text:
# Eigenvalues of time lagged covariance matrix - should be always real and positive?

Consider two random variables $$X_t$$ and $$X_{t+\tau}$$; where both come from random variable $$X$$ but are lagged by $$\tau$$. Assume that both are mean-free. If we want calculate covariance matrix of them, then we do:

$$\Sigma = X_t^T X_{t+\tau}$$ if we perform eigendecomposition

$$A\Sigma = \Sigma\Lambda$$

Then if the time lagged covariance is properly converged, all the eigenvalues are going to be positive, non-complex ?

• What do you mean by "properly converged"? Nov 25 at 19:07
• @mhdadk There's enough samples to calculate it, so that it does not change much if we add more samples Nov 25 at 20:12
• One could check for diagonal dominance, see math.stackexchange.com/questions/87528/…. By intuition, if autocorrelation $<X_{t}, X_{t+\tau}>$ is much higher than the rest, it is probable that $\Sigma$ is positive definite. Nov 25 at 23:36


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# 0.2 Practice tests (1-4) and final exams  (Page 16/36)

 Page 16 / 36

83 . $\Sigma X\sim N\left(n{\mu }_{x},\left(\sqrt{n}\right)\left({\sigma }_{x}\right)\right)$ so $\Sigma X\sim N\left(4000,28.3\right)$

84 .The probability is 0.50, because 5,000 is the mean of the sampling distribution of sums of size 40 from this population. Sums of random variables computed from a sample of sufficient size are normally distributed, and in a normal distribution, half the values lie below the mean.

85 . Using the empirical rule, you would expect 95 percent of the values to be within two standard deviations of the mean. Using the formula for the standard deviation is for a sample sum: $\left(\sqrt{n}\right)\left({\sigma }_{x}\right)=\left(\sqrt{40}\right)\left(7\right)=44.3$ so you would expect 95 percent of the values to be between 5,000 + (2)(44.3) and 5,000 – (2)(44.3), or between 4,911.4 and 588.6.

86 . $\mu -\left(\sqrt{n}\right)\left({\sigma }_{x}\right)=5000-\left(\sqrt{40}\right)\left(7\right)=4955.7$

87 . $5000+\left(2.2\right)\left(\sqrt{40}\right)\left(7\right)=5097.4$

## 7.3: using the central limit theorem

88 . The law of large numbers says that as sample size increases, the sample mean tends to get nearer and nearer to the population mean.

89 . You would expect the mean from a sample of size 100 to be nearer to the population mean, because the law of large numbers says that as sample size increases, the sample mean tends to approach the population mea.

90 . X ~ N (0.10, 0.20)

91 . $\overline{X}\sim N\left({\mu }_{x},\frac{{\sigma }_{x}}{\sqrt{n}}\right)$ and the standard deviation of a uniform distribution is $\frac{b-a}{\sqrt{12}}$ . In this example, the standard deviation of the distribution is $\frac{b-a}{\sqrt{12}}=\frac{0.10}{\sqrt{12}}=0.03$
so $\overline{X}\sim N\left(0.15,0.003\right)$

92 .

## 8.1: confidence interval, single population mean, population standard deviation known, normal

Use the following information to answer the next seven exercises. You draw a sample of size 30 from a normally distributed population with a standard deviation of four.

1 . What is the standard error of the sample mean in this scenario, rounded to two decimal places?

2 . What is the distribution of the sample mean?

3 . If you want to construct a two-sided 95% confidence interval, how much probability will be in each tail of the distribution?

4 . What is the appropriate z -score and error bound or margin of error ( EBM ) for a 95% confidence interval for this data?

5 . Rounding to two decimal places, what is the 95% confidence interval if the sample mean is 41?

6 . What is the 90% confidence interval if the sample mean is 41? Round to two decimal places

7 . Suppose the sample size in this study had been 50, rather than 30. What would the 95% confidence interval be if the sample mean is 41? Round your answer to two decimal places.

8 . For any given data set and sampling situation, which would you expect to be wider: a 95% confidence interval or a 99% confidence interval?

## 8.2: confidence interval, single population mean, standard deviation unknown, student’s t

9 . Comparing graphs of the standard normal distribution ( z -distribution) and a t -distribution with 15 degrees of freedom ( df ), how do they differ?

10 . Comparing graphs of the standard normal distribution ( z -distribution) and a t -distribution with 15 degrees of freedom ( df ), how are they similar?

Use the following information to answer the next five exercises. Body temperature is known to be distributed normally among healthy adults. Because you do not know the population standard deviation, you use the t-distribution to study body temperature. You collect data from a random sample of 20 healthy adults and find that your sample temperatures have a mean of 98.4 and a sample standard deviation of 0.3 (both in degrees Fahrenheit).

IMAGESNEWSVIDEOS A Dictionary of Computing. measures of location Quantities that represent the average or typical value of a random variable (compare measures of variation). They are either properties of a probability distribution or computed statistics of a sample. Three important measures are the mean, median, and mode.
define the measures of location
IMAGESNEWSVIDEOS A Dictionary of Computing. measures of location Quantities that represent the average or typical value of a random variable (compare measures of variation). They are either properties of a probability distribution or computed statistics of a sample. Three important measures are th
Ahmed
what is confidence interval estimate and its formula in getting it
discuss the roles of vital and health statistic in the planning of health service of the community
given that the probability of
BITRUS
can man city win Liverpool ?
There are two coins on a table. When both are flipped, one coin land on heads eith probability 0.5 while the other lands on head with probability 0.6. A coin is randomly selected from the table and flipped. (a) what is probability it lands on heads? (b) given that it lands on tail, what is the Condi
0.5*0.5+0.5*0.6
Ravasz
It should be a Machine learning terms。
Mok
it is a term used in linear regression
Saurav
what are the differences between standard deviation and variancs?
Enhance
what is statistics
statistics is the collection and interpretation of data
Enhance
the science of summarization and description of numerical facts
Enhance
Is the estimation of probability
Zaini
mr. zaini..can u tell me more clearly how to calculated pair t test
Haai
do you have MG Akarwal Statistics' book Zaini?
Enhance
Haai how r u?
Enhance
maybe .... mathematics is the science of simplification and statistics is the interpretation of such values and its implications.
Miguel
can we discuss about pair test
Haai
what is outlier?
outlier is an observation point that is distant from other observations.
Gidigah
what is its effect on mode?
Usama
Outlier  have little effect on the mode of a given set of data.
Gidigah
How can you identify a possible outlier(s) in a data set.
Daniel
The best visualisation method to identify the outlier is box and wisker method or boxplot diagram. The points which are located outside the max edge of wisker(both side) are considered as outlier.
Akash
@Daniel Adunkwah - Usually you can identify an outlier visually. They lie outside the observed pattern of the other data points, thus they're called outliers.
Ron
what is completeness?
I am new to this. I am trying to learn.
Dom
I am also new Dom, welcome!
Nthabi
thanks
Dom
please my friend i want same general points about statistics. say same thing
alex
outliers do not have effect on mode
Meselu
also new
yousaf
I don't get the example
ways of collecting data at least 10 and explain
Example of discrete variable
Gbenga
I am new here, can I get someone to guide up?
alayo
dies outcome is 1, 2, 3, 4, 5, 6 nothing come outside of it. it is an example of discrete variable
jainesh
continue variable is any value value between 0 to 1 it could be 4digit values eg 0.1, 0.21, 0.13, 0.623, 0.32
jainesh
hi
Kachalla
what's up here ... am new here
Kachalla
sorry question a bit unclear...do you mean how do you analyze quantitative data? If yes, it depends on the specific question(s) you set in the beginning as well as on the data you collected. So the method of data analysis will be dependent on the data collecter and questions asked.
Bheka
how to solve for degree of freedom
saliou
Quantitative data is the data in numeric form. For eg: Income of persons asked is 10,000. This data is quantitative data on the other hand data collected for either make or female is qualitative data.
Rohan
*male
Rohan
Degree of freedom is the unconditionality. For example if you have total number of observations n, and you have to calculate variance, obviously you will need mean for that. Here mean is a condition, without which you cannot calculate variance. Therefore degree of freedom for variance will be n-1.
Rohan
data that is best presented in categories like haircolor, food taste (good, bad, fair, terrible) constitutes qualitative data
Bheka
vegetation types (grasslands, forests etc) qualitative data
Bheka
I don't understand how you solved it can you teach me
solve what?
Ambo
mean
Vanarith
What is the end points of a confidence interval called?
lower and upper endpoints
Bheka

format: plain


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# 问题 L: 数组矩阵操作

#### 样例输入 复制

1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1

#### 样例输出 复制

   11    1    1    1    1   11
1   11    1    1   11    1
1    1   11   11    1    1
1    1   11   11    1    1
1   11    1    1   11    1
11    1    1    1    1   11


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using the mean-value property for holomorphic functions

The mean-value property for holomorphic functions states that if f is holomorphic in the neighbourhood of the closed disc centered at $z0$ of radius $R$, then

$f(z_{0}) = \frac{1}{2\pi}\int_{0}^{2\pi}f(z_{0} + re^{i\theta})\, d\theta$

if $f(z)=\sum_{n=0}^{+\infty}a_nz^n$, We have by dominated convergence:

$\int_{0}^{2\pi}f(z_{0} + re^{i\theta})d\theta = \int_{0}^{2\pi}\sum_{n=0}^{+\infty}a_n(z_{0} + re^{i\theta})^nd\theta=\sum_{n=0}^{+\infty}a_n\int_{0}^{2\pi}(z_{0} + re^{i\theta})^nd\theta =[\frac{(z_0+re^{i\theta})^{n+1}}{ir(n+1)}]_0^{2\pi}=0$

Which means evry holomorphic function is equal to 0, a non sense. I can't find my mistake, any help much appreciated.

• Your antiderivative is incorrect. – Mark Viola Apr 21 '18 at 13:09

$$(z_0+re^{i\theta})^n =\sum_{k=0}^n \binom{n}{k} z_0^{n-k}e^{ik\theta}$$
then exploit the fact that $\int_0^{2\pi}e^{ik\theta}\,d\theta=0$ for $k\ne 0$.
• Thank you. using the binomial theorem I found: $\int_{0}^{2\pi}(z_{0} + re^{i\theta})^nd\theta = z_0^n$ so that makes sense! – PerelMan Apr 21 '18 at 13:21
• Yes, and hence you recover $f(z_0)=\sum_{n=0}^\infty a_nz_0^n$, which is the Taylor series of $f$ – Mark Viola Apr 21 '18 at 13:24


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# Second Lie algebra tutorial

In this second tutorial, we continue the introduction on how to use the package GradedLieAlgebras.

By default, the scalars are assumed to be rational numbers, but it is also possible to define the coefficient field with the option Field, which is one out of four options that can be used in the constructor lieAlgebra.

 i1 : F = frac(ZZ/7[x]) o1 = F o1 : FractionField i2 : L = lieAlgebra({a_1,a_2,a_3},Field => F)/ {3 a_1 a_2 a_3+(5*x) a_2 a_1 a_3, a_3 a_2 a_3 a_1} o2 = L o2 : LieAlgebra

The example above specifies the Lie algebra over the fraction field of $\mathbb Z/7$ [ $x$ ] on three generators $a_1, a_2, a_3$, modulo the ideal generated by $3$ [$a_1$, [ $a_2$, $a_3$ ]] + $5x$ [ $a_2$, [ $a_1$, $a_3$ ]] and [$a_3$, [ $a_2$, [ $a_3$, $a_1$ ]]]. As input coefficients one may always use an integer or a quotient of integers with non-zero denominator in the field.

To specify weights of the generators, use the option lieAlgebra(...,Weights=>...). The weights are lists of integers of the same length, where the first degree is positive, also just called the degree, see firstDegree(LieElement). The degree is the two first inputs for the function dims(ZZ,ZZ,LieAlgebra). If the option lieAlgebra(...,LastWeightHomological=>...) is true, then the last degree is the homological degree, which is non-negative and less than the first degree. If the option lieAlgebra(...,LastWeightHomological=>...) is false, which is the default value, then the program defines the homological degree to be zero, and adds a last component 0 to the existing degrees. In this case the weights can also be given as a list of positive integers that will be the degrees. See Differential Lie algebra tutorial how to define a differential. Use weight to compute the weight of an arbitrary homogeneous Lie expression.

 i3 : L = lieAlgebra({a,b,c},Weights => {{1,-1},{1,4},{2,3}}) o3 = L o3 : LieAlgebra i4 : weight b o4 = {1, 4, 0} o4 : List i5 : ex = a b+(1/3) c o5 = (1/3)c - (b a) o5 : L i6 : weight ex o6 = {2, 3, 0} o6 : List i7 : firstDegree ex o7 = 2 i8 : L1 = lieAlgebra({a,b,c},Weights => {1,1,2}) o8 = L1 o8 : LieAlgebra i9 : describe L1 o9 = generators => {a, b, c} Weights => {{1, 0}, {1, 0}, {2, 0}} Signs => {0, 0, 0} ideal => {} ambient => L1 diff => {} Field => QQ computedDegree => 0

To specify signs of the generators, use the option lieAlgebra(...,Signs=>...). The sign of a generator is either 0 (even) or 1 (odd). By default, the sign of all generators are 0. Writing the option as "Signs => 1" defines the signs of all the generators to be 1. The signs affect the axioms of a Lie superalgebra, see LieAlgebra. Use sign to compute the sign of an arbitrary homogeneous Lie expression.

 i10 : L = lieAlgebra({a,b,c},Weights => {{1,-1},{1,4},{2,3}}, Signs=>{0,1,1}) o10 = L o10 : LieAlgebra i11 : ex = a b+(1/3) c o11 = (1/3)c - (b a) o11 : L i12 : sign ex o12 = 1

For each finitely presented Lie algebra $L$, ambient(L) represents a free Lie algebra $M$ such that $L$ (without differential) is a quotient of $M$, see also Quotient Lie algebras and subspaces. Moreover, the relations belong to $M$, even if $L$ is defined as a quotient in two or more steps from $M$. This is somewhat different from the situation for rings.

 i13 : M = lieAlgebra{a,b,c} o13 = M o13 : LieAlgebra i14 : L = M/{a b} o14 = L o14 : LieAlgebra i15 : a c o15 = - (c a) o15 : L i16 : Q = L/{a c} o16 = Q o16 : LieAlgebra i17 : ideal(Q) o17 = { - (b a), - (c a)} o17 : List i18 : oo_1 o18 = - (c a) o18 : M i19 : ambient Q o19 = M o19 : LieAlgebra

The relations may have a non-zero linear part. Use minimalPresentation(ZZ,LieAlgebra) to get a minimal presentation. Use describe(LieAlgebra) to get information about the Lie algebra. This contains the value of the three options lieAlgebra(...,Weights=>...), lieAlgebra(...,Signs=>...), lieAlgebra(...,Field=>...), and also the values generators(LieAlgebra), ideal(LieAlgebra), ambient(LieAlgebra), diff(LieAlgebra) and computedDegree(LieAlgebra)

 i20 : F = lieAlgebra({a,b,c},Weights => {1,1,2}, Signs=>{1,0,0}) o20 = F o20 : LieAlgebra i21 : L = F/{-c+a a,(a b) a} o21 = L o21 : LieAlgebra i22 : dims(1,4,L) o22 = {2, 2, 1, 2} o22 : List i23 : describe L o23 = generators => {a, b, c} Weights => {{1, 0}, {1, 0}, {2, 0}} Signs => {1, 0, 0} ideal => { - c + (a a), - (1/2)(b a a)} ambient => F diff => {} Field => QQ computedDegree => 4 i24 : M = minimalPresentation(4,L) o24 = M o24 : LieAlgebra i25 : describe M o25 = generators => {a, b} Weights => {{1, 0}, {1, 0}} Signs => {1, 0} ideal => {(b a a)} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0

In L#cache there is more information about $L$ collected during a computation. Some parts of the information are visible to the user, e.g., lieRing and mbRing. The ring L#cache.lieRing is the internal polynomial ring representation of Lie elements. The Lie monomials are represented as commutative monomials in this ring. The program produces internally an ideal of relations and a basis, which may be seen as L#cache.gb and L#cache.basis. The number of generators in lieRing is the number of generators in the Lie algebra times the internal counter L#cache.max, which initially is set to $5$, and is changed to $n+5$ if the computation of $L$ is performed up to degree $n$ with $n\ >$ L#cache.max.

 i26 : L = lieAlgebra{a,b}/{a a a b,b b b a} o26 = L o26 : LieAlgebra i27 : dims(1,4,L) o27 = {2, 1, 2, 1} o27 : List i28 : peek L#cache o28 = CacheTable{ambient => LieAlgebra{...10...} } basis => MutableHashTable{...15...} degree => 4 degrees => MutableHashTable{...4...} dim => MutableHashTable{...5...} gb => MutableHashTable{...4...} generators => {a, b} lieRing => QQ[aR ..aR ] 0 9 max => 5 mbRing => QQ[mb , mb , mb , mb , mb , mb ] {1, 0} {1, 1} {2, 0} {3, 0} {3, 1} {4, 0} opL => MutableHashTable{} i29 : L#cache.basis#4 o29 = {aR aR aR aR } 0 3 4 7 o29 : List i30 : basis(4,L) o30 = {(b a b a)} o30 : List i31 : L#cache.gb#4 o31 = ideal (aR aR aR aR , aR aR aR aR - aR aR aR aR , aR aR aR aR ) 0 3 5 7 0 3 5 6 0 3 4 7 0 3 4 6 o31 : Ideal of QQ[aR ..aR ] 0 9 i32 : a b b a o32 = (b a b a) o32 : L i33 : dims(1,6,L) o33 = {2, 1, 2, 1, 2, 1} o33 : List i34 : peek L#cache o34 = CacheTable{ambient => LieAlgebra{...10...} } basis => MutableHashTable{...28...} degree => 6 degrees => MutableHashTable{...6...} dim => MutableHashTable{...7...} gb => MutableHashTable{...6...} generators => {a, b} lieRing => QQ[aR ..aR ] 0 21 max => 11 mbRing => QQ[mb , mb , mb , mb , mb , mb , mb , mb , mb ] {1, 0} {1, 1} {2, 0} {3, 0} {3, 1} {4, 0} {5, 0} {5, 1} {6, 0} opL => MutableHashTable{...10...}

The ring L#cache.mbRing is used to get an output of Lie elements with indexed basis elements, which sometimes is better to use than the iterated Lie products of generators, especially in high degrees. Use indexForm to get the output in mbRing and standardForm(RingElement,LieAlgebra) to get back the standard output. The ring mbRing is very large; it has as many generators as the total dimension of the computed Lie algebra. For this reason, you should give the ring a name to avoid a large output.

 i35 : L = lieAlgebra{a,b,c} o35 = L o35 : LieAlgebra i36 : x = (a b c a) (b c) o36 = (b c a c b a) - (b c b a c a) - (b c c a b a) - (c b a c b a) + (c b b a c a) + (c b c a b a) o36 : L i37 : Q = L#cache.mbRing o37 = Q o37 : PolynomialRing i38 : numgens Q o38 = 196 i39 : indexForm x o39 = mb - mb - mb + mb + mb - mb {6, 22} {6, 24} {6, 43} {6, 45} {6, 74} {6, 76} o39 : Q i40 : standardForm(oo,L) o40 = (b c a c b a) - (b c b a c a) - (b c c a b a) - (c b a c b a) + (c b b a c a) + (c b c a b a) o40 : L


text:
# What is a “lateral fringe displacement”?

I encountered the term "lateral fridge displacement" in my optics homework for a problem about inserting a thin plate of glass over one of Young's double splits.

So what does "lateral fringe displacement" mean in this context? I Googled but found nothing quite useful. Most of the results are about how this phenomenon emerges, but not specific about the exact definition.

-

@AlexSu: every fringe spacing move is a phase shift of 2$\pi$. You can measure the displacement as a distance on the screen, an angular displacement or a phase change displacement. They are all different ways of measuring the same thing. –  John Rennie Jun 3 '13 at 5:36


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# Complicated with the quotient topology (Möbius strip) two distinct quotients

Let follow this definition of manifold. An n-manifold is a Hausdorff Topological space, Such That Each point you have an open neighborhood homeomorphic to the open disc $$U^n = \left\{ {x \in R^n :\left| x \right| < 1} \right\}$$

Let this set:

$$X = \left\{ {\left( {x,y} \right) \in R^2 /\,\,x \in \left[ { - 10,10} \right],\,y \in \left[ { - 1,1} \right]} \right\}$$

Define the quotient (10, y) related to (-10,-y) for -1

In the book ( Massey) comes out that if we consider the edges $y=1 ,y=-1$ Would not be a manifold under our definition. I do not understand why anyway would say that a manifold with boundary, if it refers not to ask more things or completely different. I'm just learning this quotient topology and I can not imagine the folds and stuff) =. If someone can give me advice on what to do with that, I really appreciate it

Edit: (TB) In order to give some context, let me quote the relevant passage from Massey, (assuming A basic course in algebraic topology, Springer GTM 127 was meant).

From the bottom of page 3:

The simplest example of a $2$-dimensional manifold exhibiting this phenomenon [non-orientability] is the well-known Möbius strip. As the reader probably knows, we construct a model of a Möbius strip by taking a long, narrow rectangular strip of paper and gluing the ends together with a half twist (see Figure 1.1). Mathematically, a Möbius strip is a topological space that is described as follows. Let $X$ denote the following rectangle in the plane: $$X = \{(x,y) \in \mathbf{R}^2 : -10 \leqq x \leqq +10, \;-1 \lt y \lt +1\}.$$ We then form a quotient space of $X$ by identifying the points $(10,y)$ and $(-10,-y)$ for $-1 \lt y \lt +1$. Note that the two boundaries of the rectangle corresponding to $y=+1$ and $y=-1$ were omitted. This omission is crucial; otherwise the result would not be a manifold (it would be a “manifold with boundary,” a concept we will take up in Chapter XIV [more precisely, XIV.§7, p.375ff]). Alternatively, we could specify a certain subset of $\mathbf{R}^3$ which is homeomorphic to the quotient space just described.

Unfortunately, Google managed to garble Figure 1.1, so here it is in full:

-
Are you asking for the definition of a manifold with boundary? Or are you asking why this quotient is not a manifold (without boundary)? –  Dylan Moreland Sep 4 '11 at 0:57
The difference is that the manifold with boundary it´s locally homeomorphic to $$R^n$$ or to $$R^n _ +$$ but how can i see geometrically this two folds )=? sorry for this stupids questions T_T –  Daniel Sep 4 '11 at 1:07
My new interpretation is that you want to know how to show that $X$ is a $2$-manifold with boundary. –  Dylan Moreland Sep 4 '11 at 1:49
Ok , That will help me )= –  Daniel Sep 4 '11 at 1:54
What's the question? –  Ryan Budney Sep 4 '11 at 1:57

So to show this is a 2 manifold with boundary you have to show that around each point there is a neighborhood that is either homeomorphic to $D^2$ or $D^2_+= \{(x,y)\in \mathbb{R} | \,\,\,\, y\geq 0, \,\,\,\, |(x,y)|<1 \}$.
Let $X$ be the described set $X / \sim$ the quotient and $\pi$ the quotient homomorphisim.
For $x \in \pi( \text{int} \, ( X )) = \text{int} \, (X)$ we are done, this set is homeomorphic to the disk. On $\text{int} \,(X)$, $\pi$ is a homeomorphism.
For $x \in \pi( (-10, 10) \times \{1\})$ consider $\pi((-10, 10) \times [1,-1))$. Similarly for the other side.
For $x \in \pi( \{10\} \times (-1,1) )$ it is more difficult. Here we have to somehow work with the twist. Let $f: [-10,-9) \cup (9,10] \times (-1,1) / \sim \,\, \to (-1,1)^2$ be: $$f(x,y) = \left\{ \begin{array}{lr} (x-10,y) & : x \in (9,10] \\ (x+10,-y) & : x \in [-10,-9) \end{array} \right.$$ I claim that this is continuous and bijective. Pulling $f$ back to $X$, i.e. considering $f \circ \pi : X \to (-1,1)^2$, it is continuous (this is the universal property of quotients). And it is bijective as $f \circ \pi$ is 1 to 1 except for the points that are identified where is is 2 to 1. But those points are identified so $f$ is 1 to 1 and onto. $f$ is also an open map, any open set in $X / \sim$ is the union of the images of an open sets from $X$ and $f \circ \pi$ is clearly an open map.


text:
# `x=arcsint , y=ln(sqrt(1-t^2)) , 0<=t<=1/2` Find the arc length of the curve on the given interval.

Arc length of a curve C described by the parametric equations x=f(t) and y=g(t), `a<=t<=b` where f' and g' are continuous on [a,b] and C is traversed exactly once as t increases from a to b, then the length of the curve is given by,

`L=int_a^bsqrt((dx/dt)^2+(dy/dt)^2)dt`

We are given:`x=arcsin(t),y=ln(sqrt(1-t^2)), 0<=t<=1/2`

...

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Arc length of a curve C described by the parametric equations x=f(t) and y=g(t), `a<=t<=b` where f' and g' are continuous on [a,b] and C is traversed exactly once as t increases from a to b, then the length of the curve is given by,

`L=int_a^bsqrt((dx/dt)^2+(dy/dt)^2)dt`

We are given:`x=arcsin(t),y=ln(sqrt(1-t^2)), 0<=t<=1/2`

`x=arcsin(t)`

`dx/dt=1/sqrt(1-t^2)`

`y=lnsqrt(1-t^2)`

`dy/dt=1/sqrt(1-t^2)d/dt(sqrt(1-t^2))`

`dy/dt=1/sqrt(1-t^2)(1/2)(1-t^2)^(1/2-1)(-2t)`

`dy/dt=-t/(1-t^2)`

Now let's evaluate arc length by using the stated formula,

`L=int_0^(1/2)sqrt((1/sqrt(1-t^2))^2+(-t/(1-t^2))^2)dt`

`L=int_0^(1/2)sqrt(1/(1-t^2)+t^2/(1-t^2)^2)dt`

`L=int_0^(1/2)sqrt((1-t^2+t^2)/(1-t^2)^2)dt`

`L=int_0^(1/2)sqrt(1/(1-t^2)^2)dt`

`L=int_0^(1/2)1/(1-t^2)dt`

`L=int_0^(1/2)1/((1+t)(1-t))dt`

Using partial fractions integrand can be written as :

`L=int_0^(1/2)1/2(1/(1+t)+1/(1-t))dt`

Take the constant out and use the standard integral:`int1/xdx=ln|x|+C`

`L=1/2int_0^(1/2)(1/(1+t)+1/(1-t))dt`

`L=1/2[ln|1+t|+ln|1-t|]_0^(1/2)`

`L=1/2{[ln|1+1/2|+ln|1-1/2|]-[ln1+ln1]}`

`L=1/2[ln|3/2|+ln|1/2|]`

`L=1/2[0.4054651081+0.69314718056]`

`L=1/2[1.09861228867]`

`L=0.54930614433`

Arc length of the curve on the given interval is `~~0.549`

Approved by eNotes Editorial Team


text:
Open Neighbourhoods of Points in a Topological Space Examples 2

The Open Neighbourhoods of Points in a Topological Space Examples 2

Recall from The Open Neighbourhoods of Points in a Topological Space page that if $(X, \tau)$ is a topological space then an open neighbourhood of the point $x \in X$ is any open set $U \in \tau$ such that $x \in U$.

We will now look at some more examples of open neighbourhoods of points in topological spaces.

Example 1

Consider the topological space $(\mathbb{R}, \tau)$ where $\tau$ is the countable complement topology. Give two examples of open neighbourhoods of $1 \in \mathbb{R}$.

The open neighbourhoods of $1 \in \mathbb{R}$ are the open sets containing $1$.

Consider the set $A = ( \mathbb{R} \setminus \mathbb{N} ) \cup \{ 1 \}$. Then $A^c = \mathbb{N} \setminus \{ 1 \}$ which is a countable set. Therefore $A \in \tau$ and $1 \in A$, so $A$ is an open neighbourhood of $1$.

For another example, consider the set $B = \mathbb{R} \setminus \{ 2 \}$. Then $B^c = \{ 2 \}$ which is a countable set. Therefore $B \in \tau$ and $1 \in B$, so $B$ is an open neighbourhood of $1$.

In fact, for any $b \in \mathbb{R}$ where $b \neq 1$, then if we define $B_b = \mathbb{R} \setminus \{ b \}$ then $B_b^c = \{ b \}$ which is countable so $B_b \in \tau$, and so $B_b$ is an open neighbourhood of $1$ since $1 \in B_b$.

Example 2

Consider the set $X = \{ a, b, c, d, e \}$. Find a non-discrete topology $\tau$ on $X$] such that [[$a, b \in X$ share exactly $3$ open neighbourhoods.

If $a$ and $b$ share exactly $3$ open neighbourhoods then there must be $3$ open sets containing both $a$ and $b$. Consider the following topology:

(1)
\begin{align} \quad \tau = \{ \emptyset, \{d \}, \{a, b, c \}, \{a, b, c, d \}, \{ X \} \} \end{align}

Then $a, b \in U_1 = \{a, b, c \} \in \tau$, $a, b \in U_2 = \{a, b, c, d \} \in \tau$ and $a, b \in U_3 = X \in \tau$ so $a$ and $b$ share exactly $3$ open neighbourhoods.

Example 3

Let $X$ be a set and let $\tau_1$ and $\tau_2$ both be topologies on $X$. Consider the topological space $(X, \tau_1 \cap \tau_2)$. Show that if $U$ is an open neighbourhood of $x \in X$ with respect to the topology $\tau_1 \cap \tau_2$ then $U$ is also an open neighbourhood of $x$ with respect to $\tau_1$ and with respect to $\tau_2$.

Suppose that $U$ is an open neighbourhood of $x \in X$ with respect to the topology $\tau_1 \cap \tau_2$. Then $x \in U \in \tau_1 \cap \tau_2$. Since $U \in \tau_1 \cap \tau_2$ we have that $U \in \tau_1$ and $U \in \tau_2$. Therefore $x \in U \in \tau_1$ and $x \in U \in \tau_2$. So $U$ is an open neighbourhood of $x$ with respect to $\tau_1$ and with respect to $\tau_2$.


text:
## Thomas' Calculus 13th Edition

$-1$
Here, we have $\lim\limits_{x \to -\infty} f(x)=\lim\limits_{x \to -\infty} \dfrac{2^x+4^x}{5^x-2^x}$ Re-arrange as: $\lim\limits_{x \to -\infty} \dfrac{2^x+4^x}{5^x-2^x}(\dfrac{\dfrac{1}{2^x}}{\dfrac{1}{2^x}})=\lim\limits_{x \to -\infty} \dfrac{1+(\dfrac{4}{2})^x}{(\dfrac{5}{2})^x-1}$ and $\dfrac{1+0}{0-1}=-1$


text:
Last edited by Vurn
Monday, October 5, 2020 | History

12 edition of Groups, representations, and physics found in the catalog.

# Groups, representations, and physics

## by H. F. Jones

Written in English

Subjects:
• Group theory.,
• Representations of groups.,
• Finite groups.,
• Continuous groups.,
• Transformations (Mathematics),
• Mathematical physics.

• Edition Notes

Includes bibliographical references (p. [291]-292) and index.

Classifications The Physical Object Statement H.F. Jones. LC Classifications QC20.7.G76 J66 1998 Pagination xiii, 326 p. : Number of Pages 326 Open Library OL345230M ISBN 10 0750305053 LC Control Number 98004227

thereby giving representations of the group on the homology groups of the space. If there is torsion in the homology these representations require something other than ordinary character theory to be understood. This book is written for students who are studying nite group representation theory beyond the level of a rst course in abstract Size: 1MB. This is an elementary introduction to the representation theory of real and complex matrix groups. The text is written for students in mathematics and physics who have a good knowledge of differential/integral calculus and linear algebra and are familiar with basic facts .

Thankfully, I eventually ran into “Groups, Representations and Physics,” 2nd ed. by H. F. Jones, Institute of Physics Publishing, Bristol and Philadelphia, There is an old, high school book which can serve as a great introduction: “Groups and Their Graphs” by Israel Grossman and Wilhelm Magnus, The Mathematical Association of. In fact, such groups play a large role in quantum mechanics and elementary particle physics, and some facts about them should be noted here. Their representations share many of the properties of the representations of finite groups, but the methods of working with the groups is somewhat different.

Space Groups and Their Representations - Ebook written by Gertjan Koster. Read this book using Google Play Books app on your PC, android, iOS devices. Download for offline reading, highlight, bookmark or take notes while you read Space Groups and Their : Gertjan Koster. Lie algebras Lie groups quantization quantum fields quantum mechanics representation theory Standard Model of particle physics unitary group representations two-state systems Lie algebra representations rotation and spin groups momentum and free particle fourier analysis and free particle Schroedinger representation Heisenberg group Poisson bracket and symplectic geometry Hamiltonian vector.

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### Groups, representations, and physics by H. F. Jones Download PDF EPUB FB2

Illustrating the fascinating interplay between physics and mathematics, Groups, Representations and Physics, Second Edition provides a solid foundation in the theory of groups, particularly group representations. For this new, fully revised edition, the author has enhanced the book's usefulness and widened its appeal by adding a chapter on the Cited by: This good book is recommended for mathematicians, physicists, philosophers of physics, researchers, and advanced students in mathematics and physics, as well as for readers with some elementary physics, multivariate calculus and linear algebra courses.” (Michael M.

5/5(3). Groups, Representations and Physics book. Read reviews from world’s largest community for readers. Illustrating the fascinating interplay between physics /5.

Δ It is important to keep in mind which vector space is meant in a given situation; e.g. in the context of a representation of $${SO(n)}$$ on an object in $${\mathbb{R}^{m}}$$, there is the space $${\mathbb{R}^{n}}$$ used to define the group, the possibly different Euclidean space $${\mathbb{R}^{m}}$$ the representation is acting on, and the space $${\mathbb{R}^{n\left(n-1\right)/2}}$$ that.

Illustrating the fascinating interplay between physics and mathematics, Groups, Representations and Physics, Second Edition provides a solid foundation in the theory of groups, particularly group representations.

For this new, fully revised edition, the author has enhanced the book's usefulness and widened its appeal by adding a chapter on the Cartan-Dynkin treatment of Lie algebras. Groups are one of the simplest and most prevalent algebraic objects in physics.

Geometry, which forms the foundation of many physical models, is concerned with spaces and structures that are preserved under transformations of these spaces.

Quantum Theory, Groups and Representations: An Introduction Peter Woit Department of Mathematics, Columbia University [email protected] Illustrating the fascinating interplay between physics and mathematics, Groups, Representations and Physics, Second Edition provides a solid foundation in the theory of groups, particularly group representations.

For this new, fully revised edition, the author has enhanced the book's usefulness and. Buy Groups, Representations and Physics 1 by Jones, H.F. (ISBN: ) from Amazon's Book Store.

Everyday low prices and free delivery on eligible orders/5(5). Download PDF Groups, Representations and Physics, by H.F Jones. What type of book Groups, Representations And Physics, By H.F Jones you will like to. Now, you will not take the published publication. It is your time to get soft documents book Groups, Representations And Physics, By H.F Jones rather the published papers.

Quantum Theory, Groups and Representations An Introduction. Authors: Woit, Peter While an elementary physics course of some kind would be helpful to the reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra.

Many exercises are provided to Brand: Springer International Publishing. Groups, Representations and Physics by H. Jones,available at Book Depository with free delivery worldwide/5(8). This is an elementary introduction to the representation theory of real and complex matrix groups.

The text is written for students in mathematics and physics who have a good knowledge of differential/integral calculus and linear algebra and are familiar with basic facts Brand: Vieweg+Teubner Verlag. Quantum Theory, Groups and Representations book. Read 3 reviews from the world's largest community for readers.

PDF available for free download here/5. Illustrating the fascinating interplay between physics and mathematics, Groups, Representations and Physics, Second Edition provides a solid foundation in the theory of groups, particularly group representations. For this new, fully revised edition, the author has enhanced the book's usefulness and widened its appeal by adding a chapter on the /5(5).

Get this from a library. Groups, representations, and physics. [H F Jones] -- Illustrating the fascinating interplay between physics and mathematics, this book provides a solid grounding in the theory of groups, and particularly of group representations. It gives the reader a. There is a book titled "Group theory and Physics" by Sternberg that covers the basics, including crystal groups, Lie groups, representations.

I think it's a good introduction to the topic. To quote a review on Amazon (albeit the only one): "This book is an excellent introduction to the use of group theory in physics, especially in crystallography, special relativity and particle physics.

Symmetry, Groups, And Representations In Physics by Dimitri D. Vvedensky,available at Book Depository with free delivery worldwide/5(2).

This book presents the study of symmetry groups in Physics from a practical perspective, i.e. emphasising the explicit methods and algorithms useful for the practitioner and profusely illustrating by examples.

The first half reviews the algebraic, geometrical and topological notions underlying the. This book is an introduction to symmetry in physics based on discrete and continuous groups. No knowledge of algebra is assumed and the book is suitable for both beginning and advanced graduate students.

Illustrating the fascinating interplay between physics and mathematics, Groups, Representations and Physics, Second Edition provides a solid foundation in the theory of groups, particularly group representations.

For this new, fully revised edition, the author has enhanced the book's Price: \$There is a natural connection between particle physics and representation theory, as first noted in the s by Eugene Wigner. It links the properties of elementary particles to the structure of Lie groups and Lie ing to this connection, the different quantum states of an elementary particle give rise to an irreducible representation of the Poincaré group.

Download PDF Abstract: These notes give an elementary introduction to Lie groups, Lie algebras, and their representations.

Designed to be accessible to graduate students in mathematics or physics, they have a minimum of prerequisites. Topics include definitions and examples of Lie groups and Lie algebras, the relationship between Lie groups and Lie algebras via the exponential mapping, the Cited by:


text:
# zbMATH — the first resource for mathematics

Scalable semidefinite programming. (English) Zbl 07368784
##### MSC:
 90C22 Semidefinite programming 65K05 Numerical mathematical programming methods 65F99 Numerical linear algebra
##### Software:
SDPLR; SDPT3; Manopt; Algorithm 971; DIMACS; Mosek; TSPLIB; SeDuMi; QAPLIB; ARPACK
Full Text:
##### References:
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Liang, Semidefinite relaxations for certifying robustness to adversarial examples, in Advances in Neural Information Processing Systems 31, 2018, pp. 10900-10910. [83] G. Reinelt, TSPLIB, http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/. [84] D. M. Rosen, L. Carlone, A. S. Bandeira, and J. J. Leonard, SE-sync: A certifiably correct algorithm for synchronization over the special euclidean group, Int. J. Robot. Res., 38 (2019), pp. 95-125. [85] M. F. Sahin, A. Eftekhari, A. Alacaoglu, F. Latorre, and V. Cevher, An inexact augmented Lagrangian framework for nonconvex optimization with nonlinear constraints, in Advances in Neural Information Processing Systems 32, 2019, pp. 13965-13977. [86] S. Sahni and T. Gonzalez, P-complete approximation problems, J. ACM, 23 (1976), pp. 555-565. · Zbl 0348.90152 [87] A. Silveti-Falls, C. Molinari, and J. Fadili, Generalized Conditional Gradient with Augmented Lagrangian for Composite Minimization, https://arxiv.org/abs/1901.01287, 2019. · Zbl 1450.65054 [88] M. Souto, J. D. Garcia, and A. Veiga, Exploiting Low-Rank Structure in Semidefinite Programming by Approximate Operator Splitting, https://arxiv.org/abs/1810.05231, 2018. [89] N. Srebro, Learning with Matrix Factorizations, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, 2004. [90] J. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11-12 (1999), pp. 625-653. · Zbl 0973.90526 [91] Y. Sun, Y. Guo, J. A. Tropp, and M. Udell, Tensor random projection for low memory dimension reduction, in NeurIPS Workshop on Relational Representation Learning, 2018. [92] K. A. Toh, M. J. Todd, and R. H. Tütüncü, SDPT3-a Matlab software package for semidefinite programming, optimization methods and software, Optim. 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Fercoq, F. Locatello, and V. Cevher, A conditional gradient framework for composite convex minimization with applications to semidefinite programming, in Proceedings of the 35th International Conference on Machine Learning, PMLR 80, 2018, pp. 5727-5736. [104] A. Yurtsever, Y.-P. Hsieh, and V. Cevher, Scalable convex methods for phase retrieval, in Proceedings of the 6th International IEEE Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2015, pp. 381-384. [105] A. Yurtsever, Q. Tran Dinh, and V. Cevher, A universal primal-dual convex optimization framework, in Advances in Neural Information Processing Systems 28, 2015, pp. 3150-3158. [106] A. Yurtsever, M. Udell, J. Tropp, and V. Cevher, Sketchy decisions: Convex low-rank matrix optimization with optimal storage, in Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR, 54, 2017, pp. 1188-1196. [107] M. Zaslavskiy, F. Bach, and J. 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text:
# Internal Energy and entropy in a open system

$$U= TS-PV+\sum_i \mu_i N_i$$

Let's suppose i=1 and a ideal gas. We know: $$U=N/N_A c_v T$$;

$$PV=NKT$$;

$$\mu=\frac{\partial U}{\partial N}= c_v T/N_A$$;

so $S=\frac{U+PV-\mu N}{T}$ would be $NK$

but it's not true. Where is the mistake?

-
I am having a hard time figuring out what each symbol means. Could you explicitly identify them, please? –  Mark Eichenlaub Mar 28 '11 at 2:11

The problem here is that the thermodynamic potentials are functions of three thermodynamic variables each. Now, each thermodynamic potential has a set of natural variables. For the internal energy $U$, these are S, V and N.
Now, your partial derivatives should explicitly state which other variables are held constant. For example, $$\mu = (\frac{\partial U}{\partial N})_{S,V}$$ i.e. you keep $S$ and $V$ constant.
You are, of course, free to express the internal energy as a function of other quantities, such as temperature instead of entropy. This would be achieved by expressing $S$ as a function of $V$, $N$ and $T$, so $S = S(T,V,N)$ or $T = T(S,V,N)$.
Thus, if you want to use the equation $U = N/N_A c_v T$ to compute $\mu$, you get $1/N_A c_v T$ from the $N$-dependency, but also some contribution from $$(\frac{\partial T}{\partial N})_{S,V}$$.


text:
# Why does tension not do work in this pulley system? etc

I have a slight difficulty understanding the solution to the following problem:

A light inextensible string with a mass $M$ at one end passes over a pulley at a distance $a$ from a vertically fixed rod. At the other end of the string is a ring of mass $m(M>m)$ which slides smoothly on the vertical rod as shown in the figure. The ring is released from rest at the same level as the point from which the pulley hangs. If $b$ is the maximum distance the ring will fall, determine $b$ using the principle of virtual work.

The solution is:

Let $l$ be the length of string. Therefore,

$$x+(a^2+b^2)^{1/2}=l\tag{1}$$ Imagine a vertical displacement $\delta b$ of the ring along the rod.

$$\delta x+b(a^2+b^2)^{-1/2}\delta b=0\tag{2}$$ Therefore, $$\delta x=-b(a^2+b^2)^{-1/2}\delta b$$

The constraints over the pulley and rod do no work. By the principle of virtual work, $$Mg\delta x + mg \delta b=0$$ Substituting value of $\delta x$: $$-Mgb(a^2+b^2)^{-1/2}\delta b = -mg\delta b$$ Since $\delta b$ is arbitrary, $$b^2=\frac{m^2}{M^2}(a^2+b^2)$$ $$b=\frac{ma}{(M^2-m^2)^{1/2}}$$

My questions are:

1. How is Equation 2 obtained from Equation 1? (especially with regard to the intermediate steps)
2. When I attempted the problem, I had the tension in the string doing work, because the displacement of the string is in line with the tension in the string. Why is this not being considered? Thanks in advance.

-

For your first question: obtaining equation (2) from (1) is basically just a matter of taking the derivative. Consider this expression that relates an infinitesimal change in the value of a function $f(x,b)$ to the infinitesimal changes in the values of its arguments:

$$\delta f(x,b) = \frac{\partial f(x,b)}{\partial x}\delta x + \frac{\partial f(x,b)}{\partial b}\delta b$$

If you've ever done anything with uncertainty analysis, in particular error propagation, you should be familiar with this sort of thing, but if not, it's still not too complicated - it's just the chain rule of multivariable calculus. Try plugging in the left side of equation (1) for $f(x,b)$ and see that it works out.

A different way to do the same derivation is to consider the right triangle whose legs are formed by the rod and the "ceiling" and whose hypotenuse is formed by the string. Originally, the triangle has side lengths $a$, $b$, and $l - x$; after an infinitesimal displacement of the system, it has side lengths $a$, $b + \delta b$, and $l - x - \delta x$. Using the Pythagorean theorem on the first set of lengths, you get

$$a^2 + b^2 = (l - x)^2$$

and on the second set of lengths, you get

$$a^2 + (b + \delta b)^2 = (l - x + \delta x)^2$$

Try combining these to see that you get equation (2) out of it. Remember that because the $\delta$ quantities are infinitesimal, you can ignore terms involving $\delta x^2$ and $\delta b^2$.

As for the second question: probably the easiest way to think about it is that for a static system, the principle of virtual work is equivalent to potential energy minimization. To find the state of the system, you find what values of the coordinates minimize the potential energy, which in this case is $-Mgx - mgb$. Ordinarily, when you want to minimize a function, you take the derivative and set it equal to zero, but you can just as well calculate an infinitesimal change using the formula above and set that equal to zero. Try setting $f(x,b)$ in that formula equal to the potential energy.

If you're still concerned about tension, though, you could make the following argument, staying true to the spirit of virtual work. As the ring moves downward by a small amount $\delta b$, the work done on it is

$$\delta W_m = mg\delta b - T\sin\theta\,\delta b$$

where the factor $\sin\theta = \frac{b}{\sqrt{a^2 + b^2}}$ picks out the vertical component of the tension.

As this happens, the string gets pulled through the pulley, thus raising the larger mass by a displacement $\delta x$. The work done on the larger mass is going to be

$$\delta W_M = -Mg\delta x + T\delta x$$

When the system is in equilibrium, it will be in a state such that the no work gets done if it shifts a tiny amount in either direction. That means that the total work for an infinitesimal displacement should be zero,

$$\delta W_m + \delta W_M = 0$$

Using this and equation (2), namely $\delta x = -\frac{b}{\sqrt{a^2 + b^2}}\delta b$, you can show that the two tension terms, one from $\delta W_m$ and one from $\delta W_M$, cancel out. Any work done by the tension on one mass is canceled out by work done by tension on the other mass. So even if you don't actually take the tension into account, you still wind up getting the right result.

-
Thanks! It makes more sense now. –  Joebevo Jul 26 '12 at 10:22

What I recommend to you is, please come up with your solution and put it here. Then we can let you know where you had done the mistake. However, since you also said that - you had a problem in understanding the solution of the above problem...I will try my best to make you understand this solution.

When you look into the problem carefully (as he was asking you to determine $b$, you should understand and write the equations in which $b$ is involved as the parameter. By taking the Pythagoras Law,

$\text{Sum of the square of the Hypotenuse = Sum of the squares of the other two adjacent sides}$

Here in the problem, it is given that $a$ and $b$ are the lengths of the two adjacent sides of the triangles. And the third side can be determined by using the Pythagoras theorem. (i.e) $(a^2+b^2)^{1/2}$.

But, you should understand that there is an also extra length in addition to that, which is connected to the block of Mass $M$ and this length was given as $x$. $$\implies x + (a^2+b^2)^{1/2} = l$$.

But a small displacement in length $b$, should strike in your mind about partial differentiation. And in addition to this, you should also understand that the parameters which change during the SMALL vertical displacement are $x$ and $b$. So you should differentiate the above equation with respect to $f(x,b)$ [function with respect to x and b, where the function is the equation].

And the rest is obvious. I hope you might have understood this solution now.

-
Hi Sai, Welcome to Physics.SE. First of all, this is a site which deals with conceptual Q&A. As a new user, you should consider reading our FAQ and also our homework policy. Our site also supports a new Latex feature - "MathJax" equations. You could have a look at my revision of your post :-) –  Waffle's Crazy Peanut Dec 22 '12 at 12:49


text:
My Math Forum Integration. About -ve area and stationary value.

 Calculus Calculus Math Forum

 February 16th, 2011, 11:46 PM #1 Newbie   Joined: Jan 2011 Posts: 17 Thanks: 0 Integration. About -ve area and stationary value. Greetings to all. Okay, I have 2 simple questions about integration. They are summarized in topic title. 1) This is the figure: It shows a curve y=x^2 and the line y=3+2x. Point A is (3,9) and point B is (-1,1). We have to calculate the area of the region enclosed between the curve and the line segment AB. I have used the formula A=$integration$ (y2-y1) dx. Correct me if the formula is wrong. I've used the limits 3 and -1. Now, I have solved this and obtained the answer, but the problem is that the Area I'm getting is coming in negative. The answer to this question is 32/3 which is what I'm getting but only that it is -32/3. Is this something to do with the figure or have I done something wrong? 2) This is the second part of a question. First part is done. We were asked to show the Volume, V cm^3, was given by: V=12$pi$r^2-2$pi$r^3 [This is correct BTW]. Okay, now the second part of the question says that Given that r varies, find the stationary value of V. What I did here is, because the stat.value says that dV/dx = 0, I used: 0(dV/dx)=12$pi$r^2-2$pi$r^3. 0=24$pi$r-6$pi$r^2. After this, I don't know what else to do. The answer is coming as 64$pi$. But how is this coming? Any help? :
 February 17th, 2011, 12:10 AM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Integration. About -ve area and stationary value. Question 1: $A=\int_{-1}\,^{3}3+2x-x^2\,dx$ $A=$3x+x^2-\frac{1}{3}x^3$_{-1}^{3}=$$3(3)+(3)^2-\frac{1}{3}(3)^3$$-$$3(-1)+(-1)^2-\frac{1}{3}(-1)^3$$=$ $(9)-$$-\frac{5}{3}$$=\frac{32}{3}$ Question 2: $V=12\pi r^2-2\pi r^3$ $\frac{dV}{dr}=24\pi r-6\pi r^2=0$ Divide through by $-6\pi$ : $r^2-4r=0$ Factor: $r(r-4)=0$ Since $\frac{d^2V}{dr^2}=24\pi$ at r = 0, and $\frac{d^2V}{dr^2}=-24\pi$ at r = 4 we determine that V has local minimum at r = 0 and a local maximum at r = 4. Thus $V(4)=12\pi (4)^2-2\pi \(4)^3=64\pi$
 February 17th, 2011, 01:11 AM #3 Newbie   Joined: Jan 2011 Posts: 17 Thanks: 0 Re: Integration. About -ve area and stationary value. Woah! Can't believe I messed up the equation's arrangement. Gotta remember that about finding area in integration. About the second question, just one thing. You got r(r-4)=0 which would give us either r=0 or r=4. If we use r=0 in the original equation, we get left with 0. When we use r=4 in the original equation, we get 64.pi. So, we only take this value as the answer. But what if it was say (r-3)(r-4)=0? We'd have r=4 and r=3. So if plugged in r=3, we'd get : 12.pi(3)^2 - 2.pi(3)^3. This would give us 54.pi. Would we have 2 answers then? How would we write it? A little more help with the last one and this problem is done. Thanks for the help.
 February 17th, 2011, 07:00 AM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Integration. About -ve area and stationary value. It would depend on what type of extrema we were interested in finding. In this problem we were presumably looking for 0 < r < 6 that maximized V. You basically look at the behavior of the function over the domain of the independent variable and choose which extrema meets the criteria, if any at all. If maximizing or minimizing, you also need to look at the end-points to determine if those values are greater than or less than the local extrema.
 February 17th, 2011, 09:12 AM #5 Newbie   Joined: Jan 2011 Posts: 17 Thanks: 0 Re: Integration. About -ve area and stationary value. Thanks for the explanation. Guess this one is solved.

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# Feynmf — Two loop correction

How to draw a two (or even more) loop diagrams using Feynmf?

I tried

\documentclass{article}
\usepackage{feynmp}
\begin{document}
\unitlength = 1mm
\begin{fmffile}{twoloop}
\begin{fmfgraph}(100,60)
\fmfleft{i}
\fmfright{o}
\fmf{fermion}{i,v1,o}
\fmf{fermion}{v1,v2,v1}
\fmf{fermion}{v2,v2}
\end{fmfgraph}
\end{fmffile}
\end{document}

but it didn't work.

I plotted it using LaTeXit. The one loop diagram worked just fine so I guess it's the problem with my usage of Feynmf.

• Could you please make a full compilable example? – egreg Feb 15 '14 at 21:50

I have been able to solve this in a fashion:

\begin{fmfgraph*}(120,75)
\fmfleft{i1,i2,i3}
\fmfright{o1,o2,o3}
\fmf{plain}{i1,v1,o1}
\fmffreeze
\fmf{phantom}{i2,v2,o2}
\fmf{phantom}{i3,v3,o3}
\fmf{plain,left}{v1,v2,v1}
\fmf{plain,left}{v2,v3,v2}
\fmfdot{v1,v2}
\end{fmfgraph*}

This produces circles instead of egg shaped loops, and differently sized ones at that. I suppose one can fix the sizes, I'll update if I manage to get to that.


text:
Chapter 7

Modern surgery has been redefined by powered instruments, technological tools that in many ways have revolutionized the delicacy, precision, and accuracy of the various operations performed. Yet many people who use these implements every day have very little understanding of the technology behind these tools. Although a complete treatise on electromagnetic generation of heat and the physics of current generation are beyond the scope of this chapter (and are available elsewhere), understanding some fundamental rules governing the behavior of electrical currents and some relatively straightforward principles helps guide the use of these technologies.

### Principles of Electricity

An electrical circuit is any pathway that allows the uninterrupted flow of electrons. Electrical current is the flow of electricity (the number of electrons) in a given circuit over a constant period of time and is measured in amperes (A). Current can be supplied either as direct current (DC) with constant positive and negative terminals or as alternating current (AC) with constantly reversing poles. The electromotive force, or voltage, is a measurement of the force that propels the current of electrons and is related to the difference in potential energy between two terminals. The resistance is the tendency of any component of a circuit to resist the flow of electrons and applies to DC circuits. The equivalent of this tendency in an AC circuit is known as impedance. Any electromagnetic wave, from household electricity to radio broadcasts to visible light, can be described by three components: speed, frequency, and wavelength. Because all electromagnetic waves travel at the speed of light, which is a constant, these waves depend on the relationship between their frequency and wavelength. Since these three characteristics are defined by the equation:

c = fλ:

(where c is the speed of light, 2.998 × 108 m/s)

frequency (f) and wavelength (λ) are inversely related; ie, as frequency increases, wavelength decreases, and vice versa. The ability to pass high-frequency current through the human body without causing excess damage makes electrosurgery possible.

### Electrocautery

Electrosurgery is often incorrectly termed electrocautery, which is a separate technique. Electrocautery is a closed-circuit DC device in which current is passed through an exposed wire offering resistance to the current (Figure 7–1). The resistance causes some of the electrical energy to be dissipated as heat, increasing the temperature of the wire, which then heats tissue. In true electrocautery, no current passes through the patient. Electrocautery is primarily applied for microsurgery, such as ophthalmologic procedures, where a very small amount of heat will produce the desired effect or where more heat or current may be dangerous.

###### Figure 7–1.

In electrocautery, current passes through a wire loop and heats it. This heat cauterizes tissue. No current passes through the patient.

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# LaTeX notes

## Making nice PDFs (including from old .ps files)

If possible, use pdflatex over latex for hassle-free PDF generation. Include \usepackage{microtype} and get more sophisticated typesetting for free.

For normal latex, the newest recommended tool for making pdf seems to be dvipdfmx, however that won’t work with all packages. The switches I used to use with dvips are below. These avoid bitmap fonts, ensure ligatures don’t turn into pound-sterling symbols and force the page size to a4.

#!/bin/sh
dvips -Ppdf -G0 -o "${1%.dvi}"_pdf.ps "$1" ;
ps2pdf -sPAPERSIZE=a4 "${1%.dvi}"_pdf.ps "${1%.dvi}".pdf;

You may find that the above mangles some of your colour figures, as it might jpeg compress them (sometimes aggressively). The following switches to ps2pdf:

-dAutoFilterColorImages=false -sColorImageFilter=FlateEncode

losslessly compresses your images. You may experience an increase in file size.

What if you no longer have the .tex or .dvi source? If you have a document made with an old version of tex/latex, it is still possible to make a decent .pdf as long as you have a .ps file made by dvips:

pkfix-helper input.ps tmp.ps
pkfix tmp.ps output.ps

output.ps will now have Type 1 fonts, and can be converted to a good PDF with ps2pdf. The pkfix-helper man page says how to check and tweak the font substitutions.

## Making small PDF files

pdfsizeopt is a tool that can rewrite graphics, compress your PDF stream, and dramatically cut down the overhead of your fonts. Although it’s sensible to compress graphics once, before TeXing. It’s pdfsizeopt’s font handling I find really useful: documents made with pdflatex can now be as small or smaller than those from xelatex and lualatex.

Here are the arguments I use.

pdfsizeopt.py --do-unify-fonts=false --do-unify-pages=false infile.pdf

The first option is important: several of my PDFs are horribly corrupted without it. The second option prevents a (benign?) error message with some PDFs in xpdf/poppler. The following options are implicitly set and can be changed to false if you don’t have the required helper program:

--use-pngout=true --use-jbig2=true --use-multivalent=true

Fonts will still be reorganized if any of these options are set to false.

## Spacing issues

Somehow I missed out on knowing this stuff for a long time. lacheck and chktex are programs that will point out problems in your LaTeX markup (possibly with a load of false positives). Those tools know about these issues.

• If a period (full-stop) does not end a sentence then the space needs escaping so that it does not end up too long: Hello Dr.\ Murray.
• If a sentence ends with a capital letter then you need to tell LaTeX that it really is the end of a sentence: I modeled the data with a GP\@. The \@ is still needed if there is a closing parenthesis between the word and the period: The result is a Gaussian Process~(GP)\@.
• Use \textit{this is in italics} for italic text (or \emph{...} for more semantic markup). Otherwise you have to know about italic correction: {\it this is in italics\/} and {\it so is this}.
• Similarly \textbf{...} is sometimes different from {\bf ...}. Compare {\bf f}' to \textbf{f}' or {\bf f\/}' (including the quote marks).

## Italic issues

As explained above, use \textit{...} for italic text, not {\it ...}. However, you might not really want italic text. A long block of special text, for example a theorem statement, can be hard to read in italic. Consider using slanted Roman text, \textsl{...}, which is often easier to read, and mathematics stands out better.

Math mode should not be used to write words. For example, $a_{arg}$ looks awful, as the subscript is rendered as “a times r times g”. Instead use $a_\text{arg}$ or $a_\mathrm{arg}$. If you really want italic text, then use $a_\mathit{arg}$.

## More spacing issues — hitting page limits

Here are some notes on tricks for squeezing a paper to within conference page limits. There’s always PSSQ, but I like to stay more within the rules than that. Many “9 page” papers can be squashed into 8 pages without changing the font size or main spacing parameters.

• Look for suspiciously large vertical whitespaces between paragraphs. This is often due to TeX not wanting to split a paragraph over a page break, or separate it from a heading. Convincing TeX to pull some text back a page into this white space can have dramatic effects on the page count.
• There are various parameters documented in the TeXbook that make TeX more willing to do ugly things.
• If a paragraph has only a word or two on its last line try adding \looseness=-1 to the end of it. If possible TeX will reduce the length of the paragraph by a line. This won’t always work because there is a limit to how close TeX will move words.
• Moving figures around is a frustrating but often fruitful activity.
• In desperate situations negative kerns, e.g. \kern-1pt can enable words to be scrunched more than normally possible. Although rewording is probably a better option.
• bibhacks.sty makes the references heading third level and optionally scrunches the font size and spacing of references. See comments inside for more information.
• Move footnotes and short unreferenced equations into the main text. Consider turning third level headings into bold words at the beginning of a paragraph.
• It's usually pointless to engage in space scrunching until finishing the paper. And it's worth keeping a copy of the unscrunched version.

## Recommended packages

The following are packages I use regularly, some of which I didn’t discover for a long time:

• microtype — makes justification look better, especially with narrow columns. Requires pdflatex. Users of normal latex can get a subset of microtype’s features with: pdflatex -output-format=dvi
• booktabs — much nicer rules and spacing in tables.
• amsmath — makes a lot of common math constructs prettier and easier. Check out the good documentation.
• natbib — flexible referencing system.
• subfigure — allows you to create sub-figures optionally labelled with subcaptions preceded by (a), (b), etc. Use the [tight] option for better spacing. You can \ref and \subref labels within subfigures to get links to figure 1a) and a). As instructed by the subfigure documentation I now use subfig instead. Having a recent version of the caption package (which subfig includes) is recommended.
• url — better than putting URLs inside \texttt{...}. Line-wrapping works better and you don’t have to escape tildes.
• hyperref — creates clickable links for all types of cross-references. But there are issues (see below) and lots of options to read through and get right.
• artikel3 — an alternative class to article.
• parskip — a hack to get any style to have vertical space between paragraphs rather than indenting them (as artikel3 already does).
• textpos — allows absolute positioning on a page. Can be useful when press-ganging LaTeX into doing slides or a poster. I sometimes use this to put a DRAFT notice, or publication details in the top margin of a paper.

## hyperref problems

I have wasted a lot of time pinning down problems when using the hyperref package. Partly because I didn’t RTFM, which it turns out is more important than normal for this package.

• Links go to wrong page? Try putting \clearpage before the references section and \phantomsection before any \addcontentsline commands.
• Complaints about footnotes? Is hyperref included after all other packages, particularly setspace? If you cannot solve the problem then passing the hyperfootnotes=false option turns off footnote links.
• Complaints about duplicate entries? If it is a duplicate page then try passing plainpages=false,pdfpagelabels. If it is complaining about other objects then maybe you have altered the counters and need to be more careful somehow(?). Or maybe (again) it is due to including hyperref before other packages; try including hyperref later in your header.
• Package hyperref Warning: Token not allowed in a PDFDocEncoded string... removing math shift' on input line XXX. Solution: provide alternative plain text for maths that will end up in links. E.g. change \section{The joy of $\pi$} to \section{The joy of \texorpdfstring{$\pi$}{pi}}.
• Include the algorithm package after hyperref, but manually include the float package (which algorithm will normally include for you) before hyperref. (This is in the README of recent hyperref versions).
• Links to figures point to the caption, so the figure itself goes off the top of the screen. As in the hyperref README use the hypcap package:
\usepackage[all]{hypcap}
But! If you are using the subfig package make sure you have a recent version of the caption package from CTAN (subfig uses caption). Otherwise references to subfigures can go wrong.

If you have problems with a particular package see if hyperref’s README file says how to deal with it.

## Upside-down slides

If dvips produces upside-down slides, put this in your document’s preamble:

% This for broken installs that flip page wrong way
\special{! TeXDict begin /landplus90{true}store end }

This doesn’t seem to do any harm in general.

## pspresent

If you want to do fullscreen postscript instead of pdf—which is easy to fullscreen with xpdf or acroread—I’ve found pspresent to be a neat and tiny utility, which compiled and worked first time for me.

## Centering stuff vertically within a line

I currently use:

\newcommand{\myvcenter}[1]{\ensuremath{\vcenter{\hbox{#1}}}}

when I want graphics or other over-sized oddities to be vertically centered. Normally all objects have their baselines at the same height. Alternatives include fiddling with the baseline or using \raisebox.

I still haven’t got around to working out why plain \vcenter doesn’t work in LaTeX.

## Gaps to match

To create an empty box with the same size as some particular text use the \phantom command. Horizontal or vertical struts of the same width or height are made with \hphantom and \vphantom. See p178 of the TeXbook and more help on alignment with boxes.

## Fonts

It’s easiest to stick with using Computer Modern. But alternatives are good, especially for those that don’t like ‘modern’ typefaces.

I often do \usepackage{palatino} as a quick way of having something different, but there are other options. See The LaTeX Font Catalogue, and maybe this older survey of TeX fonts.

For non-math documents, I’m currently playing with “Linux Libertine”, with xelatex:

\usepackage{xltxtra}
\defaultfontfeatures{Mapping=tex-text}
\setromanfont [Ligatures={Common}]{Linux Libertine O}

One could also try getting the font in pdflatex with:

\usepackage[T1]{fontenc}
\usepackage{libertine}

The PDF file will be bigger (which is fixable), and the typesetting isn’t quite the same as xelatex’s.

## Embedding LaTeX text in diagrams

fragmaster will make .eps and .pdf files from a .eps file with plain text placeholders and a separate file with LaTeX markup. This strategy means you can create a range of diagrams in different packages and have a uniform way of typesetting text in them. It’s worked pretty well for me.

Of course many programs have their own ways of calling LaTeX. This is how I used to put LaTeX in Matlab figures (now I use fragmaster):

% The magic line is:
set(0,'defaulttextinterpreter','latex');

% Example:
plot(1:3,-(1:3))
legend('${\mathcal F}_1(x)$')
print(gcf, 'picture.eps', '-depsc');
system('epstopdf picture.eps');

% To get a small .eps file, convert back again:
system('pdftops -eps picture.pdf picture.eps');


text:
# Competition time: Design an error page!

Or a 404 or the "captcha" image.

To go with our amazing new design, we need some amazing pictures to go on the error page, the 404 page, and the captcha page. They should have a lighthearted theme, vaguely related to the purpose of the page, and - of course - be TeX-related (the TeX-purists would say that it has to be actually generated by TeX).

For a couple of examples from other sites, see this post on meta by our amazing designer.

So ... keyboards at the ready ... and ... go!

-
Notes: this is on the main site to garner a greater range of suggestions. It's CW so that people are encouraged to put up weird and wacky suggestions without fear of being voted down into oblivion. Of course, only post images that we can legally (ethically, morally, technically,...) use on the site. If suggestions don't flood in, I'll consider putting a bonus up for grabs (assuming I can do so on a CW question..). –  Loop Space Nov 12 '10 at 9:04
You can put bounties on CW qns. See meta.stackexchange.com/questions/20931/… –  Charles Stewart Nov 12 '10 at 11:02
I sense some TikZ-fu would be appropriate... –  Seamus Nov 12 '10 at 11:11

## 12 Answers

What about asking Duane Bibby to contribute some art? His drawings are iconic, and would help associate the site with the TeX "brand" (if an open source project can be said to have a brand). New users would be more inclined to trust the site.

CTAN lion drawing by Duane Bibby courtesy of CTAN.org.

We could get the lion scratching his head for the 404 page, or squinting at a screen for the captcha page, etc.

-
+1 for Duane Bibby –  erjiang Nov 14 '10 at 16:25
I've posted a bug about this on meta. –  SamB Jan 25 '11 at 0:10

(with apologies to Don Knuth)

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Haha, good one! –  ℝaphink Aug 23 '11 at 20:49

I actually liked @Juan's idea for using typesetting pieces to spell "Not Found". Being keen on 3D art, here's a quick mockup I put together just now:

Please excuse the shoddy appearance, I wanted just to give an idea. The final thing will be rendered photorealistically, so all the usual stuff like materials, textures, reflections, etc. will be in place. Also, no problem to use any typeface (that I have installed), or comes as OTF, TrueType or what have you. I think something serif-y would look great, with different typeface for each letter (combining thin and thick letters of different sizes, etc. For added appearance, the typeface bits could be of different "materials", and discernible difference in height, so that it's obvious they come from different typographical sets.

I'm expecting comments, suggestions and critiques, and if enough people think this is a good idea, I'll proceed and create something more finished.

P.S. For the curious, the software is SolidWorks 2009 (no affiliation).

Update: If this gets into double-digit upvotes, I promise a new draft rendered in full glory until Monday.

Now I notice that the letters are actually in the wrong order. Oh well.

Update 2. Here's a second (rendered) version. Quite bland at the moment, I need some extra textures for the floor, I'm thinking of something paper-like, non-reflective, maybe with the printed output of the typography set. The light certainly needs adjustments. The typographic sorts are probably a bit too thick (couldn't find correct measures so I improvised from images). Now, enjoy (or be appalled):

Typeface list upon request. Font suggestions welcome. Comments highly appreciated.

Update 3: Yet another version, with softer tints. But that's not a big deal, those kind of things can be handled with some post-processing, and I'm sure Jin would help with some final touches.

-
Looks really nice! Nice job ! –  mpg Nov 12 '10 at 13:48
Am I alone in thinking that it ought to be the correct way around? At the moment, that would typeset "DNUOF TON". Can we see it the other way around for comparison? That's making my head spin as it is. (But I like the idea very much) –  Loop Space Nov 12 '10 at 14:15
This is cool but I don't think it jibes with the current design of the site. –  Matthew Leingang Nov 12 '10 at 16:10
@Matthew: I think it does, or could. The error pages should be a bit jarring. Martin's idea of making the texture paper-like should reduce the design distance... –  Charles Stewart Nov 12 '10 at 18:10
I like it, but I agree with @Matthew that it looks a bit out of place with the site design. Maybe less 3D rendering, more illustration like? –  Jin Nov 12 '10 at 19:12

For me, something like $n_{0}(t) f_{0}(u^{n}d) = 404$ or 404 = \frac{n_{0}(t)}{f_{0}(u^{n}d)}, or even a commutative diagram like (using the old amscd package)

\begin{CD}
\dots @>>> N_{0}(T) @>{f_{0}}>> U_{n}(D) @>>> \dots \\
@. @VVV @VVV \\
\dots @>>> E_{r}(r) @>{o}>> R_{404} @>>> \dots
\end{CD}

or this one (using xypic)

\xymatrix@!0{
& N \ar@{->}[rr] \ar@{-}'[d][dd]
& & o \ar@{-}[dd]
\\
F \ar@{-}[ur] \ar@{<-}[rr] \ar@{->}[dd]
& & T \ar@{<-}[ur] \ar@{-}[dd]
\\
& u \ar@{->}'[r][rr]
& & n
\\
o \ar@{-}[rr] \ar@{->}[ur]
& & d \ar@{<-}[ur]
}

captures at least some of the essence of TeX...

Edit: These produce the following:

-
@Juan, thanks for posting the pictures. I've since changed the second diagram so that most of the "arrows" are just line segments, and the remaining ones form a path spelling "NotFound". It also looks nice with the line segments "\ar@{-}" replaced by "\ar@{..}". –  John Palmieri Nov 12 '10 at 22:39
I updated the picture. (Btw, code like this is why I love TikZ.) –  Caramdir Nov 15 '10 at 2:34
@Caramdir: Amen on the superiority of TikZ syntax. –  Matthew Leingang Nov 15 '10 at 14:50
+1 for the commuting box - I love it! –  David Roberts Dec 2 '10 at 6:46
While these are certainly clever, I suspect they're a bit too cryptic for most people. (I'll admit, when I first saw them, I didn't notice that it spelled things. I just saw commutative diagrams.) –  TH. Dec 28 '10 at 17:20

I already posted a suggestion on meta, but will repeat it here as well:

For a 404, there could be a play on some of most common TeX error messages. I think the Undefined control sequence is most appropriate.

As for an image, something simple like this:

\404{Page not found}

! Undefined control sequence.
l.6 \404

Or some variation thereof. Probably rendered in different styles, to distinguish user input from TeX complaints.

No time for a mockup today, but I'll try to post update later with an actual image.

-
By the way, \404 is not a valid control sequence (at least, it will not be parsed so with the usual catcodes). Not sure if this is a problem. I like the idea anyway. –  mpg Nov 12 '10 at 13:51
@mpg of course not! It's like meta-404 (btw LaTeX does produce the same error message when I tried it in code). –  Martin Tapankov Nov 12 '10 at 13:54
@mpg: To revive a joke best left alone, if you \catcode\4=11, then \def\4#14#2{\undefined} works. It won't give exactly that message, but that could be arranged... –  Charles Stewart Nov 12 '10 at 22:34
@Martin: Sure, but it wasn't clear to me whether this was intended. –  mpg Nov 13 '10 at 1:28
@Charles: No need for the catcode. \def\404 works just fine (for reasons that might not be totally clear). Give pdftex '\def\404{Hi!}\404\bye' a shot. –  TH. Nov 15 '10 at 15:00

For the 404/captcha/error images, I think their appearances should compliment the overall site design. This way, the TeX site looks more "complete." I think what'd work great with the site design, is a combination of TeX related insider joke/lingo, and old school illustration artwork. Since I know nothing about TeX I can only contribute to the visuals. As for old book illustrations, here are two great resources with public domain images:

http://www.grandmasgraphics.com/index.php

http://www.oldbookillustrations.com

For example, the rabbit from Alice in Wonderland makes a good Captcha image IMO.

-
To me, this just screams for a Duane Bibby cartoon –  Taco Hoekwater Nov 14 '10 at 10:10
@Taco I had to google who Duane bibby is. After seeing his work, I agree with you. –  Jin Nov 14 '10 at 10:16

Is there anything to defining 404 as \TeX is defined for something like ...

Obviously it could be enlarged and repeated or whatever, but I think its a fun start. The TeX is just:

\def\my404{4\kern-.025em\lower.5ex\hbox{0}\kern-.05em4}

Error \my404\ Not Found
\bye

Someone run with it if you like

-
ERROR: Undefined control sequence.

--- TeX said ---
l.4 \404 (Page not found)

--- HELP ---
TeX encountered an unknown page name. You probably misspelled the
url. If this message occurs when a LaTeX command is being processed,
the command is probably in the wrong place---for example, the error
can be produced by an \item command that's not inside a list-making
environment. The error can also be caused by a missing \documentclass
command.

-
Hmm, no. The aim is to be witty, not informative. –  Charles Stewart Nov 12 '10 at 11:30
It doesn't match the light-hearted tone of Knuth's error messages well. For an example, run tex '\relax}' and press h <RETURN> twice. –  Philipp Nov 12 '10 at 14:16

Here's a vague idea that floated into my head regarding the captcha page. This is (I'm told) meant to have something robotic about it, so I thought about something referring to the fact that TeX is Turing complete.

Either a terminal with a standard TeX initial output ending with some error message reading: "I'm Turing complete, are you?". Or the TeX lion sitting at a terminal with some relevant output on it (some Turing problem or other) with that phrase as a caption.

-

I think that giving TeX error messages for 404 could be confusing, specially because there already is a description of why the error is happening, see for example our current 404 page, so I would stay away from that.

Now an idea I have, continuing with the letter pressed effects (like the current “TeX” logo in the header), is to have a letter pressed “404” with the last “4” missing or fallen. Like if one was trying to “press” the “4” into place but somehow the type piece failed to properly print the image and is just lying there in defeat.

Hope that makes some sense to someone. :P

Edit: Along similar lines, how about a scrambled box of typesetting pieces spelling “Not Found” or “404” like some of the pictures here. I don't know, though, how could we create such a photo.

-
I see no problem with a TeX-like error message (that also plays on the 404/Page not found theme). Not less confusing than a bunch of eggs with numbers on it (see the meta post Andrew linked to in the question). And cooking enthusiasts are most likely less knowledgeable on what 404 is than TeX users... –  Martin Tapankov Nov 12 '10 at 12:42
I like the idea of the typesetting pieces. Liek Martin, I see no problem with a TeX error messages, and we can even combine both. –  mpg Nov 12 '10 at 13:44

Theme: The LaTeX-ing Robot

\documentclass[dvipsnames,dvips,rgb]{article}
\usepackage[margin=8cm]{geometry}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{showexpl}
\usepackage{etoolbox}
\makeatletter
\patchcmd{\SX@codeInput}{xleftmargin=0pt,xrightmargin=0pt}{}
{\typeout{***Successfully patched \protect\SX@codeInput***}}
{\typeout{***ERROR! Failed to patch \protect\SX@codeInput***}}
\makeatother
\lstset{%
basicstyle=\tt\scriptsize,
keywordstyle=\color{blue}\bf,
breaklines=true,
language={[LaTeX]TeX},
xleftmargin=3.4pt,
xrightmargin=3.4pt,
rulecolor=\color{blue},
backgroundcolor=\color{yellow!20},
frame=single,
explpreset={%
pos=b,
vsep=1pt%
}%
}
\usepackage{amsbsy}
\begin{document}
\begin{LTXexample}
\color{red}
$$\boldsymbol{\lim_{x\to 8}\frac{1}{(x-8)^2}=+\;\rotatebox{90}{8}}$$
$$\boldsymbol{\lim_{x\to 7}\frac{1}{(x-7)^2}=+\;\rotatebox{90}{7}}$$
$$\boldsymbol{\lim_{x\to 6}\frac{1}{(x-6)^2}=+\;\rotatebox{90}{6}}$$
\end{LTXexample}
\end{document}
`
-

Abridged version, since this answer is unpopular and clearly inferior to Martin's "hot metal" answer, and shouldn't take up so much HTML screen estate in this thread — look at the revisions page to see the original answer in all of its gruesome glory.

One could have some Tex-resembling error message written with a vertical orientation, and typeset it with the kind of digital rain effect one sey, for instance, in the Matrix credits. Is there a font suitable for such a digital rain effect? Cf. Zawinski's GLmatrix and xmatrix from xscreensaver.

-