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# If 45 people built of a pyramid

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If 45 people built of a pyramid [#permalink]  16 Sep 2018, 07:44
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Question Stats:

80% (02:16) correct 20% (03:28) wrong based on 5 sessions
If 45 people built$$\frac{1}{2}$$ of a pyramid in 288 days, how many days did it take 65 people to build the next $$\frac{1}{6}$$ the pyramid, rounded to the nearest integer, assuming each person works at the same constant rate?

[Reveal] Spoiler: OA
66
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Re: If 45 people built of a pyramid [#permalink]  16 Sep 2018, 10:55
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KUDOS
AchyuthReddy wrote:
If 45 people built$$\frac{1}{2}$$ of a pyramid in 288 days, how many days did it take 65 people to build the next $$\frac{1}{6}$$ the pyramid, rounded to the nearest integer, assuming each person works at the same constant rate?

The official answer is 66

With solution as follows:

To solve a Rates & Work problem with multiple workers, modify the standard formula Work = Rate × Time.
Solve for an individual worker’s rate, plugging in the fact that 45 people built ½ of a pyramid in 288 days:
1/2 pyramid = (R)(45)(288 days) R = 1/25,920 pyramid per day
That is, each person can build of a pyramid in 1 day.
Now put this rate into the work equation, together with the remaining details in the problem:
1/6 = (1/25,920)(65)(T) T ≈ 66.46 days
Rounded to the nearest integer, the answer is 66 days

The problems states next $$\frac{1}{6}$$ but solution considered normal$$\frac{1}{6}$$ why?

Can I plz request you to READ THE RULES OF POSTING
https://greprepclub.com/forum/rules-for ... -1083.html

As you have already 31 post under your belt plz read it as it will help to the forum. Moreover plz put your reasoning in separate reply
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Re: If 45 people built of a pyramid [#permalink]  16 Sep 2018, 11:10
2
KUDOS
AchyuthReddy wrote:
If 45 people built$$\frac{1}{2}$$ of a pyramid in 288 days, how many days did it take 65 people to build the next $$\frac{1}{6}$$ the pyramid, rounded to the nearest integer, assuming each person works at the same constant rate?

[Reveal] Spoiler: OA
66

We Know,

Work = Individual Rate × Number of Workers × Time

Now plugging the values we have,

$$\frac{1}{2}$$ pyramid = Individual Rate * 45 * 288

or Individual Rate = $$\frac{1}{25920}$$ pyramid per day

Now we need to find the no. of days for 65 people to build the $$\frac{1}{6}$$of pyramid

We know,

Work = Individual Rate × Number of Workers × Time

$$\frac{1}{6} = \frac{1}{25920} *65 * T$$

Time = 66.46 days i.e. 66 days rounded to nearest integer
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Re: If 45 people built of a pyramid   [#permalink] 16 Sep 2018, 11:10
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# If 45 people built of a pyramid

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# Integer Combination of Coprime Integers/General Result

## Theorem

Let $a_1, a_2, \ldots, a_n$ be integers.

Then $\gcd \set {a_1, a_2, \ldots, a_n} = 1$ if and only if there exists an integer combination of them equal to $1$:

$\exists m_1, m_2, \ldots, m_n \in \Z: \ds \sum_{k \mathop = 1}^n m_k a_k = 1$

## Proof

First let $\exists m_1, m_2, \ldots, m_n \in \Z: \ds \sum_{k \mathop = 1}^n m_k a_k = 1$.

Let $\gcd \set {a_1, a_2, \ldots, a_n} = d$.

Then $\ds \sum_{k \mathop = 1}^n m_k a_k$ has $d$ as a divisor.

That means $d$ is a divisor of $1$.

Thus $\gcd \set {a_1, a_2, \ldots, a_n} = 1$.

$\Box$

It remains to be shown that if $\gcd \set {a_1, a_2, \ldots, a_n} = 1$, then $\exists m_1, m_2, \ldots, m_n \in \Z: \ds \sum_{k \mathop = 1}^n m_k a_k = 1$.

The proof proceeds by induction.

For all $n \in \Z_{\ge 2}$, let $\map P n$ be the proposition:

$\gcd \set {a_1, a_2, \ldots, a_n} = 1 \iff \exists m_1, m_2, \ldots, m_n \in \Z: \ds \sum_{k \mathop = 1}^n m_k a_k = 1$

### Basis for the Induction

$\map P 2$ is the case:

$\gcd \set {a_1, a_2} = 1 \iff \exists m_1, m_2 \in \Z: m_1 a_1 + m_2 a_2 = 1$

This is demonstrated in Integer Combination of Coprime Integers.

Thus $\map P 2$ is seen to hold.

This is the basis for the induction.

### Induction Hypothesis

Now it needs to be shown that, if $\map P r$ is true, where $r \ge 2$, then it logically follows that $\map P {r + 1}$ is true.

So this is the induction hypothesis:

$\gcd \set {a_1, a_2, \ldots, a_r} = 1 \iff \exists m_1, m_2, \ldots, m_r \in \Z: \ds \sum_{k \mathop = 1}^r m_k a_k = 1$

from which it is to be shown that:

$\gcd \set {a_1, a_2, \ldots, a_r, a_{r + 1} } = 1 \iff \exists m_1, m_2, \ldots, m_r, m_{r + 1} \in \Z: \ds \sum_{k \mathop = 1}^{r + 1} m_k a_k = 1$

### Induction Step

This is the induction step:

Let $\gcd \set {a_1, a_2, \ldots, a_r, a_{r + 1} } = 1$.

Let $\gcd \set {a_1, a_2, \ldots, a_r} = d$.

Dividing through by $d$, we have:

$\gcd \set {\dfrac {a_1} d, \dfrac {a_2} d, \ldots, \dfrac {a_d} d} = 1$

By the induction hypothesis:

$\exists m_1, m_2, \ldots, m_r \in \Z: \ds \sum_{k \mathop = 1}^r m_k a_k = 1$

whence multiplying through by $d$:

$\exists m'_1, m'_2, \ldots, m'_r \in \Z: \ds \sum_{k \mathop = 1}^r m'_k a_k = d$

where $m'_k = m_k d$ throughout.

Then by hypothesis:

$\gcd \set {d, a_{r + 1} } = 1$

From the basis for the induction:

$\exists p, q \in \Z: p d + 1 a_{r + 1} = 1$

We take:

$m_1 = m'_1 p, m_2 = m'_2 p, \ldots, m_r = m'_r p$

and:

$m_{r + 1} = q$

and thus we have:

$\gcd \set {a_1, a_2, \ldots, a_r, a_{r + 1} } = 1 \iff \exists m_1, m_2, \ldots, m_r, m_{r + 1} \in \Z: \ds \sum_{k \mathop = 1}^{r + 1} m_k a_k = 1$

$\blacksquare$

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# Learning From a Piece of Pie: The Empirical Content of Nash Bargaining

## Author

Listed:
• Pierre-André Chiappori

(Department of Economics, University of Chicago)

• Olivier Donni

(THEMA, Université de Cergy-Pontoise)

## Abstract

Consider a model of bargaining, in which two players, 1 and 2, share a pie of size $y$. The bargaining environment is described by a set of parameters $\lambda$ that may affect agents' preferences over the agreement sharing, the status quo outcome, or both. The outcomes (i.e., whether an agreement is reached, and if so the individual shares) and the environment (including the size of the pie) are known, but neither the agents' utilities nor their threat points. Assuming that the agents adopt a Nash bargaining solution, we investigate the empirical content of this assumption. We first show that in the most general framework, any outcome can be rationalized as a Nash solution. However, if (i) the size of the pie $y$ does not influence the players' threat points and (ii) there exist (at least) two parameters $\lambda_1$ and $\lambda_2$ that are player-specific, in the sense that $\lambda_i$ does not influence the utility or the threat point of player $j \neq i$, then Nash bargaining generates strong testable restrictions. Moreover, the underlying structure of the bargaining, i.e., the players? utility and threat point functions, can be recovered under slightly more demanding conditions.

## Suggested Citation

• Pierre-André Chiappori & Olivier Donni, 2005. "Learning From a Piece of Pie: The Empirical Content of Nash Bargaining," THEMA Working Papers 2006-07, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
• Handle: RePEc:ema:worpap:2006-07
as

File URL: http://www.u-cergy.fr/IMG/documents//2006-07.pdf

## References listed on IDEAS

as
1. Svejnar, Jan, 1986. "Bargaining Power, Fear of Disagreement, and Wage Settlements: Theory and Evidence from U.S. Industry," Econometrica, Econometric Society, vol. 54(5), pages 1055-1078, September.
2. Ray, Indrajit & Zhou, Lin, 2001. "Game Theory via Revealed Preferences," Games and Economic Behavior, Elsevier, vol. 37(2), pages 415-424, November.
3. Ken Binmore & Ariel Rubinstein & Asher Wolinsky, 1986. "The Nash Bargaining Solution in Economic Modelling," RAND Journal of Economics, The RAND Corporation, vol. 17(2), pages 176-188, Summer.
4. Melvyn G. Coles & Andrew K. G. Hildreth, 2000. "Wage Bargaining, Inventories, and Union Legislation," Review of Economic Studies, Oxford University Press, vol. 67(2), pages 273-293.
5. Bossert, Walter & Sprumont, Yves, 2003. "Efficient and non-deteriorating choice," Mathematical Social Sciences, Elsevier, vol. 45(2), pages 131-142, April.
6. Chiappori, Pierre-Andre, 1991. "Nash-Bargained Household Decisions: A Rejoinder," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 32(3), pages 761-762, August.
7. Nash, John, 1950. "The Bargaining Problem," Econometrica, Econometric Society, vol. 18(2), pages 155-162, April.
8. McElroy, Marjorie B & Horney, Mary Jean, 1990. "Nash-Bargained Household Decisions: Reply," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 31(1), pages 237-242, February.
9. Walter Bossert & Yves Sprumont, 2002. "Core rationalizability in two-agent exchange economies," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 20(4), pages 777-791.
10. Lundberg, Shelly & Pollak, Robert A, 1993. "Separate Spheres Bargaining and the Marriage Market," Journal of Political Economy, University of Chicago Press, vol. 101(6), pages 988-1010, December.
11. Thomson, William, 1994. "Cooperative models of bargaining," Handbook of Game Theory with Economic Applications,in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 35, pages 1237-1284 Elsevier.
12. Ray, Indrajit & Snyder, Susan, 2013. "Observable implications of Nash and subgame-perfect behavior in extensive games," Journal of Mathematical Economics, Elsevier, vol. 49(6), pages 471-477.
13. Ray, Indrajit & Snyder, Susan, 2013. "Observable implications of Nash and subgame-perfect behavior in extensive games," Journal of Mathematical Economics, Elsevier, vol. 49(6), pages 471-477.
14. Sprumont, Yves, 2000. "On the Testable Implications of Collective Choice Theories," Journal of Economic Theory, Elsevier, vol. 93(2), pages 205-232, August.
15. Chiappori, Pierre-Andre, 1988. "Nash-Bargained Households Decisions: A Comment," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 29(4), pages 791-796, November.
16. Rubinstein, Ariel & Safra, Zvi & Thomson, William, 1992. "On the Interpretation of the Nash Bargaining Solution and Its Extension to Non-expected Utility Preferences," Econometrica, Econometric Society, vol. 60(5), pages 1171-1186, September.
17. Chiappori, Pierre-Andre, 1988. "Rational Household Labor Supply," Econometrica, Econometric Society, vol. 56(1), pages 63-90, January.
18. Jan Svejnar, 1980. "On the Empirical Testing of the Nash-Zeuthen Bargaining Solution," ILR Review, Cornell University, ILR School, vol. 33(4), pages 536-542, July.
19. Chiappori, P.A. & Ekeland, I., 2006. "The micro economics of group behavior: General characterization," Journal of Economic Theory, Elsevier, vol. 130(1), pages 1-26, September.
20. Chiappori, Pierre-Andre, 1992. "Collective Labor Supply and Welfare," Journal of Political Economy, University of Chicago Press, vol. 100(3), pages 437-467, June.
21. Carvajal, Andres & Ray, Indrajit & Snyder, Susan, 2004. "Equilibrium behavior in markets and games: testable restrictions and identification," Journal of Mathematical Economics, Elsevier, vol. 40(1-2), pages 1-40, February.
22. McElroy, Marjorie B & Horney, Mary Jean, 1981. "Nash-Bargained Household Decisions: Toward a Generalization of the Theory of Demand," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 22(2), pages 333-349, June.
Full references (including those not matched with items on IDEAS)

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
as

Cited by:

1. Carvajal, Andrés & González, Natalia, 2014. "On refutability of the Nash bargaining solution," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 177-186.
2. Antonio Merlo & Xun Tang, 2010. "Identification and Estimation of Stochastic Bargaining Models, Third Version," PIER Working Paper Archive 11-008, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 11 Mar 2011.
3. Chiappori, P.A. & Ekeland, I., 2006. "The micro economics of group behavior: General characterization," Journal of Economic Theory, Elsevier, vol. 130(1), pages 1-26, September.
4. Richard Blundell & Pierre-Andre Chiappori & Thierry Magnac & Costas Meghir, 2007. "Collective Labour Supply: Heterogeneity and Non-Participation," Review of Economic Studies, Oxford University Press, vol. 74(2), pages 417-445.
5. Majlesi, Kaveh, 2016. "Labor market opportunities and women's decision making power within households," Journal of Development Economics, Elsevier, vol. 119(C), pages 34-47.
6. Arianna Degan & Antonio Merlo, 2006. "Do Voters Vote Sincerely? Second Version," PIER Working Paper Archive 07-006, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 03 Jan 2007.
7. de Clippel, Geoffroy & Eliaz, Kfir, 2012. "Reason-based choice: a bargaining rationale for the attraction and compromise effects," Theoretical Economics, Econometric Society, vol. 7(1), January.
8. Arianna Degan & Antonio Merlo, 2006. "Do Voters Vote Sincerely?," PIER Working Paper Archive 06-008, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
9. Chiappori, Pierre-André & Donni, Olivier, 2009. "Non-unitary Models of Household Behavior: A Survey of the Literature," IZA Discussion Papers 4603, Institute for the Study of Labor (IZA).
10. Antonio Merlo & Xun Tang, 2011. "Identification and Estimation of Stochastic Bargaining Models, Fourth Version," PIER Working Paper Archive 11-035, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 19 Oct 2011.
11. Chiappori, Pierre-André & Donni, Olivier, 2006. "Les modèles non unitaires de comportement du ménage : un survol de la littérature," L'Actualité Economique, Société Canadienne de Science Economique, vol. 82(1), pages 9-52, mars-juin.
12. Arianna Degan & Antonio Merlo, 2007. "Do Voters Vote Ideologically?, Third Version," PIER Working Paper Archive 08-034, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 01 Aug 2008.

### Keywords

Keywords: Bargaining Game; Nash Solution; Testability; Identifiability; Cardinal Utility;

### JEL classification:

• C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
• C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

### NEP fields

This paper has been announced in the following NEP Reports:

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# Tag Info

3

The document @Sverre wrote, but with LuaLaTeX gives a correct output. However, as @UlrikeFischer said, it seems to be “wrong” in many ways. %!TEX TS-program = lualatex \documentclass{article} \usepackage[utf8]{luainputenc} \usepackage[T1]{fontenc} \usepackage{mathptmx} % Times font \usepackage{helvet} % Helvetica font \usepackage{courier} % Courier font ...

4

There are many way to write with your preferred fonts. Just one simple example with XeLaTeX/LuaLaTeX % !TEX program = xelatex % !TEX encoding = utf8 \documentclass[journal,a4paper]{IEEEtran} \usepackage{fontspec} \defaultfontfeatures+[\sffamily,\ttfamily]{Scale=MatchLowercase} \setmainfont{FreeSerif} \setsansfont{FreeSans} \setmonofont{FreeMono} ...

2

Replacing \usepackage[cross,center,width=216mm,height=303mm]{crop} % bleed is 6mm by \usepackage[mirror,cross,center,width=216mm,height=303mm]{crop} % bleed is 6mm \makeatletter \newsavebox\WholePageBigBox \renewcommand*\CROP@genreflect[1]{% \leavevmode \savebox\WholePageBigBox{\hb@xt@\paperwidth{\vbox ...

5

The syntax is not \ifx<firsttoken><secondtoken>{<true code>}\else{<false code>}\fi The correct one is \ifx<firsttoken><secondtoken><true code>\else<false code>\fi So you can remove the braces around {#1} and {#2} (which, in the case of \usepackage{fontspec} act as a group delimiter).

0

On OSX, I was able to get things working in the following way: Install to default user location (/Users/john/Library/Fonts) mkluatexfontdb --force --verbose=-1 -vvv That's all it took!

0

Does this work (assuming the MinionPro package) \def\@biblabel#1{[{\figureversion{lining}#1}]}

7

My development version of fontspec might be out of sync with the current TeX Live version on this matter — I know I recently fixed an issue that involved \addfontfeatures incorrectly creating new font families. In my development version, the 2nd use of \addfontfeatures line in the log file doesn't lead into the ‘Font Family (…) created’ message. However, I ...

5

The fonts used by mathdesign are not compatible with XeLaTeX and fontspec, because they're neither OpenType nor TrueType fonts. Replace \usepackage{xltxtra} with \usepackage[utf8]{inputenc} and compile with pdflatex.

1

The fact that the warning diappeard if you get rid of \LaTeX from some title, means that the theme has been created either with an old definition of that command, or thet the definition is incompatible with LuaLaTeX. In facts "someone" specified that some text had to be typeset with the standard Old TeX encoding OT!; normally the default encoding for open ...

1

You may use \newunicodechar in a more complicated way to check whether the next character is the same. We need a different active character (here ?) because \newunicodechar already uses ~ internally. \documentclass[twoside]{report} \usepackage{fontspec} \usepackage{polyglossia} \setdefaultlanguage{hindi} \setotherlanguage{english} ...

2

You can define a specific font command in fontspec just for punctuation. In this example I create an \englishfont family for Latin script (your commands for this were not quite right), then I create a command \punct just for punctuation that uses the \englishfont. It takes one argument, which could be any string you want in the other font, such as ...

5

This is due to bad font design, in my opinion. Instead of letting S overshoot its bounding box, the font developers decided to define a bounding box that vertically covers the whole inked area and has some sidebearings, as it becomes apparent when setting the parameter \fboxsep to zero: \documentclass{article} \usepackage{fontspec} \begin{document} ...

10

I'm not sure I qualify as experienced and/or more successful :-). But here's an example of the setup I used for my thesis. Adobe Garamond Pro was used as the text font. Sadly, I needed bold to appease the thesis arbiters at my university, so I used BoldFont=AGaramondPro-Semibold as it looks slightly less out of place than the standard bold weight. For ...

2

I no longer have the Mona Lisa font, but here’s an example of modifying the code for fonts other than those Peter Wilson used. The code is simpler with fontspec: % xetex or luatex \documentclass[12pt,twoside,openright]{memoir} \usepackage{fontspec,xcolor} \newfontface\bmb{Bembo Book MT Pro}[Numbers=OldStyle]% or use \setmainfont if the main matter will be ...

1

you have to define a main font with \setmainfont: \documentclass{article} \usepackage{fontspec} \usepackage{arabxetex} \newfontfamily\arabicfont[Script=Arabic, Scale=1.5]{Scheherazade} \setmainfont{Linux Libertine O} %\SetTranslitConvention{dmg} \begin{document} ...

3

NB this is an answer to the original question. I described my experience with adding a new ttf font. I use lualatex, first installed tex-live, on my new Ubuntu 14 lts, I want to use from XP font constantia. I copied all CONSTANT*.TTF-files from XP in this map: /usr/share/fonts/truetype/msttcorefonts analogically(!) like other already available fonts. NB ...

0

(As requested.) One alternative might be Source Sans Pro by Adobe, which gave me the following fonts: SourceSansPro-BlackIt.otf SourceSansPro-It.otf SourceSansPro-Black.otf SourceSansPro-LightIt.otf SourceSansPro-BoldIt.otf SourceSansPro-Light.otf SourceSansPro-Bold.otf SourceSansPro-Regular.otf SourceSansPro-ExtraLightIt.otf ...

4

In according to unimath-symbols.pdf, the best fonts for math symbols are M Latin Modern Math (1588) X XITS Math (2437) C Cambria Math (2189) A Asana Math (2259) P TeX Gyre Pagella Math (1612) E Neo Euler (579) (IMHO, this is very wrong ...) There are many Times-compatible font with fontspec, like TeX Gyre Termes, FreeSerif ...

3

I had some experience with this in my thesis... The Times clone in the TeX Gyre fonts (Termes) can be used quite simply with the unicode-math package. In fact, any of the TeX Gyre fonts can be used this way (at least, the ones with corresponding math fonts) and this should work for XeLaTeX as well. \documentclass{article} \usepackage{unicode-math} ...

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Multi angle  de Rham theorem in non-Archimedean analytic geometry Berkovich, Vladimir (Auteur de la Conférence) | CIRM (Editeur )

In my work in progress on complex analytic vanishing cycles for formal schemes, I have defined integral "etale" cohomology groups of a compact strictly analytic space over the field of Laurent power series with complex coefficients. These are finitely generated abelian groups provided with a quasi-unipotent action of the fundamental group of the punctured complex plane, and they give rise to all $l$-adic etale cohomology groups of the space. After a short survey of this work, I will explain a theorem which, in the case when the space is rig-smooth, compares those groups and the de Rham cohomology groups of the space. The latter are provided with the Gauss-Manin connection and an additional structure which allow one to recover from them the "etale" cohomology groups with complex coefficients. In my work in progress on complex analytic vanishing cycles for formal schemes, I have defined integral "etale" cohomology groups of a compact strictly analytic space over the field of Laurent power series with complex coefficients. These are finitely generated abelian groups provided with a quasi-unipotent action of the fundamental group of the punctured complex plane, and they give rise to all $l$-adic etale cohomology groups of the space. ...

Multi angle  $D$-modules and $p$-curvatures Esnault, Hélène (Auteur de la Conférence) | CIRM (Editeur )

We show relations between rigidity of connections in characteristic 0 and nilpotency of their $p$-curvatures (a consequence of a conjecture by Simpson and of a generalization of Grothendieck's $p$-curvature conjecture).
Work in progress with Michael Groechenig.

Multi angle  Arithmetic $D$-modules and existence of crystalline companion Abe, Tomoyuki (Auteur de la Conférence) | CIRM (Editeur )

We will show that there exists a correspondence between smooth $l$-adic sheaves and overconvergent $F$-isocrystals over a curve preserving the Frobenius eigenvalues. Moreover, we show the existence of $l$-adic companions associated to overconvergent $F$-isocrystals for smooth varieties.
Some part of the work is done jointly with Esnault.

Multi angle  Explosion of Lyapunov exponents using non-Archimedean geometry Favre, Charles (Auteur de la Conférence) | CIRM (Editeur )

We consider a meromorphic family of endomorphisms of the complex projective space parameterized by the unit disk, and show that the blow-up of the Lyapunov exponent near the origin is controlled by a non-Archimedean quantity.

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Volume 363 - 37th International Symposium on Lattice Field Theory (LATTICE2019) - Main session
Radiative leptonic decays on the lattice
C. Kane, C. Lehner, S. Meinel,* A. Soni
*corresponding author
Full text: pdf
Pre-published on: 2019 December 05
Published on:
Abstract
Adding a hard photon to the final state of a leptonic pseudoscalar-meson decay lifts the helicity suppression and can provide sensitivity to a larger set of operators in the weak effective Hamiltonian. Furthermore, radiative leptonic $B$ decays at high photon energy are well suited to constrain the first inverse moment of the $B$-meson light-cone distribution amplitude, an important parameter in the theory of nonleptonic $B$ decays. We demonstrate that the calculation of radiative leptonic decays is possible using Euclidean lattice QCD, and present preliminary numerical results for $D_s^+ \to \ell^+ \nu\gamma$ and $K^- \to \ell^-\bar{\nu}\gamma$.
Open Access


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## Azimuth Backup Project (Part 5)

5 October, 2017

I haven’t spoken much about the Azimuth Climate Data Backup Project, but it’s going well, and I’ll be speaking about it soon, here:

International Open Access Week, Wednesday 25 October 2017, 9:30–11:00 a.m., University of California, Riverside, Orbach Science Library, Room 240.

“Open in Order to Save Data for Future Research” is the 2017 event theme.

Open Access Week is an opportunity for the academic and research community to learn about the potential benefits of sharing what they’ve learned with colleagues, and to help inspire wider participation in helping to make “open access” a new norm in scholarship, research and data planning and preservation.

The Open Access movement is made of up advocates (librarians, publishers, university repositories, etc.) who promote the free, immediate, and online publication of research.

The program will provide information on issues related to saving open data, including climate change and scientific data. The panelists also will describe open access projects in which they have participated to save climate data and to preserve end-of-term presidential data, information likely to be and utilized by the university community for research and scholarship.

The program includes:

• Brianna Marshall, Director of Research Services, UCR Library: Brianna welcomes guests and introduces panelists.

• John Baez, Professor of Mathematics, UCR: John will describe his activities to save US government climate data through his collaborative effort, the Azimuth Climate Data Backup Project. All of the saved data is now open access for everyone to utilize for research and scholarship.

• Perry Willett, Digital Preservation Projects Manager, California Digital Library: Perry will discuss the open data initiatives in which CDL participates, including the end-of-term presidential web archiving that is done in partnership with the Library of Congress, Internet Archive and University of North Texas.

• Kat Koziar, Data Librarian, UCR Library: Kat will give an overview of DASH, the UC system data repository, and provide suggestions for researchers interested in making their data open.

This will be the eighth International Open Access Week program hosted by the UCR Library.

The event is free and open to the public. Light refreshments will be served.

## Complex Adaptive System Design (Part 4)

22 August, 2017

Last time I introduced typed operads. A typed operad has a bunch of operations for putting together things of various types and getting new things of various types. This is a very general idea! But in the CASCADE project we’re interested in something more specific: networks. So we want operads whose operations are ways to put together networks and get new networks.

That’s what our team came up with: John Foley of Metron, my graduate students Blake Pollard and Joseph Moeller, and myself. We’re writing a couple of papers on this, and I’ll let you know when they’re ready. These blog articles are kind of sneak preview—and a gentle introduction, where you can ask questions.

For example: I’m talking a lot about networks. But what is a ‘network’, exactly?

There are many kinds. At the crudest level, we can model a network as a simple graph, which is something like this:

There are some restrictions on what counts as a simple graph. If the vertices are agents of some sort and the edges are communication channels, these restrictions imply:

• We allow at most one channel between any pair of agents, since there’s at most one edge between any two vertices of our graph.

• The channels do not have a favored direction, since there are no arrows on the edges of our graph.

• We don’t allow a channel from an agent to itself, since an edge can’t start and end at the same vertex.

For other purposes we may want to drop some or all of these restrictions. There is an appalling diversity of options! We might want to allow multiple channels between a pair of agents. For this we could use multigraphs. We might want to allow directed channels, where the sender and receiver have different capabilities: for example, signals may only be able to flow in one direction. For this we could use directed graphs. And so on.

We will also want to consider graphs with colored vertices, to specify different types of agents—or colored edges, to specify different types of channels. Even more complicated variants are likely to become important as we proceed.

To avoid sinking into a mire of special cases, we need the full power of modern mathematics. Instead of separately studying all these various kinds of networks, we need a unified notion that subsumes all of them.

To do this, the Metron team came up with something called a ‘network model’. There is a network model for simple graphs, a network model for multigraphs, a network model for directed graphs, a network model for directed graphs with 3 colors of vertex and 15 colors of edge, and more.

You should think of a network model as a kind of network. Not a specific network, just a kind of network.

Our team proved that for each network model $G$ there is an operad $O_G$ whose operations describe how to put together networks of that kind. We call such operads ‘network operads’.

I want to make all this precise, but today let me just show you one example. Let’s take $G$ to be the network model for simple graphs, and look at the network operad $O_G.$ I won’t tell you what kind of thing $G$ is yet! But I’ll tell you about the operad $O_G$.

Types. Remember from last time that an operad has a set of ‘types’. For $O_G$ this is the set of natural numbers, $\mathbb{N}.$ The reason is that a simple graph can have any number of vertices.

Operations. Remember that an operad has sets of ‘operations’. In our case we have a set of operations $O_G(t_1,\dots,t_n ; t)$ for each choice of $t_1,\dots,t_n, t \in \mathbb{N}.$

An operation $f \in O_G(t_1,\dots,t_n; t)$ is a way of taking a simple graph with $t_1$ vertices, a simple graph with $t_2$ vertices,… and so on, and sticking them together, perhaps adding new edges, to get a simple graph with

$t = t_1 + \cdots + t_n$

vertices.

Let me show you an operation

$f \in O_G(3,4,2;9)$

This will be a way of taking three simple graphs—one with 3 vertices, one with 4, and one with 2—and sticking them together, perhaps adding edges, to get one with 9 vertices.

Here’s what $f$ looks like:

It’s a simple graph with vertices numbered from 1 to 9, with the vertices in bunches: {1,2,3}, {4,5,6,7} and {8,9}. It could be any such graph. This one happens to have an edge from 3 to 6 and an edge from 1 to 2.

Here’s how we can actually use our operation. Say we have three simple graphs like this:

Then we can use our operation to stick them together and get this:

Notice that we added a new edge from 3 to 6, connecting two of our three simple graphs. We also added an edge from 1 to 2… but this had no effect, since there was already an edge there! The reason is that simple graphs have at most one edge between vertices.

But what if we didn’t already have an edge from 1 to 2? What if we applied our operation $f$ to the following simple graphs?

Well, now we’d get this:

This time adding the edge from 1 to 2 had an effect, since there wasn’t already an edge there!

In short, we can use this operad to stick together simple graphs, but also to add new edges within the simple graphs we’re sticking together!

When I’m telling you how we ‘actually use’ our operad to stick together graphs, I’m secretly describing an algebra of our operad. Remember, an operad describes ways of sticking together things together, but an ‘algebra’ of the operad gives a particular specification of these things and describes how we stick them together.

Our operad $O_G$ has lots of interesting algebras, but I’ve just shown you the simplest one. More precisely:

Things. Remember from last time that for each type, an algebra specifies a set of things of that type. In this example our types are natural numbers, and for each natural number $t \in \mathbb{N}$ I’m letting the set of things $A(t)$ consist of all simple graphs with vertices $\{1, \dots, t\}.$

Action. Remember that our operad $O_G$ should have an action on $A$, meaning a bunch of maps

$\alpha : O_G(t_1,...,t_n ; t) \times A(t_1) \times \cdots \times A(t_n) \to A(t)$

I just described how this works in some examples. Some rules should hold… and they do.

To make sure you understand, try these puzzles:

Puzzle 1. In the example I just explained, what is the set $O_G(t_1,\dots,t_n ; t)$ if $t \ne t_1 + \cdots + t_n?$

Puzzle 2. In this example, how many elements does $O_G(1,1;2)$ have?

Puzzle 3. In this example, how many elements does $O_G(1,2;3)$ have?

Puzzle 4. In this example, how many elements does $O_G(1,1,1;3)$ have?

Puzzle 5. In the particular algebra $A$ that I explained, how many elements does $A(3)$ have?

Next time I’ll describe some more interesting algebras of this operad $O_G.$ These let us describe networks of mobile agents with range-limited communication channels!

## Complex Adaptive System Design (Part 3)

17 August, 2017

It’s been a long time since I’ve blogged about the Complex Adaptive System Composition and Design Environment or CASCADE project run by John Paschkewitz. For a reminder, read these:

Complex adaptive system design (part 1), Azimuth, 2 October 2016.

Complex adaptive system design (part 2), Azimuth, 18 October 2016.

A lot has happened since then, and I want to explain it.

I’m working with Metron Scientific Solutions to develop new techniques for designing complex networks.

The particular problem we began cutting our teeth on is a search and rescue mission where a bunch of boats, planes and drones have to locate and save people who fall overboard during a boat race in the Caribbean Sea. Subsequently the Metron team expanded the scope to other search and rescue tasks. But the real goal is to develop very generally applicable new ideas on designing and ‘tasking’ networks of mobile agents—that is, designing these networks and telling the agents what to do.

We’re using the mathematics of ‘operads’, in part because Spivak’s work on operads has drawn a lot of attention and raised a lot of hopes:

An operad is a bunch of operations for sticking together smaller things to create bigger ones—I’ll explain this in detail later, but that’s the core idea. Spivak described some specific operads called ‘operads of wiring diagrams’ and illustrated some of their potential applications. But when we got going on our project, we wound up using a different class of operads, which I’ll call ‘network operads’.

Here’s our dream, which we’re still trying to make into a reality:

Network operads should make it easy to build a big network from smaller ones and have every agent know what to do. You should be able to ‘slap together’ a network, throwing in more agents and more links between them, and automatically have it do something reasonable. This should be more flexible than an approach where you need to know ahead of time exactly how many agents you have, and how they’re connected, before you can tell them what to do.

You don’t want a network to malfunction horribly because you forgot to hook it up correctly. You want to focus your attention on optimizing the network, not getting it to work at all. And you want everything to work so smoothly that it’s easy for the network to adapt to changing conditions.

To achieve this we’re using network operads, which are certain special ‘typed operads’. So before getting into the details of our approach, I should say a bit about typed operads. And I think that will be enough for today’s post: I don’t want to overwhelm you with too much information at once.

In general, a ‘typed operad’ describes ways of sticking together things of various types to get new things of various types. An ‘algebra’ of the operad gives a particular specification of these things and the results of sticking them together. For now I’ll skip the full definition of a typed operad and only highlight the most important features. A typed operad $O$ has:

• a set $T$ of types.

• collections of operations $O(t_1,...,t_n ; t)$ where $t_i, t \in T$. Here $t_1, \dots, t_n$ are the types of the inputs, while $t$ is the type of the output.

• ways to compose operations. Given an operation
$f \in O(t_1,\dots,t_n ;t)$ and $n$ operations

$g_1 \in O(t_{11},\dots,t_{1 k_1}; t_1),\dots, g_n \in O(t_{n1},\dots,t_{n k_n};t_n)$

we can compose them to get

$f \circ (g_1,\dots,g_n) \in O(t_{11}, \dots, t_{nk_n};t)$

These must obey some rules.

But if you haven’t seen operads before, you’re probably reeling in horror—so I need to rush in and save you by showing you the all-important pictures that help explain what’s going on!

First of all, you should visualize an operation $f \in O(t_1, \dots, t_n; t)$ as a little gizmo like this:

It has $n$ inputs at top and one output at bottom. Each input, and the output, has a ‘type’ taken from the set $T.$ So, for example, if you operation takes two real numbers, adds them and spits out the closest integer, both input types would be ‘real’, while the output type would be ‘integer’.

The main thing we do with operations is compose them. Given an an operation $f \in O(t_1,\dots,t_n ;t),$ we can compose it with $n$ operations

$g_1 \in O(t_{11},\dots,t_{1 k_1}; t_1), \quad \dots, \quad g_n \in O(t_{n1},\dots,t_{n k_n};t_n)$

by feeding their outputs into the inputs of $f,$ like this:

The result is an operation we call

$f \circ (g_1, \dots, g_n)$

Note that the input types of $f$ have to match the output types of the $g_i$ for this to work! This is the whole point of types: they forbid us from composing operations in ways that don’t make sense.

This avoids certain stupid mistakes. For example, you can take the square root of a positive number, but you may not want to take the square root of a negative number, and you definitely don’t want to take the square root of a hamburger. While you can land a plane on an airstrip, you probably don’t want to land a plane on a person.

The operations in an operad are quite abstract: they aren’t really operating on anything. To render them concrete, we need another idea: operads have ‘algebras’.

An algebra $A$ of the operad $O$ specifies a set of things of each type $t \in T$ such that the operations of $O$ act on these sets. A bit more precisely, an algebra consists of:

• for each type $t \in T,$ a set $A(t)$ of things of type $t$

• an action of $O$ on $A,$ that is, a collection of maps

$\alpha : O(t_1,...,t_n ; t) \times A(t_1) \times \cdots \times A(t_n) \to A(t)$

obeying some rules.

In other words, an algebra turns each operation $f \in O(t_1,...,t_n ; t)$ into a function that eats things of types $t_1, \dots, t_n$ and spits out a thing of type $t.$

When we get to designing systems with operads, the fact that the same operad can have many algebras will be useful. Our operad will have operations describing abstractly how to hook up networks to form larger networks. An algebra will give a specific implementation of these operations. We can use one algebra that’s fairly fine-grained and detailed about what the operations actually do, and another that’s less detailed. There will then be a map between from the first algebra to the second, called an ‘algebra homomorphism’, that forgets some fine-grained details.

There’s a lot more to say—all this is just the mathematical equivalent of clearing my throat before a speech—but I’ll stop here for now.

And as I do—since it also takes me time to stop talking—I should make it clear yet again that I haven’t even given the full definition of typed operads and their algebras! Besides the laws I didn’t write down, there’s other stuff I omitted. Most notably, there’s a way to permute the inputs of an operation in an operad, and operads have identity operations, one for each type.

To see the full definition of an ‘untyped’ operad, which is really an operad with just one type, go here:

• Wikipedia, Operad theory.

They just call it an ‘operad’. Note that they first explain ‘non-symmetric operads’, where you can’t permute the inputs of operations, and then explain operads, where you can.

If you’re mathematically sophisticated, you can easily guess the laws obeyed by a typed operad just by looking at this article and inserting the missing types. You can also see the laws written down in Spivak’s paper, but with some different terminology: he calls types ‘objects’, he calls operations ‘morphisms’, and he calls typed operads ‘symmetric colored operads’—or once he gets going, just ‘operads’.

You can also see the definition of a typed operad in Section 2.1 here:

• Donald Yau, Operads of wiring diagrams.

What I would call a typed operad with $S$ as its set of types, he calls an ‘$S$-colored operad’.

I guess it’s already evident, but I’ll warn you that the terminology in this subject varies quite a lot from author to author: for example, a certain community calls typed operads ‘symmetric multicategories’. This is annoying at first but once you understand the subject it’s as ignorable as the fact that mathematicians have many different accents. The main thing to remember is that operads come in four main flavors, since they can either be typed or untyped, and they can either let you permute inputs or not. I’ll always be working with typed operads where you can permute inputs.

Finally, I’ll say that while the definition of operad looks lengthy and cumbersome at first, it becomes lean and elegant if you use more category theory.

Next time I’ll give you an example of an operad: the simplest ‘network

## Saving Climate Data (Part 6)

23 February, 2017

Scott Pruitt, who filed legal challenges against Environmental Protection Agency rules fourteen times, working hand in hand with oil and gas companies, is now head of that agency. What does that mean about the safety of climate data on the EPA’s websites? Here is an inside report:

• Dawn Reeves, EPA preserves Obama-Era website but climate change data doubts remain, InsideEPA.com, 21 February 2017.

For those of us who are backing up climate data, the really important stuff is in red near the bottom.

The EPA has posted a link to an archived version of its website from Jan. 19, the day before President Donald Trump was inaugurated and the agency began removing climate change-related information from its official site, saying the move comes in response to concerns that it would permanently scrub such data.

However, the archived version notes that links to climate and other environmental databases will go to current versions of them—continuing the fears that the Trump EPA will remove or destroy crucial greenhouse gas and other data.

The archived version was put in place and linked to the main page in response to “numerous [Freedom of Information Act (FOIA)] requests regarding historic versions of the EPA website,” says an email to agency staff shared by the press office. “The Agency is making its best reasonable effort to 1) preserve agency records that are the subject of a request; 2) produce requested agency records in the format requested; and 3) post frequently requested agency records in electronic format for public inspection. To meet these goals, EPA has re-posted a snapshot of the EPA website as it existed on January 19, 2017.”

The email adds that the action is similar to the snapshot taken of the Obama White House website.

The archived version of EPA’s website includes a “more information” link that offers more explanation.

For example, it says the page is “not the current EPA website” and that the archive includes “static content, such as webpages and reports in Portable Document Format (PDF), as that content appeared on EPA’s website as of January 19, 2017.”

It cites technical limits for the database exclusions. “For example, many of the links contained on EPA’s website are to databases that are updated with the new information on a regular basis. These databases are not part of the static content that comprises the Web Snapshot.” Searches of the databases from the archive “will take you to the current version of the database,” the agency says.

“In addition, links may have been broken in the website as it appeared” on Jan. 19 and those will remain broken on the snapshot. Links that are no longer active will also appear as broken in the snapshot.

“Finally, certain extremely large collections of content… were not included in the Snapshot due to their size” such as AirNow images, radiation network graphs, historic air technology transfer network information, and EPA’s searchable news releases.”

#### ‘Smart’ Move

One source urging the preservation of the data says the snapshot appears to be a “smart” move on EPA’s behalf, given the FOIA requests it has received, and notes that even though other groups like NextGen Climate and scientists have been working to capture EPA’s online information, having it on EPA’s site makes it official.

But it could also be a signal that big changes are coming to the official Trump EPA site, and it is unclear how long the agency will maintain the archived version.

The source says while it is disappointing that the archive may signal the imminent removal of EPA’s climate site, “at least they are trying to accommodate public concerns” to preserve the information.

A second source adds that while it is good that EPA is seeking “to address the widespread concern” that the information will be removed by an administration that does not believe in human-caused climate change, “on the other hand, it doesn’t address the primary concern of the data. It is snapshots of the web text.” Also, information “not included,” such as climate databases, is what is difficult to capture by outside groups and is what really must be preserved.

“If they take [information] down” that groups have been trying to preserve, then the underlying concern about access to data remains. “Web crawlers and programs can do things that are easy,” such as taking snapshots of text, “but getting the data inside the database is much more challenging,” the source says.

The first source notes that EPA’s searchable databases, such as those maintained by its Clean Air Markets Division, are used by the public “all the time.”

The agency’s Office of General Counsel (OGC) Jan. 25 began a review of the implications of taking down the climate page—a planned wholesale removal that was temporarily suspended to allow for the OGC review.

But EPA did remove some specific climate information, including links to the Clean Power Plan and references to President Barack Obama’s Climate Action Plan. Inside EPA captured this screenshot of the “What EPA Is Doing” page regarding climate change. Those links are missing on the Trump EPA site. The archive includes the same version of the page as captured by our screenshot.

Inside EPA first reported the plans to take down the climate information on Jan. 17.

After the OGC investigation began, a source close to the Trump administration said Jan. 31 that climate “propaganda” would be taken down from the EPA site, but that the agency is not expected to remove databases on GHG emissions or climate science. “Eventually… the propaganda will get removed…. Most of what is there is not data. Most of what is there is interpretation.”

The Sierra Club and Environmental Defense Fund both filed FOIA requests asking the agency to preserve its climate data, while attorneys representing youth plaintiffs in a federal climate change lawsuit against the government have also asked the Department of Justice to ensure the data related to its claims is preserved.

The Azimuth Climate Data Backup Project and other groups are making copies of actual databases, not just the visible portions of websites.

## Azimuth Backup Project (Part 4)

18 February, 2017

The Azimuth Climate Data Backup Project is going well! Our Kickstarter campaign ended on January 31st and the money has recently reached us. Our original goal was $5000. We got$20,427 of donations, and after Kickstarter took its cut we received $18,590.96. Next time I’ll tell you what our project has actually been doing. This time I just want to give a huge “thank you!” to all 627 people who contributed money on Kickstarter! I sent out thank you notes to everyone, updating them on our progress and asking if they wanted their names listed. The blanks in the following list represent people who either didn’t reply, didn’t want their names listed, or backed out and decided not to give money. I’ll list people in chronological order: first contributors first. Only 12 people backed out; the vast majority of blanks on this list are people who haven’t replied to my email. I noticed some interesting but obvious patterns. For example, people who contributed later are less likely to have answered my email yet—I’ll update this list later. People who contributed more money were more likely to answer my email. The magnitude of contributions ranged from$2000 to $1. A few people offered to help in other ways. The response was international—this was really heartwarming! People from the US were more likely than others to ask not to be listed. But instead of continuing to list statistical patterns, let me just thank everyone who contributed. Daniel Estrada Ahmed Amer Saeed Masroor Jodi Kaplan John Wehrle Bob Calder Andrea Borgia L Gardner Uche Eke Keith Warner Dean Kalahan James Benson Dianne Hackborn Walter Hahn Thomas Savarino Noah Friedman Eric Willisson Jeffrey Gilmore John Bennett Glenn McDavid Brian Turner Peter Bagaric Martin Dahl Nielsen Broc Stenman Gabriel Scherer Roice Nelson Felipe Pait Kenneth Hertz Luis Bruno Andrew Lottmann Alex Morse Mads Bach Villadsen Noam Zeilberger Buffy Lyon Josh Wilcox Danny Borg Krishna Bhogaonker Harald Tveit Alvestrand Tarek A. Hijaz, MD Jouni Pohjola Chavdar Petkov Markus Jöbstl Bjørn Borud Sarah G William Straub Frank Harper Carsten Führmann Rick Angel Drew Armstrong Jesimpson Valeria de Paiva Ron Prater David Tanzer Rafael Laguna Miguel Esteves dos Santos Sophie Dennison-Gibby Randy Drexler Peter Haggstrom Jerzy Michał Pawlak Santini Basra Jenny Meyer John Iskra Bruce Jones Māris Ozols Everett Rubel Mike D Manik Uppal Todd Trimble Federer Fanatic Forrest Samuel, Harmos Consulting Annie Wynn Norman and Marcia Dresner Daniel Mattingly James W. Crosby Jennifer Booth Greg Randolph Dave and Karen Deeter Sarah Truebe Tieg Zaharia Jeffrey Salfen Birian Abelson Logan McDonald Brian Truebe Jon Leland Nicole Sarah Lim James Turnbull John Huerta Katie Mandel Bruce Bethany Summer Heather Tilert Anna C. Gladstone Naom Hart Aaron Riley Giampiero Campa Julie A. Sylvia Pace Willisson Bangskij Peter Herschberg Alaistair Farrugia Conor Hennessy Stephanie Mohr Torinthiel Lincoln Muri Anet Ferwerda Hanna Michelle Lee Guiney Ben Doherty Trace Hagemann Ryan Mannion Penni and Terry O'Hearn Brian Bassham Caitlin Murphy John Verran Susan Alexander Hawson Fabrizio Mafessoni Anita Phagan Nicolas Acuña Niklas Brunberg Adam Luptak V. Lazaro Zamora Branford Werner Niklas Starck Westerberg Luca Zenti and Marta Veneziano Ilja Preuß Christopher Flint George Read Courtney Leigh Katharina Spoerri Daniel Risse Hanna Charles-Etienne Jamme rhackman41 Jeff Leggett RKBookman Aaron Paul Mike Metzler Patrick Leiser Melinda Ryan Vaughn Kent Crispin Michael Teague Ben Fabian Bach Steven Canning Betsy McCall John Rees Mary Peters Shane Claridge Thomas Negovan Tom Grace Justin Jones Jason Mitchell Josh Weber Rebecca Lynne Hanginger Kirby Dawn Conniff Michael T. Astolfi Kristeva Erik Keith Uber Elaine Mazerolle Matthieu Walraet Linda Penfold Lujia Liu Keith Samar Tareem Henrik Almén Michael Deakin Rutger Ockhorst Erin Bassett James Crook Junior Eluhu Dan Laufer Carl Robert Solovay Silica Magazine Leonard Saers Alfredo Arroyo García Larry Yu John Behemonth Eric Humphrey Svein Halvor Halvorsen Karim Issa Øystein Risan Borgersen David Anderson Bell III Ole-Morten Duesend Adam North and Gabrielle Falquero Robert Biegler Qu Wenhao Steffen Dittmar Shanna Germain Adam Blinkinsop John WS Marvin (Dread Unicorn Games) Bill Carter Darth Chronis Lawrence Stewart Gareth Hodges Colin Backhurst Christopher Metzger Rachel Gumper Mariah Thompson Falk Alexander Glade Johnathan Salter Maggie Unkefer Shawna Maryanovich Wilhelm Fitzpatrick Dylan “ExoByte” Mayo Lynda Lee Scott Carpenter Charles D, Payet Vince Rostkowski Tim Brown Raven Daegmorgan Zak Brueckner Christian Page Adi Shavit Steven Greenberg Chuck Lunney Adriel Bustamente Natasha Anicich Bram De Bie Edward L Gray Detrick Robert Sarah Russell Sam Leavin Abilash Pulicken Isabel Olondriz James Pierce James Morrison April Daniels José Tremblay Champagne Chris Edmonds Hans & Maria Cummings Bart Gasiewiski Andy Chamard Andrew Jackson Christopher Wright Crystal Collins ichimonji10 Alan Stern Alison W Dag Henrik Bråtane Martin Nilsson William Schrade ## Give the Earth a Present: Help Us Save Climate Data 28 December, 2016 We’ve been busy backing up climate data before Trump becomes President. Now you can help too, with some money to pay for servers and storage space. Please give what you can at our Kickstarter campaign here: If we get$5000 by the end of January, we can save this data until we convince bigger organizations to take over. If we don’t get that much, we get nothing. That’s how Kickstarter works. Also, if you donate now, you won’t be billed until January 31st.

I will make public how we spend this money. And if we get more than \$5000, I’ll make sure it’s put to good use. There’s a lot of work we could do to make sure the data is authenticated, made easily accessible, and so on.

### The idea

The safety of US government climate data is at risk. Trump plans to have climate change deniers running every agency concerned with climate change. So, scientists are rushing to back up the many climate databases held by US government agencies before he takes office.

We hope he won’t be rash enough to delete these precious records. But: better safe than sorry!

The Azimuth Climate Data Backup Project is part of this effort. So far our volunteers have backed up nearly 1 terabyte of climate data from NASA and other agencies. We’ll do a lot more! We just need some funds to pay for storage space and a server until larger institutions take over this task.

### The team

Jan Galkowski is a statistician with a strong interest in climate science. He works at Akamai Technologies, a company responsible for serving at least 15% of all web traffic. He began downloading climate data on the 11th of December.

• Shortly thereafter John Baez, a mathematician and science blogger at U. C. Riverside, joined in to publicize the project. He’d already founded an organization called the Azimuth Project, which helps scientists and engineers cooperate on environmental issues.

• When Jan started running out of storage space, Scott Maxwell jumped in. He used to work for NASA—driving a Mars rover among other things—and now he works for Google. He set up a 10-terabyte account on Google Drive and started backing up data himself.

• A couple of days later Sakari Maaranen joined the team. He’s a systems architect at Ubisecure, a Finnish firm, with access to a high-bandwidth connection. He set up a server, he’s downloading lots of data, he showed us how to authenticate it with SHA-256 hashes, and he’s managing many other technical aspects of this project.

There are other people involved too. You can watch the nitty-gritty details of our progress here:

## Azimuth Backup Project (Part 1)

16 December, 2016

This blog page is to help organize the Azimuth Environmental Data Backup Project, or Azimuth Backup Project for short. This is part of the larger but decentralized, frantic and somewhat disorganized project discussed elsewhere:

Saving Climate Data (Part 2), Azimuth, 15 December 2016.

Here I’ll just say what we’re doing at Azimuth.

Jan Galkowski is a statistician and engineer at Akamai Technologies, a company in Cambridge Massachusetts whose content delivery network is one of the world’s largest distributed computing platforms, responsible for serving at least 15% of all web traffic. He has begun copying some of the publicly accessible US government climate databases. On 11 December he wrote:

John, so I have just started trying to mirror all of CDIAC [the Carbon Dioxide Information Analysis Center]. We’ll see. I’ll put it in a tarball, and then throw it up on Google. It should keep everything intact. Using WinHTTrack. I have coordinated with Eric Holthaus via Twitter, creating, per your suggestion, a new personal account which I am using exclusively to follow the principals.

Once CDIAC is done, and checked over, I’ll move on to other sites.

There are things beyond our control, such as paper records, or records which are online but are not within visibility of the public.

Oh, and I’ve formally requested time off from work for latter half of December so I can work this on vacation. (I have a number of other projects I want to work in parallel, anyway.)

By 14 December he was wanting some more storage space. He asked David Tanzer and me:

Do either of you have a large Google account, or the “unlimited storage” option at Amazon?

I’m using WebDrive, a commercial product. What I’m (now) doing is defining an FTP map at a .gov server, and then a map to my Amazon Cloud Drive. I’m using Windows 7, so these appear as standard drives (or mounts, in *nix terms). I navigate to an appropriate place on the Amazon Drive, and then I proceed to copy from .gov to Amazon.

There is no compression, and, in order to be sure I don’t abuse the .gov site, I’m deliberately passing this over a wireless network in my home, which limits the transfer rate. If necessary, and if the .gov site permits, I could hardwire the workstation to our FIOS router and get appreciably faster transfer. (I often do that for large work files.)

The nice thing is I get to work from home 3 days a week, so I can keep an eye on this. And I’m taking days off just to do this.

I’m thinking about how I might get a second workstation in the act.

The Web sites themselves I’m downloading, as mentioned, using HTTrack. I intended to tarball-up the site structure and then upload to Amazon. I’m still working on CDIAC at ORNL. For future sites, I’m going to try to get HTTrack to mirror directly to Amazon using one of the mounts.

I asked around for more storage space, and my request was kindly answer by Scott Maxwell. Scott lives in Pasadena California and he used to work for NASA: he even had a job driving a Mars rover! He is now a site reliability engineer at Google, and he works on Google Drive. Scott is setting up a 10-terabyte account on Google Drive, which Jan and others will be able to use.

Meanwhile, Jan noticed some interesting technical problems: for some reason WebDrive is barely using the capacity of his network connection, so things are moving much more slowly than they could in theory.

Most recently, Sakari Maaranen offered his assistance. Sakari is a systems architect at Ubisecure, a firm in Finland that specializes in identity management, advanced user authentication, authorization, single sign-on, and federation. He wrote:

I have several terabytes worth in Helsinki (can get more) and a gigabit connection. I registered my offer but they [the DataRefuge people] didn’t reply though. I’m glad if that means you have help already and don’t need a copy in Helsinki.

I replied saying that the absence of a reply probably means that they’re overwhelmed by offers of help and are struggling to figure out exactly what to do. Scott said:

Hey, Sakari! Thank you for the generous offer!

I’m setting these guys up with Google Drive storage, as at least a short-term solution.

IMHO, our first order of business is just to get a copy of the data into a location we control—one that can’t easily be taken away from us. That’s the rationale for Google Drive: it fits into Jan’s existing workflow, so it’s the lowest-friction path to getting a copy of the data that’s under our control.

How about if I propose this: we let Jan go ahead with the plan of backing up the data in Drive. Then I’ll look evaluate moving it from there to whatever other location we come up with. (Or copying instead of moving! More copies is better. :-) How does that sound to you?

I admit I haven’t gotten as far as thinking about Web serving at all—and it’s not my area of expertise anyway. Maybe you’d be kind enough to elaborate on your thoughts there.

Sakari responded with some information about servers. In late January, U. C. Riverside may help me with this—until then they are busy trying to get more storage space, for wholly unrelated reasons. But right now it seems the main job is to identify important data and get it under our control.

There are a lot of ways you could help.

Computer skills. Personally I’m not much use with anything technical about computers, but the rest of the Azimuth Data Backup gang probably has technical questions that some of you out there could answer… so, I encourage discussion of those questions. (Clearly some discussions are best done privately, and at some point we may encounter unfriendly forces, but this is a good place for roaming experts to answer questions.)

Security. Having a backup of climate data is not very useful if there are also fake databases floating around and you can’t prove yours is authentic. How can we create a kind of digital certificate that our database matches what was on a specific website at a specific time? We should do this if someone here has the expertise.

Money. If we wind up wanting to set up a permanent database with a nice front end, accessible from the web, we may want money. We could do a Kickstarter campaign. People may be more interested in giving money now than later, unless the political situation immediately gets worse after January 20th.

Strategy. We should talk a bit about what to do next, though too much talk tends to prevent action. Eventually, if all goes well, our homegrown effort will be overshadowed by others, at least in sheer quantity. About 3 hours ago Eric Holthaus tweeted “we just got a donation of several petabytes”. If it becomes clear that others are putting together huge, secure databases with nice front ends, we can either quit or—better—cooperate with them, and specialize on something we’re good at and enjoy.


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# Tag Info

7

Okay so let me give you the intuition. Now you would've heard of cases of a pendulum made of a rod instead of a string. (You can checkout this: http://hyperphysics.phy-astr.gsu.edu/hbase/penrod.html) Okay, so in this case you can clearly see that the torque is taken about the hinge. And you apply gravitational force at the centre of mass and take its torque ...

4

It has to do with angular force balance [torque = R * force(R)] around the center of mass of a rigid body. Those objects that have wider bases have, by definition, larger R at the base to compensate for external torque applied. Those objects that have narrower bases have smaller R at the base to compensate for torque applied, even torque applied by the body'...

4

Why a body does not rotate if force is applied on the centre of mass? The question is: Why does it rotate if it is not applied to the center of a mass? First we look at two rules that students of mechanical engineering (I don't know about physics) learn at university: Multiple forces applied to the same point of a solid body have the same effect as a ...

3

This is easily explained by Newton's second law. If there is no net force applied to a body then the center of mass will not accelerate. it will either be stationary or move in a straight line. The only allowed motion is a rotation about the center of mass. Changes in entropy, arise from the exchange of energy at some temperature and has nothing to do with ...

3

Based on your free body diagram, yes, the center of mass would not experience any acceleration, but the object would rotate about the center due to the torque from force $f$. However, based on what you have drawn, perhaps you have misunderstood how the forces are at work here. It looks like to me that you have a ball, (or disk, etc.) that is on an incline. ...

3

The 3D motion of a rigid body requires some familiarity with vector and matrix operations. The rotation about the center of mass does not have an analytical solution. Suppose at $t=0$ the body has orientation matrix $\mathbf{R}_0$ as well as rotational velocity $\boldsymbol{\omega}_0$. This means the 3×3 mass moment of inertia tensor initially is calculated ...

2

In this particular type of problem, the mechanical energy, E, is constant because no external work is done on the system. What you seem to quote ("They write ...") is not an accurate copy. Rather, they write $$E = \frac{1}{2}m\left(\dot{r}^2+r^2\dot{\phi}^2\right)-\frac{GmM}{r}.$$ That makes all the difference in the world because, when you change ...

2

In this answer I am referring to top position only.Similar arguments can be made for any position. Well think it like this.You know that radius of curvature of circular motion is given as: R= F/(m ω^2) where F is net force. Notice if we keep angular speed decreases, radius decreases. However you know that such a thing is not happening here.The reason of this ...

2

Basically, if we know the concrete is traveling in a circle with frequency $\omega$, we know its trajectory - that is, its position vector as a function of time. So, by taking the second derivative of that function, we know its acceleration. For circular motion, that works out to be a vector of magnitude $R\omega^2$, always pointing toward the center of ...

2

One solution is the above. The other one is from the rotation at the point when the spool touching the ground. For simplicity, let $\alpha = 0$: kinetic relation: $a_{||} = R \frac{d \omega}{dt}$ Newton law in rotation: $F(R-r) = (\gamma + 1) R^2\frac{d \omega}{dt}$ So that: $a_{||} = \frac{1- \frac{r}{R}}{\gamma + 1} \frac{F}{m}$

2

You appear to be missing the concept of static friction and thinking only of kinetic friction. Kinetic friction is the friction between two objects in relative motion, also called slipping. This friction force has a magnitude $F_{kinetic}=\mu_k N$ where $N$ is the magnitude of the normal force and $\mu_k$ is the coefficient of kinetic friction. The direction ...

2

Your statements do seem to apply to a rolling object. If there is acceleration, the friction provides the torque required for the corresponding angular acceleration.

2

The short answer is that when you write $$T = \frac{1}{2}M v_{CM}^2 + \frac{1}{2}I\omega^2$$ you are assuming that the moment of inertia is being calculated about the center of mass. It's true that the moment of inertia can be calculated about whatever axis you'd like; however, if you don't choose the center of mass, then the expression for the total ...

2

The rotation of a body depends upon the point at which it is hinged. So if a force applied at the center of mass (COM) then it may cause rotation about the axis (if it's not the one through COM) at which it is hinged by causing a torque to act. But what about the case when it isn't hinged at any point with the force still acting on the COM? A torque still ...

2

Although the quaternion has 4 parameters, it really has 3 degrees of freedom, since it must obey the unity condition $\sqrt{x^2+y^2+z^2+w^2}=1$. Also note that $\boldsymbol{\omega} \neq \boldsymbol{\dot q}$ and the same applied for their derivatives also. Baseline Method So the process goes like this Given known orientation $\boldsymbol{q}$ calculate the 3×...

1

With a large base, the force from the supporting surface can shift further away from a point directly below the center of gravity when necessary to oppose any external tipping torque. Any good physics text has a section on “Statics”.

1

Because force is the time derivative of momentum, and momentum is linked to the motion of the center of mass. If you consider a rigid body as a collection of particles glued together and their position split into the position of the center of mass $\boldsymbol{r}_{\rm COM}$ plus some other relative position $\boldsymbol{d}_i$, then $$\boldsymbol{r}_i = \... 1 Let \vec r_1 denote the position of the centre of mass of an object of mass m, given by the formula below.$$\vec r_1 = \frac{1}{m}\int \rho \vec r^\prime \mathrm{d^3} \vec r^\prime$$If an object is not rotating, then all of its points must have the same acceleration. Therefore, if a single force applied to the object does not cause rotation, it must be ... 1 They should get displaced by same angle, as the torque equations would be same in both cases. 1 After sliding down through an angle θ without friction, the kinetic energy of each bead is: (1/2)mv^2 = mgR(1 - cos θ). To move along the circular ring requires a centripetal force: mv^2/R = mg cos(θ) + N. The reaction force on the ring is N = 2mg(1 – cos θ) – mg cos(θ). The lift provided by the vertical components of N from both beads (while θ is ... 1 I don't think you can avoid this singularity by using Quaternions because they are good for rotation . but for numerical simulation you can use this concept: your geodetic equations are:$$\ddot{\theta}=-2\,{\frac {\dot{\theta}\dot{\phi}\cos \left( \phi \right) }{\sin \left( \phi \right) }} \tag 1$$and$$\ddot{\phi}={\frac {\sin \left( \phi \right) \left( ...

1

It's not just pendulums. It's a major issue for spacecraft navigation, especially if, for instance, your trajectory involves the deployment of a supersonic parachute on Mars. Hence: quaternions are the standard. They are also computationally faster, which can be a factor with space-qualified CPUs, which are not fast.

1

Consider the bicycle and cyclist as a system. If the bike is on a flat road, there are two relevant external forces: Friction with the ground pushes the bicycle forward. Because the velocity of the wheel's contact point with the ground is zero, the power supplied by friction is exactly zero. Sources of dissipation, such as air resistance, do negative work ...

1

It's a matter of identifying your systems. As you mentioned, power is derived from force; hence, just like force, power is delivered from one system to another system. You can talk about the power from the person or the ground or both together, and to the wheels or the bicycle as a whole. Since the wheels are rotating, you apply the torque formula to them, ...

1

we cannot apply it about centre of mass as external torque about centre of mass is not 0. Is my reasoning correct? Once the ball enters pure rolling friction force stops acting hence you can apply conservation of angular momentum but before that you can't. Now after starting rolling, it collides with a wall inelastically so that linear velocity becomes say ...

1

In the formula $$\vec{L} = \vec{L'}+M\vec{r}_{cm}\times\vec{v}_{cm}$$ Which represents the total angular momentum we have two parts/components: First let's look at the second term, which is $M\vec{r}_{cm}\times\vec{v}_{cm}$: This represents only the linear motion of the center of mass. Take the vector from the origin to the COM and call it $r_{cm}$ and take ...

1

The formula $$\mathbf L=\mathbf L' +M\mathbf r\times\mathbf v_{\rm com}$$ is used only for finding the value of angular momentum with respect from different locations. It is not a dynamical formula explaining any dynamics of the system. So it can't give you any relation between the torque applied and the subsequent angular momentum. So the answer of How ...

1

The motion of the body can be modeled by the system of first-order ordinary differential equations in vector/matrix: \begin{align} J&\frac{d\vec{\Omega}}{dt} \, = \, J\,\vec{\Omega} \times \vec{\Omega} \, + \, \vec{X}_F\times \vec{F}\\ &\\ &\frac{d}{dt} U \, = \, U\, \big(\vec{\Omega} \times \cdot\big)\\ &\\ &\frac{d\vec{x}_G}{dt} \, = \,...

1

When it will be in equilibrium, it will have some angular velocity and due to inertia it will continue to rotate till the work done is completely converted to potential energy.

Only top voted, non community-wiki answers of a minimum length are eligible


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Journal de Théorie des Nombres de Bordeaux Institut de Mathématiques de Bordeaux Université Bordeaux 1 351, cours de la Libération F-33405 TALENCE Cedex 5 Version française | JTNB Home |
 ISSN 1246-7405

 Electronic version The summary and abstracts are put online as soon as an issue is published. About eight months later, the full text of papers becomes freely accessible and can be downloaded. CONTENTS OF VOLUME 16 no. 1 (2004) D. Barsky, B. Benzaghou Nombres de Bell et somme de factorielles - Abstract K. Belabas Topics in computational algebraic number theory - Abstract J.-F. Burnol Two complete and minimal systems associated with the zeros of the Riemann zeta function - Abstract I. Chen, B. De Smit, M. Grabitz Relations between jacobians of modular curves of level $p2$ - Abstract J. von zur Gathen, A. Knopfmacher, F. Luca, L. Lucht, I. E. Schparlinski Average order in cyclic groups - Abstract J.-P. Gazeau, J.-L. Verger-Gaugry Geometric study of the beta-integers for a Perron number and mathematical quasicrystals - Abstract A. Heinis Languages under substitutions and balanced words - Abstract J. Holden Abelian varieties over finite fields with a specified characteristic polynomial modulo l - Abstract G. Khare Modularity of p-adic Galois representations via p-adic approximations - Abstract G. Nebe Strongly modular lattices with long shadow - Abstract W.G. Nowak, J. Schoissengeier The distribution of powers of integers in algebraic number fields - Abstract B.K. Spearman, K.S. Williams Normal integral bases for Emma Lehmer's parametric family of cyclic quintics - Abstract J. Steuding Extremal values of Dirichlet L-functions in the half-plane of absolute convergence - Abstract J.F. Voloch On some subgroups of the multiplicative group of finite rings - Abstract T. Zaimi On an approximation property of Pisot numbers II - Abstract W. Zudilin Arithmetic of linear forms involving odd zeta values - Abstract Contents of the previous issue Contents of the next issue


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# Height of atmosphere!!

Classical Mechanics Level pending

Assume that atmosphere of a planet is in hydro-static equilibrium and all air process occurring in the atmosphere are adiabatic.The gas of the atmosphere behaves as an ideal gas. The gravitation constant is $$g=9.8\text{ms}^{-2}$$.

The temperature of the surface of planet is $$T_{0} = 300$$ K. Assume the atmosphere is made of diatomic gas of molar mass $$M=29$$.

Find the height of the atmosphere of the planet in meters.

Details

• Take $$R=8.314\text{JK}^{1-}\text{mol}^{-1}$$
×


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This article presents a study on common design challenges of large and small turbofans. Turbofan engines powering large transport aircraft have demonstrated much different design objectives than business-jet turbofans including thrust, range, mission type, development cost, unit price, maintainability standards, and production quantities. Prolific use of ‘thermal barrier coating’ has helped turbine designers compensate for the inability to distribute a large quantity of small diameter film holes over the turbine blade surface. The historical trends in overall pressure ratio observed for both large and small turbofans have parallel slopes. Small turbofans lag behind the larger engines due to the miniaturization required for low flowrates characteristic of the smaller engines. These trends are qualitatively demonstrated, showing the growth in both the overall engine pressure ratio and turbine inlet temperature for several decades. It has been noted in this paper that the importance of high-performance impeller designs and intricate turbine blade cooling concepts for very low compressor exit corrected flows has not yet been fully appreciated.

## Article

Historically, turbofan engines powering large transport aircraft have demonstrated much different design objectives than business-jet turbofans including thrust, range, mission-type, development cost, unit-price, maintainability standards, and production quantities. In the 1960’s and 1970’s, commercial transports incorporated 3 or 4 engines; they competed with propliners, targeted transcontinental or trans-Atlantic markets, and operated under the expectation of low fuel costs without emissions restrictions. During that era, business travel largely utilized “repurposed” military transports—even bombers! After the 1973 Arab oil embargo, and with the advent of government scrutiny on auto and aircraft emissions, jet engine manufacturers began to focus heavily on fuel economy and emissions. The Environmental Protection Agency was founded in 1970, and the first aircraft engine smoke standards were defined in 1973. In the mid 1960’s, two functional parameters became the “guideposts” of turbofan engine architecture: “Bypass Ratio” (BPR) and “Thrust-Specific-Fuel-Consumption” (TSFC). By definition, “bypass ratio” is simply the net mass flowrate of air entering the fan which does not enter the engine core divided by the mass flowrate of air which does enter the core. “TSFC” is defined as the engine fuel mass flowrate divided by the propulsive thrust. Typically, engines with larger bypass ratios deliver superior TSFC and, incidentally, improved (reduced) emissions.

The earliest propulsion gas turbines were referred to as “turbojets”. All of the engine inlet air was directed to the core, with thrust being generated by the change in momentum resulting from the change in gas velocity from the engine inlet to the core exhaust nozzle. The engine itself was a “core”; there was no bypass flow. Therefore, the bypass ratio of these turbojets was zero. Brayton Cycle efficiency is largely influenced by driving higher cycle pressure ratios and turbine inlet temperatures. Due to thermal and stress limitations for turbine components, internal pressures and temperatures for these early turbojets were very modest, such that all the work derived by the turbomachinery was consumed by driving the compressor without any excess power available to drive a fan. TSFC for a P&W JT3C (Boeing 707-100) engine at cruise conditions was about 0.9. Interestingly, the last generation of aircraft piston engines (e.g. the British Napier Nomad Diesel), with gear-driven slow- turning propellers, utilized bypass ratios ~ 100 and could demonstrate an effective TSFC ~ 0.45 at cruise conditions (assuming propeller propulsive efficiencies of about 0.85). In the mid 1950’s, advanced aircraft piston engines weighed twice as much as the emerging turbojets. However, since the Boeing 707 maintained a long-range cruise speed almost “double” that of the Boeing 377, its piston-engine-powered predecessor, the range and total fuel consumption of the jet versus the propliner were about the same. But the 707 got there twice as fast! If the core of the jet engine could be made more “energetic”, i.e., more powerful, then, possibly, the bypass ratio could be increased and TSFC reduced. From Figure 1, the TSFC benefits of higher bypass ratios can be visualized. A documentary could be written describing the relentless quest for increased bypass ratio enabled by engine cores capable of sustaining elevated compressor pressure ratios and turbine inlet temperatures.

But how have the technologies required to deliver these energetic cores differed for large commercial transports compared to small business jets? Increasing compressor pressure ratio and/or “intercooling” drives a reduction in flowpath annular area, meaning compressor-discharge flow areas are reduced. At any thermodynamic station in the gas turbine the “corrected” flow is described by the relation:

$Wcor=Wmass X Tgas°R/5190.5Pstaticlbf/in2/14.696$

For a given mass flowrate (Wmass) a reduction in gas temperature (Tgas) or an increase in static pressure (Pstatic) will reduce the local corrected flow, Wcor. This effect is great for making engines smaller, improving cycle efficiency, and for enabling more energetic engine cores to allow higher fan bypass ratios. Conversely, though, as flowpath components become more “miniature”, a heavy burden results on such design features as rotor-blade-tip-clearance and turbine- blade-cooling-passages. Manufacturing tolerances and the nature of “surface-to-volume” ratios (for turbine blade cooling) become critical for preserving aerodynamic performance and component durability. To prevent dangerous and efficiency-robbing compressor surge-and- stall phenomena, manufacturers of large, modern, high- pressure-ratio-engines have resorted to complex, active clearance control features in the last stages of the compressor and in the high-pressure turbine. But manufacturers of smaller, less-costly and less-sophisticated gas turbines, such as APU’s and small business-jet engines, have advanced the implementation of surge-resistant centrifugal compressors (impellers) to replace the last 2 or 3 stages of a multi-stage axial compressor system. These “small” gas turbines typically operate with Wcor<5 lbm/sec.

Added to the requirement for managing surge and stall, flowpath area reductions present challenges to the design, fabrication and durability aspects of cooled high pressure turbine airfoils and disks. Surface-to-volume ratio scales approximately by L(-20) where “L” relates to the flowpath length scale. Figure 2 depicts how aggressively the surface- to-volume varies with turbine corrected flow. At constant turbine inlet temperature, increasing the core pressure ratio results in an increase in the turbine blade surface-to- volume ratio. Higher surface-to-volume ratio, in turn, leads to a requirement for a higher percentage of cooling flow to maintain blade metal temperature. Further complicating the turbine cooling challenge is the fact that film cooling holes distributed over the surface of the airfoil suffer from blockage effects caused by sand ingestion if cooling hole diameters are smaller than about 0.012”. So, to avoid allocating too much cooling air, an economical distribution of small film holes is desirable but restricted by hole diameter. Prolific use of “thermal barrier coating” (TBC) has helped turbine designers compensate for the inability to distribute a large quantity of small diameter film holes over the turbine blade surface.

The historical trends in overall pressure ratio observed for both large and small turbofans have parallel slopes. Small turbofans lag behind the larger engines due to the miniaturization required for low flowrates characteristic of the smaller engines. These trends are qualitatively demonstrated in Figures 3 and 4, showing the growth in both the overall engine pressure ratio and turbine inlet temperature for several decades.

In Figures 3 and 4 we find the latest large commercial turbofan engines demonstrating Overall Pressure Ratios (OPR) exceeding 50:1, with engines in the 2020’s anticipating OPR’s greater that 60:1! Compressor technology developments at GE/CFM with LEAP, at Pratt and Whitney with the Geared Turbofan (GTF), and at Rolls Royce with Advance and UltraFan may push OPR beyond 70:1! These 70,000 to 100,000 lbf thrust engines will have compressor-exit corrected flows that are actually driving down to levels currently sustained by much smaller business jet engines. Figure 5 captures these trends. Bypass ratios on these high OPR engines are expected to approach 15:1.

The importance of high performance impeller designs and intricate turbine blade cooling concepts for very low compressor- exit corrected flows has not yet been fully appreciated!

This article was inspired by a presentation delivered by Mr. Jim Kroeger, at the 2015ISABE conference. Mr. Kroeger is the Director of Military and Commercial Aircraft Engine Projects at Honeywell Aerospace.