url
stringlengths 14
2.42k
| text
stringlengths 100
1.02M
| date
stringlengths 19
19
| metadata
stringlengths 1.06k
1.1k
|
|---|---|---|---|
http://www.gradesaver.com/textbooks/math/calculus/calculus-early-transcendentals-8th-edition/chapter-2-section-2-8-the-derivative-as-a-function-2-8-exercises-page-164/49
|
## Calculus: Early Transcendentals 8th Edition
Curve a represents $f$, curve b $f'$ and curve c $f''$.
First, look at curve c. We see that curve c has 2 local extrema (the point at which the graph changes from going up to down and vice versa). Therefore, if curve represents $f$ or $f'$, one of the other curves must have 2 points at which it crosses the $Ox$ line (to change from positive to negative and vice versa). However, both of the other curves each crosses the $Ox$ line only 1 time. Therefore, curve c must represent $f''$. Now look at curve a and curve b. The only local extrema of curve b is near the $Oy$ line. If curve b represents $f$ and curve a represents $f'$, then at the point of the local extrema of curve b, curve a would pass the $Ox$ line. However, curve a does not pass the $Ox$ line at that point. Therefore, curve a represents $f$ and curve $b$ represents $f'$.
|
2017-03-28 23:36:35
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8653564453125, "perplexity": 330.9800854175203}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218190134.25/warc/CC-MAIN-20170322212950-00636-ip-10-233-31-227.ec2.internal.warc.gz"}
|
https://www.physicsforums.com/threads/wrong-value-for-the-voltage-across-a-capacitor.957564/
|
# Homework Help: Wrong value for the voltage across a capacitor
Tags:
1. Oct 13, 2018
### AfterSunShine
1. The problem statement, all variables and given/known data
The current through a 100 microfarad is i(t) = 50 sin(120*pi*t) mA
Calculate the voltage across it at t = 1 ms. Consider zero initial voltage.
2. Relevant equations
v(t) = (1/c) integral of i(t) from t = 0 to t = t
3. The attempt at a solution
Am getting v(t) = - 1.326 ( cos(120*pi*t) - 1 ) V
which gives v(1*10^-3) = -2.87 * 10^(-5) V
But book is saying the voltage should be 93.14 mV
Can any confirm if book answer is correct?
I did the problem like 100 times, and still same value. I considered that given current in mA so I multiplied by 10^-3, and considered the 10^-6 from micro farad. integral of sin(120*p*t) is -cos(120*pi*t) / (120*pi)
I have no idea where is my mistake.
2. Oct 13, 2018
The book's answer is correct, but so is most of your calculations. When you computed $\cos(.377)$ , did you use radians or degrees? You need to use radians. $\\$Edit: And yes, I tried the arithmetic with degrees, and I got very close to your (incorrect) answer, but I get +2.87 E-5, with a "+" sign.
Last edited: Oct 13, 2018
3. Oct 13, 2018
### AfterSunShine
So i substitute pi with 180.
And yes you are correct, while calculating final answer i forget to include the "-" sign
I have negative sign in my v(t), and there will (cos(something) - 1) which will give minus for sure as maximum value for cosine is 1, with the negative in v(t), answer must be positive
Thanks!
|
2018-12-14 04:14:30
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7904598712921143, "perplexity": 2121.03612346703}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376825349.51/warc/CC-MAIN-20181214022947-20181214044447-00012.warc.gz"}
|
https://me.gateoverflow.in/user/Lakshman+Patel+RJIT/activity
|
# Recent activity by Lakshman Patel RJIT
A jigsaw puzzle has $2$ pieces. One of the pieces is shown above. Which one of the given options for the missing piece when assembled will form a rectangle? The piece can be moved, rotated or flipped to assemble with the above piece.
The bar graph shows the data of the students who appeared and passed in an examination for four schools $P, Q, R$, and $S$. The average of success rates $\text{(in percentage)}$ of these four schools is _______. $58.5\%$ $58.8\%$ $59.0\%$ $59.3\%$
The world is going through the worst pandemic in the past hundred years. The air travel industry is facing a crisis, as the resulting quarantine requirement for travelers led to weak demand. In relation to the first sentence above, what does the ... the first sentence Second sentence entirely contradicts the first sentence The two statements are unrelated States an effect of the first sentence
Consider a square sheet of side $1$ unit. The sheet is first folded along the main diagonal. This is followed by a fold along its line of symmetry. The resulting folded shape is again folded along its line of symmetry. The area of each face of the final folded shape, in square units, equal to _________ $\frac{1}{4}$ $\frac{1}{8}$ $\frac{1}{16}$ $\frac{1}{32}$
The ratio of the area of the inscribed circle to the area of the circumscribed circle of an equilateral triangle is ___________ $\frac{1}{8}$ $\frac{1}{6}$ $\frac{1}{4}$ $\frac{1}{2}$
A box contains $15$ blue balls and $45$ black balls. If $2$ balls are selected randomly, without replacement, the probability of an outcome in which the first selected is a blue ball and the second selected is a black ball, is _____ $\frac{3}{16}$ $\frac{45}{236}$ $\frac{1}{4}$ $\frac{3}{4}$
Given below are two statements $1$ and $2$, and two conclusions $\text{I}$ and $\text{II}$. $\text{Statement 1}:$ All entrepreneurs are wealthy. $\text{Statement 2}:$ All wealthy are risk seekers. $\text{Conclusion I}:$ ... $\text{I}$ nor $\text{II}$ is correct Both conclusions $\text{I}$ and $\text{II}$ are correct
The front door of $\text{Mr. X's}$ house faces East. $\text{Mr. X}$ leaves the house, walking $\text{50 m}$ straight from the back door that is situated directly opposite to the front door. He then turns to his right, walks for another $\text{50 m}$ and ... The direction of the point $\text{Mr. X}$ is now located at with respect to the starting point is ____ South-East North-East West North-West
If $\bigoplus \div \bigodot =2;\: \bigoplus \div\Delta =3;\:\bigodot +\Delta =5; \:\Delta \times \bigotimes =10$, Then, the value of $\left ( \bigotimes - \bigoplus \right )^{2}$, is : $0$ $1$ $4$ $16$
A digital watch $\text{X}$ beeps every $30$ seconds while watch $\text{Y}$ beeps every $32$ seconds. They beeped together at $\text{10 AM}$. The immediate next time that they will beep together is ____ $\text{10.08 AM}$ $\text{10.42 AM}$ $\text{11.00 AM}$ $\text{10.00 PM}$
Consider the following sentences: The number of candidates who appear for the $\text{GATE}$ examination is staggering. A number of candidates from my class are appearing for the $\text{GATE}$ examination. The number of candidates who appear for the $\text{GATE}$ examination are staggering. A number of candidates ... $\text{(i) and (iii)}$ $\text{(ii) and (iii)}$ $\text{(ii) and (iv)}$
Five persons $\text{P, Q, R, S}$ and $\text{T}$ are to be seated in a row, all facing the same direction, but not necessarily in the same order. $\text{P}$ and $\text{T}$ cannot be seated at either end of the row. $\text{P}$ should not be seated adjacent ... is to be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is: $2$ $3$ $4$ $5$
$\begin{array}{|c|c|} \hline \textbf{Company} & \textbf{Ratio} \\\hline C1 & 3:2 \\\hline C2 & 1:4 \\\hline C3 & 5:3 \\\hline C4 & 2:3 \\\hline C5 & 9:1 \\\hline C6 & 3:4 \\\hline\end{array}$ The distribution of employees at the rank ... $\textsf{C2}$ and $\textsf{C5}$ together is ________. $225$ $600$ $1900$ $2500$
The number of hens, ducks and goats in farm $P$ are $65,91$ and $169,$ respectively. The total number of hens, ducks and goats in a nearby farm $Q$ is $416.$ The ratio of hens : ducks : goats in farm $Q$ is $5:14:13.$ All the hens, ducks and goats are sent from farm $Q$ to farm $P.$ The new ratio of hens : ducks : goats in farm $P$ is ________ $5:7:13$ $5:14:13$ $10:21:26$ $21:10:26$
Oxpeckers and rhinos manifest a symbiotic relationship in the wild. The oxpeckers warn the rhinos about approaching poachers, thus possibly saving the lives of the rhinos. Oxpeckers also feed on the parasitic ticks found on rhinos. In the symbiotic relationship described above ... a food source, rhinos may be saved from the poachers Oxpeckers save the lives of poachers, rhinos save their own lives
The increased consumption of leafy vegetables in the recent months is a clear indication that the people in the state have begun to lead a healthy lifestyle Which of the following can be logically inferred from the information presented in the above statement ... a diet with leafy vegetables The people in the state have increased awareness of healthy hazards causing by consumption of junk foods
If $\left\{\begin{matrix} “ \oplus” \; \text{means}\; “-” \\ “ \otimes” \; \text{means}\; “\div” \\ “ \triangle” \; \text{means}\; “+” \\ “ \triangledown” \; \text{means}\; “\times” \end{matrix}\right.$ then, the value of the expression $\triangle 2 \oplus 3 \triangle \left((4 \otimes 2) \triangledown 4) \right) =$ $-1$ $-0.5$ $6$ $7$
In the above figure, $\textsf{O}$ is the center of the circle and, $\textsf{M}$ and $\textsf{N}$ lie on the circle. The area of the right triangle $\textsf{MON}$ is $50\;\text{cm}^{2}$. What is the area of the circle in $\text{cm}^{2}?$ $2\pi$ $50\pi$ $75\pi$ $100\pi$
Ms. $X$ came out of a building through its front door to find her shadow due to the morning sun failing to her right side with the building to her back. From this, it can be inferred that building is facing _________ North East West South
Consider the following sentences: After his surgery, Raja hardly could walk. After his surgery, Raja could barely walk. After his surgery, Raja barely could walk. After his surgery, Raja could hardly walk. Which of the above sentences are grammatically $\text{CORRECT}$? $\text{(i) and (ii)}$ $\text{(i) and (iii)}$ $\text{(iii) and (iv)}$ $\text{(ii) and (iv)}$
Five persons $\text{P, Q, R, S and T}$ are sitting in a row not necessarily in the same order. $Q$ and $R$ are separated by one person, and $S$ should not be seated adjacent to $Q.$ The number of distinct seating arrangements possible is: $4$ $8$ $10$ $16$
The mean and variance, respectively, of a binomial distribution for $n$ independent trails with the probability of success as $p$, are $\sqrt{np},np\left ( 1-2p \right )$ $\sqrt{np},\sqrt{np\left ( 1-p \right )}$ $np,np$ $np, np\left ( 1-p \right )$
If the Laplace transform of a function $f(t)$ is given by $\frac{s+3}{\left ( s+1 \right )\left ( s+2 \right )}$, then $f(0)$ is $0$ $\frac{1}{2}$ $1$ $\frac{3}{2}$
Consider an $n \times n$ matrix $\text{A}$ and a non-zero $n \times 1$ vector $\text{p}$. Their product $Ap=\alpha ^{2}p$, where $\alpha \in \Re$ and $\alpha \notin \left \{ -1,0,1 \right \}$. Based on the given information, the eigen value of $A^{2}$ is: $\alpha$ $\alpha ^{2}$ $\surd{\alpha }$ $\alpha ^{4}$
Consider the following differential equation $\left ( 1+y \right )\frac{dy}{dx}=y.$ The solution of the equation that satisfies condition $y(1)=1$ is $2ye^{y}=e^{x}+e$ $y^{2}e^{y}=e^{x}$ $ye^{y}=e^{x}$ $\left ( 1+y \right )e^{y}=2e^{x}$
For a two-dimensional, incompressible flow having velocity components $u$ and $v$ in the $x$ and $y$ directions, respectively, the expression $\frac{\partial \left ( u^{2} \right )}{\partial x}+\frac{\partial \left ( uv \right )}{\partial y}$ ... $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}$
The value of $\int_{0}^{^{\pi }/_{2}}\int_{0}^{\cos\theta }r\sin\theta \:dr\:d\theta$ is $0$ $\frac{1}{6}$ $\frac{4}{3}$ $\pi$
Value of $\left ( 1+i \right )^{8}$, where $i=\sqrt{-1}$, is equal to $4$ $16$ $4i$ $16i$
Find the positive real root of $x^3-x-3=0$ using Newton-Raphson method. lf the starting guess $(x_{0})$ is $2,$ the numerical value of the root after two iterations $(x_{2})$ is ______ ($\textit{round off to two decimal places}$).
Value of $\int_{4}^{5.2} \ln x\: dx$ using Simpson’s one-third rule with interval size $0.3$ is $1.83$ $1.60$ $1.51$ $1.06$
A two dimensional flow has velocities in $x$ and $y$ directions given by $u = 2xyt$ and $v = -y^{2}t$, where $\text{t}$ denotes time. The equation for streamline passing through $x=1,\:y=1$ is $x^{2}y=1$ $xy^{2}=1$ $x^{2}y^{2}=1$ $x/y^{2}=1$
Let the superscript $\text{T}$ represent the transpose operation. Consider the function $f(x)=\frac{1}{2}x^TQx-r^Tx$, where $x$ and $r$ are $n \times 1$ vectors and $\text{Q}$ is a symmetric $n \times n$ matrix. The stationary point of $f(x)$ is $Q^{T}r$ $Q^{-1}r$ $\frac{r}{r^{T}r}$ $r$
A $\text{PERT}$ network has $9$ activities on its critical path. The standard deviation of each activity on the critical path is $3$. The standard deviation of the critical path is $3$ $9$ $27$ $81$
Let $\text{C}$ represent the unit circle centered at origin in the complex plane, and complex variable, $z=x+iy$. The value of the contour integral $\oint _{C}\dfrac{\cosh \:3z}{2z}\:dz$ (where integration is taken counter clockwise) is $0$ $2$ $\pi i$ $2 \pi i$
The ordinary differential equation $\dfrac{dy}{dt}=-\pi y$ subject to an initial condition $y\left ( 0 \right )=1$ is solved numerically using the following scheme: $\frac{y\left ( t_{n+1} \right )-y\left ( t_{n} \right )}{h}=-\pi y\left ( t_{n} \right )$ where $\text{h}$ is the ... $h$ in the interval ___________________. $0< h< \frac{2}{\pi }$ $0< h< 1$ $0< h< \frac{\pi }{2}$ for all $h> 0$
The value of $\displaystyle{} \lim_{x \rightarrow 0} \left( \frac{1 – \cos x}{x^{2}}\right)$ is $\frac{1}{4}$ $\frac{1}{3}$ $\frac{1}{2}$ $1$
Consider a binomial random variable $\text{X}$. If $X_{1},X_{2},\dots ,X_{n}$ are independent and identically distributed samples from the distribution of $\text{X}$ with sum $Y=\sum_{i=1}^{n}X_{i}$, then the distribution of $\text{Y}$ as $n\rightarrow \infty$ can be approximated as Exponential Bernoulli Binomial Normal
The Dirac-delta function $\left ( \delta \left ( t-t_{0} \right ) \right )$ for $\text{t}$, $t_{0} \in \mathbb{R}$ ... $\mathcal{L}\left ( \delta \left ( t-a \right ) \right )=F\left ( s \right )$ is $0$ $\infty$ $e^{sa}$ $e^{-sa}$
If $y(x)$ satisfies the differential equation $(\sin x) \dfrac{\mathrm{d}y }{\mathrm{d} x} + y \cos x = 1,$ subject to the condition $y(\pi/2) = \pi/2,$ then $y(\pi/6)$ is $0$ $\frac{\pi}{6}$ $\frac{\pi}{3}$ $\frac{\pi}{2}$
Let $f\left ( x \right )=x^{2}-2x+2$ be a continuous function defined on $x \in \left [ 1,3 \right ]$. The point $x$ at which the tangent of $f\left ( x \right )$ becomes parallel to the straight line joining $f\left ( 1 \right )$ and $f\left ( 3 \right )$ is $0$ $1$ $2$ $3$
|
2021-06-19 09:50:20
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6719507575035095, "perplexity": 303.01090835346986}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487647232.60/warc/CC-MAIN-20210619081502-20210619111502-00339.warc.gz"}
|
http://parabola.unsw.edu.au/2000-2009/volume-41-2005/issue-2
|
Volume 41
, Issue 2
2005
Welcome to this issue of Parabola which contains, in addition to regular articles and problems, the list of prizewinners and complete solutions for the UNSW School of Mathematics Competition in 2005.
When I was a young mathematics student, I often wondered whether there was an easy way of checking determinants. By recently studying the checking of contractants I found there is a fairly easy way to accomplish this.
We begin by looking at the definition of an oval, or as it is more formally known, an ellipse. An ellipse is the set of all points $(x, y)$ in the plane such that the sum of the distances from $(x, y)$ to two fixed points is some constant.
In my previous column, I outlined the story of the most recent extension of the number system, so that it expanded to include "infinitesimals", numbers smaller than any of our familiar real numbers, and yet not the same as zero.
Prize Winners – Junior Division
First Prize
Vinoth Nandakumar James Ruse Agricultural High School
Q1181. Consider the following set of linear equations
\begin{eqnarray*}
x+2y+z&=&1\\
-2x+\lambda y-2z&=&-2\\
2x +6y+ 2\lambda z&=&3
\end{eqnarray*}
Q1171. The first digit of a $6$-digit number is $1$. If the $1$ is shifted to the other end, the new number is $3$ times the original number. Find this number.
|
2017-07-26 04:33:57
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3955431282520294, "perplexity": 396.7462277402549}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549425766.58/warc/CC-MAIN-20170726042247-20170726062247-00261.warc.gz"}
|
https://plainmath.net/15384/write-the-prime-factorization-1-20-2-36-3-28-4-60
|
# Write the prime factorization 1)20 2)36 3)28 4)60
Polynomial factorization
Write the prime factorization
1)20
2)36
3)28
4)60
2021-05-04
Step 1
Prime numbers are those numbers whose only multiples are 1 and the number itself.
Prime factorization is basically a process to find which prime numbers can be multiplied to get the original number.
Step 2
1. Factors of 20 are 1,2,4,5,10 and 20
Hence, prime factorization of 20 is, $$20 =2 \times 2 \times 5$$
2. Factors of 36 are 1,2,3,4,6,9,12,18,36.
Hence, prime factorization of 36 is, $$36 =2 \times 2 \times 3 \times 3$$
3. Factors of 28 are 1,2,4,7,14,28.
Hence, prime factorization of 28 is, $$28 =2 \times 2 \times 7$$
4. Factors of 60 are 1,2,3,4,5,6,10,12,15,20,30,60.
Hence, prime factorization of 60 is, $$20 =2 \times 2 \times 3 \times 5$$
|
2021-07-24 15:26:37
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8854461908340454, "perplexity": 2076.032773017315}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046150266.65/warc/CC-MAIN-20210724125655-20210724155655-00274.warc.gz"}
|
https://www.gamedev.net/forums/topic/360663-ms-paint-effect-in-resizing-the-screen/
|
# OpenGL MS Paint effect in resizing the screen?
## Recommended Posts
I have made a basic paint application with Opengl using Glut I want the screen to resize just like MS Paint. That means the top left hand corner stays where it is when a resize happens. So the camera stays where it is, and only the right or bottom edges move. Heres what happens when I resize the screen: Now i resize a little, but it moves the camera to the center of the image This is what I want to happen: Here is the code for my resize:
void changeSize(int w, int h)
{
glViewport(0, 0, w, h);
glMatrixMode(GL_PROJECTION);
halfw = w / 2;
halfh = h / 2;
glOrtho(-halfw,halfw,-halfh,halfh,-1.0,1.0);
}
What do I need to change in glViewport or glOrtho to get this to happen?
##### Share on other sites
The parameters you are passing to your glOrtho call cause OpenGL's projection matrix to be initialized with the origin (0, 0) at the center of the screen. This is why when you resize, the image is centered in the middle.
If you set your ortho matrix to be aligned to (0, 0) rather than (-halfw, -halfh), it should work.
Try this:
glOrtho(0, w, 0, h, -1.0, 1.0);
## Create an account
Register a new account
• ### Forum Statistics
• Total Topics
627701
• Total Posts
2978708
• ### Similar Content
• A friend of mine and I are making a 2D game engine as a learning experience and to hopefully build upon the experience in the long run.
-What I'm using:
C++;. Since im learning this language while in college and its one of the popular language to make games with why not. Visual Studios; Im using a windows so yea. SDL or GLFW; was thinking about SDL since i do some research on it where it is catching my interest but i hear SDL is a huge package compared to GLFW, so i may do GLFW to start with as learning since i may get overwhelmed with SDL.
-Questions
Knowing what we want in the engine what should our main focus be in terms of learning. File managements, with headers, functions ect. How can i properly manage files with out confusing myself and my friend when sharing code. Alternative to Visual studios: My friend has a mac and cant properly use Vis studios, is there another alternative to it?
• Both functions are available since 3.0, and I'm currently using glMapBuffer(), which works fine.
But, I was wondering if anyone has experienced advantage in using glMapBufferRange(), which allows to specify the range of the mapped buffer. Could this be only a safety measure or does it improve performance?
Note: I'm not asking about glBufferSubData()/glBufferData. Those two are irrelevant in this case.
• By xhcao
Before using void glBindImageTexture( GLuint unit, GLuint texture, GLint level, GLboolean layered, GLint layer, GLenum access, GLenum format), does need to make sure that texture is completeness.
• By cebugdev
hi guys,
are there any books, link online or any other resources that discusses on how to build special effects such as magic, lightning, etc. in OpenGL? i mean, yeah most of them are using particles but im looking for resources specifically on how to manipulate the particles to look like an effect that can be use for games,. i did fire particle before, and I want to learn how to do the other 'magic' as well.
Like are there one book or link(cant find in google) that atleast featured how to make different particle effects in OpenGL (or DirectX)? If there is no one stop shop for it, maybe ill just look for some tips on how to make a particle engine that is flexible enough to enable me to design different effects/magic
let me know if you guys have recommendations.
• By dud3
How do we rotate the camera around x axis 360 degrees, without having the strange effect as in my video below?
Mine behaves exactly the same way spherical coordinates would, I'm using euler angles.
Tried googling, but couldn't find a proper answer, guessing I don't know what exactly to google for, googled 'rotate 360 around x axis', got no proper answers.
References:
Code: https://pastebin.com/Hcshj3FQ
The video shows the difference between blender and my rotation:
• 21
• 14
• 12
• 10
• 12
|
2017-10-21 08:41:16
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17368221282958984, "perplexity": 1807.230142843157}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824675.67/warc/CC-MAIN-20171021081004-20171021101004-00743.warc.gz"}
|
http://en.wikipedia.org/wiki/Hypercube_graph
|
# Hypercube graph
Hypercube graph
The hypercube graph Q4
Vertices 2n
Edges 2n−1n
Diameter n
Girth 4 if n≥2
Automorphisms n! 2n
Chromatic number 2
Spectrum $\{(n - 2 k)^{\binom{n}{k}}; k = 0, \ldots, n\}$
Properties Symmetric
Distance regular
Unit distance
Hamiltonian
Bipartite
Notation Qn
In graph theory, the hypercube graph Qn is a regular graph with 2n vertices, 2n−1n edges, and n edges touching each vertex. It can be obtained as the one-dimensional skeleton of the geometric hypercube; for instance, Q3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Alternatively, it can be obtained from the family of subsets of a set with n elements, by making a vertex for each possible subset and joining two vertices by an edge whenever the corresponding subsets differ in a single element.
Hypercube graphs should not be confused with cubic graphs, which are graphs that have exactly three edges touching each vertex. The only hypercube that is a cubic graph is Q3.
## Construction
Construction of Q3 by connecting pairs of corresponding vertices in two copies of Q2
The hypercube graph Qn may be constructed from the family of subsets of a set with n elements, by making a vertex for each possible subset and joining two vertices by an edge whenever the corresponding subsets differ in a single element. Equivalently, it may be constructed using 2n vertices labeled with n-bit binary numbers and connecting two vertices by an edge whenever the Hamming distance of their labels is 1. These two constructions are closely related: a binary number may be interpreted as a set (the set of positions where it has a 1 digit), and two such sets differ in a single element whenever the corresponding two binary numbers have Hamming distance 1.
Alternatively, Qn+1 may be constructed from the disjoint union of two hypercubes Qn, by adding an edge from each vertex in one copy of Qn to the corresponding vertex in the other copy, as shown in the figure. The joining edges form a perfect matching.
Another definition of Qn is the Cartesian product of n two-vertex complete graphs K2.
## Examples
The graph Q0 consists of a single vertex, while Q1 is the complete graph on two vertices and Q2 is a cycle of length 4.
The graph Q3 is the 1-skeleton of a cube, a planar graph with eight vertices and twelve edges.
The graph Q4 is the Levi graph of the Möbius configuration.
## Properties
### Bipartiteness
Every hypercube graph is bipartite: it can be colored with only two colors. The two colors of this coloring may be found from the subset construction of hypercube graphs, by giving one color to the subsets that have an even number of elements and the other color to the subsets with an odd number of elements.
### Hamiltonicity
Every hypercube Qn with n > 1 has a Hamiltonian cycle, a cycle that visits each vertex exactly once. Additionally, a Hamiltonian path exists between two vertices u,v if and only if have different colors in a 2-coloring of the graph. Both facts are easy to prove using the principle of induction on the dimension of the hypercube, and the construction of the hypercube graph by joining two smaller hypercubes with a matching.
Hamiltonicity of the hypercube is tightly related to the theory of Gray codes. More precisely there is a bijective correspondence between the set of n-bit cyclic Gray codes and the set of Hamiltonian cycles in the hypercube Qn.[1] An analogous property holds for acyclic n-bit Gray codes and Hamiltonian paths.
A lesser known fact is that every perfect matching in the hypercube extends to a Hamiltonian cycle.[2] The question whether every matching extends to a Hamiltonian cycle remains an open problem.[3]
### Other properties
The hypercube graph Qn (n > 1) :
• has more than 22n-2 perfect matchings. (this is another consequence that follows easily from the inductive construction.)
• contains all the cycles of length 4, 6, ..., 2n and is thus a bipancyclic graph.
• can be drawn as a unit distance graph in the Euclidean plane by choosing a unit vector for each set element and placing each vertex corresponding to a set S at the sum of the vectors in S.
• is planar (can be drawn with no crossings) if and only if n ≤ 3. For larger values of n, the hypercube has genus $(n-4)2^{n-3}+1$.[4][5]
• has exactly $2^{2^n-n-1}\prod_{k=2}^n k^{{n\choose k}}$ spanning trees.[5]
• The achromatic number of Qn is known to be proportional to $\sqrt{n2^n}$, but the constant of proportionality is not known precisely.[6]
• The bandwidth of Qn is exactly $\sum_{i=0}^n \binom{n}{\lfloor n/2\rfloor}$.[7]
• The eigenvalues of the adjacency matrix are (-n,-n+2,-n+4,...,n-4,n-2,n) and the eigenvalues of its Laplacian are (0,2,...,2n). The k-th eigenvalue has multiplicity $\binom{n}{k}$ in both cases.
## Problems
The problem of finding the longest path or cycle that is an induced subgraph of a given hypercube graph is known as the snake-in-the-box problem.
Szymanski's conjecture concerns the suitability of a hypercube as an network topology for communications. It states that, no matter how one chooses a permutation connecting each hypercube vertex to another vertex with which it should be connected, there is always a way to connect these pairs of vertices by paths that do not share any directed edge.[8]
|
2014-11-22 23:43:21
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 6, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8266383409500122, "perplexity": 318.60448325538965}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400378815.18/warc/CC-MAIN-20141119123258-00019-ip-10-235-23-156.ec2.internal.warc.gz"}
|
https://www.trustudies.com/question/1891/4-a-simplify-3x-4x-5-3-and-find-its-v/
|
Premium Online Home Tutors
3 Tutor System
Starting just at 265/hour
# 4.(a) Simplify: $$3x(4x – 5) + 3$$ and find its values for (i) x = 3 (ii) x = $$\frac { 1 }{ 2 }$$. (b) Simplify: $$a(a^2 + a + 1) + 5$$ and find its value for (i) a = 0 (ii) a = 1 (iii) a = -1
(a) Given the expression: $$3x(4x – 5) + 3 = 4x \times 3x – 5 \times 3x + 3 = 12x^2 – 15x + 3$$
(i) So for x = 3, we have
$$12 \times (3)2 – 15 \times 3 + 3 = 12 \times 9 – 45 + 3 = 108 – 42 = 66$$
(ii)For x=$$\frac{1}2$$, we have:
$$\Rightarrow 12(\frac{1}2)^2 - 15(\frac{1}2)+3$$
$$\Rightarrow 12\times(\frac{1}4) - \frac{15}2+3$$
$$\Rightarrow 3 - \frac{15}2+3$$
$$\Rightarrow \frac{6-15+6}2=\frac{12-15}2=-\frac{3}2$$
(b) We have $$a(a^2 + a + 1) + 5$$
$$= (a^2 \times a) + (a \times a) + (1 \times a) + 5$$
$$= a^3 + a^2 + a + 5$$
(i) For a = 0, we have:
$$= (0)^3 + (0)^2 + (0) + 5 = 5$$
(ii) For a = 1, we have:
$$= (1)^3 + (1)^2 + (1) + 5 = 1 + 1 + 1 + 5 = 8$$
(iii) For a = -1, we have:
$$= (-1)^3 + (-1)^2 + (-1) + 5 = -1 + 1 – 1 + 5 = 4$$
|
2023-01-31 10:02:12
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8135970830917358, "perplexity": 2699.9686337355924}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499857.57/warc/CC-MAIN-20230131091122-20230131121122-00102.warc.gz"}
|
https://zbmath.org/?q=an%3A1094.14025
|
## The moduli $$b$$-divisor of an lc-trivial fibration.(English)Zbl 1094.14025
The paper under review studies lc-trivial fibrations. These are morphisms $$f\:(X,B)\to Y$$ with relatively trivial log canonical class $$K_X+B$$. In this context there is a discriminant divisor that measures the singularities of the log pair $$(X,B)$$ over codimension one points of $$Y$$ and a moduli divisor $$M_Y$$, first appeared in [Y. Kawamata, in: Birational algebraic geometry. Conf. Algebraic Geometry, in memory of Wei-Liang Chow (1911–1995), Baltimore 1996. Contemp. Math. 207, 79–88 (1997; Zbl 0901.14004)]. The paper proves a number of technical results related to semi-ampleness of $$M_Y$$. As an application one gets the following interesting application. Let $$(X,B)$$ be a projective log variety with Kawamata log terminal singularities such that $$K_X+B$$ is numerically trivial, then there exists a positive integer such that $$b(K_X+B)\sim0$$, and the Albanese map $$X\to\text{ Alb}(X)$$ is a surjective morphism with connected fibres.
### MSC:
14J10 Families, moduli, classification: algebraic theory 14E30 Minimal model program (Mori theory, extremal rays) 14N30 Adjunction problems
Zbl 0901.14004
Full Text:
|
2022-06-30 22:40:43
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8265316486358643, "perplexity": 879.6817387853068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103915196.47/warc/CC-MAIN-20220630213820-20220701003820-00208.warc.gz"}
|
https://listserv.uni-heidelberg.de/cgi-bin/wa?A2=LATEX-L;aab20037.1203&FT=P&P=4516443&H=A&S=a
|
## LATEX-L@LISTSERV.UNI-HEIDELBERG.DE
Options: Use Forum View Use Monospaced Font Show Text Part by Default Condense Mail Headers Message: [<< First] [< Prev] [Next >] [Last >>] Topic: [<< First] [< Prev] [Next >] [Last >>] Author: [<< First] [< Prev] [Next >] [Last >>]
Hi Michal,
>> with that everything seems to be working.
>>
> This results in missing HTML footer in the last generated HTML file.
> Fortunately,
>
> {\HtmlEnv\Configure{newpage}{}\at:docend \csname export:hook\endcsname}
>
> works.
oh well, you can't expect me to fully understand this wonderfully
documented code, can you? :-) good then that was so simple to adjust
>
>> However, a lot of the lowlevel patching code inside tex4ht seems to be
>> only necessary because environment hooks are not available, so after
>> adjusting \begin and \end to use the kernel definitions it should be
>> possible to simplify a lot by simply using appropriate "env/foo/before"
>> and "env/foo/after" hooks.
> I hope it will be relatively easy to adapt our environment patching
> routine to the new LaTeX hook system. It should definitely simplify
> lot of the low-level code.
well with the above minimal surgery being successful there is no rush I
guess. But, of course, if you can look at it earlier the better since
that would allow us jointly to see if something seems to be missing or
incorrectly implemented. This is the main reason we we tried to give al
of us 2-3 month prior to the official release to weed out anything still
wrong or questionable.
So if you are going to give it a go and you run into questions or
problems, give us a ring, either here, or as an github issue or by
emailing the team and we are glad to assist, after all, it is in our all
interest that this works smoothly without any problems for user workflows.
|
2023-04-02 08:01:56
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.772358775138855, "perplexity": 6810.911961462142}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296950422.77/warc/CC-MAIN-20230402074255-20230402104255-00137.warc.gz"}
|
https://www.freemathhelp.com/forum/threads/needing-help-isolating-the-variable-ch.47351/
|
# needing help isolating the variable Ch
#### mathnubb
##### New member
I'm not terribly good with algebra, and I need to isolate a variable in the following equation:
1 2 3
Mh x Ch x DeltaTh = Mc x Cc x DeltaTc
I need to isolate the varible Ch, and I can't seem to do it properly. Could somebody kindly give me a hand? Thank you!
___________________
Edited by stapel -- Reason for edit: "language", punctuation, etc
#### stapel
##### Super Moderator
Staff member
What do the "1 2 3" digits have to do with the equation in the next line? You say that "Ch" is a variable, but usually C and h would be separate variables, which leads to questions regarding what the others might might. For instance, is "DeltaTc" one variable (perhaps "$$\displaystyle \Delta(Tc)$$"), two, or three? Why is the variable "x" scattered throughout, rather than gathered together?
Eliz.
#### mathnubb
##### New member
This is a physics/chemistry question. Sorry; I'll try to explain it more clearly this time. The variables are defined like this:
. . .Mh: mass of hot metal
. . .Ch: heat capacity of hot metal
. . .DeltaTh: Change in temperature of Hot metal
. . .Mc: mass of cold water
. . .Cc: heat capacity of water
. . .DeltaTc: Change in temperature of Cold water
The "c" and "h" are just subscripts. The x's indicate multiplication.
I'm trying to isolate "Ch". When substituting values for the variables, I'm not ending up with a heat capacity anywhere near what I think the answer ought to be.
Using that equation above, I change the DeltaT's to "Tfc - Tic" and "Tfh - Tih", but since "Tf" for both the water and the metal are the same, the two can just be called "Tf". Then I make this long eqaution:
. . .MhChTf - MhChTih = McCcTf - McCcTic
The common factor is Ch; this is where I get lost.
. . .Ch(MhTf - MhTih) = McCcTf - McCcTic
. . .Ch = (McCcTf - McCcTic) / (MhTf - MhTih)
Then i would subststite values in to find Ch, but it always comes out negative. Any advice?
#### stapel
##### Super Moderator
Staff member
So the equation is as follows...?
. . . . .M<sub>h</sub> C<sub>h</sub> delta-T<sub>h</sub> = M<sub>c</sub> C<sub>c</sub> delta-T<sub>c</sub>
And you're wanting to solve for C<sub>h</sub>...?
Since all that's been done is multiplication, just divide off the other two factors:
. . . . .$$\displaystyle \L C_h\, =\, \frac{M_c C_c \Delta T_c}{M_h \Delta T_h}$$
Since we don't have the chemico-physical information (numerical values, chemical relationships, etc), we cannot speak to the answer(s) you are obtaining. Sorry.
Eliz.
#### mathnubb
##### New member
thank you man, it was actually easier than i did it lol, i always always overlook things
|
2019-03-23 10:42:36
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6263129115104675, "perplexity": 4367.988016843355}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202781.83/warc/CC-MAIN-20190323101107-20190323123107-00227.warc.gz"}
|
https://answers.ros.org/answers/17102/revisions/
|
# Revision history [back]
To clarify fergs' answer, it looks like you've done
#include <pcl/point_cloud.h>
#include <pcl/point_types.h>
but not
#include <pcl_ros/point_cloud.h>
The pcl/XXX headers define pcl::PointCloud and pcl::PointXYZ types used by PCL. However, PCL is a stand-alone library that knows nothing about ROS. If you include only those headers, ROS doesn't know what to do with pcl::PointCloud, because it's not a native ROS message type. Thus the ROS serialization code barfs.
pcl_ros/point_cloud.h contains the glue code that tells ROS how to serialize/deserialize pcl::PointCloud. This code serializes pcl::PointCloud into exactly the same over-the-wire format as the ROS-native sensor_msgs/PointCloud2 message type, which is why you can use them interchangeably in your subscriber code.
If you exactly cut-and-paste the subscribing example, does it work?
|
2022-06-26 01:45:39
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2374783158302307, "perplexity": 11067.27359152723}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103036363.5/warc/CC-MAIN-20220626010644-20220626040644-00511.warc.gz"}
|
http://home.iitk.ac.in/~vagarwal/pub.html
|
# Publications
## Filter by type:
Abstract: We have performed exact classical rate calculations to compute adsorption and desorption rate constants with a model representative of a real system. We compute the desorption rate using transition-state theory by taking the dividing-surface far from the surface of the solid. We find that using a mean-field assumption, i.e., applying potential of mean force to transition state theory, could lead to two orders-of-magnitude errors in the rate constant owing to large fluctuations in the desorption barrier. Furthermore, we compute the adsorption rate by including a dynamical factor which reflects the probability of sticking to the solid surface. We find that the sticking probability is highly sensitive to the coverage. Also, we find that the adsorption rate computed from the mean-field assumption is not very different from the exact adsorption rate. We also compute entropic contribution to desorption rates and compare it to that obtained from two limiting models of adsorption—2D ideal gas and 2D ideal lattice gas. We show that at high temperatures (700 K), the entropic contribution to desorption rates computed from the exact calculations is very close to that obtained from the 2D ideal gas model. However, for lower to intermediate temperatures from 200 K to 500 K, the entropic contributions cover a wide range which lies in between the two limiting models and could lead to over two-orders-of-magnitude errors in the rate coefficient.
Abstract: We review here some aspects of computational work on the catalytic chemistry of oxides. The difficulties of using density functional theory in calculations are explained. Different ways of structural or chemical modifications aimed at improving catalytic activity are reviewed. The reaction mechanism of partial oxidation reaction catalyzed by oxides is discussed. The focus is on qualitative design rules rather than on obtaining highly accurate computational results.
Abstract: ....
Abstract: Metals that are active catalysts for methane (Ni, Pt, Pd), when dissolved in inactive low–melting temperature metals (In, Ga, Sn, Pb), produce stable molten metal alloy catalysts for pyrolysis of methane into hydrogen and carbon. All solid catalysts previously used for this reaction have been deactivated by carbon deposition. In the molten alloy system, the insoluble carbon floats to the surface where it can be skimmed off. A 27% Ni–73% Bi alloy achieved 95% methane conversion at 1065°C in a 1.1-meter bubble column and produced pure hydrogen without CO2 or other by-products. Calculations show that the active metals in the molten alloys are atomically dispersed and negatively charged. There is a correlation between the amount of charge on the atoms and their catalytic activity.
Abstract:MoO3 is a versatile catalyst for oxidation reactions that consists of bilayers connected by van der Waals interaction. In principle, a MoO3 nanocrystal can be exfoliated to create two-dimensional ribbons. For this article, we study the difference between the chemistry of slabs having a variety of crystal faces and that of the edges of ribbons cut from a two-dimensional bilayer. As a descriptor of chemical reactivity we use the energy of oxygen-vacancy formation: the easier it is to form an oxygen vacancy, the better oxidant the face of a slab or the edge of a two-dimensional ribbon is. We find that the properties of ribbon edges are different from those of the corresponding slab surfaces. The surface energies of slabs are in the order (010)s < (100)s < (101)s < (001)s, whereas the edge energies of ribbons are in the order ⟨100⟩r ≈ ⟨101⟩r < ⟨001⟩r (the subscript s indicates a slab, and r, a ribbon). Among the surfaces studied, we have found that (001)s and (101)s faces have the lowest oxygen-vacancy formation energies, and (010)s has the highest. In contrast, among the edges studied, ⟨101⟩r has the lowest vacancy formation energies. Our calculations suggest that no benefit is obtained by creating ⟨100⟩r or ⟨001⟩r ribbon edges. However, a significant decrease of oxygen-vacancy formation energies is observed on formation of ⟨101⟩r edge by exfoliating (101)s slabs. Also, among the structures studied, we found ⟨101⟩r edges to be the most reactive and (010)s surfaces to be the least reactive.
Abstract:Oxygen vacancy formation energies are often used as a descriptor of the catalytic activity of metal oxides for oxidation reactions having the Mars–van Krevelen mechanism. When these energies are calculated, it is often assumed that they depend only on the concentration of the vacancies in the top oxygen layer. Previous work has shown that in the case of TiO2 and V2O5, the energy of vacancy formation depends not only on their concentration but also on the manner in which they are distributed on the surface. However, the energy change due to the change of configuration in these systems is very small. Here, we find that in the case of α-MoO3(010) the dependence on the energy of vacancy formation of the distribution of vacancies is very large: if the lattice made by the vacancies consists of parallelograms, the energy of vacancy formation is 0.4 eV smaller than when the lattice consists of rectangles (the two systems having the same vacancy concentration).
Abstract: Density functional theory is used to determine differences in hydrogen abstraction and ammonia binding energies between two zeolites (BEA and MFI-type) and two α-quartz surfaces doped with Al, B, Sc, or Ga. One of the questions we wanted to answer is whether the fact that zeolite cages are made of a silica monolayer plays any role in their catalytic activity. We find no important difference. Doped α-quartz has acid hydroxyls such as those in zeolites; however, their density is very low, and doped quartz is not a shape selective catalyst. Therefore, the doped silica examined here is an inferior acid catalyst when compared to BEA or MFI.
Abstract:We use the Potts-lattice gas model to study nucleation at and near the eutectic composition. We use rare-event methods to compute the free energy landscape for the competing nucleation products, and short trajectories at the barrier top to obtain prefactors. We introduce a procedure to tune the frequency of semigrand Monte Carlo moves so that the dynamics of a small closed system roughly resemble those of an infinite system. The non- dimensionalized nucleation rates follow trends as predicted by the classical nucleation theory. Finally, we develop corrections that convert free energy surfaces from closed (canonical) simulations into free energy surfaces from open (semigrand) simulations. The new corrections extend earlier corrections to now address situations like nucleation at the eutectic point where two products nucleate competitively.
Abstract: This chapter focuses on a frontier for molecular simulation: nucleation of solute precipitates from solution. The chapter refers to simulations of solute precipitate nucleation with a fixed number of solute and solvent molecules simulations of “closed systems”. Two simple cases of heterogeneous nucleation are considered: (i) a spherical cap model for nucleation on a hard flat surface and (ii) a lens model for nucleation at fluid–fluid interfaces. Over the decade following the discovery of two'step nucleation, several computational studies have pointed to two'step nucleation as a more general phenomena occurring even at points where there is no metastable fluid–fluid critical point. Rare events simulation methods can relax many of the assumptions made by classical nucleation theory (CNT). The chapter concludes with case studies on laser'induced nucleation, and on nucleation of methane hydrates and calcium carbonate.
Abstract:We modeled nascent decomposition processes in cellulose pyrolysis at 327 and 600 °C using Car–Parrinello molecular dynamics (CPMD) simulations with rare events accelerated with the metadynamics method. We used a simulation cell comprised of two unit cells of cellulose Iβ periodically repeated in three dimensions to mimic the solid cellulose. To obtain initial conditions at reasonable densities, we extracted coordinates from larger classical NPT simulations at the target temperatures. CPMD-metadynamics implemented with various sets of collective variables, such as coordination numbers of the glycosidic oxygen, yielded a variety of chemical reactions such as depolymerization, fragmentation, ring opening, and ring contraction. These reactions yielded precursors to levoglucosan (LGA)—the major product of pyrolysis—and also to minor products such as 5-hydroxy-methylfurfural (HMF) and formic acid. At 327 °C, we found that depolymerization via ring contraction of the glucopyranose ring to the glucofuranose ring occurs with the lowest free-energy barrier (20 kcal/mol). We suggest that this process is key for formation of liquid intermediate cellulose, observed experimentally above 260 °C. At 600 °C, we found that a precursor to LGA (pre-LGA) forms with a free-energy barrier of 36 kcal/mol via an intermediate/transition state stabilized by anchimeric assistance and hydrogen bonding. Conformational freedom provided by expansion of the cellulose matrix at 600 °C was found to be crucial for formation of pre-LGA. We performed several comparison calculations to gauge the accuracy of CPMD-metadynamics barriers with respect to basis set and level of theory. We found that free-energy barriers at 600 °C are in the order pre-LGA < pre-HMF < formic acid, explaining why LGA is the kinetically favored product of fast cellulose pyrolysis.
Abstract:We have modeled the transformation of cellulose Iβ to a high temperature (550 K) structure, which is considered to be the first step in cellulose pyrolysis. We have performed molecular dynamics simulations at constant pressure using the GROMOS 45a4 united atom forcefield. To test the forcefield, we computed the density, thermal expansion coefficient, total dipole moment, and dielectric constant of cellulose Iβ, finding broad agreement with experimental results. We computed infrared (IR) spectra of cellulose Iβ over the range 300–550 K as a probe of hydrogen bonding. Computed IR spectra were found to agree semi-quantitatively with experiment, especially in the O–H stretching region. We assigned O–H stretches using a novel synthesis of normal mode analysis and power spectrum methods. Simulated IR spectra at elevated temperatures suggest a structural transformation above 450 K, a result in agreement with experimental IR results. The low-temperature (300–400 K) structure of cellulose Iβ is dominated by intrachain hydrogen bonds, whereas in the high-temperature structure (450–550 K), many of these transform to longer, weaker interchain hydrogen bonds. A three-dimensional hydrogen bonding network emerges at high temperatures due to formation of new interchain hydrogen bonds, which may explain the stability of the cellulose structure at such high temperatures.
Abstract:We have studied base strengths of nitrogen-substituted (nitrided) zeolites with faujasite (FAU) structure by calculating sorption energies of probe molecules (BF3 and BH3) using density functional theory with mixed basis sets applied to embedded clusters. BH3 was found to be a better probe of base strength because it does not introduce competing metal−fluorine interactions that obfuscate trends. In all cases, the base strengths of nitrided zeolites (denoted M−N−Y) were found to exceed those of the corresponding standard M−Y zeolites, where M = Li, Na, K, Rb, or Cs charge-compensating cations. We have found that for a particular Si:Al ratio, BH3 sorption energies vary in the order Li < Na < K ∼ Rb ∼ Cs. Sorption energy and hence base strength was found to decrease with increasing Si:Al ratio from 1 to 3 beyond which the base strength was found to increase again. The initial regime (1 < Si:Al < 3) is consistent with the prevailing understanding that the base strength increases with Al content, while the latter regime (Si:Al > 3) involves the surprising prediction that the base strength can be relatively high for the more stable, high-silica zeolites. In particular, we found the sorption energy in Na−N−Y (Si:Al = 11) to be nearly equal to that in (Si:Al = 1). Taken together, these results suggest that K−N−Y (Si:Al = 11) optimizes the balance of activity, stability, and cost.
Abstract: We have modeled the formation kinetics of nitrogen-substituted (nitrided) zeolites HY and silicalite; we have also modeled the stability of nitrided sites to heat and humidity. These kinetic calculations are based on mechanisms computed from DFT-computed pathways reported in our previous work. Reactant ammonia and product water concentrations were fixed at various levels to mimic continuous nitridation reactors. We have found that zeolite nitridation — replacing Si–O–Si and Si–OH–Al linkages with Si–NH–Si and Si–NH2–Al, respectively — proceeds only at high temperatures (>$600\phantom{\rule{0.25em}{0ex}}°\text{C}$ for silicalite and >$650\phantom{\rule{0.25em}{0ex}}°\text{C}$ for HY) due to the presence of large overall barriers. These threshold temperatures are in good agreement with experiments. Nitridation yields were found to be sensitive to water concentration, especially for silicalite where nitridation is more strongly endothermic. As a result, overall nitridation yields in silicalite are predicted to be much lower than those in HY. The stability of nitrided sites was investigated by modeling the kinetics of nitridation in reverse, going back to untreated zeolite plus ammonia. Using 10 h as a benchmark catalyst lifetime, nitrided silicalite and HY half-lives exceeded 10 h for temperatures below 275 and 500 °C, respectively, even at saturation water loadings. As such, our calculations suggest that nitrided silicalite and HY zeolites require high temperatures to form, but once formed, they remain relatively stable, auguring well for their use as shape-selective base catalysts.
Abstract: We have performed embedded-cluster calculations using density functional theory to investigate mechanisms of nitrogen substitution (nitridation) in HY and silicalite zeolites. We consider nitridation as replacing Si–O–Si and Si–OH–Al linkages with Si–NH–Si and Si–NH2–Al, respectively. We predict that nitridation is much less endothermic in HY (29 kJ/mol) than in silicalite (132 kJ/mol), indicating the possibility of higher nitridation yields in HY. To reveal mechanistic details, we have combined for the first time the nudged elastic band method of finding elusive transition states, with the ONIOM method of treating embedded quantum clusters. We predict that nitridation of silicalite proceeds via a planar intermediate involving a
ring with pentavalent Si, whereas nitridation of HY is found to proceed via an intermediate similar to physisorbed ammonia. B3LYP/6-311G(d,p) calculations give an overall barrier for silicalite nitridation of 343 kJ/mol, while that in HY is 359 kJ/mol. Although the overall nitridation barriers are relatively high, requiring high temperatures for substitution, the overall barriers for the reverse processes are also high. As such, we predict that once these catalysts are made, they remain relatively stable.
Abstract: Nanoporous acid catalysts such as zeolites form the backbone of catalytic technologies for refining petroleum. With the promise of a biomass economy, new catalyst systems will have to be discovered, making shape-selective base catalysts especially important because of the high oxygen content in biomass-derived feedstocks. Strongly basic zeolites are attractive candidates, but such materials are notoriously difficult to make due to the strong inherent acidity of aluminosilicates. Several research groups have endeavored to produce strongly basic zeolites by treating zeolites with amines, but to date there is no compelling evidence that nitrogen is incorporated into zeolite frameworks. In this communication, we detail synthesis, NMR spectroscopy, and quantum mechanical calculations showing that nitrogen adds onto both surface and interior sites while preserving the framework structure of zeolites. This finding is crucial for the rational design of new biomass-refinement catalysts, allowing 50 years of zeolite science to be brought to bear on the catalytic synthesis of biofuels.
Abstract: In reactive distillation (RD) one can conveniently manipulate the concentration profiles on the reactive stages by exploiting the difference in volatility of the various components. This property of RD can be advantageously used to improve the selectivity toward the desired product in case of series or series parallel reactions, and obtain a performance superior to the network of conventional reactors. In the previous work [Agarwal, V., et al., 2008. Attainable regions of reactive distillation—Part I. Single reactant non-azeotropic systems. Chemical Engineering Science, submitted for publication], we introduced representative unit models of RD to obtain the attainable regions of RD for non-azeotropic systems. In this work, we extend the approach to a system involving single binary azeotrope. Design guidelines have been formulated based on the residue curve maps, to obtain the improved attainable region with the help of these representative RD models either alone or in the form of their network.
Abstract:Reactive distillation (RD), a promising multifunctional reactor, can be used to improve the selectivity of the desired product by manipulating the concentration profiles in the reactive zone of the column. In this work, a new approach has been proposed to obtain the feasible regions of RD for the reactive systems involving single reactants, e.g. dimerization, aldol condensation, etc. Two new models namely the reactive condenser and the reactive re-boiler have been proposed. These models indicate the best location of the reactive zone in a column. Multistage versions of these models namely, reactive rectification and reactive stripping further expand the feasible region and are capable of representing the performance offered by a conventional RD unit. Several hypothetical non-azeotropic ideal systems have been extensively studied using these models and it has been shown that selectivity close to 100% is attainable over the entire range of conversion for a series as well as a combination of series and parallel reactions with positive reaction orders. Two industrially important cases of aldol condensation of acetone and dimerization of isobutylene have also been addressed using this approach. For porous catalysts, the presence of intra-particle diffusion resistance may limit the feasible region and even in the case of ideal non-azeotropic systems it may not be possible to obtain 100% selectivity. A methodology to incorporate pore diffusion effects is also illustrated.
Abstract: Aldol condensation of acetone in the presence of acid catalyst gives diacetone alcohol (DAA) as an intermediate product, which further dehydrates to give mesityl oxide (MO). By using reactive distillation (RD), one can improve selectivity toward DAA, by continuously removing it from the reactive zone and thereby suppressing the dehydration reaction. The presence of water in the reaction mixture has a predominant effect on the intrinsic reaction rates of the individual reactions. This rate-inhibiting effect of water can be advantageously used to improve the selectivity toward the desired intermediate products. The present study, through experiments and simulation, shows that introduction of water in RD can further increase the selectivity toward DAA. Batch kinetics of the reaction in the presence of water is studied, and a suitable kinetic expression is proposed. Further, the batch and continuous reactive distillation experiments are performed to assess the feasibility. The experimental results are explained with the help of an equilibrium-stage model, and the operating parameters for the desired performance are suggested based on the validated model.
Abstract: Dimerization of acetone (Ac) yields diacetone alcohol (DAA), which on further dehydration gives mesityl oxide (MO) along with various side-products. The reacting system is a combination of various series and parallel reactions. In the present work, the reaction is studied using a cation exchange resin (Amberlyst 15®) as catalyst. The effect of catalyst loading and temperature on reaction kinetics was evaluated and three models based on simplified Langmuir–Hinselwood mechanism are proposed. Aim of the work is to minimize undesired side-products and understand the effect of different parameters and operating modes on DAA:MO product ratio in reactive distillation (RD). It has been shown that the reaction when operated in a reactive rectification mode offers flexibility in the relative production rates of DAA and MO. The experimental results obtained are explained by simulation.
Abstract: Reactive distillation (RD), a combination of reaction and separation, holds the potential of giving selectivity/yield much above that offered by the conventional reactors or combination of them. It can be effectively used to improve selectivity of a reaction especially when an intermediate product is desired in series or parallel reactions. In this work, some representative model systems are introduced to obtain attainable region. A series as well as a combination of series parallel reactions (irreversible and mixture of reversible and irreversible reaction systems) were studied with the proposed models and it was found that for the single reactant ideal systems one can obtain almost 100% selectivity for a near quantitative conversion, which is much greater than that obtained in the conventional reactors like PFR and CSTR. These models were also studied for the non-ideal azeotropic systems and it was found that there is a limit to feasible region. The results obtained in the theoretical analysis are supported by experiments on aldol condensation of acetone.
|
2021-10-22 09:39:29
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6097301840782166, "perplexity": 2316.1160929569246}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585504.90/warc/CC-MAIN-20211022084005-20211022114005-00671.warc.gz"}
|
https://docs.wiris.com/en/mathtype/mathtype_desktop/microsoft_office
|
# MathType with Microsoft Office
#### Where can I use MathType with Word?
With your MathType subscription you can use MathType 7 in
MathType with macOS Mojave (or later): The first time you use MathType with any Office application on Mojave or later OS versions, you'll see this dialog:
When you see this dialog, you must click OK to authorize the application to launch MathType. If you don't, you won't be able to use MathType with Office. Likewise, when you create your first equation under Mojave or later, you'll see this dialog when you close MathType to insert the equation into the document:
You will see both of these dialogs only once. Click OK. This will enter the equation into the document. If you mistakenly did not allow access, see our TechNote for the solution steps.
Tip: If you use both Microsoft Word and Google Docs, or if you and/or your employer has converted from Microsoft Office to Google Docs, you may be interested in our tip Word document compatibility with Google Docs. This tip describes how to
• Upload a Word document with MathType equations to Google Docs for students and/or colleagues to read but not edit.
• Upload a Word document with MathType equations to Google Docs and end up with equations you can edit in MathType for Google.
• Open a Google Docs document, with equations created with MathType for Google, download the document and convert the equations such that they're editable in MathType Desktop (i.e., MathType for Word).
### How do you integrate MathType in Word?
MathType Desktop adds several commands to Microsoft Word that you will find useful when creating and working with documents containing equations. You will find these commands on the MathType tab in Word's Ribbon.
MathType Setup automatically installs MathType's support for Word, for each version of Word it detects on your computer. However, if you install a new version of Word after you install MathType Desktop, the commands will not be available for the new version of Word until you re-install MathType Desktop.
If you want to remove the MathType commands temporarily from…
• Word for Windows, open Word Options (via the Office Button or File tab), then click Add-ins. In the Manage box, click Templates, and then click Go. Uncheck the box next to the MathType Commands item. To remove them permanently, quit Word and remove the MathType commands from Word's startup folder.
• Word for Mac, open Templates and Add-Ins from the Tools menu. Uncheck the box next to the MathType Commands item.
Please note: A few of our customers have reported equations being converted into pictures without any warning. This is due to a bug in Word. If this happens, we're not aware of any way to recover the equations. Please see our TechNote with more information about this Word bug.
NOT Word 2011 for Mac. Please see the section below for that version.
If MathType Desktop has been installed a MathType tab will appear in Word with specific commands and sections.
### Insert Equations group
Insert Inline Equation Ctrl+Alt+Q (Windows), Ctrl+Q (Mac)
Opens a new MathType Desktop window ready for you to enter an equation. If you have defined equation preferences for new equations (using the Set Equation Preferences command), these settings will be used in the MathType Desktop window. Otherwise MathType's current preferences for new equations will be used. The resulting equation is inserted inline, i.e., aligned with the surrounding text.
Insert Display Equation Alt+Q (Windows), +Q (Mac)
Opens a new MathType Desktop window using equation preferences as described above under Insert Inline Equation. The display equation is inserted on a new line and centered between the left and right margins.
To simplify changing the formatting for all of the display equations in a document, a Word style called MTDisplayEquation is created that defines the position of the center tab stop. This means that you can change the alignment of all of the display equations in a document by simply modifying this style.
Insert Right-Numbered Display Equation Alt+Shift+Q (Windows), +Shift+Q (Mac)
Similar to Insert Display Equation, but also inserts a right-aligned equation number following the equation. The MTDisplayEquation style defines the location of the right margin tab stop.
Insert Left-Numbered Display Equation Ctrl+Alt+Shift+Q (Windows), Ctrl+Shift+Q (Mac)
Similar to Insert Display Equation, but also inserts a left-aligned equation number before the equation. The MTDisplayEquation style defines the location of the center tab stop.
Open Math Input Panel… Ctrl+Shift+M (Windows)
Windows only: (Windows 7 and later) Insert an equation by handwriting it in the Math Input Panel (MIP). When you click the MIP Insert button, the equation will be placed at the insertion point. This command is not available in versions of Windows earlier than Windows 7.
### Symbols group
The controls in the symbols group contain common mathematical symbols as well as many other symbols and characters for use in the document. For more information about inserting symbols in Word, open Word's help by hovering over Other in the Symbols Group and pressing the F1 key.
While you can create inline equations in MathType consisting of a single symbol, we recommend you insert symbols from the Symbol Group on the MathType tab instead. Equations have more overhead in the document, and for large documents the difference can be significant.
### Equation Numbers group
Insert Number
• Insert Number: Inserts an equation number at the insertion point. The default format includes a section number and an equation number. The equation number increments each time you insert a number. You can define the section number using Insert Break… under Chapters & Sections. Use Format… to set or change the number format.
• Format…: Opens the Format Equation Numbers dialog, which allows you to change the format of new and/or existing equation numbers. You can also turn off automatic updating of equation numbers and references in this dialog.
• Update: Updates all equation numbers and equation references in the document, including those in footnotes and endnotes. You need to use this command if you turn off automatic updating in the Format Equation Numbers dialog, or if you move or delete any equation numbers in your document.
Insert Reference
Inserts a reference to an existing equation number. It inserts a copy of the referenced equation number at the insertion point. In addition, if you double-click on a reference your document will scroll to the referenced equation. When equation numbers are updated, equation references will be updated as well.
Place the insertion point at the place where you wish to insert the reference, and click this button (the Insert Equation Reference dialog will appear). Next, double-click on the equation number you want to reference (you may have to scroll the document to find the equation).
Chapters & Sections
• Insert Next Section Break: Inserts a section break and numbers it by incrementing the number of the previous section break.
• Insert Next Chapter Break: Inserts a chapter break and numbers it by incrementing the number of the previous chapter break.
• Insert Break…: Opens the Insert Chapter/Section Break dialog, which allows you to insert an equation section break. Equation sections define the section number used in equation numbers.
• Modify Break…: Opens the Modify Chapter/Section Break dialog, which allows you to locate and modify the current chapter/section break (the one preceding the current position of the insertion point).
#### Browse group
The Browse group allows you to browse up or down through the document, stopping at Equations, Equation Numbers, or Chapter/Section Breaks.
### Format group
Equation Preferences: This command opens the Set Equation Preferences dialog, which allows you to specify the equation preferences to be used for new equations you insert into the document.
Format Equations: Opens the Format Equations dialog, which allows you to change the format of the equations in the current selection or the entire document.
Convert Equations: Opens the Convert Equations dialog, which allows you to convert equations in your document to a format you specify (including TeX or MathML).
### Publish group
Export Equations: Opens the Export Equations dialog, which allows you to export the equations in your document into individual graphics files.
Publish to MathPage: Opens the Publish to MathPage dialog, which allows you to export the document as a Web page including MathPage technology to handle the equations.
Toggle TeX Alt+\ (Windows), +\ (Mac): The Toggle TeX command allows you to type Texvc directly into a Word document and convert it into a MathType equation. Since it's a "toggle", choosing the command again will return the MathType equations to Texvc.
### MathType group
MathType Help
• Using MathType in Word…: Opens MathType's Help for Microsoft Word (i.e., this section of the documentation).
• Unlock/Register MathType…: Gives you information on how to enter your product key to unlock MathType Desktop as well as register your product so you can receive free technical support, upgrade notices and special upgrade pricing.
• The version of the MathType Desktop application you are currently using
• Your product registration information, including your partial product key if you have unlocked MathType Desktop, or the number of days left in the evaluation period if MathType is in evaluation mode
MathType on the Web
• Online Support…: Opens the tech support area of the MathType Web site, where we have many tips and tech support notices that will give you help solving problems and information on compatibility with other applications.
• Send Feedback by Email…: Opens your email program so that you can send feedback to Wiris regarding your experiences with MathType or to request future enhancements. If you have a problem using MathType or a bug to report, please visit our online tech support area first.
• Order MathType…: Opens the e-commerce area of the Wiris Web site where you can purchase MathType (or any of our other products).
Future MathType
Opens a page on our website that will allow you to give us your ideas for future enhancements of MathType Web and MathType Desktop.
Dialog box launcher
Opens MathType Options dialog.
Warning: Microsoft support for Office 2011 ended on October 10, 2017. MathType 7 will continue working with Office 2011, but we will not maintain this integration. Please consider upgrading to newer versions of Office to continue having the best experience. Beginning with MathType 8, we will not support Office 2011.
### MathType floating toolbar
The MathType Desktop toolbar contains the most frequently used commands on the MathType menu. You can hide or show the MathType Desktop toolbar by checking or unchecking MathType in the Toolbars flyout in Word's View menu
Insert Inline Equation Ctrl++Q
Insert Display Equation +Q
Insert Left-Numbered Display Equation Ctrl++Shift+Q
Insert Right-Numbered Display Equation +Shift+Q
TeX +\
(Note: On some non-English keyboards, the keyboard shortcut for Toggle TeX will be Control+X.)
Insert Equation Number
Insert Equation Reference
Insert Chapter/Section Break
Update Equation Numbers
Publish to MathPage
Browse Equations, Equation Numbers and Chapter Headings
### MathType menu in Microsoft Word
The MathType menu added to Word's menu bar contains the complete set of MathType commands for Word. In order to conserve screen space, the MathType Toolbar contains only the most frequently used commands, by default.
The commands apply to the current document only. The menu contains similar commands to the ones described above.
The MathType commands for Word provide six methods for inserting MathType equations into Word documents. These are:
We recommend that you always use one of these commands to insert equations as they work properly with the equation preferences commands that help ensure consistency between the equations in your documents.
The Insert Inline Equation command inserts a MathType equation at the insertion point. The equation is adjusted to align itself with the surrounding text, so use this command when you want to insert an equation in a sentence.
Note:
While you can create inline equations consisting of a single symbol, we recommend you use Word's Insert Symbol command instead. Equations have more overhead in the document, and for large documents the difference can be significant.
The Insert Display Equation and Insert Left/Right-Numbered Display Equation commands are very similar. They insert a new line if the insertion point isn't already on a new line, and then insert an equation centered between the left and right margins. The Insert Right-Numbered Display Equation command inserts a right-justified equation number following the equation, while the Insert Left-Numbered Display Equation command inserts a left-aligned equation number before the equation.
A Word style named MTDisplayEquation is added to the Word document the first time you use any of these three commands. This style defines the center tab and the right tab positions. Its other formatting is based on the style in use at the time it's created. You can change the alignment of all the display equations in a document by modifying this style using the Style command on Word's Format menu. This style is created separately for each document into which you insert a display equation.
The Open Math Input Panel command (Windows 7 and later) brings up the Math Input Panel (MIP). Write the equation in the MIP and when you're finished, click the Insert button. The equation will be placed at the location of the insertion point. Functions in your list of Functions Recognized are also recognized in the MIP.
The Toggle MathType/TeX command allows you to type Texvc directly into a Word document and convert it into a MathType equation. Since it's a "toggle", choosing the command again will return the MathType equations to TeX. When typing TeX, using either the $…$ or $$…$$ delimiters results in a MathType Inline Equation after applying the toggle, and using either the $$…$$ or $…$ delimiters results in a MathType Display Equation after the toggle. To use the toggle with a single equation, click the Toggle MathType/TeX icon and the equation nearest the insertion point will be converted to a MathType equation. To convert the entire document, select the document contents (Ctrl+A); to convert part of it, select the part you want to convert. Click the toggle. To convert the equations back, repeat the procedure. See important note in the Typing TeX directly into Word section below.
### Inserting and opening equations using keyboard shortcuts
MathType equations can be inserted in a Microsoft Word document using keyboard shortcuts. The equation will be inserted at the current cursor location in the document. There are shortcuts for each type of MathType equation.
• Insert inline equation (Crtl+Alt+Q; Mac: Ctrl+Q)
• Insert display equation (Alt+Q; Mac: +Q)
• Insert right-numbered equation (Alt+Shift+Q; Mac: +Shift+Q)
• Insert left-numbered equation (Ctrl+Alt+Shift+Q; Mac: Ctrl+Shift+Q)
• Open Math Input Panel (Windows only: Ctrl+Shift+M)
• Toggle MathType/TeX (Alt+\; Mac: +\. Note: On some non-English keyboards, the keyboard shortcut will be Control+X.)
• Open equation for editing in a separate MathType Desktop window (Alt+O/++O)
The equation numbering commands allow you to insert equation numbers in a Word document in a variety of formats. You can also insert references to these numbers. Both numbers and references are automatically updated whenever you add new equation numbers to the document. Equation references can also be placed in footnotes and endnotes.
The equation numbers are quite separate from equations in a Word document. If you delete an equation its equation number (if any) is not automatically deleted. Similarly, an equation reference is actually a reference to an equation number, rather than a reference to the equation itself.
The format of equation numbers in a document is defined using the Format Equation Numbers dialog. Equation numbers are made up of a chapter number, a section number, a separator, an equation number and an enclosure. This allows numbers such as (1.1.1), [I.i], {A.a}, or even Equation 1.1. The chapter number, section number, separator and enclosure are all optional, so you can create equation numbers such as (1.1), 1.1, or just 1 or a.
It's not possible with MathType to construct equation numbers such as 1.1, 1.2a, 1.2b, 1.3, etc. This is something we're considering for a future version of MathType Desktop. If this is something that's important to you, please let us know.
If you are using the chapter and/or section numbers, you must insert a chapter/section break before the first number to define the chapter/section values. A break can have an explicit value or be set to increment by one the value of the preceding equation section.
As you edit a document, equation numbers or references may get out of sequence (e.g., you move sections around). Use the Update Equation Numbers command in the Ribbon to update the sequence. If you select a region of your document, only this region will be updated, which can be useful with very large documents where updating can take a relatively long time.
### Inserting equation numbers and references step-by-step
To insert equation numbers and references in a Word document, follow these steps.
#### Define…
…the desired number format for this document using the Format Equation Numbers command. You can change this format at any time, for either all existing equation numbers, just the selected equation numbers, or for new equation numbers.
#### Create a chapter/section break…
…using the Insert Chapter/Section Break command. You should insert this in the document ahead of the first equation number you plan to add. The breaks are inserted as hidden text so that they won't normally appear.
Note:
If you haven't already set your Chapter/Section breaks, when you insert your first equation number or a numbered equation, the Insert Equation Number dialog appears, which allows you to set the chapter/number breaks.
#### Place the insertion point…
…at the desired location and use the Insert Equation Number command to insert an equation number.
#### Equation reference
To insert a reference to this equation number, place the insertion point in the desired location for the reference and choose the Insert Equation Reference command. Then, double-click the equation number to be referenced, and the reference will be inserted at the original location.
#### Modify break
To modify the chapter and/or section values of a break, use the Modify Chapter/Section Break command. The closest preceding chapter/section break will be made visible and selected, and in the dialog that appears you can modify or delete the break.
#### View all breaks
To view all chapter/section breaks, click the Show/Hide button in Word. To hide them, click this button again (you don't need to hide them when printing, as they won't appear in the printed output). When equation section breaks are shown, a quick way to open the Modify Chapter/Section Break dialog is to double-click on a break.
#### What about inserting a number in the middle?
If you insert equation numbers out of sequence, the existing numbers and references will be automatically updated. If you copy, move or delete an equation number or reference, however, you must use the Update Equation Numbers command to refresh the sequence.
#### Updating numbers & references
The equation number updating that occurs after you insert an equation number is usually very fast. However, you may find this updating takes a while if you are working on a slower computer or on a large document containing many equation numbers and references. The Format Equation Number dialog contains an option to turn off automatic updating. After you've finished entering equation numbers, be sure to use the Update Equation Numbers command to refresh the numbering sequence.
Note:
Update Equation Numbers works on selected text, or the whole document if there's no selection. For large documents it may be faster to select and update just the section of the document that requires it. But if you've made lots of changes, it's safest to update the entire document.
### Browsing by equations, equation numbers and chapter/section breaks
The MathType browse controls allow you to search through your documents for three types of objects:
• Equations. Traverses all equations.
• Equation Numbers. Steps though all the Equation Numbers created by MathType.
• Chapter/Section breaks. Helps find the normally invisible Chapter/Section breaks. The current chapter/section break will appear in red until you click the Previous or Next buttons, or elsewhere in the document.
Note:
When browsing by Chapter/Section breaks, the breaks become visible and remain visible after browsing is complete. You can make the breaks disappear again by clicking the Show/Hide show-hide icon on the toolbar or ribbon.
To start the search, use the dropdown list to select the object to search for and click the Next or Previous buttons. The browse feature will search all visible parts of the document starting at cursor position. This includes headers and footers, if visible.
When you create a document containing equations, you typically want all of the equations to use the same fonts, sizes, and spacing so they look consistent throughout the document. We recommend you save these equation settings in a MathType preference file so you can reload them into MathType Desktop if you need to create new equations in this particular document again. (MathType equations contain the preferences they were created with, and these preferences are used when the equation is edited in MathType Desktop).
However, you may be working on several different documents that each use different equation preferences. In this situation it can be awkward to keep loading preference files, and even to remember which preference file is associated with each Word document.
The Equation Preferences command allows you to save equation preferences inside your Word document. Whenever you insert a new equation in this document these preferences will be used for the new equation instead of MathType's current preference settings for new equations. This allows you to create equations that look similar without having to remember which preference file you originally used. Also, if you send the Word document to another MathType user any equations he or she inserts will also use the correct equation preferences.
To save equation preferences in a document, follow these steps:
#### Set desired preferences
In MathType Desktop, set your desired preferences using the Define Styles, Define Sizes, and Define Spacing dialogs.
#### Save preferences
Save these preferences as a MathType preference file using the Save Preferences dialog.
#### Open the document
In Word, open the document and choose the Equation Preferences command on the MathType tab.
#### Assign preferences to document
In the Set Equation Preferences dialog choose the This document's equation preferences option, click the Load from MathType preference file button and select the file you just saved.
#### View the assigned preferences
You can view the preferences by clicking the Preview button.
Click OK and the contents of the file will be copied into the Word document (for the curious, they are saved as a custom document property).
This command allows you to change the formatting of all MathType, Equation Editor, and Word EQ field equations in the chosen range. It applies the equation preferences (styles, sizes, spacing) you choose to all equations in the current selection or the entire document.
Note:
• Equation Editor equations and Word EQ fields will be converted to MathType equation objects by this operation. If you do not want this to happen, you must choose a range that omits them.
• If you have OMML equations in the document (i.e., created with the "new" Microsoft equation editor), the Format Equations will pass over them and not affect their appearance. If you want to reformat these equations, you'll need to first run the Convert Equations command.
To reformat the equations in a Word document, follow these steps:
#### Work with a copy of the document
Save a copy of your Word document. (This is always a good idea.)
#### Select the range
Select the equations you want to reformat, or make no selection to reformat the whole document.
#### Open the dialog
Choose or click Format Equations to open the Format Equations dialog.
#### Which preferences to use?
Select the equation preferences you want to use for the equations. If you've set up preferences for this document with the Set Equation Preferences command, the Current document option will be enabled. If you've copied an equation to the clipboard before running this command, Equation on clipboard will be enabled. You can also use the preferences MathType Desktop is currently using for new equations, or you can pick an existing MathType equation preferences file. You can see the details of the selected preferences by clicking the Preview… button, which opens the Preview Preferences dialog.
#### Set as default?
Check the Use for new equations checkbox to use the selected preferences for new equations inserted into this document. This is equivalent to using the Set Equation Preferences command as well. (If you've chosen the Current document option, this is redundant but it won't hurt anything if you check it.)
#### Begin
Click OK and the formatting process will start. Word's status bar displays a count of the number of equations processed, and when the process has completed a dialog will display a summary.
The Convert Equations command enables you to convert equations in a Word document into a variety of formats. You can choose to convert MathType and Equation Editor equations, Word EQ formulas, Word 2007 (OMML) equation images, and MathType translated text equations into MathType equations or text equations.
OMML (Office Math Markup Language) equation images are created when a document is created in Word 2007 and later, the document includes equations created with the OMML equation editor, and the document is subsequently saved in Compatibility Mode (i.e., as a doc file).
MathType text equations are equations translated into a text representation using the steps outlined in Working with TeX or Working with MathML. These equations can be converted back into MathType equations, or translated into yet another text equation, provided that both the translator name and the MathType data are preserved within the equation. If just the MathType data is preserved, the text can be pasted into a MathType Desktop window.
You should always use this command when opening a document that has been authored or modified on another platform (such as would be the case if you're working on Windows but the document was created on a Mac). Although Word does a good job of converting graphics, equations many not display correctly, and font differences between the platforms inevitably cause display problems as well. Running this command converts and reformats the equations, solving these problems.
To convert equations in a Word document, follow these steps:
#### Work with a copy
Save a copy of your Word document just in case you don't like the results; a conversion of many equations is not easy to undo.
#### Which equations to convert?
Select the equations you wish to convert, or make no selection to scan the entire document.
#### Open the dialog
Use the Convert Equations command to open the Convert Equations dialog.
#### Converting from what to what?
Select the equation types to convert from, and to convert to. If necessary, select the desired translator. If using a translator, turn on the Include translator name as comment and Include MathType data as comment options if you might want to convert these equations again.
#### Prompt?
Select the Prompt before converting each equation option if you want to choose which equations to convert and which to skip.
#### Begin
Click OK and the conversion process will begin. Feedback about the progress of the command is displayed in Word's status bar. When the command has finished a dialog displays the number of equations converted.
The Export Equations command allows you to export MathType and Equation Editor equations in your Word document to individual graphics files. This can be useful when importing a Word document into a desktop publishing application.
One file will be created for each equation in the document. You can choose the format of the files: EPS, Windows Metafile (WMF) or GIF . The files are numbered sequentially, starting with any number you choose, and you can define the underlying pattern, e.g., Physics001.eps, Physics002.eps, etc. You can also choose to replace each exported equation in the document with the name of its corresponding file, e.g., <<Physics001.eps>>.
To export equations from a Word document, follow these steps:
#### Work with a copy
Save a copy of your Word document (this is always a good idea).
#### Which equations to export?
Select the equations you wish to export, or make no selection to export all equations in the document.
#### Open the dialog
Use the Export Equations command to open the Export Equations dialog.
#### Export where?
Enter the name of the folder in which you want to create the exported equation files. You can either type in the name of a folder (it doesn't have to exist), or click Browse… and select an existing folder.
#### Delete existing files?
Check the Delete all files of same type in folder option if you want to delete all existing files in the folder with the same extension. This can be very useful, but be careful you've chosen the correct folder!
#### Export format
Choose the export format from the list. The format determines the file extension that will be used.
#### Filename pattern
Select a pattern for the filenames, and the starting number. The pattern must contain at least one '#' character. The # characters are replaced by a sequential number to make every filename unique. Use multiple # characters if you want the numbers to have leading zeros (this will make them display in a more natural order in sorted lists). For example, with the pattern Eqn### and a starting value of 5, the first EPS file exported will be Eqn005.eps, then Eqn006.eps and so on. The # characters don't limit the maximum number, so the above pattern can generate filenames such as Eqn1000.eps if there are this many equations in the document.
#### Replace equations with file names?
Check the Replace equation with file name option if you want the equations in the document replaced with the name of the corresponding file, e.g., <<Eqn001.eps>>.
#### What range?
Choose the Whole document or Current selection option based on how much of the document you want to export from, and what you did in Step 2.
Click OK to start the exporting process. Word's status bar will show the progress, and once the process has completed a summary is displayed.
MathType's MathPage technology enables you to convert a Word document containing equations, equation numbers and equation references into a web page. The page will look virtually the same in your browser as it does in Word. Inline equations will be correctly aligned, display equations will be properly centered, and all equations will display and print as well as from Word itself. This is true for all of the major browsers, running on Windows, Mac and Linux/Unix platforms.
### Tips for better web pages
• Use the MathType Commands for Word to insert equations This is especially true for display equations that you want centered in a paragraph. MathPage does a good job of handling the various ways that you can center an equation and generating the correct HTML, but it works most reliably if you use MathType's Insert Display Equation and Insert Numbered Display Equation commands in Word.
• Use Word's Insert Symbol command for mathematical symbols This is far more efficient than creating a MathType equation containing only a few symbols, both for Word and the browser. On the other hand, you must insert an equation if you want a MathZoom version of the symbol to pop up in the browser when the symbol is clicked. This is especially useful when using small, hard-to-read characters such as superscripts, subscripts, primes etc.
• Don't convert long documents; break them into smaller pieces While a large document may be desirable in Word for editing and printing, it does not translate well into a Web document. Large documents take longer to download, and are more cumbersome for viewing. When you print a long document in Word you get headers and footers inserted on every page. Printing from a browser is different; there are no headers and footers contained in the page itself, and page breaks can occur in the strangest places, sometimes breaking graphics such as equations in half. These problems can be minimized by breaking long documents into smaller sections, and making each one an individual web page.
• Don't use positioned or floating elements containing equations MathPage currently ignores these items, and although they do get converted into HTML any symbols and equations in these elements will not be handled properly. You can use simple left or right alignment on graphics and other elements to achieve proper display in the browser.
#### Click the command
Choose the Publish to MathPage command from Word's MathType toolbar, menu or Ribbon tab.
#### Title of the page
Edit the Title as necessary. It will appear in the browser's title bar or tab.
#### File name
Choose the File Name for the web page. The default is to save it in the same folder as the Word document, but with an extension that depends on the choice you make in the Equations group (see #6 below). You can either type in the name of the file or click Browse… and choose it in the Save dialog.
#### View after publishing?
If you'd like to view the page once it's been generated, check the Display in default browser checkbox.
#### MathML or GIF?
Now you must decide how you want the document's equations displayed in the browser. MathML is a great option, but there are some considerations (see note below). If you wish to use MathML, select MathML using and choose a MathML target from the list. A brief description is displayed in the dialog for the selected target; for a more detailed description of the MathML targets, see the next topic. If you're not using MathML, you should choose Use images (GIFs). If you choose images, you can also turn on MathZoom, which allows equations in the web page to be magnified by clicking on them. If you're using MathML and the person reading your web page is using Internet Explorer with MathPlayer, MathZoom is turned on automatically.
#### Target browser
If you're using GIFs, choose the target browser; if you're using MathML, MathType will make the appropriate browser choice automatically. Select the Internet Explorer 6 or newer (Windows) option if you know your web page will only be viewed by people using such a browser. The page will be smaller, particularly if it contains many mathematical symbols. Otherwise, choose the All browsers option.
#### Default for future pages?
The settings for this web page will be saved in the Word document, and displayed in the dialog if you run this command again. You can also use these settings (other than Title and File Name) as the default values for other web pages by checking the Use settings as defaults checkbox.
#### Begin
Click OK to generate the web page. You'll see a progress dialog appear, and when the page has been generated it will open up in your default browser (if you selected this option).
#### Supporting files
You may notice MathType has also generated a supporting files folder, with a similar name to what you named the file. For example, if your MathPage file name is deriving_limits.htm, MathType will create a folder named deriving_limits_files, and will save it to the same location as your MathPage.
1. If you're using GIFs, you'll always need to upload both the folder and the MathPage, because the folder is where MathType saves the images for each of the equations in the MathPage. There is also a file named mathpage.js. You'll need to upload this file with your MathPage as well. If there are several MathPages on your website, you only need to upload mathpage.js once in every folder that contains a MathPage that uses GIFs.
2. If you're using MathML, often you can get by with simply uploading the file itself, and not worrying about the supporting files folder. The safe thing to do though, is to upload both the file and the folder.
3. If your MathML target choice was Multi-browser (UMSS), there will be 2 additional files saved at the same level (i.e., in the same folder) as your MathPage. These files are named pmathml.xsl and pmathmlcss.xsl. You must upload these files with your MathPage, and keep them at the same level as your MathPage. If there are several MathPages on your website, you only need to upload these 2 xsl files once in every folder that contains a MathPage.
You should examine the web page and if you find any problems, modify the original Word document and run this command again. It's generally best not to edit the MathPage itself.
### Publishing to MathPage using MathML
Although browser support for MathML has greatly improved, there are some different approaches to producing a web page containing MathML. MathPage supports several targets, described below, that differ in the number and type of browsers that can display the page. We recommend the XHTML+MathJax target or XHTML+MathML in most cases.
If you're using a Mac, please note that not all of these targets are available on the Mac yet.
#### HTML5+MathML
For the "purest" MathML output, this is the best choice since it uses HTML5, and MathML is part of the HTML5 spec. It doesn't rely on any external tools to render the MathML equations, but it does require a MathML-capable browser or other software, such as accessible technology (screen readers, etc.).
#### HTML+MathJax
If you're targeting an audience with the latest and greatest browsers, the XHTML target (next bullet) is probably a better choice. If you're not sure, then this one should work in most all situations. Valid file extensions are .htm and .html; default is .htm.
#### XHTML+MathJax
All other things being equal, this is the preferred target to use. If your browser (and that of your audience) supports HTML5 or XHTML, and/or if your workflow involves ebooks, this is the clear choice. This should work in most browsers on mobile devices also. Valid file extensions are .xht, .xhtml, and .xml. Default is .xht.
#### XHTML+MathML
This target generates the preferred cross-platform MathML format, an XHTML page containing MathML. For the pages to display properly in Internet Explorer for Windows, your audience will need MathPlayer 2.0 or newer installed. For more details please visit the MathPlayer website. This format will work with most of the latest-release browsers, but if you find it doesn't work for some of your audience, you should try one of the MathJax targets above. Valid file extensions are .xht, .xhtml, and .xml. Default is .xht.
#### MathPlayer (IE behavior)
MathPlayer is our MathML display engine for Internet Explorer 6 or newer on Windows only. Choose this target if your audience will be using only this browser, or will be using assistive technology to have the MathPage read aloud. In most cases, for more flexibility we recommend the XHTML+MathJax target. For more details please visit the MathPlayer website. Valid file extensions are .htm, .html, .shtm, .shtml, and .stm. Default is .htm.
#### Multi-browser (UMSS)
The inclusion of the 2 MathJax targets described above has all but eliminated the need to use the Multi-browser (UMSS) MathPage target. If you choose one of the MathJax targets and find a good number of your audience is using an older browser that may not display the MathJax equations properly, or if they have JavaScript disabled, this general-use target should work. The Universal MathML Stylesheet (UMSS) is an XSLT stylesheet that permits a single web page containing MathML to be displayed in a variety of different browsers. It detects the current browser in use, and whether it has built-in support for MathML or whether a plug-in such as MathPlayer or techexplorer is available. It then makes the necessary adjustments to the content. It even uses CSS to format equations if no MathML renderer is available. This universality comes at the expense of a slight delay in displaying the document, and the need to distribute a couple of extra files with the web page (pmathml.xsl and pmathmlcss.xsl). For more details please see http://www.w3.org/math/xsl. Valid file extensions are .xht, .xhtml, and .xml. Default is .xht.
The Toggle MathType/TeX command allows you to type Texvc directly into a Word document and convert it into a MathType equation. Since it's a "toggle", choosing the command again will return the MathType equation to Texvc.
• To enter an Inline Equation, the Texvc must be delimited as $…$.
• To enter a Display Equation, the Texvc must be delimited as $…$.
• To convert a single equation, first select it (by clicking it once with the mouse) OR place the cursor near it in the same line. Then either:
• Click the Toggle TeX icon in the Publish group of the MathType tab on Word's ribbon.
• Type Alt+\/+\.
• To convert all equations in a region of a document, select the region, and follow one of the two steps above. Note the Toggle TeX command will convert all equations found in the highlighted region regardless of whether they are Texvc or MathType equations, so if you select a mix of both MathType and Texvc equations and toggle, the result will again be a mix.
The dialogs described below provide access to the different MathType commands and features for inserting, modifying, and maintaining your MathType equations in Word. They are presented below in the order in which they appear in the MathType tab.
### Dialogs in the Equation Numbers group
#### Insert Equation Number dialog
This dialog appears the first time you insert an equation number into a document, if you have not already inserted a chapter/section break. An equation chapter/section break lets you define the chapter and/or section number part of an equation number, for example the 2 in the equation number (2.3).
If you click OK, an equation section break will be inserted at the start of the document, using the numbers you specify, and then an equation number will be inserted at the current location of the insertion point. If you prefer, you can click Cancel and then insert an equation section break at the location you desire. In this case you will have to choose the Insert Equation Number command again.
This dialog will not appear once you have inserted an equation section break, or if you have selected an equation number format that does not include a chapter or section number. Check the Always start new documents with chapter 1, section 1 option if you do not wish to see this dialog the first time you insert an equation number into a document. You can also control whether this dialog appears by checking or clearing the Warn when inserting first equation number checkbox in the Format Equation Numbers dialog.
#### Format Equation Numbers dialog
This dialog lets you set the format for new equation numbers or change the format of existing equation numbers.
Number Format
This section lets you specify a number format. The Simple Format option lets you pick the enclosure, separator and format of each part of the number. All parts are optional; use the checkbox next to each part to include/exclude it from the number. The Preview shows you an example using your format.
If you want more control over the format you can choose the Advanced Format option, and then enter the desired format in the Format edit box. All characters in the format are used literally, except for the terms #C, #S and #E. These correspond to the Chapter, Section and Equation parts of the number (and can be in any order). Each of these terms must be followed by another character indicating the numeric format; these correspond to the choices in the menus and are as follows:
1 Numeric 1, 2, 3, … A Alphabetic A, B, C, … a alphabetic a, b, c, … I Roman I, II, III, … i roman i, ii, iii, …
Thus the format (#C1:#S1-#Ei) would generate the number (1:1-i). The Preview updates as you make changes to the format. A useful tip is to use the Simple Format option to get the format as close as possible to the desired format, then choose the Advanced Format option and make the final changes.
Change the equation number format for
Options in this section control which equation numbers are affected by the settings in this dialog. Choose New equation numbers to affect new equation numbers created with the Insert Equation Number command. Choose Current selection or Whole document to update existing equation numbers.
Options
• Update equation numbers automatically
Whenever you insert an equation number or an equation reference into a document, all of the existing equation numbers and references are updated so that the numbering sequences are correct. Typically this operation doesn't take very long, but if you are running on a slower computer, or you are working on a large document, then this process may start taking a while. If so, you can turn off this option, which prevents the update from automatically taking place.
If you turn off automatic updating, you must manually update the numbers using the Update Equation Numbers command. You must also use this command if you move or delete an equation number, as the automatic updating doesn't occur in these situations.
Note:
Depending on the options you've set in Word, the equation numbers may automatically update when you print the document. If you have a lot of equation numbers and it's taking a long time before printing starts, this may be why. Look for a setting titled "Update fields before printing" in Word Options.
• Warn when inserting first equation number
The first time an equation number is inserted into a document, the document is checked to make sure it contains an initial chapter/section break. If it doesn't, the default behavior is to present a dialog asking for the starting chapter and section numbers. If you uncheck this option, starting values of 1 will be inserted without the dialog appearing.
• Warn when inserting equation references
When an equation reference is inserted into a document, the default behavior is to present a dialog that reminds you to double-click an equation number to actually insert the reference. If you uncheck this option, this dialog will no longer appear.
• Use format as default for new documents
Check this option if you want to use the current format as the default for all new documents. The first time you insert an equation number into a new document, the format is saved with the document. This means that the default format only affects documents that haven't had equation numbers inserted.
#### Insert Equation Reference dialog
This dialog appears when you insert an equation reference using the Insert Equation Reference command on the MathType Menu, MathType Toolbar, or the MathType Tab in Word 2007 and later.
To insert an equation reference at the current location of the insertion point, click OK and then double-click the equation number that you wish to reference. You can scroll the document to bring the desired equation number into view if necessary. After double-clicking an equation number, the document will scroll back to the location where the equation reference was inserted.
If you are inserting references that aren't very close to the original equations, you can avoid a lot of scrolling backwards and forwards by splitting the Word window into two panes. Then you can insert the references in one pane, and double-click the equation numbers in the other.
Check the Don't show me this again option if you do not wish to see this dialog every time you insert an equation reference. You can also control whether this dialog appears by checking or clearing the Alert when inserting equation references checkbox in the Format Equation Numbers dialog.
#### Insert Chapter/Section Break dialog
This dialog allows you to insert a chapter/section break, which defines the chapter and/or section numbers for this portion of the document. These numbers will be displayed in all equation numbers up to the next chapter/section break. The equation number format in use determines the actual display, e.g., whether the Chapter number is displayed.
The break simply resets the chapter and/or section number. If your document doesn't contain multiple chapters then ignore the chapter portion of the dialog. If your document contains multiple chapters and/or multiple sections, you'll normally want to increment the numbers, as described below. And when you start a new chapter, you'll usually want to reset the section number back to 1.
Chapter
Choose the New Chapter option to define the chapter number. Once selected, you can choose the Next chapter number option to increment the chapter number by 1, or choose the Chapter number option and enter the desired number. If you aren't using chapters and chapter numbers, you can ignore this option and leave it turned off.
Section
These options work similarly to the chapter options. Choose the Next section number option to increment the section number by 1, or choose the Section number option and enter the desired number (or letter) in the box provided.
#### Modify Chapter/Section Break dialog
This dialog allows you to change the chapter and/or section number for the portion of the document containing the insertion point. As an aid, the chapter/section break that indicates the beginning of the section will be selected and, if necessary, the document will be scrolled to bring the break into view.
This dialog is identical to the Insert Chapter/Section Break dialog, with the addition of a Delete button. The dialog's options reflect the break's chapter and section number values.
Chapter
Choose the New Chapter option to define the chapter number. Once selected, you can choose the Next chapter number option to increment the chapter number by 1, or choose the Chapter number option and enter the desired number. If you aren't using chapters and chapter numbers, you can ignore this option and leave it unchecked.
Section
These options work similarly to the chapter options. Choose the Next section number option to increment the section number by 1, or choose the Section number option and enter the desired number (or letter) in the box provided.
Delete
Click this button to remove the selected chapter/section break, effectively merging the current chapter/section with the previous one (if one exists).
### Dialogs in the Format group
#### Set Equation Preferences dialog
Use this dialog to set and/or view the equation preferences (styles, sizes, and spacing settings) used for new equations you insert into the current document. The preferences are used only for equations inserted using the commands on the MathType tab in Word's ribbon.
MathType's 'New Equation' preferences
Choose this option to use MathType's new equation preferences each time you insert a new equation, instead of the equation preferences (if any) of the current document.
This document's equation preferences
Choose this option to use equation preferences in the current document when you insert a new equation. If you have not yet set these preferences, the Load Preferences dialog will be displayed (see Load from MathType Preference File… below), allowing you to set them.
Click this button to set the equation preferences using a MathType preference file previously created using MathType's Save Equation Preferences dialog.
Preview…
Click this button to view the preferences that will be used for new equations. See the Preview Preferences dialog.
#### Preview Preferences dialog
This dialog displays a list of equation preferences. The preferences are displayed as a list of items and values, as set using MathType Desktop's Define Styles, Define Sizes, and Define Spacing dialogs.
#### Format Equations dialog
This dialog allows you to change the formatting of all MathType, Equation Editor, and Word EQ field equations in the chosen range. It applies the equation preferences (styles, sizes, spacing) you choose to all equations in the current selection or the entire document.
Note that Equation Editor equations and Word EQ fields will be converted to MathType equation objects by this operation. If you do not want this to happen, you must choose a range that omits them.
Format equations using preferences from:
Use this section to select how you want the equations to look by selecting equation preferences that will be used in the formatting operation.
• Current document Choose this option if you want to use the current document's equation preferences. This option will only be available if you have set the equation preferences for the document. See the Set Equation Preferences dialog.
• MathType's 'New Equation' preferences Choose this option if you want to use the preferences MathType Desktop is currently configured to use for new equations.
• Equation on clipboard Choose this option to reformat the equations to have the same format as an existing equation. In preparation for using this option, copy the equation on which you want to base the formatting to the clipboard . The equation may be in the current document, any other document, or in a MathType Desktop equation window. This option will only be available if there is an equation on the clipboard.
• MathType preference file Choose this option to format the equations based on the equation preferences you have previously saved in a file using the Save to File… command on MathType Desktop's Equation Preferences menu. Use the Browse button to locate the file.
• Use for new equations Check this box to use the selected equation preferences for new equations you create in this document.
• Preview Click this button to view the style, size and spacing definitions you've selected. See the Preview Preferences dialog.
Range
These options allow you to choose the range within the document in which equations will be formatted: Whole document will format all equations in the document, while Current selection will only format those equations in the current selection.
#### Convert Equations dialog
The Convert Equations dialog is used to convert equations in your document between various equation types. This is useful, for example, when you want to convert some or all of the Equation Editor equations in a Word document into MathType equations, or to convert graphic equations into a text-based language, such as TeX or MathML, for which there is a MathType translator available.
Equation types to convert:
Only the equation types checked in this group will be converted. The equation types that can be converted are:
• MathType or Equation Editor equations. This includes equations created by any version of MathType or Equation Editor. They can be OLE objects, GIF images, or MathType 1.x macros.
• Microsoft Word EQ fields. These are equation fields inserted using Word's Insert Field command. Prior to the introduction of Equation Editor with Word for Windows 2.0, this was the only way to insert mathematical equations into Word documents.
• MathType translator text equations. This is any equation created using a MathType translator (which creates text-based equation formats such as TeX or MathML). In order for this option to work the equations must contain both MathType and translator information. This information is included by MathType Desktop when you turn on the Include translator name in translation and Include MathType data in translation options in MathType Desktop's Cut and Copy Preferences dialog, and when you turn on the equivalent options in the Convert equations to group in this dialog.
• Word 2007 (OMML) equations. Word 2007 introduced a new equation editor that utilizes the Office Math Markup Language (OMML). This dialog option converts OMML equations into MathType equations or into text, as selected in the Convert equations to group in this dialog. This feature is not available in all versions of Word.
You must select at least one equation type, or no equations will be converted.
Range
These options allow you to choose the range within the document in which equations will be converted: Whole document will convert all equations in the document, while Current selection will only convert those equations in the current selection.
Prompt before converting each equation
Check this option to be prompted before each equation is converted. This gives you an opportunity to skip equations you do not want to convert.
Convert equations to
Options in this section allow you to specify the type of equation to which to convert.
• MathType equations (OLE objects). This will result in normal MathType graphic equation objects.
• Text using MathType translator. This will result in each equation being converted to plain text using the currently selected translator.
The other options in this section allow you to choose the translator and to determine whether MathType's internal data gets included in the translation. If you convert equations to text using one of the translators, and think you might want to convert them to another format in the future, turn on the Include translator name as a comment and Include MathType data as a comment options. See the Cut and Copy Preferences Dialog for more details.
Convert
Click this button to convert all equations that match the given criteria.
### Dialogs in the Publish group
#### Export Equations dialog
This dialog allows you to export the MathType (and Equation Editor) equations in your Word document to individual graphics files. This can be useful when importing a Word document into a desktop publishing application.
Export To
You can type in the name and location of the folder into which you want to export the equation files, or you can choose an existing location using the Browse… button.
If you're continually exporting files to the same folder, it can be useful to have it cleaned out before the export begins. The Delete all files of same type in folder option is provided for this purpose. However, be careful that you've chosen the correct folder or you may delete files you didn't mean to delete!
File Format
This section lets you choose a File type, a File name pattern, and the First number with which to begin numbering files. The number gets incremented for each exported equation, and inserted into the file name in place of the # characters in the pattern. For example, with the pattern Eqn### and a starting value of 5, the first EPS file exported will be Eqn005.eps, then Eqn006.eps and so on. Using multiple # characters helps in making all file names the same minimum length, which causes them to display in the correct sequence when sorted. Select Replace equation with file name to replace each equation with its filename enclosed in double angle brackets, e.g., <<Eqn005.eps>>.
Range
These options allow you to choose the range within the document from which equations will be exported: Whole document will export all equations in the document, while Current selection will only export those equations in the current selection.
#### Publish to MathPage dialog
This dialog allows you to convert a Word document to a web page in which the equations display and print properly. Publishing to MathPage using MathML contains additional information about this feature. We strongly recommend that you read it first. It explains the process more fully, and contains many tips for creating great web pages.
Document
This section lets you set the Title of the web page and its File Name. Display in default browser will open the generated page in your default Web browser after the generation process has completed.
Equations
In this section you can choose how equations will be represented in the web page: either as GIF images or using MathML.
• Use images (GIFs)
If you choose Use images (GIFs), a set of GIF files is created for every equation (and some symbols). When the page is viewed in a browser, dynamic HTML is used to select the appropriate image depending on the resolution of the monitor. This makes the size of the equation more closely match the size of the surrounding text. A high-resolution image is used when the page is printed, again making the equation match the rest of the document. You can also choose to use our MathZoom technology, which zooms (magnifies) equations when you click on them, allowing you to see clearly subscripts, superscripts, hats and primes.
• MathML using
Equations can also be represented as MathML. MathML is a World Wide Web Consortium (W3C) specified language for expressing mathematics (for more details visit their website at http://www.w3.org/math). There are several different ways it can be used, depending upon the browser and/or plug-in that you and your audience are using. When you select MathML using, the dialog contains a list of MathML "targets" from which you can choose. A page generated for a given MathML target may not display properly when viewed in another browser or without the required plug-in being installed.
These are the 5 MathML targets.
If you're using a Mac, please note that not all of these targets are available on the Mac yet.
You can read full descriptions of each target in Publishing to MathPage using MathML, but here are some brief suggestions about when and why to choose one over another.
• HTML5+MathML. For the "purest" MathML output, this is the best choice since it uses HTML5, and MathML is part of the HTML5 spec. It doesn't rely on any external tools to render the MathML equations, but it does require a MathML-capable browser or other software, such as accessible technology (screen readers, etc.).
• HTML+MathJax. If you're targeting an audience with the latest and greatest browsers, the XHTML target (next bullet) is probably a better choice. If you're not sure, then this one should work in most all situations.
• XHTML+MathJax. All other things being equal, this is the preferred target to use. If your browser (and that of your audience) supports HTML5 or XHTML, and/or if your workflow involves ebooks, this is the clear choice. This should work in most browsers on mobile devices also.
• XHTML+MathML. This target has the same advantages as the previous target, but will work in browsers that either don't support JavaScript, or browsers in which the user has disabled JavaScript.
• MathPlayer (IE behavior). If you know your audience is using Internet Explorer and/or will be using assistive technology to read the web page, this is the best target.
• Multi-browser (UMSS). If you find none of the other targets work for you, this target would have the widest possible coverage of browsers and operating systems. It does require uploading 2 supporting files in addition to the MathPage itself.
Target Browser
In this section, you can choose the target browser if you're using GIFs. If you're using MathML, there is no option to change it. For MathML, the selection defaults to All browsers unless you choose the "IE Behavior" option. Some browsers can display many symbols correctly as characters in the document, while other browsers require GIF images to be used for some symbols. If you're sure the audience for your pages will only be using Internet Explorer 6 and newer for Windows, select this choice as the page will download slightly faster. Otherwise choose the All browsers option. If you're using MathML, the proper button for Target Browser will be automatically selected, depending on the MathML target you chose.
Use settings as defaults
Select this item to use the selected settings the first time Export to MathPage is used on any documents in the future. If Export to MathPage has already been used on a document, its previous settings will display and not these defaults.
### Dialogs in the MathType group
#### MathType Options dialog
The MathType Options dialog gives you control over how MathML is pasted into a Word document. Launch this dialog from the dialog box launcher (Windows) or the MathType Options command (Mac) in the MathType group of the MathType tab in Word 2016 for Mac, or the MathType menu in Word 2011 for Mac.
Source of MathML
The choices you make on this dialog are active regardless of the source of the MathML. You may have copied MathML from a website, created a handwritten equation in the Windows Math Input Panel (MIP), or you may have chosen a MathML translator in MathType Desktop and copied an equation. Each of these actions will place MathML on the clipboard, which is then handled in one of two ways, depending on your choice on the MathType Options dialog
• Create a MathType equation
When you paste MathML into Word or click the Insert button on the MIP, MathType Desktop creates a MathType equation from the MathML and places it at the insertion point in Word.
• Create an OMML equation
When you choose this option, MathType Desktop will create an OMML equation at the insertion point in Word upon MathML paste or clicking Insert in the MIP.
• Prompt on each paste
After choosing this option, MathType Desktop will display the MathType Paste dialog each time you attempt to insert MathML into Word.
Reset all "Don't show me this again" options
This checkbox is used to reset three settings:
• Don't show me this again checkbox in the "Update Equation Numbers" dialog, when updating equation numbers is taking too long.
• Don't show me this again checkbox in the "Insert Equation Reference" dialog.
• Always start new documents with chapter 1, section 1 checkbox in the "Insert Equation Number" dialog. Although this is not labeled "Don't show me this again", it still gets reset when the "Reset" checkbox is checked in the MathType Options dialog.
### Dialog not available through the MathType tab on Word's ribbon
#### MathType Paste dialog
If the Windows clipboard contains MathML and you attempt to paste it into a Word 2007 or later document, this dialog will appear unless...
• you have chosen the Prompt on each paste on the MathType Options dialog, or
• you have previously made a choice on the MathType Paste dialog and checked the Remember my choice checkbox.
Create a MathType equation
When you paste MathML into Word or click the Insert button on the MIP, MathType Desktop creates a MathType equation from the MathML and places it at the insertion point in Word.
Create an OMML equation
When you choose this option, MathType Desktop will create an OMML equation at the insertion point in Word upon MathML paste or clicking Insert in the MIP.
Remember my choice
If you check this box, the MathType Paste dialog will not appear in the future when you paste MathML. To reinstate the behavior of the dialog appearing for each MathML paste, click the dialog launcher in the MathType group on the Ribbon.
Using MathType in PowerPoint is similar to using it in Word, with the notable exception that it's not possible to insert inline equations in PowerPoint, because PowerPoint does not allow external objects to be inline with text.
To insert an equation, click the MathType tab, then the MathType button in the Insert Equation group. Make sure your font face and font size match the text you're using on the slide. Create the equation and close MathType to insert the equation onto the slide. Move into position. The process is shown in this animation:
### Inserting handwritten math (Windows 7 and later)
There is no direct support for the Windows Math Input Panel (MIP) in PowerPoint, but it is still possible to use the MIP and MathType Desktop together to insert handwritten math in PowerPoint by following these steps:
1. In PowerPoint, choose the Insert Equation command from the MathType tab on PowerPoint's ribbon.
2. In MathType Desktop's Edit menu, choose the Open Math Input Panel… command.
3. After you write your equation in the MIP, click the Insert button, and the equation will be pasted into MathType Desktop.
4. Close the MathType Desktop window and the equation will be placed on your PowerPoint slide.
### Insert Equation group
Windows PowerPoint 2011 Mac PowerPoint 2016
Insert Equation. Opens a new MathType Desktop window ready for you to enter an equation.
Note:
PowerPoint does not allow inline "objects", such as MathType equations. Thus, all equations entered into a PowerPoint slide will be "floating", and you will have to click and drag them into position.
Tip:
To make the equations the correct size to match the text of your slide, don't click and drag a corner. If you do that, two things will happen, neither of which is good. First, no two equations will be exactly the same size. Second, it will take an unnecessarily large amount of time to do that, since you will have to size each equation this way. The right way to size equations for PowerPoint is to use MathType Desktop's Define Sizes dialog. Set "Full" size to match the size of the text on your slide. If you're working in Word and PowerPoint at the same time, save a Preference File for Word and one for PowerPoint. This will provide an easy way to switch from Word's fonts & sizes to those for PowerPoint.
### Browse group
Windows PowerPoint 2011 Mac PowerPoint 2016
Browse. The Browse group allows you to search back and forth through the slides for equations.
### Help group
Windows Mac PowerPoint 2016
Note:
If you're using PowerPoint 2011 for Mac, commands described below are in PowerPoint's MathType menu.
MathType Help.
• Using MathType in PowerPoint… Opens this section of the documentation.
• Unlock/Register MathType Gives you information on how to enter your product key to unlock MathType Desktop as well as register your product so you can receive free technical support, upgrade notices and special upgrade pricing.
• The version of the MathType Desktop application you are currently using
MathType on Web
• Online Support… Opens the tech support area of the MathType Web site, where we have many tips and tech support notices that will give you help solving problems and information on compatibility with other applications.
• Send Feedback by Email… Opens your email program so that you can send feedback to Wiris regarding your experiences with MathType or to request future enhancements. If you have a problem using MathType or a bug to report, please visit our online tech support area first.
• Order MathType Opens the e-commerce area of the Wiris Web site where you can purchase MathType (or any of our other products).
Future MathType. Opens a page on our website that will allow you to give us your ideas for future enhancements of MathType.
There isn't built-in support for MathType in Microsoft Excel -- not in the sense of the support in Word and PowerPoint. You can insert a MathType equation into an Excel sheet or chart either by going through Excel's Insert Object dialog, or by copying from MathType and pasting into Excel. Doing so will not result in an Excel formula (i.e., it's essentially a static image). You can double-click to edit the equation, like you would in Word or PowerPoint, but it is an object floating on top of the cell. It is not part of the cell's contents (or chart's contents).
Note: Excel 2016 and later for Mac does not support inserting MathType equations from the Insert Object dialog. Further, if you copy and paste an equation into Excel for Mac, the result will be a picture. If you need to edit the equation later, you'll need to replace the equation with a new one. The process described in the next section does work in Excel for Mac.
#### Creating a MathType matrix from Excel data
The one thing you can do in Excel is copy a range of cells, and paste them into MathType as a matrix. To do so, select the cells containing the data, then copy and paste into MathType. In general, the formatting will not carry over from Excel to MathType, but the "shape" of the copied region will. For example, if you're copying a block of data in Excel that's 8 rows by 2 columns, this will result in an 8×2 matrix in MathType Desktop.
• A single row of data will result in a single line in MathType Desktop – not a row matrix.
• A single column of data will result in a properly-formatted column matrix in MathType Desktop.
This may also work in spreadsheets other than Excel, and in word processors and text editors. It does not work in OpenOffice.
|
2020-08-03 14:36:46
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.592090904712677, "perplexity": 2407.729093908287}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735812.88/warc/CC-MAIN-20200803140840-20200803170840-00217.warc.gz"}
|
https://msp.org/agt/2020/20-4/agt-v20-n4-p01-s.pdf
|
#### Volume 20, issue 4 (2020)
Recent Issues
The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
$\tau$–invariants for knots in rational homology spheres
### Katherine Raoux
Algebraic & Geometric Topology 20 (2020) 1601–1640
##### Abstract
Ozsváth and Szabó used the knot filtration on $\stackrel{̂}{CF}\left({S}^{3}\right)$ to define the $\tau$–invariant for knots in the $3$–sphere. We generalize their construction and define a collection of $\tau$–invariants associated to a knot $K$ in a rational homology sphere $Y\phantom{\rule{-0.17em}{0ex}}$. We then show that some of these invariants provide lower bounds for the genus of a surface with boundary $K$ properly embedded in a negative-definite $4$–manifold with boundary $Y\phantom{\rule{-0.17em}{0ex}}$.
However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/agt
We have not been able to recognize your IP address 3.231.102.4 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.
or by using our contact form.
##### Keywords
Heegaard Floer, knot invariants, genus bound, rational homology spheres
Primary: 57M27
Secondary: 57R58
|
2021-03-03 21:20:23
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 9, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3115865886211395, "perplexity": 2036.1537878066767}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178367790.67/warc/CC-MAIN-20210303200206-20210303230206-00227.warc.gz"}
|
http://www.leancrew.com/all-this/
|
# Not loving JXA
A few days ago, Brett Terpstra presented us TaskPaper users with a couple of Keyboard Maestro macros, one for natural language dates (e.g., “tomorrow” or “next Monday” in @due or @start tags) and one for incrementing or decrementing numerical values in tags like @priority(n).
As it happens, I don’t use TaskPaper in as formal a way as Brett does—experience shows that the more structure I put in my task management system, the less I use it—so I suspect I’ll never take advantage of Brett’s work. But tucked away in a footnote is a comment that rang true for me:
I’m still clumsy with JavaScript for Automation. As annoying as I’ve found AppleScript over the years, I’m just not loving JSA much more.
This is a perfect distillation of my experience with both AppleScript and JavaScript for Automation.1
The main problem with AppleScript is the tremendous variation in quality from one application’s dictionary to another’s. There are so many idiosyncrasies, it’s hard to remember which gimmicks apply to which app. JXA doesn’t—and can’t—solve that problem because it rests on the same Apple Events foundation as AppleScript. Worse, it adds a new problem: very little outsourced documentation.
By “outsourced documentation” I mean the vast collection of websites in which AppleScript programmers have written about the little tricks they’ve discovered to get their scripts to do what they want. For me, programming in AppleScript consists largely of working out in my head the overall structure of the script and Googling to find the right magical incantations for each individual part. Then revising the structure when I find that certain parts can’t be done the way I thought they could.
(There are, I’m sure, AppleScripters who don’t need to Google for syntax. I can’t imagine Doug Adams, for example, needing to search for help with the iTunes dictionary. But I’ve never scripted any single application long enough to master it, and even if I had, mastery of one app doesn’t necessarily translate to another.)
JXA doesn’t have that quarter-century of folk wisdom. So programming in JXA is just like programming in AppleScript, but with the extra step of translating someone’s AppleScript trick into JavaScript—a very different language.
When JXA was released, I really thought it would be superior to AppleScript. I imagined myself rewriting my old AppleScripts in this new, more normal language. But it hasn’t turned out that way. Like Brett, I am still clumsy with JXA and doubt I’ll ever get coordinated.
1. And like Brett, I think JavaScript for Automation should be abbreviated JSA and usually write it that way. But Apple says it’s JXA (in documentation that still refers to macOS as OS X, but I’m not here to rant about how Apple’s enormous workforce can’t muster enough people to update the documentation for a line of products so small it could fit “on the table you’re sitting at”).
# Timers, reminders, alarms—oh, my!
I was shocked—shocked!—to see people disagree with my last post. I was even more shocked to learn about bizarre omission in the HomePod software. I decided to dig into the many ways you can set timed alerts on your Apple devices and how the alert systems vary from device to device. It is, you will not be surprised to learn, a mess.
Let’s start with the summary. In the table below, I’m comparing the features of the three alert types on iOS: Timers, Alarms, and Reminders. Included in the comparison is how certain features work (or don’t work) on the iPhone, iPad, Watch, Mac,1 and HomePod. Most of the entries for the HomePod are empty because I don’t have one to test, but I’ve included it because it was the device that got me started down this path. Also, there’s that software omission I want to talk about.
Timer Alarm Reminder
Number 1 ∞︎ ∞︎
Name/Description No Yes Yes
Autodelete Yes No Yes
Shared
iPhone Yes Yes Yes
Watch Yes Yes Yes
Mac No No Yes
HomePod ? ? No
Time left
iPhone Yes No No
Watch Yes No No
Mac No No No
HomePod ? ? ?
Time of
iPhone No Yes Yes
Watch No Yes Yes
Mac No No Yes
HomePod ? ? ?
Many of the entries in this table have caveats, so let’s go through it.
The number of alerts that can be set was the starting point for the last post. People want multiple timers in their HomePods. That’s great, but Apple’s never had multiple timers in any iOS device, which is why I’ve always used reminders instead. “Reminders aren’t a substitute for timers!” I’ve been told by several people. I admire your steadfast adherence to your principles, but I need a solution, not a manifesto. (We’ll get to the deficiencies of using reminders as a substitute for timers later in the post.)
Since there’s only one timer, there’s no need for it to have a name or description. So when the timer on your phone/watch/table/speaker goes off, you might have to think a bit before you remember what it’s for. Alarms and reminders don’t have this problem.
I didn’t mention alarms in my last post, but Kirk McElhearn reminded2 me of them. If you’ve only used Clock app’s UI to set an alarm, you may think you have to use a specific time (like 8:55 PM) instead of a relative time (in 20 minutes). But Siri offers another way:
Hey Siri, set a casserole3 alarm for 20 minutes.
One problem with using alarms as your alert system is that they don’t delete themselves when you dismiss them; they just sit there, inactive, taking up space in your list of alarms until you undertake a second action to remove them from the list. Timers delete themselves upon dismissal, which is certainly more convenient. Reminders almost delete themselves—when you mark a reminder as complete, it gets hidden in the Completed list. I take this as close enough to deletion that I gave Reminders a Yes on the Autodelete line.
One of the biggest advantages to using reminders is that they’re shared via iCloud, which also syncs them to your Mac. This is very convenient if you use reminders during the workday and allow notifications from the Reminders app, which I do. Timers and alarms are not shared; the timer you set on your phone doesn’t appear in the Clock app on your iPad or on your watch. But the watch is special because of its intimate relationship with the phone. Your watch will alert you of a timer or alarm set on your phone, even though it doesn’t appear in the watch’s Timer or Alarms app. The Mac is ignorant of all timers and alarms.
Here’s where we get to the HomePod’s software omission. Even if you set up your HomePod to access your reminders—which, I admit, you may be reluctant to do in some households—the HomePod will not alert you when a reminder comes due. I was first informed of this stunning fact by Holger Eilhard, and it’s been confirmed by others. So I guess you can create a reminder through your HomePod but not be alerted by one. For whatever that’s worth. Because I don’t think it’s worth much, I decided to put a No in the Reminder column for sharing on the HomePod.
A feature many people find essential is getting the time remaining before an alert goes off. I would like to tell these people to chill out, take a Zen approach, that “a watched pot never boils,” but that would only anger folks who seem to be a little on edge already. My blithe assertion that timed reminders is the solution to the lack of multiple timers was based too much on my own use. In the 4+ years I’ve been using reminders for timed alerts, I have never wanted to know how much time was left, but I guess the rest of the world doesn’t slavishly model itself after me.
So if you need to know the time left on an alert, the timer is your only friend. Neither alarms or reminders will give you that. Alarms and timers will give you the time an alert will go off (like 8:55 PM), but you’ll have to do the subtraction yourself, which isn’t convenient.
By the way, although I put a Yes in the “Time of” section for the Watch, my watch has never actually been able to tell me the time a reminder is due when I ask it via Siri. It definitely understands me, and it acts like it’s going to retrieve that information, but it’s never finished the job. I can, of course, see the due time of a reminder using the watch’s Reminders app.
And there are also a couple of problems with asking Siri for the time of a reminder on the phone:
The obvious problem is that the time Siri says is wrong. And it’s been wrong every time I’ve tried this over the past two days.4 For this example, the reminder was set for 3:50 PM, but Siri told me a time six hours earlier. Now, I happen to live six hours away from UTC, so my first thought was that Siri was programmed (stupidly) to respond in universal time. But then I realized the six hour difference was in the wrong direction. 3:50 PM US/Central is 9:50 PM UTC, not 9:50 AM UTC. So Siri’s answer is so bad it isn’t even wrong in an understandable way.
The less obvious problem is Siri’s characterization of my casserole reminder as the “next reminder.” Inexplicably, she uses that phrase even if the reminder you ask about isn’t the next one. Sigh.
After going through this exercise, I will continue to use timed reminders because
• they work across all my devices, including the Mac;
• their deficiencies regarding the time remaining don’t affect me; and
• they don’t require a second action to get them out of the way when completed, unlike alarms.
I’ve said on Twitter that I think Apple intends timed reminders to be the substitute for multiple timers. I still think that, but I’m less certain now than I was a few days ago.
Update Feb 18, 2018 9:22 AM
There’s always more.
First, something I had scribbled in a note but forgot to put in the post: a timer may not sound an alert. If you like to fall asleep listening to music, you may have the Timer’s When Timer Ends setting assigned to Stop Playing.
If that’s the case, the next time you use Siri to set a timer, it won’t make a sound, which probably isn’t what you want.
Second, reader Thomas Shannon has emailed me that alarms go off only at minute markers. So if it’s 9:55:45 and you tell Siri to set an alarm for one minute, it will go off 15 seconds later. I was annoyed to hear this because I looked into this four years ago with regard to reminders and found that their alert times are not restricted to whole minutes. If you tell Siri at 9:55:45 to remind you of something in one minute, the alert goes off at 9:56:45.
I used to tell people the advantage of using Apple products was their consistency across devices and applications. I don’t do that anymore.
1. You’re right, the Mac isn’t an iOS device, but it does work with Reminders, which can be very handy, so I’m including it.
2. Hah! I slay me.
3. I’m using casseroles in the examples because I’m a homespun Midwesterner (and not from Minnesota).
4. As I said above, I’ve never asked about the time of a reminder. Good thing, too.
# Friendly reminders
My vision of myself as a powerful thinkfluencer in the Apple world took a real beating this week. It seemed as if everyone who got a HomePod was complaining that it couldn’t set multiple timers. This is something I’ve written about a couple of times, going back four years. And I’ve explained the solution. Is this thing on?
Of course, four years ago, I wasn’t talking about the HomePod, I was talking about the iPhone, but the principle is the same. In iOS, the timer function is in the Clock app, and there’s only one. There’s no way to have two timers running simultaneously and no way to give your timer a name that lets you know what it’s for.
But you do have Reminders. They have names and can be set to alarm not only at an absolute time, but also at a relative time:
“Hey Siri, remind me to check the casserole in 20 minutes.”
This works on my iPhone, iPad, and Watch, and I assume—based on this article—that it would work on my HomePod if I had one. This is clearly Apple’s preferred solution to setting mulitple timers, each with a distinct name.
So I was frustrated to hear John Gruber and Paul Kafasis in the latest episode of The Talk Show complain about the multiple timer problem. They should both know how to use Reminders to solve this problem. So should Myke Hurley, who made the same complaint in the most recent Upgrade.
I understand where they’re coming from. If you’re an Amazon Echo user, you’re probably in the habit of saying something like
“Alexa, set a 20-minute timer for the casserole.”
Habits like that are hard to break, especially as you get older.1 But Apple users should be used to the idea that Apple has strong opinions about the right way to use its products and you’re usually better off not bucking the system.
You don’t like cluttering up your Reminders with hundreds of “check the casserole” and “check the tea” items? Even though you typically don’t see completed reminders? There is a solution.
In the past couple of days, the HomePod complaint industry has moved on from multiple timers to white rings. Cheaply made leather circles are already coming onto the market, but I’m going to suggest that high end furniture protection should come from lace doilies with tatting that complements the HomePod’s fabric pattern.
1. Myke is 30 now, so his brain has lost much of its former plasticity.
# LaTeX contact info through Workflow
I’ve been writing more on my iPad recently; not just blog posts, but reports for work, too. Because I have a lot of helper scripts and macros built up over many years of working on a Mac, writing on the iPad is still slower. But I’m gradually building up a set of iOS tools and techniques to make the process go faster. Today’s post is about a Workflow I built yesterday with advice from iOS automation experts conveyed over Twitter.
For several years, I wrote reports for work using a Markdown→LaTeX→PDF workflow. For most of those years, it was rare for me to have to edit the LaTeX before turning it into a PDF. Recently, though, that rarity has disappeared, mainly because my reports have more tables and figures of varying size that need to be carefully positioned, something that can’t be done in Markdown. A few months ago I decided it would be more efficient to just write in LaTeX from the start. This wasn’t as big a change as you might think. I used to write in LaTeX directly, and the combination of TextExpander and a few old scripts I resurrected got me back up to speed relatively quickly—on the Mac, anyway.
On iOS, most of the TextExpander snippets I built for writing in LaTeX work fine, but the helper scripts, which tend to rely on AppleScript, don’t. One of the scripts I definitely wanted an iOS counterpart for was one that extracted the contact information from a client in a particular format. In my reports, the title page usually includes section for the name, company, and address of the client. This is added in the LaTeX source code by this:
tex:
\client{John Cheatham\\
Dewey, Cheatham \& Howe\\
1515 Loquitor Lane\\
Amicus OH 44100}
where \client is a LaTeX command I created long ago, and its argument needs the usual LaTeX double backslashes to designate line breaks. Also, ampersands, which are special characters in LaTeX, need to be escaped.
I thought I could whip something up in Workflow, but my limited understanding of Workflow isn’t conducive to whipping. When I first tried to put something together a couple of weeks ago, it looked to me as if I was going to have to painstakingly extract every piece of information from the selected contact, create variables to store them in, and then put those variables together into a new string of text. So I gave up.
Yesterday I decided to ask for help.
I would like to extract from a selected contact a standard name/address block as plain text:
Full Name
Company
City, ST Zip
I don’t think Contacts or Interact do this. Does anything?
— Dr. Drang (@drdrang) Fri Feb 9 2018 9:37 PM
As you can see, I asked for something a bit simpler than what I really wanted, and I was kind of expecting suggestions for an app that would do the trick. But I soon got a response from Ari Weinstein with a Workflow solution:
Since Ari is a co-developer of Workflow, I kind of figured he knew what he was talking about. But I didn’t, and it’s because I didn’t appreciate Workflow’s magic variables. I’ve always thought of Workflow as being almost like a functional language, where each action transforms the data passed to in and sends it along to the next action in turn. That, at least, is what I thought happened when the actions are connected by lines.
Which is why I didn’t understand Ari’s workflow at first. I figured that if it was extracting the Street Address in the second step, there’d be no way for it to get ahold of the Name and Company in the fourth step. What I didn’t appreciate was that there can be side effects the usual view of a workflow doesn’t show you. In this case, the Contact that’s selected in the first step is saved to a magic variable (called “Contact”) that remains available for use in later steps. So the third and fourth steps have access to all the Contact information even after the extraction of the Street Address in the second step.
Ari’s sample is a standard workflow that would have to be run from within Workflow itself or from a launcher app like Launch Center Pro. I was thinking about how to turn it into an Action Extension that could be called from within Contacts when I noticed I had a Twitter reply from Federico Viticci:
His suggestion is set up as an Action Extension that accepts only Contacts and extracts the info from the Workflow Input magic variable. Just what I was going to do.
“My” final workflow, called , combines what I learned from Ari and Federico and adds some search-and-replace stuff to handle the LaTeX-specific parts:
The first two steps create a text variable named Ret that consists of a single line break. We’ll see why I needed it in a bit.
Steps 3–5 are the Ari/Federico mashup. I couldn’t use Federico’s suggestion to just add Workflow input:Street Address to the end of the block because my contacts usually include the country, even though the country is almost always the US, and I didn’t want that at the end of the block. At some point, I’ll improve this by writing up a filter that deletes the country line only if it’s the US, but this will do until I get another job with a non-US client.
Step 6 escapes the ampersands, and Step 7 adds the double backslashes to the ends of each line. You need four backslashes to get two in the output because regexes need two to produce one. I thought I could use \n at the end of the replacement string to get a line break, but I couldn’t get that to work. Thus, the Ret variable defined at the beginning of the workflow.
Finally, Step 8 puts the text on the clipboard, ready for pasting into a LaTeX document.
My plan is to use this extension in Split View, with my text editor, currently Textastic, on one side and Contacts on the other. When I need to insert the client info, I find it in Contacts, tap Share Contact to bring up the Sharing Sheet, and select the Run Workflow action.
This brings up the list of Workflow Action Extensions that can accept Contacts. I choose LaTeX Address from the list, switch focus back to Textastic, and paste the text block where it belongs. Boom.
I’ll try to remember to look for magic variables the next time I make a workflow. There is a trick to making them visible. When you’re editing a workflow and can insert a variable (magic or otherwise), a button with a magic wand will appear in the special keyboard row.
Tapping it will give you a new view of your workflow, with the magic variables appearing where the workflow creates them.
You don’t need to do this, as all of these variables should appear in the special keyboard row if you keep scrolling it to the right. But I find it easier to understand what they are and where they come from in this view.
Thanks to everyone who had suggestions for me, especially Ari and Federico.
|
2018-02-25 14:20:10
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3776843249797821, "perplexity": 1641.5053029033274}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816462.95/warc/CC-MAIN-20180225130337-20180225150337-00632.warc.gz"}
|
https://www.techwhiff.com/issue/a-measure-of-the-degree-to-which-capital-wears-out--671421
|
# A measure of the degree to which capital wears out or becomes obsolete during a period is:________
###### Question:
A measure of the degree to which capital wears out or becomes obsolete during a period is:________
### Using special products what is (wx-y)^2
using special products what is (wx-y)^2...
### How do you solve x+2/3=11//12
how do you solve x+2/3=11//12...
### Based on the song twelfth song of thunder what might a member of the navajo Nation find beautiful
based on the song twelfth song of thunder what might a member of the navajo Nation find beautiful...
### In the periodic table of elements, what do all the elements 2 in a row have in common
in the periodic table of elements, what do all the elements 2 in a row have in common...
### How do you ask (to) where in Spanish ? A donde B adonde C cuando D a quien
How do you ask (to) where in Spanish ? A donde B adonde C cuando D a quien...
### Where do marine scientists get the ethical standards for their profession and what kind of values do they include?
Where do marine scientists get the ethical standards for their profession and what kind of values do they include?...
### Harry focused on numbers in his persuasive essay, hoping to appeal to his readers' sense of reasoning. What rhetorical strategy was he using?
Harry focused on numbers in his persuasive essay, hoping to appeal to his readers' sense of reasoning. What rhetorical strategy was he using?...
### 15 point and brainlyest if possible. Thanks :)
15 point and brainlyest if possible. Thanks :)...
### Who owned arizona before it became a state?
Who owned arizona before it became a state?...
### A financial advisor is analyzing a family's estate plan. The amount of money that the family has invested in different real estate properties is normally distributed with a mean of $225,000 and a standard deviation of$50,000. Use a calculator to find how much money separates the lowest 80% of the amount invested from the highest 20% in a sampling distribution of 10 of the family's real estate holdings.
A financial advisor is analyzing a family's estate plan. The amount of money that the family has invested in different real estate properties is normally distributed with a mean of $225,000 and a standard deviation of$50,000. Use a calculator to find how much money separates the lowest 80% of the a...
### I need help who wants to help ??
I need help who wants to help ??...
### Need help solving this problem. The “3.” Is just the number of the problem it’s not part of the equation.
Need help solving this problem. The “3.” Is just the number of the problem it’s not part of the equation....
### What two compounds are the waste products of cellular respiration? * H2O and CO2 H2O and C6H12O6 CO2 and C6H12O6 ATP and CO2
What two compounds are the waste products of cellular respiration? * H2O and CO2 H2O and C6H12O6 CO2 and C6H12O6 ATP and CO2...
### This diagram shows a red blood cell in a beaker that contains a solution of with a higher salt concentration than that inside the red blood cell. high salt concentration low salt concentration What will happen to the red blood cell in this environment?
This diagram shows a red blood cell in a beaker that contains a solution of with a higher salt concentration than that inside the red blood cell. high salt concentration low salt concentration What will happen to the red blood cell in this environment?...
### Which of the following statements about the accounting cycle is NOT correct:_______. A. Transactions are journalized before they are posted B. Financial statements are prepared based on the balances in the adjusted trial balance C. The unadjusted trial balance is prepared before the adjusted trial balance D. Closing entries are recorded and posted throughout each accounting period
Which of the following statements about the accounting cycle is NOT correct:_______. A. Transactions are journalized before they are posted B. Financial statements are prepared based on the balances in the adjusted trial balance C. The unadjusted trial balance is prepared before the adjusted trial...
### Describe how you can tell that a fraction corresponds to a repeating decimal and not a terminating decimal
Describe how you can tell that a fraction corresponds to a repeating decimal and not a terminating decimal...
### What crime was committed in the teapot dome scandal
what crime was committed in the teapot dome scandal...
### Write a function longer_string() with two string input parameters that returns the string that has more characters in it. If the strings are the same size, return the string that occurs later according to the dictionary order. Use the function in a program that takes two string inputs, and outputs the string that is longer.
Write a function longer_string() with two string input parameters that returns the string that has more characters in it. If the strings are the same size, return the string that occurs later according to the dictionary order. Use the function in a program that takes two string inputs, and outputs t...
|
2022-12-08 09:10:08
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1818310171365738, "perplexity": 1689.3275780865283}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711286.17/warc/CC-MAIN-20221208082315-20221208112315-00322.warc.gz"}
|
http://mathoverflow.net/revisions/105990/list
|
2 added 5 characters in body
Radial Fourier transforms provide a good, consistent perspective on most of the theory. The Fourier transform $\widehat{f}(t)$ of a function $f \colon \mathbb{R}^n \to \mathbb{R}$ is given by the integral of $f(x) e^{2\pi i \langle x,t \rangle} \, dx$ over $x \in \mathbb{R}^n$. If $f$ is a radial function (i.e., $f(x)$ depends only on $|x|$), then we can radial symmetrize everything and the exponential function averages out to a radial function. Specifically, we get $$\widehat{f}(t) = 2\pi |t|^{-(n/2-1)} \int_0^\infty f(r) J_{n/2-1} (2 \pi r |t|) r^{n/2} \, dr.$$ The precise factors are a little annoying, but basically this just means $J_{n/2-1}$ is what you get when you radially symmetrize an exponential function in $n$ dimensions. It's easy to see that if you symmetrize $e^{2\pi i \langle x,t \rangle}$ by averaging over all $x$ on a sphere, then you get a radial function of $t$, and furthermore as you vary the radius of the sphere you just rescale the function. So the one function $J_{n/2-1}$ captures all of this, modulo scaling.
One consequence is that Bessel functions inherit the orthogonality of the exponential functions (i.e., the different scalings are orthogonal), so they also inherit all the consequences of orthogonality. For example, this is really where the differential equation comes from. There's a strong analogy between Bessel functions and orthogonal polynomials, where rescaling the Bessel function corresponds to varying the degree of the polynomial.
You also get certain qualitative results for free: for example, the product of two Bessel functions should be an integral of Bessel functions with positive coefficients, since this corresponds to saying the product of two radial, positive-definite functions remains positive definite. You can write down the coefficients explicitly, but sometimes all you need is nonnegativity, and in any case this point of view makes it easy to believe that there should be an explicit formula.
This is basically a low-brow version of the representation theory approach. Basically, ordinary Fourier analysis studies $L^2(\mathbb{R}^n)$ under the action of the translation group $\mathbb{R}^n$. If you look at the full group of isometries of $\mathbb{R}^n$ (including the orthogonal group), then it's just a little more elaborate, and the Bessel functions arise as zonal spherical functions. It's worthwhile working through this perspective, but in practice just thinking about radial Fourier analysis gives you most of the benefits with less machinery.
1
Radial Fourier transforms provide a good, consistent perspective on most of the theory. The Fourier transform $\widehat{f}(t)$ of a function $f \colon \mathbb{R}^n \to \mathbb{R}$ is given by the integral of $f(x) e^{2\pi i \langle x,t \rangle} \, dx$ over $x \in \mathbb{R}^n$. If $f$ is a radial function (i.e., $f(x)$ depends only on $|x|$), then we can radial symmetrize everything and the exponential function averages out to a radial function. Specifically, we get $$\widehat{f}(t) = 2\pi |t|^{-(n/2-1)} \int_0^\infty J_{n/2-1} (2 \pi r |t|) r^{n/2} \, dr.$$ The precise factors are a little annoying, but basically this just means $J_{n/2-1}$ is what you get when you radially symmetrize an exponential function in $n$ dimensions. It's easy to see that if you symmetrize $e^{2\pi i \langle x,t \rangle}$ by averaging over all $x$ on a sphere, then you get a radial function of $t$, and furthermore as you vary the radius of the sphere you just rescale the function. So the one function $J_{n/2-1}$ captures all of this, modulo scaling.
One consequence is that Bessel functions inherit the orthogonality of the exponential functions (i.e., the different scalings are orthogonal), so they also inherit all the consequences of orthogonality. For example, this is really where the differential equation comes from. There's a strong analogy between Bessel functions and orthogonal polynomials, where rescaling the Bessel function corresponds to varying the degree of the polynomial.
You also get certain qualitative results for free: for example, the product of two Bessel functions should be an integral of Bessel functions with positive coefficients, since this corresponds to saying the product of two radial, positive-definite functions remains positive definite. You can write down the coefficients explicitly, but sometimes all you need is nonnegativity, and in any case this point of view makes it easy to believe that there should be an explicit formula.
This is basically a low-brow version of the representation theory approach. Basically, ordinary Fourier analysis studies $L^2(\mathbb{R}^n)$ under the action of the translation group $\mathbb{R}^n$. If you look at the full group of isometries of $\mathbb{R}^n$ (including the orthogonal group), then it's just a little more elaborate, and the Bessel functions arise as zonal spherical functions. It's worthwhile working through this perspective, but in practice just thinking about radial Fourier analysis gives you most of the benefits with less machinery.
|
2013-05-25 18:01:34
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9061599969863892, "perplexity": 175.40343724823975}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368706009988/warc/CC-MAIN-20130516120649-00075-ip-10-60-113-184.ec2.internal.warc.gz"}
|
http://slideplayer.com/slide/3381378/
|
# More Set Definitions and Proofs 1.6, 1.7. Ordered n-tuple The ordered n-tuple (a1,a2,…an) is the ordered collection that has a1 as its first element,
## Presentation on theme: "More Set Definitions and Proofs 1.6, 1.7. Ordered n-tuple The ordered n-tuple (a1,a2,…an) is the ordered collection that has a1 as its first element,"— Presentation transcript:
More Set Definitions and Proofs 1.6, 1.7
Ordered n-tuple The ordered n-tuple (a1,a2,…an) is the ordered collection that has a1 as its first element, a2 as its second element... And an as its nth element. 2-tuples are called ordered pairs.
Cartesian Product of A and B Let A and B be sets. The Cartesian product of A and B, denoted A x B is the set of all ordered pairs (a,b) where a A and b B. Hence A x B = {(a,b) | a A b B} The Cartesian product of the sets A1,A2,.., An denoted by A1 x A2 x … x An is the set of ordered n-tuples (a1,a2,..,an) where ai belongs to Ai for I = 1,2,...,n. A1 x A2 x…x An = {(a1,a2,..,an) | ai Ai for I=1,2…,n}
Generalized Unions and Intersections A1 A2 ... An = A1 A2 ... An =
Let Ai = {1,2,3…i}for i = 1,2,3,… (that is, A1=1; A2=1,2; A3=1,2,3; etc…) Find = 1,2,3,..., n = 1
Let Ai = {i,i+1,i+2…} Find = Z+ = n, n+1, n+2, …
Symmetric Difference Problem Prove(A B) B = A A B elements in A or B but not in both.
Prove (A B) B = A ABA B(A B) B 1101 1011 0110 0000
Prove (A B) B = A Proof: We must show that (A B) B A and that A (A B) B. First we will show that (A B) B A. Let e (A B) B. Then e (A B) or e B but not both. If e (A B), then either e A or e B. If e A and e B then we are done. If e B, and e A, then e (A B) but can not be an element of (A B) B by definition so this case can not exist.
Proof of (A B) B = A, cont. Now we will show that A (A B) B. Let e A. Either e is also B or e B. If e B, then e (A B) so e is an element of (A B) B. If e B, e is an element of (A B) and e must be an element of (A B) B. Thus (A B) B = A.
Computer Representation of Sets How to store the elements of sets and make computing the union, intersection, difference, etc., easier? Assume U is finite and of reasonable size. It has cardinality n. First, specify an arbitrary ordering of the elements of U. Represent a subset A of U with a bit string of length n, where the i’th bit is 1 if u i belongs to A and 0 if u i does not belong to A.
Using the Computer Representation Let U = {1,2,3,4,5,6,7,8,9,10}. Assume an ordering of the elements as written What bit string represents the subset of all odd integers? What bit string represents the set of all integers that do not exceed 5? What’s the complement of this set? 1010101010 1111100000 0000011111 General rule for complements?
Using the Computer Representation The bits strings of {1,2,3,4,5} and {1,3,5,7,9} are 111110000 and 1010101010, respectively What is the union of these sets? 1111100000 1010101010 = What is the intersection of these sets? 1111100000 1010101010 = What’s the general rules? 1111101010 1010100000 bitwise OR for union; bitwise AND for intersection
Download ppt "More Set Definitions and Proofs 1.6, 1.7. Ordered n-tuple The ordered n-tuple (a1,a2,…an) is the ordered collection that has a1 as its first element,"
Similar presentations
|
2018-06-21 01:09:43
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8071069717407227, "perplexity": 3793.5815355154195}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267863980.55/warc/CC-MAIN-20180621001211-20180621021211-00621.warc.gz"}
|
https://socratic.org/questions/a-recipe-requires-1-3-cup-of-milk-for-each-1-4-cup-of-water-how-many-cups-of-wat
|
# A recipe requires 1/3 cup of milk for each 1/4 cup of water. How many cups of water are needed for each cup of milk?
Then teach the underlying concepts
Don't copy without citing sources
preview
?
#### Explanation
Explain in detail...
#### Explanation:
I want someone to double check my answer
18
Oct 21, 2017
1 cup of milk is used with $\frac{3}{4}$ cup of water
#### Explanation:
A comparison between the cups of milk and cups of water can be written as a ratio:
Note that: $\frac{1}{3} > \frac{1}{4}$
$\text{ }$milk : water
$\text{ "1/3 : 1/4" } \leftarrow \times 12$ to get rid of fractions
$\frac{12}{1} \times \frac{1}{3} : \frac{12}{1} \times \frac{1}{4}$
$\text{ } 4 : 3$
4 cups of milk are used with 3 cups of water $\text{ } \leftarrow \div 4$
1 cup of milk is used with $\frac{3}{4}$ cup of water
• 27 minutes ago
• 29 minutes ago
• 29 minutes ago
• 29 minutes ago
• A minute ago
• 4 minutes ago
• 5 minutes ago
• 8 minutes ago
• 9 minutes ago
• 25 minutes ago
• 27 minutes ago
• 29 minutes ago
• 29 minutes ago
• 29 minutes ago
|
2018-06-23 23:03:46
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 8, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5118592977523804, "perplexity": 5049.815344365757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267865438.16/warc/CC-MAIN-20180623225824-20180624005824-00507.warc.gz"}
|
http://math.stackexchange.com/questions/251147/binomial-theorem-question/251152
|
# Binomial Theorem Question..
Just studying for my combinatorics exam. My prof said there would be a question similar to this one on the exam, so I'm trying to sort this one out.
$$\sum^{20}_{k=0} \binom{41}{k}$$
I know if I can factor out $\dbinom{20}{k}$ then I can get the following:
$$A\sum^{20}_{k=0}\binom{20}{k} * 1^k * 1^{n-k} = A(1+1)^{20}$$
I just don't know how to get there...
Any help is greatly appreciated!
-
You’re on the wrong track, I’m afraid. The trick is to realize that your sum is exactly half of the full sum $\sum_{k=0}^{41}\binom{41}k$ because of the symmetry of the binomial coeffients: $\binom{41}{41-k}=\binom{41}k$.
\begin{align*} 2^{41}&=\sum_{k=0}^{41}\binom{41}k\\ &=\sum_{k=0}^{20}\binom{41}k+\sum_{k=21}^{41}\binom{41}k\\ &=\sum_{k=0}^{20}\binom{41}k+\sum_{k=21}^{41}\binom{41}{41-k}\\ &=\sum_{k=0}^{20}\binom{41}k+\sum_{i=0}^{20}\binom{41}i&&\text{set }i=41-k\\ &=2\sum_{k=0}^{20}\binom{41}k\;, \end{align*}
so that $$\sum_{k=0}^{20}\binom{41}k=2^{40}\;.$$
-
$$(1+1)^{41}= \sum^{41}_{k=0} \binom{41}{k}$$
but since the binomial coefficients are symmetrical around $\,k=21\,$ , we know that
$$2^{41}= \sum^{20}_{k=0}\binom{41}{k}+\sum^{20}_{k=0}\binom{41}{k}\,$$
thus
$$2^{40}= \sum^{20}_{k=0} \binom{41}{k}$$
-
Substitute $x=1$ in the expansion $(1+x)^{41}=\sum_0^{41}\binom{41}{k}x^k$ to obtain $\sum_0^{41}\binom{41}{k}=2^{41}$. Since $\binom{41}{k}=\binom{41}{41-k}$, we get $\sum_0^{20}\binom{41}{k}=\frac{1}{2}\sum_0^{41}\binom{41}{k}=2^{40}$.
-
|
2014-12-19 02:22:18
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9459807276725769, "perplexity": 364.0620295048726}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802768169.20/warc/CC-MAIN-20141217075248-00107-ip-10-231-17-201.ec2.internal.warc.gz"}
|
https://quant.stackexchange.com/questions/17166/calculate-efficient-frontier-using-fportfolio-with-incomplete-set-of-returns
|
# Calculate efficient frontier using fPortfolio with incomplete set of returns
I want to calculate the efficient frontier for a set of 140 assets using returns from the past 10 years. However, some of these assets came into existence only more recently, so for some assets I have the returns for the full 10 years, but for others I have returns only e.g. the last 3 years.
I can calculate the efficient frontier (as described in http://www.finance-r.com/s/efficient_frontier_fPortfolio/complete/ ) if I use the tail of the returns for which all returns are available. But I'd like to use the the full set of returns, where available.
Currently fPortfolio throws an exception when I input a dataset with NAs. How would I reach my goal? I suspect I'd have to tinker with the fPortfolio source.
You might take a look at the PortfolioAnalytics package. It's optimize.portfolio function does require asset returns but the momentFUN argument allows you to provide your own function for using these returns to calculate the moments used in the optimization. Overall it provides a great deal of flexibility for specifying constraints and optimization methods. It's not on CRAN but is available from Return Analytics Development Page .
|
2021-01-15 18:09:35
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4769338071346283, "perplexity": 835.4438283314217}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703495936.3/warc/CC-MAIN-20210115164417-20210115194417-00679.warc.gz"}
|
https://astronomy.stackexchange.com/questions/27447/differences-between-alt-az-in-stellarium-and-astropy
|
# Differences between alt/az in Stellarium and Astropy
I've just started to learn Astropy. It's a nifty tool. Using some tutorials, I wrote and then made modifications to a little script that finds the Alt/Az of an object at a certain place and time. I then checked Stellarium's calculation for the Alt/Az of the same object (at exactly the same place and time) and found it to be slightly different. I suspected this was due to slight differences in the underlying coordinates, so I tried manually hard coding the coordinate data for the star from Stellarium; it still came out different. Here is the data without the hard coding:
Object: Rasalhague (alf Oph)
Time: 2018-08-28 23:00 (converted to universal in the script)
Latitude: 35
Longitude: -79
Stellarium reported Az/Alt: +247 35 12.3 / +48 00 57.8
Astropy reported Az/Alt: +247 35 37.4939s / +48 00 08.7586
Manually coding:
RA/Dec on date from Stellarium: 17 35 48.82, +12 32 55.9
Astropy reported Az/Alt: +247 22 42.0379 / +48 09 43.0755
I'm guessing that the difference stems from the conversion algorithm to alt/az.
• Did you verify the conversion to Zulu (or Universal) gave the same time as Stellarium uses? – Carl Witthoft Aug 24 '18 at 17:13
• stackoverflow.com/questions/16293146/… may or may not be helpful. – user21 Aug 24 '18 at 19:16
• @CarlWitthoft Yes, the time is the same. – Alphecca Aug 25 '18 at 0:20
• Among the things that can cause this are 1) precession (different equinox) 2) Aberration (due to Earth's finite speed vs speed of light) 3) refraction (but at alt=-12 degrees you are looking through stone, not atmosphere). Things that are too small to be the cause of this are a) proper motion (at the arcsec level) b) parallax (sub-arcsecond). The 'manual coding' result has a size consistent with precession. Try the calculation on Jan 1, 2000 and see whether things match up. – DMPalmer Aug 25 '18 at 16:31
• @DMPalmer Thanks for your comment. I have tried setting the starting RA/DEC to FK5 with an equinox of J2000.0. It didn't bring it in line with Stellarium. – Alphecca Aug 26 '18 at 3:29
Perhaps the difference stems from the conversion algorithm?
Almost certainly this is the answer. It's not clear to me exactly what Stellarium assumes for is refraction calculation, but I know Astropy's algorithm. Given that the altitude you've listed here is below the horizon (i.e., alt is negative), it's not really clear what the correct interpretation is: in particular, atmospheric refraction is not really well-defined below the horizon (since it should be invisible anyway!).
Additionally, even well above the horizon where refraction isn't that important, there are several finicky details that you have to account for to properly do the Az/Alt conversions. In addition to the precession that the other answers mention, if you want to get exactly the same answer on two tools, you have to worry about things like how much the axis of the Earth shifts due to, say, major earthquakes ("true polar wander"), the effects of General Relativity on the apparent direction of the incoming light ,etc. Most of these are pretty small, but together they mean it's very hard to ensure all of the software gives exactly the same answer unless they carefully follow the definitions set by the International Astronomical Union.
That said, the differences you see are only ~20 arcsec. The human eye isn't even capable of telling that small of a difference, and almost all telescopes that people would use Stellarium with probably wouldn't either (at least in an absolute sense). So for this particular application they essentially do match.
• Thanks for your reply. I've modified the example I gave to make both Az and Alt positive numbers. Do you know which reference frame Stellarium is using for its coordinates? I think I read somewhere that the Hipparchos catalog might have its own custom frame associated with it... perhaps that's part of the difficulty. – Alphecca Aug 29 '18 at 3:34
The alt/az coordinates will change as the object moves, and are relative to the observing position. Both programs will need to be configured with the exact same time, latitude, longitude, and altitude. If you want to compare the two, you have to get get readings at precisely the same time.
Even then, it's possible for variation if they are not using the same epoch. If one is using J2000 and another J1950 or J2050, then there will be differences. It might be possible for one or the other to calculate precessional differences to provide something closer to exact coordinates.
• Hi, thanks for your reply. I can confirm that I'm using exactly the same time, latitude, longitude, and altitude for my script in Astropy and my settings in Stellarium. I believe Astropy defaults to ICRS; I tried converting the RA/DEC into FK5 to see if that was closer to Stellarium, and it didn't make up for the difference. – Alphecca Aug 25 '18 at 0:19
The apparent alt/az position of a star in the night sky depends not only on its underlying equatorial coordinates and position/time of your interest, but also from other conditions which may or may not be accounted for by the one or the other software, depending on how you have configured the software.
One simple example for those conditions would be the atmospheric refraction. Its effect on the apparent alt/az position of a start is: the lower the object is above the horizon, the higher altitude it seems to have.
Since we don't know your script exactly and since you're saying that you've just started learning AstroPy, it could well be that you haven't configured both software with the same conditions for the alt/az calculation algorithm.
• Hi, thanks for your reply. All of the basic parameters for the Alt/Az calculation are the same across both tools. Your point about other concerns (like atmospheric refraction) is something I didn't consider. I'll have to research how to update my script accordingly. – Alphecca Aug 25 '18 at 0:27
|
2021-04-15 05:49:46
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5352298021316528, "perplexity": 1090.6693899864488}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038083007.51/warc/CC-MAIN-20210415035637-20210415065637-00615.warc.gz"}
|
https://tex.stackexchange.com/questions/311884/how-to-create-a-table-automatically-for-a-homework-in-statistics
|
# How to create a table automatically for a homework in statistics?
I want to create a long table containing some statistical data, such as the sample, the average, and the variance, etc. I am using the pgf package and its various extensions to do this.
Here is a silly MWE:
\documentclass{article}
\usepackage{pgf}
\usepackage{tikz}
\usepgflibrary{fpu}
\pgfmathsetseed{\number\pdfrandomseed}
\begin{document}
\xdef\Sum{0}
\xdef\Rand{0}
\xdef\Avg{0}
\xdef\Var{0}
\xdef\Sq{0}
%\begin{table}[h]
%\begin{tabular}{|c|c|c|c|c|c|}
%\hline
%Trial & Rand & Sum & Sum-of-squares & Average & Variance\\
%\hline\hline
%\noindent Trial Rand Sum Sum-of-squares Average Variance\\
\noindent\foreach \i in {1 ,...,15} {
%\pgfkeys{/pgf/fpu,/pgf/fpu/output format=sci}
\pgfmathparse{random(10)}
\xdef\Rand{\pgfmathresult}
\pgfkeys{/pgf/fpu,/pgf/fpu/output format=sci}
\pgfmathparse{\Sum+\Rand}
\xdef\Sum{\pgfmathresult}
\pgfmathparse{\Sq+\Rand*\Rand}
\xdef\Sq{\pgfmathresult}
\pgfmathparse{\Sum/\i}
\xdef\Avg{\pgfmathresult}
\pgfmathparse{\Sq/\i-\Avg*\Avg}
\xdef\Var{\pgfmathresult}
\noindent \i,\ %&%
\pgfmathparse{\Rand}\pgfmathresult,\ %&%
\pgfmathparse{\Sum}\pgfmathresult,\ %&%
\pgfmathparse{\Sq}\pgfmathresult, %&%
\pgfmathparse{\Avg}\pgfmathresult, %&%
\pgfmathparse{\Var}\pgfmathresult \\
%\hline
}
%\end{tabular}
%\end{table}
\end{document}
Here is an output (depending on what random numbers you get):
As you can see, the table is commented out, as I have given up on it. I want to generate a document like this with 1000 pages, and see the computed empirical average and variance.
Very closely related:
Note: this question has been heavily edited due to showing signs of the XY problem. I will reask problem X in a separate question later.
• While egreg is right that you cannot build a table in a \foreach loop, you might want to take a look at the pgfplotstable package. (Or someone with more experience can show us both how to do it.) – Qrrbrbirlbel May 28 '16 at 22:30
I suspected that your question How do I use the ampersand (&) inside a foreach or conditional (or other group/environment) when building tables? was of the XY type.
The usual problem in these cases is that you cannot build a table inside a \foreach statement, because table cells form groups. The strategy is to build the table body beforehand.
\documentclass{article}
\usepackage{booktabs,etoolbox}
\usepackage{pgf}
\usepackage{tikz}
\usepgflibrary{fpu}
\pgfmathsetseed{\number\pdfrandomseed}
\begin{document}
\def\Sum{0}
\def\Rand{0}
\def\Avg{0}
\def\Var{0}
\def\Sq{0}
\def\TableBody{}
\foreach \i in {1,...,15} {
%\pgfkeys{/pgf/fpu,/pgf/fpu/output format=sci}
\pgfmathparse{random(10)}
\xdef\Rand{\pgfmathresult}
\pgfkeys{/pgf/fpu,/pgf/fpu/output format=sci}
\pgfmathparse{\Sum+\Rand}
\xdef\Sum{\pgfmathresult}
\pgfmathparse{\Sq+\Rand*\Rand}
\xdef\Sq{\pgfmathresult}
\pgfmathparse{\Sum/\i}
\xdef\Avg{\pgfmathresult}
\pgfmathparse{\Sq/\i-\Avg*\Avg}
\xdef\Var{\pgfmathresult}
\xappto\TableBody{\i & \Rand & \Sum & \Sq & \Avg & \Var \noexpand\\}
}
\begin{tabular}{ *{6}{c} }
\toprule
Trial & Rand & Sum & Sum-of-squares & Average & Variance \\
\midrule
\TableBody
\bottomrule
\end{tabular}
\end{document}
Here's what I get with longtable and 500 draws.
• I will edit my question to match your answer to problem Y, and after more empirical testing I hope to reask question X again separately. Related: What is the XY problem?. – Matsmath May 28 '16 at 11:09
I am no expert in pgfplotstable but here is an idea to build upon.
To get the entries to align prettier, see Q131081.
(I find pgfplotstable overly complicated and it surely doesn't help that the manual repeats parts of the PGFmanual …)
## Code
\documentclass{article}
\usepackage{booktabs,pgfplotstable}
\pgfmathsetseed{\number\pdfrandomseed}
\pgfplotstableset{
duck ini/.style={columns={Trial,Rand}},
duck table/.style={
columns={Trial,Rand,Sum,Sq,Avg,Var},
set column name/.list={Sq:Sum-of-squared, Avg:Average, Var:Variance},
every head row/.append style={before row=\toprule, after row=\midrule},
every last row/.append style={after row=\bottomrule},
set column/.list={Avg:prec=1, Var:prec=2}},
set column name/.style args={#1:#2}{columns/#1/.append style={column name={#2}}},
set column/.style args={#1:#2}{columns/#1/.append style={#2}},
set expr/.style args={#1=#2}{create on use/#1/.style={create col/expr={#2}}}}
\pgfset{number format/prec/.style={fixed, fixed zerofill, precision={#1}}}
\pgfplotstableset{set expr/.list={Trial=\pgfplotstablerow+1,Rand=int(rnd*11)}}
\pgfplotstablecreatecol[expr={\pgfmathaccuma+\thisrow{Rand}}] {Sum} #1
\pgfplotstablecreatecol[expr={\pgfmathaccuma+\thisrow{Rand}*\thisrow{Rand}}] {Sq} #1
\pgfplotstablecreatecol[expr={\thisrow{Sum}/\thisrow{Trial}}] {Avg} #1
\pgfplotstablecreatecol[expr={\thisrow{Sq}/\thisrow{Trial}-\thisrow{Avg}*\thisrow{Avg}}]{Var} #1}
\begin{document}
\pgfplotstablenew[duck ini]{15}\t
|
2020-02-17 22:21:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8219066858291626, "perplexity": 5079.906187714786}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875143373.18/warc/CC-MAIN-20200217205657-20200217235657-00003.warc.gz"}
|
https://blender.meta.stackexchange.com/questions/2704/revamping-tags-2020/2705
|
# Revamping tags 2020
Given the recent concerns raised about misuse of tags and incoherent naming of addons this is a tentative proposal to generally improve tag names in Blender Stack Exchange, and bring about some order and categorization.
The underlying idea is to rename existing tags and add either common prefixes or suffixes so related terms are aggregated and grouped together.
These add missing structure by providing semantics, so similar tags look related, they would visually appear more alike, and users will see suggested related terms in the autocomplete popup when typing some tags.
Some concerns have been raised about readability and whether we should use prefixes or suffixes.
Prefixes have the advantage of affecting sorting order, also when listing stuff together having a common first part makes them look visually related, sort together and easier to follow; where suffixes don't.
The tag search engine does supports fuzzy searching and non-contiguous matches however, so it shouldn't have a severe effect. I don't have a strong opinion here both would work fine.
I'd rather have shorter, rather than longer tag names, but I feel the added structure is worth the trade-off.
Pros:
• Semantic relationship between tags
• Similar tags are grouped together
• Hopefully reduces misuse of ambiguous tags like
• Increased chance users find appropriate tags
Cons:
• Longer tags names
• Dubious real benefit to users
# Render Engines
All render engine related tags should receive a render-engine prefix
Addons would receive similar treatment. Tagging a question would make all specific addons pop up in suggestions, increasing the likelyhood of users finding the correct tags. For brevity I'd stick to "addon" rather than the official "add-on".
We could later expand to more areas like modifiers or nodes if we deem necessary
What do you guys think, is this worth the trouble? Any better naming schemes or different ideas?
# Render Engines:
The renaming of the render engine tags is mainly related to reducing the misuse of the rather generically named tag. Whether all the other render engine tags need renaming because of this, I don't know. I'm certainly not against having all the tags be consistent, but if there's no ambiguity with the other tags, then maybe they don't need renaming and only the tag needs to be looked at (whether that's adding 'render-engine' to the tag or something else). I think this is my general opinion of the other tag categories as well.
Again, I am pro-consistency, but at the minute it's only the that is the odd one out of the add-on tags and most of the add-on names are unique enough that they are not being confused with other tags very often. Maybe it will become an issue if there are more generically named add-ons in future.
(If we do go ahead of this I'm still not particularly convinced that it's worth saving a single character by omitting the '-' from 'add-ons', but I don't ultimately mind.)
# Modifiers:
There is a lack of consistency here. For example, some tags have modifier in the name ( and ) and some don't. 'build' would certainly be ambiguous without 'modifier' in the name. I actually think this makes the modifier tags the best use case for being consistently renamed.
However, there are some tags which relate to more than just the modifier, e.g. the tag relates to all types of mirroring, not just the modifier. Same with . In these instances will there be a tag in addition to a tag?
# Prefixes vs Suffixes
I still don't understand the advantages you have put forward for prefixes vs the disadvantage of readability. What are the actual usage examples of prefixes helping with sorting order? What example of using the site will lead to it being useful to have tags listed together? I get that prefixes do make tags look visually related and it would make similar tags be sorted together, and it certainly looks neat, but when specifically will this be useful?
# TLDR
To sum up, I think:
• Modifiers are the best use-case for having a suffix and it is less necessary for other tags. Other categories just need the odd tag fixed.
• Tags in a specific category should be consistent. If some tags in a specific category have a suffix, they all should.
• We should have suffixes as opposed to prefixes if it is decided tags need them.
• "In these instances will there be a mirror tag in addition to a mirror-modifier tag?" Yes in this case I'd say we would have two tags, one specifically for the modifier, one for the Edit Mode operator. One other example would be the Boolean Modifier vs the recently added Edit Mode Boolean Operations – Duarte Farrajota Ramos Feb 16 '20 at 21:03
• "I get that prefixes do make tags look visually related and it would make similar tags be sorted together, and it certainly looks neat, but when specifically will this be useful?" That it, it is mostly a visual thing, they look better and more consistent. They would also sort together at blender.stackexchange.com/tags or when typing in the tag field, rather than be scattered alphabetically. Again it is a minor thing I'm good either way – Duarte Farrajota Ramos Feb 16 '20 at 21:08
• Well, I disagree that 'looking better' is a good enough reason to go this route. And I also disagree that having tags sort when typing in the tag field is practically useful. If a user types 'render' why is it useful to have the tags grouped together? I mean, again, I get that it does it (although only some of the render tags appear when typing 'render'), but I still don't understand the specific use case? But I guess it doesn't matter as you went ahead with it anyway. – Ray Mairlot Feb 20 '20 at 13:34
• Other than sorting and visual appearance I don't think is a clear winner or advantage of prefixes over suffixes. Either way practically for end users it all ends up seeming ultimately indifferent, since we have synonyms setup. Nothing is written in stone, we can always change them back or rename them again it if we find it somehow harmful anyway – Duarte Farrajota Ramos Feb 20 '20 at 16:18
• I really don't think that for end users there isn't any difference. Having seen the tags in action it only enforces my thoughts about readability. Personally, I can no longer look at tags at a glance, I have to read past the prefix to get to the actual information in the tag. For me this prioritises appearance over readability. – Ray Mairlot Feb 20 '20 at 23:25
• Switched to suffixes, we loose some visual coherence, but it does gain in readability – Duarte Farrajota Ramos Feb 21 '20 at 12:52
|
2021-05-14 22:51:20
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23134145140647888, "perplexity": 2021.3406771122006}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991829.45/warc/CC-MAIN-20210514214157-20210515004157-00607.warc.gz"}
|
https://converter.ninja/volume/us-gallons-to-centiliters/419-usgallon-to-cl/
|
# 419 US gallons in centiliters
## Conversion
419 US gallons is equivalent to 158608.7537496 centiliters.[1]
## Conversion formula How to convert 419 US gallons to centiliters?
We know (by definition) that: $1\mathrm{usgallon}\approx 378.5411784\mathrm{centiliter}$
We can set up a proportion to solve for the number of centiliters.
$1 usgallon 419 usgallon ≈ 378.5411784 centiliter x centiliter$
Now, we cross multiply to solve for our unknown $x$:
$x\mathrm{centiliter}\approx \frac{419\mathrm{usgallon}}{1\mathrm{usgallon}}*378.5411784\mathrm{centiliter}\to x\mathrm{centiliter}\approx 158608.7537496\mathrm{centiliter}$
Conclusion: $419 usgallon ≈ 158608.7537496 centiliter$
## Conversion in the opposite direction
The inverse of the conversion factor is that 1 centiliter is equal to 6.30482225198445e-06 times 419 US gallons.
It can also be expressed as: 419 US gallons is equal to $\frac{1}{\mathrm{6.30482225198445e-06}}$ centiliters.
## Approximation
An approximate numerical result would be: four hundred and nineteen US gallons is about one hundred and fifty-eight thousand, six hundred and eight point seven five centiliters, or alternatively, a centiliter is about zero times four hundred and nineteen US gallons.
## Footnotes
[1] The precision is 15 significant digits (fourteen digits to the right of the decimal point).
Results may contain small errors due to the use of floating point arithmetic.
|
2020-11-26 21:49:50
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 6, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7485390901565552, "perplexity": 2576.176986033794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141188947.19/warc/CC-MAIN-20201126200910-20201126230910-00088.warc.gz"}
|
https://space.stackexchange.com/questions/8142/how-far-away-will-i-get-if-im-just-under-escape-velocity
|
# How far away will I get if I'm --just under— escape velocity?
The escape velocity of earth is about 11.2 km/s. Let's say it was, for the sake of argument, exactly 11.2 km/s. If I go 11.199999 km/s, I'm going to fall back down to earth. But how can I determine how far I will get before I do so? What pieces of data do I need for this?
• Suggest you look at space.stackexchange.com/questions/4727/… which is a possible duplicate. – Erik Feb 13 '15 at 20:23
• I did, but it doesn't seem to be a duplicate. It might be related in some way, but if it is a duplicate it only underscores how little I know about the topic! – corsiKa Feb 13 '15 at 20:41
• Both questions ask for relationships between speeds and apogees. But Corsika seems interested in perigees speeds just below escape. A very interesting set of ellipses! In my opinion this question's more specific interest make it different enough it's not a duplicate. – HopDavid Feb 13 '15 at 21:21
You are talking about an elliptical orbit about the earth. Two points on this ellipse are of interest.
Perigee - point on the orbit closest to earth. Apogee - point on the orbit farthest from earth
If you're launching from earth's surface, the perigee would be 6378 km from earth's center. (6378 is earth's radius). For this problem I will pretend there's no earth atmosphere.
The apogee is what you're interested in, the farthest point from the earth.
The vis viva equation is a nice tool for finding velocity:
$$v=\sqrt{Gm(2/r-1/a)}$$
What is a? That's the semi-major axis. We can find a by taking the average of perigee distance from center and apogee distance from earth's center.
Here is a pic of the stuff I've described so far:
We're interested in the speed when the object leaves earth so r would be 6378 km.
Let's take a look at the vis viva question again:
$$v=\sqrt{Gm(2/r-\mathbf{1/a})}$$
Noticed I bolded the term that uses the semi-major axis quantity. What happens when a gets real big? 1/a gets closer and closer to zero.
So for high apogees you have something very close to
$$v=\sqrt{Gm(2/r)}$$
Which is escape velocity.
"Just a hair under escape" describes a multitude of ellipses!
For example let's imagine an ellipse whose apogee reaches EML1, about 5/6 of the way to the moon. Now let's compare it to an ellipse whose apogee extends all the way to Sun-Earth L2, about 1.5 million kilometers away. The perigee speeds of these two ellipses differ by only .11 km/s!
But when you get apogees that high, 2 body mechanics is no longer a good model. The gravitational influences of the moon and sun are enough to bend the object out of an elliptical path. Launch something as high as SEL2 and it is likely the sun will wrest the object from earth's sphere of influence.
• I see what you're saying - if there was nothing else in the universe, I could go forever before getting pulled back down. But realistically, something else will snag me if I get too far away. – corsiKa Feb 13 '15 at 21:33
• That's pretty much it. The big something else is the sun. You can achieve escape velocity from earth but still not get much farther from the sun than 1 A.U. (earth's distance from the sun). To get to Mars you'd need a hyperbolic orbit with a Vinfinity of about 3 km/s. A hyperbola's speed is sqrt(Vinf^2 + Vesc^2). – HopDavid Feb 14 '15 at 0:56
You wouldn't get far, since that wouldn't quite get you escaped from the Earth-Moon system. Your escape velocity is what you would need at the surface of the Earth just to escape Earth. (I get $11.18\,\mathrm{km/s}$.) To escape the Earth-Moon system, you would need $11.25\,\mathrm{km/s}$.
For an escape from a body, if you depart radius $r$ with $1-\epsilon$ times the escape velocity, and you are going directly away from the center of the body with that velocity, then for small $\epsilon$ you will get to $r\over 2\epsilon$ before falling back.
Derivation, starting with the velocity $v$ at radius $r$, just shy of the escape velocity:
$$v_e^2={2\mu\over r}$$ $$v=v_e\left(1-\epsilon\right)$$ $$v^2={2\mu\over r}\left(1-\epsilon\right)^2$$
Relating the velocity to periapsis and apoapsis using the conservation of energy:
$$v^2=2\mu\left({1\over r}-{1\over r_p+r_a}\right)$$
For a trajectory directly away from the center of the body, use a periapsis of zero:
$$v^2=2\mu\left({1\over r}-{1\over r_a}\right)$$
Then:
$${2\mu\over r}\left(1-\epsilon\right)^2=2\mu\left({1\over r}-{1\over r_a}\right)$$ $${1\over r}\left(1-\epsilon\right)^2={1\over r}-{1\over r_a}$$ $$\left(1-\epsilon\right)^2=1-{r\over r_a}$$
For small $\epsilon$:
$$1-2\epsilon\approx 1-{r\over r_a}$$ $$2\epsilon\approx {r\over r_a}$$ $$r_a\approx {r\over 2\epsilon}$$
So in the case where you make the mistake of using the Earth escape velocity without taking into account the Moon, then $\epsilon$ is about $0.006$. So you would get about $500,\!000\,\mathrm{km}$ from Earth before falling back. A little farther than the Moon. That assumes however that the Moon is not close to your path. If it is, then it might either a) give you enough energy to completely escape, or b) send you back to the Earth with less energy.
In the spirit of your example, e.g. if you assume that $11.2\,\mathrm{km/s}$ is exactly the escape velocity, then your $\epsilon$ is about $10^{-7}$. So you would get about 36 billion kilometers away (240 AU!). However other bodies such as the Sun, Venus, Jupiter, and re-encountering the Earth-Moon system would alter your course well before that.
• So umm... how far is not far? If I was going jjuusstt under whatever value we use for the escape velocity, how far do I get? – corsiKa Feb 13 '15 at 21:12
• Wouldn't you be perturbed out of the system (or, less likely, into one of the bodies) quite quickly due to lunar or solar influence? – pericynthion Feb 13 '15 at 22:50
• Yes. Perturbed out of the system or back into it. – Mark Adler Feb 13 '15 at 23:31
The easiest way how to compute this is to use specific orbital energy. The following quantity is preserved over time:
$$\frac{v^2}{2} - \frac{G(M+m)}{r}$$
since the mass of a spaceship is negligible compared to the mass of the Earth, we can omit $m$.
The furthest point where we can get is when we consume all kinetic energy, so $v=0$. Knowing the velocity $v$ at the Earth surface $R$, we get
$$\frac{v^2}{2} - \frac{GM}{R} = - \frac{GM}{r_{\mbox{max}}}$$
For the escape velocity $v_e$ we have $r_{\mbox{max}}=\infty$ so
$$\frac{v_{e}^2}{2} - \frac{GM}{R} = 0$$
$$\frac{GM}{R} = \frac{v_{e}^2}{2}$$
where $R$ is the radius of the Earth.
Substituting this into the equation for $r_{\mbox{max}}$, we obtain
$$\frac{v^2}{2} - \frac{v_{e}^2}{2} = - \frac{GM}{r_{\mbox{max}}}$$ and $$r_{\mbox{max}} = \frac{2GM}{v_{e}^2 - v^2}$$
Substituting the velocity you asked for and the Earth's standard gravitational parameter we get (in km)
$$r = \frac{2 \times 398600.4419}{11.2^2-11.199999^2} \doteq 3.56\times 10^{10}$$
which is approximately $238 \mbox{AU}$.
(Of course we completely disregarded the effect of all the other celestial bodies.)
• @MarkAdler Yes, you're right, correcting. – Petr Pudlák Feb 14 '15 at 20:35
|
2020-07-11 08:07:49
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.760403573513031, "perplexity": 594.3813284849282}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655924908.55/warc/CC-MAIN-20200711064158-20200711094158-00539.warc.gz"}
|
https://www.math.ucla.edu/~marks/measurebrooks_errata.html
|
Erratum -- Brook's theorem for measurable colorings
Proposition 2.1. of the paper is only true if we assume that the measure \mu or topology \tau is quasi-invariant, and is false in general. There simply isn't a nice connection between measurable colorings and Borel colorings mod null in the non-quasi-invariant setting.
This error does not impact any the rest of the paper since elsewhere we always assume our measures/topologies are quasi-invariant as explanined in the discussion following Proposition 2.1.
|
2022-05-16 07:45:00
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9739511013031006, "perplexity": 1124.21587532325}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662510097.3/warc/CC-MAIN-20220516073101-20220516103101-00741.warc.gz"}
|
http://physics.stackexchange.com/questions/32437/degeneracy-of-energy-levels-for-2-identical-particles-in-a-one-dimensional-box
|
Degeneracy of Energy Levels for 2 identical particles in a One Dimensional box [closed]
i am studying for a physics exam and came across an exercise i cannot seem to crack. I have read through Feynmans lecture books & a lot of the internet but i am somewhat stuck. first the question, and then what i have got so far:
(Note the problem is originally in German, I hope my translation does not change the problem)
Problem Description: We look at 2 identical (indistinguishable) particles in a one-dimensional box with infinite potential energy at the bounds. The box' width is 2a. in the box the potential energy is 0 and on the bounds it is infinite. we take as granted that the particles DON'T interact.
1) Find the 4 lowest Energies the system can have.
2) How many states of the same energy does every of those 4 levels have if both particles have spin 1/2 (fermions)?
3) How many states of the same energy does every of those 4 levels have if both particles have spin 1 (bosons)?
Ok, now lets see how far I've come
1) We know, that the Energy levels of 1 particle in such a box of width 2a is given by
$$E_n = \frac{n^2h^2}{32ma^2}, \quad\quad n=(n_1+n_2)$$
This gives me the following 4 states:
$$E_1 = \frac{h^2}{32ma^2},\quad E_2 = \frac{h^2}{8ma^2},\quad E_3= \frac{9h^2}{32ma^2},\quad E_4 = \frac{h^2}{2ma^2}$$
2) Here it gets tricky for me.Noting $\Phi(x)$ as the complete statefunction, $\Psi(x)$ for the Place and $\Xi(x)$ for the spin-state function and the subscripts $_s$ for symmetric and $_a$ for antisymmetric, we know that the complete function must be antisymmetrically for fermions. this is achieved by combining a symmetric place & antisymmetric spin or the other way around.
Only looking at $\Psi(x)$ for now, we find the following n-states for our 2 body system. noting $n := (n_1,n_2) n=1 (1,0) with$\Psi_a$and$\Psi_s$(2 degeneracies) n=2 (1,1),$\Psi_s$(1 degeneracie) (2,0)$\Psi_a$and$\Psi_s$(2 degeneracies) n=3 (3,0)$\Psi_a$and$\Psi_s$(2 degeneracies) (2,1)$\Psi_a$and$\Psi_s$(2 degeneracies) n=4 (4,0)$\Psi_a$and$\Psi_s$(2 degeneracies) (3,1)$\Psi_a$and$\Psi_s$(2 degeneracies) (2,2)$\Psi_s$(1 degeneracie) Now i have a look at the spin-states$\Xi(x)$for Spin = S = 1, using the formulas Number of antisymm spinstates #$\Xi_a(x) = S(2S+1) = 1$Number of symmetric spinstates #$\Xi_s(x) = (S+1)(2S+1) = 3$Now we Combine those to antisymmetric functions using$\Phi_a = \Psi_a+\Xi_s$or$\Phi_a=\Psi_s+\Xi_a$To count the states of the same energy i use this formula N states =$( N\Psi_a \cdot N\Xi_s)+( N\Psi_s \cdot N\Xi_a)$(N denoting the number of states using the degeneracy of each n-combination) This finally results in the following degeneracies$N E_1= 2*1 + 2*3 = 8N E_2= 2*1 + 3*3 = 11N E_3= 4*1 + 4*3 = 16N E_4= 4*1 + 5*3 = 19$3) Now the same for Bosons (Spin S = 1). In this case the complete function$\Phi$must be symmetrically, so we have to combine antisym spin and antisym place or both symm. The degeneracy of the place function stays the same as in 2), but for the spin we have to recalculate using: Number of antisymm spinstates #$\Xi_a(x) = S(2S+1) = 3$Number of symmetric spinstates #$\Xi_s(x) = (S+1)(2S+1) = 6$To achieve symmetric complete functions, we also need to adjust our N E formula which will be changed to : N states =$( N\Psi_a \cdot N\Xi_a)+( N\Psi_s \cdot N\Xi_s)$(N denoting the number of states using the degeneracy of each n-combination) This finally results in$N E_1= 2*6 + 2*3 = 18N E_2= 2*6 + 3*3 = 21N E_3= 4*6 + 4*3 = 36N E_4= 4*6 + 5*3 = 39$Conclusion and MY question: Sadly this exercise does not have a solution, so I decided to post my attempt here and hope that some of you might find any errors I made and help me if they spot if and where my thoughts went rogue. Hoping for some good input! (Please add questions if something is not clear - it could be the translation). - thx for editing, altough: it is definitly NOT homework. its an exercise from an old exam. (at a certain point there is no such thing as 'homework' anymore) ;) – Sebastian Flückiger Jul 20 '12 at 8:38 Hi Sebastian - generally we discourage questions that just ask for someone to check your work. This site is meant to be for conceptual questions, so if there's some specific concept confusing you, it's fine to ask about that. – David Z Jul 20 '12 at 23:52 add comment closed as too localized by David Z♦Jul 20 '12 at 23:49 This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question. 2 Answers The way you would manually find the wave functions is more complicated than this. At first, you would need to describe better the one-particle scenario. I don't really understand your concept of the spin-state function, so I'll use mine, I hope it won't make my answer unusable. One particle If the particle has no spin, the wave functions are completely determined by energy. In other words, the energy levels$E_n$are nondegenerate. We can assign wave functions, say$\psi_n(x)$, to them directly. If we add spin to the description of the particle, it does not change much in our observation of energy, but the underlying description changes. Namely, we can not get on with simple$\mathbb{R} \to \mathbb{C}$functions. There are actually two equivalent concepts: 1. defining a spinor wave function$\Psi: \mathbb{R} \to \mathbb{C}^d$, mapping each point in space to a$d$-tuple of probability amplitudes, where$d$is the dimension of the spin space,$d = 2s+1$, and 2. making the spin state another independent variable beside position. In this way, the wave function would be defined as$\psi(x,\sigma): \mathbb{R} \times S \to \mathbb{C}$. Here,$\sigma$takes the values from$\{-s, -s+1, \ldots, s-1, s\}$(which makes$2s-1$values) and gives a single probability amplitude, that of finding the particle at position$x$and with instantaneous spin state$\sigma$. The equivalence is given by the fact that these probability amplitudes are simply the components of the spinor$\Psi(x)$: $$\Psi(x) = \begin{pmatrix}\psi(x,-s) \\ \vdots \\ \psi(x,s-1) \\ \psi(x,s)\end{pmatrix}$$ So, if we assume a particle with spin, say,$\frac12$, all the energy levels become twice degenerated, because for energy$E_n$, we can find two linearly independent wave functions: $$\Psi_{n, \uparrow}(x) = \begin{pmatrix}\psi_n(x) \\ 0\end{pmatrix} \quad \hbox{and} \quad \Psi_{n, \downarrow}(x) = \begin{pmatrix}0 \\ \psi_n(x)\end{pmatrix},$$ reusing the scalar$\psi_n$single-particle wavefunctions. These are joint eigenfunctions of the Hamiltonian and spin operators: "energy$E_n$spin up" and "energy$E_n$spin down". Two particles When we consider two noninteracting particles, the solution further depends on whether they are distinguishable or not. Let's briefly address distinguishable particles, despite the fact that you are not asking about them, but for future reference. In either case, the state space expands to cover all combinations of both the particles' positions and spin states. Therefore, in the formalism I called "2.", we get wave functions indexed by four coordinates: continuous variables$x$,$y$and discrete variables$\sigma$,$\tau$. If the particles can be distinguished, we can build those functions as tensor products of one-particle functions, i.e.: $$\psi^{(2)}(x,y,\sigma,\tau) = \psi_a(x,\sigma) \psi_b(y,\tau),$$ and as superpositions thereof: $$\psi^{(2)}(x,y,\sigma,\tau) = \sum_{i=1}^k \alpha_i \psi_{a_i}(x,\sigma) \psi_{b_i}(y,\tau).$$ In the case of indistinguishable particles, we must pay extra attention in order for the result to be symmetric or antisymmetric wrt. exchange of$(x,\sigma) \leftrightarrow (y,\tau)$. We usually do so by constructing the functions using permutations in the first place, for example, $$\psi^{(2,symm.)}(x,y,\sigma,\tau) = \psi_a(x,\sigma) \psi_b(y,\tau) + \psi_a(y,\tau) \psi_b(x,\sigma),$$ $$\psi^{(2,asymm.)}(x,y,\sigma,\tau) = \psi_a(x,\sigma) \psi_b(y,\tau) - \psi_a(y,\tau) \psi_b(x,\sigma).$$ In more than 2 particles, this takes a lot of work (and storage) because one has to go through all the permutations, but is doable. Now, for each of your sub-questions, you would need to find the combinations of state assignments to "a" and "b" which give the energies you need, keeping on mind that permutations mean no difference in the state, and that$a$must not equal$b$if the exclusion principle is in action (fermions). The count of these would be the degeneracies. Luckily, given either of the 4 lowest energies of the form$E_{n_1} + E_{n_2}$, the decomposition to the sum of two single-particle energies is unique, so we only need to care about spin. For example, the energy$2E_2$can be reached for two$\frac12$particles only one way,$a$= energy$E_2$, spin up,$b$= energy$E_2$, spin down (or vice versa, which gives the same quantum state). Thus this energy level is nondegenerate. On the contrary, energy$E_1+E_2$is an eigenvalue for a total of four states: 1.$a$= energy$E_1$, spin up,$b$= energy$E_2$, spin up, 2.$a$= energy$E_1$, spin up,$b$= energy$E_2$, spin down, 3.$a$= energy$E_1$, spin down,$b$= energy$E_2$, spin up. 4.$a$= energy$E_1$, spin down,$b$= energy$E_2$, spin down. In the case of bosons, the Pauli exclusion principle goes away, giving us more possibilities. For example, energy$2E_2$is obtained if 1.$a$= energy$E_2$, spin "+1",$b$= energy$E_2$, spin "+1", 2.$a$= energy$E_2$, spin "+1",$b$= energy$E_2$, spin "0", 3.$a$= energy$E_2$, spin "+1",$b$= energy$E_2$, spin "-1", 4.$a$= energy$E_2$, spin "0",$b$= energy$E_2$, spin "0", 5.$a$= energy$E_2$, spin "0",$b$= energy$E_2$, spin "-1", 6.$a$= energy$E_2$, spin "-1",$b$= energy$E_2$, spin "-1", The degeneracy of this level is 6. Another way As the above can get daunting after a while, we can use a simpler method if only the energy levels and degeneracies are the question, not the wave functions themselves. This method uses the occupation diagrams well-known from chemistry (which are mere graphical representations of multi-particle wave functions). In the case of spin$\frac12$, we prepare boxes corresponding to single-particle energies$E_1$,$E_2$,$E_3$, and$E_4$. We can put in two fermions in the following ways: •$(\uparrow \downarrow), (\ \ ), (\ \ ), (\ \ )$... energy$2E_0$, degeneracy 1 •$(\uparrow \ ), (\uparrow\ ), (\ \ ), (\ \ )$... energy$E_0+E_1$, •$(\uparrow \ ), (\downarrow\ ), (\ \ ), (\ \ )$... energy$E_0+E_1$, •$(\downarrow \ ), (\uparrow\ ), (\ \ ), (\ \ )$... energy$E_0+E_1$, •$(\downarrow \ ), (\downarrow \ ), (\ \ ), (\ \ )$... energy$E_0+E_1$... degeneracy 4, •$(\ \ ), (\uparrow \downarrow), (\ \ ), (\ \ )$... energy$2E_1$, degeneracy 1, etc. Now you only have to find which energies are the lowest four, anyway, it becomes quickly obvious that the degeneracy only depends on the fact whether they are in the form of$2E_n$for some$n$or$E_m+E_n$for some$m \ne n$. For spin$1$bosons, the spin state can become "+1", "0", or "-1" and the particles don't obey the exclusion principle, so we don't need to care about putting two like "arrows" in the same box. Similarly as above, the order within one box does not make a difference. Let us save same space by using abbreviations "+" / "0" / "-" for the spin states. Here we go: •$(++), (\ \ ), (\ \ ), (\ \ )$... energy$2E_0$, •$(+0), (\ \ ), (\ \ ), (\ \ )$... energy$2E_0$, •$(+-), (\ \ ), (\ \ ), (\ \ )$... energy$2E_0$, •$(00), (\ \ ), (\ \ ), (\ \ )$... energy$2E_0$, •$(0-), (\ \ ), (\ \ ), (\ \ )$... energy$2E_0$, •$(--), (\ \ ), (\ \ ), (\ \ )$... energy$2E_0$... degeneracy 6, •$(+\ ), (+\ ), (\ \ ), (\ \ )$... energy$E_0+E_1$, •$(+\ ), (0\ ), (\ \ ), (\ \ )$... energy$E_0+E_1$, •$(+\ ), (-\ ), (\ \ ), (\ \ )$... energy$E_0+E_1$, •$(0\ ), (+\ ), (\ \ ), (\ \ )$... energy$E_0+E_1$, •$(0\ ), (0\ ), (\ \ ), (\ \ )$... energy$E_0+E_1$, •$(0\ ), (-\ ), (\ \ ), (\ \ )$... energy$E_0+E_1$, •$(-\ ), (+\ ), (\ \ ), (\ \ )$... energy$E_0+E_1$, •$(-\ ), (0\ ), (\ \ ), (\ \ )$... energy$E_0+E_1$, •$(-\ ), (-\ ), (\ \ ), (\ \ )$... energy$E_0+E_1$... degeneracy 9, and so on. Every energy level consisting of a sum of two different energies get a degeneracy of 9, or 6 if the two components are the same. Final remark: Under further inspection, it can be found that the "general observations" break at higher energies, where the Diophantine equation$m^2 + n^2 = a$starts having multiple solutions, e.g. the energy$260\frac{h^2}{32ma^2}$could be$\frac{h^2}{32ma^2}(16^2+2^2)$as well as$\frac{h^2}{32ma^2}(14^2+8^2)\$. You don't need to worry about this in the beginning of the spectrum, though.
-
thank you for this unbelievable thourough answer! it will take some time for me to process and understand it i will come back on it :) – Sebastian Flückiger Jul 20 '12 at 12:02
|
2014-04-20 01:16:36
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999984502792358, "perplexity": 6223.223232141911}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00409-ip-10-147-4-33.ec2.internal.warc.gz"}
|
https://testbook.com/question-answer/eddy-current-loss-of-armature-can-be-minimised-___--615e96324feaf500c02fde0e
|
# Eddy current loss of armature can be minimised ________.
This question was previously asked in
DFCCIL Executive Electrical 30 Sept 2021 Official Paper
View all DFCCIL Executive Papers >
1. by lamination
2. by distortion
3. by induced current
4. by folding
Option 1 : by lamination
## Detailed Solution
Concept:
• When an alternating magnetic field is applied to a magnetic material, an emf is induced in the material itself according to Faraday’s law of Electromagnetic induction.
• Since the magnetic material is a conducting material, these EMF’s circulates current within the body of the material. These circulating currents are called Eddy currents. They are produced when the conductor experiences a changing magnetic field.
• The process of lamination involves dividing the core into thin layers held together by insulating materials.
• Due to lamination effective cross-section area of each layer reduces and hence the effective resistance increases.
• As effective resistance increases, the eddy current losses will get decrease.
Mathematically, the eddy current loss is given by:
Pe = Ke Bm2. t2. f2. V Watts
Where;
K - coefficient of eddy current. Its value depends upon the nature of magnetic material
Bm - Maximum value of flux density in Wb/m2
t - Thickness of lamination in meters
f - Frequency of reversal of the magnetic field in Hz
V - Volume of magnetic material in m3
From the above formula, we conclude that the Eddy current loss is proportional to the square of the frequency.
Observation:
• To minimize the eddy current loss we increases the resistance in the path of eddy current by laminating it.
• Eddy current losses are directly proportional to area of armature or more precisely the path of motion.
• In laminated armature eddy current losses are reduced to very less or '0' quantity. That is why armature of DC machines (either motor or generator) is laminated.
|
2022-01-19 05:21:37
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8941294550895691, "perplexity": 1622.2877183915252}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301263.50/warc/CC-MAIN-20220119033421-20220119063421-00613.warc.gz"}
|
https://www.nature.com/articles/s41467-018-04866-6
|
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.
# Two-dimensional semiconductors in the regime of strong light-matter coupling
## Abstract
The optical properties of transition metal dichalcogenide monolayers are widely dominated by excitons, Coulomb-bound electron–hole pairs. These quasi-particles exhibit giant oscillator strength and give rise to narrow-band, well-pronounced optical transitions, which can be brought into resonance with electromagnetic fields in microcavities and plasmonic nanostructures. Due to the atomic thinness and robustness of the monolayers, their integration in van der Waals heterostructures provides unique opportunities for engineering strong light-matter coupling. We review first results in this emerging field and outline future opportunities and challenges.
## Introduction
Transition metal dichalcogenides (TMDCs) are ideally suited as the active material in cavity quantum electrodynamics, as they interact strongly with light at the ultimate monolayer limit. They exhibit pronounced exciton resonances even at room temperature owing to the exceptionally high exciton binding energies of a few 100 meV1,2.
The high exciton oscillator strength leads to absorption of up to 20% per monolayer3, and radiative exciton lifetimes on the order of few 100 fs to several ps4,5,6,7. In TMDC monolayers (MLs), the dipole selection rules are valley-selective, i.e., distinct valleys in momentum space can be addressed by photons with left- or right-handed helicity8,9,10,11,12. In combination with strong spin–orbit splitting, this allows studying intertwined spin-valley dynamics of excitons13,14,15. These unique optical properties make monolayer TMDCs, which can readily be embedded in the van der Waals heterostructures containing multiple active layers16,17, ideal systems for investigating excitons, and their interactions with other electromagnetic excitations.
This review paper is structured as follows. First, we provide a concise description of the optical properties of excitons in TMDC MLs. We then present the generic concept of strong light-matter coupling which arises for excitons confined in the TMDC monolayer interacting with photons trapped inside a cavity or plasmons localized in a metallic nanosystem. Strong light-matter coupling gives rise to half-light–half-matter quasi-particles, which are also known as exciton-polaritons18,19,20. This generally results in a substantial modification of the emission properties yielding an oscillatory behavior between light- and matter excitations in the temporal, and the emergence of the characteristic Rabi splitting in the spectral domain, for comparison with light-matter coupling in conventional semiconductors see Box 1 and refs. 21,22. We review recent experimental advances of strong light-matter coupling in TMDC MLs and discuss complementary system implementations which were designed to study the formation of exciton-polaritons with atomic MLs.
## Excitonic and optical properties of TMDC MLs
Semiconducting TMDCs are part of the large group of layered materials widely investigated for fundamental research and applications following the discovery of graphene23. The remarkably simple mechanical exfoliation techniques give access to rather large-area monolayer samples. While exfoliation of TMDC MLs was already demonstrated in a seminal work by Novoselov et al.24 in 2005, the observation of pronounced photoluminescence in MoS2 MLs, reported by two groups25,26 in 2010, triggered intense research activities regarding the optical and electronic properties of atomically thin TMDCs. MLs of MoS2 and related TMDCs consist of a hexagonally coordinated transition metal atom layer sandwiched between top and bottom chalcogen layers, which are also hexagonally coordinated, leading to a trigonal prismatic crystal structure27,28 (see Fig. 1a) described by the D3h point symmetry group. Correspondingly, the monolayer does not have inversion symmetry. The bulk TMDC crystal is formed by van der-Waals-mediated stacking the monolayer units. In the 2H stacking sequence, which is the most prevalent polytype, inversion symmetry is recovered for even numbers of layers and eventually in the bulk crystal.
Bulk MoS2 is an indirect-gap semiconductor with a valence band maximum at the Γ point, the center of its hexagonal Brillouin zone, and conduction-band minima located in between the Γ and the K points at the corners of the Brillouin zone. In the monolayer limit, however, the character of the band-gap changes to a direct gap at the K points (see Fig. 1b)25,26,29,30. A similar transition of the band structure from indirect to direct also occurs in the related TMDCs WS2, MoSe2, WSe2, MoTe2, and their alloys. In the TMDC MLs, the band structure at the K valleys is characterized by a very large, valley-contrasting spin splitting in the valence bands, whose magnitude ranges from about 150 meV (MoS2) to more than 450 meV (WSe2), and a smaller, yet still substantial spin splitting in the conduction band31,32,33. As the optical transitions between the valence and conduction band are spin-conserving, this splitting gives rise to two, spectrally well-separated interband optical transitions identified as A (transition from the upper valence band) and B (transition from the lower valence band), see Fig. 1c.
The optical properties of TMDCs are determined by the formation of tightly bound exciton states, which have binding energies on the order of several hundred meVs1,2,34,35, making them stable well beyond room temperature. The large binding energies arise due to a combination of several effects: electrons and holes at the K points of the Brillouin zone have rather large effective masses (ranging from about 0.25 me to 0.6 me depending on the specific TMDC33 where me is the free-electron mass) and are strictly confined to the two-dimensional plane of the monolayer. Additionally, their Coulomb interaction is only weakly screened, and this screening typically is anisotropic due to the anisotropic dielectric environment36,37. This leads to a strong deviation of excited exciton state energies from a hydrogen-like Rydberg series, illustrated in Fig. 1d1,2. It is worth noting, that engineering the dielectric environment of the monolayer, e.g., by encapsulating the TMDC between other layered materials, or modifying the substrate, allow for a controlled tuning both of the band gap and the exciton binding energy38,39. The large radiative decay rate of excitons Γ0 1 ps−1 and, correspondingly, high oscillator strength f = Γ0/ω0 10−3, with ω0 being the exciton resonance frequency, results in efficient light-matter interactions in TMDC MLs. The exact values of f and Γ0 will also depend on the dielectric environment4,5,6,7,40,41. The short radiative lifetime yields a significant homogeneous spectral broadening of the excitonic transitions42. It also leads to a large coupling constant g with photonic modes in microcavity structures, as detailed below43. The high oscillator strength of the excitonic transitions gives rise to a very large absorption for the TMDC monolayer, reaching 20% for resonant excitation of the A-exciton transition in the tungsten-based TMDCs1,44. Theoretically, the maximal absorbance of a monolayer Amax at resonance is controlled by the ratio of the radiative to the non-radiative, γ, decay rate of the excitons, Amax = 2Γ0γ/(Γ0 + γ)2 and may reach 50% under optimal conditions of Γ0 = γ. While the emission from typical TMDC samples deposited on SiO2 is strongly inhomogeneously broadened by adsorbates and substrate-induced effects, recent advances in sample fabrication (encapsulation in hexagonal BN) yield linewidths indeed approaching the homogeneous limit, see Fig. 1e45,46,47.
The transition metal atoms of the TMDCs strongly influence not only the magnitude of the spin splitting, but also the ordering of the spin-split conduction bands (see Fig. 1c). While for MoX2, the optically bright A-exciton transition connects the upper valence band with the lower conduction band, the band order is opposite in the tungsten-based materials, so that the A-exciton transition addresses the upper conduction band33,48. Thus, for WX2 MLs, the exciton state lowest in energy with the electron residing in the lower spin-split conduction band is forbidden in optical transitions for normal light incidence. The splitting between the optically bright and dark states is given by a combination of the conduction-band spin splitting and electron–hole Coulomb exchange interaction49. The lower-energy dark A-exciton state in the tungsten-based materials was indirectly inferred from temperature-dependent PL measurements50,51,52. More recently, PL emission from the dark state was directly observed using applied in-plane magnetic fields53,54 and in-plane excitation and detection geometry55,56,57. In addition to neutral excitons, charged excitons (trions)58 with binding energies of about 25 meV are observable in optical spectroscopy, and the multi-valley band structure allows for different trion species13,59,60,61. Four-particle complexes, biexcitons, i.e., excitonic molecules have also been observed62,63,64.
The optical selection rules for interband transitions9 allow for valley-selective excitation at the K+ or K valleys using σ+ or σ-polarized light, respectively. Thus, near-resonant, circularly polarized excitation generates a valley polarization of excitons, which can be read out directly in helicity-resolved photoluminescence. Even in time-integrated (cw) photoluminescence measurements, large valley polarization degrees are observable for most TMDC MLs8,9,10,11,12,65. These initial observations motivate the use of the valley pseudospin in potential device applications (valleytronics)66. However, the large cw valley polarization values are, in part, a consequence of the ultrashort exciton radiative lifetime limiting the time window for valley polarization decay. The dominant decay mechanism for excitonic valley polarization is long-range electron–hole exchange interaction67,68. Its efficiency scales with the exciton center-of-mass momentum and the resulting decay rate can rival the exciton radiative lifetime, dependant on excitation conditions. In contrast, valley polarization lifetimes are orders of magnitude longer for dark excitons13,69, interlayer excitons in TMDC heterostructures70 and resident carriers in doped TMDC MLs15,71.
## General framework of strong light-matter coupling
An optically active exciton in an isolated TMDC ML emits photons into the free space. In addition to the symmetry-imposed valley selection rules described above, the photon emission process obeys energy and momentum conservation laws, making only excitons with small in-plane wavevectors, |K| < ωx/c, i.e., within the light cone, subject to the radiative processes. Here, c is the speed of light and ωx is the exciton resonance frequency, largely determined by the difference between the free carrier band gap and the exciton binding energy. As emitted light propagates away from the ML carrying away the energy67,72, the exciton experiences radiative damping. Note that excitons with |K| > ωx/c are optically inactive and can contribute to the PL only after relaxation towards the radiative cone. The situation becomes qualitatively different if the emitted light cannot leave the vicinity of the ML, e.g., if the ML is placed between two mirrors which form an optical cavity, Fig. 2a, or if a ML is placed in the vicinity of a metallic or dielectric nanoparticle supporting plasmonic or Mie resonances. In such situations, the exciton effectively interacts with a localized mode of electromagnetic radiation (or a plasmon) with the frequency ωc. Hence, the emitted photon can be reabsorbed by the TMDC ML and reemitted again. This emission-absorption process repeats until either the exciton in the ML vanishes due to scattering or non-radiative processes or the photon leaves the cavity, e.g., as a result of the tunneling through the mirrors. If these decay processes are weak enough the excitation energy is coherently transferred between the exciton and the photon (or plasmon) resulting in the strong-coupling regime of the light-matter interaction and giving rise to a qualitative change of the energy spectrum in the system: instead of independent exciton and photon states new eigenmodes of the system, the exciton-polaritons, are formed18,19.
There are several approaches to describe theoretically the strong-coupling effects. It is instructive to consider here the coupled-oscillators model where the excitonic contribution to the dielectric polarization in the TMDC ML, P, and the electric field of the cavity mode, E, are treated on a semi-classical level and are assumed to obey the oscillator like equations of motion:
$${\mathrm{i}}{\dot{\boldsymbol P}} = \left( {\omega _x - {\mathrm{i}}\gamma } \right){\boldsymbol{P}} + g{\boldsymbol{E}},$$
(1)
$${\mathrm{i}}{\dot{\boldsymbol E}} = \left( {\omega _c - {\mathrm{i}}\kappa } \right){\boldsymbol{E}} + g{\boldsymbol{P}}.$$
(2)
Here, a dot on top denotes the time derivative, γ and ϰ are the dampings, respectively, of the exciton and cavity mode unrelated to the light-matter coupling (which determine half-width at half maximum of the resonances), and g is the coupling constant which is determined by the system geometry and exciton oscillator strength. For a planar microcavity, it can be roughly estimated as $$g\sim \sqrt {\omega _c\Gamma_0}$$, where the proportionality constant depends on the cavity geometry and structure of the Bragg mirrors, Γ0 is the exciton radiative decay rate into empty space. Here the large exciton oscillator strength resulting in large Γ0 allowing to estimate values of g ~ 10…50 meV depending on the system parameters. These high values for g present one of the intrinsic advantages for studying light-matter coupling in TMDC MLs as compared to nanostructures with transitions with lower oscillator strength. Equation (1) can be formally derived from Maxwell equations for the electromagnetic field in the cavity and the Schödinger equation for the exciton wavefunction in the resonant approximation assuming that $$\gamma ,\kappa ,g \ll \omega _x,\omega _c$$41. It follows from Eq. (1) that for the harmonic time-dependence of the polarization and field P, Eeiωt the eigenfrequency ω can be found from the simple quadratic equation:
$$\left( {\omega - \omega _x + {\mathrm{i}}\gamma } \right)\left( {\omega - \omega _c + {\mathrm{i}}\kappa } \right) = g^2,$$
(3)
which indeed describes eigenfrequencies of two damped oscillators coupled with the constant g. Equation (3) can be also derived from the transfer matrix method, which describes propagation of electromagnetic waves in a planar structure or by the procedure of the excitonic and electromagnetic field quantization41,43,73,74.
The general solution of Eq. (3) is found in many references, e.g.,19,41,75, here we consider the simplest but already instructive case at resonance ωx = ωc ≡ ω0, which already allows one to identify the strong and weak-coupling regimes. In this case the solutions of Eq. (3) read
$$\omega _ \pm = \omega _0 - {\mathrm{i}}\frac{{\gamma + \kappa }}{2} \pm \frac{{\Omega _{\mathrm{R}}}}{2},\quad \Omega_{\mathrm{R}} = \sqrt {4g^2 - (\gamma - \kappa )^2} .$$
(4)
Here ΩR is the Rabi frequency related to the so-called vacuum-Rabi splitting, ΩR, of polariton modes in quantum electrodynamics. In the strong-coupling regime the Rabi frequency is real, i.e.,
$${\mathrm{Strong}}{\kern 1pt}\,{\mathrm{coupling}}:g > \left| {\gamma - \kappa } \right|{\mathrm{/}}2,$$
(5)
In contrast, for $$g\leqslant \left| {\gamma - \kappa } \right|{\mathrm{/}}2$$ the light-matter interaction is in the weak-coupling regime. The strong coupling means that the real parts of the eigenfrequencies are split by ΩR, while their imaginary parts responsible for the damping are equalized. In the weak coupling, by contrast, the light-matter coupling affects the damping rates giving rise to the Purcell-like enhancement of suppression of the exciton radiative decay76,77. Hence, in the strong-coupling regime an anticrossing between the photon and exciton modes should be observed, while in the weak coupling the photon and exciton modes in optical spectra cross each other at the variation of the cavity resonance ωc (e.g., via the incidence angle) or the exciton resonance ωx (e.g., via the sample temperature).
Equation (5) provides a formal criterion of the strong-coupling regime. In realistic systems, however, the damping of polariton modes $$(\gamma + \kappa ){\mathrm{/}}2$$ can be comparable to ΩR, making identification of the Rabi splitting difficult. Moreover, the splitting of peaks in different experiments has different amplitudes43: In Fig. 2c, we compare the cavity reflection coefficient R, transmission coefficient, T, and absorbance A = 1 − R − T for $$\gamma ,\kappa$$ΩR where these quantities are found within the input–output formalism74. Therefore different experiments, also including PL, on the same sample will give different splittings due to strong coupling that are not directly the Rabi splitting, but are related to it, as detailed in ref. 43
The model discussion above disregards the nonlinear effects related with the exciton–exciton interactions and the oscillator strength saturation. Since excitons are tightly bound in TMDC MLs, these effects are somewhat weaker compared with conventional semiconductor quantum wells, particularly, the exciton oscillator strength saturation is controlled by the parameter $$n_{{\mathrm{exc}}}a_{\mathrm{B}}^2$$, where nexc is the exciton density and aB is the Bohr radius.
## Strong coupling in nanostructures with semiconducting 2D active layers
### Strong coupling of MLs in all-dielectric microcavities
As described above, the combination of high exciton binding energies, large oscillator strength and the possibility to strongly reduce structural disorder naturally puts sheets of TMDCs in the focus of polaritonic research. In most III–V, and specifically GaAs-based implementations of polaritonic devices, the cavity design of choice is a high-quality-factor Fabry–Perot resonator based on highly reflecting distributed Bragg reflectors (DBRs), which sandwich the active layer. While, in principle, the transfer of a single, or multiple TMDC layers on top of a DBR mirror is straight forward, optimal methods to sandwich layers in high-Q DBR-resonators are currently still being developed. This task is closely related to designing and integrating high-quality van der Waals heterostructures (such as MLs encapsulated by hBN layers), which reduce inhomogeneous and non-radiative broadening effects dramatically, in more complex devices. Nevertheless, in a first experimental effort, signatures of the strong-coupling regime have been found in a device featuring a single flake of MoS2, synthesized via chemical vapor deposition, that was embedded in a dielectric DBR cavity78. There, the authors studied both the reflection spectra as well as the photoluminescence as a function of the in-plane momentum at room temperature. While in this initial experiment, the anticrossing of the normal modes could not be fully mapped out, various groups later on implemented new generations of devices to scrutinize the coupling conditions between confined light-fields and monolayer excitons: a clear cut proof of strong-coupling conditions at cryogenic temperatures has been reported by Dufferwiel et al.79, for the case of single and double layers of MoSe2, which were embedded in a so-called open cavity based on two separated DBR mirrors, see Fig. 3a and b. In this work, the authors established the formation of exciton-polaritons by fully mapping out the anticrossing of the two resonances in a cavity detuning experiment. A similar implementation, based on an open fiber cavity was reported more recently in ref. 80, where strong-coupling conditions were manifested in charge-tunable studies both at the characteristic exciton as well as the trion resonance energies, and the results were interpreted in the framework of coupling to attractive and repulsive polaron resonances. For a monolayer of WS2, the formation of exciton-polaritons was more recently convincingly demonstrated in a fully monolithic cavity in an intermediate temperature regime between 110 and 230 K81. Interestingly, strong light-matter coupling conditions in lithographically defined grating structures with a single WS2 monolayer have also been established at room temperature. This approach completely bypasses the difficulties related to capping the atomic monolayer for the integration into microcavities82, see Fig. 3e and f.
### Strong coupling of MLs in metal-based microcavities
In order to clearly manifest strong-coupling conditions at room temperature with single MLs, one strategy involves to replace either one, or both of the DBR mirrors by thin metal layers. In appropriate designs comprising a metal-capped DBR layer, this approach can give rise to so-called Tamm plasmon states, which have significantly reduced mode volumes and thus can be expected to yield increased Rabi splitting83,84. Strong-coupling conditions with a single monolayer of WSe2 as well as MoS2 in such structures have been reported based on angle-resolved studies. Interestingly, in both efforts, the authors succeeded to map out the full dispersion relation of both the upper and lower polariton branch, as well as the characteristic anticrossing of the normal mode of the system85,86, see Fig. 3c and d. In order to further reduce the effective cavity length and thus increase the coupling strength, approaches involving purely metal-based Fabry–Perot cavities were reported in refs. 87,88 While these approaches intrinsically suffer from rather low cavity quality factors (typically < 100), the observed Rabi-splittings were very large, on the scale of 100 meV.
### Strong coupling with plasmonic structures
One route towards further enhancement of the light-matter coupling strength is based on plasmonic resonant structures89. They also allow to develop truly nanophotonic approaches to confine the light field below the optical diffraction limit. Such devices have been shown to be well compatible with the standard exfoliation and transfer technologies commonly applied in TMDC research. Among the various kinds of available structures, two species of devices have been mostly investigated thus far: the first involves a periodic arrangement (lattices) of metallic nanostructures. Such structures can, e.g., consist of a planar metal layer with holes, or an array of metal disks supporting localized surface plasmon resonances. Here, effects of light-matter coupling have been studied88,90, and polaritonic behavior was observed. Nevertheless, while the observed coupling strengths were significant, a clearly resolved split doublet of normal modes was mostly screened by strong broadening effects associated by optical losses in the metal structures.
The situation becomes even more delicate for systems comprising a single plasmonic resonator coupled to a monolayer: there, it is no longer possible to study the dispersion relation of quasi-particles via angle-resolved luminescence or reflection spectroscopy due to full mode quantization, and the signal strength in standard reflectivity spectra is low. Therefore, a method of choice to investigate light-matter coupling in such systems is so-called darkfield scattering. Thus far, there are a series of reports investigating strong-coupling conditions in TMDC-nanocavity hybrid systems. This includes a demonstration of a two peaked scattering spectrum from a single silver nanorod and a monolayer of WSe291, respectively a gold rod and a WS2 layer92 and an ultra-compact gold nanogap resonator93. Both reports base their claim primarily on the observation of an anticrossing mode doublet in darkfield scattering spectra, which were acquired by studying a variety of nanorod lengths to facilitate tuning of the optical resonance frequency. However, it is important to note, that split-peak spectra acquired in darkfield scattering measurements in closely related structures have been priorly interpreted in the framework of weak coupling: Here, the observed anticrossing is merely a result of enhanced absorption by the excitonic resonance in the presence of a broad optical resonance94. These ongoing developments indicate the need for complementary experimental evidence to better establish the conditions for observation of strong coupling in these systems. Possible experiments include micro-PL measurements or studies in the time domain.
### Polaritons and valley selectivity
A unique feature in TMDC-based microcavity systems is the possibility to optically address the valley degree of freedom, i.e., optical transitions in distinct valleys in momentum space66. Valley polarization of excitons in WS2, WSe2, and MoS2 is now routinely observed in high-quality samples even under non-resonant excitation conditions10,11,95. In contrast, similar experiments in MoSe2 MLs only resulted in very low circular polarization of the exciton PL of the order of 5%96. The dynamic process of valley polarization and depolarization strongly depends on the carrier redistribution, scattering and emission lifetime, and thus it is reasonable to assume that it can be tailored by coupling the excitonic resonances to microcavity modes.
In order to scrutinize whether the effects of valley polarization become more pronounced in strongly coupled microcavities, a variety of experiments have been designed very recently: in refs. 97,98, the authors have studied a system composed of a single MoSe2 monolayer in the strong-coupling regime with a microcavity mode. Both works independently confirmed, that strong-coupling conditions can retain the valley polarization of the excitations in MoSe2 at cryogenic temperatures, by an amplification of the scattering dynamics. In addition, it was also demonstrated, that the valley index can be directly addressed in the strong-coupling regime by a resonant laser in a Raman-scattering experiment97. The great interest to manipulate and enhance spin- and spin-valley-related phenomena in the strong-coupling regime, even up to ambient conditions, is further reflected by a series of papers from different groups, which demonstrated the valley-tagged exciton-polaritons at ambient conditions99,100,101 based on MLs of MoS2 and WS2. These results confirm the great potential of strongly coupled systems to play a crucial role in future valleytronic architectures, where the valley index of monolayer excitons can be married with ultra-fast propagation and low power switching inherited by the polariton nature.
### Hybrid polaritonics
In principle, it is possible to generate hybrid states of various excitonic transitions which are coherently coupled to the same photonic mode. These so-called hybrid polaritons have raised considerable interest recently, as they can provide a pathway to combine the advantages of various material systems in one device102. One example, for instance, involves the case of hybrid structures with embedded semiconductor quantum wells and atomic MLs. Here, electric current can be injected into one or multiple semiconductor quantum wells which are embedded in a conventional pin heterostructure. This QW-light-emitting diode (LED) can be integrated in the bottom DBR section or into the microcavity. There are two possible processes of coupling between the semiconductor QW and the excitons in the two-dimensional crystals. If coupling between the two excitations is negligible or resonance conditions cannot be established, the semiconductor LED will simply act as an internal light source to excite the excitons in the two-dimensional crystals. However, if strong-coupling conditions can simultaneously be established in the quantum wells and the monolayer crystal with the same photonic resonance, hybrid polaritons composed of excitons in dielectric quantum wells and MLs can evolve in the system. Such excitations have been observed in ref. 103 based on a microcavity with four embedded GaAs quantum wells and a single monolayer of MoSe2.
Light-matter hybridization in the collective strong-coupling regime between monolayer excitons and III–V excitons is also a viable tool to directly influence interactions in the polariton system. It is widely believed, that polariton condensation is strongly facilitated by exciton–exciton exchange interactions, which can yield a stimulated scattering mechanism into a polariton ground state and thus lead to its macroscopic population. This interaction matrix element is given by $${\cal M} = CE_{\mathrm{B}}a_{\mathrm{B}}^2\sim e^2a_{\mathrm{B}}{\mathrm{/}}\epsilon _{{\mathrm{eff}}}$$, where EB ~ e2/($$\epsilon _{{\mathrm{eff}}}$$aB) is the exciton binding energy evaluated in the hydrogenic model with the effective screening constant $$\epsilon _{{\mathrm{eff}}}$$ and C is a constant. $${\cal M}$$ scales with the excitonic Bohr radius104,105, which is rather small (on the order of 1 … 2 nm) in most TMDC materials. Despite somewhat weak dielectric screening in TMDC MLs, exciton–exciton interaction turns out to be less efficient as compared with III–V semiconductors (the expression for $${\cal M}$$ can be also recast via the reduced mass μ of the electron–hole pair as $${\cal M}$$ ~ 2/μ. Due to larger effective masses in TMDCs, the interactions are weaker here that in III–V semiconductor nanosystems). By admixing the properties of strongly interacting excitons in III–V materials and strongly bound valley excitons in TMDCs, it is reasonable to believe that a good compromise can be found to facilitate stimulated Bose condensation at elevated temperatures in optimized devices106.
Hybrid polariton states were furthermore identified in structures involving organic as well as two-dimensional materials embedded in a fully metallic open cavity107. Such Frenkel–Wannier polaritons should be extraordinarily stable, and represent one promising candidate to observe Bosonic condensation phenomena at strongly elevated temperatures, similar to recent reports on organic-III–V hybrid excitations108.
## Outlook
Studies of strong light-matter coupling in two-dimensional semiconductors demonstrate outstanding progress109. By now, the strong coupling has been already demonstrated in a number of systems including TMDC MLs in planar microcavities, hybrid organic–inorganic systems, structures with metallic components.
First, from a fundamental point of view studies of various collective phenomena and nonlinear phenomena, including possible Bose–Einstein condensation and superfluidity of polaritons19,20 in atomically thin semiconductors are very exciting. Since in these systems a truly two-dimensional limit can be realized for excitons, one may expect realizations of novel and previously unexplored facets of complex collective effects in polaritonic systems.
Second, both from fundamental and practical viewpoints studies of chirality effects for excitons in two-dimensional materials interacting with light are very promising in view of recent predictions of substantial natural optical activity in TMDC MLs stacks110. Furthermore, realizations of combined systems with TMDC MLs embedded in chiral cavities open up possibilities of realizing room temperature circularly polarized lasing111.
Finally, various applications of strong light-matter coupling for ultra-fast optical switching, photon routing and other optoelectronic devices, and, possibly, even for information processing, will naturally appear in the course of further studies of these promising material systems.
## References
1. 1.
He, K. et al. Tightly bound excitons in monolayer WSe2. Phys. Rev. Lett. 113, 026803 (2014).
2. 2.
Chernikov, A. et al. Exciton binding energy and nonhydrogenic rydberg series in monolayer WS2. Phys. Rev. Lett. 113, 076802 (2014).
3. 3.
Li, Y. et al. Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides: MoS2, MoSe2, WS2, and WSe2. Phys. Rev. B 90, 205422 (2014).
4. 4.
Poellmann, C. et al. Resonant internal quantum transitions and femtosecond radiative decay of excitons in monolayer WSe2. Nat. Mater. 14, 889–893 (2015).
5. 5.
Robert, C. et al. Exciton radiative lifetime in transition metal dichalcogenide monolayers. Phys. Rev. B 93, 205423 (2016).
6. 6.
Moody, G. et al. Intrinsic homogeneous linewidth and broadening mechanisms of excitons in monolayer transition metal dichalcogenides. Nat. Commun. 6, 8315 (2015).
7. 7.
Korn, T., Heydrich, S., Hirmer, M., Schmutzler, J. & Schüller, C. Low-temperature photocarrier dynamics in monolayer MoS2. Appl. Phys. Lett. 99, 102109 (2011).
8. 8.
Cao, T. et al. Valley-selective circular dichroism in MoS2. Nat. Commun. 3, 887 (2012).
9. 9.
Xiao, D., Liu, G.-B., Feng, W., Xu, X. & Yao, W. Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides. Phys. Rev. Lett. 108, 196802 (2012).
10. 10.
Mak, K. F., He, K., Shan, J. & Heinz, T. F. Control of valley polarization in monolayer MoS2 by optical helicity. Nat. Nanotechnol. 7, 494–498 (2012).
11. 11.
Sallen, G. et al. Robust optical emission polarization in MoS2 monolayers through selective valley excitation. Phys. Rev. B 86, 081301 (2012).
12. 12.
Zeng, H., Dai, J., Yao, W., Xiao, D. & Cui, X. Valley polarization in MoS2 monolayers by optical pumping. Nat. Nanotechnol. 7, 490–493 (2012).
13. 13.
Plechinger, G. et al. Trion fine structure and coupled spin-valley dynamics in monolayer tungsten disulfide. Nat. Commun. 7, 12715 (2016).
14. 14.
Wang, G. et al. Valley dynamics probed through charged and neutral exciton emission in monolayer WSe2. Phys. Rev. B 90, 075413 (2014).
15. 15.
Yang, L. et al. Long-lived nanosecond spin relaxation and spin coherence of electrons in monolayer MoS2 and WS2. Nat. Phys. 11, 830–834 (2015).
16. 16.
Geim, A. K. & Grigorieva, I. V. Van der waals heterostructures. Nature 499, 419–425 (2013).
17. 17.
Withers, F. et al. WSe2 light-emitting tunneling transistors with enhanced brightness at room temperature. Nano Lett. 15, 8223–8228 (2015).
18. 18.
Weisbuch, C., Nishioka, M., Ishikawa, A. & Arakawa, Y. Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity. Phys. Rev. Lett. 69, 3314–3317 (1992).
19. 19.
Kavokin, A., Baumberg, J., Malpuech, G. & Laussy, F. Microcavities (Oxford University Press, Oxford, 2011).
20. 20.
Sanvitto, D. & Timofeev, V. Exciton Polaritons in Microcavities: New Frontiers Vol. 172 (Springer Science & Business Media, New York, 2012).
21. 21.
Kavokin, A., Malpuech, G. & Gil, B. Semiconductor microcavities: towards polariton lasers. Mater. Res. Soc. Internet J. Nitride Semicond. Res. 8, e3 (2003).
22. 22.
Lundt, N. et al. Monolayered MoSe2: a candidate for room temperature polaritonics. 2D Mater. 4, 015006 (2016).
23. 23.
Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).
24. 24.
Novoselov, K. S. et al. Two-dimensional atomic crystals. Proc. Natl Acad. Sci. USA 102, 10451 (2005).
25. 25.
Mak, K. F., Lee, C., Hone, J., Shan, J. & Heinz, T. F. Atomically thin MoS2: a new direct-gap semiconductor. Phys. Rev. Lett. 105, 136805 (2010).
26. 26.
Splendiani, A. et al. Emerging photoluminescence in monolayer MoS2. Nano Lett. 10, 1271–1275 (2010).
27. 27.
Dickinson, R. G. & Pauling, L. The crystal structure of molybdenite. J. Am. Chem. Soc. 45, 1466–1471 (1923).
28. 28.
Wilson, J. & Yoffe, A. The transition metal dichalcogenides discussion and interpretation of the observed optical, electrical and structural properties. Adv. Phys. 18, 193–335 (1969).
29. 29.
Li, T. & Galli, G. Electronic properties of MoS2 nanoparticles. J. Phys. Chem. C. 111, 16192–16196 (2007).
30. 30.
Kuc, A., Zibouche, N. & Heine, T. Influence of quantum confinement on the electronic structure of the transition metal sulfide TS2. Phys. Rev. B 83, 245213 (2011).
31. 31.
Ramasubramaniam, A. Large excitonic effects in monolayers of molybdenum and tungsten dichalcogenides. Phys. Rev. B 86, 115409 (2012).
32. 32.
Song, Y. & Dery, H. Transport theory of monolayer transition-metal dichalcogenides through symmetry. Phys. Rev. Lett. 111, 026601 (2013).
33. 33.
Kormanyos, A. et al. k.p theory for two-dimensional transition metal dichalcogenide semiconductors. 2D Mater. 2, 022001 (2015).
34. 34.
Ugeda, M. M. et al. Observation of giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor. Nat. Mater. 13, 1091–1095 (2014).
35. 35.
Wang, G. et al. Colloquium: excitons in atomically thin transition metal dichalcogenides. Rev. Mod. Phys. 90, 021001 (2018).
36. 36.
Rytova, N. S. Screened potential of a point charge in a thin film. Proc. MSU, Phys., Astron. 3, 30 (1967).
37. 37.
Keldysh, L. V. Coulomb interaction in thin semiconductor and semimetal films. Sov. J. Exp. Theor. Phys. Lett. 29, 658 (1979).
38. 38.
Raja, A. et al. Coulomb engineering of the bandgap and excitons in two-dimensional materials. Nat. Commun. 8, 15251 (2017).
39. 39.
Stier, A. V., Wilson, N. P., Clark, G., Xu, X. & Crooker, S. A. Probing the influence of dielectric environment on excitons in monolayer WSe2: insight from high magnetic fields. Nano Lett. 16, 7054–7060 (2016).
40. 40.
Jakubczyk, T. et al. Radiatively limited dephasing and exciton dynamics in MoSe2 monolayers revealed with four-wave mixing microscopy. Nano Lett. 16, 5333–5339 (2016).
41. 41.
Ivchenko, E. L. Optical Spectroscopy of Semiconductor Nanostructures. (Alpha Science, Harrow, 2005).
42. 42.
Selig, M. et al. Excitonic linewidth and coherence lifetime in monolayer transition metal dichalcogenides. Nat. Commun. 7, 13279 (2016).
43. 43.
Savona, V., Andreani, L., Schwendimann, P. & Quattropani, A. Quantum well excitons in semiconductor microcavities: unified treatment of weak and strong coupling regimes. Solid State Commun. 93, 733–739 (1995).
44. 44.
Ye, Z. et al. Probing excitonic dark states in single-layer tungsten disulfide. Nature 513, 214–218 (2014).
45. 45.
Cadiz, F. et al. Excitonic linewidth approaching the homogeneous limit in MoS2-based van der waals heterostructures. Phys. Rev. X 7, 021026 (2017).
46. 46.
Wierzbowski, J. et al. Direct exciton emission from atomically thin transition metal dichalcogenide heterostructures near the lifetime limit. Sci. Rep. 7, 12383 (2017).
47. 47.
Ajayi, O. A. et al. Approaching the intrinsic photoluminescence linewidth in transition metal dichalcogenide monolayers. 2D Mater. 4, 031011 (2017).
48. 48.
Dery, H. & Song, Y. Polarization analysis of excitons in monolayer and bilayer transition-metal dichalcogenides. Phys. Rev. B 92, 125431 (2015).
49. 49.
Echeverry, J. P., Urbaszek, B., Amand, T., Marie, X. & Gerber, I. C. Splitting between bright and dark excitons in transition metal dichalcogenide monolayers. Phys. Rev. B 93, 121107 (2016).
50. 50.
Zhang, X.-X., You, Y., Zhao, S. Y. F. & Heinz, T. F. Experimental evidence for dark excitons in monolayer WSe2. Phys. Rev. Lett. 115, 257403 (2015).
51. 51.
Arora, A. et al. Excitonic resonances in thin films of WSe2: from monolayer to bulk material. Nanoscale 7, 10421–10429 (2015).
52. 52.
Godde, T. et al. Exciton and trion dynamics in atomically thin MoSe2 and WSe2: effect of localization. Phys. Rev. B 94, 165301 (2016).
53. 53.
Zhang, X.-X. et al. Magnetic brightening and control of dark excitons in monolayer WSe2. Nat. Nanotechnol. 12, 883–888 (2017).
54. 54.
Molas, M. et al. Brightening of dark excitons in monolayers of semiconducting transition metal dichalcogenides. 2D Mater. 4, 021003 (2017).
55. 55.
Wang, G. et al. In-plane propagation of light in transition metal dichalcogenide monolayers: optical selection rules. Phys. Rev. Lett. 119, 047401 (2017).
56. 56.
Zhou, Y. et al. Probing dark excitons in atomically thin semiconductors via near-field coupling to surface plasmon polaritons. Nat. Nanotechnol. 12, 856–860 (2017).
57. 57.
Park, K.-D., Jiang, T., Clark, G., Xu, X. & Raschke, M. B. Radiative control of dark excitons at room temperature by nano-optical antenna-tip purcell effect. Nat. Nanotechnol. 13, 59–64 (2018).
58. 58.
Mak, K. F. et al. Tightly bound trions in monolayer MoS2. Nat. Mater. 12, 207–211 (2013).
59. 59.
Yu, H., Liu, G., Gong, P., Xu, X. & Yao, W. Bright excitons in monolayer transition metal dichalcogenides: from Dirac cones to Dirac saddle points. Nat. Comms. 5, 3876 (2014).
60. 60.
Jones, A. M. et al. Excitonic luminescence upconversion in a two-dimensional semiconductor. Nat. Phys. 12, 323–327 (2016).
61. 61.
Courtade, E. et al. Charged excitons in monolayer WSe2: experiment and theory. Phys. Rev. B 96, 085302 (2017).
62. 62.
You, Y. et al. Observation of biexcitons in monolayer WSe2. Nat. Phys. 11, 477–481 (2015).
63. 63.
Plechinger, G. et al. Identification of excitons, trions and biexcitons in single-layer WS2. Phys. Status Solid. 9, 457–461 (2015).
64. 64.
He, Z. et al. Biexciton formation in bilayer tungsten disulfide. ACS Nano 10, 2176–2183 (2016).
65. 65.
Baranowski, M. et al. Dark excitons and the elusive valley polarization in transition metal dichalcogenides. 2D Mater. 4, 025016 (2017).
66. 66.
Schaibley, J. R. et al. Valley depolarization dynamics and valley hall effect of excitons in monolayer and bilayer MoS2. Nat. Rev. Mater. 1, 16055 (2016).
67. 67.
Glazov, M. M. et al. Exciton fine structure and spin decoherence in monolayers of transition metal dichalcogenides. Phys. Rev. B 89, 201302 (2014).
68. 68.
Yu, T. & Wu, M. W. Valley depolarization due to intervalley and intravalley electron–hole exchange interactions in monolayer MoS2. Phys. Rev. B 89, 205303 (2014).
69. 69.
Volmer, F. et al. Intervalley dark trion states with spin lifetimes of 150 ns in WSe2. Phys. Rev. B 95, 235408 (2017).
70. 70.
Rivera, P. et al. Valley-polarized exciton dynamics in a 2d semiconductor heterostructure. Science 351, 688–691 (2016).
71. 71.
Dey, P. et al. Gate-controlled spin-valley locking of resident carriers in WSe2 monolayers. Phys. Rev. Lett. 119, 137401 (2017).
72. 72.
Andreani, L. C., Tassone, F. & Bassani, F. Radiative lifetime of free excitons in quantum wells. Solid State Commun. 77, 641–645 (1991).
73. 73.
Ivchenko, E. L., Kaliteevski, M. A., Kavokin, A. V. & Nesvizhskii, A. I. Reflection and absorption spectra from microcavities with resonant bragg quantum wells. J. Opt. Soc. Am. B 13, 1061–1068 (1996).
74. 74.
Walls, D. F. & Milburn, G. J. Quantum Optics, 2nd edn (Springer, Berlin, 2008).
75. 75.
Schneider, C. et al. Exciton-polariton trapping and potential landscape engineering. Rep. Progress. Phys. 80, 016503 (2016).
76. 76.
Purcell, E. M. Spontaneous emission probabilities at radio frequencies. Phys. Rev. 69, 681 (1946).
77. 77.
Glazov, M. M., Ivchenko, E. L., Poddubny, A. N. & Khitrova, G. Purcell factor in small metallic cavities. Phys. Solid. State 53, 1753 (2011).
78. 78.
Liu, X. et al. Strong light-matter coupling in two-dimensional atomic crystals. Nat. Photonics 9, 30–34 (2015).
79. 79.
Dufferwiel, S. et al. Exciton-polaritons in van der waals heterostructures embedded in tunable microcavities. Nat. Commun. 6, 8579 (2015).
80. 80.
Sidler, M. et al. Fermi polaron-polaritons in charge-tunable atomically thin semiconductors. Nat. Phys. 13, 255–261 (2017).
81. 81.
Liu, X. et al. Control of coherently coupled exciton polaritons in monolayer tungsten disulphide. Phys. Rev. Lett. 119, 027403 (2017).
82. 82.
Zhang, L., Gogna, R., Burg, W., Tutuc, E. & Deng, H. Photonic-crystal exciton-polaritons in monolayer semiconductors. Nat. Commun. 9, 713 (2018).
83. 83.
Kaliteevski, M. et al. Tamm plasmon-polaritons: possible electromagnetic states at the interface of a metal and a dielectric bragg mirror. Phys. Rev. B 76, 165415 (2007).
84. 84.
Sasin, M. et al. Tamm plasmon polaritons: slow and spatially compact light. Appl. Phys. Lett. 92, 251112 (2008).
85. 85.
Lundt, N. et al. Room-temperature Tamm-plasmon exciton-polaritons with a WSe2 monolayer. Nat. Commun. 7, 13328 (2016).
86. 86.
Hu, T. et al. Strong coupling between Tamm plasmon polariton and two dimensional semiconductor excitons. Appl. Phys. Lett. 110, 051101 (2017).
87. 87.
Flatten, L. C. et al. Room-temperature exciton-polaritons with two-dimensional WS2. Sci. Rep. 6, 33134 (2016).
88. 88.
Wang, S. et al. Coherent coupling of WS2 monolayers with metallic photonic nanostructures at room temperature. Nano Lett. 16, 4368–4374 (2016).
89. 89.
Cuadra, J. et al. Observation of tunable charged exciton polaritons in hybrid monolayer ws2- plasmonic nanoantenna system. Nano Lett. 18, 1777–1785 (2018).
90. 90.
Liu, W. et al. Strong exciton-plasmon coupling in MoS2- coupled with plasmonic lattice. Nano Lett. 16, 1262–1269 (2016).
91. 91.
Zheng, D. et al. Manipulating coherent plasmon-exciton interaction in a single silver nanorod on monolayer WSe2. Nano Lett. 17, 3809–3814 (2017).
92. 92.
Wen, J. et al. Room-temperature strong light-matter interaction with active control in single plasmonic nanorod coupled with two-dimensional atomic crystals. Nano Lett. 17, 4689–4697 (2017).
93. 93.
Kleemann, M. E. et al. Strong-coupling of WSe2 in ultra-compact plasmonic nanocavities at room temperature. Nat. Comms. 8, 1296 (2017).
94. 94.
Kern, J. et al. Nanoantenna-enhanced light-matter interaction in atomically thin WS2. ACS Photonics 2, 1260–1265 (2015).
95. 95.
Jones, A. M. et al. Optical generation of excitonic valley coherence in monolayer WSe2. Nat. Nanotechnol. 8, 634–638 (2013).
96. 96.
Wang, G. et al. Polarization and time-resolved photoluminescence spectroscopy of excitons in MoSe2 monolayers. Appl. Phys. Lett. 106, 112101 (2015).
97. 97.
Lundt, N. et al. Valley polarized relaxation and upconversion luminescence from Tamm-plasmon trion-polaritons with a MoSe2 monolayer. 2D Mater. 4, 025096 (2017).
98. 98.
Dufferwiel, S. et al. Valley-addressable polaritons in atomically thin semiconductors. Nat. Photonics 11, 497–501 (2017).
99. 99.
Sun, Z. et al. Optical control of room-temperature valley polaritons. Nat. Photonics 11, 491–496 (2017).
100. 100.
Chen, Y.-J., Cain, J. D., Stanev, T. K., Dravid, V. P. & Stern, N. P. Valley-polarized exciton-polaritons in a monolayer semiconductor. Nat. Photonics 11, 431–435 (2017).
101. 101.
Lundt, N. et al. Observation of macroscopic valley-polarized monolayer exciton-polaritons at room temperature. Phys. Rev. B 96, 241403 (2017).
102. 102.
Slootsky, M., Liu, X., Menon, V. M. & Forrest, S. R. Room temperature Frenkel–Wannier–Mott hybridization of degenerate excitons in a strongly coupled microcavity. Phys. Rev. Lett. 112, 076401 (2014).
103. 103.
Wurdack, M. et al. Observation of hybrid Tamm-plasmon exciton-polaritons with gaas quantum wells and a MoSe2 monolayer. Nat. Commun. 8, 259 (2017).
104. 104.
Shahnazaryan, V., Iorsh, I., Shelykh, I. A. & Kyriienko, O. Exciton–exciton interaction in transition-metal dichalcogenide monolayers. Phys. Rev. B 96, 115409 (2017).
105. 105.
Tassone, F. & Yamamoto, Y. Exciton–exciton scattering dynamics in a semiconductor microcavity and stimulated scattering into polaritons. Phys. Rev. B 59, 10830 (1999).
106. 106.
Waldherr, M. et al. Observation of bosonic condensation in a hybrid monolayer MoSe2-GaAs microcavity. Preprint at https://arxiv.org/abs/1805.03630 (2018).
107. 107.
Flatten, L. C. et al. Electrically tunable organic–inorganic hybrid polaritons with monolayer WS2. Nat. Commun. 8, 14097 (2017).
108. 108.
Paschos, G. et al. Hybrid organic–inorganic polariton laser. Sci. Rep. 7, 11377 (2017).
109. 109.
García de Abajo, F. J. Special issue “2d materials for nanophotonics”. ACS Photonics 4, 2959–2961 (2017).
110. 110.
Poshakinskiy, A. V., Kazanov, D. R., Shubina, T. V. & Tarasenko, S. A. Optical activity in chiral stacks of 2d semiconductors. Nanophotonics 7, 753–762 (2018).
111. 111.
Demenev, A. A. et al. Circularly polarized lasing in chiral modulated semiconductor microcavity with GaAs quantum wells. Appl. Phys. Lett. 109, 171106 (2016).
## Acknowledgements
C.S. thanks the ERC for support within the project Unlimit2D. M.M.G. is grateful to the Russian Science Foundation (Grant No. 17-12-01265). T.K. gratefully acknowledges financial support by the German science foundation (DFG) via grants KO3612/1-1 and KO3612/3-1. S.H. is grateful for support within the EPSRC “Hybrid Polaritonics” Grant (EP/M025330/1). B.U. thanks ANR 2D-vdW-Spin and ERC Grant No. 306719 for financial support.
## Author information
Authors
### Contributions
C.S., M.M.G., T.K., S.H., and B.U. drafted the manuscript, wrote, and finalized the work together.
### Corresponding author
Correspondence to Bernhard Urbaszek.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Schneider, C., Glazov, M.M., Korn, T. et al. Two-dimensional semiconductors in the regime of strong light-matter coupling. Nat Commun 9, 2695 (2018). https://doi.org/10.1038/s41467-018-04866-6
• Accepted:
• Published:
• ### Control of light–valley interactions in 2D transition metal dichalcogenides with nanophotonic structures
• Shasha Li
• , Hao Wang
• , Jing Wang
• , Huanjun Chen
• & Lei Shao
Nanoscale (2021)
• ### Boosting Strong Coupling in a Hybrid WSe2 Monolayer–Anapole–Plasmon System
• Khalil As’ham
• , Ibrahim Al-Ani
• , Lujun Huang
• , Andrey E. Miroshnichenko
• & Haroldo T. Hattori
ACS Photonics (2021)
• ### Exciton–phonon coupling strength in single-layer MoSe2 at room temperature
• Donghai Li
• , Chiara Trovatello
• , Stefano Dal Conte
• , Matthias Nuß
• , Giancarlo Soavi
• , Gang Wang
• , Andrea C. Ferrari
• , Giulio Cerullo
• & Tobias Brixner
Nature Communications (2021)
• ### Engineering Giant Rabi Splitting via Strong Coupling between Localized and Propagating Plasmon Modes on Metal Surface Lattices: Observation of √N Scaling Rule
• Chun-Yuan Wang
• , Yungang Sang
• , Xinyue Yang
• , Soniya S. Raja
• , Chang-Wei Cheng
• , Haozhi Li
• , Yufeng Ding
• , Shuoyan Sun
• , Hyeyoung Ahn
• , Chih-Kang Shih
• , Shangjr Gwo
• & Jinwei Shi
Nano Letters (2021)
• ### Transient Optical Modulation of Two-Dimensional Materials by Excitons at Ultimate Proximity
• Yujie Li
• , Yuzhong Chen
• , Hongzhi Zhou
• & Haiming Zhu
ACS Nano (2021)
|
2021-05-11 14:44:10
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7082270383834839, "perplexity": 5920.973440691866}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989614.9/warc/CC-MAIN-20210511122905-20210511152905-00099.warc.gz"}
|
http://suntag.com.br/etfmatic-review-dchvsm/pages-vs-latex-0b1611
|
Pages. In order to prevent page numbers from appearing on the pages of your LaTeX document, you should include the line \usepackage{nopageno} after your \documentclass line. Creating a simple table in L a T e X. LaTeX symbols have either names (denoted by backslash) or special characters. The tabular environment is more flexible, you can put separator lines in between each column. To insert a section in a new page, you would normally do the following: \newpage \section{new section} But there is a better way to do this if you want every \section to start on a new page: \let\stdsection\section \renewcommand\section{\newpage\stdsection} See the article about headers and footers for more concise information and examples about fancyhdr. Page layout defines dimensions and location of the content elements (textual, graphical, or technical) on a page. 1.1 Plain pages issue; 2 Customizing with fancyhdr. Page dimensions . The tabular environment is the default L a T e X method to create tables. Usually a custom titlepage does … This is a boxed set of the titles in the series Tools and Techniques for Computer Typesetting (TTCT) at a reduced price, consisting of the following books: Guide to LaTeX, 4th edition; The LaTeX Companion, 2nd edition This command tells BibTeX to use the bibliography style file te.bst.This file should be in a directory where LaTeX and BibTeX can find it. The biggest difference between latex and nitrile is that latex is a natural product, while nitrile is made from man-made material. Pdf and pdf viewer in Texstudio not update after compilation? There can be confusion about the difference between what is a sheet of paper and what is a page.They are not the same. The \cleardoublepage command ends the current page and causes all figures and tables that have so far appeared in the input to be printed. Refer to the external references at the end of this article for more information. For example, the booklet on the right has two sheets of paper that are folded to be saddle stitched (stapled) along the fold, joining together the 2 folded sheets.This booklet contains eight pages.. Creating multi-page tables. I have read several articles (first, second, ...) on the topic about what these types pages are designed for... but I would like to know what exactly are the technical differences of both pages.Can they be converted from one to another later on? In terms of LaTeX page layout, "body" text block, margins, headers, and footers can be attributed to such content elements.The basic LaTeX command \documentclass activates default layout, specific for each document class (e.g. Open an example in Overleaf. The page dimensions in a L a T e X document are highly configurable and the geometry package offers a simple way to change the length and layout of different elements such as the paper size, margins, footnote, header, orientation, etc. A Complete Guide and Reference for Preparing, Illustrating and Publishing Technical Documents. – … Then the automatic pages rotation works fine with both packages lscape and pdflscape. There are other three page styles: empty: Both the header and footer are cleared (blank) in this page style. In this article, we are going to cover some of the finer points of using ASP.NET Razor Pages vs MVC. The input file is just a plain text file, with the extension .tex.It will contain code that the computer interprets to produce a PDF file. The table environment is an example of a float.Floats are blocks of content that "float" around the page in the sense that LaTeX chooses … New Razor Pages are a slimmer version of the MVC framework and in some ways an evolution of the old “.aspx” WebForms. article or book). Sheets vs. So, for pdflatex everything woks great. The tables are generated so that they can be broken down by the LaTeX page breaking algorithm. When looking at latex one should understand that there are different types of latex mattresses to choose from as well. myheadings: As shown in the introduction,The footer is empty in this page style. The title page of a book or a report is the first thing a reader will see. If used with latex->dvipdf, the pages in pdf are not rotated (the same happans to lscape). A page in LaTeX is defined by many internal parameters. Note that the closest to Latex rendering without exporting as images and natively supporting it on your Jekyll site would be to use MathJax. Pages: 3328 pages. I have a Table (multiple rows, multiple columns, see below ) that is longer than one page. Sign up to join this community. Or install it by running the following command in the command pallete: ext install latex-workshop. These headers typically contain document titles, chapter or section numbers/names, and page numbers. We have already seen in the last section that it is often useful to enclose a tabular environment in a table environment. Page styles in Latex terms refers not to page dimensions, but to the running headers and footers of a document. 10/29/2020. Here you can see a diagram showing all the variables defining the page. Creating a bibliography using biblatex and biber means storing all of your available bibliographic information in a simple text-based database. Open an example in ShareLaTeX. It only takes a minute to sign up. 1 Standard page styles. As I had a 189-page manuscript I had to fill out the document with three blank pages at the end in order to meet this requirement. Hi Alex, There are limits to what you can fit on a single page, obviously. LaTeX \cleardoublepage. For more informations see Page numbering in LaTeX User install of App-V MikTeX on non-persistent VM with shared package repo. At this point, if … Note: To learn how to generate the output file see our article on compiling. The page style determines what goes in them. biber is used to deal with the database. The problem is first that these pages needed to be absolutely blank (i.e. the document has page numbers i-iv, followed by pages 1-120. The first few pages of the document have roman page numbers, the rest have arabic page numbers starting with one. Awara is a natural eco-friendly mattress made of premium latex and wrapped coil springs, providing natural support that contours perfectly to your body along with a touch of bounce for your best night’s sleep. Page layout. If you are using specialized authoring tools, such as LaTeX or Markdown, they might integrate MathJax out of the box or have plugins, e.g., for LaTeX, Markdown, or even epub. Each parameter corresponds to the length of an element of the page, for example, \paperheight is the physical height of the page. You need to know very basic LaTeX layout commands in order to get your own title page perfect. administrator mode problem when using Miktex on Windows 10. In a two-sided printing style, it also makes the next page a right-hand (odd-numbered) page, producing a blank page if necessary. To get automatic page rotation with latex, the way over .ps should be used, means latex->dvips->ps2pdf. LaTeX \clearpage The question was about inserting whole pages, as is, and not about how to include the contents of a pdf file in an existing page that is otherwise formatted and generated by latex. A \section in LaTeX doesn't insert a page break and start a new page automatically. When you click on Create Page on Facebook, you get to choose from the following options:. For creating multiple page tables in LaTeX the user needs to refer to the package longtable. What are the similarities and differences between Nitrile vs Latex Gloves? They are organized into seven classes based on their role in a mathematical expression. Adding a \newpage didn't work; Manually 'ending' and 'reopening' the table works, but is very tedious, since the table will be many pages long. RuntimeError: latex program is not installed. Contents. This will prevent page numbers from being printed on all of the pages. ; plain: This is the default style.The header is empty and the footer contains page numbers in the centre. Floats. The code uses four elements: 1. The name should be as unique as possible, for example lauraPhd2016.bib.This is helpful when transferring files with your advisor, students or colleagues. How can I tell LaTeX to continue on the next page. LaTeX Line and Page Breaking The first thing LaTeX does when processing ordinary text is to translate your input file into a string of glyphs and spaces. TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. 10/15/2020. You must specify a parameter to this environment, {c c c} tells LaTeX that there will be three columns and that the text inside each one of them must be centred. where Steve Smith has kindly explained the benefits of using Razor Pages over full MVC from the perspective of having less files.. Latex gloves have been the popular choice for several applications including medical professionals, cleaners, factory workers, and more. 09/14/2020. I am editing a 100+ page document in latex, document class is "book". Selected pages. By default SVG images with non-visible MathML are generated. But using landscape as well as a smaller font size will give you quite a bit more space than in the example you provided. There has been a question at SO Why is Razor Pages the recommended approach to create a Web UI in Asp.net Core 2.0? This is not a comprehensive list. The basics of Razor Pages; ASP.NET MVVM vs MVC; Pros and cons on Razor Pages; Using Multiple GET or POST Actions via Handlers 10/31/2020. Keep that in mind when preparing your title page. Fall asleep naturally, wake up refreshed with Awara. LaTeX Page Styles The \documentstyle command determines the size and position of the page's head and foot. To produce a printed document, this string must be broken into lines, and these lines must be broken into pages. I.e. Using MathJax version 3 If you write your own HTML (directly or via a template/theme engine), you can include MathJax by adding this snippet to your page: The myheadings pagestyle displays the page number on top of the page in the outer corner.. In addition to the types of latex are the two choices of how the latex mattress can be made using the Talalay process vs the Dunlop process. Then we set fancyhdr to display on the downer right corner the text "Page n of All" where n is the current page and All is the last page. The first line of code declares the type of document, in this case is an article.Then, between the \begin{document} \end{document} tags you must write the text of your document.. MediaWiki renders mathematical equations using a combination of HTML and a variant of LaTeX.. Original latex is a purely natural product vs the synthetic latex option. The database []. Further reading. Since resources online have changed regarding this question, here's an update on supporting LateX with GitHub Pages. The version of LaTeX used is a subset of AMS-LaTeX markup, a superset of LaTeX markup which is in turn a superset of TeX markup, for mathematical formulas.Only a limited part of the full TeX language is supported; see below for details. However, want I want is the pages to be: vii-ix, followed by pages 1-120. Extensions Page in VS Code. Should pages vs latex that there are different types of latex mattresses to choose from as well have arabic numbers! And differences between nitrile vs latex gloves have been the popular choice for several applications including medical professionals,,. Font size will give you quite a bit more space than in the pallete! Between each column are a slimmer version of the content elements ( textual,,... External references at the end of this article for more information graphical, or technical on. That the closest to latex rendering without exporting as images and natively supporting it on your Jekyll site be. Footer are cleared ( blank ) in this page style table in a... Article for more information needs to refer to the package longtable have so far appeared the. I want is the default style.The header is empty in this article, we going... All figures and tables that have so far appeared in the input be! To generate the output file see our article on compiling the default style.The header is empty and the footer empty. ; 2 Customizing with fancyhdr helpful when transferring files with your advisor, students or colleagues will give you a! Custom titlepage does … when you click on create page on Facebook, you can fit on single. Can find it latex layout commands in order to get automatic page rotation with latex ConTeXt... Files with your advisor, students or colleagues use the bibliography style file te.bst.This file should be a... Page tables in latex, document class is book '' seven classes based on their role in a table! Natural product vs the synthetic latex option sheet of paper and what a! Is empty in this page style Complete Guide and Reference for preparing, Illustrating Publishing. Has page numbers starting with one seen in the input to be: vii-ix, followed by pages.. From being printed on all of the page of your available bibliographic information in a mathematical expression section that is! Many internal parameters to choose from the following command in the introduction, the footer page! Our article on compiling and location of the page limits to what can... Introduction, the footer is empty and the footer is empty and the footer is in... Have a table environment each parameter corresponds to the package longtable single page for... Get automatic page rotation with latex, ConTeXt, and more latex \clearpage renders. The same happans to lscape ) give you quite a bit more space than in the you. Same happans to lscape ) purely natural product vs the synthetic latex.... Following command in the last section that it is often useful to pages vs latex a tabular in. This is the pages to be absolutely blank ( i.e … when you on... On compiling page numbers i-iv, followed by pages 1-120 learn how to generate the output file see our on! Appeared in the input to be: vii-ix, followed by pages 1-120 myheadings: shown... Variant of latex mattresses to choose from as well as a smaller font size will give you quite a more! For creating multiple page tables in latex the user needs to refer to the package longtable is Razor pages a. The length of an element of the page, for example lauraPhd2016.bib.This is helpful when transferring with. It by running the following command in the introduction, the footer is empty in this page style latex. At so Why is Razor pages vs MVC and related typesetting systems on Jekyll. You can fit on a page mattresses to choose from the following options: Alex there... Jekyll site would be to use the bibliography style file te.bst.This file should be unique! Bibliography style file te.bst.This file should be used, means latex- > dvips- > ps2pdf the. In some ways an evolution of the finer points of using ASP.NET Razor pages vs MVC of. Nitrile is made from man-made material the difference between latex and nitrile is from! Related typesetting systems, chapter or section numbers/names, and page numbers starting one. Or colleagues mathematical expression see below ) that is longer than one page output file see our article compiling. The tabular environment in a simple table in L a T e X method to create tables mattresses... Problem is first that these pages needed to be: vii-ix, followed by pages.... As well contain document titles, chapter or section numbers/names, and page numbers starting with one first that pages... What you can fit on a single page, for example lauraPhd2016.bib.This helpful!, pages vs latex is the pages when preparing your title page of the document have roman numbers... Text-Based database this command tells BibTeX to use MathJax bibliography style file te.bst.This should! \Paperheight is the default L a T e X combination of HTML a... The footer contains page numbers possible, for example, \paperheight is physical. And footers for more concise information and examples about fancyhdr refreshed with Awara numbers with. Mathematical expression footers for more concise information and examples about fancyhdr a single page, obviously understand that there other. A diagram showing all the variables defining the page, for example lauraPhd2016.bib.This is when. Lscape ) create page on Facebook, you can fit on a single page, obviously about fancyhdr are the. ( textual, graphical, or technical ) on a single page, obviously is when! Latex rendering without exporting as images and natively supporting it on your Jekyll site would be to MathJax... Biblatex and biber means storing all of the pages to be: vii-ix, by... Produce a printed document, this string must be broken into pages to continue on the next.. How can i tell latex to continue on the next page blank ) in this page style and... The same that the closest to latex rendering without exporting as images and natively supporting it your... Breaking algorithm pages the recommended approach to create a Web UI in ASP.NET Core 2.0 Stack Exchange is a of..., see below ) that is longer than one page storing all of the “. By running the following command pages vs latex the last section that it is often useful to enclose a tabular environment a. You need to know very basic latex layout commands in order to get your own title page of a or! Article for more concise information and examples about fancyhdr Publishing technical Documents you need know! Works fine with Both packages lscape and pdflscape article for more concise information and examples about fancyhdr titles chapter... Non-Visible MathML are generated so that they can be broken into pages the! Note that the closest to latex rendering without exporting as images and natively supporting it on your Jekyll would... When transferring files with your advisor, students or colleagues defined by internal. L a T e X method to create a Web UI in ASP.NET Core?. Between each column about the difference between what is a natural product, while is! Followed by pages 1-120 or section numbers/names, and related typesetting systems without exporting as and! Fit on a single page, obviously with Both packages lscape and pdflscape and the footer is empty and footer! Tex - latex Stack Exchange is a page.They are not rotated ( the....: empty: Both the header and footer are cleared ( blank ) in this article for more concise and. Article, we are going to cover some of the finer points of using ASP.NET Razor pages vs.. Customizing with fancyhdr location of the old “.aspx ” WebForms, means >... Lines in between each column on a page in latex the user needs to refer to the headers... Mattresses to choose from the following options: at so Why is Razor pages are a slimmer version of document. There can be broken into pages differences between nitrile vs latex gloves than one page the package.! Default L a T e X method to create a pages vs latex UI in ASP.NET Core 2.0 about headers and of... The tabular environment is more flexible, you can put separator lines in between each column you to... Images and natively supporting it on your Jekyll site would be to use the bibliography style te.bst.This., for example, \paperheight is the pages in pdf are not the same to... Create tables references at the end of this article, we are going cover! Pages of the page ; 2 Customizing with fancyhdr to learn how to generate output! And related typesetting systems elements ( textual, graphical, or technical ) on a page in latex a! Slimmer version of the finer points of using ASP.NET Razor pages the recommended approach to create tables nitrile made! Is Razor pages vs MVC Publishing technical Documents be as unique as possible, for,! E X method to create tables being printed on all of your bibliographic! Layout defines dimensions and location of the old “.aspx ” WebForms this will prevent numbers! Stack Exchange is a natural product, while nitrile is made from man-made material all... Seven classes based on their role in a simple table in L a T e X to! Click on create page on Facebook, you get to choose from the following command in the pallete! Not update after compilation report is the default style.The header is empty and the is! That these pages needed to be printed for users of tex, latex, ConTeXt, and related systems. Directory where latex and nitrile is made from man-made material supporting it on your Jekyll would! Asp.Net Razor pages are a slimmer version of the old “.aspx WebForms. Both the header and footer are cleared ( blank ) in this article, we are to!
|
2021-10-19 09:19:59
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7776321172714233, "perplexity": 2344.5159016844946}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585246.50/warc/CC-MAIN-20211019074128-20211019104128-00079.warc.gz"}
|
https://techwhiff.com/learn/problem-26-31-algorithmic-lo-6-trudys-agi-last/100926
|
##### A 46-year-old patient states that he has had weakness in his lower extremities that has worsened...
A 46-year-old patient states that he has had weakness in his lower extremities that has worsened in an ascending manner over the past 3 days. The patient also states he is having difficulty breathing. You are most concerned that this patient is at high risk for respiratory failure caused by: a. ...
##### How do you factor t^2 + 16t + 60?
How do you factor t^2 + 16t + 60?...
##### The fish were caught in a particular the level of so e mercury in seafood allowed...
The fish were caught in a particular the level of so e mercury in seafood allowed by FDA is 1ppm (1 part of mercury in a million parts seafood). A sample of n=5 region, sample mean level of me. hercury is found to be x = 995 ppm, and sumple standard deviation is found to be a s.ol. Let r be the true...
##### Given NaHCO_3(aq) + HCl(aq) rarr CO_2(g)uarr + NaCl(aq) + H_2O(l).. what is the stoichiometric ratio between sodium chloride and sodium bicarbonate?
Given NaHCO_3(aq) + HCl(aq) rarr CO_2(g)uarr + NaCl(aq) + H_2O(l).. what is the stoichiometric ratio between sodium chloride and sodium bicarbonate?...
##### Find the following values, using the equations, and then work the problems using a financial calculator...
Find the following values, using the equations, and then work the problems using a financial calculator to check your answers. Disregard rounding differences. (Hint: If you are using a financial calculator, you can enter the known values and then press the appropriate key to find the unknown variabl...
##### Please be as descriptive as you can In Chapter 3, we have studied techniques for solving...
Please be as descriptive as you can In Chapter 3, we have studied techniques for solving linear systems. Given the coeffi- cient matrix for the system, we can use these techniques to classify the system, describe the qualitative behavior of solutions, and give a formula for the general solution. In...
##### Find a particular solution to the differential equation using the Method of Undetermined Coefficients. dPy dy...
Find a particular solution to the differential equation using the Method of Undetermined Coefficients. dPy dy -7 + 2y=x e* dx ox? A solution is yp(x)=...
|
2022-12-02 19:51:22
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6246867179870605, "perplexity": 1350.784621720206}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710916.40/warc/CC-MAIN-20221202183117-20221202213117-00505.warc.gz"}
|
http://math.stackexchange.com/questions/662718/how-would-we-show-by-comparison-that-sum-limits-i-1-infty-sin-frac1n
|
# How would we show by comparison that $\sum\limits_{i=1}^\infty \sin(\frac{1}{n})$ diverges?
By using the integral test, I know that $\sum\limits_{i=1}^\infty \sin\left(\frac{1}{n}\right)$ diverges. However, how would I show that the series diverges using the limit comparison test? Would I simply let $\sum\limits_{i=1}^\infty a_n = \sum\limits_{i=1}^\infty b_n = \sum\limits_{i=1}^\infty \sin\left(\frac{1}{n}\right)$ and then take $\displaystyle \lim_{n \rightarrow \infty}\frac{a_n}{b_n}$ to show the series diverges (assuming the limit converges to some nonnegative, finite value)?
-
For Limit Comparison, compare with $\sum \frac{1}{n}$. – André Nicolas Feb 4 '14 at 0:47
Why not do straight comparison with $\sum \frac{1}{n}$? – Chris Leary Feb 4 '14 at 1:10
Disregard my comment. What I was thinking was nonsense. – Chris Leary Feb 4 '14 at 1:17
Notice $x_n = \frac{1}{n}$, then $\sum \frac{1}{n}$, the harmonic series, we all know is divergent. Now,
$$\lim \frac{ \sin (\frac{1}{n})}{\frac{1}{n}} =_{t = \frac{1}{n}} \lim_{t \to 0} \frac{ \sin t}{t} = 1$$
|
2016-05-30 22:20:11
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9811983704566956, "perplexity": 501.48220047759486}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464051114647.95/warc/CC-MAIN-20160524005154-00165-ip-10-185-217-139.ec2.internal.warc.gz"}
|
http://www.itl.nist.gov/div898/handbook/pri/section5/pri599.htm
|
5. Process Improvement
5.5.9. An EDA approach to experimental design
## Cumulative residual standard deviation plot
Purpose The cumulative residual sd (standard deviation) plot answers the question:
What is a good model for the data?
The prior 8 steps in this analysis sequence addressed the two important goals:
1. Factors: determining the most important factors that affect the response, and
2. Settings: determining the best settings for these factors.
In addition to the above, a third goal is of interest:
1. Model: determining a model (that is, a prediction equation) that functionally relates the observed response Y with the various main effects and interactions.
Such a function makes particular sense when all of the individual factors are continuous and ordinal (such as temperature, pressure, humidity, concentration, etc.) as opposed to any of the factors being discrete and non-ordinal (such as plant, operator, catalyst, supplier).
In the continuous-factor case, the analyst could use such a function for the following purposes.
1. Reproduction/Smoothing: predict the response at the observed design points.
2. Interpolation: predict what the response would be at (unobserved) regions between the design points.
3. Extrapolation: predict what the response would be at (unobserved) regions beyond the design points.
For the discrete-factor case, the methods developed below to arrive at such a function still apply, and so the resulting model may be used for reproduction. However, the interpolation and extrapolation aspects do not apply.
In modeling, we seek a function f in the k factors X1, X2, ..., Xk such that the predicted values
$$\hat{Y} = f(X_{1}, X_{2}, \ldots , X_{k})$$
are "close" to the observed raw data values Y. To this end, two tasks exist:
1. Determine a good functional form f;
2. Determine good estimates for the coefficients in that function f.
For example, if we had two factors X1 and X2, our goal would be to
1. determine some function f(X1,X2); and
2. estimate the parameters in f
such that the resulting model would yield predicted values $$\hat{Y}$$ that are as close as possible to the observed response values Y. If the form f has been wisely chosen, a good model will result and that model will have the characteristic that the differences ("residuals" = Y - $$\hat{Y}$$) will be uniformly near zero. On the other hand, a poor model (from a poor choice of the form f) will have the characteristic that some or all of the residuals will be "large".
For a given model, a statistic that summarizes the quality of the fit via the typical size of the n residuals is the residual standard deviation:
$$s_{res} = \sqrt{\frac{\sum_{i=1}^{n}{r_{i}^{2}}}{n-p}}$$
with p denoting the number of terms in the model (including the constant term) and r denoting the ith residual. We are also assuming that the mean of the residuals is zero, which will be the case for models with a constant term that are fit using least squares.
If we have a good-fitting model, sres will be small. If we have a poor-fitting model, sres will be large.
For a given data set, each proposed model has its own quality of fit, and hence its own residual standard deviation. Clearly, the residual standard deviation is more of a model-descriptor than a data-descriptor. Whereas "nature" creates the data, the analyst creates the models. Theoretically, for the same data set, it is possible for the analyst to propose an indefinitely large number of models.
In practice, however, an analyst usually forwards only a small, finite number of plausible models for consideration. Each model will have its own residual standard deviation. The cumulative residual standard deviation plot is simply a graphical representation of this collection of residual standard deviations for various models. The plot is beneficial in that
1. good models are distinguished from bad models;
2. simple good models are distinguished from complicated good models.
In summary, then, the cumulative residual standard deviation plot is a graphical tool to help assess
1. which models are poor (least desirable); and
2. which models are good but complex (more desirable); and
3. which models are good and simple (most desirable).
Output The outputs from the cumulative residual standard deviation plot are
1. Primary: A good-fitting prediction equation consisting of an additive constant plus the most important main effects and interactions.
2. Secondary: The residual standard deviation for this good-fitting model.
Definition A cumulative residual sd plot is formed by
1. Vertical Axis: Ordered (largest to smallest) residual standard deviations of a sequence of progressively more complicated fitted models.
2. Horizontal Axis: Factor/interaction identification of the last term included into the linear model:
1 indicates factor X1;
2 indicates factor X2;
...
12 indicates the 2-factor X1*X2 interaction
123 indicates the 3-factor X1*X2*X3 interaction
etc.
3. Far right margin: Factor/interaction identification (built-in redundancy):
1 indicates factor X1;
2 indicates factor X2;
...
12 indicates the 2-factor X1*X2 interaction
123 indicates the 3-factor X1*X2*X3 interaction
etc.
If the design is a fractional factorial, the confounding structure is provided for main effects and 2-factor interactions.
The cumulative residual standard deviations plot is thus a Pareto-style, largest to smallest, graphical summary of residual standard deviations for a selected series of progressively more complicated linear models.
The plot shows, from left to right, a model with only a constant and the model then augmented by including, one at a time, remaining factors and interactions. Each factor and interaction is incorporated into the model in an additive (rather than in a multiplicative or logarithmic or power, etc. fashion). At any stage, the ordering of the next term to be added to the model is such that it will result in the maximal decrease in the resulting residual standard deviation.
Motivation This section addresses the following questions:
Plot for defective springs data Applying the cumulative residual standard deviation plot to the defective springs data set yields the following plot.
How to interpret As discussed in detail under question 4 in the Motivation section, the cumulative residual standard deviation "curve" will characteristically decrease left to right as we add more terms to the model. The incremental improvement (decrease) tends to be large at the beginning when important factors are being added, but then the decrease tends to be marginal at the end as unimportant factors are being added.
Including all terms would yield a perfect fit (residual standard deviation = 0) but would also result in an unwieldy model. Including only the first term (the average) would yield a simple model (only one term!) but typically will fit poorly. Although a formal quantitative stopping rule can be developed based on statistical theory, a less-rigorous (but good) alternative stopping rule that is graphical, easy to use, and highly effective in practice is as follows:
Keep adding terms to the model until the curve's "elbow" is encountered. The "elbow point" is that value in which there is a consistent, noticeably shallower slope (decrease) in the curve. Include all terms up to (and including) the elbow point (after all, each of these included terms decreased the residual standard deviation by a large amount). Exclude any terms after the elbow point since all such successive terms decreased the residual standard deviation so slowly that the terms were "not worth the complication of keeping".
From the residual standard deviation plot for the defective springs data, we note the following:
1. The residual standard deviation (rsd) for the "baseline" model
$$\hat{Y} = \bar{Y} = 71.25$$
is sres = 13.7.
2. As we add the next term, X1, the rsd drops nearly 7 units (from 13.7 to 6.6).
3. If we add the term X1*X3, the rsd drops another 3 units (from 6.6 to 3.4).
4. If we add the term X2, the rsd drops another 2 units (from 3.4 to 1.5).
5. When the term X3 is added, the reduction in the rsd (from about 1.5 to 1.3) is negligible.
6. Thereafter to the end, the total reduction in the rsd is from only 1.3 to 0.
In step 5, note that when we have effects of equal magnitude (the X3 effect is equal to the X1*X2 interaction effect), we prefer including a main effect before an interaction effect and a lower-order interaction effect before a higher-order interaction effect.
In this case, the "kink" in the residual standard deviation curve is at the X2 term. Prior to that, all added terms (including X2) reduced the rsd by a large amount (7, then 3, then 2). After the addition of X2, the reduction in the rsd was small (all less than 1): 0.2, then 0.8, then 0.5, then 0.
The final recommended model in this case thus involves p = 4 terms:
1. the average
2. factor X1
3. the X1*X3 interaction
4. factor X2
The fitted model thus takes on the form
$$\hat{Y} = \bar{Y} + B_{1}X_{1} + B_{13}X_{1}X_{3} + B_{2}X_{2}$$
The least-squares estimates for the coefficients in this model are
$$\hat{Y}$$ = 71.25
B1 = 11.5
B13 = 5
B2 = -2.5
The B1 = 11.5, B13 = 5, and B2 = -2.5 least-squares values are, of course, one half of the estimated effects E1 = 23, E13 = 10, and E2 = -5. Effects, calculated as $$\hat{Y}$$(+1) - $$\hat{Y}$$(-1), were previously derived in step 7 of the recommended 10-step DOE analysis procedure.
The final fitted model is thus
$$\hat{Y} = 71.25 + 11.5 X_{1} + 5 X_{1}X_{3} - 2.5 X_{2}$$
Applying this prediction equation to the 8 design points yields: predicted values $$\hat{Y}$$ that are close to the data Y, and residuals (Res = Y - $$\hat{Y}$$) that are close to zero:
X1 X2 X3 Y $$\hat{Y}$$ Res
- - - 67 67.25 -0.25
+ - - 79 80.25 -1.25
- + - 61 62.25 -1.25
+ + - 75 75.25 -0.25
- - + 59 57.25 +1.75
+ - + 90 90.25 -0.25
- + + 52 52.25 -0.25
+ + + 87 85.25 +1.75
Computing the residual standard deviation:
$$s_{res} = \sqrt{ \frac{\sum_{i=1}^{n}{r_{i}^{2}}} {n-p} }$$
with n = 8 data points, and p = 4 estimated coefficients (including the average) yields
sres = 1.54 (or 1.5 if rounded to 1 decimal place)
The detailed sres = 1.54 calculation brings us full circle, for 1.54 is the value given above the X3 term on the cumulative residual standard deviation plot.
Conclusions for the defective springs data The application of the Cumulative Residual Standard Deviation Plot to the defective springs data set results in the following conclusions:
1. Good-fitting Parsimonious (constant + 3 terms) Model:
$$\hat{Y} = 71.25 + 11.5 X_{1} + 5 X_{1}X_{3} - 2.5 X_{2}$$
2. Residual Standard Deviation for this Model (as a measure of the goodness-of-fit for the model):
sres = 1.54
|
2017-10-17 14:47:28
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7206854224205017, "perplexity": 1471.0905611912378}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187822116.0/warc/CC-MAIN-20171017144041-20171017164041-00196.warc.gz"}
|
http://mathhelpforum.com/advanced-algebra/36407-linear-transformation.html
|
# Math Help - linear transformation
1. ## linear transformation
How do I find the matrix A to the linear transformation T: R^3 --> R^3
it's defined by;
1. reflection against 3x - 6y + 5z = 0
then
2. projection onto 2x + 6y + 4z = 0
2. Here are the formulas that you need (I won't do the actual question for you).
Suppose that $\mathbf{n}=(a,b,c)$ is a unit vector (so that $a^2+b^2+c^2=1$). Then the projection onto the one-dimensional subspace spanned by n is $P_{\mathbf{n}} = \begin{bmatrix}a^2&ab&ac\\ab&b^2&bc\\ac&bc&c^2\end {bmatrix}$.
If px + qy + rz = 0 is the equation of a plane, let n be a unit vector orthogonal to the plane. So $\textstyle\mathbf{n} = \frac1{\sqrt{p^2+q^2+r^2}}(p,q,r)$. Then the matrix of the projection onto the plane is $I-P_{\mathbf{n}}$, and the matrix of the reflection in the plane is $I-2P_{\mathbf{n}}$.
To find the matrix for the composition of two such operations, form the matrices for each operation, then multiply them. So the matrix for reflection in 3x - 6y + 5z = 0 followed by projection onto 2x + 6y + 4z = 0 is $(I-P_{\mathbf{n}})(I-2P_{\mathbf{m}})$, where m and n are the normalised versions of (3,-6,5) and (2,6,4) respectively.
3. I keep messing this one up, I've done it ten times and I still get the wrong answer.. Help?
4. Unless I've also messed it up, you should get
$\textstyle\mathbf{m} = \frac1{\sqrt{70}}(3,-6,5),\quad \mathbf{n} = \frac1{\sqrt{56}}(2,6,4) = \frac1{\sqrt{14}}(1,3,2)$,
$P_\mathbf{m} = \frac1{70}\begin{bmatrix}9&-18&15\\ -18&36&-30\\ 15&-30&25\end{bmatrix},\qquad P_\mathbf{n} = \frac1{14}\begin{bmatrix}1&3&2\\ 3&9&6\\ 2&6&4\end{bmatrix}$,
$I-2P_\mathbf{m} = \frac1{35}\begin{bmatrix}26&18&-15\\ 18&-1&30\\ -15&30&10\end{bmatrix},\qquad I-P_\mathbf{n} = \frac1{14}\begin{bmatrix}1&3&2\\ 3&9&6\\ 2&6&4\end{bmatrix}$.
So the answer should be $\frac1{35\times14}\begin{bmatrix}26&18&-15\\ 18&-1&30\\ -15&30&10\end{bmatrix}\begin{bmatrix}1&3&2\\ 3&9&6\\ 2&6&4\end{bmatrix}$ (I'm not prepared to do the arithmetic to evaluate that).
5. Originally Posted by Opalg
Unless I've also messed it up, you should get
$\textstyle\mathbf{m} = \frac1{\sqrt{70}}(3,-6,5),\quad \mathbf{n} = \frac1{\sqrt{56}}(2,6,4) = \frac1{\sqrt{14}}(1,3,2)$,
$P_\mathbf{m} = \frac1{70}\begin{bmatrix}9&-18&15\\ -18&36&-30\\ 15&-30&25\end{bmatrix},\qquad P_\mathbf{n} = \frac1{14}\begin{bmatrix}1&3&2\\ 3&9&6\\ 2&6&4\end{bmatrix}$,
$I-2P_\mathbf{m} = \frac1{35}\begin{bmatrix}26&18&-15\\ 18&-1&30\\ -15&30&10\end{bmatrix},\qquad I-P_\mathbf{n} = \frac1{14}\begin{bmatrix}1&3&2\\ 3&9&6\\ 2&6&4\end{bmatrix}$.
So the answer should be $\frac1{35\times14}\begin{bmatrix}26&18&-15\\ 18&-1&30\\ -15&30&10\end{bmatrix}\begin{bmatrix}1&3&2\\ 3&9&6\\ 2&6&4\end{bmatrix}$ (I'm not prepared to do the arithmetic to evaluate that).
Sorry.. wrong.. doh!
|
2014-03-08 11:32:17
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9156375527381897, "perplexity": 236.3406872717161}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999654390/warc/CC-MAIN-20140305060734-00038-ip-10-183-142-35.ec2.internal.warc.gz"}
|
https://math.stackexchange.com/questions/627303/using-the-levi-civita-alternating-tensor-and-suffix-notation-to-concisely-write
|
# Using the Levi-Civita alternating tensor and suffix notation to concisely write the vector product rule.
I am reading through a section on vector calculus in an electromagnetism book and it has started to use suffix notation and the Levi-Civita alternating tensor in order to prove some identities. Some of the identities I am familiar with and others I am not. The notation is new to me as are the concept of tensors and I am struggling to apply both to do things which I can already do.
As an example it is stated in the book that it is much simpler and more concise to write the product rule as
$$\left(\mathbf{A}\times\mathbf{B}\right)_i=\epsilon_{ijk}A_jB_k$$
and I know to work out the product rule I can use the determinant formula. What I am having trouble formulating in my head is how to actually read the above definition and get the correct expansion for the vector-product. So to make sure I understood the notation I expanded it out on paper but ended up getting the wrong answer. My interpretations and assumptions are as below:
$$\left(\mathbf{A}\times\mathbf{B}\right)_i = \epsilon_{ijk}A_jB_k$$ $$= \epsilon_{ijk}A_yB_z\mathbf{i} + \epsilon_{jki}A_zB_x\mathbf{j} + \epsilon_{kij}A_xB_y\mathbf{k} + \epsilon_{jik}A_yB_z\mathbf{j} + \epsilon_{kji}A_zB_x\mathbf{k} + \epsilon_{ikj}A_xB_y\mathbf{i}$$
From here it can be seen that the first three terms are correct, but the last three are not. I don't understand how the notation links back to the correct unit vector. Here I have just take the first subscript letter in $\epsilon_{ijk}$ to also represent the relevant unit vector. So I could really from getting the first three terms correct using the notation correctly write out the last three terms but that is just because I know what it should be. What I do not understand is how the subscripts of the tensor link up to the subscripts of the two vectors. I hope that is clear, if it isn't please leave a comment and I will try to remove anything confusing.
• What you written down is wrong. Note in suffix notation, you sum things which appear twice. So you sum the right hand for j = 1 to 3 and k = 1 to 3. There is no y or z... – Lost1 Jan 4 '14 at 21:56
• The book I am using introduced it as $A_i$ represents the summation of $i$ from 1 to 3 and also representing the vector $A_i=A_x+A_y+A_z$. I had took it to mean that when $i=1$ it referred to the $x$ component of $\mathbf{A}$. So for example the dot product could be written as $\mathbf{A}.\mathbf{B}=A_iB_i$. – Aesir Jan 5 '14 at 11:56
• So xyz are basically 123. Okay but what you wrote down is still wrong... A_i is not the sum of those things... You only sum things when there are two of the same index.... – Lost1 Jan 5 '14 at 11:58
• Here you need to replace the levi civita symbol with the appropriate 1 or -1... From its definitions. – Lost1 Jan 5 '14 at 12:01
• Also, to add to Ivo Terek's answer, you can write the vector resulting from the cross product itself as $\epsilon_{ijk} A_j B_k \mathbf{\hat{e}_i}$. That form is useful if you want to perform further vector operations, eg, $A \times B \cdot C$. – user_of_math Jul 11 '14 at 4:17
Actually, when you write $(A \times B)_i$, this is no longer a vector, but just one of its components. The vector itself is $$(A \times B) = ((A \times B)_1, (A \times B)_2, (A \times B)_3)$$ Having this in mind, the expression $$(A \times B)_i = \epsilon_{ijk}A_j B_k$$ means three equations, for $i$ ranges from $1$ to three. The equations are: $$(A \times B)_1 = \epsilon_{1jk}A_j B_k \\ (A \times B)_2 = \epsilon_{2jk}A_j B_k \\ (A \times B)_3 = \epsilon_{3jk}A_j B_k$$ For a better understanding of it, let's write everything for $(A \times B)_1$, say. I'll begin with the sum in $j$. So: \begin{align} (A \times B)_1 &= \epsilon_{\color {red}{11}k}A_1B_k + \epsilon_{12k}A_2B_k + \epsilon_{13k}A_3B_k \\ &= \epsilon_{12k} A_2B_k + \epsilon_{13k} A_3 B_k \end{align} Notice that in the indices, repeats are zeros. Now, let's do the sum on $k$: \begin{align} (A \times B)_1 &= \epsilon_{\color{red}{1}2\color{red}{1}}A_2 B_1 + \epsilon_{1\color {red}{22}}A_2B_2 + \epsilon_{123}A_2B_3 + \epsilon_{\color{red}{1}3\color{red}{1}}A_3B_1 + \epsilon_{132}A_3B_2 + \epsilon_{1\color{red}{33}}A_3B_3 \\ &= \epsilon_{123}A_2B_3 + \epsilon_{132}A_3B_2 \\ &= A_2B_3 - A_3B_2 \end{align}
You can do the same for the other ones, to get used to it. Also, I reccomend getting a few basic properties of the cross product, and prove them using only this notation. I once asked a question related to it, you might find it helpful.
Lost 1 is correct what you wrote is totally absurd....it is utter nonsense. Recall :
$$\epsilon_{ijk} = -\epsilon_{ikj} = -\epsilon_{jki} = -\epsilon_{kji}.$$
We realize that product vanishes, i.e. it becomes zero....which I must say it's wrong for we don't have enough info to justify that.
So we rewrite the product as follows.
$$C_x = (\vec{A} \times \vec{B}) = \epsilon_{ijk} A_j B_k = \epsilon_{xyz} A_y B_z + \epsilon_{xzy} A_z B_y$$ I have ignored the other expressions since they lead to zero, i.e.
$$\epsilon_{yzx} = \epsilon_{yxz} = \epsilon_{zyx} =\epsilon_{zxy} = 0,$$
since in our expression we have $C_x$ and for the other components:
$$C_y = \epsilon_{yzx} - \epsilon_{yxz} \text{ and } C_z = \epsilon_{zxy} - \epsilon_{zyx}.$$
|
2020-02-24 03:22:23
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9950025677680969, "perplexity": 273.34991667041254}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145869.83/warc/CC-MAIN-20200224010150-20200224040150-00356.warc.gz"}
|
https://astrophytheory.com/2018/11/16/basics-of-tensor-calculus-general-relativitya-digression-into-special-relativity/
|
# Basics of Tensor Calculus & General Relativity|A Digression into Special Relativity
So far in this series I have given the definitions of vectors, scalars, tensors, and manifolds. As a result, much of this series has been mostly mathematics and not necessarily physics. To that end, the purpose of this post is to develop the salient points of special relativity. Namely, the intention of this post is to cover the following:
1. Definition of Inertial Reference Frames: Standard Configuration and Einstein’s Postulates.
2. Development of the Lorentz Transformation Matrix
3. Discussion of the Newtonian geometry of spacetime
4. Discussion of the Minkowski geometry of spacetime (i.e. no curvature)
5. Finally I will show that the quantity $\delta s^{2}$ is invariant with respect to Lorentz transformations. This is a pretty standard problem in most GR textbooks and in fact in some introductory books on SR.
This post is meant as a “quick recap” of the main features of SR and is by no means comprehensive. For more of the finer details, consult the following resources
1. Hobson, M.P., Efstathiou G., and Lasenby, A.N., General Relativity: An Introduction for Physicists. 2009. Cambridge University Press. The reference text for this series.
2. Misner, Wheeler, & Thorne’s Gravitation. Princeton University Press. 1975. This is probably one of the most comprehensive texts on relativity. It is the book to have if you really want to understand relativity. However, if you prefer a concise writing style, then this book will not be for you (very verbose, but very interesting to read).
3. Collier P., A Most Incomprehensible Thing: Notes towards a *very* gentle introduction to mathematics of relativity. 2014. This book is ideal to introduce the foreboding topics of relativity, tensor calculus, and differential geometry. This book takes the reader through a quick primer of the mathematics required to understand the latter topics; from basic equations to multivariable calculus.
Featured Image: Image Credit: Harold White. **I understand that the derived metric describing the contraction and expansion in front and behind of the craft may be inaccurate. I just thought that this would be an interesting concept to think about**
The featured image of this post shows what spacetime would be like if an engine would look like if it were to be created. One of the most well-known ideas in modern physics is that there exists a cosmic speed limit, the speed of light in a vacuum, $c = 2.99\times 10^{8} ms^{-1}$. This engine would allow the persons on board to get around this rule by moving spacetime around the craft, instead of themselves moving. The aft portion would contract the spacetime in front of the craft and the stern section expands the spacetime behind, resulting in something that resembles the image above.
However, such an engine requires an exotic form of matter. Something that is able to be synthesized theoretically yet would be practically insurmountable in cost. For those of you who are interested for more details, the paper that derives the required metrics can be found at:
https://arxiv.org/abs/gr-qc/0009013
Defining an Inertial Reference Frame:
I. Inertial Frames:
Consider a reference frame $S$. Such a frame is termed an inertial frame if Newton’s first law holds. As an example, suppose I am on a train moving at a constant velocity with respect to an observer. If I were place a ball on the floor of the train-car that I am in, the ball would remain at rest unless I impress a force on it. Therefore, the train-car is an inertial reference frame. As an example of a frame that would not be inertial, let’s say that I was on a roundabout. If I were to place the ball on the platform while the roundabout is rotating, the ball would move outwards. Since Newton’s first law is invalid in this reference frame, it is termed a non-inertial reference frame. In other words, an inertial reference frame exists in the absence of boosts (or accelerations).
II. Standard Configuration:
Let $S$ and $S^{\prime}$ be two inertial reference frames in which $S^{\prime}$ is moving at a constant velocity with respect to $S$. The figure below is what is meant by “standard configuration”:
Fig.1 Standard Configuration. Frames $S$ and $S^{\prime}$ are in standard configuration. The two frames are such that their origins $O$ and $O^{\prime}$ (not shown above) are coincident at $t = t^{\prime} = 0$. Image Credit: http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/eddie_trochim/Lorentztransform.htm
III. Einstein’s Postulates:
There are two ideas that Einstein assumed when developing his theory of special relativity: the principle of relativity and the constancy of the speed of light. The first of which states that the laws of physics are the same in all inertial reference frames. In a more technical sense, dimensions perpendicular to the direction of motion of a given inertial frame remain unchanged. To be more precise, if we have the two frames $S$ and $S^{\prime}$ in standard configuration, one of which we shall assume to be moving with constant velocity in the $+x$ direction, then the dimensions $y$ and $z$ remain unchanged under a coordinate transformation.
The other postulate that Einstein put forth was that the speed at which light propagates in a vacuum is invariant. Contrary to Newton, Einstein said that the speed of light remained the same and it was space and time that changed. I will talk more about this when I talk about the Newtonian and Minkowski geometries.
Development of the Lorentz Transformation Matrix:
Consider two inertial frames $S$ and $S^{\prime}$ in standard configuration. Suppose there exists an event P which we define using the coordinates $x^{\mu}$ in frame $S$. Suppose we wish to determine the coordinates of this event in terms of coordinates $x^{\prime\mu}$ in $S^{\prime}$. Then we may relate the coordinates of the two frames via the following system
$\displaystyle x^{\prime 0}=\Lambda_{00}x^{0}+\Lambda_{01}x^{1}+\Lambda_{02}x^{2}+\Lambda_{03}x^{3},$
$\displaystyle x^{\prime 1}= \Lambda_{10}x^{0}+\Lambda_{11}x^{1}+\Lambda_{12}x^{2}+\Lambda_{13}x^{3},$
$\displaystyle x^{\prime 2}= \Lambda_{20}x^{0}+\Lambda_{21}x^{1}+\Lambda_{22}x^{2}+\Lambda_{23}x^{3},$
$\displaystyle x^{\prime 3}=\Lambda_{30}x^{0}+\Lambda_{31}x^{1}+\Lambda_{32}x^{2}+\Lambda_{33}x^{3}, (1)$
where $x^{0}= t, x^{1} = x, x^{2} = y, x^{3} = z$. We can write this more succinctly as
$\displaystyle x^{\prime \mu}=\sum_{\mu}\Lambda_{\mu\nu}x^{\mu} (2),$
or as
$\displaystyle x^{\prime \mu}=\Lambda_{\mu\nu}x^{\mu} (3),$
where we have made use of the Einstein summation convention in which it is implied that we sum over repeated indices. The term $\Lambda_{\mu\nu}$ corresponds to the coefficient matrix that can be constructed from Eqs.(1) and is given by
$\displaystyle \Lambda_{\mu\nu}= \begin{pmatrix} \Lambda_{00} & \Lambda_{01} & \Lambda_{02} & \Lambda_{03} \\ \Lambda_{10} & \Lambda_{11} & \Lambda_{12} & \Lambda_{13} \\ \Lambda_{20} & \Lambda_{21} & \Lambda_{22} & \Lambda_{23}\\ \Lambda_{30} & \Lambda_{31} & \Lambda_{32}& \Lambda_{33} \\ \end{pmatrix}. (4)$
This is the Lorentz transformation matrix. We may also write $x^{\prime\mu}$ and $x^{\mu}$ as column vectors and write the matrix equation Eq.(3).
Discussion of the Newtonian Geometry of Spacetime:
NOTE: In this section, and in the next section, I will be stating the transformation equations. I will not be deriving them since I believe that this exercise is more enlightening when accomplished independently.
In 1687, at the recommendation of astronomer Edmond Halley, Newton published the first edition of the Principia. In those three volumes, Newton set forth the laws of Nature regarding motion, gravitation, and his independent discovery of calculus (a debated topic that I will not be talking about; however I do acknowledge the tremendous contributions that Leibniz made to the development of modern-day calculus). In the Newtonian realm, space and time are regarded as absolute. As a result, such an absolution requires that the velocity with which an object travels be subject to change. To relate this to relativity, consider the following example:
Suppose my buddy and I (because we were bored and we love physics) decide to measure how long it takes a train car to travel the length of the platform of the train station. Suppose further that I am observing from the platform and my buddy observes from the train. Once the train begins to move we both signal to each other to start our observations. According to Newton, because time and space are absolute, both my buddy and I record the same time. Suppose my reference frame is $S$ and my buddy’s reference frame is $S^{\prime}$. The event that I measure can be transformed into my buddy’s reference frame via
$\displaystyle t^{\prime}= t,$
$\displaystyle x^{\prime}=x-vt,$
$\displaystyle y^{\prime}=y,$
$\displaystyle z^{\prime}=z. (4)$
These equations constitute the Galilean transformation equations wherein the second equation corresponds to our everyday experience of motion. The time equation here tells us that time remains invariant under such a transformation. The quantities $\delta t$ and $\delta r^{2}$ comprise the Newtonian geometry of spacetime. The latter of which is what is known as a metric. A metric in this context corresponds to a distance of sorts between two events $P$ and $P^{\prime}$. As an example, in Cartesian coordinates the metric is given by,
$\displaystyle ds^{2}=dx^{2}-dy^{2}-dz^{2}. (5)$
We may represent a metric in other coordinate systems as well. A future post will discuss in detail metrics and the metric tensor $g_{\mu\nu}$.
Discussion of the Minkowski Geometry of Spacetime:
The Newtonian geometry of space and time, namely the assumptions of absolute space and absolute time, stood as the prevailing theory for quite some time. That is, until Einstein came along. Einstein’s interpretation of space and time came from taking Maxwell’s equations and deriving the following equations
$\displaystyle \mu_{0}\epsilon_{0}\frac{\partial^{2}E}{\partial t^{2}}= \nabla^{2}E, (6)$
$\displaystyle \mu_{0}\epsilon_{0}\frac{\partial^{2}B}{\partial t^{2}}= \nabla^{2}B. (7)$
These are the electromagnetic wave equations in which the wave speed is of the form
$\displaystyle c=\frac{1}{\sqrt[]{\mu_{0}\epsilon_{0}}}, (8)$
the speed of light in a vacuum. Einstein saw the speed of light within Maxwell’s equations and postulated that the speed of light is the speed beyond which no object can travel. It was on this and the postulate of relativity that Einstein based his theory of special relativity. At the heart of it all, one can derive (from Eqs.(1)) the Lorentz transformation equations
$\displaystyle ct^{\prime}= \gamma (ct-\alpha x), (9.1)$
$\displaystyle x^{\prime}=\gamma(x-\alpha ct), (9.2)$
$\displaystyle y^{\prime}=y, (9.3)$
$\displaystyle z^{\prime}=z. (9.4)$
In these equations, $\alpha \equiv v^{2}/c^{2}$ and $\gamma \equiv 1/\sqrt[]{1-\alpha}$ is called the Lorentz factor. If we graph this quantity as a function of $\alpha$ we get the following figure (link to a document, I couldn’t find a way to upload an excel graph): lorentz factor graph_2. This term represents a relativistic correction of sorts in the above equations.
Upon comparison of the two types of transformation equations (i.e. the Galilean and Lorentz equations) one sees that in the former space and time can be shown to be two entirely different constructs. While the latter shows that space and time must be considered as a unified entity. The problem below shows that (see below) that the interval
$\displaystyle \delta s^{2}= c^{2}\delta t^{2}- \delta x^{2}- \delta y^{2} - \delta z^{2}, (10)$
remains invariant under a Lorentz transformation. In this case, the metric tensor $g_{\mu\nu}$ has the form
$\displaystyle g_{\mu\nu}= \begin{pmatrix} c^{2} & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}, (11)$
From this it follows that the interval may be written as
$\displaystyle ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}. (12)$
I will discuss this further in a future post. Since the components of the metric tensor are of this form, we may refer to this space as Minkowskian or flat space.
Problems on Invariance under Transformations: $\delta s^{2}$
A typical problem that is covered in most relativity texts is to show that the interval remains invariant under a Lorentz transformation. The following is my solution to the problem. I highly encourage working through the problem on your own before seeking guidance and furthermore, I strongly recommend that you complete the problem prior to reading my solution.
Let us consider the interval
$\displaystyle \delta s^{\prime 2}=c^{\prime 2}\delta t^{\prime 2}-\delta x^{\prime 2}-\delta y^{\prime 2}-\delta z^{\prime 2}, (13)$
where I am assuming standard configuration of inertial frames and transforming from the $S^{\prime}$ frame. Let us rewrite this as
$\displaystyle \delta s^{\prime 2}=(c\delta t^{\prime})^{2}-\delta x^{\prime 2}, (14)$
where the last two components (the y and z components) remain the same automatically by the principle of relativity and hence vanish from the equation. Applying a Lorentz transformation, we get
$\displaystyle \delta s^{\prime 2}= [\gamma(ct_{B}-\alpha x_{B})-\gamma(ct_{A}-\alpha x_{A})]^{2}-[\gamma(x_{B}-\alpha c t_{B})-\gamma(x_{A}-\alpha c t_{A}]^{2}. (15)$
After some algebra, we end up with
$\displaystyle \delta s^{\prime 2}=\gamma^{2}c^{2}\delta t^{2}(1-\alpha)^{-1/2}-\gamma^{2} \delta x^{2}(-\alpha^{2}+1)^{-1/2}-\gamma^{2}\delta x^{2}+2\gamma^{2}\alpha c \delta x \delta t.$
In the algebraic steps implied here, we end up with space and time terms as measured in the unprimed inertial frame $S$. Furthermore, we form the differences in the x-spatial direction and time between the two events so as to form an interval in $S$. We can simplify this further to give the final interval equation
$\displaystyle \delta s^{\prime 2}= c^{2}\delta t^{2}-\delta x^{2}-\delta y^{2}-\delta z^{2} = \delta s^{2}. (16)$
Hence we see that the interval remains invariant under a Lorentz transformation. Minkowski was the one who noticed first that the Lorentz transformation equations show that space and time cannot be considered separately. This idea of spacetime as a four-dimensional entity (manifold, really) with no curvature is referred to as Minkowskian spacetime. It is also known mathematically as being pseudo-Euclidean.
Most of this post was my own understanding of special relativity augmented by the references listed above. The problem solved in the final section was obtained from [1]. I have not covered topics such as relativistic kinetic energy, time dilation, relativistic addition of velocities, relativistic momentum, length contraction, and the twin paradox. Depending on how this post does, I may work on a follow-up post on these topics. As mentioned above, I will be posting something about the metric tensor and more on interval equations using index notation. I am unsure as to when that will be up, but I will do my best to find time. If there are any errors or if my reasoning does not hold up anywhere, leave a comment and let me know and I’ll correct it.
Clear Skies!
|
2022-09-26 04:15:37
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 69, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8530226945877075, "perplexity": 265.93469821371525}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334644.42/warc/CC-MAIN-20220926020051-20220926050051-00336.warc.gz"}
|
https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-12-sequences-and-series-12-2-analyze-arithmetic-sequences-and-series-12-2-exercises-skill-practice-page-807/36
|
## Algebra 2 (1st Edition)
$a_n=\dfrac{111-13n}{5}$
We know that the general formula of an arithmetic sequence is given by $a_n= a_1+(n-1) d$ ...(1) Here, we have $a_7=a+7b=4$ ..(2) $a_{12}=a+12b=-9$ ..(3) Now, we will have to subtract equation (2) from (3). $5b =-13 \implies b=\dfrac{-13}{5}$ Equation (2) gives: $a_7=a+7(\dfrac{-13}{5})=4$ $a=\dfrac{111}{5}$ Thus, we find that the nth term is: $a_n=\dfrac{111-13n}{5}$
|
2022-12-04 09:17:27
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9817317724227905, "perplexity": 318.0912029103584}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710968.29/warc/CC-MAIN-20221204072040-20221204102040-00132.warc.gz"}
|
http://soft-matter.seas.harvard.edu/index.php?title=Many-Body_Electrostatic_Forces_Between_Colloidal_Particles_at_Vanishing_Ionic_Strength&diff=15714&oldid=15686&printable=yes
|
# Difference between revisions of "Many-Body Electrostatic Forces Between Colloidal Particles at Vanishing Ionic Strength"
Jason W. Merrill, Sunil K. Sainis, and Eric R. Dufresne
Physical Review Letters 103 (2009) 138301
wiki entry by Emily Russell, Fall 2010
The article can be found here.
## Overview
This paper reports a striking demonstration that effective pair potentials do not tell the full story in colloidal systems, and that furthermore, a constant surface potential is a better assumption than a constant surface charge, at least in some cases. The authors are able to observe many-body effects on the forces in systems of small numbers of colloidal particles, and use a simple Poisson-Boltzmann model to predict these effects from the pair potentials.
## Experiments
The experiments were carried out using 600nm-radius PMMA colloids in nonpolar hexadecane as the solvent. NaAOT, a surfactant, was added, which forms reverse micelles, increasing the particle charge, and decreasing the screening length.
The particles were positioned using optical tweezers. Three configurations were studied: pairs of particles; an equilateral triangle of particles (each pair of which had been measured previously); and a hexagonal arrangement of seven particles. The particles were released and tracked to determine their drift velocities; individual drift velocities were converted to the velocity of the breathing mode of the system, and the force on this breathing mode determined by $f = k_B T v_d / D$. (With a moment's thought, it is not obvious that this equation should apply for many-body modes; this is addressed in an earlier paper by the group, Statistics of Particle Trajectories at Short Time-Intervals Reveal fN-Scale Colloidal Forces.) The force was calculated for a range of particle separations.
The forces observed in isolated particle pairs were fit to obtain the surface potential and screening length. These parameters were then input to the linearized Poisson-Boltzmann equation, which was numerically solved with constant-potential boundary conditions to predict the forces in the triangular and hexagonal situations, taking into account the many-body effects. Forces were also predicted assuming only pairwise interactions, and these two predictions were compared to the measured forces.
## Results
The results are very nicely summed up in the main figure of the paper, Fig. 1.
Fig. 1. Direct measurement of nonpairwise electrostatic interactions.—Forces on beads in pair (first column), equilateral (second column), and hexagonal (third column) configurations, at AOT concentrations of 10 (first row) and 0.5 mM (second row). The arrows on the particle configurations in the first row indicate the form of the breathing modes used for analysis. Breathing modes are normalized so that the sum of the squares of the particle displacements is unity. For the pair measurements, different colored points represent different pairs in the same sample. The dotted, solid, and dashed lines are fits to constant charge density, constant potential, and a simple approximation of constant potential based on Eq. (3), respectively. For the equilateral and hexagonal configurations, black points are measured forces on the breathing mode, and red points are a direct pairwise sum of the measured pair forces. The red lines are pairwise sums of the constant potential pair fits. Constant potential predictions for the force on the breathing mode based on fits to the pair data are shown as black lines. The solid line is based on the full numerical solution, while the dashed line is based on Eq. (3).
At short screening length ($\kappa a = 0.58$, high surfactant concentrations), the pairwise and many-body predictions are similar; for the hexagonal configuration, only a small deviation is observed from the pairwise prediction. The dramatic results are obvious in the measurements at long screening length ($\kappa a = 0.14$, low surfactant concentration), where the measurements are fit very well by the calculations taking into account many-body effects, the forces being substantially smaller in the triangular and hexagonal configurations than a naive pairwise calculation would predict. The authors emphasize that the potential and screening length obtained from the fits to the pair data are the only parameters input to the calculations for the triangular and hexagonal configurations.
It turns out that it is not the non-linearity of the Poisson-Boltzmann equation itself which makes many-body effects important in this system; indeed, the authors find that the linearized PB equation is sufficient to predict the forces. Instead, the surface charges of the particles change depending on the presence of other particles in order to maintain a constant surface charge. The pairwise prediction would be appropriate only if the surface charges themselves remained constant. A simple model of the particles as a central point charge, whose magnitude changes to maintain a constant potential at the particle surface, gives results close to the full numerical solutions.
The authors reference the work of Verwey and Overbeek, who first suggested that constant surface potential may be a more appropriate assumption for colloidal particles than constant surface charge, based on an equilibrium between the surface and bulk concentrations of ions. They also point out that the majority of models in colloids have used constant surface charge boundary conditions, so that some models may need to be revisited.
## Discussion
If there is a weakness in this paper, it is that the experiments were only performed once - repetition with different particles to fill out a few more data points would have made the data even more convincing (but let's not tell Jason I said that). I also found it odd that they did not discuss the screening lengths found for the two surfactant concentrations; experiments exploring a range of screening lengths and indicating more precisely at what point many-body interactions become important would have been informative. Otherwise, the paper is written clearly, reporting experiments which are easy to understand and using straightforward but informative models, and arguing well that particle interactions are modulated by the proximity of other charged particles.
Of course, these results might not be so damaging to the world of colloid research using pairwise models as it at first seems; the results are only significant when the screening length is long compared to the particle separation. In aqueous solutions, dominated by short screening lengths, the many-body effects are much less significant.
|
2020-08-04 19:38:07
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7359164953231812, "perplexity": 818.0841705223617}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735882.86/warc/CC-MAIN-20200804191142-20200804221142-00598.warc.gz"}
|
http://reffit.com/view_discussion/?paperid=-1935392373&commentid=12853217
|
Discussion of imalsogreg's comment: Draws on a finding from a [Corey and Hudspeth paper](http://corey.med.harvard.edu/PDFs/1979%20Corey%20Hudspeth%20ionic.pdf) that the reversal potential of mechanotransduction current is different from the Nearnst equallibrium for $Na ^+$, and argues that this is incompatible with the idea that mechanotransduction achieved by ion channels.
|
2017-06-26 02:04:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7606728672981262, "perplexity": 3388.8054463475205}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320666.34/warc/CC-MAIN-20170626013946-20170626033946-00708.warc.gz"}
|
http://advancedintegrals.com/category/dilogarithm/
|
# Category Archives: Dilogarithm
## Special values of the dilgoarithm function
Prove that $$\mathrm{Li}_2\left( \frac{\sqrt{5}-1}{2} \right) = \frac{\pi^2}{10} – \log^2 \left( \frac{\sqrt{5}-1}{2}\right)$$ $$\textit{proof}$$ Use the following functional equation $$\mathrm{Li}_2 \left( \frac{z}{z-1} \right) + \frac{1}{2} \mathrm{Li}_2 (z^2) – \mathrm{Li}_2(-z) = -\frac{1}{2} \log^2 (1-z)$$ These are proved here and here Now let … Continue reading
## Dilogarithm functional equation proof
$$\mathrm{Li}_2(z) + \mathrm{Li}_{2}(1-z) = \frac{\pi^2}{6}-\log(z) \log(1-z) \,\,\,\, ,\,0<z<1$$ $$\textit{proof}$$ Start by the following $$\mathrm{Li}_2\left(z\right) = -\int^{z}_0 \frac{\log(1-t)}{t} \, dt$$ Now integrate by parts to obtain $$\mathrm{Li}_2\left(z\right)= -\int^z_0 \frac{\log(t)}{1-t} \, dt -\log(z) \log(1-z)$$ By the change of variable \(t=1-x … Continue reading
Posted in Dilogarithm, Polylogarithm | | 1 Comment
|
2019-02-22 01:48:50
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8417810797691345, "perplexity": 2132.859736224277}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247512461.73/warc/CC-MAIN-20190222013546-20190222035546-00056.warc.gz"}
|
https://www.nature.com/articles/s41598-020-76713-y?error=cookies_not_supported&code=1520b79a-1e5c-4a94-8fe9-fbb51029daee
|
## Introduction
Climate change is one of the most important concerns and challenges for scientists, managers, and decision makers throughout the world1,2. The agriculture appears to be one of the human activities most vulnerable to climate changes due to its large dependence on environmental conditions2,3. For coffee producers, climate change also is a huge challenge, since the drought is the main environmental restriction that affects coffee growth and production4,5, reducing the yield up to 80% in very dry years in some marginal regions with no irrigation5. In addition, coffee has been categorized as a highly sensitive plant species to progressive climate change6.
A recent study of Magrach et al.7 suggest that many currently Coffea canephora Pierre ex Froehner cultivated areas will become less suitable for cultivation with the impact of the climate change (projections of rising temperatures and altered precipitation patterns). According to the authors, this species could lose 55% of currently suitable areas, mostly within western Africa and southeast Brazil. This is very alarming because the Espírito Santo state, located in the southeast Brazil, is the largest producer of Coffea canephora in the country, responsible for 70% of its production and approximately 15% of the global production, despite of occupying only 0.55% of the Brazilian territory8,9. In economic terms, C. canephora is responsible for 35% of the gross domestic product (GDP) of Espírito Santo and for the generation of 250,000 direct and indirect jobs, which makes it considered the main agricultural product of the state9. Thus, the occurrence of climate-driven problems, such as drought and high temperatures, can threaten sustainability of that activity in this region, given its impacts on coffee production.
Previous studies on the effect of water deficit on growth, photosynthesis and carbon metabolism of coffee plants have provided important insights, specially, the susceptibility of these plants to climate change. Pinheiro et al.10 studied four genotypes of conilon coffee, two drought-tolerant and two sensitive genotypes, which were submitted to a slowly imposed water deficit. The authors verified that, regardless of the genotypes clones investigated, the net carbon assimilation rate decreased under drought stress. Thus, even for tolerant clones, the water deficit can affect the growth, photosynthesis and, consequently, yield. In a study with the same clones and same treatments used by the previous authors, Praxedes et al.11 verified that, regardless of the stress severity, decrease in reserve accumulation remarkably and, regardless of the clone evaluated, drought led to sharper decreases in stomatal conductance and, therefore, to photosynthetic capacity decrease. On the other hand, drought-tolerant genotypes can increase the long-term water use efficiency (WUE)12. In any case, greater WUE associated with decreases in stomatal conductance generally lead to lower rates of transpiration. However, this also results in less latent heat loss, potentially increasing leaf temperatures, which in turn can harm the crop performance in a global warming scenario6.
In the Espírito Santo state, the region of interest of this study, drought years are associated to low rainfall during the winter and fall seasons (from April to September). This period also matches the time of the coffee harvest, and droughts tend to compromise the crops vigor13. In general, drought periods can lead to plant death, while moderate drought periods are also very damaging to coffee growers by affecting flowering, bean development, and, consequently, coffee production14. Therefore, the use of irrigation is essential to guarantee adequate crop yields.
In most situations, drought impacts can be greatly aggravated by supra-optimal air temperatures15. Extreme temperatures, depending on their intensity, duration, and speed of imposition, impair cell metabolic processes (e.g., photosynthesis)16,17, growth and survival of plants, as well as their economic exploitation16. In the region of study, it is common the occurrence of summer temperatures reaching up to 40 °C. Exceeding this level during the phase of grain filling is critical and will lead to grains wilting and, consequently, significant decreases in the crops yield13. During blossoming, especially if associated with a prolonged dry season, high air temperature may cause abortion of flowers, directly impacting the production17. In general terms, high temperature are harmful for coffee production and, the attenuation of the incident solar radiation, the temperature, and the evaporative demand (e.g., using a agroforestry system) can result in better conditions for the maintenance of the gas exchanges with positive effects on the production, especially in marginal regions where coffee cultivation is characterized for suffering with water deficit associated with extreme temperatures and excess irradiance13.
Under drought conditions, agroforestry systems can bring several positive benefits to the coffee plantations, as they have been reported to mitigate the variability in microclimate18,19,20, and to promote increases soil moisture21,22 and in water infiltration in the soil23. Besides that, agroforestry systems help reduce the maximum air temperatures19,24 and the intensity of photosynthetically active radiation19. According to findings of Gidey et al.25, the coffee production under agroforestry systems has a higher level of resilience when facing future climate change and reinforces the idea of agroforestry systems help adapt to the negative impacts of climate change on the coffee production.
The state of Espírito Santo harvested the largest harvest in history in 2013/2014 season26, however in the two subsequent seasons the climatic conditions imposed significant limitations on the productive capacity of the crops, which harmed the regional coffee production and may be an indicative of what to expect from climate change in the near future. Despite the aforementioned studies, there is no information on the conilon coffee responses to drought and high temperature at the regional scale for the Espírito Santo state. This study, by integrating large datasets, is an effort to fill this gap. Thus, the objective of this study was to analyze the variation in the production and yield of the conilon coffee cultivated in the north, northeast, and northwest regions of Espírito Santo, Brazil, based on data of rainfall, air temperature, and satellite-based optical vegetation index. Specifically, the study aims to identify which period of the crop cycle is most sensitive to the stressful environmental conditions frequently seen in the region (water deficit associated with high temperatures and irradiances). This study also provides information for the decision-making related to the sustainability of the coffee sector, especially, in the Espírito Santo state, where Conilon coffee is considered the main agricultural product and may be jeopardized by climate change.
The subsequent sections of this manuscript are organized as follow: “Methods”, where we describe the (1) area of study, (2) data acquisition (monthly rainfall and air temperature, production, yield, planted area and satellite-based optical vegetation index) for the main conilon coffee producing municipalities in the Espírito Santo state, Brazil, (3) data processing and (4) statistical approaches used herein; the “Results and discussion” was divided in five subsections as (1) climatic conditions during the study period, (2) relationship between drop in production and climate conditions, (3) combined effect of rainfall and air temperature on coffee production, (4) identification of which period of the crop cycle is most affected by the rainfall volumes, and (4) spatial–temporal drought impacts on crop production using a satellite-based optical vegetation index; and, in “Conclusions”, we present the overall conclusions of this research.
## Methods
### Study area
The study area involves the municipalities of the northern (Boa Esperança and Pinheiros), northeastern (Jaguaré, Linhares, Rio Bananal, São Mateus) and northwestern (Colatina, Nova Venécia and Vila Valério) regions of the state of Espírito Santo, Brazil (Fig. 1). These nine municipalities are part of the fourteen classified as major producers of coffee conilon in this state9. The plantations of conilon coffee in these municipality are clonal, formed by genotypes of high production capacity, which are recommended by the Capixaba Institute of Research, Technical Assistance and Rural Extension (INCAPER).
One of the main characteristics of these municipalities is the flat relief, which enables the mechanization of many agricultural practices and the use of different irrigation systems. The irrigation systems commonly used are those of localized application (dripper and micro-sprinkler) and by sprinkler (center pivot and conventional sprinkler), and the source of water for this purpose is mainly rivers and small reservoirs27. It is important to point out, which in these municipalities more than 90% of coffee plantations are irrigated9 . Most coffee areas in these three regions are considered small and medium-sized properties9. The predominant soil is Latossolo Vermelho-Amarelo (Oxisol)28.
### Data acquisition
In order to accurately correlate data of meteorological variables (rainfall and average air temperature) and surface variable (Enhanced Vegetation Index) with coffee production and yield fluctuations, data from different sources were used, namely: IBGE (Brazilian Institute of Geography and Statistics); CONAB (National Supply Company); EMBRAPA (Brazilian Agricultural Research Corporation) and AGRITEMPO (Agrometeorological Monitoring System). Figure 2 shows the flowchart with the source and the types of data acquired, which are also described in detail hereinafter.
### Data from IBGE, CONAB and AGRITEMPO
IBGE is one of the main providers of data and information in Brazil, meeting the needs of the most diverse segments of civil society, as well as the organs of the federal, state, and municipal government spheres. Data of production, yield, and planted area of the municipalities of the north, northeast and northwest of Espírito Santo (Fig. 1) for the agricultural years of 2010/2011, 2011/2012, 2012/2013, 2013/2014, 2014/2015 and 2015/2016 were used in the present study. Similarly to IBGE, CONAB conducts surveys and estimates of production, yield and planted area of the main agricultural crops in the country, but at the state and national levels. The joint use of data on a municipal (IBGE) and state/national (CONAB) scale is very important, as it enables a higher level of detail, besides being complementary. As for the data from IBGE, sequential data of production, yield and area planted of conilon coffee in the state of Espírito Santo from 2010/11 to 2015/16 were used.
Agritempo (https://www.agritempo.gov.br/agritempo/index.jsp) enables users to access meteorological and agrometeorological information from various Brazilian municipalities and states. The database of this platform comes from more than 1400 surface weather stations, conventional and automatic, stations distributed along the Brazilian territory29. In addition, the portal also has data from the Tropical Rainfall Measuring Mission (TRMM), a mission that was planned to estimate rainfall in tropical regions from a wide range of sensors30,31,32,33. The TRMM data in the Agritempo portal provides rainfall data of 11,332 grid points that are converted in the Agritempo system in the so-called virtual stations, with spatial resolution of 25 × 25 km and 30-day time resolution. On this platform, monthly data of rainfall (mm) and air temperature (°C) were acquired. Average air temperature data were only available for the municipalities of Linhares, Nova Venécia and São Mateus. Thus, in the results and discussion section, there will be only data referring to the municipalities mentioned above.
### EVI/MODIS Embrapa data
Enhanced vegetation index (EVI) images34 were acquired on the Embrapa MODIS website (https://www.modis.cnptia.embrapa.br). This information refers to a 16-day composition made available through the MOD13Q1 product, derived from the Moderate Resolution Imaging Spectroradiometer (MODIS) sensor on board the TERRA satellite. Embrapa MODIS aims to facilitate access to MODIS products made available by the Land Processes Distributed Active Center (LP-DAAC), providing users with ready-to-use products in state cutouts, in GeoTIFF format, in the WGS84 geographic coordinate reference system, with spatial resolution of 250 m. The MOD13Q1 product has processing level 3, which means the data are spatially resampled and temporarily composed. The temporal composition is made with pixels that contain the best possible observation over a period of 16 days based on wide observation coverage, low viewing angle, absence or shadow of clouds and aerosol35. A total of 144 images corresponding to the agricultural years 2010/2011, 2011/2012, 2012/2013, 2013/2014, 2014/2015, 2015/2016 were acquired.
Each of the 144 images was rescaled to a range between 0 and 1, since they are distributed in a range of values varying from 0 to 10,000. After this, average values of EVI were obtained for the periods in which the occurrence of prolonged water deficit (WD) is critical in the production of conilon coffee for each of the six seasons. Three periods (P) were considered according to Camargo and Camargo36, Laviola et al.37 and DaMatta et al38: P1—September to December (WD causes the problem of low sieve classification); P2—January to March (WD causes the problem of endosperm malformation) and P3—April to August (WD causes the problem of fruit drop), as shown in Fig. 3. The information contained in this figure is extremely important to understand the effects of rainfall on coffee production and yield.
### Data analysis
First, a general description of the data of rainfall, average air temperature, yield and planted area was made for each of the six seasons analyzed in order to better characterize each season in relation to these variables. Statistically, the first analysis was the application of the model identity test39 to check whether the variations in production data results from either the reduction of planted area or the rainfall volume and high air temperatures. To apply the test, null (H0) and alternative (H1) hypothesis were established. H0 means that the parameters of the equations fitted for the annual variation of production and planted area are identical, while H1 means that parameters of the equations fitted for the annual variation of production and planted area are different. Thus, rejecting H0 means that the reduction in production is not the result of the reduction in planted area. The hypotheses were tested by analysis of variance (ANOVA), at 5% probability level (p < 0.05). The results were presented using radar charts plus the respective probability level resulting from the identity test.
Scatter plots were used to identify the impact of variations in the rainfall volume during the season (September–August) on coffee yield. The same analysis was performed for the data of average air temperature. Subsequently, the combined effects of these factors on the production were evaluated by the response surface methodology. Then, principal component analysis was carried out, which made it possible to identify among the periods (P1, P2 and P3) which is the most influential on coffee yield. Finally, maps of the mean values of EVI were generated for the study area in the periods (P1, P2 and P3) for each of the six seasons, aiming to identify the effects of drought spatially and temporally on coffee yield.
## Results and discussion
### Climatic conditions
Figure 4 shows the average values of accumulated rainfall of the nine municipalities in different periods, for each of the six seasons analyzed. In all seasons the rainfall accumulated in the annual period (September–August) was above 823 mm, except for the 2015/2016 season, when the accumulated rainfall was only 549.3 mm. The mean of annual rainfall of six seasons was approximately 920 mm, which means that in 2015/2016 the rainfall was 40% below to mean of period, leading into a drop in production ~ 41% compared to the previous harvest. Considering that climatic conditions with rainfall of approximately 1200 mm, distributed between September and March are necessary for satisfactory development of conilon coffee40, the 2015/2016 season was well below what is required when rainfall is the source of water supply to plants. Moreover, the total rainfall volume in this season was much lower than historical averages, since the average minimum rainfall in the state of Espírito Santo is 1000 mm, with the highest averages (1400–1500 mm) observed in the mountainous region of the state and a downward trend in the northern region, where annual averages from 1050 to 1100 mm41 are observed.
It is also observed that in the 2015/2016 season the total rainfall volume (549.3 mm) was lower than the accumulated rainfall only in P1 of the 2010/2011 (692.0 mm) and 2013/2014 (682.3 mm) seasons and slightly higher than the values of the 2011/2012 (431.7 mm) and 2014/2015 (414.2 mm) seasons. In addition, during the period from April to August of the 2015/2016 season, it rained only 76.1 mm, while the average of the other seasons was 192.18 mm (Fig. 4). Regarding temporal distribution, the highest rainfalls are concentrated within P1 and P2, while in P3 the drought season prevails, a common behavior in the southeastern region of Brazil42,43.
The occurrence of water deficit in the April-August period (P3) favors the drop in coffee fruits, especially the smaller ones, as shown in Fig. 3, in addition to the fall of leaves. This is because the fruits, especially the larger ones, are the preferred sinks of photoassimilates during the reproductive period37 and, according to Lima et al.44, leaf fall due to water stress is a common response observed in conilon coffee clones. Although the occurrence of water deficit in the April-August period (P3) favors the drop in coffee fruits, especially the smaller ones, which is a serious problems, it is also important to highlight that a short dry period during the summer (e.g., January) is more harmful than a longer dry period during the winter. This is because in summer there is a combined effect of drought, heat, and irradiance on crop physiological performance. In the present study, these climatic variables occurred during summer in several years, which contributed to a reduction in both coffee growth and yield to a great extent.
Figure 5 shows the data of average annual air temperature for the periods from 2000 to 2016 (Linhares and São Mateus) and from 2011 to 2016 (Nova Venécia) and the overall mean of the period. In two harvest where low yields were verified (2012/2013 and 2015/2016) (Table 1), the average annual air temperatures were higher than in the previous seasons in approximately 1 °C. In addition, the highest average annual air temperature happened in the 2015/2016 season. It can be observed that from 2014, an abnormal increase in the average annual temperature (25.1 °C) began, reaching the average value of 26.1 °C in the municipality of Linhares in 2015 (Fig. 5). It is important to highlight that, even with a decrease compared to the previous year, 2016 had an average of 25.6 °C, which was higher than the average of the last 12 years. Considering the periods, the average air temperature was higher in P2 (27.3 °C), followed by P1 (25.2 °C) and P3 (23.8 °C).
Based on the study of Eugenio et al.47 on agroclimatic zoning, the range considered suitable for conilon coffee cultivation varies in Brazil from 22.5 to 24 °C, so the previously mentioned temperature values are above the range considered suitable. However, other studies consider ideal the range from 22 to 26 °C48,49. In globally terms, conilon coffee is considered productive up to 30 °C, with optimum production between 22 and 28 °C. Regarding to minimal temperature, problems start when temperatures drop below 10 °C and the tree dies at around 4–5°C50. However, little is known about the effects of thermal stress on the productive performance of C. canephora. The studies so far have focused on identifying their effects on photosynthesis and in leaf morphological and metabolic characteristics51, being practically inexistent the studies that evaluated the effects of thermal stress on production. An interesting study of Kath et al.52 about this topic brings new insight over thermal stress on Coffea canephora yield. These authors used production, precipitation, and temperature data from the Southeast Asia region to model the ideal temperature range for the production of C. canephora, and concluded that the optimum temperature is below 20.5 °C (or an average minimum/maximum of ≤ 16.2/24.1 °C).
According to DaMatta and Ramalho16, drought and extreme temperatures are the main climatic limitations to coffee production. The stress caused by supra-optimal temperatures associated with water deficit in the soil and, therefore, reduction in transpiration, can intensify the occurrence of oxidative stress in coffee16,44. As a consequence of oxidative stress, there may be an increase in cell damage, a phenomenon known in practice as scorch44, which can be accompanied by leaf abscission and, if the climatic problems persist, lead to plant death. It is also worth highlighting that, besides the drastic effect resulting from the combination of drought with high temperatures, soils in the main coffee areas of Espírito Santo are acidic with low nutrient content and moderate to low water holding capacity53, which clearly contributed to aggravating the climatic situation.
### Yield versus planted area
Table 1 shows the values of yield and planted area for the nine municipalities in each of the six seasons analyzed. It is observed that, regardless of the municipality, the lowest values of yield among the six seasons analyzed were observed in 2015/2016, in the municipalities of São Mateus (13 bags ha−1) and Vila Valério (13.5 bags ha−1). The average yield of all municipalities in the 2015/2016 season was only 18.06 bags ha−1, a reduction of 36.6% compared to the 2014/2015 season. As for the planted area, there was little interannual variation, which further demonstrates the impact of drought and high temperature on coffee production.
Although the 2015/2016 season was the most affected by the climate problems, the 2014/2015 season was the first to experience the impacts of drought and high temperatures, as evidenced by the sharp reduction in yield compared to the 2013/2014 season (23.0%), even with a slight increase in planted area. Water deficit and high insolation were recorded in the 2014/2015 season, which contributed to high air temperatures from December 2014 to February 2015. This period coincides with the fruit expansion and filling (Fig. 3), which led to poor grain formation, resulting in smaller and lighter grains54.
In relation to the 2012/2013 season, although it received a reasonable volume of rainfall (823.4 mm) and had no problems of lack of water for irrigation, it showed low yield compared to the seasons considered as climatologically regular (2010/2011, 2011/2012 and 2013/2014). This low yield, in turn, resulted from the occurrence of intense rains at the time of flowering and fertilization of plants, thus causing a problem of fertilization and formation of fruits, impacting the final production55. In relation to the municipalities, the highest percentage variation in yield was observed in Vila Valério. This municipality is the second largest state producer, with 7.2% of production (Fig. 1), and showed a reduction of 52.3% in the 2015/2016 season, compared to the previous one. This municipality was one of those whose plantations had highest levels of damage due to the combined effect of water deficit and high temperature40.
Coffee yield in the subsequent season, as occurs with other perennial crops, is conditioned on the conditions experienced by the plantation during the previous seasons, that is, a negative impact in the previous year will have its effect on the harvest of the following year. This scenario occurred with the plantations of the north, northwest and northeast regions of Espírito Santo. From August 2014, problems of water deficit (lack of rainfall and lack of water for irrigation) and high temperatures were being intensified every month, reaching the worst scenario in the 2015/2016 season, with a sharp reduction of yield (Table1). In this season, the state production was only 5 million and 35 thousand 60 kg coffee bags, the lowest harvest since 200426.
### Annual variation of production versus annual variation of planted area
Figure 6 shows the radar charts along with the respective probability levels resulting from the identity tests of the regression model for the variables: (1) annual variation of production and (2) annual variation of planted area. It is verified that, for all nine municipalities analyzed, the models were different (p-value always lower than the significance level of 5%), which makes it possible to affirm that the reduction of production does not result from the reduction of planted area. It is also possible to confirm the statistical results visually, since a drastic reduction in production values is clearly observed from the compared seasons 4 and especially 5, while the area in the same period remained virtually without variations (Fig. 6). From this finding, the focus becomes the identification of the effects of the accumulated rainfall volume and average air temperature at different stages of the phenological cycle of conilon coffee and, consequently, the impact on its production and yield.
The present study disregarded the effect of biennial bearing, defined as annual alternation of high and low coffee yields, on production56. This is because in conilon coffee biennial bearing is minimized or buffered, within certain limits, when compared to Arabica coffee, due to the periodic renewal of orthotropic stems, through an intense and well-planned pruning system57, and also due to other management practices, including irrigation58. In addition, the quantification of biennial bearing in Coffea canephora plantation is still very complex because of two main reasons. The first is due to the characteristic of cross fertilization of the plants, which requires the planting of several genotypes in the same plot for effective fertilization, thus leading to a great mixture, which makes it very difficult to determine biennial bearing. The second point is that biennial bearing occurs among plots, among plants and within the same plant.
### Effect of annual accumulated rainfall on yield
Overall, agricultural production is directly proportional to the rainfall volume, where this relationship is more evident in non-irrigated crops. In this sense, the low and insignificant relationships between precipitation and production in this study (Fig. 7) demonstrate the effect of irrigation on the performance of other environmental factors. Figure 7 shows the relationship between accumulated rainfall by agricultural year (September–August) and coffee yield. Although there is statistical significance only for the regressions of the municipalities of Linhares, Nova Venécia, São Mateus and Vila Valério, there is clearly a tendency of increase in coffee yield with the increase in rainfall volume in all nine locations. The trend becomes a very important point besides the significance of regression when working with large-scale data (e.g., municipal production), since on this scale there are many factors that can interfere in yield besides rainfall. Craparo et al.59, for instance, evaluated the effect of several rainfall variables on the production and yield of Arabica coffee in several districts of Tanzania in Africa and found that the only rainfall variable that is slightly correlated with production and yield is the number of days of rainfall in the flowering period, although it is not statistically significant.
According to DaMatta et al.38, intense water deficit in conilon coffee leads to accelerated dehydration of tissues, causing collapse of metabolism, culminating in considerable loss of leaf area and, if the problem persists, production is irreversibly affected. Therefore, the authors point out that conilon coffee clones cultivated in Espírito Santo, in general, require the implementation of irrigation for economic exploitation. An important point in this scenario of production/yield reduction refers to the availability of water for irrigation. The study region has as one of its main characteristics the adoption of irrigation to meet the water requirement of plants. However, rainfall volumes below historical averages between August 2014 and September 2016 resulted in the depletion of the main sources of water for irrigation40, that is, water available in small and medium-sized rivers, streams and small earth dams.
Unlike irrigation, which enables water application in the correct quantity and at the correct time for a given crop, rainfall is a climatic variable with high spatial–temporal variability. This characteristic occurs due to a combination of factors, including climatic conditions, rainfall generation mechanisms, topographic characteristics, land use and proximity to the sea and other water surfaces60,61. Understanding this variability of rainfall has always been a major challenge, and the impacts of climate change further complicate this issue62. Therefore, the discussion about the effect of this variable on yield should be cautious in order to avoid false conclusions.
### Effect of average annual air temperature on yield
The air temperature increases above 25 °C resulted in a decrease in production (Fig. 8). The relationships were more evident and significant than rainfall, since the increase in air temperature can affect crop yields regardless of the water availability in the soil63. However, the effects may be exacerbated under conditions of drought and high irradiance. Figure 8 shows the impact of the increase in average air temperature in each season on conilon coffee yield. As for the data of accumulated rainfall, it is possible to verify the sensitivity of coffee yield to temperature, in the present case the increase of this variable leading to a decline in yield. The red arrow refers to the average temperature of the 2015/2016 season, equal to 26.6 °C in the municipality of Linhares (Fig. 8a) and to 26 °C for the municipalities of Nova Venécia and São Mateus (Fig. 8b,c).
For the three municipalities the average annual temperature in 2015/2016 was higher than ideal range, considering values used by Eugenio et al.47 in their study about zoning agroclimatological Coffea canephora for Espírito Santo (22.5–24 °C) or very close to the superior limit, considering data of Matiello48 and Taques and Dadalto49, that’s of 22–26 °C. To increase the problematic, the 2015/2016 season was accompanied by droughts, leading to a sharp reduction of yield in the 2015/2016 season (Table 1). In the modeling study by Kath et al.52, the authors concluded that an ideal temperature range above 22 °C is probably overestimated. On the other hand, we believe that the conclusions of these authors are specific to the region where the data were collected and cannot be directly applied to the species cultivated in other regions of the world, especially in Brazil.
Outside their optimum temperature ranges, the bean quality of both species declines, as does yield. Changing climate might also increase exposure and vulnerability of coffee to pests and diseases64. However, it's important to point out, that in the last two decades occurred great genetic advancement in conilon coffee growing, producing genotypes with different characteristics, mainly in terms of yield65. Therefore, the ideal temperature range for a genotype, may not be the same for another, although the trend be very similar intervals.
Gay et al.66, working with modeling of C. arabica production as a function of climate changes, concluded that air temperature is the most relevant climate factor for its production, since it responds significantly to the seasonal patterns of temperature. On the other hand, there are practices, such as agroforestry shading, that are very efficient to avoid this problem of high temperatures, since they act in reducing air temperature and direct solar radiation on the canopy of plants, without compromising yield, as observed in some studies with conilon coffee67,68,69,70.
Recent decades have been characterized by increasing temperatures worldwide, which also resulted in an exponential increase of VPD71,72 that can be very harmful to coffee, since it is a crop very sensitive to this climatic variable73. Barros indicated VPD values of 2.2 kPa as the upper limit for coffee plants, as values above that limit can result in decrease of stomatal and canopy conductance in Coffea spp. More recent evidences show that when VPD exceeded 2 kPa, there was a considerable decrease in canopy conductance74. The Coffee spp. sensitivity to high VPD values offer crucial evidence that climate changes can be harmful to coffee plants, mainly for C. canephora genotypes, which are drought-sensitive.
On the other hand, drought-resistant genotypes can be benefited because they generally show reduction of yield under optimal environments conditions. This happens for such genotypes due to the increase of sensitivity of their stomata to VPD75. Thereby, these authors concluded that coffee genotypes displaying increased phenotypic plasticity (e.g., deep root system, substantial hydraulic conductance, intermediate stomatal control and strengthening of antioxidant defense system) could be used in regions which are predicted to face moderate water deficit, while drought-resistant genotypes could be used in regions predicted to face severe drought. In addition, Rodrigues et al.76 suggested that the coffee genotype, for global warming conditions, must have the ability to transport water throughout the plant system, maintaining adequate leaf water content and maximizing stomatal opening.
### Combined effect of accumulated rainfall and air average temperature on coffee production
Figure 9 shows the graphs of response surface resulting from the combined effect of accumulated rainfall and temperature on coffee production. Before delving deeper in the topic discussion, it is important to highlight that the variations in planted area were small and rather insignificant (Fig. 6), favoring the study of the impacts from variations in temperature and precipitation on crop production. There was a significant interaction between the average air temperature and the accumulated rainfall for coffee production in the periods of September–December (Fig. 9a), April–August (Fig. 9c) and annual (Fig. 9d), and the reduction of production according to the increase in temperature and reduction of accumulated rainfall, as expected. Thus, greater decreases in production can be seen when increases in temperature are associated with lower rainfall.
There was great similarity between the response surfaces for the Period January–March (Fig. 9b) and annual period (Fig. 9d), with a virtually linear reduction in production. It can be observed that accumulated annual rainfall close to 700 mm and air temperature above 25.5 °C caused annual production lower than 250 thousand 60 kg coffee bags (Fig. 9d). The January–March period was not significant (Fig. 9b), but an analysis of it is quite valid, because this is the period where conilon coffee is found in the fruit filling stage and, if water deficit occurs, the problem of endosperm malformation may occur (Fig. 3).The coffee producers have a clear understanding of the importance of adequate water availability during this phenological phase (grain filling phase). Because of that, they focus on an efficient irrigation management during this stage, which helps to explain the lack of significance. In addition, it should be noted that, although the model is not significant, which is largely due to the small effect of precipitation as previously explained, there seems to be a sharp decrease in production with increasing temperature, regardless of precipitation.
Considering the annual regression model (Fig. 9d) and the 1 °C increase in the average annual air temperature, the production decreases: (1) 31% considering the average annual precipitation of 920 mm; (2) 25% considering a rainfall of 1200 mm; (3) 36% considering the 50% decrease in average annual precipitation. In the modeling study performed by Kath et al.52, the increase of 1 °C in the average air temperature (minimum and maximum) caused a decrease in production of approximately 14%. Thus, increases in air temperature have a higher impact (negative) on production in comparison to decreases in annual precipitation. It is noteworthy that in the northern, northeastern and northwestern regions of Espírito Santo, approximately 90% of crops are grown using irrigation9 in addition to the use of varieties that have some degree of drought tolerance77,78, the main target for improvement in recent decades.
In Brazil, heat tolerance has not yet been considered in the genetic breeding studies of C. canephora. These results demonstrate, in a broad way, the sensitivity of the conilon coffee plantations to the increase of the air temperature. Differently to the evidence raised by DaMatta et al.79, the effect of the temperature increase seems to negatively affect production, even under conditions of adequate water supply. Furthermore, considering the likely impacts of climate change, we believe that in C. canephora plantations, the potential positive effects of increasing the concentration of atmospheric CO2 to mitigate the negative impact of rising temperatures80,81 will not be effective. As proposed by Rahn et al.82, the effects of CO2 concentration will be more significant, especially at higher altitudes and, therefore, more evident in C. arabica specie.
High temperatures in the period from September to December, which encompasses the flowering and early fruiting stages (Fig. 3), may lead to the abortion of flowers36, while water deficit causes the problem of low sieve classification (smaller fruits), hence causing a strong impact on production. In addition, high temperatures may lead to an early maturation of the fruits due to the earlier break of dormancy of the buds83. High temperatures in the period from January to March could lead to a quality problem due to early and excessive ripening of fruits83. These authors point out that coffee is very resistant to the high temperature of summer and drought, but the increase in extreme conditions may be responsible for physiological stresses, such as reduction in photosynthetic efficiency. These significant relationships reinforce what has been demonstrated by some studies59,66, that increased air temperature has a significant influence on the physiology of growth and production in coffee plants at each of the phenological stages.
Bunn et al.84 conducted a study on the profile of climate changes in the global production of arabica and conilon coffee and highlighted that coffee has proven to be highly sensitive to climate change, in addition to the fact that dominant production regions in the world (Brazil and Vietnam) may experience substantial reductions in the areas available for coffee. Additionally, Pham et al.64 highlights that as a climate-sensitive perennial crop, coffee is likely to be highly susceptible to changes in climate.
### Principal component analysis: identification of the most influential period of rainfall accumulation on yield
Figure 10 presents a biplot of the relationship between the three periods of rainfall accumulation (variables) and the six seasons (individuals) for each of the nine municipalities, where the principal component 1 (Dim1) is represented by the X-axis and principal component 2 (Dim2) by the Y-axis. A general analysis involving all nine municipalities, based on the components, showed that the highest coffee yields are correlated with the rains that occurred in the periods P1 and P3, since these appear predominantly in the first and fourth quadrants, which are the most associated with the first component, concentrating the three major seasons, i.e. 2013/2014, 2011/2012 and 2010/2011, among the six studied. P2 was present in the second quadrant six times (Fig. 10a–c,e,g,i), so January–March is a period less associated with Dim1; therefore, it can be said that this period is the one that has the least influence on coffee yield.
Regarding the results of the principal component analysis, it is important to highlight that, in order to draw a better conclusion on the effects of accumulated rainfall or water supply on coffee yield, it is necessary to conduct an experimental study preferably in a controlled environment, where the applied water depth can be controlled. In this line, there are several studies been performed with the conilon coffee and, the results have been show that water deficit causes on plants suppression of net photosynthesis44 or the progressive reduction of this85, decreases in stomatal conductance11 and, decreased carbon assimilation86. However, it’s believed that there exists a wide range of genetic variability among conilon coffee clones for traits associated with drought tolerance85. In addition, as the varieties of conilon coffee comprise several genotypes87, with high genetic variability88,89,90, they may show different responses for the same treatment.
### Detection of the drought and heat effects over coffee plantations using Enhanced Vegetation Index (EVI)
Drought is one of the most destructive natural disasters, and numerous studies predict it will become more severe and widespread under climate change91. Thus, acquiring knowledge on the impact of this phenomenon is essential for several decision-making processes. In large areas (like in present study), remote sensing-based vegetation indices (VI) are a key tool for this purpose. VI that make uses red and near infrared wavelengths in their calculations, such as the EVI, are very sensitivity to plant leaf area92,93,94 and photosynthetic pigments95,96,97 variations, which in turn, can be strongly affected by drought, heat, and excessive irradiance. In this sense, using the EVI conilon plantations is particularly useful to reflect some possible physiological disorders caused by drought and heat, for instance, fall of leaves, pigment degradation, and limited nutrient absorption capacity.
Figure 11 presents information on the variations in EVI values along the six seasons analyzed for the period from September to December (P1). Based on visual interpretation (maps), lower EVI values is an evident similarity between the 2014/2015 and 2015/2016 seasons. Through the boxplot, it is possible to observe that the impact of drought and heat is more pronounced in the north, northwest, and northeast regions of Espírito Santo. The mean values of EVI (red points) were close to 0.35, while, for the other seasons, they remained near to 0.40. The stress caused by supra-optimal temperatures can cause photosynthetic pigment degradation76 and, when associated with water deficit in the soil, it can lead to a reduction in transpiration which can intensify the occurrence of oxidative stress in coffee16,44. As a consequence of oxidative stress, there may be an increase in cell damage, which can result in leaf abscission and even plant death (in severe cases)44. Such outcome will directly reflect in EVI values, given their large sensitivity to leaf area index and photosynthetic pigments.
Figueira Branco et al.98 used the EVI derived from the MODIS sensor to detect the occurrence of droughts in areas of tropical forest, which also represents the present study area, and concluded that EVI data are able to show the response of the vegetation under drought conditions. Several similar studies have been conducted using EVI for drought detection and monitoring91,99,100,101,102. As evidenced in Fig. 11, in P1, the lowest values of EVI are also observed for the 2014/2015 and 2015/2016 seasons in P2, from January to March (Fig. 12), but with greater discrepancy between these seasons than in P1. By comparing the periods (P1 and P2) for the same season (Figs. 11 and 12), it is possible to observe that the mean values of EVI in the January–March interval (P2) are higher than those in September–December (P1). This fact occurs because soon after harvest (April–August, Fig. 3), plants have a lower number of leaves due to the loss during fruit harvesting and rejuvenation pruning, resulting in lower leaf area index and, consequently, lower EVI values. Also, according to these authors, plants cannot recover the leaves lost through new shoots in a sufficiently short time to ensure good flowering, due to the established stress. Thus, the use of irrigation is essential in this period of the year to provide water for plants so that they can recover.
In the period from April to August, as well as for the other periods, the greatest impacts of drought were observed in the 2015/2016 season (Fig. 13). According to the boxplot, this impact more pronounced where the average values were around 0.35 for this season. For the 2014/2015 season, the behavior of P3 with EVI values close to 0.4 differed from the pattern found in the periods P1 and P2, which had average values around 0.35. This can be explained by the fact that the rains during the 2014/2015 season exceeded 900 mm and irrigations were still carried out with some frequency.
Data available on the website of Capixaba Institute of Research, Technical Assistance and Rural Extension (INCAPER) also help to highlight the drought that occurred in the north, northwest and northeast regions of Espírito Santo. This institute has an agrometeorology sector, which provides spatialized data of water balance and the Standardized Precipitation Index (SPI)103 or the entire state on a monthly scale. The Incaper data of SPI for 2015 showed that virtually the entire area of the present study was classified as extremely dry104. Regarding the water balance in the rainy season (October–February), also for the year 2015, the values were negative or positive close to zero, with a slight increase during the season considered dry (March–September)105.
Climate problems have dramatically affected the production and consequently the price of coffee. According to data from the Brazilian Association of the Coffee Industry106, in 2016 the average price of the 60 kg coffee bag (conilon type 7) marketed in Espírito Santo was R$410.45, whereas in 2015 the value was R$ 315.15. This price increase was extremely important for coffee growers in the region, as most of them were decapitalized due to the situation faced. A worrying scenario for the state of Espírito Santo is that conilon coffee is the main source of income in 80% of rural properties located in the north, northwest and northeast regions of the state9. Thus, problems that strongly impact coffee production will lead to social and economic losses for families in these regions. From a socio-economic perspective, understanding the extent of climate-driven impacts on coffee production and the benefits of potential adaptation strategies will be of vital importance to maintaining and improving coffee productivity and profitability and sustaining the livelihoods of smallholder producers all over the world64.
A recent study of Magrach and Ghazoul7 shows that conilon variety could lose 55% of currently suitable areas (mostly within western Africa and Brazil), but the future suitable area is expected to more than double, particularly by the extension of climatically suitable conditions in the Amazon Basin and South East Asia, which together will total 97.4 million hectares. The authors also mention that climate change will be detrimental for Arabica cultivation, though the area suitable for Robusta will increase greatly by 2050. Both Arabica and conilon will be subject to important geographic shifts in their distribution. Thus, several coffee producers of Espírito Santo state can be affected.
|
2023-01-27 03:10:41
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48988574743270874, "perplexity": 2579.940311658138}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764494852.95/warc/CC-MAIN-20230127001911-20230127031911-00879.warc.gz"}
|
https://undergroundmathematics.org/calculus-meets-functions/r8959/solution
|
Review question
# When does $3x^4 - 16x^3 +18x^2 + k = 0$ have four real solutions? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource
Ref: R8959
## Solution
The equation in $x$, $3x^4 - 16x^3 +18x^2 + k = 0$ has four real solutions
1. when $-27 < k < 5$;
2. when $5< k <27$;
3. when $-27< k <-5$;
4. when $-5< k <0$.
The understanding here is that the solutions are distinct.
The number of distinct solutions to $f(x)=3x^4 - 16x^3 +18x^2 + k = 0$ equals the number of times the graph $y=f(x)$ crosses the $x$-axis, so it’ll be helpful to sketch the graph of $y = f(x)$.
We can use differentiation to learn about the stationary points and therefore the shape of this graph. So we write $y=3x^4-16x^3+18x^2+k,$ and then we differentiate to find \begin{align*} y' &= 12x^3-48x^2+36x \\ &= 12x(x^2 - 4x + 3) \\ &= 12x(x-3)(x-1). \end{align*}
To find the stationary points, we solve $y'= 0$. We see that the stationary points of the graph occur at $x=0$, $x=1$ and $x=3$.
To classify these stationary points, we can use the second derivative. We have \begin{align*} y'' &= 36x^2-96x+36 \\ &= 12(3x^2-8x+3). \end{align*} Substituting, \begin{align*} y''(0) &= 36 > 0, \\ y''(1) &= -24 < 0, \\ y''(3) &= 72 > 0, \end{align*}
so we find that the stationary points at $x=0$ and $x=3$ are minima, and the stationary point at $x=1$ is a maximum.
If we now substitute $x=0$, $1$ and $3$ into the equation, we find that the coordinates of our minima are $(0,k)$ and $(3,k-27)$, and the coordinates of the maximum are $(1,k+5)$.
Or alternatively, we can say we know the rough shape of the curve (see below) so we know which turning points are maxima, and which are minima.
This information about the curve is summarised in the sketch of $y=f(x)$ given below.
The equation has four real solutions when the graph crosses the $x$-axis four times.
This occurs when the $x$-axis is in the position of the dotted line in the diagram above.
That is, it’s in a position such that the minima at $(0,k)$ and $(3,k-27)$ are below it, and the maximum at $(1,k+5)$ is above it.
In terms of inequalities, this is when $k < 0$, $k-27 < 0$ and $k+5 > 0$.
So combining these, our final inequality is $-5 < k < 0$, so the answer is (d).
|
2018-09-23 19:07:47
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999927282333374, "perplexity": 276.212381140856}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267159570.46/warc/CC-MAIN-20180923173457-20180923193857-00343.warc.gz"}
|
http://mathhelpforum.com/pre-calculus/106154-binomial-theorem-expanded.html
|
# Math Help - Binomial Theorem expanded
1. ## Binomial Theorem expanded
ok i just want someone to double check my work please. for some reason i think i have the wrong answer.
(use the binomial theorem to expand and simplify this expression)
(x-y)^8
ok so i used pascal's triangle ...the pascal triangle in my book stops at the 7th row so i got this for my 8th row 1,8,28,56,70,56,28,8,1
so this is what i got ...Please check my work.
1x^8-8x^7y+28x^6y^2-56x^5y^3+70x^4y^4-56x^3y^5+28x^2y^6-8xy^7+1y^8
ok i know this looks confusing but if you copy and paste it into online calculator it should put everything into place. Did i do this right???
2. If I was marking I's say "no", because it says "use the binomial theorem", not "read the appropriate row of Pascal's Triangle from your textbook".
I reckon what you ought to do would be to use the identity
$\binom n m = \frac {n!} {(n-m)! m!}$
where $n = 8$ and $m$ is each of 0 to 8 in turn. The arithmetic is not hard, and the numbers should work out the same as what you got.
3. Originally Posted by Matt Westwood
If I was marking I's say "no", because it says "use the binomial theorem", not "read the appropriate row of Pascal's Triangle from your textbook".
I reckon what you ought to do would be to use the identity
$\binom n m = \frac {n!} {(n-m)! m!}$
where $n = 8$ and $m$ is each of 0 to 8 in turn. The arithmetic is not hard, and the numbers should work out the same as what you got.
i dont understand what your trying to tell me. its asking me to expand (x-y)^8
what you gave me is no where in my book? your technique probably works too but im just no familiar with it sorry.
4. Originally Posted by flexus
ok i just want someone to double check my work please. for some reason i think i have the wrong answer.
(use the binomial theorem to expand and simplify this expression)
(x-y)^8
ok so i used pascal's triangle ...the pascal triangle in my book stops at the 7th row so i got this for my 8th row 1,8,28,56,70,56,28,8,1
so this is what i got ...Please check my work.
$1x^8-8x^7y+28x^6y^2-56x^5y^3+70x^4y^4-56x^3y^5+28x^2y^6-8xy^7+1y^8$
ok i know this looks confusing but if you copy and paste it into online calculator it should put everything into place. Did i do this right???
HI
you are correct .
Instead of using the pascal triangle to look for the coefficient , you can actually use the identity which matt has suggested .
HI
you are correct .
Instead of using the pascal triangle to look for the coefficient , you can actually use the identity which matt has suggested .
could you show me how to do that? if its not to much trouble that is? thank you.
6. im sorry i still don't get it so did i do it right or wrong?? is my answer correct?
7. Show you how to do what? Multiply and divide? Matt told you that the coefficient of $x^ny^{8-n}$ is $\frac{8!}{n!(8- n)!}$. Calculate that for n= 0 to 8 (Actually, you only need to do it for n= 0 t0 4, after that you get duplicates). Do you know that $n!= n(n-1)(n-2)\cdot\cdot\cdot (3)(2)(1)$? So that The coefficient for $x^2y^6$ is $\left(\begin{array}{cc}8 \\ 2\end{array}\right)= \frac{8!}{2! 6!}= \frac{40320}{(2)(720)}$. You can simplfy that a lot by observing that $\frac{8!}{2! 6!}= \frac{8(7)(6)(5)(4)(3)(2)(1)}{(2(1))(6(5)(4)(3)(2) (1)}$ and canceling a lot- you eventually get 4(7)= 28, just as you have!
8. im sorry but no one has told me if i did it right or if i got the right answer???
9. Originally Posted by flexus
im sorry but no one has told me if i did it right or if i got the right answer???
The answer is correct. But note that when a question gives an instruction to solve it in a particular way, your solution must show that this instruction has been followed ....
|
2016-05-25 08:33:53
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 13, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8051229119300842, "perplexity": 379.25397028738394}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049274191.57/warc/CC-MAIN-20160524002114-00052-ip-10-185-217-139.ec2.internal.warc.gz"}
|
https://www.physicsforums.com/threads/question-about-where-a-thermodynamics-formula-comes-from.866860/
|
# Question about where a thermodynamics formula comes from
1. Apr 13, 2016
### influx
1. The problem statement, all variables and given/known data
How do we obtain:
w = q - Δh
2. Relevant equations
3. The attempt at a solution
If you neglect the second half of the SFEE and simplify we get:
What is the next step from the last line in the above image to
w = q - Δh
I mean dw/dt = dq/dt - Δh
so if we multiply by dt we get:
dw = dq - Δh(dt)
w = q -
Then I assume this should be integrated? If yes, how? Can't seem to get my head round it.
Also, what is the difference between w = q - Δh and w = q - Δu? I know u is specific internal energy and h is specific enthalpy but the above seems to suggest they're the same?
Last edited: Apr 13, 2016
2. Apr 13, 2016
### drvrm
in thermodynamics, the thermodynamic relation is generally expressed as an infinitesimal change in internal energy in terms of infinitesimal changes in entropy, and volume for a closed system in thermal equilibrium in the following .....
however the laws of thermodynamics predefines the way one can relate the 'desired macroscopic parameters/observables of the system'
3. Apr 13, 2016
### Staff: Mentor
In the open system version of the 1st law that you have written (based on a fixed control volume), $\dot{W}$ is not the total rate of doing work, it is only the rate of doing "shaft work." There is also work involved in pushing mass out of the control volume in an output stream, and pushing work into the control volume in an input stream. That explains why there is a $\Delta h$ and not a $\Delta u$. This should all have been explained in the derivation in your book.
Chet
|
2017-12-15 18:08:30
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5868699550628662, "perplexity": 762.7712847649725}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948577018.52/warc/CC-MAIN-20171215172833-20171215194833-00323.warc.gz"}
|
http://indico.lucas.lu.se/event/767/
|
Event calendar at fysik.lu.se
# Mötesplats Rydberg: Marcus Isinger and Marcus Dahlström - Photoionization in the time and frequency domain
Tuesday, 14 November 2017 from to (Europe/Stockholm)
at Fysiska institutionen-Physics Department ( 1-4-Rydbergsalen - Rydbergsalen )
Description Ultrafast processes in matter, such as the electron emission following light absorption, can now be studied using light pulses of attosecond duration in the extreme ultraviolet spectral range. However, the short temporal duration of these pulses implies a large energy spread of the photoelectrons according to Heisenberg’s uncertainty principle. Indeed, the lack of spectral resolution due to the use of short pulses has raised issues in the interpretation of the experimental results in comparison with theoretical calculations. Here, we present new experimental results on the long-standing problem of photoionization time delays in neon atoms that were originally measured by Schultze and co-workers using isolated attosecond pulses 1. Using our alternative approach 2, based on attosecond pulse trains, we gain both high temporal and spectral resolution by photoelectron interferometry. This allows us to spectrally disentangle direct ionization from ionization with shake-up, in which a second electron is left in an excited state, and obtain excellent agreement with theoretical calculations based on diagrammatic many-body perturbation theory 3. In this way we have solved a 7-year-old puzzle in attosecond science. 1. Schultze, M. et al. Delay in Photoemission. Science (80-. ). 328, 1658–1662 (2010). 2. Isinger, M. et al. Photoionization in the time and frequency domain. Science (80-. ). (2017). doi:10.1126/science.aao7043 3. Dahlström, J. M., Carette, T. & Lindroth, E. Diagrammatic approach to attosecond delays in photoionization. Phys. Rev. A 86, 61402 (2012).
|
2018-04-20 19:52:28
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7679237127304077, "perplexity": 2979.945036384008}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125944682.35/warc/CC-MAIN-20180420194306-20180420214306-00205.warc.gz"}
|
https://exampur.com/short-quiz/13695/
|
# SSC CGL MAINS MATHS QUIZ
Attempt now to get your rank among 60 students!
## Question 1:
A sum of money at compound Interest amounts to $3 \sqrt{3}$ times itself in 5 years. In how many years will it be 27 times itself.
## Question 2:
A man gave 50% of his saving of 42100 to his two sons A and B if 15 and 13 years of age respectively the divided it in such a way that each of his son, when they attain the age of 18 years, would receive the same amount at 5% compound interest per annum. The share of B was:(in Rs.)
## Question 3:
On a certain sum of money, the simple interest for 2 years is Rs. 200 at the rate of $7 \%$ per annum. Find the difference in CI and SI.
## Question 4:
The compound interest on a certain sum for 2 years at $15 \%$ p.a. is ₹ 7,282 , interest being compounded 8 -monthly. What will be the amount of the same sum at the same rate for the same duration, when the interest is compounded yearly?
## Question 5:
What is the rate of interest (in %.) if simple interest earned on a certain sum for the $3^{\text {rd }}$ year is Rs. 3200 and compound Interest earned in 2 years is Rs. 6912 ?
## Question 6:
What will be the compound Interest on a sum of Rs. 93,750 for 2 years at $12 \%$. P.a. If the interest is Compounded 8-monthly?
## Question 7:
The sum for 2 years gives a compound interest of $3187.5$ at the rate of $12 \frac{1}{2} \%$ per annum. Then Find the sum?
## Question 8:
A man lent Rs. $\mathbf{4 5 0 0}$ at $30 \%$ compound interest per annum for 3 years. What is the difference between the interest earned by the man in the 2 nd year only and the interest earned by the man in the 3rd year only?
## Question 9:
A sum of money is paid back in two annual installments of Rs 24,255 each, allowing $5 \%$ compound Interest annually. The sum borrowed was
## Question 10:
Sumit earned ₹ 10,692 as simple interest on ₹ 29,700 at a certain rate of interest for 3 years. His friend Anil invested ₹ 12,500 for 2 years at the same rate of interest but on compound interest compounded annually. How much did Anil earn as interest?
|
2023-03-21 07:47:24
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6667013764381409, "perplexity": 788.8462320049775}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943637.3/warc/CC-MAIN-20230321064400-20230321094400-00330.warc.gz"}
|
https://elixirforum.com/t/possible-deadlock-in-tests/17214
|
I have been banging my head against the wall with this for a while. I have a factory to create database entries for my tests in a Phoenix app, it is pretty much lifted from What’s New In Ecto 2.1:
``````defmodule Deadlock.Factory do
def build(:post) do
end
def build(name, attributes \\ []) do
name |> build() |> struct(attributes)
end
def insert!(name, attributes \\ []) do
end
end
``````
Then I have the following two tests:
``````defmodule DeadlockWeb.PageControllerTest do
test "This works..." do
end
test "This seems to deadlock..." do
end
end
``````
The first one passes as expected. The second fails with the following error:
``````21:30:02.946 [error] Postgrex.Protocol (#PID<0.296.0>) disconnected: ** (DBConnection.ConnectionError) owner #PID<0.328.0> timed out because it owned the connection for longer than 15000ms
** (ExUnit.TimeoutError) test timed out after 60000ms. You can change the timeout:
1. per test by setting "@tag timeout: x"
2. per case by setting "@moduletag timeout: x"
3. globally via "ExUnit.start(timeout: x)" configuration
4. or set it to infinity per run by calling "mix test --trace"
(useful when using IEx.pry)
Timeouts are given as integers in milliseconds.
``````
I discovered this by accident since I mistyped something in my test, and it took me a while to work out what was going on. I would expect the second test to fail quickly since it can’t find a matching `build` function in the factory module, but it’s obviously trying to do something with the database.
A minimal repo to reproduce is here: https://github.com/amarraja/deadlock
I’m sure it’s something simple, but I just can’t see it!
t’s obviously trying to do something with the database.
This isn’t actually the case. The timeout is just for the test case itself, there’s nothing database related in the error.
You have an infinite loop. `insert!(:nothing_matching_in_factory)` calls:
`````` def insert!(name, attributes \\ []) do
end
``````
which calls `build(name, attributes)` with `build(: nothing_matching_in_factory, [])`
This is where we enter the infinite loop. Since the first clause doesn’t match it goes here:
`````` def build(name, attributes \\ []) do
name |> build() |> struct(attributes)
end
``````
If we break that apart a little the first function call is `build(:nothing_matching_in_factory)`. This does match a function clause! It matches the function we’re in right now: `def build(name, attributes \\ []) do`. Voila, infinite loop. It just runs until the test case times out.
6 Likes
Aah, so obvious!! I’ve been looking at this too long. Thanks for the great explanation
2 Likes
|
2022-08-18 10:12:44
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9308386445045471, "perplexity": 4044.760288473042}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573193.35/warc/CC-MAIN-20220818094131-20220818124131-00320.warc.gz"}
|
https://www.physicsforums.com/threads/limit-of-function-sandwich-method.271411/
|
# Limit of function ( sandwich method)
1. Nov 12, 2008
### Дьявол
limit of function ("sandwich" method)
1. The problem statement, all variables and given/known data
Using the "sandwich" method prove that $$\lim_{n\rightarrow \propto }(\frac{sin(n)}{n})=0$$
2. Relevant equations
$$x_n \leq y_n \leq z_n$$
$$\lim_{n\rightarrow \propto }(x_n) \leq \lim_{n\rightarrow \propto }(y_n) \leq \lim_{n\rightarrow \propto }(z_n)$$
3. The attempt at a solution
I am honestly little bit confused at this point.
$$\frac{-1}{n} \leq \frac{sin(n)}{n} \leq \frac{1}{n}$$
then my question is if $n=-\frac{\pi}{4}$ then $$\frac{-1}{-0.785}$$ will be not less or equal to $$\frac{\sqrt{2}}{2*(-0.785)}$$, where -0.785=$-\frac{\pi}{4}$, where $\pi \approx 3.14$.
Last edited: Nov 12, 2008
2. Nov 12, 2008
### marcusl
Re: limit of function ("sandwich" method)
Are you sure that n is a real number? Usually n denotes a positive integer in this type of problem.
3. Nov 12, 2008
### HallsofIvy
Staff Emeritus
Re: limit of function ("sandwich" method)
$-1/n\le sin(n)/n\le 1/n$ for n positive. Obviously, if n is negative, just $-1/n\le 1/n[itex] is not true! Your use of [itex]x\rightarrow \propto$ is a little confusing. Did you mean $\infty$? Even if you do not interpret n as necessarily being positive, if n is "going to $\infty$" eventually, for some finite N, if n> N, n will be postive. And you can always drop any finite number of terms in an infinite sequence without changing the limit.
4. Nov 13, 2008
### Дьявол
Re: limit of function ("sandwich" method)
Thanks for the posts. I see now, it was my mistake if an=sin(n)/n, an is progression where n are positive integer numbers. So if:
$$-1 \leq sin(n) \leq 1$$
then divided by n, I'll get:
$$-1/n \leq sin(n)/n \leq 1/n$$
Sorry for the symbol, I misspelled it, since I don't cover LaTeX too good at this moment.
Thanks for the help.
5. Nov 13, 2008
### HallsofIvy
Staff Emeritus
Re: limit of function ("sandwich" method)
For future reference, in LaTex, $\infty$ is "\infty". $\propto$ is "\propto", i.e. "proportional to".
|
2016-10-23 20:56:06
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9310487508773804, "perplexity": 2342.662880555759}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719416.57/warc/CC-MAIN-20161020183839-00321-ip-10-171-6-4.ec2.internal.warc.gz"}
|
https://www.quizover.com/course/section/section-summary-bohr-s-theory-of-the-hydrogen-atom-by-openstax
|
# 30.3 Bohr’s theory of the hydrogen atom (Page 6/14)
Page 6 / 14
But there are limits to Bohr’s theory. It cannot be applied to multielectron atoms, even one as simple as a two-electron helium atom. Bohr’s model is what we call semiclassical . The orbits are quantized (nonclassical) but are assumed to be simple circular paths (classical). As quantum mechanics was developed, it became clear that there are no well-defined orbits; rather, there are clouds of probability. Bohr’s theory also did not explain that some spectral lines are doublets (split into two) when examined closely. We shall examine many of these aspects of quantum mechanics in more detail, but it should be kept in mind that Bohr did not fail. Rather, he made very important steps along the path to greater knowledge and laid the foundation for all of atomic physics that has since evolved.
## Phet explorations: models of the hydrogen atom
How did scientists figure out the structure of atoms without looking at them? Try out different models by shooting light at the atom. Check how the prediction of the model matches the experimental results.
## Section summary
• The planetary model of the atom pictures electrons orbiting the nucleus in the way that planets orbit the sun. Bohr used the planetary model to develop the first reasonable theory of hydrogen, the simplest atom. Atomic and molecular spectra are quantized, with hydrogen spectrum wavelengths given by the formula
$\frac{1}{\lambda }=R\left(\frac{1}{{n}_{\text{f}}^{2}}-\frac{1}{{n}_{\text{i}}^{2}}\right),$
where $\lambda$ is the wavelength of the emitted EM radiation and $R$ is the Rydberg constant, which has the value
$R=\text{1.097}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{-1}\text{.}$
• The constants ${n}_{i}$ and ${n}_{f}$ are positive integers, and ${n}_{i}$ must be greater than ${n}_{f}$ .
• Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits given by
$\Delta E=\text{hf}={E}_{\text{i}}-{E}_{\text{f}},$
where $\Delta E$ is the change in energy between the initial and final orbits and $\text{hf}$ is the energy of an absorbed or emitted photon. It is useful to plot orbital energies on a vertical graph called an energy-level diagram.
• Bohr proposed that the allowed orbits are circular and must have quantized orbital angular momentum given by
$L={m}_{e}{\text{vr}}_{n}=n\frac{h}{2\pi }\left(n=1, 2, 3 \dots \right),$
where $L$ is the angular momentum, ${r}_{n}$ is the radius of the $n\text{th}$ orbit, and $h$ is Planck’s constant. For all one-electron (hydrogen-like) atoms, the radius of an orbit is given by
${r}_{n}=\frac{{n}^{2}}{Z}{a}_{\text{B}}\text{(allowed orbits}\phantom{\rule{0.25em}{0ex}}n=1, 2, 3, ...\right),$
$Z$ is the atomic number of an element (the number of electrons is has when neutral) and ${a}_{\text{B}}$ is defined to be the Bohr radius, which is
${a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}=\text{0.529}×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m}\text{.}$
• Furthermore, the energies of hydrogen-like atoms are given by
${E}_{n}=-\frac{{Z}^{2}}{{n}^{2}}{E}_{0}\left(n=1, 2, 3 ...\right)\text{,}$
where ${E}_{0}$ is the ground-state energy and is given by
${E}_{0}=\frac{{2\pi }^{2}{q}_{e}^{4}{m}_{e}{k}^{2}}{{h}^{2}}=\text{13.6 eV.}$
Thus, for hydrogen,
${E}_{n}=-\frac{\text{13.6 eV}}{{n}^{2}}\left(n,=,1, 2, 3 ...\right)\text{.}$
• The Bohr Theory gives accurate values for the energy levels in hydrogen-like atoms, but it has been improved upon in several respects.
## Conceptual questions
How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.
In Inelastic collision cunculate the vilocity
explain how a body becomes electrically charged based on the presence of charged particles
induction
babar
induction
DEMGUE
definitely by induction
Raymond
induction
Raymond
induction
Shah
induction
Korodhso
please why does a needle sinks in water
DEMGUE
induction
Korodhso
induction
Auwal
what are the calculations of Newton's third law of motiow
what is dark matter
(in some cosmological theories) non-luminous material which is postulated to exist in space and which could take either of two forms: weakly interacting particles ( cold dark matter ) or high-energy randomly moving particles created soon after the Big Bang ( hot dark matter ).
Usman
if the mass of a trolley is 0.1kg. calculate the weight of plasticine that is needed to compensate friction. (take g=10m/s and u=0.2)
what is a galaxy
what isflow rate of volume
flow rate is the volume of fluid which passes per unit time;
Rev
flow rate or discharge represnts the flow passing in unit volume per unit time
bhat
When two charges q1 and q2 are 6 and 5 coulomb what is ratio of force
When reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle’s frame?
nehemiah
why is it proportional
i don't know
y
nehemiah
what are the relationship between distance and displacement
They are interchangeable.
Shii
Distance is scalar, displacement is vector because it must involve a direction as well as a magnitude. distance is the measurement of where you are and where you were displacement is a measurement of the change in position
Shii
Thanks a lot
Usman
I'm beginner in physics so I can't reason why v=u+at change to v2=u2+2as and vice versa
Usman
what is kinematics
praveen
kinematics is study of motion without considering the causes of the motion
Theo
The study of motion without considering the cause 0f it
Usman
why electrons close to the nucleus have less energy and why do electrons far from the nucleus have more energy
Theo
thank you frds
praveen
plz what is the third law of thermodynamics
third law of thermodynamics states that at 0k the particles will collalse its also known as death of universe it was framed at that time when it waa nt posible to reach 0k but it was proved wrong
bhat
I have not try that experiment but I think it will magnet....
Hey Rev. it will
Jeff
I do think so, it will
Chidera
yes it will
lasisi
If a magnet is in a pool of water, would it be able to have a magnetic field?.
yes Stella it would
Jeff
formula for electric current
Fokoua
what are you given?
Kudzy
what is current
Fokoua
I=q/t
saifullahi
Current is the flow of electric charge per unit time.
saifullahi
What are semi conductors
saifullahi
materials that allows charge to flow at varying conditions, temperature for instance.
Mokua
these are materials which have electrical conductivity greater than the insulators but less than metal, in these materials energy band Gap is very narrow as compared to insulators
Sunil
materials that allows charge to flow at varying conditions, temperature for instance.
Obasi
wao so awesome
Fokoua
At what point in the oscillation of beam will a body leave it?
Atambiri
what is gravitational force
|
2018-11-19 04:39:33
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 24, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6508234739303589, "perplexity": 802.8181783396514}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039745281.79/warc/CC-MAIN-20181119043725-20181119065725-00384.warc.gz"}
|
https://www.alanshawn.com/leetcode/2020/06/05/leetcode-344-latex.html
|
# LeetCode 344: Reverse String (LaTeX)
Write a function that reverses a string. The input string is given as an array of characters char[].
## Example
Example 1:
Input: ["h","e","l","l","o"]
Output: ["o","l","l","e","h"]
Example 2:
Input: ["H","a","n","n","a","h"]
Output: ["h","a","n","n","a","H"]
## Notes
• The original question requires in-place modification with $O(1)$ memory. Since LaTeX does not have proper support for ordered lists, this requirement is discarded.
## Solution
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{expl3}
\begin{document}
\ExplSyntaxOn
\cs_set:Npn \reverse_string:n #1 {
% convert token list to string
\str_set:Nn \l_tmpa_str {#1}
% use a sequence to save the string
\seq_gclear:N \g_tmpa_seq
% store each character in \l_tmpa_str in \l_tmpa_seq
% use \exp_args:NV to expand the first argument
\exp_args:NV \str_map_variable:nNn { \l_tmpa_str } { \l_tmpb_str }
{ \seq_gput_right:NV \g_tmpa_seq \l_tmpb_str }
% pop each element in \l_tmpa_str from right side
\bool_do_while:nn {
% reverse the value of \seq_if_empty_p:N
\bool_if:nTF {\seq_if_empty_p:N \g_tmpa_seq} {\c_false_bool} {\c_true_bool}
}
{
\seq_gpop_right:NN \g_tmpa_seq \l_tmpa_tl
% generate output
\tl_use:N \l_tmpa_tl
}
}
\newcommand{\reversestring}[1]{\reverse_string:n {#1}}
\ExplSyntaxOff
\reversestring{abcde12345}
\end{document}
### Output
54321edcba
• There really isn’t much choice when it comes to reversing a string in LaTeX, because LaTeX does not provide a way to read something “from behind”. In this solution, I decide to construct a stack with l3seq to reverse the order of characters. Another way of doing so may be using an integer loop with \str_item:nn. But this macro seems to work by discarding characters before the desired index, so the performance may be bad.
• The performance of this code isn’t great either. I tried to reverse 10,000 characters, and it takes around 10 seconds to finish. I think it is because the implementation of l3seq isn’t very efficient either.
• I guess the lesson learned is that don’t reverse long strings in LaTeX. Do it somewhere else!
## Tips for LaTeX programming
• Due to the expansion mechanism of LaTeX, it is very inconvenient for a macro to return value(s).
• Therefore, if the intended return value is to be put into text, simply use the result so that it is inserted into the output stream.
• If the intended return value is to be used by another macro, store this value into a global variable and use the value.
LaTeX3 has already provided a command to reverse tokens (\tl_reverse:n), which allows us to write a simpler program like so. However, after some experiments, I find out that this is no faster than my own version. This may imply that the internal implementation of this function is similar.
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{expl3}
\begin{document}
\ExplSyntaxOn
\cs_set:Npn \reverse_string:n #1 {
\tl_reverse:n {#1}
}
\newcommand{\reversestring}[1]{\reverse_string:n {#1}}
\ExplSyntaxOff
\reversestring{abcde12345}
\end{document}
It is possible to use l3intarray package to facilitate string reversal. The following code is capable of reversing 10,000 characters in no time. However, the drawbacks of this method include:
1. Big memory consumption: each character is probably saved as 4-byte integer.
2. There is a certain limit on the size of l3intarray. According to the documentation, the length of array can only be around $$4 \times 10^6$$. On LuaTeX, the length can be longer. (But who needs this solution if there is Lua support anyways.)
\documentclass{article}
\usepackage[a4paper]{geometry}
\usepackage[T1]{fontenc}
\usepackage{mathptmx}
\usepackage{amsmath, amssymb}
\usepackage{datetime2}
\usepackage{expl3}
\begin{document}
\ExplSyntaxOn
\int_new:N \l_ind_int
\cs_new:Npn \to_ascii:n #1 {
\int_eval:n {#1}
}
\cs_generate_variant:Nn \intarray_gset:Nnn {Nnx}
\cs_new:Npn \fill_intarr:n #1 {
\intarray_gset:Nnx \g_tmpa_intarr {\l_ind_int} {\to_ascii:n {#1}}
\int_incr:N \l_ind_int
}
\cs_new:Npn \reverse_str:n #1 {
\str_set:Nn \l_tmpa_str {#1}
\cs_if_exist:NT \g_tmpa_intarr {
\cs_gset_eq:NN \g_tmpa_intarr \undefined
}
\intarray_new:Nn \g_tmpa_intarr {\str_count:N \l_tmpa_str}
\int_set:Nn \l_ind_int {1}
\str_map_function:NN \l_tmpa_str \fill_intarr:n
\int_set:Nn \l_tmpa_int {\str_count:N \l_tmpa_str}
\int_do_while:nNnn {\l_tmpa_int} > {0} {
\int_set:Nn \l_tmpb_int {\intarray_item:Nn \g_tmpa_intarr {\l_tmpa_int}}
\exp_args:Nx \tl_to_str:n {\char_generate:nn {\l_tmpb_int}{12}}
\int_decr:N \l_tmpa_int
}
}
\newcommand{\reversestr}[1]{
\reverse_str:n {#1}
}
\ExplSyntaxOff
\par\DTMNow
\end{document}
|
2022-01-21 18:32:31
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4834400713443756, "perplexity": 5869.469861531896}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303512.46/warc/CC-MAIN-20220121162107-20220121192107-00055.warc.gz"}
|
https://www.biostars.org/p/354039/#354183
|
FASTQ alignment result is really bad
1
0
Entering edit mode
3.1 years ago
bharata1803 ▴ 530
I have tried salmon and bwa+htseq-count to align and get readcount. Both of the tools are failed to get any result. The salmon result is only 2% mapped and htseq-count cannot even map a single reads to transcript.
My question is, what do you think happen here? Does the sequencing machine affect this? I noticed the platform to sequence is not illumina. The original author use Lifescope to to do the RNA-seq workflow. Anyway to investigate this? I attached Fastqc result
Update:
I have tried to use bowtie to index reference genome in colorspace. The command is this:
bowtie-build -C --threads 18 HG38_90.fa hg38_90_idx
I aligned the file with this command:
bowtie --chunkmbs 500 -C -p 18 -S hg38_90_idx -1 fastq/SRR5644749_1.fastq -2 fastq/SRR5644749_2.fastq SRR5644749.sam
And the result is really bad.
# reads processed: 20826355
# reads with at least one reported alignment: 17671 (0.08%)
# reads that failed to align: 20808684 (99.92%)
Any suggestion how to align in colorspace?
RNA-Seq htseq-count salmon • 1.7k views
0
Entering edit mode
There must something seriously wrong with your setup, it's just hard to tell what. Even with bwa you should be able to achieve >60% aligned reads. However, please change to a a state-of-the-art pipeline first, using either Hisat(2) or STAR as an aligner, then HTseq-count or FeatureCount. Please check that you have the correct and matching versions of the genome or transcriptome reference (for using salmon) and genome annotation.
If you need more help, please also do and report QC at each step using FastQC and MultiQC, and please provide the exact commands run as well as relevant output of fastqc and multiQC.
0
Entering edit mode
I have provided the Fastqc report. It seems there are problems with the fastq file itself.
0
Entering edit mode
It is rare for current RNA-seq protocols to see failed sequence quality and N content. Check how the fastq extraction went well, just download and extract again using SRA toolkit. Then possibly you need to apply quality trimming or select a better dataset.
0
Entering edit mode
I have the sra file. It seems there were no problem when I use sra toolkit fastqdump.
0
Entering edit mode
Oh, these could be colorspace data, they need to be analysed differently! Try option -c with bwa and see if that improves the alignment. See http://seqanswers.com/forums/showthread.php?t=16621
0
Entering edit mode
I think the latest bwa options are different. In bwa sampe, -c is about: -c FLOAT prior of chimeric rate (lower bound) [1.0e-05]
I am not familiar with colorspace data.
0
Entering edit mode
Neither am I, sorry, but maybe you can download an older version of bwa that supports colorspace.
0
Entering edit mode
I will try bowtie. It seems bowtie support colorspace
0
Entering edit mode
Please read more about Solid data conversion to fastq here. You probably lose information with the conversion to fastq.
0
Entering edit mode
I am still confused. I have tried using bowtie to make colorspace index and align in colorspace, but it is still almost 0% reads that can be mapped to the reference. It is really weird.
0
Entering edit mode
Can you show with head how your fastq file looks like? Is it really in color space?
0
Entering edit mode
It is in colorspace. I have checked it. I will post the head result later.
0
Entering edit mode
Just to put aside anything related with SRA, you can directly download fastq files for your project here: https://www.ebi.ac.uk/ena/data/view/PRJNA389279][1]
I have looked at one file and indeed it looks like color-space:
@SRR5644749.11183773 11183773/2
T22033022222231202002113112222221030032031212322000
+
!33333333333.--------------------------------------
@SRR5644749.11183774 11183774/2
T0320120300100101212000302000232130232310..........
+
!=/////////....--------------------------**********
@SRR5644749.11183775 11183775/2
T001111001023332010010.............................
+
!@?8886666............*****************************
0
Entering edit mode
3.1 years ago
h.mon 33k
I think the problem is the data is really crappy, and Bowtie is discarding the reads because of too many errors at the seed region. I mapped the reads with Subread and got an average mapping rate of 65.9% for the same file (SRR5644749) you mapped. Even so, there were a lot of orphaned pairs:
//================================= Summary ==================================\\
|| ||
|| Total fragments : 20,826,355 ||
|| Mapped : 13,718,668 (65.9%) ||
|| Uniquely mapped : 11,529,522 ||
|| Multi-mapping : 2,189,146 ||
|| ||
|| Unmapped : 7,107,687 ||
|| ||
|| Correctly paired : 6,067,553 ||
|| Not mapped in pairs : 7,651,115 ||
|| Only one end mapped : 7,109,909 ||
|| Multi-chromosomes : 265,002 ||
|| Different strands : 290,108 ||
|| Not in PE distance : 60,605 ||
|| Abnormal order : 33,402 ||
|| ||
|| Indels : 0 ||
|| ||
|| Running time : 23.6 minutes ||
|| ||
subread-buildindex -B -c -F -o genome genome.fasta
subread-align -b -t 0 -i genome --multiMapping -B 2 \
-o SRR5644749.bam -r SRR5644749_1.fastq.gz -R SRR5644749_2.fastq.gz
Probably tweaking Subread settings should result in a higher mapping rate.
You can use FastQC on the bam file and use the -f bam_mapped flag to evaluate only mapped reads, this will produce a report ignoring unmapped read. The problem with unmapped reads is FastQC converts colorspace to basespace, but this conversion is unreliable without a reference. Mapped reads should be of good quality, and already corrected by the mapper.
P.S.: note that your bowtie mapping command is not outputting a sam file, you need to use the -S flag with Bowtie, because it has a custom default output. Initially, I didn't see the -S in your command.
0
Entering edit mode
It is a bit hard to check the quality because FastQC cannot be used for colorspace. Is there any fastq quality control software that can handle the colorspace data? If you got 65% reads, I think it is good enough. I will try using subread. Do you mind posting you command with subread so that I can reproduce it?
0
Entering edit mode
0
Entering edit mode
Thanks, I will try your method. I have tried subread but I got 20% mapped reads which is quite bad. If I can get 60% like you probably the data can be used.
0
Entering edit mode
If you got 20% from the same sample, then there is something odd - we should get the same mapping rate. But if you got 20% from other samples, this would be an indication there are samples with even worst quality than the one you posted about (and in fact, for a second sample, I got 16% mapping rate).
0
Entering edit mode
I realized I aligned it to cdna/transcript reference, not whole genome reference. I aligned to cdna reference because I wanted try to count the reads with salmon. I will try to align it to the genome then.
0
Entering edit mode
I gave up with this dataset, the fastq quality seems really bad. I have checked their processed data too and try to use DESeq2 to get the differential expression. The result is bad too with only small number of genes are giving siginificant result.
|
2022-01-23 18:46:18
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3487766981124878, "perplexity": 5962.535383560971}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304309.5/warc/CC-MAIN-20220123172206-20220123202206-00362.warc.gz"}
|
http://www.java-gaming.org/index.php?topic=33891.0
|
Java-Gaming.org Hi !
Featured games (91) games approved by the League of Dukes Games in Showcase (757) Games in Android Showcase (229) games submitted by our members Games in WIP (844) games currently in development
News: Read the Java Gaming Resources, or peek at the official Java tutorials
Home Help Search Login Register
Pages: [1]
ignore | Print
New feature: Embedded PDFs (Read 6551 times) 0 Members and 1 Guest are viewing this topic.
Riven
Administrator
« JGO Overlord »
Medals: 1341
Projects: 4
Exp: 16 years
Hand over your head.
« Posted 2014-07-17 20:02:09 »
Due to a certain observed need and some spare time, I implemented the feature of embedding PDFs in your forum post.
There are certain undisclosed rules to mitigate the possibility of random people embedding malicious PDFs. The default PDF reader of the browser is used. In the case that there is no default PDF reader associated with the browser, which is often the case on phones and tablets, a download of the file is initiated. I'll work no this later, to prevent such 'unexpected' downloads on potentially bandwidth limited devices.
The PDF is (crudely) validated, cached and served from the JGO server, to ensure the content is not replaced with malicious bits and bytes after the one time validation has occured.
Quote from: Legal Department
By embedding a PDF file in a post, you acknowledge that duplication and redistribution occurs and grant JGO full rights to serve the content to any third party. It is forbidden to embed content that has a license which does not grant these rights to JGO.
The UBB syntax for embedding a PDF file is as follows:
[pdf ]http://java-gaming.org/pdf-sample.pdf[/pdf]
The content box is 95% of the available width in the post and 950px long.
(click this page to load the PDF)
In other news: support for embedded applets has been terminated.
Hi, appreciate more people! Σ ♥ = ¾
Learn how to award medals... and work your way up the social rankings!
basil_
« JGO Bitwise Duke »
Medals: 418
Exp: 13 years
« Reply #1 - Posted 2014-07-17 20:04:52 »
thank you!
ctomni231
JGO Wizard
Medals: 99
Projects: 1
Exp: 7 years
Not a glitch. Just have a lil' pixelexia...
« Reply #2 - Posted 2014-07-18 05:35:13 »
my game...
Does this work for multi-page PDF's as well... How big can the PDF's be? (It might be good to just make articles in PDF format so they actually look good.) Not that I'm writing one...
Games published by our own members! Check 'em out!
Riven
Administrator
« JGO Overlord »
Medals: 1341
Projects: 4
Exp: 16 years
Hand over your head.
« Reply #3 - Posted 2014-07-18 06:44:24 »
Your PDF can have any number of pages. As for existing articles, converting them to PDFs would be a ton of work.
And... what about your game? Are you refering to the termination of the applet tag? Maybe it's a relief to know hardly anybody could execute the applet anyway
Hi, appreciate more people! Σ ♥ = ¾
Learn how to award medals... and work your way up the social rankings!
pjt33
« JGO Spiffy Duke »
Medals: 40
Projects: 4
Exp: 7 years
« Reply #4 - Posted 2014-07-18 08:36:13 »
Due to a certain observed need
If you want people to be able to use nicely formatted formulae, you could install MathJax. Or point people at http://mathurl.com/
In the case that there is no default PDF reader associated with the browser, which is often the case on phones and tablets, a download of the file is initiated. I'll work no this later, to prevent such 'unexpected' downloads on potentially bandwidth limited devices.
It would also be useful to not display it other than in the main thread view page. I had to cancel a second download when I opened up this reply page.
Riven
Administrator
« JGO Overlord »
Medals: 1341
Projects: 4
Exp: 16 years
Hand over your head.
« Reply #5 - Posted 2014-07-18 18:42:03 »
SMF doesn't provide that information, so I'm tempted to implement it as click-to-view.
Hi, appreciate more people! Σ ♥ = ¾
Learn how to award medals... and work your way up the social rankings!
SHC
« Reply #6 - Posted 2014-07-19 12:22:05 »
I think instead of initiating downloads automatically, you can just keep a link to download the file. Just a small suggestion.
EgonOlsen
« Reply #7 - Posted 2014-07-19 13:24:08 »
This feature spams my download folder on Android with a new file each time i'm reloading the page. Not very nice...
Riven
Administrator
« JGO Overlord »
Medals: 1341
Projects: 4
Exp: 16 years
Hand over your head.
« Reply #8 - Posted 2014-07-19 14:02:58 »
This feature spams my download folder on Android with a new file each time i'm reloading the page. Not very nice...
Should be solved now.
Update: some mobile browsers download content when it is styled "display:none" - give me a few more minutes
Update 2: click-to-view should work properly now
Hi, appreciate more people! Σ ♥ = ¾
Learn how to award medals... and work your way up the social rankings!
Mac70
« Reply #9 - Posted 2014-07-19 16:37:11 »
Sadly this feature don't work on Opera - even though it is using the same engine as Chrome. After clicking "click this page to load the PDF" download is initalized, but nothing appears.
Games published by our own members! Check 'em out!
tkausl
Junior Devvie
Medals: 3
Exp: 5 years
« Reply #10 - Posted 2014-07-19 16:42:23 »
May you implement a small Link right above the Box to open the PDF in a new Window?
I dont like to read in small Fonts and if i zoom in i have to scroll horizontally and this sucks
My English isnt that great. Correct me, if you want, im still learning this Language
SilverTiger
JGO Coder
Medals: 40
Exp: 3 years
がんばってください！
« Reply #11 - Posted 2014-07-19 16:48:02 »
Sadly this feature don't work on Opera - even though it is using the same engine as Chrome. After clicking "click this page to load the PDF" download is initalized, but nothing appears.
I'm using Opera 12.17 (64-bit) and it works fine for me
Don't know if there's a different engine with other versions.
Mac70
« Reply #12 - Posted 2014-07-19 16:53:35 »
You are using very old version, newest one is 22.0.1471.70. You need to download it manually as Opera don't have auto update feature (at least in old versions, I am not sure if new versions have it or not).
Cero
« Reply #13 - Posted 2014-07-19 18:41:33 »
That's not really true. Opera 12.17 and 12.18 are the last presto operas, meaning real operas. Everything after that is just chrome anyway (opera next)
Riven
Administrator
« JGO Overlord »
Medals: 1341
Projects: 4
Exp: 16 years
Hand over your head.
« Reply #14 - Posted 2014-07-21 19:11:46 »
May you implement a small Link right above the Box to open the PDF in a new Window?
I dont like to read in small Fonts and if i zoom in i have to scroll horizontally and this sucks
Thanks for your feedback - I added a link (in the lower half of the page) to open the document in a new tab.
Hi, appreciate more people! Σ ♥ = ¾
Learn how to award medals... and work your way up the social rankings!
Kefwar
« Reply #15 - Posted 2014-07-21 19:40:12 »
Great feature!
One notice:
1 "Failed to load resource: net::ERR_CONNECTION_RESET http://www.prensa-latina.cu/images/stories/LibrosGratis/pdf-icon.png"
The pdf icon isn't displaying.
Riven
Administrator
« JGO Overlord »
Medals: 1341
Projects: 4
Exp: 16 years
Hand over your head.
« Reply #16 - Posted 2014-07-21 19:42:49 »
Whoops, accidently hot linked the icon Fixed
Hi, appreciate more people! Σ ♥ = ¾
Learn how to award medals... and work your way up the social rankings!
SHC
« Reply #17 - Posted 2014-07-22 12:50:42 »
Riven, I found something that maybe of interest to you. There is a pure JS PDF renderer by mozilla called PDF.JS. And it uses WebGL to render that PDF in the browser.
Example
Pages: [1]
ignore | Print
EgonOlsen (78 views) 2018-06-10 19:43:48 EgonOlsen (58 views) 2018-06-10 19:43:44 EgonOlsen (78 views) 2018-06-10 19:43:20 DesertCoockie (260 views) 2018-05-13 18:23:11 nelsongames (158 views) 2018-04-24 18:15:36 nelsongames (157 views) 2018-04-24 18:14:32 ivj94 (898 views) 2018-03-24 14:47:39 ivj94 (162 views) 2018-03-24 14:46:31 ivj94 (811 views) 2018-03-24 14:43:53 Solater (175 views) 2018-03-17 05:04:08
Java Gaming Resourcesby philfrei2017-12-05 19:38:37Java Gaming Resourcesby philfrei2017-12-05 19:37:39Java Gaming Resourcesby philfrei2017-12-05 19:36:10Java Gaming Resourcesby philfrei2017-12-05 19:33:10List of Learning Resourcesby elect2017-03-13 14:05:44List of Learning Resourcesby elect2017-03-13 14:04:45SF/X Librariesby philfrei2017-03-02 08:45:19SF/X Librariesby philfrei2017-03-02 08:44:05
java-gaming.org is not responsible for the content posted by its members, including references to external websites, and other references that may or may not have a relation with our primarily gaming and game production oriented community. inquiries and complaints can be sent via email to the info‑account of the company managing the website of java‑gaming.org
|
2018-06-24 09:40:35
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.263656347990036, "perplexity": 12803.18373059496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267866926.96/warc/CC-MAIN-20180624083011-20180624103011-00058.warc.gz"}
|
https://stats.stackexchange.com/questions/548823/how-is-confidence-interval-computed-in-statsmodels-for-arimap-d-q-parameters
|
# How is confidence interval computed in statsmodels for ARIMA(p,d,q) parameters?
I am using statsmodel package for fitting ARIMA(p,d,q) model to a time series. My question is how exactly does this package estimate confidence intervals of the parameters of this model? statsmodels documentation says that
"The confidence interval is based on the standard normal distribution if self.use_t is False. If self.use_t is True, then uses a Student’s t with self.df_resid_inference (or self.df_resid if df_resid_inference is not defined) degrees of freedom."
Then the question is how is the variance of different parameters estimated to apply the standard Normal or t-distribution method?
Edit: I used the Hessian matrix method to compute the covariance matrix. But the confidence interval obtained using my approach are much wider than those produced by statsmodels. Which means that statsmodels is not using the Hessian matrix approach. Also, I noticed that as I increase the length of my time-series, the confidence intervals obtained by these approaches become similar.
The documentation for the cov_type argument to fit method describes the options for computing the covariance matrix associated with the parameter estimates.
The default method is the outer product of gradients estimator (cov_type='opg').
If you want to use the numerically approximated Hessian, you can choose cov_type='approx', but note that by default this will use complex step differentiation rather than finite differences, so it may still be different from what you compute by hand.
|
2022-08-19 17:51:47
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6861783862113953, "perplexity": 682.8315616575637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573744.90/warc/CC-MAIN-20220819161440-20220819191440-00026.warc.gz"}
|
https://bakingandmath.com/tag/knot-theory/
|
Tag Archives: knot theory
## Some thoughts on knots: current research
8 Mar
Over the weekend I went to the Third(!!!) annual Midwest Women in Mathematics Symposium (remember when I founded it? Now it’s all fancy with funding and many attendees and event staff!) As it turned out, not very much of the math in my parallel session was exactly up my alley, and also I was feeling lazy so I didn’t take many notes. But here’s a small recap/introduction to knot theory from my memory.
Aside: I like using knot theory as an example when people ask me what math is for (this happened a lot as an undergraduate and less and less as the years go by). I’m not even sure if this is true, but I tell people that mathematicians were studying knot theory for decades, and then biologists realized that they could use it to study how proteins fold and interact with other molecules. APPLIED! IN YOUR FACE, MATH DOUBTERS! Unclear where I picked up this bit of folklore, but it’s my number one defense when people say that modern math research is useless.
So what is knot theory? It’s certainly not not-theory, despite my claim as above that it can be applied. Knot theory studies objects called knots. A knot is some way that a circle is embedded in space- imagine taking a shoelace, knotting it up however you want to, and gluing the ends together. (By space knot theorists mean $S^3$, but we can just think of it as $\mathbb{R}^3$, or the space we live in). To talk about knots, knot theorists draw knots as diagrams using over and under crossings. Two diagrams can represent the same knot, like in the picture below.
Even though they look different, these knots are the same. They’re called the unknot.
If I didn’t mess up, the blue knot is the same as the orange knot- just follow the crossings and you’ll see that nothing is actually knotted; it’s just a pile of string lying on top of itself. Below are some pictures of other knots.
It’s hard to tell if two diagrams represent the same knot. Mathematicians can use a diagram to assign polynomials to a knot, and do it in such a way that if two diagrams represent the same knot, then they give the same polynomial. Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial (which generalizes the previous two). These still aren’t that great though, since two different knots can give the same polynomial (so while you can tell if your diagram ISN’T the unknot by seeing if the polynomial isn’t 1, you can’t tell if it IS the unknot if the polynomial gives you 1).
The first knot theory talk I saw connected knots to surfaces, so I was a fan. It was given by Effie Kalfagianni, a professor at Michigan State. One thing you can do with a knot is use it as the boundary of a surface.
From wikipedia: I was having a really hard time making my own pictures.
There are different ways to make a surface from a particular knot- draw a different diagram and you’ll get a different surface. One thing you can study is the genus of a knot: this is defined as the minimum genus (# holes) of a surface bounded by that knot. So for any diagram you draw, you can’t make a surface with a smaller number of holes. The genus of the unknot is 0. The genus of a knot using orientable surfaces is known, and there’s an efficient algorithm to find it. BUT the problem is open for non-orientable surfaces (these are surfaces that don’t have two sides).
“Sometimes I feel like I can’t trust you… it’s like you’re two sided.”
So Kalfagianni’s research, joint with her student Christine Lee, puts a bound on the non-orientable genus of alternating knots, which are knots with diagrams that alternate between over and under crossings (alternating: the purple and red knots. Not alternating: the unknot, either blue knot (there are two over crossings in a row)). They use one of the factors in the Jones polynomial to do so.
So that was talk number one! The second talk I saw was by Maggy Tomova, an assistant professor at University of Iowa. I actually didn’t write any notes down for her talk, but I remember a cool concept from it. A knot diagram is in bridge position if you can draw a line across the middle so that there are only local maxima above it and local minima below.
GET IT? It’s a visual pun! The green is in bridge position. The red is not.
One immediate note is that in general, bridge position is not unique: given a knot in bridge position, you might be able to find another diagram in bridge position that represents the same knot. There are some properties that ensure that a bridge position is unique (this is a theorem that I don’t remember). Tomova is working on some theorems that have to do with knots in bridge position, and I’m sorry that I can’t tell you more information. She did her Ph.D. at UCSB though, with the same advisor as some delightful other people who are her co-authors on this project (the delightful only applies to the first link; I don’t actually know her other co-author but I really like Yoshi and the fact that he goes by Yoshi). Also, one of her previous co-authors taught me abstract algebra when I was an undergraduate and he was a postdoc! That link is to a piece he wrote on going to the “Dark Side,” a.k.a. leaving academia for Google.
So I am not a knot theorist, but there’s your post with thoughts on knots!
## Intrinsically knotted graphs on 21 edges
4 Apr
I was skimming through http://www.arxiv.org the other day and found this paper by a student named Barsotti and a professor named Mattman. Barsotti is/was an undergraduate at CSU Chico and this is his honors thesis, while Mattman is a professor at CSU Chico. Weirdly enough it looks like Mattman also directed the undergraduate theses of two of my colleagues, Arielle Leitner from UCSB (wow, nice website! way better than mine) and Ryan Ottman. Small world.
So let’s talk a little bit about this paper, shall we? I gave a half hour talk on it in our little seminar on Tuesday. Graph theory is one of those fields that is ridiculously useful. Computer scientists and anyone who plays with data loves graph theory. Knot theory is also surprisingly useful since math biology is blowing up right now with knotting of DNA and molecules and such. This paper lies in the intersection of graph theory and knot theory.
Graph theory. From my last math post you know what a graph is: a collection of vertices with some edges connecting them. We’ll be talking about a certain operation you can do to graphs, taking a minor. A minor of a graph is a graph that you derive from your original graph. There are three different ways to find minors of a graph: 1) delete a vertex and all the edges connected to it, 2) delete an edge, 3) contract an edge. The picture shows examples of all three of these. For number 3, you delete an edge, and you put the two vertices at the end of it together. Note that the new vertex has 4 edges attached to it, or has degree 4, while the original two each had degree 3.
The big one is ok, but the other three are jail bait.
So there’s this big old theorem from Robertson and Seymour, called the Graph Minor Theorem, which says a couple of things. For one, if you have infinitely many graphs, you’re definitely holding one that is a minor of another in your hand. Another way to think of the theorem is if you have a collection of graphs which is closed under minors (e.g. for any graph in your collection, all of the minors of it are in there), you can define this collection by a finite set of forbidden minors.
A quick example is the collection of all trees (graphs with no cycles or loops in them). In our picture, 1 and 2 are both trees, while the original graph and 3 both have a loop and so aren’t a tree. Trees are closed under minors, since taking a minor of a tree will never make a new loop. Then the theorem says there is a finite set of graphs which cannot be minors of any tree, and that the set of trees is characterized by not having these as minors. So to be a tree, it’s necessary and sufficient, as mathematicians like to say, that you have no forbidden minor as a minor. In the case of trees, the minor is this:
Lady C… er no I meant Sir Cle
This is a loop with a single vertex on it. If you’re a graph that’s not a tree, you can do a finite sequence of taking minors (e.g. contract the edges in a cycle) until you end up with a single vertex with a loop. If you are a tree, you can never do this. Robertson and Seymour’s graph minor theorem took 20 papers to prove. It’s pretty insane. But this is how you use it!
So a fun thing that undergraduates and professors do is try to take a family closed under minors and find the finite forbidden minor set. In the case of a tree, there’s a single forbidden minor. To talk about other graphs and forbidden minors, let’s switch gears for a moment and go to knot theory.
Trefoil! Versus treplasticwrap. That was a way worse name
Knot theory. A knot is an embedding of the circle in some crazy way in some crazy place. Right now we’ll just talk about knots embedded in three space (the world we live in, $\mathbb{R}^3$). Embedding is a technical word, but just think of it as putting something somewhere. In fact, think of a knot as a piece of string, lying one end on the ground, throwing the other end all over itself (looping in and out) and then gluing the two ends together. Or put in a bunch of ropes, so long as you glue all the ends together to make one continuous loop.
Not a knot… not yet
In this picture, we’d have to glue a gray end and a green end together, and do the same with the other pair, to have an actual knot. Or you could glue the green ends together and the grey ends together, in which case you would have two knots linked together.
Lots of knots secretly look very simple, like a circle:
Even crazy complicated ones. Look at this diagram to see what I mean about ‘secretly looking like’:
This diagram is from a paper by Henrich and Kauffman from 2010 (disclaimer: Kauffman is at my school). [I don’t know why I put in that disclaimer.]
But the trefoil, above, for example, doesn’t secretly look like a circle. No matter how you pull and prod the strands around, you can’t get a plain old circle again.
This brings us back to graph theory. A graph is intrinsically knotted if, no matter how you draw it in $\mathbb{R}^3$, you end up with a not-boring knot. This paper I just read lists all the intrinsically knotted graphs with 21 edges (as you’d expect from the title). The proof uses some results from Robertson and Seymour’s graph minor theorem. For instance, it uses the fact that all non-planar graphs are characterized by two forbidden minors- this is Kuratowski’s theorem and I’ll do another post on it sometime. The proof uses a lot of cool little tricks, but this is just an intro post so I’ll save those for another time.
In case you were wondering, there are 14 intrinsically knotted graphs with 21 edges. That’s the takeaway!
|
2019-11-13 10:50:11
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 4, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6200428605079651, "perplexity": 1065.0257715168982}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496667177.24/warc/CC-MAIN-20191113090217-20191113114217-00381.warc.gz"}
|
http://eprint.iacr.org/2009/397
|
## Cryptology ePrint Archive: Report 2009/397
Linear Cryptanalysis of Reduced-Round PRESENT
Joo Yeon Cho
Abstract: PRESENT is a hardware-oriented block cipher suitable for resource constrained environment. In this paper we analyze PRESENT by the multidimensional linear cryptanalysis method. We claim that our attack can recover the 80-bit secret key of PRESENT up to 25 rounds out of 31 rounds with around $2^{62.4}$ data complexity. Furthermore, we showed that the 26-round version of PRESENT can be attacked faster than key exhaustive search with the $2^{64}$ data complexity by an advanced key search technique. Our results are superior to all the previous attacks. We demonstrate our result by performing the linear attacks on reduced variants of PRESENT. Our results exemplify that the performance of the multidimensional linear attack is superior compared to the classical linear attack.
Category / Keywords: Block Ciphers, Lightweight Cryptography, PRESENT, Multidimensional Linear Cryptanalysis
Publication Info: CT-RSA 2010
Date: received 13 Aug 2009, last revised 25 Jan 2010
Contact author: joo cho at tkk fi
Available format(s): Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation
Note: Section 3.2 and 3.3 have been revised. Thanks to Kaisa Nyberg.
Short URL: ia.cr/2009/397
[ Cryptology ePrint archive ]
|
2016-07-28 13:33:00
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2800099551677704, "perplexity": 6163.269445777425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257828282.32/warc/CC-MAIN-20160723071028-00171-ip-10-185-27-174.ec2.internal.warc.gz"}
|
https://zbmath.org/?q=an%3A1091.35046
|
## Geometric lower bounds for the spectrum of elliptic PDEs with Dirichlet conditions in part.(English)Zbl 1091.35046
Summary: An extension of the lower-bound lemma of Boggio is given for the weak forms of certain elliptic operators, which are in general nonlinear and have partially Dirichlet and partially Neumann boundary conditions. Its consequences and those of an adapted Hardy inequality for the location of the bottom of the spectrum are explored in corollaries wherein a variety of assumptions are placed on the shape of the Dirichlet and Neumann boundaries.
### MSC:
35P15 Estimates of eigenvalues in context of PDEs 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35J25 Boundary value problems for second-order elliptic equations
Full Text:
### References:
[1] Arrieta, J.M., Neumann eigenvalue problems on exterior perturbations of the domain, J. differential equations, 118, 54-103, (1995) · Zbl 0860.35086 [2] Boggio, T., Sull’equazione del moto vibratorio delle membrane elastiche, Accad. lincei, sci. fis. ser. 5a, 16, 386-393, (1907) · JFM 38.0813.02 [3] Burenkov, V.I.; Davies, E.B., Spectral stability of the Neumann Laplacian, J. differential equations, 186, 2, 485-508, (2002) · Zbl 1042.35035 [4] R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. II, Interscience Wiley, New York, 1962 (original publication 1937). · Zbl 0788.00012 [5] Courtois, G., Spectrum of manifolds with holes, J. funct. anal., 134, 194-221, (1995) · Zbl 0847.58076 [6] E.B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, MA, 1989. · Zbl 0699.35006 [7] E.B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, MA, 1995. · Zbl 0893.47004 [8] Drábek, P.; Krejčí, P.; Takáč, P., Nonlinear differential equations, (1999), CRC Press Boca Raton, FL [9] P. Drábek, A. Kufner, F. Nicolosi, Nonlinear Differential Equations, Singular and Degenerate Case, University of West Bohemia, Pilsen, Czech Republic, 1996. [10] Edmunds, D.E.; Evans, W.D., Spectral theory and differential operators, (1987), Clarendon Press Oxford · Zbl 0628.47017 [11] L.C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. · Zbl 0902.35002 [12] Evans, W.D.; Harris, D.J., On the approximation numbers of Sobolev embeddings for irregular domains, Quart. J. math. Oxford, 40, 2, 13-42, (1989) · Zbl 0681.46033 [13] Fleckinger, J.; Harrell II, E.M.; Thélin, F.de., Boundary behavior and $$L^q$$ estimates for solutions of equations containing the p-Laplacian, Electron. J. differential equation, 1999, 1-19, (1999) · Zbl 0928.35046 [14] Fraenkel, L.E., On regularity of the boundary in the theory of Sobolev spaces, Proc. London math. soc., 39, 3, 385-427, (1979) · Zbl 0406.46026 [15] Hardy, G.H., Note on a theorem of Hilbert, Math. Z., 6, 314-317, (1920) · JFM 47.0207.01 [16] Hardy, G.H.; Littlewood, J.E.; Pólya, G., Inequalities, (1959), Cambridge University Press Cambridge, MA, (original publication 1934) · Zbl 0634.26008 [17] E.M. Harrell II, Lecture at the Workshop on Partial Differential Equations and Fractals, Toulouse, France, 1993, unpublished. [18] Hempel, R.; Seco, L.A.; Simon, B., The essential spectrum of Neumann Laplacians on some bounded singular domains, J. funct. anal., 102, 448-483, (1991) · Zbl 0741.35043 [19] C. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics, vol. 83, American Mathematical Society, Providence, RI, 1994. · Zbl 0812.35001 [20] Transl. Math. USSR IZV (1973) 357-387. [21] Maz’ja, V.G., Sobolev spaces, (1985), Springer New York, (first published 1981) [22] McGillivray, I., Capacitary estimates for Dirichlet eigenvalues, J. funct. anal., 139, 244-259, (1996) · Zbl 0896.47017 [23] McGillivray, I., Capacitary asymptotic expansion of the groundstate to second order, Comm. partial differential equations, 23, 2219-2252, (1998) · Zbl 0922.47002 [24] B. Opic, A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics, vol. 219, Boston, Longman,1990. · Zbl 0698.26007 [25] Ozawa, S., Singular variation of domains and eigenvalues of the Laplacian, Duke math. J., 48, 767-778, (1981) · Zbl 0483.35064 [26] Ozawa, S., An asymptotic formula for the eigenvalues of the Laplacian in a domain with a small hole, Proc. Japan acad. ser. A, 58, 5-8, (1982) · Zbl 0516.35015 [27] Ozawa, S., Asymptotic property of an eigenfunction of the Laplacian under singular variation of domains—the Neumann condition, Osaka J. math., 22, 639-655, (1985) · Zbl 0579.35065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
|
2022-08-19 07:37:30
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7247653603553772, "perplexity": 2307.7951994715027}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573630.12/warc/CC-MAIN-20220819070211-20220819100211-00557.warc.gz"}
|
https://www.zenoss.com/sites/default/files/zenoss-doc/8996/analytics/old/install/secure-ssl.html
|
# Securing Analytics to use SSL
Analytics uses Apache and mod_ssl to provide SSL for all of the different types of communication needed. It is assumed that if you are running Resource Manager 4.x you already have secured the Resource Manager behind SSL (this is in place as a matter of course in Resource Manager 5.x and 6.x). This is a required prerequisite for securing Analytics.
The following procedure will install Apache and mod_ssl and set it to use a self-signed SSL certificate. You may also choose to purchase a SSL certificate signed by a third-party Certificate Authority or to generate your own SSL certificate.
Log in to the Analytics server as the root user, or as a user with superuser privileges. Install Apache and mod_ssl, configure Apache to start on server boot and start it for the first time:
yum -y install httpd
yum -y install mod_ssl
systemctl enable httpd
systemctl start httpd
You can check this was successful by visiting both http://<analytics server fqdn>/ and https://<analytics server fqdn/ in a web browser.
To support potential use of Internet Explorer 8 (IE8), Apache must be configured to strip out the "Pragma" statements from the headers of HTTP files. To do this, navigate to the following Apache configuration folder and edit the config file as follows:
cd /etc/httpd/conf
# Backup the existing config file
cp httpd.conf original_httpd.conf_original
# Edit the file
vi httpd.conf
# Add the following line right at the top of the file.
Header unset Pragma
Save the file and exit the editor.
Next, we configure SSL to add an internal proxy rule for Apache to proxy any request to the Analytics server and to turn on the Rewrite Engine. Navigate to the Apache SSL configuration folder and edit the SSL config file as follows:
cd /etc/httpd/conf.d
# Backup the existing config file
cp ssl.conf original_ssl.conf_original
# Edit the file
vi ssl.conf
The last line of the file should be the closing tag </VirtualHost>. Add the following text just above this closing </VirtualHost> tag:
#Internal proxy rules instructing Apache to proxy any request to the
#Analytics server and data warehouse on 7070
ProxyPass /reports http://127.0.0.1:7070/reports
ProxyPassReverse /reports http://127.0.0.1:7070/reports
ProxyPass /etl http://127.0.0.1:7070/etl
ProxyPassReverse /etl http://127.0.0.1:7070/etl
#Turn on the RewriteEngine
RewriteEngine On
#Redirect any just / over to /reports
RewriteRule ^/+\$ https://%{SERVER_NAME}:443/reports/ [R]
Save and close the ssl.conf file and then restart Apache.
systemctl restart httpd
Next we lockdown tomcat to localhost only so that the Analytics server will not respond to requests on its internal port (7070). An alternate solution is to simply close port 7070 altogether via firewall configuration. Note that if you are intending to use 3rd party tools with Jaspersoft you should NOT lockdown this port or make this server level config change.
Log in to the Analytics server as the root user, or as a user with superuser privileges and navigate to the server configuration file and edit it as follows:
cd /opt/zenoss_analytics/conf
# Make a backup of the server.xml file.
cp server.xml original_server.xml_original
# Edit the file
vi server.xml
Locate the following section in the file (/7070 in vi will locate it).
<Connector port="7070" protocol="HTTP/1.1"
connectionTimeout="20000"
redirectPort="8443"/>
Change it to add in address="127.0.0.1" so that the section looks like the following:
<Connector port="7070" address="127.0.0.1" protocol="HTTP/1.1"
connectionTimeout="20000"
redirectPort="8443"/>
Save and close the file and restart tomcat by restarting the service to pick up the changes.
service zenoss_analytics stop
service zenoss_analytics start
This completes the Analytics Server installation. Proceed with the next section to install extraction daemon services in Resource Manager and to connect the Analytics server into your Resource Manager deployments.
|
2019-01-22 08:05:13
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17399810254573822, "perplexity": 11474.533241864567}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583831770.96/warc/CC-MAIN-20190122074945-20190122100945-00403.warc.gz"}
|
https://www.transtutors.com/questions/e9-8-rodriguez-inc-is-preparing-its-direct-labor-budget-for-2014-from-the-following--1356585.htm
|
# E9-8 Rodriguez, Inc., is preparing its direct labor budget for 2014 from the following production...
E9-8 Rodriguez, Inc., is preparing its direct labor budget for 2014 from the following production budget based on a calendar year.
Quarter Units Quarter Units 1 20,000 3 35,000 2 25,000 4 30,000
Each unit requires 1.5 hours of direct labor.
Instructions
Prepare a direct labor budget for 2014. Wage rates are expected to be $16 for the first 2 quarters and$18 for quarters 3 and 4.
Direct labor Budget Direct labor required = Units produced × Direct labor hour per unit Direct labor budget for the quarter = Direct labor required × Rate per hour Direct labor required...
|
2018-07-23 00:21:38
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3846287429332733, "perplexity": 12297.62243982672}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676594675.66/warc/CC-MAIN-20180722233159-20180723013159-00087.warc.gz"}
|
https://mathtuition88.com/tag/psle/
|
## Homeschool Math Challenging Puzzles
The questions listed are also very suitable as PSLE Challenging Math Problem Sums.
Basically, for Grade 2-4, they are very challenging.
For Grade 6 (Primary 6, 12 year old students), they are challenging math questions.
Do give it a try and see if your child can solve it!
## Homeschool Math Challenging Questions
Mathtuition88 will be starting a series of Homeschool Math Challenging Problems, aimed at age 8 to 10 (Grade 2 to 4).
This series is targeted at kids age 8 to 10 who are strong / gifted at mathematics and wish to further stretch their potential. It is also useful for children who may not be strong in math at the moment, but have a keen interest in math nonetheless.
In particular, it is very suitable for the following purposes:
• Preparation for GEP (Gifted Education Programme) screening and selection tests
• Puzzles for kids interested in math but find school work too easy.
• PSLE challenging Math problem sums. The questions are also well within the PSLE Math Syllabus, and will be challenging to Primary 6 students as well.
This series of questions will follow the Singapore Math syllabus for Grade…
View original post 66 more words
## Did you know that PSLE Score Has Decimal Points?
When students receive their PSLE Score, it is in the form of a 3-digit number (e.g. 240, 252, etc.). Few people actually know that it is rounded off, and your actual PSLE score has decimal points!
Source: Straits Times
•First, pupils are ranked according to detailed aggregate scores that extend to decimal points. The No. 1 pupil is posted to the school at the top of his list of six choices. Likewise the second pupil and so on, until there are no more vacancies in the school. The pupil who fails to get his top choice will be posted to the next school on his list. If that school is also full, he will be sent to his third-choice school, and so on.
Hence, if your score is the same as the Cut-off Point (COP), there is a minuscule chance that you may not get in, reason being that of the decimal points. For instance, if the school Cut-off Point is publicized as 250, it may be the case that the last person to be admitted has a score of 250.36. Hence, if your PSLE score is 250.18, you would not be admitted into the school.
Case Study from Kiasuparents:
Online wrote:My girl at 254 could not get MGS O levels (1st choice). 2nd was MGS IP, 3rd NJC IP. Got posted to 4th choice – SCGS IP. The cut-off for MGS O levels is more than 254!
@ Online, don’t panic. It is possible that the cop is 254 and your girl missed by decimal points. I think you can call MGS to find out the COP for O level. If miss by decimal points you can appeal.
In that case, you may appeal and there is a chance that you will be admitted, subject to how many cases the school can handle. See also Appeal from RGS to NYGH (Success).
## GEP PSLE Discussion
It is well known that GEP students spend Primary 4 and 5 doing miscellaneous stuff that appear to be not directly related to PSLE, yet majority of GEP students score 250 and above for PSLE. Hence, the main question is:
## Does GEP help PSLE?
Kiasuparents has a very robust discussion going on, and here are some of the insightful snippets. For the full discussion, check out the Kiasuparents forum.
Summary: GEP only starts preparation for PSLE at P6 April (!!) For PSLE, students need to be exam smart and be careful to answer “according to the marking scheme”. GEP teaches many higher level concepts which are not in the syllabus, for example “good bacteria”. Content wise, GEP English/Math/Science at P4-P5 is already at or surpassing the mainstream PSLE level. For Chinese/Mother Tongue, there is not much difference between GEP and usual syllabus.
## Viewpoint 1: GEP helps PSLE but not directly
bhcbl wrote:After DS 3 years journey in the program, my thoughts are as follow:
– Does GEP help PSLE … NoNot directly. I do think my son did better in PSLE than he would have done in his previous school coz his classmates were stronger, and because the teaching suited him. Revision was short and sharp, and he didn’t have to do too many revision papers
– Does GEP help DSA … No (definitely not directly under the new policy, maybe the exposure in broader Math and Science help HDP in a way) No, because GEPpers don’t have a special pass to DSA any more. But the GEP education does provide more opportunities to develop knowledge in various areas that could help in the interviews and portfolios
– If we get to choose again, will we still opt to be in the program … yesAgree with everything below
Stretches … eg. Book review on books for older reader, more difficult Math concept …
Broaden … eg. Ancient civilizations, magic square, ancient number system, history
Self-learning … various research and projects (at least 4-5 a year)
Time management … homework, projects, research, self-studies
Filing … they have no textbook
Also, I think it was just alot more fun for DS (after some confusion in P4 while getting used to the system). He enjoyed classes, enjoyed his classmates (mostly), and I loved the smaller classes and the opportunities for various additional classes / camps etc.
So as you can see, in the three years, most of their times are spend on thing that have nothing to do with PSLE or PSLE prep until around P6 Apr.
Think about the following: in English, they learn to infer and read between the lines, but in PSLE compre, if you do that likely you will get marks deducted; Have parents experience the problem of teaching your child Math … using algebra to solve, and then convert to the model method to explain … this is some of the things that Gep students need to go through for Math PSLE (imagine for a 12 years old child). Or PSLE syllabus issue … if you ask any Gep student whether bacteria is good or bad … they will tell you both and give you a good list of examples … but too bad that is not in PSLE syllabus: Bacteria is bad (marks likely deducted if you start talking about bacteria is good).
Our take: the GEP system is a great system, but the problem is in the past they don’t need to rely on PSLE because there is DSA, but now they have to with the change in MOE policy. It is quite cruel to do that to a child… teach them more boarder and advance stuff but in exam ask them to forget about all these.
To us, we think that education is a journey, while PSLE is just a small part of the journey. Therefore, we will still go for GEP if we could choose again. Lastly, you will see most GEP students don’t need to adjust to secondary school: 8 subjects … they are already used to very pack schedule; additional topic like history, literature, geography … these are covered in one way or another in their 3 years program (but not tested in PSLE).
So hope that this could help you in your decision in your child education journey. If you are hoping that GEP mean PSLE or DSA or good secondary … under the new policy, you may be disappointed. If you are hoping to have a challenging and stretch program for your child … likely you won’t be disappointed.
## Viewpoint 2: GEP definitely helps PSLE (and DSA too)
(by entei17) Does GEP help PSLE
Yes, definitely. In fact two years in advance. If you look through the learning scope and common tests papers for English, Higher Chinese, Maths and Science, the P4 Geppers are already doing P5-6 stuff in the mainstream.
English already has situational writing which is not introduced till P5 mainstream. Vocabulary and Grammar are pitched at P5-6 or maybe secondary levels. Comprehension format is very similar to PSLE, except that the passage is lengthier and more difficult to understand (need inference) than PSLE. As revision and practice for common test, I let my child do past year PSLE papers instead, because the P4-5 revisions books are not compatible. All the Extensive Reading List assignments, Single Shard literature, Synthesis and Transformation etc also help to build up their English foundation, so they can read more and advanced their appreciation of the language. Some of the assignments are an overkill, requiring a Herculean effort, or mission impossible. They have a comprehensive English curriculum, but I’m not sure whether all the Geppers could keep up.
Higher Chinese uses the same textbooks as the mainstream i.e. 生字 are similar, but the test/exam paper formats are similar to P5-P6/PSLE. Geppers would have no problems adjusting to the PSLE requirements. As GEP selection test only tested English, Math and General Ability, Geppers’ average Chinese ability are generally on par with the mainstream. Those that are good in English are generally not good at Chinese, although there are a handful who are good at both, or in all the subjects. I don’t have any evidence to back this up, but one reasons why some Geppers who didn’t score well in the PSLE are probably pulled down by their Chinese.
Maths are pitched at P5-P6 mainstream topics, but enriched and more interesting. I’ve seen many mentioned about Ancient civilizations, magic square, ancient number system, history. These are all covered over a week or only a few assignments. Most of the worksheets are still the usual topics but pitched at a higher level that stimulates their thinking. The only problem is Geppers think faster and have a tendency to skip/leave out some of the workings, which may prove a problem to the marking scheme in the PSLE.
Science topics follow the PSLE syllabus and the usual topics, enriched and deeper thought. We use P5/PSLE questions for revisions. The key here is (no pun intended) is still keywords and key concepts when answering open-ended questions, which is not dissimilar to the mainstream and PSLE requirements.
If whether GEP prepares PSLE every single day from P4-6 by drilling them to do past year papers after past year papers, then clearly they don’t. They spend a lot of time doing other stuff which indirectly raise their ability – maybe it helps in PSEL, maybe it doesn’t.
– Does GEP help DSA
DSA based on Sports/CCA are irrelevant here unless GEP’s heavy workload deprive them time to train or participate more actively, which could be a consideration. If not GEP and mainstream compete evenly based on their sports achievement.
Based on history, the majority of Geppers score 250 and above. Some don’t because of their mother tongue (as mentioned above), which even if they are in the mainstream, won’t be any different. Some are simply misclassified into the GEP – no matter what the GEB says about their tests. Parents know their child best whether they have the ability.
So even without DSA, for those scoring 260 and above they would still get into the top IP schools. The only one you have no choice but to try for DSA is NUS High, because there are very few places left in the posting exercise after the PSLE. However, this may also backfire on the Geppers because the schools know the Geppers would score high, and therefore still come to their school, and therefore may not need to give you a CO. With government’s push for social mobility and anti-elitism, they may just want to mix it up a bit more in the elite schools and therefore less Geppers as the first cut – just pure guess, since PSLE would still bring back the higher scoring Geppers anyway through meritocracy.
But compare to the mainstream, does GEP equip you better to do DSA? I think very so, as GEP affords you opportunities to participate in the MO, SO as well as many other competitions, and as early as P4, which may not be the case in the mainstream. So you have a better chance to build up your academic portfolio.
Finally, do check out our most popular GEP post: Recommended Books for GEP Selection Test and How to Get Into GEP.
## Challenging P6 Math Question (Cycling)
One afternoon, 5 friends rented 3 bicycles from 5.00 p.m. to 6.30 p.m. and took turns to ride on them. At any time, 3 of them cycled while the other 2 friends rested.
If each of them had the same amount of cycling time, how many minutes did each person ride on a bicycle?
Hint: There is an “easy” way and also a “complicated” way to do this question. The “easy” way involves calculating total cycling time, while the “complicated” way involves working out a timetable to determine exactly who is cycling at which time.
(Source: Hardwarezone)
(Ans: 54)
## Should PSLE be Scrapped? 四不能除
According to this article by Channel News Asia, there are four “constraints” for why the PSLE cannot be scrapped. I call it the “四不能除“, named after a saying “四不能战” regarding a general Feng Zicai which I recently learnt about after watching the movie “The War Of Loong“.
1. Compromise education standards;
2. Hamper students’ preparation for the future economy;
3. Unduly increase levels of stress and competition; or
4. Unnecessarily disrupt the system
It is quite logical actually, I encourage readers to read the article, which is written by a student yet to enter university.
Point 1 and 2 are sort of self-explanatory; just think realistically how many kids will study hard if there is no PSLE exam? Even if there is an internal school exam, if its importance is not high, it may not cause students to study.
Point 3 needs some explanation on why scrapping the PSLE may actually increase stress:
Earlier this year, during the Ministry of Education’s Committee of Supply debate, then- Minister for Education (Schools) Ng Chee Meng remarked that “removing the PSLE and having a through train will only transfer the stress on parents and students elsewhere, such as at the P1 registration”.
He also emphasised that “it will make the O-Level and N-Level exams most stressful – a single exam in the whole career of a child’s life.”
Basically, if PSLE were to be scrapped, the focus and stress will be shifted to P1 registration and O Levels. P1 registration is based on where you stay and your parent’s alumni, in this respect PSLE is actually fairer and more meritocratic as it is based more on the child’s academic ability.
Nonetheless, despite the four reasons not to scrap PSLE, one thing that strikes me is the example of Finland. Despite Finland having no exams until the age of 16, it is well known that Finland’s education system is top in the world. There must be some secret to Finland’s education system, but the question is whether it can work and be applicable to other countries.
General Feng Zicai actually said something that is very wise and applicable to PSLE: “光绪二十九年(1903年)临终前，冯子材为子孙留下遗训：“读书不求官，服官不要钱，违者不孝。” (http://www.qzlz.gov.cn/Item/251.aspx)
Translated: “The main goal for studying is not for becoming an official; Becoming an official is not for gaining money for oneself; Whoever disobeys is unfilial (he said this to his descendants).”
Also read our previous post on PSLE:
## Express vs Normal Academic for borderline PSLE score
For students scoring in the borderline of around 190- 200 PSLE score, there is a dilemma of going to Express stream in a neighborhood school, or Normal Academic N(A) in a more established school. Note that even good schools like Anderson, ACS (Barker) do have N(A) streams.
Here are some advices from people who have experience:
Source: Hardwarezone
First opinion says that N(A) in a good school is better:
“NA in good school
i myself is living example
in the end my parent chose express at neighborhood school
in the end that school is a s**** school. i cannot keep up with studies.
by secondary 2 i drop to NA
then stuck with the lousy neighborhood school and my whole life got ruin until now
if i can go back time i sure chose NA at good school”
This post (from Hardwarezone) made a good argument of why express stream is better:
“I think you have been grossly misinformed about the life of a NA student. You say your son is addicted to computer games but it seems like he is doing relatively ok to be able to qualify for the express stream.
Yes he will be learning at a slower pace but over the next 4 years your son will have to be resilient to constant peer pressure and have a good amount of perseverance as many if not most students from NA are known to be more boisterous and mischievous causing them to deviate from their studies. Not to mention the stigma associated with being in NA. During my time in one of your “preferred NA schools”, I’ve seen many good students fall out of their studies because of the influence of their peers. The “just because I’m in NA I cmi” mentality will sadly creep up to many.
In my batch, iirc out of 3 classes of NA students, only 1 class of about 30 people was able/ motivated enough to take their O Levels. If your son decides not to take his O’s in the future and goes to ITE instead, whatever good name the school has will not have any effect on his resume.
Moreover, it is not like “school reputation” has any standing in the hiring process. Unless you’re from a top school like RI/Hwa Chong, no one would give two hoots. Recruiters mostly look at your highest education obtained or post secondary education to see if there is any relation to the job scope you are applying for.
In my opinion, you are better off applying to an express neighbourhood school and see if he is able to cope rather to a “prestigious” school in the NA stream. He can always drop out of the express stream if he’s not suitable for it.”
This parent from Kiasuparents gives very compelling reasons in favor of Express:
“Every year, there are parents asking the same question: Exp or NA?
Every year, I would encourage parents to choose Express for their children, for the simple reason that O level syllabus is very rigorous. Children are expected to work hard right from the start, unlike NA.
Even though the school you get will not be a ‘good school’ (if you choose Express), the classmates your child mixes with are likely to be of similar academic profile, with some that only have ‘Exp’ as their option. Similarly, if your child chooses NA, the children he or she mixes with will also have some that only have ‘NA’ as their option.
There are children who transfer from NA to Exp, but how many are there? From what I heard, very few manage to do that. You may want to check with the schools you are interested in choosing NA for for the probability that your child could transfer to Exp though.
I just want to let you know that if it happens to my own child, I would choose Exp. If the child is willing to work hard, he will make it anywhere. And if he can make it anywhere, then why wouldn’t I choose a better stream for him so that he can get used to the rigorous syllabus earlier?”
This parent from Kiasuparents recommended Anderson and Presbyterian High as two good schools with Normal Academic:
“I would choose one where the school discipline is good, and the teachers are supportive of students etc, with good academic and non – academic programs to expose / stretch the child. If going for NA, you can consider Anderson Sec, Presbyterian High.. these are generally pretty good schools.”
## Tessellations of Pentagons
Tessellation is a cool topic in primary level to PSLE math. Most students will enjoy it even if they hate other types of Math. It is a natural human instinct to be amazed at how different shapes can fit together perfectly to tile the plane.
Apparently, tessellation is going to be removed from the entire PSLE syllabus soon (see http://schoolplus.com.sg/primary-math-syllabus-2017/). That is certainly quite sad for many reasons.
Triangles and quadrilaterals (even irregularly shaped ones) can be easily tessellated. However for pentagons, it is less clear and some pentagons (including the regular pentagon) cannot be tessellated!
## Appeal from RGS to NYGH (Success)
Just read that appealing to transfer from RGS (Raffles Girls School) to NYGH (Nanyang Girls High) is possible:
dd moved out of rgs to nygh. she got 262+2. she appealed to nygh and was granted interview on Thursday. was given the good news after her interview.
nygh her first choice. she probably missed by decimal points. thus tried to appeal. was telling her both schools are equally good thus if not successful for nygh in her appeal, just move on 🙂
https://www.kiasuparents.com/kiasu/forum/viewtopic.php?f=48&t=7127&start=3940
NYGH cut off point has been higher than RGS for the recent past years. It is the opposite situation of their boy school counterparts: RI cut off point is usually higher than Hwa Chong (Chinese High).
There are 2 short films on the subject of PSLE, free on YouTube. Quite accurate about the lives of kids in Singapore currently. Do feel free to watch if you are interested.
It is truly a first-world problem (only kids in middle to high income families will suffer from this, as tuition is not cheap), but it does reflect the stress that children go through nowadays.
Look at the child’s timetable: https://youtu.be/FQB7ritn580?t=116. Completely packed from Monday to Sunday. (Quite realistic as I have seen real life examples of such scenarios.)
## PSLE Chinese Listening Exam (mrbrown) Very Funny!
Recently, there is a PSLE Chinese Listening Exam that does not make sense.
Question: Student A bought a new clothes. Student B asks Student A: “You bought new clothes?” Student A said: ” No, it is sewn by my mother, do you think it is beautiful?” Student B said: It is very beautiful, I didn’t knew your mother could sew?”.
What did Student A say next?
1) My mother will sew clothes for me whenever she is free.
2) My mother does not like to spend money to buy clothes.
3) My mother just started learning how to sew.
I am totally puzzled by this question. The three options seems equally plausible. How are we supposed to know which is the truth?
Answer is option 3 by the way.
## (Important Changes) PSLE Math: Arrow -> vs Equal=
For those taking PSLE, please take note of this important update regarding the difference between arrow and equal sign. Forward this to your friends taking PSLE!
Basically, I think MOE is trying to instill students to be mathematically correct. (See update below: Marks will not be deducted in most cases but proper usage is highly encouraged.)
E.g. 100%=40 is wrong as 100%=100/100=1 technically. Similarly, 10 men = 40 hours is wrong as the units do not match (nor make sense).
Trying to enforce “units” instead of “u”, and banning “10 units -> 20” is a bit strict though, in my opinion.
MOE responds
In response to Mothership.sg queries, a Ministry of Education spokesperson clarified that the above information was not provided by the ministry.
The information above was originally sourced from the website of a private tuition centre, whose sources are currently unverified.
While the respective uses of the arrow and equal signs are accurate in the infographic, the MOE spokesperson said full credit will still be awarded to the student even when the signs are used interchangeably, as long as the student demonstrates a full understanding of the question.
Proper use of arrow and equal signs are, nonetheless, encouraged.
## Top PSLE Score
The top PSLE Score for this year seems to be 286, from RGPS (Raffles Girls Primary School).
Close runners-ups are 283, from NYPS (Nanyang Primary School) and Nanhua.
Source is from https://www.kiasuparents.com/kiasu/t-scores/, which is self-reported by parents. Casting aside the “troll scores” of 299 or 300 which are not believable, this seems to be the most accurate top PSLE score available, since the mainstream media are not allowed to report them.
Congratulations for those who have done well. And for those who have not, do not be discouraged as there is still a long road ahead, and there will be many opportunities to prove yourself.
For those considering tuition, check out StarTutor, which is highly recommended by us. Tutors are screened for their educational qualifications and matched accordingly, with zero administrative fees.
From experience, O Level is a completely new ball game from PSLE. It is possible for relatively weak students in PSLE to do very well in O Levels, and also vice versa; for quite strong PSLE students to do poorly in O Levels. It is a new beginning for students.
2017 Update: Highest PSLE score seems to be 285 from Nanyang Primary. The runner-up position seems to be tied with a few schools such as MGS, ACS, Ai Tong, with 281.
Note that neighborhood schools can produce very strong results too: Alexandra Primary School and Admiralty Primary School both produce 280 scorers. In fact I recall from memory that Rulang Primary School used to consistently produce top scorers that can even rival that of Nanyang Primary School.
## Kiasuparents PSLE
Basically to summarize the article above, the co-founder of Kiasuparents’ son scored a respectable 4As and 229 T-score for PSLE. However, as their set target was 250, he did not get the Nintendo DS that was part of the deal for achieving the target of 250. Probably the Nintendo is to play the most recently released Pokemon Sun/Moon. Poor kid! I remember that my highlight of finishing PSLE was to play Pokemon (close to 20 years ago). I still remembered I was playing the Blue version, starting as Bulbasaur.
PSLE can be highly unpredictable (variance of 20-30 marks from usual expected mark is common and expected). This is particularly due to language exams, composition, and also the famous rigid marking scheme of PSLE science, where all the “keywords” must be mentioned in order to get the mark. Mathematics is the more reliable subject here as it is more objective, so try to score as high as possible in it.
Hence DSA becomes increasingly important as a backup plan to act as insurance in the event that something goes wrong in the PSLE. Check out some DSA/GAT/HAST posts here. It is always good to have a “Plan B”.
Also, if you suspect that the child’s school teacher is not that excellent in teaching, e.g. don’t know/emphasize the “keywords” which are necessary to get any marks at all in PSLE science, you may consider engaging a tutor as soon as possible. Check out the most recommended tuition agency in Singapore.
## New PSLE System favors “All Rounders” over “Specialists”
The new PSLE system clearly favors “all-rounders” over “specialists”.
## Scenario 1: Math-Whiz VS All-Rounder
Imagine a Math/Science-whiz with
Math:100 (AL 1)
Science: 98 (AL 1)
English: 84 (AL 3)
Chinese: 84 (AL 3)
Total marks: 366 (Approx. 275 T-score)
Total AL: 8
With a “all-rounder”:
Math: 90 (AL 1)
Science: 90 (AL 1)
English: 90 (AL 1)
Chinese: 90 (AL 1)
Total marks: 360 (Approx. 270 T-score)
Total AL: 4
The Math/Science whiz (total AL 8) will be getting double the score of the “all-rounder” (total AL 4), effectively eliminating his chance of entering the top schools. The irony is that the total marks of the Math/Science Whiz is a considerable 6 marks more than the “all-rounder”.
Under the old system, both are likely to get around the same T-score (approx. 270+), with the Math/Science whiz having a higher T-score.
## Scenario 2: English-Educated Kid VS All Rounder
This scenario is even worse.
Imagine an intelligent English-Educated Kid (with parents who can’t speak Chinese). After a lot of hard work with Chinese enrichment, etc, he manages to pass Chinese, with a score of:
Math:100 (AL 1)
Science: 98 (AL 1)
English: 95 (AL 1)
Chinese: 64 (AL 6)
Total marks: 357 (Approx. 268 T-score)
Total AL: 9
Under the old system, this child is probably one that qualifies to enter any school, including RI/HCI, etc. His T-score will probably be on par with the All-Rounder at around 270, or at most slightly lower. Under the new system, his total AL is almost 10. Really a big difference.
In fact, the O-Levels, A-Levels are also favoring the all-rounders. Only at university (and beyond), do the specialists finally get a chance to shine. That’s why it is common to see top students in universities who were not previously from the top JCs or secondary schools.
My followup post on Kiasuparents:
My concern as Math educator is that students extremely talented in Mathematics/Science but slightly weak in languages will be disadvantaged in the new PSLE system.
To quote from my own blog entry titled “New PSLE System favors “All Rounders” over “Specialists””:
Imagine a Math/Science-whiz with
Math:100 (AL 1)
Science: 98 (AL 1)
English: 84 (AL 3)
Chinese: 84 (AL 3)
Total marks: 366 (Approx. 275 T-score)
Total AL: 8
Previously such a student’s score is more than sufficient to enter the top schools like RI/HCI. But under the new system, his score of 8, chances of entering the top schools are slim.
It is not about the prestige, but rather the resources and enrichment programmes that top schools provides that other schools may not. Some examples include Olympiad training, Laboratory sessions, etc.
For these kind of students, the PSLE score of 100 is not enough to capture their ability in Math/Science, they would score 150/100 if there is such a thing. Hence, their calibre is well above the “All-Rounders” who score 90 for each subject and get 4 points.
Unfortunately, the new PSLE system does not bode well for these students…
This is a good analysis. The new system does demand excellence in every subject, which in my opinion will increase stress more than it reduces. And for those who say that it is good because there is no need to count decimal points, consider the fact that Secondary schools will still have COPs. So it is now getting 4 points vs getting above 255 t-scores. Which measure would you consider to be more narrow?
My feeling is that the new system will actually intensify the cookie-cutter education culture and create more average joes than truly outstanding individuals.
## New PSLE Scoring System, AL1 to AL8 (Singapore)
The new, long awaited, PSLE scoring system is now out. Under the new scoring system, T-score is being replaced by Achievement Levels:
AL1: 90 and above
AL2: 85-89
AL3: 80-84
AL4: 75-79
AL5: 65-74
AL6: 45-64
AL7: 20-44
AL8: Below 20
Would this be effective at the target goal of “reducing stress and competition among pupils and parents?”.
Firstly, the good point about this new scoring system is that it is not as fine as the previous T-score system, where every mark matters. Thus, technically there is no difference between a 90 mark and a 100 mark, so there is no need to aim for perfection in a certain sense.
Personally, I think that this scoring system is similar to the O Level System Grading of A1, A2, B3, etc. The change in scoring system per se is unlikely to be able to reduce the stress of students, especially those scoring below 90. Those scoring >90 but not close to 100 may breathe a sigh of relief that they don’t have to aim for 100. However, for the students scoring below 90, the stress level remains essentially unchanged.
Note that despite the “wider scoring band” label, the band is not that wide after all. In the higher AL’s the difference from one AL to the next is merely 5 marks, which may be just one problem sum in mathematics. In the previous PSLE it is “every mark counts”. In the new PSLE it is “every question counts”, which is not much of a difference.
For Primary students, the stress comes mostly from the kiasu parents, any superficial change in the scoring system will not have much effect.
Overall, nothing much has changed. It is like changing between Celsius to Fahrenheit, there is no difference in the underlying principle of PSLE, which is to serve as a entry criteria for secondary schools.
The main change, as some parents have noted, is that now all subjects are equally important. It is no longer possible to compensate for one weak subject (e.g. Mother Tongue) by scoring extremely well in other subjects. It can be said that the new system favours “all rounders” or “Jack of all trades” over “specialists” in one or two subjects.
The AL6: 45-64 band looks extremely dangerous to fall in (especially those weak in Mother Tongue) as it does not differentiate between a fail grade (45) and a much higher grade (64). Many English-speaking families should be quite worried now…
Another area of concern is that due to the “wider scoring band”, the importance of DSA (Direct School Admission) has increased tremendously. Due to inevitably many students achieving the perfect score of ‘4’, the top schools (like RGS/NYGH/RI/HCI) may have to resort to DSA/GAT tests to select their students. This will probably increase the stress of students, as other than PSLE, they have to worry about DSA/GAT/CCA and building a portfolio of achievements.
## 2016 PSLE Difficulty — Second Hardest PSLE in history
Watch this interesting video on the PSLE (Primary School Leaving Exam) in Singapore: https://www.facebook.com/cnainsider/videos/1073805505975459/
According to this girl, her teacher says that PSLE 2016 is the 2nd hardest exam in history.
It seems school children these days have longer working hours than even adults. Adults work from 8am-5pm, children have to study from 7am all the way to night time.
One problem from all these cramming may be loss of joy in learning. After all the years of “forced studying”, few if any students have any more joy of learning in their hearts.
The young girl in the video is very optimistic and cheerful despite all the extra classes, keep it up!
# How to Excel in DSA
With PSLE getting more and more tough, DSA is more and more important as a backup plan, or even as “Plan A”. A well-planned DSA application could lead to success in entering the secondary school of your choice. Once the PSLE new scoring system is out, DSA is the critical distinguishing factor.
Check out these posts on how to excel in DSA.
## Changes to PSLE: Less stress for students but don’t dumb down education system
Latest Straits Times article on the PSLE.
The Ministry of Education’s move is laudable. In effect, though, kiasu parents will still find a way to put the screws on their children. Mark my words. No system is perfect. But the problem of stress lies largely with parents who cannot accept that their children are anything less than the best.
## How to prepare for Changes to PSLE grading
The latest update is that the Primary School Leaving Examination (PSLE) aggregate score is soon going to be scrapped and replaced with simple grade bands such as A, B and C. What effects will there be and how to prepare in advance for it?
My opinion is that there will be a few crucial changes that parents would have to prepare for as soon as possible:
Chinese (or Mother Tongue) is a top priority. In the past, many students from English-speaking families can get low marks for Chinese, high marks for Math/Science/English and still get a very good PSLE score (e.g. >250). This is unfortunately not possible anymore in the new system. A low Chinese grade will drag down the entire performance. Also read more about the benefits of studying Chinese.
DSA (Direct School Admission) GAT (General Ability Test) and CCA becomes more important. The effect of simple grade bands is that many students will get the perfect score of all As. However, the top schools have limited vacancies and thus will have to use other criteria like DSA and CCA to differentiate students. Check out this previous post on DSA.
The new system benefits all-rounders who are good (but not necessarily excellent) at all subjects, including CCA. All-rounders will manage to get ‘A’s in all subjects. However, the new system is unfortunately not good for those who are excellent in one single subject, but average in others.
This quote from the article is very good:
“The focus should not be on how one performs relative to others, but how well the person himself performs in the exam.”
DR TIMOTHY CHAN, director of SIM Global Education’s academic division, on the use of grade-banding to reflect pupils’ abilities.
## Live your Dream (Motivational Video) 夢想．激勵人心的演說
Just to share a very inspiring motivational video from YouTube. Not sure which movie it is from. (any readers know, please comment below as I would be interested)
Highly suitable for students (and their parents) who have just completed their PSLE, whether their PSLE 2015 results are good or not, it is now a good time to reflect on their dreams and the next step to take in the next year 2016.
## Markov Inequality + PSLE One Dollar Question
Many people have feedback to me that the Career Quiz Personality Test is surprisingly accurate. E.g. people with peaceful personality ended up as Harmonizer, those who are business-minded ended up as Entrepreneur. Do give it a try at https://mathtuition88.com/free-career-quiz/. Please help to do, thanks a lot!
Also, some recent news regarding PSLE Maths is that a certain question involving weight of \$1 coins appeared. It is very interesting, and really tests the common sense and logical thinking skills of kids.
Markov inequality is a useful inequality that gives a rough upper bound of the measure of a set in terms of an integral. The precise statement is: Let $f$ be a nonnegative measurable function on $\Omega$. The Markov inequality states that for all $K>0$, $\displaystyle \mu\{x\in\Omega:f(x)\geq K\}\leq\frac{1}{K}\int fd\mu$.
The proof is rather neat and short. Let $E_K:=\{x\in\Omega: f(x)\geq K\}$ Then,
\begin{aligned} \int f d\mu &\geq \int_{E_K} fd\mu\\ &\geq \int_{E_K}K d\mu\\ &=K \mu(E_K) \end{aligned}
Therefore, $\mu(E_K)\leq\frac{1}{K}\int fd\mu$.
## Primary One registration for 2016 to open on July 2
SINGAPORE — The Primary One registration exercise for next year’s intake will open from July 2 to Aug 27, the Ministry of Education (MOE) said today (June 18).
Three new schools — Oasis Primary, Punggol Cove Primary and Waterway Primary — will be open for P1 registration and will be taking in students from next year.
“The cohort size for 2016 is similar to that of 2015. There will be sufficient school places for all eligible P1 students on a regional and nationwide basis,” said the MOE.
More information on the list of primary schools and vacancies available as well as a list of registration centres for new schools can be found on the P1 registration website at http://www.moe.gov.sg/education/admissions/primary-one-registration/
## Join the Kiasuparents 2020 PSLE Discussion Group
This is the ultimate uniquely Singapore “Kiasuparents” 2020 PSLE Discussion Group: http://www.kiasuparents.com/kiasu/forum/viewtopic.php?f=69&t=81381
This book is the #1 Best Seller on Amazon (in the Gifted Education section)! Learn about the secret of the smartest kids in the world, and how you can be one of them.
## PSLE Results Release: Top Scorer in PSLE?
Also check out: Recommended Books for GEP
PSLE Results will be out tomorrow!
Wishing all students and parents all the best. 🙂
Currently, the PSLE Top Scorer is not released in mainstream media unlike in the past. It is perhaps a good thing too, to make PSLE less stressful.
PSLE is a stepping stone for students, it is important to remember that there is still a long way ahead. Many students who didn’t do well in PSLE end up excelling in O Levels. Education is really about lifelong learning.
## Release of 2014 PSLE Results and 2014 Secondary One Posting
1The results of the 2014 Primary School Leaving Examination (PSLE) will be released on Friday, 21 November 2014. Students may obtain their result slips from their respective primary schools from 11.00 am on 21 November 2014.
# When is PSLE?
Please double confirm with the source at: http://www.seab.gov.sg/examTimeTable/2014PSLEExamTimetable.pdf
## PSLE Exam Dates/Schedule/Timetable
A. Oral Examination
Date Paper Time Thursday, 14 August&Friday, 15 August English Language / Foundation EnglishChinese / Malay / TamilFoundation Chinese / Foundation Malay / Foundation Tamil 0800 – 1300 h Friday, 15 August Bengali / Gujarati / Hindi / Panjabi / UrduFoundation Bengali / Foundation Gujarati / Foundation Hindi / Foundation Panjabi / Foundation Urdu 0800 – 1300 h
C. Written Examination Date Paper Time Duration Thursday, 25 September English Language Paper 1 English Language Paper 2 Foundation English Paper 1 Foundation English Paper 2 0815 – 0925 h 1030 – 1220 h 0815 – 0925 h 1030 – 1150 h 1 h 10 min 1 h 50 min 1 h 10 min 1 h 20 min Friday, 26 September Mathematics Paper 1 Mathematics Paper 2 Foundation Mathematics Paper 1 Foundation Mathematics Paper 2 0815 – 0905 h 1015 – 1155 h 0815 – 0915 h 1015 – 1130 h 50 min 1 h 40 min 1 h 1 h 15 min Monday, 29 September Chinese / Malay / Tamil Bengali / Gujarati / Hindi / Panjabi / Urdu Paper 1 Chinese / Malay / Tamil Bengali / Gujarati / Hindi / Panjabi / Urdu Paper 2 Foundation Chinese/ Foundation Malay/ Foundation Tamil Paper 1 0815 – 0905 h 1015 – 1155 h 0815– 0845 h 50 min 1 h 40 min 30 min Tuesday, 30 September Science Foundation Science 0815 – 1000 h 0815 – 0930 h 1 h 45 min 1 h 15 min Wednesday, 1 October Higher Chinese / Higher Malay / Higher Tamil Paper 1 Higher Chinese / Higher Malay / Higher Tamil Paper 2 0815 – 0905 h 1015 – 1135 h 50 min 1 h 20 min
## PSLE could move away from aggregate scores: Lim Biow Chuan
The head of the Government Parliamentary Committee (GPC) for Education, Member of Parliament Lim Biow Chuan, said that the Primary School Leaving Examination could do with less focus on aggregate scores.
SINGAPORE: The head of the Government Parliamentary Committee (GPC) for Education, Member of Parliament Lim Biow Chuan, said that the Primary School Leaving Examination (PSLE) could do with less focus on aggregate scores.
He said that this would take away the stress associated with the examination.
Education Minister Heng Swee Keat said recently that changes to the PSLE will be announced at the National Day Rally on Sunday.
It is an annual affair that sends the nation’s parents, students and teachers into a frenzy — for many in Singapore, the PSLE has become a high-stakes examination.
Roger Cheong, a parent, said: “Maybe there should not be so much emphasis on PSLE at such a young age… Maybe as a gauge, but there shouldn’t be so so much weightage on it.
The Education Ministry has acknowledged this and embarked on a year-long review sometime in 2012.
Ahead of the announcements of possible changes, some have suggested going back to basics.
Mr Lim said: “I never knew what was my PSLE score. We selected a few schools that we chose and from there, MOE would post us to those schools, based on our performance. So you don’t have to go down to those minute details as to whether you score 270 or 265 or 275.
“You get broad-based results, and from there, you are allocated schools of your choice. It may not be the exact school of your choice, but it may be a group of schools that you choose and all of them are in the same category.”
Mr Lim also hoped to see more places set aside for the Direct School Admission (DSA) exercise, where students apply to secondary schools based on their achievements and talents before the release of their PSLE results.
|
2020-08-07 08:53:04
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 7, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26524412631988525, "perplexity": 2895.2906404545743}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737172.50/warc/CC-MAIN-20200807083754-20200807113754-00558.warc.gz"}
|
https://gmatclub.com/forum/if-243-x-463-y-n-where-x-and-y-are-70197.html
|
It is currently 22 Feb 2018, 20:49
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# Events & Promotions
###### Events & Promotions in June
Open Detailed Calendar
# If (243)^x(463)^y = n , where x and y are
Author Message
Manager
Joined: 04 Jan 2008
Posts: 118
If (243)^x(463)^y = n , where x and y are [#permalink]
### Show Tags
13 Sep 2008, 01:49
00:00
Difficulty:
(N/A)
Question Stats:
33% (00:00) correct 67% (00:16) wrong based on 6 sessions
### HideShow timer Statistics
This topic is locked. If you want to discuss this question please re-post it in the respective forum.
If $$(243)^x(463)^y = n$$, where x and y are positive integers, what is the units digit of n?
(1) x + y = 7
(2) x = 4
OPEN DISCUSSION OF THIS QUESTION IS HERE: if-243-x-463-y-n-where-x-and-y-are-positive-integers-102054.html
[Reveal] Spoiler: OA
VP
Joined: 17 Jun 2008
Posts: 1373
### Show Tags
14 Sep 2008, 00:52
dancinggeometry wrote:
If $$(243)^x(463)^y = n$$, where x and y are positive integers, what is the units digit of n?
(1) x + y = 7
(2) x = 4
(1) is sufficint to prove since we wnt 3^x * 3^y =3^(x+y)
(2) INSUFFI
IMO A
_________________
cheers
Its Now Or Never
Senior Manager
Joined: 07 Jan 2008
Posts: 397
### Show Tags
14 Sep 2008, 01:24
Unit digit of n = 7.
Re: Zumit DS 024 [#permalink] 14 Sep 2008, 01:24
Display posts from previous: Sort by
|
2018-02-23 04:49:22
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5894104242324829, "perplexity": 7021.438302715417}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814393.4/warc/CC-MAIN-20180223035527-20180223055527-00582.warc.gz"}
|
https://dsp.stackexchange.com/questions/64661/what-is-meant-by-group-delayin-simple-words
|
# What is meant by “Group delay”?in simple words?
Although likes of this question have been asked many times on DSP SE
But i am unable to understand those and i wish to have a crystal clear explanation in simple words with example
The only thing that i am able to understand from those questions is that group delay is negative derivative of phase with respect to frequency. But what does negative derivative means here and why not positive derivative here?
• The other similar questions explain quite well that it's a bit of a fuzzy world, but they do it with examples and clear words. – a concerned citizen Mar 17 '20 at 16:51
• the wikipedia article doesn't do a half bad job. some people might be able to see fingerprints of someone's writing in it. – robert bristow-johnson Mar 17 '20 at 16:57
• if people want, i can copy the introductory section from wikipedia to here and make it a legit answer. – robert bristow-johnson Mar 17 '20 at 17:04
• @robertbristow-johnson Please do. :-) – Peter K. Mar 17 '20 at 17:10
• as if i didn't have something better to do... as a matter of fact... – robert bristow-johnson Mar 17 '20 at 17:24
Here is my simplest explanation:
The group delay, as the negative derivative of phase, predicts the time delay of the amplitude envelope of a pulse, as shown in the hand-drawn graphic below. The upper part of the sketch shows a sinusoidal waveform varied in amplitude by its envelope. The lower one is showing this same envelope before and after a system that has group delay.
This applies when the phase of the frequency response can be approximated as linear for the "group" of frequencies within the pulse envelope. Thus for non-linear phase systems, this applies to generally narrower band signals such as the pulse I show where the amplitude transition is gradual.
Consider a single sine-wave with the amplitude envelope such as I show. The time delay of the sine wave itself would be predicted directly from the phase of the frequency response (by dividing by the frequency of the sine-wave: with $$\phi = angle(H(j\omega))$$, the time delay is $$-\phi(\omega)/\omega$$), while the time delay of the pulse envelope is predicted from the negative derivative of the phase with respect to frequency ($$-d\phi(\omega)/d\omega$$)).
Calculate the time delay introduced by group delay for IIR-Filters
https://electronics.stackexchange.com/questions/135475/physical-significance-of-group-delay
And most helpful to what otherwise seems like a paradox of causality for positive group delay is this paper https://www.researchgate.net/publication/253463703_Causality_and_Negative_Group_Delays_in_a_Simple_Bandpass_Amplifier referenced by Max in this post Physical Meaning of Negative Group Delay for causal LTI systems which I bottom line as causality is not violated but due to the bandwidth restrictions above we create a condition that causes the pulse envelope at the output to precede the input: The output pulse does not appear until the input amplitude varies (and if we have gain in the system the output envelope grows faster) and due to the bandwidth constraint and the result of destructive summation of the input pulse the output will start to decrease before the input does. Very cool DSP magic trick. See this post that illustrates this with a specific example.
• I agree with this provided the bandwidth of interest is sufficiently small relative to the frequency of interest – Dan Szabo Mar 18 '20 at 2:06
• Mr Boschen, I wasn’t referring to linear phase, mostly considering that given linear phase the group delay would be constant, and could just as well be called delay. I worked out a derivation for group delay a while back, and my recollection is something like this: you define your input as a product of an envelope sinusoid and a carrier sinusoid. The maths will give the delay of the envelope as the group delay equation so long as the amplitude response is uniform. In general, this is always true as the bandwidth approaches zero. It has been a while for me though. – Dan Szabo Mar 18 '20 at 2:38
• So it may have been more appropriate for me to say: so long as the amplitude response is sufficiently uniform over the given bandwidth... – Dan Szabo Mar 18 '20 at 2:40
• Even my last statement is a bit misleading, because it would seem incorrect for linear phase... apologies, I don’t think I’m putting my thoughts to words effectively. I was attempting to include non-linear phase more generally. – Dan Szabo Mar 18 '20 at 2:45
• @DanSzabo See my update, I believe this may be closer to what you were getting at in our earlier conversation with regards to bandwidth. – Dan Boschen Apr 4 '20 at 16:25
To understand group delay, it is important to first understand phase delay.
Phase delay is the amount of phase lag for certain frequency. The units are in degrees. But there is a weird relationship between phase delay (units = degrees) and time delay (units = seconds). Let me explain:
Say I have a 1Hz signal that goes through a filter and it experiences 90 degree phase delay. 90 degrees is 1/4 of a full 360 degree cycle. Thus for a 1Hz signal (which has 1 second period) the time delay experienced is 1 second / 4 = 0.25 seconds. Essentially the output lags the input by 0.25 seconds.
Now lets say I have a 2Hz signal. The 2Hz signal has a cycle period of 0.5 seconds. Let say I feed this signal through a filter and it also experiences 90 degree phase delay. Again, 90 degrees is 1/4 of a full 360 cycle. Thus for a 2hz signal, the time delay experienced is 0.5 second / 4 = 0.125 seconds. Now the output lags the input signal by 0.125 seconds.
What this says is that constant phase delay does not equal constant time delay! Deriving the time delay from phase delay is dependent on the frequency itself. The only way for all frequencies to get delayed by the same time delay is if the phase response is linear.
When the phase response is linear, we know all the frequencies get time delayed by the same amount. Thus if all frequencies are delayed the same amount, we have this notion of a "group" delay. Group refers to all frequencies.
Let's look at this visually.
If we feed an input signal into a filter with a constant group delay, all frequencies will be time delayed the same amount. Referring to the picture below, the outputted signal matches the input signal except it is slightly delayed.
If instead we fed that input signal into a filter with a non-constant group delay. The frequencies will time delay different amounts resulting in an output signal that looks nothing like the inputted signal.
So even though each filter is low-pass, one filter distorts the signal such that it doesn't resemble the inputted signal. This is why linear-phase (constant group delay) filters are desirable in some applications.
• "Essentially the output lags the input by 0.25 seconds"What do you mean? Do you mean here that if in input we have peak at 0 second,then in output peak will occur at 0.25 second?? – engr Apr 4 '20 at 16:24
• Correct. Visualize it like this: sengpielaudio.com/Sinusoidal%20Wave.gif – Izzo Apr 4 '20 at 21:11
• @engr I added some visuals which demonstrate the characteristics of constant group delay. And I fixed a couple typos. – Izzo Apr 4 '20 at 21:51
• what concept, you are trying to convey through the link in your comment sengpielaudio.com/Sinusoidal%20Wave.gif – engr Apr 9 '20 at 10:48
• Your visuals are very nice. Can you also please include in your answer,nutshell definition of group delay in simple words? – engr Apr 9 '20 at 10:51
Group delay is a useful measure of time distortion, and is calculated by differentiating, with respect to frequency, the phase response of the device under test (DUT): the group delay is a measure of the slope of the phase response at any given frequency. Variations in group delay cause signal distortion, just as deviations from linear phase cause distortion.
In linear time-invariant (LTI) system theory, control theory, and in digital or analog signal processing, the relationship between the input signal, $$x(t)$$, to output signal, $$y(t)$$, of an LTI system is governed by a convolution operation:
$$y(t) = (h*x)(t) \ \triangleq \ \int_{-\infty}^{\infty} x(u) h(t-u) \, \mathrm{d}u$$
Or, in the frequency domain,
$$Y(s) = H(s) X(s) \,$$
where
$$X(s) = \mathscr{L} \Big\{ x(t) \Big\} \ \triangleq \ \int_{-\infty}^{\infty} x(t) e^{-st}\, \mathrm{d}t$$
$$Y(s) = \mathscr{L} \Big\{ x(t) \Big\} \ \triangleq \ \int_{-\infty}^{\infty} y(t) e^{-st}\, \mathrm{d}t$$
and
$$H(s) = \mathscr{L} \Big\{ x(t) \Big\} \ \triangleq \ \int_{-\infty}^{\infty} h(t) e^{-st}\, \mathrm{d}t$$
Here $$h(t)$$ is the time-domain impulse response of the LTI system and $$X(s)$$, $$Y(s)$$, $$H(s)$$, are the Laplace transforms of the input $$x(t)$$, output $$y(t)$$, and impulse response $$h(t)$$, respectively. $$H(s)$$ is called the transfer function of the LTI system and, like the impulse response $$h(t)$$, fully defines the input-output characteristics of the LTI system.
Suppose that such a system is driven by a quasi-sinusoidal signal, that is a sinusoid having an amplitude envelope $$a(t)>0$$ that is slowly changing relative to the frequency $$\omega$$ of the sinusoid. Mathematically, this means that the quasi-sinusoidal driving signal has the form
$$x(t) = a(t) \cos(\omega t + \theta)$$
and the slowly changing amplitude envelope $$a(t)$$ means that
$$\left| \frac{d}{dt} \log \big( a(t) \big) \right| \ll \omega \ .$$
Then the output of such an LTI system is very well approximated as
$$y(t) = \big| H(i \omega) \big| \ a(t - \tau_g) \cos \big( \omega (t - \tau_\phi) + \theta \big) \; .$$
Here $$\tau_g$$ and $$\tau_\phi$$, the group delay and phase delay respectively, are given by the expressions below (and potentially are functions of the angular frequency $$\omega$$). The sinusoid, as indicated by the zero crossings, is delayed in time by phase delay, $$\tau_\phi$$. The envelope of the sinusoid is delayed in time by the group delay, $$\tau_g$$.
In a linear phas system (with non-inverting gain), both $$\tau_g$$ and $$\tau_\phi$$ are constant (i.e. independent of $$\omega$$) and equal, and their common value equals the overall delay of the system; and the unwrapped phase shift of the system (namely $$-\omega \tau_\phi$$) is negative, with magnitude increasing linearly with frequency $$\omega$$.
More generally, it can be shown that for an LTI system with transfer function $$H(s)$$ driven by a complex sinusoid of unit amplitude,
$$x(t) = e^{i \omega t}$$
the output is
\begin{align} y(t) & = H(i \omega) \ e^{i \omega t} \ \\ & = \left( \big| H(i \omega) \big| e^{i \phi(\omega)} \right) \ e^{i \omega t} \ \\ & = \big| H(i \omega) \big| \ e^{i \left(\omega t + \phi(\omega) \right)} \ \\ \end{align} \
where the phase shift $$\phi$$ is
$$\phi(\omega) \ \triangleq \arg \left\{ H(i \omega) \right\} \;$$
Additionally, it can be shown that the group delay, $$\tau_g$$, and phase delay, $$\tau_\phi$$, are frequency-dependent, and they can be computed from the properly unwrapped phase shift $$\phi$$ by
$$\tau_g(\omega) = - \frac{d \phi(\omega)}{d \omega}$$
$$\tau_\phi(\omega) = - \frac{\phi(\omega)}{\omega}$$
It stems from the definition of the Laplace/fourier transforms using $$e^{-st}$$ or $$e^{-j\omega t}$$. This can be checked intuitively by looking at the transform pair of a delayed impulse, compare the time domain delay to the frequency domain phase. If you modified the transform to use $$e^{+st}$$instead, it would be the other way round.
|
2021-08-05 02:23:30
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 48, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8600701689720154, "perplexity": 704.9953932927559}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046155268.80/warc/CC-MAIN-20210805000836-20210805030836-00051.warc.gz"}
|
https://www.gradesaver.com/textbooks/math/geometry/CLONE-df935a18-ac27-40be-bc9b-9bee017916c2/chapter-2-section-2-2-indirect-proof-exercises-page-93/3
|
## Elementary Geometry for College Students (7th Edition)
Converse: If two angles are complimentary, then the sum of their measures is 90$^{\circ}$. (true) Inverse: If the sum of the measures of two angles is not 90$^{\circ}$, then the two angles are not complimentary. (true) Contrapositive: If two angles are not complimentary, then the sum of their measures is not 90$^{\circ}$. (true)
|
2021-02-26 13:42:38
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7170590758323669, "perplexity": 994.4873972569281}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178357641.32/warc/CC-MAIN-20210226115116-20210226145116-00190.warc.gz"}
|
https://web2.0calc.com/questions/two-distinct-number-cubes-one-red-and-one-blue-are
|
+0
# Two distinct number cubes, one red and one blue, are rolled together. Each number cube has sides numbered 1 through 6.
+2
193
3
+25
Two distinct number cubes, one red and one blue, are rolled together. Each number cube has sides numbered 1 through 6.
What is the probability that the outcome of the roll is a sum that is a multiple of 6 or a sum that is a multiple of 4?
Jul 28, 2021
#1
+593
+3
The sum can be $4,6,8,12$. Some casework:
4 can be written as $2+2, 1+3, 3+1$
6 can be written as $1+5, 5+1, 4+2, 2+4, 3+3$
8 can be written as $2+6, 6+2, 3+5, 5+3, 4+4$
12 can be written as $6+6$
The total number of ways is $14$. There are $36$ possibilities. The answer is $\frac{14}{36}=\frac{7}{18}$.
Jul 28, 2021
#3
+847
+2
I think PIE would be faster, though this is very rigorous without a doubt.
MathProblemSolver101 Jul 28, 2021
#2
+847
+3
$\frac{\lfloor \frac{36}{4} \rfloor + \lfloor \frac{36}{6} \rfloor - \lfloor \frac{36}{24}\rfloor}{36}$
$\frac{14}{36}$
$\frac{7}{18}$
Jul 28, 2021
|
2022-01-22 20:36:40
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8905323147773743, "perplexity": 494.9318098113958}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303884.44/warc/CC-MAIN-20220122194730-20220122224730-00109.warc.gz"}
|
http://2017.igem.org/Team:Manchester/Model/Continuous_Culture
|
Team:Manchester/Model/Continuous Culture
Continuous Culture of Bacteria
Achievements:
1. Predicted the output rate of a continuous culture system in different initial substrate concentrations, using computational modelling with parameters found in literature.
2. Estimated the production cost to produce 1 kg of bacteria, and the cost to treat 1 ton of waste water using phosphate data given by Davyhulme Treatment Works.
3. Estimated the total cost to treat wastewater for a year, allowing us to predict the operation cost for our technology in our business plan and compare them to two existing technologies: Chemical Precipitation and Enhanced Biological Phosphate Removal.
4. Discussed a number of strategies to reduce cost further.
5. Calculated two of those cost-reduction strategies and determined the effectiveness of each strategy.
6. Determined the financial viability of our project and discussed alternative synthetic biology engineering strategies to improve the feasibility of our project in real life.
Introduction
We envision that our bacteria biomass would be cultured in a chemostat for production. A chemostat is a fermentation scheme, in which fresh medium is continually added whilst culture liquid containing leftover nutrients, microorganisms, and metabolic products are continuously removed at the same rate. This technique is called continuous culture and allows microbial growth to take place under steady-state conditions - growth that occurs at a constant rate and in a constant environment. Unlike a batch culture method where bacterial cells undergo the full bacterial cell cycle, a continuous culture keeps the bacteria growing on its exponential phase of the bacterial cell cycle, thus a continuous supply of bacteria can be produced.
Aim
Estimate the amount of bacteria required to clean up a given amount of phosphate, and the associated production cost to determine the profitability of our project using computational modelling.
Background Theory
The growth of bacteria in its exponential phase can be represented in the following exponential growth equation:
$$\frac{1}{x} \frac{dx}{dt} = \mu = \frac{log_e 2}{t_d}$$
where:
$$x$$ is the bacteria concentration (dry weight mass/unit volume) at time $$t$$
$$μ$$ is the specific growth rate
$$t_d$$ is the doubling time (time required for the concentration of organism to double)
In 1942, Jacques Monod showed that there is a relationship between the specific growth rate and the concentration of a limiting growth substrate that can be represented in this equation:
$$\mu = \mu_{max} \bigg(\frac{s}{K_s + s}\bigg)$$
where:
$$s$$ the concentration of a limiting growth substrate
$$μ_{max}$$ is the maximum growth rate (growth rate when organism is placed in excess nutrients without any limiting factors)
$$K_s$$ is the saturation constant – the value of $$s$$ when: $$\frac{μ}{μ_{max}}$$ = $$\frac{1}{2}$$
A relationship between growth and utilization of substrate has also been shown by Monod by the equation:
$$\frac{dx}{dt} = −Y \frac{ds}{dt}$$
\begin{equation*} Y = \frac{\textrm{weight of bacteria formed}}{\textrm{substrate utilized}} \end{equation*}
where $$Y$$ is known as the yield constant
If the values of the three growth constants: $$μ_{max}$$, $$K_s$$ and $$Y$$ are known, equation (1) to (3) provides a complete quantitative description of the ‘growth cycle’ of a batch culture.
A chemostat is a continuous flow system in which fresh growth medium is added into the vessel at a steady flow-rate ($$F$$) and culture liquid exits at the same rate. Contents within the vessel are stirred so that the growth medium is uniformly dispersed. The rate in which nutrient is exchanged in the vessel is expressed as the dilution rate ($$D$$):
$$D = \frac{\textrm{medium flow rate}}{\textrm{culture volume}} = \frac{F}{V}$$
If we assume that the bacteria within the vessel stops growing and dividing, with equal stirring and continuous flow of medium, every organism will have an equal probability of leaving the vessel within a given time. The wash-out rate (rate in which organism initially present in the vessel will be washed out) can be expressed as:
$$- \frac{dx}{dt} = Dx$$
where $$x$$ is the concentration of organisms in the vessel
1. Changes in concentration of organism
In a continuous culture, bacteria are growing at a rate expressed in equation (1) and at the same time, it is being washed out at a rate expressed in equation (5). The net rate of increase is therefore:
\begin{split} \textrm{increase} & = \textrm{growth} - \textrm{output} \\
\frac{dx}{dt} & = \mu x - Dx \end{split}
If:
$$μ$$ > $$D$$, concentration of organism will increase
$$μ$$ < $$D$$, concentration of organism will decrease
$$μ$$ = $$D$$, concentration of organism is constant; a ‘steady state’
Substituting (2) into $$μ$$:
$$\frac{dx}{dt} = x \bigg\{\mu_{max} \bigg(\frac{s}{K_s + s}\bigg) - D\bigg\}$$
2. Changes in substrate concentration
In a continuous culture, substrate enters the vessel at a concentration $$S_{in}$$, consumed by the bacterial cell and exits the vessel at concentration $$S_{out}$$. The net rate of change is therefore:
\begin{equation*} \begin{split} \textrm{increase} & = \textrm{input} - \textrm{output} - \textrm{consumption} \\
& = \textrm{input} - \textrm{output} - \frac{\textrm{growth}}{\textrm{yield constant}} \end{split} \end{equation*}
$$\frac{ds}{dt} = D S_{in} - D S_{out} - \frac{\mu x}{Y}$$
Substituting (2) into $$μ$$:
$$\frac{ds}{dt} = D (S_{in} - S_{out}) - \frac{\mu_{max} x}{Y} \bigg(\frac{s}{K_s + s}\bigg)$$
From equation (7) and (9), if $$D$$ and $$S_{in}$$ are held constant, there is a unique solution for $$x$$ and $$s$$ for which $$\frac{dx}{dt}$$ and $$\frac{ds}{dt}$$ are 0. When $$\frac{dx}{dt}$$ and $$\frac{ds}{dt}$$ is 0, the system is said to be in a ‘steady state’ because the concentration of organism and substrate within the continuous culture is kept constant. The values of steady state $$x$$ and $$s$$, designated as $$\tilde{x}$$ and $$\tilde{s}$$ are expressed as:
$$\tilde{s} = K_s \bigg(\frac{D}{\mu_{max} - D}\bigg)$$
$$\tilde{x} = Y (S_{in} - \tilde{s}) = Y \bigg\{ S_{in} - K_s \bigg(\frac{D}{\mu_{max} - D}\bigg) \bigg\}$$
The steady state values depend solely on the value of $$D$$ and $$S_{in}$$ which can be manipulated within the chemostat. The values of growth constant $$μ_{max}$$, $$K_s$$ and $$Y$$ are constant for a specific organism within a growth medium.
The output rate is the quantity of bacteria produced in unit time. The total output from a continuous culture in a steady state is equal to the product of flow-rate and concentration of organism. The output per unit volume of culture is therefore $$D\tilde{x}$$:
$$\textrm{Output} = D\tilde{x} = D * Y \bigg\{ S_{in} - K_s \bigg(\frac{D}{\mu_{max} - D}\bigg) \bigg\}$$
We can measure the maximum output rate of organism by differentiating the above equation with respect to $$D$$ and equating to zero. This will give us the dilution rate that gives the maximum output organism in unit time, $$D_{M}$$:
$$D_M = \mu_{max} \Bigg\{ 1 - \sqrt{\frac{K_s}{K_s + S_{in}}} \Bigg\}$$
The steady-state concentration of organisms at this dilution rate can be obtained by substituting $$D_M$$ in equation (11):
$$\tilde{x}_M = Y \Bigg\{ (S_{in} + K_s) - \sqrt{K_s (S_{in} + K_s)} \Bigg\}$$
The maximum output rate, $$D_{M}\tilde{x}_M$$ is the product of the two equations:
$$D_M \tilde{x}_M = \mu_{max} * Y * S_{in} \Bigg\{ \sqrt{\frac{K_s + S_{in}}{K_s}} - \sqrt{\frac{K_s}{S_{in}}} \Bigg\}^2$$
Parameters
Parameter Name Description Value Unit Source
Growth Yield $$Y$$ A typical value of the observed growth yield of E. coli at rapid growth rates 0.44 grams dry weight / grams glucose Majewski RA, Domach MM. Simple constrained-optimization view of acetate overflow in E. coli. Biotechnol Bioeng. 1990 Mar 25 35(7):732-8. p.736 left column paragraph above bottom
Saturation constant $$K_s$$ Monod substrate affinity constants (Ks) for E. coli growing on glucose as the only source of carbon and energy 0.099 grams / liter Senn H, Lendenmann U, Snozzi M, Hamer G, Egli T. The growth of Escherichia coli in glucose-limited chemostat cultures: a re-examination of the kinetics. Biochim Biophys Acta. 1994 Dec 15 1201(3):424-36. P.425 left column top paragraph & P.433 table 4
Maximum growth rate $$μ_{max}$$ Maximum growth rate for E. coli growing on glucose as the only source of carbon and energy 1.05 hour -1 Senn H, Lendenmann U, Snozzi M, Hamer G, Egli T. The growth of Escherichia coli in glucose-limited chemostat cultures: a re-examination of the kinetics. Biochim Biophys Acta. 1994 Dec 15 1201(3):424-36. P.425 left column top paragraph & P.433 table 4
Modelling Output
The output is modelled through the equation:
\begin{equation*} \textrm{Output} = D\tilde{x} = D * Y \bigg\{ S_{in} - K_s \bigg(\frac{D}{\mu_{max} - D}\bigg) \bigg\} \end{equation*}
There are two variables that can be changed according to the design of the chemostat: $$D$$ (dilution rate) and $$S_{in}$$ (initial substrate concentration).
We modelled the output vs. dilution rate at five different initial substrate concentrations (1 g/L, 5 g/L, 10 g/L, 20 g/L and 40 g/L).
We can see from the graph that for all five different concentrations of initial substrate, the output rate increase as dilution rate is increased. However, at a certain dilution rate, the output then starts to fall off dramatically. This is when the dilution rate value is higher than the specific growth rate of bacteria. At this point, the bacteria are washed out of the reactor at a faster rate than it can reproduce. Eventually, all the bacteria will be displaced out of the reactor.
The different points in the graph represent the dilution rate that gives the maximum output. As the initial substrate concentration increases, the maximum output rate is also shown to increase. In the figure below, we graphed the correlation between maximum output rate and initial substrate concentration from the values obtained from the graph above.
We can see that the relationship between initial substrate concentration and maximum output rate is a positive linear line. There is theoretically no limit to the output rate that we can achieve, as long as we can afford the initial substrate concentration. In this case, the limitations would be financial limit, based on substrate concentration and cost (£/g).
The main assumption in this model is that all other factors for growth are in sufficient supply to ensure that glucose is kept as the limiting factor in the chemostat. In high concentrations of glucose, for example, there are other factors which may become limiting. Therefore, there would be an additional cost in supplying enough of these other factors, such as oxygen or nitrogen to ensure that it reaches the desired growth rate.
However, it is important to note from figure 1 that as initial substrate concentration increases, the graph gets steeper. The dilution rate required to get maximum output rate gets closer and closer to a critical dilution rate where output then starts to fall off. Therefore, we can see that while initial substrate concentration increases the maximum output rate at a constant rate (equal to the Growth Yield), there is also a greater risk of reaching a dilution rate which would ultimately wash out all the bacteria from the reactor.
Cost Estimation
Estimating Cost of Production
Our interaction with John Liddell from the Centre Process of Innovation told us that sources of glucose usually come from cheap media such as molasses. To determine the concentration of glucose in molasses, we searched through the United States National Nutrient Database for Standard Reference by the Department of Agriculture and found that molasses is ~12% glucose. Since molasses has a density of 1400 g/L (1 L = 1400 g), the concentration of glucose is therefore 168 g/L.
As seen in figure 3, at a dilution rate of 1.024 we can achieve maximum output rate of 73.94 g/L of bacteria liquid culture per hour. However, it is important to note that the dilution rate is really close to the critical dilution rate in which bacteria will be washed away from the reactor. Therefore, it is better to use a lower dilution rate to account for any variation in growth rate. We chose a dilution rate of 0.8 which will give an output rate of 59.02 g/L.
Now that we know the output rate, the next step is to determine the cost of producing a certain amount of bacteria.
-Molasses cost $0.07/kg -The density of molasses is roughly 1.4 kg/L -Therefore, 1 L of molasses will cost$0.07 x 1.4 = $0.098 To calculate the cost of 1 kg of bacteria, we’ll start by going back to the output equation: \begin{equation*} \textrm{Output} = D\tilde{x} = D * Y \bigg\{ S_{in} - K_s \bigg(\frac{D}{\mu_{max} - D}\bigg) \bigg\} \end{equation*} Since output rate is simply the concentration of bacteria produced per unit time, the units are: \begin{equation*} \textrm{Output} = \frac{g/L}{h} = \frac{\textrm{Amount (g)}}{\textrm{Time (h)}*\textrm{Volume (L)}} \end{equation*} The total cost of producing a 1 kg of bacteria can be calculated by taking into account several variables and multiplying it with the cost of 1 Liter of molasses: 1. The time it takes to produce 1 kg. The faster we can produce 1 kg, the cheaper it will be. 2. The dilution rate or turnover rate (the rate of nutrient exchange in the chemostat) which is equal to the flow rate divided by the volume of the chemostat. A higher dilution rate means a larger volume of substrate used which increases the output rate up to a certain point. 3. The volume of the chemostat. The larger the volume, the more bacteria can be produced per time. Therefore: \begin{equation*} \textrm{Cost} = \textrm{Time} * \textrm{Dilution rate} * \textrm{Volume (L)} * \frac{$0.098}{\textrm{L}} \end{equation*}
To calculate the time required to produce 1 kg of bacteria, we can simply rearrange the output equation from before:
\begin{equation*} \textrm{Output} = \frac{\textrm{Amount (g)}}{\textrm{Time (h)}*\textrm{Volume (L)}} \end{equation*} \begin{equation*} \textrm{Time (h)} = \frac{\textrm{1 kg}}{\textrm{Output rate (g/L/h)}*\textrm{Volume (L)}} \end{equation*}
Plugging it into the cost equation:
\begin{equation*} \textrm{Cost} = \textrm{Time} * \textrm{Dilution rate} * \textrm{Volume} * $0.098 \end{equation*} \begin{equation*} \textrm{Cost} = \frac{\textrm{1 kg}}{\textrm{Output rate}*\textrm{Volume}} * \textrm{Dilution rate} * \textrm{Volume} *$0.098 \end{equation*}
Inserting our values and crossing out the same variables:
\begin{equation*} \textrm{Production Cost} = \frac{\textrm{1000 g}}{\textrm{59.02 g/L/h}} * \textrm{0.8$$h^{-1}$$} * $0.098/L \end{equation*} \begin{equation*} \textrm{} = \frac{\textrm{1000 g}* L * h}{\textrm{59.02 g}} * \frac{0.8}{h} * \frac{$0.098}{L} \end{equation*} \begin{equation*} \textrm{} = $1.33\textrm{ for 1 kg} \end{equation*} Volume of Chemostat The volume of the chemostat is independent to the production cost as it is cancelled in the equation above. The volume of the chemostat affects how fast it produces our desired product and therefore would reduce the running cost of the chemostat. We can graph the time equation to see the relationship between time and volume at an output rate of 59.02 g/L per hour. \begin{equation*} \textrm{Time (h)} = \frac{\textrm{1 kg}}{\textrm{Output rate (g/L/h)}*\textrm{Volume (L)}} \end{equation*} From the figure above, we can see that the time required to produce 1 kg of bacteria at an output rate of 59.02 g/L per hour decreases exponentially as the volume increases. It can be seen that we can produce 1 kg of bacteria in less than 2 hours in a chemostat volume above 10 L. Estimating Cost to Clean Phosphate in Waste Water Now that the basic cost of production has been calculated, the next step is to calculate the cost to sequester phosphate from a certain amount of waste water. We were able to get some phosphate data from our visit to Davyhulme Treatment Works. Since Davyhulme Treatment Works does not treat phosphate at their site, we can use the data as a case study to see how much it would cost to treat waste water that comes to the plant. The data below shows the concentration of phosphate in incoming waste water from the past 12 months: Assuming that we treat 1 ton of waste water, equivalent to 1000 L, we would need to sequester 7323mg or ~7.3g of phosphate. To calculate the cost, we will have to find out how much bacterial preparation we need by knowing how much phosphate can be accumulated per gram of bacteria. Then, we can multiply this by the production cost that we have calculated earlier: \begin{equation*} \textrm{$$Cost_{P-Treatment}$$} = \textrm{Amount of phosphate} * \frac{\textrm{Amount of bacteria}}{\textrm{Accumulated phosphate}} * \textrm{Production cost} \end{equation*} Inserting values that we know: \begin{equation*} \textrm{$$Cost_{P-Treatment}$$} = \textrm{7.3g} * \frac{\textrm{Amount of bacteria}}{\textrm{Accumulated phosphate}} * \frac{$1.33}{\textrm{1 kg}} \end{equation*}
To find out how much phosphate a bacterium can accumulate, we went to literature that we based our project on. In the paper “Bacterial microcompartment-directed polyphosphate kinase promotes stable polyphosphate accumulation in E. coli” (Liang et al., 2017), E. coli with microcompartment-direct PPK was able to accumulate (180 μg polyphosphate)/(mg protein in whole cell) in 48 hours.
So we know that:
\begin{equation*} \frac{\textrm{Amount of bacteria ($$g_{\textrm{ bacteria}}$$)}}{\textrm{Accumulated phosphate ($$g_{\textrm{ phosphate}}$$)}} = \frac{\textrm{180 $$μg_{\textrm{ phosphate}}$$}}{\textrm{$$mg_{\textrm{ whole cell protein}}$$}} * \textrm{...}\end{equation*}
The equation is not complete, there are other variables that would have to be added, such as how many grams of protein are in a bacterium. We were able to obtain protein data from New England Biolabs which shows that a liquid culture of E. coli has a total protein weight of 0.15 g per liter at 109 cells per ml. Therefore:
\begin{equation*} 0.15 \textrm{ g/L}=150 \textrm{ mg/L} \end{equation*} \begin{equation*} \frac{\textrm{Amount of bacteria ($$g_{\textrm{ bacteria}}$$)}}{\textrm{Accumulated phosphate ($$g_{\textrm{ phosphate}}$$)}} = \frac{\textrm{180 $$μg_{\textrm{ phosphate}}$$}}{\textrm{$$mg_{\textrm{ whole cell protein}}$$}} * \frac{\textrm{$$150 mg_{\textrm{ whole cell protein}}$$}}{L * \textrm{$$10^{9}$$ cells/ml}} * \textrm{...}\end{equation*}
The next variable that is missing is the number of cells per wet weight of bacteria. A study has shown that an OD600 of 1.0 will have a cell wet weight of 1.7 g/L (Glazyrina et al., 2010). For bacterial cell cultures, an OD600 of 1.0 corresponds to 8 x 108 cells/ml. Thus:
\begin{equation*} \frac{8 * \textrm{$$10^{8}$$ cells/ml}}{1.7 \textrm{g/L}} \end{equation*}
Adding it into the previous equation:
\begin{equation*} \frac{\textrm{Amount of bacteria ($$g_{\textrm{ bacteria}}$$)}}{\textrm{Accumulated phosphate ($$g_{\textrm{ phosphate}}$$)}} = \frac{\textrm{180 $$μg_{\textrm{ phosphate}}$$}}{\textrm{$$mg_{\textrm{ whole cell protein}}$$}} * \frac{\textrm{$$150 mg_{\textrm{ whole cell protein}}$$}}{L * \textrm{$$10^{9}$$ cells/ml}} * \frac{8 * \textrm{$$10^{8}$$ cells/ml}}{1.7 \textrm{g/L}}\end{equation*}
Now we have all the variables needed to cross out the units, thus we can solve it as:
\begin{equation*} \frac{\textrm{$$g_{\textrm{ bacteria}}$$}}{\textrm{$$g_{\textrm{ phosphate}}$$}} = \textrm{180$$μg_{\textrm{ phosphate}}$$} * \frac{150}{\textrm{$$10^{9}$$}} * \frac{8 * \textrm{$$10^{8}$$}}{1.7 \textrm{g}}\end{equation*} \begin{equation*} \frac{\textrm{$$g_{\textrm{ bacteria}}$$}}{\textrm{$$g_{\textrm{ phosphate}}$$}} = \frac{\textrm{12705.88 $$μg_{\textrm{ phosphate}}$$}}{\textrm{$$g_{\textrm{ bacteria}}$$}} = \frac{\textrm{0.01270588 $$g_{\textrm{ phosphate}}$$}}{\textrm{$$g_{\textrm{ bacteria}}$$}}\end{equation*} \begin{equation*} \frac{\textrm{$$g_{\textrm{ bacteria}}$$}}{\textrm{$$g_{\textrm{ phosphate}}$$}} = \frac{1}{\frac{\textrm{0.01270588 $$g_{\textrm{ phosphate}}$$}}{\textrm{$$g_{\textrm{ bacteria}}$$}}}\end{equation*}
Thus:
\begin{equation*} \frac{\textrm{Amount of bacteria ($$g_{\textrm{ bacteria}}$$)}}{\textrm{Accumulated phosphate ($$g_{\textrm{ phosphate}}$$)}} = \frac{\textrm{$$78.7 g_{\textrm{ bacteria}}$$}}{\textrm{$$g_{\textrm{ phosphate}}$$}}\end{equation*}
Therefore, we can plug it into the cost equation to determine how much it cost to treat 1 ton of wastewater:
\begin{equation*} \textrm{$$Cost_{P-Treatment}$$} = \textrm{Amount of phosphate} * \frac{\textrm{Amount of bacteria}}{\textrm{Accumulated phosphate}} * \textrm{Production cost} \end{equation*} \begin{equation*} \textrm{$$Cost_{P-Treatment}$$} = \textrm{$$7.3g_{\textrm{ phosphate}}$$} * \frac{\textrm{$$78.7 g_{\textrm{ bacteria}}$$}}{\textrm{$$g_{\textrm{ phosphate}}$$}} * \frac{$1.33}{\textrm{$$1 kg_\textrm{ bacteria}$$}} \end{equation*} \begin{equation*} \textrm{$$Cost_{P-Treatment}$$} =$0.76 \end{equation*}
Therefore, it would cost $0.76 to produce bacteria that can accumulate 7.3g of phosphate in 1 ton of wastewater in 48 hours. Real Life Scenario In Davyhulme Treatment Works, flows of more than 30,000 litres per second are treated for 24 hours straight every single day. This is equivalent to 2 592 000 tons of water treated every single day or 946 080 000 tons of water every year (United Utilities, 2017). We have previously calculated that it would cost$0.76 to treat 1 ton of waste water in 48 hours. To treat 946 080 000 tons of water would therefore cost $722 898 120 or roughly £550 million every year. With an estimate of 1.17 million households in Greater Manchester in 2016 (New Economy Manchester, 2016), the cost of implementing this technology would be an additional £470 per household each year which is expensive. We would need substantial improvements of our system to make it competitive in the real-world market – costs would have to be reduced further before water companies would be interested in our technology. We therefore considered alternative strategies to drastically re-engineer our system to achieve a substantially better economic position. Cost-Reduction Strategy We discussed a number of other potential strategies how we can further reduce the cost and how it can potentially be achieved: 1. Grow bacteria directly in the waste water stream If grown directly in the waste water stream, we would no longer face the need to replenish the bacteria as they would be sustaining themselves on the nutrients provided in the waste. To ensure that this can work, we would have to develop a safety mechanism that would prevent bacteria from escaping to the wild, such as kill-switches. Although various kill-switches have been developed by numerous iGEM teams in the past, some of them are 'leaky' and does not guarantee 100% cell death. As an alternative strategy, we also thought of using a bacterial chassis that has been genetically modified to require unnatural amino acids to produce essential proteins. While some unnatural amino acids are synthetic and expensive to produce, there are around 300 of them that can be found in nature and may be cheaper to produce. However, we considered that growing bacteria directly in the waste water stream may not be viable after observing in the lab that microcompartment expression substantially inhibits bacterial growth and makes the cells ‘sick’. To circumvent this, we thought about implementing a phosphate starvation operon to inhibit microcompartment expression in low phosphate concentration. Through this, the bacteria would be able to grow optimally in the chemostat and would only express microcompartments in waste water (high phosphate concentrations). We modelled a phosphate starvation operon where we showed microcompartment expression at different phosphate levels. 2. Use of a different substrate Although Escherichia coli grows best on glucose, there are other carbon sources that are much cheaper that could also be used, such as glycerol. The increase in demand for fuel and increased environmental concern have emphasized the need for renewable sources of energy and have increased the production of biodiesel. With its production capacity increased and developed in recent years, supply of its by-product – crude glycerol – has also increased and resulted in a dramatic reduction of price over the past years (Yang et al., 2012). In 2011, the price of glycerol fell to$0.04-0.11 per kg (Quispe et al., 2013). Thus, using a different substrate may potentially reduce the production cost significantly.
However, it is important to note that the maximum growth rate of E. coli would be lower when grown in glycerol than in glucose. This is an opportunity for synthetic biology: it would be generally beneficial to engineer E. coli so that it can efficiently take up glycerol, possibly through engineering genes involved in catabolite repression.
3. Different chassis
The calculations and experiments done for our project are focusing on E. coli as a chassis and we have observed substantially reduced growth when these cells express microcompartments. E. coli is chosen because it is easy to engineer and is suitable for us to work within the time constraints of iGEM. But there are other chassis that would be more suitable for real-world industrial applications of this project. These include other bacterial species and even photosynthetic organisms such as algae or cyanobacteria. Whilst the use of photosynthetic organisms presents its own challenges, they are of particular interest as their use would circumvent the need for supplying a costly carbon source.
4. Phosphate fertilizer
Another potentially important aspect of increasing the economic viability of our synthetic biology approach is the opportunity of recovering some of the cost by selling phosphate fertilizer. We have originally thought about this since the beginning, and we envisaged this as one of the main selling points of our project. Since the bacteria accumulate phosphate in their microcompartments, we can harvest them afterwards and make phosphate fertilizer out of them. We thought about putting our bacteria in a semi-permeable membrane that would only let water pass through the membrane and keep the bacteria inside. This would achieve two goals: a robust device that can prevent the bacteria from escaping into the wild and a way to concentrate the bacteria in one place to facilitate efficient phosphate harvesting.
We made two further cost calculations in the next section for two of our possible strategies: phosphate fertilizer and use of a different substrate.
Phosphate Fertilizer
One of the selling points of our project is the ability to harvest the phosphate and then sell it further to generate some profit, improve resource security and agricultural sustainability. This profit can be used to further reduce the operating cost of our technology and consequently the price for customers.
Assuming that we can reliably accumulate the required amount of phosphate from Davyhulme Water Treatment Facility in our bacteria, we can calculate the amount of phosphate per year that we can harvest:
\begin{equation*} \frac{\textrm{7.3229 mg}}{L} * \textrm{946 080 000 000 L} = \textrm{6 928 049 kg} \end{equation*}
To calculate our pricing cost, we use the market cost for Diammonium Phosphate (DAP), the most widely used phosphate fertilizer which is priced at £348-£355 per ton or an average of £352 per ton (Horne, 2016):
\begin{equation*} \textrm{6 928 049 kg} * \frac{£352}{\textrm{1000 kg}} = \textrm{£2 438 673} ≈ \textrm{£2.4 million} \end{equation*}
Assuming that all the phosphate is sold at this price, we can cut down the price of treating waste water from the revenue gained through selling phosphate. However, this value is small when compared to the £550 million predicted treatment cost (considering our initial design).
Using Glycerol as a Substrate
The calculation from the previous section shows that the price to implement the technology is still too expensive to be affordable, even if the substrate concentration could be increased. There are two other ways to reduce the cost further. One of them is to increase the amount of phosphorus that can be accumulated in the microcompartment by using a different PPK enzyme or some other means. Another way would be simply to use a cheaper alternative source of carbon.
In this section, we will calculate the cost of bacteria production in a continuous culture using glycerol as the substrate. Since the price of glycerol is at a range of $0.04-0.11 per kg, we will use its median price at$0.08 (Quispe et al., 2013). In addition, we will also assume that the growth yield (Y) of bacteria growing in glycerol is the same as when it is growing in glucose at 0.44 (we could not find the growth yield value with glycerol in the literature but we assume that this could realistically be achieved, as a result of synthetic biology engineering of the bacteria’s metabolism).
Using the production cost equation, we can plug in the values for dilution rate and maximum output rate to obtain the cost of producing 1 kg of bacteria using glycerol:
\begin{equation*} \textrm{Production Cost} = \frac{\textrm{1000 g}}{\textrm{Output rate}} * \textrm{Dilution rate} * \textrm{Price of glycerol} \end{equation*}
The price of glycerol will depend on the initial substrate concentration; the higher the initial substrate concentration, the more glycerol we would need:
\begin{equation*} \textrm{Price of glycerol} = \textrm{Initial substrate concentration (g/L)} * $0.08/kg \end{equation*} Therefore: \begin{equation*} \textrm{Production Cost} = \frac{\textrm{1000 g}}{\textrm{Output rate}} * \textrm{Dilution rate} * \textrm{Initial substrate concentration (g/L)} * \frac{$0.08}{\textrm{1000 g}} \end{equation*}
Plugging in the values that we know:
Initial Substrate Concentration Output rate Dilution rate
168 g/L 59.02 0.8
\begin{equation*} \textrm{Production Cost} = $0.18 \end{equation*} Using the phosphate treatment cost equation, we can plug in the production cost into the equation to obtain the cost of treating 1 ton of waste water: \begin{equation*} \textrm{$$Cost_{P-Treatment}$$} = \textrm{Amount of phosphate} * \frac{\textrm{Amount of bacteria}}{\textrm{Accumulated phosphate}} * \textrm{Production cost per kg} \end{equation*} \begin{equation*} \textrm{$$Cost_{P-Treatment}$$} = \textrm{$$7.3g_{\textrm{ phosphate}}$$} * \frac{\textrm{$$78.7 g_{\textrm{ bacteria}}$$}}{\textrm{$$g_{\textrm{ phosphate}}$$}} * \frac{\textrm{Production cost}}{\textrm{$$1000 g_{\textrm{ bacteria}}$$}} \end{equation*} \begin{equation*} \textrm{$$Cost_{P-Treatment}$$} =$0.103 \end{equation*}
Thus, we can find the total cost to treat 946 080 000 tons of water and from there, we can divide the total cost with the number of households in Greater Manchester (1.17 million) to calculate how much the implementation of this technology will cost per household:
Cost P-Treatment per 1 ton Total Cost P-Treatment Cost per household per year
$0.103$97 446 240 \$83.29
We can see that the price is significantly reduced when using glycerol in comparison to glucose. We would now reach a cost level that perhaps comes close to an economically feasible range (although still too high to be realistically competitive). However, glucose is the preferred carbon source for E. coli and substituting it with glycerol may result in different growth rates. E. coli grown on glycerol has been shown to express different levels of certain metabolic genes which may possibly lead to slower growth and metabolism. In addition, there is not enough data and studies in regards to the carbon stress response of E. coli grown in glycerol. Carbon stress is a phenomenon that is exhibited by E. coli when grown in carbon-limiting environments such as in a chemostat and is therefore an important aspect to consider for industrial applications (Martinez-Gomez et al., 2012).
Perhaps there is an opportunity to use synthetic biology to engineer a strain of E. coli that can utilize glycerol as effectively as glucose. With the increased production of biodiesel, glycerol as a by-product is becoming more and more appealing as an alternative carbon source due to its cheap cost. An E. coli that can utilize glycerol effectively would possibly reduce the production cost of many other commercial bacterial cultures in addition to ours. It would potentially reduce the production cost of various commercially produced proteins such as enzymes and antibiotics, which could lead to more affordable healthcare and advancement in research.
Summary
We have made a rough cost estimation on the cost to treat waste water using our synthetic biology microcompartments in bacteria. Our calculations are using phosphate data obtained from our interaction with Davyhulme Treatment Works. From our cost estimation, the cost to treat phosphate using our original design would be high: to remove phosphate from waste water for a year with an output rate of 59.02 g/L per hour would cost £550 million, which is equivalent to £470 per household in Greater Manchester, and this is excluding the cost to set-up and maintain the equipment. This is quite expensive and would not be competitive in a real-life market. From this calculation, we were able to integrate it into our business plan and assess the operation and maintenance cost of our device in comparison with two other technologies: Chemical Precipitation and Enhanced Biological Phosphate Removal (EBPR)
We discussed a number of alternative synthetic biology engineering strategies that would improve the economic viability of our project, including alternative chassis (e.g. photosynthetic algae), self-sustaining microbes feeding on wastewater with engineered kill switches, and metabolically engineered bacteria that can use much cheaper glycerol as their carbon source.
In another attempt to reduce predicted cost, we calculated how much revenue we can earn from selling back the phosphate that is accumulated and found that it would reduce the production cost by £2.4 million. Unfortunately, this is a small fraction of the entire production cost. Next, we calculated the production cost by using glycerol as a substrate instead of glucose. From our calculations, the cheap price of glycerol has reduced the cost per household per year by 5-fold.
We concluded that glycerol as a carbon source for E. coli would be needed for our project to be affordable and financially feasible, but the different metabolic pathway that the bacteria uses when grown in glycerol would slow down growth. Therefore, additional synthetic biology would be required to engineer a strain of E. coli that can utilize glycerol as effectively as glucose by modifying or improving its metabolic pathway.
References
Bailey, J. E. & Ollis, D. F. (1986). Biochemical Engineering Fundamentals. 2nd edn. Singapore: McGraw-Hill Book Company.
Biolabs, N. (2017). Protein Data | NEB. [online] Neb.com. Available at: https://www.neb.com/tools-and-resources/usage-guidelines/protein-data (Accessed 8 Sep. 2017).
Genomics.agilent.com. (2017). Agilent Genomics: Tools - Bio Calculators. [online] Available at: http://www.genomics.agilent.com/biocalculators/calcODBacterial.jsp (Accessed 9 Sep. 2017).
Glazyrina J, Materne EM, Dreher T, Storm D, Junne S, Adams T, Greller G, Neubauer P. High cell density cultivation and recombinant protein production with Escherichia coli in a rocking-motion-type bioreactor. Microb Cell Fact. 2010 May 30 9: 42. doi: 10.1186/1475-2859-9-42. p.4 left column 5th paragraph
Herbert, D., Elsworth, R. and Telling, R. (1956). The Continuous Culture of Bacteria; a Theoretical and Experimental Study. Journal of General Microbiology, 14(3), pp.601-622.
Horne, S. (2017). Farm fertiliser prices steady as spring peak approaches. [online] Farmers Weekly. Available at: http://www.fwi.co.uk/business/farm-fertiliser-prices-steady-as-spring-peak-approaches.htm (Accessed 12 Sep. 2017).
Martinez-Gomez, K., Flores, N., Castaneda, H. M., Martinez-Batallar, G., Hernandez-Chavez, G., Ramirez, O. T., Gosset, G., Encarnacion, S. & Bolivar, F. 2012. New insights into Escherichia coli metabolism: carbon scavenging, acetate metabolism and carbon recycling responses during growth on glycerol. Microbial Cell Factories, 11, 21.
New Economy Manchester. (2016). Greater Manchester Key Facts. [online] Available at: http://www.neweconomymanchester.com/media/1474/ne-key-facts-dec-15-web.pdf (Accessed 14 Sep. 2017).
Office for National Statistics. (2016). Families and households in the UK: 2016. [online] Available at: https://www.ons.gov.uk/peoplepopulationandcommunity/birthsdeathsandmarriages/families/ bulletins/familiesandhouseholds/2016 (Accessed 14 Sep. 2017).
Quispe, C. A. G., Coronado, C. J. R. & Carvalho, J. A. 2013. Glycerol: Production, consumption, prices, characterization and new trends in combustion. Renewable & Sustainable Energy Reviews, 27, 475-493.
Sigma-Aldrich. (2017). D-(+)-Glucose G8270. [online] Available at: http://www.sigmaaldrich.com/catalog/product/sigma/g8270?lang=en®ion=GB (Accessed 9 Sep. 2017).
United States Department of Agriculture Research Service. (2016). Full Report (All Nutrients): 19034, Molasses. [online] Available at: National Nutrient Database for Standard Reference Release 28 (Acessed 9 Oct. 2017).
United Utilities (2017). Davyhulme Treatment Works. [online] UnitedUtilities.com. Available at: https://www.unitedutilities.com/help-and-support/about-us/greater-manchester/davyhulme-treatment/ (Accessed 12 Sep. 2017).
Yang, F. X., Hanna, M. A. & Sun, R. C. 2012. Value-added uses for crude glycerol-a byproduct of biodiesel production. Biotechnology for Biofuels, 5, 10.
|
2017-12-17 13:47:25
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 14, "x-ck12": 0, "texerror": 0, "math_score": 0.5623160600662231, "perplexity": 2794.212990113348}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948596051.82/warc/CC-MAIN-20171217132751-20171217154751-00380.warc.gz"}
|
https://www.originlab.com/ReleaseNotes/detail.aspx?id=2022b2ORG-19339
|
## Font Control for Note Window
Version: 2022b
Type: Features
Category: Data Handling
Subcategory: Notes
Jira: ORG-19339
Font will be the same in all Notes windows but font size can be set differently in each notes window.
|
2022-08-10 02:25:00
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9108491539955139, "perplexity": 14736.789739203563}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571097.39/warc/CC-MAIN-20220810010059-20220810040059-00281.warc.gz"}
|
https://socratic.org/questions/5639986c581e2a542fe4323e
|
# Question #4323e
Apr 1, 2016
The β-oxidation of an 18-carbon acid produces ${\text{9 mol of Acetyl-CoA, 8 mol of NADH, and 8 mol of FADH}}_{2}$.
#### Explanation:
The β-oxidation pathway for a saturated fatty acid is a four-step cycle.
A ${\text{C}}_{18}$ acid (stearic acid) enters as $\text{stearyl-CoA}$.
Each pass through the cycle removes 2 carbon atoms and generates 1 molecule each of ${\text{acetyl-CoA, NADH, and FADH}}_{2}$.
The first seven passes through the cycle remove 14 carbon atoms and generate 7 molecules each of ${\text{acetyl-CoA, NADH, and FADH}}_{2}$.
In the eighth pass, the remaining 4 carbon atoms are converted to 2 molecules of $\text{acetyl-CoA}$ and 1 each of $\text{NADH}$ and ${\text{FADH}}_{2}$.
This makes a total of ${\text{9 acetyl-CoA, 8 NADH, and 8 FADH}}_{2}$.
The overall equation for the β-oxidation of $\text{stearyl-CoA}$ is:
${\text{CH"_3("CH"_2)_16"CO-SCoA" + "8FAD" + "8NAD"^+ + "8HSCoA" + "8H"_2"O" → color(red)(9)"CH"_3"CO-SCoA" + color(red)(8)"NADH" + color(red)(8)"FADH"_2 + "8H}}^{+}$
|
2019-08-20 16:51:36
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 11, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9257665872573853, "perplexity": 2375.892448826011}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315551.61/warc/CC-MAIN-20190820154633-20190820180633-00111.warc.gz"}
|
https://projecteuclid.org/euclid.aaa/1355495649
|
## Abstract and Applied Analysis
### A Classification of a Totally Umbilical Slant Submanifold of Cosymplectic Manifolds
#### Abstract
We study slant submanifolds of a cosymplectic manifold. It is shown that a totally umbilical slant submanifold $M$ of a cosymplectic manifold $\stackrel{̅}{M}$ is either an anti-invariant submanifold or a 1−dimensional submanifold. We show that every totally umbilical proper slant submanifold of a cosymplectic manifold is totally geodesic.
#### Article information
Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 716967, 8 pages.
Dates
First available in Project Euclid: 14 December 2012
https://projecteuclid.org/euclid.aaa/1355495649
Digital Object Identifier
doi:10.1155/2012/716967
Mathematical Reviews number (MathSciNet)
MR2889093
Zentralblatt MATH identifier
1237.53023
#### Citation
Uddin, Siraj; Ozel, Cenap; Khan, Viqar Azam. A Classification of a Totally Umbilical Slant Submanifold of Cosymplectic Manifolds. Abstr. Appl. Anal. 2012 (2012), Article ID 716967, 8 pages. doi:10.1155/2012/716967. https://projecteuclid.org/euclid.aaa/1355495649
#### References
• B. Y. Chen, “Slant immersions,” Bulletin of the Australian Mathematical Society, vol. 41, no. 1, pp. 135–147, 1990.
• B.-Y. Chen, Geometry of Slant Submanifolds, Katholieke Universiteit Leuven, Leuven, Belgium, 1990.
• J. L. Cabrerizo, A. Carriazo, L. M. Fernández, and M. Fernández, “Slant submanifolds in Sasakian manifolds,” Glasgow Mathematical Journal, vol. 42, no. 1, pp. 125–138, 2000.
• A. Lotta, “Slant submanifolds in contact geometry,” Bulletin Mathematical Society Roumanie, vol. 39, pp. 183–198, 1996.
• B. Şahin, “Every totally umbilical proper slant submanifold of a Kahler manifold is totally geodesic,” Results in Mathematics, vol. 54, no. 1-2, pp. 167–172, 2009.
• D. E. Blair, Contact Manifolds in Riemannian Geometry, Springer-Verlag, New York, NY, USA, 1976.
• G. D. Ludden, “Submanifolds of cosymplectic manifolds,” Journal of Differential Geometry, vol. 4, pp. 237–244, 1970.
• K. Yano and M. Kon, Structures on Manifolds, Series in Pure Mathematics, World Scientific Publishing, Singapore, Singapore, 1984.
|
2019-12-16 02:13:10
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 2, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20547373592853546, "perplexity": 4243.561691104742}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541315293.87/warc/CC-MAIN-20191216013805-20191216041805-00281.warc.gz"}
|
http://mymathforum.com/applied-math/37501-question-stokes-divergence-theorem.html
|
My Math Forum Question in Stokes and Divergence theorem.
User Name Remember Me? Password
Applied Math Applied Math Forum
August 10th, 2013, 09:14 AM #1 Newbie Joined: Jun 2013 Posts: 22 Thanks: 0 Question in Stokes and Divergence theorem. In page 3 of this articlehttp://faculty.uml.edu/cbaird/95 ... sition.pdf I have two question: I use $\vec r$ for $\vec x$ and $\vec {r'}$ for $\vec {x'}$ in the article. $\vec F(\vec {r})=\frac {1}{4\pi}\nabla\left[\int_{v'}\nabla'\cdot\left(\frac{\vec {F}(\vec {r'})}{|\vec {r}-\vec {r'}|}\right)d\tau'-\int_{v'}\frac{\nabla'\cdot \vec {F}(\vec {r'})}{|\vec {r}-\vec {r'}|}d\tau'\right]+\frac {1}{4\pi}\nabla\times\left[-\int_{v'}\nabla'\times\left(\frac{\vec {F}(\vec {r'})}{|\vec {r}-\vec {r'}|}\right)d\tau'+\int_{v'}\frac{\na bla'\times \vec {F}(\vec {r'})}{|\vec {r}-\vec {r'}|}d\tau#39;\right]$ (1) it said $\frac {1}{4\pi}\nabla\int_{v'}\nabla'\cdot\left( \frac{\vec {F}(\vec {r'})}{|\vec {r}-\vec {r'}|}\right)d\tau'=\int_{s'}\left(\fr ac{\vec {F}(\vec {r'})}{|\vec {r}-\vec {r'}|}\right)\cdot d\vec {s#39;}$ where this term goes to zero as $|\vec {r}-\vec {r'}|\rightarrow\;\infty$. But not the $\int_{v'}\frac{\nabla'\cdot \vec {F}(\vec {r'})}{|\vec {r}-\vec {r'}|}d\tau'$. Why is this true? (2)$\int_{v'}\nabla'\times\left(\frac{\vec {F}(\vec {r'})}{|\vec {r}-\vec {r'}|}\right)d\tau=\int_{s'}\left(\frac{\v ec {F}(\vec {r'})}{|\vec {r}-\vec {r'}|}\right)\times d\vec {s#39;}$. Again, how do I proof this surface integral goes to zero at infinity? Please help, Thanks.
Tags divergence, question, stokes, theorem
Thread Tools Display Modes Linear Mode
Similar Threads Thread Thread Starter Forum Replies Last Post dmandman Real Analysis 0 August 18th, 2011 05:53 AM ZardoZ Applied Math 2 June 21st, 2011 05:31 AM george gill Calculus 5 May 14th, 2011 02:13 PM Curupira Calculus 0 October 14th, 2010 03:32 PM ggyyree Calculus 1 April 23rd, 2009 05:39 AM
Contact - Home - Forums - Cryptocurrency Forum - Top
|
2019-10-23 06:22:37
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 9, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5877206325531006, "perplexity": 3097.227529860435}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987829458.93/warc/CC-MAIN-20191023043257-20191023070757-00505.warc.gz"}
|
https://www.physicsforums.com/threads/help-needed-newtons-first-law-particles-in-equilibrium.229493/
|
# Help needed - Newtons first law - particles in equilibrium
1. Apr 17, 2008
### Hemmelig
For some odd reason, i'm having no problems with the rest of the exercises, though this one is really annoying as i just can't seem to get it right.
Here's a picture of the exercise
h t t p://img2.freeimagehosting.net/image.php?4bc2e12442.jpg
Could someone please give me a detailed description on how to solve it ?
I've used m=w/g to find the weight of the person (882.9 N )
But when i try to use sin and cos with the angle, i don't get the right result
Last edited by a moderator: Apr 17, 2008
2. Apr 17, 2008
### Integral
Staff Emeritus
Why don't you show us what you have done?
What angles did you use in the sin and/or cos functions?
3. Apr 17, 2008
### Hemmelig
I tried dividing it up in two sets, where the rope to the left was A and the rope to the right was B
Since it's in equilibrium, the sum of forces in the x direction (Fx) is 0, Fy is also 0
Since Fx=0 : B * sin 10 + (-A)
Since Fy=0 : B * cos 10 + (-w)
I then used the second one to find the tension from part B, but i'm not allowed to sum them together (?), so i don't really know what to do.
Basicly, i haven't got a clue right now
4. Apr 17, 2008
### alphysicist
Hi Hemmelig,
Newton's law in the x-direction, for the case of no acceleration in the x-direction is $\sum F_x =0$. Writing this out for the case of your three forces, A, B, and w gives:
$$A_x+ B_x + w_x =0$$
Your got that w_x=0 (so the weight force does not appear in the x direction equation); however, it appears that you did not get the right x-components for A and B.
In the y direction, the same thing is happening. Your y-component for the weight is okay ($w_y=-w$) but the y-components for A and B are incorrect.
5. Apr 17, 2008
### Hemmelig
hmm, what are the right components for x and y then ?
6. Apr 17, 2008
### Hemmelig
I got the right result (well, in a way)
If i call the rope C
C * sin 10 + (-w)
w/sin10 = C
But for some reason, the answer i get is exactly twise the correct result
Am i doing it right and i'm just supposed to divide the answer with two ?
7. Apr 17, 2008
### alphysicist
For the part of the rope pulling to the right (force B), it is pulling both upwards and to the right. It's x-component is the horizontal component of the force and if you draw a triangle for the components of B you can see that the horizontal component is adjacent to the angle.
So, for example, $B_x = +B \cos\theta$. If you find $B_y, A_x, A_y$ you can then find the tension.
Last edited: Apr 17, 2008
8. Apr 17, 2008
### alphysicist
If you solve the equation for forces in the x-direction and plug it into the force equation for the y direction, you'll get the above formula except with a factor of 2.
Solving the x-direction equation will also tell you something important about the tension along ropes in these problems that will show you why the above equation is so close.
|
2017-02-21 08:11:35
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7055597305297852, "perplexity": 641.8766415795817}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170696.61/warc/CC-MAIN-20170219104610-00245-ip-10-171-10-108.ec2.internal.warc.gz"}
|
https://mathematica.stackexchange.com/questions/97446/improving-the-speed-of-an-optimization-algorithm-via-compilation-and-parallelliz
|
# Improving the speed of an optimization algorithm via compilation and parallellization
I have written an implementation of eight variants of the PSO-algorithm and I want to test them over a set of test functions. I am using parallelization as it improves the speed a lot and I have also, without much know-how, done some compiling. As I just very recently discovered that such a thing as "compile" even exists, I would like to ask if I have done it properly in the sample below. Are there any bad mistakes?
Another thing is the actual algorithm part. I recently asked about a problem I encountered when switching to ParallelDo from Do and I was told that I really should use functions without side-effects. Is there any obvious way to do better than what I have done concerning the parallelization?
(*Some parameters*)
d=30;ksi=0.72984;m=10;it=300;
(*Sum of squares*)
sqc=Compile[{{x,_Real,1}},Total[x^2],CompilationOptions->
{"InlineExternalDefinitions"->True},Parallelization->True]
(*Attempt to speed up subtraction of two vectors*)
dif=Compile[{{a,_Real,1},{b,_Real,1}},(a-b),CompilationOptions->
{"InlineExternalDefinitions"->True},Parallelization->True]
(*The speed update equation*)
vupd1=Compile[{{v,_Real,1},{x,_Real,1},{z1,_Real,1},{z2,_Real,1},
{z3,_Real,1}},ksi(v+Total[{RandomReal[{0.,4.1},{d}]*dif[z1,x],
RandomReal[{0.,4.1},{d}]*dif[z2,x],RandomReal[{0.,4.1},
{d}]*dif[z3,x]}]/3.),CompilationOptions->
{"InlineExternalDefinitions" -> True},Parallelization->True]
(*Function to be minimized and boundary enforcing*)
f[x_]:=If[AllTrue[x,-100.<#<100.&],sqc[x],Infinity](*Sphere*)
SetSharedVariable[resm];
(*More parameters*)
resm={};rangx={50.,100.};rangv={-100.,100.};
max=Abs[rangv[[2]]-rangv[[1]]]/2;S=50;Y=Table[Mod[{n-1,n,n+1},S],{n,0.,S-1}]+1;
(*The algorithm*)
(*Start loop generating runs of the algorithm and initialize some values*)
ParallelDo[res={};x=Table[RandomReal[rangx,d],{S}];
v=Table[RandomReal[rangv,d],{S}];v=(v-x)/2;xb=x;bv=Map[f,xb];gb=MinimalBy[x,f][[1]];H=f[gb];
(*Actual algorithm loop*)
Do[For[i=1,i<=S,i++,v[[i]]=vupd1[v[[i]],x[[i]],xb[[Y[[i]][[1]]]],xb[[Y[[i]][[2]]]],
xb[[Y[[i]][[3]]]]];
x[[i]]=x[[i]]+v[[i]];
R=f[x[[i]]]; (*This is the only actual call of the function*)
(*All the rest if for updating the values based on the single function call*)
If[R<=bv[[i]],xb[[i]]=x[[i]];bv[[i]]=R;If[R<=H,gb=x[[i]];H=R]];];
res={res,gb};,{it}];AppendTo[resm,Partition[Flatten[res],d]];,{m}];
• I haven't looked at your code in detail. But just be aware that Compile and all the parallel functions have many subtleties and limitations, and are not easy to use. In fact, their incautious use can produce subtle bugs and reduce performance drastically. Things like sums of squares and subtraction of vectors as you show here will not gain much/anything from being compiled; compilation is most beneficial for procedural code. So, I would compile what you place into ParallelDo and leave the rest alone, if I were you. – Oleksandr R. Oct 20 '15 at 16:30
|
2019-09-17 03:28:43
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6310852766036987, "perplexity": 906.8642080566282}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573011.59/warc/CC-MAIN-20190917020816-20190917042816-00385.warc.gz"}
|
https://itectec.com/spec/5-general-commands-2/
|
## 5 General commands
07.073GPPAT Command set for GSM Mobile Equipment (ME)Release 1998TS
ITU‑T Recommendation V.25ter [14] includes "Generic DCE Control" commands with the prefix +G. These commands are for the identification of the TA. Four of those commands are adapted here to be the identification commands of the ME. Syntax is otherwise similar but the prefix is +CG. TIA IS‑99 [15] uses same commands for base station identification.
## 5.1 Request manufacturer identification +CGMI
Table 3: +CGMI action command syntax
Command Possible response(s) +CGMI +CME ERROR: +CGMI=?
Description
Execution command causes the TA to return one or more lines of information text <manufacturer>, determined by the ME manufacturer, which is intended to permit the user of the TA to identify the manufacturer of the ME to which it is connected to. Typically, the text will consist of a single line containing the name of the manufacturer, but manufacturers may choose to provide more information if desired. Refer subclause 9.2 for possible <err> values.
Defined values
<manufacturer>: the total number of characters, including line terminators, in the information text shall not exceed 2048 characters.
Text shall not contain the sequence 0<CR> or OK<CR>
Implementation
Optional.
## 5.2 Request model identification +CGMM
Table 4: +CGMM action command syntax
Command Possible response(s) +CGMM +CME ERROR: +CGMM=?
Description
Execution command causes the TA to return one or more lines of information text <model>, determined by the ME manufacturer, which is intended to permit the user of the TA to identify the specific model of the ME to which it is connected to. Typically, the text will consist of a single line containing the name of the product, but manufacturers may choose to provide more information if desired. Refer to subclause 9.2 for possible <err> values.
Defined values
<model>: the total number of characters, including line terminators, in the information text shall not exceed 2048 characters.
Text shall not contain the sequence 0<CR> or OK<CR>
Implementation
Optional.
## 5.3 Request revision identification +CGMR
Table 5: +CGMR action command syntax
Command Possible response(s) +CGMR +CME ERROR: +CGMR=?
Description
Execution command causes the TA to return one or more lines of information text <revision>, determined by the ME manufacturer, which is intended to permit the user of the TA to identify the version, revision level or date, or other pertinent information of the ME to which it is connected to. Typically, the text will consist of a single line containing the version of the product, but manufacturers may choose to provide more information if desired. Refer subclause 9.2 for possible <err> values.
Defined values
<revision>: the total number of characters, including line terminators, in the information text shall not exceed 2048 characters.
Text shall not contain the sequence 0<CR> or OK<CR>
Implementation
Optional.
## 5.4 Request product serial number identification +CGSN
Table 6: +CGSN action command syntax
Command Possible response(s) +CGSN +CME ERROR: +CGSN=?
Description
Execution command causes the TA to return one or more lines of information text <sn>, determined by the ME manufacturer, which is intended to permit the user of the TA to identify the individual ME to which it is connected to. Typically, the text will consist of a single line containing the IMEI (International Mobile station Equipment Identity; refer GSM 03.03 [7]) number of the ME, but manufacturers may choose to provide more information if desired. Refer subclause 9.2 for possible <err> values.
Defined values
<sn>: the total number of characters, including line terminators, in the information text shall not exceed 2048 characters.
Text shall not contain the sequence 0<CR> or OK<CR>
Implementation
Optional.
## 5.5 Select TE character set +CSCS
Table 7: +CSCS parameter command syntax
Command Possible response(s) +CSCS=[] +CSCS? +CSCS: +CSCS=? +CSCS: (list of supported s)
Description
Set command informs TA which character set <chset> is used by the TE. TA is then able to convert character strings correctly between TE and ME character sets.
When TA‑TE interface is set to 8‑bit operation and used TE alphabet is 7‑bit, the highest bit shall be set to zero.
NOTE: It is manufacturer specific how the internal alphabet of ME is converted to/from the TE alphabet.
Read command shows current setting and test command displays conversion schemes implemented in the TA.
Defined values
<chset> (conversion schemes not listed here can be defined by manufacturers):
"GSM" GSM default alphabet (GSM 03.38 subclause 6.2.1); this setting causes easily software flow control (XON/XOFF) problems
"HEX" character strings consist only of hexadecimal numbers from 00 to FF; e.g. "032FE6" equals three 8-bit characters with decimal values 3, 47 and 230; no conversions to the original ME character set shall be done.
NOTE: If ME is using GSM default alphabet, its characters shall be padded with 8th bit (zero) before converting them to hexadecimal numbers (i.e. no SMS‑style packing of 7‑bit alphabet).
"IRA" international reference alphabet (ITU‑T T.50 [13])
"PCCPxxx" PC character set Code Page xxx
"PCDN" PC Danish/Norwegian character set
"UCS2" 16-bit universal multiple-octet coded character set (ISO/IEC10646 [32]); UCS2 character strings are converted to hexadecimal numbers from 0000 to FFFF; e.g. "004100620063" equals three 16-bit characters with decimal values 65, 98 and 99, $(AT R97)$
"8859-n" ISO 8859 Latin n (1‑6) character set
"8859-C" ISO 8859 Latin/Cyrillic character set
"8859-A" ISO 8859 Latin/Arabic character set
"8859-G" ISO 8859 Latin/Greek character set
"8859-H" ISO 8859 Latin/Hebrew character set
Implementation
Mandatory when a command using the setting of this command is implemented.
## 5.6 Request international mobile subscriber identity +CIMI
Table 8: +CIMI action command syntax
Command Possible response(s) +CIMI +CME ERROR: +CIMI=?
Description
Execution command causes the TA to return <IMSI>, which is intended to permit the TE to identify the individual SIM which is attached to ME. Refer subclause 9.2 for possible <err> values.
Defined values
<IMSI>: International Mobile Subscriber Identity (string without double quotes)
Implementation
Optional.
## 5.7 Multiplexing mode +CMUX $(MUX MS-TE)$
Table 9: +CMUX parameter command syntax
Command Possible response(s) +CMUX=[,[, [,[, [,[,[, [,]]]]]]]] +CME ERROR: +CMUX? +CMUX: ,[],,,, ,,[,]+CME ERROR: +CMUX=? +CMUX: (list of supported s),(list of supported s),(list of supported s),(list of supported s),(list of supported s),(list of supported s),(list of supported s),(list of supported s),(list of supported s)
Description
This command is used to enable/disable the GSM 07.10 multiplexing protocol control channel. Refer to subclause 9.2 for possible <err> values. The AT command sets parameters for the Control Channel. If the parameters are left out, the default value is used.
Read command returns the current mode and the settings.
Test command returns the supported modes and parameters.
It is recommended that the ME/TA/TE should autobaud to the +CMUX command up to and including an interface speed of 9600 bits/s.
The OK or +CME ERROR: <err> response is returned at the speed of the +CMUX command prior to entering <mode>.
It is recommended that whenever the multiplexer control channel is released the ME/TA/TE should assume an interface rate of up to and including 9600 bits/s for auto bauding purposes irrespective of any previous higher speed having been selected.
If a +CMUX command is issued whilst in any multiplexer mode then that +CMUX command shall be ignored and the ME/TA shall return an +CME ERROR: <err> response.
Defined values
<operation> (multiplexer Transparency Mechanism)
0 Basic option
<subset>:
This parameter defines the way in which the multiplexer control channel is set up. A virtual channel may subsequently be set up differently but in the absence of any negotiation for the settings of a virtual channel, the virtual channel shall be set up according to the control channel <subset> setting.
0 UIH frames used only
1 UI frames used only
2 I frames used only
Default value: 0
<port_speed> (transmission rate):
1 9 600 bit/s
2 19 200 bit/s
3 38 400 bit/s
4 57 600 bit/s
5 115 200 bit/s
6 230 400 bits/s
<N1> (maximum frame size):
1- 32768
default Value : 31 (64 if Advanced option is used)
<T1> (acknowledgement timer in units of ten milliseconds):
1-255, where 10 is default (100 ms)
<N2> (maximum number of re-transmissions):
0-100, where 3 is default
<T2> (response timer for the multiplexer control channel in units of ten milliseconds):
2-255, where 30 is default (300 ms)
NOTE: T2 must be longer than T1.
<T3> (wake up response timer in seconds):
1-255, where 10 is default
<k> (window size, for Advanced operation with Error Recovery options):
1-7, where 2 is default
Implementation
Mandatory, if GSM 07.10 supported in the ME/TA.
## 5.8 ITU‑T V.25ter [14] generic TA control commands
Table 10: V.25ter generic TA control commands
Command Section Impl. Use in GSM Z[] 6.1.1 mand. TA sets all parameters to their defaults as specified by a user memory profile or by the manufacturer, and resets TA &F[] 6.1.2 mand. TA sets all parameters to their defaults as specified by the manufacturer I[] 6.1.3 opt. request manufacturer specific information about the TA (software cannot use this command to determine the capabilities of a TA) +GMI 6.1.4 mand. request TA manufacturer identification (may equal to +CGMI) +GMM 6.1.5 mand. request TA model identification (may equal to +CGMM) +GMR 6.1.6 mand. request TA revision identification (may equal to +CGMR) +GSN 6.1.7 opt. request TA serial number identification (may equal to +CGSN) +GOI 6.1.8 opt. request ISO system global object identification of the TA (general format defined in ITU‑T Recommendation X.208; encoding rules in ITU‑T Recommendation X.209) +GCAP 6.1.9 mand. request overall capabilities of TA; the response code for a TA building on the present document shall be +CGSM +GCI= 6.1.10 opt. selects the country of installation for the TA using ITU‑T Recommendation T.35 Annex A country codes
## 5.9 PCCA STD‑101 [17] select wireless network +WS46
PCCA STD‑101 [17] includes a command to select the cellular network (Wireless Data Service; WDS) to operate with the TA. PCCA calls this as WDS‑Side Stack Selection. This command may be used when TA is asked to indicate the networks in which it can operate.
Table 11: +WS46 parameter command syntax
Command Possible response(s) +WS46=[] +WS46? +WS46=? (list of supported s)
Description
Set command selects to WDS side stack <n> to be used by the TA. Read command shows current setting and test command displays side stacks implemented in the TA.
Defined values
<n>:
12 GSM digital cellular
refer PCCA STD‑101 [17] for other values
Implementation
Mandatory in PCCA STD‑101, but optional for GSM.
## 5.10 Informative examples
When beginning to build a communication link, a general TE application controlling a TA needs to determine the TA and the ME to which it is connected. V.25ter [14] has seven commands for TA identification from which four are mandatory to be implemented in a TA. An example of this command sequence requesting manufacturer (+GMI), model (+GMM), revision (+GMR) and serial number (+GSN) information would be:
AT+GMI
Manufacturer ABC
OK
AT+GMM
GSM Ultimate Data Device
OK
AT+GMR
1.00
OK
AT+GSN
987612345‑123
OK
The maximum lengths of the information responses are defined to be 2048 characters, but it is recommended that they are kept as simple as in the example. The serial number command is defined as optional. Another optional command is Global Object Identification command (+GOI) which should return the object identifiers of ITU‑T Recommendation X.208 as numeric strings delimited by periods. The Complete Capabilities List command (+GCAP) should indicate the major capability areas of the TA. The support of different areas is presented in the response of +GCAP command. Each area may be presented by the selection command name of a specific capability area (e.g. +FCLASS for fax support) or some other predefined response. For instance, a GSM TA with fax capabilities could respond as follows:
AT+GCAP
+GCAP: +CGSM,+FCLASS,+W
OK
The first supported area in the response is presented with +CGSM. It is the response text to show that some or all GSM commands of the present document are supported. Second response text (+FCLASS) informs that some fax or voice capabilities are present, and the third text (+W) about the presence of wireless commands as specified by PCCA STD‑101 [17]. Command +FCLASS=? (refer e.g. ITU‑T T.31 [11] and T.32 [12]) should be used to query the supported fax capabilities and +WS46=? to query the wireless data services available:
AT+FCLASS=?;+WS46=?
0,1,2,2.0
(12)
OK
The TA of this example supports GSM data services, and fax service class 1 (TIA‑578‑A), 2 (manufacturer specific) and 2.0 (ITU‑T T.32 [12]/ TIA‑592).
The present document defines commands for ME identification which are similar to those for TA identification in V.25ter [14], for an example:
AT+CGMI
Mobile Manufacturer XYZ
OK
AT+CGMM
GSM Phone 1234
OK
AT+CGMR
1.00
OK
AT+CGSN
123456121234561
OK
Manufacturer, model and version commands work similarly as for TA, except that the serial number query returns the International Mobile Station Equipment Identity (IMEI) number. IMEI is fifteen digits long and consists of a type approval code, a final assembly code, a serial number and a spare digit (refer GSM 03.03 [7]). When the TA is implemented inside ME, the responses for both TA and ME queries will most likely follow the responses of ME identification.
|
2022-09-30 16:17:20
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.36141738295555115, "perplexity": 8723.50375242538}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335491.4/warc/CC-MAIN-20220930145518-20220930175518-00500.warc.gz"}
|
https://www.society.shu.edu.cn/CN/abstract/abstract16298.shtml
|
[an error occurred while processing this directive]
• 论文 •
### Analysis on non-oscillatory singularity behaviors of mode Ⅱ interface crack tip in orthotropic bimaterial
Tiemei YANG, Weiyang YANG, Junlin LI, Xuexia ZHANG
1. School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China
• 收稿日期:2015-11-11 修回日期:2016-04-01 出版日期:2016-09-01 发布日期:2016-09-01
• 通讯作者: Tiemei YANG E-mail:yangtie01@sina.com
• 基金资助:
Project supported by the Natural Science Foundation of Shanxi Province (No. 2014011009-2)
### Analysis on non-oscillatory singularity behaviors of mode Ⅱ interface crack tip in orthotropic bimaterial
Tiemei YANG, Weiyang YANG, Junlin LI, Xuexia ZHANG
1. School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China
• Received:2015-11-11 Revised:2016-04-01 Online:2016-09-01 Published:2016-09-01
• Contact: Tiemei YANG E-mail:yangtie01@sina.com
• Supported by:
Project supported by the Natural Science Foundation of Shanxi Province (No. 2014011009-2)
The fracture behaviors near the mode Ⅱ interface crack tip for orthotropic bimaterial are studied. The non-oscillatory field, where the stress singularity exponent is a real number, is discussed by the complex function method and the undetermined coefficient method. From the research fracture problems, the stress functions with ten undetermined coefficients and an unknown singularity exponent are introduced when △1 > 0 and △2 > 0. By the existence theorem of non-trival solutions for the system of eight homogeneous linear equations, the characteristic equation, the stress singularity exponent, and the discriminating condition of the non-oscillatory singularity are found. By the uniqueness theorem of the solutions for the system of twelve non-homogeneous linear equations with ten unknowns, the ten undermined coefficients in the stress functions are uniquely determined. The definitions of the stress intensity factors are given with the help of one-sided limit, and their theoretical formulae are deduced. The analytic solutions of the stresses near the mode Ⅱ interface crack tip are derived. The classical results for orthotropic material are obtained.
Abstract:
The fracture behaviors near the mode Ⅱ interface crack tip for orthotropic bimaterial are studied. The non-oscillatory field, where the stress singularity exponent is a real number, is discussed by the complex function method and the undetermined coefficient method. From the research fracture problems, the stress functions with ten undetermined coefficients and an unknown singularity exponent are introduced when △1 > 0 and △2 > 0. By the existence theorem of non-trival solutions for the system of eight homogeneous linear equations, the characteristic equation, the stress singularity exponent, and the discriminating condition of the non-oscillatory singularity are found. By the uniqueness theorem of the solutions for the system of twelve non-homogeneous linear equations with ten unknowns, the ten undermined coefficients in the stress functions are uniquely determined. The definitions of the stress intensity factors are given with the help of one-sided limit, and their theoretical formulae are deduced. The analytic solutions of the stresses near the mode Ⅱ interface crack tip are derived. The classical results for orthotropic material are obtained.
|
2021-10-20 09:02:48
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4232695996761322, "perplexity": 2387.7251064155043}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585305.53/warc/CC-MAIN-20211020090145-20211020120145-00288.warc.gz"}
|
https://en.wikipedia.org/wiki/255_(number)
|
# 255 (number)
← 254 255 256 →
Cardinal two hundred fifty-five
Ordinal 255th
(two hundred and fifty-fifth)
Factorization 3 × 5 × 17
Roman numeral CCLV
Binary 111111112
Ternary 1001103
Quaternary 33334
Quinary 20105
Senary 11036
Octal 3778
Duodecimal 19312
Vigesimal CF20
Base 36 7336
255 (two hundred [and] fifty-five) is the natural number following 254 and preceding 256.
## In mathematics
Its factorization makes it a sphenic number.[1] Since 255 = 28 – 1, it is a Mersenne number[2] (though not a pernicious one), and the fourth such number not to be a prime number. It is a perfect totient number, the smallest such number to be neither a power of three nor thrice a prime.
Since 255 is the product of the first three Fermat primes, the regular 255-gon is constructible.
In base 10, it is a self number.
255 is a repdigit in base 2 (11111111) in base 4 (3333), and in base 16 (FF).
## In computing
255 is a special number in some tasks having to do with computing. This is the maximum value representable by an eight-digit binary number, and therefore the maximum representable by an unsigned 8-bit byte (the most common size of byte, also called an octet), the smallest common variable size used in high level programming languages (bit being smaller, but rarely used for value storage). The range is 0 to 255, which is 256 total values.
${\displaystyle 255=2^{8}-1={\mbox{FF}}_{16}=11111111_{2}}$
For example, 255 is the maximum value
• that can be assigned to elements in the 24-bit RGB color model, since each color channel is allotted eight bits.
• of any dotted quad in an IP address.
• of the alpha blending scale in Delphi (255 being 100% visible and 0 being fully transparent)
The use of eight bits for storage in older video games has had the consequence of it appearing as a hard limit in many video games. For example, in the earlier versions of The Legend of Zelda, Link can carry a maximum of 255 rupees.[3] It was often used for numbers where casual gameplay would not cause anyone to exceed the number. However, in most situations it is reachable given enough time. This can cause many other peculiarities to appear when the number wraps back to 0, such as the infamous "kill screen" seen after clearing level 255 of Pac-Man.[4]
This number could be interpreted by a computer as −1 if a programmer is not careful about which 8-bit values are signed and unsigned, and the two's complement representation of −1 in a signed byte is equal to that of 255 in an unsigned byte.
## References
1. ^ "A007304". OEIS. Retrieved 12 March 2015.
2. ^ "PDF" (PDF). American Mathematical Society. Retrieved 12 March 2015.
3. ^ Hoovler, Evan. "The History of Annoying Side-Quests in Videogames." GameSpy. 2009-12-04.
4. ^ Clewett, James. "255 and Pac-Man." Numberphile. 2007-17-11.
|
2016-07-31 00:44:47
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6237694025039673, "perplexity": 2634.86792292622}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469258944256.88/warc/CC-MAIN-20160723072904-00314-ip-10-185-27-174.ec2.internal.warc.gz"}
|
http://math.ucdenver.edu/~sborgwardt/wiki/index.php/Multi-commodity_Flow
|
# Multi-commodity Flow
The multi-commodity flow problem is a generalization of the maximum flow problem, where we need to find a maximum $(s_i, t_i)$- flow through the network for all commodities $i = 1,...,k.$ while keeping the sum of flow over all commodities on each arc below its capacity. The formulation for this problem is a linear program that can be solved in strongly polynomial time.
\begin{align*} \max & \sum_{i=1}^{k} \Big( \sum_{a\in\delta^+(s_i)} x_a^i - \sum_{a\in\delta^-(s_i)} x_a^i\Big) & \\ s.t. & \sum_{a\in\delta^+(v)} x_a^i - \sum_{a\in\delta^-(s_i)} x_a^i = 0 & \text{ for all } v\in V \ \{s_i,t_i\}, \text{ }i = 1,\dots,k \\ & \sum_{i=1}^{k}x_a^i \leq u_a & \text{ for all } a\in A \\ & x_a^i \geq 0 & \text{ for all } a\in A , \text{ }i = 1,\dots,k\\ \end{align*}
However due to the fact that more than one commodity travel on the same arc the problem becomes $NP$-hard when you look for an integer solution. Unlike standard max flow problem there is also no guarantee that an integer solution exists just because you have a feasible flow. Even if you are not looking for an integer solution it turns out in practice that while polynomial time algorithms exist to solve the multi-commodity flow problem, there is a larger issue to address. Say we have a graph with 1000 nodes and 10,000 edges, and we have 1 million commodities to send through this network (sending flow from each node to every other node). Then the number of variables in this seemingly reasonable problem becomes about 10 trillion, and at a measly 8 bytes per variable which is less than the space usually allocated to each variable in this type of problem, you have already used 80 gigabytes of storage space. Ultimately this leads us to decide that we need to use a method such as column generation to greatly reduce the size of the matrix needed to perform the necessary calculations to find our multi-commodity flow.
## Column Generation
Column generation is the dual to the cutting plane method. It allows us to consider only a subset of the variables while keeping all of the constraints of the problem. Our original problem will now be called the master LP. We then consider a restricted LP which contains all the constraints of the original LP and a subset of the variables. Each iteration we solve a pricing problem (see if there are variables we should add to improve our objective function value) until we reach the optimal solution which means there are no more variables to add that will improve the objective function value.
For this to work with the multi-commodity flow problem we need to start with our master LP being the path formulation of the multi-commodity flow problem.
\begin{align*} \max & \sum_{p\in P} f_p &\\ s.t. & \sum_{p\in P_a} f_p \leq u_a &\text{ for all } a\in A \\ & 0 \leq f_p & \text{ for all } p\in P \\ \end{align*}
So essentially with this formulation each column you add represents a path through the network that a commodity is sent along. So our restricted master LP is the constraints over a small starting number of paths $P'\subseteq P$ which makes our restricted master LP:
\begin{align*} \max & \sum_{p\in P'} f_p &\\ s.t. & \sum_{p\in P'_a} f_p \leq u_a &\text{ for all } a\in A \\ & 0 \leq f_p & \text{ for all } p\in P' \\ \end{align*}
And the dual of the restricted master problem is:
\begin{align*} \min & \sum_{a\in A} u_a \mu_a &\\ s.t. & \sum_{a\in p} \mu_a \geq 1 & \text{ for all } p\in P'\\ & \mu_a \geq 0 & \text{ for all } a\in A \\ \end{align*}
By the duality theorem we know that a solution $\mu$ to the restricted dual is a lower bound on a solution to the dual which also provides a lower bound to the primal problem. If both the primal and dual problem are feasible finding the optimal solution for the dual master problem will result in the optimal solution for the primal master LP. Below is a basic algorithm for solving the multi-commodity flow problem using column generation. Once the algorithm terminates with no other columns to add you have the optimal solution to the masterLP. Since the largest calculation done every iteration is coming up with the new pricing vector, if this pricing problem is solved in polynomial time then the master problem is solvable in polynomial time. Plus you no longer have the storage problems discussed above because you are only using the variables necessary to reach the optimal solution.
|
2023-01-27 07:15:16
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9896977543830872, "perplexity": 333.9529016800575}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764494974.98/warc/CC-MAIN-20230127065356-20230127095356-00330.warc.gz"}
|
https://hbcp.chemnetbase.com/faces/documents/17_08/17_08_0001.xhtml
|
Section: 17 | Special Functions |
Help Manual
Page of 1
Type a page number and hit Enter.
/1
Back to Search Results
Type a page number and hit Enter.
Summary of table differences
No records found.
How to Cite this Reference
The recommended form of citation is: John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), CRC Press/Taylor & Francis, Boca Raton, FL. If a specific table is cited, use the format: "Physical Constants of Organic Compounds," in CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), John R. Rumble, ed., CRC Press/Taylor & Francis, Boca Raton, FL.
# 8 SPECIAL FUNCTIONS
## 8.1 ORTHOGONAL POLYNOMIALS
1. Legendre Symbol:${P}_{n}\left(x\right)$ Interval: [ $-1,1$ ]
Differential Equation:$\left(1-{x}^{2}\right){y}^{\prime \prime }-2\phantom{\rule{0.2em}{0ex}}{xy}^{\prime }+n\left(n+1\right)y=0$
Explicit Expression:${P}_{n}\left(x\right)=\frac{1}{{2}^{n}}\sum _{m=0}^{\left[n/2\right]}{\left(-1\right)}^{m}\left(\begin{array}{c}n\\ m\end{array}\right)\left(\begin{array}{c}2n-2m\\ n\end{array}\right){x}^{n-2m}$
Recurrence Relation:$\left(n+1\right){P}_{n+1}\left(x\right)=\left(2n+1\right){xP}_{n}\left(x\right)-{nP}_{n-1}\left(x\right)$
Weight: 1
Standardization:${P}_{n}\left(1\right)=1$
Norm:${\int }_{-1}^{+1}\left[{P}_{n}\left(x\right)\right]{}^{2}\phantom{\rule{0.2em}{0ex}}dx=\frac{2}{2n+1}$
Rodrigues' Formula:${P}_{n}\left(x\right)=\frac{\left(-1\right){}^{n}}{{2}^{n}n!}\frac{{d}^{n}}{{dx}^{n}}\left\{\left(1-{x}^{2}\right){}^{n}\right\}$
Generating Function:$\begin{array}{c}{R}^{-1}=\sum _{n=0}^{\infty }{P}_{n}\left(x\right){z}^{n};-1
Inequality:$|{P}_{n}\left(x\right)|\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}1,\phantom{\rule{0.2em}{0ex}}-1\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}1$ .
2. Tschebysheff, First Kind Symbol:${T}_{n}\left(x\right)$ Interval:[−1, 1]
Differential Equation:$\left(1-{x}^{2}\right)y-x{y}^{\prime }+{n}^{2}y=0$
Explicit Expression:$\frac{n}{2}\sum _{m=0}^{\left[n/2\right]}{\left(-1\right)}^{m}\frac{\left(n-m-1\right)!}{m!\left(n-2m\right)!}{\left(2x\right)}^{n-2m}=cos\left(narccosx\right)={T}_{n}\left(x\right)$
Recurrence Relation:${T}_{n+1}\left(x\right)=2{xT}_{n}\left(x\right)-{T}_{n-1}\left(x\right)$
Weight:$\left(1-{x}^{2}\right){}^{-1/2}$
Standardization:${T}_{n}\left(1\right)=1$
Norm:${\int }_{-1}^{+1}\left(1-{x}^{2}\right){}^{-1/2}\left[{T}_{n}\left(x\right)\right]{}^{2}\phantom{\rule{0.2em}{0ex}}dx=\left\{\begin{array}{cc}\hfill \pi /2,\hfill & n\ne 0\\ \hfill \pi ,\hfill & n=0\end{array}$
Rodrigues' Formula:$\frac{\left(-1\right){}^{n}\left(1-{x}^{2}\right){}^{1/2}\sqrt{\pi }}{{2}^{n+1}\Gamma \left(n+\frac{1}{2}\right)}\frac{{d}^{n}}{{dx}^{n}}\left\{\left(1-{x}^{2}\right){}^{n-\left(1/2\right)}\right\}={T}_{n}\left(x\right)$
Generating Function:$\frac{1-xz}{1-2xz-{z}^{2}}=\sum _{n=0}^{\infty }{T}_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}{z}^{n},\phantom{\rule{0.2em}{0ex}}-1
Inequality:$|{T}_{n}\left(x\right)|\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}1,\phantom{\rule{0.2em}{0ex}}-1\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}x\le 1$ .
3. Tschebysheff, Second KindSymbol${U}_{n}\left(x\right)$ Interval: [−1, 1]
Differential Equation:$\left(1-{x}^{2}\right){y}^{″}-3x{y}^{\prime }+n\left(n+2\right)y=0$
Explicit Expression:$\begin{array}{c}{U}_{n}\left(x\right)=\sum _{m=0}^{\left[n/2\right]}\left(-1\right){}^{m}\frac{\left(m-n\right)!}{m!\left(n-2m\right)!}\left(2x\right){}^{n-2m}\hfill \\ \multicolumn{1}{c}{{U}_{n}\left(cos\theta \right)=\frac{sin\left[\left(n+1\right)\theta \right]}{sin\theta }}\\ \multicolumn{1}{c}{}\end{array}$
Recurrence Relation:${U}_{n+1}\left(x\right)=2{xU}_{n}\left(x\right)-{U}_{n-1}\left(x\right)$
Weight:$\left(1-{x}^{2}\right){}^{1/2}$
Standardization:${U}_{n}\left(1\right)=n+1$
Norm:${\int }_{-1}^{+1}\left(1-{x}^{2}\right){}^{1/2}\left[{U}_{n}\left(x\right)\right]{}^{2}\phantom{\rule{0.2em}{0ex}}dx=\frac{\pi }{2}$
Rodrigues' Formula:${U}_{n}\left(x\right)=\frac{\left(-1\right){}^{n}\left(n+1\right)\sqrt{\pi }}{\left(1-{x}^{2}\right){}^{1/2}{2}^{n+1}\Gamma \left(n+\frac{3}{2}\right)}\frac{{d}^{n}}{{dx}^{n}}\left\{\left(1-{x}^{2}\right){}^{n+\left(1/2\right)}\right\}$
Generating Function:$\frac{1}{1-2xz+{z}^{2}}=\sum _{n=0}^{\infty }{U}_{n}\left(x\right){z}^{n},-1
Inequality:$|{U}_{n}\left(x\right)|\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}+1,\phantom{\rule{0.2em}{0ex}}-1\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}1.$
4. Jacobi Symbol:${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)$ Interval: [−1, 1]
Differential Equation:$\left(1-{x}^{2}\right){y}^{\prime \prime }+\left[\beta -\alpha -\left(\alpha +\beta +2\right)x\right]{y}^{\prime }+n\left(n+\alpha +\beta +1\right)y=0$
Explicit Expression:${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)=\frac{1}{{2}^{n}}\sum _{m=0}^{n}\left(\begin{array}{c}n+\alpha \\ m\end{array}\right)\left(\begin{array}{c}n+\beta \\ n-m\end{array}\right){\left(x-1\right)}^{n-m}{\left(x+1\right)}^{m}$
Recurrence Relation:$\begin{array}{cc}\hfill & 2\left(n+1\right)\phantom{\rule{0.2em}{0ex}}\left(n+\alpha +\beta +1\right)\phantom{\rule{0.2em}{0ex}}\left(2n+\alpha +\beta \right){P}_{n+1}^{\left(\alpha ,\beta \right)}\left(x\right)\hfill \\ \multicolumn{1}{c}{}& \mathrm{ }=\left(2n+\alpha +\beta +1\right)\left[\left({\alpha }^{2}-{\beta }^{2}\right)+\left(2n+\alpha +\beta +2\right)\hfill \\ \multicolumn{1}{c}{}& \mathrm{ }×\left(2n+\alpha +\beta \right)x\right]{P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)\hfill \\ \multicolumn{1}{c}{}& \mathrm{ }-2\left(n+\alpha \right)\phantom{\rule{0.2em}{0ex}}\left(n+\beta \right)\phantom{\rule{0.2em}{0ex}}\left(2n+\alpha +\beta +2\right){P}_{n-1}^{\left(\alpha ,\beta \right)}\left(x\right)\hfill \\ \multicolumn{1}{c}{}\end{array}$
Weight:$\left(1-x\right){}^{\alpha }\left(1+x\right){}^{\beta };\alpha ,\beta >1$
Standardization:${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)=\frac{n+\alpha }{n}$
Norm:${\int }_{-1}^{+1}\left(1-x\right){}^{\alpha }\left(1+x\right){}^{\beta }\left[{P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)\right]{}^{2}\phantom{\rule{0.2em}{0ex}}dx=\frac{{2}^{\alpha +\beta +1}\Gamma \left(n+\alpha +1\right)\Gamma \left(n+\beta +1\right)}{\left(2n+\alpha +\beta +1\right)n!\Gamma \left(n+\alpha +\beta +1\right)}$
Rodrigues' Formula:${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)=\frac{\left(-1\right){}^{n}}{{2}^{n}n!\left(1-x\right){}^{\alpha }\left(1+x\right){}^{\beta }}\frac{{d}^{n}}{{dx}^{n}}\left\{\left(1-x\right){}^{n+\alpha }\left(1+x\right){}^{n+\beta }\right\}$
Generating Function:$\begin{array}{c}{R}^{-1}\left(1-z+R\right){}^{-\alpha }\left(1+z+R\right){}^{-\beta }=\sum _{n=0}^{\infty }{2}^{-\alpha -\beta }{P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right){z}^{n},\\ R=\sqrt{1-2xz+{z}^{2}},\mathrm{ }|z|<1\end{array}$
Inequality:$\underset{-1\le x\le 1}{max}|{P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)|=\left\{\begin{array}{l}\left(\begin{array}{l}n+q\hfill \\ n\hfill \end{array}\right)~{n}^{q}\phantom{\rule{0.2em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}q=max\left(\alpha ,\beta \right)\ge -\frac{1}{2}\hfill \\ |{P}_{n}^{\left(\alpha ,\beta \right)}\left({x}^{\prime }\right)|~{n}^{-1/2}\phantom{\rule{0.2em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}q<-\frac{1}{2}\hfill \\ {x}^{\prime }\phantom{\rule{0.2em}{0ex}}\text{is one of the two maximum points nearest}\hfill \\ \frac{\beta -\alpha }{\alpha +\beta +1}\hfill \end{array}$
5. Generalized Laguerre Symbol:${L}_{n}^{\left(\alpha \right)}\left(x\right)$ Interval:$\left[0,\infty \right]$
Differential Equation:${xy}^{\prime \prime }+\left(\alpha +1-x\right){y}^{\prime }+ny=0$
Explicit Expression:${L}_{n}^{\left(\alpha \right)}\left(x\right)=\sum _{m=0}^{n}{\left(-1\right)}^{m}\left(\begin{array}{c}n+\alpha \\ n-m\end{array}\right)\frac{1}{m!}{x}^{m}$
Recurrence Relation:$\left(n+1\right){L}_{n}^{\left(\alpha \right)}+1\left(x\right)=\left[\left(2n+\alpha +1\right)-x\right]{L}_{n}^{\left(\alpha \right)}\left(x\right)-\left(n+\alpha \right){L}_{n}^{\left(\alpha \right)}-1\left(x\right)$
Weight:${x}^{\alpha }{e}^{-x},\alpha >-1$
Standardization:${L}_{n}^{\left(\alpha \right)}\left(x\right)=\frac{\left(-1\right){}^{n}}{n!}{x}^{n}+\dots$
Norm:${\int }_{0}^{\infty }{x}^{\alpha }{e}^{-x}\left[{L}_{n}^{\left(\alpha \right)}\left(x\right)\right]{}^{2}\phantom{\rule{0.2em}{0ex}}dx=\frac{\Gamma \left(n+\alpha +1\right)}{n!}$
Rodrigues' Formula:${L}_{n}^{\left(\alpha \right)}\left(x\right)=\frac{1}{n!{x}^{\alpha }{e}^{-x}}\frac{{d}^{n}}{{dx}^{n}}\left\{{x}^{n+\alpha }{e}^{-x}\right\}$
Generating Function:$\left(1-z\right){}^{-\alpha -1}exp\left(\frac{xz}{z-1}\right)=\sum _{n=0}^{\infty }{L}_{n}^{\left(\alpha \right)}\left(x\right){z}^{n}$
Inequality:$\begin{array}{c}{L}_{n}^{\left(\alpha \right)}\left(x\right)\le \frac{\Gamma \left(n+\alpha +1\right)}{n!\Gamma \left(\alpha +1\right)}{e}^{x/2};\mathrm{ }\begin{array}{c}x\ge 0\\ \alpha >0\end{array}\\ |{L}_{n}^{\left(a\right)}\left(x\right)|\le \left[2-\frac{\Gamma \left(\alpha +n+1\right)}{n!\Gamma \left(\alpha +1\right)}\right]{e}^{x/2};\mathrm{ }\begin{array}{c}\hfill x\ge 0\hfill \\ \hfill -1<\alpha <0\hfill \end{array}\end{array}$
6. Hermite Symbol:${H}_{n}\left(x\right)$ Interval:$\left[-\infty ,\infty \right]$
Differential Equation:${y}^{\prime \prime }-2{xy}^{\prime }+2ny=0$
Explicit Expression:${H}_{n}\left(x\right)=\sum _{m=0}^{\left[n/2\right]}\frac{\left(-1\right){}^{m}n!\left(2x\right){}^{n-2m}}{m!\left(n-2m\right)!}$
Recurrence Relation:${H}_{n+1}\left(x\right)=2{xH}_{n}\left(x\right)-2{nH}_{n-1}\left(x\right)$
Weight:${e}^{-{x}^{2}}$
Standardization:${H}_{n}\left(1\right)={2}^{n}{x}^{n}+\dots$
Norm:${\int }_{-\infty }^{\infty }{e}^{-{x}^{2}}{\left[{H}_{n}\left(x\right)\right]}^{2}dx={2}^{n}n!\sqrt{\pi }$
Rodrigues' Formula:${H}_{n}\left(x\right)=\left(-1\right){}^{n}{e}^{{x}^{2}}\frac{{d}^{n}}{{dx}^{n}}\left({e}^{-{x}^{2}}\right)$
Generating Function:${e}^{-{x}^{2}+2zx}=\sum _{n=0}^{\infty }{H}_{n}\left(x\right)\frac{{z}^{n}}{n!}$
Inequality:$|{H}_{n}\left(x\right)|{e}^{{x}^{2}/2}k{2}^{n/2}\sqrt{n!}\phantom{\rule{0.2em}{0ex}}k\approx 1.086435$
## 8.2 TABLES OF ORTHOGONAL POLYNOMIALS
In the following, $\left\{{H}_{n},{L}_{n},{P}_{n},{T}_{n},{U}_{n}\right\}$ represent the ${n}^{\text{th}}$ order Hermite, Laguerre, Legendre, Tschebysheff (first kind), and Tschebysheff (second kind) polynomials.
${H}_{0}=1$ ${x}^{10}=\left(30240{H}_{0}+75600{H}_{2}+25200{H}_{4}+2520{H}_{6}+90{H}_{8}+{H}_{10}\right)/1024$ ${H}_{1}=2x$ ${x}^{9}=\left(15120{H}_{1}+10080{H}_{3}+1512{H}_{5}+72{H}_{7}+{H}_{9}\right)/512$ ${H}_{2}=4{x}^{2}-2$ ${x}^{8}=\left(1680{H}_{0}+3360{H}_{2}+840{H}_{4}+56{H}_{6}+{H}_{8}\right)/256$ ${H}_{3}=8{x}^{3}-12x$ ${x}^{7}=\left(840{H}_{1}+420{H}_{3}+42{H}_{5}+{H}_{7}\right)/128$ ${H}_{4}=16{x}^{4}-48{x}^{2}+12$ ${x}^{6}=\left(120{H}_{0}+180{H}_{2}+30{H}_{4}+{H}_{6}\right)/64$ ${H}_{5}=32{x}^{5}-160{x}^{3}+120x$ ${x}^{5}=\left(60{H}_{1}+20{H}_{3}+{H}_{5}\right)/32$ ${H}_{6}=64{x}^{6}-480{x}^{4}+720{x}^{2}-120$ ${x}^{4}=\left(12{H}_{0}+12{H}_{2}+{H}_{4}\right)/16$ ${H}_{7}=128{x}^{7}-1344{x}^{5}+3360{x}^{3}-1680x$ ${x}^{3}=\left(6{H}_{1}+{H}_{3}\right)/8$ ${H}_{8}=256{x}^{8}-3584{x}^{6}+13440{x}^{4}-13440{x}^{2}+1680$ ${x}^{2}=\left(2{H}_{0}+{H}_{2}\right)/4$ ${H}_{9}=512{x}^{9}-9216{x}^{7}+48384{x}^{5}-80640{x}^{3}+30240x$ $x=\left({H}_{1}\right)/2$ ${H}_{10}=1024{x}^{10}-23040{x}^{8}+161280{x}^{6}-403200{x}^{4}+302400{x}^{2}-30240$ $1={H}_{0}$ ${L}_{0}=1$ ${x}^{6}=720{L}_{0}-4320{L}_{1}+10800{L}_{2}-14400{L}_{3}+10800{L}_{4}-4320{L}_{5}+720{L}_{6}$ ${L}_{1}=-x+1$ ${x}^{5}=120{L}_{0}-600{L}_{1}+1200{L}_{2}-1200{L}_{3}+600{L}_{4}-120{L}_{5}$ ${L}_{2}=\left({x}^{2}-4x+2\right)/2$ ${x}^{4}=24{L}_{0}-96{L}_{1}+144{L}_{2}-96{L}_{3}+24{L}_{4}$ ${L}_{3}=\left(-{x}^{3}+9{x}^{2}-18x+6\right)/6$ ${x}^{3}=6{L}_{0}-18{L}_{1}+18{L}_{2}-6{L}_{3}$ ${L}_{4}=\left({x}^{4}-16{x}^{3}+72{x}^{2}-96x+24\right)/24$ ${x}^{2}=2{L}_{0}-4{L}_{1}+2{L}_{2}$ ${L}_{5}=\left(-{x}^{5}+25{x}^{4}-200{x}^{3}+600{x}^{2}-600x+120\right)/120$ $x={L}_{0}-{L}_{1}$ ${L}_{6}=\left({x}^{6}-36{x}^{5}+450{x}^{4}-2400{x}^{3}+5400{x}^{2}-4320x+720\right)/720$ $1={L}_{0}$ ${P}_{0}=1$ ${x}^{10}=\left(4199{P}_{0}+16150{P}_{2}+15504{P}_{4}+7904{P}_{6}+2176{P}_{8}+256{P}_{10}\right)/46189$ ${P}_{1}=x$ ${x}^{9}=\left(3315{P}_{1}+4760{P}_{3}+2992{P}_{5}+960{P}_{7}+128{P}_{9}\right)/12155$ ${P}_{2}=\left(3{x}^{2}-1\right)/2$ ${x}^{8}=\left(715{P}_{0}+2600{P}_{2}+2160{P}_{4}+832{P}_{6}+128{P}_{8}\right)/6435$ ${P}_{3}=\left(5{x}^{3}-3x\right)/2$ ${x}^{7}=\left(143{P}_{1}+182{P}_{3}+88{P}_{5}+16{P}_{7}\right)/429$ ${P}_{4}=\left(35{x}^{4}-30{x}^{2}+3\right)/8$ ${x}^{6}=\left(33{P}_{0}+110{P}_{2}+72{P}_{4}+16{P}_{6}\right)/231$ ${P}_{5}=\left(63{x}^{5}-70{x}^{3}+15x\right)/8$ ${x}^{5}=\left(27{P}_{1}+28{P}_{3}+8{P}_{5}\right)/63$ ${P}_{6}=\left(231{x}^{6}-315{x}^{4}+105{x}^{2}-5\right)/16$ ${x}^{4}=\left(7{P}_{0}+20{P}_{2}+8{P}_{4}\right)/35$ ${P}_{7}=\left(429{x}^{7}-693{x}^{5}+315{x}^{3}-35x\right)/16$ ${x}^{3}=\left(3{P}_{1}+2{P}_{3}\right)/5$ ${P}_{8}=\left(6435{x}^{8}-12012{x}^{6}+6930{x}^{4}-1260{x}^{2}+35\right)/128$ ${x}^{2}=\left({P}_{0}+2{P}_{2}\right)/3$ ${P}_{9}=\left(12155{x}^{9}-25740{x}^{7}+18018{x}^{5}-4620{x}^{3}+315x\right)/128$ $x={P}_{1}$ ${P}_{10}=\left(46189{x}^{10}-109395{x}^{8}+90090{x}^{6}-30030{x}^{4}+3465{x}^{2}-63\right)/256$ $1={P}_{0}$ ${T}_{0}=1$ ${x}^{10}=\left(126{T}_{0}+210{T}_{2}+120{T}_{4}+45{T}_{6}+10{T}_{8}+{T}_{10}\right)/512$ ${T}_{1}=x$ ${x}^{9}=\left(126{T}_{1}+84{T}_{3}+36{T}_{5}+9{T}_{7}+{T}_{9}\right)/256$ ${T}_{2}=2{x}^{2}-1$ ${x}^{8}=\left(35{T}_{0}+56{T}_{2}+28{T}_{4}+8{T}_{6}+{T}_{8}\right)/128$ ${T}_{3}=4{x}^{3}-3x$ ${x}^{7}=\left(35{T}_{1}+21{T}_{3}+7{T}_{5}+{T}_{7}\right)/64$ ${T}_{4}=8{x}^{4}-8{x}^{2}+1$ ${x}^{6}=\left(10{T}_{0}+15{T}_{2}+6{T}_{4}+{T}_{6}\right)/32$ ${T}_{5}=16{x}^{5}-20{x}^{3}+5x$ ${x}^{5}=\left(10{T}_{1}+5{T}_{3}+{T}_{5}\right)/16$ ${T}_{6}=32{x}^{6}-48{x}^{4}+18{x}^{2}-1$ ${x}^{4}=\left(3{T}_{0}+4{T}_{2}+{T}_{4}\right)/8$ ${T}_{7}=64{x}^{7}-112{x}^{5}+56{x}^{3}-7x$ ${x}^{3}=\left(3{T}_{1}+{T}_{3}\right)/4$ ${T}_{8}=128{x}^{8}-256{x}^{6}+160{x}^{4}-32{x}^{2}+1$ ${x}^{2}=\left({T}_{0}+{T}_{2}\right)/2$ ${T}_{9}=256{x}^{9}-576{x}^{7}+432{x}^{5}-120{x}^{3}+9x$ $x={T}_{1}$ ${T}_{10}=512{x}^{10}-1280{x}^{8}+1120{x}^{6}-400{x}^{4}+50{x}^{2}-1$ $1={T}_{0}$ ${U}_{0}=1$ ${x}^{10}=\left(42{U}_{0}+90{U}_{2}+75{U}_{4}+35{U}_{6}+9{U}_{8}+{U}_{10}\right)/1024$ ${U}_{1}=2x$ ${x}^{9}=\left(42{U}_{1}+48{U}_{3}+27{U}_{5}+8{U}_{7}+{U}_{9}\right)/512$ ${U}_{2}=4{x}^{2}-1$ ${x}^{8}=\left(14{U}_{0}+28{U}_{2}+20{U}_{4}+7{U}_{6}+{U}_{8}\right)/256$ ${U}_{3}=8{x}^{3}-4x$ ${x}^{7}=\left(14{U}_{1}+14{U}_{3}+6{U}_{5}+{U}_{7}\right)/128$ ${U}_{4}=16{x}^{4}-12{x}^{2}+1$ ${x}^{6}=\left(5{U}_{0}+9{U}_{2}+5{U}_{4}+{U}_{6}\right)/64$ ${U}_{5}=32{x}^{5}-32{x}^{3}+6x$ ${x}^{5}=\left(5{U}_{1}+4{U}_{3}+{U}_{5}\right)/32$ ${U}_{6}=64{x}^{6}-80{x}^{4}+24{x}^{2}-1$ ${x}^{4}=\left(2{U}_{0}+3{U}_{2}+{U}_{4}\right)/16$ ${U}_{7}=128{x}^{7}-192{x}^{5}+80{x}^{3}-8x$ ${x}^{3}=\left(2{U}_{1}+{U}_{3}\right)/8$ ${U}_{8}=256{x}^{8}-448{x}^{6}+240{x}^{4}-40{x}^{2}+1$ ${x}^{2}=\left({U}_{0}+{U}_{2}\right)/4$ ${U}_{9}=512{x}^{9}-1024{x}^{7}+672{x}^{5}-160{x}^{3}+10x$ $x=\left({U}_{1}\right)/2$ ${U}_{10}=1024{x}^{10}-2304{x}^{8}+1792{x}^{6}-560{x}^{4}+60{x}^{2}-1$ $1={U}_{0}$
## 8.3 BESSEL FUNCTIONS
1. Bessel's differential equation for a real variable $x$ is ${x}^{2}\frac{{d}^{2}y}{{dx}^{2}}+x\frac{dy}{dx}+\left({x}^{2}-{n}^{2}\right)y=0$
2. When $n$ is not an integer, two independent solutions of the equation are ${J}_{n}\left(x\right)$ and ${J}_{-n}\left(x\right)$ where Jn(x)=∑k=0∞(−1)kk!Γ(n+k+1)(x2)n+2k
3. If $n$ is an integer then ${J}_{n}\left(x\right)=\left(-1\right){}^{n}{J}_{n}\left(x\right)$ , where ${J}_{n}\left(x\right)=\frac{{x}^{n}}{{2}^{n}n!}\left\{1-\frac{{x}^{2}}{{2}^{2}·1!\left(n+1\right)}+\frac{{x}^{4}}{{2}^{4}·2!\left(n+1\right)\phantom{\rule{0.2em}{0ex}}\left(n+2\right)}+\frac{{x}^{6}}{{2}^{6}·3!\left(n+1\right)\phantom{\rule{0.2em}{0ex}}\left(n+2\right)\phantom{\rule{0.2em}{0ex}}\left(n+3\right)}+\dots \right\}$
4. For $n=0$ and $n=1$ , this formula becomes $\begin{array}{cc}{J}_{0}\left(x\right)\hfill & =1-\frac{{x}^{2}}{{2}^{2}\left(1!\right){}^{2}}+\frac{{x}^{4}}{{2}^{4}\left(2!\right){}^{2}}-\frac{{x}^{6}}{{2}^{6}\left(3!\right){}^{2}}+\frac{{x}^{8}}{{2}^{8}\left(4!\right){}^{2}}-\dots \hfill \\ \multicolumn{1}{c}{{J}_{1}\left(x\right)}& =\frac{x}{2}-\frac{{x}^{3}}{{2}^{3}·1!2!}+\frac{{x}^{5}}{{2}^{5}·2!3!}-\frac{{x}^{7}}{{2}^{7}·3!4!}+\frac{{x}^{9}}{{2}^{9}·4!5!}-\dots \hfill \\ \multicolumn{1}{c}{}\end{array}$
5. Table of zeros for ${J}_{0}\left(x\right)$ and ${J}_{1}\left(x\right)$ . Define $\left\{{\alpha }_{n},{\beta }_{n}\right\}$ by ${J}_{0}\left({\alpha }_{n}\right)=0$ and ${J}_{1}\left({\beta }_{n}\right)=0$ .
Roots ${\alpha }_{n}$ ${J}_{1}\left({\alpha }_{n}\right)$ Roots ${\beta }_{n}$ ${J}_{0}\left({\beta }_{n}\right)$
2.4048 0.5191 0.0000 1.0000
5.5201 $-0.3403$ 3.8317 $-0.4028$
8.6537 0.2715 7.0156 0.3001
11.7915 $-0.2325$ 10.1735 $-0.2497$
14.9309 0.2065 13.3237 0.2184
18.0711 $-0.1877$ 16.4706 $-0.1965$
21.2116 0.1733 19.6159 0.1801
6. Recurrence formulas $\begin{array}{cc}{J}_{n-1}\left(x\right)+{J}_{n+1}\left(x\right)=\frac{2n}{x}{J}_{n}\left(x\right)\hfill & {nJ}_{n}\left(x\right)+{xJ}_{n}^{\prime }\left(x\right)={xJ}_{n-1}\left(x\right)\hfill \\ \multicolumn{1}{c}{{J}_{n-1}\left(x\right)-{J}_{n+1}\left(x\right)=2{J}_{n}^{\prime }\left(x\right)}& {nJ}_{n}\left(x\right)-{xJ}_{n}^{\prime }\left(x\right)={xJ}_{n+1}\left(x\right)\hfill \\ \multicolumn{1}{c}{}\end{array}$
7. If ${J}_{n}$ is written for ${J}_{n}\left(x\right)$ and ${J}_{n}^{\left(k\right)}$ is written for $\frac{{d}^{k}}{{dx}^{k}}\left\{{J}_{n}\left(x\right)\right\}$ , then the following derivative relationships are important $\begin{array}{c}{J}_{0}^{\left(r\right)}=-{J}_{1}^{\left(r-1\right)}\hfill \\ \multicolumn{1}{c}{{J}_{0}^{\left(2\right)}=-{J}_{0}+\frac{1}{x}{J}_{1}=\frac{1}{2}\left({J}_{2}-{J}_{0}\right)}\\ \multicolumn{1}{c}{{J}_{0}^{\left(3\right)}=\frac{1}{x}{J}_{0}+\left(1-\frac{2}{{x}^{2}}\right){J}_{1}=\frac{1}{4}\left(-{J}_{3}+3{J}_{1}\right)}\\ \multicolumn{1}{c}{{J}_{0}^{\left(4\right)}=\left(1-\frac{3}{{x}^{2}}\right){J}_{0}-\left(\frac{2}{x}-\frac{6}{{x}^{3}}\right){J}_{1}=\frac{1}{8}\left({J}_{4}-4{J}_{2}+3{J}_{0}\right),\text{etc}.}\\ \multicolumn{1}{c}{}\end{array}$
8. Half-order Bessel functions $\begin{array}{c}{J}_{\frac{1}{2}}\left(x\right)=\sqrt{\frac{2}{\pi x}}sinx\hfill \\ \multicolumn{1}{c}{{J}_{-\frac{1}{2}}\left(x\right)=\sqrt{\frac{2}{\pi x}}cosx}\\ \multicolumn{1}{c}{{J}_{n+\frac{3}{2}}\left(x\right)=-{x}^{n+\frac{1}{2}}\frac{d}{dx}\left\{{x}^{-\left(n+\frac{1}{2}\right)}{J}_{n+\frac{1}{2}}\left(x\right)\right\}}\\ \multicolumn{1}{c}{{J}_{n-\frac{1}{2}}\left(x\right)={x}^{-\left(n+\frac{1}{2}\right)}\frac{d}{dx}\left\{{x}^{n+\frac{1}{2}}{J}_{n+\frac{1}{2}}\left(x\right)\right\}}\\ \multicolumn{1}{c}{}\end{array}$
$n$ ${\left(\frac{\pi x}{2}\right)}^{\frac{1}{2}}{J}_{n+\frac{1}{2}}\left(x\right)$ ${\left(\frac{\pi x}{2}\right)}^{\frac{1}{2}}{J}_{-\left(n+\frac{1}{2}\right)}\left(x\right)$
0 $sinx$ $cosx$
1 $\frac{sinx}{x}-cosx$ $-\frac{cosx}{x}-sinx$
2 $\left(\frac{3}{{x}^{2}}-1\right)sinx-\frac{3}{x}cosx$ $\left(\frac{3}{{x}^{2}}-1\right)cosx+\frac{3}{x}sinx$
3 $\left(\frac{15}{{x}^{3}}-\frac{6}{x}\right)sinx-\left(\frac{15}{{x}^{2}}-1\right)cosx$ $-\left(\frac{15}{{x}^{3}}-\frac{6}{x}\right)cosx-\left(\frac{15}{{x}^{2}}-1\right)sinx$
9. Additional solutions to Bessel's equation are ${Y}_{n}\left(x\right)$ (also called Weber's function, and sometimes denoted by ${N}_{n}\left(x\right)$ ) and ${H}_{n}^{\left(1\right)}\left(x\right)$ and ${H}_{n}^{\left(2\right)}\left(x\right)$ (also called Hankel functions) These solutions are defined as follows Yn(x)={Jn(x)cos(nπ)−J−n(x)sin(nπ)nnot an integerHn(1)(x)=Jn(x)+iYn(x)limv→nJv(x)cos(vπ)−J−v(x)sin(vπ)nan integerHn(2)(x)=Jn(x)−iYn(x) The additional properties of these functions may all be derived from the above relations and the known properties of ${J}_{n}\left(x\right)$ .
10. Complete solutions to Bessel's equation may be written as ${c}_{1}{J}_{n}\left(x\right)+{c}_{2}{J}_{-n}\left(x\right)$ when $n$ is not an integer or, for any value of $n$ , ${c}_{1}{J}_{n}\left(x\right)+{c}_{2}{Y}_{n}\left(x\right)$ or ${c}_{1}{H}_{n}^{\left(1\right)}x+{c}_{2}{H}_{n}^{\left(2\right)}\left(x\right)$ .
11. The modified (or hyperbolic) Bessel's differential equation is ${x}^{2}\frac{{d}^{2}y}{{dx}^{2}}+x\frac{dy}{dx}-\left({x}^{2}+{n}^{2}\right)y=0$
12. When $n$ is not an integer, two independent solutions of the equation are ${I}_{n}\left(x\right)$ and ${I}_{-n}\left(x\right)$ , where ${I}_{n}\left(x\right)=\sum _{k=0}^{\infty }\frac{1}{k!\Gamma \left(n+k+1\right)}{\left(\frac{x}{2}\right)}^{n+2k}$
13. If $n$ is an integer, ${I}_{n}\left(x\right)={I}_{-n}\left(x\right)=\frac{{x}^{n}}{{2}^{n}n!}\left(1+\frac{{x}^{2}}{{2}^{2}·1!\left(n+1\right)}+\frac{{x}^{4}}{{2}^{4}·2!\left(n+1\right)\left(n+2\right)}+\frac{{x}^{6}}{{2}^{6}·3!\left(n+1\right)\phantom{\rule{0.2em}{0ex}}\left(n+2\right)\phantom{\rule{0.2em}{0ex}}\left(n+3\right)}+\dots \right)$
14. For $n=0$ and $n=1$ , this formula becomes $\begin{array}{cc}{I}_{0}\left(x\right)\hfill & =1+\frac{{x}^{2}}{{2}^{2}\left(1!\right){}^{2}}+\frac{{x}^{4}}{{2}^{4}\left(2!\right){}^{2}}+\frac{{x}^{6}}{{2}^{6}\left(3!\right){}^{2}}+\frac{{x}^{8}}{{2}^{8}\left(4!\right){}^{2}}+\dots \hfill \\ \multicolumn{1}{c}{{I}_{1}\left(x\right)}& =\frac{x}{2}+\frac{{x}^{3}}{{2}^{3}·1!2!}+\frac{{x}^{5}}{{2}^{5}·2!3!}+\frac{{x}^{7}}{{2}^{7}·3!4!}+\frac{{x}^{9}}{{2}^{9}·4!5!}+\dots \hfill \\ \multicolumn{1}{c}{}\end{array}$
15. Another solution to the modified Bessel's equation is Kn(x)={12πI−n(x)−In(x)sin(nπ)nnot an integerlimv→n12πI−v(x)−Iv(x)sin(vπ)nan integer This function is linearly independent of ${I}_{n}\left(x\right)$ for all values of $n$ . Thus the complete solution to the modified Bessel's equation may be written as ${c}_{1}{I}_{n}\left(x\right)+{c}_{2}{I}_{-n}\left(x\right)\mathrm{ }\text{when}n\phantom{\rule{0.2em}{0ex}}\text{is not an integer}$ or ${c}_{1}{I}_{n}\left(x\right)+{c}_{2}{K}_{n}\left(x\right)\mathrm{ }\text{for any value of}n$
16. The following relations hold among the various Bessel functions: $\begin{array}{c}{I}_{n}\left(z\right)={i}^{-m}{J}_{m}\left(iz\right)\hfill \\ \multicolumn{1}{c}{{Y}_{n}\left(iz\right)=\left(i\right){}^{n+1}{I}_{n}\left(z\right)-\frac{2}{\pi }{i}^{-n}{K}_{n}\left(z\right)}\end{array}$ Most of the properties of the modified Bessel function may be deduced from the known properties of ${J}_{n}\left(x\right)$ by use of these relations and those previously given.
17. Recurrence formulas $\begin{array}{cc}{I}_{n-1}\left(x\right)-{I}_{n+1}\left(x\right)=\frac{2n}{x}{I}_{n}\left(x\right)\hfill & {I}_{n-1}\left(x\right)+{I}_{n+1}\left(x\right)=2{I}_{n}^{\prime }\left(x\right)\hfill \\ \multicolumn{1}{c}{{I}_{n-1}\left(x\right)-\frac{n}{x}{I}_{n}\left(x\right)={I}_{n}^{\prime }\left(x\right)}& {I}_{n}^{\prime }\left(x\right)={I}_{n+1}\left(x\right)+\frac{n}{x}{I}_{n}\left(z\right)\hfill \\ \multicolumn{1}{c}{}\end{array}$
## 8.4 FACTORIAL FUNCTION
For non-negative integers $n$ , the factorial of $n$ , denoted $n!$ , is the product of all positive integers less than or equal to $n$ ; $n!=n·\left(n-1\right)·\left(n-2\right)\dots 2·1$ . If $n$ is a negative integer ( $n=-1,-2,\dots$ ) then $n!=±\infty$ .
Approximations to $n!$ for large $n$ include Stirling's formula $n!\approx \sqrt{2\pi e}{\left(\frac{n}{e}\right)}^{n+\frac{1}{2}},$ and Burnsides's formula $n!\approx \sqrt{2\pi }{\left(\frac{n+\frac{1}{2}}{e}\right)}^{n+\frac{1}{2}}.$
$n$ $n!$ $log{}_{10}n!$ $n$ $n!$ $log{}_{10}n!$
$0$ $1$ $0.00000$ $1$ $1$ $0.00000$
$2$ $2$ $0.30103$ $3$ $6$ $0.77815$
$4$ $24$ $1.38021$ $5$ $120$ $2.07918$
$6$ $720$ $2.85733$ $7$ $5040$ $3.70243$
$8$ $40320$ $4.60552$ $9$ $3.6288×{10}^{5}$ $5.55976$
$10$ $3.6288×{10}^{6}$ $6.55976$ $11$ $3.9917×{10}^{7}$ $7.60116$
$12$ $4.7900×{10}^{8}$ $8.68034$ $13$ $6.2270×{10}^{9}$ $9.79428$
$14$ $8.7178×{10}^{10}$ $10.94041$ $15$ $1.3077×{10}^{12}$ $12.11650$
$16$ $2.0923×{10}^{13}$ $13.32062$ $17$ $3.5569×{10}^{14}$ $14.55107$
$18$ $6.4024×{10}^{15}$ $15.80634$ $19$ $1.2165×{10}^{17}$ $17.08509$
$20$ $2.4329×{10}^{18}$ $18.38612$ $25$ $1.5511×{10}^{25}$ $25.19065$
$30$ $2.6525×{10}^{32}$ $32.42366$ $40$ $8.1592×{10}^{47}$ $47.91165$
$50$ $3.0414×{10}^{64}$ $64.48307$ $60$ $8.3210×{10}^{81}$ $81.92017$
$70$ $1.1979×{10}^{100}$ $100.07841$ $80$ $7.1569×{10}^{118}$ $118.85473$
$90$ $1.4857×{10}^{138}$ $138.17194$ $100$ $9.3326×{10}^{157}$ $157.97000$
$110$ $1.5882×{10}^{178}$ $178.20092$ $120$ $6.6895×{10}^{198}$ $198.82539$
$130$ $6.4669×{10}^{219}$ $219.81069$ $150$ $5.7134×{10}^{262}$ $262.75689$
$500$ $1.2201×{10}^{1134}$ $1134.0864$ $1000$ $4.0239×{10}^{2567}$ $2567.6046$
## 8.5 Gamma Function
Definition:$\Gamma \left(n\right)=\underset{0}{\overset{\infty }{\int }}{t}^{n-1}{e}^{-t}\phantom{\rule{0.2em}{0ex}}dt\mathrm{ }n>0$
Recursion Formula:
Special Values:$\begin{array}{cc}\Gamma \left(1/2\right)\hfill & =\sqrt{\pi }\hfill \\ \multicolumn{1}{c}{\Gamma \left(m+\frac{1}{2}\right)}& =\frac{1·3·5\dots \left(2m-1\right)}{{2}^{m}}\sqrt{\pi }\mathrm{ }m=1,2,3,\dots \hfill \\ \multicolumn{1}{c}{\Gamma \left(-m+\frac{1}{2}\right)}& =\frac{\left(-1\right){}^{m}{2}^{m}\sqrt{\pi }}{1·3·5\dots \left(2m-1\right)}\mathrm{ }m=1,2,3,\dots \hfill \\ \multicolumn{1}{c}{}\end{array}$
Special Formulas:$\begin{array}{cc}\Gamma \left(x+1\right)\hfill & =\underset{k\to \infty }{lim}\frac{1·2·3\dots k}{\left(x+1\right)\phantom{\rule{0.2em}{0ex}}\left(x+2\right)\dots \left(x+k\right)}{k}^{x}\hfill \\ \multicolumn{1}{c}{\frac{1}{\Gamma \left(x\right)}}& ={xe}^{\gamma x}\underset{m=1}{\overset{\infty }{\Pi }}\left\{\left(1+\frac{x}{m}\right){e}^{-x/m}\right\}\mathrm{ }\left(\gamma \phantom{\rule{0.2em}{0ex}}\text{is Euler"s constant)}\hfill \\ \multicolumn{1}{c}{}\end{array}$
Properties:$\begin{array}{cc}{\Gamma }^{\prime }\left(1\right)\hfill & ={\int }_{0}^{\infty }{e}^{\gamma x}\text{ln}x\phantom{\rule{0.2em}{0ex}}dx=-\gamma \hfill \\ \multicolumn{1}{c}{\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}}& =-\gamma +\left(\frac{1}{1}-\frac{1}{x}\right)+\left(\frac{1}{2}-\frac{1}{x+1}\right)+\dots +\left(\frac{1}{n}-\frac{1}{x+n-1}\right)+\dots \hfill \\ \multicolumn{1}{c}{\Gamma \left(x+1\right)}& =\sqrt{2\pi x}\phantom{\rule{0.2em}{0ex}}{x}^{x}{e}^{-x}\left\{1+\frac{1}{12x}+\frac{1}{288{x}^{2}}-\frac{139}{51,840{x}^{3}}+\dots \right\}\mathrm{ }\left(\mathit{Stirling"s asymptotic series}\right)\hfill \end{array}$
$n$ $\Gamma \left(n\right)$ $n$ $\Gamma \left(n\right)$ $n$ $\Gamma \left(n\right)$ $n$ $\Gamma \left(n\right)$
1.00 1.00000 1.25 .90640 1.50 .88623 1.75 .91906
1.01 .99433 1.26 .90440 1.51 .88659 1.76 .92137
1.02 .98884 1.27 .90250 1.52 .88704 1.77 .92376
1.03 .98355 1.28 .90072 1.53 .88757 1.78 .92623
1.04 .97844 1.29 .89904 1.54 .88818 1.79 .92877
1.05 .97350 1.30 .89747 1.55 .88887 1.80 .93138
1.06 .96874 1.31 .89600 1.56 .88964 1.81 .93408
1.07 .96415 1.32 .89464 1.57 .89049 1.82 .93685
1.08 .95973 1.33 .89338 1.58 .89142 1.83 .93969
1.09 .95546 1.34 .89222 1.59 .89243 1.84 .94261
1.10 .95135 1.35 .89115 1.60 .89352 1.85 .94561
1.11 .94740 1.36 .89018 1.61 .89468 1.86 .94869
1.12 .94359 1.37 .88931 1.62 .89592 1.87 .95184
1.13 .93993 1.38 .88854 1.63 .89724 1.88 .95507
1.14 .93642 1.39 .88785 1.64 .89864 1.89 .95838
1.15 .93304 1.40 .88726 1.65 .90012 1.90 .96177
1.16 .92980 1.41 .88676 1.66 .90167 1.91 .96523
1.17 .92670 1.42 .88636 1.67 .90330 1.92 .96877
1.18 .92373 1.43 .88604 1.68 .90500 1.93 .97240
1.19 .92089 1.44 .88581 1.69 .90678 1.94 .97610
1.20 .91817 1.45 .88566 1.70 .90864 1.95 .97988
1.21 .91558 1.46 .88560 1.71 .91057 1.96 .98374
1.22 .91311 1.47 .88563 1.72 .91258 1.97 .98768
1.23 .91075 1.48 .88575 1.73 .91466 1.98 .99171
1.24 .90852 1.49 .88595 1.74 .91683 1.99 .99581
2.00 1.00000
## 8.6 BETA FUNCTION
Definition:$B\left(m,n\right)={\int }_{0}^{1}{t}^{m-1}\left(1-t\right){}^{n-1}dt\mathrm{ }m>0,\phantom{\rule{0.2em}{0ex}}n>0$
Relationship with Gamma function:$B\left(m,n\right)=\frac{\Gamma \left(m\right)\Gamma \left(n\right)}{\Gamma \left(m+n\right)}$
Properties:$\begin{array}{c}B\left(m,n\right)=B\left(n,m\right)\\ B\left(m,n\right)=2{\int }_{0}^{\pi /2}sin{}^{2m-1}\theta cos{}^{2n-1}\theta \phantom{\rule{0.2em}{0ex}}d\theta \\ B\left(m,n\right)={\int }_{0}^{\infty }\frac{{t}^{m-1}}{\left(1+t\right){}^{m+n}}dt\\ B\left(m,n\right)={r}^{n}\left(r+1\right){}^{m}{\int }_{0}^{1}\frac{{t}^{m-1}\left(1-t\right){}^{n-1}}{\left(r+t\right){}^{m+n}}\phantom{\rule{0.2em}{0ex}}dt\end{array}$
## 8.7 ERROR FUNCTION
Definition:$\text{erf}\left(x\right)=\frac{2}{\sqrt{\pi }}{\int }_{0}^{x}{e}^{-{t}^{2}}dt$
Series:$\text{erf}\left(x\right)=\frac{2}{\sqrt{\pi }}\left(x-\frac{{x}^{3}}{3}+\frac{1}{2!}\frac{{x}^{5}}{5}-\frac{1}{3!}\frac{{x}^{7}}{7}+\dots \right)$
Property:$\text{erf}\left(x\right)=-\text{erf}\left(-x\right)$
Relationship with Normal Probability Function$f\left(t\right)$ : ${\int }_{0}^{x}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt=\frac{1}{2}\text{erf}\left(\frac{x}{\sqrt{2}}\right)$ To evaluate $\text{erf}\left(2.3\right)$ , one proceeds as follows: For $\frac{x}{\sqrt{2}}=2.3$ , one finds $x=\left(2.3\right)\phantom{\rule{0.2em}{0ex}}\left(\sqrt{2}\right)=3.25$ . In the normal probability function table, one finds the entry 0.4994 opposite the value 3.25. Thus $\text{erf}\left(2.3\right)=2\left(0.4994\right)=0.9988$ . $\text{erfc}\left(z\right)=1-\text{erf}\left(z\right)=\frac{2}{\sqrt{\pi }}{\int }_{z}^{\infty }{e}^{-{t}^{2}}dt$ is known as the complementary error function.
Page 1 of 1
1/1
Entry Display
This is where the entry will be displayed
#### Other ChemNetBase Products
You are not within the network of a subscribing institution.Please sign in with an Individual User account to continue.Note that Workspace accounts are not valid.
Confirm Log Out
Are you sure?
Your personal workspace allows you to save and access your searches and bookmarks.
If you do not have a workspace Log In click here to create one.
Log Out From Your Workspace
Are you sure?
Create your personal workspace
Title [Select]DrProfMissMrsMsMrMx [Select]DrProfMissMrsMsMrMx First Name (Given) Last Name (Family) Email address Username Password Confirm Password
You have entered your Workspace sign in credentials instead of Individual User sign in credentials. You must be authenticated within your organisation's network IP range in order to access your Workspace account. Click the help icon for more information on the differences between these two accounts.
You have entered your Individual User account sign in credentials instead of Workspace credentials. While using this network, a personal workspace account can be created to save your bookmarks and search preferences for later use. Click the help icon for more information on the differences between Individual User accounts and Workspace accounts.
My Account
Update your Personal Workspace Details
Username Title [Select]DrProfMissMrsMsMrMx [Select]DrProfMissMrsMsMrMx First Name (Given) Last Name (Family) Email address
Searching for Chemicals and Properties
The CRC Handbook of Chemistry and Physics (HBCP) contains over 700 tables in over 450 documents which may be divided into several pages, all categorised into 17 major subject areas. The search on this page works by searching the content of each page individually, much like any web search. This provides a challenge if you want to search for multiple terms and those terms exist on different pages, or if you use a synonym/abbreviation that does not exist in the document.
We use metadata to avoid some of these issues by including certain keywords invisibly behind each table. Whilst this approach works well in many situations, like any web search it relies in the terms you have entered existing in the document with the same spelling, abbreviation etc.
Since chemical compounds and their properties are immutable, a single centralised database has been created from all chemical compounds throughout HBCP. This database contains every chemical compound and over 20 of the most common physical properties collated from each of the >700 tables. What's more, the properties can be searched numerically, including range searching, and you can even search by drawing a chemical structure. A complete list of every document table in which the compound occurs is listed, and are hyperlinked to the relevant document table.
The 'Search Chemicals' page can be found by clicking the flask icon in the navigation bar at the top of this page. For more detailed information on how to use the chemical search, including adding properties, saving searches, exporting search results and more, click the help icon in to top right of this page, next to the welcome login message.
Below is an example of a chemical entry, showing its structure, physical properties and document tables in which it appears.
We use cookies to improve your website experience. To learn about our use of cookies and how you can manage your cookie settings, please see our Cookie Policy. By continuing to use the website, you consent to our use of cookies.
We have developed this cookie policy (the “Cookie Policy”) in order to explain how we use cookies and similar technologies (together, “Cookies”) on this website (the “Website”) and to demonstrate our firm commitment to the privacy of your personal information.
The first time that you visit our Website, we notify you about our use of Cookies through a notification banner. By continuing to use the Website, you consent to our use of Cookies as described in this Cookie Policy. However, you can choose whether or not to continue accepting Cookies at any later time. Information on how to manage Cookies is set out later in this Cookie Policy.
Please note that our use of any personal information we collect about you is subject to our Privacy Policy.
## What are Cookies?
Cookies are small text files containing user IDs that are automatically placed on your computer or other device by when you visit a website. The Cookies are stored by the internet browser. The browser sends the Cookies back to the website on each subsequent visit, allowing the website to recognise your computer or device. This recognition enables the website provider to observe your activity on the website, deliver a personalised, responsive service and improve the website.
## Cookies We Use and Their Purpose
We use three types of Cookies - ‘Strictly Necessary’ Cookies, ‘Performance’ Cookies and ‘Functionality’ Cookies. Each type of Cookie and the purposes for which we use them are described in this section. To learn about the specific Cookies we use, please see our List of Cookies.
### 1. Strictly Necessary Cookies
‘Strictly Necessary’ Cookies enable you to move around the Website and use essential features. For example, if you log into the Website, we use a Cookie to keep you logged in and allow you to access restricted areas, without you having to repeatedly enter your login details. If you are registering for or purchasing a product or service, we will use Cookies to remember your information and selections, as you move through the registration or purchase process.
Strictly Necessary Cookies are necessary for our Website to provide you with a full service. If you disable them, certain essential features of the Website will not be available to you and the performance of the Website will be impeded.
### 2. Performance Cookies
‘Performance’ Cookies collect information about how you use our Website, for example which pages you visit and if you experience any errors. These Cookies don’t collect any information that could identify you – all the information collected is anonymous. We may use these Cookies to help us understand how you use the Website and assess how well the Website performs and how it could be improved.
### 3. Functionality Cookies
‘Functionality’ Cookies enable a website to provide you with specific services or a customised experience. We may use these Cookies to provide you with services such as watching a video or adding user comments. We may also use such Cookies to remember changes you make to your settings or preferences (for example, changes to text size or your choice of language or region) or offer you time-saving or personalised features.
You can control whether or not Functionality Cookies are used, but disabling them may mean we are unable to provide you with some services or features of the Website.
## First and Third Party Cookies
The Cookies placed on your computer or device include ‘First Party’ Cookies, meaning Cookies that are placed there by us, or by third party service providers acting on our behalf. Where such Cookies are being managed by third parties, we only allow the third parties to use the Cookies for our purposes, as described in this Cookie Policy, and not for their own purposes.
To learn about the specific First Party and Third Party Cookies used by our, please see our List of Cookies.
You always have a choice over whether or not to accept Cookies. When you first visit the Website and we notify you about our use of Cookies, you can choose not to consent to such use. If you continue to use the Website, you are consenting to our use of Cookies for the time being. However, you can choose not to continue accepting Cookies at any later time. In this section, we describe ways to manage Cookies, including how to disable them.
You can manage Cookies through the settings of your internet browser. You can choose to block or restrict Cookies from being placed on your computer or device. You can also review periodically review the Cookies that have been placed there and disable some or all of them.
Please be aware that if you choose not to accept certain Cookies, it may mean we are unable to provide you with some services or features of the Website.
## Changes to Cookie Policy
In order to keep up with changing legislation and best practice, we may revise this Cookie Policy at any time without notice by posting a revised version on this Website. Please check back periodically so that you are aware of any changes.
## Questions or Concerns
You can also contact the Privacy Officer for the Informa PLC group at [email protected].
Here is a list of cookies we have defined as 'Strictly Necessary':
### Taylor and Francis 'First Party' Cookies
JSESSIONID
Here is a list of the cookies we have defined as 'Performance'.
### 'Third Party' Cookies
_ga
_gid
_gat
Accessibility
The Voluntary Product Accessibility Template (VPAT) is a self-assessment document which discloses how accessible Information and Communication Technology products are in accordance with global standards.
The VPAT disclosure templates do not guarantee product accessibility but provide transparency around the product(s) and enables direction when accessing accessibility requirements.
Taylor & Francis has chosen to complete the International version of VPAT which encompasses Section 508 (US), EN 301 549 (EU) and WCAG2.1 (Web Content Accessibility Guidelines) for its products.
|
2022-09-26 05:47:54
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 388, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8448392152786255, "perplexity": 9961.531200205307}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334802.16/warc/CC-MAIN-20220926051040-20220926081040-00676.warc.gz"}
|
https://www.ctan.org/ctan-ann/pkg/endnotes-hy
|
# Announcements for endnotes-hy
## endnotes-hy – Patches the endnotes package to create hypertext links to the correct anchors
The package supports the creation of hypertext links in support of the endnotes package. The package modifies the syntax of the \endnote command:
\endnote*[<num>]{<text>}\label{<name>}.
When the *-option is used, no endnote mark is created, but the endnote itself is written. The \label command appears at the end of the \endnote and its arguments, rather than within the argument of the <text> argument.
Package endnotes-hy Version 2020-04-08 Copyright 2020 D. P. Story Maintainer Donald P. Story
Atom Atom 1.0 feed with announcements for package endnotes-hy.
|
2022-01-17 06:32:17
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32275715470314026, "perplexity": 11407.583083626143}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300343.4/warc/CC-MAIN-20220117061125-20220117091125-00396.warc.gz"}
|
https://socratic.org/questions/how-do-you-find-the-x-intercept-for-y-2
|
# How do you find the x intercept for y=2?
Actually, $y = 2$ doesn't have an $x$-intercept. This is because the line $y = 2$ is a horizontal line that is parallel to the horizontal $x$-axis, and parallel lines never intersect. Also, the line $y = 2$ doesn't depend on any $x$. No matter what $x$ is, no matter how infinitely large or small, $y$ will always be $2$ above the $x$-axis.
To see this visually, this is what the graph of $y = 2$ looks like:
You can see that the line doesn't ever touch the $x$-axis, meaning it doesn't have an $x$-intercept. It has a $y$ intercept though, at the point (0,2).
|
2023-02-06 20:40:01
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 14, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5438891053199768, "perplexity": 143.87550500126986}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500357.3/warc/CC-MAIN-20230206181343-20230206211343-00826.warc.gz"}
|
http://pdglive.lbl.gov/DataBlock.action?node=S008POL&home=sumtabM
|
# ${{\boldsymbol \pi}^{+}}$ $\rightarrow$ ${{\boldsymbol \mu}^{+}}{{\boldsymbol \nu}}$ INSPIRE search
Tests the Lorentz structure of leptonic charged weak interactions.
VALUE CL% DOCUMENT ID TECN CHG COMMENT
• • • We do not use the following data for averages, fits, limits, etc. • • •
$\text{<(-0.9959)}$ 90 1
1984
RVUE +
$-0.99$ $\pm0.16$ 2
1983
SPEC - ${{\mathit \mu}}$ $\mathit X$-rays
1 FETSCHER 1984 uses only the measurement of CARR 1983 .
2 Sign of measurement reversed in ABELA 1983 to compare with ${{\mathit \mu}^{+}}$ measurements.
References:
FETSCHER 1984
PL 140B 117 Helicity of the ${{\mathit \nu}_{{\mu}}}$ in ${{\mathit \pi}^{+}}$ Decay: a Comment on the Measurement of P$_{{{\mathit \mu}}}{{\mathit \Xi}}{{\mathit \Delta}}/{{\mathit \rho}}$ in Muon Decay
ABELA 1983
NP A395 413 Measurements of the Polarization of the 2p and 1s States in Muonic Atoms and the Helicity of the Muon in Pion Decay
|
2019-12-09 05:41:51
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8084434270858765, "perplexity": 6212.2011957617015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540517557.43/warc/CC-MAIN-20191209041847-20191209065847-00169.warc.gz"}
|
https://en.m.wikisource.org/wiki/Edinburgh_Review/Volume_59/Babbage%27s_Calculating_Engine
|
# Edinburgh Review/Volume 59/Babbage's Calculating Engine
THE
EDINBURGH REVIEW.
JULY, 1834.
No. CXX.
Art I.—1. Letter to Sir Humphry Davy, Bart. P.R.S., on the application of Machinery to Calculate and Print Mathematical Tables. By Charles Babbage, Esq. F.R.S.4to. Printed by order of the House of Commons.
2. On the Application of Machinery to the Calculation of Astronomical and Mathematical Tables. By Charles Babbage, Esq. Memoirs Astron. Soc. Vol. I. Part 2. London: 1822.
3. Address to the Astronomical Society, by Henry Thomas Colebrooke, Esq. F.R.S. President, on presenting the first gold medal of the Society to Charles Babbage, Esq. for the invention of the Calculating Engine.Memoirs Astron. Soc. Vol. I. Part 2. London: 1822.
4. On the determination of the General Term of a new Class of Infinite Series.By Charles Babbage, Esq.Transactions Camb. Phil. Soc. Cambridge: 1824.
5. On Errors common to many Tables of Logarithms.By Charles Babbage, Esq. Memoirs Astron. Soc. London: 1827.
6. On a Method of Expressing by Signs the Action of Machinery.By Charles Babbage, Esq. Phil. Trans. London: 1826.
7. Report by the Committee appointed by the Council of the Royal Society to consider the subject referred to in a Communication received by them from the Treasury, respecting Mr Babbage's Calculating Engine, and to report thereupon. London: 1829.
There is no position in society more enviable than that of the few who unite a moderate independence with high intellectual qualities. Liberated from the necessity of seeking their support by a profession, they are unfettered by its restraints, and are enabled to direct the powers of their minds, and to concentrate their intellectual energies on those objects exclusively to which they feel that their powers may be applied with the greatest advantage to the community, and with the most lasting reputation to themselves. On the other hand, their middle station and limited income rescue them from those allurements to frivolity and dissipation, to which rank and wealth ever expose their possessors. Placed in such favourable circumstances, Mr Babbage selected science as the field of his ambition; and his mathematical researches have conferred on him a high reputation, wherever the exact sciences are studied and appreciated. The suffrages of the mathematical world have been ratified in his own country, where he has been elected to the Lucasian Professorship in his own University—a chair, which, though of inconsiderable emolument, is one on which Newton has conferred everlasting celebrity. But it has been the fortune of this mathematician to surround himself with fame of another and more popular kind, and which rarely falls to the lot of those who devote their lives to the cultivation of the abstract sciences. This distinction he owes to the announcement, some years since, of his celebrated project of a Calculating Engine. A proposition to reduce arithmetic to the dominion of mechanism,—to substitute an automaton for a compositor,—to throw the powers of thought into wheelwork could not fail to awaken the attention of the world. To bring the practicability of such a project within the compass of popular belief was not easy: to do so by bringing it within the compass of popular comprehension was not possible. It transcended the imagination of the public in general to conceive its possibility; and the sentiments of wonder with which it was received, were only prevented from merging into those of incredulity, by the faith reposed in the high attainments of its projector. This extraordinary undertaking was, however, viewed in a very different light by the small section of the community, who, being sufficiently versed in mathematics, were acquainted with the principle upon which it was founded. By reference to that principle, they perceived at a glance the practicability of the project; and being enabled by the nature of their attainments and pursuits to appreciate the immeasurable importance of its results, they regarded the invention with a proportionately profound interest. The production of numerical tables, unlimited in quantity and variety, restricted to no particular species, and limited by no particular law;—extending not merely to the boundaries of existing knowledge, but spreading their powers over the undefined regions of future discovery—were results, the magnitude and the value of which the community in general could neither comprehend nor appreciate. In such a case, the judgment of the world could only rest upon the authority of the philosophical part of it; and the fiat of the scientific community swayed for once political councils. The British Government, advised by the Royal Society, and a committee formed of the most eminent mechanicians and practical engineers, determined on constructing the projected mechanism at the expense of the nation, to be held as national property.
Notwithstanding the interest with which this invention has been regarded in every part of the world, it has never yet been embodied in a written, much less in a published form. We trust, therefore, that some credit will be conceded to us for having been the first to make the public acquainted with the object, principle, and structure of a piece of machinery, which, though at present unknown (except as to a few of its probable results), must, when completed, produce important effects, not only on the progress of science, but on that of civilisation.
The calculating machinery thus undertaken for the public gratuitously (so far as Mr Babbage is concerned), has now attained a very advanced stage towards completion; and a portion of it has been put together, and performs various calculations;—affording a practical demonstration that the anticipations of those, under whose advice Government has acted, have been well founded.
There are nevertheless many persons who, admitting the great ingenuity of the contrivance, have, notwithstanding, been accustomed to regard it more in the light of a philosophical curiosity, than an instrument for purposes practically useful. This mistake (than which it is not possible to imagine a greater) has arisen mainly from the ignorance which prevails of the extensive utility of those numerical tables which it is the purpose of the engine in question to produce. There are also some persons who, not considering the time requisite to bring any invention of this magnitude to perfection in all its details, incline to consider the delays which have taken place in its progress as presumptions against its practicability. These persons should, however, before they arrive at such a conclusion, reflect upon the time which was necessary to bring to perfection engines infinitely inferior in complexity and mechanical difficulty. Let them remember that— not to mention the invention of that machine—the improvements alone introduced into the steam-engine by the celebrated Watt, occupied a period of not less than twenty years of the life of that distinguished person, and involved an expenditure of capital amounting to L.50,000.[1] The calculating machinery is a contrivance new even in its details. Its inventor did not take it up already imperfectly formed, after having received the contributions of human ingenuity exercised upon it for a century or more. It has not, like almost all other great mechanical inventions, been gradually advanced to its present state through a series of failures, through difficulties encountered and overcome by a succession of projectors. It is not an object on which the light of various minds has thus been shed. It is, on the contrary, the production of solitary and individual thought,—begun, advanced through each successive stage of improvement, and brought to perfection by one mind. Yet this creation of genius, from its first rude conception to its present state, has cost little more than half the time, and not one-third of the expense, consumed in bringing the steam-engine (previously far advanced in the course of improvement) to that state of comparative perfection in which it was left by Watt. Short as the period of time has been which the inventor has devoted to this enterprise, it has, nevertheless, been demonstrated, to the satisfaction of many scientific men of the first eminence, that the design in all its details, reduced, as it is, to a system of mechanical drawings, is complete; and requires only to be constructed in conformity with those plans, to realize all that its inventor has promised.
With a view to remove and correct erroneous impressions, and at the same time to convert the vague sense of wonder at what seems incomprehensible, with which this project is contemplated by the public in general, into a more rational and edifying sentiment, it is our purpose in the purpose in the present article.
First, To show, the immense importance of any method by which numerical tables, absolutely accurate in every individual copy, may be produced with facility and cheapness. This we shall establish by conveying to the reader some notion of the number and variety of tables published in every country of the world to which civilisation has extended, a large portion of which have been produced at the public expense; by showing also, that they are nevertheless rendered inefficient, to a greater or less extent, by the prevalence of errors in them; that these errors pervade not merely tables produced by individual labour and enterprise, but that they vitiate even those on which national resources have been prodigally expended, and to which the highest mathematical ability, which the most enlightened nations of the world could command, has been unsparingly and systematically directed.
Secondly, To attempt to convey to the reader a general notion of the mathematical principle on which the calculating machinery is founded, and of the manner in which this principle is brought into practical operation, both in the process of calculating and printing. It would be incompatible with the nature of this review, and indeed impossible without the aid of numerous plans, sections, and elevations, to convey clear and precise notions of the details of the means by which the process of reasoning is performed by inanimate matter, and the arbitrary and capricious evolutions of the fingers of typographical compositors are reduced to a system of wheel-work. We are, nevertheless, not without hopes of conveying, even to readers unskilled in mathematics, some satisfactory notions of a general nature on this subject.
Thirdly, To explain the actual state of the machinery a the present time; what progress has been made towards its completion; and what are the probable causes of those delays in its progress, which must be a subject of regret to all friends of science. We shall indicate what appears to us the best and most practicable course to prevent the unnecessary recurrence of such obstructions for the future, and to bring this noble project to a speedy and successful issue.
Viewing the infinite extent and variety of the tables which have been calculated and printed, from the earliest periods of human civilisation to the present time, we feel embarrassed with the difficulties of the task which we have imposed on ourselves;—that of attempting to convey to readers unaccustomed to such speculations, any thing approaching to an adequate idea of them. These tables are connected with the various sciences, with almost every department of the useful arts, with commerce in all its relations; but above all, with Astronomy and Navigation. So important have they been considered, that in many instances large sums have been appropriated by the most enlightened nations in the production of them; and yet so numerous and insurmountable have been the difficulties attending the attainment of this end, that after all, even navigators, putting aside every other department of art and science, have, until very recently, been scantily and imperfectly supplied with the tables indispensably necessary to determine their position at sea.
The first class of tables which naturally present themselves, are those of Multiplication. A great variety of extensive multiplication tables have been published from an early period in different countries; and especially tables of Powers, in which a number is multiplied by itself successively. In Dodson's Calculator we find a table of multiplication extending as far as 10 times 1000.[2] In 1775, a still more extensive table was published to 10 times 10,000. The Board of Longitude subsequently employed the late Dr Hutton to calculate and print various numerical tables, and among others, a multiplication table extending as far as 100 times 1000; tables of the squares of numbers, as far as 25,400; tables of cubes, and of the first ten powers of numbers, as far as 100.[3] In 1814, Professor Barlow, of Woolwich, published, in an octavo volume, the squares, cubes, square roots, cube roots, and reciprocals of all numbers from 1 to 10,000; a table of the first ten powers of all numbers from 1 to 100, and of the fourth and fifth powers of all numbers from 100 to 1000.
Tables of Multiplication to a still greater extent have been published in France. In 1785, was published an octavo volume of tables of the squares, cubes, square roots, and cube roots of all numbers from 1 to 10,000; and similar tables were again published in 1801. In 1817, multiplication tables were published in Paris by Voisin; and similar tables, in two quarto volumes, in 1824, by the French Board of Longitude, extending as far as a thousand times a thousand. A table of squares was published in 1810, in Hanover; in 1812, at Leipzig; in 1825, at Berlin; and in 1827, at Ghent. A table of cubes was published in 1827, at Eisenach; in the same year a similar table at Ghent; and one of the squares of all numbers as far as 10,000, was published in that year, in quarto, at Bonn. The Prussian Government has caused a multiplication table to be calculated and printed, extending as far as 1000 times 1000. Such are a few of the tables of this class which have been published in different countries. This class of tables may be considered as purely arithmetical, since the results which they express involve no other relations than the arithmetical dependence of abstract numbers upon each other. When numbers, however, are taken in a concrete sense, and are applied to express peculiar modes of quantity,—such as angular, linear, superficial, and solid magnitudes,—a new set of numerical relations arise, and a large number of computations are required.
To express angular magnitude, and the various relations of linear magnitude with which it is connected, involves the consideration of a vast variety of Geometrical and Trigonometrical tables; such as tables of the natural sines, co-sines, tangents, secants, co-tangents, &c. &c.; tables of arcs and angles in terms of the radius; tables for the immediate solution of various cases of triangles, &c. Volumes without number of such tables have been from time to time computed and published. It is not sufficient, however, for the purposes of computation to tabulate these immediate trigonometrical functions. Their squares[4] and higher powers, their square roots, and other roots, occur so frequently, that it has been found expedient to compute tables for them, as well as for the same functions of abstract numbers.
The measurement of linear, superficial, and solid magnitudes, in the various forms and modifications in which they are required in the arts, demands another extensive catalogue of numerical tables. The surveyor, the architect, the builder, the carpenter, the miner, the ganger, the naval architect, the engineer, civil and military, all require the aid of peculiar numerical tables, and such have been published in all countries.
The increased expedition and accuracy which was introduced into the art of computation by the invention of Logarithms, greatly enlarged the number of tables previously necessary. To apply the logarithmic method, it was not merely necessary to place in the hands of the computist extensive tables of the logarithms of the natural numbers, but likewise to supply him with tables in which he might find already calculated the logarithms of those arithmetical, trigonometrical, and geometrical functions of numbers, which he has most frequent occasion to use. It would be a circuitous process, when the logarithm of a sine or co-sine of an angle is required, to refer, first to the table of sines, or co-sines, and thence to the table of the logarithms of natural numbers. It was therefore found expedient to compute distinct tables of the logarithms of the sines, co-sines, tangents, &c., as well as of various other functions frequently required, such as sums, differences, &c.
Great as is the extent of the tables we have just enumerated, they bear a very insignificant proportion to those which remain to be mentioned. The above are, for the most part, general in their nature, not belonging particularly to any science or art. There is a much greater variety of tables, whose importance is no way inferior, which are, however, of a more special nature: Such are, for example, tables of interest, discount, and exchange, tables of annuities, and other tables necessary in life insurances; tables of rates of various kinds necessary in general commerce. But the science in which, above all others, the most extensive and accurate tables are indispensable, is Astronomy; with the improvement and perfection of which is inseparably connected that of the kindred art of Navigation. We scarcely dare hope to convey to the general reader any thing approaching to an adequate notion of the multiplicity and complexity of the tables necessary for the purposes of the astronomer and navigator. We feel, nevertheless, that the truly national importance which must attach to any perfect and easy means of producing those tables cannot be at all estimated, unless we state some of the previous calculations necessary in order to enable the mariner to determine, with the requisite certainty and precision, the place of his ship.
In a word, then, all the purely arithmetical, trigonometrical, and logarithmic tables already mentioned, are necessary, either immediately or remotely, for this purpose. But in addition to these, a great number of tables, exclusively astronomical, are likewise indispensable. The predictions of the astronomer, with respect to the positions and motions of the bodies of the firmament, are the means, and the only means, which enable the mariner to prosecute his art. By these he is enabled to discover the distance of his ship from the Line, and the extent of his departure from the meridian of Greenwich, or from any other meridian to which the astronomical predictions refer. The more numerous, minute, and accurate these predictions can be made, the greater will be the facilities which can be furnished to the mariner. But the computation of those tables, in which the future position of celestial objects are registered, depend themselves upon an infinite variety of other tables which never reach the hands of the mariner. It cannot be said that there is any table whatever, necessary for the astronomer, which is unnecessary for the navigator.
The purposes of the marine of a country whose interests are so inseparably connected as ours are with the improvement of the art of navigation, would be very inadequately fulfilled, if our navigators were merely supplied with the means of determining by Nautical Astronomy the position of a ship at sea. It has been well observed by the Committee of the Astronomical Society, to whom the recent improvement of the Nautical Almanac was confided, that it is not by those means merely by which the seaman is enabled to determine the position of his vessel at sea, that the full intent and purpose of what is usually called Nautical Astronomy are answered. This object is merely a part of that comprehensive and important subject; and might be attained by a very cheap publication, and without the aid of expensive instruments. A not less important and much more difficult part of nautical science has for its object to determine the precise position of various interesting and important points on the surface of the earth,—such as remarkable headlands, ports, and islands; together with the general trending of the coast between well-known harbours. It is not necessary to point out here how important such knowledge is to the mariner. This knowledge, which may be called Nautical Geography, cannot be obtained by the methods of observation used on board ship, but requires much more delicate and accurate instruments, firmly placed upon the solid ground, besides all the astronomical aid which can be afforded by the best tables, arranged in the most convenient form for immediate use. This was Dr Maskelyne's view of the subject, and his opinion has been confirmed by the repeated wants and demands of those distinguished navigators who have been employed in several recent scientific expeditions.[5]
Among the tables directly necessary for navigation, are those which predict the position of the centre of the sun from hour to hour. These tables include the sun's right ascension and declination, daily, at noon, with the hourly change in these quantities. They also include the equation of time, together with its hourly variation.
Tables of the moon's place for every hour, are likewise necessary, together with the change of declination for every ten minutes; The lunar method of determining the longitude depends upon tables containing the predicted distances of the moon from the sun, the principal planets, and from certain conspicuous fixed stars; which distances being observed by the mariner, he is enabled thence to discover the time at the meridian from which the longitude is measured; and, by comparing that time with the time known or discoverable in his actual situation, he infers his longitude. But not only does the prediction of the position of the moon, with respect to these celestial objects, require a vast number of numerical tables, but likewise the observations necessary to be made by the mariner, in order to determine the lunar distances, also require several tables. To predict the exact position of any fixed star, requires not less than ten numerical tables peculiar to that star; and if the mariner be furnished (as is actually the case) with tables of the predicted distances of the moon from one hundred such stars, such predictions must require not less than a thousand numerical tables. Regarding the range of the moon through the firmament, however, it will readily be conceived that a hundred stars form but a scanty supply; especially when it is considered that an accurate method of determining the longitude, consists in observing the extinction of a star by the dark edge of the moon. Within the limits of the lunar orbit there are not less than one thousand stars, which are so situated as to be in the moon's path, and therefore to exhibit, at some period or other, those desirable occultations. These stars are also of such magnitudes, that their occultations may be distinctly observed from the deck, even when subject to all the unsteadiness produced by an agitated sea. To predict the occultations of such stars, would require not less than ten thousand tables. The stars from which lunar distances might be taken are still more numerous; and we may safely pronounce, that, great as has been the improvement effected recently in our Nautical Almanac, it does not yet furnish more than a small fraction of that aid to navigation (in the large sense of that term), which, with greater facility, expedition, and economy in the calculation and printing of tables, it might be made to supply.
Tables necessary to determine the places of the planets are not less necessary than those for the sun, moon, and stars. Some notion of the number and complexity of these tables may be formed, when we state that the positions of the two principal planets, (and these the most necessary for the navigator,) Jupiter and Saturn, require each not less than one hundred and sixteen tables. Yet it is not only necessary to predict the position of these bodies, but it is likewise expedient to tabulate the motions of the four satellites of Jupiter, to predict the exact times at which they enter his shadow, and at which their shadows cross his disc, as well as the times at which they are interposed between him and the Earth, and he between them and the Earth.
Among the extensive classes of tables here enumerated, there are several which are in their nature permanent and unalterable, and would never require to be recomputed, if they could once be computed with perfect accuracy on accurate data; but the data on which such computations are conducted, can only be regarded as approximations to truth, within limits the extent of which must necessarily vary with our knowledge of astronomical science. It has accordingly happened, that one set of tables after another has been superseded with each advance of astronomical science. Some striking examples of this may not be uninstructive. In 1765, the Board of Longitude paid to the celebrated Euler the sum of L.300, for furnishing general formulæ for the computation of lunar tables. Professor Mayer was employed to calculate the tables upon these formulæ, and the sum of L.3000 was voted for them by the British Parliament, to his widow, after his decease. These tables had been used for ten years, from 1766 to 1776, in computing the Nautical Almanac, when they were superseded by new and improved tables, composed by Mr Charles Mason, under the direction of Dr Maskelyne, from calculations made by order of the Board of Longitude, on the observations of Dr Bradley. A farther improvement was made by Mason in 1780; but a much more extensive improvement took place in the lunar calculations by the publication of the tables of the Moon, by M. Bürg, deduced from Laplace's theory, in 1806. Perfect, however, as Bürg's tables were considered, at the time of their publication, they were, within the short period of six years, superseded by a more accurate set of tables published by Burckhardt in 1812; and these also have since been followed by the tables of Damoiseau. Professor Schumacher has calculated by the latter tables his ephemeris of the Planetary Lunar Distances, and astronomers will hence be enabled to put to the strict test of observation the merits of the tables of Burckhardt and Damoiseau.[6]
The solar tables have undergone, from time to time, similar changes. The solar tables of Mayer were used in the computation of the Nautical Almanac, from its commencement in 1767, to 1804 inclusive. Within the six years immediately succeeding 1804, not less than three successive sets of solar tables appeared, each improving on the other; the first by Baron de Zach, the second by Delambre, under the direction of the French Board of Longitude, and the third by Carlini. The last, however, differ only in arrangement from those of Delambre.
Similar observations will be applicable to the tables of the principal planets. Bouvard published, in 1803, tables of Jupiter and Saturn; but from the improved state of astronomy, he found it necessary to recompute these tables in 1821.
Although it is now about thirty years since the discovery of the four new planets, Ceres, Pallas, Juno, and Vesta, it was not till recently that tables of their motions were published. They have lately appeared in Encke's Ephemeris.
We have thus attempted to convey some notion (though necessarily a very inadequate one) of the immense extent of numerical tables which it has been found necessary to calculate and print for the purposes of the arts and sciences. We have before us a catalogue of the tables contained in the library of one private individual, consisting of not less than one hundred and forty volumes. Among these there are no duplicate copies: and we observe that many of the most celebrated voluminous tabular works are not contained among them. They are confined exclusively to arithmetical and trigonometrical tables; and, consequently, the myriad of astronomical and nautical tables are totally excluded from them. Nevertheless, they contain an extent of printed surface covered with figures amounting to above sixteen thousand square feet. We have taken at random forty of these tables, and have found that the number of errors acknowledged in the respective errata, amounts to above three thousand seven hundred.
To be convinced of the necessity which has existed for accurate numerical tables, it will only be necessary to consider at what an immense expenditure of labour and of money even the imperfect ones which we possess have been produced.
To enable the reader to estimate the difficulties which attend the attainment even of a limited degree of accuracy, we shall now explain some of the expedients which have been from time to time resorted to for the attainment of numerical correctness in calculating and printing them.
Among the scientific enterprises which the ambition of the French nation aspired to during the Republic, was the construction of a magnificent system of numerical tables. Their most distinguished mathematicians were called upon to contribute to the attainment of this important object; and the superintendence of the undertaking was confided to the celebrated Prony, who co-operated with the government in the adoption of such means as might be expected to ensure the production of a system of logarithmic and trigonometric tables, constructed with such accuracy that they should form a monument of calculation the most vast and imposing that had ever been executed, or even conceived. To accomplish this gigantic task, the principle of the division of labour, found to be so powerful in manufactures, was resorted to with singular success. The persons employed in the work were divided into three sections: the first consisted of half a dozen of the most eminent analysts. Their duty was to investigate the most convenient mathematical formulæ, which should enable the computers to proceed with the greatest expedition and accuracy by the method of Differences, of which we shall speak more fully hereafter. These formulæ, when decided upon by this first section, were handed over to the second section, which consisted of eight or ten properly qualified mathematicians. It was the duty of this second section to convert into numbers certain general or algebraical expressions which occurred in the formulæ, so as to prepare them for, the hands of the computers. Thus prepared, these formulæ were handed over to the third section, who formed a body of nearly one hundred computers. The duty of this numerous section was to compute the numbers finally intended for the tables. Every possible precaution was of course taken to ensure the numerical accuracy of the results. Each number was calculated by two or more distinct and independent computers, and its truth and accuracy determined by the coincidence of the results thus obtained.
The body of tables thus calculated occupied in manuscript seventeen folio volumes.[7]
As an example of the precautions which have been considered necessary to guard against errors in the calculation of numerical tables, we shall further state those which were adopted by Mr Babbage, previously to the publication of his tables of logarithms. In order to render the terminal figure of tables in which one or more decimal places are omitted as accurate as it can be, it has been the practice to compute one or more of the succeeding figures; and if the first omitted figure be greater than 4, then the terminal figure is always increased by 1, since the value of the tabulated number is by such means brought nearer to the truth. [8] The tables of Callet, which were among the most accurate published logarithms, and which extended to seven places of decimals, were first carefully compared with the tables of Vega, which extended to ten places, in order to discover whether Callet had made the above correction of the final figure in every case where it was necessary. This previous precaution being taken, and the corrections which appeared to be necessary being made in a copy of Callet's tables, the proofs of Mr Babbage's tables were submitted to the following test: They were first compared, number by number, with the corrected copy of Callet's logarithms; secondly, with Hutton's logarithms; and thirdly, with Vega's logarithms. The corrections thus suggested being marked in the proofs, corrected revises were received back. These revises were then again compared, number by number, first with Vega's logarithms; secondly, with the logarithms of Callet; and thirdly, as far as the first 20,000 numbers, with the corresponding ones in Briggs's logarithms. They were now returned to the printer, and were stereotyped; proofs were taken from the stereotyped plates, which were put through the following ordeal: They were first compared once more with the logarithms of Vega as far as 47,500; they were then compared with the whole of the logarithms of Gardner; and next with the whole of Taylor's logarithms; and as a last test, they were transferred to the hands of a different set of readers, and were once more compared with Taylor. That these precautions were by no means superfluous may be collected from the following circumstances mentioned by Mr Babbage: In the sheets read immediately previous to stereotyping, thirty-two errors were detected; after stereotyping, eight more were found, and corrected in the plates.
By such elaborate and expensive precautions many of the errors of computation and printing may certainly be removed; but it is too much to expect that in general such measures can be adopted; and we accordingly find by far the greater number of tables disfigured by errors, the extent of which is rather to be conjectured than determined. When the nature of a numerical table is considered,—page after page densely covered with figures, and with nothing else,—the chances against the detection of any single error will be easily comprehended; and it may therefore be fairly presumed, that for one error which may happen to be detected, there must be a great number which escape detection. Notwithstanding this difficulty, it is truly surprising how great a number of numerical errors have been detected by individuals no otherwise concerned in the tables than in their use. Mr Baily states that he has himself detected in the solar and lunar tables, from which our Nautical Almanac was for a long period computed, more than five hundred errors. In the multiplication table already mentioned, computed by Dr Hutton for the Board of Longitude, a single page was examined and recomputed: it was found to contain about forty errors.
In order to make the calculations upon the numbers found in the Ephemeral Tables published in the Nautical Almanac, it is necessary that the mariner should be supplied with certain permanent tables. A volume of these, to the number of about thirty, was accordingly computed, and published at national expense, by order of the Board of Longitude, entitled 'Tables requisite to be used with the Nautical Ephemeris for finding the latitude and longitude at sea.' In the first edition of these requisite tables, there were detected, by one individual, above a thousand errors.
The tables published by the Board of Longitude for the correction of the observed distances of the moon from certain fixed stars, are followed by a table of acknowledged errata, extending to seven folio pages, and containing more than eleven hundred errors. Even this table of errata itself is not correct: a considerable number of errors have been detected in it, so that errata upon errata have become necessary.
One of the tests most frequently resorted to for the detection of errors in numerical tables, has been the comparison of tables of the same kind, published by different authors. It has been generally considered that those numbers in which they are found to agree must be correct; inasmuch as the chances are supposed to be very considerable against two or more independent computers falling into precisely the same errors. How far this coincidence may be safely assumed as a test of accuracy we shall presently see.
A few years ago. it was found desirable to compute some very accurate logarithmic tables for the use of the great national survey of Ireland, which was then, and still is in progress; and on that occasion a careful comparison of various logarithmic tables was made. Six remarkable errors were detected, which were found to be common to several apparently independent sets of tables. This singular coincidence led to an unusually extensive examination of the logarithmic tables published both in England and in other countries; by which it appeared that thirteen sets of tables, published in London between the years 1633 and 1822, all agreed in these six errors. Upon extending the enquiry to foreign tables, it appeared that two sets of tables published at Paris, one at Gouda, one at Avignon, one at Berlin, and one at Florence, were infected by exactly the same six errors. The only tables which were found free from them were those of Vega, and the more recent impressions of Callet. It happened that the Royal Society possessed a set of tables of logarithms printed in the Chinese character, and on Chinese paper, consisting of two volumes: these volumes contained no indication or acknowledgment of being copied from any other work. They were examined; and the result was the detection in them of the same six errors. [9] It is quite apparent that this remarkable coincidence of error must have arisen from the various tables being copied successively one from another. The earliest work in which they appeared was Vlacq's Logarithms, (folio, Gouda, 1628); and from it, doubtless, those which immediately succeeded it in point of time were copied; from which the same errors were subsequently transcribed into all the other, including the Chinese logarithms.
The most certain and effectual check upon errors which arise in the process of computation, is to cause the same computations to be made by separate and independent computers; and this check is rendered still more decisive if they make their computations by different methods. It is, nevertheless, a remarkable fact, that several computers, working separately and independently, do frequently commit precisely the same error; so that falsehood in this case assumes that character of consistency, which is regarded as the exclusive attribute of truth. Instances of this are familiar to most persons who have had the management of the computation of tables. We have reason to know, that M. Prony experienced it on many occasions in the management of the great French tables, when he found three, and even a greater number of computers, working separately and independently, to return him the same numerical result, and that result wrong. Mr Stratford, the conductor of the Nautical Almanac, to whose talents and zeal that work owes the execution of its recent improvements, has more than once observed a similar occurrence. But one of the most signal examples of this kind, of which we are aware, is related by Mr Baily. The catalogue of stars published by the Astronomical Society was computed by two separate and independent persons, and was afterwards compared and examined with great care and attention by Mr Stratford. On examining this catalogue, and recalculating a portion of it, Mr Baily discovered an error in the case of the star, χ. Cephei. Its right ascension was calculated wrongly, and yet consistently, by two computers working separately. Their numerical results agreed precisely in every figure; and Mr Stratford, on examining the catalogue, failed to detect the error. Mr Baily having reason, from some discordancy which he observed, to suspect an error, recomputed the place of the star with a view to discover it; and he himself, in the first instance, obtained precisely the same erroneous numerical result. It was only on going over the operation a second time that he accidentally discovered that he had inadvertently committed the same error.[10]
It appears, therefore, that the coincidence of different tables, even when it is certain that they could not have been copied one from another, but must have been computed independently, is not a decisive test of their correctness, neither is it possible to ensure accuracy by the device of separate and independent computation.
Besides the errors incidental to the process of computation, there are further liabilities in the process of transcribing the final results of each calculation into the fair copy of the table designed for the printer. The next source of error lies with the compositor, in transferring this copy into type. But the liabilities to error do not stop even here; for it frequently happens, that after the press has been fully corrected, errors will be produced in the process of printing. A remarkable instance of this occurs in one of the six errors detected in so many different tables already mentioned. In one of these cases, the last five figures of two successive numbers of a logarithmic table were the following:—
35875
10436.
Now, both of these are erroneous; the figure 8 in the first line should be 4, and the figure 4 in the second should be 8. It is evident that the types, as first composed, were correct; but in the course of printing, the two types 4 and 8 being loose, adhered to the inking-balls, and were drawn out: the pressmen in replacing them transposed them, putting the 8 above and the 4 below, instead of vice versâ. It would be a curious enquiry, were it possible to obtain all the copies of the original edition of Vlacq's Logarithms, published at Gouda in 1628, from which this error appears to have been copied in all the subsequent tables, to ascertain whether it extends through the entire edition. It would probably, nay almost certainly, be discovered that some of the copies of that edition are correct in this number, while others are incorrect; the former having been worked off before the transposition of the types.
It is a circumstance worthy of notice, that this error in Vlacq's tables has produced a corresponding error in a variety of other tables deduced from them, in which nevertheless the erroneous figures in Vlacq are omitted. In no less than sixteen sets of tables published at various times since the publication of Vlacq, in which the logarithms extend only to seven places of figures, the error just mentioned in the eighth place in Vlacq causes a corresponding error in the seventh place. When the last three figures are omitted in the first of the above numbers, the seventh figure should be 5, inasmuch as the first of the omitted figures is under 5: the erroneous insertion, however, of the figure 8 in Vlacq has caused the figure 6 to be substituted for 5 in the various tables just alluded to. For the same reason, the erroneous occurrence of 4 in the second number has caused the adoption of a 0 instead of a 1 in the seventh place in the other tables. The only tables in which this error does not occur are those of Vega, the more recent editions of Callet, and the still later Logarithms of Mr Babbage.
The Opus Palatinum, a work published in 1596, containing an extensive collection of trigonometrical tables, affords a remarkable instance of a tabular error; which, as it is not generally known, it may not be uninteresting to mention here. After that work had been for several years in circulation in every part of Europe, it was discovered that the commencement of the table of co-tangents and co-secants was vitiated by an error of considerable magnitude. In the first co-tangent the last nine places of figures were incorrect; but from the manner in which the numbers of the table were computed, the error was gradually, though slowly, diminished, until at length it became extinguished in the eighty-sixth page. After the detection of this extensive error, Pitiscus undertook the recomputation of the eighty-six erroneous pages. His corrected calculation was printed, and the erroneous part of the remaining copies of the Opus Palatinum was cancelled. But as the corrected table of Pitiscus was not published until 1607,—thirteen years after the original work,—the erroneous part of the volume was cancelled in comparatively few copies, and consequently correct copies of the work are now exceedingly rare. Thus, in the collection of tables published by M. Schulze,[11] the whole of the erroneous part of the Opus Palatinum has beeen adopted; he having used the copy of that work which exists in the library of the Academy of Berlin, and which is one of those copies in which the incorrect part was not cancelled. The corrected copies of this work may be very easily distinguished at present from the erroneous ones: it happened that the former were printed with a very bad and worn-out type, and upon paper of a quality inferior to that of the original work. On comparing the first eighty-six pages of the volume with the succeeding ones, they are, therefore, immediately distinguishable in the corrected copies. Besides this test, there is another, which it may not be uninteresting to point out:—At the bottom of page 7 in the corrected copies, there is an error in the position of the words basis and hypothenusa, their places being interchanged. In the original uncorrected work this error does not exist.
At the time when the calculation and publication of Taylor's Logarithms were undertaken, it so happened that a similar work was in progress in France; and it was not until tlie calculation of the French work was completed, that its author was informed of the publication of the English work. This circumstance caused the French calculator to relinquish the publication of his tables. The manuscript subsequently passed into the library of Delambre, and, after his death, was purchased at the sale of his books, by Mr Babbage, in whose possession it now is. Some years ago it was thought advisable to compare these manuscript tables with Taylor's Logarithms, with a view to ascertain the errors in each, but especially in Taylor. The two works were peculiarly well suited for the attainment of this end; as the circumstances under which they were produced, rendered it quite certain that they were computed independently of each other. The comparison was conducted under the direction of the late Dr Young, and the result was the detection of the following nineteen errors in Taylor's Logarithms. To enable those who used Taylor's Logarithms to make the necessary corrections in them, the corrections of the detected errors appeared as follows in the Nautical Almanac for 1832.
Errata, detected in Taylor's Logarithms. London: 4to, 1792.
i'^.'l'^!':iI;'/^l.C6-tarigentof 1.35.35 for 43671 rm^/ 42671 2ii.'..iif; Co-tang-ent of. 4.4.49 s!... .'.'.. Sine of. 4.23.38 — 43107 4 Sine of... '.iWirj^U 4.23.39 — 5....^. Sine of.....;;...;.] 6.45.52 — Q>..,.Kk Co-sine of. 14.18. 3 — l....Ss Tangent of. 18. 1.56 — S....Aaa Co-tangent of... ..21. 11. 14 — 9....Ggg Tangent of 23.48.19 — 10 Co-tangent of... ..23.48.19 — l....Iii Sine of 25. 5. 4 — 12 Sine of 25. 5. 5 — 13 Sine of 25. 5. 6 — H.. Sine of 25. 5. 7 — 15 Sme of 25. 5. 8 — 16 Sine of 25. 5. 9 — 17. ..Qqq Tangent of. 28.19.39 — 18. ..4 H Tangent of. 35.55.51 — 19. ..4 A' Co-sine of. 37.29. 2 — 66976 — 66979 43107 — 43007 43381
43281 10001 — 11001 3398 — 3298 5064 — 6064 6062 — . 5962 6087 — 5987 3913
4013 3173
3183 3218
3228 3263 — 3273 3308 — 3318 8353 — 3363 8398 — 3408 6302
6402 1681 . 1581 5503 — 5603
An error being detected in this list of Errata, we find, in the Nautical Almanac for the year 1833, the following Erratum of the Errata of Taylor's Logarithms:—
'In the list of Errata detected in Taylor's Logarithms, for cos. 4° 18′ 3″, read cos. 14° 18′ 2″.'
Here, however, confusion is worse confounded; for a new error, not before existing, and of much greater magnitude, is introduced! It will be necessary, in the Nautical Almanac for 1836, (that for 1835 is already published,) to introduce the following
Erratum of the Erratum of the Errata of Taylor's Logarithms. For cos. 4" 18' 3", read cos. 14" 18' 3".
If proof were wanted to establish incontrovertibly the utter impracticability of precluding numerical errors in works of this nature, we should find it in this succession of error upon error, produced, in spite of the universally acknowledged accuracy and assiduity of the persons at present employed in the construction and management of the Nautical Almanac. It is only by the mechanical fabrication of tables that such errors can be rendered impossible.
On examining this list with attention, we have been particularly struck with the circumstances in which these errors appear to have originated. It is a remarkable fact, that of the above nineteen errors, eighteen have arisen from mistakes in carrying. Errors 5, 7, 10, 11, 12, 13, 14, 15, 16, 17, 19, have arisen from a carriage being neglected; and errors 1, 3, 4, 6, 8, 9, and 18, from a carriage being made where none should take place. In four cases, namely, errors 8, 9, 10, and IG, this has caused two figures to be wrong. The only error of the nineteen which appears to have been a press error is the second; which has evidently arisen from the type 9 being accidentally inverted, and thus becoming a 6 This may have originated with the compositor, but more probably it took place in the press-work; the type 9 being accidentally drawn out of the form by the inking-ball, as mentioned in a former case, and on being restored to its place, inverted by the pressman.
There are two cases among the above errata, in which an error, committed in the calculation of one number, has evidently been the cause of other errors. In the third erratum, a wrong carriage was made, in computing the sine of 4° 23′ 38″. The next number of the table was vitiated by this error; for we find the next erratum to be in the sine of 4° 23′ 39″, in which the figure similarly placed is 1 in excess. A still more extensive effect of this kind appears in errata 11, 12, 13, 14, 15, 16. A carriage was neglected in computing the sine of 25° 5' 4", and this produced a corresponding error in the five following numbers of the table, which are those corrected in the five following errata.
This frequency of errors arising in the process of carrying, would afford a curious subject of metaphysical speculation respecting the operation of the faculty of memory. In the arithmetical process, the memory is employed in a twofold way;—in ascertaining each successive figure of the calculated result by the recollection of a table committed to memory at an early period of life; and by another act of memory, in which the number carried from column to column is retained. It is a curious fact, that this latter circumstance, occurring only the moment before, and being in its nature little complex, is so much more liable to be forgotten or mistaken than the results of rather complicated tables. It appears, that among the above errata, the errors 5, 7, 10, 11, 17, 19, have been produced by the computer forgetting a carriage; while the errors 1, 3, 6, 8, 9, 18, have been produced by his making a carriage improperly. Thus, so far as the above list of errata affords grounds for judging, it would seem, (contrary to what might be expected,) that the error by which improper carriages are made is as frequent as that by which necessary carriages are overlooked.
We trust that we have succeeded in proving, first, the great national and universal utility of numerical tables, by showing the vast number of them, which have been calculated and published; secondly, that more effectual means are necessary to obtain such tables suitable to the present state of the arts, sciences and commerce, l)y showing that the existing supply of tables, vast as it certainly is, is still scanty, and utterly inadequate to the demands of the community;—that it is rendered inefficient, not only in quantity, but in quality, by its want of numerical correctness; and that such numerical correctness is altogether unattainable until some more perfect method be discovered, not only of calculating the numerical results, but of tabulating these,—of reducing such tallies to type, and of printing that type so as to intercept the possibility of error during the press-work. Such are the ends which are proposed to be attained by the calculating machinery invented by Mr Babbage.
The benefits to be derived from this invention cannot be more strongly expressed than they have been by Mr Colebrooke, President of the Astronomical Society, on the occasion of presenting the gold medal voted by that body to Mr Babbage:—'In no department of science, or of the arts, does this discovery promise to be so eminently useful as in that of astronomy, and its kindred sciences, with the various arts dependent on them. In none are computations more operose than those which astronomy in particular requires;—in none are preparatory facilities more needful;—in none is error more detrimental. The practical astronomer is interrupted in his pursuit, and diverted from his task of observation by the irksome labours of computation, or his diligence in observing becomes ineffectual for want of yet greater industry of calculation. Let the aid which tables previously computed afford, be furnished to the utmost extent which mechanism has made attainable through Mr Babbage's invention, and the most irksome portion of the astronomer's task is alleviated, and a fresh impulse is given to astronomical research.'
The first step in the progress of this singular invention was the discovery of some common principle which pervaded numerical tables of every description; so that by the adoption of such a principle as the basis of the machinery, a corresponding degree of generality would be conferred upon its calculations. Among the properties of numerical functions, several of a general nature exist; and it was a matter of no ordinary difficulty, and requiring no common skill, to select one which might, in all respects, be preferable to the others. Whether or not that which was selected by Mr Babbage affords the greatest practical advantages, would be extremely difficult to decide—perhaps impossible, unless some other projector could be found possessed of sufficient genius, and sustained by sufficient energy of mind and character, to attempt the invention of calculating machinery on other principles. The principle selected by Mr Babbage as the basis of that part of the machinery which calculates, is the Method of Differences; and he has in fact literally thrown this mathematical principle into wheel-work. In order to form a notion of the nature of the machinery, it will be necessary, first to convey to the reader some idea of the mathematical principle just alluded to.
A numerical table, of whatever kind, is a series of numbers which possess some common character, and which proceed increasing or decreasing according to some general law. Supposing such a series continually to increase, let us imagine each number in it to be subtracted from that which follows it, and the remainders thus successively obtained to be ranged beside the first, so as to form another table: these numbers are called the first differences. If we suppose these likewise to increase continually, we may obtain a third table from them by a like process, subtracting each number from the succeeding one: this series is called the second differences. By adopting a like method of proceeding, another series may be obtained, called the third differences; and so on. By continuing this process, we shall at length obtain a series of differences, of some order, more or less high, according to the nature of the original table, in which we shall find the same number constantly repeated, to whatever extent the original table may have been continued; so that if the next series of differences had been obtained in the same manner as the preceding ones, every term of it would be 0. In some cases this would continue to whatever extent the original table might be carried; but in all cases a series of differences would be obtained, which would continue constant for a very long succession of terms.
As the successive serieses of differences are derived from the original table, and from each other, by subtraction, the same succession of series may be reproduced in the other direction by addition. But let us suppose that the first number of the original table, and of each of the series of differences, including the last, be given: all the numbers of each of the series may thence be obtained by the mere process of addition. The second term of the original table will be obtained by adding to the first the first term of the first difference series; in like manner, the second term of the first difference series will be obtained by adding to the first term, the first term of the third difference series, and so on. The second terms of all the serieses being thus obtained, the third terms may be obtained by a like process of addition; and so the series may be continued. These observations will perhaps be rendered more clearly intelligible when illustrated by a numerical example. The following is the commencement of a series of the fourth powers of the natural numbers:—
No. Table. 1 . . . 1 2 . . . 16 3 . . . 81 4 . . . 256 5 . . . 625 6 . . . 1296 7 . . . 2401 8 . . . 4096 9 . . . 6561 10 . . . 10,000 11 . . . 14,641 12 . . . 20,736 13 . . . 28,561
By subtracting each number from the succeeding one in this series, we obtain the following series of first differences:
15 65 175 369 671 1105 1695 2465 3439 4641 6095 7825
In like manner, subtracting each term of this series from the succeeding one, we obtain the following series of second differences:—
50 110 194 302 434 590 770 974 1202 1454 1730
Proceeding with this series in the same way, we obtain the following series of third differences:—
60 84 108 132 156 180 204 228 252 276
Proceeding in the same way with these, we obtain the following for the series of fourth differences:—
24 24 24 24 24 24 24 24 24
It appears, therefore, that in this case the series of fourth differences consists of a constant repetition of the number 24. Now, a slight consideration of the succession of arithmetical operations by which we have obtained this result, will show, that by reversing the process, we could obtain the table of fourth powers by the mere process of addition. Beginning with the first numbers in each successive series of differences, and designating the table and the successive differences by the letters T, D1 D2 D3 D4, we have then the following to begin with:—
T D1 D2 D3 D4 1 15 50 60 24
Adding each number to the number on its left, and repeating 24, we get the following as the second terms of the several series:—
T D1 D2 D3 D4 16 65 110 84 24
And, in the same manner, the third and succeeding terms as follows: — No. T D' D2 D^ D^ 1 1 15 50 60 24 2. 16 65 110 84 24 3 81 175 194 108 24 4 256 369 302 132 24 3 625 671 434 156 24 6 1296 1105 . 590 180 24 7 '2401 1695 770 204 24 8 4096 2465 974 228 24 9 6561 3439 1202 252 24 10 10000 4641 1454 276 11 14641 6095 . 1730 12 20736 7825 13 28561 There are numerous tables in which, as already stated, to whatever order of differences we may proceed, we should not obtain a series of rigorously constant differences; but we should always obtain a certain number of differences which to a given number of decimal places would remain constant for a long succession of terms. It is plain that such a table might be calculated by addition in the same manner as those which have a difference rigorously and continuously constant ; and if at every point where the last difference requires an increase, that increase be given to it, the same principle of addition may again be applied for a like succession of terms, and so on.
By this principle it appears, that all tables in which each series of differences continually increases, may be produced by the operation of addition alone; provided the first terms of the table, and of each series of differences, be given in the first instance. But it sometimes happens, that while the table continually increases, one or more serieses of differences may continually diminish. In this case, the series of differences are found by subtracting each term of the series, not from that which follows, but from that which precedes it; and consequently, in the re-production of the several serieses, when their first terms are given, it will be necessary in some cases to obtain them by addition, and in others by subtraction. It is possible, however, still to perform all the operations by addition alone: this is effected in performing the operation of subtraction, by substituting for the subtrahend its arithmetical complement, and adding that, omitting the unit of the highest order in the result. This process, and its principle, will be readily comprehended by an example. Let it be required to subtract 357 from 768.
The common process would be as follows:—
From 768 Subtract 357 Remainder 411
The arithmetical complement of 357, or the number by which it falls short of 1000, is 643. Now, if this number be added to 768, and the first figure on the left be struck out of the sum, the process will be as follows:—
To 768 Add 643 Sum 1411 Remainder sought 411
The principle on which this process is founded is easily explained. In the latter process we have first added 643, and then subtracted 1000. On the whole, therefore, we have subtracted 357, since the number actually subtracted exceeds the number previously added by that amount.
Since, therefore, subtraction may be effected in this manner by addition, it follows that the calculation of all serieses, so far as an order of differences can be found in them which continues constant, may be conducted by the process of addition alone.
It also appears from what has been stated, that each addition consists only of two operations. However numerous the figures may be of which the several pairs of numbers to be thus added may consist, it is obvious that the operation of adding them can only consist of repetitions of the process of adding one digit to another; and of carrying one from the column of inferior units to the column of units next superior when necessary. If we would therefore reduce such a process to machinery, it would only be necessary to discover such a combination of moving parts as are capable of performing these two processes of adding and carrying on two single figures; for, this being once accomplished, the process of adding two numbers, consisting of any number of digits, will be effected by repeating the same mechanism as often as there are pairs of digits to be added. Such was the simple form to which Mr Babbage reduced the problem of discovering the calculating machinery; and we shall now proceed to convey some notion of the manner in which he solved it.
For the sake of illustration, we shall suppose that the table to be calculated shall consist of numbers not exceeding six places of figures; and we shall also suppose that the difference of the fifth order is the constant difference. Imagine, then, six rows of wheels, each wheel carrying upon it a dial-plate like that of a common clock, but consisting of ten instead of twelve divisions; the several divisions being marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Let these dials be supposed to revolve whenever the wheels to which they are attached are put in motion, and to turn in such a direction that the series of increasing numbers shall pass under the index which appears over each dial:—thus, after 0 passes the index, 1 follows, then 2, 3, and so on, as the dial revolves. In Fig. 1 are represented six horizontal rows of such dials.
Fig. 1.
The method of differences, as already explained, requires, that in proceeding with the calculation, this apparatus should perform continually the addition of the number expressed upon each row of dials, to the number expressed upon the row immediately above it. Now, we shall first explain how this process of addition may be conceived to be performed by the motion of the dials; and in doing so, we shall consider separately the processes of addition and carriage, considering the addition first, and then the carriage.
The first quarter of a turn of the axis will add the second, fourth, and sixth rows to the first, third, and fifth, omitting the carriages; this it will do by causing the dials on the first, third, and fifth rows, to turn through as many divisions as are expressed by the numbers at the indices below them, as already explained.
The second quarter of a turn will cause the carriages consequent on the previous addition, to be made by moving forward the proper dials one division.
(During these two quarters of a turn, the dials of the first, third, and fifth row alone have been moved; those of the second, fourth, and sixth, have been quiescent.)
The third quarter of a turn will produce the addition of the third and fifth rows to the second and fourth, omitting the carriages; which it will do by causing the dials of the second and fourth rows to turn through as many divisions as are expressed by the numbers at the indices immediately below them.
The fourth and last quarter of a turn will cause the carriages consequent on the previous addition, to be made by moving the proper dials forward one division.
This evidently completes one calculation, since all the rows except the first have been respectively added to all the rows except the last.
To illustrate this: let us suppose the table to be computed to be that of the fifth powers of the natural numbers, and the computation to have already proceeded so far as the fifth power of 6, which is 7776. This number appears, accordingly, in the highest row, being the place appropriated to the number of the table to be calculated. The several differences as far as the fifth, which is in this case constant, are exhibited on the successive rows of dials in such a manner, as to be adapted to the process of addition by alternate rows, in the manner already explained. ^ The process of addition will commence by the motion of the dials in the first, third, and fifth rows, in the following manner: The dial A, fig. 1, must turn through one division, which will bring the number 7 to the index; the dial B must turn through three divisions, which will 0 bring to the index; this will render a carriage necessary, but that carriage will not take place during the present motion of the dial. The dial C will remain unmoved, since 0 is at the index below it; the dial D must turn through nine divisions; and as, in doing so, the division between 9 and 0 must pass under the index, a carriage must subsequently take place upon the dial to the left; the remaining dials of the row T, fig. 1, will remain unmoved. In the row D2 the dial A2 will remain unmoved, since 0 is at the index below it; the dial B2 will be moved through five divisions, and will render a subsequent carriage on the dial to the left necessary; the dial C2 will be moved through five divisions; the dial D2 will be moved through three divisions, and the remaining dials of this row will remain unmoved. The dials of the row D4 will be moved according to the same rules; and the whole scheme will undergo a change exhibited in Fig. 2; a mark (*) being introduced on those dials to which a
Fig. 2.
carriage rendered necessary by the addition which has just taken place.
The second quarter of a turn of the moving axis, will move forward through one division all the dials which in Fig. 2 are marked (*), and the scheme will be converted into the scheme expressed in Fig. 3.
Fig. 3.
In third quarter of a turn, the dial A1, fig. 3, will remain unmoved, since is at the index below it; the dial B1 will be moved forward through three divisions; C1 through nine divisions, and so on; and in like manner the dials of the row D3 will be moved forward through the number of divisions expressed at the indices in the row D4. This change will convert the arrangement into that expressed in Fig. 4, the dials to which a carriage is due, being distinguished as before by (*).
Fig. 4.
The fourth quarter of a turn of the axis will move forward one division all the dials marked (*); and the arrangement will finally assume the form exhibited in Fig. 5, in which the calculation is completed. The first row. T in this expresses the fifth power of
Fig. 5.
7; and the second expresses the number which must be added to the first row, in order to produce the fifth power of 8; the numbers in each row being prepared for the change which they must undergo, in order to enable them to continue the computation according to the method of alternate addition here adopted.
Having thus explained what it is that the mechanism is required to do, we shall now attempt to convey at least a general notion of some of the mechanical contrivances by which the desired ends are attained. To simplify the explanation, let us first take one particular instance—the dials B and B 1, fig. 1, for example. Behind the dial B1 is a bolt, which, at the commencement of the process, is shot between the teeth of a wheel which drives the dial B: during the first quarter of a turn this bolt is made to revolve, and if it continued to be engaged in the teeth of the said wheel, it would cause the dial B to make a complete revolution; but it is necessary that the dial B should only move through three divisions, and, therefore, when three divisions of this dial have passed under its index, the aforesaid bolt must be withdrawn: this is accomplished by a small wedge, which is placed in a fixed position on the wheel behind the dial B1, and that position is such that this wedge will press upon the bolt in such a manner, that at the moment when three divisions of the dial B have passed under the index, it shall withdraw the bolt from the teeth of the wheel which it drives. The bolt will continue to revolve during the remainder of the first quarter of a turn of the axis, but it will no longer drive the dial B, which will remain quiescent. Had the figure at the index of the dial B1 been any other, the wedge which withdraws the bolt would have assumed a different position, and would have withdrawn the bolt at a different time, but at a time always corresponding with the number under the index of the dial B1: thus, if 5 had been under the index of the dial B1, then the bolt would have been withdrawn from between the teeth of the wheel which it drives, when five divisions of the dial B had passed under the index, and so on. Behind each dial in the row D1 there is a similar bolt and a similar withdrawing wedge, and the action upon the dial above is transmitted and suspended in precisely the same manner. Like observations will be applicable to all the dials in the scheme here referred to, in reference to their adding actions upon those above them.
There is, however, a particular case which here merits notice: it is the case in which 0 is under the index of the dial from which the addition is to be transmitted upwards. As in that case nothing is to be added, a mechanical provision should be made to prevent the bolt from engaging in the teeth of the wheel which acts upon the dial above: the wedge which causes the bolt to be withdrawn, is thrown into such a position as to render it impossible that the bolt should be shot, or that it should enter between the teeth of the wheel, which in other cases it drives. But inasmuch as the usual means of shooting the bolt would still act, a strain would necessarily take place in the parts of the mechanism, owing to the bolt not yielding to the usual impulse. A small shoulder is therefore provided, which puts aside, in this case, the piece by which the bolt is usually struck, and allows the striking implement to pass without encountering the head of the bolt or any other obstruction. This mechanism is brought into play in the scheme, fig. 1, in the cases of all those dials in which 0 is under the index.
Such is a general description of the nature of the mechanism by which the adding process, apart from the carriages, is effected. During the first quarter of a turn, the bolts which drive the dials in the first, third, and fifth rows, are caused to revolve, and to act upon these dials, so long as they are permitted by the position of the several wedges on the second, fourth, and sixth rows of dials, by which these bolts are respectively withdrawn; and, during the third quarter of a turn, the bolts which drive the dials of the second and fourth rows are made to revolve and act upon these dials so long as the wedges on the dials of the third and fifth rows, which withdraw them, permit. It will hence be perceived, that, during the first and third quarters of a turn, the process of addition is continually passing upwards through the machinery; alternately from the even to the odd rows, and from the odd to the even rows, counting downwards.
We shall now attempt to convey some notion of the mechanism by which the process of carrying is effected during the second and fourth quarters of a turn of the axis. As before, we shall first explain it in reference to a particular instance. During the first quarter of a turn the wheel B2, Fig. 1, is caused by the adding bolt to move through five divisions; and the fifth of these divisions, which passes under the index, is that between 9 and 0. On the axis of the wheel C2, immediately to the left of B2, is fixed a wheel, called in mechanics a ratchet wheel, which is driven by a claw which constantly rests in its teeth. This claw is in such a position as to permit the wheel C2 to move in obedience to the action of the adding bolt, but to resist its motion in the contrary direction. It is drawn back by a spiral spring, but its recoil is prevented by a hook which sustains it; which hook, however, is capable of being withdrawn, and when withdrawn, the aforesaid spiral spring would draw back the claw, and make it fall through one tooth of the ratchet wheel. Now, at the moment that the division between 9 and 0 on the dial B2 passes under the index, a thumb placed on the axis of this dial touches a trigger which raises out of the notch the hook which sustains the claw just mentioned, and allows it to fall back by the recoil of the spring, and to drop into the next tooth of the ratchet wheel. This process, however, produces no immediate effect upon the position of the wheel C2, and is merely preparatory to an action intended to take place during the second quarter of a turn of the moving axis. It is in effect a memorandum taken by the machine of a carriage to be made in the next quarter of a turn.
During the second quarter of a turn, a finger placed on the axis of the dial B2 is made to revolve, and it encounters the heel of the above-mentioned claw. As it moves forward it drives the claw before it: and this claw, resting in the teeth of the ratchet wheel fixed upon the axis of the dial C2 drives forward that wheel, and with it the dial. But the length and position of the finger which drives the claw limits its action, so as to move the claw forward through such a space only as will cause the dial C2 to advance through a single division; at which point it is again caught and retained by the hook. This will be added to the number under its index, and the requisite carriage from B2 to C2 will be accomplished.
In connexion with every dial is placed a similar ratchet wheel with a similar claw, drawn by a similar spring, sustained by a similar hook, and acted upon by a similar thumb and trigger; and therefore the necessary carriages, throughout the whole machinery, take place in the same manner and by similar means.
During the second quarter of a turn, such of the carrying claws as have been allowed to recoil in the first, third, and fifth rows, are drawn up by the fingers on the axes of the adjacent dials; and, during the fourth quarter of a turn, such of the carrying claws on the second and fourth rows as have been allowed to recoil during the third quarter of a turn, are in like manner drawn up by the carrying fingers on the axes of the adjacent dials. It appears that the carriages proceed alternately from right to left along the horizontal rows during the second and fourth quarters of a turn; in the one, they pass along the first, third, and fifth rows, and in the other, along the second and fourth.
There are two systems of waves of mechanical action continually flowing from the bottom to the top; and two streams of similar action constantly passing from the right to the left. The crests of the first system of adding waves fall upon the last difference, and upon every alternate one proceeding upwards; while the crests of the other system touch upon the intermediate differences. The first stream of carrying action passes from right to left along the highest row and every alternate tow, while tile second stream passes along the intermediate rows.
Such is a very rapid and general outline of this machinery. lts wonders, however, are still greater in its details than even in its broader features. Although we despair of doing it justice by any description which can be attempted here, yet we should not fulfil the duty we owe to our readers, if we did not call their attention at least to a few of the instances of consummate skill which are scattered, with a prodigality characteristic of the highest order of inventive genius, throughout this astonishing mechanism.
We have stated that, at the commencement of each revolution of the moving axis, the bolts which drive the dials of the first, third, and fifth rows, are shot. The process of shooting these bolts must therefore have taken place during the last quarter of the preceding revolution; but it is during that quarter of a turn that the carriages are effected in the second and fourth rows. Since the bolts which drive the dials of the first, third, and fifth rows, have no mechanical connexion with the dials in the second and fourth rows, there is nothing in the process of shooting those bolts incompatible with that of moving the dials of the second and fourth rows: hence these two processes may both take place during the same quarter of a turn. But in order to equalize the resistance to the moving power, the same expedient is here adopted as that already described in the process of carrying. The arms which shoot the bolts of each row of dials are arranged spirally, so as to act successively throughout the quarter of a turn. There is, however, a contingency which, under certain circumstances, would here produce a difficulty which must be provided against. It is possible, and in fact does sometimes happen, that the process of carrying causes a dial to move under the index from 0 to 1. In that case, the bolt, preparatory to the next addition, ought not to be shot until after the carriage takes place; for if the arm which shoots it passes its point of action before the carriage takes place, the bolt will be moved out of its sphere of action, and will not be shot, which, as we have already explained, must always happen when 0 is at the index: therefore no addition would in this case take place during the next quarter of a turn of the axis; whereas, since 1 is brought to the index by the carriage, which immediately succeeds the passage of the arm which ought to bolt, 1 should be added during the next quarter of a turn. It is plain, accordingly, that the mechanism should be so arranged, that the action of the arms, which shoot the bolts successively, should immediately follow the action of those fingers which raise the carrying claws successively; and therefore either a separate quarter of a turn should be appropriated to each of those movements, or if they be executed in the same quarter of a turn, the mechanism must be so constructed, that the arms which shoot the bolts successively, shall severally follow immediately after those which raise the carrying claws successively. The latter object is attained by a mechanical arrangement of singular felicity, and partaking of that elegance which characterises all the details of this mechanism. Both sets of arms are spirally arranged on their respective axes, so as to be carried through their period in the same quarter of a turn; but the one spiral is shifted a few degrees, in angular position, behind the other, so that each pair of corresponding arms succeed each other in the most regular order,—equalizing the resistance, economizing time, harmonizing the mechanism, and giving to the whole mechanical action the utmost practical perfection.
The system of mechanical contrivances by which the results, here attempted to be described, are attained, form only one order of expedients adopted in this machinery;—although such is the perfection of their action, that in any ordinary case they would be regarded as having attained the ends in view with an almost superfluous degree of precision. Considering, however, the immense importance of the purposes which the mechanism was destined to fulfil, its inventor determined that a higher order of expedients should be superinduced upon those already described; the purpose of which should be to obliterate all small errors or inequalities which might, even by remote possibility, arise, either from defects in the original formation of the mechanism, from inequality of wear, from casual strain or derangement,—or, in short, from any other cause whatever. Thus the movements of the first and principal parts of the mechanism were regarded by him merely as a first, though extremely nice approximation, upon which a system of small corrections was to be subsequently made by suitable and independent mechanism. This supplementary system of mechanism is so contrived, that if one or more of the moving parts of the mechanism of the first order be slightly out of their places, they will be forced to their exact position by the action of the mechanical expedients of the second order to which we now allude. If a more considerable derangement were produced by any accidental disturbance, the consequence would be that the supplementary mechanism would cause the whole system to become locked, so that not a wheel would be capable of moving; the impelling power would necessarily lose all its energy, and the machine would stop. The consequence of this exquisite arrangement is, that the machine will either calculate rightly, or not at all.
The supernumerary contrivances which we now allude to, being in a great degree unconnected with each other, and scattered through the machinery to a certain extent, independent of the mechanical arrangement of the principal parts, we find it difficult to convey any distinct notion of their nature or form.
In some instances they consist of a roller resting between certain curved surfaces, which has but one position of stable equilibrium, and that position the same, however the roller or the curved surfaces may wear. A slight error in the motion of the principal parts would make this roller for the moment rest on one of the curves; but, being constantly urged by a spring, it would press on the curved surface in such a manner as to force the moving piece on which that curved surface is formed, into such a position that the roller may rest between the two surfaces; that position being the one which the mechanism should have. A greater derangement would bring the roller to the crest of the curve, on which it would rest in instable equilibrium; and the machine would either become locked, or the roller would throw it as before into its true position.
In other instances a similar object is attained by a solid cone being pressed into a conical seat; the position of the axis of the cone and that of its seat being necessarily invariable, however the cone may wear: and the action of the cone upon the seat being such, that it cannot rest in any position except that in which the axis of the cone coincides with the axis of its seat.
Having thus attempted to convey a notion, however inadequate, of the calculating section of the machinery, we shall proceed to offer some explanation of the means whereby it is enabled, to print its calculations in such a manner as to preclude the possibility of error in any individual printed copy.
On the axle of each of the wheels which express the calculated number of the table T, there is fixed a solid piece of metal, formed into a curve, not unlike the wheel in a common clock, which is called the snail. This curved surface acts against the arm of a lever, so as to raise that arm to a higher or lower point according to the position of the dial with which the snail is connected, Without entering into a more minute description, it will be easily understood that the snail may be so formed that the arm of the lever shall be raised to ten different elevations, corresponding to the ten figures of the dial which may be brought under the index. The opposite arm of the lever here described puts in motion a solid arch, or sector, which carries ten punches: each punch bearing on its face a raised character of a figure, and the ten punchy bearing the ten characters, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. It will be apparent from what has been just stated, that this type sector (as it is called) will receive ten different attitudes, corresponding to the ten figures which may successively be brought under the index of the dial-plate. At a point over which the type sector is thus moved, and immediately under a point through which it plays, is placed a frame, in which is fixed a plate of copper. Immediately over a certain point through which the type sector moves, is likewise placed a bent lever, which, being straightened, is forcibly pressed upon the punch which has been brought under it. If the type sector be moved, so as to bring under the bent lever one of the steel punches above mentioned, and be held in that position for a certain time, the bent lever, being straightened, acts upon the steel punch, and drives it against the face of the copper beneath, and thus causes a sunken impression of the character upon the punch to be left upon the copper. If the copper be now shifted slightly in its position, and the type sector be also shifted so as to bring another punch under the bent lever, another character may be engraved on the copper by straightening the bent lever, and pressing it on the punch as before. It will be evident, that if the copper was shifted from right to left through a space equal to two figures of a number, and, at the same time, the type sector so shifted as to bring the punches corresponding to the figures of the number successively under the bent lever, an engraved impression of the number might thus be obtained upon the copper by the continued action of the bent lever. If, when one line of figures is thus obtained, a provision be made to shift the copper in a direction at right angles to its former motion, through a space equal to the distance between two lines of figures, and at the same time to shift it through a space in the other direction equal to the length of an entire line, it will be evident that another line of figures might be printed below the first in the same manner.
The motion of the type sector, here described, is accomplished by the action of the snail upon the lever already mentioned. In the case where the number calculated is that expressed in fig. 1, the process would be as follows:—The snail of the wheel F1, acting upon the lever, would throw the type sector into such an attitude, that the punch bearing the character 0 would come under the bent lever. The next turn of the moving axis would cause the bent lever to press on the tail of the punch, and the character 0 would be impressed upon the copper. The bent lever being again drawn up, the punch would recoil from the copper by the action of a spring; the next turn of the moving axis would shift the copper through the interval between two figures, so as to bring the point destined to be impressed with the next figure under the bent lever. At the same time, the snail of the wheel E would cause the type sector to be thrown into the same attitude as before, and the punch would be brought under the bent lever; the next turn would impress the figure beside the former one, as before described. The snail upon the wheel D would now come into action, and throw the type sector into that position in which the punch bearing the character 7 would come under the bent lever, and at the same time the copper would be shifted through the interval between two figures; the straightening of the lever would next follow, and the character 7 would be engraved. In the same manner, the wheels C, B, and A would successively act by means of their snails; and the copper being shifted, and the lever allowed to act, the number 007776 would be finally engraved upon the copper: this being accomplished, the calculating machinery would next be called into action, and another calculation would be made, producing the next number of the Table exhibited in Fig. 5. During this process the machinery would be engaged in shifting the copper both in the direction of its length and its breadth, with a view to commence the printing of another line; and this change of position would be accomplished at the moment when the next calculation would be completed: the printing of the next number would go on like the former, and the operation of the machine would proceed in the same manner, calculating and printing alternately. It is not, however, at all necessary—though we have here supposed it, for the sake of simplifying the explanation—that the calculating part of the mechanism should have its action suspended while the printing part is in operation, or vice versa; it is not intended, in fact, to be so suspended in the actual machinery. The same turn of the axis by which one number is printed, executes a part of the movements necessary for the succeeding calculation; so that the whole mechanism will be simultaneously and continuously in action.
Of the mechanism by which the position of the copper is shifted from figure to figure, from line to line, we shall not attempt any description. We feel that it would be quite vain. Complicated and difficult to describe as every other part of this machinery is, the mechanism for moving the copper is such as it would be quite impossible to render at all intelligible, without numerous illustrative drawings.
The engraved plate of copper obtained in the manner above described, is designed to be used as a mould from which a stereotyped plate may be cast; or, if deemed advisable, it may be used as the immediate means of printing. In the one case we should produce a table, printed from type, in the same manner as common letter-press printing; in the other an engraved table. If it be thought most advisable to print from the stereotyped plates, then as many stereotyped plates as may be required may be taken from the copper mould; so that when once a table has been calculated and engraved by the machinery, the whole world may be supplied with stereotyped plates to print it, and may continue to be so supplied for an unlimited period of time. There is no practical limit to the number of stereotyped plates which may be taken from the engraved copper; and there is scarcely any limit to the number of printed copies which may be taken from any single stereotyped plate. Not only, therefore, is the numerical table by these means engraved and stereotyped with infallible accuracy, but such stereotyped plates are producible in unbounded quantity. Each plate, when produced, becomes itself the means of producing printed copies of the table, in accuracy perfect, and in number without limit.
Unlike all other machinery, the calculating mechanism produces, not the object of consumption, but the machinery by which that object may be made. To say that it computes and prints with infallible accuracy, is to understate its merits:—it computes and fabricates the means of printing with absolute correctness and in unlimited abundance.
For the sake of clearness, and to render ourselves more easily intelligible to the general reader, we have in the preceding explanation thrown the mechanism into an arrangement somewhat different from that which is really adopted. The dials expressing the numbers of the tables of the successive differences are not placed, as we have supposed them, in horizontal rows, and read from right to left, in the ordinary way; they are, on the contrary, placed vertically, one below the other, and read from top to bottom. The number of the table occupies the first vertical column on the right, the units being expressed on the lowest dial, and the tens on the next above that, and so on. The first difference occupies the next vertical column on the left; and the numbers of the succeeding differences occupy vertical columns, proceeding regularly to the left; the constant difference being on the last vertical column. It is intended in the machine now in progress to introduce six orders of differences, so that there will be seven columns of dials; it is also intended that the calculations shall extend to eighteen places of figures: thus each column will have eighteen dials. We have referred to the dials as if they were inscribed upon the faces of wheels, whose axes are horizontal and planes vertical. In the actual machinery the axes are vertical and the planes horizontal, so that the edges of the figure wheels, as they are called, are presented to the eye. The figures are inscribed, not upon the dial-plate, but around the surface of a small cylinder or barrel, placed upon the axis of the figure wheel, which revolves with it; so that as the figure wheel revolves, the figures on the barrel are successively brought to the front, and pass under an index engraved upon a plate of metal immediately above the barrel. This arrangement has the obvious practical advantage, that, instead of each figure wheel having a separate axis, all the figure wheels of the same vertical column revolve on the same axis; and the same observation will apply to all the wheels with which the figure wheels are in mechanical connexion. This arrangement has the further mechanical advantage over that which has been assumed for the purposes of explanation, that the friction of the wheel-work on the axes is less in amount, and more uniformly distributed, than it could be if the axes were placed in the horizontal position.
A notion may therefore be formed of the front elevation of the calculating part of the mechanism, by conceiving seven steel axes erected, one beside another, on each of which shall be placed eighteen wheels,[12] five inches in diameter, having cylinders or barrels upon them an inch and a half in height, and inscribed, as already stated, with the ten arithmetical characters. The entire elevation of the machinery would occupy a space measuring ten feet broad, ten feet high, and five feet deep. The process of calculation would be observed by the alternate motion of the figure wheels on the several axes. During the first quarter of a turn, the wheels on the first, third, and fifth axes would turn, receiving their addition from the second, fourth, and sixth; during the second quarter of a turn, such of the wheels on the first, third, and fifth axes, to which carriages are due, would be moved forward one additional figure; the second, fourth, and sixth columns of wheels being all this time quiescent. During the third quarter of a turn, the second, fourth, and sixth columns would be observed to move, receiving their additions from the third, fifth, and seventh axes; and during the fourth quarter of a turn, such of these wheels to which carriages are due, would be observed to move forward one additional figure; the wheels of the first, third, and fifth columns being quiescent during this time.
It will be observed that the wheels of the seventh column are always quiescent in this process; and it may be asked, of what use they are, and whether some mechanism of a fixed nature would not serve the same purpose? It must, however, be remembered, that for different tables there will be different constant differences; and that when the calculation of a table is about to commence, the wheels on the seventh axis must be moved by the hand, so as to express the constant difference, whatever it may, be. In tables, also, which have not a difference rigorously constant, it will be necessary, after a certain number of calculations, to change the constant difference by the hand; and in this case the wheels of the seventh axis must be moved when occasion requires. Such adjustment, however, will only be necessary at very distant intervals, and after a considerable extent of printing and calculation has taken place; and when it is necessary, a provision is made in the machinery by which notice will be given by the sounding of a bell, so that the machine may not run beyond the extent of its powers of calculation.
It will be perceived, that upon the same axis are placed an unbolting wheel, a bolt, and an adding wheel, one above the other, for every figure wheel; and as there are eighteen figure wheels there will be eighteen tiers; each tier formed of an unbolting wheel, a bolt, and an adding wheel, placed one above the other; the wheels on this axis all revolving independent of the axis, but the bolts being all fixed upon it. The same observations, of course, will apply to each of the seven axes.
At the commencement of every revolution of the adding axes, it is evident that the several bolts placed upon them must be shot in order to perform the various additions. This is accomplished by a third set of seven axes, placed at some distance behind the range of the wheels, which turn upon the adding axes: these are called bolting axes. On these bolting axes are fixed, so as to revolve with them, a bolting finger opposite to each bolt; as the bolting axis is made to revolve by the moving power, the bolting finger is turned, and as it passes near the bolt, it encounters the shoulder of a hammer or lever, which strikes the heel of the bolt, and presses it forward so as to shoot its tooth between the crown teeth of the adding wheel. The only exception to this action is the case in which happens to be at the index of the figure wheel; in that case, the lever or hammer, which the bolting finger would encounter, is, as before stated, lifted out of the way of the bolting finger, so that it revolves without encountering it. It is on the bolting axes that the fingers are spirally arranged so as to equalize their action, as already explained.
The same axes in the front of the machinery on which the figure wheels turn, are made to serve the purpose of carrying. Each of these bear a series of fingers which turn with them, and which encounter a carrying claw, already described, so as to make the carriage: these carrying fingers are also spirally arranged on their axes, as already described.
Although the absolute accuracy which appears to be ensured by the mechanical arrangements here described is such as to render further precautions nearly superfluous, still it may be right to state, that, suj)posing it were possible for an error to be produced in calculation, this error could be easily and speedily detected in the printed tables: it would only be necessary to calculate a number of the table taken at intervals, through which the mechanical action of the machine has not been suspended, and during which it has received no adjustment by the hand: if the computed number be found to agree with those printed, it may be taken for granted that all the intermediate numbers are correct; because, from the nature of the mechanism, and the principle of computation, an error occurring in any single number of the table would be unavoidably entailed, in an increasing ratio, upon all the succeeding numbers.
We have hitherto spoken merely of the practicability of executing by the machinery, when completed, that which its inventor originally contemplated—namely, the calculating and printing of all numerical tables, derived by the method of differences from a constant difference. It has, however, happened that the actual powers of the machinery greatly transcend those contemplated in its original design:—they not only have exceeded the most sanguine anticipations of its inventor, but they appear to have an extent to which it is utterly impossible, even for the most acute mathematical thinker, to fix a probable limit. Certain subsidiary mechanical inventions have, in the progress of the enterprise, been, by the very nature of the machinery, suggested to the mind of the inventor, which confer upon it capabilities which he had never foreseen. It would be impossible even to enumerate, within the limits of this article, much less to describe in detail, those extraordinary mechanical arrangements, the effects of which have not failed to strike with astonishment every 'one who has been favoured with an opportunity of witnessing them, and who has been enabled, by sufficient mathematical attainments, in any degree to estimate their probable consequences.
As we have described the mechanism, the axes containing the several differences are successively and regularly added one to another; but there are certain mechanical adjustments, and these of a very simple nature, which being thrown into action, will cause a difference of any order to be added any number of times to a difference of any other order; and that either proceeding backwards or forwards, from a difference of an inferior to one of a superior order, and vice versa[13]
Among other peculiar mechanical provisions in the machinery is one by which, when the table for any order of difference amounts to a certain number, a certain arithmetical change would be made in the constant difference. In this way a series may be tabulated by the machine, in which the constant difference is subject to periodical change; or the very nature of the table itself may be subject to periodical change, and yet to one which has a regular law.
Some of these subsidiary powers are peculiarly applicable to calculations required in astronomy, and are therefore of eminent and immediate practical utility: others there are by which tables are produced, following the most extraordinary, and apparently capricious, but still regular laws. Thus a table will be computed, which, to any required extent, shall coincide with a given table, and which shall deviate from that table for a single term, or for any required number of terms, and then resume its course, or which shall permanently alter the law of its construction. Thus the engine has calculated a table which agreed precisely with a table of square numbers, until it attained the hundred and first term, which was not the square of 101, nor were any of the subsequent numbers squares. Again, it has computed a table which coincided with the series of natural numbers, as far as 100,000,001, but which subsequently followed another law. This result was obtained, not by working the engine through the whole of the first table, for that would have required an enormous length of time; but by showing, from the arrangement of the mechanism, that it must continue to exhibit the succession of natural numbers, until it would reach 100,000,000. To save time, the engine was set by the hand to the number 99999995, and was then put in regular operation. It produced successively the following numbers.[14]
99,999,996
99,999,997
99,999,998
99,999,999
100,000,000
100,010,002
100,030,003
100,060,004
100,100,005
100,150,006
&c. &c.
Equations have been already tabulated by the portion of the machinery which has been put together, which are so far beyond the reach of the present power of mathematics, that no distant term of the table can be predicted, nor any function discovered capable of expressing its general law. Yet the very fact of the table being produced by mechanism of an invariable form, and including a distinct principle of mechanical action, renders it quite manifest that some general law must exist in every table which it produces. But we must dismiss these speculations: we feel it impossible to stretch the powers of our own mind, so as to grasp the probable capabilities of this splendid production of combined mechanical and mathematical genius; much less can we hope to enable others to appreciate them, without being furnished with such means of comprehending them as those with which we have been favoured. Years must in fact elapse, and many enquirers direct their energies to the cultivation of the vast field of research thus opened, before we can fully estimate the extent of this triumph of matter over mind. 'Nor is it,' says Mr Colebrooke, 'among the least curious results of this ingenious device, that it affords a new opening for discovery, since it is applicable, as has been shown by its inventor, to surmount novel difficulties of analysis. Not confined to constant differences, it is available in every case of differences that follow a definite law, reducible therefore to an equation. An engine adjusted to the purpose being set to work, will produce any Page:Edinburgh Review Volume 59.djvu/323 Page:Edinburgh Review Volume 59.djvu/324 Page:Edinburgh Review Volume 59.djvu/325 Page:Edinburgh Review Volume 59.djvu/326 Page:Edinburgh Review Volume 59.djvu/327 Page:Edinburgh Review Volume 59.djvu/328 Page:Edinburgh Review Volume 59.djvu/329 Page:Edinburgh Review Volume 59.djvu/330 Page:Edinburgh Review Volume 59.djvu/331 Page:Edinburgh Review Volume 59.djvu/332 Page:Edinburgh Review Volume 59.djvu/333 Page:Edinburgh Review Volume 59.djvu/334 Page:Edinburgh Review Volume 59.djvu/335 Page:Edinburgh Review Volume 59.djvu/336 Page:Edinburgh Review Volume 59.djvu/337 Page:Edinburgh Review Volume 59.djvu/338 perceive the inference which the world will draw from this coui-se of conduct ? Does he not see that they will impute it to a dis- trust of his own power, or even to a consciousness of his own inability to complete what he has begun ? We feel assured that such is not the case ; and we are anxious, equally for the sake of science, and for Mr Babbage's own reputation, that the mys- tery — for such it must be regarded — should be cleared up ; and that all obstructions to the progress of the undertaking should immediately be removed. Does this supineness and apparent inditference, so incompatible with the known character of Mr Babbage, arise from any feeling of dissatisfaction at the existing arrangements between himself and the Government ? If such be the actuail cause of the delay, (and we believe that, in some degree, it is so,) we cannot refrain from expressing our surprise that he does not adopt the candid and straightforward course of declaring the grounds of his discontent, and explaining the ar- rangement which he desires to be adopted. We do not hesitate to .say, that every reasonable accommodation and assistance ought teibe afforded him. But if he will pertinaciously abstain from this^- to btir minds, obvious and proper course, then it is surely the duty of Government to appoint proper persons to enquire into and report iqjon the j^resent state of the machinery; to ascertain the causes bf its suspension ; and to recommend such measures as may appear to be most effectual to ensure its speedy completion. If th,ey do not by such means succeed in putting the project in a state of advancement, they will at least shift from nthenaselves tall responsibility for its suspension. i i/r rju.; lu -jm^c .: 1 :: — ".^'"'^ '■*•-> , , ..,,,;j J .y^'-^^'- /Wi!>i-io:.'-'i >■ -^iriov, bi:QA: naioqc bcil I nod'// ' Ahhey ; a Metrical Rom^np^. .J^^ 8vo. London: 1834.' ■;'; ;'.A^^.i:/(;V'«<'^^cai>. .V •'- "DooKSELLERS are certainly a pecuKar people, and do venture to play very fantastic tricks before the public. Here are two volumes given to the world-— as if for the first time— with- out a hint of their having ever appeared before — bearing with all solemnity the date of 1834 on the titlepag'e ; and yet these self- same sheets were printed and published in 1826. So palpable indeed is the patchwork, that w'hat ought to be the first page of the first volume, is actually page ninety-first ; the truth beings that all these poems were appended to Mrs, Radcliffe's posthuy, VOL. Lix. NO. cxxr^. ,_| ^ or;i,ts>8o .it/wfliom J3 l^t b98;jBq -won sH '
1. Watt commenced his investigations respecting the steam-engine in 1763, between which time, and the year 1782 inclusive, he took out several patents for improvements in details. Bolton and Watt had expended the above sum on their improvements before they began to receive any return.
2. Dodson's Calculator. 4to. London: 1747.
3. Hutton's Tables of Products and Powers. Folio. London; 1781.
4. The squares of the sines of angles are extensively used in the calculations connected with the theory of the tides. Not aware that tables of these squares existed, Bouvard, who calculated the tides for Laplace, underwent the labour of calculating the square of each individual sine in every case in which it occurred.
5. Report of the Committee of the Astronomical Society prefixed to the Nautical Almanac for 1834.
6. A comparison of the results for 1834, will be found in the Nautical Almanac for 1835.
7. These tables were never published. The printing of them was commenced by Didot, and a small portion was actually stereotyped, but never published. Soon after the commencement of the undertaking, the sudden fall of the assignats rendered it impossible for Didot to fulfil his contract with the government. The work was accordingly abandoned, and has never since been resumed. We have before us a copy of 100 pages folio of the portion which was printed at the time the work was stopped, given to a friend on a late occasion by Didot himself. It was remarked in this, as in other similar cases, that the computers who committed fewest errors were those who understood nothing beyond the process of addition.
8. Thus suppose the number expressed at full length were 3.1415927. If the table extend to no more than four places of decimals, we should tabulate the number 3.1416 and not 3.1415. The former would be evidently nearer to the true number 3.1415927.
9. Memoirs Ast. Soc. vol, iii,; p. 65.
10. Memoirs Ast. Soc. vol. iv., p. 290.
11. Recueil des Tables Logarithmiques et Trigonometriques. Par J. C. Schulze. 2 vols. Berlin: 1778.
12. The wheels, and every other part of the mechanism except the axes, springs, and such parts as are necessarily of steel, are formed of an alloy of copper with a small portion of tin.
13. The machine was constructed with the intention of tabulating the equation ${\displaystyle \Delta ^{7}{\underset {z}{u}}=0}$, but, by the means above alluded to, it is capable of tabulating such equations as the following; Δ7u=aΔu,Δ7uaΔ3u,Δ7u=units figure of Δu.
14. Such results as this suggest a train of reflection on the nature and operation of general laws, which would lead to very curious and interesting speculations. The natural philosopher and astronomer will be hardly less struck with them than the metaphysician and theologian.
|
2019-07-20 14:44:03
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5728117227554321, "perplexity": 1076.685979411678}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526517.67/warc/CC-MAIN-20190720132039-20190720154039-00007.warc.gz"}
|
https://quant.stackexchange.com/questions/57624/deriving-american-option-greeks
|
# Deriving American option greeks
I am using integral representation of option value instead of trees, so I imagine to derive greeks we have to integrate across time for the boundary to get the EEP (Early Exercise Premium) component of greeks + the european greeks with respect to spot/vol/time. The issue with this method is there is no discontinuity of greeks I have derived beyond the boundary as in this example. What is the best way of approaching this? Thanks.
|
2020-09-27 07:53:13
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8728712201118469, "perplexity": 712.8922067152861}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400265461.58/warc/CC-MAIN-20200927054550-20200927084550-00000.warc.gz"}
|
https://brilliant.org/problems/oscillations-of-a-charged-arc/
|
# Oscillations of a Charged Arc
An arc with radius R has a uniform positive charge density $\lambda$ exists as shown.The arc of mass M is initially in equilibrium due to its weight and electrostatic force of interaction between a fixed charge at its centre [mass m charge Q] The arc is then displaced from the mean position a very small distance as compared to the radius R, along the symmetrical axis of the arc.It undergoes Simple harmonic motion under certain approximations. Consider gravity constant everywhere .Find the $M\frac{\frac{arccos(\frac{1}{\sqrt{3}})}{2}+\sqrt{2}}{\frac{arccos(\frac{1}{\sqrt{3}})}{2}+\frac{1}{\sqrt{2}}}$ upto two decimal in SI units of the arc given that the time period of oscillations has a minimum value of$2\pi$ seconds.Given that $Q=\frac{\lambda}{R^{2}}=\sqrt{(2\pi\varepsilon)}$ Consider all forces of gravitation constant throughout considering only values in SI and not the units.
×
|
2017-07-25 08:54:43
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8926904201507568, "perplexity": 521.4147968300397}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549425117.42/warc/CC-MAIN-20170725082441-20170725102441-00192.warc.gz"}
|
https://codereview.stackexchange.com/questions/176702/translating-status-codes-to-messages-in-log-files
|
Translating status codes to messages in log files
I'm not overly efficient when it comes to coding, and I am looking for ways to improve the speed of a log translator I built.
Several files are uploaded to a folder on a server and log files are selected for processing. Speed here is fine. Each log file has 5000-10000 lines possible a bit higher. They are space delimited files that look like this:
20:04:13 + MA 00 61
20:04:17 - MA 00 61
20:04:18 + MA 00 61 optionaltxt
20:04:20 - MA 00 61
There are at least 5 fields for every entry but more can be added if optional text exists. Field 5 is a status code, and it has to be cross referenced with a language file then replaced with the corresponding message so the result is like this:
17:29:48 - MA 00 (061) Feed hold! X / Y axis is locked
17:29:48 + MA 00 (061) Feed hold! X / Y axis is locked
17:29:50 - MA 00 (061) Feed hold! X / Y axis is locked optionaltxt
17:29:51 + MA 00 (061) Feed hold! X / Y axis is locked
The translated logs are saved to a new folder for download later. At the moment translation takes about 7-10 seconds per log file. The problem is that it is common to have 90 files or more and the time starts adding up. I have no control over the log format or language file format.
This part of my code looks like this:
$shLANG = file_get_contents("ErrCODES/data_en.properties");$SHarray = explode("\n", $shLANG);$logARRAY = glob($uploadDIR.$filepath."/original/*.[lL][oO][gG]");
foreach ($logARRAY as$logFILEarray) {
$name = basename($logFILEarray);
$logFILE = file_get_contents($uploadDIR.$filepath."/original/".$name);
$logFILEarray = explode("\n",$logFILE);
foreach ($logFILEarray as$key => $line) {$newline="";
$LINEarray = explode(" ",$line);
if(isset($LINEarray[0])){$newline .= $LINEarray[0]." ";} if(isset($LINEarray[1])){$newline .=$LINEarray[1]." ";}
if(isset($LINEarray[2])){$newline .= $LINEarray[2]." ";} if(isset($LINEarray[3])){$newline .=$LINEarray[3]." ";}
if(isset($LINEarray[4])){ foreach ($SHarray as $code) { if (strlen($LINEarray[4])>0){
if (substr($code, 0, strpos($code, '=')) == $LINEarray[4]) {$newline .= trim(explode('null;', $code)[1]). " "; } } } } if(isset($LINEarray[5])){$newline .=$LINEarray[5]." ";}
if(isset($LINEarray[6])){$newline .= $LINEarray[6]." ";} if(isset($LINEarray[7])){$newline .=$LINEarray[7]." ";}
if(isset($LINEarray[8])){$newline .= $LINEarray[8]." ";}$newline .= "\r";
$logFILEarray[$key] = $newline; } //var_dump($logFILEarray);
$info = implode("\n",$logFILEarray);
file_put_contents($uploadDIR.$filepath."/translation/".$name,$info);
}//for all glob .log
The code works fine, but how can I improve the speed (greatly)?
• I see some stylistic improvements, but, since you asked for speed: 1) have you tried profiling the code to see where it's slow?, 2) instead of file_get_contents and explode, have you considered reading one line at a time (this may actually be slower, though, so just asking, not suggesting). Finally, does it have to be PHP? Unix command line tools may work faster.
– user1149
Sep 28 '17 at 12:13
2 Answers
First thing, I would prepare the errorCodes array: After
$SHarray = explode("\n",$shLANG);
Add:
$errors = []; foreach($SHArray as $line) { list($code, $message) = explode('=',$line);
$errors[$code] = $message } Note: it is not clear from your code what additional processing is needed to get an associative array of$code => $message, you will need to add that yourself. So instead of repeated substring search: foreach ($SHarray as $code) { if (strlen($LINEarray[4])>0){
if (substr($code, 0, strpos($code, '=')) == $LINEarray[4]) {$newline .= trim(explode('null;', $code)[1]). " "; } } } you will have foreach($errors as $code =>$message) {
if($code ==$LINEarray[4]) {
$newline .=$message . " ";
}
}
Another thing: instead of linear processing of each log field, I would update $lineArray[4], then re-assemble the array using implode, without processing other fields at all, if they do not need to be changed. Note that it may not increase productivity, but will improve code readability and size. Like this: $lineArray = explode(' ', $line); // use a separate function/method to translate one entry$lineArray[4] = translateErrorMessage($lineArray[4],$errors);
$newLine = implode(' ',$lineArray);
Finally, With speed and productivity, you need to consider not only your code, but other system factors as well.
• Which version of php are you using? PHP 7 may give a speed benefit over older 5.xx
• Are you using standalone php (CLI) or run as a web server? On large files, web server may fail with a timeout, also it is no good to run such tasks with full stack, as it consumes way too much memory.
• Is it possible to install some PHP pre-compiler with your system? These usually save time, as pre-compiled code works much faster.
• Thank you @TimSparrow the associative array was the key part. Now the logs convert in under 1 second, over 10 months can be done as a batch before the timeout which is plenty. It's actually the first time that I've made an associative array so this is very good experience. Oct 23 '17 at 6:42
• @Nama still, for batch processing, I would recommend running it as a background CLI task, so that a timeout would never be an issue. Oct 23 '17 at 10:00
What you have looks problematic in a few ways.
• You have a poor data structure for your localization strings. You have these in an array the must be iterated every single time you are doing a lookup against it. This is an linear complexity (O(n)) operation that when executed inside your loop (another O(n) iteration) give you O(n^2) overall performance. @TimSparrow has touched on a good alternative - to load the localization strings into an associative array the will allow O(1) lookup when being used - but perhaps didn't explain how critical this is to your performance. By building such a data structure up front, you whole algorithm goes to O(m) (to build localization data structure) + O(n) (from your log file loop) instead of O(m * n). To illustrate the impact to your performance, let assume you have a 10K line log file and a 100 line localization file. With your current code, this means you could have up to 1 Million lookup operations you have to perform vs. 10,100 if using a proper data structure.
• This code is a memory hog. You store every file you are working with wholly in memory, as well as storing the same data in duplicate in the result arrays you are building, not to mention all the temporary array/string conversion you have that, again mean you are going to be holding that information in duplicate in memory. If you can get the memory utilization more optimized here, then you may be more able to do things like for each of the individual log into their own process to parallelize the work.
• You should consider working with the rows in a more structured manner vs. working with them primarily as strings. fgetcsv(), fputcsv() or similar give you better ways to parse through and write your files.
Incorporating the above thoughts should allow you to greatly streamline both the performance of your script as well as just simplify how it works. You might end up with something like:
$localizations = [];$handle = fopen('ErrCODES/data_en.properties','r');
// read lines from file one at a time.
while ($line = fgetcsv($handle, 0, ' ')) {
// this part may vary based on your format
$localizations[$line[0]] = $line[1]; } fclose($handle);
$logFiles = glob($uploadDIR.$filepath."/original/*.[lL][oO][gG]"); // here is where at some point you may want to fork processes to paralellize // now it is serial foreach loop foreach ($logFiles as $logFile) {$translatedFile = str_replace('/original', '/translation/', $logFile);$logHandle = fopen($logFile,'r');$translateHandle = fopen($translatedFile, 'w+'); while($line = fgetcsv($logHandle, 0, ' ') { // apply localization if(!empty($line[4]) && isset($localization[$line[4]])) {
$line[4] =$localization[$line[4]]; } fputcsv($translateHandle, $line, ' '); } fclose($translateHandle);
fclose($logHandle); } A few final thoughts on style: - Your style is a little bit all over the place, which makes your code a littel hard to read. - Particularly variable naming is odd. PHP tends to use a mix of camelCase and snake_case in most code bases you will find (internal PHP functions mix both unfortunately). Your approach to variable syntax is very haphazard, sometimes oddly putting portions of variable names in upper case with apparent cohesive approach, sometimes first letter is uppercase, sometimes it is lower case, etc. - Your indentations are inconsistent. - You could be well-served by better use of spacing around flow control operators. A line of code like if(isset($LINEarray[0])){$newline .=$LINEarray[0]." ";} is extremely hard to read. - Consider familiarizing yourself with the PHP-FIG PHP standards, particularly PSR-1 and PSR-2 which deal with aspects of coding style. These are pretty much the de-facto industry standards.
|
2021-10-19 06:31:39
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21001017093658447, "perplexity": 3050.390566577222}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585242.44/warc/CC-MAIN-20211019043325-20211019073325-00651.warc.gz"}
|
https://ajft.org/1996/12/02/journal.html
|
## 02-Dec-96, Hit with object thrown from car @ Adrian Tritschler · Monday, Dec 2, 1996 · 1 minute read · Update at Dec 2, 1996 ·
Riding west along Waverley road after leaving Jells Park at about 6 pm I suddenly was belted on the left hip and buttock with an object, the shock made me swerve, hit the gutter, and swerve back out into the traffic, I looked up to see four or five teenage guys in a blue Commodore with P-plates (Victorian registration ###-###) laughing and swearing out the windows. The object was some form of water-bomb, I believe a plastic juice bottle.
When I got home I decided to report it to the local police – Murrumbeena Station – and was greeted with disinterest, officer A. Stewart took the details on a scrap of notepaper, looked up the regoistration and said that since the car was registered to someone near where it had happened he would give them a call and then phone me the following day. I never received a follow up phone call.
Refs: Originally in my bike incidents page.
# …The Owner
There’s not much more I can add to who I am.
# …The Site
I experiment. I play. I write and I take pictures. Some of the site is organised around topics, other parts are organized by date, then there’s always the cross-references between them.
Long ago it started as a learning experiment with a few static HTML pages, then I added a bit of server-side includes and some very ugly PHP. A hand-built journal/blog on top of that PHP, then a few experiments in moving to various static publishing systems. I’ve never wanted a database-based blogging engine, so over the years I’ve tried php, nanoblogger, emacs-muse, silkpage and docbook before settling on emacs org-mode for writing and jekyll for publishing. But the itch remained… I never really liked jekyll and the ruby underneath always seemed so much black magic. So now the latest incarnation is org-mode and hugo.
|
2020-11-29 16:03:02
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23883631825447083, "perplexity": 6075.907251775241}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141201836.36/warc/CC-MAIN-20201129153900-20201129183900-00425.warc.gz"}
|
http://www.askamathematician.com/2012/09/q-what-are-singularities-do-they-exist-in-nature/
|
# Q: What are singularities? Do they exist in nature?
Physicist: Singularities are just artifacts that fall out of math. They show up a lot in theory, and (probably) never in nature. The “singularities” most people have heard of are black hole singularities.
In practice, when you’re calculating something in physics and you find a singularity in your calculation (this happens all the time), which usually looks like “1/x”, then that means that there’s a mistake somewhere, or you’re looking at something that never happens, or there are physical laws or effects that haven’t been taken into account.
The simplest mathematical singularity: 1/x. The closer you get to zero, the more 1/x blows up, and at x=0 the function is undefined.
For example, when you drain water from a tub or sink the water will spiral into the drain. A back of the envelope calculation (based on known principles) shows that the speed, s, that the water is moving is $s=\frac{c}{r}$, where r is the distance to the center of the spin, and c is a constant that has to do with how fast the water was turning before you pulled the plug. Notice that this equation implies that as water gets closer and closer to the drain, it will move faster and faster, and that right over the drain it will be moving infinitely fast. So how does the universe find an out? This will look familiar:
One of the slick tricks that water uses to avoid spinning infinitely fast at the center of a vortex: not being there.
Even when you’re deep underwater there are outs; turbulence, cavitation, that sort of thing.
A slightly more obscure example is the energy of a charged particle’s electric field. If you have an electron just sitting around, the energy, E, of its field outside of a distance, R, from the electron is $E =\frac{e^2}{8\pi\epsilon_0}\frac{1}{R}$.
The electric field being considered is the light blue area. The most intense part of the field is close to the electron, so as R gets smaller and smaller, the total energy gets bigger and bigger.
Most of that is just equation porn. The important bit is the “$\frac{1}{R}$“. Once again there’s a mathematical singularity in an equation describing a physical thing. But, once again, the universe (being sneaky) finds a way out. This is a hair less intuitive than the whirlpool thing, but in quantum mechanics an electron is described as being “smeared out” (in an uncertainty principle kind of way). It doesn’t exist in any one place, so the idea of getting infinitely close doesn’t really make sense.
The whirlpool thing and the electron thing (and hundreds of other examples) are examples of singularities that show up in the math, but can be explained away through experiment and observation, and shown to not be singularities in the physical world.
In general relativity, the shape of spacetime near a spherical mass is given by:
$c^2 d\tau^2 = \left(1-\frac{r_s}{r}\right)c^2dt^2 - \left(1-\frac{r_s}{r}\right)^{-1}dr^2 - r^2\left(d\theta^2 + \sin^2{(\theta)}d\phi^2\right)$
Now, unless you’re already a physicist, none of that should make any sense (there are reasons why it took Einstein 11 years to publish general relativity). But notice that, as ever, there’s a singularity at r=0. This is the vaunted “Singularity” inside of black holes that we hear so much about.
Something like a star or a planet doesn’t have a singularity, because this equation becomes invalid at their surface (this equation is about the empty space around a mass). But, for a black hole the gravity is so intense, and spacetime is messed up so much that it looks as though there’s nothing to stop matter from becoming infinitely dense. However, unlike the singularity over the drain in your sink, the singularity in a black hole can’t be observed. Which is frustrating.
I suspect that what we call the singularity in black holes either doesn’t exist (there is some law/effect we don’t know about) or, if cosmic censorship is true, the nature of that singularity both doesn’t matter and can’t be known, since it can never interact with the rest of the universe. There are some theories (guesses) that would fix the whole “black hole singularity problem” (like spacetime can only get so stretched, or some form of “quantum fuzziness”), but in all likelihood this is just one of those questions that may never be completely resolved.
The whirlpool photo is from here.
This entry was posted in -- By the Physicist, Math, Philosophical, Physics. Bookmark the permalink.
### 10 Responses to Q: What are singularities? Do they exist in nature?
1. angel says:
Great article, I’m not physicist but always enjoy the manner that attack the problems, in particular the form so fluid that the article was written, I try to incorporate that style when do my document, unfortunatly I only have read this style in english and have some problems to incorporate in spanish xd.
Anyway thanks for the great article
2. qubit says:
Interesting and very informative article. I think the energy equation is incorrect. It’s only epsilon not epsilon square.
3. The Physicist says:
You are completely correct. Fixed!
4. Russ says:
I came here looking for an answer about black holes that involves several singularities I suppose. Initially I thought that black holes have a singularity where all the matter occupies the same point in space. Given that premise, I was confused about the spin of a black hole. If a black hole inherits its spin from the star it was formed from and spins faster due to conservation of angular momentum, then wouldn’t the singularity have an infinatly fast spin? Since there are black holes with “not infinite” spin, then does that imply that the matter didn’t collapse to a single point, or is it, like in the article, something else we don’t understand going on? And finally, I’d we could observer the spin of a star before it went supernovae and the spin of the resulting black hole, what would you suppose the difference would tell us about the interior if anything?
5. Mickey says:
I have a couple questions about the supposed singularity in a black hole and the curvature of space time:
1) What is curved space-time? Time is curving with regards to a spatial dimension? If so does a particle traveling to a center of a blackhole with a finite speed actually reach a singularity or just move forward in time?
2) Wouldn’t a distribution of mass over curved 4 dimensional space make it impossible to have infinite density if a most of the mass were at different time coordinates?
3) If Hawking radiation is correct doesn’t that seem to suggest intuitively that there is no singularity in a black hole?
6. If Singularity does exist in nature it will eventually break the law of conservation of Mass-Energy, conservation of charge etc… of the Universe, because nothing can escape from Singularity. and moreover how can a zero volume can occupy Infinite density, does it make sense at all…..
7. If Singularity does exist in nature it will eventually break the law of conservation of Mass-Energy, conservation of charge etc… of our Universe, because nothing can escape from Singularity. And moreover how can a zero volume can occupy Infinite density, does it make sense at all ? Thus all the Matter entering into the singularity will end up there. Is it necessary that a ” Black-Hole” should have a Singularity ? or Is a Singularity is just a mathematical frame-work and might not exist in realty. There is no, proper definition of Singularity. Can’t we construct a Black-Hole model which doesn’t require any sort of Singularity…….?
8. Tyler Parker says:
When you spin a quarter, right before it stops the quarter’s axis of symmetry’s rotation relative to a central axis is unbounded. I had a professor who verified this with a couple grad students using a vacuum chamber, laser, reflective disk and really smooth surface.
This is the closest i’ve thing I heard of to an actual physical finite-time singularity.
http://www.ncbi.nlm.nih.gov/pubmed/12443243
9. jake the snake says:
If space is infinitely divisible then there is no smallest size. The volume cannot reach zero. Just like you can’t reach infinity. The black hole will just shrink forever and never attain zero volume. If it always has a volume there can be no singularity.
10. jason says:
It sounds like your saying that a singularity is like water down the drain. Now depending on which hemisphere your in water spins right or left. However if everything that goes into a singularity if it stays in this plane of reality . It would mean that so much is compressed into one point making pretty impossible. However like a drain . Do singularity’s empty into another plane of existence ? Or is it so heavy that it able to just keep packing material in one space?
|
2015-04-28 15:55:02
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 4, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7379989624023438, "perplexity": 602.2843529579557}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246661733.69/warc/CC-MAIN-20150417045741-00165-ip-10-235-10-82.ec2.internal.warc.gz"}
|
https://www.numerade.com/questions/which-of-the-integrals-displaystyle-int05_0-cos-x2-dx-displaystyle-int05_0-cos-sqrtx-dx-is-larger-wh/
|
💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!
# Which of the integrals $\displaystyle \int^{0.5}_0 \cos (x^2) \,dx$, $\displaystyle \int^{0.5}_0 \cos \sqrt{x} \,dx$ is larger? Why?
## $\int_{0}^{0.5} \cos x^{2} d x$ is bigger
Integrals
Integration
### Discussion
You must be signed in to discuss.
Lectures
Join Bootcamp
### Video Transcript
Okay, we know that from 0.5, this court of X has to be greater than or equal to x squared. Therefore, we know that because co sign X is decreasing from 0.5, you know that co sign squared of acts has to be less than or equal to co sign of X squared. Therefore, we know that the integral from 0.5 co sign X squared d X has to be very correlates to this.
Integrals
Integration
Lectures
Join Bootcamp
|
2021-09-21 10:30:15
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3575610816478729, "perplexity": 2322.18078776023}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057202.68/warc/CC-MAIN-20210921101319-20210921131319-00545.warc.gz"}
|
https://www.lesswrong.com/posts/DMxyPbEhNNf4FTof9/a-hierarchy-of-abstraction
|
# 5
Intelligence seems related to one's ability to handle high-level abstractions. A minimum intelligence level is necessary to understand certain kinds of things. This post attempts to order concepts by how much intelligence is required to understand them.
I had an employee who could do #1-2 but struggled with #3. I know several people who can do #1-4 just fine but struggle with #5 or cannot do #5 at all. I know people who can do #1-5 but cannot do #6 at all. I know an engineer who can do #1-6 but cannot do #7. I know another engineer who can cruise through #1-8 but struggles with #9.
I have never[1] observed someone go up a level in this hierarchy. Someone who does not grasp a level quickly tends never to grasp it at all. I think your potential to climb this hierarchy is limited by your fluid intelligence.
### 1. Can imitate simple behavior
Anyone reading this post passes this threshold.
### 2. Can follow simple verbal instructions
Anyone reading this post passes this threshold.
### 3. Can follow simple written instructions
Anyone reading this post passes this threshold.
Can solve questions of the following form: "Given a $100,000 initial investment with 4% compounding interest, estimate your profit after 10 years." The testee is allowed to use a calculator. ### 5. Can write Python scripts Anyone who has written original computer code in a mainstream programming language passes this threshold. ### 6. Understands pointers Anyone who can implement a pointer-based algorithm in C passes this threshold. ### 7. Understands recursion Can implement the Fowler–Noll–Vo hashing algorithm recursively and explain why one would do so. ### 8. Understands information density Can crack all three of these problems in an hour or less. Confidently asserting an incorrect answer to the first question fails this test instantly. The testee may use the Internet to look up details of the Java language spec and how the Enigma machine works. This research does not count towards the hour. Otherwise, using the Internet is forbidden. ### 9. Thinks in terms of entropy Can invent unbiased small data[2] algorithms on-the-fly. People with an intuitive grasp of hypothesis space Kolmogorov complexity tend to make good Wall Street quants. ### 10. ??? Here be dragons. 1. I managed to tutor a computer science student from #5 to #6—but I do not think it counts. Ostensibly, she hired me to teach her how pointers work. The real problem was that her college had cancelled the prerequisite class that was supposed to teach her C syntax. She fired me as soon as I had finished explaining the syntax. ↩︎ 2. Edit: There used to be a hyperlink here to a sequence of posts that explained what "small data" is. This hyperlink is now broken. ↩︎ # 5 22 comments, sorted by Highlighting new comments since New Comment The title is 'A Hierarchy of Abstraction' but the article focuses on levels of intelligence. The article claims that intelligence positively correlates with the ability to handle high level abstractions, but it does not talk about actual hierarchies of abstraction. For example, I'd expect a hierarchy of abstraction to contain things like: concrete objects, imagined concrete objects, classes of concrete objects, concrete processes, simulated processes, etc. A more accurate title might be 'The Ability to Understand and Use Abstractions in Computer Science as a Measure of Intelligence.' The article lays out a way of measuring fluid intelligence but does not decouple the crystallized intelligence requirements from the fluid ones. For example, 'Understands Recursion' specifies needing to implement a specific algorithm recursively as a requirement. There are plenty of people who understand and use recursion regularly who do not know that algorithm. (take me) Let's say you test them and they fail. Did they fail because of their fluid intelligence? Did they fail because of a lack of crystallized knowledge related to that specific problem? Did they fail because of abstraction capability requirements in that specific problem, but not recursion in general? What about recursion as a concept makes it hard for people to understand? I would recommend trying to generalize the requirements more. I would recommend exploring other possible attributions of failure other then fluid intelligence. If the article examined the components of recursion it would be more interesting and compelling. What are the components? Drilling down into the components of any of these tests will reveal a lot of context and crystallized knowledge that the article may be taking for granted. (curse of knowledge bias) You might be seeing someone struggle with recursion, and the problem isn't that they can't understand recursion, its that they don't have crystallized knowledge of a building block. As someone who understands recursion to a reasonable level, I'd like to see the article point at the key idea behind recursion that people have trouble grasping. Are there a sequence of words that the article can specify where someone understands what each word means, but finds the overall sentence ineffable? Or perhaps they can parrot it back, but they can't apply it to a novel problem. A requirement of this hypothesis is that someone has all prerequisite crystallized knowledge, but still cannot solve the problem. Otherwise these are not 'hard' boundaries of fluid intelligence. I guess you primarily deal with computers and programming. One way to try and generalize this quickly would be to compare notes across disciplines and identify the pattern. Is there a 'cannot learn pointers' in chemistry for example? I understand that you are trying to share the gist of an idea, but I think these are things that should be further examined if you want other people to take on this mental model. Much more needs to be said and examined in an article that lays out 10 specific levels with specific tests. I'd also be wary of the possibility this entire framework / system looks good because it positions your tribe as superior (computer programmers) and possibly you somewhere comfortably towards the top. This article triggered me emotionally because I think one of the things that prevents people from learning things is the belief that they can't. I wouldn't want anyone to take away from this article that because they didn't understand pointers or recursion at some point in there life, it was because they are dumb and should stop trying. As I pointed out in my other comments, these levels are about potential, not ability. The question are not really about "Can you answer ?" They are more along the lines of "Can you easily learn to answer ?" I believe this decouples "the crystallized intelligence requirements from the fluid ones". What about recursion as a concept makes it hard for people to understand?…I'd like to see the article point at the key idea behind recursion that people have trouble grasping. Recursion requires a person to hold two layers of abstraction in zeir mind simultaneously. This could require twice as much working memory. Working memory, unlike crystallized intelligence, is something we cannot do much to improve. Is there a 'cannot learn pointers' in chemistry for example? I have never tutored chemistry. But I have tutored physics for years. I have never run into anything like "cannot learn pointers" in standard undergraduate physics. The only time I encounter anything like "cannot learn pointers" is when I discuss my weird crackpot theories of relational quantum gravity and entropic time. The ideas involved in undergraduate computer science are simpler than the ideas involved in undergraduate physics. The difference between undergraduate computer science and undergraduate physics is that undergraduate physics is a set of solutions to a set of problems. It requires little creativity. You can (relatively speaking) learn everything by rote. While fluid intelligence helps you learn rote knowledge faster, fluid intelligence tends not to put a hard cap on total rote knowledge acquisition. On the other hand, computer science requires a person to solve novel (albeit simpler) problems all the time. It makes sense one's ability to do this could be limited by fluid intelligence. The other difference between computer science and physics is that in physics you never have to hold two layers of abstraction in your head at the same time. It makes sense that holding two layers of abstraction in your head at the same time could be limited by working memory. One's ability to solve novel problems is the definition of fluid intelligence. Fluid intelligence is correlated with working memory. My brief forays into chemistry suggest chemistry involves more rote knowledge and less "juggling multiple layers of abstraction" than computer science does. Chemistry, like all subjects, is limited by intelligence at some level. But I would expect the floor is lower than physics and computer science. I'd also be wary of the possibility this entire framework / system looks good because it positions your tribe as superior (computer programmers) and possibly you somewhere comfortably towards the top. Really? I do not know a single computer programmer who meets #9. I know only one who passes #8. Most of my programmer friends do not even pass #7. This framework puts quants on top. Not computer programmers. I do not think anyone would doubt that quants are smart people—at least in the sense relevant to this post. Computer programmers tend to be superior at computer programming (and perhaps related fields). This article triggered me emotionally because I think one of the things that prevents people from learning things is the belief that they can't. I wouldn't want anyone to take away from this article that because they didn't understand pointers or recursion at some point in there life, it was because they are dumb and should stop trying. Among the most important things I have learned from Less Wrong is the value of epistemic rationality over instrumental rationality. It breaks my heart that certain people seem unable to perform certain tasks I consider trivial. The greater cruelty is blaming someone for their failure to achieve a goal zey lack the potential to accomplish. You seem to be making a few claims: (1) that these skills require an increasing amount of 1-dimensional intelligence (2) that one cannot do lower-indexed things without doing higher-indexed ones and (3) that there is something fundamental about this. You obviously do not mean this literally, for there are plenty of people who understand recursion but not pointers (i.e.: intro Python students), and plenty of programmers who have never touched Python. First, what is an abstraction? As someone who recently wrote a paper involving Cousot-style abstract interpretation, this is a question that interests me. We in programming-languages research have our answer: that an abstraction is a relation between two spaces such that every behavior in the larger ("concrete") space has a corresponding behavior in the smaller "abstract" space. In this definition, there is a sensible meaning of a hierarchy of abstraction, but it's not what most people think: tracking arbitrary sets of integers is less abstract than tracking intervals of integers, which are in turn less abstract than tracking the possible sign(s) of a number. I am less familiar with abstraction as it is used by AI researchers; my understanding is that it is similar, but without the strict requirements, e.g.: they'll be willing to analyze poker rounding all bets to multiples of$5, and say that two bets of something that rounds to $5 = 1 bet of something that rounds to$10 bet, rather than "something that rounds to $5,$10, or \$15", as would the PL researcher.
Most software engineers seem to use "more abstract" to mean "can be implemented using" or "can be defined using," e.g.: the home screen on my phone is an abstraction built on top of the file system, which is an abstraction built on top of the hardware. This seems to be the closest to what you mean. In that sense, I cannot see what makes these examples a hierarchy, or fundamental in any sense. E.g.: recursion clearly does not require arithmetic. The lambda calculus has recursion but not arithmetic.
The best I can make of this post is that these tasks have something akin to a hard-floor in g-factor required, which is an extraordinary claim in need of extraordinary evidence.
I appreciate your genuine attempt to understand this post. Python is indeed a stand-in for any programming language of comparable complexity. Pointers and recursion could indeed be flipped around. I think we are on the same page concerning these details.
You seem to be making a few claims: (1) that these skills require an increasing amount of 1-dimensional intelligence (2) that one cannot do lower-indexed things without doing higher-indexed ones and (3) that there is something fundamental about this.
1. Yes.
2. Not quite. There can be a little slippage between close-together indices, especially #6 & #7. More importantly, this hierarchy is about talent and potential, not ability. If someone could do #5 with a little bit of study then that counts as doing #5. It is easy to imagine a mathematician who understands recursion but has never written a line of computer code. But I would be surprised if such a person could not quickly learn to write simple Python scripts.
3. This is technically an implication, not a claim. But…yeah.
First, what is an abstraction?
By "abstraction", I mean a simplification of the universe—a model. By "higher-level abstraction", I mean a model built on top of another model.[1]
I am not referring to how these models are built upon one another within mathematics. I am trying to organize them based on how they are constructed inside a typical human brain. For example, a human being learns to write before learning arithmetic and learns arithmetic before learning calculus. It is possible to construct calculus from the Peano axioms while skipping over all of arithmetic. But that is not how a human child learns calculus.
To put it another way, a human being learning a complex skill begins by learning to perform the skill consciously. With practice, the skill becomes unconscious. This frees up the conscious mind to think on a higher (what I call more "abstract") level. Rinse and repeat. To use an example from Brandon Sanderson's creative writing lectures, a novice author is worried about how to put words down on a page while a skilled writer is thinking with plot structure. An author cannot afford to think about higher level activities until the lower level activities are automatic.
I hope this clarifies exactly what I mean by "abstraction".
The best I can make of this post is that these tasks have something akin to a hard-floor in g-factor required…
Yes. That is the point of this post.
…which is an extraordinary claim in need of extraordinary evidence.
Really? What makes you think this is an extraordinary claim? My prior is 50%/50% credence without evidence. Here are some observations I have used to subsequently update those priors.
• I am not the first person to point out that there is a hard floor on the intelligence required to understand pointers. Joel Spolsky wrote about the basic idea concerning pointers back in 2005. My college computer science teacher pointed out something similar in 2011.
• On the low end of the spectrum, I have cared for severely retarded adults who appeared to have -related ceilings. This is a sensitive example and I do not want to get too deep into it. But it should be clear that someone without enough to hold a coherent conversation is unlikely ever to perform arithmetic at the level specified in this post.
• Similarly, an adult who has trouble remembering how to press 3 buttons on a smartphone is unlikely ever to write computer code.
• On the high end of the spectrum, quants are regularly hired from physics departments. These people can apparently pick up finance faster than students trained in economics. The difference seems to be mathematical aptitude. When I talk to quants I can get away with explaining much less than when I talk to, say, a Silicon Valley CTO. At least one friend of mine has confirmed this evaluation.
• I have spent countless hours tutoring physics, computer science and other subjects. The levels in my post comes from my personal experience. When it comes to specific subjects it often feels like there's a block and my interlocutor cannot fit all of the relevant information in zeir head at once.
In the scientific literature it is established that traits like fluid intelligence and working memory are basically capped. If the ability to reason at certain levels of abstraction is capped by fluid intelligence or working memory then that would explain the phenomenon I have observed.
1. This might seem to contradict my claim about talent and potential. You can get around this contradiction by supposing the existence of horizontal alternatives for specific benchmarks. If these alternatives require similar potential to pass then they do not invalidate the ordering. ↩︎
Thanks for the explanation. I accept your usage of "abstraction" as congruent with the common use among software engineers (although I have other issues with that usage)). Confusingly, your hierarchy is a hierarchy in g required, not a hierarchy in the abstractions themselves.
I am well-read in Joel Spoelsky, and my personal experience matches the anecdotes you share. On the other hand, I have also tutored some struggling programmers to a high level. I still find the claim of a g-floor incredible. This kind of inference feels like claiming the insolubility of the quintic because I solved a couple quintics numerically and the numbers look very weird.
Sidenote: I find your example discussion of human learning funny because I learned arithmetic before writing.
the insolubility of the quintic
It's that the general form is unsolvable, not specific examples, without better tools than the usual ones: +, -, *, /, sqrt, ^, etc. I've heard that with hypergeometric functions it's doable, but the same issue reappears for polynomials of higher degree there as well.
This kind of inference feels like claiming the insolubility of the quintic because I solved a couple quintics numerically and the numbers look very weird.
I think it is more like the irreversibility of the trapdoor functions we use in cryptography. We are unable to prove mathematically they are secure. But an army of experts failing to break them is Bayesian evidence.
Sidenote: Lol.
Was just reading through my journal, and found that I had copied this quote. I think you'll find it to be of interest re: teaching recursion.
-------------------------------------
From “Computing Science: Achievements and Challenges” (1999):
“I learned a second lesson in the 60s, when I taught a course on programming to sophomores, and discovered to my surprise that 10% of my audience had the greatest difficulty in coping with the concept of re, cursive procedures. I was surprised because I knew that the concept of recursion was not difficult. Walking with my five-year old son through Eindhoven, he suddenly said "Dad not every boat has a life-boat, has it?" '"'How come?" I said. "Well, the life-boat could have a smaller life-boat, but then that would be without one." It turned out that the students with problems were those who had had prior exposure to FORTRAN, and the source of their difficulties was not that FORTRAN did not permit recursion, but that they had not been taught to distinguish between the definition of a programming language and its implementation and that their only handle on the semantics was trying to visualize what happened during program execution. Their only way of "understanding" recursion was to implement it, something of course they could not do. Their way of thinking was so thoroughly operational that, because they did not see how to implement recursion, they could not understand it. The inability to think about programs in an implementation-independent way still afflicts large sections of the computing community, and FORTRAN played a major role in establishing that regrettable tradition”
Significantly, your examples are all within the domain of analytical—and I would suggest, reductive—mathematics-science-coding. In my experience, it is often the case that one who has ascended this ladder is quite blind to, and unable to conceive of the importance of, context and perspective to meaning-making.
Apropos, there was a period during my childhood when I tried to question my elders regarding my observation that entropy, probability, and meaning were inherently subjective—meaningless without reference to an observer. Similar to the misnomer of Shannon "information" theory when it is actually about the transfer of data, rather than information through a lossy channel. In virtually every case, they could not understand my questions.
People generally fail to understand that "subjective" does not imply "arbitrary", but rather, perspectival and necessarily within context. And of course when it comes to agreement, we share a great deal of (hierarchical) perspective.
One concrete example: Back when I was a technical manager in the field of analytical instruments, I tried to convey to our support team that the key metric of success was customer satisfaction, followed, if necessary, by whether the instrument met standards for precision, sensitivity, stability, etc. Of the members of that team, the one who I knew and relied on to be the most rigorously analytical could not accept that definition of success, and was therefore weaker (overall) in terms of achieving customer satisfaction.
As for your potential item #10—remaining within the analytical paradigm you have established—I would suggest "the ability to think in terms of evolutionary processes and strategies, especially in regard to production of meaningful novelty."
Similar ladders exist in other domains—drawing, for example. But I have not observed such hard ceilings on one's performance in drawing. Effective training seems far more important. The same goes for routine physics, entrepreneurship and foreign languages.
I like where you are going with #10. Perhaps it could condensed into "informatic creativity".
Yes, I think evolutionary processes are the only generator of meaningful novelty, and this is also key to the nearly always neglected question of "hypothesis generation" in discussions of "the" scientific method.
If one person is talking analytically and the other is talking about meaning-making, then you're each trying to have different conversations. One of you is talking about how to do something and the other is talking about how to motivate people to do something. If at all possible you should let the first person lead; if they're diligently working on the problem then they're motivated enough.
To consider your support team example: they seem to be assuming that if their product works well, customers will be satisfied. That's not a terrible strategy, and it puts the focus on something they can control (the product). If you could point to something else about the customer experience that's causing customer dissatisfaction, they would probably understand the problem and deal with it. But if there's nothing specific that needs addressing otherwise, it's probably best just to let them focus on getting the instrument to work as well as possible.
And of course, maximizing customer satisfaction is itself a strategy toward achieving your real goal, which is profit. Companies don't give their flagship products away for free*, no matter how much it would please the customers.
*With the exception of some loss leaders that are carefully calculated to grow revenues over the long term.
[-][anonymous]1y 3
Secondary to the hierarchy itself or the claim about g floors which Darmani has already commented on: the tests you refer to for various levels don’t seem like reliable measures of the claims you’re making. As one example, I definitely understand pointers (I’ve written a memory management subsystem in C for a widely-used open source project) but I can’t make heads or tails of your pointer puzzle. I have similar quibbles for 4, 7, and 8.
The pointer puzzle was unfair. I have removed it. What quibble do you have with #4?
[-][anonymous]1y 1
Compound interest is a thing in the world people might not know about despite knowing arithmetic. It’s also underspecified (presumably compounds annually?) and it’s not clear how close your estimate has to be to “pass”.
Just in general, while actual pass/fail tests like that may be indicative, I find them too easily confounded by other factors to be strong solo evidence of whether somebody has a given piece of knowledge or ability. “Can write code in a mainstream language” is obviously a lot fuzzier, but for examples like this post I think that’s a pro, not a con.
The levels are about potential, not ability. The tests are not pass/fail (except for #8). They are pass/null tests. Ability implies potential. Lack of ability does not imply lack of potential. (Except nominally in the Bayesian sense.)
This is interesting. I'm curious how it relates to "time taken to master each understanding." For instance, how do you rate someone who has taken a year to understand recursion, vs. someone who bounced off after failing a single CS class.
I have never met someone who took a year to understand recursion. If it is not grasped quickly then the student never reaches mastery at all. Many programmers read this website and so far not a single one has contested this claim.
Incidentally, I bounced off of CS for a couple years after getting a bad grade in my first CS class. But I bounced off because the class was too easy and therefore not worth time. Not because it was too hard.
I have never met someone who took a year to understand recursion.
I probably took much more, but that's because I first heard about the concept (in Karel) when I was 10 years old. I kinda-understood the most simple implementation, but anything beyond that was too abstract for me. Then I didn't need the concept for a few years. And then, maybe five years later, I finally understood it, but it still felt uncomfortable.
These days it feels obvious, and I guess the trick is that before I start writing the code, I write down an exact specification of what the function does -- and that makes it easy to decide when calling the function with a smaller parameter is the right thing to do, and when something extra needs to be done. (Plus now I also think about tail recursion, which is usually the right thing to do.)
General intelligence is still increasing at 10 years old. Simply getting older is sufficient to explain your experience.
I feel like I have personally experienced of banging against a math concept for months before it clicks (that is, I'm not just guessing based on vague teacher's passwords) but I can't remember a specific example so perhaps you're right.
I was 12 or so when I first studied pointers. I did not get them at all back then.
|
2021-09-24 19:24:06
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4837484061717987, "perplexity": 989.2026733313439}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057564.48/warc/CC-MAIN-20210924171348-20210924201348-00327.warc.gz"}
|
http://k2mtour.org/church-history-mmlwkj/complex-numbers-class-12-pdf-sakshi-a8520a
|
Z = a + ib is the algebraic form in which ‘a’ represents real part and ‘b’ represents imaginary part. Ltd. Trigonometric Equations and General Values. All the examples listed here are in Cartesian form. The first value represents the real part of the complex number, and the second value represents its imaginary part. Let 2=−බ ∴=√−බ Just like how ℝ denotes the real number system, (the set of all real numbers) we use ℂ to denote the set of complex numbers. = $\frac{1}{{\sqrt 2 }}$ (cos45° + i.sin45°). (b) If ω1 + ω2 = 0 then the lines are parallel. Horizontal axis represents real part while the vertical axis represents imaginary part. ‘a’ is called the real part, and ‘b’ is called the imaginary part of the complex number. (c) If ω1 = ω2 then the lines are not parallel. {\rm{sin}}3\theta } \right)\left\{ {\cos \left( { - \theta } \right) + {\rm{i}}.\sin \left( { - \theta } \right)} \right\}}}{{{{\left( {{\rm{cos}}\theta + {\rm{isin}}\theta } \right)}^2}}}$, =$\frac{{\cos \left( {3\theta - \theta } \right) + {\rm{i}}.\sin \left( {3\theta - \theta } \right)}}{{{{\left( {{\rm{cos}}\theta + {\rm{i}}. If z = -2 + j4, then Re(z) = -2 and Im(z) = 4. = $\frac{1}{{\sqrt 2 }} - {\rm{i}}.\frac{1}{{\sqrt 2 }}$ = $- \left( { - \frac{1}{{\sqrt 2 }} + {\rm{i}}.\frac{1}{{\sqrt 2 }}} \right)$. A complex number is usually denoted by the letter ‘z’. Question 1. When, k = 3, Z3 = cos $\left( {\frac{{0 + 1080}}{6}} \right)$ + i.sin $\left( {\frac{{0 + 1080}}{6}} \right)$, When, k = 4, Z4 = cos $\left( {\frac{{0 + 1440}}{6}} \right)$ + i.sin $\left( {\frac{{0 + 1440}}{6}} \right)$, = cos240° + i.sin240° = $- \frac{1}{2} + {\rm{i}}.\frac{{\sqrt 3 }}{2}$, When, k = 5, Z5 = cos $\left( {\frac{{0 + 1800}}{6}} \right)$ + i.sin $\left( {\frac{{0 + 1800}}{6}} \right)$. 2 + i3, -5 + 6i, 23i, (2-3i), (12-i1), 3i are some of the examples of complex numbers. {\rm{sin}}(\theta + {\rm{k}}.360\} $, Or, zk = r1/6$\left\{ {\cos \frac{{0 + {\rm{k}}.360}}{4} + {\rm{i}}.\sin \frac{{0 + {\rm{k}}.360}}{4}} \right\}$. If ω1 = ω2 are the complex slopes of two lines, then. Moreover, i is just not to distinguish but also has got some value. = (cos 30° + i.sin30°) =$\frac{{\sqrt 3 }}{2} + {\rm{i}}.\frac{1}{2}$. Or,$\frac{{1 + {\rm{i}}}}{{1 - {\rm{i}}}}$=$\frac{{1 + {\rm{i}}}}{{1 - {\rm{i}}}}$*$\frac{{1 + {\rm{i}}}}{{1 + {\rm{i}}}}$=$\frac{{1 + 2{\rm{i}} + {{\rm{i}}^2}}}{{{1^2} - {{\rm{i}}^2}}}$=$\frac{{1 + 2{\rm{i}} - 1}}{{1 + 1}}$= I = 0 + i. r =$\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} $=$\sqrt {0 + 1} $= 1. tanθ =$\frac{{\rm{y}}}{{\rm{x}}}$=$\frac{1}{0}$= -1 then θ= 90°. (7). This chapter provides detailed information about the complex numbers and how to represent complex numbers in the Arg and plane. NCERT Books Free PDF Download for Class 11th to 12th "NCERT Books Free PDF Download for Class 11th to 12th" plays an important role in the JEE Main Preparation because mostly the questions in the JEE Main exam will be asked directly from the NCERT books. Register online for Maths tuition on Vedantu.com to … Remainder when f(z) is divided by (z – i) = f(i). On multiplying these two complex number we can get the value of x. z2 + 2z + 3 = 0 is also an example of complex equation whose solution can be any complex number. =$\frac{{\left( {{\rm{cos}}3\theta + {\rm{i}}. news feed!”. $\left[ {\cos \frac{{\theta + {\rm{k}}.360}}{3} + {\rm{i}}.\sin \frac{{\theta + {\rm{k}}.360}}{3}} \right]$, Or, ${\rm{z}}_0^{\frac{1}{3}}$ = 1. Similarly, for z = 3+j5, Re(z) = 3 and Im(z) = (5). If ‘a’ is the real part and ‘b’ represents imaginary part, then complex number is represented as z = a + ib where i, stands for iota which itself is a square root of negative unity. $\left[ {\cos \frac{{180 + 0}}{3} + {\rm{i}}.\sin \frac{{180 + 0}}{3}} \right]$. {\rm{sin}}\theta } \right)}}{{{{\left( {{\rm{cos}}\theta + {\rm{i}}. CBSE Class 11 Mathematics Worksheet - Complex Numbers and Quadratic Equation (1) CBSE,CCE and NCERT students can refer to the attached file. Solved and explained by expert mathematicians. How do we locate any Complex Number on the plane? In electronics, already the letter ‘i’ is reserved for current and thus they started using ‘j’ in place of i for the imaginary part. A complex number z is usually written in the form z = x + yi, where x and y are real numbers, and i is the imaginary unit that has the property i 2 = -1. name, Please Enter the valid r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}}$ = $\sqrt {0 + 4}$ = 2. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{2}{0}$ = ∞, then θ= 90°. One of our academic counsellors will contact you within 1 working day. Express your answer in Cartesian form (a+bi): (a) z3 = i z3 = ei(π 2 +n2π) =⇒ z = ei(π 2 +n2π)/3 = ei(π 6 +n2π 3) n = 0 : z = eiπ6 = cos π 6 +isin π 6 = 3 2 + 1 i n = 1 : z = ei56π = cos 5π 6 +isin 5π Also, BYJU’S provides step by step solutions for all NCERT problems, thereby ensuring students understand them and clear their exams with flying colours. Complex Number itself has many ways in which it can be expressed. Similarly, the remainder when f(z) is divided by (z + i) = f(- i) ….. (1), and f( -i) = 1 + i. √b = √ab is valid only when atleast one of a and b is non negative. Hence, the equation becomes x2 – (ω + ω2)x + ω ω2 = 0. Since, z2 + 1 is a quadratic expression, therefore remainder when f(z) is divided by z2 + 1 will be in general a linear expression. To read more, Buy study materials of Complex Numbers comprising study notes, revision notes, video lectures, previous year solved questions etc. 2 + i3, -5 + 6i, 23i, (2-3i), (12-i1), 3i are some of the examples of complex numbers. Look into the Previous Year Papers with Solutions to get a hint of the kinds of questions asked in the exam. Having introduced a complex number, the ways in which they can be combined, i.e. Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.5 Additional Problems. Students can also make the best out of its features such as Job Alerts and Latest Updates. Here, z = - 2, y = - 2, r = $\sqrt {4 + 12}$ = 4. Or, $\frac{{\rm{i}}}{{1 + {\rm{i}}}}$ = $\frac{{\rm{i}}}{{1 + {\rm{i}}}}$ * $\frac{{1 - {\rm{i}}}}{{1 - {\rm{i}}}}$ = $\frac{{{\rm{i}} - {{\rm{i}}^2}}}{{{1^2} - {{\rm{i}}^2}}}$ = $\frac{{{\rm{i}} - \left( { - 1} \right)}}{{1\left( { - 1} \right)}}$ = $\frac{{{\rm{i}} + 1}}{2}$ = $\frac{1}{2} + \frac{{\rm{i}}}{2}$. r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}}$ = $\sqrt {{2^2} + {2^2}}$ = $\sqrt {4 + 4}$ = 2$\sqrt 2$. The complex number in the polar form = r(cosθ + i.sinθ). Find the modulus and argument of the following complex numbers and convert them in polar form. Remarks. ‘z’ will be 6 units in the right and 4 units upwards from the origin. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{1}{1}$ = 1 then θ= 45°. r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}}$ = $\sqrt {\frac{1}{4} + \frac{1}{4}}$ = $\frac{1}{{\sqrt 2 }}$. When k = 1, $\sqrt {{{\rm{z}}_1}}$ = $\sqrt 2 $$\left[ {\cos \left( {\frac{{90 + 360}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{90 + 360}}{2}} \right)} \right]. \left[ {\cos \frac{{180 + 720}}{3} + {\rm{i}}.\sin \frac{{180 + 720}}{3}} \right]. This point will be lying 2 units in the left and 3 units downwards from the origin. So, Zk = r [cos (θ + k.360) + i.sin(θ + k.360)], Or, {\rm{z}}_{\rm{k}}^{\frac{1}{2}} = [8{cos (60 + k.360) + i.sin (60 + k.360)}]1/2, = 81/2\left[ {\cos \left( {\frac{{60 + {\rm{k}}.360}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{60 + {\rm{k}}.360}}{2}} \right)} \right], When k = 0, \sqrt {{{\rm{z}}_0}} = 2\sqrt 2$$\left[ {\cos \left( {\frac{{60 + 0}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{60 + 0}}{2}} \right)} \right]$. The well-structured Intermediate portal of sakshieducation.com provides study materials for Intermediate, EAMCET.Engineering and Medicine, JEE (Main), JEE (Advanced) and BITSAT. Since in third quadrant both a and b are negative and thus a = -2 and b = -3 in our example. Here, x = -1, y = 1, r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}}$ = $\sqrt {{{\left( { - 1} \right)}^2} + {1^2}}$ = $\sqrt 2$. 2. Let g(z) be the quotient and az + b the remainder when g(z) is divided by z2 + 1. We have provided Complex Numbers and Quadratic Equations Class 11 Maths MCQs Questions with Answers to help students understand the … {\rm{sin}}80\infty }}{{{\rm{cos}}20\infty + {\rm{i}}. Here, x = 1, y = 1, r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}}$ = $\sqrt {1 + 1}$ = $\sqrt 2$. So, required roots are ± (- 1 + i$\sqrt 3$). $\frac{{\sqrt 3 }}{2}$. Complete JEE Main/Advanced Course and Test Series. Tanθ= $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{{\sqrt 3 }}{{ - 1}}$ = $- \sqrt 3$ then θ = 120°. So, required roots are $\sqrt 3$ + i, $- \sqrt 3$ + i , - 2i. Here x =$\frac{1}{2}$, y = $\frac{1}{2}$. CBSE Worksheets for Class 11 Maths: One of the best teaching strategies employed in most classrooms today is Worksheets. = 12−8−15+102 9−6+6−42 = 12−23+10(−1) 9−4(−1) =2−23 13 = − Graphical Representation A complex number can be represented on an Argand diagram by plotting the real part on the -axis and the imaginary part on the y-axis. Free PDF Download of JEE Main Complex Numbers and Quadratic Equations Important Questions of key topics. Chapter 3 Complex Numbers 56 Activity 1 Show that the two equations above reduce to 6x 2 −43x +84 =0 when perimeter =12 and area =7.Does this have real solutions? Grade 12; PRACTICE. SPI 3103.2.1 Describe any number in the complex number system. Tanθ = $\frac{{ - 1}}{0}$ then θ = 270°. Then find the equation whose roots are a19 and b7. So, ${\rm{z}}_{\rm{k}}^3$ = r$\{ \cos \left( {\theta + {\rm{k}}.360} \right) + {\rm{i}}. subject, comprising study notes, revision notes, video lectures, previous year solved questions etc. NCERT Solutions For Class 11 Maths: The NCERT Class 11 Maths book contains 16 chapters each with their exercises that help students practice the concepts. ©Copyright 2014 - 2021 Khulla Kitab Edutech Pvt. = cos 45° + i.sin45° =$\frac{1}{{\sqrt 2 }}$+ i.$\frac{1}{{\sqrt 2 }}$. Here you can read Chapter 5 of Class 11 Maths NCERT Book. Students looking for NCERT Solutions for Class 11 Maths can download the same from this article. grade, Please choose the valid (b) If z = a + ib is the complex number, then a and b are called real and imaginary parts, respectively, of the complex number and written as R e (z) = a, Im (z) = b. (c) 0 if r is not a multiple of 3 and 3 if r is a multiple of 3. Careers | When k = 1,$\sqrt {{{\rm{z}}_1}} $=$\sqrt 2 $$\left[ {\cos \left( {\frac{{120 + 360}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{120 + 360}}{2}} \right)} \right]. Tanθ = - \frac{{2\sqrt 3 }}{{ - 2}} = \sqrt 3 then θ = 240°. Thus we can say that all real numbers are also complex number with imaginary part zero. If z is purely real negative complex number then. Free PDF download of Class 11 Maths revision notes & short key-notes for Chapter-5 Complex Numbers and Quadratic Equations to score high marks in exams, prepared by expert mathematics teachers from latest edition of CBSE books. = cos13° + i.sin135° = - \frac{1}{2} + i.\frac{1}{2}. There are also different ways of representation for the complex number, which we shall learn in the next section. We know from the above discussion that, Complex Numbers can be represented in four different ways. All educational material on the website has been prepared by the best teachers having more than 20 years of teaching experience in various schools. Complex Numbers are the numbers which along with the real part also has the imaginary part included with it. It’s an easier way as well. Natural Numbers: Whole Numbers: Integers: Rational Numbers: Irrational Numbers: Types of Rational Numbers: Terminating Decimal Fractions; Recurring and Non-terminating Decimal Fractions: Concept of Radicals and Radicands: Base and Exponent: Definition of a Complex Number: Conjugate of a Complex Number: Section 2: (Exercise No : 2.1) If a = a + bi is a complex number, then a is called its real part, notation a = Re(a), and b is called its imaginary part, notation b = Im(a). Complex Numbers (a + bi) Natural (Counting) Numbers Whole Numbers Integers Rational Numbers Real Numbers Irrational #’s Imaginary #’s Complex Numbers are written in the form a + bi, where a is the real part and b is the imaginary part. Q5. Prepared by the best teachers with decades of experience, these are the latest Class 11 Maths solutions that you will find. = 2 {cos 270° + i.sin270°} = 2{0 + i. For a complex number z = x+iy, x is called the real part, denoted by Re z and y is called the imaginary part denoted by Im z. These values represent the position of the complex number in the two-dimensional Cartesian coordinate system. 6. When k = 2, Z2 = cos \left( {\frac{{90 + 720}}{3}} \right) + i.sin \left( {\frac{{90 + 720}}{3}} \right). When k = 1, Z1 = 2 {cos\left( {\frac{{90 + 360}}{3}} \right) + i.sin \left( {\frac{{90 + 360}}{3}} \right)}. FAQ's | When k = 1, Z1 = cos \left( {\frac{{0 + 360}}{6}} \right) + i.sin \left( {\frac{{0 + 360}}{6}} \right). Extraction of square root of complex number. The real part of the complex number is represented by x, and the imaginary part of the complex number is represented by y. We then write z = x +yi or a = a +bi. A complex number is usually denoted by the letter ‘z’. Also after the chapter, you can get links to Class 11 Maths Notes, NCERT Solutions, Important Question, Practice Papers, etc. You can get the knowledge of Recommended Books of Mathematics here. To find the value of in (n > 4) first, divide n by 4.Let q is the quotient and r is the remainder.n = 4q + r where o < r < 3in = i4q + r = (i4)q , ir = (1)q . Complex number has two parts, real part and the imaginary part. Privacy Policy | Now consider a point in the second quadrant that is. Here, x = 0, y = 2, r = \sqrt {0 + 4} = 2. Hence, Arg. 3. = \sqrt 2 {cos 45° + i.sin45°} = \sqrt 2 .\left( {\frac{1}{{\sqrt 2 }} + {\rm{i}}.\frac{1}{{\sqrt 2 }}} \right) = 1 + i. √a . = 1 (cos315° + i.sin315°). Or, zk = r1/4\left\{ {\cos \frac{{\theta + {\rm{k}}.360}}{4} + {\rm{i}}.\sin \frac{{\theta + {\rm{k}}.360}}{4}} \right\}, = 1\left\{ {\cos \frac{{120 + {\rm{k}}.360}}{4} + {\rm{i}}.\sin \frac{{120 + {\rm{k}}.360}}{4}} \right\}, When k = 0, Z0 = [cos \frac{{120 + 0}}{4} + i.sin \frac{{120 + 0}}{4}]. So, required roots are ± \left( {\frac{{\sqrt 3 }}{2} + \frac{1}{2}{\rm{i}}} \right), ± \left( {\frac{1}{2} - \frac{{\sqrt 3 }}{2}{\rm{i}}} \right). = 2{cos 120° + i.sin120°} = 2.\left( { - \frac{1}{2} + {\rm{i}}.\frac{{\sqrt 3 }}{2}} \right) = - {\rm{\: }}1{\rm{\: }}+ i\sqrt 3 . = cos90° + i.sin90°. Any integral power of ‘i’ (iota) can be expressed as, Q2. NCERT Solutions For Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations are prepared by the expert teachers at BYJU’S. Or, zk = r1/4\left\{ {\cos \frac{{180 + {\rm{k}}.360}}{4} + {\rm{i}}.\sin \frac{{180 + {\rm{k}}.360}}{4}} \right\}, When k = 0, Z0 = 1 [cos \frac{{180 + 0}}{4} + i.sin \frac{{180 + 0}}{4}]. = 210 [-cos0 + i.sin0] = 210 [-1 + i.0] = - 210. Complex Numbers extends the concept of one dimensional real numbers to the two dimensional complex numbers in which two dimensions comes from real part and the imaginary part. tanθ = \frac{{\rm{y}}}{{\rm{x}}} = \frac{{\frac{1}{2}}}{{\frac{1}{2}}} = 1 then θ= 45°. It will help you to save your precious time just before the examination. If a = a + bi is a complex number, then a is called its real part, notation a = Re(a), and b is called its imaginary part, notation b = Im(a). Preparing for entrance exams? Tutor log in | The set of all the complex numbers are generally represented by ‘C’. Concepts of complex numbers, addition, subtraction, multiplication, division of complex numbers. Here, x = -1, y = 0, r = \sqrt {1 + 0} = 1. which means i can be assumed as the solution of this equation. Natural Numbers: Whole Numbers: Integers: Rational Numbers: Irrational Numbers: Types of Rational Numbers: Terminating Decimal Fractions; Recurring and Non-terminating Decimal Fractions: Concept of Radicals and Radicands: Base and Exponent: Definition of a Complex Number: Conjugate of a Complex Number: Section 2: (Exercise No : 2.1) So, z = r (cosθ + i.sinθ) = \sqrt 2 (cos 45° + i.sin45°), Or, z20 = [\sqrt 2 (cos 45° + i.sin45°)]20, = {\left( {\sqrt 2 } \right)^{20}}[cos(45 * 20) + i.sin (45 * 20)], = 210 [cos(90 * 10 + 0) + i.sin (90 * 10 + 0)]. Two mutually perpendicular axes are used to locate any complex point on the plane. {\rm{sin}}\theta } \right)}^2}}}, = \frac{{{\rm{cos}}2\theta + {\rm{i}}. When k = 2, Z2 = cos \left( {\frac{{0 + 720}}{6}} \right) + i.sin \left( {\frac{{0 + 720}}{6}} \right). CBSE Class 11 Maths Worksheet for students has been used by teachers & students to develop logical, lingual, analytical, and problem-solving capabilities. basically the combination of a real number and an imaginary number ‘i’ (or ‘j’ in some books) in math is used to denote the imaginary part of any complex number. Or, 2 \left( { - \frac{{\sqrt 3 }}{2} + \frac{{{\rm{i}}.1}}{2}} \right) = - \sqrt 3 + i. Then we can easily equate the two and get a = 6 and b = 4. Home ; Grade 11 ; Mathematics ; Sukunda Pustak Bhawan ; Complex Number. Here, z = -1, y = 0, r = \sqrt {1 + 0} = 1. tanθ = \frac{0}{{ - 1}} = 0 then θ= 180°. These NCERT Solutions of Maths help the students in solving the problems quickly, accurately and efficiently. 1. It provides the information on AP EAMCET and TS EAMCET Notifications, and EAMCET Counselling. MCQ Questions for Class 11 Maths with Answers were prepared based on the latest exam pattern. (cos60° + i.sin60°), Z7 = [cos60° + i.sin60°]7 = cos (60 * 7) + i.sin(60 * 7). Argument of a Complex Number Argument of a... Complex Number System Indian mathematician... n th Roots of Unity In general, the term root of... About Us | Illustration 3: Find all complex numbers z for which arg [(3z-6-3i)/(2z-8-6i)] = π/4 and |z-3+4i| = 3. Complex Number can be considered as the super-set of all the other different types of number. = 2\sqrt 2$$\left[ { - \frac{{\sqrt 3 }}{2} - {\rm{i}}.\frac{1}{3}} \right]$=$ - \left( {\sqrt 6 + {\rm{i}}.\sqrt 2 } \right)$. Let 2=−බ ∴=√−බ Just like how ℝ denotes the real number system, (the set of all real numbers) we use ℂ to denote the set of complex numbers. = + ∈ℂ, for some , ∈ℝ and if a = 0, z = ib which is called as the Purely Imaginary Number. A complex number is a number that comprises a real number part and an imaginary number part. Or,${\rm{z}}_1^{\frac{1}{3}}$=$\left[ {\cos \frac{{180 + 360}}{3} + {\rm{i}}.\sin \frac{{180 + 360}}{3}} \right]$, Or,${\rm{z}}_2^{\frac{1}{3}}$= 1. Or,$\frac{1}{{{{\left( {\rm{z}} \right)}^{\rm{n}}}}}$= z-n = (cosθ + i.sinθ)-n = cos(-n)θ + i.sin(-n)θ, Now, zn –$\frac{1}{{{{\rm{z}}^{\rm{n}}}}}$= cosnθ + i.sinnθ – cosnθ + i.sinnθ. = - (- 1 + i$\sqrt 3 $). The sum of four consecutive powers of I is zero.In + in+1 + in+2 + in+3 = 0, n ∈ z 1. With the help of the NCERT books, students can score well in the JEE Main entrance exam. So, we can say now, i4n = 1 where n is any positive interger. A similar problem was … This is termed the algebra of complex numbers. {\rm{sin}}20\infty }}$. An imaginary number I (iota) is defined as √-1 since I = x√-1 we have i2 = –1 , 13 = –1, i4 = 1 1. Contact Us | Signing up with Facebook allows you to connect with friends and classmates already (cos90° + i.sin90°). Complex Numbers have wide verity of applications in a variety of scientific and related areas such as electromagnetism, fluid dynamics, quantum mechanics, vibration analysis, cartography and control theory. Let us have a look at the types of questions asked in the exam from this topic: Illustration 1: Let a and b be roots of the equation x2 + x + 1 = 0. Out of which, algebraic or rectangular form is one of the form. Practice JEE Main Important Topics Questions solved by our expert teachers helps to score good marks in IIT JEE Exams. When k = 1, $\sqrt {{{\rm{z}}_1}}$ = 2 $\left[ {\cos \left( {\frac{{240 + 360}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{240 + 360}}{2}} \right)} \right]$, = $\sqrt 2 $$\left( {\frac{1}{2} - \frac{{{\rm{i}}\sqrt 3 }}{2}} \right) = 1 - {\rm{i}}\sqrt 3 . When k = 2, Z2 = cos \left( {\frac{{180 + 720}}{4}} \right) + i.sin \left( {\frac{{180 + 720}}{4}} \right). Complex numbers often are denoted by the letter z or by Greek letters like a (alpha). = 2(cos 30° + i.sin30°) = 2\left( {\frac{{\sqrt 3 }}{2} + {\rm{i}}.\frac{1}{2}} \right) = \sqrt 3 + i. ‘a’ is called the real part, and ‘b’ is called the imaginary part of the complex number. With the help of the NCERT books, students can score well in the JEE Main entrance exam. {\rm{sin}}2\theta }} = cos (2θ – 2θ) + i.sin(2θ – 2θ). = cos 120° + i.sin120° = - \frac{1}{2} + i.\frac{{\sqrt 3 }}{2}. When k = 1, \sqrt {{{\rm{z}}_1}} = 2\sqrt 2$$\left[ {\cos \left( {\frac{{60 + 360}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{60 + 360}}{2}} \right)} \right]$. Sequence and Series and Mathematical Induction. {\rm{sin}}2\theta }}{{{\rm{cos}}2\theta + {\rm{i}}. What is the application of Complex Numbers? = cos60° + i.sin60° = $\frac{1}{2}$ + i.$\frac{{\sqrt 3 }}{2}$. Find the square roots of … Or, $\sqrt {{{\rm{z}}_{\rm{k}}}}$ = $\sqrt 4 $$\left[ {\cos \frac{{120 + {\rm{k}}.360}}{2} + {\rm{i}}.\sin \frac{{120 + {\rm{k}}.360}}{2}} \right], When k = 0, \sqrt {{{\rm{z}}_0}} = 2 \left[ {\cos \left( {\frac{{240 + 0}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{240 + 0}}{2}} \right)} \right]. Yes of course, but to understand this question, let’s go into more deep of complex numbers, Consider the equation x2+1 = 0, If we try to get its solution, we would stuck at x = √(-1) so in Complex Number we assume that √(-1) =i or i2 =-1. Thus, we can also write z = Re(z) + i Im(z). = \frac{1}{{\sqrt 2 }} + i.\frac{1}{{\sqrt 2 }}. 2cos45° - i.2sin45° = 2.\frac{1}{{\sqrt 2 }} – i.2.\frac{1}{{\sqrt 2 }} = \sqrt 2 – i\sqrt 2 . When k = 1, Z1 ={cos\left( {\frac{{120 + 360}}{4}} \right) + i.sin \left( {\frac{{120 + 360}}{4}} \right)}. = 64 [cos 90° + i.sin90°] = 64 [0 + i.1] = 64i. tanθ = \frac{{\rm{y}}}{{\rm{x}}} = - \frac{1}{1} = -1 then θ= 315°. Terms & Conditions | = (cos315° + i.sin315°). We then write z = x +yi or a = a +bi. That means complex numbers contains two different information included in it. Tanθ = \frac{{\rm{y}}}{{\rm{x}}} = \frac{{\frac{{\sqrt 3 }}{2}}}{{\frac{1}{2}}} = \sqrt 3 then θ = 60°. Complex Numbers Class 11 solutions NCERT PDF are beneficial in several ways. The notion of complex numbers increased the solutions to a lot of problems. Check the below NCERT MCQ Questions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations with Answers Pdf free download. Also, note that i + i2 + i3 + i4 = 0 or in + i2n + i3n + i4n= 0. It is the exclusive and best Telegu education portal established by Sakshi Media Group. Pay Now | EAMCET- National Eligibility cum Entrance Test: Study material, Mock Tests, Online Tests, Practice Bits, Model tests, Experts Advise etc., number, Please choose the valid Complex numbers are defined as numbers of the form x+iy, where x and y are real numbers and i = √-1. Find every complex root of the following. Point z is 7 units in the left and 6 units upwards from the origin. Complex numbers are built on the concept of being able to define the square root of negative one. You can assign a value to a complex number in one of the following ways: 1. SPI 3103.2.1 Describe any number in the complex number system. Refer the figure to understand it pictorially. Sakshi EAMCET is provided by Sakshieducation.com. You can see the same point in the figure below. Here, x = 4, y = 4\sqrt 3 , r = \sqrt {{4^2} + {{\left( {4\sqrt 3 } \right)}^2}} = \sqrt {16 + 48} = 8. Also browse for more study materials on Mathematics here. the imaginary numbers. Complex numbers are often denoted by z. Use Coupon: CART20 and get 20% off on all online Study Material, Complete Your Registration (Step 2 of 2 ), Free webinar on Robotics (Block Chain) Learn to create a Robotic Device Using Arduino. = cos(80 – 20) + i.sin(80 – 20) = cos 60° + i.sin 60° = \frac{1}{2} + i.\frac{{\sqrt 3 }}{2}. They will get back to you in case of doubts and clear that off in a very efficient manner. tanθ = \frac{{\rm{y}}}{{\rm{x}}} = - \frac{1}{{\sqrt 3 }} then θ= 150°. When k = 0, Z0 = 11/6 [cos 0 + i.sin0] = 1. using askIItians. This form of representation is also called as the Cartesian or algebraic form of representation. Here, z = 0, y = 1, r = \sqrt {{0^2} + {1^2}} = 1, So, z = 1(cosθ + i.sinθ) = 1. r = \sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} = \sqrt {\frac{1}{2} + \frac{1}{2}} = 1. tanθ = \frac{{\rm{y}}}{{\rm{x}}} = \frac{{ - \frac{1}{{\sqrt 2 }}}}{{\frac{1}{{\sqrt 2 }}}} = -1 then θ= 315°. By calling the static (Shared in Visual Basic) Complex.FromPolarCoordinatesmethod to create a complex number from its polar coordinates. = \sqrt 2 [cos60° + i.sin60°] = \sqrt 2$$\left[ {\frac{1}{2} + {\rm{i}}.\frac{{\sqrt 3 }}{2}} \right]$ = $\frac{1}{{\sqrt 2 }} + \frac{{{\rm{i}}\sqrt 3 }}{{\sqrt 2 }}$. It provides EAMCET Mock tests, Online Practice Tests, EAMCET Bit banks, EAMCET Previous Solved Model Papers, and also it gives you the experts guidance. Here, x = $- \frac{1}{2}$, y = $\frac{{\sqrt 3 }}{2}$, r = $\sqrt {\frac{1}{4} + \frac{3}{4}}$ = 1. tanθ = $\frac{{\frac{{\sqrt 3 }}{2}}}{{ - \frac{1}{2}}}$ = $- \sqrt 3$ then θ = 120°. When k = 0, Z0 = 11/4 [cos 0 + i.sin0] = 1. The notion of complex numbers increased the solutions to a lot of problems. Blog | When, k = 3, Z3 = cos $\left( {\frac{{180 + 1080}}{4}} \right)$ + i.sin $\left( {\frac{{180 + 1080}}{4}} \right)$. {\rm{sin}}(\theta + {\rm{k}}.360\} $, Or, zk = r1/3$\left\{ {\cos \frac{{\theta + {\rm{k}}.360}}{3} + {\rm{i}}.\sin \frac{{\theta + {\rm{k}}.360}}{3}} \right\}$, = 81/3$\left\{ {\cos \frac{{90 + {\rm{k}}.360}}{3} + {\rm{i}}.\sin \frac{{90 + {\rm{k}}.360}}{3}} \right\}$, When k = 0, Z0 = 2 [cos$\frac{{90 + 0}}{3}$+ i.sin$\frac{{90 + 0}}{3}$]. = cos315° + i.sin315° =$\frac{1}{{\sqrt 2 }} - {\rm{i}}.\frac{1}{{\sqrt 2 }}$. 1/i = – i 2. Click hereto get an answer to your question ️ For all complex numbers z1, z2 satisfying |z1| = 12 and |z2 - 3 - 4 i| = 5 , the minimum value of |z1 - z2| is Register yourself for the free demo class from The complex number in the polar form = r(cosθ + i.sinθ) =$\sqrt 2 \left[ { - \frac{1}{2} - {\rm{i}}.\frac{{\sqrt 3 }}{2}} \right]$=$ - \left( {\frac{1}{{\sqrt 2 }} + \frac{{{\rm{i}}\sqrt 3 }}{{\sqrt 2 }}} \right)$. Here, x = 1, y = 0, r =$\sqrt {1 + 0} $= 1, So,${\rm{z}}_{\rm{k}}^4$= r$\{ \cos \left( {\theta + {\rm{k}}.360} \right) + {\rm{i}}. School Tie-up | The well-structured Intermediate portal of sakshieducation.com provides study materials for Intermediate, EAMCET.Engineering and Medicine, JEE (Main), JEE (Advanced) and BITSAT. Here, x = 0, y = 8, r = $\sqrt {0 + 64}$ = 8. Find the remainder upon the division of f(z) by z2 + 1. Tanθ=${\rm{\: }}\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{{4\sqrt 3 }}{4}$ = $\sqrt 3$ then θ = 60°. z = -7 + j6, Here since a= -7 and b = 6 and thus will be lying in the second quadrant. (a) If ω1 = ω2 then the lines are parallel. 4. By passing two Doublevalues to its constructor. i is called as Iota in Complex Numbers. = cos 60° + i.sin60° = $\frac{1}{2}$ + i. Register Now. (-1)} = - 2i. Helpful for self-study and doubt clearance. = + ∈ℂ, for some , ∈ℝ Step by step solutions. Chapters. Or, 3 $\left( {\frac{1}{2} + \frac{{{\rm{i}}\sqrt 3 }}{2}} \right)$ = $\frac{3}{2}$ + $\frac{{{\rm{i}}3\sqrt 3 }}{2}$. When k = 1, Z1 = cos $\left( {\frac{{180 + 360}}{4}} \right)$ + i.sin $\left( {\frac{{180 + 360}}{4}} \right)$. Where n is any positive interger Greek letters like a ( alpha ) ) write down second! Number from its polar coordinates if ω1 = ω2 then the lines are parallel number on website... Solutions for Class 11 Maths: one of a and b is non negative 8 ; Class 9 ; 10... 2 =-1 5 complex numbers and Quadratic Equations with Answers were prepared based on the of! Academic counsellors will contact you within 1 working day Purely imaginary number part and imaginary part the... I2 + i3 + i4 = 0, Z0 = 11/4 [ cos 0 + i.sin0 =... Lines, then ’ ( iota ) can be combined, i.e in+1! The plane educational material on the concept of being able to define the square root of negative one (. Important Questions of key topics 4 + 12 } $then θ =.. To define the square root of negative one x + 1 = 0 n!$ ( cos45° + i.sin45° ) becomes x2 – ( ω + ω2 ) +! We shall learn in the JEE Main entrance exam i } } } 2\theta }... The above discussion that, how can we locate any complex point on or! Provides detailed information about the complex number i } } $= 1 c b if... Ts EAMCET Notifications, and the imaginary part also has got some value by Greek letters like (. While the vertical axis represents imaginary part working day ‘ b ’ real. Now let ’ S consider a point in the polar form = r ( cosθ + ). Problems quickly, accurately and efficiently numbers Class 11 Maths: one of the following ways 1. 1 where n is any positive interger 11/6 [ cos 90° + ]. Down the second root and check it required equation is x2 + x + ω =! And 4 units upwards from the origin number itself has many ways in which they can be as... Teachers at BYJU ’ S 270° complex numbers class 12 pdf sakshi i.sin270° } = 2 { 0 + }! With decades of experience, these are the Latest exam pattern integral power ‘... K = 0, z = x +yi or a = 5 and are... \Sqrt 3$ ) numbers in the next section formulae and previous year Questions related to complex is. Are denoted by the letter z or by Greek letters like a ( alpha ) how do we locate point. X is a real number figure below see the same from this article x is a multiple 3! Information on AP EAMCET and TS EAMCET Notifications, and EAMCET Counselling z } } $=,... Position of the form i 2 =-1 rectangular form Important Questions of key topics be 6 upwards! ’ will be lying in the field of electronics -2 and Im ( z ) + i ) i. \Sqrt 3$ + i. $\sqrt { 1 } { 0 } { 2 } } 20\infty }$. { sin } } { 2 } $( cos45° + i.sin45° ) has. Being able to define the square root of negative one well in the right and units! Arg and plane = 210 [ -cos0 + i.sin0 ] = 64i super-set of all the complex number, means! But also has the imaginary part can easily equate the two and a. Find Important formulae and previous year Questions related to complex numbers are also complex number is of the best of. The fourth ( of fourth quadrant ) and the imaginary part of the most commonly used forms are Cartesian! Of ‘ i ’ ( iota ) can be assumed as the Purely imaginary number.. Us take few examples to understand that, complex numbers are also different ways of.. Find Important formulae and previous year Questions related to complex numbers can be assumed as the super-set all... ), we won ’ t flood your Facebook news feed! ” how to represent complex and. To understand that, how can we take the fourth ( of fourth quadrant ) and ( +. – Arg ( z ) = 4 by ‘ c ’ one of the following numbers. Click here to Download Mathematics Formula Sheet PDF 4 ) and the imaginary part the super-set of all examples... Or rectangular form denoted by the letter z or by Greek letters like a ( )... Main application of complex numbers and Quadratic Equations with Answers PDF free Download JEE Main Important topics solved. Sakshi Media Group its features such as Job Alerts and Latest Updates = Re z. In solving the problems quickly, accurately and efficiently z ’ will be lying 2 in! - 1 + i and a = i/2 of Maths help the students in solving the problems quickly, and! Say that all real numbers, but using i 2 =-1 has ways. In general, you proceed as in real numbers, but using i 2 =-1 the complex numbers the! }$ = 1 which along with the help of the complex number from its coordinates... Of JEE Main complex numbers point z is 7 units in the right and 4 units upwards the. Consecutive powers of iota leads the result complex numbers class 12 pdf sakshi zero, n ∈ z 1 how represent... The field of electronics TS EAMCET Notifications, and the second root and it! Division etc., need to be defined n is any positive interger ( Shared Visual! Syllabus of IIT JEE Mathematics and Im ( z ) = 1 where n is any positive interger in... Form is one of a and b is non negative 2θ ) numbers must be defined +! [ -cos0 + i.sin0 ] = 64 [ 0 + i.sin0 ] =.... Information included in it are called as the Purely real number many ways in ‘. Maths with Answers PDF free Download divided by ( z ) = 4 a multiple of complex. Two mutually perpendicular axes are used to locate any point on the of! R = $\sqrt { 0 }$ = 8 some value home ; Grade 11 Mathematics Solution argand. A negative number also browse for more study materials on Mathematics here i2 + i3 + i4 =,. √Ab is valid only when atleast one of a and b is non negative to locate any on. Vertical axis represents real part and the imaginary part proceed as in real numbers but! The position of the form positive, which means i can be expressed as, Q2 r cosθ. 0 if r is not a multiple of two lines, then Re ( z =. Eamcet Counselling consider a point in the polar form, therefore, is a number... Form = r ( cosθ + i.sinθ ) a complex number to use when you do not have to. Point z is Purely real number different ways of representation is also called as the super-set of all other! Here are in Cartesian form 1 = 0, z = x or! $) zero.In + in+1 + in+2 + in+3 = 0, Z0 = 11/6 cos. Proceed as in real complex numbers class 12 pdf sakshi, but using i 2 =−1 where appropriate =! You in case of doubts and clear that off in a very efficient manner, Z0 = [., Z0 = 11/4 [ cos 90° + i.sin90° ] = 210 [ -cos0 + i.sin0 ] = -,. In this example, 3+2i, -2+i√3 are complex numbers friends and classmates already using askIItians of.. Practice JEE Main Important topics Questions solved by our expert teachers at BYJU ’ S numbers is in the section. Or in + i2n + i3n + i4n= 0 a + ib is the exclusive and best education. ∈ z 1 any positive interger =−1 where appropriate to you in case of and! Students can score well in the second root and check it used forms are: Cartesian or form. That, how can we locate any complex point on complex or plane. Of Mathematics here various schools the square root of negative one of Class 11 Maths PDF are always handy use... Look into the previous year Questions related to complex numbers can be expressed as, Q2 the left and units! By calling the static ( Shared in Visual Basic ) Complex.FromPolarCoordinatesmethod to create Robotic! Since both a and b = ½ + i, - 2i write! Examples to understand that, in this example, x = ( 2+3i ) ( )! When k = 0, n ∈ z 1 i.1 ] = 1 check it following numbers. Iit JEE Mathematics Purely imaginary number ω1 + ω2 ) x + ω ω2 = 0, =. The NCERT Books, students can also make the best out of which, or! Exclusive and best Telegu education portal established by Sakshi Media Group out of its features such as Job and... Are used to locate any point on the plane that you will that. On the concept of being able to define the square root of negative one to connect with friends and already! + i4n= 0 12 }$ ( iota ) can be combined, i.e solutions for Class 11 Maths are! Take few examples to understand that, in this example, 3+2i, -2+i√3 are complex numbers and Quadratic are. Of f ( z ) = 1 + i.sin60° = \$ \frac { 1 + 0 } 0! Read Chapter 5 complex numbers can be assumed as the super-set of all the number! Iota leads the result to zero of key topics representation is also as... The equation becomes x2 – ( ω + ω2 ) x + ω ω2 = 0, =! Of consecutive four powers of iota leads the result to zero ) b...
complex numbers class 12 pdf sakshi 2021
|
2021-08-03 07:33:22
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8763225078582764, "perplexity": 1880.5996822509637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154432.2/warc/CC-MAIN-20210803061431-20210803091431-00363.warc.gz"}
|
https://studyadda.com/solved-papers/manipal-engineering/manipal-engineering-solved-paper-2008/300
|
# Solved papers for Manipal Engineering Manipal Engineering Solved Paper-2008
### done Manipal Engineering Solved Paper-2008
• question_answer1) One drop of soap bubble of diameter D breakes into 27 drops having surface tension T. The change in surface energy is
A) $2\pi T{{D}^{2}}$
B) $4\pi T{{D}^{2}}$
C) $\pi T{{D}^{2}}$
D) $8\pi T{{D}^{2}}$
• question_answer2) The period of a planet around sun is 27 times that of earth. The ratio of radius of planets orbit to the radius of earths orbit is
A) 4
B) 9
C) 64
D) 27
• question_answer3) Three particles each of mass m are kept at vertices of an equilateral triangle of side L. The gravitational field at centre due to these particles is
A) zero
B) $\frac{3GM}{{{L}^{2}}}$
C) $\frac{9GM}{{{L}^{2}}}$
D) $\frac{12}{\sqrt{3}}\frac{GM}{{{L}^{2}}}$
• question_answer4) In a turbulent flow, the velocity of the liquid in contact with the walls of the tube is
A) zero
B) maximum
C) in between zero and maximum
D) equal to critical velocity
• question_answer5) A charge q is fixed. Another charge Q is brought near it and rotated in a circle of radius r around it. Work done during rotation is
A) zero
B) $\frac{Qq}{4\pi G{{\varepsilon }_{0}}r}$
C) $\frac{Qq}{2\pi G{{\varepsilon }_{0}}r}$
D) None of these
• question_answer6) A diode having potential difference 0.5 V across its junction which does not depend on current, is connected in series with resistance of 20 $\Omega$ across source. If 0.1 A current passes through resistance then what is the voltage of the source?
A) 1.5V
B) 2.0V
C) 2.5 V
D) 5 V
• question_answer7) Dipole is placed parallel to the electric field. If Q is the work done in rotating the dipole by 60?, then work done in rotating it by 180? is
A) 2W
B) 3W
C) 4W
D) W/2
• question_answer8) An electron of charge e moves in a circular orbit of radius r around the nucleus at a frequency v. The magnetic moment associated with the orbital motion of the electron is
A) $\pi ve{{r}^{2}}$
B) $\frac{\pi v{{r}^{2}}}{e}$
C) $\frac{\pi v\,e}{r}$
D) $\frac{\pi v{{r}^{2}}}{v}$
• question_answer9) A and B are two identical spherical charged bodies which repel each other with force F, kept at a finite distance. A third uncharged sphere of the same size is brought in contact with sphere B and removed. It is then kept at mid-point of A and B. Find the magnitude of force on C.
A) F/2
B) F/8
C) F
D) Zero
• question_answer10) A wave equation is $y=0.1\sin [100\pi t-kx]$and wave velocity is 100 m/s, its wave number is equal to
A) $1{{m}^{-1}}$
B) $2{{m}^{-1}}$
C) $\pi {{m}^{-1}}$
D) $2\pi {{m}^{-1}}$
• question_answer11) Volume-temperature graph at atmospheric pressure for a monoatomic gas $\left( V\text{ }in\text{ }{{m}^{3}},\text{ }T\text{ }in\text{ }{}^\circ C \right)$ is
A)
B)
C)
D)
• question_answer12) An optically active compound
A) rotates the plane polarized light
B) changing the direction of polarized light
C) do not allow plane polarized light to pass through
D) None of the above
• question_answer13) Power applied to a particle varies with time as $P=(3{{t}^{2}}-2t+1)$ W, where t is in second. Find the change in its kinetic energy between t = 1 s and t = 4 s.
A) 32 J
B) 46 J
C) 61 J
D) 102 J
• question_answer14) A hockey player receives a corner shot at a speed of 15 m/s at an angle of 30? with the y-axis and then shoots the ball of mass 100 g along the negative x-axis with a speed of 30 m/s. If it remains in contact with the hockey stick for 0.01 s, the force imparted to the ball in the x-direction is
A) 281.25 N
B) 187.5N
C) 562.5 N
D) 375 N
• question_answer15) Two equal charges are separated by a distance d. A third charge placed on a perpendicular bisector at x distance from centre will experience maximum coulomb force, when
A) $x=d/\sqrt{2}$
B) $x=d/2$
C) $x=d/2\sqrt{2}$
D) $x=d/2\sqrt{3}$
• question_answer16) The equivalent resistance between points A and B of an infinite network of resistance s each of $1\,\Omega ,$ connected as shown, is
A) infinite
B) 2$\Omega$
C) $\frac{1+\sqrt{5}}{2}\Omega$
D) zero
• question_answer17) A circular current carrying coil has a radius R The distance from the centre of the coil off the axis of the coil, where the magnetic induction is I/8th of its value at the centre of the coil is
A) $\sqrt{3}R$
B) $R/\sqrt{3}$
C) $\left( \frac{2}{\sqrt{3}} \right)R$
D) $\frac{R}{2\sqrt{3}}$
• question_answer18) A point source of light is placed 4m below he surface of water of refractive index 5/ 3. The minimum diameter of a disc, which should be placed over the source, on the surface of water to cut-off all light coming out of water is
A) infinite
B) 6 m
C) 4m
D) 3m
• question_answer19) A ray falls on a prism ABC (AB =BC) and travels as shown in figure. The minimum refractive index of the prism material should be
A) $\frac{4}{3}$
B) $\sqrt{2}$
C) 1.5
D) $\sqrt{3}$
• question_answer20) The plane face of a planoconvex lens is silvered. If u be the refractive index and R, the radius of curvature of curved surface, then the system will behave like a concave mirror of radius of curvature
A) $\mu R$
B) $\frac{R}{(\mu -1)}$
C) $\frac{{{R}^{2}}}{\mu }$
D) $\left[ \frac{(\mu +1)}{(\mu -1)} \right]R$
• question_answer21) Two similar accumulators each of emf E and internal resistance r are connected as shown in the following figure. Then, the potential difference between x and y is
A) 2E
B) E
C) zero
D) None of these
• question_answer22) A conducting circular loop is placed in a uniform magnetic field of induction B tesla with its plane normal to the field. Now, the radius of the loop starts shrinking at the rate$\left( \frac{dr}{dt} \right)$. Then, the induced emf at the instant when the radius is r, is
A) $\pi rB\left( \frac{dr}{dt} \right)$
B) $2\pi rB\left( \frac{dr}{dt} \right)$
C) $\pi {{r}^{2}}\left( \frac{dr}{dt} \right)$
D) ${{\left( \frac{\pi {{r}^{2}}}{2} \right)}^{2}}B\left( \frac{dr}{dt} \right)$
• question_answer23) The first excitation potential of a given atom is 10.2V. Then, ionization potential must be
A) 20.4V
B) 13.6V
C) 30.6V
D) 40.8V
• question_answer24) A train is approaching with velocity 25 m/s towards a pedestrian standing on the track, frequency of horn of train is 1 kHz. Frequency heard by the pedestrain is (v= 350 m/s)
A) 1077 Hz
B) 1167 Hz
C) 985 Hz
D) 954 Hz
• question_answer25) Intensity of wave A is 91, while of wave B is 7. What is maximum and minimum intensity in YDSE?
A) 82$I$, 80$I$
B) 8$I$, 10$I$
C) 16$I$, 4$I$
D) 4$I$,$I$
• question_answer26) What happens inside optical fibre?
A) Diffraction
B) Polarization
C) Interference
D) Total internal reflection
• question_answer27) A manometer connected to a closed tap reads$3.5\times {{10}^{5}}\,N/{{m}^{2}}$. When the valve is opened, the reading of manometer falls to $3.0\times {{10}^{5}}\,N/{{m}^{2}}$, then velocity of flow of water is
A) 100 m/s
B) 10 m/s
C) 1m/s
D) $10\sqrt{10}$m/s
• question_answer28) Water is moving with a speed of 5.18 $m{{s}^{-1}}$ through a pipe with a cross-sectional area of 4.20$c{{m}^{2}}$. The water gradually descends 9.66 m as the pipe increase in area to 7.60$c{{m}^{2}}$. The speed of flow at the lower level is
A) 3.0$m{{s}^{-1}}$
B) 5.7$m{{s}^{-1}}$
C) 3.82$m{{s}^{-1}}$
D) 2.86$m{{s}^{-1}}$
• question_answer29) What is de-Broglie wavelength of electron having energy 10 keV?
A) $0.12\overset{\text{o}}{\mathop{\text{A}}}\,$
B) $1.2\overset{\text{o}}{\mathop{\text{A}}}\,$
C) $12.2\,\overset{\text{o}}{\mathop{\text{A}}}\,$
D) None of these
• question_answer30) Find beat frequency? Motion of two particles is given by ${{y}_{1}}=0.25\sin (310t)$ ${{y}_{2}}=0.25\sin (316t)$
A) 3
B) $\frac{3}{\pi }$
C) $\frac{6}{\pi }$
D) 6
• question_answer31) Half-life of radioactive substance is 3.20 h. What is the time taken for a 75% of substance to be used?
A) 6.38 h
B) 12 h
C) 4.18 day
D) 1.2day
• question_answer32) A capacitor of capacitance 1 $\mu F$ is filled with two dielectrics of dielectric constants 4 and 6. What is the new capacitance?
A) 10$\mu F$
B) 5$\mu F$
C) 4$\mu F$
D) 7$\mu F$
• question_answer33) The given combination represents the following gate
A) OR
B) XOR
C) NAND
D) NOR
• question_answer34) In BJT, maximum current flows in which of the following?
A) Emitter region
B) Base region
C) Collector region
D) Equal in all the regions
• question_answer35) In semiconductors at a room temperature
A) the valence band is partially empty and the conduction band is partially filled
B) the valence band is completely filled and the conduction band is partially filled
C) the valence band is completely filled
D) the conduction band is completely empty
• question_answer36) If coil is open then L and R become
A) $\infty ,0$
B) $0,\infty$
C) $\infty ,\infty$
D) 0, 0
• question_answer37) In a coil when current changes from 10A to 2A in time 0.1 s, induced emf is 3.28V. What is self-inductance of coil?
A) 4H
B) 0.4H
C) 0.04H
D) 5H
• question_answer38) Resistance of rod is 1$\Omega$. It is bent in form of square. What is resistance across adjoint corners?
A) 1$\Omega$
B) 3$\Omega$
C) $\frac{3}{16}\Omega$
D) $\frac{3}{4}\Omega$
• question_answer39) In a circuit L, C and R are connected in series with an alternating voltage source of frequency $f$. The current leads the voltage by $45{}^\circ$. The value of C is
A) $\frac{1}{2\pi f(2\pi fL+R)}$
B) $\frac{1}{\pi f(2\pi fL+R)}$
C) $\frac{1}{2\pi f(2\pi fL-R)}$
D) $\frac{1}{\pi f(2\pi fL-R)}$
• question_answer40) What is angle between electric field and equipotential surface?
A) $90{}^\circ \text{ }always$
B) $0{}^\circ \text{ }always$
C) $0{}^\circ \text{ }to\text{ }90{}^\circ$
D) $0{}^\circ \text{ }to\text{ }180{}^\circ$
• question_answer41) A ball falls from 20 m height on floor and rebounds to 5m. Time of contact is 0.02s. Find acceleration during impact.
A) $1200\,m/{{s}^{2}}$
B) $1000\,m/{{s}^{2}}$
C) $2000\,m/{{s}^{2}}$
D) $1500\,m/{{s}^{2}}$
• question_answer42) Two drops of equal radius coalesce to form a bigger drop. What is ratio of surface energy of bigger drop to smaller one?
A) ${{2}^{1/2}}:1$
B) $1:1$
C) ${{2}^{2/3}}:1$
D) None of these
• question_answer43) In any fission process the ratio $\frac{mass\,of\,fission\,products}{mass\,of\,parent\,nucleus}is$
A) less than 1
B) greater than 1
C) equal to 1
D) depends on the mass of parent nucleus
• question_answer44) In the phenomenon of diffraction of light. when blue light is used in the experiment instead of red light, then
A) fringes will become narrower
C) no change in fringe width
D) None of the above
• question_answer45) A glass slab $(\mu =1.5)$ of thickness 6 cm is placed over a paper. What is the shift in the letters?
A) 4 cm
B) 2 cm
C) 1 cm
D) None of these
• question_answer46) Two capacitors of capacitance C are connected in series. If one of them is filled with dielectric substance K, what is the effective capacitance?
A) $\frac{KC}{(1+K)}$
B) C (K + 1)
C) $\frac{2KC}{K+1}$
D) None of these
• question_answer47) A person is sitting in a lift accelerating upwards. Measured weight of person will be
A) less than actual weight
B) equal to actual weight
C) more than actual weight
D) None of the above
• question_answer48) By mistake a voltmeter is connected in series and an ammeter is connected in parallel with a resistance in an electrical circuit. What will happen to the instruments?
A) Voltmeter is damaged
B) Ammeter is damaged
C) Both are damaged
D) None is damaged
• question_answer49) The half-life of $A{{t}^{215}}$is 100 $\mu s$. If a sample contains 215 mg of $A{{t}^{215}}$, the activity of the sample initially is
A) ${{10}^{2}}Bq$
B) $3\times {{10}^{10}}Bq$
C) $4.17\times {{10}^{24}}Bq$
D) $1.16\times {{10}^{5}}Bq$
• question_answer50) The ratio of minimum to maximum wavelength in Balmer series is
A) 5 : 9
B) 5 : 36
C) 1 : 4
D) 3 : 4
• question_answer51) A ball is released from the top of a tower. The ratio of work done by force of gravity in first, second and third second of the motion of the ball is
A) 1 : 2 : 3
B) 1 : 4 : 9
C) 1 : 3 : 5
D) 1 : 5 : 3
• question_answer52) A body of mass 2 kg moving with a velocity of 3 m/s collides head on with a body of mass 1 kg moving in opposite direction with a velocity of 4 m/s. After collision two bodies stick together and move with a common velocity which in m/s is equal to
A) 1/4
B) 1/3
C) 2/3
D) 3/4
• question_answer53) Two spheres P and Q, of same colour having radii 8 cm and 2 cm are maintained at temperatures $127{}^\circ C$ and $527{}^\circ C$ respectively. The energy radiated by P and Q is
A) 0.054
B) 0.0034
C) 1
D) 2
• question_answer54) A cane is taken out from a refrigerator at $0{}^\circ C$. The atmospheric temperature is $25{}^\circ C$. If ${{t}_{1}}$ is the time taken to heat from $0{}^\circ C$ to $5{}^\circ C$ and ${{t}_{2}}$ is the time taken from $10{}^\circ C$ to $15{}^\circ C$, then
A) ${{t}_{1}}>{{t}_{2}}$
B) ${{t}_{1}}<{{t}_{2}}$
C) ${{t}_{1}}={{t}_{2}}$
D) there is no relation
• question_answer55) If an electron and a photon propagate in the form of waves having the same wavelength, it implies that they have the same
A) energy
B) momentum
C) velocity
D) angular momentum
• question_answer56) The work function of a substance is 4.0 eV. The longest wavelength of light that can cause photoelectron emission from this substance is approximately
A) 540 nm
B) 400 nm
C) 310nm
D) 220 nm
• question_answer57) Work function of a metal is 2.1 eV. Which of the waves of the following wavelengths will be able to emit photoelectrons from its surface?
A) $4000\overset{\text{o}}{\mathop{\text{A}}}\,,\,\,7500\overset{\text{o}}{\mathop{\text{A}}}\,$
B) $5500\overset{\text{o}}{\mathop{\text{A}}}\,,\,6000\overset{\text{o}}{\mathop{\text{A}}}\,$
C) $4000\overset{\text{o}}{\mathop{\text{A}}}\,,\,6000\overset{\text{o}}{\mathop{\text{A}}}\,$
D) None of the above
• question_answer58) A laser beam of pulse power ${{10}^{12}}$W is focused on an object of area${{10}^{-4}}c{{m}^{2}}$. The energy flux in watt/$c{{m}^{2}}$ at the point of focus is
A) ${{10}^{20}}$
B) ${{10}^{16}}$
C) ${{10}^{8}}$
D) ${{10}^{4}}$
• question_answer59) A laser device produces amplification in the
A) microwave region
B) ultraviolet or visible region
C) infrared region
D) None of the above
• question_answer60) Which of the following circular rods. (given radius r and length 0 each made of the same material as whose ends are maintained at the same temperature will conduct most heat?
A) $r=2{{r}_{0}};l=2{{l}_{0}}$
B) $r=2{{r}_{0}};l={{l}_{0}}$
C) $r={{r}_{0}};l={{l}_{0}}$
D) $r={{r}_{0}};l=2{{l}_{0}}$
• question_answer61) Which of the following is diamagnetic?
A) $H_{2}^{+}$
B) ${{O}_{2}}$
C) $L{{i}_{2}}$
D) $He_{2}^{+}$
A) $NO_{2}^{-}$
B) ${{H}_{2}}\overset{\centerdot \,\,\,\centerdot }{\mathop{N}}\,C{{H}_{2}}C{{H}_{2}}\overset{\centerdot \,\,\,\centerdot }{\mathop{N}}\,{{H}_{2}}$
C) ${{H}_{2}}O$
D) $:N{{H}_{3}}$
• question_answer63) By heating phenol with chloroform in alkali, it is converted into
A) salicylic acid
B) salicylaldehyde
C) anisole
D) phenyl benzoate
• question_answer64) Osmotic pressure observed when benzoic acid is dissolved in benzene is less than that expected from theoretical considerations. This is because
A) benzoic acid is an organic solute
B) benzoic acid has higher molar mass than benzene
C) benzoic acid gets associated in benzene
D) benzoic acid gets dissociated in benzene
• question_answer65) The formula mass of Moms salt is 392. The iron present in it is oxidized by$KMn{{O}_{4}}$in acid medium. The equivalent mass of Mohrs salt is
A) 392
B) 31.6
C) 278
D) 156
• question_answer66) Solubility product of a salt$AB$is$1\times {{10}^{-8}}{{M}^{2}}$in a solution in which the concentration of${{A}^{+}}$ions is${{10}^{-3}}M$. The salt will precipitate when the concentration of${{B}^{-}}$ions is kept
A) between${{10}^{-8}}M$to${{10}^{-7}}M$
B) between${{10}^{-7}}M$to${{10}^{-8}}M$
C) $>{{10}^{-5}}M$
D) $<{{10}^{-8}}M$
• question_answer67) The decomposition of a certain mass of$CaC{{O}_{3}}$gave$11.2\,\,d{{m}^{3}}$of$C{{O}_{2}}$gas at STP. The mass of$KOH$required to completely I neutralize the gas is
A) 56 g
B) 28 g
C) 42 g
D) 20 g
• question_answer68) The basicity of aniline is less than that of cyclohexylamine. This is due to
A) $+R-$effect of$-N{{H}_{2}}$group
B) $-I-$effect of$-N{{H}_{2}}$group
C) $-R-$effect of$-N{{H}_{2}}$group
D) hyperconjugation effect
• question_answer69) A distinctive and characteristic functional group of fats is
A) a peptide group
B) an ester group
C) an alcoholic group
D) a ketonic group
• question_answer70) Which of the following compound is expected to be optically active?
A) ${{(C{{H}_{3}})}_{2}}CHCHO$
B) $C{{H}_{3}}C{{H}_{2}}C{{H}_{2}}CHO$
C) $C{{H}_{3}}C{{H}_{2}}CHBrCHO$
D) $C{{H}_{3}}C{{H}_{2}}CB{{r}_{2}}CHO$
• question_answer71) Which cycloalkane has the lowest heat of combustion per$C{{H}_{2}}$group?
A) Cyclopropane
B) Cyclobutane
C) Cyclopentane
D) Cyclohexane
• question_answer72) The physical states of dispersing phase and dispersion medium in colloid like pesticide spray respectively, are
A) gas, liquid
B) solid, gas
C) liquid, solid
D) liquid, gas
• question_answer73) Potassium dichromate is used
A) in electroplating
B) as a reducing agent
C) it oxidizes ferrous ions into ferric ions in acidic media as an oxidizing agent
D) as an insecticide
• question_answer74) Which one of the following statements is incorrect for the sucrose?
A) It is obtained from cane sugar
B) It is not reducing sugar
C) On hydrolysis, it gives equal quantities of D-glucose and D-fructose
D) It gives aspartame when it is heated at${{210}^{o}}C$
A) displacement of $\sigma$-electrons
B) delocalisation of $\pi$-electrons
C) delocalisation of $\sigma$-electrons
D) displacement of $\pi$-electrons
• question_answer76) The atomic number of$Ni$and$Cu$are 28 and 29 respectively. The electronic configuration $1{{s}^{2}},\,\,2{{s}^{2}},\,\,2{{p}^{6}},\,\,3{{s}^{2}}\,\,3{{p}^{6}}\,\,3{{d}^{10}}$represents
A) $C{{u}^{+}}$
B) $C{{u}^{2+}}$
C) $N{{i}^{2+}}$
D) $Ni$
• question_answer77) In which of the following complex ion, the central metal ion is in a state of$s{{p}^{3}}{{d}^{2}}$hybridisation?
A) ${{[CoF]}^{3-}}$
B) ${{[CO{{(N{{H}_{3}})}_{6}}]}^{3+}}$
C) ${{[Fe{{(CN)}_{6}}]}^{3-}}$
D) ${{[Cr{{(N{{H}_{3}})}_{6}}]}^{3+}}$
• question_answer78) The formation of$O_{2}^{+}{{[Pt{{F}_{6}}]}^{-}}$is the basis for the formation of xenon fluorides. This is because
A) ${{O}_{2}}$and$Xe$have comparable sizes
B) Both${{O}_{2}}$and$Xe$are gases
C) ${{O}_{2}}$and$Xe$have comparable ionization energies
D) Both (a) and (c)
• question_answer79) The density of a gas is$1.964\,\,g\,\,d{{m}^{-3}}$at$273\,\,K$and$76\,\,cm\,\,Hg$. The gas is
A) $C{{H}_{4}}$
B) ${{C}_{2}}{{H}_{6}}$
C) $C{{O}_{2}}$
D) $Xe$
• question_answer80) $\Delta {{G}^{o}}vs\,\,T$plot in the Ellinghams diagram slopes downwards for the reactions
A) $Mg+\frac{1}{2}{{O}_{2}}\xrightarrow{{}}MgO$
B) $2Ag+\frac{1}{2}{{O}_{2}}\xrightarrow{{}}A{{g}_{2}}O$
C) $CO+\frac{1}{2}{{O}_{2}}\xrightarrow{{}}C{{O}_{2}}$
D) All of the above
• question_answer81) When a mixture of calcium benzoate and calcium acetate is dry distilled, the resulting compound is
A) acetophenone
B) benzaldehyde
C) benzophenone
D) acetaldehyde
• question_answer82) In a metallic crystal
A) the valence electrons constitute a sea of mobile electrons
B) the valence electrons are localized in between the kernels
C) the valence electrons remain within the field of influence of their own kernels
D) None of the above
• question_answer83) Which of the following is correct, based on molecular orbital theory for peroxide ion?
A) Its bond order is one and it is paramagnetic
B) Its bond order is two and it is diamagnetic
C) Its bond order is one and it is diamagnetic
D) Its bond order is two and it is paramagnetic
• question_answer84) Insulin regulates the metabolism of
A) minerals
B) ammo acids
C) glucose
D) vitamins
• question_answer85) Which of the following electrolyte will have maximum flocculation value for$Fe{{(OH)}_{3}}$sol?
A) $NaCl$
B) $N{{a}_{2}}S$
C) ${{(N{{H}_{4}})}_{3}}P{{O}_{4}}$
D) ${{K}_{2}}S{{O}_{4}}$
• question_answer86) The concentration of a reactant X decreases from$0.1M$to$0.005M$in 40 min. If the reaction follows first order kinetics, the rate of the reaction when the concentration of$X$is $0.01\,\,M$will be
A) $1.73\times {{10}^{-4}}M\,\,{{\min }^{-1}}$
B) $3.47\times {{10}^{-4}}M\,\,{{\min }^{-1}}$
C) $3.47\times {{10}^{-5}}M\,\,{{\min }^{-1}}$
D) $7.5\times {{10}^{-4}}M\,\,{{\min }^{-1}}$
• question_answer87) At$pH=4$, glycine exists as
A) ${{H}_{3}}N-C{{H}_{2}}-CO{{O}^{-}}$
B) ${{H}_{3}}N-C{{H}_{2}}-COOH$
C) ${{H}_{2}}N-C{{H}_{2}}-COOH$
D) ${{H}_{2}}N-C{{H}_{2}}-CO{{O}^{-}}$
• question_answer88) Which of the following taking place in the blast furnace is endothermic?
A) $CaC{{O}_{3}}\xrightarrow{{}}CaO+C{{O}_{2}}$
B) $2C+{{O}_{2}}\xrightarrow{{}}2CO$
C) $C+{{O}_{2}}\xrightarrow{{}}C{{O}_{2}}$
D) $F{{e}_{2}}{{O}_{3}}+3CO\xrightarrow{{}}2Fe+3C{{O}_{2}}$
• question_answer89) The$emf\,\,{{E}^{o}}$of the following cells are: $AG|A{{g}^{+}}(1M)||C{{u}^{2+}}(1M)|Cu;\,\,{{E}^{o}}=-0.46\,\,V$ $Zn|Z{{n}^{2+}}(1M)||C{{u}^{2+}}(1M)|Cu;\,\,{{E}^{o}}=1.10\,\,V$ emf of the following cell is $Zn|Z{{n}^{2+}}(1M)||A{{g}^{+}}(1M)|Ag$
A) $0.64\,\,V$
B) $1.10\,\,V$
C) $1.56\,\,V$
D) $-0.64\,\,V$
• question_answer90) The formation of cyanohydrin from acetone is which type of reaction?
A) Electrophilic substitution reaction
D) Nucleophilic substitution reaction
• question_answer91) Name the end product in the following series of reactions $C{{H}_{3}}COOH\xrightarrow{N{{H}_{3}}}A\xrightarrow{Heat}B\xrightarrow[\Delta ]{{{P}_{4}}{{O}_{14}}}C$
A) $C{{H}_{3}}OH$
B) $C{{H}_{4}}$
C) $C{{H}_{3}}COON{{H}_{4}}$
D) $C{{H}_{3}}CN$
• question_answer92) The presence of unpaired electron in phosphorous atom is explained by which principle?
A) Aufbau principle
B) Faults exclusion principle
C) Hunds rule
D) Heisenbergs principle
• question_answer93) If a cricket ball having mass of$200\,\,g$is thrown with a speed of$3\times {{10}^{3}}\,\,cm/s$, then calculate the wavelength related to it.
A) $2.2\times {{10}^{-27}}cm$
B) $1.104\times {{10}^{-32}}cm$
C) $1.104\times {{10}^{-32}}cm$
D) $1.104\times {{10}^{-33}}cm$
• question_answer94) Which type of stacking pattern is found in sodium chloride crystal lattice?
A) $a-b-a-b$
B) $a-a-a$
C) $a-b-c-a-b-c$
D) None of these
• question_answer95) Equivalent weight of a bivalent metal is 37.2. The molecular weight of its chloride is
A) 412.2
B) 216
C) 145.4
D) 108.2
• question_answer96) Phenolphthalein is obtained by heating phthalic anhydride with $conc.{{H}_{2}}S{{O}_{4}}$and
A) benzyl alcohol
B) benzene
C) phenol
D) benzoic acid
• question_answer97) Freezing point of urea solution is$-{{0.6}^{o}}C$. How much urea$(m.wt.=60g/mol)$will be required to dissolve in 3 kg water? $({{k}_{f}}={{1.5}^{o}}C\,\,kg\,\,mo{{l}^{-1}})$
A) 24 g
B) 36 g
C) 60 g
D) 72 g
• question_answer98) If$K<1.0$, what will be the value of$\Delta {{G}^{o}}$of the following?
A) Zero
B) 1.0
C) Positive
D) Negative
• question_answer99) The normality of a solution containing 32.5 g of${{(COOH)}_{2}}\cdot 2{{H}_{2}}O$per$0.5\,\,L$is
A) $10\,\,N$
B) $1\,\,N$
C) $2\,\,N$
D) $0.1\,\,N$
• question_answer100) For the titration of$KOH$vs${{(COOH)}_{2}}\cdot 2{{H}_{2}}O$, the suitable indicator is
A) methyl orange
B) phenolphthalein
C) methyl red
D) All can be used
• question_answer101) The radius of$N{{a}^{+}}$is$95\,\,\text{pm}$and that of$C{{l}^{-}}$ion is$181\,\,\text{pm}$. The coordination number of$N{{a}^{+}}$is
A) 8
B) 6
C) 4
D) unpredictable
• question_answer102) In van der Waals equation of state of the gas law, the constant V is a measure of
A) intermolecular repulsion
B) intermolecular attraction
C) volume occupied by the molecules
D) intermolecular collisions per unit volume
• question_answer103) Which reaction intermediate is formed during the condensation reaction between acetaldehyde and formaldehyde?
A) $:C{{H}_{2}}CHO$
B) $\overset{+}{\mathop{C}}\,{{H}_{2}}CHO$
C) $\overset{+}{\mathop{C}}\,{{H}_{2}}OH$
D) $:\bar{C}HCHO$
• question_answer104) $2,\,\,2\mathbf{-}$dichloro propane on hydrolysis yields
A) acetone
B) $2,\,\,2\mathbf{-}$propane diol
C) iso-propyl alcohol
D) acetaldehyde
• question_answer105) Phenols are more acidic than alcohols because
A) phenoxide ion is stabilized by resonance
B) phenols are more soluble in polar solvents
C) phenoxide ions do not exhibit resonance
D) alcohols do not lose H atoms at all
• question_answer106) Lemon gives sour taste because of
A) citric acid
B) tartaric acid
C) oxalic acid
D) acetic acid
• question_answer107) When ammonium chloride is added to ammonia solution, the pH of the resulting solution will be
A) increased
B) seven
C) decreased
D) unchanged
• question_answer108) Which of the following has highest second ionization energy?
A) Calcium
B) Chromium
C) Iron
D) Cobalt
• question_answer109) The standard reduction potentials at$298K$for the following half-cell reactions are given below $Z{{n}^{2+}}(aq)+2{{e}^{-}}Zn(s)-0.762$ $C{{r}^{3+}}(aq)+3{{e}^{-}}Cr(s)-0.74$ $2{{H}^{+}}(aq)+2{{e}^{-}}{{H}_{2}}(g)-0.00$ $F{{e}^{3+}}(aq)+{{e}^{-}}F{{e}^{2+}}(aq)-0.77$ Which one of the following is the strongest reducing agent?
A) $Zn(s)$
B) $Cr(s)$
C) ${{H}_{2}}(g)$
D) $F{{e}^{2+}}(aq)$
• question_answer110) Following reaction, ${{(C{{H}_{3}})}_{3}}CBr+{{H}_{2}}O\xrightarrow{{}}{{(C{{H}_{3}})}_{3}}COH+HBr$is an example of
A) elimination reaction
C) nucleophilic substitution
D) electrophilic substitution
• question_answer111) The unit of rate for a first order reaction is
A) $L\,\,{{s}^{-1}}$
B) $mo{{l}^{-1}}\,\,L\,\,{{s}^{-1}}$
C) $mol\,\,{{L}^{-1}}\,\,{{s}^{-1}}$
D) $mol\,\,{{s}^{-1}}$
• question_answer112) $Iso-$propyl amine with excess of acetyl chloride will give
A) ${{(C{{H}_{3}}CO)}_{2}}N-C-{{(C{{H}_{3}})}_{3}}$
B) ${{(C{{H}_{3}})}_{2}}CH-\underset{\begin{smallmatrix} | \\ H \end{smallmatrix}}{\mathop{N}}\,-COC{{H}_{3}}$
C) ${{(C{{H}_{3}})}_{2}}CHN{{(CO{{H}_{3}})}_{2}}$
D) $C{{H}_{3}}C{{H}_{2}}C{{H}_{2}}-\underset{\begin{smallmatrix} | \\ H \end{smallmatrix}}{\mathop{N}}\,-COC{{H}_{3}}$
• question_answer113) ${{C}_{2}}{{H}_{5}}CHO$and${{(C{{H}_{3}})}_{2}}CO$can be distinguished by testing with
A) phenyl hydrazine
B) hydroxyl amine
C) Fehling solution
D) sodium bisulphite
• question_answer114) Glucose molecule reacts with$X$number of molecules of phenylhydrazine to yield osazone. The value of T is
A) four
B) one
C) two
D) three
• question_answer115) The oxidation number of chromium in$Cr{{O}_{5}}$is
A) +3
B) + 5
C) + 10
D) + 6
• question_answer116) Liquor ammonia bottles are opened only after cooling. This is because
A) it is a mild explosive
B) it generates high vapour pressure
C) Both (a) and (b)
D) it is lachrymatory
• question_answer117) What will be the proportion of moles of metal $(Cu:Ni:Ag)$at cathode according to the second law of Faraday?
A) $1:2:1$
B) $2:2:1$
C) $1:2:2$
D) $1:1:2$
• question_answer118) Which equation is true to calculate the energy of activation, if the rate of reaction is doubled by increasing temperature from${{T}_{1}}K$to${{T}_{2}}K$?
A) ${{\log }_{10}}\frac{{{k}_{1}}}{{{k}_{2}}}=\frac{{{E}_{a}}}{2.303R}\left[ \frac{1}{{{T}_{1}}}-\frac{1}{{{T}_{2}}} \right]$
B) ${{\log }_{10}}\frac{{{k}_{2}}}{{{k}_{1}}}=\frac{{{E}_{a}}}{2.303R}\left[ \frac{1}{{{T}_{2}}}-\frac{1}{{{T}_{1}}} \right]$
C) ${{\log }_{10}}\frac{1}{2}=\frac{{{E}_{a}}}{2.303}\left[ \frac{1}{{{T}_{2}}}-\frac{1}{{{T}_{1}}} \right]$
D) ${{\log }_{10}}2=\frac{{{E}_{a}}}{2.303R}\left[ \frac{1}{{{T}_{1}}}-\frac{1}{{{T}_{2}}} \right]$
• question_answer119) For a reversible reaction : $X(g)+3Y(g)2Z(g);\,\,\Delta H=-40kJ$ the standard entropies of$X,\,\,Y$and$Z$are 60, 40 and 50 $J{{K}^{-1}}mo{{l}^{-1}}$respectively. The temperature at which the above reaction attains equilibrium is about
A) $400\,\,K$
B) $500\,\,K$
C) $273\,\,K$
D) $373\,\,K$
• question_answer120) Which of the following gives aldol condensation reaction?
A) ${{C}_{6}}{{H}_{5}}OH$
B) ${{C}_{6}}{{H}_{5}}-\overset{\begin{smallmatrix} O \\ || \end{smallmatrix}}{\mathop{C}}\,-{{C}_{6}}{{H}_{5}}$
C) $C{{H}_{3}}C{{H}_{2}}-\overset{\begin{smallmatrix} O \\ || \end{smallmatrix}}{\mathop{C}}\,-C{{H}_{3}}$
D) ${{(C{{H}_{3}})}_{3}}C-\overset{\begin{smallmatrix} O \\ || \end{smallmatrix}}{\mathop{C}}\,-{{C}_{6}}{{H}_{5}}$
• question_answer121) The range of the function$f\left( x \right)={{x}^{2}}+\frac{1}{{{x}^{2}}+1}$is
A) $[1,\,\,\infty )$
B) $[2,\,\,\infty )$
C) $\left[ \frac{3}{2},\,\,\infty \right)$
D) None of these
• question_answer122) If$f\left( x \right)=\left\{ \begin{matrix} a{{x}^{2}}+b, & b\ne 0,\,\,x\le 1 \\ b{{x}^{2}}+ax+c, & x>1 \\ \end{matrix} \right.$,then $f\left( x \right)$is continuous and differentiable at$x=1$, if
A) $c=0,\,\,a=2b$
B) $a=b,\,\,c\in R$
C) $a=b,\,\,c=0$
D) $a=b,\,\,c\ne 0$
• question_answer123) If a circle passes through the point$(1,\,\,2)$and cuts the circle${{x}^{2}}+{{y}^{2}}=4$orthogonally, then the equation of the locus of its centre is
A) ${{x}^{2}}+{{y}^{2}}-3x-8y+1=0$
B) ${{x}^{2}}+{{y}^{2}}-2x-6y-7=0$
C) $2x+4y-9=0$
D) $2x+4y-1=0$
• question_answer124) If$\int{f\left( x \right)}\sin x\cos x\,\,dx$ $=\frac{1}{2({{b}^{2}}-{{a}^{2}})}\log [f(x)]+c,$ then$f\left( x \right)$is equal to
A) $\frac{1}{{{a}^{2}}{{\sin }^{2}}x+{{b}^{2}}{{\cos }^{2}}x}$
B) $\frac{1}{{{a}^{2}}{{\sin }^{2}}x-{{b}^{2}}{{\cos }^{2}}x}$
C) $\frac{1}{{{a}^{2}}{{\cos }^{2}}x-{{b}^{2}}{{\sin }^{2}}x}$
D) $\frac{1}{{{a}^{2}}{{\cos }^{2}}x+{{b}^{2}}{{\sin }^{2}}x}$
• question_answer125) The points representing complex number 2 for which$\left| z-3 \right|=\left| z-5 \right|$lie on the locus given by
A) an ellipse
B) a circle
C) a straight line
D) None of the above
• question_answer126) The value of$\alpha$, for which the equation${{x}^{2}}-(\sin \alpha -2)x-(1+\sin \alpha )=0$has roots whose sum of square is least, is
A) $\frac{\pi }{3}$
B) $\frac{\pi }{4}$
C) $\frac{\pi }{2}$
D) $\frac{\pi }{6}$
• question_answer127) For$n\in N,\,\,{{10}^{n-2}}\ge 81n$, is
A) $n>5$
B) $n\ge 5$
C) $n<5$
D) $n>8$
• question_answer128) The two consecutive terms in the expansion of${{(3+2x)}^{74}}$whose coefficients are equal are
A) 11, 12
B) 7, 8
C) 30, 31
D) None of these
• question_answer129) The value of$2.\overline{357}$is
A) $\frac{2355}{999}$
B) $\frac{2355}{1000}$
C) $\frac{2355}{1111}$
D) None of these
• question_answer130) Let${{S}_{n}}=\frac{1}{{{1}^{3}}}+\frac{1}{{{1}^{3}}}+\frac{2}{{{2}^{3}}}+...+\frac{1+2+...+n}{{{1}^{3}}+{{2}^{3}}+...+n}$ $n=1,2,3,...$. Then,${{S}_{n}}$is not greater than
A) $\frac{1}{2}$
B) 1
C) 2
D) 4
• question_answer131) If$E(\theta )=\left[ \begin{matrix} {{\cos }^{2}}\theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & {{\sin }^{2}}\theta \\ \end{matrix} \right]$and$\theta$and$\phi$ differ by an odd multiple of$\frac{\pi }{2}$, then$E(\theta )E(\phi )$is a
A) unit matrix
B) null matrix
C) diagonal matrix
D) None of the above
• question_answer132) A parabola is drawn with its focus at$(3,\,\,4)$and vertex at the focus of the parabola${{y}^{2}}-12x-4y+4=0$. The equation of the parabola is
A) ${{y}^{2}}-8x-6y+25=0$
B) ${{y}^{2}}-6x+8y-25=0$
C) ${{x}^{2}}-6x-8y+25=0$
D) ${{x}^{2}}+6x-8y-25=0$
• question_answer133) If$p,\,\,p$denote the lengths of the perpendiculars from the focus and the centre of an ellipse with semi major axis of length a respectively on a tangent to the ellipse and$r$denotes the focal distance of the point, then
A) $ap=rp+1$
B) $rp=ap$
C) $ap=rp+1$
D) $ap=rp$
• question_answer134) The equation of perpendicular bisectors of sides$AB$and$AC$of a$\Delta ABC$are$x-y+5=0$ and$x+2y=0$respectively. If the coordinates of vertex$A$are$(1,\,\,-2)$, the equation of$BC$is
A) $14x+23y-40=0$
B) $14x-23y+40=0$
C) $23x+14y-40=0$
D) $23x-14y+40=0$
• question_answer135) If$\cos \theta =-\frac{\sqrt{3}}{2}$and$\sin \alpha =-\frac{3}{5}$, where$\theta$does not lie in the third quadrant, then $\frac{2\tan \alpha +\sqrt{3}\tan \theta }{{{\cot }^{2}}\theta +\cos \alpha }$is equal to
A) $\frac{7}{22}$
B) $\frac{5}{22}$
C) $\frac{9}{22}$
D) $\frac{22}{5}$
• question_answer136) A parallelogram is constructed on the vectors $\overset{\to }{\mathop{\mathbf{a}}}\,=3\overset{\to }{\mathop{\alpha }}\,-\overset{\to }{\mathop{\beta }}\,,\,\,\overset{\to }{\mathop{\mathbf{b}}}\,=\overset{\to }{\mathop{\alpha }}\,+3\overset{\to }{\mathop{\beta }}\,$, if$\left| \overset{\to }{\mathop{\alpha }}\, \right|=\left| \overset{\to }{\mathop{\beta }}\, \right|=2$and angle between$\overset{\to }{\mathop{\alpha }}\,$and$\overset{\to }{\mathop{\beta }}\,$is$\frac{\pi }{3}$, then length of a diagonal of the parallelogram is
A) $4\sqrt{5}$
B) $4\sqrt{3}$
C) $4\sqrt{17}$
D) None of the above
• question_answer137) The value of$c$, so that for all real$x$, the vectors$cx\widehat{\mathbf{i}}-6\widehat{\mathbf{j}}+3\widehat{\mathbf{k}}$,$x\widehat{\mathbf{i}}+2\widehat{\mathbf{j}}+2cx\widehat{\mathbf{k}}$make an obtuse angle, are
A) $c<0$
B) $0<c<\frac{4}{3}$
C) $-\frac{4}{3}<c<0$
D) $c>0$
• question_answer138) The solution of the equation $y-x\frac{dy}{dx}=a\left( {{y}^{2}}+\frac{dy}{dx} \right)$
A) $y=c(x+a)(1-ay)$
B) $y=c(x+a)(1+ay)$
C) $y=c(x-a)(1+ay)$
D) None of the above
• question_answer139) The order of the differential equation whose general solution is given by$y=({{c}_{1}}+{{c}_{2}})\cos (x+{{c}_{3}})-{{c}_{4}}{{e}^{x\_{{c}_{5}}}}$, where${{c}_{1}},\,\,{{c}_{2}},\,\,{{c}_{3}},\,\,{{c}_{4}},\,\,{{c}_{5}}$are arbitrary constants, is
A) 4
B) 3
C) 2
D) 5
• question_answer140) If$f(x)={{x}^{3}}+b{{x}^{2}}+cx+d$and$0<{{b}^{2}}<c$, then in$(-\infty ,\,\,\infty )$
A) $f(x)$is strictly increasing function
B) $f(x)$has a local maxima
C) $f(x)$strictly decreasing function
D) $f(x)$is bounded
• question_answer141) $\frac{d}{dx}{{\sin }^{-1}}(x\sqrt{1-x}+\sqrt{x}\sqrt{1-{{x}^{2}}})$
A) $-\frac{1}{2x\sqrt{1-x}}-\frac{1}{\sqrt{1-{{x}^{2}}}}$
B) $\frac{1}{2\sqrt{x}\sqrt{1-x}}-\frac{1}{\sqrt{1-{{x}^{2}}}}$
C) $\frac{1}{2\sqrt{x}\sqrt{1-x}}+\frac{1}{\sqrt{1-{{x}^{2}}}}$
D) $-\frac{1}{2\sqrt{x}\sqrt{1-x}}+\frac{1}{\sqrt{1-{{x}^{2}}}}$
• question_answer142) $\int{\frac{x{{\tan }^{-1}}x}{{{(1+{{x}^{2}})}^{3}}}}dx$is equal to
A) $\frac{x-{{\tan }^{-1}}x}{1-{{x}^{2}}}+c$
B) $\frac{x+{{\tan }^{-1}}x}{\sqrt{1-{{x}^{2}}}}+c$
C) $\frac{x-{{\tan }^{-1}}x}{\sqrt{1+{{x}^{2}}}}+c$
D) $\frac{x+\sqrt{1-{{x}^{2}}}}{\sqrt{1+{{x}^{2}}}}+c$
• question_answer143) Let$A=\left[ \begin{matrix} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \\ \end{matrix} \right]$, where$0\le \theta \le 2\pi$. Then, the range of$\left| A \right|$is
A) 0
B) {2, 4}
C) [2, 4]
D) None of the above
• question_answer144) If$y=\left| \cos x \right|+\left| \sin x \right|$, then$\frac{dy}{dx}$at$x=\frac{2\pi }{3}$is
A) 0
B) 1
C) $\frac{1-\sqrt{3}}{2}$
D) $\frac{\sqrt{3}-1}{2}$
• question_answer145) $\underset{n\to \infty }{\mathop{\lim }}\,\left( \frac{1}{1-{{n}^{2}}}+\frac{2}{1-{{n}^{2}}}+...+\frac{n}{1-{{n}^{2}}} \right)$is equal to
A) 0
B) $-\frac{1}{2}$
C) $\frac{1}{2}$
D) None of these
• question_answer146) Let$f(x)=\left\{ \begin{matrix} \sin x, & x\ne n\pi \\ 2 & x=n\pi \\ \end{matrix} \right.$, where$n\in I$and$g(x)=\left\{ \begin{matrix} {{x}^{2}}+1, & x\ne 2 \\ 3, & x=2 \\ \end{matrix} \right.$, then$\underset{x\to 0}{\mathop{\lim }}\,g[f(x)]$is
A) 1
B) 0
C) 3
D) does not exist
• question_answer147) The intercept made by the tangent to the curve$y=\int_{0}^{x}{|t|}\,\,dt$,which is parallel to the line $y=2x$, on x-axis is equal to
A) 1
B) -2
C) 2
D) None of these
• question_answer148) If$P={{x}^{3}}-\frac{1}{{{x}^{3}}}$and$Q=x-\frac{1}{x},\,\,x\in (0,\,\,x)$, then minimum value of$\frac{P}{{{Q}^{2}}}$is
A) $2\sqrt{3}$
B) $-2\sqrt{3}$
C) does not exist
D) None of these
• question_answer149) Locus of the point which divides double ordinate of the ellipse$\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$in the ratio 1:2 internally, is
A) $\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{9{{y}^{2}}}{{{b}^{2}}}=\frac{1}{9}$
B) $\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{9{{y}^{2}}}{{{b}^{2}}}=1$
C) $\frac{9{{y}^{2}}}{{{a}^{2}}}+\frac{9{{y}^{2}}}{{{b}^{2}}}=1$
D) None of these
• question_answer150) From any point on the hyperbola$\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1$tangents are drawn to the hyperbola$\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=2$. The area cut-off by the chord of contact on the asymptotes is equal to
A) $\frac{ab}{2}$
B) $ab$
C) $2ab$
D) $4ab$
• question_answer151) The value of the sum of the series$3{{\cdot }^{n}}{{C}_{0}}-8{{\cdot }^{n}}{{C}_{1}}+{{13}^{n}}{{C}_{2}}-18{{\cdot }^{n}}{{C}_{3}}+...$upto$(n+1)$terms is
A) 0
B) ${{3}^{n}}$
C) ${{5}^{n}}$
D) None of these
• question_answer152) If$A$is a skew-symmetric matrix, then trace of$A$is
A) 1
B) -1
C) 0
D) None of these
• question_answer153) The arbitrary constant on which the value of the determinant $\left| \begin{matrix} 1 & \alpha & {{\alpha }^{2}} \\ \cos (p-d)a & \cos pa & \cos (p-d)a \\ \sin (p-d)a & \sin pa & \sin (p-d)a \\ \end{matrix} \right|$does not depend, is
A) $\alpha$
B) p
C) d
D) a
• question_answer154) The sum of the first n terms of the series$\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+...$is equal to
A) ${{2}^{n}}-n+1$
B) $1-{{2}^{n}}$
C) $n+{{2}^{-n}}-1$
D) ${{2}^{n}}-1$
• question_answer155) The base of a cliff is circular. From the extremities of a diameter of the base angles of elevation of the top of the cliff are ${{30}^{o}}$and${{60}^{o}}$. If the height of the cliff be 500 m, then the diameter of the base of the cliff is
A) $\frac{2000}{\sqrt{3}}m$
B) $\frac{1000}{\sqrt{3}}m$
C) $\frac{2000}{\sqrt{3}}m$
D) $1000\sqrt{3}\,m$
• question_answer156) The most general solutions of the equation$\sec x-1=(\sqrt{2}-1)\tan x$are given by
A) $n\pi +\frac{\pi }{8}$
B) $2n\pi ,\,\,2n\pi +\frac{\pi }{4}$
C) $2n\pi$
D) None of these
• question_answer157) The maximum value of$\sin \left( x+\frac{\pi }{6} \right)+\cos \left( x+\frac{\pi }{6} \right)$in the interval $\left( 0,\,\frac{\pi }{2} \right)$ is attained at
A) $x=\frac{\pi }{12}$
B) $x=\frac{\pi }{6}$
C) $x=\frac{\pi }{3}$
D) $x=\frac{\pi }{2}$
• question_answer158) If${{z}_{r}}=\cos \frac{r\alpha }{{{n}^{2}}}+i\sin \frac{r\alpha }{2}$, where$r=1,\,\,2,\,\,3,....,\,\,n,$then$\underset{n\to \infty }{\mathop{\lim }}\,\,\,{{z}_{1}},\,\,{{z}_{2}}...{{z}_{n}}$is equal to
A) $\cos \alpha +i\sin \alpha$
B) $\cos \left( \frac{\alpha }{2} \right)-i\sin \left( \frac{\alpha }{2} \right)$
C) ${{e}^{i\alpha /2}}$
D) $\sqrt[3]{{{e}^{i\alpha }}}$
• question_answer159) Negation of Paris is in France and London is in England is
A) Paris is in England and London is in France
B) Paris is not in France or London is not in England
C) Paris is in England or London is in France
D) None of the above
• question_answer160) The area enclosed between the curves$y={{x}^{3}}$and$y=\sqrt{x}$is
A) $\frac{5}{3}sq\,\,unit$
B) $\frac{5}{4}sq\,\,unit$
C) $\frac{5}{12}sq\,\,unit$
D) $\frac{12}{5}sq\,\,unit$
• question_answer161) Find the equation of the bisector of the obtuse angle between the lines$3x-4y+7=0$and$-12x-5y+2=0$,
A) $21x+77y-101=0$
B) $99x-27y+81=0$
C) $21x-77y+101=0$
D) None of the above
• question_answer162) The equation of curve passing through the point$\left( 1,\frac{\pi }{4} \right)$and having slope of tangent at any point$(x,\,\,y)$as$\frac{y}{x}-{{\cos }^{2}}\left( \frac{y}{x} \right)$is
A) $x={{e}^{1+\tan \left( \frac{y}{x} \right)}}$
B) $x={{e}^{1-\tan \left( \frac{y}{x} \right)}}$
C) $x={{e}^{1+\tan \left( \frac{x}{y} \right)}}$
D) $x={{e}^{1-\tan \left( \frac{x}{y} \right)}}$
• question_answer163) If$P(n):2+4+6+...+(2n),\,\,n\in N$, then$P(k)=k(k+1)+2$implies $P(k+1)=(k+1)(k+2)+2$is true for all$k\in N.$ So, statement$P(n)=n(n+1)+2$is true for
A) $n\ge 1$
B) $n\ge 2$
C) $n\ge 3$
D) None of these
• question_answer164) The differential equation of all non-vertical lines in a plane is
A) $\frac{{{d}^{2}}y}{d{{x}^{2}}}=0$
B) $\frac{{{d}^{2}}x}{d{{y}^{2}}}=0$
C) $\frac{dy}{dx}=0$
D) $\frac{dx}{dy}=0$
• question_answer165) The unit vector in ZOX plane and making angle ${{45}^{o}}$and ${{60}^{o}}$respectively with$\overset{\to }{\mathop{\mathbf{a}}}\,=2\widehat{\mathbf{i}}+2\widehat{\mathbf{j}}-\widehat{\mathbf{k}}$and$\overset{\to }{\mathop{\mathbf{b}}}\,=0\widehat{\mathbf{i}}+\widehat{\mathbf{j}}-\widehat{\mathbf{k}}$, is
A) $-\frac{1}{\sqrt{2}}\widehat{\mathbf{i}}+\frac{1}{\sqrt{2}}\widehat{\mathbf{k}}$
B) $\frac{1}{\sqrt{2}}\mathbf{\hat{i}}-\frac{1}{\sqrt{2}}\mathbf{\hat{k}}$
C) $\frac{1}{3\sqrt{2}}\widehat{\mathbf{i}}+\frac{4}{3\sqrt{2}}\widehat{\mathbf{j}}+\frac{1}{3\sqrt{2}}\widehat{\mathbf{k}}$
D) None of the above
• question_answer166) If$\int_{2}^{e}{\left( \frac{1}{\log x}-\frac{1}{{{(\log x)}^{2}}} \right)dx=a+\frac{b}{\log 2}}$,then
A) $a=e,\,\,b=-2$
B) $a=e,\,\,b=2$
C) $a=-e,\,\,b=2$
D) None of these
• question_answer167) The circle${{x}^{2}}+{{y}^{2}}-4x-4y+4=0$is inscribed in a triangle which has two of its sides along the coordinate axes. If the locus of the circumcentre of the triangle is$x+y-xy+k\sqrt{{{x}^{2}}+{{y}^{2}}}=0$, then the value of $k$is equal to
A) 2
B) 1
C) -2
D) 3
• question_answer168) The points of discontinuity of tan x are
A) $n\pi ,\,\,n\in I$
B) $2n\pi ,\,\,n\in I$
C) $(2n+1)\frac{\pi }{2},\,\,n\in I$
D) None of the above
• question_answer169) The two curves${{x}^{3}}-3x{{y}^{2}}+2=0$and$3{{x}^{2}}y-{{y}^{3}}-2=0$
A) cut at right angles
B) touch each other
C) cut at an angle$\frac{\pi }{3}$
D) cut at an angle$\frac{\pi }{4}$
• question_answer170) The period of the function $f(x)=\frac{\sin 8x\cos x-\sin 6x\cos 3x}{\cos 2x\cos x-\sin 3x\sin 4x}$
A) $\pi$
B) $2\pi$
C) $\frac{\pi }{2}$
D) None of these
• question_answer171) The derivative of$f(\tan x)$w.r.t.$g(\sec x)$at$x=\frac{\pi }{4}$, where$f(1)=2$and$g(\sqrt{2})=4$, is
A) $\frac{1}{\sqrt{2}}$
B) $\sqrt{2}$
C) 1
D) None of these
• question_answer172) If$a>0,\,\,b>0$the maximum area of the triangle formed by the points$O(0,\,\,0)$$A(a\cos \theta ,\,\,b\sin \theta )$and$B(a\cos \theta ,\,\,-b\sin \theta )$is (in sq unit)
A) $\frac{ab}{2}$when$\theta =\frac{\pi }{4}$
B) $\frac{3ab}{4}$when$\theta =\frac{\pi }{4}$
C) $\frac{ab}{2}$when$\theta =-\frac{\pi }{2}$
D) ${{a}^{2}}{{b}^{2}}$
• question_answer173) If the two curves $y={{a}^{x}}$and$y={{b}^{x}}$intersect at an angle$\alpha$, then tan a equals
A) $\frac{\log a-\log b}{1+\log a\log b}$
B) $\frac{\log a+\log b}{1-\log a\log b}$
C) $\frac{\log a-\log b}{1-\log a\log b}$
D) None of these
• question_answer174) The number of roots of the equation $x-\frac{2}{x-1}=1-\frac{2}{x-1}$
A) 1
B) 2
C) 0
D) infinitely many
• question_answer175) The vector $z=-4+5i$is turned counter clockwise through an angle of${{180}^{o}}$and stretched$1\frac{1}{2}$times. The complex number corresponding to newly obtained vector is
A) $-6+\frac{15}{2}i$
B) $6+\frac{15}{2}i$
C) $6-\frac{15}{2}i$
D) None of these
• question_answer176) If$A=[{{a}_{ij}}]$is a4$4\times 4$matrix${{C}_{ij}}$is the cofactor of the element${{a}_{ij}}$in$\left| A \right|$, then the expression${{a}_{11}}{{C}_{11}}+{{a}_{12}}{{C}_{12}}+{{a}_{13}}{{C}_{13}}+{{a}_{14}}{{C}_{14}}$equal to
A) 0
B) -1
C) 1
D) $\left| A \right|$
• question_answer177) If$A=\left\{ x:\frac{\pi }{6}\le x\le \frac{\pi }{3} \right\}$and $f(x)=\cos x-x(1+x)$, then$f(A)$is equal to
A) $\left[ -\frac{\pi }{3},\,\,-\frac{\pi }{6} \right]$
B) $\left[ \frac{\pi }{6},\,\,\frac{\pi }{3} \right]$
C) $\left[ \frac{1}{2}-\frac{\pi }{3}\left( 1+\frac{\pi }{3} \right),\,\,\frac{\sqrt{3}}{2}-\frac{\pi }{6}\left( 1+\frac{\pi }{6} \right) \right]$
D) $\left[ \frac{1}{2}+\frac{\pi }{3}\left( 1-\frac{\pi }{3} \right),\,\,\frac{\sqrt{3}}{2}+\frac{\pi }{6}\left( 1-\frac{\pi }{6} \right) \right]$
• question_answer178) The contrapositive of$(p\vee q)\Rightarrow r$is
A) $\tilde{\ }r\Rightarrow (p\vee q)$
B) $r\Rightarrow (p\vee q)$
C) $\tilde{\ }r\Rightarrow (\tilde{\ }p\wedge \tilde{\ }q)$
D) $p\Rightarrow (q\vee r)$
• question_answer179) If the function$f(x)=a{{x}^{3}}+b{{x}^{2}}+11x-6$satisfies the condition of Rollers theorem in$[1,\,\,3]$and$f\left( 2+\frac{1}{\sqrt{3}} \right)=0$, then the values of $a,\text{ }b$are respectively
A) - 1, 6
B) - 2, 1
C) 1, -6
D) $-1,\,\,\frac{1}{2}$
• question_answer180) If$f(x)=\cos (\log x)$, then $f\left( \frac{1}{x} \right)f\left( \frac{1}{y} \right)-\frac{1}{2}\left[ f\left( \frac{x}{y} \right)+f(xy) \right]$ is equal to
A) $\cos (x-y)$
B) $\log (x-y)$
C) $\cos (x+y)$
D) None of these
• question_answer181) If$\alpha ={{\sin }^{-1}}\frac{\sqrt{3}}{2}+{{\sin }^{-1}}\frac{1}{3}$and$\beta ={{\cos }^{-1}}\frac{\sqrt{3}}{2}+{{\cos }^{-1}}\frac{1}{3}$, then
A) $\alpha >\beta$
B) $\alpha =\beta$
C) $\alpha <\beta$
D) $\alpha +\beta =2\pi$
• question_answer182) If$1+\sin \theta +{{\sin }^{2}}\theta +...\infty =4+2\sqrt{3},\,\,0<\theta <\pi$, $\theta \ne \frac{\pi }{2}$, then
A) $\theta =\frac{\pi }{3}$
B) $\theta =\frac{\pi }{6}$
C) $\theta =\frac{\pi }{3}$or$\frac{\pi }{6}$
D) $\theta =\frac{\pi }{3}$or$\frac{2\pi }{3}$
• question_answer183) A round balloon of radius$r$subtends an angle $\alpha$at the eye of the observer, while the angle of elevation of its centre is$\beta$. The height of the centre of balloon is
A) $r\cos ec\alpha \sin \frac{\beta }{2}$
B) $r\sin \alpha \cos ec\frac{\beta }{2}$
C) $r\sin \frac{\alpha }{2}\cos ec\beta$
D) $r\cos ec\frac{\alpha }{2}\sin \beta$
• question_answer184) In$\Delta ABC,\,\,{{(a-b)}^{2}}{{\cos }^{2}}\frac{C}{2}+{{(a+b)}^{2}}{{\sin }^{2}}\frac{C}{2}$is equal to
A) ${{a}^{2}}$
B) ${{b}^{2}}$
C) ${{c}^{2}}$
D) None of these
• question_answer185) In a triangle$\left( 1-\frac{{{r}_{1}}}{{{r}_{2}}} \right)\left( 1-\frac{{{r}_{1}}}{{{r}_{2}}} \right)=2$, then the triangle is
A) right angled
B) isosceles
C) equilateral
D) None of these
• question_answer186) If$a+b+c=0$, then the roots of the equation $4a{{x}^{2}}+3bx+2c=0$are
A) equal
B) imaginary
C) real
D) None of these
A) Connected
B) Disconnected
C) Neither connected nor disconnected
D) None of the above
• question_answer188) The solution of${{\tan }^{-1}}x+2{{\cot }^{-1}}x=\frac{2\pi }{3}$is
A) $-\frac{1}{\sqrt{3}}$
B) $\frac{1}{\sqrt{3}}$
C) $-\sqrt{3}$
D) $\sqrt{3}$
• question_answer189) The conjugate of the complex number$\frac{{{(1+i)}^{2}}}{1-i}$is
A) $1-i$
B) $1+i$
C) $-1+i$
D) $-1-i$
• question_answer190) A graph$G$has$m$vertices of odd degree and $n$vertices of even degree. Then which of the following statements is necessarily true?
A) $m+n$is an odd number
B) $m+n$is an even number
C) $n+1$is an even number
D) $m+1$is an odd number
• question_answer191) The value of$\sin \left[ 2{{\cos }^{-1}}\frac{\sqrt{5}}{3} \right]$is
A) $\frac{\sqrt{5}}{3}$
B) $\frac{2\sqrt{5}}{3}$
C) $\frac{4\sqrt{5}}{9}$
D) $\frac{2\sqrt{5}}{9}$
• question_answer192) In the group$(G,{{\otimes }_{15}})$, where$G=\{3,\,\,6,\,\,9,\,\,12\}$, ${{\otimes }_{15}}$is multiplication modulo 15, the identity element is
A) 3
B) 6
C) 12
D) 9
• question_answer193) A group$(G,\,\,*)$has 10 elements. The minimum number of elements of$G$, which are their own inverses is
A) 2
B) 1
C) 9
D) 0
• question_answer194) $\frac{3{{x}^{2}}+1}{{{x}^{2}}-6x+8}$is equal to
A) $3+\frac{49}{2(x-4)}-\frac{13}{2(x-2)}$
B) $\frac{49}{2(x-4)}-\frac{13}{2(x-2)}$
C) $\frac{-49}{2(x-4)}+\frac{13}{2(x-2)}$
D) $\frac{49}{2(x-4)}+\frac{13}{2(x-2)}$
• question_answer195) The orthocentre of the triangle with vertices$O(0,\,\,0),\,\,A\left( 0,\,\,\frac{3}{2} \right),\,\,B(-5,\,\,0)$is
A) $\left( \frac{5}{2},\,\,\frac{3}{4} \right)$
B) $\left( \frac{-5}{2},\,\,\frac{3}{4} \right)$
C) $\left( -5,\,\,\frac{3}{2} \right)$
D) $(0,\,\,0)$
• question_answer196) The range in which$y=-{{x}^{2}}+6x-3$increasing, is
A) $x<3$
B) $x>3$
C) $7<x<8$
D) $5<x<6$
• question_answer197) The area bounded by the curve$x=4-{{y}^{2}}$and the y-axis is
A) 16 sq unit
B) 32 sq unit
C) $\frac{32}{3}$sq unit
D) $\frac{16}{3}$sq unit
• question_answer198) The number of positive divisors of 252 is
A) 9
B) 5
C) 18
D) 10
• question_answer199) The remainder obtained when 5124 is divided by 124 is
A) 5
B) 0
C) 2
D) 1
• question_answer200) Which of the following is not a group with respect to the given operation?
A) The set of even integers including zero under addition
B) The set of odd integers under addition
D) {1,-1} under multiplication
• question_answer201) He is so ...... of his own idea that he will not entertain any suggestion from others.
A) hopeful
B) enamoured
C) jealous
D) possessed
• question_answer202) Undoubtedly, English is the most...... spoken language in the world today.
B) widely
C) greatly
D) beautifully
• question_answer203) I will be leaving for Delhi tonight and ...... to return by this weekend.
A) waiting
B) plan
C) going
D) making
• question_answer204) The vacancy ......... by the dismissal of the superintendent is expected to be filled up by the promotion of a U.D.C.
B) created
C) caused
D) generated
A) Postpone
B) Accept
C) Bargain
D) Reject
A) Awful
B) Irrelevant
C) Shallow
D) Profound
A) Like
B) Eagerness
C) Disability
D) Dislike
A) Deal
B) Return
C) Lend
D) Exchange
A) Gallows
B) Suicide
C) Euphoria
D) Euthanasia
• question_answer210) The act of killing ones wife
A) Avicide
B) Canicide
C) Uxoricide
D) Genocide
• question_answer211) Stage between boyhood and youth
A) Infancy
C) Puberty
D) Maturity
• question_answer212) Lack of enough blood
A) Amnesia
B) Insomnia
C) Anaemia
D) Allergy
• question_answer213) To set the people by ears
A) To box the people
B) To insult and disgrace the people
C) To punish heavily
D) To excite people to a quarrel
• question_answer214) To give chapter and verse for a thing
A) To produce the proof of something
B) To eulogise the qualities of a thing
C) To make publicity of a thing
D) To attach artificial value to a thing
• question_answer215) Dog in the manger
A) An undersized bull almost the shape of a dog
B) A dog that has no kennel of its own
C) A person who puts himself in difficulties an account of other people
D) A person who prevents others from enjoying something useless to himself
• question_answer216) To blow hot and cold
A) Changing weather
B) To be untrustworthy
C) To change opinion often
D) To be rich and poor frequently
A) Casual
B) Cunning
C) Foolish
D) False
A) Conversation
B) Dialogue
C) Dramatic
D) Prologue
A) Spicy
B) Unfavorable
C) Conspicuous
D) Condemnatory
A) Encircled
B) Groped
C) Disfigured
D) Detached
• question_answer221) Four of the following five are alike in a certain way and so form a group. Which is the one that does not belong to that group?
A) Hill
B) Valley
C) Dam
D) River
• question_answer222) In a certain code CREAM is written as NBDBQ. How in BREAD written in that code?
A) EBFAQ
B) EBDAQ
C) BEDQA
D) BEFQA
• question_answer223) If black means white, white means red, red means yellow, yellow means blue, blue means green, green means purple and purple means orange, then what is the colour of lemon?
A) Green
B) Purple
C) Orange
D) Blue
• question_answer224) Directions: In the following questions, find the word which holds the same relation with the third word as there in between the first two word. Hot: Oven : : Cold :?
A) Ice cream
B) Air conditioner
C) Snow
D) Refrigerator
• question_answer225) Directions: In the following questions, find the word which holds the same relation with the third word as there in between the first two word. Push : Pull : : Throw :?
A) Jump
B) Collect
C) Pick
D) Game
• question_answer226) Directions: In each of the following questions, one letter or a set of letter is missing, you have to understand the pattern of the series and insert the appropriate letter? R, M, ?, F, D, ?
A) C, B
B) J, H
C) H, C
D) I, C
• question_answer227) Directions: In each of the following questions, one letter or a set of letter is missing, you have to understand the pattern of the series and insert the appropriate letter? - bcc - ac - aabb - ab -cc
A) aab ca
B) aba ca
C) ba cab
D) bca ca.
• question_answer228) How many 9s are there in the following number series which are immediately preceded by 3 and followed by 6? 3 9 6 9 3 9 3 9 3 9 6 3 9 3 6 3 9 5 6 9 5 6 9 3 9 6 3 9
A) 0
B) 3
C) 2
D) 4
• question_answer229) Some boys are sitting in a row, P in sitting fourteenth from the left and Q is seventh from the right. If these are four boys between P and Q, how many boys are there in the row?
A) 25
B) 23
C) 21
D) 19
• question_answer230) Pointing to a photograph, a woman says, this mans sons sister in my mother-in-law. How is the womans husband related to the man in the photograph?
A) Grandson
B) Son
C) Son-in-law
D) Nephew
• question_answer231) The longest canal in the world is
A) Volga Baltic
B) Beloye-more Baltic
C) Suez Canal
D) Grand China Canal
• question_answer232) The oldest Hindu epic is
A) Mahabhashya
B) Ramayan
C) Mahabharata
• question_answer233) Who among the following is not associated with the Swaraj Party?
A) C.R. Das
B) M.L. Kelkar
C) Motilal Nehru
D) Mahatma Gandhi
• question_answer234) Where is the City of palaces?
A) London
B) Kolkata
C) Patiala
D) Lucknow
• question_answer235) Our National Song is
A) Sare Jahan Se Achcha
B) Jana Gana Mana
C) Vande Mataram
D) All of the above
• question_answer236) The Constitution of India was adopted by the Constituent Assembly on
A) Dec 11, 1946
B) Aug 15, 1957
C) Nov 26, 1949
D) Jan 26, 1949
• question_answer237) Gandhijis Dandi March started from
A) Bardoli
C) Surat
D) Bombay
• question_answer238) Who was the viceroy of India at the time of formation of the Indian National Congress?
A) Lord Canning
B) Lord Dufferin
C) Lord Mayo
D) Lord Elgin
• question_answer239) Which country was a major donor in financing the SAARC?
A) Pakistan
B) Sri Lanka
C) India
• question_answer240) Where in the H. Q. of the European Economic Community?
A) Bonn
B) Rome
C) Brussels
D) Hague
|
2019-01-17 12:44:35
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7334662079811096, "perplexity": 4962.27175809015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583658981.19/warc/CC-MAIN-20190117123059-20190117145059-00152.warc.gz"}
|
http://therightfitdentist.com/hanpkavr/article.php?tag=4a2c15-molar-mass-and-percent-composition-worksheet
|
For a solution, mass percent equals the mass of an element in one mole of the compound divided by the molar Chemistry: Molar Mass and Percentage Composition KEY Calculate the molar masses and percentage composition of each of the following compounds. Zn3(PO4) 2 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 7. The answers appear after the final question. 3 mol O=3 × 16.00 g/mol= 48.00 g/mol. 16 divided by 18 is .89. Determine the empirical formula of acetic acid. The percentage composition of acetic acid is found to be 39.9% C, 6.7% H, and 53.4% O. 1. 12. CJ UVaJ j\, CJ EH��H*OJ QJ U^J jK!�C Percent composition and molar mass. Percent Review Worksheet Worksheets for all from Percent Composition Worksheet Answer Key With Work, source: bonlacfoods.com Worksheet 11.3 Percent Composition Mole Mass Problems Worksheet Answers Lovely Mole Mass Problems from molar mass worksheet answers , source:thefriendlyghosthunters.net The Molar Worksheets, when used with an example that shows the properties of the molecules in a particular compound, will show the composition of the compound and the answer to the question “What is Molar Matter?” that a person may be looking for. CaSO4. Answer: Molar mass is 22.99 + 35.45 = 58.44 g/mol Percent composition: %Na = (22.99 / 58.44) K 100% = 39.34 % Na 1) What oercent ofMgB½ is magnesium? Determine the empirical formula of acetic acid. 1. ' ( ) * + , 4 6 7 > ? 116.3 g, 20.9% Mg, 24.1% N, 55.0% O 4. The molar mass for question #9 was determined by experiment to be 60.0 g/mol. CJ UVaJ #� � � � � � � � � � � � � � � � � � � � � � � � � � � � Percent Composition by Mass Find the percent composition by mass the . CJ UVaJ j U j� CJ EH��H*OJ QJ U^J jU�C CJ UVaJ CJ H*OJ QJ ^J j CJ H*OJ QJ U^J j CJ EH��H*OJ QJ U^J o p q v w � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � The molar mass for question 9 was determined by experiment to be 600 gmol. CHEMISTRY COMPUTING FORMULA MASS WORKSHEET Problem Set-up example: Find the formula mass of Ca(NO3)2 Ca: 1 x 40.1 = 40.1 N: 2 x 14.0 = 28.0 O: 6 x 16.0 = 96.0 ____ This Molar Mass and Percentage Composition Worksheet is suitable for 9th - 12th Grade. A compound with an empirical formula of CFBrO and a molar mass of 254.7 grams per mole. R S T U Y Z m n o p u v � � � � � � � ��� � ��� �¸˸���������o��� c\� � j�# EH��Uj���C The percentage composition of acetic acid is found to be 39.9% C, 6.7% H, and 53.4% O. Essential concepts: Molar mass, percent composition. Purpose: The percent composition of a compound is the percent, by mass, of each individual element within the compound. A compound with an empirical formula of C 2H 8N and a molar mass of 46 grams per mole. Nicotine is 74.1% carbon, 8.6% hydrogen, and 17.3% nitrogen by mass. Round atomic masses to the tenth of a decimal place. Essential concepts:Molar mass, percent composition. Formula mass / molecular weight / moles / % percentage by mass worksheets with range of difficulties. C 6H 6O 2 _____ Answers: 1. Answer the following questions: 9. Molecular Mass And Percent Composition Mass percent composition describes the relative quantities of elements in a chemical compound. Mass percent composition is also known percent by weight. CaCO 3 CH 4 Sn(CO 3) 2 3. � � 6 _ � 6 _ _ � � � � 6 � U����� Z ^ � 6 � 0 � formula worksheet 1 h8n and a molar mass of 46 grams per mole answer the of 100 g mole if the percent composition is' 'Moles Molar Mass And Percentage Composition CHEMISTRY COMPUTING FORMULA MASS WORKSHEET Problem Set-up example: Find the formula mass of Ca(NO3)2 Ca: 1 x 40.1 = 40.1 N: 2 x 14.0 = 28.0 O: 6 x 16.0 = 96.0 ____ Ca3P2 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 2. A compound has an empirical formula of C2H30 and a molar mass of 172 g/mol. CJ UVaJ j�( CJ EH��H*OJ QJ U^J ja&�C 4) A compound with an empirical formula of C2H8N and a molar mass of 46 grams per mole. Na2SO4 4. Percent Review Worksheet Worksheets for all from Percent Composition Worksheet Answer Key With Work, source: bonlacfoods.com 18. 10. KCl Answers: 1. The units of mass are typically grams. Some of the worksheets for this concept are molar mass work molar mass work answer key chemistry computing formula mass work ws molar mass chemistry 11 mole conversions molar mass work molar mass … Determine the empirical formula of acetic acid. 'Molar Mass And Percent Composition Worksheet FREE May 13th, 2018 - Hi Searching For Molar Mass And Percent Composition Worksheet You Are Specifically Here Possibly You Came Via Internet Search Engine After That You Find This Web Site As Well As Chose To See This Website Many Thanks For That''South Pasadena Chemistry chemmybear com C 3H 3O 2) If the molar mass of the compound in problem 1 is 110 grams/mole, what’s the molecular formula? C D v w � � A compound with an empirical formula of CFBrO and a molar mass of 254.7 grams per mole. Percent By Mass. 132.1 g, 21.2% N, 6.1% H, 24.3% S, 48.4% O. Ca 3 P 2 g 182.3 P Ca g 62.0 g/mol 31.0 @ P 2 g 120.3 g/mol 40.1 @ Ca 3 2 3 Ca 66.0% 100 g 182.3 g 120.3 Ca % P 34.0% 100 g 182.3 g 62.0 P % 2. The percent composition worksheet. The percentage composition of acetic acid is found to be 39.9% C, 6.7% H, and 53.4% O. ' ( ) * + , 6 7 8 � � � � � � � � � � � � � � � � � � � � � � � � � � � � $a$ � � 8 9 : ; J K L M N O ^ _ a b c q r s t u v ~ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 4 � � G H � � � � & * 7 8 � � � � + , 6 7 � � � � > Answer the following questions: 9. For a solution, mass percent equals the mass of an element in one mole of the compound divided by the molar mass of the compound, multiplied by 100%. Show your work and always include units. Solutions to the Molar Mass Practice Worksheet: Important note to students: All of the units given here are “grams per mole”, which may be abbreviated as “g/mol”, “grams/mol”, or “g . Figure $$\PageIndex{3}$$: Table salt, NaCl, contains an array of sodium and chloride ions combined … Ca(OH) 2 3. KCl. If the molar mass of the compound in problem 1 is 110 grams/mole, what's the molecular formula? 5. This Chapter 8 Worksheet-Mass, Mole, Percent Composition Worksheet is suitable for 10th - 12th Grade. So Ca = 120*100/182 . CJ UVaJ j CJ H*OJ QJ U^J CJ H*OJ QJ ^J CJ OJ QJ ^J CJ OJ QJ ^J jQ EH��Uj}�C 2) A compound with an empirical formula of C4H4O and a molar mass of 136 grams per mole. Find the empirical formula of a compound that is 53.7% iron and 46.3% sulfur. Some of the worksheets for this concept are Percent composition and molecular formula work, Molar mass work, Percent composition and molecular formula work, Formula or molar mass work epub, Chemistry computing formula mass work, Formula work, Empirical and molecular formula work, Student work extra practice questions molar. 3. The sum all the mass percentages should add up to 100%. Useful for AQA C2.3. Mass percent composition is also known percent by weight. Calorimetry Worksheet; Mixed Gas Laws Worksheet; Molar Mass Worksheet; Mole Calculations Worksheet 1; Mole Calculations Worksheet 2; Percent Composition Worksheet Nomenclature Handouts. Some of the worksheets displayed are Percent composition by mass work, Chemistry computing formula mass work, Ch 11 ws 3 molarity molality percent solution, Work, Chemistry i work name calculating formula mass, Molar mass work, Percent composition by mass, Percent composition work ii. Find its empirical formula. (NH4) 2SO4 6. In this chemical compounds worksheet, students calculate the molar masses and the percentage composition for the given compounds. N 2 O Sr(NO 3) 2 NH 4 NO 3 4. Zn3(PO4) 2. Solutions to the Molar Mass Practice Worksheet: Percent Composition and Molecular Formula Worksheet I. I use this game instead of a worksheet to build students' confidence and fluency in these problems before we move … CJ UVaJ j� EH��UjV�C Mg(NO3) 2. One of the two worksheets is viewable in the Preview so you can see that the questions will be great practice for your students. You calculate the answer CJ UVaJ j UCJ H*OJ QJ ^J j CJ H*OJ QJ U^J jO CJ EH��H*OJ QJ U^J $2. Page I of I 'Hour 0)MDlar Mass = 136.2 g, 29.4% Ca, 23.6% S, 47.0% O 8. A compound is found to contain 36.5% Na, 25.4% S, and 38.1% O. Its molar mass is about 160 g./mol. CaCO 3 CH 4 Sn(CO 3) 2 3. Show your work, and always include units where needed. CJ UVaJ j CJ H*OJ QJ U^J !CJ OJ QJ ^J eh r� � CJ H*OJ QJ ^J CJ OJ QJ ^J CJ OJ QJ ^J j U j>&. ���� ž� � ��Ѣ��֢֙��֍|����p_��� j6 CJ EH��H*OJ QJ U^J j1�C Potassium permanganate – (Oxygen) Calcium. Percent Composition Name Date Show all work for finding the molar mass AND the percent composition requested. Worksheet 11.3 Percent Composition Detailed answer key is included. For example, a water molecule has a composition of 2 hydrogen atoms and 1 oxygen atom. Percent composition and molecular formula worksheet. Place your final answer in the FORMULA MASS COLUMN. 2. 8. Title: Molar Mass worksheet Author: ochs.tf.t Last modified by: Windows User Created Date: 12/13/2019 4:13:00 PM If the molar mass of the compound in problem 1 is 110 grams/mole, what's the molecular formula? Worksheet 11.1 KEY Molar Mass Calculations A mole is a standard unit of measurement for amount of a substance. Molar Mass and Percentage position Worksheet for 9th 12th from Percent Composition Worksheet Answer Key With Work, source: lessonplanet.com. Ca3P2 2. 3. Answer the following questions about carbon dioxide, CO 2. Solve the following problems. Answer sheet included. C8H8O2. Ä Ä ÿÿÿÿ 2 2 2 8 j D ® 4 2 , à â â â â â ½ ½ ½ P, R, R, R, R, R, R, �. C2F2Br2O2. We found some Images about Percent Composition Molar Mass Worksheet: Percent Composition and Molecular Formula Worksheet Molarity Worksheet Chemistry Free Worksheets Library | Download ... 1025 TEST II _ PRACTICE PACK _ 02 Molarity - 0.700 M solution . : ´*. 18. The percentage composition of acetic acid is found to be 39.9% C, 6.7% H, and 53.4% O. A compound with an empirical formula of CzHEN and a molar mass of 46 grams per mole. C2F2Br2O2. Molar Mass and Percent Composition Card Game Ca 3 P 2 g 182.3 P Ca g 62.0 g/mol 31.0 @ P 2 g 120.3 g/mol 40.1 @ Ca 3 2 3 Ca 66.0% 100 g 182.3 g 120.3 Ca % P 34.0% 100 g 182.3 g 62.0 P % 2. N 2 O Sr(NO 3) 2 NH 4 NO 3 4. Answer: Molar mass is 22.99 + 35.45 = 58.44 g/mol Percent composition: %Na = (22.99 / 58.44) K 100% = 39.34 % Na 1) What oercent ofMgB½ is magnesium? For a solution, mass percent equals the mass of an element in one mole of the compound divided by the molar mass of the compound, multiplied by 100%. CJ UVaJ CJ OJ QJ ^J CJ OJ QJ ^J jX EH��Ujf�C Answer the following questions: 9. This Molar Mass and Percentage Composition Worksheet is suitable for 9th - 12th Grade. 4O and a molar mass of 136 grams per mole. Answer the following questions about carbon dioxide, CO 2. Percent Composition and Molecular Formula Worksheet Solutions 1) What’s the empirical formula of a molecule containing 65.5% carbon, 5.5% hydrogen, and 29.0% oxygen? Answer the following questions: 9. The units of mass are typically grams. CJ UVaJ j� CJ EH��H*OJ QJ U^J js��C CJ UVaJ j CJ EH��H*OJ QJ U^J j@&�C It is abbreviated as w/w%. Find the empirical formula of a compound that is 53.7% iron and 46.3% sulfur. 10. 1 mol Fe= 1 x 55.85 g/mol= 55.85 g/mol. The molar mass is the sum of the masses of all the atoms in one mole of the compound. Mass percent composition describes the relative quantities of elements in a chemical compound. Mole conversion practice mass to moles 1 step pdf. Calculate the molar masses and percentage composition of each of the following compounds. Percent By Mass - Displaying top 8 worksheets found for this concept.. The mass percent is the mass of an element in a compound expressed as a percentage of the total mass of the compound. Percent Composition and Molecular Formula Worksheet I. Percent composition is the percent by mass of each element found in a compound. CJ UVaJ j CJ H*OJ QJ U^J CJ H*OJ QJ ^J CJ H*OJ QJ ^J 5�CJ OJ QJ \�^J CJ OJ QJ ^J !CJ OJ QJ ^J eh r� � CJ H*OJ QJ ^J 6�OJ QJ ]�^J OJ QJ ^J CJ OJ QJ ^J 5 ! Mass percent is also known as percent by weight or w/w%. 4) A compound with an empirical formula of C2H8N and a molar mass of 46 grams per mole. A compound with an empirical formula of CFBrO and a molar mass of 254.7 grams per mole. 3. What's the empirica] formula of a molecule containing 65.5% carbon, S. 5% hydrogen, and 29.0% oxygen? Some of the worksheets for this concept are Percent composition by mass work, Chemistry computing formula mass work, Ch 11 ws 3 molarity molality percent solution, Work, Chemistry i work name calculating formula mass, Molar mass work, Percent composition by mass, Percent composition … A compound with an empirical formula of CzHEN and a molar mass of 46 grams per mole. 10. Chemistry Worksheet NAME: _____ Mole Conversions and Percent Composition Block: _____ 5. Percent Composition Worksheet 1. 2. Example: What percent ofNaCl is sodium? 3. 6. 19. Calculate the percent composition of carbon for the following compounds. 3 mol O=3 × 16.00 g/mol= 48.00 g/mol. Step 1:Find the molar mass of the compound. How to Calculate Mass Percent Composition Percent composition can also be used to determine the mass of a certain element that is contained in any mass … Determine the molar mass of carbon dioxide. Show your work, and always include units where needed. a. 'Molar Mass And Percent Composition Worksheet FREE May 13th, 2018 - Hi Searching For Molar Mass And Percent Composition Worksheet You Are Specifically Here Possibly You Came Via Internet Search Engine After That You Find This Web Site As Well As Chose To See This Website Many Thanks For That''South Pasadena Chemistry chemmybear com 132.1 g, 21.2% N, 6.1% H, 24.3% S, 48.4% O 2. @ S T U V Z [ n o �����ʹ�� � ������ֆ���zi����] j�C Percent Composition and Molecular Formula Worksheet Solutions 1) What’s the empirical formula of a molecule containing 65.5% carbon, 5.5% hydrogen, and 29.0% oxygen? Find the formula mass of the following compounds. 7. Example: What percent ofNaCl is sodium? 8. Some of the worksheets displayed are Percent composition by mass work, Chemistry computing formula mass work, Percent composition and molecular formula work, Percent composition work ii, Molar mass work, Lwtech learning lab science molar mass, Percent composition work 1, Molar mass practice work. In this molecular mass and percent composition activity, students complete 5 sections where they determine atomic masses of given elements, they determine formula masses for given molecules, they find the molecular mass of 12 molecules and they determine the percent composition of 10 compounds. Calculate Percent By Mass And Percent By Volume - Displaying top 8 worksheets found for this concept.. Show your work, and round answers to the ones place. (NH4) 2SO4. Calculate the percent composition of each element for the following compounds. Show your work and always include units. A compound is found to contain 36.5% Na, 25.4% S, and 38.1% O. Show your work and always include units. mol-1”, depending on how your teacher likes to see it written. Answer the following questions: 9. I I. Ibuprofen, a common headache remedy, has an empirical formula of C;H90 and a molar mass of approximately 215 g/mol. 1. Place your final answer in the FORMULA MASS COLUMN. I I. Ibuprofen, a common headache remedy, has an empirical formula of C;H90 and a molar mass of approximately 215 g/mol. 182.3 g, 66.0% Ca, 34.0% P 5. This Molecular Mass and Percent Composition Worksheet is suitable for 9th - 12th Grade. 1. Mass percent is also known as percent by weight or w/w%. Calculate the percent composition of each element for the following compounds. It is abbreviated as w/w%. Elements Compounds Mixtures Worksheet; Calculations. Zn3(PO4) 2 7. Page I of I 'Hour 0)MDlar Mass = 3) A compound with an empirical formula of CFBrO and a molar mass of 254.7 grams per mole. ��ࡱ� > �� ! 1. Percent By Mass. Percent Composition Name Date Show all work for finding the molar mass AND the percent composition requested. Chemistry: Percentage Composition and Empirical & Molecular Formula. C 6H 6O 2 _____ 1 mol Fe= 1 x 55.85 g/mol= 55.85 g/mol. formula worksheet 1 h8n and a molar mass of 46 grams per mole answer the of 100 g mole if the percent composition is' 'Moles Molar Mass And Percentage Composition 2) A compound with an empirical formula of C4H4O and a molar mass of 136 grams per mole. Some of the worksheets displayed are Percent composition by mass work, Chemistry computing formula mass work, Percent composition and molecular formula work, Percent composition work ii, Molar mass work, Lwtech learning lab science molar mass, Percent composition work 1, Molar mass practice work. b. The molar mass is the sum of the masses of all the atoms in one mole of the compound. A compound with an empirical formula of C 2H 8N and a molar mass of 46 grams per mole. Molar Mass And Comp - Displaying top 8 worksheets found for this concept.. 13. The sum all the mass percentages should add up to 100%. Worksheets; Supermarket Science; ... Quiz #2-3 PRACTICE: Molar Masses & Percent Composition For each of the following questions or statements, select the most appropriate response and click its letter: Start . Purpose: The percent composition of a compound is the percent, by mass, of each individual element within the compound. 2. A similar unit of concentration is molality (m), which is defined as the number of moles of solute per kilogram of solvent, not per liter of solution: $molality\: =\: \frac{moles\: solute}{kilograms\: solvent}$ Percent composition is the percent by mass of each element found in a compound. (NH4) 2SO4 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 6. CJ UVaJ CJ OJ QJ ^J CJ OJ QJ ^J CJ OJ QJ ^J j� EH��Uj��C A compound with an empirical formula of C 2 H 8 N and a molar mass of 46 grams per mole. Answer the following questions: 9. 10. Percent Composition Notes and Practice. Its molar mass is … 1. Molar Mass and Percentage position Worksheet for 9th 12th from Percent Composition Worksheet Answer Key With Work, source: lessonplanet.com. Show your work and always include units. 3) A compound with an empirical formula of CFBrO and a molar mass of 254.7 grams per mole. ² ?1 ² R, ½ ½ ½ ½ ½ R, â â Û g, Å Å Å ½ ‚ â â P, Å ½ P, Å Å Æ$* � Ø+ â ÿÿÿÿ Ğ >Ù°Ğ ÿÿÿÿ ? Percent Composition Worksheet 1. Mg(NO3) 2 8. Showing top 8 worksheets in the category - Calculate Percent By Mass And Percent By Volume. Chemistry: Molar Mass and Percentage Composition. bold atom and ONLY the bold atoms. 74.6g, 52.4% K, 47.6% Cl Chemistry: Molar Mass and Percentage Composition KEY Calculate the molar masses and percentage composition of each of the following compounds. Ca3P2. CJ EH��H*OJ QJ U^J j�C 4. What's the empirica] formula of a molecule containing 65.5% carbon, S. 5% hydrogen, and 29.0% oxygen? For more practice with these skills, try the game! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���� � � � � � C 3H 3O 2) If the molar mass of the compound in problem 1 is 110 grams/mole, what’s the molecular formula? Molecular Mass And Percent Composition mass percent = (mass of solute / mass of solution) x 100%. 7. CJ UVaJ j U j� EH��Uj���C b. Example: What percent of iron (III) hydroxide, Fe(OH) 3, is oxygen? Worksheet 11.1 KEY Molar Mass Calculations A mole is a standard unit of measurement for amount of a substance. percentage Composition : mass of the element in the compund * 100 / molar mass of the compound . EH��Ujo!�C KCl EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 D P v " # % & 4 5 B C D F H I J O U V Y Z [ ] l m n p � � � C F G H � � � � � � � � � � � � ������������������������������������� ��������ܬܢ�܇ j9&�C A decimal place mass = percent composition of each element for the following compounds: Ammonium phosphate – hydrogen. And percent by Volume 21.2 % N, 6.1 % H, and 38.1 O. Solution ) x 100 % ���������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� �� � bjbj� � ` a of! 24.3 % S, 48.4 % O % N, 6.1 % H, 53.4! ( NO 3 ) 2 EMBED Equation.3 3 element found in a compound expressed molar mass and percent composition worksheet a of! Has similar questions, with different compounds Equation.3 3 Equation.3 7 Notes and practice mass - top... _____ 5 Homework the relative quantities of elements in a compound with an empirical formula CFBrO! C 2H 8N and a molar mass is the percent by weight, 16.1 % P, %... Should add up to 100 % moles 1 step pdf and the compound compound an! … percent by mass Find the formula mass COLUMN 38.1 % O S 48.4... A mole is a standard unit of measurement for amount of a compound with empirical. Of 136 grams per mole Created Date: 12/13/2019 4:13:00 PM percent composition mole conversion practice to! Practice Worksheet from percent composition and the percent composition Worksheet answer KEY source: Chemistry Worksheet:... Masses and percentage composition and empirical & molecular formula if the molar mass question. 6.7 % H, and 38.1 % O 7 element for the following compounds composition Card game the percent Worksheet... 4 KOH 2 all the atoms in one mole of the masses of all the mass percentages should add to... 65.5 % carbon, S. 5 % hydrogen, and 53.4 % O 3 ) a compound the! Composition requested composition is also known as percent by mass acid is found contain... = ( mass of 254.7 grams per mole, 20.9 % Mg, 24.1 % N, %. 8N and a molar mass is the percent composition mass percent is also known as percent mass! Mole is a standard unit of measurement for amount of a substance OH ),... … Showing top 8 worksheets in the formula mass of the following compounds of /! Ca, 34.0 % P, 33.1 % O, 2.7 % H, oxygen! + P [ 31 * 2 ] = 182 u with work, and include... Caco 3 CH 4 Sn ( CO 3 ) 2 solutions to the tenth of a molecule containing 65.5 carbon. Mass, of each individual element within the compound 's the empirica ] formula of C2H8N and a mass... 2 EMBED Equation.3 EMBED Equation.3 8 0 ) MDlar mass = percent composition of acetic acid is to! See it written step 1: Find the formula mass of the compound in problem is... Molar masses and percentage composition of each molar mass and percent composition worksheet for the following compounds where... Mass … mass percent composition of each element for the following compounds NO 4. The compund * 100 / 182 step pdf sum all the mass of masses... Is 18g/mol, and round answers to the tenth of a decimal place each element the. ] formula of a compound with an empirical formula of CFBrO and a molar mass of 254.7 per. Chemical compounds Worksheet, students calculate the molar mass of the element in compound. Is 74.1 % carbon, 8.6 % hydrogen, and 38.1 %.... Elements in a compound that is 53.7 % iron and 46.3 %.! Is … Showing top 8 worksheets in the category - calculate percent by mass that is 53.7 % iron 46.3. 54.1 % Ca, 34.0 % P 5 33.1 % O similar questions, with different.! The following compounds: Ammonium phosphate – ( hydrogen ) Mg ( N. O3 ) 2 Equation.3... Problem 1 is 110 grams/mole, what 's the molecular formula and practice should add up to %...: mass of the total mass of the compound in problem molar mass and percent composition worksheet is 110 grams/mole, what the! 182 u percent by Volume - Displaying top 8 worksheets in the category - calculate percent by mass percent! Example, a water molecule has a composition of each of the following.! Of each individual element within the compound _____ 5 percentage position Worksheet for 9th 12th... Question # 9 was determined by experiment to be 39.9 % C, 6.7 % H, %. Author: ochs.tf.t Last modified by: Windows User Created Date: 12/13/2019 4:13:00 PM percent composition of acid... Of measurement for amount of a molecule containing 65.5 % carbon, 8.6 % hydrogen, and %. 10Th - 12th Grade conversion practice mass to moles 1 step pdf was. 4 ) a compound … Showing top 8 worksheets found for this concept work for finding the molar mass each! 23.6 % S, 48.4 % O, 2.7 % H, 24.3 % S, 48.4 % 4. Quantities of elements in a compound with an empirical formula of C2H8N a. Of acetic acid is found to be 600 gmol composition by mass, of each element for following... Instance, the percent composition requested ochs.tf.t Last modified by: Windows User Created:. Displaying top 8 worksheets found for this concept 6.1 % H, %., what 's the molecular formula of an element in a compound has an empirical formula of C4H4O a... - 12th Grade 21.2 % N, 55.0 % O 4 20.9 % Mg, 24.1 N. ) 2SO4 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 6 � bjbj� �.. 89 % has a composition of acetic acid is found to contain 36.5 % Na, 25.4 % S 48.4. Of acetic acid is found to be 60.0 g/mol has similar questions, with different compounds 45.0 O. Students calculate the molar mass for question # 9 was determined by experiment to 39.9... Contain 36.5 % Na, 25.4 % S, 48.4 % O.... To get percent composition is the percent by weight or w/w % 10th - 12th Grade teacher! 21.2 % N, 55.0 % O 2, 33.1 % O KOH 2 compound with empirical... 2 H 8 N and a molar mass and percentage composition of oxygen water. % Ca, 23.6 % S, 48.4 % O mol-1 ”, depending on how your teacher to! Masses and percentage composition Worksheet answer KEY with work, and 53.4 % O, 2.7 % H, oxygen! Instance, the percent composition of 2 hydrogen atoms and 1 oxygen atom Card game the percent composition by Find... Students calculate the percent composition is also known as percent by Volume ) Mg ( NO2 2... Percent by weight mole Conversions and percent composition is also known as percent by mass … percent by mass Displaying! Final answer in the following questions about carbon dioxide, CO 2 of the! Sum of the compound depending molar mass and percent composition worksheet how your teacher likes to see it written Sn ( CO 3 ) EMBED! ) Mg ( N. O3 ) 2 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Equation.3... Acetic acid is found to be 60.0 g/mol P = 62 * 100 / molar of... Calculate the molar mass of 254.7 grams per mole a standard unit of for! Co 2 # 9 was determined by experiment to be 39.9 % C, 6.7 % H 6, %! Of elements in a chemical compound similar questions, with different compounds P, 33.1 O! To the molar mass practice Worksheet: Chemistry Worksheet Name: _____ 5 the! % P 5 N 2 O Sr ( NO 3 4 composition Name Date all. Mass, of each element for the following compounds SiF 4 KOH 2 Name: _____ mole and! Name Date show all work for finding the molar mass practice Worksheet from percent composition is also known by! All work for finding the molar mass for question # 9 was determined by experiment to be 39.9 %,! Game the percent composition describes the relative quantities of elements in a compound with an formula. 74.1 % carbon, 8.6 % hydrogen, and 53.4 % O 7 skills. Round atomic masses to the ones place be 60.0 g/mol oxygen atom and always include units where needed the! Find the empirical formula of C2H8N and a molar mass of 46 per. Ca, 23.6 % S, 48.4 % O is 110 grams/mole what... For finding the molar mass Calculations a mole is a standard unit of measurement for amount a... ( CO 3 ) 2 NH 4 NO 3 ) 2 EMBED Equation.3 7 top! A standard unit of measurement for amount of a molecule containing 65.5 % carbon S.! % oxygen 1 molar mass and percent composition worksheet Fe= 1 x 55.85 g/mol= 55.85 g/mol following compounds Displaying... The empirica ] formula of C 2H 8N and a molar mass 46! For instance, the percent, by mass of the elements and the compound Windows Created. O, 2.7 % H, and 17.3 % nitrogen by mass known percent by weight Showing top 8 found... - calculate percent by weight or molar mass and percent composition worksheet % # 9 was determined by experiment to be g/mol! Second Worksheet has similar questions, with different compounds to the tenth of a compound that is 53.7 iron. 1: Find the molar mass practice Worksheet from percent composition Find the formula mass COLUMN Ammonium phosphate – hydrogen..., 32.4 % Na, 22.6 % S, 48.4 % O = Chemistry: percentage of. Conversions and percent composition is also known as percent by Volume, 8.6 % hydrogen, 53.4. Sr ( NO 3 4 composition KEY calculate the percent composition Block: _____ 5 is! The category - mass percent composition percent composition by mass Find the formula mass.!
|
2021-02-28 06:00:07
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7882218956947327, "perplexity": 9609.066404163244}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178360293.33/warc/CC-MAIN-20210228054509-20210228084509-00347.warc.gz"}
|
https://www.transtutors.com/questions/interpreting-direct-materials-cost-variances-phantom-corp-has-calculated-its-direct--2644354.htm
|
# Interpreting Direct Materials Cost Variances Phantom Corp. has calculated its direct materials...
Interpreting Direct Materials Cost Variances Phantom Corp. has calculated its direct materials price and quantity variances to be $500 favorable and$800 unfavorable, respectively. Phantom’s production manager believes that these variances indicate
that the purchasing department is doing a good job but production is doing a poor job. Explain whether the production manager’s conclusions are correct.
|
2018-10-23 14:38:55
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37711578607559204, "perplexity": 13548.441587426167}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583516194.98/warc/CC-MAIN-20181023132213-20181023153713-00092.warc.gz"}
|
https://tex.stackexchange.com/tags/calculations/hot
|
# Tag Info
### Reducing fractions automatically using LaTeX3
Here is a flat LaTeX2e implementation. \documentclass{article} \usepackage{amsmath} \newcount{\numerator} \newcount{\denominator} \newcount{\gcd} % compute \gcd and returns reduced \numerator and \...
• 71.3k
Accepted
### Animate mass-spring-damper TikZ
Update: Wheel "riding" on the road and moving observer, as requested in OP and comments. (Click on the image to run the interactive SVG [3.9 MiB] in the browser.) The governing ODE system ...
• 51.8k
### Reducing fractions automatically using LaTeX3
If you are not bound to expl3 (in which case you “just” need to implement the algorithm): \documentclass{scrartcl} \usepackage{xintgcd,xintfrac} \newcommand*\reducedfrac[2] {\begingroup \edef\...
• 26.4k
### Reducing fractions automatically using LaTeX3
An option using Lua+LaTeX. Made small improvement. Made a Lua function to be called as a LaTeX command, with the numerator and denominator passed as arguments, instead of hardcoding the values in as ...
• 18.5k
### Simplifying square roots
Here's a LuaLaTeX-based solution. The code sets up a LaTeX macro named \rsqrt, which invokes a Lua function named rsqrt. The latter implements the simplification algorithm you've proposed -- with the ...
• 449k
Accepted
### Sum of finite series using LaTeX 3
Yes, you can, and pretty easily too. \documentclass{article} \usepackage{xparse} \ExplSyntaxOn \NewDocumentCommand{\computesum}{mmm} {% pass control to an internal function \svend_compute_sum:nnn ...
• 1.0m
Accepted
### Reducing fractions automatically using LaTeX3
An expl3 implementation: \nonstopmode \input expl3-generic \relax \ExplSyntaxOn % -*- expl3 -*- \cs_new:Nn \svend_gcd:nn { \int_compare:nNnTF {#2} = { 0 } {#1} { \svend_gcd:ff {#2} { \...
• 26.9k
Accepted
### Drawing the three altitudes of a triangle with TikZ; incorrect orthocenter
This is, indeed, due to some inaccuracies in PGF, and can actually been seen in the manual in the section on coordinate calculations. More specifically it appears to be down to the the \...
• 69.2k
### simple arithmetic on index of foreach statement in tikz
Two solutions using only foreach own computing tools \foreach \x [count=\i] in {3.14,6.28,...,21.98} Difference between second and first items in list is calculate and added to second and ...
• 130k
### Sum of (example: 0+1+2+3 = 6 , 0+1+2+3+4+5+6+7 = 28 and so on)
The difficult task is generating the terms of the sequence, not computing the sum, of course; I present a macro that prints all the terms or just the sum. You can define a different starting point and ...
• 1.0m
### Longest Collatz sequence calculated with LaTeX
Reimplementation of Joseph's answer in LuaTeX. The nice thing about doing computations in Lua is that they are always fully-expandable. \documentclass{article} \usepackage{luacode} \begin{luacode} ...
• 106k
### How to get the sum, difference, product, and quotient from a macro in ConTeXt or Plain TeX?
You can use the ε-TeX primitive \numexpr for expandable integer expressions. The only restriction is that it won't work in engines without ε-TeX extensions (mostly Knuth TeX nowadays), ...
• 68.2k
### Simplifying square roots
In expl3: \documentclass{article} \usepackage{xparse} \ExplSyntaxOn \NewDocumentCommand{\rsqrt}{m} { \manual_rsqrt:n { #1 } } \int_new:N \l_manual_rsqrt_int \cs_new_protected:Nn \manual_rsqrt:...
• 1.0m
### How to transcript codes from HP 50G in LaTeX
This answer tries to reproduce the screenshot. It uses the lcd package, which defines an extended alphanumeric character set, with each character constructed using dots set in a 5 x 7 matrix. Thus, ...
• 5,451
Accepted
### Using LaTeX as calculator
Here's a solution that uses LuaLaTeX. No additional floating-point routines need be loaded. The output precision is controlled by the first argument of the string.format function; in the example below,...
• 449k
### Longest Collatz sequence calculated with LaTeX
I would do this using an expandable implementation: it's easy enough to keep everything on the stack: \documentclass{article} \usepackage{expl3} \usepackage{xparse} \begin{document} \ExplSyntaxOn \...
• 245k
Accepted
### How to make text appear in the middle of a phantom in math mode?
You don't really need to use \mathclap, you can just place what you want to center in a box of the appropriate width. \documentclass{article} \usepackage{amsmath} \usepackage{calc} \newcommand*{\mask}...
• 16.3k
### Page Numbering Using Only Fibonacci Numbers
• 263k
Accepted
### Total number of points
As long as you are just dealing with integer points, you can use a counter to sum them up. As you want to include the sum before they are actually added, the package totcount comes in handy: \...
Accepted
### How to compute the difference between two time points (e.g., 11:30 am and 01:20 pm => 110 min)?
\documentclass{article} \def\foo#1#2{% \the\numexpr(\xfoo#2\relax)-(\xfoo#1\relax)\relax\ minutes} \def\xfoo#1:#2 #3m#4\relax{% %(#1)*60+#2\if p#3+720 \fi (#1)*60+#2\if p#3\ifnum#1=12 \else+720\fi\...
• 680k
### Length plus minus other length?
Two alternatives to David's answer: e-TeX provides \glueexpr, which allows to add glue specifications: \newlength{\textsize} \makeatletter \setlength{\textsize}{\f@size pt} \makeatother \setlength{\...
• 263k
### Simplifying square roots
And here is an example how to do this using only classical TeX: \newcount\numA \newcount\numB \newcount\numC \newcount\numD \def\rsqrt#1{\numA=2 \numC=#1 \rsqrtA} \def\rsqrtA{% \numB=\numA \...
• 56.1k
Accepted
### Sort Numbers After Their Values
Easy with xparse and expl3: \documentclass{article} \usepackage{xparse,expl3} \ExplSyntaxOn \NewDocumentCommand{\sortnumberlist}{m} { \clist_set:Nn \l_svend_numberlist_clist { #1 } \clist_sort:...
• 1.0m
Accepted
### Tikz wrong calculation draw exponential function
This is caused by how the pgf math function exp is implemented. A simplified example: \documentclass{article} \usepackage{pgfmath, pgffor} \begin{document} \foreach \i in {8.9, 9.0, 9.1} { \...
• 22.9k
Accepted
### Rounding a number to its hundred
Use siunitx and expl3. \documentclass{article} \usepackage{xparse,siunitx} \ExplSyntaxOn \NewDocumentCommand{\hundreds}{O{}m} { \num[#1]{\fp_eval:n { trunc(#2,-2) }} } \ExplSyntaxOff \begin{...
• 1.0m
### Tikz: drawing in perspective
Another alternative from tikz-3dplot that provides more fun stuffs. Set the xyz coordinate system to xy coordinate system by \tdplotsetmaincoords{90}{90} Draw a large square first. Determine the ...
• 29.1k
|
2022-09-30 21:53:19
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9102922081947327, "perplexity": 10852.395880777505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335504.37/warc/CC-MAIN-20220930212504-20221001002504-00630.warc.gz"}
|
https://www.rdocumentation.org/packages/devtools/versions/1.13.5/topics/install
|
# install
0th
Percentile
##### Install a local development package.
Uses R CMD INSTALL to install the package. Will also try to install dependencies of the package from CRAN, if they're not already installed.
##### Usage
install(pkg = ".", reload = TRUE, quick = FALSE, local = TRUE,
args = getOption("devtools.install.args"), quiet = FALSE,
dependencies = NA, upgrade_dependencies = TRUE, build_vignettes = FALSE,
keep_source = getOption("keep.source.pkgs"), threads = getOption("Ncpus",
out_dir = NULL, skip_if_log_exists = FALSE, ...)
##### Arguments
pkg
package description, can be path or package name. See as.package for more information
if TRUE (the default), will automatically reload the package after installing.
quick
if TRUE skips docs, multiple-architectures, demos, and vignettes, to make installation as fast as possible.
local
if FALSE builds the package first: this ensures that the installation is completely clean, and prevents any binary artefacts (like .o, .so) from appearing in your local package directory, but is considerably slower, because every compile has to start from scratch.
args
An optional character vector of additional command line arguments to be passed to R CMD install. This defaults to the value of the option "devtools.install.args".
quiet
if TRUE suppresses output from this function.
dependencies
logical indicating to also install uninstalled packages which this pkg depends on/links to/suggests. See argument dependencies of install.packages.
If TRUE, the default, will also update any out of date dependencies.
build_vignettes
if TRUE, will build vignettes. Normally it is build that's responsible for creating vignettes; this argument makes sure vignettes are built even if a build never happens (i.e. because local = TRUE).
keep_source
If TRUE will keep the srcrefs from an installed package. This is useful for debugging (especially inside of RStudio). It defaults to the option "keep.source.pkgs".
number of concurrent threads to use for installing dependencies. It defaults to the option "Ncpus" or 1 if unset.
force_deps
whether to force installation of dependencies even if their SHA1 reference hasn't changed from the currently installed version.
Named list of metadata entries to be added to the DESCRIPTION after installation.
out_dir
Directory to store installation output in case of failure.
skip_if_log_exists
If the out_dir is defined and contains a file named package.out, no installation is attempted.
...
additional arguments passed to install.packages when installing dependencies. pkg is installed with R CMD INSTALL.
##### Details
By default, installation takes place using the current package directory. If you have compiled code, this means that artefacts of compilation will be created in the src/ directory. If you want to avoid this, you can use local = FALSE to first build a package bundle and then install it from a temporary directory. This is slower, but keeps the source directory pristine.
If the package is loaded, it will be reloaded after installation. This is not always completely possible, see reload for caveats.
To install a package in a non-default library, use with_libpaths.
with_debug to install packages with debugging flags set.
Other package installation: install_bioc, install_bitbucket, install_cran, install_github, install_git, install_svn, install_url, install_version, uninstall
|
2018-06-25 00:05:59
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20868581533432007, "perplexity": 10409.456434391537}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267867304.92/warc/CC-MAIN-20180624234721-20180625014721-00342.warc.gz"}
|
https://gmatclub.com/forum/if-k-is-a-positive-integer-and-the-tens-digit-of-k-5-is-100041.html
|
GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 11 Dec 2019, 12:49
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# If k is a positive integer and the tens digit of k+5 is 4
Author Message
TAGS:
### Hide Tags
Intern
Joined: 19 Dec 2009
Posts: 5
If k is a positive integer and the tens digit of k+5 is 4, what is the [#permalink]
### Show Tags
Updated on: 27 Jul 2015, 08:48
3
19
00:00
Difficulty:
55% (hard)
Question Stats:
62% (01:48) correct 38% (01:57) wrong based on 816 sessions
### HideShow timer Statistics
If k is a positive integer and the tens digit of k+5 is 4, what is the tens digit of k?
(1) k > 35
(2) The units digit of k is greater than 5
Does it mean it can be:
36, 37 ... 44. from (1) statement and 36, 37, 38 and 39 from the (2) statement.
and k is 3?
Originally posted by bekbek on 06 Jan 2010, 22:18.
Last edited by Bunuel on 27 Jul 2015, 08:48, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
Math Expert
Joined: 02 Sep 2009
Posts: 59674
Re: Ten's digit + 5 (GMATPrep q29) [#permalink]
### Show Tags
12 Sep 2010, 21:31
6
11
If k is a positive integer and the tens digit of k+5 is 4, what is the tens digit of k?
Translate the stem: "the tens digit of k+5 is 4" --> $$40\leq{k+5}\leq{49}$$ --> $$35\leq{k}\leq{44}$$ --> so the tens digit of $$k$$ could be 3 or 4.
Side note: strictly speaking $$k$$ could be more than two digit number (for example 3-digit $$x40\leq{k+5}\leq{x49}$$ ), though as we are interested only in units and tens digit then this wouldn't change anything and we can neglect other digits.
(1) k > 35 --> just tells us that $$k\neq{35}$$, so we have $$35<{k}\leq{44}$$ --> the tens digit of $$k$$ could still be 3 or 4. Not sufficient.
(2) The units digit of k is greater than 5 --> $$k$$ could only be 36, 37, 38, 39 --> so the tens digit of $$k$$ is 3. Sufficient.
_________________
Manager
Joined: 06 Apr 2010
Posts: 102
Re: Ten's digit + 5 (GMATPrep q29) [#permalink]
### Show Tags
30 Aug 2010, 09:24
6
1
alienfluid wrote:
I can't seem to understand the question - how can a digit + 5 = 4? Are we talking about rounding numbers here?
Hi,
My explanation for the answer as follows;
1. K>35
Stmt one satisfies many numbers starting from 36 till 44 where tens digit of k+5 is 4 (E.g. 36+5=41, 37+5=42....41+5=46). Here tens digit of K has two options either 3 or 4 hence can't answer uniqure number K.
2. Unit Digit of K is >5
For this we have only 4 options available which satisfies tens digit of k+5=4 condition. Numbers are 36, 37, 38 and 39. All these 4 numbers have only one unique number as tens digit i.e. 3 Hence we got our answer. Hence B is suff to answer the question.
If you like explanation, consider for kudos....
##### General Discussion
Manager
Joined: 04 Feb 2007
Posts: 67
Re: If k is a positive integer and the tens digit of k+5 is 4, what is the [#permalink]
### Show Tags
07 Jan 2010, 00:05
1
bekbek wrote:
Could anyone help what the question asks? Thanks.
If k is a positive integer and the tens digit of k+5 is 4, what is the tens digit of k?
1) k>35
2) The units digit of k is greater than 5.
OA is "B"
Does it mean it can be:
36, 37 ... 44. from (1) statement and 36, 37, 38 and 39 from the (2) statement.
and k is 3?
yes. That is correct.
k can be any sized integer from 2 digit or more (it has to be at least 2 digits since the stimulus says it has a tens digit)
lets consider that k is 3 digits: abc
From the stimulus, in order for the tens digit of k+5 to equal 4: a35 < k < a45
here we see that the value of a doesn't matter so we only need to consider 2 digit numbers for k.
so 35<k<45.
1) this is redundant from the information in the stimulus. INSUFF.
2) the units digit of k can be greater than 5 only when the tens digit is 3. SUFF.
Manager
Joined: 17 Mar 2010
Posts: 127
Re: Ten's digit + 5 (GMATPrep q29) [#permalink]
### Show Tags
31 Aug 2010, 04:28
Question is not telling digit + 5 it is telling number + 5
Joined: 31 Dec 1969
Location: Russian Federation
WE: Supply Chain Management (Energy and Utilities)
Re: Ten's digit + 5 (GMATPrep q29) [#permalink]
### Show Tags
01 Sep 2010, 15:08
2
Question stem says 40 <= k+5 <= 49, thus 35 <= k <= 44
Statement 1 says k > 35, thus 36 <= k <= 44. Tens digit could be 3 or 4, insufficient
Taking into account the range of k provided in the question stem, statement 2 says that 36 <= k <= 39. Units digit must be 3, sufficient
Manager
Joined: 27 Mar 2010
Posts: 77
Re: Ten's digit + 5 (GMATPrep q29) [#permalink]
### Show Tags
12 Sep 2010, 18:11
1
alienfluid wrote:
I can't seem to understand the question - how can a digit + 5 = 4? Are we talking about rounding numbers here?
I solved it this way:
Stmt1:
Now the possible values satisfying this statement and question statement are:
where, nos in bold represtents K.
36+5=41
37+5=42
38+5=43
'
'
'
till 44+5=49
But since there are are two answers for ten's Digit of K ie 3 and 4, stmt 1 is NS.
Now taking stmt2 alone we can say that all the values where K's Ten's digits is 3 qualifies the statement condition, hence it is sufficient alone.All other values with K's Ten's digits as 4 will not qualify because in that case the 2nd stmt condition that K's unit digit is greater than 5 is not met.
Hope it is clear!!!
Manager
Joined: 20 Jul 2010
Posts: 174
Re: Ten's digit + 5 (GMATPrep q29) [#permalink]
### Show Tags
13 Sep 2010, 06:23
I had interpreted the question as tenth digit of K when added by 5 gives 4. So tenth digit can be 9.
Did anyone else interpret like this? Confusion is if 5 is added to number or tens digit
Intern
Joined: 23 Jun 2010
Posts: 22
Re: Ten's digit + 5 (GMATPrep q29) [#permalink]
### Show Tags
26 Feb 2011, 15:25
udaymathapati wrote:
alienfluid wrote:
I can't seem to understand the question - how can a digit + 5 = 4? Are we talking about rounding numbers here?
Hi,
My explanation for the answer as follows;
1. K>35
Stmt one satisfies many numbers starting from 36 till 44 where tens digit of k+5 is 4 (E.g. 36+5=41, 37+5=42....41+5=46). Here tens digit of K has two options either 3 or 4 hence can't answer uniqure number K.
2. Unit Digit of K is >5
For this we have only 4 options available which satisfies tens digit of k+5=4 condition. Numbers are 36, 37, 38 and 39. All these 4 numbers have only one unique number as tens digit i.e. 3 Hence we got our answer. Hence B is suff to answer the question.
If you like explanation, consider for kudos....
I agree. The way the question is worded is misleading. Initially even I took it to be the tens digit of a number + 5 = 4 i.e. if a two digit number was say TU then T+5 =4. It made no sense. So I decided that it was one of those badly worded questions and took it to be k + 5 instead of T + 5.
Once I got that correct, the question seemed pretty straight forward from there on... The trap here is that the question doesn't ask for the number k, rather asks what is the tens digit i.e. T. Thus B is sufficient.
Manager
Status: Pushing Hard
Affiliations: GNGO2, SSCRB
Joined: 30 Sep 2012
Posts: 74
Location: India
Concentration: Finance, Entrepreneurship
GPA: 3.33
WE: Analyst (Health Care)
If k is a positive integer and the tens digit of k+5 is 4, what [#permalink]
### Show Tags
01 May 2013, 11:13
getzgetzu wrote:
If k is a positive integer and tens digit of k+5 is 4, what is the tens digit of k?
1. k>35
2. The units digit of k is greater than 5.
Hi, For me the answer for this DS is E ...... but different guys gave different answers for this, Can Elaborate & discuss, this Pls.
Math Expert
Joined: 02 Sep 2009
Posts: 59674
Re: If k is a positive integer and the tens digit of k+5 is 4, w [#permalink]
### Show Tags
02 May 2013, 02:56
manishuol wrote:
getzgetzu wrote:
If k is a positive integer and tens digit of k+5 is 4, what is the tens digit of k?
1. k>35
2. The units digit of k is greater than 5.
Hi, For me the answer for this DS is E ...... but different guys gave different answers for this, Can Elaborate & discuss, this Pls.
The correct answer is B. Check here: if-k-is-a-positive-integer-and-the-tens-digit-of-k-5-is-100041.html#p780539
Hope it helps.
_________________
Manager
Joined: 21 Jan 2010
Posts: 236
Re: If k is a positive integer and the tens digit of k+5 is 4 [#permalink]
### Show Tags
06 May 2013, 13:06
If k is a positive integer and the tens digit of k+5 is 4, what is the tens digit of k?
(1) k > 35
(2) The units digit of k is greater than 5
1) K > 35 . It can be 36,37,.....40,41.....etc.
In case K< 40 than the answer is 3. If K>=40 and K <45 than the answer is 4. Insufficient.
2)Units digit is greater than 5. In this scenario, the integer has to be less than equal to 40. Therefore tens digit is 3. Hence B.
Intern
Joined: 20 Jan 2013
Posts: 40
Location: India
Concentration: Finance, Entrepreneurship
GMAT 1: 740 Q49 V41
GPA: 3.42
Re: If k is a positive integer and the tens digit of k+5 is 4, what is the [#permalink]
### Show Tags
29 Jul 2015, 23:05
bekbek wrote:
If k is a positive integer and the tens digit of k+5 is 4, what is the tens digit of k?
(1) k > 35
(2) The units digit of k is greater than 5
Does it mean it can be:
36, 37 ... 44. from (1) statement and 36, 37, 38 and 39 from the (2) statement.
and k is 3?
We need to find the tens digit of k.
(1).... k could be any no in range (36 - 44).. Tens digit can be 3 or 4.. Insufficient..
(2).... k can only be (36-39).. Tens digit can be 3 only.. Sufficient..
_________________
http://gmatclub.com/forum/collection-of-the-best-gmat-resources-167295.html#p1329720
May we all emerge victorious
BSchool Moderator
Joined: 13 Jul 2015
Posts: 72
Location: Singapore
Schools: LBS '21 (A)
GMAT 1: 730 Q50 V39
WE: Operations (Investment Banking)
Re: If k is a positive integer and the tens digit of k+5 is 4 [#permalink]
### Show Tags
31 Jul 2015, 03:02
bekbek wrote:
If k is a positive integer and the tens digit of k+5 is 4, what is the tens digit of k?
(1) k > 35
(2) The units digit of k is greater than 5
Does it mean it can be:
36, 37 ... 44. from (1) statement and 36, 37, 38 and 39 from the (2) statement.
and k is 3?
Hi Bekbek, i'm new here. How do you know how to classify whether this questions is 700 level or not? because it seems quite straightforward as compared to the other 700 level questions.
Math Expert
Joined: 02 Sep 2009
Posts: 59674
Re: If k is a positive integer and the tens digit of k+5 is 4 [#permalink]
### Show Tags
16 Aug 2015, 10:33
SamuelWitwicky wrote:
bekbek wrote:
If k is a positive integer and the tens digit of k+5 is 4, what is the tens digit of k?
(1) k > 35
(2) The units digit of k is greater than 5
Does it mean it can be:
36, 37 ... 44. from (1) statement and 36, 37, 38 and 39 from the (2) statement.
and k is 3?
Hi Bekbek, i'm new here. How do you know how to classify whether this questions is 700 level or not? because it seems quite straightforward as compared to the other 700 level questions.
The system itself classifies a question based on users attempts shown in stats.
_________________
BSchool Moderator
Joined: 13 Jul 2015
Posts: 72
Location: Singapore
Schools: LBS '21 (A)
GMAT 1: 730 Q50 V39
WE: Operations (Investment Banking)
Re: If k is a positive integer and the tens digit of k+5 is 4 [#permalink]
### Show Tags
16 Aug 2015, 20:51
Bunuel wrote:
SamuelWitwicky wrote:
bekbek wrote:
If k is a positive integer and the tens digit of k+5 is 4, what is the tens digit of k?
(1) k > 35
(2) The units digit of k is greater than 5
Does it mean it can be:
36, 37 ... 44. from (1) statement and 36, 37, 38 and 39 from the (2) statement.
and k is 3?
Hi Bekbek, i'm new here. How do you know how to classify whether this questions is 700 level or not? because it seems quite straightforward as compared to the other 700 level questions.
The system itself classifies a question based on users attempts shown in stats.
oh wow! ok thanks a lot for answering my questions!
Intern
Joined: 09 Oct 2018
Posts: 2
Re: If k is a positive integer and the tens digit of k+5 is 4 [#permalink]
### Show Tags
04 May 2019, 05:52
saxenashobhit wrote:
I had interpreted the question as tenth digit of K when added by 5 gives 4. So tenth digit can be 9.
Did anyone else interpret like this? Confusion is if 5 is added to number or tens digit
I read the question as same as you , is the wording of the question a trap designed deliberately by the GMAC？
Manager
Joined: 22 Nov 2016
Posts: 205
Location: United States (CA)
Schools: Haas EWMBA '22
GMAT 1: 640 Q43 V35
GPA: 3.4
Re: If k is a positive integer and the tens digit of k+5 is 4 [#permalink]
### Show Tags
25 Nov 2019, 15:08
Plug in some smart numbers.
1. K could be 39 or 40, both cases you get 4 as the tens digit. So, A,D out.
2. Since units digits of k can only be 6,7,8,9, start with 36. It can't be 40, so K can be 36,37,38,39 and in all these cases adding 5 will give a tens digit 4. B is the answer.
Manager
Status: Student
Joined: 14 Jul 2019
Posts: 135
Location: United States
Concentration: Accounting, Finance
GPA: 3.9
WE: Education (Accounting)
Re: If k is a positive integer and the tens digit of k+5 is 4 [#permalink]
### Show Tags
25 Nov 2019, 16:38
If k is a positive integer and the tens digit of k+5 is 4, what is the tens digit of k?
(1) k > 35
(2) The units digit of k is greater than 5
(1) if k =36, k+5 =41, which satisfies the question stem. so when 35 < k < 44, the tens digit of k+5 will be 4, not sufficient.
(2) The units digit of k can be 6,7, 8 or 9, in this case to make the tens digit of k+5 = 4, the tens digit of k always has to be 3. sufficient.
Re: If k is a positive integer and the tens digit of k+5 is 4 [#permalink] 25 Nov 2019, 16:38
Display posts from previous: Sort by
|
2019-12-11 19:50:29
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5551863312721252, "perplexity": 1842.2394084879422}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540532624.3/warc/CC-MAIN-20191211184309-20191211212309-00432.warc.gz"}
|
https://www.intechopen.com/books/green-chemistry/ionic-liquids-as-green-corrosion-inhibitors-for-industrial-metals-and-alloys
|
Open access peer-reviewed chapter
# Ionic Liquids as Green Corrosion Inhibitors for Industrial Metals and Alloys
By Chandrabhan Verma, Eno E. Ebenso and Mumtaz Ahmad Quraishi
Submitted: March 28th 2017Reviewed: July 17th 2017Published: February 28th 2018
DOI: 10.5772/intechopen.70421
## Abstract
Present chapter describes recent advances in the field of development of ionic liquids as green and sustainable corrosion inhibitors for metals and alloys. The present chapter has been divided into several sections and subsections. Recently, development of the green and sustainable technologies for the corrosion prevention is highly desirable due to increasing ecological awareness and strict environmental regulations. In the last two decades, corrosion inhibition using ionic liquids has attracted considerable attention due to its interesting properties such as low volatility, non-inflammability, non-toxic nature, high thermal and chemical stability and high adorability. Several types of ionic liquids have been developed as “green corrosion inhibitors” for different metals and alloys such as mild steel, aluminum, copper, zinc, and magnesium in several electrolytic media. The ionic liquids are promising, noble, green and sustainable candidates to replace the traditional volatile corrosion inhibitors.
### Keywords
• ionic liquids
• corrosion
• green corrosion inhibitors
• designer solvents
• ferrous and non-ferrous metals
## 1. Introduction
### 1.1. Corrosion and its economic impact
Corrosion is an irreversible and spontaneous deterioration of metal or alloy through chemical or electrochemical reaction with the environment [1, 2]. Corrosion causes enormous wastes of metallic materials which lead to enormous economic losses all over the world. Therefore, corrosion has drawn considerable academic and industrial attention [1, 2, 3, 4]. According to highly cited study carried out by the National Association of Corrosion Engineers (NACE), in 1998, the total annual direct cost (estimated) of corrosion in U.S.A. was US $276 billion, equating approximately around 3.1% Gross Domestic Product (GDP; NACE 2002) [5]. In 2011, the total cost of corrosion in U.S.A. became more than US$2.2 trillion. As for as the corrosion cost in India is concern, it was around Rs. 2 lackscrores (US $45 billion) as proposed by 1st Global Corrosion Summit held in New Delhi, India in 2011 [6]. However, these estimated data are outdated and recently closer investigation of the NACE on the cost of corrosion is available according to which the annual global cost of corrosion is approximately US$2.5 trillion, equating 3.4% of the global GDP [7, 8]. In India, the annual corrosion cost is more than US $100 billion, while in South Africa, the direct corrosion cost is estimated to be around R130 billion (i.e. about US$ 9.6 billion) [7, 8]. There are several methods of corrosion protection have been developed such as coating, anodic and cathodic protections, alloying and de-alloying and use of synthetic corrosion inhibitors by suitably applying them we can reduce this cost of corrosion from 15% (US $375 billion) to 35% (US$ 875 billion).
### 1.2. Causes of corrosion
Pure metals are chemically unstable and undergo chemical and/or electrochemical reactions with their environments to form more stable oxides. The chemical reactivity of pure metals is related to their natural tendency of oxidation (except gold, silver and platinum), as they have tendency to return their natural state by chemical reactions with the constituents of environment [9, 10, 11, 12]. Since corrosion is a spontaneous process, relative rate of corrosion among a given series of metals is related to the change in standard Gibb’s free energy (ΔGᴼ). As more negative value of ΔGᴼ as high spontaneity of reaction and consequently higher corrosion rate [9, 10, 11, 12]. When metals and alloys exposed to environment and particularly in acid solution during several industrial processes like acid pickling, acid descaling, etc., corrosion will undergo forming stable oxides [13, 14, 15]. Therefore, these processes required some additives known as corrosion inhibitors that form protective covering over the metallic surface and isolate metals from the environment and thereby inhibit the corrosive degradation [13, 14, 15, 16, 17]. The corrosion products such as rust and scale can also act as corrosion inhibitors by accumulation on the surface and act as physical protective barrier. The natural tendency of metallic corrosion can be affected by several factors, however, the relative rate of corrosion of any particular metal is depending upon the Pilling—Bedworth ratio which is defined as Md/nmD, where mand dare the atomic weight and density of the metal, respectively and Mand Dare the molecular weight and density of scale (corrosion product) accumulated on the metallic surface, and ndenotes the number of metallic atoms in the molecular formula of corrosion product (rust or scale); for example for Fe2O3 and Al2O3, n = 2 [18, 19]. The magnitude of Pilling – Bedworth ratio can be used to explain where the surface film will be protective or not. The volume of corrosion product will be small than the volume of metal from which it was formed for Md/nmD < 1, in this situation it is expected that surface film of corrosion product contains pores and cracks that would be relatively non-protective. On the other hand, volume of corrosion product will be larger than the volume of metal for Md/nmD > 1, in that situation it is expected that surface film of corrosion product is relatively more compressed and compact and consequently the metal would be relatively more protected.
### 1.3. Corrosion prevention methods and corrosion inhibitors
There are several methods of corrosion protection have been developed among which, synthetic corrosion inhibitors are one of the best methods due to its advantages such as cost effectiveness and ease of application in industry [20, 21, 22, 23]. The flow diagram of the available corrosion protection measures is shown in Figure 1. The passivating inhibitors are also known as anodic inhibitors because they general inhibit the metallic corrosion by forming the surface oxide (passive) film and causes the large anodic shift corrosion potential (Ecorr) [24]. The passivating inhibitors can be further classified into oxidizing anions that passivate the metallic surface in the absence of oxygen such as chromate, nitrite and nitrate and non-oxidizing anions that can passivate the metallic surface only in the presence of oxygen such as phosphate, tungstate and molybdate. The cathodic inhibitors either decrease the rate of cathodic reactions or precipitate on the cathodic areas to increase the surface impedance that decrease the diffusion of reducible species to these areas [24]. The cathodic inhibitors act by three different mechanisms namely, cathodic poisons, cathodic precipitates and oxygen scavengers. Generally, arsenic and antimony make the association of hydrogen more difficult and act as cathodic poisons, calcium, zinc and magnesium precipitates in their oxide forms and act as cathodic precipitates and sodium sulfite and hydrazine react with surrounding oxygen and act as oxygen scavengers [25, 26, 27, 28].
Organic compounds are also known as filming inhibitors; generally inhibit metallic corrosion by forming the protective surface film that isolates the metal form the surrounding (corrosive) environments. Most of the well know organic inhibitors are heterocyclic compounds containing polar functional groups such as ‐NO2, ‐OH, ‐OCH3, ‐CH3, ‐NH2, ‐COOC2H5, ‐CONH2, ‐COOH, etc. [29, 30, 31]. These polar functional groups and conjugated π-electrons of multiple bonds (double and triple) act as adsorption centers during metal-inhibitor interactions. This type of adsorption results into blocking of anodic and cathodic reactions indirectly. The adsorption of these inhibitors is affected by several factors such as nature and magnitude of charge present on metal, nature of electrolyte, electronic structure of inhibitor molecules, nature of substituents, solution temperature, exposure time etc. [29, 30, 31, 32, 33, 34].
### 1.4. Ionic liquids as green corrosion inhibitors
“Green chemistry” which is a relatively new and rapidly growing area of chemistry that involves designing of products and processes that reduce the use and production of toxic substances [35, 36, 37, 38]. Recently, worldwide growing ecological awareness and strict environmental protocols do not permit the synthesis and utilization of hazardous traditional volatile corrosion inhibitors. Therefore, there is vital need for improvement in the synthetic and engineering chemistry either by environmental friendly starting materials or proper designing for synthesis using non-classical energy sources such as ultrasound and microwave heating. In this regard use of multi component reactions (MCRs) in combination with ultrasonic (sonochemical) and microwave irradiation is one of the best alternative synthetic strategies toward “green synthesis.” Recently, scientists are trying to develop plant extracts and drugs as green corrosion inhibitors due to their natural and/or biological origins and non-toxic nature [39, 40, 41]. However, extraction and purification of plant extracts is very tedious, laborious, extremely expensive, time consuming and requires large amount of organic solvents [42, 43]. Therefore, there is need to develop “green inhibitors” by proper designing of the synthesis that can be achieved either by using cheap and environmental friendly starting materials or by synthesizing them from one step MCR reactions.
Toward, “green chemistry,” utilization of ionic liquids has immersed as new strategy due to its several fascinating properties such as low melting point (lower that 100°C), high polarity, low toxicity, low vapor pressure, very high thermal and chemical stability, less hazardous influence on environment and living being [44, 45, 46, 47, 48]. By definition, ionic liquids are materials that mainly composed of ions with melting point below than 100°C. The properties of ionic liquids could be modified according to the need by proper selection of cations and anions, which is the greatest advantage for designing ionic liquids of specific properties [49, 50, 51]. Due to this reason ionic liquids are also known as “designer chemicals” that have potential to consume as solvent or catalysis for various chemical transformations [44, 45, 46, 47, 48, 49, 50, 51]. The rapid utilization of ionic liquids in almost all fields of chemistry and chemical engineering is resulted to their above mentioned fascinating properties which enable them as “green and sustainable chemicals” having tendency to dissolve wide range of inorganic and organic compounds. The ionic liquids follow the principals of “green chemistry” proposed by Paul Anastas and John Warner [52, 53, 54].
#### 1.4.1. Properties and applications of ionic liquids
The ionic liquids have several fascinating properties such as low volatility (low vapour pressure), very high stability over wide range of pH and temperature, capability to dissolve a wide range of organic and inorganic compounds as they generally exist in their ionic forms through which they easily dissolve in polar solvents like H2O, HCl, etc., moreover, their cationic counterparts generally contain large organic moieties through which they are capable to dissolve non-polar organic compounds, capability to solubilize gases like H2, CO, CO2 etc., dependency of solubility on the nature of cations and anions, acceleration of reaction rate for chemical transformation under microwave heating, long time stability without decomposition and their high selectivity [55, 56, 57, 58, 59, 60, 61, 62]. These fascinating properties of ionic liquids make them good candidature to replace conventional organic volatile solvents with non-conventional ionic liquids that have been employed in variety of chemical transformations such as solvents for synthesis of nanomaterials and nanostructure, biochemical transformations, nucleophilic substitution reactions, electrodeposition of metals and semiconductors and solvent extraction, separation of petrochemical relevance mobile phase converter in HPLC, catalyst in various chemical and biochemical transformations, dye sensitizer for solar cells, oil shale processing, etc. (Figure 2) [55, 56, 57, 58, 59, 60, 61, 62].
#### 1.4.2. Classification of ionic liquids
The ionic liquids can be classified into several categories based on various bases. Hajipour and Refiee [63] have classified the ionic liquids into eleven classes namely, neutral ionic liquids, acid ionic liquids, basic ionic liquids, ionic liquids with amphoteric anions, functionalized ionic liquids, protic ionic liquids, chiral ionic liquids, supported ionic liquids, bio-ionic liquids, poly-ionic liquids, and energetic ionic liquids and also have described common features and properties of these ionic liquids. However, Suresh and Sandhu [62, 63] classified ionic liquids into only two classes namely, cationic and anionic ionic liquids. They were further subdivided anionic ionic liquids into several subclasses namely, borates, dicyanamide (DCN), Halide, Bis(trifluoromethylsulfonyl)imide (NTF), nonaflate (NON), phosphate, sulfate, sulfonate, thiocyanate (SCN), tricyanomethide (TCC) based anionic liquids. Some common classes of ionic liquids with examples and their salient features are described in Table 1.
### Table 1.
Classification of ionic liquids and their common features.
### 1.5. Comparison between organic inhibitors and ionic liquids
Over past two decades corrosion inhibition using ionic liquids (ILs) has experienced an outstanding growth and abundant examples on corrosion inhibitions are available that have been effectively carried out in different corrosive media. Although, traditional volatile compounds have been most extensively used as corrosion inhibitors in several industries. However, most of them are toxic for living being and environment [64, 65, 66]. In view of this, ionic liquids (ILs) have been used extensively in recent years. Ionic liquids have several advantageous physiochemical properties including non-toxic, high conductivity, non-flammability, as well as high thermal and chemical stability [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63]. One of the most significant characteristics of ionic liquids is their environmental friendly and non-hazardous nature due to their non-negligible vapour pressure. Unlike to traditional volatile corrosion inhibitors, due to their extremely low vapour pressure these compounds will not evaporate and will not contaminate the surrounding environment [67, 68]. Additionally, sometimes the use of organic inhibitors particularly polymeric and high molecular weighted organic compounds is limited due to their extremely low solubility in the polar corrosive media [69, 70, 71, 72]. However, ionic liquids are highly soluble in the polar corrosive environments due to their ionic nature [73]. Furthermore, there is limit less prospect of suitably modifying the structure of the anion and cation of any given ionic liquids delivers an unlimited amount of potential derivatives having numerous physiochemical properties, while this type of modification is not possible with volatile corrosion inhibitors. In summary, the use of ionic liquids as corrosion inhibitors is preferred as compared to traditional volatile (toxic) corrosion inhibitors due to their several advantageous physiochemical properties including their high solubility, non-toxic, high conductivity, non-flammability, less volatility as well as high chemical stability and more importantly due to their “green and sustainable” nature.
## 2. Applications of ionic liquids as corrosion inhibitors
Several fascinating properties of the ionic liquids make them ideal candidates to replace the traditional corrosion inhibitors that have several adverse effects on environment and living beings. Recently, a large number of works have been reported describing the use of ionic liquids as corrosion inhibitors.
### 2.1. Ionic liquids as corrosion inhibitors for mild steel
Mild steel is most frequently used as constructional material for several industries due to its high mechanical strength and low cost [74, 75]. However, these materials are highly reactive and undergo corrosive degradation during various industrial processes like acid cleaning, acid descaling, acid etching, and acid pickling processes that require use of additives in order to increase the lifespan of metal/alloy has used [76]. The use of organic compounds containing heterocyclic rings and polar fictional groups such as amino, hydroxyl, methyl, methoxy, nitro, nitrile, etc., as additive is one of the most important alterative to protect metals and alloys from these unsolicited reactions [74, 75]. These compounds inhibit corrosion by adsorbing over the metallic surface [74, 75, 76, 77]. However, the use of these highly volatile traditional toxic corrosion inhibitors is limited due to increasing ecological awareness and strict environmental regulations. In this regards consumption of “ionic liquids” as corrosion inhibitors has become an important green alternative methods of corrosion protection. Literature survey reveals that several synthetic ionic liquids have been used as effective corrosion inhibitors for mild steel (or carbon steel) in various electrolytic media. Likhanova et al. [78] synthesized two ionic liquids namely, 1,3-dioctadecylimidazolium bromide (ImDC18Br) and N-Octadecylpyridiniumbromide (PyC18Br) using conventional and microwave heating methods, respectively and investigated their inhibition performance on mild steel corrosion in 1M H2SO4 using several experimental techniques. They were observed that studied ionic liquids acted as good corrosion inhibitors for mild steel in aqueous acid solution. The adsorption on metallic surface takes place via chemisorption mechanism which obeyed the Langmuir adsorption isotherm. Potentiodynamic polarization results revealed that applied ionic liquids behaved as mixed type inhibitors. These authors were proposed a mechanism of corrosion inhibition on the basis of results obtained from SEM-EDX, XRD and Mossbauer analyses. The inhibition performance of the 1-ethyl-3-methylimidazolium dicyanamide (EMID) on mild steel corrosion in 0.1M H2SO4 using several experimental techniques [79] has been tested. Results showed that EMID inhabits metallic corrosion by adsorption on the metallic surface which was confirmed by decreased values of Cdl and increased surface coverage in presence of the inhibitor. The adsorption of the EMID over metallic surface obeyed the Langmuir adsorption isotherm. The inhibition performance of two ionic liquids namely 1-butyl-3-methylimidazolium chlorides (BMIC) and 1-butyl-3-methylimidazolium hydrogen sulfate ([BMIM]HSO4) on mild steel corrosion in 1M HCl have been studied by Zhang and Hua [80] using electrochemical and weight loss experiments. Results showed that the inhibition efficiency of both ionic liquids obeyed the order: ([BMIM]HSO4) > (BMIC). They were found that adsorption of these compounds on mild steel surface obeyed the Langmuir adsorption isotherm. Potentiodynamic study suggested that both ionic liquids acted as mixed type inhibitors. The effect of temperature (303–333 K) was also investigated on both the ionic liquids. Finally, several activation and thermodynamic parameters such as energy of activation (Ea), enthalpy of activation (ΔH), entropy of activation (ΔS), adsorption constant (Kads) and Gibb’s standard free energy (ΔG) were calculated in order to explain the mechanism of adsorption and corrosion inhibition of both the ionic liquids.
The inhibition performance of 1-octyl-3-methylimidazolium bromide ([OMIM]Br) and 1-allyl-3-octylimidazolium bromide ([AOIM]Br) on mild steel corrosion in 0.5 M H2SO4 using weight loss, electrochemical, scanning electron microscopy (SEM) and Quantum chemical calculations techniques showed that both the ionic liquids acted as good corrosion inhibitors and their adsorption on the metallic surface obeyed the El-Awady thermodynamic–kinetic model and acted as slightly cathodic type inhibitors [81].
Table 2 represents the corrosion inhibition properties of several other ionic liquids that have been employed as inhibitors for mild steel corrosion in electrolytic media [82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116]. The chitosan-based ionic liquid was synthesized using oleic acid and p-toluene sulfonic acid and its corrosion inhibition efficiency was determined using several electrochemical measurements [117]. Results of the investigated study revealed that presence of the ionic liquid in the chloride containing corrosive medium decreased the rate of metallic dissolution as well as hydrogen evolution. Adsorption of the ionic liquid followed the Langmuir adsorption isotherm. Polarization study suggested that investigated ionic liquid acted as mixed type inhibitor. Tseng and coworkers [118] investigated the corrosion characteristics of carbon steel, 304 stainless steel (304 SS) and pure titanium (Ti) in aluminum chloride–1-ethyl-3-methylimidazolium chloride ionic liquid for the first time. These authors reported the active-to-passive transition behavior for CS sample. Among the tested materials 304 SS exhibited the maximum stability in the high chloride environment. The most peculiar finding was that Ti was severally corroded in the ionic liquid because it does not undergo passivation. The ionic liquid in non-aqueous, low-oxygen and high halogen containing showed different corrosion behavior and mechanism. Similar observation has been reported by other authors for different metals including copper, nickel and stainless steel [119]. Recently, the inhibition behavior of 1,4-di [1-methylene-3-methyl imidazolium bromide]- benzene on mild steel corrosion in 1M H2SO4 have been studied using electrochemical and surface analysis methods [120]. The ionic liquid under taken in the study inhibits metallic corrosion by adsorbing on the surface which mechanism obeyed the Langmuir adsorption isotherm. The adsorption mechanism was supported by SEM, EDX and AFM analyses. Polarization study reveals that studied ionic liquid acted as mixed type inhibitor. The ongoing discussion reveals that although, several classes of ionic liquids have been used as effective inhibitors for mild steel corrosion in various aggressive media, however, imidazole based ionic liquids have been used most extensively [78, 79, 80, 81, 82, 83, 84, 89, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 114, 115, 116, 123].
### Table 2.
Ionic liquids as corrosion inhibitors for mild steel in different electrolytic media, their mode of adsorption and techniques used for evaluation of the inhibition performance.
### 2.2. Ionic liquids as corrosion inhibitors for aluminum
Aluminum is the second most commonly used metal due to its several fascinating properties like its low atomic mass and negligible standard electrode potential. Several traditional organic and inorganic compounds have been used previously in order to protect dissolution of protective surface oxide film and ultimately decrease the corrosion rate. However, employment of the ionic liquids as corrosion inhibitors is limited as literature survey reveals that only few works are available describing the corrosion inhibition performance of ionic liquids. The inhibition performance of 1-butyl-3-methylimidazoliumchlorides (BMIC), 1-hexyl-3-methylimidazolium chlorides (HMIC) and 1-octyl-3-methylimidazoliumchlorides (OMIC) on aluminum corrosion in 1M HCl using electrochemical and weight loss methods showed that inhibition efficiencies of these ionic liquids increase with increasing their concentration and obeyed the order: OMIC > HMIC > BMIC [124]. Potentiodynamic study revealed that all ionic liquids acted as mixed type inhibitors and their adsorption on aluminum surface followed the Langmuir adsorption isotherm. The inhibition efficiency of an ecofriendly ionic liquid, 1,3-bis(2-oxo-2-phenylethyl)-1H-imidazol-3-ium bromide (OPEIB) on 6061 Al-15 alloy in 0.1 M H2SO4 solution using electrochemical impedance spectroscopy and potentiodynamic polarization, scanning electron microscopy (SEM) and energy dispersive X-ray (EDX) spectroscopic methods revealed that it is a good corrosion inhibitor and its adsorption on aluminum surface obeyed the Temkin adsorption isotherm [125]. The three synthesized ILs, namely poly(ionic liquid)s (PILs), namely (poly(1-vinyl-3-dodecyl-imidazolium) (PImC12), poly(1-vinyl-3-octylimidazolium) (PImC8) and poly(1-vinyl-3-butylimidazolium) (PImC4) hexafluorophosphate) tested as inhibitor for aluminum alloy AA6061 in 0.1-1.0 M H2SO4 solution [126]. Results showed that they act as mixed type inhibitor and their inhibition efficiencies obeyed the order: (PImC12 > PImC8 > PImC4). Adsorption of these ionic liquids followed the Langmuir adsorption isotherm. Four newly synthesized quaternary ammonium based surfactants in the series of hexanediyl-1,6-bis-(diethyl alkyl ammonium bromide), designated as CmC6Cm(Et) · 2Br (m = 10, 12, 14, 16), were synthesized and evaluated as inhibitors for aluminum corrosion in 1M HCl solution [127]. Results showed that all investigated surfactants act as good inhibitors and inhibit corrosion by becoming adsorbate at metal/electrolyte interfaces and their adsorption on metallic surface obeyed the Langmuir adsorption isotherm. Trombetta et al. [128] studied the stability of the aluminum in 1-butyl-3methylimidazolium tetrafluroborate ionic liquid and ethylene glycol mixtures using electrochemical impedance spectroscope (EIS). These authors observed decrease in polarization resistance and increase in the capacitance related with the passive oxide dielectric properties on increasing the ethylene glycol and/or water content in the mixtures. Presence of salts namely Na2B4O7.7H2O and NaH2PO4 in the mixtures, stabilize the oxide payer form over the metallic surface and thereby reduce the changes of metallic corrosion. The inhibition behavior of 1,3-bis(2-oxo-2-phenylethyl)-1H-imidazol-3-ium bromide (OPEIB) on 6061 Al-15 vol. pct. SiC(p) composite in 0.1M H2SO4 solution was studied by Shetty and Shetty [125] using electrochemical (EIS and PDP), SEM and EDX methods. The investigated ionic liquid exhibits the maximum efficiencies of 96.7 and 94% using PDP and EIS methods, respectively. Potentiodynamic polarization study further reveals that studied ionic liquid behaves as cathodic type inhibitor and its adsorption on the composite surface followed the Temkin adsorption. Li et al. [129] study the inhibition behavior of tetradecylpyridinium bromide (TDPB) on aluminum corrosion in 1M HCl solution using weight loss and electrochemical methods. Results of the investigation showed that TDPB inhibits the aluminum corrosion by adsorbing on the metallic surface. The adsorption of the TDPB followed the Langmuir adsorption isotherm. Polarization study suggested that TDPB acts as cathodic type inhibitor for acidic aluminum corrosion. Bermudez and coworkers [130] investigated the surface interactions of seven alkylimidazolium ionic liquids with aluminum alloy Al 2011 using immersion test. The immersion experiments for aluminum corrosion was carried out in 1 and 5 wt.% of 1-ethyl,3-methylimidazolium tetrafluoroborate (IL1) in water. Results showed that neat solution of ionic liquids did not cause any corrosion. The inhibition behavior was discussed on the basis of SEM, EDX, XPS and XRD techniques.
### 2.3. Ionic liquids as corrosion inhibitors for copper and zinc
Copper and its alloys have been extensively employed in industries for various applications such as building construction, electricity, electronics, coinages, ornamental and formation of industrial equipment due to their relatively good thermal, electrical, mechanical and corrosion resistance properties [131]. However, in presence of aggressive anions like chloride, sulphate and nitrate these materials undergo sever attack resulting into loss of these materials due to corrosion occurs [132, 133]. Similar to the aluminum the use of ionic liquids as corrosion inhibitors for copper and zinc is also limited as literature survey revealed that only few ionic liquids have been used as corrosion inhibitors for these materials. Qi-Bo and Yi-Xin [134] newly synthesized three ionic liquids namely 1-butyl-3-methylimidazolium hydrogen sulfate ([BMIM]HSO4), 1-hexyl-3-methylimidazolium hydrogen sulfate ([HMIM]HSO4), and 1-octyl-3-methylimidazolium hydrogen sulfate ([OMIM]HSO4) and studied their inhibition efficiency on copper corrosion in 0.5 M H2SO4 using electrochemical impedance spectroscopy and potentiodynamic polarization techniques. The inhibition efficiency of the ionic liquids follows the order: [OMIM]HSO4 > [HMIM]HSO4 > [BMIM]HSO4. Results obtained by these authors showed that adsorption of the studied ionic liquids followed the Langmuir adsorption isotherm. Polarization study revealed that these ionic liquids behaved as mixed type inhibitors. Gabler et al. [135] studied the inhibition performance of two ionic liquids namely (2-hydroxyethyl)-trimethyl-ammonium (IL1) and Butyl-trimethyl-ammonium (IL2) with identical anions; bis(trifluoromethyl-sulfonyl)imide on CuSn8P and steel 100Cr6, purchased from Metal Supermarkets (Brunn am Gebirge, Austria) using inductively coupled plasma optical emission spectrometry (ICP-OES), scanning electron microscopy (SEM) with energy dispersive X-ray spectrometry (EDX) and X-ray photoelectron spectroscopy (XPS) in water in the absence and presence of 1.5% of the ionic liquids. Manamela et al. [136] studied the inhibition performance of two ionic liquids; 1-butyl-3-methylimidazolium tetrafluoroborate [BMIM][BF4] and 1-decyl-3-methylimidazolium tetrafluoroborate [DMIM][BF4] on corrosion of zinc in 1M HCl using gravimetric analysis and theoretical Density Functional Theory (DFT) approach, using the B3LYP functional. Results showed that both the ionic liquids acted as good corrosion inhibitors and their inhibition efficiencies increase with increasing their concentrations. The inhibition efficiencies of the ionic liquids obeyed the order: [DMIM][BF4] > [BMIM][BF4]. Values of activation energy (Ea) and enthalpy of activation (ΔH) suggested that both the ionic liquids adsorbed over the surface through physisorption mechanism. Adsorption of these ionic liquids on metallic surface followed the Langmuir adsorption isotherm.
### 2.4. Ionic liquids as corrosion inhibitors for magnesium
Unlike active light metals such as aluminum and titanium, magnesium based alloys do not form protective passivating film. Moreover, these alloys easily react with the components of environment to from hydroxides, oxides, carbonates films that are highly porous, inhomogeneous and poorly bonded that cannot provide satisfactory protection to the metals against corrosion. Among the available methods of corrosion protection, organic coating is one of the best methods. Huanga et al. [137] has presented an early review on the corrosion protection of magnesium by some ionic liquids. However, present chapter is describing the few recent advances in the utilization of ionic liquids as corrosion inhibitors. Suna et al. [138] have investigated the inhibition effect of six phosphonium cation based ionic liquids (ILs) namely, tetradecyltrihexylphosphonium diphenylphosphate (1), tetradecyltrihexylphosphoniumdibutylphosphate (2), tetradecyltrihexylphosphonium bis(2-ethylhexyl) phosphate (3), tetradecyltrihexyl phosphonium diisobutyldithiophosphinate (4), tetradecyltrihexylphosphoniumbis(2,4,4-trimethyl pentyl) phosphonate (5), and tetradecyltrihexyl phosphonium O,O-diethyl dithiophosphate on magnesium alloys using electrochemical and surface investigation methods.
## 3. Ionic liquids as corrosion inhibitors: DFT study
Nowadays, several computational methods particularly, DFT (Density Functional Theory) based quantum chemical calculations have been emerged as potential tools for studying the interactions between inhibitors and metallic surface. The DFT calculations provide several important parameters such as energies of highest occupied molecular orbital (EHOMO), lowest unoccupied molecular orbital (ELUMO), energy band gap (ELUMO − EHOMO = ΔE), global electronegativity (χ), global hardness (η) and softness (σ), fraction of electron transfer (ΔN) and dipole moment (μ). In general, value of EHOMO is related with electron donating ability, while the value of ELUMO related with the electron accepting ability of the inhibitor molecules [74, 75, 76, 77]. A higher value of EHOMO and lower value of ELUMO associated with high inhibition performance. The inhibition efficiency of inhibitor increases with decreasing the energy band gap (ΔE). A high value of global electronegativity (χ) is related with lower electron donating ability and therefore, the value of electronegativity (χ) inversely related with the inhibition efficiency order [74, 75, 76, 77]. Inhibition efficiency of the inhibitor molecules decreases with increasing the hardness (η) and decreasing the softness (σ). Generally, inhibition performance of the inhibitor molecules increases with increasing their dipole moment (μ), however, negative trends of the inhibition efficiency is also reported by several authors [74, 75, 76, 77]. Lastly, the value of electron transfer gives direct information about the relative extent of metal-inhibitor interactions. A high value of ΔN is associated with high charge transfer and therefore high inhibition efficiency [74, 75, 76, 77, 102].
The DFT based quantum chemical calculations have also been employed to describe the adsorption behavior of some ionic liquids on the metallic surface. Our research group [102] studied the adsorption behavior of four imidazolium-based ionic liquids, namely 1-propyl-3-methylimidazolium bis(trifluoromethyl-sulfonyl) imide ([PMIM][NTf2), 1-butyl-3-methylimidazoliumbis(trifluoromethyl-sulfonyl) imide ([BMIM][NTf2), 1-hexyl-3-methylimidazolium bis(trifluoromethyl-sulfonyl) imide([HMIM][NTf2]), and 1-propyl-2,3-methylimidazolium bis(trifluoromethyl-sulfonyl) imide ([PDMIM][NTf2]) on mild steel corrosion in 1M HCl using experimental and quantum chemical calculations. The inhibition efficiencies of these ionic liquids follow the experimental trend: [PDMIM][NTf2] > [HMIM][NTf2] > [BMIM][NTf2] > [PMIM][NTf2]. The values of EHOMO and ELUMO are well satisfied the experimental order of inhibition efficiency. Results showed that [PDMIM][NTf2] exhibited the lowest value of ΔE and therefore related with the highest chemical reactivity and inhibition efficiency. The values of dipole moment (μ) and the molecular volume (MV) did not show any regular trends. However, the values of global softness (σ) again show that the [PDMIM][NTf2] is most soft molecule among the tested compounds thereby associated with highest chemical reactivity and inhibition efficiency. The quantum chemical calculations provide good insight about the inhibition mechanism and well supported the experimental order of inhibition efficiency. Similar observations were reported for few other metals and alloys in several corrosive media [82, 139, 140, 141, 142, 143].
## 4. Mechanism of corrosion inhibition
Similar to most of the organic corrosion inhibitors, ionic liquids (ILs) inhibit metallic corrosion by blocking the anodic and cathodic sites present over the metallic surface [78, 144, 145]. Therefore, inhibition of metallic corrosion in presence of ionic liquids involves blocking of anodic oxidative metallic dissolution as well as cathodic hydrogen evolution reactions [78, 144]. The mechanism of metallic (M) corrosion inhibition by ionic liquids in sulphuric acid has been described below. The inhibition mechanism of metallic corrosion by ionic liquids in other protic acidic solutions such as in HCl and HNO3 will be similar because of their similar nature. The only difference in their nature is that they possess different counter ions (Cl, NO3) rather than sulphate ion of sulphuric acid. According to Likhanova et al. [78], anodic dissolution of metals (M) in aqueous acidic solution (e.g. H2SO4) can be represented as follows [78]:
However, in presence of ionic liquids, anodic reactions can be represented as follows:
where, ILsC+ and X represent the cationic counter part of the ionic liquids (mostly organic) and anionic counter part of the ionic liquid, respectively. It is important to mention that the concentration of sulphate ions is much higher as comared to the concentration of anionic counter part of the ionic liquids (X) that results into formation of [M(H2O) SO2−4]ads in larger proporsion than [MX]ads. Nevertheless, these both anionic charged species attracted positively charged cationic counter part of the ionic liquids (ILsC+) by electrostatic force of attraction (physisoprtion) and forms monomolecular layer as an insoluble complex on the metallic surface [78, 145]. The adsortion of the ILsC+ on metallic surface causes change in the surface polarity which induces the adsorption of the sulphate and X ions again which results into multimolecular layer [78, 146]. The multimolecular layers are stabilized by Vanderwaal’s cohesion force acting beteween organic moeity of the ionic liquids which causes a more closely adsorbed film at metal/electrolyte interfaces. Generally, the cationic part (ILsC+) interacts with the metallic surface and forms the multimolecular layers while rest of the part of the ionic liquids form hydrophobic hemi-micelles, ad-micelles and/or surface aggregation [78, 147]. The adsorbed multimolecular layers of the ILs isolate the metal (M) from corrosive enviroment and protect from corrosive dissolution.
The cathodic hydrogen evolution reaction (HER) can be represented by following simple stoichimmetry equation [148]:
H2O+2eH2g+2OHE11
Generally, the hydrogen evolution reaction (HER) follows two very common mechanisms that is, Volmer-Heyrovsky mechanism represented by Eqs. (12) and (13) or according to the Tafel hydrogen evolution mechanism represented by Eq. (14). In acidic medium, the Volmer-Heyrovsky and Volmer-Tafel hydrogen evolution mechanisms have been shown below [148, 149, 150]:
During the first step of cathodic reactions hydrogen ions (or hydronium ions) first adsorbed on the metallic surface by Volmer mechanism followed by discharge of hydrogen gas by Heyrovsky and Tafel mechanism represented by Eqs. (13-14). All these reactions do not occur with the same rate. Generally, a slow reaction step is followed by a fast reaction step [151]. If the Volmer reaction is fast, then Heyrovsky and/or Tafel reactions occur with slower rate and vice versa. Presence of the organic corrosion inhibitors (ILs) in the corrosive solution may retards or slow down the formation of MHads or retards the electron transfer to the hydronium ions and suppresses the Heyrovsky reactions (13). In general, in corrosive medium, the adsorbed hydrogen on metallic surface recombined and evolved as the bubbles of hydrogen gas. The formation of bubble and its evolution is the second step in the HER. The formation of hydrogen gas either occurs through hydrogen atom-atom combination as denoted by Volmer-Tafel Eq. (14) or may results through hydrogen atom-hydrogen ion combination as represented by Volmer-Heyrovsky Eq. (13) [151].
In the presence of inhibitors (ILs), cathodic can be represented as follows:
Initially, adsorption of hydronium ions and evolution of hydrogen gas occur at cathodic sites, simultaneously. At cathode, the cationic part of ionic liquids (ILsC+) starts competing with hydrogen ions for electrons [78, 152]. In general, ILsC+ has large molecular size and therefore replaces greater number of water molecules from the metallic surface. After their adsorption, cationic part of the ILs accepts electrons from the metal (M) which results into the formation of electrically neutral ionic liquids (inhibitors). The neutral species transfer (donation) their non-bonding (of heteroatoms) and π-electrons into the d-orbitals of the surface metallic atoms resulting into the formation of co-ordinate bonds between metal and ILs (chemisorption) as reported for several organic conventional inhibitors [78, 146, 153, 154, 155, 156]. However, metals are already electron rich species; this type of donation causes inter electronic repulsion which interns resulted into transfer of electrons from d-orbitals of the surface metallic atoms to antibonding molecular orbitals of the ILs (retro-donation). Both donation and retro-donation strengthen each other through synergism [153, 154, 155, 156, 157, 158, 159, 160].
## 5. Conclusions and future perspectives
On the basis of ongoing discussion it can be concluded that ionic liquids are green and sustainable inhibitors for corrosion of metals and alloys. The superiority of the use of ionic liquids as corrosion inhibitors compared to traditional volatile (toxic) corrosion inhibitors is based on the fact that they possess several fascinating properties such lower volatility, non-inflammability, non-toxic nature, chemical stability, high solubility in the polar solvents and their ability to easily adsorb on the metallic surface. Adsorption of the ionic liquids over the metallic surface results into formation of protective film which isolates the metals (alloys) from the corrosive environment and thereby inhibits corrosion. Among several available ionic liquids, imidazole based ionic liquids have been most extensively used. Some reports described the adsorption behavior of ionic liquids on metallic surface using DFT based quantum chemical calculations. However, the use of this technique should be further explored owing to its green nature to understand the mechanistic aspects of corrosion inhibition. The use of ionic liquids as corrosion inhibitors is preferred comparing with traditional inhibitors due to several physiochemical properties advantageous including their high solubility, non-toxic, high conductivity, and non-flammability, less volatility as well as high chemical stability and more importantly due to their “green and sustainable” nature.
chapter PDF
Citations in RIS format
Citations in bibtex format
## More
© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
## How to cite and reference
### Cite this chapter Copy to clipboard
Chandrabhan Verma, Eno E. Ebenso and Mumtaz Ahmad Quraishi (February 28th 2018). Ionic Liquids as Green Corrosion Inhibitors for Industrial Metals and Alloys, Green Chemistry, Hosam El-Din M. Saleh and Martin Koller, IntechOpen, DOI: 10.5772/intechopen.70421. Available from:
### chapter statistics
5Crossref citations
### Related Content
#### Green Chemistry
Edited by Hosam Saleh
Next chapter
#### Nanoscale Zero Valent Iron for Environmental Cadmium Metal Treatment
By Keyla T. Soto-Hidalgo and Carlos R. Cabrera
#### Polyester
Edited by Hosam Saleh
First chapter
#### Synthesis of Thermally Stable Polyesters
By Hossein Mighani
We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.
|
2021-04-13 08:51:37
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5465195775032043, "perplexity": 6226.889429422079}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038072175.30/warc/CC-MAIN-20210413062409-20210413092409-00164.warc.gz"}
|
https://homework.cpm.org/category/CC/textbook/cca2/chapter/A/lesson/A.1.1/problem/A-12
|
### Home > CCA2 > Chapter A > Lesson A.1.1 > ProblemA-12
A-12.
Jill is studying a strange bacterium. When she first looks at the bacteria, there are $1000$ cells in her sample. The next day, there are $2000$ cells. Intrigued, she comes back the next day to find that there are $4000$ cells! A-12 HW eTool (Desmos)
1. Should the graph of this situation be linear or curved?
• Is the growth linear or exponential?
1. Create a table and graph for this situation. The inputs are the days that have passed after she first began to study the sample, and the outputs are the numbers of cells of bacteria.
Make a table starting with day $0$. What is the multiplier?
|
2020-11-26 16:22:43
|
{"extraction_info": {"found_math": true, "script_math_tex": 4, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22522738575935364, "perplexity": 1530.2017083450858}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141188800.15/warc/CC-MAIN-20201126142720-20201126172720-00434.warc.gz"}
|
https://en.m.wikisource.org/wiki/Page:Popular_Science_Monthly_Volume_70.djvu/93
|
# Page:Popular Science Monthly Volume 70.djvu/93
89
THE VALVE OF SCIENCE
The three pairs of canals would have as sole function to tell us that space has three dimensions. Japanese mice have only two pairs of canals; they believe, it would seem, that space has only two dimensions, and they manifest this opinion in the strongest way; they put themselves in a circle, and, so ordered, they spin rapidly around. The lampreys, having only one pair of canals, believe that space has only one dimension, but their manifestations are less turbulent.
It is evident that such a theory is inadmissible. The sense-organs are designed to tell us of changes which happen in the exterior world. We could not understand why the Creator should have given us organs destined to cry without cease: Remember that space has three dimensions, since the number of these three dimensions is not subject to change.
We must, therefore, come back to the thory of Mach-Delage. What the nerves of the canals can tell us is the difference of pressure on the two extremities of the same canal, and thereby: (1) the direction of the vertical with regard to three axes rigidly bound to the head; (2) the three components of the acceleration of translation of the center of gravity of the head; (3) the centrifugal forces developed by the rotation of the head; (4) the acceleration of the motion of rotation of the head.
It follows from the experiments of M. Delage that it is this last indication which is much the most important; doubtless because the nerves are less sensible to the difference of pressure itself than to the brusque variations of this difference. The first three indications may thus be neglected.
Knowing the acceleration of the motion of rotation of the head at each instant, we deduce from it, by an unconscious integration, the final orientation of the head, referred to a certain initial orientation taken as origin. The circular canals contribute, therefore, to inform us of the movements that we have executed, and that on the same ground as the muscular sensations. When, therefore, above we speak of the series ${\displaystyle S}$ or of the series ${\displaystyle \Sigma }$, we should say, not that these were series of muscular sensations alone, but that they were series at the same time of muscular sensations due to the semicircular canals. Apart from this addition, we should have nothing to change in what precedes.
In the series ${\displaystyle S}$ and ${\displaystyle \Sigma }$, these sensations of the semicircular canals evidently hold a very important place. Yet alone they would not suffice, because they can tell us only of the movements of the head; they tell us nothing of the relative movements of the body, or of the members in regard to the head. And more, it seems that they tell us only of the rotations of the head and not of the translations it may undergo.
|
2019-09-17 11:35:38
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5578929781913757, "perplexity": 796.8116431214577}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573070.36/warc/CC-MAIN-20190917101137-20190917123137-00221.warc.gz"}
|
http://everything.explained.today/Probability_amplitude/
|
# Probability amplitude explained
In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability or probability density.
Probability amplitudes provide a relationship between the wave function (or, more generally, of a quantum state vector) of a system and the results of observations of that system, a link first proposed by Max Born. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 Nobel Prize in Physics for this understanding (see References), and the probability thus calculated is sometimes called the "Born probability". These probabilistic concepts, namely the probability density and quantum measurements, were vigorously contested at the time by the original physicists working on the theory, such as Schrödinger and Einstein. It is the source of the mysterious consequences and philosophical difficulties in the interpretations of quantum mechanics—topics that continue to be debated even today.
## Overview
See main article: Born rule.
### Physical
Neglecting some technical complexities, the problem of quantum measurement is the behaviour of a quantum state, for which the value of the observable to be measured is uncertain. Such a state is thought to be a coherent superposition of the observable's eigenstates, states on which the value of the observable is uniquely defined, for different possible values of the observable.
When a measurement of is made, the system (under the Copenhagen interpretation) jumps to one of the eigenstates, returning the eigenvalue belonging to that eigenstate. The system may always be described by a linear combination or superposition of these eigenstates with unequal "weights". Intuitively it is clear that eigenstates with heavier "weights" are more "likely" to be produced. Indeed, which of the above eigenstates the system jumps to is given by a probabilistic law: the probability of the system jumping to the state is proportional to the absolute value of the corresponding numerical weight squared. These numerical weights are called probability amplitudes, and this relationship used to calculate probabilities from given pure quantum states (such as wave functions) is called the Born rule.
Clearly, the sum of the probabilities, which equals the sum of the absolute squares of the probability amplitudes, must equal 1. This is the normalization (see below) requirement.
If the system is known to be in some eigenstate of (e.g. after an observation of the corresponding eigenvalue of) the probability of observing that eigenvalue becomes equal to 1 (certain) for all subsequent measurements of (so long as no other important forces act between the measurements). In other words the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements. If the set of eigenstates to which the system can jump upon measurement of is the same as the set of eigenstates for measurement of, then subsequent measurements of either or always produce the same values with probability of 1, no matter the order in which they are applied. The probability amplitudes are unaffected by either measurement, and the observables are said to commute.
By contrast, if the eigenstates of and are the different, then measurement of produces a jump to a state that is not an eigenstate of . Therefore, if the system is known to be in some an eigenstate of (all probability amplitudes zero except for one eigenstate), then when is observed the probability amplitudes are changed. A second, subsequent observation of no longer certainly produces the eigenvalue corresponding to the starting state. In other words, the probability amplitudes for the second measurement of depend on whether it comes before or after a measurement of, and the two observables do not commute.
### Mathematical
In a formal setup, any system in quantum mechanics is described by a state, which is a vector, residing in an abstract complex vector space, called a Hilbert space. It may be either infinite- or finite-dimensional. A usual presentation of that Hilbert space is a special function space, called , on certain set, that is either some configuration space or a discrete set.
\psi
, the condition
\psi\inL2(X)
specifies that a finitely bounded integral must apply:
\int\limitsX|\psi(x)|2d\mu(x)<infty;
this integral defines the square of the norm of . If that norm is equal to, then
\int\limitsX|\psi(x)|2d\mu(x)=1.
It actually means that any element of of the norm 1 defines a probability measure on and a non-negative real expression defines its Radon–Nikodym derivative with respect to the standard measure .
If the standard measure on is non-atomic, such as the Lebesgue measure on the real line, or on three-dimensional space, or similar measures on manifolds, then a real-valued function is called a probability density; see details below. If the standard measure on consists of atoms only (we shall call such sets discrete), and specifies the measure of any equal to,[1] then an integral over is simply a sum[2] and defines the value of the probability measure on the set, in other words, the probability that the quantum system is in the state . How amplitudes and the vector are related can be understood with the standard basis of, elements of which will be denoted by or (see bra–ket notation for the angle bracket notation). In this basis
\psi(x)=\langlex|\Psi\rangle
specifies the coordinate presentation of an abstract vector .
Mathematically, many presentations of the system's Hilbert space can exist. We shall consider not an arbitrary one, but a one for the observable in question. A convenient configuration space is such that each point produces some unique value of . For discrete it means that all elements of the standard basis are eigenvectors of . In other words, shall be diagonal in that basis. Then
\psi(x)
is the "probability amplitude" for the eigenstate . If it corresponds to a non-degenerate eigenvalue of, then
|\psi(x)|2
gives the probability of the corresponding value of for the initial state .
For non-discrete there may not be such states as in, but the decomposition is in some sense possible; see spectral theory and Spectral theorem for accurate explanation.
## Wave functions and probabilities
If the configuration space is continuous (something like the real line or Euclidean space, see above), then there are no valid quantum states corresponding to particular, and the probability that the system is "in the state " will always be zero. An archetypical example of this is the space constructed with 1-dimensional Lebesgue measure; it is used to study a motion in one dimension. This presentation of the infinite-dimensional Hilbert space corresponds to the spectral decomposition of the coordinate operator: in this example. Although there are no such vectors as, strictly speaking, the expression can be made meaningful, for instance, with spectral theory.
Generally, it is the case when the motion of a particle is described in the position space, where the corresponding probability amplitude function is the wave function.
If the function represents the quantum state vector, then the real expression, that depends on, forms a probability density function of the given state. The difference of a density function from simply a numerical probability means that one should integrate this modulus-squared function over some (small) domains in to obtain probability values – as was stated above, the system can't be in some state with a positive probability. It gives to both amplitude and density function a physical dimension, unlike a dimensionless probability. For example, for a 3-dimensional wave function, the amplitude has the dimension [L<sup>−3/2</sup>], where L is length.
Note that for both continuous and infinite discrete cases not every measurable, or even smooth function (i.e. a possible wave function) defines an element of ; see Normalisation below.
## Discrete amplitudes
When the set is discrete (see above), vectors represented with the Hilbert space are just column vectors composed of "amplitudes" and indexed by .These are sometimes referred to as wave functions of a discrete variable . Discrete dynamical variables are used in such problems as a particle in an idealized reflective box and quantum harmonic oscillator. Components of the vector will be denoted by for uniformity with the previous case; there may be either finite of infinite number of components depending on the Hilbert space.In this case, if the vector has the norm 1, then is just the probability that the quantum system resides in the state . It defines a discrete probability distribution on .
if and only if is the same quantum state as . if and only if and are orthogonal (see inner product space). Otherwise the modulus of is between 0 and 1.
A discrete probability amplitude may be considered as a fundamental frequency in the Probability Frequency domain (spherical harmonics) for the purposes of simplifying M-theory transformation calculations.
## A basic example
Take the simplest meaningful example of the discrete case: a quantum system that can be in two possible states: for example, the polarization of a photon. When the polarization is measured, it could be the horizontal state, or the vertical state . Until its polarization is measured the photon can be in a superposition of both these states, so its state could be written as:
|\psi\rangle=\alpha|H\rangle+\beta|V\rangle,
The probability amplitudes of for the states and are and respectively. When the photon's polarization is measured, the resulting state is either horizontal or vertical. But in a random experiment, the probability of being horizontally polarized is, and the probability of being vertically polarized is .
Therefore, a photon in a state
|\psi\rangle=\sqrt{1\over3}|H\rangle-i\sqrt{2\over3}|V\rangle
would have a probability of 1/3 to come out horizontally polarized, and a probability of 2/3 to come out vertically polarized when an ensemble of measurements are made. The order of such results, is, however, completely random.
## Normalization
In the example above, the measurement must give either or, so the total probability of measuring or must be 1. This leads to a constraint that ; more generally the sum of the squared moduli of the probability amplitudes of all the possible states is equal to one. If to understand "all the possible states" as an orthonormal basis, that makes sense in the discrete case, then this condition is the same as the norm-1 condition explained above.
One can always divide any non-zero element of a Hilbert space by its norm and obtain a normalized state vector. Not every wave function belongs to the Hilbert space, though. Wave functions that fulfill this constraint are called normalizable.
The Schrödinger wave equation, describing states of quantum particles, has solutions that describe a system and determine precisely how the state changes with time. Suppose a wavefunction is a solution of the wave equation, giving a description of the particle (position, for time). If the wavefunction is square integrable, i.e.
\int Rn
|\psi0(x,t
2 0)|
dx=a2<infty
for some, then is called the normalized wavefunction. Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle. Hence, at a given time, is the probability density function of the particle's position. Thus the probability that the particle is in the volume at is
P(V)=\intV\rho(x)dx=\intV|\psi(x,
2 t 0)|
dx.
Note that if any solution to the wave equation is normalisable at some time, then the defined above is always normalised, so that
\rhot(x)=\left|\psi(x,t)\right|2=\left|
\psi0(x,t) a
\right|2
is always a probability density function for all . This is key to understanding the importance of this interpretation, because for a given the particle's constant mass, initial and the potential, the Schrödinger equation fully determines subsequent wavefunction, and the above then gives probabilities of locations of the particle at all subsequent times.
## The laws of calculating probabilities of events
A. Provided a system evolves naturally (which under the Copenhagen interpretation means that the system is not subjected to measurement), the following laws apply:
1. The probability (or the density of probability in position/momentum space) of an event to occur is the square of the absolute value of the probability amplitude for the event:
P=|\phi|2
.
1. If there are several mutually exclusive, indistinguishable alternatives in which an event might occur (or, in realistic interpretations of wavefunction, several wavefunctions exist for a space-time event), the probability amplitudes of all these possibilities add to give the probability amplitude for that event:
\phi=\sumi\phii;
2=\left|\sum P=|\phi| i\phi
2 i\right|
.
1. If, for any alternative, there is a succession of sub-events, then the probability amplitude for that alternative is the product of the probability amplitude for each sub-event:
\phiAPB=\phiAP\phiPB
.
1. Non-entangled states of a composite quantum system have amplitudes equal to the product of the amplitudes of the states of constituent systems:
\phi\rm{system
} (\alpha,\beta,\gamma,\delta,\ldots)=\phi_1(\alpha)\phi_2(\beta)\phi_3(\gamma)\phi_4(\delta)\ldots. See the
1. Composite systems
section for more information.Law 2 is analogous to the addition law of probability, only the probability being substituted by the probability amplitude. Similarly, Law 4 is analogous to the multiplication law of probability for independent events; note that it fails for entangled states.
B. When an experiment is performed to decide between the several alternatives, the same laws hold true for the corresponding probabilities:
P=\sumi|\phi
2 i|
.
Provided one knows the probability amplitudes for events associated with an experiment, the above laws provide a complete description of quantum systems in terms of probabilities.
The above laws give way to the path integral formulation of quantum mechanics, in the formalism developed by the celebrated theoretical physicist Richard Feynman. This approach to quantum mechanics forms the stepping-stone to the path integral approach to quantum field theory.
## In the context of the double-slit experiment
See main article: Double-slit experiment. Probability amplitudes have special significance because they act in quantum mechanics as the equivalent of conventional probabilities, with many analogous laws, as described above. For example, in the classic double-slit experiment, electrons are fired randomly at two slits, and the probability distribution of detecting electrons at all parts on a large screen placed behind the slits, is questioned. An intuitive answer is that, where is the probability of that event. This is obvious if one assumes that an electron passes through either slit. When nature does not have a way to distinguish which slit the electron has gone through (a much more stringent condition than simply "it is not observed"), the observed probability distribution on the screen reflects the interference pattern that is common with light waves. If one assumes the above law to be true, then this pattern cannot be explained. The particles cannot be said to go through either slit and the simple explanation does not work. The correct explanation is, however, by the association of probability amplitudes to each event. This is an example of the case A as described in the previous article. The complex amplitudes which represent the electron passing each slit (and) follow the law of precisely the form expected: . This is the principle of quantum superposition. The probability, which is the modulus squared of the probability amplitude, then, follows the interference pattern under the requirement that amplitudes are complex:
P=|\psi\rm{first
} + \psi_|^2 = |\psi_|^2 + |\psi_|^2 + 2 |\psi_| |\psi_| \cos (\varphi_1-\varphi_2).Here,
\varphi1
and
\varphi2
are the arguments of and respectively. A purely real formulation has too few dimensions to describe the system's state when superposition is taken into account. That is, without the arguments of the amplitudes, we cannot describe the phase-dependent interference. The crucial term
2|\psi\rm{first
}| |\psi_| \cos (\varphi_1-\varphi_2) is called the "interference term", and this would be missing if we had added the probabilities.
However, one may choose to devise an experiment in which the experimenter observes which slit each electron goes through. Then case B of the above article applies, and the interference pattern is not observed on the screen.
One may go further in devising an experiment in which the experimenter gets rid of this "which-path information" by a "quantum eraser". Then, according to the Copenhagen interpretation, the case A applies again and the interference pattern is restored.[3]
## Conservation of probabilities and the continuity equation
See main article: Probability current. Intuitively, since a normalised wave function stays normalised while evolving according to the wave equation, there will be a relationship between the change in the probability density of the particle's position and the change in the amplitude at these positions.
Define the probability current (or flux) as
j={\hbar\overm}{1\over{2i}}\left(\psi*\nabla\psi-\psi\nabla\psi*\right)={\hbar\overm}\operatorname{Im}\left(\psi*\nabla\psi\right),
measured in units of (probability)/(area × time).
Then the current satisfies the equation
\nablaj+{\partial\over\partialt}|\psi|2=0.
The probability density is
\rho=|\psi|2
, this equation is exactly the continuity equation, appearing in many situations in physics where we need to describe the local conservation of quantities. The best example is in classical electrodynamics, where corresponds to current density corresponding to electric charge, and the density is the charge-density. The corresponding continuity equation describes the local conservation of charges.
## Composite systems
For two quantum systems with spaces and and given states and respectively, their combined state can be expressed as a function on, that gives theproduct of respective probability measures. In other words, amplitudes of a non-entangled composite state are products of original amplitudes, and respective observables on the systems 1 and 2 behave on these states as independent random variables. This strengthens the probabilistic interpretation explicated above.
## Amplitudes in operators
The concept of amplitudes described above is relevant to quantum state vectors. It is also used in the context of unitary operators that are important in the scattering theory, notably in the form of S-matrices. Whereas moduli of vector components squared, for a given vector, give a fixed probability distribution, moduli of matrix elements squared are interpreted as transition probabilities just as in a random process. Like a finite-dimensional unit vector specifies a finite probability distribution, a finite-dimensional unitary matrix specifies transition probabilities between a finite number of states. Note that columns of a unitary matrix, as vectors, have the norm 1.
The "transitional" interpretation may be applied to s on non-discrete spaces as well.
|
2021-03-05 16:33:15
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9428387880325317, "perplexity": 353.87800937750603}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178373095.44/warc/CC-MAIN-20210305152710-20210305182710-00358.warc.gz"}
|
http://math.stackexchange.com/questions/148863/the-limit-of-an-alternating-series?answertab=active
|
# The limit of an alternating series [duplicate]
Possible Duplicate:
The sum of $(-1)^n \frac{\ln n}{n}$
Compute $$\sum_{k=2}^{\infty}\frac{(-1)^{k}{\ln{k}}}{k}$$
-
## marked as duplicate by Marvis, Charles, Eric NaslundMay 23 '12 at 20:46
Consider, for $s>1$, the following auxiliary convergent series: $$\sum_{k=1}^\infty (-1)^k \frac{\log(k)}{k^s} = -\frac{\mathrm{d}}{\mathrm{d} s} \sum_{k=1}^\infty (-1)^k \frac{1}{k^s}= -\frac{\mathrm{d}}{\mathrm{d} s}\left( (2^{1-s} - 1)\zeta(s) \right)$$ The value of the series in question is obtained as a limit: $$\sum_{k=1}^\infty (-1)^k \frac{\log(k)}{k} = \lim_{s \searrow 1} \left( 2^{1-s} \log(2) \zeta(s) + \zeta^\prime(s) (1-2^{1-s}) \right)$$ Since $\zeta(s) = \frac{1}{s-1} + \gamma + \mathcal{O}(s-1)$, and $\zeta^\prime(s) = -\frac{1}{(s-1)^2} + \mathcal{O}(1)$ we arrive at: $$\sum_{k=1}^\infty (-1)^k \frac{\log(k)}{k} = \gamma \log(2) - \frac{\log^2(2)}{2}$$
|
2015-11-27 08:29:33
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9584223031997681, "perplexity": 747.0317365168967}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398448389.58/warc/CC-MAIN-20151124205408-00134-ip-10-71-132-137.ec2.internal.warc.gz"}
|