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https://chemistry.stackexchange.com/questions/150187/to-calculate-total-vapour-pressure-of-a-solution/150268 | # To calculate total vapour pressure of a solution
Question: 1 mole each of A and B were mixed to obtain an ideal solution of which one mole went to the vapour phase at equilibrium. If $$P°_\mathrm{A} = \pu{100 torr}$$ and $$P°_\mathrm{B} = \pu{900 torr}$$, then $$P_\mathrm{S}$$ would be?
For the above question, I first calculated vapour pressure of A and B which came out to be $$\pu{50 torr}$$ and $$\pu{450 torr}$$ respectively (using $$P°_\mathrm{A} × X_\mathrm{A}$$ where $$X_\mathrm{A}$$ is mole fraction of A) and then I calculated vapour pressure for A and B in vapour phase which came out to be $$0.1$$ and $$0.9$$ respectively, so the amount remaining in the liquid phase will be $$\pu{0.9 mole}$$ for A and $$\pu{0.1 mole}$$ for B, according to which the $$P_\mathrm{S}$$ should be $$\pu{180 torr}$$, but the answer provided is $$\pu{300 torr}$$, am I missing something? (also see the image below for better understanding of my approach).
Any help is appreciated! Thanks in advance!
• Could you please type out what you have written instead of posting an image? This is because images are not searchable, and harder to read. – Shoubhik R Maiti Apr 19 at 9:19
• I'm really sorry for that, actually i am pretty bad with message formatting, and i was running out of time so i decided to add an image instead, i'll redo that with a pen and replace the image as i get time. Thanks for pointing out. – kanishk6103 Apr 19 at 16:04
• Images are discouraged. Please use mathjax/mhchem, see Notation basics and How can I format math/chemistry expressions on Chemistry SE. See also Math SE MathJax tutorial. – Buck Thorn Apr 20 at 6:55
The starting point to solve the problem is Raoult's law. For each component we can write that the vapour pressure (related by Dalton's law to the total pressure) is equal to the product of the mole fraction in the solution and the vapour pressure of the pure component:
$$y_iP_S = \chi _i P^{\circ} _i$$
You can exploit the fact that the molar amounts in the problem are particularly tidy:
\begin{align} n_{1l} + n_{1g} &=1 \\ n_{2l} + n_{2g} &=1 \\ n_{1g} + n_{2g} &=1 \end{align}
So that for the first component: \begin{align} y_1 &= n_{1g} \\ \chi _1 &= n_{1l} \\ \rightarrow n_{1g}P_S &= n_{1l}P^{\circ}_1 \\ \rightarrow n_{1g}P_S &= (1-n_{1g})P^{\circ}_1 \\ \rightarrow P_S &= \frac{(1-n_{1g})P^{\circ}_1}{n_{1g}} = \frac{n_{2g}P^{\circ}_1}{(1-n_{2g})} \tag{1}\label{eq:firstcomponent} \end{align}
For the second component we can similarly write \begin{align} y_2 &= n_{2g} \\ \chi_2 &= n_{2l} \\ \rightarrow n_{2g}P_S &= n_{2l}P^{\circ}_2 \\ \rightarrow n_{2g}P_S &=(1-n_{2g})P^{\circ}_2 \\ \rightarrow P_S &=\frac{(1-n_{2g})P^{\circ}_2}{n_{2g}} \tag{2}\label{eq:secondcomponent} \end{align}
Equating the expressions for $$P_S$$ in \eqref{eq:firstcomponent} and \eqref{eq:secondcomponent} and rearranging terms we can write a quadratic equation in $$n_{2g}$$ whose solution can be inserted back into the expression for $$P_S$$. | 2021-06-21 22:43:30 | {"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": 21, "wp-katex-eq": 0, "align": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000079870224, "perplexity": 670.5807069383525}, "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/1623488504838.98/warc/CC-MAIN-20210621212241-20210622002241-00322.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-8-polynomials-and-factoring-8-7-factoring-special-cases-lesson-check-page-514/8 | # Chapter 8 - Polynomials and Factoring - 8-7 Factoring Special Cases - Lesson Check: 8
(2x+1)(x+6)
#### Work Step by Step
Given the polynomial $2x^{2}$ + 13x + 6 *** We break of the middle term into two factors that add to give +13 and multiply to give +12. The two numbers are +12 and +1. $2x^{2}$ + 1x + 12x + 6 We take the GCD of the first two and the GCD of the last two terms. x(2x+1) + 6(2x+1) We take (2x+1) and factor it out which gives us. (2x+1)(x+6)
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2018-08-17 16:09:24 | {"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": 0.6453101634979248, "perplexity": 1070.0178286608527}, "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-2018-34/segments/1534221212598.67/warc/CC-MAIN-20180817143416-20180817163416-00335.warc.gz"} |
http://en.wikipedia.org/wiki/Baud | # Baud
For other uses, see Baud (disambiguation).
In telecommunication and electronics, baud (, unit symbol "Bd") is synonymous with symbols per second or pulses per second. It is the unit of symbol rate, also known as baud or modulation rate; the number of distinct symbol changes (signaling events) made to the transmission medium per second in a digitally modulated signal or a line code. Baud is related to but should not be confused with gross bit rate expressed as bits per second, bps, b/s, bit/s or bits/s. However, though technically incorrect, in the case of modem manufacturers baud commonly refers to bits per second. They make a distinction by also using the term characters per second (CPS). In these anomalous cases, refer to the modem manufacturer's documentation to ensure an understanding of their use of the term "baud". An example would be the 1996 User's guide for the U.S. Robotics Sportster modem, which includes these definitions.
The symbol duration time, also known as unit interval, can be directly measured as the time between transitions by looking into an eye diagram of an oscilloscope. The symbol duration time Ts can be calculated as:
$T_s = {1 \over f_s},$
where fs is the symbol rate. There is also a chance of miscommunication which leads to ambiguity.
A simple example: A baud of 1 kBd = 1,000 Bd is synonymous to a symbol rate of 1,000 symbols per second. In case of a modem, this corresponds to 1,000 tones per second, and in case of a line code, this corresponds to 1,000 pulses per second. The symbol duration time is 1/1,000 second = 1 millisecond.
In digital systems (i.e., using discrete/discontinuous values) with binary code, 1 Bd = 1 bit/s. By contrast, non-digital (or analog) systems use a continuous range of values to represent information and in these systems the exact informational size of 1 Bd varies.
The baud unit is named after Émile Baudot, the inventor of the Baudot code for telegraphy, and is represented in accordance with the rules for SI units. That is, the first letter of its symbol is uppercase (Bd), but when the unit is spelled out, it should be written in lowercase (baud) except when it begins a sentence.
## Relationship to gross bit rate
Main article: Gross bit rate
The symbol rate is related to but should not be confused with gross bit rate expressed in bit/s. The term baud has sometimes incorrectly been used to mean bit rate,[citation needed] since these rates are the same in old modems as well as in the simplest digital communication links using only one bit per symbol, such that binary "0" is represented by one symbol, and binary "1" by another symbol. In more advanced modems and data transmission techniques, a symbol may have more than two states, so it may represent more than one bit (a bit (binary digit) always represents one of exactly two states).
If N bits are conveyed per symbol, and the gross bit rate is R, inclusive of channel coding overhead, the symbol rate fs can be calculated as
$f_\mathrm{s} = {R \over N}.$
In that case M=2N different symbols are used. In a modem, these may be sinewave tones with unique combinations of amplitude, phase and/or frequency. For example, in a 64QAM modem, M=64, and so the bit rate is N=6 ( 6=log2(64) ) times the baud. In a line code, these may be M different voltage levels.
The ratio might not even be an integer; in 4B3T coding, the bit rate is 4/3 baud. (A typical basic rate interface with a 160 kbit/s raw data rate operates at 120 kbaud.) On the other hand, Manchester coding has a bit rate equal to 1/2 the baud.
By taking information per pulse N in bit/pulse to be the base-2-logarithm of the number of distinct messages M that could be sent, Hartley[1] constructed a measure of the gross bitrate R as
$R = f_\mathrm{s} \log_2(M)\quad$ where $\quad N = \log_2(M).$ | 2014-07-11 17:50:08 | {"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": 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": 0.7580064535140991, "perplexity": 1658.6436529759428}, "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-2014-23/segments/1404776428273.44/warc/CC-MAIN-20140707234028-00015-ip-10-180-212-248.ec2.internal.warc.gz"} |
https://stats.stackexchange.com/questions/181283/when-2-variables-are-highly-correlated-can-one-be-significant-and-the-other-not | # When 2 variables are highly correlated can one be significant and the other not in a regression?
In regression, when 2 parameters are correlated and added to a model separately, how likely is it that one parameter will be a significant predictor of the response variable while the other is not? To me this seems unlikely but I've encountered it in a publication. Based on my understanding and experience of multi-collinearity, if one variable is highly correlated with another, both should be influential in a model. Under what conditions could this occur?
Context: For pedagogical purposes I am replicating an GLMM analysis from a 2004 Science paper that used observational data which was posted with the paper. They are interested in causal interpretations of 2 parameters that are correlated (Pearson's=0.67, p<0.01). They state that the 1st parameter is significant in their regression (beta=0.24,SE=0.05,p<0.01) while the other is not (b=0.02,SE=0.07,p=0.51). The language of the paper implies that they ran two separate models, one with each predictor on its own (but see edit below). A model with just the 1st predictor also has a lower AIC than a model with just the 2nd predictor (335.9 vs 341.0).
I can replicate the much of the analysis, including the correlation, the coefficients of the 1st parameter, and qualitatively the difference in AIC.
Beyond not being able to replicate their analysis (perhaps I am missing some modeling detail), I don't understand how they could have two correlated predictors yet end up with only one being significant in the model.
**EDIT:**The paper implies that they are reporting coefficients for two separate models that had only one or the other predictor. However, the coefficients that they report possibly came from a model that had both predictors in at the same time. I can now replicate the beta and SE for the 2nd parameter. They appear to have ignored issues of multi-colliearity in running and interpreting their models.
I am still wondering if my original intuition about their being a problem with the reported values is generally correct, or are their conditions where two correlated variables will behave differently in a fitted model if entered in separately.
• Because significance depends on sample size as much as anything else, there is no universal relationship between correlation of regressors and significance of their coefficients. It is possible for two regressors to have arbitrarily high correlation (but still less than $1$, so they are not perfectly correlated) and for one to have a high p-value (above 50%) and the other to have an arbitrarily low p-value. – whuber Nov 11 '15 at 18:19
The effect of two predictors being correlated is to increase the uncertainty of each's contribution to the effect. For example, say that $Y$ increases with $X_1$, but $X_1$ and $X_2$ are correlated. Does $Y$ only appear to increase with $X_1$ because $Y$ actually increases with $X_2$ and $X_1$ correlated with $X_2$ (and vice versa)? The difficulty in teasing these apart is reflected in the width of the standard errors of your predictors. (The SE is a measure of the uncertainty of your estimate.)
We can determine how much wider the variance of your predictors' sampling distributions are as a result of the correlation by using the Variance Inflation Factor (VIF). For two variables, you just square their correlation, then compute:
$$VIF = \frac{1}{1-r^2}$$ In your case the VIF is $2.23$, meaning that the SEs are $1.5$ times as wide. It is possible that this will make only one still significant, neither, or even that both are still significant, depending on how far the point estimate is from the null value and how wide the SE would have been without any correlation.
To get a stronger sense of how the individual contributions of correlated variables can appear different when both are included vs. not, it may help to read my answer here: Is there a difference between 'controlling for' and 'ignoring' other variables in multiple regression?
To see a demonstration showing that whether the variables are significant can vary, you may want to check out my answer here: How seriously should I consider the effects of multicollinearity in my regression model?
• Thanks. This helps with my general understanding of multi-coll. Your answer seems to relate to when you have a model with two correlated predictors in it, eg y ~ x1 + x2. However, my main interest is when you run two models M1=y ~ x1 and M2=y ~ x2. If x1 and x2 are correlated, how likely will it be that x1 will have a significant association with y but x2 will not given that x1 and x2 are correlated. – N Brouwer Nov 11 '15 at 18:06
• Perhaps I misunderstood, @NBrouwer. It depends on how several factors (eg, the correlation, but also the size of the true effect, N, etc) trade off. Your correlation, r = .67, is not really that strong 45% of the variance in 1 can be predicted using the other. It is certainly possible that a model w/ 1 would be significant & a model w/ the other wouldn't be. – gung Nov 11 '15 at 19:24
• To be clear about your main response, if the model y ~ x1 + x2 was fit and x1 and x2 have are correlated with VIF of 2.23, then the SE of each should be multiplied by 2.23? The author's main parameter of interest had beta=0.24 with SE=0.05 and p<0.01. The SE should therefore be 2.23*0.05=0.11. Constructing a simple 95% CI 1.96*0.11 = 0.23. The p value should therefore be about 0.06 not <0.01. Is this all accurate? Thanks for all the assistance. – N Brouwer Nov 11 '15 at 20:08
• @NBrouwer, no. The variance has been computed correctly by the model. The variance is already 2.23 times larger than it would have been if the variables had been perfectly uncorrelated. The standard error is the square root of the variance (again, the increased width has already been incorporated correctly into the SE). – gung Nov 11 '15 at 21:11 | 2019-07-18 20:12:51 | {"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": 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": 0.6531441807746887, "perplexity": 513.6211594950099}, "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-2019-30/segments/1563195525793.19/warc/CC-MAIN-20190718190635-20190718212635-00401.warc.gz"} |
https://homework.study.com/explanation/how-many-moles-of-naoh-are-required-to-react-with-0-50-moles-of-co2-in-the-following-reaction-naoh-aq-plus-co2-g-arrow-na2co3-aq-plus-h2o-l.html | # How many moles of NaOH are required to react with 0.50 moles of CO2 in the following reaction?...
## Question:
How many moles of {eq}NaOH {/eq} are required to react with 0.50 moles of {eq}CO_2 {/eq} in the following reaction?
{eq}NaOH(aq) + CO_2(g) \to Na_2CO_3(aq) + H_2O(l) {/eq}
## Moles:
The quantify of a group of species like atoms, molecules, etc can be expressed using moles. This is the SI unit for the amount of substance.
## Answer and Explanation: 1
Become a Study.com member to unlock this answer!
Given data:
• moles of carbon dioxide: 0.50 moles
The reaction given is:
{eq}\rm NaOH\left( {aq} \right) + C{O_2}\left( g \right) \to...
See full answer below. | 2023-03-22 02:26:43 | {"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": 0.9603826403617859, "perplexity": 9382.137135513622}, "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-2023-14/segments/1679296943749.68/warc/CC-MAIN-20230322020215-20230322050215-00769.warc.gz"} |
https://math.stackexchange.com/questions/1010591/what-is-the-number-of-intersections-of-diagonals-in-a-convex-equilateral-polygon | # What is the number of intersections of diagonals in a convex equilateral polygon?
Question: [See here for definitions]. Consider an arbitrary convex regular polygon with $$n$$-vertexes ($$n\geq 4$$) and the $$n$$-sequence $$\langle \alpha_i~|~i of its angles which $$\alpha_i$$ is the angle corresponding to vertex labeled by $$i$$ and vertexes labeled by successive numbers are neighbors. What is the number of intersection points of diagonals of such a shape with respect to $$n,\alpha_1,\cdots,\alpha_n$$?
Note that it is always less than or equal to $$\frac{n(n-1)(n-2)(n-3)}{24}$$ because in an ideal case when each intersection point is corresponding to a unique pair of diagonals (not two or more pairs) then by choosing $$4$$ vertexes of $$n$$ we will have two diagonals and this pair of diagonals determines an intersection point and this point is not an intersection point for any other pair of diagonals.
But the difficulty is that even in the simplest case which our convex equilateral polygon is a convex regular polygon there are cases which more than one pairs of diagonals share a same point as an intersection and this decreases the total number of intersection points of diagonals. For example diagonals of a regular convex polygon with $$6$$ vertexes have only $$13$$ intersection points but $$\frac{6\times 5\times 4\times 3}{24}=15$$ because three pairs of diagonals shared a single point in the center as their intersection. Note that in more complicated regular convex polygons with $$n\geq 6$$ these shared points will occur in many other places as well as center of the shape. Thus:
Any special formula for calculating the number of intersections of diagonals of a regular convex $$n$$-polygon is also welcome.
For regular polygons this is OEIS A006561, where for odd $n$ it is $(n^4-6n^3+11n^2-6n)/24$. For even $n$ there are corrections of subtracting ${k \choose 2}-1$ for each $k$ crossing, which implies there are no 3-crossings in odd polygons except the center point. The best paper referenced seems to be here in arXiv
• Very useful! Thanks. Are you aware of any known result about the general form of the question? – user180918 Nov 8 '14 at 3:44
• No, I know nothing about more general polygons. Amusingly, the number in OEIS, $6561=3^8$ – Ross Millikan Nov 8 '14 at 3:49
If $$n$$ is odd, therefore the diagonals don't intersect each other at the center. Note that if the two diagonals intersect, it will create one intersection. So the problem is similar to find how many quadrilaterals can be formed by connecting the vertices of the polygon, which is $$^nC_4$$. If $$n$$ is even, the diagonals will intersect at the centre. So we need to find the numbers of diagonals which pass through the centre. Note further that we just need to find one point, and find another point which is symmetry to the first point. Connect these two points will give us a diagonal. Thus the number of diagonals pass through the centre is $$^{\frac{n}{2}}C_1$$. But one of them is to be included, so, the total intersection is $$^nC_4 - ^{\frac{n}{2}}C_1 + 1$$.
A formula for a regular polygon's (n$$\ge$$3) pts of intersection is
The image of formula
Here $$\delta_m(n)=1$$ if n is a multiple of m, $$0$$ otherwise.
Note that for odd m, it is simply $$^nC_4$$. | 2020-04-04 07:59:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 24, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.938505232334137, "perplexity": 2337.5425629898205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370521574.59/warc/CC-MAIN-20200404073139-20200404103139-00417.warc.gz"} |
https://zbmath.org/?q=an%3A1157.60065 | # zbMATH — the first resource for mathematics
Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays. (English) Zbl 1157.60065
The exponential stability in the $$p$$-th mean and the exponential pathwise stability of mild solutions of SPDEs with delays are investigated in this paper. This is achieved by fixed-point methods, so the author makes the following assumptions: Lipschitz conditions on coefficients, exponential estimation of $$C_0$$-semigroup and certain technical condition connecting constants from previous assumptions. These conditions do not require the monotone decreasing behavior of the delays.
##### MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35B35 Stability in context of PDEs 93C15 Control/observation systems governed by ordinary differential equations
##### Keywords:
SPDE with delays; mild solution; exponential stability
Full Text:
##### References:
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https://www.intechopen.com/chapters/74120 | Open access peer-reviewed chapter
Downflow Hanging Sponge System: A Self-Sustaining Option for Wastewater Treatment
By Namita Maharjan, Choolaka Hewawasam, Masashi Hatamoto, Takashi Yamaguchi, Hideki Harada and Nobuo Araki
Submitted: August 30th 2020Reviewed: October 1st 2020Published: November 19th 2020
DOI: 10.5772/intechopen.94287
Abstract
Need of self-sustaining wastewater treatment plants (WWTPs) has become critical to cope up with dynamics of the environmental regulations and rapid advancements in the contemporary technologies. At present there are limited number of self-sustaining WWTPs around the world. The aim of this chapter is to present state -of- art of Downflow Hanging Sponge (DHS) system which was developed as a post treatment unit of Upflow Anaerobic Sludge Blanket (UASB) from sustainability perspective. DHS system is a non-submerged fixed bed trickling filter (TF) that employs a core technology of polyurethane sponges as a media where the microorganisms thrive and major treatment processes take place. This chapter reviews the introduction of DHS system (UASB+DHS) summarizes the quantitative analysis of environmental, economic and social sustainability using indicators. Furthermore, self-sustaining prospects of DHS system are assessed and discussed by comparing with conventional TF (UASB+TF).
Keywords
• downflow hanging sponge
• trickling filter
• self-sustainability
• indicators
1. Introduction
Wastewater treatment plants (WWTPs) are integral part of our society. Lately, WWTPs and its management have become an important issue in the world and also listed in many of sustainable development goals [1]. The WWTPs were initially designed to remove the pollutants depending on the flow variability to meet the certain discharge standards. However, emerging concepts and practices in WWTPs field have extended its application to energy recovery, reuse and nutrient recycling. Also, majority of WWTPs around the world are not designed with these multiple functions in mind and depend only on conventional technologies to solve these problems and fails to strike the balance between demand and supply of water. Especially, this is quite evident from the water scarcity clock which shows that there are about 2 billion of world population still living in water scarce areas [2]. Moreover, the changing dynamics of world such as accelerating energy dependent lifestyles, sudden global pandemics, economic fragmentations, climate change patterns and other major environmental concerns have affected the selection of suitable WWTPs. Since many WWTPs have life cycle of 50–100 years, or even longer, the selection of WWTP will affect the development of that particular area. Several studies have shown that selecting and deciding the WWTPs are mostly based on their technical, economic, environment and social aspects [3]. Though extensive researches are being conducted and major strategies have been formulated, the search for long lasting technology in the wastewater treatment field is still on-going.
2. Downflow hanging sponge system
Aerobic biological treatment process can be traced back to the late nineteenth century. The biological process uses oxygen to break down organic contaminants and nutrients from wastewater. Oxygen is continuously mixed using aeration device (air blower or compressor) into the wastewater. Aerobic microorganisms then feed on the wastewater’s organic matter converting it into carbon dioxide and biomass which is later removed. There are several types of aerobic treatment processes based on their designs such as fixed film system, continuous flow suspended growth aerobic system, retrofit aerobic system and composting toilets [4]. For this chapter, fixed film aerobic treatment systems called Downflow Hanging Sponge (DHS) and Trickling filter (TF) have been chosen and discussed. The rationale behind choosing these systems is the similarity in their working principle.
TFs is the second most widely used aerobic biological wastewater treatment system after activated sludge process (ASP) around the world [5]. TF is non-submerged fixed-bed, aerobic biological reactor which was applied for sewage treatment for the first time in England in 1893 [6]. Pre-settled wastewater is continuously trickled or sprayed from the top with the help of a rotating sprinkler. As the water moves through the pores of the filter, organics are aerobically degraded by the biofilm covering the filter material. The trickling filter consists of a cylindrical tank and is filled with different packing material such as stones, rocks, gravels or special pre-formed plastic filter media. Since couple of decades, various improvements have been made in TF and it has found its application as a combination unit with other treatment systems. There are 129 TFs in Latin America being operated as a post treatment unit of Upflow Anaerobic Sludge Blanket (UASB) [7]. UASB is an anaerobic treatment system originally developed for industrial wastewater treatment. With due course of time, it became popular in developing countries for domestic wastewater treatment due to its affordability, simple construction, easy operation and maintenance [6, 7, 8]. Recent studies on TF following anaerobic sewage treatment system revealed that 25% of UASB reactor employed TF as post treatment system [8, 9, 10]. The combination of anaerobic and aerobic treatment is advantageous, and this system is simpler than those involving ASP, and leads to much lower energy consumption [11]. The combination of UASB and TF exhibits the high treatability and also is economically advantageous over other treatment systems in developing countries. It has been adopted whenever compact systems were required.
Employing the working concept of TF, in 1995 a research team of Professor Hideki Harada came up with the first concept of promising sewage treatment technology referred to as a DHS system in Japan [12]. DHS system is comparatively a new aerobic, post-treatment process where a simple polyurethane sponge act as a medium for all biological removal processes. DHS system consists of the sponge modules arranged along its height unlike the TF which use gravel, plastics, rocks as supporting media. There are six variations of sponges developed and tested through the rigorous improvement in its shape, arrangements and packing method [13]. The first generation was cube type DHS (G1), second was curtain type DHS (G2), third was similar to TF with sponge supported by polypyrene plastic net (G3), fourth was arrayed sponge type (G4), fifth was improved design of G2 and sixth was similar with G3 but with hardened sponges (G6). The other technical details of the configurations are discussed in other study [14].
The working principle of DHS system is similar to TF. The wastewater is supplied to the top of the DHS system with the help of distributor, which trickles down through the sponge module and finally exits the system through a clarifier at the bottom. The influent with its organic matter is trapped and flows down through the sponge modules in the reactor where the biomass within the sponge degrades the organic matter. External aeration is not necessary for the operation since there is natural diffusion of air as it flows through the DHS system.
For almost two decades, DHS system was researched as the post treatment of UASB for domestic wastewater and implementation of its full-scale have justified it [15, 16, 17, 18]. Modifications to DHS systems are mainly conducted to eliminate the shortcoming of reactor and improve the nutrient and pathogen removal efficiencies [19, 20]. At early stages, DHS reactor was developed to treat domestic wastewater, however the potential of DHS reactor for treating different types of wastewater such as aquaculture [21], industrial wastewater [22, 23, 24] (textile, arsenic, rubber processing etc.) leachates [25] are other trends observed from literatures. The full scale and pilot scale DHS systems are being operating in Japan [10, 16], Thailand [26], India [18, 27, 28], Egypt [20], and Vietnam [22, 23] for various kinds of wastewater. Besides, DHS as a standalone bioreactor for rare metal recovery [29] phosphate recovery [30], gas scrubbing [31] and methane recovery [32] have also been investigated. Since, tremendous amount of researches are being carried out, more emphasis on its self-sustainability would help to validate its application for developing countries for domestic wastewater treatment.
3. Concept of wastewater treatment sustainability
With the on-going stress on selection of WWTPs, sustainability assessment of wastewater has become standard in developed countries and aspiration for the developing countries [33]. The concept of sustainability of wastewater treatment plants is based on the observation that economy, environment and social well-being are interlinked. The term sustainability has various interpretations, however, the World Commission on Environment and Development (WCED, 1987) quoted “Sustainable Development is the development that meets the needs of the present generation without compromising the ability of future generations to meet their own needs” [34]. To assess the sustainability of a system, various dimensions based on the short- or long-term goals have been taken into consideration. From the classic definition of sustainability indicators, it should always incorporate the three main pillars of sustainability i.e. economic, environmental and societal for holistic assessment [35, 36]. In case of developing countries, the studies were more focused on the economic affordability, convenience of end user and stakeholder, health risks, technology sustainability, environmental impacts by products, natural resource optimization and sanitation [37, 38, 39]. This pertains to the fact that choosing sustainability indicators should be contextualized to the local requirements for the decision makers to ascertain WWTPs for specific areas.
Since there is no comprehensive definition of self - sustainable WWTP, it could be defined as “a state of treatment system which can sustain itself without or less use of energy or resources from external source without causing harm or less harm to the surrounding environment”. This definition is restated with reference to the definition of appropriate technology for water sanitation for developing provided in these studies [40, 41]. The following could be some of the features that indicate the self-sustainability of WWTPs for developing countries;
• Simple design and construction
• Less carbon footprint and economic costs
• Simple operation and maintenance
• Least amount or no chemical use
• Stable and reliable performance meeting all the discharge standards
• Having energy sufficiency potential or energy recovery potential
• Productive reuse of biosolids and treated wastewater
• Promote institutional development (environmental agencies, policy makers and regulation agencies, service providers)
Merely saying DHS is a ‘sustainable’ system is not possible until and unless sustainability indicators indicate progress towards or away from sustainability [42]. The main goal of this chapter is to present the state-of- - art of DHS system based on sustainability indicators. Additionally, the self - sustainability potential of DHS system was compared and discussed with the similar kind of technology i.e. TF for future application of this technology in developing countries. TF is a well-known technology since ages and comparing DHS system with TF would assist in its proof of concept, scalability and deployment for its validation in the field of the sustainability science. So far, there is only one study which has addressed the sustainability of the full-scale DHS system [43]. However, self-sustainability of DHS system has not been explored yet. From here on wards, DHS system is rephrased as UASB+DHS system as majority of researches on DHS system are presented as post treatment unit of UASB system. Similarly TF is also rephrased as UASB+TF. Apparently, the literatures on performance of UASB+TF systems are scarce so some discussions are presented with only TF data.
This chapter collects and analyzes the pre-requisites of self-sustainability indicators for UASB+DHS system. To address the self-sustainability of UASB+DHS system multiple indicators are considered from literature review for the holistic assessment which is guided predominantly by these studies [44, 45]. The indicators considered for this review are discussed henceforth and are summarized in Table 1.
3.1 Treatment performance
Right from the first prototype of UASB+DHS system, its treatment efficiency for organic, nitrogen and pathogen have shown impressive results for domestic wastewater treatment [10, 12, 13, 14, 15, 16, 17, 18, 19, 20]. There are plethora of studies reporting the treatment performance of UASB+DHS system. For comparison, the treatment efficiencies for the parameters such as Total suspended solids (TSS), biological oxygen demand (BOD), ammonia (NH4+-N) and fecal coliform (FC) are collected and tabulated in Table 2. Full scale UASB+DHS system till date have shown significant TSS and BOD efficiencies of 94% and 96% respectively [18, 27]. While some of the selected UASB +TF system indicated a slightly reduced efficiency i.e. (TSS: 88–93% and BOD:89–93%). For most of the developing countries, BOD standards are regarded as the basic compliance discharge standard [12] which might have increased the popularity of UASB+DHS system. The data clearly shows that UASB+DHS system has benefits over UASB+TF system attributed by its unique sponges, improving the quality of effluent in terms of organic matter. Similarly, a noticeable ammonia removal efficiency ranging from 79–83% was showcased by UASB+DHS system whereas UASB+TF displayed decreased efficiency below 50%. Studies on molecular microbiology of UASB+DHS have highlighted slow growers such as nitrifiers, denitrifies and even active annamox bacteria in the inner aerobic niches of the sponges facilitating the nitrification and denitrification reaction for nitrogen removal [16, 51]. The other studies also reported that TFs have poor consistency in the removal of nitrogen and phosphorus compared to other conventional treatment systems [6]. Likewise, UASB+DHS system is also efficient for removing pathogenic bacteria from wastewater which was due to high DO condition which prevented growth of bacteria and allowed the higher micro-organisms (protozoa and metazoan) to predate on pathogens such as E. coliand total coliforms [52]. Moreover, the other factor for removal of pathogen in the UASB+DHS system reported was adsorption onto biomass [20]. While on the other hand, pathogen removal by UASB+TF system is promising in this case. However, the pathogen removal capacity in TFs have been observed inconsistent and varied from 1.0 log to 3 logs, depending on the operating conditions when compared to ASP [53].
Environmental SustainabilityTreatment performanceContaminant removal efficiency to mitigate environmental and health risksCinf.Ceff.Cinf.×100Removal % (log removal)
Cinf. = Influent concentration
Ceff. = Effluent concentration
Land areaLand area required for the wastewater treatment facilityTotal area occupied /Population equivalencem2/p.e.
CO2 emissions from COD oxidationCOD removal (kg COD m−3d−1) × 0.08 kg CO2 / kg COD
CO2 emissions from CH4 combustion
Global warming potentialEmissions of greenhouse gases (CO2 and N2O) into the atmosphereN2O emissionsCH4 emission (m3 CH4) × 3.5 (kg CO2 / m3 CH4(kg CO2 equivalent m−3d−1) Atkins et al. (2005) [46, 47]
Ninflent·EFeffluent·44/28 × 298 (Emission Factor effluent =0.005)Campos et al. (2016); IPCC [48, 49]
Carbon FootprintEnergy sufficiencyEnergy consumed during emissionCO2 emissions from energy consumptionEnergy consumption (kWhm−3 d−1) × 0.391 (kg CO2 equivalent / kWh)(kg CO2 equivalent m−3d−1)
Reduced Sludge reduction potentialSludge amount produced, treated water for irrigation, nutrientsSs = Qef(Win-Weff)/Q(Cin-Ceff)(kg SS kg−1 COD removed) [18]
Qef(= flow rate to settler (m−3d−1)
Q = flow rate to the reactor (m−3d−1)
Win = SS influent conc. (kg SS m−3)
Weff = SS effluent conc. (kg SS m−3)
Cin = COD influent conc. (kg COD m−3)
Ceff = COD effluent conc. (kg COD m−3)
Economic sustainabilityCapexCost of construction and installation of the WWTPTAEC=r1+rt1+rt1Capex+Opex$/m3 t = expenses at time t$/p.e.-year
r = discount rate (5%)
OpexOperating costs per volume unit of wastewater treatedt=1TOpex1+rt$/p.e. t = expenses at time t r = discount rate (5%) Social sustainabilityPublic acceptanceOpinion of the local population affected by the plant.Qualitative EstheticsMeasured level of nuisance deriving from e.g. odor, noise, visual impact, insects and other pests.Qualitative System manageabilityEase of construction, complexity of O & M; professional skills required for O & MQualitative Table 1. Multiple indicators chosen for assessing the sustainability. Adapted from [44, 45]. Support mediaLand areaHRT (h)Influent (mgL−1)Removal efficiency (%)References m2/p.eBODTSSNH4+-NFC(MPN/100 ml)BODTSSNH4+-NFC (log) UASB + TFSponge bed0.3312379308988443.5[6] UASB+DHSG3 sponge0.031.5151228257.71 × 1069694793.0[27] UASB+DHSG3.4 sponge0.051.5161228161.22 × 1099694833.0[43] UASB+TFHigh rate stones0.22.0250150201.8× 1089393502.8[50] Table 2. Treatment performance and land required for UASB+DHS system and UASB+TF system. 3.2 Land requirements The increasing land prices and scarcity of available land resources are becoming one of the bottlenecks for WWTP management issues [54]. Land requirement for the WWTPs directly affects its performance and costs. The land requirement per m2/p.e. for UASB+DHS and UASB+TF systems are shown in Table 2. The available literatures show that the land required for the construction of UASB+DHS is almost 10 times less than UASB+TF. Though having the same external design, this difference could be explained by the supporting media. The DHS sponges are comparatively smaller and light weight in comparison to the most frequently used media such as stones, gravel, plastics etc. which implies much higher tank volumes and areas. Nonetheless, having the similar working principle, the packing of the media in DHS system resulted in smaller footprint. 3.3 Carbon footprint Carbon footprint is relatively a new measure of sustainability in WWTPs to determine its overall impact on climate change and as a result WWTPs performance has recently been evaluated based on carbon minimization [55, 56]. To address sustainability, carbon footprint minimization has become an important environmental indicator [57]. For carbon footprint, assessment, all relevant forms of the energy demand in WWTPs, sludge production and common GHGs emissions are accounted. This review aims to investigate previously unexplored relationships between carbon footprint and sustainability in the context of UASB+DHS system, focusing particularly on the impact of energy minimization measures. 3.3.1 GHGs emission Global Warming Potential (GWP) is generally used as a metric for weighting the climatic impact of emissions of different greenhouse gases [58]. Among GHGs stated by Kyoto Protocol, the most common GHGs emitted during operation and on-site anthropogenic activities in WWTPs are carbon dioxide (CO2), methane (CH4), nitrous oxide (N2O) [49]. According to USEPA, WWTPs are the 7th largest contributors of CH4 and nitrous N2O emissions in the atmosphere [59]. Particularly, WWTPs produce GHGs during the biological wastewater treatment processes. For calculation, all GHGs emission can be expressed as CO2 equivalents (CO2e) with respect to their GWP. CH4 and N2O have 28 and 265 times greater GWP compared to CO2 in a 100-year time horizon [60]. Therefore, more stringent regulatory efforts, mandatory reports and measurements on GHGs emissions from WWTPs are being enforced to control GHGs emissions. Since, UASB+DHS system is also an anaerobic and aerobic biological treatment process, this information would be vital for the wastewater specialists. For almost all WWTPs, CO2 production is attributed to two main factors: biological treatment process and electricity consumption. In UASB+DHS system, CO2 is emitted during the production of the energy required for the plant operation. Emission of N2O is generated by nitrification and denitrification processes used to remove nitrogenous compounds from wastewater. Similar to the mainstream WWTPs, the organic carbon of wastewater is either incorporated into biomass or oxidized to CO2. During anaerobic digestion in UASB, it is mainly converted to CO2 and CH4. It is assumed that all the CH4 produced is oxidized to CO2 during biogas combustion. Estimation of (CO2e) is attained using units and equations summarized in Table 1. For the calculation of GHGs, considering the total treatment process is important. Therefore, GHGs emissions of both the system were estimated based on CO2 emission from COD oxidation, CH4 combustion and N2O emission as presented in Table 3. The data for GHGs calculation for UASB+DHS system and UASB+TF were taken from these studies [27, 60, 62]. The value for GWP by UASB+DHS system was 0.59 kg CO2 equivalent m−3 d−1 and that for UASB+TF was 0.50 kg CO2 equivalent m−3 d−1. It is to be noted that for the calculation of GWP of UASB+TF, N2O emissions value was not available as there were no literatures reporting its values. Nevertheless, other studies associated with GHGs emission of TF + ASP system and TF+ Lagoon system showed GWP values of 1.232 kg CO2 equivalent m−3 d−1 and 0.898 kg CO2 equivalent m−3 d−1 respectively. Therefore, it could be assumed that the UASB+TF system might show fairly higher values compared to UASB+DHS system. The another reason behind assuming the lower GWP values by UASB+DHS system could be justified by its higher solid retention time (SRT) values of almost 92–101 days [17] compared to 2–4 days of TF [65]. Higher values of SRT supports endogenous respiration of biomass which increases the amount of COD oxidized to CO2 thus decreasing the overall sludge production [17]. This decrease of sludge production reduces the methane production and therefore, a decrease in CO2 emissions is associated with its combustion. Similarly, higher SRT capacity of DHS system helps to maintain low ammonia and nitrite concentrations in the media which leads to minimum N2O emissions to the atmosphere. Despite the accuracy of estimated GWP value is not exact, conclusive potential of operating UASB+DHS system at low GHGs emission levels has been assured. Hence, the analysis of GWP revealed the potential of UASB+DHS system to become a sustainable option in the future of wastewater treatment. UnitsUASB+DHS systemUASB+TF Global warming Potential(kg CO2 equivalent m−3d−1)0.59 [27, 61]0.50 [62] Sludge production(kg SS kg−1 COD removed)0.06 [10]0.38 [63] Energy consumption(kWhm−3)0.12 [27]0.65 [64] Table 3. Carbon footprint assessment of UASB+DHS system and UASB+TF. 3.3.2 Sludge production For most of the WWTPs, one of the biggest challenges is its sludge production, its post treatment and disposal. Being an aerobic system, DHS system has advantage over other biological treatment system for sludge management [61]. Any sludge accumulated in the clarifier of DHS is called as excess sludge. The sludge production in DHS reactor is calculated by taking the sum of SS volumes in the DHS effluent and the settled excess sludge in the clarifier and relative to the COD or BOD removed by the system. For bench scale experiment, the excess sludge produced by UASB+DHS system was 0.02 kg SS/kg COD removed which is basically 2.5% of the total COD removed or 7% of the total SS load removed [17]. Further, excess sludge from UASB usually varied from 0.03 to 0. 2 kg SS/kg COD removed [8]. Therefore, a total sludge from UASB+DHS system was 0.06 kg SS/kg COD removed. While, excess sludge production from UASB+TF system was 0.38 kgSS/kgCOD removed [63]. The sludge production from UASB+TF system was almost 6 times higher than UASB+DHS system. The DHS sponges are designed with the high void ratio and reticulated structure which cater as a favorable site for the attachment, adsorption and growth of active biomass [25, 61]. Further, the profiling data from same researchers stated that the majority of organic removal especially SS occurred at highest part of the reactor, however after attaining stable state, uniform distribution of sludge was observed along its height. In real scale DHS, for every liter of wastewater treated, about 0.04 kg-COD was wasted as excess sludge which is quite negligible as compared to the other treatment systems. The basic mechanism for the sludge removal in DHS is the physical entrapment of the sludge inside and outside of the sponge which lengthens the solid retention time and provide ample time for self-degradation of sludge minimizing the excess sludge production [61, 66]. 3.3.3 Energy consumption Nowadays, for developing countries energy efficiency has become the first priorities in the WWTPs hierarchy [67]. Minimizing net energy consumption for WWTPs has become mandatory [68]. Generally, for aerobic treatment processes, the aeration is the highest energy consuming process of the wastewater treatment technology which can account upto 50–60% of all electricity consumption followed by 15–25% of energy by sludge treatment and 15% by secondary sedimentation including recirculation pumps [69]. The energy consumed in UASB+DHS system is through electricity required for pumping [27]. The pumps are used for supplying UASB effluent to the top of DHS system. It is usually estimated on the basis of treatment performance and electricity utilized by pumps. Comparison of energy consumption of UASB+DHS system [27] with UASB+TF [64] is summarized in Table 3. From the data, it is evident that the energy consumption of UASB+TF is approximately 5 times higher than that of UASB+DHS system. For UASB+DHS system, 0.05 kWhm−3 of energy was consumed by main pumping from UASB unit and 0.07 kWhm-3 for the pump of the DHS system, which sums up the total energy consumption for the system of 0.12 kWh per m3 of wastewater treated. It is noteworthy that the energy consumption for both these systems was solely by pumping. The UASB+DHS system has likelihood of becoming energy sufficient system. The energy sufficiency of UASB+DHS system can be explained by its minimized energy consumption. In addition, when constructing a UASB+DHS system, the energy sufficiency or neutrality can be achieved if UASB is designed in such a way where the outlet is positioned above the DHS distributor or maintained through gravity. Considering the overall arguments, environmental indicators suggest that the UASB+DHS system is considerably superior in terms of high treatability, less land requirement and reduced carbon footprint. This information could assist the planners and stakeholders in developing nations for good decision making while selecting WWTPs in future. Advertisement 4. Economic sustainability Economic sustainability of WWTPs refers to the economic factors affecting social, environmental and cultural aspects of the treatment systems. Economic efficiency of WWTPs presents the scenario of investments in terms of input and effluent quality as the output [70]. Decision and policy makers in developing countries are challenged with the fact that poor urban residents cannot afford costly conventional sewage treatment systems [71]. Fortunately, a broad range of cost-effective technological options are extensively being studied to cater this category of people. Therefore, economic factors become vital to address these issues. For any WWTPs, economic indicators generally represent the costs associated with the construction and the operation of treatment management during its life time [72]. These are driving factors for decision making while selecting a technology in a practical situation. For the economic assessment, the two most common indicators called capital expenditure (Capex) and operational expenditure (Opex) were calculated using the equations provided by [73]. The capital expenditure included cost of construction and life cycle costs. Operational expenditure included number of mechanical equipment, skilled workers, power consumption, labor, chemicals, and consumables. The data were taken from the state-of-art literatures [43, 74, 75] for calculating the construction and operating costs for economic evaluation for both the systems. For the comparison in the same scale few considerations were made: Capex is annualized by taking the means of the initial investment costs by the life period of the project accounting for the time value of money while Opex is total discounted lifetime operational expenses. • life span of each treatment plant was considered as 20 years. • All costs were expressed in US dollars ($) /Population equivalence (p.e.).
• The cost of implementing and operating the WWTPs are adjusted to 2015 as the base year and was discounted to 2015 values using equations given in Table 1.
The economic costs comparison Table 4 showed that capex and opex costs of UASB+DHS system of 9740 p.e. is almost equal to the UASB+TF of 50,000 pe. While for the UASB+TF of 2337 p.e. UASB+DHS system expressed significantly less capex and opex values. For most of the UASB+TF, among the various costs, cost of personnel is the maximum for opex [50]. This could be also the reason for the decreased economic costs for UASB+DHS system. The another rationale for reduced economic costs of UASB+DHS system could be less manpower required due to the simple O&M processes including cleaning of the mechanical parts, chemical free operation and easy handling of sludge [76]. It is interesting to note that DHS inclination is towards negative value for economic assessment which means it is economically fit for the developing countries as capex and opex costs are negative indicators of sustainability and qualifies the criteria to be considered as self-sustainable WWTP system.
Process systemCountryTreatment volume (PE)CAPEX US$/PEOPEX US$/PE/yearReferences
UASB+DHSIndia974086.20.36[43]
UASB+TFEgypt50,00092.142.07[74]
UASB+TFEgypt2337519.318.25[75]
Table 4.
Economic assessment of UASB+DHS system and UASB+TF system.
5. Social sustainability
Social assessment based on indicators portrays a big picture of multidimensional issues and facilitate decision making. Many key aspects such as community management aspects, satisfaction and opinions of users, service quality, materials and personnel managements, etc. have to be profoundly analyzed before and after the establishment of WWTPs [45]. In this light, social indicators are rapidly becoming the preferred tools for policymakers and public communicators for disseminating the information on the advantages and disadvantages of the WWTPs [77]. However, societal indicators are generally difficult to quantify and often their meaning and relevance is based on the local stakeholders [78]. The data on social indicators for UASB+TF in this review is lacking since data availability on TF. This does not impair our comparison as it’s combinations with other systems is a disadvantage identified at the global level in developing countries [79]. Hence, assessment of the chosen indicators is based on TF studied [43]. Caution should be exercised that all the data for social assessment do not represent any generic weighting. The chosen indicators for this assessment are (i) simplicity of the system (ii) esthetics, and (iii) public acceptance of the technology.
5.1 System manageability
System manageability includes ease of construction, complexity of O&M issues and professional skills required for the troubleshooting of the issues during the O&M of the WWTPs. Studies on the social sustainability of WWTPs in India demonstrated that the UASB+DHS system has fewer mechanical parts than TF. The simple configuration of UASB+DHS system makes it easy for construction and several intensive studies have followed up the ramifications of sponge designs for the easy packaging and enhanced efficiency [76]. For developing countries, simplicity of the system might be a key factor in the selection of the WWTP. Due to simple construction, there are few mechanical parts for O&M issues and UASB+DHS system has already been proven to be no laborious maintenance system [80]. Supplementary to this, the operators do not need to have a high technical knowledge.
5.2 Esthetics
UASB+DHS system has a slightly better stance on esthetics than the other TFs. Hydrogen sulphide (H2S) and ammonia (NH3) are the predominant objectionable odors in wastewater [81]. TF has been reported to have limited and passive aeration which might create an anaerobic condition which degrades organic matter and nutrients to release malodorous by-products. Also, TF is very sensitive to temperature change especially high temperatures which generate odors [5]. However, till this date, there are no evidences of odor problems in UASB+DHS system though it was operated at the high temperature in India (∼40°C). Nonetheless, both the systems have one common problem i.e. flies and snails which might affect the visual aethetics. Whilst none of the insects found have been found as a nuisance to the surrounding people and there is no evidences of any diseases caused by these insects. However, this issue could be resolved by the installation of nets or covers and cleaning with water sprays [82].
5.3 Public acceptance of the technology
One of the major problems faced during the establishment of WWTPs is its location. Very often resistance and protests from the local people significantly impact on the implementation of any social infrastructure plan [83, 84]. Therefore, “Public acceptance” is a key component when it comes to establishing a new WWTP [85]. In most of the researches, public acceptance of WWTPs facilities are based on concepts such as LULU (locally unwanted land use) and NIMBY (not in my backyard [86, 87]. These studies have highlighted the preferability of TF over activated sludge process. Besides, the result of investigation of Indian WWTPs has also exhibited that moderate value for the public acceptance of TF over UASB+DHS system and suggested WWTPs’ location far from the settlement zones. Besides, the sludge and treated water from UASB+DHS system was used by the nearby farmers for agriculture and did not show any social concerns. It is clear that further studies need to be undertaken analyzing local conditions in a stepwise manner towards the acceptance of these technologies.
The above discussions on sustainability indicators assessment support the notion that DHS system could have a wide range of commercial applications for different kinds of wastewater. In most of the developing countries, centralized WWTP are limited to urban areas due to the several financial and social constraints. Likewise, many institutions such as large-scale apartments, complexes, hospitals, hotels, etc. need to maintain their own onsite WWTPs. Until now, the most preferred WWTP option was ASP based treatment plants. However, these systems are expensive due to its high operational and maintenance cost. On this verge, there is a potential use of DHS system as a substitute for ASP which would lead to huge commercial benefit. The technology validation of DHS system over ASP for developing countries has already been reported in several research investigations [12, 13, 14, 15, 16, 17, 18, 43]. Another area of commercial application of this system could be in aquaculture industry. Most of the aquariums and fish farms require frequent exchange of water leading to huge financial burden and increased workload. Current progressive researches on the application of DHS system for aquarium water treatment and live seafood transportation by minimizing the exchange of water and decreasing workload have broaden its scalability for aquaculture industry [21, 88]. In line with the environmental, economic and social sustainability of DHS system, it could have prospective applications in industrial wastewater treatment specifically for developing nations. Moreover, DHS reactor has commercial applications in the industries such as food processing industry, beverage industries, rubber processing industries and many agriculture products processing industries. Therefore, it could be concluded that with more researches and real scale implementation of this system, there could be a huge commercial demand for DHS system in near future.
6. Conclusion
From the retrospection of the state-of-art of DHS system, it has always been considered a sustainable system for developing countries. Even though, there are extensive researches being carried out for the performance improvement and application of DHS for other types of wastewater, few efforts have been made for testing and validating the sustainability of DHS system. This chapter introduced and analyzed the sustainability indicators of DHS system based on environment, economic and social indicators. By assessing the range of environmental performance indicators, including treatment performance, land requirement, carbon footprint along with economic costs and social factors, this review provides information on DHS system pursuing positively towards sustainability than UASB+TF. However, the availability of data is still an issue in this context for both the systems. It is recommended to conceptualize the sustainability assessment framework that will also encourage and support data collection for better and more quantitative analysis to ensure the applicability and usefulness of DHS technology. Considering the outcomes from sustainability assessment, regardless of data insufficiency, the DHS system fulfilled the criteria of self-sustaining WWTP to a greater extent. Nevertheless, more comprehensive studies are suggested for understanding the other aspects of self-sustainability which are not discussed in this chapter.
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Namita Maharjan, Choolaka Hewawasam, Masashi Hatamoto, Takashi Yamaguchi, Hideki Harada and Nobuo Araki (November 19th 2020). Downflow Hanging Sponge System: A Self-Sustaining Option for Wastewater Treatment, Promising Techniques for Wastewater Treatment and Water Quality Assessment, Iqbal Ahmed Moujdin and J. Kevin Summers, IntechOpen, DOI: 10.5772/intechopen.94287. Available from:
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https://lfortran.org/download/ | # Binaries
The easiest way to install LFortran is to install the lfortran Conda package from conda-forge. See the installation instructions in the Documentation.
# Source Tarballs
When installing from source, the recommended way is to use the release tarballs. For installation instructions consult the installation page in the Documentation.
Latest release:
All releases:
## Development Versions
Latest development version in master:
All development versions:
# Git Repository
To develop LFortran, one has to install from the main git repository:
https://gitlab.com/lfortran/lfortran
When installing from git, in addition to the dependencies for the source tarballs, one has to also install Java and ANTLR4. The souce tarballs do not depend on Java or ANTLR4, and so they are the recommended way to install LFortran for end users. For more details, see the installation instructions in our Documentation. | 2021-06-23 18:52:40 | {"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": 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": 0.19806282222270966, "perplexity": 6831.016115047412}, "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/1623488539764.83/warc/CC-MAIN-20210623165014-20210623195014-00550.warc.gz"} |
https://rylanschaeffer.github.io/content/learning/statistics.html | # Rylan Schaeffer
Notation: I was schooled using the statistician notation $$\mathbb{E}_{p(x)}[f(x)]$$ to denote the expected value of $$f(x)$$ with respect to the density $$p(x)$$. More recently, I’ve picked up the physicist notation $$\langle f(x) \rangle_x$$. I may move interchangably between the two. | 2021-07-31 02:59:01 | {"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": 0.9949801564216614, "perplexity": 441.12490398857517}, "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-2021-31/segments/1627046154042.23/warc/CC-MAIN-20210731011529-20210731041529-00304.warc.gz"} |
https://learn.careers360.com/ncert/question-a-solid-cylinder-of-mass-20-kg-rotates-about-its-axis-with-angular-speed-100-rad-s-to-the-power-minus-1/ | # Q7.12 A solid cylinder of mass $20 kg$ rotates about its axis with angular speed $100 \: rad\: s^{-1}$. The radius of the cylinder is $0.25m$ . What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of angular momentum of the cylinder about its axis?
Firstly we will calculate moment of inertia of the solid cyliner :
$I_c\ =\ \frac{1}{2}mr^2$
$=\ \frac{1}{2}(20)(0.25)^2\ =\ 0.625\ Kg\ m^2$
So the kinetic energy is given by :
$E_k\ =\ \frac{1}{2}I\omega ^2$
$=\ \frac{1}{2}\times (6.25)\times (100) ^2$
$=\ 3125\ J$
And the angular momentum is given by : $=\ I\omega$
$=\ 0.625\times 100$
$=\ 62.5\ Js$
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₹ 4999/- ₹ 2999/- | 2020-07-05 20:26:08 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "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": 0.9217475652694702, "perplexity": 5132.877846090864}, "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/1593655888561.21/warc/CC-MAIN-20200705184325-20200705214325-00015.warc.gz"} |
https://nrich.maths.org/6552/solution | ### Generally Geometric
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
### Bend
What is the longest stick that can be carried horizontally along a narrow corridor and around a right-angled bend?
### Exponential Trend
Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.
# Calculus Countdown
##### Age 16 to 18 Challenge Level:
Game a): $\textrm{Target} = 8$
$\mathrm{D} \left( x^2 \right) = 2x$
$\mathrm{D} \left( 2x \right) = 2$
$\mathrm{P} \left( 4,2 \right) = 8$
Game b): $\textrm{Target} = x^4$
$\mathrm{P} \left( x, x^2 \right) = x^3$
$\mathrm{I} \left( x^3 \right) = \frac{x^4}{4}$
$\mathrm{P} \left( \frac{x^4}{4}, 4 \right) = x^4$
Game c): $\textrm{Target} = \frac{1}{2}$
$\mathrm{D} \left( x^2 \right) = 2x$
$\mathrm{D} \left( 2x \right) = 2$
$\mathrm{R} \left( 2 \right) = \frac{1}{2}$
Game d): $\textrm{Target} = \frac{x^6}{36}$
Method 1:
$\mathrm{I} \left( \mathrm{I}(x) \right) = \frac{x^3}{6}$
$\mathrm{P} \left( x^2,\frac{x^3}{6} \right) = \frac{x^5}{6}$
$\mathrm{I} \left( \frac{x^5}{6} \right) = \frac{x^6}{36}$
Method 2:
$\mathrm{I} \left( x^2 \right) = \frac{x^3}{3}$
$\mathrm{D} \left( \ln(x) \right) = \frac{1}{x}$
$\mathrm{R} \left( \frac{1}{x} \right) = x$
$\mathrm{P} \left( x, x \right) = x^2$
$\mathrm{I} \left( x^2 \right) = \frac{x^3}{3}$
$\mathrm{P} \left( \frac{x^3}{3}, \frac{x^3}{3} \right) = \frac{x^6}{9}$
$\mathrm{R} \left( 4 \right) = 0.25$
$\mathrm{P} \left( 0.25,\frac{ x^6}{9} \right) =\frac{x^6}{36}$
Game e): $\textrm{Target} = \frac{-32}{x^5}$
$\mathrm{D} \left( \mathrm{D} \left( \mathrm{D} \left( \ln(x) \right) \right) \right) = 2x^{-3}$
$\mathrm{R} \left( x \right) = \frac{1}{x}$
$\mathrm{P} \left( x^{-1}, 2x^{-3} \right) = 2x^{-4}$
$\mathrm{P} \left( 4, 2x^{-4} \right) = 8x^{-4}$
$\mathrm{D} \left( 8x^{-4} \right) = -32x^{-5}$
Game f): $\textrm{Target} = x(2 - x)$
$\mathrm{P} \left( x^2, \mathrm{R}(\exp(x)) \right) =x^2 \exp(-x))$
$\mathrm{D} \left( x^2 \exp(-x) \right) = 2x \exp(-x) - x^2 \exp(-x)$
$\mathrm{P} \left( \exp(x),2x \exp(-x) - x^2 \exp(-x) \right) = 2x-x^2$ | 2019-09-15 16:21:59 | {"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": 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": 0.30849072337150574, "perplexity": 5589.026918052868}, "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/1568514571651.9/warc/CC-MAIN-20190915155225-20190915181225-00026.warc.gz"} |
https://indico.cern.ch/event/648004/contributions/3030088/ | # XIIIth Quark Confinement and the Hadron Spectrum
31 July 2018 to 6 August 2018
Maynooth University
Europe/Dublin timezone
## Equation of state in 2 + 1 flavor QCD at high temperatures
1 Aug 2018, 14:30
30m
Hall F (Arts Bldg.)
### Hall F
#### Arts Bldg.
Talk D: Deconfinement
### Speaker
Dr Johannes Heinrich Weber (Michigan State University)
### Description
We calculate the equation of state at high temperatures in 2+1 flavor QCD using the highly improved staggered quark (HISQ) action. We study the lattice spacing dependence of the pressure at high temperatures using lattices with temporal extent (N_\tau= 6,\ 8,\ 10) and (12) and perform continuum extrapolations.
We also give a continuum estimate for the equation of state up to temperatures (T = 2) GeV, which are then compared with results of the weak-coupling calculations. We find a reasonably good agreement with the weak-coupling calculations at the highest temperatures.
### Primary authors
Dr Johannes Heinrich Weber (Michigan State University) Peter Petreczky (BNL) Alexei Bazavov (Michigan State University) | 2019-10-16 09:19:42 | {"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": 0.9192652106285095, "perplexity": 8278.57330925865}, "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-2019-43/segments/1570986666959.47/warc/CC-MAIN-20191016090425-20191016113925-00275.warc.gz"} |
http://openstudy.com/updates/4dbc0b04b0ab8b0bec69808b | • anonymous
In the Problem Set of Unit 4, in the part of surface Area, problem 4G-5 I don't understand why the limits of integration go from 0 to 2 when the integration is made by "dy" and in the y-axis the upper limit is 4.
OCW Scholar - Single Variable Calculus
Looking for something else?
Not the answer you are looking for? Search for more explanations. | 2017-03-28 18:14:17 | {"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": 0.8045600652694702, "perplexity": 652.8738385745528}, "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-2017-13/segments/1490218189802.72/warc/CC-MAIN-20170322212949-00539-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://leanprover-community.github.io/archive/stream/144837-PR-reviews/topic/3083.20finset.20lattice.html | ## Stream: PR reviews
### Topic: 3083 finset lattice
#### Yury G. Kudryashov (Jun 15 2020 at 20:24):
#3083 makes finset use lattice operations instead of "set" operations. What do you think about this? data.finset now compiles but I don't want to waste time on porting the rest of mathlib if this is not going to be merged.
#### Bhavik Mehta (Jun 15 2020 at 20:27):
What's the advantage of this? I think it would be slightly more confusing for newbies who want to use finsets but don't want to think about a lattice, and if I remember correctly the lattice operations already work on finsets anyway
#### Chris Hughes (Jun 15 2020 at 20:30):
I think it's a good idea. It just means you don't need duplicates of all lemmas on lattice operations and set operations. Occasionally I can't rw with a lattice lemma because I've got set operations or vice versa.
#### Patrick Massot (Jun 15 2020 at 20:51):
Does it mean you want to use the obscure lattice notations for set/finset intersection and union?
#### Patrick Massot (Jun 15 2020 at 20:52):
If you want to unify everything I would rather drop the lattice notations and use round intersection and union everywhere instead of the lattice square versions
#### Patrick Massot (Jun 15 2020 at 20:52):
Oh I see Gabriel also likes round notations.
#### Patrick Massot (Jun 15 2020 at 20:54):
Either way it sounds like a drastic change that is not drastically motivated.
#### Patrick Massot (Jun 15 2020 at 20:55):
I would very much prefer having integrals that work.
#### Kevin Buzzard (Jun 15 2020 at 21:07):
We have the zany lattice notation for ideals and I got used to it in the end. There are advantages of a unified approach. I get annoyed that set.preimage isn't called comap nowadays
#### Yury G. Kudryashov (Jun 15 2020 at 21:25):
I was thinking about definitions, not notation.
#### Yury G. Kudryashov (Jun 15 2020 at 21:27):
The only question about using ∪ ant ∩ for lattice is what to do with ≤/</⊥ vs ⊆/⊂/∅.
#### Yury G. Kudryashov (Jun 15 2020 at 21:28):
I know only one type with different interpretation of ≤ and ⊆: multiset.
#### Scott Morrison (Jun 15 2020 at 23:09):
I think having any types with both ≤ and ⊆ is probably a bad idea: just a recipe for confusion. (I remember being dismayed stumbling on this one.) I would propose picking the important one to receive notation (whether it's ≤ and ⊆) and just dealing with being more verbose for the other.
#### Jeremy Avigad (Jun 16 2020 at 01:05):
Whenever I have to do anything with lattices I struggle a lot with typing \lub and \glb because I have to mentally reconstruct which is which. It doesn't help that we think of them as union and inter, and the lattice names are sup and inf. Maybe it would help if we could also use \lunion and \linter for "lattice union" and "lattice inter", or something like that.
#### Mario Carneiro (Jun 16 2020 at 01:27):
what about \sqcap and \sqcup?
#### Jeremy Avigad (Jun 16 2020 at 01:36):
Yeah, that works for me.
#### Kevin Buzzard (Jun 16 2020 at 01:36):
I must confess I feel the same as Jeremy about these funny words. I guess I recognise cup as union though
#### Jeremy Avigad (Jun 16 2020 at 02:42):
Oh, yeah, and when talking about lattices in ordinary mathematics I use "meet" and "join." We give the dictionary here: https://avigad.github.io/mathematics_in_lean/basics.html#proving-facts-about-algebraic-structures
I put the translations there to help myself sort them out more than anyone else.
#### Yury G. Kudryashov (Jun 16 2020 at 03:02):
Changing sup to join and inf to meet should not be too hard.
#### Yury G. Kudryashov (Jun 16 2020 at 03:02):
But I don't want to start any of these refactors until we'll agree on the goal.
#### Johan Commelin (Jun 16 2020 at 06:38):
I would like to maintain readability as much as possible. But uniformity pays off in the long run. And I think the symbols are similar enough that we can explain to anyone that \sqcup is the symbol for union of "set-like" things. Then we only need to tell them that in names
• subset becomes le
• union becomes meet
• inter becomes join
and I think all of those are pretty self-explanatory.
The hardest thing might be empty becomes bot...
And of course univ was already confusing to begin with, so replacing it with top doesn't change anything. (-;
#### Yury G. Kudryashov (Jun 16 2020 at 06:40):
We can use union and inter for meet and join though in some contexts (e.g., supr in reals or ennreals) this looks funny.
#### Gabriel Ebner (Jun 16 2020 at 07:10):
Jeremy Avigad said:
I use "meet" and "join."
I mix up meet and join pretty often always, I guess mostly because these words don't exist in German. In German we typically use supremum and infimum, so I prefer the current naming.
#### Gabriel Ebner (Jun 16 2020 at 07:19):
Yury G. Kudryashov said:
in some contexts [...] this looks funny.
$\sqrt{2} \cup \pi = \pi$ doesn't look so bad, it reminds me of Dedekind cuts.
#### Jeremy Avigad (Jun 16 2020 at 11:34):
I didn't mean to suggest that meet and join are better than sup and inf. It would help if we could reclaim \sup and \inf in VS Code. We already have \supseteq and \infty for the symbols they are assigned to now. It might help to be able to type \sqcap and \sqcup when working with finsets, because \cap and \cup are used in Latex for intersections and unions. But I can probably get used to the fact that intersection is \inf because it is smaller and union is \sup because it is bigger. It's just that \glb and \lub are horrible, because e.g. "greatest" and "lower" go in opposite directions, and I have to stop to think about which one wins.
#### Johan Commelin (Jun 16 2020 at 17:28):
Gabriel Ebner said:
Yury G. Kudryashov said:
in some contexts [...] this looks funny.
$\sqrt{2} \cup \pi = \pi$ doesn't look so bad, it reminds me of Dedekind cuts.
I guess this is tongue in cheek? I'm not a fan :wink: I guess that for linear orders we want to simp this away to max a b as soon as possible?
#### Gabriel Ebner (Jun 16 2020 at 17:34):
I'm 60% serious. $\sqrt{2} \sqcup \pi = \pi$ doesn't look any better to me. But $1 \in \pi$ would be too much, even for me.
#### Johan Commelin (Jun 16 2020 at 17:35):
Gabriel Ebner said:
I'm 60% serious. $\sqrt{2} \sqcup \pi = \pi$ doesn't look any better to me.
But it also doesn't look any worse, right? (At least to me...)
#### Gabriel Ebner (Jun 16 2020 at 17:36):
I completely agree. That's why I think ${}\cup{}$ is the best choice if we want to standardize, because it looks right half of the time (with sets, multisets, etc.).
#### Johan Commelin (Jun 16 2020 at 17:36):
But on an abstract lattice, it looks "wrongish".
#### Johan Commelin (Jun 16 2020 at 17:37):
And the problem remains with \le vs \subset, right?
#### Johan Commelin (Jun 16 2020 at 17:37):
We definitely don't want 0 \subset 1.
#### Gabriel Ebner (Jun 16 2020 at 17:38):
Yes, ⊆ vs. ≤ is the hard one. How about ∅ ≤ A ∩ B? ducks
#### Johan Commelin (Jun 16 2020 at 17:42):
I think I prefer to go full lattice notation everywhere.
#### Reid Barton (Jun 16 2020 at 17:52):
$\cup$/$\cap$ look really wrong when they are wrong, for example, filters ordered by reverse inclusion.
#### Scott Morrison (Jun 17 2020 at 04:47):
Is it insane to have all the notations mean the same thing (i.e. lattice operations), and let people use different ones in different contexts?
#### Yury G. Kudryashov (Jun 17 2020 at 04:50):
Proof state will always use one notation.
#### Bryan Gin-ge Chen (Jun 17 2020 at 05:03):
Is this sort of thing why we have to prove everything for both additive groups and multiplicative groups?
#### Yury G. Kudryashov (Jun 17 2020 at 05:05):
For multiplicative and additive groups we also have rings. They use both multiplicative and additive structures.
#### Yury G. Kudryashov (Jun 17 2020 at 05:06):
One possible solution is to write a to_additive-style attribute and add it to all the lattice lemmas.
#### Yury G. Kudryashov (Jun 17 2020 at 05:07):
But then we'll have to decide which notation is better in each case.
#### Floris van Doorn (Jun 26 2020 at 22:45):
I wouldn't mind using lattice notation everywhere.
I'm not in favor using set notation for lattices.
The current status quo means we have to redo everything about lattices separately for sets and finsets, which is not great. There have been plenty of cases where I needed a lemma that existed for exactly one of {bUnion, supr} and I needed it for the other one.
#### Yury G. Kudryashov (Jun 26 2020 at 23:30):
Even worse, we need to redo lots of stuff separately for sets, finsets, and lattices.
#### Yury G. Kudryashov (Jun 29 2020 at 19:25):
Let me revive this thread. What are the reasons against migration of sets, multisets and finsets to lattice?
#### Yury G. Kudryashov (Jun 29 2020 at 19:26):
I remember the following:
• $\sqcup$ and $\sqcap$ look worse than $\cup$ and $\cap$;
• it's hard to remember \lub for $\sqcup$.
What else?
#### Johan Commelin (Jun 29 2020 at 19:28):
The second point is no longer valid after all the new additions to the translations file.
#### Floris van Doorn (Jun 29 2020 at 19:28):
point 2 has been resolved: there are now many ways to input ⊔ and ⨆.
I think a point that is close to point 1: lattice notation is beginner unfriendly.
#### Yury G. Kudryashov (Jun 29 2020 at 19:30):
And defining $f'(x)$ using normed spaces and filters is very beginner friendly ;)
#### Yury G. Kudryashov (Jun 29 2020 at 19:31):
BTW, when I first saw Lean the fact that ⊂ means $\subsetneq$ was very surprising.
#### Floris van Doorn (Jun 29 2020 at 19:31):
There is a difference: with a good enough API you don't need to know about normed spaces & filters to work with the derivative on functions real -> real. The lattice operations on set is the API.
(just to be clear: I'm in favor of using lattice notation for sets over the status quo)
#### Sebastien Gouezel (Jun 29 2020 at 19:35):
Can we have local notation : ∪ := ⊔, and use it in the set files?
#### Sebastien Gouezel (Jun 29 2020 at 19:36):
Or even have it globally as an input notation in all files?
#### Floris van Doorn (Jun 29 2020 at 19:37):
I'm afraid that might make things even more confusing if the lemmas you use are called sup_comm and image_sup (or sup_comm and image_union??).
#### Johan Commelin (Jun 29 2020 at 19:38):
image and preimage should clearly be renamed to map and comap... ...
... :confused: :rolling_on_the_floor_laughing:
#### Sebastien Gouezel (Jun 29 2020 at 19:39):
Floris van Doorn said:
I'm afraid that might make things even more confusing if the lemmas you use are called sup_comm and image_sup (or sup_comm and image_union??).
Good point!
#### Floris van Doorn (Jun 29 2020 at 19:41):
Also, what does s ≤ t mean to a mathematician (if anything)? Is it pointwise inequalities: ∀ x ∈ s, ∀ y ∈ t, x ≤ y? Using lattice notation means that it has to mean subset...
#### Johan Commelin (Jun 29 2020 at 19:44):
I guess it could mean pointwise ineqs, but I think that's very rare.
#### Alex J. Best (Jun 29 2020 at 19:44):
I don't think the idea that sets form a partial order w.r.t. inclusion is so alien to mathematicians, so I'd say most would guess it means subset.
#### Johan Commelin (Jun 29 2020 at 19:46):
It definitely helps that the lattice notation is just a pointy-squary version of the set notation.
#### Floris van Doorn (Jun 29 2020 at 19:50):
But it might take some time to get used to names like le_image_sup : f '' (s ⊔ f ⁻¹' t) ≤ f '' s ⊔ t (or le_map_sup). I do feel like we're losing something when moving from the current notation/naming to the lattice versions...
#### Sebastien Gouezel (Jun 29 2020 at 19:53):
We would need an attribute like to_additive to create union lemmas from sup lemmas, and a notation union = sup. Then it would be mostly transparent to the user, but with better defeq properties.
#### Jeremy Avigad (Jun 29 2020 at 19:53):
How about having something like the to_additive trick, so that we can batch-duplicate lattice theorems with set names and set notation (for types that have that notation defined)?
I was faster :)
#### Alex J. Best (Jun 29 2020 at 19:53):
As a small experiment, I just asked two grad students (in number theory) what X \le Y for X, Y sets means with no hints, one guessed the partial order by inclusion, and one guessed an inequality of cardinals (i.e. exists an injection from X \to Y. It might have been more realistic to pose it as X, Y subsets of Z, what does X\le Y mean.
#### Yury G. Kudryashov (Jun 29 2020 at 19:55):
If we try the to_additive trick, then which notation should we use for submonoids etc?
#### Sebastien Gouezel (Jun 29 2020 at 19:58):
I would say: don't touch existing files. Both notations should be completely equivalent, so you can even use \le at one line and \subseteq at the next one.
#### Floris van Doorn (Jun 29 2020 at 19:58):
I guess lattice operations are fine on submonoids, opens, ...
For submonoids you don't even have ↑(s ⊔ t) = ↑s ∪ ↑t in general
#### Floris van Doorn (Jun 29 2020 at 20:00):
@Sebastien Gouezel As an temporary solution? Eventually we will want to name our lemmas in a consistent way (which probably means using _sup_ instead of _union_ everywhere)
#### Floris van Doorn (Jun 29 2020 at 20:01):
But if we use ∪ for set, we probably also want it for finset...
#### Johan Commelin (Jun 29 2020 at 20:01):
I think that @Sebastien Gouezel is proposing to throw away set/basic.lean and instead tag every lemma in order/lattice.lean with @[to_set].
Exactly.
#### Sebastien Gouezel (Jun 29 2020 at 20:03):
In more specialized domains, we should settle on one of the choices. I'd say use union and subset for sets and finsets (and more generally if there is a coercion to sets satisfying coe (a sup b) = coe a union coe b), and order notation otherwise.
#### Floris van Doorn (Jun 29 2020 at 20:04):
is the statement of lemma union_comm the same as sup_comm, or is it specialized to sets?
#### Sebastien Gouezel (Jun 29 2020 at 20:05):
It is exactly the same statement.
#### Sebastien Gouezel (Jun 29 2020 at 20:05):
(Possibly with a different pretty printer setting so that the sup is shown as a union in the tooltip)
#### Floris van Doorn (Jun 29 2020 at 20:06):
ah ok. that would make it easier.
#### Floris van Doorn (Jun 29 2020 at 20:06):
yeah, you probably still want different has_union and has_sup classes, and let to_set modify the classes as well, so that the goal/lemmas are printed the way you wrote them.
I guess that could work.
#### Kevin Buzzard (Jun 29 2020 at 20:07):
@Alex J. Best I agree: if you don't know X and Y are subsets of Z then $X\subseteq Y$ doesn't make much sense. If they're both subsets of Z then there's at most one canonical injection from X to Y and so now you're more likely to guess the inclusion.
#### Johan Commelin (Jun 29 2020 at 20:08):
This is going to be a refactor so big that :head_bandage: :head_bandage: :head_bandage:
#### Alex J. Best (Jun 29 2020 at 20:08):
I agree, it was a flawed experiment! But someone still got it right!
#### Yury G. Kudryashov (Jun 29 2020 at 20:17):
If we have different has_sup and has_union classes, then how can union_comm and sup_comm be the same?
#### Sebastien Gouezel (Jun 29 2020 at 20:18):
I think we should just have one class.
#### Yury G. Kudryashov (Jun 29 2020 at 20:19):
Then what should @[to_set] do?
#### Yury G. Kudryashov (Jun 29 2020 at 20:19):
And I guess we won't be able to tell pretty-printer to consistently use ∪ for sets and finsets.
#### Sebastien Gouezel (Jun 29 2020 at 20:21):
@[to_set] theorem le_sup_right : b ≤ a ⊔ b
should create a theorem subset_union_rightwhich is defeq to le_sup_right, but in which notation union := sup and notation subseteq := \le are activated so that the tooltip shows right.
#### Sebastien Gouezel (Jun 29 2020 at 20:22):
Yury G. Kudryashov said:
And I guess we won't be able to tell pretty-printer to consistently use ∪ for sets and finsets.
I agree, but I think this is only a minor annoyance. (And it might be solvable in Lean 4).
#### Yury G. Kudryashov (Jun 29 2020 at 20:25):
What do you mean by "notation ... are activated"? AFAIK, Lean doesn't store notation with lemma statements. It uses currently active notation in #print etc.
#### Floris van Doorn (Jun 29 2020 at 20:25):
Yury G. Kudryashov said:
If we have different has_sup and has_union classes, then how can union_comm and sup_comm be the same?
with two different classes, they are not syntactically equal.
My idea (to get the pretty printing right) is to have separate has_sup and has_union classes. Given a lattice you have instances to both has_sup and has_union, and the underlying maps are definitionally equal.
Then @[to_set] replaces all sups with unions, in the same way that to_additive does it.
#### Floris van Doorn (Jun 29 2020 at 20:25):
The lemmas will be definitionally equal, but not syntactically equal.
#### Sebastien Gouezel (Jun 29 2020 at 20:26):
Yury G. Kudryashov said:
What do you mean by "notation ... are activated"? AFAIK, Lean doesn't store notation with lemma statements. It uses currently active notation in #print etc.
ok, too bad
#### Yury G. Kudryashov (Jun 29 2020 at 20:27):
If we use different classes, why should we care about defeq? We can make @[to_set] do the same as @[to_additive].
#### Floris van Doorn (Jun 29 2020 at 20:29):
we don't necessarily care about defeq.
#### Sebastien Gouezel (Jun 29 2020 at 20:32):
I don't remember the start of the thread: wasn't your goal to get union defeq to to \sup to start with?
#### Yury G. Kudryashov (Jun 29 2020 at 20:33):
Let's say we consider different options. The goal is to reduce code duplication.
#### Yury G. Kudryashov (Jun 29 2020 at 20:34):
Currently we prove quite a few lattice lemmas for (a) sets; (b) finsets; (c) all other lattices.
#### Yury G. Kudryashov (Jun 29 2020 at 20:36):
Technically the simplest solution is to start using ≤, $\sqcap$ etc everywhere but this is bad for new users.
#### Yury G. Kudryashov (Jun 29 2020 at 20:41):
Other solutions include: (a) use a @[to_additive]-like attribute to transfer lemmas from lattice operations to set operations; (b) allow ∪ as a (local?) alternative syntax for $\sqcup$ and use it for sets and finsets.
#### Sebastien Gouezel (Jun 29 2020 at 20:42):
From the discussion, (a) looks like the nicest solution, albeit not the easiest to implement.
#### Yury G. Kudryashov (Jun 29 2020 at 20:44):
The main drawback of (a) is that we'll need class names for set versions of everything up to complete_boolean_algebra and, e.g., monotone won't work automatically unless we define has_le as well.
#### Yury G. Kudryashov (Jun 29 2020 at 20:45):
I hope that with Lean 4 we'll be able to define a pretty-printer that uses ∪ for some types and $\sqcup$ for all other types.
#### Yury G. Kudryashov (Jun 29 2020 at 20:48):
So we have option (c): wait for Lean 4.
#### Sebastien Gouezel (Jun 29 2020 at 20:50):
Even with Lean 4, we would need @[to_set] to create lemma names with union and subset instead of sup and le.
#### Jeremy Avigad (Jun 29 2020 at 20:51):
Is it a problem that order.complete_lattice depends on properties of sets, through order.bounds? On a quick look, order.bounds only uses finite unions and intersections, subset, empty and univ. So maybe it is o.k.
#### Yury G. Kudryashov (Jun 29 2020 at 20:52):
It's easy to reorder imports so that data/set/basic will know the definition of complete_boolean_algebra and lemmas about bounded_lattice.
(done locally)
#### Yury G. Kudryashov (Jun 29 2020 at 21:01):
Sebastien Gouezel said:
Even with Lean 4, we would need @[to_set] to create lemma names with union and subset instead of sup and le.
It's easier to add aliases with custom names than to modify types.
#### Johan Commelin (Jun 30 2020 at 04:31):
Yury G. Kudryashov said:
The main drawback of (a) is that we'll need class names for set versions of everything up to complete_boolean_algebra and, e.g., monotone won't work automatically unless we define has_le as well.
Well, to_set wouldn't have to replay the proofs (like to_additive) does, if we make sure that things are defeq. Then it just has to modify the name and statement, and can prove the resulting lemma by directly calling the lattice version.
So it generates
lemma subset_union_right : the-statement :=
le_sup_right _ _ _
To make this work, we have to give set some low-priority has_le and has_sup instances, but it would save us from duplicating the entire lattice class hierarchy for set-like notation.
#### Yury G. Kudryashov (Jun 30 2020 at 06:54):
What's wrong with the approach "make \cup and \sqcup two notations for the same operation and make @[to_set] work as alias + autogen name"? This way we can't have good notation in the proof state but we can be sure that any lemma about lattices work for sets even without a @[to_set] alias.
#### Yury G. Kudryashov (Jun 30 2020 at 06:55):
We still can have good notation in the input.
#### Yury G. Kudryashov (Jun 30 2020 at 06:55):
(and I hope that we'll be able to have a good notation in the proof state in Lean 4)
#### Yury G. Kudryashov (Jun 30 2020 at 07:06):
How exactly should the "defeq" based @[to_set] work? Should we add some instances like has_subset.to_has_le and use them when transforming the statement? Will it play nice with diamonds?
#### Sebastien Gouezel (Jun 30 2020 at 07:09):
Shouldn't we remove completely has_subset, and use subset as a notation for le?
#### Yury G. Kudryashov (Jun 30 2020 at 07:12):
With the "make \cup and \sqcup two notations for the same operation" approach yes, we should. But I'm trying to understand what other options do we have, and can't understand the "defeq" approach @Johan Commelin is talking about.
#### Kevin Buzzard (Jun 30 2020 at 07:22):
Sebastien Gouezel said:
Shouldn't we remove completely has_subset, and use subset as a notation for le?
Shouldn't we completely remove has_mul, and use * as a notation for has_add?
#### Kevin Buzzard (Jun 30 2020 at 07:23):
I guess there might be some edge cases where a type has two monoid structures
#### Yury G. Kudryashov (Jun 30 2020 at 14:36):
@Kevin Buzzard Let's discuss lattices, not monoids. Currently there is only one rarely used type (multiset) with both has_subset and has_le and has_le.le ≠ has_subset.subset.
#### Kevin Buzzard (Jun 30 2020 at 14:42):
So rather fewer edge cases than in the */+ case.
#### Yury G. Kudryashov (Jun 30 2020 at 14:57):
And I'm not aware of good structures involving has_subset on multiset, so we can just add a custom notation for this.
#### Yury G. Kudryashov (Jun 30 2020 at 16:29):
One more type with sensible has_subset different from has_le: list.
#### Gabriel Ebner (Jun 30 2020 at 16:33):
IIRC there is more than one reasonable (non-subset) definition of ≤ on lists. And only one of them is well-founded (if the order on the elements is well-founded). Maybe these should be different relations (i.e., not ≤) anyhow.
Last updated: May 09 2021 at 13:20 UTC | 2021-05-09 14:12:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 19, "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": 0.6078280806541443, "perplexity": 4209.037006171523}, "config": {"markdown_headings": true, "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-2021-21/segments/1620243988986.98/warc/CC-MAIN-20210509122756-20210509152756-00191.warc.gz"} |
https://stats.stackexchange.com/questions/374533/summation-of-squared-x-i-if-summation-of-x-i-is-1/374553 | # Summation of squared x_i if summation of x_i is 1
How to prove "If $$\sum_{i=1}^n x_i=1$$, then $$\sum_{i=1}^n x_i^2>1/n$$"? I'm thinking about $$Var(x_i)=E(x_i^2)-[E(x_i)]^2=\frac{1}{n}\sum_{i=1}^n x_i^2-1/n^2\ge0$$. Is that correct?
• Can you please add the self-study tag and read its wiki? – kjetil b halvorsen Oct 30 '18 at 22:53
• There is a pleasant geometrical interpretation: No point on the hyperplane passing through the point $P=(1/n, 1/n,\ldots, 1/n)$ with normal direction $nP=(1,1,\ldots, 1)$ lies in the interior of the origin-centered ball of squared radius $|P|^2 = 1/n^2 + \cdots 1/n^2 = 1/n.$ The proof is that this hyperplane is (obviously) tangent to the boundary of that ball, because its normal vector is parallel to the radius vector at one point of intersection ($P$). – whuber Oct 31 '18 at 3:03
Try writing variance as $$Var(x_i)=E(x^2_i)−[E(x_i)]^2$$
substitute $$[E(x_i)]^2 = 1/n^2$$
$$1/n\Sigma (x_i - \bar x)^2 = 1/n\Sigma x_i^2 - 1/n^2$$
from there
$$1/n\Sigma x_i^2 = 1/n\Sigma(x_i - \bar x)^2 + 1/n^2)$$
$$\Sigma x_i^2 = (\Sigma(x_i - \bar x)^2 + 1/n)$$
which is the same as
$$\Sigma x_i^2 = (\Sigma(x_i - \bar x)^2 + 1/n)$$
or $$\Sigma x_i^2 = n*Var(x_i) + 1/n$$
Since variance and n are greater than 0
$$n*Var(x_i) + 1/n > 1/n$$
$$\Sigma x_i^2 > 1/n$$
Hint: Can you calculate the (arithmetic) mean of the $$x_i$$, $$\bar{x}=\frac1{n}\sum_{i=1}^n x_i$$? Then write $$\sum_{i=1}^n x_i^2 = \sum_{i=1}^n\left( (x_i-\bar{x})+ \bar{x}\right)^2$$ and you should be able to conclude. | 2020-01-24 22:23:29 | {"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": 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": 15, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8918890953063965, "perplexity": 506.7451647259948}, "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-2020-05/segments/1579250626449.79/warc/CC-MAIN-20200124221147-20200125010147-00380.warc.gz"} |
http://math.stackexchange.com/questions/151953/regular-wreath-product-of-nilpotent-groups | # (Regular) wreath product of nilpotent groups
Is the wreath product of two nilpotent groups always nilpotent?
I know the answer is no due to a condition "The regular wreath product A wr B of a group A by a group B is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the same prime p ", but I can't easily construct a counter example to show it.
-
Take any p-group $P$ and q-group $Q$, and then $P\wr Q$ is not nilpotent, as it has no normal Sylow q-subgroup. – user641 May 31 '12 at 11:12
@SteveD: You should mention your implicit assumption $p\ne q$ ;-). To user31899: To be even more concrete than Steve, take $A$ and $B$ as simple as possible, but violating the condition after "if and only if", e.g., take something like $A=C_3$, $B=C_2$. – j.p. May 31 '12 at 14:25
Following Jug's suggestion: let $\,\,A:=C_3=\langle a\rangle\,,\,B:=C_2=\langle c\rangle\,$ , with the regular action of $\,B\,$ on itself, and form the (regular) wreath product $$A\wr B\cong \left(C_3\times C_3\right)\rtimes_R C_2$$ Take the elements $$\pi=((1,1),c))\,\,,\,\,\sigma=((a,a^2),1)$$It's now easy to check that$$\pi^2=\sigma^3=1\,\,,\,\,\pi\sigma\pi=\sigma^2$$ so we get that $\,\,\langle \pi\,,\,\sigma\rangle\cong S_3\,\,$ and thus $\,\,A\wr B\,\,$ can't be nilpotent, though both $\,\,A,B\,\,$ are (they're even abelian...) | 2015-08-28 17:55:49 | {"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": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8699293732643127, "perplexity": 284.3321538118702}, "config": {"markdown_headings": true, "markdown_code": false, "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-2015-35/segments/1440644063825.9/warc/CC-MAIN-20150827025423-00085-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://www.mindspring.com/~toxiccow/13_0074.html | Subject: Re: Don't you just hate it?
Date: 25 Mar 1997 00:00:00 GMT
Organization: MotPU: Where Binary Moodswings are ALWAYS on the Menu
Newsgroups: alt.slack
References: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
: Previously, MegaLiz at MotPU: Where Binary Moodswings are ALWAYS on
: :
: : : I have this nagging fear of becoming "the cloying couple".
: :
: : You mean you have a couple of cloying fears of being nagged, SNOOKUMS.
:
: Remember last week when we tried to diss each other here? That SUCKED.
: I'd be even more cloying here but I'm still bummed about Madame Flora.
OKAY. OKAY. We shouldn't PRETEND. How would it be if I SHARED an
Actual Taunt? Is that too CLOYING? Like when I said, "Ya know, I guy
like you deserves a world-class blowjob, and right now, somewhere in
the world, a guy just like you is getting one!"
Is that BETTER?
------------------------------------------------------------------
* Asterisk of sounding serious, I may prefer to remain mysterious.
Fat-free Feet: Not just a tasty breakfast treat. | 2017-10-22 04:28:44 | {"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": 0.8778664469718933, "perplexity": 7420.073735053194}, "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/1508187825141.95/warc/CC-MAIN-20171022041437-20171022061437-00509.warc.gz"} |
https://www.esn-analysis.net/wiki/closeness.html | Warning: This website has been archived and some features are not available anymore.
# Closeness Centrality
The closeness centrality measures the average shortest path, also known as the geodesic distance, through a network between two vertices (Newman, 2010). It was first published by Sabidussi (1966). Since the closeness is the opposite of the distance, the value of the average shortest path is inverted, to get the value for the closeness centrality. A higher value for the closeness centrality means that a vertex is closer to all other vertices (Scott, 2012) and quicker to interact with all others vertices (Wasserman et al., 1994).
Because the closeness centrality measures the distance from one specific node to all others, it is of ego-centric scope. This metric was proposed by Smith et al. (2009), Hacker et al. (2015), Viol et al. (2016) and Berger et al. (2014).
## Calculation
The calculation is based on Wasserman (1994). To calculate the closeness centrality, the number of edges between the two nodes $v_i \in V$ and $v_j \in V$ for the shortest path is defined as the $distance(v_i, v_j)$ -- this equates to the shortest path length or the geodesic distance.
To get the closeness centrality $c$, the sum of all distances is calculated and inverted: $$c(v_i) = \left[\sum_{v_j \in V}distance(v_i, v_j)\right]^{-1}\quad\text{, where }v_j \neq v_i .$$
For comparison of networks with different sizes, Wasserman (1994) as well as Freeman (1979) picked up the suggestion by Beauchamp (1965) to standardize the metric through multiplying with the network size $g$ minus one: $$c^{'}(v_i) = (g-1) * \left[\sum_{v_j \in V}distance(v_i, v_j)\right]^{-1}\quad\text{, where }v_j \neq v_i .$$
Because the formula yields an infinite value for disconnected nodes, it is problematic. While Newman (2010) suggests to use the inverse distance instead of the distance, he acknowledges that this is rarely used in practice. Instead the formula variant from the igraph package is used, which states that "if there is no path between vertex $v_i$ and $v_j$ then the total number of vertices is used in the formula instead of the path length"^igc.
Therefore we define a new distance function as: $$distance^{'}(v_i, v_j) = \begin{cases} distance(v_i, v_j) & \text{, if there is a path} \\ g & \text{, otherwise}. \end{cases}$$
The final formula looks like this: $$c^{'}(v_i) = (g-1) * \left[\sum_{v_j \in V}distance^{'}(v_i, v_j)\right]^{-1}\quad\text{, where }v_j \neq v_i .$$
## Interpretation
The value can be interpreted in the context of Social Networks. Newman (2010) mentions that a user with high closeness centrality is able to quickly respond and interact with other users due to the short distance. Such a user can efficiently disseminate information through the network because of the short communication paths to others. This argument is confirmed by Berger et al. (2014), who state that users with high closeness centrality are able to spread information easily. Similar to the interpretation of degree centrality, Berger et al. (2014) further claim that a high closeness centrality is related to being a key user. Key users are knowledge hubs, meaning they contribute and help other users to solve their daily problems. They are able to diffuse innovative ideas quickly to other people.
Smith et al. (2009) relate high closeness centrality to users, who regularly spawn new discussions and ideas as well as take part in other users' threads. Contrary, a low closeness centrality is related to people, who tend to reply in other people threads only, but do not initiate discussions on their own. The notion of engagement is introduced by Hacker et al. (2015). On the one hand a high closeness centrality indicates high levels of continuous engagement by a user in the Enterprise Social Network, but on the other hand such a user is not very focused i.e. he does talk about multiple topics compared to being an expert in one topic.
Hacker et al. (2015) draw connections to Viegas (2004), who researched that the closeness centrality can be related to frequency of posts. A high closeness centrality and degree centrality indicate a high post count, which is discussed in the metric messages created. Furthermore Hacker et al. (2015) links to Holtzblatt (2013) and their results on valuable themes of social platform experience. They claim a high engagement supports collaboration and facilitates cooperation with staff in other locations. It also strengthens social connection, expanding a user's network and tracking other people's activities. Since a high closeness centrality implies that you can easily communicate with all other users, both of these statements are reasonable to make.
Viol et al. (2016) picks up the interpretation of continuous engagement in an Enterprise Social Network. A high closeness centrality indicates that a user is well connected within the network. They are always online and active and therefore can initiate and take part in multiple discussions. This is also means that they are not focused on one topic, but rather dispersed across a lot of discussions and threads (Viol, 2016), which fits to the interpretation in the other literature.
Similar to the degree centrality: if multiple users exhibit a high closeness centrality, it leads to a dense and cohesive network. The short distance between all users results in strong ties between the users. Strong ties are a reason for Social Capital and the formation of effective norms and trust (Riemer, 2005). According to Coleman (1990) effective norms and trust among the users allow for successful collaboration.
The interpretation that information can quickly be disseminated, fits to the effect of Social Capital described by Nahapiet et al. (1998). They argue that "Social Capital constitutes a valuable source of information benefits" (p. 252). It manifests itself in the distribution of information, making information readily available. Because of the established trust through strong ties, specifically the distribution of complex or sensitive information is positively influenced by Social Capital as noted by Riemer (2005) and Koka et al. (2002). Whereas the dissemination of arbitrary and small information is more positively influenced by weak ties and thus a low closeness centrality. | 2021-05-16 15:39:08 | {"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": 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": 0.8086174726486206, "perplexity": 789.9634456243363}, "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-2021-21/segments/1620243991224.58/warc/CC-MAIN-20210516140441-20210516170441-00281.warc.gz"} |
https://new.rosettacommons.org/docs/latest/rosetta_basics/file_types/Bin-transition-probabilities-file | Author: Vikram K. Mulligan, Baker laboratory (vmullig@uw.edu)
Bin transition probabilities (.bin_params) files are used to define the probabilities of transitioning from one mainchain torsion bin at residue i to another bin at residue i+1. For our purposes, a mainchain torsion bin is defined as a region in mainchain torsion space lying within well-defined, rectangular boundaries. For example, bin "B" from the ABEGO definitions is defined as any mainchain conformation for which phi is between -180 and 0 degrees, psi is greater than 50 or less than -125 degrees, and omega is greater than 90 or less than -90 degrees.
The defined transition probabilities are used by certain sampling schemes, and could be used for scoring as well. Note that the format is intended to be completely general, so that it could be applied to alpha-amino acids, beta-amino acids, nucleic acids, other noncanonical building blocks, or mixed heteropolymers.
## Bin transition probabilities file format
Pre-defined bin transition probabilities files are located in the database/protocol_data/generalizedKIC/bin_params directory. A sample bin transition probability file is shown below:
#ABBA.binparams
#Created by Vikram K. Mulligan (vmullig@uw.edu), 2 February 2015
#This file defines transition probabilities for the A, B, B', A', O, and O' bins.
#Bins A and B are defined as they are for ABEGO bins (phi < 0, psi -125 to 50, and
#omega 90 to 180 or -180 to -90 for A; phi < 0, psi -180 to -125 or 50 to 180, and
#omega 90 to 180 or -180 to -90 for B). Bins A' and B' are the mirror image,
#rotated about phi=0, psi=0 (so phi > 0, psi -50 to 125, omega 90 to 180 or -180
#to 90 for A'; phi > 0, psi 125 to 180 or -180 to 50, omega 90 to 180 or -180 to -90
#for B'). Bins O and O' have cis omega angles (omega -90 to 90), and are in the
#negative or positive phi space, respectively.
#Note the defined bins must cover the full mainchain torsion space!
BEGIN
MAINCHAIN_TORSIONS_I 3
MAINCHAIN_TORSIONS_IPLUS1 3
BIN_COUNT_I 6
BIN_COUNT_IPLUS1 6
I_BIN A -180.0 0.0 -125.0 50.0 90.0 -90.0
I_BIN B -180.0 0.0 50.0 -125.0 90.0 -90.0
I_BIN Aprime 0.0 180.0 -50.0 125.0 90.0 -90.0
I_BIN Bprime 0.0 180.0 125 -50.0 90.0 -90.0
I_BIN O -180.0 0.0 -180.0 180.0 -90.0 90.0
I_BIN Oprime 0.0 180.0 -180.0 180.0 -90.0 90.0
SUB_BINS_I L_AA
IPLUS1_BINS_COPY_I
PROPERTIES_I L_AA ALPHA_AA
PROPERTIES_IPLUS1 L_AA ALPHA_AA
NOT_PROPERTIES_I D_AA
NOT_PROPERTIES_IPLUS1 D_AA
NOT_RES_I GLY
NOT_RES_IPLUS1 GLY
MATRIX 1670055 313251 87373 26201 3915 682
MATRIX 363556 1205782 71936 34623 9406 1044
MATRIX 39384 118400 17691 2634 705 63
MATRIX 25528 35553 1949 2328 493 64
MATRIX 3121 5916 176 183 171 70
MATRIX 215 742 75 42 51 20
END
Bin transition probabilities files may be commented with the pound sign (#). Anything past a pound sign character is ignored.
Each bin transition probability file must define at least one bin transition probability matrix. Each probability matrix defines a set of bins for position i, a set of bins for position i+1, a set of rules for what types of residues could be at positions i and i+1, and the actual (un-normalized) transition probabilities or counts. Multiple matrices may be defined to allow different transition probabilities given different types of residues at positions i and i+1. For example, the probability of transitioning to the normally prohibited regions of Ramachandran space is very low unless residue i+1 is a glycine. Each defined transition matrix must be flanked with BEGIN and END lines.
After the BEGIN line, the next two lines define the number of mainchain torsions (rotatable bonds) in residues i and i+1. This is necessary since there is no assumption that this is being applied solely to alpha-amino acids. In our example, though, we are defining transitions for alpha-amino acids, so the number of mainchain torsions at both positions is 3.
MAINCHAIN_TORSIONS_I 3
MAINCHAIN_TORSIONS_IPLUS1 3
The next two lines specify the number of bins that will be defined at positions i and i+1. In this example, both positions will have six bins defined.
BIN_COUNT_I 6
BIN_COUNT_IPLUS1 6
Having specified that there will be six bins, we now need to define the bin names and boundaries. We do so with I_BIN and IPLUS1_BIN lines, each with the following syntax:
I_BIN <bin_name> <torsion_1_start_of_range> <torsion_1_end_of_range> <torsion_2_start_of_range> <torsion_2_end_of_range> ... <torsion_n_start_of_range> <torsion_n_end_of_range>
Torsion values must lie between -180 degrees and 180 degrees. If the end of range value is less than the start of range value for any torsion range, it is assumed that the bin runs from the start of the range to 180, then wraps back to -180 and up to the end of range value. Note that bins must not overlap, and must cover the entire torsion space.
Within each bin, the relative probability of a particular set of mainchain torsion values might not be equal. In the case of alpha-amino acids, sub-bins may be defined automatically based on the Ramachandran map (and these permit Ramachandran-biased sampling within each bin). The SUB_BINS_I and SUB_BINS_IPLUS1 lines define how sub-bins will be set up. Current options are:
• "NONE" (i.e. uniform probability across the bin)
• "L_AA" (uses the Ramachandran map for L-alanine to set up the sub-bin probability distribution)
• "D_AA" (uses the Ramachandran map for D-alanine)
• "L_PRO" (uses the Ramachandran map for L-proline)
• "D_PRO" (uses the Ramachandran map for D-proline)
• "GLY" (uses the Ramachandran map for glycine).
The number of I_BIN lines must match the BIN_COUNT_I line, and the number of IPLUS1_BIN lines must match the BIN_COUNT_IPLUS1 line. The exception is if the bins for the i+1 position are identical to the bins for the i position, in which case a shorthand is to include a IPLUS1_BINS_COPY_I line. If an IPLUS1_BINS_COPY_I line is included, then IPLUS1_BIN and SUB_BINS_IPLUS1 lines will not be required.
So for the six bins in our example, we have the following lines:
I_BIN A -180.0 0.0 -125.0 50.0 90.0 -90.0
I_BIN B -180.0 0.0 50.0 -125.0 90.0 -90.0
I_BIN Aprime 0.0 180.0 -50.0 125.0 90.0 -90.0
I_BIN Bprime 0.0 180.0 125 -50.0 90.0 -90.0
I_BIN O -180.0 0.0 -180.0 180.0 -90.0 90.0
I_BIN Oprime 0.0 180.0 -180.0 180.0 -90.0 90.0
SUB_BINS_I L_AA
IPLUS1_BINS_COPY_I
The next few lines define properties that residues at positions i or i+1 MUST have (PROPERTIES_I and PROPERTIES_IPLUS1 lines), and properties that they must NOT have (NOT_PROPERTIES_I and NOT_PROPERTIES_IPLUS1 lines), for the rules defined by the current bin transition probabilities to apply to them. In the present example, we require that the i and i+1 positions are alpha-amino acids and L-amino acids, and that they cannot be D-amino acids (which is redundant, but that's fine).
PROPERTIES_I L_AA ALPHA_AA
PROPERTIES_IPLUS1 L_AA ALPHA_AA
NOT_PROPERTIES_I D_AA
NOT_PROPERTIES_IPLUS1 D_AA
The next few lines define residue identities for positions i or i+1. If RES_I or RES_IPLUS1 lines are present, each with a list of three-letter codes, then the residue at position i or i+1 MUST be one of the ones in the list in order for the bin transition rules to apply. In this case, we have no list of required residues. If NOT_RES_I or NOT_RES_IPLUS1 lines are included, then the residue at position i or i+1 must NOT be one of the ones in the list in order for the bin transition rules to apply. Here, we exclude glycine (even though this is redundant, since we have already required that the i and i+1 positions be L-amino acids).
NOT_RES_I GLY
NOT_RES_IPLUS1 GLY
Finally, we actually define the transition matrix, with one line corresponding to each bin for position i (with the order the same as the I_BIN definition lines), and one column corresponding to each bin for position i+1 (with the order the same as the IPLUS1_BIN definition lines). The matrix values are counts or un-normalized transition probabilities. The algorithm will automatically normalize these and generate cumulative distribution functions for random sampling.
MATRIX 1670055 313251 87373 26201 3915 682
MATRIX 363556 1205782 71936 34623 9406 1044
MATRIX 39384 118400 17691 2634 705 63
MATRIX 25528 35553 1949 2328 493 64
MATRIX 3121 5916 176 183 171 70
MATRIX 215 742 75 42 51 20
Additional BEGIN ... END blocks may be defined for as many bin transitions probability matrices as one wishes to define. If a particular i/i+1 pair of residues matches the properties and residue identity criteria for more than one bin transition probability matrix, the first one encountered that matches is used (so it is best for the bin transition probability matrices to define non-overlapping criteria). | 2021-12-08 13:03:44 | {"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": 0.8000604510307312, "perplexity": 3736.590926152738}, "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-2021-49/segments/1637964363510.40/warc/CC-MAIN-20211208114112-20211208144112-00371.warc.gz"} |
https://solvable.group/posts/secret-rng/ | # TLDR
This was a very nice challenge in the Donjon CTF. The goal was to reverse engineer the state of the PRNG math/rand in the Go standard library to guess the private key of a signature scheme and sign a given message. I solved this challenge by using the SMT solver Z3 to automagically recover the RNG state.
# Description
Connecting to nc ots-sig.donjon-ctf.io 4001, we get:
Public key: djX6vqVMj0fTZGIhJFgN1VHLgGgaBxWEhvxp1MzB09s=
Enter signature: test
We also get a single Go script:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 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 package main import ( "bufio" "crypto/sha256" "encoding/base64" "fmt" "io/ioutil" "os" "strings" "time" "./math/rand" // same as default implementation, with different rngCooked array "github.com/dchest/wots" ) const defaultFlag string = "CTF{xxx}" func main() { var message = []byte("Sign me if you can") // Not so secure seed, but prng internals are secret rng := rand.New(rand.NewSource(time.Now().UnixNano())) for { var ots = wots.NewScheme(sha256.New, rng) priv, pub, _ := ots.GenerateKeyPair() fmt.Println("Public key:", base64.StdEncoding.EncodeToString(pub)) reader := bufio.NewReader(os.Stdin) fmt.Print("Enter signature: ") text, err := reader.ReadString('\n') if err != nil { fmt.Println("Error occurred. Please try again later.") return } text = strings.TrimSuffix(text, "\n") signature, err := base64.StdEncoding.DecodeString(text) if err != nil { return } if ots.Verify(pub, message, signature) { fmt.Print("Congratulations! Flag: ") flag, err := ioutil.ReadFile("secret") if err != nil { fmt.Println(defaultFlag) } else { fmt.Println(string(flag)) } } else { fmt.Println("You failed! Private key was:", base64.StdEncoding.EncodeToString(priv)) fmt.Println() } } }
It is clear that the service on ots-sig.donjon-ctf.io:4001 is the Go script. The goal of the challenge is then to find a valid signature of the message Sign me if you can given only the public key and possibly some previous private keys.
The signature scheme is the Lamport signature scheme implemented in the wots Go package. I assumed that this implementation is secure (it’s a bit unethical to exploit a 0-day in an open-source project).
# Exploit
To start, I looked at the code for generating a PrivateKey from a RNG. The answer is straightforward, the private key just holds the next $N$ bytes of the RNG, where $N=B*(B+2)$ is the size of the private key, and $B$ is the size of the hash function. This challenge uses SHA256, so we have $B=32$ and $N=1088$.
So the challenge allows us to get as many subsequent private keys (from a single seed) as we want. As we have just seen, this is equivalent to being able to stream the RNG rand.New(rand.NewSource(time.Now().UnixNano())). Therefore, if we break the RNG, meaning we’re able to guess the following 1088 bytes given an arbitrary number or previous bytes, we win.
At this point, it is worth looking back at the challenge description and also line 13 in the Go script:
13 import "./math/rand" // same as default implementation, with different rngCooked array
So the RNG used is the same as the standard math/rand but with different constants (otherwise the challenge would be straightforward since we could just bruteforce the seed). rngCooked and the RNG logic in general are defined here.
After staring at this file for a while, I found the potential vulnerability1. Here is the function to generate uint64 (which is used to generate random bytes):
237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 // Uint64 returns a non-negative pseudo-random 64-bit integer as an uint64. func (rng *rngSource) Uint64() uint64 { rng.tap-- if rng.tap < 0 { rng.tap += rngLen } rng.feed-- if rng.feed < 0 { rng.feed += rngLen } x := rng.vec[rng.feed] + rng.vec[rng.tap] rng.vec[rng.feed] = x return uint64(x) }
For some context, rng.vec is the RNG state represented by a vector of $\texttt{rngLen}=607$ int64 elements. rng.tap and rng.feed are two counters ranging from 0 to rngLen and originally set at 333 and 606 respectively. The “vulnerability” is at line 250. Every time the function is called, the state element rng.vec[rng.feed] is replaced by the returned value2. So after having observed rngLen calls to this function Uint64, we can deduce the RNG state and thus compute the RNG’s next values, without knowing anything about the original state nor the seed.
However, in this challenge we do not get raw uint64 values from the RNG, but bytes. So, let’s see how random bytes are generated from the RNG. This is implemented here:
267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 func read(p []byte, src Source, readVal *int64, readPos *int8) (n int, err error) { pos := *readPos val := *readVal rng, _ := src.(*rngSource) for n = 0; n < len(p); n++ { if pos == 0 { if rng != nil { val = rng.Int63() } else { val = src.Int63() } pos = 7 } p[n] = byte(val) val >>= 8 pos-- } *readPos = pos *readVal = val return }
A few things to note:
• rng.Int63 is just the signed version of rng.Uint64 that we analyzed above.
• The function read works by iteratively generating a Int63, and writing the first (little-endian) 7 bytes of this value to the buffer.
• After the 7 bytes have been written, the rest is discarded, a new Int63 is generated and the process continues.
• This means that we get back the first $7*8=56$ bits of every value generated by the RNG.
I think this is the appropriate time to introduce our good friend Z3.
# Modelling the problem with Z3
To recapitulate:
1. After observing 7*rngLen bytes, we can deduce the first 56 bits of every element of the RNG’s state.
2. Afterwards, we can observe 56 bits of an arbitrary number of linear combinations of state elements.
The state is a vector of $\texttt{rngLen}$ int64 elements. We can thus model them with a list of $\texttt{rngLen}$ z3.BitVec('state_i', 65) elements:
state = [z3.BitVec(f'state_{i}', 65) for i in range(rngLen)]
Then, by 1., after observing 7*rngLen bytes, I create a list masked_state of rngLen 56-bits integers and set the constraints:
s = z3.Solver()
for i in range(rngLen):
where mask = (1<<56)-1 is the 56 bits mask. Then, by 2., whenever we observe 7 bytes seven_bytes, we can create a (little-endian) 56-bits integer from them and do the following:
x = int.from_bytes(seven_bytes, 'little')
y = state[feed] + state[tap]
state[feed] = y
We can then repeat this process infinitely (by also decrementing the counters tap and feed).
It turns out that after 2*rngLen*7 observed bytes, Z3 manages to correctly recover the state. After that, it’s easy to solve the challenge by generating the next 1088 bytes from the RNG, give them to the wots Go package to sign the message and get the flag.
# Full solution
Here is the full solution in Python:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 from pwn import remote import re import base64 import z3 import subprocess # Open socket and recover 13 private keys. def open_socket(): private_keys = [] r = remote('ots-sig.donjon-ctf.io', 4001) r.recvuntil("Public key") for i in range(13): r.sendline('bG9sCg==') s = r.recvuntil("Public key") sk = re.findall(b'.*Private key was:\s*(.*)', s)[0] private_keys.append(base64.b64decode(sk)) return private_keys, r # Reverse the RNG state given only the RNG's first N raw bytes. def reverse_rng(raw_bytes): blockSize = 1088 rngLen = 607 rngTap = 273 mask = (1 << (7 * 8)) - 1 masked_state = [0 for _ in range(rngLen)] feed = rngLen - rngTap - 1 tap = rngLen - 1 for i in range(rngLen): x = int.from_bytes(raw_bytes[i * 7:(i + 1) * 7], 'little') masked_state[feed] = x feed -= 1 if feed < 0: feed += rngLen s = z3.Solver() state = [z3.BitVec(f'v_{i}', 65) for i in range(rngLen)] for i in range(rngLen): s.add(state[i] & mask == masked_state[i]) for i in range(rngLen, 3 * rngLen): x = int.from_bytes(raw_bytes[i * 7:(i + 1) * 7], 'little') y = state[feed] + state[tap] s.add((y & mask) == x) state[feed] = y feed -= 1 if feed < 0: feed += rngLen tap -= 1 if tap < 0: tap += rngLen print("The Z3 model is: ", s.check()) model = s.model() further_bytes = b'' for i in range(3 * rngLen, 5 * rngLen): x = int.from_bytes(raw_bytes[i * 7:(i + 1) * 7], 'little') y = state[feed] + state[tap] yl = model.eval(state[feed] + state[tap]).as_long() further_bytes += int.to_bytes(yl & mask, 7, 'little') state[feed] = y feed -= 1 if feed < 0: feed += rngLen tap -= 1 if tap < 0: tap += rngLen return further_bytes[13 * blockSize - 21 * rngLen:14 * blockSize - 21 * rngLen] private_keys, socket = open_socket() raw_bytes = b''.join(private_keys) next_private_key = reverse_rng(raw_bytes) signature = subprocess.check_output( ["../go/bin/go", "run", "sol.go", base64.b64encode(next_private_key)]) socket.sendline(signature.strip()) socket.recvuntil("Enter signature: ") print(socket.recvline().decode())
I call the following Go script on line 75 to generate the signature:
package main
import (
"crypto/sha256"
"encoding/base64"
"fmt"
"github.com/dchest/wots"
"math/rand"
"os"
"time"
)
func main() {
var message = []byte("Sign me if you can")
rng := rand.New(rand.NewSource(time.Now().UnixNano()))
var ots = wots.NewScheme(sha256.New, rng)
priv, _ := base64.StdEncoding.DecodeString(os.Args[1])
s, _ := ots.Sign(priv, message)
fmt.Println(base64.StdEncoding.EncodeToString(s))
}
Running the Python script then outputs the flag:
The Z3 model is: sat
Congratulations! Flag: CTF{m4th_RanD_1s_s0_pr3d1cT4bl3}
# Conclusion
This was a really nice challenge, especially when seen as a Z3 exercise. It’s also cool to reverse a real-world RNG that can probably be found in the wild if people forget to use crypto/rand. Also during the challenge, googling for variations of “reversing golang math/rand” didn’t get me anything, so solving this challenge kind of felt like finding a 0-day, which was nice :)
1. Of course this isn’t a vulnerability per se, as the math/rand RNG does not pretend to be cryptographically secure. ↩︎
2. Well, that last sentence isn’t totally true, since we get back the uint64-casted value of this state element. But as we’ll see, it doesn’t matter. ↩︎ | 2022-05-28 01:38:23 | {"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": 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": 0.22308504581451416, "perplexity": 2606.5237338171514}, "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-2022-21/segments/1652663011588.83/warc/CC-MAIN-20220528000300-20220528030300-00535.warc.gz"} |
https://mathoverflow.net/questions/322981/is-restricting-replacement-and-separation-enough-to-make-qi-sigma-n-bi-interp/323034 | # Is restricting Replacement and Separation enough to make $Q+I\Sigma_n$ bi-interpretable with Set Theory?
We have the result that $$\mathsf{ZFCfin}$$, the usual $$\mathsf{ZFC}$$ axioms with the axiom of infinity replaced by its negation, is bi-interpretable with $$\mathsf{PA}$$, first order Peano Arithmetic. We also know of a natural way to weaken $$\mathsf{PA}$$ into fragments, by restricting the induction axiom schema to forumlae of a specific complexity, so $$\mathsf{Q+I\Sigma_3}$$ is the non inductive axioms of $$\mathsf{PA}$$ plus induction restricted to formulae of at most $$\mathsf{\Sigma_3^0}$$ complexity in the language of first order arithmetic.
Does weakening the axiom of separation and the axiom of replacement in $$\mathsf{ZFCfin}$$ result in the above outlined fragments of $$\mathsf{PA}$$? For example, does weakening the two axiom schema to formulae of $$\mathsf{\Sigma_3}$$ complexity in the language of set theory give a set theory bi-interpretable with $$\mathsf{Q+I\Sigma_3}$$? Or does the axiom of powerset also need to be dropped and then replacement replaced with collection?*
If restricting separation and replacement is the correct way of getting bi-interpretable theories, does a theory bi-interpretable with $$\mathsf{Q}$$, Robinson Arithmetic, result from dropping both axiom schemes entirely?
*The reason I say this is because I know that $$\mathsf{KP}$$, Kripke-Platek set theory, does away with powerset and without powerset collection and replacement are not equivalent. I am wondering if that plays a factor and if it isn't too much for one question, if anyone can explain the interplay between getting rid of powerset and the strength of fragments of arithmetic.
First let me note that one should be careful with formulation of $$\mathsf{ZFCfin}$$, for it to be bi-interpretable with $$\mathsf{PA}$$ (see the paper "On interpretations of arithmetic and set theory" by Richard Kaye and Tin Lok Wong and the paper "$$\omega$$-models of finite set theory" by Ali Enayat, James H. Schmerl, and Albert Visser). Basically the issue is that for this bi-interpretation to work fine one either need to explicitly add to $$\mathsf{ZFCfin}$$ the axiom that every set is contained in a transitive set ($$\mathsf{TC}$$), or alternatively start with the axiomatization of $$\mathsf{ZFC}$$ where we have scheme of foundation instead of the axiom of regularity.
For fragments the situation is roughly speaking that the scheme $$\Sigma_n\mbox{-}\mathsf{Sep}$$ in set theory corresponds to the scheme $$\Sigma_n\mbox{-}\mathsf{Ind}$$ in arithmetic and the scheme $$\Sigma_n\mbox{-}\mathsf{Rep}$$ in set theory corresponds to the scheme $$\mathsf{B}\Sigma_n$$ in arithmetic. More formally, let us choose the following base system of set theory $$\mathsf{ZFfin}_1=\mathsf{Ext}+\mathsf{Pair}+\mathsf{Union}+\mathsf{TC}+\mathsf{Reg}+\Sigma_1\mbox{-}\mathsf{Sep}+\Sigma_1\mbox{-}\mathsf{Rep}+\lnot\mathsf{Inf}.$$ Note that although I haven't included powerset axiom in $$\mathsf{ZFfin}_1$$, it is provable there. This system is interpretable in $$\mathsf{I}\Sigma_1$$ by the Ackermann membership $$\in_{\mathsf{Ack}}$$ that is defined as follows: $$n\in_{\mathsf{Ack}} m \mbox{ iff the n-th bit in the binary expansion of m is equal to 1}.$$ In the other direction $$\mathsf{ZFfin}_1$$ interpretes $$\mathsf{I}\Sigma_1$$ by the ordinal arithmetics. With some efforts one could show that this two interpretations form a bi-interpretation. Further, for $$n\ge 1$$, it is easy to show that the theory $$\mathsf{I}\Sigma_n=\mathsf{I}\Sigma_1+\mathsf{I}\Sigma_n$$ proves that $$\Sigma_n\mbox{-}\mathsf{Sep}$$ holds in Ackermann interpretation and that the theory $$\mathsf{ZFfin}_1+\Sigma_n\mbox{-}\mathsf{Sep}$$ proves that $$\Sigma_n\mbox{-}\mathsf{Sep}$$ holds in ordinal arithmetic. This verifies the fact that $$\mathsf{I}\Sigma_n$$ and $$\mathsf{ZFfin}_1+\Sigma_n\mbox{-}\mathsf{Sep}$$ are bi-interpretable. By the same kind of argument $$\mathsf{I}\Sigma_1+\mathsf{B}\Sigma_n$$ and $$\mathsf{ZFfin}_1+\Sigma_n\mbox{-}\mathsf{Rep}$$ are bi-interpretable.
Note that the connection that I have outlined above really required using relatively strong theories: we need totality of exponentiation in arithmetic to prove even very basic facts about $$\in_{\mathsf{Ack}}$$ and due to this the approach wouldn't work if our base set theory wouldn't be able to prove totality of exponentiation in ordinal arithmetic. With a more refined approach it is possible to show bi-interpretability $$\mathsf{I}\Delta_0+\mathsf{Exp}$$ ($$\mathsf{Exp}$$ states totality of binary exponentiation function) and certain set theory that includes powerset axiom (see R. Pettigrew "On Interpretations of Bounded Arithmetic and Bounded Set Theory"). Although, strictly speaking, I don't know whether $$\mathsf{Q}$$ is bi-interpretable with $$\mathsf{ZFCfin}-\mathsf{Sep}-\mathsf{Rep}$$, it would be very strange for it to be the case. | 2021-12-04 05:07:45 | {"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": 44, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.890630841255188, "perplexity": 192.17389317194824}, "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-2021-49/segments/1637964362930.53/warc/CC-MAIN-20211204033320-20211204063320-00216.warc.gz"} |
https://doc.dovecot.org/settings/plugin/fts-plugin/ | # Full Text Search (FTS) plugin¶
## fts_autoindex¶
• Default: no
• Values: Boolean
If enabled, index mail as it is delivered or appended.
## fts_autoindex_max_recent_msgs¶
New in version 2.2.9.
To exclude infrequently accessed mailboxes from automatic indexing, set this value to the maximum number of Recent flagged messages that exist in the mailbox.
Mailboxes with more flagged Recent messages than this value will not be autoindexed, even though they get deliveries or appends. This is useful for, e.g., inactive Junk folders.
Any folders excluded from automatic indexing will still be indexed, if a search on them is performed.
Example Setting:
plugin {
fts_autoindex_max_recent_msgs = 999
}
## fts_autoindex_exclude¶
• Default: <empty>
• Values: String
To exclude a mailbox from automatic indexing, it can be listed in this setting. To exclude additional mailboxes, add sequential numbers to the end of the plugin name. It’s possible to use either mailbox names or refer to them using special-use flags (e.g. Trash)
For example:
fts_autoindex_exclude = \Junk
fts_autoindex_exclude2 = \Trash
fts_autoindex_exclude3 = External Accounts/*
## fts_index_timeout¶
When the full text search backend detects that the index isn’t up-to-date, the indexer is told to index the messages and is given this much time to do so. If this time limit is reached, an error is returned, indicating that the search timed out during waiting for the indexing to complete: NO [INUSE] Timeout while waiting for indexing to finish
Example Setting:
plugin {
fts_index_timeout = 60s
}
## fts_enforced¶
• Default: no
• Values: yes, no or body
Require FTS indexes to perform a search?
If disabled, and searching using FTS fails, Dovecot will fall back on using the built in search which does not have indexes for mail bodies. This may timeout for large mailboxes and/or slow storage.
Value
Search type
FTS index updated
Error handling
New in version
yes
yes
Fail search
New in version v2.2.19.
yes
body
yes
Fail search
New in version v2.2.19.
no
no
Search without FTS: Try to use dovecot.index.cache, or open all mails
New in version v2.2.19.
no
body
yes
Search without FTS by opening all mails
New in version v2.2.19.
body
no
Fail search
New in version v2.3.7.
body
body
yes
Fail search
New in version v2.3.7.
## fts_filters¶
• Default: <none>
The list of filters to apply.
Language specific filter chains can be specified with fts_filters_<lang> (e.g. fts_filters_en).
List of available filters:
lowercase:
Change all text to lower case. Supports UTF8, when compiled with libicu and the library is installed. Otherwise only ASCII characters are lower cased.
stopwords:
Filter certain common and short words, which are usually useless for searching.
Settings: stopwords_dir, path to the directory containing stopword files.
Stopword files are looked up in ”<path>”/stopwords_<lang>.txt.
See FTS languages for list of stopword files that are currently distributed with Dovecot. More can be obtained from the Apache Lucene project or the snowball stemmer source.
Stopword language files are also available from https://github.com/stopwords-iso/.
snowball:
Stemming tries to convert words to a common base form. A simple example is converting “cars” to “car”.
This stemmer is based on the Snowball stemmer library.
See ref:fts_languages
normalizer-icu:
Normalize text using libicu. This is potentially very resource intensive.
Caveat for Norwegian: The default normalizer filter does not modify U+00F8 (Latin Small Letter O with Stroke). In some configurations it might be desirable to rewrite it to e.g. o. Same goes for the upper case version. This can be done by passing a modified “id” setting to the normalizer filter. Similar cases can exists for other languages as well.
Settings: id, description of the normalizing/transliterating rules to use. See http://userguide.icu-project.org/transforms/general#TOC-Transliterator-Identifiers for syntax. Defaults to Any-Lower; NFKD; [: Nonspacing Mark :] Remove; [\\x20] Remove
english-possessive:
Remove trailing 's from english possessive form tokens. Any trailing single ' characters are already removed by tokenizing, whether this filter is used or not.
The snowball filter also removes possessive suffixes from English, so when using snowball, english-possessive is not needed. Snowball quite likely produces better results, so english-possessive is advisable only when snowball is not available or can not be used due to extreme CPU performance requirements.
contractions:
Removes certain contractions that can prefix words. The idea is to only index the part of the token that conveys the core meaning.
Only works with the French language, so the language of the input needs to be recognized by textcat as French.
It filters “qu’”, “c’”, “d’”, “l’”, “m’”, “n’”, “s’” and “t’”.
Do not use at the same time as generic tokenizer with algorithm=tr29 wb5a=yes.
Example:
fts_filters = normalizer-icu snowball stopwords
fts_filters_en = lowercase snowball english-possessive stopwords
## fts_language_config¶
• Default: <textcat dir>
Path to the textcat/exttextcat configuration file, which lists the supported languages. For example /usr/share/libexttextcat/fpdb.conf. This is recommended to be changed to point to a minimal version of a configuration that supports only the languages listed in fts_languages. Doing this improves language detection performance during indexing and also makes the detection more accurate.
## fts_languages¶
• Default: <empty>
A space-separated list of languages that the full text search should detect. At least one language must be specified. The language listed first is the default and is used when language recognition fails.
For better performance it’s recommended to synchronize this setting with the textcat configuration file, see fts_language_config.
The filters used for stemming and stopwords are language dependent.
Example setting:
plugin {
fts_languages = en de
}
### FTS languages¶
Language names are given as ISO 639-1 alpha 2 codes. Stemming support indicates whether the “snowball” filter can be used. Stopwords support indicates whether a stopwords file is distributed with Dovecot. Currently supported languages:
Language Code
Language
Stemming
Stopwords
da
Danish
Yes
Yes
de
German
Yes
Yes
en
English
Yes
Yes
es
Spanish
Yes
Yes
fi
Finnish
Yes
Yes
fr
French
Yes
Yes
it
Italian
Yes
Yes
jp
No
No
nl
Dutch
Yes
Yes
no
Norwegian (Bokmal & Nynorsk detected)
Yes
Yes
pt
Portuguese
Yes
Yes
ro
Romanian
Yes
Yes
ru
Russian
Yes
Yes
sv
Swedish
Yes
Yes
tr
Turkish
Yes
Yes
## fts_tokenizers¶
• Default: generic email-address
The list of tokenizers to use. This setting can be overridden for specific languages by using fts_tokenizers_<lang> (e.g. fts_tokenizers_en).
List of tokenizers:
generic:
Input data, such as email text and headers, need to be divided into words suitable for indexing and searching. The generic tokenizer does this.
Settings:
maxlen: Maximum length of token, before an arbitrary cut off is made.
Defaults to FTS_DEFAULT_TOKEN_MAX_LENGTH. The default is probably OK.
algorithm: Accepted values are simple or tr29. It defines the
method for looking for word boundaries. Simple is faster and will work for many texts, especially those using latin alphabets, but leaves corner cases. The tr29 implements a version of Unicode technical report 29 word boundary lookup. It might work better with e.g. texts containing Katakana or Hebrew characters, but it is not possible to use a single algorithm for all existing languages. The default is simple.
wb5a: Unicode TR29 rule WB5a setting to the tr29 tokenizer. Splits
prefixing contracted words from base word. E.g. “l’homme” → “l” “homme”. Together with a language specific stopword list unnecessary contractions can thus be filtered away. This is disabled by default and only works with the TR29 algorithm. Enable by fts_tokenizer_generic = algorithm=tr29 wb5a=yes.
email-address:
This tokenizer preserves email addresses as complete search tokens, by bypassing the generic tokenizer, when it finds an address. It will only work as intended if it is listed after other tokenizers.
kuromoji:
This tokenizer is used for Japanese text. This tokenizer utilizes Atilika Kuromoji tokenizer library to tokenize Japanese text. This tokenizer also does NFKC normalization before tokenization. What NFKC normalization does is half-width and full-width character normalizations, such as:
• transform half-width Katakana letters to full-width.
• transform full-width number letters to half-width
• transform those special letters (e.g, 1 will be transformed to 1, and 平成 to 平成)
Settings:
maxlen: Maximum length of token, before an arbitrary cut off is made.
The default value for the kuromoji tokenizer is 1024.
kuromoji_split_compounds: This setting enables “search mode” in the
Atilika Kuromoji library. The setting defaults to enabled (i.e .1) and should not be changed unless there is a compelling reason. To disable, set the value to 0. NB If this setting is changed, existing FTS indexes will produce unexpected results. The FTS indexes should be recreated in this case.
id: Description of the normalizing/transliterating rules to use.
See http://userguide.icu-project.org/transforms/general#TOC-Transliterator-Identifiers for syntax. Defaults to “Any-NFKC” which is quite good for CJK text mixed with latix alphabet languages. It transforms CJK characters to full-width encoding and transforms latin ones to half-width. The NFKC transformation is described above. NB In case this setting is changed, existing FTS indexes will produce unexpected results. The FTS indexes should be recreated.
We use the predefined set of stopwords which is recommended by Atilika. Those stopwords are reasonable and they have been made by tokenizing Japanese Wikipedia and have been reviewed by us. This set of stopwords is also included in the Apache Lucene and Solr projects and it is used by many Japanese search implementations.
## fts_tika¶
New in version 2.2.13.
http://tikahost:9998/tika/: This URL needs to be running Apache Tika server (e.g. started with java -jar tika-server/target/tika-server-1.5.jar)
URL for TIKA decoder for attachments.
## fts_decoder¶
New in version 2.1.
Decode attachments to plaintext using this service and index the resulting plaintext. See the decode2text.sh script included in Dovecot for how to use this.
Example on both:
plugin {
fts_decoder = decode2text
fts_tika = http://tikahost:9998/tika/
}
service decode2text {
executable = script /usr/lib/dovecot/decode2text.sh
user = vmail
unix_listener decode2text {
mode = 0666
}
} | 2020-09-22 22:42:39 | {"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": 0.2804347276687622, "perplexity": 10067.216345936135}, "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/1600400208095.31/warc/CC-MAIN-20200922224013-20200923014013-00659.warc.gz"} |
https://www.physicsforums.com/threads/moment-of-inertia.212639/ | Homework Help: Moment of inertia
1. Feb 2, 2008
supercali
[SOLVED] moment of inertia
1. The problem statement, all variables and given/known data
what is the moment of interia of these 2 rods (picture attached) about their center of mass
each rod mass is m with length L
2. Relevant equations
one rod i know is $$\frac{L^2m}{12}$$
3. The attempt at a solution
File size:
2 KB
Views:
99
2. Feb 2, 2008
Staff: Mentor
First things first: Find the center of mass.
3. Feb 2, 2008
supercali
good question
i thought the answer is $$\frac{L^2m}{12}*2+\frac{L^2m}{4}*2$$
obviously it isnt
i think center of mass is at $$\frac{L}{4}\hat{x}-\frac{L}{4}\hat{y}$$
Last edited: Feb 2, 2008
4. Feb 2, 2008
Staff: Mentor
Good--it's right in the middle between the two rod centers. So what's the distance from the center of each rod to that center of mass? To find the rotational inertia of the rods about that center of mass, use the parallel axis theorem.
5. Feb 2, 2008
supercali
yeah i got it it is $$\frac{L^2m}{12}*2+\frac{L^2m}{8}*2$$ and one more thing if it was a square or rods i would have to multiply by 4?
6. Feb 2, 2008
Staff: Mentor
Good.
If it were a square of rods, the distance from each rod to the center of mass would be different. (But there would be a factor of 4 involved.) | 2018-07-17 06:20:11 | {"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": 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": 0.5306119322776794, "perplexity": 1338.8142014641803}, "config": {"markdown_headings": false, "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-2018-30/segments/1531676589573.27/warc/CC-MAIN-20180717051134-20180717071134-00089.warc.gz"} |
https://davetbutler.github.io/publication/2009-10-01-paper-title-number-1 | # How to Simulate it in Isabelle: Towards Formal Verification of Multi-Party Computation
Published in ITP 2017, 2017
Recommended citation: David Butler, David Aspinall, Adria Gascon, . How to Simulate it in Isabelle: Towards Formal Verification of Multi-Party Computation. ITP 2017. https://arxiv.org/pdf/1805.12482.pdf
In this work we provided the first formalisation of MPC in an theorem prover. | 2021-07-24 23:43:59 | {"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": 0.9409849643707275, "perplexity": 10084.987235192319}, "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-2021-31/segments/1627046151531.67/warc/CC-MAIN-20210724223025-20210725013025-00136.warc.gz"} |
https://stats.stackexchange.com/questions/545113/coherence-of-conditional-probabilities | # Coherence of conditional probabilities
Dennis Lindley's paper The Philosophy of Statistics in 2001 includes the following 'simple' example of statistical coherence: "A set of uncertainty statements is said to be coherent if they satisfy the rules of the probability calculus. Thus, the pair of statements p(A|B) = 0.7 and p(A| ~B) = 0.4 do not cohere with the pair p(B|A)= 0.5 and p(B|~ A) = 0.3. (Here ~B denotes the complement of B.) Think of A as a statement about data x and B as a statement about parameter theta. The first pair refers to uncertainties in the data and coheres with the first parameter statement, p(B|A)= 0.5, for data A. (Take p(B) = 0.4/1.1 = 0.36.) But all three do not cohere with the second parameter statement for data A, that p(B| A) = 0.3. With p(B) = 0.36, the coherent value is 0.22."
How is the coherent value of 0.22 calculated?
$$p(B|A)=\frac{p(A|B)p(B)}{p(A|B)p(B))+p(A|\neg B)p(\neg B)} =\frac{p(A|B)}{p(A|B)+p(A|\neg B)p(\neg B)/p(B)}$$ leads to $$0.5=\frac{0.7}{0.7+0.4p(\neg B)/p(B)}$$ hence to $$p(B)=0.4/1.1=0.36$$. This means that $$P(B|\neg A)=\frac{p(\neg A|B)p(B)}{p(\neg A|B)p(B)+p(\neg A|\neg B)p(\neg B)}$$ is equal to $$\frac{0.3\cdot 0.36}{0.3\cdot 0.36+0.6\cdot 0.64}\approx 0.22$$ And using instead $$p(B|\neg A)=\frac{p(\neg A|B)p(B)}{p(\neg A|B)p(B))+p(\neg A|\neg B)p(\neg B)} =\frac{p(\neg A|B)}{p(A|B)+p(\neg A|\sim B)p(\neg B)/p(B)}$$ leads to $$0.3=\frac{0.3}{0.3+0.6p(\neg B)/p(B)}$$ hence to $$p(B)=0.6/1.3\approx 0.46$$. | 2021-10-24 13:27:24 | {"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": 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": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5580862760543823, "perplexity": 995.1393129969099}, "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-2021-43/segments/1634323585997.77/warc/CC-MAIN-20211024111905-20211024141905-00063.warc.gz"} |
https://www.aminer.cn/pub/5f86c64d91e011dbc7eba25e/to-be-robust-or-to-be-fair-towards-fairness-in-adversarial-training | # To be Robust or to be Fair: Towards Fairness in Adversarial Training
Han Xu
Yaxin Li
Cited by: 1|Views30
Abstract:
Adversarial training algorithms have been proven to be reliable to improve machine learning models' robustness against adversarial examples. However, we find that adversarial training algorithms tend to introduce severe disparity of accuracy and robustness between different groups of data. For instance, PGD adversarially trained ResNet1...More
Code:
Data:
Full Text
Bibtex | 2021-05-06 19:40:44 | {"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": 0.8229357004165649, "perplexity": 2632.887521002014}, "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/1620243988759.29/warc/CC-MAIN-20210506175146-20210506205146-00588.warc.gz"} |
https://math.stackexchange.com/questions/770584/troubles-figuring-out-how-to-calculate-the-centroid-for-a-defuzzification-proces | # Troubles figuring out how to calculate the centroid for a defuzzification process
I am trying to implement a fuzzy logic system, but am having serious issues finding the centroid for the defuzzification process.
This is what my output sets look like:
My reference source gives this to me as an example:
This is going to sound silly, but unfortunately I cannot understand why he uses certain values for the calculation, I mean, how does he chose which values from the x axis to multiply with the ones from the y axis. For example, he has at some point (0.75*5), now how come he can't have (0.75*4)? There is no rule telling me I have to calculate every x axis member that is a multiplication of 5, but yet all the examples show me to do this, without really explaining.
A second example I found, tells me to do this:
This is, a much simpler calculation without the values in between. Unfortunately whenever I try to recreate the examples using both these methods, my results aren't matching what they are supposed to.
So to sum it all up, can someone please explain me exactly how you calculate the centroid for a deffuzification process?
Thank you very much for your time.
The first answer is that using ultra-precision on fuzzy calculations is a waste of time. Approximate answers are good enough for fuzzy sets.
In your first example, the author has taken a set of evenly spaced samples and approximated the centroid of the full region with the centroid of the samples (as if the region were not distributed, but were just masses sitting at 0, 5, 10, 15, ...).
In your second example, the author has again approximated. This approximation is not very good. Note that the second example's estimate of the centroid of just the region over $[x_1,x_2]$ is as if all the mass were concentrated at $x_2$.
It is worth pointing out that both of these are approximations, both have drawbacks, and both are generally fine for getting useful results in practical systems. This does not mean that they are the best possible results.
The summation in your second example hints to the correct method. The centroid of the region bounded by $I=[x_1,x_2]$, the function $y(x)$ taking points of $I$ to real numbers, and the $x$-axis is $$\frac{\int_{x_1}^{x_2} x f(x) \mathrm{d}x}{\int_{x_1}^{x_2} f(x) \mathrm{d}x}$$. This is overkill, especially if your functions are piecewise linear (which your output and examples are). For your output set and both examples, it would be better to follow this recipe:
• For each rectangle, replace it with a mass at its (horizontal) midpoint equal to its width times its height. (At $\frac{1}{2}x_2 + \frac{1}{2}x_3$ we have mass $(x_3-x_2)u(x_2)$. We could use $u(x_3)$ as well since they're equal.)
• For each triangle, replace it with a mass two-thirds of the way from the tip to the spine. In the second example, this would be at $\frac{2}{3}x_2 + \frac{1}{3}x_1$. The mass is one half the width ("the base") times the height.
• Decompose each trapezoid (see $[x_3,x_4]$ in the second example) into the rectangle sitting on the $x$-axis and the triangle sitting above the rectangle. Replace the pair with the mass for the triangle and the mass for the rectangle. (I.e., we get two contributions -- one for the rectangle at the midpoint and one for the triangle $2/3$ of the way along.)
Use these locations as the $x_i$ and weights as the $u(x_i)$ in either of your formulae.
This overprecise method computes the exact centroids of piece-wise linear regions.
• Thank you so much! This finally cleared things up to me. But I would like some clarification on the recipe you gave me: basically I just use the mass of the shapes? And how do I add them all up in the end to find the centroid of the whole shape? Apr 28 '14 at 19:30
• Yup. You sum the moment (mass times displacement) of the shapes (then normalize by the total mass). The formula "$g$" in your second example does exactly this. Apr 28 '14 at 20:59
• I am so so sorry. But I've been trying to use that recipe, but my results still are no where near the matlab results. My time is running out so I am starting to become a bit desperate. Firstly, I decomposed the trapezoids into two triangles on the sides with a rectangle in between them, because it made it easier, my assumption is that it wouldn't make a difference. Could I perhaps argue that my results are valid but just different from matlabs because of the precisions used? Apr 29 '14 at 16:21
• What does Matlab get? I get $\frac{1}{2} \cdot 25 \cdot 1$ at $\frac{2}{3}50+\frac{1}{3}25$, $25\cdot 1$ at $\frac{50+75}{2}$, and $\frac{1}{2}\cdot 25 \cdot 1$ at $\frac{2}{3}75+\frac{1}{3}100$. The result would then be $\frac{3125}{50} = 62.5$ for your output set labeled "average". Apr 29 '14 at 20:37
• That is indeed matlab's result. I must be missing something then... Because with what I did, I get 29.47. Lets go by parts. For the places that are labelled insufficient, you did first the rectangle, which is (1/2)⋅25⋅1 and then the triangle which is ((2/3)⋅50 + (1/3)⋅25)⋅1. Now for the average label, I am not sure where you got the 25⋅1, wouldn't it have been ((2/3)⋅50 + (1/3)⋅25)⋅1 again? Because of the second triangle (the left side of the trapezoid). From there on I lose myself. Once more, I am sorry for my troubles. But thank you so much for your time! Apr 30 '14 at 16:46 | 2021-09-22 20:25:46 | {"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": 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": 0.790343701839447, "perplexity": 323.88991187290134}, "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/1631780057388.12/warc/CC-MAIN-20210922193630-20210922223630-00375.warc.gz"} |
https://zenodo.org/record/4319670/export/json | Conference paper Open Access
# An empirical methodology for rating building thermal mass as energy storage system
Antonis Peppas; C. Politi; K. Kollias; M. Taxiarchou; D. Mantelis; L. Karalis
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"description": "<p>Cities are facing unprecedented challenges as the pace of urbanization, the last decades, has faced a significant increase. These challenges appear in several fields, such as supply, exchange, and consumption of energy. At the same time, the increasing demand for electrification by end-use sectors can lead to higher power fluctuations across the daily demand profile. In fact, it is well documented that the building sector demands approximately 40% of primary energy used within the European borders. Managing energy consumption is a multifaceted challenge; introducing grid flexibility and offering various innovative approaches for optimal use in both building and district level are the keys for a sustainable urbanization in the upcoming years.</p>\n\n<p>According to research, Decentralized Energy Systems (DES) using Renewable Energy Sources (RES) for energy production, offer an optimum solution for energy savings and grid flexibility, especially in stand-alone systems. However, due to the intermittent nature of RES and for meeting the load demand at any time, the need for energy storage systems is essential. Building thermal mass is a key parameter to mitigate inside temperature variations. Used with an optimized control strategy, a thermal mass increase is a solution to maintain a better thermal comfort, to stabilize heating and cooling loads and mitigate peak power demand. This study introduces both an efficient and flexible way to rate the thermal storage capability and through that the exploitation potential as a short-term energy storage system, by mainly using room temperature data as well as basic information of the building construction along with its Heating, Ventilation, and Air Conditioning (HVAC) equipment. So far, complex procedures are followed, which require extensive input of historical data, human efforts and time by developing theoretical models on simulation software, used by high experienced personnel. This study presents an empirical methodology for rating and exploiting the building thermal inertia in order to enhance RES penetration by evaluating its performance. The research is based on real data, harvested by an intelligent monitored building in Lavrion Technological and Cultural Park operated solely for research activities. The methodology will provide a tool for real time quantification and evaluation of building thermal mass, which could be integrated to intelligent control algorithms. The whole system through commercial monitoring technologies and Building Energy Management Systems (BEMS) will deliver to the market a low cost, reliable tool for efficient and precise control of the HVAC equipment aiming to maximize RES penetration without compromising occupants’ comfort levels.</p>",
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https://encyclopediaofmath.org/wiki/Well-ordered_set | # Well-ordered set
A set $\mathbb{P}$ equipped with a binary relation $\leq$ that satisfies the following conditions:
1. For any $x,y \in \mathbb{P}$, either $x \leq y$ or $y \leq x$.
2. For any $x,y \in \mathbb{P}$, if $x \leq y$ and $y \leq x$, then $x = y$.
3. For any $x,y,z \in \mathbb{P}$, if $x \leq y$ and $y \leq z$, then $x \leq z$.
4. In any non-empty subset $X \subseteq \mathbb{P}$, there exists an element $a$ such that $a \leq x$ for all $x \in X$.
Thus, a well-ordered set is a totally ordered set satisfying the minimum condition.
The concept of a well-ordered set was introduced by G. Cantor ([1]). An example of a well-ordered set is the naturally ordered set of natural numbers. On the other hand, the interval of real numbers $[0,1]$ with the natural order is not well-ordered. Any subset of a well-ordered set is itself well-ordered. The Cartesian product of a finite number of well-ordered sets is well-ordered by the relation of lexicographic order. A totally ordered set is well-ordered if and only if it contains no subset that is anti-isomorphic to the set of natural numbers.
The smallest element of a well-ordered set $\mathbb{P}$ is denoted by zero (the symbol $0$). For any element $a \in \mathbb{P}$, the set $$[0,a) \stackrel{\text{df}}{=} \{ x \mid x \in \mathbb{P}, x < a \}$$ is called an initial segment of $\mathbb{P}$. For any element $a$ that is not the largest element in $\mathbb{P}$, there exists an element immediately following it; it is usually denoted by $a + 1$. An element of a well-ordered set that has no element immediately preceding it is called a limit element.
The Comparison Theorem. For any two well-ordered sets $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$, one and only one of the following situations occurs: (a) $\mathbb{P}_{1}$ is isomorphic to $\mathbb{P}_{2}$; (b) $\mathbb{P}_{1}$ is isomorphic to an initial segment of $\mathbb{P}_{2}$; or (c) $\mathbb{P}_{2}$ is isomorphic to an initial segment of $\mathbb{P}_{1}$.
If the axiom of choice is included in the axioms of set theory, it may be shown that it is possible to impose on any non-empty set an order relation that converts it into a well-ordered set (i.e., any non-empty set can be well-ordered). This theorem, known as Zermelo’s Well-Ordering Theorem, is in fact equivalent to the axiom of choice. Zermelo’s Well-Ordering Theorem and the Comparison Theorem form the basis for the comparison between cardinalities of sets. Order types of well-ordered sets are called ordinal numbers.
#### References
[1] G. Cantor, “Über unendliche, lineaire Punktmannigfaltigkeiten”, Math. Ann., 21 (1883), pp. 51–58. [2] P.S. Aleksandrov, “Einführung in die Mengenlehre und die Theorie der reellen Funktionen”, Deutsch. Verlag Wissenschaft. (1956). (Translated from Russian) [3] F. Hausdorff, “Grundzüge der Mengenlehre”, Leipzig (1914). (Reprinted (incomplete) English translation: Set theory, Chelsea (1978)) [4] N. Bourbaki, “Elements of mathematics. Theory of sets”, Addison-Wesley (1968). (Translated from French) [5] K. Kuratowski, A. Mostowski, “Set theory”, North-Holland (1968).
In the definition above, Condition (3) (the transitivity of the order relation) is in fact redundant: It follows from the existence of a least element in the subset $\{ x,y,z \}$. | 2022-05-17 05:30:03 | {"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": 0.9536868333816528, "perplexity": 203.1900180968678}, "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/1652662515501.4/warc/CC-MAIN-20220517031843-20220517061843-00006.warc.gz"} |
https://www.azdictionary.com/definition/final%20examination | # final examination definition
• noun:
• A test offered after a period of coursework to find out whether or not the pupil features adequately discovered the material covered in that duration.
• A test provided at the end of a period of training to find out perhaps the student features adequately discovered the materials covered for the reason that duration.
• A test provided at the conclusion of a period of training to find out if the student features adequately discovered the materials covered for the reason that period.
• an examination administered after an academic term
• an examination administered at the end of an academic term
• an examination administered at the end of an academic term
• A test provided at the end of a period of coursework to determine perhaps the pupil features acceptably learned the materials covered in that duration.
• an examination administered at the conclusion of an academic term
• A test offered after a time period of training to find out perhaps the pupil features properly learned the materials covered because period.
• an examination administered at the conclusion of an academic term
• A test given at the end of a period of training to find out if the student has acceptably discovered the materials covered for the reason that duration.
• A test given at the conclusion of a period of training to determine whether the student has adequately learned the material covered for the reason that period.
• A test offered at the end of a period of training to find out perhaps the student has acceptably learned the materials covered in that period.
• an examination administered at the conclusion of an academic term
• an examination administered at the end of an academic term
• an examination administered after an academic term
## Related Sources
• Definition for "final examination"
• A test offered after a period of coursework…
• Quotes for "final examination"
• "I had nearly finished school because…"
• Hypernym for "final examination"
• examination
• Etymologically Related for "final examination"
• final paper
• Form for "final examination"
• final exam | 2017-12-15 06:53:33 | {"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": 0.8303138613700867, "perplexity": 2350.289765585402}, "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/1512948567042.50/warc/CC-MAIN-20171215060102-20171215080102-00430.warc.gz"} |
http://georgehernandez.com/h/xComputers/zVoc/J.asp | Java
Java started out in 1990 as a project headed by James Gosling at Sun Microsystems called Oak. Java is a medium-low level programming language similar to C/C++ but more reliable because of its Oak-origin. Java is "author once, use often", i.e. it is cross-platform since it can be run on any machine that has a Java Virtual Machine or Java enabled browser, eg Netscape Navigator, or a virtual machine or a Java interpreter such as JIT or HotJava.
According to the Sun Microsystems, Java is "a simple, robust, object-oriented, platform-independent multi-threaded, dynamic general-purpose programming environment. It's best for creating applets and applications for the Internet, intranets and any other complex, distributed network." It is a distributed, interpreted, secure, architecture-neutral, portable, and high-performance language.
The best tools for Java development include Symantec's Visual Cafe, Powersoft's NetImpact Studio, Sun's Java Workshop, and Microsoft's Visual J++
Right now Java is competing with Windows CE for the embedded OS market.
Java Server Pages
See JSP.
JavaBean
A Sun Microsystems specification that defines how Java objects interact, especially Java component software. It is also the name for the objects that conform to this specification. JavaBean technology is analogous to Microsoft's ActiveX technology.
Interestingly, JavaBeans can only be developed using Java, but can run on any platform; in contrast, ActiveX can be developed using any programming language but can run only on a Windows platform!
JavaScript
.js. A scripting programming language developed by Netscape Communications as Livescript 1.0. JavaScript and Microsoft's JScript are tied together by ECMAScript, the vendor-neutral version. It is frequently used directly on web pages to provide more features and even execute small programs. It doesn't have to be compiled but can be included directly into an HTML web page and can be interpreted by a JavaScript interpreter like a web browser. This also implies that JavaScript is dynamically bound instead of statically bound like Java, i.e. when a program makes references to other objects, the existence of those objects is checked at compilation for Java but for JavaScript the check is only done at execution. See script.
JavaScript Object Notation
See JSON.
JBOD
Just a Bunch Of Disks Refers to a group of hard disks that are not RAID configured.
JDBC
Java Database Connectivity. Sun Microsystems's protocol that allows Java programs to connect and access any database that is SQL compliant. Since most database management systems (DBMSs) are SQL compliant, and since Java runs on most platforms, Java and JDBC allows the possibility to create a program that can run on different platforms and interact with different DBMSs. See also ODBC.
JDK
Java Development Kit. Sun Microsystems's software development kit (SDK) used to produce Java programs. The JDK supports JavaBeans software components and JDBC.
JET
Joint Engine Technology. The database engine behind Microsoft Access.
JFIF
.jfif. JPEG File Interchange Format. A standard format for exchanging JPEG file between computers.
Jini
(pronounced gee-nee.) Sun software that allows devices to dynamically and interactively connect together. A device attaching to a network via Jini would announce itself, explain its capabilities, and then would be immediately accessible to other devices on the network as well as having access to other devices on the network. Jini allows a network wide sharing of resources, storage, memory, and processing power.
Jini works by passing applets (little programs) back and forth among devices that can run Java.
JK
Just Kidding.
job object
See process.
join
To relate datasheets of information in a database. Two related tables will have a common column (field) that is the key to the relationship.
• Inner joins (aka equi-joins) bring together rows from two tables whose common field match.
• Outer joins also include unmatched rows.
• Left joins include all the rows from the first/left table.
• Right joins include all the rows from the second/right table. See also relationships.
Joint Engine Technology
See JET.
Joint Photographic Experts Group
See JPEG.
JPEG
.jpeg or .jpg. Joint Photographic Experts Group. (Often pronounced "jay-peg".) A graphic file format established by photographers. It does lossy compression (roughly 25:1) of graphic intensive files. It has 24 bit color. A format common on the WWW.
JPG
See JPEG.
JPEG File Interchange Format
See JFIF.
JSON
JavaScript Object Notation. A light-weight, human-readable data exchange format that is also the way that JavaScript creates literal objects.
JSON RFC: 4627. JSON MIME: application/json. JSON basic data types: Number (integer or decimal), String (Unicode with the usual backslash escapes), Boolean, Array (in square brackets), null, and Object (as JSON of course).
JSON essentially consists of name:value pairs delimited by commas and enclosed in curly brackets. EG:
{
"Entity": "Foo",
"Site": "http://www.foo.com/",
"Valid": true,
"Founders": ["Tom","Terry","Bill","George"]
"City": "Chicago",
"State": "IL",
"Zip": 60661
}
}
The above example could easily be made into a JS object like this: var x=eval("("+strJSON+")");. It could then be used like x.Entity, x.Founders[0], or x.Address.Zip.
JSP
Java Server Pages. Sun's equivalent of Microsoft's ASP (Active Server Pages). Where as ASP pages use server side script, JSP calls upon server side Java servlets. | 2017-09-23 16:27:32 | {"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": 0.20465786755084991, "perplexity": 10043.453044244801}, "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-2017-39/segments/1505818689752.21/warc/CC-MAIN-20170923160736-20170923180736-00106.warc.gz"} |
https://www.wikihow.com/Graph-a-Function | How to Graph a Function
A graph of a function is a visual representation of a function's behavior on an x-y plane. Graphs help us understand different aspects of the function, which would be difficult to understand by just looking at the function itself. You can graph thousands of equations, and there are different formulas for each one. That said, there are always ways to graph a function if you forget the exact steps for the specific type of function.
Method 1
Method 1 of 3:
Graphing Linear Equations with Slope
1. 1
Recognize linear functions as simple, easily-graphed lines, like . There is one variable and one constant, written as ${\displaystyle F(x)ory=a+bx}$ in a linear function, with no exponents, radicals, etc. If you've got a simple equation like this, then graphing the function is easy.[1] Other examples of linear functions include:
• ${\displaystyle F(n)=4-2n}$
• ${\displaystyle y=3t-120}$
• ${\displaystyle F(x)={\frac {2}{3}}x+3}$
2. 2
Use the constant to mark your y-intercept. The y-intercept is where the function crosses the y-axis on your graph. In other words, it is the point where ${\displaystyle x=0}$. So, to find it, you simply set x to zero, leaving the constant in the equation alone. For the earlier example, ${\displaystyle y=2x+5}$, your y-intercept is 5, or the point (0,5). On your graph, mark this spot with a dot.[2]
3. 3
Find the slope of your line with the number right before the variable. In your example, ${\displaystyle y=2x+5}$, the slope is "2." That is because 2 is right before the variable in the equation, the "x." Slope is how steep a line is, or how high the line goes before going to the right or left. Bigger slopes mean steeper lines.
4. 4
Break the slope into a fraction. Slope is about steepness, and steepness is simply the difference between movement up and down and movement left and right. Slope is a fraction of rise over run. How much does the line "rise" (go up) before it "runs" (goes to the side)? For the example, the slope of "2" could be read as ${\displaystyle {\frac {2{\text{ }}up}{1{\text{ }}over}}}$.[3]
• If the slope is negative, that means the line goes down as you move to the right.
5. 5
Starting at your y-intercept, follow your "rise" and "run" to graph more points. Once you know your slope, use it to plot out your linear function. Start at your y-intercept, here (0,5), and then move up 2, over 1. Mark this point (1,7) as well. Find 1-2 more points to create an outline of your line.[4]
6. 6
Use a ruler to connect your dots and graph your linear function. To prevent mistakes or rough graphs, find and connect at least three separate points, though two will do in a pinch. This is the graph of your linear equation![5]
Method 2
Method 2 of 3:
Estimating Points on a Graph
1. 1
Determine the function. Get the function of the form like f(x), where y would represent the range, x would represent the domain, and f would represent the function. As an example, we'll use y = x+2, where f(x) = x+2.[6]
2. 2
Draw two lines in a + shape on a piece of paper. The horizontal line is your x axis. The vertical line is your y axis.
3. 3
Number your graph. Mark both the x axis and the y axis with equally-spaced numbers. For the x axis, the numbers are positive on the right side and negative on the left side. For the y axis, the numbers are positive on the upper side and negative on the lower side.[7]
4. 4
Calculate a y value for 2-3 x values. Take your function f(x) = x+2. Calculate a few values for y by putting the corresponding values for x visible on the axis into the function. For more complicated equations, you may want to simplify the function by getting one variable isolated first.[8]
• -1: -1 + 2 = 1
• 0: 0 +2 = 2
• 1: 1 + 2 = 3
5. 5
Draw the graph point for each pair. Simply sketch imaginary lines vertically for each x axis value and horizontally for each y axis value. The point where these lines intersect is a graph point.[9]
6. 6
Remove the imaginary lines. Once you have drawn all the graph points, you can erase the imaginary lines. Note: the graph of f(x) = x would be a line parallel to this one passing through the origin (0,0), but f(x) = x+2 is shifted two units up (along the y-axis) on the grid because of the +2 in the equation.[10]
Method 3
Method 3 of 3:
Graphing Complicated Functions by Hand
1. 1
Understand how to graph common equation types. There are as many different graphing strategies out there as there are types of functions, far too many to completely cover here. If you're struggling, and estimations won't work, check out articles on:
2. 2
Find any zeros first. Zeros, also called x-intercepts, are the points where the graph crosses the horizontal line on the graph. While not all graphs even have zeros, most do, and it is the first step you should take to get everything on track. To find zeros, simply the entire function to zero and solve. For example:
• ${\displaystyle F(x)=2x^{2}-18}$
• Set F(x) equal to zero: ${\displaystyle 0=2x^{2}-18}$
• Solve: ${\displaystyle 0=2x^{2}-18}$
• ${\displaystyle 18=2x^{2}}$
• ${\displaystyle 9=x^{2}}$
• ${\displaystyle x=3,-3}$[11]
3. 3
Find and mark any horizontal asymptotes, or places where it is impossible for the function to go, with a dotted line. This is usually points where the graph does not exist, like where you are dividing by zero. If your equation has a variable in a fraction, like ${\displaystyle y={\frac {1}{4-x^{2}}}}$, start by setting the bottom of the fraction to zero. Any places where it equals zero can be dotted off (in this example, a dotted line at x=2 and x=-2), since you cannot ever divided by zero. Fractions, however, are not the only places you can find asymptotes. Usually, all you need is some common sense:
• Some squared functions, like ${\displaystyle F(n)=n^{2}}$ can never be negative. Thus there is an asymptote at 0.
• Unless you're working with imaginary numbers, you cannot have ${\displaystyle {\sqrt {-1}}}$[12]
• For equations with complex exponents, you may have many asymptotes.
4. 4
Plug in and graph several points. Simply pick a few values for x and solve the function. Then graph the points on your graph. The more complicated the graph, the more points you'll need. In general, -1, 0, and 1 are the easiest points to get, though you'll want 2-3 more on either side of zero to get a good graph.[13]
• For the equation ${\displaystyle y=5x^{2}+6}$, you might plug in -1,0,1, -2, 2, -10, and 10. This gives you a nice range of numbers to compare.
• Be smart selecting numbers. In the example, you'll quickly realize that having a negative sign doesn't matter -- you can stop testing -10, for example, because it will be the same as 10.
5. 5
Map the end behavior of the function to see what happens when it is really huge. This gives you an idea of the general direction of a function, usually as a vertical asymptote. For example -- you know that eventually, ${\displaystyle y=x^{2}}$ gets really, really big. Just one additional "x" (one million vs. one million and one) makes y much bigger. There are a few ways to test end behavior, including:
• Plug in 2-4 large values of x, half negative and half positive, and plot the points.
• What happens if you plugged in "infinity" for one variable? Does the function get infinitely bigger or smaller?
• If the degrees are the same in a fraction, like ${\displaystyle F(x)={\frac {x^{3}}{-2x^{3}+4}}}$, simply divide the first two coefficients (${\displaystyle {\frac {1}{-2}}}$ to get your ending asymptote (-.5).[14]
• If the degrees are different in a fraction, you must divide the equation in the numerator by the equation in denominator by Polynomial Long Division.
6. 6
Connect the dots, avoiding asymptotic and following the end behavior to graph an estimate of the function. Once you have 5-6 points, asymptotes, and a general idea of end behavior, plug it all in to get an estimated version of the graph.[15]
7. 7
Get perfect graphs using a graphing calculator. Graphing calculators are powerful pocket computers that can give exact graphs for any equation. They allow you to search exact points, find slope lines, and visualize difficult equations with ease. Simply input the exact equation into the graphing section (usually a button labeled "F(x) = ") and hit graph to see your function at work.
Community Q&A
Search
• Question
How do I sketch a graph of a square root function?
Donagan
The process is the same as shown in the article above except, of course, it involves calculating (or estimating) the square roots of certain values.
• Question
How do I graph function y = -2 sin(2/3x)?
Donagan
Choose a value for x. Find 2/3 of that value. Then use a trigonometry table to find the sine of that last value. Then multiply the sine by -2. That gives you the value of y that corresponds to the chosen value of x. Do this again for other x values, and you will then have several x-y pairs to form the graph of the function.
200 characters left
Tips
• If you are ever completely lost with what to do, start plugging in points. You could technically graph the entire function like this if you tried infinite combinations of numbers.
⧼thumbs_response⧽
• Graphing calculators are a great way to practice. Try to graph by hand, then use the calculator to get a perfect image of the graph and see how you did.
⧼thumbs_response⧽
Co-authored by:
This article was co-authored by David Jia. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. This article has been viewed 133,493 times.
Co-authors: 13
Updated: March 4, 2023
Views: 133,493
Categories: Algebra | Graphs
Article SummaryX
To graph a function, start by plugging in 0 for x and then solving the equation to find y. Then, mark that spot on the y-axis with a dot. Next, find the slope of the line, which is the number that's right before the variable. Once you know your slope, write it as a fraction over 1 and then use the rise over run to plot the rest of the points from the spot you marked on the y-axis. Finally, use a ruler to draw a line connecting all of the points on your graph. To learn how to graph complicated functions by hand, scroll down! | 2023-03-21 21:32:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 21, "math_alttext": 1, "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": 0.5675862431526184, "perplexity": 619.3225570260968}, "config": {"markdown_headings": false, "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-2023-14/segments/1679296943746.73/warc/CC-MAIN-20230321193811-20230321223811-00646.warc.gz"} |
https://etheses.bham.ac.uk/id/eprint/4300/ | # Nanocarbon-hydride composites for hydrogen storage
Vines, Joshua Edwin (2013). Nanocarbon-hydride composites for hydrogen storage. University of Birmingham. M.Res.
Preview
VinesJ13MRes.pdf
PDF - Redacted Version
## Abstract
Graphite is considered a possible material for solid state hydrogen storage. In this thesis graphite was ball milled for 8hrs under 3 bar H$$_2$$, followed by the addition of LiBH$$_4$$ at a ratio of 2:1 (graphite:LiBH$$_4$$) and milled for a further 2hrs. Characterization of the as milled material showed the formation of a nanocrsytalline/amorphous mixture of graphite and LiBH$$_4$$.
Decomposition was performed by heating to 500°C at 2°Cmin$$^{-1}$$. Graphite milled for 8 hrs released a small amount of hydrogen at 350°C, though no reaction was observed using DSC. The graphite+LiBH$$_4$$ sample, however, exhibited 4 endothermic peaks consistent with a modified LiBH$$_4$$ decomposition. The application of an over pressure of hydrogen during heating of the graphite+LiBH$$_4$$ was found to suppress the decomposition of LiBH$$_4$$.
In-situ XRD measurements demonstrated that Li$$_2$$C$$_2$$ was formed. Li$$_2$$C$$_2$$ is known to be a key decomposition product in making a graphite based system reversible. It was not possible to identify the exact decomposition path due to partial oxidation of the sample during the measurement.
Partial reversibility was observed at non optimized conditions of 500°C under 100bar H$$_2$$ for 20 minutes.
Type of Work: Thesis (Masters by Research > M.Res.)
Award Type: Masters by Research > M.Res.
Supervisor(s):
Supervisor(s)EmailORCID
Book, DavidUNSPECIFIEDUNSPECIFIED
Licence:
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Metallurgy and Materials
Funders: Engineering and Physical Sciences Research Council
Subjects: Q Science > QC Physics
Q Science > QD Chemistry
T Technology > TA Engineering (General). Civil engineering (General)
URI: http://etheses.bham.ac.uk/id/eprint/4300
### Actions
Request a Correction View Item | 2023-03-20 17:09:56 | {"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": 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": 0.5208203792572021, "perplexity": 10555.451216663854}, "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/1679296943484.34/warc/CC-MAIN-20230320144934-20230320174934-00349.warc.gz"} |
https://www.amplifiedparts.com/products/vacuum_tubes?page=31&filters=1a646 | # Vacuum Tubes
The first electronic amplification of sound was done with Vacuum Tubes. We have pre-amp, power and rectifier vacuum tubes among other types in New Old Stock from the golden era of American & European manufacturing as well as current production tubes from today.
#### Vacuum Tubes
The first electronic amplification of sound was done with Vacuum Tubes. We have pre-amp, power and rectifier vacuum tubes among other types in New Old Stock from the golden era of American & European manufacturing as well as current production tubes from today.
Vacuum Tube - 6X5G, Rectifier, Full Wave
The 6X5G is an indirectly heated full-wave rectifiers for use in equipment where the current drain does not exceed 70 mA.
$14.80 Out of Stock Vacuum Tube - 7308 / E188CC, Dual Triode Special quality double triode designed for use as cascode amplifier, cathode follower, etc. in RF and AF circuits.$99.90
Vacuum Tube - 7408 / 6V6GT, Beam Power Amplifier
Beam power pentode primarily designed for use in audio frequency power amplifier applications. The tube is a direct replacement for the 6V6Gt, but features additional controlled zero-bias characteristics.
$79.90 Vacuum Tube - 7581A / KT66, Beam Power Amplifier Beam-power pentode primarily designed for use in audio-frequency power-amplifier applications. The 7581A is unilaterally interchangeable with the 7581 and the 6L6GC.$99.90
Vacuum Tube - 809, Triode, Power Amplifier
Transmitting triode
$49.90 Vacuum Tube - 81, Rectifier, ST Glass Half-wave high-vacuum rectifier which may be used with either choke or condenser input to the filter.$27.35
Out of Stock
Vacuum Tube - 8233 / E55L, Pentode, Power Amplifier
Special quality pentode designed for use as a wide band output tube.
$51.70 Out of Stock Vacuum Tube - 826, Triode, Power Amplifier Transmitting triode for operation at maximum ratings up to 250 Mc.$48.80
Vacuum Tube - 841, Triode, Power Amplifier
The 841 is an RF power amplifier, oscillator, and AF voltage amplifier.
$69.90 Vacuum Tube - 8950, Pentode, Power Amplifier Compactron beam power pentode primarily designed for RF power output applications. Features of the 8950 are dual cathode and grid connections for lower lead inductance, and a 13 volt heater.$199.90
Vacuum Tube - EL84 / 6BQ5, Amperex Bugle Boy, Pentode, Power
Amperex Bugle Boy, Pentode, Power
$149.90 Vacuum Tube - EL84 / 6BQ5, Mullard, Pentode, Power The EL84 is an output pentode rated for 12W anode dissipation, primarily intended for use in a.c. mains operated equipment.$149.90
Vacuum Tube - EL84 / 6BQ5, Telefunken, Pentode, Power
Telefunken, Pentode, Power
$199.90 Out of Stock Vacuum Tube - 6MJ6, Beam Power Pentode Novar type beam power tube used as a horizontal-deflection amplifier in color and black-and-white television receivers.$99.90
Vacuum Tube - 7027, Sovtek
Sovtek 7027 beam pentode power tube. Made in Russia.
Replacement for all original 7027's.
$23.95 Vacuum Tube - 25CD6G, Beam Pentode New Old Stock (N.O.S.) tube. 25CD6GB is identical to the 6CD6GA except with different heater ratings. Beam power tube, octal base.$8.05
Vacuum Tube - 7189, Tung-Sol Reissue
Tung-Sol has reissued the rare 7189, a high fidelity audio specific variation on the EL84 tube. In addition to the increased fidelity, the 7189 features a higher maximum plate voltage rating of 400V. These tubes are great for both guitar and HiFi amplifiers and can be used in any EL84 or 6BQ5 position.
Starting at $29.95 Vacuum Tube - 12AX7B, China 9 pin miniature preamp tube (Amplification Factor = 100). The Chinese 12AX7B is very similar in tone to its standard Chinese cousin, but the Bs seem to average in at slightly lower gain and noise.$11.25
Vacuum Tube - 5751, JJ Electronics
A dual triode that can be used in place of most 12AX7 circuits. The rugged build of this tube and reduced gain (30% less than a 12AX7) can improve headroom and give better control in high gain stages allowing users to more easily dial in sweet spots. The JJ 5751 features a smooth and balanced tone and a detailed response with low noise and low microphonics.
$13.95 Vacuum Tube - E83CC Frame Grid, JJ Electronics Alternative to the 12AX7 or ECC83. The JJ Electronic E83CC is a high quality dual triode that rivals the NOS version of this tube. The E83CC was originally designed as an audio specific dual triode with very low noise and microphonics. The JJ E83CC stays true to the original design with the signature thick frame grid and boxed plate. The rigid frame allows for tighter tolerances and closer construction which results in lower noise and increased stability. The JJ Electronic E83CC is a great choice for high fidelity amplifiers as well as guitar amps and studio equipment. The E83CC has the same pinout and amplification factor as a standard 12AX7 and can be used in any 12AX7 or ECC83 position. Guitarist Description: This frame grid tube has an incredibly rich yet balanced tone. The E83CC works well for clean sounds but also breaks up with a satisfying warmth and plenty of articulation.$21.95
Don't see what you're looking for? Send us your product suggestions! | 2020-02-19 01:12:45 | {"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": 0.2115056812763214, "perplexity": 12519.661987356634}, "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-2020-10/segments/1581875143963.79/warc/CC-MAIN-20200219000604-20200219030604-00037.warc.gz"} |
http://math.stackexchange.com/questions/200856/differential-inverse-matrix | # differential inverse matrix
How to show that the application $$f:U=GL(\mathbb{R}^{2})\subset\mathbb{R}^{n^{2}}\longrightarrow \mathbb{R}^{n^{2}},$$ defined by $$f(A)=A^{-1}$$ is differentiable and and its derivative at point $A\in U$ is the linear transformation $$f'(A):\mathbb{R}^{n^{2}}\longrightarrow \mathbb{R}^{n^{2}},$$ defined by $f'(A)\cdot V=-A\cdot V\cdot A^{-1}$.
-
Fix $N$ a sub-multiplicative norm on $\Bbb R^{n^2}$. We have $$f(A+H)-f(A)=(A+H)^{-1}-A^{-1}=(A(I+A^{—1}H))^{—1}-A^{—1}=\left((I+A^{—1}H)^{—1}-I\right)A^{—1}.$$ This gives, for $N(H)<\frac 1{2N(A^{—1})}$, \begin{align} f(A+H)-f(A)+A^{-1}HA^{—1}&=\left((I+A^{—1}H)^{—1}-I+A^{—1}H\right)A^{-1}\\ &=\left(\sum_{j=0}^{+\infty}(-1)^j(A^{—1}H)^j-I+A^{—1}H\right)A^{—1}\\ &=\sum_{j=2}^{+\infty}(-1)^j(A^{—1}H)^jA^{-1}, \end{align} hence \begin{align} N(f(A+H)-f(A)+A^{—1}HA^{—1})&\leq \sum_{j=2}^{+\infty}N(A^{-1})^jN(H)^jN(A^{—1})\\ &=N(H)^2\sum_{k\geq 0}N(A^{—1})^{k+3}N(H)^k\\ &\leq N(H)^2N(A^{—1})^3\frac 1{1-1/2}\\ &=2N(H)^2N(A^{—1})^3. \end{align} This proves that $f'(A)\cdot H=-A^{-1}HA^{—1}$.
-
I can't see how the $j=1$ term cancels, but I think this is because the result should be $f'(A).H=-A^{-1}HA^{-1}$: think about the case $n=1$! This form for $f'$ yields the required cancellation at $j=1$. – user12477 Sep 22 '12 at 20:50
@user12477 You are perfectly right. I've edited. – Davide Giraudo Sep 22 '12 at 20:52 | 2016-02-10 09:21:17 | {"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": 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": 0.9928480982780457, "perplexity": 933.3573335830557}, "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-2016-07/segments/1454701159031.19/warc/CC-MAIN-20160205193919-00226-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://jacoxu.com/statistical-pattern-recognition-chapter-1-2-notes/ | # Statistical Pattern Recognition Chapter 1-2 Notes
Statistical Pattern Recognition Chapter 1-2 Notes:
1.4 Approaches to statistical pattern recognition
Some important points to make about the design process: a). Finite design set, (overfit — underfit) -> generalization performance; b). Optimality; c). Representative data.
Supervised classification (or discrimination), unsupervised classification (sometimes in the statistics literature simply referred to as classification or clustering)
Decision rule, decision boundaries or decision surfaces .
Two approaches to discrimination: The first assumes a knowledge of the underlying class-conditional probability density functions (the probability density function of the feature vectors for a given class); the second approach introduced in the next section develops decision rules that use the data to estimate the decision boundaries directly, without explicit calculation of the probability density functions.
1.5.1 Bayes’ decision rule for minimum error
A decision rule based on probabilities is to assign $${\bf{x}}$$ (here we refer to an object in terms of its measurement vector) to class $${\omega _j}$$ if the probability of class $${\omega _j}$$ given the observation $${\bf{x}}$$, that is $$p({\omega _j}|{\bf{x}})$$, is greatest over all classes $${\omega _1}$$, …, $${\omega _C}$$.
Bayes’ rule for minimum error: $$p({\bf{x}}|{\omega _j})p({\omega _j}) > p({\bf{x}}|{\omega _k})p({\omega _k})\begin{array}{*{20}{c}} {}&{k = 1,…,C;k \ne j} \end{array}$$.
a posteriori probabilities, a priori probabilities.
1.5.2 Bayes’ decision rule for minimum error – reject option
As we have stated above, an error or misrecognition occurs when the classifier assigns a pattern to one class when it actually belongs to another. In this section we consider the reject option. Usually it is the uncertain classifications (often close to the decision boundaries) that contribute mainly to the error rate. Therefore, rejecting a pattern (withholding a decision) may lead to a reduction in the error rate. This rejected pattern may be discarded, or set aside until further information allows a decision to be made. Although the option to reject may alleviate or remove the problem of a high misrecognition rate, some otherwise correct classifications are also converted into rejects. Here we consider the trade-offs between error rate and reject rate.
1.5.3 Bayes’ decision rule for minimum risk
Bayes’ rule developed assumes that the a priori distributions and the class-conditional distributions are known. In a real-world task, this is unlikely to be so. Therefore approximations must be made based on the data available.
[参考] 1. Statistical Pattern Recognition.[M] Webb. 2011 | 2018-12-19 07:05:16 | {"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": 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": 0.7616409063339233, "perplexity": 1271.701437687638}, "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-2018-51/segments/1544376831715.98/warc/CC-MAIN-20181219065932-20181219091932-00501.warc.gz"} |
http://mymathforum.com/number-theory/344627-euler-s-totient-little-trick.html | My Math Forum Euler's Totient Little Trick
Number Theory Number Theory Math Forum
August 1st, 2018, 04:36 AM #1 Member Joined: Aug 2015 From: Montenegro (Podgorica) Posts: 37 Thanks: 4 Euler's Totient Little Trick Euler's totient function uses this formula to count the numbers which are relatively prime to $n$: $$\varphi(n)=n \prod_{p|n} \bigg(1-\frac{1}{p}\bigg)$$ example: $n=50=2\cdot5\cdot5$ $$\varphi(50)=50\cdot\bigg(1-\frac{1}{2}\bigg)\cdot\bigg(1-\frac{1}{5}\bigg)=(\cancel{2}\cdot \cancel{5} \cdot5) \cdot \frac{1}{\cancel{2}} \cdot \frac{4}{\cancel{5}}=5\cdot1\cdot4=20$$ since $$\bigg(1-\frac{1}{p}\bigg)=\frac{p-1}{p}$$ every prime factor $p_i^1$ with power $1$ will just be $p_i-1$ and prime factors with power greater than $1$, $p^k$ can be written as $p^1 \cdot p^{k-1}$ and result of a totient function for that prime will be $(p-1) \cdot p^{k-1}$. So in example above we have $50=2^1\cdot5^2$, $\varphi(50)=(2-1)\cdot(5^1-1)\cdot5^{2-1}=1\cdot4\cdot5=20$ another example, number $1350=2^1\cdot3^3\cdot5^2$ $\varphi(1350)=(2-1)\cdot(3-1)\cdot3^{3-1}\cdot(5-1)\cdot5^{2-1}=1\cdot(2\cdot3^2)\cdot(4\cdot5^1)=360$ Thanks from 2sly4U
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http://cran.ma.ic.ac.uk/web/packages/FielDHub/vignettes/partially_replicated.html | # Partially Replicated Design
This vignette shows how to generate a partially replicated design using both the FielDHub Shiny App and the scripting function partially_replicated() from the FielDHub R package.
## Overview
Partially replicated designs are commonly employed in early generation field trials. This type of design is characterized by replication of a portion of the entries, with the remaining entries only appearing once in the experiment. Commonly, the part of treatments with reps is due to an arbitrary decision by the research, also in some cases, it is due to technical reasons. In this design, it is recommended to replicate at least 1/3 of the treatments.
In this design you can set the number of entries that will have reps, as well as the number of entries that will only appear once. You can also choose to run the same experiment over multiple locations.
## Use case
Consider a plant breeding field trial with 300 plots containing 75 entries appearing two times each, and 130 entries only appearing once. This field trial is arranged in a field of 15 rows by 20 columns. In this case, the breeder decided to replicate the genotypes that do not share significant generic information with each other (75), as well as leave with just one copy the genotypes that are siblings (130).
## Running the Shiny App
To launch the app you need to run either
FielDHub::run_app()
or
library(FielDHub)
run_app()
## 1. Using the FielDHub Shiny App
Once the app is running, go to the tab Partially Replicated Design
Then, follow the following steps where we will show how to generate a partially replicated design.
## Inputs
1. Import entries’ list? Choose whether to import a list with entry numbers and names for genotypes or treatments.
• If the selection is No, that means the app is going to generate synthetic data for entries and names of the treatment/genotypes based on the user inputs.
• If the selection is Yes, the entries list must fulfill a specific format and must be a .csv file. The file must have the columns ENTRY, NAME, and REPS. The ENTRY column must have a unique entry integer number for each treatment/genotype. The column NAME must have a unique name that identifies each treatment/genotype. The REPS column must have an integer number for the replications of the groups. Both ENTRY and NAME must be unique, duplicates are not allowed. In the following table, we show an example of the entries list format. This example has an entry list with four treatments/genotypes that will appear twice and 8 that appear just once.
ENTRY NAME REPS
1 GenotypeA 2
2 GenotypeB 2
3 GenotypeC 2
4 GenotypeD 2
5 GenotypeE 1
6 GenotypeF 1
7 GenotypeG 1
8 GenotypeH 1
9 GenotypeI 1
10 GenotypeJ 1
11 GenotypeK 1
12 GenotypeL 1
1. Enter the number of entries per replicate group in the # of Entries per Rep Group box as a comma separated list. In our example we will have 2 groups with 85 and 130 entries. So, we enter 75, 130 in the box for our sample experiment.
2. Enter the number of replications per group in the # of Rep per Group box. In our example we will have 2 and 1 replications for the 2 groups, so we enter 2, 1 in this box.
3. Enter the number of locations in Input # of Locations. We will run this experiment over a single location, so set Input # of Locations to 1.
4. Select serpentine or cartesian in the Plot Order Layout. For this example we will use the default serpentine layout.
5. To ensure that randomizations are consistent across sessions, we can set a seed number in the box labeled Seed Number. In this example, we will set it to 1245.
6. Enter the starting plot number in the Starting Plot Number box. If the experiment has multiple locations, you must enter a comma separated list of numbers the length of the number of locations for the input to be valid.
7. Once we have entered the information for our experiment on the left side panel, click the Run! button to run the design.
8. You will then be prompted to select the dimensions of the field from the list of options in the drop down in the middle of the screen with the box labeled Select dimensions of field. In our case, we will select 15 x 20.
9. Click the Randomize! button to randomize the experiment with the set field dimensions and to see the output plots. If you change the dimensions again, you must re-randomize.
If you change any of the inputs on the left side panel after running an experiment initially, you have to click the Run and Randomize buttons again, to re-run with the new inputs.
## Outputs
After you run a single diagonal arrangement in FielDHub and set the dimensions of the field, there are several ways to display the information contained in the field book. The first tab, Get Random, shows the option to change the dimensions of the field and re-randomize, as well as a reference guide for experiment design.
### Data Input
On the second tab, Data Input, you can see all the entries in the randomization in a list, as well as a table of the checks with the number of times they appear in the field. In the list of entries, the reps for each check is included as well.
### Randomized Field
The Randomized Field tab displays a graphical representation of the randomization of the entries in a field of the specified dimensions. The checks are the green colored cells, with the The display includes numbered labels for the rows and columns. You can copy the field as a table or save it directly as an Excel file with the Copy and Excel buttons at the top.
### Plot Number Field
On the Plot Number Field tab, there is a table display of the field with the plots numbered according to the Plot Order Layout specified, either serpentine or cartesian. You can see the corresponding entries for each plot number in the field book. Like the Randomized Field tab, you can copy the table or save it as an Excel file with the Copy and Excel buttons.
### Field Book
The Field Book displays all the information on the experimental design in a table format. It contains the specific plot number and the row and column address of each entry, as well as the corresponding treatment on that plot. This table is searchable, and we can filter the data in relevant columns.
## 2. Using the FielDHub function: partially_replicated().
You can run the same design with the function partially_replicated() in the FielDHub package.
First, you need to load the FielDHub package typing,
library(FielDHub)
Then, you can enter the information describing the above design like this:
prep <- partially_replicated(
nrows = 15,
ncols = 20,
repGens = c(75,150),
repUnits = c(2,1),
planter = "serpentine",
plotNumber = 101,
l = 1,
exptName = "Expt1",
locationNames = "PALMIRA",
seed = 1245,
)
#### Details on the inputs entered in optimized_arrangement() above
The description for the inputs that we used to generate the design,
• nrows = 15 is the number of rows in the field.
• ncols = 20 is the number of columns in the field.
• repGens = c(75,150) are the values for the groups to replicate
• repUnits = c(2,1) are the values for representing respective replicates of each group.
• planter = "serpentine" is the layout order.
• plotNumber = 101 is the starting plot number for the experiment.
• l = 1 is the number of locations.
• exptName = "Expt1" is an optional name for experiment.
• locationNames = "PALMIRA" is the optional name for the locations.
• seed = 1245 is the seed number to replicate identical randomizations.
### Access to prep output
The partially_replicated() function returns a list consisting of all the information displayed in the output tabs in the FielDHub app: design information, plot layout, plot numbering, entries list, and field book. These are Accessible by the $ operator, i.e. prep$layoutRandom or prep$fieldBook. prep$fieldBook is a list containing information about every plot in the field, with information about the location of the plot and the treatment in each plot. As seen in the output below, the field book has columns for ID, EXPT, LOCATION, YEAR, PLOT, ROW, COLUMN, CHECKS, ENTRY, and TREATMENT.
Let us see the first 10 rows of the field book for this experiment.
field_book <- prep\$fieldBook
head(field_book, 10)
ID EXPT LOCATION YEAR PLOT ROW COLUMN CHECKS ENTRY TREATMENT
1 1 Expt1 PALMIRA 2022 101 1 1 49 49 G49
2 2 Expt1 PALMIRA 2022 102 1 2 47 47 G47
3 3 Expt1 PALMIRA 2022 103 1 3 15 15 G15
4 4 Expt1 PALMIRA 2022 104 1 4 0 146 G146
5 5 Expt1 PALMIRA 2022 105 1 5 59 59 G59
6 6 Expt1 PALMIRA 2022 106 1 6 5 5 G5
7 7 Expt1 PALMIRA 2022 107 1 7 0 136 G136
8 8 Expt1 PALMIRA 2022 108 1 8 29 29 G29
9 9 Expt1 PALMIRA 2022 109 1 9 0 102 G102
10 10 Expt1 PALMIRA 2022 110 1 10 0 177 G177
### Plot field layout
For plotting the layout in function of the coordinates ROW and COLUMN in the field book object we can use the generic function plot() as follow,
plot(prep)
In the figure above, green plots contain replicated entries, and gray plots contain entries that only appear once. | 2022-09-25 11:26:34 | {"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": 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": 0.3364614248275757, "perplexity": 1472.8140072733345}, "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/1664030334528.24/warc/CC-MAIN-20220925101046-20220925131046-00514.warc.gz"} |
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=vuu&paperid=324&option_lang=eng | Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki
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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki: Year: Volume: Issue: Page: Find
Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, Issue 2, Pages 84–99 (Mi vuu324)
MATHEMATICS
On Volterra type generalization of monotonization method for nonlinear functional operator equations
A. V. Chernov
Department of Mathematical Physics, Nizhni Novgorod State University, Nizhni Novgorod, Russia
Abstract: Let $n,m,\ell,s\in\mathbb N$ be given numbers, $\Pi\subset\mathbb R^n$ be a set measurable by Lebesgue and $\mathcal{X,Z}$ be some Banach ideal spaces of functions measurable on $\Pi$. We consider a nonlinear operator equation of the form as follows:
$$x=\theta+AF[x],\quad x\in\mathcal X^\ell, \tag{1}$$
where $A\colon\mathcal Z^m\to\mathcal X^\ell$ is bounded linear operator, $F\colon\mathcal X^\ell\to\mathcal Z^m$ is some operator. Equation (1) is a natural form of lumped and distributed parameter systems from a wide enough class. Formerly, by V. P. Polityukov it was suggested monotonization method for justification of solvability of equation (1) and obtaining pointwise estimations for solutions. The matter of this method consisted in that solvability of equation (1) was proved (besides other conditions) under following: I) operator $F$ allows some correction of the form $G=\lambda I$ to monotone operator $\mathcal F[x]=F[\theta+x]+G[x]$ such that II) $(I+A G)^{-1}A\geq0$ ($\lambda>0$, $I$ is identity operator). As our examples show, conditions I) and II) may be contradictory to each other, that narrows a sphere of application of the method. The main result of the paper is that for the case of operator $A$, possessing the Volterra property, which is natural for evolutionary equations, the requirement I) of ability to be monotonized can be replaced by the requirement of some upper and lower estimates for operator $F$ on some cone segment through linear operator $G$ and additional fixed element. We prove that for global solvability of a boundary value problem associated with a semilinear evolutionary equation it is sufficient that analogous boundary value problem associated with linear equation, derived from the original equation by estimating of a right-hand side on some cone segment, have a positive solution. The application of results obtained is illustrated by Goursat–Darboux system, Cauchy problem associated with wave equation and first boundary value problem associated with diffusion equation.
Keywords: nonlinear operator equation, solvability, monotonization method, Volterra property.
Full text: PDF file (279 kB)
References: PDF file HTML file
UDC: 517.988.63
MSC: 47J05, 47J35
Citation: A. V. Chernov, “On Volterra type generalization of monotonization method for nonlinear functional operator equations”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 2, 84–99
Citation in format AMSBIB
\Bibitem{Che12} \by A.~V.~Chernov \paper On Volterra type generalization of monotonization method for nonlinear functional operator equations \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2012 \issue 2 \pages 84--99 \mathnet{http://mi.mathnet.ru/vuu324} \elib{https://elibrary.ru/item.asp?id=17790056}
• http://mi.mathnet.ru/eng/vuu324
• http://mi.mathnet.ru/eng/vuu/y2012/i2/p84
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles
This publication is cited in the following articles:
1. A. V. Chernov, “Ob $\varepsilon$-ravnovesii v beskoalitsionnykh funktsionalno-operatornykh igrakh so mnogimi uchastnikami”, Tr. IMM UrO RAN, 19, no. 1, 2013, 316–328
2. A. V. Chernov, “Uniformly continuous dependence of a solution to a controlled functional operator equation on a shift of control”, Russian Math. (Iz. VUZ), 57:5 (2013), 29–41
3. A. V. Chernov, “Ob upravlyaemosti nelineinykh raspredelennykh sistem na mnozhestve konechnomernykh approksimatsii upravleniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2013, no. 1, 83–98
4. A. V. Chernov, “On the structure of a solution set of controlled initial-boundary value problems”, Russian Math. (Iz. VUZ), 60:2 (2016), 62–71
5. A. V. Chernov, “On a majorant-minorant criterion for the total preservation of global solvability of distributed controlled systems”, Differ. Equ., 52:1 (2016), 111–121
6. A. V. Chernov, “On total preservation of solvability for a controlled Hammerstein type equation with non-isotone and non-majorized operator”, Russian Math. (Iz. VUZ), 61:6 (2017), 72–81
7. A. V. Chernov, “Differentiation of the functional in a parametric optimization problem for a coefficient of a semilinear elliptic equation”, Differ. Equ., 53:4 (2017), 551–562
• Number of views: This page: 342 Full text: 108 References: 35 First page: 1 | 2021-12-07 00:27:15 | {"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": 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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.2902894914150238, "perplexity": 7753.738584456481}, "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-2021-49/segments/1637964363327.64/warc/CC-MAIN-20211206224536-20211207014536-00177.warc.gz"} |
https://math.stackexchange.com/questions/419072/finding-the-cumulative-distribution-function-of-the-minimum-of-a-sample | # Finding the cumulative distribution function of the minimum of a sample
The Weibull pdf is given as the following.
$$f(x) = \begin{cases} \frac{1}{\alpha}mx^{m-1}e^{-x^m/\alpha} \quad \text{if } x > 0 \\ 0 \quad \text{else} \end{cases}$$
If a random sample of size $n$ is taken from a Weibull distribution, what is the pdf and cdf of the minimum of the sample, $Y_i$?
I realize $g_1(y_1) = n[1-F(y_1)]^{n-1}f(y_1)$; however how do I find the cdf $F(y_1)$? I tried integrating the pdf from 0 to $y$ but am having difficulty.
Let $X_1,X_2,\dots,X_n$ be iid Weibull, and let $W$ be the minimum of the $X_i$.
Then $W\gt w$ if and only if all the $X_i$ are $\gt w$. This has probability $(1-F_X(w))^n$, where $F_X$ is the cdf of one of the Weibulls. Thus the cdf of $W$ is given by $$F_W(w)=1-\Pr(W\gt w)=1-(1-F_X(w))^n$$ (for positive $w$).
To find $1-F_X(w)$, we need to calculate $$\int_w^\infty \frac{1}{\alpha}mx^{m-1}\exp(-x^m/\alpha)\,dx.$$ Make the substitution $\frac{x^m}{\alpha}=t$. Then $dt=\frac{1}{\alpha}mx^{m-1}\,dx$, and the integral becomes the easy $$\int_{t=w^m/\alpha}^\infty e^{-t}\,dt.$$
• I have shown you how to find the cdf (cumulative distribution function). To find the probability density function, differentiate. We get $nf_X(w)(1-F_X(w))^{n-1}$. But your post seemed to indicate you knew that. – André Nicolas Jun 13 '13 at 6:42 | 2019-07-21 20:58:26 | {"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": 0.9796772003173828, "perplexity": 82.42779751811658}, "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-2019-30/segments/1563195527204.71/warc/CC-MAIN-20190721205413-20190721231413-00090.warc.gz"} |
https://labs.tib.eu/arxiv/?author=Oph%C3%A9lia%20Fabre | • ### Primordial magnetogenesis before recombination(1504.07853)
Jan. 21, 2016 gr-qc, astro-ph.CO, astro-ph.GA
The origin of large magnetic fields in the Universe remains currently unknown. We investigate here a mechanism before recombination based on known physics. The source of the vorticity is due to the changes in the photon distribution function caused by the fluctuations in the background photons. We show that the magnetic field generated in the MHD limit, due to the Coulomb scattering, is of the order $10^{-49}$~G on a coherence scale of 10 kpc. We explicitly show that the magnetic fields generated from this process are sustainable and are not erased by resistive diffusion. We compare the results with current observations and discuss the implications. Our seed magnetic fields are generated on small scales whereas the main mechanisms studied in the literature are on scale bigger than 1 Mpc. However, compared to more exotic theories generating seed magnetic fields on similar scales, the strength of our fields are generally smaller.
• ### Cosmological perturbation of Unimodular Gravity and General Relativity are identical(1511.01805)
Nov. 5, 2015 hep-ph, gr-qc, astro-ph.CO
Unimodular Gravity (UG) is a restricted version of General Relativity (GR) in which the determinant of the metric is a fixed function and the field equations are given by the trace-free part of the full Einstein equations. The background equations in UG and GR are identical. It was recently claimed that, the first order contribution in the temperature fluctuation of the Cosmic Microwave Background (CMB) in UG is different from GR. In this work, we calculate the first order perturbation equations in UG and show that the Sachs-Wolfe effect in UG, in terms of gauge invariant variables, is identical to GR. We also show that the second order perturbation equation of Mukhnanov-Sasaki variable in UG, is identical to GR. The only difference comes from the gauge choices due the constraint on the metric determinant. Hence, UG and GR are identical and indistinguishable in CMB data on large scales.
• ### Topology beyond the horizon: how far can it be probed?(1311.3509)
June 8, 2015 astro-ph.CO
The standard cosmological model does not determine the spatial topology of the universe. This article revisits the signature of a non-trivial topology on the properties of the cosmic microwave background anisotropies. We show that the correlation function of the coefficients of the expansion of the temperature and polarization anisotropies in spherical harmonics, encodes a topological signature that can be used to distinguish a multi-connected space from an infinite space on sizes larger than the last scattering surface. The effect of the instrumental noise and of a galactic cut are estimated. We thus establish boundaries for the size of the biggest torus dintinguisable with temperature and polarization CMB data. We also describe the imprint of the spatial topology on the 3-point function and on non-Gaussianity. | 2019-11-15 04:14:35 | {"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": 0.653670072555542, "perplexity": 526.6547895385814}, "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-2019-47/segments/1573496668561.61/warc/CC-MAIN-20191115015509-20191115043509-00144.warc.gz"} |
http://etheses.bham.ac.uk/3696/ | eTheses Repository
# Mixed anion amides for hydrogen storage
Hewett, David R. (2012)
Ph.D. thesis, University of Birmingham.
PDF (15Mb)Accepted Version
## Abstract
Metal hydride materials have attracted much interest for their potential use as hydrogen storage materials. Complex hydrides are amongst the most promising due to their high gravimetric storage capacities and favourable de/rehydrogenation conditions. Here, mixed anion complex hydrides are investigated both through halide doping of LiNH$$_2$$ and Li$$_2$$NH, and though a mixed LiNH$$_2$$-LiBH$$_4$$ system.
The reaction of LiNH$$_2$$ and Li$$_2$$NH with lithium or magnesium chloride, bromide and iodide has been shown to form a series of amide- and imide-halide phases. The structures of these phases were investigated through powder diffraction methods as well as Raman spectroscopy. The hydrogen releasing properties of these materials were investigated through reaction with LiH and MgH$$_2$$; while the equivalent hydrogenation reactions were also tested. In both cases these materials performed more favourably than the pure LiNH$$_2$$-LiH system. The lithium ion conductivity of these materials was also investigated; it was shown that the most conducting materials were also the quickest to release and uptake hydrogen.
The LiNH$$_2$$-LiBH$$_4$$ system was studied, with particular focus on the decomposition product, Li$$_3$$BN$$_2$$. All three known polymorphs of this compound were shown to be able to form by the reaction of $$_2$$LiNH$$_2$$ + LiBH$$_4$$ by carefully changing the reaction conditions. Further stages of this system were investigated through reaction of Li$$_2$$NH and Li$$_3$$N with LiBH$$_4$$. Here the products from these reactions were studied along with the thermal desorption properties of the systems.
Type of Work: Ph.D. thesis. Anderson, P.A. (Paul Alexander) (1965-) Colleges (2008 onwards) > College of Engineering & Physical Sciences School of Chemistry QC PhysicsQD Chemistry University of Birmingham 3696
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
Repository Staff Only: item control page | 2016-04-29 14:04:52 | {"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": 0.3318829834461212, "perplexity": 5115.714079906954}, "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-2016-18/segments/1461860111365.36/warc/CC-MAIN-20160428161511-00024-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://trac-hacks.org/ticket/839 | Opened 11 years ago
Closed 11 years ago
Defective time value in manual page
Reported by: Owned by: James Mills Russ Tyndall normal TimingAndEstimationPlugin normal time, column 0.10
Description
Hi,
It took a while to discover this and thanks to coderanger we found out what the problem was.
A defect in the TimingAndEstimationPluginUserManual page's time value in teh wiki database was causing the BlogShow macro to fail with StopIteration.
SQLite version 3.2.6
Enter ".help" for instructions
sqlite> .mode columns
sqlite> select time from wiki where name="TimingAndEstimationPluginUserManual";
time
----------
%s
sqlite>
Changing this to 0 or any integral value fixes the problem.
comment:1 Changed 11 years ago by Russ Tyndall
Thank you very much for the excellent bug report.
This was supposed to be a parameter to the insert statement that is being set to the current time. I am trying to figure out why this parameter is not being substituted.
Russ
comment:2 Changed 11 years ago by Russ Tyndall
Resolution: → fixed new → closed
I think this actually got fixed a little bit ago while we were making the plugin work with postgres.
Hopefully I have a solution in place that will not corrupt the data any further and will set all of the times set to %s to 0. See this changeset for more info
Thanks again, Russ
Modify Ticket
Change Properties | 2018-02-23 08:41:13 | {"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": 0.48142170906066895, "perplexity": 4356.858733661404}, "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-2018-09/segments/1518891814538.66/warc/CC-MAIN-20180223075134-20180223095134-00703.warc.gz"} |
https://forum.allaboutcircuits.com/threads/ac-gear-motor-reversing-cycle-with-trip-switches.183705/ | # AC gear motor reversing cycle with trip switches
#### JeffN
Joined Dec 14, 2021
14
I'm building a small panel saw set up using a ball screw powered by an AC 120 volt reversible gear motor. I'd like to run a circuit that has the following.
1. a start button that powers the motor to travel about 32"
2. then tripping a (micro?) switch that stops the travel,
3. and then reverses the motor to return to its starting point and stopping.
4. in a ready state to begin again upon pushing the start button.
I have a basic working knowledge of AC circuits with simple switches. I have no idea what type of switches and or controls, or how to wire the circuit to accomplish what I need.
Any help, suggestions, or redirection would be greater appreciated. If possible a simple drawing of the needed circuit would be extremely helpful.
Thank you,
Jeff
#### LowQCab
Joined Nov 6, 2012
2,615
This is a very simple Project .... but .......
The Mechanical bits are very important to get right,
that is, the type of Limit-Switches, their Mounting and Protection, and Protection of the Wiring.
If You can provide a drawing of Your Jig,
I'll come up with a reliable Limit-Switch arrangement.
You also need to supply any and all info on exactly how your Motor Reverses Direction.
Preferably with Pictures to go along with it.
Keep in mind that what You want to do may be EXTREMELY DANGEROUS,
and create projectiles flying across the room at ~200 MPH.
.
.
.
#### LowQCab
Joined Nov 6, 2012
2,615
What is the size and thickness of the Panels You are cutting ?
This is starting to sound like a bad idea.
.
.
.
#### BobTPH
Joined Jun 5, 2013
6,035
You may regret this when your hand gets stuck in the way the saw.
Bob
#### JeffN
Joined Dec 14, 2021
14
This is a very simple Project .... but .......
The Mechanical bits are very important to get right,
that is, the type of Limit-Switches, their Mounting and Protection, and Protection of the Wiring.
If You can provide a drawing of Your Jig,
I'll come up with a reliable Limit-Switch arrangement.
You also need to supply any and all info on exactly how your Motor Reverses Direction.
Preferably with Pictures to go along with it.
Keep in mind that what You want to do may be EXTREMELY DANGEROUS,
and create projectiles flying across the room at ~200 MPH.
.
.
.
First of all I appreciate your concern for safety, I have the same concerns and I want to be aware of all the implications and necessary protections. Let me give you a bit of context.
I have a panel production shop and one of the products requires the production of 3/8" thick plywood panels cut to various sizes and quantities. The sizes range from 16" x 16" to 24" x 24". The material is first cut to 16" x 96" runs, or 24" x 96" runs. Sometimes 50 pcs. or 100's. Prior to my new project I have been cutting them on a very large sliding table saw. This works, but is quite labor intensive. The new saw would be used for cross cut only.
The project I'm building is comprised of a circular saw mounted and captured on sliding rails and ball bearings. The saw cannot just fly off the rails there are heavy stops at either end of the 32" travel that would prevent this. Also the blade is located in such a way that keeps it away from the operator. It's not fully completed yet, but I'm going to run some tests without it being automated and see what tweaks and adjustments need to be made before adding any automation.
Other than a person purposely trying to put their hand in the way of the blade, which is easier to do on the table saw, what concerns do you have? What "things can fly across the room at 200 mph?" Why is this a "bad idea?" Appropriate guards will be added to insure safety.
I'll try to send some pictures of the set up so far, but again, it's not complete yet. I think this might help address the safety concerns.
With the correct set up, this would keep the operator further away from any danger by not needing to touch the saw motor or components. But again, any cautions etc. are welcome. I'd just like to understand what they are, and address them.
Thank you for your time and consideration.
Jeff
Last edited:
Joined Jul 18, 2013
26,229
You most likely should not reverse immediately in this set up, so you will need some kind of delay before reverse takes place.
I assume this is a AC motor (Induction)?
A simple controller such as a smart relay could be employed ?
Also don't forget the big RED button, the hardest one to find in an emergency!
#### LowQCab
Joined Nov 6, 2012
2,615
Waiting on some pictures or accurate-drawings .
It would probably be best to have the start of Travel,
both forward and reverse,
be manually initiated with 2 separate Push-Buttons.
It sounds like You are describing a "Hand-Held" Circular-Saw that runs on solidly mounted Rails.
If this is the case, it sounds like you're going in the right direction.
I would use a continuous Steel-Cable, with a Pulley on the far-end,
and a V-Belt-Pulley mounted to the Shaft of a Gear-Motor,
with the Cable wrapped 1.5 turns around the Pulley,
this will facilitate adjusting Travel-Speed by changing Pulley sizes.
Have You selected a Gear-Motor yet ?
.
.
.
#### GetDeviceInfo
Joined Jun 7, 2009
2,122
With the correct set up, this would keep the operator further away from any danger by not needing to touch the saw motor or components. But again, any cautions etc. are welcome. I'd just like to understand what they are, and address them.
When you feed a part through fixed point tooling, the guarding is typically compact around the tooling. With traverse tooling, the guarding ‘envelope’ expands along with it. That guarding should prevent intrusion, while retaining expulsion. As an employer, you want to ensure not only the safety of the worker, but your exposure to claim. I trust you are aware, and I say this only because I’ve seen the chit that can go down. With that being said;
Many saws I’ve worked with retract the blade for the return stroke. This allows rapid return, and maintains cut quality.
I have often added an ‘arbor running’ interlock to the feed drive. Arbor run contactor engagement is often not enough, with a current ‘window’ covering most conditions.
With an auto cycle circuit, one must be able to ‘home’ the carriage.
The ‘big red button’ should extend the perimeter of the safety envelope. A
Variable speed feeds allow a range of blades and materials.
A lockout procedure needs to be implemented.
An auto cycle circuit is relatively simple, if all goes well. It’s when things don’t go well is when the aforementioned chit happens.
Do you have feed motor/ drive selected, and what is it’s description, and what is your electrical source
Last edited:
#### JeffN
Joined Dec 14, 2021
14
Here are some pictures of what I'm working on. I can work out the details of guards, movement of the material, etc. The circuit and trip switches, etc. is over my head. I'm going to work with a ball screw that is not hooked up yet, but you can see the chrome shaft to the left of the saw. Once I add the stops and guides for the material the operator will not need to touch the saw or the material. The material will be held in place. The big read "STOP" button sounds like a very good idea.
I have not yet selected the gear motor. I want to run some test cuts and get a sense for the rate of feed. This does not have to be lightening fast, that is not my goal.
My goals are 1. Safety 2. Accuracy, and 3. Efficiency
#### Attachments
• 1.5 MB Views: 9
• 1.2 MB Views: 9
Joined Jul 18, 2013
26,229
I would say you are going to need some kind of protective cover over that ball-screw!
Particularly if it is the precision type.
In fact, most of the detection devices etc may need it also, may be better to go proximity sensors?
#### JeffN
Joined Dec 14, 2021
14
I would say you are going to need some kind of protective cover over that ball-screw!
Particularly if it is the precision type.
Yes that's correct, planning on that. It's not even mounted yet.
Joined Jul 18, 2013
26,229
I have a basic working knowledge of AC circuits with simple switches. I have no idea what type of switches and or controls, or how to wire the circuit to accomplish what I need.
Any help, suggestions, or redirection would be greater appreciated. If possible a simple drawing of the needed circuit would be extremely helpful.
It should really conform to NFPA79, (code), there is a copy of the older edition out there in PDF if you want to look it over.
How did you intend wiring it? flexible/cabling type etc?
#### JeffN
Joined Dec 14, 2021
14
Waiting on some pictures or accurate-drawings .
It would probably be best to have the start of Travel,
both forward and reverse,
be manually initiated with 2 separate Push-Buttons.
It sounds like You are describing a "Hand-Held" Circular-Saw that runs on solidly mounted Rails.
If this is the case, it sounds like you're going in the right direction.
I would use a continuous Steel-Cable, with a Pulley on the far-end,
and a V-Belt-Pulley mounted to the Shaft of a Gear-Motor,
with the Cable wrapped 1.5 turns around the Pulley,
this will facilitate adjusting Travel-Speed by changing Pulley sizes.
Have You selected a Gear-Motor yet ?
.
.
.
I'm pondering your cable idea vs. my use of the ball screw. I agree that adjusting the speed will be much easier. Since this is not a CNC set up, the exact position of the saw is not required. I also like the idea of a v-belt pulley to the motor, will allow for initially a loose adjustment at the belt to allow for slippage in initial testing just to insure things are working correctly on the first runs.
#### LowQCab
Joined Nov 6, 2012
2,615
You don't want any "slippage" of the Cable, it needs to be spring-tensioned, and very-tight,
probably around ~50-lbs of Spring-tension.
Speed of the Cut needs to be adjustable,
but adjusting the Speed of the Motor adds a lot of expense to the project,
and Pulley selection for speed adjustment may get You close enough.
The Saw should be mounted to the Sled by the front-edge of the Shoe, with a strong/wide hinge.
The rear of the Shoe should have a mechanical latch mechanism that
holds the rear of the Saw down against spring pressure.
This Latch should be easy to release by hand.
This Latch should also have a release mechanism that can be set up to release the Latch
when any part of the Saw or Sled bumps into an object.
The Spring must be strong enough to be capable of lifting the Blade clear of the Work reliably,
but be soft enough to allow compression by hand,
and also soft enough to not cause any binding in the Latch/Release mechanism.
The Latch will be mechanically released,
via a metal-rod, or similar,
shortly before the end of the Rail travel.
This will cause the Saw-Motor to stop by opening a Switch mounted to the Shoe of the Saw.
This Switch will be spliced into the Power-Cord of the Saw.
This Switch will prevent the Saw from running unless
the rear of the Saw-Shoe is mechanically latched-down securely.
When the Power is stopped to the Saw-Motor, ( by any means ),
a Current-Sensor monitoring the power going to the Saw-Motor
will electrically switch the Gear-Motor direction,
this will bring the "not running" Saw back to the start-position.
The Saw can not "return-Home" with the Motor running, and,
the Saw can not move "Forward" if the Saw-Motor is NOT running.
When the Saw-Motor is running, the Gear-Motor can only run in the "forward" direction,
when the Saw-Motor stops, the Gear-Motor will only run in the "reverse", or "return" direction.
The Gear-Motor runs at all times after the "Start-Button(s)" are pressed,
until it returns to the "Home" or "Park" position,
where it is turned-OFF by a a Mechanical-Limit-Switch, or an Optical-Limit-Switch.
An Optical-Switch is preferred.
The Saw must be manually latched-down into cutting position,
and the "Twin-Start-Buttons" pushed,
before the Saw-Motor and the Gear-Motor will run, and begin a new "Cutting-Sequence".
The "Twin-Start-Buttons" must be mounted far enough apart to insure that
2-Hands are required to start the "Cutting-Sequence",
and both of the Buttons must be a safe distance from the Saw.
A schematic is not realistic until the type of Gear-Motor, and its Wiring requirements, are known.
Do You need help with selecting a Gear-Motor ?, or do You already have one in mind ?
.
.
.
#### JeffN
Joined Dec 14, 2021
14
You don't want any "slippage" of the Cable, it needs to be spring-tensioned, and very-tight,
probably around ~50-lbs of Spring-tension.
Speed of the Cut needs to be adjustable,
but adjusting the Speed of the Motor adds a lot of expense to the project,
and Pulley selection for speed adjustment may get You close enough.
The Saw should be mounted to the Sled by the front-edge of the Shoe, with a strong/wide hinge.
The rear of the Shoe should have a mechanical latch mechanism that
holds the rear of the Saw down against spring pressure.
This Latch should be easy to release by hand.
This Latch should also have a release mechanism that can be set up to release the Latch
when any part of the Saw or Sled bumps into an object.
The Spring must be strong enough to be capable of lifting the Blade clear of the Work reliably,
but be soft enough to allow compression by hand,
and also soft enough to not cause any binding in the Latch/Release mechanism.
The Latch will be mechanically released,
via a metal-rod, or similar,
shortly before the end of the Rail travel.
This will cause the Saw-Motor to stop by opening a Switch mounted to the Shoe of the Saw.
This Switch will be spliced into the Power-Cord of the Saw.
This Switch will prevent the Saw from running unless
the rear of the Saw-Shoe is mechanically latched-down securely.
When the Power is stopped to the Saw-Motor, ( by any means ),
a Current-Sensor monitoring the power going to the Saw-Motor
will electrically switch the Gear-Motor direction,
this will bring the "not running" Saw back to the start-position.
The Saw can not "return-Home" with the Motor running, and,
the Saw can not move "Forward" if the Saw-Motor is NOT running.
When the Saw-Motor is running, the Gear-Motor can only run in the "forward" direction,
when the Saw-Motor stops, the Gear-Motor will only run in the "reverse", or "return" direction.
The Gear-Motor runs at all times after the "Start-Button(s)" are pressed,
until it returns to the "Home" or "Park" position,
where it is turned-OFF by a a Mechanical-Limit-Switch, or an Optical-Limit-Switch.
An Optical-Switch is preferred.
The Saw must be manually latched-down into cutting position,
and the "Twin-Start-Buttons" pushed,
before the Saw-Motor and the Gear-Motor will run, and begin a new "Cutting-Sequence".
The "Twin-Start-Buttons" must be mounted far enough apart to insure that
2-Hands are required to start the "Cutting-Sequence",
and both of the Buttons must be a safe distance from the Saw.
A schematic is not realistic until the type of Gear-Motor, and its Wiring requirements, are known.
Do You need help with selecting a Gear-Motor ?, or do You already have one in mind ?
.
.
.
Though I understand your suggestion of lifting the saw up from the cut material, I don't want to add that to the mix. Here's why.
1. The purpose of this cross cut saw is to facilitate the cutting of same size panels usually 16" x 16" or less from pre-cut runs of the appropriate width stock. For example if I'm making 16 x 16 inch panels, I'm feed 16" wide runs of 96" material into the saw. These runs are stacked, 3-4 sheets high. Keep in mind the sheets are 3/8" thick.
2. Having the saw hinge up from the stock would require it to lift much too high, and expose the blade to the operator, I don't want to do that.
3. Additionally, the saw motor does not have a brake it will continue to spin to a stop once the power is cut off anyway.
4. Because the cycle of repeated passes will be relatively "quick" I don't want to turn off the saw motor upon each pass. The action will mimic a radial arm saw, or a chop saw that is being used repetitively. We mill literally 100's of these size wood tiles therefore both safety and being efficient is needed. Both the saw and the wood material will be captured preventing any possible kick back.
5. The way I'm building the set up, the spinning blade will be encapsulated keeping the operator safe and away from the blade. I also think the constant on and off of the saw motor will limit the duty cycle of the motor. Having said all of that. A safety cutoff switch that de energizes everything in case of something going wrong is a requirement.
I have a gear motor on hand with a speed control I think it will work quite well. Attached are some pictures. If you can help with the circuit that would be greatly appreciated.
#### Attachments
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Joined Jul 18, 2013
26,229
That appears to be a small AC induction motor with TRIAC control worm & pinion GB, not the best set up, but for your use, it may be worth a try, as you already have it.
The wiring appears in need of a little clean up, however!
#### LowQCab
Joined Nov 6, 2012
2,615
1)
You need a ~14"-Blade and a 5-Horsepower-Motor.
2)
The idea is that the Blade-Guard will cover the Blade when the Saw lifts at the end of a run.
3)
The proposed "Kill-Switch" is a common Single-Pole-Double-Throw configuration.
When it changes position, the power is cut, and the Motor is Shorted-Out,
causing it to come to a complete stop within roughly 1-Second.
4)
Cycles won't be very quick using a "Screw-Drive", unless it can withstand full Motor-RPM.
"Quick" through 1.5" of material is a recipe for burning-up a Motor.
5)
Starting and Stopping the Motor is nothing compared to pushing it hard through 1.5" of material.
Your Motor may well work, but will be very slow-moving with a "Screw-Drive" arrangement.
For faster Feed-Rates, ( maybe ~4" per second ), You will need at least a 1/2-Horsepower Gear-Motor,
( and a more powerful Saw ).
What You want normally starts out at around ~$20,000.oo, for a reason. Multiple Emergency-Stop-Buttons are routine work for any decent Industrial/Commercial-Electrical Company. Personally, I would prefer a single "Dead-Man" Push-Button, which must be held down for the Saw to run. Cheaper, Safer, and quicker responding, than standard Emergency-Stop-Buttons. . . . #### MaxHeadRoom Joined Jul 18, 2013 26,229 Your set up should stop pretty quickly with that gear box, being a worm & pinion version, they cannot be back-fed, so virtually no over-run. Thread Starter #### JeffN Joined Dec 14, 2021 14 1) You need a ~14"-Blade and a 5-Horsepower-Motor. 2) The idea is that the Blade-Guard will cover the Blade when the Saw lifts at the end of a run. 3) The proposed "Kill-Switch" is a common Single-Pole-Double-Throw configuration. When it changes position, the power is cut, and the Motor is Shorted-Out, causing it to come to a complete stop within roughly 1-Second. 4) Cycles won't be very quick using a "Screw-Drive", unless it can withstand full Motor-RPM. "Quick" through 1.5" of material is a recipe for burning-up a Motor. 5) Starting and Stopping the Motor is nothing compared to pushing it hard through 1.5" of material. Your Motor may well work, but will be very slow-moving with a "Screw-Drive" arrangement. For faster Feed-Rates, ( maybe ~4" per second ), You will need at least a 1/2-Horsepower Gear-Motor, ( and a more powerful Saw ). What You want normally starts out at around ~$20,000.oo, for a reason.
Multiple Emergency-Stop-Buttons are routine work for any
decent Industrial/Commercial-Electrical Company.
Personally, I would prefer a single "Dead-Man" Push-Button,
which must be held down for the Saw to run.
Cheaper, Safer, and quicker responding, than standard Emergency-Stop-Buttons.
.
.
.
Can you work out a schematic for the set up without lifting the saw? I've run some tests and will run more, but the current saw seems to work fine. I've run a production shop for 30 years, I don't overkill my motors and though this is a prototype, and I will learn even if the idea ends up not working, I want to give it a try.
Perhaps my use of "quick" implied faster than the saw can handle, I'm not going to do that.
#### GetDeviceInfo
Joined Jun 7, 2009
2,122
Have you performed a time study on the activities? Working on similar, but larger scale work centers, I had found that efficiencies aren't gained on the saw cut, because it is what it is due to blade and feed rates. The largest gains come from handling the product into and out of the the saw cut. One place I worked cut 'books' of OSB as web for I joists. Pallets of OSB were raised on a hoist, and a stack of 10 were pushed off by a carriage, squared, then advanced through the saw. Fully programmable to account for a variety of joist widths and maximize yield. Fine tuning and some modifications to the hoist/carriage made substantial gains, while the saw cycle itself was allowed to slow somewhat to extend the life of the 32" blades, which would drift as they dulled.
I would seriously be looking at auto feeding the stock, manually stroke the saw, then auto cycling the feed. Once that was perfected, I'd auto cycle the saw stroke. That approach would then deploy a PLC , micro, or computer for a range of products. Add a printer for part ID and your on the heels of mass producers.
Im meeting with a fellow next week who is setting up his cabinet shop. Should be interesting to see what he has and where he wants to go with it,
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https://projecteuclid.org/euclid.aaa/1365168369 | Abstract and Applied Analysis
Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations
Abstract
We discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations $({\mathcal{D}}^{\alpha }-\rho t{\mathcal{D}}^{\beta })x(t)=f(t,x(t),{\mathcal{D}}^{\gamma }x(t))$, $t\in (0,1)$ with boundary conditions $x(0)={x}_{0},\mathrm{ }x(1)={x}_{1}$ or satisfying the initial conditions $x(0)=0,\mathrm{ }{x}^{\prime }(0)=1$, where ${\mathcal{D}}^{\alpha }$ denotes Caputo fractional derivative, $\rho$ is constant, $1<\alpha <2,$ and $0<\beta +\gamma \le \alpha$. Schauder's fixed-point theorem was used to establish the existence of the solution. Banach contraction principle was used to show the uniqueness of the solution under certain conditions on $f$.
Article information
Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 632681, 14 pages.
Dates
First available in Project Euclid: 5 April 2013
https://projecteuclid.org/euclid.aaa/1365168369
Digital Object Identifier
doi:10.1155/2012/632681
Mathematical Reviews number (MathSciNet)
MR2947672
Zentralblatt MATH identifier
1251.34010
Citation
Babakhani, Azizollah; Baleanu, Dumitru. Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 632681, 14 pages. doi:10.1155/2012/632681. https://projecteuclid.org/euclid.aaa/1365168369
References
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• X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 168, pp. 398–410, 2005. | 2019-07-21 21:59:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 9, "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": 0.2248508185148239, "perplexity": 1572.4530743427529}, "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-2019-30/segments/1563195527204.71/warc/CC-MAIN-20190721205413-20190721231413-00144.warc.gz"} |
https://docs.astropy.org/en/latest/api/astropy.timeseries.BasePeriodogram.html | # BasePeriodogram¶
class astropy.timeseries.BasePeriodogram(t, y, dy=None)[source]
Bases: object
Methods Summary
from_timeseries(timeseries[, …]) Initialize a periodogram from a time series object.
Methods Documentation
classmethod from_timeseries(timeseries, signal_column_name=None, uncertainty=None, **kwargs)[source]
Initialize a periodogram from a time series object.
If a binned time series is passed, the time at the center of the bins is used. Also note that this method automatically gets rid of NaN/undefined values when initalizing the periodogram.
Parameters
signal_column_namestr
The name of the column containing the signal values to use.
uncertaintystr or float or Quantity, optional
The name of the column containing the errors on the signal, or the value to use for the error, if a scalar.
**kwargs
Additional keyword arguments are passed to the initializer for this periodogram class. | 2020-02-29 14:22:18 | {"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": 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": 0.2671230137348175, "perplexity": 3675.983056340368}, "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/1581875149238.97/warc/CC-MAIN-20200229114448-20200229144448-00350.warc.gz"} |
http://cms.math.ca/cmb/msc/17A20?fromjnl=cmb&jnl=CMB | Conjugacy Classes of Subalgebras of the Real Sedenions By applying the Cayley--Dickson process to the division algebra of real octonions, one obtains a 16-dimensional real algebra known as (real) sedenions. We denote this algebra by $\bA_4$. It is a flexible quadratic algebra (with unit element 1) but not a division algebra. We classify the subalgebras of $\bA_4$ up to conjugacy (\emph{i.e.,} up to the action of the automorphism group $G$ of $\bA_4$) with one exception: we leave aside the more complicated case of classifying the quaternion subalgebras. Any nonzero subalgebra contains 1 and we show that there are no proper subalgebras of dimension 5, 7 or $>8$. The proper non-division subalgebras have dimensions 3, 6 and 8. We show that in each of these dimensions there is exactly one conjugacy class of such subalgebras. There are infinitely many conjugacy classes of subalgebras in dimensions 2 and 4, but only 4 conjugacy classes in dimension 8. Categories:17A45, 17A36, 17A20 | 2014-11-28 19:25:01 | {"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": 0.9259918332099915, "perplexity": 179.64726360393163}, "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/1416931010792.55/warc/CC-MAIN-20141125155650-00098-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://fawkes.data-imaginist.com/reference/axi_options.html | This function encapsulates defining general settings for the AxiDraw. Input will be checked against valid values.
axi_options(
speed_down = 25,
speed_up = 75,
acceleration = 75,
position_down = 40,
position_up = 60,
rate_lower = 50,
rate_raise = 75,
delay_down = 0,
delay_up = 0,
const_speed = FALSE,
model = NULL,
port = NULL,
ignore_port = FALSE
)
## Arguments
speed_down The maximum movement speed while the pen is lowered as a percentage of the maximum cariage speed. Values must lie between 1 and 110. Unless const_speed = TRUE the carriage uses a smooth accelaration meaning that the maximum speed may only be reached during long straight draws. There is little gain in plotting time when increasing this value, but it may adversely affect quality. As speed_down but for travel when the pen is raised. Movement when the pen is raised will always use acceleration and increasing this value can have a large effect and plotting time in cases where the pen jumps around a lot. Gives the acceleration and decelaretion speed as a percentage of the maximum rate. Values must lie between 1 and 100. Changing this value will not affect the maximum speed, but how fast it will reach it. The height of the pen when lowered as a percentage of vertical travel. Value must lie between 1 and 100. As position_down but setting the height of the pen when raised. The rate of vertical movement when the pen is lowered given as a percentage of maximum vertical travel speed. Value must lie between 1 and 100. As rate_lower but for specifying relative speed when raising the pen. A delay in ms before commencing movement after the pen has been lowered. Value must lie between -500 and 500. A delay is automatically applied based on the position and rate, but this can be modified with this setting. As delay_down but for commencing movement after the pen has been raised. A boolean given whether the carriage should travel with maximum speed while the pen is down, or use the smooth acceleration. The AxiDraw model to use. This will affect the travel limits imposed on the carriage. Valid values are 'A4', 'A3', or 'XL'. If NULL the value will be taken from the 'fawkes.model' option or 'A4' if missing. The port connection to use. If NULL it will be resolved automatically. If set to TRUE it will always communicate with the first device located even if it is not the one given by port
## Value
An axi_options object | 2020-07-07 05:51:01 | {"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": 0.400389701128006, "perplexity": 1334.972662515395}, "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-2020-29/segments/1593655891654.18/warc/CC-MAIN-20200707044954-20200707074954-00487.warc.gz"} |
https://www.physicsforums.com/threads/the-simpsons-mathematic-geek-received-no-attention.241727/ | # The Simpsons mathematic geek received no attention
1. Jun 23, 2008
### Dark Fire
I remember an episode of The Simpsons, where a mathematic geek received no attention of his fellow peers, so he shouted: PI IS EXACTLY THREE!
I just couldn't help laughing:D
OK, over to my recent low-leveled mathematics-discovery, regarding whether i is not equal to 1, -1 nor anything like that.
I first wrote this, to ensure I understood the principle:
9^0.5 = 3
9^0.5*9 = 3*9
27 = 27
Now, over to the subject:
i^4 = 1
1^0.25 = i
(1^0.25)*1*1*1 = (i*1*1*1) //here's the issue
1 = i
I didn't even get here earlier, I was trapped in my own confusing thinking, though when I ended up here - I saw the flaw at once..
But now, I forgot it
Anyone'd like to refresh me?
I can't use the Google calculator either, since it doesn't show flaws, it only calculates what you put in on one side, not both.
2. Jun 23, 2008
### Cyrus
What, that makes no sense.$$i=\sqrt{-1}$$
1^0.25=1 not i.
3. Jun 23, 2008
### DaveC426913
What exactly is the problem?
i^4=1 for i=1
i^.25=1 for i=1
So?
4. Jun 23, 2008
### DaveC426913
Not necessarily. I mean, he might be referring to i, but who says he isn't simply using the variable i?
5. Jun 23, 2008
Right.
i is *one* of the fourth roots of 1, the others being 1, -1 and -i.
Okay, so you multiply each side by 1 three times...
Does not follow from the previous line. The previous line just returns the same equation you had before you started: 1^0.25 = i.
I think the issue is that you're mixing roots of 1 here. There are four fourth roots of 1: 1, -1, i and -i. If you take any of these to the fourth power, you'll get one. But that does NOT mean that any product of four of these terms gives 1, nor that they're equal to one another. It seems to me that the fundamental error here is confusion of the multiplicity of roots.
6. Jun 23, 2008
### BryanP
Or you can just think about it visually. I don't really get what the post is for, but just visualize a unit circle on the complex plane.
Then it's just going to alternate between 1 and -1 for each even exponent of i with i^0 = 1. This comes from Euler's formula.
7. Jun 23, 2008
### Staff: Mentor
How many times do we have to go over this with you?
You can use the search tool to find your own posts, and thence find the answer.
The problem is that you are ignoring that 1^0.25 has four roots. There is nothing special about imaginary numbers going on here. The exact same faulty reasoning can lead you to conclude that -1=1 because (-1)^2=(1)^2. It is faulty reasoning. You can conclude $$a^r=b^r \,\Rightarrow\,a=b$$ if a and b are positive real numbers and r is real. You cannot conclude this if any of the values is complex or if a or b is negative.
8. Jun 23, 2008
### DaveC426913
Question to the OP for clarity:
Is this about complex numbers and the root of -1, or not?
9. Jun 23, 2008
### Staff: Mentor
10. Jun 24, 2008
### Dark Fire
Re: i=1
With real positive numbers, like I used in my first example, you can solve this:
a^0.x by multiplying the number with itself, relative to its exponent, for example:
81^0.5=9
81^0.5*81=9*81
729=729
Then with a lower exponent:
81^0.25=3
81^0.25*81*81*81=3*81*81*81
1594323 = 1594323
Therefor it would follow if it was a real positive number. <-- with a negative, it becomes complex again
I guess my point/the conclusion is that there's several 'formulas' for real numbers, that doesn't work with complex numbers...
And that just.. not surprises, but amazes me; it's interesting, since i is actually somehow a "real number" (I've been trying to figuring this out as well).
i is (-)^0.5 1 or whatever you'd like to call it.
It's not negative nor positive, until multiplied with itself, which is just an absurd idea from the beginning of, and again interesting.
@Dave
It is.
Also I didn't know there were a way to say "for i=1" ;o
Thanks, now I know that too x)
I'm not very educated, 17 years old, done primary school for 9 years, stopped with homework @ 5th grade, I'm just trying to get back :D
And D H
Last edited: Jun 24, 2008
11. Jun 24, 2008
### Gokul43201
Staff Emeritus
And what happens if I write 81^0.5=-9?
Your question has been answered in posts #5 & #7. Please read them carefully. This has nothing do do with i "being special" and has everything to do with multiple roots and principal values.
12. Jun 24, 2008
### HallsofIvy
Staff Emeritus
Re: i=1
No, it doesn't follow. In the real numbers, with x a positive real number, $x^{1/n}$ is defined as the positive real number whose nth power is x (the "principal" root). That is NOT the definition for a negative real number which has NO "principal" root.
Last edited by a moderator: Jun 24, 2008
13. Jun 24, 2008
### LukeD
Re: i=1
Dark Fire: Be very careful. With your personality, you're at risk of becoming what most people would call a crack pot. It's not a bad thing (and it's certainly not inevitable), I'm the same way some times, but you should be aware of it. To avoid this, you need to listen to what other people are saying. Don't just post something so that you can get validity; read what people are saying and try to understand the mistakes that people are pointing out. Also, try not to immediately trust your own results (I've seen you with other similar posts where you'll write a "proof" that 1 = -1 and ask why it is the case. However, it's not. 1 = -1 is completely false.) because you can certainly be wrong. If you come up with some result that seems wrong, there's a very good chance that it is and that you've made some mistake somewhere. Even if you come up with something that is correct, it does not mean that the logic you used to arrive there is correct.
Most importantly: Don't be afraid to make mistakes, but accept that you can, have, and will continue to make them. Most people in any profession (especially science and mathematics) would be more surprised to learn that they haven't made a mistake than to find that they made several. With practice though, people learn to avoid the larger mistakes and how to spot others so that they know that their mistakes are probably small. But even then, most people know that even small mistakes can completely ruin the result. You're not yet at this point with mathematics, so try to be aware of this.
Last edited: Jun 24, 2008
14. Jun 24, 2008
### Dark Fire
Re: i=1
This is deeper than that I'm not aware of "denial"/to ignore others statements.
"nth power of blabla" <-- wtf..? power of = exponent, n = real number?
I'm living in a very emotional place, with emotional people, I'm trying hard not to burst out in anger or sadness, and instead of going mad, and then possibly come with shallow, angry statements, I just ignore whatever I need to, to not burst out~
I need a clearcut, easy-English statement that explains not the obvious, but the complex - the part of the equation I don't get, or I'm having trouble with~
I've probably written this a 100 times, but I live not in an English-speaking country, also I never payed attention to school; what seem to be a possibly great vocabulary (or not) is nothing but what I've learned the last year through Googling and games, I could barely write and say simple words such as: yes, no, bye, what - a couple of years ago.
While my life's gotten just worse and worse, I've gotten less and less space of learning, and I've ended up barely Googling anything, though I'm still philosophizing sometimes, and solving math puzzles, since it steals no energy for me to philosophize, rather than figuring out, for example how a Buffer Overflow actually works by source (not just the idea, but in practice).
I'm just trying to focus at something else, since there's no way I can solve my problems IRL, and I just have to keep acting shallow, a "crack pot".
Oh, and another thing: I also got bad experiences about people that sighs @ me all the time, mom, teachers, I guess we 'all' experience that, but since I've never felt hope - I quit acting out, I guess.
Rather just change the subject than experiencing a million sighs, like they're truly my superior, and I'm worthless.
Thanks for replies.
15. Jun 24, 2008
### matt grime
Re: i=1
Well, if you are asking 'where have I gone wrong', then the answer is there. (The line you wrote after this also makes no sense.)
There is nothing to justify that this assertion is true; it isn't as it happens since the principal (not principle) root is taken to be the positive real root anyway.
16. Jun 24, 2008
### cristo
Staff Emeritus
Re: i=1
So what are you trying to say? You have been given the answer to your problems several times in this thread. Are you just intending on ignoring them?
17. Jun 24, 2008
### Dark Fire
Re: i=1
@up
1^0.25=i is wrong?
The line after is what I'm using to make a number with an exponent lower than 1 to become 1, though with complex numbers, as you can see, doesn't work.
If you watch the first line (9^0.5=3)-thingy, u can find the method~
18. Jun 24, 2008
### Dark Fire
Re: i=1
If not written in a clearcut, satisfying way, then yes.
19. Jun 24, 2008
### Staff: Mentor
Re: i=1
What's even frustrating is that he has raised this same issue in three other threads, has been answered multiple times in each and every one of those threads, and has ignored all of those answers. Sigh.
20. Jun 24, 2008
### matt grime
Re: i=1
Yes.
And it doesn't make any sense. At best you're claiming (by adding powers in the exponent on the left hand side) that
1^{3.25}=i
Since you're arbitrarily choosing roots at will there is no logical reason to suppose that this equality holds, or even makes sense. It is, in fact, equivalent to you asserting that since 1 and -1 square to the same number, then 1=-1. As is frequently the case, light can be shed on the problem without recourse to the overly complicated ideas initially used. The infamous James Whatisisname (honestly can't remember) was a prince of this on sci.math with his "over-interpretation of galois theory". | 2016-10-23 18:03:12 | {"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": 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": 0.5857038497924805, "perplexity": 1354.4119982314878}, "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/1476988719397.0/warc/CC-MAIN-20161020183839-00349-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://docs.blender.org/manual/ko/dev/modeling/meshes/editing/misc.html | # Miscellaneous Editing Tools¶
## Sort Elements¶
Reference
Mode: Edit Mode Mesh ‣ Sort Elements...
This tool (available from the Specials, Vertices, Edges and Faces menus) allows you to reorder the matching selected mesh elements, following various methods. Note that when called from the Specials menu, the affected element types are the same as the active select modes.
View Z Axis
Sort along the active view's Z axis, from farthest to nearest by default (use Reverse if you want it the other way).
View X Axis
Sort along the active view's X axis, from left to right by default (again, there is the Reverse option).
Cursor Distance
Sort from nearest to farthest away from the 3D cursor position (Reverse also available).
Material
Sort faces, and faces only, from those having the lowest material's index to those having the highest. Order of faces inside each of those "material groups" remains unchanged. Note that the Reverse option only reverses the order of the materials, not the order of the faces inside them.
Selected
Move all selected elements to the beginning (or end, if Reverse enabled), without affecting their relative orders. Warning: This option will also affect unselected elements' indices!
Randomize
Randomizes indices of selected elements (without affecting those of unselected ones). The seed option allows you to get another randomization -- the same seed over the same mesh/set of selected elements will always give the same result!
Reverse
Simply reverse the order of the selected elements.
힌트
Enabling the Display Indices Option
Type bpy.app.debug = True into the Python Console and a checkbox will appear in the Properties region under Mesh Display ‣ Edge Info ‣ Indices.
## Separate¶
Reference
Mode: Edit Mode Mesh ‣ Vertices ‣ Separate P
At some point, you will come to a time when you need to cut parts away from a mesh to be separate.
To separate an object, the vertices (or faces) must be selected and then separated, though there are several different ways to do this.
Selected
This option separates the selection to a new object.
All Loose Parts
Separates the mesh in its unconnected parts.
By Material
Creates separate mesh objects for each material.
더 보기 | 2019-03-21 03:50:47 | {"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": 0.29088738560676575, "perplexity": 3796.445288001758}, "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/1552912202484.31/warc/CC-MAIN-20190321030925-20190321052925-00078.warc.gz"} |
http://mymathforum.com/number-theory/45616-messing-around-relating-sets-dyadics-reals.html | My Math Forum Messing Around - Relating Sets of Dyadics to Reals
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Number Theory Number Theory Math Forum
August 6th, 2014, 10:08 AM #1 Senior Member Joined: Jun 2014 From: USA Posts: 422 Thanks: 26 Messing Around - Relating Sets of Dyadics to Reals I haven't seen any new threads here for a while. As always, I promise this is Fields Medal material , but perhaps someone will find it interesting anyways. To start, in my hopeless quest to find a discrete uniform distribution over $\mathbb{N}$, I was thinking in terms of base 2 on the unit interval and came across the following: Let $D = \{d \in (0,1]: d \text{ is a dyadic rational } \}$ Let $\alpha(x) = \{ d \in D : d < x \}$ Conjecture: There exists $x,y \in (0,1]$ where $x \not= y$ and $\alpha(x) = \alpha(y)$. Can anyone give an example where this is true? I'm thinking no, though it better be true if $|\mathbb{R}_{(0,1]}| > |D|$, so the lack of a real-world example isn't going to deter me here. Assuming the conjecture is true, we can create a unique interval on (0,1] for each dyadic in $D$: Let $\beta(x) = d \rightarrow (d \in D \wedge \alpha(x) = \alpha(d))$. Each dyadic $d \in D$ then defines its own interval equal to $\{ x \in (0,1] : \beta(x) = d \}$. These intervals are interesting. Their measure would have to be 0, because the sum of the measures would equal $\infty$ otherwise, but at the same time we must assume that the set of elements in each interval has a cardinality equal to $2^{\aleph_0}$ and is continuous. Is this possible? (clearly, it is, but ) Let $z$ be a randomly selected element of (0,1] given a uniform distribution. As CRGreathouse pointed out in an earlier thread, we can do this by randomly choosing each bit $x_i$ of $0.x_1x_2x_3...$ (temporarily ignoring the patchable problem of each dyadic having two binary representations, except 1, and all other elements of (0,1] having only one representation). This is the paragraph where CRGreathouse will have to come save the day (ie... potential crankery warning)... That said, we are guaranteed to have $\beta(z)$ equal to a dyadic in (0,1]. All dyadics in $D$ had an equal probability of being equal to $\beta(z)$, having selected $z$ uniformly at random, so we must conclude that our dyadic $\beta(z)$ is selected uniformly at random as well. ... Might as well finish it. Let $\psi$ be a bijection from $D$ onto $\mathbb{N}$. Then we have a unique positive integer, $\psi(\beta(z))$, where again all positive integers had an equal chance of being equal to $\psi(\beta(z))$, so I conclude that $\psi(\beta(z))$ is also selected uniformly at random. Last edited by AplanisTophet; August 6th, 2014 at 10:16 AM.
August 6th, 2014, 10:49 AM #2
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For those of you following along: "dyadic rational" is the name for rational numbers with a denominator (in lowest terms) which is a power of 2.
Quote:
Originally Posted by AplanisTophet Let $D = \{d \in (0,1]: d \text{ is a dyadic rational } \}$ Let $\alpha(x) = \{ d \in D : d < x \}$ Conjecture: There exists $x,y \in (0,1]$ where $x \not= y$ and $\alpha(x) = \alpha(y)$.
The conjecture is false. There is a dyadic rational in (min(x,y), max(x,y)) unless x = y.
Quote:
Originally Posted by AplanisTophet it better be true if $|\mathbb{R}_{(0,1]}| > |D|$
I have no idea why you think these two are related.
Quote:
Originally Posted by AplanisTophet Let $z$ be a randomly selected element of (0,1] given a uniform distribution. As CRGreathouse pointed out in an earlier thread, we can do this by randomly choosing each bit $x_i$ of $0.x_1x_2x_3...$ (temporarily ignoring the patchable problem of each dyadic having two binary representations, except 1, and all other elements of (0,1] having only one representation).
Right... but be sure to specify that you want z to be selected uniformly at random (not just "a randomly selected element of (0, 1]" which could have some other distribution).
Quote:
Originally Posted by AplanisTophet Assuming the conjecture is true
I skipped this part since it isn't.
August 6th, 2014, 01:56 PM #3
Senior Member
Joined: Jun 2014
From: USA
Posts: 422
Thanks: 26
Quote:
Originally Posted by AplanisTophet Can anyone give an example where [the conjecture from the op] is true? I'm thinking no, though it better be true if $|\mathbb{R}_{(0,1]}| > |D|$, so the lack of a real-world example isn't going to deter me here.
Quote:
Originally Posted by CRGreathouse I have no idea why you think these two are related.
Well, let $\gamma(x) = \alpha(x) \setminus \bigcup_{y \in (0,x)} \alpha(y)$
The set $\gamma(x)$ will then contain at least one element if the conjecture is false (and $x \leq 1$). Further, the collection $\{ \gamma(x) : x \in (0,1] \}$ must be pairwise disjoint.
Let $\delta(d) = x \rightarrow d \in \gamma(x)$
Then, $\delta$ is a function from (a subset of) $D$ onto $\mathbb{R}_{(0,1]}$, and the cardinality of $D$ must be equal to or greater than the cardinality of $\mathbb{R}_{(0,1]}$.
Last edited by AplanisTophet; August 6th, 2014 at 02:17 PM.
August 7th, 2014, 05:45 AM #4
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Quote:
Originally Posted by AplanisTophet Well, let $\gamma(x) = \alpha(x) \setminus \bigcup_{y \in (0,x)} \alpha(y)$
Let's simplify that. $\alpha(x)=(0,x)\cap D,$ so
$$\gamma(x)=\left[(0,x)\setminus\bigcup_{y<x}(0,y)\right]\cap D=\{z:\ z<x\wedge\forall y<x,\ z>y\}\cap D=\emptyset\cap D=\emptyset$$
Quote:
Originally Posted by AplanisTophet The set $\gamma(x)$ will then contain at least one element if the conjecture is false (and $x \leq 1$).
No, the conjecture is false and $\gamma(x)$ is uniformly empty.
August 7th, 2014, 07:55 AM #5
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Quote:
Originally Posted by CRGreathouse No, the conjecture is false and $\gamma(x)$ is uniformly empty.
I agree that $\gamma(x)$ is uniformly empty as implied by the op, but then I don't see how the conjecture is not true if that is the case.
Specifically, if the conjecture is false, we have $y < x \rightarrow \alpha(y) \subset \alpha(x)$. This should hold true for every possible value of $y < x$. It doesn't though, so at what point do we hit a $y$ where it doesn't?
August 7th, 2014, 08:08 AM #6
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Quote:
Originally Posted by AplanisTophet I agree that $\gamma(x)$ is uniformly empty as implied by the op, but then I don't see how the conjecture is not true if that is the case.
I'm sorry for your confusion, but I don't understand it.
Quote:
Originally Posted by AplanisTophet Specifically, if the conjecture is false, we have $y < x \rightarrow \alpha(y) \subset \alpha(x)$.
I think you mean $0<y<x<1\rightarrow\alpha(y)\subsetneq\alpha(x).$ Right?
Expanding, this is $0<y<x<1\rightarrow(0,y)\cap D\subsetneq(0,x)\cap D.$ In other words, $(y,x)\cap D\ne\emptyset$ when $0<y<x<1$ ("there is some dyadic rational between y and x when y < x"). This is pretty obvious, yes? It's a standard undergraduate exercise to prove this for $\mathbb{Q}$ in place of D, and this version is no harder.
August 7th, 2014, 08:27 AM #7
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Quote:
Originally Posted by CRGreathouse Expanding, this is $0 It's only obvious when$y$is not thought of as approaching$x$infinitesimally. I compare this to standing in a field and declaring the world is flat based on what is observable from a point in the field. If$\gamma(x)$is empty and$y$never equals$x$, then there must exist a$y$and$x$where$\alpha(y) = \alpha(x)$. I don't see any way around this. August 7th, 2014, 08:57 AM #8 Global Moderator Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms That I may properly understand you: you're claiming that there are real numbers$y
August 7th, 2014, 10:45 AM #9
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Quote:
Originally Posted by CRGreathouse That I may properly understand you: you're claiming that there are real numbers $y No. We'll have to leave that alone or we'll get stuck on this. We know that as$x$increases,$\alpha(x)$changes (grows, I'll say). Let$dx$be an infinitesimal change in$x$. Does$\alpha(x) = \alpha(x + dx)$? If$x \in D$, the answer is no, it doesn't. If$x \not\in D\$, then I believe the answer is yes.
Now what say you?
August 7th, 2014, 11:32 AM #10
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Quote:
Originally Posted by AplanisTophet Now what say you?
The only real infinitesimal is 0.
Tags dyadics, messing, reals, relating, sets
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Copyright © 2018 My Math Forum. All rights reserved. | 2018-12-15 09:28:17 | {"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": 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": 0.8705265522003174, "perplexity": 1217.0071935423414}, "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-2018-51/segments/1544376826842.56/warc/CC-MAIN-20181215083318-20181215105318-00601.warc.gz"} |
https://www.physicsforums.com/tags/liquid/ | # Liquid Definition and 33 Discussions
A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, and plasma), and is the only state with a definite volume but no fixed shape. A liquid is made up of tiny vibrating particles of matter, such as atoms, held together by intermolecular bonds. Like a gas, a liquid is able to flow and take the shape of a container. Most liquids resist compression, although others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly constant density. A distinctive property of the liquid state is surface tension, leading to wetting phenomena. Water is, by far, the most common liquid on Earth.
The density of a liquid is usually close to that of a solid, and much higher than in a gas. Therefore, liquid and solid are both termed condensed matter. On the other hand, as liquids and gases share the ability to flow, they are both called fluids. Although liquid water is abundant on Earth, this state of matter is actually the least common in the known universe, because liquids require a relatively narrow temperature/pressure range to exist. Most known matter in the universe is in gaseous form (with traces of detectable solid matter) as interstellar clouds or in plasma from within stars.
View More On Wikipedia.org
1. ### Study of harmonic motion of a liquid in a V shaped tube
A V-shaped tube with a cross-section A contains a perfect liquid with mass density and length L plus and the angles between the horizontal plane and the tube arms as shown in the attached figure. We displace the liquid from its equilibrium position with a distance and without any initial...
2. ### An exercise with Pascal's Law
Answer : Using Pascal's law, this is my answer : ##\color{blue}{\boxed{\vec F_a = \vec F_c < \vec F_b}}##. Reasoning : Forces ##F_a## and ##F_c## are equal because the pressures required at the two cylinders for case (c) is the same as that required in (a). It doesn't matter how many of those...
3. ### Reason for Calculating Surface tension
When calculating force due to surface tension across a hemispherical drop, we look at only the circumference and multiply it by the value of surface tension. When we know that it is the surface tension which is responsible for the curved surface of the liquid drop, why don't we calculate the...
4. ### Surface Tension and Capillary Rise
According to this definition I am unable to understand why does surface tension acts tangentially to surface of contact of liquid and capillary tube. And is the force of surface tension balancing the adhesive forces which lead to capillary rise OR it is the reason behind the capillary rise?
5. ### Undergraduate physics: Body flows inside liquid [ v(x)=? and V(lim)=? ]
Homework Statement: Object inside liquid Homework Equations: T=k*v^2 F=m*a We hold an object with a mass (m) inside a liquid. On t=0 we free the object. Except the weight there is another one force, the friction of the liquid, witch is T=k*v^2 ( v=instant speed, and k=constant > 0). Also...
6. ### Flow (liquid or gas) across a rotating surface's face
<< Mentor Note -- Two threads on the same question merged into one thread >> How does the maximum Power equation change if there's an angle to the way the wind falls into the wind turbine's blades? Example, when it falls vertically to the blades, it's Pmax= 8/27Sρu13 But if there's for...
7. ### I Pascal's law (fluids): Derivable from fundamental laws?
Hi. Pascal's law states that static pressure in a confined incompressible fluid without gravity is the same everywhere. Is this law derivable from more fundamental laws? Some thoughts: Is Pascal's law part of the definition of the liquid state? If the liquid operates between two hydraulic...
8. ### B How does liquid starts moving when there is Pascal's law
Consider a conventional U-tube with both the vertical tubes having the same uniform cross section area A and the horizontal tube of length L, connecting those tubes containing an ideal liquid. Now the free surfaces in both the vertical tubes will be at the same height and will have pressure...
9. ### B Direction of buoyant force on sunken object?
We say that buyont force act upwards (in usual cases) and that the normal force exerted by the base of a container (of liquid) on a object is less than its true weight, so a weghing machine will give smaller reading (in terms of value) than expected. But suppose a cube sinks in water. Now water...
10. ### Pressure and boiling
Hi everyone, I have been mulling over the relationship between pressure and boiling for some time, and I am still slightly confused. I shall attempt to provide an overview of my current understanding in the hope that I can get some corrections/clarification on my current conceptual...
11. ### What do surface tension vectors mean in this quote?
I was reading Fundamentals of Inket Printing and it said the following: "The surface tension in a liquid causes a force to act in the plane of the free surface perpendicularly to a free edge in that surface." Can someone explain to me what this means? What's the direction of the force? I have...
12. ### Why do bubbles of air in water move up?
What I am confused about is why do bubbles of air in water move up. I understand why solids and liquids would move up in water if they are less dense. I get the idea that the deeper you go in water the more pressure there is because of the more water weighing down on the water and so there would...
13. ### The acceleration of a bubble in water
Hi, I have a question about a rising bubble. I read that the initial acceleration of a bubble (with negligible mass) in water is 2g, where g is the gravitational acceleration. I understand that if a bubble rise then the water move around it, but I can't derive this equation. Could someone help...
14. ### Question about vapor pressure
Hello everyone, before I start I just want to mention that I am not an expert in physics whatsoever, so please be as specific as you can get if you wish to provide an answer. (The question itself might be considered stupid to be honest) I read the definition of the boiling point recently and...
15. ### I Liquid boiling and Evaporation
In the normal conditions (sea level) water evaporates at 100 C. In thermodynamics, we say: the amount of energy Q, can raise temperature of the liquid by the formula Q1=cm(t2-t1); when the liquid reaches the boiling point (100 C), we write Q2=Lm. Q2 is entirely spent on changing liquid state...
16. ### Thermal expansion of liquid into gas void
I have a question about thermal expansion of liquid into a gas void. Imagine a closed upright cylinder filled mostly with water – say 99%, and the remaining 1% is gas. Now imagine that you heat the cylinder and its contents. The water will expand by ΔV owing to thermal expansion. The gas...
17. ### I Dissolved gas concentration in undersaturated liquid column
Hi, In an enclosed system - of say Methane & water - in which the water column is sufficiently large to have significant pressure and some modest temperature difference due to gravity and geothermal effects, how would one calculate / predict the changing methane concentration (or partial...
18. ### B What is the concept of Physics of Cymatics (of a liquid)
Hi, Could someone please help me about what is the physics concept behind Cymatics. Is there a formula which establishes the patterns form on a liquid when the frequency or amplitude are changed? Thanks.
19. ### Floating block with objects that are thrown in a liquid
Homework Statement I'm confused about the following kind of situation. Consider a block of density ##\rho_b##, mass ##M_b## and section ##S_b## that floats on a liquid of density ##\rho_l##, in a tank of section ##\mathcal{S}##. On the block there are some objects (all equal), of density...
20. ### How can i calculate the time delay in arduino?
hi every one , I'm a student in university , i have a projet it's about calculate speed of waves in liquid by to piezo electrique , one workd like emmeter and the other receppter , the distance betwin the two piezo is fixe , the probleme is i nedd to calculat the time betwin the moment who i...
21. ### A paint for magician's trick?
Greetings Forum Members, I am quite new to this & this is my first post. To tell you guys a little bit about me I am a magician that creates effects and I figured that when I am stuck I should go somewhere and ask people that are smarter than me :-). First Question: Is there any kind of...
22. ### Fluid Dynamics - Pressure at the end of a syringe
Hello there Physics Forum! This is my first post here. This problem has been boggling me all day and I'm need of help in piecing together what I've come up with so far. I'm currently working on a personal project related to paint spraying. In this project I'm using a syringe & needle to feed...
23. ### Standard Enthelpy of water
Dear PF Forum, As per wiki: https://en.wikipedia.org/wiki/Standard_enthalpy_change_of_formation_(data_table) Standard enthalpy of water: Gas: -241.818 kJ/mol Liquid: -285.8 kj/mol What does it means? That to produce H2O per mol at 1000C, it releases 241.818 kJ. That to produce H2O per mol in...
24. ### Liquids and Electricity
Hello everyone, I am currently working on a project to design a more efficient method of electricity production in the nuclear industry. I haven't been able to find anything online (and chemistry isn't exactly my strong suit most of the time), but I wanted to know if anyone knows of a liquid or...
25. ### What causes the liquid to rise in this picture?
This is a diagram of a pitot-static tube. My question is however not related to its applications but rather, what causes the liquid to rise up the static tube? The static tube is at right angles to the fluid flow. I understand that this is a very basic question but I can't seem to get my head...
26. ### Interpreting Kinetic Temperature for Solids / Liquids
Hi all, I'm brushing up on some thermodynamics, and having been reading up on the interpretation of temperature as derived from kinetic theory. I can follow the derivation for an ideal, monatomic gas which relates temperature to the average, translational kinetic energy of the molecules. Most...
27. ### Aqueous Solution
What are we referring to when we denote, say, NaCl (aq)? Are we referring to the dissolved NaCl (Na+ and Cl-)? Are we speaking of the solution as a whole (the dissolved NaCl and water)? Also, how does the liquid state differ from an aqueous?
28. ### Liquid Hydrogen Boil-off
I've read a fews post about liquid hydrogen boil off, but could use a bit more clarification; insight on the following thought experiment would be really helpful: What happens if I take a 1 L steel vessel, put 1 L of liquid H2 into it, seal it off, and let it sit in a room at 25C? The density...
29. ### Fluids and thermodynamics
Homework Statement Homework Equations height of liquid above water level ##h=\frac{2T}{R\rho g}## for isothermal process :##PV##=constant And if ##P_0## is atm. pressure, and P is pressure just below the water level in capillary tube, then $$P=P_0-\frac{2T}{R}$$ The Attempt at a Solution...
30. ### Magnetic liquid
Hey Is there any liquid available that is magnetic at room temprature? I know that liquid helium is magnetic but at very low tempratures not at room temperature. Thanks
31. ### Calculation of the force exerted by a liquid
Question: The height of the given vessel is h,and the width of the given vessel is b (as given in the diagram). The density of the liquid is r.Find the force exerted by the liquid on the slant wall. Relevant formulae : P = F/A F = Vdg An attempt at the solution...
32. ### Ferromagnetic powder question!
Hello! I am trying to understand a few properties of the ferromagnetic powder. I could not get the answers to these questions on the web, since any kind of phrasing i used, popped up search results that related to iron powder or ferromagnetic powder, which are manipulated and shape-shift...
33. ### Exam question that I have no idea how to solve (regarding forces and mass)
1. The problem statement Peter threw 10 (same) coins into an empty container (a tube/drum), which was floating in A LIQUID. The diameter of the container was 2,257 cm, and so because the container is a tube the surface of the tube was equal to π times the diameter squared. He measured, after 4... | 2023-03-26 22:55:51 | {"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": 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": 0.6094774007797241, "perplexity": 891.7606346130683}, "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-2023-14/segments/1679296946535.82/warc/CC-MAIN-20230326204136-20230326234136-00519.warc.gz"} |
https://mrchasemath.com/category/math-in-the-news/ | # National Math Festival 2017
There was mathematical mayhem in DC on Saturday!
Did you miss it? Let me try to capture the day with some photos:
That’s just ONE room, just one part of a very large and increasingly popular National Math Festival.
This was the second festival which is held every two years (alternating with the the US Science and Engineering Festival). The festival was a huge success and was very well attended. I was a little cautious about attendance predictions, given that the festival moved to the convention center from the DC Mall–a location which benefited from wandering foot-traffic.
This year, however, we benefited from the rain. It was dark and rainy all day long, but the National Math Festival provided a wonderful rainy-day escape from the dreary weather. See? Look at all the fun we’re having!
The photos you’re seeing here are all from the travelling exhibits brought to us by the Museum of Mathematics in NYC. I helped MoMATH coordinate volunteers this year, just as I did two years ago. And our volunteers were AWESOME!
We engaged thousands of people throughout the course of the day in meaningful mathematical play. There is a great need for this kind of popular-focus on mathematics, illuminating the beauty, joy, and fun of mathematics, rather than the impression people have of difficulty and drudgery.
All my photos are MoMATH-focused, since that’s where I spent my day. You can find even more of my photos here. And you can see more coverage in my twitter feed. For example, here’s a little clip of some juggling-math:
Did you miss this year’s festival? Mark your calendars for April 2019 and make it a priority!
# I’m back
Hey everyone.
I took a two year hiatus from blogging. Life got busy and I let the blog slide. I’m sorry.
But I’m back, and my New Year’s Resolution for 2017 is to post at least once a month!
Here’s what I’ve been up to over the last two years:
• Twitter. When people ask why I haven’t blogged, I say “twitter ate my blog.” It’s true. Twitter keeps feeding me brilliant things to read, engaging me in wonderful conversations, and providing the amazing fellowship of the MTBoS.
• James Key. I consistently receive mathematical distractions from my colleague and friend, James, who has a revolutionary view on math education and a keen love for geometry. This won’t be the last time I mention his work. Go check out his blog and let’s start the revolution.
with my nerdy friends named James
• My Masters. I finally finished my 5-year long masters program at Johns Hopkins. I now have a MS in Applied and Computational Mathematics…whatever that means!
• Life. My wife and I had our second daughter, Heidi. We’re super involved in our church. I tutor two nights a week. Sue me for having a life! 🙂
family photo
• New curriculum. In our district, like many others, we’ve been rolling out new Common Core aligned curriculum. This has been good for our district, but also a monumental chore. I’m a huge fan of the new math standards, and I’d love to chat with you about the positive transitions that come with the CCSS.
• Curriculum development. I’ve been working with our district, helping review curriculum, write assessments, and I even helped James Key make some video resources for teachers.
• Books. Here are a few I’ve read in the last few months: The Joy of x, Mathematical Mindsets, The Mathematical Tourist, Principles to Actions
• Math Newsletters. Do you get the newsletters from Chris Smith or James Tanton (did you know he’s pushing three essays on us these days?). Email these guys and they’ll put you on their mailing list immediately.
• Growing. I’ve grown a lot as a teacher in the last two years. For example, my desks are finally in groups. See?
my classroom
• Pi day puzzle hunt! Two years ago we started a new annual tradition. To correspond with the “big” pi-day back in 2015, we launched a giant puzzle hunt that involves dozens of teams of players in a multi-day scavenger hunt. Each year we outdo ourselves. Check out some of the puzzles we’ve done in the last two years.
• Quora. This question/answer site is awesome, but careful. You’ll be on the site and an hour later you’ll look up and wonder what happened. Here are some of the answers I’ve written recently, most of which are math-related. I know, I know, I should have been pouring that energy into blog posts. I promise I won’t do it again.
• National Math Festival. Two years ago we had the first ever National Math Festival on the mall in DC. It was a huge success. I helped coordinate volunteers for MoMATH and I’ll be doing it again this year. See you downtown on April 22!
famous mathematicians you might run into at the National Math Festival
Now you’ll hopefully find me more regularly hanging out here on my blog. I have some posts in mind that I think you’ll like, and I also invited my colleague Will Rose to write some guest posts here on the blog. Please give him a warm welcome.
Thanks for all the love and comments on recent posts. Be assured that Random Walks is back in business!
# MAA Distinguished Lecture Series
If you live in the DC area and you like math, you have no excuse! Come to the MAA Distinguished Lecture Series.
These are one-hour talks, complete with refreshments, all for free due to the generous sponsorship of the NSA. The talks are at the Carriage House, at the MAA headquarters near Dupont Circle.
Here are some of the great talks that are on the schedule in the next few months (I’m especially excited to hear Francis Su on May 14th).
I’ve been to many of these lectures and always enjoyed them. Robert Ghrist‘s lecture was out of this world (here’s the recap, but no video, audio, or slides yet) and was so very accessible and entertaining, despite the abstract nature of his expertise–algebraic topology.
And that’s the wonderful thing about all these talks: Even though these are very bright mathematicians, they go out of their way to give lectures that engage a broad audience.
Here’s another great one from William Dunham, who spoke about Newton (Dunham is probably the world’s leading expert on Newton’s letters). Recap here, and a short youtube clip here:
(full talk also available)
So, if you’re a DC mathophile, stop by sometime. I’ll see you there!
# USA Science and Engineering Festival
If you’re local, you should go check out the USA Science and Engineering Festival this weekend. It’s on the mall in DC and everything is free.
They will have tons of booths, free stuff, demonstrations, presentations, and performances. Go check it out!
For my report on the fest from two years ago, see this post. The USA Science and Engineering Festival is also responsible for bringing to our school, free of charge, the amazing James Tanton!
# I ♥ Icosahedra
Do you love icosahedra?
I do. On Sunday, I talked with a friend about an icosahedron for over an hour. Icosahedra, along with other polyhedra, are a wonderfully accessible entry point into math–and not just simple math, but deep math that gets you pretty far into geometry and topology, too! Just see my previous post about Matthew Wright’s guest lecture.)
A regular icosahedron is one of the five regular surfaces (“Platonic Solids”). It has twenty sides, all congruent, equilateral triangles. Here are three icosahedra:
Here’s a question which is easy to ask but hard to answer:
How many ways can you color an icosahedron with one of n colors per face?
If you think the answer is $n^{20}$, that’s a good start–there are $n$ choices of color for 20 faces, so you just multiply, right?–but that’s not correct. Here we’re talking about an unoriented icosahedron that is free to rotate in space. For example, do the three icosahedra above have the same coloring? It’s hard to tell, right?
Solving this problem requires taking the symmetry of the icosahedron into account. In particular, it requires a result known as Burnside’s Lemma.
For the full solution to this problem, I’ll refer you to my article, authored together with friends Matthew Wright and Brian Bargh, which appears in this month’s issue of MAA’s Math Horizons Magazine here (JSTOR access required).
I’m very excited that I’m a published author!
# Welcome James Tanton!
image stolen directly from mathcircles.org
Today we have the special privilege of hosting the one and only, Dr. James Tanton. He will be our guest speaker today and he’ll be talking with our students about his love for math, and hopefully spark in them an appreciation for mathematical play.
We’ll have 800 students at the assembly. And James will be armed with nothing but paper and pen (and a document camera). Bold man! 🙂
If you’ve never checked out James’ materials, go visit his website, take a look at his prolific youtube channel, or follow him on twitter @jamestanton.
James is the author of 10 books on mathematics and math education. He is currently a Mathematician in Residence at the Mathematical Association of America, right here in Washington DC. He comes to us by way of the USA Science & Engineering Festival and its sponsors (Lockheed Martin, Northrop Grumman, Scientific American, Popular Science, and others). Thank you, USA Science & Engineering Festival!
We’re very excited to have James with us!
# Math on the web
Here are two items that have been shared with me in the last 24 hours:
Item 1: Want To Be Better At Math? Use Hand Gestures! Jeremy Shere of Indiana Public Media. Check out this very short audio news that suggests that math instruction has been shown more effective with gestures. I flail around in front of my classroom all the time, so I guess that makes me a good teacher, right? I’d sure like to think so! 🙂 (HT: Tim Chase)
Item 2: How to Fall in Love With Math. Manil Suri, professor at a small school down the road from me (University of Maryland…maybe you’ve heard of it?), has a very nice piece on why math is a worthy object for our affection. It’s been heavily shared in the circles I travel–and for good reason. He reminds us that people fall susceptible to two very common errors when casually speaking about math: (1) We reduce it to arithmetic, as in “come on guys, do the math” or (2) we elevate it to something so ethereal that it’s impossible to grasp, as in “that mathematician talks and I don’t understand a word he says. I never was good at math.” Math, Suri says, is much more than arithmetic and much more accessible than people give it credit for. People can appreciate it without understanding every difficult nuance, just as they do art. (HT: Beth Budesheim)
# Progress Toward Twin-Prime Conjecture
This nice article came through on wired today:
# Unknown Mathematician Proves Surprising Property of Prime Numbers
By Erica Klarreich, Simons Science News
Image: bwright923/Flickr
On April 17, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.
Editors of prominent mathematics journals are used to fielding grandiose claims from obscure authors, but this paper was different. Written with crystalline clarity and a total command of the topic’s current state of the art, it was evidently a serious piece of work, and the Annals editors decided to put it on the fast track.
Just three weeks later — a blink of an eye compared to the usual pace of mathematics journals — Zhang received the referee report on his paper.
“The main results are of the first rank,” one of the referees wrote. The author had proved “a landmark theorem in the distribution of prime numbers.”
(more)
This is very exciting news, and the whole story has a fantastic David & Goliath feel–“little known mathematician delivers a crushing blow to a centuries old problem” (not a fatal blow, but a crushing one). It’s such a feel-good story, almost like Andrew Wiles and Fermat’s Last Theorem. Here’s my favorite part of the article:
…during a half-hour lull in his friend’s backyard before leaving for a concert, the solution suddenly came to him. “I immediately realized that it would work,” he said.
Just chillin’ in his friend’s backyard…and it came to him! Anyone who has worked on math problems or puzzles has had this experience, right? It seems like an experience common to all people. This has definitely happened to me lots of times–an insight hits me out of nowhere and unlocks a problem I’ve been working on for weeks. It’s one of the reasons we do mathematics!
# TI Calculator Emulators
Online Emulator
Check out this online TI-83 Plus emulator! This just came across my radar from Hackaday.
It requires that you upload a (legally acquired) rom, but once you do, this seems like it would be a very good ‘on-the-go’ resource for presentations, teaching, or just any other time and place when you might need at graphing calculator.
I don’t have a TI-83 plus rom lying around, and I tried a regular TI-83 rom (which I did happen to have) but it didn’t seem to work for me. Hmm.
Mobile Devices
[updated] I now recommend Wabbitemu as the best emulator on the computer and for Android devices. It accepts a very wide range of rom files and has a nice feature set. The whole process is pretty user friendly. Here’s a link to the app in the Google Play store and here’s their website where you can download the desktop app.
Another great emulator for Android is Andie Graph which can be obtained in the following way (instructions come from our student, Jim Best):
5. Go to settings by pushing the little icon on the phone itself that looks like a garage door or a tool box.
6. Go down to ROM and select the ROM you downloaded. If the app doesn’t find the ROM, then you can search for it from the app in the phone.
Jim also suggests this calculator if you have an Apple product:
There is an app that is a type of TI-83. It is called RK-83 on the app store for apple products such as the iPhone and iPod touch. This is a \$0.99 app that has the same functionality as a TI-83. It does not have the best of reviews but for \$0.99, its worth a shot. There is also an app by the same creator that has better reviews but it is an 89.
Of course, there are scads of other great calculators out there if you’re willing to give up the look and feel of the TI experience. Desmos is a popular choice and works nicely on all platforms but isn’t a powerhouse of a calculator.
Plain-old Software
And as far as plain-old desktop software goes, here are some great emulators:
I actually prefer the Wabbitemu and Rusty Wagner emulators to the TI-SmartView emulator, even though our school has purchased copies for all of the math teachers.
Rom Files
In almost all the above cases, you’ll need to obtain a rom file for the calculator you’re interested in emulating. This is like the brains of the calculator. The emulator is just the pretty buttons and interface that run on top of the rom. | 2021-05-06 00:14:03 | {"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": 3, "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": 0.18307386338710785, "perplexity": 2304.6423649276353}, "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/1620243988724.75/warc/CC-MAIN-20210505234449-20210506024449-00327.warc.gz"} |
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# x is an integer such that –x|x| ≥ 4.
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x is an integer such that –x|x| ≥ 4. [#permalink] 08 Aug 2018, 16:37
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x is an integer such that $$- x|x| ≥ 4$$.
Quantity A Quantity B x 2
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
Kudos for R.A.E
[Reveal] Spoiler: OA
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Re: x is an integer such that –x|x| ≥ 4. [#permalink] 09 Aug 2018, 15:15
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Carcass wrote:
x is an integer such that $$- x|x| ≥ 4$$.
Quantity A Quantity B x 2
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
Kudos for R.A.E
Given
$$- x|x| ≥ 4$$
Note:
|x| is always positive.
we have a negative sign with x but ultimate value is positive. So x have to be negative. Negative times negative is positive.
The best answer is B. 2 is greater than x, a negative integer.
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Re: x is an integer such that –x|x| ≥ 4. [#permalink] 23 Oct 2018, 03:02
Expert's post
Carcass wrote:
x is an integer such that $$- x|x| ≥ 4$$.
Quantity A Quantity B x 2
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
Kudos for R.A.E
Now $$- x|x| ≥ 4$$ which means $$- x|x| > 0$$ which is possible when both -x and |x| have same sign. |x|>0 so -x>0...
Multiply the sides if equation by '-', so -(-x)<-(0)...x<0
Since x<0, x has to be less than 2..
Thus B>A
B
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html
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Re: x is an integer such that –x|x| ≥ 4. [#permalink] 23 Oct 2018, 08:41
it was interesting.
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Re: x is an integer such that –x|x| ≥ 4. [#permalink] 23 Oct 2018, 20:00
Divide both sides by -x gives you |x| (< or =) 4/-x noting that you will be flipping the direction of the greater than sign since you are dividing by a negative. While you can then go and solve as you would any absolute value, having isolated the absolute value onto one side at this point, you can also see that the only way for the absolute value to be equal to a positive number (absolute values cannot be negative) is if the x value itself is negative. Since any negative number (x > 0) is going to be less than 2, we know that the answer will be B.
Re: x is an integer such that –x|x| ≥ 4. [#permalink] 23 Oct 2018, 20:00
Display posts from previous: Sort by | 2019-03-24 13:18:47 | {"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": 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": 0.5458630919456482, "perplexity": 2140.8649276787633}, "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/1552912203448.17/warc/CC-MAIN-20190324124545-20190324150545-00413.warc.gz"} |
https://rbasics.netlify.app/3-rstudiobasics.html | # 3 R and RStudio Basics
## 3.1 What is R?
In Chapter 2, I discussed many of the reasons why you should begin doing your analyses (especially those of the data type) using R. If you skipped over that chapter in the hopes of just diving in to learning about R, I suggest you go back and read it over carefully. As you begin building fluency in working with R, it is especially important to review that introductory chapter from time to time.
### 3.1.1 R beginnings
R was developed by a group of statisticians who wanted an open-source alternative to the costly proprietary options that were (and still are) popular. Because it was created by statisticians (instead of computer scientists), R has some quirky aspects to it that take some time to get used to. We’ll see that many packages have been developed to help with this, and these days, you don’t need an advanced degree in statistics to work with R.
Getting back to the development of R… R was created by Ross Ihaka and Robert Gentleman in New Zealand at the University of Auckland. It is a spin-off of the S programming language and was named partly after the first names of its developers (as you can see from the emphasis above). The beginning ideas for creating R came in 1992, and the first version of R was released in 1994. You can find much more about the background of R, its features, and its connections to the S language on Wikipedia.
### 3.1.2 R packages
I first learned to use R as a graduate student at Northern Arizona University from Dr. Philip Turk in 2007. At the time, I never thought that R could have exploded in users as we have seen since 2011. I never would have thought that students taking an introductory statistics course would be encouraged to learn to use R.
In 2007, R was still largely an esoteric and tricky language used by statisticians to do analyses. Getting used to the syntax for producing plots and working with data was especially tricky for those with little to no programming experience. So what has changed since 2007 about learning R?
I believe one of the biggest developments has been the creation of packages to make R easier to work with for newbies. Packages are add-ons created by users of R to increase the functionality of the base R installation. Packages created by Hadley Wickham and others recently have greatly expanded the capabilities of R, while also working to make beginning with R simpler. As of April 2017, more than 10,400 packages were available on common R repositories.1
Another great development is the graphical user interface called RStudio and a package developed by RStudio, Inc. called rmarkdown. We will discuss rmarkdown (also referred to as R Markdown) in a Chapter 4, and will now focus on discussing RStudio.
## 3.2 What is RStudio?
RStudio is a powerful, free, open-source integrated development environment for R. Development on RStudio began in 2010, and the first beta was released in February 2011. It is available in two editions: RStudio Desktop and RStudio Server. This book will focus mostly on RStudio Server, but both versions are nearly identical to work with.
Instructions for downloading and installing R and RStudio on Windows and Mac machines are linked below. If you are using RStudio Server, your professor or members of your organization’s IT department have done these steps for you. For RStudio Server, you log on using a web browser to an account on the cloud. There are many advantages to using the RStudio Server for beginning users, including sharing of R projects to help with feedback and error resolution. Installation of software can also cause its own headaches, which are eliminated by using the RStudio Server.
Note for advanced users: You can also install your very own RStudio Server for around $5 per month on Digital Ocean. Instructions to do so can be found from Dean Attali here and on the Digital Ocean site here. After you complete a few months of work with the RStudio Server, it is recommended that you download RStudio Desktop to your computer. The instructions to do so are below. ### 3.2.1 Installing R and RStudio Desktop It is worth noting that you can’t just install RStudio Desktop without first installing R, as RStudio needs to have R installed to run. Step-by-step guides to installing R and RStudio Desktop with screenshots are available Unless you plan to create PDF documents (which requires a multiple gigabyte download of LaTeX), you can skip some of the later steps of the installation. It is recommended that you select HTML as the Default Output Format for R Markdown. You’ll see more about this in Chapter 4. ## 3.3 Working in RStudio Server ### 3.3.1 Logging in and initial screen The RStudio Server provides a web-based interface to run analyses in R. This means that you will only need an internet connection and a web browser to run your analyses. Your professor or administrator will provide you with a link to the web location of your RStudio Server. After entering the link, you’ll see a page that looks something like: After logging in with your username and password, you should see a layout similar to what follows. For reference, a screenshot of RStudio Desktop looks similar: This makes switching between the two RStudio set-ups painless. A discussion of each of the different RStudio panes and their corresponding tabs is in Chapter 4. You’ll find that a lot of what follows also applies to RStudio Desktop (except for the Shared Projects feature), but it is always recommended to create an RStudio project regardless of whether you are on the cloud or working locally. ### 3.3.2 Basic Workflow with RStudio When starting a new R project, it is good practice to create a new RStudio project to go along with it. RStudio project files have the extension .Rproj and store metadata and information about the R environment you are working in. More information about RStudio projects is available from RStudio, Inc. If you are sharing homework or lab assignments with your instructor using RStudio Server, for example, it might make sense to create an RStudio project, share it with your instructor, and then create new folders for each lab. We will follow this example below. The video below shows you how to create a new RStudio project called initial and add your first R Markdown file. Note that you also may see a description about what version of R is running on your initial login, as shown in the screencast below in the Console pane. We have our first_rmarkdown.Rmd file set up. ### 3.3.3 Sharing Projects on RStudio Server Pro You will now see an example of how to share this project with another user. This will enable you and collaborators (other students, your instructor, etc.) to work on the Rmd file at the same time. This is similar to working collaboratively on a Google Doc. RStudio Server is available in multiple formats, so you’ll need to make sure you (or your IT administrator) have installed RStudio Server Pro to use the Shared Projects feature. You can find more information on this process from RStudio, Inc. The video below illustrates the process of sharing the initial.Rproj project file we just created with another user on the same installation of RStudio Server. Both myself and bottk can now work together on this project. We can type comments and code into first_rmarkdown.Rmd or other files in the project and save files to the common folder where initial.Rproj resides. In Chapter 4, you’ll see why it is recommended you work in R Markdown files and you’ll also begin to see some examples of how R works with R Markdown. ## 3.4 RStudio Layout Initially, you may be a little overwhelmed by all the different panes and tabs that are available in RStudio, but you will soon learn to appreciate this layout. We will begin with the top left pane and proceed clockwise. The layout of these panes can be customized, but it is recommended that beginning users keep the standard layout. ### 3.4.1 Code Editor / View Window The pane in which you will likely spend a majority of your time is the pane on the top left. When you first login, this pane isn’t there, but it appeared when we made the first_rmarkdown.Rmd. This pane serves as a place to view the contents of files and objects in R. In the screencast below, you can see that we can change the text in this file and then save the file. Note that, when you edit the file, the tab with the file name changes from black to red, and an asterisk appears after the file name, indicating that the latest changes have not been saved. You should get in the habit of saving files frequently. You can have multiple tabs open and view different files in this pane. In Chapter 4, you’ll also see that you can View datasets in this pane. It may not be clear what these additions are really doing just yet. That’s ok! When finished, you’ll be pressing the Knit HTML button near the top of the pane to put all of your text, code, and its output together. We aren’t quite there just yet though! ### 3.4.2 Environment / History By default, the top right pane includes an Environment tab and a History tab. To get a sense of what these tabs provide, we’ll need to also use the bottom left pane and the Console tab. I’ll show you how to create multiple objects in R using the Console. Initially, you will see that the Environment tab tells us that the “Environment is empty.” This means there are no objects (e.g., data) yet. If you click on the History tab, you should also see a blank screen with a few icons. We won’t go over all of these buttons here, but I encourage you to hover over them and click on them to get a sense for what they do. As I enter code into the Console, watch to see how the Environment and History tabs change. You can think of the Console as a place to play around. It is your R sandbox. You can test your code to make sure it is working and then copy that text into your Rmd file once you are satisfied with it. We’ll see more examples of this in the chapters to come. Note that, by default, when you enter the name of an object into the console like I did with sum_1_2 it displays the result. You’ve also been shown what is called the assignment operator denoted by <-. You can read this as putting the contents of the right-hand-side into an object named whatever appears on the left-hand-side. In the example, num1 is the name of an object that stores the value 7. A powerful feature of the R language has been introduced in the sum_1_2 <- sum(num1, num2) line. sum is a function. Functions are denoted by their name and then a parenthesis, followed by one or more arguments separated by commas, and then a closing parenthesis. You’ll see many examples of these going forward. Of course, we’ll be using R for more than just a basic calculator shown here but this should give you an idea of what the Environment and History tabs store. ### 3.4.3 Console You’ll frequently use the Console as a way to check your work or experiment with how to solve a problem using R. Before RStudio, most users of R just had a window like the Console provides where they entered their commands and then looked at the results in different windows. We will see that the Console and the Code Editor/View Window will allow us to store all of our code in a file and then “run” that code through the Console to check that it works. The example video below shows how this is done. Rules for naming objects It is good practice to get in the habit of naming variables corresponding to what they actually represent. If you are dividing two different sums of numbers, you might want to choose a name like ratio_of_sums to refer to that object. R has a few restrictions regarding what can be included in the name of R objects: 1. Object names cannot begin with a number. 2. Object names cannot contain symbols used for mathematics or to denote other operations native to R. These symbols include $, @, !, ^, +, -, /, and *.
Another important property of R is that is case-sensitive. You’ll see what this means in the video below that also includes examples of invalid names of objects. Note that we will continue to work inside the R chunk as we did in the last video. You’ll see that these code chunks provide a nice way to keep track of our important analyses.
You may have noticed that R will do some checks and alert you to potential errors by placing a red X to the left of lines of code with common syntax errors. These checks won’t catch all errors, but this can be helpful. Note also that Name, name, and nAme refer to three different values.
Another thing to notice is that you can store numbers into an object with a given name like we did earlier with num1. If we’d like to store a character string such as “Chester”, we can provide the name of the object on the left-hand-side of <- and the string in quotation marks on the right-hand-side of <-. There are more complex object types that you will see in Chapter 5, but it is always important to think about the difference between a number and a character in R.
It’s also a good idea to not call objects the names of functions that are built into R. You may want to call the addition of two numbers sum and R will allow this, but it is HIGHLY recommended that you create more descriptive names and not choose to name objects the same as common R functions. Something like sum_densities is better and less likely to be the name of a function.
### 3.4.4 The help function (?)
Of course, you won’t know what all of the names of built-in functions are until you practice, but it is something to think about. If you are ever wondering if a function is built-in or is in a package you have included, you can use the ? function in the Console to check. Some examples are shown below as a screencast.
### 3.4.5 The bottom right pane
The bottom right pane in RStudio contains the most tabs by default and is a useful place to view a variety of miscellaneous information about your RStudio project and its files.
Files
The leftmost tab here shows the file and folder structure for the current working directory. In an RStudio project, this is the folder in which the project file was saved. This shows you where the files are stored, what they are called, and any folders that may exist in your project folder. This can be thought of as similar to going to My Computer on a PC or opening Finder on a Mac. Similarly, this tab lists the file and directory structure either on the cloud for RStudio Server or on your local machine for RStudio Desktop.
Plots / Viewer
You’ll see clearer examples of what the Plots and Viewer tabs provide in Chapter 4. As you likely guessed Plots will show you the resulting graphs/figures that your R code has generated. The Viewer tab can show you the resulting HTML file created from an R Markdown Knit.
Packages
The Packages tab lists all packages installed on your computer or cloud server. You can see which packages are loaded in the current working environment by looking to see if a check-mark exists next to the package name. Note that you may not have all of the packages loaded onto your machine that I do below in the video. That’s OK. This is just an example of what you may expect.
You’ll also notice a Description of the package as well as the Version number here. Packages are frequently updated and improved, so this is a way to check whether you have the most up-to-date version of a package. Remember, if you are using RStudio Server, this is likely taken care of for you. If you are using RStudio Desktop, you might find the Install and Update buttons useful for downloading new packages or updating currently installed ones.
Help
We also saw an example of using the Help tab when we invoked the ? function. This will show you documentation on R functions, datasets, and packages. When (not if) you come across code and you aren’t sure what it does, it is often helpful to enter a question mark followed by the name, and see if the built-in documentation can help you out.
1. You’ll see how to download these packages via install.packages("dplyr") and load them into your current R working environment via library("dplyr"), for example, in Chapter 5. | 2020-07-02 08:15:28 | {"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": 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": 0.23145540058612823, "perplexity": 893.1056050207495}, "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/1593655878639.9/warc/CC-MAIN-20200702080623-20200702110623-00212.warc.gz"} |
https://puzzling.stackexchange.com/questions/61198/pick-seven-numbers-so-that-every-operation-gives-same-result | # Pick seven numbers so that every operation gives same result
Pick seven numbers so that every operation (addition, subtraction, multiplication, but not division) gives the same result.
• For subtraction, you may take the result's absolute value.
• Numbers cannot be $0$.
Example:
• $a, b, c, d, e, f, g$ are the numbers.
• $a + b + c + d + e + f + g = x$
• $| a - b - c - d - e - f - g | = x$ or $a - b - c - d - e - f - g = x$
• $a \times b \times c \times d \times e \times f \times g = x$
How should I choose the numbers?
• Should all the numbers be integers? – athin Mar 1 '18 at 6:54
• as a=x, there have to be fractions involved, and some of them have to be negative – smriti Mar 1 '18 at 6:59
• $a,b,c,d,e,f,g$ should be distinct numbers or not neccessarily? – Oray Mar 1 '18 at 7:13
• Not sure if feasible, but if you want to add to the challenge: division, or inverse of division is equal to x. – Dennis Jaheruddin Mar 1 '18 at 12:01
If we choose the values so that the absolute value is necessary for the subtraction, then: $$a+b+c+d+e+f+g = -(a-b-c-d-e-f-g)\\2a=0$$ Which makes $a$ zero, which is not allowed.
So we have $$a+b+c+d+e+f+g = a-b-c-d-e-f-g\\b+c+d+e+f+g=0$$ and $$a+b+c+d+e+f+g = a\cdot b \cdot c \cdot d \cdot e \cdot f \cdot g\\ a = a\cdot b \cdot c \cdot d \cdot e \cdot f \cdot g\\ b \cdot c \cdot d \cdot e \cdot f \cdot g=1$$ There are many solutions to these two equations for $b,c,d,e,f,g$. We can choose $b=c$ and $d=e=f=g$ to reduce it to two equations with two variables.
(Side remark: I tried putting $c=d=e=f=g$ first, but that leads to complex numbers, as did trying $b=c=d, e=f=g$.)
So we now get: $$2b+4d=0$$ and $$b^2\cdot d^4=1$$ Solving this gives $$4d^2 \cdot d^4 = 1\\ d=2^{-\frac{1}{3}} \text{ and } b=-2\cdot 2^{-\frac{1}{3}}$$ So one solution is:
$a$ has any value
$b=c= -2\cdot 2^{-\frac{1}{3}}$
$d=e=f=g=2^{-\frac{1}{3}}$
• Great solution, the complex numbers arise when the powers (i.e. the number of equal variables) are odd. Other simplifications are possible: eg (b+c+d+e+f)=-1 gives us g. – Bee157 Mar 1 '18 at 11:56
From the first two equations we get
\begin{align}b+c+d+e+f+g &= -(b+c+d+e+f+g) \\ \Rightarrow \, b+c+d+e+f+g &= 0\end{align} As Athin already posted in his answer the second equation written with the absolute values would imply $a = 0$ and is therefore not allowed.
In order to also fulfill the third equation we need
$b \cdot c \cdot d \cdot e \cdot f \cdot g = 1$, which is solved for example by setting $b = c = d = e = \frac{1}{A}$ and $f = g = A^2$.
Together with the condition for the first two equations we then have
\begin{align}4 \cdot \frac{1}{A} + 2 \cdot A^2 &= 0 \\ \Rightarrow \, A &= -\sqrt[3]{2}\end{align}
Altogether, one of the many solutions is
$a = 42 \\ b = c = d = e = -2^{-\frac{1}{3}} \\ f = g = 2^{\frac{2}{3}}$
so $x = 42$. | 2019-08-21 01:35:42 | {"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": 0.9991434216499329, "perplexity": 526.2310367081184}, "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/1566027315695.36/warc/CC-MAIN-20190821001802-20190821023802-00045.warc.gz"} |
http://math.stackexchange.com/questions/126653/can-someone-point-out-what-im-doing-wrong-here-with-my-solids-of-revolution | # Can someone point out what I'm doing wrong here with my solids of revolution?
This is very simple but I can't figure out what I'm doing wrong.
Let's say I'm rotating $f(x)=x^2$ around the y-axis. I have limits on the x-axis of a=1 and b=2. I want the volume under the curve, so I use the shells method, which gives me:
$2\pi \int_a^b f(x)x\delta x$ = $\frac{15}{2}\pi$
Now let's say I do it another way. I'll find the volume of the hollow cylinder with outer radius $b$ and inner radius $a$ and height f(b). This is $12\pi$. Now I want to subtract what's above the curve rotated around the y-axis.
I rotate function f around the y-axis, this time getting the volume above the curve. I can do this by integrating $f^{-1}(y)$ with limits $f(a)$ and $f(b)$. That gives me:
$\pi\int_{f(a)}^{f(b)}(f^{-1}(y))^2dy = \pi\int_1^4(\sqrt(y))^2dy = \pi\int_1^4y dy=\frac{15}{2}\pi$.
I'm sure at this point I've already made my mistake since the last answer gave me my final answer before I was expecting it.
Where's my mistake?
-
There is a mild typo, you wrote $\pi\int_1^4x\, dy$ instead of $\pi\int_1^4y\, dy$, but that's not the problem, since you calculated as if it had been $y$.
In your second way, when you are trying to find out what to subtract, you are subtracting too much. You are using slicing to find the volume of the piece removed. So you need to find $$\int_1^4 \pi(y-1)\,dy.$$ This is because your slice also has a hole in it. The cross-section of the slice has a hole of radius $1$. So the area of the slice is $\pi((\sqrt{y})^2-1^2)$.
-
Yeah thanks for pointing out the typo. In my head, I mix it up a little: I imagine 2 different $y$s, hence the confusion :) – Korgan Rivera Mar 31 '12 at 22:51
Thanks, I see it now. I need to be careful about that. So I suppose it was just a coincidence that $2\pi \int_a^b f(x)x\delta x = \pi\int_{f(a)}^{f(b)}(f^{-1}(y))^2dy$. – Korgan Rivera Mar 31 '12 at 22:55
In calculating the volume generated above the curve, you forgot that you have washers with outer radius $\sqrt{y}$ and inner radius $1$, not disks of radius $\sqrt{y}$: you should have
\begin{align*} \pi\int_1^4\left(\left(\sqrt{y}\right)^2-1^2\right)dy&=\pi\int_1^4(y-1)dy\\ &=\pi\left[\frac12y^2-y\right]_1^4\\ &=\pi\left(\frac{15}2-3\right)\\ &=\frac92\pi\;. \end{align*}
Now when you subtract this from $12\pi$ youl’ll get exactly what you expect.
-
@David: Eep! Yes. – Brian M. Scott Mar 31 '12 at 20:13 | 2015-04-28 00:46:48 | {"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": 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": 0.9609155058860779, "perplexity": 247.78022846916414}, "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/1429246660448.47/warc/CC-MAIN-20150417045740-00018-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://paulsfishandchickenbar.com/article55.php | # Wiki Latex Symbols Guide
• List of LaTeX symbols | LaTeX Wiki | Fandom
• Wikipedia:LaTeX symbols - Wikipedia
• The Comprehensive LaTeX Symbol List
• Aiuto:Formule matematiche TeX - Wikipedia
• A simple guide to LaTeX - Step by Step
• ## List of LaTeX symbols | LaTeX Wiki | Fandom
LaTeX symbols have either names (denoted by backslash) or special characters. They are organized into seven classes based on their role in a mathematical expression. This is not a comprehensive list. Refer to the external references at the end of this article for more information. Letters are... An online LaTeX editor that's easy to use. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article!This article doesn't yet, but we're working on it! See more info or our list of citable articles.
### The Great, Big List of LaTeX Symbols
(Where two symbols are present, the left one is the “faked” symbol that LATEX2εprovides by default, and the right one is the “true” symbol that textcomp makes available.) 7. Table 19: AMS Delimiters p \ulcorner q \urcorner x \llcorner y \lrcorner ... The Great, Big List of LaTeX Symbols ... Robbie and her fellow 'Birds of Prey' stars talk to Fandom about bringing Harley Quinn, Huntress, Black Canary and more together on the big screen.
### latex - backslash and tilde symbols in latex - By ...
backslash and tilde symbols in latex - The
### List of LaTeX mathematical symbols - OeisWiki
List of LaTeX mathematical symbols. From OeisWiki. There are no approved revisions of this page, so it may not have been reviewed. Jump to: navigation, search. All the predefined mathematical symbols from the T e X package are listed below. More symbols are available from extra packages. Contents. There are two major modes of typesetting math in LaTeX one is embedding the math directly into your text by encapsulating your formula in dollar signs and the other is using a predefined math environment. You can follow along and try the code in the sandbox below. I also prepared a quick reference of math symbols. LaTeX (uitspraak: la-tech; traditioneel geschreven als ) is een softwaresysteem voor het zetten van documenten. Het is populair in de wetenschappelijke wereld omdat het uitblinkt in het zetten van technische documenten, en beschikbaar is voor bijna alle computersystemen.
## Wikipedia:LaTeX symbols - Wikipedia
### LaTeX/Mathematics - Wikibooks, open books for an open world
If you are writing a scientific document that contains numerous complicated formulas, the amsmath package introduces several new commands that are more powerful and flexible than the ones provided by basic LaTeX. The mathtools package fixes some amsmath quirks and adds some useful settings, symbols, and environments to amsmath. This page contains references to core documentation about LaTeX written by the LaTeX team.Articles on specific topics, talks, etc. can be found on the publication page indexed by topic. In addition pointer to documentation in other languages (usually developed and maintained by user groups) is given. The following is a complete list of both the kanji and the various symbols that appear in the Dragon Ball series. Trivia Many of these symbols are available to put on your customized characters clothing or skin in the video game Dragon Ball Z: Ultimate Tenkaichi, such as the Ginyu Force symbol, the Demon mark, and many others., In Dragon Ball Xenoverse 2, there are several Gis, Battle Suits ...
### LaTeX – Wikipedia
LaTeX ( [ˈlaːtɛç]), in Eigenschreibweise , ist ein Softwarepaket, das die Benutzung des Textsatzsystems TeX mit Hilfe von Makros vereinfacht. LaTeX liegt derzeit in der Version 2 ε vor. LaTeX wurde Anfang der 1980er Jahre von Leslie Lamport entwickelt. Der Name bedeutet so viel wie Lamport TeX.Die Entwicklung wurde seit den 1990er Jahren von einer Anzahl Entwicklern weitergeführt. LaTeX (parfois typographié L A T E X) est un langage et un système de composition de documents. Il s'agit d'une collection de macro-commandes destinées à faciliter l'utilisation du « processeur de texte » TeX de Donald Knuth.. LaTeX permet de rédiger des documents dont la mise en page est réalisée automatiquement en se conformant du mieux possible à des normes typographiques.
### LaTeX - OeisWiki
You can use \dots in both text and math mode and LaTeX will replace it with three dots "…" but it will decide according to the context whether to put it on the bottom (like \ldots) or centered (like \cdots). Dots. LaTeX gives you several commands to insert dots in your formulae. pitch), and an extensive set of mathematical symbols. It’s also possible to use other font families, such as Times, Palatino, etc. •TEX is also a programming language, making it possible to create commands that simplify its use. 1.2 What is LATEX? •LATEX is a TEX macro package, originally written by Leslie Lamport, that simplifies the ...
## The Comprehensive LaTeX Symbol List
1 Introduction Welcome to the Comprehensive LATEX Symbol List!This document strives to be your primary source of LATEX symbol information: font samples, LATEX commands, packages, usage details, caveats—everything needed to put thousands of different symbols at your disposal. LaTeX (pronúncia em inglês: [ˈlɑːtɛx] ou uma abreviação de Lamport TeX) é um sistema de preparação de documentos.Ao escrever, o escritor usa texto simples, ao invés do texto formatado encontrado em processadores de texto WYSIWYG como Microsoft Word, LibreOffice Writer e Apple Pages.O escritor utiliza convenções de tagging de marcação para definir a estrutura geral do documento ...
### Beginner’s LaTeX Guide
Beginner’s LaTeX Guide This guide should be used as a starting point for learning Latex (pronounced \lay-tek"), but is in no way a full manual. LATEX Mathematical Symbols The more unusual symbols are not defined in base LATEX (NFSS) and require \usepackage{amssymb} 1 Greek and Hebrew letters α \alpha κ \kappa ψ \psi z \digamma ∆ \Delta Θ \Theta β \beta λ \lambda ρ \rho ε \varepsilon Γ \Gamma Υ \Upsilon
The LaTeX Wiki has two main branches. It is a user's guide and language reference for LaTeX, a markup language for documents and web pages. This site tries to achieve two distinct goals: to provide specific information about LaTeX quickly and efficiently, and to teach the LaTeX language and how... Ponctuation. Selon le Lexique des règles typographiques en usage à l'Imprimerie nationale éd. 2002, p. 110, la ponctuation s'applique aux formules mathématiques, y compris celles qui sont centrées. Elles doivent donc notamment comporter un point si c'est la fin d'une phrase. escrito LaTeX en texto plano) es un sistema de composición de textos, orientado a la creación de documentos escritos que presenten una alta calidad tipográfica.Por sus características y posibilidades, es usado de forma especialmente intensa en la generación de artículos y libros científicos que incluyen, entre otros elementos, expresiones matemáticas.
### LaTeX - Wikipedia
LaTeX is widely used in academia for the communication and publication of scientific documents in many fields, including mathematics, statistics, computer science, engineering, chemistry, physics, economics, linguistics, quantitative psychology, philosophy, and political science. It also has a prominent role in the preparation and publication of books and articles that contain complex ... LaTeX is a typesetting system that allows Campuswire users to conveniently type all the symbols and functions they’ll need to correctly format mathematical expressions (no more handwritten ...
### Art of Problem Solving
Latex, or , (pronounced lah-TEK or lay-TEK) is a typesetting markup language that is useful to produce properly formatted mathematical and scientific expressions.. Using LaTeX in the Community, the AoPSWiki, or the Classroom. If you are just learning for use on the AoPS site, then go to the LaTeX on AoPS page. If you would like to produce full documents of your own LaTeX, please follow the ... This document is an unofficial reference manual for LATEX, a document preparation system, version of October 2018. This manual was originally translated from LATEX.HLP v1.0a in the VMS Help Library. In this tutorial I will show you how to create a very basic Latex document. The code from this tutorial is posted on the page for you to try it out. The video is here uploaded on Youtube with ...
## Aiuto:Formule matematiche TeX - Wikipedia
Questa pagina è la traduzione della pagina inglese meta:Help:Formula.Verrà aggiornata di tanto in tanto, ma la pagina inglese resta la guida di riferimento. Dal gennaio 2003, è stata aggiunta la possibilità di usare su Wikipedia dei comandi TeX per formule matematiche.. Ogni markup matematico deve rientrare all'interno dei due tag $\dots$ Le interruzioni fisiche di linea all ... The Carleton LaTeX workshop offers several guides to help you get started using LaTeX. Getting started with LaTeX (View on WriteLaTeX) If you've never used LaTeX before, this is the place to start. This guide will walk you through everything you need to do to get started, including: What LaTeX is (and isn't!) How to use LaTeX on your computer
### List of mathematical symbols by subject - Wikipedia
This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are included. The guide LATEX2" Font Selection describes the LATEX font selection scheme for class- and package-writers; it is in fntguide.tex. Support for Cyrillic languages in LATEX is described in Cyrillic languages support in LATEX. The documented source code (from the les used to produce the kernel format via latex.ltx) is now available as The LATEX2 ... Most import, this post is showing you the basics about math symbols in Latex. This what wikipedia said about Latex: One of the greatest motivating forces for Donald Knuth when he began developing the original TeX system was to create something that allowed simple construction of mathematical formulas, while looking professional when printed.
### LaTeX/Text Formatting - Wikibooks, open books for an open ...
This section will guide you through text-formatting techniques. ... LaTeX is so flexible that we will actually only skim the surface, as you can have much more control over the presentation of your document if you wish. ... No special characters or symbols are allowed in the argument. Example: \hyphenation {FORTRAN Hy-phen-a-tion} A playlist to help get you started using LaTeX, a free and powerful typesetting program used most often for composing documents about mathematics and science...
## A simple guide to LaTeX - Step by Step
A simple guide to LaTeX - Step by Step Learn about LaTeX in short lessons with full code examples. A comprehensive guide to basic and advanced features. 1 Leslie Lamport. LATEX—A Document Preparation System—User’s Guide and Reference Manu-al. Addison-Wesley, Reading, MA, USA, 1985. 2 Donald E. Knuth. The TEXbook, volume A of Computers and Typesetting. Addison-Wesley, Reading, MA, USA, 1986. 3 Michel Goossens, Frank Mittelbach and Alexander Samarin. The LATEX Companion Addison-
### Symbols - Art of Problem Solving
Finding Other Symbols. Here are some external resources for finding less commonly used symbols: Detexify is an app which allows you to draw the symbol you'd like and shows you the code for it!; MathJax (what allows us to use on the web) maintains a list of supported commands.; The Comprehensive LaTeX Symbol List. AMS-LaTeX Prerequisites. A working TeX system TeX is not an AMS ... amsfonts is a collection of fonts that extends the Computer Modern family with additional symbols and alphabets intended for use in mathematical ... Short Math Guide for LaTeX This guide is a concise summary of the essential features in LaTeX for writing ... please, add support for comparision symbols! those symbols are in unicode and a way much easier to input than use google to find some indirect way to math-syntax help! symbols are "≤" and "≥" Tex was designed for old-time punch cards, that lacked maaaany symbols. Now that symbols are given, please make use of them!
### The Comprehensive LaTeX Symbol List - CTAN
The Comprehensive LATEX Symbol List Scott Pakin ∗ 19 January 2017 Abstract This document lists 14283 symbols and the corresponding LATEX ... LaTeX (scritto anche L A T E X; pronunciato /ˈlatek/, e non /ˈlateks/, perché la X è in realtà una chi maiuscola) è un linguaggio di markup per la preparazione di testi, basato sul programma di composizione tipografica T E X; la versione attuale è chiamata LaTeX2ε, LaTeX3 è in corso di sviluppo. introduction. Wikidot uses a markup language called LaTeX (pronounced "lay-tech") along with jsMath to generate properly typeset mathematics. $\LaTeX$ requires memorizing (or looking up) "commands" for creating certain kinds of characters. Since you'll be posting to the forum and creating wiki pages with serious mathematical content, it's good that you get some practice writing mathematical ...
### Math formatting guide - Developers Wiki | HackerEarth
Math formatting guide. We now support latex throughout the site. ... To include math symbols in new line, use newline latex. \$ is the delimiter for both opening and closing delimiters for newline latex. Check the example below. Refer to wiki for complete list of latex symbols. si vous utilisez un codage unicode pour le fichier (comme utf8) : l'utilisation des caractères Unicode comme ×, ÷, ∀, ∃, ∈, … est actuellement déconseillée en mode mathématiques : les caractères ont des propriétés différentes (espacement, alignement vertical) selon que ce sont des quantificateurs, des opérateurs, … et LaTeX considère — pour l'instant — ces caractères ... | 2020-07-15 18:23:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "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": 0.9232552647590637, "perplexity": 9068.768824500665}, "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/1593657170639.97/warc/CC-MAIN-20200715164155-20200715194155-00599.warc.gz"} |
https://intl.siyavula.com/read/maths/grade-12/trigonometry/04-trigonometry-01 | We think you are located in United States. Is this correct?
# Chapter 4: Trigonometry
• Emphasize the value and importance of making sketches, where appropriate.
• It is very important for learners to understand that it is incorrect to apply the distributive law to the trigonometric ratios of compound angles and that $$\cos (\alpha - \beta) \ne \cos \alpha - \cos \beta$$.
• Emphasize that the area, sine and cosine rules do not require right-angled triangles.
• Remind learners that angles in the Cartesian plane are always measured from the positive $$x$$-axis.
• Important to note that $$(\text{270}° \pm x)$$ is not included in the curriculum.
• Note that the co-function for tangent is also not included in the curriculum.
• Remind learners to check that their answers are within the required interval.
• For the general solution, determine the solution in the correct quadrants and within the required interval.
• To prove that an identity is true, remind learners that they are only allowed to manipulate one side at a time.
• To prove identities, we usually manipulate the more complicated expression so that it looks the same as the more simple expression.
## 4.1 Revision (EMCG9)
Trigonometric ratios
We defined the basic trigonometric ratios using the lengths of the sides of a right-angled triangle.
$$\begin{array}{r@{\;}l@{\;}l@{\;}l@{\;}cr@{\;}l@{\;}l@{\;}l@{\;}} \sin \hat{A}&=&\frac{\text{opposite}}{\text{hypotenuse}} &= \frac{a}{c} & \qquad \qquad & \qquad \sin \hat{B}&=&\frac{\text{opposite}}{\text{hypotenuse}} &= \frac{b}{c} \\ & & & & & & & \\ \cos \hat{A}&=&\frac{\text{adjacent}}{\text{hypotenuse}} &= \frac{b}{c} &\qquad \qquad & \qquad \cos \hat{B}&=&\frac{\text{adjacent}}{\text{hypotenuse}} &= \frac{a}{c} \\ & & & & & & & \\ \tan \hat{A}&=&\frac{\text{opposite}}{\text{adjacent}} &= \frac{a}{b} & \qquad \qquad & \qquad \tan \hat{B}&=&\frac{\text{opposite}}{\text{adjacent}} & = \frac{b}{a} \end{array}$$
Trigonometric ratios in the Cartesian plane
We also defined the trigonometric ratios with respect to any point in the Cartesian plane in terms of $$x$$, $$y$$ and $$r$$. Using the theorem of Pythagoras, $$r^{2} = x^{2} + y^{2}$$.
CAST diagram
The sign of a trigonometric ratio depends on the signs of $$x$$ and $$y$$:
Reduction formulae and co-functions:
1. The reduction formulae hold for any angle $$\theta$$. For convenience, we assume $$\theta$$ is an acute angle ($$\text{0}°<\theta <\text{90}°$$).
2. When determining function values of ($$\text{180}°±\theta$$), ($$\text{360}°±\theta$$) and ($$-\theta$$) the function does not change.
3. When determining function values of ($$\text{90}°±\theta$$) and ($$\theta ±\text{90}°$$) the function changes to its co-function.
Negative angles
\begin{align*} \sin (- \theta) &= - \sin \theta \\ \cos (- \theta) &= \cos \theta \\ \tan (- \theta) &= - \tan \theta \end{align*}
Special angle triangles
These values are useful when we need to solve a problem involving trigonometric functions without using a calculator. Remember that the lengths of the sides of a right-angled triangle obey the theorem of Pythagoras.
θ $$\text{0}$$° $$\text{30}$$° $$\text{45}$$° $$\text{60}$$° $$\text{90}$$° $$\cos θ$$ $$\text{1}$$ $$\frac{\sqrt{3}}{2}$$ $$\frac{1}{\sqrt{2}}$$ $$\frac{1}{2}$$ $$\text{0}$$ $$\sin θ$$ $$\text{0}$$ $$\frac{1}{2}$$ $$\frac{1}{\sqrt{2}}$$ $$\frac{\sqrt{3}}{2}$$ $$\text{1}$$ $$\tan θ$$ $$\text{0}$$ $$\frac{1}{\sqrt{3}}$$ $$\text{1}$$ $$\sqrt{3}$$ undef
Trigonometric identities
Quotient identity:
$\tan \theta = \frac{\sin \theta}{\cos \theta} \qquad ( \cos \theta \ne 0)$
Square identity:
$\sin^{2} \theta + \cos^{2} \theta = 1$
It also follows that:
\begin{align*} \sin^2 \theta &= 1 - \cos^2 \theta \\ \cos^2 \theta &= 1 - \sin^2 \theta \\ & \\ \sin \theta &= \pm \sqrt{1 - \cos^2 \theta} \\ \cos \theta &= \pm \sqrt{1 - \sin^2 \theta} \end{align*}
All these relationships and identities are very useful for simplifying trigonometric expressions.
## Worked example 1: Revision
Determine the value of the expression, without using a calculator:
$\frac{\cos \text{420} ° - \sin \text{225} ° \cos ( - \text{45} °)}{\tan \text{315} ° }$
### Use reduction formulae to express each trigonometric ratio in terms of an acute angle
\begin{align*} &\frac{\cos \text{420} ° - \sin \text{225} ° \cos ( - \text{45} °)}{\tan \text{315} ° } \\ &= \frac{\cos ( \text{360} ° + \text{60} °) - \sin ( \text{180} ° + \text{45} °) \cos ( - \text{45} °)}{\tan ( \text{360} ° - \text{45} °) } \\ &= \frac{\cos \text{60} ° - ( - \sin \text{45} ° ) ( \cos \text{45} °)}{- \tan \text{45} ° } \\ &= \frac{ \cos \text{60} ° + \sin \text{45} ° \cos \text{45} ° }{- \tan \text{45} ° } \end{align*}
Now use special angles to evaluate the simplified expression:
\begin{align*} &= \frac{ \cos \text{60} ° + \sin \text{45} ° \cos \text{45} ° }{- \tan \text{45} ° } \\ &= \frac{ \frac{1}{2} + \left( \frac{1}{\sqrt{2}} \right)\left( \frac{1}{\sqrt{2}} \right) }{- 1} \\ &= - \left( \frac{1}{2} + \frac{1}{2} \right) \\ &= -1 \end{align*}
## Worked example 2: Revision
Prove:
$\sin^{2} \alpha - (\tan \alpha - \cos \alpha)(\tan \alpha + \cos \alpha) = \frac{\cos^{2} \alpha - \sin^{2} \alpha}{\cos^{2} \alpha}$
State restrictions where applicable.
### Use trigonometric identities to simplify each side separately
Simplify the left-hand side of the identity:
\begin{align*} \text{LHS} &= \sin^{2} \alpha - (\tan \alpha - \cos \alpha)(\tan \alpha + \cos \alpha) \\ &= \sin^{2} \alpha - (\tan^{2} \alpha - \cos^{2} \alpha) \\ &= \sin^{2} \alpha - \tan^{2} \alpha + \cos^{2} \alpha \\ &= \left( \sin^{2} \alpha + \cos^{2} \alpha \right) - \tan^{2} \alpha \\ &= 1 - \tan^{2} \alpha \end{align*}
Simplify the right-hand side of the identity so that it equals the left-hand side:
\begin{align*} \text{RHS} &= \frac{\cos^{2} \alpha - \sin^{2} \alpha}{\cos^{2} \alpha} \\ &= \frac{\cos^{2} \alpha}{\cos^{2} \alpha} - \frac{\sin^{2} \alpha}{\cos^{2} \alpha} \\ &= 1 - \tan^{2} \alpha \\ & \\ \therefore \text{LHS} &= \text{RHS} \end{align*}
Alternative method: we could also have started with the left-hand side of the identity and substituted $$\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$$ and simplified to get the right-hand side.
Restrictions
We need to determine the values of $$\alpha$$ for which any of the terms in the identity will be undefined:
\begin{align*} \cos^{2} \alpha &= 0 \\ \therefore \cos \alpha &= 0 \\ \therefore \alpha &= \text{90} ° \text{ or } \text{270} ° \end{align*}
We must also consider the values of $$\alpha$$ for which $$\tan \alpha$$ is undefined. Therefore, the identity is undefined for $$\alpha = \text{90} ° + k \cdot \text{180} °$$.
Useful tips:
• It is sometimes useful to write $$\tan \theta$$ in terms of $$\sin \theta$$ and $$\cos \theta$$.
• Never write a trigonometric ratio without an angle. For example, $$\tan = \frac{\sin }{ \cos}$$ has no meaning.
• For proving identities, only simplify one side of the identity at a time.
• As seen in the worked example above, sometimes both sides of the identity need to be simplified.
• Remember to write down restrictions:
• the values for which any of the trigonometric ratios are not defined;
• the values of the variable which make any of the denominators in the identity equal to zero.
# Test yourself now
High marks in maths are the key to your success and future plans. Test yourself and learn more on Siyavula Practice.
## Revision - reduction formulae, co-functions and identities
Exercise 4.1
Given: $$\sin \text{31} ° = A$$
Write each of the following expressions in terms of $$A$$:
$$\sin \text{149} °$$
\begin{align*} \sin \text{149} ° &= \sin ( \text{180} ° - \text{31} °) \\ &= \sin \text{31} ° \\ &= A \end{align*}
$$\cos ( - \text{59} °)$$
\begin{align*} \cos ( - \text{59} °) &= \cos \text{59} ° \\ &= \cos ( \text{90} ° - \text{31} °) \\ &= \sin \text{31} ° \\ &= A \end{align*}
$$\cos \text{329} °$$
\begin{align*} \cos \text{329} ° &= \cos ( \text{360} ° - \text{31} °) \\ &= \cos \text{31} ° \\ &= \sqrt{1 - \sin^{2} \text{31} °} \\ &= \sqrt{1 - A^{2}} \end{align*}
$$\tan \text{211} ° \cos \text{211} °$$
\begin{align*} \tan \text{211} ° \cos \text{211} ° &= \left( \frac{\sin \text{211} °}{\cos \text{211} °} \right) \cos \text{211} ° \\ &= \sin \text{211} ° \\ &= \sin ( \text{180} ° + \text{31} °) \\ &= - \sin \text{31} ° \\ &= - A \end{align*}
$$\tan \text{31} °$$
\begin{align*} \tan \text{31} ° &= \frac{\sin \text{31} °}{\cos \text{31} °} \\ &= \frac{A}{\sqrt{1 - A^{2}}} \end{align*}
Simplify $$P$$ to a single trigonometric ratio:
$P = \sin ( \text{360} ° + \theta) \cos ( \text{180} ° + \theta) \tan( \text{360} ° + \theta)$
\begin{align*} P &= \sin ( \text{360} ° + \theta) \cos ( \text{180} ° + \theta) \tan( \text{360} ° + \theta) \\ &= \sin \theta (- \cos \theta) ( \tan \theta) \\ &= \sin \theta \cos \theta \left( \frac{\sin \theta}{\cos \theta} \right) \\ &= \sin^{2} \theta \end{align*}
Simplify $$Q$$ to a single trigonometric ratio:
$Q = \frac{ \cos (\theta - \text{360} °) \sin ( \text{90} ° + \theta) \sin (- \theta)}{\sin (\theta + \text{180} °)}$
\begin{align*} \text{Note: } & \cos (\theta - \text{360} °) \\ &=\cos [- ( \text{360} ° - \theta)] \\ &=\cos ( \text{360} ° - \theta) \\ &=\cos \theta \\ & \\ Q &= \frac{ \cos (\theta - \text{360} °) \sin ( \text{90} ° + \theta) \sin (- \theta)}{\sin (\theta + \text{180} °)} \\ &= \frac{\cos \theta \cos \theta (- \sin \theta)}{- \sin \theta} \\ &= \cos^{2} \theta \end{align*}
Hence, determine:
1. $$P + Q$$
2. $$\dfrac{Q}{P}$$
1. \begin{align*} P + Q &= \sin^{2} \theta + \cos^{2} \theta \\ &= 1 \end{align*}
2. \begin{align*} \frac{Q}{P} &= \frac{\cos^{2} \theta}{\sin^{2} \theta} \\ &= \frac{1}{\tan^{2} \theta} \end{align*}
If $$p = \sin \beta$$, express the following in terms of $$p$$: $\frac{\cos (\beta + \text{360} °) \tan (\beta - \text{360} °) \cos (\beta + \text{90} °)}{\sin^{2}(\beta + \text{180} °) \cos(\beta - \text{90} °)}$
\begin{align*} \text{Note: } & \tan (\beta - \text{360} °) \\ &= \tan [- ( \text{360} ° - \beta)] \\ &= - \tan ( \text{360} ° - \beta) \\ &= - ( - \tan \beta ) \\ &= \tan \beta \\ \text{And } & \cos (\beta - \text{90} °) \\ &= \cos [- ( \text{90} ° - \beta)] \\ &= \cos ( \text{90} ° - \beta) \\ &= \sin \beta \\ & \\ & \frac{\cos (\beta + \text{360} °) \tan (\beta - \text{360} °) \cos (\beta + \text{90} °)}{\sin^{2}(\beta + \text{180} °) \cos(\beta - \text{90} °)} \\ &= \frac{ \cos \beta \tan \beta (- \sin \beta)}{(- \sin \beta )^{2} \sin \beta} \\ &= - \frac{ \cos \beta \left( \frac{\sin \beta}{\cos \beta} \right) \sin \beta}{( \sin^{2} \beta ) \sin \beta} \\ &= - \frac{ \sin^{2} \beta}{ \sin^{2} \beta \sin \beta } \\ &= - \frac{1}{ \sin \beta } \\ &= - \frac{1}{p} \end{align*}
Evaluate the following without the use of a calculator:
$$\dfrac{\cos (- \text{120} °)}{\tan \text{150} °} + \cos \text{390} °$$
\begin{align*} & \dfrac{\cos ( \text{120} °)}{\tan \text{150} °} + \cos \text{390} ° \\ &= \frac{ \cos( \text{180} ° - \text{60} °)}{ \tan ( \text{180} ° - \text{30} °)} + \cos( \text{360} ° + \text{30} °) \\ &= \frac{ - \cos \text{60} °}{ - \tan \text{30} °} + \cos \text{30} ° \\ &= \frac{ \sin \text{30} °}{\frac{ \sin \text{30} °}{\cos \text{30} °}} + \cos \text{30} ° \\ &= \cos \text{30} ° + \cos \text{30} ° \\ &= 2 \cos \text{30} ° \\ &= 2 \left( \frac{\sqrt{3}}{2} \right) \\ &= \sqrt{3} \end{align*}
$$(1 - \sin \text{45} °)(1 - \sin \text{225} °)$$
\begin{align*} & (1 - \sin \text{45} °)(1 - \sin \text{225} °) \\ &= 1 - \sin \text{45} ° - \sin \text{225} ° + (\sin \text{45} °)(\sin \text{225} °) \\ &= 1 - \sin \text{45} ° - \sin ( \text{180} ° + \text{45} °) + (\sin \text{45} °)(\sin ( \text{180} ° + \text{45} °)) \\ &= 1 - \sin \text{45} ° + \sin \text{45} ° - \sin^{2} \text{45} ° \\ &= 1 - \sin^{2} \text{45} ° \\ &= 1 - \left( \frac{1}{\sqrt{2}} \right)^{2} \\ &= 1 - \frac{1}{2} \\ &= \frac{1}{2} \end{align*}
Reduce the following to one trigonometric ratio:
$$\tan^{2} \beta - \dfrac{1}{\cos^{2} \beta}$$
\begin{align*} \tan^{2} \beta - \dfrac{1}{\cos^{2} \beta} &= \dfrac{\sin^{2} \beta}{\cos^{2} \beta} - \dfrac{1}{\cos^{2} \beta} \\ &= \dfrac{\sin^{2} \beta - 1}{\cos^{2} \beta} \\ &= \dfrac{-( 1 - \sin^{2} \beta)}{\cos^{2} \beta} \\ &= \dfrac{- \cos^{2} \beta}{\cos^{2} \beta} \\ &= -\text{1} \end{align*}
$$\sin^{2} ( \text{90} ° + \theta) \tan^{2} \theta + \tan^{2} \theta \cos^{2} ( \text{90} ° - \theta)$$
\begin{align*} &\sin^{2} ( \text{90} ° + \theta) \tan^{2} \theta + \tan^{2} \theta \cos^{2} ( \text{90} ° - \theta) \\ &= \cos^{2}\theta \tan^{2}\theta + \tan^{2}\theta \sin^{2}\theta \\ &= \tan^{2}\theta \left( \cos^{2}\theta + \sin^{2}\theta \right) \\ &= \tan^{2}\theta \left( 1 \right) \\ &= \tan^{2}\theta \end{align*}
$$\sin \alpha \cos \alpha \tan \alpha - 1$$
\begin{align*} \sin \alpha \cos \alpha \tan \alpha -1 &= \sin \alpha \cos \alpha \left( \frac{\sin\alpha}{\cos\alpha} \right) - 1 \\ &= \sin^{2} \alpha - 1 \\ &= - \left( 1 - \sin^{2} \alpha \right) \\ &= - \cos^{2} \alpha \end{align*}
$$\tan^2 \theta + \dfrac{\cos^2 \theta - 1}{\cos^2 \theta}$$
\begin{align*} \tan^2 \theta + \dfrac{\cos^2 \theta - 1}{\cos^2 \theta} &= \tan^2 \theta - \dfrac{ (1 - \cos^2 \theta)}{\cos^2 \theta} \\ &= \tan^2 \theta - \dfrac{ \sin^2 \theta}{\cos^2 \theta} \\ &= \tan^2 \theta - \tan^2 \theta \\ &= \text{0} \end{align*}
Use reduction formulae and special angles to show that
$\frac{\sin ( \text{180} ° + \theta) \tan ( \text{720} ° + \theta) \cos (- \theta)}{\cos ( \text{90} ° + \theta) }$
can be simplified to $$\sin \theta$$.
Use reduction formulae and co-functions to simplify the expression
\begin{align*} & \frac{\sin ( \text{180} ° + \theta) \tan ( \text{720} ° + \theta) \cos (- \theta)}{\cos ( \text{90} ° + \theta) }\\ &= \frac{ - \sin \theta \tan \left( 2( \text{360} °) + \theta \right) \cos \theta}{ - \sin \theta } \\ &= \tan \theta \cos \theta \\ &= \left( \frac{\sin \theta }{\cos \theta} \right) \cos \theta \\ &= \sin \theta \end{align*}
Without using a calculator, determine the value of $$\sin \text{570} °$$.
Use special angles to determine the value of the expression
\begin{align*} \sin \text{570} ° &= \sin ( \text{360} ° + \text{210} °) \\ &= \sin ( \text{210} °) \\ &= \sin ( \text{180} ° + \text{30} °) \\ &= - \sin \text{30} ° \\ &= - \frac{1}{2} \end{align*}
Question:
Prove that $$\dfrac{\cos\theta}{1+\sin\theta} = \dfrac{1-\sin\theta}{\cos\theta}$$.
\begin{align*} \dfrac{\cos\theta}{1+\sin\theta} &= \dfrac{1-\sin\theta}{\cos\theta}\\ (\cos \theta)(\cos \theta) &= (1+\sin\theta)(1-\sin\theta) \\ \cos^{2} \theta &= 1 -\sin^{2} \theta \\ \cos^{2} \theta &= \cos^{2} \theta \\ \therefore \text{LHS} &= \text{RHS} \end{align*}
Comment on Troy's answer and show the correct method for proving this identity.
The question requires that Troy prove the identity. However, by working with both sides of the identity at the same time, he accepted that it was true. The correct method for proving an identity is to work with only one side at a time and to show that one side equals the other. Sometimes it is necessary to first simplify one side of the identity, and then also to simplify the other side in order to show that they are equal. Troy also should have stated restrictions.
Correct method:
\begin{align*} \text{RHS} &= \dfrac{1-\sin\theta}{\cos\theta} \\ &= \dfrac{1-\sin\theta}{\cos\theta} \times \dfrac{1+\sin\theta}{1+\sin\theta} \\ &= \dfrac{1-\sin^{2}\theta}{\cos\theta(1+\sin\theta)} \\ &= \dfrac{\cos^{2}\theta}{\cos\theta(1+\sin\theta)} \\ &= \dfrac{\cos\theta}{1+\sin\theta} \\ &= \text{LHS} \end{align*}
Restrictions: undefined where $$\cos \theta = \text{0}, \text{ and } \sin \theta = - \text{1}$$.
So then $$\theta \ne \text{90} + k \cdot \text{180} ° \text{ and } \theta \ne - \text{90} + k \cdot \text{360} °$$.
Therefore $$\theta \ne \text{90} ° + k \cdot \text{180} °, \quad k \in \mathbb{Z}$$.
Prove the following identities:
(State any restricted values in the interval $$[ \text{0} °; \text{360} °]$$, where appropriate.)
$${\sin}^{2}\alpha + \left(\cos\alpha - \tan\alpha \right)\left(\cos\alpha +\tan\alpha \right)=1-{\tan}^{2}\alpha$$
\begin{align*} \text{LHS } &= \sin^2 \alpha + (\cos\alpha-\tan\alpha) (\cos\alpha+\tan\alpha) \\ &=\sin^2\alpha+\cos^2\alpha-\tan^2\alpha \\ &=1 - \tan^2\alpha \\ &= \text{RHS} \end{align*}
Restrictions: undefined where $$\tan \alpha$$ is undefined.
Therefore $$\alpha \ne \text{90} °; \text{270} °$$.
$$\dfrac{1}{\cos\theta}-\dfrac{\cos\theta {\tan}^{2}\theta }{1}=\cos\theta$$
\begin{align*} \text{LHS } &= \dfrac{1}{\cos\theta}-\dfrac{\cos\theta \tan^2\theta}{1}\\ &=\dfrac{1-\cos^2\theta \times \tan^2\theta}{\cos\theta}\\ &=\dfrac{1-\cos^2\theta \times \dfrac{\sin^2\theta}{\cos^2\theta}}{\cos\theta}\\ &=\dfrac{1-\sin^2\theta}{\cos\theta}\\ &=\dfrac{\cos^2\theta}{\cos\theta}\\ &=\cos\theta\\ &= \text{RHS} \end{align*}
Restrictions: undefined where $$\cos \theta = \text{0}$$ and where $$\tan \theta$$ is undefined.
Therefore $$\theta \ne \text{90} °; \text{270} °$$.
$$\dfrac{2\sin\theta\cos\theta}{\sin\theta +\cos\theta }=\sin\theta +\cos\theta -\dfrac{1}{\sin\theta +\cos\theta }$$
\begin{align*} \text{RHS } &= \sin\theta+\cos\theta-\dfrac{1}{\sin\theta+\cos\theta} \\ &=\dfrac{\sin^2\theta+\sin\theta \cos\theta+\cos\theta \sin\theta+\cos^2\theta-1}{\sin\theta+\cos\theta} \\ &=\dfrac{1+2\sin\theta \cos\theta-1}{\sin\theta+\cos\theta} \\ &=\dfrac{2\sin\theta \cos\theta}{\sin\theta+\cos\theta} \\ &= \text{LHS} \end{align*}
Restrictions: undefined where $$\sin \theta = \text{0}, \enspace \cos \theta = \text{0}$$.
Therefore $$\theta \ne \text{0} °; \text{90} °; \text{180} °; \text{270} °; \text{360} °$$.
$$\left(\dfrac{\cos\beta}{\sin\beta }+\tan\beta \right) \cos\beta =\dfrac{1}{\sin\beta }$$
\begin{align*} \text{ LHS} &= \left ( \dfrac{\cos\beta}{\sin\beta}+\dfrac{\sin\beta}{\cos\beta} \right ) \cos\beta \\ &=\left ( \dfrac{\cos^2\beta + \sin^2\beta}{\sin\beta\cos\beta} \right )\cos\beta\\ &=\dfrac{1}{\sin\beta}\\ &= \text{RHS } \end{align*}
Restrictions: undefined where $$\sin \beta = \text{0}, \enspace \cos \beta = \text{0}$$ and where $$\tan \beta$$ is undefined.
Therefore $$\beta \ne \text{0} °; \text{90} °; \text{180} °; \text{270} °; \text{360} °$$.
$$\dfrac{1}{1 + \sin \theta} + \dfrac{1}{1 - \sin \theta} = \dfrac{2 \tan \theta}{\sin \theta \cos \theta}$$
\begin{align*} \text{ LHS} &= \dfrac{1-\sin\theta+1+\sin\theta}{(1+\sin\theta)(1-\sin\theta)} \\ &=\dfrac{2}{1-\sin^2\theta} \\ &=\dfrac{2}{\cos^2\theta} \end{align*} \begin{align*} \text{ RHS} &= \dfrac{2 \tan \theta}{\sin \theta \cos \theta} \\ &=\dfrac{2 \sin \theta}{\sin \theta \cos \theta \cos \theta} \\ &=\dfrac{2}{\cos^2\theta} \\ \therefore \text{LHS } &= \text{RHS } \end{align*}
Restrictions: undefined where $$\sin \theta = \pm \text{1}, \enspace \sin \theta = \text{0}, \enspace \cos \theta = \text{0}$$.
Restrictions also include the values of $$\theta$$ for which $$\tan \theta$$ is undefined.
Therefore $$\theta \ne \text{0} °; \text{90} °; \text{180} °; \text{270} °; \text{360} °$$.
$$(1 + \tan^2 \alpha) \cos \alpha = \dfrac{1 - \tan \alpha}{\cos \alpha - \sin \alpha}$$
\begin{align*} \text{LHS} &= (1 + \tan^2 \alpha) \cos \alpha \\ &= \left(1+\frac{\sin^2\alpha}{\cos^2\alpha} \right) \cos\alpha \\ &= \left( \frac{\cos^2\alpha+\sin^2\alpha}{\cos^2\alpha} \right) \cos\alpha \\ &=\dfrac{1}{\cos^{2}\alpha} \times \cos\alpha \\ &=\dfrac{1}{\cos\alpha} \end{align*} \begin{align*} \text{RHS} &= \dfrac{1 - \tan \alpha}{\cos \alpha - \sin \alpha} \\ &= \dfrac{1 - \frac{\sin \alpha}{\cos \alpha}}{\cos \alpha - \sin \alpha} \\ &= \dfrac{\frac{\cos \alpha - \sin \alpha}{\cos \alpha}}{\cos \alpha - \sin \alpha} \\ &= \dfrac{\cos \alpha - \sin \alpha}{\cos \alpha(\cos \alpha - \sin \alpha)} \\ &= \dfrac{1}{\cos\alpha} \\ &= \text{LHS } \end{align*}
Restrictions: where $$\sin \alpha = \cos \alpha$$ and where $$\tan \alpha$$ is undefined.
Therefore $$\alpha \ne \text{45} °; \text{90} °; \text{270} °; \text{225} °$$.
Determine whether the following statements are true or false.
If the statement is false, choose a suitable value between $$\text{0}$$ ° and $$\text{90}$$ ° to confirm your answer.
$$\cos ( \text{180} ° - \theta) = -1 - \cos \theta$$
False
\begin{align*} \text{Let } \theta &= \text{35} ° \\ \text{LHS} &= \cos ( \text{180} ° - \text{35} °) \\ &= \cos \text{145} ° \\ &= -\text{0,819} \\ & \\ \text{RHS} &= - 1 - \cos \text{35} ° \\ &= -\text{1,819} \\ & \\ \text{LHS} & \ne \text{RHS} \end{align*}
$$\sin (\alpha + \beta) = \sin \alpha + \sin \beta$$
False
\begin{align*} \text{Let } \alpha &= \text{62} ° \\ \text{Let } \beta &= \text{20} ° \\ \text{LHS} &= \sin( \text{62} ° + \text{20} ° ) \\ &= \sin \text{82} ° \\ &= \text{0,990} \\ & \\ \text{RHS} &= \sin \text{62} ° + \sin \text{20} ° \\ &= \text{1,224} \\ & \\ \text{LHS} &\ne \text{RHS} \end{align*}
$$\sin \alpha = 2 \sin \frac{\alpha}{2} \sin \frac{\alpha}{2}$$
True
$$\frac{1}{3} \sin 3\alpha = \sin \alpha$$
False
\begin{align*} \text{Let } \alpha &= \text{62} ° \\ \text{LHS} &= \frac{1}{3} \sin 3 ( \text{62} ° ) \\ &= -\text{0,034} \\ & \\ \text{RHS} &= \sin \text{62} ° \\ &= \text{0,882} \\ & \\ \text{LHS} &\ne \text{RHS} \end{align*}
$$\cos \beta = \sqrt{1 - \sin^{2} \beta}$$
True
$$\sin \theta = \tan \theta \cos \theta$$
True | 2021-04-10 23:00:16 | {"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": 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": 1.0000100135803223, "perplexity": 4063.4905036459427}, "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/1618038059348.9/warc/CC-MAIN-20210410210053-20210411000053-00050.warc.gz"} |
http://www.perimeterinstitute.ca/people/keith-copsey | Keith Copsey
University of Waterloo
Areas of Research:
Research Interests
My research has focused on the gravitational aspects of string theory in both AdS/CFT and cosmology. In AdS/CFT I have generally been involved in examining and testing the correspondence beyond the protection of supersymmetry by both modifying boundary conditions and studying non-supersymmetric spacetimes with striking properties using the usual supersymmetric boundary conditions. In particular I have recently been involved in showing that the current constructions of asymptotically Lifshitz spacetimes, conjectured to be dual descriptions of relatively common condensed matter systems, suffer from a subtle inner singularity that can not be resolved by standard string theory effects and that imposing the usual boundary conditions forbids generic perturbations to the spacetime deep in the bulk. Cosmologically I am interested in singularity resolution, inflation and possible alternatives to inflation, de Sitter in string theory, and understanding the extent to which a string theory landscape exists. I have recently pointed out that the usual methods constructing purportedly long-lived de Sitter vacua actually result in relatively rapidly decaying states, once gravity and some technical subtleties are taken into account. As a result, it is not clear that any constructions currently understood result in long-lived de Sitter vacua, let alone large families of such solutions.
Positions Held
• 2007-2010 Postdoctoral Fellow, University of Waterloo
Recent Publications
• K. Copsey and R. B. Mann, Pathologies in Asymptotically Lifshitz Spacetimes,'' arXiv: 1011.3502 JHEP {\bf 1103} (2011) 039
• K. Copsey, Gravitation and tunnelling: Subtleties of the thin-wall approximation and rapid decays,'' arXiv: 1108.2255
Seminars
• Lifshitz spacetimes: a cautionary tale,'' Black Holes VIII, Niagara Falls
• New puzzles for AdS/CFT: Bubble Trouble,'' USC
• New puzzles for AdS/CFT: Bubble Trouble,'' UCLA
• New puzzles for AdS/CFT: Bubble Trouble,'' Durham University
• New puzzles for AdS/CFT: Bubble Trouble,'' Imperial College, London
• Higher dimensional gravity, bubbles, and new puzzles for AdS/CFT,'' DAMTP, Cambridge
• PIRSA:11080145, Gravity and Rapid Tunnelling, 2011-08-24, Cosmology Talks
• Gravity and Rapid Tunnelling'', 2011-08-26, Strings group meeting
• "Gravitation and tunneling: Subtleties of the thin wall approximation and rapid decays" 2011-05-05, Cosmology group meeting
• Pathologies in Asymptotically Lifshitz Spacetimes,'' 2010-11-18, Strings group meeting
• PIRSA:08060167, Bubbles of Nothing and Violations of the Energy Conditions in AdS-CFT, 2008-06-06, PASCOS 08 | 2017-02-28 13:47:19 | {"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": 0.35939595103263855, "perplexity": 6828.105780534483}, "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-2017-09/segments/1487501174159.38/warc/CC-MAIN-20170219104614-00318-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://docutils.sourceforge.net/test/functional/output/math_output_mathjax.html | # Mathematics
Docutils supports inline math with the prefix or postfix :math: role specificator, $$n! + \sin(x_n^2)$$ and $$A_\text{c} = \frac{\pi}{4} d^2$$ , as well as displayed math via the math directive:
\begin{equation*} f(\epsilon) = \frac{1}{1 + \exp\left(\frac{\varepsilon}{k_\text{B}T}\right)} \end{equation*}
Content may start on the first line of the directive, e.g.
\begin{equation*} N = \frac{\text{number of apples}}{7} \end{equation*}
Equations can be labeled with a reference name using the :name: option. See eq:M and eq:schrödinger below.
The determinant of the matrix
\begin{equation*} \mathbf{M} = \left(\begin{matrix}a&b\\c&d\end{matrix}\right) \end{equation*}
is $$|\mathbf{M}| = ad - bc$$ .
More than one display math block can be put in one math directive. For example, the following sum and integral with limits:
\begin{equation*} \int_0^1 x^n dx = \frac{1}{n + 1} \end{equation*}
\begin{equation*} \sum_{n=1}^m n = \frac{m(m+1)}{2} \end{equation*}
LaTeX-supported Unicode math symbols can be used in math roles and directives:
The Schrödinger equation
\begin{equation*} i\hbar \frac{\partial }{\partial t}\Psi = \hat{H}\Psi , \end{equation*}
with the wave function $$\Psi$$ , describes how the quantum state of a physical system changes in time.
Math-Accents:
$$\acute{a}$$ \acute{a} $$\dot{t}$$ \dot{t} $$\hat{\gamma}$$ \hat{\gamma} $$\grave{a}$$ \grave{a} $$\ddot{t}$$ \ddot{t} $$\tilde{\alpha}$$ \tilde{\alpha} $$\breve{x}$$ \breve{x} $$\dddot{t}$$ \dddot{t} $$\vec{\imath}$$ \vec{\imath} $$\check{a}$$ \check{a} $$\bar{a}$$ \bar{a} $$\vec{R}$$ \vec{R}
Modulation Transfer Function:
\begin{equation*} \text{MTF} = \left|\frac{\mathcal{F}\{s(x)\}} {\mathcal{F}\{ s(x)\} |_{\omega _{x}=0}}\right| = \mathrm{abs}\left(\frac {\int _{-\infty }^{\infty }s(x) \mathrm{e}^{\mathrm{i}\omega _{x}x}\mathrm{d}{x}} {\int _{-\infty }^{\infty }s(x)\mathrm{d}{x}} \right). \end{equation*}
Math split over two lines: If a double backslash is detected outside a \begin{...} \end{...} pair, the math code is wrapped in an AMSmath align environment:
\begin{align*} s_{\mathrm{out}}(x) & = s_{\mathrm{in}}(x') * s_\delta (x-x') \\ & = \int s_{\mathrm{in}}(x')s_\delta (x-x')\mathrm{d}x' \end{align*}
Cases ("manually", with matrix environment):
\begin{equation*} \mathrm{sgn}(x) = \left\{\begin{matrix} -1 & x<0\\ 1 & x>0 \end{matrix}\right. \end{equation*}
Cases with the AMSmath cases environment (not (yet) supported by HTML writers with --math-output=MathML):
\begin{equation*} \mathrm{sgn}(x) = \begin{cases} -1 & x<0\\ 1 & x>0 \end{cases} \end{equation*} | 2018-01-24 01:40:00 | {"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": 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": 0.9825558662414551, "perplexity": 11849.666967976595}, "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-2018-05/segments/1516084892892.86/warc/CC-MAIN-20180124010853-20180124030853-00270.warc.gz"} |
http://ferocioustruth.com/ | Let Us Be Curious Disciples — of — Ferocious Truth
March 29, 2017
# Book Review: Freezing People is (Not) Easy
Views and opinions expressed are those of the author only.
Recently I read Freezing People is (Not) Easy: My Adventures in Cryonics by Bob Nelson.1 It recounts his time as president of the Cryonics Society of California (CSC) during which he cryopreserved and then attempted to maintain the cryopreservation of a number of early cryonics patients. I’ve been thinking about and investigating cryonics of late, so I went looking for any and all related material I could get my hands on. Since this book was the only one I could source from my local library, it ended up being the first one I read. I was looking to gain insight into why cryonics has not become more popular and to learn some of the challenges faced when attempting to run a cryonics organization. What I found instead was a sad tale of typical human failings. It should be noted that the author has been a very controversial figure and some might claim that the veracity of this book is questionable. He has been called a fraud and a con man. If his memoir is to be believed, he was simply seeking to help people fulfill their desires to be cryopreserved, but foolishly failed to adequately plan, prepare, or ensure monetary provision for the task.
It’s often said that one should start a story at the beginning. The question is: what is the beginning of the story one is telling? The first chapter describes Mr. Nelson’s difficult childhood and very early marriage under challenging circumstances. I found this chapter boring and extraneous. Even after finishing the whole book, I’m not sure what the purpose of including this material was. Perhaps it was to show why he cared about others so much that he fell into his later mistakes? If so, it did a poor job of that besides one anecdote of him helping a severely injured teenager when he was himself a teenage runaway. More likely it was in the hope that the reader might take pity on him, viewing him as a tragic hero.
Nelson first learned of cryonics in 19652 when an acquaintance pointed out a newspaper article discussing the recently published book The Prospect of Immortality by Robert Ettinger.3 He was immediately consumed by the idea of cryonics: that a person’s death could be postponed until technology advanced to the point where they could be healed. Searching local bookstores for a copy, he finally located a store that had them. Upon arrival, he found they hadn’t even removed the books from the shipping box because they had received them so recently. Reading through Mr. Ettinger’s book his excitement grew. Nelson loved astronomy and space travel, and was immediately thrilled by the potential of using cryonics technology to place astronauts in suspended animation for long-term spaceflight. He envisioned a new race, in the vein of the space race, to be the first country to revive an individual from cryopreservation. As soon as he became convinced of the scientific validity of the possibility of cryonics by consulting with several friends and the recognition of natural examples of suspended animation, he was a true convert. The strength of his faith in the advancement of technology to achieve near-miraculous results is evident.
I find it fascinating that a small number of individuals take immediately to the idea of cryonics upon learning of it while the vast majority dismiss it or are repulsed. What is it that sets these people apart? Nelson himself wonders about this in the book but offers no insights. Understanding this would seem to be a crucial first step to anyone seeking to encourage others to sign up for cryonics. This information is even more important for a cryonics organization whose success depends on it.
Nelson’s excitement led him to attend the first meeting of the Life Extension Society in California on May 13, 19664. He found the attendees’ scientific credentials and financial resources underwhelming. So, it should not have been so surprising to him when, at the second meeting, he was elected president despite being a TV repairman. He then established the nonprofit Cryonics Society of California (CSC) whose initial plan was to promote research into cryonics. A scientific advisory council was organized, and the CSC was to provide funding. The council informed them that if the CSC cryopreserved someone then the council would walk away and have nothing to do with them. The book is unclear on this point, but it appears the council, being composed of researchers in existing fields like cryobiology, thought that more direct involvement of the CSC with cryopreservation would hurt their reputations. This restriction was not concerning to Nelson as he felt more research was needed and didn’t plan to cryopreserve anyone. He was giddy from the success of forming the council and receiving the validation of scientists. Indeed, his excitement at this stage seems incredible. He states that “in December 1966, cryonics was a bright golden vista with limitless possibilities on the horizon.”5
I would have liked to hear more about the CSC’s structure and plans. Later we learn that they had membership dues, yet it is never stated how much the dues were or approximately how many members they had. The funding for the scientific advisory council was to come from CSC funds, and the researchers would apply for external grants as well. Details of how Nelson expected this to develop would have been helpful.
Early in 1967, Nelson was unexpectedly approached by the son of Dr. James Bedford, a psychology professor at the University of California, with a request to cryopreserve his father upon legal death. Despite having agreed with the advisory council not to cryopreserve anyone and being entirely unprepared, they decided to proceed with the cryopreservation of Dr. Bedford. The CSC scrambled to acquire the necessary equipment. Bedford’s son stated that there was “three hundred thousand dollars in a foundation for cryobiological research, and [he was] director of that foundation.”6 However, Bedford died before they had collected any money and almost before they had the necessary chemicals. They performed the cryopreservation regardless and promptly turned his body over to his son. The CSC never received a dime for their efforts.
Much later Nelson learned that the money from the estate had been spent on legal fees as Bedford’s wife and son defended his will and cryopreservation from lawsuits brought by other relatives. Bedford is widely considered to be the first genuine cryonics patient because his body was preserved immediately after his death and cryoprotection measures were taken. Over the years, Bedford’s son diligently maintained his father’s cryopreservation. Long-term support from relatives like this was rare, and Bedford is the only patient prior to 1974 that remains cryopreserved.7 His body is now in the care and custody of the Alcor Life Extension Foundation. While the cryopreservation of Dr. Bedford generated copious publicity for the CSC and cryonics, it also immediately led to the scientific advisory council withdrawing, just as they’d warned.
This began a series of further cryopreservations by Nelson and his associates. Three people were cryopreserved and their bodies held at a mortuary while arrangements were made to keep them at a cemetery. (The patients were legally regarded as dead and as such their bodies were required to be stored at a mortuary or cemetery.) In each case, they collected no money up front, and the CSC received only $10,300 in donations after the procedures. By treating all fees as donations and handling the transfer of the body under the Anatomical Gift Act, the CSC avoided creating any contractual obligation to care for the patients and hoped to shelter themselves from any legal problems. However, this did not prevent patients’ relatives from later suing. In all these cases, the plan was that relatives would pay for care after cryopreservation. This approach was predictably an egregious mistake, yet the reason for adopting it is never stated. These patients needed potentially indefinite care that relatives might lack the resolve or means to provide. What if relatives passed away before the patient was revived? I wish there had been some discussion of why this approach was chosen and considered viable. Nevertheless, it appears to have been the assumed practice at the time as all three organizations that performed cryopreservations followed it.7 In taking on these patients it seems Nelson was acting out of what he felt was a duty to care for these people. Yet, as any hospital will attest, one can’t give away unlimited free care. Doing so means an end to one’s ability to care for anyone. In addition to the three patients CSC cryopreserved, they took over care of a fourth, who had been frozen by Cryo-Care Equipment Corporation. At the time, this business was one of only three organizations performing cryonics, but it made no attempt at cryoprotection, advertising its services for cosmetic purposes rather than eventual revival.7 Nelson was given charge of this patient because he offered the family a lower rate for ongoing care. He made this offer partially because he wanted the cryonics capsule the body was in as he had not been able to acquire a usable one himself. Despite promising to pay$150 per month for storage, the family never paid any money and instead explicitly left his fate to the CSC.
Nelson’s “predicted boom in cryonics patients never materialized.”8 Without more paying patients, his failure to collect adequate fees or establish a revenue stream to pay for the patients’ ongoing care left the CSC in dire financial straights. He found himself paying out of pocket for the ongoing patient care. In 1970, Nelson was unable to continue the care of these patients and was left no other option than to allow them to thaw.9 According to Nelson, this was a very emotional moment for him, and he went out into the desert to decide if it was the right course of action. This is how he lost all four patients in his care at the time.
Despite this failure to care for the patients in his charge, Nelson took on the cryopreservation of two additional patients at a price of $10,000 for cryopreservation and a suggested donation of$100–$300 per month for storage costs. The$10,000 donation did not cover the costs of a cryonics capsule and was not collected from most patients. As he had done previously, he also took charge of another patient who was cryopreserved by a separate organization, the Cryonics Society of New York (CSNY). Nelson repeated all his mistakes. He failed to collect money in advance and had no better plan for ongoing care when, again, relatives failed to pay. He continued to pay for patient care out of pocket and using the CSC membership dues. Eventually, when he was out of town for a week, having left the patients under the watch of the cemetery’s groundskeeper, disaster struck. While Nelson was away, there was a failure of the vacuum pump that was maintaining the insulation of the poorly constructed capsule. The groundskeeper, speaking in heavily-accented, broken English, was unable to communicate the problem to the liquid nitrogen supplier, who had contracted to maintain the temperature of the capsule. Nelson returned to find that the bodies had completely thawed.
After some time spent in despair, Nelson refroze them. He did this despite knowing that they were almost certainly no longer viable for revival since he estimated they may have been without liquid nitrogen for five days. To his surprise, when he contacted the families they expressed the belief that there was nothing to be done but to keep them frozen and hope for the best. He had clearly explained to them when they initially signed up the consequences of a cryopreservation failure. Still, the families’ reactions are understandable given their emotional involvement. On the other hand, Nelson’s decision is inexplicable, especially since some of the families were not paying for ongoing care.
While all this was going on, Nelson also assisted a man named Nicholas DeBlasio in setting up a storage location on the grounds of a cemetery in New York for his wife Ann who had been cryopreserved by the CSNY. Mr. DeBlasio was unhappy with his dealings with the CSNY and wanted to handle everything himself. He did in fact successfully maintain his wife’s cryopreservation for some years, and even took charge of another patient, storing them in the same capsule. However, infrequent checking of and issues with the capsule eventually led to a partial thaw and the subsequent decision to terminate the patients. This case highlights that long-term cryopreservation necessitates a dedicated professional team.
Nelson’s wife had divorced him. Exhausted and disillusioned by his efforts and failures to maintain the patients, he eventually decided to leave the field of cryonics. Without funding to continue patient care he contacted the families of the patients and with their consent removed the patients from cryopreservation. However, Nelson’s story was not over, as several family members sued him and the cemetery he had worked with to store the bodies. Even though Nelson was broke, they sought damages from the cemetery’s liability insurance and needed him as part of their case to improve their chances of winning. At the time, there was also a lot of bad publicity as news organizations picked up the story of the thawing and trial. The, at times farcical, trial is described at some length. It did make for a good climax to the story but was ultimately not very enlightening.
The book concludes by describing Nelson’s life since that time and how he came to the decision to write his story. As with the first chapter describing his life before his involvement with cryonics, this felt extraneous to the story. It was made somewhat more palatable by having come to see him as a pitiable figure so that it was nice to hear he was able to move on with his life.
The image sometimes painted of Nelson is that of a fraud and swindler. That was how the attorneys of the families suing him attempted to portray him. In this book, he comes off as a sincere man trying to do his best to help others but ultimately failing. By not acting as a professional he repeatedly set himself up to fail. No one can provide a service without taking adequate payment. Indefinitely maintaining care of cryonics patients is a tremendous undertaking and not to be taken on without significant planning. Organizations with long-term financial obligations such as cemeteries and insurances companies have been known to fail. With cryonics one hopes that no room for failure would be left open. Despite this, Nelson and all the cryonics organizations operating at the time foolishly relied on family members for ongoing patient care funding. To me, this is simply another example of human irrationality. Modern cryonics organizations are, by comparison, much better managed. Still, it is not clear to me if they have adequately planned for the rigors of perpetual care.
Overall, the book was an enjoyable read for someone interested in cryonics. I think it serves as valuable documentation of the history of cryonics. However, it would have served that purpose much better had it detailed more the financial, organizational, and legal structure of the CSC; and explained the thinking behind having relatives fund patient care and not demanding payment up front for cryopreservation and the cryonics capsule. While I think it is good this book was written, I do worry that it will hurt the cryonics movement by dredging up the early failures. Readers may not make the distinction between the CSC and contemporary cryonics organizations.
1. Nelson, Robert F. Freezing People Is (Not) Easy: My Adventures in Cryonics. Guilford: Lyons Press, 2014. Print.
2. Freezing People Is (Not) Easy, p. 21.
3. Ettinger, Robert C.W. The Prospect of Immortality. New York: Doubleday & Co, 1964. Print.
4. Freezing People Is (Not) Easy, p. 30
5. Freezing People Is (Not) Easy, p. 41
6. Freezing People Is (Not) Easy, p. 45
7. Freezing People Is (Not) Easy, p. 82
8. Freezing People Is (Not) Easy, p. 84
March 15, 2017
# Automoderation
Views and opinions expressed are those of the author only.
[For a quick review, jump to the summary or chart of signs.]
Group discussions can easily become unwieldy. Certain individuals have a tendency to dominate the conversation, limiting others’ ability to contribute. When the goal is collaborative truth seeking, this is counter productive. Those who are excluded from the discussion may be exactly the ones with the necessary insights to advance the dialogue toward the truth. To deal with this and other issues, a moderator is sometimes appointed. However, traditional moderation may introduce too much friction and is geared toward potentially combative dialogue. A lighter-weight system intended for more collaborative discussions is helpful. To address this, a system of hand signals called automoderation was created. Automoderation has been used in the rationality community in Columbus, Ohio, for almost two years now with good success.
The rationality community should ostensibly be seeking the truth in any discussion. One might imagine that rationalists, cognizant of the pitfalls of group discussion and genuinely seeking truth, would manage their discussions better than others. However, human nature still dominates and there are often individuals involved who have underdeveloped social skills. For example, they may struggle to signal that they wish to speak or fail to notice the social signals that they are inappropriately dominating the conversation. Additionally, a focus on thinking through the subject and developing genuine insight often draws cognitive resources away from preparing a quick response as might be needed to jump into the middle of a conversation. It can be helpful, in that situation, to have a method of signaling one’s desire to speak, and be given adequate time by the group to formulate one’s thoughts.
In larger and more formal settings a moderator can be appointed. They become responsible for determining who may speak and for how long. They facilitate discussion, ensuring that all members with standing are given a chance to speak and no one inappropriately dominates the conversation. A moderator can also facilitate metadiscussion about procedure, topics, and participant needs while ensuring the debate isn’t derailed by them. However, in smaller or less formal settings a traditional moderator may not be called for and there may be no one willing to fill the role. Yet, the smaller or less formal setting does not always obviate the need for some degree of moderation.
The rationality community in Columbus, Ohio, found itself in the position of needing a system of moderation for their discussions, in particular for a rationality dojo. A little over two years ago Max Harms along with another member of the community created a system of hand signals supporting moderation in smaller, less formal settings. This system was inspired by the Occupy movement hand signals. When all participants know the hand signals, a moderator may not even be needed. A moderator is still useful, but often does little besides clarifying the system and consequently introduces very little friction. This system of hand signals is called automoderation. It has been used successfully in groups as small as 3 to 4 people and as large as 15 to 20.
Automoderation embodies a wait rather than an interrupt culture. Interrupt culture may be fine for a causal fun conversation, but collaborative truth seeking is aided by a wait culture. Traditional moderation systems such as Robert’s Rules of Order are intended to handle parliamentary proceedings and combative discussion. This system is intended for use in a more cooperative group discussion.
## Automoderation Hand Signals
The automoderation system requires participants to sit in a circle. The flow of who speaks proceeds clockwise based on hand signals used to indicate desire for and purpose of speaking. There are five hand signals each having a different priority. Higher priority signals take precedence. The primary signal is simply raising the hand indicating the desire to expand on the current discussion topic. People new to automoderation are encouraged to use expand if they are in doubt as to the correct signal to use. Each signal indicates a different intent of the speaker. In addition, some signals modify the flow of the discussion. In order from lowest to highest priority, they are a raised fist indicating change topic, the pinkie finger extended indicating a probing question, a raised hand indicating a desire to expand on the topic, the pointer finger extended indicating a clarifying question, and two hands forming a triangle with the thumbs and forefingers indicating meta point. The hand signals are illustrated in the chart below in order from lowest to highest priority. A high-resolution version of this chart suitable for printing as a poster can also be downloaded.
The meta signal is used to raise points regarding the nature or situation of the discussion. Meta signals are relatively rare but often important. They include such things as: keeping the discourse focused on the topic at hand; clarifying the rules of automoderation; requesting changes to the environment or situation such as getting a drink, taking a break, or going to the bathroom; requesting information about the discussion, for example, how long it is planned to run; addressing concerns of tone and undesirable behavior; and dealing with strong emotional reactions.
A clarifying question asks a speaker to clarify something they said. This should be confined to instances of genuine ambiguity or confusion about what someone meant. After the question has been asked, the person is given a chance to briefly respond to the question but should stay focused on answering the question and avoid making additional statements. Typically, a clarifying question is raised while someone is speaking and the question is asked of them. Technically though, a clarifying question could be asked of anyone, not just the last speaker. This can happen when the confusion is not noticed until after another speaker has started.
Expand signals a desire to continue the conversation and expand on what has been said. This includes sharing ideas, opinions, experiences, and facts that are relevant. It is also used to respond to other speakers’ points and raise further questions or areas to be discussed. The majority of the signals used in a discussion are typically expand. Do not waste time making statements indicating agreement with other speakers; instead, use nonverbal hand signals described later to indicate agreement or disagreement.
A probing question is any question directed at a particular participant that is not a clarifying question. Probing questions can be used to explore a point in more depth. As with clarifying questions, the person the question is directed to is given a chance to respond. A probing question could be asked of anyone but is normally directed at the last speaker. Often, it makes more sense to transform a probing question into a statement of confusion or curiosity that can be directed to the whole group using the expand signal instead. This avoids the lower priority of the probing question signal and the potential for creating too much back and forth. It also gives other participants who may be better able to address the question the chance to weigh in.
Change topic signals that a participant wishes to change the topic. This can be because the current topic seems to have been discussed fully or there is simply a more interesting topic that they would like to discuss instead. Sometimes it is clear to multiple people that a topic is drawing to a close and one signal for a change topic will elicit multiple change topic signals in response. Generally, after a change in topic is proposed, a group decision is made, often using the approve and disapprove hand signals described later. One more pass around the circle may be allowed on the current topic so people can make closing remarks. If multiple people are requesting a change of topic, it is often worth seeing if there are multiple suggestions for the next topic and allowing the group to decide amongst them. There are situations when it may be appropriate to use meta to signal a desire to change topic instead. This might be the case if the current topic is too emotional for a participant to continue with or if, for reasons of time or agenda, it is important to move on to another topic even though the current topic is not complete. A change topic signal does not have to include a suggestion for the new topic.
## Variations
The automoderation system should be taken as a set of guidelines rather than rules. Each community and group should adapt it to their needs. Someday it may become more codified and it might make sense to be more strict. Automoderation is still new and certainly has room for improvements to be discovered through experimentation. Additionally, there are situations it doesn’t cover. Those are instances where a moderator may need to step in and make a judgment call.
As originally created, the flow of who speaks always continued clockwise from whoever the last speaker was, regardless of which hand signal led to their speaking. If multiple people were signaling, the one with the highest priority was selected. Ties between signals of the same priority were broken by selecting the first person clockwise from the last speaker. This has the virtue of being very simple and clear. However, it means that people can be skipped over. For example, someone wishing to expand might be skipped because a person after them is using the meta signal. The skipped person would then not be able to speak until the flow came back around the circle. The same problem can arise when a question is asked since the flow continues from the respondent.
### Simple Rules
1. When someone is done speaking, call on people who are signaling a desire to speak.
2. If two or more people are signaling, call on the one with the highest priority signal; break ties by going in a circle, clockwise from the last speaker.
3. If someone asks a question (probing or clarifying), the person they ask should respond; flow continues from the question answerer.
In practice, the Columbus rationality community has not followed the simple rules but rather continues flow from a point that allows the skipped participants to speak next. This works well but is more complicated to keep track of. In this variation, flow may continue either from the last speaker who was not responding to a question or from a previous speaker depending on the relative priority of the signals involved. When multiple people are signaling, the one with the highest priority is selected. Ties are broken by selecting the first person clockwise from the appropriate participant. When the last speaker was expanding or requesting a change of topic, flow continues from their left. When the last speaker was responding to a probing question, flow continues from the left of the questioner. When first going up to the priority of a clarifying question or meta point from a lower priority signal, note the last speaker or if the last speaker was responding to a probing question, note the questioner. Flow continues from the noted person when returning to lower a priority. When the last speaker was making a meta point, flow continues to their left if there are more meta signals; otherwise, flow continues from the noted speaker. When the last speaker was responding to a clarifying question, flow continues from the questioner’s left if there are more clarifying questions or meta points; otherwise, flow continues from the noted speaker. If your group feels comfortable with this approach, it is the one currently recommended by the Columbus rationality community.
### Standard Rules
1. When someone is done speaking, call on people who are signaling a desire to speak.
2. If two or more people are signaling, call on the one with the highest priority signal; break ties by going in a circle, clockwise from the appropriate participant.
3. When the last speaker was expanding or changing the topic, flow continues from their left. If the next speaker is asking a clarifying question or raising a meta point, note the current speaker. This is the point flow will continue from when returning to a lower priority.
4. When the last speaker was responding to a probing question, flow continues from the questioner’s left. If the next speaker is asking a clarifying question or raising a meta point, note the questioner. This is the point flow will continue from when returning to a lower priority.
5. When the last speaker was making a meta point, flow continues from their left if the next speaker is also making a meta point; otherwise, flow continues from the previously noted participant.
6. When the last speaker was responding to a clarifying question, flow continues from the questioner’s left if the next speaker is asking a clarifying question or making a meta point; otherwise, flow continues from the previously noted participant.
7. If someone asks a question (probing or clarifying), the person they ask should respond.
Other variations on the system have been suggested but not tried to determine whether they are better. Some community members have suggested eliminating the probing question signal as it is rarely used. Its priority below expand discourages use. It can lead to back and forth discussion, and expand is typically a better signal to use instead. Other community members have suggested making the change topic signal not be a request to speak. Like the signals described below, it would be used only to communicate nonverbally to the group. If one wished to talk about the change of topic, they could raise a hand at the same time. This would go well with the removal of probing question as it would eliminate all signals with priority below expand.
In practice in Columbus, the change topic and probing question signals are sometimes treated as having the same priority as expand. This simplifies things slightly and doesn’t seem to cause problems. However, it does open up the possibility that someone could force discussion on a change of topic before other people are ready.
## Other Hand Signals
In addition to the automoderation hand signals, the community has found it helpful to have a few other hand signals to facilitate communication without adding any auditory distraction to the discussion. The first few of these are so useful that they could be considered for official inclusion in the automoderation system. The rest of the signals should probably be taken as possible suggestions if the situation arises or a community finds the need for them.
A thumbs up is used to signal approval or agreement and a thumbs down to signal disapproval or disagreement. These can be used for informal voting when a change of topic or meta change is proposed. They may be used when someone makes a particularly good or enlightening point. It is important that they be interpreted and used to signal about the content of a speakers message and never as a judgment of the speaker themselves.
Two more signals are in occasional use. The first is the OK hand sign, which is used to indicate that you are listening and interested in what someone has to say. The second is to point the flat hand, palm down toward the speaker and wiggle the finger tips. This sign was imported by a few community members from CFAR and means “I feel you” or “This resonates with me”. That is subtly different from the thumbs up which implies something more like agreement. These signals have rarely been used in practice and may not be worth the complexity of having them.
It may be helpful to add a hand signal to keep track of where flow should start from when continuing after a meta or clarifying question. The noted speaker would make this sign so that others don’t have to mentally keep track of this. Flow would then continue from their left. Two suggestions for this signal have been made. The first is the “shaka sign” with the thumb and pinkie extended and the other fingers curled closed. The second suggestion is pointing to the left.
## Examples
To help clarify the rules, here are some examples following the standard rules described above. Imagine Amy, Ben, Cora, and Dan wish to hold a discussion. They sit in a circle in that order. Here are how a number of scenarios would play out. For clarity, each scenario starts with Amy speaking.
While Amy is speaking, Cora raises her hand to expand. When Amy is done speaking, Cora speaks next because she is the only one with a hand raised. Flow continues from Cora when she is done.
While Amy is speaking, Ben and Dan raise their hands to expand and Cora signals a meta point. When Amy is done speaking, there are signals of priority higher than expand, so it is noted that flow will continue from Amy in the future. Cora speaks next because her signal is higher priority. When Cora is done, flow continues from Amy, so Ben will speak next.
While Amy is speaking, Ben signals a probing question and Dan wishes to expand. When Amy is done speaking, Dan speaks next because expand is higher priority than a probing question. When Dan is done speaking, since no other people wish to expand, Ben may ask his probing question. He asks it of Amy, who then responds. Flow then continues from Dan because flow continues from the questioner rather than the respondent.
On the whole, the automoderation system has proved very successful and continues to be used in the Columbus, Ohio, rationality community. It is mainly used at the regularly scheduled meetings that are open to rationalists only; at these, everyone present knows the hand signals or can be expected to learn them as part of their initiation into the community. The advantages of using automoderation include:
• It ensures that everyone is given an opportunity to speak. This is particularly valuable for those members who are more introverted, take longer to formulate a response, or have underdeveloped social skills. Women have also reported that this helps them participate in the discussion as they may have been socialized to wait longer before speaking or to be less aggressive in entering a conversation. When women are trying to participate in a male-dominated discussion, the automoderation system highlights the shift to a wait culture for all involved.
• Having the system as a community norm enables members to initiate it when they see value in doing so. It is not uncommon, if there is a discussion going on in which someone feels they are not able to interject, for them to raise a hand. This serves as a reminder to the others present to make sure everyone is included in the discussion and often immediately triggers the use of automoderation.
• A group of people who know the system often need no moderator at all as it is clear to all participants who the next speaker is.
• The hand signals provide an easy nonverbal channel of communication to moderate the discussion that does not interfere in any way with the current speaker.
• Many raised hands and other signals can indicate to a speaker that others wish to speak and it may be a good idea to bring their current remarks to a close to allow others to speak. This is made more palatable by the knowledge they will have another chance to speak when it comes around the circle again.
• The requirement to select the appropriate hand signal can force one to clarify in their own mind the purpose of speaking and better plan their remarks.
• The hand signals can be useful in situations where automoderation is not being used. In particular, the meta symbol can be used to good effect outside automoderation to indicate a desire to interrupt a conversation with a meta item. Often this is something the conversation members will feel constitutes a reasonable interruption.
• By promoting wait culture, automoderation can help to train members of the community away from speaking as much as possible or as quickly as possible simply for the, often unconscious, purpose of gaining social status.
While automoderation is a useful social tool, it is not without its drawbacks. It is important to be aware of these to help mitigate them.
• There are times it would be better to allow a back and forth discussion between a few participants. Automoderation does not support this. This is a situation where a moderator can step in if needed.
• Discussion points can pile up, so that by the time one has an opportunity to speak, there are now many things to respond to. It can be hard to keep track of all the threads that are now being discussed in parallel.
• It is easy to lose track of what one was planning to say while waiting for others to speak.
• The pile up of points can encourage people to move on rather than continuing to explore a point that it might be valuable to delve into more.
• The topic can naturally drift as participants expand on each other’s statements.
• If a participant is long winded or doesn’t make positive contributions to the discussion, it can be difficult to limit their disruption of the discussion.
• Overuse of automoderation removes opportunity to practice valuable social discussion skills. Often, a conversation with fellow rationalists would otherwise be an ideal situation for this practice since they are likely to be more forgiving and supportive of one’s efforts to grow.
## Applicability
There are situations it may not be appropriate to use automoderation. It is good to be aware of these so that one can judge when it would be more appropriate to simply have a discussion or to use another moderation system.
• Automoderation breaks down in large groups. While it has been used with some success in groups of 15 to 20 that was only because not many participants actually wished to speak. Had all of them wished to speak, it likely would not have gone well.
• It should not be the default for casual conversations amongst groups of friends.
• Don’t use it for small groups of 3 to 4 unless the need becomes apparent or it is a more formal meeting.
• Automoderation is probably best suited for groups of 5 to 10 people.
• Automoderation is best suited for groups where all members are cognizant of the degree to which they can genuinely contribute to the discussion.
• Don’t use automoderation when not enough participants are familiar with the system.
## Etiquette
Some etiquette can help the automoderation system run more smoothly and ensure all participants feel included.
• If someone has had their hand up for a while either because they have been skipped or for a change of topic or probing question, it is polite to avoid raising one’s hand so that they may speak.
• If someone is wishing to speak, it is polite for the speaker to try and finish their point rather than going on at length, especially if they have a clarifying question.
• Don’t waste time using expand to state agreement; instead, use the nonverbal hand signal of a thumbs up.
## Summary
The hand signals in priority order, low to high, are:
• Change Topic (Fist) – request to move on to another topic.
• Probing Question (Pinkie Extended) – ask a question and receive a response.
• Expand (Raised Hand) – make further statements about the topic.
• Clarifying Question (Pointer Finger Extended) – ask a question to clear up confusion or ambiguity and receive a response.
• Meta (Triangle) – raise an issue about the nature or context of the discussion.
Hand signals that are not a request to speak:
• Approve/Agree (Thumbs Up)
• Disapprove/Disagree (Thumbs Down)
### Simple Variant
1. When someone is done speaking, call on people who are signaling a desire to speak.
2. If two or more people are signaling, call on the one with the highest priority signal; break ties by going in a circle, clockwise from the last speaker.
3. If someone asks a question (probing or clarifying), the person they ask should respond; then flow continues from the question answerer.
### Standard Variant
1. When someone is done speaking, call on people who are signaling a desire to speak.
2. If two or more people are signaling, call on the one with the highest priority signal; break ties by going in a circle, clockwise from the appropriate participant.
3. When the last speaker was expanding or changing the topic, flow continues from their left. If the next speaker is asking a clarifying question or raising a meta point, note the current speaker. This is the point flow will continue from when returning to a lower priority.
4. When the last speaker was responding to a probing question, flow continues from the questioner’s left. If the next speaker is asking a clarifying question or raising a meta point, note the questioner. This is the point flow will continue from when returning to a lower priority.
5. When the last speaker was making a meta point, flow continues from their left if the next speaker is also making a meta point; otherwise, flow continues from the previously noted participant.
6. When the last speaker was responding to a clarifying question, flow continues from the questioner’s left if the next speaker is asking a clarifying question or making a meta point; otherwise, flow continues from the previously noted participant.
7. If someone asks a question (probing or clarifying), the person they ask should respond.
### Experimental Variant
This experimental variant is based on feedback from members in the Columbus rationality community and has not been tried yet. It is like the standard variant with the following changes. Remove the probing question signal. The change topic signal is no longer a request to speak but is simply a nonverbal signal like approve or disapprove. Use the flow marker signal of pointing to the person to one’s left when going to a meta or clarifying question to indicate where flow should continue from.
## Conclusion
Automoderation has been a very useful addition to the social toolbox of the Columbus rationality community. It has helped many discussions go smoothly and people to contribute and feel more comfortable participating in the dialogue. This post was written in the hope that other rationalist communities will find it useful and spread and improve the system. Take whatever parts seem useful and apply them as seems best.
## Reactions
EDIT 2017-04-21: Added reactions, clarified chart order.
March 2, 2017
# Ferocious Truth
Views and opinions expressed are those of the author only.
Humans frequently fail to make their beliefs match what is true. There are many areas of life where this is typically not the case. When asked what a ball thrown in the air will do or whether the keys will be where they were left, almost all people will express beliefs that match reality. However, in many other areas of life they hold beliefs that bear no relation to what is true. This is most common in the areas of religion, morality, and philosophy. It is not limited to those though, but appears often where feedback on the veracity of belief is absent or indirect. Examples include astrology, crystal healing, and conspiracy theories. False beliefs are very widespread. According to a 2016 Gallup Poll, 79% of American’s believe in God.1 Another survey shows that 72% of American’s believe in an afterlife.2 Indeed, even a small percentage of “atheists” claim to believe in an afterlife according to a pew research poll.3 Large percentages of Americans believe in various conspiracy theories. A 2012 survey showed that 10% of Americans believe astrology is “very scientific” and 32% thought it was “sort of scientific”.4 Those are only a few of the more egregious examples of false beliefs. Everyone holds many minor beliefs that don’t comport with reality.
What are we to make of the fact that so many people hold false beliefs when most profess to be competent to determine the truth? Perhaps a better question to ask is: what reason is there to think that humans will believe true things? Humans are evolved animals, and the process of natural selection does not inherently select for individuals that hold true beliefs. Evolution selects for individuals that propagate their genes. In many instances, that is actually a reasonably good approximation for selecting for true beliefs. An individual holding proper beliefs on which animals are its predators and what is edible, will be more likely to survive and pass on its genes. Thus, there is reason to believe that in some areas human beliefs should tend to be true. However, in other areas there may have been no selection pressure or indeed have been selection for a tendency to false belief. It has been speculated that widespread belief in God is a result of over active agency detectors in the human brain. That is, a bias toward seeing external events as the result of the actions of another individual or animal rather than as mere coincidence. A human progenitor might be more likely to survive if they ascribe the rustle of some nearby grass to a predator rather than the wind. Even if doing so means they run for safety when it is not necessary, it ensures they will run truly when needed. Consistent with the hypothesis that we evolved to often hold false beliefs, cognitive psychology has identified many biases common to humans.
We humans have many reasons for holding the beliefs we do besides that the beliefs are the best explanation of the evidence. Holding a belief may be beneficial for fitting in with a social group. One may believe that they “ought” to hold a certain belief. For example, that it is a right or proper belief to hold, or that virtuous people hold that belief. Beliefs may have been learned from an authority and not questioned. Often, people do not have the spare cognitive resources to spend on questioning a belief, or feel that getting to the truth may be beyond their capabilities and so not worth the attempt. Frequently beliefs aren’t selected for truth, but for expediency or benefit. Cognitive science finds that humans are prone to engage in motivated reasoning. Where the desired belief has already been selected for some other purpose and the reasoning mind is engaged to provide justification to others or to oneself. Often motivated reasoning is used because facing the actual truth has lots of negative emotions attached to it. Ultimately, too many simply fear the truth. They find truth to be ferocious.
It is not so much the mundane beliefs and facts of everyday life that people find ferocious, but the answers to the great questions of morality, meaning, religion, and philosophy. These are what they find too ferocious to face. Yet, they already live in the world as it actually is. Psychologist Eugene Gendlin wrote about this in his book Focusing.
What is true is already so.
Owning up to it doesn’t make it worse.
Not being open about it doesn’t make it go away.
And because it’s true, it is what is there to be interacted with.
Anything untrue isn’t there to be lived.
People can stand what is true,
for they are already enduring it.
Eugene Gendlin
This quote is referred to as the “Litany of Gendlin” in the rationality community.5 This community grew on the website Less Wrong started in 2009 using material written by Eliezer Yudkowsky for the blog Overcoming Bias. In this community, rationality is considered to have two aspects: epistemic rationality and instrumental rationality. Epistemic rationality is having accurate beliefs about the world. Instrumental rationality is taking the best actions to achieve your goals, whatever those may be. In pursuit of being more rational, they focus a great deal on learning about and counteracting their own cognitive biases. When thinking about whether their beliefs are accurate, they like to use the metaphor of the map and the territory.
This metaphor originated with amateur philosopher Alfred Korzybski who remarked that “the map is not the territory.”6 The actual reality is compared to a territory and one’s beliefs to a map of the territory. This serves as an important reminder that our beliefs don’t always match reality, just as a map may not always be an accurate reflection of the territory it is meant to depict. Embedded in this metaphor is the view of truth philosophers term the correspondence theory of truth. That what makes something true or false is whether it properly corresponds to reality.
## Map Problems
Using the map and territory metaphor, we can more easily see that there are a wide variety of ways our beliefs can fail to match reality. Put another way, it is not simply that we hold false beliefs, but that there are many ways our beliefs can fail to correspond to reality.
### Gaps
One of the least problematic ways a map can fail to match the territory is to simply have gaps or blank spaces. Modern maps generally don’t have this. However, historical maps often show distant lands fade off into blank spaces or the unexplored interiors of regions are blank. Here, the map makes it clear that there is something there, but does not indicate what it is. We experience this in our beliefs when we consider something and find that we don’t know the answer. We are knowingly ignorant of a topic. This is generally less dangerous than other kinds of map/territory errors because at least we know that we don’t know.
### Omissions
A more serious issue is an omission from a map. As when a map fails to indicate the presence of a hazard or barrier. Reading the map, it seems there are no gaps; that nothing is missing. There is no indication of the omission. In our beliefs this is experienced as unknown ignorance. We believe we know about a given subject, but, unbeknownst to us, our beliefs have important omissions. This can be problematic because we make decisions believing we are adequately informed when we are not. We do not know to seek out more information to help us make a better decision, and so may make the wrong one.
### Mismatch
Before modern cartographic methods it was more common for a map to show the shape of a coastline or other feature incorrectly. The map indicates the presence of a feature, but describes that feature incorrectly. This can be almost as insidious as an omission. Again, we do not recognize the error in our map, in our beliefs. Yet, our beliefs are subtle distortions of reality, leading us astray.
### Mythical Lands
Perhaps the most insidious of errors is a map that shows features that don’t exist in the territory, as when ancient maps showed mythical lands. We may, like bearers of maps of lost pirate treasure, set off on hopeless quests. Beliefs that bear no correspondence to anything in the world are more likely to cause active pursuit of incorrect goals. The other map/territory errors tend to manifest in failure to act. This error, tends to manifest in incorrect action. Great harm can be caused when people act on this, often attempting to force others to conform to their false beliefs about the world.
## Correcting Maps
Given the many ways our maps can fail to match the territory and the potentially high price of the resulting mistakes, it behooves us to attempt to correct our maps. Indeed, that is the very project of epistemic rationality. Yet has the rationality movement actually made its members more rational? Consider that in a 2016 survey of Less Wrong users, only 48 of 1,660 or 2.9% of respondents answering the question said that they were “signed up or just finishing up paperwork” for cryonics.7 This despite the fact that Eliezer Yudkowsky has argued in the strongest terms possible that it is the rational thing to do, saying “I want you to live.”8 While this is certainly a much higher portion than the essentially 0% of Americans who are signed up for cryonics based on published membership numbers,9 it is still a tiny percentage when considering that cryonics is the most direct action one can take to increase the probability of living past one’s natural lifespan. If “rationality is systematized winning”10 then it would seem that involvement in the rationality community hasn’t been able to increase rationality very much. It has been objected to this characterization that the problem may not be a failure of epistemic rationality, but rather of instrumental rationality. That the Less Wrong site does not focus on this aspect of rationality. This objection is consistent with the fact that 515 or 31% of respondents to the question answered that they “would like to sign up,” but haven’t for various reasons. Beyond that, when asked “Do you think cryonics, as currently practiced by Alcor/Cryonics Institute will work?”, 71% of respondents answered yes or maybe.7 I will concede that Less Wrong does not focus on training instrumental rationality and there is a disconnect between beliefs and actions. However, the distinction between instrumental and epistemic rationality is not so clear cut.
If a reliable and trustworthy source said that for the entire day, a major company or government was giving out $100,000 checks to everyone who showed up at a nearby location, what would be the rational course of action? It might be argued that it is more likely the source is mistaken or lying, but assume for the sake of the argument that one does not believe this to be the case. Any course of action not involving going down and collecting the$100,000 would likely not be rational. To do so would be an indication that one didn’t actually believe they would receive the money upon going to claim it. Likewise, when offered a chance to live beyond our natural lifespan, responses that fail to result in actually taking that opportunity must call into question whether the opportunity is believed to be real. Yet, there is a large gap between those claiming to believe and those acting as we would expect based on that belief. This is the divide between professed and actual belief that too often occurs. So, we must find ways to correct not only our professed beliefs, but our actual beliefs.
Very few individuals are able to routinely seek and discover the truth. It is my experience that those who do place a great value on knowing the truth. They hold curiosity as a virtue. Curiosity admits its ignorance and seeks to replace it with knowledge. But curiosity is not sufficient unless it is paired with diligent study. Otherwise, it is simply idle curiosity. Diligent study comes when we whole heartedly follow after something. Personally, I place a high value on knowing the truth and learning new things. I wish this value was more widely shared. And so I say to all: let us be curious disciples of ferocious truth. | 2017-05-24 09:56:53 | {"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": 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": 0.3272775411605835, "perplexity": 1525.651856843155}, "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-2017-22/segments/1495463607811.15/warc/CC-MAIN-20170524093528-20170524113528-00096.warc.gz"} |
https://scribesoftimbuktu.com/find-the-mean-arithmetic-6-842-0/ | # Find the Mean (Arithmetic) 6 , 842 , 0
6 , 842 , 0
The mean of a set of numbers is the sum divided by the number of terms.
6+842+03
Simplify the numerator.
6+8423 | 2022-10-03 01:57:57 | {"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": 0.8886083960533142, "perplexity": 328.9185500298824}, "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/1664030337371.9/warc/CC-MAIN-20221003003804-20221003033804-00621.warc.gz"} |
https://codereview.stackexchange.com/questions/226448/count-the-frequency-of-integers-in-an-array | Count the frequency of integers in an array
I recently reviewed a question here on Code Review. The problem statement is
Write a program that prompts the user to input ten values between 80 and 85 and stores them in an array. Your program must be able to count the frequency of each value appears in the array.
I coded my own solution for the problem statement. My questions are
- Can this be optimized more?
- Did I miss anything in C++14 or C++17 that might improve the code?
- Are the variable and function names good or can they be improved?
My goals were to write the best C++ I could, remove any magic numbers and allow the solution to scale for different sets of numbers.
#include "pch.h"
#include <iostream>
#include <array>
const size_t INPUTSIZE = 10;
const size_t FREQUENCYSIZE = 6;
const int MINLEGALVALUE = 80;
const int MAXLEGALVALUE = 85;
std::array<int, INPUTSIZE> getUserInput()
{
std::array<int, INPUTSIZE> inputValues;
size_t i = 0;
do
{
int inputValue = 0;
std::cout << "Please enter a number between " << MINLEGALVALUE << " and " << MAXLEGALVALUE << ":" ;
std::cin >> inputValue;
if (inputValue >= MINLEGALVALUE && inputValue <= MAXLEGALVALUE)
{
inputValues[i] = inputValue;
i++;
}
else
{
std::cout << "The number must be between" << MINLEGALVALUE << " and " << MAXLEGALVALUE << "\n";
}
} while (i < INPUTSIZE);
return inputValues;
}
std::array<unsigned, FREQUENCYSIZE> getFrequencyCounts(std::array<int, INPUTSIZE> inputValues)
{
std::array<unsigned, FREQUENCYSIZE> freqs = { 0 };
for (auto inputs : inputValues)
{
freqs[inputs - MINLEGALVALUE]++;
}
return freqs;
}
void printFrequencies(std::array<unsigned, FREQUENCYSIZE> freqs)
{
unsigned rowLabel = MINLEGALVALUE;
for (auto frequency : freqs)
{
std::cout << rowLabel << " " << frequency << "\n";
rowLabel++;
}
}
int main()
{
std::array<int, INPUTSIZE> inputValues = getUserInput();
std::array<unsigned, FREQUENCYSIZE> freqs = getFrequencyCounts(inputValues);
std::cout << "\n";
printFrequencies(freqs);
}
• You've asked for optimizations, but the input size is too small for the usual histogramming optimizations to matter, and anyway the program spends all of its time doing IO. Do you want to know about them anyway? – harold Aug 19 at 19:56
• Can this be optimized more? The answer is basically always "yes". :P The real questions are "how", "how much", and go on from there into what speed vs. machine-code size footprint tradeoff you want to hit, and details about which inputs to optimize for (worst case vs. best case vs. average). (And of course for which ISA). Also optimizing for latency vs. throughput on an out-of-order execution CPU if the problem is small enough. – Peter Cordes Aug 20 at 11:13
• @PeterCordes You're right, I should have specified speed vs. size. I generally look for speed optimizations and accept the memory size trade off. – pacmaninbw Aug 20 at 11:19
• Sometimes you can shrink code-size without hurting speed. And for performance as a small part of a large program, smaller L1i cache footprint is an advantage. So sometimes the best thing for overall speed is to put some effort into code-size. (Often not, though, if there's anything you can usefully do with SIMD. But for histograms there isn't much until AVX512 scatter/gather + conflict-detection on x86, and even then it's not always worth it, and certainly not for a small problem size like 10 elements, not even one full vector of 32-bit counters.) – Peter Cordes Aug 20 at 14:22
Names
All-caps names are typically reserved for macros. They don't seem to me to make much sense for const variables. In fact, they only make minimal sense for object-like macros--they were originally used for function-like macros as kind of a warning that you should be cautious about passing an argument with side-effects, because they might happen more than once.
Minimize Magic
I'd typically try to keep the magic numbers to a minimum. For example, instead of defining FREQUENCYSIZE by itself, I'd probably do something like this:
const int lower_bound = 80;
count int upper_bound = 85;
const int frequency_size = upper_bound - lower_bound + 1;
Separation of Concerns
I'd at least consider separating validating data from reading the data. I'd prefer to have a function on the general order of:
bool valid(int val) {
return val >= lower_bound && val < upper_bound;
}
Class Usage
We have a number of different things related to reading and working with numbers in a specified range. It might be worth considering wrapping those bits and pieces into a coherent class for dealing with a value in a range, and let the outside world create and use objects of that class.
template <class T, T lower_bound, T upper_bound>
class bounded {
public:
static bool valid(T val) { return val >= lower_bound && val < upper_bound; }
friend std::istream &operator>>(std::istream &is, bounded &b) {
T val;
is >> val;
if (valid(val))
b.val = val;
else
is.setstate(std::ios_base::failbit);
return is;
}
friend std::ostream &operator<<(std::ostream &os, bounded const &b) {
return os << b.val;
}
size_t index() { return size_t(val - lower_bound); }
static constexpr size_t range() { return upper_bound - lower_bound + 1; }
private:
T val;
};
That let's us simplify the rest of the code a bit, something on this general order:
int main() {
using integer = bounded<int, 80, 85>;
std::array<integer, 10> inputs;
std::array<size_t, integer::range()> freqs {};
for (integer &i : inputs) {
std::cin >> i;
++freqs[i.index()];
}
for (auto freq : freqs)
std::cout << freq << "\n";
}
Technically, this doesn't meet the requirements as-is (e.g., it doesn't print out a prompt to tell the user to enter data), but I think it gives at least some idea of a direction things could go.
If you go for modern C++ the static variables should be marked as constexpr instead of plain old const.
As was said in the other question, it should be beneficial to create an array of length MAXLEGALVALUE - MINLEGALVALUE and directly index into that array. That way there is probably less memory consumed and we count automatically.
Personally I would use std::size_t or a well specified integer type like std::uint32_t rather than unsigned, which depends on the implementation.
In range based for loops where the type is unambiguous I am not really a fan of auto.
for (auto inputs : inputValues)
How do you know that copying it is cheap here? You have to check the type of the container. Also you should consider const correctness so rather use const int or const auto if you prefere that.
for (const int inputs : inputValues)
Note that you have a truncation warning here as MINLEGALVALUE is of type int:
unsigned rowLabel = MINLEGALVALUE;
• unsigned is actually a reasonable choice: the max count you might need to hold depends on how big your input array is. int / unsigned int are normally one of the fastest integer types that the machine can handle efficiently, unless it's an 8-bit implementation where unsigned char would be even faster. Obviously with a huge array of the same value repeated, with more than UINT_MAX entries, you could wrap around, but the OP is aiming for portable performance. Apparently even if that means a tradeoff in wrapping around in weird edge-case inputs (huge and unevenly distributed). – Peter Cordes Aug 20 at 11:22
One idea for simplification is to count frequencies directly instead of putting all the values in an array that you only use for counting frequencies. You can then remove the getFrequencyCounts function and the whole thing gets a little more efficient. This works well when you have a set of values in a dense range like [80,85]. If you have a lot of values far between eachother, using a set or unordered_set is probably a better choice. I found the code easy enough to read. Here are some ideas with comments in the code:
#include <array>
#include <iostream>
constexpr size_t INPUTSIZE = 10;
constexpr int MINLEGALVALUE = 80;
constexpr int MAXLEGALVALUE = 85;
constexpr size_t FREQUENCYSIZE = MAXLEGALVALUE - MINLEGALVALUE + 1;
std::array<unsigned, FREQUENCYSIZE> getUserInputFreq() {
std::array<unsigned, FREQUENCYSIZE> inputValues{0}; // initialize with 0
size_t i = 0;
do {
int inputValue = 0;
std::cout << "Please enter a number between " << MINLEGALVALUE << " and "
<< MAXLEGALVALUE << ":";
// make sure the istream you read from succeeded in extracting
if(std::cin >> inputValue) {
if(inputValue >= MINLEGALVALUE && inputValue <= MAXLEGALVALUE) {
// count frequencies directly if you don't actually need the
// input values
++inputValues[static_cast<size_t>(inputValue - MINLEGALVALUE)];
++i; // prefer prefix operator++
} else {
std::cout << "The number must be between" << MINLEGALVALUE << " and "
<< MAXLEGALVALUE << "\n";
}
} else
break; // erroneous input or EOF
} while(i < INPUTSIZE);
return inputValues;
}
void printFrequencies(std::array<unsigned, FREQUENCYSIZE> freqs) {
int rowLabel = MINLEGALVALUE;
for(auto frequency : freqs) {
std::cout << rowLabel << " " << frequency << "\n";
++rowLabel; // prefer prefix operator++
}
}
int main() {
std::array<unsigned, FREQUENCYSIZE> freqs = getUserInputFreq();
std::cout << "\n";
printFrequencies(freqs);
}
• Yes counting on the fly as you read data can be good (hiding store-forwarding latency for a run of the same number), but now you're hard-coding reading it from std::cin. That's even less generic. Plus, I think the OP was just using GetInput as part of the caller for the histogram function to make a complete example. Anyway, you could take an InputIterator as a function arg or something like that to let the caller specify where the input comes from. – Peter Cordes Aug 20 at 11:29
• @PeterCordes "now you're hard-coding reading it from std::cin" - That part was actually not changed from OP's implementation. I only changed the function name from getUserInput to getUserInputFreq and continued to use std::cin since the whole thing looked pretty focused on user interaction. – Ted Lyngmo Aug 20 at 12:06
• The original had one function that reads input, and a separate function that histogrammed a container (with an unfortunate hard-coding of the type). You're making it worse, not better, for separability / reusability. After inlining + optimization of an iterator, you can still have basically the same machine code you could get from your function, but with clean source. – Peter Cordes Aug 20 at 13:05
• @PeterCordes "The original had one function that reads input" - yes, from std::cin, with a prompt-like text. What I did was to reduce the number of steps and also to add status checking where appropriate. The only thing not saved is the order in which the numbers were entered. The original post didn't show any demand for it. if I were to make a generic input function, I'd make it read from a std::istream& but that didn't seem worth the effort since the whole getUserInput function has focus on input from a user. – Ted Lyngmo Aug 20 at 13:52 | 2019-09-18 15:00:22 | {"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": 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": 0.26240262389183044, "perplexity": 4559.72234438235}, "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-2019-39/segments/1568514573289.83/warc/CC-MAIN-20190918131429-20190918153429-00323.warc.gz"} |
https://pi2-docs.readthedocs.io/en/latest/reference/cleanmaxima.html | # cleanmaxima¶
Syntax: cleanmaxima(image, maxima, radius multiplier)
Removes all maxima that are smaller in radius than neighbouring maximum. Maximum is neighbour to another maximum if distance between them is less than radius of the larger maximum multiplied by radiusMultiplier. Removes all maxima $$m$$ that satisfy $$distance(m, n) < radiusMultiplier * radius(n)$$ and $$radius(n) > radius(m)$$ for some $$n$$. The distance is measured between centroids of the maxima. The maxima are removed by combining them to the larger maxima.
This command cannot be used in the distributed processing mode. If you need it, please contact the authors.
## Arguments¶
### image [input]¶
Data type: uint8 image, uint16 image, uint32 image, uint64 image, int8 image, int16 image, int32 image, int64 image, float32 image
The image where the maxima have been extracted (by localmaxima command).
### maxima [input & output]¶
Data type: int32 image
Image that contains the maxima. See output from localmaxima command.
Data type: real
Default value: 1
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https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/625/2/e/a/ | # Properties
Label 625.2.e.a Level $625$ Weight $2$ Character orbit 625.e Analytic conductor $4.991$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$
# Related objects
## Newspace parameters
Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.e (of order $$10$$, degree $$4$$, not minimal)
## Newform invariants
Self dual: no Analytic conductor: $$4.99065012633$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.58140625.2 Defining polynomial: $$x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$
## $q$-expansion
Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.
$$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{2} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{3} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{4} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{6} + ( 1 - 2 \beta_{2} - \beta_{4} - \beta_{6} ) q^{7} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{8} + ( 1 + \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{2} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{3} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{4} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{6} + ( 1 - 2 \beta_{2} - \beta_{4} - \beta_{6} ) q^{7} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{8} + ( 1 + \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{9} + 2 \beta_{3} q^{11} + ( 2 - \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{12} + ( -1 + \beta_{4} - \beta_{7} ) q^{13} + ( -2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{14} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{16} + ( 1 + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{7} ) q^{18} + ( 3 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} + ( 1 - \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{7} ) q^{21} + ( 2 - 2 \beta_{1} ) q^{22} + ( 2 - \beta_{1} + 2 \beta_{4} - \beta_{6} ) q^{23} + ( -4 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{24} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{26} + ( 2 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{27} + ( -1 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{5} - 3 \beta_{6} ) q^{28} + ( 1 + \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{29} + ( -3 + \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{31} + ( 2 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 6 \beta_{4} + \beta_{6} + \beta_{7} ) q^{32} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{33} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{34} + ( -4 + \beta_{1} - 3 \beta_{3} + 4 \beta_{4} - \beta_{6} ) q^{36} + ( -1 - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{37} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{38} + ( 2 - \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{39} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{41} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{42} + ( -3 - \beta_{1} + \beta_{2} - 5 \beta_{3} + 6 \beta_{4} + \beta_{6} - \beta_{7} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{44} + ( 3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{46} + ( -5 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{47} + ( 2 - 6 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{48} + ( 4 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{49} + ( -4 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{51} + ( -3 + 2 \beta_{1} - 3 \beta_{4} + 2 \beta_{6} ) q^{52} + ( 2 - \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{53} + ( -1 + \beta_{2} - 4 \beta_{4} - \beta_{5} + \beta_{6} ) q^{54} + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{56} + ( -1 - \beta_{1} - 3 \beta_{2} - 5 \beta_{3} + 3 \beta_{6} - \beta_{7} ) q^{57} + ( -3 + 4 \beta_{1} + \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{58} + ( -1 - 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{59} + ( 1 - 3 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{61} + ( 9 - 4 \beta_{2} + 7 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{62} + ( -4 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{63} + ( -4 - 2 \beta_{1} + \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{64} + ( -2 + 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{66} + ( 2 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{6} + 4 \beta_{7} ) q^{67} + ( 4 - 4 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - \beta_{6} - 4 \beta_{7} ) q^{68} + ( -3 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{69} + ( -5 + \beta_{1} + 6 \beta_{2} - 5 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( 2 + \beta_{1} - \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{72} + ( -3 + 3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{73} + ( -3 + \beta_{1} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{74} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{76} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{77} + ( -4 + 5 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{78} + ( -1 - 4 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 8 \beta_{7} ) q^{79} + ( 1 + 2 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} ) q^{81} + ( 1 - 3 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{82} + ( -1 + \beta_{1} + 5 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{83} + ( 1 - 2 \beta_{1} - 6 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - \beta_{5} - \beta_{7} ) q^{84} + ( 4 + \beta_{1} + 3 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{86} + ( -11 + 6 \beta_{2} - 5 \beta_{3} + 9 \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{87} + ( 4 - 2 \beta_{2} + 4 \beta_{3} ) q^{88} + ( 7 - \beta_{1} - 7 \beta_{4} - \beta_{5} + \beta_{7} ) q^{89} + ( 1 - 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{91} + ( -2 + 4 \beta_{1} - \beta_{2} - \beta_{3} + 5 \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{92} + ( 5 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} - \beta_{4} - 4 \beta_{6} + 5 \beta_{7} ) q^{93} + ( -3 - 5 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{94} + ( 1 - 3 \beta_{1} - 4 \beta_{2} + 7 \beta_{4} - 6 \beta_{7} ) q^{96} + ( -1 - 8 \beta_{1} - \beta_{2} - 8 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{97} + ( -6 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{98} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 5q^{3} + 4q^{4} + 6q^{6} + 10q^{8} + q^{9} + O(q^{10})$$ $$8q - 5q^{3} + 4q^{4} + 6q^{6} + 10q^{8} + q^{9} - 4q^{11} + 10q^{12} - 5q^{13} - 7q^{14} - 2q^{16} + 15q^{17} + 10q^{19} + q^{21} + 10q^{22} + 15q^{23} - 20q^{24} + 6q^{26} - 5q^{27} - 20q^{28} + 15q^{29} + q^{31} + 10q^{33} - 12q^{34} - 17q^{36} - 5q^{37} + 12q^{39} - 9q^{41} + 5q^{42} + 8q^{44} + 16q^{46} - 15q^{47} - 5q^{48} + 14q^{49} - 4q^{51} - 20q^{52} + 35q^{53} - 10q^{54} - 15q^{56} - 20q^{58} + 15q^{59} + 6q^{61} + 45q^{62} - 20q^{63} - 26q^{64} - 18q^{66} - 13q^{69} - 29q^{71} + 5q^{72} + 10q^{73} - 12q^{74} - 20q^{76} + 20q^{77} - 25q^{78} - 10q^{79} - 12q^{81} + 15q^{83} - 27q^{84} + 16q^{86} - 55q^{87} + 20q^{88} + 40q^{89} + q^{91} - 5q^{92} - 7q^{94} + 11q^{96} - 10q^{97} - 40q^{98} - 8q^{99} + O(q^{100})$$
Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25$$:
$$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$406 \nu^{7} - 714 \nu^{6} + 747 \nu^{5} - 1896 \nu^{4} + 2103 \nu^{3} + 4949 \nu^{2} + 1065 \nu - 7800$$$$)/1355$$ $$\beta_{3}$$ $$=$$ $$($$$$420 \nu^{7} - 776 \nu^{6} + 698 \nu^{5} - 1924 \nu^{4} + 2297 \nu^{3} + 5129 \nu^{2} + 1055 \nu - 10265$$$$)/1355$$ $$\beta_{4}$$ $$=$$ $$($$$$728 \nu^{7} - 1327 \nu^{6} + 1246 \nu^{5} - 3353 \nu^{4} + 3584 \nu^{3} + 8547 \nu^{2} + 2190 \nu - 15715$$$$)/1355$$ $$\beta_{5}$$ $$=$$ $$($$$$-857 \nu^{7} + 1666 \nu^{6} - 1743 \nu^{5} + 4424 \nu^{4} - 4907 \nu^{3} - 9470 \nu^{2} - 2485 \nu + 18200$$$$)/1355$$ $$\beta_{6}$$ $$=$$ $$($$$$891 \nu^{7} - 1623 \nu^{6} + 1624 \nu^{5} - 4492 \nu^{4} + 4991 \nu^{3} + 9520 \nu^{2} + 3235 \nu - 18960$$$$)/1355$$ $$\beta_{7}$$ $$=$$ $$($$$$955 \nu^{7} - 1829 \nu^{6} + 1942 \nu^{5} - 4891 \nu^{4} + 5723 \nu^{3} + 9646 \nu^{2} + 2415 \nu - 20550$$$$)/1355$$
$$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{5} - 3 \beta_{4} + 4 \beta_{3} + \beta_{2} + \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{7} - 5 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 2$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{7} - 6 \beta_{6} - 7 \beta_{3} + 11 \beta_{2} + 4 \beta_{1} - 13$$ $$\nu^{6}$$ $$=$$ $$7 \beta_{6} + 7 \beta_{5} - 8 \beta_{4} - 7 \beta_{3} + 21 \beta_{2} - 2 \beta_{1} - 21$$ $$\nu^{7}$$ $$=$$ $$23 \beta_{7} + 38 \beta_{5} + 23 \beta_{4} - 23 \beta_{3} + 12 \beta_{2}$$
## Character values
We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$$.
$$n$$ $$2$$ $$\chi(n)$$ $$\beta_{4}$$
## Embeddings
For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.
For more information on an embedded modular form you can click on its label.
Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
124.1
1.66637 − 0.917186i −0.357358 + 1.86824i −0.983224 + 0.644389i 1.17421 − 0.0566033i −0.983224 − 0.644389i 1.17421 + 0.0566033i 1.66637 + 0.917186i −0.357358 − 1.86824i
−1.07822 0.350334i −1.52988 + 2.10569i −0.578217 0.420099i 0 2.38723 1.73443i 0.407162i 1.80902 + 2.48990i −1.16637 3.58973i 0
124.2 2.19625 + 0.713605i −0.279141 + 0.384204i 2.69625 + 1.95894i 0 −0.887234 + 0.644613i 3.03582i 1.80902 + 2.48990i 0.857358 + 2.63868i 0
249.1 −1.22570 + 1.68703i −2.09089 + 0.679371i −0.725700 2.23347i 0 1.41668 4.36010i 0.992398i 0.690983 + 0.224514i 1.48322 1.07763i 0
249.2 0.107666 0.148189i 1.39991 0.454857i 0.607666 + 1.87020i 0 0.0833172 0.256424i 3.26086i 0.690983 + 0.224514i −0.674207 + 0.489840i 0
374.1 −1.22570 1.68703i −2.09089 0.679371i −0.725700 + 2.23347i 0 1.41668 + 4.36010i 0.992398i 0.690983 0.224514i 1.48322 + 1.07763i 0
374.2 0.107666 + 0.148189i 1.39991 + 0.454857i 0.607666 1.87020i 0 0.0833172 + 0.256424i 3.26086i 0.690983 0.224514i −0.674207 0.489840i 0
499.1 −1.07822 + 0.350334i −1.52988 2.10569i −0.578217 + 0.420099i 0 2.38723 + 1.73443i 0.407162i 1.80902 2.48990i −1.16637 + 3.58973i 0
499.2 2.19625 0.713605i −0.279141 0.384204i 2.69625 1.95894i 0 −0.887234 0.644613i 3.03582i 1.80902 2.48990i 0.857358 2.63868i 0
$$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 499.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles
## Inner twists
Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner
## Twists
By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.e.a 8
5.b even 2 1 625.2.e.i 8
5.c odd 4 2 625.2.d.o 16
25.d even 5 1 25.2.e.a 8
25.d even 5 1 125.2.e.b 8
25.d even 5 1 625.2.b.c 8
25.d even 5 1 625.2.e.i 8
25.e even 10 1 25.2.e.a 8
25.e even 10 1 125.2.e.b 8
25.e even 10 1 625.2.b.c 8
25.e even 10 1 inner 625.2.e.a 8
25.f odd 20 4 125.2.d.b 16
25.f odd 20 2 625.2.a.f 8
25.f odd 20 2 625.2.d.o 16
75.h odd 10 1 225.2.m.a 8
75.j odd 10 1 225.2.m.a 8
75.l even 20 2 5625.2.a.x 8
100.h odd 10 1 400.2.y.c 8
100.j odd 10 1 400.2.y.c 8
100.l even 20 2 10000.2.a.bj 8
By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 25.d even 5 1
25.2.e.a 8 25.e even 10 1
125.2.d.b 16 25.f odd 20 4
125.2.e.b 8 25.d even 5 1
125.2.e.b 8 25.e even 10 1
225.2.m.a 8 75.h odd 10 1
225.2.m.a 8 75.j odd 10 1
400.2.y.c 8 100.h odd 10 1
400.2.y.c 8 100.j odd 10 1
625.2.a.f 8 25.f odd 20 2
625.2.b.c 8 25.d even 5 1
625.2.b.c 8 25.e even 10 1
625.2.d.o 16 5.c odd 4 2
625.2.d.o 16 25.f odd 20 2
625.2.e.a 8 1.a even 1 1 trivial
625.2.e.a 8 25.e even 10 1 inner
625.2.e.i 8 5.b even 2 1
625.2.e.i 8 25.d even 5 1
5625.2.a.x 8 75.l even 20 2
10000.2.a.bj 8 100.l even 20 2
## Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(625, [\chi])$$:
$$T_{2}^{8} - 4 T_{2}^{6} - 10 T_{2}^{5} + 11 T_{2}^{4} + 40 T_{2}^{3} + 21 T_{2}^{2} - 5 T_{2} + 1$$ $$T_{3}^{8} + \cdots$$
## Hecke characteristic polynomials
$p$ $F_p(T)$
$2$ $$1 - 5 T + 21 T^{2} + 40 T^{3} + 11 T^{4} - 10 T^{5} - 4 T^{6} + T^{8}$$
$3$ $$16 + 40 T + 64 T^{2} - 20 T^{3} - 39 T^{4} - 5 T^{5} + 9 T^{6} + 5 T^{7} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$16 + 116 T^{2} + 121 T^{4} + 21 T^{6} + T^{8}$$
$11$ $$( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$13$ $$1 - 5 T + 14 T^{2} - 5 T^{3} - 19 T^{4} + 5 T^{5} + 14 T^{6} + 5 T^{7} + T^{8}$$
$17$ $$1936 - 2200 T + 1416 T^{2} - 580 T^{3} + 341 T^{4} - 230 T^{5} + 86 T^{6} - 15 T^{7} + T^{8}$$
$19$ $$400 - 800 T + 800 T^{2} - 450 T^{3} + 335 T^{4} - 135 T^{5} + 50 T^{6} - 10 T^{7} + T^{8}$$
$23$ $$256 - 960 T + 1744 T^{2} - 1860 T^{3} + 1201 T^{4} - 465 T^{5} + 109 T^{6} - 15 T^{7} + T^{8}$$
$29$ $$483025 - 344025 T + 132750 T^{2} - 33775 T^{3} + 6885 T^{4} - 1105 T^{5} + 150 T^{6} - 15 T^{7} + T^{8}$$
$31$ $$1936 - 9064 T + 105252 T^{2} - 16322 T^{3} + 2255 T^{4} + 67 T^{5} - 3 T^{6} - T^{7} + T^{8}$$
$37$ $$116281 + 20460 T + 116 T^{2} + 8530 T^{3} + 3631 T^{4} + 380 T^{5} + T^{6} + 5 T^{7} + T^{8}$$
$41$ $$13456 - 15544 T + 8332 T^{2} - 1222 T^{3} + 3805 T^{4} + 382 T^{5} + 62 T^{6} + 9 T^{7} + T^{8}$$
$43$ $$246016 + 56784 T^{2} + 4421 T^{4} + 129 T^{6} + T^{8}$$
$47$ $$65536 - 61440 T + 38656 T^{2} + 1200 T^{3} - 939 T^{4} - 450 T^{5} + 26 T^{6} + 15 T^{7} + T^{8}$$
$53$ $$8755681 - 3314080 T + 530284 T^{2} - 92710 T^{3} + 25131 T^{4} - 4840 T^{5} + 549 T^{6} - 35 T^{7} + T^{8}$$
$59$ $$4080400 - 2605800 T + 851400 T^{2} - 172200 T^{3} + 24985 T^{4} - 2610 T^{5} + 220 T^{6} - 15 T^{7} + T^{8}$$
$61$ $$116281 + 140151 T + 38437 T^{2} - 51672 T^{3} + 17305 T^{4} - 318 T^{5} + 252 T^{6} - 6 T^{7} + T^{8}$$
$67$ $$246016 - 79360 T - 24384 T^{2} - 3520 T^{3} + 4976 T^{4} + 880 T^{5} - 4 T^{6} + T^{8}$$
$71$ $$24245776 + 9818456 T + 2542392 T^{2} + 427928 T^{3} + 54105 T^{4} + 5172 T^{5} + 462 T^{6} + 29 T^{7} + T^{8}$$
$73$ $$1 + 5 T - T^{2} - 30 T^{3} + T^{4} + 30 T^{5} + 74 T^{6} - 10 T^{7} + T^{8}$$
$79$ $$33408400 - 4913000 T + 433500 T^{2} + 36550 T^{3} + 7935 T^{4} + 375 T^{5} + 270 T^{6} + 10 T^{7} + T^{8}$$
$83$ $$99856 + 9480 T + 7024 T^{2} - 74940 T^{3} + 36361 T^{4} - 1185 T^{5} - 11 T^{6} - 15 T^{7} + T^{8}$$
$89$ $$1392400 - 1014800 T + 629000 T^{2} - 206550 T^{3} + 48335 T^{4} - 7485 T^{5} + 740 T^{6} - 40 T^{7} + T^{8}$$
$97$ $$301334881 + 110403240 T + 15337196 T^{2} + 795650 T^{3} - 28154 T^{4} - 6200 T^{5} - 119 T^{6} + 10 T^{7} + T^{8}$$ | 2021-10-16 21:49:11 | {"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": 0.9993312358856201, "perplexity": 12684.216374821579}, "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/1634323585025.23/warc/CC-MAIN-20211016200444-20211016230444-00576.warc.gz"} |
http://tex.stackexchange.com/questions/26220/control-spacing-around-table-caption | # Control spacing around table caption
I want to put several tables in the same table float like this:
\documentclass[a4paper, 12pt]{article}
\usepackage[T1]{fontenc}
\usepackage{subfig}
\begin{document}
\begin{table}
\caption{Internal exon scores}\label{tab:internal}
\begin{tabular}{|c|l|l|}
\hline
Ranking&Exon Coverage&Splice Site Support\\
\hline
E1&Complete coverage by a single transcript&Both splice sites\\
E2&Complete coverage by more than a single transcript&Both splice sites\\
E3&Partial coverage&Both splice sites\\
E4&Partial coverage&One splice site\\
E5&Complete or partial coverage&No splice sites\\
E6&No coverage&No splice sites\\
\hline
\end{tabular}
\caption{External exon scores}\label{tab:external}
\begin{tabular}{|c|l|l|}
\hline
Ranking&Exon Coverage&Internal Splice Site Coverage\\
\hline
X1&Complete coverage by one or more transcripts&Splice site covered\\
X2&Exon coverage >= 80\%&Splice site covered\\
X3&Exon coverage < 80\%&Splice site covered\\
X4&Exon coverage < 80\%&Splice site not covered\\
X5&No coverage&Splice site not covered\\
\hline
\end{tabular}
\end{table}
\end{document}
When I build the document, table 2's caption is closer to the bottom of table 1, which makes things confusing. hspace{} doesn't seem to work in the table environment. I've worked out that the subfloat package appears to be causing the spacing behaviour of the captions. Is there anything else that I can do to introduce a blank horizontal break to space out the captions?
-
It looks fine with the article document class. Perhaps you could turn your code into a complete compilable file, a so-called minimal working example (MWE) that shows the behaviour? – Jake Aug 21 '11 at 10:47
Yup, you're right. I probably should have stripped it down to the basics before asking. I found that the problem was caused by using the subfloat package, which I require for a couple of graph floats in my document. I updated the example to a MWE that reproduces the issue. – kikumbob Aug 21 '11 at 11:09
Simply use \vspace between the tabular environements, or something like \medskip or \bigskip. – Axel Sommerfeldt Aug 21 '11 at 11:32
The spacing is caused by the caption package, which is loaded by subfig. You can adjust the two parameters aboveskip and belowskip provided by caption using something like \captionsetup{belowskip=12pt,aboveskip=4pt} in your preamble. Note that below and above have the meaning as it would make sense for figure captions below a figure: aboveskip is the space between the content and the caption (which would be above the caption for a figure caption that's set underneath the figure, but below it for a table caption that's set above the table), belowskip is the space between the caption and the surrounding text.
Here's your example with the aboveskip and belowskip adjusted:
\documentclass[a4paper, 12pt]{article}
\usepackage[T1]{fontenc}
\usepackage{subfig}
\captionsetup{belowskip=12pt,aboveskip=4pt}
\begin{document}
\begin{table}
\caption{Internal exon scores}\label{tab:internal}
\begin{tabular}{|c|l|l|}
\hline
Ranking&Exon Coverage&Splice Site Support\\
\hline
E1&Complete coverage by a single transcript&Both splice sites\\
E2&Complete coverage by more than a single transcript&Both splice sites\\
E3&Partial coverage&Both splice sites\\
E4&Partial coverage&One splice site\\
E5&Complete or partial coverage&No splice sites\\
E6&No coverage&No splice sites\\
\hline
\end{tabular}
\caption{External exon scores}\label{tab:external}
\begin{tabular}{|c|l|l|}
\hline
Ranking&Exon Coverage&Internal Splice Site Coverage\\
\hline
X1&Complete coverage by one or more transcripts&Splice site covered\\
X2&Exon coverage >= 80\%&Splice site covered\\
X3&Exon coverage < 80\%&Splice site covered\\
X4&Exon coverage < 80\%&Splice site not covered\\
X5&No coverage&Splice site not covered\\
\hline
\end{tabular}
\end{table}
\end{document}
-
Since you use subfig, you can adjust this by \captionsetup of the caption package, for example:
\captionsetup[table]{aboveskip=0pt}
\captionsetup[table]{belowskip=10pt}
With these adjustments, you will get this output:
-
\usepackage{subfig}
\let\belowcaptionskip\abovecaptionskip
- | 2014-07-28 14:58:55 | {"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": 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": 0.852836549282074, "perplexity": 9078.595556274468}, "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-2014-23/segments/1406510260734.19/warc/CC-MAIN-20140728011740-00357-ip-10-146-231-18.ec2.internal.warc.gz"} |
http://smlnj.org/dist/working/110.98/110.98-README.html | Standard ML of New Jersey
Release Notes
Version 110.98
July 16, 2020
## Summary
This release includes substantial improvements to the documentation (the SML/NJ Library is now mostly documented) as well as many small improvements to the SML/NJ Library, various improvements in other parts of the system, and bug fixes.
## Future Plans
We are reworking the back-end of the compiler in preparation for migrating from our current MLRISC code generator to one based on LLVM. With that migration, we expect to drop support for all processors other than the x86-64 (aka amd64), but we plan to add support for the arm64 (aka AArch64).
## Details
### Compiler
Made the `LambdaVar.lvar` type abstract. This is an internal change that should not affect compiler behavior.
We have started a project to migrate the backend of SML/NJ to use the LLVM infrastructure for code generation. The prepare the ground for this migration, we have made a number of changes to the compiler internals:
• Support for mapping the trigonometry functions `sin`, `cos`, and `tan` to hardware instructions on the x86 architecture has been removed. This change was made to simplify the code generator as we work on migrating to a new LLVM backend.
• The SML operators `div` and `mod` have a floor rounding semantics instead of the truncation semantics supported by hardware. Previously, we relied on MLRISC to handle the implementation of these operators, but we now do that in a new lowering pass. We also added optimizations for when the second argument to `div` or `mod` is a power of two to the CPS contraction phase.
• The lowering pass also lowers trapping conversions (CPS primops `TEST` and `TESTU`) such that they only involve conversions that can be checked using trapping add operations.
### MLRISC
The MLRISC instruction selector for the x86 and amd64 targets erroneously assumed that the `idiv` instruction sets the `OF` (overflow) condition code when dividing the largest negative number by `~1`. In fact, such a division operation traps, which is okay, because the runtime system maps the trap to the `Overflow` exception. Since the check for overflow is unnecessary, it has been removed from the files `MLRISC/amd64/mltree/amd64-gen.sml` and `MLRISC/x86/mltree/x86.sml`.
### Basis Library
This version implements the following Basis Library proposal:
[2020-001]
Basis Library proposal 2020-001 (Addition of Universal module) — This proposal adds the `Universal` structure found in Poly/ML to the Basis Library.
### SML/NJ Library
The SML/NJ Library is now mostly documented; see `doc/html/smlnj-lib/index.html` in the distribution or the online documentation.
The `HTMLDev` structure in the pretty-printing library (`$/pp-lib.cm`) has been renamed as `HTML3Dev` and moved into its own library (`pp-extras-lib.cm`). The renaming is in anticipation of renaming the HTML Library to "HTML3" and the moving it to its own library removes a dependency from the compiler on `$/html-lib.cm`.
There were many small improvements (and a couple of bug fixes) to various parts of the SML/NJ Library; see the `smlnj-lib/CHANGES` file for details.
### ML-LPT
Changed the semantics of the `--debug` command-line option for ml-antlr. Previously this option replaced the actions with a print expression, but that limited its usefulness because of type errors in the generated code. The new behavior is to preserve the existing actions and just add the printing code.
### Documentation
This release contains a substantial amount of new documentation for the SML/NJ Library (see `doc/html/smlnj-lib`). While the documentation is not complete, it does cover the most commonly used components (with the exception of the pretty-printing library).
### Installation
The default installation for machines that report “`x86_64`” as their hardware is now 64 bits. See the installation instructions for more details.
### 32-bit macOS issues
While the x86 installer for 110.98 works on macOs 10.14 Mojave, building from source requires some extra steps because the version of Xcode distributed for Mojave does not include a 32-bit SDK.
Another issue that you may encounter when building on macOs 10.14 Mojave is an error message for a shell script of the form
` /bin/sh: bad interpreter: Operation not permitted`
This error arises because the `com.apple.quarantine` attribute is set on the shell script. To fix the problem, remove the attribute using the command
`` xattr -d com.apple.quarantine shell-script``
and resume the build.
## Bugs
Here is a list of tracked bugs fixed (or closed) with this release, please see the bug tracker for more details.
238 Cross compilation from amd64 to x86 does not work 256 `Ref.exchange` incorrect 257 `HASH_TABLE` `listItems` error in reference manual 260 Perform divide on `Position.int` crashes with FPE on Linux 261 Weird "calc_strictness" message being printed 262 JSON parser fails on empty object 263 JSON parser ignores suffixes
## Supported systems
We believe that SML/NJ will build and run on the following systems, but have only tested some of them:
Architecture Operating System Status
AMD64
FreeBSD 12.0
Tested
macOS 10.14 (Mojave)
Tested
macOS 10.15 (Catalina)
Tested
macOS 11.0 (Big Sur)
Tested
Ubuntu 16.04.3 LTS
Tested
Ubuntu 18.04.3 LTS
Tested
Power PC
Mac OS X 10.5 (Leopard)
AIX
Sparc
Solaris
Linux
x86 (32-bit)
Mac OS X 10.6 (Snow Leopard)
Mac OS X 10.7 (Lion)
Mac OS X 10.8 (Mountain Lion)
Mac OS X 10.9 (Mavericks)
Mac OS X 10.10 (Yosemite)
Mac OS X 10.11 (El Capitan)
macOS 10.12 (Sierra)
Tested
macOS 10.13 (High Sierra)
Tested
macOS 10.14 (Mojave)
Tested
Ubuntu 16.04.3 LTS
Tested
Other Linux variants
FreeBSD 12.0
Tested
Other BSD variants
Windows 7
Windows 10
Cygwin (32-bit) | 2022-11-30 22:58:36 | {"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": 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": 0.34204918146133423, "perplexity": 5667.640969210177}, "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/1669446710777.20/warc/CC-MAIN-20221130225142-20221201015142-00706.warc.gz"} |
https://www.lesswrong.com/posts/KwbJFexa4MEdhJbs4/classifying-games-like-the-prisoner-s-dilemma | # 82
Note: the math and the picture didn't transfer. I may try to fix it in future, but for now you might want to just read it at the original site. [Mod/Edit note: Should all be fixed now!]
Consider games with the following payoff matrix:
One such game is the Prisoner's Dilemma (in which strategy "Krump" is usually called "Cooperate", and "Flitz" is usually called "Defect"). But the Prisoner's Dilemma has additional structure. Specifically, to qualify as a PD, we must have gives the motivation to defect if the other player cooperates, and gives that motivation if the other player defects. With these two constraints, the Nash equilibrium is always going to be Flitz/Flitz for a payoff of is what gives the dilemma its teeth; if instead , then that equilibrium is a perfectly fine outcome, possibly the optimal one.
I usually think of a Prisoner's Dilemma as also having . That specifies that mutual cooperation has the highest total return - it's "socially optimal" in a meaningful sense1 - while mutual defection has the lowest. It also means you can model the "defect" action as "take some value for yourself, but destroy value in the process". (Alternatively, "cooperate" as "give some of your value to your playmate2, adding to that value in the process".) We might consider instead:
• If , then defecting while your playmate cooperates creates value (relative to cooperating). From a social perspective, Krump/Flitz or Flitz/Krump is preferable to Krump/Krump; and in an iterated game of this sort, you'd prefer to alternate with than to get a constant . Wikipedia still classes this as a Prisoner's Dilemma, but I think that's dubious terminology, and I don't think it's standard. I might offhand suggest calling it the Too Many Cooks game. (This name assumes that you'd rather go hungry than cook, and that spoiled broth is better than no broth.)
• If , then defecting while your playmate defects creates value. I have no issue thinking of this as a Prisoner's Dilemma; my instinct is that most analyses of the central case will also apply to this.
By assigning different values to the various numbers, what other games can we get?
As far as I can tell, we can classify games according to the ordering of (which determine individual outcomes) and of (which determine the social outcomes). Sometimes we'll want to consider the case when two values are equal, but for simplicity I'm going to classify them assuming there are no equalities. Naively there would be possible games, but
• Reversing the order of everything doesn't change the analysis, it just swaps the labels Krump and Flitz. So we can assume without loss of generality that . That eliminates half the combinations.
• Obviously , so it's just a question of where falls in comparison to them. That eliminates another half.
• If then . That eliminates another four combinations.
• If then , eliminating another four.
• If then , eliminating four.
• If then , eliminating two.
• If then , eliminating two.
That brings us down to just 20 combinations, and we've already looked at three of them, so this seems tractable. In the following, I've grouped games together mostly according to how interesting I think it is to distinguish them, and I've given them names when I didn't know an existing name. Both the names and the grouping should be considered tentative.
## Cake Eating: W>∙>∙>Z (two games)
In this game, you can either Eat Cake or Go Hungry. You like eating cake. You like when your playmate eats cake. There's enough cake for everyone, and no reason to go hungry. The only Nash equilibrium is the one where everyone eats cake, and this is the socially optimal result. Great game! We should play it more often.
(If , then if you had to choose between yourself and your playmate eating cake, you'd eat it yourself. If , then in that situation you'd give it to them. Equalities between and signify indifference to (yourself, your playmate) eating cake in various situations.)
## Let's Party: W>Z>∙>∙ (two games)
In this game, you can either go to a Party or stay Home. If you both go to a party, great! If you both stay home, that's cool too. If either of you goes to a party while the other stays home, you'd both be super bummed about that.
Home/Home is a Nash equilibrium, but it's not optimal either individually or socially.
In the case , this is a pure coordination game, which doesn't have the benefit of an obvious choice that you can make without communicating.
(Wikipedia calls this the assurance game on that page, but uses that name for the Stag Hunt on the page for that, so I'm not using that name.)
## Studying For a Test: W>X>Z>Y (two games)
You can either Study or Bunk Off. No matter what your playmate does, you're better off Studying, and if you Study together you can help each other. If you Bunk Off, then it's more fun if your playmate Bunks Off with you; but better still for you if you just start Studying.
The only Nash equilibrium is Study/Study, which is also socially optimal.
## Stag hunt: W>Y>Z>X (two games)
You can either hunt Stag or Hare (sometimes "Rabbit"). If you both hunt Stag, you successfully catch a stag between you, which is great. If you both hunt Hare, you each catch a hare, which is fine. You can catch a hare by yourself, but if you hunt Stag and your playmate hunts Hare, you get nothing.
This also works with . If then two people hunting Hare get in each other's way.
The Nash equilibria are at Stag/Stag and Hare/Hare, and Stag/Stag is socially optimal. Hare/Hare might be the worst possible social result, though I think this game is usually described with .
## The Abundant Commons: X>W>∙>∙ (five games)
You can Take some resource from the commons, or you can Leave it alone. There's plenty of resource to be taken, and you'll always be better off taking it. But if you and your playmate both play Take, you get in each other's way and reduce efficiency (unless ).
If then you don't intefere with each other significantly; the socially optimal result is also the Nash equilibrium. But if then the total cost of interfering is more than the value of resource either of you can take, and some means of coordinating one person to Take and one to Leave would be socially valuable.
If then if (for whatever reason) you Leave the resource, you'd prefer your partner Takes it. If you'd prefer them to also Leave it.
An interesting case here is and . Take/Leave and Leave/Take are social optimal, but the Leave player would prefer literally any other outcome.
Take/Take is the only Nash equilibrium.
## Farmer's Dilemma: Y>W>X>Z (two games)
In this game, you can Work (pitch in to help build a mutual resource) or Shirk (not do that). If either of you Works, it provides more than its cost to both of you. Ideally, you want to Shirk while your playmate Works; but if your playmate Shirks, you'd rather Work than leave the work undone. The Nash equilibria are at Work/Shirk and Shirk/Work.
If then the socially optimal outcome is Work/Work, and a means to coordinate on that outcome would be socially useful. If , the socially optimal outcome is for one player to Work while the other Shirks, but with no obvious choice for which one of you it should be.
Also known as Chicken, Hawk/Dove and Snowdrift.
## Anti-coordination: ∙>∙>W>Z (two games)
In this game, the goal is to play a different move than your playmate. If then there's no reason to prefer one move over another, but if they're not equal there'll be some maneuvering around who gets which reward. If you're not happy with the outcome, then changing the move you play will harm your playmate more than it harms you. The Nash equilibria are when you play different moves, and these are socially optimal.
## Prisoner's Dilemma/Too Many Cooks: Y>W>Z>X (three games)
Covered in preamble.
(I'm a little surprised that this is the only case where I've wanted to rename the game depending on the social preference of the outcomes. That said, the only other games where isn't forced to be greater or less than are the Farmer's Dilemma and the Abundant Commons, and those are the ones I'd most expect to want to split in future.)
## A graph
I made a graph of these games. I only classified them according to ordering of (i.e. I lumped Prisoner's Dilemma with Too Many Cooks), and I drew an edge whenever two games were the same apart from swapping two adjacent values. It looks like this:
The lines are colored according to which pair of values is swapped (red first two, blue middle two, green last two). I'm not sure we learn much from it, but I find the symmetry pleasing.
## A change of basis?
I don't want to look too deep into this right now, but here's a transformation we could apply. Instead of thinking about these games in terms of the numbers , we think in terms of "the value of Player 2 playing Flitz over Krump":
• , the value to Player 1, if Player 1 plays Krump.
• , the value to Player 2, if Player 1 plays Krump.
• , the value to Player 1, if Player 1 plays Flitz.
• , the value to Player 2, if Player 1 plays Flitz.
These four numbers determine , up to adding a constant value to all of them, which doesn't change the games. For example, Prisoner's Dilemma and Too Many Cooks both have . A Prisoner's Dilemma also has while Too Many Cooks has .
So what happens if we start thinking about these games in terms of instead? Does this give us useful insights? I don't know.
Of course, for these numbers to point at one of the games studied in this post, we must have . I think if you relax that constraint, you start looking into games slightly more general than these. But I haven't thought about it too hard.
## Footnotes
[1] My use of the phrase comes from Ellickson's Order Without Law. Part of why I'm writing this is to help clarify my thinking about that book. I don't mean to imply anything in particular by it, I just like the ring of it better than alternatives like "welfare maximizing".
[2] Calling them your "opponent" assumes a level of antagonism that may not be present.
# 82
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I'd heard another friend discuss this idea a few months back, and thought it was a useful thing for someone to write up.
Something I found a bit difficult reading this was the arbitrariness of W, X, Y and Z (I had trouble remembering which was which). I think I'd have found it a little easier to parse the examples if they used something like XX, XY, YX, and YY. (Honestly, CC, CD, DC, DD would have been easiest to map onto my existing models, although I get part of the point here was to break from the Cooperate/Defect concept)
Ah, yeah. I had the same trouble and that would have been way better.
Curated.
A year ago, a mathematician friend of mine commented that "as far as I can tell, nobody has published a paper that just outlines all the different types of 2x2 quadrant-game payoffs", and they spent a weekend charting out the different payoff matrixes, and just meditating on how they felt about each one, and how it fit into their various game theory intuitions. But, they didn't get around to publishing it AFAICT.
This seems like a really obvious thing to do, but prior to this post I don't think anyone had written it up publicly. (if someone does know of an existing article, feel free to link here). But regardless I think a good writeup of this is useful to have in the LessWrong body of knowledge.
Real life is obviously more complicated than 2x2 payoff games, but having a set of crisp formulations is helpful for orientation on more complex issues. And, the fact that many people default to using prisoner's dilemma all the time even when it's not really appropriate seems like an actual problem that needed fixing.
I have some sense that the pedagogy of this post could be improved. I'd previously commented that using different symbols would be helpful for me. I have a nagging sense that there is other more useful feedback I could give on how to articulate some of the games, but don't have clear examples.
Those concerns are relatively minor though. Overall, thanks for the great post. :)
if someone does know of an existing article
Herbert Gintis's Game Theory Evolving (2nd edition) offers the following exercise. (Bolding and hyperlinks mine.)
#### 6.15 Characterizing 2 x 2 Normal Form Games I
We say a normal form game is generic if no two payoffs for the same player are equal. Suppose and are the payoff matrices for Alice and Bob, so the payoff to Alice's strategy against Bob's strategy is for Alice and for Bob. We say two generic 2 x 2 games with payoff matrices (A, B) and (C, D) are equivalent if, for all i, j, k, l = 1, 2:
and
In particular, if a constant is added to the payoffs to all the pure strategies of one player when played against a given pure strategy of the other player, the resulting game is equivalent to the original.
Show that equivalent 2 x 2 generic games have the same number of pure Nash equilibria and the same number of strictly mixed Nash equilibria. Show also that every generic 2 x 2 game is equivalent to either the prisoner's dilemma (§3.11), the battle of the sexes (§3.9), or the hawk-dove (§3.10). Note that this list does not include throwing fingers (§3.8), which is not generic.
Thanks for finding this! I'm a bit confused, though; it suggests that the game with payoffs
3,3 2,1
1,2 0,0
(an instance of Cake Eating), is equivalent to one of those named games. But... which? It only has one pure Nash equilibrium, so it can't be either hawk-dove or BOS, which both have two. And it can't be equivalent to PD - an instance of that would be
3,3 1,4
4,1 0,0
and these aren't equivalent. We have (3 > 1) but (3 < 4). So what am I missing?
(I had intended to try look this up myself, but I'm unlikely to do that in a timely manner, so I'm just leaving a comment. No obligation on you, of course.)
Von Neumann and Morgenstern also classify the two-player games, but they get only two games, up to equivalence. The reason is they assume the players get to negotiate beforehand. The only properties that matter for this are:
• The maximin value , which represents each player's best alternative to negotiated agreement (BATNA).
• The maximum total utility .
There are two cases:
1. The inessential case, . This includes the Abundant Commons with . No player has any incentive to negotiate, because the BATNA is Pareto-optimal.
2. The essential case, . This includes all other games in the OP.
It might seem strange that VNM consider, say, Cake Eating to be equivalent to Prisoner's Dilemma. But in the VNM framework, Player 1 can threaten not to eat cake in order to extract a side payment from Player 2, and this is equivalent to threatening to defect.
Nicely done.
In my experience, "Farmer's Dilemma" (aka Chicken/Snowdrift) is both more common than PD, and harder for human players to coordinate away from pathological outcomes. I think it should be the prototypical "nasty" game instead of PD. We've all been assigned a group project at school; we have not all been interrogated by the police in a situation where the payoffs are truly PD.
(Under "farmer's dilemma":)
If 2W>X+Y then the socially optimal outcome is Work/Work, and a means to coordinate on that outcome would be socially useful. If 2W<X+Y, the socially optimal outcome is for one player to Work while the other Shirks, but with no obvious choice for which one of you it should be.
(Under PD / too many cooks:)
(I'm a little surprised that this is the only case where I've wanted to rename the game depending on the social preference of the outcomes. That said, the only other games where X+Y isn't forced to be greater or less than 2X are the Farmer's Dilemma and the Abundant Commons, and those are the ones I'd most expect to want to split in future.)
To my personal taste, "chicken" is the game where 2W<X+Y; I think of chicken as fundamentally being a game where there isn't a fair and socially optimal outcome w/o correlated randomness (ie in correlated equilibria, we have the "stop light" solution where one player gets a signal to go straight, with fair odds of who gets to go). It becomes "hawk/dove" when 2WX+Y, representing the idea that fighting can't increase the amount of resources (at best the hawk strategy doesn't destroy anything unless there's another hawk; realistically it burns some resources either way, just a lot more when there are two hawks). However, I realize this isn't standard -- it seems like a lot of people consider both games to be "chicken".
(By the way, I was surprised you labelled the section "farmer's dilemma" -- just from my experience, "chicken" is far more common, and "hawk/dove" is also quite common, but not as common as "chicken"; whereas I'd never actually heard "farmer's dilemma" before.)
[I also want to briefly point out that it's questionable to assume that the utility functions of the two players are comparable, and hence, that social welfare is necessarily captured by the sum of the two payoffs. Utility functions are always "up to a scalar", so really "symmetric game" should mean something more complicated ... but all that being said, I think you handled this the right way in the end, because we don't want to deal with that extra complexity in a classification like this.]
To my personal taste, “chicken” is the game where 2W<X+Y; I think of chicken as fundamentally being a game where there isn’t a fair and socially optimal outcome w/o correlated randomness (ie in correlated equilibria, we have the “stop light” solution where one player gets a signal to go straight, with fair odds of who gets to go). It becomes “hawk/dove” when 2W≥X+Y, representing the idea that fighting can’t increase the amount of resources (at best the hawk strategy doesn’t destroy anything unless there’s another hawk; realistically it burns some resources either way, just a lot more when there are two hawks).
Hm. I agree that Hawk/Dove feels weird with . Chicken... I think I don't have strong feelings on the question. To the extent that I do, it's hard to disentangle from "actually, the outcome where you crash into each other really is super super bad", but that's not actually necessarily true. If I think in terms of "if you both swerve, then you're both cowards", then yeah, does feel natural. For the Farmer's Dilemma I feel like it can go both ways - there can be gains from working together, but also inefficiencies if e.g. you only have one bulldozer.
Farmer's Dilemma is definitely the less common name. (I didn't think I coined it myself but couldn't find any other references to it. From memory, I wrote most of the linked article without realizing it was the same as Chicken.) But I also just like it better as a name, and I have the vague sense that people equipped with that name will have an easier time recognizing instances of the problem. It's probably relevant that Order Without Law talks about problems that it calls Prisoner's Dilemmas, that I think are often Farmer's Dilemmas, in the context of... well, ranching, but close enough.
Yep, very interesting :) The different names/stories definitely make me think about the games in very different ways, such that it feels quite non-obvious that the Farmer's Dilemma is Chicken (or Hawk/Dove), intuitively.
(Actually, my mental tag for this is going to be the problem of cleaning common spaces/dishes in an apartment -- it's better if someone does it, but no one wants to do it individually.)
Also, I've probably heard people refer to the commons game as a PD, too. I think PD informally becomes a tag for a coming-apart of the individual and social good.
If , then defecting while your playmate cooperates creates value (relative to cooperating). From a social perspective, Krump/Flitz or Flitz/Krump is preferable to Krump/Krump; and in an iterated game of this sort, you'd prefer to alternate with , than to get a constant . Wikipedia still classes this as a Prisoner's Dilemma, but I think that's dubious terminology, and I don't think it's standard. I might offhand suggest calling it the Too Many Cooks game.
But the only Nash is still , and so there is an incentive to defect from Krump/Flitz. Furthermore, that defection reduces total social welfare, so I think it makes sense to call this variant "PD" still.
I absolutely think it makes sense, I just don't know if I endorse it. Maybe it's context dependent; someone who just wants to write about the Nash equilibria of games won't have any reason to distinguish these cases, and lumping them all under "Prisoner's Dilemma" seems absolutely fine. But Iterated Prisoner's Dilemma (in my sense) is probably a very different game from Iterated Too Many Cooks, but I'm not sure I've ever heard anyone talk about ITMC. And from a social perspective, "punish anyone who defects in a PD" makes sense but "punish anyone who doesn't cook in TMC" risks leaving value on the table.
Maybe it would make sense to call them a "(something) Prisoner's Dilemma" and a "(something else) Prisoner's Dilemma" but I'm not sure what the somethings would be.
I just cleaned up all the formatting and converted the LaTeX. Sorry for the inconvenience!
Thanks!
Would love to see similar for public goods games as there are so many simulations based off of them.
Very tempting to meme the graph of relations into the kabbalah tree of life diagram.
It does seem like the "best" game, Cake Eating, takes the "most holy" slot and the "worst" one, PD, takes the "most profane" slot. TINACBNIEAC.
I've been wanting to do something like this for a while, so it's good to see it properly worked out here.
If you wanted to expand this you could look at games which weren't symmetrical in the players. So you'd have eight variables, W, X, Y and Z, and w, x, y and z. But you'd only have to look at the possible orderings within each set of four, since it's not necessarily valid to compare utilities between people. You'd also be able to reduce the number of games by using the swap-the-players symmetry.
Oh, I hadn't seen it before, but this file on wikipedia seems like it might be roughly that expanded version? Info Direct link to pdf I haven't looked closely at it though. The pdf doesn't render in firefox for me, but does render in evince, my external pdf viewer.
I don't in general think there's anything wrong with comparing utilities between people in these things - that's what I'm doing when I talk about whether - but it would be simpler not to do so. Still, even then I think extending to all 8 would give far too many possibilities to be manually tractable - I make it .
But it wouldn't be too hard to write a program to classify them according to Nash equilibria, if someone wanted to do that. That might be a decent start.
I'm reminded of this paper, which discusses a smaller set of two-player games. What you call "Cake Eating" they call the "Harmony Game". They also use the more suggestive variable names -- which I believe come from existing literature -- R (reward), S (sucker's payoff), T (temptation), P (punishment) instead of (W, X, Y, Z). Note that in addition to R > P (W > Z) they also added the restrictions T > P (Y > Z) and R > S (W > X) so that the two options could be meaningfully labeled "cooperate" and "defect" instead of "Krump" and "Flitz" (the cooperate option is always better for the other player, regardless of whether it's better or worse for you). (I'm ignoring cases of things being equal, just like you are.)
(Of course, the paper isn't actually about classifying games, it's an empirical study of how people actually play these games! But I remember it for being the first place I saw such a classification...)
With these additional restrictions, there are only four games: Harmony Game (Cake Eating), Chicken (Hawk-Dove/Snowdrift/Farmer's Dilemma), Stag Hunt, and Prisoner's Dilemma (Too Many Cooks).
I'd basically been using that as my way of thinking about two-player games, but this broader set might be useful. Thanks for taking the time to do this and assign names to these.
I do have to wonder about that result that Zack_M_Davis mentions... as you mentioned, where's the Harmony Game in it? Also, isn't Battle of the Sexes more like Chicken than like Stag Hunt? I would expect to see Chicken and Stag Hunt, not Battle of the Sexes and Chicken, which sounds like the same thing twice and seems to leave out Stag Hunt. But maybe Battle of the Sexes is actually equivalent, in the sense described, to Stag Hunt rather than Chicken? That would be surprising, but I didn't set down to check whether the definition is satsified or not...
Note: the math and the picture didn't transfer. I may try to fix it in future, but for now you might want to just read it at the original site.
Ah, GreaterWrong makes the link very obvious, but I couldn't see it on LW. Have done, thanks.
I might get around to editing the post at some point, but unless/until I do, a note:
Anti-coordination: (two games)
In this game, the goal is to play a different move than your playmate. If then there’s no reason to prefer one move over another, but if they’re not equal there’ll be some maneuvering around who gets which reward. If you’re not happy with the outcome, then changing the move you play will harm your playmate more than it harms you. The Nash equilibria are when you play different moves, and these are socially optimal.
Correction: if you're not happy with the outcome, changing your move will harm your playmate more than it harms you if , and harm you more than them if .
Also, Battle of the Sexes is an instance of this game if you relabel the moves. If you label them "do the thing you/they prefer", you get . ( is "we both do my thing", is "we both do their thing", is "I do my thing and they do theirs" and is "I do their thing and they do mine". if going alone to either is equally bad.) If you label them "do the thing they/you prefer", you get . You won't get . (That would imply payoffs like: I most want to do "my thing" together. Failing that I want to do "their thing" together. But if we do go separately, I'd rather do "their thing" than "my thing". Which wouldn't be a BoS.)
X>W>Z>Y is interesting. It's similar to a PD, and so it appears that many of the same systems that evolved to enforce cooperation in PDs misfire in those cases and end up with Z (based on some observation, not any good evidence). However, unlike in a PD, Z is a worse outcome than the purely selfish X.
If a PD has the option to transfer value to your playmate and create more value, a XWZY (aka Deadlock) has the option to transfer value to your playmate inefficiently, so, in the inefficient state Z, both people are sacrificing to benefit the other, and yet each would be better off if the norm of sacrifice were not present.
EDIT: All of this assumes 2W > X + Y.
I like the analysis, but W, X, Y, Z is confusing terminology- it'd be easier to read and process if I didn't have to constantly remind myself which situation each variable corresponds to | 2021-12-01 08:52:19 | {"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": 0.5488234758377075, "perplexity": 1180.6441936499818}, "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-2021-49/segments/1637964359976.94/warc/CC-MAIN-20211201083001-20211201113001-00318.warc.gz"} |
https://mathoverflow.net/questions/153824/an-open-problem-in-convex-geometry | # An open problem in convex geometry
Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist?
An equivalent statement:
Is it possible to find four $0$-symmetric convex bodies in the plane such that a three-dimensional $0$-symmetric convex body containing four central sections linearly equivalent to these four bodies does not exist?
This seems to be an old problem, but I'm not sure of its origin. The earliest reference I've found is
Rolewicz, S., Plane sections of centrally symmetric convex bodies. Israel J. Math. 4 1966 135–138.
Does anyone know the status of this problem? Some earlier references?
Just for completeness, let me add that Bessaga proved (C. Bessaga,A note on universal Banach spaces of a finite dimension, Bull. Acad. Polon. Sci.6 (1958), 97–101.) that
Given $n > 2$, it is possible to find $N = N(n)$ norms $\| \cdot\|_k$ $( 1 \leq k \leq N)$ on the plane such that an n-dimensional normed space containing $N$ subspaces isometric to these normed planes does not exist.
Rolewicz mentions that it is not clear whether we can take $N(n) = n+1$.
Added on 08/01/2014. Trying to guess what possible four normed planes cannot be linearly and isometrically embedded in any one three-dimensional normed space: Does there exist a three-dimensional centrally symmetric convex body that contains four central sections that are linearly equivalent to a circle, a square, a regular hexagon, and a regular octagon?
• Do you think that these bodies need to be smooth? – Suvrit Jan 7 '14 at 23:28
• @Suvrit: No, I don't think so. In fact, it is not clear to me how "stable" the problem is. Suppose you find the four normed planes, will other four planes very close (Banach-Mazur distance) to them satisfy the same thing? If so, we can stick to smooth bodies or to polygons and it won't make a difference. – alvarezpaiva Jan 8 '14 at 12:48
• The problem is stable, Alvarez. That follows from the fact that the three dimensional spaces are compact in the Banach-Mazur distance. – Bill Johnson Jan 8 '14 at 20:59
• I'm missing something: why there should be a 3-dimensional normed space containing two linear subspaces isometric say to the Euclidean norm $\|\cdot\|$ and to $10000\|\cdot\|$ ? – Pietro Majer Aug 15 '16 at 16:08
• @PietroMajer: the isometries for different subspaces can be different. The Euclidean space is then a 3-space containing two linear subspaces isometric say to the Euclidean norm on a plane and to 10000 the Euclidean norm on another plane. – alvarezpaiva Aug 21 '16 at 17:23 | 2019-10-22 22:43:31 | {"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": 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": 0.5735998749732971, "perplexity": 518.2791942239836}, "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-2019-43/segments/1570987824701.89/warc/CC-MAIN-20191022205851-20191022233351-00109.warc.gz"} |
https://zbmath.org/?q=an%3A1180.35413 | # zbMATH — the first resource for mathematics
Resolvent estimates for the linearized compressible Navier-Stokes equation in an infinite layer. (English) Zbl 1180.35413
The author deals with the following resolvent problem: (1) $$(\lambda+L) u=f$$ in an infinite layer $$\Omega=\mathbb{R}^{n-1}x(0,a)$$, $$n\geq 2$$, where $$\lambda \in\mathbb{C}$$ is a parameter, $$f=f(x)$$ is a given function with values in $$\mathbb{R}^{n+1}$$, $$u=^T(\varphi,m)$$ is the unknown function with $$\varphi= \varphi (x)\in\mathbb{R}$$ and $$m=^T(m^1(x),\dots,m^n(x))\in\mathbb{R}^n$$, and $$L$$ is an operator defined by $L=\left(\begin{matrix} 0 & \gamma\text{div} \\ \gamma\nabla & -\nu\Delta I_n-\widetilde\nu\nabla\text{div}\end{matrix}\right)$ with positive constants $$\nu$$ and $$\gamma$$ and a nonnegative constant $$\widetilde \nu$$. Here $$x=^T(x',x_n)\in\Omega$$ with $$x'\in\mathbb{R}^{n-1}$$, $$x_n\in (0,a)$$; the superscript $$T(x',x_n)$$ stands for the transposition; $$I_n$$ is the $$(n\times n)$$ identity matrix; and $$\text{div},\nabla$$ and $$\Delta$$ are the usual divergence, gradient and Laplacian with respect to $$x$$. The author considers (1) under the boundary condition $$m|_{\partial \Omega}=0(2)$$. The author establishes the $$L^p$$ estimates for the solution of (1)–(2) for $$1\leq p\leq\infty$$. The estimates show that $$-L$$ generates an analytic semigroup in $$W^{1,p}\times L^p$$ for $$1<p< \infty$$. Based on the estimates the author obtains short-time estimates for the semigroup in $$L^p$$ norms for all $$1\leq p\leq\infty$$. Moreover, the author establishes the estimates for the high frequency part of the resolvent, which lead to the exponential decay of the corresponding part of the semigroup.
##### MSC:
35Q30 Navier-Stokes equations 76N15 Gas dynamics (general theory)
Full Text: | 2021-09-25 00:26:10 | {"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": 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": 0.851848840713501, "perplexity": 146.4401926727035}, "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/1631780057584.91/warc/CC-MAIN-20210924231621-20210925021621-00002.warc.gz"} |
https://www.kanroshobo.com/2016/09/28/%E4%B8%80%E6%96%B0%E4%BC%9A%E5%A4%A7%E5%B8%82%E4%BC%9A-7/ | # 一新会大市会
くもり空。
4227歩 2.53km 39分 211.4kcal 5.0g
ILCE-7II SMC-TAKUMAR28mmF3.5
2016-09-28 ｜ Posted in 甘露日記Comments Closed | 2018-08-20 10:40:38 | {"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": 0.9410703182220459, "perplexity": 11080.713957184174}, "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-2018-34/segments/1534221216333.66/warc/CC-MAIN-20180820101554-20180820121554-00041.warc.gz"} |
https://forum.allaboutcircuits.com/threads/wtb-1-4w-5-resistors-in-small-lots.158027/page-2 | # WTB 1/4W 5% RESISTORS IN SMALL LOTS
#### KeepItSimpleStupid
Joined Mar 4, 2014
4,631
US postage is nuts. It's like $9.00 to$15.00; Express mail is a little cheaper.
Overseas: China Post incoming to US most of the time is 0. It' costs like $26.00 USD fpr me to mail a small parcel overseas, I just received my fist package from Sweeden and it cost me$16.00 USD. A free item for the price of postage.
When cost is the only parameter, free postage is possible at Digi-key.
I had a project, laundry filter, that I was doing for home and I needed multiple vendors
V1: An aquarium supply store
V2: The sock filter
V3: A special clamp and material to cut a gasket.
Had: 1/4-20 Tap, drill and clearance drill
V4: Appliance place
V5: local for various pipe/hose fittings
I also needed to use a lathe and drill press. The lathe option isn't available now. It tok a couple of iterations to get the right sock filter.
#### Wolframore
Joined Jan 21, 2019
2,249
Tayda uses usps and normally shipping is inexpensive. They specialize in small orders and it ships out of Colorado. They’re great for breadboarding. Examples from about a year ago. One was for a project that went into production the final device is SMD and purchased through Arrow electronics.
Digikey and Mouser shipping is higher and specialize in larger orders. Arrow has handling changes for orders under \$50. These guys have helped me to get projects finished at a low cost when I need a few little things. They do not have a huge inventory or components just a decent collection of standardize parts for the hobbyist. Usually takes 2-3 days for delivery to Chicago.
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#### DickCappels
Joined Aug 21, 2008
7,248
Thank you for this! I think I can order some long-missing resistor values from Tayda and have them sent to my dear brother who can bring them to me from the U.S. These days through-hole parts are becoming scarce. Very scarce.
#### USMC_Spike
Joined Jan 23, 2017
14
Can't you just buy them at Mouser or Digikey?
They will even back them for you.
#### USMC_Spike
Joined Jan 23, 2017
14
And of course, now that I replied, I see that other forumite's have said the same thing.
Sorry 'bout that.
#### USMC_Spike
Joined Jan 23, 2017
14
To anyone in the Dallas Fort Worth area, even if you are visiting,
take a trip over to Tanner's Electronics, in Carrollton, it's a 3rd generation
#### Wolframore
Joined Jan 21, 2019
2,249
Digi-Key and Mouser has their place, hard to get stuff, high volume, and absolutely must be destroyed over night. They‘re vital for commercial usage... they might even lose money on the one off orders that hobbyists need. Luckily we have other resources these days with the end of Radio Shack.
#### Wendy
Joined Mar 24, 2008
22,360
Well, since my 3D printer resistor storage project has been temporarily derailed, my next step is to make a nw storage system using snack bags and scotch tape my old resistor storage was lost in the aftermath of the stroke, So I am truly starting from scratch. I will let everyone know how it goes.
I see this thread has been a spammer magnet. Figures.
#### schmitt trigger
Joined Jul 12, 2010
422
When I lived in the DFW Metroplex 30 years ago, Tanners was already there. In the corner of I-35 and Beltline Road, if I remember correctly.
I am glad to hear that they are still in business! As someone mentioned, it was a candy store for electronic hobbyists.
#### Wolframore
Joined Jan 21, 2019
2,249
I just ordered 2 of these If they're good maybe more in the future:
#### schmitt trigger
Joined Jul 12, 2010
422
Right after I posted above, I decided to Google Tanner's Electronics. What I found really shocked me:
#### Wendy
Joined Mar 24, 2008
22,360
I am paralyzed and sheltering in place, Tanner's is not an option any longer, I used them to create my 1st resistor kit. a lot of what I've acquired over the years simply dissipated away. I might find them again, I might not.
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#### Wendy
Joined Mar 24, 2008
22,360
Order placed w/ taydaelectonics.com. I'll let y'all know how it goes.
#### Wolframore
Joined Jan 21, 2019
2,249
I just got a nice shipment from Tayda last week for our end of semester project, they are now packing in pink poly bags... it’s all THT and again no problems.
#### djsfantasi
Joined Apr 11, 2010
7,619
#### Wendy
Joined Mar 24, 2008
22,360
I have moved off topic posts over here.
#### Wendy
Joined Mar 24, 2008
22,360
My order arrived Friday, I spent most Saturday sorting the component to my new Resistor Kit. I was quite impressed with these folks and told them so.
#### Wolframore
Joined Jan 21, 2019
2,249
Yeah I like them when I have one off projects mostly with thruhole. Nothing exotic but they have a decent supply of fun stuff.
#### Forgeralex
Joined May 13, 2020
1
I know of someone who has this.....reach him on <Mod:deleted email> Avoid you being Spammed. | 2021-05-07 16:40:30 | {"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": 0.2289378046989441, "perplexity": 7636.142439690126}, "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/1620243988796.88/warc/CC-MAIN-20210507150814-20210507180814-00206.warc.gz"} |
https://www.physicsforums.com/threads/problem-involving-rates-of-change-of-distance.513458/ | # Problem involving rates of change of distance.
1. Jul 11, 2011
### Stochastic13
1. The problem statement, all variables and given/known data
Two cars start moving form the same place. One goes north at a rate of 50 mi/hr, while the other heads east at a rate of 30 mi/hr. At what rate is the distance between the two cars changing exactly two hours later?
2. Relevant equations
3. The attempt at a solution
I thought that the equation for the velocity vectors is:
r(x) = 30xi + 50yj
and thus distance d two hours later is given by:
d = √(30x² + 50x² )
and
dd/dt = ∂x/dt + ∂y/dt
and after solving the equation I just plug in 2 for x and y to get what time the distance changes at the two hour mark.
Is this right?
2. Jul 11, 2011
### Skins
Were you asked to do this using vectors ? You don't really need to use vectors to solve this.
3. Jul 11, 2011
### Stochastic13
No, I wasn't. How would you solve it without vectors?
4. Jul 12, 2011
### Skins
You can do it with and without vectors. To do it without vectors you would set it up as a standard related rate problem. Lets say you let x = the distance traveled by the car heading east and let y = the distance traveled by the other car heading north. Let s = the distance between the two cars. Since the cars are traveling at right angles to each other we have the relationship
(*) $$s^{2} \, = \, x^{2} \, + y^{2}$$
Since x, s, and y change with respect to time t, can't we differentiate the above expression with respect to t ?
Furthermore we are given that
$$\frac{dx}{dt} \, = \, 30 mi/hr \mbox{ and } \frac{dy}{dt} \, = \, 50 mi/hr$$
We want to find
$$\frac{ds}{dt}$$
So now all you need to do is implicitly differentiate (*) , determine the appropriate values for x, y, and s when t = 2 and you should be able to solve and get the desired result.
You could also use vectors. Using vectors is in this case extremely easy since both cars travel at uniform speeds and depart from the same point and travel in directions perpendicular.
5. Jul 12, 2011
### Stochastic13
Thank you.
6. Jul 12, 2011
### Skins
Merci.. Glad to be of help.
7. Jul 12, 2011
### Ray Vickson
Now that you already have the solution, let's look at an easier way. Measuring time in hours, the positions t hours after the start are x = 30*t and y = 50*t, so the distance between the cars is s = c*t, where c = sqrt(30^2 + 50^2). So, for *any* t the rate is ds/dt = c.
RGV | 2017-09-21 03:37:43 | {"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": 0.7914072275161743, "perplexity": 738.7764568043341}, "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-2017-39/segments/1505818687606.9/warc/CC-MAIN-20170921025857-20170921045857-00170.warc.gz"} |
https://solvedlib.com/n/tell-the-maximum-number-of-real-zeros-that-each-polynomial,13283461 | ##### 3. Assuming hybrid orbital theory describes these molecules, indicate the hybrid orbitals on the center atom...
3. Assuming hybrid orbital theory describes these molecules, indicate the hybrid orbitals on the center atom of the following molecules: (eg, sp’ for a carbon on a H,C=CH, molecule) a. XeF4 b. PFS c. 4. Draw the hybrid orbital diagram for the carbon on CH2O. Include the unhybridized p orbitals...
##### 1. Do you think it is ethical to use customer information across multiple divisions of the...
1. Do you think it is ethical to use customer information across multiple divisions of the same company? Explain 2. Can a company be successful without using a portfolio matrix as a strategic planning tool? 3. Why would a course in marketing be helpful even if you don't pursue marketing as a car...
##### LDt4 Question E elexoc0r and Nedi: 1 ma*es during thc Bame? 1 Id"LJRcdick shoot I0 ree [Nox 0f 19tionothc
LDt4 Question E elexoc0r and Nedi: 1 ma*es during thc Bame? 1 Id"LJRcdick shoot I0 ree [Nox 0f 19 tionothc...
##### 1. Given the following time series data, compute the forecast (for 7) using: • Unweighted Moving...
1. Given the following time series data, compute the forecast (for 7) using: • Unweighted Moving Average using N = 3. • Weighted Moving Average using N = 3. • Exponential Smooting with a = 0.2 where F411 = OY: + (1 -a) Ft. Do NOT compute any MSE's. Weck Value UMA (N=3) WMA (N=3) E...
##### Order the reagents to provide a reasonable synthesis for the conversion of 2-bromo-2-methylbutane into 3-methyl-but-1-yne: Hint: This involves a combination of "moving the position of a pi bond;" and "interconverting between alkenes and alkynes:BrSlep 1 Zatsev E2Step 4BrStep 2Step 3anti-Markov additionHHoffman E2Hethoxide (Eto )t-butoxide (tBuO )HBr; ROOR1) Brz, CCl42) xs NaNHz3) H2O
Order the reagents to provide a reasonable synthesis for the conversion of 2-bromo-2-methylbutane into 3-methyl-but-1-yne: Hint: This involves a combination of "moving the position of a pi bond;" and "interconverting between alkenes and alkynes: Br Slep 1 Zatsev E2 Step 4 Br Step 2... | 2023-03-24 13:52:56 | {"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": 2, "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": 0.5922752618789673, "perplexity": 14473.542088764425}, "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-2023-14/segments/1679296945282.33/warc/CC-MAIN-20230324113500-20230324143500-00431.warc.gz"} |
https://brilliant.org/problems/a-cool-aboration-problem/ | # A cool-aboration Problem
Geometry Level 3
Figure above shows 3 circles each with radius of 1 touching the other two circles externally.
Find the area bounded by the three circles. Give your answer to 3 decimal places.
Use the approximation $\pi=\dfrac{22}{7}$.
× | 2019-11-15 17:05:18 | {"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": 0.9157876372337341, "perplexity": 1584.775086716775}, "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/1573496668682.16/warc/CC-MAIN-20191115144109-20191115172109-00507.warc.gz"} |
http://spot.pcc.edu/math/orcca/knowl/example-quadratic-equation-flying-object.html | Example8.4.5
A physics class launches a tennis ball from a rooftop that is $$90.2$$ feet above the ground. They fire it directly upward at a speed of $$14.4$$ feet per second and measure the time it takes for the ball to hit the ground below. We can model the height of the tennis ball, $$h\text{,}$$ in feet, with the quadratic equation $$h=-16x^2+14.4x+90.2\text{,}$$ where $$x$$ represents the time in seconds after the launch. According to the model, when should the ball hit the ground? Round the time to one decimal place.
The ground has a height of $$0$$ feet. Substituting $$0$$ for $$h$$ in the equation, we have this quadratic equation:
\begin{equation*} 0=-16x^2+14.4x+90.2 \end{equation*}
We cannot solve this equation with factoring or the square root property, so we will use the quadratic formula. First we will identify that $$\highlight{a=-16}\text{,}$$ $$\highlight{b=14.4}$$ and $$\highlight{c=90.2}\text{,}$$ and substitute them into the formula:
\begin{align*} x\amp=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\ x\amp=\frac{-(\substitute{14.4})\pm\sqrt{(\substitute{14.4})^2-4(\substitute{-16})(\substitute{90.2})}}{2(\substitute{-16})}\\ x\amp=\frac{-14.4\pm\sqrt{207.36-(-5772.8)}}{-32}\\ x\amp=\frac{-14.4\pm\sqrt{207.36+5772.8}}{-32}\\ x\amp=\frac{-14.4\pm\sqrt{5980.16}}{-32}\\ \end{align*}
These are the exact solutions but because we have a context we want to approximate the solutions with decimals.
\begin{align*} x\amp\approx-2.0\text{ or }x\approx2.9 \end{align*}
We don't use the negative solution because a negative time does not make sense in this context. The ball will hit the ground approximately $$2.9$$ seconds after it is launched.
in-context | 2018-07-17 19:35:50 | {"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": 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": 0.7317134141921997, "perplexity": 275.9335087018268}, "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-2018-30/segments/1531676589892.87/warc/CC-MAIN-20180717183929-20180717203929-00337.warc.gz"} |
https://general.chemistrysteps.com/electrochemistry-practice-problems/ | ## General Chemistry
Electrochemistry practice problems include questions on balancing redox reactions in acidic and basic solutions, calculating the cell potential, Eo/E at standard conditions using the table of standard reduction potentials, calculating the free energy change, ΔG based on the cell potential, calculating the cell potential under nonstandard conditions using the Nernst equation, calculating the equilibrium constant of the reaction based on the cell potential.
#### Practice
1.
Balance the following redox reactions occurring in acidic aqueous solution:
a) Al(s) + Fe2+(aq) → Al3+(aq) + Fe(s)
b) SO32(aq) + MnO4(aq) → SO42(aq) + Mn2+(aq)
c) Cr2O72-(aq) + C2O22-(aq) → Cr3+(aq) + CO2(g)
d) PbO2(s) + Mn2+(aq) + SO42-(aq) → PbSO4(s) + MnO4(aq)
e) MnO4(aq) + H2O2(aq) → Mn2+(aq) + O2(g)
a)
2Al(s) + 3Fe2+(aq) → 2Al3+(aq) + 3Fe(s)
b)
5SO32(aq) + 2MnO4(aq) + 6H+(aq) → 5SO42(aq) + 2Mn2+(aq) + 3H2O(l)
c)
3C2O22- + Cr2O72- + 14H+(aq) → 6CO2 + 2Cr3+ + 7H2O(l)
d)
2Mn2+ + 5PbO2 + 5SO42+ 4H+(aq) → 2MnO4+ 5PbSO4 + 2H2O(l)
e)
5H2O2(aq) + 2MnO4(aq) + 6H+(aq) → 5O2(g) + 2Mn2+(aq) + 8H2O(l)
Solution
a) Al(s) + Fe2+(aq) → Al3+(aq) + Fe(s)
1) Separate the half-reactions:
Al(s) → Al3+(aq) (oxidation)
Fe2+(aq) → Fe(s) (reduction)
2) Balance the elements. In this case, they are balanced, so proceed to the next step.
3) Balance the charges by adding electrons:
Al(s) → Al3+(aq) + 3e
Fe2+(aq) + 2e→ Fe(s)
4) Balance the number of electrons by multiplying the equation(s) with a whole number.
2Al(s) → 2Al3+(aq) + 6e
3Fe2+(aq) + 6e→ 3Fe(s)
5) Once everything is balanced, add the half-reactions together and check if all the atoms and charges are balanced again.
2Al(s) → 2Al3+(aq) + 6e
3Fe2+(aq) + 6e→ 3Fe(s)
_____________________
2Al(s) + 3Fe2+(aq) + 6e→ 2Al3+(aq) + 6e+ 3Fe(s)
6) Cancel any species that appear in equal quantitates on both sides of the equation:
2Al(s) + 3Fe2+(aq) + 6e → 2Al3+(aq) + 6e + 3Fe(s)
2Al(s) + 3Fe2+(aq) → 2Al3+(aq) + 3Fe(s)
b) SO32(aq) + MnO4(aq) → SO42(aq) + Mn2+(aq)
1) Separate the half-reactions:
SO32(aq) → SO42(aq)
MnO4(aq) → Mn2+(aq)
2) Balance the elements. Add H2O to balance the O atoms, and H+ to balance the H atoms.
H2O(l) + SO32(aq) → SO42(aq) + 2H+(aq)
MnO4(aq) + 8H+(aq) → Mn2+(aq) + 4H2O(l)
3) Balance the charges by adding electrons:
H2O(l) + SO32(aq) → SO42(aq) + 2H+(aq) + 2e
MnO4(aq) + 8H+(aq) + 5e → Mn2+(aq) + 2H2O(l)
4) Balance the number of electrons by multiplying the equation(s) with a whole number.
5H2O + 5SO32(aq) → 5SO42(aq) + 10H+(aq) + 10e
2MnO4(aq) + 16H+(aq) + 10e- → 2Mn2+(aq) + 4H2O(l)
5) Once everything is balanced, add the half-reactions together and check if all the atoms and charges are balanced again.
5H2O + 5SO32(aq) → 5SO42(aq) + 10H+(aq) + 10e
2MnO4(aq) + 16H+(aq) + 10e → 2Mn2+(aq) + 8H2O(l)
______________________________________________
5H2O + 5SO32(aq) + 2MnO4(aq) + 16H+(aq) + 10e→ 5SO42(aq) + 10H+(aq) + 10e+ 2Mn2+(aq) + 8H2O(l)
6) Cancel any species that appear in equal quantitates on both sides of the equation:
5H2O(l) + 5SO32(aq) + 2MnO4(aq) + 16 6H+(aq) + 10e → 5SO42(aq) + 10H+(aq) + 10e + 2Mn2+(aq) + 8 3H2O(l)
5SO32(aq) + 2MnO4(aq) + 6H+(aq) → 5SO42(aq) + 2Mn2+(aq) + 3H2O(l)
We can rearrange the positions of species just to make the equation look a more conventional way:
2MnO4(aq) + 5SO32(aq) + 6H+(aq) → 2Mn2+(aq) + 5SO42(aq) + 3H2O(l)
c) Cr2O72- + C2O22- → Cr3+ + CO2
1) Separate the half-reactions:
Cr2O72- → Cr3+
C2O22- → CO2
2) Balance the elements. Add H2O to balance the O atoms, and H+ to balance the H atoms.
Cr2O72- + 14H+(aq) → 2Cr3+ + 7H2O
C2O22-2CO2
3) Balance the charges by adding electrons. We need 6e on the left side of the first equation because the charges are +12 vs +6. The second equation needs two electrons on the right side to make the charges -2 on both sides:
Cr2O72- + 14H+(aq) + 6e → 2Cr3+ + 7H2O(l)
C2O22- → 2CO2 + 2e
4) Balance the number of electrons by multiplying the equation(s) with a whole number.
Cr2O72- + 14H+(aq) + 6e → 2Cr3+ + 7H2O(l)
3C2O22- → 6CO2 + 6e
5) Once everything is balanced, add the half-reactions together and check if all the atoms and charges are balanced again.
Cr2O72- + 14H+(aq) + 6e → 2Cr3+ + 7H2O(l)
3C2O22- → 6CO2 + 6e
______________________________________________
Cr2O72- + 14H+(aq) + 6e + 3C2O22- → 2Cr3+ + 7H2O + 6CO2 + 6e
6) Cancel any species that appear in equal quantitates on both sides of the equation:
Cr2O72- + 14H+(aq) + 6e + 3C2O22- → 2Cr3+ + 7H2O(l) + 6CO2 + 6e
3C2O22- + Cr2O72- + 14H+(aq) → 6CO2 + 2Cr3+ + 7H2O(l)
d) PbO2 + Mn2+ + SO42 → PbSO4 + MnO4
1) Separate the half-reactions:
Mn2+ → MnO4
PbO2 + SO42 → PbSO4
2) Balance the elements. Add H2O to balance the O atoms, and H+ to balance the H atoms.
Mn2+ + 4H2O(l) → MnO4+ 8H+(aq)
PbO2 + SO42 + 4H+(aq) → PbSO4 + 2H2O(l)
3) Balance the charges by adding electrons.
Mn2+ + 4H2O(l) → MnO4+ 8H+(aq) + 5e
PbO2 + SO42 + 4H+(aq) + 2e→ PbSO4 + 2H2O(l)
4) Balance the number of electrons by multiplying the equation(s) with a whole number. We need to multiply the first half-reaction (oxidation half-reaction) by 2 and the reduction half-reaction by 5:
2Mn2+ + 8H2O → 2MnO4+ 16H+(aq) + 10e
5PbO2 + 5SO42 + 20H+(aq) + 10e→ 5PbSO4 + 10H2O(l)
5) Once everything is balanced, add the half-reactions together and check if all the atoms and charges are balanced again.
2Mn2+ + 8H2O → 2MnO4+ 16H+(aq) + 10e
5PbO2 + 5SO42 + 20H+(aq) + 10e→ 5PbSO4 + 10H2O(l)
______________________________________________
2Mn2+ + 8H2O(l) + 5PbO2 + 5SO42 + 20 4H+(aq) + 10e → 2MnO4+ 16H+(aq) + 10e + 5PbSO4 + 10 2H2O(l)
2Mn2+ + 5PbO2 + 5SO42+ 4H+(aq) → 2MnO4+ 5PbSO4 + 2H2O(l)
e) MnO4(aq) + H2O2(aq) → Mn2+(aq) + O2(g)
1) Separate the half-reactions:
H2O2(aq) → O2(g)
MnO4(aq) → Mn2+(aq)
2) Balance the elements. Add H2O to balance the O atoms, and H+ to balance the H atoms.
H2O2(aq) → O2(g) + 2H+(aq)
MnO4(aq) + 8H+(aq) + 5e→ Mn2+(aq) + 4H2O(l)
3) Balance the charges by adding electrons. The first reaction needs two electrons on the right, and for the second, we need 5 electrons on the left side:
H2O2(aq) → O2(g) + 2H+(aq) + 2e
MnO4(aq) + 8H+(aq) + 5e→ Mn2+(aq) + 4H2O(l)
4) Balance the number of electrons by multiplying the equation(s) with a whole number.
5H2O2(aq) → 5O2(g) + 10H+(aq) + 10e
2MnO4(aq) + 16H+(aq) + 10e→ 2Mn2+(aq) + 8H2O(l)
5) Once everything is balanced, add the half-reactions together and check if all the atoms and charges are balanced again.
5H2O2(aq) → 5O2(g) + 10H+(aq) + 10e
2MnO4(aq) + 16H+(aq) + 10e→ 2Mn2+(aq) + 8H2O(l)
_____________________________________________
5H2O2(aq) + 2MnO4(aq) + 16 6H+(aq) + 10e → 5O2(g) + 10H+(aq) + 10e + 2Mn2+(aq) + 8H2O(l)
5H2O2(aq) + 2MnO4(aq) + 6H+(aq) → 5O2(g) + 2Mn2+(aq) + 8H2O(l)
2.
Balance the following redox reactions occurring in basic aqueous solution:
a) Mn2+(aq) + ClO3(aq) → MnO2(s) + ClO2(g)
b) MnO4(aq) + Br(aq) → MnO2 + BrO3(aq)
c) MnO4(aq) + CN(aq) → CNO(aq) + MnO2(s)
d) H2O2(aq) + ClO2(aq) → ClO2(aq) + O2(g)
e) MnO4(aq) + Fe(OH)2(s) → MnO2(s) + Fe(OH)3(s)
a)
Mn2+ + ClO3 → MnO2 + ClO2
b)
Br(aq) + 2MnO4(aq) → BrO3(aq) + 2MnO2 + 2OH(aq)
c)
3CN(aq) + 2MnO4(aq) + H2O(l) → 3CNO(aq) + 2MnO2(s) + 2OH(aq)
d)
H2O2(aq) + 2ClO2(aq) + 2OH(aq) → O2(g) + 2ClO2(aq) + 2H2O(l)
e)
MnO4(aq) + 3Fe(OH)2(s) + 2H2O(l) → MnO2(s) + 3Fe(OH)3(s) + OH(aq)
Solution
a) Mn2+ + ClO3 → MnO2 + ClO2
1) Separate the half-reactions:
Mn2+ → MnO2
ClO3 → ClO2
2) Balance the elements. Add H2O to balance the O atoms, and H+ to balance the H atoms.
Mn2+ + 2H2O → MnO2 + 4H+(aq)
ClO3 + 2H+(aq) → ClO2 + H2O(l)
3) Balance the charges by adding electrons.
Mn2+ + 2H2O → MnO2 + 4H+(aq) + 2e
ClO3 + 2H+(aq) + e→ ClO2 + H2O(l)
4) Balance the number of electrons by multiplying the equation(s) with a whole number.
Mn2+ + 2H2O → MnO2 + 4H+(aq) + 2e
2ClO3 + 4H+(aq) + 2e→ 2ClO2 + 2H2O(l)
5) Once everything is balanced, add the half-reactions together and cancel any species that appear in equal quantitates on both sides of the equation.
Mn2+ + 2H2O → MnO2 + 4H+(aq) + 2e
2ClO3 + 4H+(aq) + 2e→ 2ClO2 + 2H2O(l)
_____________________________________________
Mn2+ + 2H2O + 2ClO3 + 4H+ + 2e → MnO2 + 4H+ + 2e + 2ClO2 + 2H2O
Mn2+ + ClO3 → MnO2 + ClO2
We don’t need to add any OH ions on any side of the equation, since there are no H+ ions. Therefore, for this reaction, the balanced equation in basic and acidic solutions is identical.
b) MnO4(aq) + Br(aq) → MnO2 + BrO3(aq)
1) Separate the half-reactions:
Br(aq) → BrO3(aq)
MnO4(aq) → MnO2
2) Balance the elements. Add H2O to balance the O atoms, and H+ to balance the H atoms.
Br(aq) + 3H2O(l) → BrO3(aq) + 6H+(aq)
MnO4(aq) + 4H+(aq) → MnO2 + 2H2O(l)
3) Balance the charges by adding electrons.
Br(aq) + 3H2O(l) → BrO3(aq) + 6H+(aq) + 6e
MnO4(aq) + 4H+(aq) + 3e→ MnO2 + 2H2O(l)
4) Balance the number of electrons by multiplying the equation(s) with a whole number.
Br(aq) + 3H2O(l) → BrO3(aq) + 6H+(aq) + 6e
2MnO4(aq) + 8H+(aq) + 6e→ 2MnO2 + 4H2O(l)
5) Once everything is balanced, add the half-reactions together and cancel any species that appear in equal quantitates on both sides of the equation.
Br(aq) + 3H2O(l) → BrO3(aq) + 6H+(aq) + 6e
2MnO4(aq) + 8H+(aq) + 6e→ 2MnO2 + 4H2O(l)
_____________________________________________
Br(aq) + 3H2O + 2MnO4(aq) + 8 2H+ + 6e→ BrO3(aq) + 6H+ + 6e + 2MnO2 + 4H2O(l)
Br(aq) + 2MnO4(aq) + 2H+(aq) → BrO3(aq) + 2MnO2 + H2O(l)
6) This is the balanced red-ox reaction in acidic media, so to obtain the one in a basic solution, we need to add 2OH on both sides of the equation:
Br(aq) + 2MnO4(aq) + 2H+(aq) + 2OH(aq) → BrO3(aq) + 2MnO2 + H2O(l) + 2OH(aq)
The two H+ and OH ions are written as two water molecules, and one of them is canceled with the one on the right side:
Br(aq) + 2MnO4(aq) + 2H2O(l) → BrO3(aq) + 2MnO2 + H2O(l) + 2OH(aq)
Br(aq) + 2MnO4(aq) → BrO3(aq) + 2MnO2 + 2OH(aq)
c) MnO4(aq) + CN(aq) → CNO-(aq) + MnO2(s)
1) Separate the half-reactions:
CN(aq) → CNO(aq)
MnO4(aq) → MnO2(s)
2) Balance the elements. Add H2O to balance the O atoms, and H+ to balance the H atoms.
CN(aq) + H2O(l) → CNO(aq) + 2H+(aq)
MnO4(aq) + 4H+(aq) → MnO2(s) + 2H2O(l)
3) Balance the charges by adding electrons.
CN(aq) + H2O(l) → CNO(aq) + 2H+(aq) + 2e
MnO4(aq) + 4H+(aq) + 3e→ MnO2(s) + 2H2O(l)
4) Balance the number of electrons by multiplying the equation(s) with a whole number.
3CN(aq) + 3H2O(l) → 3CNO(aq) + 6H+(aq) + 6e
2MnO4(aq) + 8H+(aq) + 6e→ 2MnO2(s) + 4H2O(l)
5) Once everything is balanced, add the half-reactions together and cancel any species that appear in equal quantitates on both sides of the equation.
3CN(aq) + 3H2O(l) → 3CNO(aq) + 6H+ + 6e
2MnO4(aq) + 8H+(aq) + 6e→ 2MnO2(s) + 4H2O(l)
_______________________________________
3CN(aq) + 3H2O(l) + 2MnO4(aq) + 8 2H+(aq) + 6e → 3CNO(aq) + 6H+ + 6e + 2MnO2(s) + 4H2O(l)
3CN(aq) + 2MnO4(aq) + 2H+(aq) → 3CNO(aq) + 2MnO2(s) + H2O(l)
6) This is the balanced red-ox reaction in acidic media, so to obtain the one in a basic solution, we need to add 2OH on both sides of the equation:
3CN(aq) + 2MnO4(aq) + 2H+(aq) + 2OH(aq) → 3CNO(aq) + 2MnO2(s) + H2O(l) + 2OH(aq)
The two H+ and OH ions are written as two water molecules, and one of them is canceled with the one on the right side:
3CN(aq) + 2MnO4(aq) + 2H2O(l) → 3CNO(aq) + 2MnO2(s) + H2O + 2OH(aq)
3CN(aq) + 2MnO4(aq) + H2O(l) → 3CNO(aq) + 2MnO2(s) + 2OH(aq)
d) H2O2(aq) + ClO2(aq) → ClO2(aq) + O2(g)
1) Separate the half-reactions:
H2O2(aq) → O2(g)
ClO2(aq) → ClO2(aq)
2) Balance the elements. Add H2O to balance the O atoms, and H+ to balance the H atoms.
H2O2(aq) → O2(g) + 2H+(aq)
ClO2(aq) → ClO2(aq)
3) Balance the charges by adding electrons.
H2O2(aq) → O2(g) + 2H+(aq) + 2e
ClO2(aq) + e → ClO2(aq)
4) Balance the number of electrons by multiplying the equation(s) with a whole number.
H2O2(aq) → O2(g) + 2H+(aq) + 2e
2ClO2(aq) + 2e → 2ClO2(aq)
5) Once everything is balanced, add the half-reactions together and cancel any species that appear in equal quantitates on both sides of the equation.
H2O2(aq) → O2(g) + 2H+ + 2e
2ClO2(aq) + 2e → 2ClO2(aq)
_______________________________________
H2O2(aq) + 2ClO2(aq) + 2e → O2(g) + 2H+ + 2e + 2ClO2(aq)
H2O2(aq) + 2ClO2(aq) → O2(g) + 2H+ + 2ClO2(aq)
6) This is the balanced red-ox reaction in acidic media, so to obtain the one in a basic solution, we need to add 2OH on both sides of the equation:
H2O2(aq) + 2ClO2(aq) + 2OH(aq) → O2(g) + 2H+ + 2ClO2(aq) + 2OH(aq)
The two H+ and OH ions are written as two water molecules:
H2O2(aq) + 2ClO2(aq) + 2OH(aq) → O2(g) + 2ClO2(aq) + 2H2O(l)
e) MnO4(aq) + Fe(OH)2(s) → MnO2(s) + Fe(OH)3(s)
1) Separate the half-reactions:
MnO4(aq) → MnO2(s)
Fe(OH)2(s) → Fe(OH)3(s)
2) Balance the elements. Add H2O to balance the O atoms, and H+ to balance the H atoms.
MnO4(aq) + 4H+(aq) → MnO2(s) + 2H2O(l)
Fe(OH)2(s) + H2O(l) → Fe(OH)3(s) + H+(aq)
3) Balance the charges by adding electrons.
MnO4(aq) + 4H+(aq) + 3e→ MnO2(s) + 2H2O(l)
Fe(OH)2(s) + H2O(l) → Fe(OH)3(s) + H+(aq) + e
4) Balance the number of electrons by multiplying the equation(s) with a whole number.
MnO4(aq) + 4H+(aq) + 3e→ MnO2(s) + 2H2O(l)
3Fe(OH)2(s) + 3H2O(l) → 3Fe(OH)3(s) + 3H+(aq) + 3e
5) Once everything is balanced, add the half-reactions together and cancel any species that appear in equal quantitates on both sides of the equation.
MnO4(aq) + 4H+(aq) + 3e→ MnO2(s) + 2H2O(l)
3Fe(OH)2(s) + 3H2O(l) → 3Fe(OH)3(s) + 3H+(aq) + 3e
_______________________________________
MnO4(aq) + 4H+(aq) + 3e + 3Fe(OH)2(s) + 3H2O(l) → MnO2(s) + 2H2O(l) + 3Fe(OH)3(s) + 3H+ + 3e
MnO4(aq) + H+(aq) + 3Fe(OH)2(s) + H2O(l) → MnO2(s) + 3Fe(OH)3(s)
6) This is the balanced red-ox reaction in acidic media, so to obtain the one in a basic solution, we need one OH on both sides of the equation:
MnO4(aq) + H+(aq) + OH(aq) + 3Fe(OH)2(s) + H2O(l) → MnO2(s) + 3Fe(OH)3(s) + OH(aq)
The H+ and OH ions are written as a water molecule:
MnO4(aq) + 3Fe(OH)2(s) + 2H2O(l) → MnO2(s) + 3Fe(OH)3(s) + OH(aq)
3.
For each redox reaction, sketch a voltaic cell, label the anode and cathode, and indicate the half-reaction that occurs at each electrode. Show the direction of electron flow and the species present in each solution.
a) Cu2+(aq) + 2e → Cu(s)
Sn(s) → Sn2+(aq) + 2e
b) Cd2+(aq) + 2e → Cd(s)
Mn(s) → Mn2+(aq) + 2e
a)
b)
4.
Given the values of standard reduction potentials, determine the redox reaction that will occur in a galvanic cell. Sketch a voltaic cell, label the anode and cathode, and indicate the half-reaction that occurs at each electrode. Show the direction of electron flow and the species present in each solution.
Pb2+(aq) + 2e → Pb(s) Eo = -0.13 V
Ag+(aq) + e → Ag(s) Eo = 0.80 V
Solution
In a galvanic cell, the Eo must be greater than 0. Therefore, the only possibility is to reverse the half-reaction of the Pb2+.
Pb(s) → Pb2+(aq) + 2e Eo = +0.13 V
The overall balanced equation for the redox reaction would be:
2Ag+(aq) + Pb(s) → 2Ag(s) + Pb2+(aq)
Eo = +0.13 V + 0.80 V = 0.93 V
Agis the oxidizing agent, and it is being reduced, while Pb(s) is the reducing agent and it is being oxidized. This corresponds to what we know from the table of standard reduction potentials: The species standing higher in the table are a stronger oxidizing agent
5.
Calculate the E°cell for each balanced redox reaction and determine if the reaction is spontaneous as written.
a) Ni(s) + Mg2+(aq) → Ni2+(aq) + Mg(s)
b) 2Cl(aq) + Pb2+ (aq) → Cl2(l) + Pb(s)
c) Cd(s) + Pb2+(aq) → Cd2+(aq) + Pb(s)
d) Fe(s) + 3Ag+(aq) → Fe3+(aq) + 3Ag(s)
e) Ca2+(aq) + Fe(s) → Ca(s) + Fe2+(aq)
a)
Nonspontaneous
b)
Nonspontaneous
c)
Spontaneous
d)
Spontaneous
e)
Nonspontaneous
Solution
a) Separate the half-reactions and look up the Eo values from a table of standard reduction potentials. When reversing the reaction, change the sign for Eo.
Ni(s) → Ni2+(aq) + 2eEo = +0.23 V
Mg2+(aq) + 2e→ Mg(s) Eo = -2.37 V
The overall cell potential is the sum of the potentials for the half-reactions.
Eo = +0.23 V + (-2.37 V) = -2.14 V
The cell potential is negative, therefore, the reaction is nonspontaneous.
An alternative approach is to write the reactions without changing the sign of Eo, and then use the formula for the total cell potential when the Eo of the anode is subtracted from the Eo of the anode.
In this case, the cathode would work on the reaction Mg2+(aq) + 2e→ Mg(s) since it is a reduction reaction, and the anode is the oxidation of Ni: Ni(s) → Ni2+(aq) + 2e.
The cell potential is equal to:
Eo = EocathodeEoanode = -2.37 V – (-0.23 V) = -2.14 V
Both approaches are correct, however, I found my students like the method of changing the sign of Eo when the reaction is reversed more since, in this case, we only need to add the potentials without worrying about what is being subtracted from what.
Therefore, we will follow this strategy from now on.
b) 2Cl(aq) + Pb2+ (aq) → Cl2(l) + Pb(s)
Separate the half-reactions and look up the Eo values from a table of standard reduction potentials. When reversing the reaction, change the sign for Eo.
2Cl(aq) → Cl2(l) + 2eEo = 1.36 V
Pb2+ (aq) + 2e→ Pb(s) Eo = -0.13 V
The overall cell potential is the sum of the potentials for the half-reactions.
Eo = -1.36 V + (-0.13 V) = -1.49 V
The cell potential is negative, therefore, the reaction is nonspontaneous.
c) Cd(s) + Pb2+(aq) → Cd2+(aq) + Pb(s)
Separate the half-reactions and look up the Eo values from a table of standard reduction potentials. When reversing the reaction, change the sign for Eo.
Cd(s) → Cd2+(aq) + 2eEo = +0.40 V
Pb2+ (aq) + 2e→ Pb(s) Eo = -0.13 V
The overall cell potential is the sum of the potentials for the half-reactions.
Eo = 0.40 V + (-0.13 V) = 0.27 V
The cell potential is positive, therefore, the reaction is spontaneous.
d) Fe(s) + 3Ag+(aq) → Fe3+(aq) + 3Ag(s)
Separate the half-reactions and look up the Eo values from a table of standard reduction potentials. When reversing the reaction, change the sign for Eo.
Fe(s) → Fe3+(aq) + 3eEo = +0.036 V
Ag+ (aq) + e→ Ag(s) Eo = +0.80 V
The overall cell potential is the sum of the potentials for the half-reactions.
Eo = +0.036 V + 0.80 V = 0.836 V
The cell potential is positive, therefore, the reaction is spontaneous.
Notice that we did not balance the number of electrons and it is simply because multiplying a half-reaction with a coefficient does not change the potential of that reaction.
e) Ca2+(aq) + Fe(s) → Ca(s) + Fe2+(aq)
Separate the half-reactions and look up the Eo values from a table of standard reduction potentials. When reversing the reaction, change the sign for Eo.
Fe(s) → Fe2+(aq) + 2eEo = +0.45 V
Ca2+(aq) + 2e→ Ca(s) Eo = -2.76 V
The overall cell potential is the sum of the potentials for the half-reactions.
Eo = +0.45 V + (-2.76 V) = -2.31 V
The cell potential is negative, therefore, the reaction is nonspontaneous.
6.
Balance the following redox reaction occurring in acidic media, and using a table for standard electrode potentials, determine the Eocell at 25 oC.
Cd(s) + Cr2O72−(aq) → Cd2+(aq) + Cr3+(aq)
Eocell = +1.73 V
Solution
1) Separate the half-reactions:
Cr2O72- → Cr3+
Cd → Cd2+
2) Balance the elements. Add H2O to balance the O atoms, and H+ to balance the H atoms.
Cr2O72- + 14H+(aq) → 2Cr3+ + 7H2O
Cd → Cd2+
3) Balance the charges by adding electrons. We need 6e on the left side of the first equation because the charges are +12 vs +6. The second equation needs two electrons on the right side to make the charges 0 on both sides:
Cr2O72- + 14H+(aq) + 6e → 2Cr3+ + 7H2O(l)
Cd → Cd2+ + 2e
4) Balance the number of electrons by multiplying the equation(s) with a whole number.
Cr2O72- + 14H+(aq) + 6e → 2Cr3+ + 7H2O(l)
3Cd → 3Cd2+ + 6e
5) Once everything is balanced, add the half-reactions together and check if all the atoms and charges are balanced again.
Cr2O72- + 14H+(aq) + 6e → 2Cr3+ + 7H2O(l)
3Cd → 3Cd2+ + 6e
______________________________________________
Cr2O72- + 14H+(aq) + 6e + 3Cd → 2Cr3+ + 7H2O(l) + 3Cd2+ + 6e
6) Cancel any species that appear in equal quantitates on both sides of the equation:
Cr2O72- + 14H+(aq) + 6e + 3Cd → 2Cr3+ + 7H2O(l) + 3Cd2+ + 6e
Cr2O72- + 14H+(aq) + 3Cd → 2Cr3+ + 7H2O(l) + 3Cd2+
Now that we have the overall reaction, we need to separate it into the oxidation and reduction half-reactions to look up their potentials in the table:
Oxidation: 3Cd → 3Cd2+ + 6e– Eo = +0.40 V
Reduction: Cr2O72- + 14H+(aq) + + 6e → 2Cr3+ + 7H2O(l) Eo = +1.33 V
Eocell = 0.40 V + +1.33 V = +1.73 V
The cell potential is positive, and therefore, the reaction is spontaneous.
7.
Determine the balanced net reaction from the following two half-reactions and calculate the Eocell at 25 oC.
NO(g) + 2H2O(l) ⟶ NO3(aq) + 4H+(aq) + 3e E° = -0.96 V
MnO2(s) + 4H+(aq) + 2e ⟶ Mn2+(aq) + 2H2O(l) E° = 1.21 V
E°cell = 0.25 V
Solution
The first reaction is an oxidation reaction as the nitrogen goes from +2 to +5 oxidation state, while the second is a reduction reaction where the manganese goes from +4 to +2. Therefore, there is no need to reverse any of the reactions. We only need to balance the number of electrons by multiplying the first equation by 2 and the second equation by 3.
2NO(g) + 4H2O(l) ⟶ 2NO3(aq) + 8H+(aq) + 6e E° = -0.96 V
3MnO2(s) + 12H+(aq) + 6e ⟶ 3Mn2+(aq) + 6H2O(l) E° = 1.21 V
Notice that the reaction potentials do not change with coefficients. The net equation would be:
2NO(g) + 4H2O(l) ⟶ 2NO3(aq) + 8H+(aq) + 6e E° = 0.96 V
3MnO2(s) + 12H+(aq) + 6e ⟶ 3Mn2+(aq) + 6H2O(l) E° = 1.21 V
___________________________________________________
2NO(g) + 4H2O(l) + 3MnO2(s) + 12 4H+(aq) + 6e ⟶ 2NO3(aq) + 8H+(aq) + 6e + 3Mn2+(aq) + 6 2H2O(l)
2NO(g) + 3MnO2(s) + 4H+(aq) ⟶ 2NO3(aq) + 3Mn2+(aq) + 2H2O(l)
E°cell = 1.21 V – 0.96 V = 0.25 V
8.
Using a table for standard electrode potentials, find a metal that can be used to reduce Al3+ ions but not Na+ ions.
Mg
Solution
The strength of reducing power increasing as we go down the table of standard reduction potentials. So, metals that are below Al3+ can reduce it to Al, and since we don’t want to reduce the Na+ ions, the metal should be above Na. The only metal that is below Al3+ and above Na+ is Mg.
You can also test this by writing the half-reactions. The Eo must be positive in order for the reaction to occur:
We switch the sign of Efor the reversed reaction:
Al3+(aq) + 2e– → Al(s) E= -1.66 V
Mg(s) → Mg2+(aq) + 2e– E+2.37 V
Eocell = 2.37 – 1.66 = 0.71 V
9.
Using a table for standard electrode potentials, determine if Fe can be reduced by Sn2+ ions.
Fe can reduce Sn2+ ions.
Solution
The strength of reducing power increasing as we go down the table of standard reduction potentials. So, metals that are below Sn2+ can reduce it to Sn, and because Fe is below Sn, it can reduce Sn2+ ions.
You can also test this by writing the half-reactions. The Eo must be positive in order for the reaction to occur:
We switch the sign of Efor the reversed reaction:
Sn2+(aq) + 2e– → Sn(s) E= -0.14 V
Fe(s) → Fe2+(aq) + 2e– E= +0.45 V
Eocell = 0.45 – 0.14 = 0.31 V
Notice that the iron cannot be oxidized to Fe3because the Eocell won’t be positive in that case.
10.
Using a table for standard electrode potentials, determine if H2(g) is capable of reducing Cd2+(aq).
Hcannot reduce Cd2+.
Solution
Hcannot reduce Cd2+ because it stands above in the table of reduction potentials.
You can also test this by writing the half-reactions. The Eo must be positive in order for the reaction to occur:
We switch the sign of Efor the reversed reaction:
Cd2+(aq) + 2e– → Cd(s) E= -0.14 V
H2(g) → 2H+(aq) + 2e E° = 0 V
Eocell = -0.14 + 0 = -0.14 V
11.
Cell Potential, Free Energy, and the Equilibrium Constant
Use tabulated electrode potentials to calculate ΔG°rxn for each reaction at 25 °C.
a) Zn(s) + Pb2+(aq) → Zn2+(aq) + Pb(s)
b) Sn2+(aq) + Mg(s) → Sn(s) + Mg2+(aq)
c) Br2(l) + 2I(aq) → 2Br(aq) + I2(s)
d) 2Al(s) + 3Cu2+(aq) → 2Al3+(aq) + 3Cu(s)
e) O2(g) + 4H+(aq) + Cu(s) → Cu2+(aq) + 2H2O(l)
f) MnO2(s) + 4H+(aq) + Cd(s) → Mn2+(aq) + 2H2O(l ) + Cd2+(aq)
a)
ΔG° = -120 kJ
b)
ΔG° = -430 kJ
c)
ΔG° = -110 kJ
d)
ΔG° = -1.16 x 103 kJ
e)
ΔG° = -340 kJ
f)
ΔG° = -310 kJ
Solution
ΔG° is correlated to Eocell with the following formula:
ΔG° = –nFEocell
Where n is the moles of electrons and F is the Faraday’s constant which is equal to 96,485 C. Therefore, the plan is to first calculate the cell potential and then use it to determine the ΔG°.
a) Zn(s) + Pb2+(aq) → Zn2+(aq) + Pb(s)
Separate the half-reactions:
Zn(s) → Zn2+(aq) + 2e– Eo = +0.76 V
Pb2+(aq) + 2e→ Pb(s) Eo = -0.13 V
Eocell = +0.76 V + (-0.13 V) = 0.63 V
Once again, what we did here is looked up the values of Eo and changed the sign of Eo for the reaction that is reversed. In this case, it is the oxidation of Zn. The Eo (Zn2+ → Zn(s)) is -0.76 V and because we have the reverse reaction, we changed the sign and simply added the Eo values to find the cell potential. This method seemed to be less confusing to my students, and that is how will do in these practice problems.
Alternatively, and this is what you will see in most textbooks, the signs of Eo are not changed for reversed reactions, and instead, a minus sign is introduced in the formula of Eocell:
Eocell = Eocathode Eoanode
In this example, Zn(s) → Zn2+(aq) + 2e– is the oxidation, and therefore, it occurs on the anode. The Eoanode = -0.76 V as per the table of standard reduction potentials.
Eocathode, then is equal to -0.13 V, and the cell potential would be:
Eocell = Eocathode Eoanode = -0.13 V – (-0.76 V) = 0.63 V
Now that we have the value of Eocell, we can use it in the formula for ΔG°:
ΔG° = –nFEocell
$\Delta G^\circ {\rm{ }} = {\rm{ }} – nF{E^o}_{cell}\; = \, – 2\,\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}\,{\rm{ \times }}\,\frac{{{\rm{96,485}}\,{\rm{C}}}}{{\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}}}\;{\rm{ \times }}\,{\rm{0}}{\rm{.63}}\,{\rm{V}}\,{\rm{ = }}\, – {\rm{2}}\;{\rm{ \times }}\,{\rm{96,485}}\,\cancel{{\rm{C}}}\,{\rm{ \times }}\,{\rm{0}}{\rm{.63}}\frac{{\rm{J}}}{{\cancel{{\rm{C}}}}}\,{\rm{ = }}\, – 1.2\,{\rm{ \times }}\,{10^5}\,{\rm{J}}\,{\rm{ = }}\, – 1.2\,{\rm{ \times }}\,{10^2}\,{\rm{kJ}}$
b) Sn2+(aq) + Mg(s) → Sn(s) + Mg2+(aq)
1) Separate the half-reactions, and switch the sign of Eo for the reversed reaction:
Sn2+(aq) + 2e→ Sn(s) Eo = -0.14 V
Mg(s) → Mg2+(aq) + 2e– Eo = +2.37 V
2) Calculate the Eocell by adding the potentials of the two reactions:
Eocell = -0.14 V + 2.37 V = 2.23 V
3) Use the value of Eocell to determine the ΔG°:
ΔG° = –nFEocell
$\Delta G^\circ {\rm{ }} = {\rm{ }} – nF{E^o}_{cell}\; = \, – 2\,\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}\,{\rm{ \times }}\,\frac{{{\rm{96,485}}\,{\rm{C}}}}{{\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}}}\;{\rm{ \times }}\,{\rm{2}}{\rm{.23}}\,{\rm{V}}\,{\rm{ = }}\, – {\rm{2}}\;{\rm{ \times }}\,{\rm{96,485}}\,\cancel{{\rm{C}}}\,{\rm{ \times }}\,{\rm{2}}{\rm{.23}}\frac{{\rm{J}}}{{\cancel{{\rm{C}}}}}\,{\rm{ = }}\, – 4.30\,{\rm{ \times }}\,{10^5}\,{\rm{J}}\,{\rm{ = }}\, – 430\,{\rm{kJ}}$
c) Br2(l) + 2I(aq) → 2Br(aq) + I2(s)
1) Separate the half-reactions, and switch the sign of Eo for the reversed reaction:
Br2(l) + 2e → 2Br(aq) Eo = +1.09 V
2I(aq) → I2(s) + 2e– Eo = 0.54 V
2) Calculate the Eocell by adding the potentials of the two reactions:
Eocell = 1.09 V + (-0.54 V) = 0.55 V
3) Use the value of Eocell to determine the ΔG°:
ΔG° = –nFEocell
$\Delta G^\circ {\rm{ }} = {\rm{ }} – nF{E^o}_{cell}\; = \, – 2\,\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}\,{\rm{ \times }}\,\frac{{{\rm{96,485}}\,{\rm{C}}}}{{\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}}}\;{\rm{ \times }}\,{\rm{0}}{\rm{.55}}\,{\rm{V}}\,{\rm{ = }}\, – {\rm{2}}\;{\rm{ \times }}\,{\rm{96,485}}\,\cancel{{\rm{C}}}\,{\rm{ \times }}\,{\rm{0}}{\rm{.55}}\frac{{\rm{J}}}{{\cancel{{\rm{C}}}}}\,{\rm{ = }}\, – 1.1\,{\rm{ \times }}\,{10^5}\,{\rm{J}}\,{\rm{ = }}\, – 110\,{\rm{kJ}}$
d) 2Al(s) + 3Cu2+(aq) → 2Al3+(aq) + 3Cu(s)
1) Separate the half-reactions, and switch the sign of Eo for the reversed reaction:
Al(s) → Al3+(aq) + 3e Eo = +1.66 V
Cu2+(aq) + 2e– → Cu(s) Eo = +0.34 V
Even though adding coefficients does not change the potential of a reaction, it is important to do it when you need to calculate the ΔG° because the formula includes the number of moles of electrons. To balance the number of electrons in both equations, we multiply the first by two, and the second by three, and the number of electrons will be six:
2Al(s) → 2Al3+(aq) + 6e Eo = +1.66 V
3Cu2+(aq) + 6e– → 3Cu(s) Eo = +0.34 V
2) Calculate the Eocell by adding the potentials of the two reactions:
Eocell = 1.66 V + 0.34 V = 2.00 V
3) Use the value of Eocell to determine the ΔG°:
ΔG° = –nFEocell
$\Delta G^\circ {\rm{ }} = {\rm{ }} – nF{E^o}_{cell}\; = \, – 6\,\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}\,{\rm{ \times }}\,\frac{{{\rm{96,485}}\,{\rm{C}}}}{{\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}}}\;{\rm{ \times }}\,{\rm{2}}{\rm{.00}}\,{\rm{V}}\,{\rm{ = }}\, – {\rm{6}}\;{\rm{ \times }}\,{\rm{96,485}}\,\cancel{{\rm{C}}}\,{\rm{ \times }}\,{\rm{2}}{\rm{.00}}\frac{{\rm{J}}}{{\cancel{{\rm{C}}}}}\,{\rm{ = }}\, – 1.16\;{\rm{ \times }}\,{10^6}\,{\rm{J}}\,{\rm{ = }}\, – 1.16\;{\rm{ \times }}\,{10^3}\,{\rm{kJ}}$
e) O2(g) + 4H+(aq) + Cu(s) → Cu2+(aq) + 2H2O(l)
1) Separate the half-reactions, and switch the sign of Eo for the reversed reaction:
O2(g) + 4H+(aq) + 4e→ 2H2O(lEo = +1.23 V
Cu(s) → Cu2+(aq) + 2e– Eo = 0.34 V
Balance the number of electrons:
O2(g) + 4H+(aq) + 4e→ 2H2O(lEo = +1.23 V
2Cu(s) → 2Cu2+(aq) + 4e– Eo = -0.34 V
2) Calculate the Eocell by adding the potentials of the two reactions:
Eocell = 1.23 V + (-0.34 V) = 0.89 V
3) Use the value of Eocell to determine the ΔG°:
ΔG° = –nFEocell
$\Delta G^\circ {\rm{ }} = {\rm{ }} – nF{E^o}_{cell}\; = \, – 4\,\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}\,{\rm{ \times }}\,\frac{{{\rm{96,485}}\,{\rm{C}}}}{{\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}}}\;{\rm{ \times }}\,0.89\,{\rm{V}}\,{\rm{ = }}\, – {\rm{4}}\;{\rm{ \times }}\,{\rm{96,485}}\,\cancel{{\rm{C}}}\,{\rm{ \times }}\,{\rm{0}}{\rm{.89}}\frac{{\rm{J}}}{{\cancel{{\rm{C}}}}}\,{\rm{ = }}\, – 3.4\,{\rm{ \times }}\,{10^5}\,{\rm{J}}\,{\rm{ = }}\, – 340\,{\rm{kJ}}$
f) MnO2(s) + 4H+(aq) + Cd(s) → Mn2+(aq) + 2H2O(l) + Cd2+(aq)
1) Separate the half-reactions, and switch the sign of Eo for the reversed reaction:
MnO2(s) + 4H+(aq) + 2e– → Mn2+(aq) + 2H2O(lEo = +1.21 V
Cd(s) → Cd2+(aq) + 2e– Eo = +0.40 V
2) Calculate the Eocell by adding the potentials of the two reactions:
Eocell = 1.21 V + (0.40 V) = 1.61 V
3) Use the value of Eocell to determine the ΔG°:
ΔG° = –nFEocell
$\Delta G^\circ {\rm{ }} = {\rm{ }} – nF{E^o}_{cell}\; = \, – 2\,\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}\,{\rm{ \times }}\,\frac{{{\rm{96,485}}\,{\rm{C}}}}{{\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}}}\;{\rm{ \times }}\,1.61\,{\rm{V}}\,{\rm{ = }}\, – {\rm{2}}\;{\rm{ \times }}\,{\rm{96,485}}\,\cancel{{\rm{C}}}\,{\rm{ \times }}\,{\rm{1}}{\rm{.61}}\frac{{\rm{J}}}{{\cancel{{\rm{C}}}}}\,{\rm{ = }}\, – 3.1\,{\rm{ \times }}\,{10^5}\,{\rm{J}}\,{\rm{ = }}\, – 310\,{\rm{kJ}}$
Notice that all the reactions had a positive Eo indicating that the reactions are spontaneous and can be used to run a galvanic cell. Importantly, when Eo is positive, the ΔG° is negative which is also an indicator of a spontaneous process.
12.
Calculate the equilibrium constant for each redox reaction in problem 11.
a) Zn(s) + Pb2+(aq) → Zn2+(aq) + Pb(s)
b) Sn2+(aq) + Mg(s) → Sn(s) + Mg2+(aq)
c) Br2(l) + 2I(aq) → 2Br(aq) + I2(s)
d) 2Al(s) + 3Cu2+(aq) → 2Al3+(aq) + 3Cu(s)
e) O2(g) + 4H+(aq) + Cu(s) → Cu2+(aq) + 2H2O(l)
f) MnO2(s) + 4H+(aq) + Cd(s) → Mn2+(aq) + 2H2O(l ) + Cd2+(aq)
a)
K = 1.91 x 1021
b)
K = 2.33 x 1075
c)
K = 3.88 x 1018
d)
K = 6.07 x 10202
e)
K = 1.44 x 1060
f)
K = 2.59 x 1054
Solution
There are two equations we can use to calculate the equilibrium constant. One is linked through the Eocell, and the other, through the ΔG° which is also calculated through the Eocell. Yes, we can calculate the ΔG° using the standard free energies of formations, however, this is not related to electrochemistry and we won’t focus on that here.
${E^o}_{{\rm{cell}}}\, = \,\frac{{0.0257\;{\rm{V}}}}{n}\,\ln \,K$
$K\; = \;{e^{\frac{{nE}}{{0.0257}}}}$
For reaction (a), Eocell = 0.63 V, therefore, K would be:
$K\; = \;{e^{\frac{{2\, \cdot \,0.63}}{{0.0257}}}}\, = \,{e^{49.0}}\, = \;1.91\, \times \,{10^{21}}$
The second approach:
The equilibrium constant is correlated with the standard free energy change (ΔG°) by this formula:
ΔG° = –RT ln K
Therefore, the equilibrium constant would be equal to:
$\ln \,K\; = \;\frac{{ – \Delta G^\circ }}{{RT}}$
$K\; = \;{e^{\frac{{ – \Delta G^\circ }}{{RT}}}}$
a) Zn(s) + Pb2+(aq) → Zn2+(aq) + Pb(s)
We determined that ΔG° = -120 x 103 J, however, this is after rounding off the significant figures. When calculating the K, you should not round off the value of ΔG° but rather use it as it is and round off the value of K since being in the exponent, it changes the value of K significantly:
$K\; = \;{e^{\frac{{{\rm{ – }}\left( {{\rm{ – 121,571}}{\rm{.1}}\,{\rm{J}}} \right)}}{{{\rm{8}}{\rm{.314}}\,{\rm{J/K}} \cdot {\rm{mol}}\, \cdot {\rm{298}}\;{\rm{K}}}}}}\, = \,{e^{49.0}}\; = \;1.90\, \times \,{10^{21}}\,$
And now, compare to what we’d get when using the rounded value of ΔG°:
$K\; = \;{e^{\frac{{{\rm{ – }}\left( {{\rm{ – 120,000}}\,{\rm{J}}} \right)}}{{{\rm{8}}{\rm{.314}}\,{\rm{J/K}} \cdot {\rm{mol}}\, \cdot {\rm{298}}\;{\rm{K}}}}}}\, = \,{e^{48.4}}\; = \;1.08\, \times \,{10^{21}}\,$
For the next problems, we will use the first approach to calculate the K.
b) Sn2+(aq) + Mg(s) → Sn(s) + Mg2+(aq)
For reaction (b), Eocell = 2.23 V, therefore, K would be:
$K\; = \;{e^{\frac{{2\, \cdot \,2.23}}{{0.0257}}}}\, = \;2.33\, \times \,{10^{75}}$
c) Br2(l) + 2I(aq) → 2Br(aq) + I2(s)
For reaction (c), Eocell = 0.55 V, therefore, K would be:
$K\; = \;{e^{\frac{{2\, \cdot \,0.55}}{{0.0257}}}}\, = \;3.88\, \times \,{10^{18}}$
d) 2Al(s) + 3Cu2+(aq) → 2Al3+(aq) + 3Cu(s)
For reaction (d), Eocell = 2.00 V, and the number of electron moles is 6, therefore, K would be:
$K\; = \;{e^{\frac{{6\, \cdot \,2.00}}{{0.0257}}}}\, = \;6.07\, \times \,{10^{202}}$
e) O2(g) + 4H+(aq) + Cu(s) → Cu2+(aq) + 2H2O(l)
For reaction (e), Eocell = 0.89 V, and the number of electron moles is 4, therefore, K would be:
$K\; = \;{e^{\frac{{4\, \cdot \,0.89}}{{0.0257}}}}\, = \;1.44\, \times \,{10^{60}}$
f) MnO2(s) + 4H+(aq) + Cd(s) → Mn2+(aq) + 2H2O(l ) + Cd2+(aq)
For reaction (f), Eocell = 1.61 V, therefore, K would be:
$K\; = \;{e^{\frac{{2\, \cdot \,1.61}}{{0.0257}}}}\, = \;2.59\, \times \,{10^{54}}$
13.
Use an appendix for the free energies of formation to calculate the standard cell potential for the oxidation reaction of ethylene, C2H4 by permanganate ion, MnO4.
5C2H4(g) + 12MnO4(aq) + 36H+(aq) → 10CO2(g) + 12Mn2+(aq) + 28H2O(l)
The standard free energy of formation for Mn2+ ions is -228 kJ/mol.
1.43 V
Solution
The plan is to calculate the ΔG° rxn and use in the expression ΔG° = –nFEocell to determine the Eocell.
To calculate 𝚫G°rxn, we need to subtract the standard free energies of formations of the reactants from the standard free energies of formations of products, both multiplied by their stoichiometric coefficients:
ΔG°rxn = ΣnpΔGof (products) – ΣnrΔGo(reactants)
Where np and nr are the molar coefficients of the products and reactants in the balanced chemical equation.
Remember, the standard free energies of formations of elements or their molecules in standard states are equal to zero.
5C2H4(g) + 12MnO4(aq) + 36H+(aq) → 10CO2(g) + 12Mn2+(aq) + 28H2O(l)
ΔG°rxn = [10 ΔGo CO2(g) + 12 ΔGo Mn2+(aq) + 28 ΔGo H2O(l)] – [5 ΔGo C2H4(g) + 12 ΔGo MnO4(aq) + 36 ΔGo H+(aq)]
ΔG°rxn = [10 (-394) + 12 (-228) + 28 x (-237)] – [5 x 68 + 12 (-449) + 36 x 0]
ΔG°rxn = [-3940 -2736 – 6636] – [340 – 5388 + 0] = -8264 kJ = -8.264 x 106 J
From the expression ΔG° = –nFEocell, we can write that:
${E^o}_{{\rm{cell}}}\, = \, – \frac{{\Delta G{^\circ _{rxn}}\;}}{{nF}}$
So, at this point, we need to obtain the number of electrons by separating the half-reactions. First separate, then balance the half-reaction equations, and add electrons when needed as we did for balancing redox reactions:
5C2H4(g) + 12MnO4(aq) + 36H+(aq) → 10CO2(g) + 12Mn2+(aq) + 28H2O(l)
____________________________________________________________
12MnO4(aq) + 36H+(aq) → 12Mn2+(aq) + 28H2O(l)
5C2H4(g) → 10CO2(g)
Balance the atoms and charges:
12MnO4(aq) + 96H+(aq) + 60e → 12Mn2+(aq) + 48H2O(l)
5C2H4(g) + 20H2O(l) → 10CO2(g) + 60H+(aq) + 60e
There are 60 electrons in each half-reaction, therefore, n = 60. To match the units, we will write the value of ΔG° in C·V.
${E^o}_{{\rm{cell}}}\, = \, – \frac{{ – 8.264\; \times \,{{10}^6}\,{\rm{C}}\,{\rm{ \times }}\;{\rm{V\;}}}}{{60\, \times 96,485\,{\rm{C}}}} = \,1.43\,{\rm{V}}$
14.
The Nernst Equation
A voltaic cell runs the following redox reaction:
Fe2+(aq) + Mg(s) → Mg2+(aq) + Fe(s)
Determine whether Ecell is larger or smaller than E°cell and determine the Ecell for the following concentrations:
a) when [Fe2+] = 1.0 M and [Mg2+] = 2.5 M
b) when [Fe2+] = 3.0 M and [Mg2+] = 1.0 M
a)
1.908 V
b)
1.934 V
Solution
There are two parts to this practice problem; one is to determine if Ecell is larger or smaller than E°cell, and the second is to actually calculate the Ecell at the given concentrations.
On the bases of both is the Nernst equation:
$E\; = \,{E^o}\; – \,\frac{{0.0257\;{\rm{V}}}}{n}\,\ln \,Q$
Remember, V is just a unit here, and if it confuses you, you can write the equation as:
$E\; = \,{E^o}\; – \,\frac{{0.0257}}{n}\,\ln \,Q$
Where Q is the reaction quotient, and it defines, based on the concentrations of products and reactants, whether Ecell is larger or smaller than E°cell.
However, before moving on to that part, we want to make sure that this is a galvanic cell i.e. the reaction is spontaneous.
Fe2+(aq) + 2e→ Fe(s) E° = -0.45 V
Mg(s) → Mg2+(aq) + 2e– E° = +2.37 V
Ecell = -0.45 V + 2.37 V = 1.92 V
The cell potential is positive so, at standard conditions, it is a spontaneous reaction. This means any changes in concentrations that increase the tendency of the forward reaction to occur will increase the cell potential.
Consequently, if changing the concentrations decreases the tendency of the forward reaction to occur, it
will decrease the cell potential and increases the tendency of the reverse reaction to occur.
So, the next part is to use the Le Chatelier’s principle and determine how each of the concentration set will affect the tendency of the reaction and therefore, the cell potential.
a) when [Fe2+] = 1.0 M and [Mg2+] = 2.5 M
Fe2+(aq) + Mg(s) → Mg2+(aq) + Fe(s)
The concentration of the product, Mg2+ is higher than what it is at standard conditions (1 M), therefore, according to the Le Chatelier’s principle, the reaction will tend to occur in the reverse direction. This also means that Ecell is smaller than E°cell.
Let’s also plug the numbers into the Nernst equation and determine the Ecell:
$E\; = \,{E^o}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,\frac{{{\rm{[M}}{{\rm{g}}^{{\rm{2 + }}}}{\rm{]}}}}{{{\rm{[F}}{{\rm{e}}^{{\rm{2 + }}}}{\rm{]}}}}$
$E\; = \,1.92\,{\rm{V}}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,\frac{{{\rm{[2}}{\rm{.5]}}}}{{{\rm{[1}}{\rm{.0]}}}}\; = \;1.908\,{\rm{V}}$
Because Q = 2.5, the difference of Ecell and E°cell is not large. However, it is important to remember the pattern: if Q > 1, Ecell < E°cell, and if Q < 1, Ecell > E°cell.
Review the principle of reaction quotient if this is still confusing.
b) when [Fe2+] = 3.0 M and [Mg2+] = 1.0 M
There is more reactant in the system than there is at standard conditions, so the reaction will tend to shift forward and consume some of it. Therefore, Ecell should be larger than E°cell.
$E\; = \,{E^o}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,\frac{{{\rm{[M}}{{\rm{g}}^{{\rm{2 + }}}}{\rm{]}}}}{{{\rm{[F}}{{\rm{e}}^{{\rm{2 + }}}}{\rm{]}}}}$
$E\; = \,1.92\,{\rm{V}}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,\frac{{{\rm{[1]}}}}{{{\rm{[3}}{\rm{.0]}}}}\; = \;1.934\,{\rm{V}}$
15.
Calculate E°, E, and ΔG for the following cell reaction:
Mn(s) + Pb2+(aq) → Mn2+(aq) + Pb(s)
[Mn2+] = 0.164 M, [Pb2+] = 0.038
E°cell = 1.05 V
Ecell = 1.03 V
ΔG = -199 kJ
Solution
To calculate the E°, we separate the half-reactions, and look up the standard potentials:
Mn(s) → Mn2+(aq) + 2e E° = +1.18 V
Pb2+(aq) + 2e→ Pb(s) E° = -0.13 V
E°cell = +1.18 V – 0.13 V = 1.05 V
To calculate the E, we use the Nernst equation:
$E\; = \,{E^o}\; – \,\frac{{0.0257\;{\rm{V}}}}{n}\,\ln \,Q$
$E\; = \,{E^o}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,\frac{{{\rm{[M}}{{\rm{n}}^{{\rm{2 + }}}}{\rm{]}}}}{{{\rm{[P}}{{\rm{b}}^{{\rm{2 + }}}}{\rm{]}}}}$
$E\; = \,1.05\,{\rm{V}}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,\frac{{{\rm{[0}}{\rm{.164]}}}}{{{\rm{[0}}{\rm{.038]}}}}\; = \,1.031\,{\rm{V}}$
And now, we use the value of Ecell to determine the ΔG:
ΔG = –nFEcell
$\Delta G{\rm{ }} = {\rm{ }} – nF{E_{cell}}\; = \, – 2\,\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}\,{\rm{ \times }}\,\frac{{{\rm{96,485}}\,{\rm{C}}}}{{\cancel{{{\rm{mol }}{{\rm{e}}^{\rm{ – }}}}}}}\;{\rm{ \times }}\,{\rm{1}}{\rm{.031V}}\,{\rm{ = }}\, – {\rm{2}}\;{\rm{ \times }}\,{\rm{96,485}}\,\cancel{{\rm{C}}}\,{\rm{ \times }}\,{\rm{1}}{\rm{.031}}\frac{{\rm{J}}}{{\cancel{{\rm{C}}}}}\,{\rm{ = }}\, – 1.99\,{\rm{ \times }}\,{10^5}\,{\rm{J}}\,{\rm{ = }}\, – 199\,{\rm{kJ}}$
16.
A galvanic cell operates on the following redox reaction:
2MnO4(aq) + 10Br(aq) + 16H+(aq) → 2Mn2+(aq) + 5Br2(l) + 8H2O(l)
Calculate the cell potential at 25 oC, given the concentrations of the aqueous components are: [MnO4] = 0.0250 M, [Br] = 0.0150 M, [Mn2+] = 0.380 M, and [H+] = 0.64 M.
1.37 V
Solution
To calculate the cell potential, we again use the Nernst equation because the concentrations are not 1 M.
$E\; = \,{E^o}\; – \,\frac{{0.0257\;{\rm{V}}}}{n}\,\ln \,Q$
We need to first calculate the Eo by separating the half-reactions to find the potential of each electrode in the table:
2MnO4(aq) + 16H+(aq) + 10e → 2Mn2+(aq) + 8H2O(l) Eo = +1.51 V
10Br(aq) → 5Br2(l) + 10e Eo = -1.09 V
Eocell = 1.51 – 1.09 = 0.42 V
The quotient is equal to:
$Q\; = \,\frac{{{{{\rm{[M}}{{\rm{n}}^{{\rm{2 + }}}}{\rm{]}}}^{\rm{2}}}}}{{{{{\rm{[Mn}}{{\rm{O}}_{\rm{4}}}^{\rm{ – }}{\rm{]}}}^{\rm{2}}}{{{\rm{[B}}{{\rm{r}}^{\rm{ – }}}{\rm{]}}}^{{\rm{10}}}}{{{\rm{[}}{{\rm{H}}^{\rm{ + }}}{\rm{]}}}^{{\rm{16}}}}}}$
$Q\; = \,\frac{{{{\left( {{\rm{0}}{\rm{.380}}} \right)}^{\rm{2}}}}}{{{{\left( {{\rm{0}}{\rm{.0250}}} \right)}^{\rm{2}}}{{\left( {{\rm{0}}{\rm{.0150}}} \right)}^{{\rm{10}}}}{{\left( {{\rm{0}}{\rm{.64}}} \right)}^{{\rm{16}}}}}}\; = \;5.06\; \times \;{10^{23}}$
There are 10 electrons involved in the reaction, therefore, for the Nernst equation, we have:
$E\; = \,1.51\,{\rm{V}}\; – \,\frac{{0.0257\;{\rm{V}}}}{{10}}\,\ln \,{\rm{5}}{\rm{.06}}\, \times \,{\rm{1}}{{\rm{0}}^{{\rm{23}}}}\, = \,1.37\,{\rm{V}}$
17.
A voltaic cell consists of a Mg/Mg2+ half-cell and a Sn/Sn2+ half-cell at 25 °C. The initial concentrations of Sn2+ and Mg2+ are 1.80 M and 0.250 M, respectively.
a) Calculate the initial cell potential.
b) What is the cell potential when the concentration of Sn2+ dropped to 0.650 M?
c) Determine the concentrations of Sn2+ and Mg2+ ions when the cell potential falls to 2.15 V.
a)
E = 2.26 V
b)
E = 2.22 V
c)
[Mg2+] = 2.04595 M
[Sn2+] = 0.00405 M
Solution
a) To calculate the initial cell potential, we determine the E°, by separating the half-reactions such E° > 0. Because Sn/Sn2+ has a higher potential, it will be the oxidizing agent, and therefore, taking the electrons from the Mg2+ ion:
Mg(s) → Mg2+(aq) + 2e E° = +2.37 V
Sn2+(aq) + 2e→ Sn(s) E° = -0.14 V
The overall reaction is the sum of the two half-reactions:
Mg(s) + Sn2+(aq) → Mg2+(aq) + Sn(s)
E°cell = +2.37 V – 0.14 V = 2.23 V
Now, this is the standard cell potential which is when the concentrations are 1 M. However, we have [Sn2+] = 1.80 M, and [Mg2+] = 0.25 M, and therefore, we use the Nernst equation to calculate the Ecell:
$E\; = \,2.23\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,\frac{{{\rm{[0}}{\rm{.25]}}}}{{{\rm{[1}}{\rm{.80]}}}}\; = \,2.26\,{\rm{V}}$
Ecell > Eo and this is what we expect when Q < 1.
b) To calculate the E, we again use the Nernst equation.
$E\; = \,{E^o}\; – \,\frac{{0.0257\;{\rm{V}}}}{n}\,\ln \,Q$
$E\; = \,{E^o}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,\frac{{{\rm{[M}}{{\rm{g}}^{{\rm{2 + }}}}{\rm{]}}}}{{{\rm{[S}}{{\rm{n}}^{{\rm{2 + }}}}{\rm{]}}}}$
We are given the concentration of Sn2+ ions (0.650 M), so to use the Nernst equation, we also need the [Mg2+]. Because of the 1:1 stochiometric ratio, [Mg2+] increases by as much as [Sn2+] decreases. The initial concentration of Sn2+ is 1.80 M 1 molar, so if [Sn2+] drops to 0.650 M, 1.15 M (1.8 – 0.65) of it must be reacted and this is by how much [Mg2+] increases. Therefore, [Mg2+] = 0.250 + 1.15 = 1.4 M.
$E\; = \,2.23\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,\frac{{{\rm{[1}}{\rm{.40]}}}}{{{\rm{[0}}{\rm{.65]}}}}\; = \;2.22\,{\rm{V}}$
c) Here, we need to determine the concentrations of the ions given the values of E° = 2.23 V, and E = 2.15 V.
The total concentration of the ions is 1.80 M + 0.250 M = 2.05 M, and it will stay constant throughout the reaction only that [Sn2+] will be decreasing, and [Mg2+] increase by the same amount. So, we can assign [Mg2+] = x M, and [Sn2+] = (2.05 -x) M and use the Nernst equation to determine the x:
$E\; = \,2.23\,{\rm{V}}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,\frac{{\rm{x}}}{{{\rm{2}}{\rm{.05}}\,{\rm{ – }}\,{\rm{x}}}}\; = \;2.15\,{\rm{V}}$
$E\; = \,2.23\,{\rm{V}}\; – \,2.15\,{\rm{V}}\; = \;\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,\frac{{\rm{x}}}{{{\rm{2}}{\rm{.05}}\,{\rm{ – }}\,{\rm{x}}}}$
$\frac{{2\,\left( {2.23\,{\rm{V}}\; – \,2.15\,{\rm{V}}} \right)\,}}{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}\; = \,{\rm{ln}}\,\frac{{\rm{x}}}{{{\rm{2}}{\rm{.05}}\,{\rm{ – }}\,{\rm{x}}}}$
$6.2257\; = \,{\rm{ln}}\,\frac{{\rm{x}}}{{{\rm{2}}{\rm{.05}}\,{\rm{ – }}\,{\rm{x}}}}$
${e^{6.2257}}\; = \,\frac{{\rm{x}}}{{{\rm{2}}{\rm{.05}}\,{\rm{ – }}\,{\rm{x}}}}\; = \,505.6$
Solving for x, we find that x = 2.04595, and this is the [Mg2+].
The concentration of Sn2+ ions would then be: 2.05 – 2.04595 = 0.00405 M
Note: You can also use the notation [Mg2+] instead of x if that helps keeping track of things easier:
$E\; = \,2.23\,{\rm{V}}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,\frac{{{\rm{[M}}{{\rm{g}}^{{\rm{2 + }}}}{\rm{]}}}}{{{\rm{2}}{\rm{.05}}\,{\rm{ – }}\,{\rm{[M}}{{\rm{g}}^{{\rm{2 + }}}}{\rm{]}}}}\; = \;2.15\,{\rm{V}}$
Alternatively, we could assign the concentration of Sn2+ as (1.80 – x) M, and [Mg2+] = (0.25 + x) M for the time when E drops to 2.15, and the Nernst equation would look like this:
$E\; = \,2.23\,{\rm{V}}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,\frac{{{\rm{0}}{\rm{.25}}\,{\rm{ + }}\,{\rm{x}}}}{{{\rm{1}}{\rm{.80}}\,{\rm{ – }}\,{\rm{x}}}}\; = \;2.15\,{\rm{V}}$
Solving for x, we find that x = 1.79595. Therefore,
[Mg2+] = (0.25 + 1.79595) = 2.0459 M
[Sn2+] = (1.80 – x) = 0.00405 M
If the fraction in the ln term is not what you want to see at all, just write Q and use the unknowns once the value of Q is calculated.
$E\; = \,2.23\,{\rm{V}}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,Q\; = \;2.15\,{\rm{V}}$
18.
A voltaic cell consists of a Zn/Zn2+ half-cell and a H2/H+ half-cell at 25 °C. Calculate the initial cell potential if [Zn2+] = 0.0250 M, [H+] = 2.80 M, P(H2) = 0.45 atm.
E = 0.84 V
Solution
The first step is to obtain a correct net reaction such that the cell potential is positive. Zn is below H2 in the reduction potential table, therefore, it will reduce the H+ ions to hydrogen gas.
Zn(s) → Zn2+(aq) + 2e E° = +0.76 V
2H+(aq) + 2e→ H2(g) E° = 0 V
The overall reaction is the sum of the two half-reactions:
Zn(s) + 2H+(aq) → Zn2+(aq) + H2(g)
E°cell = +0.76 V + 0 V = 0.76 V
Next, we calculate the quotient to use it in the Nernst equation.
$Q = \frac{{{P_{{{\rm{H}}_{\rm{2}}}}}\, \times \,{\rm{[Z}}{{\rm{n}}^{{\rm{2 + }}}}{\rm{]}}}}{{{{{\rm{[}}{{\rm{H}}^{\rm{ + }}}{\rm{]}}}^{\rm{2}}}}}\; = \,\frac{{0.45\, \times \,0.0250}}{{{{2.80}^2}}}\; = \,0.001435$
$E\; = \,{E^o}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,Q$
$E\; = \,0.76\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,{\rm{0}}{\rm{.001435}}\,{\rm{ = }}\,{\rm{0}}{\rm{.84 V}}$
19.
Determine the emf of a cell consisting of a Sn2+/Sn half-cell and a Pt/H+/H2 half-cell if [Sn2+] = 0.12 M, [H+] = 0.0650 M, and P(H2) = 1.25 atm?
E = 0.0941 V
Solution
The first step is to obtain a correct net reaction such that the cell potential is positive. Sn is below H2 in the reduction potential table, therefore, it will reduce the H+ ions to hydrogen gas.
Sn(s) → Sn2+(aq) + 2e E° = +0.14 V
2H+(aq) + 2e→ H2(g) E° = 0 V
The overall reaction is the sum of the two half-reactions:
Sn(s) + 2H+(aq) → Sn2+(aq) + H2(g)
E°cell = +0.14 V + 0 V = 0.14 V
Next, we calculate the quotient to use it in the Nernst equation.
$Q = \frac{{{P_{{{\rm{H}}_{\rm{2}}}}}\, \times \,{\rm{[S}}{{\rm{n}}^{{\rm{2 + }}}}{\rm{]}}}}{{{{{\rm{[}}{{\rm{H}}^{\rm{ + }}}{\rm{]}}}^{\rm{2}}}}}\; = \,\frac{{1.25\, \times \,0.120}}{{{{0.0650}^2}}}\; = \,35.5$
$E\; = \,{E^o}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,Q$
$E\; = \,0.14\,{\rm{V}}\; – \,\frac{{{\rm{0}}{\rm{.0257}}\;{\rm{V}}}}{{\rm{2}}}\,{\rm{ln}}\,{\rm{35}}{\rm{.5}}\,{\rm{ = }}\,{\rm{0}}{\rm{.0941 V}}$ | 2022-09-27 20:45:13 | {"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": 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": 0.6479389667510986, "perplexity": 7219.485341131123}, "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/1664030335058.80/warc/CC-MAIN-20220927194248-20220927224248-00411.warc.gz"} |
http://clay6.com/qa/51205/which-of-the-following-orders-regarding-the-melting-points-is-correct- | Browse Questions
# Which of the following orders regarding the melting points is correct?
$\begin{array}{1 1}C > Si > Ge > Sn\\C < Si < Ge < Sn\\C < Si < Ge < Sn\\C < Si > Ge > Sn\end{array}$
Can you answer this question?
Answer : C < Si < Ge < Sn
Because of catenation of carbon,its melting point is greater than silicon.
answered Jul 24, 2014 | 2017-01-23 12:42:26 | {"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": 0.6076568961143494, "perplexity": 6428.943370007709}, "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-2017-04/segments/1484560282926.64/warc/CC-MAIN-20170116095122-00372-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://advancedintegrals.com/2017/01/integral-representation-of-gauss-hypergeometric-function-proof/ | # Integral representation of Gauss Hypergeometric function proof
$$\beta(c-b,b) \, _2F_1(a,b;c;z)=\int_0^1 \frac{t^{b-1}(1-t)^{c-b-1}}{(1-tz)^a}\, dt$$
$$\textit{proof}$$
Start by the RHS
$$\int_0^1 t^{b-1}(1-t)^{c-b-1} \, (1-tz)^{-a}\, dt$$
Using the expansion of $(1-tz)^{-a}$ we have
$$\int_0^1 t^{b-1}(1-t)^{c-b-1} \sum_{k=0}^\infty\frac{(a)_k}{k!}\, (tz)^k$$
Interchanging the integral with the series
$$\sum_{k=0}^\infty\frac{(a)_k}{k!}\, z^k \, \int_0^1 t^{k+b-1}(1-t)^{c-b-1}\, dt$$
Recalling the beta function we have
$$\sum_{n=0}^\infty\frac{(a)_k \Gamma(k+b) \Gamma(c-b)}{ \Gamma(k+c)}\, \frac{z^k}{k!}$$
Using the identity that
$$\beta(c-b,b) = \frac{\Gamma(b)\Gamma(c-b)}{\Gamma(c)}$$
and
$$\frac{\Gamma(z+k)}{\Gamma(z)} = (z)_k$$
We deduce that
$$\beta(c-b,b)\sum_{k=0}^\infty\frac{(a)_k \Gamma(k+b) \Gamma(c)}{\Gamma(b) \Gamma(k+c)}\, \frac{z^k}{k!}= \beta(c-b, b) \sum_{k=0}^\infty \frac{(a)_k (b)_k}{(c)_k} \frac{z^k}{k!} \,\,$$
This entry was posted in Hypergeoemtric function and tagged , , , , . Bookmark the permalink. | 2018-10-18 00:14:24 | {"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": 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": 0.9920666813850403, "perplexity": 2097.6721260490017}, "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/1539583511365.50/warc/CC-MAIN-20181018001031-20181018022531-00134.warc.gz"} |
https://scifi.stackexchange.com/questions/37388/are-the-pod-born-humans-of-the-matrix-universe-sterile/37403 | # Are the pod-born humans of The Matrix universe sterile?
I recently re-watched the movies. Afterward, I sat and mused on various aspects that I had come across. Things I had been unware of the first few times.
One of these things was whether the "grown" humans in the series retained the ability to have children; the answer would, I think, provide an interesting insight into the worldview of those born so. Also how they would view themselves alongside the "naturals", those who were born naturally in Zion.
Could anyone help shed some light on this, please?
• If they were that would have presented a problem for the Architect and the previous Ones. – Xantec Jun 25 '13 at 16:19
• By definition, the Humans of the Matrix universe are NOT sterile, otherwise, the redeployment of Zion would be impossible without the ability to have live birth. The only way this would even be plausible is if the area of Zion is just another layer of the Matrix where the Humans are led to believe they have escaped the lower level of the Matrix. A strange thought but considered by many to be plausible. – Thaddeus Howze Jun 25 '13 at 16:28
• possible duplicate of Where Do Babies Come From (In the Matrix, That Is)? – phantom42 Jun 25 '13 at 17:49
• Oh yeah...I guess they're brother and sister now. – Michael Brown Jun 26 '13 at 13:40
Logically, there would be no reason to sterilize the 'grown' humans, as they aren't able to breed while isolated in pods. In Matrix 1 we see humans being grown from embryos in the harvester fields, which implies that the grown human females don't have their bodies used for gestation, so someone who was freed from the Matrix after experiencing pregnancy and childbirth within it may find themselves in a body that had not, in fact, ever been pregnant; however, this would not prevent them from becoming so using natural methods.
The other evidence point supporting non-sterilization of grown humans is that the Architect specified that new populations of Zion were seeded by grown humans from the Matrix. If those people were unable to reproduce naturally, there would be no Zion and thus, no Matrix rebooting protocol.
It is mentioned that Neo is unusually old to be successfully freed from the Matrix. This suggests that the scenario described above, of a woman being freed after having experienced virtual childbirth, would be unusual (but not impossible if she were freed in, say, her early to mid-teens). Now, this is interesting because the structure of the Matrix dictates that everyone experiences it from birth onward, so an infant, once harvested, would have to have been assigned to a woman who had experienced a virtual pregnancy. Therefore, the woman's 'child' could still be plugged into the Matrix after she was freed.
This highlights one of the more poignant issues that the movies never addressed: freeing someone from the Matrix leaves a person knowing that all the people they have loved - parents, children, siblings, friends - are still trapped in the Matrix. It would be only human for those people to want to visit their loved ones when entering the Matrix, and to work to free those who are young enough to bear the transition. But since the machines are capable of overwriting any consciousness in the Matrix with Agents, it would be exceedingly dangerous to draw their attention. I shall leave further speculation on the emotional ramifications to the reader.
• Very thought provoking. +1! – Donald.McLean Jun 25 '13 at 17:57
• Given that people outside Matrix looks like their avatars it is likely that pod-grown humans have the gene seeded from virtual parents to get the matching fenotypes/genotypes. – Maciej Piechotka Jun 26 '13 at 11:58
• issues that the movies never addressed disagree. From a certain point of view. Cypher was originally meant to want to go back to the Matrix because his loved one (girlfriend?) was still inside. It's in the earlier scripts if I remember well. Even in the final version this is a quite possible past for him, and his feelings for Trinity could be just his next failed attempt to find solution. Also, Trinity (supposedly) liking to watch Neo has a very similar emotional charge, and their romance is of central importance to the movie and the plot. – n611x007 Jun 26 '13 at 18:16
Since all of the initial residents of Zion were themselves pod-grown, I think the answer has to be, "no, they're not sterile", since we've seen that they have children.
• (in Jeff Goldblum voice): uh, life, uh... finds a way. – Lèse majesté Jun 26 '13 at 2:08
Neo, "The One", is responsible for rebuilding humans after a wipeout.
http://matrix.wikia.com/wiki/Neo
Neo is then ultimately presented with a choice: there are two doors; the door on the right leads to the Source, where he would select 23 individuals (16 females and 7 males) who would rebuild Zion after its destruction, with him reinserting his Prime Program to the Machine Mainframe and rebooting the Matrix
There are many Operators born outside of the Matrix. Tank and Link as mentioned in the article are the two that come immediately to mind, for me.
Since "Humanity" has been wiped out on at-least 5 occasions (Neo had '5 predecessors'... and you could probably count the previous iterations of the matrix as at least 6), the only explanation for natural born humans is that pod-people can bear children since individuals like Tank had to have been born from pod-people. | 2021-04-22 04:06:42 | {"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": 0.41582420468330383, "perplexity": 2140.514729075987}, "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/1618039560245.87/warc/CC-MAIN-20210422013104-20210422043104-00543.warc.gz"} |
https://howlingpixel.com/i-en/Stefan%E2%80%93Boltzmann_constant | # Stefan–Boltzmann constant
The Stefan–Boltzmann constant (also Stefan's constant), a physical constant denoted by the Greek letter σ (sigma), is the constant of proportionality in the Stefan–Boltzmann law: "the total intensity radiated over all wavelengths increases as the temperature increases", of a black body which is proportional to the fourth power of the thermodynamic temperature.[1] The theory of thermal radiation lays down the theory of quantum mechanics, by using physics to relate to molecular, atomic and sub-atomic levels. Slovenian physicist Josef Stefan formulated the constant in 1879, and it was later derived in 1884 by Austrian physicist Ludwig Boltzmann.[2] The equation can also be derived from Planck's law, by integrating over all wavelengths at a given temperature, which will represent a small flat black body box.[3] "The amount of thermal radiation emitted increases rapidly and the principal frequency of the radiation becomes higher with increasing temperatures".[4] The Stefan–Boltzmann constant can be used to measure the amount of heat that is emitted by a blackbody, which absorbs all of the radiant energy that hits it, and will emit all the radiant energy. Furthermore, the Stefan–Boltzmann constant allows for temperature (K) to be converted to units for intensity (W m−2), which is power per unit area.
The value of the Stefan–Boltzmann constant is given in SI units by
σ = 5.670367(13)×10−8 W⋅m−2⋅K−4.[5]
In cgs units the Stefan–Boltzmann constant is:
σ5.6704×10−5 erg cm−2 s−1 K−4.
In thermochemistry the Stefan–Boltzmann constant is often expressed in cal cm−2 day−1 K−4:
σ11.7×10−8 cal cm−2 day−1 K−4.
In US customary units the Stefan–Boltzmann constant is:[6]
σ1.714×10−9 BTU hr−1 ft−2 °R−4.
The value of the Stefan–Boltzmann constant is derivable as well as experimentally determinable; see Stefan–Boltzmann law for details. It can be defined in terms of the Boltzmann constant as:
${\displaystyle \sigma ={\frac {2\pi ^{5}k_{\rm {B}}^{4}}{15h^{3}c^{2}}}={\frac {\pi ^{2}k_{\rm {B}}^{4}}{60\hbar ^{3}c^{2}}}=5.670367(13)\,\times 10^{-8}\ {\textrm {J}}\,{\textrm {m}}^{-2}\,{\textrm {s}}^{-1}\,{\textrm {K}}^{-4}}$
where:
The CODATA recommended value is calculated from the measured value of the gas constant:
${\displaystyle \sigma ={\frac {2\pi ^{5}R^{4}}{15h^{3}c^{2}N_{\rm {A}}^{4}}}={\frac {32\pi ^{5}hR^{4}R_{\infty }^{4}}{15A_{\rm {r}}({\rm {e}})^{4}M_{\rm {u}}^{4}c^{6}\alpha ^{8}}}}$
where:
Dimensional formula: [ M1 T−3 Θ−4]
A related constant is the radiation constant (or radiation density constant) a which is given by:[7]
${\displaystyle a={\frac {4\sigma }{c}}=7.5657\times 10^{-15}{\textrm {erg}}\,{\textrm {cm}}^{-3}\,{\textrm {K}}^{-4}=7.5657\times 10^{-16}{\textrm {J}}\,{\textrm {m}}^{-3}\,{\textrm {K}}^{-4}.}$
Log-log graphs of peak emission wavelength and radiant exitance vs black-body temperature – red arrows show that 5780 K black bodies have 501 nm peak wavelength and 63.3 MW/m² radiant exitance
## References
1. ^ Krane, Kenneth (2012). Modern Physics. John Wiley & Sons. p. 81.
2. ^ "Stefan-Boltzmann Law". Encyclopædia Britannica.
3. ^ Halliday & Resnick (2014). Fundamentals of Physics (10th Ed). John Wiley and Sons. p. 1166.
4. ^ Eisberg, Resnick, Robert, Robert (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Ed) (PDF). John Wiley & Sons. Archived from the original (PDF) on 2014-02-26.
5. ^ "CODATA Value: Stefan-Boltzmann constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
6. ^ Heat and Mass Transfer: a Practical Approach, 3rd Ed. Yunus A. Çengel, McGraw Hill, 2007
7. ^ Radiation constant from ScienceWorld
Boltzmann (disambiguation)
Ludwig Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics.
Boltzmann may also refer to:
24712 Boltzmann, a main-belt asteroid
Boltzmann constant
Boltzmann (crater), an old lunar crater
Boltzmann distribution
Boltzmann equation
Boltzmann's entropy formula
Boltzmann relation
Stefan–Boltzmann law
Stefan–Boltzmann constant
Cloud forcing
Cloud forcing (sometimes described as cloud radiative forcing or cloud radiative effect) is, in meteorology, the difference between the radiation budget components for average cloud conditions and cloud-free conditions. Much of the interest in cloud forcing relates to its role as a feedback process in the present period of global warming.
Effect of radiation on perceived temperature
The “radiation effect” results from radiation heat exchange between human bodies and surrounding surfaces, such as walls and ceilings. It may lead to phenomena such as houses feeling cooler in the winter and warmer in the summer at the same temperature. For example, in a room in which air temperature is maintained at 22° Celsius at all times, but in which the inner surfaces of the house is estimated to be an average temperature of 10° Celsius in the winter or 25° Celsius in the summer, heat transfer from the surfaces to the individual will occur, resulting in a difference in the perceived temperature.
We can observe and compare the rate of radiation heat transfer between a person and the surrounding surfaces if we first make a few simplifying assumptions:
The heat exchange in the environment is in a “steady state”, meaning that there is a constant flow of heat either into or out of the house.
The person is completely surrounded by the interior surfaces of the room.
Heat transfer by convection is not considered.
The walls, ceiling, and floor are all at the same temperature.For an average person, the outer surface area is 1.4 m², the surface temperature is 30° Celsius, and the emissivity (ε) is 0.95. Emissivity is the ability of a surface to emit radiant energy compared to that of a black body at the same temperature.
We will be using the following equation to find out how much heat is lost by a person standing in the same room in summertime as compared to the winter, at exactly the same thermostat reading temperature:
Q ̇=εσA_s (T_s^4-T_surr^4)
Where Q ̇ is the rate of heat loss (W), ε is the emissivity (or the ability of an objects surface to emit energy by radiation) of a person, σ is the Stefan-Boltzmann constant (5.670x〖10〗^(-8 )W/m2∙K4), As is the surface area of a person, Ts is the surface temperature of a person (K), and Tsurr is the surface temperature of the walls, ceiling, and floor (K). Please note that this equation is only valid for an object standing in a completely enclosed room, box, etc.In the winter, the amount of heat loss from a person, when the inner surfaces of the room were 10 degrees Celsius, was found to be 152 Watts.
(Q ̇=(0.95)(5.67x〖10〗^(-8) )(1.4)[(30+273)^4-(10+273)^4 ]=152) ̇
In the summer, the amount of heat loss from a person, when the inner surfaces of the room were 25 degrees Celsius, was found to be 40.9 Watts.
(Q ̇=(0.95)(5.67x〖10〗^(-8) )(1.4)[(30+273)^4-(25+273)^4 ]=40.9) ̇
Thermal radiation is the form of radiation emitted by bodies because of their temperature.
It differs from other forms of electromagnetic radiation such as x-rays, gamma rays, microwaves, radio waves, and television rays that are not related to temperature. Scientists have found that all bodies at a temperature above absolute zero emit thermal radiation. People are constantly radiating their body heat, but at different rates. From these values, the rate of heat loss from a person is almost four times as large in the winter than in the summer, which explains the “chill” we feel in the winter even if the thermostat setting is kept the same.
HD 85512
HD 85512 is a solitary K-type main-sequence star located approximately 36 light-years away in the constellation Vela. It is approximately one billion years older than the Sun. It is extremely chromospherically inactive, only slightly more active than Tau Ceti. The star is known to host one low-mass planet.
Heat current
A heat current is a kinetic exchange rate between molecules, relative to the material in which the kinesis occurs. It is defined as ${\displaystyle {\frac {dQ}{dt}}}$, where ${\displaystyle Q}$ is heat and ${\displaystyle t}$ is time.
For conduction, heat current is defined by Fourier's law as
${\displaystyle {\frac {\partial Q}{\partial t}}=-k\oint _{S}{{\overrightarrow {\nabla }}T\cdot \,{\overrightarrow {dS}}}}$
where
${\displaystyle {\big .}{\frac {\partial Q}{\partial t}}{\big .}}$ is the amount of heat transferred per unit time [W] and
${\displaystyle {\overrightarrow {dS}}}$ is an oriented surface area element [m2]
The above differential equation, when integrated for a homogeneous material of 1-D geometry between two endpoints at constant temperature, gives the heat flow rate as:
${\displaystyle {\big .}{\frac {\Delta Q}{\Delta t}}=-kA{\frac {\Delta T}{\Delta x}}}$
where
A is the cross-sectional surface area,
${\displaystyle \Delta T}$ is the temperature difference between the ends,
${\displaystyle \Delta x}$ is the distance between the ends.
For thermal radiation, heat current is defined as
${\displaystyle W=\sigma \cdot A\cdot T^{4}}$
where the constant of proportionality ${\displaystyle \sigma }$ is the Stefan–Boltzmann constant, ${\displaystyle A}$ is the radiating surface area, and ${\displaystyle T}$ is temperature.
Heat current can also be thought of as the total phonon distribution multiplied by the energy of one phonon, times the group velocity of the phonons. The phonon distribution of a particular phonon mode is given by the Bose-Einstein factor, which is dependent on temperature and phonon energy.
Josef Stefan
Josef Stefan (Slovene: Jožef Štefan; 24 March 1835 – 7 January 1893) was an ethnic Carinthian Slovene physicist, mathematician, and poet of the Austrian Empire.
Lacaille 9352
Lacaille 9352 (Lac 9352) is a star in the southern constellation of Piscis Austrinus. With an apparent visual magnitude of 7.34, this star is too faint to be viewed with the naked eye even under excellent seeing conditions. Parallax measurements place it at a distance of about 10.74 light-years (3.29 parsecs) from Earth. It is the eleventh closest star system to the Solar System and is the closest star in the constellation Piscis Austrinus. The ChView simulation shows that its closest neighbour is the EZ Aquarii
triple star system at about 4.1 ly from Lacaille 9352.
List of letters used in mathematics and science
the symbol of time is seconds and minutes.Latin and Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities.
Some common conventions:Intensive quantities in physics are usually denoted with minusculeswhile extensive are denoted with capital letters.
Most symbols are written in italics.
Vectors can be denoted in boldface.
Sets of numbers are typically bold or blackboard bold.
List of scientific constants named after people
This is a list of physical and mathematical constants named after people.Eponymous constants and their influence on scientific citations have been discussed in the literature.
• Reduced Planck constant or Dirac constant (${\displaystyle h}$-bar, ħ) – Max Planck, Paul Dirac
List of scientists whose names are used in physical constants
Some of the constants used in science are named after great scientists. By this convention, their names are immortalised. Below is the list of the scientists whose names are used in physical constants.
List of things named after Ludwig Boltzmann
This refers to a list of things named after physicist Ludwig Eduard Boltzmann (February 20, 1844 – September 5, 1906)
Luminosity
In astronomy, luminosity is the total amount of energy emitted per unit of time by a star, galaxy, or other astronomical object. As a term for energy emitted per unit time, luminosity is synonymous with power.In SI units luminosity is measured in joules per second or watts. Values for luminosity are often given in the terms of the luminosity of the Sun, L⊙. Luminosity can also be given in terms of the astronomical magnitude system: the absolute bolometric magnitude (Mbol) of an object is a logarithmic measure of its total energy emission rate, while absolute magnitude is a logarithmic measure of the luminosity within some specific wavelength range or filter band.
In contrast, the term brightness in astronomy is generally used to refer to an object's apparent brightness: that is, how bright an object appears to an observer. Apparent brightness depends on both the luminosity of the object and the distance between the object and observer, and also on any absorption of light along the path from object to observer. Apparent magnitude is a logarithmic measure of apparent brightness.
Opacity (optics)
Opacity is the measure of impenetrability to electromagnetic or other kinds of radiation, especially visible light. In radiative transfer, it describes the absorption and scattering of radiation in a medium, such as a plasma, dielectric, shielding material, glass, etc. An opaque object is neither transparent (allowing all light to pass through) nor translucent (allowing some light to pass through). When light strikes an interface between two substances, in general some may be reflected, some absorbed, some scattered, and the rest transmitted (also see refraction). Reflection can be diffuse, for example light reflecting off a white wall, or specular, for example light reflecting off a mirror. An opaque substance transmits no light, and therefore reflects, scatters, or absorbs all of it. Both mirrors and carbon black are opaque. Opacity depends on the frequency of the light being considered. For instance, some kinds of glass, while transparent in the visual range, are largely opaque to ultraviolet light. More extreme frequency-dependence is visible in the absorption lines of cold gases. Opacity can be quantified in many ways; for example, see the article mathematical descriptions of opacity.
Different processes can lead to opacity including absorption, reflection, and scattering.
Planetary equilibrium temperature
The planetary equilibrium temperature is a theoretical temperature that a planet would be at when considered simply as if it were a black body being heated only by its parent star. In this model, the presence or absence of an atmosphere (and therefore any greenhouse effect) is not considered, and one treats the theoretical black body temperature as if it came from an idealized surface of the planet.
Other authors use different names for this concept, such as equivalent blackbody temperature of a planet, or the effective radiation emission temperature of the planet. Similar concepts include the global mean temperature, global radiative equilibrium, and global-mean surface air temperature, which includes the effects of global warming.
The first and second radiation constants c1 and c2 – see Planck's Law
The radiation density constant a – see Stefan–Boltzmann constant
Standard asteroid physical characteristics
For the majority of numbered asteroids, almost nothing is known apart from a few physical parameters and orbital elements and some physical characteristics are often only estimated. The physical data is determined by making certain standard assumptions.
Stefan–Boltzmann law
The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time ${\displaystyle j^{\star }}$ (also known as the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature T:
${\displaystyle j^{\star }=\sigma T^{4}.}$
The constant of proportionality σ, called the Stefan–Boltzmann constant, is derived from other known physical constants. The value of the constant is
${\displaystyle \sigma ={\frac {2\pi ^{5}k^{4}}{15c^{2}h^{3}}}=5.670373\times 10^{-8}\,\mathrm {W\,m^{-2}K^{-4}} ,}$
where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum. The radiance (watts per square metre per steradian) is given by
${\displaystyle L={\frac {j^{\star }}{\pi }}={\frac {\sigma }{\pi }}T^{4}.}$
A body that does not absorb all incident radiation (sometimes known as a grey body) emits less total energy than a black body and is characterized by an emissivity, ${\displaystyle \varepsilon <1}$:
${\displaystyle j^{\star }=\varepsilon \sigma T^{4}.}$
The radiant emittance ${\displaystyle j^{\star }}$ has dimensions of energy flux (energy per time per area), and the SI units of measure are joules per second per square metre, or equivalently, watts per square metre. The SI unit for absolute temperature T is the kelvin. ${\displaystyle \varepsilon }$ is the emissivity of the grey body; if it is a perfect blackbody, ${\displaystyle \varepsilon =1}$. In the still more general (and realistic) case, the emissivity depends on the wavelength, ${\displaystyle \varepsilon =\varepsilon (\lambda )}$.
To find the total power radiated from an object, multiply by its surface area, ${\displaystyle A}$:
${\displaystyle P=Aj^{\star }=A\varepsilon \sigma T^{4}.}$
Wavelength- and subwavelength-scale particles, metamaterials, and other nanostructures are not subject to ray-optical limits and may be designed to exceed the Stefan–Boltzmann law. | 2019-01-21 22:31:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 29, "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": 0.7801626324653625, "perplexity": 944.9238987369371}, "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/1547583814455.32/warc/CC-MAIN-20190121213506-20190121235506-00537.warc.gz"} |
https://plainmath.net/73753/let-mtable-rowspacing | # Let [ <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd>
Let
$\left[\begin{array}{ccc}-2& 1& 0\\ 0& -2& 1\\ 0& 0& -2\end{array}\right]$
and
$|x\left(t\right)|=\left({x}_{1}^{2}\left(t\right)+{x}_{2}^{2}\left(t\right)+{x}_{3}^{2}\left(t\right){\right)}^{1/2}$
Then any solution of the first order system of the ordinary differential equation
$\left\{\begin{array}{r}{x}^{\prime }\left(t\right)=Ax\left(t\right)\\ x\left(0\right)={x}_{0}\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}$
satisfies
1. $\underset{t\to \mathrm{\infty }|x\left(t\right)|=0}{lim}$
2. $\underset{t\to \mathrm{\infty }|x\left(t\right)|=\mathrm{\infty }}{lim}$
3. $\underset{t\to \mathrm{\infty }|x\left(t\right)|=2}{lim}$
4. $\underset{t\to \mathrm{\infty }|x\left(t\right)|=12}{lim}$
You can still ask an expert for help
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Since −2 is the only eigenvalue of $A$, the general solution of ${x}^{\prime }\left(t\right)=Ax\left(t\right)$ has the form
$x\left(t\right)={e}^{-2t}\left({c}_{1}{v}_{1}\left(t\right)+{c}_{2}{v}_{2}\left(t\right)+{c}_{3}{v}_{3}\left(t\right)$
where ${c}_{1}$,${c}_{2}$ and ${c}_{3}$ are constants and ${v}_{j}\left(t\right)$ is a vector in ${\mathbb{R}}^{3}$ whose coordinates are polynomials with grade $\le j-1$.
Hence $|x\left(t\right)|\to 0$ for $t\to \mathrm{\infty }$. | 2022-08-09 10:36:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 45, "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": 0.9517585635185242, "perplexity": 634.4955439666429}, "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/1659882570921.9/warc/CC-MAIN-20220809094531-20220809124531-00431.warc.gz"} |
https://math.stackexchange.com/questions/1126860/question-about-tensor-products-decomposable-tensors | # Question about tensor products, decomposable tensors, …
I need some help with the following problem:
Let $V_1,\ldots,V_m$ be finite dimensional vector spaces over $\mathbb{K}$.
Let $\varphi \in L(V_1,\ldots,V_m;U)$ such that $Im(\varphi)=U$.
Show that there exists a subspace $K$ of $V_1 \otimes \cdots \otimes V_m$ such that every class of $\frac{V_1 \otimes \cdots \otimes V_m}{K}$ contains a decomposable tensor.
I have some ideas but none of them work, such as:
$$U = Im(\varphi) \simeq \frac{V_1 \times \cdots \times V_m}{Ker(\varphi)}$$
which is wrong because $\varphi$ is multilinear and not linear, so I have no idea how to solve this problem.
Any ideas? Hints? Thanks!
If you know where I can find this problem (maybe in some book) I would be very grateful.
• Your last equation doesn't make sense because $\varphi$ is not a homomorphism. – Matt Samuel Jan 30 '15 at 18:48
• $V_1 \times \cdots \times V_m$ and $U$ are vector spaces and $\varphi$ is linear (i.e. is a homomorphism of $\mathbb{K}$-modules (or $\mathbb{K}$-vector spaces), so I can apply the homomorphism theorem. Am I wrong? – Leafar Feb 1 '15 at 17:01
• The part that is wrong: $\varphi$ is not linear, it is multilinear. – Matt Samuel Feb 1 '15 at 17:02
• Ok, I will erase then – Leafar Feb 1 '15 at 17:04
• As written the problem has a gap: the "Show that..." part makes no use of $\varphi$, so $\varphi$ doesn't look relevant to the problem. You could give a stupid answer like $K = V_1 \otimes \cdots \otimes V_n$, for which the quotient space is zero. – KCd Feb 1 '15 at 17:53
By the universal property of the tensor product, $\varphi$ induces a linear map $\psi\colon V_1 \otimes \cdots \otimes V_m \to U$ which is also surjective (why?). And in fact we know more: for any $u \in U$, because $\varphi$ is surjective there are $v_i \in V_i$ such that $\varphi(v_1, \dots, v_m) = \psi(v_1 \otimes \cdots \otimes v_m) = u$. I'm hoping that from here you can fill in a gap. Thinking about the first isomorphism theorem is a good idea, though I don't think you need to apply it directly.
• Thanks! I will think about it. Actually I also tried to use the universal factorization property of $\otimes$. I am going to try again then... – Leafar Feb 1 '15 at 17:22
• Ok, we have $U \simeq \frac{V_1 \otimes \cdots \otimes V_m}{Ker(\psi)}$, so I let $K = Ker(\psi)$. We have $\frac{V_1 \otimes \cdots \otimes V_m}{Ker(\psi)} = \{v_1 \otimes \cdots \otimes v_m + Ker(\psi) : v_1 \otimes \cdots \otimes v_m \in V_1 \otimes \cdots \otimes V_m\}$ – Leafar Feb 2 '15 at 18:48
• We know that for any $u \in U$, $u$ is the image of a decomposable tensor, so each class of $\frac{V_1 \otimes \cdots \otimes V_m}{Ker(\psi)}$ is of the form $v_1 \otimes \cdots \otimes v_m + Ker(\psi)$. As $0 \in Ker(\psi)$, $v_1 \otimes \cdots \otimes v_m$ is in the class, so we can conclude that that each class of $\frac{V_1 \otimes \cdots \otimes V_m}{Ker(\psi)}$ contains a decomposable tensor. Right? – Leafar Feb 4 '15 at 2:24
• @Leafar Yes, that sounds right. – Hoot Feb 4 '15 at 18:26
By the universal factorization property of $\otimes$, there exists a unique linear map $\psi: V_1 \otimes \cdots \otimes V_m \longrightarrow U$ such that $\varphi = \psi \circ \otimes$.
As $\varphi$ is surjective, $\psi$ is surjective: given $u \in U$, $u = \varphi(v_1,\ldots,v_m)$ for some $(v_1,\ldots,v_m) \in V_1 \times \cdots \times V_m$, then $u = \psi \circ \otimes (v_1,\ldots,v_m) = \psi(v_1 \otimes \cdots \otimes v_m)$, then $u \in Im(\psi)$.
So given $u \in U$, there exists $(v_1,\ldots,v_m) \in V_1 \times \cdots \times V_m$ such that \begin{align*} u = \psi(v_1 \otimes \cdots \otimes v_m) = \varphi(v_1,\ldots,v_m). \end{align*}
Let $K=Ker(\psi)$.
Then, \begin{align*} \pi: \frac{V_1 \otimes \cdots \otimes V_m}{K} &\to U\\ z + Ker(\psi) &\mapsto \psi(z) \end{align*} is an isomorphism (easy to check).
Let us prove that each class of $\frac{V_1 \otimes \cdots \otimes V_m}{K}$ contains a decomposable tensor.
If $z + K = K$ it is trivial because $0 \in K$ and $0$ is decomposable.
Suppose that there exists a class $z_0 + K \neq K$ such that $z_0 + K$ does not contain a decomposable tensor.
Then, it does not exist $z \in z_0 + K$ decomposable such that $\pi(z + K) = \psi(z) = u$, but this is false, since each $u \in U$ is of the form $\psi(v_1 \otimes \cdots \otimes v_m)$ for some $(v_1,\ldots,v_m) \in V_1 \times \cdots \times V_m$ as we saw above.
So we conclude that each class of $\frac{V_1 \otimes \cdots \otimes V_m}{K}$ contains a decomposable tensor. | 2019-06-16 17:35:52 | {"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": 0.9526365399360657, "perplexity": 170.50750041695952}, "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-2019-26/segments/1560627998288.34/warc/CC-MAIN-20190616162745-20190616184745-00541.warc.gz"} |
https://www.physicsforums.com/threads/how-to-write-a-proposoal-for-funding.184044/ | # How to write a proposoal for funding
1. Sep 11, 2007
### mattmns
So I did a Math REU this past summer and have been invited to present at the undergraduate poster session at the joint AMS-MAA meeting in January, but being an undergraduate funding is a little difficult, and something of which I know nothing about, so I have a few questions.
The school I did the REU at will reimburse me for at least $200 for anything related to this meeting (flight, hotel, food, etc.). I asked my department about funding, and they said it might be possible, but that they want me to write a 1 page proposal for funding. The lady I talked to gave me some guidelines about what to put in the proposal: specifically, my name and information, the research topic, where the presentation is at, who sponsored the research, dates, and the amount requested. Now I have a few questions. 1. Formatting: Should I present my name in some special way, should I double space, or anything else? 2. Should I add that the school I did the REU is offering$200 of support? Also our math club will likely give $50. The expected cost of this trip is approximately$600.
3. Should I make it sound as though the funding will be of extreme help? For example, that without funding I would not be able to go?
4. Should I ask for more than I want? That is, I would be thrilled if they payed for just my plane ticket, but I still have a little more cost. Should I ask that they pay for the ticket and some extra for hotel/food cost? Basically, should I ask for a lot and hope they give me some, or ask for just the minimum and hope they give me it all?
5. Anything else that I should definitely add, or definitely not put in this request?
Thanks!
Last edited: Sep 12, 2007 | 2016-12-08 10:32:13 | {"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": 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": 0.5171319246292114, "perplexity": 1019.6887576776772}, "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-2016-50/segments/1480698542520.47/warc/CC-MAIN-20161202170902-00030-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.nature.com/articles/s41467-021-21082-x | 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.
# Ligand-directed two-step labeling to quantify neuronal glutamate receptor trafficking
## Abstract
The regulation of glutamate receptor localization is critical for development and synaptic plasticity in the central nervous system. Conventional biochemical and molecular biological approaches have been widely used to analyze glutamate receptor trafficking, especially for α-amino-3-hydroxy-5-methyl-4-isoxazole-propionate-type glutamate receptors (AMPARs). However, conflicting findings have been reported because of a lack of useful tools for analyzing endogenous AMPARs. Here, we develop a method for the rapid and selective labeling of AMPARs with chemical probes, by combining affinity-based protein labeling and bioorthogonal click chemistry under physiological temperature in culture medium. This method allows us to quantify AMPAR distribution and trafficking, which reveals some unique features of AMPARs, such as a long lifetime and a rapid recycling in neurons. This method is also successfully expanded to selectively label N-methyl-D-aspartate-type glutamate receptors. Thus, bioorthogonal two-step labeling may be a versatile tool for investigating the physiological and pathophysiological roles of glutamate receptors in neurons.
## Introduction
In the central nervous system, ionotropic glutamate receptors (iGluRs) mediate fast excitatory neurotransmission. iGluRs are categorized into distinct classes based on their pharmacology and structural homology, including the α-amino-3-hydroxy-5-methyl-4-isoxazole-propionate (AMPA) receptor (GluA1–4), kainate receptor (GluK1–5), N-methyl-D-aspartate (NMDA) receptor (GluN1, GluN2A–D, GluN3A–B), and δ receptors (GluD1–2)1. iGluRs assemble as tetramers, and functional receptors are formed exclusively by the assembly of subunits within the same functional receptor class.
AMPA receptors (AMPARs), which are mainly permeable to monovalent cations (Na+ and K+), mediate the majority of excitatory synaptic transmission. AMPARs can form homotetramers or heterotetramers, and subunit compositions are dependent on brain regions. In hippocampal CA1 neurons, the majority of AMPARs are made up of GluA1/A2 and GluA2/A3 subunit combinations, with a small contribution of GluA1 homomers2,3. Recent studies have revealed that AMPARs are constitutively cycled in and out of the postsynaptic membrane through endocytosis and exocytosis. The precise regulation of this process is critical for synaptic plasticity, which is the basis of learning, memory, and development in neural circuits2,3. Although AMPARs and kainate receptors are activated by glutamate binding, NMDA receptors (NMDARs), which have high permeability to Ca2+, require depolarization as well as agonist binding for their activation. Functional NMDARs require the assembly of two GluN1 subunits together with either two GluN2 subunits, or a combination of GluN2 and GluN3 subunits1. An NMDAR-dependent Ca2+ influx triggers intracellular signal transduction cascades, and the precise targeting of NMDARs to synapses is essential for controlling neuronal connectivity or neuroplasticity4. Thus, to understand the molecular mechanisms of learning and memory, it is critical to analyze the membrane localization and trafficking of iGluRs.
Biochemical approaches, such as surface biotinylation assays or related methods, have been widely used to analyze membrane protein localizations, and these methods have been successfully applied to AMPARs5,6,7. Although they are powerful tools for the analysis of AMPAR trafficking, cell-surface proteins are randomly labeled with biotin using these methods. As a result, purification of biotin-labeled AMPAR is required, which hampers quantitative analyses of trafficking. In contrast, to selectively visualize glutamate receptors, fluorescent proteins are fused to the receptors using genetically encoded approaches. For example, a pH-sensitive variant of GFP (super-ecliptic pHluorin [SEP]) can be fused to the extracellular region of receptors to visualize cell-surface receptors in live neurons8,9. Instead of fluorescent proteins, protein tags such as SNAP- or Halo-tags are fused to the receptors for the covalent labeling of small chemical probes at the time that the probes are added10,11,12. The downsizing of these protein tags has been successfully demonstrated by using a short peptide tag (1–3 kDa) and its probe pair13,14,15. More recently, genetic code expansion in combination with bioorthogonal click chemistry has been reported for the fluorescent labeling of iGluRs in HEK293T cells, in which chemical probes are covalently attached to the side chain of an unnatural amino acid residue16,17. These genetically encoded approaches have been widely used in trafficking studies of iGluRs, especially for AMPARs. However, in most cases, these methods largely rely on the overexpression of target iGluR subunits. Given the formation of heterotetramers consisting of different subunits in iGluRs, the overexpression of a single iGluR subunit may interfere with the localization and/or trafficking of native iGluRs in neurons. Ideally, endogenously expressed iGluRs should be tagged with small chemical probes18,19.
In situ chemical protein labeling is ideal for analyzing native proteins in live cells. Affinity-based protein labeling is a powerful technique for the selective modification of target proteins20,21,22,23,24,25,26. As a traceless affinity-based labeling method for cell-surface proteins, our group has reported ligand-directed acyl imidazole (LDAI) chemistry24,25. With this technique, small chemical probes including fluorophores are covalently attached to nucleophilic amino acid residues located near the ligand-binding site. Recently, we have developed an AMPAR-selective LDAI reagent, termed “chemical AMPAR modification 2” (CAM2) reagents, which allows us to label chemical probes to AMPARs endogenously expressed in cultured neurons or acutely prepared brain slices26. Although this technique is powerful for the selective modification of chemical probes to AMPARs, there are some restrictions for visualizing or analyzing cell-surface AMPARs. First, live cells need to be kept at low temperatures (e.g., 17 °C) during CAM2 labeling (1–4 h) to suppress the internalization of labeled AMPARs27. Second, the neuronal culture medium needs to be exchanged for serum-free medium or buffered saline during labeling to decrease non-specific labeling of serum proteins such as albumin. The relatively long-term exposure (1–4 h) to these non-physiological conditions may interfere with neuronal activity or survival28,29,30. Ideally, neurons should be kept under physiological conditions during chemical labeling.
Here, we show a method for the rapid and selective labeling of AMPARs under physiological temperature in culture medium by combining LDAI-based protein labeling and the inverse electron demand Diels–Alder (IEDDA) reaction, a form of fast click chemistry31,32,33. This two-step labeling allows the quantitative analyses of distribution and/or trafficking of endogenous AMPARs from short to long periods in cultured neurons. In addition, we successfully apply this technique to chemically label and study the trafficking of endogenous NMDARs in neurons.
## Results
### Rapid labeling of surface AMPARs by bioorthogonal two-step labeling
We propose a bioorthogonal two-step labeling technique, which combines LDAI-based protein labeling with the bioorthogonal IEDDA reaction for the rapid and selective modification of chemical probes to cell-surface iGluRs. For the first step, a strained alkene is covalently attached to iGluRs using LDAI chemistry, where the acyl substitution reaction to nucleophilic amino acid residues is facilitated by selective ligand–protein recognition (first step in Fig. 1a). Next, the labeled alkene group is rapidly modified with tetrazine-conjugated probes (Tz-probes) on the cell surface, as a result of the high selectivity and high reaction rate of the IEDDA reaction (second step in Fig. 1a).
For the selective labeling of a strained alkene to AMPARs in the first step, we designed a CAM2 reagent bearing trans-cyclooctene (TCO), which we termed CAM2(TCO) (Fig. 1b). TCO was selected as the strained alkene because of its extremely fast cycloaddition kinetics in the IEDDA reaction. Compared with the original CAM2 reagents (e.g., CAM2(Ax488)) that bear aromatic fluorophores (see Supplementary Fig. 1a, b), an ethylene glycol linker is added between the reactive acyl imidazole unit and the TCO group in CAM2(TCO) to increase its hydrophilicity. Hydrophobic or aromatic groups have high affinity to albumin abundantly contained in serum34; therefore, this improvement decreases the undesired labeling of albumin, which allows the chemical labeling of AMPARs to be conducted in cell culture medium containing serum or substitutes. In addition, the first labeling is conducted at a physiological temperature (37 °C). Although some of the labeled AMPARs are likely to be internalized in this condition, this is not problematic with the two-step labeling technique. This is because the chemical probes are selectively tethered to cell-surface AMPARs in the second step reaction (Fig. 1d).
Regarding the second step (the IEDDA reaction), the reaction rate is highly dependent on the chemical structure of the tetrazine group. We selected monoaryl tetrazine, which has both a fast reaction rate and high bioorthogonality, and prepared cell-impermeable Tz-probes bearing hydrophilic and anionic fluorophores or biotin for cell-surface labeling (Fig. 1c and Supplementary Fig. 2). Shortening the reaction time of the probe labeling not only contributes to cell-surface specific labeling, but also decreases the adsorption of the chemical probes to cells, culture dishes, or coverslips. Moreover, some tetrazine-fluorophore conjugates have a “turn-on” response upon the IEDDA cycloaddition35,36,37, which contributes to a high signal-to-noise (S/N) ratio in fluorescence imaging.
### Chemical labeling of surface AMPARs ectopically expressed in HEK293T cells
The designed two-step labeling method was initially examined in HEK293T cells transiently expressing GluA2, a main subunit of AMPARs. For the first step reaction, CAM2(TCO) was added to the culture medium, which included 10% fetal bovine serum (FBS), and the culture dish was incubated at 37 °C for 4 h. The second step reaction was performed for 5 min by adding membrane-impermeable Tz(Fl) for fluorescein labeling on the cell surface. As shown in Fig. 2a, western blotting of the cell lysate using anti-fluorescein (anti-Fl) antibodies showed a strong band around 110 kDa (lane 1). This band was not observed in the cells co-treated with a competitive ligand (NBQX) or in any other control conditions (lanes 2–5). With regard to the molecular weight of the labeled band, the anti-Fl signal corresponded to the highest signal among multiple bands that were detected using anti-GluA2/3 antibodies (Fig. 2b). The multiple GluA2 bands converged into a single lower band after treatment with peptide-N-glycosidase F (PNGase F), which is consistent with previous reports showing that GluA2 is highly glycosylated with N-linked sugars38. Importantly, in the PNGase F-treated samples, the shifted anti-GluA2 band merged with the anti-Fl signal (Fig. 2b). These findings indicate that highly glycosylated GluA2 is selectively labeled using our methods. Furthermore, in the case of direct fluorescein labeling using the original CAM2(Fl) under the same conditions (see Supplementary Fig. 1b for its structure), there was a strong band around 70 kDa as well as the 110 kDa band (lanes 6–7 in Fig. 2a and Supplementary Fig. 3). The 70 kDa band, whose intensity did not change even in the presence of NBQX, corresponds to albumin contained in serum (for details, see Fig. 2a legend). These results therefore indicate the high selectivity of the two-step labeling technique using CAM2(TCO) compared with the original CAM2(Fl) under cell culture conditions.
For visualizing fluorescently labeled AMPARs on the cell surface, confocal microscopic live imaging was performed after the two-step labeling process under cell culture conditions. Here, Tz(Ax488) was used in the second step of labeling. Alexa 488 has bright fluorescence that is unaffected under endosomal acidic conditions; in contrast, fluorescein has weakened fluorescence under acidic conditions. Thus, Alexa 488 is more suitable to quantify the cellular distribution or trafficking of labeled AMPARs using fluorescent imaging. As shown in Fig. 2c, prominent fluorescence was observed exclusively from the cell surface in cells co-transfected with mCherry-F, a membrane-targeted transfection marker. In contrast, fluorescent signals were not observed in control conditions, such as in CAM2(TCO)-untreated or NBQX-co-treated cells (Supplementary Fig. 4). In the case of direct Alexa 488 labeling using CAM2(Ax488) in the same cell culture conditions, labeled signals were observed not only from the cell surface but also from the intracellular space (Supplementary Fig. 1c). This suggests that the two-step labeling technique is superior for the fluorescent visualization of cell-surface AMPARs under cell culture conditions. We also determined the reaction kinetics of rapid fluorophore labeling of cell-surface AMPARs with the help of the turn-on fluorescent property of Tz(Ax488) upon the IEDDA reaction (Supplementary Fig. 5). Immediately after adding Tz(Ax488), prominent fluorescent signals were observed from the cells co-transfected with the transfection marker mCherry-F (Fig. 2d), and the fluorescent signals were saturated within 3 min. Thus, cell-surface AMPARs can be labeled by the fluorophore with fast kinetics.
By taking advantage of the high bioorthogonality of the IEDDA reaction, cell-surface AMPARs were successfully labeled with various kinds of chemical probes, ranging from small molecules to middle-sized molecules such as SeTau-647, a squaraine rotaxane dye that has high photostability and a long fluorescence lifetime39,40 (Supplementary Figs. 2 and 6). This probe flexibility is another feature of the two-step labeling technique that is superior compared with the direct labeling of probes using original CAM2 reagents, where each probe-tethered CAM2 needs to be synthesized (Supplementary Fig. 1b).
With regard to labeling efficacy, quantification of the remaining unlabeled GluA2 fraction showed that 35 ± 3% of surface AMPARs were visualized in the two-step labeling (Supplementary Fig. 7). Moreover, AMPAR function was not visibly affected by the two-step labeling (Supplementary Fig. 8), which is consistent with our previous analyses that showed minimal disturbance of AMPAR ion channel properties by CAM2 labeling26.
### Analyses of AMPAR trafficking in HEK293T cells
Once we had a potential labeling method for cell-surface AMPARs under cell culture conditions, we analyzed receptor trafficking using both live imaging and biochemical approaches. First, we analyzed cell-surface AMPAR trafficking in HEK293T cells using confocal live imaging. After incubating the cells with CAM2(TCO) under physiological temperature in culture medium, Tz(Ax488) was added to the culture medium to selectively visualize cell-surface AMPARs and cells were incubated for each period (0–8 h) (Fig. 3a). As shown in Fig. 3b, the labeled fluorescence on the cell surface decreased after incubation at 37 °C. Fluorescent granules were instead observed in the intracellular area, and most of the fluorescent signals were from intracellular granules after 8 h of incubation. The half-time of cell-surface AMPARs (t1/2surface), which includes both the remaining and recycled fractions, was calculated to be 5.7 ± 0.7 h from the fluorescent intensity on the cell surface (Fig. 3b). In addition, the intracellular punctate signals merged with a fluorescent lysosome marker (LysoTracker) after 8 h of incubation, suggesting that internalized AMPARs were transported to lysosomes (Fig. 3c). Similar internalization behavior was observed when AMPARs were labeled with different fluorophores using Tz(Ax647) or Tz(ST647) (Supplementary Fig. 9).
Quantitative analyses of the fates of cell-surface AMPARs were examined using biochemical approaches. To quantify the t1/2surface of AMPARs, HEK293T cells were incubated for each period (0–36 h) after treatment with CAM2(TCO) (path-1 in Fig. 3d). Next, Tz(Fl) was added for the selective modification of fluorescein to cell-surface AMPARs. Using western blotting of the cell lysate, the t1/2surface of AMPARs was determined to be 5.3 ± 0.3 h (Fig. 3e), which was similar to the value that was determined using confocal imaging (Fig. 3b). The half-time of the degradation (t1/2life) of cell-surface receptors was evaluated by modifying the protocols, where Tz(Fl) was added after CAM2(TCO) labeling (path-2 in Fig. 3d). The cells were then incubated for each period (0–36 h), and the cell lysates were subjected to western blotting. As shown in Fig. 3f, t1/2life was determined to be 8.1 ± 0.7 h, which was slightly longer than the t1/2surface of cell-surface AMPARs (p < 0.05). Considering the colocalization of internalized AMPARs and lysosomes (Fig. 3c), the internalized AMPARs are likely decomposed via lysosomal degradation in HEK293T cells.
We next determined the intracellular versus surface percentages of TCO-labeled AMPARs (TCO-AMPARs) in each period, which can provide valuable information regarding the fate of cell-surface AMPARs. To quantify these percentages, cell-surface AMPARs in HEK293T cells treated with CAM2(TCO) were selectively labeled with fluorescein by adding cell-impermeable Tz(Fl) to the medium under live cell conditions (path-1 in Fig. 3g). To label intracellular TCO-AMPARs, surface TCO-AMPARs were first masked with cell-impermeable Tz(Ax647) (path-2 in Fig. 3g). After lysis of the cells, Tz(Fl) was added to the cell lysate to label intracellular TCO-AMPARs. The whole-cell-labeling fraction, where both cell-surface and intracellular TCO-AMPARs were labeled with fluorescein, was prepared by adding Tz(Fl) after cell lysis (path-3 in Fig. 3g). Prior to these analyses, we first investigated whether the second step reaction using Tz(Fl) proceeds rapidly and/or selectively in cell lysate. Western blotting analyses revealed that the covalent modification of fluorescein was selective to AMPARs in cell lysate (Supplementary Fig. 10), and the reaction was saturated after 15 min when either 0.1 or 0.3 μM Tz(Fl) was added. The intracellular and cell-surface percentages of TCO-AMPARs after 4 h of incubation with CAM2(TCO) were analyzed using this protocol and determined to be 4.1 ± 0.9% and 94.4 ± 3.3%, respectively (Fig. 3h), indicating that intracellular TCO-AMPAR levels were quite low.
### Rapid labeling of endogenous AMPARs in neurons
We next examined the applicability of the bioorthogonal two-step labeling technique for the rapid modification of cell-surface AMPARs that are endogenously expressed in neurons. Primary cultured neurons from the cerebral cortex were incubated with CAM2(TCO) for 10 h under neuronal culture conditions, and Tz(Fl) was then added for 5 min for cell-surface labeling. Western blotting analyses of the cell lysate showed a single strong band corresponding to the molecular weight of AMPARs (see lane 1 in Fig. 4a). This band was not detected in the co-presence of the competitive ligand NBQX, or in other control conditions (see lanes 2–4 in Fig. 4a). As observed with the AMPARs expressed in HEK293T cells (Fig. 2a), smeared bands were detected using anti-GluA2 antibodies; the anti-Fl band corresponded to the highest band in the smeared anti-GluA2 signals. After the removal of N-linked sugars by PNGase treatment, the smeared anti-GluA2 bands converged into a single lower band, which merged with the anti-Fl signal (Supplementary Fig. 11). These results suggest that the highly glycosylated fraction of endogenous AMPARs were selectively labeled with fluorescein by the rapid labeling.
Of the AMPAR subunits (GluA1–4), GluA1, GluA2, and GluA3 are highly expressed in cultured cortical neurons41. We next examined the efficacy of our methods for visualizing tetrameric AMPARs by quantifying the remaining unlabeled GluA2 fraction. As shown in Supplementary Fig. 12a, 44 ± 4% of GluA2-containing AMPARs were recognized by the two-step labeling method. Similarly, we calculated that 37 ± 7% of GluA1- and 43 ± 5% of GluA3-containing AMPARs were recognized. However, considering the heterotetrameric formation of AMPAR subunits, we also needed to examine whether each subunit was covalently labeled with the probe or not. In this context, the immunoprecipitation assay in the denatured condition revealed that GluA2 and GluA3, but not GluA1, were covalently labeled with CAM2(TCO) (Supplementary Fig. 12b). This selectivity is consistent with our previous results26 in HEK293T cells. With regard to the efficacy of CAM2(TCO) labeling, the time-course of the labeling clearly indicated that chemical labeling occurred more efficiently at 37 °C than in the previous condition at 17 °C (Supplementary Fig. 13a). In addition, the concentration dependency of CAM2(TCO) revealed the EC50 value (0.90 ± 0.10 µM) of two-step labeling at 37 °C in neurons (Supplementary Fig. 13b).
Using the two-step labeling technique in primary hippocampal neurons, fluorescently labeled AMPARs were visualized by confocal microscopy. At 5 min after the addition of Tz(Ax488), confocal live imaging showed punctate fluorescent signals from the CAM2(TCO)-treated neurons, and these signals were not observed in neurons co-treated with NBQX (Fig. 4b). To characterize the fluorescent signals in detail, Tz(Ax488)-treated neurons were fixed with paraformaldehyde (PFA) and immunostained with anti-MAP2 or anti-PSD95 antibodies for dendritic or postsynaptic staining, respectively. As shown in Fig. 4c and Supplementary Fig. 14a, labeled Alexa 488 signals were observed alongside the anti-MAP2 signals, and merged well with the anti-PSD95 signals. Considering the short incubation time with Tz(Ax488), the Alexa 488 signal likely corresponds to cell-surface AMPARs that are endogenously expressed in neurons. In addition, the endogenous AMPARs were successfully visualized using probes such as Alexa 647 or SeTau-647, using Tz(Ax647) or Tz(ST647), respectively (Supplementary Fig. 14b, c).
Next, we quantified the surface distribution of AMPARs in neurons using fluorescence lifetime imaging microscopy (FLIM). In this method, the fluorescence decay curve in each pixel is analyzed by fitting it to a multi-exponential function, and the target fluorescence lifetime (τ) component can be extracted to quantify the fluorescence of interest without any background. Here, we used Tz(ST647) to visualize surface AMPARs because SeTau-647 has a unique fluorescence lifetime and high photostability39,40. The typical FLIM image for a lifetime fraction (τ = 2.4 ± 0.1 ns) corresponding to SeTau-647 is shown in Fig. 4d, which revealed that surface AMPARs in spines are 3.3 times more concentrated than those in dendrites.
### Trafficking analyses of endogenous AMPARs in neurons
AMPARs are dynamically regulated at synapses, which underlie activity-dependent neuronal plasticity. Molecular biology or biochemical methods, such as the genetic incorporation of fluorescent proteins, surface biotinylation assays, and metabolic incorporation of radioisotopes, have revealed the diffusion dynamics42, recycling process5, and half-life43,44 of AMPARs, respectively. Although powerful, these methods are highly specialized for analyzing each process (see the Discussion for detail). However, we now have a rapid method to selectively label cell-surface AMPARs in neurons under physiological temperature in culture medium. We therefore applied the two-step labeling method to analyze AMPAR trafficking over a long period.
Prior to analyzing the AMPAR trafficking, we examined the influence of the two-step labeling process on the viability of primary cultured neurons by comparing it with our original CAM2 labeling. In immature neurons (day in vitro [DIV] 4), neither the original CAM2 labeling nor the two-step labeling affected neuronal viability after 24 h of labeling (Supplementary Fig. 15). However, in mature neurons (DIV 12), where iGluRs are highly expressed, viability was decreased after 3 h of labeling with the original CAM2 labeling, and neurons were severely damaged after 24 h of the original labeling. The main reason for the lowered viability may be ascribed to the experimental procedure rather than the CAM2 reagents, because similar neuronal damage was observed when the growth medium was exchanged for serum-free medium. In contrast, in the case of the two-step labeling method, where CAM2(TCO) was directly added into the growth medium at 37 °C, negligible damage was observed even after 24 h of the two-step labeling, indicating that this method is suitable for analyzing AMPARs over long periods in neurons. Besides, we examined the effects of CAM2(TCO) labeling on neuronal function. Neither the surface amount nor the synaptic fraction of AMPARs and NMDARs were affected by CAM2(TCO) labeling for 10 h (Supplementary Fig. 17a, b). Association between GluA2 and the accessary protein, TARPγ8 (ref. 45) was also unaffected by CAM2(TCO) labeling (Supplementary Fig. 17c). Although the homeostatic phosphorylation of ERK and CREB was not significantly but slightly decreased, phosphorylation levels of AMPAR (GluA1) and NMDAR (GluN1) were not influenced by the CAM2(TCO) labeling (Supplementary Fig. 17d).
We then analyzed the trafficking of endogenous AMPARs in neurons over a long period using the two-step labeling method. As described in the previous section, confocal microscopic imaging was able to clearly visualize cell-surface AMPARs in neurons in the two-step labeling method (Fig. 4c). However, synapses are very narrow (200–500 nm diameter), and it is thus difficult to analyze AMPAR trafficking in detail using optical microscopy. We therefore applied biochemical methods for analyzing the t1/2surface of AMPARs, including remaining and recycled components (Fig. 5a). As shown in Fig. 5b, the t1/2surface was calculated to be 33.2 ± 5.1 h, which was markedly longer than the t1/2surface in HEK293T cells (5.3 ± 0.3 h).
To explore the difference in the t1/2surface values, we focused on the trafficking of internalized AMPARs. We analyzed the intracellular and surface percentages of TCO-AMPARs after 10 h of incubation with CAM2(TCO). The intracellular percentage (10.3 ± 1.7%) was markedly smaller than the surface percentage (90.4 ± 0.6%), and these values were largely similar to those obtained after 10 h of labeling in HEK293T cells (Fig. 5c and Supplementary Fig. 18). We next evaluated the recycling process of internalized AMPARs, as follows. The cultured neurons were labeled with CAM2(TCO) for 10 h in the culture medium at 37 °C, and Tz(Fl) was then added to the culture medium for 5 min as a first pulse to mask the surface TCO-AMPARs (Fig. 5d). The neurons were further incubated for 15 min in the culture medium at 37 °C, and then Tz(Ax647) was added to the culture medium as a second pulse to label the recycled TCO-AMPARs. As shown in Fig. 5e, western blotting using anti-Alexa 647 antibodies clearly detected recycled AMPARs. The percentage was determined to be 12.5 ± 2.8% compared with the surface AMPARs, suggesting that most of internalized AMPARs were recycled during this short period. In contrast, the recycled fraction was almost undetectable in HEK293T cells (Supplementary Fig. 20). These data indicate that AMPARs are constantly recycled via endocytosis and exocytosis in neurons, which may be the molecular basis for the long lifetimes of AMPARs (Fig. 5f, g).
### Two-step labeling and trafficking analyses of NMDARs
The NMDARs are another essential family of iGluRs, and form heterotetramers composed of GluN1, GluN2A–D, and GluN3 in neurons. Pharmacological features of NMDARs are different from those of AMPARs. GluN2A–D recognize glutamate, whereas GluN1 and GluN3 recognize glycine or D-serine, as endogenous ligands4. Here, we designed a ligand-directed two-step labeling reagent targeted for GluN2A using the selective antagonist perzinfotel (Pzf) as the ligand moiety (Fig. 6a)46. This reagent was termed “chemical NMDAR modification (TCO)” (CNM(TCO)).
To confirm the selective labeling of GluN2A, the two-step labeling method was first examined in HEK293T cells transiently co-expressed with GluN1 and GluN2A. Western blotting analyses of cell lysate showed a prominent band around 180 kDa, corresponding to the molecular weight of GluN2A (see lane 1 in Fig. 6b). The 180 kDa band was not detected under the control conditions (see lanes 2–5 in Fig. 6b and Supplementary Fig. 21), suggesting that this band corresponds to labeled GluN2A. The t1/2surface of cell-surface NMDARs was quantitatively analyzed using western blotting by focusing on the labeled GluN2A band. As shown in Fig. 6c, the t1/2surface was calculated to be 4.3 ± 0.4 h, which was slightly shorter than that of AMPARs in HEK293T cells.
Next, we examined the chemical labeling of endogenous NMDARs in cultured neurons using CNM(TCO). As shown in Fig. 6d, western blotting showed a prominent band around 180 kDa, which was negligible in the co-presence of Pzf, indicating the successful labeling of endogenous NMDARs. Similar to AMPAR labeling, the time-course of NMDAR labeling clearly indicated that chemical labeling occurred more efficiently at 37 °C than at 17 °C (Supplementary Fig. 22a). However, the concentration dependency of CNM(TCO) for NMDAR labeling was different from that of CAM2(TCO) for AMPARs. As shown in Supplementary Fig. 22b, the labeled bands were not saturated in the 0–10 µM range, indicating that the affinity of CNM(TCO) was lower than that of CAM2(TCO). Importantly, CNM(TCO) labeled NMDAR with minimal disturbance to neuronal functions, including the constitutive phosphorylation of ERK and CREB (Supplementary Fig. 23). We next analyzed NMDAR trafficking using two-step labeling via biochemical methods in neurons. As shown in Fig. 6e, the t1/2surface of endogenous NMDARs in cultured neurons was 22.6 ± 7.2 h, which was substantially longer than in HEK293T cells. However, the value was shorter than that of endogenous AMPARs in neurons (33.2 ± 5.1 h). Thus, two-step labeling was successfully applied to NMDARs, another important family of iGluRs, by changing the selective ligands. In addition, we revealed that the lifetime of surface NMDARs is shorter than that of AMPARs in neurons.
## Discussion
We described a bioorthogonal two-step labeling method for the selective modification of chemical probes to cell-surface AMPARs. This method can be used under neuronal culture conditions. In our previous method, fluorescein was directly labeled to cell-surface AMPARs using CAM2(Fl) in serum-free medium at 17 °C (ref. 26). Although the dynamics of cell-surface AMPARs have been successfully tracked over a short period after labeling, the present study revealed that this labeling condition affects the viability of mature neurons after 24 h of labeling. In contrast, the present bioorthogonal two-step labeling method negligibly affected the cell viability of neurons. Our method therefore allows the quantitative analysis of AMPAR trafficking for over 120 h after labeling. The present investigation also revealed that the homeostatic phosphorylation of ERK and CREB was slightly decreased by CAM2(TCO) but not CNM(TCO) treatment. Considering the high affinity of CAM2(TCO) for AMPAR, this influence may be reduced when neurons are treated with low concentration of CAM2(TCO).
Many studies have focused on analyzing AMPAR trafficking in neurons using biochemical methods. To study trafficking over short periods (less than 30 min), surface biotinylation assays, where cleavable (disulfide-linked) biotin is randomly labeled on the cell surface, have conventionally been used. To date, both endocytosis and exocytosis of AMPARs have been investigated in cultured neurons using this method. However, this method is not suitable for analyzing AMPAR trafficking over a long period, because the biotinylation reaction must be conducted in serum-free medium at 4 °C; this affects neuronal viability, as we demonstrated in the present study. The long-term process of AMPAR trafficking, including decomposition, has been previously analyzed using the metabotropic incorporation of a radioisotope [35S]-labeled amino acid, or SILAC (stable isotope labeling with amino acids in cell culture), for mass spectrometry analyses43,44,47. Although isotope labeling can be conducted under cell culture conditions, these methods are not suitable for analyzing AMPAR trafficking over short periods. In contrast, our bioorthogonal two-step labeling selectively and rapidly modifies target iGluRs under cell culture conditions, and this allows us to analyze receptor trafficking over both short and long periods. Moreover, both biotinylation assays and metabotropic isotope labeling methods require the solubilization of iGluRs by mild detergents for pull-down or immunoprecipitation assays, because cellular proteins are randomly labeled in these methods. This step is problematic for the quantitative analysis of iGluRs, because iGluRs such as NMDARs are mainly localized in the postsynaptic density (PSD), where it is difficult for proteins to be solubilized by mild detergents. This peculiar feature hampers the quantitative analysis of iGluR trafficking by conventional biochemical methods. In contrast, purification steps are not required after labeling in our two-step labeling method. That is, in our method, all labeled proteins in the neurons can be analyzed quantitatively after denaturing in Laemmli sample buffer. This allows for the quantitative analysis of iGluRs contained in the PSD fraction.
The selective visualization of cell-surface iGluRs is essential for analyzing their trafficking and distribution. One potential method involves the use of antibodies that selectively recognize the extracellular regions of iGluRs. However, these antibodies are very limited, and in most cases their selectivity is insufficient. Positron emission tomography imaging would be another candidate for visualizing native AMPARs48. However, this method would not be useful for trafficking studies due to its low resolution (1–2 mm). Here, we demonstrated that various kinds of chemical probes could be used to selectively and rapidly label endogenous cell-surface iGluRs in neurons using bioorthogonal two-step labeling. The selective labeling of SeTau-647, a middle-sized molecule with a long fluorescence lifetime and high photostability, allowed the quantitative analyses of cell-surface AMPAR distribution in neurons using FLIM. These analyses revealed a three-fold concentration of cell-surface AMPARs in spines compared with dendrites. As a future direction, cell-surface SeTau-647 labeling would be utilized for single molecule tracking40 or super resolution imaging of synaptic AMPARs. Moreover, the second step IEDDA reaction can be applied to label polymers or nanoparticles, to allow the distribution of iGluRs to be analyzed in more detail using electron microscopy in the future.
Previous studies using radiolabeling methods indicated that the half-lives of synaptic proteins are 1–2 days43,44. Consistent with previous results, our present investigation demonstrated that the half-lives of AMPARs and NMDARs are 33.2 and 22.6 h, respectively, in cultured neurons. However, these values were significantly longer than those obtained in HEK293T cells. A plausible explanation for the differences between neurons and HEK293T cells may involve the formation of the macromolecular protein complexes of iGluRs. In the case of AMPARs, many binding partners such as transmembrane AMPAR regulatory proteins, synapse-associated protein 97 kDa, and glutamate receptor-interacting protein, which are selectively expressed in neurons, control the recycling of AMPARs and/or their stabilization at synapses45,49,50. Differences in phosphorylation levels may be another possible explanation. Activity-dependent phosphorylation by CaMKII or PKA contributes to the recycling and cell-surface insertion of AMPARs51,52. In addition, ubiquitination or deubiquitination via Nedd4 or USP46, respectively, may be considered53,54. In most cases, the contributions of these accessory proteins and post-translational modifications to AMPAR trafficking have been evaluated using genetic approaches, such as overexpression of an AMPAR subunit tagged with pH-sensitive SEP on the N-terminus. However, in some cases, complementary genetic experiments using knock-in or knock-out mice of the target gene have not supported the data19. Considering the heterotetrameric formation of AMPARs by the GluA1–4 subunits, and considering that each subunit has inherent roles, conflicting findings may be ascribed to the formation of non-native tetramers by the overexpression of single AMPAR subunits in neurons. In contrast, the present two-step labeling method can be used to visualize native iGluRs under physiological temperature in culture medium. Thus, this method can contribute to our understanding of the physiological and pathophysiological roles of iGluR trafficking in neurons.
## Methods
### Synthesis
All synthesis procedures and compound characterizations are described in Supplementary Methods.
### General methods for biochemical and biological experiments
SDS-PAGE and western blotting were carried out using a BIO-RAD Mini-Protean III electrophoresis apparatus. Samples were applied to SDS-PAGE and electrotransferred onto polyvinylidene fluoride membranes (BIO-RAD), followed by blocking with 5% nonfat dry milk in Tris-buffered saline containing 0.05% Tween 20. Primary antibody was indicated in each experimental procedure, and anti-rabbit IgG-HRP conjugate (CST, 7074S, 1:3,000) or anti-mouse IgG-HRP conjugate (CST, 7076S, 1:3,000) was utilized as the secondary antibody. Chemiluminescent signals generated with ECL Prime (GE Healthcare) were detected with a Fusion Solo S imaging system (Vilber Lourmat).
### Animals
Pregnant ICR mice and pregnant Sprague Dawley rats maintained under specific pathogen-free conditions were purchased from Japan SLC, Inc (Shizuoka, Japan). The animals were housed in a controlled environment (23 ± 1 °C, 12 h light/dark cycle) and had free access to food and water, according to the regulations of the Guidance for Proper Conduct of Animal Experiments by the Ministry of Education, Culture, Sports, Science, and Technology of Japan. All experimental procedures were performed in accordance with the National Institute of Health Guide for the Care and Use of Laboratory Animals, and were approved by the Institutional Animal Use Committees of Kyoto University or Nagoya University.
### Expression of AMPARs or NMDARs in HEK293T cells
HEK293T cells (ATCC) were cultured in Dulbecco’s modified Eagle’s medium (DMEM)-GlutaMAX (Invitrogen) supplemented with 10% dialyzed FBS (Invitrogen), penicillin (100 units ml–1), and streptomycin (100 µg ml–1), and incubated in a 5% CO2 humidified chamber at 37 °C. Cells were transfected with a plasmid encoding rat GluA2 (GluA2flip(Q))26 or the control vector pCAGGS (kindly provided by Dr. H. Niwa from RIKEN) using Lipofectamine 2000 (Invitrogen) according to the manufacturer’s instructions, and subjected to labeling experiments after 36–48 h of the transfection. For NMDAR expression, cells were transfected with plasmids encoding rat GluN1-155 and rat GluN2A55, and 30 µM MK-801 (Funakoshi) was added to the culture medium to suppress cell death.
### Two-step labeling of AMPARs or NMDARs in HEK293T cells
For the first step labeling of AMPARs, HEK293T cells transfected with GluA2 were treated with 2 µM CAM2(TCO) in the absence or presence of 50 µM NBQX in the culture medium at 37 °C for 4 h. For the second step labeling, the culture medium was removed, and 1 µM Tz(Fl) in PBS was added for 5 min at room temperature. To quench excess Tz(Fl), 1 µM TCO-PEG4-COOH in PBS was added.
For western blot analyses of labeled AMPARs, labeled cells were washed three times with PBS, lysed with radio immunoprecipitation assay (RIPA) buffer containing 1% protease inhibitor cocktail (Nacalai tesque), and mixed with 5× Laemmli sample buffer containing 250 mM DTT. Western blotting analyses were performed as described in “General methods for biochemical and biological experiments.” The Fl-labeled GluA2 was detected using rabbit anti-fluorescein antibody (abcam, ab19491, 1:3,000). GluA2 was detected using a rabbit anti-GluA2/3 antibody (Millipore, 07-598, 1:3,000).
In the case of two-step labeling of NMDARs, HEK293T cells transfected with GluN1-1 and GluN2A were treated with 10 µM CNM(TCO) in the absence or presence of 250 µM Pzf in the culture medium at 37 °C for 4 h. The second step labeling and subsequent western blotting were performed as described above. Immunodetection of GluN2A was performed with a rabbit anti-NR2A antibody (Millipore, 07-632, 1:1,000).
CAM2(TCO), CNM(TCO) and Tz-probes were stored in DMSO solution. The stock solutions were kept in deep freezer (–80 °C) to prevent decomposition.
### Enzymatic deglycosylation of AMPARs expressed in HEK293T cells
GluA2-expressing HEK293T cells were labeled as described in “Two-step labeling of AMPARs or NMDARs in HEK293T cells.” The labeled cells were washed three times with PBS and lysed in PBS containing 1% triton X-100, 0.6% SDS, and 1% protease inhibitor cocktail for 30 min at 37 °C. The lysates were diluted (2.0-fold) in sodium phosphate buffer (50 mM, pH 7.5) containing 2% NP40 and 100 mM DTT. PNGase F (New England Biolabs) were used at 1,000 units/100 µl of lysate and incubated overnight at 37 °C. The samples were subjected to western blotting analyses as described in “Two-step labeling of AMPARs or NMDARs in HEK293T cells.” In this experiment, after western blotting using anti-Fl antibody, the membrane was stripped with stripping buffer (250 mM glycine (pH = 2.5) and 1% SDS) and reprobed with the anti-GluA2/3 antibody.
### Confocal live cell imaging of labeled AMPARs in HEK293T cells
HEK293T cells were co-transfected with GluA2flip(Q) and mCherry-F26 as a transfection marker. First step labeling was performed as describe in “Two-step labeling of AMPARs or NMDARs in HEK293T cells.” For the second step labeling, after removal of the culture medium, 100 nM Tz(Ax488) was treated for 5 min in HBS (20 mM HEPES, 107 mM NaCl, 6 mM KCl, 2 mM CaCl2, and 1.2 mM MgSO4 at pH 7.4) at room temperature and washed three times with HBS. Confocal live imaging was performed with a confocal microscope (LSM900, Carl Zeiss) equipped with a 63×, numerical aperture (NA) = 1.4 oil-immersion objective. Fluorescence images were acquired by excitation at 405, 488, 561, or 640 nm derived from diode lasers.
For studying the reaction kinetics of tetrazine ligation to cell surface AMPARs, after first step labeling, cells were then incubated with 300 nM Tz(Ax488) at room temperature and imaged at specified time points by confocal microscopy. To quantify the fluorescence intensity of the membrane at each time point, mCherry-F positive cells (n = 6) were selected and the average signal intensity of ROIs set on the membrane was calculated by ZEN blue software (Carl Zeiss). After subtracting background fluorescence, the averaged membrane intensity was defined as F at each time point (FMAX was defined as F at 20 min). The membrane intensity was fitted with KaleidaGraph (Synergy software) using following equation (1): F = a + b (1 – e−ct).
For trafficking analyses, after first step labeling, 100 nM Tz(Ax488) was added in DMEM-GlutaMAX for 5 min. The cells were washed three times in DMEM-GlutaMAX and incubated for 0, 1, 4, and 8 h in growth medium at 37 °C. Live cell imaging was performed with a confocal microscope. To quantify the fluorescence intensity of the membrane at each time point, mCherry-F positive cells were selected and the average signal intensity of ROIs set on the membrane was calculated by ZEN blue software after subtracting background fluorescence. The averaged membrane intensity was defined as F at each time point. The membrane intensity was fitted with KaleidaGraph using following equation (2): F = a + b·e−ct, and the offset value (a) was set equal to zero. The t1/2 was defined as t1/2 = ln(2)/c.
For co-staining with LysoTracker Red DND-99 (Invitrogen), after first step labeling, the cells were treated with 50 nM LysoTracker Red DND-99 for 30 min at 37 °C in culture medium, and then treated with 100 nM Tz(Ax488) in culture medium for 5 min. After washing three times with culture medium or subsequent incubation for 8 h, cells were imaged using a confocal microscope.
### Half-life studies of AMPARs by western blotting in HEK293T cells
Schematic illustration of the experiments is shown in Fig. 3d. For determining t1/2surface (path-1 in Fig. 3d), first step labeling was conducted as described in “Two-step labeling of AMPARs or NMDARs in HEK293T cells.” After medium exchange for removal of the labeling reagents, the cells were incubated for 0, 1,4, 6, 12, 24, and 36 h. The cells were treated with 1 µM Tz(Fl) in PBS for 5 min, and excess Tz(Fl) was quenched by addition of 1 µM TCO-PEG4-COOH in PBS.
For determining t1/2life (path-2 in Fig. 3d), after first step labeling, cells were washed three times with culture medium and treated with 1 µM Tz(Fl) in culture medium for 5 min. Excess Tz(Fl) was quenched by addition of 1 µM TCO-PEG4-COOH in culture medium. Cells were then incubated for 0, 1,4, 6, 12, 24, and 36 h and washed three times with PBS. Cell lysis and western blotting were performed as described in “Two-step labeling of AMPARs or NMDARs in HEK293T cells.”
The target bands were manually selected, and the intensity were calculated with Fusion software (Vilber Lourmat), background intensity was manually subtracted by cutting the minimal intensity in the selected area. The half-life was calculated by curve fitting using KaleidaGraph and following equation (3): I = a + b·e−ct, and the offset value (a) was set equal to zero. The t1/2 was defined as t1/2 = ln(2)/c.
### Intracellular and surface ratio of labeled AMPARs in HEK293T cells
Schematic illustration of the experiments is shown in Fig. 3g. For determining labeled AMPARs on cell surface (path-1 in Fig. 3g), after first step labeling as describe in “Two-step labeling of AMPARs or NMDARs in HEK293T cells,” the cells were treated with 1 µM Tz(Fl) for 5 min in PBS at room temperature. To quench excess Tz(Fl), 1 µM TCO-PEG4-COOH in PBS was added and lysed with RIPA buffer containing 1% protease inhibitor cocktail for 30 min at 4 °C.
For determining intracellular labeled AMPARs (path-2 in Fig. 3g), after first step labeling, 1 µM Tz(Ax647) was treated for 5 min for masking of cell-surface TCO-labeled AMPARs. After cell lysis using RIPA buffer, the lysate was reacted with 0.3 µM Tz(Fl) for 30 min at room temperature. Excess Tz(Fl) was quenched by addition of 1 µM TCO-PEG4-COOH in the cell lysate.
For preparing whole-cell-labeling fraction (path-3 in Fig. 3g), after first step labeling, the cells were lysed with RIPA buffer containing 1% protease inhibitor cocktail for 30 min at 4 °C. The lysate was reacted with 0.3 µM Tz(Fl) for 30 min at room temperature. Excess Tz(Fl) was quenched by addition of 1 µM TCO-PEG4-COOH in the cell lysate.
Western blotting was performed as described in “Half-life studies of AMPARs by western blotting in HEK293T cells.” The target bands were manually selected, and the intensity were calculated with ImageJ software, background intensity was manually subtracted by selecting a region with no bands from the same lane. In more detail, the band intensity was determined as described below:
$$\left( {{\mathrm{target}}\;{\mathrm{intensity}}} \right)-\left( {{\mathrm{target}}\;{\mathrm{area}}} \right)/\left( {{\mathrm{background}}\;{\mathrm{area}}} \right) \times \left( {{\mathrm{background}}\;{\mathrm{intensity}}} \right)$$
### Preparation of primary cortical neuronal culture
Twenty-four-well plates (BD Falcon) were coated with poly-D-lysine (Sigma-Aldrich), and washed with sterile dH2O three times. Cerebral cortices from 16-day-old ICR mouse embryos were aseptically dissected and digested with 0.25 w/v% trypsin (Nacalai tesque) for 20 min at 37 °C. The cells were re-suspended in Neurobasal Plus medium supplemented with 10% FBS, penicillin (100 units/ml), and streptomycin (100 µg/ml) and filtered by Cell Strainer (100 µm, Falcon) and centrifuged at 1,000 rpm for 5 min. The cells were re-suspended in Neurobasal Plus medium supplemented with 2% of B-27 Plus Supplement, 1.25 mM GlutaMAX I (Invitrogen), penicillin (100 units/ml), and streptomycin (100 µg/ml) and plated at a density of 2 × 105 cells on the 24-well plate. The cultures were maintained at 37 °C in a 95% air and 5% CO2 humidified incubator. Culture medium was replaced every 3 or 4 days and the neurons were used at 12–15 DIV.
### Two-step labeling of AMPARs or NMDARs in cultured neurons
To label endogenous AMPARs, 12 µM CAM2(TCO) in 100 µl of the culture medium with or without 300 µM NBQX was gently added to the cortical neurons cultured in 500 µl medium on 24-well plates to a final concentration of 2 µM CAM2(TCO) and 50 µM NBQX. The cells were incubated for 10 h at 37 °C. For the second step, the culture medium was removed and the cells were treated with 1 µM Tz(Fl) for 5 min in PBS at room temperature. To quench excess Tz(Fl), 1 µM TCO-PEG4-COOH in PBS was added. Western blot analyses of labeled AMPARs were performed as described in “Two-step labeling of AMPARs or NMDARs in HEK293T cells.”
To label endogenous NMDARs, 60 µM CNM(TCO) in 100 µl of growth medium with or without 1.5 mM Pzf was gently added to the cortical neurons cultured in 500 µl medium on 24-well plates to a final concentration of 10 µM CNM(TCO) and 50 µM Pzf. The cells were incubated for 10 h at 37 °C. The second step labeling and subsequent western blotting was performed as described above.
### Half-life studies of endogenous AMPARs or NMDARs in cultured neurons
Schematic illustration of the experiments is shown in Fig. 5a. For determining t1/2surface, after first step labeling as describe in “Two-step labeling of AMPARs or NMDARs in cultured neurons,” the cells were incubated for 0, 2, 4, 6, 12, 24, 48, 72, and 120 h. For the second step, the culture medium was removed and the cells were treated with 1 µM Tz(Fl) for 5 min in PBS at room temperature. To quench excess Tz(Fl), 1 µM TCO-PEG4-COOH in PBS was added and washed three times with PBS. Cell lysis and western blotting were performed as described in “Two-step labeling of AMPARs or NMDARs in HEK293T cells.” The immunodetection of GluA2 was conducted with a rabbit anti-GluA2 antibody (abcam, ab20673, 1:3,000). Quantification of the band intensity and calculation of the half-time was calculated described as in “Half-life studies of AMPARs by western blotting in HEK293T cells.”
### Intracellular and surface ratio of labeled AMPARs in cultured neurons
Schematic illustration of the experiments is shown in Fig. 3g. For determining labeled AMPARs on cell surface (path-1 in Fig. 3g), after first step labeling as describe in “Two-step labeling of AMPARs or NMDARs in cultured neurons,” the cells were treated with 1 µM Tz(Fl) for 5 min in PBS at room temperature. To quench excess Tz(Fl), 1 µM TCO-PEG4-COOH in PBS was added and lysed with RIPA buffer containing 1% protease inhibitor cocktail for 30 min at 4 °C.
For determining intracellular labeled AMPARs (path-2 in Fig. 3g), after first step labeling, 1 µM Tz(Ax647) was treated for 5 min for masking of cell-surface TCO-labeled AMPARs. After cell lysis using RIPA buffer containing 1% protease inhibitor cocktail for 30 min at 4 °C, the lysate was reacted with 0.3 µM Tz(Fl) for 30 min at room temperature. Excess Tz(Fl) was quenched by addition of 1 µM TCO-PEG4-COOH in the cell lysate.
For preparing whole-cell-labeling fraction (path-3 in Fig. 3g), after first step labeling, the cells were lysed with RIPA buffer containing 1% protease inhibitor cocktail for 30 min at 4 °C. The lysate was reacted with 0.3 µM Tz(Fl) for 30 min at room temperature. Western blotting was performed as described in “Two-step labeling of AMPARs or NMDARs in HEK293T cells.” Quantification of the band intensity and calculation of the ratio was conducted as described in “Intracellular and surface ratio of labeled AMPARs in HEK293T cells.”
### Quantification of recycled AMPARs in cultured neurons
The first step labeling was performed as describe in “Two-step labeling of AMPARs or NMDARs in cultured neurons.” For the second step, the culture medium was removed and the cells were treated with 1 µM Tz(Fl) for 5 min in the culture medium at 37 °C. To quench excess Tz(Fl), 1 µM TCO-PEG4-COOH in the culture medium was added. After incubation at 37 °C for 15 min, recycled AMPARs were labeled with 1 µM Tz(Ax647) for 5 min in PBS. To quench excess Tz(Ax647), 1 µM TCO-PEG4-COOH in PBS was added. Cell lysis and western blotting were performed as described in “Two-step labeling of AMPARs or NMDARs in HEK293T cells” using anti-Alexa 647 antibody. Quantification of the band intensity and calculation of the ratio was conducted as described in “Intracellular and surface ratio of labeled AMPARs in HEK293T cells.”
Anti-Alexa 647 antibody was prepared from the sera of a rabbit immunized with an antigen which was a conjugate of Alexa 647-NHS and KLH (Sigma), and the antibody was affinity-purified using Alexa 647-conjugated agarose. Alexa 647-conjugated agarose was prepared from CarboxyLink Coupling Resin (Thermo Fisher) and Alexa 647 NHS ester (Invitrogen). The anti-sera (1:2,000) or the purified antibody (1:1,000) was used for the western blotting.
### Preparation of primary hippocampal neuronal culture
Glass bottom dishes (IWAKI) or coverslips (diameter, 13 mm, Matsunami) were coated with poly-d-lysine (Sigma-Aldrich), and washed with sterile dH2O three times. Hippocampi from 18-day-old Sprague Dawley rat embryos were aseptically dissected and digested with 0.25 w/v% trypsin (Nacalai tesque) for 20 min at 37 °C. The cells were re-suspended in Neurobasal Plus medium supplemented with 10% FBS, penicillin (100 units/ml) and streptomycin (100 µg/ml) and filtered by Cell Strainer (100 µm, Falcon) and centrifuged at 1,000 rpm for 5 min. The cells were re-suspended in Neurobasal Plus medium supplemented with 2% of B-27 Plus Supplement, 1.25 mM GlutaMAX I (Invitrogen), penicillin(100 units/ml), and streptomycin (100 µg/ml) and plated at a density of 2 × 104 cells on glass coverslips inside 24-well plates (BD Falcon) or glass bottom dishes. Cultures were maintained at 37 °C in a 95% air and 5% CO2 humidified incubator. Culture medium was replaced every 7 days and the neurons were used at 16–18 DIV.
### Live cell imaging of AMPARs in cultured neurons
To label endogenous AMPARs, 12 µM CAM2(TCO) in 300 µl of growth medium with or without 300 µM NBQX was gently added to the hippocampal neurons cultured in 1.5 ml medium on glass bottom dishes to a final concentration of 2 µM CAM2(TCO) and 50 µM NBQX. After removal of the culture medium, neurons were treated with 100 nM Tz(Ax488) for 5 min in HBS at room temperature and washed three times with HBS. Confocal live imaging was performed with a confocal microscope.
### Immunostaining of cultured neurons after labeling
Primary cultures of hippocampal neurons were labeled by 2 µM CAM2(TCO) and followed by 100 nM Tz(Ax488) as described above. The cells were fixed with 4% PFA in PBS at room temperature for 30 min and washed three times with PBS. PFA-fixed cells were permeabilized for 15 min with PBS containing 0.1% Triton X-100 at room temperature. The cells were washed three times in PBS and incubated in 10% normal goat serum for 1 h at room temperature. After blocking, the cells were incubated overnight at 4 °C with primary antibodies in PBS containing 1% normal goat serum. The cells were then washed three times with PBS and incubated for 1 h at room temperature with secondary antibodies in PBS containing 1% normal goat serum. The following primary antibodies were used: mouse anti-PSD95 (abcam, ab2723, 1:1,000) or rabbit anti-MAP2 (Millipore, AB5622, 1:1,000). Secondary antibodies were used goat anti-mouse Alexa 647 (abcam, ab150115, 1:1,000) and goat anti-rabbit Alexa633 (Invitrogen, A21070, 1:2,000). Imaging of immunostained hippocampal neurons was performed with a confocal microscope.
### Fluorescence lifetime imaging of AMPARs in cultured hippocampal neurons
Primary cultures of hippocampal neurons were labeled by 2 µM CAM2(TCO), followed by 100 nM Tz(ST647) and fixed with 4% PFA in PBS. The cells were immunolabeled with PSD95 and MAP2 primary antibodies, and stained with Alexa 488 and Alexa405 secondary antibodies, respectively. Confocal and lifetime imaging of immunostained hippocampal neurons was performed by TCS SP8 FALCON (Leica microsystems) equipped with a white light laser and 63×, NA = 1.4 oil-immersion objective. SeTau-647 was imaged using 640 nm exc. (laser power 100, emission collected at 653–700 nm, using 0–12.5 ns time gate).
FLIM images were processed in LAS X 3.5.5 software (Leica microsystems) to fit the lifetime decay curves using an n-exponential reconvolution model with the number of components that χ2 value is closest to 1. In our data, a three-component fit was utilized. The component of the lifetime corresponding to SeTau-647 (τ = 2.4 ± 0.1 ns) was used for calculating the intensity of FLIM images. The synaptic or dendric region was selected ROI on PSD95 or MAP2 signals and the background intensities were subtracted by selecting a region of no cells.
### Statistics and reproducibility
All graphs were generated using Microsoft Excel. All data are expressed as mean ± s.e.m. We accumulated the data for each condition from at least three independent experiments. We evaluated statistical significance with Student’s t-test for comparisons between two mean values. A value of P < 0.05 was considered significant.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
## Data availability
The authors declare that the data supporting the findings of this study are available with the paper and its Supplementary information files. The data that support the findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.
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## Acknowledgements
The authors thank Mr. Hideyuki Yamaguchi (Leica Microsystems) for technical supports in FLIM imaging and Dr. Bronwen Gardner (Edanz Group) for editing a draft of this manuscript. This work was funded by Grants-in-Aid for Scientific Research (KAKENHI) (Grant Number 18J22952 to K.O., 17H06348 to I.H., 16H03290, 19H05778, and 20H02877 to S.K.), Daiichi Sankyo Foundation of Life Science, the Takeda Science Foundation, and the Mochida Memorial Foundation for Medical and Pharmaceutical Research to S.K., and supported by JST CREST (JPMJCR1854) to M.Y. and JST ERATO Grant Number JPMJER1802 to I.H.
## Author information
Authors
### Contributions
S.K. and I.H. initiated and designed the project. K.O., K.Shiraiwa, K.Soga, T.D., M.T. and K.K. performed synthesis and chemical labeling in HEK293T cells. K.O., K.Shiraiwa, M.Y. and S.K. performed chemical labeling in cultured neurons. K.O. and S.K. wrote the manuscript. All authors discussed and commented on the manuscript.
### Corresponding authors
Correspondence to Itaru Hamachi or Shigeki Kiyonaka.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Peer review information Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.
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Ojima, K., Shiraiwa, K., Soga, K. et al. Ligand-directed two-step labeling to quantify neuronal glutamate receptor trafficking. Nat Commun 12, 831 (2021). https://doi.org/10.1038/s41467-021-21082-x
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• ### Organocatalytic atroposelective construction of axially chiral N, N- and N, S-1,2-azoles through novel ring formation approach
• Yu Chang
• Chuandong Xie
• Hailong Yan
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https://earthscience.stackexchange.com/questions/10617/what-produces-these-distinct-shapes-in-the-rub-al-khali-seen-from-space | What produces these distinct shapes in the Rub' al Khali seen from space?
update: Searching "Rub' al Khali Empty Quarter" found "Q2: What are sabkhas?" in https://www.geocaching.com/geocache/GC6BYQ0_rub-al-khali-the-empty-quarter which seems to be related.
From the NASA image gallery; and also Catalog Page for PIA11084 where the caption says:
Original Caption Released with Image: The Rub' al Khali is one of the largest sand deserts in the world, encompassing most of the southern third of the Arabian Peninsula. It includes parts of Oman, United Arab Emirates, and Yemen. The desert covers 650,000 square kilometers, more than the area of France. Largely unexplored until recently, the desert is 1000 km long and 500 km wide. The first documented journeys made by Westerners were those of Bertram Thomas in 1931 and St. John Philby in 1932. With daytime temperatures reaching 55 degrees Celsius, and dunes taller than 330 meters, the desert may be one of the most forbidding places on Earth.
The image was acquired December 2, 2005, covers an area of 54.8 x 61.9 km, and is located near 20.7 degrees north latitude, 53.6 degrees east longitude.
The U.S. science team is located at NASA's Jet Propulsion Laboratory, Pasadena, Calif. The Terra mission is part of NASA's Science Mission Directorate.
Image Credit: NASA/GSFC/METI/ERSDAC/JAROS, and U.S./Japan ASTER Science Team
Question: The pattern shown in this image has an amazing amount of approximate repetition of shapes; long straight segments with a periodicity of about 2 kilometers, "stair steps" northward and sharp, pointy extensions southward. Is there any understanding how such a distinct pattern could be formed?
below: "Rub' al Khali (Arabian Empty Quarter) sand dunes imaged by Terra (EOS AM-1)" cropped and full sized (reduced resolution) images from here.
below: screen shot from Google Maps confirming orientation. Marker is at 20.7N, 53.6E.
• Suggestions for tags welcomed. – uhoh Jun 19 '17 at 5:24
• what are the light-colored/reflective areas? – Colin Jun 19 '17 at 7:19
• @ColinZwanziger I'm out of my depth here. I would think the flat areas are sand and the raised areas are rock, but the NASA caption calls the raised areas "dunes". It's a really good question! – uhoh Jun 19 '17 at 7:26
• I'm reasonably certain that those are dunes (ergs) where evaporite salt has formed in the troughs between the dunes, thus giving that somewhat reflective white color. Reasonably certain, but not certain enough to post as an answer. – kingledion Jun 19 '17 at 11:27
• @kingledion I'm wondering if this should be two separate questions - what is this stuff, and why the highly distinct, repeating pattern. But I suppose the answer to the latter requires the former. The Google Earth image is probably fairly close to the visible appearance. The satellite image is from ASTER and with 14 spectral channels from visible all the way to thermal IR, this might be a false color (thus the google Earth image for reference/comparison). Erg, I learned a new word today, thanks! ;) – uhoh Jun 19 '17 at 12:58
NASA has just published (16 June 2018) a similar picture on its Earth Observatory website.
The pictures in the question were taken after Tropical Cyclone Mekunu passed over the region in May 2018.
The grey, metallic looking shapes are interdune sand flats filled with rain water.
This a great example of "self-organization" in a geomorphological system; this one an oscillating system at a fairly large scale. On a much smaller scale, you find similar structures on sand dune surfaces themselves as well as dry dusty dirt roads:
Self organization (and its scaling) is a complex process delicately dependent on just a few variables, in the case of desert dunes, wind carrying capacity, and critically, bed surface. (You can also find examples of oscillating chemical self organization in rocks when, for example, you see "tree-rings" around plagioclase crystals in anorthosites.)
The hidden (and surprising) complexity of self organization in natural systems is reflected in the difficulty in modelling these behaviors. In the case of small sand dunes:
...a complex process of ripple coalescence is responsible for this evolution of the ripple field. Small, fast-moving ripples catch up with and are absorbed by larger, slower forms, each such merger resulting in a growth of the mean wavelength and a decline in the relative dispersion of the wavelengths.
This mechanism may figure in modelling the formation of large dune fields:
Because dunes reorient only at their crest terminations with a change in wind regime, the rate of formation of a new pattern of small dunes is typically faster than the rate of reorientation of the existing pattern, resulting in the superposition of simple patterns to give rise to complex patterns. Complex patterns are distinct from spatial changes in a simple pattern, and from the type of superposition that characterizes compound/complex dunes.
There are literally tons of examples of self organization in earth systems, at all kinds of scales, but that NASA image is a stunner!
• Ugh, anyone who's driven a car over a washboard road will cringe seeing that photo :) This is a great introduction to self-organization and periodicity. +1 However, I'm still interested if there has been any kind of possible explanation for, or discussions of the "stair steps" northward and sharp, pointy extensions southward. So I've asked Is there any understanding how such a distinct pattern could be formed? about the stair steps and pointy extensions. – uhoh Jun 20 '17 at 3:06
• I'll stop by the library in a few days and take a look at the papers. Thanks for the links! – uhoh Jun 20 '17 at 3:12
• Pointy ends are a standard feature of many dunes (even on Mars!), particularly Barchan dunes: google.com/…. If you can imagine winds shifting to one direction for a prolongued period, then shifting back, you can easily imagine these pointy tails smeared and superposed in complex geometries. – Knob Scratcher Jun 20 '17 at 3:15
• OK, that's great! Do you think it's possible to add that back into the answer? I thought the points and steps were really unique. If they are not, then that information in the answer could wrap this question up nicely. – uhoh Jun 20 '17 at 3:17
• No edit necessary; it's already in there! ......."Because dunes reorient only at their crest terminations with a change in wind regime, the rate of formation of a new pattern of small dunes is typically faster than the rate of reorientation of the existing pattern, resulting in the superposition of simple patterns to give rise to complex patterns. Complex patterns are distinct from spatial changes in a simple pattern, and from the type of superposition that characterizes compound/complex dunes." – Knob Scratcher Jun 20 '17 at 3:31 | 2021-07-30 07:37:23 | {"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": 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": 0.3619600534439087, "perplexity": 2638.550035141567}, "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-2021-31/segments/1627046153934.85/warc/CC-MAIN-20210730060435-20210730090435-00283.warc.gz"} |
https://www.physicsforums.com/threads/the-difference-between-1-mole-of-c-and-1-molecule-of-c.944840/ | # The difference between 1 mole of C and 1 molecule of C?
1. Apr 15, 2018 at 6:08 AM
### Indranil
As we know, in the case of an atom, 1 mole of C = 6.023 X 10^23 C atoms and in the case of a molecule, 1 mole of C molecule = 6.023 X 10^23 molecules.
as we know, atoms are used to make molecules, So can I say 1 mole of C atoms make 1 molecule of C because what we get If we break 1 molecule of C I think the answer is 6.023 X 10^23 atoms. So can I say 1 mole of C is the same as 1 molecule of C? please point out where I am wrong above.
2. Apr 15, 2018 at 6:31 AM
### Staff: Mentor
Stop posting half your questions here and half at CF, starting multiple threads. This way you won't move forward and you just combine all possible confusions from answers posted to different questions. Besides, seems like you ask the questions without reading the answers, as you repeat the same errors several times. As much as I wan't to help you I am more and more wondering if you are not just trolling.
Mole and molecule are completely different things. Molecule contains bonded atoms, mole doesn't. You were told multiple times mole is similar to dozen, it is just a collection of 6.02×1023 objects.
3. Apr 15, 2018 at 9:59 AM
### Indranil
Ok, thank you for pointing out my mistakes. Could you tell me please what to do myself If I am stuck or can't clear my concept in the same thread? will I keep asking different questions related to the same topic in the same thread until I clear my concept or start multiple threads? Please suggest me.
4. Apr 15, 2018 at 10:17 AM
### Staff: Mentor
Keeping to one thread is typically much better. Not always, and it sometimes makes sense to start a new one when the old one drifted in a wrong direction (you can always ask Mentors whether it will be appropriate or whether they can clean up the thread from side discussions).
5. Apr 15, 2018 at 10:48 AM
### Indranil
1. say, I start a thread and received some answers but did not clear my concept. I waited for 2 or 3 days for the replies to my question but did not get any replies. then what should I do? to start a new thread or to ask a mentor as you said above 'whether it will be appropriate...;
2. Where to ask the mentors?
6. Apr 15, 2018 at 11:18 AM
### Drakkith
Staff Emeritus
If C means the element carbon, then you cannot have molecules of C. You would need a C4 molecule or something.
No, not at all. Let's assume that we have 1 mole of CH4 molecules (also known as methane). One mole of methane contains one mole of carbon atoms and four moles of hydrogen atoms, because each individual molecule is composed of one carbon atom and four hydrogen atoms. Does that make sense?
7. Apr 15, 2018 at 11:20 AM
### Staff: Mentor
You can always bump a thread after 24h.
Either using report function (link below a post) or sending a direct PM.
8. Apr 15, 2018 at 12:34 PM
### Indranil
Say, I post a thread and I received some replies but still, I did not clear my concept and I waited for 24 hours but did not get any replies. Should I use 'report' to get replies regarding that concept in the same thread or I will start a new thread?
9. Apr 15, 2018 at 12:42 PM
### Indranil
1. ''Each individual molecule is composed of one carbon atom and four hydrogen atoms'' Here one carbon atom and four hydrogen atoms mean 'one mole of carbon atoms and four moles of hydrogen atoms'?
2. Is it possible to get one molecule of C because you pointed earlier that 'If C means the element carbon, then you cannot have molecules of C. You would need a C4 molecule or something.'? Could you clarify it, please?
10. Apr 15, 2018 at 12:58 PM
### Staff: Mentor
Keep your discussions of a topic in one thread. Period. Do not "report" it and do not start a new thread. If you don't get any new replies after a couple of days, simply "bump" it by replying to one of the responses you got and saying what, specifically, you are still confused about.
11. Apr 15, 2018 at 12:59 PM
### Staff: Mentor
No, individual molecule is made of individual atoms.
Molecule is made of several atoms. C is either a symbol of an element (and as such doesn't refer to neither atom or molecule) or a symbol of a C atom. In the latter case, as molecule is made of several atoms, just C is not a molecule, C2 or C3 would be.
12. Apr 15, 2018 at 8:33 PM
### Indranil
1. 'Each individual molecule is composed of one carbon atom and four hydrogen atoms'' Here one carbon atom and four hydrogen atoms mean 'one mole of carbon atoms and four moles of hydrogen atoms'?--------- Here one carbon atom means what? What should I consider 'one carbon atom''? one single carbon atom or one mole of carbon atoms and same question ''four single hydrogen atoms or four moles of hydrogen atoms. in the above context?
2.' Molecule is made of several atoms. C is either a symbol of an element (and as such doesn't refer to neither atom or molecule) or a symbol of a C atom. In the latter case, as molecule is made of several atoms, just C is not a molecule, C2 or C3 would be.'------------You said above that 'just C is not a molecule' but below in the rection
C + O2 = 2CO
Here in the reactants above, what should I consider 'C'? 'a molecule of C or what"
13. Apr 15, 2018 at 8:37 PM
### Drakkith
Staff Emeritus
One carbon atom is exactly what it sounds like. A single carbon atom.
It's a single atom of carbon.
14. Apr 15, 2018 at 10:21 PM
### symbolipoint
One carbon atom is not a molecule. One carbon atom is not a mole of carbon. One carbon atom is not an element of carbon. Maybe someone like Borek will help clarify any distinction between Element and Atom.
Graphite is made of carbon. Element? As a compound?
15. Apr 15, 2018 at 11:01 PM
### Staff: Mentor
The wording says precisely what is meant. Nothing more, nothing less. If you, yourself are adding words, you are changing the meaning.
16. Apr 16, 2018 at 1:37 AM
### DrDu
In the case of e.g. a diamond, you could argue that a diamond of 12 grams is in fact a single macromolecule composed of 6 x 10^23 atoms of C. So you could say that 1 diamond molecule $\mathrm{C_{6\times 10^{23}}}$ equals 1 mole of C atoms. But take in mind that usually when speaking about solid elements and no other specification is made, when speaking of one mole of an element, we are referring to 1 mole of atoms of which the element is made up. In former times, in the case of elements, one used to speak of "gram-atoms" instead of moles, to make this distinction clearer. However, as, with the exception of radiological units, we decided to break with the babylonian tradition to use different units for apples and pears, in both cases mole is is used as a unit and it is left to the user to make clear whether he is referring to atoms or some kind of molecules.
17. Apr 16, 2018 at 1:44 AM
### jishnu
Carbon in elemental state does not exist as a molecule,
So, now considering generally any single species formed by group of atoms then, you can call it a molecule(eg:methane, carbon dioxide, dioxygen etc..)
In case of mole, its the general term used to denote a group of avagadro number(N A =6.022 x 10^23 particles) of particles(which may be atoms, ions, or molecules)
so, 1 mole C contains 6.022 x 10^23 individual carbon atoms in it( its really huge quantity :p)
18. Apr 16, 2018 at 4:53 AM
### Indranil
1. How many questions (including homework) should I ask for 24 hours?
2. Where to PM to any mentor If I want some help?
19. Apr 16, 2018 at 5:33 AM
### Staff: Mentor
Stop this. You are being dense, whether you know it or not. | 2018-04-21 23:36:09 | {"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": 0.33867862820625305, "perplexity": 1196.4196854060733}, "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/1524125945459.17/warc/CC-MAIN-20180421223015-20180422003015-00149.warc.gz"} |
https://advice.thisoldhouse.com/showthread.php?4611-ceiling-crack&p=16933 | 1. Junior Member
Join Date
Mar 2008
Posts
1
## ceiling-crack
I step on the drywall above my kitchen and it cracked all the way thru and ran for about 3 feet and now you can see where that cracked area is. I put spackle in crack but you can see where the ceilng is lower.
do i have to replace the drywall or how can I push the ceiling back up even with the rest of the ceiling?
2. Junior Member
Join Date
Jan 2008
Location
huntersville NC
Posts
51
## Re: ceiling-crack
you will probably need to cut out the area between the 2x material and then put \in a new piece. add nailers to the 2x so you have something to screw the piece to. then tape and use drywall mud couple of coats and sand and prime then paint. this is the correct way to do it.
otherwise it will crack and never look right.
#### Posting Permissions
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• | 2017-02-21 22:20:11 | {"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": 0.8032324910163879, "perplexity": 3288.0245366310946}, "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-2017-09/segments/1487501170839.31/warc/CC-MAIN-20170219104610-00191-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/3757936/pretty-conjecture-x-left-fracyx-rightny-left-fracxy-rightn | # Pretty conjecture $x^{\left(\frac{y}{x}\right)^n}+y^{\left(\frac{x}{y}\right)^n}\leq 1$
inspired (again) by an inequality of Vasile Cirtoaje I propose my own conjecture :
Let $$x,y>0$$ such that $$x+y=1$$ and $$n\geq 1$$ a natural number then we have : $$x^{\left(\frac{y}{x}\right)^n}+y^{\left(\frac{x}{y}\right)^n}\leq 1$$
First I find it very nice because all the coefficient are $$1$$ .
I have tested with Geogebra until $$n=50$$ without any counter-examples.
Furthermore we have an equality case as $$x=y=0.5$$ or $$x=1$$ and $$y=0$$ and vice versa .
To solve it I have tried all the ideas here
My main idea was to make a link with this inequality (my inspiration) see here
So if you can help me to solve it or give me an approach...
...Thanks for all your contributions !
## Little update
I think there is also an invariance as in question here Conjecture $a^{(\frac{a}{b})^p}+b^{(\frac{b}{a})^p}+c\geq 1$
## Theoretical method
Well,Well this method is very simple but the result is a little bit crazy (for me (and you ?))
Well ,I know that if we put $$n=2$$ we can find (using parabola) an upper bound like
$$x^{\left(\frac{1-x}{x}\right)^2}\leq ax^2+bx+c=p(x)$$ And $$(1-x)^{\left(\frac{x}{1-x}\right)^2}\leq ux^2+vx+w=q(x)$$
on $$[\alpha,\frac{1}{2}]$$ with $$\alpha>0$$ and such that $$p(x)+q(x)<1$$
In the neightborhood of $$0$$ we can use a cubic .
Well,now we have (summing) :
$$x^{\left(\frac{1-x}{x}\right)^2}+(1-x)^{\left(\frac{x}{1-x}\right)^2}\leq p(x)+q(x)$$
We add a variable $$\varepsilon$$ such that $$(p(x)+\varepsilon)+q(x)=1$$
Now we want an inequality of the kind ($$k\geq 2$$):
$$x^{\left(\frac{1-x}{x}\right)^{2k}}+(1-x)^{\left(\frac{x}{1-x}\right)^{2k}}\leq (p(x)+\varepsilon)^{\left(\frac{1-x}{x}\right)^{2k-2}}+q(x)^{\left(\frac{x}{1-x}\right)^{2k-2}}$$
Now and it's a crucial idea we want something like :
$$\left(\frac{x}{1-x}\right)^{2k-2}\geq \left(\frac{1-(p(x)+\varepsilon)}{q(x)}\right)^y$$
AND :
$$\left(\frac{1-x}{x}\right)^{2k-2}\geq \left(\frac{1-q(x)}{p(x)+\varepsilon}\right)^y$$
Now it's not hard to find a such $$y$$ using logarithm .
We get someting like :
$$x^{\left(\frac{1-x}{x}\right)^{2k}}+(1-x)^{\left(\frac{x}{1-x}\right)^{2k}}\leq q(x)^{\left(\frac{1-q(x)}{q(x)}\right)^{y}}+(1-q(x))^{\left(\frac{q(x)}{1-q(x)}\right)^{y}}$$
Furthermore the successive iterations of this method conducts to $$1$$ because the values of the differents polynomials (wich are an approximation of the initial curve) tend to zero or one (as abscissa).
The extra-thing (and a little bit crazy) we can make an order on all the values.
## My second question
Is it unusable as theoretical\practical method ?
• Have you already tried the limit cases around 1 and 0.5? Jul 15, 2020 at 17:15
• @Biggus Dickus Python, Does this help you to prove your conjecture ? Jul 15, 2020 at 20:57
• @zeraoulia rafik interesting but it seems to be not sufficient to prove the entire inequality.Anyway with this I have a special case .(+1) for your question. Jul 17, 2020 at 13:37
• Oups this not real number but natural number ah ! Anyway have a good day. :-) Jul 17, 2020 at 13:43
• @ErikSatie: I would replace $n$ with a positive variable and try Method of Lagrange Multipliers to see if we can extract some thing. Dec 6, 2020 at 3:00
I am sorry for not proving your conjecture, but I thought of writing up my thoughts as it might help you.
Without loss of generality, we can say that $$y \geq 0.5$$. Let $$q=y/x$$ and because $$y \geq 0.5$$ we have $$q\geq 1$$ (you can do everything in reverse with $$x \geq 0.5$$ and $$q\leq 1$$). Let $$a_0=x$$ and $$b_0=y$$. Then we can write: $$x^{\left( \frac{y}{x} \right)^n} + y^{\left(\frac{x}{y}\right)^n}=x^{q^n}+y^{\left(\frac{1}{q}\right)^n}=a_n+b_n,$$ with \begin{align} a_n &= a_{n-1}^q,\\ b_n &= b_{n-1}^{1/q}. \end{align}
It is easy to verify that with $$b_n\geq0.5$$, also $$b_{n+1}\geq 0.5$$. Now, let us assume that $$a_{n-1} \leq 1-b_{n-1}$$. For $$n=1$$, we have $$a_0 \leq 1-b_0$$ (more specifically, $$a_0=1-b_0$$). Because of that, we can write (remember that $$q \geq 1$$): \begin{align} a_{n+1} + b_{n+1} &= a_n^q+b_n^{1/q} \\ &\leq (1-b_n)^q + b_n^{1/q}. \end{align}
What is left, is to prove that $$(1-b_n)^q+b_n^{1/q} \leq 1, \quad \forall \,\,\, b_n \in [0.5, 1], q\geq1$$
This is where I got stuck, but perhaps you know how to continue from here. In the figure below, I plotted $$(1-b)^q+b^{1/q}$$ with $$b$$ on the interval $$[0.5,1]$$ for various values of $$q$$ and it seems that the result is always equal or below $$1$$.
Let us prove the case when $$0 < x \le \frac{14}{33}$$.
By Bernoulli inequality, we have $$y^{(x/y)^n} = (1 - x)^{(x/y)^n} \le 1 - x(x/y)^n.$$ It suffices to prove that $$x^{(y/x)^n} + 1 - x(x/y)^n \le 1$$ or $$\ln x \le \frac{- \ln[(y/x)^n]}{(y/x)^n - 1}.$$ Since $$u \mapsto \frac{-\ln u}{u - 1}$$ is strictly increasing on $$u > 1$$, it suffices to prove that $$\ln x \le \frac{- \ln[(y/x)]}{(y/x) - 1}.$$ With the substitution $$x = \frac{1}{z}$$, it suffices to prove that, for all $$z \ge \frac{33}{14}$$, $$z\ln(z) - \ln(z - 1) - 2\ln z \ge 0$$ which is true (easy). | 2022-09-27 08:08:29 | {"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": 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": 60, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9850107431411743, "perplexity": 291.83027867920055}, "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/1664030334992.20/warc/CC-MAIN-20220927064738-20220927094738-00477.warc.gz"} |
https://www.physicsforums.com/threads/is-this-function-continous.143924/ | # Is this function continous?
1. Nov 15, 2006
### pivoxa15
Is
f(t)= {0, -2<t<0
{t, 0<=t<2
continous everywhere in -2<t<2. I am a bit concerned at t=0
I think yes, after applying the epsilon delta argument but I rememer that a similar function
f(t) = {-t, -2<t<0
{t, 0<=t<2
is not differentiable hence not continous at t=0.
Have I made a mistake somwhere?
Last edited: Nov 15, 2006
2. Nov 15, 2006
### matt grime
It is perfectly possible for a function to be continuous and not differentiable.
3. Nov 15, 2006
### pivoxa15
So
f(t)= {0, -2<t<0
{t, 0<=t<2
is continous at t=0?
4. Nov 15, 2006
### HallsofIvy
Staff Emeritus
What is $\lim_{t\rightarrow0}f(t)$?
5. Nov 15, 2006
### pivoxa15
I have to say it is 0 because from the left, it must be 0 since the function is 0. From the right, it is 0 since the function is t. f(0)=0 is defined so it should be continous. But is it differentiable at t=0?
This means
f(t) = {-t, -2<t<0
{t, 0<=t<2
is continous as well. But it is not differentiable at t=0 is it? How come?
6. Nov 15, 2006
### HallsofIvy
Staff Emeritus
Yes, the limit from both left and right is 0 so the limit is 0. Since the limit there exists and is equal to the value of the function, the function is continuous at 0.
Your f(t) is, of course, |t| between -2 and 2. It is well know that |x|, while continuous, does not have a derivative at x= 0. Essentially, it is not "smooth": there is a corner at x= 0 so the tangent line there is not well defined. More precisely, if h< 0 then f(0+h)= f(h)= -h so
(f(0+h)- f(0))/h= -h/h= -1 while if h> 0, (f(0+h)- f(0))/h= h/h= 1. The two one-sided limits are different so the limit itself, and therefore the derivative at t=0, does not exist.
It's easy to see that the derivative of f is 1 for t> 0 and -1 for t< 0. While the derivative of a function is not necessarily continuous, it does satisfy the "intermediate value property" so the derivative cannot exist at 0.
7. Nov 15, 2006
### pivoxa15
So |x| is everywhere continous but not differentiable at x=0 because the derivative function is undefined at x=0.
Back to
f(t)= {0, -2<t<0
{t, 0<=t<2
If I wanted to represent this function as a fourier series than I am assured that the fourier series is uniformly convergent in -2<t<2 since f(t) is continous everywhere in this domain as have been shown. Or could this not happen because if an infinite series of functions is uniformly convergent than the function it represents is continous everywhere in the domain. But the function being continous everywhere dosen't always mean uniform convergence of the series representing it. In other words is it 'if' or 'if and only if'?
Last edited: Nov 15, 2006 | 2016-12-06 16:07:47 | {"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": 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": 0.9002861380577087, "perplexity": 997.1525897927393}, "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-2016-50/segments/1480698541910.39/warc/CC-MAIN-20161202170901-00371-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.talkunafraid.co.uk/category/eve/page/5/ | ## Looking to the future of EVE Metrics
EVE Metrics has become a disorganised, sprawling project. A year or two ago I set out with the idea in my head to build something like EVE Central, but for as much data as possible. I’m a data junkie in a way- I’m the sort of person who gets a bit of a kick from being able to see complex relationships between seemingly unrelated data and do that sort of analysis on a large scale. In short- <3 databases. I set out to do this with very little Ruby/Rails experience, a fairly solid MySQL experience after my earlier projects doing high-volume event logging using Garry’s Mod for Half-Life 2 to provide an audit trail for an FPS-based roleplaying environment, and a modicum of webapp experience.
What I hadn’t really considered was the market. Read on to read a short bit of background and find out where the site’s heading.
## Passive alignment and other myths
There are some things in EVE that people seem to take as fact which are quite simply not.
My pet hate on this front has got to be passive alignment. For those who aren’t in the know, this is when you align to a point in space (say, another stargate), at zero speed. The idea is that when you hit the warp button, you will enter warp faster than if you were pointing in another direction.
Now, this doesn’t work. If you’re pointing exactly 180 degrees away from the warp-out point you’ll warp as quickly as if you were pointing at it. If you don’t believe me, go undock and try it.
The exact equation involved looks like this:
$TimeToWarp=-ln(0.25) * Mass * Agility / 1000000$
A little explanation- the mass and agility of your ship are equal contributors to the warp time of your ship. The other element- the natural logarithm of 0.25- is why you warp when you’re 3/4 of the way round your ship’s speed indicator, and not at maximum velocity. This equation will give the absolutely correct (not accounting for latency) time to enter warp no matter the orientation of the ship involved.
Some of you might be screaming “But the ship has to be pointing in the right direction!”. Well, you’d be right in a way. EVE models, fairly accurately, ships and everything else in space as balls. They’re spheres with a fixed radius, and that’s how collision and all that stuff is worked out- just simple sphere interactions. Now, your ship is obviously not ball-shaped. You’ve got those stonking great big engines on there! And they only fire in one (or three, if it’s a Kitsune) direction(s). So logically, the time it takes for those engines to get to where they’re pushing in the right direction must have an impact, right? WRONG! EVE doesn’t model that! EVE assumes that your engines are pushing you towards wherever you want to go, not in the opposite direction to where they appear to be pointing. The model of your ship- that is, the bit you see- is actually only attempting to respond to the actual vector of your ship to give you a smooth experience.
Direction is important to your ship’s vector, but not the model. What you see is, most of the time during warping, crap. If your ship is stationary it has effectively (and this is where most people give up) no direction. It is directionless. The vector’s magnitude is zero and as such any addition to that vector (by your engines) result in that vector instantly snapping to that direction. An analogy, if you will. Let’s say I have a marble. I put it on a flat surface. If it is stationary then it takes me no extra effort to push it to the left or right. If it’s rolling to the left and I want to push it right it takes extra effort, if I want to push it to the left.. it’s already rolling to the left. This is about as good an analogy I can come up with for alignment.
This is why your alignment time as reported by EFT and so on is always correct, no matter where you’re facing. The only thing passive alignment will do is tip off the enemy as to where you’re going to go if you need to run, making the whole practice doubly useless.
I know there’s been a few notes on how passive alignment doesn’t work, but I’ve seen a lot of people talk about this recently and it’s really been bugging me, so there you have it- the definitive rant, compiled from both my own notes and experimental efforts as well as Entity’s post on the topic.
## Why ECM ships don’t need a nerf, and how to fix ECM properly
CCP, you may have spotted, are planning an ECM nerf. Now, I’m biased in this- I fly support. I’m rarely seen in fleet fights because I’m in a buzzard 250km from anyone else, but if I’m there I’m in a Scorpion or a Kitsune. I’ve not gotten a killmail since 2008, for crying out loud. And therein lies my point.
The changes to Scorp, Falcon and Rook assume that ECM pilots just fit ECM as a side benefit of their guns. This is simply not the case in the vast majority of situations. None of the EW ships are particularly strong with a decent quantity of ECM- using racial jammers you need 4 midslots of ECM, and trying to generate a tank out of your remaining mids doesn’t tend to work well. The goal of the ECM pilot is to reduce the damage output of the opposing force by disrupting target locks and removing one or more ships from the battle in terms of actually dealing damage to your team.
Now, there’s one situation and one ship that has received a lot of attention.
Falcons are fairly legendary for having the ability to lock out to ~200km and to jam from that distance with good strength. This is a great asset in fleet fights and smaller engagements like gate camps where having a few distributed ECM platforms around the place to break up hostile fire concentration can swing the battle to the defenders. However, in smaller fights it can lead to ‘problems’ where the hostiles will be set up for close range high DPS situations with a Falcon or two dotted around the battlefield at long range, well out of the attacker’s range. Problems like the attackers not being able to shoot anything.
What is the solution to this, you might ask? Well, CCP’s answer is to turn the Falcon- a paper-thin, untankable (This is a Caldari ship- shields are all about the midslots, which is where your ECM goes. One or the other, chaps) fairly nippy cruiser that can cloak- into a close-range brawler. Otherwise known as ‘primary’. Unless you jam 100% of the targets 100% of the time, you’ve popped already in 99% of situations. Your cloaking advantage is useless, your tank doesn’t exist, your DPS is tiny compared to, say, a HAC.
Then there’s the Rook, which CCP want to turn into a longish-range platform. But wait- drone bay? Why does a long-range platform need a frakking drone bay? It’s enough for one sentry drone, but that’s about all you could concieveably find useful as a Rook pilot. At 80k or thereabouts, you’ll still die nice and fast but you’ll at least be able to do some damage with your one sentry drone and heavy missiles.
No nerf would be complete without planned changes for every good ship, though- the Scorpion gets messed around with too! No more optimal range bonus for the Scorp- it’s getting brawlerfied too. Now, I’ve got a Scorpion set up for W-Space. Strong tank, a little ECM, and cruises for contributing to the longer-range targets DPS-wise. Now, a torpscorp might work well if there were some more hardpoints/highslots added, but your tank would again destroy your ECM. I’ve only got 3 multispecs fitted, along with two SDAs. It’s not a PvP-worthy fitting, by any means, and reduces the Scorp’s usefulness greatly.
Now- enough ranting. How do we fix ECM properly, to give smaller gangs a chance against these obviously overpowered ships? Fix ECCM. Give Remote ECCM a huge boost, either in strength or by making it an area-of-effect module. Of course RECCM is useless- RECCM providers will simply be jammed. Why not make RECCM a shield effect, similar to a heavy interdictor, and a highslot module instead of a midslot module? Pick a ship class that has a grey area in terms of it’s role and choose that as a specialist platform for RECCM. RECCM shielding could give all ships within the bubble a boost to their sensor strength, making them more resistant to ECM.
By making RECCM a more viable option for players and making it a clearly defined fun role to play, ECM gets more interesting, other pilots get jammed less and are happier, and balance can be restored to small gangs with the addition of a RECCM pilot or some extra ECCM modules on snipers. Nothing needs nerfing; the battlefield just needs evening up to give RECCM pilots a chance to swing the battle back to their side.
Edit: Dev update- they’re not going to nerf the Falcon to short range, and are making the Rook the brawler instead. Unfortunately they still seem to think Scorpions can survive at anything less than sniper range, and that they should not be able to fight beyond 140k. What the fuck? | 2019-02-20 17:30:06 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "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": 0.4545908272266388, "perplexity": 1955.9951420100515}, "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-2019-09/segments/1550247495367.60/warc/CC-MAIN-20190220170405-20190220192405-00504.warc.gz"} |
https://tex.stackexchange.com/questions/248581/footruleskip-already-defined-error-when-remove-header-line-abntex | I'm trying to remove the line of the header. Doing some searches, the best answer was \renewcommand{\headrulewidth}{0pt} but when I add the package fancyhdr I get an error saying that the footruleskip command is already defined.
I'm using \pagestyle{myheadings} to remove the Chapter title from the header.
\include{fixos/instituicao/sistemas}
%\renewcommand{\baselinestretch}{1.5}
\usepackage[compact]{titlesec}
\titleformat{\section}{\bfseries}{\thesection\space }{12pt}{}
\titlespacing*{\chapter}{0pt}{0pt}{20pt}
\usepackage{fancyhdr}
\renewcommand{\sfdefault}{\rmdefault}
\begin{document}
.
.
.
the template that Im using https://github.com/guylhermetabosa/TCC/tree/master/modelo-tcc-abntex2 TCC.tex is the main file
• abntex2 uses memoir under the hood and it uses its own mechanism for headers and footers and is incompatible with fancyhdr. – Johannes_B Jun 4 '15 at 14:25
The repository shown seems to be a university template. Documented code is in a language i don't speak, but doing the regular algorithm of making a minimal working example, it boils down to /textual which is the mainmatter in abntex2. The pagestyle is set to abntheadings along with
\makeheadrule{abntheadings}{\textwidth}{\normalrulethickness}. We can see, that the key is to make the width of that line equal to zero point, i.e. having an invisible rule.
\documentclass{abntex2}
\usepackage[alf]{abntex2cite}
\usepackage{blindtext}
\renewcommand{\normalrulethickness}{0pt}
\begin{document}
\textual
\blindtext[10]
\end{document}
As this is a university template adhering to the ABNT, it might not be a good idea to redefine their layout of pages. memoir internally uses normalrulethickness in various occasions, so redefining it might lead to unexpected outcome.
EDIT: Instead of changing the \normalrulethickness, it is possible to modify the abntheadings style: \makeheadrule{abntheadings}{\textwidth}{0pt} in the preamble, by doing so only the abntex2 heading is changed. (Guilherme Z. Santos)
• Care to elaborate/explain? This isn't a very good answer at the moment IMHO. – Paul Gessler Jun 4 '15 at 15:00
• @PaulGessler Because i don't understand any of the stuff that is going on in that template. There is a lot of crazy stuff going on, but it all boils down to setting the normalrulethickness. – Johannes_B Jun 4 '15 at 15:01
• @Johannes_B that comment is already more of an explanation than the answer gave... why not just put it in the answer? – Paul Gessler Jun 4 '15 at 15:03
• Give me a moment @paul :-) – Johannes_B Jun 4 '15 at 15:05
• @ItaloPessoa -- if the university can't or won't accept the output from using this template/document class without the kinds of adjustments they're requesting, then it is not an appropriate tool for the job. this should be taken up with the university authorities. i'm not suggesting that you should undertake this, but if you could make clear to someone with "connections" that it is causing unnecessary difficulties unrelated to reporting your research, then maybe someone else in the future could benefit. (and good luck. i hope everything else goes well.) – barbara beeton Jun 5 '15 at 11:59 | 2021-01-27 13:28:25 | {"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": 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": 0.5099318027496338, "perplexity": 1704.0420744056644}, "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-2021-04/segments/1610704824728.92/warc/CC-MAIN-20210127121330-20210127151330-00414.warc.gz"} |
https://www.physicsforums.com/threads/fluid-mechanics-navier-stokes-flow-around-geometry.721587/ | # Fluid mechanics navier stokes flow around geometry
1. Nov 8, 2013
### joshmccraney
hey pf!
i am studying fluid mechanics and was wondering if any of you are familiar with a flow around some geometry? for example, perhaps a 2-D fluid flowing around a circle?
if so please reply, as i am wondering how to model the navier-stokes equations. i'll be happy to post the equations and my thoughts if you would like me to?
for the record, i am using white's book, and while it is great, i have not seen any examples dealing with 2-D flow around geometry. so far it's been flow in between plates and against a wall.
thanks so much for your support and interset!
josh
2. Nov 10, 2013
### njoysci
3. Nov 10, 2013
The problem is that in general you can't find a solution to the Navier-Stokes equations analytically, especially external flows. There are some that are possible, like laminar flow over a flat plate (the Blasius solution), but these are relatively few and far between.
Except at very low Reynolds number, the 2-D flow around a circle (or cylinder, for example) is one such flow where an analytical solution is not possible for a viscous flow. You have to deal with flow separation and an unsteady wake (for example, try Googling von Kármán vortex street).
4. Nov 10, 2013
### joshmccraney
thanks you guys! hey bonehead, what do we consider "very low Re number"? would it be possible to model, say, syrup sliding over a plate with a slight angle as a low reynolds number, or is this too fast?
5. Nov 10, 2013
That depends on the context. With a cylinder in a viscous flow, very low Reynolds number in regards to whether Stokes flow is valid is typically considered to be $Re \ll 1$. It doesn't really make sense to just ask what constitutes very low Reynolds number in a random situation since you aren't really specifying in that case what physical phenomenon you are hoping to capture or avoid. For example, in Stokes flow, very low Reynolds number represents the region where the assumptions used in deriving it are valid. | 2016-02-06 03:06:15 | {"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": 0.6651249527931213, "perplexity": 502.06786302710145}, "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-2016-07/segments/1454701145751.1/warc/CC-MAIN-20160205193905-00065-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://tcs.nju.edu.cn/wiki/index.php/%E9%9A%8F%E6%9C%BA%E7%AE%97%E6%B3%95_(Fall_2011)/Card_Shuffling | # Riffle Shuffle
The following is the Gilbert-Shannon-Reeds model of riffle shuffle introduced by Gilbert and Shannon in 1955 and independently by Reeds in 1985.
Riffle Shuffle (Gilbert-Shannon 1955; Reeds 1985) Given a deck of ${\displaystyle n}$ cards, at each round, do as follows. Split the original deck into two decks according to the binomial distribution ${\displaystyle \mathrm {Bin} (n,1/2)}$. Cut off the first ${\displaystyle k}$ cards with probability ${\displaystyle {\frac {n \choose k}{2^{n}}}}$, put into the left deck, and put the rest ${\displaystyle n-k}$ cards into the right deck. Drop cards in sequence, where the next card comes from one of the two decks with probability proportional to the size of the deck at that time. Suppose at a step there are ${\displaystyle L}$ cards in the left deck and ${\displaystyle R}$ cards in the right deck. A card is dropped from left with probability ${\displaystyle {\frac {L}{L+R}}}$, and from right otherwise.
The second step actually samples a uniform interleaving of left and right deck. The magician and mathematician Diaconis in 1988 found evidence showing that this mathematical model reasonably approximate the riffle shuffling acted by human.
The above model is equivalent to the following model of inverse riffle shuffle.
Inverse Riffle Shuffle Label each card with a bit from ${\displaystyle \{0,1\}}$ uniformly and independently at random. Place all cards labeled with 0 above all cards labeled with 1, keeping the cards with the same label in the same the relative order as before.
The process of inverse riffle shuffle is depicted as follows.
For each card, the random bits used for labeling the card compose a binary code ${\displaystyle x}$, with the ${\displaystyle t}$-th bit (from lower to higher radix) ${\displaystyle x_{t}}$ being the random bit chosen in round ${\displaystyle t}$ to label the card.
Lemma After every round of the inverse riffle shuffle, the cards are sorted in non-decreasing order of their corresponding binary codes.
Proof.
By induction. After the first round, all ${\displaystyle 0}$ are above all ${\displaystyle 1}$s. Assume that the hypothesis is true for the first ${\displaystyle t-1}$ rounds. After the ${\displaystyle t}$-th round, all cards with the highest coordinate ${\displaystyle x_{t}=0}$ are above all cards with ${\displaystyle x_{t}=1}$, and by the definition of inverse riffle shuffle, cards with the same highest bit ${\displaystyle x_{t}}$ are kept in the same relative order as they are after the ${\displaystyle (t-1)}$-th round of the shuffling, which due to the hypothesis, means that the cards with the same highest bit ${\displaystyle x_{t}}$ are sorted in the non-decreasing order of ${\displaystyle x_{t-1}x_{t-2}\cdots x_{1}}$. By the definition of binary representation, all cards are sorted.
${\displaystyle \square }$
# Rapid Mixing of Riffle Shuffle by Coupling
Theorem The mixing time of inverse riffle shuffle of a deck of ${\displaystyle n}$ cards is ${\displaystyle \tau _{\mathrm {mix} }=O(\log n)}$.
Consider two instance of shuffling, started with two decks, each in an arbitrary order. We couple the two shuffling processes by coupling their random choices of labelings:
• In each round, we label the same cards (same content, not the same order) in the two decks with the same random bit.
Formally, for each ${\displaystyle i\in [n]}$, we uniformly and independently choose a random ${\displaystyle b_{i}\in \{0,1\}}$ as the label of the card ${\displaystyle i}$ in each of the two decks in that round. It is easy to see that, each of the two shuffling processes viewed individually follows exactly the rules of inverse riffle shuffle.
Lemma With the above coupling rule, started with arbitrary two decks of cards, the two decks are the same once all cards in the first (or the second) deck have distinct labels.
Proof.
By the definition of the coupling rule, any two identical cards in the two decks always have the same label, since in every round they choose the same random bit. And we have shown that after each round, the labels are sorted in non-decreasing order. Thus, we can conclude: after each round, the sequences of the labels of the two decks are identical (and sorted in non-decreasing order). Therefore, when all cards in the first deck have distinct labels, the second deck have the same set of distinct labels, and the cards are sorted increasingly in their labels in the respective decks. Since any two identical cards have the same labels, the cards in the two decks must be in the same order.
${\displaystyle \square }$
We denote by ${\displaystyle T}$ the random variable representing the time that in an inverse riffle shuffle, all cards in the deck have distinct labels. Due to the coupling lemma for Markov chains,
${\displaystyle \Delta (t)\leq \Pr[T\geq t]}$.
This bound can be deduced from the birthday problem. After ${\displaystyle t}$ round, each of the ${\displaystyle n}$ cards is randomly assigned one of the ${\displaystyle 2^{t}}$ possible labels. We know that for some ${\displaystyle 2^{t}=O(n^{2})}$, i.e. ${\displaystyle t=2\log n+O(1)}$, the probability that no two cards have the same label is ${\displaystyle >1-1/2\mathrm {e} }$, i.e. ${\displaystyle \Delta (t)<1/2\mathrm {e} }$, thus
${\displaystyle \tau _{\mathrm {mix} }\leq 2\log n+O(1)}$.
Note that the state space is the set of all permutations, which is of size ${\displaystyle n!}$, whose logarithm is still as large as ${\displaystyle \log(n!)=O(n\ln n)}$ due to Stirling's approximation. Thus the riffle shuffle is mixing extremely fast.
This estimation of mixing time is fairly tight. For the case that ${\displaystyle n=52}$, the real mixing rate is as follows.
${\displaystyle t}$ 1 2 3 4 5 6 7 8 9 ${\displaystyle \Delta (t)}$ 1 1 1 1 0.924 0.614 0.334 0.167 0.003 | 2018-12-19 04:09:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 45, "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": 0.7939617037773132, "perplexity": 293.6580050117454}, "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-2018-51/segments/1544376830479.82/warc/CC-MAIN-20181219025453-20181219051453-00564.warc.gz"} |
https://chemistry.stackexchange.com/questions/50615/testing-for-nitrate-ions-with-nano3-in-devardas-test | # Testing for nitrate ions with NaNO3 in Devarda's test
When we use Devarda's test on $\ce{NaNO3}$ solution (we add $\ce{NaOH}$ and $\ce{Al}$ powder, heat the mixture, and produce ammonia), what is the equation of the process?
I understand that $\ce{Al}$ reduces $\ce{NO3-}$ to $\ce{NH4+}$ then $\ce{NH4+}$ and $\ce{OH-}$ have a reversible reaction to produce $\ce{NH3}$, and water, but I'm not sure how to summarize this in an equation. In particular, what happens to the aluminium?
• Aluminium oxidizes, as you may guess. To find its final form, keep in mind that $\ce{Al}$ is amphoteric. – Ivan Neretin May 3 '16 at 19:22
$$\ce{3NO3- + 8Al + 5OH- + 18H2O ->3NH3 ^ + 8[Al(OH)4]-}$$ | 2019-10-20 16:14:40 | {"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": 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": 0.8420546054840088, "perplexity": 976.8403901313104}, "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-2019-43/segments/1570986717235.56/warc/CC-MAIN-20191020160500-20191020184000-00140.warc.gz"} |
https://chemistry.stackexchange.com/questions/98864/why-is-this-redox-reaction-possible | Why is this redox reaction possible?
I have the redox reaction $\ce{N_2H_4 {(g)} + N_2O_4 {(g)} -> N_2 {(g)} + H_2O {(g)}}$.
In $\ce{N_2O_4 {(g)}}$, the oxidation state of nitrogen is $+4$. In $\ce{N_2 {(g)}}$, the oxidation state of $\ce{N}$ is $0$. Thus, $\ce{N_2O_4 {(g)}}$ is reduced.
In $\ce{N_2H_4 {(g)}}$, the oxidation state of nitrogen is $+2$, because the oxidation state of hydrogen when bonded to nonmetals is $-1$. In $\ce{N_2 {(g)}}$, the oxidation state of $\ce{N}$ is $0$. Thus, $\ce{N_2H_4 {(g)}}$ is also reduced.
Clearly, both $\ce{N_2O_4 {(g)}}$ and $\ce{N_2H_4 {(g)}}$ cannot both be reduced. What is wrong with this?
• Well first you need to balance the equation. $\ce{2N2H4_{(g)} + N2O4_{(g)} -> 3N2{(g)} + 4H2O{(g)}}$. Then what are the half cell reactions? – MaxW Jun 29 '18 at 2:02
• I'm not sure, because in both reactants, the nitrogen atom has a positive oxidation state. They can't both be reduced. – coder Jun 29 '18 at 2:15
• Again stop thinking about "oxidation states" and think half cells. Add both $\ce{H+ \text{and} e-}$ to one side to create a balanced equation. Start with hydrazine. $\ce{N2H4 -> \text{?}}$ – MaxW Jun 29 '18 at 2:20
• On an unrelated note, what oxidation state would you assign to F in HF? – Ivan Neretin Jun 29 '18 at 5:07
• The oxidation state of hydrogen is $+1$ when bonded to non-metals and not $-1$. – PolarBear Jul 14 '18 at 4:14 | 2019-04-19 01:00:24 | {"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": 0.6252754330635071, "perplexity": 672.6331024475428}, "config": {"markdown_headings": false, "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-2019-18/segments/1555578526923.39/warc/CC-MAIN-20190419001419-20190419023419-00332.warc.gz"} |
https://electronicsreference.com/module1/electric-charge/ | # Electric Charge
Electric charge is the property of matter that causes any matter (objects or particles) with charge to experience a force exerted by other charged matter.
Electric charge is one of the most important concepts in physics, chemistry, and electrical engineering. It is responsible for the structure of atoms and of bulk materials. Electric charge both holds atoms together and also prevents atoms from entering each others’ spaces.
You have probably never thought to ask why you don’t sink into the floor when you walk, or why you can use an object without your hand going through it, but the answer is electric charge. The electrons in your body repels the electrons in the objects that you touch.
Electric charge is fundamental to every aspect of electricity and electronics. By understanding electric charge, we can continue to build on our understanding of electricity and begin to explore the fundamental interactions that will allow us to learn about electrical current at a deeper level.
## The Property of Electric Charge
The main action of electric charge is that like-charged objects repel (called electrostatic repulsion) and opposite-charged objects attract (called electrostatic attraction).
Charge can either be positive or negative. When we say that two objects are ‘like-charged’, we mean that the net charge of the two objects have the same sign; either they are both negatively charged or both positively charged. When we say that two objects are ‘opposite-charged’, we mean that one object has a net positive charge and the other object has a net negative charge.
### Electrostatic Repulsion
Two electrons (or protons) have the same charge, so they repel each other:
This is called electrostatic repulsion. It occurs with all ‘like charged’ objects. For instance, two ions with negative charge (i.e. atoms that have gained an electron) will repel each other. The same will happen for two ions with positive charge.
### Electrostatic Attraction
Opposites attract.
Protons and electrons have the opposite charge, so they attract each other.
This is called electrostatic attraction. It occurs between any two objects that are oppositely charged. An ion with a positive charge will be attracted to an ion with a negative charge.
Protons have a charge of (positive) + 1.602 x 10-19 coulomb, and electrons have a charge of (negative) -1.602 x 10-19 coulomb. The charges have the opposite sign, so they attract one another. A coulomb is just the SI unit for charge (i.e. how we measure and compare quantities of charge). In terms of physical size, protons are much bigger than electrons (i.e. protons have more mass than electrons), but the force exerted by each is the same. In other words, electrons have a much higher charge to mass ratio than protons.
The force pulling electrons and protons together depends on the charge of the particle, not the mass. However, because the mass of the proton is much greater, it will move less distance than the electron when experiencing the same force.
## Electric Charge is Deceptively Strong
Electric charge exerts a deceptively strong force. The technical name for the attraction or repulsion between charged objects is called the electromagnetic force.
### How strong is the electromagnetic force?
For comparison, the electromagnetic force is 1036 times stronger than gravity.
It’s actually really important that the force of electric is so strong, otherwise electrons could be pulled away from atoms by gravity, and atoms would collapse into each other because the repulsion between electrons would be too weak to prevent it.
### If Electric Charge is So Strong, Why Does it Seem Weak?
The strength of the force exerted by electric charge seems weak because the typical atom is electrically neutral. The attraction between protons and electrons help the atom to stick together, and the repulsion between electrons does help prevent atoms from collapsing into each other.
But at a large distance away from an atom, the positive and negative charges within it balance out and exert a net zero force. So while there’s a low of charged particles, like protons and electrons in the earth, the net electromagnetic force on our bodies is zero. This is due to to the fact that charge can be either positive or negative, and they cancel each other out.
In contrast, gravity doesn’t have a ‘negative gravity’ counterpart to cancel out its’ effect. The gravity exerted by a single particle is very small, but it adds on to the gravity exerted by every other particle that it is near. So every atom within the earth contributes to the overall force of gravity, which feels really strong. Yet the electromagnetic force is still strong enough to allow us to walk around the earth’s surface without being pulled down into the center of the earth.
## SI Unit of Electric Charge
The term SI stands for Système international, which is a worldwide system of standardized measurements. The SI system basically extends the metric system to lots of measurable properties like force, pressure, time, voltage, and electric charge. For reference, you can find a table of standard unit measurements here.
The SI unit for charge is the Coulomb, with an abbreviation of ‘C’. For example, a charge of 5 Coulombs is written as ‘5 C’.
An interesting aspect of electric charge is that because the charge of an electron (or proton) is an inherent property of particles in nature, it makes sense to define measurement units using this naturally occurring baseline.
Thus the unit of Coulomb is defined by the charge of a certain number of electrons. Originally, the Coulomb was defined as the amount of charge carried by 1 ampere of current in one second, which is a lot of electrons. This was sue to the fact that current was discovered before charge.
For this reason, the official exact definition of one Coulomb is the charge of 6, 241, 509, 074, 460, 762, 607.776 electrons.
Of course, a perfect Coulomb of charge is impossible to achieve because it includes a decimal of .776 elementary charges, and neither electrons nor protons can be split into a fraction.
However, this doesn’t stop the Coulomb from doing its’ job as a unit of measurement.
### Elementary Charge: The Electric Charge of a Single Proton
Measured in the SI Unit of the Coulomb, a single proton has a charge of about 1.6 x 10-19 Coulomb. This is called an elementary charge, which is abbreviated by the letter ‘e‘:
e = 1.60217662 \times 10^{-19} C
A single proton has a charge of e, and a single electron has a charge of –e.
Keep in mind, charge is not a unit that most humans are very familiar with. In terms of electricity, it is much easier for use to understand voltage and current. For instance, a single AA battery will discharge about 5000 Coulombs over its life, whereas a lightning bolt may only transmit a handful! Yet, a AA battery is only 1.5 volts whereas a lightning bolt can strike at 1 Billion volts. We understand that the lightning bolt is much more dangerous than the battery because it’s voltage and current are much greater than that produced by the battery. The total charge doesn’t seem to matter much, but it is important as it determines the battery’s capacity.
### Charge is Quantized
Electric charge only occurs in integer multiples of e; fractional charges are never found.
This is called charge quantization.
### Total Charge vs. Elementary Charge
When we work with charge in equations, we typically use the abbreviation ‘q‘. For example, the total charge of 5 electrons would be written as:
q = -5e
If we wanted to determine how many Coulombs this is, we would write the following:
q = -5e = -5(1.6\times 10^{-19} C) = 8 \times 10^{-19} C
## How Electric Charge Relates to Electricity
Electric charge is central to all forms of electricity and all systems that use electricity. When we investigate different aspects of electricity or electric circuits, we can always understand them in terms of charge. Some of the most important properties are voltage and current.
Voltage is the amount of energy required to move an elementary charge between two points. It is measured in units of Joules (energy) per Coulomb (charge).
Current is the rate of flow of electric charge. It is measured in units of Coulombs (charge) per second (time). The more electric charge (i.e. electrons) moves per second, the higher the current.
Module 1 – Introduction to Electrical Theory:
Lesson 0: Module 1 Introduction
Lesson 1: What is Electricity?
Lesson 2: History of Electricity
Lesson 4: Of Atoms and Electrons
Lesson 5: Electrical Properties of Materials
Lesson 6: Properties of Electric Charge
Lesson 7: Electric Field | 2023-02-08 09:52:44 | {"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": 0.7271559238433838, "perplexity": 486.8337228600517}, "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/1674764500758.20/warc/CC-MAIN-20230208092053-20230208122053-00717.warc.gz"} |
https://math.stackexchange.com/questions/1560496/how-do-i-find-all-of-the-orbits-and-stabilisers-of-x | How do I find all of the orbits and stabilisers of X?
Consider $D_{10}$ The group of symmetries of the regular pentagon. Let $\sigma= (12345)$ and $\tau =(13)(45)$ being rotation by $72^{\circ}$ and reflection (with 2 being the fixed point) respectively. Therefore $D_{10} = ({1, \sigma, \sigma^2, \sigma^3, \sigma^4, \tau, \sigma\tau, \sigma^2\tau, \sigma^3\tau, \sigma^4\tau})$.
Let $D_{10}$ act on the set $X =\left \{ 1,2,3,4,5 \right \}$ of vertices by permutation.
How do I find all the orbits of $X$?
How do I find all the stabilisers of $X$?
I'd like to thank you in advance for any help/guidance you can give me.
• Consider the subgroup of $D_{10}$ consisting of the rotations. Is there any pair of vertices $v,w\in X$ such that $v$ cannot be rotated to $w$? – Bungo Dec 5 '15 at 2:07
• P.S. $360/5 = 72$, not $108$ :-) – Bungo Dec 5 '15 at 2:11
• @Bungo thanks man, I was being silly. I'm still not too sure. Surely $v$ can be rotated to any $w$? – Geometry Dec 5 '15 at 2:16
• Yes, exactly. We can move any vertex to any other vertex by some rotation. So there is just one orbit, namely all of $X$. Now, how does the size of the orbit relate to the size of the stabilizer of a particular vertex in the orbit? – Bungo Dec 5 '15 at 2:18
• @Bungo I'm really not sure :( – Geometry Dec 5 '15 at 2:25
Summarizing the comments and discussion in chat:
Given any vertices $v,w \in X$, there is a rotation that takes $v$ to $w$, so there is just one orbit, namely all of $X$.
Let's denote the group by $G = D_{10}$.
Now choose any vertex $v \in X$. The orbit containing $v$ is all of $X$, as noted above. This orbit has size $|X| = 5$. Let us denote the stabilizer of $v$ by $G_v$. By the orbit-stabilizer theorem, the index of $G_v$ equals the size of the orbit containing $v$. In other words, $|G:G_v| = |X| = 5$. Since $|G:G_v| = |G| / |G_v|$, this means that $|G| / |G_V| = 5$. Since $|G| = |D_{10}| = 10$, it follows that $|G_v| = 2$.
Now $G_v$ is a group, so it contains the identity. (This makes sense since clearly the identity stabilizes every vertex.) Since $|G_v| = 2$, there is one additional element of $G$ which stabilizes $v$. None of the nontrivial rotations stabilize any vertex, so it must be one of the "flips". In particular, if we draw a line from the vertex $v$ to the midpoint of the opposite side, and flip the pentagon along that axis, then $v$ is fixed (stabilized) by that flip.
To summarize, each vertex of the pentagon is fixed by two elements of $D_{10}$: one is the identity, and the other is the flip along the axis which connects the vertex to the opposite side.
I'll leave it to you to work out which element of $D_{10}$ (in terms of $\tau$ and $\sigma$) corresponds to the flip that fixes each vertex. | 2019-07-24 08:25:48 | {"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": 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": 0.9422096610069275, "perplexity": 156.6438393519449}, "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-2019-30/segments/1563195532251.99/warc/CC-MAIN-20190724082321-20190724104321-00284.warc.gz"} |