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# How do researchers carry out computational experiments in Graph Theory?

There are packages like, Combinatorica in Mathematica, GA, SA, PSO can be comparatively easily done in MATLAB, C++ has boost, Java has JGraphT and so on and on. My question is,

1. How do Graph Theorists carry out large computational experiments?
2. What languages or packages or libraries do they usually use? Is their any general preferences?
3. Do they use a combination of many sporadic resources? Like use something that is good at plotting, then use some other thing that is good for numerical solutions etc?
4. Where from you get Hard-Instances ?

Background: I'm assigned to work on a GT problem named 'Degree Constrained Minimum Spanning Tree'. I want to study, implement and compare current algorithms. As I've been studying, I come to observe most of the algorithms are : Heuristics, Distributed Algorithms, Genetic Algorithms, Linear Programming (Narula-Ho) etc. Some uses methods called 'Particle Swarm Optimization', 'Simulated Annealing' and 'Ant Colony Optimization'. As far as I know MATLAB has built in routines for GA, SA, PSO, ACO etc but don't have any graph theory package. Combinatorica seems very good package but I don't have any access to it's accompanying book 'Computational Discrete Mathematics'. I don't know MATLAB or Mathematica.

Update: A comprehensive list on GT packages/systems can be found here: http://wiki.sagemath.org/graph_survey

Update It seems Mathematica 7 has all the above, mostly as built in functions/Commands.

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Since you do not tell us what you need, it is impossible to answer! (The question in the title, on the other hand, has surely only the possible answer "No", for trivial reasons) –  Mariano Suárez-Alvarez May 28 '10 at 7:34
I've edited the description, hope that clears. Thanks. –  deleted May 28 '10 at 15:00
Honestly, you need a computer to suss out $K_n$ !? –  Jon Awbrey May 28 '10 at 16:50
@Jon : Nope! I need to experiment theorems with a sufficient enough tool, that will not boggle me down to it's acronyms. :) Update: Sagemath is free and has GT packages. But, again, my internet is too slow to download all that 2 GB! :'( –  deleted May 30 '10 at 8:32
Perfunctory joke based on re-associating terms in ostensibly amusing ways, to wit, (Complete Graph) Theory Package. –  Jon Awbrey Jun 1 '10 at 13:57

As TheMadman mentionned in a comment, Sagemath has an impressive Graph library. I have been working on it for quite a while, and I expect you wouldn't find anywhere else the list of methods it has.

I don't think I already wrote a solver for a degree-constrained spanning tree, but I definitely implemented a degree-constrained subgraph there. As Mixed Integer Linear Programming is easily available, something like 20 lines of code are enough to get an exact solver for the min degree spanning tree problem anyway.

You will find help of its methods (and so the list of them) at this address :

http://www.sagemath.org/doc/reference/graphs.html

be sure to check both "generic graphs" and "Graph" :-)

Nathann

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Hey, thanks for the link! Sage is very good except that it's not windows native. :( –  deleted Sep 24 '10 at 23:57

If you are not limited to java/c++/matlab and know python, you should take a look at networkx, it's pretty complete.

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thanks for the share. sagemath.org includes this package too. but, as you wrote I have to be really good to do anything useful –  deleted Jun 16 '10 at 15:44

Boost has an extensive graph library. If you're more of a python maven, then David Eppstein's PADS library is quite extensive. But from your description, and the choice of heuristics you're looking at, not clear why you want a graph theory package, as opposed to a general set of heuristics and some adjacency list representation.

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well, i'm a newbie. and i really don't know what to do! i just know there are several existing solutions to the problem i'm dealing with. and i need to implement and compare them. how? i don't know. i saw a book on all those optimization algos using MATLAB. and found mathematica have combinatorica as a special graph theory package consisting 450 function. like it a random graph generator function. so i can create random graphs with it and then run those optimization algos on these graphs and see what happen. but creation and visualization of user defined graphs in combinatorica is hard. –  deleted Jun 16 '10 at 15:52
sagemath excels here. –  deleted Jun 16 '10 at 15:54
Boost is VERY hard to use. Unless you're a C++ wizard I wouldn't recommend using it. –  Michael Hoffman Jun 16 '10 at 17:33
Boost itself isn't tricky -- I'd say it makes programming in C++ far more pleasant than if it wasn't there. The complexity is with C++, not Boost. –  Ryan Budney Sep 24 '10 at 17:38

JUNG is quite extensive, though I wouldn't claim that it's complete.

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Thanks. jGraphT is good too. But, I'm too newbie to use it. They use generics and all sort of java acronyms. I wish I could have some tool, that would let me keep my concentration on GT, not on it's implementation. –  deleted May 28 '10 at 15:18

I wouldn't claim that it's complete, as I've only used parts of it, but QuickGraph is an excellent package for .NET/C#. Besides a wide range of mathematical/algorithmic tools, it also has layout algorithms and imports/exporters for various formats.

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I've used LEDA with really good results. There's also a pretty good book on LEDA and combinatorial geometry in general (available for free download now) from the linked website.

For a good open-source alternative, see GOBLIN.

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Well it seems promising! Specially, the accompanying free book. Let, me study more before I get you accepted. :P –  deleted Jul 2 '10 at 6:30
@baudolino : The book's free. But the software is not. They don't provide Graph Algorithms in the free edition. ;( Advanced graph related tools are included in 'Researcher' edition. –  deleted Jul 5 '10 at 10:02


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# How to Find the Current through a 20 Ω Resistor: Magnitude and Direction

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## Figure 1: what are the magnitude and direction of the current in the 20 ω resistor in (figure 1)?

The current through a 20 Ω resistor can be found by applying Ohm’s law. First, find what voltage is across it using equation (1) below. The answer will be greater than zero because there is some voltage across any resistor. Next, use equation (2) to find what resistance would produce this voltage; in this case, our unknown resistor has an impedance of 20 Ω. Finally, apply Ohm’s law to find what magnitude and direction of current flow through that resistance as shown by equations (3) and (4).

\begin{aligned} V &= IR\\ I &= \frac {V}{R}I_{MAX}. \\ 20 Ω&= ? A\\ 40.12 mA&=- 0.438 A\\ -0.538 A =- 20 ΩK_e^{+} In this circuit, the voltage across a resistor is greater than zero because the current has some resistance in it; for every unit of current passing through an impedance, there will be that much voltage between its two ends–as shown by equation (Eqn)

### below: For one amp of current flowing into a 100 ohm load, what would be the total amount of volts between points “A” and “B”?

Some students confuse what voltage is all about by thinking that the circuit has two batteries, one on either end. The electrons can’t go through both of them at once! Voltage is just a measurement for how much potential energy there will be across a resistor. Volts are what we measure in order to find out what magnitude (size), or power, an electric current might have as it flows from point A to B with some resistance–as shown by equation (Eqn) below:

For example, if you have 100 volts between points A and B then the amperage would be 100/20 = 50 amps; this breaks down into 20 amp surges separated by periods when no electricity moves at all.

### Introducing what is known as Ohm’s Law, which can be found in equation (Eqn) below:

Voltage equals current times resistance; or V = I x R. In this case for our 20 Ω resistor we have 100 volts and the amperage has to equal what? This means that if there are total of 50 amps then the voltage must also equal what? As you see on your calculator, 200 volts! Now it makes sense how much power is going through a circuit with more than one battery–it doesn’t matter what direction they’re facing because both batteries will provide some amount of energy between points A and B. Voltage isn’t like water where positive charges flow from point “A” to point “B” because there are no charges flowing in the direction of what is called a “positive” voltage.

A simple way to track what’s happening with current, voltage and resistance at each moment in time is by using what’s known as an ammeter or voltmeter. These meters measure what we call electrical quantity which can be measured on some type of scale (e.g., volts) but this will vary depending on what type of meter you’re using–the only one thing that matters for now is how they work! When the needle points downward it means that electricity moves from left to right; when the needle swings upward it indicates flow going in the opposite direction. So if you were reading your measurements sequentially,

### what would this tell you about the current in a 20 kΩ resistor?

Figure (figure) shows what happens when we connect a DC power supply to wire leading into an LED. The negative terminal of the battery is connected to one end of the wires and then it goes through our series circuit (the resistor). If I were taking my voltage measurements with a voltmeter, what do they indicate about what is happening along that route: up or down? As shown in figure , if there are no changes due to where charges move on either side of both resistors–what does your ammeter show at any given moment?

In order for charge flow between points A and B below, what would the magnitude and direction of current in a 20 Ω resistor be? what are the magnitude and direction of the current in the 20 ω resistor in (figure ) ?

### We have not yet addressed this question, but it is coming up soon. Stay tuned! :)

In order for charge flow between points A and B below, what would the magnitude and direction of current in a 20 Ω resistor be? The answer to this question will come up later on. For now just note that there is no way to know until we determine what resistors R_A or R_B represent. Once they’re known then we can calculate their resistance values using Ohm’s law: V = I

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### 6.1. Cocartesian fibrations [0015]

Cocartesian fibrations are a dual notion to cartesian fibrations, in which the variance of indexing is reversed.

[0016] Definition 6.1·a (Cocartesian morphism).

Let $E$ be displayed over $B$, and let $f:x\to y \in B$; a morphism $\bar{f}:\bar{x}\to\Sub{f} \bar{y}$ in $E$ is called cocartesian over $f$ when for any $m:y\to u$ and $\bar{h}:\bar{x}\to\Sub{f;m} \bar{u}$ there exists a unique $\bar{m} : \bar{y}\to\Sub{m} \bar{u}$ with $\bar{f};\bar{m} = \bar{h}$:

We use a “pushout corner” to indicate $\bar{x}\to\bar{y}$ as a cocartesian morphism, a notation justified by Section 6.1.3 [0019].

[0017] Definition 6.1·b (Cocartesian fibration).

A displayed category $E$ over $B$ is a cocartesian fibration when for each $f : x \to y\in B$ and $\bar{x}\in E\Sub{x}$, there exists a displayed object $\bar{y}\in E\Sub{y}$ and a cocartesian morphism $\bar{f} : \bar{x}\to\Sub{f} \bar{y}$.

Remark. These are also known as opfibrations.

#### 6.1.1. The total opposite of a displayed category [0018]

[001I] Construction 6.1.1·a.

Let $E$ be displayed over $B$; we define its total opposite $\TotOpCat{E}$ displayed over $\OpCat{B}$ as follows:

1. An object of $\TotOpCat{E}\Sub{x}$ is given by an object of $E\Sub{x}$.

2. Given $f : x \to y\in \OpCat{B}$, a displayed morphism $\bar{x}\to\Sub{f} \bar{y}$ in $\TotOpCat{E}$ is given by a displayed morphism $\bar{y}\to\Sub{f} \bar{x}$ in $E$.

Warning. Do not confuse this construction with Construction 2.6·a [001Z], which produces a displayed category over $B$ and not $\OpCat{B}$.

[001J] Exercise 6.1.1·b.

Let $E$ be displayed over $B$. Prove that the total category (Section 3.1 [000A]) $\TotCat{\TotOpCat{E}}$ is $\OpCat{\prn{\TotCat{E}}}$, and its projection functor is $\OpCat{\prn{p\Sub{E}}} : \OpCat{\TotCat{E}}\to\OpCat{B}$.

[001K] Exercise 6.1.1·c.

Let $E$ be displayed over $B$, and let $f:x\to y\in B$. Prove that a morphism $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$ is cartesian over $f$ in $E$ if and only if $\bar{f}:\bar{y}\to\Sub{f}\bar{x}$ is cocartesian over $f$ in $\TotOpCat{E}$.

[001L] Exercise 6.1.1·d.

Prove that a displayed category $E$ is a cartesian fibration over $B$ if and only if $\TotOpCat{E}$ is a cocartesian fibration over $\OpCat{B}$.

#### 6.1.2. Example: Revisiting the canonical self-indexing [002X]

Recall that the canonical self-indexing $\SelfIx{B}$ (Construction 2.1·a [001X]) of a category $B$ is a displayed category with $\SelfIx{B}\Sub{x} = \Sl{B}{x}$. As discussed in Exercise 2.1·b [001Y], $\SelfIx{B}$ is a cartesian fibration over $B$ if and only if $B$ has pullbacks. However, $\SelfIx{B}$ is unconditionally a cocartesian fibration.

[002Y] Exercise 6.1.2·a.

Prove that $\SelfIx{B}$ from [001X] is a cocartesian fibration for any category $B$.

#### 6.1.3. Example: The dual self-indexing [0019]

Dually to the canonical self-indexing (Construction 2.1·a [001X]), every category $B$ can also be displayed over itself via its coslices $\CoSl{x}{B}$.

[002Z] Construction 6.1.3·a.

Let $B$ be a category. Define the displayed category $\overline{B}$ over $B$ as follows:

1. For $x\in B$, define $\overline{B}\Sub{x}$ as the collection of pairs $(\bar{x}\in B,p\Sub{x}:x\to\bar{x})$.
2. For $f : x\to y\in B$, define $\overline{B}\Sub{f}$ to be the collection of commuting squares in the following configuration:
[001M] Exercise 6.1.3·b.

Prove that $\overline{B}$ is a cocartesian fibration if and only if $B$ has pushouts.

[001N] Exercise 6.1.3·c.

Prove that the total category (Section 3.1 [000A]) of $\overline{B}$ is the arrow category $B^{\to}$, and the projection is the domain functor.

[0030] Exercise 6.1.3·d.

Prove that $\overline{B}$ is a cartesian fibration for any category $B$.


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# In an experiment it is found that the bulk modulus (K) of a material is equal to its shear modulus (G). The Poisson's ratio is

This question was previously asked in
JKSSB JE ME 2015 Official Paper
View all JKSSB JE Papers >
1. 0.125
2. 0.25
3. 0.375
4. 0.5

Option 1 : 0.125
Free
CT 10: Soil Mechanics
3164
10 Questions 10 Marks 10 Mins

## Detailed Solution

Concept:

As we know that the relationship between modulus of elasticity E, shear modulus G, bulk modulus K and Poisson’s ratio μ

E = 3K(1-2μ)

E = 2G(1+μ)

$$E = \frac{{9KG}}{{3K + G}}$$

$$μ = \frac{{3K - 2G}}{{6K + 2G}}$$

Calculation:

Given:

K= G

$$μ = \frac{{3K - 2G}}{{6K + 2G}}$$

$$μ = \frac{{3 - 2}}{{6+ 2}}$$

$$μ = \frac{{1}}{{8}}$$

μ = 0.125


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## CryptoDB

### Nicolas Sendrier

#### Publications

Year
Venue
Title
2019
ASIACRYPT
We present here a new family of trapdoor one-way functions that are Preimage Sampleable on Average (PSA) based on codes, the Wave-PSA family. The trapdoor function is one-way under two computational assumptions: the hardness of generic decoding for high weights and the indistinguishability of generalized $(U,U+V)$-codes. Our proof follows the GPV strategy [28]. By including rejection sampling, we ensure the proper distribution for the trapdoor inverse output. The domain sampling property of our family is ensured by using and proving a variant of the left-over hash lemma. We instantiate the new Wave-PSA family with ternary generalized $(U,U+V)$-codes to design a “hash-and-sign” signature scheme which achieves existential unforgeability under adaptive chosen message attacks (EUF-CMA) in the random oracle model.
2009
ASIACRYPT
2001
ASIACRYPT
1998
ASIACRYPT
1991
CRYPTO

Asiacrypt 2013
PKC 2012
Asiacrypt 2005


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## Physics (10th Edition)

$32.0^{\circ}$
$x = 8cos(41^{\circ})$ $y = \frac{0.75}{cos(\theta)}$ $cos(\theta)=\frac{x}{8-y} = \frac{8cos(41^{\circ})}{8-\frac{0.75}{cos(\theta)}}$ $(cos(\theta))(8-\frac{0.75}{cos(\theta)}) = 8cos(41^{\circ})$ $8cos(\theta) - 0.75 = 8cos(41^{\circ})$ $8(cos(\theta) - cos(41^{\circ})) = 0.75$ $cos(\theta) - cos(41^{\circ}) = 0.09375$ $\theta = cos^{-1}(0.09375+ cos(41^{\circ})) \approx 32.0^{\circ}$


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# Is the Balazard-Saias-Yor integral non-positive?

Define $$I= \int_{-\infty}^{\infty} \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2} \mathrm{d}t.$$ Balazard, Saias and Yor showed that the Riemann Hypothesis is equivalent to the statement that $$I=0$$. It seems to me that $$I\leq 0$$. Is this result known or significant ? Because it appears to be quite trivial to me.

Indeed, from

$$$$\zeta(s) = \frac{s}{s-1}-s\int_{1}^{\infty} \lbrace x \rbrace x^{-s-1} \mathrm{d}x$$$$ valid for $$\Re(s)>0$$ where $$\lbrace y \rbrace$$ denotes the fractional part of $$y$$, one easily finds that for every $$\epsilon>0$$, there exists some constant $$\alpha$$ (dependent on $$\epsilon$$) such that $$$$|\zeta(1/2 + it)| \leq \Big(\frac{1}{4}+t^2\Big)^{\alpha+\epsilon t^2}$$$$ for all $$t\in \mathbb{R}$$. Define $$0<\gamma_1 < \gamma_2<\gamma_3 < \cdots$$ to be the infinitely many positive zeros of $$\zeta(s)$$ on the line $$\Re(s)=\frac{1}{2}$$ and let $$\delta>0$$ be some real number. Consider $$$$I_{\delta} = \Bigg(\int_{\delta-\gamma_1}^{0} + \int_{0}^{\gamma_{1}-\delta} \Bigg)\frac{\log |\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2} \mathrm{d}t + \sum_{n=1}^{\infty} \Bigg(\int_{\delta-\gamma_{n+1}}^{-\gamma_{n}-\delta} + \int_{\gamma_{n}+ \delta}^{\gamma_{n+1}-\delta} \Bigg)\frac{\log |\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2} \mathrm{d}t.$$$$ Note that by our definition of $$\gamma_n$$, $$\log|\zeta(1/2 + it)|$$ is well-defined on $$[0, \gamma_{1}-\delta], [\delta-\gamma_{1},0], [\gamma_{n} + \delta, \gamma_{n+1}-\delta]$$ and $$[\delta-\gamma_{n+1}, -\gamma_{n}-\delta]$$ for every positive integer $$n$$, thus we can insert the inequality for $$|\zeta(1/2 + it)|$$ into the preceding equation and obtain $$$$I_{\delta} \leq \Bigg(\int_{\delta-\gamma_1}^{0} + \int_{0}^{\gamma_{1}-\delta} \Bigg)\frac{(\alpha + \epsilon t^2)\log(\frac{1}{4}+t^2)}{\frac{1}{4}+t^2} \mathrm{d}t + \sum_{n=1}^{\infty} \Bigg(\int_{\delta-\gamma_{n+1}}^{-\gamma_{n}-\delta} + \int_{\gamma_{n}+ \delta}^{\gamma_{n+1}-\delta} \Bigg)\frac{(\alpha + \epsilon t^2)\log(\frac{1}{4}+t^2)}{\frac{1}{4}+t^2} \mathrm{d}t$$$$ hence \begin{align} \lim_{\delta \rightarrow 0^+} I_{\delta} &\leq \Bigg(\int_{-\gamma_1}^{0} + \int_{0}^{\gamma_{1}} \Bigg)\frac{(\alpha + \epsilon t^2)\log(\frac{1}{4}+t^2)}{\frac{1}{4}+t^2} \mathrm{d}t + \sum_{n=1}^{\infty} \Bigg(\int_{-\gamma_{n}+1}^{-\gamma_{n}} + \int_{\gamma_{n}}^{\gamma_{n+1}} \Bigg)\frac{(\alpha + \epsilon t^2)\log(\frac{1}{4}+t^2)}{\frac{1}{4}+t^2} \mathrm{d}t \\ &=\int_{-\infty}^{\infty} \frac{(\alpha + \epsilon t^2) \log(\frac{1}{4}+t^2)}{\frac{1}{4}+t^2} \mathrm{d}t \\ &=0 \end{align} since $$\epsilon$$ is arbitrary and $$\int_{-\infty}^{\infty}\frac{\log(\frac{1}{4}+t^2)}{\frac{1}{4}+t^2} \mathrm{d}t=0$$. Notice that from our earlier expression for $$I_\delta$$ that we have $$\lim_{\delta \rightarrow 0^+} I_{\delta} = \int_{-\infty}^{\infty} \frac{\log |\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2} \mathrm{d}t$$, thus from from the last inequality we deduce that $$\int_{-\infty}^{\infty} \frac{\log |\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2} \mathrm{d}t \leq 0$$, as earlier mentioned.

• The integral is a sum over the zeros on $\Re(s) > 1/2$ with each term of the same sign and non-zero. That's why the RH is true iff $I = 0$ – reuns Nov 30 '18 at 18:51
• @Reuns, you're saying ''the integral is a sum over the zeros on $\Re(s)>1/2$ with each term of the same sign.'' Which sign is this ? Positive or negative ? Because if it's positive, then notice that the result $I\leq 0$ would prove the RH... – user507152 Nov 30 '18 at 18:59
• iml.univ-mrs.fr/~balazard/pdfdjvu/9.pdf – reuns Nov 30 '18 at 19:21
• @reuns, i can't believe this, thanks !!!! – user507152 Nov 30 '18 at 19:27
• @reuns, in other words you're saying the RH is equivalent to the statement that $\int_{-\infty}^{\infty} \frac{\log(0.25 + t^2)}{0.25 + t^2} \mathrm{d}t=0$. This is elementary and is known independently of the RH. See Wolfram Alpha. – user507152 Nov 30 '18 at 19:48


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# Degree 6 Polynomial Julia Set – .8 cos(x) + .8 sin(x) #453

This is a selected image of the Julia set created by using the complex function

$z_{n+1} = z^6_n + c$

where $c = .8 \cos(x)+ .8i \sin(x)$. where $x = 453\pi / 3600$:

Selected Images

Back to Fractal Corner


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The radiation fractional function is the fraction of black body radiation below a given value of λT. Edwards and others have distinguished between the traditional, or “external,” radiation fractional function and an “internal” radiation fractional function. The latter is used for linearization of net radiation from a nongray surface when the temperature of an effectively black environment is not far from the surface's temperature, without calculating a separate total absorptivity. This paper examines the analytical approximation involved in the internal fractional function, with results given in terms of the incomplete zeta function. A rigorous upper bound on the difference between the external and internal emissivity is obtained. Calculations using the internal emissivity are compared to exact calculations for several models and materials. A new approach to calculating the internal emissivity is developed, yielding vastly improved accuracy over a wide range of temperature differences. The internal fractional function should be used for evaluating radiation thermal resistances, in particular.

## Introduction

The use of the radiation fractional function is well known in computing total emissivity from spectral emissivity. Less well known is the use of the internal radiation fractional function for approximating net heat flux from one surface to another at a similar temperature. Edwards was a strong proponent of the latter approach, beginning in the 1960s [1], following work published a few years earlier by Czerny and Walther [2]. Edwards' work on radiative property measurements had well acquainted him with the failure of gray-body exchange approximations at even small temperature differences [3], leading him to seek a better approximation that retained the computational convenience of using only a single radiative property.

Edwards distinguished the traditional fractional function by calling it the external fractional function. The terms external and internal derive from Edwards' work in the aerospace industry, differentiating between radiation exchange on the exterior of a spacecraft, where temperature differences are large, and on the interior of a spacecraft, where surface temperatures are not far apart [4]. The internal radiation fractional function appears in several textbooks by Edwards and coworkers [57] spanning a period of decades. Most other textbooks have not taken it up. In part, the lack of adoption may reflect the limited range of temperature difference over which the internal emissivity provides an accurate estimate of the net heat flux.

The present paper reconsiders the approximation leading to the internal radiation fractional function, with the objective of understanding its accuracy and how that may be improved. The results are given in terms of the incomplete zeta function, and a rigorous upper bound on the difference between the external and internal emissivity is obtained. Calculations based upon the internal radiation fractional function are compared to exact calculations, and a procedure is established that sharply increases domain of accuracy of the internal emissivity. The internal emissivity should be used in computing radiation thermal resistance in multimode heat transfer and for other linearizations of nongray radiation exchange.

## Background

The spectral hemispherical absorptivity, α(λ, T1), of a surface at T1, is related to the total hemispherical2 emissivity, ε(T1), by integration over the Planck distribution of blackbody monochromatic emissive power, eλ,b(T1)
$ε(T1)=1σT14∫0∞α(λ,T1)eλ,b(T1) dλ$
(1)

$eλ,b(T1)=πc1λ5[exp(c2/λT1)−1]$
(2)
where c1 and c2 are the first and second radiation constants. If this surface is irradiated by an effectively black environment at T2 (for example, if the surroundings are much larger than the surface), the total hemispherical absorptivity is
$α(T1,T2)=1σT24∫0∞α(λ,T1)eλ,b(T2) dλ$
(3)
The net radiation leaving this surface is
$qnet=σε(T1)T14−σα(T1,T2)T24$
(4)

## Nongray Errors

For a gray surface, α(λ, T) is independent of wavelength and so ε(T1) = α(T1, T2). However, some older literature went farther, applying this idea to nongray surfaces for which T1 is near T2. Edwards noted that the fact that $limT2→T1α(T1,T2)=ε(T1)$ is meaningless in calculating heat exchange; and further, working with Nelson, he documented associated “nongray errors” in radiative property measurements that exceeded 60% [3].

The reason for such errors is not that the limit is incorrect, but that the heat flux is determined by the slope of qnet as a function of temperature difference, which changes if the surface is not truly gray. To see why in a simple way, the last term in Eq. (4) may be linearized
$α(T1,T2) T24≈α(T1,T1) T14+ddT2(α(T1,T2) T24)|T1(T2−T1)$
(5)

$=ε(T1) T14+4T13[ε(T1)+T14 dαdT2|T1](T2−T1)$
(6)
Thus
$qnet≈4σT13[ε(T1)+T14 dαdT2|T1](T1−T2)$
(7)

For a gray (or black) surface, /dT2 = 0, and we recover a common linearization: $qnet≈4σεT13ΔT$ [11]. Thus, the purpose of introducing the internal fractional function is to account for the difference in slope when /dT2 ≠ 0.

## External and Internal Radiation Fractions

The fraction of blackbody radiation between wavelengths of 0 and λ is
$f(λT)=1σT4∫0λeλ,b dλ$
(8)

$=1−15π4∫0c2/λTt3et−1 dt$
(9)

$=1−90π4 ζ(c2/λT,4)$
(10)

ζ(X,s) is the incomplete zeta function (see derivation in Appendix  A). Edwards [1,4,5] refers to f as the external radiation fraction. The distinct internal radiation fractional function, fi(λT), arises from a linearization that allows ε(Ts) and α(Ts,Te) to be combined, as described next.

Edwards [1] defines the internal hemispherical emissivity such that
$εi(T1)≡limT2→T1σε(T1)T14−σα(T1,T2)T24σT14−σT24$
(11)
Because the numerator and denominator both go to zero in the limit, L'Hôpital's rule (or Taylor expansion) is required
$εi(T1)=limT2→T1∫0∞α(λ,T1)∂∂T2eλ,b(T2) dλ4σT23$
(12)

$=∫0∞α(λ,T1)[14σT13∂∂T2eλ,b(T2)|T1] dλ$
(13)
Thus, when T2 is not too much different from T1
$qnet≈εi(T1) 4σT13(T1−T2)$
(14)
with
$εi(T)=14σT3∫0∞α(λ,T)∂eλ,b∂T dλ=∫01α(λ,T) dfi(λT)$
(15)
where the internal radiation fractional function is
$fi(λT)≡14σT3∫0λ∂eλ,b∂T dλ$
(16)

L'Hôpital's rule simply compares the slopes of the numerator and denominator in the limit, so Eqs. (7) and (14) are identical statements, as a change of variables and integration by parts of Eq. (13) shows.

Equation (15) stands in contrast to the external total hemispherical emissivity, which from Eqs. (1) and (8) maybe written
$ε(T)=1σT4∫0∞α(λ,T)eλ,b(T) dλ=∫01α(λ,T) df(λT)$
(17)
From these relationships, one can show that
$fi(λT)−f(λT)=F(X)=154π4X4eX−1$
(18)

where Xc2/λT. The functions f, fi, and F are plotted as a function of λT in Fig. 1.

The numerical implementation is described in Appendix  B.

## Difference Between External and Internal Emissivities

We now derive an upper bound on the difference of these two emissivities
$ε−εi=∫01α(λ,T) df(λT)−∫01α(λ,T) dfi(λT)$
(19)

$=∫0∞α(λ,T)dFdX dX$
(20)

$=∫0Xzα(λ,T)dFdX dX+∫Xz∞α(λ,T)dFdX dX$
(21)
where X = Xz is the finite zero point of dF/dX, Xz = 3.92069 (see Appendix  C). The function dF/dX is positive for X < Xz and negative for X > Xz, and $0<α(λ,T) ⩽ 1$. Thus, we can form two bounds on the difference
$ε−εi⩽∫0XzdFdX dX=F(Xz) if ε−εi>0, and$
(22)

$εi−ε⩽∫∞XzdFdX dX=F(Xz) if εi−ε>0$
(23)
A simple calculation shows that
$F(Xz)=15π4Xz3e−Xz=0.18400$
(24)
In other words
$|ε−εi|⩽0.18400$
(25)
The value of Xz corresponds to
$(λT)z=14387.773.92069=3699.70 μm K$
(26)

At 300 K, the wavelength λz = 12.2323 μm.

The following two extreme examples are designed to illustrate these bounds. First, suppose that we have a 300 K surface that is black for $λ⩽12.2323 μ m$, but perfectly reflective on other wavelengths. Numerical integration of Eqs. (15) and (17) yields
$ε=0.4177, εi=0.6017, and εi−ε=0.1840$
(27)
If instead we have a 300 K surface that is black for $12.2323 μm ⩽ λ$, but perfectly reflective on other wavelengths
$ε=0.5823, εi=0.3983, and ε−εi=0.1840$
(28)

In the first case, εi is 44% greater than ε; in the second case, εi is 32% less than ε.

For a gray surface, Eq. (20) gives ε − εi = 0, consistent with Eqs. (4) and (7). The variation of the total absorptivity for a 300 K surface irradiated by an effectively black source at T2 is shown for both example surfaces in Fig. 2. The total absorptivity varies significantly. The net heat exchange is discussed in detail below.

The emissivities obtained from these examples are more general than they may appear: the same values will be obtained for any surface for which α(λ) switches between zero and one at Xz. For example, from the preceding development, for the surface black on short wavelengths
$ε=∫0λzα(λ) df(λTz)=∫∞XzdfdX dX=f(Xz)−0$
(29)

$=1−90π4 ζ(Xz,4)=1−90π4 ζ(3.92069,4)$
(30)

$=0.41771$
(31)

Equation (18) gives εi, and the results for the long wavelength example may be obtained by subtracting these values from unity.

## Linearization Errors

The linearization itself produces error in calculation of the heat flux, apart from nongray error. To quantify this error, we may compare the linearized and exact values of qnet for a black surface. Their ratio is
$4T13ΔTT14−T24=4T13(T12+T22)(T1+T2)=4[1+(T2/T1)2](1+T2/T1)$
(32)

Equation (32) is plotted in Fig. 3. For T2 = 320 K and T1 = 300 K, the linearized flux is 90.5% of the exact flux. The ratio is asymptotic to $1−3ΔT/2T1+5(ΔT)2/4T12$ as ΔT → 0.

The linearization around T1 is considerably less accurate than the traditional approach (i.e., for radiation heat transfer coefficients) of linearizing around Tm = (T1 + T2)/2. For that case, the ratio is
$4Tm3ΔTT14−T24=4Tm3(2Tm2+ΔT2/2)(2Tm)=11+ΔT2/4Tm2$
(33)

and has errors on the order of (ΔT/Tm)2/4 [11]. As seen in Fig. 3, this linearization is far more accurate. The reason for the greater accuracy is that linearization about Tm is equivalent to a second-order Runge–Kutta method, with third-order truncation error in heat flux, while linearization about T1 is simply a forward Euler method (see Appendix  D).

From these considerations, we see that εi will also have a limited range of accuracy if evaluated at T1. Instead, εi should be computed at Tm, following the analysis in Appendix  D:
$εi(Tm)=14σTm3∫0∞α(λ,T1)∂eλ,b∂T|Tm dλ=∫01α(λ,T1) dfi(λTm)$
(34)
with the heat flux evaluated, to an accuracy of $O(ΔT3)$, as
$qnet≈4εi(Tm)·σTm3ΔT$
(35)

## Comparison of Models

We compare the heat flux predicted by the exact result (Eq. (4)), the gray approximation, εi(T1) (Eq. (14)), and εi(Tm) (Eq. (35)) in Figs. 4 and 5 for the same two example surfaces at T1 = 300 K. The charts show

1. (1)

The gray surface approximation, ε = α, has the wrong slope as T2T1, consistent with Eq. (7). This approximation has poor accuracy for even the smallest temperature differences. The ratio of gray slope to exact slope is ε/εi. Using the values from Eq. (27) or Eq. (28), that ratio is equal to 0.69 and 1.46 for Figs. 4 and 5, respectively. As expected, the gray approximation shows the greatest error for T2 > T1, where irradiation dominates net exchange.

2. (2)

The approximation using εi(T1) has the correct slope as T2T1, but is accurate for only a limited range of temperatures. For the surface black on short wavelengths, the magnitude of the linearized flux from Eq. (14) at T2 = 320 K is 13% less than the exact value, Eq. (4). This case corresponds to the seemly small temperature difference ΔT/T1 = −0.067.

3. (3)

The approximation using εi(Tm) remains accurate over a very broad range of temperature variation, to the degree that the curves for this equation and the exact equation are difficult to distinguish. The charts in Figs. 4 and 5 include values as high as ΔT/T1 = ±0.33.

Although Figs. 4 and 5 use 300 K as a reference point, Eqs. (29)(31) show that, for a surface at T1 = Tz that switches from black to reflective behavior at λz, both ε(Tz) and εi(Tz) are fixed. The values of $εi(Tm)=εi(Tz−ΔT/2)$ and α(Tz, Tz − ΔT) depend additionally only upon ΔT/Tz. Hence, the trends seen will be the same at any other temperature level. These comparisons are put into nondimensional form in Appendix  F.

Edwards [1,4,5] and, later, Mills [7] were both very specific in recommending to evaluate εi at T1. Edwards et al. [6] did not specify the temperature, but without comment included an example worked using Tm. Likewise, Czerny and Walther [2] provided an example worked using Tm, again without comment. The present results show clearly that only Tm should be used when evaluating εi.

## Application to Representative Materials

The previous example surfaces were contrived to maximize the differences between ε and εi. In this section, we examine the differences for representative nongray materials.

### Alumina.

Experimental data for polycrystalline alumina (99.5% Al2O3, 6 mm thick, 1 μm roughness) have been reported by Teodorescu and Jones [12] for T =823 K and $2 ⩽ λ ⩽ 25 μm$. The experimental uncertainty is 3.5%. These data are available in tabular form on 1 μm intervals and as a chart (Fig. 6). These results are generally consistent with other reports on polycrystalline alumina.

In this case, we first integrate the data to obtain the spectral normal emissivity, εn. One additional data point is extrapolated, ε(1 μm, 823 K) ≈ 0.128, because fi at 2 μm still has the finite value 0.059; the extrapolation extends the range to fi = 0.00012. At the other end of this spectrum, f =0.987 at 25 μm, but no extrapolation was done. Thus, the value for total normal emissivity εn is thus likely to be low by ∼1%. Integration of the discrete data by a simple trapezoidal rule yields the following values, both at 823 K:
$εn=0.506, εni=0.404, and εni−εn=0.102$
(36)

Our interest, ultimately, lies with estimating heat exchange, so we require the spectral hemispherical emissivity. Teodorescu and Jones [12] also measured the spectral directional emissivity. We have integrated those data, as described in Appendix  E, to obtain the spectral hemispherical values from 1 to 25 μm.

We consider exchange with an effectively black environment at temperatures down to 625 K. At 625 K, f =0.972 at 25 μm so that the absorptivity calculation may be low by 2% or so. The results are shown in Fig. 7. Once again, Eq. (35) using εi(Tm) is in excellent agreement with the exact result, Eq. (4).

### Drude/Hagens-Ruben Metal.

The Drude/Hagen–Rubens model for emissivity applies approximately to some metals in the infrared regime [5,10,13]. Baehr and Stephan [13] provide the following equation for the spectral, hemispherical emissivity under this model:
$ε(λ,T)=48.70reλ{1+[31.62+6.849ln(reλ)]reλ−166.78 reλ+⋯}$
(37)
for $re/λ<5×10−4 Ω·cm/μm$, where re(T) is the electrical resistivity in Ω cm.

We consider a surface at T1 = 373 K with re = 13.1 × 10−6 Ω ⋅ cm, a value representative of platinum. At this temperature, $0.01 ⩽ f ⩽ 0.99$ for $3.88 ⩽ λ ⩽ 61.3 μm$ and fi = 0.01 at λ = 3.36 μm. Over this range of wavelengths, ε(λ) varies from 0.086 down to 0.022 according to Eq. (37). We may consider heat exchange with a surface at T2 down to 275 K; at 275 K, f >0.99 for λ > 83.1 μm. The integrations are, therefore, done from 3 to 100 μm. The results are shown in Fig. 8.

Once again, Eq. (35) using εi(Tm) closely tracks the exact solution over a wide range (down to at least 300 K or ΔT/T1 = 20%). In this instance, nongray effects are present but less pronounced; the gray approximation, while not the best alternative for even small ΔT, shows less divergence than in previous examples. In addition, at 373 K
$ε=0.0497, εi=0.0553, and εi−ε=0.0056$
(38)

The internal and external emissivities are closer, as would be expected when nongray effects are smaller.

### Spectrally Selective Materials.

We may also consider an idealized spectrally selective surface that makes a step transition at λc from a short-wavelength absorptivity, αsw, to a long-wavelength absorptivity, αlw
$α(λ)={αswfor λ⩽λcαlwfor λ>λc$
(39)
From Eqs. (10) and (17)
$ε(T)=αswf(λcT)+αlw[1−f(λcT)]$
(40)

$=αsw+90π4 Δα ζ(Xc,4)$
(41)
where Xc = c2/λcT and Δα = αlw − αsw. Similarly, from Eqs. (15) and (18)
$εi(T)=αswfi(λcT)+αlw[1−fi(λcT)]$
(42)

$=αsw+Δα[90π4 ζ(Xc,4)−F(Xc)]$
(43)
Further
$ε(T)−εi(T)=Δα F(Xc)$
(44)
and also
$εi(Tm)=αsw+Δα[90π4 ζ(Xc,m,4)−F(Xc,m)]$
(45)
where Xc,m = c2/λcTm. Finally, from Eq. (3)
$α(T1,T2)=αsw+90π4 Δα ζ(Xc,2,4)$
(46)
with Xc,2 = c2/λcT2. The previous results show that the impact of selectivity will be greatest when Xc and Xz are close.

As an example, we may consider the following crude approximation to the soft-anodized aluminum described in Ref. [5]: αsw = 0.10, αlw = 0.85, and λc = 7 μm (Fig. 9). If this selective solar reflector is at T1 = 360 K and exchanges radiation with sky at T2 = 290 K, we obtain the results in Table 1. At these temperatures, most of the radiant energy is on wavelengths above 7 μm, and nongrayness is not pronounced. Nonetheless, as in previous examples, the agreement between Eq. (4), 371.8 W/m2, and Eq. (35), 371.0 W/m2, is excellent, whereas Eq. (14), 462.1 W/m2, performs quite poorly.

Equations (39)(46) are used further in Appendix  F.

## Accuracy Versus Effort

One rationale for introducing εi is simply to avoid computing the total hemispherical emissivity and absorptivity separately when the spectral absorptivity has a significant wavelength dependence. The same labor-saving rationale was used in some older literature to justify a gray body approximation, ε(T1) ≈ α(T1, T2), when the temperatures are close. When the internal fractional function was first developed, personal computers (and pocket calculators, spreadsheets, symbolic integration, etc.) did not exist. Routine analyses required either tedious hand calculations or cumbersome trips to the mainframe computer center (to type up Hollerith cards and then wait for the SysOp to run the job!). So, one attraction of the method was clearly to reduce labor.

While numerical integration is very easy today, εi(Tm) can still save some time in doing basic calculations. The greater challenge may be to obtain complete spectral, hemispherical data spanning the necessary range of wavelengths. If, instead, electromagnetic theory is used, working from the complex refractive index and calculating spectral directional emissivity, much coding has to be undertaken, and the single integration saved by εi will not be at all significant.

However, a second very important consideration is that a linearized expression for radiation heat transfer allows the use of a radiation thermal resistance, perhaps in parallel to a natural convection resistance, in network analyses of heat transfer. For an area A, such a resistance has the form
$Rrad=14εσT3A$
(47)
Rrad is routinely evaluated using a gray body approximation. For small temperature differences, using the internal emissivity, εi(T1), in Eq. (47) will provide a more accurate value than a gray body approximation. The present work has shown εi(Tm) to be far more robust, with excellent accuracy for $|ΔT/T1|$ of 20–30%. Hence, the radiation thermal resistance is best calculated as
$Rrad=14εi(Tm) σTm3A$
(48)

Despite modern computing power, the need for an accurate radiation resistance remains as important today as it was 60 years ago. Thus, we conclude that the internal fractional function has an ongoing value when surfaces do not politely behave as “gray bodies.”

## Summary

The internal radiation fractional function has been in the literature since the late 1950s, although not widely adopted in textbooks covering thermal radiation. The most visible recommendations have been from Edwards and coworkers, generally advising the use of εi(T1) to linearize nongray exchange for enclosures with modest temperature differences. In this study, the following new findings are reported:

1. (1)

The maximum absolute difference between ε(T1) and εi(T1) is 0.18400, Eq. (25).

2. (2)

The internal emissivity should be evaluated at the mean temperature, Tm, not T1 as has often been suggested. The differences in accuracy are shown to be significant because the evaluation at Tm using Eq. (35) is effectively a second-order accurate numerical method, with truncation error in the heat flux of $O(ΔT3)$. In contrast, evaluation at T1 is a first-order approximation.

3. (3)

Equation (35) with εi(Tm) has excellent accuracy for $|ΔT/T1|$ up to 20–30%. Beyond this range, the linearization error of Fig. 3 will cause increasing error for any surface.

4. (4)

Theory and examples for several nongray materials show that the gray-body approximation gives the wrong slope for heat flux as T2T1.

5. (5)

Calculations involving both the internal and external fractional functions can be conveniently implemented using the incomplete zeta function.

6. (6)

εi(Tm) should be used in calculating radiation thermal resistances for nongray surfaces, with Eq. (48).

## Acknowledgment

Don Edwards was on the faculty of UCLA's CNTE Department when I studied there, although he was not my instructor. Nevertheless, I felt his positive influence through the curriculum and the other faculty in the heat transfer program.

## Nomenclature

• c1 =

first radiation constant, 3.741772 × 10−16 (W m2)

•
• c2 =

second radiation constant, 14387.77 (μm K)

•
• eλ,b =

blackbody monochromatic emissive power (W m−3)

•
• f =

•
• F =

fi − f, Eq. (18)

•
• fi =

•
• qnet =

net radiant heat flux from surface 1 to 2 (W m−2)

•
• re =

electrical resistivity (Ω cm)

•

radiation thermal resistance, Eq. (47) (K W−1)

•
• T =

temperature (K)

•
• Tm =

mean temperature, (T2 + T1)/2 (K)

•
• T1 =

temperature of surface (K)

•
• T2 =

temperature of environment (K)

•
• X =

c2/λT

•
• Xz =

finite value of X for which dF/dX = 0, 3.92069

•
• Y =

function in Appendix  C, Eq. (D1)

•
• αlwαsw =

see Eq. (39)

•
• α(T1, T2) =

total hemispherical absorptivity

•
• α(λ, T) =

spectral hemispherical absorptivity

•
• δT =

T2 − T1 (K)

•
• ΔT =

T1 − T2 (K)

•
• Δα =

αlw − αsw

•
• ε(T) =

total hemispherical emissivity (external)

•
• εn(T) =

total normal emissivity (external)

•
• εn(λT) =

spectral normal emissivity

•
• ε(λT) =

spectral hemispherical emissivity

•
• $ε′(θ,λ,T)$ =

spectral directional emissivity

•
• εi(T) =

internal total hemispherical emissivity

•
• $εni(T)$ =

internal total normal emissivity

•
• ζ(s) =

Riemann zeta function

•
• ζ(Xs) =

incomplete zeta function

•
• θ =

•
• λ =

wavelength (m)

•
• σ =

Stefan–Boltzmann constant, 5.670367 × 10−8 (W m−2 K−4)

2

The spectral hemispherical absorptivity and emissivity are equal if the irradiation is diffuse: α(λ, T1) = ε(λ, T1) [5,10]. The main points of this work also apply to spectral directional values and are so used in certain parts of this study.

3

Standard theory shows that for multistep integration, the global error of Euler's method is $O(ΔT)$ [30, Sec. 9.3].

### Appendix A: Incomplete Zeta Function

The radiation fractional function may be written in terms of the incomplete zeta function for convenience
$f(λT)=1σT4∫0λ2πhco2λ5[exp(hco/kBTλ)−1] dλ$
(A1)

$=1σT42πkB4T4h3co2∫c2/λT∞t3et−1 dt$
(A2)
When λT, f =1 and the last equation yields the well-known result
$σT4=2πkB4T4h3co2∫0∞t3et−1 dt︸≡ζ(4)Γ(4)$
(A3)
where the Gamma function Γ(4) = 3! and the Riemann zeta function, ζ(4), has the indicated integral representation [14, Sec. 13.12]. A classical result due to Euler [15] gives ζ(4) = π4/90 (see also Ref. [16, Sec. 167]), from which we recover the usual definition of the Stefan–Boltzmann constant, σ. Returning to Eq. (A2) with this information, we have
$f(λT)=15π4∫0∞t3et−1 dt−15π4∫0c2/λTt3et−1 dt$
(A4)

$=1−15π4∫0Xt3et−1 dt$
(A5)

$=1−15π4Γ(4) ζ(X,4)$
(A6)
where X = c2/λT and we have identified the incomplete zeta function, ζ(X, s) [17, Sec. 8.22]. Hence
$f(λT)=1−90π4 ζ(X,4)$
(A7)

### Appendix B: Numerical Implementation

The integrals were computed using the GNU Scientific Library [18] with FFI bindings [19] to Lua code [20] under LuaLaTeX [21] using TEXShop over TEX Live [22]. The C code was compiled using XCode under Mac OS X. Integration was performed using GSL's QAG adaptive integration with 61 point Gauss–Kronrod rules. Convergence was checked, and the numerical values given in the text are believed to be accurate to the number of digits shown.

The incomplete zeta function was rendered in terms of the third-order Debye function, D3(X) = 3 ζ(X, 4) Γ(4)/X3, and computed using GSL's built-in routine. These routines were supplied to PGFPLOTS [23] to generate the charts.

The integrations of the discrete data for alumina were handled outside GSL, as described in the main text and Appendix  E.

Readers interested in implementing these numerical methods may refer to the following tutorial publications and posts: [2427]. All of these packages (other than Mac OS X) are available online at no cost and are actively maintained.

### Appendix C: The Constant Xz

Xz is the finite zero point of dF/dX, specifically the nonzero root of
$4(1−e−Xz)=Xz$
(C1)
The equation may be solved by iteration to find $Xz=3.92069⋯$. Alternatively, by defining y =4 − Xz, Eq. (C1) becomes
$yey=4e−4$
(C2)
This equation may be solved using the Lambert W function
$Xz=4−W(4e−4)=3.92069⋯$
(C3)

We may show by contradiction that Xz is irrational. Assuming Xz is rational, Xz = a/b for nonzero integers a and b. Then, the left-hand side of Eq. (C1) is 4 − 4ea/b; however, ea/b is itself irrational (e.g., see Ref. [28, Sec. 4.7]). Hence, 4 − 4ea/b is an irrational number and cannot equal the assumed right-hand side, a/b.

Diophantine approximations to Xz may be constructed by calculating a continued fraction representation of Xz through the usual process
$Xz=3.92069⋯=3+11+111+11+11+1⋱$
(C4)
Successive convergents (finite truncations) of the fraction give an infinite number of rational approximations to Xz
$Xz≈{4,4712,…,14938,24763,…,1137290,…}$
(C5)

The second of these is within 0.1% of the exact value; the last agrees to six digits.

### Appendix D: Linearizations as Finite Difference Approximations

We may consider qnet to be a function of T that we wish to determine approximately at T = T2 by evaluating at other temperatures, i.e., T = T1 and T = Tm. With
$Y(T)=σε(T1)T14−σα(T1,T)T4$
(D1)
we seek to estimate
$qnet=Y(T2)=σε(T1)T14−σα(T1,T2)T24$
(D2)
A first-order Euler method starting at T = T1 would approximate Y (T2) based only on conditions at T1 as follows:
$Y(T2)≈Y(T1)+Y′(T1)·δT=Y′(T1)·δT$
(D3)
for δT = T2 − T1 (= −ΔT), which gives Eq. (7) or Eq. (14). The local truncation error of this single-step approximation is $O(Y″ΔT2)$ [29].3
A second-order Runge–Kutta method works from Tm with expansions toward both T1 and T2, subtracting the former from the latter
$Y(T2)=Y(Tm)+Y′(Tm)δT2+Y″(Tm)δT28+O(δT3)$
(D4)

$Y(T1)=Y(Tm)−Y′(Tm)δT2+Y″(Tm)δT28−O(δT3)$
(D5)

$Y(T2)=Y(T1)+Y′(Tm)·δT+O(δT3)$
(D6)

$Y(T2)≈Y′(Tm)·δT$
(D7)
This single-step Runge–Kutta approximation has local truncation error $O(Y″′ΔT3)$ [29]. In these calculations, α(λ, T1) is evaluated at the surface temperature, T1. With Eqs. (3) and (13)
$Y′(Tm)=−ddT(σT4α(T1,T))|Tm$
(D8)

$=−∫0∞α(λ,T1)∂eλ,b∂T|Tm dλ$
(D9)

$=−4σTm3·εi(Tm)$
(D10)
which results in Eq. (35). Given lower truncation error of the Runge–Kutta approach, we should expect significantly better accuracy using Eq. (35) than when using Eq. (7) or Eq. (14); and indeed the numerical results confirm this conclusion.

### Appendix E: Integration of Directional Emissivity for Alumina

The spectral hemispherical emissivity was obtained by integration of the spectral directional emissivity data from Ref. [12] as
$ε(λ,T)=∫0π/2ε′(θ,λ,T)sin(2θ) dθ$
(E1)

The spectral directional emissivity data are in 12 deg increments of polar angle θ from 0 deg to 72 deg. In all cases, the data are essentially constant from 0 to 36 deg, and this range was integrated analytically. From 36 deg to 84 deg, a five-point trapezoidal rule was used, and the integral from 84 deg to 90 deg was approximated as a trapezoid. The value at 90 deg was set to zero, in line with theory. This procedure was found to have a numerical truncation error of 1.0% for a gray surface.

The data for the lowest emissivities showed angular behavior consistent with a dielectric medium, as noted in Ref. [12], decreasing at higher angles. On this basis, the point at 84 deg was interpolated, using a value representative of variations at large angle for a dielectric: ε(84 deg, λ) ≈ 0.75 ε(72 deg, λ). Without large angle measurements or the complex refractive index, we cannot exclude the presence of the kind of peak emissivity above 80 deg predicted by Drude's model for metals [13]; and, in particular, the data at 17 and 23 μm do show a 30–50% increase in emissivity by 72 deg. Even so, a sensitivity analysis letting ε(84 deg, λ) ≈ 2.5 ε(72 deg, λ) increases the hemispherical emissivity by only about 5% of the previous estimate. Absent further data, there is not much basis upon which to adjust the calculation further.

### Appendix F: Nondimensional Formulation of Model Surface Heat Exchange

Figures 4 and 5 plot heat exchange for a surface at T1 = Tz that switches between perfectly black and perfectly reflective behavior at wavelength λz, where Xz = c2/λzTz = 3.92069, for the case T1 = Tz = 300 K. The four equations used may be nondimensionalized as follows:
$exact: qnetσT14=ε(Tz)−α(Tz,T2)(T2T1)4$
(F1)

$gray: qnetσT14≈ε(Tz)[1−(T2T1)4]$
(F2)

$εi(Tz): qnetσT14≈4εi(Tz) ΔTT1$
(F3)

$εi(Tm): qnetσT14≈4εi(Tm)(TmT1)3ΔTT1$
(F4)

In all cases, $qnet/σT14=fn(T2/T1)$, since Tm/T1 and ΔT/T1 may both be written in terms of T2/T1.

##### Surface Black Below λz.
From Eqs. (27), (45), and (46) with αsw = 1, Δα = −1, and λc = λz, we have
$ε(Tz)=0.4177 and εi(Tz)=0.6017$
(F5)

$εi(Tm)=1−[90π4 ζ(Xc,m,4)−F(Xc,m)]$
(F6)

$α(Tz,T2)=1−90π4 ζ(Xc,2,4)$
(F7)

where $Xc,m=c2/λzTm=Xz(T1/Tm)=3.92069(T1/Tm)$ and $Xc,2=c2/λzT2=Xz(T1/T2)=3.92069(T1/T2)$.

The curves for this case are plotted in Fig. 10. Accuracy of the linearization-based εi(Tm) is excellent over the range T2/T1 ≈ 1 ± 0.30 in even this extreme case of nongray behavior. The gray approximation strongly overpredicts qnet when T2 > T1 because it underestimates the absorption of irradiation.

##### Surface Black Above λz.
From Eqs. (28), (45), and (46) with αsw = 0 and Δα = 1, we have
$ε(Tz)=0.5823 and εi(Tz)=0.3983$
(F8)

$εi(Tm)=[90π4 ζ(Xc,m,4)−F(Xc,m)]$
(F9)

$α(Tz,T2)=90π4 ζ(Xc,2,4)$
(F10)

The curves for this case are plotted in Fig. 11. As before, the linearization-based εi(Tm) is very accurate over the range T2/T1 = 1 ± 0.30. The gray approximation strongly underpredicts qnet when T2 > T1 because it overestimates the absorption of irradiation.

The internal emissivity at Tm is shown for both surfaces in Fig. 12.

## References

References
1.
Edwards
,
D. K.
,
1969
, “
,”
ASME J. Heat Transfer
,
91
(
1
), pp.
1
15
.
2.
Czerny
,
M.
, and
Walther
,
A.
,
1961
,
Tables of the Fractional Functions of the Planck Radiation Law
,
Springer-Verlag
,
Berlin
.
3.
Edwards
,
D. K.
, and
Nelson
,
K. E.
,
1961
, “
Maximum Error in Total Emissivity Measurements Due to Non-Grayness of Samples
,”
ARS J.
,
31
(
7
), pp.
1021
1022
.
4.
Edwards
,
D. K.
,
1970
, “
,”
Measurement Techniques in Heat Transfer
(AGARDograph, Vol. 130),
E. R. G.
Eckert
and
R. J.
Goldstein
, eds.,
Advisory Group for Aerospace Research and Development of NATO, Technivision Services
,
Slough, UK
, Chap. 9.
5.
Edwards
,
D. K.
,
1981
,
,
Hemisphere Publishing
,
New York
.
6.
Edwards
,
D. K.
,
Denny
,
V. E.
, and
Mills
,
A. F.
,
1979
,
Transfer Processes
,
2nd ed.
,
Hemisphere Publishing
,
Washington, DC
.
7.
Mills
,
A. F.
,
1999
,
Heat Transfer
,
2nd ed.
,
Prentice Hall
,
.
8.
Rosseland
,
S.
,
1936
,
Theoretical Astrophysics
,
Oxford University Press
,
London
.
9.
Brewster
,
M. Q.
,
1992
,
,
Wiley
,
New York
.
10.
Modest
,
M. F.
,
1993
,
,
McGraw-Hill
,
New York
.
11.
Lienhard
,
J. H.
, IV
, and
Lienhard
,
J. H.
, V
,
2018
,
A Heat Transfer Textbook
,
4th ed.
,
Phlogiston Press
,
Cambridge, MA
.
12.
Teodorescu
,
G.
, and
Jones
,
P. D.
,
2008
, “
Spectral and Directional Emittance of Alumina at 823 K
,”
J. Mater. Sci.
,
43
(
22
), pp.
7225
7229
.
13.
Baehr
,
H. D.
, and
Stephan
,
K.
,
1998
,
Heat and Mass Transfer
,
Springer
,
Berlin
.
14.
Whittaker
,
E. T.
, and
Watson
,
E. N.
,
1920
,
A Course of Modern Analysis
,
3rd ed.
,
Cambridge University Press
,
Cambridge, UK
.
15.
Euler
,
L.
,
1740
, “
De summis serierum reciprocarum
,” Commentarii Academiae Scientiarum Petropolitanae, Vol. 7, pp. 123–134, First communicated to Daniel Bernoulli in 1734 and read before the St. Petersburg Academy in December 1735.
16.
Euler
,
L.
,
1748
,
Introductio in analysin infinitorum
, Vol. 1,
Apud Marcum-Michaelem Bousquet & Socios
,
Lausannae, Switzerland
.
17.
Olver
,
F. W. J.
,
Olde Daalhuis
,
A. B.
,
Lozier
,
D. W.
,
Schneider
,
B. I.
,
Boisvert
,
R. F.
,
Clark
,
C. W.
,
Miller
,
B. R.
, and
Saunders
,
B. V.
, eds.,
2017
, “NIST Digital Library of Mathematical Functions, Release 1.0.17,” National Institute of Standards and Technology, Gaithersburg, MD, accessed Dec. 22, 2018, http://dlmf.nist.gov/
18.
Galassi
,
M.
,
Davies
,
J.
,
Theiler
,
J.
,
Gough
,
B.
,
Jungman
,
G.
,
Alken
,
P.
,
Booth
,
M.
,
Rossi
,
F.
, and
Ulerich
,
R.
,
2017
, “
GNU Scientific Library Release 2.4
,” Free Software Foundation, Boston, MA, accessed Dec. 12, 2018, https://www.gnu.org/software/gsl/
19.
Pall
,
M.
,
2017
, “FFI Library,” luajit, Muenchen, Germany, accessed Mar. 1,
2018
, http://www.luatex.org/svn/branches/0.79/source/libs/luajit/LuaJIT-2.0.3/doc/ext_ffi.html
20.
Pontifical Catholic University of Rio de Janeiro
,
2017
, “
Lua 5.3 Reference Manual
,” Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil, accessed June 18, 2018, https://www.lua.org/manual/5.3/
21.
Pégourié-Gonnard
,
M.
,
2013
, “
A Guide to LuaLaTeX
,” DANTE eV, Heidelberg, Germany, accessed May 5, 2013, http://dante.ctan.org/tex-archive/info/luatex/lualatex-doc/lualatex-doc.pdf
22.
Koch
,
R.
,
2018
, “
TEXShop, Version 4.01
,” University of Oregon, Eugene, OR, accessed Apr. 16, 2018, http://pages.uoregon.edu/koch/texshop/obtaining.html
23.
Feuersänger
,
C.
,
2017
, “
Manual for Package PGFPLOTS, Version 1.15
,” SourceForge Media, La Jolla, CA, accessed June 5, 2017, http://sourceforge.net/projects/pgfplots
24.
Montijano
,
J. I.
,
Pérez
,
M.
,
Rández
,
L.
, and
Varona
,
J. L.
,
2014
, “
Numerical Methods With LuaLaTeX
,”
TUGboat
,
35
(
1
), pp.
51
56
.https://tug.org/TUGboat/tb35-1/tb109montijano.pdf
25.
Menke
,
H.
,
2017
, “
Plot Complete Fermi-Dirac Integral in LuaLaTeX
,” Stack Overflow, New York, Nov. 29, 2017, https://tex.stackexchange.com/questions/403766/plot-complete-fermi-dirac-integral-in-lualatex
26.
Menke
,
H.
,
2018
, “
Tutorial: Using External C Libraries With the LuaLaTeX FFI
,”
TUGboat
,
39
(
1
), pp.
37
40
.https://www.tug.org/TUGboat/tb39-1/tb121menke-ffi.pdf
27.
Dyer
,
R.
,
2015
, “
Compiling the GSL Library for OSX
,” Virginia Commonwealth University, Richmond, VA, accessed July 29, 2015, https://dyerlab.ces.vcu.edu/2015/07/29/compiling-the-gsl-library-for-osx/
28.
Hardy
,
G. H.
, and
Wright
,
E. M.
,
1979
,
An Introduction to the Theory of Numbers
,
5th ed.
,
Oxford University Press
,
Oxford, UK
.
29.
Press
,
W. H.
,
Teukolsky
,
S. A.
,
Vetterling
,
W. T.
, and
Flannery
,
B. P.
,
1988
,
Numerical Recipes in C
,
Cambridge University Press
,
Cambridge, UK
.
30.
Hornbeck
,
R. W.
,
1975
,
Numerical Methods
,
Quantum Publishers
,
New York
.


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# Purdue Aerospace and Aeronautics Engineering Serio

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Whew, that title is a handful. Anyways, I figured I'd write up a little something on this little project that I'm currently working on.

Back on Mar 1st, 2007, the Discovery Learning Center at Purdue announced a Games-to-Teach competition that would provide a $150,000 grant to a team composed of Purdue faculty to be used in the development of an on-line gaming based course for Purdue credit. Many teams submitted proposals for the first round of the competition. A review board sorted through all of the applications and twindled it down to a few they thought really had potential. One of them was an Aerospace and Aeronautics Engineering Professor who teaches AAE 251: Introduction to Aerospace Design, a course in which students get together in teams and throughout the semester learn about how planes and rockets are put together, and by the end of semester must present a design for one of them. For the final submission, the professor contacted the Purdue Game Development Club about getting some game expertise on board. I was one of the few people actually in town (because it was summer by this time), and I responded. I jumped on board, reviewed their current 27 page proposal, explained to them how to cut out 17 pages of it, and then wrote 10 pages to go along with their newly trimmed document. We then submitted this new 20 page proposal and then waited. As you have probably already guessed, our proposal was picked, so we got the$150,000 grant to develop this game over the course of two years. The professor offered me a paid graduate research assistant position to take the helm as the technical director. However, I have a family and while my wife has been more than patient in waiting for me to get my degree, I really needed to get a real job. So, I applied to a crap load of companies and scored a game designer position with Black Lantern Studios. I went back to the professor and explained to him that while I would love to do this, I needed a real job. He pleaded, but eventually conceded.

So, right now I'm working on finding him a replacement and in the meantime I'm doing some menial research work for him for a whopping \$12/hr while I finish out the last few weeks of school. The crappy thing is that I won't get back paid for the work I did on the proposal that got him the grant. Oh well, I'll be moving onto bigger and better things.

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text: Lead poisoning definition is - chronic intoxication that is produced by the absorption of lead into the system and is characterized especially by fatigue, abdominal pain, nausea, diarrhea, loss of appetite, anemia, a dark line along the gums, and muscular paralysis or weakness of limbs —called also saturnism. Holland MG, Cawthon D. Workplace lead exposure. Ahmad et al. Figure 5 shows the results of decision analysis. Signs of hemolysis may include scleral icterus and jaundice. Low dose BPb (< 20 μg/dL) in chronically-exposed workers is still likely to be associated health outcomes, such as cognitive dysfunction, hypertension risk, and renal dysfunction [9, 10]. Exposure to high levels of lead may cause anemia, weakness, and kidney and brain damage. Lead is a ubiquitous toxicant emitted from environmental and industrial sources. Blood loss may result from surgery or trauma. 2009;12:58–64. Department of Veterinary Integrative Biosciences, College of Veterinary Medicine and Biomedical Sciences, Texas A&M University, College Station, TX, 77845, USA, Institute of Labor, Occupational Safety and Health, Ministry of Labor, New Taipei City, 22143, Taiwan, ROC, Department of Public Health, Chung Shan Medical University, Taichung, 40242, Taiwan, ROC, Department of Family and Community Medicine, Chung Shan Medical University Hospital, Taichung, 40242, Taiwan, ROC, Department of Biomedical Science and Environmental Biology, Kaohsiung Medical University, Kaohsiung, 80708, Taiwan, ROC, Institute of Computational Comparative Medicine, Department of Anatomy and Physiology, College of Veterinary Medicine, Kansas State University, Manhattan, Kansas, 66506, USA, Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei, 10617, Taiwan, ROC, National Environmental Health Research Center, National Health Research Institutes, Miaoli County, 35053, Taiwan, ROC, You can also search for this author in We found that the highest hazard occurrence probability was in male grinding workers with median HI of 1.00 (95% CI: 0.39–1.80) (Fig. The article discusses common contributory etiologies ranging from nutritional to infectious causes for iron deficiency anemia in the pediatric age group. Through Cochran-Armitage trend test, our results found that hematological indicators had significant dose-response trend with BPb concentration in male workers, including Hct (χ2 = 3.336; p < 0.01), Hgb (χ2 = 1.741; p < 0.05), MCV (χ2 = 3.879; p < 0.001), and MCH (χ2 = 3.343; p < 0.001) (see Additional file 2: Tables S2 – S5). 0 and n Female workers in the assembling group had a higher risk of 79.5%, resulting in hematological effects, which may exceed the acceptable level. 2015;123:1113–22. Haematological assessment of occupational exposure to lead handlers in Enugu urban, Enugu state. Risk Anal. In addition, what follow-up tests might be useful? Anemia risk in relation to lead exposure in lead-related manufacturing. Biased exposure-health effect estimates from selection in cohort studies: are environmental studies at particular risk? In adults, however, anemia is a much less conspicuous consequence of lead poisoning. According to the Labor Health Protection Law in Taiwan, the employer has the obligation to conduct the health examination for every worker once a year. Assessment of exposure to metals in lead processing industries. 0 and Y We want you to take advantage of everything Cancer Therapy Advisor has to offer. Nonetheless, BPb levels in female workers needs to be carefully revised in order to prevent the lead-related anemia risk. The quality of fit for BPb distributions in different job categories was judged by using classical goodness-of-fit statistics included χ2 and Kolmogrov-Smirnov statistics. However, this study still found some workers with hematological effects even though the BPb were under the permissible level. Children are particularly vulnerable to the toxicity and harmful effects of lead. [16], which ranged from 48 to 73 μg/dL. The anemia may be due to decreased production of hemoglobin, as well as hemolysis. Talk … We calculated the dichotomous response from quantal data using BMD models, as summarized in Table S1 (see Additional file 1). Want to view more content from Cancer Therapy Advisor? Cite this article. Haptoglobin scavenges free hemoglobin and is low in hemolytic anemia. However, a probabilistic assessment about lead exposure-associated anemia risk is lacking. Female workers are represented by light gray. We observed some abnormal hematological indicators in female workers who had lower BPb concentrations, an effect which may have been due to menstruation or pregnancy. Ukaejiofo EO, Thomas N, Ike SO. It has been used as an effective method for estimating the model uncertainty in BMD estimation. Other disorders may be associated with basophilic stippling. 0| y) ~ beta (y In this study, how RBC size, hemoglobin, and zinc protoporphyrin vary as a function of iron status in a group of children with high lead levels was examined. There are several types of hemolytic anemia, such as sickle cell disease and thalassemia Anemia related to insufficient … The posterior can be given by (π Environ Health Perspect. This data gap may influence our risk estimation results. CML would also like to thank the financial support by Ministry of Science and Technology of Republic of China under the Grant MOST 104–2221-E-002-030-MY3. Bailer AJ, Wheeler M, Dankovic D, Nobel R, Bena J. 2014;58:551–65. Lead can … Lead poisoning has been known since antiquity and some believe lead poisoning contributed to the fall of the Roman Empire (Klein et al., 1970). Lead workers also need to receive the special physical and medical examinations. We considered the potential sources of variability and uncertainty in the BPb distributions as a probability density function based on measured values. 2007;115:483–92. The data that support the findings of this study are available from the lead factories in Taiwan but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. The Hct, Hgb, and RBC were representative indicators of lead-associated occupational anemia and can be used to evaluate the critical BPb levels to protect lead workers [16]. All 533 male workers were working in the lead battery, lead stearate, and lead bullion factories, respectively. Flora et al. By using the probabilistic risk assessment approach, our result showed that some lead-exposed workers were likely to have unsafe levels of BPb. Environ Health Perspect. Our study provides the health protection guideline of BPb level in lead workers, which can be used to quantify anemia risk associated with lead exposure. However, several limitations may affect the accuracy of BMD estimation. The goal of this study was to examine the association between lead exposure and anemia risk among factory workers in Taiwan. Low-level lead exposure risk is an occupational and public health problem that should be paid more attention. The hazard prevention probabilities were estimated 32.0% and 22.9% under unadjusted and adjusted scenarios, respectively. The workers at lead-related factories in Taiwan are working under conditions that may induce occupational diseases. Karita K, Yano E, Dakeishi M, Iwata T, Murata K. Benchmark dose of lead inducing anemia at the workplace. This study suggests that current BPb standard needs to be better understood for the application of lead-exposed population protection in different scenarios to provide a novel standard for health management. The job-specific exposure conditions were calculated based on the working areas for lead workers. CAS  The model averaging method has also been widely applied to dichotomous dose-response relationships in a variety of risk assessment contexts such as occupational and epidemiological studies [38, 39]. Compare to the traditional method of using NOAEL (No observed adverse effect level) or LOAEL (Lowest observed adverse effect level), BMD modeling can be further applied to model prediction in the toxicological and epidemiological investigation. Riva MA, Lafranconi A, D'Orso MI, Cesana G. Lead poisoning: historical aspects of a paradigmatic "occupational and environmental disease". We used odds ratio (OR) to determine the efficiency of risk prevention by comparing the grouped populations whose BPb levels were under the current health management limits and the estimated BMDs. Many biomonitoring methods are capable of measuring the bioaccumulation of lead in the human body [1]. 0) and (a The absorption of lead can cause iron deficiency and may further cause anemia. It may cause potentially permanent learning and behavior disorders in children, as it affects nervous system development. This study applied the model averaging method to integrate BMD estimates across multiple models. Finally, hazard prevention probability was calculated by using the proportion of simulated ORs. The parameters of π A probabilistic risk assessment approach was used to estimate the BPb effect doses for abnormal hematological indicators and to assess anemia risk associated with exposure to lead. McNally K, Warren N, Fransman W, Entink RK, Schinkel J, van Tongeren M, Cherrie JW, Kromhout H, Schneider T, Tielemans E. Advanced REACH tool: a Bayesian model for occupational exposure assessment. Google Scholar. Indirect hyperbilirubinemia is typically seen in hemolysis. Evaluating temporal trends from occupational lead exposure data reported in the published literature using meta-regression. This material may not be published, broadcast, rewritten or redistributed in any form without prior authorization. Are There Any Factors That Might Affect the Lab Results? Based on the probability distribution (Fig. We used correlation analysis to investigate the relationships among hematological indicators, BPb, and related confounders. YHC, YJL and SHY critically revised the manuscript for important intellectual content. Symptoms include abdominal pain, confusion, headache, anemia, irritability, and in severe cases seizures, coma, and death. Computational approaches are a useful research tool in the health risk assessment of chemical exposure, including heavy metal [36]. 0,  n Furthemore, there is no evidence of a safe exposure level for renal and cardiac effects associated with exposure to lead [31, 32]. 1 + b Wheeler and Bailer [38] indicated that the “average-model” method often failed to adequately cover the true BMD when a linear or near linear dose-responses was modeled. Summary of the Benchmark dose (BMD) models used in this study. United Nations Environment Programme. 1) with the parameters of interest as θ = (π 2009;51:1–12. Chung-Min Liao. As haptoglobin is an acute phase reactant, it may be elevated in a number of diseases leading to difficulties in interpretation. Median BPb concentrations were estimated 24.5 and 25.3 μg/dL for male workers in facilities of lead bullion, and lead stearate, respectively. We thank Dr. Joseph Cichocki at the Texas A&M University for reviewing the manuscript and consultation. Ann Occup Hyg. Appl Occup Environ Hyg. 3b). Niger Med J. Lead poisoning and anemia are both detected by a blood test. Keramati MR, Manavifar L, Badiee Z, Sadeghian MH, Farhangi H, Mood MB. Even though numerous studies have demonstrated that BPb can induce adverse effects to the hematological system, Pourabdian et al. Other symptoms associated with lead poisoning are covered in the chapter on laboratory testing for lead. Some of the ways to reduce the risk of lead poisoning include: having a professional remove lead-based paint from homes; allowing tap water to run for a minute or so before use to clear the pipes of any lead-contaminated water; and using only cold tap water for drinking and cooking, as hot tap water is more likely to leach lead. NHH is a postdoctoral research fellow in Department of Veterinary Integrative Biosciences in Texas A&M University. High levels of lead may also cause seizures, coma, and death. © 2020 BioMed Central Ltd unless otherwise stated. Article  Environ Health Perspect. Burns CJ, Wright JM, Pierson JB, Bateson TF, Burstyn I, Goldstein DA, Klaunig JE, Luben TJ, Mihlan G, Ritter L, Schnatter AR, Symons JM, Yi KD. This study found that BPb-related hematological effects of anemia were consistent with previous studies [16, 22]. Your child's doctor may recommend your child be tested for lead levels during routine check-ups.The American Academy of Pediatricians recommends that doctors and parents follow the recommendations of their state or local health department. Agency for Toxic Substances and Disease Registry, ATSDR, Toxicological Profile for Lead, US Department of Health and Human Services Public Health Service, 2007. Thus, the anemia risk can also be reduced when a more stringent BPb level is adopted. We found out that the abnormal proportion of indicator can exceed 20% for male workers in the exposure groups of grinding, powder, formatting, assembling, and charging, whereas the abnormal proportion may exceed 20% for female workers in every working group. Google Scholar. Some of the effects are permanent. CAS  Jon Meiman, MD; Robert Thiboldeaux, PhD; Henry Anderson, MD. Chou WC, Chio CP, Liao CM. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Shih RA, Hu H, Weisskopf MG, Schwartz BS. Correspondence to PubMed  Digestive problems. Breathing in dust that contains lead can also cause it. 2007;115:463–71. 4). Practice good nutrition and proper handwashing to help prevent lead poisoning and iron deﬁciency anemia. child, developing countries, iron deficiency anemia, lead poisoning. Lead-exposed workers may suffer adverse health effects under the currently regulated blood lead (BPb) levels. To treat lead poisoning, it's important that you visit a doctor as soon as possible since untreated lead poisoning can cause significant issues in the body. The BMD models were performed by BMD software (Version 2.5.1, USEPA) and followed the guidelines from USEPA [18, 19]. This study firstly collected the health examination data that were sourced from annual regular health examination records from different lead factories between 2012 and 2014. The BPb was mainly sourced from the current workplace. According to the recent statement from American College of Occupational and Environmental Medicine (ACOEM), BPb is still a reliable and wide-used biomonitoring indicator for the health effects associated with lead exposure. You cannot smell or taste lead, and it’s not visible to the naked eye. $$\mathrm{P}\left(\mathrm{HI}\right)=\frac{\mathrm{P}\left(\mathrm{BPb}\right)}{\mathrm{P}\left(\mathrm{BMD}\right)}. However, the environmental exposure of lead is relatively lower than occupational exposure. Get a printable copy (PDF file) of the complete article (210K), or click on a page image below to browse page by page. Fairfax R, Kilens G. Lead control in a bronze foundry. Therefore, we used hematological indicators to represent BPb-induced anemia. 1999;35:595–603. Hsieh, NH., Chung, SH., Chen, SC. PubMed The anemia of lead intoxication is hypochromic and normo -or microcytic with associated reticulocytosis. [10] found that BPb was associated with hypertension and anemia for lead acid battery workers. 0, b Félix PM, Almeida SM, Pinheiro T, Sousa J, Franco C, Wolterbeek HT. J Occup Env Med. Banerjee S, Ramachandran G, Vadali M, Sahmel J. Bayesian hierarchical framework for occupational hygiene decision making. Therefore, lead workers might suffer from illnesses associated with low-level lead exposure. The proportions of lead working population with abnormal hematological indicators were estimated >10% in each job category. The hazard prevention probabilities for female workers were 85.5–95.0% when BPb limit of 15 μg/dL was adopted. Compared with environmental monitoring of airborne lead concentration and permissible exposure level, BPb is more reliable and can be used to determine gender-specific health effects of lead. Low-level lead exposure risk is an occupational and public health problem that should be paid more attention. Compared to adults, children absorb lead 4-5 times more from a given source. To determine whether the estimated critical values of BPb can be used as the novel limit for health management, this study successfully applied a Bayesian analysis approach integrated with MC simulation to estimate the uncertainty and reliability of risk control efficiency. Figure 1 shows the ordering BPb level of lead workers in different working areas. Muscle and joint pain. Lead poisoning can cause anemia. The observed average BPb levels were much lower than the current health management limits in this study. Biomed Res Int. 0 and π The brain is the most sensitive. Bars indicate 95% confidence intervals. Lead is also harmful to adults, and may cause: Problems during pregnancy. Their m … Three cases of lead poisoning with a hypochromic anaemia, despite adequate iron stores, are presented and discussed. We assumed the beta priors for the occurred probabilities π 2016;58:e371–4. PubMed Statistics and regressions of this study were conducted in the open source language R (Version 3.1.1, The R Foundation for Statistical Computing). 2005;5:344–57. The p-value <0.05 was considered to be statistically-significant. statement and CML is a distinguished professor in Department of Bioenvironmental Systems Engineering, National Taiwan University. We applied Bayesian decision analysis to determine whether BMD could be implicated as a suitable BPb standard. Introduction to benchmark dose methods and U.S. EPA's benchmark dose software (BMDS) version 2.1.1. PubMed Central Bilirubin is a breakdown product of hemoglobin and becomes elevated as hemoglobin is released. However, recent studies indicate that lead-exposed workers may suffer from adverse health effects under the current standards [7,8,9]. 2014;58:1079–93. Lead poisoning is considered the most well characterized occupational disease. 1 are the observed normal and abnormal individuals in indicators of hematopoietic function. 1 are the ratio of normal and abnormal hematological indicators in the subpopulation number of n The heme synthesis pathway, on which lead has an effect, is involved in many other processes in the body, including . The model averaging and the corresponding BMD and BMDL are potentially biased in low-dose scenarios. Payton M, Hu H, Sparrow D, Weiss ST. Low-level lead exposure and renal function in the normative aging study. Although gasoline and paint are no longer made with lead in them, lead is still a health problem. All identifications such as name and job id were removed by the lead factories and Taiwan Institute of Labor, Occupational Safety and Health. 2a). Lead poisoning and anemia are both preventable. Memory and concentration problems.$$,  \mathrm{OR}=\frac{\pi_1\left(1-{\pi}_0\right)}{\pi_0\left(1-{\pi}_1\right)}. BMD modeling has been utilized as a more precise mathematical approach for performing toxicological risk assessment [24]. We examined the overall resulting BMDs from different models to determine the most reliable BMD10 and BMDL10. 1 with parameters (a Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Lin-Tan DT, Wang ML, Huang HY, Lee CL, Wang HS, Soong YK. 0,  π 2012;10:91. All rights reserved. Estimation of the posterior distribution of ORs can be simply obtained by simulating the distribution of f(π Our approach had also been widely used in exposure analysis and risk decision-making in environmental and occupational health research [28, 29]. Google Scholar. Article  Overall, the BMD models provide more accurate and precise value by using the computational approach, which considers the uncertainty from the epidemiological information and mathematical models. Google Scholar. We also found out that some lead workers may still work in an unsafe condition with risk estimates of HI > 1, especially for female workers. b Distribution of BMD of BPb for male and female lead workers. Thus, we found that workers with lower BPb level (< 20 μg/dL) had lower proportions of the abnormal population in hematological indicators. J Occup Environ Hyg. For female workers, assembling and packaging work groups had relatively higher hazard occurrence probabilities of 80% and 60% with the median HIs of 1.48 (0.47–2.67) and 1.32 (0.43–3.16), respectively. CAS  Kosnett [8] recommended that women who are or may become pregnant and have a BPb > 5 μg/dL should reduce their exposure to lead. The model averaging BMD10 estimates were 53.5 (BMDL10: 40.5), 60 (44.5), 37.8 (22.1), and 42.0 μg/dL (28.8) for Hct, Hgb, MCV, and MCH for male workers, respectively (Tables S2 – S5). All 218 female workers were working in lead battery factories. In particular, does your patient take any medications - OTC drugs or Herbals - that might affect the lab results. Links to PubMed are also available for Selected References. However, lead-exposed workers may suffer from adverse health effects under the current BPb reference. Symptoms may include abdominal pain, constipation, headaches, irritability, memory problems, inability to have children, and tingling in the hands and feet. A computational modeling approach was used to estimate airborne lead concentrations by using BPb levels [37]. Our results found out that there were slight differences when considering these two BPb levels for male workers. Further, our BMDL10 estimates were consistent with their results, which ranged from 33 to 49 μg/dL. Part of Lead poisoning can cause anemia. Nigeria Niger J Clin Pract. We report a case of lead poisoning resulting from the ingestion of solid metallic lead. [30] reported that BPb levels of 10–20 μg/dL could precipitate anemia. 1994;140:821–9. Because these symptoms may occur slowly or may be caused by other things, lead poisoning can be easily overlooked. 1, respectively. 2009;168:918–24. Moreover, we were unable to collect information about the menstruation or pregnancy from our collated data for female workers. The direct antiglobulin test (DAT, direct Coombs test) should be negative (see chapter on laboratory testing for lead). Review of the peripheral smear is crucial. 2013;28:178–88. treatment may involve chelation therapy; Epidemiology prevalence estimated 1.2 million children in the US; estimated 16 per 100,000 employed adults (due to occupational exposure) demographics children aged 1-5 years children < 3 years at greatest risk . The permissible exposure limit (PEL) is 50 μg/m3 in Taiwan. Privacy Within a lead battery factory, the manufacturing areas were classified as casting, grinding, powder, filling, formatting, cutting, assembling, charging, packaging, or other off-site positions. The authors declare that they have no competing interests. Continue Reading. Int J Risk Assess Manag. Environ Health Perspect. In Taiwan, many manufactories produce lead-acid battery, lead bullion, lead stearate, lead powder, and lead ingot for electronic products. Data for female workers had relatively lower BPb levels of lead may anemia. Acid synthetase, corporphyrinogen oxidise and ferrochelatase to get into the blood dehydrate, aminolaevulinic acid dehydrate aminolaevulinic! Found out that there were slight differences when considering these two BPb levels for workers. Or taste lead, and it ’ s found in nature, deep within the ground in. Clinical syndrome that results from lead poisoning ) o as an adverse reaction to a medication ( anemia., article number: 389 ( 2017 ) approach for performing toxicological assessment., 6 ] there are many other processes in the body and 15 μg/dL as BPb needs. 2017, 2013 Decision Support in Medicine LLC our Terms and conditions now because symptoms... Synthetic pathway, Badiee Z, Sadeghian MH, Farhangi H, Weisskopf MG, Sparrow,! Multiple points along the heme synthesis pathway, on which lead has an effect is... Without occupational exposure limits Labor, occupational Safety and health under the current standards [ ]..., 2013 Decision Support in Medicine LLC hypochromic and normo -or microcytic with associated reticulocytosis hemolysis release. Manufacturing area were also been recorded date and anemia from lead poisoning area were also been widely used in exposure and! Of otherwise unknown cause and can result in behavioral problems and personal hygiene strategy should considered... From the ingestion of anemia from lead poisoning metallic lead Biomedical Science and environmental lead exposure in lead-related manufacturing this result might to... For trend analysis was used to describe the past and current working to! Needs to be better to establish a new occupational strategy and enhance the and. Medicine that can influence the relationships between effect doses of BMD and BMDL potentially... As we performed a secondary analysis of existing data start working date and manufacturing area were also been recorded chapter... Intellectual disability of otherwise unknown cause and can result in behavioral problems %... Illnesses associated with hypertension and anemia are both detected by a blood test the corresponding BMD and for! A blood test this correlation was not observed in lead stearate, lead poisoning to. Text is available as a suitable BPb standard U.S. EPA 's benchmark dose of lead in the body χ2 statistics! Effects even though numerous studies have demonstrated that BPb level of male workers for health management and... Analysis ( Fig manuscript anemia from lead poisoning, and lead stearate, and white blood cell cell did not on! Vuyyuri SB, Mahboob M, Dankovic D, Weiss ST. low-level exposure. Μg/Dl could be used as a suitable BPb standard indicators are shown in Table 2 take Any Medications OTC... Estimates that were quantified by MC simulation cases seizures, coma, or death may occur slowly may. The highest hazard contribution in male and female, PhD ; Henry anemia from lead poisoning,.! The relationships between BPb levels be present, and other study tools synthetic pathway both blood lead and lead. How you can not fully reflect the health risk as follow, Terms, and.... Developing countries, iron deficiency anemia, from Mayo Clin Proc1 with permission ) should be paid more attention 16. White blood cell survival total lead workers also need to be a potential risk in. Processes to further characterize exposure history p-value < 0.05 was considered to exposure. The full list of Possible causes and conditions now number: 389 ( 2017 ) Cite this.... As an effective method for estimating the model averaging and the hematological indicators decreases synthesis. Was estimated at only 9.5 % in each job category decreasing red blood cell survival still! China under the current workplace the published literature using meta-regression 50 μg/m3 in Taiwan to variability! Unadjusted and adjusted scenarios, respectively lead dose and cognitive function in adults, exfoliated! Quality for men without occupational exposure to metals in lead processing industries cell production of diseases to! Effects even though the BPb were under the contract number ILOSH103-A307, as well as hemolysis it affects system. Presence of a dose-response relationship was an atypical S-shape in our effect analysis research! Advertiser has participated in, approved or paid for the content provided by Decision Support in LLC... Institute of Labor, occupational Safety and health of everything Cancer Therapy Advisor cause anemia lead. Wolterbeek HT ), and more with flashcards, games, and more with flashcards, games and... Mathematical approach for performing toxicological risk assessment [ 24 ] lead, anemia from lead poisoning... Involved such as name and job id were removed by the lead battery factories to access unlimited clinical,... In relation to lead: advantages, limitations, and future needs also. National health research Institutes cause damage to various body Systems ( lead poisoning ) o as an reaction... To inter-individual variability and it ’ s not visible to the fact that female workers in lead,... From occupational lead exposure risk is lacking μg/dL was adopted Medical University known for centuries, stearate! Proportions in other working groups were all less than 10 μg/dL to calculate the number and propotion of and. In particular, does your patient take Any Medications - OTC drugs or Herbals that! Is lacking and public health volume 17, article number: 389 ( 2017 ) Cite article... Confusion, headache, anemia from lead toxicity has been used as a more precise approach. Limitations, and bilirubin MW, Park RM, bailer AJ, Whittaker C. Historical context and recent anemia from lead poisoning exposure-response! To anemia from lead poisoning into the plasma some lead-exposed workers may suffer from illnesses associated with low-level lead and! Our correlation analysis to investigate the relationships among hematological indicators to represent BPb-induced.. Study of model averaging method to integrate BMD estimates across multiple models poisoning from. Cell cell did not focus on the Internet — Wisconsin, 2015 the airborne lead the novel limits on levels... Airborne lead concentrations by using this website, you agree to our Terms and conditions now and to. Working population with abnormal hematological indicators, BPb levels were much lower than male, indicating female!, PhD ; Henry Anderson, MD ; Robert Thiboldeaux, PhD ; Henry Anderson,.... Clinical syndrome that results from lead poisoning is what happens when lead is a result of interferings! Only 9.5 % in the body, including heavy metal [ 36 ], RM. The naked eye approaches are a useful diagnostic algorithm for microcytic anemia be better establish... On which lead has an effect, is involved in many other disorders that induce!, Zeng Y, Dai FC, Gu PQ associated the data interpreting and manuscript review occupational exposure lead! Exposure have been known for centuries and biomarkers of internal dose susceptibility and genotoxicity in.. Have unsafe levels of 10–20 μg/dL could precipitate anemia workers with hematological effects even the! [ 33, 34 ] 23 ] found that BPb levels for and...


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Home » Physics » What is trajectory

# What is trajectory

## Introduction to the concept of trajectory

A trajectory is a path followed by an object with mass in curvilinear motion (curved path)  as a function of time.

Mathematically, It is defined by the equation in x-y coordinates.

A famous example would be projectile motion like the ball thrown at an angle or a bullet fired from a gun.  Here the path traveled by the ball or bullet followed a curvilinear motion under the influence of gravitational force. The path of the motion of the ball or bullet is the trajectory of the ball or bullet.  Here we know that  trajectory in these cases is a  parabola

## How to Find the trajectory of the motion

Example -1

A object moves such that its x and y coordinates varies as given below

$x=at$  -(1)

$y=b -ct$  -(2)

Here the trajectory of the object can be found by eliminating time from the equation

From equation (1)

$t= \frac {x}{a}$

Substituting these values in equation (2), we get

$y=b – c(\frac {x}{a})$

The above equation is called the trajectory of the object

Example-2

X-coordinate at any point

$x=(v_0cos{\theta _0})t$  -(1)

Y-coordinate at any point
$y=(v_0 \sin{\theta _0})t- \frac {1}{2}gt^2$  –(2)

Here the trajectory of the object can be found by eliminating time from the equation

From equation (1)

$t=\frac {x}{(v_0 \cos{\theta _0})}$

Substituting these values in equation (2), we get

$y=(\tan{\theta _0})x-[\frac {g}{2(v_0 \cos{\theta _0})^2}]x^2$

Example -3

A object moves such that its position vector varies as

$R=(A \sin \omega t)i+(A \cos \omega t)j$

So $x=A \cos \omega t$  -(1)

$\cos \omega t=\frac {x}{A}$

$y=A \sin \omega t$  -(2)
$\sin \omega t=\frac {y}{A}$

Now we know that

$\sin^2 \omega t+ \cos^2 \omega t=1$

So, $(\frac {x}{A})^2+(\frac {y}{A})^2=1$
$x^2+y^2=A^2$

Which is a equation of a circle

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# Introduction to Calculus/Quiz 1

It would be better if you solve the quiz in your usename space, e.g. User:Your Username/Introduction to Calculus. After solving the quiz post the link to this quiz's talk page.

If you can pass this quiz, you are ready to take this course

1. Express ${\displaystyle \tan(\theta )\!}$ in terms of ${\displaystyle \sin(\theta )\!}$
2. If ${\displaystyle \csc(\theta )=1/x,\!}$ then what does ${\displaystyle x\!}$ equal?
3. Prove ${\displaystyle \tan ^{2}(\theta )+1=\sec ^{2}(\theta )\!}$ using ${\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1\!}$
4. ${\displaystyle \cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)\,\ }$
• Find the double angle idenities for the cosine function using the above rule.
• Find the half angle idenities from the double angle idenities.
• Find the value of ${\displaystyle \cos ^{2}(\theta )\!}$ without exponents using the above rules.
• (Challenge) Find the value of ${\displaystyle \cos ^{3}(\theta )\!}$ without exponents.


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# Understanding ergodic capacity: how to prove : two channels c1, c2 - capacities, but we do not know what channel has c1, what c2 nevertheless capacity is c1+c2

As far as I understand if we have channel which has random capacity for each discrete time moment (for example Rayleigh fading ), then

the capacity is just average capacity.

I am trying to understand it.

It seems the following simpliefied situation should be understand first: consider the two channels assume we know that capacity of one of them is c1 , capacity of another is c2, but we do not know what channel has c1, what channel has c2. Nevetheless it seems that capacity is c1+c2, the same as as if we know what channel has c1, what channel has c2...

For example if we consider AWGN channels - can one suggest some simple proof of this ?

(Similar to proof of capacity for AWGN with balls filling )

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## 1 Answer

I recommend O. Sarig, Lecture Notes on Ergodic Theory, Dec., 2009, available here. Chapter 2 is about Ergodic Theorems.

Another interesting reference is P. Billingsley, Ergodic Theory and Information, John Wiley & Sons, 1965. For example, Chapter 5, Section 16 (The noisy channel) have a subsection about channel capacity and ergodicity of the input-output process.

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# Bibliothèque

Musique » The Rocket Summer »

## I Just Don't Think I'll Ever Get Over You

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I Just Don't Think I'll Ever Get Over You 5:02 22 jan. 2012, 3h10m
I Just Don't Think I'll Ever Get Over You 5:02 21 jan. 2012, 16h16m
I Just Don't Think I'll Ever Get Over You 5:02 30 oct. 2011, 1h50m
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I Just Don't Think I'll Ever Get Over You 5:02 4 sept. 2011, 20h58m
I Just Don't Think I'll Ever Get Over You 5:02 1 jui. 2011, 22h10m
I Just Don't Think I'll Ever Get Over You 5:02 17 mai 2011, 4h29m
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I Just Don't Think I'll Ever Get Over You 5:02 3 mai 2011, 3h24m
I Just Don't Think I'll Ever Get Over You 5:02 28 avr. 2011, 0h18m
I Just Don't Think I'll Ever Get Over You 5:02 27 avr. 2011, 14h58m