Decoding, Conferencing, Cooperation, Coordination, Cribbing, Double Binning, Duality, Forward Encoding, Joint Typicality, Successive Refinement.

## I Introduction

Cooperation can dramatically boost the performance of a network. The literature abounds with models for cooperation, when communication between nodes of a network is over a noisy channel. In multiple access channels, the setting of cribbing was introduced by Willems and Van der Muelen in [1], where one encoder obtains the channel input symbols of the other encoder (referred to as “crib”) and uses it for coding over a multiple access channel (MAC). This was further generalized to deterministic function cribbing (where an encoder obtains a deterministic function of the channel input symbols of another encoder) and to cribbing with actions (where one encoder can control the quality and availability of the “crib” by taking cost constrained actions) by Permuter and Asnani in [2]. Cooperation can also be modeled as information exchange among the transmitters and receivers via rate limited links, generally referred to as conferencing in the literature. Such a model was introduced in the context of the MAC by Willems in [3], and subsequently studied by by Bross, Lapidoth and Wigger [4], Wiese et al. [5], Simeone et al. [6], and Maric, Yates and Kramer [7]. Cooperation has also been modeled via conferencing/cribbing in cognitive interference channels, such as the settings in Bross, Steinberg and Tinguely [8] and Prabhakaran and Vishwanath [9]-[10]. We refer to Ng and Goldsmith [11] for a survey of various cooperation strategies and their fundamental limits in wireless networks.

In multi terminal source coding, cooperation is generally modeled as a rate limited link such as in the cascade source coding setting of Yamamoto [12], Cuff, Su and El Gamal [13], Permuter and Weissman [14], Chia, Permuter and Weissman [15], as well as the triangular source coding problems of Yamamoto [16], Chia, Permuter and Weissman [15]. In cascade source coding (Fig. 1), Decoder 1 sends a description () to Decoder 2, which does not receive any direct description from the encoder, while in triangular source coding (Fig. 2), Decoder 1 provides a description () to Decoder 2 in addition to the direct description () from the encoder.

The contribution of this paper is to introduce new models of cooperation in multi terminal source coding, inspired by the cribbing of Willems and Van der Muelen [1] and by the implicit communication model of Cuff and Zhao [17]. Specifically, we consider cooperation between decoders in a successive refinement setting (introduced in Equitz and Cover [18]). In successive refinement, a single encoder describes a common rate to both the decoders and a private rate to only one of the decoders. We generalize this model to accommodate cooperation among the decoders as follows :

1. Cooperation via Conferencing : One such cooperation model considered is that shown in Fig. 3, where the encoder provides a common description () to both the decoders and a refined description () to Decoder 1, Decoder 1 cooperates with Decoder 2 by providing an additional description which is the function of its own private description (), as well as the common description (). This setting is inspired by the conferencing problem in channel coding described earlier. The region of achievable rates and distortions for this problem is given by,

 R0+R1 ≥ I(X;^X1,^X2) (1) R0+R12 ≥ I(X;^X2), (2)

for some joint probability distribution such that , for , where refers to the distortion function and are the distortion constraints, as is formally explained in Section II. The direct part of this characterization, namely that this region is achievable, follows standard arguments that generalize those used in the original successive refinement problem [18] (cf. Appendix A).

2. Cooperation via Cribbing : The main setting analyzed in this paper is shown in Fig. 4. A single encoder describes a common message to both decoders and a refined message to only Decoder 1. Instead of cooperating via a rate limited link, as in Fig. 3, Decoder 2 “cribs” (in the spirit of Willems and Van der Muelen [1]) a deterministic function of the reconstruction symbols of Decoder 1, non-causally, strictly-causally, or causally. Note a trivial function corresponds to the original successive refinement setting characterized in Equitz and Cover [18]. The goal is to find the optimal encoding and decoding strategy and to characterize the optimal encoding rate region which is defined as the set of achievable rate tuples such that the distortion constraints are satisfied at both the decoders. Cuff and Zhao [17], considered the problem of characterizing the coordination region (non-causal, strictly causal and causal coordination) in our setting of Fig. 4, for a specific function, , such that and for a specific rate tuple , that is Decoder 1 has access to the source sequence while Decoder 2 uses the reconstruction symbols of Decoder 1 (non-causally, strictly-causally or causally) to estimate the source. We use a new source coding scheme which we refer to as Forward Encoding and Block Markov Decoding, and show that it achieves the optimal rate region for strictly causal and causal cribbing. It draws on the achievability ideas (for causal coordination) introduced in Cuff and Zhao [17]. This scheme operates in blocks, where in the current block, the encoder encodes for the source sequence of the future block, (hence the name Forward Encoding) and the decoders rely on the decodability in the previous block to decode in the current block (hence the name Block Markov Decoding). More details about this scheme are deferred to Section III.

The general motivation for our work is an attempt to understand fundamental limits in source coding scenarios involving the availability of side information in the form of a lossily compressed version of the source. This is a departure from the standard and well studied models where side information is merely a correlated “noisy version” of the source, and is challenging because the effective ‘channel’ from source to side information is now induced by a compression scheme. Thus, rather than dictated by nature, the side information is now another degree of freedom in the design. There is no shortage of practical scenarios that motivate our models.

One such scenario may arise in the context of video coding, as considered by Aaron, Varodayan and Girod in ([19]). Consider two consecutive frames in a video file, denoted by Frame 1 and Frame 2, respectively. The video encoder starts by encoding Frame 1, and then it encodes the difference between Frame 1 and Frame 2. Decoder 1 represents decoding of Frame 1, while Decoder 2 uses the knowledge of decoded Frame 1 (via cribbing) to estimate the next frame, Frame 2.

Our problem setting is equally natural for capturing noncooperation as it is for capturing cooperation, by requiring the relevant distortions to be bounded from below rather than above (which, in turn, can be converted to our standard form of an upper bound on the distortion by changing the sign of the distortion criterion). For instance, Decoder 1 can represent an end-user with refined information (common and private rate) about a secret document, the source in our problem, while Decoder 2 has a crude information about the document (via the common rate). Decoder 1 is required to publicly announce an approximate version of the document, but due to privacy issues would like to remain somewhat cryptic about the source (as measured in terms of distortion with respect to the source) while also helping (via conferencing or cribbing) Decoder 2 to better estimate the source. For example, Decoder 1 can represent a Government agency required by law to publicly reveal features of the data, while on the other hand there are agents who make use of this publicly announced information, along with crude information about the source that they too, not only the government, are allowed to access, to decipher or get a good estimate of the classified information (the source).

The contribution of this paper is two-fold. First, we introduce new models of decoder cooperation in source coding problems such as successive refinement, where decoders cooperate via cribbing, and we characterize the fundamental limits on performance for these problems using new classes of schemes for the achievability part. Second, we leverage the insights gained from these problems to introduce and solve a new class of channel coding scenarios that are dual to the source coding ones. Specifically, we consider the MAC with cribbing and a common message, where there are two encoders who want to communicate messages over the MAC, one has access to its own private message, there is a common message between the two encoders, and the encoders cooperate via cribbing (non-causally, strictly causally or causally).

The paper is organized as follows. Section II gives a formal description of the problem and the main results. Section III presents achievability and converses, with non-causal, causal and strictly-causal cribbing. Some special cases of our setting and numerical examples, are studied in Section IV. Channel coding duals are considered in Section V. Finally, the paper is concluded in Section VI.

## Ii Problem Definitions and Main Results

We begin by explaining the notation to be used throughout this paper. Let upper case, lower case, and calligraphic letters denote, respectively, random variables, specific or deterministic values which random variables may assume, and their alphabets. For two jointly distributed random variables, and , let , and respectively denote the marginal of , joint distribution of and conditional distribution of given . is a shorthand for the tuple . We impose the assumption of finiteness of cardinality on all alphabets, unless otherwise indicated.

In this section we formally define the problem considered in this paper (cf. Fig. 4). The source sequence is drawn i.i.d. . Let and denote the reconstruction alphabets, and , for denote single letter distortion measures. Distortion between sequences is defined in the usual way,

 di(xn,^xni)=1nn∑j=1di(xj,^xi,j), for i=1,2. (3)
###### Definition 1.

A () rate-distortion code consists of the following,

1. Encoder, , .

2. Decoder 1, .

3. Decoder 2 (depending on in Fig. 4, the decoder mapping changes as below),

 gnc2,i:{1,...,2nR0}×^Xn1 → ^X2{\ \ \ \ non-causal cribbing, d=n} (4) gsc2,i:{1,...,2nR0}×^Xi−11 → ^X2{\ \ \ \ strictly-causal cribbing, d=i−1}, (5) gc2,i:{1,...,2nR0}×^Xi1 → ^X2{\ \ \ \ causal cribbing, d=i} (6)

.

###### Definition 2.

A rate-distortion tuple is said to be achievable if , and () rate-distortion code such that the expected distortion for decoders are bounded as,

 E[di(Xni,^Xni)]≤Di+ϵ,\ i=1,2. (7)
###### Definition 3.

The rate-distortion region is defined as the closure of the set to all achievable rate-distortion tuples .

Our main results for this setting are presented in the Table I. Note that in all the rate regions in the table, we use the notation for , and we omit the distortion condition , for the sake of brevity. These results will be derived later in Section III. As another contribution, in Section V, we establish duality between the problem of successive refinement with cribbing decoders and communication over multiple access channels with cribbing encoders and a common message. We establish a complete duality between the settings (in a sense that is detailed in Section V) and rate regions of one can be obtained from those of the other by listed transformations.

###### Lemma 1 (Equivalence to Cascade Source Coding with Cribbing Decoders).

The setup in Fig. 4 is equivalent to a cascade source coding setup with cribbing decoders as in Fig. 5 in the following way : fix a distortion pair and let denote the rate region for the problem of successive refinement with cribbing with achievable rate pairs . Let denote the closure of rate pairs, and denote the rate region for the problem of cascade source coding with cribbing (closure of achievable rate pairs ). We then have the equivalence, .

###### Proof.

Proof is similar to the proof of Theorem 3 in Vasudevan, Tian and Diggavi [20]. We state it in Appendix B for quick reference. ∎

We use certain standard techniques such as Typical Average Lemma, Covering Lemma and Packing Lemma which are stated and established in [21]. Herein, we state them for the sake of quick reference. For typical sets we use the definition as in chapter 2 of [21]. Henceforth, we omit the alphabets from the notation of typical set when it is clear from context, e.g. is denoted by .

###### Lemma 2 (Typical Average Lemma, Chapter 2, [21]).

Let . Then for any nonnegative function on ,

 (1−ϵ)E[g(X)]≤1nn∑i=1g(xi)≤(1+ϵ)E[g(X)]. (8)
###### Lemma 3 (Covering Lemma, Chapter 3, [21]).

Let . Let be a pair of arbitrarily distributed random sequences such that as and let , where , be random sequences, conditionally independent of each other and of given , each distributed according to . Then, there exists such that as , if .

###### Lemma 4 (Packing Lemma, Chapter 3, [21]).

Let . Let be a pair of arbitrarily distributed random sequences (not necessarily according to ). Let , where , be random sequences, each distributed according to . Assume that , is pairwise conditionally independent of given , but is arbitrarily dependent on other sequences. Then, there exists such that as , if .

## Iii Successive Refinement with Cribbing Decoders

In this section we analyze the main settings considered in this paper and derive rate regions. In the various subsections to follow we will respectively study the problem of successive refinement with non-causal, strictly causal and causal cribbing. For clarity, in each subsection, we will first study the setting of “perfect” cribbing where and then generalize it to cribbing with any deterministic function .

### Iii-a Non-causal Cribbing

#### Iii-A1 Perfect Cribbing

###### Theorem 1.

The rate region for the setting in Fig. 6 with perfect (non-causal) cribbing is given as the closure of the set of all the rate tuples such that,

 R0+R1 ≥ I(X;^X1,^X2) (9) R0 ≥ {I(X;^X1,^X2)−H(^X1)}+, (10)

for some joint probability distribution such that , for .

###### Proof.

Achievability :

“Double Binning” scheme

Before delving into the details, we first provide a high level understanding of the achievability scheme. Consider the simplified setup where , that is only Decoder 1 has access to the description of the source, and Decoder 2 gets the reconstruction symbols of Decoder 1 (“crib”). The intuition is to reveal a lossy description of source to the Decoder 2 through the “crib”. So we first generate codewords, and index them as bins. In each bin, we generate a superimposed codebook of codewords. Thus total rate of is needed to describe to Decoder 1. Decoder 2 knows via the crib, it then needs to infer the unique bin index which was sent, as then it would infer . The only issue to verify is that the codeword known via cribbing should not lie in two bins. We upper bound the probability of occurrence of such an event by , as there are overall codewords, and the probability that a particular lies in two bins is . This event has a vanishing probability so long as . Thus the achieved rate region is such that the constraint and distortion constraints are satisfied.

The general coding scheme when is depicted in Fig. 7 and has a “doubly-binned” structure. Non-zero helps reduce by providing an extra dimension of binning. We first generate codewords, the indexes of which are the rows (or horizontal bins), and then in each row, we generate codewords. For each row, these codewords are then binned uniformly into vertical bins, which are the columns of our “doubly-binned” structure. Thus each bin is “doubly-indexed” (row and column index) and has a uniform number of codewords (as in Fig. 7). Note that this extra or independent dimension of vertical binning was not there when . Intuition is that column indexing with common rate is independent or orthogonal to the row indexing, and hence it helps to reduce the private rate . The column or vertical bin index is described to both the decoders via common rate and thus reduces to to describe to Decoder 1. Here again, from knowledge of the crib, and the column index, Decoder 2, infers the unique row index, which now will require .

We now describe the achievability in full detail.

• Codebook Generation : Fix the distribution , such that and . Generate codebook consisting of codewords generated i.i.d , . For each , first generate a codebook consisting of codewords generated i.i.d. , then bin them all uniformly in vertical bins , and in each bin index them accordingly with . As outlined earlier, corresponds to the row or horizontal index and corresponds to the column or vertical index in our “doubly-binned” structure, while indexes codewords within a “doubly-indexed” bin. Thus for each row and column index pair, , there are codewords. can therefore be indexed by the triple . The codebooks are revealed to the encoder and both the decoders.

• Encoding : Given source sequence , first the encoder finds from such that . Then the encoder finds pair such that . Thus . Encoder describes column or vertical bin index as to both the decoders, and the tuple to the Decoder 1 as rate . Thus

 R1≥I(X;^X2)+I(X;^X1|^X2)−R0=I(X;^X1,^X2)−R0. (11)
• Decoding : Decoder 1 knows all the indices , and it constructs . Decoder 2 receives from the non-causal cribbing and it also knows the column index through rate . It then checks inside the column or vertical bin of index , to find the unique row or horizontal bin index such that for some . The reconstruction of the Decoder 2 is then .

• Distortion Analysis : Consider the following events :

1.  E1 = No ^Xn2 is jointly typical to a given Xn (12) = {(Xn,^Xn2(mh))∉Tnϵ,∀ mh∈[1:2nI(X;^X2)]}. (13)

The probability of this event vanishes as there are codewords. (cf. Covering Lemma, Lemma 3).

2.  E2 = No ^Xn1 is jointly typical to a typical pair (Xn,^Xn2) = {(Xn,^Xn2(mh))∈Tnϵ} ∩{(Xn,^Xn1(mh,mv,l),^Xn2(mh))∉Tnϵ,∀ mv∈[1:2nR0],

The probability of this event vanishes as corresponding to each there are codewords, (cf. Covering Lemma, Lemma 3). Without loss of generality, now suppose that encoder does the encoding, . Decoder 2 receives via non-causal cribbing. The next two events are with respect to Decoder 2.

3.  E3 = ^Xn1 does not lie in bin indexed by mh=1 and mv=1 (16) = {^Xn1≠^Xn1(1,1,~l),∀ ~l∈[1:2n(I(X;^X1|^X2)−R0)]}. (17)

But the probability of this event goes to zero, because due to our encoding procedure, .

4.  E4 = ^Xn1 lies in a bin with row index, ^mh≠1 and column index mv=1. = {^Xn1=^Xn1(^mh,1,~l), ^mh≠1 for some ~l∈[1:2n(I(X;^X1|^X2)−R0)]}.

Since , this event is equivalent to finding lying in two different rows or horizontal bins, but with the same column or vertical bin index (). The probability of a single codeword occurring repeatedly in two horizontal bins indexed with different row index is , while knowing the column index, , total number of codewords with a particular column index are, , so the probability of event vanishes so long as,

 I(^X1;^X1,^X2)−R0

Thus consider the event, , using the rate constraints from Eq. (11) and Eq. (30), the probability of the event vanishes if,

 R0+R1 ≥ I(X;^X1,^X2) (21) R0 ≥ {I(X;^X1,^X2)−H(^X1)}+. (22)

We will now bound the distortion. Assume without loss of generality that, , for . For both the decoders, (),

 E[d(Xn,^Xni)] = P(E)E[d(Xn,^Xni)|E]+P(Ec)E[d(Xn,^Xni)|Ec] (23) \lx@stackrel(a)≤ P(E)Dmax+(1+ϵ)E[d(X,^Xi)] (24) ≤ P(E)Dmax+Di, (25)

where is via typical average lemma (cf. Typical Average Lemma 2). Proof is completed by letting when .

Converse : Converse for this setting follows by substituting in the converse for the deterministic function cribbing in the next subsection.

###### Note 1 (Joint Typicality Decoding).

Note that here our decoding for Decoder 2 relies on finding a unique bin index in which (obtained via cribbing) lies, and there is an error if two different bins have the same . An alternative based on joint typicality decoding can also be used to achieve the same region as follows : Decoder 2 receives via non-causal cribbing and it also knows the column index through rate . It then finds the unique row or horizontal bin index such that for some . The reconstruction of the Decoder 2 is then . We analyze the following two events, assuming without loss of generality that encoder does the encoding .

•  Ed,1 = Decoder 2 finds no jointly typical ^Xn1 indexed by mh=1 and mv=1 (26) = {(^Xn1,^Xn1(1,1,~l),^Xn2(1))∉Tnϵ,∀ ~l∈[1:2n(I(X;^X1|^X2)−R0)]}. (27)

But the probability of this event goes to zero, because due to our encoding procedure, with high probability, . As this implies,


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$$\require{cancel}$$

# 27.8: Sample lab report (Measuring g using a pendulum)

## Abstract

In this experiment, we measured $$g$$ by measuring the period of a pendulum of a known length. We measured $$g = 7.65\pm 0.378\text{m/s}^{2}$$. This correspond to a relative difference of $$22$$% with the accepted value ($$9.8\text{m/s}^{2}$$), and our result is not consistent with the accepted value.

## Theory

A pendulum exhibits simple harmonic motion (SHM), which allowed us to measure the gravitational constant by measuring the period of the pendulum. The period, $$T$$, of a pendulum of length $$L$$ undergoing simple harmonic motion is given by: \begin{aligned} T=2\pi \sqrt {\frac{L}{g}}\end{aligned}

Thus, by measuring the period of a pendulum as well as its length, we can determine the value of $$g$$: \begin{aligned} g=\frac{4\pi^{2}L}{T^{2}}\end{aligned} We assumed that the frequency and period of the pendulum depend on the length of the pendulum string, rather than the angle from which it was dropped.

## Predictions

We built the pendulum with a length $$L=1.0000\pm 0.0005\text{m}$$ that was measured with a ruler with $$1\text{mm}$$ graduations (thus a negligible uncertainty in $$L$$). We plan to measure the period of one oscillation by measuring the time to it takes the pendulum to go through 20 oscillations and dividing that by 20. The period for one oscillation, based on our value of $$L$$ and the accepted value for $$g$$, is expected to be $$T=2.0\text{s}$$. We expect that we can measure the time for $$20$$ oscillations with an uncertainty of $$0.5\text{s}$$. We thus expect to measure one oscillation with an uncertainty of $$0.025\text{s}$$ (about $$1$$% relative uncertainty on the period). We thus expect that we should be able to measure $$g$$ with a relative uncertainty of the order of $$1$$%

## Procedure

The experiment was conducted in a laboratory indoors.

1. Construction of the pendulum

We constructed the pendulum by attaching a inextensible string to a stand on one end and to a mass on the other end. The mass, string and stand were attached together with knots. We adjusted the knots so that the length of the pendulum was $$1.0000\pm0.0005\text{m}$$. The uncertainty is given by half of the smallest division of the ruler that we used.

2. Measurement of the period

The pendulum was released from $$90$$ and its period was measured by filming the pendulum with a cell-phone camera and using the phone’s built-in time. In order to minimize the uncertainty in the period, we measured the time for the pendulum to make $$20$$ oscillations, and divided that time by $$20$$. We repeated this measurement five times. We transcribed the measurements from the cell-phone into a Jupyter Notebook.

## Data and Analysis

Using a $$100\text{g}$$ mass and $$1.0\text{m}$$ ruler stick, the period of $$20$$ oscillations was measured over $$5$$ trials. The corresponding value of $$g$$ for each of these trials was calculated. The following data for each trial and corresponding value of $$g$$ are shown in the table below.

Trial Angle (Degrees) Measured Period (s) Value of g $$m/s^{2}$$
$$1$$ $$90$$ $$2.24$$ $$7.87$$
$$2$$ $$90$$ $$2.37$$ $$7.03$$
$$3$$ $$90$$ $$2.28$$ $$7.59$$
$$4$$ $$90$$ $$2.26$$ $$7.73$$
$$5$$ $$90$$ $$2.22$$ $$8.01$$

Table A3.8.1

Our final measured value of $$g$$ is $$(7.65\pm 0.378)\text{m/s}^{2}$$. This was calculated using the mean of the values of g from the last column and the corresponding standard deviation. The relative uncertainty on our measured value of $$g$$ is $$4.9$$% and the relative difference with the accepted value of $$9.8\text{m/s}^{2}$$ is $$22$$%, well above our relative uncertainty.

## Discussion and Conclusion

In this experiment, we measured $$g=(7.65\pm 0.378)\text{m/s}^{2}$$. This has a relative difference of $$22$$% with the accepted value and our measured value is not consistent with the accepted value. All of our measured values were systematically lower than expected, as our measured periods were all systematically higher than the §$$2.0\text{s}$$ that we expected from our prediction. We also found that our measurement of $$g$$ had a much larger uncertainty (as determined from the spread in values that we obtained), compared to the $$1$$% relative uncertainty that we predicted.

We suspect that by using $$20$$ oscillations, the pendulum slowed down due to friction, and this resulted in a deviation from simple harmonic motion. This is consistent with the fact that our measured periods are systematically higher. We also worry that we were not able to accurately measure the angle from which the pendulum was released, as we did not use a protractor.

If this experiment could be redone, measuring $$10$$ oscillations of the pendulum, rather than $$20$$ oscillations, could provide a more precise value of $$g$$. Additionally, a protractor could be taped to the top of the pendulum stand, with the ruler taped to the protractor. This way, the pendulum could be dropped from a near-perfect $$90^{\circ}$$ rather than a rough estimate.


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# zbMATH — the first resource for mathematics

On divisibility of the class number of real octic fields of a prime conductor $$p=n^ 4+16$$ by $$p$$. (English) Zbl 0818.11042
Let $$K$$ be a real octic subfield of the $$p$$-th cyclotomic field, where $$p$$ is a prime of the form $$n^ 4+16$$. The author proves that $$p$$ divides the class number of $$K$$ if and only if $$p$$ divides the numerator of at least one of the Bernoulli numbers $$B_{j(p- 1)/8}$$ for $$j=1,3,5,7$$. In a previous work [Abh. Math. Semin. Univ. Hamb. 64, 105- 124 (1994)] he used a similar method to obtain an analogous result for quintic fields.
##### MSC:
 11R29 Class numbers, class groups, discriminants 11R18 Cyclotomic extensions 11B68 Bernoulli and Euler numbers and polynomials 11R20 Other abelian and metabelian extensions
##### Keywords:
cyclotomic field; Bernoulli numbers
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