text:
I can't find a way to remove only the page index of sections while keeping the page index of the subsections

• Welcome to tex.sx. The setting of section information is defined in the document class, so if anyone is to be able to help you, we need to know that. For best results, please include a complete small example, beginning with \documentclass and ending with \end{document} that demonstrates what you're getting now. Aug 16, 2021 at 15:31

Assuming you're using one of the "standard" LaTeX document classes (or a document class that's based on one of the standard classes), you could employ the machinery of the tocloft package to achieve your formatting objective.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[italian]{babel}
\usepackage{setspace}
\setstretch{1.5}  % just for this example

\setcounter{secnumdepth}{0}
\setcounter{tocdepth}{2} % or some higher number

\usepackage{tocloft} % modify appearance of ToC, LoF, and/or LoT
\makeatletter
% Don't show section page numbers in ToC:
\renewcommand{\cftsecpagefont}{\@gobble}
\makeatother

\begin{document}
\tableofcontents

\clearpage
\section{Dati dell'opera}
\subsection{Informazioni generali}
\subsection{Storia dell'opera}
\clearpage
\stepcounter{page}
\subsection{Studi per l'opera}

\clearpage
\section{Leonardo da Vinci}
\subsection{Biografia}
\clearpage
\stepcounter{page}
\subsection{Movimento artistico: Il Rinascimento}
\end{document}

• Thank you, I tried to apply your suggestion but (the numbers disappeared as i wanted) but now the same dots that were to the side of subsections appeared also to the side of sections. Aug 16, 2021 at 16:54
• @MichaelB - Sadly, but probably not surprisingly, my mind-reading and psychic divination skillls are pure rubbish. Please feel free to tell us a lot more about your document setup, including which document class you employ. Without this information, chances that anyone will be able to guess correctly what's going on in your document, let alone come up with a solution, will be slim to nonexistent.
– Mico
Aug 16, 2021 at 18:19


text:
# Prove $(A \triangle B) \cap (B\triangle C) \cap (C\triangle A) = \emptyset$

This can be proved by assuming that there exists some $x \in (A \triangle B) \cap (B\triangle C) \cap (C\triangle A)$ and then deriving a contradiction by considering each of the cases that arise.

[$X\triangle Y$ is the symmetric difference of $X$ and $Y$]

Now, this proof ultimately breaks down into six cases which makes it a bit long considering the simple goal. So I was wondering if there was a better way of doing this?

My six case proof goes something like this:

Suppose to the contrary that $x \in (A \triangle B) \cap (B\triangle C) \cap (C\triangle A)$. Then $x \in (A \triangle B)$ and $x \in (B \triangle C)$ and $x \in (C\triangle A)$. Since $x \in (C \triangle A)$, $x \in A\backslash C$ or $x \in C\backslash A$.

Case 1: $x \in A\backslash C$. Since $x \in (B \triangle C)$, $x \in B\backslash C$ or $x \in C\backslash B$. Case 1.1: $x\in B\backslash C$...

Case 2: $x \in C\backslash A$. Since $x \in (B \triangle C)$, $x \in B\backslash C$ or $x \in C\backslash B$. Case 2.1: $x\in B\backslash C$...

-
By symmetry there is only one case. Suppose $x$ is in the first symmetric difference. Without loss of generality $x$ is in $A$ but not in $B$. so by the second term it is in $C$, which contradicts the last symmetric difference. –  André Nicolas Apr 6 '13 at 16:04

In principle there are two cases, but symmetry brings it down to one.

Suppose $x$ is in our set. Then $x$ is in $A\triangle B$. Without loss of generality we may assume that $x$ is in $A$ but not in $B$.

So since $x$ is in $B\triangle C$, we conclude that $x$ must be in $C$. But then $x$ cannot be in $A\triangle C$. So our set is empty.

-

Indicator functions are perhaps a good way to go if you want to avoid cases or symmetry arguments. First prove that $$\chi_{A \triangle B} = (\chi_A+\chi_B \bmod 2)$$ Then show that if $S = (A \triangle B) \cap (B \triangle C) \cap (C \triangle A)$ then we have \begin{align} \chi_S &= (\chi_A+\chi_B)(\chi_B+\chi_C)(\chi_C+\chi_A) \bmod 2 \\ &= \chi_A\chi_B\chi_C + \chi_A^2\chi_B+\chi_A^2\chi_C+\chi_A^2\chi_C\\ & \qquad +\chi_B^2\chi_C+\chi_B^2\chi_A+\chi_B\chi_C^2+\chi_B\chi_C\chi_A \bmod 2 \\ &= \chi_A\chi_B\chi_C + \chi_A\chi_B + \chi_A\chi_C+\chi_A\chi_C \\ & \qquad +\chi_B\chi_C + \chi_B\chi_A + \chi_B \chi_C + \chi_B\chi_C\chi_A \bmod 2\\ &= 2(\chi_A\chi_B\chi_C + \chi_A\chi_B + \chi_A\chi_C + \chi_B\chi_C) \bmod 2 \\ &= 0 \end{align}

-
Actually, we can just work $\mod 2$ so ${\bf 1}_{A\triangle B}\equiv {\bf 1 }_A+{\bf 1 }_B$, right? Maybe it makes calculations shorter. –  Pedro Tamaroff Apr 6 '13 at 16:16
@PeterTamaroff: D'oh, I learnt that trick in my first year, should have used that... thanks! I'll update the answer now. –  Clive Newstead Apr 6 '13 at 16:18

It's possible to make only one case. Because $$x\in (A \triangle B)\iff (x\in A \wedge x\notin B ) \vee (x\notin A \wedge x\in B)$$

Then we have $$x\in (A\triangle B) \cap (B\triangle C) \cap (C\triangle A)$$ using the definition above we see that $x$ can't be in $A,B$ or $C$. Hence the intersection is empty.

-

Your equality $$(A \triangle B) \cap (B\triangle C) \cap (C\triangle A) = \varnothing \tag{\clubsuit}$$

can be interpreted as: $$\color{blue}{for\ any\ three\ numbers,\ some\ two\ have\ the\ same\ parity,}$$ which is obviously true. To get the details, you can consider $\clubsuit$ element-wise and translate into logic, that is, it is equivalent to $$(x_A \oplus x_B) \land (x_B \oplus x_C) \land (x_C \oplus x_A) = \mathtt{false}.$$

However, the last equality might be interpreted in $\mathbb{Z}_2$, that is, in numbers modulo 2. From this perspective we get $$(y_A + y_B)(y_B + y_C)(y_C+y_A) \equiv 0 \pmod 2 \tag{\spadesuit}$$ for any $y_A,y_B,y_C \in \mathbb{Z}$. But $\spadesuit$ literally means: some two of any three numbers have same parity, and this concludes the proof.

I hope this helps ;-)

-
Very nice. Working modulo makes things much easier! –  Pedro Tamaroff Apr 6 '13 at 16:36
@PeterTamaroff Yeah, it does. When I approach XOR, I immediately think of $\mathbb{F}_2$ field. I've just noticed, that because of your comment the old Clive's answer looks now almost exactly like mine :-P –  dtldarek Apr 6 '13 at 16:41
\begin{align} & x \in (A \triangle B) \cap (B\triangle C) \cap (C\triangle A) \\ \equiv & \;\;\;\;\;\text{"expand definition of $\;\cap\;$ twice, and of $\;\triangle\;$ three times"} \\ & (x \in A \not\equiv x \in B) \land (x \in B \not\equiv x \in C) \land (x \in C \not\equiv x \in A) \\ \equiv & \;\;\;\;\;\text{"rewrite third part to $\;x \in C \equiv x \not\in A\;$; use that to substitute in second part"} \\ & (x \in A \not\equiv x \in B) \land (x \in B \not\equiv x \not\in A) \land (x \in C \equiv x \not\in A) \\ \equiv & \;\;\;\;\;\text{"simplify second part to $\;x \in B \equiv x \in A\;$; use that to substitute in first part"} \\ & (x \in A \not\equiv x \in A) \land (x \in B \equiv x \in A) \land (x \in C \equiv x \not\in A) \\ \equiv & \;\;\;\;\;\text{"first part is false"} \\ & \textrm{false} \\ \equiv & \;\;\;\;\;\text{"definition of $\;\emptyset\;$"} \\ & x \in \emptyset \\ \end{align}

format: plain


text:
# Determining which scale items to use for further analysis

From a questionnaire survey I have 44 statements (items) which were supposed to measure 11 bases of power (factors). Four items per factor. Since this survey was only slightly altered from its well established original source, it is expected that eleven factors are actually measured. The number of observations is 258.

Due to the origin of the survey instrument and only slight adaptations made to the new context, I am unsure how to analyze the validity of the scales. Until now I have done the following:

• Grouped the descriptives of the 44 statements by the factor they should measure and looked for anomalies (describe() from the psych package).

• Created boxplots for the 11 bases of power - or rather their underlying scale items - and looked for anomalies (see here).

• Observed whether there is a drop in $\alpha$ when removing any of the scale items (alpha from the psych package).

• Observed whether there is an increase in average inter-item correlation when removing any of the scale items (alpha() from the psych package).

• Created an iclust plot for all 11 factors to see whether more items should be removed based on an increase of $\beta$ (iclust() from the psych package, see here).

By following this approach I end up removing five items due to an increase in $\alpha$, and one more item due to an increase in average inter-item correlation (and .01 decrease in $\alpha$). Some more items could be removed if I were strict with the increase in $\beta$.

Question 1: Should I remove items if this brings an increase in $\beta$ whilst the $\alpha$ remains unchanged?

Question 2: Should I remove items for a little increase in $\alpha$ or $\beta$ if the removal leads to less than three items for this factor?

Now having removed these items which are obviously not measure what they are supposed to, I tried to find a way to quantify the overall suitability of this measurement instrument. Since the number of factors is known and concluded from previous literature, I thought a confimatory factor analysis (CFA) would be a good idea (sem package). However no matter how hard I tried (parcelling based on the above iclust analysis, covariance between latent variables, modIndices() ) the best results were:

Model                   X²          DF      GFI     AGFI    RMSEA   NNFI    CFI     SRMR
38 items                1935.132    643     0.72    0.6773  0.0884  0.6678  0.6962  0.1661
Parcelling - 23 items   766.373     202     0.8168  0.7534  0.1032  0.7151  0.7691  0.1714

Now I wonder two things:

Question 3: Is it neccessary to report these findings?

Question 4: Should I ditch the idea of eleven factors and start over with exploratory factor analysis (EFA)?

I would face some challenges if the questionnaire hadn't measured what it was supposed to, hence the effort in keeping the eleven factors.

First of all I would say that you should definitely think about a latent variable approach. I don't know about the 11 bases of power you are talkng about. But it's definitely a psychological and as such a latent construct. This means that the questionaire cannot perfectly measure 11 distinguishable factors, even if it does indeed measure them. To account for this measurement error, you should do a confirmatory factor analysis, as you have already considered.

As far as your first two questions are concerned, this is also a better way to determine, if the items in a scale do measure the same construct. You should definitely not use Cronbach's $\alpha$, as it is not related to the internal structure of a questionaire. See the article of Sijtsma (2009) for a detailed account. This might come as a surprose to you - I know it certainly came as a surpise for me when I first learned about it. Especially since in the psychological literature, Cronbach's $\alpha$ is used all over the place.

If you do a confirmatory factor analysis, you can test different measurement models for your survey. It is relatively unlikely that a $\tau$-parallel model will fit the data. But an essential $\tau$-parallel model or a $\tau$-congeneric model might provide a decent or even a good fit. And there are a lot more possibilites to try. You can go a pretty long way, if you want.

Note that this does not have to do with the validity of the questionaire. Validity concerns the question, if the test is measuring what it is supposed to do. There are different ways to investigate validity. Usually you try to find other questionaires, that are supposed to measure something similar. Your questionaire should correlate highly with those. Likewise it should not correlate with questionaires that mesure something completely unrelated. For a very interesting critique of this approach see Borsboom et al. (2004).

As to your third question, yes, I would definitely report all your results, even if you wish they were different. This is also related to the fourth question. Maybe you have to give up the idea of 11 factors, because you just can't find support for it. I understand that that's not exactly what you wish for. But this would be a progress in knowledge as well. Coming from psychology myself, I know that there is a tendency to postulate some structure consisting of a number of - possibly even - independent factors. But very often, the assumptions just don't hold in empirical tests.

References:

Sijtsma, K. (2009). On the use, the misuse, and the very limited usefulness of Cronbach’s alpha. Psychometrika, 74(1), 107-120. LINK
Borsboom, D., Mellenbergh, G. J., & van Heerden, J. (2004). The concept of validity. Psychological review, 111(4), 1061. LINK

• +1 and thanks for the links. One thing though I can't help frowing at is "internal consistency", a vague term which I'd not recommend to use at all. For sometimes it is used for items-construct cohesion (validity) and sometimes for inter-items interchangeability (an aspect of reliability) – ttnphns Aug 14 '13 at 11:52
• @ttnphns I see your point and changed my wording. – Jens Kouros Aug 14 '13 at 12:06
• Thanks for the ideas and links Jens. I will look into $\tau$ congeneric models starting here, since I have no idea what this really means. The paper from Sijtsma (2009) was very enlightening about the usefulness of Cronbach's $\alpha$. I suppose for my purpose I better stay away of this measure. I'm very happy to accept your answer as a solution, as it opens me some more doors where to look at and which road to go down from here. Thanks. – Roman Aug 14 '13 at 12:11
• @Roman You're welcome. You should learn about Classical Test Theory, that's what this is about. From the website on your profile I get the impression that you speak german. Is that true? If so, I could refer you to this site which has excellent lectures on this topic. – Jens Kouros Aug 14 '13 at 12:22
• @Roman Every link on the site will lead to a whole semester worth of lectures. I know it's a lot, but they are very good. You can get a really good understanding of the theory that way. Also, you might not have to watch all of them. But there is a good chance you will want to. Unfortunately all the texts that I have read are from textbooks that are not freely available. I like this one and this one a lot. – Jens Kouros Aug 14 '13 at 12:45


text:
# Efficiency of rail

Calculate how much must cost rail ticket (x) in today's ticket price € 11 and if the government did not subsidize the trains if it is known that it would cost three times today's ticket without subsidies prices.

Calculate the value of today's ticket subsidies (y).

x =  33
y =  22

### Step-by-step explanation:

We will be pleased if You send us any improvements to this math problem. Thank you!

Tips to related online calculators
Check out our ratio calculator.

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text:
# ARMA when ARIMA should be used

(Note: I am taking a first course in time series -- correct me where I am wrong.)

What happens when we fit an ARMA model to a time series when a differenced model (ARIMA with nontrivial $d$), should be used instead?

To be precise, consider the following simulation in R:

example <- arima.sim(n=2000,list(order=c(1,1,3),ar=0.9,ma=c(-1,0.1,0.01)))

When I plot the acf of this function, I get the following slowly decaying ACF, which is exactly what we expect from an ARIMA process.

I have been taught that this indicates the time series should be differenced. Suppose that I don't and fit an ARMA(1,1) to the data instead.

fit <- arima(example, order=c(1,0,1))

Now, if I plot the residuals, they look stationary and well-behaved. Also, here is the ACF and PACF of the residuals for "fit":

So the residuals look like white noise, which is what we expect from a good'' model.

My question is: What was the foul that I made by doing this ARMA fit when I should have differenced the series immediately? I am too inexperienced to detect a flaw in this proposed ARMA(1,1) model. (Maybe there isn't one...)

## 1 Answer

Any ARIMA($p,d,q$) model can be approximated arbitrarily closely by an ARMA($p+d, q$) model. To do this, all you need to do is to set the auto-regressive characteristic polynomial of the latter to have $d$ roots that are set very close to one (but still less than one to allow stationarity).

Details: Under an ARIMA($p,d,q$) model you have auto-regressive characteristic polynomial:

$$\phi(B) = (1-B)^d(1 - \phi_1 B - \cdots - \phi_p B^p).$$

Taking the root $r = 1 + \epsilon$ for some small value $\epsilon > 0$, you can approximate this arbitrarily closely by the atuo-regressive characteristic polynomial:

$$\phi^*(B) = \Big( 1- \frac{B}{r} \Big)^d (1 - \phi_1 B - \cdots - \phi_p B^p) = (1 - \phi_1^* B - \cdots - \phi_{p+d}^* B^{p+d}).$$

As $\epsilon \rightarrow 0$ you have $\phi^* \rightarrow \phi$, so the ARMA approximation converges to the ARIMA form.

Yours results: The above relationship means that it is unsurprising that you can get a good fit to ARIMA data using an ARMA model. In your particular example you have only used $p = 1$ values in your ARMA model, not $p+d=2$. The result is that you get a second auto-correlation line in your PACF. If you instead fit an ARMA($2,3$) model, you should get a good fit to your data, and you will probably find that the estimated auto-regressive characteristic polynomial has $d = 1$ roots that are close to unity.

You have not used set.seed in your example, so your results are not replicable. You have also fit an ARMA model with lower degree than would be required to approximate the ARIMA data ideally. This means that the above issue is conflated with use of insufficient parameters in your model. To rectify this, here is a replicable example using an ARMA($2,3$) model to approximate data from an ARIMA($1,1,3$) process:

set.seed(1);

#Generate time-series data from ARIMA(1,1,3) model
arima.model <- list(order = c(1,1,3), ar = 0.3, ma = c(-1, 0.1, 0.01));
example <- arima.sim(n = 10000, arima.model);

#Fit data to ARMA(2,3) model
arima(example, order = c(2,0,3));

Call:
arima(x = example, order = c(2, 0, 3))

Coefficients:
ar1     ar2     ma1      ma2      ma3  intercept
0.0286  0.9705  0.2782  -0.8137  -0.1422    -5.3551
s.e.  0.0893  0.0892  0.0899   0.0644   0.0140     2.9047

sigma^2 estimated as 1.026:  log likelihood = -14323.05,  aic = 28660.11

As you can see from this output, the estimated auto-regressive characteristic polynomial has a root that is very close to one. The ARMA model has absorbed the differencing operation in the ARIMA data and approximated this by an additional degree in the AR part, with a root that is close to one.


text:
Rectangle With Sigma With Pi

Algebra Level 1

The lenth of a rectangle is $$\sum_{n=1}^3 n$$ and its area is $$\prod_{k=1}^4 k$$


text:
# François G. Dorais

## Santa Exists!

Here is a theorem that I like to mention during the first week when I teach set theory.

Theorem. Santa exists!

Proof. Consider the set

Then $S \in S \liff (S \in S \lthen \text{Santa exists}).\tag{🎅}\label{hohoho}$ The forward implication of \eqref{hohoho} gives that

But then, the backward implication of $(*)$ gives that $S \in S.$

Knowing that $S \in S$ and $S \in S \lthen \text{Santa exists}$, we conclude that Santa does exist! QED

As you probably noticed, this is actually a disguised version of Russell’s Paradox, which is due to Haskell Curry. Should Santa fail to exist, $S$ would simply become the usual paradoxical set $R = \set{X : X \notin X}$ (but we all know that $S$ simply consists of all sets).

What I like about this variant of Russell’s Paradox is that it does not rely on the law of excluded middle.1 The usual argument for Russell’s Paradox proceeds by reaching a contradiction from $R \in R$ and another contradiction from $R \notin R$. In order to reach a final contradiction, this argument assumes that $R \in R \lor R \notin R$ is true, so all of your intuitionist students will immediately be disappointed.

On the other hand, Curry’s Paradox works even in minimal logic! The only sketchy step is the deduction of $S \in S \lthen \text{Santa exists}$ from \eqref{hohoho}. The forward implication actually is

and the desired conclusion is reached by contraction (the fact that assuming $S \in S$ twice is not more powerful than assuming it once). Odds are that very few students will bother counting how many times the assumption $S \in S$ is used in this step.

This shows that self-reference is not a problem unique to classical logic. It applies equally well to intuitionistic and minimal logic. In fact, to avoid self-referential paradoxes through logic it seems necessary to reject the contraction rule (as in linear logic). This is rather extreme from the foundational perspective, so we’re probably better off resolving self-referential issues by other means.

My personal conclusion from this story is that logicians will never be out of work… so Santa does exist!!!

1. As Andreas Blass pointed out in the comments, one can derive Russell’s Paradox in intuitionistic logic. (By the same argument, replacing Santa exists by $\bot$.) The real strength of Curry’s Paradox is that it does not even use negation!

Originally posted on by François G. Dorais. To the extent possible under law, François G. Dorais has waived all copyright and neigboring rights to this work.

• Andreas Blass wrote

I don’t think the Russell paradox needs the law of the excluded middle. Once you’ve derived (as you said) a contradiction from $R \in R$ and another contradiction from $R \notin R$, the first proves (even intuitionistically) that $R \notin R$ (thanks to the intuitionistic meaning of negation as implying a contradiction), and then the second gives you an outright contradiction.

• François G. Dorais replied to Andreas Blass

Right! The same argument works replacing “Santa exists” by $\bot$

I was thinking of a different argument (which I think is how the paradox is usually described): Since $R \in R \lthen (R \notin R \land R \in R)$ and $R \notin R \lthen (R \notin R \land R \in R)$, from $R \in R \lor R \notin R$ we conclude $R \notin R \land R \in R$.

I guess the real difference is that Curry’s version doesn’t require negation (or $\bot$).

• Buschi Sergio wrote

This is simply (seem to me) a wrong proof (no a different version of Russel paradox).

When you write: “The forward implication of (∗) gives that…” forget to mention $if\ S\in S$

• François G. Dorais replied to Buschi Sergio

This is explained later on in the post. The forward implication is $S \in S \lthen (S \in S \lthen \text{Santa exists})$ which gives $S \in S \lthen \text{Santa exists}$ by contraction.

Of course, this is an invalid proof in linear logic, which does not admit the contraction rule, but then I would have to be more precise about what implication means…


text:
Select libraries are now open to Wayne State students, faculty and staff. See what services we're offering online and in person

# How to Use LaTeX

A guide for installing and using the typesetting program $$\LaTeX$$.

## Writing a $$\LaTeX$$ Document

LaTeX allows users to create a near-endless variety of documents, so it's impossible to give anything like a comprehensive introduction to writing in LaTeX. However, nearly all LaTeX documents follow the same, fairly simple structure:

## Basic LaTeX Document Structure

All LaTeX documents are divided into two basic sections

1. The preamble, which provides its visual style and structure
2. The body, which provides the content

### 1. Preamble

The preamble specifies the class of document being written (e.g., article, book, slides, etc.), any packages required, and any options for the document class and packages:

\documentclass[options]{class}  \usepackage[options]{package1}  \usepackage[options]{package2}  ⋮

### 2. Body

The body directly follows the last line of the preamble. Its general format is fairly simple, with all of the document's visible content written between two lines of code which specify the beginning and end of the document's body:

\begin{document}  ⋮  \end{document}

## LaTeX Editors

Writing a LaTeX document can either be done in a general text editor or in a LaTeX-specific editor. Below are just a few examples of what's available.

## Browser-Based LaTeX

Because of the sheer volume of packages available for LaTeX and difficulties translating them to an HTML environment, browser-based options are less flexible but more accessible, with the added benefit that they allow collaboration on documents.


text:
# Solve this following

Question:

Find the values of $\lambda$ and $\mu$ so that the points $A(3,2,-4), B(9,8,-10)$ and $C(\lambda, \mu-6)$ are collinear.

Solution:


text:
Visitor

# Windows 11

I just upgraded to windows 11 today and my pc hasnt been able to start up properly, even after a few resets. All i see is taskbar disappearing and appearing with the background, i cant even see my icons.
0 Replies


text:
# OpenGL ES 2.0: Calculated Cube Vertex Normals Verification

Could I kindly ask to confirm, that the calculated normals are correct, please? I have calculated them on my own, but my testcube is still strangely lighted within OpenGLES 2.0.

• The vertices were exported from 3D authoring application.
• The vertex normals were calculated using matrix cross product.
• The vertex and fragment shaders I were exact copies from the book, so there should not be a bug.

The output in OpenGL looks like the sphere is lighted from its bottom. Moreover, it looks, like the box is visible from inside, as the top part is always transparent. Any idea, where can be a problem?

## Added:

The vertex normals are calculated based on triangulated quad-polygons. The triangulation is performed internally, by the 3d application. Vertices/ indices are generated by calling API calls of that application.

#begin 8 vertices
vs 24
v -25.000000 -25.000000 -25.000000
v 25.000000 -25.000000 -25.000000
v -25.000000 25.000000 -25.000000
v 25.000000 25.000000 -25.000000
v -25.000000 -25.000000 25.000000
v 25.000000 -25.000000 25.000000
v -25.000000 25.000000 25.000000
v 25.000000 25.000000 25.000000
#end 8 vertices

#begin 12 faces
fs 36
f 0 2 3
f 0 3 1
f 0 1 5
f 0 5 4
f 0 4 6
f 0 6 2
f 1 3 7
f 1 7 5
f 2 6 7
f 2 7 3
f 4 5 7
f 4 7 6
#end 12 faces

#begin 8 calculated normals
ns 24
n -0.577350 -0.577350 -0.577350
n 0.816497 -0.408248 -0.408248
n -0.408248 0.816497 -0.408248
n 0.408248 0.408248 -0.816497
n -0.408248 -0.408248 0.816497
n 0.408248 -0.816497 0.408248
n -0.816497 0.408248 0.408248
n 0.577350 0.577350 0.577350
#end 8 calculated normals

I use OpenGL ES 2.0, but that should not be relevant at this moment.

## Update:

I can confirm now, that the calculated vertex normals are correct, and can therefore be used by others, to verify correctness of their triangulated cube vertex normals calculations.

However, I still have a problem that all the vertices, which are in positive z axis, are not rendered. Not sure if they are culled, but I have pretty large frustum. Vertices in the -z axis are kept, so the rendered mesh looks like cut in half, as it is position in 0 z-axis.

This lines below show that I should not concsicously setup small render area:

mProjection = glm::perspective (45.0f, (float)WINDOW_WIDTH / (float)WINDOW_HEIGHT, 1130.0f, -1130.0f);

mViewTransformation = glm::lookAt(glm::vec3(vIntCamera.x, vIntCamera.y, 1130.0f), glm::vec3(vIntCamera.x, vIntCamera.y, 0.0f), glm::vec3(0.0f, 1.0f, 0.0f));

Any ideas, please, what else should I check?

## Solved:

Eye position of the camera (in glm::lookAt() function) should be above the top of the perspective frustum, defined by glm::perspective().

Thanks for your support guys, especially with the upper part of my question.

• possible duplicate of Any reliable polygon normal calculation code? – bummzack May 14 '11 at 9:30
• How do you define your cubes triangles? Are you sure the triangle vertices are all defined in counter-clockwise orientation? – bummzack May 14 '11 at 9:38
• @bummzack: Hi, please, do not mark this trhead as a duplication. It is not difficult to find vertex normal generation algorithm. This question is where the probem can be (it might not be in normals), and can help others to check correctness of their algorithms against my cube normals (if proven as correct), as the cube is very standard mesh for initial testings. – Bunkai.Satori May 14 '11 at 15:26
• @bummzack: to answer your second question: I use 3D authoring application, and I have developed a plug in. The plug in uses API calls to output vertices, triangulated indices and facet normals. I generated the above normals in two ways: Firstly, by summing and averating the facet normals as returned from the software. Secondly, I calculated facet normals from scratch, by generating vertices cross product. Both results were the same. As the application uses OpenGL internaly, I that th order of triangle vertices is counter-clockwise, but I am going to verify that now. – Bunkai.Satori May 14 '11 at 15:33
• @all: the probem si partially solved: I can confirm now, that the calculated normals are correct. The problem I had, which took me more than 24 hours to solve, was in one sign. I used: mProjection = glm::perspective (45.0f, (float)WINDOW_WIDTH / (float)WINDOW_HEIGHT, -1130.0f, 1130.0f); instead of correct mProjection = glm::perspective (45.0f, (float)WINDOW_WIDTH / (float)WINDOW_HEIGHT, 1130.0f, -1130.0f); However, there is one more problem: the mesh gets rendered only if it is moved into negative -z. The vertices that are in positive z are ignored/culled. Any idea, please? – Bunkai.Satori May 14 '11 at 17:16

## 2 Answers

Without a picture I'll have a rough guess on some of your problems.

Normals: If this is a cube, I'd expect all the normal component values to be +/- 0.577350 in different combinations depending on the vertex. Where you have other values the normal is no longer being properly averaged from the three faces that share it and it'll be tending more in one direction. Because you're using shared vertices for the faces then despite it being a cube, it's trying to be lit as if it's a sphere. If you wanted it to look like an actual cube you'll need to duplicate up the vertices and have unique normals for them.

"visible from inside": This sounds like a winding order issue. Graphics hardware tends to have Backface Culling enabled as a speed optimisation, depending on the order of the vertices given to the hardware it makes a decision as to whether it thinks a polygon faces towards or away from the viewer. It's possible you can fix this problem by changing the order of the vertices in your faces, so the first face would become "0 3 2" instead of "0 2 3".

A picture of the output would help to be more specific, as would a more accurate description of what problems you actually have, for example, you say it's lit from below, is this a problem or was that just stating how it looks?

Quick update:

You can display the normals manually by drawing some simple lines, just use the vertex position as the start point, then add the normal value (scaled by some constant factor to ensure it's the length you want) and add this to the original vertex position to generate an end point.

• +1 for the first paragraph here. Bunkai needs 24 vertices to make a properly shaded cube here. 3 vertices located at each corner with normals orthogonal to the face of the triangle the vertex will be used for. – Steve H May 14 '11 at 11:12
• +1 for valuable answer. Hi Roger, thanks for your reply. I understand the issue of trippling vertices to have properly shaded cube. However, at this moment, I think the higher priority is solving the issues I described above: it looks like I always see the inside of the cube, and despite the light is positioned in front of the cube, it looks like the cube is lit from its back. I process triangulated cube, therefore, there are some irregularities in normals, assuming the normals are calculated correctly. – Bunkai.Satori May 14 '11 at 15:54
• @Steve H: thanks for your comment. I did not realize this, but because I process triangulated cubes I might need 48 vertices, is that correct? – Bunkai.Satori May 14 '11 at 15:57
• since this answer was more detailed and provided me with additional information, I am marking it as the Acceptable Answer. The problem is solved, although it was elsewhere. I described it in the update of my question above. – Bunkai.Satori May 14 '11 at 18:32

I doubt GL ES has geometry shaders but just in case it does (or if you can test your data on something with a full OpenGL 3 and above), my answer to this question on SO provides a geometry shader to visualize your normals.

• Unfortunatelly, GL ES has only vertex and fragment shaders. I do not use a full OpenGL 3. Therefore, dipping into OpenGL 3+ would probably be more consuming than finding answer to my question. Maybe, if anybody can generate vertex normals for a cube, so we could compare the results. The normals should be the same. – Bunkai.Satori May 14 '11 at 7:28
• If anyone has idea, that would really help. – Bunkai.Satori May 14 '11 at 9:18


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# If sin A =3/4 , calculate cos A and tan A. - CBSE Class 10 - Mathematics

#### Question

If sin A =3/4 , calculate cos A and tan A.

#### Solution

Let ΔABC be a right-angled triangle, right-angled at point B.

Given that,

sin A = 3/4

(BC)/(AC) = 3/4

Let BC be 3k. Therefore, AC will be 4k, where k is a positive integer.

Applying Pythagoras theorem in ΔABC, we obtain

AC2 = AB2 + BC2

(4k)2 = AB2 + (3k)2

16k 2 − 9k 2 = AB2

7k 2 = AB2

AB = sqrt7k

cos A = ("Side adjacent to"angleA)/"Hypotenuse"

=(AB)/(AC) = sqrt(7k)/(4k) = sqrt7/4

tan A = ("Side adjacent to"angleA)/("Side adjacent to"angleA)

= (BC)/(AB) = (3k)/(sqrt7k) = 3/sqrt7

Is there an error in this question or solution?

#### APPEARS IN

NCERT Solution for Mathematics Textbook for Class 10 (2019 to Current)
Chapter 8: Introduction to Trigonometry
Ex. 8.10 | Q: 3 | Page no. 181
Solution If sin A =3/4 , calculate cos A and tan A. Concept: Trigonometric Ratios.
S


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# Screw

Screw
Screws come in a variety of shapes and sizes for different purposes. U.S. quarter coin (diameter 24 mm) shown for scale.

A screw, or bolt, is a type of fastener characterized by a helical ridge, known as an external thread or just thread, wrapped around a cylinder. Some screw threads are designed to mate with a complementary thread, known as an internal thread, often in the form of a nut or an object that has the internal thread formed into it. Other screw threads are designed to cut a helical groove in a softer material as the screw is inserted. The most common uses of screws are to hold objects together and to position objects.

Often screws have a head, which is a specially formed section on one end of the screw that allows it to be turned, or driven. Common tools for driving screws include screwdrivers and wrenches. The head is usually larger than the body of the screw, which keeps the screw from being driven deeper than the length of the screw and to provide a bearing surface. There are exceptions; for instance, carriage bolts have a domed head that is not designed to be driven; set screws have a head smaller than the outer diameter of the screw; J-bolts have a J-shaped head which is not designed to be driven, but rather is usually sunk into concrete allowing it to be used as an anchor bolt. The cylindrical portion of the screw from the underside of the head to the tip is known as the shank; it may be fully threaded or partially threaded.[1]

The majority of screws are tightened by clockwise rotation, which is termed a right-hand thread. Screws with left-hand threads are used in exceptional cases. For example, when the screw will be subject to counterclockwise torque (which would work to undo a right-hand thread), a left-hand-threaded screw would be an appropriate choice. The left side pedal of a bicycle has a left-hand thread.

## Differentiation between bolt and screw

A carriage bolt with square nut
A structural bolt with a hex nut and washer.

There is no universally accepted distinction between a screw and a bolt. Machinery's Handbook describes the distinction as follows:

A bolt is an externally threaded fastener designed for insertion through holes in assembled parts, and is normally intended to be tightened or released by torquing a nut. A screw is an externally threaded fastener capable of being inserted into holes in assembled parts, of mating with a preformed internal thread or forming its own thread, and of being tightened or released by torquing the head. An externally threaded fastener which is prevented from being turned during assembly and which can be tightened or released only by torquing a nut is a bolt. (Example: round head bolts, track bolts, plow bolts.) An externally threaded fastener that has thread form which prohibits assembly with a nut having a straight thread of multiple pitch length is a screw. (Example: wood screws, tapping screws.)[2]

This distinction is consistent with ASME B18.2.1 and some dictionary definitions for screw[3][4] and bolt.[5][6][7]

The issue of what is a screw and what is a bolt is not completely resolved with Machinery's Handbook distinction, however, because of confounding terms, the ambiguous nature of some parts of the distinction, and usage variations.[8] Some of these issues are discussed below:

### Machine screws

ASME standards specify a variety of "Machine Screws"[9] in diameters ranging up to 0.75 in (19.05 mm). These fasteners are often used with nuts as well as often driven into tapped holes. They might be considered a screw or a bolt based on the Machinery's Handbook distinction. In practice, they tend to be mostly available in smaller sizes and the smaller sizes are referred to as screws or less ambiguously as machine screws, although some kinds of machine screws can be referred to as stove bolts.

### Hex cap screws

ASME standard B18.2.1 -1996 specifies Hex Cap Screws that range in size from 0.25–3 in (6.35–76.20 mm) in diameter. These fasteners are very similar to hex bolts. They differ mostly in that they are manufactured to tighter tolerances than the corresponding bolts. The Machinery's Handbook refers parenthetically to these fasteners as "Finished Hex Bolts".[10] Reasonably, these fasteners might be referred to as bolts but based on the US government document, Distinguishing Bolts from Screws, the US government might classify them as screws because of the tighter tolerance.[11] In 1991 responding to an influx of counterfeit fasteners Congress passed PL 101-592[12] "Fastener Quality Act" This resulted in the rewriting of specifications by the ASME B18 committee. B18.2.1[13] was re-written and as a result they eliminated the "Finished Hex Bolts" and renamed them the "Hex Cap Screw"—a term that had existed in common usage long before, but was now also being codified as an official name for the ASME B18 standard.

### Lug bolts & head bolts

These terms refer to fasteners that are designed to be threaded into a tapped hole that is in part of the assembly and so based on the Machinery's Handbook distinction they would be screws. Here common terms are at variance with Machinery's Handbook distinction.[14][15]

### Lag bolt

Lag screws, also called lag bolts.
Another view.

Lag bolts, also called lag screws, are basically "large wood screws". A typical lag bolt is a 150 mm long, 13 mm diameter screw (6-inch-long, half-inch-diameter screw) with coarse threads of a wood-screw or sheet-metal-screw threadform (but larger). The head is typically an external hex. The materials are usually carbon steel substrate with a coating of zinc galvanization (for corrosion resistance). The zinc coating may be bright (electroplated), yellow (electroplated), or dull gray hot-dip galvanized. Lag bolts are used to lag together lumber framing, to lag machinery feet to wood floors, and other heavy carpentry applications. These fasteners are clearly "screws" when defined by the Machinery's Handbook distinction. The term "lag bolt" has been replaced by "lag screw" in the Machinery's Handbook.[16] However, in the minds of most tradesmen, they are "bolts", simply because they are large, with external-hex heads.

### Government standards

The US government made an effort to formalize the difference between a bolt and a screw because different tariffs apply to each.[17] The document seems to have no significant effect on common usage and does not eliminate the ambiguous nature of the distinction between screws and bolts for some threaded fasteners. The document also reflects (although it probably did not originate) significant confusion of terminology usage that differs between the legal/statutory/regulatory community and the fastener industry. The legal/statutory/regulatory wording uses the terms "coarse" and "fine" to refer to the tightness of the tolerance range, referring basically to "high-quality" or "low-quality", but this is a poor choice of terms, because those terms in the fastener industry have a different meaning (referring to the steepness of the helix's lead).

### Historical issue

Old USS and SAE standards defined cap screws as fasteners with shanks that were threaded to the head and bolts as fasteners with shanks that were partially unthreaded.[18] This is now an obsolete distinction.

### Controlled vocabulary versus natural language

The distinctions delineated above are enforced in the controlled vocabulary of standards organizations. Nevertheless, there are sometimes differences between the controlled vocabulary and the natural language usage of the words among machinists, auto mechanics, and other workers. These differences reflect linguistic evolution shaped by the changing of technology over centuries. The words bolt and screw have both existed since before today's modern mix of fastener types existed, and the natural usage of those words has evolved retronymously in response to the technological change. (That is, the use of words as names for objects changes as the objects themselves change.) Nonthreaded fasteners predominated in fastening technology until the advent of practical, inexpensive screw-cutting in the early 19th century. The basic meaning of the word screw has long involved the idea of a helical screw thread, but the Archimedes screw and the screw gimlet (like a corkscrew) preceded the fastener.

The word bolt is also a very old word, and it was used for centuries to refer to metal rods that passed through the substrate to be fastened on the other side, often via nonthreaded means (clinching, forge welding, pinning, wedging, etc.). The connection of this sense to the sense of a door bolt or the crossbow bolt is apparent. In the 19th century, bolts fastened via screw threads were often called screw bolts in contradistinction to clench bolts.

In common usage, the distinction is often that screws are smaller than bolts, and that screws are generally tapered and bolts are not. (This distinction is not rigorous.) For example, cylinder head bolts are called "bolts" (at least in North American usage) despite the fact that by some definitions they ought to be called "screws". Their size and their similarity to a bolt that would take a nut seem linguistically to overrule any other factors in this natural word choice proclivity.

Other distinctions: Bolts have been defined as headed fasteners having external threads that meet an exacting, uniform bolt thread specification (such as ISO metric screw thread M, MJ, Unified Thread Standard UN, UNR, and UNJ) such that they can accept a nontapered nut. Screws are then defined as headed, externally threaded fasteners that do not meet the above definition of bolts.[citation needed] These definitions of screw and bolt eliminate the ambiguity of the Machinery's handbook distinction. And it is for that reason, perhaps, that some people favor them. However, they are neither compliant with common usage of the two words nor are they compliant with formal specifications.

## Types of screws and bolts

Threaded fasteners either have a tapered shank or a non-tapered shank. Fasteners with tapered shanks are designed to either be driven into a substrate directly or into a pilot hole in a substrate. Mating threads are formed in the substrate as these fasteners are driven in. Fasteners with a non-tapered shank are designed to mate with a nut or to be driven into a tapped hole.

### Fasteners with a tapered shank (self-threading screws)

American name British name Description
chipboard screw
particle board screw
Similar to a drywall screw except that it has a thinner shaft and provides better resistance to pull-out in particle board, while offset against a lower shear strength. The threads on particle board screws are asymmetrical.
concrete screw
Tapcons
masonry screw
confast screw
blue screw
self-tapping screw
Titen
A stainless or carbon steel screw for fastening wood, metal, or other materials into concrete or masonry. Concrete screws are commonly blue in color, with or without corrosion coating.[19] They may either have a Phillips flat head or a slotted hex washer head. Heads sizes range from 0.1875 to 0.375 in (4.763 to 9.525 mm) and lengths from 1.25 to 5 in (32 to 127 mm). Typically an installer uses a hammer drill to make a pilot hole for each concrete screw.
deck screw Similar to drywall screw except that it has improved corrosion resistance and is generally supplied in a larger gauge. Most deck screws have a type-17 (auger type) thread cutting tip for installation into decking materials.
double ended screw
dowel screw
hanger bolt
Similar to a wood screw but with two pointed ends and no head, used for making hidden joints between two pieces of wood.
A hanger bolt has wood screw threads on one end and machine threads on the other. A hanger bolt is used when it is necessary to fasten a metal part to a wood surface.
drywall screw Specialized screw with a bugle head that is designed to attach drywall to wood or metal studs, however it is a versatile construction fastener with many uses. The diameter of drywall screw threads is larger than the shaft diameter.
eye screw
screw eye
Screw with a looped head. Larger ones are sometimes called lag eye screws. Designed to be used as attachment point, particularly for something that is hung from it.
lag bolt
lag screw[20]
coach screw Similar to a wood screw except that it is generally much larger running to lengths up to 15 in (381 mm) with diameters from 0.25–0.5 in (6.35–12.70 mm) in commonly available (hardware store) sizes (not counting larger mining and civil engineering lags and lag bolts) and it generally has a hexagonal drive head. Lag bolts are designed for securely fastening heavy timbers (post and beams, timber railway trestles and bridges) to one another, or to fasten wood to masonry or concrete.

Lag bolts are usually used with an expanding insert called a lag in masonry or concrete walls, the lag manufactured with a hard metal jacket that bites into the sides of the drilled hole, and the inner metal in the lag being a softer alloy of lead, or zinc alloyed with soft iron. The coarse thread of a lag bolt and lag mesh and deform slightly making a secure near water tight anti-corroding mechanically strong fastening.

mirror screw This is a flat-head wood screw with a tapped hole in the head, which receives a screw-in chrome-plated cover. It is usually used to mount a mirror.
sheet metal screw Has sharp threads that cut into a material such as sheet metal, plastic or wood. They are sometimes notched at the tip to aid in chip removal during thread cutting. The shank is usually threaded up to the head. Sheet metal screws make excellent fasteners for attaching metal hardware to wood because the fully threaded shank provides good retention in wood.
Twinfast screw A Twinfast screw is a type of screw with two threads (i.e. a lead of 2), so that it can be driven twice as fast.[21] Dry wall screws designated as fine are the most common screws to use the twinfast style of threads.[22]
wood screw wood screw A metal screw with a sharp point designed to attach two pieces of wood together. Wood screws are commonly available with flat, pan or oval-heads. A wood screw generally has a partially unthreaded shank below the head. The unthreaded portion of the shank is designed to slide through the top board (closest to the screw head) so that it can be pulled tight to the board it is being attached to.
Security head screw These screws are use for security purpose. The head of these screw is impossible to reverse. It requires special tool or mechanism to operate like spanner,tri-wing,torx,square driver etc. In some screw, the head can be removed out by breaking it after installing the screw.

### Fasteners with a non-tapered shank

American name British name Description
anchor bolt A special type of screw that is set in wet concrete, with the screw threads protruding above the concrete surface.
breakaway bolt A breakaway bolt is a bolt with a hollow threaded shank, which is designed to break away upon impact. Typically used to fasten fire hydrants, so they will break away when hit by a car. Also used in aircraft to reduce weight.
Narrow definition
Wide definition

A socket cap screw, also known as a socket head cap screw, socket screw or Allen bolt, this is a type of cap screw with a hexagonal recessed drive. The most common types in use have a cylindrical head whose diameter is nominally 1.5 times (1960 series design) that of the screw shank (major) diameter. Counterbored holes in parts allow the screw head to be flush with the surface or recessed. Other head designs include button head and flat head, the latter designed to be seated into countersunk holes. A hex key (sometimes referred to as an Allen wrench or Allen key) or hex driver is required to tighten or loosen a socket screw. Socket screws are commonly used in assemblies that do not provide sufficient clearance for a conventional wrench or socket.

carriage bolt A carriage bolt, also known as a coach bolt, has a domed or countersunk head, and the shank is topped by a short square section under the head. The square section grips into the part being fixed (typically wood), preventing the bolt from turning when the nut is tightened. A rib neck carriage bolt has several longitudinal ribs instead of the square section, to grip into a metal part being fixed.
elevator bolt An elevator bolt is a bolt similar to a carriage bolt, except the head is thin and flat. There are many variations. Some do not have a square base, but rather triangular sections of the flat head are folded down to form "fangs" that cut into wood and hold it secure.[27]
eye bolt An eye bolt is a bolt with a looped head.
hex cap screw
hex bolt
A hex cap screw is a cap screw with a hexagonal head, designed to be driven by a wrench (spanner). An ASME B18.2.1 compliant cap screw has somewhat tighter tolerances than a hex bolt for the head height and the shank length. The nature of the tolerance difference allows an ASME B18.2.1 hex cap screw to always fit where a hex bolt is installed but a hex bolt could be slightly too large to be used where a hex cap screw is designed in.
machine screw A machine screw is generally a smaller fastener (less than 14 inches (6.35 mm) in diameter) threaded the entire length of its shank that usually has a recessed drive type (slotted, Phillips, etc.). Machine screws are also made with socket heads (see above), in which case they may be referred to as socket head machine screws.
plow bolt A plow bolt is bolt similar to a carriage bolt, except the head is flat or concave, and the underside of the head is a cone designed to fit in a countersunk recess. There are many variations, with some not using a square base, but rather a key, a locking slot, or other means. The recess in the mating part must be designed to accept the particular plow bolt.[28][29][30]
self-drilling screw
Teks screw
Similar to a sheet metal screw, but it has a drill-shaped point to cut through the substrate to eliminate the need for drilling a pilot hole. Designed for use in soft steel or other metals. The points are numbered from 1 through 5, the larger the number, the thicker metal it can go through without a pilot hole. A 5 point can drill a 0.5 in (12.7 mm) of steel, for example.
self-tapping machine screw A self-tapping machine screw is similar to a machine screw except the lower part of the shank is designed to cut threads as the screw is driven into an untapped hole. The advantage of this screw type over a self-tapping screw is that, if the screw is reinstalled, new threads are not cut as the screw is driven.
set bolt tap bolt A bolt that is threaded all the way to the head. An ASME B18.2.1 compliant set/tap bolt has the same tolerances as an ASME B18.2.1 compliant hex cap screw.
set screw grub screw A set screw is generally a headless screw but can be any screw used to fix a rotating part to a shaft. The set screw is driven through a threaded hole in the rotating part until it is tight against the shaft. The most often used type is the socket set screw, which is tightened or loosened with a hex key.
shoulder bolt
shoulder screw
stripper bolt A shoulder screw differs from machine screws in that the shank is ground to a precise diameter, known as the shoulder, and the threaded portion is smaller in diameter than the shoulder. Shoulder bolt specifications call out the shoulder diameter, shoulder length, and threaded diameter; the threaded length is fixed, based on the threaded diameter, and usually quite short. It is usually used for revolving joints in mechanisms and linkages; when used as a guide for the stripper plate in a die set its called a stripper bolt.
stove bolt A stove bolt is a type of machine screw that has a round or flat head and is threaded to the head. They are usually made of low grade steel, have a slot or Phillips drive, and are used to join sheet metal parts using a hex or square nut.[31]
tension control bolt A tension control bolt (TC bolt) is a heavy duty bolt used in steel frame construction. The head is usually domed and is not designed to be driven. The end of the shank has a spline on it which is engaged by a special power wrench which prevents the bolt from turning while the nut is tightened. When the appropriate torque is reached the spline shears off.
thread rolling screws These have a lobed (usually triangular) cross-section. They form threads in a pre-drilled hole in the mating workpiece by pushing the material outward during installation.

#### Superbolt, or multi-jackbolt tensioner

A superbolt, or multi-jackbolt tensioner is an alternative type of fastener that retrofits or replaces existing nuts, bolts, or studs. Tension in the bolt is developed by torquing individual jackbolts, which are threaded through the body of the nut and push against a hardened washer. Because of this, the amount of torque required to achieve a given preload is reduced. Installation and removal of any size tensioner is achieved with hand tools, which can be advantageous when dealing with large diameter bolting applications.

#### Hanger screw or hanger bolt

A hanger screw is a headless fastener that has machine screw threads on one end and self-tapping threads on the other designed to be driven into wood or another soft substrate. Often used for mounting legs to tables.

## Materials

Screws and bolts are made from a wide range of materials, with steel being perhaps the most common, in many varieties. Where great resistance to weather or corrosion is required, stainless steel, titanium, brass (steel screws can discolor oak and other woods), bronze, monel or silicon bronze may be used, or a coating such as brass, zinc or chromium applied. Electrolytic action from dissimilar metals can be prevented with aluminium screws for double-glazing tracks, for example. Some types of plastic, such as nylon or polytetrafluoroethylene (PTFE), can be threaded and used for fastening requiring moderate strength and great resistance to corrosion or for the purpose of electrical insulation.Material selection criteria is decided on Temperature,required strength,resistance to corrosion,joint material and cost.

## Bolted joints

The American Institute of Steel Construction (AISC) 13th Edition Steel Design Manual section 16.1 chapter J-3 specifies the requirements for bolted structural connections. Structural bolts replaced rivets due to decreasing cost and increasing strength of structural bolts in the 20th century. Connections are formed with two types of joints: slip-critical connections and bearing connections. In slip-critical connections, movement of the connected parts is a serviceability condition and bolts are tightened to a minimum required pretension. Slip is prevented through friction of the "faying" surface, that is the plane of shear for the bolt and where two members make contact. Because friction is proportional to the normal force, connections must be sized with bolts numerous and large enough to provide the required load capacity. However, this greatly decreases the shear capacity of each bolt in the connection. The second type and more common connection is a bearing connection. In this type of connection the bolts carry the load through shear and are only tightened to a "snug-fit." These connections require fewer bolts than slip-critical connections and therefore are a less expensive alternative. Slip-critical connections are more common on flange plates for beam and column splices and moment critical connections. Bearing type connections are used in light weight structures and in member connections where slip is not important and prevention of structural failure is the design constraint. Common bearing type connections include: shear tabs, beam supports, gusset plates in trusses.

## Mechanical classifications

The numbers stamped on the head of the bolt are referred to the grade of the bolt used in certain application with the strength of a bolt. High-strength steel bolts usually have a hexagonal head with an ISO strength rating (called property class) stamped on the head. And the absence of marking/number indicates a lower grade bolt with low strength. The property classes most often used are 5.8, 8.8, and 10.9. The number before the point is the tensile ultimate strength in MPa divided by 100. The number after the point is 10 times the ratio of tensile yield strength to tensile ultimate strength. For example, a property class 5.8 bolt has a nominal (minimum) tensile ultimate strength of 500 MPa, and a tensile yield strength of 0.8 times tensile ultimate strength or 0.8(500) = 400 MPa.

Tensile ultimate strength is the stress at which the bolt fails. Tensile yield strength is the stress at which the bolt will receive a permanent set (an elongation from which it will not recover when the force is removed) of 0.2 % offset strain. When elongating a fastener prior to reaching the yield point, the fastener is said to be operating in the elastic region; whereas elongation beyond the yield point is referred to as operating in the plastic region, since the fastener has suffered permanent plastic deformation.

Mild steel bolts have property class 4.6. High-strength steel bolts have property class 8.8 or above.

The same type of screw or bolt can be made in many different grades of material. For critical high-tensile-strength applications, low-grade bolts may fail, resulting in damage or injury. On SAE-standard bolts, a distinctive pattern of marking is impressed on the heads to allow inspection and validation of the strength of the bolt. However, low-cost counterfeit fasteners may be found with actual strength far less than indicated by the markings. Such inferior fasteners are a danger to life and property when used in aircraft, automobiles, heavy trucks, and similar critical applications.

### Inch

SAE J429 defines the bolt grades for inch-system sized bolts and screws. It defines them by grade, which ranges from 0 to 8, with 8 being the strongest. Higher grades do not exist within the specification.[32][33] SAE grades 5 and 8 are the most common.

### Metric

The international standard for metric screws is defined by ISO 898, specifically ISO 898-1. SAE J1199 and ASTM F568M are two North American metric standards that closely mimic the ISO standard. In case of inch sizes the grade is dictated by the number of radial shapes plus a value of two. Inch-system bolts use integer values to indicate grades but metric bolts use numbers with one decimal. The two North American standards use the same property class markings as defined by ISO 898.[41] The ASTM standard only includes the following property classes from the ISO standard: 4.6, 4.8, 5.8, 8.8, 9.8, 10.9, and 12.9; it also includes two extra property classes: 8.8.3 and 10.9.3.[42] ASTM property classes are to be stamped on the top of screws and it is preferred that the marking is raised.[43]

(a) pan, (b) button, (c) round, (d) truss, (e) flat (countersunk), (f) oval
Combination flanged-hex/Phillips-head screw used in computers
A low disc with chamfered outer edge
Cylindrical with a rounded top
A dome-shaped head used for decoration.[48]
Lower-profile dome designed to prevent tampering
Conical, with flat outer face and tapering inner face allowing it to sink into the material. The angle of the screw is measured as the full angle of the cone.
A decorative screw head with a countersunk bottom and rounded top[48]. Also known as "raised countersunk" (UK)
Similar to countersunk, but there is a smooth progression from the shank to the angle of the head, similar to the bell of a bugle
Disc with cylindrical outer edge, height approximately half the head diameter
Cylindrical, but with a slightly convex top surface. Height to diameter ratio is larger than cheese head.
A flanged head can be any of the above head styles with the addition of an integrated flange at the base of the head. This eliminates the need for a flat washer.

Some varieties of screw are manufactured with a break-away head, which snaps off when adequate torque is applied. This prevents tampering and also provides an easily inspectable joint to guarantee proper assembly. An example of this is the shear bolts used on vehicle steering columns, to secure the ignition switch.

## Types of screw drives

Part of a series on
Screw drive types
Slot (flat)
Phillips
PH
PZ
Square
Robertson (square)
Hex
Hex socket (Allen)
Security hex socket (pin-in-hex-socket)
Torx
T & TX
Security Torx
TR
Tri-Wing
Torq-set
(Snake-eye)
Triple square
XZN
Polydrive
One-way
Spline drive
Double hex
Bristol
Pentalobular

This box:

Modern screws employ a wide variety of drive designs, each requiring a different kind of tool to drive in or extract them. The most common screw drives are the slotted and Phillips; hex, Robertson, and torx are also common in some applications. Some types of drive are intended for automatic assembly in mass-production of such items as automobiles. More exotic screw drive types may be used in situations where tampering is undesirable, such as in electronic appliances that should not be serviced by the home repair person.

## Tools

An electric driver screws a self-tapping phillips head screw into wood

The hand tool used to drive in most screws is called a screwdriver. A power tool that does the same job is a power screwdriver; power drills may also be used with screw-driving attachments. Where the holding power of the screwed joint is critical, torque-measuring and torque-limiting screwdrivers are used to ensure sufficient but not excessive force is developed by the screw. The hand tool for driving hex head threaded fasteners is a spanner (UK usage) or wrench (US usage).

There are many systems for specifying the dimensions of screws, but in much of the world the ISO metric screw thread preferred series has displaced the many older systems. Other relatively common systems include the British Standard Whitworth, BA system (British Association), and the Unified Thread Standard.

The basic principles of the ISO metric screw thread are defined in international standard ISO 68-1 and preferred combinations of diameter and pitch are listed in ISO 261. The smaller subset of diameter and pitch combinations commonly used in screws, nuts and bolts is given in ISO 262. The most commonly used pitch value for each diameter is the coarse pitch. For some diameters, one or two additional fine pitch variants are also specified, for special applications such as threads in thin-walled pipes. ISO metric screw threads are designated by the letter M followed by the major diameter of the thread in millimeters (e.g., M8). If the thread does not use the normal coarse pitch (e.g., 1.25 mm in the case of M8), then the pitch in millimeters is also appended with a multiplication sign (e.g. "M8×1" if the screw thread has an outer diameter of 8 mm and advances by 1 mm per 360° rotation).

The nominal diameter of a metric screw is the outer diameter of the thread. The tapped hole (or nut) into which the screw fits, has an internal diameter which is the size of the screw minus the pitch of the thread. Thus, an M6 screw, which has a pitch of 1 mm, is made by threading a 6 mm shank, and the nut or threaded hole is made by tapping threads into a hole of 5 mm diameter (6 mm - 1 mm).

Metric hexagon bolts, screws and nuts are specified, for example, in British Standard BS 4190 (general purpose screws) and BS 3692 (precision screws). The following table lists the relationship given in these standards between the thread size and the maximal width across the hexagonal flats (wrench size):

 ISO metric thread Wrench size (mm) M1.6 M2 M2.5 M3 M4 M5 M6 M8 M10 M12 M16 M20 M24 M30 M36 M42 M48 M56 M64 3.2 4 5 5.5 7 8 10 13 17 19 24 30 36 46 55 65 75 85 95

In addition, the following non-preferred intermediate sizes are specified:

 ISO metric thread Wrench size (mm) M7 M14 M18 M22 M27 M33 M39 M45 M52 M60 M68 11 22 27 32 41 50 60 70 80 90 100

### Whitworth

The first person to create a standard (in about 1841) was the English engineer Sir Joseph Whitworth. Whitworth screw sizes are still used, both for repairing old machinery and where a coarser thread than the metric fastener thread is required. Whitworth became British Standard Whitworth, abbreviated to BSW (BS 84:1956) and the British Standard Fine (BSF) thread was introduced in 1908 because the Whitworth thread was too coarse for some applications. The thread angle was 55° and a depth and pitch of thread that varied with the diameter of the thread (i.e., the bigger the bolt, the coarser the thread). The spanner size is determined by the size of the bolt, not the distance between the flats.

The most common use of a Whitworth pitch nowadays is in all UK scaffolding. Additionally, the standard photographic tripod thread, which for small cameras is 1/4" Whitworth (20 tpi) and for medium/large format cameras is 3/8" Whitworth (16 tpi). It is also used for microphone stands and their appropriate clips, again in both sizes, along with "thread adapters" to allow the smaller size to attach to items requiring the larger thread. Note that while 1/4" UNC bolts fit 1/4" BSW camera tripod bushes, yield strength is reduced by the different thread angles of 60° and 55° respectively.

A later standard established in the United Kingdom was the British Association (BA) screw threads, named after the British Association for Advancement of Science. Screws were described as "2BA", "4BA" etc., the odd numbers being rarely used, except in equipment made prior to the 1970s for telephone exchanges in the UK. This equipment made extensive use of odd-numbered BA screws, in order—it may be suspected—to reduce theft. BA threads are specified by British Standard BS 93:1951 "Specification for British Association (B.A.) screw threads with tolerances for sizes 0 B.A. to 16 B.A."

While not related to ISO metric screws, the sizes were actually defined in metric terms, a 0BA thread having a 6 mm diameter and 1 mm pitch. Other threads in the BA series are related to 0BA in a geometric series with the common factors 0.9 and 1.2. For example, a 4BA thread has pitch $\scriptstyle p=0.9^4$ mm (0.65mm) and diameter $\scriptstyle 6p^{1.2}$ mm (3.62mm). Although 0BA has the same diameter and pitch as ISO M6, the threads have different forms and are not compatible.

BA threads are still common in some niche applications. Certain types of fine machinery, such as moving-coil meters and clocks, tend to have BA threads wherever they are manufactured. BA sizes were also used extensively in aircraft, especially those manufactured in the United Kingdom. BA sizing is still used in railway signalling, mainly for the termination of electrical equipment and cabling.

BA threads are extensively used in Model Engineering where the smaller hex head sizes make scale fastenings easier to represent. As a result many UK Model Engineering suppliers still carry stocks of BA fasteners up to typically 8BA and 10BA. 5BA is also commonly used as it can be threaded onto 1/8 rod.

The Unified Thread Standard (UTS) is most commonly used in the United States of America, but is also extensively used in Canada and occasionally in other countries. The size of a UTS screw is described using the following format: X-Y, where X is the nominal size (the hole or slot size in standard manufacturing practice through which the shaft of the screw can easily be pushed) and Y is the threads per inch (TPI). For sizes 14 inch and larger the size is given as a fraction; for sizes less than this an integer is used, ranging from 0 to 16. For most size screws there are multiple TPI available, with the most common being designated a Unified Coarse Thread (UNC or UN) and Unified Fine Thread (UNF or UF).

## Manufacture

The blanks are then polished[citation needed] again prior to threading. The threads are usually produced via thread rolling, however some are cut. The workpiece is then tumble finished with wood and leather media to do final cleaning and polishing.[citation needed] For most screws a coating, such as hot-dip galvanizing or blackening, is applied to prevent corrosion.

## History

A lathe of 1871, equipped with leadscrew and change gears for single-point screw-cutting.
A Brown & Sharpe single-spindle screw machine.

While a recent hypothesis attributes the Archimedes' screw to Sennacherib, King of Assyria, archaeological finds and pictorial evidence only appear in the Hellenistic period and the standard view holds the device to be a Greek invention, most probably by the 3rd century BC polymath Archimedes himself.[49][dubious ]

The screw was later described by the Greek mathematician Archytas of Tarentum (428 – 350 BC). By the 1st century BC, wooden screws were commonly used throughout the Mediterranean world in devices such as oil and wine presses. Metal screws used as fasteners did not appear in Europe until the 15th century.[50][dubious ]

Rybczynski has shown[51] that handheld screwdrivers (formerly called "turnscrews" in English, in more direct parallel to their original French name, tournevis[52]) have existed since medieval times (the 1580s at the latest), although they probably did not become truly widespread until after 1800, once threaded fasteners themselves had become commodified, as detailed below.

There were many forms of fastening in use before threaded fasteners became widespread. They tended to involve carpentry and smithing rather than machining, and they involved concepts such as dowels and pins, wedging, mortises and tenons, dovetails, nailing (with or without clenching), forge welding, and many kinds of binding with cord made of leather or fiber, using many kinds of knots. Prior to the mid-19th century, cotter pins or pin bolts, and "clinch bolts" (now called rivets), were used in shipbuilding.

These developments of the 1760–1800 era, with the Wyatts and Maudslay being arguably the most important drivers, caused great increase in the use of threaded fasteners. Standardization of threadforms began almost immediately, but it was not quickly completed; it has been an evolving process ever since. Further improvements to the mass production of screws continued to push unit prices lower and lower for decades to come, throughout the 19th century.[58]

The development of the turret lathe (1840s) and of automatic screw machines derived from it (1870s) drastically reduced the unit cost of threaded fasteners by increasingly automating the machine tool control. This cost reduction spurred ever greater use of screws.

Throughout the 19th century, the most commonly used forms of screw head (that is, drive types) were simple internal-wrenching straight slots and external-wrenching squares and hexagons. These were easy to machine and served most applications adequately. Rybczynski describes a flurry of patents for alternative drive types in the 1860s through 1890s,[59] but explains that these were patented but not manufactured due to the difficulties and expense of doing so at the time. In 1908, Canadian P. L. Robertson was the first to make the internal-wrenching square socket drive a practical reality by developing just the right design (slight taper angles and overall proportions) to allow the head to be stamped easily but successfully, with the metal cold forming as desired rather than being sheared or displaced in unwanted ways.[59] The internal-wrenching hexagon drive (hex socket) shortly followed in 1911.[60] In the early 1930s, the Phillips-head screw was invented by Henry F. Phillips.

Threadform standardization further improved in the late 1940s, when the ISO metric screw thread and the Unified Thread Standard were defined.

## Other fastening methods

Alternative fastening methods are nails, rivets, roll pins, pinned shafts, welding, soldering, brazing, and gluing (including taping), and clinch fastening.

## References

1. ^ Smith 1990, p. 39.
2. ^ Oberg et al. 2000, p. 1492.
3. ^ "Cambridge Dictionary of American English". Cambridge University Press. Retrieved 2008-12-03.
4. ^
5. ^ "Merriam Webster Dictionary bolt". Retrieved 2008-12-03.
6. ^ "Compact Oxford English Dictionary bolt". Oxford. Retrieved 2008-12-03.
7. ^ "Cambridge Advanced Learner's Dictionary bolt". Cambridge University Press. Retrieved 2008-12-03.
8. ^ "The Fastener Resource Center - Know your Bolts". Retrieved 2011-03-13.
9. ^ Oberg et al. 2000, pp. 1568–1598.
10. ^ Oberg et al. 2000, p. 1496.
11. ^
12. ^ Fastener Quality Act (FQA): Text of the Fastener Quality Act - fqaregs2
13. ^ B18.2.1 - 1996 Square and Hex Bolts and Screws, Inch Series - Print-Book
14. ^ "autorepair.com Glossary - lug bolt". Retrieved 2009-01-13.
15. ^
16. ^ Oberg et al. 2000, p. 1497.
17. ^ U.S. Customs and Border Protection Agency (CBP) (2011-02), What Every Member of the Trade Community Should Know About: Distinguishing Bolts from Screws, An Informed Compliance Publication (2011-02 ed.), Washington, DC, USA: CBP.gov.
18. ^ a b Dyke's Automobile and Gasoline Engine Encyclopedia page 701, A.L. Dyke, 1919, retrieved 2009-01-13.
19. ^ Source: http://www.confast.com/articles/tapcon-screw.aspx
20. ^ "coach screw definition". dictionary.com. Retrieved 2010-01-19.
21. ^ Soled, Julius (1957), Fasteners handbooks, Reinhold, p. 151.
22. ^ "Fine thread drywall screws". Mutual Screw & Fastener Supply. Retrieved 20011-03-16.
23. ^ Oberg 2000, pp. 1599–1605.
24. ^ Samuel, Andrew (1999), Introduction to Engineering Design, Oxford: Butterworth-Heinemann, p. 213, ISBN 0750642823
25. ^ Anthony, Gardner Chase (1910), Machine Drawing, D. C. Heath, p. 16.
26. ^ Woolley, Joseph William; Meredith, Roy Brodhead (1913), Shop sketching, McGraw-Hill, pp. 40–41.
27. ^ The Meaning of "elevator head"
28. ^ Colvin & Stanley 1914, p. 569.
29. ^ Plow bolts, retrieved 2008-12-25.
30. ^ The Meaning of "plow head, plow bolt" at MyWord.info
31. ^ Huth, pp. 166–167.
32. ^ a b
33. ^ Smith 1990, p. 54.
34. ^ a b c
35. ^ a b c d e f g h i Grade Markings: Carbon Steel Bolts, retrieved 2009-05-30.
36. ^ a b c d e f
37. ^ a b c d e f g h
38. ^ a b c
39. ^ a b Other markings may be used to denote atmospheric corrosion resistant material
40. ^ a b c
41. ^ Bickford & Nassar 1998, p. 154.
42. ^ a b c ASTM F568M - 07, 2007, retrieved 2009-06-06.
43. ^ a b Metric Handbook, archived from the original on 2007-10-31, retrieved 2009-06-06.
44. ^
45. ^ a b c d Metric structural fasteners, retrieved 2009-06-06.
46. ^ a b ASTM A325M - 09, retrieved 2009-06-13.
47. ^ a b ASTM A490M - 09, 2009, retrieved 2009-06-06.
48. ^ a b Mitchell, George (1995), Carpentry and Joinery (3rd ed.), Cengage Learning, p. 205, ISBN 9781844800797.
49. ^ Stephanie Dalley and John Peter Oleson (January 2003). "Sennacherib, Archimedes, and the Water Screw: The Context of Invention in the Ancient World", Technology and Culture 44 (1).
50. ^ Am_Wood_Screws, retrieved 2010-04-30.
51. ^ Rybczynski 2000, pp. 34, 66, 90.
52. ^ Rybczynski 2000, pp. 32–36, 44.
53. ^ Rybczynski 2000, pp. 75–99.
54. ^ Rybczynski 2000, p. 99.
55. ^ a b c Rybczynski 2000, p. 75.
56. ^ Rybczynski 2000, p. 76.
57. ^ Rybczynski 2000, pp. 97–99.
58. ^ Rybczynski 2000, pp. 76–78.
59. ^ a b Rybczynski 2000, pp. 79–81.
60. ^ Hallowell 1951, pp. 51–59.

### Bibliography

Wikimedia Foundation. 2010.

Synonyms:

### Look at other dictionaries:

• Screw — (skr[udd]), n. [OE. scrue, OF. escroue, escroe, female screw, F. [ e]crou, L. scrobis a ditch, trench, in LL., the hole made by swine in rooting; cf. D. schroef a screw, G. schraube, Icel. skr[=u]fa.] 1. A cylinder, or a cylindrical perforation,… …   The Collaborative International Dictionary of English

• screw — [skro͞o] n. [ME screwe < MFr escroue, hole in which the screw turns < L scrofa, sow, infl. by scrobis, vulva] 1. a) a mechanical device for fastening things together, consisting essentially of a cylindrical or conical piece of metal… …   English World dictionary

• Screw — Screw, v. t. [imp. & p. p. {Screwed}; p. pr. & vb. n. {Screwing}.] 1. To turn, as a screw; to apply a screw to; to press, fasten, or make firm, by means of a screw or screws; as, to screw a lock on a door; to screw a press. [1913 Webster] 2. To… …   The Collaborative International Dictionary of English

• Screw — (kurz für: screwing chopping) ist eine Remix Technik der Hip Hop Musik, die aus dem Süden der USA stammt. Dabei wird ein Lied (meist ein Hip Hop Track) in langsamer Geschwindigkeit abgespielt (screwed zu deutsch etwa: heruntergeschraubt), Teile… …   Deutsch Wikipedia

• screw — screw·able; screw; screw·drive; screw·driv·er; screw·er; screw·less; screw·man; un·screw; cork·screw; …   English syllables

• screw — ► NOUN 1) a thin, sharp pointed metal pin with a raised spiral thread running around it and a slotted head, used to join things together by being rotated in under pressure. 2) a cylinder with a spiral ridge or thread running round the outside… …   English terms dictionary

• screw up — {v. phr.}, {slang}, {semi vulgar}, {best avoided} 1. To make a mess of, to make an error which causes confusion. * /The treasurer screwed up the accounts of the Society so badly that he had to be fired./ 2. To cause someone to be neurotic or… …   Dictionary of American idioms

• screw up — {v. phr.}, {slang}, {semi vulgar}, {best avoided} 1. To make a mess of, to make an error which causes confusion. * /The treasurer screwed up the accounts of the Society so badly that he had to be fired./ 2. To cause someone to be neurotic or… …   Dictionary of American idioms

• Screw 32 — was a Berkeley/ East Bay area punk rock band. Their name is derived from an anti skateboarding measure on the ballots in Concord, California. They were noted for self publicity. For example, they ordered stickers by the thousands and stuck them… …   Wikipedia

• screw-up — noun count usually singular INFORMAL a situation in which someone makes a big mistake: There s been a big screw up with the mailing list no one s received the invitations. another government screw up …   Usage of the words and phrases in modern English

• screw up — (something) to spoil or damage something. You couldn t screw up much worse than I did. Somehow the lawyer screwed up my appointment again. This is detailed work, and people screw it up once in a while …   New idioms dictionary


text:
# Project Euler Problem 33 - digit canceling fractions

Problem:

The fraction $(\dfrac{49}{98})$ is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that $(\dfrac{49}{98}) = (\dfrac{4}{8})$, which is correct, is obtained by cancelling the 9s.

We shall consider fractions like, $(\dfrac{30}{50}) = (\dfrac{3}{5})$, to be trivial examples.

There are exactly four non-trivial examples of this type of fraction, less than one in value, and containing two digits in the numerator and denominator.

If the product of these four fractions is given in its lowest common terms, find the value of the denominator.

This is some rough code -- this one was a lot harder than I expected it to be. Please help me optimize.

from timeit import default_timer as timer

start = timer()
fractions_that_work = []

def a_common_element(list_a, list_b):
count = 0
common = 0
for m in list_a:
for n in list_b:
if m == n:
count += 1
common = m
if count == 1:
return common
return False

for numerator in range(10, 100):
for denominator in range(numerator + 1, 100): # denom must be > num
n_digits = sorted([int(x) for x in str(numerator)])
d_digits = sorted([int(x) for x in str(denominator)])
common = a_common_element(n_digits, d_digits)
if common:
n_digits.remove(common)
d_digits.remove(common)
n_rem = n_digits[0]
d_rem = d_digits[0]
if 0 not in n_digits and 0 not in d_digits:
if float(numerator) / denominator == float(n_rem) / d_rem:
fractions_that_work.append([numerator, denominator])
product_of_fractions = [1, 1]
for frac in fractions_that_work:
product_of_fractions[0] *= frac[0]
product_of_fractions[1] *= frac[1]

start = timer()
ans = product_of_fractions[1] / product_of_fractions[0]
elapsed_time = (timer() - start) * 1000 # s --> ms

print "Found %d in %r ms." % (ans, elapsed_time)

• Note for reviewers : the start = timer() is a mistake. Keeping the code in the answer as it is unless a mod thinks it's better to change it. Apr 25, 2014 at 13:22

Firstly, you set start at the top of the script (which is OK). However, you reset start right before your final calulation:

# This will only time the execution of one statement!
start = timer()
ans = product_of_fractions[1] / product_of_fractions[0]
elapsed_time = (timer() - start) * 1000 # s --> ms

Remove the above start = timer() to get a more accurate timing. Also, I would recommend moving the first start declaration to just before the nested for loops. That was the start time is more spatially related to what you are actually timing.

Secondly, your a_common_element function's name is a little misleading. Currently, it insinuates it will return positively if the two lists have any number of common elements. Changing the function name to something like has_single_common_element gives a better description of what the function actually does.

Also, a_common_element can be simplified a tad. By using list comprehension we can combine the nested for loops into one comprehension:

def a_common_element(list_a, list_b):
common_list = [value for value in list_a if value in list_b]

if len(common_list) == 1:
return common_list[0]
return False

Your next section is actually quite interesting: you used a neat way to separate the digits. However, the downfall (I believe) to your method is the back-and-forth casting that it requires. I would, instead, recommend doing to calculations to get your digits:

for numerator in range(10, 100):
for denomerator in range(numerator+1, 100):
# Division will strip off the first digit
# Modulus will strip off the 0th digit
n_digits = sorted([int(numerator/10), numerator % 10])
d_digits = sorted([int(denomerator/10), denomerator % 10])

NOTE: This recommendation is based on the project description you provided, that is all numbers will have exactly 2 digits. If you wanted to extend this to greater sets of numbers (3-digits, 4-digits, etc) I would keep your current implementation.

This next recommendation, like the previous, is based off of 2-digit numbers only.

Once you remove the common digit from the lists, you assign the remaining digits to variables; which is good. However, you then go check if the lists contain 0 instead of comparing the variables:

# I imagine this is what Python does in the backgroud when you say:
#    if 0 not in list
if n_rem != 0 and d_rem != 0:

Your next section is nice. Its simple and intuitive. If you wanted to make the code more Pythonic, you could instead do it this way:

product_of_fractions = [1, 1]
for frac in fractions_that_work:
product_of_fractions = [i*j for i, j in zip(product_of_fractions, frac)]

If you wanted to go a bit more towards functional programming you could use this single line of code:

from functools import reduce   # Needed for Python 3, works in Python 2.6+
import operator

product_of_fractions = [reduce(operator.mul, value) for value in zip(*fractions_that_work)]

Even though it may be more complex (and thus slightly less readable), I prefer this last way because of its brevity.


text:
Elementary Statistics (12th Edition)

By using the Central Limit Theorem, the sample mean has a mean of $\mu$ and standard deviation of $\frac{\sigma}{\sqrt n}$. $z=\frac{value-mean}{standard \ deviation}=\frac{4.444-4.457}{\frac{0.382}{\sqrt{25}}}=-1.74.$ Using the table, the probability of z being less than -0.52 is: 0.0409.


text:
mirage

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• (Wake Forest University, 2013)
This thesis studies deterministic movement over toroidal grids (and more generally, Cayley graphs on groups), integrating local information, bounded memory and choice at individual nodes. The research is motivated by recent ...
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We explore the concept of dimension using mathematical tools. We start by providing definitions, examples, and basic properties for two types of topological dimension: small inductive dimension and large inductive dimension. ...
• (Wake Forest University, 2016)
The fibonomial triangle has been shown by Chen and Sagan to have a fractal nature mod 2 and 3. Both these primes have the property that the Fibonacci entry point of $p$ is $p+1$. We study the fibonomial triangle mod 5, ...


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# Need help with probabilities

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Need help with probabilities [#permalink]  09 Oct 2013, 22:31
Will be gratefull for help in solving 2 questions:
1. When a coin is tossed for 6 times, what is the probability that after the first tossing, every outcome will be different from previous one?
2. If n is an integer from 1 to 96, what is the probability for n*(n+1)*(n+2) being divisible by 8?
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Re: Need help with probabilities [#permalink]  11 Oct 2013, 02:08
Expert's post
DDM wrote:
Will be gratefull for help in solving 2 questions:
1. When a coin is tossed for 6 times, what is the probability that after the first tossing, every outcome will be different from previous one?

The first toss can be anything H or T. Probability = 1
The second toss must be the opposite. Say if first was H, the second must be T. Probability = 1/2
The third must be opposite again i.e. H now. Probability = 1/2
The fourth must be T. Probability = 1/2
Fifth must be H. Probability = 1/2
Sixth must be T. Probability = 1/2

So overall probability = $$1*(1/2)*(1/2)*(1/2)*(1/2)*(1/2) = 1/2^5$$

Or think of it this way.
The favorable outcomes are only two: HTHTHT or THTHTH
The total outcomes = $$2^6$$
Probability $$= 2/2^6 = 1/2^5$$
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5678 Location: Pune, India Followers: 1412 Kudos [?]: 7304 [0], given: 186 Re: Need help with probabilities [#permalink] 11 Oct 2013, 02:21 Expert's post DDM wrote: 2. If n is an integer from 1 to 96, what is the probability for n*(n+1)*(n+2) being divisible by 8? This one has been discussed here: if-an-integer-n-is-to-be-chosen-at-random-from-the-integers-126654.html _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: Need help with probabilities   [#permalink] 11 Oct 2013, 02:21
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