text:
# trigonometry

• July 31st 2009, 11:17 PM
Rose Wanjohi
trigonometry
it is given that y=sin^-1,that is, y=arc sinx. by expressing x as a function of y, determine dx/dy and hence determine dy/dx as a function of x only(Cool)
• July 31st 2009, 11:49 PM
Chris L T521
Quote:

Originally Posted by Rose Wanjohi
it is given that y=sin^-1,that is, y=arc sinx. by expressing x as a function of y, determine dx/dy and hence determine dy/dx as a function of x only(Cool)

Note that $y=\arcsin x\implies\sin y =x$

Now, differentiate w.r.t. y to get $\cos y=\frac{\,dx}{\,dy}$.

From here, it's evident that $\frac{\,dy}{\,dx}=\frac{1}{\cos y}$

I leave it for you to show that if $\sin y=x$, then $\cos y = \sqrt{1-x^2}$.

Thus, it follows that if $y=\arcsin x$, then $\frac{\,dy}{\,dx}=\frac{1}{\cos y}=\frac{1}{\sqrt{1-x^2}}$

Does this make sense?
• July 31st 2009, 11:53 PM
alexmahone
Quote:

Originally Posted by Rose Wanjohi
it is given that y=sin^-1,that is, y=arc sinx. by expressing x as a function of y, determine dx/dy and hence determine dy/dx as a function of x only(Cool)

$y=arcsin x$

$x=sin y$ -------- (1)

$\frac{dx}{dy}=cos y$

$\frac{dy}{dx}=\frac{1}{cos y}$

Using (1), $cos^2y=1-x^2$

$\frac{dy}{dx}=\frac{1}{\sqrt{1-x^2}}$
• August 1st 2009, 01:11 AM
tom@ballooncalculus
Just in case a picture helps...

http://www.ballooncalculus.org/asy/t...arcsinDiff.png

This isn't quite to express x as a function of y - I'll put a balloon version of that further down the post (later)... but this here is the picture I find best of all for this problem. .

As usual, straight continuous lines differentiate downwards (integrate up) with respect to the explicit variable (x or theta) and the straight dashed line similarly but with respect to the dashed balloon expression.

For enlightenment, work from the top left corner - the given expression arcsin x. Then consider arcsin as the outer function of a chain rule differentiation...

http://www.ballooncalculus.org/asy/chain.png

If the inner function is sin, then the derivative with respect to sin will be multiplied by the derivative of sin...

http://www.ballooncalculus.org/asy/t...rcsinDiff1.png

But then arcsin(sin(theta)) is of course theta, and the derivative of that is 1, so the whole of the bottom of the chain-rule shape wants to come to 1, so we know it must be...

http://www.ballooncalculus.org/asy/t...rcsinDiff2.png

Which we just re-write using Pythag and then - swapping sin theta back for x - we finish at the bottom left corner...

_______________________________

And if you do need to express x as a function of y...

http://www.ballooncalculus.org/asy/dXdY/arcsinDiff.png

... and proceed as per the other 2 posts.

Don't integrate - balloontegrate!

Balloon Calculus Forum


text:
## Encyclopedia > Ideal gas law

Article Content

# Gas laws

Redirected from Ideal gas law

The gas laws include Boyle's Law, Charles' Law and Graham's Law and describe the relationship between temperature, pressure and volume of gases. They are collectively generalized by the universal gas equation.

A gas which obeys gas laws exactly is hypothetical, and is known as an ideal gas (or perfect gas).

Boyle's Law, named afer Robert Boyle, states that the volume occupied by a gas at constant temperature is inversely proportional to the pressure applied.

Charles Law, named after Jacques Charles, states that the pressure that a gas exerts on the walls of its container is determined by the momentum of the atoms and molecules of the gas, which in turn is determined by the temperature. As the temperature increases the atoms and molecules move faster, and so exert a greater pressure on the walls. If the walls are rigid, such that the volume of the container is held constant, then the relationship between pressure P and temperature T is given by Charles' Law:

$P = \mathrm{constant} \times T$

Graham's Law, named after Thomas Graham, states that the kinetic energy of two samples of different gases at the same temperature is identical.

merge needed:

### Ideal Gas Law

$PV = nRT$

which states that pressure P multiplied by the volume V is equal to the number of moles of the gas n, multiplied by the temperature T and a constant R. In SI units, the pressure is expressed in pascals, the volume in litres, and the temperature in Kelvin.

The ideal gas law incorporates both Charles Law and Boyle's Law.

format: plain


text:
# WEICOM - Editorial

#1

Author: Trung Nguyen

Tester: Oleksandr Kulkov

Editorialist: Oleksandr Kulkov

MEDIUM

None

### PROBLEM:

n players participated in tournaments. Each player compete with each other once. Winner earns 1 point, loser earns 0 points. After the tournament player that won g_i games is awarded by g_i^2 money. You have to check if it is possible that overall earns of players equals k.

### QUICK EXPLANATION:

Construct answer with the minimum total score and then iteratively increase it till you get k.

### EXPLANATION:

Let’s think of what score at all can we gain? First of all we should note that parity of sum of squares is the same as parity of the sum since x^2 =x \mod 2. Also sum of victories for all players is fixed and always equals to \dfrac{n(n-1)}{2}. Thus gain’s parity should be the same as of this value.

Let’s think of what’s the maximum gain we can earn. If there is match in which player with score a won against player with score b such that b\geq a then if we change the result of match we will gain (b+1)^2+(a-1)^2-b^2-a^2=2(b-a+1)>0. Thus for maximum gain player with score won match against every player with lower score and there are no two players with equal score. This situation is equivalent to one when player k won matches against all players with number less than k and gain k-1 score. Thus maximum total gain is \sum\limits_{k=0}^{n-1} k^2. Note that from that equality yields that flipping the result of game between players with same score increases the answer by 2.

And what about minimum gain? We can see that if there are two numbers with b-a\geq 2 then we can lower the score by decreasing b and increasing a. Thus we can set the lower bound as the situation in which every player scored either \left\lfloor\dfrac n 2\right\rfloor or \left\lfloor\dfrac {n-1}{2}\right\rfloor. Luckily this situation can be obtained.

Consider the following algorithm which builds the matrix a_{ij} with minimum gain for odd n. Let player i win the battle against all players (i-k) \mod n where k is from 1 to \dfrac{n+1}{2} and lose to the others. As you can see i won against j iff j didn’t win against i thus matrix is well defined.

\begin{pmatrix}0\end{pmatrix} o \begin{pmatrix}0 & 0 & 1 \\ 1 & 0 & 0\\ 0 & 1 & 0 \end{pmatrix} o \begin{pmatrix} 0 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 \end{pmatrix} o \begin{pmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 \end{pmatrix}

It turns out that to get matrix for even n it’s enough to construct it for n+1 and remove (n+1)^{th} player.

As was mentioned earlier, if we flip the result of game for players who scored same number of games, the total gain will be increased by 2. Keeping doing this optimal way we can get to the answer and maybe fit TL. However it still will be O(n^3) with small constant. To achieve O(n^2) complexity we should do the following thing. While it doesn’t make our gain more than k, choose a player which have maximum score now and make him win all games. Now subtract (n-1)^2 from k and consider only other (n-1) players. Otherwise k is not greater than (n-1)^2 which was obtained in at most O(n) steps and now we can in a fair way do the algorithm which chooses two players with same score and flip their game, it will need only O(n^2) iterations to finish.

### AUTHOR’S AND TESTER’S SOLUTIONS:

Author’s solution can be found here.

Tester’s solution can be found here.

### RELATED PROBLEMS:

#2

Authors’s and Testers’s solutions aren’t working

#3

“Otherwise kk is not greater than (n−1)2(n−1)2 which was obtained in at most O(n)O(n) steps and now we can in a fair way do the algorithm which chooses two players with same score and flip their game, it will need only O(n2)O(n2) iterations to finish.”

can u please explain a bit why it is O(n) now ?

#4

@admin takes wrong solution :). Correct one: https://www.codechef.com/viewsolution/16014971 .

#5

Hello Guys I am stuck with this problem … Can anyone explain it in more simple ways ??
I understood that maximum sum can be found by Σn^2=n*(n+1)(2n+1)/2 . Now put k-1 in position of n ,we get (k-1)(k)(2*k-1)/2 . I could not understand minimum part . Please someone help !!

#6

I want to ask a question. This question is not related to the editorial. Please give me some karma points.
By the way, I don’t like that we should have some karma points to post questions

#7

Hello Guys I am stuck with this problem … Can anyone explain it in more simple ways ?? I understood that maximum sum can be found by Σn^2=n(n+1)(2n+1)/2 . Now put k-1 in position of n ,we get (k-1)(k)(2k-1)/2 . I could not understand minimum part . Please someone help !!


text:
Complex Analysis Problem #1

1. Sep 21, 2007

BSMSMSTMSPHD

1. The problem statement, all variables and given/known data

Let $$f(z) = \frac{1-iz}{1+iz}$$ and let $$\mathbb{D} = \{z : |z| < 1 \}$$.

Prove that $$f$$ is a one-to-one function and $$f(\mathbb{D}) = \{w : Re(w) > 0 \}$$.

2. The attempt at a solution

I've already shown the first part: Assume $$f(z_1) = f(z_2)$$ for some $$z_1, z_2 \in \mathbb{C}$$, then $$z_1 = z_2$$. (I worked this out).

But for the second part, I'm not sure what to do. I've written the function in rectangular coordinates $$(z = x + iy)$$ and the real part of the simplified fraction is:

$$\frac{1 - (x^2 + y^2)}{1 - 2y + x^2 + y^2}$$.

Now, I know that the numerator is nonnegative (since $$z \in \mathbb{D}, |z| < 1$$, so, $$x^2 + y^2 < 1)$$. But, I am not certain about the sign of the denominator in the case where $$y > 0$$. Any ideas? And, if I can show this, will I have finished the proof, or do I have to show reverse inclusion?

2. Sep 21, 2007

Dick

This may be complex analysis, but do you still remember (y-1)^2=y^2-2y+1?

3. Sep 21, 2007

BSMSMSTMSPHD

Man, it's always something really obvious. Thanks.


text:
Thread: Low-pops primorial multiple View Single Post
2020-07-29, 03:12   #13
Citrix

Jun 2003

62C16 Posts

Quote:
 Originally Posted by preda Hi, I have this little question: given a bound N, taking the primorial N# (the product of all the primes up to N), we count the number of 1-bits in the binary representation of N#: pops(N#) I would expect the pops to be around 1/2 * log2(N#), coming from a roughly equal chances of bits being 0/1 in the binary representation of N#. I would like to find a low-pops multiple of that primorial. i.e.: find X such that pops(N# * X) is [a lot] smaller then pops(N#). Let's define the gain in bits: gain(X)=pops(N#) - pops(N# * X) How much "gain" can I expect, and how to find a X multiplier that produces a good gain? What if, additionally, I bound the magnitude of X, let's say by log2(X) <= 10 * log2(N#) -- does this restriction change anything? My intuition says that this is a "hopeless" problem, meaning that it's extremely hard to find an X that produces a gain of some significant ratio of log2(N#), e.g. a gain of 1/4*log2(N#). Is there some mathematical argument in this area? thanks! PS: maybe the problem can be generalized to any number M (not only a primorial), becoming: given a number M, find a low-pops multiple of it.
I am not sure if I understand your question correctly.
From what I understand you are looking for X*N# where the number of '1' bits are very low.

Another way of phrasing this problem would be:
Find 2^x such that 2^x==y (mod N#) and y is small and negative
This becomes a discrete log problem
With N# consisting of all the small primes this can be further be divided for each prime under N. This can be computed extremely fast.
For a random number M similar method can be used assuming p-1 of each factor is smooth to some extent.

If you want to keep x small then we can try solving 2^x+2^z==y (mod N#) and so on...


text:
## anonymous one year ago PLZ HELP Find the area of sector POM of circle O. Give your answer in terms of pi.

1. anonymous

|dw:1439517703911:dw|

2. anonymous

im not good at angles

3. anonymous

lol ok

4. anonymous

We know that the area of a circle with radius r and angle theta is $\Large \frac{ \pi r ^{2} \theta }{ 360 }$ replacing with theta=54 degrees and r=4 we get $\Large \frac{ \pi \times4^{2} \times54 }{ 360 }=\frac{ 12 }{ 5 }\pi$

5. anonymous

thanks so much!!


text:
# Minimal polynomials and invertibility

1. Mar 1, 2006

Let $$T\in L(V)$$. Let $$g(x)\in F[x]$$ and let $$m(x)$$ be the minimal polynomial of $$T$$. Show that $$g(T)$$ is invertible $$\Leftrightarrow$$ $$\gcd (m(x),g(x))=1$$.

Backwards is easy. For forwards, suppose I say that $$g(T)$$ is invertible implies that $$g(T)(v)=0 \Rightarrow v=0$$ and therefore $$g(x)$$ prime, therefore it is not divisible, and therefore $$\gcd (g,m)=1$$. Is that correct?

Why is latex not showing up?

Last edited: Mar 1, 2006
2. Mar 1, 2006

### topsquark

They had some sort of server disaster yesterday, I take it.

-Dan


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# Illustrator script - create objects, rotated and centred with a selected object

Edit:

Oops, it's working!
It seems that the problem has been solved by loading the updated script via File / Script / Other File... option. Before the stroke width change wasn't affecting the result. I guess it was some caching. I'll post the answer anyway, maybe it will be useful for someone.

Original question:

The purpose of the following Illustrator JavaScript is to create a rotated rectangle that spans the selected line and place these object in the middle of it:

1. a small circle
2. a vertical line rotated the same way as the rectangle
3. a vertical line rotated the same way as the rectangle

I'm doing it by rotating the selected line with a reversed angle, creating objects and rotate everything back.

The problem is that the objects are not centered, as you can see below. It seems that the degree of misalignmet depends on a with of the selected line, but even with a zero width there is a problem. I tried to apply a zero width stroke to each object but it doesn't seem to work. They have some predefined width that I don't know how to change. Please help.

#target Illustrator

// Define the new color 1
var newRGBColor = new RGBColor();
newRGBColor.red = 255;
newRGBColor.green = 0;
newRGBColor.blue = 0;
app.activeDocument.defaultFillColor = newRGBColor;

// Define the new color 2
var newRGBColor_Green = new RGBColor();
newRGBColor_Green.red = 0;
newRGBColor_Green.green = 255;
newRGBColor_Green.blue = 0;

// Define the new color 3
var newRGBColor_Black = new RGBColor();
newRGBColor_Black.red = 0;
newRGBColor_Black.green = 0;
newRGBColor_Black.blue = 0;

var idoc = app.activeDocument;
// This doesn't seem to work
idoc.defaultStrokeWidth = 55;

// My desired angle
var ang = 11.947;

// Stroke withs and element sizes.
// 1 point = 1p = 1/2660 m = 0.37594 mm
var zeroStrokeWidth = 0.0001;

var mySelection = new Object;

// Get active selection
selections = idoc.selection;

for(i=0;i<selections.length;i++)
{
mySelection = idoc.selection[i];

mySelection.rotate(-ang);

var newRectWidth = mySelection.width; // width is Geometric width, so we need to make it smaller...to accomodate the visible portion.
var newRectHeight = mySelection.height;
var newRectTop = mySelection.top; // Top is actually Visible top
var newRectLeft = mySelection.left; // dito for Left

mySelection.rotate(ang);
//mySelection.remove();container.pathItems. (top, left, width, height, undefined, true);

// Create a rectangle
var rect1Full = idoc.pathItems.rectangle(newRectTop, newRectLeft, newRectWidth, newRectHeight);
rect1Full.strokeWidth = zeroStrokeWidth;
rect1Full.strokeColor = newRGBColor; // defined above
rect1Full.fillColor = new NoColor();
rect1Full.rotate(ang);

// Create a vertical line (by reducing rectangle's width)
var rect2Vert = idoc.pathItems.rectangle(newRectTop, newRectLeft, newRectWidth, newRectHeight);
rect2Vert.strokeWidth = zeroStrokeWidth;
rect2Vert.width=0.0;
rect2Vert.left=rect1Full.left+rect1Full.width/2.0;
rect2Vert.strokeColor = newRGBColor; // defined above
rect2Vert.fillColor = new NoColor();
rect2Vert.rotate(ang);

// Create a horizontal line (by reducing rectangle's height)
var rect3Horiz = idoc.pathItems.rectangle(newRectTop, newRectLeft, newRectWidth, newRectHeight);
rect3Horiz.strokeWidth = zeroStrokeWidth;
rect3Horiz.height=0.0;
rect3Horiz.top=rect1Full.top-rect1Full.height/2.0;
rect3Horiz.strokeColor = newRGBColor; // defined above
rect3Horiz.fillColor = new NoColor();
rect3Horiz.rotate(ang);

// Create an ellipse
var ellip1 = idoc.pathItems.ellipse(rect1Full.top-rect1Full.height/2 +ellRadius, rect1Full.left+rect1Full.width/2.0 - ellRadius, ellDiameter, ellDiameter, undefined, true);
ellip1.strokeWidth = zeroStrokeWidth;
ellip1.strokeColor = newRGBColor_Black; // defined above
ellip1.fillColor = newRGBColor_Green;
ellip1.strokeWidth = ellStrokeWidth;

}

• @CAI Oops, I'm sorry, I've just done it. – ellockie Apr 5 '16 at 11:55

// This doesn't seem to work

Before I put this huge number to check if it makes any difference, but there wasn't any. Later, after loading the script via "File / Script / Other File..." option the defaultStrokeWidth suddenly the new stroke started to be visible. And that seemed to be the root of the original problem. Like the visual boundaries are taken into consideration by Illustrator with the operations used, not path points positions.


text:
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in the society. Two possible ways of measuring well
being based on utilities are:

finalized_jax”>$\(+\sum_{1}^{N}U_{i}\)$
,where i = 1,2…n

2. Usociety = min(u1, u2…un)

The first measure adds up individual utilities (ui)
across the entire society. The other

measure uses a minimum function to do the same.
Interpret these two measures.

Criticise both these measures using the capabilities
approach.

Hello, Welcome to our WhatsApp support. Reply to this message to start a chat.


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# What type of trading is most profitable? – Swing Trading For Dummies

I don’t see the question that way. The question that emerges from the above is how much of each type of trading is profitable? To quantify this we can use the following two-sided test.

First, suppose that in the time-step we chose our trade execution strategy, only 90% of all positions have produced profits. Given this outcome is a reasonable probability given expected returns, what is the expected number of profit-producing orders? In other words, can that $1.5$2.5 $4$6 $8$10 $12$1 $10^2 =$1.5 $2.5$4 $6$8 $10$12 $1 \cdot (1/2)*10^{2/2}$ orders be executed over time to produce such a “fair” estimate of the expected number of profits produced by a trader?

Let’s see how this test works. Suppose we are running a trading strategy with the following parameters:[2]

D = 10^2 D = 10^2 D = 10^2 D = 10^2

Given the two-sided difference between expected returns and the expected number of orders with a trade execution strategy, we find that D = 0.45. If we use the model of the stock market, D should come out to about $0.0075. Here, when D = 0.0075 it means the actual expected return on the position is less than what we would predict from the price at which that stock traded in the past. Thus, we can conclude that the stock is overpriced relative to the price where it traded. Now suppose we are running the stock market model with the parameters: D = 10 D = 10 D = 10 D = 10 D = 10 The model gives reasonable results for D =$0.25 given the market price of the stock. Given this, our expected number of profits should be relatively large. But if we do not take the stock price out of the picture, this still means that the number of profits should be relatively small.

How do you get the stock to trade at a price far above its “market price”? How do you change the price? The answer is that the market price for the stock will vary over time. This means if you wish to increase the stock price, you need to adjust the position parameters. For instance, the stock price could change by a factor of (10^4^

swing-trade-stocks.com review, bullish swing setup, swing trade scanner settings, swing trade stocks and forex with the ichimoku cloud by udemy, udemy day trading for beginners


text:
# Set of discontinuities

I know that the functions $f$ and $g$ are Riemann-integrable on some closed set. Also, $f$ has a min ($m>0$) on the closed set.
Let $h(x) = f(x)^{g(x)}$

I want to prove that $h$ is Riemann-integrable.
I've been trying to use the Lebesgue's Theorem but I'm not sure how to prove the set of discontinuities of $h$ is measure zero without assuming it is Riemann-integrable.

I've tried breaking up the integral of $f$ into it's continuous sections, then the discontinuities of $f^g$ for each section is equal to the discontinuities of $g$. Hence, we have a countable union of measure zero sets, which is measure zero. But, am I allowed to assume the integral of $f^g$ exists for each section?

-
I assume when you say "$f$ has a min ($m>0$)" we mean $f(x)>0$ on this set. Let $E_g$ and $E_f$ be the set of all points of discontinuity for $g$, $f$ respectively. (which have Lebesgue measure 0) Then let $$U=E_f^c\cap E_g^c.$$ Both $f,g$ must be continuous on $U$. Since $x^y$ is a continuous function from $(0,\infty)\times\mathbb{R}\rightarrow \mathbb{R}$ we see $f(x)^{g(x)}$ will be continuous on $U$ as the composition of continuous functions is continuous. (Note: For the range/domains to work out, we need $f(x)>0$). Now, as $m (U^c)=0$, that is the Lebesgue measure of $U^c$ is 0, we conclude $f(x)^{g(x)}$ is Riemannn integrable as desired. (The set of discontinuities is a subset of $U^c$)


text:
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 作者：Maryam Khordad , Robert E. Mercer 来源：[J].Journal of Biomedical Semantics, 2017, Vol.8 (1) 摘要：One important type of information contained in biomedical research literature is the newly discovered relationships between phenotypes and genotypes. Because of the large quantity of literature, a reliable automatic system to identify this information for future curation is essen...
 作者：Lijuan Wang , Zhihao Wu , Mengxia Liu ... 来源：[J].Thalassas: An International Journal of Marine Sciences(IF 0.368), 2017, Vol.33 (1), pp.57-63 摘要：In this study, we analyzed the length-weight relationships (LWRs), total length (TL)—standard length (SL) relationships (LLRs), and Fulton’s condition factor ( K ) of male and female black rockfish Sebastes schlegelii Hilgendorf, 1880 in Lidao Bay, Yellow Sea, China. Among ...
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 作者：Marike Palmer , Emma T. Steenkamp , Martin P. A. Coetzee ... 来源：[J].Antonie van Leeuwenhoek(IF 2.072), 2017, Vol.110 (10), pp.1287-1309 摘要：Investigation of the evolutionary relationships between related bacterial species and genera with a variety of lifestyles have gained popularity in recent years. For analysing the evolution of specific traits, however, a robust phylogeny is essential. In this study we examined th...
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 作者：Ioannis Angelakis , Patricia Gooding 来源：[J].Psychiatry Research(IF 2.456), 2020, Vol.285 摘要：Abstract(#br)There is robust evidence for relationships between adverse events experienced in childhood and mental health problems experienced as an adult. However, the measurement of perceptions of different types of adverse relationships in childhood, especially those that are ...
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text:
# Documentation

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# kaiser

## Syntax

w = kaiser(L,beta)

## Description

w = kaiser(L,beta) returns an L-point Kaiser window in the column vector w. beta is the Kaiser window parameter that affects the sidelobe attenuation of the Fourier transform of the window. The default value for beta is 0.5.

## Examples

collapse all

Create a 200-point Kaiser window with a beta of 2.5. Display the result using wvtool.

w = kaiser(200,2.5);
wvtool(w)

## Algorithms

The coefficients of a Kaiser window are computed from the following equation:

$w\left(n\right)=\frac{{I}_{0}\left(\beta \sqrt{1-{\left(\frac{n-N/2}{N/2}\right)}^{2}}\right)}{{I}_{0}\left(\beta \right)},\text{ }0\le n\le N,$

where I0 is the zeroth-order modified Bessel function of the first kind. The length L = N + 1. Thus kaiser(L,beta) is equivalent to

besseli(0,beta*sqrt(1-(((0:L-1)-(L-1)/2)/((L-1)/2)).^2))/besseli(0,beta).

To obtain a Kaiser window that designs an FIR filter with sidelobe attenuation of α dB, use the following β.

$\beta =\left\{\begin{array}{ll}0.1102\left(\alpha -8.7\right),\hfill & \alpha >50\hfill \\ 0.5842{\left(\alpha -21\right)}^{0.4}+0.07886\left(\alpha -21\right),\hfill & 50\ge \alpha \ge 21\hfill \\ 0,\hfill & \alpha <21\hfill \end{array}$

Increasing β widens the mainlobe and decreases the amplitude of the sidelobes (i.e., increases the attenuation).

## References

[1] Kaiser, James F. “Nonrecursive Digital Filter Design Using the I0-Sinh Window Function.” Proceedings of the 1974 IEEE® International Symposium on Circuits and Systems. April, 1974, pp. 20–23.

[2] Digital Signal Processing Committee of the IEEE Acoustics, Speech, and Signal Processing Society, eds. Selected Papers in Digital Signal Processing. Vol. II. New York: IEEE Press, 1976.

[3] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1999, p. 474.


text:
## Basic College Mathematics (9th Edition)

Published by Pearson

# Chapter 6 - Percent - Review Exercises: 84

#### Answer

$\frac{21}{50}$

#### Work Step by Step

In order to write a percent as a fraction, we can drop the percent sign and then divide by 100. Next, we must simplify the resulting fraction to lowest terms. 42%=$42\div100=\frac{42}{100}=\frac{42\div2}{100\div2}=\frac{21}{50}$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Advanced data manipulation in R

## 1. Introduction

You might have some experience in the basics of data manipulation and you know what pipes (%>%) are, but you are eager to learn some new variations of the functions you already know AND learn new ones? Then this tutorial is definitely for you!

The idea behind it is to make some more advanced aspects of data manipulation clearer, and to take a moment to learn a few new functions. The package dplyr (one of the packages from tidyverse) is your best friend for solving various challenges that you can encounter doing data manipulation.

We’ll go through dplyr functions together to make you more aware of their diversity. You will also become more efficient with the functions you already know by making small changes in the code, these will be e.g. my favorite functions select() and mutate(). You will probably spend an hour or so going through the tutorial - this will depend on how well you already know dplyr! However, in the long term, you will save yourself time from having to read through the endless information on the internet on why you keep getting errors or simply not getting the answer you want with the dplyr functions you thought you already knew (which has been my struggle for some time!). We’ll also cover a few other functions from other tidyverse packages to improve your confidence in data manipulation skills.

In PART I, we’ll be working with a fictional ocean animals dataset - to make the learning of combining tables less scary than it would be with large data frames. Then, in PART II, we’ll dive into the manipulation of variables and cases based on Living Planet dataset on marine organisms in Oceania. So, before you put on your scuba diving equipment for PART II, grab your snorkelling mask and we’re ready to start!

All the files you need to complete this tutorial can be downloaded from thisrepository. Click on Code -> Download ZIP, and unzip the files into your working directory.

## Part I: Ocean Animals

Create a new, blank script in RStudio and add tutorial title, your name, and the date at the top. When going through the tutorial, copy the individual code chunks and paste them to your script. Use # when adding comments.

Set the working directory, add the libraries, and load the multiple datasets.

# Title: Creative Data Manipulation: familiarising with the diversity of dplyr
# Date

# Set working directory to where you saved the folder with tutorial materials on your computer
setwd("file-path")

# Libraries ----
library(dplyr)
# when we add new libraries throughout the tutorial, you can paste them here

# If you've never used a given package, install it with install.packages("name")

# Most of them are a part of tidyverse but we will be loading them separately so that you are aware of the contexts in which they are used

### PART I: OCEAN ANIMALS ----

To give you some background on PART I: imagine you have collected this data with your research partner when visiting a coral reef at Palmyra Atoll, a small island in the middle of the Pacific (fig. 1). You successfully managed to identify 10 animals of four different types (going down to the species level was not expected because you don’t necessarily need to be a marine biologist to do this tutorial!). Our aim is to combine different data tables, which we’ll then use to display average weight and meal type for each animal type!

Figure 1. Coral Reef at Palmyra Atoll National Wildlife Refuge. Photo credit: U.S. Fish and Wildlife Service Headquarters/Jim Maragos (CC BY 2.0).

## 2. Combining tables

In dplyr there are various functions for combining tables.

### a) Combining rows with bind_rows

Each animal was given a unique id and weighted. To start with, you have your datasets in two parts: animal_p1 in which you described octopuses and fish; and animal_p2 where you’ve got turtles only.

The datasets are in the same format (columns are in the order; id, animal, weight) so you can simply put them on top of each other with bind_rows().

# 2. Combining tables ----
# a) Combining rows ----

# Firstly view the p1 and p2 datasets, which will appear in your console
animal_p1
animal_p2

# Now, bind them on top of each other
# Adding brackets around the expression prints out the result
(animal <- bind_rows(animal_p1, animal_p2))  # 8 observations

Oh no, it looks like we are missing some observations - we only have 8 in total, and we also can’t find any squids (I remember you told me about seeing them when you had been snorkelling yesterday!). We’ll have to compare your data with your research partner (rp) to find out what is missing.

Your rp’s data are in a separate table (animal_rp) so we will need to compare and hopefully combine them with animal. Together, they can be called relational data - the relations between them are important when we want to connect them. Most importantly, we need a key - a variable which is common for two datasets and thus can be used as a point of reference. This can be the animal’s id because it is unique for each of them. We will use it to combine tables in the following sections.

### b) Comparing data with set operations

We’ll first find out how the data tables are different. We can see that both animal and animal_rp have 8 observations in the Environment window in RStudio, but we have no idea if they are the same. While visual comparison would be possible here, if we had thousands of rows of data, that would be much more difficult. Therefore, we can use set operations to compare the data with code.

#### Set operations

setqual(x, y) returns TRUE if all observations in x and y are identical.

intersect(x, y) finds observations present in both x and y.

setdiff(x, y) finds observations present in x, but not in y.

union(x, y) finds unique observations in x and y (or use union_all() to also retain duplicates).

# b) Set operations ----

setequal(animal_p1, animal_p2)  # FALSE is returned, so the tables are not identical

# Since they differ, let's see how many observations you and your rp have in common.
# You're not creating an object yet so the output will be shown in the console without having to add brackets

intersect(animal, animal_rp)  # 6 observations in common

# View which observations you have and your rp doesn't

setdiff(animal, animal_rp)  # id no. 2 and 5

# Now view which ones your rp has that you don't

setdiff(animal_rp, animal)  # id no. 6 and 10

# Let's now connect them with union() but removing any duplicated rows

(animal_weight <- union(animal, animal_rp) %>%
arrange(id))  # puts the id in numerical order (useful function for rearranging cases!)

Perfect, now we have the animal_weight dataset with 10 unique observations (in the range of 1 to 10). You normally wouldn’t need to use all set operations like above, we could simply use union(). However, the other functions might be useful if you have a different aim of the study (e.g. you could use intersect() to find only observations which are confirmed by two researchers), or simply to get to know your data better.

### c) Combining tables with mutating joins

Now, we will want to combine animal_weight with another dataset - animal_meal which contains information on the last meal that you observed for each animal.

# c) Mutating joins ----

animal_meal  # there are 21 observations!

We no longer have the name of the animal, but we still have the key - unique id - based on which we will be able to combine the two tables by adding columns to each other.

Note: if we didn’t have any key, but e.g. we knew that rows in both tables are in the same order (OR the order didn’t matter and we simply wanted to add random meal to each animal), we could use bind_cols(). Here, it would generate various NAs since the number of observations for two tables is not equal.

There are many different functions in dplyr to combine tables, but each is different and might be better suited for your needs. The following are called mutating joins and are used to combine variables from two tables in slightly different ways. Don’t get put off by the number of functions, I have spent quite a lot of time trying to understand them! Hopefully, I will make it easier for you to understand with the diagram (fig. 2) and explanations below.

Figure 2. Basic depiction of mutating joins with a Venn diagram. Source: R for Data Science (CC BY-NC-ND 3.0 US).

#### Inner join:

• inner_join(x, y) keeps observations appearing in both tables.

#### Outer joins:

• left_join(x, y) keeps all observations in x and only adds matches from y.
• right_join(x, y) keeps all observations in y and only adds matches from x. (Note: it is the same as left_join(y, x).)
• full_join(x, y) keeps all observations in x and y; if there’s no match, it returns NAs.

So each join will return different information. In the case of our ocean animals, we have a table of animal_weight which we checked thoroughly, and now we simply want to add meal information for each individual animal, recognised by its id.

(animal_joined <- left_join(animal_weight, animal_meal,
by = c("id" = "IDs")))  # indicate which columns in two tables refer to the same id variable

# We can also write it differently by using the pipe operator
(animal_joined <- animal_weight %>%
left_join(animal_meal, by = c("id" = "IDs")))

What happened here? We used left_join() to keep animal_weight as the “base” to which meal information was added. That way we kept only the 10 animal id’s (from 1 to 10) that we were interested in, and only for them we added meals. We can see that turtle of id == 2 was given <NA> which means that there was no id match in animal_meal.

Note: We have specified that tables should be connected by = c("id" = "IDs") so that RStudio knows what is our key for matching both tables. Since in animal_meal the id column was named IDs, we had to tell RStudio that both tables are describing the same key. But even if both datasets were named id from the beginning, it is always a good practice to specify by = so that we are 100% sure what is happening to our data!

Now, let’s check the other joins!

inner_join(animal_weight, animal_meal, by = c("id" = "IDs"))  # only rows with id's appearing in both tables were left (turtle of id = 2 is now missing)

right_join(animal_weight, animal_meal, by = c("id" = "IDs"))  # we have all meal id's but various NAs for animal and weight columns were introduced because there was no match for them in animal_weight

full_join(animal_weight, animal_meal, by = c("id" = "IDs"))  # all possible id's from both tables are retained and various NAs are introduced

You might have noticed that we could also have used a mutating join full_join() when adding the rows on top of each other in part “a) connect rows of data with bind_rows()”.

full_join(animal_p1, animal_p2, by = c("id", "animal", "weight"))

Yes, that’s perfectly fine! I think it is great to get to know all the functions which might later prove in different contexts. When using full_join() for adding rows on top of each other, remember to always specify all the columns by which the function should join the two tables (otherwise they might be added next to each other instead).

### d) Combining tables with filtering joins

Another group of joins which is useful to know about are filtering joins.

semi_join(x, y) finds all the observations in x that have a match in y.

anti_join(x, y) removes all the observations in x that have a match in y.

# d) Filtering joins ----

semi_join(animal_weight, animal_meal, by = c("id" = "IDs"))  # returns animal_weight dataset apart from rows of which id is not present in animal_meal

anti_join(animal_weight, animal_meal, by = c("id" = "IDs"))  # returns only one row from animal_weight of which id was not present in animal_meal

So, as you have seen with the code above, filtering joins allow you to filter the x dataset based on the matches with y dataset. We’re not going to use them any further, but remember that semi_join() and anti_join() can be useful to identify matches or mismatches in the datasets.

### e) Challenge

Well done! You should now have some understanding of the different types of joining tables and of the diverse functions that can be used to do that in R!

Actually, while you were testing all the functions, your research partner collected 10 more observations!

# e) Challenge ----

str(animal_new)  # check the dataset

Now, take the challenge to create a final_animal table by combining animal_new with your table animal_joined!

#### Solution:

animal_final <- animal_joined %>%
full_join(animal_new,
by = c("id" = "ID", "animal" = "Animals", "weight", "meal" = "Meal"))

Notice that bind_rows() was not appropriate here because the column order was different between the two tables. full_join() allowed us to specify which columns are corresponding to each other between the two tables, and also to indicate that despite of the different spelling, the names were still referring to the same variables.

Now let’s visualize what is the meal type and average weight for each animal type you found (fig. 3).

# Libraries
library(ggplot2)    # one of the tidyverse packages for beautiful graphs
library(gridExtra)  # for the panel

# Barplot of diet
(barplot <- ggplot(animal_final, aes(animal, fill = meal)) +
geom_bar(alpha = 0.8) +
labs(title = "Diversity of meals", x = NULL) +
scale_fill_brewer(palette = "Set3", type = "seq", na.value = "grey") +
theme_minimal() +
theme(plot.title = element_text(size = 14, hjust = 0.5, face = "bold"),
plot.margin = unit(c(0.5, 0.5, 0.5, 0.5), units = , "cm")))

# Boxplot of weight
(boxplot <- ggplot(animal_final) +
geom_boxplot(aes(animal, weight, fill = animal), alpha = 0.5, position = "dodge2") +
scale_y_continuous(limits = c(0, 30)) +
labs(title = "Mean weights of animals", x = NULL, y = "Weight (kg)") +
theme_minimal() +
theme(plot.title = element_text(size = 14, hjust = 0.5, face = "bold"),
plot.margin = unit(c(0.5, 0.5, 0.5, 0.5), units = , "cm")))

# Combine the plots into one panel
animal_panel <- grid.arrange(barplot, boxplot, ncol = 2)

# Save the panel if you want! You will have to create images folder where you've set your working directory (or change filename to "animal_panel.png")
ggsave(filename = "images/animal_panel.png", plot = animal_panel, width = 10, height = 5)

Figure 3. Results of the study on ocean animals. Left: the varying meal types of the ocean animals. Right: significant differences in animals’ weight, with the highest mean for squids and the lowest for fish.

## Part II: Oceania LPI data

Now that we know some useful tricks for combining tables, we’ll have a closer look at manipulating variables and cases. As promised, we’ll now get a grasp of real-world dataset - Living Planet data. It might be more complicated because we will be handling thousands of rows of data, but this will give you a great hands-on experience which you can use for your future projects!

We’re going to have a look at Oceania’s marine species populations between 1970 and 2014.

A quick reminder on pipes: x %>% f(y) is the same as f(x, y). So if you provide the data x at the beginning, you don’t recall it within the following functions (f()) - the pipes would not like to see you putting x multiple times! Go to Efficient data manipulation tutorial for a more thorough introduction.

### PART II: OCEANIA LPI DATA ----

# Libraries
# We'll need some extra functions from tidyverse package for initial data wrangling, but we will then come back to dplyr!
library(tidyr)

marine <- read.csv("data/LPI_marine.csv")  # you can also use readr::read_csv() - much faster for large files

# Now let's tidy the data a little
marine2 <- marine %>%
# change to long format with tidyr::gather()
gather(key = year, value = pop, c(25:69)) %>%
# get rid of X's for year with readr::parse_number()
mutate(year = parse_number(as.character(year)),
# define pop to be numeric, missing data/character entries will become NAs
pop = as.numeric(pop)) %>%
# drop all missing values
drop_na(pop)

# Ignore the warning message on NAs, that's because we kept only numeric values for pop

glimpse(marine2)  # have a look at the dataset
View(marine2)  # you can also scroll through the table

We have changed the data to long format so that it is easier to work with (each observation is in a separate row), we also got rid of some X-s for year variable and dropped missing or incorrect values for population variable. We’ll now get to the details of dplyr functions, but if you want to know more on basic data tidying (like changing data to long format), head to the Basic data manipulation tutorial.

Note: in the following sections we’ll try to create as few objects as possible, not to overload your environment in RStudio. To learn different functions, we’ll be using a dplyr function - glimpse(), which is a nice alternative to str() and allows you to view all variables’ names, data type, and some values for each (in a very tidy way!).

## 3. Manipulating variables

With dplyr, you can easily manipulate variables by extracting entire columns, rename them or create new ones.

### a) Extracting variables: select() and others

Firstly, let’s extract a column which can be transferred into a new vector (with pull()) or a table (with select()).

# 3. Manipulating variables ----
# a) Extracting variables ----

marine2 %>%
pull(Species) %>%
glimpse()  # returns a vector of values

marine2 %>%
select(Species) %>%
glimpse()  # returns a new table with one column

select() is especially useful because it is quite flexible in its use to create new tables.

We can create a new table with chosen columns and in the preferred order.

# Select only columns you need, in the order you want
marine2 %>%
select(id, pop, year, Country.list) %>%
glimpse()

As well as you can give the new names to the columns of the new table.

# Rename columns using the format new_name = previous_name
marine2 %>%
select("Country list" = Country.list,  # you can use names than include spaces or other symbols that can create errors by putting them in ""
method = Sampling.method) %>%
glimpse()

If you want to reorder some columns and leave the rest unchanged, you can use everything().

# Use everything() to return all unselected columns
marine2 %>%
select(id, year, pop, everything()) %>%
glimpse()  # notice that id, year & pop went to the front but the rest is unchanged

You can also indicate the range of columns you want to keep using star_col:end_col (by using column names or numbers).

# Add ranges of columns with star_col:end_col
marine2 %>%
select(Family:Species, 24:26) %>%
glimpse()

Delete columns you don’t need with - (remember, if you delete multiple columns use select(-c()) so that - is applied to all of them).

# Delete columns with -
marine2 %>%
select(-c(2:22, 24)) %>%
glimpse()

Define the chosen columns in a vector beforehand and then recall it with !!.

# Call a vector of column names with !!
marine_cols <- c("Genus", "Species", "year", "pop", "id")

marine2 %>%
select(!!marine_cols) %>%
glimpse()

Moreover, you can use select() with these functions:

• starts_with("x") matches names starting with “x”
• ends_with("x") matches names ending with “x”
• contains("x") matches names containing “x”
# Choose columns with names starting with "Decimal"
marine2 %>%
select(starts_with("Decimal")) %>%
glimpse()

You can also select columns based on their data type using select_if(). The common data types to be called are: is.character, is.double, is.factor, is.integer, is.logical, is.numeric.

marine2 %>%
select_if(is.numeric) %>%
glimpse()  # numeric data types only selected (here: integer or double)

You can also mix various ways to call columns within select():

marine2 %>% select(id,  # put id first
Class:Family,  # add columns between Class and Family
genus = Genus,  # rename Genus to lowercase
starts_with("Decimal"),  # add columns starting with "Decimal"
everything(),  # add all the other columns
-c(6:9, system:Data.transformed)) %>%  # delete columns in these ranges
glimpse()
# OK, maybe it wasn't the most effective way of reaching the end state here but it proves the point of large flexibility!

Now that we have learned all the varieties of extracting variables with select(), let’s just keep the columns we want to use in further tasks (and do it efficiently!).

# Save a new object marine3 that we'll keep
marine3 <- marine2 %>%
select(id, Class, Genus, Species, year, pop,
location = Location.of.population,
lat = Decimal.Latitude,
lon = Decimal.Longitude) %>%
glimpse()

### b) Renaming variables: rename() and others

We still have some uppercase letters in marine3. I prefer to have all variable names in lowercase for higher clarity (and for a lower chance of mistyping them!). Since we didn’t do it with select() when creating marine3, we can now just rename them with rename(), or rename_with().

# b) Renaming variables ----

# Renaming to lowercase
marine3 %>%
rename(class = Class,
genus = Genus,
species = Species) %>%  # renames only chosen columns
glimpse()

If we just want column names to be changed with a function, we can use rename_with which is useful in this case since we can rename them with tolower.

marine3 %>%
rename_with(tolower) %>%
glimpse()
# If you didn't want all of them renamed, you could specify columns with .cols =

We could have achieved the same as above by selecting all columns and applying tolower.

marine4 <- marine3 %>%  # let's save this one!
select_all(tolower) %>%
glimpse()

If we wanted only some of them renamed and kept, we could have used select_at() which specifies columns with vars().

# Select and rename a couple of columns only
marine3 %>%
select_at(vars(Genus, Species), tolower) %>%
glimpse()

You may have noticed that if you knew just one of these functions, you would manage to rename these columns. However, this was just to show you the diversity of dplyr functions which might sometimes cut you a line of code, or make it easier when you want to rename a number of them efficiently. If you’re unsure about any aspect of these functions in a month’s time, e.g. what is this vars() or what is the difference between select_with() or rename_with(), you can always check it easily in RStudio. When you start writing the function, it will give you suggestions what these functions can be - simply put the cursor on the chosen function and you will easily manage!

And remember, these are not complicated and you can usually guess the right function by thinking about the exact action you want to undertake (have a look below!).

• _all() if you want to apply the function to all columns
• _at() if you want to apply the function to specific columns (specify them with vars())
• _if() if you want to apply the function to columns of a certain characteristic (e.g. data type)
• _with() if you want to apply the function to columns and include another function within it

These variants are quite flexible, and keep changing for individual functions (e.g. RStudio tells me now that rename_with() has superseded previously used rename_all(), rename_at(), rename_if()). This will happen with time, so stay open-minded about these functions and use help from RStudio to make your code efficient!

### c) Making new variables: mutate() and others

The family of mutate() functions can be used to create new variables by applying vectorized functions to entire columns. It’s a yet another amazing function because you can do pretty much everything with it (of course in terms of making new variables)!

Let’s firstly create a genus_species variable by connecting genus and species with an underscore.

# c) Making new variables ----

# New column appears at the end of the table
marine5 <- marine4 %>%
mutate(genus_species = paste(genus, species, sep = "_")) %>%
glimpse()

We can also use ifelse() or case_when(), and the logical operators. The mutate() function takes each row of the table and calculates the new value in the new column with the function of our choice!

marine6 <- marine5 %>%
mutate(region = case_when(lat > 0 & lon >= 0 ~ "NE",
lat <= 0 & lon >= 0 ~ "SE",
lat > 0 & lon < 0 ~ "NW",
lat <= 0 & lon < 0 ~ "SW")) %>%
glimpse()

unique(marine6\$region)  # we've got populations from both sides of equator and meridian 180!

Another function is transmute() which creates the new columns and drops the rest. Let’s use it with the examples of variables added above (genus_species and region).

# A table of only two variables: genus_species and region
marine4 %>%
transmute(genus_species = paste(genus, species, sep = "_"),
region = case_when(lat > 0 & lon >= 0 ~ "NE",
lat <= 0 & lon >= 0 ~ "SE",
lat > 0 & lon < 0 ~ "NW",
lat <= 0 & lon < 0 ~ "SW")) %>%
glimpse()

Similarly to select(), mutate() also has _at, _all and _if variations. The mechanism is generally the same so we won’t go through all the examples. Let’s just have a look at mutate_at(). We will indicate the variables with vars(), and then change all values for these variables to lowercase. Notice that with the earlier example of lowercase, select() dealt with column names (i.e. changed names to lowercase) while mutate() deals specifically with the values of chosen columns.

# Change values to lowercase
marine6 %>%
mutate_at(vars(class, genus, location), tolower) %>%
glimpse()  # column names stayed the same, but individual values within them are lowercase

Apart from mutate(), another way to create new variables can be with add_ functions, e.g. add_column().

# Adding a column (add_column() is from a different package - tibble)
library(tibble)  # load tibble package

marine6 %>%
add_column(observation_num = 1:4456) %>%  # we gave each row an observation number
glimpse()

There are also summarising functions count() and tally() which have a mutating variation as add_count() and add_tally(). Here we will see how many unique annual observations we have for each species.

# Summarising functions
marine6 %>%
select(genus_species, year) %>%
group_by(genus_species) %>%
glimpse()

marine6 %>%
select(genus_species, year) %>%
# add_count() includes the grouping variable (here genus_species) inside the function
glimpse()

Nice! Now you know quite a variety of possibilities for creating new variables!

## 4. Manipulating cases

Now, we’ll talk about manipulating cases - so that’s working with rows.

### a) Extracting cases: filter() and others

In this section, we’ll talk about how to filter the dataset to return a subset of all rows. We can filter to a specific category or a few categories with, conveniently named, filter().

With filter(), you can make great use of logical operators which can be found in Table 1 below. You can find more information on them from ETH Zurich or Towards Data Science websites.

 Operator Description > greater than >= greater than or equal to < less than <= less than or equal to == exactly equal to != not equal to a | b a OR b xor(a, b) only a OR only b a & b a AND b is.na() only NAs !is.na() all but NAs %in% in one of the specified values

Table 1. Common logical operators.

The most common use is of == when we want to e.g. filter the table for a specific class or species.

# 4. Manipulating cases ----
# a) Extracting cases ----

# Here we use comparison operator ==
marine6 %>%
filter(class == "Mammalia") %>%
glimpse()  # mammals only

We can also filter it for a few matching values.

# Match one of the specified values with %in%
marine6 %>%
filter(class %in% c("Mammalia", "Aves")) %>%
glimpse()  # mammals and aves

# Same result as above but using |
marine6 %>%
filter(class == "Mammalia" | class == "Aves") %>%
glimpse()

Or we can omit some categories with !.

# Omit a category
marine6 %>%
filter(class != "Actinopteri") %>%
glimpse()  # all classes but Actinopteri

marine6 %>%
filter(!class %in% c("Mammalia", "Aves")) %>%
glimpse()  # remove classes of mammals and aves

We can also filter based on numeric variables.

# Filter numeric variables
marine6 %>%
filter(pop >= 10 & pop <= 100) %>%
glimpse()  # observations for population size between 10 and 100

marine6 %>%
filter(between(pop, 10, 100)) %>%
glimpse()  # same as above but more aesthetically pleasing!

We can also use filter() to drop NAs.

# Delete NAs from pop
marine6 %>%
filter(!is.na(pop)) %>%
glimpse()  # there's already none because we deleted them earlier with mutate()!

Wow, you can do a lot within this one function! But remember to always understand your specified conditions and how they interact with each other in the function. When using multiple logical operators, you might easily become trapped in too much complexity. In the example below, I will only add brackets to the code but this largely changes the meaning of conditions!

# Be careful with logical operators
marine6 %>%
filter((class == "Mammalia" | pop > 100) & region != "SE") %>%
glimpse()  # 38 rows

# Argument 1: class is Mammalia or population is larger than 100
# AND
# Argument 2: in each case the region cannot be SE

marine6 %>%
filter(class == "Mammalia" | (pop > 100 & region != "SE")) %>%
glimpse()  # 96 rows!

# Argument 1: class is Mammalia
# OR
# Argument 2: population is larger than 100 and region is not SE

Again, filter() has the _at(), _all(), and _if() variations, but I don’t want to overload you with more examples. These can be however quite useful, especially if you want to e.g. detect strings within the values! I will redirect you to Suzan Baert’s tutorial on filter() where you can find these variations explained very well!

Apart from filter() family, we also have a few other friendly functions to extract cases.

distinct() can be used to drop all duplicated rows (warning: make sure you want to do that, you can check duplicates with base R function duplicated()).

marine6 %>%
distinct() %>%
glimpse()  # still 4456 rows so there were no duplicates!

# You can also check a number of distinct rows directly with n_distinct() - equivalent of length(unique(x))
marine6 %>%
n_distinct()  # 4456 distinct rows

slice() can be used to select which rows we want to keep.

marine6 %>%
select(id:species) %>%  # we'll keep less columns for clarity
slice(2:4)  # keep only rows 2, 3 and 4! (they get new numbers though - 1, 2, 3)

And top_n() can be used to select and order top entries.

marine6 %>%
top_n(5, pop) %>%  # 5 highest population values
glimpse()

### b) Adding cases: add_row()

Lastly, if you would like to manually add an observation, you can use add_row(). Although it is actually a function from tibble package, it’s also important to learn!

But first, quick exercise to keep you fresh! Can you take the challenge to create marine7 from marine6 which you will need to complete this section? Create a new table for Chrysophrys auratus population with the id number of 2077. Then, keep only the columns of id, full species name (in Latin!), year of measurement and size of the population.

#### Solution:

# b) Adding cases ----

# Create a table for id == "2077"
(marine7 <- marine6 %>%
filter(id == "2077") %>%
select(id, genus_species, year, pop))  # we can see 27 observations for 1970-1996

Notice that we only had to specify the id and not the full name - Chrysophrys auratus. That’s because each id is unique for a population (although you might want to go through the table to make sure that each genus_species is actually Chrysophrys auratus - maybe someone has messed up and gave two populations the same id by accident!).

Nice! Now to the new function. Let’s say we wanted to add an observation for the year 1997 at the bottom of the table.

# Add a row at the end
(marine7 %>%
add_row(id = 2077, genus_species = "Chrysophrys_auratus", year = 1997, pop = 39000))

You can also add a row at a specified location with .before =.

# Add a row at a specified location
(marine7 %>%
add_row(id = 2077, genus_species = "Chrysophrys_auratus", year = 1969, pop = 39000,
.before = 1))

Great! Manipulating cases should no longer be a mystery to you :)

## 5. Extra: Green Sea Turtle population trend

Let’s make use of our LPI dataset and visualise population trends over time of Green Sea Turtle (Chelonia mydas). We won’t go into detail of how to produce this graph, but if you want to learn more on data visualisation, have a look at these Coding Club tutorials: Data vis. 1 & Data vis. 2.

# 5. Extra: Green Sea Turtle population trend ----

marine_final <- marine6 %>%
filter(genus_species == "Chelonia_mydas") %>%
# change id to factor (otherwise it would display as a continuous variable on the plot)
mutate(id = as.factor(id))

library(ggplot2)  # you don't have to load it again if you've done it in part 2e

(chelonia_trends <- ggplot(marine_final, aes(x = year, y = pop, colour = location)) +
geom_point(size = 2, alpha = 0.7) +
geom_smooth(method = "lm", colour = "black", fill = "lightgrey") +
scale_x_continuous(limits = c(1970, 2005), breaks = c(1970, 1980, 1990, 2000)) +
labs(x = NULL, y = "Population count\n",
title = "Positive trend of Green Sea Turtle population in Australia\n",
colour = "Location") +
theme_minimal() +
theme(plot.title = element_text(size = 14, hjust = 0.5, face = "bold"),
plot.margin = unit(c(0.5, 0.5, 0.5, 0.5), units = , "cm")))

ggsave(chelonia_trends, filename = "images/chelonia_trends.png", width = 8, height = 6)

Figure 4. Population trend of Green Sea Turtle seems to be increasing in the past decades. The trend is marked with a bold black line along with an error band (shaded area). Data source: Living Planet Index.

We’re finishing off with a positive message of the increase in the measured Chelonia mydas populations across Australia (fig. 4). I hope you have enjoyed today’s journey with dplyr and will be able to make use of these creative functions in your future data manipulations tasks!


text:
# Find the Intercepts Cut off by the Plane 2x + Y – Z = 5. - Mathematics

Find the intercepts cut off by the plane 2x + y – z = 5.

Find the sum of the intercepts cut off by the plane $2x + y - z = 5,$on the coordinate axes.

#### Solution

2x + y – z = 5  ...(1)

Dividing both sides of equation (1) by 5, we obtain

Thus, the intercepts cut off by the plane are 5/2, 5 and -5

∴ Required sum of intercepts = $\frac{5}{2} + 5 + \left( - 5 \right) = \frac{5}{2}$ .

Is there an error in this question or solution?

#### APPEARS IN

NCERT Class 12 Maths
Chapter 11 Three Dimensional Geometry
Q 7 | Page 493


text:
# NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5

The topics and sub-topics included in Chapter 13 Probability the following:

 Section Name Topic Name 13 Probability 13.1 Introduction 13.2 Conditional Probability 13.3 Multiplication Theorem on Probability 13.4 Independent Events 13.5 Bayes’ Theorem 13.6 Random Variables and its Probability Distributions 13.7 Bernoulli Trials and Binomial Distribution

NCERT Solutions for Class 12 Maths Chapter 13 Probability 13.5 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Probability Ex 13.5

Question 1.
A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of
(i) 5 successes?
(ii) at least 5 successes?
(iii) at most 5 successes?
Solution:

Question 2.
There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item ?
Solution:

Question 3.
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability, of two successes.
Solution:

Question 4.
Five cards are drawn successively with replacement from a well- shuffled deck of 52 cards. What is the probability that
(i) all the five cards are spades?
(ii) only 3 cards are spades?
(iii) none is spade?
Solution:

Question 5.
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs.
(i) none
(ii) not more than one
(iii) more than one
(iv) at least one will fuse after 150 days of use
Solution:

Question 6.
A bag consists of 10 balls each marked with one of the digits 0 to 9. If four bails are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?
Solution:

Question 7.
In an examination, 20 questions of true – false type are asked. Suppose a student tosses fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true,’ if it falls tails, he answers “ false’. Find the probability that he answers at least 12 questions correctly.
Solution:

Question 8
Suppose X has a binomial distribution $B\left( 6,\frac { 1 }{ 2 } \right)$

Question 9.
On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
Solution:

Question 10.
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is $\frac { 1 }{ 100 }$

Question 11.
Find the probability of getting 5 exactly twice in 7 throws of a die.
Solution:

Question 12.
Find the probability of throwing at most 2 sixes in 6 throws of a single die.
Solution:

Question 13.
It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles 9 are defective?
Solution:

Question 14.
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is
(a) ${ 10 }^{ -1 }$

Question 15.
The probability that a student is not a swimmer is $\frac { 1 }{ 5 }$

+


text:
## Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs and Tracts

 Short Title: Pure Appl. Math., Wiley-Intersci. Ser. Texts Monogr. Tracts Publisher: John Wiley & Sons, Hoboken, NJ Predecessor: Interscience Tracts in Pure and Applied Mathematics Successor: Pure and Applied Mathematics. A Wiley Series of Texts, Monographs, and Tracts Comments: Book series; No longer indexed
 Documents Indexed: 52 Publications (1974–2012)

all top 5

### Authors / Editors

 2 Bridger, Mark 2 Castillo, Enrique F. 2 Cox, David A. 2 Sewell, Granville 2 Whitham, Gerald Beresford 1 Adamowicz, Zofia 1 Ainsworth, Mark 1 Alguacil, Natalia 1 Allen, Myron Bartlett III 1 Aubin, Jean-Pierre 1 Barthelemy, Jean-Pierre 1 Basener, William F. 1 Bellman, Richard Ernest 1 Berg, Michael C. 1 Berkovitz, Leonard David 1 Boyarintsev, Yurij Eremeevich 1 Burk, Frank E. 1 Cobo, Angel 1 Conejo, Antonio J. 1 Cornet, Bernard 1 Dinculeanu, Nicolae 1 Dinitz, Jeffrey H. 1 Faticoni, Theodore G. 1 Folland, Gerald B. 1 Fröberg, Ralf 1 Frölicher, Alfred 1 García, Ricardo 1 Greene, Robert Everist 1 Grillet, Pierre Antoine 1 Grove, Larry C. 1 Guénoche, Alain 1 Isaacson, Eli L. 1 Jubete, Francisco 1 Khamsi, Mohamed Amine 1 Kirk, William Arthur 1 Koshy, Thomas 1 Krantz, Steven George 1 Kriegl, Andreas 1 Lasry, Jean-Michel 1 Lax, Peter David 1 Mahmoud, Hosam M. 1 Markley, Nelson G. 1 Mikhlin, Solomon Grigor’evich 1 Morrison, Terry J. 1 Oden, John Tinsley 1 O’Neil, Peter V. 1 Page, Warren 1 Pedregal, Pablo 1 Petkov, Vesselin M. 1 Promislov, S. David 1 Pruneda, Rosa Eva 1 Rassias, Themistocles Michael 1 Romanowska, Anna B. 1 Rossi, Richard J. 1 Samelson, Hans 1 Schulze, Bert-Wolfgang 1 Shick, Paul L. 1 Šimša, Jaromír 1 Smith, Jonathan Dallas Hayden 1 Šolín, Pavel 1 Stade, Eric 1 Stahl, Saul 1 Stakgold, Ivar 1 Stinson, Douglas Robert 1 Stoyanov, Luchezar N. 1 Wing, George Milton 1 Zauderer, Erich 1 Zbierski, Pawel
all top 5

### Fields

 14 Partial differential equations (35-XX) 9 Functional analysis (46-XX) 8 General and overarching topics; collections (00-XX) 8 Numerical analysis (65-XX) 6 Operator theory (47-XX) 5 Mathematical logic and foundations (03-XX) 5 Linear and multilinear algebra; matrix theory (15-XX) 5 Ordinary differential equations (34-XX) 5 Mechanics of deformable solids (74-XX) 4 Measure and integration (28-XX) 4 General topology (54-XX) 4 Operations research, mathematical programming (90-XX) 3 Number theory (11-XX) 3 Real functions (26-XX) 3 Global analysis, analysis on manifolds (58-XX) 3 Fluid mechanics (76-XX) 2 Combinatorics (05-XX) 2 General algebraic systems (08-XX) 2 Field theory and polynomials (12-XX) 2 Commutative algebra (13-XX) 2 Category theory; homological algebra (18-XX) 2 Group theory and generalizations (20-XX) 2 Topological groups, Lie groups (22-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Integral equations (45-XX) 2 Algebraic topology (55-XX) 2 Probability theory and stochastic processes (60-XX) 2 Computer science (68-XX) 1 History and biography (01-XX) 1 Functions of a complex variable (30-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Approximations and expansions (41-XX) 1 Abstract harmonic analysis (43-XX) 1 Integral transforms, operational calculus (44-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Convex and discrete geometry (52-XX) 1 Differential geometry (53-XX) 1 Manifolds and cell complexes (57-XX) 1 Optics, electromagnetic theory (78-XX) 1 Quantum theory (81-XX)

### Citations contained in zbMATH Open

47 Publications have been cited 5,609 times in 5,569 Documents Cited by Year
Linear and nonlinear waves. Zbl 0373.76001
Whitham, G. B.
1974
Real analysis. Modern techniques and their applications. 2nd ed. Zbl 0924.28001
Folland, Gerald B.
1999
A posteriori error estimation in finite element analysis. Zbl 1008.65076
Ainsworth, Mark; Oden, J. Tinsley
2000
Fibonacci and Lucas numbers with applications. Volume I. Zbl 0984.11010
Koshy, Thomas
2001
Functional analysis. Zbl 1009.47001
Lax, Peter D.
2002
An introduction to metric spaces and fixed point theory. Zbl 1318.47001
Khamsi, Mohamed A.; Kirk, William A.
2001
Linear and nonlinear waves. Paperback ed. Zbl 0940.76002
Whitham, Gerald B.
1999
Evolution of random search trees. Zbl 0762.68033
Mahmoud, Hosam M.
1992
Green’s functions and boundary value problems. 2nd ed. Zbl 0897.35001
Stakgold, Ivar
1998
Linear spaces and differentiation theory. Zbl 0657.46034
Frölicher, Alfred; Kriegl, Andreas
1988
An introduction to invariant imbedding. Zbl 0325.34001
Bellman, R.; Wing, G. M.
1975
Boundary value problems and singular pseudo-differential operators. Zbl 0907.35146
Schulze, Bert-Wolfgang
1998
Vector integration and stochastic integration in Banach spaces. Zbl 0974.28006
Dinculeanu, Nicolae
2000
Post-modern algebra. Zbl 0946.00001
Smith, Jonathan D. H.; Romanowska, Anna B.
1999
Geometry of reflecting rays and inverse spectral problems. Zbl 0761.35077
Petkov, Vesselin M.; Stoyanov, Luchezar N.
1992
Contemporary design theory. A collection of surveys. Zbl 0746.00028
1992
Partial differential equations of applied mathematics. 3rd ed. Zbl 1103.35001
Zauderer, Erich
2006
Partial differential equations and finite element method. Zbl 1092.65080
Šolín, Pavel
2006
Convexity and optimization in $${\mathbb R^n}$$. Zbl 0991.90002
Berkovitz, Leonard D.
2002
Numerical analysis for applied science. Zbl 0899.65001
Allen, Myron B. III; Isaacson, Eli L.
1998
Topology and its applications. Zbl 1113.54001
Basener, William F.
2006
An introduction to Gröbner bases. Zbl 0997.13500
Fröberg, Ralf
1997
Applied functional analysis. Exercises by Bernard Cornet and Jean-Michel Lasry. Transl. by Carole Labrousse. 2nd ed. Zbl 0946.46001
Aubin, Jean-Pierre
2000
Topological uniform structures. Zbl 0397.46001
Page, Warren
1978
The numerical solution of ordinary and partial differential equations. 2nd ed. Zbl 1089.65053
Sewell, Granville
2005
Trees and proximity representations. With a preface by Michel Minoux. Transl. from the French by Gregor Lawden. Zbl 0790.05021
Barthélémy, Jean Pierre; Guénoche, Alain
1991
Groups and characters. Zbl 0896.20001
Grove, Larry C.
1997
Logic of mathematics: a modern course of classical logic. Zbl 0874.03003
1997
Galois theory. Zbl 1057.12002
Cox, David A.
2004
Functional analysis. An introduction to Banach space theory. Zbl 1005.46004
Morrison, Terry J.
2001
Orthogonal sets and polar methods in linear algebra. Applications to matrix calculations, systems of equations, inequalities, and linear programming. Zbl 0916.15001
Castillo, Enrique; Cobo, Angel; Jubete, Francisco; Pruneda, Rosa Eva
1999
Building and solving mathematical programming models in engineering and science. Zbl 1029.90001
Castillo, Enrique; Conejo, Antonio J.; Pedregal, Pablo; Garciá, Ricardo; Alguacil, Natalia
2002
Primes of the form $$x^2+ny^2$$. Fermat, class field theory, and complex multiplication. Paperback ed. Zbl 0956.11500
Cox, David A.
1997
Principles of differential equations. Zbl 1057.34001
Markley, Nelson G.
2004
Error analysis in numerical processes. Transl. and rev. from the German. Transl. by Reinhard Lehmann. Zbl 0786.65038
Mikhlin, Solomon G.
1991
Finite sums decompositions in mathematical analysis. Zbl 0859.26005
Rassias, Themistocles M.; Šimša, Jaromir
1995
Function theory of one complex variable. Zbl 0880.30003
Greene, Robert E.; Krantz, Steven G.
1997
Methods of solving singular systems of ordinary differential equations. Transl. from the Russian by Vassily Michalkowski. Zbl 0748.65060
Boyarintsev, Y.
1992
Lebesgue measure and integration. An introduction. Zbl 0886.28001
Burk, Frank
1998
Algebra. Zbl 1018.00001
Grillet, Pierre Antoine
1999
Fourier analysis. Zbl 1084.42002
2005
Topology. Point-set and geometric. Zbl 1120.54001
Shick, Paul L.
2007
Computational methods for linear algebra. 2nd ed. Zbl 1081.65029
Sewell, Granville
2005
The Fourier-analytic proof of quadratic reciprocity. Zbl 1004.11064
Berg, Michael C.
2000
Real analysis. A constructive approach. Zbl 1113.26001
Bridger, Mark
2007
An introduction to linear algebra. Zbl 0299.15001
Samelson, Hans
1974
A first course in functional analysis. Zbl 1189.46001
Promislow, S. David
2008
A first course in functional analysis. Zbl 1189.46001
Promislow, S. David
2008
Topology. Point-set and geometric. Zbl 1120.54001
Shick, Paul L.
2007
Real analysis. A constructive approach. Zbl 1113.26001
Bridger, Mark
2007
Partial differential equations of applied mathematics. 3rd ed. Zbl 1103.35001
Zauderer, Erich
2006
Partial differential equations and finite element method. Zbl 1092.65080
Šolín, Pavel
2006
Topology and its applications. Zbl 1113.54001
Basener, William F.
2006
The numerical solution of ordinary and partial differential equations. 2nd ed. Zbl 1089.65053
Sewell, Granville
2005
Fourier analysis. Zbl 1084.42002
2005
Computational methods for linear algebra. 2nd ed. Zbl 1081.65029
Sewell, Granville
2005
Galois theory. Zbl 1057.12002
Cox, David A.
2004
Principles of differential equations. Zbl 1057.34001
Markley, Nelson G.
2004
Functional analysis. Zbl 1009.47001
Lax, Peter D.
2002
Convexity and optimization in $${\mathbb R^n}$$. Zbl 0991.90002
Berkovitz, Leonard D.
2002
Building and solving mathematical programming models in engineering and science. Zbl 1029.90001
Castillo, Enrique; Conejo, Antonio J.; Pedregal, Pablo; Garciá, Ricardo; Alguacil, Natalia
2002
Fibonacci and Lucas numbers with applications. Volume I. Zbl 0984.11010
Koshy, Thomas
2001
An introduction to metric spaces and fixed point theory. Zbl 1318.47001
Khamsi, Mohamed A.; Kirk, William A.
2001
Functional analysis. An introduction to Banach space theory. Zbl 1005.46004
Morrison, Terry J.
2001
A posteriori error estimation in finite element analysis. Zbl 1008.65076
Ainsworth, Mark; Oden, J. Tinsley
2000
Vector integration and stochastic integration in Banach spaces. Zbl 0974.28006
Dinculeanu, Nicolae
2000
Applied functional analysis. Exercises by Bernard Cornet and Jean-Michel Lasry. Transl. by Carole Labrousse. 2nd ed. Zbl 0946.46001
Aubin, Jean-Pierre
2000
The Fourier-analytic proof of quadratic reciprocity. Zbl 1004.11064
Berg, Michael C.
2000
Real analysis. Modern techniques and their applications. 2nd ed. Zbl 0924.28001
Folland, Gerald B.
1999
Linear and nonlinear waves. Paperback ed. Zbl 0940.76002
Whitham, Gerald B.
1999
Post-modern algebra. Zbl 0946.00001
Smith, Jonathan D. H.; Romanowska, Anna B.
1999
Orthogonal sets and polar methods in linear algebra. Applications to matrix calculations, systems of equations, inequalities, and linear programming. Zbl 0916.15001
Castillo, Enrique; Cobo, Angel; Jubete, Francisco; Pruneda, Rosa Eva
1999
Algebra. Zbl 1018.00001
Grillet, Pierre Antoine
1999
Green’s functions and boundary value problems. 2nd ed. Zbl 0897.35001
Stakgold, Ivar
1998
Boundary value problems and singular pseudo-differential operators. Zbl 0907.35146
Schulze, Bert-Wolfgang
1998
Numerical analysis for applied science. Zbl 0899.65001
Allen, Myron B. III; Isaacson, Eli L.
1998
Lebesgue measure and integration. An introduction. Zbl 0886.28001
Burk, Frank
1998
An introduction to Gröbner bases. Zbl 0997.13500
Fröberg, Ralf
1997
Groups and characters. Zbl 0896.20001
Grove, Larry C.
1997
Logic of mathematics: a modern course of classical logic. Zbl 0874.03003
1997
Primes of the form $$x^2+ny^2$$. Fermat, class field theory, and complex multiplication. Paperback ed. Zbl 0956.11500
Cox, David A.
1997
Function theory of one complex variable. Zbl 0880.30003
Greene, Robert E.; Krantz, Steven G.
1997
Finite sums decompositions in mathematical analysis. Zbl 0859.26005
Rassias, Themistocles M.; Šimša, Jaromir
1995
Evolution of random search trees. Zbl 0762.68033
Mahmoud, Hosam M.
1992
Geometry of reflecting rays and inverse spectral problems. Zbl 0761.35077
Petkov, Vesselin M.; Stoyanov, Luchezar N.
1992
Contemporary design theory. A collection of surveys. Zbl 0746.00028
1992
Methods of solving singular systems of ordinary differential equations. Transl. from the Russian by Vassily Michalkowski. Zbl 0748.65060
Boyarintsev, Y.
1992
Trees and proximity representations. With a preface by Michel Minoux. Transl. from the French by Gregor Lawden. Zbl 0790.05021
Barthélémy, Jean Pierre; Guénoche, Alain
1991
Error analysis in numerical processes. Transl. and rev. from the German. Transl. by Reinhard Lehmann. Zbl 0786.65038
Mikhlin, Solomon G.
1991
Linear spaces and differentiation theory. Zbl 0657.46034
Frölicher, Alfred; Kriegl, Andreas
1988
Topological uniform structures. Zbl 0397.46001
Page, Warren
1978
An introduction to invariant imbedding. Zbl 0325.34001
Bellman, R.; Wing, G. M.
1975
Linear and nonlinear waves. Zbl 0373.76001
Whitham, G. B.
1974
An introduction to linear algebra. Zbl 0299.15001
Samelson, Hans
1974
all top 5

### Cited by 7,279 Authors

 38 Khamsi, Mohamed Amine 32 Kalisch, Henrik 28 Schulze, Bert-Wolfgang 27 Smith, Jonathan Dallas Hayden 22 Carstensen, Carsten 22 Liu, Yue 22 Yin, Zhaoyang 21 El, Gennady A. 21 Mahmoud, Hosam M. 21 Wazwaz, Abdul-Majid Abdul-Rahman 19 Díez, Pedro 19 Smyth, Noel Frederick 18 Engelbrecht, Jüri 18 Nowak, Marian 17 Baccouch, Mahboub 17 Praetorius, Dirk 16 Grimshaw, Roger Hamilton James 15 Bridges, Thomas J. 15 Gabeleh, Moosa 15 Maugin, Gérard A. 15 Michor, Peter Wolfram 14 Jordan, Pedro M. 14 Nicaise, Serge 13 Hussain, Nawab 13 Kamchatnov, Anatoly M. 13 Liu, Hailiang 13 Neininger, Ralph 13 Sharma, Vishnu Dutt 13 Zhang, Zhimin 12 Castillo, Enrique F. 12 Kraenkel, Roberto A. 12 Oden, John Tinsley 12 Otárola, Enrique 12 Repin, Sergeĭ Igorevich 12 Taşkara, Necati 11 Hoefer, Mark A. 11 Huerta, Antonio 11 Parés, Núria 11 Russo, Giovanni 11 Yazlik, Yasin 11 Zhang, Peng 10 Abdou, Afrah Ahmad Noan 10 Alfuraidan, Monther Rashed 10 Allendes, Alejandro 10 Banks, Jeffrey W. 10 Biondini, Gino 10 Boyd, John Philip 10 Chang, Der-Chen E. 10 Gudi, Thirupathi 10 Guo, Hailong 10 Hwang, Hsien-Kuei 10 Juang, Jonq 10 Kriegl, Andreas 10 Minzoni, Antonmaria Alessio 10 Shen, Chunyu 10 Voigtlaender, Felix 10 Yan, Kai 10 Zhang, Weiguo 9 Ainsworth, Mark 9 Chen, Gui-Qiang G. 9 Devroye, Luc P. J. A. 9 Flajolet, Philippe 9 Henshaw, William D. 9 Himonas, A. Alexandrou 9 Hoppe, Ronald H. W. 9 Janson, Svante 9 Johnson, Mathew A. 9 Kharif, Christian 9 Menegatto, Valdir Antonio 9 Oliveri, Francesco 9 Rasulov, Mahir A. 9 Ratliff, Daniel J. 9 Rodrigues, Luis Miguel 9 Romanowska, Anna B. 9 Salupere, Andrus 9 Saut, Jean-Claude 9 Vohralík, Martin 9 Zenchuk, Alexandre I. 8 Adjerid, Slimane 8 Babuška, Ivo 8 Bordas, Stéphane Pierre Alain 8 Chaudhry, Jehanzeb Hameed 8 Chen, Hua 8 Das, Ashok 8 Dougalis, Vassilios A. 8 Ercolani, Nicholas M. 8 Escher, Joachim 8 Espínola García, Rafael 8 Fusco, Domenico 8 Grübel, Rudolf 8 Haddow, J. B. 8 Kaya, Doǧan 8 Lepskiĭ, Oleg Vital’evich 8 Luo, Tao 8 McLaughlin, David W. 8 Mortell, Michael Philip 8 Noble, Pascal 8 Olela Otafudu, Olivier 8 Prodinger, Helmut 8 Rainer, Armin ...and 7,179 more Authors
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### Cited in 733 Journals

 168 Physica D 163 Computer Methods in Applied Mechanics and Engineering 156 Wave Motion 142 Journal of Computational Physics 128 Journal of Mathematical Analysis and Applications 128 Applied Mathematics and Computation 119 Journal of Differential Equations 100 Journal of Computational and Applied Mathematics 89 Physics Letters. A 84 Computers & Mathematics with Applications 68 Journal of Scientific Computing 66 Applied Numerical Mathematics 64 Mathematics of Computation 62 Journal of Fluid Mechanics 61 Journal of Mathematical Physics 56 Numerische Mathematik 54 Chaos, Solitons and Fractals 51 Communications in Mathematical Physics 48 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 42 European Journal of Mechanics. B. Fluids 41 ZAMP. Zeitschrift für angewandte Mathematik und Physik 40 Journal of Functional Analysis 37 Fixed Point Theory and Applications 35 Acta Mechanica 33 Journal of Statistical Physics 33 The Annals of Applied Probability 31 International Journal for Numerical Methods in Engineering 31 Mathematics and Computers in Simulation 31 Transactions of the American Mathematical Society 30 SIAM Journal on Numerical Analysis 30 Mathematical and Computer Modelling 29 Archive for Rational Mechanics and Analysis 29 International Journal of Engineering Science 29 Proceedings of the American Mathematical Society 29 Physics of Fluids 29 Journal of Mathematical Sciences (New York) 28 Journal of Engineering Mathematics 28 Physics of Fluids 28 Journal of Nonlinear Science 28 SIAM Journal on Scientific Computing 27 Advances in Mathematics 27 Random Structures & Algorithms 27 Communications in Nonlinear Science and Numerical Simulation 27 Nonlinear Analysis. Real World Applications 26 Computational Mechanics 26 Applied Mathematics Letters 24 Computers and Fluids 24 Acta Applicandae Mathematicae 24 Abstract and Applied Analysis 23 Theoretical Computer Science 22 SIAM Journal on Applied Mathematics 22 Nonlinear Dynamics 22 Journal of Pseudo-Differential Operators and Applications 21 Advances in Applied Clifford Algebras 20 Journal of Applied Mathematics and Mechanics 20 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 19 Physica A 19 SIAM Journal on Mathematical Analysis 19 Stochastic Processes and their Applications 19 Discrete and Continuous Dynamical Systems 18 Numerical Methods for Partial Differential Equations 18 Applied Mathematical Modelling 17 Theoretical and Mathematical Physics 17 Numerical Functional Analysis and Optimization 17 International Journal of Computer Mathematics 17 Linear Algebra and its Applications 17 Advances in Computational Mathematics 17 Journal of Inequalities and Applications 17 Journal of Fixed Point Theory and Applications 17 Asian-European Journal of Mathematics 16 Studies in Applied Mathematics 16 Applied and Computational Harmonic Analysis 16 Advances in Difference Equations 15 Fluid Dynamics 15 Communications in Partial Differential Equations 15 Mathematical Problems in Engineering 15 Positivity 15 Journal of Nonlinear Mathematical Physics 14 Astrophysics and Space Science 14 Journal of Soviet Mathematics 14 Mathematische Zeitschrift 14 Monatshefte für Mathematik 14 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 14 Filomat 14 Journal of Integer Sequences 13 The Annals of Probability 13 Quarterly of Applied Mathematics 13 SIAM Journal on Control and Optimization 13 Annals of Global Analysis and Geometry 13 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 13 The Journal of Geometric Analysis 13 Journal de Mathématiques Pures et Appliquées. Neuvième Série 13 Bernoulli 13 Central European Journal of Mathematics 13 Journal of Hyperbolic Differential Equations 13 Nonlinear Analysis. Theory, Methods & Applications 12 Nonlinearity 12 BIT 12 Calcolo 12 Integral Equations and Operator Theory ...and 633 more Journals
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### Cited in 63 Fields

 2,206 Partial differential equations (35-XX) 1,318 Numerical analysis (65-XX) 1,314 Fluid mechanics (76-XX) 428 Mechanics of deformable solids (74-XX) 366 Probability theory and stochastic processes (60-XX) 360 Dynamical systems and ergodic theory (37-XX) 350 Operator theory (47-XX) 299 Number theory (11-XX) 278 Functional analysis (46-XX) 202 Ordinary differential equations (34-XX) 184 Computer science (68-XX) 178 Combinatorics (05-XX) 174 Statistical mechanics, structure of matter (82-XX) 167 Operations research, mathematical programming (90-XX) 164 Calculus of variations and optimal control; optimization (49-XX) 151 Quantum theory (81-XX) 148 Global analysis, analysis on manifolds (58-XX) 147 General topology (54-XX) 127 Statistics (62-XX) 127 Optics, electromagnetic theory (78-XX) 120 Harmonic analysis on Euclidean spaces (42-XX) 104 Geophysics (86-XX) 104 Biology and other natural sciences (92-XX) 91 Integral equations (45-XX) 86 Linear and multilinear algebra; matrix theory (15-XX) 85 Systems theory; control (93-XX) 73 Real functions (26-XX) 67 Measure and integration (28-XX) 67 Approximations and expansions (41-XX) 65 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 62 Differential geometry (53-XX) 57 Mechanics of particles and systems (70-XX) 56 Functions of a complex variable (30-XX) 51 Information and communication theory, circuits (94-XX) 50 Classical thermodynamics, heat transfer (80-XX) 45 Group theory and generalizations (20-XX) 40 Special functions (33-XX) 40 Difference and functional equations (39-XX) 38 Topological groups, Lie groups (22-XX) 37 Mathematical logic and foundations (03-XX) 33 Category theory; homological algebra (18-XX) 33 Potential theory (31-XX) 33 Convex and discrete geometry (52-XX) 31 Integral transforms, operational calculus (44-XX) 30 Several complex variables and analytic spaces (32-XX) 30 Abstract harmonic analysis (43-XX) 28 Relativity and gravitational theory (83-XX) 25 Algebraic geometry (14-XX) 19 Associative rings and algebras (16-XX) 19 Astronomy and astrophysics (85-XX) 18 Sequences, series, summability (40-XX) 16 General and overarching topics; collections (00-XX) 16 Commutative algebra (13-XX) 15 Order, lattices, ordered algebraic structures (06-XX) 14 Geometry (51-XX) 13 Nonassociative rings and algebras (17-XX) 13 Manifolds and cell complexes (57-XX) 12 General algebraic systems (08-XX) 10 History and biography (01-XX) 8 Algebraic topology (55-XX) 7 Field theory and polynomials (12-XX) 5 $$K$$-theory (19-XX) 5 Mathematics education (97-XX)