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Learn about walking abnormalities and what causes them. Meaning of ice. Anyone who has been raised solely in Orre knows to avoid the deserts during the night, not just because of the sheer cold temperature, but because these "Dune Demons", as they're called, will jump up from their sandy burrows and attack and maim anything that moves, sometimes even their own out of blind, unbridled aggression. Like the fill … The site has become a favorite resource of teachers of reading, spelling, and English as a second language. The water displacement method conveniently solves this problem. Tomato; Peppers; Eggplant; Cucumber; Squash; Watermelon; Melon; Rootstock; Cabbage Oxford Word Skills Idioms and Phrasal Verbs Advanced. +90 539 522 4054; [email protected]; Toggle navigation. v Glossar ii Second foreword Welcome to the IOM Glossary on Migration. The curvature measures how fast a curve is changing direction at a given point. The record, which is called an electrocardiogram (ECG or EKG), shows the series of waves that relate to the electrical impulses which occur during each beat of the heart. haunch: The hip, buttock, and upper thigh in humans and certain other animals. As you can see, smooth works as both a verb and an adjective. Follow Chris Ciaccia on Twitter @Chris_Ciaccia. Section 1-10 : Curvature. How to use ramble in a sentence. However, the volume of irregularly shaped objects cannot be easily determined. Here's information on their symptoms, diagnosis, treatment, and prevention. Candle flame spiritual meanings.While burning, candles also speak a language of their own, which you can learn to interpret with some effort. Ultima Thule is located in the Kuiper belt in the outermost regions of the Solar System, beyond the orbit of Neptune. It’s not recommended to have more than 400 mg of caffeine in a day. If your wooden table is giving you splinters, use a sander to smooth it down and make it smooth. 19. An irregularly shaped pool filled with //fill. The C81 QLED’s key screen specifications make it difficult to predict how it will perform. Choppy definition, (of the sea, a lake, etc.) 8. You’re also more prone to fall as it’s hard to keep your balance. Source: Adapted from International Detention Coalition, There Are Alternatives: A Handbook for Preventing Unnecessary Immigration Detention (revised edition, 2015) p. What does ice mean? Not knowing how Oxford Word Skills Idioms and Phrasal Verbs Advanced You’ll need something to reduce the vibrations because after a while your feet will start to hurt and you might feel a tingling sensation. Turntables and Vinyl: Part 7. The spectral (or frequency) distribution of the energy in this particular sound wave is represented by the frequency spectrum of Figure 1.4d. Elliott Surveying provided free estimates to to the following … ... the magnet and the coils remain stationary, and the stylus moves a small piece of iron instead. It is professional enough to satisfy academic standards, but accessible enough to be used by anyone. Macula definition, a spot or blotch, especially on one's skin; macule. 200 Pages. The online etymology dictionary is the internet's go-to source for quick and reliable accounts of the origin and history of English words, phrases, and idioms. Tusk(タスク (牙),Tasuku, sometimes written as 爪) is the Stand of Johnny Joestar, featured in Steel Ball Run. would definition: 1. used to refer to future time from the point of view of the past: 2. used to refer back to a…. For such complex processes, controlled experiments help clarifying the roles of chemical, mechanical and biological cues. forming short, irregular, broken waves. Rheological and geometric effects in cementing of irregularly shaped wells Alondra Renteria1, Amir Maleki1, Ian Frigaard1,2 ... meaning that to get good cov-erage of cement ... moves the narrow side mud much more ef-fectively. See more. Ramble definition is - to move aimlessly from place to place. Put like that, it sounds as if this document would solve all issues related to how we speak about, address and find solutions to concerns around migration. beyond-archigram-the-structure-of-circulation 1/1 Downloaded from www.gettinguxdone.com on January 21, 2021 by guest [Books] Beyond Archigram The Structure Of Circulation As recognized, adventure as skillfully as experience practically lesson, amusement, as skillfully as pact can be gotten by just checking out a book beyond archigram the structure of … The stone is often referred to as a piece of glass. It is artificial, Smoking See more. Information and translations of ice in the most comprehensive dictionary definitions resource on the web. There are several formulas for determining the curvature for a curve. Finding irregular bowel movement treatment can be quite troublesome, especially when it is far too late to do anything about your symptoms. With the image formatting and layout tools in Microsoft Word, it’s possible to add and manipulate pictures in your document—even if you’re a beginner. Electrocardiography is the process of producing an electrocardiogram (ECG or EKG).It is a graph of voltage versus time of the electrical activity of the heart using electrodes placed on the skin. The amount by which the water level rises when an object is submerged is equivalent to the volume of the object. Pregnant women should either avoid it or restrict consumption to 300 mg in a day. Opalite is one of the most efficient stones in terms of pricing and availability. You must have wondered why that candle is acting so strange, well below is the explanation of what a flame does.. Keep an eye on! Learn more. NASA has renamed the distant space rock after it was criticized over the old name’s Nazi connections. Definition of ice in the Definitions.net dictionary. Sometimes you can even see the pulse as it moves the skin in a more forceful way. It moves through the aquatic environment similarly to arsenic and antimony; its water-soluble salts, in higher concentrations, have a similar toxicological profile to that of arsenic. Back to Part 6. Similar to Part 4's Echoes, Tusk takes various forms, or "ACT"s. Like other Stands such as Dragon's Dream or Cinderella, Tusk's abilities are intimately linked with a supernatural phenomenon inherent to the world of JoJo's Bizarre Adventure, namely the Spin. In this section we want to briefly discuss the curvature of a smooth curve (recall that for a smooth curve we require $$\vec r'\left( t \right)$$ is continuous and $$\vec r'\left( t \right) \ne 0$$). Something smooth is free of roughness, stubble, or other imperfections that you can feel with your hands. It will not. Alternatives to detention – Any legislation, policy or practice, formal or informal, aimed at preventing the unnecessary detention of persons for reasons relating to their migration status. A wood shaper is a functionally different woodworking tool, typically with a powered rotating cutting head and manually fed workpiece, usually known simply as a shaper in North America and spindle moulder in the UK.. A metalworking shaper is somewhat analogous to a metalworking planer, with the cutter riding a ram that moves relative to a stationary workpiece, rather than the … Russell Elliott, provides quality land surveys, property boundary surveys, subdivision surveys, mortgage surveys, topographic maps and construction surveys. These electrodes detect the small electrical changes that are a consequence of cardiac muscle depolarization followed by repolarization during each cardiac cycle (heartbeat). NASA has renamed the distant space rock after it was criticized over the old name’s Nazi connections. Excessive consumption of caffeine can cause your heart to beat irregularly. If you can’t ride smoothly it’s harder to do some tricks as your skateboard If the piston moves irregularly but cyclically, for example, so that it produces the waveform shown in Figure 1.4c, the resulting sound field will consist of a combination of sinusoids of several frequencies. electrocardiography: Definition Electrocardiography is a commonly used, noninvasive procedure for recording electrical changes in the heart. For your reference, an 8-ounce cup of coffee has around 95 to 200 mg of caffeine. Home; Products. We also survey sections in the public land survey system, PLSS and ALTA, American Land and Title Association, SIR, Survey Inspection Reports and As-built surveys. a very irregularly-updated blog on random topics. More than ever, pictures are critical to the effectiveness of your communication materials. 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text:
# Labeling rows/columns of a matrix with text and braces

I would like to label rows and columns of my matrix with formulas and text. I've started with a version using bordermatrix.

$\underbrace{ \bordermatrix{ & \underset{\downarrow}{x} & \underset{\downarrow}{c} \cr (17') \rightarrow & \dfrac{n}{\eta \cdot \delta} & \dfrac{1}{\lambda} \cr (16') \rightarrow & \dfrac{1-s_R}{\delta} & - \dfrac{\frac{\lambda-1}{\lambda}}{\rho + \frac{n}{\eta} - n} } }_{\text{coefficient matrix } \Delta} \cdot \begin{pmatrix} \widetilde{x} \\ \widetilde{c} \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$

But furthermore I need to

• only apply the 1 big underbrace for the main matrix
• apply similar braces for single rows and columns (additional or instead of the rightarrow/downarrow)

Thank you very much for advice.

Here is one option using \bordermatrix that provides what you're after:

\documentclass{article}

\usepackage{mathtools}

\begin{document}
$\phantom{(17') \rightarrow\hspace{\arraycolsep}}% For correct horizontal spacing within display math \[ ...$
\underbrace{
\bordermatrix{\hspace{-\arraycolsep} &
\overbrace{\hphantom{\dfrac{1-s_R}{\delta}}}^{x} &
\overbrace{\hphantom{- \dfrac{\frac{\lambda-1}{\lambda}}{\rho + \frac{n}{\eta} - n}}}^{c} \cr
\hspace{-\arraycolsep}\mathllap{(17') \rightarrow} & \dfrac{n}{\eta \cdot \delta} & \dfrac{1}{\lambda}   \cr
\hspace{-\arraycolsep}\mathllap{(16') \rightarrow} & \dfrac{1-s_R}{\delta} & - \dfrac{\frac{\lambda-1}{\lambda}}{\rho + \frac{n}{\eta} - n}
}
}_{\text{coefficient matrix $\Delta$}}
\cdot
\begin{pmatrix}
\widetilde{x} \\
\widetilde{c}
\end{pmatrix} =
\begin{pmatrix}
1 \\ 0
\end{pmatrix}
\]

\end{document}

\overbraces are stretched to the widest element in each column using \hphantom, while care is taken to remove the left-most row indices from the \underbrace yet restoring their horizontal space outside of the \bordermatrix construction.

For having left braces to identify/mark the rows, you could use the following:

$\phantom{(17') \left\{\vphantom{\dfrac{n}{\eta \cdot \delta}}\right.\hspace{\dimexpr\arraycolsep-\nulldelimiterspace}}% For correct horizontal spacing within display math \[ ...$
\underbrace{
\bordermatrix{\hspace{-\arraycolsep} &
\overbrace{\hphantom{\dfrac{1-s_R}{\delta}}}^{x} &
\overbrace{\hphantom{- \dfrac{\frac{\lambda-1}{\lambda}}{\rho + \frac{n}{\eta} - n}}}^{c} \cr
\hspace{-\arraycolsep}\mathllap{(17') \left\{\vphantom{\dfrac{n}{\eta \cdot \delta}}\right.\kern-\nulldelimiterspace} &
\dfrac{n}{\eta \cdot \delta} & \dfrac{1}{\lambda}   \cr
\hspace{-\arraycolsep}\mathllap{(16') \left\{\vphantom{- \dfrac{\frac{\lambda-1}{\lambda}}{\rho + \frac{n}{\eta} - n}}\right.\kern-\nulldelimiterspace} &
\dfrac{1-s_R}{\delta} & - \dfrac{\frac{\lambda-1}{\lambda}}{\rho + \frac{n}{\eta} - n}
}
}_{\text{coefficient matrix $\Delta$}}
\cdot
\begin{pmatrix}
\widetilde{x} \\
\widetilde{c}
\end{pmatrix} =
\begin{pmatrix}
1 \\ 0
\end{pmatrix}
\]

• Wow ok, this looks good for the colomns. It will take me some time to bring those phantom-braces to the rows :)
– Mac
Mar 8, 2016 at 21:14
• @Mac: See my updated answer...
– Werner
Mar 8, 2016 at 21:21
• Wow perfect, this even troubles me with more headache :) If I copy your code the lower big brace doesn't fit yet, and I need some time to figure out the problem. But the left and upper braces are perfect.
– Mac
Mar 8, 2016 at 21:29
• Sorry my mistake, my current .tex file missed the mathtools packages. Now everything is fine and the learning process will begin :) thx
– Mac
Mar 8, 2016 at 21:32

format: plain


text:
## Results (1-50 of at least 1000)

Next
Label $\alpha$ $A$ $d$ $N$ $\chi$ $\mu$ $\nu$ $w$ prim arith $\mathbb{Q}$ self-dual $\operatorname{Arg}(\epsilon)$ $r$ First zero Origin
2-980-1.1-c1-0-0 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 1.1 $$1.0 1 0 0 0.595071 Elliptic curve 980.a Modular form 980.2.a.a Modular form 980.2.a.a.1.1 2-980-1.1-c1-0-1 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 1.1$$ $1.0$ $1$ $0$ $0$ $0.763760$ Elliptic curve 980.b Modular form 980.2.a.b Modular form 980.2.a.b.1.1
2-980-1.1-c1-0-10 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 1.1 $$1.0 1 0.5 1 1.73203 Elliptic curve 980.c Modular form 980.2.a.c Modular form 980.2.a.c.1.1 2-980-1.1-c1-0-11 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 1.1$$ $1.0$ $1$ $0.5$ $1$ $2.16393$ Elliptic curve 980.f Modular form 980.2.a.f Modular form 980.2.a.f.1.1
2-980-1.1-c1-0-12 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 1.1 $$1.0 1 0.5 1 2.24009 Modular form 980.2.a.j.1.2 2-980-1.1-c1-0-2 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 1.1$$ $1.0$ $1$ $0$ $0$ $0.894208$ Modular form 980.2.a.k.1.1
2-980-1.1-c1-0-3 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 1.1 $$1.0 1 0 0 0.982426 Elliptic curve 980.d Modular form 980.2.a.d Modular form 980.2.a.d.1.1 2-980-1.1-c1-0-4 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 1.1$$ $1.0$ $1$ $0$ $0$ $1.01545$ Elliptic curve 980.e Modular form 980.2.a.e Modular form 980.2.a.e.1.1
2-980-1.1-c1-0-5 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 1.1 $$1.0 1 0 0 1.14605 Elliptic curve 980.g Modular form 980.2.a.g Modular form 980.2.a.g.1.1 2-980-1.1-c1-0-6 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 1.1$$ $1.0$ $1$ $0$ $0$ $1.36532$ Modular form 980.2.a.k.1.2
2-980-1.1-c1-0-7 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 1.1 $$1.0 1 0 0 1.37767 Elliptic curve 980.h Modular form 980.2.a.h Modular form 980.2.a.h.1.1 2-980-1.1-c1-0-8 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 1.1$$ $1.0$ $1$ $0$ $0$ $1.60608$ Elliptic curve 980.i Modular form 980.2.a.i Modular form 980.2.a.i.1.1
2-980-1.1-c1-0-9 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 1.1 $$1.0 1 0.5 1 1.63176 Modular form 980.2.a.j.1.1 2-980-140.107-c1-0-0 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $0.299$ $0$ $0.00930197$ Modular form 980.2.x.n.667.5
2-980-140.107-c1-0-1 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 -0.480 0 0.0286113 Modular form 980.2.x.k.667.13 2-980-140.107-c1-0-10 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $0.387$ $0$ $0.234247$ Modular form 980.2.x.l.667.6
2-980-140.107-c1-0-100 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 0.470 0 1.89508 Modular form 980.2.x.n.667.2 2-980-140.107-c1-0-101 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $0.181$ $0$ $2.04838$ Modular form 980.2.x.n.667.24
2-980-140.107-c1-0-102 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 -0.385 0 2.09925 Modular form 980.2.x.k.667.6 2-980-140.107-c1-0-103 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $0.217$ $0$ $2.13842$ Modular form 980.2.x.m.667.18
2-980-140.107-c1-0-104 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 0.450 0 2.34835 Modular form 980.2.x.l.667.14 2-980-140.107-c1-0-105 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $0.459$ $0$ $2.42894$ Modular form 980.2.x.j.667.14
2-980-140.107-c1-0-106 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 0.420 0 2.46214 Modular form 980.2.x.k.667.16 2-980-140.107-c1-0-107 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $0.389$ $0$ $2.53423$ Modular form 980.2.x.l.667.18
2-980-140.107-c1-0-108 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 -0.450 0 2.60490 Modular form 980.2.x.j.667.4 2-980-140.107-c1-0-109 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $-0.447$ $0$ $2.74162$ Modular form 980.2.x.n.667.22
2-980-140.107-c1-0-11 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 -0.371 0 0.241017 Modular form 980.2.x.n.667.17 2-980-140.107-c1-0-110 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $-0.315$ $0$ $2.84692$ Modular form 980.2.x.m.667.11
2-980-140.107-c1-0-111 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 -0.247 0 2.98062 Modular form 980.2.x.n.667.18 2-980-140.107-c1-0-12 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $-0.402$ $0$ $0.257769$ Modular form 980.2.x.l.667.10
2-980-140.107-c1-0-13 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 0.321 0 0.310896 Modular form 980.2.x.m.667.15 2-980-140.107-c1-0-14 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $0.255$ $0$ $0.321615$ Modular form 980.2.x.k.667.12
2-980-140.107-c1-0-15 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 -0.322 0 0.323639 Modular form 980.2.x.k.667.14 2-980-140.107-c1-0-16 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $0.493$ $0$ $0.334694$ Modular form 980.2.x.m.667.8
2-980-140.107-c1-0-17 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 -0.264 0 0.353858 Modular form 980.2.x.k.667.1 2-980-140.107-c1-0-18 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $-0.312$ $0$ $0.361349$ Modular form 980.2.x.k.667.5
2-980-140.107-c1-0-19 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 -0.0651 0 0.396784 Modular form 980.2.x.l.667.11 2-980-140.107-c1-0-2 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $0.0411$ $0$ $0.0765190$ Modular form 980.2.x.n.667.13
2-980-140.107-c1-0-20 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 -0.102 0 0.410889 Modular form 980.2.x.j.667.10 2-980-140.107-c1-0-21 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $-0.262$ $0$ $0.447043$ Modular form 980.2.x.n.667.11
2-980-140.107-c1-0-22 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 -0.324 0 0.461756 Modular form 980.2.x.n.667.21 2-980-140.107-c1-0-23 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $0.304$ $0$ $0.518849$ Modular form 980.2.x.n.667.23
2-980-140.107-c1-0-24 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 0.447 0 0.520399 Modular form 980.2.x.j.667.7 2-980-140.107-c1-0-25 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $0.292$ $0$ $0.526950$ Modular form 980.2.x.l.667.13
2-980-140.107-c1-0-26 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 -0.410 0 0.530898 Modular form 980.2.x.m.667.2 2-980-140.107-c1-0-27 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $-0.182$ $0$ $0.555012$ Modular form 980.2.x.j.667.5
2-980-140.107-c1-0-28 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 -0.349 0 0.561132 Modular form 980.2.x.l.667.4 2-980-140.107-c1-0-29 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $-0.211$ $0$ $0.562601$ Modular form 980.2.x.g.667.2
2-980-140.107-c1-0-3 $2.79$ $7.82$ $2$ $2^{2} \cdot 5 \cdot 7^{2}$ 140.107 $$1.0 1 -0.491 0 0.122712 Modular form 980.2.x.l.667.1 2-980-140.107-c1-0-30 2.79 7.82 2 2^{2} \cdot 5 \cdot 7^{2} 140.107$$ $1.0$ $1$ $-0.376$ $0$ $0.566612$ Modular form 980.2.x.j.667.15
Next


text:
7R10. Applied Linear Optimal Control: Examples and Algorithms. - AE Bryson (Dept of Aeronaut and Astronaut, Stanford Univ, Stanford CA). Cambridge UP, Cambridge, UK. 2002. 362 pp. Softcover, CD-Rom incl. ISBN 0-521-01231-7. $45.00. (Also available in Hardcover ISBN 0-521-81285-2,$120.00.)

Reviewed by M Reyhanoglu (Dept of Phys Sci, Embry-Riddle Aeronaut Univ, 600 S. Clyde Morris Blvd, Daytona Beach FL 32114).

This book updates and extends part of the material in Applied Optimal Control by Bryson and Ho 1. In particular, it focuses on linear optimal control in the presence of uncertainties, including random inputs, measurement errors and model uncertainties. It completes the picture that has begun with the author’s book Dynamic Optimization2, a successor of the first part of 1, which deals with the deterministic (nonrandom) case.

The book commences in Chapter 1 with an overview of random variables and static estimation. In Chapter 2, discrete and continuous linear Gauss-Markov processes are introduced. Chapter 3 treats discrete and continuous filtering. Both time-invariant and time-varying Kalman and Kalman-Bucy filters are discussed and new software codes are presented. Chapter 4 covers smoothing without control inputs and discusses both batch and recursive algorithms. Chapter 5 is devoted to the synthesis of time-varying linear quadratic followers and terminal controllers for deterministic systems using state feedback. Chapter 6 treats linear-quadratic-Gaussian (LQG) controllers, which use output measurements instead of full state feedback. Chapter 7 extends the ideas in Chapter 4 to include smoothing of data from runs of controlled plants. Both batch and recursive algorithms are presented and demonstrated on examples. Chapter 8 presents discrete and continuous time-invariant filters. Chapters 9 and 10 treat time-invariant linear-quadratic (LQ) state feedback controllers and LQG controllers, respectively. Chapter 11 presents discrete and continuous linear-quadratic worst-case (LQW) controllers and estimators. Chapter 12 develops a parameter-robust LQG controller design method based on minimizing the maximum quadratic performance index at the corners of a specified plant parameter space. The resulting controllers are shown to achieve best performance for a specified plant-parameter robustness. The book concludes with two appendices: one appendix covers filters and controllers with colored measurement noise, and the other appendix contains mathematical models of plants used in the examples and problems.

The book is written as a theoretical and practical tool for anyone involved in optimal control. The author has carefully collected many realistic examples drawn from diverse engineering fields. The examples are very helpful in explaining new concepts and ideas behind theories. Intriguing problems are available within each chapter. In an effort to close the gap between theory and practice, computational algorithms are included for solving practical optimization problems. The algorithms are coded in MATLAB. Students are asked to write simple MATLAB programs as they progress through the book, to convince themselves that they have confidence in the theory and understand its practical implications.

The author stated the goal at the beginning of the book as “to aid readers in utilizing the theory of optimal control to solve practical problems in the face of uncertainty.” It appears this goal has been achieved. Applied Linear Optimal Control: Examples and Algorithms should be a good addition to the optimal control community. It makes an excellent text for engineering and applied mathematics students who already have some optimal control background. It is strongly recommended as a helpful guide for anyone who desires to learn and apply many of the current state of the art results in optimal control.

1.
Bryson AE and Ho YC, 1975, Applied Optimal Control, Hemisphere, New York.
2.
Bryson AE, 1999, Dynamic Optimization, Addison Wesley Longman, Menlo Park CA.


text:
# Random Function Bias Tester

I wanted to create a short little piece of code so that I could see the biases in the random function.

import random
import statistics
import time

num = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
average = []

while True:
randomNum = random.randint(0, 9)
num[randomNum] += 1
average.append(randomNum)
print(f"\nThe average for all the numbers is {round(statistics.mean(average), round(sum(num) / 20))}")  # The second part of the round function is to keep as few digits on screen while still showing changes with every number. 20 is the arbitrary number I found worked best.
print(f"The most common number is {num.index(max(num))}\n")
time.sleep(0.5)

Since I am new-ish to python I wanted to know if, even though its only 15 lines, that I am following good coding practices. That way when my code gets more complicated, I don't have to worry about that as much

num = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

The shorthand to initialize this is [0] * 10; it can be used for strings too (bonus fact.)

round(sum(num) / 20))

This should be written as round(sum(num), 20) as the division operator adds in error, and is present through many iterations. Changing it to the latter reduced the bias.

time.sleep(0.5)

Usually the user doesn't want to wait if they don't have to; if this is run as a console program and if the user wants to inspect the output (before it disappears), it can be managed in the IDE. If the user must press a key, calling input() waits.

I would also avoid hardcoding the start and end values for random, and the array size and instead use variables.

The program will eventually run out of memory because the average list is being appended to every iteration and not being cleared. This can be prevented by instead summing the numbers (keeping a single numeric counter) and dividing by the total number of iterations. This has been an exercise left for the author.

The final code becomes:

import random
import statistics

start = 0
end = 9

num = [0] * abs(end - start + 1)
average = []

while True:
randomNum = random.randint(start, end)
num[randomNum] += 1
average.append(randomNum)
print(
f"\nThe average for all the numbers is {round(statistics.mean(average), round(sum(num), 20))}"
)
print(f"The most common number is {num.index(max(num))}\n")

Hard coding your data can be lengthy and it can restrict flexibility when carrying out further analysis. You can shorten your code with list comprehension.

import random

n = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

def main(val):
ls = [i for i in random.choices(n, k=val)]
ns = sum(ls)/len(ls)
print('Length of Data : {}\nSum of Data    : {}\nAverage of Data: {}\nRandom Gen Data: {}'.format(len(ls), sum(ls), ns, ls))

main(100) #Enter amount of data you want to pass through the function

This returns:

Length of Data : 100
Sum of Data    : 399
Average of Data: 3.99
Random Gen Data: [7, 4, 0, 8, 8, 0, 6, 7, 6, 4, 7, 0, 0, 0, 9, 4, 3, 0, 0, 7, 0, 4, 6, 5, 3, 0, 6, 0, 7, 0, 0, 4, 3, 7, 7, 9, 5, 3, 5, 2, 0, 4, 1, 1, 5, 8, 9, 0, 3, 2, 7, 5, 3, 3, 3, 3, 7, 5, 7, 5, 7, 1, 6, 2, 7, 7, 5, 1,6, 6, 4, 7, 8, 2, 0, 1, 5, 3, 5, 4, 9, 7, 7, 2, 0, 3, 4, 6, 3, 4, 3, 5, 6, 3, 0, 1, 3, 8, 0, 1]

It isn't necessary to use a while loop in this instance because you will still need to break your loop which is the same as giving a start and end value to a random range reference.

As noted in the answer by @alexyorke, you keep on adding to the data list. This means that your program can run out of memory. One way to reduce this risk greatly is to not save every data point, but use the data of how often each number appears.

Here are two different ways to do it, one uses a list as an array, like your code:

def mean(numbers):
return sum(i * n for i, n in enumerate(numbers)) / sum(numbers)

def most_common(numbers):
return max((n, i) for i, n in enumerate(numbers))[1]

if __name__ == "__main__":
high = 10
counter = [0] * high
for _ in range(1000):
counter[random.randrange(high)] += 1
average = round(mean(counter), round(sum(counter) / 20))
print(f"\nThe average for all the numbers is {average}")
print(f"The most common number is {most_common(counter)}\n")

This has the disadvantage that it only works for integers (because only integers can be indices for a list). Instead you could use a collections.Counter to keep the counts:

from collections import Counter

def mean(counts):
return sum(i * n for n, i in counts.items()) / sum(counts.values())

def most_common(counts):
return counts.most_common(1)[0][0]

if __name__ == "__main__":
high = 10
counter = Counter()
for _ in range(1000):
counter[random.randrange(high)] += 1
average = round(mean(counter), round(sum(counter.values()) / 20))
print(f"\nThe average for all the numbers is {average}")
print(f"The most common number is {most_common(counter)}\n")

Note that in both cases I added a if __name__ == "__main__": guard to allow importing from this script from other scripts, which is a good practice to use.

As a bonus, making a rudimentary histogram is also quite easy with the counter and some format string magic:

width = 60
norm = sum(counter.values())
for i, n in sorted(counter.items()):
print(f"{i:<2} | {'#' * int((n * width / norm)):<{width}}| {n / norm:>7.2%}")

0  | #####                                                       |   9.80%
1  | ######                                                      |  10.40%
2  | ######                                                      |  10.20%
3  | #####                                                       |   8.40%
4  | ######                                                      |  10.70%
5  | #####                                                       |   9.30%
6  | ######                                                      |  11.00%
7  | #####                                                       |   9.40%
8  | ######                                                      |  10.80%
9  | ######                                                      |  10.00%


text:
# Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)

I'd love to learn about further basic or central examples and I think such examples serve as good invitations to various areas. (Which is why a bounty was offered.)

To make this question and the various examples a more useful source there is a designated answer to point out connections between the various examples we collected.

In order to make it a more useful source, I list all the answers in categories, and added (for most) a date and (for 2/5) a link to the answer which often offers more details. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery, year? means that I am not sure if this is the correct year and ? means that I do not know the date. Please edit and correct.) Of course, if you see some important example missing, add it!

Logic and foundations: $\aleph_\omega$ (~1890), Russell's paradox (1901), Halting problem (1936), Goedel constructible universe L (1938), McKinsey formula in modal logic (~1941), 3SAT (*1970), The theory of Algebraically closed fields (ACF) (?),

Physics: Brachistochrone problem (1696), Ising model (1925), The harmonic oscillator,(?) Dirac's delta function (1927), Heisenberg model of 1-D chain of spin 1/2 atoms, (~1928), Feynman path integral (1948),

Real and Complex Analysis: Harmonic series (14th Cen.) {and Riemann zeta function (1859)}, the Gamma function (1720), li(x), The elliptic integral that launched Riemann surfaces (*1854?), Chebyshev polynomials (?1854) punctured open set in C^n (Hartog's theorem *1906 ?)

Partial differential equations: Laplace equation (1773), the heat equation, wave equation, Navier-Stokes equation (1822),KdV equations (1877),

Functional analysis: Unilateral shift, The spaces $\ell_p$, $L_p$ and $C(k)$, Tsirelson spaces (1974), Cuntz algebra,

Algebra: Polynomials (ancient?), Z (ancient?) and Z/6Z (Middle Ages?), symmetric and alternating groups (*1832), Gaussian integers ($Z[\sqrt -1]$) (1832), $Z[\sqrt(-5)]$,$su_3$ ($su_2)$, full matrix ring over a ring, $\operatorname{SL}_2(\mathbb{Z})$ and SU(2), quaternions (1843), p-adic numbers (1897), Young tableaux (1900) and Schur polynomials, cyclotomic fields, Hopf algebras (1941) Fischer-Griess monster (1973), Heisenberg group, ADE-classification (and Dynkin diagrams), Prufer p-groups,

Number Theory: conics and pythagorean triples (ancient), Fermat equation (1637), Riemann zeta function (1859) elliptic curves, transcendental numbers, Fermat hypersurfaces,

Probability: Normal distribution (1733), Brownian motion (1827), The percolation model (1957), The Gaussian Orthogonal Ensemble, the Gaussian Unitary Ensemble, and the Gaussian Symplectic Ensemble, SLE (1999),

Dynamics: Logistic map (1845?), Smale's horseshoe map(1960). Mandelbrot set (1978/80) (Julia set), cat map, (Anosov diffeomorphism)

Geometry: Platonic solids (ancient), the Euclidean ball (ancient), The configuration of 27 lines on a cubic surface, The configurations of Desargues and Pappus, construction of regular heptadecagon (*1796), Hyperbolic geometry (1830), Reuleaux triangle (19th century), Fano plane (early 20th century ??), cyclic polytopes (1902), Delaunay triangulation (1934) Leech lattice (1965), Penrose tiling (1974), noncommutative torus, cone of positive semidefinite matrices, the associahedron (1961)

Topology: Spheres, Figure-eight knot (ancient), trefoil knot (ancient?) (Borromean rings (ancient?)), the torus (ancient?), Mobius strip (1858), Cantor set (1883), Projective spaces (complex, real, quanterionic..), Poincare dodecahedral sphere (1904), Homotopy group of spheres, Alexander polynomial (1923), Hopf fibration (1931), The standard embedding of the torus in R^3 (*1934 in Morse theory), pseudo-arcs (1948), Discrete metric spaces, Sorgenfrey line, Complex projective space, the cotangent bundle (?), The Grassmannian variety,homotopy group of spheres (*1951), Milnor exotic spheres (1965)

Graph theory: The seven bridges of Koenigsberg (1735), Petersen Graph (1886), two edge-colorings of K_6 (Ramsey's theorem 1930), K_33 and K_5 (Kuratowski's theorem 1930), Tutte graph (1946), Margulis's expanders (1973) and Ramanujan graphs (1986),

Combinatorics: tic-tac-toe (ancient Egypt(?)) (The game of nim (ancient China(?))), Pascal's triangle (China and Europe 17th), Catalan numbers (18th century), (Fibonacci sequence (12th century; probably ancient), Kirkman's schoolgirl problem (1850), surreal numbers (1969), alternating sign matrices (1982)

Algorithms and Computer Science: Newton Raphson method (17th century), Turing machine (1937), RSA (1977), universal quantum computer (1985)

Social Science: Prisoner's dilemma (1950) (and also the chicken game, chain store game, and centipede game), the model of exchange economy, second price auction (1961)

Statistics: the Lady Tasting Tea (?1920), Agricultural Field Experiments (Randomized Block Design, Analysis of Variance) (?1920), Neyman-Pearson lemma (?1930), Decision Theory (?1940), the Likelihood Function (?1920), Bootstrapping (?1975)

• I think that this should be community wiki. – Loop Space Nov 11 '09 at 7:55
• @Jose: Hard to say exactly. My instinct is that the kind of answers that this question will garner are those that didn't involve much actual thought, and the votes up or down will be more an assessment of whether the voter liked the example rather than whether the voter liked the answer (which, ideally, should contain an explanation of why that example shaped the discipline); both of these indicate that the answerers should not gain reputation for their answers, hence community wiki. – Loop Space Nov 11 '09 at 9:50
• I've hit this with the wiki hammer. – Scott Morrison Nov 11 '09 at 19:34
• I can't imagine a counterexample to the following rule: Any question whose purpose is to produce a sorted list of resources (i.e. the question includes, or should include, "one per post please") should be community wiki. – Anton Geraschenko Nov 12 '09 at 8:03
• Why does this question have a bounty anyway? – Kevin H. Lin Nov 21 '09 at 17:33

## 141 Answers

Heisenberg model of 1-D chain of spin 1/2 atoms, solved exactly by Bethe in 1931, is where Bethe Ansatz was born, and with it the field of integrable models in statistical and quantum mechanics.

Answered by Mio

The Lorenz system of ordinary differential equations: $$\dot x=\sigma(y-x)$$ $$\dot y = rx-y-xz$$ $$\dot z = xy-bz$$ ($\sigma$, $r$, $b$ are parameters) is a good example in dynamical system. It is an example of a deterministic system displaying chaotic behaviour. Also the Lorenz attractor. Date: 1963.

Dirichlet Function is a fundamental example in Calculus where Riemann integral does not work. It is also a function which is discontinuous everywhere. The function D(x) is defined as D(x) = 1, if x is a rational number; otherwise D(x) = 0.

For more information, see for example: http://mathworld.wolfram.com/DirichletFunction.html

In $C^*$-algebras, the Cuntz Algebra is a fundamental example of a separable unital $C^*$-algebra. Its appearance has reshaped much of the theory of $C^*$-algebras.

• You probably want to say something about it being simple and infinite and nuclear, as well... – Yemon Choi Nov 22 '09 at 14:10

The image of a torus embedded in $\mathbb{R}^3$ tilted on its side is the fundamental example of Morse Theory. In some sense it shows why you should "believe" all of the Morse lemmas before you sit down to prove them.

I think the question of whether or not $li(x) := \int_{2}^{x} \frac{dt}{\ln(t)}$ was contained in $\mathbb{C}\left(x, \ln\left(x\right)\right)$ spawned the field of galois theory of differential equations. One key question in this field, is what sort of extensions arise from adjoining solutions to a differential equation with coefficients in some ring (for example $\mathbb{C}(x)$?

In the case of the original question it can be posed as what extension arises from the differential equation $$\ln(x)y' = 1?$$

This field developed in parallel to galois theory of number fields (and other algebraic geometry, and arithmetic geometry). A good reference for the field is the aptly titled "Galois Theory of Linear Differential Equations" by Marius Van der Put and Michael Singer.

• In case it is not clear why this function is interesting, it is asymptotic to $\pi(x)$, the number of primes less than x. – Ben Weiss Nov 24 '09 at 5:44
• This reminds me that we did not include any number in our list. Probably pi and e are too fundamental, but the Euler constant could be a candidate. – Gil Kalai Nov 24 '09 at 6:05
• The question of whether li(x) is contained in C(x,ln(x)) reduces at once to the transcendence of ln(x) over C(x), which is easy to prove. So perhaps you mean the question of whether li(x) is an elementary function. Questions similar to this were indeed the root of Liouville's theory of elementary integrals. – lhf Nov 25 '09 at 1:31

Theorem on Friends and strangers in Ramsey Theory.

• In other words, $R(3,3)$, starting point for all of Ramsey Theory. We should probably include the closely-related Happy Ending Problem ( en.m.wikipedia.org/wiki/Happy_ending_problem ) for its additional role in kicking off Discrete Geometry. – anonymous_coward May 16 '19 at 18:05

Some of the examples above have indeed shaped whole "disciplines" but others, as striking as they are, are less sweeping in scope.

For me, a very important and thought provoking example, and historically important for a variety of reasons is the Tutte graph. This graph shows a 3-valent 3-polytopal graph which has no hamiltonian circuit. This example, in particular, doomed attempts to prove the 4-color theorem based on ideas related to hamiltonian circuits.

http://en.wikipedia.org/wiki/Tutte_graph

http://mathworld.wolfram.com/TuttesGraph.html

It inspired a great variety of conjectures and work related to hamiltonian circuits for polytopes.

• A very nice example. It is true that the examples vary in terms of importance and also in terms of how large we understand the concept of an "example". – Gil Kalai Dec 6 '09 at 6:49

To make this question and the various examples a more useful source this is a designated answer to point out connections between the various examples we collected. please indicate only strong, definite, nontrivial, and clear connections.

1) The Petersen graph is obtained by identifying antipodal vertices and edges in the graph of the dodecahedron - one of the five platonic solids. Such an identification gives a polyhedral complex realizing the real projective plane. Applying this operation to the icosahedron leads to a 6-vertex triangulation of the real projective plane.

The cyclotomic field $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ is the most basic example of a field extension in which splitting of primes depends on an obvious congruence condition. Specifically, if $\ell$ is another prime, then the Frobenius of $\ell$ is $\ell \mod p \in (\mathbb{Z}/p)^\times = \mathrm{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$. In particular, $\ell$ splits in the field iff its Frobenius is trivial, and this is true iff $\ell \equiv 1 \mod p$. We can then relate other congruences to splitting in subfields of $\mathbb{Q}(\zeta_p)$, etc. The theorems of global class field theory show that this basic concept holds in a very general case, although the general case is much harder to prove. This basic example, does, however, motivate the ideas in class field theory, which have greatly influenced modern number theory and related areas. (As an added note, the fact that the Artin reciprocity law is true for cyclotomic fields is actually a key ingredient in the proof for general abelian extensions!)

Answered by Davidac897

The semicircular law and the Marchenko-Pastur distribution are fundamental examples of probability distributions in random matrix theory.

The hyperfinite $\mathrm{II}_1$ factor $\mathcal{R}$ and its ultrapower $\mathcal{R}^{\omega}$ are fundamental examples in von Neumann algebras and Connes' embedding problem.

The free group factors $L(\mathbb{F}_{n})$, which are the closer in the weak operator topology of the left regular representation of the free group $\mathbb{F}_n$, are fundamental examples in von Neumann algebras. The isomorphism question is the root of the so important Free Probability theory of Voiculescu.

The example that launched category theory: (co)homology, for example simplicial homology and Čech cohomology. The various maps linking (co)homology groups for different 'resolutions' of a topological space (by triangulation or open sets resp.) were I think the first examples of natural transformations. This necessitated defining functors, and hence defining categories.

Linear Algebra: In linear algebra the symmetric group $S_n=Bij(\{1,,n\})$ is a classical and important example if you learn basics in group theory.

Operator-algebra/ functional-analysis: Fundamental examples of $C^*$-algebras are $C(X)$ ( continuous functions on a compact Haudorff space X), $C_0(X)$ ( continuous functions vanishing at infinity on a localcompact Haudorff space X), $B(H)$ (bounded linear maps on a Hilbert space H). By Gelfand-Naimark you know a lot of abstract $C^*$algebras if you know these concrete examples.

Margulis's expanders: This class of 8 regular graphs is the first explicit example for a family of expanders. The vertices are pairs of integer modulo m, the neighbors of (x,y) are (x+y,y), (x-y,y), (x,y+x), (x,y-x), (x+y+1,y), (x-y+1,y), (x, y+x+1), (x,y-x+1). All operations are modulo m.

Expanders were first discovered and constucted probabilistically by Pinsker. The Ramanujan graphs of Lubotzky, Philips and Sarnak are expanders with extremely good properties. This paper by Hoory, Linial and Wigderson contains much more information.

Category theory: There is an isomorphism between a vector space and its double-dual which does not depend on choice of basis. It is natural in the sense that every vector space has such an isomorphism, and these isomorphisms commute with every linear transformation.

This should be contrasted between the isomorphisms between a finite-dimensional vector space and its dual. These depend on a choice of basis and are not natural in this sense.

This example constitutes the first two paragraphs of the first paper in category theory! Eilenberg-Mac Lane: General theory of natural equivalences.

In Categories for the working mathematician, Mac Lane writes that the purpose of discussing categories is to discuss functors, and that the purpose of discussing functors is to discuss natural transformations.

Weierstrass's example of a continuous, nowhere differentiable real function -- apparently was a real surprise at the time and is still surprising to students.

The existence of dense open sets of small measure -- very counterintuitive and allows for all sorts of mischief.

In number theory: the diophantine equation

$a^2-b^3=c$.

This equation can be seen as an epitome of modern number theory, despite its deceptive simplicity.

This equation is not 'fully' understood to this day. But much about it is understood.

This equation is a recurring theme in the award-winning article

Ravi Vakil gives interesting examples in algebraic geometry: "The existence of some of these pathologies is common knowledge'', but I had never known what they were.".

Although this may fall foul of the criticism that it perhaps it has not shaped a subject yet, I'll give it the benefit of the doubt that it may still shape a future subject. In a way the answer touches two answers already given: the Platonic solids and also the quaternions.

I am talking about the ADE classification, which appears in the theory of Lie algebras, finite subgroups of $SU(2)$ (McKay correspondence), representation theory of quivers (Gabriel's theorem), singularity theory (Du Val), classification of conformal field theories,...

The arithmetics of conics with pythagorean triples is since long been used as toy model for the beautifull combination of arithmetics, analysis and geometry in the study of algebraic curves, but Lemmermeyer's "Conics - a Poor Man's Elliptic Curves" and his subsequent arxiv articles pushes the "toy" into the direction of a "fundamental example" for some fascinating issues.

Spheres (of various dimensions) are the fundamental examples of (compact) Riemannian manifolds (or even Alexandrov spaces) of curvature > 0. Several major theorems of Riemannian geometry were motivated by the question of how to recognize a sphere. Most recently this culminated in Brendle and Schoen's proof of the differentiable sphere theorem.

Three related fundamental examples in random matrix theory (in mathematical physics and probability) are the Gaussian Orthogonal Ensemble, the Gaussian Unitary Ensemble, and the Gaussian Symplectic Ensemble. Much of random matrix theory has been devoted to determining the properties of these families of random matrices and proving that other families exhibit the same behaviors.

The discovery of Transcendental numbers, or numbers that are not the root of any finite polynomial with rational coefficients.
Also, the proof that e and π were transcendental, the latter via the proof that ea is only algebraic for transcendental values of a (and e*i*π = -1 is algebraic, as is i, so therefore π must be transcendental). Their discovery, as well as the first explicitly created example, the Liouville number, sparked what's called "Transcendence theory".
And as it turns out, any randomly chosen real number is "almost surely" transcendental. In other words, the density of transcendental numbers among the real numbers is 1!!

$\mathbb{Q}_{p}$. The field of p-adic numbers brings the study of local methods. Hensel's lemma is a great example. It is also interesting that p-adic integers is the projective limit of the rings $\mathbb{Z}/p^{n}\mathbb{Z}$.

The (complex analytic) proof of the Prime Number Theorem is the first major use of complex analysis to prove results about asymptotic behavior of prime numbers, which, at first glance, do not at all seem to be tied to complex numbers. This has led to an enormous amount of mathematics.

The field extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is the most basic example of an algebraic extension which is not Galois. In particular, one notices that it has no non-trivial automorphisms, and that this is related to the fact that not enough of the roots of $x^3-2$ are in this field. This leads to the key concept of Galois extensions and the relation between automorphisms and roots. It also led Galois to develop the concept of normal subgroup.

I think no one's pointed Lorenz equations.

• Can you explain or add a link, please? – Gil Kalai Feb 21 '11 at 19:54


text:
# Encoding of categorical variables#

In this notebook, we will present typical ways of dealing with categorical variables by encoding them, namely ordinal encoding and one-hot encoding.

Let’s first load the entire adult dataset containing both numerical and categorical data.

import pandas as pd

# drop the duplicated column "education-num" as stated in the first notebook

target_name = "class"

## Identify categorical variables#

As we saw in the previous section, a numerical variable is a quantity represented by a real or integer number. These variables can be naturally handled by machine learning algorithms that are typically composed of a sequence of arithmetic instructions such as additions and multiplications.

In contrast, categorical variables have discrete values, typically represented by string labels (but not only) taken from a finite list of possible choices. For instance, the variable native-country in our dataset is a categorical variable because it encodes the data using a finite list of possible countries (along with the ? symbol when this information is missing):

data["native-country"].value_counts().sort_index()

 ?                               857
Cambodia                         28
China                           122
Columbia                         85
Cuba                            138
Dominican-Republic              103
England                         127
France                           38
Germany                         206
Greece                           49
Guatemala                        88
Haiti                            75
Holand-Netherlands                1
Honduras                         20
Hong                             30
Hungary                          19
India                           151
Iran                             59
Ireland                          37
Italy                           105
Jamaica                         106
Japan                            92
Laos                             23
Mexico                          951
Nicaragua                        49
Outlying-US(Guam-USVI-etc)       23
Peru                             46
Philippines                     295
Poland                           87
Portugal                         67
Puerto-Rico                     184
Scotland                         21
South                           115
Taiwan                           65
Thailand                         30
United-States                 43832
Vietnam                          86
Yugoslavia                       23
Name: native-country, dtype: int64

How can we easily recognize categorical columns among the dataset? Part of the answer lies in the columns’ data type:

data.dtypes

age                int64
workclass         object
education         object
marital-status    object
occupation        object
relationship      object
race              object
sex               object
capital-gain       int64
capital-loss       int64
hours-per-week     int64
native-country    object
dtype: object

If we look at the "native-country" column, we observe its data type is object, meaning it contains string values.

## Select features based on their data type#

In the previous notebook, we manually defined the numerical columns. We could do a similar approach. Instead, we will use the scikit-learn helper function make_column_selector, which allows us to select columns based on their data type. We will illustrate how to use this helper.

from sklearn.compose import make_column_selector as selector

categorical_columns_selector = selector(dtype_include=object)
categorical_columns = categorical_columns_selector(data)
categorical_columns

['workclass',
'education',
'marital-status',
'occupation',
'relationship',
'race',
'sex',
'native-country']

Here, we created the selector by passing the data type to include; we then passed the input dataset to the selector object, which returned a list of column names that have the requested data type. We can now filter out the unwanted columns:

data_categorical = data[categorical_columns]

workclass education marital-status occupation relationship race sex native-country
0 Private 11th Never-married Machine-op-inspct Own-child Black Male United-States
1 Private HS-grad Married-civ-spouse Farming-fishing Husband White Male United-States
2 Local-gov Assoc-acdm Married-civ-spouse Protective-serv Husband White Male United-States
3 Private Some-college Married-civ-spouse Machine-op-inspct Husband Black Male United-States
4 ? Some-college Never-married ? Own-child White Female United-States
print(f"The dataset is composed of {data_categorical.shape[1]} features")

The dataset is composed of 8 features

In the remainder of this section, we will present different strategies to encode categorical data into numerical data which can be used by a machine-learning algorithm.

## Strategies to encode categories#

### Encoding ordinal categories#

The most intuitive strategy is to encode each category with a different number. The OrdinalEncoder will transform the data in such manner. We will start by encoding a single column to understand how the encoding works.

from sklearn.preprocessing import OrdinalEncoder

education_column = data_categorical[["education"]]

encoder = OrdinalEncoder()
education_encoded = encoder.fit_transform(education_column)
education_encoded

array([[ 1.],
[11.],
[ 7.],
...,
[11.],
[11.],
[11.]])

We see that each category in "education" has been replaced by a numeric value. We could check the mapping between the categories and the numerical values by checking the fitted attribute categories_.

encoder.categories_

[array([' 10th', ' 11th', ' 12th', ' 1st-4th', ' 5th-6th', ' 7th-8th',
' 9th', ' Assoc-acdm', ' Assoc-voc', ' Bachelors', ' Doctorate',
' HS-grad', ' Masters', ' Preschool', ' Prof-school',
' Some-college'], dtype=object)]

Now, we can check the encoding applied on all categorical features.

data_encoded = encoder.fit_transform(data_categorical)
data_encoded[:5]

array([[ 4.,  1.,  4.,  7.,  3.,  2.,  1., 39.],
[ 4., 11.,  2.,  5.,  0.,  4.,  1., 39.],
[ 2.,  7.,  2., 11.,  0.,  4.,  1., 39.],
[ 4., 15.,  2.,  7.,  0.,  2.,  1., 39.],
[ 0., 15.,  4.,  0.,  3.,  4.,  0., 39.]])

print(
f"The dataset encoded contains {data_encoded.shape[1]} features")

The dataset encoded contains 8 features

We see that the categories have been encoded for each feature (column) independently. We also note that the number of features before and after the encoding is the same.

However, be careful when applying this encoding strategy: using this integer representation leads downstream predictive models to assume that the values are ordered (0 < 1 < 2 < 3… for instance).

By default, OrdinalEncoder uses a lexicographical strategy to map string category labels to integers. This strategy is arbitrary and often meaningless. For instance, suppose the dataset has a categorical variable named "size" with categories such as “S”, “M”, “L”, “XL”. We would like the integer representation to respect the meaning of the sizes by mapping them to increasing integers such as 0, 1, 2, 3. However, the lexicographical strategy used by default would map the labels “S”, “M”, “L”, “XL” to 2, 1, 0, 3, by following the alphabetical order.

The OrdinalEncoder class accepts a categories constructor argument to pass categories in the expected ordering explicitly. You can find more information in the scikit-learn documentation if needed.

If a categorical variable does not carry any meaningful order information then this encoding might be misleading to downstream statistical models and you might consider using one-hot encoding instead (see below).

### Encoding nominal categories (without assuming any order)#

OneHotEncoder is an alternative encoder that prevents the downstream models to make a false assumption about the ordering of categories. For a given feature, it will create as many new columns as there are possible categories. For a given sample, the value of the column corresponding to the category will be set to 1 while all the columns of the other categories will be set to 0.

We will start by encoding a single feature (e.g. "education") to illustrate how the encoding works.

from sklearn.preprocessing import OneHotEncoder

encoder = OneHotEncoder(sparse=False)
education_encoded = encoder.fit_transform(education_column)
education_encoded

/opt/hostedtoolcache/Python/3.9.16/x64/lib/python3.9/site-packages/sklearn/preprocessing/_encoders.py:808: FutureWarning: sparse was renamed to sparse_output in version 1.2 and will be removed in 1.4. sparse_output is ignored unless you leave sparse to its default value.
warnings.warn(

array([[0., 1., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.],
...,
[0., 0., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.]])

Note

sparse=False is used in the OneHotEncoder for didactic purposes, namely easier visualization of the data.

Sparse matrices are efficient data structures when most of your matrix elements are zero. They won’t be covered in detail in this course. If you want more details about them, you can look at this.

We see that encoding a single feature will give a NumPy array full of zeros and ones. We can get a better understanding using the associated feature names resulting from the transformation.

feature_names = encoder.get_feature_names_out(input_features=["education"])
education_encoded = pd.DataFrame(education_encoded, columns=feature_names)
education_encoded

education_ 10th education_ 11th education_ 12th education_ 1st-4th education_ 5th-6th education_ 7th-8th education_ 9th education_ Assoc-acdm education_ Assoc-voc education_ Bachelors education_ Doctorate education_ HS-grad education_ Masters education_ Preschool education_ Prof-school education_ Some-college
0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0
2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0
4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
48837 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
48838 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0
48839 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0
48840 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0
48841 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0

48842 rows × 16 columns

As we can see, each category (unique value) became a column; the encoding returned, for each sample, a 1 to specify which category it belongs to.

Let’s apply this encoding on the full dataset.

print(
f"The dataset is composed of {data_categorical.shape[1]} features")

The dataset is composed of 8 features

workclass education marital-status occupation relationship race sex native-country
0 Private 11th Never-married Machine-op-inspct Own-child Black Male United-States
1 Private HS-grad Married-civ-spouse Farming-fishing Husband White Male United-States
2 Local-gov Assoc-acdm Married-civ-spouse Protective-serv Husband White Male United-States
3 Private Some-college Married-civ-spouse Machine-op-inspct Husband Black Male United-States
4 ? Some-college Never-married ? Own-child White Female United-States
data_encoded = encoder.fit_transform(data_categorical)
data_encoded[:5]

/opt/hostedtoolcache/Python/3.9.16/x64/lib/python3.9/site-packages/sklearn/preprocessing/_encoders.py:808: FutureWarning: sparse was renamed to sparse_output in version 1.2 and will be removed in 1.4. sparse_output is ignored unless you leave sparse to its default value.
warnings.warn(

array([[0., 0., 0., 0., 1., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 1., 0., 0., 0., 0., 1., 0., 0., 0., 1., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 1., 0., 0.],
[0., 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 1., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 1., 0., 0.],
[0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 1.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 1., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 1., 0., 0.],
[0., 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 1., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 1.,
0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 1., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 1., 0., 0.],
[1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0., 1., 0., 0.,
1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 1., 0., 0., 0., 0., 0., 0., 1., 1., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 1., 0., 0.]])

print(
f"The encoded dataset contains {data_encoded.shape[1]} features")

The encoded dataset contains 102 features

Let’s wrap this NumPy array in a dataframe with informative column names as provided by the encoder object:

columns_encoded = encoder.get_feature_names_out(data_categorical.columns)

workclass_ ? workclass_ Federal-gov workclass_ Local-gov workclass_ Never-worked workclass_ Private workclass_ Self-emp-inc workclass_ Self-emp-not-inc workclass_ State-gov workclass_ Without-pay education_ 10th ... native-country_ Portugal native-country_ Puerto-Rico native-country_ Scotland native-country_ South native-country_ Taiwan native-country_ Thailand native-country_ Trinadad&Tobago native-country_ United-States native-country_ Vietnam native-country_ Yugoslavia
0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 ... 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0
1 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 ... 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0
2 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ... 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0
3 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 ... 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0
4 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ... 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0

5 rows × 102 columns

Look at how the "workclass" variable of the 3 first records has been encoded and compare this to the original string representation.

The number of features after the encoding is more than 10 times larger than in the original data because some variables such as occupation and native-country have many possible categories.

### Choosing an encoding strategy#

Choosing an encoding strategy will depend on the underlying models and the type of categories (i.e. ordinal vs. nominal).

Note

In general OneHotEncoder is the encoding strategy used when the downstream models are linear models while OrdinalEncoder is often a good strategy with tree-based models.

Using an OrdinalEncoder will output ordinal categories. This means that there is an order in the resulting categories (e.g. 0 < 1 < 2). The impact of violating this ordering assumption is really dependent on the downstream models. Linear models will be impacted by misordered categories while tree-based models will not.

You can still use an OrdinalEncoder with linear models but you need to be sure that:

• the original categories (before encoding) have an ordering;

• the encoded categories follow the same ordering than the original categories.

The next exercise shows what can happen when using an OrdinalEncoder with a liner model and the conditions above are not met.

One-hot encoding categorical variables with high cardinality can cause computational inefficiency in tree-based models. Because of this, it is not recommended to use OneHotEncoder in such cases even if the original categories do not have a given order. We will show this in the final exercise of this sequence.

## Evaluate our predictive pipeline#

We can now integrate this encoder inside a machine learning pipeline like we did with numerical data: let’s train a linear classifier on the encoded data and check the generalization performance of this machine learning pipeline using cross-validation.

Before we create the pipeline, we have to linger on the native-country. Let’s recall some statistics regarding this column.

data["native-country"].value_counts()

 United-States                 43832
Mexico                          951
?                               857
Philippines                     295
Germany                         206
Puerto-Rico                     184
India                           151
Cuba                            138
England                         127
China                           122
South                           115
Jamaica                         106
Italy                           105
Dominican-Republic              103
Japan                            92
Guatemala                        88
Poland                           87
Vietnam                          86
Columbia                         85
Haiti                            75
Portugal                         67
Taiwan                           65
Iran                             59
Greece                           49
Nicaragua                        49
Peru                             46
France                           38
Ireland                          37
Hong                             30
Thailand                         30
Cambodia                         28
Laos                             23
Yugoslavia                       23
Outlying-US(Guam-USVI-etc)       23
Scotland                         21
Honduras                         20
Hungary                          19
Holand-Netherlands                1
Name: native-country, dtype: int64

We see that the Holand-Netherlands category is occurring rarely. This will be a problem during cross-validation: if the sample ends up in the test set during splitting then the classifier would not have seen the category during training and will not be able to encode it.

In scikit-learn, there are two solutions to bypass this issue:

• list all the possible categories and provide it to the encoder via the keyword argument categories;

• use the parameter handle_unknown, i.e. if an unknown category is encountered during transform, the resulting one-hot encoded columns for this feature will be all zeros.

Here, we will use the latter solution for simplicity.

Tip

Be aware the OrdinalEncoder exposes as well a parameter handle_unknown. It can be set to use_encoded_value. If that option is chosen, you can define a fixed value to which all unknowns will be set to during transform. For example, OrdinalEncoder(handle_unknown='use_encoded_value', unknown_value=42) will set all values encountered during transform to 42 which are not part of the data encountered during the fit call. You are going to use these parameters in the next exercise.

We can now create our machine learning pipeline.

from sklearn.pipeline import make_pipeline
from sklearn.linear_model import LogisticRegression

model = make_pipeline(
OneHotEncoder(handle_unknown="ignore"), LogisticRegression(max_iter=500)
)

Note

Here, we need to increase the maximum number of iterations to obtain a fully converged LogisticRegression and silence a ConvergenceWarning. Contrary to the numerical features, the one-hot encoded categorical features are all on the same scale (values are 0 or 1), so they would not benefit from scaling. In this case, increasing max_iter is the right thing to do.

Finally, we can check the model’s generalization performance only using the categorical columns.

from sklearn.model_selection import cross_validate
cv_results = cross_validate(model, data_categorical, target)
cv_results

{'fit_time': array([0.84990811, 0.79812551, 0.88143182, 0.78920507, 0.7831552 ]),
'score_time': array([0.03406978, 0.03360724, 0.03421521, 0.03650665, 0.03367352]),
'test_score': array([0.83222438, 0.83560242, 0.82872645, 0.83312858, 0.83466421])}

scores = cv_results["test_score"]
print(f"The accuracy is: {scores.mean():.3f} ± {scores.std():.3f}")

The accuracy is: 0.833 ± 0.002

As you can see, this representation of the categorical variables is slightly more predictive of the revenue than the numerical variables that we used previously.

In this notebook we have:

• seen two common strategies for encoding categorical features: ordinal encoding and one-hot encoding;

• used a pipeline to use a one-hot encoder before fitting a logistic regression.


text:
# How to “Copy as Unicode” from a Notebook?

I spent some time manually editing a post replacing Mathematica ASCII \[Alpha] with Unicode α. I did this by laboriously choosing Copy as LaTeX, pasting into the edit box, and then copying the Unicode symbol from the preview below. This made me realize I am lacking a "Copy as Unicode string" function in Mathematica.

How can I most easily copy an expression such as:

In Unicode:

αβ + Mod[δΨ, 2 ρ^2]

-

Since a native method is not forthcoming, I shall post my file based circumvention, for Windows.

You will need to have this utility in the command path (it apparently is stock with Windows 7).

copyUnicode[expr_] := Run["clip <",
Export["$Clipboard.temp", ToString[expr, InputForm], "Text", CharacterEncoding -> "Unicode"] ];  Usage: expr = \[Alpha]\[Beta] + Mod[\[Delta]\[CapitalPsi], 2\[InvisibleTimes]\[Rho]^2]; copyUnicode[expr]  This leaves the following text in the Windows Clipboard: αβ + Mod[δΨ, 2*ρ^2]  - I think clip is native in at least win-7. Perhaps you should note that this places the output in the system clipboard. The output of your copyUnicode function is just a 0 (at least, on my PC). You have to do a ctrl-v paste afterwards. A MMA Paste doesn't seem to work. – Sjoerd C. de Vries Feb 2 '12 at 21:10 Very useful function! Why not make it available for any code sample? (Now it works only for single expression…) – xzczd Nov 23 '12 at 14:09 @xzczd I'll see if I can improve it. – Mr.Wizard Nov 23 '12 at 23:56 To circumvent Mathematica's internal representation, I decided to use the operating system. Of course, this means it's only going to work on Mac OS X because it uses Cocoa bindings in the built-in Python interpreter: copyAsUnicode[t_] := Module[{ out = FileNameJoin[{$TemporaryDirectory,
"MathematicaOutput" <> StringJoin[Map[ToString, DateList[]]] <>
".rtf"}]
},
Export[out, t];
Run["printf \"from AppKit import *\n\
board=NSPasteboard.generalPasteboard()\n\
content=NSData.dataWithContentsOfFile_('" <> out <>
"')\nboard.declareTypes_owner_([NSRTFPboardType], None)\n\
board.setData_forType_(content, NSRTFPboardType)\n\" | \
/usr/bin/python"];
DeleteFile[out]
]

The idea is to export to RTF and read the result to the clipboard outside of Mathematica. The function is invoked for example as copyAsUnicode["αβ+Mod[δΨ+ρ2]"]. This example itself was copied that way, too, i.e., I typed copyAsUnicode["copyAsUnicode[\"αβ+Mod[δΨ+ρ2]\"]"], which I again copied the same way... OK, I think you get the idea.

Of course the next step would be to make this into a Palette that acts on the NotebookSelection, but the above is the main step. Maybe someone else knows how to do something like this in other operating systems (I don't).

-
 This looks good, but since I cannot test it on Windows I will leave it to Mac users to upvote. Thank you for your answer. – Mr.Wizard♦ Feb 2 '12 at 4:27


text:
12,000

We have over 12,000 students, from over 100 countries, within one of the safest campuses in the UK

97%

97% of Lancaster students go into work or further study within six months of graduating

Home > Research > Publications & Outputs > Measurement of the $W$ mass and width in $e^+ e... ### Text available via DOI: ## Measurement of the$W$mass and width in$e^+ e^-$collisions at 189-GeV Research output: Contribution to journalJournal article Published Journal publication date 10/2000 European Physical Journal C: Particles and Fields 2 17 21 241-261 English ### Abstract The mass of the W boson is determined in e + e − collisions at LEP by the direct reconstruction of W decays in WW→qq¯qq¯ and WW→ℓνqq¯ events, supplemented by measurements using the kinematic properties of the leptons in the \mathrm{W}\mathrm{W}\rightarrow\mbox{\ell\nu }\mbox{$\ell\nu \$} decay channel. The main sample of W pairs is selected from an integrated luminosity of 174 pb −1 collected with the ALEPH detector in 1998 at a centre-of-mass energy of 188.63 GeV. The combined result from all channels is
mW=80.432±0.072(stat.)±0.041(syst.)±0.019(FSI)±0.017(LEP)GeV/c2,
where FSI represents the possible effects of final state interactions in the qq¯qq¯ channel. In a second two-parameter fit to the qq¯qq¯ , \mathrmeνqq¯ and μνqq¯ channels, where the W mass and width are decoupled, the average W width is found to be { 2.24±0.20(stat.)±0.13(syst.) GeV/ c2 }, consistent with the Standard Model prediction. The combination of the mass measurement presented in this paper together with those derived previously from the W pair cross section at 161 and 172 GeV and direct reconstruction at 172 and 183 GeV gives
mW=80.418±0.061(stat.)±0.040(syst.)±0.019(FSI)±0.017(LEP)GeV/c2.


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University Physics Volume 1

# Challenge Problems

University Physics Volume 1Challenge Problems

### Challenge Problems

120.

The pressure on the dam shown early in the problems section increases with depth. Therefore, there is a net torque on the dam. Find the net torque.

121.

The temperature of the atmosphere is not always constant and can increase or decrease with height. In a neutral atmosphere, where there is not a significant amount of vertical mixing, the temperature decreases at a rate of approximately 6.5 K per km. The magnitude of the decrease in temperature as height increases is known as the lapse rate $(Γ).(Γ).$ (The symbol is the upper case Greek letter gamma.) Assume that the surface pressure is $p0=1.013×105Pap0=1.013×105Pa$ where $T=293KT=293K$ and the lapse rate is $(−Γ=6.5Kkm)(−Γ=6.5Kkm)$. Estimate the pressure 3.0 km above the surface of Earth.

122.

A submarine is stranded on the bottom of the ocean with its hatch 25.0 m below the surface. Calculate the force needed to open the hatch from the inside, given it is circular and 0.450 m in diameter. Air pressure inside the submarine is 1.00 atm.

123.

Logs sometimes float vertically in a lake because one end has become water-logged and denser than the other. What is the average density of a uniform-diameter log that floats with 20.0% of its length above water?

124.

Scurrilous con artists have been known to represent gold-plated tungsten ingots as pure gold and sell them at prices much below gold value but high above the cost of tungsten. With what accuracy must you be able to measure the mass of such an ingot in and out of water to tell that it is almost pure tungsten rather than pure gold?

125.

The inside volume of a house is equivalent to that of a rectangular solid 13.0 m wide by 20.0 m long by 2.75 m high. The house is heated by a forced air gas heater. The main uptake air duct of the heater is 0.300 m in diameter. What is the average speed of air in the duct if it carries a volume equal to that of the house’s interior every 15 minutes?

126.

A garden hose with a diameter of 2.0 cm is used to fill a bucket, which has a volume of 0.10 cubic meters. It takes 1.2 minutes to fill. An adjustable nozzle is attached to the hose to decrease the diameter of the opening, which increases the speed of the water. The hose is held level to the ground at a height of 1.0 meters and the diameter is decreased until a flower bed 3.0 meters away is reached. (a) What is the volume flow rate of the water through the nozzle when the diameter is 2.0 cm? (b) What is the speed of the water coming out of the hose? (c) What does the speed of the water coming out of the hose need to be to reach the flower bed 3.0 meters away? (d) What is the diameter of the nozzle needed to reach the flower bed?

127.

A frequently quoted rule of thumb in aircraft design is that wings should produce about 1000 N of lift per square meter of wing. (The fact that a wing has a top and bottom surface does not double its area.) (a) At takeoff, an aircraft travels at 60.0 m/s, so that the air speed relative to the bottom of the wing is 60.0 m/s. Given the sea level density of air as $1.29kg/m31.29kg/m3$, how fast must it move over the upper surface to create the ideal lift? (b) How fast must air move over the upper surface at a cruising speed of 245 m/s and at an altitude where air density is one-fourth that at sea level? (Note that this is not all of the aircraft’s lift—some comes from the body of the plane, some from engine thrust, and so on. Furthermore, Bernoulli’s principle gives an approximate answer because flow over the wing creates turbulence.)

128.

Two pipes of equal and constant diameter leave a water pumping station and dump water out of an open end that is open to the atmosphere (see the following figure). The water enters at a pressure of two atmospheres and a speed of $(v1=1.0m/s)(v1=1.0m/s)$. One pipe drops a height of 10 m. What is the velocity of the water as the water leaves each pipe?

129.

Fluid originally flows through a tube at a rate of $100cm3/s100cm3/s$. To illustrate the sensitivity of flow rate to various factors, calculate the new flow rate for the following changes with all other factors remaining the same as in the original conditions. (a) Pressure difference increases by a factor of 1.50. (b) A new fluid with 3.00 times greater viscosity is substituted. (c) The tube is replaced by one having 4.00 times the length. (d) Another tube is used with a radius 0.100 times the original. (e) Yet another tube is substituted with a radius 0.100 times the original and half the length, and the pressure difference is increased by a factor of 1.50.

130.

During a marathon race, a runner’s blood flow increases to 10.0 times her resting rate. Her blood’s viscosity has dropped to 95.0% of its normal value, and the blood pressure difference across the circulatory system has increased by 50.0%. By what factor has the average radii of her blood vessels increased?

131.

Water supplied to a house by a water main has a pressure of $3.00×105N/m23.00×105N/m2$ early on a summer day when neighborhood use is low. This pressure produces a flow of 20.0 L/min through a garden hose. Later in the day, pressure at the exit of the water main and entrance to the house drops, and a flow of only 8.00 L/min is obtained through the same hose. (a) What pressure is now being supplied to the house, assuming resistance is constant? (b) By what factor did the flow rate in the water main increase in order to cause this decrease in delivered pressure? The pressure at the entrance of the water main is $5.00×105N/m25.00×105N/m2$, and the original flow rate was 200 L/min. (c) How many more users are there, assuming each would consume 20.0 L/min in the morning?

132.

Gasoline is piped underground from refineries to major users. The flow rate is $3.00×10−2m3/s3.00×10−2m3/s$ (about 500 gal/min), the viscosity of gasoline is $1.00×10−3(N/m2)⋅s1.00×10−3(N/m2)⋅s$, and its density is $680kg/m3.680kg/m3.$ (a) What minimum diameter must the pipe have if the Reynolds number is to be less than 2000? (b) What pressure difference must be maintained along each kilometer of the pipe to maintain this flow rate?

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