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https://cs.stackexchange.com/questions/108967/do-all-recursive-problems-have-optimal-substructure
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# Do all recursive problems have optimal substructure?
I am reading about dynamic programming and I understand the overlapping subproblem requirement but not sure why optimal substructure is explicitly stated. Are there problems that can be solved recursively that do not have an optimal substructure?
• A simple simplified answer could be "problems that can be solved recursively should have recursive substructures". Optimal substructure? That seems off the point. – Apass.Jack May 5 at 4:38
• @Apass.Jack resources.saylor.org/wwwresources/archived/site/wp-content/… In the overview section, near the end of the paragraph, it says " Likewise, in computer science, a problem that can be broken down recursively is said to have optimal substructure" – Sazz May 5 at 5:31
• Where is that article published? No author is listed. No date is given. It looks like its author is pretty familiar with dynamic programming. However, the article itself might not be the best reference. May I ask why you are reading that article? Are you learning dynamic programming for the first time or you are experienced in dynamic programming? – Apass.Jack May 5 at 12:55
• The article is from Wikipedia. Go one step further and have a look at en.wikipedia.org/wiki/Optimal_substructure. The idea is that the optimal solution of a subproblem can be extended to the optimal solution of the problem. This is not always the case. – Marcus Ritt May 5 at 15:01
• @Apass.Jack Partially, but wanted to point out Sazz that I believe that Wikipedia's article on optimal substructure responds his question. On the article: it seems to be and old version, see the source on the last page. – Marcus Ritt May 5 at 19:23
Every problem can be solved recursively. Recall first that every problem can be solved using an algorithm which uses a WHILE loop and conditionals. Indeed, the operation of a CPU can be described in this way. We can adapt this solution to a (tail) recursive procedure as follows.
The parameters are the instruction pointer, the content of the registers, and the content of memory. The procedure performs the instruction that the instruction pointer is pointing at, modifying the instruction pointer, the registers, and the memory. Then (unless the instruction was HALT), it calls itself recursively with the updated instruction pointer, register contents, and memory contents.
When solving an optimization problem recursively, optimal substructure is the requirement that the optimal solution of a problem can be obtained by extending the optimal solution of a subproblem (see for example, Cormen et al. 3ed, ch. 15.3).
Consider for example the problem of finding a simple shortest $$s$$-$$t$$ path in a directed graph with non-negative distances. Such a path must visit a predecessor $$u$$ of $$t$$. So if you are able to recursively find the shortest $$s$$-$$u$$ path for all predecessors $$u$$ of $$t$$, you can find a shortest $$s$$-$$t$$ path. In this case you have an optimal substructure.
Contrast this with finding a simple longest $$s$$-$$t$$ path. Such a path must also visit a predecessor $$u$$ of $$t$$. However, it isn't true that the longest $$s$$-$$t$$ path can be obtained by extending some longest $$s$$-$$u$$ path for a predecessor $$u$$ of $$t$$, since such a path may already visit $$t$$, and so its extension may not be a simple path, and you cannot remove the resulting cycle to make the path simple, since this can make it shorter. In this case you don't have an optimal substructure.
The longest $$s$$-$$t$$ path can be found by choosing a different kind of subproblem that has optimal substructure. If you define $$C(s,t,V)$$ as the longest path from $$s$$ to $$t$$ without visiting the vertices in $$V$$, you get \begin{align*} C(s,t,V)= \begin{cases} \max_{u\in N^{-}(t)\setminus V} C(s,u,V\cup\{t\})+d_{ut}, & \text{if s\neq t and N^{-}(t)\setminus V\neq \emptyset},\\ -\infty, & \text{if s\neq t and N^{-}(t)\setminus V=\emptyset},\\ 0, & \text{if s=t}, \end{cases} \end{align*} where $$N^-(t)$$ is the set of predecessors of vertex $$t$$, and $$d_{uv}$$ is the distance between $$u$$ and $$v$$.
The article you were reading might not be not good enough for educational purpose, especially for a beginner in dynamic programming. It is not accurate enough, either.
It is apparent, as noticed by Marcus Ritt that article copies from the Wikipedia article on dynamic programming, a lot.
However, that article is not up to date with the following statement at the latest version of Wikipedia,
Likewise, in computer science, if a problem can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have optimal substructure.
We can compare the statement above with the following statement in the article, where "recursively" instead of "optimally" is used to define optimal substructure, which is not correct.
Likewise, in computer science, a problem that can be broken down recursively is said to have optimal substructure."
Are there problems that can be solved recursively that do not have an optimal substructure?
This question is more or less off the point in the sense that being recursive is not related to being optimal. Not directly.
The truism would be that problems that can be solved recursively should have recursive substructures. Likewise, if an optimization problem can be solved by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have optimal substructure.
If we have to answer that question head-on, then the answer is, of course, yes and plenty of them. We can compute Fibonacci numbers recursively, but there is no apparent optimal substructure in Fibonacci numbers. We can compute recursively the factorials which does not have substructure that can be called optimal reasonably.
In fact, Yuval's answer presents a common view that explained why most programs can be considered or adapted as solving problems recursively.
It might have been better had you followed an excellent textbook, such as the most popular textbook Introduction to Algorithm by CLRS, in case you have not read it yet. Its chapter 15, Dynamic Programming explains the basic concepts and techniques about dynamic programming clearly. Section 3 of that chapter, elements of dynamic programming, elaborates on the components of dynamic programming, "Optimal substructure", "Overlapping subproblems", "Reconstructing an optimal solution" and "Memoization".
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2019-09-18 11:23:20
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http://cvgmt.sns.it/paper/2245/
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# Spectral optimization problems for potentials and measures
created by buttazzo on 03 Oct 2013
modified by velichkov on 27 Sep 2015
[BibTeX]
Published Paper
Inserted: 3 oct 2013
Last Updated: 27 sep 2015
Journal: SIMA
Year: 2014
Abstract:
In the present paper we consider spectral optimization problems involving the Schr\"odinger operator $-\Delta +\mu$ on ${\bf R}^d$, the prototype being the minimization of the $k$ the eigenvalue $\lambda_k(\mu)$. Here $\mu$ may be a capacitary measure with prescribed torsional rigidity (like in the Kohler-Jobin problem) or a classical nonnegative potential $V$ which satisfies the integral constraint $\int V^{-p}dx \le m$ with $0<p<1$. We prove the existence of global solutions in ${\bf R}^d$ and that the optimal potentials or measures are equal to $+\infty$ outside a compact set.
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2017-07-22 04:35:20
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https://plainmath.net/7016/magazine-consumer-reports-publishes-information-automobile-variables
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# The magazine Consumer Reports publishes information on automobile gas mileage and variables that affect gas mileage. In one issue, data on gas mileage
The magazine Consumer Reports publishes information on automobile gas mileage and variables that affect gas mileage. In one issue, data on gas mileage (in miles per gallon) and engine displacement (in liters) were published for 121 vehicles. a) Obtain a scatterplot for the data. b) Decide whether finding a regression line for the data is reasonable. If so, then also do parts (c)-(f). c) Determine and interpret the regression equation for the data. d) Identify potential outliers and influential observations. e) In case a potential outlier is present, remove it and discuss the effect. f) In case a potential influential observation is present, remove it and discuss the effect.
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Given: a) DISP is on the horizontal axis and MPG is on the vertical axis. b) It is reasonable to find a regression line for the data if there is no strong curvature present in the scatterplot. We note that there is strong curvature in the scatterplot of part (a) and thus it is not reasonable to find a regression line for the data. c) Not applicable, because it is not reasonable to find a regression line by part (b). d) Not applicable, because it is not reasonable to find a regression line by part (b). e) Not applicable, because it is not reasonable to find a regression line by part (b). f) Not applicable, because it is not reasonable to find a regression line by part (b).
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2022-06-28 04:01:46
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https://www.zigya.com/study/book?class=11&board=bsem&subject=Physics&book=Physics+Part+I&chapter=Motion+in+A+Plane&q_type=&q_topic=Motion+In+A+Plane&q_category=&question_id=PHEN11039786
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 Two trains 120 m and 80 m in length are running in opposite directions with velocities 42 km/hr and 30 km/hr. In what time they will completely cross each other? from Physics Motion in A Plane Class 11 Manipur Board
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Class 10 Class 12
Two trains 120 m and 80 m in length are running in opposite directions with velocities 42 km/hr and 30 km/hr. In what time they will completely cross each other?
Relative velocity of one train w.r.to second is,Â
= 42 - (-30)
= 72 km/hr = 20 m/s
Total distance to be travelled = 120 +80 = 200 m
Time taken, t =Â Â
In 10 sec, the two trains will completely cross each other.Â
156 Views
Give three examples of vector quantities.
Force, impulse and momentum.
865 Views
Give three examples of scalar quantities.
Mass, temperature and energy
769 Views
What is a scalar quantity?
A physical quantity that requires only magnitude for its complete specification is called a scalar quantity.
1212 Views
What are the basic characteristics that a quantity must possess so that it may be a vector quantity?
A quantity must possess the direction and must follow the vector axioms. Any quantity that follows the vector axioms are classified as vectors.Â
814 Views
What is a vector quantity?
A physical quantity that requires direction along with magnitude, for its complete specification is called a vector quantity.
835 Views
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2018-06-22 15:14:10
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http://jdh.hamkins.org/tag/psa/
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# Is the dream solution of the continuum hypothesis attainable?
• J. D. Hamkins, “Is the dream solution of the continuum hypothesis attainable?,” Notre Dame J. Form. Log., vol. 56, iss. 1, pp. 135-145, 2015.
@article {Hamkins2015:IsTheDreamSolutionToTheContinuumHypothesisAttainable,
AUTHOR = {Hamkins, Joel David},
TITLE = {Is the dream solution of the continuum hypothesis attainable?},
JOURNAL = {Notre Dame J. Form. Log.},
FJOURNAL = {Notre Dame Journal of Formal Logic},
VOLUME = {56},
YEAR = {2015},
NUMBER = {1},
PAGES = {135--145},
ISSN = {0029-4527},
MRCLASS = {03E50},
MRNUMBER = {3326592},
MRREVIEWER = {Marek Balcerzak},
DOI = {10.1215/00294527-2835047},
eprint = {1203.4026},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/dream-solution-of-ch},
}
Many set theorists yearn for a definitive solution of the continuum problem, what I call a dream solution, one by which we settle the continuum hypothesis (CH) on the basis of a new fundamental principle of set theory, a missing axiom, widely regarded as true, which determines the truth value of CH. In an earlier article, I have described the dream solution template as proceeding in two steps: first, one introduces the new set-theoretic principle, considered obviously true for sets in the same way that many mathematicians find the axiom of choice or the axiom of replacement to be true; and second, one proves the CH or its negation from this new axiom and the other axioms of set theory. Such a situation would resemble Zermelo’s proof of the ponderous well-order principle on the basis of the comparatively natural axiom of choice and the other Zermelo axioms. If achieved, a dream solution to the continuum problem would be remarkable, a cause for celebration.
In this article, however, I argue that a dream solution of CH has become impossible to achieve. Specifically, what I claim is that our extensive experience in the set-theoretic worlds in which CH is true and others in which CH is false prevents us from looking upon any statement settling CH as being obviously true. We simply have had too much experience by now with the contrary situation. Even if set theorists initially find a proposed new principle to be a natural, obvious truth, nevertheless once it is learned that the principle settles CH, then this preliminary judgement will evaporate in the face of deep experience with the contrary, and set-theorists will look upon the statement merely as an intriguing generalization or curious formulation of CH or $\neg$CH, rather than as a new fundamental truth. In short, success in the second step of the dream solution will inevitably undermine success in the first step.
This article is based upon an argument I gave during the course of a three-lecture tutorial on set-theoretic geology at the summer school Set Theory and Higher-Order Logic: Foundational Issues and Mathematical Development, at the University of London, Birkbeck in August 2011. Much of the article is adapted from and expands upon the corresponding section of material in my article The set-theoretic multiverse.
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2017-09-19 11:29:29
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http://dakota.tensen.net/wiki/index.php/LaTeX
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# LaTeX
Typesetting mathematics with LaTeX can be frustrating. This page contains commonly needed, but surprisingly hard to find, LaTeX commands and procedures.
## Starring equations
Use the \tag command to change what \ref returns for a given equation.
\begin{align}
e^{\pi i} + 1 = 0
\tag{$*$}
\ref{euler}
\end{align}
## LaTeX on wordpress.com
LaTeX is enabled by default on wordpress.com. Use $latex to begin mathmode and$ to end it.
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2013-06-19 05:54:53
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https://docs.mantidproject.org/nightly/fitting/fitfunctions/Gaussian.html
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$$\renewcommand\AA{\unicode{x212B}}$$
# Gaussian¶
## Description¶
A Gaussian function (also referred to as a normal distribution) is defined as:
$\mbox{Height}*\exp \left( -0.5*\frac{(x-\mbox{PeakCentre})^2}{\mbox{Sigma}^2} \right)$
where
• Height - height of peak
• PeakCentre - centre of peak
• Sigma - Gaussian width parameter
Note that the FWHM (Full Width Half Maximum) of a Gaussian equals $$2\sqrt{2\ln 2}*\mbox{Sigma}$$.
The integrated peak intensity for the Gaussian is given by $$\mbox{height} * \mbox{sigma} * \sqrt{2\pi}$$.
The uncertainty for the intensity is: $$\mbox{intensity} * \sqrt{\left(\frac{\delta \mbox{height}}{\mbox{height}}\right)^2 + \left(\frac{\delta \mbox{sigma}}{\mbox{sigma}}\right)^2}$$.
The figure below illustrate this symmetric peakshape function fitted to a TOF peak:
## Properties (fitting parameters)¶
Name
Default
Description
Height
0.0
Height of peak
PeakCentre
0.0
Centre of peak
Sigma
0.0
Width parameter
Categories: FitFunctions | Peak | Muon\MuonModelling
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2023-03-27 14:40:47
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https://www.jobilize.com/online/course/11-1-temperature-by-openstax-temperature-and-heat?qcr=www.quizover.com&page=1
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# 11.1 Temperature (Page 2/14)
Page 2 / 14
## Temperature scales
Thermometers are used to measure temperature according to well-defined scales of measurement, which use pre-defined reference points to help compare quantities. The three most common temperature scales are the Fahrenheit, Celsius, and Kelvin scales. A temperature scale can be created by identifying two easily reproducible temperatures. The freezing and boiling temperatures of water at standard atmospheric pressure are commonly used.
The Celsius scale (which replaced the slightly different centigrade scale) has the freezing point of water at $0\text{º}\text{C}$ and the boiling point at $\text{100}\text{º}\text{C}$ . Its unit is the degree Celsius $\left(\text{º}\text{C}\right)$ . On the Fahrenheit scale (still the most frequently used in the United States), the freezing point of water is at $\text{32}\text{º}\text{F}$ and the boiling point is at $\text{212}\text{º}\text{F}$ . The unit of temperature on this scale is the degree Fahrenheit $\left(\text{º}\text{F}\right)$ . Note that a temperature difference of one degree Celsius is greater than a temperature difference of one degree Fahrenheit. Only 100 Celsius degrees span the same range as 180 Fahrenheit degrees, thus one degree on the Celsius scale is 1.8 times larger than one degree on the Fahrenheit scale $\text{180}/\text{100}=9/5\text{.}$
The Kelvin scale is the temperature scale that is commonly used in science. It is an absolute temperature scale defined to have 0 K at the lowest possible temperature, called absolute zero . The official temperature unit on this scale is the kelvin , which is abbreviated K, and is not accompanied by a degree sign. The freezing and boiling points of water are 273.15 K and 373.15 K, respectively. Thus, the magnitude of temperature differences is the same in units of kelvins and degrees Celsius. Unlike other temperature scales, the Kelvin scale is an absolute scale. It is used extensively in scientific work because a number of physical quantities, such as the volume of an ideal gas, are directly related to absolute temperature. The kelvin is the SI unit used in scientific work.
price elasticity of demand is the degree of responsiveness of a quantity demanded to the change in price of the commodity in question.
what is the importance of learning economics?
it helps to make the correct choice
it helps firm to produce products that will bring more profit
the difference between needs and wants
needs are things that we basically can't live without wants are just luxury things
Thelma
needs are things without them we can't live but want are things without we can live
KP
what is education
KP
it's a process in which we give or receiving methodical instructions
Thelma
what is mixed economy
Amex
who are u?
Lamine
haha
Cleaford
scarm
nura
what it this
Cleaford
hi y'all
Dope
how does group chat help y'all 🤔
Dope
hi y'all
Dope
how does group chat help y'all 🤔
Dope
how does group chat help y'all 🤔
Dope
to learn from one another
Lamine
oh okay
Dope
😟
Creative
Yes
Lamine
what is type of economic
how to understand basics of economics
what is demand schedle
When you make a Scedule of the demand you made
Rodeen
What is macroeconomics
It's one of the two branches of Economics that deal with the aggregate economy.
Mayen
it's about inflation, occupation, gdp and so on
alberto
What is differences between Microeconomics and Macroeconomic?
Bethrand
microeconomics focuses on the action of individual agents in the economy such as businesses, workers and household. while macroeconomics looks at the economy as a whole. it focuses on broad issues in the economy such as government deficit, economy growth, levels of exports and imports, and
Thelma
inflationary increase in prices
Thelma
a price floor of 24 imposed
monopolistic competition
yap
nura
any one there to answer my question
Fixed Costs per week Variable Costs per bear Rent & Rates of Factory Hire & machines Heating & Lighting Repayment of Bank Loan K100.00 K45.00 K5.00 K50.00 Materials Foam Wages K6.00 K1.00 K1.00 Total K200.00 K8.00
Richard
one of the scarce resources that constrain our behaviour is time. each of us has only 24 hours in a day. how do you go about allocating your time in a given day among completing alternatives? once you choose a most important use of time. why do you not spend all your time to it. use the notion of op
mohsina mala..Bangla app hobe na
mani Baba. First learn the spelling of Economics
Economics- The study of how people use their limited resources to tey and satisfy unlimited wants.
Kelly
hmmm
Mani
etar bangla apps hobe na?
Mohsina
Difference between extinct and extici spicies
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
How we can toraidal magnetic field
How we can create polaidal magnetic field
4
Because I'm writing a report and I would like to be really precise for the references
where did you find the research and the first image (ECG and Blood pressure synchronized)? Thank you!!
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2020-09-20 08:53:31
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https://web2.0calc.com/questions/consider-the-right-triangle-illustrated-with-sides
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+0
# Consider the right triangle illustrated with sides of length a = 5 and b = 1.
-5
220
1
+7
Consider the right triangle illustrated with sides of length a = 5 and b = 1. First, determine the length c of the hypotenuse and then fill in the table of values. Express all answers using exact arithmetic. NO DECIMALS. If the answer involves a square root it should be entered as sqrt. E.g. the square root of 2 should be written as sqrt(2)).
Sep 22, 2021
#1
+13571
+2
First, determine the length c of the hypotenuse and then fill in the table of values.
Hello GamemasterX!
$$\ c\ =\sqrt{26}$$
$$\ \delta$$ $$\ A$$ $$\ B$$
$$\ sin(\delta)$$ $$\dfrac{5}{\sqrt{26}}=\dfrac{5\sqrt{26}}{26}$$ $$\dfrac{1}{\sqrt{26}}=\dfrac{\sqrt{26}}{26}$$
$$\ cos(\delta)$$ $$\dfrac{1}{\sqrt{26}}=\dfrac{\sqrt{26}}{26}$$ $$\dfrac{5}{\sqrt{26}}=\dfrac{5\sqrt{26}}{26}$$
$$\ tan(\delta)$$ $$\ 5$$ $$\ \dfrac{1}{5}$$
!
Sep 22, 2021
edited by asinus Sep 22, 2021
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2022-05-20 04:55:20
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https://mathoverflow.net/questions/275775/mysterious-relationship-between-central-charges-of-conformal-field-theories-and
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Mysterious relationship between central charges of conformal field theories and the Beraha numbers
Background:
• Conformal field theories (CFTs) in two dimensions are partially characterized by a so-called central charge (characterizing the central extension of the Virasoro algebra which defines it). Under a condition of unitarity and minimality, all CFTs with $c<1$ have been classified. These make up the so-called minimal models $\mathcal M(m+1,m)$, where $m$ can be any integer $\geq 2$. The central charge $c_m$ of $\mathcal M(m+1,m)$ is given by $$c_m = 1 - \frac{6}{m(m+1)}$$
• Seemingly unrelated, there is the notion of the Beraha numbers. These are important numbers in algebraic graph theory, appearing in the study of the roots of so-called chromatic polynomials, which count the ways you can color graphs with a given set of colors. They are given by $$B_m = 4\cos^2 \left( \frac{\pi}{m} \right)$$
I am somewhat struck by a curious relationship between the central charge $c_m$ of the minimal model $\mathcal M(m+1,m)$ and the Beraha number $B_{m+1}$. To let the numbers speak for themselves:
$\begin{array}{c|c|c} m & 4^{c_m} & B_{m+1} \\ \hline 2 & 1 & 1 \\ 3 & 2 & 2 \\ 4 & 4^\frac{7}{10} \approx 2.639 & 1+\textrm{golden ratio} \approx 2.618 \\ 5 & 4^\frac{4}{5} \approx 3.031 & 3 \\ 6 & 4^\frac{6}{7} \approx 3.281 & \textrm{silver constant} \approx 3.247\\ 7 & 4^\frac{25}{28} \approx 3.448 & 2 + \sqrt{2} \approx 3.414\\ 8 & 4^\frac{11}{12} \approx 3.564 & 2+2\cos\left(\frac{2\pi}{9}\right)\approx 3.532 \\ 9 & 4^\frac{14}{15} \approx 3.647 & \frac{1}{2} \left( 5+\sqrt{5} \right) \approx 3.618 \\ 10 & 4^\frac{52}{55} \approx 3.709 & 2+2\cos\left(\frac{2\pi}{11}\right)\approx 3.683 \\ 11 & 4^\frac{21}{22} \approx 3.756 & 2+\sqrt{3} \approx 3.732 \\ 12 & 4^\frac{25}{26} \approx 3.792 & 2+2\cos\left(\frac{2\pi}{13}\right) \approx 3.771 \\ \vdots & \vdots & \vdots \\ \infty & 4 & 4 \end{array}$
As $m$ increases, the relationship gets obscured (or to phrase it more pessimistically, becomes perhaps less meaningful) since all the values get bunched up near each other (although it is curious to note that always $4^{c_m} \geq B_{m+1}$). Nevertheless the above table contains various apparently serendipitous relationships, suggesting --at least to me-- that there must be some conceptual link between $4^{c_m}$ and $B_{m+1}$. One possible relationship, might be if $4^{c_{m+1}}$ occurs as the zero of a chromatic polynomial, since the Beraha numbers are known to be accumulation points of such zeros (in the infinite graph limit).
How to explain' the above table?
To be clear, both numbers $c_m$ and $B_{m+1}$ have been known to appear together, e.g. in the discussion of Potts models (of which my knowledge is limited), but no direct relationship (of the type that I am hinting at) has been discussed.
EDIT: The inequality $4^{c_m} \geq B_{m+1}$ can be given a physical interpretation and justification. This is described in a recent manuscript of ours, but I will sketch the idea here. It is clearer in the equivalent form $c_m \geq \log_2 d_m$ where $d_m = \sqrt{B_{m+1}}$. The minimal model $\mathcal M(m+1,m)$ describes the phase transition between a so-called trivial phase and a topological' phase of quantum matter (in one spatial dimension), the latter exhibiting topological edge modes with quantum dimension $d_m$. Since the transition is exactly described by delocalized edge modes, and the central charge is known to be proportional to the massless degrees of freedom, one expects a relationship between $c$ and $d$. This has lead to the conjecture $c \geq \log_2 d$ (for the general case of a transition between a trivial and a topological phase) of which this is but a special case. That being said, it does not explain why this lower bound on the central chargea is almost-but-not-quite saturated. Indeed, this is how we noticed this funny relationship between $c_m$ and $B_{m+1}$.
• This pair of papers by Fendley and Krushkal might be related: arxiv.org/abs/0711.0016 arxiv.org/abs/0806.3484 . – j.c. Jul 19 '17 at 0:22
• It could be related to: mathoverflow.net/q/70575/34538 – Sebastien Palcoux Jul 19 '17 at 11:59
• There is nothing deep or mysterious here, $c_m$ and $\mathrm{log}_4 B_m$ just have approximately equal Taylor expansions in $1/m$. Honestly, both functions are so elementary that some relation between them isn't surprising. – Anton Fetisov Jul 20 '17 at 15:59
• @AntonFetisov Why should there be a relationship at all? I agree that if I had sampled from the space of all functions and I searched for (and found) two seemingly different functions that have vaguely similar values, of course that would not be surprising. In this case, however, as evidenced by the comments by Sebastian Palcoux and j.c. (thanks!), there are clearly deep links between the two topics in which these numbers appear. The fact that their numerical values subsequently turn out to be linked by a seemingly haphazard exponentiation involving $c_m$ makes a link all the more tantalizing. – Ruben Verresen Jul 20 '17 at 16:14
• (note that my 'EDIT' above moreover indicates that the exponentiation involved is in fact not as random as it might seem on first sight, at least from a physical perspective) – Ruben Verresen Jul 20 '17 at 16:16
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2019-04-20 11:16:46
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https://en.wikipedia.org/wiki/Steinberg_group_(K-theory)
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# Steinberg group (K-theory)
In algebraic K-theory, a field of mathematics, the Steinberg group ${\displaystyle \operatorname {St} (A)}$ of a ring ${\displaystyle A}$ is the universal central extension of the commutator subgroup of the stable general linear group of ${\displaystyle A}$.
It is named after Robert Steinberg, and it is connected with lower ${\displaystyle K}$-groups, notably ${\displaystyle K_{2}}$ and ${\displaystyle K_{3}}$.
## Definition
Abstractly, given a ring ${\displaystyle A}$, the Steinberg group ${\displaystyle \operatorname {St} (A)}$ is the universal central extension of the commutator subgroup of the stable general linear group (the commutator subgroup is perfect and so has a universal central extension).
Concretely, it can be described using generators and relations.
### Steinberg relations
Elementary matrices — i.e. matrices of the form ${\displaystyle {e_{pq}}(\lambda ):=\mathbf {1} +{a_{pq}}(\lambda )}$, where ${\displaystyle \mathbf {1} }$ is the identity matrix, ${\displaystyle {a_{pq}}(\lambda )}$ is the matrix with ${\displaystyle \lambda }$ in the ${\displaystyle (p,q)}$-entry and zeros elsewhere, and ${\displaystyle p\neq q}$ — satisfy the following relations, called the Steinberg relations:
{\displaystyle {\begin{aligned}e_{ij}(\lambda )e_{ij}(\mu )&=e_{ij}(\lambda +\mu );&&\\\left[e_{ij}(\lambda ),e_{jk}(\mu )\right]&=e_{ik}(\lambda \mu ),&&{\text{for }}i\neq k;\\\left[e_{ij}(\lambda ),e_{kl}(\mu )\right]&=\mathbf {1} ,&&{\text{for }}i\neq l{\text{ and }}j\neq k.\end{aligned}}}
The unstable Steinberg group of order ${\displaystyle r}$ over ${\displaystyle A}$, denoted by ${\displaystyle {\operatorname {St} _{r}}(A)}$, is defined by the generators ${\displaystyle {x_{ij}}(\lambda )}$, where ${\displaystyle 1\leq i\neq j\leq r}$ and ${\displaystyle \lambda \in A}$, these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by ${\displaystyle \operatorname {St} (A)}$, is the direct limit of the system ${\displaystyle {\operatorname {St} _{r}}(A)\to {\operatorname {St} _{r+1}}(A)}$. It can also be thought of as the Steinberg group of infinite order.
Mapping ${\displaystyle {x_{ij}}(\lambda )\mapsto {e_{ij}}(\lambda )}$ yields a group homomorphism ${\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)}$. As the elementary matrices generate the commutator subgroup, this mapping is surjective onto the commutator subgroup.
## Relation to ${\displaystyle K}$-theory
### ${\displaystyle K_{1}}$
${\displaystyle {K_{1}}(A)}$ is the cokernel of the map ${\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)}$, as ${\displaystyle K_{1}}$ is the abelianization of ${\displaystyle {\operatorname {GL} _{\infty }}(A)}$ and the mapping ${\displaystyle \varphi }$ is surjective onto the commutator subgroup.
### ${\displaystyle K_{2}}$
${\displaystyle {K_{2}}(A)}$ is the center of the Steinberg group. This was Milnor's definition, and it also follows from more general definitions of higher ${\displaystyle K}$-groups.
It is also the kernel of the mapping ${\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)}$. Indeed, there is an exact sequence
${\displaystyle 1\to {K_{2}}(A)\to \operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)\to {K_{1}}(A)\to 1.}$
Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group: ${\displaystyle {K_{2}}(A)={H_{2}}(E(A);\mathbb {Z} )}$.
### ${\displaystyle K_{3}}$
Gersten (1973) showed that ${\displaystyle {K_{3}}(A)={H_{3}}(\operatorname {St} (A);\mathbb {Z} )}$.
## References
• Gersten, S. M. (1973), "${\displaystyle K_{3}}$ of a Ring is ${\displaystyle H_{3}}$ of the Steinberg Group", Proceedings of the American Mathematical Society, American Mathematical Society, 37 (2): 366–368, doi:10.2307/2039440, JSTOR 2039440
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2017-02-24 05:04:30
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https://www.semanticscholar.org/paper/Cosmological-Simulations-of-Number-Counts-Lepori-Adamek/bc560243e7ec99d3c872959377bb4f562b397f46
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Corpus ID: 235293918
# Cosmological Simulations of Number Counts
@inproceedings{Lepori2021CosmologicalSO,
title={Cosmological Simulations of Number Counts},
author={Francesca Lepori and J. Adamek and R. Durrer},
year={2021}
}
• Published 2021
• Physics
In this paper we present for the first time the angular power spectra C`(z, z′) for number counts from relativistic N-body simulations. We use the relativistic N-body code gevolution with its exact integration of lightlike geodesics which include all relativistic scalar contributions to the number counts. We compare our non-perturbative numerical results with the results from class using the Halofit approximation for the non-linear matter power spectrum. We find that the Halofit approximation… Expand
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Fast estimation of polarization power spectra using correlation functions
• Physics
• 2004
We present a fast method for estimating the cosmic microwave background polarization power spectra using unbiased estimates of heuristically weighted correlation functions. This extends the O(N 3/2Expand
Detection of cosmic magnification via galaxy shear-galaxy number density correlation from HSC survey data
• Xiangkun Liu, +5 authors Z. Fan
• Physics
• 2021
Xiangkun Liu, ∗ Dezi Liu, Zucheng Gao, Chengliang Wei, Guoliang Li, 4 Liping Fu, Toshifumi Futamase, and Zuhui Fan † South-Western Institute for Astronomy Research, Yunnan University, Kunming 650500,Expand
Beware of commonly used approximations
• Part I. Errors in forecasts, JCAP 10
• 2020
Beware of commonly used approximations. Part I. Errors in forecasts
• Physics
• 2020
In the era of precision cosmology, establishing the correct magnitude of statistical errors in cosmological parameters is of crucial importance. However, widely used approximations in galaxy surveysExpand
Modeling relativistic contributions to the halo power spectrum dipole
• Physics
• 2020
We study the power spectrum dipole of an N-body simulation which includes relativistic effects through ray-tracing and covers the low redshift Universe up to $z_{\rm max} = 0.465$ (RayGalGroupExpand
Nonlinear contributions to angular power spectra
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Future galaxy clustering surveys will probe small scales where non-linearities become important. Since the number of modes accessible on intermediate to small scales is very high, having a preciseExpand
Nonlinear redshift-space distortions in the harmonic-space galaxy power spectrum
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Future high spectroscopic resolution galaxy surveys will observe galaxies with nearly full-sky footprints. Modeling the galaxy clustering for these surveys, therefore, must include the wide-angleExpand
Observing relativistic features in large-scale structure surveys – I. Multipoles of the power spectrum
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Planned efforts to probe the largest observable distance scales in future cosmological surveys are motivated by a desire to detect relic correlations left over from inflation, and the possibility ofExpand
Observing relativistic features in large-scale structure surveys – II. Doppler magnification in an ensemble of relativistic simulations
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• 2020
The standard cosmological model is inherently relativistic, and yet a wide range of cosmological observations can be predicted accurately from essentially Newtonian theory. This is not the case onExpand
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2021-09-28 13:18:13
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https://www.ask-math.com/implicit-differentiation.html
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# Implicit Differentiation
Covid-19 has led the world to go through a phenomenal transition .
E-learning is the future today.
Stay Home , Stay Safe and keep learning!!!
A function can be implicit or explicit.
Explicit form When 'y' is expressed as function of x then such a function is called Explicit function. For example : 1) y = $x^{2}$ +1 2) y = sin(x) + $x^{3}$ 3) y = $\sqrt{x+1}$ Implicit form When the function is expressed in terms of x and y then such equations are called Implicit form. For example : 1) $x^{2} + y^{2}$= 5 2) $x^{2}$ + xy = 0 3) $y^{3} -y^{2} + x^{2}$= 0
Implicit differentiation : So far we have discussed derivatives of the function y = f(x). If the derivatives x and y are connected by a relation of the form f(x,y)= 0 and sometimes it is not possible to express y as a function of x in the form of y = g(x) then y is said to an implicit function of x. To find dy/dx in such a case, we differentiate both sides of the given relation with respect to x. So to differentiate such functions we use a chain rule.
## Examples on Implicit differentiation
Example 1 : If $x^{2} + 2xy +y^{3}$= 42, find $\frac{dy}{dx}$
Solution : We have $x^{2} + 2xy +y^{3}$= 42
Differentiating both sides with respect to x, we get
$\frac{d}{dx}(x^{2})+ 2 \frac{d}{dx}(xy) + \frac{d}{dx}(y^{3}) = \frac{d}{dx}(42)$
Apply the power rule, product rule and chain rule for differentiation
2x + 2(x.$\frac{dy}{dx} + y.1) + 3y^{2}\frac{dy}{dx}$ = 0
2x + 2x. $\frac{dy}{dx} + 2y + 3y^{2}\frac{dy}{dx}$ = 0
2x + 2y + $\frac{dy}{dx}(2x + 3y^{2})$ = 0
$\frac{dy}{dx}(2x + 3y^{2})$ = -2x - 2y
$\frac{dy}{dx}(2x + 3y^{2})$ = -(2x + 2y)
$\frac{dy}{dx} = \frac{-(2x + 2y)}{(2x + 3y^{2})}$
Example 2 : If sin(y) = x.sin(a + y) , find $\frac{dy}{dx}$
Solution : We have sin(y) = x.sin(a + y)
Differentiating both sides with respect to x, we get
$\frac{d}{dx}(sin(y))= \frac{d}{dx}[x.sin(a + y)]$
Apply the product rule and chain rule for differentiation
cos(y)$\frac{dy}{dx}= 1.sin(a + y) + x.cos(a+ y) \frac{d}{dx}(a + y)$
cos(y)$\frac{dy}{dx}= sin(a + y) + x.cos(a+ y)[ \frac{d}{dx}(a) + \frac{dy}{dx}]$
cos(y)$\frac{dy}{dx}= sin(a + y) + x.cos(a+ y) . \frac{dy}{dx}$
cos(y)$\frac{dy}{dx} - x.cos(a+ y) . \frac{dy}{dx}$ = sin(a + y)
$\frac{dy}{dx}[cos(y) - x.cos(a + y)]$ = sin(a + y)
$\frac{dy}{dx} = \frac{sin(a + y)}{[cos(y) - x.cos(a + y)]}$
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2022-01-16 19:19:04
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https://papers.nips.cc/paper/2014/file/670e8a43b246801ca1eaca97b3e19189-Reviews.html
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Paper ID: 811 Title: Top Rank Optimization in Linear Time
Current Reviews
Submitted by Assigned_Reviewer_2
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
The authors present a novel approach to learning to rank. In contrast to traditional approaches, the idea is to focus on the number of positive instances that are ranked before the first negative one. Following a large-margin approach leads to primal and dual representations. Compared to similar approaches, the complexity is only linear in the number of instances.
This is a nice paper! Particularly the technical contribution is strong.
Apart from the url data set, the rest of the experiments is pretty small-scale. Adding experiments on larger-scales would certainly strengthen the paper as baseline competitors will at some point drop out. Consider having a figure showing performance vs number of training instances to showcase the benefit of processing more data than the baselines. The same figure for time instead of performance would also be interesting.
The divide and conquer schema should at least be sketched in the final paper.
Strong technical contribution. Good paper.
Submitted by Assigned_Reviewer_25
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
In this paper, the authors consider bipartite ranking and specifically, they consider optimizing the ranking at the top.
The authors first define a loss function that penalizes positive
example whenever its predicted score is less than that of the negative example with the highest
score. The main technical contribution is to propose a regularized optimization problem with
the above loss and to show that it can be solved efficiently via the dual formulation. The authors
give an optimization algorithm to solve the proposed formulation. They also give a theoretical result
that gives an upper bound on the probability that any positive example is ranked below delta fraction
of negative examples. The main advantage of the proposed formulation is that it brings down the computational
complexity of to rank optimization to linear in the number of examples.
Finally, the authors give experimental results comparing their approach with several other approaches. The proposed method clearly outperforms most other methods both in terms of speed and accuracy.
Minor comment:
The definitions of loss used in this paper is non-standard. Typically e^{-z} is the exponential
loss and [1-z]_{+}^2 is the truncated exponential loss. However, there is no issue since the
authors have also flipped the sign of the argument in Eqn (3). For clarity, I think that it is
better to modify this to match standard definitions.
The paper seems quite novel to me and the contributions in this paper seem non trivial. I do not
The authors propose an elegant approach to reduce the time complexity of bipartite ranking
to linear in the number of examples. The experimental results are quite compelling. I strongly recommend
accepting this paper.
Submitted by Assigned_Reviewer_31
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
The paper addresses the computational issue of bipartite ranking. The authors propose a new algorithm whose computational complexity is linear in number of training instances, and provide theoretical analysis of generalization error. The paper is rounded of with extensive experiments.
Strong points- The paper is clearly written and might be of some value in case of bipartite ranking with large datasets. The generalization bound is novel and experiments section is detailed.
1. Though bipartite ranking is well studied, it is restricted in scope, in the sense that it is in the domain of ranking but cannot handle queries. Considering there are already well established algorithms for bipartite ranking which have been well studied theoretically and tested empirically, is the study really very valuable? For eg., this will only be useful when m,n are really large. Is that very practical in domain of bipartite ranking? Admittedly, this is just my thought and i would like to hear authors' view on this (citing example or something).
2. The main reason the paper gets an O(m+n/\sqrt(e)) error bound is because of the new defined target loss (2) and using a smooth convex surrogate which allows standard primal-dual trick to get a smooth dual, thereby allowing standard accelerated gradient descent optimization. According to target loss (2), a negative instance followed by all positive instances has greater loss than a positive instance followed by all negative instances. Can this be considered promoting good ranking at top, especially since there is no position based discount factor?
Moreover, in the supplement it is stated that maximizing positives at top cannot be achieved by AUC optimization, but target loss (2) is an upper bound on rank loss. So why should someone try to optimize an upper bound on rank loss, if AUC optimization itself is not suitable for the purpose of pushing positives at top?
3. Once the convex surrogate is taken to be smooth, conversion to dual and applying Nestrov technique is neat but i do not think it is extremely novel.
4. Looking at empirical section, i am confused as to what TopPush is gaining on LR (logistic regression). Computational power of the new algorithm is the USP of the paper; it does not seem to be doing any better than LR; infact LR takes less time than TopPush in 4/7 experiments. Nor is it gaining anything significant in AP and NDCG metrics, effectively metrics which are popular; it gains a little in position at top but loses in AUC. So why should we consider TopPush over LR? Or am i reading the experimental results wrong?
A side point: On comparing the computational complexity with SVM (Rank,MAP); it can be seen that all of them scale linearly with training data size. The gain in computation time in TopPush is because SVM consider hinge loss while TopPush considers a smooth surrogate. So computational complexity linear in number of training instances is not unique to TopPush.
5. Theory- In the generalization bound, shouldn't the focus be on the cases when there are a large number of negative instances and few positive instances? The other way round is less practical and even an average ranking function would put a few positive instances on the top. However, if we focus on the negative instances, i am not sure what the bound is relaying. With growing n, the empirical loss is much more likely to keep increasing, since the normalizing factor is only 1/m (no dependence on n). Since \delta will become small, the L.H.S probability is likely to grow but the R.H.S is also likely to grow. Maybe i am not being able to understand the significance of the bound, from a more useful n >> m point of view.
My ratings later on are provisional. I would like the authors to address the questions i have raised. Specifically i would like the authors to shed more light on the comparison between TopPush and LR ( 4, question on empirical section). In my opinion, the accept/reject hinges on clarifying how TopPush gains on LR. I will be happy to review my decision after author feedback.
Update after Author Feedback-
1. "Example of m,n large"- I am not an expert in bipartite ranking, so i will take the authors' words for it. However, from my knowledge of online advertisement, is it not the case that ranking online advertisements is in the learning to rank framework? (i.e query dependent?). I completely agree with the first reviewer that showing experiments for large datasets (possibly real datasets used in bi-partite ranking) will be very useful.
2. "AUC optimization"- The authors dont really answer the question. They talk about advantage of the new loss, in terms of optimization and gen. bounds. However, it has nothing to do with AUC. In fact, independent of how the new loss compares with AUC, the advantages will hold. From the point of view that the new loss is an upper bound on AUC and AUC is not useful for the objective of "pushing positives at top", why should someone optimize the new loss?
3."comparison with LR"- This is critical. I agree that TopPush is doing better than LR in Pos@Top metric. I think in the revised version, the authors should modify the introduction slightly. The USP of the paper is the computational advantage of TopPush over other algorithms. This is overselling the paper a little bit. TopPush has no (visible) computational advantage over LR. It can be seen as an alternate, with advantage when it comes to performance on a specific metric (and disadvantage on some others).
4."SVM"- I believe the advantage over SVM based methods is the quadratic convergence rate (O(1/T^2) as opposed to O(1/T)), not linear in "m+n"? Both SVM based methods and top push are linear in "m+n", as the authors have clearly shown in Table 1.
5."Gen bound"- Please include the discussion in the revised draft. This is critical.
Overall, i like this paper. With revision, it will certainly be a very good paper. I have updated my decision to an accept.
The problem addressed is well known with neat techniques used and might be of potential interest. However, there are some questions about the practical significance and the theoretical results. I do think it is an interesting paper and i will be happy to reconsider after authors' feedback; but right now, based on the nature of the highly competitive venue, i do not believe it will be a loss if NIPS gives it a miss.
Author Feedback
Author Feedback
Q1:Author rebuttal: Please respond to any concerns raised in the reviews. There are no constraints on how you want to argue your case, except for the fact that your text should be limited to a maximum of 6000 characters. Note however, that reviewers and area chairs are busy and may not read long vague rebuttals. It is in your own interest to be concise and to the point.
We want to thank reviewers and will improve the paper by incorporating the suggestions. In the following we will focus on technical questions.
Q1: examples where both m,n are large in practical bipartite ranking problems.
A1: Bipartite ranking has wide applications in information retrieval, recommender systems, etc. For example, in online advertisement system, we need to rank the advertisements that are more likely to be clicked at the top. In this case, both m and n (i.e. the number of clicked and non-clicked events) are very large.
Q2:“According to target loss (2) … Can this be considered promoting good ranking at top”
A2: Yes, in this paper, we focus on promoting ranking quality for the instances at the top of ranked list, which is different from AUC optimization. Specifically, we achieve this by the loss (2).
Q3:“why should someone try to optimize an upper bound on rank loss, if AUC optimization itself is not suitable for the purpose of pushing positives at top”
A3: The advantage of using loss in (2) is twofold. First, using loss function (2) will result in a dual problem with only $m+n$ dual variables, which is significantly smaller than $m*n$, the number of dual variables when using other loss function. Second, it will lead to a generalization error bound that emphasizes the error at the top of the ranking list.
Q4:“Once the convex surrogate is taken to be smooth, conversion to dual and applying Nestrov technique is neat but i do not think it is extremely novel.”
A4: There is an important technical trick we used to transform the dual problem into an optimization problem with only $m+n$ variables. We will make this clear in the revised draft.
Q5:“why should we consider TopPush over LR?” (emphasized by the 3rd reviewer)
A5: As shown in [Kotlowski et al. ICML (2011)], the logistic loss of LR is consistent to ranking loss, and as a result, LR can be seen as a method for optimizing AUC. In contrast, we emphasize optimizing the instances ranked at the top of a ranking list, which is beyond the optimization of AUC. This is demonstrated by our empirical study which shows that LR performs significantly worse than the proposed method if we measure the ranking results only based on the top ranked instances (i.e. Pos@Top).
Q6:“The gain in computation time in TopPush is because SVM(Rank, MAP) consider hinge loss while TopPush considers a smooth surrogate. So computational complexity linear in number of training instances is not unique to TopPush.”
A6: Hinge loss is critical to SVM(Rank, MAP) because it will result in a sparse dual solution, a key to make the cutting-plane method efficient. Replacing hinge loss in SVM(Rank, MAP) with a smooth loss will make it impossible to implement an efficient cutting plane method because the number of constraints is combinatorial in the number of training examples. Thus, complexity linear in the number of training examples is very unique to our approach.
Q7:“shouldn't the focus be on the cases when there are a large number of negative instances and few positive instances? …, if we focus on the negative instances, i am not sure what the bound is relaying. With growing n, the empirical loss is much more likely to keep increasing,…”
A7: We thank the reviewer for the insightful comment. Since the empirical loss only compares the positive instances to the negative instance with the largest score, it usually grows significantly slower with increasing $n$ (i.e. the number of negative instances). For instance, the largest absolute value of Gaussian random samples grows in $\log n$. Thus, we believe that the main effect of increasing $n$ in our bound is to reduce $\delta$ (decrease at the rate of $1/\sqrt{n}$), especially when $n$ is already very large.
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2021-10-25 13:56:09
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http://jkms.kms.or.kr/journal/list.html?Vol=9&Num=2&mod=vol&book=JKMS&aut_box=Y&sub_box=Y&pub_box=Y
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- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnline Submission ㆍMy Manuscript - For Reviewers - For Editors
<< Previous Issue Journal of the Korean Mathematical Society (Vol. 9, No. 2) Next Issue >>
J. Korean Math. Soc. 1972 Vol. 9, No. 2, 49—104
Invariant submanifolds of a manifold with quasi-normal $(f,g,u,v,\lambda)$-structure Jae Kyu Lim and U-Hang Ki J. Korean Math. Soc. 1972 Vol. 9, No. 2, 49—57
On $(f,g,e,u,v,\lambda)$-structure Yong Bai Baik J. Korean Math. Soc. 1972 Vol. 9, No. 2, 59—74
On a criterion for obtaining full information about the unknown state of nature Hyun-Chun Shin and Jae-Joo Kim J. Korean Math. Soc. 1972 Vol. 9, No. 2, 75—84
A natural approach to Fredholm structures on Banach manifolds Yong Tae Shin J. Korean Math. Soc. 1972 Vol. 9, No. 2, 85—89
On convergence of semigroups of operators in Banach spaces Ki Sik Ha J. Korean Math. Soc. 1972 Vol. 9, No. 2, 91—99
On the Gross domain of a meromorphic function Un Haing Choi J. Korean Math. Soc. 1972 Vol. 9, No. 2, 101—104
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2017-10-17 16:53:09
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https://asf.microchip.com/docs/latest/uc3l/html/trans__dsp16__twiddle__factors_8c.html
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trans_dsp16_twiddle_factors.c File Reference
#include "trans_dsp16_twiddle_factors.h"
## Macros
#define DSP16_Q_CONVERT(x) DSP16_Q((x)/2.)
This macro is used to convert the raw data according to the algorithm's optimization we need. More...
## Variables
1024 elements twiddle factors table
A_ALIGNED
TWIDDLE_FACTORS_PREFIX_TAB
dsp16_t
dsp16_twiddle_factors [DSP16_N_TWIDDLE_FACTORS/2+2]
This table has been generated using the following algorithm: More...
A_ALIGNED
TWIDDLE_FACTORS_PREFIX_TAB
dsp16_t
dsp16_twiddle_factors2 [DSP16_N_TWIDDLE_FACTORS]
This table has been generated using the following algorithm: More...
#define DSP16_Q_CONVERT ( x ) DSP16_Q((x)/2.)
This macro is used to convert the raw data according to the algorithm's optimization we need.
A_ALIGNED TWIDDLE_FACTORS_PREFIX_TAB dsp16_t dsp16_twiddle_factors[DSP16_N_TWIDDLE_FACTORS/2+2]
This table has been generated using the following algorithm:
w = exp(-2*PI*%i*k);
end;
It is a one dimensional array containing the real parts (even parts) and imaginary parts (odd parts) of the w value.
Referenced by dsp16_trans_complexfft(), dsp16_trans_complexifft(), and dsp16_trans_realcomplexfft().
A_ALIGNED TWIDDLE_FACTORS_PREFIX_TAB dsp16_t dsp16_twiddle_factors2[DSP16_N_TWIDDLE_FACTORS]
This table has been generated using the following algorithm:
w2 = exp(-2*PI*%i*k*2);
w3 = exp(-2*PI*%i*k*3);
end;
It is a one dimensional array containing the real parts (4*i parts) and imaginary parts (4*i+1 parts) of the w2 value and the real parts (4*i+2 parts) and imaginary parts (4*i+3 parts) of the w3 value.
Referenced by dsp16_trans_complexfft(), dsp16_trans_complexifft(), and dsp16_trans_realcomplexfft().
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2022-06-26 22:58:23
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https://math.stackexchange.com/questions/495227/is-it-true-that-a-dihedral-group-is-nonabelian
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# Is it true that a dihedral group is nonabelian?
Is it true that a dihedral group is nonabelian?
I'm not sure if the result is true. I checked it for some lower order and I think the result may correct.
But I failed to prove/disprove the result.
• What happens if you first do the smallest possible rotation and then reflect in some line? What happens if you first reflect in that same line and then do that rotation? – Tobias Kildetoft Sep 16 '13 at 7:46
• There are many equivalent definition about dihedral group. What definition did you have learn? – D. N. Sep 16 '13 at 8:27
• The easier the question, the more identical answers. What a nonsense ... – Martin Brandenburg Sep 16 '13 at 10:04
Yes, the dihedral groups $D_n$ are nonabelian for $n\ge 3$. It is generated by a rotation $r$ with $r^n=1$ and a reflection $s$ with $s^2=1$. However, you can easily check that a rotation and a reflection will not commute in general. We have $sr=r^{-1}s$ instead for $D_n$ with this presentation.
The dihedral groups for $n=1$ and $n=2$ are abelian; for $n\geq 3$, the dihedral groups are nonabelian (this is mentioned on Wikipedia).
$D_3$, i.e, the dihedral group of a triangle is isomorphic to $S_3$ which is non-abelian. It can be shown that this is true for $n \geq 3$.
Remark: Some people denote the dihedral group by $D_{2n}$ which is based on the fact the order is $2n$, while some people denote it by $D_n$.
• That's a good point to mention about notation! – David Ward Sep 16 '13 at 7:55
Dihedral group of order $2n$ has the presentation
$$\langle x,y \mid x^n=y^2=e,yxy=x^{-1}\rangle$$
When $n=1$, we have $x=e$ and thus it is a group of order 2.
When $n=2$, we have $yxy=x$ and so $xy=yx$. Thus it is abelian.
But when $n\geq 3$, then $yxy=x^{n-1}$ and so $xy=yx^{n-1} \neq yx$. Thus it is not abelian.
• You need to explain why the inequality holds. – Zhen Lin Sep 16 '13 at 8:12
• $x$ and $y$ are two element of the group and $xy\neq yx$, means they do not commute, and so it is not an abelian group. – D. N. Sep 16 '13 at 8:19
• @deibor: You cannot conclude from a group presentation that certain relations do not hold. For example, even when a presentation contains something like $x^3=1$, it doesn't mean that the order of $x$ is three. In fact, other relations might imply that $x=1$. The general word problem for groups is undecidable. – Martin Brandenburg Sep 16 '13 at 10:03
• @Martin What you say is true, but in this specific case we are fine. This is because we know that these relations hold and we know that $x$ has order $n$. If $yx^{n-1}=yx$ then we would have $x^2=1$ so $n\leq 2$. – user1729 Sep 16 '13 at 10:06
• The presentation $\langle x,y∣x^n=y^2=e,yxy=x−1\rangle$, has been taken as an equivalent definition of dihedral group of order $2n$ by many authors of many text book. Also it is an equivalent definition on wikipedia and groupprop. – D. N. Sep 16 '13 at 10:34
My picture for $D_4$ shows $fr = r^{-1}f$ and not $rf$.
See Zev Chonoles's exquisite pictures at https://math.stackexchange.com/a/686175/53934 too.
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2019-09-15 10:13:16
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https://www.physicsforums.com/threads/maxwells-equations-and-qft.948044/
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Maxwell's equations and QFT
• I
• Featured
Hello,
I have been wondering about the validity of Maxwell's equations in quantum physics. I looked in the internet and it seems from what I understood that: Maxwell's equations are valid for any situation, classical or quantum. In fact, maybe it holds more legitimacy than Schroedinger equation since it is a relativistic (invariant) set of equations.
Yet, I am really baffled! The equations said to be valid, yet I don't see any wave function in it. (Ok, this might be hilarious. But, any equation I see in quantum physics have a wave function and probability distribution!). Are the electric and magnetic fields alongside the functions ##\rho \ \& \ J## probabilistic!
My question is, how Maxwell's equations are implemented/related to quantum physics? Why it is not usually in use?
*Disclaimer: I am undergraduate student, and I don't have that much experience with quantum physics.
**I would appreciate it if you supported your replies with references of books and papers.
Last edited:
hilbert2
Gold Member
The same Lagrangian density function ##\mathcal{L}## that is used in deriving the classical Maxwell equations is also the basis of quantum electrodynamics. A "wave function" describing a quantum field would actually need to be a set of infinitely many functions, one for every point in space, so that you could form a probability density for the field strengths at any point.
Phylosopher
"But it wasn't until 1884 that Oliver Heaviside, concurrently with similar work by Josiah Willard Gibbs and Heinrich Hertz, grouped the twenty equations together into a set of only four, via vector notation.[3] This group of four equations was known variously as the Hertz–Heaviside equations and the Maxwell–Hertz equations, but are now universally known as Maxwell's equations.[4] Heaviside's equations, which are taught in textbooks and universities as Maxwell's equations are not exactly the same as the ones due to Maxwell, and, in fact, the latter are more easily forced into the mold of quantum physics.[5] This very subtle and paradoxical sounding situation can perhaps be most easily understood in terms of the similar situation that exists with respect to Newton's second law of motion. In textbooks and in classrooms the law F = ma is attributed to Newton, but his second law was in fact F = p', where p' is the time derivative of the momentum p. This seems a trivial enough fact until you realize that F = p' remains true in the context of Special relativity." (my italics)
en.wikipedia.org/wiki/History_of_Maxwell's_equations (The term Maxwell's equations)
David Lewis, dextercioby and Phylosopher
A. Neumaier
The equations said to be valid, yet I don't see any wave function in it.
The Maxwell equations are equations both for classical fields and for quantum fields. The latter represent observables, while wave functions represent states.
Both classically and in quantum mechanics, the free Maxwell equations (for electromagnetic fields without sources) are linear, hence free electromagnetic waves and free photons don't interact.
In the presence of matter, one has to use a bigger coupled system consisting of the Maxwell equations coupled to moving sources, which are themselves coupled to the electromagnetic field, and these coupled equations are nonlinear, hence interacting - again both in classical and in quantum mechanics.
Last edited:
bhobba, Spinnor and Phylosopher
DrDu
The point is that up to now, satisfactory QFT's for which you can write down a Hamiltonian and a Schrödinger equation, have only been found in 0, 1 and 2 dimensions. In 3 dimensions, QFT is little more than a rather ill defined set of rules to do perturbation expansions.
Phylosopher
DrDu
It is a mathematically well defined set of rules. Yet it is not clear at all whether this perturbation series is even an asymptotic expansion of some more fundamental underlying theory. Of course it is a very successful construct. However, I think that many, especially non-specialist have the idea that QED or QFT is an albeit very complex but complete theory which allows to derive all other observations in nature from it.
samalkhaiat
Hello,
I have been wondering about the validity of Maxwell's equations in quantum physics. I looked in the internet and it seems from what I understood that: Maxwell's equations are valid for any situation, classical or quantum.
I am afraid, your understanding is not correct. The classical Maxwell equations (like the equations of all other gauge field theories) must be modified at quantization if we want to maintain Poincare’ covariance. Bellow, I will provide two proofs for my statement. The first one is at the undergraduate level (i.e., for you), and the second proof is for those who know few things about the Wightman formalism of QFT.
1) Recall that in the (covariant) Lorenz gauge, the free Maxwell equations are $$\partial^{2}A_{\mu}(x) = 0 , \ \ \ \ \ \ \ \ \ \ (1.1)$$$$\partial^{\mu}A_{\mu}(x) = 0 . \ \ \ \ \ \ \ \ \ \ (1.2)$$
Now, suppose that eq(1.1) and eq(1.2) are valid operator equations in the quantized theory. Also notice that eq(1.1) is a set of 4 massless Klein-Gordon operator field equations. So, we may use the covariant (4-dimentional) commutation relations of the Klein-Gordon operators to set up the following commutation relations for the Maxwell’s field operator $A_{\mu}(x)$:
$$\left[ A_{\mu}(x) , A_{\nu}(y) \right] = - i \eta_{\mu\nu} \ \Delta (x - y \ ; 0) , \ \ \ \ \ \ \ (1.3)$$ where $\Delta (x ; 0)$ is the massless limit of the Lorentz invariant $\Delta$-function $$\Delta (x ; 0) = \lim_{m \to 0} \left( \frac{-1}{(2 \pi )^{3}} \int d^{3} \vec{k} \ \frac{\sin kx}{\omega_{\vec{k}}}\right) .$$ Now, you can see at once that eq(1.2) can not hold as operator equation in the quantized theory because it is incompatible with the covariant commutation relations (1.3): $$\left[ \partial^{\mu}A_{\mu}(x) , A_{\nu}(0) \right] = - i \ \partial_{\nu} \Delta (x ; 0) . \ \ \ \ \ \ \ (1.4)$$ If you interpret the Lorenz condition (1.2) as operator equation, then the LHS of (1.4) is identically zero while the RHS is certainly not identically zero. This inconsistency can be resolved by replacing the Lorenz condition (1.2) (hence modifying Maxwell’s equations) by the weaker (Gupta-Bleuler) condition $$\partial^{\mu}A^{+}_{\mu}(x) | \Psi \rangle = 0 , \ \ \ \ \ \ \ \ \ \ (1.5)$$ involving annihilation operators only. So, the allowed states of the quantized Maxwell theory must be restricted to satisfy the condition (1.5) instead of (1.2). From the condition (1.5) and its adjoint, it follows that the Lorenz condition (1.2) holds (as it should) for expectation values $$\langle \Psi | \partial^{\mu}A^{+}_{\mu} + \partial^{\mu}A^{-}_{\mu}| \Psi \rangle = \langle \Psi | \partial^{\mu}A_{\mu} | \Psi \rangle = 0 .$$ This ensures that the Lorenz condition and hence the full Maxwell’s equations hold as the classical limit of the quantum theory of electromagnetic field.
****
2) Proposition: The free Maxwell equation $$\left( \delta^{\nu}_{\mu} \partial^{2} - \partial_{\mu}\partial^{\nu}\right) A_{\nu}(x) = 0 , \ \ \ \ (2.1)$$ is incompatible with manifest Poincare’ covariance if $A_{\mu}(x)$ is a nontrivial quantized vector field.
Proof: Consider the 2-point function $\langle 0 | A_{\nu}(x) A_{\rho}(y) | 0 \rangle$. Covariance with respect to the Poincare’ group allows us to express the 2-point function in the form $$\langle 0 | A_{\nu}(x) A_{\rho}(y) | 0 \rangle = \eta_{\nu\rho} F_{1}(x-y) + \partial_{\nu}\partial_{\rho}F_{2}(x-y) , \ \ \ \ \ (2.2)$$ for some Lorentz invariant functions $F_{1}$ and $F_{2}$. Applying the Maxwell differential operator $\left( \delta^{\nu}_{\mu} \partial^{2} - \partial_{\mu}\partial^{\nu}\right)$ to both sides of (2.2) and using (2.1), we get $$\eta_{\mu\rho}\partial^{2}F_{1}(x) - \partial_{\mu}\partial_{\rho}F_{1}(x) = 0 . \ \ \ \ \ \ \ \ (2.3)$$ Contracting $\mu$ and $\rho$, we obtain $$\partial^{2}F_{1}(x) = 0 . \ \ \ \ \ \ \ \ \ \ \ \ \ (2.4)$$ Substituting (2.4) back in (2.3), we find $$\partial_{\mu}\partial_{\rho}F_{1}(x) = 0 . \ \ \ \ \ \ \ \ \ \ \ \ \ (2.5)$$ Since $F_{1}(x)$ is Lorentz invariant, a constant $C$ is the only solution of (2.5). Thus, the form of the 2-point function (2.2) reduces to $$\langle 0 | A_{\nu}(x) A_{\rho}(y) | 0 \rangle = \eta_{\nu\rho} C + \partial_{\nu}\partial_{\rho}F_{2}(x-y) . \ \ \ \ \ \ (2.6)$$ Now, from (2.6) and the definition of the physical field $F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$, we obtain $$\langle 0 | F_{\mu\nu}(y) F_{\rho\sigma}(x) | 0 \rangle = 0 . \ \ \ \ \ \ \ \ \ \ \ \ (2.7)$$ Smearing the local physical field $F_{\mu\nu}$ with an arbitrary real test function $f^{\mu\nu}(x)$, we conclude that the state $$\int d^{4}x \ f^{\rho\sigma}(x) F_{\rho\sigma}(x) \ | 0 \rangle , \ \ \ \ \ \ \ \ \ \ \ (2.8)$$ must belong to a positive-metric Hilbert space. Thus, eq(2.7) implies that (2.8) vanishes identically. Then, due to the arbitrariness of the test function $f^{\mu\nu}(x)$, we get $$F_{\rho\sigma}(x) | 0 \rangle = 0 .$$ Now, using the separating property of the vacuum with respect to local fields, we conclude that $F_{\mu\nu}(x) = 0$, meaning that $A_{\mu}(x)$ is trivial. As we have seen in part(1), this difficulty cannot be resolved by adding the Lorenz condition (1.2). Thus, the classical gauge field equation cannot survive in quantum theory as a local operator equation.
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bhobba, nrqed, Delta2 and 4 others
A. Neumaier
1) Recall that in the (covariant) Lorenz gauge, the free Maxwell equations are $$\partial^{2}A_{\mu}(x) = 0 , \ \ \ \ \ \ \ \ \ \ (1.1)$$$$\partial^{\mu}A_{\mu}(x) = 0 . \ \ \ \ \ \ \ \ \ \ (1.2)$$
No. The free Maxwell equations are not about the vector potential but about the electromagnetic field, and these hold also on the operator level!
Peter Morgan
samalkhaiat
No. The free Maxwell equations are not about the vector potential but about the electromagnetic field, and these hold also on the operator level!
One solves the Maxwell equation in terms of $A_{\mu}$; quantizes $A_{\mu}$; classical and quantum interaction is given by $A_{\mu}J^{\mu}$; the successes of QED is achieved by imposing subsidiary condition on $A_{\mu}$, and you now come and tell me it is “ not about the vector potential”.
protonsarecool and Demystifier
Demystifier
Gold Member
The free Maxwell equations are not about the vector potential but about the electromagnetic field
Then what is classical general relativity about? Perhaps only about Ricci scalar ##R##, because tensors like ##R_{\mu\nu}## and ##g_{\mu\nu}## are not invariant under diffeomorphisms?
DrDu
Then what is classical general relativity about? Perhaps only about Ricci scalar ##R##, because tensors like ##R_{\mu\nu}## and ##g_{\mu\nu}## are not invariant under diffeomorphisms?
But they are observables, in contrast to the vectorpotential.
Demystifier
Gold Member
But they are observables, in contrast to the vector potential.
They are observables only when one fixes coordinates. Similarly, the vector potential is also an observable when one fixes a gauge.
bhobba
DrDu
First there is a difference between coordinates and gauge. Second, a tensor is not coordinate dependent, only its components with respect to a basis are.
What would be more appropriate to compare is the vector potential A in QED and the Christoffel connection in GR.
A. Neumaier
One solves the Maxwell equation in terms of $A_{\mu}$; quantizes $A_{\mu}$; classical and quantum interaction is given by $A_{\mu}J^{\mu}$; the successes of QED is achieved by imposing subsidiary condition on $A_{\mu}$, and you now come and tell me it is “ not about the vector potential”.
The fields that occur in the Maxwell equations do not determine the vector potential. Thus the equations for the vector potential are not the same as those for the electromagnetic field. The latter are called the Maxwell equations.The free massless spin 1 Wightman theory is about the free electromagnetic field, and it satisfies precisely the Maxwell equations in operator form.
QED is not about the Maxwell equations.
samalkhaiat
The fields that occur in the Maxwell equations do not determine the vector potential. Thus the equations for the vector potential are not the same as those for the electromagnetic field.The latter are called the Maxwell equations.
Maxwell’s electromagnetism is an Abelian gauge theory. So, up to gauge transformation $\mbox{A} \to \mbox{A} + \mbox{d}\lambda$, the free Maxwell equations $\mbox{d}\mbox{F} = 0, \ \ \delta \mbox{F} \equiv {}^{*} \mbox{d}^{*}\mbox{F} = 0$ (on the topologically trivial spacetime $\mathbb{R}^{4}$) are completely equivalent to the free Maxwell equations $\mbox{d}^{2}\mbox{A} = 0, \ \ \delta \mbox{d}\mbox{A} = 0$. In other words, All electromagnetic phenomena are describable by some $A_{\mu}(x)$. Is this news to you?
The free massless spin 1 Wightman theory is about the free electromagnetic field, and it satisfies precisely the Maxwell equations in operator form.
Therefore, by the above equivalence, what is true for the Maxwell equation $\partial^{\nu}F_{\mu\nu} = 0$ is also true for the Maxwell equation $( \eta^{\mu\nu}\partial^{2} - \partial^{\mu}\partial^{\nu})A_{\nu} = 0$. Thus, by proposition (2) of my previous post, neither the latter nor the former form of Maxwell equation can survive quantization. A modification of the Maxwell equation, such as for example, $$\partial_{\nu}F^{\mu\nu} + \partial^{\mu}B = J^{\mu}, \ \ \ \ \ (1)$$$$\partial^{\mu}A_{\mu} + \alpha B = 0, \ \ \ \ \ (2)$$ is indispensable in quantized theory. In this B-field formalism, (1) is called the quantum Maxwell equation. And, eq(2) tells you that the Lorenz condition holds as operator equation only in the Landau-gauge $\alpha = 0$.
So, In any (non-abelian) gauge theory, the classical (non-abelian) Gauss’ law (constraint) $$G^{a}(x) \equiv - (D_{i}E_{i})^{a}(x) = 0,$$ which is one of the (non-abelian) Maxwell equations, cannot hold as an operator equation since it cannot be made compatible with any acceptable commutation relations, rather it is implemented as a restriction on the allowed physical states: $G^{a}| \Psi \rangle = 0$.
Rigorous treatment of this subject can be found in the good book of F. Strocchi, “An introduction to non-perturbative foundations of quantum field theory”, Oxford, 2013.
QED is not about the Maxwell equations.
Don’t throw ambiguous statements at me. Write down for us a classical, Poincare invariant action integral describing the interaction between Maxwell’s field and some charged matter field, then answer the following: (i) Can you do it without using the vector potential? (ii) What is the name of the equation which follows from the action principle? Do you call it The Cucumber equation, or the Maxwell equation?
dextercioby
A. Neumaier
are completely equivalent to the free Maxwell equations [...] Is this news to you?
Again you are calling the free Maxwell equations the equations in the vector potential A, while Maxwell wrote down equations in E and B, later combined into the electromagnetic field tensor F.
It is indeed news to me that your equations are completely equivalent to those of Maxwell, since they aren't. From the A's you get the F's, yes, but this does not yet make an equivalence. From the F's you get an infinite collection of possible A's, so the equations are already not equivalent classically, but the A's contain redundant information that must be gauged away. A is not a field, as - in your description - it is determined by the observable fields only up to a residual gauge freedom.
Thus your arguments about the A-equation in QFT do not imply anything about the F-equations. Indeed, these hold in QFT on the operator level, completely unchanged!
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A. Neumaier
Write down for us a classical, Poincare invariant action integral describing the interaction between Maxwell’s field and some charged matter field
I didn't claim anything about your setup. QED is not about the Maxwell equations but about a bigger system of equations involving a fermionic field not known before 1925. Such a field does not figure in Maxwell's equations. Neither do Maxwell's equations demand a derivation from an action principle; they stand for themselves.
A. Neumaier
Not an expert in QM/QFT but the Aharonov–Bohm effect certainly illustrates that the EM vector A exists and is physically relevant.
https://en.wikipedia.org/wiki/Aharonov–Bohm_effect
We were discussing primarily the free electromagnetic field in Minkowski space, and in that case there is no Aharonov–Bohm effect.
In any case, physical information is only in the corresponding Wilson loop, not in A itself.
We were discussing primarily the free electromagnetic field in Minkowski space, and in that case there is no Aharonov–Bohm effect.
In any case, physical information is only in the corresponding Wilson loop, not in A itself.
I'll have to think about that, and what "physical information" means. It seems to me that all "physical" effects are indirectly measured.
A. Neumaier
I'll have to think about that, and what "physical information" means. It seems to me that all "physical" effects are indirectly measured.
A is defined only up to a gauge transformation, and all physical information must be gauge invariant.
The same Lagrangian density function ##\mathcal{L}## that is used in deriving the classical Maxwell equations is also the basis of quantum electrodynamics. A "wave function" describing a quantum field would actually need to be a set of infinitely many functions, one for every point in space, so that you could form a probability density for the field strengths at any point.
I don't think he was looking for an exact answer. :-)
samalkhaiat
Not an expert in QM/QFT but the Aharonov–Bohm effect certainly illustrates that the EM vector A exists and is physically relevant.
https://en.wikipedia.org/wiki/Aharonov–Bohm_effect
The dilemma is the following:
A) The tensor $F_{\mu\nu}$ is observable physical field. However, as dynamical variables $F_{\mu\nu}$ gives incomplete description in the quantum theory.
B) The vector potential $A_{\mu}$ is not an observable. But, as dynamical variables, it was found to give a full (classical and quantum) description of the physical phenomena.
Indeed, this state of affair was demonstrated nicely by the Aharonov-Bohm effect:
Classical electrodynamics can be described entirely in terms of $F_{\mu\nu}$: Once the value of $F_{\mu\nu}(x)$ at a point $x$ is given, we know exactly how a charged particle placed at $x$ will behave. We simply solve the Lorentz force equation. This is no longer the case in the quantum theory. Indeed, in the A-B effect, the knowledge of $F_{\mu\nu}$ throughout the region traversed by the electron is not sufficient for determining the phase of the electron wave function, without which our description will be incomplete. In other words, $F_{\mu\nu}$ under-describes the quantum theory of a charged particle moving in an electromagnetic field. This is why we use the vector potential $A_{\mu}$ as dynamical variable in the A-B effect as well as in QFT. However, the vector potential has the disadvantage of over-describing the system in the sense that different values of $A_{\mu}$ can describe the same physical conditions. Indeed, if you replace $A_{\mu}$ by $A_{\mu} + \partial_{\mu}f$ for any function $f$, you will still see the same diffraction pattern on the screen in the A-B experiment. This shows that the potentials $A_{\mu}(x)$, which we use as dynamical variables, are not physically observable quantities. In fact, even the phase difference at a point is not an observable, a change by an integral multiple of $2\pi$ leaves the diffraction pattern unchanged. The real observable in the A-B effect is the Dirac phase factor $$\Phi (C) = \exp \left( i e \oint_{C} dx^{\mu} A_{\mu}(x) \right) .$$ Just like $F_{\mu\nu}$, $\Phi$ is gauge invariant, but unlike $F_{\mu\nu}$, it gives correctly the phase effect of the electron wave function.
bhobba, dextercioby and Boing3000
samalkhaiat
Again you are calling the free Maxwell equations the equations in the vector potential A, while Maxwell wrote down equations in E and B, later combined into the electromagnetic field tensor F.
That was long time ago. We now regard Maxwell theory as an abelian gauge theory and study it in terms of $A_{\mu}$.
... that your equations are completely equivalent to those of Maxwell,
They are not “my equations”. They are the Maxwell equations written in terms of the gauge field. They are also called Maxwell’s equations in the literatures.
From the A's you get the F's, yes, but this does not yet make an equivalence. From the F's you get an infinite collection of possible A's,
This is just the poor man version of what I have already said. Let me repeat: Minkowski spacetime $M = \mathbb{R}^{(1,3)}$ is topologically trivial. This means that all de Rham cohomology groups are trivial:
$$H^{p}(M) \equiv \frac{\{ \mbox{closed p-forms} \}}{\{ \mbox{exact p-forms} \}} = 0.$$
Thus, on $M$, a form is exact if and only if it is closed (Poincare Lemma). So, up to gauge transformation, $\{ dF = \delta F = 0 \}$ if and only if $\{ d^{2}A = \delta dA = 0 \}$.
so the equations are already not equivalent classically,
They are, because $A$ and $A + d\lambda$ describe the same physics. Mathematically, we speak of equivalence classes with $A$ and $A + d\lambda$ are identified.
Thus your arguments about the A-equation in QFT do not imply anything about the F-equations. Indeed, these hold in QFT on the operator level, completely unchanged!
Almost all textbooks on QFT quantize the vector potential $A_{\mu}$ not the field tensor $F_{\mu\nu}$. Open one of those textbooks and find the expansion $$A_{\mu}(x) = \int \frac{d^{3}k}{2k_{0}(2 \pi)^{3}} \sum_{\beta = 0}^{3} a^{(\beta)}(k) \epsilon_{\mu}^{(\beta)}(k) e^{-ikx} + \mbox{H.C.} \ .$$ Now, if you calculate $F_{\mu\nu}$ from the above $A_{\mu}$, you find that $\epsilon^{\mu\nu\rho\sigma}\partial_{\nu}F_{\rho\sigma} = 0$ holds identically. However, you also find that the remaining Maxwell equations fail to hold. Indeed, you obtain $$\partial^{\mu}F_{\mu\nu} = - \partial_{\nu}(\partial \cdot A) \neq 0 \ .$$
So, I can summarise my “argument” in #8 by the following: The free Maxwell equation $\partial^{\mu}F_{\mu\nu}=0$ does not hold as operator equation in the usual covariant quantization of the em-field that one can find in almost all usual textbooks. And that is a complete answer to the remarks raised in #1.
The same thing happens in QED. To see that, consider a classical theory described by $\mathcal{L}(\varphi_{a} , A_{\mu})$ where $\varphi_{a} = ( \varphi , \varphi^{\ast})$ is a complex scalar field and $A_{\mu}$ is a massless vector field. Assume that our theory is invariant under the local (infinitesimal) transformations $$\delta \varphi_{a}(x) = i \epsilon (x) \varphi_{a}(x), \ \ \ \delta A_{\mu}(x) = \partial_{\mu}\epsilon (x),$$ with arbitrary spacetime-dependent function $\epsilon (x)$. Now we define the objects $$J^{\mu} \equiv \frac{\partial \mathcal{L}}{\partial (\partial_{\mu}\varphi_{a})} (i\varphi)_{a} \equiv \frac{\partial \mathcal{L}}{\partial (\partial_{\mu}\varphi)} (i\varphi) + \frac{\partial \mathcal{L}}{\partial (\partial_{\mu}\varphi^{\ast})}(-i\varphi^{\ast}) ,$$ and $$F^{\mu\nu} \equiv - \frac{\partial \mathcal{L}}{\partial (\partial_{\mu}A_{\nu})} .$$ With simple algebra, we find
\begin{align*}\delta \mathcal{L} & = \left( \mathcal{E}^{a}(\varphi) i\varphi_{a} + \partial_{\mu}J^{\mu} \right) \epsilon \\ & + \left( \mathcal{E}^{\mu}(A) + J^{\mu} - \partial_{\nu}F^{\nu\mu}\right) \partial_{\mu}\epsilon \\ & - \frac{1}{2} \left( F^{\mu\nu} + F^{\nu\mu}\right) \partial_{\mu}\partial_{\nu}\epsilon , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1) \end{align*} where $\mathcal{E}^{a}(\varphi)$ and $\mathcal{E}^{\mu}(A)$ are the Euler derivatives of the fields:
$$\mathcal{E}^{a}(\varphi) = \frac{\partial \mathcal{L}}{\partial \varphi_{a}} - \partial_{\mu}\left( \frac{\partial \mathcal{L}}{\partial (\partial_{\mu}\varphi_{a})}\right) ,$$ $$\mathcal{E}^{\mu}(A) = \frac{\partial \mathcal{L}}{\partial A_{\mu}} - \partial_{\nu}\left( \frac{\partial \mathcal{L}}{\partial (\partial_{\nu}A_{\mu})}\right) \equiv \frac{\partial \mathcal{L}}{\partial A_{\mu}} + \partial_{\nu}F^{\nu\mu} .$$ Thus, $\delta \mathcal{L} = 0$ if and only if each coefficient of $\epsilon$, $\partial_{\mu}\epsilon$ and $\partial_{\mu}\partial_{\nu}\epsilon$ vanishes identically.
So, from the first line in (1), we obtain the familiar Noether identity $$\mathcal{E}^{a}(\varphi) \ (i\varphi)_{a} + \partial_{\mu}J^{\mu} \equiv 0,$$ associated with invariance under the global $U(1)$ transformations $$\delta \varphi_{a}(x) = i\epsilon \varphi_{a}(x) \ , \ \ \ \ \delta A_{\mu}(x) = 0 .$$ Thus, on actual “trajectories”, i.e., when $\mathcal{E}^{a}(\varphi) = 0$, we have a locally-conserved current $\partial_{\mu}J^{\mu} = 0$ and time-independent global-charge $$Q = q \int d^{3}x \ J^{0}(x) \ . \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$$
The third line in (1) gives us $F_{\mu\nu} = - F_{\nu\mu}$, and the second line gives us the identity $$\mathcal{E}^{\mu}(A) + J^{\mu} - \partial_{\nu}F^{\nu\mu} \equiv \frac{\partial \mathcal{L}}{\partial A_{\mu}} + J^{\mu} \equiv 0 .$$ This shows that our Lagrangian $\mathcal{L}(\varphi , A)$ must contain a term proportional to $(-A_{\mu}J^{\mu})$ which shows that $\mathcal{L}(\varphi_{a} , A_{\mu})$ describes an interacting theory of massless vector field $A_{\mu}$ and an electrically charged scalar field $\varphi_{a}$. Of course, this is just a natural consequence of local gauge invariance. The identity also shows that the equation of motion followed by the field $A_{\mu}$, $$\mathcal{E}^{\mu}(A) = 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$$ is equivalent to the Maxwell equation $$\partial_{\nu}F^{\nu\mu} = J^{\mu} . \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)$$
With this, we end our discussions of the classical (Lagrangian) scalar electrodynamics. Also, the perturbative solution of (Lagrangian) scalar QED is discussed in many textbooks, so we don’t need to do that here. We will not bother ourselves with the vector potential $A_{\mu}$ or the underline Lagrangian. Instead, we consider a local QFT consisting of the electrically charged field $\varphi$, the Maxwell’s field tensor $F_{\mu\nu}$ and the conserved (electric) vector current $J^{\mu}$. Of course, all these fields are operator-valued distributions. The question we need to answer is the following: Are these fields together with the “operator” Maxwell’s equations (4) sufficient for non-trivial QED? The answer is negative and it is known for long time.
Theorem (Ferrari, Picasso & Strocchi): In any local QFT in which the Noether charge $$Q_{R} \equiv q J^{0}(f_{T}, f_{R}) = q \int d^{4}x \ f_{T}(x^{0})f_{R}(\vec{x}) J^{0}(x^{0}, \vec{x}) ,$$ generates non-trivial automorphism on the local field algebra, $$\lim_{R \to \infty} [ Q_{R} , \varphi (f)] = q \varphi (f),$$ the Maxwell equations $$\partial_{\nu}F^{\nu\mu} = J^{\mu},$$ cannot be valid.
The proof is very easy and can be done formally with no need for all those smearing test functions. Just substitute $J^{0} = \partial^{j}F_{j0}$ and use Stokes theorem, then the commutator vanishes by locality. This contradicts the assumption that $\varphi$ is a charged local field (i.e., transforms non-trivially under the group generated by $Q$). So, in order to keep Maxwell’s equations as valid operator equations, one has to abandon locality. This is the so called Coulomb gauge quantization. Thus, an unphysical local field operator, $$B^{\mu} = \partial_{\nu}F^{\nu\mu} - J^{\mu},$$ must necessarily be introduced in the local formulation of QED. When $B^{\mu} = - \partial^{\mu}(\partial \cdot A)$, this gives the usual Gupta-Bleuler formulation.
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Spinnor, bhobba, protonsarecool and 1 other person
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2021-04-19 11:25:46
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https://mathematica.stackexchange.com/questions/57526/creating-a-conditional-table
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# Creating a conditional table
I'm trying to create a conditional table. Let's say I want to have such result: {1,2,3,4,5,0,0,0,0,0}.
The idea is to create a table of n elements (10 in a given example), but when one element takes a specific value (5 in my example), then all of the remaining elements must take provided value (let's say zero).
It's important not to use IF checking every element whether it satisfies provided condition.
You wrote:
It's important not to use IF checking every element whether it satisfies provided condition.
I cannot agree with this, unless you mean that once the sought element is found the rest of the elements should not be checked (possibly) using If. What I mean is that even if not using If itself there is going to be some kind of by-element checking until the target value is found.
One approach to what I believe you want:
SeedRandom[0]
a = RandomInteger[9, 10]
{7, 0, 8, 2, 1, 5, 8, 0, 6, 7}
p = FirstPosition[a, 5][[1]]
Join[Take[a, p], ConstantArray[0, Length@a - p]]
{7, 0, 8, 2, 1, 5, 0, 0, 0, 0}
Or more concise but less efficient:
Join[Take[a, p], 0 Drop[a, p]]
{7, 0, 8, 2, 1, 5, 0, 0, 0, 0}
## Update
Based on your comments I believe this should be of use to you:
cTable[f_, n_] := FoldList[If[# == 0, 0, f @ #2] &, f @ 1, 2 ~Range~ n]
Example:
f = Mod[2 # + 1, 9] &;
cTable[f, 10]
{3, 5, 7, 0, 0, 0, 0, 0, 0, 0}
Note that f is only called four times here, not once for each element in the output. As proof we can add a Pause to it:
f = (Pause[1]; Mod[2 # + 1, 9]) &;
cTable[f, 10] // AbsoluteTiming
{4.010006, {3, 5, 7, 0, 0, 0, 0, 0, 0, 0}}
Because FoldList auto-compiles (by default for lists 100 or longer) this method should be acceptably fast. For example a list with nearly 5,000,000 zeros takes only a fraction of a second on my machine:
cTable[Mod[2 # + 1, 9] &, 5000000]; // AbsoluteTiming
{0.360001, Null}
• Yes, it's important not to check every element. I'm trying to create such table: Table[f[i],[i,1,N]]. The function f[i] is complicated and it takes long time to calculate it's value. However I know that if it takes the first value 0 as "i" varies from 1 to N, then the next values of it is also zero. If f[3]=0, then the result should be {f[1],f[2],0,..,0} - N elments in the list. – Fancier of Mathematica Aug 17 '14 at 18:59
• Also important that I'm not creating list from a list. – Fancier of Mathematica Aug 17 '14 at 19:02
• @Fan please see my updated answer. – Mr.Wizard Aug 18 '14 at 1:32
This is a more general pattern solution that doesn't require each value after five to be larger than five:
list = {9, 4, 9, 1, 2, 9, 5, 4, 4, 6};
list /. {a___, 5, b___} :> {a, 5, Sequence @@ ConstantArray[0, Length@{b}]}
(* Out: {9, 4, 9, 1, 2, 9, 5, 0, 0, 0} *)
• this seems to most accurately implement what the text describes +1 – ubpdqn Aug 17 '14 at 12:01
• @Artes The way I see it it's the only solution. – C. E. Aug 17 '14 at 15:15
• Let's say we do so: Table[f[i],{i,1,10}], with some given function f. It might be Sin[i], Cos[i] and etc. I want to do so: if f[i]=0, then f[i+1],..,f[10]=0. Not checking whether f[i+1],..,f[10]=0. If, for example, f[1]>0, f[2]>0, but f[3]=0, then the result should be {f[1],f[2],0,0,0,0,0,0,0,0}. – Fancier of Mathematica Aug 17 '14 at 15:51
• @Pickett Ok I provided another solution. – Artes Aug 17 '14 at 16:02
• @FancierofMathematica With this solution you can do that, and also with the new solution by Artes. – C. E. Aug 17 '14 at 16:12
This solves the problem as it has been posed:
list = {1, 2, 3, 7, 9, 11, 5, 3, 5, 9};
Join[ TakeWhile[ list, # != 5 &], {5},
ConstantArray[0, Length[list] - FirstPosition[ list, 5]]]
{1, 2, 3, 7, 9, 11, 5, 0, 0, 0}
In case the list consitst of consecutive elements:
Range @ 10 // # UnitStep[5 - #]&
{1, 2, 3, 4, 5, 0, 0, 0, 0, 0}
If we are to find larger values we can use Threashold
Threshold[ Range @ 10, {"LargestValues", 5}]
{0, 0, 0, 0, 0, 6, 7, 8, 9, 10}
• ... or Threshold[Range[10], {"Hard", 5}]. Pity that Treshhold doesn't find smallest values. Anyway, +1 – eldo Aug 17 '14 at 12:18
• @eldo Thanks, it seems the OP asked for something else, thus I updated the anser. – Artes Aug 17 '14 at 16:05
• I haven't benchmarked it but it looks costly to evaluate both FirstPosition and TakeWhile, another option: With[{fp = First@FirstPosition[list, 5]}, Join[list[[1 ;; fp]], ConstantArray[0, Length@list - fp]]] – C. E. Aug 17 '14 at 16:22
• Perhaps TakeWhile[list, # != 5 &] // Join[#, {5}, ConstantArray[0, Length@list - Length@# - 1]] & so that list is not crawled twice. – seismatica Aug 17 '14 at 19:26
• Clear[f1, f2, listTest]; listTest = Range[1, 10, 0.0001]; f1[list_, n_] := Join[TakeWhile[list, # != n &], {n}, ConstantArray[0, Length[list - FirstPosition[list, n]]]] // AbsoluteTiming // First; f2[list_, n_] := TakeWhile[list, # != n &] // Join[#, {n}, ConstantArray[0, Length@list - Length@# - 1]] & // AbsoluteTiming // First; Mean@Table[#[listTest, 5] & /@ {f1, f2}, {10}] screenshot – seismatica Aug 17 '14 at 19:54
The simplest code I can think of is:
Range@10 /. (x_ /; x > 5 :> 0)
{1, 2, 3, 4, 5, 0, 0, 0, 0, 0}
• Thank You. It's simple when you know :) – Fancier of Mathematica Aug 17 '14 at 11:27
ClearAll[f1, f2, f3, f4];
list = {1, 2, 3, 7, 9, 11, 5, 3, 5, 9};
SetAttributes[f1, {Listable}]
(* redefine f1 to 0& when an input with value t is processed: *)
f1[t_, x_] := Piecewise[{{f1 = 0 &; x, x == t}}, x]
f1[5 , list]
(* {1,2,3,7,9,11,5,0,0,0} *)
f2 = MapAt[0 &, #2, {1 + Position[#2, #1, 1, 1][[1, 1]] ;;}] &;
f2[5, list]
(* {1,2,3,7,9,11,5,0,0,0} *)
f3 = Function[{t, lst},
Module[{ca = ConstantArray[0, {Length@lst}],
lw = ;; 1 + LengthWhile[lst, # != t &]}, ca[[lw]] = lst[[lw]]; ca]];
f3[5, list]
(* {1,2,3,7,9,11,5,0,0,0} *)
f4 = Function[{t, lst}, Module[{splt = Split[lst, # != t &]},
splt[[2 ;;]] = 0 splt[[2 ;;]]; Join @@ splt]];
f4[5, list]
(* {1,2,3,7,9,11,5,0,0,0} *)
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2019-11-14 19:02:02
|
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|
https://www.rdocumentation.org/packages/memisc/versions/0.99.27.3/topics/collect
|
# collect
0th
Percentile
##### Collect Objects
collect gathers several objects into one, matching the elements or subsets of the objects by names or dimnames.
Keywords
manip, utilities
##### Usage
collect(…,names=NULL,inclusive=TRUE)
# S3 method for default
collect(…,names=NULL,inclusive=TRUE)
# S3 method for array
collect(…,names=NULL,inclusive=TRUE)
# S3 method for matrix
collect(…,names=NULL,inclusive=TRUE)
# S3 method for table
collect(…,names=NULL,sourcename=".origin",fill=0)
# S3 method for data.frame
collect(…,names=NULL,inclusive=TRUE,
fussy=FALSE,warn=TRUE,sourcename=".origin")
# S3 method for data.set
collect(…,names=NULL,inclusive=TRUE,
fussy=FALSE,warn=TRUE,sourcename=".origin")
##### Arguments
more atomic vectors, arrays, matrices, tables, data.frames or data.sets
names
optional character vector; in case of the default and array methods, giving dimnames for the new dimension that identifies the collected objects; in case of the data.frame and data.set methods, levels of a factor indentifying the collected objects.
inclusive
logical, defaults to TRUE; should unmatched elements included? See details below.
fussy
logical, defaults to FALSE; should it count as an error, if variables with same names of collected data.frames/data.sets have different attributes?
warn
logical, defaults to TRUE; should an warning be given, if variables with same names of collected data.frames/data.sets have different attributes?
sourcename
name of the factor that identifies the collected data.frames or data.sets
fill
numeric; with what to fill empty table cells, defaults to zero, assuming the table contains counts
##### Value
If x and all following … arguments are vectors of the same mode (numeric,character, or logical) the result is a matrix with as many columns as vectors. If argument inclusive is TRUE, then the number of rows equals the number of names that appear at least once in each of the vector names and the matrix is filled with NA where necessary, otherwise the number of rows equals the number of names that are present in all vector names.
If x and all … arguments are matrices or arrays of the same mode (numeric,character, or logical) and $$n$$ dimension the result will be a $$n+1$$ dimensional array or table. The extend of the $$n+1$$th dimension equals the number of matrix, array or table arguments, the extends of the lower dimension depends on the inclusive argument: either they equal to the number of dimnames that appear at least once for each given dimension and the array is filled with NA where necessary, or they equal to the number of dimnames that appear in all arguments for each given dimension.
If x and all … arguments are data frames or data sets, the result is a data frame or data set. The number of variables of the resulting data frame or data set depends on the inclusive argument. If it is true, the number of variables equals the number of variables that appear in each of the arguments at least once and variables are filled with NA where necessary, otherwise the number of variables equals the number of variables that are present in all arguments.
##### Aliases
• collect
• collect.default
• collect.array
• collect.matrix
• collect.table
• collect.data.frame
• collect.data.set
##### Examples
# NOT RUN {
x <- c(a=1,b=2)
y <- c(a=10,c=30)
x
y
collect(x,y)
collect(x,y,inclusive=FALSE)
X <- matrix(1,nrow=2,ncol=2,dimnames=list(letters[1:2],LETTERS[1:2]))
Y <- matrix(2,nrow=3,ncol=2,dimnames=list(letters[1:3],LETTERS[1:2]))
Z <- matrix(3,nrow=2,ncol=3,dimnames=list(letters[1:2],LETTERS[1:3]))
X
Y
Z
collect(X,Y,Z)
collect(X,Y,Z,inclusive=FALSE)
X <- matrix(1,nrow=2,ncol=2,dimnames=list(a=letters[1:2],b=LETTERS[1:2]))
Y <- matrix(2,nrow=3,ncol=2,dimnames=list(a=letters[1:3],c=LETTERS[1:2]))
Z <- matrix(3,nrow=2,ncol=3,dimnames=list(a=letters[1:2],c=LETTERS[1:3]))
collect(X,Y,Z)
collect(X,Y,Z,inclusive=FALSE)
df1 <- data.frame(a=rep(1,5),b=rep(1,5))
df2 <- data.frame(a=rep(2,5),b=rep(2,5),c=rep(2,5))
collect(df1,df2)
collect(df1,df2,inclusive=FALSE)
collect(Male,Female,sourcename="Gender")
collect(unclass(Male),unclass(Female))
collect(Male=Male1,Female=Female2,sourcename="Gender")
collect(Male=Male1,Female=Female3,sourcename="Gender")
collect(Male=Male1,Female=Female3,sourcename="Gender",fill=NA)
f1 <- gl(3,5,labels=letters[1:3])
f2 <- gl(3,6,labels=letters[1:3])
collect(f1=table(f1),f2=table(f2))
ds1 <- data.set(x = 1:3)
ds2 <- data.set(x = 4:9,
y = 1:6)
collect(ds1,ds2)
# }
Documentation reproduced from package memisc, version 0.99.27.3, License: GPL-2
### Community examples
Looks like there are no examples yet.
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2021-01-22 10:42:37
|
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|
https://forum.allaboutcircuits.com/threads/finding-the-energy-of-a-sequence.59152/
|
# Finding the Energy of a Sequence
#### tquiva
Joined Oct 19, 2010
176
I am given the following sequences:
I know the process in finding a, b, & c:
- (f(x))^2 where f(x) is the sequence
- Integrate in terms of x from limits 0 to n
However, when trying to integrate through Matlab or a calculator, I am unable to do so. The exponential of these sequences can't seem to be integrated? Am I doing this correctly?
#### Attachments
• 25 KB Views: 36
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2020-02-26 20:14:03
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https://zbmath.org/?q=an:0517.16011
|
# zbMATH — the first resource for mathematics
##### Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.
##### Operators
a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Units in Whitehead groups of finite groups. (English) Zbl 0517.16011
##### MSC:
16S34 Group rings (associative rings), Laurent polynomial rings 16U60 Units, groups of units (associative rings and algebras) 16E20 Grothendieck groups and $K$-theory of noncommutative rings 18F25 Algebraic $K$-theory and $L$-theory 11R23 Iwasawa theory
Full Text:
##### References:
[1] R. C. Alperin, R. K. Dennis, R. Oliver, and M. R. Stein, SK1 of finite abelian groups, II, to appear. [2] Bass, H.: Algebraic K-theory. (1968) [3] Hasse, H.: Uber die klassenzahl abelschen zahlkorper. (1952) [4] Higman, G.: The units of group rings. Proc. London math. Soc. 46, 231-248 (1940) · Zbl 0025.24302 [5] Jajodia, S.; Magurn, B. A.: Surjective stability of units and simple homotopy type. J. pure appl. Algebra 18, 45-58 (1980) · Zbl 0438.57008 [6] Keating, M. E.: On the K-theory of the quaternion group. Mathematika 20, 59-62 (1973) · Zbl 0267.18016 [7] Koblitz, N.: P-adic numbers, p-adic analysis, and zeta-functions. Gtm 58 (1977) · Zbl 0364.12015 [8] Lang, S.: Algebraic number theory. (1970) · Zbl 0211.38404 [9] Magurn, B. A.: SK1 of dihedral groups. J. algebra 51, No. 2, 399-415 (1978) · Zbl 0376.16026 [10] Magurn, B. A.: Images of SK1ZG. Pacific J. Math. 79, No. 2, 531-539 (1978) · Zbl 0398.16007 [11] Oliver, R.: See correction in invent. Math.. Invent. math. 64, 167-169 (1981) · Zbl 0455.18007 [12] Oliver, R.: SK1 for finite group rings, II. Math. scand. 47, 195-231 (1980) · Zbl 0456.16027 [13] Oliver, R.: SK1 for finite group rings, III. Lecture notes in mathematics no. 854, 299-337 (1981) [14] Oliver, R.: SK1 for finite group rings, IV. Proc. London math. Soc. 46, 1-37 (1983) · Zbl 0499.16017 [15] Reiner, I.: Maximal orders. (1975) · Zbl 0305.16001 [16] Swan, R. G.: Strong approximation and locally free modules. Ring theory and algebra III, Proceedings of the third oklahoma conference, 153-223 (1980) [17] Vaserstein, L. N.: Math. USSR sb.. 8, No. No. 3, 383-400 (1969) [18] Vaserstein, L. N.: The structure of classical arithmetic groups of rank greater than one. Math. USSR sb. 20, 465-491 (1973) · Zbl 0291.14016 [19] Vaserstein, L. N.: Stable rank of rings and dimensionality of topological spaces. Funct. anal. Appl. 5, 102-110 (1971) · Zbl 0239.16028 [20] Wall, C. T. C: On the classification of Hermitian forms: III: Complete semilocal rings. Invent. math. 19, 59-71 (1973) · Zbl 0259.16013 [21] Wall, C. T. C: Norms of units in group rings. Proc. London math. Soc. 29, 593-632 (1974) · Zbl 0302.16013 [22] Weil, A.: Basic number theory. Die grundlehren der math. Wissenschaften 144 (1967) [23] Weiss, E.: Algebraic number theory. (1963) · Zbl 0115.03601 [24] Williamson, S.: Crossed products and hereditary orders. Nagoya math. J. 23, 103-120 (1963) · Zbl 0152.02002 [25] Wilson, S. M. J: Reduced norms in the K-theory of orders. J. algebra 46, 1-11 (1977) · Zbl 0358.16021
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2016-05-06 15:07:17
|
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https://socratic.org/questions/how-do-you-name-ticl-4
|
# How do you name TiCl_4?
Jul 9, 2016
$\text{Titanic chloride}$
#### Explanation:
Also known as $\text{Titanium (IV) chloride}$. It is a distillable liquid.
You don't see it around much anymore (probably because of environmental concerns), but titanic chloride was the chemical that was used for sky-writing, i.e. writing messages in the sky, by light aircraft.
$T i C {l}_{4} \left(l\right) + 2 {H}_{2} O \rightarrow T i {O}_{2} \left(s\right) + 4 H C l \left(g\right)$
Vast quantities of titanium dioxide (or $\text{titanic oxide}$) are used in the paints and pigments industry. It gives the colour that is whiter than white.
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2022-01-18 15:55:45
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https://wenbo.tv/pte%E5%8F%A3%E8%AF%AD%E6%9C%80%E6%96%B0%E7%9C%9F%E9%A2%98%E9%9F%B3%E9%A2%91%E8%A7%A3%E8%AF%BB%E7%B3%BB%E5%88%97-di-floorplan-of-a-school/
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# PTE口语最新真题音频解读系列-DI-Floorplan of a school
The picture gives information about the floorplan of a school. From the picture, we can find that there are two stairs and many labs. We may climb up the stairs from the bottom left corner of the picture, and find a staff office on the left-hand side and an OMVPE lab on the right-hand side. It is also worth noticing that there are two photolithography labs if we move further down the corridor. If we climb up the stairs from the top left corner of the picture, we can find a student office and a water plant. Overall, we can draw a conclusion that the layout of the school is crystal-clear.
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2019-09-22 14:59:00
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https://jameshoward.us/2015/04/14/why-i-bought-a-mega-millions-ticket/
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Why I Bought a Mega Millions Ticket | James Howard Why I Bought a Mega Millions Ticket | James Howard
# Why I Bought a Mega Millions Ticket
Yesterday’s post needed a picture of a Maryland lottery ticket for illustration. I couldn’t find a good one online, so I dropped a dollar and picked one up. So why did I pick a Mega Millions ticket, rather than Powerball? Powerball sounds cooler. What gives?
Someone’s gonna win, and it might as well be me. If I must purchase a ticket, I want to maximize my potential earnings. Here’s a spreadsheet showing the expected returns from the two major jackpot games in Maryland. There are three substantial concerns I’ve left out:
1. Accepting the lump sum versus the annuity,
2. The income reduction due to taxes, and
3. The effects of parimutuel splitting.
Business Insider has covered this thoroughly in the case of Powerball. However, these concerns affect the returns to each proportionally, so these can be assumed away. Due to the structure and payout of each game, the different ticket prices, and the different jackpot amounts (Mega Millions is $47 million whereas Powerball is$40 million), the Mega Millions ticket was a better ticket to buy. But not just because it is cheaper.
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2021-07-29 21:40:57
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http://mymathforum.com/linear-algebra/12191-computing-rank-matrix-quickly.html
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My Math Forum Computing the rank of this matrix quickly?
Linear Algebra Linear Algebra Math Forum
April 9th, 2010, 02:59 PM #1 Senior Member Joined: Dec 2009 Posts: 150 Thanks: 0 Computing the rank of this matrix quickly? How do I quickly find the rank of this matrix? A = $\left( \begin{array}{ccc} 1 & 2 & 3 & 4 & 5 \\ 6 & 7 & 8 & 9 & 10 \\ 11 & 12 & 13 & 14 & 15 \\ 16 & 17 & 18 & 19 & 20 \\ 21 & 22 & 23 & 24 & 25 \end{array} \right)$ element $a_{ij}= j + 5(i-1)$ and subtracting from each of the rows i > 1 a multiple of row 1 that gives 0 in the first column for each row after i >1 we have that $a'_{ij} = a_{ij} - a_{i1}a_{1j} = j + 5(i-1) - (1 + 5(i-1))*j = 5(i-1)(1-j)$ for i > 2. This gives similar matrix A' A' = $\left( \begin{array}{ccc} 1 & 2 & 3 & 4 & 5 \\ 0 & -5 & -10 & -15 & -20 \\ 0 & -10 & -20 & -30 & -40 \\ 0 & -15 & -30 & -45 & -60 \\ 0 & -20 & -40 & -60 & -80 \end{array} \right)$ Divide each of these elements by -5(i-1) = 5(1- i) (which you can do because you're dividing each element of the same row by the same number) and you get that (for each row after row 1) the elements are just j-1 , row 2,3,4, and 5 become [0 1 2 3 4]. So the last 3 rows are redundant, and thus the rank is 2. That is A' becomes similar matrix A'': A'' = $\left( \begin{array}{ccc} 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 & 4 \end{array} \right)$
April 10th, 2010, 11:39 PM #2 Member Joined: Nov 2009 From: France Posts: 98 Thanks: 0 Re: Computing the rank of this matrix quickly? You can take the line $i$ and substract the line $i-1$ for $i$ from $2$ to $5$. You will obtain : $A' = \begin{pmatrix} 1&2&3&4&5\\ 5&5&5&5&5\\ 5&5&5&5&5\\ 5&5&5&5&5\\ 5&5&5&5&5 \end{pmatrix}$
April 12th, 2010, 02:39 PM #3 Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0 Re: Computing the rank of this matrix quickly? Oh, nicely done.
April 16th, 2010, 12:30 PM #4 Newbie Joined: Apr 2010 Posts: 11 Thanks: 0 Re: Computing the rank of this matrix quickly? smart
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2017-09-21 01:37:37
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http://timothyandrewbarber.blogspot.com/2010/12/latex-line-and-page-breaking.html
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## Sunday, December 12, 2010
### LaTeX Line and Page Breaking
I did not write any of this by the way. I have only reposted from the following site:
did not write any of this by the way. I have only reposted from the following site:
LaTeX Line and Page Breaking
The first thing LaTeX does when processing ordinary text is to translate your input file into a string of glyphs and spaces. To produce a printed document, this string must be broken into lines, and these lines must be broken into pages. In some environments, you do the line breaking yourself with the \\ command, but LaTeX usually does it for you. The available commands are
\\ start a new paragraph.
\\* start a new line but not a new paragraph.
\- OK to hyphenate a word here.
\cleardoublepage flush all material and start a new page, start new odd numbered page.
\clearpage plush all material and start a new page.
\hyphenation enter a sequence pf exceptional hyphenations.
\linebreak allow to break the line here.
\newline request a new line.
\newpage request a new page.
\nolinebreak no line break should happen here.
\nopagebreak no page break should happen here.
\pagebreak encourage page break.
\\
\\[*][extra-space]
The \\ command tells LaTeX to start a new line. It has an optional argument, extra-space, that specifies how much extra vertical space is to be inserted before the next line. This can be a negative amount. The \\* command is the same as the ordinary \\ command except that it tells LaTeX not to start a new page after the line.
\-
The \- command tells LaTeX that it may hyphenate the word at that point. LaTeX is very good at hyphenating, and it will usually find all correct hyphenation points. The \- command is used for the exceptional cases, as e.g.
man\-u\-script
\cleardoublepage
The \cleardoublepage command ends the current page and causes all figures and tables that have so far appeared in the input to be printed. In a two-sided printing style, it also makes the next page a right-hand (odd-numbered) page, producing a blank page if necessary.
\clearpage
The \clearpage command ends the current page and causes all figures and tables that have so far appeared in the input to be printed.
\hyphenation
\hyphenation{words}
The \hyphenation command declares allowed hyphenation points, where words is a list of words, separated by spaces, in which each hyphenation point is indicated by a - character, e.g.
\hyphenation{man-u-script man-u-stripts ap-pen-dix}
\linebreak
\linebreak[number]
The \linebreak command tells LaTeX to break the current line at the point of the command. With the optional argument, number, you can convert the \linebreak command from a demand to a request. The number must be a number from 0 to 4. The higher the number, the more insistent the request is.The \linebreak command causes LaTeX to stretch the line so it extends to the right margin.
\newline
The \newline command breaks the line right where it is. The \newline command can be used only in paragraph mode.
\newpage
The \newpage command ends the current page.
\nolinebreak
\nolinebreak[number]
The \nolinebreak command prevents LaTeX from breaking the current line at the point of the command. With the optional argument, number, you can convert the \nolinebreak command from a demand to a request. The number must be a number from 0 to 4. The higher the number, the more insistent the request is.
\nopagebreak
\nopagebreak[number]
The \nopagebreak command prevents LaTeX form breaking the current page at the point of the command. With the optional argument, number, you can convert the \nopagebreak command from a demand to a request. The number must be a number from 0 to 4. The higher the number, the more insistent the request is.
\pagebreak
\pagebreak[number]
The \pagebreak command tells LaTeX to break the current page at the point of the command. With the optional argument, number, you can convert the \pagebreak command from a demand to a request. The number must be a number from 0 to 4. The higher the number, the more insistent the request is.
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2017-12-14 20:35:49
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http://jaoa.org/article.aspx?articleid=2498826
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Review | March 2016
The Glymphatic-Lymphatic Continuum: Opportunities for Osteopathic Manipulative Medicine
Author Notes
• From the Department of Biomedical Sciences (Student Doctors Hitscherich, Smith, Cuoco, and Ruvolo and Drs Leheste and Torres) and the Department of Osteopathic Manipulative Medicine (Dr Mancini) at the New York Institute of Technology College of Osteopathic Medicine (NYITCOM) in Old Westbury.
• Support: Financial support for this work was provided in part by the Department of Biomedical Sciences at NYITCOM.
• *Address correspondence to German Torres, PhD, Department of Biomedical Sciences, Division of Preclinical Medical Education, NYITCOM, PO Box 8000, Old Westbury, NY 11568-8000. E-mail: torresg@NYIT.edu
Article Information
Neuromusculoskeletal Disorders / Osteopathic Manipulative Treatment
Review | March 2016
The Glymphatic-Lymphatic Continuum: Opportunities for Osteopathic Manipulative Medicine
The Journal of the American Osteopathic Association, March 2016, Vol. 116, 170-177. doi:10.7556/jaoa.2016.033
The Journal of the American Osteopathic Association, March 2016, Vol. 116, 170-177. doi:10.7556/jaoa.2016.033
Web of Science® Times Cited: 1
Abstract
The brain has long been thought to lack a lymphatic drainage system. Recent studies, however, show the presence of a brain-wide paravascular system appropriately named the glymphatic system based on its similarity to the lymphatic system in function and its dependence on astroglial water flux. Besides the clearance of cerebrospinal fluid and interstitial fluid, the glymphatic system also facilitates the clearance of interstitial solutes such as amyloid-β and tau from the brain. As cerebrospinal fluid and interstitial fluid are cleared through the glymphatic system, eventually draining into the lymphatic vessels of the neck, this continuous fluid circuit offers a paradigm shift in osteopathic manipulative medicine. For instance, manipulation of the glymphatic-lymphatic continuum could be used to promote experimental initiatives for nonpharmacologic, noninvasive management of neurologic disorders. In the present review, the authors describe what is known about the glymphatic system and identify several osteopathic experimental strategies rooted in a mechanistic understanding of the glymphatic-lymphatic continuum.
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2016-12-08 20:00:51
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http://en.wikipedia.org/wiki/Effective_radius
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Half light radius Re encloses half of the total light emitted by an object
The effective radius ($R_e$) of a galaxy is the radius at which half of the total light of the system is emitted.[1][2] This assumes the galaxy has either intrinsic spherical symmetry or is at least circularly symmetric as viewed in the plane of the sky. Alternatively, a half-light contour, or isophote, may be used for spherically and circularly asymmetric objects.
$R_e$ is an important length scale in de Vaucouleurs $\sqrt[4] R$ law, which characterizes a specific rate at which surface brightness decreases as a function of radius:
$I(R) = I_e \cdot e^{-7.67 \left( \sqrt[4]{\frac R {R_e}} - 1 \right)}$
where $I_e$ is the surface brightness at $R = R_e$. Note that at $R = 0$,
$I(R=0) = I_e \cdot e^{7.67} \approx 2000 \cdot I_e$
Thus, the central surface brightness is approximately $2000 \cdot I_e$.
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2015-03-31 03:35:41
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https://gateoverflow.in/2302/gate1993-8-4
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870 views
Let A be a finite set of size n. The number of elements in the power set of $A\times A$ is:
1. $2^{2^n}$
2. $2^{n^2}$
3. $(2^n)^2$
4. $(2^2)^n$
5. None of the above
edited | 870 views
Cardinality of $A\times A = n^2$
Cardinality of power set of $A\times A = 2^{n^2}$
selected by
+3
Say set is A= {1,2,3}=n
Subset of set A ={phi, {1} , {2} , {3} , {1,2} , {2,3}, {1,3} , {1,2,3}}=2n
Now, $A\times A=\left \{ 1,2,3 \right \}\times \left \{ 1,2,3 \right \}$
$=\left \{ \left \{ 1,1 \right \},\left \{ 1,2 \right \} \left \{ 1,3 \right \}\left \{ 2,1 \right \}\left \{ 2,2 \right \}\left \{ 2,3 \right \}\left \{ 3,1 \right \}\left \{ 3,2 \right \}\left \{ 3,3 \right \}\right \}$
So, number of subsets will be$2^{n^{2}}$
0
This is same as the total number of relations on Set A
$|A| = n$ , $|p(A)| = 2^n$
$| A×A| =n^2$
$| p(A×A)|$ =$2^{n^{2}}$
edited
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2018-09-18 20:02:07
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https://www.nature.com/articles/s41598-022-23722-8?error=cookies_not_supported&code=35c6f1b9-9afb-428f-9f2e-6e8b800f2a89
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## Introduction
Activated carbon has been recognized as one of the widely applied adsorbents for spray coating1, food processing2, biomass3, pharmaceuticals4, chemicals5, waste water treatment6 petroleum7, and nuclear industries8, as well as the glycol treatment for removing organic and volatile organic compounds (VOCs) pollutants in natural gas industries9. Activated carbon is used in natural gas sweetening and dehydration systems with fluids such as amines and glycols10,11. The conventional techniques for the regeneration of activated carbon include thermal volatilization12, chemical extraction13, ultrasound14, microwave15, electrochemical16, and bio-regeneration. These methods have several disadvantages such as loss of carbon, damage of its porous structure , treatment of exhaust gases, chemical regenerations using solvents are not necessarily acceptable because additional separation and environmental problems and bio-regeneration requires long reaction time for regeneration. Recently, supercritical fluids (SCFs) has attracted widespread attention in many fields and regeneration of activated carbon as one of the applications of this technology has been studied17. The unique characteristics of SCFs have made these solvents attractive. Particularly, solvent density, and hence its dissolvent properties, can be controlled modifying pressure and temperature. Besides, liquid-like density and gas-like viscosity, coupled with diffusion coefficients that are at least an order of magnitude higher than those of liquids, contribute to the enhancement of the mass transfer processes18,19,20,21,22,23. Among the different substances, carbon dioxide is the best choice as it is an environment-friendly solvent providing such advantages as nontoxicity and high chemical stability.
Regeneration of activated carbon using SCFs has been studied by several researchers. DeFilippi et al. observed that the supercritical regeneration was economical even though the operating temperature and pressure were above 387 K and 150 atm, respectively. They proposed a local equilibrium model (Freundlich isotherm) fitting well with the experimental data24. Regeneration of activated carbon loaded with phenol using supercritical carbon dioxide was studied by Kander and Paulaitis25. They found that supercritical carbon dioxide offered no significant advantages for the regeneration of carbon loaded with phenol. However, they suggested that for organic compounds which are not adsorbed strongly onto activated carbon, supercritical carbon dioxide would be a powerful adsorbent. Tan and Liou investigated desorption by supercritical carbon dioxide of activated carbon loaded with either ethyl acetate or toluene. They presented that this regeneration method would give better results than the steam regeneration method and accordingly, presented a linear desorption kinetics model that was found to fit experimental data quite well17,26.
Madras et al.27 studied the desorption of several nonvolatile solids such as hexachlorobenzene and pentachlorophenol via break through experiments at a fixed carbon dioxide density. Macnaughton and Foster28 measured the adsorption equilibrium of 1,1,1-trichloro-2,2-bis (p-chlorophenyl) ethane (DDT) on an activated carbon and illustrated that one limiting factor on supercritical desorption process is adsorption equilibrium at a low CO2 flow rate. Using a three-parameter mode, the adsorption breakthrough and equilibrium of ethylbenzene on an activated carbon from supercritical carbon dioxide was investigated by Harikrishnan et al.29. Tan and Liou26 studied desorption behavior of ethyl acetate from activated carbon in CO2 at temperatures from 300 to 338 K and pressures from 8.7 to 12.9 MPa. They observed that increasing pressure and decreasing temperature increased the desorption rate. Benkhedda et al.30 also presented similar temperature and pressure dependence for m-xylene desorption behavior using SC-CO2 at temperatures from 313 to 333 K and pressures from 10.0 to 15.0 MPa. They suggested that desorption behavior of VOCs from activated carbons using SC-CO2 largely depends on temperature and pressure conditions. Salvador et al.31 studied the regeneration of activated carbons contaminated with phenol using supercritical water at 260 bar and temperature ranging between 400 and 500 °C. Regeneration temperature and time were the variables that most affected the process. Ikuo Ushiki et al.9 investigated the desorption behavior of various VOCs (toluene, acetone, n-hexane, n-octane, methanol, ethanol, 2-propanol, and propylene glycol monomethyl ether) from activated carbon using a fixed-bed method employing SC-CO2 at temperature values ranging between 313 and 353 K and pressure values ranging between 10.0 and 15.0 MPa.
Several factors such as pressure, temperature, CO2 flow rate, time, particle size, among others, simultaneous influenced the supercritical regeneration of activated carbons. In most of the previously published studies, the process conditions have just been optimized by conducting one-factor- at-a-time experiments. The results of one-factor-at-a-time experiments do not reflect real changes in the environment as they ignore interactions between the different variables of the process. Thus, to avoid this problem, the design, optimization and assessment of the process using design of experiment (DOE) seems to be essential32. Response surface methodology (RSM) is a mathematical tool to specify the effect of each variable along with their interactions, on the yield of process, and to predict its behavior under a given sets of conditions. The main advantage of the RSM is the reduced number of experiments required to optimize a process. RSM including Central Composite Design (CCD) and Box–Behnken Design (BBD) have been applied to optimize a supercritical fluid extraction (SFE) process. CCD is usually more expensive than BBD due to requires a higher number of runs than BBD33.
However, despite the fact that there can be various contaminants that need to be removed to regenerate activated carbons, most of these works investigated polluting substances intentionally added to the activated carbon. To our knowledge this is the first study that propose the use of SC-CO2 to regenerate activated carbon used in the gas industry, developing a comprehensive study to optimize and modeling the process. CCD design as a powerful experimental tool was applied to evaluate the influence of operating parameters on the regeneration process. In addition, the process was modeled via mathematical modeling by correlating the experimental data. Moreover, the extract obtained from the SC-CO2 method was characterized by GC and the physical properties of activated carbon were studied by SEM and iodine number methods, respectively.
## Material and methods
### Materials
The activated carbon samples used in this study were taken, at summer 2017 during three months, from a gas dehydration unit of the South Zagros Oil and Gas Company (Iran). It should be remarked that, in gas the gas dehydration units activated carbon is used to remove hydrocarbons and aromatic compounds in triethylene glycol. CO2 (99.90% purity) was purchased from Aboughadareh Gas Factory.
### Dehydration of natural gas using triethylene glycol
Raw natural gas is fully saturated with water vapor when produced from an underground reservoir. Because most of the water vapor has to be removed from natural gas before it can be commercially marketed, natural gas is subjected to a dehydration process. One of the most substances used for removing the water from produced gas is glycol. Figure 1 shows a scheme of the typical equipment used for the dehydration process using glycol. While the overall process equipment is similar for all glycol dehydration units, there can be considerable variation among installations. The gas flows through a separator to remove condensed liquids or any solids that might be in the gas. Some absorbers incorporate the separator at its bottom section, in which case the gas then flows upward through a chimney tray into the glycol absorber portion of the vessel. The glycol contactor provides the close contact between the gas and the glycol. The glycol is highly hygroscopic, and most of the water vapor in the gas is absorbed by the glycol. The water-rich glycol is withdrawn from the contactor near the bottom of the vessel above the chimney tray through a liquid level control valve and passes to the regeneration section. The treated gas leaves the contactor at the top through a mist eliminator and usually meets the specified water content. The water-rich glycol can be routed through a heat exchange coil in the top of the reboiler column called the steel column. The heat exchange generates some reflux for the separation of the water from the glycol in the top of the steel and heats the rich glycol somewhat. In some installations, the rich solution passes to a flash tank operating pressure at about 15 to 50 psig, which allows to the absorbed hydrocarbon gas to separate from the glycol. The glycol then flows into the still through an activated carbon filter and a heat exchanger, exchanging heat with the regenerated glycol. Then, it drops through a packed section in the still into the glycol reboiler vessel where it is heated to the necessary high regeneration temperature to near atmospheric pressure. At the high temperature, the glycol loses its ability to hold water; the water is vaporized and leaves through the top of the still. The regenerated glycol flows to the surge tank, from which it is routed through the lean/rich heat exchanger to the glycol pump. The pump boosts the pressure of the lean glycol to the contactor pressure. Prior to entering the contactor, it exchanges heat with the dry gas leaving the contactor or some other heat exchange medium34.
### Supercritical fluid extraction method
The main components of the apparatus were a CO2 cylinder (E-1), needle valve (E-2), molecular sieve filter (E-3), refrigerator unit (E-4), air compressor (E-5), high-pressure pump (air-driven liquid pump, Haskel, USA) (E-6), oven (Memert) (E-7), Surge tank (E-8), stainless-steel impregnation cell (E-9), back-pressure valve (Xi'an Shelok Instrument Technology Co., Shaanxi, China) (E-10), Flowmeter (E-11), Sampler (E-12), and automation system (E-13). At first, CO2 from the reservoir passed through a filter that is rated as a 1µ nominal size to avoid passing any impurity to the process path. Then, CO2 was guided towards a refrigerator unit, in which its temperature was drop to -10 °C and 60 Bar. The liquefied CO2 was pressurized using a reciprocating pump, and translated towards shell and tube configuration surge Drum. At this drum the fluid temperature was set to proper temperature to reach the supercritical fluid condition. This drum has been designed to reduce the fluctuations and the heat exchange happens by circulating water using 1000 W elements with a temperature accuracy setting of 1 K. A pressure gauge was installed at the output of surge drum to illustrate the outlet fluid pressure. Afterward the fluid was guided towards an extraction cell loaded with 10 g of activated carbon along with glass beads. At the input and output of the extraction cell two filters having Porosity of 1 micron have been installed to prevent the escaping of particles. Existence of glass beads in cell allowed the homogenous distribution of CO2 inside the extraction cell. After loading the extraction cell with SC-CO2, the cell was set at appropriate temperature and pressure for about 45 min. Following the static stage, the dynamic one begins by opening the sample valve. To prevent the valve from freezing, it was wrapped in a heating element. During the dynamic time, sample collection temperature was kept below 0 °C by using ice booth. Each experiment was done in triplicate35. Figure 2 shows the details of supercritical fluid apparatus. The yield of supercritical recovery was calculated through dividing the solute weight per the initial activated carbon. Weight of initial solute in Eq. (1) was determined as follows, first, a known mass of activated carbon was loaded into the filter. Then, after three months, the active carbon is going out from the filter. The difference in the primary and secondary weight of activated carbon indicates the initial value in Eq. (1).
$$Yield \,(\% ) = \frac{Weight\,of\,extracted\,solute }{{Weight\,of\, initial\,solute }} \times 100$$
(1)
### Design of experiments
Response surface methodology (RSM) was employed to evaluate the effects of pressure (A), temperature (B), CO2 flow rate (C) and time (D) on the regeneration yield. The coded and uncoded independent parameters applied in the RSM and their respective levels were tabulated in Table 1. The remaining independent parameters (e.g., particle size and volume of cell) were kept constant during the experimental procedures. The experimental design was based on the central composite design (CCD) using a 30 factorial and star design with six central points as shown in Table 2. Although the cost of this method is more than the Box Behnken method, the authors used this method to evaluate the effect of their work (According to the request of oil industry).
All experimental runs, which included 8 factorial points, 6 axial points and 5 center points, were based on run order performed. Furthermore, second-order polynomial equation was used to express the yield as a function of the independent parameters.
\begin{aligned} Y & = \beta_{0} + \beta_{1} A + \beta_{2} B + \beta_{3} C + \beta_{4} D \\ & \quad + \beta_{12} AB + \beta_{13} AC + \beta_{14} AD + \beta_{23} BC + \beta_{24} BD + \beta_{34} CD + \beta_{11} A^{2} + \beta_{22} B^{2} + \beta_{33} C^{2} + \beta_{44} D^{2} \\ \end{aligned}
(2)
Design expert software (Trial version 7.1) was applied for statistical treatment of the results. Analysis of variance (ANOVA) was employed to determine the statistically significant parameters and interactions using Fisher’s test and its associated probability p(F). The determination coefficients, R2, and their adjusted values, R2,adj, were used to evaluate the goodness of fit of the regression models.
### Gas chromatography Analysis
A Varian CP-3800 gas chromatograph (Varian instruments) with an FID detector and CP-sil 9 CB capillary column (100 m 0.25 mm, 0.25 μm film thickness) was used to accomplish chemical analysis in this work. The oven temperature was maintained at 35 °C for 7 min before being raised up to 250 °C at 3 °C/min. Injector and detector temperatures were set at 275 °C and 275 °C, respectively. Helium was used as carrier gas at a flow rate of 1.6 ml/min with the samples being injected manually under a split ratio of 1:80. Peaks’ area percentages were used to obtain quantitative data by using of the DHA software.
### Iodine number
The iodine number (IN) was determined according to the ASTM D4607-94 method36,37. The iodine number is defined as the milligrams of iodine adsorbed by 1.0 g of carbon when the iodine concentration of the filtrate is 0.02 N (0.02 mol/L). This method is based upon a three-point isotherm. A standard iodine solution is treated with three different weights of activated carbon under specified conditions. The experiment consists of treating the activated carbon sample with 10.0 mL of 5% HCl. This mixture is boiled for 30 s and then cooled. Soon afterwards, 100.0 mL of 0.1 N (0.1 mol L-1) iodine solution is added to the mixture and stirred for 30 s. The resulting solution is filtered and 50.0 mL of the filtrate is titrated with 0.1 N (0.1 mol L-1) sodium thiosulfate, using starch as indicator. The iodine amount adsorbed per gram of carbon (X/M) is plotted against the iodine concentration in the filtrate (C), using logarithmic axes.
The iodine number is the X/M value when the residual concentration (C) is 0.02 N (0.02 mol.L-1). The X/M and C values are calculated by the Eqs. 3 and 4 respectively.
$${\text{X}}/{\text{M}} = \left\{ {\left( {{\text{N}}_{1} \times 126.93 \times {\text{V}}_{1} } \right) - \left[ {\left( {{\text{V}}_{1} + {\text{V}}_{{{\text{Hcl}}}} } \right)/{\text{V}}_{{\text{F}}} } \right] \times \left( {{\text{N}}_{{{\text{Na2S2O3}}}} \times 126.93} \right) \times {\text{V}}_{{{\text{Na2S2O3}}}}}\right\}\,/ \,{\text{Mc}}$$
(3)
$$C = \left( {{\text{N}}_{{{\text{Na2S2O3}}}} \times {\text{V}}_{{{\text{Na2S2O3}}}} } \right)$$
(4)
where N1 is the iodine solution normality, V1 is the added volume of iodine solution, VHCl is the added volume of 5% HCl, VF is the filtrate volume used in titration, NNa2S2O3 is the sodium thiosulfate solution normality, VNa2S2O3 is the consumed volume of sodium thiosulfate solution and MC is the mass of activated carbon.
### Differential evolution algorithm
Differential evolution algorithm (DEA) is a population-based algorithm like genetic algorithms by similar operators; crossover, mutation, and selection. The main difference in constructing better solutions is that genetic algorithms depend on crossover while DE relies on mutation operation. This main operation is found on the differences of randomly sampled pairs of solutions in the population. The algorithm uses mutation operation as a seek mechanism and selection operation to direct the search toward the probable regions in the search space. The DE algorithm also uses a non-uniform crossover that can take child vector parameters from one parent more often than it does from others. Among the DE’s advantages its simple structure, ease of use, speed, and robustness can be mentioned38,39,40.
## Results and discussion
### Analysis of the response surface design
CCD was used for the design of experiments and process optimization. In this way, four coded operating parameters were defined in five levels, namely − 2, − 1, 0, 1 and 2, to obtain the optimal regeneration yield. Pressure (100–300) bar, temperature (313–333 K), flow rate (2–6 g/min), and dynamic time (30–150 min) were considered as the operating parameters of the process to be studied. Table 2 reports the experiments responses at the different values of operating conditions maintaining constant the particle size at 1 mm. Among models presented in CCD, the quadratic model represents the best model for the prediction and optimization of the regeneration yield. Equation (5) represents the obtained quadratic polynomial model through which the yield was correlated, as a response to the coded independent parameters. Therefore, the RSM predictive model was obtained as follows:
\begin{aligned} Yield \,(\% ) & = 59.72 + 8.36 A + 1.30 B + 5.67 C + 7.88 D \\ & \quad - 0.13 AB - 0.74 AC - 1.24 AD - 0.20 BC + 0.29 BD - 1.06CD \\ & \quad - 1.58 A^{2} + 1.58 B^{2} - 2.89 C^{2} - 1.66 C^{2} \\ \end{aligned}
(5)
This model was found to be significant at 95% level of confidence while the corresponding F-value and p-value were calculated to be 92.58 and 0.0001, respectively. Also, accuracy of the model can be tested by investigating the determination coefficient (R2). The values of the coefficient of determination (R2) and adjusted coefficient of determination (R2adj.) were calculated to be 98.86 and 97.79%, respectively. The value of R2 indicates a good agreement between the experimental and predicted response values. The value of adjusted R2 revealed that only 2.21% of total variations failed to be explained by the model. The lack-of-fit measures the failure of the model to represent data in the experimental domain at points which are not included in the regression. The p-value of the lack-of-fit was higher than 0.05 indicating an excellent fit. Furthermore, Signal-to-noise ratio (SNR) is known as an adequate precision measure. In fact, SNR compares the range of predicted values at design points to the average prediction error. As a SNR greater than 4 represents desirable results, the achieved ratio (35.30) served as an adequate signal as it was much smaller than the actual effect size. Low C.V. (3.32%) indicates the reliability of the carried-out experiments. The ANOVA of quadratic model was carried out using the data tabulated in Table 3 with the purpose of examining significance of the variables as linear, quadratic and interaction coefficients of the RSM. Those variables and their interactions with higher regression coefficient and smaller p-value (p < 0.05) have a significant influence on the regeneration yield41,42. Analysis results reported in Table 3, as well as, the parameters which statistically indicated highly significant impact on the yield were linear term of pressure, CO2 flow rate and time (p < 0.0001), followed using the quadratic term of flow rate(p < 0.0001). The quadratic term of time (p = 0.0003), the linear term of temperature (p = 0.0037) as well as the quadratic term of pressure and temperature (p = 0.0005), represent significant influence on the yield.
### Effect of process parameters on regeneration yield
In this section the effect of the different operational parameters on the supercritical regeneration of activated carbon were examined with three dimensional graphs, while the other variables were maintained at its respective fixed middle level(Pressure, 200 bar, Temperature, 318 K, flow rate,4 g/min and time 90 min), corresponding to zero code.
Figure 3 shows the 3D plot for the influence of pressure and temperature. It can be observed that the regeneration yield increased with increasing pressure and temperature from 100 to 300 bar and from 313 to 333 K, respectively. Based on CCD, pressure had the most important influence on the regeneration yield. Particularly, the increase of pressure leads to increase solvent power of CO2 and therefore solubility of the solutes increased as well9,43. On another hand, the increase in temperature had a slight positive effect on the regeneration yield (Fig. 4). At a pressure of 100 to 300 bar, with the increase in temperature, the solubility of solute increases, and it is due to the increment of the solute vapor pressure effect. therefore, the solubility of solute increases with the increase in temperature. On the other hand, the extract vapor pressure is raised with increasing temperature, which leads to an enhancement in the supercritical fluid diffusivity. Figure 5 denotes the 3D plot of the yield as a function of pressure and flow rate at 323 K and 90 min. Increasing the CO2 flow rate from 1 (code = − 2) to 4 g/min (code = 1) reduces the film thickness around the particles leading to lower resistance for mass transfer around the particles and accordingly enhancing the regeneration of the activated carbon. Whereas at more than 4 g/min with reduction of residence time (SC-CO2- particles contact time) a negative effect of flow rate was observed on the regeneration yield. This contrast effect was also observed in Figs. 6, 8 and 9.
Figures 4, 7, and 8 illustrate the effect of dynamic time and flow rate on the regeneration yield. As can be seen in Figs. 4, 7, and 8 the regeneration yields gradually increase with increasing the dynamic time, achieving its highest value about 150 min. This behavior has been reported by other researchers and explained in terms of the increase of the ratio between SC-CO2 and the solute as the dynamic time increase35. On another hand, the residence time decrease as flow rate increase, while the external mass transfer coefficient increased, so these opposite phenomena canceled their effects out leading to the almost constant yield. Moreover, the increase of the flow rate could reduce the contact time between the solvent and the activated carbon. This phenomenon could be related to the channeling effect, where SC-CO2 at high flow rates would just flow around the samples with no ability to diffuse through the pores within the samples. Furthermore, the increased flow rate could cause the sample compaction limiting the amount of CO2 able to being in direct contact with the sample matrix. Similar results were reported by some researchers33,44,45,46,47,48. In addition, the perturbation plot (Fig. 9) revealed the significant effect of all process variables on the extraction. A perturbation plot does not show the effect of interactions and it is like one factor-at-a-time experimentation. The perturbation plot helps to compare the effect of all independent variables at a particular point in the design space. The response is plotted by changing only one factor over its range while holding the other factors constant.
The optimal conditions to obtain the highest yield of regeneration from activated carbon were determined at 285 bar, 333 K, 4 g/min, and 147 min and the predicted yield was 93.75%. Average actual yield (94.25%) was in good agreement with the estimated value, revealing the ability of the developed model in terms of extraction process prediction and optimization.
### Gas chromatography analysis
The solute composition determined for the extract, at the optimum condition, by gaseous chromatography (GC) were shown in Table 4. Thirty-two components were identified in the extract obtained using SC-CO2 which comprised 95.2% of the extract. The main components extracted by SC-CO2 were N-C6 (4.90), N-C7 (9.45%), N-C8 (9.04%), N-C9 (11.90%), N-C10 (12.40%) and N-C11 (4.73%).
### Iodine results
Iodine number is a measure of iodine molecules adsorbed in the pores of a particle, which indicates the pore volume capacity and extent of micro pore distribution in the activated carbon. Iodine number can be correlated with the adsorbent ability to adsorb low molecular weight substances. Iodine adsorbed results in extracted and non-extracted activated carbon samples were shown in Table 5. Results of physical and adsorptive characteristics denote that SC-CO2 appears to be a suitable way for the regeneration of activated carbon.
### Evaluating the structure of activated carbon
As was previously mentioned the structure and surface morphology of non-extracted and extracted activated carbon samples was evaluated by scanning electron microscopy (SEM). Figure 10a,b presents two micrographs of activated carbon before (a) and after supercritical regeneration (b). The micrograph of the unprocessed sample shows an uneven and rough surface covered uniformly with a layer of solute. The structure of the treated sample is more porous and the surface is clearly deflated and depleted of solute due to the high recovery efficiency obtained by the extraction with supercritical carbon dioxide.
### Mathematical modeling
The experimental desorption was correlated with a kinetic model proposed by Tan and Liou17,49. This model assumes linear desorption kinetics in the adsorbed phase, and has a high applicability for correlating desorption behavior using SC-CO2 with only one fitting parameter. The following assumptions were considered in the formulation of the model:
1. 1.
The system was isothermal and isobaric.
2. 2.
The physical properties of the SC-CO2 were constant during the extraction process.
3. 3.
The extraction model was expressed as an irreversible desorption process.
4. 4.
All particles were considered to be spherical and the solutes were uniformly distributed in their structures.
5. 5.
The volume of the solid matrix (particles) was not changed during the extraction process.
6. 6.
The solvent flow rate was constant along the bed and uniformly distributed without radial dispersion.
7. 7.
Axial dispersion was neglected
A schematic diagram of the particles and bed was presented in Fig. 11. According to the above assumptions, the material balances for the solute in the solid and bulk phases are as follows:
Bulk phase:
$$\varepsilon \frac{\partial C}{{\partial t}} + u\frac{\partial C}{{\partial z}} = - \left( {1 - \varepsilon } \right)\frac{{\partial C_{P} }}{\partial t}$$
(6)
$$C = 0 \to \left( {t = 0, z} \right)$$
(7)
$$C = 0 \to \left( { z = 0, t } \right)$$
(8)
Solid phase:
$$\frac{{\partial C_{P} }}{\partial t} = - KC_{P}$$
(9)
$$C_{P} = C_{P0} \to \left( {t = 0} \right)$$
(10)
The concentration at the exit of the packed bed can then be obtained by:
$$C_{e} = C_{P0} .\frac{1 - \varepsilon }{\varepsilon }\left\{ {exp\left[ { - k\left( {t - \frac{\varepsilon L}{u}} \right)} \right] - exp\left( { - kt} \right)} \right\}$$
(11)
K is the adjustable parameter of the kinetic model.
The adjustable parameter was computed by minimizing the errors between experimental and calculated yield values. The average absolute relative deviation (AARD), described using the following equation was applied to evaluate the adjustable parameter:
$$AARD\% = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left( {\left| {\frac{{y_{i,cal} - y_{i,exp} }}{{y_{i,exp} }}} \right|} \right) \times 100$$
(12)
The mathematical modeling was applied to study the impact of pressure, temperature and flow rate as on the regeneration process. Figure 12 and Table 6 show the modeling results by using the kinetics model. Fig 12a shows a positive effect of the pressure on the regeneration process at fixed temperature, flow rate, particle size and dynamic time. This effect was related to the increase of the SC-CO2 density as pressure increase. As was expected, the increase of density leads to an increase in the solubility of the solutes in SC-CO2. On the other side, increasing pressure decreased the diffusion coefficient of CO2. The decreasing effect of mass transfer coefficient may lead decreasing the yield. Nevertheless, the effect of increasing density and solubility overcomes the decreasing effect of diffusivity on the final yield of extraction22,50. The adjustable parameter of the model (K) and AARD values for the different extraction runs were reported in Table 6. The effect of temperature on the regeneration yield of activated carbon was shown in Fig. 12b. The yield increased with increasing the temperature at fixed pressure and flow rate. As previously mentioned, this may be due to the increase of the vapor pressure of the solute. In the present section, the effect of temperature on the regeneration yield at points of 318, 328 K and optimum temperature were modeled. The flow rate of carbon dioxide was another parameter that was investigated. The experimental data and results of model in three flow rates including 1, 2 and 4 (optimum) were shown in Fig. 12. Based on the results indicated in Fig. 12c, the yield was higher at optimal points. As can be seen in Fig. 12 the experimental data were well described with the kinetics model. Values of average absolute relative deviation are in the range 5.91-9.04%.
### A single‑sphere model (SSM)
Single sphere model (SSM) proposed by Crank51 with an assumption that particle size is one of the main factors on the diffusivity step in the extraction process. In addition, diffusion is assumed to occur in the sphere surface area of a particle solute. Dissemination of SC-CO2 in sphere equations is conducted to determine the diffusiveness of solvent to dissolve in matrices. Other assumptions made for an SSM model are: (1) The resistance of mass transfer is zero, (2) all of the particle sizes are homogenous, (3) the main factor in the extraction process is intraparticle of mass transfer, (4) the solute is in the inert porous sphere, and lastly, (5) all of the solutes in the bed will be extracted, and the extracted component will be dissolved in the particles by a process similar to diffusion. Equation 13 exhibits the diffusion equation for a constant coefficient.
$$Y = \frac{{M_{t} }}{{M_{\infty } }} = 1 - \frac{6}{{\pi^{2} }} \mathop \sum \limits_{n - 1}^{\infty } \frac{1}{{n^{2} }}exp\frac{{D_{eff } t \pi^{2} n^{2} }}{{R^{2} }}$$
(13)
where Mt is the total amount of the diffusing substances entered on the sheet at a specific time, M∞ is the particular quantity after countless time, Deff is the diffusivity, R is the radius of particle solute, and t is the time.
The single sphere model is usually applied to determine the diffusivity coefficient and mass transfer between solvent and solute. Compared to another kinetic model, the single sphere model is easily applied, particularly the shrinking core model52 and the broken intact cell53 due to one adjustable parameter. This is because one adjustable parameter is fit enough to determine the mass transfer process of experimental data. Furthermore, a single-sphere model can assess the effect of the parameter on the diffusivity process between solvent and solute54.
The single sphere model was fitted to the experimental data with effective diffusivity as a fitting parameter, using average absolute deviation (AARD). Table 7 shows the AARD and effective diffusivity. The minimum %AARD was 2.08% at 285 bar and 333 K, while the maximum %AARD was 19.04% at 200 bar and 323 K. As shown in Fig. 13 the model successfully fitted the exponential trends of the extraction process. Putra et al.1 applied single sphere model for fitting the experimental data of modified supercritical carbon dioxide with error below 5% with the highest diffusivity coefficient was 6.794 × 10‐12 m2/s at operating condition was 10 MPa, 40 °C and particle size 425 µm. Moreover, Aris et al.2 determined Momordica charantia extract yield with different mean particle size as well as diffusion coefficient, De, in the extraction process with and without co-extractant. Based on the results, mean particle size of 0.3 mm gave the highest extract yield, 3.32% and 1.34% with and without co-extractant respectively. Whereas, the value of De at 0.3 mm mean particle size, with and without co-extractant are 8.820 × 10−12 and 7.920 × 10−12 m2/s respectively.
## Conclusion
Response surface methodology (RSM) was used in order to find out the effect of pressure, temperature, flow rate and dynamic time on the regeneration yield of activated carbon using SC-CO2. The optimal conditions for regeneration were found at 285 bar, 333 K, 5 g/min and 147 min which resulted in a regeneration yield equal to 93.71%. The regeneration effectiveness was revealed by the iodine number and visualized by SEM micrographs. The kinetic models proposed by Crank and Tan et al. successfully described the regeneration process using SC-CO2 and the estimated regeneration yields using the model adequately agreed with the experimental data for all studied conditions. The model parameters (K) and effective diffusivity was obtained by best fitting procedure applying differential evolution (DE) algorithm. Gas chromatography analysis was performed to identify the composition of solutes at the optimal extraction condition.
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2023-04-01 05:03:55
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{"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": 0.6312095522880554, "perplexity": 1859.4789996771447}, "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/1679296949701.0/warc/CC-MAIN-20230401032604-20230401062604-00797.warc.gz"}
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http://www.lmfdb.org/knowledge/show/artin.dimension
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show · artin.dimension all knowls · up · search:
The dimension of an Artin representation $\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}(V)$ is the dimension of the associated vector space $V$.
Authors:
Knowl status:
• Review status: reviewed
• Last edited by Andrew Booker on 2011-09-07 08:59:17
Referred to by:
History:
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2019-06-17 20:37:40
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{"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": 0.6346042156219482, "perplexity": 11497.300004057797}, "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/1560627998580.10/warc/CC-MAIN-20190617203228-20190617225228-00237.warc.gz"}
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http://mathhelpforum.com/trigonometry/33926-identity.html
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1. ## Identity
This is a question that Thelema asked me:
cos(s+t)cos(s-t)=cos^2s +cos^2t-1
so far i am at
[cos(s)cos(t)-sin(s)sin(t)][cos(s)cos(t)+sin(s)sin(t)] = cos^2(s)+cos^2(t)-1
I can multiply the left side but then i dont know
how to get it to equal the right. Can u plz help?
I don't have time to help at the moment so I'm asking if someone else can. Thanks!
-Dan
2. From:
$[cos~s * cos~t-sin~s*sin~t][cos~s*cos~t+sin~s*sin~t] = cos^2s+cos^2t-1$
Distribute (FOIL) and simplify:
$cos^2s*cos^2t - sin^2s*sin^2t~~~~=~~~~ cos^2s+cos^2t-1$
Use $sin^2x = 1-cos^2x$. We choose this one because our answer has cosines in it, and so we know we need to get rid of our sines.
$cos^2s*cos^2t - (1-cos^2s)(1-cos^2t)~~~~=~~~~ cos^2s+cos^2t-1$
Distribute:
$cos^2s*cos^2t - (1-cos^2t-cos^2s+cos^2s*cos^2t)~~~~=~~~~ cos^2s+cos^2t-1$
Distribute the negative sign:
$cos^2s*cos^2t -1+cos^2t+cos^2s-cos^2s*cos^2t~~~~=~~~~ cos^2s+cos^2t-1$
Simplify:
$cos^2s+cos^2t-1~~~~=~~~~ cos^2s+cos^2t-1$
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2016-10-23 06:39:59
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https://vulkan.lunarg.com/doc/view/1.0.33.0/linux/vkspec.chunked/ch12.html
|
# Chapter 12. Samplers
VkSampler objects represent the state of an image sampler which is used by the implementation to read image data and apply filtering and other transformations for the shader.
Samplers are represented by VkSampler handles:
VK_DEFINE_NON_DISPATCHABLE_HANDLE(VkSampler)
To create a sampler object, call:
VkResult vkCreateSampler(
VkDevice device,
const VkSamplerCreateInfo* pCreateInfo,
const VkAllocationCallbacks* pAllocator,
VkSampler* pSampler);
• device is the logical device that creates the sampler.
• pCreateInfo is a pointer to an instance of the VkSamplerCreateInfo structure specifying the state of the sampler object.
• pAllocator controls host memory allocation as described in the Memory Allocation chapter.
• pSampler points to a VkSampler handle in which the resulting sampler object is returned.
The VkSamplerCreateInfo structure is defined as:
typedef struct VkSamplerCreateInfo {
VkStructureType sType;
const void* pNext;
VkSamplerCreateFlags flags;
VkFilter magFilter;
VkFilter minFilter;
VkSamplerMipmapMode mipmapMode;
float mipLodBias;
VkBool32 anisotropyEnable;
float maxAnisotropy;
VkBool32 compareEnable;
VkCompareOp compareOp;
float minLod;
float maxLod;
VkBorderColor borderColor;
VkBool32 unnormalizedCoordinates;
} VkSamplerCreateInfo;
• sType is the type of this structure.
• pNext is NULL or a pointer to an extension-specific structure.
• flags is reserved for future use.
• magFilter is the magnification filter to apply to lookups, and is of type:
typedef enum VkFilter {
VK_FILTER_NEAREST = 0,
VK_FILTER_LINEAR = 1,
} VkFilter;
• minFilter is the minification filter to apply to lookups, and is of type VkFilter.
• mipmapMode is the mipmap filter to apply to lookups as described in the Texel Filtering section, and is of type:
typedef enum VkSamplerMipmapMode {
VK_SAMPLER_MIPMAP_MODE_NEAREST = 0,
VK_SAMPLER_MIPMAP_MODE_LINEAR = 1,
} VkSamplerMipmapMode;
• addressModeU is the addressing mode for outside [0..1] range for U coordinate. See VkSamplerAddressMode.
• addressModeV is the addressing mode for outside [0..1] range for V coordinate. See VkSamplerAddressMode.
• addressModeW is the addressing mode for outside [0..1] range for W coordinate. See VkSamplerAddressMode.
• mipLodBias is the bias to be added to mipmap LOD calculation and bias provided by image sampling functions in SPIR-V, as described in the Level-of-Detail Operation section.
• anisotropyEnable is VK_TRUE to enable anisotropic filtering, as described in the Texel Anisotropic Filtering section, or VK_FALSE otherwise.
• maxAnisotropy is the anisotropy value clamp.
• compareEnable is VK_TRUE to enable comparison against a reference value during lookups, or VK_FALSE otherwise.
• Note: Some implementations will default to shader state if this member does not match.
• compareOp is the comparison function to apply to fetched data before filtering as described in the Depth Compare Operation section. See VkCompareOp.
• minLod and maxLod are the values used to clamp the computed level-of-detail value, as described in the Level-of-Detail Operation section. maxLod must be greater than or equal to minLod.
• borderColor is the predefined border color to use, as described in the Texel Replacement section, and is of type:
typedef enum VkBorderColor {
VK_BORDER_COLOR_FLOAT_TRANSPARENT_BLACK = 0,
VK_BORDER_COLOR_INT_TRANSPARENT_BLACK = 1,
VK_BORDER_COLOR_FLOAT_OPAQUE_BLACK = 2,
VK_BORDER_COLOR_INT_OPAQUE_BLACK = 3,
VK_BORDER_COLOR_FLOAT_OPAQUE_WHITE = 4,
VK_BORDER_COLOR_INT_OPAQUE_WHITE = 5,
} VkBorderColor;
• unnormalizedCoordinates controls whether to use unnormalized or normalized texel coordinates to address texels of the image. When set to VK_TRUE, the range of the image coordinates used to lookup the texel is in the range of zero to the image dimensions for x, y and z. When set to VK_FALSE the range of image coordinates is zero to one. When unnormalizedCoordinates is VK_TRUE, samplers have the following requirements:
• minFilter and magFilter must be equal.
• mipmapMode must be VK_SAMPLER_MIPMAP_MODE_NEAREST.
• minLod and maxLod must be zero.
• addressModeU and addressModeV must each be either VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_EDGE or VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_BORDER.
• anisotropyEnable must be VK_FALSE.
• compareEnable must be VK_FALSE.
• When unnormalizedCoordinates is VK_TRUE, images the sampler is used with in the shader have the following requirements:
• The viewType must be either VK_IMAGE_VIEW_TYPE_1D or VK_IMAGE_VIEW_TYPE_2D.
• The image view must have a single layer and a single mip level.
• When unnormalizedCoordinates is VK_TRUE, image built-in functions in the shader that use the sampler have the following requirements:
• The functions must not use projection.
• The functions must not use offsets.
Mapping of OpenGL to Vulkan filter modes magFilter values of VK_FILTER_NEAREST and VK_FILTER_LINEAR directly correspond to GL_NEAREST and GL_LINEAR magnification filters. minFilter and mipmapMode combine to correspond to the similarly named OpenGL minification filter of GL_minFilter_MIPMAP_mipmapMode (e.g. minFilter of VK_FILTER_LINEAR and mipmapMode of VK_SAMPLER_MIPMAP_MODE_NEAREST correspond to GL_LINEAR_MIPMAP_NEAREST).There are no Vulkan filter modes that directly correspond to OpenGL minification filters of GL_LINEAR or GL_NEAREST, but they can be emulated using VK_SAMPLER_MIPMAP_MODE_NEAREST, minLod = 0, and maxLod = 0.25, and using minFilter = VK_FILTER_LINEAR or minFilter = VK_FILTER_NEAREST, respectively.Note that using a maxLod of zero would cause magnification to always be performed, and the magFilter to always be used. This is valid, just not an exact match for OpenGL behavior. Clamping the maximum LOD to 0.25 allows the λ value to be non-zero and minification to be performed, while still always rounding down to the base level. If the minFilter and magFilter are equal, then using a maxLod of zero also works.
addressModeU, addressModeV, and addressModeW must each have one of the following values:
typedef enum VkSamplerAddressMode {
} VkSamplerAddressMode;
These values control the behavior of sampling with coordinates outside the range [0,1] for the respective u, v, or w coordinate as defined in the Wrapping Operation section.
• VK_SAMPLER_ADDRESS_MODE_REPEAT indicates that the repeat wrap mode will be used.
• VK_SAMPLER_ADDRESS_MODE_MIRRORED_REPEAT indicates that the mirrored repeat wrap mode will be used.
• VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_EDGE indicates that the clamp to edge wrap mode will be used.
• VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_BORDER indicates that the clamp to border wrap mode will be used.
• VK_SAMPLER_ADDRESS_MODE_MIRROR_CLAMP_TO_EDGE indicates that the mirror clamp to edge wrap mode will be used. This is only valid if the VK_KHR_mirror_clamp_to_edge extension is enabled.
The maximum number of sampler objects which can be simultaneously created on a device is implementation-dependent and specified by the maxSamplerAllocationCount member of the VkPhysicalDeviceLimits structure. If maxSamplerAllocationCount is exceeded, vkCreateSampler will return VK_ERROR_TOO_MANY_OBJECTS.
Since VkSampler is a non-dispatchable handle type, implementations may return the same handle for sampler state vectors that are identical. In such cases, all such objects would only count once against the maxSamplerAllocationCount limit.
To destroy a sampler, call:
void vkDestroySampler(
VkDevice device,
VkSampler sampler,
const VkAllocationCallbacks* pAllocator);
• device is the logical device that destroys the sampler.
• sampler is the sampler to destroy.
• pAllocator controls host memory allocation as described in the Memory Allocation chapter.
|
2020-01-24 11:13:39
|
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|
https://www.techwhiff.com/issue/the-scores-of-players-on-a-golf-team-are-shown-in-the--385569
|
# The scores of players on a golf team are shown in the table. The team’s combined score was 0. What was Travis’s score?
###### Question:
The scores of players on a golf team are shown in the table. The team’s combined score was 0. What was Travis’s score?
### English 2 V18, Cynthia Kautz (4844/8) - (MC) Passage from Metamorphosis by Franz Kafka (1) One morning, when Gregor Samsa woke from troubled dreams, he found himself transformed in his bed into a horrible vermin. He lay on his armour-like back, and if he lifted his head a little he could see his brown belly, slightly domed and divided by arches into stiff sections. The bedding was hardly able to cover it and seemed ready to slide off any moment. His many legs, pitifully thin compared with the si
English 2 V18, Cynthia Kautz (4844/8) - (MC) Passage from Metamorphosis by Franz Kafka (1) One morning, when Gregor Samsa woke from troubled dreams, he found himself transformed in his bed into a horrible vermin. He lay on his armour-like back, and if he lifted his head a little he could see his bro...
### Many firms in the United States file for bankruptcy every year, yet they still continue operating. Why would they do this instead of completely shutting down?
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### What would happen if grass is over fertilized and why
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### How can migration affect the density of an area
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### Read these sentences from a message to a friend: I'm super excited for the concert! All the critics are saying this group puts on an electrifying show! Which are the two strongest appeals used in these sentences? A. Ethos and kairos O O B. Pathos and ethos O C. Ethos and logos O D. Logos and pathos
Read these sentences from a message to a friend: I'm super excited for the concert! All the critics are saying this group puts on an electrifying show! Which are the two strongest appeals used in these sentences? A. Ethos and kairos O O B. Pathos and ethos O C. Ethos and logos O D. Logos and pathos...
### Atomic Structure 1. Answer. Protons are found in the: The number of protons determines: The number of protons is called: Where you look to find the number of protons: Protons are charged: Neutrons are found in the: Neutrons are charged: They help protons by: The mass of protons and neutrons are: Electrons mass is: Electrons are charged: In a neutral atom: Electrons are responsible for:
Atomic Structure 1. Answer. Protons are found in the: The number of protons determines: The number of protons is called: Where you look to find the number of protons: Protons are charged: Neutrons are found in the: Neutrons are charged: They help protons by: The mass of protons and neutrons are: Ele...
### A professional baseball team won 84 games this season
a professional baseball team won 84 games this season...
### 3) Look carefully at the following structural formula, and predict if this compound would turn black if heated like those in the above experiments. a. It would turn black. b. It would not turn black. H O H
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### Which are true about the area of a circle? Check all that apply
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### What is the final amount if 1885 is increased by 5% followed by a further 18% increase
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### Determine the area of the shaded region (grey) in the quadratic form. Hint: Shaded area = (gray rectangle area) – (white rectangle area)
Determine the area of the shaded region (grey) in the quadratic form. Hint: Shaded area = (gray rectangle area) – (white rectangle area)...
### If you dont know the answer dont answer. I will give 30 points!!!!!
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### Who calculated the volume of a sphere by comparing it to a cylinder? Pythagoras Euclid Archimedes Socrates
Who calculated the volume of a sphere by comparing it to a cylinder? Pythagoras Euclid Archimedes Socrates...
### After having read through The Declaration of Independence, carefully read through each question. If it is multiple choice Which of the following best expresses the author’s main purpose in this document? 1.To end the war between the colonies and Great Britain by declaring a revolution 2.To criticize the King of England’s reign over the colonies and advocate for his removal from power 3.To declare the thirteen colonies free of Great Britain’s rule and illustrate why they are declaring independenc
After having read through The Declaration of Independence, carefully read through each question. If it is multiple choice Which of the following best expresses the author’s main purpose in this document? 1.To end the war between the colonies and Great Britain by declaring a revolution 2.To critici...
### Kathy became addicted to alcohol in her teen years, which factors influenced her drinking? check all that apply
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### I am in 8th grade and I want to start studying for the sat and I have a psat for my school in about 20 days. Could someone refer me to practice websites and resources I could use. (For reference I am practicing Algebra 1 and Chemistry for 8th grade at my school) Thank you!
I am in 8th grade and I want to start studying for the sat and I have a psat for my school in about 20 days. Could someone refer me to practice websites and resources I could use. (For reference I am practicing Algebra 1 and Chemistry for 8th grade at my school) Thank you!...
### Over the past two years, Green Caterpillar Garden Supplies Inc. has relied more on the use of short-term debt than on long-term debt financing. This statement is , because: a. Green Caterpillar’s total current liabilities increased by $547 million, while its use of long-term debt increased by$1,640 million. b. Green Caterpillar’s total current liabilities decreased by $547 million, while its long-term debt account decreased by$1,640 million. c. Green Caterpillar’s total notes payable increa
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### PLEASE HELP(2 questions)!!!!! BRAINLIEST IF FIRST AND CORRECTLY ANSWERED!!! ____________ have a general property of being hydrophobic, meaning they are insoluble-will not dissolve in water. All other organic compounds are water soluble. A) Carbohydrates B) Nucleic Acids C) Proteins D) Lipids KEY-b=boron c=carbon cl=chlorine h=hydrogen n=nitrogen na=sodium o=oxygen s=sulfur The compounds shown, in the answer choices below, a
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The data set represents the prices for a full tank of gasoline for different motorcycles: $19,$18, $15,$17, $19,$12, $19 and$15 What is the range of prices? a $7.00 b$17.50 c $16.75 d$19.00...
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2023-03-31 07:10:32
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https://www.r-bloggers.com/2012/11/page/9/
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# Monthly Archives: November 2012
## Fun with coin flips
November 21, 2012
By
We all know that the odds of flipping an unbiased coin is 50% heads, 50% tails. But what happens if you do this a lot of times. Do you expect the same number of heads and tails? What if we took a cumulative sum where heads = +1 and tails = -1. What wou...
## Video: SimpleR tricks and tools: Help, debugging, git, LaTeX, and workflow with R by Prof Rob Hyndman
November 21, 2012
By
This post shares the video from a talk presented on 20th November 2012 by Professor Rob Hyndman at Melbourne R Users. The talk provides an introduction to: Getting R help Debugging R functions R style guides Making good use of … Continue reading →
## Rcpp attributes: A simple example ‘making pi’
November 20, 2012
By
We introduced Rcpp 0.10.0 with a number of very nice new features a few days ago, and the activity on the rcpp-devel mailing list has been pretty responsive which is awesome. But because few things beat a nice example, this post tries to build some more excitement. We will illustrate how Rcpp attributes makes it really easy to add C++ code...
## R User Conference in Spain: Call for Tutorials
November 20, 2012
By
I'm really looking forward to useR! 2013 (the international conference for R users), and not just because it's being held in Spain next year (July 10-12). The program is already coming together, with a great lineup of invited speakers, including R-core member Duncan Murdoch and prolific package authoR Hadley Wickham. You too can be part of the program, by...
## optimising accept-reject
November 20, 2012
By
$optimising accept-reject$
I spotted on R-bloggers a post discussing optimising the efficiency of programming accept-reject algorithms. While it is about SAS programming, and apparently supported by the SAS company, there are two interesting features with this discussion. The first one is about avoiding the dreaded loop in accept-reject algorithms. For instance, taking the case of the truncated-at-one
## Functional programming with lambda.r
November 20, 2012
By
$Functional programming with lambda.r$
After a four month simmer on various back burners and package conflicts, I’m pleased to announce that the successor to …Continue reading »
## SimpleR tips, tricks and tools
November 20, 2012
By
I gave this talk last night to the Melbourne Users of R Network. Examples
## Claims reserving in R: ChainLadder 0.1.5-4 released
November 20, 2012
By
Last week we released version 0.1.5-4 of the ChainLadder package on CRAN. The R package provides methods which are typically used in insurance claims reserving. If you are new to R or insurance check out my recent talk on Using R in Insurance.The chain-ladder method which is a popular method in the insurance industry to forecast future...
## Heteroskedastic GLM in R
November 20, 2012
By
A commenter on my previous blog entry has drawn my attention to an R function called hetglm() that estimates heteroskedastic probit models. This function is contained in the glmx package. The glmx package is not available on CRAN yet, but thankfully can be downloaded here. The hetglm() function has a number of computational advantages compared with
## Prime Factorization Visualization with R and Shiny
November 20, 2012
By
Quite a lot of people have had fun recently with prime factorization. It all started on The Math Less Traveled, then various versions of the prime factorization diagrams appeared (here, here, this animated one, etc., they are actually more or less listed here). So I wanted to have fun too and give a try...
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2016-10-26 23:07:32
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https://par.nsf.gov/biblio/10219983-periodicity-random-walks-dynamic-networks
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On the Periodicity of Random Walks in Dynamic Networks
We investigate random walks in graphs whose edges change over time as a function of the current probability distribution of the walk. We show that such systems can be chaotic and can exhibit hyper-torpid" mixing. Our main result is that, if each graph is strongly connected, then the dynamics is asymptotically periodic almost surely.
Authors:
Editors:
Award ID(s):
Publication Date:
NSF-PAR ID:
10219983
Journal Name:
IEEE transactions on network science and engineering
Volume:
7
Issue:
3
Page Range or eLocation-ID:
1337 - 1343
ISSN:
2327-4697
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2023-02-08 11:40:18
|
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http://kitchingroup.cheme.cmu.edu/blog/2013/10/26/Git-archives-for-data-sharing/
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## Git archives for data sharing
| categories: data | tags: | View Comments
in some past posts we have looked at constructing JSON files for data sharing. While functional, that approach requires some extra work to create the data files for sharing, and may not be useful for all sorts of data. For instance you may not want to store electron density in a JSON file.
Here we consider using git archives for packaging exactly the data you used in the same hierarchy as on your file system. The idea is to store your work in a git repository. You commit any data files you would want to share, and then create an archive of that data. This enables you to control what gets shared, while keeping the data that should not be shared out of the archive.
We will run a few VASP calculations, and summarize each one in a JSON file. We will commit those JSON files to the git repository, and finally make a small archive that contains them.
## 1 A molecule
Calculate the total energy of a CO molecule.
from ase import Atoms, Atom
from jasp import *
import numpy as np
import json
np.set_printoptions(precision=3, suppress=True)
co = Atoms([Atom('C',[0, 0, 0]),
Atom('O',[1.2, 0, 0])],
cell=(6., 6., 6.))
with jasp('molecules/simple-co', #output dir
xc='PBE', # the exchange-correlation functional
nbands=6, # number of bands
encut=350, # planewave cutoff
ismear=1, # Methfessel-Paxton smearing
sigma=0.01,# very small smearing factor for a molecule
atoms=co) as calc:
print 'energy = {0} eV'.format(co.get_potential_energy())
print co.get_forces()
with open('JSON', 'wb') as f:
f.write(json.dumps(calc.dict))
energy = -14.687906 eV
[[ 5.095 0. 0. ]
[-5.095 0. 0. ]]
## 2 A bulk calculation
Now we run a bulk calculation
from jasp import *
from ase import Atom, Atoms
atoms = Atoms([Atom('Cu', [0.000, 0.000, 0.000])],
cell= [[ 1.818, 0.000, 1.818],
[ 1.818, 1.818, 0.000],
[ 0.000, 1.818, 1.818]])
with jasp('bulk/alloy/cu',
xc='PBE',
encut=350,
kpts=(13,13,13),
nbands=9,
ibrion=2,
isif=4,
nsw=10,
atoms=atoms) as calc:
print atoms.get_potential_energy()
with open('JSON', 'wb') as f:
f.write(json.dumps(calc.dict))
-3.723306
## 3 Analysis via the JSON files
This analysis is independent of jasp and therefore is portable
We can retrieve the bulk data
import json
with open('bulk/alloy/cu/JSON', 'rb') as f:
print d['data']['total_energy']
-3.723306
Or the molecule data:
import json
with open('molecules/simple-co/JSON', 'rb') as f:
print d['data']['total_energy']
-14.687906
## 4 Create the archive file
As you do your work, you add and commit files as needed. For this project all that needs to be shared are the JSON files, and the scripts (which are in this document) that we used to run the calculations and do the analysis. If we are satisfied with the state of the git repository, we create an archive like this:
git archive --format zip HEAD -o archive.zip
Here is the result: archive.zip .
You can download the zip file, unzip it, and rerun the analysis to extract the total energies on any system with a modern Python installation.
## 5 Summary
This seems to be an easy way to share data from a single project, i.e. a single git repository. It isn't obvious how you would package data from multiple projects, or how you would run multiple projects in a single directory.
|
2017-09-26 16:31:02
|
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|
https://www.zbmath.org/?q=ut%3Afinite-measure+subsets
|
# zbMATH — the first resource for mathematics
Subsets with finite measure of multifractal Hausdorff measures. (English) Zbl 0963.28004
Summary: Let $$\mu$$ be a Borel probability measure on $$\mathbb{R}^d$$, $$q,t\in\mathbb{R}$$. Let $${\mathcal H}^{q,t}_\mu$$ denote the multifractal Hausdorff measure. We prove that, when $$\mu$$ satisfies the so-called Federer condition for a closed subset $$E\in\mathbb{R}^n$$ such that $${\mathcal H}^{q,t}_\mu(E)> 0$$, there exists a compact subset $$F$$ of $$E$$ with $$0<{\mathcal H}^{q,t}_\mu(F)< \infty$$, i.e., finite-measure subsets of multifractal Hausdorff measure exist.
##### MSC:
28A78 Hausdorff and packing measures
|
2021-03-07 04:50:02
|
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|
https://paperswithcode.com/paper/megba-a-high-performance-and-distributed?from=n22
|
# MegBA: A GPU-Based Distributed Library for Large-Scale Bundle Adjustment
2 Dec 2021 · , , , , , ·
Large-scale Bundle Adjustment (BA) requires massive memory and computation resources which are difficult to be fulfilled by existing BA libraries. In this paper, we propose MegBA, a GPU-based distributed BA library. MegBA can provide massive aggregated memory by automatically partitioning large BA problems, and assigning the solvers of sub-problems to parallel nodes. The parallel solvers adopt distributed Precondition Conjugate Gradient and distributed Schur Elimination, so that an effective solution, which can match the precision of those computed by a single node, can be efficiently computed. To accelerate BA computation, we implement end-to-end BA computation using high-performance primitives available on commodity GPUs. MegBA exposes easy-to-use APIs that are compatible with existing popular BA libraries. Experiments show that MegBA can significantly outperform state-of-the-art BA libraries: Ceres (41.45$\times$), RootBA (64.576$\times$) and DeepLM (6.769$\times$) in several large-scale BA benchmarks. The code of MegBA is available at https://github.com/MegviiRobot/MegBA.
PDF Abstract
## Datasets
Add Datasets introduced or used in this paper
## Results from the Paper Edit
Submit results from this paper to get state-of-the-art GitHub badges and help the community compare results to other papers.
|
2023-01-27 17:45:28
|
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|
https://www.nature.com/articles/s41598-020-69671-y?error=cookies_not_supported&code=6aaaa01d-acc7-47db-9061-a7a055944b08
|
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.
# Characterization of pressure fluctuations within a controlled-diffusion blade boundary layer using the equilibrium wall-modelled LES
## Abstract
In this study, the generation of airfoil trailing edge broadband noise that arises from the interaction of turbulent boundary layer with the airfoil trailing edge is investigated. The primary objectives of this work are: (i) to apply a wall-modelled large-eddy simulation (WMLES) approach to predict the flow of air passing a controlled-diffusion blade, and (ii) to study the blade broadband noise that is generated from the interaction of a turbulent boundary layer with a lifting surface trailing edge. This study is carried out for two values of the Mach number, $${{\rm Ma}}_{\infty } = 0.3$$ and 0.5, two values of the chord Reynolds number, $${{\rm Re}}=8.30 \times 10^5$$ and $$2.29 \times 10^6$$, and two angles of attack, AoA $$=4^\circ$$ and $$5^\circ$$. To examine the influence of the grid resolution on aerodynamic and aeroacoustic quantities, we compare our results with experimental data available in the literature. We also compare our results with two in-house numerical solutions generated from two wall-resolved LES (WRLES) calculations, one of which has a DNS-like resolution. We show that WMLES accurately predicts the mean pressure coefficient distribution, velocity statistics (including the mean velocity), and the traces of Reynolds tensor components. Furthermore, we observe that the instantaneous flow structures computed by the WMLES resemble those found in the reference WMLES database, except near the leading edge region. Some of the differences observed in these structures are associated with tripping and the transition to a turbulence mechanism near the leading edge, which are significantly affected by the grid resolution. The aeroacoustic noise calculations indicate that the power spectral density profiles obtained using the WMLES compare well with the experimental data.
## Introduction
The noise generated by the flow of air passing over a lifting surface is a major issue for a wide range of engineering applications. Examples include the noise generated by propellers, rotors, wind turbines, fans, wings, and hydrofoils. Even in the absence of disturbances in the incoming stream, an airfoil can generate noise due to an unsteady turbulent boundary layer and wake, or through interactions with the lifting surface, particularly near the trailing edge region. This so-called self-noise or trailing edge noise is a major contributor to the overall noise generated by rotating machines and, in general, defines the lower bound of noise1. Reducing self-noise is challenging and requires an accurate and efficient method for describing and understanding the sources of noise near the trailing edge region. During recent decades, a variety of computational methods and theoretical models have been developed to predict the noise generated by unsteady flows passing over fixed or moving surfaces2. These approaches can be subdivided into three classes: semi-empirical, direct, and hybrid methods. The semi-empirical method, based on a set of acoustic data, is the most widely adopted for acoustic prediction when realistic aircraft designs must be used. However, the accuracy of the semi-empirical method is limited to standard flight conditions, and cannot be used for aircraft with unconventional airfoil profiles or operating conditions3. In contrast, the direct method uses a fully numerical method for both acoustic propagation and turbulence simulation and thus can provide accurate and reliable noise predictions4. However, these high-fidelity methods are prohibitively expensive and memory-intensive when used as a design and optimization tool. Hybrid methods offer an attractive compromise in terms of accuracy and computational cost because they combine (i) fluid dynamic calculations in the near field to resolve unsteady near-flowfields and (ii) the acoustic analogy for the propagation of sound to the far field5. Currently, the most popular method used for the aeroacoustic analogy is based on Ffowcs Williams and Hal’s seminal work6.
The power of model-based noise prediction depends on high spatial and temporal turbulent flow statistics, such as the length scales, spanwise and streamwise correlation length scales, degree of anisotropy, and acoustic spectral shape, which must be known a priori. large-eddy simulation (LES) is a promising high-fidelity approach that resolves the majority of energetic eddies in a turbulent flow while accounting for the effect of unresolved motions by subgrid scale (SGS) models7,29. Wall-resolved large-eddy simulation (WRLES) is extremely expensive because it must resolve the small but energetic structures in the near-wall region of the boundary layer. For instance, Terracol and Manoha8 employed 2.6 billion grid cells and six million core hours in their WRLES for a three-element airfoil with a chord Reynolds number of $${{\rm Re}}=1.23\times 10^6$$. Because the small-scale energetic structures near the surface must be accurately captured, a large fraction of the cells in a typical computational domain are used to resolve the near-wall region of the boundary layer. Choi and Moin9 showed that the number of cells needed to resolve the inner layer (defined as $$y^+=y u_\tau /\nu <100$$, where y is the wall-normal direction) scales to $${{\rm Re}}^{13/7}$$, which is a nearly quadratic dependence on the Reynolds number. The cost of a DNS calculation is even more prohibitive as the grid size scales to $${{\rm Re}}^{37/14}$$9. A more affordable alternative to DNS and WRLES calculations is provided by the so-called wall-modeled LES (WMLES). This technique resolves large-scale flow structures in the outer region of the boundary layer only and models the effect of near-wall turbulence using a Reynolds-averaged Navier–Stokes (RANS) type model in the inner part of the boundary layer10,11. In contrast to detached eddy simulation (DES) methods12, which typically treat the entire attached boundary layer with a RANS approach, WMLES resolves most parts of the boundary layer with LES.
Bodart and Larsson13 used WMLES to achieve good agreement with the experimental data for the flow-field statistics of a McDonnell–Douglas 30P/30N multi-element airfoil at $${{\rm Re}}=9\times 10^6$$ and estimated that the computational cost was reduced by two orders of magnitude compared to WRLES. Kawai and Larsson14 used a WMLES approach to simulate a supersonic flat-plate boundary layer at a Reynolds number based on the boundary layer thickness of $${{\rm Re}}_{\delta } = 6.1 \times 10^5$$ and a free-stream Mach number of $${{\rm Ma}}_{\infty }=1.69$$. They obtained a converged and predictive solution when compared with experimental results under the same flow conditions. George and Lele15 also performed WMLES to predict self-noise at a Reynolds number ranging from $$1.0 \times 10^6$$ to $$1.5 \times 10^6$$. In addition, they investigated the capability of the equilibrium WMLES10 to predict the flow around a wind turbine airfoil under stalled conditions, a NACA0012 airfoil in the near-stall regime, and a NACA64-618 airfoil in the post-stall regime.
All these applications using the WMLES approach highlight the potential of this emerging methodology for predicting aerodynamic and aeroacoustic fields and solving canonical problems. Less is known about using WMLES in complex flows, especially for the study of sound generation.
The present work has been carried out within the SCONE (Simulation of Contra Rotating Open Rotor and fan broadband NoisE with reduced order modeling) project16, which is part of the FP7 Clean Sky Joint Undertaking of the European Union that aims to investigate the noise generated by CROR/UHBR fan technologies and to achieve optimal noise reduction. Since it is difficult to experimentally measure flow pressure fluctuations on a surface without significantly affecting its motion, we instead employed a computational approach. Accurately predicting the airfoil self-noise generated by the interaction of a turbulent boundary layer with an airfoil trailing edge will provide enhanced inputs for current semi-analytic models. For this reason, the first objective of the SCONE project is to describe the flow over a controlled-diffusion blade with the ultimate aim to develop ‘quiet’ CROR and UHBR engines. The blade section geometry used here was originally part of an air-conditioning unit developed by Valeo that we slightly re-designed to impose a load on it similar to that of a CROR blade. We carried out the simulations for different values of Mach number ($${{\rm Ma}}_{\infty } = 0.3$$ and 0.5), Reynolds number ($${{\rm Re}}=8.30 \times 10^5$$ and $$2.29 \times 10^6$$), and angle of attack (AoA $$=4^\circ$$ and $$5^\circ$$). The configuration was a replica of the experiments performed within the CRORTET Clean Sky project, the computational results of which are compared to this study. Note that experimental aerodynamic data alone are not sufficient to provide the boundary layer parameters needed to normalize the wall-pressure spectra obtained in the experiment, which highlights the importance of LES for robust wall pressure modeling.
Numerical simulations of high Reynolds and Mach number flows around a controlled diffusion airfoil, which provide wall-pressure statistics, are scarce in the literature. However, to the best of our knowledge, convection velocity profiles and cross-spectra or coherence functions have not been analyzed in any previous studies that have used a WMLES approach. Therefore, our understanding of the multivariate statistics of the wall-pressure fluctuations induced by the interaction of a turbulent boundary layer and airfoil trailing edge in such flow conditions is still quite limited and thus needs to be improved.
This study represents a first step towards a deeper quantitative investigation of the effect of the Reynolds number, the angle of attack, and the upstream Mach number on the chord-wise development of the boundary layer characteristics of a controlled-diffusion blade. To this end, we use the compressible WMLES approach to compute the flowfield passing over a lifting surface. We validate the WMLES results against experimental data. In addition, we evaluate the efficacy of the WMLES approach for predicting the noise generated by a blade and investigate the source of noise and its generation mechanisms.
## Methods
### Governing equations
In this paper, the following notation is adopted: $$x_1$$ or x is the streamwise coordinate; $$x_2$$ or y is the wall-normal coordinate; and $$x_3$$ or z is the spanwise coordinate. The governing equations used in this study are the full Favre-filtered compressible Navier–Stokes which, for a calorically perfect gas, read
\begin{aligned}{\left\{ \begin{array}{ll} \frac{\partial{\overline{\rho }}}{\partial t}+ \frac{\partial{\overline{\rho }}\widetilde{{\mathscr{U}}_j}}{\partial x_j}=0, \\ \frac{\partial{\overline{\rho }}\widetilde{{\mathscr{U}}_i}}{\partial t}+ \frac{\partial \rho \widetilde{{\mathscr{U}}_j}\widetilde{{\mathscr{U}}_i}}{\partial x_j}+ \frac{\partial \overline{{\mathscr{P}}}}{\partial x_i}= \frac{\partial \breve{\tau }_{ij}}{\partial x_j}, \\ \frac{\partial{\overline{\rho }}\widetilde{{\mathscr{E}}}}{\partial t}+ \frac{\partial ({\overline{\rho }}\widetilde{{\mathscr{E}}}+ \overline{{\mathscr{P}}})\widetilde{{\mathscr{U}}_j}}{\partial x_j}= \frac{\partial \tau _{ij}\widetilde{{\mathscr{U}}_i}}{\partial x_j}- \frac{\partial \breve{{\mathscr{Q}}}_j}{\partial x_j}, \\ \overline{{\mathscr{P}}}={\overline{\rho }}{\rm R}\widetilde{{\mathscr{T}}}, \end{array}\right. } \end{aligned}
(1)
where $$\rho$$ is the density, $${\mathscr{U}}_i$$ is the velocity component in the $$x_i$$-direction, $${\mathscr{P}}$$ is the pressure, $$\widetilde{{\mathscr{E}}}=\overline{{\mathscr{P}}}/[{\overline{\rho }}(\gamma -1)]+ \widetilde{{\mathscr{U}}_i}\widetilde{{\mathscr{U}}_i/2}$$ is the total energy, $${\rm R}$$ is the gas constant, $${\mathscr{T}}$$ is the temperature, and $$\gamma =c_{{\mathscr{P}}}/c_{\rm V}$$ is the ratio of specific heats, which is kept constant at 1.4. The overline (resp. tilde) denotes the filtered (resp. Favre filtered) value. The variables are either spatially filtered or ensemble-averaged quantities depending on the use of LES or RANS equations. The latter model is used for the wall-model part. It is assumed that both the filtered stress tensor $$\breve{\tau }_{ij}$$ and the filtered heat flux $$\breve{{\mathscr{Q}}}_j$$ can be expressed in a way similar to their instantaneous counterparts, but applied to filtered quantities
\begin{aligned} \tau _{ij}=2(\mu +\mu _t)\widetilde{{\mathscr{S}}_{ij}}^{{\rm d}},\quad{\mathscr{Q}}_j=- (\lambda +\lambda _t)\frac{\partial \widetilde{{\mathscr{T}}}}{\partial x_j}, \end{aligned}
(2)
where $$\mu$$ is the molecular viscosity that has a power-law dependence on temperature
\begin{aligned} \mu =\mu _0(\widetilde{{\mathscr{T}}}/{\mathscr{T}}_0)^{0.76}. \end{aligned}
(3)
The parameter $$\lambda$$ in (2) is the molecular thermal conductivity, $${\rm Pr}=\mu c_{{\mathscr{P}}}/\lambda$$ is the molecular Prandtl number kept constant at 0.7, and $$\widetilde{{\mathscr{S}}_{ij}}^{{\rm d}}=\widetilde{{\mathscr{S}}_{ij}}- \delta _{ij}\widetilde{{\mathscr{S}}_{ij}}/3$$ is the deviatoric part of the rate-of-strain tensor, which is defined as $${\mathscr{S}}_{ij}=(\partial \widetilde{{\mathscr{U}}}_i/\partial x_j+ \partial \widetilde{{\mathscr{U}}_j}/\partial x_i)/2$$, where $$\delta _{ij}$$ is the Kronecker delta symbol. The other parameters $$\mu _t$$ and $$\lambda _t$$ in (2) are the turbulent eddy viscosity and conductivity, respectively. The LES solution is formally defined everywhere in the computational domain, and the wall-model equations are solved on a separate embedded grid near the wall. The only difference between the LES and the wall-model equations is in the computation of $$\mu _t$$ and $$\lambda _t$$. In the LES framework, $$\mu _t=\mu _{{SGS}}$$ and $$\lambda _t=\lambda _{{SGS}}$$ are the SGS eddy viscosity and conductivity, respectively. Note that the isotropic part of the modeled turbulent stress tensor is neglected in (2). This is often the case in the LES approach for low Mach number flows. The same assumption is made in the zero-equation RANS modeling context, but is less justified.
### Equilibrium wall-model
The equilibrium wall-model is derived from the compressible Reynolds-averaged Navier–Stokes equations with the boundary layer scaling approximations and neglecting unsteady and convective terms such as pressure gradient. The equilibrium model essentially relies on a grid resolution based on the outer layer requirements using the largest scales of the boundary layer thickness. The inner layer lies within the first cells normal to the wall and its behavior is modeled through the momentum wall normal flux. A typical resolution is about 20 cells per turbulent boundary layer thickness, $$\delta$$, in each spatial direction10. However, in this study, a fundamental question arises when considering the pressure fluctuations that are responsible for noise generation. In fact, to the best of our knowledge, there are no studies in the literature that guarantee that using a WMLES approach with a resolution of 20 cell per $$\delta$$ results in an accurate prediction of the turbulent structures responsible for the pressure fluctuations and hence, that can achieve an overall discretization that can predict the noise sources. However, because the location of the maximum pressure fluctuations is within the outer layer, this resolution appears to generate reasonable and satisfying results. The verification of this assumption is one of the primary goals of this paper.
In the following, we briefly review the equilibrium model used herein combined with the WMLES approach. Hereafter, we use the subscript “$${{\rm wm}}$$” to denote a quantity at the wall. Thus, given the instantaneous magnitude of the wall-parallel velocity $${\mathscr{U}}_{||}$$ and the instantaneous temperature $${\mathscr{T}}$$ in the LES at height $$x_2=l_{{{\rm wm}}}$$ perpendicular to the wall, we estimate the instantaneous wall shear stress vector $$\tau _{w,i}$$ by solving the following system using two ordinary differential equations (ODEs)10
\begin{aligned} \frac{d }{d x_2} \left[ \left( \mu + \mu _{t,{\rm wm}} \right) \frac{d{\mathscr{U}}_{||}}{d x_2}\right]= &{} 0, \end{aligned}
(4)
\begin{aligned} \frac{d }{d x_2} \left[ c_{\mathscr{P}}\left( \frac{\mu }{{\rm Pr}} + \frac{\mu _{t,{\rm wm}}}{{\rm Pr}_{t,{\rm wm}}}\right) \frac{d{\mathscr{T}}}{d x_2}\right]= &{} - \frac{d }{d x_2} \left[ \left( \mu + \mu _{t,{\rm wm}} \right){\mathscr{U}}\frac{d{\mathscr{U}}_{||}}{d x_2}\right] , \end{aligned}
(5)
where the wall-model eddy viscosity is taken as
\begin{aligned} \mu _{t,{\rm wm}} = \kappa \sqrt{\rho |\tau _w|} \, x_2 \left[ 1 - \exp \left( -\frac{x_2^+}{A^+}\right) \right] ^2. \end{aligned}
(6)
The values of the constants appearing (5) and (6) are $$\kappa =0.41$$, $$A^+=17$$ and $${\rm Pr}_{t,{\rm wm}}=0.9$$. For compressible flows, the scaled distance from the wall in the Van Driest damping factor ($$x_2^+$$ in (6)) is computed using a semi-local scaling given by $$x_2^+=x_2\sqrt{\rho |\tau _w|}$$17. These two coupled ODEs are solved using the tridiagonal matrix or Thomas algorithm (TDMA) applied in a segregated manner, i.e., by alternating TDMA sweeps of the momentum and energy equations with updated eddy-viscosity $$\mu _{t,{\rm wm}}$$ in between. The pressure is assumed to be constant in the $$x_2$$ direction and is obtained from the LES solution at the exchange location, while the density is computed using the temperature via the equation of state.
One major difficulty in the parallel implementation of the model for unstructured grids is when the boundary condition for the wall-stress model (exchange location) and the wall location are located on different processors in terms of the domain decomposition associated with parallel computations. This issue is fixed by linking each wall face with its exchange location during the pre-processing step using parallel communicators, which are then used during the computation. This last point is important in terms of numerical errors that can arise within the first cell of the LES mesh. Indeed, these errors may significantly decrease the accuracy of the solution even though the model is correct. The recommended choice of the exchange location above the wall boundary is typically $$\sim 0.1\delta$$14. The grid spacing in the wall normal direction must be smaller than the resolution needed for fully turbulent simulations, which is a limitation of the WMLES approach for transitional flow. In fact, the typical wall normal grid spacings for WMLES based on turbulent boundary layer thickness, $$\delta$$, might be too coarse to resolve the pre-transitional laminar boundary layer and its instability. Park and Moin18 reported that, for WMLES in transitional cases, at least the size of the first grid cell near the wall $$\Delta x_2^+<20$$ is required to marginally resolve the integral length scales of the pressure-producing eddies near the wall. However, when using such a fine wall normal spacing, it may not always be possible to have the LES input taken at a distance even close to $$\sim \, 0.1\delta$$19. In the present study, because the flows are often transitional, the exchange location is taken at least three cells away from the wall, which lies at approximately 0.08–0.1$$\delta$$.
### Computational setup
The compressible Navier–Stokes equations are solved numerically using the massively parallel CharLES$$^{{\rm X}}$$ solver. This solver implements a cell-centered finite volume scheme that is minimally dissipative. The Euler flux is computed by a blend of a non-dissipative central scheme and a dissipative upwind scheme
\begin{aligned}{\mathfrak{F}} = \frac{1}{V} \int _{\partial V}{\mathbf{F}} \left({\mathbf{q}} \right) \cdot d{\mathbf{S}} = \sum _{f\in \text{faces}} (1-f_{\alpha f}){\mathfrak{F}}_{{CENTRAL}} + \, \, f_{\alpha f}{\mathfrak{F}}_{{UPWIND}}, \end{aligned}
(7)
where the blending parameter $$0\le f_{\alpha f}\le$$ is precomputed based on the local grid quality. To avoid numerical instabilities, the dissipative upwind-flux contribution is significant only in the region of relatively poor grid quality20. Indeed, the proportion of the upwind flux $$f_{\alpha f}$$ is not a static parameter, but it scales with the local departure of the global advection matrix (constructed with the central scheme) from a skew-symmetric matrix. For all the grids used in this study, the upwind proportion is less than 1.5% ($$f_{\alpha f}<0.015$$) in the regions with non-zero Reynolds stresses. Thus, the small-scale near-wall eddies and the large eddies passing through the separated shear layer, which are important in the dynamics of the separating and reattaching mechanisms, are largely unaffected by the numerical dissipation.
The numerical method is formally second-order accurate in space, although it achieves fourth-order accuracy on a uniform mesh containing only hexahedral cells. Time integration is performed using a third-order low-storage Rung–Kutta–Wray scheme21. Unresolved turbulent scales are modeled using the constant-coefficient Vreman subgrid scale model22.
The computational grid around the controlled-diffusion airfoil is topologically an O-type mesh due to the round leading and trailing edges, with boundary layer clustering at the airfoil walls. The boundary layer clustering blocks are structured blocks, whereas the rest are unstructured to allow for a quick outward coarsening. The computational grid is also equipped with wake blocks with a large angle of opening. Note that independently of the angle of attack to be simulated, the blocks are not rotated; hence, the mesh is not changed, as the inlet velocity angle is instead adapted. The computational domain extends 20c in both the streamwise, $$x_1$$, and wall-normal, $$x_2$$, directions, to allow for a velocity inlet boundary condition to act as free-stream (see Fig. 1). Unphysical numerical reflections at the computational boundaries are avoided by the choice of appropriate boundary conditions. Characteristic boundary conditions are used at all inflow and outflow boundary conditions23,24. At the airfoil surface, an adiabatic, no-slip condition is applied. Periodic boundary conditions are imposed in the spanwise direction.
Although there are no experimental data dealing with the spanwise correlation length, Grebert et al.25 found in their WRLES that a spanwise extent $$L_{x_3}$$ of at least $$2\delta _{\max }$$ showed reasonable agreement with experimental data for an accurate estimate of the correlation length $$\ell _z$$.
Three flow conditions from the matrix of computation proposed by the SCONE project16 are studied here, and their details are summarized in Table 1. For $${{\mathcal{C}}}_{*,{\mathfrak{c}}_1}$$ cases, the resolution level is very similar to the standard DNS resolution criteria26,27; see Table 2. For these simulations, the mesh resolution quality was also verified by Pope’s criterion27, which monitors the quantity $$IQ_k={\mathscr{K}}/({\mathscr{K}}+{\mathscr{K}}_{{SGS}})$$, where $${\mathscr{K}}$$ and $${\mathscr{K}}_{{SGS}}$$ are the resolved and the SGS turbulent kinetic energies, respectively. On the one hand, the resolved turbulent kinetic energy is evaluated as $${\mathscr{K}}=({\mathscr{U}}_1^{'2}+{\mathscr{U}}_2^{'2}+{\mathscr{U}}_3^{'2})/2$$, where $${\mathscr{U}}_i^{'}$$ is the root-mean square (RMS) value of fluctuating part of the velocity component $${\mathscr{U}}_i$$. On the other hand, the SGS turbulent kinetic energy is estimated by $${\mathscr{K}}_{{SGS}}=(\nu _t/(C_M\Delta ))^2$$, where $$\nu _t$$ is the kinematic turbulent viscosity, $$C_M$$ a constant set to 0.06928, and $$\Delta$$ is estimated as the cubic root of the elements volume. For the grids indicated in Table 2, we found that $$IQ_k\ge 0.94$$ for the three WRLES cases, which means that the resolution level is close to that of a DNS mesh.
With the wall-model configuration, no near-wall streaks can be captured. In fact, the friction momentum flux is imposed by the model. Therefore, there is no need to follow any resolution requirement imposed by these structures. Note that when considering the $$x_2$$ direction, for example, the number of points remain essentially unchanged compared to WRLES. Instead, the first cell close to the wall increases in size. This ensures a proper resolution in the outer layer and drastically increases the time step because of the Courant–Friedrichs–Lewy (CFL) limit, which is directly dependent on this length scale. The overall cost ratio between the WRLES and the WMLES simulations is close to 50. The grid resolution used for the WMLES is chosen to resolve the flow scales in the outer layer and thus, the grid spacing is scaled by the local boundary layer thickness. As a consequence, the mesh does not strongly depend on the Reynolds number. As indicated in Table 2, the same grid is used for all WMLES computations. The cases labeled with $${{\mathcal{C}}}_{\star ,{\mathfrak{c}}_3}$$ all have the same grid resolution but the wall-model is not activated to investigate its effect at an iso-mesh resolution. The grid distributions in wall units for these computations along the chord are shown in Fig. 2. The grid sizes for all the WMLES cases have $$\Delta x_1\approx \Delta x_3$$. In order to capture the boundary layer transition on the upper surface of the blade and achieve as smooth a flow as possible at the airfoil trailing edge, the grid spacing in the wall normal direction is fine, i.e., $$\Delta y^+$$ is less than 20 for the three cases. To allow errors due to the subgrid modeling and numerics to be made arbitrarily small, as shown in the study by Kawai and Larsson14, five grid points ($$l_{\text{wm}}=y_5$$) off the wall in the LES mesh are matched to the wall-model top boundary in this work. The variations of the height of the exchange location in viscous units are included in Fig. 2d. In the present study, the local wall-model layer thickness is set to a maximum of 4 times the local wall-normal grid-spacing and a user-defined minimum thickness $$l_{\text{ud}}$$. The effects of the exchange location on the flow field, especially on the velocity profiles, will be investigated in future simulations.
Finally, it is important to emphasize that for the WMLES calculations, the non-dimensional time-step size $$\Delta t=\Delta t^{\star }{\mathscr{U}}_{\infty }/c$$ (where the superscripts $$^{\star }$$ denote dimensional quantities) is larger due to the coarse grid resolution near the wall, and approximately two orders of magnitude bigger than that used for the WRLES computations. Furthermore, the maximum CFL number is $$\sim \, 0.4$$.
## Results
In this section, we simultaneously: (i) investigate the generation of airfoil trailing edge broadband noise that arises from the interaction of a turbulent boundary layer with an airfoil trailing edge and (ii) study the performance of the equilibrium boundary layer wall-model presented in “Equilibrium wall-model”. Describing a flow as it passes a controlled-diffusion blade is challenging because it involves complex physical processes: laminar separation, turbulent transition, turbulent reattachment, and turbulent separation near the trailing edge on the suction side. We compare our numerical results with the experimental database generated within the framework of the CRORTET Clean Sky project29.
### Flow topology
The three-dimensional features of vortex structures and their breakdown processes in the laminar-turbulent transition region near the leading edge are visualized by the instantaneous iso-surfaces of the second invariant of velocity gradient tensor, $$Q=(\omega ^2+2{\mathscr{S}}_{ij}{\mathscr{S}}_{ij})/4$$, in Fig. 3. The Q iso-surfaces are colored by streamwise velocity to approximately identify the height of these vortex structures. According to the definition of Q-criterion, a vortical structure is identified in a region with positive Q, i.e, a region where vorticity overcomes the strain.
A fully laminar boundary layer is present on the lower (pressure) side up to the trailing edge of the airfoil, as well as a transitional and turbulent boundary layer on the upper (suction) side. A transition from a laminar to a turbulent state occurs in the shear layer resulting from the flow separation in the leading edge region, which results in the massive generation of vorticity downstream of the leading edge, with large vortices shed from the suction side of the airfoil. However, some smaller vortices still remain attached to the wall and roll over the airfoil suction side, grazing the trailing edge. As the curvature of the controlled-diffusion airfoil changes, the adverse pressure gradient leads to an increase of the boundary layer thickness. At the trailing edge, the laminar boundary layer coming from the pressure side destabilizes into a small vortex-shedding. This Von Kármán street then interacts with the fully turbulent vortical structures issuing from the upper side.
Figure 4 depicts the level of vorticity and the size of turbulent structures in the flow at the same instant for all three $${{\mathcal{C}}}_1$$ simulations. We observe that the vorticity and the vortex shedding is more intense for the $${{\mathcal{C}}}_{1,{\mathfrak{c}}_1}$$ case (i.e., the WRLES) than for $${{\mathcal{C}}}_{1,{\mathfrak{c}}_2}$$ and $${{\mathcal{C}}}_{1,{\mathfrak{c}}_3}$$ cases. Furthermore, in the $${{\mathcal{C}}}_{1,{\mathfrak{c}}_1}$$ case, many more structures are resolved. The nature of the developed turbulent structures is broadly influenced by the size and the structure of the recirculation bubble near the leading edge (see Fig. 4). Furthermore, the WMLES (i.e., the $${{\mathcal{C}}}_{1,{\mathfrak{c}}_2}$$ case) does not show a laminar separation, and the flow becomes turbulent without the clear $$2{{\rm D}}$$ vortex breakdown, as is shown in the $${{\mathcal{C}}}_{1,{\mathfrak{c}}_1}$$ and $${{\mathcal{C}}}_{1,{\mathfrak{c}}_3}$$ cases. Indeed, the vortices are first generated very close to the wall due to the flow instability induced by the sudden increase in the wall shear stress $$\tau _w$$ (which is fed into the LES as flux boundary conditions) by activating $$\mu _{t,{\rm wm}}$$ in the wall-model at the leading edge. As shown in Fig. 5, by increasing both the Reynolds and the Mach numbers, the flow topology observed in the former case $${{\mathcal{C}}}_1$$ is also reproduced in the $${{\mathcal{C}}}_2$$ case, and we can see that, even when the flow seems to be similar at $$x_1/c>0.2$$, the processes that reach the turbulent state are completely different. The comparison of the WMLES results with the reference WRLES configuration (the $${{\mathcal{C}}}_{2,{\mathfrak{c}}_1}$$ case) indicates that the equilibrium dynamic wall-model does not adequately capture the formation of two-dimensional intermittent laminar separation vortices, which are induced by the gradual adverse pressure gradient near the leading edge. Therefore, the wall-model failure directly translated into an incorrect estimate of the momentum flux and thus a different boundary layer thickening. This interesting feature is presented in Fig. 6, which shows that the wall-model simulation cannot capture the leading edge bubble properly. In fact, the laminar part of the boundary layer is treated with a wall-model, which enhances the friction and hence leads to an artificial growth of the boundary layer. Therefore, because the turbulent boundary layer has a weaker resistance against separation, the separation occurs earlier. For a very small recirculation region as in the current cases, the wall-model pushes the flow to recover a turbulent boundary layer. This numerical behavior is problematic and is the cause of the very small size of the recirculation region. However, the vortical structures at the trailing edge and the wake are similar to the reference WRLES case; see Figs. 4 and 5.
### Mean flow profiles
In Fig. 7, we present the time- and spanwise-averaged pressure coefficient, $$C_{{\rm {\mathscr{P}}}}$$, computed as
\begin{aligned} C_{{\mathscr{P}}}= \frac{\langle{\mathscr{P}}\rangle -{\mathscr{P}}_\infty }{\frac{1}{2}\rho _\infty{\mathscr{U}}_\infty ^2}, \end{aligned}
(8)
where $$\langle {\rm P}\rangle$$ is the time- and spanwise-averaged wall pressure and $${\mathscr{P}}_\infty$$ is the reference pressure taken at the outlet boundary. The whole pressure side of the blade is subjected to a favorable pressure gradient and the boundary layer is completely attached. On the suction side, the separation bubble creates a strong pressure drop close to the leading edge and large fluctuations of the wall shear stress are observed in the reattachment region. Downstream of the separation bubble, the mean pressure increases slightly up to mid-chord, then an adverse pressure gradient is observed up to the trailing edge. The mean flow features described in the previous section are recovered quantitatively with the numerical results, with a slight mismatch for $$C_{{\mathscr{P}}}$$ at the beginning of the separation zone. The results obtained with the WMLES approach are very close to the results of the reference WRLES configuration. However, note that the discrepancies observed at the leading edge are not surprising since our numerical simulations do not account for the jet shear layers and the possible introduction of the free-stream turbulence that are present in the experiments30. We observe a small difference in the leading edge region between the WRLES and WMLES cases. However, we found very good agreement between the WMLES calculation and the reference WRLES for the surface pressure coefficient near the trailing edge region, and we found significant improvements between the iso-resolution configurations $${{\mathcal{C}}}_{3,{\mathfrak{c}}_2}$$ (with the wall-model) and $${{\mathcal{C}}}_{3,{\mathfrak{c}}_3}$$ (without the wall-model). Therefore, we can conclude that the WMLES approach is particularly suitable for the estimation of the leading edge turbulent boundary layer characteristics. At this stage, we must acknowledge that the accurate prediction of $$C_{{\mathscr{P}}}$$ is a common finding in most external aerodynamics calculations. This result is most probably attributed to the outer-layer nature of the mean pressure, which appears to be less sensitive to the flow details of the near-wall turbulence31.
The profiles of the mean velocity normal to wall $${\mathscr{U}}_{\bot }$$ are shown in Fig. 8 for the cases $${{\mathcal{C}}}_1$$ and $${{\mathcal{C}}}_2$$. The profiles of the case $${{\mathcal{C}}}_3$$ are roughly similar to the case $${{\mathcal{C}}}_2$$; unless otherwise stated, only the results of $${{\mathcal{C}}}_2$$ are shown. The velocity profiles show almost no deviation compared to the reference WRLES. The relative errors of the three mean velocities are less than 2% for the five locations considered. The turbulent boundary layer over the airfoil is about 10–20 mm thick, which, with our grid resolution, results in roughly 20–40 points per boundary later thickness, $$\delta$$. Assuming that the flow at a given station can be approximated by a local canonical zero-pressure-gradient turbulent boundary layer, the expected error in the mean velocities can be estimated as $$\varepsilon _m=0.4\Delta /\delta$$ (with $$\Delta$$ an isotropic grid size), which yields values of 2% for the current grid resolutions. The previous estimation is strictly valid for the zero-pressure-gradient turbulent boundary layer and, as such, it should be understood only as representative of the errors in complex geometries. However, this estimation provides a useful reference for the expected performance of WMLES in the absence of 3D and non-equilibrium effects for the current grid resolutions.
In Fig. 9, the Reynolds stress component at five different locations are also plotted. The trend followed by all the stress components is correctly captured, although their magnitudes are systematically underpredicted by $$\sim \, 18\%$$. In fact, they present a slight deviation consistent with the modeling approach, which supports a limited fraction of the turbulent fluctuations. This behavior is typical for WMLES, which usually tends to overestimate the streamwise fluctuations while underestimating the transverse components. This is a direct effect of the grid resolution; the turbulent flow cannot carry identical turbulent structures in the near-wall region, which affects the Reynolds stress structures. However, the difference in the Reynolds stresses is mainly localized in a narrow region very close to the wall. Thus, it should not significantly affect the pressure fluctuations whose maximum value is located further away from the solid wall. The prediction capability of the WMLES reduces close to the trailing edge for the streamwise component of Reynolds stress. Note that at iso-resolution, the resolved Reynolds stress computed by WMLES are in good agreement with the reference WRLES simulations compared to the case where the wall-model is not activated.
In Fig. 10, the mean wall-parallel velocity profiles on the suction side are presented in log scale. The most basic wall-models are based on analytical laws that provide a direct link between the velocity normal to the wall $${\mathscr{U}}_{\bot }$$ at a certain wall distance $$x_2$$ and the wall-shear stress $$\tau _w=\rho{\mathscr{U}}_\tau ^2$$. The most widely used law is the Reichardt law-of-the-wall32:
\begin{aligned}{\mathscr{U}}_{\bot }^+=\frac{1}{\kappa }\ln (1+y^+\kappa )+ \left( C-\frac{1}{\kappa }\ln (\kappa )\right) \left( 1-e^{-\frac{y^+}{11}}- \frac{y^+}{11}e^{-\frac{y^+}{11}}\right) , \end{aligned}
(9)
with $${\mathscr{U}}_{\bot }^+={\mathscr{U}}_{\bot }/{\mathscr{U}}_\tau$$ and $$y^+=y{\mathscr{U}}_\tau /\nu$$. Although this law is dedicated to equilibrium flows, it should be applicable to out-of-equilibrium flows as long as the input position is chosen below the end of the log-layer region. The constant $$\kappa$$ and C are the same than those of the typical von Kármán classical law of the wall33
\begin{aligned}{\mathscr{U}}_{\bot }^+=\frac{1}{\kappa }\ln (y^+)+C. \end{aligned}
(10)
Both analytical approaches are compared in Fig. 10. Notice that the wall-parallel velocity component profiles show a fast increasing boundary layer thickness after 50% of the chord, due to a strong adverse pressure gradient on the blade. At $$x_1/c=0.9$$, the flow is nearly at the edge of the separation, but is still attached ($$\partial{\mathscr{P}}/\partial x_2$$ is positive). All WMLES profiles show quite a large log-region, which collapse on the Reichardt law-of-the-wall for $$y^+\le 300$$. If the wall-model input is chosen below that threshold, the equilibrium wall-model in this study appears to be sufficient. Nagib and Chauhan34 have emphasized the non-universality of the Kármán constant $$\kappa$$ through various experiments on canonical flows and showed that FPG leads to higher values of $$\kappa$$ while adverse pressure gradient gives lower $$\kappa$$. They also showed that the constant C can be negative. Therefore, the comparability between our results and the two proposed analytical approaches could be improved by finding an optimal choice of $$\kappa$$ and C values.
In Fig. 11, we show the variation of the shape factor ($${\rm H}_{12}=\delta ^{\star }/\theta$$), the momentum thickness $$\theta$$, and the displacement thickness $$\delta ^\star$$ against the chordwise distance $$x_1/c$$ along the aerofoil upper surface. In the case of aerofoil flows, the traditional definition of boundary layer parameters as used for flat plates must be adapted to be consistent with the spatially varying potential flow. In the present study, the definitions used for the displacement thickness $$\delta ^\star$$ and the momentum thickness $$\theta$$ follow the recommendation of Wagner et al.35. Near the leading edge, the flow is laminar and the shape factor is approximately 2.6 for $${{\mathcal{C}}}_{*,{\mathfrak{c}}_1}$$ cases, which is a typical value of the Blasius profile, while the $${{\mathcal{C}}}_{*,{\mathfrak{c}}_2}$$ cases feature a higher value $$\sim 2.9$$ (zoom not shown). The displacement thickness reaches a local maximum in the region of the separation bubble while the momentum thickness is nearly zero, which results in a very large shape factor. Downstream of the reattachment region ($$x_1/c\simeq 0.03$$ for all cases), the shape factor decreases abruptly, and then close to the mid-chord where there is close to a zero pressure gradient, a plateau is observed at $$\sim 1.5$$ for $${{\mathcal{C}}}_1$$, 1.35 for $${{\mathcal{C}}}_2$$ and $$\sim 1.28$$ for $${{\mathcal{C}}}_3$$, which is smaller than the typical value of the Klebanoff profile $${\rm H}_{12}\simeq 1.4$$ observed for turbulent boundary layers). Along the second half of the chord, both the two thicknesses and the shape factor increase because of the adverse pressure gradient. A larger difference is observed between $${{\mathcal{C}}}_2$$ and $${{\mathcal{C}}}_3$$. Those differences depend both on the Reynolds number and the pressure gradient36.
### Acoustic field
Analyzing the divergence of the velocity field is particularly useful to delineate the sources and the strength of the acoustic waves in the flow field. An instantaneous picture of the field of $$\nabla \cdot{\mathscr{U}}$$ on the midspan plane is shown in Figs. 12 and 13 along with contours of iso-entropy $$s=\log ({\mathscr{P}}/\rho ^\gamma )$$. The $${{\mathcal{C}}}_3$$ simulation is found to be qualitatively similar to the $${{\mathcal{C}}}_2$$ simulation. Two major acoustic source regions can be visually identified: the trailing edge and the transition/reattachment zones on the upper surface. Contrary to the study of Wu et al.37, the near wake does not appear to contribute significantly to the self-noise. The transition/reattachment noise source appears to be even stronger than the trailing edge noise for this flow configuration. This is likely caused by the high local Mach number due to the strong flow acceleration near the leading edge where the blade curvature is higher. An additional sound source is visible near the leading edge on the suction side of the lifting surface. This sound source is mainly caused by the massively accelerated vortex structures near the leading edge38. The transition/reattachment noise source radiates mostly downstream as it is shielded by the upper blade surface (with only a slight diffraction at the leading edge), whereas the trailing edge source shows an anti-symmetric pattern on each side of the blade, as expected from classical trailing edge noise theories39. Note the existence of a noise source near the trailing edge. The vortical structures in the turbulent boundary layer pass the trailing edge and generate sound. Visually, the WMLES approach seems to reproduce a qualitatively similar acoustic source field to the reference WRLES simulations. For the higher Mach number cases $${\mathscr{C}}_2$$$${\mathscr{C}}_3$$, we notice a stronger radiation in the presence of the separating shear layer, which generate additional noise sources and affect the propagation of acoustic waves upstream by the refraction through the separated shear layer. The secondary source, at higher frequency (closer wave fringes), appears less on the suction side, near the leading edge, close to the reattachment point of the recirculation bubble when using WMLES, which results from the fact that the current equilibrium WMLES does not show properly the laminar separation.
### Frequency spectra of surface pressure fluctuations
Here we assess the capability of wall-modeled LES to predict surface pressure fluctuations on the suction side of the airfoil surface at $$x_1/c=0.976$$, as shown in Fig. 14. The frequency spectra of pressure fluctuations (PSD) as well as the cross-spectral and auto-spectra have been estimated numerically based on the Welch method of periodogram40, which minimizes the variance of the PSDs estimator41. The overall pressure signals from our computations, extracted from probes that are placed in the first wall-normal cell along the airfoil, are subdivided into $${\mathscr{N}}_{\rm s}/16$$ sub-blocks, where $${\mathscr{N}}_{\rm s}$$ represents the number of pressure recordings of each case. We adopt the Welch method combined with a Hanning window throughout the spectral analysis while using a FFT. The overlapping of the sub-blocks is set to 50%. The frequency spectrum is then obtained by averaging the periodograms of all the sub-blocks. The normalized sampling frequency is set to 3,000, i.e., we collect 3,000 pressure signals per each probe and convective time scale. The PSD is plotted as a function of the streamwise location and the normalized frequency (or Strouhal number $${{\rm St}}=fc/{\mathscr{U}}_{\infty }$$). The trailing edge noise displays a low frequency broadband spectrum with most of the energy of the signal below $${{\rm St}}=9$$. The PSD of each case features noticeable differences, as indicated by the slopes that are superimposed onto each dimensionless PSD. In general, the WMLES is found to underpredict the spectral level of the experimental data by 2–4 dBs in the low to intermediate range of frequency. The high range of frequency of the experimental pressure spectra is more accurately captured by the WMLES compared the reference WRLES’s. The most striking finding in this figure is that at iso-resolution, WMLES cases are in better agreement than the cases without the wall-model, mainly at the low frequency range. Overall, all of the spectra in Fig. 14 demonstrate that the broadband content is predicted reasonably well by the WMLES, which provides a good baseline study to assess and quantify noise prediction. The tonal peaks that arise from the experimental spectra on Fig. 14 are attributable to acoustically untreated duct models of the wind-tunnel facility. Moreover, the observed disparities at low Strouhal number may be partly caused by the installation effects of the experiment that are not accounted for in the numerical simulations, but also to a lack of statistical convergence of the WRLES spectra as a result of the time series being short.
The wall-pressure statistics computed from the time-domain analysis are deduced from the temporal pressure cross-correlation calculated as
\begin{aligned}{\mathscr{R}}_{{a,b}}(\xi _1,0,\xi _3,\tau )=\langle{\mathscr{P}}'(x_1,0,x_3,t){\mathscr{P}}'(x_1+\xi _1,0,x_3+\xi _3,t+\tau )\rangle , \end{aligned}
(11)
where $${\mathscr{P}}^\prime$$ is the surface pressure fluctuations, a and b represent two arbitrary points on the airfoil separated by distance $$\xi _1$$, $$\xi _3$$ in the streamwise and spanwise directions, respectively. The brackets $$\langle \cdot \rangle$$ indicate an ensemble average. The time delay between two signals $$\tau$$ is normalized on the chord c, and the inflow velocity $${\mathscr{U}}_\infty$$. The effect of streamwise separation is depicted in Fig. 15. The cross-correlation $${\mathscr{R}}_{{a,b}}(\xi _1,0,0,\tau )$$ is computed using the probe located at $$x_1/c=0.975$$ as a reference for the computation of correlation with upstream probes that are separated by $$\xi _1$$. We note the diminution of the correlation between the two signals with increasing separation between the signals. The decreasing peak of cross-correlation is due to the change in the pressure signature in the turbulent boundary layer, since the greater the distance between the probes, the greater the distance over which the structures can change and evolve, which yields a lower correlation between the pressure signals. For the WMLES cases, we notice that the pressure signals are correlated over a larger separation distance, and thus for a higher time delay relative to the reference WRLES configurations. This can be explained by the fact that the structures evolve less rapidly when using the wall model, i.e., the structures are constrained by using the wall-model, leading to much lower correlation value of $${\mathscr{R}}_{{a,b}}(\xi _1,0,0,\tau )$$.
Using the function $${\mathscr{R}}_{{a,b}}(\xi _1,0,0,\tau )$$, it is possible to estimate the value of convection velocity of the vortical structures $${\mathscr{U}}_c(\xi _1)$$ as a function of the separation distance $$\xi _1$$ in the streamwise direction between probes as
\begin{aligned} \frac{{\mathscr{U}}_c(\xi _1)}{{\mathscr{U}}_{{\rm edge}}}= \frac{\xi _1/\delta ^\star }{\left[ \tau{\mathscr{U}}_{{\rm edge}}/\delta ^\star \right] _{\max }}, \end{aligned}
(12)
where $$\left[ \tau{\mathscr{U}}_{{\rm edge}}/\delta ^\star \right] _{\max }$$ represents the time lag, which corresponds to the maximum of the cross-correlation $${\mathscr{R}}_{{a,b}}(\xi _1,0,0,\tau )$$ between two pressure signals separated by a normalized distance $$\xi _1/\delta ^\star$$.
In Fig. 16, the normalized convection velocity is extracted as a function of the longitudinal separation $$\xi _1$$, taken as the reference probe located at $$x_1/c=0.978$$, for both the reference WRLES and WMLES computation. The convective velocity ranges from $$0.55\cdot{\mathscr{U}}_\infty$$ to $$0.80\cdot{\mathscr{U}}_\infty$$ with an apparent dependency on the Reynolds number or angle of attack. WMLES seems to cause a higher streamwise convective velocity $${\mathscr{U}}_c$$ with respect to the baseline WRLES configuration.
### Cross-spectral density and coherence
One of the main quantities generally used to describe wall-pressure statistics is the cross-spectral density in the spanwise direction, which is defined by
\begin{aligned} \Gamma _{{{\text{P}}_{a}}{{\textsf{P}}_{b}}}(\xi _{1},0,\xi _{3},\omega ) =\langle \widehat{\mathscr{P}}'^{*}(x_{1},0,x_{3},\omega ), \widehat{\mathscr{P}}'(x_{1}+\xi _{1},0,x_{3}+\xi _{3},\omega )\rangle , \end{aligned}
(13)
where $$\widehat{\mathscr{P}'}$$ is the time FFT transform of the fluctuated wall pressure. The complex conjugate is denoted by $$\cdot ^{*}$$. According to the coherence function, which is used to indicate the strength of the relation between the pressure fluctuations at two separate locations as the frequency varies, as the disturbed flow is convected in the streamwise or in the spanwise direction, it is possible to express the cross-spectrum in non-dimensional form. Hence, the coherence function can be expressed as
\begin{aligned}{\gamma }_{{a,b}}(\xi _{1},0,\xi _{3},\omega ) =\frac{|\Gamma _{{{\textsf{P}}_{a}}{{\textsf{P}}_{b}}}(\xi _{1},0,\xi _{3},\omega )|}{\sqrt{\Gamma _{{{\textsf{P}}_{a}}{{\textsf{P}}_{a}}}(\omega ) \Gamma _{{{\textsf{P}}_{b}}{{\textsf{P}}_{b}}}(\omega )}}, \end{aligned}
(14)
where $$\Gamma _{{{\textsf{P}}_{a}}{{\textsf{P}}_{a}}}$$, $$\Gamma _{{{\textsf{P}}_{b}}{{\textsf{P}}_{b}}}$$ are the pressure auto-spectra at the two positions a and b, respectively. The different analytical expression coherence functions proposed in the literature are a function of the angular frequency $$\omega$$ (or equivalently the Strouhal number $${\rm St}=\omega \delta /u_\tau$$), and normalized with respect to $$\xi _3$$ and $${\mathscr{U}}_c$$, which are computed from the cross-correlation functions.
Despite its simplicity, the Corcos model42 is still extensively used in many practical applications, since it provides a simple analytical form and clear physical significance. This model is based on the assumption that the correlation lengths in both the streamwise and spanwise directions are statistically independent, and thus the coherence takes the form
\begin{aligned}{\gamma }_{{a,b}}(\xi _1,0,\xi _3,\omega )= e^{-\alpha _1k_c|\xi _1|}e^{-\alpha _3k_c|\xi _3|}e^{-\alpha _1k_c\xi _1}, \end{aligned}
(15)
where $$k_c=\omega /U_c$$ is the convective wavenumber. In the formulation, $$\alpha _1$$ and $$\alpha _3$$ are constants that measure the loss of coherence in the streamwise and spanwise directions, for which different values can be found in the literature.
A second semi-empirical model, which is an extension of the Corcos model, has been proposed by Efimtsov43 and improved by Salze et al.44. This model explicitly considers the compressibility effects by introducing a number of additional tunable constants. The coherence function is expressed for this model, according to the multiplicative hypothesis (cf. (15)), as
\begin{aligned}{\gamma }_{{a,b}}(\xi _1,0,\xi _3,\omega )= e^{-|\xi _1|/\Lambda _1}e^{-|\xi _3|/\Lambda _3}e^{-\alpha _1k_c\xi _1}, \end{aligned}
(16)
where $$\Lambda _1$$ and $$\Lambda _3$$ are the correlation lengths in the streamwise ($$\Lambda _1={\mathscr{U}}_c/\omega \alpha _1$$) and spanwise direction ($$\Lambda _3={\mathscr{U}}_c/\omega \alpha _3$$), respectively. For a free stream Mach number below 0.75, the spanwise correlation length is modeled as follows44
\begin{aligned} \frac{\Lambda _3}{\delta }=\left[ \left( a_4\frac{\omega \xi _3}{{\mathscr{U}}_c}\frac{\delta }{\xi _3}\right) ^2+ \frac{\left( a_5\frac{u_\tau }{{\mathscr{U}}_c}\right) ^2}{\left( \frac{\omega \xi _3}{{\mathscr{U}}_c} \frac{\delta }{\xi _3}\right) ^2+\left( \frac{a_5}{a_6}\frac{u_\tau }{{\mathscr{U}}_c}\right) ^2} \right] ^{-1/2}, \end{aligned}
(17)
where the empirical parameters $$a_{4,5,6}$$ are adjustable through proper fit of experimental or numerical data. According to Palumbo45, the Efimtsov–Salze’s constant $$a_4$$ constrains the amplitude of mid- to high-frequency coherence lengths, similar to the constants in the Corcos model. The frequency at which the Efimtsov–Salze model breaks away from the Corcos model is governed by the parameter $$a_5$$, while the low-frequency roll off is controlled by the parameter $$a_6$$.
The major limitations of the Salze–Efimtsov model are the tunable constants $$a_{4,5,6}$$, which generally depend on the Mach and Reynolds numbers, as well as on the separation distance. These parameters are estimated in this study at $$x_1/c=0.976$$ by fitting the coherence data points obtained from the reference WRLES configurations, by means of a nonlinear least-squares optimization procedure. The coefficients thus obtained in the range of Mach numbers considered are $$a_4=0.47$$, $$a_5=17.80$$ and $$a_6=1.0$$. We depict in Fig. 17 the spanwise coherence length at different locations close to the trailing edge for the three cases. We see that the collapse of the curves for both the Corcos or Salze–Efimtsov models is achieved at high frequencies for all the tested computations at $$x_1/c=0.976$$. We obtain less satisfactory agreement between the theoretical predictions and the numerical data far upstream $$x_1/c=0.976$$ (for which the parameters of Salze–Efimtsov model are computed). Nevertheless, by comparing spectra obtained for the two locations $$x_1/c=0.909$$ and $$x_1/c=0.976$$ , it can be concluded that very close to the trailing edge, the wall-pressure statistics are well established and almost stationary justifying the use of radiation models based on a wall-pressure statistics at a single point close to the trailing-edge46,47. The Salze–Efimtsov model predicts the peak in coherence lengths and compares reasonably well to the Corcos model. The Salze–Efimtsov also achieves a much better agreement with the present simulated dataset at lower frequencies. The lack of agreement for the Corcos model at low frequencies is predicted, since the Corcos model is not suited for this range of frequencies.
Moreover, good agreement between the predictions of the Salze–Efimtsov model and the numerical data is obtained in all cases. Unlike the Corcos approximation, the Salze–Efimtsov model is able to provide a good agreement in the low-frequency range of the coherence functions, especially at $$x_1/c=0.976$$ for all the matrix computation. We note that the comparison of coherence length distribution at $${{\rm Re}}=2.29\times 10^6$$ between both models and the simulated results is better than the case $${{\rm Re}}=8.30\times 10^5$$, suggesting a Reynolds number dependence of the shape of these models. We can also see that the WMLES curves fit the reference WRLES computations reasonably well, regardless of the studied case.
## Conclusion
The prohibitive computational cost of resolving the inner region of turbulent boundary layers prevents using the WRLES approach to study high-Reynolds number turbulent flows in complex geometries. However, using the wall-modeled LES approach can reduce this computational cost. The major objective of this work was to evaluate the feasibility and accuracy of the WMLES approach for the prediction of broadband noise generated by a controlled-diffusion blade. To this end, we carried out detailed comparisons between the results obtained with the equilibrium WMLES approach, experimental measurements, and the wall-resolved LES reference solutions. LES with wall-modeling accurately predicted the pressure coefficient distribution, velocity statistics (including the mean velocity), and the trace of Reynolds tensor components. We also found that, for the wall-modeled LES cases, the instantaneous flow structures resembled those observed in the reference wall-resolved LES, except near the leading edge region. In addition, the boundary layer thickness, displacement thickness, and momentum thickness along the airfoil chord showed a convergent behavior towards those obtained using the wall-resolved LES. Surface pressure fluctuations, which act as the sources of the broadband noise at the far-field, were also extracted in the form of surface pressure spectral densities along the airfoil chord for both the wall-modeled LES and wall-resolved LES calculations. Our results indicate that the pressure spectral density profiles from the wall-modeled LES compare well with the experimental profiles. Furthermore, we found that wall-modeled LES is more comparable with the experimental data in the high-frequency region than the WRLES reference case. The convection velocity showed an increase towards an asymptotic value as the separation distance, $$\xi _1$$, increased, and we found that the WMLES results overestimated the WRLES computations. We compared the spanwise correlation length to the Corcos42 model, which assumes a frequency-independent correlation length, and the Salze–Efimtsov model44, which, in contrast, introduces a frequency dependence. We found both WMLES and WRLES computations compare well with both models over the higher range of frequency, which accounted reasonably well for the effects of the pressure gradients. Nevertheless, the Salze–Efimtsov model was in better agreement with the computational results for the lower range of frequency, but two parameters needed to be tuned. Overall, the equilibrium WMLES presents an interesting approach to (i) investigate in more detail the turbulent pressure field induced by turbulent boundary layer over the controlled-diffusion airfoil, and (ii) advance the development of new versions of trailing edge noise models on coarser grids, which is needed for fast simulations in industrial design.
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19. Mettu, B. R. & Subbareddy, P. K., Modeling non-equilibrium effects in wall modeled LES of high-speed flows. In AIAA Aviation 2019 Forum, 3699 (2019).
20. Khalighi, Y., Ham, F., Nichols, J., Lele, S. & Moin, P. Unstructured large eddy simulation for prediction of noise issued from turbulent jets in various configurations. In 17th AIAA/CEAS Aeroacoustics Conference, 2886 (2011).
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31. Lozano-Duran, A., Bose, S. T. & Moin, P. Prediction of trailing edge separation on the NASA Juncture Flow using wall–modeled LES. In AIAA Scitech 2020 Forum, 1776 (2020).
32. Reichardt, H. Vollständige darstellung der turbulenten geschwindigkeitsverteilung in glatten leitungen. ZAMM J. Appl. Math. Mech./Z. Angew. Math. Mech. 31, 208–219 (1951).
33. Von Kármán, T. H. Mechanical similitude and turbulence. Ph.D. thesis, National Advisory Committee on Aeronautics, Washington, DC. (1931).
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35. Wagner, G. A., Deuse, M., Illingworth, S. J. & Sandberg, R. D. Resolvent analysis-based pressure modeling for trailing-edge noise prediction. In 25th AIAA/CEAS Aeroacoustics Conference, 2720 (2019).
36. Cousteix, J. Aérodynamique: Turbulence et Couche Limite (Cépaduès-Editions, Toulouse, 1989).
37. Wu, H., Sanjose, M., Moreau, S. & Sandberg, R. D. Direct numerical simulation of the self–noise radiated by the installed controlled-diffusion airfoil at transitional Reynolds number. In 25th AIAA/CEAS Aeroacoustics Conference, 3797 (2018).
38. Sano, A., Abreu, L. I., Cavalieri, A. V. G. & Wolf, W. R. Trailing-edge noise from the scattering of spanwise-coherent structures. Phys. Rev. Fluids 4, 094602 (2019).
39. Doolan, C. J. & Moreau, D. J. A review of airfoil trailing edge noise with some implications for wind turbines. Int. J. Aeroacoust. 14, 811–832 (2015).
40. Solomon, O. M. Jr. PSD computations using Welch’s method. STIN 92, 23584 (1991).
41. Di Marco, A., Camussi, R., Bernardini, M. & Pirozzoli, S. Wall pressure coherence in supersonic turbulent boundary layers. J. Fluid Mech. 732, 445–456 (2013).
42. Corcos, G. The structure of the turbulent pressure field in boundary-layer flows. J. Fluid Mech. 18, 353–378 (1964).
43. Efimtsov, B. Characteristics of the field of turbulent wall pressure-fluctuations at large Reynolds-numbers. Sov. Phys. Acoust. USSR 28, 289–292 (1982).
44. Salze, É., Bailly, C., Marsden, O., Jondeau, E. & Juvé, D. An experimental characterisation of wall pressure wavevector-frequency spectra in the presence of pressure gradients. In 20th AIAA/CEAS Aeroacoustics Conference, 2014–2909 (2014).
45. Palumbo, D. Determining correlation and coherence lengths in turbulent boundary layer flight data. J. Sound Vib. 331, 3721–3737 (2012).
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## Acknowledgements
The authors acknowledge the computing resources provided by the Barcelona Supercomputing Center (PRACE (16th call) award 2017174204), and the Supercomputing Laboratory and the Extreme Computing Research Center at KAUST. The first author benefited from interesting discussions with Professors Laurent Joly and Marc C. Jacob.
## Author information
Authors
### Contributions
J.B. conceived the study. R.B. generated the numerical database and the post-prcessing of the numerical results. J.B. particpated in the design of the model. R.B. and J.B analysed the data. R.B. wrote the manuscript with support from M.P. and J.B. M.P. was in charge of overall direction and planning.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
### Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Boukharfane, R., Parsani, M. & Bodart, J. Characterization of pressure fluctuations within a controlled-diffusion blade boundary layer using the equilibrium wall-modelled LES. Sci Rep 10, 12735 (2020). https://doi.org/10.1038/s41598-020-69671-y
• Accepted:
• Published:
• DOI: https://doi.org/10.1038/s41598-020-69671-y
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2022-07-01 15:14:23
|
{"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": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8186805248260498, "perplexity": 1290.7962762991688}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103941562.52/warc/CC-MAIN-20220701125452-20220701155452-00025.warc.gz"}
|
https://topnursingessay.com/algebra-exponential-and-sums-difference-of-logarithmic-questions/
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# Algebra Exponential and Sums & Difference of Logarithmic Questions
1. Determine whether the given functions are one-to-one. It they are one-to-one, find a formula for the inverse.(a)
(
x
)
=
x
+
5
(b)
(
x
)
=
x
2
4
(c)
(
x
)
=
2
x
3
5
(d)
(
x
)
=
x
5
(e)
(
x
)
=
6
x
+
5
2. Answer the following exponential and logarithmic questions(a) Use properties of logarithms to expand the logarithmic expression,
(
10
,
000
x
2
)
Then simplify.(b) Solve the equation
${}^{}$
3
x
=
81
(c) Solve the equation
${}_{}$
log
6
(
x
+
2
)
=
2
(d) Express in terms of sums and difference of logarithms
${}_{}$
log
b
(
6
x
5
y
z
2
)
(e) Expand and simplify
Maybe one or two problems on graphing
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2021-09-24 22:18:54
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https://physics.stackexchange.com/questions/207298/focussing-a-laser-beam-through-a-sapphire-substrate
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# Focussing a laser beam through a sapphire substrate
I am in the middle of focusing a blue (400nm) laser beam through a sapphire substrate, of thickness 430 microns. To do this, I am using a LD-Plan Neofluar 63x objective lens, which is equipped with a coverslip correction collar.
If this were glass, I would merely adjust the coverslip correction to 430 microns, and the thickness would be compensated for.
However, how should I adjust the correction for a substrate of different refractive index? (Assuming for simplicity sapphire is optically isotropic)
Is there any equation which allows you to find the optimum correction for such a system?
I have tried using a correction of 430 microns, and the optical spot was outside the sample completely. In contrast, a correction of 170 microns (which I tried first by mistake) gives a relatively good focus, so the change in refractive index is obviously very important!
Many thanks in advance!
Well, what is the index for sapphire? ... http://refractiveindex.info/?shelf=main&book=Al2O3&page=Malitson says it's 1.7866 at 400 nm. BK-7 glass is 1.53, so if the correction collar were linear, you'd adjust the correction by at $\frac{1.7866}{1.53}$ .
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2019-10-23 10:13:02
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https://socratic.org/questions/how-do-you-find-the-vertex-and-intercepts-for-y-12x-2x-2-15
|
# How do you find the vertex and intercepts for y+12x-2x^2=15?
Dec 12, 2017
$\text{see explanation}$
#### Explanation:
$\text{express in standard form}$
$\Rightarrow y = 2 {x}^{2} - 12 x + 15$
$\text{the equation of a parabola in "color(blue)"vertex form}$ is.
$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a {\left(x - h\right)}^{2} + k} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
$\text{where "(h,k)" are the coordinates of the vertex and a}$
$\text{is a multiplier}$
$\text{to obtain this form use the method of "color(blue)"completing the square}$
• " the coefficient of the "x^2" term must be 1"
$\Rightarrow 2 \left({x}^{2} - 6 x\right) + 15$
• " add/subtract "(1/2"coefficient of x-term")^2"to"
${x}^{2} - 6 x$
$2 \left({x}^{2} + 2 \left(- 3\right) x \textcolor{red}{+ 9} \textcolor{red}{- 9}\right) + 15$
$= 2 {\left(x - 3\right)}^{2} - 18 + 15$
$\Rightarrow y = 2 {\left(x - 3\right)}^{2} - 3 \leftarrow \textcolor{red}{\text{in vertex form}}$
$\Rightarrow \textcolor{m a \ge n t a}{\text{vertex }} = \left(3 , - 3\right)$
$\textcolor{b l u e}{\text{Intercepts}}$
• " let x = 0, in equation for y-intercept"
• " let y = 0, in equation for x-intercepts"
$x = 0 \to y = 2 {\left(- 3\right)}^{2} - 3 = 15 \leftarrow \textcolor{red}{\text{y-intercept}}$
$y = 0 \to 2 {\left(x - 3\right)}^{2} - 3 = 0$
$\Rightarrow {\left(x - 3\right)}^{2} = \frac{3}{2}$
$\textcolor{b l u e}{\text{take the square root of both sides}}$
$\Rightarrow x - 3 = \pm \sqrt{\frac{3}{2}} \leftarrow \textcolor{b l u e}{\text{note plus or minus}}$
$\Rightarrow x = 3 \pm \sqrt{\frac{3}{2}} \leftarrow \textcolor{red}{\text{x-intercepts}}$
$\Rightarrow x \approx 1.78 , x \approx 4.22 \text{ to 2 dec. places}$
|
2020-07-15 08:17:30
|
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|
https://math.stackexchange.com/questions/2999651/is-there-a-typo-in-this-text/2999656
|
# Is there a typo in this text?
I would like to receive some help with the next problem:
The Problem:
I have the problem with the part of the text from the book i am currently studying from. I am confused about some notations. This is the text:
"Let $$a_1$$, $$a_2$$, $$a_3$$, ... be an array of real numbers. Expresion
$$(1)\sum_{n = 1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdot \cdot \cdot + a_n + \cdot \cdot \cdot,$$
is called infinite real series with general member $$a_n$$, or shorter real series. Sums
$$s_1 = a_1,$$ $$s_2 = a_1 + a_2,$$ $$\cdot \cdot \cdot \cdot \cdot \cdot \cdot$$ $$s_n = a_1 + a_2 + \cdot \cdot \cdot + a_n,$$ $$\cdot \cdot \cdot \cdot \cdot \cdot \cdot$$
are called partial sums of series (1).
Definition 1
If there is a finite limes $$\lim_{n \to \infty} s_n = s$$ of the array $$(s_n)_{n \in \mathbb{N}}$$ of the partial sums of series (1), then we can say that that series converge and that its sum is equal $$s$$. In that case we write $$s = \sum_{n = 0}^{\infty} a_n$$. This notation is also used when it is $$\lim_{n \to \infty} s_n = \pm \infty$$. For the series that don't converge (either because $$\lim_{n \to \infty} s_n$$ is infinite or because it doesn't exist) we say that the series diverge."
My question:
Please, could you help me understand why in the paragraph before the definition, the sum goes from $$n = 1$$, but in the definition it goes from $$n = 0$$? Could you tell me what is the reason behind these diferent notations?
This is just the literal begining of the chapter, so i think that it is important to understand this.
• Just a typo I think. The sum in the definition should start from $n=1$ to be consistent with the rest of the section. – gandalf61 Nov 15 '18 at 12:37
It's a pure typo. Given that all other sums start with $$n=1$$, you can easily replace
In that case we write $$s=\sum_{n=0}^\infty a_n$$
with
In that case we write $$s=\sum_{n=1}^\infty a_n$$
The thing is that the first few elements of a series are usually not what interests us, in the sense that, so long as $$a_0$$ is defined, the sum $$\sum_{n=1}^\infty a_n$$
converges if and only if the sum
$$\sum_{n=0}^\infty a_n$$ converges, and the two sums only differ by $$a_0$$, so we can investigate either one, it doesn't really matter. That's why authors sometimes get sloppy. It's not an excuse, just a reason, but there you go.
• Interestingly, all other sums in this lesson in my book(in examples, theorems and propositions) begin with $n = 0$. Can it be that $n = 1$ was a typo and that i should always consider that given sum starts form $n = 0$, excpet when it's told differently? – MathsLearner Nov 15 '18 at 12:46
• @OgnjenMojovic A given sum always starts from the number that is given (unless it's a typo). In a mathematical text, the starting indices are almost always given. But yeah, if every other example starts the sum at $0$, then probably, the definition should do that too. – 5xum Nov 15 '18 at 12:48
• It's confusing me, because when they define series (for the first time in the book) they use sum from $n = 1$, and then in all other instances in this lesson, they use sum from $n = 0$. So, it creates a little trust issue (why defining it that way?). At the end, i guess this notation problem isn't going to be a big problem? – MathsLearner Nov 15 '18 at 12:52
• @OgnjenMojovic Like I said, practically any property that's of interest when analyzing series is shared among the two sums (one starting at $0$, one at $1$). So the authors got a little sloppy. You can clearly, from the definition of $\sum_{n=1}^\infty$, see what the sum $\sum_{n=0}^\infty$ should be defined as (i.e., the limit of a sequence of partial sums). – 5xum Nov 15 '18 at 12:56
• @OgnjenMojovic Exactly. And it's precisely because the two cases are so similar that they messed up. Because people are sloppy with the easy tasks. You are welcome, and remember, if the answer is useful to you and is what you needed, you can always accept it. – 5xum Nov 15 '18 at 13:03
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2019-08-19 21:10:01
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https://greprepclub.com/forum/a-distillate-flows-into-an-empty-64-gallon-drum-at-spout-a-a-10591.html
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It is currently 10 Dec 2018, 22:25
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A distillate flows into an empty 64-gallon drum at spout A a
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A distillate flows into an empty 64-gallon drum at spout A a [#permalink] 26 Aug 2018, 04:41
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Question Stats:
63% (01:46) correct 36% (01:49) wrong based on 11 sessions
A distillate flows into an empty 64-gallon drum at spout A and out of the drum at spout B. If the rate of flow through A is 2 gallons per hour, how many gallons per hour must flow out at spout B so that the drum is full in exactly 96 hours?
(A) $$\frac{3}{8}$$
(B) $$\frac{1}{2}$$
(C) $$\frac{2}{3}$$
(D) $$\frac{4}{3}$$
(E) $$\frac{8}{3}$$
[Reveal] Spoiler: OA
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Re: A distillate flows into an empty 64-gallon drum at spout A a [#permalink] 26 Aug 2018, 06:10
1
KUDOS
Carcass wrote:
A distillate flows into an empty 64-gallon drum at spout A and out of the drum at spout B. If the rate of flow through A is 2 gallons per hour, how many gallons per hour must flow out at spout B so that the drum is full in exactly 96 hours?
(A) $$\frac{3}{8}$$
(B) $$\frac{1}{2}$$
(C) $$\frac{2}{3}$$
(D) $$\frac{4}{3}$$
(E) $$\frac{8}{3}$$
Let x = rate at which water is flowing out from B
so Net rate =$$2 - x$$
SInce we know
Amount of work = Rate * time
or $$64 = (2-x) * 96$$
or $$64 = 192 - 96x$$
or$$x = \frac{128}{96} = \frac{4}{3}$$
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Re: A distillate flows into an empty 64-gallon drum at spout A a [#permalink] 26 Aug 2018, 06:10
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2018-12-11 06:25:10
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https://docs.quantumatk.com/tutorials/diffusion_liquid_copper/diffusion_liquid_copper.html
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# Diffusion in Liquids from Molecular Dynamics Simulations¶
|V2019.03|
In this tutorial, you will learn how to calculate the atomic diffusion coefficient in liquid metals using molecular dynamics (MD) simulations. In general, an accurate measurement of the diffusion coefficient in liquids is a challenging task. The present tutorial demonstrates how the QuantumATK tools allow you to efficiently study this fundamental phenomenon in a straightforward manner on a computer instead.
We adopt classical molecular dynamics to be able to generate rather long MD trajectories of liquid copper. This is a computationally efficient approach to obtain statistically meaningful averages of the physical quantities needed for calculating the diffusion coefficient of a single Cu atom in this liquid metal. The entire MD procedure is divided into three major steps:
1. heating crystalline copper to above its melting point,
2. annealing liquid copper at high temperature to bring the system to equilibrium,
3. further annealing of this equilibrated liquid copper to get more statistical data for accurate calculation of the atomic diffusion coefficient, using the time-dependent mean-square displacement derived from the corresponding MD trajectory.
## Theory¶
Diffusion in liquid metals differs from atomic diffusion in solids, as it takes place at a much shorter time scale, allowing for a brute force approach based on molecular dynamics simulations to calculate an atomic diffusion coefficient, denoted $$D$$. Knowing this fundamental physical quantity is needed to understand liquid dynamics, as well as many other related phenomena, such as nucleation and crystal growth. Unfortunately, experimental diffusion data for liquids are often rather inaccurate for various reasons [Mey10]. Instead, liquid diffusion can be rather accurately studied theoretically as demonstrated in this tutorial.
Tip
If you are interested in atomic diffusion in solids, please refer to the following two tutorials on Boron diffusion in bulk silicon and Modeling Vacancy Diffusion in Si0.5 Ge0.5 with AKMC as studying diffusion in solids usually requires a more sophisticated computational approach compared to the molecular dynamics approach.
Given that the MD simulation of the equilibrated liquid has been run for time $$T_{\rm MD}$$, the diffusion coefficient, $$D$$, can then be expressed in terms of the mean-square displacement of atoms in this MD run
$6 t D = \left\langle X^{2}(t) \right\rangle,$
where $$\left\langle X^{2}(t) \right\rangle$$ is the mean-square displacement of Cu atoms in liquid copper, calculated at the observation time $$t$$
$\begin{split}\left\langle X^{2}(t) \right\rangle &=& \lim_{T\to\infty} \frac{1}{T} \int_{0}^{T} dt^{\prime} \frac{1}{N_{\rm at}} \sum_{j=1}^{N_{\rm at}} \left[ {\bf r}_{j}( t^{\prime} + t ) - {\bf r}_{j}( t^{\prime} ) \right]^{2}\\ &\approx& \frac{1}{T_{\rm MD}-t} \int_{0}^{T_{\rm MD}-t} dt^{\prime} \frac{1}{N_{\rm at}} \sum_{j=1}^{N_{\rm at}} \left[ {\bf r}_{j}( t^{\prime} + t ) - {\bf r}_{j}( t^{\prime} ) \right]^{2},\end{split}$
and $$N_{\rm at}$$ is the total number of Cu atoms in the supercell, and $${\bf r}_{j}(t^{\prime}+t)$$ and $${\bf r}_{j}(t^{\prime})$$, where $$j \in [1,\ldots,N_{\rm at}]$$, are atomic position coordinates at time $$t^{\prime}+t$$ and $$t^{\prime}$$, respectively. In this tutorial, $$t$$ is the time interval of observation that is adopted to extract the atomic diffusion coefficient from the time-dependent mean-square displacement, $$\left\langle X^{2}(t) \right\rangle$$, assuming that $$\left\langle X^{2}(t) \right\rangle$$ shows a linear behavior with respect to $$t$$. This observation time $$t$$ cannot exceed the total time $$T_{\rm MD}$$ of the MD simulation of liquid. The mean-square displacement may become quite noisy, not showing a linear dependence with respect to the time interval $$t$$ for the values of $$t$$ close to $$T_{\rm MD}$$. In some cases, only those values of $$t$$ that are much smaller than the simulation time, i.e., when $$\, \, t\ll T_{\rm MD}$$, allow for accurate estimates of the diffusion coefficient from the linear fit of $$\left\langle X^{2}(t) \right\rangle$$.
Strictly speaking, the formula given in the previous paragraph defines the self-diffusion coefficient of liquid copper. That is the atomic diffusion coefficient averaged over all the copper atoms in the liquid. This ensemble averaging is possible in this case because all these copper atoms are assumed to be identical, having the same atomic mass and other physical characteristics that may affect diffusion of atoms. If a foreign (impurity) atom or a copper isotope with a different atomic mass was introduced in liquid copper, we would then have an additional diffusion process, which is diffusion of this foreign atom in liquid copper. To quantify this process, we can calculate the atomic diffusion coefficient, $$D_{\rm imp}$$, of this single atom as follows
$6 t D_{\rm imp} = \left\langle X^{2}(t) \right\rangle \approx \frac{1}{T_{\rm MD}-t} \int_{0}^{T_{\rm MD}-t} dt^{\prime} \left[ {\bf r}_{\rm imp}( t^{\prime} + t ) - {\bf r}_{\rm imp}( t^{\prime} ) \right]^{2},$
where the individual atom trajectory $$r_{\rm imp}(t)$$ is taken from the MD trajectory of the entire system of many atoms.
One should bear in mind that diffusion properties of materials depend on the temperature, and that also holds true for the atomic diffusion coefficient. In this tutorial, we will study liquid copper at T=2000 K, which is a temperature well above the melting point of crystalline copper.
## Computational Procedure¶
### Building a Copper Supercell Structure¶
• Open the Builder and add a face-centered crystal structure of Copper from Database to the Stash.
• In Bulk Tools, choose Supercell and click on Conventional.
• Then click on Transform to convert the primitive cell of bulk copper with 1 Cu atom to the conventional unit cell with 4 Cu atoms.
• In Bulk Tools, choose Repeat and set the repetition of the conventional unit cell along the A, B, and C axes to 6.
• Click on Apply to create a 6x6x6 cubic supercell of bulk copper, which is the initial structure for MD simulations.
• Send the copper supercell structure to the Script Generator with the Send To icon .
### Setting up MD Simulations in the Scripter¶
• Add a Calculators ‣ ForceFieldCalculator .
Note
For QuantumATK-versions older than 2017, the ATK-ForceField calculator can be found under the name ATK-Classical.
• In the Potential Settings, set the classical potential in Parameter set to EAM_Cu_2004 [ZJW04].
• Add an Optimization ‣ MolecularDynamics object to the Script to set up the first step of the MD procedure, which is heating crystalline copper above its melting point.
Set the computational parameters for MolecularDynamics to heat up the supercell structure of crystalline copper from 1000 to 2000 K, using an NPT barostat. This procedure allows us to increase the temperature of the system gradually above its melting point. It means that the ordered solid will eventually become a liquid, which only exhibits a short-range order as demonstrated using the radial distribution function (RDF) in the Analysis section.
• Set Molecular Dynamics Type to NPT Martyna Tobias Klein, Steps to 100000, Log interval to 1000, and Save trajectory to the trajectory_heating.hdf5 file.
• Set Initial Velocity Type to Maxwell-Boltzmann, and Temperature to 1000 K.
• In the NPT Martyna Tobias Klein barostat settings, set Reservoir temperature to 1000 K, and Final temperature to 2000 K.
• All the other MD parameters are kept at default.
• Add a second Optimization ‣ MolecularDynamics object to the script to set up the second step of the MD procedure, which is annealing liquid copper to bring the system to equilibrium.
We should set the computational parameters for MolecularDynamics to equilibrate the supercell structure of liquid copper at the constant temperature 2000 K, using an NPT barostat. This step is needed to eliminate any memory effect of the initial copper structure (solid copper) on the physical properties of liquid copper obtained from this particular initial structure.
• Set Molecular Dynamics Type to NPT Martyna Tobias Klein, Steps to 100000, Log interval to 1000, and Save trajectory to the trajectory_equilibrating.hdf5 file.
• Set Initial Velocity Type to Maxwell-Boltzmann, and Temperature to 2000 K.
• In the NPT Martyna Tobias Klein barostat settings, set Reservoir temperature and Final temperature to 2000 K.
• All the other MD parameters are kept at default.
• Add a third Optimization ‣ MolecularDynamics object to the script to set up the third step of the MD procedure to collect sufficient statistical data for calculating the self-diffusion coefficient of liquid copper.
Set the computational parameters for MolecularDynamics to run the MD simulation for the supercell structure of equilibrated liquid copper at the constant temperature 2000 K, using an NPT barostat. In this step, we aim at collecting a good statistical data (a set of MD frames) for calculating atomic diffusion coefficient of liquid copper.
• Set Molecular Dynamics Type to NPT Martyna Tobias Klein, Steps to 100000, Log interval to 1000, and Save trajectory to the trajectory_data_equilibrium.hdf5 file.
• Set Initial Velocity Type to ConfigurationVelocities, so that the velocities from the previous (second) MD run will be used as initial velocities for this (third) MD simulation.
• In the NPT Martyna Tobias Klein barostat settings, set Reservoir temperature and Final temperature to 2000 K.
• All the other MD parameters are kept at default.
• In Output settings, rename Results file to Cu_diffusion.hdf5, and save the Script to File ‣ Save as Cu_diffusion.py.
• Send the job to the Job Manager .
• Depending on your machine, it will take about half an hour to do the classical MD simulations for the supercell structure with 864 copper atoms adopted in this tutorial.
Download the scripts here: Cu_diffusion.py and analysis.py.
## Analysis¶
### Self-diffusion of Liquid Copper¶
Once the calculations are done, the requested HDF5 data files should appear in the QuantumATK Project Files list, and their contents should be visible on the LabFloor.
• Select the trajectory named trajectory_heating.hdf5 and open the Movie Tool from the right-hand plugins bar.
The left-hand plot shows the time-evolution of the MD quantities: kinetic energy, potential energy, total energy, and temperature, all as a function of MD simulation time. On the right you can see a movie of the MD trajectory.
• Use mouse right-click to tick the option Wrap bulk atoms, and then click on the icon to start the movie.
As shown in the animation of the MD trajectory, the average velocity of the copper atoms increases at elevated temperatures, and then the phase transition from the crystalline to liquid copper occurs about half-way through the heating MD run.
Tip
You can easily save such an animated GIF file yourself: Right-click the movie and select Export animated GIF.
You can also use the Movie Tool to inspect the trajectory for the equilibration MD run, trajectory_equilibrating.hdf5. Note that the temperature and energy are fairly constant throughout the equilibration of the system:
As mentioned in the Computational procedure section, an ordered solid transforms into a liquid phase with a short-range order preserved upon melting. To demonstrate that liquid copper exhibits some structural ordering, select the final MD trajectory, trajectory_data_equilibrium.hdf5, and open the MD Analyzer from the right-hand plugins bar. Select Radial Distribution in the drop-down menu, and click on Plot to generate a plot of the radial distribution function $$g(r)$$ for the liquid copper. The distribution function shows a pronounced peak at 2.5 Å, indicating that this is the dominant nearest-neighbor distance between atoms in the liquid copper. This value is comparable to the nearest-neighbor bond distance of 2.55 Å in crystalline copper at room temperature, suggesting that there still exists some short-range order in the liquid phase of copper.
As described in the Theory section, the diffusion coefficient is related to the mean-square displacement (MSD) of the Cu atoms as the final MD simulation progresses. You can plot the MSD using the MD Analyzer: Select Mean Square Displacement as the quantity to be plotted, and then click on Plot.
The MSD increases linearly with simulation time, and all you need to compute the diffusion coefficient is the slope of the line. Right-click on the line, select “Add fit”, and click “Polynomial fit”. Close the Plot editor.
The atomic diffusion coefficient in liquid copper calculated at 2000 K is 0.85 Å2/ps.
### Diffusion of a Single Atom in Liquid Copper¶
To demonstrate how the diffusion coefficient, $$D_{\rm imp}$$, of an individual atom in liquid can be obtained within the framework of the MD approach, we will look at the diffusion of a single copper atom in the liquid copper, pretending that this (randomly-chosen) atom differs from all the other copper atoms in the system. In this particular case, one would expect that the value of the atomic diffusion coefficient for this individual atom should be close to the self-diffusion coefficient value of liquid copper.
The MD trajectory that has already been obtained for liquid copper can be directly adopted for calculating atomic diffusion of a single copper atom. The MSD of a single atom also increases linearly with simulation time, at least on the short-time scale as seen in the figure given below, and all you need to compute the diffusion coefficient is the slope of the line.
Use the script analysis_single_atom.py for computing the diffusion coefficient. The content of the script is shown below. The script takes as an argument the name of the HDF5 file trajectory_data_equilibrium.hdf5 from which it extracts the MD trajectory of a single copper atom (atom #7 as chosen in the script) in the equilibrated liquid copper, then uses the built-in function MeanSquareDisplacement() to calculate the MSD of the copper atom #7, and uses NumPy to do the linear fitting. Finally, a plot is produced using pylab.
# Get the name of the MD trajectory (hdf5) file given in the system arguments
filename = sys.argv[1]
#filename = 'trajectory_data_equilibrium.hdf5'
# Get the frames of the MD trajectory from the file
# Get Mean Square Displacement (MSD) for a single Cu atom (#7) from the MD trajectory
msd = MeanSquareDisplacement(md_trajectory,atom_selection=[7])
# Get the times in ps and the MSD values in Ang**2.
t = msd.times().inUnitsOf(ps)
msd_data = msd.data().inUnitsOf(Angstrom**2)
# Fit the slope of the MSD to estimate the diffusion coefficient.
# If you discover non-linear behavior at small times, you should then discard this initial part in the fit.
for i in range(10,110,10):
first_image = 0
a = numpy.polyfit(t[first_image:i], msd_data[first_image:i], deg=1)
print(i, a[0]/6)
# Setting the endpoint to 25 is based on a visual inspection of the plot for MSD(t).
# Note that this kind of plots may have some noise for large time-values.
endpoint = 25
a = numpy.polyfit(t[first_image:endpoint], msd_data[first_image:endpoint], deg=1)
line_fit = numpy.poly1d(a)
# Calculate the diffusion coefficient of the copper atom #7 in Ang**2/ps.
diffusion_coefficient = a[0]/6.0
print('Diffusion coefficient for a single atom in liquid copper at T=2000K is: ',diffusion_coefficient, ' Ang**2/ps')
# Plot the data using pylab.
import pylab
pylab.plot(t, msd_data, linewidth=2,label='MSD of a copper atom')
pylab.plot(t[first_image:endpoint], line_fit(t[first_image:endpoint]), linewidth=4,color='k', linestyle='--', label='Linear fit of MSD of a copper atom')
pylab.xlabel('t (ps)')
pylab.ylabel('MSD(t) ($\AA^{2}$)')
pylab.ylim(0,600)
pylab.legend(loc='upper left')
pylab.show()
Download the script and execute it from the command line. The calculated diffusion coefficient of a single atom will be printed at the end of the logging output:
\$ atkpython analysis_single_atom.py trajectory_data_equilium.hdf5
...
...
...
Diffusion coefficient for a single atom in liquid copper at T=2000K: 1.10459718032 Ang**2/ps
Tip
This python script used for the analysis can also be executed from the Job Manager, i.e., without using the command line. For that, replace filename = sysargv[1] with filename = ‘trajectory_data_equilibrium.hdf5’ in the downloaded analysis_single_atom.py script, save it, and send it to the Job Manager to execute it as a regular QuantumATK job.
The diffusion coefficient of a single atom in liquid copper calculated at 2000 K is found to be 1.10 Å2/ps in this case. As expected, this value is comparable to that of the self-diffusion coefficient of liquid copper, 0.89 Å2/ps. The difference between these two calculated coefficients originates from the fact that we have done ensemble-averaging over all the copper atoms in liquid to calculate the self-diffusion coefficient, $$D$$, whereas there is only one data point at a given time, $$t$$, for calculation of the atomic diffusion coefficient, $$D_{\rm imp}$$. So, we may need to run MD simulations for much longer time ($$T_{\rm MD}$$) to more accurately calculate the self-diffusion coefficient of liquid copper from the MD trajectory of a single copper atom. This longer MD trajectory should allow us to get better statistics and increase the observation time interval ($$t$$) where the mean-square displacement exhibits a linear time dependence. The linear region in the following figure, which shows the time-dependent mean-square displacement, is then expected to increase from $$t=25$$ ps to a larger observation time value.
Note
When analyzing only a single atom for this amount of time, the statistics are not sufficient for obtaining the diffusion constant. If you want the diffusion constant from a single atom, the simulation needs to run much longer than if the averaging is done over many atoms.
### Temperature Dependence of Diffusion Coefficient¶
As an exercise, you may repeat all the steps of the MD procedure to obtain the MD trajectory of liquid copper equilibrated at T=1620 K, and then calculate the self-diffusion coefficient from this MD trajectory. You will find that this physical quantity calculated without any adjustable parameters, i.e., calculated from first-principles, agrees very well with the experimental value of 0.52 Å2/ps measured for liquid copper at T=1620 K. The smaller diffusion coefficient value at 1620 K compared to that at 2000 K suggests that the self-diffusion coefficient decreases upon lowering the temperature of liquid, as expected.
References
[Mey10] A. Meyer. Self-diffusion in liquid copper as seen by quasielastic neutron scattering. Phys. Rev. B, 81:012102, Jan 2010. doi:10.1103/PhysRevB.81.012102.
[ZJW04] X. W. Zhou, R. A. Johnson, and H. N. G. Wadley. Misfit-energy-increasing dislocations in vapor-deposited cofe/nife multilayers. Phys. Rev. B, 69:144113, Apr 2004. doi:10.1103/PhysRevB.69.144113.
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2019-10-16 00:12:51
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https://socratic.org/questions/is-1-9-a-rational-number
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# Is -1/9 a rational number?
rational number is a number that can be expressed as a ratio of two integer numbers. In this case, $- \frac{1}{9}$ is already expressed as a ratio of $- 1$ and $9$.
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2020-07-03 23:46:18
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|
https://codereview.stackexchange.com/questions/122241/recursive-directory-copy-program
|
# Recursive directory copy program
A little while ago, I had to write a little C# application to recover my HDD data (full context on this question)
To answer my problem I developed a console application which job was to recursively copy the entire folder tree of my HDD.
The code can be seen below :
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;
namespace CopyCat
{
class Program
{
private static long _fileCount = 0;
private static long _byteCount = 0;
private static long _byteProgress = 0;
private static ProgressBar _progressCount = new ProgressBar();
static void Main(string[] args)
{
Directory.Exists(args[0]);
Directory.Exists(args[1]);
FileDiscovery(args[0]);
FileCopy(args[0], args[1]);
}
static void FileCopy(String source, String dest)
{
try
{
foreach (var file in Directory.EnumerateFiles(source, "*", SearchOption.TopDirectoryOnly))
{
try
{
if (file == null) continue;
var dFile = File.Open(Path.Combine(dest, Path.GetFileName(file)), FileMode.Create,
oFile.CopyTo(dFile, 104857600);
oFile.Close();
dFile.Flush();
dFile.Close();
_byteProgress += new FileInfo(file).Length;
_progressCount.Report((double)_byteProgress / (double)_byteCount);
}
catch (Exception e)
{
Console.WriteLine("[COPY][ERROR] : Couldn't copy file : {0} => {1}", file, e.Message);
}
}
foreach (var directory in Directory.EnumerateDirectories(source, "*", SearchOption.TopDirectoryOnly))
{
if (directory == @"G:$RECYCLE.BIN") continue; var dir = Path.GetFileName(directory); if (!Directory.Exists(Path.Combine(dest, dir))) { Directory.CreateDirectory(Path.Combine(dest, dir)); } FileCopy(directory, Path.Combine(dest, dir)); } } catch (Exception exception) { Console.WriteLine("[COPY][WARNING] : Couldn't open directory : {0}", source); } } static void FileDiscovery(String dir) { try { foreach (var file in Directory.EnumerateFiles(dir, "*", SearchOption.TopDirectoryOnly)) { _fileCount++; _byteCount += new FileInfo(file).Length; } foreach (var directory in Directory.EnumerateDirectories(dir, "*", SearchOption.TopDirectoryOnly)) { FileDiscovery(directory); } } catch (Exception exception) { Console.WriteLine("[DISCOVERY][WARNING] : Couldn't open directory : {0}", dir); } } static String HumanReadableByteCount(long bytes, Boolean si = false, int precision = 2) { int unit = si ? 1000 : 1024; if (bytes < unit) return bytes + " B"; int exp = (int)(Math.Log(bytes) / Math.Log(unit)); String pre = (si ? "kMGTPE" : "KMGTPE")[(exp - 1)] + (si ? "" : "i"); return String.Format("{0} {1}{2}", Math.Round(bytes / Math.Pow(unit, exp), precision), pre, si ? "b" : "B"); } } } (Using a custom ProgressBar) I wanted to know specifically how I could improve the copying speed. ## 2 Answers This Directory.Exists(args[0]); Directory.Exists(args[1]); doesn't serve any purpose because you aren't evaluating the returned boolean value. In fact its more dangerous to use it with the args not checked for null at all. Your application will just crash if the application is called using only one or no argument at all. The static void FileDiscovery() method would benefit from a better name like CalculateDirectorySize() and by returning a long instead of void. Another point to mention is that the DirectoryInfo class contains the method GetFileSystemInfos() which would lead to the following change private static long CalculateDirectorySize(String dir) { long directorySize = 0; var dirInfo = new DirectoryInfo(dir); try { foreach (var fileInfo in dirInfo.GetFileSystemInfos("*", SearchOption.TopDirectoryOnly)) { directorySize += fileInfo.Length; } foreach (var directory in Directory.EnumerateDirectories(dir, "*", SearchOption.TopDirectoryOnly)) { directorySize += CalculateDirectorySize(directory); } } catch (Exception exception) { Console.WriteLine("[DISCOVERY][WARNING] : Couldn't open directory : {0}", dir); } return directorySize; } I have added private access modifier to the method, because it is a good habit to add one. Because _fileCount is no where used I have removed it. The FileCopy() method is using a strange way to copy the files by reading and writing them using FileStream's. • A more idiomatic way would be to use one of the overloaded File.Copy() methods. This methods are doing under the hood the same like your code but the streams are properly closed if any exception occurs. • Directory.CreateDirectory() can be called regardless if the directory exists or not. You can just skip the check for Exists(). • Having three times Path.Combine(dest, dir) won't be necessary if the result is stored in a variable. • Instead of calling GetFileName() for a directory you should call GetDirectoryName() this would result in the following change private static void FileCopy(String source, String dest) { try { foreach (var file in Directory.EnumerateFiles(source, "*", SearchOption.TopDirectoryOnly)) { try { File.Copy(file, Path.Combine(dest, Path.GetFileName(file))); _byteProgress += new FileInfo(file).Length; _progressCount.Report((double)_byteProgress / (double)_byteCount); } catch (Exception e) { Console.WriteLine("[COPY][ERROR] : Couldn't copy file : {0} => {1}", file, e.Message); } } foreach (var directory in Directory.EnumerateDirectories(source, "*", SearchOption.TopDirectoryOnly)) { if (directory == @"G:$RECYCLE.BIN") continue;
var dir = Path.GetDirectoryName(directory);
var destination = Path.Combine(dest, dir)
Directory.CreateDirectory(destination);
FileCopy(directory, destination);
}
}
catch (Exception exception)
{
Console.WriteLine("[COPY][WARNING] : Couldn't open directory : {0}", source);
}
}
Based on the comment
I was aware of the File.Copy() method at the time of writing the application BUT (and that's why I included the context) this method was throwing the same error as my explorer when used : that the file was not accessible and/or corrupted.
I would like to suggest to only use this kind of copy operation if the File.Copy() method throws an IOException having its own method like so
private static readonly int blockSize = 104857600;
private static bool SafeFileCopy(string source, string destination) {
try
{
using(var destinationStream = File.Open(destination, FileMode.Create,
{
sourceStream.CopyTo(destinationStream, blockSize);
return true
}
}
catch (Exception ex)
{
// do some logging
}
return false;
}
Resulting in the former foreach like so
foreach (var file in Directory.EnumerateFiles(source, "*", SearchOption.TopDirectoryOnly))
{
string destinationFile = Path.Combine(dest, Path.GetFileName(file));
try
{
File.Copy(file, destinationFile);
_byteProgress += new FileInfo(file).Length;
_progressCount.Report((double)_byteProgress / (double)_byteCount);
}
catch (IOException ioex)
{
if (!SafeFileCopy(file, destinationFile))
{
// do some logging here
}
}
catch (Exception e)
{
Console.WriteLine("[COPY][ERROR] : Couldn't copy file : {0} => {1}", file, e.Message);
}
}
Basically you should only catch exceptions which you can handle or which you only need to log/swallow. But you should always catch specific exceptions only. Catching Exception itself is a bad habit so for the SafeFileCopy() this would be for you to change.
• I was aware of the File.Copy() method at the time of writing the application BUT (and that's why I included the context) this method was throwing the same error as my explorer when used : that the file was not accessible and/or corrupted. Which was not the case when manually copying using streams. And since the performance was less than good, I came to post the question :) – Sidewinder94 Mar 8 '16 at 15:03
• Thanks for the rest of your remarks, I will be applying them. – Sidewinder94 Mar 8 '16 at 15:04
• Would you know of any way to speed up the copy between the streams ? Or is it a question better asked on SO ? – Sidewinder94 Mar 8 '16 at 16:01
• Basically you have used a high blocksize and I guess that the bottleneck is your harddisc. You can ask on SO or better search first on SO. – Heslacher Mar 8 '16 at 16:05
• – Heslacher Mar 8 '16 at 16:10
Use File.Copy. They use some lower level libraries that are much faster than copying streams. This improved performance for our application quite a bit.
Here's the code that File.Copy uses to actually copy the file (notice Win32Native.CopyFile):
/// <devdoc>
/// Note: This returns the fully qualified name of the destination file.
/// </devdoc>
[System.Security.SecuritySafeCritical] // auto-generated
[ResourceExposure(ResourceScope.Machine)]
[ResourceConsumption(ResourceScope.Machine)]
internal static String InternalCopy(String sourceFileName, String destFileName, bool overwrite)
{
Contract.Requires(sourceFileName != null);
Contract.Requires(destFileName != null);
Contract.Requires(sourceFileName.Length > 0);
Contract.Requires(destFileName.Length > 0);
String fullSourceFileName = Path.GetFullPathInternal(sourceFileName);
new FileIOPermission(FileIOPermissionAccess.Read, new String[] { fullSourceFileName }, false, false).Demand();
String fullDestFileName = Path.GetFullPathInternal(destFileName);
new FileIOPermission(FileIOPermissionAccess.Write, new String[] { fullDestFileName }, false, false).Demand();
bool r = Win32Native.CopyFile(fullSourceFileName, fullDestFileName, !overwrite);
if (!r)
{
// Save Win32 error because subsequent checks will overwrite this HRESULT.
int errorCode = Marshal.GetLastWin32Error();
String fileName = destFileName;
if (errorCode != Win32Native.ERROR_FILE_EXISTS)
{
// For a number of error codes (sharing violation, path
// not found, etc) we don't know if the problem was with
// the source or dest file. Try reading the source file.
{
if (handle.IsInvalid)
fileName = sourceFileName;
}
if (errorCode == Win32Native.ERROR_ACCESS_DENIED)
{
if (Directory.InternalExists(fullDestFileName))
throw new IOException(Environment.GetResourceString("Arg_FileIsDirectory_Name", destFileName), Win32Native.ERROR_ACCESS_DENIED, fullDestFileName);
}
}
__Error.WinIOError(errorCode, fileName);
}
return fullDestFileName;
}
• If they're using File.Copy, then why do you need to show the code for it? It would be customary to show how the code is different if you use File.Copy, not how File.Copy itself works. – mdfst13 Mar 10 '16 at 23:04
|
2019-11-19 01:33:21
|
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|
http://math.stackexchange.com/tags/banach-algebras/new
|
# Tag Info
0
Convolution really comes out of gathering like powers of things such as powers of exponentials or of powers of a complex variable. \begin{align} \sum_{n=-\infty}^{\infty}a_n e^{in\theta}\sum_{n=-\infty}^{\infty}b_ne^{in\theta}&=\sum_{n=-\infty}^{\infty}\left(\sum_{j+k=n}a_j b_k\right)e^{in\theta} \\ & = ...
2
Some important tricks/theorems: The spectrum is non-empty for Banach algebras (over $\mathbb{C}$) The spectral radius formula $$r(a) = \lim \|a^n\|^{1/n}$$ tells you that if $a$ is nilpotent, then $\sigma(a) = \{0\}$ If $A$ is commutative, then $$\sigma(a) = \{\tau(a) : \tau \in \Omega(A)\}$$ where $\Omega(A)$ denotes the set of non-zero multiplicative ...
1
No, this is false. Let $f\in H^\infty(\mathbb{D})$ be a function that is continuous on $\partial D\setminus \{1\}$ and does not have a limit as at $z\to 1$. (E.g., $f$ could be a conformal map onto a domain one with one nontrivial prime end.) If $f=g+\phi(z^2)$ with $g\in A(\mathbb{D})$, then $\phi$ must be discontinuous at $1$; but then $\phi(z^2)$ is also ...
3
That's a good proof. Here is another one. Assume that $a^m=0$. If you know that $$\sigma(a)=\{f(a):\ f \text{ is a multiplicative functional }\},$$ then for any such $f$ we have $f(a)^m=f(a^m)=f(0)=0$. So $f(a)=0$, and then $\sigma(a)=\{0\}$. Yet another proof, without machinery: if $a^m=0$, then $1-a$ is invertible: indeed, the inverse is ...
1
Yes, that is one way to look at it. Here's another: even before the Gelfand transform, we already know from Lemma 1.2.4 that $\sigma(a)$ is compact. Here it is only proven for unital Banach algebras, but that is simply because Murphy only defines the spectrum of an element in a non-unital algebra on page 13, at the very end of section 1.2. (Think about it: ...
1
No, not in general. For instance, suppose $A=\mathbb{C}$ with the standard norm. Then (via a change of basis $(a,z)\mapsto (a+z,z)$) we can identify $A_+$ with $\mathbb{C}^2$ with coordinatewise multiplication, and your norm with $\|(a,b)\|=(|a-b|^p+|b|^p)^{1/p}$. Now consider the elements $x=(1,0)$ and $y=(1,1/2)$. We have $\|x\|=1$ and ...
2
Well, nothing special: write $C \in M_{mn}(\mathbb C)$. Let $C_{ab} \in M_{mn} (\mathbb C)$, where $a,b \in \{1, 2, \cdots, n\}$ so that $$(C_{ab})_{ij} = \begin{cases} C_{ij} & \text{if } (a-1)m +1 \le i\le am, (b-1)m+1\le j\le bm,\\ 0 & \text{otherwise.}\end{cases}$$ Abusing notations, we also consider $C_{ab} \in M_m(\mathbb C)$. Then C = ...
3
This isn't true. For instance, if $X=\omega_1$ is the first uncountable ordinal, then $C_c(X)=C_0(X)$ is complete (since every continuous map $X\to \mathbb{R}$ is eventually constant), but $X$ is not compact.
0
Let $f : V \to W$ be a function between two Banach spaces. Then by definitnion, the Frechet derivative at $x$ is the only bounded (=continuous) linear operator such that... Hence $\nabla f(x) \in L(V,W)$. So $\nabla f(x) \in V'$ if and only if $W = \Bbb R$ (or $\Bbb C$). For your question 2), if $f:V\to W$, then $\nabla f(x) : V\to W$, so it doesn't ...
1
Note that $a-\lambda$ commutes with $be^{i\lambda}$. In any ring, if $x$ and $y$ commute and the product is invertible, then each is invertible. Proof: suppose that $zxy=xyz=I$. Then $zyx=zxy-I$, so $x$ has a left inverse. From $xyz=I$ we know that $x$ has a right inverse. Then $x$ is invertible.
Top 50 recent answers are included
|
2015-11-27 08:17:05
|
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|
https://www.ideals.illinois.edu/handle/2142/14828/browse?order=ASC&rpp=20&sort_by=1&etal=-1&offset=18&type=title
|
Browse Dissertations - Biophysics and Computational Biology by Title
• (1972)
application/pdf
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• (1983)
The three themes of the thesis are: (1) the search for light-activated enzymes in octopus photoreceptors analogous to those found in vertebrates' photoreceptors, (2) the question of which photointermediate(s) of photoexcited ...
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• (2014-05-30)
Thanks to the improvement of genome sequencing technology, abundant multi-species genomic data now became available and comparative genomics continues to be a fast prospering filed of biological research. Through the ...
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The last decade has seen an explosion of data arising from the development and proliferation of high-throughput data gathering and analysis pipelines. In order to transform this data into useful hypotheses and conclusions, ...
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• (2012-09-18)
Purple photosynthetic bacteria achieve remarkably high light harvesting efficiency, thus reconciling multiple competing processes in the chromatophore. The first step in photosynthesis is the capture and transport of light ...
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• (2010-05-20)
Evolutionary transitions, times at which the behavior of evolution as a dynamic system dramatically changes, have occurred many times throughout the history of life on Earth. Carl Woese proposed that one such transition ...
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• (1975)
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• (2011-01-14)
Understanding the origins of life on Earth is one of the most intriguing problems facing science today. In the research presented here, we apply computational methods to explore origins of life scenarios. In particular, ...
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• (2009)
Mechanical proteins perform a vital role in cells by transmitting and bearing mechanical forces involved in cell growth, locomotion, and adhesion. Little is known at the molecular level that explains how such proteins are ...
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• (1996)
The pH dependent properties of proteins can be successfully modeled using continuum electrostatic methods provided that a sufficiently detailed and accurate model of the protein system is used. In this work, the effects ...
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• (1994)
Magnetic Resonance Imaging (MRI) holds enormous potential for exploring the functional centers of a living brain in vivo. Among the MRI techniques a number of different approaches have provided initial success. The focus ...
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• (1996)
Two peaks consistently arise near 1.3 ppm in $\sp1$H NMR spectroscopic studies of fatiguing skeletal muscles. They are typically 15-20 Hz apart at 300 MHz. Known resonances in this region include lactate, alanine, threonine, ...
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• (2013-05-24)
Noncommunicable diseases (NCD) are currently the leading cause of death worldwide. Over 57 million deaths occur globally each year, with close to 36 million of them attributed to NCD’s, and 80% of those in low and middle ...
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• (1988)
The site of ultrasonic absorption in suspensions of negatively-charged large unilamellar vesicles (LUVs) was investigated. The ultrasonic absorption per wavelength, $\alpha\lambda$, was determined for suspensions of LUVs ...
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• (2008)
The dissertation is concerned with biophysical applications of Forster resonance energy transfer (FRET) and development and application of fluorescence lifetime-resolved imaging microscopy (FLIM). It includes work to extend ...
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• (1989)
The pH of the tumor microenvironment may be important in assessment of response to hyperthermic therapy. Little clinical in vivo data is available during such therapy due to the inherent limitations of the microelectrode ...
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• (1993)
In this thesis, the electron paramagnetic resonance (EPR) methods for simultaneous measurement of intra- and extracellular concentration of oxygen ( (O$\sb2$)) in viable cells have been developed.
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• (1992)
Bicarbonate-reversible inhibition of the electron acceptor side of photosystem II (PS II), but not of photosynthetic bacteria, has been known for a long time. In formate treated thylakoids, the (Q$\sb{\rm A}\sp-$) decays ...
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• (1991)
Motions in proteins are often an important part of their function. Structural information about the protein is hence incomplete without an understanding of the protein dynamics. X-ray crystallography is the technique used ...
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• (1994)
Improvements in the information content of nuclear magnetic resonance (NMR) images have recently been sought through the development of new contrast mechanisms for magnetic resonance imaging (MRI) and NMR microscopy. This ...
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2015-07-04 01:44:35
|
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|
https://codereview.stackexchange.com/questions/239126/return-values-from-array-a-in-the-order-they-are-in-array-b
|
# Return values from array A in the order they are in array B
So my question is the cleanest way to do #product_panels_for below:
class ProductPanel < Struct.new(:product)
end
class ProductPresenter
def product_panels_for(selected_products)
selected_products.map do |product|
product_panels.find { |panel| panel.product.to_s == product }
end.
compact
end
def product_panels
end
end
This is obviously simplified, but the point is #product_panels_for should return values from product_panels but in the order given in selected_panels. Maybe this is fine is just feels a bit gangly for something that seems conceptually simpler than that.
To be clear the order is defined by the business, it won't be alphabetical or anything nice.
In case anyone's thinking it, yes, I know in Ruby 2.7 I could replace .map...compact with .filter_map, but I'm not on 2.7 yet.
My previous implementation was:
def product_panels_for(selected_products)
product_panels.select { |pp| selected_products.include?(pp) }
end
which was nice and clean but didn't preserve the order of selected_products.
If the implementation at the top can't be improved on, is there a nice way to use this previous implementation but chain something after the select that sorts the result so the order matches selected_products? I'm just generally curious about a good way to cleanly sort A so its order matches B where B is a list of values for an attributes of A.
• Did you notice that if you have duplicate products you only get one product panel? If you are assuming a 1-to- relationship then that's probably fine... – Garrett Motzner Apr 3 at 1:29
Also, I don't think it is worth adding one more step to order it just to use select`.
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2020-05-31 20:35:12
|
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|
https://matheducators.stackexchange.com/tags/undergraduate-education/new
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# Tag Info
My friend, who is currently teaching an abstract algebra course, recently provided me with an example of an incorrect proof that I think showcases a few types of faulty reasoning that are common among students. I'm sharing it here with his permission. Proposition: Suppose $R$ is a ring and $I$ is a prime ideal in $R$. For all ideals $J$ and $K$ in $R$, if $... 1 This is simple, but effective. Consider doing a weekly, period-long test, every Friday. I remember having that in HS for pre-calc and calc. At first was surprised by the frequency. But ended up admiring the pedagogic effect. Tests are some of the most effective drill (because of the drive for prep and the higher stakes...and the learning from the ... 2 The concepts behind limits are actually very important to engineering (in the form of error/precision analysis), but are rarely phrased that way. Given a function$f$, we can imagine an engineering situation where there is some desired range of outputs from the function, but the engineer has control over the value of the inputs of the function. If$f(c) = L$,... 4 It is probably not the place of mathematics educators to decide what mathematics courses engineering majors should take. But a good reference point is ABET accreditation. Over 600 universities in the US have ABET accredited engineering programs. We should defer to the professionals who set these standard and assess outcomes. Here is a description of their ... 1 No, it's definitely not "necessary". I'm not an engineering major, but roomed with one, did a general engineering minor, and worked in/around mechanical, nuclear, mining and chemical engineering (had electrical on staff too). Passed my EIT and was at one time, about to take the PE (mechanical) exam. Most engineers in the workplace don't even use ... 4 Well it doesn't really feel right to get degrees in engineering and gain years of engineering experience without even knowing what a limit actually is. And even though many engineers will do just fine without having been exposed to the rigorous definition of a limit, some engineers will need to be familiar with rigorous definitions/proofs if they ever pursue ... 2 Expanding a bit on my comment, there is (1) a new textbook available for project-based intro stats, (2) an online syllabus describing a course based on community projects, and (3) an academic paper concluding that "the project-based course ... provides a promising model for getting students hooked on the power and excitement of applied statistics." ... 0 There's a recent book by Nicholas A. Scoville: Discrete Morse Theory, AMS Student Mathematical Library, 2019. Publisher's page: https://bookstore.ams.org/stml-90 MAA review: https://www.maa.org/press/maa-reviews/discrete-morse-theory It looks very accessible to the undergraduate - some background in proof writing is recommended as is some linear algebra, but ... 1 That's pretty normal to have a lot of ground to cover. Even with the more friendly books, it still ends up being a lot of concepts and formulas. Given this, I think you sort of have to make your peace with the idea that kids will not master everything, especially in the long term. I probably wouldn't try some fundamental change to improve things since it ... 1 That book gets ripped pretty hard on Amazon. The Dover texts by Trudeau and Chartrand are supposedly easier and friendlier, per reviews. And will be cheap, since Dover. If you want to develop familiarity and speed, I would certainly not eschew (i.e. I would do) problems that are repetitive. You'll get more practiced at the concept. Also more practiced at ... 3 I'd say yes, and I'd go with binary if you had to do any one alternative base simply because it's so relevant to computing and technology, and in my experience teaching discrete math, once you understand binary, related bases like octal and hex are pretty simple to pick up. But I don't think the converse is necessarily true. Ideally I'd like math and CS ... 4 I can't answer the OP's questions, but I'll just mention that a local 6th-grade teacher (in the U.S.) has a successful unit on base-$5\$. It is mentioned in the recent article below. Sometimes he called it "star-fish math." James Henle. "Math for Grades 1 to 5 Should Be Art." Mathematical Intelligencer. 42, pages 64–69, Dec. 2020. ...
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2021-05-13 08:58:49
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https://cantera.org/documentation/docs-2.3/doxygen/html/InterfaceKinetics_8h_source.html
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Cantera 2.3.0
InterfaceKinetics.h
Go to the documentation of this file.
1 /**
2 * @file InterfaceKinetics.h
3 * @ingroup chemkinetics
4 */
5
6 // This file is part of Cantera. See License.txt in the top-level directory or
8
9 #ifndef CT_IFACEKINETICS_H
10 #define CT_IFACEKINETICS_H
11
13 #include "Kinetics.h"
14 #include "Reaction.h"
15 #include "cantera/base/utilities.h"
16 #include "RateCoeffMgr.h"
17
18 namespace Cantera
19 {
20
21 // forward declarations
22 class SurfPhase;
23 class ImplicitSurfChem;
24
25 //! A kinetics manager for heterogeneous reaction mechanisms. The reactions are
26 //! assumed to occur at a 2D interface between two 3D phases.
27 /*!
28 * There are some important additions to the behavior of the kinetics class due
29 * to the presence of multiple phases and a heterogeneous interface. If a
30 * reactant phase doesn't exists, i.e., has a mole number of zero, a
31 * heterogeneous reaction can not proceed from reactants to products. Note it
32 * could perhaps proceed from products to reactants if all of the product phases
33 * exist.
34 *
35 * In order to make the determination of whether a phase exists or not actually
36 * involves the specification of additional information to the kinetics object.,
38 * phases (i.e., temperature pressure, and mole fraction).
39 *
40 * The extrinsic specification of whether a phase exists or not must be
41 * specified on top of the intrinsic calculation of the reaction rate. This
42 * class carries a set of booleans indicating whether a phase in the
43 * heterogeneous mechanism exists or not.
44 *
45 * Additionally, the class carries a set of booleans around indicating whether a
46 * product phase is stable or not. If a phase is not thermodynamically stable,
47 * it may be the case that a particular reaction in a heterogeneous mechanism
48 * will create a product species in the unstable phase. However, other reactions
49 * in the mechanism will destruct that species. This may cause oscillations in
50 * the formation of the unstable phase from time step to time step within a ODE
51 * solver, in practice. In order to avoid this situation, a set of booleans is
52 * tracked which sets the stability of a phase. If a phase is deemed to be
53 * unstable, then species in that phase will not be allowed to be birthed by the
54 * kinetics operator. Nonexistent phases are deemed to be unstable by default,
55 * but this can be changed.
56 *
57 * @ingroup chemkinetics
58 */
60 {
61 public:
62 //! Constructor
63 /*!
64 * @param thermo The optional parameter may be used to initialize the object
65 * with one ThermoPhase object.
66 * HKM Note -> Since the interface kinetics object will
67 * probably require multiple ThermoPhase objects, this is
68 * probably not a good idea to have this parameter.
69 */
71
72 virtual ~InterfaceKinetics();
74 InterfaceKinetics& operator=(const InterfaceKinetics& right);
75 virtual Kinetics* duplMyselfAsKinetics(const std::vector<thermo_t*> & tpVector) const;
76
77 virtual int type() const;
78
79 virtual std::string kineticsType() const {
80 return "Surf";
81 }
82
83 //! Set the electric potential in the nth phase
84 /*!
85 * @param n phase Index in this kinetics object.
86 * @param V Electric potential (volts)
87 */
88 void setElectricPotential(int n, doublereal V);
89
90 //! @name Reaction Rates Of Progress
91 //! @{
92
93 //! Equilibrium constant for all reactions including the voltage term
94 /*!
95 * Kc = exp(deltaG/RT)
96 *
97 * where deltaG is the electrochemical potential difference between
98 * products minus reactants.
99 */
100 virtual void getEquilibriumConstants(doublereal* kc);
101
102 //! values needed to convert from exchange current density to surface
103 //! reaction rate.
105
106 virtual void getDeltaGibbs(doublereal* deltaG);
107
108 virtual void getDeltaElectrochemPotentials(doublereal* deltaM);
109 virtual void getDeltaEnthalpy(doublereal* deltaH);
110 virtual void getDeltaEntropy(doublereal* deltaS);
111
112 virtual void getDeltaSSGibbs(doublereal* deltaG);
113 virtual void getDeltaSSEnthalpy(doublereal* deltaH);
114 virtual void getDeltaSSEntropy(doublereal* deltaS);
115
116 //! @}
117 //! @name Reaction Mechanism Informational Query Routines
118 //! @{
119
120 virtual void getActivityConcentrations(doublereal* const conc);
121
122 //! Return the charge transfer rxn Beta parameter for the ith reaction
123 /*!
124 * Returns the beta parameter for a charge transfer reaction. This
125 * parameter is not important for non-charge transfer reactions.
126 * Note, the parameter defaults to zero. However, a value of 0.5
127 * should be supplied for every charge transfer reaction if
128 * no information is known, as a value of 0.5 pertains to a
129 * symmetric transition state. The value can vary between 0 to 1.
130 *
131 * @param irxn Reaction number in the kinetics mechanism
132 *
133 * @return Beta parameter. This defaults to zero, even for charge
134 * transfer reactions.
135 */
136 doublereal electrochem_beta(size_t irxn) const;
137
138 virtual bool isReversible(size_t i) {
139 if (std::find(m_revindex.begin(), m_revindex.end(), i)
140 < m_revindex.end()) {
141 return true;
142 } else {
143 return false;
144 }
145 }
146
147 virtual void getFwdRateConstants(doublereal* kfwd);
148 virtual void getRevRateConstants(doublereal* krev,
149 bool doIrreversible = false);
150
151 //! Return effective preexponent for the specified reaction
152 /*!
153 * Returns effective preexponent, accounting for surface coverage
154 * dependencies.
155 *
156 * @param irxn Reaction number in the kinetics mechanism
157 * @return Effective preexponent
158 */
159 double effectivePreExponentialFactor(size_t irxn) {
160 return m_rates.effectivePreExponentialFactor(irxn);
161 }
162
163 //! Return effective activation energy for the specified reaction
164 /*!
165 * Returns effective activation energy, accounting for surface coverage
166 * dependencies.
167 *
168 * @param irxn Reaction number in the kinetics mechanism
169 * @return Effective activation energy divided by the gas constant
170 */
171 double effectiveActivationEnergy_R(size_t irxn) {
172 return m_rates.effectiveActivationEnergy_R(irxn);
173 }
174
175 //! Return effective temperature exponent for the specified reaction
176 /*!
177 * Returns effective temperature exponenty, accounting for surface coverage
178 * dependencies. Current parameterization in SurfaceArrhenius does not
179 * change this parameter with the change in surface coverages.
180 *
181 * @param irxn Reaction number in the kinetics mechanism
182 * @return Effective temperature exponent
183 */
184 double effectiveTemperatureExponent(size_t irxn) {
185 return m_rates.effectiveTemperatureExponent(irxn);
186 }
187
188 //! @}
189 //! @name Reaction Mechanism Construction
190 //! @{
191
192 //! Add a phase to the kinetics manager object.
193 /*!
194 * This must be done before the function init() is called or
195 * before any reactions are input.
196 *
197 * This function calls Kinetics::addPhase(). It also sets the following
198 * fields:
199 *
200 * m_phaseExists[]
201 *
202 * @param thermo Reference to the ThermoPhase to be added.
203 */
205
206 virtual void init();
207 virtual void resizeSpecies();
209 virtual void modifyReaction(size_t i, shared_ptr<Reaction> rNew);
210 //! @}
211
212 //! Internal routine that updates the Rates of Progress of the reactions
213 /*!
214 * This is actually the guts of the functionality of the object
215 */
216 virtual void updateROP();
217
218 //! Update properties that depend on temperature
219 /*!
220 * Current objects that this function updates:
221 * m_kdata->m_logtemp
222 * m_kdata->m_rfn
223 * m_rates.
224 * updateKc();
225 */
226 void _update_rates_T();
227
228 //! Update properties that depend on the electric potential
229 void _update_rates_phi();
230
231 //! Update properties that depend on the species mole fractions and/or
232 //! concentration,
233 /*!
234 * This method fills out the array of generalized concentrations by calling
235 * method getActivityConcentrations for each phase, which classes
236 * representing phases should overload to return the appropriate quantities.
237 */
238 void _update_rates_C();
239
240 //! Advance the surface coverages in time
241 /*!
242 * This method carries out a time-accurate advancement of the
243 * surface coverages for a specified amount of time.
244 *
245 * \f[
246 * \dot {\theta}_k = \dot s_k (\sigma_k / s_0)
247 * \f]
248 *
249 * @param tstep Time value to advance the surface coverages
250 */
252
253 //! Solve for the pseudo steady-state of the surface problem
254 /*!
255 * This is the same thing as the advanceCoverages() function,
256 * but at infinite times.
257 *
258 * Note, a direct solve is carried out under the hood here,
259 * to reduce the computational time.
260 *
261 * @param ifuncOverride One of the values defined in @ref solvesp_methods.
262 * The default is -1, which means that the program will decide.
263 * @param timeScaleOverride When a pseudo transient is selected this value
264 * can be used to override the default time scale for
265 * integration which is one. When SFLUX_TRANSIENT is used, this
266 * is equal to the time over which the equations are integrated.
267 * When SFLUX_INITIALIZE is used, this is equal to the time used
268 * in the initial transient algorithm, before the equation
269 * system is solved directly.
270 */
271 void solvePseudoSteadyStateProblem(int ifuncOverride = -1,
272 doublereal timeScaleOverride = 1.0);
273
274 void setIOFlag(int ioFlag);
275
276 //! @deprecated To be removed after Cantera 2.3.
277 void checkPartialEquil();
278
279 //! Update the standard state chemical potentials and species equilibrium
280 //! constant entries
281 /*!
282 * Virtual because it is overridden when dealing with experimental open
283 * circuit voltage overrides
284 */
285 virtual void updateMu0();
286
287 //! Update the equilibrium constants and stored electrochemical potentials
288 //! in molar units for all reversible reactions and for all species.
289 /*!
290 * Irreversible reactions have their equilibrium constant set
291 * to zero. For reactions involving charged species the equilibrium
292 * constant is adjusted according to the electrostatic potential.
293 */
294 void updateKc();
295
296 //! Apply modifications for the forward reaction rate for interfacial charge
297 //! transfer reactions
298 /*!
299 * For reactions that transfer charge across a potential difference,
300 * the activation energies are modified by the potential difference.
301 * (see, for example, ...). This method applies this correction.
302 *
303 * @param kfwd Vector of forward reaction rate constants on which to have
304 * the voltage correction applied
305 */
306 void applyVoltageKfwdCorrection(doublereal* const kfwd);
307
308 //! When an electrode reaction rate is optionally specified in terms of its
309 //! exchange current density, adjust kfwd to the standard reaction rate
310 //! constant form and units. When the BV reaction types are used, keep the
311 //! exchange current density form.
312 /*!
313 * For a reaction rate constant that was given in units of Amps/m2
314 * (exchange current density formulation with iECDFormulation == true),
315 * convert the rate to kmoles/m2/s.
316 *
317 * For a reaction rate constant that was given in units of kmol/m2/sec when
318 * the reaction type is a Butler-Volmer form, convert it to exchange
319 * current density form (amps/m2).
320 *
321 * @param kfwd Vector of forward reaction rate constants, given in either
322 * normal form or in exchange current density form.
323 */
324 void convertExchangeCurrentDensityFormulation(doublereal* const kfwd);
325
326 //! Set the existence of a phase in the reaction object
327 /*!
328 * Tell the kinetics object whether a phase in the object exists. This is
329 * actually an extrinsic specification that must be carried out on top of
330 * the intrinsic calculation of the reaction rate. The routine will also
331 * flip the IsStable boolean within the kinetics object as well.
332 *
333 * @param iphase Index of the phase. This is the order within the
334 * internal thermo vector object
335 * @param exists Boolean indicating whether the phase exists or not
336 */
337 void setPhaseExistence(const size_t iphase, const int exists);
338
339 //! Set the stability of a phase in the reaction object
340 /*!
341 * Tell the kinetics object whether a phase in the object is stable.
342 * Species in an unstable phase will not be allowed to have a positive
343 * rate of formation from this kinetics object. This is actually an
344 * extrinsic specification that must be carried out on top of the
345 * intrinsic calculation of the reaction rate.
346 *
347 * While conceptually not needed since kinetics is consistent with thermo
348 * when taken as a whole, in practice it has found to be very useful to
349 * turn off the creation of phases which shouldn't be forming. Typically
350 * this can reduce the oscillations in phase formation and destruction
351 * which are observed.
352 *
353 * @param iphase Index of the phase. This is the order within the
354 * internal thermo vector object
355 * @param isStable Flag indicating whether the phase is stable or not
356 */
357 void setPhaseStability(const size_t iphase, const int isStable);
358
359 //! Gets the phase existence int for the ith phase
360 /*!
361 * @param iphase Phase Id
362 * @return The int specifying whether the kinetics object thinks the phase
363 * exists or not. If it exists, then species in that phase can be
364 * a reactant in reactions.
365 */
366 int phaseExistence(const size_t iphase) const;
367
368 //! Gets the phase stability int for the ith phase
369 /*!
370 * @param iphase Phase Id
371 * @return The int specifying whether the kinetics object thinks the phase
372 * is stable with nonzero mole numbers. If it stable, then the
373 * kinetics object will allow for rates of production of of
374 * species in that phase that are positive.
375 */
376 int phaseStability(const size_t iphase) const;
377
378 virtual void determineFwdOrdersBV(ElectrochemicalReaction& r, vector_fp& fwdFullorders);
379
380 protected:
381 //! Build a SurfaceArrhenius object from a Reaction, taking into account
382 //! the possible sticking coefficient form and coverage dependencies
383 //! @param i Reaction number
384 //! @param r Reaction object containing rate coefficient parameters
385 //! @param replace True if replacing an existing reaction
387 bool replace);
388
389 //! Temporary work vector of length m_kk
391
392 //! List of reactions numbers which are reversible reactions
393 /*!
394 * This is a vector of reaction numbers. Each reaction in the list is
395 * reversible. Length = number of reversible reactions
396 */
397 std::vector<size_t> m_revindex;
398
399 //! Templated class containing the vector of reactions for this interface
400 /*!
401 * The templated class is described in RateCoeffMgr.h
402 * The class SurfaceArrhenius is described in RxnRates.h
403 */
405
406 bool m_redo_rates;
407
408 //! Vector of irreversible reaction numbers
409 /*!
410 * vector containing the reaction numbers of irreversible reactions.
411 */
412 std::vector<size_t> m_irrev;
413
414 //! Array of concentrations for each species in the kinetics mechanism
415 /*!
416 * An array of generalized concentrations \f$C_k \f$ that are defined
417 * such that \f$a_k = C_k / C^0_k, \f$ where \f$C^0_k \f$ is a standard
418 * concentration/ These generalized concentrations are used by this
419 * kinetics manager class to compute the forward and reverse rates of
420 * elementary reactions. The "units" for the concentrations of each phase
421 * depend upon the implementation of kinetics within that phase. The order
422 * of the species within the vector is based on the order of listed
423 * ThermoPhase objects in the class, and the order of the species within
424 * each ThermoPhase class.
425 */
427
428 //! Array of activity concentrations for each species in the kinetics object
429 /*!
430 * An array of activity concentrations \f$Ca_k \f$ that are defined
431 * such that \f$a_k = Ca_k / C^0_k, \f$ where \f$C^0_k \f$ is a standard
432 * concentration. These activity concentrations are used by this
433 * kinetics manager class to compute the forward and reverse rates of
434 * elementary reactions. The "units" for the concentrations of each phase
435 * depend upon the implementation of kinetics within that phase. The order
436 * of the species within the vector is based on the order of listed
437 * ThermoPhase objects in the class, and the order of the species within
438 * each ThermoPhase class.
439 */
441
442 //! Vector of standard state chemical potentials for all species
443 /*!
444 * This vector contains a temporary vector of standard state chemical
445 * potentials for all of the species in the kinetics object
446 *
447 * Length = m_kk. Units = J/kmol.
448 */
450
451 //! Vector of chemical potentials for all species
452 /*!
453 * This vector contains a vector of chemical potentials for all of the
454 * species in the kinetics object
455 *
456 * Length = m_kk. Units = J/kmol.
457 */
459
460 //! Vector of standard state electrochemical potentials modified by a
461 //! standard concentration term.
462 /*!
463 * This vector contains a temporary vector of standard state electrochemical
464 * potentials + RTln(Cs) for all of the species in the kinetics object
465 *
466 * In order to get the units correct for the concentration equilibrium
467 * constant, each species needs to have an RT ln(Cs) added to its
468 * contribution to the equilibrium constant Cs is the standard concentration
469 * for the species. Frequently, for solid species, Cs is equal to 1.
470 * However, for gases Cs is P/RT. Length = m_kk. Units = J/kmol.
471 */
473
474 //! Vector of phase electric potentials
475 /*!
476 * Temporary vector containing the potential of each phase in the kinetics
477 * object. length = number of phases. Units = Volts.
478 */
480
481 //! Vector of potential energies due to Voltages
482 /*!
483 * Length is the number of species in kinetics mech. It's used to store the
484 * potential energy due to the voltage.
485 */
487
488 //! Storage for the net electric energy change due to reaction.
489 /*!
490 * Length is number of reactions. It's used to store the net electric
491 * potential energy change due to the reaction.
492 *
493 * deltaElectricEnergy_[jrxn] = sum_i ( F V_i z_i nu_ij)
494 */
496
497 //! Pointer to the single surface phase
499
500 //! Pointer to the Implicit surface chemistry object
501 /*!
502 * Note this object is owned by this InterfaceKinetics object. It may only
503 * be used to solve this single InterfaceKinetics object's surface problem
504 * uncoupled from other surface phases.
505 */
507
508 //! Electrochemical transfer coefficient for the forward direction
509 /*!
510 * Electrochemical transfer coefficient for all reactions that have
511 * transfer reactions the reaction is given by m_ctrxn[i]
512 */
514
515 //! Vector of reaction indexes specifying the id of the charge transfer
516 //! reactions in the mechanism
517 /*!
518 * Vector of reaction indices which involve charge transfers. This provides
519 * an index into the m_beta and m_ctrxn_BVform array.
520 *
521 * irxn = m_ctrxn[i]
522 */
523 std::vector<size_t> m_ctrxn;
524
525 //! Vector of Reactions which follow the Butler-Volmer methodology for specifying the
526 //! exchange current density first. Then, the other forms are specified based on this form.
527 /*!
528 * Length is equal to the number of reactions with charge transfer coefficients, m_ctrxn[]
529 *
530 * m_ctrxn_BVform[i] = 0; This means that the irxn reaction is calculated via the standard forward
531 * and reverse reaction rates
532 * m_ctrxn_BVform[i] = 1; This means that the irxn reaction is calculated via the BV format
533 * directly.
534 * m_ctrxn_BVform[i] = 2; this means that the irxn reaction is calculated via the BV format
535 * directly, using concentrations instead of activity concentrations.
536 */
537 std::vector<size_t> m_ctrxn_BVform;
538
539 //! Vector of booleans indicating whether the charge transfer reaction rate constant
540 //! is described by an exchange current density rate constant expression
541 /*!
542 * Length is equal to the number of reactions with charge transfer coefficients, m_ctrxn[]
543 *
544 * m_ctrxn_ecdf[irxn] = 0 This means that the rate coefficient calculator will calculate
545 * the rate constant as a chemical forward rate constant, a standard format.
546 * m_ctrxn_ecdf[irxn] = 1 this means that the rate coefficient calculator will calculate
547 * the rate constant as an exchange current density rate constant expression.
548 */
550
551 //! Vector of standard concentrations
552 /*!
553 * Length number of kinetic species
554 * units depend on the definition of the standard concentration within each phase
555 */
557
558 //! Vector of delta G^0, the standard state Gibbs free energies for each reaction
559 /*!
560 * Length is the number of reactions
561 * units are Joule kmol-1
562 */
564
565 //! Vector of deltaG[] of reaction, the delta Gibbs free energies for each reaction
566 /*!
567 * Length is the number of reactions
568 * units are Joule kmol-1
569 */
571
572 //! Vector of the products of the standard concentrations of the reactants
573 /*!
574 * Units vary wrt what the units of the standard concentrations are
575 * Length = number of reactions.
576 */
578
579 bool m_ROP_ok;
580
581 //! Current temperature of the data
582 doublereal m_temp;
583
584 //! Current log of the temperature
585 doublereal m_logtemp;
586
587 //! Boolean flag indicating whether any reaction in the mechanism
588 //! has a coverage dependent forward reaction rate
589 /*!
590 * If this is true, then the coverage dependence is multiplied into
591 * the forward reaction rates constant
592 */
594
595 //! Boolean flag indicating whether any reaction in the mechanism
596 //! has a beta electrochemical parameter.
597 /*!
598 * If this is true, the Butler-Volmer correction is applied
599 * to the forward reaction rate for those reactions.
600 *
601 * fac = exp ( - beta * (delta_phi))
602 */
604
605 //! Boolean flag indicating whether any reaction in the mechanism
606 //! is described by an exchange current density expression
607 /*!
608 * If this is true, the standard state Gibbs free energy of the reaction
609 * and the product of the reactant standard concentrations must be
610 * precalculated in order to calculate the rate constant.
611 */
613
614 //! Int flag to indicate that some phases in the kinetics mechanism are
615 //! non-existent.
616 /*!
617 * We change the ROP vectors to make sure that non-existent phases are
618 * treated correctly in the kinetics operator. The value of this is equal
619 * to the number of phases which don't exist.
620 */
622
623 //! Vector of booleans indicating whether phases exist or not
624 /*!
625 * Vector of booleans indicating whether a phase exists or not. We use this
626 * to set the ROP's so that unphysical things don't happen. For example, a
627 * reaction can't go in the forwards direction if a phase in which a
628 * reactant is present doesn't exist. Because InterfaceKinetics deals with
629 * intrinsic quantities only normally, nowhere else is this extrinsic
630 * concept introduced except here.
631 *
632 * length = number of phases in the object. By default all phases exist.
633 */
634 std::vector<bool> m_phaseExists;
635
636 //! Vector of int indicating whether phases are stable or not
637 /*!
638 * Vector of booleans indicating whether a phase is stable or not under
639 * the current conditions. We use this to set the ROP's so that
640 * unphysical things don't happen.
641 *
642 * length = number of phases in the object. By default all phases are stable.
643 */
645
646 //! Vector of vector of booleans indicating whether a phase participates in
647 //! a reaction as a reactant
648 /*!
649 * m_rxnPhaseIsReactant[j][p] indicates whether a species in phase p
650 * participates in reaction j as a reactant.
651 */
652 std::vector<std::vector<bool> > m_rxnPhaseIsReactant;
653
654 //! Vector of vector of booleans indicating whether a phase participates in a
655 //! reaction as a product
656 /*!
657 * m_rxnPhaseIsReactant[j][p] indicates whether a species in phase p
658 * participates in reaction j as a product.
659 */
660 std::vector<std::vector<bool> > m_rxnPhaseIsProduct;
661
662 //! Values used for converting sticking coefficients into rate constants
663 struct StickData {
664 size_t index; //!< index of the sticking reaction in the full reaction list
665 double order; //!< exponent applied to site density term
666 double multiplier; //!< multiplicative factor in rate expression
667 bool use_motz_wise; //!< 'true' if Motz & Wise correction is being used
668 };
669
670 //! Data for sticking reactions
671 std::vector<StickData> m_stickingData;
672
673 void applyStickingCorrection(double T, double* kf);
674
675 int m_ioFlag;
676
677 //! Number of dimensions of reacting phase (2 for InterfaceKinetics, 1 for
678 //! EdgeKinetics)
679 size_t m_nDim;
680 };
681 }
682
683 #endif
void applyVoltageKfwdCorrection(doublereal *const kfwd)
Apply modifications for the forward reaction rate for interfacial charge transfer reactions...
void setElectricPotential(int n, doublereal V)
Set the electric potential in the nth phase.
vector_fp m_deltaG0
Vector of delta G^0, the standard state Gibbs free energies for each reaction.
virtual bool addReaction(shared_ptr< Reaction > r)
Add a single reaction to the mechanism.
doublereal electrochem_beta(size_t irxn) const
Return the charge transfer rxn Beta parameter for the ith reaction.
doublereal m_logtemp
Current log of the temperature.
virtual void getDeltaEnthalpy(doublereal *deltaH)
Return the vector of values for the reactions change in enthalpy.
Various templated functions that carry out common vector operations (see Templated Utility Functions)...
thermo_t & thermo(size_t n=0)
This method returns a reference to the nth ThermoPhase object defined in this kinetics mechanism...
Definition: Kinetics.h:276
SurfPhase * m_surf
Pointer to the single surface phase.
vector_fp m_phi
Vector of phase electric potentials.
Advance the surface coverages in time.
vector_fp m_deltaG
Vector of deltaG[] of reaction, the delta Gibbs free energies for each reaction.
virtual void getDeltaEntropy(doublereal *deltaS)
Return the vector of values for the reactions change in entropy.
vector_fp m_beta
Electrochemical transfer coefficient for the forward direction.
virtual void updateMu0()
Update the standard state chemical potentials and species equilibrium constant entries.
This rate coefficient manager supports one parameterization of the rate constant of any type...
Definition: RateCoeffMgr.h:21
virtual void getDeltaSSEntropy(doublereal *deltaS)
Return the vector of values for the change in the standard state entropies for each reaction...
virtual void getEquilibriumConstants(doublereal *kc)
Equilibrium constant for all reactions including the voltage term.
double effectiveActivationEnergy_R(size_t irxn)
Return effective activation energy for the specified reaction.
std::vector< std::vector< bool > > m_rxnPhaseIsProduct
Vector of vector of booleans indicating whether a phase participates in a reaction as a product...
vector_fp m_StandardConc
Vector of standard concentrations.
Add a phase to the kinetics manager object.
double effectiveTemperatureExponent(size_t irxn)
Return effective temperature exponent for the specified reaction.
void convertExchangeCurrentDensityFormulation(doublereal *const kfwd)
When an electrode reaction rate is optionally specified in terms of its exchange current density...
bool m_has_electrochem_rxns
Boolean flag indicating whether any reaction in the mechanism has a beta electrochemical parameter...
int m_phaseExistsCheck
Int flag to indicate that some phases in the kinetics mechanism are non-existent. ...
vector_fp m_mu0_Kc
Vector of standard state electrochemical potentials modified by a standard concentration term...
vector_fp deltaElectricEnergy_
Storage for the net electric energy change due to reaction.
SurfaceArrhenius buildSurfaceArrhenius(size_t i, InterfaceReaction &r, bool replace)
Build a SurfaceArrhenius object from a Reaction, taking into account the possible sticking coefficien...
Base class for a phase with thermodynamic properties.
Definition: ThermoPhase.h:93
Values used for converting sticking coefficients into rate constants.
void updateKc()
Update the equilibrium constants and stored electrochemical potentials in molar units for all reversi...
std::vector< int > vector_int
Vector of ints.
Definition: ct_defs.h:159
virtual void resizeSpecies()
Resize arrays with sizes that depend on the total number of species.
doublereal m_temp
Current temperature of the data.
A simple thermodynamic model for a surface phase, assuming an ideal solution model.
Definition: SurfPhase.h:143
bool m_has_coverage_dependence
Boolean flag indicating whether any reaction in the mechanism has a coverage dependent forward reacti...
vector_int m_ctrxn_ecdf
Vector of booleans indicating whether the charge transfer reaction rate constant is described by an e...
virtual void getDeltaGibbs(doublereal *deltaG)
Return the vector of values for the reaction Gibbs free energy change.
std::vector< size_t > m_ctrxn_BVform
Vector of Reactions which follow the Butler-Volmer methodology for specifying the exchange current de...
virtual void getFwdRateConstants(doublereal *kfwd)
Return the forward rate constants.
vector_int m_phaseIsStable
Vector of int indicating whether phases are stable or not.
A kinetics manager for heterogeneous reaction mechanisms.
vector_fp m_conc
Array of concentrations for each species in the kinetics mechanism.
Solve for the pseudo steady-state of the surface problem.
std::vector< bool > m_phaseExists
Vector of booleans indicating whether phases exist or not.
bool use_motz_wise
'true' if Motz & Wise correction is being used
size_t m_nDim
Number of dimensions of reacting phase (2 for InterfaceKinetics, 1 for EdgeKinetics) ...
void _update_rates_T()
Update properties that depend on temperature.
A reaction occurring on an interface (i.e. a SurfPhase or an EdgePhase)
Definition: Reaction.h:206
vector_fp m_actConc
Array of activity concentrations for each species in the kinetics object.
Public interface for kinetics managers.
Definition: Kinetics.h:111
virtual void getActivityConcentrations(doublereal *const conc)
Get the vector of activity concentrations used in the kinetics object.
void _update_rates_C()
Update properties that depend on the species mole fractions and/or concentration,.
vector_fp m_pot
Vector of potential energies due to Voltages.
int phaseExistence(const size_t iphase) const
Gets the phase existence int for the ith phase.
void setPhaseStability(const size_t iphase, const int isStable)
Set the stability of a phase in the reaction object.
Base class for kinetics managers and also contains the kineticsmgr module documentation (see Kinetics...
bool m_has_exchange_current_density_formulation
Boolean flag indicating whether any reaction in the mechanism is described by an exchange current den...
double multiplier
multiplicative factor in rate expression
std::vector< std::vector< bool > > m_rxnPhaseIsReactant
Vector of vector of booleans indicating whether a phase participates in a reaction as a reactant...
virtual void modifyReaction(size_t i, shared_ptr< Reaction > rNew)
Modify the rate expression associated with a reaction.
std::vector< size_t > m_revindex
List of reactions numbers which are reversible reactions.
int phaseStability(const size_t iphase) const
Gets the phase stability int for the ith phase.
virtual void getRevRateConstants(doublereal *krev, bool doIrreversible=false)
Return the reverse rate constants.
std::vector< StickData > m_stickingData
Data for sticking reactions.
virtual bool isReversible(size_t i)
True if reaction i has been declared to be reversible.
virtual std::string kineticsType() const
Identifies the Kinetics manager type.
void _update_rates_phi()
Update properties that depend on the electric potential.
virtual void init()
Prepare the class for the addition of reactions, after all phases have been added.
std::vector< size_t > m_irrev
Vector of irreversible reaction numbers.
Advances the surface coverages of the associated set of SurfacePhase objects in time.
InterfaceKinetics(thermo_t *thermo=0)
Constructor.
vector_fp m_mu
Vector of chemical potentials for all species.
void updateExchangeCurrentQuantities()
values needed to convert from exchange current density to surface reaction rate.
std::vector< double > vector_fp
Turn on the use of stl vectors for the basic array type within cantera Vector of doubles.
Definition: ct_defs.h:157
virtual void updateROP()
Internal routine that updates the Rates of Progress of the reactions.
double effectivePreExponentialFactor(size_t irxn)
Return effective preexponent for the specified reaction.
double order
exponent applied to site density term
Rate1< SurfaceArrhenius > m_rates
Templated class containing the vector of reactions for this interface.
std::vector< size_t > m_ctrxn
Vector of reaction indexes specifying the id of the charge transfer reactions in the mechanism...
vector_fp m_ProdStanConcReac
Vector of the products of the standard concentrations of the reactants.
void setPhaseExistence(const size_t iphase, const int exists)
Set the existence of a phase in the reaction object.
virtual int type() const
Identifies the kinetics manager type.
size_t index
index of the sticking reaction in the full reaction list
virtual void getDeltaSSGibbs(doublereal *deltaG)
Return the vector of values for the reaction standard state Gibbs free energy change.
Namespace for the Cantera kernel.
Definition: application.cpp:29
virtual void getDeltaElectrochemPotentials(doublereal *deltaM)
Return the vector of values for the reaction electrochemical free energy change.
vector_fp m_mu0
Vector of standard state chemical potentials for all species.
vector_fp m_grt
Temporary work vector of length m_kk.
virtual void getDeltaSSEnthalpy(doublereal *deltaH)
Return the vector of values for the change in the standard state enthalpies of reaction.
ImplicitSurfChem * m_integrator
Pointer to the Implicit surface chemistry object.
An Arrhenius rate with coverage-dependent terms.
Definition: RxnRates.h:118
An interface reaction which involves charged species.
Definition: Reaction.h:235
virtual Kinetics * duplMyselfAsKinetics(const std::vector< thermo_t *> &tpVector) const
Duplication routine for objects which inherit from Kinetics.
|
2023-03-27 11:11:53
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|
http://www.parresianz.com/mpi/c++/parallel-domain-decomposition-part-4/
|
Biswajit Banerjee
### Parallel domain decomposition for particle methods: Part 4
Applying the Plimpton method for migrating particles
#### Introduction
The Plimpton scheme of communicating ghost information between patches was described in Part 3 of this series. Let us now see how a similar approach can be used to migrate particles that have moved across patches.
In the animation below we just move the particles within each patch randomly. To make the identity of the particles clear, we have used solid circles for the patch particles and three-quarter circle for the particles in the ghost regions. As you can see, some of the particles have moved outside the patches and need either to be deleted (if they have left the computational domain - assuming that the domain size remains unchanged) or they need to be moved to adjacent patches.
#### Plimpton’s scheme for migrating particles
If we run Plimpton’s scheme in reverse order, we can move the particles to the appropriate patches with only four communication steps in 2D and six in 3D. Notice in the animation below that we start with a search region in the x-direction that contains the top and bottom patches along with the right (or left) patch. We relocate particles in this region first and then need to move particles only in the top and bottom patches. Note also that the ghost particles have been moved back to their original locations, indicating that we can ignore these during the migration process. Depending on the requirements of the problem, we may either delete particles that have left the domain, introduce them back in a periodic manner, or extend the domain itself.
#### MPI implementation
The implementation of the migration process is similar to that for ghost exchange. A typical migrateParticles function can have the following form:
In Part 2 we defined a PatchNeighborComm struct and a Patch struct. We can keep the PatchNeighborComm struct in the same form, with the possible addition of a method of two. However, the Patch struct becomes considerably simplified as show below.
##### Patch struct
The Patch struct described in Part 3 now has a few more methods. Let us see how some of these new functions may be implemented.
The first new function is sendRecvMigrateXMinus which is the equivalent of sendRecvGhostXMinus for th emigration process. Note that the only difference between these two function is the definition of the search box.
The next new method is combineSentParticlesX which is defined as:
The combineSentParticles method in PatchNeighborComm is defined as
One also needs to delete the sent particles from the patch, using the method deleteSentParticles; this is where the use of a hashmap becomes handy.
Finally, we add the received particles to the list of particles in the patch using addReceivedParticles:
In some special cases, we will also need to remove particles outside the domain. One approach is to use removeParticlesOutsidePatch, but this step is typically not recommended in general as it is costly and often not necessary.
#### Remarks
In the next part of this series we will explore how information about forces can be communicated across patches.
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2017-12-17 15:35:54
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https://asone.ai/polymath/index.php?title=Basic_facts_about_Bohr_sets&diff=cur&oldid=3970
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# Difference between revisions of "Basic facts about Bohr sets"
Parent page: Improving the bounds for Roth's theorem
## Definition
### Version for cyclic groups
Let $r_1,\dots,r_k$ be elements of $\mathbb{Z}_N$ and let δ>0. The Bohr set $B(r_1,\dots,r_k;\delta)$ is the set of all $x\in\mathbb{Z}_N$ such that $r_ix$ lies in the interval $[-\delta N,\delta N]$ for every i=1,2,...,k. If $K=\{r_1,\dots,r_k\}$, then it is usual to write $B(K,\delta)$ for $B(r_1,\dots,r_k;\delta)$.
### Version for more general finite Abelian groups
Let G be a finite Abelian group, let $\chi_1,\dots,\chi_k$ be characters on G and let δ>0. The Bohr set $B(\chi_1,\dots,\chi_k;\delta)$ is the set of all $g\in G$ such that $|1-\chi_i(g)|\leq\delta$ for every i=1,2,...,k.
Note that this definition does not quite coincide with the definition given above in the case $G=\mathbb{Z}_N$. In practice, the difference is not very important, and sometimes when working with $\mathbb{Z}_N$ it is in any case more convenient to replace the condition given by the inequality $|1-\exp(2\pi i r_jx/N)|\leq\delta$ for each j.
### Version for sets of integers
Needs to be written ...
### Regularity
Of considerable importance when it comes to making use of Bohr sets is the notion of regularity, introduced by Bourgain. Here we give the bare definition: below it will be explained why regularity is useful.
The formal definition (as it appears in Sanders's paper) is this. Let K be a set of size d. Then the Bohr set $B=B(K,\delta)$ is C-regular if for every $0\leq\eta\leq 1/Cd$ we have the inequality $|B(K,\delta(1+\eta))|\leq(1+Cd\eta)|B(K,\delta)|$ and also the inequality $|B(K,\delta(1-\eta))|\geq(1+Cd\eta)^{-1}|B(K,\delta)|$.
The precise numbers here are not too important. What matters is that if you slightly increase the width of a regular Bohr set, then you only slightly increase its size. Another way to think about it is this. Let B' be the "small" Bohr set $B(K,\eta)$. Then if you choose a random point in B and add to it a random point x' in B', the probability that x+x' also belongs to B is close to 1. An equivalent way of saying this is that the characteristic measure of B is approximately unchanged if you convolve it by the characteristic measure of B'.
## Ways of thinking about Bohr sets
### Approximate subgroups
If N is prime, then $\mathbb{Z}_N$ has no non-trivial subgroups. Therefore, if one wishes to translate an argument that works in $\mathbb{F}_3^n$ into one that works in $\mathbb{Z}_N$, then one must find some kind of analogue for the notion of a subgroup (or subspace) that is not actually a subgroup. Bohr sets are one way of fulfilling this role.
The key property enjoyed by a subgroup is closure: if H is a subgroup of a group G and x,y belong to H, then x+y belongs to H. Although a Bohr set is not closed under addition, a regular Bohr set has a property that can be used as a substitute for closure. Indeed, the property discussed above is exactly the one we use: that if B is a regular Bohr set $B(K,\delta)$ and $B'=B(K,\eta)$ for some suitably small η, then most elements x of B have the property that if you add any element y of B' you obtain another element of B. Thus, B is not closed under addition, but it is "mostly closed" under addition of elements of B'.
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2020-05-31 12:30:29
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|
https://docs.contango.xyz/basics/what-are-expirables
|
contango
Search…
⌃K
🤔
# What are expirables?
### Definition
An expirable is a derivative contract to buy or sell an asset at a specific date and price in the future. It’s called a derivative because it derives its value from the underlying asset.
• If the expirable price is higher than the spot price of the underlying asset, the market is in contango (that's us! 💃).
• If the expirable price is lower than the spot, then the market is in backwardation.
In TradFi, forwards are traded on many different assets such as currencies (EUR, USD, etc.) or commodities (wheat, oil, etc.), either for speculation or for hedging purposes. In crypto, forwards are normally traded on cryptocurrencies (BTC, ETH, etc.) on standardised products called futures. See the use cases section for more details.
Just like in TradFi, forward contracts on cryptocurrencies have:
• a price at which they can be bought and sold.
• an expiry date, at which the underlying asset is exchanged between the parties (physical delivery).
When buying or selling forwards, traders can make use of leverage, i.e. borrowed capital to make trades. Leverage allows traders to amplify their buying and selling power, but it also increases the risks of liquidation.
#### Example
Let’s say the spot price of ETH in January 2022 is
$S = 100$
.
The sentiment is optimistic so the market is currently in contango: the forward price of ETHDAI is
$F = 110$
.
A trader believes the price of ETHDAI will go up over the next few months. So she buys (goes long) 1 ETHDAI March 22 forward contract, at
$F = 110$
.
In March, at expiry, the trader will receive 1 ETH for the original price paid of
$110 \:DAI$
, regardless of the ongoing spot price
$S$
.
If, as expected, ETHDAI appreciates even more to
$S = 120$
and the 1 ETH is sold for DAI then she makes a profit of
$10 \:DAI$
.
If, on the contrary, ETHDAI goes to 100 then, by selling the 1 ETH, she realises a loss of
$10 \:DAI$
.
### Forward vs Futures contracts
As we like to say internally: all futures are a forward, but not all forwards are futures.
Forward and futures are both derivative contracts to buy or sell a specific asset at a set price by a certain date in the future. A forward contract settles just once at the end of the contract and has specific terms (e.g. on Contango the price of an expirable depends on the amount of collateral a trader uses to buy or sell a contract). However, futures contracts are standardized contracts. As such, they are settled on a daily basis. These arrangements come with fixed maturity dates and uniform terms.
### So what’s unique about Contango?
Contango is the first exchange to offer expirables using fixed interest rates protocols. This is possible thanks to DeFi. Contango is permissionless, decentralized and composable.
Check out the next section to see why we need expirables and why Contango is the market leader in the space.
|
2023-03-22 00:41:25
|
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|
http://mathhelpforum.com/calculus/185844-prove-limit-definition.html
|
# Math Help - prove limit by definition
1. ## prove limit by definition
prove that $lim_{x\rightarrow2}\frac{x}{x+1}\neq1$ by epsilon delta
defintion.
we need to prove that their is $\varepsilon>0$ so for every $\delta>0$
their is x $|x-2|<\delta$ and $|\frac{x}{x+1}-1|\geq\varepsilon$
so we are free to choose our epsilon and our x
i choose $\varepsilon=2$
x cannot be equal to -1 because of the denominator.
and we need to choose x for which $|\frac{x}{x+1}-1|\geq2$
$|\frac{x-x-1}{x+1}|\geq2$ -> $|\frac{1}{x+1}|\geq2$ -> $\frac{1}{2}\geq|x+1|$
-> $-\frac{1}{2}\leq x+1\leq\frac{1}{2}$ -> $-\frac{3}{2}\leq x\leq-\frac{1}{2}$
but x=-1 there so we have a problem.
where is the problem in my way???
the prof solved it by $\varepsilon=1/6$
$2>x>max\{2-\delta/2,1\}$
cant understand why he choose such x
2. ## Re: prove limit by definition
Originally Posted by transgalactic
prove that $lim_{x\rightarrow2}\frac{x}{x+1}\neq1$ by epsilon delta
defintion.
we need to prove that their is $\varepsilon>0$ so for every $\delta>0$
their is x $|x-2|<\delta$ and $|\frac{x}{x+1}-1|\geq\varepsilon$
so we are free to choose our epsilon and our x
i choose $\varepsilon=2$
x cannot be equal to -1 because of the denominator.
and we need to choose x for which $|\frac{x}{x+1}-1|\geq2$
$|\frac{x-x-1}{x+1}|\geq2$ -> $|\frac{1}{x+1}|\geq2$ -> $\frac{1}{2}\geq|x+1|$
-> $-\frac{1}{2}\leq x+1\leq\frac{1}{2}$ -> $-\frac{3}{2}\leq x\leq-\frac{1}{2}$
but x=-1 there so we have a problem.
where is the problem in my way???
the prof solved it by $\varepsilon=1/6$
$2>x>max\{2-\delta/2,1\}$
cant understand why he choose such x
Since the actual limit is 2/3, you will always be able to show that for x close enough to 2 that |x/(x+1)-1|>0.3, so you need to choose epsilon greater than zero but less than 0.3 (well actually any thing less than 1/3 will do). Then show that for all delta>0 there is always some x such that |x-1|<delta and |x/(x+1) - 1|>epsilon.
CB
3. ## Re: prove limit by definition
what is the problem of my way.
??
4. ## Re: prove limit by definition
Consider ${\delta}$ <_1. Then if x satisfy's 0< |x-2|< ${\delta}$, F(x)<3/4. Hence |F(x)-1|>1/4. Choosing ${\epsilon}$<_1/4 completes the proof.
5. ## Re: prove limit by definition
i cant because its for every delta
and can you tell me where am i wrong
?
6. ## Re: prove limit by definition
For any delta>1, x values in (1,3) will still have to satisfy |F(x)-1|< epsilon.
7. ## Re: prove limit by definition
It's not true that you are totally free to choose any ε that you like.
As CaptainBlack told you, the actual limit is 2/3.
|1 - 2/3| = 1/3. So, if you use an ε ≥ 1/3, it is possible to find a δ > 0 such that for all x for which 0 < | x - 2 | < δ it follows that | x/(x+1) - 1 | < ε .
That's why CB said you need to choose ε < 1/3 .
|
2015-03-03 09:43:29
|
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https://physics.stackexchange.com/questions/368265/zero-potential-inside-and-on-boundary
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# Zero potential inside and on boundary
In Griffiths introduction to Electrodynamics, the classic image problem is presented: There is a charge q, above a grounded conducting plane.
The boundary conditions are therefore: 1.V=0, at plane 2.v=0 at infinity
My question is, since potentials are harmonic functions and the potential is zero both inside and on boundary, shouldn't the potential therefore be zero everywhere in R^3 ?
Your statement was correct if the differential equation you have had to solve was the Laplace's equation $$\nabla^2V=0$$ which accounts for space without charges. In that case the potential was indeed an harmonic function, and in order to satisfy the maximum principle - it must have vanish identically.
However, in your particular case there are charges in the volume of interest, and thus one needs to fulfill the Poisson's equation $$\nabla^{2}V=-\frac{Q}{\varepsilon_{0}}\delta\left(x\right)\delta\left(y\right)\delta\left(z-d\right)$$ in the upper half part of $\mathbb{R}^{3}$. Here I assumed the charge $Q$ is placed at $\vec{r}=\left(0,0,d\right)$. Therefore, the function is not harmonic and the maximum principle is not applicable.
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2020-02-19 13:57:49
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https://worldbuilding.stackexchange.com/questions/109343/what-would-be-the-effect-of-a-very-very-fast-sword-strike
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# What would be the effect of a very very fast sword strike
Let's imagine some very powerful dude with absolutely ridiculous strength/speed/durability. I'm not asking about the credibility of this, let's suppose it's magic. He's got some magical sword as durable as it need to be (probably way more than the hardest thing we can think of today. Here again, let's say it's magic). It would be 115 cm long.
Now the real question : If this guy strikes air with his sword at a very high speed, what would be the effects ? Sounds, shockwaves, light, sparks, fire, plasma ? In which amplitude ?
To clear up the ambiguity left by the "very high speed" expression, please give me your advice about these 3 orders of magnitude :
1. Speed of sound (~300 m/s)
2. 10 times the speed of sound (~3000 m/s)
3. Half of the light speed (~150 000 000 m/s)
Let's say it's the instantaneous speed of the tip of the sword. We're on earth, same atmospheric pressure, same gravity, at sea level.
I'm looking for some cool effect without obliterating the entire world, so I'm looking for the best speed for that. I'm hungry for maths too, if possible.
Feel free to give me your advice about how to make things even funnier if possible. Keep in mind that I'm very well conscious of the ridiculousness of the situation.
• 115 cm makes for a pretty short sword, unless I'm mistaken. – AndreiROM Apr 12 '18 at 15:45
• I took the average value for a longsword from wikipedia. looks alright for me anyway – GlorfSf Apr 12 '18 at 15:47
• Obligatory XKCD regarding #3: what-if.xkcd.com/1 – F1Krazy Apr 12 '18 at 15:50
• speed of light is 10^8 m/s, multiply by 1000 or change to km/s – V. Sim Apr 12 '18 at 20:11
• My bad, it's fixed. – GlorfSf Apr 12 '18 at 20:54
## 2 Answers
Perhaps the best way of looking at this is to use the equation for kinetic energy:
Ke=1/2M*V^2
Since the mass of the sword is going to be held constant, the part of the equation where we see the change in the energy delivered to the target is the "V^2" part. In other words, the energy increases as the square of the velocity.
From other Stack Exchange questions we can see how the effects multiply. A longbow arrow can strike a target with @ 100J of energy. A crossbow quarrel, having more mass than the arrow and moving somewhat faster from a steel crossbow can generate @ 200J of energy. An Arquebus from the period, shooting a ball at a much higher velocity, can strike a target with 1000J of energy, an order of magnitude greater.
So a typical "arming sword" (which I am assuming you are referencing when you say "longsword" weighs 1.1Kg, or 1100g
300m/s, KE = 49500 J
3000m/s, KE = 4950000 J
150,000,000m/s, KE = 1.2375E+16 J
For context, we will look at the Atomic Rockets "Boom Table"
At 300m/s, you are striking with just under the energy of a 20mm cannon round, or @ 11 grams of TNT. In context, an anti personnel mine might be that powerful, removing limbs fairly easily.
At 3000 m/s, you are delivering just under the energy of a kilogram of TNT. This is like having an anti tank mine or IED explode on your body, expect shredded remains to go flying across the battlefield.
At 150,000,000, m/s, you are delivering a nuclear punch, of @ 3.5Mt of energy. No human or animal target will even remain (the amount of energy is so great even "pink mist" will have been rendered into its constituent atoms in a rapidly expanding ball of plasma).
Perhaps a high block won't be so effective in this case
Frankly, I'll just sit back with my sniper buddies and try to engage from @ 3 km away.
• Nice one. I did what it would sound like, you did what would happen if it hit them :) – JSCoder says Reinstate Monica Apr 12 '18 at 22:55
• Also, “under 1 kg of TNT?” 11 times 100 is over 1000... – JSCoder says Reinstate Monica Apr 13 '18 at 1:59
• Watch out, your sniper buddies might get cancer... – Hazard Apr 13 '18 at 2:09
• 4.1E10^6 J is 1Kg of TNT – Thucydides Apr 15 '18 at 5:03
Let's try to work this out for the lowest order of magnitude.
1. 300 m/s (aka speed of sound)
So this dude has absolutely great speed/strength, as you mentioned. So let's assume his arm accelerates at 46.2 g of force, which a human could definetly survive. That would take $v_f = v_o + at = 453.07t = 300$ (bad with units, so didn't include, sorry). So it's concievable that it could accelerate in less than one second to the speed of sound. Obviously, the object would produce a sonic boom (an order of magnitude louder of a bullet, I'd guess, so wear earmuffs because 150 decibels is a lot). One side effect would be any drones/air troops fighting him would literally get knocked out of the sky if the sound is at the correct resonance frequency (140 decibels to disable a craft that was up to 130 feet away, so can disable crafts up to 1/4 mile away).
The next question is what they are doing with that momentum $p=mv$ which would probably be several hundred kilogram-meters per second.
A rule of Newtonian physics states that the impulse imparted to an object is equal to the change in momentum for that object, provided no other forces or effects are involved. This requires an impulse in the reverse direction of hundreds of Newton seconds, which would also cause massive deceleration. For the impulse to not have him black out, he would have to use hundreds of Newtons of force.
Reader: Don't worry, he's that strong.
In the meantime, "light, sparks, fire, plasma" are not likely, because if you see a plane break the speed of sound, they don't go 'lights sparks fire plasma' (unless something is really wrong). So a big soundwave which could cause permanent hearing damage and knock drones out of the sky is it (unless you consider what it is that he/she's hitting). I don't dare forecast for the second/third scenario (my physics skills aren't sturdy enough).
• All I'm wondering is how his arms are attached if he accelerates to speed of sound in a second. Lets say his arms are 1m in length and travel a circle (3.14m circumference then right?). His average speed would be 650m/s or 207 times his arms would circle his shoulder and slash at the enemy before he's up to speed. Creepy anatomy that, but a human can handle the stress! XD – Demigan Apr 12 '18 at 21:18
• HANDWAVIUM MAN!!!! Now selling for 89.99 dollars :) – JSCoder says Reinstate Monica Apr 12 '18 at 22:34
• Best. Handwavium comment. Ever. – Demigan Apr 13 '18 at 6:29
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2020-06-06 17:46:52
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https://www.digi.com/resources/documentation/digidocs/embedded/dey/3.0/cc6ul/uboot_r_environment-variables.html
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The U-Boot environment is a block of memory in persistent storage. It is used to store variables in the form name=value.
## Where is the U-Boot environment?
The ConnectCore 6UL reads the U-Boot environment (including the MAC addresses) from a partition called environment in the NAND flash. If it does not find a valid environment on the NAND, U-Boot uses a hard-coded default environment:
• The environment is located at the offset determined by constant CONFIG_ENV_OFFSET in U-Boot source code at include/configs/ccimx6ulsbc.h and include/configs/ccimx6ulstarter.h.
• The size of the environment is determined by constant CONFIG_ENV_SIZE at the same header file.
• There is a back-up copy of the environment in the same NAND partition (where the original copy is) at the offset determined by constant CONFIG_ENV_OFFSET_REDUND in the same header file.
## Access U-Boot environment from Linux
You can access the U-Boot environment from a Linux application. In the U-Boot source tree, you can find the environment tools in the directory tools/env, which you can build with the following command:
``$make env`` Digi Embedded Yocto also compiles this tool by default and installs it on your target’s root file system as two executable files: • fw_printenv: to print the value of variables. • fw_setenv: to set the value of variables. The tools work with the configuration file /etc/fw_env.config, which must contain one or two entries in the form: Device name, offset, size. The first entry must point to the U-Boot environment location. The second (if present) must point to its redundant copy, for example: /etc/fw_env.config ``````# Configuration file for fw_(printenv/setenv) utility. # Up to two entries are valid, in this case the redundant # environment sector is assumed present. # Device name Offset Size /dev/mtd1 0x0 0x20000 /dev/mtd1 0x20000 0x20000`````` ## Append boot arguments to the kernel command line You can use the extra_bootargs variables to append commands to the default kernel command line. ## Important U-Boot environment variables on the ConnectCore 6UL The following U-Boot environment variables are worth mentioning: ### MAC addresses Variable Description Flags$ethaddr
MAC address of the first wired Ethernet interface
change-default
$eth1addr MAC address of the second wired Ethernet interface if there is one change-default$wlanaddr
MAC address of the Wi-Fi interface
change-default
$btaddr MAC address of the Bluetooth interface change-default About flags Variables with write-once flag are protected and will not be overwritten by setenv or env default commands (unless manually forced with -f option). Variables with change-default flag can only be written once using setenv command (unless manually forced with -f option). Digi programs the MAC addresses of the ConnectCore 6UL during manufacturing and saves them in the U-Boot environment on the NAND. You can find the Digi-assigned MAC address on the ConnectCore 6UL module label. See Determine Digi MAC addresses for more information. ### Wireless virtual MAC addresses The wireless interface on the ConnectCore 6UL module allows you to define up to three optional virtual interfaces so that the platform can run concurrently as station, p2p, and SoftAP. Digi neither reserves nor programs unique MAC addresses for these virtual interfaces. A user who wishes to assign unique MAC addresses to such virtual interfaces can use the following U-Boot environment variables: Variable Description Flags$wlan1addr
MAC address of virtual wireless interface 1
change-default
$wlan2addr MAC address of virtual wireless interface 2 change-default$wlan3addr
MAC address of virtual wireless interface 3
change-default
### Module variant
Variable Description Flags
$module_variant Variant ID code for the System-On-Chip write-once During start-up, U-Boot automatically sets this variable to the ConnectCore 6UL System-On-Chip variant ID number (a hexadecimal code programmed in the SOM one-time programmable bits). ### Carrier board version and ID Variable Description Flags$board_version
Version number of the carrier board
write-once
\$board_id
ID number of the carrier board
write-once
During start-up, U-Boot automatically sets these variables to the carrier board’s version and ID numbers (decimal numbers programmed in the SOM one-time programmable bits). See Carrier board version and ID.
## Determine Digi MAC addresses
### Sequential MAC address scheme
Digi assigns MAC addresses according to a sequential scheme. The order of assignment depends on the available interfaces, but the scheme always respects the order Ethernet 1, Ethernet 2, Wi-Fi, Bluetooth and skips non-available interfaces.
### Determine MAC addresses
You can determine the MAC addresses for your device either by reading the Ethernet MAC off the label and using the scheme to calculate subsequent addresses, or by reading them from your device. To read the stock environment variables from the device using the `printenv` command executed at the U-Boot prompt:
To get First Ethernet MAC address:
``=> printenv ethaddr``
If your SOM has a second Ethernet MAC address:
``=> printenv eth1addr``
To get Wi-Fi MAC address:
``=> printenv wlanaddr``
To get Bluetooth MAC address:
``=> printenv btaddr``
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2022-05-19 02:59:49
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http://cosmocoffee.info/viewtopic.php?t=2808&sid=7797204cf542eb59d5d35e8a71aa730f
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CosmoCoffee
FAQ Search SmartFeed Memberlist Register Profile Log in Arxiv New Filter | Bookmarks & clubs | Arxiv ref/author:
Temperature Anisotropies !
Author Message
Joined: 05 Jan 2017
Posts: 7
Affiliation: University of Alberta
Posted: March 28 2017 Hi everyone, I just have one simple question question, I want to calculate the temperature anisotropies as a function of wave number i.e. ΔT(k) / T using the list of transfer functions output from the camb code. I just want to use my own initial power spectrum Pζ(k) of initial perturbations(comoving curvature perturbations) to calculate $\langle \Delta T^2(k)/T^2 \rangle = \Delta_i^2(k) P_\zeta(k)$. Where Δi(k) would be some combination of transfer functions of the camb code. So, what precise combinations of transfer functions do I need here as Δi's? Thanks in Advance!! M Junaid
Antony Lewis
Joined: 23 Sep 2004
Posts: 1293
Affiliation: University of Sussex
Posted: March 28 2017 Are you sure you don't want to calculate the C power spectrum? That's easily done by modifying CAMB's power_tilt.f90 to use your own initial power spectrum (and increasing accuracy settings if needed).
Joined: 05 Jan 2017
Posts: 7
Affiliation: University of Alberta
Posted: March 28 2017 No we are working out some moments in momentum space so we don't need Cl or any angular part.
Joined: 05 Jan 2017
Posts: 7
Affiliation: University of Alberta
Posted: March 30 2017 My hunch is that the temperature anisotropy is to leading order equal to $\Delta T/T \approx \frac{1}{4} \hat{\Delta}_\gamma+ 2\Phi$, where Φ is Weyl potential. While, you have said in your notes that power spectrum of Weyl potential is PΦ = TΦ(k)Pζ(k) where ζ is primordial perturbations. Is the same true for other power spectra as $\Delta_\gamma^2 = \hat{\Delta}_\gamma^2 P_\zeta(k)$. So the transfer function output from the camb code are $\hat{\Delta}_\gamma^2$ that don't include the primordial power spectrum Pζ(k). Please correct me if I am wrong! M.
Antony Lewis
Joined: 23 Sep 2004
Posts: 1293
Affiliation: University of Sussex
Posted: March 30 2017 Transfer function outputs are not squared (and indeed not scaled by the primordial power). The temperature sources are quite complicated in general (calculated in output routine of equations.f90).
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2017-11-23 01:41:28
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https://dylaniki.org/Freight%20Train%20Blues
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Played by Bob Dylan on Bob Dylan (1962)
Tabbed by Eyolf Østrem
Intro:
|:C . . . |. . G C :|
|C . . . |. . . . D |G . . . |
|C . . . |F . . . |C . . G |C
C
I was born in Dixie in a boomer shack
G C
Just a little shanty by the railroad track
C
Freight train was it taught me how to cry
G C
hummin' of the driver was my lullaby
I got the freight train blues
D G
Oh Lord mama, I got them in the bottom of my rambling shoes
C F
And when the whistle blows I gotta go baby, don't you know
C G C
Well, it looks like I'm never gonna lose the freight train blues.
Well, my daddy was a fireman and my old ma here,
She was the only daugther of an engineer
My sweetheart loved a brakeman and it ain't no joke
It's a shamethe way she keeps a good man broke
I got the freight train blues
Oh Lord mama, I got them in the bottom of my rambling shoes
And when the whistle blows I gotta go oh mama, don't you know
Well, it looks like I'm never gonna lose the freight train blues.
Well, the only thing that makes me laugh again
Is a southbound whistle on a southbound train
Every place I wanna go
I never can go, because you know
I got the freight train blues
Oh Lord mama, I got them in the bottom of my rambling shoes.
[finish off with harp]
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2021-09-18 17:32:04
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https://www.nature.com/articles/s41467-018-03515-2?error=cookies_not_supported&code=0a0ffb4b-fba9-4a8a-8b80-3904e4c3a35d
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Article | Open | Published:
# Silicon and glass very large scale microfluidic droplet integration for terascale generation of polymer microparticles
## Abstract
Microfluidic chips can generate emulsions, which can be used to synthesize polymer microparticles that have superior pharmacological performance compared to particles prepared by conventional techniques. However, low production rates of microfluidics remains a challenge to successfully translate laboratory discoveries to commercial manufacturing. We present a silicon and glass device that incorporates an array of 10,260 (285 × 36) microfluidic droplet generators that uses only a single set of inlets and outlets, increasing throughput by >10,000× compared to microfluidics with a single generator. Our design breaks the tradeoff between the number of generators and the maximum throughput of individual generators by incorporating high aspect ratio flow resistors. We test these design strategies by generating hexadecane microdroplets at >1 trillion droplets per h with a coefficient of variation CV <3%. To demonstrate the synthesis of biocompatible microparticles, we generated 8–16 µm polycaprolactone particles with a CV <5% at a rate of 277 g h−1.
## Introduction
In 1959, Richard Feynman famously proposed the use of micrometer- and nanometer-scale particles for medicine, as well as the creation of enormous numbers of microfabricated “factories” to generate large quantities of these engineered materials1. In the last two decades, significant progress has been made toward accomplishing this vision. In particular, microfluidics has been used to enable precise control of multiphasic flows to generate micrometer- and nanometer-scale materials with control and uniformity not possible using conventional techniques2,3,4,5,6. These micro- and nano-engineered materials have generated particular enthusiasm in the pharmaceutical industry, as well as the food and cosmetics industries, where they have created new opportunities to generate novel drug formulations that offer unprecedented spatial and temporal control of drug delivery within the body7,8,9. In comparison to conventional particle formation techniques, such as spray drying or ball milling10, microfluidic-generated formulations have demonstrated increased particle monodispersity, more uniform composition of drug within the particles, increased drug yield, longer lasting formulations that are still injectable, and reduced burst release of drug2,6,11.
The low production rate of microfluidic devices for the generation of microparticles (<10 mL h−1 for the dispersed phase, <100 mg h−1 of particles) has remained a key challenge to successfully translate the many promising laboratory-scale results of microfluidics to commercial-scale production of microfluidic-generated materials. In previous work, architectures have been developed that make it possible to operate many microfluidic droplet generators in parallel11,12,13,14,15,16,17,18,19,20,21,22. While great progress has been made in these approaches, current chips with parallelized devices are limited to production rates ϕmax 1 L h−1, have droplet homogeneities set by three-dimensional (3D) soft-lithography fabrication15, are limited to low temperature and pressure operation, can only be used with the solvents compatible with the device’s polymer construction, or are unable to be adapted to produce higher-order emulsions and particles that require multi-step processing23,24,25.
To address these challenges, we present the all silicon and glass very large scale droplet integration (VLSDI), in which we incorporate an array of 10,260 (285 × 36) microfluidic droplet generators onto a 3D-etched single silicon wafer that is operated using only a single set of inlets and outlets. The monolithic construction from a single silicon wafer obviates the alignment and bonding challenges of prior multilayer approaches and allows high pressure use. To demonstrate the power of this approach, we generate polycaprolactone (PCL) solid microparticles, a biodegradable material approved by the United States Food and Drug Administration (US FDA), with a coefficient of variation CV <5%, and an emulsion production rate that results in 277 g h−1 particle production (2.09 L h−1 dispersed phase, 328 billion particles per h). Key to achieving this throughput lies in a design strategy that breaks the tradeoff between the number of integrated droplet generators Nmax and the maximum throughput of each droplet generator ϕimax. Our design includes a high aspect ratio flow resistor into each droplet generator, which decouples the design of the individual droplet generator from the high fluidic resistance requirement necessary for parallelization. Moreover, because of the VLSDI’s all silicon and glass construction, it can operate at high pressure (Pmax > 1000 PSI (pounds per square inch)) and high temperature (Tmax > 500 °C), use solvents prevalent in the pharmaceutical industry but that are incompatible with polymer devices, and achieve a uniformity not possible using soft-lithography based devices (CV <3%). Because of the VLSDI’s 3D fabrication strategy, it can be scaled to the 10,000 droplet generators that we demonstrate in this study and beyond using conventional semiconductor fabrication, in contrast to prior approaches that have used either two-dimensional microfluidic or two-dimensional microfluidics attached to macroscopically defined manifolds. Because our device allows arbitrary microfluidic droplet generators to be parallelized, it can produce higher-order emulsions and particles that require multi-step processing.
## Results
### Very large scale droplet integration fabrication
We fabricate the VLSDI using a single microfabricated 500 µm thick 4″ Si wafer encapsulated in glass, resulting in a robust, monolithic construction.(Supplementary Fig. 2 for step by step fabrication) (Fig. 1a, b) This design enables the production of highly monodispersed PCL solid microparticles (CV <5%), >1000× faster than previously reported parallelized microfluidic approaches (Fig. 1c, d).
We use four steps of lithography and deep reactive ion etch (DRIE) to define the droplet generators (hµF = 22.5 µm), underpasses (hUP = 26 µm), vias (hV = 130 µm), and delivery channels (hD = 360 µm) (Fig. 1e). Underpasses are channels that are required to allow fluid to pass underneath the arterial lines that deliver fluid to each of the rows (Supplementary Fig. 1b). The microfabricated Si wafer is anodically bonded to two 4″ Borofloat33 glass wafers on its top and its bottom. Anodic bonding results in microfluidic channels that can operate at a maximum pressure of >1000 PSI, enabling high-throughput operation even on highly viscous samples. To make fluid connections to the VLSDI, we drill 1.5 mm holes in the top glass plate before anodic bonding using an excimer laser micromachining tool (IPG Photonics IX-255). We use steel compression fittings to avoid the creation of debris in the device that can occur using pressure fit tubing (Supplementary Fig. 4). The fittings are bonded to our device using chemically resistant epoxy (Mater Bond Epoxy EP41S-5) and connected to polytetrafluoroethylene (PTFE) tubing (1/8″ OD (outter diameter), 1/16″ ID (inner diameter)). The dispersed and continuous phases are delivered to the device from N2 gas pressured steel pressure vessels (Alloy Products) through the PTFE tubings. All of the parts and tubing are chemically resistant, allowing our platform to generate particles using a wide range of temperatures, pressures, and solvents (Supplementary Figs 35).
### Design principles
There are three main design goals for our parallelized device: to ensure uniform flow across each of the N microfluidic droplet generators, such that each generator supplies the same shear stresses to produce droplets with the same diameter. To minimize the footprint of each individual microfluidic droplet generator, such that the greatest number N can be incorporated onto a given wafer. To maximize the production rate (droplets per s) of each individual droplet generator, while keeping the microfluidic device in a low flow velocity regime where each device produces uniform droplets (i.e., a dripping regime not a jetting regime)26. These design goals and the fluid physics of microfluidic droplet generators provide tradeoff relationships and constraints that guide the VLSDI’s design.
The droplet generators on our device use a flow-focusing geometry (Fig. 1b) and are organized in a ladder architecture13,14,15,16 (Fig. 1f) (Supplementary Fig. 1). In the ladder design, the individual generators are connected in a line along a single set of liquid distribution channels. A two-dimensional array of microfluidic droplet generators is created by connecting multiple rows of droplet generators, also in a ladder geometry, using a single set of arterial lines. In previous work, it has been shown that liquids can be uniformly distributed over N flow-focusing generators (FFGs) connected using a single set of distribution channels, such that the flow resistance in the distribution channel between droplet generators RD is small relative to that of each individual microfluidic droplet generator RDev, resulting in a design rule16:
$$2N(R_{\mathrm{D}}/R_{\mathrm{Dev}}) < 0.01.$$
(1)
In previous work, this design rule has been satisfied for large numbers of microfluidic droplet generators by taking advantage of the hydrodynamic resistance of a microfluidic channel’s R1/h3 dependence on the height h of the microfluidic channel, using delivery channels with a height hD > 100 µm and droplet generators with a height hDev ~10 µm13,14,15,16. Using these design principles for parallelizing microfluidic droplet generators, there is a tradeoff between the number of droplet generators that can be incorporated onto a given chip area N and the maximum throughput ϕ of each individual droplet generator. The origin of this tradeoff comes from two competing goals: a to keep the fluid velocity low at high volumetric flow rates, thus decreasing the device’s fluidic resistance RDev, such that each device produces uniform droplets (i.e., a dripping regime not a jetting regime)26 and b to increase the fluidic resistance RDev of each droplet generator to satisfy Eq. 1 for the largest possible number of droplet generators N.
The transition from dripping to jetting is a well-studied phenomenon, and depends on the capillary number of the continuous phase of each individual flow-focusing droplet generator Cao = µv/σ (where µ is the flow viscosity, v is the velocity, and σ is the interfacial tension), representing the relative importance of surface tension to viscous forces, as well as the Weber number WeD of the dispersed phase, which represents the relative magnitudes of inertial and surface tension forces. When Ca and We <O(100), surface tension force dominates the droplet break-up and uniform droplets are formed via the dripping mechanism26. For the application of high-throughput particle production, the dripping to jetting transition defines a maximum throughput ϕimax for the individual droplet generators.
To overcome the tradeoff between increasing the number of droplet generators N and the maximum flow rate at which each device can be operated ϕimax, we incorporated flow resistors for both the dispersed and continuous phase upstream of the droplet generators (Fig. 2a). Each of these flow resistors has a width less than their height w < h (w = 10 µm, h = 22.5 µm), such that the dependence of resistance on the channel dimensions become RR1/hw3. By adding these flow resistors, the resistance of the individual droplet generators RDev decreases as a function of h at a rate less than that of a traditional parallelization design RDev1/wh313,14,16. Thus, the minimum resistance $$R_{{\mathrm{dev}}}^{{\it{min}}}$$ necessary to incorporate 10,260 FFGs to satisfy Eq. 1 can be achieved at a greater height h (Fig. 2e) and thus enable the droplet generators to operate at a higher volumetric flow rate before transitioning from the dripping to jetting regime. This work, wherein we have decoupled the design of individual flow-focusing droplet generators in massively parallelized chips, builds on earlier reports where flow resistors were used to control the flow rate and decouple droplet generators in chips that contain several droplet generators27,28.
### Validation of design principles
To validate the design principles described above, we analyzed the performance of three VLSDI devices whose channel dimensions satisfy the design rule for 10,260 FFGs (Eq. 1) (Fig. 2a). For each design, the footprint of the individual droplet generator is identical (175 µm × 1475 µm) and 10,260 FFGs are incorporated using the same ladder geometry. Device I: we built this device without the benefit of our decoupling strategy, using upstream flow resistors, with height h = 10 µm and the width w = 20 µm immediately downstream of the droplet generator and w = 10 µm upstream. Device II: here, we incorporated flow resistors and increased the dimensions of the droplet generators to h = 22.5 µm and w = 80 µm immediately downstream of the droplet generator and w = 40 µm upstream in the dispersed and continuous phases, and Device III: we increased the dimensions of the droplet generator further, such that h = 22.5 µm and w = 140 µm immediately downstream of the droplet generator and w = 40 µm upstream in the dispersed and continuous phases. The flow resistors, with width w = 10 µm and height h = 22.5 µm, are incorporated in the same layer as the microfluidic droplet generators. The channel length between the flow resistor and the droplet generator is defined by the low Reynold’s number entrance length (length = 0.06 × Re)29.
We first evaluated these devices by generating hexadecane droplets in water (2 wt% Tween 80). We confirmed that at all flow rates droplets are generated in every one of the 10,260 droplet generators (Supplementary Movie 1). As expected, we found that Device I transitioned from making uniform droplets to polydisperse droplets at a low dispersed flow rate of ϕdmax = 0.35 L h−1 (Fig. 2f), Device II with its increased channel dimensions produced uniform droplets at a maximum dispersed flow rate of ϕdmax = 3.5 L h−1, and Device III with its even larger downstream channel dimensions produced uniform droplets up to a maximum dispersed flow rate of ϕdmax = 7.3 L h−1. For each of the three devices, at flow rates where the device was in the dripping regime, the droplets were highly monodispersed (CV <5%) (Fig. 2g). And, at flow rates where the droplet generator were in the jetting regime, the droplets became highly polydisperse (CV »5%) (Fig. 2h). All three devices transitioned from dripping to jetting at approximately the same Capillary number Ca immediately downstream of the droplet generator orifice (Ca0.08).
We tested the mass production of oil-in-water emulsion by using pressure-driven flow (Supplementary Figs. 4 and 5 for experimental setup) and the size of emulsion droplets could be changed by varying the ratio of the flow rates of the dispersed and continuous phases (Supplementary Movie 2). For example, by changing the dispersed oil phase flow rate over the range of ϕd = 0.5 L h−1 (10 PSI) to 7.3 L h−1 (59 PSI) and the continuous aqueous phase over the range of ϕc = 0.8 L h−1 (12 PSI) to 9.2 L h−1 (62 PSI), the average droplet size could be controlled over a range of d = 22.5–37.5 μm (Fig. 3a, Supplementary Fig. 6). The generated droplets were highly monodisperse at all flow rates, with the coefficient of variation CV <3% (Fig. 3b) at a throughput >1 trillion droplets per h (Supplementary Movie 3). Because our device is fabricated using anodically bonded silicon and glass, it can operate at extremely high pressures (>1000 PSI), making it very well suited for high-throughput processing of highly viscous fluids. We demonstrated the high-throughput production of mineral oil droplets (30 cP), which is an order of magnitude more viscous than the hexadecane (3.0 cP) and dichloromethane (DCM) (0.41 cP). We achieved a dispersed phase throughput of ϕD = 2.2 L h−1 (d = 24.3 µm droplets, CV <3%) (Supplementary Fig. 7 and Supplementary Movie 4). The size of the droplets that could be produced by each of the chips were comparable to that of the orifice dimensions, as has been shown in previous work30. Using our Device III chip, which had orifice dimensions of height h = 22.5 µm and width w = 20 µm, we were able to produce droplets with diameters d = 22.5–37.5 μm. With the Device I chip that had orifice dimensions of height h = 10 µm and width w = 10 µm, we were able to produce droplets with diameters d = 14.5–18.4 μm.
### VLSDI integration of T-junctions
To demonstrate the modularity of the VLSDI architecture, we designed, fabricated, and tested a version of the chip that incorporates 10,260 (285 × 36) T-junction devices (Fig. 4a). We designed flow resistors (width w = 10 µm and height h = 18 µm) upstream of the T-junctions, following the same design rules used on our FFG device (Fig. 4b). We first evaluated these devices by generating hexadecane droplets in water (2 wt% Tween 80) and confirmed that at all flow rates droplets are generated in every one of the 10,260 droplet generators (Supplementary Movie 5). We found that the T-junction device could generate droplets at a maximum dispersed flow rate of ϕdmax = 1.5 L h−1 (Fig. 4c). We mass produced oil-in-water emulsion using this device by using pressure-driven flow and the size of emulsion droplets could be changed by varying the ratio of the flow rates of the dispersed and continuous phases. Droplets were generated with sizes that range from 46.2 µm (ϕd = 0.5 L h−1, ϕc = 4.5 L h−1) (Fig. 4d) to 33.2 µm (ϕd = 1.5 L h−1, ϕc = 12.8 L h−1) (Fig. 4e). The generated droplets were highly monodisperse with the coefficient of variation CV <4% at a throughput of 80 billion droplets per h.
### Large-scale manufacturing of polymer microparticles
To demonstrate the power of this approach for the industrial scale manufacturing of uniform solid particles, appropriate to use as an injectable drug delivery system5,6, we generated micrometer scale PCL microparticles, a biodegradable material approved by the US FDA31. We achieved a production rate of 277 g h−1 (328 billion particles per h) for particles with diameters ranging from 8–15 µm with CV <5% (Supplementary Movie 6, Supplementary Movie 7). Emulsion templates for the solid particles were fabricated on our chip using a dispersed phase of DCM with either 4 wt% (ρE = 53.2 g L−1) or 10 wt% (ρE = 133 g L−1) of PCL. The continuous phase was deionized water with 2 wt/vol% of polyvinyl alcohol (PVA) (Fig. 5a). These two phases were driven through the FFG-based VLDSI chip using pressure-driven flow, collected, and then further processed using roto-evaporation and lyophilization prior to being analyzed. Uniform DCM droplets were generated with sizes ranging from 23 to 42 µm with CV <3% at a production rate of ϕ 2.09 L h−1 (Fig. 5b). After the DCM was extracted (Fig. 5c), spherical, highly monodispersed solid PCL polymer particles remain (Fig. 5d, e).
To test our chip’s ability to produce highly monodispersed particles with a determined particle diameter, we generated four different particle formulations. Using a dispersed phase of DCM with 10 wt% PCL (ρE = 133 g L−1), we generated two pools of particles, one with a diameter of dp = 11.2 µm (CV = 4.4%) (Fig. 5f) and one with a diameter dp = 16.1 µm (CV = 4.4%) (Fig. 5g), beginning with droplet templates with diameters of dT = 23.2 µm (CV = 2.5%) and dT = 35.9 µm (CV = 1.4%), respectively. Using a dispersed phase of DCM with 4 wt% PCL (ρE = 53.2 g L−1), we generated two populations of particles, one with a diameter of dp = 8.4 µm (CV = 4.0%) (Fig. 5h) and one with a diameter dp = 15.2 µm (CV = 3.6%) (Fig. 5i), beginning with droplet templates with diameters dT = 24.5 µm (CV = 2.4%) and dT = 42.7 µm (CV = 2.0%), respectively. The slight increase in CV from droplets to particles came primarily from rare particles that were deformed during post-processing off chip, and we postulate that CV can be further improved by translating our process to a continuous flow liquid-liquid extraction system32. We have produced as much as two gallons of DCM-PCL in water emulsion droplets at a flow rate of ϕd = 2.09 L h−1 with a run time of 75 min without a single device failing. These tests were limited by the size of our pressure vessels—1 gallon for dispersed, 3 gallons for continuous phase (Supplementary Fig. 5).
We compared the measured diameter of our microparticles dp with their expected diameter d3p = dT3 (ρE/ρPCL) based on the diameter of the emulsion template dT, the density of solid PCL microparticle ρPCL = 1.143 g/mL, and the weight per volume concentration of PCL in our emulsions ρE. We considered four separate formulations, generated using a dispersed phase with both ρE = 133 g L−1 (Fig. 5j) and ρE = 53.2 g L−1 (Fig. 5k). Excellent agreement was found between the measured particle diameters and the predicted diameters (R2 = 0.99), suggesting the particles prepared from the emulsions were non-porous. (Fig. 5l).
## Discussion
We present a new platform to mass produce highly uniform microparticles using a highly parallelized microfluidic device. By developing a new 3D architecture, implemented entirely in a monolithically fabricated silicon wafer with glass encapsulation, we can generate polymer solid microparticles at a rate >1000× faster than existing parallelized devices12,21,34. Moreover, if commercialized and implemented using a 12-inch wafer34, a production rate of 10 trillion droplets per h is feasible. In this paper, we systematically studied the effect of flow rate on droplet uniformity and demonstrated that the addition of high aspect ratio flow resistors, allows microfluidic droplet generators that can operate at high production rates to be successfully parallelized without redesign.
Given the high value of therapeutics and the large mass of drug particles that each VLSDI device could produce, we postulate that these devices could be economically feasible for use in the pharmaceutical industry21. We estimate that if these chips were fabricated at-scale (>1000), then each wafer would cost on the order of a hundred dollars. As an example, the current market for anti-retroviral therapy (ART), for the 37 million people living with HIV worldwide, is ~\$24 billion35. To manufacture the entire world’s supply of ART in the form of long-lasting microparticle-based injectables, assuming 200 mg drug administration per day per patient and a 200 g per h production rate and a fraction of active ingredient in each particle of 10–70%, <100 of our chips continuously running 24 h a day could provide the world’s supply. Moreover, due to the low cost and automated use of the VLSDI design, it can be used for point-of-demand pharmaceutical production at locations closer to the patient. An improved solution for automated manufacturing of high-quality pharmaceutical formulations closer to the point of patient care can help address current challenges, such as the shortage of generic injectables36 associated with high manufacturing and storage costs, which currently limit access to essential therapies for sepsis, cancer, and other life threatening conditions.
## Methods
### VLSDI device fabrication
The VLSDI was fabricated at The Singh Center at The University of Pennsylvania (Supplementary Fig. 2). The designs for all layers of the VLSDI chips were designed in DraftSight (Dassault Systems). Four mask layers were designed, droplet makers (Layer-1), underpass channels (Layer-2), vias (Layer-3), and delivery channels (Layer-4). The designs files were written on chrome-coated soda lime photomasks (AZ1500) using Heidelberg 66 plus mask writer with a 10 mm write head. After exposure, the photomasks were developed in MF 319 for 1 min and then in Chrome etchant for 1 min. Finally, the remaining photoresist on the photomask is removed by plasma oxidation for 10 min in Anatech SCE-106-barrel Asher.
All layers of the VLSDI chip were lithographically patterned and etched in a single 4-inch double-side polished silicon wafer using DRIE (SPTS Rapier Si DRIE). In the first etch step, 16 µm of positive photoresist SPR 220.7 is spin-coated on the front side of the silicon wafer, and exposed with the delivery channels (Layer-4) photomask. After exposure, the wafer is left at room temperature for rehydration for 24 h, then it is developed in MF 319 for 1 min. The wafer is then etched in DRIE for an etch depth of 370 µm. The wafer is then cleaned in Acetone and placed in nanostrip for 30 min and cleaned in DI water and dried in N2 gas. In the second etch step, the wafer is flipped and spin-coated with 12 µm SPR 220.7 positive photoresist. The wafer is then exposed with Via layer (Layer-3) photomask, and left at room temperature for 12 h for rehydration, and then developed in MF-319 for 1 min. The wafer is bonded to another silicon wafer (carrier wafer) using crystal bond adhesive. The wafer is etched in DRIE for through vias in the silicon wafer. The wafer is then placed on a hotplate at 65 °C, and the carrier wafer is removed. The wafer is cleaned in acetone, and then nanostrip, and then in DI water, and then dried in N2 gas. In the third etch step, the wafer is spray-coated with 8 µm S1805 resist. The wafer is then exposed with underpass channel (Layer-2) photomask, and then developed in MF 319. The wafer is again bonded to carrier wafer using crystal bond. Underpass channels are etched in DRIE for a 30 µm deep etch. The wafer is then cleaned in acetone, and then nanostrip. In the final etch step, the wafer is spray-coated with 4 µm S1805 resist, and then exposed with the droplet maker photomask (Layer-1), and developed in MF 319 for 1 min. The wafer is placed on a carrier wafer, and then etched for 23 µm deep in DRIE. The carrier wafer is removed by placing it on hotplate at 65 °C. The 3D-etched silicon wafer is then placed in acetone and in nanostrip for 1 h. Borofloat33 glass wafers are laser micromachined with 1 mm holes using IPG Photonics excimer laser. These vias serve as inlets and outlet connections for the VLSDI chip.
The completely etched silicon wafer and plain Borofloat33 glass wafer is cleaned in acetone, Isopropyl alcohol (IPA), and deionzied water (DI) water for 10 min each. The wafers are then dried in a Spin Rinse dryer. The wafers are then placed in nanostrip for half an hour and then placed in piranha solution for 2 h. Then, the wafers are cleaned in water, dried in spin rinse dryer, and then bonded in an EVG 510 wafer bonder by applying 600 V, and a pressure of 600 N for 1 h. Once front side of wafers are bonded, the bonded wafer and laser machined Borofloat33 wafer are kept in Nanostrip for half an hour and then in piranha solution for 2 h. The wafers are then placed in DI water for 12 h to completely remove the the acid solution that may filled in the channels. The wafers are then dried in spin rinse drier and bonded in EVG 510 wafer bonder with pressure of 600 N for 2 h.
Stainless steel compressed tube fittings (1/8 tube OD) form McMaster Carr (52245K609) are bonded to the glass wafer using chemically resistant epoxy from master bond (EP41S-5). The epoxy is left to cure at room temperature for 4 days. PTFE tubes of 1/8 OD were connected to the fittings.
### Experimental setup
Pressure-driven flow is used to conduct the experiments. Nitrogen pressure tanks were connected to 1-gallon and 3-gallon stainless steel pressure vessels (Alloy products). The 1-gallon vessel is used for dispersed phase and the 3-gallon vessel is used for continuous phase. The VLSDI chip is connected to the pressure vessels using PTFE tubings. The VLSDI chip is housed in a custom-built acrylic box and then mounted on to an xyz stage. Inline filters (McMaster Carr: 9816K72) are used to filter debris in the continuous and dispersed phases. An inline flow meter (McMaster Carr: 5084K23) was used to measure the flow rates for the aqueous phase. To test hexadecane droplets in water, hexadecane of viscosity µ = 3 cps was purchased from Alfa Aesar (Stock number: 43283-LW). Tween 80 surfactant was purchased from Fisher Scientific (catalog number: AC278630025). For mineral oil droplets in water, mineral oil with viscosity of µ = 30 cps was purchased from Sigma-Aldrich (product number: M3516).
### Polymer microparticle synthesis
For the generation of polymer microparticles, DCM solvent of viscosity µ = 0.43 cps was purchased from fisher scientific (catalog number: AC406920040), PCL was purchased from Sigma-Aldrich (product number: 440752) and PVA was purchased from Sigma-Aldrich (product number: 363170). The dispersed phase consisted of 10 wt% (133.3 g/L) or 4 wt% (53.2 g/L) PCL in DCM. DCM of 2 L was mixed with the corresponding amount of PCL and mixed on a magnetic stirrer for 1 h. For the continuous phase, 2 wt% of PVA (87% hydrolyzed) was mixed with water at 95 °C for 12 h. Continuous phase of 9 L was used for each experiment. The generated emulsion templates were collected in 4 L glass beakers. To post-process the emulsion templates, 10 mL of emulsion templates were dispersed in 30 mL of water with PVA and placed in Rotovap for 5 min. The precipitated polymer particles were washed thrice in water using centrifuge and then dried in lyophilization unit for 24 h. Scanning electron microscope images were obtained using JEOL 7500F HRSEM and Quanta 600 FEG SEM. An accelerating voltage of 5 kV is used to collect the SEM images.
### Data availability
The authors declare that the data supporting the findings of this study are available within the paper and its supplementary information files.
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Anna, S. L., Bontoux, N. & Stone, H. A. Formation of dispersions using flow focusing in microchannels. Appl. Phys. Lett. 82, 364–366 (2003).
31. 31.
Gref, R. et al. Biodegradable long-circulating polymeric nanospheres. Science 263, 1600–1604 (1994).
32. 32.
Ramstack, J. M. U.S. Patent No. 6,830,737 (U.S. Patent and Trademark Office, Washington, DC, 2004).
33. 33.
Keohane, K., Brennan, D., Galvin, P. & Griffin, B. T. Silicon microfluidic flow focusing devices for the production of size-controlled PLGA based drug loaded microparticles. Int. J. Pharm. 467, 60–69 (2014).
34. 34.
Selvaraja, S. K. et al. Highly uniform and low-loss passive silicon photonics devices using a 300mm CMOS platform. Opt. Fiber Commun. Conf. Exhib. 2014, 1–3 (2014).
35. 35.
Gubernick, S. I., Felix, N., Lee, D., Xu, J. J. & Hamad, B. The HIV therapy market. Nat. Rev. Drug Discov. 15, 451–453 (2016).
36. 36.
Donohue, J. M. & Angus, D. C. National shortages of generic sterile injectable drugs: norepinephrine as a case study of potential harm. J. Am. Med. Assoc. 317, 1415–1417 (2017).
## Acknowledgements
We thank Noah Clay (Director), Meredith Metzlerm, and all QNF Staff at University of Pennsylvania for their help in device fabrication. We also thank Ravi Yellesarapu, Hari Katepalli, Martin F. Hasse, Jessica Liu, and Harsha Kalluru for helpful discussions. We are grateful to Dr. Andrew Tsourkas lab—Ahmad Amirshaghaghi, Kido Nwe, and Elizabeth Higbee—for their help in post-processing of polymer emulsion templates. We would like to acknowledge support from The National Science Foundation (1554200) and Glaxo Smith Kline, in particular from David Lai, Sonja Sharpe, and the entire GSK microfluidics team. D.L. acknowledge the support from NSF CBET 1604536.
## Author information
### Author notes
• Heon-Ho Jeong
Present address: Department of Chemical and Biomolecular Engineering, Chonnam National University, Jeonnam, Yeosu, 59626, Republic of Korea
### Affiliations
2. #### Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, PA, 19104, USA
• Heon-Ho Jeong
• , Daeyeon Lee
### Contributions
S.Y. conceived and performed all designs, fabrication, experiments, and characteriation in this study, as well as prepared the manuscript and figures. S.Y. and H.-H.J. performed hexadecane in water experiments. D.L. and D.I. conceived and oversaw all aspects of this study, and prepared the manuscript.
### Competing interests
David Issadore is the founder of, and currently holds shares of, Chip Diagnostics. The remaining authors declare no competing interests.
## Electronic supplementary material
### DOI
https://doi.org/10.1038/s41467-018-03515-2
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2018-10-15 21:09:05
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https://www.edaboard.com/threads/which-one-will-dominate-systemverilog-or-systemc.2181/
|
# Which one will dominate SystemVerilog or SystemC ?
## What would yoy prefer: SystemC or SystemVerilog?
• ### SystemVerilog
• Total voters
0
Status
Not open for further replies.
#### thecat
##### Member level 1
SystemVerilog
Hello,
All of us have heard about SystemC and that you can write synthesysable code in it as well as advanced tesbenches using the power of C. In the same time, now we learn about SystemVerilog which will contain Verilog2001 + a C extension to help system level design (not for synthesys I think) and verification, as well as writing software for the chip.
In my opinion SystemVerilog will be much better since all of ASIC designers are very used with Verilog and do not have to learn how to design synthesysable RTL code in C. What do you think?
For SystemC there is a tool from S-y-no-psssys CoCentric System Studio which I've heard it's the best (it also have a simulator for the SystemC code). For SystemVerilog I don't know of any tool yet. Do you?
#### joe2moon
##### Full Member level 5
Re: SystemVerilog
thecat said:
Hello,
. For SystemVerilog I don't know of any tool yet. Do you?
From my point of view, SystemVerilog is better than SystemC in verification domain. Because Accellera has already accepted it as the SystemVerilog "standard" :!:
SystemVerilog (next-generation version of the Verilog) was introduced by Co-Design Automation, Inc. And this company has provided the simulator, named "SYSTEMSIM" to run the simulation. It also provides 'SYSTEMEX" to expand Superlog (now SystemVerilog) into the synthesizable subset syntax which can be accepted be the current logic synthesizer, such as $ynopsys' Des!gn Compiler. (You can go to its website www.c0-design.com for more detail.) Just a few weeks ago, the Co-Design Automation, Inc. has been acquired by$ynopsys. Good or bad ? Who knows ? But, one thing can be sure is $ynopsys has admitted the power of the Superlog and decide to support it ! ----------------------------------------------------------------------------------- By the way, if you have experience about running the Verilog simulation with c-model, please reply the topic "VC$' Direct C or M0delsim's c-debug" on System-On-Chip forum to share it !
#### thecat
##### Member level 1
joe2moon,
Just to understand better: Systemlog was developed by Co-Design and then accepted by Accelera who renamed it into SystemVerilog?
I will aslo orefer SystemVerilog, but I see on the polll that there are more people that prefer SystemC.
In my opinion it's good that Syn o psys aquired Co-Design and I wonder if their tool CoCentric System Studio will also support SystemVerilog or they will make a new tool for that language.
#### joe2moon
##### Full Member level 5
Sorry, I have made a mistake.
Co-Design Automation has donated its SUPERLOG Extended Synthesizable Subset (ESS) and Design Assertion language Subset (DAS) to Accellera.
So SUPERLOG is a superset of SystemVerilog.
Status
Not open for further replies.
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2022-07-03 20:29:53
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https://studyadda.com/solved-papers/10th-class/mathematics/solved-paper-mathematics-2017-outside-delhi-set-iii/583
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Solved papers for 10th Class Mathematics Solved Paper - Mathematics 2017 Outside Delhi Set-III
done Solved Paper - Mathematics 2017 Outside Delhi Set-III
• question_answer1) For what value of n, are the nth terms of two A.Ps 63, 65, 67,.... and 3, 10, 17,.... equal ?
• question_answer2) A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius on its circular face. The total height of the toy is 15.5 cm. Find the total surface area of the toy.
• question_answer3) How many terms of an A.P. 9, 17, 25, .., must be taken to give a sum of 636?
• question_answer4) If the roots of the equation $\left( {{a}^{2}}+{{b}^{2}} \right){{x}^{2}}-2(ac+bd)x+\left( {{c}^{2}}+{{d}^{2}} \right)=0$ are equal, prove that $\frac{a}{c}=\frac{c}{d}$.
• question_answer5) Solve for x: $\frac{x-1}{2x+1}+\frac{2x+1}{x-1}=2$, where $x\ne -\frac{1}{2},1$
• question_answer6) A takes 6 days less than B to do a work. If both A and B working together can do it in 4 days, how many days will B take to finish it?
• question_answer7) From the top of a tower, 100 m high, a man observes two cars on the opposite sides of the tower and in same straight line with its base, with angles of depression $30{}^\circ$ and $45{}^\circ$. Find the distance between the cars. [Take $\sqrt{3}=1.732$]
In the given figure, O is the centre of the circle with $AC=24\text{ }cm,\,\,AB=7\text{ }cm$ and $\angle BOD=90{}^\circ$. Find the area of the shaded region.
Study Package
15 10
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2019-03-21 21:46:33
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https://www.mathreference.org/index/page/id/424/lg/en
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General Information
Symbols, relationships
Keywords:
$\left|a\right|$ The absolute value of the number a
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2019-08-21 23:21:28
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http://hackage-origin.haskell.org/package/dep-t-dynamic
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# dep-t-dynamic: A dynamic environment for dependency injection.
[ bsd3, control, library ] [ Propose Tags ]
This library is a companion to "dep-t". It provides "environments" into which you can register record values parameterized by a monad.
The environments are dynamically typed in the sense that the types of the contained records are not reflected in the type of the environment, and in that searching for a component record might fail at runtime if the record hasn't been previously inserted.
The main purpose of the library is to support dependency injection.
## Modules
[Index] [Quick Jump]
• Dep
#### Maintainer's Corner
Package maintainers
For package maintainers and hackage trustees
Candidates
• No Candidates
Versions [RSS] 0.1.0.0, 0.1.0.1, 0.1.0.2, 0.1.1.0 CHANGELOG.md algebraic-graphs (>=0.6 && <0.7), base (>=4.10.0.0 && <5), dep-t (>=0.6.4.0 && <0.7), hashable (>=1.0.1.1), sop-core (>=0.5.0.0 && <0.6), transformers (>=0.5.0.0), unordered-containers (>=0.2.14) [details] BSD-3-Clause Daniel Diaz diaz_carrete@yahoo.com Control head: git clone https://github.com/danidiaz/dep-t-dynamic.git by DanielDiazCarrete at 2022-11-08T20:56:55Z NixOS:0.1.1.0 141 total (12 in the last 30 days) (no votes yet) [estimated by Bayesian average] λ λ λ Docs available Last success reported on 2022-11-08
[back to package description]
# dep-t-dynamic
This library is a compation to dep-t and in particular it complements the Dep.Env module. It provides various types of dependency injection environments that are dynamically typed to some degree: missing dependencies are not detected at compilation time. Static checks are sacrificed for advantages like faster compilation.
• Dep.Dynamic is the simplest dynamic environment, but it doesn't give many guarantees.
• Dep.Checked and Dep.SimpleChecked give greater guarantees at the cost of more ceremony and explicitness. Dep.Checked can only be used with the DepT monad.
## Rationale
In dep-t, functions list their dependences on "injectable" components by means of Has constraints. One step when creating your application is defining a big environment record whose fields are the components, and giving it suitable Has instances.
Environments often have two type parameters:
• One is an Applicative "phase" that wraps each field and describes how far along we are in the process of constructing the environment (the Identity function correspond to a "finished" environment, ready to be used).
• The other is the effect monad which parameterizes each component in the environment.
Usually environments will be vanilla Haskell records. It has the advantage that "missing" dependencies are caught at compile-time. But using records might be undesirable is some cases:
• For environments with a big number of components, compiling the environment type might be slow.
• Defining the required Has instances for the environment might be a chore, even with the helpers from Dep.Env.
### How Dep.Dynamic helps
DynamicEnv from Dep.Dynamic allows registering any component at runtime. Because there aren't static fields to check, compilation is faster.
DynamicEnv also has a Has instance for any component! However, if the component hasn't been previously registered, dep will throw an exception.
### Isn't that a wee bit too unsafe?
Yeah, pretty much. It means that you can forget to add some seldomly-used dependency and then have an exception pop up at an inconvenient time when that dependency is first exercised.
That's where Dep.Checked and Dep.SimpleChecked may help.
### How can the -Checked modules help?
They define wrappers around DynamicEnv that require you to list each component's dependencies as you add them to the environment. Then, before putting the environment to use, they let you check at runtime that the dependencies for all components are present in the environment.
It's more verbose and explicit, but safer. It makes easy to check in a unit test that the environment has been set up correctly.
As a side benefit, the -Checked modules give you the graph of dependencies as a value that you can analyze and export as a DOT file.
Dep.Checked can only be used when your dependencies live in the DepT monad. Use Dep.SimpleChecked otherwise.
## Relationship with the "registry" package
This library is heavily inspired in the registry package, which provides a Typeable-based method for performing dependency injection.
This library is more restrictive and less ergonomic in some aspects, but it fits better with how dep-t works.
Some notable differences with registry:
• Dep.Dynamic only reports missing dependencies when the program logic first searches for them, while registry reports them when calling withRegistry.
• Dep.Checked and Dep.SimpleChecked do allow you to find missing dependencies before running the program logic, but they force you to explicitly list the dependencies of each new component you add to the environment, something that registry doesn't require.
• Unlike in registry, there are no specific warmup combinators. Allocating the resources required by a component must be done in an Applicative "phase" of a Phased environment.
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2023-03-23 19:00:25
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https://physics.stackexchange.com/questions/342672/is-electromagnetism-generally-covariant
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# Is Electromagnetism Generally Covariant?
I'm sure there's a good explanation for the issues leading to my question so please read on:
Classically, we can represent Electromagnetism using tensorial quantities such as the Faraday tensor $F^{\alpha\beta}$ from which (with the help of the metric $g^{\mu\nu}$) we can construct the Maxwell Stress energy tensor $T^{\alpha\beta}$.
This is all well and good and I'v never had an issue with the covariance of electromagnetism, until I was reading about the “paradox” of a charge in a gravitational field.
(the Wiki page is a decent introduction: https://en.wikipedia.org/wiki/Paradox_of_a_charge_in_a_gravitational_field)
Rorlich's resolution to this (referenced on the wiki page) was to calculate that in the free falling frame (but rest frame of the charge) there is no radiation emitted from the charge. Meanwhile on the supported observer's frame (say at rest on the Earth's surface), one would observe radiation being emitted from the charge. It is argued that the coordinate transformation between the two frames is NOT a Lorentz transformation and hence the radiation observed in one frame vanishes upon transformation to the inertial frame.
How does this mesh with general covariance if we can choose a frame (or class of inertial frames) in which the radiation is zero?
Via the same arguments used for the gravitational stress-energy pseudotensor (if a tensor vanishes in one frame it vanishes in all frames) the radiation cannot be a tensorial object I would think.
Note that this argument applies even more simply to the concept of Unruh radiation (i.e we can choose frames in which it is zero). This makes me think Electromagnetism only respects Lorentz covariance which indicates it's not represented by true tensors (but rather some pseudotensor-like object akin to gravitational energy)?
I'm a huge fan of GR and reading about this “paradox” immediately reminded me of the gravitational pseudotensor vanishing in an inertial frame and hence the question. Perhaps there's a simple answer I'm missing.
EDIT: While the question is flagged as a duplicate, My question is more general. I'm curious about how the radiation stress energy tensor can change in non inertial frames. the radiating charge in a gravitational field is just one example, where the radiation appears to exist in one frame and not the other. From quantum physics the unruh and hawking radiations are other examples of radiations appearing in one frame and not in another. I get that a tensorial object shouldn't be able to vanish in one frame. I'm just curious about what we can say about making an arbitrary electromagnetic wave vanish in some frame. since it can be done in some cases, in what other cases can this be done? While we can say that the quantum and classical cases are totally different, note that they both involve radiation vanishing under a noninertial (nonlorenztian) coordinate transformation.
• As any tensor EM is generally covariant, although of course you have to use the covariant version of maxwell's equations. – Slereah Jul 3 '17 at 8:24
• @Slereah I don't doubt it, but how can you transform electromagnetic radiation "away" (granted only in strange cases) and still maintain general covariance of the electromagnetic Stress energy tensor? – R. Rankin Jul 3 '17 at 8:34
• You can't. The Unruh effect is a quantum effect without any analogy to the classical case. In the classical case you can't transform a $0$ EM field into one that isn't (except up to gauge). – Slereah Jul 3 '17 at 8:35
• Possible duplicate of Does a charged particle accelerating in a gravitational field radiate? – tparker Mar 31 '18 at 3:23
• The falling charge paradox has been discussed to death on this site. – tparker Mar 31 '18 at 3:24
General covariance means that the 'form' of the fundamental laws of physics is independent of the coordinate system chosen - it does not mean that observations of observers using different coordinate systems necessarily agree. This statement is quite obvious but important. For example, in the context of SR, two observers might measure different elapsed times and distances for a particular event (say, $t/T$ and $x/X$). Even though x does not equal X and t does not equal T, if both observers construct the invariant proper time $t^2-x^2$ and $T^2-X^2$, they will get the same answer (c=1 here). Thus, the 'law of physics' that is invariant wrt Lorentz transformations is that the proper time elapsed for a particular event will be the same for all inertial observers.
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2019-10-16 18:03:21
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https://socratic.org/questions/what-can-you-add-to-x-2-9x-to-get-a-perfect-square-trinomial
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# What can you add to x^2-9x to get a perfect square trinomial?
May 12, 2018
$20.25$
#### Explanation:
In a perfect square, the equation is $a {x}^{2} + b x + c$, where $\left\mid b \right\mid = 2 a \sqrt{c}$. Therefore, to find $c$, plug in $a$ and $b$.
$\left\mid b \right\mid = 2 a \sqrt{c}$
$\left\mid - 9 \right\mid = 2 \cdot 1 \cdot \sqrt{c}$
$9 = 2 \cdot \sqrt{c}$
$4.5 = \sqrt{c}$
$20.25 = c$
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2021-06-15 08:02:57
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https://www.physicsforums.com/threads/energy-stored-in-a-toroid.112826/
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# Energy stored in a toroid
1. Mar 3, 2006
### Reshma
Calculate the energy stored in a toroidal coil.
The magnetic field in a toroid is give by: $B = \frac{\mu_0 n I}{2\pi r}$
Energy stored is given by:
$$W = {1\over 2\mu_0}\int B^2 d\tau$$
$$W = {1\over 2\mu_0} \frac{\mu_0^2 n^2 I^2}{4\pi^2}\int {1\over r^2}d\tau$$
How do I determine the volume element for a toroid?
2. Mar 3, 2006
### Meir Achuz
The toroid can usually be approximated by a solenoid of length 2\pi R,
with r constant inside the toroid. Then V=(2\pi L)*(\pi a^2), with B constant.
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2017-12-16 15:27:42
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https://napsterinblue.github.io/notes/stats/basics/sampling_distributions/
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# Sampling Distributions
Populations follow distributions, which are likelihoods of values we expect to see. Not to be confused with an individual sample, a sampling distribution is the distribution of a particular statistic (mean, median, etc) across multiple repeated samples of the same size from the same population.
But what effect does sampling have on what you can infer that from your population?
Does the ratio of “4 out of 5 dentists” that recommend something extrapolate to all dentists or would it be more accurate to say “4 out of the 5 dentists that we asked”?
## Ex: Coin Flipping
There’s no real “population” of coin flips– you can perform the flip an infinite number of times, so it’d be impossible to gather all values and calculate the true population parameters. Nevertheless, we know that the distribution of a fair coin is 5050.
But, for the sake of argument, say you only flipped one coin and recored that it landed on heads. That heads result obviously doesn’t extrapolate to every other coin. Or to put it differently, we can’t then infer that the “probability of getting heads is 100%,” based on the result of our one coin-flip.
### Simulating a Lot of Coin Flips
Here, let’s assign the value 1 to any coin that lands on heads, and 0 for tails.
import numpy as np
%pylab inline
Populating the interactive namespace from numpy and matplotlib
So we have a function that will give the sample proportion of a coin that was flipped n times
def coinflip_prop(n):
return np.random.randint(2, size=n).mean()
coinflip_prop(10)
0.40000000000000002
And build a way to sample that flip multiple times
def samples_of_coinflips(samples, nFlips):
return np.array([coinflip_prop(nFlips) for x in range(samples)])
samples_of_coinflips(5, 10)
array([ 0.5, 0.6, 0.3, 0.5, 0.4])
### A Few Samples
We can take a look at what happens to the distribution of results, when we monitor the sample proportion.
For 25 samples of “fair coin flipped 10 times”, we get
tenFlips = samples_of_coinflips(25, 10)
_ = plt.hist(tenFlips, range=(0, 1))
tenFlips.mean(), tenFlips.std()
(0.436, 0.18736061485808589)
And again
tenFlips = samples_of_coinflips(25, 10)
_ = plt.hist(tenFlips, range=(0, 1))
tenFlips.mean(), tenFlips.std()
(0.47599999999999992, 0.110562199688682)
### Big Sample Size
Whereas if we did 1000 flips, 25 times, we see that the average value for all flips is perfectly centered around the value we expect, .5
thousandFlips = samples_of_coinflips(25, 1000)
_ = plt.hist(thousandFlips, range=(0, 1))
thousandFlips.mean(), thousandFlips.std()
(0.50087999999999999, 0.013865987162838432)
Thus, the larger your sample size, the less variability in your sampling distribution
### A Lot of Samples
Conversely, if we only ever look at 10 flips at a time, but repeat that sample thousands of times
tenFlips = samples_of_coinflips(250000, 10)
_ = plt.hist(tenFlips, range=(0, 1))
tenFlips.mean(), tenFlips.std()
(0.4997975999999999, 0.15809452563020646)
we find that the sampling distribution starts to look like a normal curve
### A Lot of Both
Indeed, when we take a ton of samples, each sampling a lot of coin flips, we can see that the distribution of the sample mean follows a normal distribution
import pandas as pd
hundredFlips = samples_of_coinflips(250000, 100)
pd.Series(hundredFlips).plot(kind='density')
hundredFlips.mean(), hundredFlips.std()
(0.50011791999999988, 0.049848230609256333)
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2021-01-16 05:31:28
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https://api-project-1022638073839.appspot.com/questions/how-do-you-integrate-2x-x-2-25-using-partial-fractions
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# How do you integrate (2x)/(x^2-25) using partial fractions?
Sep 2, 2016
This could be integrated by substitution, but the question specifies partial fractions, so see below.
#### Explanation:
Factor the denominator:
${x}^{2} - 25 = \left(x + 5\right) \left(x - 5\right)$
solve for $A$ and $B$
$\frac{A}{x + 5} + \frac{B}{x - 5} = \frac{2 x}{{x}^{2} - 25}$
$A \left(x - 5\right) + B \left(x + 5\right) = 2 x$
$A x - 5 A + B x + 5 B = 2 x + 0$
$A + B = 2$
$- 5 A + 5 B = 0$
$A = B = 1$
$\int \frac{2 x}{{x}^{2} - 25} \mathrm{dx} = \int \left(\frac{1}{x + 5} + \frac{1}{x - 5}\right) \mathrm{dx}$
$= \ln \left\mid x + 5 \right\mid + \ln \left\mid x - 5 \right\mid + C$
$= \ln \left\mid {x}^{2} - 25 \right\mid + C$
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2021-10-21 01:24:55
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https://www.physicsforums.com/threads/simple-rotational-motion-problem.68055/
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# Simple Rotational Motion Problem
1. Mar 20, 2005
### veraction
Ok, this problem has been driving me crazy. I was thinking that the force is zero since the magnitude of the angular acceleration is zero, however that is not the case.
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2017-10-20 09:16:44
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https://www.shaalaa.com/question-bank-solutions/if-b-c-are-mutually-perpendicular-show-that-c-b-0-converse-true-physics_66286
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Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
# If → a , → B , → C Are Mutually Perpendicular, Show that → C × ( → a × → B ) = 0 is the Converse True? - Physics
Sum
If $\vec{A} , \vec{B} , \vec{C}$ are mutually perpendicular, show that $\vec{C} \times \left( \vec{A} \times \vec{B} \right) = 0$ Is the converse true?
#### Solution
Given: $\vec{A} , \vec{B} \text{ and }\vec{C}$ are mutually perpendicular. $\vec{A} \times \vec{B}$ is a vector with its direction perpendicular to the plane containing $\vec{A} \text{ and } \vec{B}$
∴ The angle between $\vec{C} \text{ and } \vec{A} \times \vec{B}$ is either 0° or 180°.
i.e., $\vec{C} \times \left( \vec{A} \times \vec{B} \right) = 0$ However, the converse is not true. For example, if two of the vectors are parallel, then also, $\vec{C} \times \left( \vec{A} \times \vec{B} \right) = 0$
So, they need not be mutually perpendicular.
Concept: What is Physics?
Is there an error in this question or solution?
#### APPEARS IN
HC Verma Class 11, Class 12 Concepts of Physics Vol. 1
Chapter 2 Physics and Mathematics
Exercise | Q 16 | Page 29
Share
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2023-03-31 19:31:22
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https://rpg.stackexchange.com/questions/102300/stacking-ki-pool-from-multiple-sources-in-pathfinder?noredirect=1
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# Stacking Ki pool from multiple sources in pathfinder
Does Ki pool Stack from multiple sources in pathfinder? I'm looking to multiclass monk/ninja and want to know if I can combine those two ki pools into one.
• Depends what sources. Got any in mind? Jun 26, 2017 at 22:19
• Monk and Ninja @mxyzplk Jun 28, 2017 at 4:55
## Whether or not the ki pools stack depends on the sources.
### And some of them "stack" multiple pools, but get a smaller total size.
By default, even if two abilities or class features have the same name, they do not stack. If something does stack, then it must say so in the ability description. There are multiple possibilities of stacking ki pools in Pathfinder; they only stack if at least one specifically says they stack.
• The Ninja's ki pool (level 2+) stacks with other ki pools to form a shared pool. Yay! However, for calculating the pool size, you only include one ability score modifier.
If the ninja possesses levels in another class that grants points to a ki pool, ninja levels stack with the levels of that class to determine the total number of ki points in the combined pool, but only one ability score modifier is added to the total. The choice of which score to use is made when the second class ability is gained, and once made, the choice is set. The ninja can now use ki points from this pool to power the abilities of every class she possesses that grants a ki pool.
• The Monk's ki pool (level 4+) does not mention stacking.
• The unchained Monk's ki pool (level 3+) does not mention stacking either.
• The Enlightened Paladin archetype gets a ki pool at level 4 that functions like the monk ability, which does not mention stacking, so it does not stack either.
• The Sacred Fist Warpriest archetype also gains a monk-like ki pool at level 7. No stacking.
• The Ki Pool rogue talent is bizarre. It gives a ki pool feature similar to the ninja's feature, yet it does not scale with level. It "stacks" in the sense that you share it between classes, but the size of the pool doesn't actually increase. I explain this below.
Here are some examples of how different combinations stack with each other. Below are some class combinations that can share ki pools between class features.
• Ninja (Lvl 2+) / Monk (Lvl 4+). Or, Ninja (Lvl 2+) / Unchained Monk (Lvl 3+). The pools stack, and scale with both class levels:
Total ki points = (Ninja level + Monk level)/2 + (either Cha or Wis modifier)
• Ninja (Lvl 2+) / Enlightened Paladin (Lvl 4+). The pools stack, and scale with both class levels:
Total ki points = (Ninja level + Paladin level)/2 + (Cha modifier)
• Ninja (Lvl 2+) / Sacred Fist (Lvl 7+). The pools stack, and scale with both class levels:
Total ki points = (Ninja level + Warpriest level - 3)/2 + (either Cha or Wis modifier)
So far, so good. Now to explain this rogue talent...
• Monk (Lvl 4+) and Ki Pool Rogue talent. You share the pool between both classes, but you only count the Wisdom modifier once, and so the pool size only increases with monk level:
Total ki points = (Monk level)/2 + (Wis modifier)
• Ninja (Lvl 2+) and Ki Pool Rogue talent. Once again, the pools are shared, but only increase with ninja level, so you don't actually gain any more points than a ninja would:
Total ki points = (Ninja level)/2 + (either Cha or Wis modifier)
### Does that mean it's better to not stack?
That depends on the character build you want. If you have two sources of ki pool and neither of them stack, then you can consider them to be completely separate abilities. In other words, you have a tradeoff between:
• One shared pool between classes, counting only one ability score modifier; or
• Multiple separate pools, each including an ability score modifier.
For example, a Monk + Sacred Fist warpriest would have two completely separate ki pools. Each pool would include your Wisdom modifier and half of the corresponding class level. However, although they are both called "ki pool", they are completely separate; the points in your Sacred Fist's pool cannot be used for Monk abilities, and vice versa.
• Wish I could +2 Jun 28, 2017 at 5:05
According to the Paizo FAQ, abilities gained from multiple sources don't stack unless they specifically say they stack.
The question is about Channel Energy, but it states a generalized point.
Channel Energy: If I have this ability from more than one class, do they stack? No—unless an ability specifically says it stacks with similar abilities (such as an assassin's sneak attack), or adds in some way based on the character's total class levels (such as improved uncanny dodge), the abilities don't stack and you have to use them separately. Therefore, cleric channeling doesn't stack with paladin channeling, necromancer channeling, oracle of life channeling, and so on.
http://paizo.com/paizo/faq/v5748nruor1fm#v5748eaic9o80
So you'd basically get multiple separate Ki pools that each do different things.
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2022-12-02 17:40:12
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https://pytorch.org/docs/stable/generated/torch.optim.lr_scheduler.ConstantLR.html
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# ConstantLR¶
class torch.optim.lr_scheduler.ConstantLR(optimizer, factor=0.3333333333333333, total_iters=5, last_epoch=- 1, verbose=False)[source]
Decays the learning rate of each parameter group by a small constant factor until the number of epoch reaches a pre-defined milestone: total_iters. Notice that such decay can happen simultaneously with other changes to the learning rate from outside this scheduler. When last_epoch=-1, sets initial lr as lr.
Parameters
• optimizer (Optimizer) – Wrapped optimizer.
• factor (float) – The number we multiply learning rate until the milestone. Default: 1./3.
• total_iters (int) – The number of steps that the scheduler decays the learning rate. Default: 5.
• last_epoch (int) – The index of the last epoch. Default: -1.
• verbose (bool) – If True, prints a message to stdout for each update. Default: False.
Example
>>> # Assuming optimizer uses lr = 0.05 for all groups
>>> # lr = 0.025 if epoch == 0
>>> # lr = 0.025 if epoch == 1
>>> # lr = 0.025 if epoch == 2
>>> # lr = 0.025 if epoch == 3
>>> # lr = 0.05 if epoch >= 4
>>> scheduler = ConstantLR(self.opt, factor=0.5, total_iters=4)
>>> for epoch in range(100):
>>> train(...)
>>> validate(...)
>>> scheduler.step()
get_last_lr()
Return last computed learning rate by current scheduler.
load_state_dict(state_dict)
Parameters
state_dict (dict) – scheduler state. Should be an object returned from a call to state_dict().
print_lr(is_verbose, group, lr, epoch=None)
Display the current learning rate.
state_dict()
Returns the state of the scheduler as a dict.
It contains an entry for every variable in self.__dict__ which is not the optimizer.
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2022-09-28 23:19:08
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https://www.khanacademy.org/math/cc-third-grade-math/cc-3rd-fractions-topic/cc-3rd-equivalent-fractions-number-line/e/equivalent-fraction-models
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# Equivalent fractions on the number line
### Problem
Use the number line to find a fraction that is equivalent to 1.
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2017-09-21 07:12:07
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http://mathhelpforum.com/calculus/97879-help-please.html
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# Math Help - Help please?
1. ## Help please?
Can someone please tell me what 1/x dx is using antiderivatives?
2. $\int \frac{1}{x} ~dx = ln(x)+c$
3. thank you for your help
4. Correction in red:
Originally Posted by pickslides
$\int \frac{1}{x} ~dx = ln{\color{red}|}x{\color{red}|}+c$
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2015-11-29 19:22:15
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https://www.physicsforums.com/threads/big-o-notation-proofs.336620/
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# Homework Help: Big O Notation Proofs
1. Sep 12, 2009
### needhelp83
O-notation
– O(g(n)) = { f (n) : there exist positive constants c and n0 such that
0 ≤ f (n) ≤ cg(n) for all n ≥ n0}
– g(n) is an upper bound of f(n), may not be tight
Ω-notation
– Ω(g(n)) = { f (n) : there exist positive constants c and n0such that
0 ≤ cg(n) ≤ f(n) for all n ≥ n0}
– g(n) is an lower bound of f(n), may not be tight Θ-notation
– Θ(g(n)) = { f (n) : there exist positive constants c1, c2, and n0such
that 0 ≤ c1g(n) ≤ f (n) ≤ c2 g(n) for all n ≥ n0}
– We write f (n) = Θ(g(n)) instead of f (n) ∈Θ(g(n))
– g(n) is an asymptotically tight bound for f(n)
1) $$n^{2} + 2n^{3} = O(n^{3})$$
$$f(n) = 2n^{3}, g(n)= (n^{3})$$
if $$f(n) \in O(g(n))$$
$$0 \leq f(n) \leq cg(n)$$
$$0 \leq 2n^{3} \leq c*n^{3} for \ n \geq 0$$
$$any \ c \geq 2 \ \ f(n) \leq g(n) \ for \ n \geq 0$$
2)$$2n + n^{2} \neq \Omega (n^{3})$$
$$f(n)=n^{2}, g^{3}$$
$$0 \leq cg(n) \leq f(n)$$
$$0 \leq c*n^{3} \leq n^{2} \ for \ n \geq 0$$
$$any \ c \leq \frac{1}{n} \ for \ n \geq 0$$
3) $$ln \ n = \Theta (lg \ n)$$
f(n) = Θ(g(n))
C1lg(n) ≤ ln(n) ≤ C2lg(n)
(1)ln(n) ≤ C2lg(n)
C2 = 1
ln(n) ≤ lg(n)
n = 1
e^x = 1 n^x = 1
x = 0
holds for all n > 0
(2)C1lg(n) ≤ ln(n)
C1 = ½
½lg(n) ≤ ln(n)
n = 1
½(0) ≤ 0
holds for all n > n/2
I am a beginner and looking for feedback. Are there any mistakes I have made or changes that need to be made when proving these?
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2018-07-18 07:25:46
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http://html.rhhz.net/ieee-jas/html/2017-4-668.htm
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IEEE/CAA Journal of Automatica Sinica 2017, Vol. 4 Issue(4): 668-676 PDF
A Facial Expression Emotion Recognition Based Human-robot Interaction System
Zhentao Liu, Min Wu, Weihua Cao, Luefeng Chen, Jianping Xu, Ri Zhang, Mengtian Zhou, Junwei Mao
School of Automation, China University of Geosciences, and with the Hubei Key Laboratory of Advanced Control and Intelligent Automation for Complex Systems, Wuhan 430074, China
Abstract: A facial expression emotion recognition based human-robot interaction (FEER-HRI) system is proposed, for which a four-layer system framework is designed. The FEERHRI system enables the robots not only to recognize human emotions, but also to generate facial expression for adapting to human emotions. A facial emotion recognition method based on 2D-Gabor, uniform local binary pattern (LBP) operator, and multiclass extreme learning machine (ELM) classifier is presented, which is applied to real-time facial expression recognition for robots. Facial expressions of robots are represented by simple cartoon symbols and displayed by a LED screen equipped in the robots, which can be easily understood by human. Four scenarios, i.e., guiding, entertainment, home service and scene simulation are performed in the human-robot interaction experiment, in which smooth communication is realized by facial expression recognition of humans and facial expression generation of robots within 2 seconds. As a few prospective applications, the FEERHRI system can be applied in home service, smart home, safe driving, and so on.
Key words: Emotion generation facial expression emotion recognition (FEER) human-robot interaction (HRI) system design
Ⅰ. INTRODUCTION
To make it easier and more natural to interact with robots, people put forward new demands to human-robot interaction (HRI) [1], [2]. It is hoped that robots can recognize human's facial expressions, understand emotions and give appropriate response [3]-[6]. Emotional intelligence robots have attracted great attention in recent years. Research on emotional robot involves many fields such as computer science and psychology [7]-[9]. However, current research is still in preliminary stage. There are only a few intelligent service systems with emotion. Mascot Robot System including five eye robots was proposed in [10]-[12], in which the eye robot can achieve a friendly interaction with human by eye-rolling and speech recognition. A face robot called KAPPA is introduced in [13] which can recognize emotions by facial expressions and generate six basic emotions. Minotaurus robot system is introduced in [14], where a smart human-robot interaction environment is built and the robot can interact with users by gestures, speech, and facial expressions. Although several human-robot interaction systems involve the emotion of robots, only a few researchers study on both emotion recognition and emotion expression by robots to facilitate smooth communication between humans and robots.
A facial expression emotion recognition based human-robot interaction (FEER-HRI) system is proposed, which is a sub-system of multi-modal emotional communication based human-robot interaction (MEC-HRI) system [15]. There are three NAO robots, two mobile robots, Kinect, workstation, server, eye tracker, portable electroencephalograph (EEG), and other intelligent devices in the MEC-HRI system. FEER-HRI system is designed primarily for two targets: one is the robot's abilities to recognize human emotions based on facial expressions, the other is the robot's abilities to generate emotions for emotional communication with humans instead of unemotional communication as the traditional one.
The operation processes of system consist of three steps. Firstly, the robot collects the human face image data through the Kinect and transmits it to the workstation. Secondly, the facial expression recognition method based on extreme learning machine (ELM) is used to recognize users' emotions and then the system generates robots' facial expressions for adapting to users. Thirdly, system transmits the affective control signal to the robot and the robot can respond to users by expressing its own facial expressions which are made up of some basic cartoon symbols.
The remainder of this paper is organized as follows. The architecture of MEC-HRI system and scenarios design are presented in Section Ⅱ. Facial expression feature extraction method is briefly introduced in Section Ⅲ. Experiment setup and experiment results are given in Section Ⅳ.
Ⅱ. ARCHITECTURE OF MEC-HRI SYSTEM
MEC-HRI system can realize multi-modal emotional communication through speech, facial expressions, body gestures, etc. Emotional robots and emotional information acquisition equipment/sensors are connected to a workstation, in which emotion recognition algorithms for facial expressions are embedded. In MEC-HRI system, hardware devices can be extended and algorithms can be improved.
A. Hierarchical Structure of MEC-HRI System
The hierarchy of MEC-HRI system is divided into four layers. From bottom to top, there are hardware layer, physical interface layer, data processing layer, and application layer, as shown in Fig. 1. The hardware layer is used to capture humans' emotional signals and express robots' emotions, in which the sensor module is responsible for data collection and pretreatment, as well as actuator module is responsible for the interaction with users. For instance, the high-resolution camera is used to capture real-time pictures of facial expressions and body gestures; microphones are used to collect speech signals; eye tracker and other motion tracking devices are used to acquire motion information. This module can be extended based on specific system requirements. For example, wearable equipment such as smart glove, intelligent heart-rate belt, and EEG can be used to detect physiological data of human body for emotion recognition. The actuator module is used to control the robot to interact with user based on the emotional analysis and behavior instruction from upper layers. Many interaction equipments like NAO, mobile robot, facial expression interactive software, and mobile terminal can be used to extend this module.
Physical interface layer provides the channel for data transmission, which is the bridge between software and hardware in MEC-HRI system. The network module is responsible for the initialization of network and the communication with each module. The data processing layer is the key part of system which can achieve following functions.
1) Correlation analysis and feature extraction of speech, facial expressions, gestures, and physiological signals.
2) Multi-modal information fusion based on the two-layer fusion structure, i.e., feature level and decision level.
3) In addition to emotion recognition, recognizing human's emotional states and other deep cognitive information during the interaction.
4) Getting the operating instruction of robot by multi-robot behavioral adaption mechanism.
Interactive application layer is the highest layer of the system, which provides a variety of interactive ways, such as speech, facial expressions, gestures, and multi-modal interaction. Besides, there are two interactive objects that the user can choose, one is the robot, and the other is virtual robot via a graphical interface.
B. Scenario Design
Four scenarios including guiding, entertainment, home service, and scene simulation are designed as shown in Fig. 2.
In guiding region, there is a mobile robot with functions of guiding and interacting with users effectively. When users enter robots' vision, robots welcome users according to their historical data. For a new user, the robot will give a happy expression by LED screen, and then talk with users and guide them.
In entertainment region, there are two NAO robots which can play a finger guessing game with users. Camera of NAO robot captures pictures and transmits it to the workstation, by which users' gestures are recognized and the game result is judged. NAO robot expresses emotions according to game results by speech, facial expressions, and gestures. Different users can also play this game with each other, and NAO acts as a spectator. NAO will cheer for the winner and encourage the loser.
In home service region, there are three NAO robots. Emotional communication between multi-human and multi-robot can be carried out here. NAO robots can provide services for the old, the disabled, and children. When an elder is watching TV, MEC-HRI system is monitoring their health condition through wearable sensors and talk to them. In addition, Kinect can recognize children's gestures and sign language of the disabled.
In scenario simulation region, scenarios can be simulated, e.g., coffee bar. Users can drink coffee here and talk with robots casually. Robots recognize users' emotions through multi-modal information. Meanwhile, MEC-HRI system can change the background music to adjust the atmosphere.
Ⅲ. FACIAL EXPRESSION EMOTION RECOGNITION
In order to communicate with users, FEER-HRI system needs to collect and analyze facial information. Facial expression images of the user can be acquired through Kinect equipped in the mobile robot, which are transmitted to the mobile robot through USB port. Then, through the WLAN, they link to the workstation for image processing. Finally, the users' emotional states can be obtained, and system will adapt to users in accordance with their emotions. Considering different cultural backgrounds and people's subjective feelings for understanding emotions, facial expressions are divided into six categories, including happy, angry, surprise, fear, disgust, and sad [16]. Furthermore, different classifications of facial expressions are compared to each other, from which above six categories of facial expressions are thought to be more universal [17]. Therefore, facial expressions are divided into seven basic categories in this paper, i.e., happy, angry, surprise, fear, disgust, sad, and neutral. An approach of facial expression recognition using multi-feature extraction can promote the accuracy rate of classifier, which includes three parts [18]. This process is summarized in Fig. 3. The main steps are feature collection, feature extraction, and emotion recognition.
Download: larger image Fig. 3 Process of emotion recognition by facial expression.
Firstly, images are preprocessed using face detection [19] and segmentation. Then, facial images are divided into three regions of interest (ROI), i.e., eyes, nose, and mouth. These three regions contain most of face emotion features. Secondly, facial expression features are extracted using 2D-Gabor filter [20] and uniform LBP operator [21]. 2D-Gabor filter is robust against illumination change and face pose rotation of human face image. Moreover, 2D-Gabor filter has less calculation and strong real-time performance, which can extract local features of different scales and different directions. Fig. 4 shows the real part of 2D-Gabor filters at five scales and eight directions. When face image is filtered by these 2D-Gabor filters, the energy of other texture features is suppressed, and only the texture features corresponding to featured frequency are passed smoothly. The texture features are composed of all 2D-Gabor filters' output. Fig. 5 shows amplitude spectrum of the segmented eye image after 2D-Gabor feature extraction.
Download: larger image Fig. 4 The real part of the 2D-Gabor filters at five scales and eight directions with the following parameters: $\sigma =2\pi$, ${{k}_{_{\max }}}=p/2$, and $f=\sqrt{2}$.
The LBP can describe image texture features, which is used in image processing. The LBP operator compares pixels with their nearby pixels and the results are stored as binary numbers. It is one of the best performing methods in texture features description. In addition, its computational efficiency is high, and it is robust against image offset and the light change. Face often moves and face image is easily affected by the light in each direction. Therefore, LBP operator is very appropriate for feature extraction of facial images. Moreover, LBP operator can well describe local features since face can be seen as the composition of local features.
However, basic LBP operator will produce too many kinds of binary patterns. As a result, the histogram of LBP is too sparse which cannot effectively describe the texture feature [22]. Excessive binary patterns will occupy more storage space and reduce the computational efficiency. To solve this problem, uniform LBP operator is used, which can reduce pattern number from $2^p$ to $p(p-1)+2$. It significantly improves the performance of LBP. Fig. 6 shows the face image processed by uniform LBP operator.
2D-Gabor cannot capture the subtle changes in each direction and frequency of the texture feature [21]. LBP operator can extract local texture features. The combination of these two methods can effectively integrate the advantages of both, which not only extracts features from multi-scale and multi-direction but also preserves local features of face image. In addition, it reduces the dimension of the data so that computational efficiency is improved. These two methods also make up for their deficiencies. The filtering process of 2D-Gabor wavelet transform can effectively reduce the influence of noise on the LBP operator, and uniform LBP operator enhances the local texture characteristics of the 2D-Gabor wavelet transform.
Figs. 7 and 8 show overall processes of facial emotion recognition. Face features are extracted using the method combining 2D-Gabor and LBP. Furthermore, principal component analysis (PCA) is used to reduce redundant features which can increase the computational efficiency. The processed facial feature is divided into two parts. One is for training and the other one is for testing. Since emotions are divided into seven categories, a multiclass classifier ELM is used for emotion recognition. In our previous works [23], it was verified that ELM which is used in facial expression recognition has its own characteristics compared with other multi-class classification methods. The computing speed of ELM is fast, the time of modeling and facial expression recognition is usually less than 0.1 second. Meanwhile, the recognition rate of facial expression is usually above 80%. As a result, ELM is adopted for FFER-HRI system, which can meet the requirement of real-time facial recognition.
Ⅳ. EXPERIMENTS ON FEER-HRI SYSTEM
The proposed facial recognition algorithm is applied in FEER-HRI system successfully and FEER-HRI system can recognize users' emotions timely and accurately.
A. Experimental Setup
MEC-HRI system consists of three NAO robots, two mobile robots, Kinect, eye tracker, two high-performance computers (i.e., a server and a workstation), portable EEG, wearable sensing devices as well as data transmission, and network-connecting devices. The topology structure of MEC-HRI system is shown in Fig. 9.
Two high-performance computers in the system are configured as HP Z840 workstations which consist of two NVIDIA Tesla K40 accelerator card. It can achieve double precision floating point 9Tflops and reach the best configuration of $1:1$ (CPU: GPU) which has faster computing speed compared to general computer. The advantages of workstations can enhance the efficiency of affective computing, and reduce the computing time to ensure smooth human-robot interaction.
When MEC-HRI system is built up, both NAO robots and mobile robots are connected to the wireless router via WIFI. The eye tracker, Kinect, and wearable sensing devices access to the mobile workstation that is responsible for capturing emotional information and controlling devices via USB interface and WIFI. Mobile workstation and wireless router are connected to the server and workstation via a hub. NAO robots can capture video images and audio data for emotion recognition of humans. In turn, NAO robots can express its own emotions by using speech, body gestures, and movement according to human emotions.
Fig. 10 shows the structure of Mobile robot, which is mainly composed of an industrial personal computer (IPC), a Kinect, a touch screen, a $16\times 32$ LED screen, and so on. The mobile robot can move around in four directions and chassis of it is equipped with some laser sensors aimed at avoiding obstacles. The speech synthesis software and microphone in IPC are used for mobile robots' emotion expression. Kinect is a 3D somatosensory camera that can capture dynamic image and conduct image recognition and voice recognition. It has RGB camera and depth camera, which provide facial recognition and 3D motion capture. The resolution of the camera is $640$ $\times$ $480$ and it outputs one face image every 4 ms-8 ms. It works well through a range limit of 1.2 m-3.5 m distance.
The LED screen is the device which can display facial expressions of mobile robot. Compared with human's seven basic expressions, nine kinds of facial expressions, i.e., angry, disgust, fear, neutral, sad, surprise, doubtful, and pitiful are designed for mobile robot. These expressions can fully reflect the emotional state of robots in the process of human-robot interaction. In the FFER-HRI system, facial expressions of robots are represented by some simple cartoon symbols which can express the expression vividly and be easily understood by human.
Fig. 11 shows nine facial expressions displayed on LED screen. Each pattern in the Fig. 11 corresponds to a facial expression, for example, a pattern with two opposite triangles represents anger; a pattern with two love images represents happiness; a pattern with two question marks represents doubt; a pattern with two symmetrical check marks represents sadness; a pattern with two inverted U-shape represents fear. In addition, two extra expressions, i.e., doubtful and pitiful, are added based on human's seven basic expressions. These two expressions are designed according to the characteristics of the robot in human-robot interaction. When the robot cannot recognize users' emotions, it can display the doubtful expression for adapting to users. When users are angry with the robot, the robot can display the pitiful expression in order to gain users' sympathy.
When system is running, Kinect and NAO robots can capture users' facial images. First of all, these image data are transmitted to the server where they are segmented into three ROI, i.e., eyes, nose and mouth. These parts contain most of the facial emotion information. After that, the method of feature extraction and expression recognition, i.e., PCA, the combination of 2D-Gabor and LBP, and ELM classifier are employed to get the final emotion state. Then, the system will make an appropriate affective decision according to the user's emotion. Finally, the server will send certain control instruction to the robots and sensors. As a result, the robot and sensors can make some emotional feedback for adapting to users. For example, mobile robot can express emotion by speech, LED display, and its movement.
B. Classification of Facial Expression
The standard face emotion corpus used in this experiment is JAFFE [24]. Fig. 12 shows the some images of this corpus. Seven emotions are included in this corpus, i.e., happy, angry, sad, surprise, neutral, disgust, and fear.
The method combining 2D-Gabor and uniform LBP is used to extract facial emotion features. As shown in Fig. 13, 800 facial features are extracted from every face image. The dimension of the features is too large, which will take a lot of computing resources. To solve this problem, PCA is adopted which can reduce features dimension from 800 to 96. These representative features are input into an ELM classifier [25] to obtain the final emotion results.
Download: larger image Fig. 13 Gray level histogram of every segmented region.
C. Application on the Mobile Robot
In order to make mobile robot recognize human emotions, feature extraction methods and classification algorithm in Section Ⅲ should be applied in it. C++ is used to program the Kinect for capturing real-time facial image. Features extraction methods and classification algorithm are realized in MATLAB. In order to combine them, MATLAB program is compiled into dll file which can be called by C++ programming. Robots are connected to workstation via WLAN. By connecting IP and port, we can operate mobile robot from another computer. Fig. 14 shows the operation interface of MEC-HRI system which can connect and control all devices in the system. This operation interfaces show some interaction information between humans and robots. For example, the mobile robot module in Fig. 15 displays real-time images captured by the Kinect equipped in the robot and shows the recognition result of facial expression.
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2020-09-28 22:00:54
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https://physics.stackexchange.com/questions/207991/does-the-poisson-bracket-f-g-have-any-meaning-if-neither-of-f-or-g-is/208021
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# Does the poisson bracket $\{f,g\}$ have any meaning if neither of $f$ or $g$ is the system's Hamiltonian?
Say one has a mechanical system with hamiltonian $H$, and two other arbitrary observables $f,g$. $H$ is super useful since $\{H, \cdot\} = \frac{d}{dt}$. But does $\{f,g\}$ give any useful information in and of itself?
I'm currently going through "Lectures on Quantum Mechanics for Mathematics Students" by Faddeev and Yakubovskii (with not terribly much background in classical physics).
• A good answer to this question could point out the connection between the Poisson bracket and the geometry of the vector fields associated to $f$ and $g$. A discussion of the physical relevance of those vector fields would be very welcome. – DanielSank Sep 20 '15 at 6:36
• – Emilio Pisanty Sep 20 '15 at 13:13
• Related: Understanding Poisson brackets – ACuriousMind Sep 20 '15 at 13:27
Well, $\{f, \cdot \}$, similarly to $\{H,\cdot\}$, computes the derivative of the argument $\cdot$ with respect to the action of the one-parameter group of canonical transformations generated by $f$ (see the note below for the complete definition) $$\phi_a^{(f)} : F \to F\:,\quad a \in \mathbb R\:,$$ satisfying $$\phi_a^{(f)} \circ \phi_b^{(f)}= \phi_{a+b}^{(f)}\:, \quad\phi_{-a}^{(f)} = (\phi_a^{(f)})^{-1} \:, \quad \phi_0^{(f)}= id$$ Here $F$ is the space of phases. Indeed it holds (see below) $$\{f,g\}(x)= \frac{d}{da}|_{a=0} g(\phi_a^{(f)}(x))\:,\tag{1}$$ where $g: F \to \mathbb R$ is sufficiently regular.
Therefore, $\{f,g\}(x)=0$ everywhere in $F$ means that $g$ is invariant under the group of transformations generated by $f$ (the fact that the derivative is computed at $a=0$ is immaterial, as the group structure implies that the derivative vanishes for every value of $a$).
In particular $\{f,H\}=0$ means that the Hamiltonian function is invariant under the action generated by $f$. This fact is remarkable because it gives rise to the Hamiltonian version of Noether theorem.
As a matter of fact, since $\{H,f\}=- \{f,H\}=0$, invariance of $H$ under the action of $f$ is equivalent to the invariance of $f$ under the action of $H$ (i.e. under time evolution). In other words,
$H$ is invariant under the action of the one-parameter group of canonical transformations generated by $f : F \to \mathbb R$ if and only if $f$ is constant along the motion of the physical system.
Finally, let $X_h$ be the vector field over $F$ tangent to the orbits of the curves $\mathbb R \ni a \mapsto \phi_a^{(h)}(x)$ for every $x\in F$ (this vector field is fully defined in the note below). Since $$[X_f,X_g]=X_{\{f,g\}} \tag{1'}\:,$$ $\{H,f\}=0$ implies that, if $t \mapsto x(t)$ solves Hamilton equations, $t \mapsto \phi^{(f)}_a(x(t))$ does for every value of $a$. In other words, $\{H,f\} =0$ also implies that the group of canonical transformations generated by $f$ transforms motions of the physical system to motions of the system as well.
(a) $\{f,g\}=0$
implies, via (1') and using $X_0=0$, that
(b) the action of the group of transformations on the states of the system (points in $F$) and on observables (real valued functions on $F$) generated by $f$ and the one generated by $g$ commute.
Since $X_h=X_l$ if and only if $h=l + const.$, the two statements (a) and (b) are not completely equivalent. This non-equivalence turns out to be fundamental in quantization procedures since it permits to deal with CCR and central extensions of groups.
NOTE regarding used definitions
[1] if $\omega$ is the symplectic form on $F$, the Hamiltonian field associated to $f\in C^\infty(F,\mathbb R)$ is defined as the unique vector field, $X_f$, such that $$\omega_x(X_f,u)= \langle df_x, u\rangle \tag{2}$$ for every vector $u \in T_xF$. $X_f$ is uniquely defined this way since $\omega$ is non-degenerate by definition.
[2] The one-parameter group of canonical diffeomorphisms $\phi^{(f)}$ generated by $f$ is properly defined as follows. $$\mathbb R \ni a \mapsto \phi_a^{(f)}(x) =: y_x(a)\in F \tag{3}\:,\quad \forall x \in F$$ where $y_x$ is the unique (maximal) solution of the Cauchy problem $$\frac{dy}{da} = X_f(y(a))\:, \quad y(0) =x \tag{4}$$ (I am assuming that the solution is complete, as it happens if $f$ is compactly supported of $F$ itself is compact, otherwise some subtleties regrading domains are to be fixed and $\phi_a^{(f)}(x)$ is only locally defined in the variable $a$.)
[3] The Poisson bracket is defined as $$\{f,g\}:= \omega(X_f,X_g) \quad f,g \in C^\infty(F,\mathbb R)\:.\tag{5}$$
With these definitions, (3) and (4) imply, as asserted in the main text, that $X_f$ is tangent to the curves $\mathbb R \ni a \mapsto \phi_a^{(f)}(x)$. Next (4) and (5) easily produce (1). An explicit expression of the action of $\phi^{(f)}$ on a function $g : F \to \mathbb R$, $$\left(\phi^{(f)*}_{a}[g]\right)(x):= g(\phi^{(f)}_{a}(x))$$ is provided by the formula $$\phi^{(f)*}_{a}[g] = \sum_{n=0}^{+\infty} \frac{a^n}{n!}\{f,\:\:\}^n g\:.$$ This identity holds if $f,g$ are real analytic and not only smooth.
It is finally worth stressing that the equations in (4) are nothing but the standard Hamilton equations if $f$ is indicated by $H$.
• This is a nice answer. It would be even easier to understand with a diagram illustrating the vector flows. – DanielSank Oct 4 '15 at 19:43
• Thanks. Unfortunately I am not able to draw diagrams in Latex! – Valter Moretti Oct 5 '15 at 7:06
• You can attach images to Stack Exchange posts. – DanielSank Oct 5 '15 at 7:54
• Well, drawing images is the problem, also I have no time to do it. – Valter Moretti Oct 5 '15 at 8:11
• Nice edit - it's definitely a lot more readable and beginner-friendly. I'll leave the bounty up the full week as free advertisement ;). If you have image drafts (using e.g. MS Paint) I might be able to pimp them up to a more presentable form. – Emilio Pisanty Oct 5 '15 at 14:20
Yes it does. In fact, it is one of the( if not the) most important conclusions of Quantum mechanics. If {f,g}= 0 it means that the variables have simultaneous eigenvalues i.e. you can measure both of them on same instant of time. but {f,g} can be non-zero which leads to the theoretical conchusion that the eigenstates of f and g are not simutaneously measurable. This is the Hisenberg Uncertainty Principal's most general form.
for the mathematical treatment you may wanna see http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/GenUncertPrinciple.htm
• But the question is about Poisson brackets in classical mechanics, not commutators in quantum mechanics... – Nathaniel Oct 6 '15 at 7:40
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2019-08-21 05:16:38
|
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https://dsp.meta.stackexchange.com/questions/1468/whats-this-stupid-thing-about-having-to-indent-all-of-your-code-with-4-spaces-f
|
# What's this stupid thing about having to indent all of your code with 4 spaces for it to be rendered as code?
isn't there some markup hook that i can use to do that?
i'm getting tired of fetching some old C file that i want to simply paste in and having to manually add spaces to many of the lines.
this seems silly.
r b-j
• Use a proper IDE and go ctrl-A tab and you're done. :-) – Peter K. Jan 6 '17 at 23:10
• used to be ctrl-]. – robert bristow-johnson Jan 7 '17 at 1:09
• you mean command-mode >. – Marcus Müller Jan 8 '17 at 12:36
You could do:
<pre>
code
code2
code3
</pre>
which gives you
code
code2
code3
I think triple backticks code should work to:
code
more code
some final code
code more code some final code
This should also work:
<code>
code
code2
code3
</code>
which renders as:
code code2 code3
The <pre> tag has trouble with HTML special characters < and >:
<pre>
#include <limits.h>
</pre>
gives:
#include
• i'll give it a try. – robert bristow-johnson Jan 8 '17 at 19:26
• the <pre> tag is the way to go – Matt L. Jan 8 '17 at 20:44
• The triple-backtick works in desktop, but mis-renders without line breaks on mobile – Marcus Müller Jan 10 '17 at 1:03
• please tell me you don't have a class pre; std::vector<pre> prepositions_vector; in your C++... – Marcus Müller Jan 12 '17 at 17:38
• @OlliNiemitalo no, seriously, I'll make this answer a community wiki. Please add an example for what doesn't work! – Marcus Müller Jan 12 '17 at 17:39
In fact, just use the curly braces when answering: the SE answer "IDE" has this function automatically. Just hit the curly brace button after selecting the code you want indented.
• Peter, i cut and pasted text from an old C file into the SE window, but i wanted this text to look like code. best that i understand, every line (even the blank lines) has to have a tab or 4 spaces preceding anything else. – robert bristow-johnson Jan 7 '17 at 1:08
• @robertbristow-johnson: The curly braces button does do that for you, is there something I'm missing? – Stefan Monov Jan 11 '17 at 21:16
• maybe it's something i am missing. i'll fiddle with it again. – robert bristow-johnson Jan 11 '17 at 21:53
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2019-09-18 07:06:09
|
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https://www.tutorialspoint.com/How-to-calculate-square-root-of-a-number-in-Python
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# How to calculate square root of a number in Python?
PythonServer Side ProgrammingProgramming
using the sqrt() function defined in math module of Python library is the easiest way to calculate square root of a number
>>> import math
>>> math.sqrt(10)
3.1622776601683795
>>> math.sqrt(3)
1.7320508075688772
Published on 06-Mar-2018 19:20:05
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2020-11-29 05:18:06
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https://www.gamedev.net/forums/topic/504710-ccollada-and-normals/
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## Recommended Posts
giugio 246
Hy. In a collada file I have a poly(a cube)with 18 normal, i dont know how normal(face or vertex or..???) <source id="Cube-Geometry-Normals"> <float_array count="18" id="Cube-Geometry-Normals-array">0.00000 0.00000 -1.00000 0.00000 0.00000 1.00000 1.00000 -0.00000 0.00000 -0.00000 -1.00000 -0.00000 -1.00000 0.00000 -0.00000 0.00000 1.00000 0.00000</float_array> <technique_common> <accessor count="6" source="#Cube-Geometry-Normals-array" stride="3"> <param type="float" name="X"></param> <param type="float" name="Y"></param> <param type="float" name="Z"></param> </accessor> </technique_common> </source> and 4 index for face: <polygons count="6" material="Material"> <input offset="0" semantic="VERTEX" source="#Cube-Geometry-Vertex"/> <input offset="1" semantic="NORMAL" source="#Cube-Geometry-Normals"/> <input offset="2" semantic="TEXCOORD" source="#Cube-Geometry-UV"/>
0 0 0 1 0 1 2 0 2 3 0 3
4 1 4 7 1 5 6 1 6 5 1 7
0 2 8 4 2 9 5 2 10 1 2 11
1 3 12 5 3 13 6 3 14 2 3 15
2 4 16 6 4 17 7 4 18 3 4 19
4 5 20 0 5 21 3 5 22 7 5 23
</polygons> what kind of normals i have? Because i have 4 normal for face,I suppose that are vertex normals but if i have a normal for vertex i would have 24 normals(6*4 = 24) and i have only 18 normals. Thanks in advance.
##### Share on other sites
gzboli 122
<accessor count="6" source="#Cube-Geometry-Normals-array" stride="3">
This tells you there are 6 normal vectors. Each normal vector has 3 components. The three components are:
<param type="float" name="X"></param><param type="float" name="Y"></param><param type="float" name="Z"></param>
The normals are stored in a float array. So the first normal is stored as the first three floats in the float array.
For the <polygons> you have, there are 6 of them, making 12 indices per polygon. Because you have 3 inputs (vertex,normal,tex), then that means the poygons are quads (12/3 = 4).
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2017-08-18 16:41:46
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https://inquiryintoinquiry.com/2015/07/06/relations-their-relatives-14/
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Relations & Their Relatives : 14
I constructed the “Ann and Bob” examples of sign relations back when I was enrolled in a Systems Engineering program and had to explain how triadic sign relations would naturally come up in building intelligent systems possessed of a capacity for inquiry. My advisor asked me for a simple, concrete, but not too trivial example of a sign relation and after cudgeling my wits for a while this is what fell out. Up till then I had never much considered finite examples before as the cases that arise in logic almost always have formal languages with infinite numbers of elements as their syntactic domains if not also infinite numbers of elements in their object domains.
The illustration at hand involves two sign relations:
• $L_\text{A}$ is the sign relation that captures how Ann interprets the signs in the set $S = I = \{ {}^{\backprime\backprime}\text{Ann}{}^{\prime\prime}, {}^{\backprime\backprime}\text{Bob}{}^{\prime\prime}, {}^{\backprime\backprime}\text{I}{}^{\prime\prime}, {}^{\backprime\backprime}\text{you}{}^{\prime\prime} \}$ to denote the objects in $O = \{ \text{Ann}, \text{Bob} \}.$
• $L_\text{B}$ is the sign relation that captures how Bob interprets the signs in the set $S = I = \{ {}^{\backprime\backprime}\text{Ann}{}^{\prime\prime}, {}^{\backprime\backprime}\text{Bob}{}^{\prime\prime}, {}^{\backprime\backprime}\text{I}{}^{\prime\prime}, {}^{\backprime\backprime}\text{you}{}^{\prime\prime} \}$ to denote the objects in $O = \{ \text{Ann}, \text{Bob} \}.$
Each of the sign relations, $L_\text{A}$ and $L_\text{B},$ contains eight triples of the form $(o, s, i)$ where $o$ is an object in the object domain $O,$ $s$ is a sign in the sign domain $S,$ and $i$ is an interpretant sign in the interpretant domain $I.$ These triples are called elementary or individual sign relations, as distinguished from the general sign relations that generally contain many sign relational triples.
If this much is clear we can move on next time to discuss the two types of reducibility and irreducibility that arise in semiotics.
To be continued …
3 Responses to Relations & Their Relatives : 14
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2018-11-13 03:17:14
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https://space.stackexchange.com/questions/22735/roughly-how-fast-do-rockets-travel
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# Roughly, how fast do rockets travel?
I know that different rockets are built with different a range of different sizes or purposes, but there must be some kind of range of typical speeds for those things when leaving the Earth.
So how fast are rockets or any other human made ship when leaving the Earth, after the engines shut down?
• I've tried to clarify the wording of your question a bit to make some things clearer. The speed is constantly changing while the engines are still running, so I added "...after the engines shut down" to it. You might consider trying to do a little investigation yourself. For example how fast does an object move in low Earth orbit? (LEO) roughly? How far is the moon, and how long did it take the Apollo missions to reach the moon once they left Earth orbit? – uhoh Aug 20 '17 at 12:12
• Speed depends on your frame of reference. The Earth is moving at some 30km/sec along its orbital path around the Sun, which is orbiting the galactic center at some 220 km/sec. To achieve a low Earth orbit, a rocket must accelerate its payload by about 7.8km/sec; Apollo had to accelerate to about 10.4 km/sec to get to the Moon, and Voyager 1 has attained speeds of about 17km/sec on its journey through the solar system and beyond. – Anthony X Aug 20 '17 at 14:20
• You'll probably need to think what you mean by"leave the earth" - do you mean go into orbit? Which orbit? Do you mean to leave orbit? – Rory Alsop Aug 20 '17 at 14:28
• @Rory Alsop I meant to leave orbit, aiming for a inter stellar trip. – Matthew Aug 20 '17 at 14:31
• In that case: 25000mph is escape velocity. – Rory Alsop Aug 20 '17 at 14:40
Your ballpark estimate, a velocity you can compare everything else to is: 8 km/s. This is the orbital speed in Low Earth Orbit (LEO); you'll see this number quite frequently. It's something to get started with. You need that much of "horizontal" speed, on top of whatever's needed to lift the rocket above the atmosphere to enter the orbit.
Circular orbit velocity: $v_o = \sqrt{{GM} \over r}$. Escape velocity $v_e = \sqrt{{2GM} \over r}$. ${v_e \over v_o} = \sqrt{2}$. So, multiply LEO speed by $\sqrt{2}$ (resulting in 11.3km/s) for escape speed of Earth. That multiplier applies to any body and any circular orbit, giving the speed needed for leaving given body. The speed for interplanetary transfers when escaping Earth will usually be higher - usually only slightly so (for Mars and Venus transfers), so that the probe reaches the target instead of escaping Earth gravity and ending up nowhere in particular.
Then more advanced orbital mechanics kicks in, varying the speed wildly, regardless of what you measure the speed relative to. Earth moves around the Sun at ~30km/s so once you escape Earth gravity, this is the baseline to which you add (or subtract) your travel speed.
In strongly eccentric elliptical orbits, the speed will vary depending on where in the orbit you are: near the body it will be close to escape speed; at apoapsis you'll crawl at scarce meters per second, taking weeks until gravity pulls you back in for something of like half an hour dashing past the planet. Low gravity bodies, like comets or asteroids will have very slow orbits, you could enter orbit of Cruithne, a 10km asteroid, by running.
High circular orbits are much slower than low orbits; Pluto's about 1km/s (vs Earth's 30km/s). Moon is also 1km/s (relative to Earth; vs LEO 8km/s.) I'd recommend playing Kerbal Space Program to understand how the speed varies in orbital mechanics - e.g. how your rocket blasting at 11km/s past Earth's upper atmosphere ends up crawling at speed of a bicycle once it approaches the edges of Earth sphere of influence.
But your ballparks are 8km/s for entering LEO, 1km/s for Moon, or trans-Neptunian objects, 30km/s for solar orbit around Earth altitude. And a scarce meter per second or less for orbiting asteroids and the likes.
Remember, multiply circular orbit by $\sqrt{2}$ for escape speed; to escape Earth, $30\sqrt{2} = 42$ km/s to escape the solar system.
• Nice answer! Where does the $\sqrt{2}$ come from? Adding a link to that would be helpful for both the OP and future readers to see how it's derived and to double-check the validity. – uhoh Aug 21 '17 at 0:54
• @uhoh: Circular orbit velocity: $v_o = \sqrt{{GM} \over r}$. Escape velocity $v_e = \sqrt{{2GM} \over r}$. ${v_e \over v_o} = \sqrt{2}$. I think deriving the equations for these two velocities is beyond the scope of this question. – SF. Aug 21 '17 at 10:17
• Comments are considered temporary. I'm suggesting if you refer to an equation in the answer, just add the link in the answer. I try to do it every single time I refer to something. This gives both the OP and future readers maximal chance of following things up in case they want to learn more. – uhoh Aug 21 '17 at 10:21
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2019-08-20 12:26:21
|
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http://lambda-the-ultimate.org/node/4376
|
Can I express variable occurence ranges in logic?
I wonder whether there is some way out to express variable occurence range in logic. I have the following query:
exists X,Y(p(X) & q(X,Y) & r(Y))
I can use quantifiers and mini-scope to express that a variable first occurs, when I view the conjunction right associative (xfy):
exists X(p(X) & exists Y(q(X,Y) & r(Y))).
In the above I see that a usage of Y starts after the p(X). Similarly I can use quantifiers and mini-scope to express that a variable last occurs, when I view the conjunction as left associative (yfx):
exists Y(exists X(p(X) & q(X,Y)) & r(Y)).
Now in the above I see that the variable X is last used in q(X,Y) and then not anymore.
But what I would like to have is a formalism that allows to express both information, start and end of a variable use. Graphically it would express:
p(X) & q(X,Y) & r(Y)
+------X---------+
+--------Y-------+
Is this possible in logic? A logic that would come with semantics, substitution rules, etc.. Somehow I have the feeling it would violate the Frege principle. On the other hand approaches such as Continuation Passing Style formulation could eventually solve the problem.
Best Regards
Comment viewing options
Cross-posted to a lot of places
Math Overflow, Math StackEchange, and it had been posted to cstheory but since removed.
The idea is not to compute the ranges
Hi,
The idea is not to compute the ranges given a formula in ordinary logic. But to have a logic that allows to express the ranges in its own language, including a semantics, substitution rule, etc.. for this new logic. Computing the ranges is easy, I have already given an algorithm in my question itself. Namely right associative mini-scope gives start and left associative mini-scope gives end.
This logic would for example allow manipulations such as renaming a variable range. Thus effectively saving one variable name. This is not possible in normal logic as one can easily figure out, i.e. p(X), q(X,Y), q(Y) is not the same as p(X), q(X,X), q(X). But why not use X inplace of Y, after X is anyway not used anymore.
I am interested into whether somebody has already come up with a corresponding calculus. The calculus would govern the manipulations, so that movements that would change the meaning, like for example by variable clashes, are not possible.
Best Regards
Yes, but likely that's not what you want.
You are asking a question that the metatheory implementing the meaning of your logic would be able to ask, not your logic itself (unless it were metacircular, a property I don't think any existing consistent logics have. Check out Type Theory Should Eat Itself for more on this topic). The metatheory has to know what binding means in terms of syntax and not just the introduction and elimination rules.
Formalization of metatheory is easy, but all known formalizations are very difficult to work with, since substitution has to be proven to commute with just about every operation on syntax. This difficulty formed what is now called the POPLmark challenge, to mechanize this burden.
Scope extrusion rules are common, so that the transformations in your post are provably allowed, but a formalization of graphical representation of syntax? Not so common. Easy to do in an unrestricted metatheory, but proving that if your representation determines some scope such that your scope extrusion rules are applicable is a non-trivial metatheoretic endeavor.
Hi,Thank you very much for
Hi,
Thank you very much for your comment and the very interesting
links. The difficulty I am facing is that I have only this
graphical idea in mind and don't know yet much about the
possible rules that would govern the concept.
Concerning some doubts whether graphical ideas can be formulized.
I think knuth was facing the same problem when he created TeX.
And I guess the resolution was that a linearisation even of
2-dimensional concepts is most often easily possible.
But compared to TeX the 2-dimensions here are not boxes and
their juxtaposition or inclusion, but rather something that
reminds me of proof nets in linear logic. If we could model
variable start and end as positive and negative literals we
would eventually be done.
I forgot to post the motivation behind this kind of logic.
My primary motivation is currently some code generation.
But today I came up with some additional possible
application areas. So my list now goes:
- Compiler Backends: Register Allocation
- Code Verification: Proof Statements
- Natural Language Processing: Anaphora
- Database Systems: Query Optimization
- Logic Programming: Partial Evaluation
- What else?
I have already looked a little bit into the register allocation
literature. There are some results, like from Chaitin, that
register allocation corresponds to graph coloring. Also if I
look at the papers I see that the authors have a refined
language to talk about the issues: live interval,
use position, etc..
When looking at the domain of register allocation, then one
might ask oneself how this domain, a sequence of expression
assignments, eventually if-then-else and while, are modelled
in a functional programming language. I guess monads are
here in use. I have even seen monads covering non-determinism.
But what I have not yet seen, these monads interspersed with
quantifiers, as the graphical intuition would suggest. Maybe
a quantifier corresponds to a variable declaration block. But
then we end up with hierarchical variable scopes again, and
not the overlapping thing I have in mind.
So if somebody has already put hands here, I would be very
The problem with scope
The problem with scope extrusion would be possibly that
the number of needed variable names increases. Something
which is counter to the intended application areas.
So if you have graphically, in this not yet fully
identified logic:
+--- Y ---+
+--- X ---+ +---- X ---+
Then when we translate it into normal logic via scope
extrusion, we can only arrive at:
+----------- Y -----------+
+----------- X' ----------+
+----------- X -----------+
We have to rename one of the X, because the quantifier
extrusion rule has a side condition:
exists X(A & B) = A & exists X B if X not in A
Right?
Although my question could give the impression
that I am interested in scope extrusion, I guess
scope extrusion will be not a dominant rule of
this not yet fully identified logic. Since the
logic should be able to deal with variable occurence
ranges, the contrary is the case. The interest could
be more characterized as "scope intrusion".
By these ranges, a new, more intrusive, way of
defining a scope should be possible. The quantifers
should be able to intrude into the conjunction
boundaries independently of each other.
Best Regards
abdmaL
I'd consider the adbmaL work as related. From the poor high-level understanding I have of it, they would represent your term as
exists X, (p(X) & (exists Y, q(X,Y) & (unlet X in r(Y))))
(the ASCI notation "unlet X in " to express removal of an assumption from context is my own.)
The scope is not "linear" as in your intuitive drawing, but covers a subtree of a tree-like expression. "unlet" allows to 'carve' subtrees out of the scoped-over region.
Thank you very much. This
Thank you very much. This could work. Would need to look into it in more detail. I guess the non-linearity is not a no-go. Since the linear case, as you showed, is simply a special case.
Little excursion: Interestingly removal of assumptions is even a sound rule, also in normal sequent logic. If we look at the assumption introduction rule, which would read (the rule is named according to forward chaining, but removal amounts to introduction in backward chaining and vice versa):
G, A |- B
----------- (Right Implication Introduction and Assumption Removal)
G |- A -> B
One could also define the following rule, which is more or less the inverse of the previous rule. It would not define a complete fusion operator & though. The rule reads:
G |- B
------------- (Right Fusion and Assumption Introduction)
G, A |- A & B
Its just a short-hand for the valid:
------ (Id)
A |- A G |- B
----------------- (Right Fusion Introduction and Assumption Fusion)
G, A |- A & B
Very currious now to look into your proposal.
Could be related to consume/preduce as well
Just noticed that the "unlet" could also be a solution to a question I already have for a long time. A consume/produce modal operator in logic. It conflicts a little bit with the way linear logic does deal with reasources. Since it would give control of the resource consumption to the query and not to the premisses.
For more detail see here:
http://mathoverflow.net/questions/78557/can-consume-produce-be-modeled-in-linear-logic
CPS option?
Is it that you would want to get rid of Z and T in:
exists Z,T(
exists X(p(X) & q(X,Z) & X=T) &
exists Y(q(T,Y) & r(Y) & Y=Z)
)
How do you propose to use continuation-passing style to solve this?
@Denis: Looks more like equational
@Denis: Looks more like equational theory too me. Where one would first use the following rule (replacement of equals for equals) together with normal predicate logic scope extrusion:
S=T & A(S) <-> S=T & A(T) (1)
To arrive at:
exists X,Y,Z,T(p(X) & q(X,Y) & q(X,Y) & r(Y) & X=T & Y=Z)
Then one would use the following rule (non-empty domain and terms have always values) together with normal predicate logic scope intrusion:
exists X (X = T) <-> true (2)
To arrive at:
exists X,Y(p(X) & q(X,Y) & q(X,Y) & r(Y))
Let's at the moment assume that we do not yet have an equality sign in this new yet to be fully identified logic. An equality sign and corrsponding rules. The absence of an equality sign would not
mean we cannot define semantics, substitution rules, etc.. for
this new logic.
But I don't exclude that something could also be done with equality signs. Do you think the formula you gave does express the variable occurrence ranges from my example? If yes how? Or were you more fascinated by the redundance of Z and T? (Well we could also deem X and Y redundant, and some other picks of variables).
For the CSP question in your response. I did not yet think about, since I would rather like to avoid having = in the first place.
Bye
Just a rewrite
I expect that q(X,Y) & q(X,Y) can be simplified to q(X,Y) isn't it?
I was just trying to rewrite your formula in a way which matches better the graphics - and which keeps X and Y separate. Redundancy, yes maybe.
X and Y are overlapping in
@Denis: X and Y are overlapping in the graphics. They share q(X,Y), that is why probably your proposal doesn't match very well the graphics. Goal would be to have a logic that reflects that q(X,Y) is only called once.
When we only look for truth then calling q(X,Y) once or twice wouldn't matter so match. Since in classical logic we have A & A = A. But for the applications that are listed it might be important to be able to express more verbatim the computation, the uterance, etc..
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2018-12-12 18:30:44
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https://www.esaral.com/q/write-the-median-class-for-the-following-frequency-distribution-68287/
|
Write the median class for the following frequency distribution:
Question:
Write the median class for the following frequency distribution:
Solution:
We are given the following table.
Here, N = 100
$\therefore \frac{N}{2}=50$
The cumulative frequency just greater than 50 is 60.
So, the median class is 40−50.
|
2022-12-01 19:56:26
|
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https://stacks.math.columbia.edu/tag/0BEV
|
Lemma 33.45.9. Let $k$ be a field. Let $X$ be proper over $k$. Let $Z \subset X$ be a closed subscheme of dimension $d$. If $\mathcal{L}_1, \ldots , \mathcal{L}_ d$ are ample, then $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z)$ is positive.
Proof. We will prove this by induction on $d$. The case $d = 0$ follows from Lemma 33.33.3. Assume $d > 0$. By Lemma 33.45.6 we may assume that $Z$ is an integral closed subscheme. In fact, we may replace $X$ by $Z$ and $\mathcal{L}_ i$ by $\mathcal{L}_ i|_ Z$ to reduce to the case $Z = X$ is a proper variety of dimension $d$. By Lemma 33.45.5 we may replace $\mathcal{L}_1$ by a positive tensor power. Thus we may assume there exists a nonzero section $s \in \Gamma (X, \mathcal{L}_1)$ such that $X_ s$ is affine (here we use the definition of ample invertible sheaf, see Properties, Definition 28.26.1). Observe that $X$ is not affine because proper and affine implies finite (Morphisms, Lemma 29.44.11) which contradicts $d > 0$. It follows that $s$ has a nonempty vanishing scheme $Z(s) \subset X$. Since $X$ is a variety, $s$ is a regular section of $\mathcal{L}_1$, so $Z(s)$ is an effective Cartier divisor, thus $Z(s)$ has codimension $1$ in $X$, and hence $Z(s)$ has dimension $d - 1$ (here we use material from Divisors, Sections 31.13, 31.14, and 31.15 and from dimension theory as in Lemma 33.20.3). By Lemma 33.45.8 we have
$(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot X) = (\mathcal{L}_2 \cdots \mathcal{L}_ d \cdot Z(s))$
By induction the right hand side is positive and the proof is complete. $\square$
Comment #6803 by Yuto Masamura on
It seems that The Stacks project does not have any lemma related to the Nakai-Moishezon criterion, which is the converse to a special case of this lemma. Is there any reason (or am I overlooking)?
Comment #6812 by Yuto Masamura on
It may be better to add a reference to Section 31.14, in particular Lemma 31.14.10, in the sentence " we use material from Divisors, Sections 31.13 and 31.15".
Comment #6946 by on
Yes, we don't have any good ampleness criterions related to intersection theory. This is on the to do list! Thanks for your second comment. The fix is here.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
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2022-07-01 00:53:33
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https://infoscience.epfl.ch/record/203826/export/hm?ln=en
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000203826 001__ 203826
000203826 005__ 20190509132516.0
000203826 0247_ $$2doi$$a10.5075/epfl-thesis-6451
000203826 02470 $$2urn$$aurn:nbn:ch:bel-epfl-thesis6451-7
000203826 02471 $$2nebis$$a10335518
000203826 037__ $$aTHESIS 000203826 041__$$aeng
000203826 088__ $$a6451 000203826 245__$$bModeling, Design, and Characterization$$aUltra-Low-Temperature Silicon and Germanium-on-Silicon Avalanche Photodiodes 000203826 269__$$a2014
000203826 260__ $$bEPFL$$c2014$$aLausanne 000203826 336__$$aTheses
000203826 502__ $$aProf. J.-Ph. Thiran (président) ; Prof. E. Charbon (directeur) ; Prof. E. Cantatore, Prof. G. De Micheli, Prof. L.K. Nanver (rapporteurs) 000203826 520__$$aIn this thesis we propose the use of photodiodes fabricated in planar technologies to address the detection problem in these applications. A number of solutions exist, optimized for these wavelengths, based on Germanium (Ge) and other III-V materials. In this thesis we focused on Ge thanks to its versatility and ease to use in the clean room. The main advantage of this material is in fact a good compatibility with Silicon and standard CMOS processes. Note that the proposed technology is not based on Silicon/Germanium (SiGe), whereby Ge is used to strain Si to achieve higher bandwidth in Si, not higher sensitivity. In our pure Ge approach, Ge is grafted onto Si (Ge-on-Si), achieving high responsivity at wavelengths of 900nm and higher. The proposed devices can operate in avalanche mode (avalanche photodiodes - APDs), and in Geiger mode (Geiger mode APDs (GAPDs) or single-photon avalanche diodes (SPADs)). To combine the advantages of Ge with single-photon sensitivity and excellent timing resolution of Si-based SPADs, this thesis proposes a new generation of SPADs, achieved in collaboration with Prof. Nanver at TUDelft, aimed at near-infrared range. The fabrication process of the Ge-on-Si SPAD approach, which we are investigating together with the TUDelft group, consists of a standard CMOS process combined with post-processing steps to grow Ge on top of a Si/SiO2 layer. In our study we have investigated the potential for a new generation of massively parallel, Ge-on-Si sensors fabricated in fully CMOS compatible technology. The objective was to address the next challenges of super-parallel pixel arrays, while exploiting the advantages of Ge substrate. The key technology developed in the thesis is a selective chemical-vapor deposition (CVD) epitaxial growth. A novel processing procedure was developed for the p+ Ge surface doping by a sequence of pure-Ga and pure-B depositions (PureGaB). The resulting p+n diodes have exceptionally good I-V characteristics with ideality factor of ~1.1 and low saturation currents. They can be operated both in proportional and in Geiger mode, and exhibit relatively low dark counts. We also looked at techniques to improve red and infrared sensitivity in conventional deep-submicron CMOS processes, by careful selection of standard layers at high depths in the Si substrate. Using the proposed approach, 12 µm-diameter SPADs were fabricated in 0.18µm CMOS technology showing low dark count rates (363 cps) at room temperature and considerably lower rates at cryogenic temperatures (77 K), while the FWHM timing jitter is as low as 76 ps. That of cryogenic SPADs is a novel research direction and in this thesis it was advocated as a significant trend for the future of optical sensing, especially in mid-infrared wavelengths. Low temperature characterizations reported in this thesis exposed how the relevant properties of fabrication materials, such as strength, thermal conductivity, ductility, and electrical resistance are changing. One of the most important properties is superconductivity in materials cooled to extreme temperatures: this is an important trend that will be pursued in the future activities of our group.
000203826 6531_ $$aAvalanche Photodiode 000203826 6531_$$aAPD
000203826 6531_ $$aCryogenic 000203826 6531_$$aCMOS compatible
000203826 6531_ $$aGermanium 000203826 6531_$$aGe-on-Si
000203826 6531_ $$aNear-Infrared 000203826 6531_$$aSingle photon avalanche diode
000203826 6531_ $$aSPAD 000203826 6531_$$aSPAD array
000203826 700__ $$0243383$$g192052$$aAminian, Mahdi 000203826 720_2$$aCharbon, Edoardo$$edir.$$g146991$$0240305 000203826 8564_$$uhttps://infoscience.epfl.ch/record/203826/files/EPFL_TH6451.pdf$$zn/a$$s23192822$$yn/a 000203826 909C0$$xU12178$$0252106$$pAQUA
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000203826 918__ $$dEDEE$$cIEL$$aSTI 000203826 919__$$aGR-SCI-IC
000203826 920__ $$b2014$$a2014-12-22
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2019-06-25 08:01:21
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https://networkengineering.stackexchange.com/questions/55855/does-time-division-duplexing-work-this-way
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# Does Time division duplexing work this way?
During the course of research, I have found a really hard time understanding this short lines below. I will appreciate a short hint or explanation from any expert in this area.
• Did any answer help you? If so, you should accept the answer so that the question doesn't keep popping up forever, looking for an answer. Alternatively, you can provide and accept your own answer. – Ron Maupin Dec 14 '19 at 18:19
• So for instance based on the modified diagram above, the relays can receive a total of $\lambda_1P_{re}$ which is shared into two halves (for reception and transmission)? – Abdulhameed Jan 4 '19 at 0:42
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2021-05-16 21:40:36
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http://www.egri-nagy.hu/post/201801quadratic/
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Image generated by Desmos
# The empowering quadratic
Most people openly hate mathematics. I did that too, before university. The question is why? I think it is often taught badly. Once grades are involved, it is very difficult to get it right.
One thing changed when I turned from a hater to a fan of math. I realized that there was a storyline in most examples. So, maybe we could put the narrative back. As a little example, here is a story of the quadrative formula.
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2018-08-17 05:19:01
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http://www.csam.or.kr/journal/view.html?uid=1788&&vmd=Full
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TEXT SIZE
CrossRef (0)
Independence and maximal volume of d-dimensional random convex hull
Won Sona, Seongoh Parka, and Johan Lim1,a
aDepartment of Statistics, Seoul National University, Korea
Correspondence to: 1Corresponding author: Department of Statistics, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Korea. E-mail: johanlim@snu.ac.kr
Received October 16, 2017; Revised December 19, 2017; Accepted December 20, 2017.
Abstract
In this paper, we study the maximal property of the volume of the convex hull of d-dimensional independent random vectors. We show that the volume of the random convex hull from a multivariate location-scale family indexed by ∑ is stochastically maximized in simple stochastic order when ∑ is diagonal. The claim can be applied to a broad class of multivariate distributions that include skewed/unskewed multivariate t-distributions. We numerically investigate the proven stochastic relationship between the dependent and independent random convex hulls with the Gaussian random convex hull. The numerical results confirm our theoretical findings and the maximal property of the volume of the independent random convex hull.
Keywords : convex hull, independence, multivariate location-scale family, simple stochastic order, stochastic geometry
1. Introduction
Random convex hull, the convex hull of independent and identically distributed random points, have been studied for decades after the seminal work by Rényi and Sulanke (1963, 1964). In particular, researchers in stochastic geometry focus on the functionals of random convex hull (where the two most important functionals are the volume and number of faces) and investigate their finite and asymptotic properties.
The random convex hull is also employed in many multivariate statistical procedures. For example, Barnett (1976) defines an ordering of the multivariate data based on the notion of convex hull peeling depth. In Cook (1979), the random convex hull generated by the data points is used to identify the influential observations in linear regression. It is also used to find an optimal classifier in machine learning literature (Fawcett and Niculescu-Mizil, 2007; Lim and Won, 2012; Son et al., 2015). Ng et al. (2014) recently developed an efficient algorithm to simulate a random convex hull on a plane, and applied it to testing the independence of a d-dimensional random vector. They consider the use of the area (volume) of the convex hull for testing independence. However, no statistical justification is given for the use of the volume of the convex hull.
In this paper, we are interested in the maximal property of the volume of the convex hull of d-dimensional random vectors under independence. This paper argues that “the volume of the random convex hull of d-dimensional vectors $Xi=(Xi1,Xi2,…,Xid)T$, i = 1, 2, …, n, is maximized in simple stochastic order (defined later) when $Xi1,Xi2,…,Xid$ are independent (or uncorrelated) to each other.” We show the claim is true for a broad class of multivariate distributions indexed by covariance matrix ∑ which includes skewed/unskewed multivariate t-distributions.
The remainder of the paper is organized as follows. In Section 2, we introduce general notations to be used in the paper. In addition, we show the invariance of the volume of the random convex hull with respect to rotational and axis-scalable transformations. We also introduce multivariate location-scale (MLS) family indexed by ∑, denoted by MLS(∑), which is invariant to the two transformations above. In Section 3, we prove the maximal property of the random convex hull under independence when the random vectors are from a distribution in MLS(∑). We then discuss the Gaussian random convex hull as an illustrative example and provide numerical illustrations of the results in the section. Finally, in Section 4, we conclude the paper with a discussion on the extension of the results to serially correlated data.
2. Preliminaries
### 2.1. Notation
Suppose we consider n data points in ℛd with d ≥ 1. The convex hull of the data points x1, x2, …, xn is defined as
$chull(x1,x2,…,xn):={∑i=1nαixi|∑i=1nαi=1, αi≥0, i=1,2,…,n}.$
A vertex of a convex set S ⊂ ℛd is a point x which cannot be written as a convex combination of the points in S {x}. The vertexes of the convex hull chull(x1, x2, …, xn) are elements of the set {x1, …, xn} and forms the vertexes of the convex hull denoted by V = {v1, v2, …, vK} (for some Kn). Finally, we find that chull(x1, x2, …, xn) equals to the polytope with vertexes v1, v2, …, vK which is denoted as pt(v1, v2, …, vK).
### 2.2. Multivariate location-scale family indexed by ∑
In this section, we introduce the MLS family that we assume for the distribution of random vectors in the paper.
The MLS family is one of the important parametric families and many important distributions are included in location-scale family. The family {P(μ,∑) : μ ∈ ℛd, ∑ ∈ ℳd} is defined to be a location-scale family on ℛd if
$P(μ,Σ)(B)=G(Σ-12(B-μ)), B∈Bd,$
where G(·) is an arbitrary given probability measure on the d-dimensional Borel field, ∑−1/2(Bμ) = {∑−1/2(xμ) : xB ⊂ ℛd}, and ℳd is a collection of d × d symmetric positive definite matrices (Shao, 2003).
Some examples of the MLS family are as follows. Elliptically symmetric distributions (or simply elliptical distributions) belong to a location-scale family (Ollila et al., 2003). A d-dimensional random vector X is said to have an elliptical distribution, denoted by X ~ ECd(μ, ∑, ξ), if it has a stochastic representation
$X=μ+ξΣ12U,$
where U is a random vector uniformly distributed on the unit sphere in ℛd, ξ ≥ 0 is a scalar random variable independent of U, and ∑1/2 is a deterministic symmetric matrix such that ∑1/2(∑1/2)T = ∑. There are many subclasses of elliptical distributions. Multivariate normal distributions, multivariate t-distributions, logistic distributions and multivariate Cauchy distributions are well-known examples of them (Fang et al., 1990). While the underlying population distributions are free of the assumption of symmetry, some distributions are also included in the MLS family. Multivariate Pareto of the second type (Asimit et al., 2010), a generalized multivariate gamma distributions (Carpenter and Diawara, 2007), and multivariate skew normal distributions and multivariate skew t-distributions (Azzalini and Capitanio, 2003) are examples of asymmetric distributions which can be classified as the MLS family. Some further extensions can also be found from Zhao and Kim (2016).
In this paper, we assume μ = 0 without loss of generality, and index the MLS family with the covariance matrix ∑. In sequel, we simply write the MLS family indexed by ∑ as MLS(∑) omitting the distribution notation of ξ of (2.3). Further, we let λjs for j = 1, 2, …, d be the eigenvalues of the covariance matrix ∑.
### 2.3. Vertex invariance to rotation and scale transformation
Let us consider n independent random samples $Xi=(Xi1,Xi2,…,Xid)T$, i = 1, 2, …, n, which are identically from the d-dimensional multivariate distribution with mean 0 and covariance matrix ∑. Let v.chull(X1, X2, …, Xn) be the volume of chull(X1, X2, …, Xn).
The following lemma shows that the vertex set of chull(X1, X2, …, Xn) is invariant to both rotation and scale (according to the axis) transformations of {Xi, i = 1, 2, …, n}.
Lemma 1
Let P be a d-dimensional orthonormal matrix and Q be the diagonal matrix with diagonal elements q1, q2, …, qd. If V = {v1, v2, …, vK} is the set of the vertexes of chull(X1, X2, …, Xn), then (i) the vertexes of chull(PX1, PX2, …, PXn) is PV = {Pv1, Pv2, …, PvK} and (ii) the vertex set of chull(QX1, QX2, …, QXn) is QV = {Qv1, Qv2, …, QvK}.
Proof
The proof of (ii) is very similar to that of (i). Thus, we only prove (i) at here.
We first show that Pvk is a vertex of chull(PX1, PX2, …, PXn). Suppose, without loss of generality, Pvk = PX1 and assume it is not a vertex. Then, it can be written as a convex combination of PX2, PX3, …, PXn as
$PX1=∑i=2nαiPXi, ∑i=2nαi=1, αi≥0.$
By multiplying PT in both sides of the above, we have
$νk=X1=∑i=2nαiXi, ∑i=2nαi=1, αi≥0,$
which implies vk is not a vertex of chull(X1, X2, …, Xn). This introduces a contradiction.
We now show that any point in chull(PX1, PX2, …, PXn) has a convex representation of PV = {Pv1, Pv2, …, PvK}. Suppose y is a vertex of the convex hull but not in PV. Since it is within a convex hull, it can be written as
$y=∑i=1nαiPXi, αi≥0, ∑i=1nαi=1.$
Again, by multiplying PT both sides,
$PTy=∑i=1nαiXi=∑k=1Kβkνk, αi≥0, ∑i=1nαi=1 and βk≥0, ∑k=1Kβk=1,$
where the second equation is from that V = {v1, v2, …, vK} is the vertex set of chull(X1, X2, …, Xn). In the above, the last equation
$PTy=∑k=1Kβkνk, βk≥0, ∑k=1Kβk=1,$
is equivalent to
$y=∑k=1KβkPνk, βk≥0, ∑k=1Kβk=1,$
which contradicts to that y is a vertex not in PV.
3. Maximal volume of random convex hull of samples from MLS(∑)
### 3.1. Main result
We now present our main results of the paper which show the volume of the convex hull of a MLS(∑) is stochastically maximized when true covariance matrix ∑ is diagonal, equivalently, X1, X2, …, Xd are uncorrelated to each other. If the data are from a multivariate normal distribution, the volume is maximized when the d-variables are independent. The stochastic order between two variables X and Z is defined as: Z is (simply) stochastically larger than X, denoted by XstZ if and only if P(Za) ≤ P(Xa) for every a ∈ ℛ.
Theorem 1
SupposeX1, X2, …, Xn are independently from a distribution from a MLS with covariance matrix, MLS(∑), andZ1, Z2, …, Zn are independently from MLS(diag(∑)), where diag(∑) is the diagonal matrix whose diagonal elements are same with those of. Then, for every n and a positive definite covariance matrix, we have
$v.chull(X1,X2,…,Xn)≼stv.chull(Z1,Z2,…,Zn).$
Proof
Suppose V = {v1, v2, …, vK} is the set of vertexes of chull(X1, X2, …, Xn) and let the singular value decomposition of ∑ be PT ΛP with an orthonormal matrix P and Λ = diag(λ1, λ2,, λd).
$v.chull(X1,X2,…,Xn)=∬I(x∈chull(X1,X2,…,Xn)) dx=∬I(y∈chull(PX1,PX2,…,PXn))·det(PT) dy=∬I(y∈pt(Pν1,Pν2,…,PνK))·1 dy,$
which is from the rotation transformation y = PTx (equivalently, Py = x) and the invariance from Lemma 1. In (3.2), pt(Pv1, Pv2, …, PvK) equals to chull(PX1, PX2, …, PXn), where Yi = PXi are independently and identically distributed (iid) as MLS(Λ). Thus,
$(3.2)=∬I(y∈chull(Y1,Y2,…Yn)) dy=v.chull(Y1,Y2,…Yn).$
We now show that v.chull(Y1, Y2, …, Yn) is stochastically smaller than v.chull(Z1, Z2, …, Zn). This is simply by applying the axis-wise scale transformation to both chull(Y1, Y2, …, Yn) and chull (Z1, Z2, …, Zn). First, for chull(Y1, Y2, …, Yn), we consider the transformation matrix QY = Λ−1/2 and have
$v.chull(Y1,Y2,…,Yn)=∬I(y∈chull(Y1,Y2,…,Yn)) dy=∬I(u∈chull(QYY1,QYY2,…,QYYn))·det(QY-1) du=(∏j=1dλj)12v.chull(U1.y,U2.y,…,Un.y),$
where Ui.y are iid from MLS(Id) with Id is the d-dimensional identity matrix. In showing (3.3), we use the invariance property proven in (ii) of Lemma 1 as in (3.2). The similar steps are applied to chull(Z1, Z2, …, Zn) with the scale transformation QZ = {diag(∑)}−1/2, and show that
$v.chull(Z1,Z2,…,Zn)=(∏j=1dσjj)12v.chull(U1.z,U2.z,…,Un.z),$
where σj j is the jth diagonal element of ∑ and Ui.z are iid from MLS(Id).
Finally, we conclude the proof using the Hadamard’s inequality (Cover and Gamal, 1983), which tells, for the covariance matrix ∑,
$(∏j=1dλj)=det(Σ)≤(∏j=1dσjj).$
3.2. Gaussian random convex hull
The Gaussian random convex hull, the random convex hull for d-dimensional Gaussian random vectors, is studied by many researchers. Suppose Z1, Z2, …, Zn are iid from the d-dimensional standard normal distribution. It is shown by Affentranger (1991) that
$E{v.chull(Z1,Z2,…,Zn)}=κd(2 log n)d2(1+o(1))={πd2Γ(d2+1)}(2 log n)d2(1+o(1)),$
where κd is the volume of the d-dimensional unit ball. It is also shown by Hug and Reitzner (2005) that, for d ≥ 1,
$var{v.chull(Z1,Z2,…,Zn)}=O((log n)d-32).$
However, Hug (2013) points out that the explicit finite sample distribution function is still unknown. Instead, some of its asymptotic are known. For example, Bárány and Vu (2007) prove the central limit theorem for the volume of the Gaussian random convex hull.
Theorem 1 in Section 3 further shows that, for the general ∑, the volume has a constant multiplicative factor $(∏j=1dλj)1/2$ which is canceled out from numerator and denominator of its standardized form. The standardized statistic
$tv.chull=v.chull(X1,X2,…,Xn)-E(v.chull(X1,X2,…,Xn))var(v.chull(X1,X2,…,Xn))$
is invariant to the scale transformation and has the same distribution with
$tv.chull=v.chull(Z1,Z2,…,Zn)-E(v.chull(Z1,Z2,…,Zn))var(v.chull(Z1,Z2,…,Zn))$
where E(v.chull(Z1, Z2, …, Zn)) and var(v.chull(Z1, Z2, …, Zn)) are those in (3.6) and (3.7), respectively.
3.3. Numerical illustration
We now numerically illustrates the findings in the previous subsection. The identity (3.3) tells that
$v.chull(X1,X2,…,Xn)=(∏j=1dλj)12v.chull(Z1,Z2,…,Zn),$
where Xis are from the multivariate normal distribution with mean 0 and variance ∑, Zis are iid d-dimensional standard normal vector, and λj, j = 1, 2, …, d, are the eigenvalues of ∑. Thus,
$Δ=log v.chull(X1,X2,…,Xn)-12log(∏j=1dλj)$
has the same distribution with
$log v.chull(Z1,Z2,…,Zn),$
and is invariant to the choice of ρ for fixed d and n. We numerically investigate this identity.
We generate samples from d-dimensional multivariate t-distribution with degrees of freedom ν = 3, 5, 10,∞(∞ corresponds to the multivariate normal distribution), μ = 0d, and two types of ∑ defined below with the dimension d = 2, 4. Two covariance matrices we consider are: (i) the compound symmetry (CS) matrix, notated as CS(ρ), is defined as
$Σcs=(1-ρ)Id+ρ1d1dT$
and its log-determinant is log det(CS(ρ)) = log{1 + (d − 1)ρ} + (d − 1) log(1 − ρ), (ii) the first order auto-regressive (AR) model, notated as AR1(ρ), is defined as
$Σar=(σij=ρ∣i-j∣,i,j=1,2,…,(d-1)),$
and its log-determinant is log det(AR1(ρ)) = (d − 1) log(1 − ρ2). The sample size is set as n = 50, 100, 300, 500. We simulate B = 1000 data sets and, in each data, we compute the area of the convex hull. To compute the area of the convex hull, we use the “convhulln” function in the R-package “geometry”, which implements the Quickhull algorithm (Barber et al., 1996).
The box plots of $Δ=log v.chull(X1,X2,…,Xn)-(1/2) log(∏j=1dλj)$ versus ρ for fixed ν, n are presented in Figures 14 for each combination of d = 2, 4 and ∑ = ∑cs, ∑ar. The figures show that the distribution of Δ is invariant to the choice of ρ for given ν and n in every combination of d = 2, 4 and ∑ = ∑cs, ∑ar. In each figure, the four box plots in each panel has the same distribution as the distribution of log v.chull(Z1, Z2, …, Zn). The mean of log v.chull(Z1, Z2, …, Zn) therefore varies according to the changes of ν and n; it tends to increase as either ν decreases or n increases.
4. Conclusion
In this paper, the maximal property of the volume of the convex hull of d-dimensional independent random vectors is investigated. In stochastic sense, the volume of the convex hull is maximized when the covariance matrix ∑ of the underlying probability distribution is diagonal. This results is true for the distribution from the multivariate location-scale family that includes skewed/unskewed multivariate t-distribution and elliptical distribution. Thus, the volume of the convex hull can be used for testing the independence of a d-dimensional vector as in Ng et al. (2014).
Possible future research direction is to extend this conclusion to the random convex hull from dependent samples including time series data. In time series data, the lagged plot, the plot (yt, yt−1, …, ytd+1), …, (ytd+1, ytd, …, yt−2d), plays an important role in exploring the data. In particular, the convex hull of the data points in the lagged plot is a key tool to find outliers and influential points. However, its theoretical property is rarely understood and further study on it is demanded.
Figures
Fig. 1. d = 2 and ∑ = ∑ar: Box plots of for different choices of ρ, ν, and n. AR = auto-regressive.
Fig. 2. d = 4 and ∑ = ∑ar: Box plots of for different choices of ρ, ν and n. AR = auto-regressive.
Fig. 3. d = 2 and ∑ = ∑cs: Box plots of for different choices of ρ, ν and n. CS = compound symmetry.
Fig. 4. d = 4 and ∑ = ∑cs: Box plots of for different choices of ρ, ν and n. CS = compound symmetry.
References
1. Affentranger, F (1991). The convex hull of random points with spherically symmetric distributions. Rendiconti del Seminario Matematico Università e Politecnico di Torino. 49, 359-383.
2. Asimit, AV, Furman, E, and Vernic, R (2010). On a multivariate Pareto distribution. Insurance: Mathematics and Economics. 46, 308-316.
3. Azzalini, A, and Capitanio, A (2003). Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. Journal of the Royal Statistical Society. Series B (Statistical Methodology). 65, 367-389.
4. Bárány, I, and Vu, V (2007). Central limit theorems for Gaussian polytopes. The Annals of Probability. 35, 1593-1621.
5. Barber, CB, Dobkin, DP, and Huhdanpaa, H (1996). The Quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software (TOMS). 22, 469-483.
6. Barnett, V (1976). The ordering of multivariate data. Journal of the Royal Statistical Society. Series A (General). 139, 318-355.
7. Carpenter, M, and Diawara, N (2007). A multivariate gamma distribution and its characterizations. American Journal of Mathematical and Management Sciences. 27, 499-507.
8. Cook, RD (1979). Influential observations in linear regression. Journal of the American Statistical Association. 74, 169-174.
9. Cover, T, and Gamal, AE (1983). An information - theoretic proof of Hadamard’s inequality (Corresp). IEEE Transactions on Information Theory. 29, 930-931.
10. Fang, KT, Kotz, S, and Ng, KW (1990). Symmetric Multivariate and Related Distributions. Boston: MA Springer US
11. Fawcett, T, and Niculescu-Mizil, A (2007). PAV and the ROC convex hull. Machine Learning. 68, 97-106.
12. Hug, D (2013). Random polytopes. Stochastic Geometry, Spatial Statistics and Random Fields, Lecture Notes in Mathematics, Spodarev, E, ed. Berlin-Heidelberg: Springer-Verlag, pp. 205-238
13. Hug, D, and Reitzner, M (2005). Gaussian polytopes: variances and limit theorems. Advances in Applied Probability. 37, 297-320.
14. Lim, J, and Won, JH (2012). ROC convex hull and nonparametric maximum likelihood estimation. Machine Learning. 88, 433-444.
15. Ng, CT, Lim, J, Lee, KE, Yu, D, and Choi, S (2014). A fast algorithm to sample the number of vertexes and the area of the random convex hull on the unit square. Computational Statistics. 29, 1187-1205.
16. Ollila, E, Oja, H, and Croux, C (2003). The affine equivariant sign covariance matrix: asymptotic behavior and efficiencies. Journal of Multivariate Analysis. 87, 328-355.
17. Rényi, A, and Sulanke, R (1963). Über die konvexe Hülle von n zufällig gewählten Punkten. Zeitschrift Für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 2, 75-84.
18. Rényi, A, and Sulanke, R (1964). Über die konvexe Hülle von n zufällig gewählten Punkten. Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete. 3, 138-147.
19. Shao, J (2003). Mathematical Statistics. New York: Springer
20. Son, W, Ng, CT, and Lim, J (2015). A new integral representation of the coverage probability of a random convex hull. Communications of Statistical Applications and Methods. 22, 69-80.
21. Zhao, J, and Kim, HM (2016). Power t distribution. Communications for Statistical Applications and Methods. 23, 321-334.
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## Women modifiers
Maddie York, "Why there are too many women doctors, women MPs, and women bosses", The Guardian 10/17/2014:
I am a subeditor at the Guardian. I am a woman. I am not a woman subeditor. But “woman” and its plural seem to be taking over the role of modifier, so that now, there is no such thing, as far as much of the media is concerned, as a female doctor, a female MP or a female chef. Instead you hear or read about a woman doctor, a woman MP and so on. […]
As far as the Guardian style guide is concerned, it is simply wrong to use “woman” and “women” in this way, because, it says,
John McIntyre responds ("Women beware 'woman'", The Baltimore Sun 10/17/2014:
Let's take a moment to unpack where Ms York is right-headed and wrong-headed.
I'm sure that a subeditor at The Guardian is aware of English's polymorphous parts of speech. You can insist that woman is a noun and not an adjective, but that doesn't make it so. English freely makes use of nouns as adjectives: have you sat in a window seat or closed a cellar door? So the "not an adjective" stuff is merely careless overstating of "I don't like using it as an adjective.
Geoff Pullum would say instead that English makes free use of nouns as modifiers, as John notes in an Addendum quoting Ian Loveless:
Believe it or not, nouns can, will, and do modify other nouns attributively. There is no need to reclassify every noun in the dictionary as an adjective just so we can explain how they can do this. Furthermore, these types of modifiers fail every test for an adjective devised by linguists, specifically modification by an adverb (she can't be a "very woman" doctor) and gradation (she can't be "womaner" than her sister). As Geoffrey Pullum says, "A noun is a noun".
John also links to a response by Dave Wilton at Wordorigins.org ("Women in the Guardian", 10/17/2014), tracing the use of woman as a modifier back to the 14th century. Some of the same points were made in a LLOG post a few years ago ("False logic and linguistic blindness: You could look it up", 10/31/2007).
Ms. York's false syntax obscures a valid point:
There would be no real problem if we used both “woman” and “man” as modifiers, but we don’t, so the implication is that a “woman manager” is a modification of the standard or natural form, or something slightly less than the full version. It behaves like “junior”: doctor, woman doctor, junior doctor, for example. Doctor – male implied – is the standard, woman and junior the variants. They are the not-quite-doctors.
In fact there have long been parallel examples with man or men as a modifier, like the KJV translation of Ecclesiastes 2:8:
I gathered me also silver and gold, and the peculiar treasure of kings and of the provinces: I gat me men singers and women singers, and the delights of the sons of men, as musical instruments, and that of all sorts.
And there are a few reasonably common current expressions in which man is used as a modifier, e.g. "man friend", "man nurse", "man witch", "man boobs". But these are cases where female gender is expected. Thus "man friend" seems mostly to be used to denote a male friend of a woman, in circumstances where female friends would be the norm, as in these NYT headlines: "Blond Teller Stole $65000 for Man Friend, Police Say"; "Three Get Estate of Miss Whitehead: Two Sisters and Man Friend to Divide Between$200,000 and \$300,000". Or these examples from The Guardian:
I am a widow, aged 60, enjoy living alone and am not looking for a partner. I do, though, have a man friend, also widowed, with whom I share many common interests.
A man friend expressed disappointment that Em D was abandoning her old wardrobe of unmitigated glamour in favour of something challenging and edgy.
In further Bel drama, her mother – a cross between Marilyn Monroe and Ab Fab's Edina Monsoon – comes to stay, but then doesn't, as she decides to go out with her man friend instead.
So Ms. York is correct that for professions where male gender is the unmarked case, people write (and say) "woman doctor" but not "man doctor", "woman president" but not "man president", etc. This is a fact about our society rather than a fact about English syntax, however — the same regrettable asymmetry applies to "female president" vs. "male president".
Still, it seems to me that Ms. York also has a valid linguistic point. It's common to find terms like "male doctor" used where gender is relevant, e.g. these web quotes:
It is prudent for women to avoid male doctors for intimate female health issues.
An American male doctor has contracted the Ebola virus while working at a hospital in Liberia, it was confirmed today as the CDC warned that the deadly disease was spiraling out of control.
Doctors of both sexes also enjoy the linguistic freedom to ask personal questions. For instance, the male doctor can openly and freely inquire about a female patient's bowel, bladder, vaginal, and rectal condition. Likewise, the female doctor can ask her male patients about the character and frequency of an erection.
And it would be unexpected to use the modifier "man" in such cases. Perhaps this is because using a noun as the modifier suggests that an unusual or marked category is being referenced, even when reference to gender is contextually relevant. And this is what bothered Ms. York in the first place.
On the other hand, as John McIntyre observes,
[T]astes vary. When women were first ordained to the priesthood in the Episcopal Church forty years ago, woman priest was the term many of them favored, because the use of female as a noun (another polymorphous part of speech) for non-human species ("the female of the species is more deadly than the male") carries disagreeable overtones.
Ms. York is, however, on to something when she sniffs for sexist condescension. After all, "woman driver" was a staple leitmotif for hack comedians of the 1950s and 1960s, and we have discarded poetess. But now that the Church of England has belatedly figured out that once you ordain a woman a priest there is no obstacle to consecrating her as a bishop, women in miters will be a novelty, and woman bishop will be a handy term for talking and writing about them.
1. ### Janne said,
October 19, 2014 @ 9:06 am
Adjectives don't have to allow adverbial modification or gradation in English, though? "Two-stroke" is an adjective, yet it'd feel a bit awkward to say it's "a very two-stroke" engine or that it's "more two-stroke" than that one.
[(myl) Why would you believe that "two-stroke" is an adjective? It seems transparently to be a modifier consisting of a cardinal number and a noun, similar to "ten gallon hat", "three dog night", "five dollar bill", "seven year itch", "thousand year egg", and so on.]
2. ### Mr Fnortner said,
October 19, 2014 @ 9:47 am
What can be said about agreement between the noun as modifier and its associated noun? We have 'woman doctor' and 'women doctors'. Why? Why not 'woman doctors'? We don't say 'windows seats', 'cellars doors' or 'eyes doctors' do we?
3. ### Jon Lennox said,
October 19, 2014 @ 10:19 am
Mr Fortner@2:
This seems to have to do with regular vs. irregular plurals? Thus "teeth marks" but not "*claws marks".
Not that plural agreement is mandatory for irregular plurals — both "woman doctors" and "tooth marks" seem fine to me as well — but it's permitted.
Example from Steven Pinker's Words and Rules.
[(myl) It seems that "tooth mark" is a noun+noun compound, rather than a modifier+noun phrase – both on the basis of stress (first-word stress) and semantics (a mark made by a tooth, not a mark that is a tooth).]
4. ### Ben said,
October 19, 2014 @ 10:48 am
"yet it'd feel a bit awkward to say it's "a very two-stroke" engine or that it's "more two-stroke" than that one."
You could just say it's three-stroke or four-stroke or thoroughly-struck if you felt the need.
5. ### John Walden said,
October 19, 2014 @ 10:51 am
Perhaps it's something to do with "and a". A window seat is not a window and a seat but a man servant is a man and a servant.
An idea which lasted the seconds it took to think of 'boy scout', 'girl guide' and 'child soldier'.
6. ### Judith Strauser said,
October 19, 2014 @ 10:51 am
The first thing that comes to my mind is that perhaps the Guardian is starting to feel wary of using the adjective "female" after so many feminists* have rightly objected to the use of "females" as a substantive instead of "women". There has been backlash and affirmations that "women" is better because it's the proper word and "females" carries too many animal, reductive connotations, and because the use of "females" seems to have spread from MRA types and other misogynistic circles. It is possible that the newspaper is simply swinging the pendulum back the other way a bit too strongly…
(I self-identify as one, I'm not using the word disparagingly – how sad that I feel this caveat is needed.)
7. ### Lazar said,
October 19, 2014 @ 11:15 am
Even having a nurse for a mother, I've never encountered the phrase "man nurse" before – although Google indicates that it does see some use. In my experience it's always been "male nurse".
8. ### Jerry Friedman said,
October 19, 2014 @ 11:29 am
John Walden: "Child soldier [actor, prodigy]" shows nicely that an irregular plural isn't enough to allow the word to become plural as a modifier. Why are there women priests but no children prodigies? No doubt Pinker gives the complete rule, if there is one
9. ### efahl said,
October 19, 2014 @ 11:52 am
@Janne:
Taking a cue from mathematics, I'd categorize "two-stroke" as a discrete adjective, which disallows (or at least discourages) grading modifiers "-er", "-est", "more", "less", whereas things like "tall" or "ugly" would be continuous adjectives.
10. ### Martha said,
October 19, 2014 @ 12:17 pm
In my social group at least, "man friend" is not "used to denote a male friend of a woman, in circumstances where female friends would be the norm," but rather someone you date but aren't officially dating, probably typically used when describing older people, as in the first "man friend" example above. It would describe a situation where people aren't "just friends." What I'd use to describe a male friend of a woman, who is just a friend who happens to be male, is "guy friend."
My mom used the phrase "man doctor" recently. It amused me.
11. ### Aaron said,
October 19, 2014 @ 1:59 pm
On the point about two-stroke,' I seem to remember that a company (BMW?) has developed an engine with a two-stroke top end and a 4-stroke-ish bottom end, so maybe the engine in my lawnmower is two-strokier than the new hybrid.
I do like the idea of discrete adjectives' though.
12. ### A. Mandible said,
October 19, 2014 @ 2:09 pm
"Man boobs" doesn't fit in the list where it appears– it's more like "man cave", "man purse", "man date" and other things which pertain to men but are not themselves men, and so for which substituting "male" would be bizarre.
[(myl) Yes, you're right — by the argument I gave above, this is clearly a noun+noun compound, as both the stress pattern and the meaning indicate.]
"Man friend", "man nurse" and "man witch" are totally unfamiliar to me, but if they're things people say, I assume the people so denoted are men– "man friend" isn't used like "I went out for barbecue and power tools with my man friend Sarah– she's great!"
13. ### Keith said,
October 19, 2014 @ 2:30 pm
Besides the idea that it is condescending to feel the need to point out that the doctor, president or police officer in question happens to be a woman as opposed to being a man as is "normal", the plural construction of "women doctors" seems to break the rule that in English adjectives are generally invariable.
A noun can function as an adjective when placed before another noun, in English, but it still obeys the rule of being invariable. This is why we have "claw marks", this is why (despite Pinker's protests) we have "rat eaters" (but then there are exceptions to this rule, of course).
As for "male nurse", this is a common term in the UK, and it is much better than "man nurse". A "male nurse" is a nurse who happens to be male. A "man nurse" would be a nurse who takes care of sick men, in the same way that a "horse doctor" is a vet who takes care especially of horses… note again how the noun "horse", when standing in for an adjective, remains in the singular.
K.
14. ### Coby Lubliner said,
October 19, 2014 @ 2:40 pm
Wasn't Woman Police Constable (WPC) a standard designation in the British constabulary for most of the 20th century? How old is Ms. York, anyway? Even if she doesn't remember WPCs, hasn't she watched Morse reruns, or Life on Mars?
15. ### Stan Carey said,
October 19, 2014 @ 2:49 pm
Jan Freeman wrote a good article about the history of commentary on this issue in the Boston Globe a few years ago.
16. ### Jerry Friedman said,
October 19, 2014 @ 3:04 pm
I almost forgot Dr. Slop the man-midwife.
17. ### Rubrick said,
October 19, 2014 @ 5:48 pm
@A. Mandible: '"Man friend", "man nurse" and "man witch" are totally unfamiliar to me'.
The important thing to remember is that a sand witch is a sand witch, but a man witch is a male.
18. ### Stuart Brown said,
October 19, 2014 @ 7:11 pm
When I were a lad (in Lancashire), 60 years ago, we would never have said "woman doctor". The correct term was "lady doctor". And the plural would clearly be "lady doctors" and not "ladies doctors".
19. ### Chingona said,
October 19, 2014 @ 8:08 pm
Why is John McIntyre sometimes called John here, but Maddie York is "Ms. York"?
[(myl) In my usage, it's because I've corresponded with John McIntyre, and met him in person a couple of times, and have no such history With Maddie York.]
20. ### ET said,
October 19, 2014 @ 8:10 pm
Why there are too many women doctors, women MPs, and women bosses – absolutely!
Unless every description is prefaced with gender, none of them should be.
There is no special school for doctors or lawyers of bosses of either gender.
If you insist on writing women doctors (in 2014!) then all doctors should be called women doctor or man doctor. Anything else is absurd.
21. ### Jerry Friedman said,
October 19, 2014 @ 9:02 pm
Keith: "Student nurse" seems to be a common term in the UK and doesn't mean a nurse who cares for students.
22. ### mollymooly said,
October 20, 2014 @ 6:58 am
@Jerry Friedman: A man-midwife was a real thing, not a Laurence Sterne joke. But that was several centuries ago.
23. ### Zubon said,
October 20, 2014 @ 7:49 am
I have heard "lady doctor" before, but only as a euphemism for an OB/GYN.
24. ### tpr said,
October 20, 2014 @ 7:49 am
I think the awkwardness of man doctor and woman doctor might be a garden path-type processing issue.
Note that we can freely use man to modify nouns that have been derived from verbs by adding -er, so we have familiar terms like man eater (someone or something that eats men), and man hater (someone or something that hates men). This appears to be a productive rule, so the same applies to expressions that couldn't be mistaken for idioms such as a man counter (someone or something that counts men).
But the word doctor sounds like it's derived from a verb + er so that a man doctor ought to be someone who docts men, but because that isn't a word, the parser would have to backtrack and try to attempt the next most plausible interpretation.
Note also that there are several masculine modifiers we can use before a noun that each have a different prototypical effect on the semantics: man, male, and men's:
1. a man hater (a hater of men)
2. a male doctor (a doctor who is a male)
3. a men's doctor (a doctor for men)
It can't be (1) because there is no verb doct, and the fact that there are better words available for the other two meanings makes it hard to know what a speaker might intend by man doctor. Is it a doctor who is a man or is it a doctor for men?
25. ### Dave K said,
October 20, 2014 @ 8:29 am
A related point is that "male" seems to be replacing "man" surprisingly often as in "I saw two males running from the crime scene".
26. ### Jerry Friedman said,
October 20, 2014 @ 9:29 am
mollymooly: I wasn't saying that "man-midwife" was a joke, though I suspect Sterne saw some humor in the term.
Dave K: At Google Books, there seems to have been an increase in "males" and then a decrease, as shown here. Of course to do it right, you'd have to look at how many uses of "males" refer to humans. I think the reason for it is to cover both men and boys.
27. ### Jerry Friedman said,
October 20, 2014 @ 9:32 am
ET: "Unless every description is prefaced with gender, none of them should be.
There is no special school for doctors or lawyers of bosses of either gender.
If you insist on writing women doctors (in 2014!) then all doctors should be called women doctor or man doctor. Anything else is absurd."
So I shouldn't say that some person prefers to go to a man/male or a woman/female doctor unless I want to specify the gender of every doctor? I think that's going too far. Or do you mean that in 2014 no one should have such a preference?
28. ### J. W. Brewer said,
October 20, 2014 @ 10:29 am
Indeed, sometimes people who seem to have a progressive/feminist/egalitarian agenda identify female professionals by sex when they wouldn't do the same for males in the same (presumably historically male-dominated) field. So, e.g., some legal trade journal recently compiled a list of 100 female lawyers ("lady lawyers" is a bit archaic-sounding) who it thought could be plausible candidates to become general counsel of a Fortune 500 company. http://www.insidecounsel.com/2014/07/24/poised-for-prominence. No one would compile a list of 100 specifically male lawyers for that purpose, and if anyone compiled a list of 100 lawyers (w/o prescreening for sex) for such a purpose and the resulting list turned out to be *too* overwhelmingly male (at a guess, more than 70 or 75ish %?), they would attract criticism. Belief that femaleness is a salient characteristic worth expressly noting in a particular context can go with all sorts of substantive views about the appropriate role(s) of women in society, including views that contradict each other.
Once upon a time in the U.S., there was indeed a law school with an all-female student body (at a time when even those other law schools that did not formally exclude women typically had female enrollments of <5%). but the experiment proved unsustainable in the long run and the school became fully co-ed by (sez wikipedia) 1938. It ultimately abandoned its distinctive original name (Portia Law School), which had the piquant irony of suggesting that the most inspiring role model for a would-be female lawyer was fictional. I don't know if there was a parallel in medical or other professional education.
29. ### Chris C. said,
October 20, 2014 @ 4:20 pm
"lady lawyers" is a bit archaic-sounding
But perhaps not incorrect, even politically, given the predilection of American attorneys for styling themselves "Esquire". I suppose this may not be as common as it once was, though.
30. ### J. W. Brewer said,
October 20, 2014 @ 5:27 pm
"Subeditor" is one of those Briticisms (or at least non-USisms) that always disorients me for a moment before I can remember what it means. The marked-for-femaleness "subeditrix" seems almost non-existent (three hits in google books, all possibly jocular). Perhaps women were excluded from the relevant function on Fleet Street so long that most gender-inflected occupational designations in -trix, -ess, etc. had already become archaic/awkward before there were any pioneers to which they might be applied?
OTOH, "lady subeditor" (or "lady sub editor") appears to be in current use in India, with some hits in contexts that suggest at first blush it is not considered beyond the pale or likely to generate lawsuits.
31. ### Michael Watts said,
October 20, 2014 @ 11:53 pm
A woman may not be describable as "womaner" than her sister, but I'm pretty sure you could call her "more woman" than her sister.
32. ### Keith said,
October 21, 2014 @ 1:45 am
@Jerry Friedman
Keith: "Student nurse" seems to be a common term in the UK and doesn't mean a nurse who cares for students.
I think that the term "student nurse" denotes somebody who is both a student and a nurse, who attends lectures and who also at times works in a hospital or clinic under fully-qualified supervisors. It's analogous to the term "student teacher".
In that sense, both the element "student" and "nurse" are nouns and perhaps it would be "more correcter" to write the compound noun as "student-nurse".
K
33. ### Alex said,
October 21, 2014 @ 9:53 am
@Jerry Friedman and Keith. For things like "student nurse," you're meant to resolve the ambiguity with context and background knowledge. We're never going to get everything fully specified. It doesn't help to say they are both nouns, because that is true of both interpretations. Likewise, the proposed hyphenation would not resolve which sense was intended unless we made up a new rule and got everyone to agree to it.
There was a classic Saturday Night Live sketch in the late 70's where Laraine Newman was a child psychologist. She was both a child and a psychologist who treated children. It was funnier than it sounds.
34. ### Jerry Friedman said,
October 21, 2014 @ 12:56 pm
Keith: Would you accept "woman-doctor" as a parallel to your (possibly wishful) suggestion of "student-nurse"? I'd say it suggests at least as strongly that the doctor treats women. In other words, I agree with Alex.
35. ### Keith said,
October 22, 2014 @ 3:21 am
@Jerry
At the risk of giving away my age, I admit that I also (like Stuart Brown) remember when we referred to GPs who happened to be women as "lady doctors".
When speaking English, I prefer to not have to qualify the sex of a person each time I refer to his or her profession. So no, I'm not about to start referring to "woman-doctors".
It's not necessary in English to make the noun agree with the biological gender of the person. French (the language I use most of the time these days) requires this for some professions, though not for all. A primary school teachers is "instituteur" (masc) or "institutrice" (fem), while a secondary school teacher of either sex is "professeur". A doctor is "docteur" or (bizarrely) "doctoresse", though this is rarely used and I prefer to say "doctrice", just to put the cat among the pigeons and see if the person I'm speaking with is paying attention.
I recently started studying Armenian, another language without grammatical gender; there are a very few nouns with a suffix to denote the feminine: "haj" for Armenian man, "hajuhi" for Armenian woman; "usuts'ich" for a male teacher, "usuts'chuhi" for a female teacher.
But I'm straying off the point.
I can imagine a situation where I might, possibly, want to refer specifically to the sex of the doctors in question (maybe there is a preponderance of women in certain specialisations, for example), but in such cases I think I'd still prefer to refer to them as "male doctors" in comparison with "female doctors".
October 22, 2014 @ 3:48 am
I'm hoping the LL comments has no objection to using obscenities, but I was wondering about 'fucking' as a modifier. In the phrase 'he's a fucking idiot', it looks to me as though the f-word is adjectivally modifying (or intensifying) the noun; but one wouldn't talk of 'a very fucking idiot'. I was going to add that one wouldn't say 'fuckingest', either; but then I thought again. I suppose I could imagine, though I haven't encountered, somebody describing another as 'the absolute fuckingest idiot I ever met'. It's an extremely adaptable word, after all (thinking of Anthony Burgess's experience in army when the phrase 'the fucking fucker's fucked' was intended and understood by all as meaning 'the annoying machine is broken').
I hope this doesn't look like derailing the thread.
37. ### Keith said,
October 22, 2014 @ 4:35 am
I'm sure I've read an explanation of this kind of thing, describing how words of certain categories can occupy certain well-defined "slots", how a particular word in a particular slot requires or forbids certain words in other slots, how slots appear in a particular order… e.g. speed slot followed by colour slot followed by noun ("quick, brown fox" and not *"brown, quick fox").
Maybe in "The Atoms of Language", or maybe in one of Pinker's books.
38. ### Martha said,
October 22, 2014 @ 10:33 pm
You can't say "He's a very total idiot," either.
For me, a "lady doctor" is an OB-GYN, a "woman doctor" describes the old-fashioned situation, and a "female doctor" is a doctor who happens to be a woman ("Sally prefers to go to a female doctor.").
39. ### Peter Erwin said,
October 23, 2014 @ 6:02 am
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2021-12-09 07:12:06
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http://latex-community.org/forum/viewtopic.php?f=5&t=2026
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LaTeX forum ⇒ General ⇒ Full context variable depth references to list labels?
LaTeX specific issues not fitting into one of the other forums of this category.
kreil
Posts: 11
Joined: Mon Jul 28, 2008 3:38 am
Full context variable depth references to list labels?
Dear Community,
What I try to do is the following: In a document, I will have enumerate lists at varying depths in the section structure, e.g., I will have some enumerate lists that occur within a section, while others occur in a subsection, etc.
I now need to cross-reference them in full context (sometimes called "legal style"), e.g., list item 3 in section 2.1 should be referenced ad "2.1.3", whereas list item 4 in subsection 2.1.1 should be referenced as "2.1.1.4". (Just the in-text reference should be "2.1.1.4". The list item label in the enumerate environment should still read "4", however, and not "2.1.1.4".)
Annoyingly, I can get Microsoft Word to do this but after a full night of google, forum posts, and digging through CTAN, all I can do is either:-
- construct the reference manually after storing the section counters for each label, e.g., through package "smartref"
- extend the label with additional counters, e.g., using the ref= option of package "enumitem"
also, all direct counter redefinitions seem to link one counter directly to a "level" above.
All the above thus do not work to link a list label to the deepest actually used sectioning command in which it is nested.
I'd be most grateful for a pointer.
Many thanks,
David.
kreil
Posts: 11
Joined: Mon Jul 28, 2008 3:38 am
I have read more of the LaTeX source (am I going in the wrong direction there?) and it seems that \@currentlabel holds exactly what I need. I'd now love to save this before starting an enumerate list and prepend it to the item labels.
\def\savecounter{\protected@edef\savedlabel\@currentlabel}
would store the current label in \savecounter but I can tell that this does not work. Clearly, I don't understand what the above code does and "coding by analogy" from the latex source was apparently not a good strategy.
I had hoped to then say something like
\item\copycounter\label{foo}
to add the saved sectioning label back in front of the new item label, with something like
\def\copycounter{\protected@edef\@currentlabel{\savedlabel-\@currentlabel}}
but, again, this does not do what I thought it should.
David.
Ted
Posts: 94
Joined: Sat Jun 23, 2007 4:11 pm
Location: Columbus, OH
Contact:
UPDATE: Make sure you see my response to this message. It's a much cleaner solution
kreil wrote:I'd be most grateful for a pointer.
See the enumitem package:
This works well for me (though I think there are more compact solutions):
\begin{enumerate} \item First item \begin{enumerate}[label={\theenumi.\arabic*.}] \item Second item \begin{enumerate}[label={\theenumii\arabic*.}] \item\label{item:last} Last item \end{enumerate} \end{enumerate}\end{enumerate} Here is a reference: \ref{item:last}
Of course, make sure
\usepackage{enumitem}
exists in the preamble.
Last edited by Ted on Mon Jul 28, 2008 4:26 pm, edited 1 time in total.
-- Ted [home/blog]
Ted
Posts: 94
Joined: Sat Jun 23, 2007 4:11 pm
Location: Columbus, OH
Contact:
Ted wrote:
kreil wrote:I'd be most grateful for a pointer.
See the enumitem package:
Consulting the documentation, it gives the legal case as an example. Put in your preamble:
\usepackage{enumitem}\newlist{legal}{enumerate}{10}\setlist[legal]{label*=\arabic*.}
That lets you use a "legal" enumeration environment:
\begin{legal} \item First item \begin{legal} \item Second item \begin{legal} \item\label{item:last} Last item \end{legal} \end{legal}\end{legal} Here is a reference: \ref{item:last}
I hope that helps.
-- Ted [home/blog]
kreil
Posts: 11
Joined: Mon Jul 28, 2008 3:38 am
Dear Ted!
Many thanks for your comments. The problem is that unless I manually construct a combined label, I cannot easily combine section and enumerate labels.
My best bet so far has been the following:
\usepackage{enumitem} \makeatletter\def\sc{\protected@edef\savedlabel{\@currentlabel\relax\relax}}\def\cc{\protected@edef\@currentlabel{\savedlabel-\@currentlabel}}\makeatother
Which allows me to "save counter" (\sc), say, after a new subsection command.
I can then recall this and add it to a particular item label by "copy counter" (\cc).
For example:
\section{foo}\subsection{bar}\sc\begin{enumerate}\item aaa\item\cc\label{mine} bbb\end{enumerate}\section{foo too}It is \ref{mine}. % will give 1.1-2 to refer to item 2 in subsection 1.1
So this works now after about 8h of fiddling (the "\relax" took me ages to figure out) but it is still very manual.
Attempts to automate further, e.g., by doing a
\def\longlabel#1{\cc\label#1}
or such still fail. Again, that's of course a result of coding by analogy rather than understanding. So if someone knows of a package that solves this better, or actually understands any of the above code then I'd be grateful for a hint!
Many thanks,
David.
Ted
Posts: 94
Joined: Sat Jun 23, 2007 4:11 pm
Location: Columbus, OH
Contact:
UPDATE: Be sure to see my response to this message. It's a much cleaner solution.
kreil wrote:Many thanks for your comments. The problem is that unless I manually construct a combined label, I cannot easily combine section and enumerate labels.
Thanks for your patience. I see that I completely overlooked the real problem. To start on a cleaner solution, have you tried
\begin{legal}[ref=\arabic{section}.\arabic*.] \item\label{item:first} First item \begin{legal}[ref=\thelegali\arabic*.] \item \label{item:second} Second item \begin{legal}[ref=\thelegalii\arabic*.] \item\label{item:last} Last item \end{legal} \end{legal}\end{legal}
(using the definition of legal given above, of course)?
[ Of course, you could change the initial \arabic{section} so that it includes subsections, etc. ]
Last edited by Ted on Mon Jul 28, 2008 5:37 pm, edited 1 time in total.
-- Ted [home/blog]
Ted
Posts: 94
Joined: Sat Jun 23, 2007 4:11 pm
Location: Columbus, OH
Contact:
UPDATE: The firstlegal environment below is NOT needed if you add a
\newcommand{\thelegal}{\thesubsection.}
in the preamble. See a later message in this thread for details. The modification allows you to use legal without any parameters at every level of your enumerated list (including the first one).
---
Ted wrote:Thanks for your patience. I see that I completely overlooked the real problem. To start on a cleaner solution, have you tried...
Here's a better idea...
First, in the preamble:
\usepackage{enumitem}\newlist{legal}{enumerate}{10}\makeatletter\setlist[legal]{label*=\arabic*.,ref=\csname the\enit@prevlabel\endcsname\arabic*.}\makeatother
Then you can do
\begin{legal}[ref=\thesubsection.\arabic*.] \item\label{item:first} First item \begin{legal} \item \label{item:second} Second item \begin{legal} \item\label{item:last} Last item \end{legal} \end{legal}\end{legal}
So the only special thing you need to do within the document is add that [ref=...] line to the first legal environment. All nested ones are handled without any special arguments.
You can simplify that first legal environment a little... In your preamble, put
\newenvironment{firstlegal}{\begin{legal}[ref=\thesubsection.\arabic*.]}{\end{legal}}
Then you can use
\begin{firstlegal} \item\label{item:first} First item \begin{legal} \item \label{item:second} Second item \begin{legal} \item\label{item:last} Last item \end{legal} \end{legal}\end{firstlegal}
How does that work for you?
Last edited by Ted on Mon Jul 28, 2008 7:17 pm, edited 2 times in total.
-- Ted [home/blog]
kreil
Posts: 11
Joined: Mon Jul 28, 2008 3:38 am
Dear Ted,
Yes, that looks nice! I tried your approach ...
Ted wrote:First, in the preamble:
\usepackage{enumitem}\newlist{legal}{enumerate}{10}\makeatletter\setlist[legal]{label*=\arabic*.,ref=\csname the\enit@prevlabel\endcsname\arabic*.}\makeatother
Then you can do
\begin{legal}[ref=\thesubsection.\arabic*.] \item\label{item:first} First item \begin{legal} \item \label{item:second} Second item \begin{legal} \item\label{item:last} Last item \end{legal} \end{legal}\end{legal}
So the only special thing you need to do within the document is add that [ref=...] line to the first legal environment. All nested ones are handled without any special arguments.
Are the inner loops supposed to inherit the ref from the outer loops? That doesn't seem to work for me. A label/ref to the outer loop works fine but the inner loops just have the "normal" legal style refs.
What works quite nicely is substituting the first line with
\begin{legal}[ref=\savedlabel-\arabic*.]
Together with the "\sc" save counter macro I posted above, this makes the first loop start from the section nesting level where the save counter macro was last applied.
So if we could get the inner loops to inherit that would be great!
Best wishes,
David.
Ted
Posts: 94
Joined: Sat Jun 23, 2007 4:11 pm
Location: Columbus, OH
Contact:
kreil wrote:Yes, that looks nice! I tried your approach ...
kreil wrote:Are the inner loops supposed to inherit the ref from the outer loops?
Yes, they are. It's in the \setlist. The \csname should cause the previous label to built into the refs.
kreil wrote:That doesn't seem to work for me. A label/ref to the outer loop works fine but the inner loops just have the "normal" legal style refs.
It works for me. Here's a (working) complete example:
\documentclass{article} \usepackage{enumitem}\newlist{legal}{enumerate}{10}\makeatletter\setlist[legal]{label*=\arabic*.,ref=\csname the\enit@prevlabel\endcsname\arabic*.}\makeatother \newenvironment{firstlegal}{\begin{legal}[ref=\thesubsection.\arabic*.]}{\end{legal}} \begin{document} \section{First section} First: \ref{item:first}\\Second: \ref{item:second}\\Third: \ref{item:last} \section{Second section}\subsection{A subsection}\subsection{Another subsection} \begin{firstlegal} \item An item \item Another item \item\label{item:first} First labeled item \begin{legal} \item \label{item:second} Second item \begin{legal} \item\label{item:last} Last item \end{legal} \end{legal}\end{firstlegal} \end{document}
That produces:
First: 2.2.3.
Second: 2.2.3.1.
Third: 2.2.3.1.1.
which is what I THINK you want. Right?
-- Ted [home/blog]
kreil
Posts: 11
Joined: Mon Jul 28, 2008 3:38 am
Perfect!
Many thanks, I don't know what I got wrong when I first tried. The below does exactly what I want (even semi-automatically picks up the section level).
Thanks a lot again for your help!
All the best,
David.
\documentclass{article} \usepackage{enumitem}\newlist{legal}{enumerate}{10}\makeatletter\def\sc{\protected@edef\savedlabel{\@currentlabel\relax\relax}}\setlist[legal]{label*=\arabic*.,ref=\csname the\enit@prevlabel\endcsname\arabic*.}\makeatother\newenvironment{firstlegal}{\begin{legal}[ref=\savedlabel-\arabic*.]}{\end{legal}} \begin{document} \section{First section} First: \ref{item:first}\\Second: \ref{item:second}\\Third: \ref{item:last} \section{Second section}\subsection{A subsection}\subsection{Another subsection}\sc \begin{firstlegal} \item An item \item Another item \item\label{item:first} First labeled item \begin{legal} \item \label{item:second} Second item \begin{legal} \item\label{item:last} Last item \end{legal} \end{legal}\end{firstlegal} \end{document}
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2017-01-16 19:20:05
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https://physics.stackexchange.com/questions/34486/reflection-positivity-in-general
|
Reflection positivity in general
In the Euclidean QFT obtained by "Wick-rotating" a unitary QFT, the correlation functions satisfy a property called reflection positivity, see e.g. this Wikipedia article for the case of a scalar field.
What's the correct formulation if you have chiral fermions and/or terms like the QCD theta angle? Could someone give the references?
This is one of the first hits for "reflection positivity" on google, so someone better answer it!
Reflection positivity is just positivity of the Hilbert space norm along with the fact that complex conjugation negates the imaginary time.
To begin the discussion, let's just consider spacetime of the form $$Y \times \mathbb{R}_t$$. We have a Hilbert space associated to $$Y \times 0$$ and some operators $$\phi(y,0)$$ which we're interested in computing correlation functions with. We define the usual time evolved operators by $$\phi(y,t) = e^{-itH} \phi(y,0) e^{itH}.$$ These have the nice property of being Hermitian if $$\phi(y,0)$$ is. We can also define the analytically continued operators by just replacing $$t$$ with $$z = t + i\tau$$. Now taking the adjoint we find (assuming $$\phi(y,0)$$ is Hermitian) $$\phi(y,z)^\dagger = \phi(y,z^*),$$ in particular $$\phi(y,i\tau)^\dagger = \phi(y,-i\tau).$$
Now let us consider the states $$\phi(y,z)|0\rangle$$ obtained from the vacuum $$|0\rangle$$. These are nonzero Hilbert space states, so they have positive norm, ie. $$\langle 0 | \phi(y,z)^\dagger \phi(y,z)|0\rangle = \langle 0 | \phi(y,z^*) \phi(y,z) | 0 \rangle > 0.$$ (Note that this is automatically properly imaginary-time-ordered as long as $$\tau>0$$, which we need anyway to have good states). In particular $$\langle 0 | \phi(y,-i\tau) \phi(y,i\tau) | 0 \rangle > 0.$$ We see the reflection principle at work here, $$i\tau \mapsto -i\tau$$, so if we imagined this was computed in a Euclidean path integral, it would be a reflection-symmetric configuration on $$Y \times \mathbb{R}_{\tau}$$.
The general statement of reflection positivity is that for all such reflection-symmetric configurations of Hermitian operators, even through other coordinates on other spacetime manifolds, the Euclidean path integral always computes something positive. In all these cases the proof is just to realize that what you're computing is the norm of some Hilbert space state.
It does not matter if there's chiral fermions or a theta angle, although these do Wick rotate in interesting ways, since (spacetime)parity-odd terms remain imaginary in the Euclidean action. Sorry I don't know any references, but I'm trying to write one!
• Sorry I don't know any references, but I'm trying to write one! – now that you got me excited, you must deliver or I will be left heartbroken. Feb 26, 2020 at 18:48
• @Prof.Legolasov Actually here is a nice reference that goes into some of the details: arxiv.org/abs/1602.07982 Feb 26, 2020 at 21:10
• CFT is nice, but I’m more interested in the axiomatic QFT. I’d kill for a good introduction to OS and Wightman with explicitly built examples such as $\phi^4_2$, $\phi^4_3$, $P(\phi)_2$, etc Feb 27, 2020 at 23:14
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2022-05-20 23:21:45
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http://openstudy.com/updates/50c0312ee4b0231994ecdfea
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## iheartfood Group Title Which of the following correctly represents the coordinates of the foci of the ellipse attached? one year ago one year ago
1. iheartfood Group Title
2. iheartfood Group Title
answer choices A,B,C,D from top to bottom :) ***idk :( please explain! THanks :D
3. Hero Group Title
There's a great tutorial on this at purplemath.com
4. iheartfood Group Title
could you please explain this to me? :)
5. Hero Group Title
I'm not very good at explaining things.
6. iheartfood Group Title
so im not quite sure... but I'm thinking B or D right? :/
7. iheartfood Group Title
aww :( okay :/ but is it B or D? thats what i have simplified it down to so far...
8. Hero Group Title
You sound like you're guessing.
9. iheartfood Group Title
no I'm not cuz i remember a little bit... i got this: The denominator to y is 36, which is > than denominator of x which is 9... so its vertical right?? so you list the y first?? like this?? (y-4)^2/36 + (x+2)^2/9=1... idk am i on the right track so far??
10. iheartfood Group Title
does that look right to you so far??
11. Hero Group Title
I thought that was a minus, it's a plus, so yes, it will be either B or D.
12. iheartfood Group Title
kk :) so what i did so far is right? :D
13. Hero Group Title
At first, I thought it was $\frac{(x+2)^2}{9} - \frac{(y-4)^2}{36} = 1$
14. iheartfood Group Title
and the center is (-2, 4) right?? which leads me to either B or D... but idk how to find that last part :(
15. iheartfood Group Title
ohhh hhaa yeah its + :)
16. Hero Group Title
I looked at it wrong
17. iheartfood Group Title
haha yeah its all good =D
18. Hero Group Title
If you made it this far, you should be able to figure out which one it is between the two. I told you that you need to be more confident in your work rather than ask for confirmation every time you worked out a problem. When I was your age, I never asked for confirmation. I was always confident in my work.
19. Hero Group Title
I didn't use calculators either. I couldn't afford one when I was in HS.
20. iheartfood Group Title
oh yeah i just don't know what to do after that.. :(
21. iheartfood Group Title
can you please show me what to do from there?? idk what to do :(
22. Hero Group Title
I told you to review the tutorial on purplemath.com
23. iheartfood Group Title
i don't know where to find it though .... :( can u pls show me what to do from here?? :)
24. iheartfood Group Title
k so i found this link... http://www.purplemath.com/modules/ellipse2.htm is that it?? I'm reading on it and from that i think the answer is B.... is that it?? am i understanding that link correctly?? is it B?? :D @hero? :)
25. Hero Group Title
As I stated before, you should be more confident in your own work. I believe you're smart enough to understand this without having to constant beg me for confirmation.
26. iheartfood Group Title
okay... well i didn't know this one :( could u pls tell me if i got it right?? =)
27. iheartfood Group Title
but the tutorial helped a bit... did i get it?? is it B? :O
28. Hero Group Title
I'm not going to answer that question.
29. iheartfood Group Title
okay :) thanks for your help though @Hero!! :) It's much appreciated :)
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2014-07-28 06:39:11
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https://planetmath.org/polyomino
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# polyomino
A polyomino consists of a number of identical connected squares placed in distinct locations in the plane so that at least one side of each square is adjacent to (i.e. completely coincides with the side of) another square (if the polyomino consists of at least two squares).
A polyomino with $n$ squares is called an n-omino. For small $n$, polyominoes have special names. A 1-omino is called a monomino, a 2-omino a domino, a 3-omino a tromino or triomino, etc. The famous Tetris video game derives its name from the fact that the bricks are tetrominoes or 4-ominoes.
Fixed polyominoes (which are also called lattice animals) are considered distinct if they cannot be translated into each other, while free polyominoes must also be distinct under rotation and reflection.
The topic of how many distinct (free or fixed) n-ominoes exist for a given $n$ has been the subject of much research. It is known that the number of free n-ominoes $A_{n}$ grows exponentially. More precisely, it can be proven that $3.72^{n}.
Polyominoes are special instances of polyforms.
Title polyomino Canonical name Polyomino Date of creation 2013-03-22 15:20:18 Last modified on 2013-03-22 15:20:18 Owner s0 (9826) Last modified by s0 (9826) Numerical id 10 Author s0 (9826) Entry type Definition Classification msc 05B50 Defines n-omino Defines domino Defines tromino Defines tetromino Defines fixed polyomino Defines lattice animal
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2020-04-09 11:27:17
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https://www.maths.cam.ac.uk/computing/laptops/print_configs/osxprint105problems
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# Troubleshooting printer detection in Mac OSX 10.5
Starting with Mac OSX 10.5 Apple changed the default way it listens for printer adverts, and the same is true in OSX 10.6. As a result the printing system will no longer see the standard CUPS/IPP printer adverts from the laptop network print server. Therefore by default no shared printers show up in the list when you try to add one.
This change in behaviour also stops it seeing printers advertised by Macs running older versions of Mac OSX and so this causes problems for many people.
To correct the problem a change needs to be made (once) to the cups configuration.
Note that upgrading Mac OSX version (e.g. from 10.5 to 10.6) will usually undo the change. Using the Reset Printing System feature will certainly undo the change.
Apple used to have a helpful Support Article about this but removed it in March 2010. That article advised people to run the following command in a terminal:
cupsctl BrowseProtocols='"cups dnssd"'
Please note the nested quotes in the command!
This tells the Apple cups server to listen for standard CUPS/IPP printer adverts or use DNSSD (Bonjour) - by default on Mac OSX 10.5/10.6 it only uses the latter.
The lapnet print server only sends out CUPS/IPP adverts - just like older versions of Mac OSX did - so the same change is needed to see our printer adverts.
Once the command has been run your Mac should start to see the CUPS/IPP printer adverts. Within 30-120 seconds it should have picked up all our printers. The next time you add a printer you should see them all.
## Freezing of Printing Preferences and/or other OSX features
A few applications (or dummy printer setups such as (some versions of) Adobe Distiller) seem to prevent the cups daemon from working once the required change has been made. If you have one of these then after running the cupsctl command cupsd to fail to run and applications like Systems Preferences may become unresponsive. To work round this you need to manually fix the configuration - since access via the usual cupsctl or GUI mechanisms will not work.
If there are "freezing" problems after making the change (cupsctl command, above) then removeing old printer definitions usually cures it. Run the following in a Terminal:
sudo mv /etc/cups/printers.conf /etc/cups/printers.conf.BAD
sudo killall cupsd
If launchd does not automatically restart cupsd after killing it, you may need to tell it to unload/load the cups configuration again, e.g. with:
sudo launchctl unload /System/Library/LaunchDaemons/org.cups.cupsd.plist
If renaming the printers.conf file fixes things then you may want to read through the old file to track down which printer definition caused the problem. Note that manually editing printers.conf may not always work since cupsd re-writes this file itself. Let us know if you need help with this.
## Apple Firewall blocking IPP
If you have the Apple Firewall set to (OSX 10.5) Allow only essential services or (OSX 10.6) Block all incoming connections then it will block the IPP printer advertisements, and so you will see no printers at all
To test if this if the problem try disabling the firewall, if that works you should be able to re-enable it with slightly less strict rules to allow cups to see the printer adverts.
The configuration options are under Advanced in the Firewall tab (From System Preferences select Security and then the Firewall tab).
The settings are different on OSX 10.5 compared to 10.6. For OSX 10.6 allowing Automatically allow signed software to receive incoming connections seems to be sufficient to allow cupsd to work.
See Apple's article about the Firewall settings for details.
## Odd lists of printers when trying to print
Normally apps should show only the printers that you have added (with + or using the Add Printer options, but sometimes it seems to get confused and shows a seemingly random set of printers as well.
If this happens it seems to help to clear out some of the caches, running in a Terminal:
sudo launchctl unload /System/Library/LaunchDaemons/org.cups.cupsd.plist
sudo rm -rf /var/spool/cups/cache/* /Library/Preferences/org.cups.printers.plist
This stops cupsd, removes the caches and then starts it again. The cache files will be regenerated over the next few seconds - up to 120 seconds to get a complete list.
You can see the current list of favourite printers by running in a Terminal:
FAV=~/Library/Preferences/com.apple.print.favorites.plist
/usr/libexec/PlistBuddy -c print \$FAV
Apple seem to provide no obvious way to remove printers from this list since OSX 10.6.
## More troubleshooting
If you still do not see printers then here are some troubleshooting steps which you can try. In a terminal window run:
grep -i '^browseprotocol' /etc/cups/cupsd.conf
/usr/bin/lpstat -v
/usr/sbin/netstat -p udp | grep ipp
sudo /usr/sbin/lsof -i udp:ipp
If things are working as expected the lpstat command will show a long list of printers, and the netstat command will show something like:
udp4 0 0 *.ipp *.*
and the lsof command ought to report something a bit like:
COMMAND PID USER FD TYPE DEVICE SIZE/OFF NODE NAME
cupsd 25 root 8u IPv4 0x1913d98 0t0 UDP *:ipp
On rare occasions some other service may have already chosen to use the CUPS/IPP udp-port (631). If this happens the cups server will be unable to see our printer adverts and you will see no printers. That can usually be corrected by simply restarting the machine.
If things still do not work please let us know and if possible include the contents of the files /etc/cups/cupsd.conf and /etc/cups/printers.conf and the result of running the lpstat, netstat and lsof commands mentioned above.
The cupsctl command only needs to be run once - unless you change it or reset the printing system settings.
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2020-03-28 14:54:56
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