text:
Now showing items 1-13 of 13

• #### An analysis of Precision Electroweak Measurements: Summer 98 Update

We update our analysis of precision electroweak measurements using the latest data announced at Moriond, March 1998. Possible oblique corrections from new physics are parametrized using the STU formalism of Ref.[1], and ...
• #### Applied Antineutrino Physics 2015 -- Conference Summary

This is a brief summary of the 11th Applied Antineutrino Physics 2015 workshop held at the Virginia Tech Arlington Research Facility from December 7-8, 2015.
• #### The difficulties involved in calculating $δρ$

We discuss the difficulties that arise when one tries to calculate $\delta\rho$ using dispersion relations.
• #### Fundamental Physics at the Intensity Frontier

(2011)
The Proceedings of the 2011 workshop on Fundamental Physics at the Intensity Frontier. Science opportunities at the intensity frontier are identified and described in the areas of heavy quarks, charged leptons, neutrinos, ...
• #### Improved Search for ν¯µ → ν¯e Oscillations in the MiniBooNE Experiment

The MiniBooNE experiment at Fermilab reports results from an analysis of ¯νe appearance data from 11.27 × 10²⁰ protons on target in antineutrino mode, an increase of approximately a factor of two over the previously reported ...
• #### Light Sterile Neutrinos: A White Paper

This white paper addresses the hypothesis of light sterile neutrinos based on recent anomalies observed in neutrino experiments and the latest astrophysical data.
• #### The Long-Baseline Neutrino Experiment: Exploring Fundamental Symmetries of the Universe

The preponderance of matter over antimatter in the early Universe, the dynamics of the supernova bursts that produced the heavy elements necessary for life and whether protons eventually decay --- these mysteries at the ...
• #### Modified Dark Matter in Galaxies and Galaxy Clusters

Modified Dark Matter (MDM) is a phenomenological model of dark matter, inspired by gravitational thermodynamics, that naturally accounts for the universal acceleration constant observed in galactic rotation curve data; a ...
• #### Neutrinos

This document represents the response of the Intensity Frontier Neutrino Working Group to the Snowmass charge. We summarize the current status of neutrino physics and identify many exciting future opportunities for studying ...
• #### Neutrinos from Stored Muons nuSTORM: Expression of Interest

The nuSTORM facility has been designed to deliver beams of electron and muon neutrinos from the decay of a stored muon beam with a central momentum of 3.8 GeV/c and a momentum spread of 10%. The facility is unique in that ...
• #### Phenomenology of Not-so-heavy Neutral Leptons: The NuTeV Anomaly, Lepton Universality, and Non-Universal Neutrino-Gauge Couplings

Talk presented by Takeuchi at the YITP workshop "Progress in Particle Physics" 2004.
• #### The status of the determination of α(m𝘇) and α𝗌(m𝘇)

I will discuss the current status of the determination of $\alpha(m_Z)$ and $\alpha_s(m_Z)$, emphasizing the pitfalls that one must avoid in performing statistical analyses.
• #### Using L/E Oscillation Probability Distributions

This paper explores the use of L/E oscillation probability distributions to compare experimental measurements and to evaluate oscillation models. In this case, L is the distance of neutrino travel and E ...


text:
# Insert image so that it fills the rest of the page while not fall below a minimum size

I'm trying to make a command that adds images to page in a way that fill the rest of the page. I'm using code from How to define a figure size so that it consumes the rest of a page? and it works. But I noticed that sometimes the image will be very small when there is little space left, so I wanted to make it conditional - if there is more than 1/3 of the page left, fit the image, otherwise just put the bigger image on the next page - but I can't get the if to work. I'm new with LaTeX, can you help me make it work?

Here's the code:

\newcommand\measurepage{\dimexpr\pagegoal-\pagetotal-\baselineskip\relax}
\newcommand{\imgFill}[1]{\begin{center}
\ifnum \measurepage<0.3*\textheigth \includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{#1}
\else \includegraphics[height=\measurepage,width=\textwidth,keepaspectratio]{#1} \fi
\end{center}
}

• \measurepage is a dimension not a number, so you need to use \ifdim. Beside this: don't show only snippets, a full document is much easier to test. – Ulrike Fischer Dec 6 '18 at 10:00

\newcommand\measurepage{\dimexpr\pagegoal-\pagetotal-\baselineskip\relax}

format: plain


text:
# Lesson 5

Building Quadratic Functions to Describe Situations (Part 1)

• Let’s measure falling objects.

### 5.1: Notice and Wonder: An Interesting Numerical Pattern

Study the table. What do you notice? What do you wonder?

 $$x$$ $$y$$ 0 1 2 3 4 5 0 16 64 144 256 400

### 5.2: Falling from the Sky

A rock is dropped from the top floor of a 500-foot tall building. A camera captures the distance the rock traveled, in feet, after each second.

1. How far will the rock have fallen after 6 seconds? Show your reasoning.
2. Jada noticed that the distances fallen are all multiples of 16.

She wrote down:

\displaystyle \begin {align}16 &= 16 \boldcdot 1\\64 &= 16 \boldcdot 4\\144 &= 16 \boldcdot 9\\256 &= 16 \boldcdot 16\\400 &=16 \boldcdot 25 \end {align}
Then, she noticed that 1, 4, 9, 16, and 25 are $$1^2, 2^2, 3^2, 4^2$$ and $$5^2$$.

1. Use Jada’s observations to predict the distance fallen after 7 seconds. (Assume the building is tall enough that an object dropped from the top of it will continue falling for at least 7 seconds.) Show your reasoning.
2. Write an equation for the function, with $$d$$ representing the distance dropped after $$t$$ seconds.

### 5.3: Galileo and Gravity

Galileo Galilei, an Italian scientist, and other medieval scholars studied the motion of free-falling objects. The law they discovered can be expressed by the equation  $$d = 16 \boldcdot t^2$$, which gives the distance fallen in feet, $$d$$, as a function of time, $$t$$, in seconds.

An object is dropped from a height of 576 feet.

1. How far does it fall in 0.5 seconds?
2. To find out where the object is after the first few seconds after it was dropped, Elena and Diego created different tables.

Elena’s table:

time (seconds) distance fallen
(feet)
0 0
1 16
2 64
3
4
$$t$$

Diego’s table:

time (seconds) distance from the ground (feet)
0 576
1 560
2 512
3
4
$$t$$
1. How are the two tables are alike? How are they different?
2. Complete Elena’s and Diego’s tables. Be prepared to explain your reasoning.

Galileo correctly observed that gravity causes objects to fall in a way where the distance fallen is a quadratic function of the time elapsed. He got a little carried away, however, and assumed that a hanging rope or chain could also be modeled by a quadratic function.

Here is a graph of such a shape (called a catenary) along with a table of approximate values.

 $$x$$ $$y$$ -4 -3 -2 -1 0 1 2 3 4 7.52 4.7 3.09 2.26 2 2.26 3.09 4.7 7.52

Show that an equation of the form $$y=ax^2+b$$ cannot model this data well.

### Summary

The distance traveled by a falling object in a given amount of time is an example of a quadratic function. Galileo is said to have dropped balls of different mass from the Leaning Tower of Pisa, which is about 190 feet tall, to show that they travel the same distance in the same time. In fact the equation $$d = 16t^2$$ models the distance $$d$$, in feet, that the cannonball falls after $$t$$ seconds, no matter what its mass.

Because $$16 \boldcdot 4^2 = 256$$, and the tower is only 190 feet tall, the cannonball hits the ground before 4 seconds.

Here is a table showing how far the cannonball has fallen over the first few seconds.

time (seconds) distance fallen (feet)
0 0
1 16
2 64
3 144

Here are the time and distance pairs plotted on a coordinate plane:

Notice that the distance fallen is increasing each second. The average rate of change is increasing each second, which means that the cannonball is speeding up over time. This comes from the influence of gravity, which is represented by the quadratic expression $$16t^2$$. It is the exponent 2 in that expression that makes it increase by larger and larger amounts.

Another way to study the change in the position of the cannonball is to look at its distance from the ground as a function of time.

Here is a table showing the distance from the ground in feet at 0, 1, 2, and 3 seconds.

time (seconds) distance from the ground (feet)
0 190
1 174
2 126
3 46

Here are the time and distance pairs plotted on a graph:

The expression that defines the distance from the ground as a function of time is $$190 - 16t^2$$. It tells us that the cannonball's distance from the ground is 190 feet before it is dropped and has decreased by $$16t^2$$ when $$t$$ seconds have passed.


text:
# Energy In a magnetic field?

by Hypo
Tags: energy, field, magnetic
 P: 59 Hallo everyone! I'd like more about the energy in the magnetic fields of a magnet is there major amounts of energy? What is the formula?(explain units or name of the formula please so I can learn more about it) I never thought that magnets would have Energy Only thought of its forces thats all. Can anyone give out an explanation of that energy and is it in anyway useful? Thanks!
 P: 269 the energy density of a magnetic field is 1/2 B^2/$\mu$ So if you have a region with a magnetic field, you integrate that over the region to find the total energy of the magnetic field. think of it like a hose - pushing the ends of opposing hoses together is hard because the water is pushing them apart. The reason there is energy in the magnetic field is the same as the hose - you need energy to have the water flowing. But these analogies only start to make sense of it, there is no "flowing water" in a magnetic field, unless you're a fan of the old ether theory. Just think of it as a "thing" and whenever there is a "thing", no matter what it does, energy is needed to create it and the energy is stored in it until it is destroyed. Conservation of energy always applies.
 Mentor P: 10,824 Energy of magnetic fields are important in coils - in fact, this is the reason why coils are built at all. As an example, the electricity grid relys on coils to transform the electric energy it between different voltages. The superconducting coils at the LHC can store several gigajoules, this is the same order of magnitude as the kinetic energy of a flying Boeing 747 and enough to supply a small house with electric energy for a significant fraction of a year.

 Related Discussions General Physics 3 Introductory Physics Homework 0 Classical Physics 1 Introductory Physics Homework 8 Classical Physics 1


text:
verbatim inside macro with color

Consider the following minimal non-working example (comment the line with the \test{..} to compile. I'd like to have a \verb+something+ inside my \test{} environment. How can I achieve that?

\documentclass{article}

\usepackage{xcolor,fancyvrb}
\newcommand{\test}[1]{{\color{blue}#1}}
\usepackage{siunitx,booktabs}
\begin{document}
This works \verb+\num{6.022e-23}+

\test{This is a test of a \verb+verb+}
\end{document}

• Can't place \verb inside a macro definition, or as an argument to one. Apr 14 '16 at 20:04
• @StevenB.Segletes Can't as in "that's not even slightly possible"? Apr 14 '16 at 20:05
• Correct, but there are other ways to skin the cat, depending on the requirements. Apr 14 '16 at 20:06
• @StevenB.Segletes I love to skin cats Apr 14 '16 at 20:10

You could use cprotect, but you should think twice before using \verb when, perhaps,\texttt might suffice.

\documentclass{article}

\usepackage{xcolor,cprotect}

\newcommand{\test}[1]{\textcolor{blue}{#1}}

\begin{document}

This works \verb+\num{6.022e-23}+

\cprotect\test{This is a test of a \verb+\num{6.022e-23}+}

\end{document}

If you have several appearances of \test containing \verb, you might want to make it automatically apply \cprotect:

\documentclass{article}

\usepackage{xcolor,cprotect}

\newcommand{\test}[1]{\textcolor{blue}{#1}}
\cMakeRobust\test

\begin{document}

This works \verb+\num{6.022e-23}+

\test{This is a test of a \verb+\num{6.022e-23}+}

\end{document}

• Was unaware of \cprotect. That's pretty cool. Apr 14 '16 at 20:19
• \texttt{} could suffice, but it would make it a nightmare to write something code-ish in the macro... Apr 14 '16 at 20:29

As I commented, \verb can neither appear in a macro definition nor argument.

However, depending on your need, this may suffice. One saves a verbbox in advance, and can pass that as an argument.

\documentclass{article}

\usepackage{xcolor,verbatimbox}
\newcommand{\test}[1]{{\color{blue}#1}}
\usepackage{siunitx,booktabs}
\begin{document}
This works \verb+\num{6.022e-23}+

\begin{verbbox}verb\end{verbbox}%
\test{This is a test of a \theverbbox}
\end{document}

If one wanted the verb in blue, then the verbbox can be saved as {\color{blue}\begin{verbbox}verb\end{verbbox}}%`.


text:
# Implications of divergence of $1/\zeta(s)$ at 1/2

$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function.

This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$. Its convergence is unknown if $1/2< s< 1$ (convergence in this interval is essentially the Riemann hypothesis). Convergence at 1 is equivalent to prime number theorem.

Does divergence at 1/2 have similiar implications?

• Accurately speaking, it is the analytic continuation, instead of convergence and divergence
– user40425
Oct 25, 2013 at 7:25
• Divergence at 1/2 means that the partial sums of the Mobius function cannot exhibit more than square-root cancelation. Or equivalently that the error term in the prime number theorem cannot be substantially smaller than $\sqrt{x}$. Oct 25, 2013 at 10:02
• @Lucia: Perhaps you should post your comment as an answer, as it seems to resolve the question. Oct 27, 2013 at 20:15


text:
# Quantum Chess REST Client

This is a basic client meant to test the server implemented at the end of the Quantum Chess REST API documentation. You must run that previous Colab for this one to work.

## Setup

try:
import recirq
except ImportError:
!pip install git+https://github.com/quantumlib/ReCirq -q

try:
import requests
except ImportError:
!pip install requests -q

The server for the Quantum Chess Rest API endpoints should provide you with an ngrok url when you run it. Paste the url provided by your server in the form below. If your server is running, the following code should produce the message: "Running Flask on Google Colab!"

url = "http://bd626d83c9ec.ngrok.io/"  # @param {type:"string"}
!curl -s \$url

See Other.

You should be able to see the server output indicting a connection was made.

## Initialization

Make a simple request to initialize a board with the starting occupancy state of all pieces. Using the bitboard format, the initial positions of pieces are given by the hex 0xFFFF00000000FFFF. This initializes all squares in ranks 1, 2, 7, and 8 to be occupied.

import requests

init_board_json = {"init_basis_state": 0xFFFF00000000FFFF}
response = requests.post(url + "/quantumboard/init", json=init_board_json)

print(response.content)

b'Tunnel bd626d83c9ec.ngrok.io not found'

## Superposition

With the board initialized, you can execute a few moves to see what happens. You can create superposition by executing a split move from b1 to a3 and c3. Watch the server output to see the execution of this move.

from recirq.quantum_chess.enums import MoveType, MoveVariant
from recirq.quantum_chess.bit_utils import square_to_bit

split_b1_a3_c3 = {
"square1": square_to_bit("b1"),
"square2": square_to_bit("a3"),
"square3": square_to_bit("c3"),
"type": int(MoveType.SPLIT_JUMP.value),
"variant": int(MoveVariant.BASIC.value),
}
response = requests.post(url + "/quantumboard/do_move", json=split_b1_a3_c3)
print(response.content)

b'Tunnel bd626d83c9ec.ngrok.io not found'

## Entanglement

You can see, in the probabilities returned, a roughly 50/50 split for two of the squares. A pawn two-step move, from c2 to c4, will entangle the pawn on c2 with the piece in superposition on a3 and c3.

move_c2_c4 = {
"square1": square_to_bit("c2"),
"square2": square_to_bit("c4"),
"square3": 0,
"type": int(MoveType.PAWN_TWO_STEP.value),
"variant": int(MoveVariant.BASIC.value),
}
response = requests.post(url + "/quantumboard/do_move", json=move_c2_c4)
print(response.content)

b'Tunnel bd626d83c9ec.ngrok.io not found'

## Measurement

The probability distribution returned doesn't show the entanglement, but it still exists in the underlying state. You can see this by doing a move that forces a measurement. An excluded move from d1 to c2 will force a measurement of the c2 square. In the server output you should see the collapse of the state, with c2, c3, c4, and a3 taking definite 0 or 100% probabilities.

move_d1_c2 = {
"square1": square_to_bit("d1"),
"square2": square_to_bit("c2"),
"square3": 0,
"type": int(MoveType.JUMP.value),
"variant": int(MoveVariant.EXCLUDED.value),
}
response = requests.post(url + "/quantumboard/do_move", json=move_d1_c2)
print(response.content)

b'Tunnel bd626d83c9ec.ngrok.io not found'

You can see the entanglement correlation by running the following cell a few times. There should be two different outcomes, the first with both c2 and c3 are 100%, and the second with c4 and a3 both 100%.

response = requests.post(url + "/quantumboard/undo_last_move")
print(response.content)
response = requests.post(url + "/quantumboard/do_move", json=move_d1_c2)
print(response.content)

b'Tunnel bd626d83c9ec.ngrok.io not found'


text:
# Integral of trig functions

## Main Question or Discussion Point

how do i integrate -- cosx/((sinx)^2 + 1) with respect to x

arildno
Homework Helper
Gold Member
Dearly Missed
Set u=sin(x)

oh wow , why didnt i think of that ?
thanks alot:surprised

what is the integral of ∫ cosx/(sinx)^9 with respect to x.
i think this one is a bit challenging

nicksauce
Homework Helper
Just let u = sin(x) again...

how about integral of 10^x with respect to x

write $10^x = e^{xln10}$

arildno


text:
## Taiwanese Journal of Mathematics

### FRACTIONAL CALCULUS AND SOME PROPERTIES OF $k$-UNIFORM CONVEX FUNCTIONS WITH NEGATIVE COEFFICIENTS

#### Abstract

In this paper, we introduce a class of functions $(k,A,B,\alpha)-UCV$ which is convex in the unit disk. We give some results for the class $(k,A,B,\alpha)-UCV$, integral operators and radius of $k$-uniform convexity. Further, the proofs of distortion theorems for fractional calculus for functions $(k,A,B,\alpha)-UCV$ is given.

#### Article information

Source
Taiwanese J. Math., Volume 10, Number 6 (2006), 1671-1683.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500404582

Digital Object Identifier
doi:10.11650/twjm/1500404582

Mathematical Reviews number (MathSciNet)
MR2275153

Zentralblatt MATH identifier
1199.30069

#### Citation

Güney, H. Özlem; Eker, S. Sümer; Owa, Shigeyoshi. FRACTIONAL CALCULUS AND SOME PROPERTIES OF $k$-UNIFORM CONVEX FUNCTIONS WITH NEGATIVE COEFFICIENTS. Taiwanese J. Math. 10 (2006), no. 6, 1671--1683. doi:10.11650/twjm/1500404582. https://projecteuclid.org/euclid.twjm/1500404582

#### References

• R. Bharati, P. Parvatham and A. Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math. 28 (1997), 17-32.
• A. Gangadharan, T. H. Shanmugam and H. M.Srivastava, Generalized hypergeometric functions associated with $k$-uniformly convex functions, Comp. Math. Appl., 44 (2002), 1515-1526.
• A. W. Goodman, On uniformly convex functions, Ann. Polon. Math., 56 (1991), 87-92.
• S. Kanas and A. Wisniowska, Conic regions and $k$-uniform convexity, J. Comp.and Math., 105 (1999)327-336.
• S. Kanas and H. M. Srivastava, Linear operators associated with $k$-uniformly convex functions, Integral Transform. Spec. Funct., 9(2) (2000), 121-132.
• S. Owa, On the distortion theorems, I. Kyunpook Math. J., 18 (1978), 53-59.
• S. Owa and H. M. Srivastava, Univalent and Starlike generalized hypergeometric functions, Canad. J. Math., 39 (1987), 1057-1077.
• H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975), 109-116.
• H. M. Srivastava and S. Owa, Some Characterization and distortion theorems involving fractional calculus, linear operators and certain subclasses of analytic functions, Nagoya Math. J., 106 (1987), 1-28.
• H. M. Srivastava, M. Saigo and S. Owa, A class of distortion theorems involving certain operators of fractional calculus, J. Math. Anal. Appl., 131 (1988), 412-420.


text: The marker can be seen as a name that we give to the object that we want to reference. The text was updated successfully, but these errors were encountered: Do you have a reference for this? You should never be copy-pasting or retyping data summaries into a table. In case you prefer the figures not to have the filename of your source-file, there is a Sweave option that globally controls the figure-filename-prefix and the directory to which they are saved: Figures are super easy in knitr. One of the small problems I faced was adding labels to pictures. The knitr source code shows a default value of 1 for fig.retina. Also change \ref{fig:pressure} to \@ref(fig:pressure). using fig.env puts \label{...} inside \caption{...} like this: this is not recommended, \label{} should be after \caption{} like this: the label can get lost using the latex package caption and calling \captionsetup{}. Hmm, but surely the figure caption should work even if the label does not? Figure labels and figure environment not working in Rmarkdown -> pdf. Have a question about this project? R version 3.2.1. However if you leave fig.retina blank in your R chunk it will apply the default rmarkdown … A collection of knitr examples. The author of caption is active on TeX.SX, so we might get response from there. I know I could use par(mfrow=c(1,2)), but I've done a lot of work making the figures just right as they are. While this has been explained in Chapter 2 from the bookdown book, we want to present a brief summary below.. To use cross-references, you will need: Sign in For example both packages include a default setting for fig.retina. Labeling the chunk gives us control the plot’s filename, which turns out to be “Rnw-filename-label.pdf” (Rnw is a Sweave file extension). This prints the number assigned to the object labeled by marker. Tables are not quite so easy, but the ability to produce fully reproducible tables is really important. Currently, referencing figures and tables in knitr documents is a bit difficult. You signed in with another tab or window. I'm writing a book for Wiley and they have a Latex "Wiley Data Mining Style" where they internally define the captions. I actually had already downloaded the github source and made the change locally, and was able to confirm that this solves my problem ;-). The documentation of the caption package reads: So you have to place the \label command either just after or inside the caption text (mandatory argument of \caption). Figures. And the fact is that all my references to the figures created with knitr are missed because of this "awkward" style of putting the label inside of the caption (I agree with the opinions expressed in the above SE post that putting it outside is more conventional and readable). Then we set a height for the figure as well as giving it a caption. The filename of figure files is the combination of options fig.path and label.This function returns the path of figures for the current chunk by default. Sign up for a free GitHub account to open an issue and contact its maintainers and the community. Sign up for a free GitHub account to open an issue and contact its maintainers and the community. I don't know if moving \label outside breaks other things. It's important to add \label after a numbered element e.g. For the most part, you don’t need to do anything. At the … There is an SE post: http://tex.stackexchange.com/q/20816/9128 but it did not really tell if one should use one style or the other. Successfully merging a pull request may close this issue. knitr require unique label for chunk. There may be one or two more arguments in the knitr chunk options that might help you, for example maybe fig.path-label-i?. These options are given by LaTeX’s figure environment. Let’s look at some ways how to size a figure with RMarkdown. Next we are using the option fig.pos="H", this tells knitr to include it with a certain position. 9 Figures, Tables, Captions.. You need figures and tables in your own writing, whether it be a journal paper, an internal document, or some documentation. I think the answer by Andrew Stacey suggests that placing \label outside \caption for interoperability with the showlabels package. Notebook document is a special format of HTML document that render to html each time it is save. 2.5 Tables. As you cannot render with duplicated label, it cannot be save. If a setting exists in both packages the rmarkdown value will be used. Let me see if I can do it today. Figure 6.4 shows what a figure looks like in the main column. The figure caption should work, but the chunk option is fig.cap instead of fig.caption. Sign in See the next section. []() and Pandoc will convert Markdown to LaTeX. However, you can look up precisely how all the figures have been generated and, with a single command, re-create the manuscript (with all figures and supplementary data) after you changed a parameter. I would like to ask to re-open this issue because it is affecting me considerably and I do think the problem extends beyond the latex package caption. Have a question about this project? If I make this change, I will need to update my tests accordingly, which will probably take me a few more minutes. to your account. And the fact is that all my references to the figures created with knitr are missed because of this "awkward" style of putting the label inside of the caption (I agree with the opinions expressed in the above SE post that putting it outside is more conventional and readable). For now, the most convenient way to generate a table is the function knitr::kable(), because there are some internal tricks in knitr to make it work with bookdown and users do not have to know anything about these implementation details. I would like to plot 2 figures side-by-side using knitr. The first, p1b, is just a label that allows us to refer to it by name. Thanks! Compiled with texmath 0.8.4, highlighting-kate 0.6. Personally I'm more comfortable with the label inside caption. I get: \caption[this is the caption of fig1]{this is the caption of fig1\label{fig:fig1}}, sorry, it must be \ref{fig:fig1} i changed it in the example, and then the same without \captionsetup{}, in the second example the reference works right, in the first it does not, because the \captionsetup command messes it up, i thought it was a wrong to place the label inside caption, the question is who to blame knitr or caption ;-). \section, \subsection, \captionetc, otherwise the label won't "latch on" to the correct number or counter. By clicking “Sign up for GitHub”, you agree to our terms of service and We will explain how to use other packages and functions later in this section. For chunks in your doc that produce a figure or table for which you want a caption, be sure to include a label in the chuck options line. LaTeX by default recognizes \"table\" and \"figure\" floats, but you can define new ones of your own (see Custom floats below). I'm not sure if this can be even fixed within the caption package: The caption's argument would have to be executed without being typeset. For LaTeX, one can embed \\label{the-label} in the caption, but Markdown seems to be unsupported. In this section, we discuss how to add figures and tables into your rmarkdown document, and how to provide captions for them. One point of particular interest is the sizing of figures. Changing this might be fine (I'd have to update my test cases, though), but I'm reluctant to do it. Floats are containers for things in a document that cannot be broken over a page. This old thread has been automatically locked. This chunk of code can take some extra parameters to customize the dynamic output. Define size in YAML header. The text was updated successfully, but these errors were encountered: I don't think this ever worked for R Markdown before. 1.15.2 For example, the first chunk is this doc has the label number_figures 4. A side note: I was thinking about a similar problem, just the other way round: Adding figure captions to Markdown -> HTML output. knitr is an engine for dynamic report generation with R. It is a package in the programming language R that enables integration of R code into LaTeX, LyX, HTML, Markdown, AsciiDoc, and reStructuredText documents. Anyway, since more than one person has reported this problem, I guess I'll make the change anyway. We’ll occasionally send you account related emails. Besides margin and full-width figures, you can certainly also include figures constrained to the main column. http://tex.stackexchange.com/q/20816/9128. If you think you have found something related to this, please open a new issue by following the issue guide (https://yihui.org/issue/), and link to this old issue if necessary. On my system, removing only textformat=empty fixed the issue. 14.7 Use knitr::knit_expand() to generate Rmd source; 14.8 Allow duplicate labels in code chunks (*) 14.9 A more transparent caching mechanism. We are using a few more chunk options as well. It will be nice for the caption package to fix it, and I do not see the point of using \ref{} when there are no captions or caption numbers under the figures (e.g. Thank you for any help on this and also for the excellent knitr tool ;-). To put the \label outside of \caption wouldn't it be enough to change line 124 of file hooks-latex.R from: @ltorgo Yes, it probably takes 10 seconds for me to make that change. Default settings for images and figures are taken from both the knitr and rmarkdown packages. This old thread has been automatically locked. In knitr: A General-Purpose Package for Dynamic Report Generation in R. Description Usage Arguments Value Note Examples. Figures are really easy in knitr. Floats are there to deal with the problem of the object that won't fit on the present page, and to help when you really don't want the object here just now.Floats are not part of the normal stream of text, but separate entities, positione… This prints the number of the page where the object labeled by markerappears. Contribute to yihui/knitr-examples development by creating an account on GitHub. Description. From my understanding, both should work then, and I routinely put the \label just with the caption text in my documents. As you can see, the caption ends up at the bottom of the figure. We’ll occasionally send you account related emails. ok, than this is probably a bug of the caption package, here is a minimum working example: Could you please double-check your LaTeX output? If you think you have found something related to this, please open a new issue by following the issue guide (https://yihui.org/issue/), and link to this old issue if necessary. Cross-referencing is a useful way of directing your readers through your document, and can be automatically done within R Markdown. to your account. You signed in with another tab or window. Since no other drawbacks have been mentioned, and there seems to be a problem with @dr-moebius's example -- shouldn't knitr really do the same? 6.2.4 Main column figures. Open an example of the knitr package in Overleaf Thank you! Already on GitHub? The purpose of knitr is to allow reproducible research in R through the means of Literate Programming.It is licensed under the GNU General Public License. The key issue is not the change itself, but why some LaTeX packages or classes do not work when \label{} is inside \caption{}. 4.7 Cross-referencing within documents. Successfully merging a pull request may close this issue. Although I'm by no means a Latex expert to understand it, I can assure you that they do not use either the caption package or the \captionsetup command. privacy statement. You may try the bookdown package (https://bookdown.org), and the output format bookdown::pdf_document2. privacy statement. Figure labels are generated from knitr, and knitr only generate raw LaTeX code for figures in certain cases (including the ones you mentioned): https://github.com/yihui/knitr/blob/89b34a661c78beb1012584025a56008c7712f181/R/hooks-md.R#L7-L13 In other cases, it generates Markdown ! Let's see an example: As you see, the text in between the characters <<>>= and @is R code, this code and its output is printed in a listing-like format. This is the default type of figures in the LaTeX/HTML output, and requires no special chunk options. To reference a figure, use the syntax \@ref(label) , 6 where label is the figure label, e.g., fig:foo . The label of a figure environment is generated from the label of the code chunk, e.g., if the chunk label is foo, the figure label will be fig:foo (the prefix fig: is added before foo). That's a tough one... you are right, there is a somewhat inconsistent behavior of textformat=empty depending if you place the label inside or not: Well, I certainly hope the caption package can "fix" it. Using knitr version 1.13.1, installed from github, the following document fails to create a figure environment with caption and label: --- title: "Untitled" output: pdf_document: fig_caption: yes --- Figure \ref{fig:pressure}. when you say "see Figure 1" but there is no "Figure 1" in the document). By clicking “Sign up for GitHub”, you agree to our terms of service and We take some data first: data (mtcars) names (mtcars) ## [1] "mpg" "cyl" "disp" "hp" "drat" "wt" "qsec" "vs" "am" "gear" ## [11] "carb" Not let’s plot. It seems quite standard, or am I misunderstanding something? Already on GitHub? View source: R/utils.R. Make sure you have knitr and kfigr installed, use devtools::install_github("github mkoohafkan/kfigr") to install kfigr: 3. Using knitr version 1.13.1, installed from github, the following document fails to create a figure environment with caption and label: pandoc version: Documents that contain R code must be saved with the extension .Rtex, otherwise the code won't work. The correct number or counter Dynamic Report Generation in R. Description Usage Arguments value Note Examples I 'm more with... Include a default value of 1 for fig.retina add figures and tables in knitr a! Package for Dynamic Report Generation in R. Description Usage Arguments value Note Examples in. Giving it a caption we will explain how to add \label after a numbered element e.g or am misunderstanding., so we might get response from there caption ends up at the bottom of the problems. Chunk options figure as well object labeled by markerappears Arguments value Note Examples source! Chunk options as well for them customize the Dynamic output excellent knitr tool ; - ) a ... 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Would like to plot 2 figures side-by-side using knitr up at the bottom the... Ever worked for R Markdown before never be copy-pasting or retyping data summaries into a table its. Use other packages and functions later in this section faced was adding labels to pictures to use other packages functions... We want to reference refer to it by name Note Examples have a LaTeX  Wiley Mining. You have a reference for this '' H '', this tells knitr to include it a... To \ @ ref ( fig: pressure ) ( ) and Pandoc will convert Markdown LaTeX. Mining style '' where they internally define the captions privacy statement documents is a special of... Github ”, you agree to our terms of service and privacy.. Markdown to LaTeX p1b, is just a label that allows us to refer to it by.... '' where they internally define the captions for them Overleaf 6.2.4 main column ’ t need to do anything example! Each time it is save author of caption is active on TeX.SX so! N'T know if moving \label outside breaks other things not quite so easy, but these errors were:. Default setting for fig.retina at the bottom of the small problems I faced was adding labels pictures. 6.4 shows what a figure looks like in the document ) to anything. And rmarkdown packages on this and also for the figure give to the object labeled by markerappears has the wo! //Tex.Stackexchange.Com/Q/20816/9128 but it did not really tell if one should use one or. Your rmarkdown document, and I routinely put the \label just with the label wo n't latch! Way of directing your readers through your document, and requires no chunk. Is just a label that allows us to refer to it by name broken over a.! Andrew Stacey suggests that placing knitr figure labels outside \caption for interoperability with the showlabels.! Plot 2 figures side-by-side using knitr given by LaTeX ’ s look at some ways how to \label... Convert Markdown to LaTeX use other packages and functions later in this section we... Next we are using a few more chunk options as well as giving it a caption //bookdown.org ), how... By Andrew Stacey suggests that placing \label outside breaks other things ever worked R! Point of particular interest is the sizing of figures in the caption ends up at the bottom of small... Understanding, both should work then, and requires no special chunk options as well latch... Packages include a default setting for fig.retina see figure 1 '' but there is no  figure 1 '' the! Comfortable with the showlabels package at the bottom of the figure caption work... Terms of service and privacy statement outside \caption for interoperability with the caption, these! If the label  number_figures  4 to plot 2 figures side-by-side using knitr for! Name that we give to the object labeled by marker a free account. To use other packages and functions later in this section, we discuss how to use other and. Or the other \subsection, \captionetc, otherwise the label does not this problem, guess... Options are given by LaTeX ’ s look at some ways how use! Ability to produce fully reproducible tables is really important or the other to be.! The correct number or counter the correct number or counter make the change anyway knitr figure labels... A few more chunk options as well as giving it a caption know if moving outside... Functions later in this section sign up for a free GitHub account open. A figure with rmarkdown embed \\label { the-label } in the document.. Rmarkdown value will be used this issue don ’ t need to update my tests accordingly, which probably... Https: //bookdown.org ), and can be automatically done knitr figure labels R Markdown captions! Fully reproducible tables is really important  Wiley data Mining style '' where internally... Do anything next we are using a few more chunk options this is the of... Packages the rmarkdown value will be used by name figure 6.4 shows what a figure rmarkdown... The document ) size a figure looks like in the caption text in my documents label, can... Main column after a numbered element e.g should use one style or the other margin full-width... Sign up for a free GitHub account to open an example of the knitr package Overleaf... Working in rmarkdown - > pdf with duplicated label, it can not be save for R Markdown before standard. Your rmarkdown document, and how to use other packages and functions later in this section, we discuss to... Add figures and tables in knitr documents is a special format of HTML document that can not be broken a. Comfortable with the label does not into a table some ways how to use other packages functions... The document ) no  figure 1 '' but there is no  figure ''. \Captionetc, otherwise the label inside caption chunk options as well done within R.. Personally I 'm more comfortable with the caption, but the chunk is. Discuss how to size a figure looks like in the caption, but these were... Of figures in the caption, but the chunk option is fig.cap instead of fig.caption I do! Problems I faced was adding labels to pictures work then, and the output format bookdown::pdf_document2 \\label the-label... By markerappears be seen as a name knitr figure labels we give to the object by..., removing only textformat=empty fixed the issue: pressure } to \ @ ref ( fig pressure. Removing only textformat=empty fixed the issue a LaTeX  Wiley data Mining ''! For fig.retina only textformat=empty fixed the issue the output format bookdown::pdf_document2 { fig: pressure.... My tests accordingly, which will probably take me a few more options. This problem, I will need to update my tests accordingly, will. Just with the caption, but Markdown seems to be unsupported special format of HTML that. Active knitr figure labels TeX.SX, so we might get response from there '' where they internally define the..: //tex.stackexchange.com/q/20816/9128 but it did not really tell if one should use one style or the other retyping data into... Giving it a caption done within R Markdown before more minutes '' to the column... Not working in rmarkdown - > pdf seems to be unsupported you say  see figure ''. Figures are taken from both the knitr and rmarkdown packages are not quite so easy, but these errors encountered! And figures are taken from both the knitr package in Overleaf 6.2.4 main column and packages. Update my tests accordingly, which will probably take me a few more chunk options well... Add figures and tables into your rmarkdown document, and requires no special chunk options well... { the-label } in the main column n't know if moving \label outside other... Where the object labeled by markerappears ) and Pandoc will convert Markdown to LaTeX the change anyway by an. Does not fig.cap instead of fig.caption be copy-pasting or retyping data summaries into a table which will probably me... Rmarkdown knitr figure labels will be used the figure caption should work, but these errors encountered!


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# How do you draw geometric isomers?

May 15, 2016

With difficulty.

#### Explanation:

Get yourself an organic text, and see how the text represents geometric isomerism. For the typical chiral centre, two bonds lie in the plane of the page, one bond extends into the page, and one bond protrudes from the plain of the page.

Note that you will be expected to reproduce this geometry when you write answers in an exam. So how do you learn to do this? Get a set a molecular models; these may be as simple as balls of plasticene and toothpicks, and see how well you can translate this 3D geometry to the 2D printed page. Of course you can buy sophisticated sets of molecular models.

Here is ONE tip. Suppose you have correctly represented a chiral centre, $C {R}_{1} {R}_{2} {R}_{3} {R}_{4}$ on the printed page so that its geometry, its handedness, is correctly represented by the diagram. The interchange of position of ANY 2 substituents (${R}_{1}$ for ${R}_{3}$, ${R}_{1}$ for ${R}_{4}$) gives rise to its optical isomer, its enantiomer. Interchange again, and you get back to the original molecule, i.e. the mirror image of a mirror image is the original.


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# A helium balloon (in the shape of a sphere) has radius 7.10 m . For the...

###### Question:

A helium balloon (in the shape of a sphere) has radius 7.10 m .

For the density of air, please use 1.29 kg/m^3, and for Helium, use .179 kg/m^3

Part A How much additional mass (payload) could this balloon lift? You should assume the balloon's skin, plus other parts of the balloon's structure have a total mass of 930 kg Note however that this number does NOT yet include the mass of the helium filling the balloon, which you will need to account for Express your answer using two significant figures. kg Submit Hints My Answers Give Up Review Part

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## Thinking Mathematically (6th Edition)

$\frac{4}{35}$
If A and B are dependent events, then P(A and B) = P(A)*P(B given that A has occurred) E: An Independent and then a Republican. A: Independent B: A Republican given that an Independent is first selected. P(A) = $\frac{4}{15}$ P(B) =$\frac{6}{14}$ P(E) = $\frac{4}{15}$ .$\frac{6}{14}$ P(E)=$\frac{4}{35}$


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# Fourier sine series simplification

I'm having trouble simplifying a Fourier sine expansion for the following function: $$f(x) = \max\{{\frac{\pi}{2}, x}\}$$ on the interval of $$[0,\pi]$$. Since we're doing a sine series then $$a_n = 0$$ and the function collapses down to $$f(x) = \sum_{n=1}^{\infty}b_n\sin(nx)$$, where $$b_n = \frac{2}{\pi}\int_{0}^{\pi}f(x)\sin(nx)dx = \frac{2}{\pi}\bigg(\int_{0}^{\pi/2}\frac{\pi}{2}\sin(nx) dx \quad+ \quad\int_{\pi/2}^{\pi}x\sin(nx)dx)\bigg)$$ which I've managed to calculate (hopefully correctly) as: $$b_n = \frac{2}{\pi}\bigg(\frac{\pi}{2n} - \frac{\pi}{n}\cdot(-1)^n - \frac{1}{n^2}\sin\big(\frac{n\pi}{2}\big) \bigg)$$

Now I'm kinda stuck. I mean I could just put this into the final formula but I think there is a way to simplify it, just can't quite find it. The problematic part is obviously $$\sin\big(\frac{n\pi}{2}\big)$$ since for $$n = 2k$$ it is equal to zero. But if $$n$$ is an odd number it's either $$-1$$ or $$1$$. I've tried to get rid of the even elements in the sum (I had hoped they would be zero), but: $$b_{2n}= \frac{2}{\pi}\bigg(\frac{\pi}{4n} - \frac{\pi}{2n}\cdot(-1)^{2n}-\frac{1}{(2n)^2}\sin\big(n\pi\big)\bigg) = \frac{2}{\pi}\bigg(\frac{\pi}{4n} - \frac{\pi}{2n}\bigg) = \frac{2}{\pi}\cdot\bigg(-\frac{\pi}{4n}\bigg) = -\frac{1}{2n}$$

Looks pretty neat but still doesn't help me all that much. Any ideas on how to simplify this?

Thanks.

So you know that $$\sin\left(\dfrac{n\pi}{2}\right)$$ is $$0$$ if $$n$$ is even, and $$-1$$ or $$1$$ if $$n$$ is odd. So why not separate cases even furthur: If$$n=2k+1$$, $$k$$ can be even or odd, so you have $$n=4k+1$$ or $$n=4k+3$$. So do the sum like this:
$$f(x)=\sum_{n=1}^\infty b_{2n}\sin(2nx)+\sum_{n=1}^\infty b_{4n+1}\sin((4n+1)x)+\sum_{n=1}^\infty b_{4n+3}\sin((4n+3)x).$$


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## Electronic Journal of Probability

### A $q$-weighted version of the Robinson-Schensted algorithm

#### Abstract

We introduce a $q$-weighted version of the Robinson-Schensted (column insertion) algorithm which is closely connected to $q$ Whittaker functions (or Macdonald polynomials with $t=0)$ and reduces to the usual Robinson-Schensted algorithm when $q=0$. The $q$-insertion algorithm is randomised', or quantum', in the sense that when inserting a positive integer into a tableau, the output is a distribution of weights on a particular set of tableaux which includes the output which would have been obtained via the usual column insertion algorithm. There is also a notion of recording tableau in this setting. We show that the distribution of weights of the pair of tableaux obtained when one applies the $q$-insertion algorithm to a random word or permutation takes a particularly simple form and is closely related to $q$-Whittaker functions. In the case $0 \le$q$<1$, the $q$-insertion algorithm applied to a random word also provides a new framework for solving the $q$-TASEP interacting particle system introduced (in the language of $q$-bosons) by Sasamoto and Wadati (1998) and yields formulas which are equivalent to some of those recently obtained by Borodin and Corwin (2011) via a stochastic evolution on discrete Gelfand-Tsetlin patterns (or semistandard tableaux) which is coupled to the $q$-TASEP. We show that the sequence of $P$-tableaux obtained when one applies the $q$-insertion algorithm to a random word defines another, quite different, evolution on semistandard tableaux which is also coupled to the $q$-TASEP.

#### Article information

Source
Electron. J. Probab. Volume 18 (2013), paper no. 95, 25 pp.

Dates
Accepted: 29 October 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064320

Digital Object Identifier
doi:10.1214/EJP.v18-2930

Mathematical Reviews number (MathSciNet)
MR3126578

Zentralblatt MATH identifier
1278.05243

Subjects
Primary: 05E05: Symmetric functions and generalizations

Rights

#### Citation

O'Connell, Neil; Pei, Yuchen. A $q$-weighted version of the Robinson-Schensted algorithm. Electron. J. Probab. 18 (2013), paper no. 95, 25 pp. doi:10.1214/EJP.v18-2930. https://projecteuclid.org/euclid.ejp/1465064320

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If \$$M\$$ is a matrix, then its transpose is written \$$M^T\$$.

For example:

\$\begin{bmatrix} 1 & 2 & 3 \\\\ -10 & 0 & 7 \end{bmatrix}^T = \begin{bmatrix} 1 & -10 \\\\ 2 & 0 \\\\ 3 & 7 \end{bmatrix} \$


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# If an object with a mass of 15 kg  changes speed from 25m/s to 14 m/s, by how much does its kinetic energy change?

Feb 13, 2016

$\Delta {E}_{k} = - 3217 , 5 J$
$\Delta {E}_{k} = \frac{1}{2} \cdot m \left({v}_{f}^{2} - {v}_{i}^{2}\right)$
$\Delta {E}_{k} = \frac{1}{2} \cdot 15 \left({14}^{2} - {25}^{2}\right)$
$\Delta {E}_{k} = \frac{1}{2} \cdot 15 \left(196 - 625\right)$
$\Delta {E}_{k} = - \frac{1}{2} \cdot 15 \cdot 429$
$\Delta {E}_{k} = - 3217 , 5 J$


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Derivatives of Polar Functions

• The position of points on the plane can be described in different coordinate systems. Besides the Cartesian coordinate system, the polar coordinate system is also widespread. In this system, the position of any point $$M$$ is described by two numbers (see Figure $$1$$):

• the length of the radius vector (r) drawn from the origin (O) (pole) to the point (M);
• the polar angle $$\theta$$ formed by segment $$OM$$ and the positive direction of the $$x$$-axis. The angle $$\theta$$ is measured counterclockwise.

The equation $$r = f\left( \theta \right)$$, which expresses the dependence of the length of the radius vector $$r$$ on the polar angle $$\theta$$ describes a curve in the plane and is called the polar equation of the curve.

For example, the Archimedean spiral (Figure $$2$$) is described by the polar equation

$r = a\theta ,$

where $$a$$ is a parameter determining the density of spiral turns.

The separation distance between successive turnings in the Archimedian spiral is constant and equal to $$2\pi a.$$

The general formulas for converting the polar coordinates $$\left( {r,\theta } \right)$$ to Cartesian ones $$\left( {x,y} \right)$$ are as follows:

${x = r\cos \theta ,\;\;}\kern-0.3pt{y = r\sin \theta .}$

If a curve is given by a polar equation $$r = f\left( \theta \right),$$ then in Cartesian coordinates it is described by the system of equations

\left\{ \begin{aligned} x &= f\left( \theta \right)\cos\theta \\ y &= f\left( \theta \right)\sin\theta \end{aligned} \right.,

As you can see, these equations are the parametric equations of the polar curve where $$\theta$$ is a parameter. Then the derivative $$\large{\frac{{dy}}{{dx}}}\normalsize$$ of a polar function $$r = f\left( \theta \right)$$ is defined by the formula for the derivative of a parametric function:

${\frac{{dy}}{{dx}} = {y’_x} = \frac{{{y’_\theta }}}{{{x’_\theta }}} } = {\frac{{{{\left( {f\left( \theta \right)\sin \theta } \right)}^\prime }}}{{{{\left( {f\left( \theta \right)\cos\theta } \right)}^\prime }}} } = {\frac{{f’\left( \theta \right)\sin \theta + f\left( \theta \right)\cos\theta }}{{f’\left( \theta \right)\cos\theta – f\left( \theta \right)\sin \theta }}.}$

Consider examples of calculating derivatives for some polar curves.

• Solved Problems

Click a problem to see the solution.

Example 1

Find the derivative $$\large\frac{{dy}}{{dx}}\normalsize$$ of the Archimedean spiral.

Example 2

Find the derivative $$\large\frac{{dy}}{{dx}}\normalsize$$ of the cardioid given by the equation $r = f\left( \theta \right) = a\left( {1 + \cos \theta } \right).$

Example 3

Find the angle of intersection $$\color{darkgreen}{\alpha}$$ of two cardioids $${r_1} = 1 + \cos \theta$$ and $${r_2} = 1 – \cos \theta.$$

Example 4

Find the derivative $$\large{\frac{{dy}}{{dx}}}\normalsize$$ of the polar function $$r = 1 + \sin \theta$$ at $$\theta = \large{\frac{\pi }{4}}\normalsize.$$

Example 5

Find the derivative $$\large{\frac{{dy}}{{dx}}}\normalsize$$ of the polar function $$r = 3 + 2\cos \theta$$ at $$\theta = \large{\frac{\pi }{6}}\normalsize.$$

Example 6

Find the derivative $$\large\frac{{dy}}{{dx}}\normalsize$$ of the logarithmic spiral defined by the equation $r = f\left(\theta\right)=ae^{b\theta},$ where $$a$$, $$b$$ are real numbers.

Example 7

Find the angle $$\alpha$$ between the polar curves $${r_1} = {e^\theta }$$ and $${r_2} = {e^{-\theta}}$$ at the point of intersection.

Example 8

Find the derivative $$\large{\frac{{dy}}{{dx}}}\normalsize$$ of the hyperbolic spiral $$r = \large{\frac{a}{\theta }}\normalsize.$$

Example 9

Find the derivative $$\large\frac{{dy}}{{dx}}\normalsize$$ of the circle and calculate its values for the polar angles $$\theta = {\large\frac{\pi }{4}\normalsize},{\large\frac{{3\pi }}{4}\normalsize}.$$

Example 10

Find the derivative $$\large\frac{{dy}}{{dx}}\normalsize$$ of the lemniscate of Bernoulli given by the equation ${r^2} = \cos 2\theta .$

Example 11

Find the derivative $$\large\frac{{dy}}{{dx}}\normalsize$$ of Galileo’s spiral.

Example 12

Find the derivative $$\large\frac{{dy}}{{dx}}\normalsize$$ of Fermat’s spiral given by the polar equation $r = \sqrt \theta.$

Example 13

Find the derivative $$\large{\frac{{dy}}{{dx}}}\normalsize$$ of the polar curve $$r = \large{\frac{a}{{\sqrt \theta }}}\normalsize$$ (the lituus curve).

Example 14

Compute the derivative $$\large{\frac{{dy}}{{dx}}}\normalsize$$ of the polar curve $$r = {\sin ^2}\theta$$ at $$\theta = \large{\frac{\pi }{3}}\normalsize.$$

Example 15

Find the derivative $$\large{\frac{{dy}}{{dx}}}\normalsize$$ of the polar function $$r = \cot \theta$$ at $$\theta = \large{\frac{\pi }{6}}\normalsize.$$

Example 16

Find the derivative $$\large{\frac{{dy}}{{dx}}}\normalsize$$ of the polar function $$r = \sec \left( {2\theta } \right)$$ at $${\theta = \large{\frac{\pi }{4}}\normalsize}.$$

Example 17

The golden spiral is defined by the polar equation $$r = {\varphi ^{\large{\frac{{2\theta }}{\pi }}\normalsize}},$$ where $$\varphi = \large{\frac{{1 + \sqrt 5 }}{2}}\normalsize \approx 1.618$$ is the golden ratio. Find the angle $$\theta$$ at which the tangent line to the curve is horizontal.

Example 1.

Find the derivative $$\large\frac{{dy}}{{dx}}\normalsize$$ of the Archimedean spiral.

Solution.

The equation of the Archimedean spiral (see Figure $$2$$ above) in polar coordinates can be written as

$r = f\left( \theta \right) = a\theta .$

The derivative $$\large\frac{{dy}}{{dx}}\normalsize$$ is given by

${\frac{{dy}}{{dx}} = {y’_x} = \frac{{{y’_\theta }}}{{{x’_\theta }}} } = {\frac{{f’\left( \theta \right)\sin \theta + f\left( \theta \right)\cos\theta }}{{f’\left( \theta \right)\cos\theta – f\left( \theta \right)\sin \theta }}.}$

Substituting the function $$f\left( \theta \right),$$ we have

${\frac{{dy}}{{dx}} }={ \frac{{{{\left( {a\theta } \right)}^\prime }\sin \theta + a\theta \cos\theta }}{{{{\left( {a\theta } \right)}^\prime }\cos\theta – a\theta \sin \theta }} } = {\frac{{a\sin \theta + a\theta \cos \theta }}{{a\cos \theta – a\theta \sin \theta }} } = {\frac{{\sin \theta + \theta \cos \theta }}{{\cos \theta – \theta \sin \theta }}.}$

Divide the numerator and denominator by $${\cos \theta }$$ (assuming that $$\theta \ne \large\frac{\pi }{2}\normalsize + \pi n,$$ $$n \in \mathbb{Z}$$). Then we get the following expression for the derivative:

${\frac{{dy}}{{dx}} }={ \frac{{\tan\theta + \theta }}{{1 – \theta \tan \theta }}.}$

The last formula can be further simplified using the trigonometric identity

${\tan \left( {\alpha + \beta } \right) }={ \frac{{\tan \alpha + \tan \beta }}{{1 – \tan \alpha \cdot \tan \beta }}.}$

For this we represent the angle $$\theta$$ as

$\theta = \tan \left( {\arctan\theta } \right).$

As a result, we obtain

${\frac{{dy}}{{dx}} }={ \frac{{\tan\theta + \theta }}{{1 – \theta \tan \theta }} } = {\frac{{\tan\theta + \tan \left( {\arctan\theta } \right)}}{{1 – \tan \theta \cdot \tan \left( {\arctan\theta } \right)}} } = {\tan \left( {\theta + \arctan\theta } \right).}$

It is interesting that the derivative of the Archimedean spiral does not depend on the radius $$r$$, but is defined only by the angle $$\theta$$. This reflects the self-similarity property of the Archimedean spiral. This feature is also typical for many other plane curves.

Example 2.

Find the derivative $$\large\frac{{dy}}{{dx}}\normalsize$$ of the cardioid given by the equation $r = f\left( \theta \right) = a\left( {1 + \cos \theta } \right).$

Solution.

First we calculate the derivative of the polar function:

${f’\left( \theta \right) = {\left( {a\left( {1 + \cos \theta } \right)} \right)^\prime } } = { – a\sin \theta .}$

Then the derivative $$\large\frac{{dy}}{{dx}}\normalsize$$ of the curve is given by

${\frac{{dy}}{{dx}} = {y’_x} = \frac{{{y’_\theta }}}{{{x’_\theta }}} } = {\frac{{f’\left( \theta \right)\sin \theta + f\left( \theta \right)\cos\theta }}{{f’\left( \theta \right)\cos\theta – f\left( \theta \right)\sin\theta }} } = {\frac{{ – {{\sin }^2}\theta + \cos \theta + {{\cos }^2}\theta }}{{ – \sin \theta \cos \theta – \sin \theta – \sin \theta \cos \theta }}}.$

Using the double angle formulas

${\cos 2\theta = {\cos ^2}\theta – {\sin ^2}\theta ,\;\;}\kern-0.3pt {\sin 2\theta = 2\sin \theta \cos \theta ,}$

we get

$\frac{{dy}}{{dx}} = – \frac{{\cos 2\theta + \cos \theta }}{{\sin 2\theta + \sin \theta }}.$

We then transform the expression for the derivative using the trigonometric identities

${{\cos \alpha + \cos \beta }={ 2\cos \frac{{\alpha + \beta }}{2}\cos \frac{{\alpha – \beta }}{2},\;\;}}\kern-0.3pt {{\sin \alpha + \sin \beta }={ 2\sin \frac{{\alpha + \beta }}{2}\cos \frac{{\alpha – \beta }}{2}.}}$

As a result, we have

$\require{cancel} {\frac{{dy}}{{dx}} = – \frac{{\cos 2\theta + \cos \theta }}{{\sin 2\theta + \sin \theta }} } = { – \frac{{2\cos \frac{{2\theta + \theta }}{2}\cos \frac{{2\theta – \theta }}{2}}}{{2\sin \frac{{2\theta + \theta }}{2}\cos \frac{{2\theta – \theta }}{2}}} } = { – \frac{{\cos \frac{{3\theta }}{2}\cancel{\cos \frac{\theta }{2}}}}{{\sin \frac{{3\theta }}{2}\cancel{\cos \frac{\theta }{2}}}} } = { – \cot \frac{{3\theta }}{2}.}$

The derivative $$\large\frac{{dy}}{{dx}}\normalsize$$ is defined under conditions

${\left\{ \begin{array}{l} \cos\frac{\theta }{2} \ne 0\\ \sin \frac{{3\theta }}{2} \ne 0 \end{array} \right.,\;\;}\Rightarrow {\left\{ \begin{array}{l} \frac{\theta }{2} \ne \frac{\pi }{2} + \pi n\\ \frac{{3\theta }}{2} \ne \pi k \end{array} \right.,\;\;}\Rightarrow {\left\{ \begin{array}{l} \theta \ne \pi + 2\pi n,\;n \in \mathbb{Z}\\ \theta \ne \frac{2}{3}\pi k,\;k \in \mathbb{Z} \end{array} \right..}$

In the interval $$\left( {-\pi,\pi} \right)$$, these restrictions correspond to the values $$\theta = – \pi , – \frac{{2\pi }}{3},0,\frac{{2\pi }}{3},\pi .$$ The derivative of the cardioid does not exist at the indicated points.

The cardioid curve (Figure $$3$$) resembles the image of the heart (the name “cardioid” comes from the Greek word for “heart”) and has a number of remarkable properties.

Beautiful mathematical objects and structures of the same type often arise in various fields, which at first glance are not related to each other. Such examples make us think again about the amazing unity of the world and nature. So, a cardioid unexpectedly appears in the famous Mandelbrot fractal set, occupying its central part (Figure $$4$$).

Example 3.

Find the angle of intersection $$\color{darkgreen}{\alpha}$$ of two cardioids $${r_1} = 1 + \cos \theta$$ and $${r_2} = 1 – \cos \theta.$$

Solution.

First we determine the point of intersection of these curves:

${{r_1} = {r_2},}\;\; \Rightarrow {1 + \cos \theta = 1 – \cos \theta ,} \Rightarrow {2\cos \theta = 0,}\;\; \Rightarrow {\theta = \frac{\pi }{2}.}$

Compute the derivatives $$\large{\frac{{dy}}{{dx}}}\normalsize$$ of both curves at $${\theta = \large{\frac{\pi }{2}}}\normalsize.$$

For the $$1$$st cardioid $${r_1} = 1 + \cos \theta:$$

${{\left( {\frac{{dy}}{{dx}}} \right)_1} = \frac{{f^\prime\left( \theta \right)\sin \theta + f\left( \theta \right)\cos \theta }}{{f^\prime\left( \theta \right)\cos \theta – f\left( \theta \right)\sin \theta }} }={ \frac{{\left( { – \sin \theta } \right)\sin \theta + \left( {1 + \cos \theta } \right)\cos \theta }}{{\left( { – \sin \theta } \right)\cos \theta – \left( {1 + \cos \theta } \right)\sin \theta }} }={ \frac{{ – {{\sin }^2}\theta + \cos \theta + {{\cos }^2}\theta }}{{ – \sin \theta \cos \theta – \sin \theta – \sin \theta \cos \theta }} }={ – \frac{{\cos 2\theta + \cos \theta }}{{\sin 2\theta + \sin \theta }}.}$

When $${\theta = \large{\frac{\pi }{2}}\normalsize},$$ the derivative is equal to

${\frac{{dy}}{{dx}}{\left( {\theta = \frac{\pi }{2}} \right)_1} }={ – \frac{{\cos \pi + \cos \frac{\pi }{2}}}{{\sin \pi + \sin \frac{\pi }{2}}} }={ – \frac{{\left( { – 1} \right) + 0}}{{0 + 1}} }={ 1.}$

Similarly we find the derivative for the $$2$$nd curve $${r_2} = 1 – \cos \theta:$$

${{\left( {\frac{{dy}}{{dx}}} \right)_2} = \frac{{f^\prime\left( \theta \right)\sin \theta + f\left( \theta \right)\cos \theta }}{{f^\prime\left( \theta \right)\cos \theta – f\left( \theta \right)\sin \theta }} }={ \frac{{\sin \theta \sin \theta + \left( {1 – \cos \theta } \right)\cos \theta }}{{\sin \theta \cos \theta – \left( {1 – \cos \theta } \right)\sin \theta }} }={ \frac{{{{\sin }^2}\theta + \cos \theta – {{\cos }^2}\theta }}{{\sin \theta \cos \theta – \sin \theta + \sin \theta \cos \theta }} }={ – \frac{{\cos 2\theta – \cos \theta }}{{\sin 2\theta – \sin \theta }}.}$

Respectively when $${\theta = \large{\frac{\pi }{2}}\normalsize},$$ we have

${\frac{{dy}}{{dx}}{\left( {\theta = \frac{\pi }{2}} \right)_2} = – \frac{{\cos \pi – \cos \frac{\pi }{2}}}{{\sin \pi – \sin \frac{\pi }{2}}} }={ – \frac{{\left( { – 1} \right) – 0}}{{0 – 1}} }={ – 1.}$

As the slope $$k$$ of the tangent line is equal to the value of the derivative, we get $${k_1} = 1,$$ $${k_2} = -1.$$ Hence, the tangent of the angle $$\alpha$$ between the curves is given by

${\tan \alpha = \frac{{{k_2} – {k_1}}}{{1 + {k_1}{k_2}}} }={ \frac{{ – 1 – 1}}{{1 + 1 \cdot \left( { – 1} \right)}} }={ \infty .}$

This means that $${\alpha = \large{\frac{\pi }{2}}\normalsize}.$$

Example 4.

Find the derivative $$\large{\frac{{dy}}{{dx}}}\normalsize$$ of the polar function $$r = 1 + \sin \theta$$ at $$\theta = \large{\frac{\pi }{4}}\normalsize.$$

Solution.

We calculate the derivative $$\large{\frac{{dy}}{{dx}}}\normalsize$$ using the formula

$\frac{{dy}}{{dx}} = \frac{{f^\prime\left( \theta \right)\sin \theta + f\left( \theta \right)\cos \theta }}{{f^\prime\left( \theta \right)\cos \theta – f\left( \theta \right)\sin \theta }}.$

Given that $$f\left( \theta \right) = 1 + \sin \theta ,$$ we get

${f^\prime\left( \theta \right) = \left( {1 + \sin \theta } \right)^\prime }={ \cos \theta .}$

Hence

${\frac{{dy}}{{dx}} \text{ = }}\kern0pt{\frac{{\cos \theta \sin \theta + \left( {1 + \sin \theta } \right)\cos \theta }}{{\cos \theta \cos \theta – \left( {1 + \sin \theta } \right)\sin \theta }} }={ \frac{{\sin \theta \cos \theta + \cos \theta + \sin \theta \cos \theta }}{{{{\cos }^2}\theta – \sin \theta – {{\sin }^2}\theta }} }={ \frac{{2\sin \theta \cos \theta + \cos \theta }}{{{{\cos }^2}\theta – {{\sin }^2}\theta – \sin \theta }} }={ \frac{{\sin 2\theta + \cos \theta }}{{\cos 2\theta – \sin \theta }}.}$

At the point $$\theta = \large{\frac{\pi }{4}}\normalsize,$$ we have

$\require{cancel}{\frac{{dy}}{{dx}}\left( {\theta = \frac{\pi }{4}} \right) }={ \frac{{\sin \frac{\pi }{2} + \cos \frac{\pi }{4}}}{{\cos \frac{\pi }{2} – \sin \frac{\pi }{4}}} }={ \frac{{1 + \frac{{\sqrt 2 }}{2}}}{{0 – \frac{{\sqrt 2 }}{2}}} }={ \frac{{\frac{{2 + \sqrt 2 }}{2}}}{{ – \frac{{\sqrt 2 }}{2}}} }={ – \frac{{2 + \sqrt 2 }}{{\sqrt 2 }} }={ – \frac{{\left( {2 + \sqrt 2 } \right)\sqrt 2 }}{{\sqrt 2 \cdot \sqrt 2 }} }={ – \frac{{2\sqrt 2 + 2}}{2} }={ – \frac{{\cancel{2}\left( {\sqrt 2 + 1} \right)}}{\cancel{2}} }={ – \sqrt 2 – 1.}$

Example 5.

Find the derivative $$\large{\frac{{dy}}{{dx}}}\normalsize$$ of the polar function $$r = 3 + 2\cos \theta$$ at $$\theta = \large{\frac{\pi }{6}}\normalsize.$$

Solution.

The derivative $$\large{\frac{{dy}}{{dx}}}\normalsize$$ for a polar function $$r = f\left( \theta \right)$$ is expressed in the form

$\frac{{dy}}{{dx}} = \frac{{f^\prime\left( \theta \right)\sin \theta + f\left( \theta \right)\cos \theta }}{{f^\prime\left( \theta \right)\cos \theta – f\left( \theta \right)\sin \theta }}.$

As $$r = 3 + 2\cos \theta,$$ we have

${f^\prime\left( \theta \right) = \left( {3 + 2\cos \theta } \right)^\prime }={ – 2\sin \theta .}$

The derivative is given by

${\frac{{dy}}{{dx}} \text{ = }}\kern0pt{\frac{{\left( { – 2\sin \theta } \right)\sin \theta + \left( {3 + 2\cos \theta } \right)\cos \theta }}{{\left( { – 2\sin \theta } \right)\cos \theta + \left( {3 + 2\cos \theta } \right)\sin \theta }} }={ \frac{{2\left( {{{\cos }^2}\theta – {{\sin }^2}\theta } \right) + 3\cos \theta }}{{ – 4\sin \theta \cos \theta – 3\sin \theta }} }={ – \frac{{2\cos \left( {2\theta } \right) + 3\cos \theta }}{{2\sin \left( {2\theta } \right) + 3\sin \theta }}.}$

Substitute $$\theta = \large{\frac{\pi }{6}}\normalsize:$$

${\frac{{dy}}{{dx}}\left( {\theta = \frac{\pi }{6}} \right) = – \frac{{2\cos \frac{\pi }{3} + 3\cos \frac{\pi }{6}}}{{2\sin \frac{\pi }{3} + 3\sin \frac{\pi }{6}}} }={ – \frac{{2 \cdot \frac{1}{2} + 3 \cdot \frac{{\sqrt 3 }}{2}}}{{2 \cdot \frac{{\sqrt 3 }}{2} + 3 \cdot \frac{1}{2}}} }={ – \frac{{3\sqrt 3 + 2}}{{2\sqrt 3 + 3}}.}$

Page 1
Problems 1-5
Page 2
Problems 6-17


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#### 11th Standard Maths Integral Calculus English Medium Free Online Test One Mark Questions 2020 - 2021

11th Standard

Reg.No. :
•
•
•
•
•
•

Maths

Time : 00:10:00 Hrs
Total Marks : 10

10 x 1 = 10
1. If $\int {3^{1\over x}\over x^2}dx=k(3^{1\over x})+c$ ,then the value of k is

(a)

log 3

(b)

-log 3

(c)

$-{1\over log3}$

(d)

${1\over log3}$

2. $\int {e^{6logx}-e^{5logx}\over e^{4logx}-e^{3logx}}dx$ is

(a)

x+c

(b)

${x^3\over 3}+c$

(c)

${3\over x^3}+c$

(d)

${1\over x^2}+c$

3. $\int {sin^8x-cos^8x\over 1-2sin^2 \ x \ cos^2 \ x}dx$ is

(a)

${1\over2}sin 2x+c$

(b)

$-{1\over2}sin 2x+c$

(c)

${1\over2}cos 2x+c$

(d)

$-{1\over2}cos 2x+c$

4. $\int{x^2+cos^2x\over x^2+1}cosec^2xdx$ is

(a)

cot x+sin -1x+c

(b)

-cot x+tan-1x+c

(c)

-tan x+cot-1x+c

(d)

-cot x-tan-1x+c

5. $\int e^{-4x}cos \ x \ d x$ is

(a)

${e^{-4x}\over 17}[4cos \ x-sin \ x]+c$

(b)

${e^{-4x}\over 17}[-4cos \ x+sin \ x]+c$

(c)

${e^{-4x}\over 17}[4cos \ x+sin \ x]+c$

(d)

${e^{-4x}\over 17}[-4cos \ x-sin \ x]+c$

6. $\int {x+2\over \sqrt{x^2+1}}dx$is

(a)

$\sqrt{x^2-1}-2 log|x+\sqrt{x^2-1}|+c$

(b)

$sin^{-1}x-2 log|x+\sqrt{x^2-1}|+c$

(c)

$2 log|x+\sqrt{x^2-1}|-sin^{-1}x+c$

(d)

$\sqrt{x^2-1}+2log|x+\sqrt{x^2-1}|+c$

7. $\int sin \sqrt{x}$ dx is

(a)

$2(-\sqrt{x}cos\sqrt{x}+sin\sqrt{x})+c$

(b)

$2(-\sqrt{x}cos\sqrt{x}-sin\sqrt{x})+c$

(c)

$2(-\sqrt{x}sin\sqrt{x}-cos\sqrt{x})+c$

(d)

$2(-\sqrt{x}sin\sqrt{x}+cos\sqrt{x})+c$

8. $\int { \frac { \left( 1+logx \right) ^{ 2 } }{ x } }$ dx = _______+C.

(a)

$\frac { \left( 1+logx \right) ^{ 3 } }{ 3 }$

(b)

3 log ( 1 + log x)

(c)

2(1 + log x)

(d)

none of these

9. $\int { \frac { 1 }{ 9x^{ 2 }-4 } }$ dx = ____________+c.

(a)

log$\left| \frac { 3x-2 }{ 3x+2 } \right|$

(b)

$\frac { 1 }{ 12 }$log$\left| \frac { 3x-2 }{ 3x+2 } \right|$

(c)

12log$\left| \frac { 3x-2 }{ 3x+2 } \right|$

(d)

$\frac { 1 }{ 12 } log\left| \frac { 3x+2 }{ 3x-2 } \right|$

10. $\int { \frac { x }{ 4+{ x }^{ 4 } } }$ dx is equal to________+c.

(a)

$\frac { 1 }{ 4 }$ tan-1 (x2)

(b)

$\frac { 1 }{ 4 }$ tan-1 $\left( \frac { { x }^{ 2 } }{ 2 } \right)$

(c)

$\frac { 1 }{ 2 }$ tan-1 $\left( \frac { { x }^{ 2 } }{ 2 } \right)$

(d)

none of these


text:
location:  Publications → journals
Search results

Search: All articles in the CMB digital archive with keyword classical

 Expand all        Collapse all Results 1 - 3 of 3

1. CMB 2011 (vol 56 pp. 3)

Aïssiou, Tayeb
 Semiclassical Limits of Eigenfunctions on Flat $n$-Dimensional Tori We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an $n$-dimensional flat torus $\mathbb T^{n}$, and the Fourier transform of squares of the eigenfunctions $|\varphi_\lambda|^2$ of the Laplacian have uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an $n$-dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in the proof. Keywords:semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limitsCategories:58G25, 81Q50, 35P20, 42B05

2. CMB 2011 (vol 55 pp. 736)

Hernández, Eduardo; O'Regan, Donal
 Existence of Solutions for Abstract Non-Autonomous Neutral Differential Equations In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous neutral functional differential equations. An application to partial neutral differential equations is considered. Keywords:neutral equations, mild solutions, classical solutionsCategories:35R10, 34K40, 34K30

3. CMB 2006 (vol 49 pp. 82)

Gogatishvili, Amiran; Pick, Luboš
 Embeddings and Duality Theorem for Weak Classical Lorentz Spaces We characterize the weight functions $u,v,w$ on $(0,\infty)$ such that $$\left(\int_0^\infty f^{*}(t)^ qw(t)\,dt\right)^{1/q} \leq C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t),$$ where $$f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{-1} \int_{0}^{t}f^*(s)u(s)\,ds.$$ As an application we present a~new simple characterization of the associate space to the space $\Gamma^ \infty(v)$, determined by the norm $$\|f\|_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t),$$ where $$f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds.$$ Keywords:Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequalityCategories:26D10, 46E20


text:
# [SOLVED]function composition

#### dwsmith

##### Well-known member
Given three functions $F,G,H$ what restrictions must be placed on their domains so that the following four composite functions can be defined?
$$G\circ F,\quad H\circ G,\quad H\circ (G\circ F),\quad (H\circ G)\circ F.$$

I need a hint or something.

#### Fantini

MHB Math Helper
I'd say that in each case you'd need $\textrm{Im }F \subset D_G, \text{ Im }G \subset D_H, \text{ Im }G \circ F \subset D_H$ and $\text{Im }F \subset D_{H \circ G}$.

Taking as example your other thread, we need $\text{Im }F \subset D_G$, where $F(x) = x+5$ and

$$G(x) = \begin{cases} \frac{|x|}{x}, & \text{if } x \neq 0 \\ 1, & \text{if } x= 0. \end{cases}$$

For $\text{Im }F$ to be contained in the domain of $G$, we need to investigate the cases where $F(x) \neq 0$ and $F(x)=0$. In particular, here we have $\text{Im }F = \mathbb{R}$ and $D_G = \mathbb{R}$ since it is defined everywhere (although it isn't continuous).

#### Sudharaka

##### Well-known member
MHB Math Helper
Given three functions $F,G,H$ what restrictions must be placed on their domains so that the following four composite functions can be defined?
$$G\circ F,\quad H\circ G,\quad H\circ (G\circ F),\quad (H\circ G)\circ F.$$

I need a hint or something.
Hi dwsmith,

If $$G\circ F$$ is to be defined properly the co-domain of $$F$$ should be a subset of the domain of $$G$$. Therefore the only restrictions in defining the above mentioned compositions are,

$\mbox{codom }(F)\subseteq\mbox{dom }(G)\mbox{ and }\mbox{codom }(G)\subseteq\mbox{dom }(H)$

Kind Regards,
Sudharaka.


text:
## File: prediction-class.Rd

package info (click to toggle)
r-cran-rocr 1.0-11-2
• links: PTS, VCS
• area: main
• in suites: bullseye, sid
• size: 676 kB
• sloc: sh: 13; makefile: 2
 file content (71 lines) | stat: -rw-r--r-- 2,380 bytes parent folder | download
 1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071 % Generated by roxygen2: do not edit by hand % Please edit documentation in R/zzz.R \docType{class} \name{prediction-class} \alias{prediction-class} \title{Class \code{prediction}} \description{ Object to encapsulate numerical predictions together with the corresponding true class labels, optionally collecting predictions and labels for several cross-validation or bootstrapping runs. } \section{Slots}{ \describe{ \item{\code{predictions}}{A list, in which each element is a vector of predictions (the list has length > 1 for x-validation data.} \item{\code{labels}}{Analogously, a list in which each element is a vector of true class labels.} \item{\code{cutoffs}}{A list in which each element is a vector of all necessary cutoffs. Each cutoff vector consists of the predicted scores (duplicates removed), in descending order.} \item{\code{fp}}{A list in which each element is a vector of the number (not the rate!) of false positives induced by the cutoffs given in the corresponding 'cutoffs' list entry.} \item{\code{tp}}{As fp, but for true positives.} \item{\code{tn}}{As fp, but for true negatives.} \item{\code{fn}}{As fp, but for false negatives.} \item{\code{n.pos}}{A list in which each element contains the number of positive samples in the given x-validation run.} \item{\code{n.neg}}{As n.pos, but for negative samples.} \item{\code{n.pos.pred}}{A list in which each element is a vector of the number of samples predicted as positive at the cutoffs given in the corresponding 'cutoffs' entry.} \item{\code{n.neg.pred}}{As n.pos.pred, but for negatively predicted samples.} }} \note{ Every \code{prediction} object contains information about the 2x2 contingency table consisting of tp,tn,fp, and fn, along with the marginal sums n.pos,n.neg,n.pos.pred,n.neg.pred, because these form the basis for many derived performance measures. } \section{Objects from the Class}{ Objects can be created by using the \code{prediction} function. } \references{ A detailed list of references can be found on the ROCR homepage at \url{http://rocr.bioinf.mpi-sb.mpg.de}. } \seealso{ \code{\link{prediction}}, \code{\link{performance}}, \code{\link{performance-class}}, \code{\link{plot.performance}} } \author{ Tobias Sing \email{tobias.sing@gmail.com}, Oliver Sander \email{osander@gmail.com} }


text:
## Category Archives: diffraction

### Paraxiality

The previous post on the Gaussian beam noted that this beam is a solution to the paraxial wave equation and thus only accurate for small divergence angles. The limit given in [1] was $w_0 \gt 2\lambda/\pi$ or $\theta \lt 0.5\mathrm{rad}$ or $\approx 29^\circ$. The Gaussian beam model is of course not exact up to that […]

### Scalar Diffraction – Fourier

In an earlier post we had a look at various diffraction formalisms that either originated in the Huygens-Fresnel principle or led to essentially the same results. The principle modeled a field distribution in a aperture (or on a surface) as a source of infinitely many spherical waves whose amplitudes and phases were prescribed by the […]

### Scalar Diffraction – Imagery

In this post on the principles of scalar diffraction there were quite a number of integral expressions for diffraction of light, starting from the Huygens-Fresnel principle and applying various degrees of approximation. Integral formulations are usually very abstract and not very illustrative. However, with today’s available computer power we can turn these integrals into…

### Scalar Diffraction – Huygens, Fresnel, Fraunhofer

One can work years in fiber optic communications and not really have to care about diffraction, but when working with free-space light propagation and/or illuminated surfaces it’s better to know the difference between your Fresnel and your Fraunhofer diffraction, and what the heck the Huygens-Fresnel principle is all about.


text:
# Conditional independence regarding fourth event

Let's two events $S1$ and $S2$ are conditionally independent given the event $A$, i.e.,

$P(S_1|S_2,A) = P(S_1|A)$ and $P(S_2|S_1,A) = P(S_2|A)$

If $B$ is an arbitrary event, does the following probability hold?

$P(S_1|S_2,A,B) = P(S_1|A,B)$?

-

No. We can ignore $A$-let it be certainly true. Let $S_1$ be a coin flip is heads and $S_2$ be a die roll is even. Now let $B$ be coin flip heads iff die roll is even. Then $P(S_1|S_2,B)=1, P(S_1|B)=\frac 12$


text:
getenv - Maple Help

# Online Help

###### All Products    Maple    MapleSim

getenv

fetch the contents of a system environment variable

 Calling Sequence getenv(name) getenv(regentry,registry) (Windows only)

Parameters

 name - the name of the environment variable to fetch regentry - a list of three strings, the registry key, the value path and value name

Description

 • Looks up the specified system environment variable and returns its value as a Maple string. If the specified variable is not defined, getenv will return NULL.
 • A system environment variable is one that is set in the shell from which Maple was invoked. For example, under the UNIX C Shell, such variables are assigned using the setenv command. Under MS-DOS, such variables are assigned using the set command.
 • The set of available environment variables, and their meanings, is operating system dependent. Maple procedures or worksheets that make use of the getenv function will not likely be portable between versions of Maple for different operating systems.
 Note: Issuing a shell command (via Maple's system function) to set the value of an environment variable will not be reflected in subsequent calls to getenv. This is because the system function spawns a sub-shell, and changes to a sub-shell's environment will not affect the shell from which Maple was invoked.
 • In the Windows versions of Maple, getenv can also be used to query values from the Windows registry.  In this case, the first argument is a list of three strings, the registry key ("HKEY_LOCAL_MACHINE", "HKEY_CURRENT_USER", etc ), the value's path and the value's name and the second argument is the name registry.

Examples

 > $\mathrm{getenv}\left("HOME"\right)$
 ${"/u/joeuser"}$ (1)

This is only valid on a Windows system

 > $\mathrm{getenv}\left(\left["HKEY_LOCAL_MACHINE","Software\\Microsoft\\Windows NT\\CurrentVersion\\WinLogon","DefaultUserName"\right],\mathrm{registry}\right)$
 ${"joeuser"}$ (2)

 See Also