text:
If you use a line-breaking command such as \\ or \newline at a time when LaTeX is not expecting it, you will generate the error shown below:

main.tex, line 12

LaTeX Error: There's no line here to end..

See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.8 \\ Your command was ignored. Type I <command> <return> to replace it with another command, or <return> to continue without it.

# Common Causes

Using a linebreak \\ when it is not appropriate:

One common cause of this error is if a line-break command such as \\ or \newline is used inappropriately. If LaTeX encounters a line-break command when it is not building a paragraph, it will generate an error. An example of this mistake is shown below:

\\ We have put a line-break in the wrong place

In the above example, a line-break has been inserted with no text before it. LaTeX doesn't know what to do since it cannot break the line which doesn't exist, so it returns an error.

Putting a line-break after a list environment:

This error can also appear if you attempt to place a line-break after a list environment as shown below

\begin{itemize}
\item This is a list item
\item This is another list item
\end{itemize}\newline

LaTeX does not see this list as a paragraph, but rather as a separate grouping of objects, and as such it does not find a line to break. If you would like there to be more space beneath all of your lists, it is best to redefine the itemize in your preamble or class file, however for a one time case, this error can be avoided by writing \leavevmode in front of the line-break command as shown below:

\begin{itemize}
\item This is a list item
\item This is another list item
\end{itemize}\leavevmode\newline

Trying to write a list item label on a line of its own:

Another common reason for this error to occur is if you are trying to write a list with labels using the description environment, and you would like the label to be on a line of its own. An example how a mistake would be made here is shown below:

\begin{description}
\item[This is our label] \\
We would like this text to be on its own line.
\end{description}

This will generate an error, as LaTeX sees the label entry as an argument of the \item command, and not as part of a paragraph of text. If you would like to have the labels in your list be on separate lines to the list entries, you would have to redefine the description environment. The easiest way to do this is by using the enumitem package, and including \setlist[description]{style=nextline} in your preamble as shown below:

% In your preamble

\usepackage{enumitem}
\setlist[description]{style=nextline}

% In the main body of your document

\begin{description}
\item[This is our label]
We would like this text to be on its own line.
\end{description}

This will give the desired result, by making all labels appear on a separate line to the list entry.

If you would prefer to simply have only one label be on a separate line, you can do this by using the \leavevmode command, as shown below:

\begin{description}
\item[This is a label] \leavevmode \\
This text will appear on its own line
\item[This is another label] This text will be on the same line as the label.
\end{description}

This is because LaTeX encounters such errors when it is trying to skip lines in vertical mode. The \leavevmode command causes the compiler to temporarily leave vertical mode, so the error is avoided.

Creating whitespace when using the center, flushleft or flushright environments:

Another cause of this error appearing is when you have tried to include multiple line-breaks by writing

• \\\\
• \\\newline
• \newline\newline
• etc.

while in the center, flushleft or flushright environments as shown below

\begin{center}
We would like there to be a one line gap between this line \\\\
and this line
\end{center}

This is not allowed in LaTeX, and will generate an error. The correct way to include multiple line-breaks here is to write \\[length] where length is the distance you would like between the lines. An example is shown below:

\begin{center}
We would like there to be a one line gap between this line \\[2\baselineskip]
and this line
\end{center}

Here, the \\[2\baselineskip] command tells latex to skip two lines at that point. The different measurement options for the distances you can input in the length option are listed here.


text:
# Delaunay Triangulation of Parallelepiped

Given $n$ linearly independent vectors $v_1, v_2, \ldots, v_n$ in $n$-dimensional space. Let $V$ be the set of $2^n$ points of the form $x_1 v_1 + x_2v_2 + \ldots + x_nv_n$, in which $x_i$ can be $0$ or $1$. $V$ is the set of vertices of an n-dimensional parallelepiped. Is there any way to find the Delaunay triangulation of $V$ using subexponential space with respect to $n$? I understand that the size of the output is big, and we can't store the whole output. But is there any way to output the Delaunay edges one by one without storing the previous result?

Testing whether a pair of points $p_i$ and $p_j$ are the endpoints of a Delaunay edge can be solved as a linear programming feasability problem: Lift each point to one higher dimension by making its last coordinate be the sum of squares of the other coordinates. Then look for a hyperplane passing through $p_i$ and $p_j$, such that all the other points are on one side of it. This can be represented in linear inequality constraints by seeking a vector $v$ and scalar $c$ such that the dot product of $v$ with each point is at least $c$, and is exactly $c$ in the case of $p_i$ and $p_j$.

So for any set of points in any dimension, it's possible to find the Delaunay edges one by one, by solving a number of linear programs that is polynomial in the number of points. The reason Delaunay triangulation is hard is not because of the edges, it's because there may be too many higher-dimensional features.

• Thanks David. The idea of lifting point to one higher dimension is interesting. By the way, I have another question. Is it possible to find all Delaunay edges in $2^{O(n)}$ using subexponential space?
– Jinx
Nov 5 '13 at 3:17
• The idea of lifting to one dimension higher is extremely standard when working with Delaunay triangulations. And I'm confused by your followup question: doesn't the algorithm I just described find them all in polynomial time and polynomial space? Nov 5 '13 at 6:29
• it's definitely in polynomial time and space with respect to the number of points. Here we have $2^n$ vertices given by $n$ linearly independent vectors, and I ask for an algorithm running in subexponential space in $n$. If it's possible to solve a linear programming of $2^n$ inequalities in $2^{o(n)}$ space then your algorithm will works. But then the running time will probably be bad, so I wonder if it's possible to find an algorihm that runs in $2^{O(n)}$ or not.
– Jinx
Nov 5 '13 at 15:06
• I'm thinking of doing gift wrapping in $n+1$ dimensions, since we don't have to store previous results with gift wrapping. If we can somehow bound the number of open ridges, then the space would not be too big.
– Jinx
Nov 5 '13 at 15:17
• Oh, I see what you mean. The ellipsoid method for linear programming runs in polynomial time given only a separation oracle (a black box that can tell whether a point is inside or outside the feasible polytope, and if outside returns a separating hyperplane). See "Geometric Algorithms and Combinatorial Optimization" by Grötschel, Lovász, and Schrijver. And in the parallelotope case, such an oracle is easy to implement in low space — just loop through all the parallelotope vertices testing each one. So I think the answer is yes, this same algorithm runs in polynomial space in terms of n. Nov 5 '13 at 16:47

format: plain


text:
# Justification of interchanging Expectation and Limit in Ito Integral Approximation

My reason for asking this question is because I can't seem to justify extending the results from the Ito Integral of elementary functions to the continuous form after taking the limit. For example, if I prove that the Ito integral of a simple function for finite $$n$$ is a martingale, I don't understand how to extend this for when we take the limit as $$n \to \infty$$. So that is my bigger picture question.

I have looked all over for a direct justification of: \begin{align*} \mathbb E\left[\int_{0}^{t}f(W_s,s) \, dW_s \right] &=\mathbb E\left[\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{f(W_{t_{i-1}},{{t}_{i-1}})(W({{t}_{i}})-W({{t}_{i-1}})})\right] \\ &= \underset{n\to \infty }{\mathop{\lim }} \mathbb E\left[\sum\limits_{i=1}^{n}{f(W_{t_{i-1}},{{t}_{i-1}})(W({{t}_{i}})-W({{t}_{i-1}})})\right] \end{align*}

where, $$f$$ is square integrable, so $$\mathbb E [ \int_s^tf^2(\omega, r)dr] \leq \infty$$, $$f$$ is adapted to the natural filtration generated by $$W$$, and also measurable with respect to the underlying probability space.

where the limit and expectation is interchanged but I haven't been able to find anything precise enough. I have seen mentions of using Dominated Convergence but not to which the bounding random variable is.

I know the sequence converges to the Ito Integral, but the Ito integral isn't necessarily with finite expectation or such that $$\left|\int_{0}^{t}f_s \, dW_s \right| \geq \sum\limits_{i=1}^{n}{f(W_{t_{i-1}},{{t}_{i-1}})(W({{t}_{i}})-W({{t}_{i-1}})})$$ for all $$n$$.

I tried using the absolute value of the Ito Integral and other types of simple functions but can't seem to pick one that definitely fits the DCT criteria.

Thanks a lot!

• What are your assumptions on $X$...? You will need some integrability and measurability condition. – saz Apr 5 at 5:12
• I'll edit them in – Slade Apr 5 at 5:14
• Finished. Basically it's the 'usual' conditions on the function, like in Oksendal's book on SDEs $\mathcal V$. I changed it to $f(W_s,s)$ to make it more general as well – Slade Apr 5 at 5:22

Interchanging the limit with the expectation is typically justified by choosing an approximating sequence of simple functions $$(f_n)_n$$ such that the corresponding stochastic integrals $$\int_0^t f_n(s) \, dW_s$$ converge to $$\int_0^t f(s) \, dW_s$$ in $$L^2(\mathbb{P})$$.

If $$f: \Omega \times [0,\infty) \to \mathbb{R}$$ is a progressively measurable function such that

$$\mathbb{E} \left( \int_0^T f(s)^2 \, ds \right) < \infty$$

for all $$T>0$$, then there exists a sequence of simple functions $$(f_n)_{n \in \mathbb{N}}$$ such that

$$\mathbb{E} \left( \int_0^t |f_n(s)-f(s)|^2 \, ds \right) \xrightarrow[]{n \to \infty} 0$$

for all $$T>0$$. By the very definition of the Itô integral, this implies that

$$\int_0^T f(s) \, dW_s = L^2(\mathbb{P})- \lim_{n \to \infty} \int_0^T f_n(s) \, dW_s \tag{1}$$

for all $$T>0$$. Note that this implies, in particular, that

$$\mathbb{E} \left( \int_0^T f(s) \, dW_s \right) = \lim_{n \to \infty} \mathbb{E}\left( \int_0^t f_n(s) \, dW_s \right). \tag{2}$$

In general, there is no explicit formula for $$f_n$$. If $$f$$ is mean-sequare continuous, i.e.

$$\lim_{s \to t} \mathbb{E}(|f(s)-f(t)|^2) = 0, \qquad t>0,$$

then it can be shown that

$$f_n(s) := \sum_{j=1}^{k(n)} f(t_{j-1}^{(n)}) 1_{[t_{j-1}^{(n)},t_j^{(n)})}(s)$$

does the job for any sequence of partitions $$\Pi_n = \{0=t_0^{(n)}<\ldots with mesh size tending to zero. In this case, $$(1)$$ reads

$$\int_0^t f(s) \, dW_s = L^2(\mathbb{P})-\lim_{n \to \infty} \sum_{j=1}^{k(n)} f(t_{j-1}^{(n)}) (W_{t_j^{(n)}}-W_{t_{j-1}^{(n)}})$$

and $$(2)$$ becomes

$$\mathbb{E} \left( \int_0^t f(s) \, dW_s \right) = \lim_{n \to \infty} \sum_{j=1}^{k(n)} \mathbb{E} \bigg[ f(t_{j-1}^{(n)}) (W_{t_j^{(n)}}-W_{t_{j-1}^{(n)}}) \bigg].$$

All the above-mentioned results can be, for instance, found in the monograph Brownian Motion - An Introduction to Stochastic Processes by Schilling & Partzsch.

• Thanks a lot! This is perfect. I have never seen the mean-square continuous condition before so I will check it out and see if it helps with the problems I am having. – Slade Apr 5 at 17:40
• Just to confirm, $L^2$ convergence implies $\lim_{n \to \infty} \|\phi_n-f\|_{L^2(\lambda_T \otimes \mathbb{P})} = 0$, and not $\|\lim_{n \to \infty} \phi_n-f\|_{L^2(\lambda_T \otimes \mathbb{P})} = 0$? – Slade Apr 5 at 18:06
• @Slade Yes, the first one is the very definition of $L^2$ convergence: $\phi_n \to f$ in $L^2(\mu)$ if and only if $$\lim_{n \to \infty} \|\phi_n-f\|_{L^2(\mu)} = 0.$$ – saz Apr 5 at 19:34


text:
# Return key statistical quantities generated from value, lower and upper confidence bounds

A function is generated that can be called with a number of keywords to return the requested quantity of interest, accordingly. The function is typically used in IQRtable.

statCI(value, lower, upper, CL = 0.95, digits = NA)

## Arguments

value numeric values. the lower confidence bound of the value. the upper confidence bound of the value. confidence level to which the provided lower and upper bounds correspond. number of significant digits (if NA (default), format_GUM is used to determine number of significant digits, otherwise, signif function is used instead of format_GUM)

## Value

Function outfn(char) that can be called with the key words

• value (value rounded to significant digits according to standard error)

• se (standard error rounded to two significant digits)

• rse (relative standard error rounded to significant digits according to standard error)

• rse% (relative standard error as percentage)

• lower.x where x is between 1 and 99 (lower confidence bound rounded to significant digits according to standard error)

• upper.x where x is between 1 and 99 (upper confidence bound rounded to significant digits according to standard error)

## Details

The function uses format_GUM to round to significant digits and format the output.

Interpolation between different requested output confidence levels is based on approximation by a left- and right-handed standard error. From the (possibly) asymmetric confidence intervals, left/right-handed standard errors are determined such that the lower/upper confidence bound corresponding to the procided CL matches the provided lower/upper bounds. Extrapolation to other confidence levels is performed separately for upper and lower bounds using the right and left standard error, respectively.

The reported standard error is the mean of left and right standard error. In case, either of the standard errors is not defined, the reported standard error is NA.

The standard error of rse is assumed to be rse^2 which is derived by Gaussian error propagation and the assumption that se is known without uncertainty.

format_GUM returns values rounded to significant digits according to the provided standard error. For asymmetric confidence intervals we use the smaller value of the approximated left and right standard errors for rounding of values.


text:
A First Course in String Theory (Second Edition, 2009)

Available from Cambridge University Press.

If you find errors/typos or wish to suggest corrections,
write to Barton Zwiebach at  zwiebach@mit.edu

Corrections for the first edition (2004) are below, but have
not been updated since 2005.

Additional material of possible interest to instructors and students:

Here is our final test in 8.251, spring term of 2009, with solutions.

Corrections were last updated on 06/27/2013.

• p.24. Below eqn. (2.60) replace (2.3) by (2.58).
• p.28. Above eqn.(2.84) replace (2.3) by (2.58).
• p.39. Second sentence below (2.121), ending as
"... sines and cosines." change to
"... sines and cosines, with $l= 0,1,2, \cdots$. " [suggestion by Y. Okawa]
• p. 51. Two lines below eqn. (3.35), replace (3.15) by (3.25).
• p. 52 A change to Eqn.(3.37): change the period for a comma, leave some space, and add " i = 1, 2, \ldots , d \,, " On the line following add "where $d$ is the number of spatial dimensions." [suggestion by Y. Okawa]
• p.56. Two lines above equation (3.68), replace "subspace" by "subset". [Thanks to Richard Lebed, 07/01/09].
• p.60. Paragraph below equation (3.84), two occurrances of "$\epsilon\cdot\left\{\cal O\right\}\left(h\right)$" that should be "{\cal O}(\epsilon, h)". [05/19/2009].
• p.66. Second sentence below (3.114) replace the part "Since the effective four-dimensional gravitational potential is V_g^{(5)}," by the following "Moreover, the four-dimensional gravitational potential, which is the average of V_g^{(5)} over x^4, is just V_g^{(5)}. Therefore, " [clarification suggested by Hong Liu]
• p.80. 7 lines from the bottom (4.5) should be (4.28). [05/19/2009].

• p.87. Problem 4.6, "m^2" should be just "m" in the first two displayed equations. [05/19/2009]
• p.114. One line below and two lines above equation (6.56), replace (6.5) by (6.54). [Thanks to Glenn Wouda, 05/19/2009].
• p.117. Eqn. (6.64). Replace the { } braces in the last right-hand side for parenthesis ( ). [suggestion by Y. Okawa]
• p.119. Eqn (6.69), second equation, left-hand side. replace X'^\mu by {X^\mu}' [Y. Okawa]
• p.145. Fourth line from the beginning of section 7.6. Replace "study" by "studied".[Barak Shoshani]
• p.148. Figure. 7.6, caption, 1st line, spelling mistake: change "obserber" to "observer". [Thanks to Richard Lebed, 07/01/09].
• p.150. Not a correction, but rather an improvement for part (c) of Problem 7.3. The new version would read: "(c) Consider a closed string which at t=0 is static and traces a closed curve \gamma of length \ell. How are \sigma_1 and \ell related? Find a time t_P >0 and smaller than \ell/c for which the closed string traces the curve \gamma again. Relate \vec{X} (t_P, \sigma) to the time equal to zero string \vec{X} (0,\sigma)."[05/19/2009].
• p.152. Problem 7.7, part (a), equation (1). Both the 1/4 and 1/12 should be replaced by 1/2. [Thanks to Uchupol Ruangsri, 05/19/2009].
• p.179. Above equation (9.13), sentence beginning with "Looking at (6.93)...". The reference to (6.93) is inappropriate since this expression assumes the static gauge. The sentence should be changed to "Looking at (6.49) we note that the numerator has two \sigma derivatives while the denominator has effectively one, making {\cal P}^{\tau \mu}transform like {d\over d\sigma} under reparameterizations."[05/19/2009].
• p.206. First line. Two changes. Change "fixed." for "fixed and positive." Change p^+ for p^+ > 0 .

• p.207. Eqn. (10.71) in the exponent replace "px" by "p\cdot x". [Y. Okawa]
• p.223. Second line below (11.39), replace (11.25) by (11.26).
• p.257. Two corrections.
-- Equation (12.131), sum should begin with "k=1".[Christopher Chamberland]
-- Six lines below (12.133) the L_1's in the commutator should be L_1^\perp [Y. Okawa]
• p.258. After last line ending with "..show." add the following sentence. Problem 12.10 shows that the above reparameterizations preserve the constraints (9.33). [Suggestion by Hong Liu]
• p.283. Add the following sentence below (13.19).
"The reality of X^\mu (\tau, \sigma) requires \alpha_0^\mu, x_0^{L\mu}, x_0^{R\mu} be real, as well as \alpha_{-n}^\mu = (\alpha_n^\mu)* and \bar{\alpha}_{-n}^\mu = (\bar{alpha}_n^\mu)*, for n \geq 1." [suggestion by Y. Okawa]
• p.285. Below eqn. (13.31) add the sentence
"Together with the hermiticity of x_0^\mu and \alpha_0^\mu, these make X^\mu (\tau, \sigma) in (13.24)
Hermitian." [suggestion by Y. Okawa]
• p.313. 2nd line after (14.31), spelling, change "parenthesis" to "parentheses". [Thanks to Richard Lebed, 07/01/09].
• p.313. Eqn (14.29), for uniformity of notation replace \delta^{IJ} by \eta^{IJ} [suggestion by Y. Okawa]
• p.315. Eqn. (14.43), for uniformity of notation replace \delta^{IJ} by eta^{IJ} [suggestion by Y. Okawa]

• p.318. Line below eqn. (14.61). Replace "physical that" for "physical than" [Y. Okawa]
• p.320. Section 14.7, second paragraph, third line, replace (14.47) by (14.46).[05/19/2009].
• p.324. Second to last paragraph from the bottom, beginning as "Finally, …" The second sentence ending with "…closed unoriented strings" change the ending to "…..closed unoriented strings, also with gauge group SO(32)". [suggestion by Y. Okawa]
• p. 363. Eq. (16.37), place a loop symbol on the integral over \Gamma (as in (16.36)). [Thanks to Richard Lebed, 07/01/09].
• p.384. There are four occurrences of the cite (17.3) that are wrong:
-- three lines above eqn. (17.35) should be (17.33)
-- line above eqn. (17.36), should (17.29).
-- line below eqn. (17.39), should be (17.33).
-- line below eqn. (17.40), should be (17.30)
• p.387. Three corrections:
-- one line above eqn. (17.49), replace (17.3) by (17.30)
-- one line above eqn. (17.53), replace (17.5) by (17.47)
-- one line below eqn. (17.53), replace (17.3) by (17.30)
• p.390. Paragraph above eqn. (17.64) replace the two occurrences of (17.6) by (17.63).

• p.392. Paragraph above (17.71), replace the two occurrences of (17.6) by (17.64).
• p.394. Section 17.8, three lines above and line below equation (17.76), replace (17.3) by (17.31). [05/19/2009].
• p.395. Third line above (17.80), replace "q" by "w". [Thanks to Richard Lebed, 07/01/09].
• p.396. Below eqn. (17.87) replace (17.3) for (17.31).
• p.403. Below eqn. (18.16) replace "The differential relations in (18.1) can" by "The above differential relations can".
• p.420. Line below eqn. (19.25), replace (18.1) by (18.16).
• p.424. Quick calculation 19.4, replace (19.3) by (19.37).
• p.430. Problem 19.2. Delete the last two sentences, beginning with "The back cover..." and ending with ".. of the strings." [05/19/2009].
• p.436. Eq. (20.18), replace "H_x" by "H_1" for consistent labeling. [Thanks to Richard Lebed, 07/01/09].
• p.475. Quick calculation 21.12, first line, replace (21.4) by (21.58).
• p.477. Paragraph below eqn. (21.66). Two corrections. Replace the first (21.4) by (21.56) and the second (21.4) by (21.62).
• p.478. Two corrections
-- Figure, 21.8, bottom, the second label N_2(Y= -1/2) should read N_3(Y= -1/2)
-- fifth line, replace (21.4) by (21.62).

• p.483. 9th line after (21.68), change "22.4" for "23.8". [Thanks to Richard Lebed, 07/01/09].
• p.484. Fifth line from the top. Delete the period after "strengths"
• p.486. Two changes.
-- Third line from top, change 21.6 for 21.5.
-- Six lines from bottom. replace "in the range (-\Lambda_P, \Lambda_P)." for "of order \Lambda_P." [Y. Okawa].
• p.488. Eq. (21.93), last displayed equation, right-hand side, the numerator should be multiplied by 2. [Thanks to Richard Lebed, 07/01/09].
• p.494. Problem 21.5. second line, replace (n,m) for (m, n). [Thomas Klose]
• p.516. Two corrections. Two appearances of (22.6) (right above eqn.(22.132) and bottom line) should be replaced by (22.131).
• p.517. Second line. Replace (22.13) by (22.131).
• p.527. 5th line above equation (23.1), spelling, change "parenthesis" to "parentheses".[Thanks to Richard Lebed, 07/01/09].
• p.531. Two corrections.
-- 2nd line above (23.29), replace "(14.51)" for "(15.51)". [Thanks to Richard Lebed, 07/01/09].
-- Line above (23.28) replace "such string" by "such a string"

• p.538. 8 lines from the bottom, change "Section 15.5" for "Section 21.2". [Thanks to Richard Lebed, 07/01/09].
• p.545. Two lines above (23.65), replace "with higher" by "with a higher"
• p.548. 4 lines below eqn. (23.82), replace 22.6 by 25.6.[Thanks to Jeremy Green, 05/19/2009].
• p.549. 4th line after (23.84), change "(23.8)" for "(23.65)". [Thanks to Richard Lebed, 07/01/09].
• p.552. 2nd line, replace "(23.8)" for "(23.65)". [Thanks to Richard Lebed, 07/01/09].
• p.553. Two corrections.
-- Figure 23.12, bottom annotation, "spatital" should be "spatial".[05/19/2009].
-- Must change equations (23.98) and (23.99) to read like [Thanks to Thomas Klose, and Curtis Asplund]
• p.554. Two lines above beginning of section 23.10, replace "such small" by "such a small"
• p.559. Quick Calculation 23.11, second line, delete the second word, "of".[Thanks to Jeremy Green, 05/19/2009].
• p.561. Third line. Replace the whole set of sentences and equation beginning with "The AdS/CFT..." and ending with "...principle (Problem 23.8)." Insert instead the following paragraph [Thanks to H. Liu]
• p.562. Problem 23.1, above last paragraph, definitions of X and \bar X in displayed equation, replace X^I by X^2 and X^2 by X^3. [Thanks to Jeremy Green, 05/22/2009].
• p.563. Two lines above equation (4), replace "areas" by "width"
• p.564. Problem 23.5, part (a), first line, replace (23.9) by (23.90). [Thanks to Jeremy Green, 05/22/2009].
• p.566. Problem 23.7. part (a) second line "Exhibit ... z=0." change to "Exhibit a timeline tangent at all points on the surface."
• p. 567. Delete the last line, "The estimate (3) ...temperature range."
• p.571. Equation (24.12), replace "x^\mu" by "x_0^\mu". [Thanks to Richard Lebed, 07/01/09].
• p.594. Figure 25.2(b), change a "P_2" for "p_2". [Thanks to Richard Lebed, 07/01/09].
• p.620. Two lines above (25.91), change (25.7) for (25.89).
• p.624. Four appearances of (25.8) that must be changed
-- Second line from top, change to (25.101)
-- Second line below eqn. (25.104), change to (25.103)
-- Second line above eqn. (25.105), change to (25.104)
-- Line above eqn. (25.107), change to (25.101).
• p.635. Figure 26.4, bottom right, label "- ln r" on axis should be "ln r" (remove the minus sign). [05/19/2009].
• p.638. Two corrections.
-- Equation (26.9), remove the loops on the first two integrals (the \gamma paths are not closed). [Thanks to Richard Lebed, 07/01/09].
-- Figure 26.6. A few radial lines do not reach the outer boundary.

A First Course in String Theory (First Edition, 2004)

Corrections were last updated on 05/06/2005. No further updates will be done.

Corrections by date:

June 2004

• p.73, below equation (4.38) delete "on the first line" and delete "on the second line"[6/14/04].
• p.117, equation (7.2), arrow (for vector) missing on top of v_\perp [Thanks to Siarhei Vaurynovich, 6/14/04].
• p118, equation (7.7), middle expression: the square root in the numerator must be squared, as in (\sqrt{...})^2     [Thanks to Siarhei Vaurynovich, 6/14/04].
• p.150, lines 1 and 2,  \lambda missing in the right hand side, equation should read  n_\mu X^\mu = \lambda \tau [Thanks to Siarhei Vaurynovich, 6/14/04].
• p.211, immediately above the unnumbered equation (which uses two lines), replace (12.25) by (12.33).[Thanks to Mikael Djurfeldt, 6/14/04].
• p.211, second line below (12.36), delete the words "in (12.2)" [6/14/04].
• p.211, line immediately above (12.37), replace "second line in (12.2) is replaced by"  by "commutator is". [6/14/04].
• p.211, line immediately below (12.38), replace "final right-hand side in (12.2)" by "result that led to (12.36)".[6/14/04]
• To facilitate the four corrections of page 211 entered above here is a pdf showing the corrected page.
• p.248,  for clarity, replace "world-sheet" by "parameter space" immediately before equation (13.8) and immediately before equation (13.9). [6/14/04].
• p.260, section 13.4, second paragraph, replace "m_e" by just  "m" (for consistency with equations below) [6/14/04].
• p.305, problem 14.10, part (a), corrections on the sentence beginning as: "Show that the orientifold action...".                 Delete the word "simultaneous" and add the words "either one of" between the words "about" and "the". [6/14/04].
• p.434, equation (19.40), the "Y" should be replaced by an "X".[6/14/04].
• p.483, figure 22.1, very minor correction: the dashing and the size of arrows should match those of figure 22.2. [6/14/04].
• p.531, Figure 23.10.  The sizes of the outgoing cylinders should match those of figures 23.11 and 23.12. So, in particular, P_4 in figure 23.10 should be a bit smaller (consequently, both P_1 and P_2 should be smaller)[6/14/04].

July 2004

• p.31, second line, replace Q^-  by Q'  (also in Figure 2.6, p.30). [Thanks to James Snyder, 7/04/04].
• p.45, equation (3.30), last term to the right: the \mu and \nu indices on the two eta's should be exchanged.  [Thanks to James Snyder, 7/04/04].
August 2004
• p.3, second paragraph, line containing "Michael Faraday", there is a double "that", remove one! [Unbelievable that this was not caught earlier. Thanks to Jan Troost, 8/16/04].
• To facilitate the corrections of page 211 entered  in June 2004 here is a pdf showing the corrected page.[8/16/04]
• p.118, equation (7.10), very minor correction: for clarity and consistency include a factor of (1/c) inside parenthesis in the middle term. [Thanks to James Snyder, 8/18/04].
• p.168, equation (10.7), the second partial derivative should be  \partial_\nu. [Thanks to James Snyder, 8/30/04].
September 2004
• p.119, two lines below eqn. (7.16). "How do we recast (7.16)" should be "How do we recast (7.15)". [Thanks to Julian Vogel and Pierre Jouvelot, 9/05/04].
• p.187, eqn. (11.2), the last  "\mu" should be replaced by "\nu". [Thanks to Pierre Jouvelot, 9/05/04].
• p.208, in the two lines above (12.17), for clarity, replace \xi(\tau)  by \xi(\tau, \sigma), and replace \xi by \xi (\sigma). [Thanks to Pierre Jouvelot, 9/05/04].
• p.243, two lines before Equation (2), "we" is duplicated. [Thanks to Pierre Jouvelot, 9/05/04].
• p.264, in the paragraph following Equation (13.95), b^I_{-r} should be b^J_{-r}. Also, three lines before Quick Calculation 13.3, the word "states" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/04].
• p.267, 4 lines before the end of the page, the word "the" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/04].
• p.285, first term on the right-hand side of the equation in (14.54): the \alpha should be primed. [Thanks to Pierre Jouvelot, 9/05/04].
• p.291, line right after eqn. (14.65), the word "to" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/04].
• p.299, one line before the end of the page, the word "the" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/04].
• p.549, in the entry for Das and Mathur, the word "pysics" should be replaced by "physics". [Thanks to Pierre Jouvelot, 9/05/04].
• p.347, last line, replace the third word, "the", by "a closed". [see motivation below, 9/05/04].
• p.335. Four lines after (15.108), an "on" is missing in the sentence "the strings end (on) a D-brane". [Thanks to Pierre Jouvelot, 9/10/04].
• p.364. equation (16.81), an arrow is missing on top of p in d^d p. [Thanks to Pierre Jouvelot, 9/10/04].
• p.375. In (16.140), the Q_2 in the intermediate step should be Q_5. At the end of the page, last  character, the "d" should be a "b", and immediately in the next page "d_f" should be "f". [Thanks to Pierre Jouvelot, 9/10/04].
• p.379, end of second line, replace 15.6 by 15.5. [Thanks to Pierre Jouvelot, 9/10/04].
• p.389 (very minor correction),  Figure 17.1,  the letters P and P' should be  lowercase (p and p') for consistency with the explanation beginning as "A small confusion ...", in p. 390. [Thanks to Pierre Jouvelot, 9/14/04].
• p.403,  line 6 of the second paragraph of Section 17.8, "radius us" should be "radius of". [Thanks to Pierre Jouvelot, 9/14/04].
• p.407, Problem 17.3, the first displayed equation should be labelled (1). [Thanks to Pierre Jouvelot, 9/14/04].
• p.383, Problem 16.6. Second line: delete "(derivatives of )" and add "(and derivatives thereof)" right after the word "functions".  Moreover, the quoted asymptotic expansion has a missing factor and is not the most useful. It is better to give the expansion for K_\nu without derivatives.  For this: 1) delete the prime to the left of the ~, 2) delete the minus sign immediately to the right of the ~, 3) add a multiplicative factor of \exp(-z), and 4) change the +3 for a -1. [9/14/04].

October 2004
• p.484,  two lines before equation (22.4), missing period (.)  before "In". [Thanks to Pierre Jouvelot, 10/24/04].
• p.511, 4th line of the 3rd paragraph, replace "x_2 and x_3" by "x_3 and x_4". [Thanks to Pierre Jouvelot, 10/24/04].
• p.524, 4 lines after equation (23.6), replace "clockwise" by "counterclockwise". [Thanks to Pierre Jouvelot, 10/24/04].
• p.530, Fig.23.9, top: the arrow to the left of  "\delta T" extends a bit too much,  it should stop right above the beginning of the lower slit.[Thanks to Pierre Jouvelot, 10/24/04].
• p.533, Fig. 23.13, caption: replace "23.12 and 23.11" by "23.11 and 23.12".[Thanks to Pierre Jouvelot, 10/24/04].
• p.536. Fig 23.16, caption, second line: the w in \eta = w/T should have a tilde.[Thanks to Pierre Jouvelot, 10/24/04].
• p.538, Fig. 23.18, a series of very minor changes:  (a) cylinder shown at lower center, top face,  the angle for \theta should be denoted  with a small curved line rather than a little square (that incorrectly suggests right angle), (b) Im(\tau) and Re(\tau) should be I(\tau) and R(\tau), with the proper font for I and Rs. [Thanks to Pierre Jouvelot, 10/24/04].
• p.177, line immediately below eqn (10.67), replace "multiples" by "multiplies".[Thanks to James Snyder, 10/24/04].

February 2005

• p.37  Problem (2.4), (b) line before last: change "rectangular" for "square". [Thanks to Ira Rothstein, 02/16/05].
• p.98, This is not a correction but rather an additional sentence that notes an interesting special case (alluded to in the remarks on Ch 6 at the bottom of this web page -- as noted by James Snyder). At the end of page 98, right after "sense."  add:  "There is a small caveat: at a point where the string moves with the speed of light there is no timelike direction (see Quick Calculation 6.3).
• p.120, Equation (7.21), left-hand side: arrow missing over the X. [Thanks to Dana Mihai, 2/19/05].
• p.121, Line immediately below (7.25); add "to" after the word "assigned".[Thanks to Dana Mihai, 2/20/05].
• p.143, A very minor improvement (for clarity). Below (8.53)  replace " indices"  by  "first index" .[Thanks to James Snyder].
• p.151 and p.154.  It was called to my attention by Ira Rothstein and Dana Mihai (02/05) that the argument for the vanishing of  $n\cdot \mathcal{P}^\sigma$ advanced in the paragraph below equation (9.18) does not prove the claim. Indeed, the correct way to frame the issue is the following:  "We assume that $n \cdot \mathcal{P}^{\sigma}=0$
at the open string endpoints since this condition naturally guarantees the conservation of $n\cdot p$ (consider equation (8.38) dotted with $n^\mu$)."  This sentence fits well in page 151, at the end of the paragraph below equation (9.4) (ending with "conservation.").  Then, when we get to p.154, paragraph below (9.18), one can eliminate the sentence  "Indeed, ....   with $n^\mu$)."  and replace it with: "As noted below equation (9.4),  we  assume $n \cdot \mathcal{P}^{\sigma}=0$ at the string endpoints since it guarantees  the conservation of $n\cdot p$."
• p. 197, Title of section 11.5: replace  "generators" by "operators" (they are *translation* generators) [02/05]
• p.202,  Second line above (11.85), add ", $x^-(\tau)$," right after $x^+(\tau)$.  Next line:  add ", (11.30), " right after (11.29) [Thanks to James Snyder].
• p. 205. This is an extra hint for a problem, which Siarhei Vaurynovich noted is otherwise fairly difficult to solve.             At the end of part (b) of  Problem 11.7 add the following line:[Hint:  The reparameterization takes the hermiticized"   form $\delta x^\mu (\tau) = {1\over 2} (\lambda \,\partial_\tau x^\mu + \partial_\tau x^\mu \, \lambda )$.]
• p.212  Equation (12.46):  add a prime to the sigma in the left-hand side [Thanks to Ira Rothstein 2/21/05].

March 2005
• p.157.  Last sentence before sect. 9.4: ${X'}^\mu$ should be replaced by ${X^\mu}'$.[Thanks to Ng Jia Tong, 03/04/05].
• p. 164. Very minor: Problem 9.3, part (b): for notational consistency, delete the parenthesis surrounding the superscripts 2 and 3 of  the X's.[Thanks to Ng Jia Tong, 03/04/05].
• p. 165.  Problem 9.4 part (c):  replace  "real"  by "positive $x^1$".[Thanks to Ira Rothstein 03/06/05].
• p. 387.  More than a correction, this is a clarification suggested by Stanley Deser, who found some comments on electromagnetic duality in p.387 somewhat misleading.  If you want an improved version, do the following changes:   Second line below (17.2), add the word "familiar"  before "Lagrangian" (this is because there is a less familiar first order Lagrangian for transverse fields that is invariant).  Third line below (17.2) up to the seventh line, delete all text from "First of all...."  up to  ".... would not be invariant."(the comments on nonlocal changes can be stated more accurately, see below).  Third line below (17.3) delete the sentence "The significant ...   Lagrangian."  Now, to add some substance insert the following paragraph in between the paragraph ending with "unchanged" and that beginning with "Consider now":  Since the dynamical variables in  electromagnetism  are the gauge potentials, one may ask:  Are there transformations of the  potentials that induce (17.2)? Not exactly, but close. One can formulate electromagnetism (without sources) using dynamical variables $\vec{E}$ and $\vec{A}$, both  divergenceless.   Duality transformations are then written as spatially nonlocal transformations of $\vec{E}$ and $\vec{A}$.  (A typical spatially nonlocal transformation expresses the output  at a point $\vec{x}$ in terms of the values of the input at all points $\vec{x}'$.)" [Thanks to Stanley Deser, 03/06/05]

April 2005

• p.53. Very minor: Equation (3.82), left hand side, superscripts on the x: reverse the order of prime and \mu. [Hyung Jun Lee, 4/20/05]
• p. 129   Line above part (c) add "with constant angular velocity"  between "rotates" and "around".[Thanks to the students at MIT].
• p. 213,  Very minor: Eqn (12.54), second equation, right-hand side, superscripts on the
symbol "a": reverse the order of the star and the \mu.
• p.231,  Equation (12.162), add a minus sign "-" right after the equal sign (top line) [Thanks to J. Fernandez-Alcazar, who re-checked the computation, 4/05]
• p. 308,  End of the second line, replace the exponent  "D-4" by  "4-D".
• p. 406   Table 17.2, first line (line that begins with $\tilde H$), first term on the right-hand side: the (two) p^i's  should have no tildes. [Ira Rothstein, 04/05]

May  2005

• p.194. Very minor, immediately below eqn (11.47). Missing " on top of the letter "o" in Schrodinger [Thanks to Karsten Koeneke, 5/6/05]
• p. 308  Minor, second paragraph, second to last line, replace "This a"  by "This is a" [Thanks to Karsten Koeneke, 5/6/05]
• p. 309  Second line, equation in parenthesis, right-hand side: replace the exponent  "D-6" by  "6-D" [Thanks to Karsten Koeneke, 5/6/05]

Corrections by page:

Chapter 1
• p.3, second paragraph, line containing "Michael Faraday", there is a double "that", remove one! [Unbelievable that this was not caught earlier. Thanks to Jan Troost, 8/16/04].
Chapter 2
• p.31, second line, replace Q^-  by Q'  (also in Figure 2.6, p.30). [Thanks to James Snyder, 7/04/04].
• p.37,  Problem (2.4), (b) line before last: change "rectangular" for "square". [Thanks to Ira Rothstein, 02/20/05].
Chapter 3
• p.45, equation (3.30), last term to the right: the \mu and \nu indices on the two eta's should be exchanged.  [Thanks to James Snyder, 7/04/04].
• p.53. Very minor: Equation (3.82), left hand side, superscripts on the x: reverse the order of prime and \mu. [Hyung Jun Lee, 4/20/05]
Chapter 4
• p.73, below equation (4.38) delete "on the first line" and delete "on the second line"[6/14/04].
Chapter 5
Chapter 6

• p.98, This is not a correction but rather an additional sentence that notes an interesting special case (alluded to in the remarks on Ch 6 at the bottom of this web page -- as noted by James Snyder). At the end of page 98, right after "sense."  add:  "There is a small caveat: at a point where the string moves with the speed of light there is no timelike direction (see Quick Calculation 6.3).

Chapter 7
• p.117, equation (7.2), arrow (for vector) missing on top of v_\perp [Thanks to Siarhei Vaurynovich, 6/14/04].
• p118, equation (7.7), middle expression: the square root in the numerator must be squared, as in (\sqrt{...})^2     [Thanks to Siarhei Vaurynovich, 6/14/04].
• p.118, equation (7.10), very minor correction: for clarity and consistency include a factor of (1/c) inside parenthesis in the middle term. [Thanks to James Snyder, 8/18/04].
• p.119, two lines below eqn. (7.16). "How do we recast (7.16)" should be "How do we recast (7.15)". [Thanks to Julian Vogel and Pierre Jouvelot, 9/05/04].
• p.120, Equation (7.21), left-hand side: arrow missing over the X. [Thanks to Dana Mihai, 2/19/05].
• p. 129   Line above part (c) add "with constant angular velocity"  between "rotates" and "around".[Thanks to the students at MIT].
Chapter 8
• p.143, A very minor improvement (for clarity). Below (8.53)  replace " indices"  by  "first index" .[Thanks to James Snyder, 2//05].
Chapter 9
• p.150, lines 1 and 2,  \lambda missing in the right hand side, equation should read  n_\mu X^\mu = \lambda \tau [Thanks to Siarhei Vaurynovich, 6/14/04]
• p.151 and p.154.  It was called to my attention by Ira Rothstein and Dana Mihai (02/05) that the argument for the vanishing of  $n\cdot \mathcal{P}^\sigma$ advanced in the paragraph below equation (9.18) does not prove the claim. Indeed, the correct way to frame the issue is the following:  "We assume that $n \cdot \mathcal{P}^{\sigma}=0$
at the open string endpoints since this condition naturally guarantees the conservation of $n\cdot p$ (consider equation (8.38) dotted with $n^\mu$)."  This sentence fits well in page 151, at the end of the paragraph below equation (9.4) (ending with "conservation.").  Then, when we get to p.154, paragraph below (9.18), one can eliminate the sentence  "Indeed, ....   with $n^\mu$)."  and replace it with: "As noted below equation (9.4),  we  assume $n \cdot \mathcal{P}^{\sigma}=0$ at the string endpoints  since it guarantees  the conservation of $n\cdot p$."
• p.157  Last sentence before sect. 9.4: ${X'}^\mu$ should be replaced by ${X^\mu}'$.[Thanks to Ng Jia Tong, 03/04/05].
• p. 164 Very minor: Problem 9.3, part (b): for notational consistency, delete the parenthesis surrounding the superscripts 2 and 3 of  the X's.[Thanks to Ng Jia Tong, 03/04/05].
• p. 165,  Problem 9.4 part (c):  replace  "real"  by "positive $x^1$".[Thanks to Ira Rothstein 03/06/05].

Chapter 10
• p.168, equation (10.7), the second partial derivative should be  \partial_\nu. [Thanks to James Snyder, 8/30/04].
• p.177, line immediately below eqn (10.67), replace "multiples" by "multiplies".[Thanks to James Snyder, 10/24/04].

Chapter 11

• p.187, eqn. (11.2), the last  "\mu" should be replaced by "\nu". [Thanks to Pierre Jouvelot, 9/05/2004].
• p.194. Very minor, immediately below eqn (11.47). Missing " on top of the letter "o" in Schrodinger [Thanks to Karsten Koeneke, 5/6/05]
• p. 197, Title of section 11.5: replace  "generators" by "operators" (they are *translation* generators) [02/05].
• p.202,  Second line above (11.85), add ", $x^-(\tau)$," right after $x^+(\tau)$.  Next line:  add ", (11.30), " right after (11.29) [Thanks to James Snyder].
• p. 205. This is an extra hint for a problem, which Siarhei Vaurynovich noted is otherwise fairly difficult to solve.             At the end of part (b) of  Problem 11.7 add the following line:[Hint:  The reparameterization takes the hermiticized"   form $\delta x^\mu (\tau) = {1\over 2} (\lambda \,\partial_\tau x^\mu + \partial_\tau x^\mu \, \lambda )$.]

Chapter 12
• p.208, in the two lines above (12.17), for clarity, replace \xi(\tau)  by \xi(\tau, \sigma), and replace \xi by \xi (\sigma). [Thanks to Pierre Jouvelot, 9/05/2004].
• p.211, immediately above the unnumbered equation (which uses two lines), replace (12.25) by (12.33).[Thanks to Mikael Djurfeldt, 6/14/04].
• p.211, second line below (12.36), delete the words "in (12.2)" [6/14/04].
• p.211, line immediately above (12.37), replace "second line in (12.2) is replaced by"  by "commutator is". [6/14/04].
• p.211, line immediately below (12.38), replace "final right-hand side in (12.2)" by "result that led to (12.36)".[6/14/04].
• p.212  Equation (12.46):  add a prime to the sigma in the left-hand side [Thanks to Ira Rothstein 2/21/05].
• p. 213,  Very minor: Eqn (12.54), second equation, right-hand side, superscripts on the
symbol "a": reverse the order of the star and the \mu.
• p.231,  Equation (12.162), add a minus sign "-" right after the equal sign (top line) [Thanks to J. Fernandez-Alcazar, who re-checked the computation, 4/05]
• p.243, two lines before Equation (2), "we" is duplicated. [Thanks to Pierre Jouvelot, 9/05/2004].
Chapter 13
• p.248,  for clarity, replace "world-sheet" by "parameter space" right before equation (13.8) and right before equation (13.9). [6/14/04].
• p.260, section 13.4, second paragraph, replace "m_e" by "m" (for consistency with equations below) [6/14/04].
• p.264, in the paragraph following Equation (13.95), b^I_{-r} should be b^J_{-r}. Also, three lines before Quick Calculation 13.3, the word "states" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/2004].
• p.267, 4 lines before the end of the page, the word "the" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/2004].
Chapter 14
• p.285, first term on the right-hand side of the equation in (14.54): the \alpha should be primed. [Thanks to Pierre Jouvelot, 9/05/2004].
• p.291, line right after eqn. (14.65), the word "to" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/2004].
• p.299, one line before the end of the page, the word "the" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/2004].
• p.305, problem 14.10, part (a), corrections on the sentence beginning as: "Show that the orientifold action...".                 Delete the word "simultaneous" and add the words "either one of" between the words "about" and "the". [6/14/04].
Chapter 15

• p. 308,  End of the second line, replace the exponent  "D-4" by  "4-D".
• p. 308  Minor, second paragraph, second to last line, replace "This a"  by "This is a" [Thanks to Karsten Koeneke, 5/6/05]
• p. 309  Second line, equation in parenthesis, right-hand side: replace the exponent  "D-6" by  "6-D" [Thanks to Karsten Koeneke, 5/6/05]
• p.335. Four lines after (15.108), an "on" is missing in the sentence "the strings end (on) a D-brane". [Thanks to Pierre Jouvelot, 9/10/04].
• p.347, last line, replace the third word, "the", by "a closed". [see motivation below, 9/05/04].

Chapter 16

• p.364. equation (16.81), an arrow is missing on top of p in d^d p. [Thanks to Pierre Jouvelot, 9/10/04].
• p.375. In (16.140), the Q_2 in the intermediate step should be Q_5. At the end of the page, last  character, the "d" should be a "b", and immediately in the next page "d_f" should be "f". [Thanks to Pierre Jouvelot, 9/10/04].
• p.379, end of second line, replace 15.6 by 15.5. [Thanks to Pierre Jouvelot, 9/10/04].
• p.383, Problem 16.6. Second line: delete "(derivatives of )" and add "(and derivatives thereof)" right after the word "functions".  Moreover, the quoted asymptotic expansion has a missing factor and is not the most useful. It is better to give the expansion for K_\nu without derivatives.  For this: 1) delete the prime to the left of the ~, 2) delete the minus sign immediately to the right of the ~, 3) add a multiplicative factor of \exp(-z), and 4) change the +3 for a -1. [9/14/04].

Chapter 17
• p. 387  More than a correction, this is a clarification suggested by Stanley Deser, who found some comments on electromagnetic duality in p.387 somewhat misleading.  If you want an improved version, do the following changes:   Second line below (17.2), add the word "familiar"  before "Lagrangian" (this is because there is a less familiar first order Lagrangian for transverse fields that is invariant).  Third line below (17.2) up to the seventh line, delete all text from "First of all...."  up to  ".... would not be invariant."(the comments on nonlocal changes can be stated more accurately, see below).  Third line below (17.3) delete the sentence "The significant ...   Lagrangian."  Now, to add some substance insert the following paragraph in between the paragraph ending with "unchanged" and that beginning with "Consider now":  Since the dynamical variables in  electromagnetism  are the gauge potentials, one may ask:  Are there transformations of the  potentials that induce (17.2)? Not exactly, but close. One can formulate electromagnetism (without sources) using dynamical variables $\vec{E}$ and $\vec{A}$, both  divergenceless.   Duality transformations are then written as spatially nonlocal transformations of $\vec{E}$ and $\vec{A}$.  (A typical spatially nonlocal transformation expresses the output  at a point $\vec{x}$ in terms of the values of the input at all points $\vec{x}'$.)" [Thanks to Stanley Deser, 03/06/05]
• p.389 (very minor correction),  Figure 17.1,  the letters P and P' should be  lowercase (p and p') for consistency with the explanation beginning as "A small confusion ...", in p. 390. [Thanks to Pierre Jouvelot, 9/14/04].
• p.403,  line 6 of the second paragraph of Section 17.8, "radius us" should be "radius of". [Thanks to Pierre Jouvelot, 9/14/04].
• p. 406   Table 17.2, first line (line that begins with $\tilde H$), first term on the right-hand side: the (two) p^i's  should have no tildes. [Ira Rothstein, 04/05]
• p.407, Problem 17.3, the first displayed equation should be labelled (1). [Thanks to Pierre Jouvelot, 9/14/04].

Chapter 18
Chapter 19
• p.434, equation (19.40), the "Y" should be replaced by an "X". [6/14/04].
Chapter 20
Chapter 21
Chapter 22
• p.483, figure 22.1, very minor correction: the dashing and the size of arrows should match those of figure 22.2. [6/14/04].
• p.484,  two lines before equation (22.4), missing period (.)  before "In". [Thanks to Pierre Jouvelot, 10/24/04].
• p.511, 4th line of the 3rd paragraph, replace "x_2 and x_3" by "x_3 and x_4". [Thanks to Pierre Jouvelot, 10/24/04].

Chapter 23
• p.524, 4 lines after equation (23.6), replace "clockwise" by "counterclockwise". [Thanks to Pierre Jouvelot, 10/24/04].
• p.530, Fig.23.9, top: the arrow to the left of  "\delta T" extends a bit too much,  it should stop right above the beginning of the lower slit.[Thanks to Pierre Jouvelot, 10/24/04].
• p.531, Figure 23.10.  The sizes of the outgoing cylinders should match those of figures 23.11 and 23.12. So, in particular, P_4 in figure 23.10 should be a bit smaller (consequently, both P_1 and P_2 should be smaller)[6/14/04].
• p.533, Fig. 23.13,  caption : replace "23.12 and 23.11" by "23.11 and 23.12".[Thanks to Pierre Jouvelot, 10/24/04].
• p.536. Fig 23.16, caption, second line: the w in \eta = w/T should have a tilde.[Thanks to Pierre Jouvelot, 10/24/04].
• p.538, Fig. 23.18, a series of very minor changes:  (a) cylinder shown at lower center, top face,  the angle for \theta should be denoted  with a small curved line rather than a little square (that incorrectly suggests right angle), (b) Im(\tau) and Re(\tau) should be I(\tau) and R(\tau), with the proper font for I and Rs. [Thanks to Pierre Jouvelot, 10/24/04].

References
• p.549, in the entry for Das and Mathur, the word "pysics" should be replaced by "physics". [Thanks to Pierre Jouvelot, 9/05/2004].

Additional material of possible interest to instructors and students:

Here is the quiz that was given to students of 8.251 in the spring term of 2004.

Chapter 1
Concerning my citation of Biot-Savart (1820) in page 3,  Blake Stacey [MIT senior] wrote:
"For history's sake, it might be worth making clear that Jean Baptiste Biot (1774-1862) and Felix Savart (1791-1841) were two distinct people--say, by changing the hyphen to an "and".   Biot sure lived well:  the mica-based mineral biotite and a lunar crater are both named after him.  It doesn't look like Savart picked up as many honors."[8/18/04]

Chapter 2

After noting that the light-cone energy is (-p_+)  in (2.91) it may be useful to  emphasize to the
students that in a Lorentz frame (-p_0) is the energy. [6/14/04]

Some students find the discussion of orbifolds in the final paragraph of section 2.7 a little insufficient
to deal with problems 2.4 and 2.7.  A nice additional example that can be discussed in class is that of
a "cone" with angle at the apex equal to "2 pi/N". This orbifold is obtained by the identification
z \sim z \exp(i 2 pi/N) on the complex coordinate z. [6/14/04]

Chapter 3

Question (by Andrew Mamo, MIT undergrad, on 12 Feb 2004):  When we saw how the gravitational constant G requires different units in different dimensions, I wondered why no similar constant is necessary for the electric field.  If the electric potential is analogous to the gravitational potential, and if the charge density has different units in different dimensions, why is there no need to have a dimension-specific constant to keep units matching, the way we had for gravity?
My reply: It is just a matter of convenience and the fact that charge is less familiar a concept than mass. The first hint for the answer is that (Quick Calculation 3.6) the units of charge *are* different in different dimensions.  So, it turns out that both sides of the equation  \laplacian Phi = - \rho    have the same units and no additional constant is needed. While the gravitational potential is defined to have the same units in all dimensions, the electric potential, which is energy per unit charge, has different units in different dimensions. Since charges have different units in different dimensions, if you are given a 5-dimensional charged particle, you have to do a nontrivial calculation to find the 4-dimensional effective charge after compactification. In gravity, we want to keep the units of mass the same in all dimensions (because of F= ma, which exists independent of gravity).  As a result, we had no choice but to have a dimensionality dependent Newton constant. [6/14/04]

A note on pedagogy.  In section 3.6 there is a very brief introduction to the language of General Relativity. I made no attempt to prove that Newtonian gravity arises as a limit of General Relativity.  This is because I did not want to delay the beginning of our study of strings. A point  relevant to the Newtonian limit comes up in problem 5.7. where the student derives the geodesic equation.  In  Chapter 10 we discuss gravity fields and graviton states in some detail.  An instructor that wishes to discuss the Newtonian limit to convince students that the h_{\mu\nu} field is the gravitational field, may find it useful to introduce this material as a digression, perhaps through problem sets. A recent pedagogical discussion can be found in Hartle's "Gravity" (see section 22.4).[8/16/04]

Chapter 4
Chapter 5
Chapter 6

It is stated at the bottom of page 98 that at any point on the world-sheet there are spacelike and timelike directions. There is, however, an exception.  For open strings any point on the boundaries of the world-sheet represents an open string endpoint. At any (free) open string endpoint we have a spacelike direction and a null direction (the tangent to the boundary), since the endpoint moves with the speed of light (section 6.9). There is no timelike direction. This is actually the situation considered in Quick calculation 6.3. [9/05/04, Thanks to James Snyder for the comment.]

Chapter 7

Problem 7.1, part (a) The hyperplane is not supposed to be a D-brane. As stated in the problem, the endpoints are completely free, as if we had a spacefilling D-brane. It is not true that an open string must lie at all times fully on a D-brane that does not fill space.[8/18/04]

An application of problem 7.3.  [03/07/05]

Consider a closed  string  that at time t=0 has zero velocity and is stretched along the
perimeter of a square of side L/4.  Choose a convenient set of coordinate axes and describe
$\vec{F}(u)$ and $\vec{F}'(u)$.  Explain why, at any time, the string is composed by a set of
piecewise linear parts.  Draw a few sketches showing the shape of the string as it contracts
down to zero size.

Chapter 8
Chapter 9
Chapter 10
Chapter 11

In the above list of "corrections" an extra hint has been added for  part (b) of  Problem 11.7 .

Chapter 12
Chapter 13
Chapter 14

Chapter 15

Problem 15.5.  The conservation of the string current in (15.11) holds only for closed strings. For open strings there are additional contributions to the string current that are needed to obtain conservation.  The statement of the problems is better modified from "..to the case of the string" to read "... to the case of a closed string".
Question [P. Jouvelot, 9/05/04]: Why is there no "Kalb-Ramond coupling constant" in the term (15.3) as there is one for electromagnetism in (15.1), via "q"?
Answer:In string theory there is no coupling that is independent of the string coupling g, so introducing a coupling with some different label in (15.3) could give a wrong impression. Any coupling that you may introduce in (15.3) is compatible with (15.4), where the second term has no coupling. This is because a rescaling of the Kalb-Ramond field B can be used to move the coupling from the second to the last term in (15.4), where it would simply affect what we call \kappa. As stated below (15.6), \kappa is a calculable function of g.  Note, incidentally, that in a similar spirit the charges at  the end of the string were conventionally chosen to be equal to plus or minus one in equation (15.54).
A little abuse of notation [9/10/04, P. Jouvelot] In page 323 there are two occurances where a repeated index is not supposed to be summed over.  In (15.68), there are three "m" indices -- the last one is not part of the sum.   On the line before (15.69), the two "m" indices are not summed over.

Chapter 16
Problem 16.6 has an important correction (see above). The partition function can be expressed in terms of the modified Bessel functions and its derivatives. The required asymptotic expansion is that of K_\nu without derivatives [9/14/04].

Chapter 17

Some modifications of p.387 were suggested above (see corrections) to clarify the discussion of electromagnetic duality. A useful (and quite brief) reference  is Deser, S. (1982).  Off-shell electromagnetic duality invariance," J. Phys.A:Math. Gen. {\bf 15} 1053. [03/07/05]

Chapter 18

Comment [suggested by P. Jouvelot, 9/14/04].  In section 18.2, to obtain equation (18.28) we must use the fact that the group elements U_1 and U_2 (and their inverses) commute.  This is the case because they are complex numbers. The group U(1) is an abelian group: the multiplication is commutative.

Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23


text:
# Algebraic Geometry Jottings 18

Prev TOC Next

Up till now, I’ve been using power series to parametrize branches:

x(t) = a0+a1t+a2t2+ ⋯,      y(t) = b0+b1t+b2t2+ ⋯

If the branch passes through the origin, then a0=b0=0. In the last post, we established Facts 4 and 5, assuming that y(t)=t for all branches, so x(t) = a0+a1y+a2y2+ ⋯.

That special assumption doesn’t always hold. As a fairly simple example, take this blue circle tangent to the x-axis:

x2+(y–2)2=4

(Ignore the red and greeen lines for a moment.) Solving for x in terms of y, we get $x=\pm\sqrt{y}\sqrt{4-y}$. While $\sqrt{4-y}$ can be expanded in a Taylor series about 0, $\sqrt{y}$ cannot. It’s clear enough why we can’t express x as a power series in y near 0: the derivative dx/dy is infinite at y=0, wreaking havoc with the Taylor series.

In this case, we could switch the roles of x and y, getting the two branches $y=2\pm\sqrt{4-x^2}$, or near the origin, $y=x^2/4+x^4/16+\cdots$. But we’re not out of the woods yet. Say we wanted to verify Fact 5 for the intersections of the blue circle with the red line. At the other intersection, the circle is tangent to the green line, signifyiing an infinite dy/dx.

Even with a single curve, we could be in trouble. Consider the semicubical parabola y2=x3 (see Kendig Example 3.11, p.55):

y2=x3

We have either $y=\pm\sqrt{x^3}$ or $x=\omega^i\sqrt[3]{y^2}$ (with ω a primitive cube root of unity, i=0,1,2); neither is expressible as a power series about 0.

The solution: fractional power series, aka Puiseux series. For the circle, we have near 0

$x=2y^{1/2}-y^{3/2}/4-y^{5/2}/16+\cdots$

and for the semicubical parabola,

$x=y^{3/2}$    or    $y=x^{2/3}$.

Perhaps you have qualms about the ambiguity of n-th roots (as I did at first). In the next post, I will try to dispel these three ways: via extension fields, via Riemann surfaces, and via the theory of fractional power series. We’ll see that the ambiguity plays an essential role.

First let’s see how Facts 4 and 5 work out with this technique. Along the way, we’ll see how it squashes the “grade inflation bug” mentioned way back in post 3.

You can paper over the fractional exponents using a so-called uniformizing parameter t. For the circle, use t=y½, giving $x=2t-t^3/4-t^5/16+\cdots$. Chew on this for a moment, chomping down on the multiplicity of the intersection at O between the circle and the x-axis. That multiplicity is 2 because if $(x_E(t),y_E(t))$ is the branch $(2t+\cdots,t^2)$ and $F(x,y)=y$ is the x-axis, then $F(t)=F(x_E(t),y_E(t))=t^2$, and the equation $F(t)=\epsilon$ has two solutions for $\epsilon\approx 0$. And furthermore, those two t‘s give two intersections because xE(t)≈2t. The “grade inflation bug” shows its feelers when we use y=t4 for the parametrization (for example).

If we stick with fractional exponents, the multiplicity calculation looks like this: the parametrization is

$(x_E(t),y_E(t))= (2y^{1/2}+\cdots,y)$

and F(x,y)=y. The equation F(x,y)=y=ϵ has only a single solution for any ϵ, but x≈½ has two possible values because of the ambiguity of the square root, when ϵ≠0. So we get two intersections when we perturb things slightly.

As an instructive comparison, let F(x,y)=y–x be the red diagonal line. Plugging in the parametrization now gives F(x,y)≈y–2y½. Before we had two values of x for one value of F. Now we have two values of x for two values of F (near but not at O), because both have y½ as their lowest power of y. One F for one x, multiplicity 1.

Let’s generalize. If E(x,y) is monic as a polynomial in x, then any branch crossing the x-axis has a parametrization

$(x_E,y_E)= (a_ky^{k/d}+a_{k+1}y^{(k+1)/d}+\cdots,y)$

with ak≠0 and all the exponents having the same denominator d (see Kendig §3.4, p.54). As usual, we care only about the lowest-order term, so assume that k and d are coprime—it doesn’t matter if higher-order exponents require a larger denominator. Clearly the lowest-order term of F(xE,yE) will have the form $cy^{r/d}$. Claim: the multiplicity of the intersection (between the branch of E and the curve F) is r. This is easiest to see when k=1. Introduce the parameter t=y1/d. For a given y≠0, there are d different values of t. Since x=a1t+a2t2+ ⋯, that means d different values of x for any y near but not equal to 0. Now, how many of these land on the level curve F(x,y)=ϵ? Since F(xE,yE)≈ctr, the answer is r: to the r different solutions to ctr=ϵ correspond r different values of x. Multiplicity r, in other words.

This is a good spot to discuss conjugate parametrizations (see Kendig p.55). The parameter t belongs to the family of d conjugate parameters t, ωt, …, ωd–1t, where ω is a primitive d-th root of unity. As t “fills out” a small disk |t|<δ in the complex plane, so do all of its conjugates (as Kendig puts it). Since y=td, all the conjugate values give the same y, but different x‘s. We get the same branch, in the sense of the totality of (x,y) values, but d different parametrizations of it.

It’s particularly easy to picture all this for the blue circle above, since d=2 and the conjugate values are ±t. Restricting attention to real values of t near 0, one parametrization traces a small arc of the circle near O from left to right, the other from right to left. For the level curves which are lines parallel to the x-axis, two values of t land on the curve; for the level curves which are lines parallel to the red line, only one (keeping t small).

We have to make almost no changes to this argument when k≠1, provided k is still coprime to d (as assumed). We still get d different values of x for the d values of t and that’s all we need. A little number theory furnishes the proof of this claim. Namely, the exponents of ω essentially belong to the ring ℤ/dℤ, and k is a unit in that ring: lk≡1 mod d for some l, since gcd(k,d)=1 implies lk+qd=1 for some l and q. That means that as j ranges from 0 to d-1, (ωjt)kkjtk takes on d different values. Incidentally, the need for k to be coprime to d prevents the grade inflation bug: replacing d with a multiple is forbidden.

Very good, we now understand multiplicity when using fractional power series. How about the proof of Fact 4? Once again we consider the formula

$\text{res}_x(E,F) = a_m^n \prod_{i=1}^m F(u_i(y))$    (2)

where we assume that the leading coefficient am is a constant, and the x-degree of E is m, of F is n. Key point: the ui(y)’s must include all the conjugate parametrizations of each branch. Otherwise we won’t get m of them! (See Kendig pp.55–56. We saw a similar situation in post 10 with the two-ellipse example: we had to include both positive and negative square roots in the factorizations.)

Among the E-branches crossing the x-axis, we might find all sorts of denominators d. (When d=1, we have the special case treated in the last post.) We need to show that an E-branch intersecting F with multiplicity r on the x-axis, contributes a factor of yr to the lowest-order term in eq.(2). Multiplicity r means that the lowest-order term of F(ui) looks like cyr/d. With all its d conjugates, we get a contribution of (yr/d)d. All’s right with the world!

Finally, Fact 5 follows from Fact 4 just as in the last post.

Prev TOC Next


text:
# Homework Help: Complex integration and residue theorem.

1. Nov 13, 2013

### peripatein

Hi,
1. The problem statement, all variables and given/known data
I was wondering whether any of you could kindly explain to me how the equation in the attachment was derived.
I mean, how could I have known that it could be separated into these two fractions?

2. Relevant equations
The attachment also specifies the integration to be performed in the rectangular region whose coordinates are given.

3. The attempt at a solution
I know that the integrand, and integral, are zero for x=R, as R->infinity.

#### Attached Files:

• ###### D.JPG
File size:
8.3 KB
Views:
116
2. Nov 13, 2013

### Staff: Mentor

Those are just the two non-vanishing parts of the contour.

3. Nov 13, 2013

### peripatein

But how were these expressions derived? I mean, how could I have obtained them myself?
Is it by using Cauchy's Theorem as shown in the attachment I just added? I have tried using that to obtain these expressions, but was unsuccessful.

#### Attached Files:

• ###### C.JPG
File size:
5.5 KB
Views:
99
4. Nov 14, 2013

### jackmell

Your attachment is sorely lacking in detail. How about a picture, labeled, a nice one. All four legs, C1 through C4 starting on the real axis and going counterclockwise. Now, treat it like a fine wine, delicately and in meticulous detail, every square mm of it. Start with the expression:

$$\int_{c1} \frac{e^{iz}}{\cosh(z)}dz$$

what is that? Well, since it's on the real axis, then z=x+iy=x, and we're integrating from -R to R it's simply:

$$\int_{-R}^{R} \frac{e^{ix}}{\cosh(x)}dx$$

let's just leave that one there for now. How about the leg going from (R,0) to (R\pi)? What is that? Well, in terms of the complex variable z=x+iy, that's just z=R+iy but don't forget that dz=idy. Now make the substitution:

$$\int_{C2}\frac{e^{iz}}{\cosh(z)}dz=\int_0^{\pi} \frac{e^{i(R+iy)}}{\cosh(R+iy)}idy$$

Ok, that's #2. How about the other two. Get those and then see what you got. Study it for a little while and see what you can do with it. Maybe things cancel, other things.

You learn what to do by working with them in meticulous detail and then sitting back and studying what you have for a little while, overlooking nothing. Often the answer does not come immediately. It has to simmer a little while in your mind. Don't rush.


text:
# Multi-line equations in latex

Is there anyway to write multi-line equations in latex automatically, I mean without using line break or other manually breaking lines. is it possible to tell latex to begin with the next line when the first line reached the edge of the right margin of your page?

You could use the breqn package for that, BUT...

The following example is taken from the breqn manual:

\documentclass{article}
\pagestyle{empty}% for cropping
\usepackage{breqn}
\begin{document}
Replace $j$ by $h-j$ and by $k-j$ in these sums to get [cf.~(26)]
\begin{dmath}[label={sna74}]
\frac{1}{6} \left(\sigma(k,h,0) +\frac{3(h-1)}{h}\right)
+\frac{1}{6} \left(\sigma(h,k,0) +\frac{3(k-1)}{k}\right)
=\frac{1}{6} \left(\frac{h}{k} +\frac{k}{h} +\frac{1}{hk}\right)
+\frac{1}{2} -\frac{1}{2h} -\frac{1}{2k},
\end{dmath}
which is equivalent to the desired result.
\end{document}


text:
# How can Tensorflow text be used with UnicodeScriptTokenizer to encode the data?

TensorflowServer Side ProgrammingProgramming

The ‘UnicodeScriptTokenizer’ can be used to tokenize the data. The start and end offsets of every word in each sentence can be obtained.

We will use the Keras Sequential API, which is helpful in building a sequential model that is used to work with a plain stack of layers, where every layer has exactly one input tensor and one output tensor.

A neural network that contains at least one layer is known as a convolutional layer. We can use the Convolutional Neural Network to build learning model.

TensorFlow Text contains collection of text related classes and ops that can be used with TensorFlow 2.0. The TensorFlow Text can be used to preprocess sequence modelling.

We are using the Google Colaboratory to run the below code. Google Colab or Colaboratory helps run Python code over the browser and requires zero configuration and free access to GPUs (Graphical Processing Units). Colaboratory has been built on top of Jupyter Notebook.

Tokenization is the method of breaking down a string into tokens. These tokens can be words, numbers, or punctuation.

The important interfaces include Tokenizer and TokenizerWithOffsets each of which have a single method tokenize and tokenize_with_offsets respectively. There are multiple tokenizers, each of which implement TokenizerWithOffsets (which extends the Tokenizer class). This includes an option to get byte offsets into the original string. This helps know the bytes in the original string the token was created from.

## Example

print("Unicode script tokenizer is being called")
tokenizer = text.UnicodeScriptTokenizer()
(tokens, start_offsets, end_offsets) = tokenizer.tokenize_with_offsets(['everything not saved will be lost.', u'Sad☹'.encode('UTF-8')])
print("The tokenized data is converted to a list")
print(tokens.to_list())
print("The beginning offsets of characters are stored in a list ")
print(start_offsets.to_list())
print("The ending offsets of characters are stored in a list ")
print(end_offsets.to_list())

## Output

Unicode script tokenizer is being called
The tokenized data is converted to a list
[[b'everything', b'not', b'saved', b'will', b'be', b'lost', b'.'], [b'Sad', b'\xe2\x98\xb9']]
The beginning offsets of characters are stored in a list
[[0, 11, 15, 21, 26, 29, 33], [0, 3]]
The ending offsets of characters are stored in a list
[[10, 14, 20, 25, 28, 33, 34], [3, 6]]

## Explanation

• When tokenizing strings, it is important to know where in the original string the token originated from.

• Hence, every tokenizer that implements TokenizerWithOffsets has a tokenize_with_offsets method.

• This method returns the byte offsets along with the tokens.

• The start_offsets will tell the number of bytes in the original string each token starts at.

• The end_offsets will tell the number of bytes right after the point where each token ends.

• It is important to note that the start offsets are inclusive and end offsets are exclusive.

Published on 12-Feb-2021 15:57:28
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text:
Opuscula Math. 37, no. 4 (2017), 597-608
http://dx.doi.org/10.7494/OpMath.2017.37.4.597

Opuscula Mathematica

# A hierarchy of maximal intersecting triple systems

Joanna Polcyn
Andrzej Ruciński

Abstract. We reach beyond the celebrated theorems of Erdȍs-Ko-Rado and Hilton-Milner, and a recent theorem of Han-Kohayakawa, and determine all maximal intersecting triples systems. It turns out that for each $$n\geq 7$$ there are exactly 15 pairwise non-isomorphic such systems (and 13 for $$n=6$$). We present our result in terms of a hierarchy of Turán numbers $$\operatorname{ex}^{(s)}(n; M_2^{3})$$, $$s\geq 1$$, where $$M_2^{3}$$ is a pair of disjoint triples. Moreover, owing to our unified approach, we provide short proofs of the above mentioned results (for triple systems only). The triangle $$C_3$$ is defined as $$C_3=\{\{x_1,y_3,x_2\},\{x_1,y_2,x_3\},\{x_2,y_1,x_3\}\}$$. Along the way we show that the largest intersecting triple system $$H$$ on $$n\geq 6$$ vertices, which is not a star and is triangle-free, consists of $$\max\{10,n\}$$ triples. This facilitates our main proof's philosophy which is to assume that $$H$$ contains a copy of the triangle and analyze how the remaining edges of $$H$$ intersect that copy.

Keywords: maximal intersecting family, 3-uniform hypergraph, triple system.

Mathematics Subject Classification: 05D05, 05C65.

Full text (pdf)

1. M. Całczyńska-Karłowicz, Theorem on families of finite sets, Bull. Acad. Pol. Sci., Ser. Sci Math. Astron. Phys. 12 (1964), 87-89.
2. R. Csákány, J. Kahn, A homological Approach to Two Problems on Finite Sets, J. Algebraic Combin. 9 (1999), 141-149.
3. P. Erdös, C. Ko, R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. 12 (1961) 2, 313-320.
4. P. Frankl, Z. Füredi, Non-trivial intersecting families, J. Combin. Theory Ser. A 41 (1986), 150-153.
5. P. Frankl, Z. Füredi, Exact solution of some Turán-type problems, J. Combin. Theory Ser. A 45 (1987), 226-262.
6. J. Han, Y. Kohayakawa, Maximum size of a non-trivial intersecting uniform family which is not a subfamily of the Hilton-Milner family, Proc. Amer. Math. Soc. 145 (2017) 1, 73-87.
7. M.A. Henning, A. Yeo, Transversals and matchings in 3-unoform hypergraphs, Europ. J. Combin. 34 (2013), 217-228.
8. A.J.W. Hilton, E.C. Milner, Some intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. 18 (1967) 2, 369-384.
9. E. Jackowska, J. Polcyn, A. Ruciński, Multicolor Ramsey numbers and restricted Turán numbers for the loose 3-uniform path of length three, submitted.
10. A. Kostochka, D. Mubayi, The structure of large intersecting families, Proc. Amer. Math. Soc., to appear.
11. Zs. Tuza, Critical hypergraphs and intersecting set-pair systems, J. Combin. Theory Ser. B 39 (1985), 134-145.
• Joanna Polcyn
• Adam Mickiewicz University, Faculty of Mathematics and Computer Science, Department of Discrete Mathematics, Umultowska 87, 61-614 Poznań, Poland
• Andrzej Ruciński
• Adam Mickiewicz University, Faculty of Mathematics and Computer Science, Department of Discrete Mathematics, Umultowska 87, 61-614 Poznań, Poland
• Communicated by Dalibor Fronček.
• Revised: 2016-12-12.
• Accepted: 2016-12-13.
• Published online: 2017-04-28.


text:
Data Accessor Variants¶

Basic data accessors¶

class tkp.accessors.dataaccessor.DataAccessor[source]

Base class for accessors used with tkp.sourcefinder.image.ImageData.

Data accessors provide a uniform way for the ImageData class (ie, generic image representation) to access the various ways in which images may be stored (FITS files, arrays in memory, potentially HDF5, etc).

This class cannot be instantiated directly, but should be subclassed and the abstract properties provided. Note that all abstract properties are required to provide a valid accessor.

Additional properties may also be provided by subclasses. However, TraP components are required to degrade gracefully in the absence of this optional properties.

beam

Restoring beam. Tuple of three floats – semi-major axis (in pixels), semi-minor axis (pixels) and position angle (radians).

centre_decl

Declination at the central pixel of the image. J2000 decimal degrees.

centre_ra

Right ascension at the central pixel of the image. J2000 decimal degrees.

data

Two dimensional numpy.ndarray of floating point pixel values. TODO: Definitive statement on orientation/transposing.

extract_metadata()[source]

Massage the class attributes into a flat dictionary with database-friendly values.

While rather tedious, this is easy to serialize and store separately to the actual image data.

May be extended by subclasses to return additional data.

freq_bw

The frequency bandwidth of this image in Hz.

freq_eff

Effective frequency of the image in Hz. That is, the mean frequency of all the visibility data which comprises this image.

pixelsize

(x, y) tuple representing the size of a pixel along each axis in units of degrees.

rms_qc()[source]

RMS for quality-control.

Root mean square value calculated from central region of this image. We sigma-clip the input-data in an attempt to exclude source-pixels and keep only background-pixels.

tau_time

Total time on sky in seconds.

taustart_ts

Timestamp of the first integration which constitutes part of this image. MJD in seconds.

url

A (string) URL representing the location of the image at time of processing.

wcs

An instance of tkp.coordinates.WCS.

The following acessors are derived from the basic DataAccessor class:

class tkp.accessors.fitsimage.FitsImage

Generic FITS data access.

class tkp.accessors.casaimage.CasaImage

Generic CASA image access.

class tkp.accessors.kat7casaimage.Kat7CasaImage

KAT-7 specific CASA image access.

LOFAR-specific data accessors¶

class tkp.accessors.lofaraccessor.LofarAccessor[source]
antenna_set

Antenna set in use during observation. String; ‘LBA_INNER’, ‘LBA_OUTER’, ‘LBA_SPARSE’, ‘LBA’ or ‘HBA’

ncore

Number of core stations in use during observation. Integer.

nintl

Number of international stations in use during observation. Integer.

nremote

Number of remote stations in use during observation. Integer.

subbands

Number of subbands.

subbandwidth

Width of a subband in Hz.

The following accessors are derived from the generic LofarAccessor:

class tkp.accessors.lofarfitsimage.LofarFitsImage

LOFAR FITS access.

class tkp.accessors.lofarcasaimage.LofarCasaImage

LOFAR CASA image access.


text:
Mexican Economic Activity Up as Currency Continues Slide

| Sept. 29, 2011 |

Published by Minyanville

Newly released economic activity figures from July show a rapid rise, spurred on by Mexico’s agriculture sector. Economic activity was up 0.88% in July from the previous month, nearly triple analysts’ median forecast in a Reuters poll which predicted just 0.33%. The figure was 3.7% higher than the same period in 2010, slightly lower than the 3.8% median estimate by Dow Jones. However, the month’s figures will please many after June’s disappointing 0.19% contraction.

Oil production in August was also up to 2.55m barrels per day, however, this is much lower than the 3.4m barrels per day the country’s output peaked at in 2004, notes Adam Thomson writing in the Financial Times. “Mexico has one of the most restricted and anachronistic energy-sector regimes in the world—even more so than that of Cuba,” he writes. This has resulted in minimal investment and a “dearth of exploration,” resulting in many missed opportunities.

International reserves were up to $136.5 billion last week, registering a net increase of$378m after a $380m decline the previous week. Reserves have risen 20.1% this year. The economy did grow 3.9% in the first half of 2011, with the government predicting the year to end at 4%, with 2012 predictions around 3.5% growth. The country’s peso continues to slide, though has gained from last week’s 14 to the USD to around 13.3810 per USD on Wednesday. Still, this adds up to a 7.7% drop this year alone or 11% over the last month. This is primarily down to the European debt crisis hitting Mexico hard, as investors lack confidence in its leaders. Many are wondering whether it’s not time for the country’s Central Bank to step in. Nomura’s Benito Berber believes that authorities will intervene either as part of a coordinated G20 plan, if the peso weakens substantially more than other emerging market currencies or if other Latin American banks begin to intervene. Mexican authorities, claims Berber, are reluctant to intervene. “They have worked hard to foster deep [foreign exchange] hedging markets and believe strongly in the model of a truly flexible exchange rate,” the strategist wrote. “Other than Chile, Mexico is the only Latin American country that really believes in the advantages of a floating regime. Intervention is considered a tool of last resort.” Drug War The country’s war on drugs is making international headlines once again, though perhaps solely as editors flock to the story as popular social-networking site Twitter gets a namecheck. As journalists in Mexico steer clear of open and honest reporting of the drug war, for justified fear of reprisal, anonymous locals are taking charge, posting messages on Twitter revealing locations of murders and—more dangerously—encouragement to turn those guilty over to the police. This has led to a number of murders, bodies left with notes alongside them warning that they had been killed for those actions. Eight journalists have been murdered in Mexico this year alone, according to the country’s Human Rights Commission. The war on the cartels has been a term-long headache for President Felipe Calderón since he took the helm in December 2006. It is likely to lead to the opposition National Action Party (PAN) taking over the country in elections next year, as many are dismayed by the huge rise in deaths over the last five years. In Brief Mexico will not be raising its sugar import quota this year, as the government believes there is enough supply to cover domestic demand. The 150,000-ton quota was brought in earlier this year. There were rumors in August that the government would be doubling the quota in order to stabilize prices. However, prices are falling and so the need for the doubling has disappeared, claims Lorenza Martinez, the Deputy Economy Minister, speaking to Reuters. World’s richest man Carlos Slim’s son Marco Antonio Slim Domit is to join the board of directors of Blackrock (BLK), the world’s biggest money manager which oversees$3.66 trillion in assets. Slim acquired shares in the company last year.


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# Math Help - find the tangent line of a parametrized curve

1. ## line

ok.

2. 1. Evaluate $\displaystyle \gamma'(t)$ at the point $\displaystyle t$ where $\displaystyle \sin{2t} = 0$ and $\displaystyle \cos{2t} = 1$.

2. Velocity = $\displaystyle \mathbf{v}(t) = \frac{d}{dt}[\mathbf{r}(t)]$.

To find the unit tangent vector, divide $\displaystyle \mathbf{v}(t)$ by its length.

Acceleration = $\displaystyle \mathbf{a}(t) = \frac{d}{dt}[\mathbf{v}(t)]$.

3. Hello, Taurus3!

$\text{1. Find the tangent line to given parameterized curve at the given point:}$

. . . . $r(t)\:=\:(2\sin t\cos t,\;\cos 2t),\;\;(0, 1)$

$\text{We have: }\;\begin{Bmatrix}x &=& \sin 2t \\ y &=& \cos 2t \end{Bmatrix} \quad\Rightarrow\quad \begin{Bmatrix}\frac{dx}{dt} &=& 2\cos2t \\ \\[-3mm]\frac{dy}{dt} &=& \text{-}2\sin2t \end{Bmatrix}$

$\text{Then: }\;\dfrac{dy}{dx} \;=\;\dfrac{\frac{dy}{dt}}{\frac{dx}{dt}} \;=\;\dfrac{\text{-}2\sin2t}{2\cos2t} \;=\;-\tan2t$

$\text{At }(0,1),\:t = 0 \quad\Rightarrow\quad \dfrac{dy}{dx} \:=\:-\tan 0 \:=\:0$

$\text{The line through }(0,1)\text{ with slope 0 is: }\:y = 1$

4. for the velocity, since x=ln(t) and y= sqrt.(t^2+1), we could write r(t) as (ln(t)+3t)i + (sqrt.(t^2+1))j right?
And v(t) would be r prime t = (1/t + 3)i + t/(sqrt.(t^2+1)) right?

5. Yes.


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# TGraph with SetBarOffset

In the documentation for THistPainter, the THn’s support bar-related drawing options including multiple histograms offset by some amount.

I would like to use this functionality to make something like Figure 4 here (Measurements of the production cross section of a $Z$ boson in association with high transverse momentum jets in $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector)

where we have asymmetric errors drawn (hence the TGraph / TGraphAsymmErrors). How can this be done? Is the option to draw a TH1 with the regular binning, and then make TGraphs with expanded binning and draw them on top of the plot?

By definition TGraph (and the classes inheriting from it) has no binning. Points are drawn at the exact positions stored in the graph. The only way to produce such plots with graph will be to modify the initial graph by offsetting the x positions with some value.

This topic was automatically closed 14 days after the last reply. New replies are no longer allowed.


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OpenStudy (mathmath333):

\large \begin{align} \color{black}{\normalsize \text{A can run around a circular track in 4 min.} \hspace{.33em}\\~\\ \normalsize \text{ B can do the same thing in 7 min. } \hspace{.33em}\\~\\ \normalsize \text{ If they start running together from same point , clock wise } \hspace{.33em}\\~\\ \normalsize \text{when will they meet for the first time } \hspace{.33em}\\~\\ \normalsize \text{ at the point diametrically opposite to} \hspace{.33em}\\~\\ \normalsize \text{ the starting point. ?}\hspace{.33em}\\~\\}\end{align}

2 years ago
OpenStudy (mathmath333):

|dw:1424450041379:dw|

2 years ago
OpenStudy (zarkon):

if their speeds are constant then, unless i'm missing something, they will never meet at the same time

2 years ago
OpenStudy (zarkon):

A will be at the opposite side at 2,6,10,... B will be at the opposite side at 3.5,11.5,17.5,... (always on a half min)

2 years ago
OpenStudy (zarkon):

3.5,10.5,17.5...

2 years ago
OpenStudy (mathmath333):

yes zarkon is right!

2 years ago
OpenStudy (mathmath333):

since the distance is not given we can also assume it

2 years ago
OpenStudy (mathmath333):

|dw:1424453973019:dw| \large \begin{align} \color{black}{t_1=\dfrac{d}{s_1} \hspace{.33em}\\~\\ =\dfrac{14}{7} \hspace{.33em}\\~\\ =2,6,10,14..... \hspace{.33em}\\~\\ t_2=\dfrac{14}{s_2} \hspace{.33em}\\~\\ =\dfrac{14}{2} \hspace{.33em}\\~\\ =7,21,35,49..... \hspace{.33em}\\~\\ }\end{align}

2 years ago


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PDESystem

PDESystem is the common symbolic PDE specification for the SciML ecosystem. It is currently being built as a component of the ModelingToolkit ecosystem,

Vision

The vision for the common PDE interface is that a user should only have to specify their PDE once, mathematically, and have instant access to everything as simple as a finite difference method with constant grid spacing, to something as complex as a distributed multi-GPU discontinuous Galerkin method.

The key to the common PDE interface is a separation of the symbolic handling from the numerical world. All of the discretizers should not "solve" the PDE, but instead be a conversion of the mathematical specification to a numerical problem. Preferably, the transformation should be to another ModelingToolkit.jl AbstractSystem, but in some cases this cannot be done or will not be performant, so a SciMLProblem is the other choice.

These elementary problems, such as solving linear systems Ax=b, solving nonlinear systems f(x)=0, ODEs, etc. are all defined by SciMLBase.jl, which then numerical solvers can all target these common forms. Thus someone who works on linear solvers doesn't necessarily need to be working on a discontinuous Galerkin or finite element library, but instead "linear solvers that are good for matrices A with properties ..." which are then accessible by every other discretization method in the common PDE interface.

Similar to the rest of the AbstractSystem types, transformation and analysis functions will allow for simplifying the PDE before solving it, and constructing block symbolic functions like Jacobians.

Constructors

ModelingToolkit.PDESystemType
struct PDESystem <: AbstractMultivariateSystem

A system of partial differential equations.

Fields

• eqs

: The equations which define the PDE

• bcs

: The boundary conditions

• domain

: The domain for the independent variables.

• ivs

: The independent variables

• dvs

: The dependent variables

• ps

: The parameters

• defaults

: defaults: The default values to use when initial conditions and/or parameters are not supplied in ODEProblem.

• connector_type

: type: type of the system

• systems

: systems: The internal systems. These are required to have unique names.

• name

: name: the name of the system

• metadata

: metadata: metadata for the system, to be used by downstream packages.

Example

using ModelingToolkit

@parameters x
@variables t u(..)
Dxx = Differential(x)^2
Dtt = Differential(t)^2
Dt = Differential(t)

#2D PDE
C=1
eq  = Dtt(u(t,x)) ~ C^2*Dxx(u(t,x))

# Initial and boundary conditions
bcs = [u(t,0) ~ 0.,# for all t > 0
u(t,1) ~ 0.,# for all t > 0
u(0,x) ~ x*(1. - x), #for all 0 < x < 1
Dt(u(0,x)) ~ 0. ] #for all  0 < x < 1]

# Space and time domains
domains = [t ∈ (0.0,1.0),
x ∈ (0.0,1.0)]

@named pde_system = PDESystem(eq,bcs,domains,[t,x],[u])
source

Domains (WIP)

Domains are specifying by saying indepvar in domain, where indepvar is a single or a collection of independent variables, and domain is the chosen domain type. A 2-tuple can be used to indicate an Interval. Thus forms for the indepvar can be like:

t ∈ (0.0,1.0)
(t,x) ∈ UnitDisk()
[v,w,x,y,z] ∈ VectorUnitBall(5)

Domain Types (WIP)

• Interval(a,b): Defines the domain of an interval from a to b (requires explicit

import from DomainSets.jl, but a 2-tuple can be used instead)

discretize and symbolic_discretize

The only functions which act on a PDESystem are the following:

• discretize(sys,discretizer): produces the outputted AbstractSystem or SciMLProblem.
• symbolic_discretize(sys,discretizer): produces a debugging symbolic description of the discretized problem.

Discretizer Ecosystem

NeuralPDE.jl: PhysicsInformedNN

NeuralPDE.jl defines the PhysicsInformedNN discretizer which uses a DiffEqFlux.jl neural network to solve the differential equation.

MethodOfLines.jl: MOLFiniteDifference

MethodOfLines.jl defines the MOLFiniteDifference discretizer which performs a finite difference discretization. Includes support for higher approximation order stencils and nonuniform grids.


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# Electronic – Power a circuit with power supply and USB port, the solution

hardwarepcb-designpowerpower supplyusb device

I want to power a circuit I have from a dedicated power supply. But at the same time, I have a USB port that can also supply power but only when I decide to upload new firmware to a atmega32u2. Here is my circuit.

J2 is where my main voltage comes in and can be anywhere from +3.3V to +5V. Q1 acts as an ideal diode. Q2 is my load switch. R22 will pull the gate to ground. J4 is the USB port my circuit uses to connect to a PC. D1 isolates the USB voltage from the circuit's operating voltage. C3 is the bypass cap for my circuit. R21 and C1 mimic a tantulum cap. R15 and R16 are my current limiting series resistors for the USB data lines.

The idea here is to disable the use of J2's voltage whenever J4 receives voltage from the USB port. In other words, when +VUSB is present, it will go through D1 and supply about +4.7V which is good enough for communicating back to the PC and enough for my circuit to operate to receive code. +VUSB will have Q2's gate to +5V causing Vgs to be positive or zero volts, turning it off, and thus having +V only sourced from J4, despite J2 still supplying voltage.

Or at least I think so! I am basically asking for a design check to see if my understanding is correct. I am aware of power mux ICs but they are too expensive and I want to use "common parts" for this. In case this is needed, Q1,Q2 = DMP3056LDM-7 and D1 = SD1206S100S1R0. Current draw of the circuit is no more than 10mA.

EDIT 11/1/20:
User Ste Kulov informed me of a cheaper kind of power mux. I like the solution. Here is the new circuit. It uses the TPS2104DBVT. Q1 is still DMP3056LDM. Note that when one of the voltage inputs are disconnected, the potential is floating. I first had a pulldown resistor but the datasheet informs to not ground the INx pin if not used. I took that as no pulldown resistor as well.

Most of this was resolved in the comments (whoops!), but I thought it's worthwhile to summarize everything.

First, is that the original solution with Q1 and Q2 won't work properly. Let's say that J2 has 3.3V attached and is connected in the correct polarity while the USB power is also present. +VUSB connects through D1 to +V. Even though Q2's gate is pulled high, it will conduct through Q2's body diode. Now, since Q1's source is a healthy voltage it is fully on due to its gate being tied low. Re-explained better in the MSPaint masterpiece below, you have a conducting path between VUSB and J2. There is also a similar conducting path when J2 is connected in the reverse polarity.

Therefore, a power MUX IC was recommended and I threw out the idea of TI's TPS2104/TPS2105 series. Your modified post includes a schematic using this solution. However, I suggest a few changes based on additional info you provided.

The TPS2104 has an active low EN, where "active" means IN1 is switched on instead of the default IN2. This $$\\overline{\text{EN}}\$$ pin has an internal pullup according to the datasheet. Therefore, your schematic will always have IN2 enabled since it is controlled by the +VUSB signal. This signal is either +5-ish volts when USB is connected or high-Z if USB is disconnected. In either case, the $$\\overline{\text{EN}}\$$ pin sees a logic high.

This power MUX series uses an N-channel MOSFET for IN1 and a P-channel MOSFET for IN2. The N-channel has the benefit of lower $$\R_{DS(on)}\$$, while the P-channel has the benefit of lower quiescent current since its drive circuitry is simpler. Since J2 has the possibility of being a battery in the future and your operating current (10mA) is low enough that $$\R_{DS(on)}\$$ on either input doesn't really matter, I suggest connecting the J2 input to IN2 instead.

Due to everything laid out above, I think you should be using the TPS2105 instead which has an active high EN pin. Also, please add the 0.22µF capacitor on the other INx pin too. See professionally MSPaint-ed schematic below for reference:


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# Math Help - 2 polar graph questions

1. ## 2 polar graph questions

1.Identify all values of theta for which r=0, and find a range of values of theta that produce one copy of the graph
a) $r=2+4cos3\theta$
my work
$0=2+4cos3\theta$
$\theta= \frac{2\pi}{9}$
$\theta= \frac{4\pi}{9}$
-however, i have no idea to do to find the range to make a copy? can someone tell me the procedure.

2.find a corresponding rectangular equation.
a) $r=3sin\theta$
my work
none
-i understand there are some equations that relate cartesian equations to polar ones. but none of the ones listed in my book list any relevant ones that relate to solving this problem.

All help is appreciated.

2. Originally Posted by dandaman
1.Identify all values of theta for which r=0, and find a range of values of theta that produce one copy of the graph
a) $r=2+4cos3\theta$
my work
$0=2+4cos3\theta$
$\theta= \frac{2\pi}{9}$
$\theta= \frac{4\pi}{9}$
-however, i have no idea to do to find the range to make a copy? can someone tell me the procedure.

2.find a corresponding rectangular equation.
a) $r=3sin\theta$
my work
none
-i understand there are some equations that relate cartesian equations to polar ones. but none of the ones listed in my book list any relevant ones that relate to solving this problem.

All help is appreciated.
For 2.

Note that $\sin\theta=\frac{y}{r}$ from the definition of the sine function, so

$r=3\sin\theta=3\frac{y}{r}$

Which implies that

$r^2=3y$

But, isn't it the case that $r^2=x^2+y^2$ ?


text:
# Sine Function

## Sine Function

In a trigonometric circle, function f of $$\mathbb{R}$$ in $$\mathbb{R}$$ under which all angles of rotation x are made to correspond to the y-coordinate of a point P of this circle.

In the illustration above, sin(x) = k, where k is the y-intercept of point P.

### Symbol

The symbol for the sine function is “sin” and the image of an angle of rotation x of this function is written as “sin(x)” which is read as “sine x” or “sine of x.”
The argument of the function is a number that represents an angle of rotation (and not a figure).

### Example

The graphic representation of the base function defined by f (x) = sin(x), x being the measure of the angle in radians, is:


text:
Math Help - Another solve for X problem help

1. Another solve for X problem help

Six times a positive number when added to the square of a number will produce 16. What is the number?

2. Originally Posted by thecapaccino
Six times a positive number when added to the square of a number will produce 16. What is the number?

let x be the number, then:

6x+x^2 =16

So x(6+x)=16,

and then trial and error gives x=2.

RonL

3. but...plugging in X to check answer

RonL.

If you plug 8 into the equation x(6+x) = 16....it does not equal 16. I get

8(6+8) = 16 (but that does not equal??)

Have I lost the plot?

4. Originally Posted by thecapaccino
RonL.

If you plug 8 into the equation x(6+x) = 16....it does not equal 16. I get

8(6+8) = 16 (but that does not equal??)

Have I lost the plot?
No its a typo should have been 2.

RonL


text:
# How do you find the indefinite integral of int 1/(x+1)?

##### 1 Answer
Aug 4, 2017

The answer is $= \ln | 1 + x | + C$

#### Explanation:

We need

$\int \frac{\mathrm{dx}}{x} = \ln | x | + C$

Therefore,

$\int \frac{\mathrm{dx}}{1 + x} = \ln | 1 + x | + C$