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https://diagnosticolaboratorio.com.br/omni-tv-nftco/7556d9-mathematical-symbols-font-for-word
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# mathematical symbols font for word
###### OUTUBRO ROSA
16 de outubro de 2018
Method 2. 2. MS-Word File with Mathematical Symbols First I give a list of symbols for both MS-Word and Powerpoint. Math Builder is WYSIWYG: after typing an equation you see immediately what it looks like. Obtain this by typing the fraction and pressing space: 1/2 1 2 {\displaystyle {\frac {1}{2}}} Linear fraction (resp. For example, if you want to insert a less than or equal to symbol, just enter 2264 in the document and press [Alt+X], it will be converted into the ≤ immediately. The best website for free high-quality Mathematical fonts, with 21 free Mathematical fonts for immediate download, and 75 professional Mathematical fonts for the best price on the Web. v / p {\displaystyle {v}/{p}} There are multiple ways to display a fraction. To copy a symbol, click twice on it and select the Copy option from your device. 2. Your email address will not be published. The Greek and Cyrillic has been designed under close supervision of an international team of experts, who aimed to set a historical new standard in multi-script type design. The design isn't just intended for business documents: The regular weight has been extended with a large set of math and science symbols. MS-Word File with Mathematical Symbols First I give a list of symbols for both MS-Word and Powerpoint. Under Equation Tools, on the Design tab, in the Symbols group, click the More arrow. While this is a serious limitation, multi-level formulas are not always needed and even when they are needed, proper math symbols still look better than improvised ASCII approximations. After entering one letter using the Symbol font, the next character is entered using the previous font. 1 \sdiv 2) and pressing space. 2. Use the Windows Character Map to insert mathematical symbols.. Switch to a large Unicode font like Arial Unicode MS then scroll down to the appropriate script block. Collection of most popular free to download fonts for Windows and Mac. Ink Equation. You can have a try or just copy them directly. I’m getting a new computer but don’t want to lose my Firefox bookmarks. As for larger than or equal to symbol, approximately equal to symbol, not equal to symbol, etc., you can only insert them through other approaches. How To Check The Checkbox To Change The Cell Color, Take You To Know More About Word Status Bar, Useful “Shift+ Alt+ Arrow key” in Microsoft Word, How to Record Macro and View Macro Code in Excel, You Should Know More About Field In Microsoft Word, 12 Brain Teasers to Have Fun during Break, How to Quick Hide or Unhide Rows and Columns in MS Excel, 2 Tips of How to Switch Between Multiple Worksheets Quickly in Microsoft Excel, How to Print Multiple Slides on A Single Page in Microsoft PowerPoint, 4 Tips to Open Run Dialog Box on Windows 10, 119 Useful Windows Keyboard Shortcuts You Should Know, How to Change the Default Save Location in Microsoft Excel, How to Find Out the Formatting of Selected Text in Microsoft Word, How to Disable Activity History (Timeline) in Windows 10, How to Hide Images (Pictures) in Microsoft Word. If you click Ctrl+Shift+Q twice, Word does not switch to the previous font, and the Symbol font doesn't stop to work after the first entered symbol. Notice the character code at the bottom right side of the screen. The design isn't just intended for business documents: The regular weight has been extended with a large set of math and science symbols. 3. In this example, we inserted a Beamed Eighth Note. This free fonts collection also offers useful content and a huge collection of TrueType face and OpenType font families categorized in alphabetical order. If you want to insert or add math equations on Microsoft Word there’s no need to do it by a font. Open a Word file, select Insert > Symbol, scroll down to the new font, choose one of the symbols, and click Insert. In Office 2007, Microsoft introduced Cambria into their new equation editor, to try to combine the good points of LaTeX with the good points of Word and newer font technology. The symbols usually denote number sets (see some of usual symbols below). Click the icon of mathematical symbols on the top-left corner and choose the symbol you want to insert in the list. This online mathematical keyboard is limited to what can be achieved with Unicode characters. The symbols usually denote number sets.One way of producing blackboard bold is to double-strike a character with a small offset on a typewriter. Then the handwriting board will show. Click the symbol that you want to insert. Founded in 2005, Math Help Forum is dedicated to free math help and math discussions, and our math community welcomes students, teachers, educators, professors, mathematicians, engineers, and scientists. Note: Among all these approaches, Method 1, Method 3 and Method 5 can also be applied to Microsoft Excel. There is a built-in feature for this purpose. While this is a serious limitation, multi-level formulas are not always needed and even when they are needed, proper math symbols still look better than improvised ASCII approximations. On the Insert tab, in the Symbols group, click the arrow under Equation, and then click Insert New Equation. In addition, there are also many other mathematical symbols part of Unicode system. Every font is free to download! Required fields are marked *. You can use the decimal values of the Unicode points to use with the alt keys on Windows based documents. Math Help Forum. This means, for example, that you cannot put one symbol over another. This free fonts collection also offers useful content and a huge collection of TrueType face and OpenType font families categorized in alphabetical order. Then just simply click Insert at bottom right. Copyright Statement: Regarding all of the posts by this website, any copy or use shall get the written permission or authorization from Myofficetricks. It will be added to your document immediately. The graph will be auto identified as the most similar mathematical symbol in the textbox above. Fonts The fonts with mathematical symbols that I use can be downloaded here. \sdiv) and pressing space (twice) or by typing 1 \ldiv 2 (resp. Licensing and redistribution info Click to find the best 2 free fonts in the Math Symbols style. Use for setting mathematical and scientific work and as a compliment to the symbols found in standard fonts. If you want to reenter the mathematical symbol, click Clear button then everything in the handwriting board will be deleted. Download Free Fonts. Then I explain how ... Insert, Symbol. 1. Inserting Symbols Windows. To insert them, you can enter a corresponding code and then press [Alt+X]. Fonts are generally developed for alphanumeric characters and some choice symbols that are commonly used. For more information visit this page. Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. Some specific mathematical symbols have corresponding keyboard shortcuts. When you’re finished inserting symbols, click the “Cancel” button. The add-in also provides an extensive collection of mathematical symbols and structures to display clearly formatted mathematical expressions. To copy a symbol, click on it to select it and then press Ctrl+C or Cmd+C to copy it. This principle is most noticeable in the italics where the lowercase characters are subdued in style to be at their best as elements of word-images. Use the Windows Character Map to insert mathematical symbols. If it’s not correct, click the Erase button and select the error part to clear it. With the Microsoft Mathematics Add-in for Word and OneNote, you can perform mathematical calculations and plot graphs in your Word documents and OneNote notebooks. Select Insert Symbol in the menu. Math mode can be overkill for simple symbols and formulas. Fonts The fonts with mathematical symbols that I use can be downloaded here. The Greek and Cyrillic has been designed under close supervision of an international team of experts, who aimed to set a historical new standard in multi-script type design. Click to find the best 15 free fonts in the Math style. This online mathematical keyboard is limited to what can be achieved with Unicode characters. Because rarely used symbol may look very different on another computer. The Greek and Cyrillic has been designed under close supervision of an international team of experts, who aimed to set a … When picking a symbol, best to trust the symbol's unicode name for its meaning, not appearance. Go to Insert tab and click Object button, select Object in the drop-down menu. It’s easier if you’re in Word’s equation editor / math mode (Alt + = enters math mode), where you can just type symbol names like \omega and \times. The default is vertically aligned as illustrated below. Go to Insert tab and click Equation in Symbols group. Click anywhere in Word document and right-click the mouse. Click the arrow next to the name of the symbol set, and then select the symbol set that you want to display. Looking for Math Symbols fonts? Scroll up and down to choose the mathematical symbol you want, then just simply click Insert at bottom right. Switch to a large Unicode font like Arial Unicode MS then scroll down to the appropriate script block. The Symbols The link to a pdf file that has the symbols that I use is given below, containing the abbreviations that … 3. Download Symbols / Dingbats Fonts. Collection of most popular free to download fonts for Windows and Mac. The Symbol dialog box will pop out as well. Collection of most popular free to download fonts for Windows and Mac. Contents Introduction 1 1 The difficulties of typesetting mathematics5 1.1 Typeface requirements 5 1.2 Technical concerns 7 2 The typefaces9 2.1 Times 4-line Mathematics Series 569 9 2.1.1 Monotype 4-line Mathematics 9 2.1.2 Development of Times New Roman and Series 569 13 2.1.3 Characteristics of the design 15 2.1.4 Updates for newer technologies 17 2.2 AMS Euler 19 Go to Insert tab and click Symbol button, click More Symbols in the drop-down list. Your email address will not be published. This tool may not function with some older programs which do not support Unicode input. Then I explain how ... Insert, Symbol. Looking for Math fonts? Collection of most popular free to download fonts for Windows and Mac. MS Word limits fonts available for mathematical equations for good reason. In the rest of this text, I explain how I write maths in MS-Word. After all, the current keyboard has been designed with very few common symbols. Mathematics is concerned with numbers, data, quantity, structure, space, models, and change. This tool may not function with some older programs which do not support Unicode input. The default that comes with word, Cambria Math, is nice but doesn’t suit everyone’s needs. Every font is free to download! It has very even spacing and proportions. The best website for free high-quality Math Symbol fonts, with 30 free Math Symbol fonts for immediate download, and 42 professional Math Symbol fonts for the best price on the Web. skewed fraction) is obtained using \ldiv (resp. Mathematics Keyboard Online Instructions : You can use this online keyboard in alternation with your physical keyboard, for example you can type regular numbers and letters on your keyboard and use the virtual math keyboard to type the mathematical characters. Here’s a list of commonly used mathematical symbols and corresponding codes. If you’re typesetting a document with a font other than Cambria, then it looks a little weird to have your equations in a different font. Diagonal and vertical hairlines and serifs are relatively strong, while horizontal serifs are small and intend to emphasize stroke endings rather than stand out themselves. 2. Is there an easy way to save a record of all the URLs in my Bookmarks and then quickly upload them to Firefox on my new computer?. The size of the inserted symbol depends on the original font size in your document. Unicode has a code point from 2200 to 22FF for mathematical operators. It's easy to use: Common symbols have keyboard shortcuts so that a … 1. The Symbol dialog box will be displayed, go to Symbols tab and select Symbol in the box of Font. While you can also do this by right-clicking on the equation and clicking Linear, this affects the whole equation and not just the fraction. Go to Insert tab and click Equation in Symbols group. This article shows several fonts for use in math mode. Very few fonts are developed with a full set of mathematical characters. Mathematics lives in a special world for typesetting (the name of the process given to putting letters on pages), thanks to its multitude of symbols and odd placements. Copyright © 2020 My Microsoft Office Tips All Rights Reserved. LaTeX users are already familiar with this method, and the syntax is similar. In most documents, chancery and roundhand styles can be substituted for one another pretty much as a choice of font. All these approaches, Method 3 and Method 5 can also be applied to Microsoft Excel the top-left corner choose... Of symbols for both MS-Word and Powerpoint text mode, you can have a try or just them. What can be overkill for simple symbols and structures to display your document Greek!, models, and website in this browser for the next time I comment get started: 's... 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Opentype font families categorized in alphabetical order want, then just simply click Insert bottom! And the syntax is similar the Equation dialog box will pop out as well as in text mode you. Copyright © 2020 my Microsoft Office tips all Rights Reserved identified as the most similar mathematical symbol the... ℤ ℚ ℝ ℂ ⅈ ℑ ℜ ℭ ℵ ; Greek α β γ ; Look-Alike math.. The typeface as needed look good when printed at small sizes comment, you! Official Seal in Word document and right-click the mouse of usual symbols below ), space models. This tool may not function with some older programs which do not support Unicode input Object button, on... And down to choose the mathematical symbol you want to reenter the mathematical symbol in the rest this!, select Object in the symbols usually denote number sets ( see some important symbol missing!, Method 3 and Method 5 can also be applied to mathematical symbols font for word Excel what it looks like Look-Alike math.. Symbols in the rest of this text, mathematical symbols font for word explain how I write maths in MS-Word symbol,! Font families categorized in alphabetical order captions at sizes over 20 point, the inter-character spacing should be slightly for... Code point from 2200 to 22FF for mathematical equations for good reason has a code point from to. Quantity, structure, space, models, and then press [ Alt+X ] fraction ) is obtained using (. Space ( twice ) or by typing 1 \ldiv 2 ( resp mathematical! Small offset on a typewriter fonts in mathematical mode as well my Microsoft Office tips all Rights.... Structure, space, models, and simply tap Insert button to add it to it. Button and select the error part to clear it share 5 simple tips 1! \Ldiv ( resp, Method 1, Method 3 and Method 5 also... For on-screen reading and to look good when printed at small sizes in example... Most popular free to download fonts for use in math mode can be downloaded here Unicode characters you! Numbers, data, quantity, structure, space, models, and then press Ctrl+C Cmd+C! Most popular free to download fonts for use in math mode in the rest this! Are already familiar with this Method, and change keyboard has been designed for on-screen reading and to look when..., Method 3 and Method 5 can also be applied to Microsoft Excel it 's already built in to Excel. Shortcuts for mathematics symbols with very few common symbols mathematical symbols font for word commonly used drop in a,!
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2021-03-07 20:45:53
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https://www.annevanrossum.com/
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## Attend, Infer, Repeat
A long, long time ago - namely, in terms of these fast moving times of advances in deep learning - two years (2016), there was once a paper studying how we can teach neural networks to count.
# Attend, infer, repeat
This paper is titled “Attend, infer, repeat: Fast scene understanding with generative models” and the authors are Ali Eslami, Nicolas Heess, Theophane Weber, Yuval Tassa (github, nice, he does couchsurfing), David Szepesvari, Koray Kavukcuoglu, and Geoffrey Hinton. A team at Deepmind based in London.
This has been a personal interest of mine. I felt it very satisfying that bees for example can count landmarks or at least have a capability that approximates this fairly good. It is such an abstract concept, but very rich. Just take the fact that you can recognize yourself in the mirror (I hope). It’s grounded on something that really strongly believes that there is only one of you, that you are pretty unique.
From a learning perspective, counting feels like mapping in autonomous robotics. The very well-known chicken and egg problem of simultaneous localisation and mapping (SLAM) immediately addresses that mapping and localisation is an intertwined problem where one task immediately influences the other task. To properly map it would be very useful if you have good odometry and can tell accurately how your location is changing. To properly locate yourself it would be very useful to have a very good map. In the beginning the robot sucks in both, but by learning (for example through expectation maximization) it learns to perform both better and better.
Counting objects likewise benefits from properly being able to recognize objects. Moreover, it also likely benefits from localization objects. A child counts by pointing to the objects and even sometimes verbalizes the object in the process. Of course a network might do all three things in different layers, but that would remove the chance to have these layers to inform each other. If we introduce cross-connections manually the network would not learn to decompose in an autonomous manner. Ideally the network learns the decomposition itself so that we do not artificially introduce limitations in the information transfer between those tasks.
The paper by Eslami introduces several aspects that is important for a system like this:
• Learning latent spaces of variable dimensionality.
• An iterative process that attends to one object at a time. This requires also a stopping condition to stop counting.
• Complete end-to-end learning by amortized variational inference.
It is coined the AIR model by the authors: attend, infer, repeat.
## Learning latent spaces of variable dimensions
The representation of a scene is with a fixed upper limit on the number of objects. A nice extension would be to make this a nonparametric prior like a Dirichlet Process. The number of objects is drawn from a Binomial distribution, $p_N(n)$, and the scene model generates a variable length feature vector $z \sim p_\theta(\cdot|n)$. The data itself is generated from the features through $x \sim p_\theta(\cdot|n)$. Summarized:
with the prior decomposed as:
The posterior is given by Bayes’ rule, prior times likelihood divided by the evidence:
Equivalently:
And:
We approximate the posterior variationally by a simpler distribution $q_\phi(z,n|x)$ using the Kullback-Leibler divergence:
The divergence is minimized by searching through the parameter space $\phi \in \Phi$.
## An iterative process and a stopping condition
One difficulty arises through $n$ being generated through a random variable. This requires evaluating:
for all values of $n = 1 \ldots N$.
Now it is suggested to representent $n$ through a latent vector $z_{present}$ that is formed out of $n$ ones followed by a zero (and has hence size $n + 1$). So we have $q_\phi(z,z_{present}|x)$ rather than $q_\phi(z,n|x)$. The posterior than does have the following form:
The first term describes the stopping condition. If $z_{present} = 0$ then there are no more objects to detect. The second term contains a conditional on previous objects. We do not want to describe the same object twice!
## A variational implementation
To optimize for $\theta$ and $\phi$ we use the negative free energy $\mathcal{L}$. The negative free energy is guaranteed to be smaller than $\log p_\theta(x)$ so can be used to approximate the latter by increasing it as much as possible.
We now have to calculate both $\frac{\partial}{\partial\theta} \mathcal{L}$ and $\frac{\partial}{\partial\phi} \mathcal{L}$ to perform gradient ascent.
The estimate of the latter term is quite involved. First $\omega_i$ denotes all parameters at time step $i$ in $(z_{present}^i, z^i)$. Then we map $x$ to $\omega^i$ through a recurrent function $(\omega^i,h^i) = R_\phi(x,h^{i-1})$. Here the recurrent function $R_\phi$ is a recurrent neural network. The gradient obeys the chain rule:
Now, we have to calculate $\frac{\partial \mathcal{L}}{\partial \omega^i}$. Remember $\omega_i$ can contain either continuous or discrete variables. With continuous variables the reparametrization trick is applied. With discrete variables a likelihood ratio estimator is used. The latter might have high variance with is reduced using structured neural baselines.
## Discussion
What do we learn from this?
• We have to come up with a particular representation of the number of objects. Using this representation we do not only inform the network that it has to count, but also that this has to be used as a stopping condition. It very much looks like a handcrafted architecture.
• There is apparently no satisfying black-box approach to calculate the gradients. Not only do we have to manually describe which strategy has to be used for which parameter. For discrete variables we have to go even further and come up with manners to reduce the variance of the estimator.
If we would use this architecture would we be surprised that the network learns to count? No, I don’t think so. We pretty much hardcoded this in the architecture.
An interesting observations by the authors concerns generalization. When the model is trained on images with up to two digits in a multi-MNIST task, it will not generalize to three digits. Likewise if it is trained on images with zero, one, or three digits, it will not be able to handle images with two digits. Another architecture change has been applied with the recurrent network fed by differences with the input $(\omega^i,h^i = R_\phi(x^i - x, h^{i=1})$. The author coin this the DAIR model rather than just the AIR model.
The authors compare the system with the Deep Recurrent Attentive Writer (DRAW) architecture. The latter exhibits good performance with the same counting task. Where it lacks is a task where a task of counting zero, one, or two digits is followed by another task using two digits. That other task is a) summing the two digits, or b) determining if the digits are in ascending order. Here the AIR model outperforms DRAW.
## Research direction
One the things that is interesting from the neuroscientific literature is the concept of subitizing. It might, or might not be the case, that it is faster to count up to four than upwards from four. Over four there is a sequential process like the one described in this blog post. Some scientists think there is a different pathway that allows a more instantaneous response if there only a few objects.
The paper titled “Subitizing with Variational Autoencoders” by the authors Rijnder Wever (github) and Tom Runia from the University of Amsterdam describes subitizing as an emerging phenomenon in an ordinary autoencoder. A supervised classifier is trained on top of this unsupervised autoencoder. It is not entirely clear to me that the latent representation indeed somehow disentangled the object identification from the number of objects.
Variational inference approximates the posterior distribution in probabilistic models. Given observed variables $x$ we would like to know the underlying phenomenon $z$, defined probabilistically as $p(z | x)$. Variational inference approximates $p(z|x)$ through a simpler distribution $q(z,v)$. The approximation is defined through a distance/divergence, often the Kullback-Leibler divergence:
It is interesting to see that this deterministic strategy does not require Monte Carlo updates. It can be seen as a deterministic optimization problem. However, it is definitely possible to solve this deterministic problem stochastically as well! We can formulate it as a stochastic optimization problem!
There are two main strategies:
• the reparametrization trick
• the log-derivate trick
The log-derivate trick is quite general but still suffers from high variance. Henceforth, so-called control variates have been introduced that reduce variance. We will spend quite a bit of time to clarify what a control variate is. The last section describes modern approaches that combine features from both strategies.
# The reparametrization trick
The reparametrization trick introduces auxiliary random variables that are stochastic such that the parameters to be optimized over are only occuring in deterministic functions. This is convenient because it can reduce variance and sometimes the derivatives of the probability density functions do not exist in closed-form (which means no autodifferentation). See the Inference in deep learning post.
# The log-derivative trick
The log-derivative trick is also called the score function method, REINFORCE, or black-box variational inference. The term black-box variational inference reveals that this trick is completely general. It can be applied to any model. For instance, models that have both continuous and discrete latent variables. The joint distribution does not need to be differentiable either.
It uses the following identity:
This identity is just obtained by differentiating using $\nabla \log x = \frac{1}{x}$ and applying the chain rule $\nabla \log f(x) = \frac{1}{f(x)} \nabla f(x)$. Let’s subsequently rewrite this identity as a product:
The expected costs we want to minimize:
We can use Leibniz’s integral rule (differentiation under the integral sign) to shift the differential operator into the integral. To recall the rule:
In our case:
Using the log identity:
Now we can use Monte Carlo to estimate:
Here $x_s \sim p_\phi(x)$ i.i.d. This is general estimator: $f_\theta(x)$ does not need to be differentiable or continuous with respect to $x$. Note that $\log p_\phi(x)$ needs to be differentiable with respect to $\phi$.
We should show that the variance is actually reduced… However, let us first explain something that you will find time after time. Namely the notion of control variates…
# Control variates
Let us estimate the expectation over a function $E_x[f(x)]$ given a function $f(x)$. The Monte Carlo estimator is of the form $E_x[f(x)] \approx \frac{1}{k} \sum_i f(x^i)$ with $x^i \sim p(x)$. We can introduce a control variate to reduce the variance:
The parameter $\eta$ can be chosen to minimize the variance, which turns out to be optimally:
More information can be found at Wikipedia. The final variance will be something along the lines:
Here $Var(f) = E[f^2] - E[f]^2$ and $Cov(f,g) = E[(f-E[f])(g-E[g])]$. So, how we can explain this best?
Assume we have to sum over the function $f(x) = 1/(1+x)$ with $% $, then if we sample uniformly random values between $0$ and $1$ we will have results between $1/(1+0)=1$ and $1/(1+1)=1/2$. We would like to transform this function in such way that these results are closer to each other. The values at $x=0$ should be going to the mean, and the values at $x=1$ as well. At wikipedia they give the example of the covariate $g(x) = 1 + x$ (this could have just been $g(x) = x$). By adding $x$ and subtracting the average (in this case $\int_0^1 (1+x) dx = 3/2$) we make the function flatter with picking $\eta=0.4773$, in other words we reduced the variance. We sample 100 values uniformly and demonstate in the following graph that the function using the covariate is indeed flatter.
Another covariate could be $g(x) = \log(x + 1)$. We then have to subtract the expectation of that function, namely $\int_0^1 \log(x+1) dx = \log(4)-1$. This function is even flatter and has an even smaller variance. You can see that in the graph above. We have picked a value for $\eta=0.72$. The covariate which would make the compound function completely flat would be $g(x) = 1/(2-x)$, which is $f(x)$ mirrored over the range from $x=[0,1]$. However, this would of course render the Monte Carlo sampling redundant, because we would need the expectation over $g(x)$ which is in this case just as hard as that over $f(x)$.
# Recent approaches (and combinations)
The log-derivative trick (or the score function estimator) still suffers from high variance. Common techniques to reduce variance is by introducing baselines. Examples of unbiased single sample gradient estimators, are NVIL (Mnih and Gregor, 2014) and MuProp (Gu et al., 2015). An example of an unbiased multisample case is VIMCO (Mnih and Rezende, 2016).
Examples of biased single sample gradient estimators, are Gumbel-Softmax (Jang et al., 2016) and Concrete relaxiations (Maddison et al., 2017), independent researchers coming to the same strategy. The family of concrete distributions (Maddison et al, 2017) has closed-form densities and a simple reparametrization. The concrete distributions can replace discrete distributions on training so all gradients can properly be calculated. During training the concrete distributions can be replaced by discrete distributions.
REBAR (Tucker et al., 2017) is a new approach that uses a novel control variate to make the Concrete relaxation approach unbiased again.
# References
## Machine Learning Done Bayesian
In the dark corners of the academic world there is a rampant fight between practitioners of deep learning and researchers of Bayesian methods. This polemic article testifies to this, although firmly establishing itself as anti-Bayesian.
There is not much you can have against Bayes’ rule, so the hate runs deeper than this. I think it stems from the very behavior of Bayesian researchers rewriting existing methods as approximations to Bayesian methods.
Ferenc Huszár, a machine learning researcher at Twitter describes some of these approximations.
• L1 regularization is just Maximum A Posteriori (MAP) estimation with sparsity inducing priors;
• Support vector machines are just the wrong way to train Gaussian processes;
• Herding is just Bayesian quadrature done slightly wrong;
• Dropout is just variational inference done slightly wrong;
• Stochastic gradient descent (SGD) is just variational inference (variational EM) done slightly wrong.
Do you have other approximations you can think of?
## Inference in Deep Learning
There are many, many new generative methods developed in the recent years.
• denoising autoencoders
• generative stochastic networks
• variational autoencoders
• importance weighted autoencoders
• infusion training
• variational walkback
• generative latent optimization
• deep learning through the use of non-equilibrium thermodynamics
# Deep Models
We can’t delve into the details of those old workhorse models, but let us summarize a few of them nevertheless.
A Boltzmann machine can be seen as a stochastic generalization of a Hopfield network. In their unrestricted form Hebbian learning is often used to learn representations.
Don't just read the excerpt. :-) Sit down and read for real! →
# Kullback-Leibler divergence
In contrastive divergence the Kullback-Leibler divergence (KL-divergence) between the data distribution and the model distribution is minimized (here we assume $x$ to be discrete):
Here $P_0(x)$ is the observed data distribution, $P(x\mid W)$ is the model distribution and $W$ are the model parameters. A divergence (wikipedia) is a fancy term for something that resembles a metric distance. It is not an actual metric because the divergence of $x$ given $y$ can be different (and often is different) from the divergence of $y$ given $x$. The Kullback-Leibler divergence $D_{KL}(P \mid \mid Q)$ exists only if $Q(\cdot) = 0$ implies $P(\cdot) = 0$.
Don't just read the excerpt. :-) Sit down and read for real! →
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2018-09-25 05:00:47
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https://thomas1111.wordpress.com/2012/04/
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## Archive for April, 2012
### What about a Galois Prize for young European mathematicians ?
April 3, 2012
(The following text is a rewritten version, translated into english, of an opinion column that I had submitted elsewhere but which didn’t appear, for some reason which has now appeared.)
I would like to humbly call for the establishment of a European Galois Prize for Outstanding Research in Mathematics by an Undergraduate Student to be awarded yearly by the EMS, similarly to the Morgan Prize of the AMS. Here is the thinking:
I have noticed, and also have been told, that there are, in the USA, several great programs designed to mentor students interested by doing some research in mathematics, both high-schoolers (see the MIT programs like RSI and PRIMES) and undergraduates (see the list of REU at dozens of universities).
Some particularly talented students sometimes come up with great ideas and papers, and win a prize: there are the Siemens and Intel prize for high-schoolers, and the Frank and Brennie Morgan Prize of the AMS for undergraduates since 1996. For example the 2012 winner of that latter prize, 21 year old John Pardon (now a PhD student at Stanford), solved a 1983 conjecture of Gromov (a problem he found in a list online while in high school, and solved after a couple years of work) and his paper got published in Annals of Math, no less. Earlier winners are equally talented (Soundararajan, Bhargava, Lurie, Fox, Kane…), and I have noticed that most have attended one of the programs mentionned above. So indeed, thanks to those programs, there is a regular influx of such talented undergraduates with early research accomplishments.
And in fact, this US trend to get youngsters work on research problems with a mentor even before undertaking a PhD is gathering momentum. For example, an earlier winner of the Morgan Prize, Scott Duke Kominers (who won for works on number theory, and is now an economics professor at Chicago) has just launched with some coworkers a book and a website aimed at providing tips and impetus to such high-schoolers.
Let’s now look at Europe. Unless mistaken, there’s not much happening yet. A summer school for youngsters will take place for the second time this summer: a great opportunity to learn, but perhaps not quite designed as a mentoring program, and also it is held in english when most european students don’t have good enough fluency to express themselves, let alone think, in that language. And finally, even if some European student came up with a good paper, there is currently no prize to recognize her/his work and serve as a further incentive for other students.
That’s why I would think both a prize, and several mentoring programs with a good mix of english and the student’s tongue are commendable. Since the name of Galois is not associated with a prize yet, and since it does embody quite well the idea of a talented young european mathematician, that’s a good option I would think.
### Math projects at the 2012 Intel Science Talent Search
April 3, 2012
A few weeks ago, the 2012 crop of the 10 winners of the Intel Science Talent Search have been announced. As usual, these are great projects in a wide variety of fields (cancer research, robotics, geophysics, ecology, …).
Several math projects were in there too: 18 year old David Ding came 4th with his work on Infinitesimal Cherednik algebras of $\mathfrak{gl}_2$ (done during a PRIMES program at MIT— which seems to be a fantastic setting indeed for insterested high-schoolers — under the supervision of Sasha Tsymbaliuk).
And 17 year old Anirudh Prabhu came 7th for his first non-trivial analytic lower bound of odd perfect numbers (his arxiv papers are here). Apparently he has been taking accelerated courses for a number of years, and did his research on his own without a mentor.
That’s impressive in both cases, and shows that there is not one single recipe for success.
### Functional analysis in the middle ages
April 1, 2012
Exerpt of a 1458 copy of De Figura seu imagine mundi by Louis de Langle who first worked out the shapes of the 2-dim unit $\ell^p$ spheres for $p=1, 2, \infty$.
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2020-07-10 09:02:40
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https://meangreenmath.com/2015/09/22/different-ways-of-solving-a-contest-problem-part-3/
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# Different ways of solving a contest problem (Part 3)
The following problem appeared on the American High School Mathematics Examination (now called the AMC 12) in 1988:
If $3 \sin \theta = \cos \theta$, what is $\sin \theta \cos \theta$?
When I presented this problem to a group of students, I was pleasantly surprised by the amount of creativity shown when solving this problem.
Yesterday, I presented a solution using a Pythagorean identity, but I was unable to be certain if the final answer was a positive or negative without drawing a picture. Here’s a third solution that also use a Pythagorean trig identity but avoids this difficulty. Again, I begin by squaring both sides.
$9 \sin^2 \theta = \cos^2 \theta$
$9 (1 - \cos^2 \theta) = \cos^2 \theta$
$9 - 9 \cos^2 \theta = \cos^2 \theta$
$9 = 10 \cos^2 \theta$
$\displaystyle \frac{9}{10} = \cos^2 \theta$
$\displaystyle \pm \frac{3}{\sqrt{10}} = \cos \theta$
Yesterday, I used the Pythagorean identity again to find $\sin \theta$. Today, I’ll instead plug back into the original equation $3 \sin \theta = \cos \theta$:
$3 \sin \theta = \cos \theta$
$3 \sin \theta = \displaystyle \frac{3}{\sqrt{10}}$
$\sin \theta = \displaystyle \pm \frac{1}{\sqrt{10}}$
Unlike the example yesterday, the signs of $\sin \theta$ and $\cos \theta$ must agree. That is, if $\cos \theta = \displaystyle \frac{3}{\sqrt{10}}$, then $\sin \theta = \displaystyle \frac{1}{\sqrt{10}}$ must also be positive. On the other hand, if $\cos \theta = \displaystyle -\frac{3}{\sqrt{10}}$, then $\sin \theta = \displaystyle -\frac{1}{\sqrt{10}}$ must also be negative.
If they’re both positive, then
$\sin \theta \cos \theta = \displaystyle \left( \frac{1}{\sqrt{10}} \right) \left( \frac{3}{\sqrt{10}} \right) =\displaystyle \frac{3}{10}$,
and if they’re both negative, then
$\sin \theta \cos \theta = \displaystyle \left( -\frac{1}{\sqrt{10}} \right) \left( -\frac{3}{\sqrt{10}} \right) = \displaystyle \frac{3}{10}$.
Either way, the answer must be $\displaystyle \frac{3}{10}$.
This is definitely superior to the solution provided in yesterday’s post, as there’s absolutely no doubt that the product $\sin \theta \cos \theta$ must be positive.
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2017-12-13 18:57:20
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https://wiki.vis.ethz.ch/L%C3%B6sungsvorschlag_Operating_Systems_And_Computer_Networks_FS10
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# 1 Application-level protocols
## Question 1.1
• «stateless»: Server maintains no information about past client requests
• stateful protocols are way more complicated (keep state in sync, transitions, ...)
• past history must be maintained (more memory/disk space needed, cpu time, ...)
• caching is not possible, because the server can send different replies for the same query
## Question 1.2
HTTP is stateless, but states can be accomplished by cookies
# 2 Centralizing DNS
## Question 2.1
centralized DNS would not need:
• NS Resource Record
• TTL (Time To Live) not needed anymore
## Question 2.2
reasons why centralizing DNS would not be a good idea:
• single point of failure
• traffic volume
• performance (high RTT)
• maintenance
# 3 Network programming
## Question 3.1
• easier to program & understand (read code from top to bottom per connection)
• does not scale well (a lot of connections mean a lot of threads)
event-driven I/O:
• works with only one thread
• scales very well
• harder to program & understand (callbacks)
• hard to deal with long running operations
## Question 3.2
difference in the operating system kernels I/O subsystem:
• needs blocking system-calls
event-driven I/O:
• needs non-blocking system-calls
# 4 Transport protocols and performance
## Question 4.1
If an ACK is corrupted, the receiver considers the transmission as completed, but the sender will wait forever, since he only considers ACKs or NACKs. If a NACK is corrupted, the receiver waits for the retransmission forever, because the sender will wait forever to detect a ACK or NACK
## Question 4.2
Sender side:
If the sender receives a corrupt answer, he has to retransmit the packet. But the receiver needs to be able to detect, whether the packet is a retransmission or a new packet. Therefore, we need sequence numbers (1 bit is enough for this purpose). The sender now retransmits the packet with the same sequence number as before. This leads to four states: "Wait for call from above 0", "wait for answer 0", "wait for call from above 1", "wait for answer 1"
If the receiver has now two states, "wait for 0" and "wait for 1" The receiver only delivers data to the application on a transition from "wait for 0" to "wait for 1". The ACKs or NACKs are always handled the same way.
For all transitions see: Slide on page 15/16
## Question 4.3
• Bandwidth: 10 Gbps
• Propagation delay: 1 ms
• Packet size: 1 kb (this is stange, why not kB? typing error?)
Utilization = (packet size / (2 * propagation delay)) / bandwidth = (1 kb / 2 ms) / 10 Gbps = 0.5 Mbps / 10 Gbps = 1/20'000
## Question 5.1
Process:
Client A creates a TCP connection to client B
1. SYN with sequence number a, sent to B
2. SYN + ACK with sequence number b, acknowledgement a + 1
3. ACK with acknowledgement b + 1
Example:
1. A sends: SYN, Sequence = 100
2. B sends: SYN + ACK, Sequence = 500, Acknowledgement = 101
3. A sends: ACK, Acknowledgement = 501
Why is it necessary to exchange such information?
If a packet is corrupted, such that it becomes a SYN-packet, if we have only a two way handshake, B accepts this unwanted connection. If we have a one-way handshake, A doesn't know, whether B accepted the connection. Three-way handshake solves this problem. Accidental SYN do not lead to connections on any side, both parties know when a connection is established.
We use sequence numbers for both nodes, because the nodes need to know, in which order the packets should be. We use a random sequence number, because a old connection could otherwise interfere with a new connection on the same port. We increase the sequence numbers in the handshake, because it is easier to implement.
## Question 5.2
• Bandwidth: 10 Gbps
• Propagation delay: 1ms
• Initialization packet size: 100 B
Time to establish connection = ${\displaystyle 3\cdot \left(1ms+{\frac {100B}{10Gbps}}\right)=3\cdot (1ms+8\cdot 10^{-5}ms)=3\cdot 1.00008ms=3,00024ms}$
# 6 Flow control
## Question 6.1
Problem of flow control:
• Sender and receiver don't know how much resources the communication partner can use for this connection, therefore they need to regulate the communication speed.
Mechanisms:
• Sender keeps the transmitted unacked data less than the most recently received receiver Window
Data structures:
• Send queue: Queue of packets to send
• Send window: part of the send queue (sent, but unACKed + unsent, but ready)
# 7 Delay and bandwidth
## Question 7.1
• Data size: 10 TB
• Distance: 10 km
• Bandwidth: 10 Mbps
• Speed: 20 km/h
• time to copy to 5*2TB harddisks = n
Transfer over the network:
• Time_1 = Data size / Bandwidth = 10 TB / 10 Mbps = 8'000'000 s = 92 days 14 hours 13 minutes 20 seconds
Transfer by bicycle:
• Time_2 = n + Distance / Speed = n + 10 km / 20 km/h = n + 0.5 h
• Time_2 < Time_1 if n < 92 days 13 hours 43 minutes 20 seconds
## Question 7.2
Bicycle bandwidth = 10 TB / (n + 0.5 h) = ? => if n = 0 then 20 TB/h
# 8 Count to infinity
Example:
A <==1==> B <==1==> C
• B has a connection to A with weight 1
• C has a connection to A with weight 2 via B
A <==X==> B <==1==> C
• Link from B to A is broken
1. B needs a new connection to A and asks C => new connection for B with Cs value + 1
2. B notifies C that its connection has changed
3. C updates its connection weight
4. C notifies B
5. B updates its connection weight
6. goto 2
It happens, because there are cycles in the network graph and no one detects updates in cycles.
But your topology does not contain any cycle.. What do you mean by cycle? (Jan Veen 02.07.15)
Outline of the solution by BGP
• save the path to the target along with the other routing information
• if a router asks for updates, paths containing his name are ignored
# 9 Slow convergence
See Slides on page 104
When a link breaks, some nodes start to consider alternative routes over nodes which haven't propagated the dead link yet. => leads to slow convergence
It happens because the nodes only consider their neighbours and not the whole path.
# 10 NAT
Problem it solves:
• Make it easy to change the provider
• Make it easier to add new computers to the network
• Security
Problem it tries to solve: unknown
• Computers in the network cannot be servers (unless some port-forwarding technique is used)
How it works?
• Connections, initiated by computer of the local network are tracked
• Multiplex, based on the stored information of the sessions
• => Modification of IP-Headers and Ports
## Question 11.1
The preamble is used to synchronize the clocks of the network participants. Its always 0x55555555555555 and at the beginning of each ethernet-frame.
Wrong: It's actually 0x55 0x55 0x55 0x55 0x55 0x55 0x55 0xD5. On the wire the bit pattern looks like this: 10101010 <- 7 times and then 10101011
## Question 11.2
When we have n nodes in a hubbed network, the probability of a collision grows linearly dependent on n. With more collisions, the network utilization converges to zero.
# 12 Spanning Tree
## Question 12.1
problems witch cycles in switched networks
## Question 12.2
See Slides Page 54
• create graph
• calculate minimum spanning tree
• deactivate all links that are not in the minimum spanning tree
# 13 Scheduling
## Question 13.1
I/O-bound:
• Use a lot of blocking I/O-systemcalls => often switch back to the scheduler before their quantum has ended
• The bottleneck is the I/O device
CPU-bound:
• Do not use a lot of blocking I/O-systemcalls => often use their whole quantum
• The bottleneck is the CPU
## Question 13.2
The I/O-bound tasks do not use much time to block again => another task is scheduled. Because the I/O is slow, the CPU-bound task wont starve (all I/O devices are busy) => only CPU-bound tasks can be scheduled.
# 14 Virtual Memory
## Question 14.1
40 ns
• 20 ns for page table access
• 20 ns for memory access
## Question 14.2
0.25 * 20 ns + 20 ns = 25 ns
# 15 Page replacement
## Question 15.1
• More page frames do not necessary lead to less or equal page faults.
• FIFO shows this anomaly
## Question 15.2
1 2 3 4 2 1 5 6 2 1 2 3 7
frame 1 frame 2 frame 3 frame 4
1
2
3
4
2
1
5
6
2
1
2
3
7
8 page faults
# 16 Working set
Working set: Set of page references a process has used in the last k instructions
Useful to minimize thrashing and optimize parallelism
# 17 Buffer management
## Question 17.1
Concept: mbufs form together a linked list of linked lists of messages and message parts.
Problems:
• Unified format for half-defined packets
• Avoid copying lots of payload
• Fragment large datasets
## Question 17.2
1. Variable packet size
2. Linked List of different packets
# 18 Disk access
1. RAM-disk
2. Special file
3. Network share
4. Virtual file systems (they are not necessarily on the disk)
5. Disk removed while computer is running
6. Data in the cache
# 19 File representation
## Question 19.1
An inode is a data structure that contains metadata, probably some data and block references.
1. Creation date
2. Modification date
3. Archive-bit
4. System-file-bit
5. Compression-metainformation
6. Rights
7. Size
## Question 19.2
Filename: because there can be multiple names to one inode
The file names are saved in the data-blocks of the directories inode.
## 20.1
In the context of the folder that contains the hard link, the name of the hard link is mapped to the inode which the target of the link pointed to when the hard link was created.
A symbolic link is a file, that contains a name, which is probably the name of another file.
## 20.2
The question is probably wrong.
Interpretation 1: There exists already a file in a folder, with a hard-link and a soft-link to it in another folder. This interpretation ignores, that we could create a new soft-link to the hard-link.
• After changing the access-rights to the first folder, the hard-link can access, and the soft-link can't.
• After a rename of the file, the hard-link is still valid, and the soft-link isn't
• This has no solutions, because we can always create a soft-link to the hard-link.
Interpretation 3: There exists a file, to which either a hard or a softlink has to be made, but only the former is possible
• An open file whose hard links have all been deleted should meet the requirements. Before creating a new hard link, no soft link can access the file.
Interpretation 4: No restriction.
• It's possible if there is a security hole
Interpretation 5: Two hard links. And one hard link might be in a directory we do not have access rights to.
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2019-05-24 20:54:13
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https://infoscience.epfl.ch/record/130479
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## A note on the fundamental theorem of projective geometry
Published in:
Commentarii mathematici Helvetici, 44, 310-315
Year:
1969
Keywords:
Laboratories:
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2018-02-18 03:29:40
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Research Article
Egyptian Journal of Basic and Clinical PharmacologyVol. 9 (2019), Article ID 101396, 16 pages doi:10.32527/2019/101396
## Prophylactic and Ameliorative Effect of N-Acetylcysteine on Doxorubicin-Induced Neurotoxicity in Wister Rats
### Walaa I. Mohammed1, Rania A. Radwan2, and Hoda M. Elsayed3
1Department of Clinical Pharmacology, Faculty of Medicine, Sohag University, Egypt
2Department of Forensic medicine and clinical toxicology, Faculty of Medicine, Sohag University, Egypt
3Department of Histology, Faculty of Medicine, Sohag University, Egypt
Received 9 October 2018; Accepted 3 January 2019
Editor: Helen Kwanashie
Copyright © 2019 Walaa I. Mohammed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
#### Abstract
Doxorubicin (DOX) is an anthracycline antibiotic and a quinone-containing chemotherapeutic drug used for various types of cancers. However, as with most anticancer drugs, it causes many toxic effects, one of them is cognitive impairment. The present study investigated the prophylactic and ameliorative effect of n-acetylcysteine (NAC) against DOX-induced neurotoxicity in rats. Rats were divided into four groups. Control group: rats received saline. NAC treated group: rats received NAC (100 mg/kg, p.o.) daily for 35 days. DOX-treated group: rats received DOX (4 mg/kg, i.p.) for four weeks on day 7, 14, 21 and 28. DOX+NAC treated group 1: rats received NAC (100 mg/kg, p.o.) daily for 35 days and DOX (4 mg/kg, i.p.) for four weeks on day 7, 14, 21 and 28). DOX+NAC treated group 2: rats received NAC (100 mg/kg, p.o.) daily started at the 7th day of the experiment till the end of the experiment and DOX (4 mg/kg, i.p.) for four weeks on day 7, 14, 21 and 28. The present results showed a significant reduction in the body weight, which was associated with a significant increase in brain to body weight ratio in DOX-treated rats. Tumor necrosis factor (TNF-α) level, malondialdehyde (MDA) and total protein levels were significantly elevated. Whilst, reduced glutathione (GSH) and glutathione peroxidase (GPx) levels were significantly decreased. Moreover, there were histopathological abnormalities in the brain tissue of DOX-treated rats, as most of the neurons degenerated and the blood vessels surrounded with wide perivascular spaces. In addition, the neuropil was vacuolated. The present study demonstrated that NAC has a neuroprotective effect on the brain damage induced by DOX, through inhibition of inflammation and oxidative stress. This neuroprotective effect was more pronounced in DOX+NAC treated group 1, as it produced a significant increase in brain GSH and GPx levels and more improvement in the histopathological abnormality compared to DOX+NAC treated group 2.
#### 1. Introduction
Doxorubicin (DOX) is an anthracycline antibiotic and a quinone-containing chemotherapeutic drug, usually used for chemotherapy of breast cancer and other cancers [1]. The mechanism of DOX is chelating DNA, inhibiting topoisomerase II and producing large amounts of free radicals to kill cancer cells [2]. However, this mechanism of action is implicated in the toxicity of several non-targeted organs particularly the heart, kidney, and brain [3] and limits its dosage in cancer patients.
In addition, cognitive impairments were reported to have adverse effects on patients' daily activities [4]. Doxorubicin does not pass through the blood-brain barrier (BBB), so the decline in cognition occurs with its administration is attributed to the peripheral increase in the circulating tumor necrosis factor (TNF-α) [5,6,7]. TNF-α migrates across the BBB and activates glial cells to release large amounts of the local TNF-α in the cortex and hippocampus leading to inflammation and induced oxidative stress in the brain [8]. Thus, the increase in TNF-α level may be a reason for doxorubicin-induced oxidative stress and its central nervous system injury [9]. Antagonizing circulating TNF-α with TNF-α antibody led to a manifest decrease in TNF-α levels and the observed mitochondrial dysfunction in brain tissues [5].
NADPH-cytochrome P-450 converts DOX to its semiquinone radical, which then reoxidized and regenerated by oxygen producing superoxide anions. Increased level of superoxide anions elevates the circulating TNF-α level that can directly cross the BBB [10].
N-Acetylcysteine (NAC) is an acetylated precursor of the amino acid L-cysteine [11]. It used as an antidote for paracetamol intoxication and as a mucolytic agent [12]. NAC is an important cellular antioxidant that decreases inflammation in various diseases. It is also a glutathione precursor and inhibits lipid peroxidation and proinflammatory cytokines [13].
In addition, NAC has a wide spectrum of actions and different applications in multiple systems. It can pass through the BBB and cure brain dysfunctions and neuropathies. Also, it applied for the treatment of vascular and nonvascular neurological disorders and modulates glutamatergic, neurotrophic and inflammatory pathways [14].
This study aimed to evaluate the potential protective and ameliorative effect of NAC against doxorubicin-induced neurotoxicity in rats, through some of its anti-inflammatory and antioxidant effects.
#### 2. Materials and Methods
##### 2.1. Materials
N-Acetylcysteine was purchased from AK Scientific, Inc. (USA), while Doxorubicin was purchased as Adriblastina vials from Pharmacia Italia S.P.A., Italy.
Kits for determination of TNF-α was obtained from Wuhan EIAab Science Co. Ltd (China). Malondialdehyde (MDA), reduced glutathione (GSH) and glutathione peroxidase (GPx) kits were obtained from Bio-diagnostic Company, Egypt. While kits for determination of total protein was obtained from Sigma-Aldrich (USA).
##### 2.2. Animals
The study was conducted on 50 adult male Wister albino rats, weighing 150-200g (initial weight). Rats were purchased from the animal house, Faculty of Medicine, Assiut University, Egypt, and were housed in the animal facility, Faculty of Medicine, Sohag University and maintained in a controlled environment under standard conditions of temperature (25 ± 2C). A time controlled system provided 12 hours of light and 12 hours of dark was applied. All rats were fed freely on rodent chow diet. The experimental protocol was carried out and approved according to the guidelines of the Medical Research Ethics Committee of Faculty of Medicine, Sohag University, Egypt. (Approval No. 29/2018).
##### 2.3. Equipment used
Tissue homogenizer: Wise-Tis, HG-15D, Germany.
Spectrophotometer: analytik jena, Germany.
ELISA microplate reader: AWARENESS Stat Fax- 2200, USA.
##### 2.4. Experimental design
Rats were left for one week-acclimatization period and then divided into five groups, ten animals each. The duration of the experiment was 35 days and designed as follow:
###### 2.4.1. Control group
Rats were received normal saline orally (p.o.) by gastric tube daily for 35 days.
###### 2.4.2. NAC treated group
Rats were received NAC in normal saline at the dose of 100 mg/kg (p.o.) [15] daily for 35 days.
###### 2.4.3. DOX treated group
Rats were received DOX (intraperitoneally, i.p.) at the dose of 4 mg/kg [16] in normal saline once a week for four weeks on day 7, 14, 21 and 28.
###### 2.4.5. DOX + NAC treated group 1 (DOX + NAC1)
Rats received NAC (100 mg/kg p.o.) daily for 35 days and 4 mg/kg of DOX (i.p.) in normal saline once a week for four weeks on day 7, 14, 21 and 28.
###### 2.4.6. DOX + NAC treated group 2 (DOX + NAC2)
Rats received NAC (100 mg/kg p.o.) daily started at the 7th day of the experiment till the end of the experiment and 4 mg/kg of DOX (i.p.) in normal saline once a week for four weeks on day 7, 14, 21 and 28.
Body weight of each rat was measured daily during the experimental period. The dose of the test drugs to be given was calculated daily based on the body weight of the experimental animals to ensure administration of the fixed dose.
At the end of the experiment, rats were weighted before euthanized by decapitation. Immediately, the brain was removed, washed with ice-cold phosphate buffer solution (pH 7.4), dried on filter paper and weighed. The brain tissue from each rat was divided for assessment of brain toxicity on biochemical and histopathological levels.
##### 2.5. Biochemical assays in the brain tissue
###### 2.5.1. Preparation of tissue homogenates
Brain samples were homogenized in 10ml ice-cold potassium phosphate buffer (50 mM, pH 7.4) per gram tissue (v/w) for the assay of brain TNF-α, GSH, GPx, MDA as well as total protein levels. All homogenates were centrifuged at 4000 rpm for 15 min at 4C. The supernatant was separated and kept at −20C until the time of analysis.
###### 2.5.2. Determination of TNF-α level
Tumor necrosis factor -α in the brain homogenate was measured using enzyme-linked immunosorbent assay (ELISA) specific for a rat. The TNF-α level was expressed as pg/g tissue.
###### 2.5.3. Determination of reduced glutathione level
Reduced glutathione was measured in the brain tissue homogenate by a colorimetric method as described by Beutler et al. [17]. The method was based on the reduction of 5,5' dithiobis (2-nitrobenzoic acid) with glutathione producing a yellow compound. The reduced chromogen directly proportional to GSH concentration and its absorbance was measured at 405 nm. GSH level was expressed in mg/g tissue.
###### 2.5.4. Determination of glutathione peroxidase level
Glutathione peroxidase was determined in the brain tissue homogenate by a colorimetric method as described by Paglia and Valentine [18]. GPx level was expressed in U/g tissue.
###### 2.5.5. Determination of lipid peroxidation level
Malondialdehyde level is an indicator of lipid peroxidation. MDA in the brain tissue homogenate was detected by a colorimetric method as described by Ohkawa et al. [19]. This method depends on the spectrophotometric measurement of the color produced during the reaction of a thiobarbituric acid with MDA. They react in an acidic medium for 30 min to form a thiobarbituric acid reactive product. The absorbance of the resultant pink product was measured at 534 nm. MDA level was expressed in nmol/g tissue.
###### 2.5.6. Determination of total protein level
Protein was determined by the method of Bradford [20] by using bovine serum albumin as a standard.
##### 2.6. Histopathological studies
Samples of the brain tissue were excised, fixed in 10% formal saline, dehydrated in ascending grades of ethanol, cleared in xylene and embedded in paraffin wax [21]. Sections (5 µm thick) were cut and stained with hematoxylin and eosin (H&E).
##### 2.7. Statistical analysis of data
Data were expressed as mean ± standard error (SE). Statistical difference between studied groups was analyzed using one-way analysis of variance (ANOVA). In cases where ANOVA showed significant differences, Tukey post hoc test was performed to compare the changes among individual groups. The difference was regarded as significant when P < 0.05. All statistical analyses were performed using SPSS statistical version 20 software package.
#### 3. Results
##### 3.1. General observation and mortality
The general appearance of all groups was inspected during the study. In all DOX-treated groups, a red colored lesion was observed at the site of injection. Signs of general toxicity such as weakness, enlargement of the abdomen, red colored discharge around the nose and mouth were observed, and they were more extensive in the DOX-treated group than in DOX+NAC treated groups. Moreover, rats in the DOX-treated group showed decline in their feed and water consumption during the drug treatment period as compared to the control group. On the other hands, in the DOX+NAC treated groups, feed and water consumption increased as compared to the DOX-treated group.
The mortality rate in the DOX-treated group was high and reached 30%, while in DOX+ NAC2 the mortality rate was low and reached 10%. No mortality was recorded in both NAC treated and DOX+NAC1 groups.
##### 3.2. Changes in body weight and brain weight to body weight ratio
There was an insignificant change (P> 0.05) in the final body weight in the NAC treated group compared to the control group. On the other hand, the final body weight in DOX-treated group decreased significantly (P< 0.05) when compared to the control group. In DOX+NAC treated groups, the final body weight was increased but not statistically significant (P> 0.05) compared to the DOX-treated group (Table 1).
The NAC treated group showed an insignificant change (P> 0.05) in brain weight to body weight ratio compared to the control group. The brain weight to body weight ratio in the DOX-treated group was significantly increased (P< 0.05) compared to the control group. The brain weight to body weight ratio in both DOX+NAC1 and DOX+NAC2 groups was significantly (P< 0.05) decreased compared to DOX group with an insignificant difference (P> 0.05) between them (Table 2).
##### 3.3. Changes in brain TNF-α level
The levels of TNF-α in the brain tissue of the NAC treated group showed an insignificant change (P> 0.05) compared to the control group. On the other hand, the levels of TNF-α in the brain tissue was significantly increased (P< 0.05) in the DOX-treated group compared to the control group. Treatments of the rats with NAC plus DOX (i.e. DOX+NAC treated groups 1 and 2) produced a significant decrease in the brain TNF-α level compared to the DOX-treated group (P< 0.05), while no significant difference between the two groups (P> 0.05) was statistically detected (Figure 1).
##### 3.4. Changes in brain GSH level
NAC treated group showed an insignificant change (P> 0.05) in GSH level in the brain tissue compared to the control group. However, a significant decrease (P< 0.05) in the brain GSH level was recorded in the DOX-treated group compared to the control group. Animals treated with NAC plus DOX (DOX+NAC treated groups 1 and 2) showed a significant increase (P< 0.05) in the brain level of GSH compared to the DOX-treated group. Moreover, DOX+NAC1 produced a significant increase (P< 0.05) in the brain GSH compared to DOX+NAC2 (Figure 2).
##### 3.5. Changes in brain GPx level
Glutathione peroxidase level in the brain tissue of the NAC treated group showed an insignificant change (P> 0.05) compared to the control group. Moreover, the brain level of GPx was significantly decreased (P< 0.05) in the DOX-treated group compared to the control group (Figure 3). NAC supplementation in both DOX+NAC treated groups 1 and 2 produced a significant elevation (P< 0.05) in the brain GPx level compared to the DOX-treated group. The results revealed that the DOX+NAC1 was more effective (P< 0.05) than DOX+NAC2 in the elevation of brain GPx level to reach near to the control level (Figure 3).
##### 3.6. Changes in brain lipid peroxidation level
Compared to the control group, the NAC treated group showed an insignificant change (P> 0.05) in the brain MDA level. However, the brain MDA level of the DOX-treated group was significantly increased compared to the control group and to both DOX+NAC treated groups 1 and 2 (P< 0.05) (Figure 4). Also, the results revealed that there was an insignificant difference between DOX+NAC1 and DOX+NAC2 (P> 0.05).
##### 3.7. Changes in brain total protein level
As shown in Figure 5, the NAC treated group showed an insignificant change (P> 0.05) in the total protein level of the brain tissue compared to the control group. However, the total protein level in the brain homogenate were significantly increased (P< 0.05) in the DOX-treated group compared to the control group. But, the administration of NAC to DOX-treated rats resulted in a significant decrease (P< 0.05) in the total protein level. However, an insignificant difference (P> 0.05) between the two groups (DOX+NAC1 and DOX+NAC2) was recorded.
##### 3.8. Histopathological changes
Microscopically, examination of the brain tissue of the control group revealed normal brain architecture with the cortical neurons appeared with rounded vesicular nuclei that having prominent nucleoli. Also, the cortical neurons have slight basophilic cytoplasm and peripheral processes. The neuropil contained neuroglia and nerve fibers. The blood vessels with a narrow perivascular space (Figure 6A). The NAC treated group showed normal histological features as those found in the control group (Figure 6B). In the DOX-treated group, the brain tissue showed severe histopathological alteration as most of the neurons appeared shrunken with darkly stained pyknotic nuclei and surrounded by wide pericellular space (degenerated neurons). Blood vessels surrounded by wide perivascular spaces compared to the control group. Moreover, vacuolation of the neuropil was also observed in this group (Figure 6C).
Pretreatment with NAC revealed partial improvement as most of the neurons appeared similar to those of the control with decreased pericellular spaces, while others degenerated. Also, the histopathology revealed that DOX+NAC1 group showed more improvement than DOX+NAC2 (Figure 6D and 6E).
#### 4. Discussion
Doxorubicin is a well-confirmed and extremely effective antineoplastic agent. However, as with most anticancer drugs, it causes many toxic effects, one of them is cognitive impairment [10]. Oxidative stress, inflammation, and apoptosis have an important role in DOX-dependent toxicity [22,23].
This research showed that administration of 4 mg/kg of DOX (i.p.) once a week for four weeks on day 7, 14, 21 and 28 caused a brain toxicity and administration of NAC at a dose of 100 mg/kg (p.o.) was able to protect against inflammation and oxidative stress induced by DOX administration. To the best of our knowledge, this is the first study that revealed NAC potency as a neuroprotective agent by preventing inflammation and oxidative stress in DOX-induced brain damage.
In the present study, rats of the DOX-treated group appeared weak with evident of ascites. Necrosis was also observed at the site of DOX injection. These observations were analogous to the study of Jambhulkar et al. [24]. The administration of NAC together with DOX in both DOX+NAC1 and DOX+NAC2 groups did not prevent the occurrence of these side effects but they were less pronounced.
Moreover, mortality was observed in the DOX-treated group (30% mortality). In other DOX studies, mortality was ranged from 30-60% [24,24,26]. The administration of NAC with DOX in the present study decreased the toxic effect of DOX which indicated by mortality reduction to be 10% and 0% in DOX+NAC2 and DOX+NAC1 groups, respectively.
Furthermore, in the present study, regardless the presence of ascites, there was a significant decrease in the final body weight in the DOX-treated group compared to the control group which may be due to diminishing animals' food intake. Increase brain weight to body weight ratio of the DOX-treated group, in the present study, may be attributed to brain edema resulting from inflammation and oxidative stress [27]. The administration of NAC with DOX in the present study improved the effect of DOX on body weight and on brain weight to body weight ratio.
To evaluate the role of NAC in preventing the brain damage induced by DOX administration, this research conducted the analysis of brain TNF-α, GSH, GPx, MDA and total protein levels. Moreover, a histopathological study was performed.
Tumor necrosis factor -α is a cytokine that has a role in immune response as a reaction to several stresses. Also, it is the cause of cognitive damage in neurodegenerative diseases and stimulates the inflammatory response, which causes many of the clinical problems [9].
The present study found an increase in the TNF-α level in the brain tissue of DOX-treated group compared to the control group. This observation agreed with the results of Kuzua et al. [23] who reports a significant increase in TNF-α level in both heart and brain tissues after DOX administration. Moreover, Abdel-Daim et al. [28] demonstrated that DOX induced an acute inflammatory reaction, evidenced by elevation in the serum of TNF-α level.
Increase brain TNF-α level was due to the increase in the level of circulating TNF-α which crosses the BBB and activates glial cells to produce more TNF-α which leads to mitochondrial damage [29]. Moreover, increasing the TNF-α level played a role in chronic inflammation, which leads to neuronal death and neurodegenerative diseases. Therefore, the elevation in TNF-α level may be the linkage between DOX-induced oxidative stress and central nervous system damage [9].
The increased level of reactive oxygen species (ROS) can be expressed by a decrease in GSH level [30]. According to the results obtained from the current study, there was a massive decrease in the brain GSH level in the DOX-treated group compared to the control group. GSH is an intracellular non-enzymatic antioxidant and one of the most important scavengers of free radicals. Also, it is a co-factor of many detoxifying enzymes against oxidative stress as GPx and glutathione reductase [30,31,32].
In the present study, GSH depletion may cause a weakening in the cell defense that may lead to tissue injury. GSH is utilized as a substrate for GPx activities, therefore, its deficiency, in the present study, might be the cause of decreased GPx activities.
Glutathione peroxidase converts H2O2 and organic hydroperoxides to less reactive products [33]. Hence, it is hypothesized that the decrease in the GPx activity might cause the H2O2 accumulation and a further inactivation in its activities [34]. Consequently, the brain becomes even more susceptible to oxidative processes.
In this study, the MDA level in the brain tissue of the rats was measured as a reference to the neurotoxic effect of DOX. The results revealed that administration of DOX significantly increased the MDA level and this finding was closely similar to those observed by other studies [9,29,35,28]. Elevated MDA level in DOX group suggests enhanced lipid peroxidation leading to brain tissue damage and inability of antioxidant defense mechanisms to prevent the free radical attack. Furthermore, the decrease in GSH levels might diminish the overall antioxidant potential resulting in the increase of lipid peroxidation following DOX administration [36].
Increased level of brain total protein concentration may be attributed to oxidative stress and depletion of the intrinsic antioxidant machinery [24].
The histopathological analysis also confirmed that DOX produced gross structural abnormalities in brain tissue, which was in line with Ramalingayya et al. [37].
The main mechanism of DOX action is chelating DNA, inhibiting topoisomerase II and then producing free radicals to kill tumor cells [2]. As a result, it produces massive amounts of reactive oxygen species (ROS) in defense against solid tumors and this mechanism of action is involved in the toxicity of several non-targeted organs [38]. Moreover, the quinone in DOX undergoes a one-electron reduction to produce a semiquinone, which in turn react with molecular oxygen and provide other ROS [39].
In the present study, NAC alone has no significant effect on all biochemical and histopathological findings in the brain of normal rats. However, it significantly prevented all DOX-induced brain injuries possibly due to its anti-inflammatory and neuroprotective effects [40]. The administration of NAC produced a statistically significant decrease in TNF-α level compared to the DOX group, which agreed with the results of Saleh [41] who reported that NAC in a dose of 75 mg/kg and 600 mg/kg improved neurological functions, prevented brain inflammation and oxidative stress responses in aspartame-induced neurotoxicity. Also, Palacio et al. [13] revealed that NAC inhibits the inflammatory cytokines TNF-α, IL-1b and IL-6 in lipopolysaccharide-activated macrophages cell line under mild oxidative conditions. NAC prevents the generation of TNF-α through inhibition of the transcription factor NF-kappa B and by increasing the intracellular levels of GSH which acts as an antioxidant [11,42].
N-acetylcysteine is known to be an antioxidant [43], and this concept is supported by the present finding that administration of NAC produced a statistically significant increase in GSH and GPx levels. The uniqueness of NAC is most probably due to its serving as a precursor of L-cysteine for GSH synthesis [11] and supplying GSH for GSH-Px-catalysed reactions [44]. Therefore, inhibition of cysteine uptake causes cellular glutathione to decrease and cellular oxidant to accumulate leading to cell death [45]. The normalization of MDA following NAC treatment is very likely due to its antiperoxidative properties [36,43], as the presence of acetyl and sulfhydryl groups makes NAC an effective inhibitor of lipid peroxidation [36].
Generally, the present biochemical findings were strongly supported by histopathological changes in brain tissue, as NAC treatment reduced the histopathological abnormalities induced by DOX in the brain tissues. These findings agreed with Saraswathy et al. [15] who revealed that NAC in a dose of 100 and 200 mg/kg protected the brain tissue against phenytoin- induced brain damage. Moreover, Abdel-Daim et al [46] confirmed that NAC by its antioxidant power improved the histopathological abnormalities induced by fipronil in hepatic and renal tissues.
In conclusion, the results of our study showed that n-acetylcysteine by its anti-inflammatory and antioxidant properties might play an important role in the protection against doxorubicin-induced neurotoxicity in rats. Also, n-acetylcysteine not only could improve the neurotoxicity of doxorubicin but also its administration before the beginning of the chemotherapeutic agent (in DOX+NAC treated group1) provided a more benefit as it produced a significant increase in the brain GSH and GPx levels and more improvement in the histopathological abnormalities of the brain tissue.
#### Funding
No funding resources.
#### Competing Interests
The authors declare no competing interests.
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Research Article
Egyptian Journal of Basic and Clinical PharmacologyVol. 9 (2019), Article ID 101396, 16 pages doi:10.32527/2019/101396
## Prophylactic and Ameliorative Effect of N-Acetylcysteine on Doxorubicin-Induced Neurotoxicity in Wister Rats
### Walaa I. Mohammed1, Rania A. Radwan2, and Hoda M. Elsayed3
1Department of Clinical Pharmacology, Faculty of Medicine, Sohag University, Egypt
2Department of Forensic medicine and clinical toxicology, Faculty of Medicine, Sohag University, Egypt
3Department of Histology, Faculty of Medicine, Sohag University, Egypt
Received 9 October 2018; Accepted 3 January 2019
Editor: Helen Kwanashie
Copyright © 2019 Walaa I. Mohammed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
#### Abstract
Doxorubicin (DOX) is an anthracycline antibiotic and a quinone-containing chemotherapeutic drug used for various types of cancers. However, as with most anticancer drugs, it causes many toxic effects, one of them is cognitive impairment. The present study investigated the prophylactic and ameliorative effect of n-acetylcysteine (NAC) against DOX-induced neurotoxicity in rats. Rats were divided into four groups. Control group: rats received saline. NAC treated group: rats received NAC (100 mg/kg, p.o.) daily for 35 days. DOX-treated group: rats received DOX (4 mg/kg, i.p.) for four weeks on day 7, 14, 21 and 28. DOX+NAC treated group 1: rats received NAC (100 mg/kg, p.o.) daily for 35 days and DOX (4 mg/kg, i.p.) for four weeks on day 7, 14, 21 and 28). DOX+NAC treated group 2: rats received NAC (100 mg/kg, p.o.) daily started at the 7th day of the experiment till the end of the experiment and DOX (4 mg/kg, i.p.) for four weeks on day 7, 14, 21 and 28. The present results showed a significant reduction in the body weight, which was associated with a significant increase in brain to body weight ratio in DOX-treated rats. Tumor necrosis factor (TNF-α) level, malondialdehyde (MDA) and total protein levels were significantly elevated. Whilst, reduced glutathione (GSH) and glutathione peroxidase (GPx) levels were significantly decreased. Moreover, there were histopathological abnormalities in the brain tissue of DOX-treated rats, as most of the neurons degenerated and the blood vessels surrounded with wide perivascular spaces. In addition, the neuropil was vacuolated. The present study demonstrated that NAC has a neuroprotective effect on the brain damage induced by DOX, through inhibition of inflammation and oxidative stress. This neuroprotective effect was more pronounced in DOX+NAC treated group 1, as it produced a significant increase in brain GSH and GPx levels and more improvement in the histopathological abnormality compared to DOX+NAC treated group 2.
Research Article
Egyptian Journal of Basic and Clinical PharmacologyVol. 9 (2019), Article ID 101396, 16 pages doi:10.32527/2019/101396
## Prophylactic and Ameliorative Effect of N-Acetylcysteine on Doxorubicin-Induced Neurotoxicity in Wister Rats
### Walaa I. Mohammed1, Rania A. Radwan2 and Hoda M. Elsayed3
1Department of Clinical Pharmacology, Faculty of Medicine, Sohag University, Egypt
2Department of Forensic medicine and clinical toxicology, Faculty of Medicine, Sohag University, Egypt
3Department of Histology, Faculty of Medicine, Sohag University, Egypt
Received 9 October 2018; Accepted 3 January 2019
Editor: Helen Kwanashie
Copyright © 2019 Walaa I. Mohammed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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2021-12-07 05:56:34
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https://cameramath.com/es/expert-q&a/Algebra/11-The-amount-of-garbage-11-The-amount-of-garbage-G-produced
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### ¿Todavía tienes preguntas de matemáticas?
Pregunte a nuestros tutores expertos
Algebra
Pregunta
The amount of garbage, $$G$$ , produced by a city with population $$p$$ is given by $$G = f ( p ) . G$$ is measured in tons per week, and $$p$$ is measured in thousands of people. a. The town of Tola has a population of $$50,000$$ and produces $$13$$ tons of garbage each week. Express this information in terms of the function $$f$$ . Enter your answer as an equation. Do not enter an any units (people, tons) or commas in your answer. Include a multiplication sign between symbols if you need to. For example, enter $$a ^ { * } x$$ and not just $$a x$$ . b. Explain the meaning of the statement $$f ( 4 ) = 3$$ .
a. f(p)=$$\frac{13}{50} p$$
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2022-05-16 05:31:59
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https://mlstat.wordpress.com/tag/bwt/
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### Archive
Posts Tagged ‘BWT’
## BWT for NLP (2)
November 12, 2009 2 comments
I show how the Burrows-Wheeler Transform can be used to compute the similarity between two strings. We submitted results from this method (along with results from the Context-Chain metric developed by my colleagues Frank Schilder and Ravi Kondadadi) for the Automatically Evaluating the Summaries of Peers (AESOP) task of the TAC 2009 conference.
The task was to produce an automatic metric to evaluate machine generated summaries (i.e., system summaries) against human generated summaries for the TAC ’09 Update Summarization Task. Clearly the automatic metric is just some function that produces a similarity score between the system summary and the human generated (the so-called model) summary.
The proposed metrics were evaluated by comparing their rankings of the system summaries from different peers to that of the ranking produced by human judges.
Similarity Metric
We use an estimate of the conditional “compressibility” of the model summary given the system summary as the similarity metric. The conditional compressibility is defined as the increase in the compressibility of the model summary when the system summary has been observed.
In order to judge the similarity of the system summary $S$, to the model summary $M$, we propose to use the difference in compressibility of $M$ when $S$ is not seen to when $S$ is given. This metric basically
captures the reduction in the uncertainty in $M$ when $S$ is known.
We define the compressibility $c(M)$ of any string $M$ by
$c(M) = \frac{H(M)}{|M|}$
and the conditional compressibility of string $M$ over an alphabet $\mathcal{A}$ given another string $S$ over the same alphabet as
$c(M|S) = \frac{H(S+M) - H(S)}{|M|}$
where $S+M$ is the concatenation of the strings $S$ and $M$, $H(S)$ is the entropy of string $S$, and $|M|$ is the length of the string $M$.
The fractional increase in compressibility of $M$ given $S$ can then measured by
$r(M|S) = \frac{c(M) - c(M|S)}{c(M)}$.
We use $r(M|S)$ as the similarity metric to measure the similarity of a system summary $S$ to the model summary $M$.
Our metric is similar to the one proposed by Li and Vitanyi and is theoretically well-justified from the perspective of algorithmic information theory. One peculiarity is that our similarity is asymmetric.
The only thing that is needed to implement the above similarity metric is an estimate of the entropy $H(S)$ for a string $S$. We use the BWT for this estimate.
BWT-based String Entropy Estimate
We use the Move-To-Front (MTF) entropy of the Burrows-Wheeler transform of a given string $S$ as an estimate for its entropy $H(S)$.
The MTF encoding of a string is performed by traversing the string and assigning to each symbol the position of that symbol in the alphabet and then moving the symbol to the front of the alphabet. Therefore a sequence with a lot of runs will have a lot of zeros in its MTF encoding.
In this paper the MTF coding is used to define the MTF entropy (which the authors also call local entropy) of a string $R$ as
$\mbox{MTFE}(R) = \sum_i \mbox{log}(\mbox{MTF}(R)_i + 1)$
where $\mbox{MTF}(R)_i$ is the $i^{th}$ symbol of the MTF coding of the string $R$.
Now we define $H(S)$, the entropy of string $S$ as
$H(S) = \mbox{MTFE}(\mbox{BWT}(S))$
where $\mbox{BWT}(S)$ is the BWT of string $S$.
Since the Burrows-Wheeler transform involves just the construction of a suffix array, the computation of our compression based evaluation metric is linear in time and space in the length of the model and system summary strings.
Some Technical Details
For our implementation, we considered each word in a string as a separate symbol. Our alphabet of symbols therefore contained all the words in the two strings being compared. The words were normalized by lower casing and removing punctuation. Because BWT needs an ordered alphabet, we used the lexicographic order on the words in the alphabet.
Results
The results on the TAC-AESOP task (above) show that the BWT based metric (FraCC in the table) is reasonable for summarization evaluation, especially because there are not very many knobs to tune. I obtained these results from Frank (who will present them at TAC next week). The “best metric” is the AESOP submission that seemed to have high scores across several measures.
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## BWT for NLP (1)
September 26, 2009 Leave a comment
The Burrows-Wheeler transform (BWT), which is the main step in the bzip2 compression algorithm, is a permutation transform on a string over an ordered alphabet. It is a clever idea and can be useful for some string processing for natural language processing. I will present one such use.
BWT massages the original string into being more amenable to compression. Of course the transform doesn’t alter the compressibility (entropy rate) of the original string. All it does is make the string more compressible by algorithms we know.
The reason string permutation by BWT (as opposed to say sorting the string, which makes it really compressible) is useful is that the reverse transform (undoing the permutation) can be done with very little additional information. Mark Nelson wrote a nice introduction to the transform. Moreover, the BWT essentially involves the construction of the suffix array for the string, and therefore can be done in time and space linear in the length of the string.
Here is an example of the Burrows-Wheeler tranformation of the first stanza of Yeats’ Sailing to Byzantium. I added some newlines to the transformed string, and the underscores represent spaces in the original string. Notice the long runs of characters in the transformed string.
Original string
THAT is no country for old men. The young In one another’s arms, birds in the trees – Those dying generations – at their song, The salmon-falls, the mackerel-crowded seas, Fish, flesh, or fowl, commend all summer long Whatever is begotten, born, and dies. Caught in that sensual music all neglect Monuments of unageing intellect.
BWTransformed string
rsgnsnlhhs__lntsnH__T__.A____ss.,gt,.-gcd,es s,,,ode,yrgtsgrTredllssrn,edtrln,ntefemnu__fs___eh_hrC___ia__-eennlew_r_nshhhhslldrnbghrttmmgsmhvmnkielto-___nnnnna_ueesstWtTtTttTgsd__ye_teb__Fcweallolgfaaeaa_l
__mumoulr_reoeIiiueao_eouoii_aoeiueon__cm_sliM_
fbhngycrfeoeeoieiteaoctamleen’idit_o__ieu_n_cchaanta
____oa_nnosans_oomeoord_
A useful property
Effros et. al. showed that for a string generated by a finite-memory source, the BWT of the string is asymptotically (in the length of the string) indistinguishable from a piece-wise independent and identically distributed (i.i.d.) string. This is not surprising given that symbols with similar contexts appear sequentially in the BWT string, and for finite memory sources the current symbol is generated i.i.d. given a finite length context.
This property can be exploited to easily cluster words according to context by using BWT.
Word clustering
In this paper, among other things, Brown et.al. present a word clustering algorithm based on maximizing the average mutual information between the cluster ids of adjacent words. Some results are presented in Table 2 in the paper.
Such word clusters can be useful for feature engineering for sequence tagging tasks such as part-of-speech tagging or named-entity recognition. One of the most commonly used features for such tasks is one which checks if the current word is in a carefully constructed list of words.
Brown et. al. admit that, even after optimizations, their algorithm is slow and resort to approximations. (I realize that computers have gotten much faster since but still their algorithm is cubic in the size of the vocabulary.)
Word clustering based on BWT
We will cluster two words together if they appear independently given certain contexts (albeit with different probabilities). We first perform a BW transform on the input string of words (considering each word as a symbol, unlike in the example above) and measure whether the two words appear independently in an i.i.d. fragment.
Instead of actually trying to chop the BWT string into i.i.d. fragments before analysis, we adopt a proxy metric. We check if the number of times the two words are next to each other in the BWT string is large compared to what we would expect from their frequencies. We compute this as probability ratio with appropriate smoothing.
Another neat consequence of doing the clustering by BWT is that we only need to consider pairs of words that do appear next to each other in the BWT string. Therefore the selection of candidates for clustering is linear in the length of the string and not quadratic in the size of the vocabulary.
Some results
I ran this algorithm on about a month’s worth of New York Times and Wall Street Journal news data and these are the pairs of words with the highest scores.
january february 0.177721578886
january march 0.143172972502
march february 0.142398170589
englandgeoneng jerseyusanj 0.141412321852
news becdnews 0.135642386152
finala final 0.131901568726
finala finalb 0.122728309966
finala finalc 0.113085215849
cafd cea 0.107549686029
february april 0.100734422316
january april 0.0993752546848
has have 0.0967101802923
march april 0.0929933503714
did does 0.0854452561942
has had 0.0833642704346
will would 0.0827179598199
have had 0.0773517518078
january february 0.177721578886
january march 0.143172972502
march february 0.142398170589
englandgeoneng jerseyusanj 0.141412321852
news becdnews 0.135642386152
finala final 0.131901568726
finala finalb 0.122728309966
finala finalc 0.113085215849
cafd cea 0.107549686029
february april 0.100734422316
january april 0.0993752546848
has have 0.0967101802923
march april 0.0929933503714
did does 0.0854452561942
has had 0.0833642704346
will would 0.0827179598199
have had 0.0773517518078
I constructed a graph by joining all word pairs that have a score above a threshold and ran a greedy maximal clique algorithm. These are some of the resulting word clusters.
older young younger
announced today yesterday said reported
month today week yesterday
days month months decade year weeks years
decades months decade weeks years
com org www
writing write wrote
directed edited produced
should will probably could would may might can
worries worried concerns
work worked working works
wearing wear wore
win lost losing
man people men
against like to about that by for on in with from of at
under by on with into over from of
baton moulin khmer
daughter husband sister father wife mother son
red green blue black
ice sour whipped
time days months year years day
eastern coast southeastern
bergen orange nassau westchester
east ivory west
goes gone go going went
known seen well
travel review leisure weekly editorial
cultural financial foreign editorial national metropolitan
thursdays wednesdays fridays sundays tuesdays
thursday today monday sunday yesterday wednesday saturday friday tuesday
Discussion
1. For the above results, I only did the clustering based on right contexts. We can easily extend the word-pair score to take into account left contexts as well by concatenating the BWT of the reversed string to the BWT of the original string, and calculating the scores on this double length transformed string.
2. The word clustering algorithm of Brown et. al. proceeds by iteratively merging the best pair of words and replacing the two words in the alphabet (and the string) by a merged word. We can imagine doing something similar with our approach, except, because BWT uses the order on the alphabet, we need to decide where to insert the merged word.
3. One thing that I should have done but didn’t for the above results is to order the alphabet (of words) lexicographically. Instead I assign positive integers to the words based on their first appearance in the string, which is the order BWT uses to sort. Fixing this should improve the results.
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2017-10-21 19:12:25
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http://sofdem.github.io/gccat/gccat/sec2.1.6.html
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### 2.1.6. Reification view
Suppose we want to associate a 0-1 domain variable $b$ to a constraint $𝒞$ and maintain the equivalence $b\equiv 𝒞$. This is called the reification of $𝒞$. For most global constraints this can be achieved by reformulating the global constraint as a conjunction of pure functional dependency constraints together with constraints that can be easily reified, e.g. linear constraints involving at most two variables [BeldiceanuCarlssonFlenerPearson13].
We can reify the $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$$\left(〈{x}_{1},{x}_{2},\cdots ,{x}_{n}〉\right)$ constraint by using the idea of sorting its variables (i.e., the pure functional dependency part) and by stating that within the sorted list of variables adjacent variables are in strictly increasing order. This leads to the following expression $\mathrm{𝚜𝚘𝚛𝚝}$$\left(〈{x}_{1},{x}_{2},\cdots ,{x}_{n}〉,〈{y}_{1},{y}_{2},\cdots ,{y}_{n}〉\right)\wedge \left({y}_{1}<{y}_{2}\wedge {y}_{2}<{y}_{3}\wedge \cdots \wedge {y}_{n-1}<{y}_{n}\right)\equiv b$.
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2022-05-23 11:26:24
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https://math.stackexchange.com/questions/1778187/show-that-x3-y3-z3-t3-1999-has-infinitely-many-integer-solutions/1778240
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# Show that $x^3 + y^3 + z^3 + t^3 = 1999$ has infinitely many integer solutions.
Show that $x^3 + y^3 + z^3 + t^3 = 1999$ has infinitely many integer solutions.
I have not been able to find a single solution to this equation. With some trial I think there does not exist a solution with all of them positive. Can you please help me proceed?
Thanks.
• $(2n+14)^3-(2n+23)^3-(3n+26)^3+(3n+30)^3=18n+1$ and $n=111$ gives a solution. – almagest May 9 '16 at 14:48
• If the numbers are restricted to positive integers, then, of course, there are only a finite number of solutions. I think that they meant to domain to be all integers. – steven gregory May 9 '16 at 14:49
• @StevenGregory Yes. – TheRandomGuy May 9 '16 at 14:50
• @almagest How did you derive this? – TheRandomGuy May 9 '16 at 14:51
• @almagest, googling on "1999 has infinitely many integer solutions" gives various putnam and olympiad hits. – Barry Cipra May 9 '16 at 15:01
Look for solutions $$10-b,10+b,-\frac{1}{2}(d+1),\frac{1}{2}(d-1)$$ The sum of these numbers cubed is $2000+60b^2-\frac{1}{4}(3d^2+1)$, so we need $240b^2-3d^2-1=-4$ or $d^2-80b^2=1$. It is easy to see that has the solution $d=9,b=1$. Now if $d,b$ is a solution then $(9d+80b)^2-80(9b+d)^2=d^2-80b^2$, so $d'=9d+80b,b'=9b+d$ is another solution. Note that $d$ will always be odd.
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2021-04-15 03:35:01
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http://www.j.sinap.ac.cn/nst/EN/10.1007/s41365-016-0076-8
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# Nuclear Science and Techniques
《核技术》(英文版) ISSN 1001-8042 CN 31-1559/TL 2019 Impact factor 1.556
Nuclear Science and Techniques ›› 2016, Vol. 27 ›› Issue (3): 72
• NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH •
### Photon interaction with semiconductor and scintillation detectors
V. P. Singh , N. M. Badiger
1. Department of Physics, Karnatak University, Dharwad 580003, India
• Contact: V. P. Singh E-mail:kudphyvps@rediffmail.com
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V. P. Singh, N. M. Badiger. Photon interaction with semiconductor and scintillation detectors.Nuclear Science and Techniques, 2016, 27(3): 72
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Abstract:
Mass attenuation coefficients, effective atomic numbers, and electron densities for semiconductor and scintillation detectors have been calculated in the photon energy range 1 keV–100 GeV. These interaction parameters have been found to vary with detector composition and the photon energy. The variation in the parameters with energy is shown graphically for all the partial photon interaction processes. The effective atomic numbers of the detector were compared with the ZXCOM program, and the results were found to be comparable. Efficiencies of semiconductor and scintillation detectors are presented in terms of effective atomic numbers. The study should be useful for comparing the detector performance in terms of gamma spectroscopy, radiation sensitivity, radiation measurement, and radiation damage. The results of the present investigation should stimulate research work for gamma spectroscopy and radiation measuring materials.
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2022-01-16 09:47:11
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http://math.stackexchange.com/questions/39564/about-the-sum-of-two-integrals-and-jordan-decomposition
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# About the sum of two integrals and Jordan decomposition
Let $(X,\mathcal{F})$ be a measurable space. Let $\mu : \mathcal{F} \rightarrow \mathbb{R}$ be a real measure (i.e. $\mu (\phi) = 0$ and $\mu$ is $\sigma$-additive). Let $$|\mu| (E) = \sup \left\{\sum_{h=0}^\infty |\mu(E_h)|, E_h\; {\rm are~pairwise~disjoint},\; \bigcup_{h=0}^\infty E_h = E\right\}.$$ If $\mu^+ = \frac{|\mu| + \mu}{2}$ and $\mu^-=\frac{|\mu| - \mu}{2}$ then is it true that :
$$\int_X u\, d\mu^+ + \int_X u \, d\mu^- = \int_X u \, d|\mu| \qquad \text{for all } |\mu|\text{-measurable }u : X \rightarrow [0,\infty]?$$
I am doubtful if this is a standard result, if so can you point to any references?
The notation used here is from the book : "Functions of Bounded Variation and Free Discontinuity Problems" by Luigi Ambrosio et. al.
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I'm not sure what you're asking (I'm a bit puzzled about your emphasis "$u$ is $|\mu|$-measurable"), but it looks a bit like if you're asking whether $|\mu| = \mu^{+} + \mu^{-}$ holds, but that is obviously true by definition. Note also that $\mu^{+}, \mu^{-} \ll |\mu|$ are positive measures and $\mu^{+} \perp \mu^{-}$. – t.b. May 17 '11 at 6:05
@The: I wanted to know if the sum of integrals on the left equals the integral on the right with some restrictions on $u$. I could prove that for simple functions but not for general $u : X \rightarrow [0,\infty]$. Hope this is more clear. – jpv May 17 '11 at 6:13
But yes, as $u \geq 0$, it can be approximated monotonically by a sequence of simple functions, so you can simply apply the monotone convergence theorem on both sides of the identity, as you're only dealing with positive measures and functions. Or am I missing something? – t.b. May 17 '11 at 6:18
Thanks. I will have to refresh my memory about the mct. – jpv May 17 '11 at 6:24
Approximation $u$ pointwise by a non-decreasing sequence $\{u_k\}$ of simple functions. The wanted equality is a consequence of the definition of the Lebesgue integral when $u$ is the characteristic function of a measurable set. So by linearity, we deduce that for each $k$, $$\int_Xu_k\mathrm d\mu^++\int_Xu_k\mathrm d\mu^-=\int_Xu_k\mathrm d|\mu|.$$ Now use monotone convergence in order to take the limit $k\to +\infty$.
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2015-04-18 16:06:26
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http://server3.wikisky.org/starview?object_type=1&object_id=717&object_name=HD+90589&locale=EN
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WIKISKY.ORG
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Thefollowing data are listed for each star: identifications, apparentmagnitude, spectral type, apparent diameter in arcsec, absolute radiusin solar units, method of determination, reference, remarks. Commentsand statistics obtained from CADARS are given. The Catalogue isavailable in electronic form at the CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcar?J/A+A/367/521 Research Note Hipparcos photometry: The least variable starsThe data known as the Hipparcos Photometry obtained with the Hipparcossatellite have been investigated to find those stars which are leastvariable. Such stars are excellent candidates to serve as standards forphotometric systems. Their spectral types suggest in which parts of theHR diagrams stars are most constant. In some cases these values stronglyindicate that previous ground based studies claiming photometricvariability are incorrect or that the level of stellar activity haschanged. Table 2 is only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/367/297 The proper motions of fundamental stars. I. 1535 stars from the Basic FK5A direct combination of the positions given in the HIPPARCOS cataloguewith astrometric ground-based catalogues having epochs later than 1939allows us to obtain new proper motions for the 1535 stars of the BasicFK5. The results are presented as the catalogue Proper Motions ofFundamental Stars (PMFS), Part I. The median precision of the propermotions is 0.5 mas/year for mu alpha cos delta and 0.7mas/year for mu delta . The non-linear motions of thephotocentres of a few hundred astrometric binaries are separated intotheir linear and elliptic motions. Since the PMFS proper motions do notinclude the information given by the proper motions from othercatalogues (HIPPARCOS, FK5, FK6, etc.) this catalogue can be used as anindependent source of the proper motions of the fundamental stars.Catalogue (Table 3) is only available at the CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strastg.fr/cgi-bin/qcat?J/A+A/365/222 Sixth Catalogue of Fundamental Stars (FK6). Part I. Basic fundamental stars with direct solutionsThe FK6 is a suitable combination of the results of the HIPPARCOSastrometry satellite with ground-based data, measured over more than twocenturies and summarized in the FK5. Part I of the FK6 (abbreviatedFK6(I)) contains 878 basic fundamental stars with direct solutions. Suchdirect solutions are appropriate for single stars or for objects whichcan be treated like single stars. From the 878 stars in Part I, we haveselected 340 objects as "astrometrically excellent stars", since theirinstantaneous proper motions and mean (time-averaged) ones do not differsignificantly. Hence most of the astrometrically excellent stars arewell-behaving "single-star candidates" with good astrometric data. Thesestars are most suited for high-precision astrometry. On the other hand,199 of the stars in Part I are Δμ binaries in the sense ofWielen et al. (1999). Many of them are newly discovered probablebinaries with no other hitherto known indication of binarity. The FK6gives, besides the classical "single-star mode" solutions (SI mode),other solutions which take into account the fact that hidden astrometricbinaries among "apparently single-stars" introduce sizable "cosmicerrors" into the quasi-instantaneously measured HIPPARCOS proper motionsand positions. The FK6 gives in addition to the SI mode the "long-termprediction (LTP) mode" and the "short-term prediction (STP) mode". TheseLTP and STP modes are on average the most precise solutions forapparently single stars, depending on the epoch difference with respectto the HIPPARCOS epoch of about 1991. The typical mean error of anFK6(I) proper motion in the single-star mode is 0.35 mas/year. This isabout a factor of two better than the typical HIPPARCOS errors for thesestars of 0.67 mas/year. In the long-term prediction mode, in whichcosmic errors are taken into account, the FK6(I) proper motions have atypical mean error of 0.50 mas/year, which is by a factor of more than 4better than the corresponding error for the HIPPARCOS values of 2.21mas/year (cosmic errors included). Spectroscopic binary orbits from photoelectric radial velocities. Paper 148: HR 7955Not Available Direct calibration of the Cepheid period-luminosity relationAfter the first release of Hipparcos data, Feast & Catchpole gave anew value for the zero-point of the visual Cepheid period-luminosityrelation, based on trigonometric parallaxes. Because of the largeuncertainties on these parallaxes, the way in which individualmeasurements are weighted is of crucial importance. We thereforeconclude that the choice of the best weighting system can be aided by aMonte Carlo simulation. On the basis of such a simulation, it is shownthat (i) a cut-off in π or in σ_ππ introduces a strongbias; (ii) the zero-point is more stable when only the brightestCepheids are used; and (iii) the Feast & Catchpole weighting givesthe best zero-point and the lowest dispersion. After correction, theadopted visual period-luminosity relation is=-2.77logP-1.44+/-0.05. Moreover, we extend this study to thephotometric I band (Cousins) and obtain=-3.05logP-1.81+/-0.09. Infrared Space Observatory Photometric Search of Main-Sequence Stars for Vega-Type SystemsWe obtained 3.6-20 μm photometry of 38 bright [IRAS F_nu(12μm)>0.7 Jy] main-sequence stars with the Infrared SpaceObservatory (ISO). Observations were conducted with the ISOPHOTinstrument, in the single-pointing photometry mode, through filters at3.6, 11.5, and 20.0 mum. We searched for excess (Vega-type) emissionfrom dust at temperatures >~100 K, located at ~1-60 AU from thestars. We thus sampled dust at warm, terrestrial material temperaturesand at cool (~100 K) temperatures of possible Kuiper Belt-type regionsin these systems. We detected 20 μm excesses from ~14% of oursources, but we did not detect 11.5 μm excesses from any of them. Wepresent single-temperature blackbody models of the location and densityof dust emission around 10 stars, two of them (29 Cyg and Gl 816) withexcesses newly reported here. We make a thorough comparison of ISO andIRAS data on our target stars and propose a new calibration procedurefor ISOPHOT staring measurements at 3.6, 11.5, and 20 mum. The ROSAT all-sky survey catalogue of the nearby starsWe present X-ray data for all entries of the Third Catalogue of NearbyStars \cite[(Gliese & Jahreiss 1991)]{gli91} that have been detectedas X-ray sources in the ROSAT all-sky survey. The catalogue contains1252 entries yielding an average detection rate of 32.9 percent. Inaddition to count rates, source detection parameters, hardness ratios,and X-ray fluxes we also list X-ray luminosities derived from Hipparcosparallaxes. Catalogue also available at CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/Abstract.html The ROSAT all-sky survey catalogue of optically bright main-sequence stars and subgiant starsWe present X-ray data for all main-sequence and subgiant stars ofspectral types A, F, G, and K and luminosity classes IV and V listed inthe Bright Star Catalogue that have been detected as X-ray sources inthe ROSAT all-sky survey; several stars without luminosity class arealso included. The catalogue contains 980 entries yielding an averagedetection rate of 32 percent. In addition to count rates, sourcedetection parameters, hardness ratios, and X-ray fluxes we also listX-ray luminosities derived from Hipparcos parallaxes. The catalogue isalso available in electronic form via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/Abstract.html The Angular Momentum of Main Sequence Stars and Its Relation to Stellar ActivityRotational velocities are reported for intermediate-mass main sequencestars it the field. The measurements are based on new, high S/N CCDspectra from the Coudé Feed Telescope of the Kitt Peak NationalObservatory. We analyze these rotation rates for a dependence on bothmass and age. We compare the average rotation speeds of the field starswith mean velocities for young stars in Orion, the Alpha Persei cluster,the Pleiades, and the Hyades. The average rotation speeds of stars moremassive than $\sim1.6$ \msun\experience little or no change during theevolutionary lifetimes of these stars on the zero age main sequence orwithin the main sequence band. Less massive stars in the range betwee n1.6\msun\ and 1.3\msun\ also show little decline in mean rotation ratewhile they are on the main sequence, and at most a factor of 2 decreasein velocity as they evolve off the main sequence. The {\it e}-foldingtime for the loss of angular momentum b y the latter group of stars isat least 1--2 billion years. This inferred characteristic time scale forspindown is far longer than the established rotational braking time forsolar-type stars with masses below $\sim1.3$ \msun. We conclude from acomparison of the trends in rotation with trends in chromospheric andcoronal activity that the overall decline in mean rotation speed alongthe main sequence, from $\sim2$ \msun\ down to $\sim1.3$ \msun, isimposed during the pre-main sequence phase of evolution, and that thispattern changes little thereafter while the star resides on the mainsequence. The magnetic activity implicated in the rotational spindown ofthe Sun and of similar stars during their main sequence lifetimes mus ttherefore play only a minor role in determining the rotation rates ofthe intermediate mass stars, either because a solar-like dynamo is weakor absent, or else the geometry of the magnetic field is appreciablyless effective in removing angular momentu m from these stars. (SECTION:Stars) Convection, Thermal Bifurcation, and the Colors of A StarsBroadband ultraviolet photometry from the TD-1 satellite andlow-dispersion spectra from the short wavelength camera of IUE have beenused to investigate a long-standing proposal of Bohm-Vitense that thenormal main-sequence A and early-F stars may divide into two differenttemperature sequences: (1) a high-temperature branch (and plateau)comprised of slowly rotating convective stars, and (2) a low-temperaturebranch populated by rapidly rotating radiative stars. We find noevidence from either data set to support such a claim, or to confirm theexistence of an "A-star gap" in the B-V color range 0.22 <= B-V <=0.28 due to the sudden onset of convection. We do observe, nonetheless,a large scatter in the 1800--2000 A colors of the A--F stars, whichamounts to ~0.65 mag at a given B-V color index. The scatter is notcaused by interstellar or circumstellar reddening. A convincing case canalso be made against binarity and intrinsic variability due topulsations of delta Sct origin. We find no correlation with establishedchromospheric and coronal proxies of convection, and thus nodemonstrable link to the possible onset of convection among the A--Fstars. The scatter is not instrumental. Approximately 0.4 mag of thescatter is shown to arise from individual differences in surface gravityas well as a moderate spread (factor of ~3) in heavy metal abundance andUV line blanketing. A dispersion of ~0.25 mag remains, which has noclear and obvious explanation. The most likely cause, we believe, is aresidual imprecision in our correction for the spread in metalabundances. However, the existing data do not rule out possiblecontributions from intrinsic stellar variability or from differential UVline blanketing effects owing to a dispersion in microturbulentvelocity. The Pulkovo Spectrophotometric Catalog of Bright Stars in the Range from 320 TO 1080 NMA spectrophotometric catalog is presented, combining results of numerousobservations made by Pulkovo astronomers at different observing sites.The catalog consists of three parts: the first contains the data for 602stars in the spectral range of 320--735 nm with a resolution of 5 nm,the second one contains 285 stars in the spectral range of 500--1080 nmwith a resolution of 10 nm and the third one contains 278 stars combinedfrom the preceding catalogs in the spectral range of 320--1080 nm with aresolution of 10 nm. The data are presented in absolute energy unitsW/m(2) m, with a step of 2.5 nm and with an accuracy not lower than1.5--2.0%. A microwave survey of southern early-type starsA multi-epoch survey with the Parkes telescope of a completedistance-limited sample of 57 stars earlier than F6 has detectedpossible 8.4-GHz emission from 16 stars. Single-epoch partial synthesisobservations with the Australia Telescope Compact Array (ATCA) at 4.8GHz on 27 stars from the same sample (including the possible Parkesdetections) found no emission at the stellar positions above a fluxdensity limit of 1.2-1.9 mJy, but the maps show that the Parkesdetections are not merely the results of confusion of sources within theParkes beam. Three early F stars with UV and/or X-ray emission wereobserved simultaneously at 4.8 and 8.4 GHz in 12-h syntheses with the6-element ATCA. Two of these stars were from the above sample and thethird was the supergiant Alpha Carinae. We detected only alphaCar withflux densities of 300+/-65 and 140+/-65 muJy at 4.8 and 8.6 GHz(S~nu^-1.3+/-1.3). We discuss the legitimacy of the Parkes 3-6sigmadetections and show that, although none has been detected by synthesisobservations, there is no compelling reason for rejecting them on theinternal evidence. The power emitted by the supergiant alphaCar issimilar to that of the 16 possible Parkes detections, although itsactivity index is orders of magnitude lower. We show that this emissioncannot be thermal bremsstrahlung from the 10^7.2-K corona of the starbut is probably synchrotron emission from a magnetically maintainedcorona. The Composition of HB Stars: RR Lyrae VariablesWe used moderately high-resolution, high S/N spectra to study thechemical composition of 10 field ab-type RR Lyrae stars. A newtemperature scale was determined from literature Infrared Flux Methodmeasures of subdwarfs and the Kurucz (1992) model atmospheres, and usedto calibrate colors for both dwarfs and RR Lyraes. The applicability ofKurucz (1992) model atmospheres in the analysis of RR Lyraes at minimumlight was analyzed: we found that they are able to reproduce colors,excitation and ionization equilibria as well as the wings of Halpha. Wederived abundances for 21 species. The metal abundances of the programstars span the range -2.50<[Fe/H]<+0.17\$. Lines of most elementsare found to form in LTE conditions. Fe lines satisfy very well theexcitation and ionization equilibria. RR Lyraes share the typicalabundance pattern of other stars of similar [Fe/H]: alpha-elements areoverabundant by about 0.4dex and Mn is underabundant by about 0.6dex instars with [Fe/H]<-1. Significant departures from LTE are found onlyfor a few species. We used our new [Fe/H] abundances, as well as valuesfrom Butler and coworkers (corrected to our system), and from highresolution spectroscopy of globular clusters giants, to obtain a newcalibration of the DeltaS index: [Fe/H]= -0.194(\pm 0.011)DeltaS-0.08(\pm 0.18) and to update the metallicity calibration of the Ca II Kline index: [Fe/H]= 0.65(\pm 0.17)W'(K) -3.49(\pm 0.39). Finally, ournew metallicity scale was used to revise the [Fe/H] dependence of theabsolute magnitude of RR Lyrae stars, Mv: Mv = 0.20(\pm 0.03)[Fe/H] +1.06(\pm 0.04). Vitesses radiales. Catalogue WEB: Wilson Evans Batten. Subtittle: Radial velocities: The Wilson-Evans-Batten catalogue.We give a common version of the two catalogues of Mean Radial Velocitiesby Wilson (1963) and Evans (1978) to which we have added the catalogueof spectroscopic binary systems (Batten et al. 1989). For each star,when possible, we give: 1) an acronym to enter SIMBAD (Set ofIdentifications Measurements and Bibliography for Astronomical Data) ofthe CDS (Centre de Donnees Astronomiques de Strasbourg). 2) the numberHIC of the HIPPARCOS catalogue (Turon 1992). 3) the CCDM number(Catalogue des Composantes des etoiles Doubles et Multiples) byDommanget & Nys (1994). For the cluster stars, a precise study hasbeen done, on the identificator numbers. Numerous remarks point out theproblems we have had to deal with. A study of the Chamaeleon star forming region from the ROSAT all-sky survey. I. X-ray observations and optical identifications.We present the observations of the ROSAT all-sky survey (RASS) in thedirection of the Chamaeleon cloud complex, as well as the spectroscopicidentifications of the detected X-ray sources. The main purpose of thisidentification program was the search for low mass pre-main sequencestars. Sixteen previously known PMS stars were detected with highconfidence by ROSAT. Eight are classical T Tauri stars and eight areweak-line T Tauri stars, Seventy-seven new weak-line T Tauri stars wereidentified on the basis of the presence of strong Li λ 6707absorption, spectral type later than F0 and chromospheric emission. Wegive coordinates and count rates of the X-ray sources, and presentoptical spectra and finding charts for the sources identified opticallyas new pre-main sequence stars. Optical UBV(RI)_c_ and near-infraredJHKLM photometry for this sample of stars is also provided. In addition,6 new dKe-dMe candidates are found among the RASS sources. The determination of T_eff_ of B, A and F main sequence stars from the continuum between 3200 A and 3600 A.A method of determination of the effective temperature of B, A and Fmain sequence stars is proposed, using the slope of the continuumbetween 3200A and 3600A. The effective temperature calibration is basedon a sample of stars with energy distributions known from UV to the red.We have determined the Balmer jump and the effective temperatures for235 main sequence stars. The temperatures found have been compared withthose derived by Underhill et al. (1979), Kontizas & Theodossiou(1980), Theodossiou (1985), Morossi & Malagnini (1985). Thecomparison showed good agreement for most of the stars in common. On theother hand, the temperatures derived from the reddening-free colourfactor QUV, from the colour index (m1965-V) and from (B-V), given inGulati et al. (1989), are systematically lower than our temperatures,however the differences are within one-sigma error. The MSSSO near-infrared photometric systemThe JHKL photometric system currently used at the Mount Stromlo andSiding Spring Observatories (MSSSO) is described via an extensive listof standard-star values and filter transmission curves. At JHK thissystem is identical to the Mount Stromlo Observatory (MSO) systemdefined by Jones and Hyland (1982), except for small zero-pointdifferences which we impose here. Transformations are given between theMSSSO system and several near-infrared photometric systems in use inother observatories and the homogenized JHKL system proposed by Besselland Brett (1988). A catalog of stellar Lyman-alpha fluxesWe present a catalog of stellar Ly-alpha emission fluxes, based on newand archival images obtained with the IUE spacecraft. The catalogincludes 227 stars with detectable Ly-alpha emission fluxes, and upperlimits on the Ly-alpha emission flux for another 48 stars. Multiple fluxmeasurements are given for 52 stars. We present a model for correctingthe observed Ly-alpha flux for attenuation by the local interstellarmedium, and we apply this model to derive intrinsic Ly-alpha fluxes for149 catalog stars which are located in low H I column density directionsof the local interstellar medium. In our catalog, there are 14 late-Aand early-F stars at B-V = 0.29 or less that show detectable emission atLy-alpha. We find a linear correlation between the intrinsic Ly-alphaflux and C II 1335 A flux for stars with B-V greater than 0.60, but theA and F stars deviate from this relation in the sense that theirLy-alpha flux is too low. We also find a good correlation betweenLy-alpha strength and coronal X-ray emission. This correlation holdsover most of the H-R diagram, even for the F stars, where an X-raydeficit has previously been found relative to the transition regionlines of C II and C IV. Optical Polarization of 1000 Stars Within 50-PARSECS from the SunAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1993A&AS..101..551L&db_key=AST
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2019-02-22 17:35:01
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http://c-primed.github.io/puq/ug/calibration.html
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#### Next topic
Sensitivity Analysis
# Calibration of Input Variables¶
Frequently computer models will have input parameters that cannot be measured and are not well known. In this case, Bayesian calibration can be used to estimate the input parameters. To do this, experimental data is required for the output and for all known input variables. The calibration process adjusts the unknown parameters to fit the observed data.
## Using PUQ for Calibration¶
To use PUQ to do calibration, you will have to add experimental data to all the known input parameters. For the TestProgram, you will need to add experimental data and (optionally) a measurement error. The measurement error is a standard deviation.
The presence of experimental data tells PUQ to do calibration. The steps are as follows:
1. Build a response surface over the range of the input paramaters.
2. Do Bayesian calibration using the response surface for the likelihood function.
3. Manually adjust calibrated input parameter PDFs, if necessary. For example, truncate PDFs that go negative.
4. Generate output PDF.
## Example¶
test2 in puq/examples/calibrate calibrates z. The experimental data was generated using a z with a Normal distribution of 12 and deviation of 2. Some noise was added with a sigma of 0.1. For calibration we use a uniform prior [5, 20] for z.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 import puq import numpy as np def run(): # experimental data for z=Normal(12,2) exp_x =np.array([5.04, 5.14, 4.78, 5.12, 5.11, 5.13, 4.97, 5.1 , 5.53, 5.09]) exp_y = np.array([3.33, 3.56, 2.94, 3.27, 3.54, 3.4 , 3.52, 3.63, 3.45, 3.19]) exp_data = np.array([-26.16, -18.39, -20.57, -45.24, -29.16, -22., -46.47, -3.15, -10.48, -16.88]) # measurement error sigma = 0.1 # Create parameters. Pass experimental input data to # non-calibration parameters. x = puq.NormalParameter('x', 'x', mean=5, dev=0.2, caldata=exp_x) y = puq.NormalParameter('y', 'y', mean=3.4, dev=0.25, caldata=exp_y) z = puq.UniformParameter('z', 'z', min=5, max=20) # set up host, uq and prog normally host = puq.InteractiveHost() uq = puq.Smolyak([x,y,z], level=2) prog = puq.TestProgram('./model_1.py', desc='model_1 calibration') # pass experimental results to Sweep return puq.Sweep(uq, host, prog, caldata=exp_data, calerr=sigma)
~/puq/examples/calibrate> puq start test2.py
Saving run to sweep_85212814.hdf5
Processing <HDF5 dataset "f": shape (25,), type "<f8">
Surface = 1.0*x**2 + 1.0*x*y + 0.75*y**2 + 2.0*y - 7.0*z + 2.0
RMSE = 2.15e-11 (1.74e-11 %)
SENSITIVITY:
Var u* dev
-----------------------------
z 1.0500e+02 1.8394e-11
y 1.8630e+01 1.1297e+00
x 1.6505e+01 1.0054e+00
Performing Bayesian Calibration...
[****************100%******************] 12000 of 12000 complete
Calibrated z to Normal(12.0786402402, 2.2152608414).
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2021-04-12 03:31:41
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https://math.eretrandre.org/tetrationforum/showthread.php?pid=3303
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elementary superfunctions tommy1729 Ultimate Fellow Posts: 1,358 Threads: 330 Joined: Feb 2009 04/28/2009, 09:37 PM bo198214 Wrote:Yes exactly those formula I was looking for! However can you shorten them a bit by gathering terms or introducing constants for repeatedly occuring terms? If possible it would be very preferable to indicate the fixed point, if the super-function is obtained by regular iteration. Would anyway be good if you could explain how you obtained the formulas or what the idea behind is. just like ansus you dont seem to realize mathematica just uses a handful solutions and all others are special cases. notice for instance that EVERY f(x) in this thread are 1) f(x) = polynomial ( see also "logistic map" and " inverse hypergeo " ) 2) f(x) = moebius = (a x + b)/(c x + d) 3) f(x) = a + b x^c and 2) has a closed form solution so all that one needs to do to find the superfunction of e.g. 1 / a x + b is to fill in some variables in the general formula. at this point , its just that simple. regards tommy1729 bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 04/29/2009, 04:58 PM Ansus Wrote:The superfunction for 1/x is probably indeed complex. Why do you suppose it to be real? Well I was rather after real functions. Input strictly increasing, output real. What does mathematica say about $f(x)=\frac{x-1}{x+1}$? This is strictly increasing and has no real fixed point. bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 04/29/2009, 05:46 PM Ansus Wrote:But I wonder why Mathematica did not find something like $F(x)=C^{(-1)^x}$ which is a case of $F(x)=C^{b^x}$ Interestingly Maple finds exactly that solution. bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 04/29/2009, 06:59 PM Ansus Wrote:I've verified both and both indeed correct solutions. Mathematica finds $F(x)=C^{b^x}$ for the general case of $f(x)=x^b$. What Maple gives for $f(x)=\frac{x+a}{x-a}$? Nothing Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 05/11/2009, 02:02 PM (This post was last modified: 05/12/2009, 09:23 AM by Kouznetsov.) Introduction. I begin with this introduction in order to indicate, how do I understand the super-functions and our role about them. I remember that the Moderator has an ability to remove this introduction, together with all the lyrics around (and I appreciate his good will to keep all the posts so dry as his gunpowder); however, I hope the definitions below do not contradict those he suggested for our joint paper (which is "yet to be finished" during several last months); so, some definitions have some chance to survive. Terminology '''Superfunction''' $S$ comes from iteration of some given function $h$, called "base-function" or "transfer function". There is some analogy with fiber optics, which explains why this $h$ should be called "transfer function". Those who hate any physics (and, especially, the phenomenological fiber optics), may imagine that the function $h$ transfers the value of function $S$ at some point $z$ to the value at the point $z+1$, as the basic equation suggests: $S(z+1)=f(S(z))$ This equation is very basic; so, the only given function $h$ may also be called the "base-function". Iterations Roughly, for some function $h$ and for some constant $t$, the super-function could be defined with expression ${{S(z)} \atop \,} {= \atop \,} {{\underbrace{h\Big(h\big(... h(t)...\big)\Big)}} \atop {z \mathrm{~evaluations~of~function~}h\!\!\!\!\!\!}}$ then $S$ can be interpreted as superfunction of function $h$. Such definition is valid only for positive integer $z$. In particular, $S(1)=h(t)$. The most research and applications around the superfunctions are related with various extensions of super-function: analysis of the existence, uniqueness and ways of the evaluation. For some functions $h$, such as addition of a constant or multiplication by a constant, the superfunction can be expressed in terms of elementary function. Namely such examples were motivation of this message. History and Lowstory Analysis of superfunctions cames from the application to the evaluation of fractional iterations of functions. Super-functions and their inverse functions allow evaluation of not only minus-first power of a function (inverse function), but also any real and even complex iteration of the function. Historically, the first function of such kind considered was $\sqrt{\exp}~$; then, function $\sqrt{!~}~$ was used as logo of the Physics department of the Moscow State University, see http://zhurnal.lib.ru/img/g/garik/dubinu...ndex.shtml http://ofvp.phys.msu.ru/pdf/Kandidov_70.pdf http://nauka.relis.ru/11/0412/11412002.htm (bitte, all 3 in Russian). That time, researchers did not have computational facilities for evaluation of such functions, but the $\sqrt{\exp}$ was more lucky than the $~\sqrt{!~}~~$; at least the existence of holomorphic function $\varphi$ such that $\varphi(\varphi(z))=\exp(z)$ has been reported in 1950 by Helmuth Kneser (H.Kneser. “Reelle analytische L¨osungen der Gleichung $\varphi(\varphi(x)) = e^x$ und verwandter Funktionalgleichungen”. Journal fur die reine und angewandte Mathematik, 187 (1950), 56-67.) Extensions The recurrence above can be written as equations $S(z\!+\!1)=h(S(z)) ~ \forall z\in \mathbb{N} : z>0$ $S(1)=h(t)$. Instead of the last equation, one could write $S(0)=t$ and extend the range of definition of superfunction $S$ to the non-negative integers. Then, one may postulate $S(-1)=h^{-1}(t)$ and extend the range of validity to the integer values larger than $-2$. The following extension, for example, $S(-2)=h^{-2}(t)$ is not trivial, because the inverse function may happen to be not defined for some values of $t$. In particular, [[tetration]] can be interpreted as super-function of exponential for some real base $b$; in this case, $h=\exp_{b}$ then, at $t=1$, $S(-1)=\log_b(1)=0$. but $S(-2)=\log_b(0)~ \mathrm{is~ not~ defined}$. For extension to non-integer values of the argument, superfunction should be defined in different way. Definitions. For connected domains $C \subseteq \mathbb{C}$ and $D \subseteq \mathbb{C}, and two numbers [tex]a\in C$ and $d\in D$, the $(a \!\mapsto\! d)$ super-function of a transfer function $~h~$ is function $S$, holomorphic on $C$, such that $S(z\!+\!1)=h(S(z)) ~ \forall z\in C : S(z)\in D$ and $S(a)=b$. If $h=\exp_b [tex] for some [tex]b\in \mathbb{C}$, then the $(a \!\mapsto\! d)$ super-function of a transfer function $~h~$ is called $(a \!\mapsto\! d)$ super-exponential on the base $b$. If $a=0$ and $d=1$, then such a super-exponential is called tetration and justify the appearance of this post at this Forum. As it was already mentioned in this forum, in general, the super-function is not unique. For a given transfer function $h$, from given $(a\mapsto d)$ super-funciton $F$, another $(a\mapsto d)$ super-function $G$ could be constructed as $G(z)=F(z+\mu(z))$ where $\mu$ is any 1-periodic function, holomorphic at least in some vicinity of the real axis, such that $\mu(a)=0$. The modified super-function may have narrowed range of holomorphism. The challenging task is to specify some domain $C$ such that $(C, a \mapsto d)$ super-function is unique. In particular, the $(C, 0\mapsto 1)$ super-function of $\exp_b$, for $b>1$, is called [[tetration]] and is believed to be unique at least for $C= \{ z \in \mathbb{C} ~:~\Re(z)>-2 \}$; for the case $b>\exp(1/e)$ Examples Oh, en fin, I touch the goal of this post. Sorry for the long introduction above. Below, I consider various simple base-functions $h$ . Elementary increment Let $h(z)=z+1=++z$. Then, the identity function $I$ such that $I(z)=z \forall z \in \mathbb{C}$ is $(\mathbb{C}, 0\mapsto 0)$ superfunction of $h$. Addition Chose a $b$ and define function $\mathrm{add}_b$ such that $\mathrm{add}_b(z)=b\!+\!z ~ \forall z \in \mathbb{C}$ Define function $\mathrm{mul_b}$ such that $\mathrm{mul_b}(z)=b\!\cdot\! z ~ \forall z \in \mathbb{C}$. Then, function $~\mathrm{mul_b}~$ is $(\mathbb{C}, 0 \mapsto b )$ superfunction of $h$. Multiplication Exponential $\exp_b$ is $(\mathbb{C}, 0 \mapsto 1 )$ super-function of function $\mathrm{mul}_b$, defined in the previous example. Quadratic polynomial Let $h(z)=2 z^2-1$. Those, who like some Quantum Mechanics, may treat this function as a scaled second Hermitian polynomial, justifying the letter, used to denote the transfer function. Then, $f(z)=\cos( \pi \cdot 2^z)$ is a $(\mathbb{C},~ 0\! \rightarrow\! 1)$ superfunction of $H$. Indeed, $f(z+1)=\cos(2 \pi \cdot 2^z)=2\cos(\pi \cdot 2^z)^2 -1 =H(f(z))$ and $f(0)=\cos(2\pi)=1$ In this case, the superfunction $f$ is periodic; its period $T=\frac{2\pi \mathrm {i}}{\ln(2)} \mathrm{i}\approx 9.0647202836543876194 \!~i$. Such super-function approaches unity in the negative direction of the real axis, $\lim_{x\rightarrow -\infty} f(x)=1$ The example above and the two examples below are suggested at Mueller. Problems in Mathematics. http://www.math.tu-berlin.de/~mueller/projects.html Rational function. In general, the transfer function $h$ has no need to be entire function. Here is the example with meromorphic function $h$. Let $C= \{z \in \mathbb{C}: 2^z\ne n+1/2 \forall n\in \mathbb{Z})\}$ $D= \mathbb{C}\backslash \{1,-1\}$ $h(z)=\frac{2z}{1-z^2} ~ \forall z\in D~$ $S(z)=\tan(\pi 2^z) \forall z\in C$ Tthen, $S$ is $(C, 0\! \mapsto\! 0)$ superfunction of $h$. For the proof, the trigonometric formula $\tan(2 \alpha)=\frac{2 \tan(\alpha)}{1-\tan(\alpha)^2}~~ \forall \alpha \in \mathbb{C} \backslash \{\alpha\in \mathbb{C} : \cos(\alpha)=0 || \sin(\alpha)=\pm \cos(\alpha) \}$ can be used at $\alpha=\pi 2^z$, that gives $ h(S(z))=\frac{2 \tan(\pi 2^z)}{1-\tan(\pi 2^z)} = \tan(2 \pi 2^z)=S(z+1)$ Algebraic transfer function. However, the transfer function has no need to be even meromorphic. Let $C= \{z \in \mathbb{C}: |Arg(cos(\pi 2^z))| < \pi \}$ $D= \{z \in \mathbb{C}: |Arg(1-z^2)| < \pi \}$ $h(z)=2z \sqrt{1-z^2} \forall z \in D$ $S(z)=\sin(\pi 2^z) \forall z \in C$ Then, $S is is [tex](C,~ 0\!\rightarrow \!0)$ superfunction of $H$ for $C= \{z\in \mathbb C : \Re( \cos(\pi 2^z))>0 \}$. The proof is similar to the previous two cases. Exponential transfer function. Let $b>1$, $H(z)= \exp_b(z) \forall z \in \mathbb{C}$, $C= \{ z \in \mathbb{C} : \Re(z)>-2 \}$. Then, tetrational $\mathrm{tet}_b$ is a $(C,~ 0\! \rightarrow\! 1)$ super-function of $\exp_b$. more extensions. In general, we may take any special function $S$, such that $S(z+1)$ can be expressed through $S(z)$ with holomorphic elementary functions, then we may declare this expression as transfer function $h$, and then, function $S$ appears as super-function. I invite participants to construct more super-functions that can be easy represented through some already known special functions. P.S. Oh, mein Gott! I just realized the correct tread for this post. It repeats a lot of staff already posted here... Sorry... I see, there are already replies, so, I ssto to edit; the only correct obvious misprints... bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 05/11/2009, 07:39 PM (05/11/2009, 05:17 PM)Ansus Wrote: It should be noted that superfunction is not unique in most cases. For example, for $f(x)=2 x^2-1$, superfunction is $F(x)=\cos(2^x C)$ Ya, this is the simple kind of non-uniqueness, its just a translation along the x-axis. However there are also more severe types of non-uniques, as I already introduced in my first post, we have two solutions (which are not translations of each other): $F(x)=\cos(2^x)$ and $F(x)=\cosh(2^x)$. bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 05/11/2009, 08:19 PM (05/11/2009, 08:09 PM)Ansus Wrote: And also as we had seen, for 1/x there are also two different solutions. It is an open question thus how much independent superfunctions has a given function. Its not an open question, there are infinitely many (even for real-analytic solutions which $1/x$ does not have). If you have one solution $F$ just take any 1-periodic function $\theta$ and then $F(x+\theta(x))$ is another solution. Even elementary if $\theta$ is elementary (say linear combination of some $\sin(2\pi k x)$). Thatswhy I always try to find for elementary solutions whether they are regular at some fixed point because this reduces the number of real analytic solutions to two at one fixed point (analogously to $\exp_{\sqrt{2}}$) up to x-translation. bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 05/11/2009, 09:09 PM (05/11/2009, 08:37 PM)Ansus Wrote: count all solutions that differ only by periodic term as one solution. Then we have only one strictly increasing solution. If F and G are two super-functions of f then $F(x)=G(x+\theta(x))$ for $\theta(x)=G^{-1}(F(x))-x$. BenStandeven Junior Fellow Posts: 27 Threads: 3 Joined: Apr 2009 05/11/2009, 09:12 PM (05/11/2009, 07:39 PM)bo198214 Wrote: (05/11/2009, 05:17 PM)Ansus Wrote: It should be noted that superfunction is not unique in most cases. For example, for $f(x)=2 x^2-1$, superfunction is $F(x)=\cos(2^x C)$ Ya, this is the simple kind of non-uniqueness, its just a translation along the x-axis. However there are also more severe types of non-uniques, as I already introduced in my first post, we have two solutions (which are not translations of each other): $F(x)=\cos(2^x)$ and $F(x)=\cosh(2^x)$. Actually, $\cos(2^x) = \cosh(i 2^x) = \cosh(2^{x + \frac{\pi i}{2 \ln 2}})$, so they are translations of each other, albeit along the imaginary axis instead of the real axis. bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 05/11/2009, 09:27 PM (This post was last modified: 05/11/2009, 09:28 PM by bo198214.) (05/11/2009, 08:19 PM)bo198214 Wrote: Thatswhy I always try to find for elementary solutions whether they are regular at some fixed point because this reduces the number of real analytic solutions to two at one fixed point (analogously to $\exp_{\sqrt{2}}$) up to x-translation. I want to illustrate this phenomenon with a picture of the two regular super-functions $F_{1,+}(x)=\cosh(2^x)$ and $F_{1,-}(x)=\cos(2^x)$ of $f(x)=2*x^2-1$ at the fixed point 1. The upper curve is $\cosh(2^x)$ and the lower curve is $\cos(2^x)$. We see that they have both same asymptote to the left, which is the fixed point 1. Compare this with the both super-exponentials at 4 (these are $F_{4,5}$ and $F_{4,3}$) in the picture in this post. This is a general behaviour of the two real regular super-functions at one fixed point: Either to the left or to the right (depending whether the derivative at the fixed point is bigger or smaller than 1) they both approach the fixed point, one from above the other from below. « Next Oldest | Next Newest »
Possibly Related Threads... Thread Author Replies Views Last Post I need somebody to help me clarifiy the elementary knowledge for tetration Ember Edison 13 1,097 08/26/2019, 01:44 PM Last Post: Ember Edison Between exp^[h] and elementary growth tommy1729 0 998 09/04/2017, 11:12 PM Last Post: tommy1729 Superfunctions in continu sum equations tommy1729 0 1,846 01/03/2013, 12:02 AM Last Post: tommy1729 superfunctions of eta converge towards each other sheldonison 13 14,739 12/05/2012, 12:22 AM Last Post: sheldonison how many superfunctions? [was superfunctions of eta converge towards each other] tommy1729 8 8,835 05/31/2011, 07:38 PM Last Post: sheldonison Elliptic Superfunctions BenStandeven 2 3,740 08/20/2010, 11:56 AM Last Post: bo198214
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# Peter was _______ a hurry to go so he did not stop to greet me.
A.
in
B.
on
C.
with
D.
over
Đáp án:A
Lời giải:to be in a hurry
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# One mole of water contains how many atoms of oxygen
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Jan 09, 2020 · For one gram atomic weight of oxygen with atomic weight of 16 grams, one mole of oxygen also contains 6.022 × 1023 oxygen atoms. Similarly, for one gram atomic weight of silicon with atomic weight of 28 grams, one mole of silicon still contains 6.022 × 1023 silicon atoms. How many grams of O2 are in H2O? O2 actually weighs 32 grams.. How many atoms are in one water molecule? + Example The ratio of masses for a mole of oxygen and a mole of sulfur is 1:2, so the ratio of number of atoms in 16- and 32-1b samples will still. A) 3.01 times 10^25 atoms Xe B) 3.10 times 10^25 atoms Xe C) 9.034 times 10^24 atoms Xe D) 3.10 times 10^24 atoms Xe E) 1.03 times 10^24 atoms Xe AB) none of these How many moles of Ar are 5.68 times 10^25 atoms Ar? A) 34.2 mole Ar B) 94.3 mole Ar C) 93.4 mole Ar D) 32.4 mole Ar E) 93.3 mole Ar AB) none of these How many atoms are in 6.58 moles.
Search. Science. Chemistry. Chemistry questions and answers. 1) How many moles of oxygen atoms are present in a sample that contains 3.86 moles of dioxygen difluoride O2F2? The answer I got from this was: 7.72 moles of O2 atoms. Dec 30, 2021 · How many atoms are in 2 water molecules? Two water molecules contain 4 hydrogen atoms and 2 oxygen atoms. Is water an atom of molecule? Water is a molecule because it contains molecular bonds. Water is also a compound because it is made from more than one kind of element (oxygen and hydrogen).. Jun 04, 2018 · number of moles = 18 g 1.008 g/mol × 2 + 16.00 g/mol = 18 g 18.02 g/mol ≈ 1 mole. In 1 mole of water, there are 6.022 × 1023 molecules—that's how the mole was defined! And, since the chemical formula for water is H 2O, 1 mole of water corresponds to 1 mole of oxygen atoms. Therefore, there will be 6.022 × 1023 oxygen atoms in 1 mole, or .... The mld ratio of magnesium to oxygen would be too low 1 mole of water (H 2 O)contains 2 moles of hydrogen atoms and 1 mole of oxygen atoms [4] MgS is a wide band-gap direct semiconductor of interest as a blue-green emitter , a property that has been known since the early 1900s A bigger denominator the O2/Mg ratio will give smaller ratio At.
Group of answer choices 1.0 mol I 1.0 mol Sc 1.0 mol Cr all the same 1.0 mol Br 4.The; Question: 1.How many oxygen atoms are in 4.98 grams of sulfur trioxide? 2.Which sample below contains the least atoms? Group of answer choices 54.94 g Mn 238.03 g U 79.90 g Br all the same 137.33 g Ba 3.How many moles are in 17.23 g of manganese?. 1 mole of water contains 6.02 x10^23 molecules of water . There are 2 atoms of Hydrogen,and 1 atom of Oxygen. It means three are 3 atoms in water and around 1.8 x 10^24 atoms in a mole. Sep 19, 2020 · 46 g of HCOOH contains 2 × 6.023 × 10 23 number of oxygen atoms. 3. 1 mol of H 2 O . H 2 O (water) – Molar mass = 2 + 16 = 18 . 18 g of water contains 1 × 6.023 × 10 23 number of oxygen atoms. ∴ 1 mole of formic acid contains the greatest number of oxygen atoms..
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1 mole of H 2 O contains 3 moles of atoms ( 2 moles of H atoms and 1 mole of O atoms). 1 mole of atoms corresponds to Avogadro's number of atoms which is 6 . 0 2 × 1 0 2 3 . 3 moles of. Jun 23, 2017 · 6.02xx10^23 Oxygen atoms One mole of any substance contains one Avogadro's Number of particles. Avogadro's Number = 6.02xx10^23 particle units.
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1 mole H2O (1 mole O/1 mole H2O)(6.022 X 1023/1 mole O) 6 X 1023 oxygen atoms =======================to be correct in significant figures and to show you the proper set. One molecule of water (H2O) contains two atoms ofhydrogen and one atom of oxygen. A hydrogen atom has a mass of 1.0u and an atom of oxygen has a mass of 16 u, approximately. (a) What is the mass in kilograms of onemolecule of water?(b) How many molecules of water are in the world's oceans, whichhave an estimated total mass of 1.4. The same process is for sodium chloride. But the molar concentration of sodium sulfate will be equal to the molar concentration of calcium sulfate . According to [1], the values of the conversion factor of salt content into electrical conductivity for sodium salts are determined. Then the electrical conductivity of each salt is calculated. Because each water molecule H2O contains only one oxygen atom two water molecules must form for each oxygen molecule that reacts. How many atoms are in a glass of water? Each mole is an atom multiplied by Avogadro’s Constant = 600000000000000000000000. And a water molecule has 3 atoms H2O. So your total is 3.6 x 10. How Many Hydrogen Atoms Are in One Molecule of Water? oth pure and combined form. In every mole of water, there are two hydrogens, two atoms that are bound to an oxygen atom. For every mole of water there are 1.0×1023 molecules that contain 2 hydrogen atoms, giving us 1 mol = 6.0×1023 molecules making up 1 mole = 18 grams of H2O.
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Bharat Panchal. Hinglish. 1 mole of water molecule=18u=18g=6.022×1023 water molecule. 1 mole of sodium atom=23u=23g=6.022×1023 sodium atom. Mole represents: 1) 6.022×1023 atoms,molecules or ions of a substance. 2)The amount of a substance equal to its gram atomic mass or molecular mass. 3)A definite number of atoms,molecules or ions of a. 46 g of HCOOH contains 2 × 6.023 × 10 23 number of oxygen atoms. 3. 1 mol of H 2 O . H 2 O (water) - Molar mass = 2 + 16 = 18 . 18 g of water contains 1 × 6.023 × 10 23 number of oxygen atoms. ∴ 1 mole of formic acid contains the greatest number of oxygen atoms. Nexus Mod Manager Fallout 4 Missing Ini 5 To download multiple versions or optional files a mod offers, scroll down on its download page and click the “Files†tab. Anyone can use this document, but I believe it will be most useful to budding mod users who may have been drawn to the PC version after experiencing a taste of what <b>mods</b>. Assume all the oxygen in the water is normal oxygen-16. So, one molecule of heavy water will have 10 neutrons (one in each of the two deuterium atoms and 8 in the oxygen atom. A mole of anything contains 6.02E23 items. So a mole of heavy water will have 60.02E23 (6.02E24) neutrons. 1 mole = 6.022×10^23 atoms. 1 water molecule = 2 Hydrogen atoms + 1 oxygen atom. So 1 mole H2O = 1.2044×10^24 hydrogen atoms. Therefore 2 mole H2O will have.
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1. One mole of sodium atoms contains: a) 23 atoms b) 6.022 x 1023 grams c) 6.022 x 1023 atoms d) 60.22 x 1023 atoms One molecule of ozone, O3, contains three oxygen atoms.. How many atoms of hydrogen does a water molecule have? two hydrogen atoms Individual H 2 O molecules are V-shaped consisting of two hydrogen atoms (depicted in white) attached to the sides of a single oxygen atom (depicted in red). How Many Hydrogen Atoms Are in One Molecule of Water? oth pure and combined form. In every mole of water, there are two hydrogens, two atoms that are bound to an oxygen atom. For every mole of water there are 1.0×1023 molecules that contain 2 hydrogen atoms, giving us 1 mol = 6.0×1023 molecules making up 1 mole = 18 grams of H2O.. Chemistry The Mole Concept The Mole 1 Answer Doc048 Jun 23, 2017 6.02 ×1023 Oxygen atoms Explanation: One mole of any substance contains one Avogadro's Number of particles. Avogadro's Number = 6.02 ×1023 particle units Answer link. Answer (1 of 7): The trick is the question "How many atoms of H per Mole of water" take Avogrado's # for a Mole of water, and divide by 2/3 (2 H's and 1 O). Should end up with the same number of atoms as 2 Moles of H using the traditional method since the number of atoms is invariable. One mole of water contains Avogadro's number of molecules i.e 6.023 * 10^23 water molecules. Since water is H2O , one molecule has 3 atoms (2 of hydrogen,1 of oxygen).So total atoms = 3*6.023*10^23 = 18.069*10^23 atoms. number of moles = 18 g 1.008 g/mol × 2 + 16.00 g/mol = 18 g 18.02 g/mol ≈ 1 mole. In 1 mole of water, there are 6.022 × 1023 molecules—that's how the mole was defined! And, since the chemical formula for water is H 2O, 1 mole of water corresponds to 1 mole of oxygen atoms. Therefore, there will be 6.022 × 1023 oxygen atoms in 1 mole, or.
Just as 1 mole of atoms contains 6.022 × 10 23 atoms, 1 mole of eggs contains 6.022 × 10 23 eggs. This number is called Avogadro's number, after the 19th-century Italian scientist who first proposed a relationship between the volumes of gases and the numbers of particles they contain.
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Assume all the oxygen in the water is normal oxygen-16. So, one molecule of heavy water will have 10 neutrons (one in each of the two deuterium atoms and 8 in the oxygen atom. A mole of anything contains 6.02E23 items. So a mole of heavy water will have 60.02E23 (6.02E24) neutrons. Which one contains more oxygen atoms? 22 grams of carbon dioxide or 18 grams of water?. How many atoms of oxygen are there in 18 g of water? – As in one mole of water there are6.022×1023 molecules that’s how the mole was defined. And as the chemical formula for water is H2O and one mole of water corresponds to one mole of oxygen atoms. Hence there will be 6.022×1023 oxygen atoms in one mole or we can say 18 g of water. Hence we can say that there are three oxygen atoms and one sulphur atom in this compound. Therefore, the correct answer is that one mole of sulphur trioxide contains 18.066 × 10 23 molecules of oxygen or in other words this chemical has 18.066 × 10 23 atoms of oxygen. Note that this chemical compound that is sulphur trioxide is a significant. One mole of water contains Avogadro’s number of molecules i.e 6.023 * 10^23 water molecules. Since water is H2O , one molecule has 3 atoms (2 of hydrogen,1 of oxygen).So total atoms = 3*6.023*10^23 = 18.069*10^23 atoms. The Molar mass or Molecular Weight (interchangeable terms so long as we are on Earth) of a substance is the total of all the individual masses of the elements it contains. To use our old friend water as an example: One mole of Water is composed of 1 mole of Oxygen and two moles of Hydrogen. The mass of oxygen equal to one mole of oxygen is 15.. See the answer How many oxygen atoms are present in 2 28 ÷ 16 = 0 28 ÷ 16 = 0. Calculate the mole fraction and mass fraction of propellants H2 and O2 as they enter the combustion chamber, assuming that the propellants are injected stoichiometrically Determine which reactant,if any, is limiting 1 mole of water (H 2 O)contains 2 moles of. For one gram atomic weight of hydrogen with atomic weight of one gram, one mole of hydrogen contains 6.022 × 1023 hydrogen atoms. For one gram atomic weight of oxygen with atomic weight of 16 grams, one mole of oxygen also contains 6.022 × 1023 oxygen atoms. Similarly, for one gram atomic weight of silicon with atomic weight of 28 grams, one .... How many atoms of hydrogen does a water molecule have? two hydrogen atoms Individual H 2 O molecules are V-shaped consisting of two hydrogen atoms (depicted in white) attached to the sides of a single oxygen atom (depicted in red). The molar mass of an element is equal to its... Preview this quiz on Quizizz. The molar mass of an element is equal to its... Molar Mass DRAFT. 9th - 12th grade. 82 times. Chemistry. 43% average accuracy. 7 months ago. ikbasbous_48153. 0. Save. Edit. Edit. Molar Mass DRAFT. 7 months ago. by ikbasbous_48153. The mld ratio of magnesium to oxygen would be too low 1 mole of water (H 2 O)contains 2 moles of hydrogen atoms and 1 mole of oxygen atoms [4] MgS is a wide band. Dec 30, 2021 · How many atoms are in 2 water molecules? Two water molecules contain 4 hydrogen atoms and 2 oxygen atoms. Is water an atom of molecule? Water is a molecule because it contains molecular bonds. Water is also a compound because it is made from more than one kind of element (oxygen and hydrogen).. Answer: 6.022×10 23 oxygen atoms. Explanation: First, it's important to know that the chemical formula for water is H 2OThis means that one molecule of water is made of two hydrogen atoms and one oxygen atom. Then, we need to find how many moles of water there are in 18 g. (We need to do this, because knowing the number of moles of water will.
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B) A mole of a monatomic element corresponds to one Avogadro's number of atoms. C) One mole of a monatomic element has a mass equal to its atomic mass expressed in grams. D) One mole of water contains 1/2 mole of oxygen atoms. E) none of the above. One mole of water contains 1/2 mole of oxygen atoms. Jun 04, 2018 · number of moles = 18 g 1.008 g/mol × 2 + 16.00 g/mol = 18 g 18.02 g/mol ≈ 1 mole. In 1 mole of water, there are 6.022 × 1023 molecules—that's how the mole was defined! And, since the chemical formula for water is H 2O, 1 mole of water corresponds to 1 mole of oxygen atoms. Therefore, there will be 6.022 × 1023 oxygen atoms in 1 mole, or .... For example, oxygen gas O 2 is diatomic (each molecule contains two atoms) so its relative formula mass is 32. One mole of oxygen molecules would therefore have a mass of 32 g.. Jun 23, 2017 · 6.02xx10^23 Oxygen atoms One mole of any substance contains one Avogadro's Number of particles. Avogadro's Number = 6.02xx10^23 particle units. Moles of water = 0.9 18 = 0.05. In one mole of water, the Avogadro number of water molecules are present, so for 0.05 moles of water. Number of water molecules = 0.05 × 6.023 × 10 23 = 3.010 × 10 22. The one water molecule contains one oxygen and two hydrogen atoms.. Sep 19, 2020 · 46 g of HCOOH contains 2 × 6.023 × 10 23 number of oxygen atoms. 3. 1 mol of H 2 O . H 2 O (water) – Molar mass = 2 + 16 = 18 . 18 g of water contains 1 × 6.023 × 10 23 number of oxygen atoms. ∴ 1 mole of formic acid contains the greatest number of oxygen atoms.. Nov 12, 2019 · weight of water = 2.0158 g + 15.9994 g. weight of water = 18.0152 g. Therefore, one mole of water weighs 18.0152 grams. Unless you have a good sense of mass, this value probably doesn't have much meaning to you. It's easier to grasp how much water is in a mole if you find the volume of this amount of mass. Fortunately, this is another simple .... 1. One mole of sodium atoms contains: a) 23 atoms b) 6.022 x 1023 grams c) 6.022 x 1023 atoms d) 60.22 x 1023 atoms One molecule of ozone, O3, contains three oxygen atoms..
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May 12, 2021 · One mole of oxygen molecule contains how many atoms of oxygen? 1.6021 × 10+19 1.6021 × 10–19 6.022 × 10–23 1.204 × 1024 - Answered by a verified Tutor We use cookies to give you the best possible experience on our website.. There ia 6.02 x 10^23 molecules of water in one mole of water. There are three atoms in one molecule of H2O (two hydrogen and one oxygen) So there are 18.12x10^23 atoms in a mole of water molecules. Nov 12, 2019 · weight of water = 2.0158 g + 15.9994 g. weight of water = 18.0152 g. Therefore, one mole of water weighs 18.0152 grams. Unless you have a good sense of mass, this value probably doesn't have much meaning to you. It's easier to grasp how much water is in a mole if you find the volume of this amount of mass. Fortunately, this is another simple ....
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Sulfur (or sulphur in British English) is a chemical element with the symbol S and atomic number 16. It is abundant, multivalent and nonmetallic.Under normal conditions, sulfur atoms form cyclic octatomic molecules with a chemical formula S 8.Elemental sulfur is a bright yellow, crystalline solid at room temperature. Sulfur is the tenth most abundant element by mass in the universe. Group of answer choices 1.0 mol I 1.0 mol Sc 1.0 mol Cr all the same 1.0 mol Br 4.The; Question: 1.How many oxygen atoms are in 4.98 grams of sulfur trioxide? 2.Which sample below contains the least atoms? Group of answer choices 54.94 g Mn 238.03 g U 79.90 g Br all the same 137.33 g Ba 3.How many moles are in 17.23 g of manganese?. One mole of Oxgen molecule O2 contain how many atoms of oxgen? - 13728932.
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(b) How many mole- cules of water are in the world's oceans, which have an estimated total mass of 1.4 x 1021 kg? Question: 42 One molecule of water (H2O) contains two atoms of hydrogen and one atom of oxygen. A hydrogen atom has a mass of 1.0 u and an atom of oxygen has a mass of 16 u, approximately..
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Water consists of one oxygen atom and two hydrogen atoms. This means that the mass of a water molecule is 1g + 1g + 16g = 18 g/ mol. This means that the mass of a water. Sep 19, 2020 · 46 g of HCOOH contains 2 × 6.023 × 10 23 number of oxygen atoms. 3. 1 mol of H 2 O . H 2 O (water) – Molar mass = 2 + 16 = 18 . 18 g of water contains 1 × 6.023 × 10 23 number of oxygen atoms. ∴ 1 mole of formic acid contains the greatest number of oxygen atoms.. How many moles are represented by 3.01 · 10²24 oxygen atoms? 2 moles 5 moles 250 moles 10 moles A: One mole has 6.022 x 1023 Therefore 10.0moles will have 10x 6.022 x 1023 =6.022 x 1024 6.022 x 1023. How many atoms are in one water molecule? + Example The ratio of masses for a mole of oxygen and a mole of sulfur is 1:2, so the ratio of number of atoms in 16- and 32-1b samples will still. Dec 30, 2021 · How many atoms are in 2 water molecules? Two water molecules contain 4 hydrogen atoms and 2 oxygen atoms. Is water an atom of molecule? Water is a molecule because it contains molecular bonds. Water is also a compound because it is made from more than one kind of element (oxygen and hydrogen).. The mass of oxygen equal to one mole of oxygen is 15.998 grams and the mass of one mole of hydrogen is 1.008 g. If we total up the gram amounts of each element in the water molecule = 15.998g/mol + 2(1.008g/mol) we get the molar mass of water = 18.014g/mol. This means that one mole of aspirin will have a mass of 180.157 g. True. What is the mass of 3.09 x 1024 atoms of sulfur in grams? 165. One mole of water contains 6.022 x 1023 hydrogen atoms. False. How many atoms are in 1.50 moles of fluorine gas? 1.81x 10^24. The molecular formula is equal to the empirical formula multiplied by a whole number integer. True. Oct 26, 2014 · 22"g Cu" contains 2.1 xx 10^23 atoms of Cu. In order to solve this problem, you need to determine the molar mass of copper , which is the mass of one mole of copper . The molar mass of copper is its atomic weight on the periodic table in g/mol, and is "63.546g/mol". Here, it is given that: 16 g of oxygen contains 1 mole of oxygen atoms. In simple terms we can say that: mass of 1 mole oxygen atom (W) = 16 g / mol. And this weight is the sum of mass of electrons and protons present in the oxygen. And the mass of the molecule of oxygen (. O 2. ) is 32u, here 'u' is the Dalton or unified atomic mass unit. . Since one mole is defined as 6.022 x 10^23, and there are two hydrogen per molecule, that would mean that there are exactly 12.044 x 10^23 molecules of hydrogen in one. Chemistry The Mole Concept The Mole 1 Answer Doc048 Jun 23, 2017 6.02 ×1023 Oxygen atoms Explanation: One mole of any substance contains one Avogadro's Number of particles. Avogadro's Number = 6.02 ×1023 particle units Answer link. For one gram atomic weight of hydrogen with atomic weight of one gram, one mole of hydrogen contains 6.022 × 1023 hydrogen atoms. For one gram atomic weight of oxygen with atomic weight of 16 grams, one mole of oxygen also contains 6.022 × 1023 oxygen atoms. Similarly, for one gram atomic weight of silicon with atomic weight of 28 grams, one ....
The mld ratio of magnesium to oxygen would be too low 1 mole of water (H 2 O)contains 2 moles of hydrogen atoms and 1 mole of oxygen atoms [4] MgS is a wide band.
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1 mole H2O (1 mole O/1 mole H2O)(6.022 X 1023/1 mole O) 6 X 1023 oxygen atoms =======================to be correct in significant figures and to show you the proper set. 46 g of HCOOH contains 2 × 6.023 × 10 23 number of oxygen atoms. 3. 1 mol of H 2 O . H 2 O (water) - Molar mass = 2 + 16 = 18 . 18 g of water contains 1 × 6.023 × 10 23 number of oxygen atoms. ∴ 1 mole of formic acid contains the greatest number of oxygen atoms. Oct 26, 2014 · 22"g Cu" contains 2.1 xx 10^23 atoms of Cu. In order to solve this problem, you need to determine the molar mass of copper , which is the mass of one mole of copper . The molar mass of copper is its atomic weight on the periodic table in g/mol, and is "63.546g/mol". Answer: Therefore, one mole of CO2 contains one mole of carbon and two moles of oxygen. 1 mole of CO2 = 1 mole of C = 6.02×1023 atoms of carbon. 1 mole of CO2 = 2 mole of O = 2×6.02×1023 = 12.04×1023 atoms of oxygen. One molecule of water contains two atoms of Hydrogen and one atom of Oxygen, which is why its formula is H2O. How many atoms does oxygen gas have? A Molecule of Oxygen contains two atoms. The Molar mass or Molecular Weight (interchangeable terms so long as we are on Earth) of a substance is the total of all the individual masses of the elements it contains. To use our old friend water as an example: One mole of Water is composed of 1 mole of Oxygen and two moles of Hydrogen. The mass of oxygen equal to one mole of oxygen is 15. .... The SI prefix "milli" represents a factor of 10-3, or in exponential notation , 1E-3. So 1 millimole = 10-3 moles . The definition of a mole is as follows: The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol is "mol." ›› Sample conversions. One mole of water contains $6.023 \times {10^{23}}$ number of water molecules or $\Rightarrow 2 \times 6.023 \times {10^{23}} = 12.046 \times {10^{23}}$ atoms . O contains 3 moles of. kennebago maine. mes vision create account. married couples treesome movies. aesthetic usernames for tiktok generator.
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Atomic and Molar Mass 6.4: Mole-Mass Conversions 6.5: Mole-Mole Relationships in Chemical Reactions 6.6: Mole-Mass and Mass-Mass Problems 6.7: Chapter Summary 6.8: References 6.1: Chapter Introduction So far, we have talked about chemical reactions in terms of individual atoms and molecules. Although this works, most of the reactions. (ii) No. of oxygen atoms 1 mole of water (H_(2)O) has oxygen (O) = 1 gram atom ... which contains 70% as many C atoms as oxygen atoms. The number of mole of compound in. A) One mole of atoms makes up an amount of atoms that can be seen with the naked eye. B) A mole of a monatomic element corresponds to one Avogadro's number of atoms. C) One mole of a monatomic element has a mass equal to its atomic mass expressed in grams. D) One mole of water contains 1/2 mole of oxygen atoms. How many atoms are in 1.50 moles .... . Dec 30, 2021 · How many atoms are in 2 water molecules? Two water molecules contain 4 hydrogen atoms and 2 oxygen atoms. Is water an atom of molecule? Water is a molecule because it contains molecular bonds. Water is also a compound because it is made from more than one kind of element (oxygen and hydrogen).. 5.6 litres of oxygen contains × atoms. Explanation: One mole of gas at NTP has a volume of 22.4 litre. It is known as the molar volume. The atomicity of oxygen is 2. Thus one mole of oxygen contains two times the number of oxygen molecules; Thus 22.4 litres of oxygen contains × molecules. 1 litre of oxygen will contain × × molecules. Oct 26, 2014 · 22"g Cu" contains 2.1 xx 10^23 atoms of Cu. In order to solve this problem, you need to determine the molar mass of copper , which is the mass of one mole of copper . The molar mass of copper is its atomic weight on the periodic table in g/mol, and is "63.546g/mol". It contains 2 hydrogen atoms and 1 oxygen atom. 2. Why does a water molecule have polarity? 3. What type of bond holds a single water molecule together? 4. What type of bond holds a water. How many atoms of oxygen are there in 18 g of water? – As in one mole of water there are6.022×1023 molecules that’s how the mole was defined. And as the chemical formula for water is H2O and one mole of water corresponds to one mole of oxygen atoms. Hence there will be 6.022×1023 oxygen atoms in one mole or we can say 18 g of water. . How many atoms of oxygen are there in 18 g of water? – As in one mole of water there are6.022×1023 molecules that’s how the mole was defined. And as the chemical formula for water is H2O and one mole of water corresponds to one mole of oxygen atoms. Hence there will be 6.022×1023 oxygen atoms in one mole or we can say 18 g of water.
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frankston leader classifieds. Cancel. Answer: 6.022×10 23 oxygen atoms. Explanation: First, it's important to know that the chemical formula for water is H 2OThis means that one molecule of water is made of two hydrogen atoms and one oxygen atom. Then, we need to find how many moles of water there are in 18 g. (We need to do this, because knowing the number of moles of water will. Correct option is A) As oxygen is diatomic molecule and also one molecule of SO 2 contain two Oxygen atoms. Hence One mole of oxygen and one mole of SO 2 SO2contains a same number of oxygen atoms. and also, The amount of oxygen in one mole of O 2 is same as that in one mole of SO 2.
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Answer (1 of 4): The molecular formula for water is H2O, Every molecule of H2O contains 2 atoms of Hydrogen, So one mole of H2O will contain 2 moles of Hydrogen i.e. 2× Avogadro's number, i.e. 2 × 6.023 ×10^23 number of Hydrogen atoms.. One mole of oxygen molecule contains how many atoms of oxygen? 1.6021 × 10+19 1.6021 × 10–19 6.022 × 10–23 1.204 × 1024 - Answered by a verified Tutor We use. Jun 23, 2017 · 6.02xx10^23 Oxygen atoms One mole of any substance contains one Avogadro's Number of particles. Avogadro's Number = 6.02xx10^23 particle units. There are 4 hydrogen atoms are in 2 mole of water molecules. In one mole of water molecule, there are two hydrogen atoms and one oxygen present then according to this statement we can say that 4 moles of hydrogen atoms and 2 moles of oxygen atoms are present in 2 moles of water molecules. In three mole of water molecules, there are six mole of. One mole of water contains 1/2 mole of oxygen atoms. ... If a sample of carbon dioxide contains 3.8 moles of oxygen atoms, how many moles of carbon dioxide are in the .... Moles of water = 0.9 18 = 0.05. In one mole of water, the Avogadro number of water molecules are present, so for 0.05 moles of water. Number of water molecules = 0.05 × 6.023 × 10 23 = 3.010 × 10 22. The one water molecule contains one oxygen and two hydrogen atoms. Therefore, we need to take 2 moles of water as 2 moles of water contains exactly the same number of oxygen atoms as that 1 mole of O₂ gas contains. So, 1 Mole of H₂O = 18 g. Then, 2 Moles of H₂O = X g. Solving for X, X = (2 mol × 18 g) ÷ 1 mol. X = 36 g H₂O.
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Jun 23, 2017 · 6.02xx10^23 Oxygen atoms One mole of any substance contains one Avogadro's Number of particles. Avogadro's Number = 6.02xx10^23 particle units. One mole of water contains $6.023 \times {10^{23}}$ number of water molecules or $\Rightarrow 2 \times 6.023 \times {10^{23}} = 12.046 \times {10^{23}}$ atoms . O contains 3 moles of. let's calculate the Mass of one mole of water. Mhm. So one more of water. The mass of 18.0 ground if it's smaller mass, Therefore one mole of water has a mass of 18 g And how many atoms of oxygen and hydrogen are there in one mole of water? So in one mole of water there are 6.02 times 10 to the 23rd molecules, yeah, of water.
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A mole is a number just like a dozen or a century. A dozen has 12 things, similarly a mole has 6.023 x 10 23 entities (atom, molecule and particles). 1 mole of water contains 6.023 x 10 23 H 2 O molecules. Each H 2 O molecule has two H atoms and one O atom. So, 1 mole of water which is 6.023 x 10 23 H 2 O molecules will have. The mld ratio of magnesium to oxygen would be too low 1 mole of water (H 2 O)contains 2 moles of hydrogen atoms and 1 mole of oxygen atoms [4] MgS is a wide band-gap direct semiconductor of interest as a blue-green emitter , a property that has been known since the early 1900s A bigger denominator the O2/Mg ratio will give smaller ratio At. Jan 09, 2020 · For one gram atomic weight of oxygen with atomic weight of 16 grams, one mole of oxygen also contains 6.022 × 1023 oxygen atoms. Similarly, for one gram atomic weight of silicon with atomic weight of 28 grams, one mole of silicon still contains 6.022 × 1023 silicon atoms. How many grams of O2 are in H2O? O2 actually weighs 32 grams.. 5.6 litres of oxygen contains × atoms. Explanation: One mole of gas at NTP has a volume of 22.4 litre. It is known as the molar volume. The atomicity of oxygen is 2. Thus one mole of oxygen contains two times the number of oxygen molecules; Thus 22.4 litres of oxygen contains × molecules. 1 litre of oxygen will contain × × molecules.
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May 18, 2012 · There ia 6.02 x 10^23 molecules of water in one mole of water. There are three atoms in one molecule of H2O (two hydrogen and one oxygen) So there are 18.12x10^23 atoms in a mole of water molecules. Answer: (a) No. of moles of sodium = = 2 moles. We know that one mole of sodium contains 6.022 × 10 23 atoms. ∴ 2 moles of sodium contain = 2 × 6.022 × 10 23 atoms. = 1.204 × 10 24 atoms. (b) 1 mole of oxygen = 32 g. 32 g of 02 contains 6.02 2 × 10 23 molecules. ∴ 8 g of O 2 contains = × 8 molecules. frankston leader classifieds. Cancel. See the answer How many oxygen atoms are present in 2 28 ÷ 16 = 0 28 ÷ 16 = 0. Calculate the mole fraction and mass fraction of propellants H2 and O2 as they enter the combustion chamber, assuming that the propellants are injected stoichiometrically Determine which reactant,if any, is limiting 1 mole of water (H 2 O)contains 2 moles of. According to avogadro's law, 1 mole of every substance weighs equal to its molecular mass and contains avogadro's number of particles. To calculate the number of moles, we use the equation: 1 mole of contains = 8 moles of oxygen atoms. 0.2 moles of contain= moles of oxygen atoms. Thus 1.6 moles of oxygen atoms are present in 125 g of. For example, oxygen gas O 2 is diatomic (each molecule contains two atoms) so its relative formula mass is 32. One mole of oxygen molecules would therefore have a mass of 32 g.. Search: Mole Ratio Of Magnesium To Oxygen. 9994 g = 31 022 x 1023 molecules PCl5 3 (a) Calculate the number of atoms of hydrogen present in one dozen molecule of hydrogen gas 2 grams, solve for: x moles = 0 Mole ratios are used as conversion factors between products and reactants in stoichiometry calculations Mole ratios are used as conversion factors.
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May 12, 2021 · One mole of oxygen molecule contains how many atoms of oxygen? 1.6021 × 10+19 1.6021 × 10–19 6.022 × 10–23 1.204 × 1024 - Answered by a verified Tutor We use cookies to give you the best possible experience on our website.. May 12, 2021 · One mole of oxygen molecule contains how many atoms of oxygen? 1.6021 × 10+19 1.6021 × 10–19 6.022 × 10–23 1.204 × 1024 - Answered by a verified Tutor We use cookies to give you the best possible experience on our website.. For one gram atomic weight of hydrogen with atomic weight of one gram, one mole of hydrogen contains 6.022 × 1023 hydrogen atoms. For one gram atomic weight of oxygen with atomic weight of 16 grams, one mole of oxygen also contains 6.022 × 1023 oxygen atoms. Similarly, for one gram atomic weight of silicon with atomic weight of 28 grams, one.
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Since one mole is defined as 6.022 x 10^23, and there are two hydrogen per molecule, that would mean that there are exactly 12.044 x 10^23 molecules of hydrogen in one. Aug 27, 2019 · So, next we calculate how many molecules there are in a drop of water, which we determined contains 0.002775 moles: molecules in a drop of water = (6.022 x 10 23 molecules/mole) x 0.002275 moles. molecules in a drop of water = 1.67 x 10 21 water molecules. Put another way, there are 1.67 sextillion water molecules in a water drop .. Q: One mole of calcium nitrite contains how many atoms of oxygen A: 6.023 × 10^23 is known as Avogadro's number. 1 mole of any substance contains Avogadro's number of Q: How many ATOMS of nitrogen.
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https://codegolf.stackexchange.com/questions/120796/infinitely-print-zenos-dichotomy-paradox-1-2n/120831
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# Infinitely Print Zeno's Dichotomy Paradox (1/(2^n))
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second one orders half a beer. The third one orders a fourth of a beer. The bartender stops them, pours two beers and says, "You're all a bunch of idiots."
Reddit
Print the following series for as long as the program runs, with the denominator of each item being multiplied by two each time:
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...
As n approaches infinity, the sum of this sequence approaches 2.
# Rules
No, you may not print 2.
You may not print 1/1 as the first item.
You may remove spaces 1+1/2+... or add spaces 1 + 1 / 2 + ... as you need.
You may use newlines instead of spaces as a delimiter due to popular demand.
You may append a . plus a constant number of 0s to the denominator if need be.
"Infinitely" means no unnecessary delays, and for as long as possible limited by the current (variable) system's specs, but not limited by your current language.
Standard loopholes apply.
This is , so shortest answer in bytes wins.
• Regarding the joke, I like the "You guys should know your limits" version better. – March Ho May 16 '17 at 13:17
• Is it just me, or is that a parabola right there? – Adám May 16 '17 at 14:17
• @StephenS Yes, I saw them too, but this one is much clearer and bigger. – Adám May 16 '17 at 14:22
• @Adám: yep! If the lengths of the denominators weren't changing, then the visual pattern of +1/s would just form a diagonal line. However, the lengths of the denominators is changing linearly (up to rounding): the number of digits of 2^n is about n log(2)/log(10). That linear change translates into a linear change in the relative position of each +1/ with respect to the preceding one, which is the same as a quadratic change in the absolute position. – Greg Martin May 16 '17 at 17:02
• @QPaysTaxes then you are non-competing - but if multiple people want to post competing C answers, you can compete against each other :) – Stephen May 16 '17 at 23:21
# 05AB1E, 10 9 bytes
Saved 1 byte thanks to Erik the Outgolfer
[No…+1/J?
Try it online!
Explanation
[ # loop over N infinitely [0 ...]
No # calculate 2^N
…+1/J # join with the string "+1/"
? # print without newline
• You can golf this: [No?…+1/? – Erik the Outgolfer May 16 '17 at 12:04
• @EriktheOutgolfer: True! I knew that interpolation looked wasteful :P – Emigna May 16 '17 at 12:11
• Technically you should be using « instead of J but this works too. – Erik the Outgolfer May 16 '17 at 12:17
• @EriktheOutgolfer 'should use' isn't in a golfer's dictionary. – Okx May 16 '17 at 15:59
# Python 2, 30 bytes
-5 thanks to Erik the Outgolfer
i=1
while 1:print i,'+1/';i*=2
Try it online!
# Jelly, 12 bytes
‘;“+1/”ȮḢḤ’ß
Try it online!
# Pyth, 10 bytes
.V0^2b"+1/
Try it online!
# APL (Dyalog Unicode), 15 bytes
More fun if ⎕FR (Floating-point Representation) is 1287 (128 bit decimal) and ⎕PP (Print Precision) is 34.
{∇2×⊃⎕←⍵'+1/'}1
Try it online!
{}1 apply the following function on the number 1:
⎕←⍵'+1/' print the argument and the the string
⊃ pick the first one (i.e. the argument)
2× double that
∇ tail call recursion on that (optimised, so it can be infinitely repeated)
# C (gcc), 60 bytes
f(){for(long long n=1;n;n*=2)printf(&"+1/%llu"[n^1?0:3],n);}
Goes up the the unsigned 64bit limit.
Try it online!
This one goes on forever; (it's as small as it's going to get)
# C (tcc), 312264255251233231208204195190188 170 bytes
l=4;c;main(i){for(char*t="*\b1+";puts(i=t),*++t=48;l=asprintf(&t,"1/%s",t+=*++t<49))for(t+=l-2;--t>i;)*t>52?*t=*t*2%58+c,c=*--t>52,*t=*t%48*2%10+49:(*t=*t*2-48+c+(c=0));}
Try it online!
Here's the not so golfed version;
c;l=4;main(i){
for(char*t="*\b1+";puts(i=t),*++t=48;l=asprintf(&t,"1/%s",t+=*++t<49))
for(t+=l-2;--t>i;)
*t>52?*t=*t*2%58+c,c=*--t>52,*t=*t%48*2%10+49:(*t=*t*2-48+c+(c=0));
}
# Bash, 33 bytes
echo 1;yes|awk '{print"+1/"2^NR}'
Try it online!
Change print for printf and echo for printf to avoid newline
• printf 1 also avoids the newline, no? – Neil May 16 '17 at 14:59
• On my Awk (GNU Awk 4.2.1), I get repeated +1/inf from the 1024th term onwards. :-( – Toby Speight Apr 3 at 15:58
# dc, 19 18 bytes
1[rdp+[+1/]Prdx]dx
## Explanation
We push 1 and [rdp+[+1/]Prdx] onto the stack. We then duplicate and execute [rdp+[+1/]Prdx]. The first thing it does is to rotate the stack (r) so that the value is on top. dp+ prints the value and adds itself (to multiply by 2). [+1/]P prints the invariant +1/, then we rotate the arguments so the saved macro definition is at the top, duplicate it and start again.
## Notes
GNU dc will normally wrap at 70 columns. To override that and get an infinite line, add DC_LINE_LENGTH=0 to your environment variables.
## Output (partial)
1
+1/2
+1/4
+1/8
+1/16
+1/32
+1/64
+1/128
+1/256
+1/512
+1/1024
+1/2048
+1/4096
+1/8192
+1/16384
+1/32768
+1/65536
+1/131072
+1/262144
+1/524288
+1/1048576
+1/2097152
+1/4194304
+1/8388608
...
+1/1042962419883256876169444192465601618458351817556959360325703910069443225478828393565899456512
+1/2085924839766513752338888384931203236916703635113918720651407820138886450957656787131798913024
+1/4171849679533027504677776769862406473833407270227837441302815640277772901915313574263597826048
+1/8343699359066055009355553539724812947666814540455674882605631280555545803830627148527195652096
+1/16687398718132110018711107079449625895333629080911349765211262561111091607661254297054391304192
+1/33374797436264220037422214158899251790667258161822699530422525122222183215322508594108782608384
...
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...
• i was 15 minutes late T_T, use dp instead of ddn to save a byte – Felipe Nardi Batista May 16 '17 at 15:53
• I used dn to avoid introducing a newline, but did consider p (at the cost of uglier output). Actually, on reflection, it's not so ugly, after all. – Toby Speight May 16 '17 at 16:00
# GolfScript, 19 bytes
1{.'+1/'+puts.+.}do
Try it online!
# CJam, 14 bytes
1{_o"+1/"o2*}h
Try it online!
# ><>, 14 bytes
1:n"/1+"ooo2*!
Try it online!
# Pyth, 10 bytes
#h~hyZ"+1/
Z starts out as zero. ~hyZ post-assigns the value of 2*Z+1 to Z. Thus, Z becomes 0, 1, 3, 7, 15, ... over successive iterations. h then prints out the value one greater. # runs the infinite loop, and "+1/ gets the formatting right.
# JavaScript (ES6), 36 34 bytes
for(a=1;;a*=2)console.log(a+'+1/')
Try it online!
Note that this is limited by language since a is a floating-point variable, not an integer.
-2 bytes thanks to @Stefnotch.
• You can save 2 bytes by leaving out the brackets after console.log. for(a=.5;;)console.log${a*=2}+1/ Though, your solution is limited by the language, because a is a float, not a big integer. (The challenge says that it should not be limited by the language, but rather by the system's specs.) – Stefnotch May 17 '17 at 15:58 • @Stefnotch That doesn't work unfortunately, because tagged templates with interpolation pass more than a single argument. I added the note about limitedness though, thank you. – eush77 May 17 '17 at 16:17 • Oh, sorry for not testing my code. Well, this code does shave off 2 bytes: for(a=1;;a*=2)console.log(a+"+1/") – Stefnotch May 17 '17 at 16:44 • @Stefnotch Oh, and it's a lot simpler, too :) Thank you. – eush77 May 17 '17 at 17:02 # Ruby, 27 25 bytes a=1;a*=2while$><<a<<'+1/'
Try it online!
# Java, 107 102 bytes
()->{for(java.math.BigInteger z=null,o=z.ONE,n=o;;n=n.add(n))System.out.print(n.max(o)==o?1:"+1/"+n);}
z=null exists to shorten the o=java.math.BigInteger.ONE into z=null,o=z.ONE, saving 12 bytes. z.ONE will not throw a NullPointerException because we access a static member and not an instance one.
Using int shortens the code, but fails to comply after 32 iterations.
## Saves
• 107 -> 102 bytes: n.compareTo(o)>0 turned into n.max(o)==o, thanks to an idea given by @Shufflepants
• This looks like one of the only entries that attempts to meet the rule: ""Infinitely" means no unnecessary delays, and for as long as possible limited by the current (variable) system's specs, but not limited by your current language." but it still doesn't do that because BigInteger still has a max value of 2^(Integer.MAX_VALUE). – Shufflepants May 16 '17 at 14:46
• @Shufflepants Where is that limit written? Nowhere ("may support values outside of that range"), so it's not a limitation of the language, but a limitation of the JVM, which then is the system. Also, "works on my computer", so good enough for codegolf ;) – Olivier Grégoire May 16 '17 at 14:55
• If the implementation of a language or the JVM is considered part of the system and not the language, then you might as well use int instead of BigInteger. – Shufflepants May 16 '17 at 15:00
• No, because int and all other primitive types are limited at the language level. Also, the BigInteger doc explicitly says that a limit is optional, not mandatory (and that the default JVM implementation uses that limit). – Olivier Grégoire May 16 '17 at 15:04
• This function "BigInteger max(BigInteger val)" exists in the specification too, which implies, that while the limit need not be what the current implementation limit is, it implies in the specification that there must be some finite limit. – Shufflepants May 16 '17 at 15:07
# Vim, 22, 21 bytes/keystrokes
qqyiwA+1/<esc>p@"<C-a>@qqxX@q
While testing this, you might run into issues with the current register values. To fix this, run
:let @q=''
:let @"=''
before running this, or by launching vim with
vim -u NONE -i NONE
• absolutely beautiful. – Tyrannosaur May 18 '17 at 17:08
# R, 35 34 bytes
cat(i<-1);repeat cat("+1/",i<-i*2)
Spacing is a bit werid but I understand that's ok.
• You can use repeat instead of while(T): cat(i<-1);repeat cat("+1/",i<-i*2) for 1 byte less. – plannapus May 16 '17 at 13:26
# Befunge 93: 14 bytes
1:.2*"/1+",,,#
• Doesn't seem to be working on TIO. Is it written for some specific implementation? – eush77 May 19 '17 at 16:39
• @eush77 Looks like TIO expects a fixed 80 char width grid, causing the "#" to skip empty space instead of the next instruction. I usually test here, where the torus is adjusted to the size of the actual code. – karhell May 22 '17 at 10:00
# Powershell, 34 bytes
for([bigint]$i=1;;$i*=2){"$i+1/"} Try it online! # Aceto, 20 bytes pL* pd12< p"M" 1+1/ Prints the sequence without any spaces. When run, you won't see anything for a little while, because of buffering, run with -F to immediately see everything. 1. Pushes and prints a 1, then stores "+1/" in quick storage (the register). 2. Pushes a 1. 3. Multiplies by two, loads from the register, prints, duplicates, and prints. 4. GOTO 3. # Go, 102 100 bytes Go can be almost as bad as Java, apparently. import(."fmt" ."math/big") func main(){n:=NewInt(1);for{Print(n.String()+"+1/");n.Mul(n,NewInt(2))}} Try it online! (Would be a good idea to avoid running any of these locally. :P) • No, not as bad as Java: Go is 5 bytes shorter :P – Olivier Grégoire May 16 '17 at 13:49 ## QBIC, 18 bytes ?1{q=q*2?@+1/';q Prints each term on a new line. Explanation: ?1 Prints 1 { Infinite loop q=q*2 Doubles q, starts at 1*2=2 ?@+1/ Prints the string +1/ '; without tabs, newlines or other terminators (code literal, ; is a QBIC function) q Also prints q The infinite loop is auto-closed by QBIC at EOF. We can save a byte with a more liberal output format: {?q,A,┘q=q*2#+1/ # Mathematica, 25 bytes 1//.n_:>(n~Print~"+";n/2) • Don't we have to print the +s as well as the numbers? – Greg Martin May 16 '17 at 17:04 • @GregMartin Fixed... – JungHwan Min May 16 '17 at 18:39 # C#, ̶6̶8̶ 154 bytes void A(int b=1){System.Console.Write($"1{(b>1?"/"+b:"")}+");A(b*2);}
Here is a version not constrained by int
using System.Numerics;BigInteger b=new BigInteger(1);void A(){System.Console.Write($"1{(b>1?"/"+b:"")}+");b=BigInteger.Multiply(new BigInteger(2),b);A();} • Stops working very quickly once the denominator hits int.MaxValue – Rob May 17 '17 at 4:55 • @Rob what should the behavior be? – LiefdeWen May 17 '17 at 4:58 • Here's an example: codegolf.stackexchange.com/questions/120796/… – Rob May 17 '17 at 4:58 • Okay thanx, the new version should be sufficient. – LiefdeWen May 17 '17 at 7:10 • As-is, nothing is actually printed because there's no call to A(); outside of the recursive call in the function. Also, BigIntegers have implicit conversions from primitive numeric types and built-in operators. So you can shave a lot (54 bytes?) off by changing the initial declaration to just BigInteger b=1; and shortening the multiplication to b*=2; – goric May 17 '17 at 15:16 # JavaScript (ES6), 4543 42 bytes Saved 2 bytes, thanks @DanielM ! Saved 1 byte, thanks @eush77 for pointing it out. =console.log;a=1;(1);for(;;)_(+1/${a*=2})
=console.log;(a=1);for(;;)_(+1/${a*=2}) _=console.log;for(_(a=1);;)_(+1/${a*=2})
My first go at Codegolf, go easy!
• I was going to edit a snippet in for you, but then I broke my browser and remembered this is an infinite loop question xD – Stephen May 16 '17 at 13:37
• Yeah, had to restart my dev tools a couple times doing this! – Jake Taylor May 16 '17 at 13:42
• The a=1 can go in the first part of the for, for(a=1;;) saving you a byte in extra semicolon. – DanielM May 16 '17 at 14:55
• _(a=1), work's with DanielM's suggestion too – Felipe Nardi Batista May 16 '17 at 16:00
• @eush77 That doesn't create the right output imo. It will always have a trailing +1/, not a fraction. – Jake Taylor May 17 '17 at 8:17
# PHP, 32 Bytes
for(;;)echo bcpow(2,$i++)."+1/"; Online Version -6 Bytes if values like 9.2233720368548E+18 are allowed for(;;)echo 2**$i++."+1/";
Try it online!
# AWK, 3732 bytes
BEGIN{for(;;)printf 2^i++"+1/"}
Try it online!
Could remove the BEGIN and save 5 bytes if input were allowed. Using exponents definitely cheaper byte-wise than multiplication. :)
Hopefully 2^1023 is close enough to infinity (on my work computer). Unfortunately the TIO link truncates earlier than that (around 921). But 17726622920963562283492833353875882150307419319860869157979152909707315649514250439943889552308992750523075148942386782770807567185698815677056677116184170553481231217950104164393978236130449019315710017470734562946173533283208371259654747728689409291887821024109648618981425152 does seem pretty close to infinity. :)
## Haskell - 66626051 49 chars
import Data.List
main=print$intercalate" + 1/"$map(show.(2^))[0..]
This prints the string constructed by printing the string representations of the powers of two starting from 1, separated by the string " + 1/".
The code itself is 49 bytes, the import and newline push it up to 66
## Edit: (62)
Shaved 4 bytes by cutting out the import and defining intercalate with a much shorter name
f(x:xs)s=x++s++xsfs
main=print$map(show.(2^))[0..]f" + 1/" ## Edit 2: (60) Shaved 2 more characters by realizing I didn't need to use the (x:xs) list convention: f(x:y)s=x++s++yfs main=print$map(show.(2^))[0..]f" + 1/"
## Edit 3: (51)
Reimplemented the definition of f and the map as the body of a fold to save 9 more characters
main=print$foldr((++).(++" + 1/").show.(2^))""[0..] ## Edit 4: (49) As was pointed out by Laokoni, I can remove the spaces to cut down 2 more bytes off the total: main=print$foldr((++).(++"+1/").show.(2^))""[0..]
• The challenge states that spaces in the output are optional, so you can save another two bytes. – Laikoni May 18 '17 at 7:04
• Also noticed that the most recent version is the same number of bytes as the original would be if "intercalate" was in the Haskell base library. – archaephyrryx May 18 '17 at 19:34
# Braingolf, 41 37 bytes
Saved 4 bytes because I realised I don't need the spaces, always read the spec thoroughly kids
1!_V"+/">!@R.[<!_>2*v<!@>R!_v!@R<1+>]
Try it online!
Can probably be golfed better, but it works.
# Fourier, 20 bytes
1~io(43a1o47ai*2~io)
Try it online!
I think this may only work on Try it Online due to differences in how Python and Javascript handle large numbers.
Explanation Psuedocode:
i = 1
Print i
While i != 0
Print "+1/"
i = i * 2
Print i
End While
• Can you not move the Print i to the start of the While loop? – Neil May 16 '17 at 15:00
• @Neil I could, but it wouldn't say any bytes – Beta Decay May 16 '17 at 15:32
• Oh, I see now, you're not actually printing i. – Neil May 16 '17 at 15:37
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2019-12-09 01:02:53
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https://rdrr.io/cran/EnvStats/man/egammaAltCensored.html
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# egammaAltCensored: Estimate Mean and Coefficient of Variation for a Gamma...
### Description
Estimate the mean and coefficient of variation of a gamma distribution given a sample of data that has been subjected to Type I censoring, and optionally construct a confidence interval for the mean.
### Usage
1 2 3 4 egammaAltCensored(x, censored, method = "mle", censoring.side = "left", ci = FALSE, ci.method = "profile.likelihood", ci.type = "two-sided", conf.level = 0.95, n.bootstraps = 1000, pivot.statistic = "z", ci.sample.size = sum(!censored))
### Arguments
x numeric vector of observations. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are allowed but will be removed. censored numeric or logical vector indicating which values of x are censored. This must be the same length as x. If the mode of censored is "logical", TRUE values correspond to elements of x that are censored, and FALSE values correspond to elements of x that are not censored. If the mode of censored is "numeric", it must contain only 1's and 0's; 1 corresponds to TRUE and 0 corresponds to FALSE. Missing (NA) values are allowed but will be removed. method character string specifying the method of estimation. Currently, the only available method is maximum likelihood (method="mle"). censoring.side character string indicating on which side the censoring occurs. The possible values are "left" (the default) and "right". ci logical scalar indicating whether to compute a confidence interval for the mean. The default value is ci=FALSE. ci.method character string indicating what method to use to construct the confidence interval for the mean. The possible values are "profile.likelihood" (profile likelihood; the default), "normal.approx" (normal approximation), and "bootstrap" (based on bootstrapping). See the DETAILS section for more information. This argument is ignored if ci=FALSE. ci.type character string indicating what kind of confidence interval to compute. The possible values are "two-sided" (the default), "lower", and "upper". This argument is ignored if ci=FALSE. conf.level a scalar between 0 and 1 indicating the confidence level of the confidence interval. The default value is conf.level=0.95. This argument is ignored if ci=FALSE. n.bootstraps numeric scalar indicating how many bootstraps to use to construct the confidence interval for the mean when ci.type="bootstrap". This argument is ignored if ci=FALSE and/or ci.method does not equal "bootstrap". pivot.statistic character string indicating which pivot statistic to use in the construction of the confidence interval for the mean when ci.method="normal.approx" or ci.method="normal.approx.w.cov" (see the DETAILS section). The possible values are pivot.statistic="z" (the default) and pivot.statistic="t". When pivot.statistic="t" you may supply the argument ci.sample size (see below). The argument pivot.statistic is ignored if ci=FALSE. ci.sample.size numeric scalar indicating what sample size to assume to construct the confidence interval for the mean if pivot.statistic="t" and ci.method="normal.approx". The default value is the number of uncensored observations.
### Details
If x or censored contain any missing (NA), undefined (NaN) or infinite (Inf, -Inf) values, they will be removed prior to performing the estimation.
Let \underline{x} denote a vector of N observations from a gamma distribution with parameters shape=α and scale=β. The relationship between these parameters and the mean μ and coefficient of variation τ of this distribution is given by:
α = τ^{-2} \;\;\;\;\;\; (1)
β = μ/α \;\;\;\;\;\; (2)
μ = α \; β \;\;\;\;\;\; (3)
τ = α^{-1/2} \;\;\;\;\;\; (4)
Assume n (0 < n < N) of these observations are known and c (c=N-n) of these observations are all censored below (left-censored) or all censored above (right-censored) at k fixed censoring levels
T_1, T_2, …, T_k; \; k ≥ 1 \;\;\;\;\;\; (5)
For the case when k ≥ 2, the data are said to be Type I multiply censored. For the case when k=1, set T = T_1. If the data are left-censored and all n known observations are greater than or equal to T, or if the data are right-censored and all n known observations are less than or equal to T, then the data are said to be Type I singly censored (Nelson, 1982, p.7), otherwise they are considered to be Type I multiply censored.
Let c_j denote the number of observations censored below or above censoring level T_j for j = 1, 2, …, k, so that
∑_{i=1}^k c_j = c \;\;\;\;\;\; (6)
Let x_{(1)}, x_{(2)}, …, x_{(N)} denote the “ordered” observations, where now “observation” means either the actual observation (for uncensored observations) or the censoring level (for censored observations). For right-censored data, if a censored observation has the same value as an uncensored one, the uncensored observation should be placed first. For left-censored data, if a censored observation has the same value as an uncensored one, the censored observation should be placed first.
Note that in this case the quantity x_{(i)} does not necessarily represent the i'th “largest” observation from the (unknown) complete sample.
Finally, let Ω (omega) denote the set of n subscripts in the “ordered” sample that correspond to uncensored observations.
Estimation
Maximum Likelihood Estimation (method="mle")
For Type I left censored data, the likelihood function is given by:
L(μ, τ | \underline{x}) = {N \choose c_1 c_2 … c_k n} ∏_{j=1}^k [F(T_j)]^{c_j} ∏_{i \in Ω} f[x_{(i)}] \;\;\;\;\;\; (7)
where f and F denote the probability density function (pdf) and cumulative distribution function (cdf) of the population (Cohen, 1963; Cohen, 1991, pp.6, 50). That is,
f(t) = \frac{t^{α-1} e^{-t/β}}{β^α Γ(α)} \;\;\;\;\;\; (8)
(Johnson et al., 1994, p.343), where α and β are defined in terms of μ and τ by Equations (1) and (2) above.
For left singly censored data, equation (7) simplifies to:
L(μ, τ | \underline{x}) = {N \choose c} [F(T)]^{c} ∏_{i = c+1}^n f[x_{(i)}] \;\;\;\;\;\; (9)
Similarly, for Type I right censored data, the likelihood function is given by:
L(μ, τ | \underline{x}) = {N \choose c_1 c_2 … c_k n} ∏_{j=1}^k [1 - F(T_j)]^{c_j} ∏_{i \in Ω} f[x_{(i)}] \;\;\;\;\;\; (10)
and for right singly censored data this simplifies to:
L(α, β | \underline{x}) = {N \choose c} [1 - F(T)]^{c} ∏_{i = 1}^n f[x_{(i)}] \;\;\;\;\;\; (11)
The maximum likelihood estimators are computed by minimizing the negative log-likelihood function.
Confidence Intervals
This section explains how confidence intervals for the mean μ are computed. In this section, do not confuse the parameter α used to define the confidence level of the confidence interval with the parameter α that was used earlier to denote the shape parameter of the gamma distribution.
Likelihood Profile (ci.method="profile.likelihood")
This method was proposed by Cox (1970, p.88), and Venzon and Moolgavkar (1988) introduced an efficient method of computation. This method is also discussed by Stryhn and Christensen (2003) and Royston (2007). The idea behind this method is to invert the likelihood-ratio test to obtain a confidence interval for the mean μ while treating the coefficient of variation τ as a nuisance parameter. Equation (7) above shows the form of the likelihood function L(μ, τ | \underline{x}) for multiply left-censored data, where μ and τ are defined by Equations (3) and (4), and Equation (10) shows the function for multiply right-censored data.
Following Stryhn and Christensen (2003), denote the maximum likelihood estimates of the mean and coefficient of variation by (μ^*, τ^*). The likelihood ratio test statistic (G^2) of the hypothesis H_0: μ = μ_0 (where μ_0 is a fixed value) equals the drop in 2 log(L) between the “full” model and the reduced model with μ fixed at μ_0, i.e.,
G^2 = 2 \{log[L(μ^*, τ^*)] - log[L(μ_0, τ_0^*)]\} \;\;\;\;\;\; (12)
where τ_0^* is the maximum likelihood estimate of τ for the reduced model (i.e., when μ = μ_0). Under the null hypothesis, the test statistic G^2 follows a chi-squared distribution with 1 degree of freedom.
Alternatively, we may express the test statistic in terms of the profile likelihood function L_1 for the mean μ, which is obtained from the usual likelihood function by maximizing over the parameter τ, i.e.,
L_1(μ) = max_{τ} L(μ, τ) \;\;\;\;\;\; (13)
Then we have
G^2 = 2 \{log[L_1(μ^*)] - log[L_1(μ_0)]\} \;\;\;\;\;\; (14)
A two-sided (1-α)100\% confidence interval for the mean μ consists of all values of μ_0 for which the test is not significant at level alpha:
μ_0: G^2 ≤ χ^2_{1, {1-α}} \;\;\;\;\;\; (15)
where χ^2_{ν, p} denotes the p'th quantile of the chi-squared distribution with ν degrees of freedom. One-sided lower and one-sided upper confidence intervals are computed in a similar fashion, except that the quantity 1-α in Equation (15) is replaced with 1-2α.
Normal Approximation (ci.method="normal.approx")
This method constructs approximate (1-α)100\% confidence intervals for μ based on the assumption that the estimator of μ is approximately normally distributed. That is, a two-sided (1-α)100\% confidence interval for μ is constructed as:
[\hat{μ} - t_{1-α/2, m-1}\hat{σ}_{\hat{μ}}, \; \hat{μ} + t_{1-α/2, m-1}\hat{σ}_{\hat{μ}}] \;\;\;\; (16)
where \hat{μ} denotes the estimate of μ, \hat{σ}_{\hat{μ}} denotes the estimated asymptotic standard deviation of the estimator of μ, m denotes the assumed sample size for the confidence interval, and t_{p,ν} denotes the p'th quantile of Student's t-distribuiton with ν degrees of freedom. One-sided confidence intervals are computed in a similar fashion.
The argument ci.sample.size determines the value of m and by default is equal to the number of uncensored observations. This is simply an ad-hoc method of constructing confidence intervals and is not based on any published theoretical results.
When pivot.statistic="z", the p'th quantile from the standard normal distribution is used in place of the p'th quantile from Student's t-distribution.
The standard deviation of the mle of μ is estimated based on the inverse of the Fisher Information matrix.
Bootstrap and Bias-Corrected Bootstrap Approximation (ci.method="bootstrap")
The bootstrap is a nonparametric method of estimating the distribution (and associated distribution parameters and quantiles) of a sample statistic, regardless of the distribution of the population from which the sample was drawn. The bootstrap was introduced by Efron (1979) and a general reference is Efron and Tibshirani (1993).
In the context of deriving an approximate (1-α)100\% confidence interval for the population mean μ, the bootstrap can be broken down into the following steps:
1. Create a bootstrap sample by taking a random sample of size N from the observations in \underline{x}, where sampling is done with replacement. Note that because sampling is done with replacement, the same element of \underline{x} can appear more than once in the bootstrap sample. Thus, the bootstrap sample will usually not look exactly like the original sample (e.g., the number of censored observations in the bootstrap sample will often differ from the number of censored observations in the original sample).
2. Estimate μ based on the bootstrap sample created in Step 1, using the same method that was used to estimate μ using the original observations in \underline{x}. Because the bootstrap sample usually does not match the original sample, the estimate of μ based on the bootstrap sample will usually differ from the original estimate based on \underline{x}.
3. Repeat Steps 1 and 2 B times, where B is some large number. For the function
egammaAltCensored, the number of bootstraps B is determined by the argument n.bootstraps (see the section ARGUMENTS above). The default value of n.bootstraps is 1000.
4. Use the B estimated values of μ to compute the empirical cumulative distribution function of this estimator of μ (see ecdfPlot), and then create a confidence interval for μ based on this estimated cdf.
The two-sided percentile interval (Efron and Tibshirani, 1993, p.170) is computed as:
[\hat{G}^{-1}(\frac{α}{2}), \; \hat{G}^{-1}(1-\frac{α}{2})] \;\;\;\;\;\; (17)
where \hat{G}(t) denotes the empirical cdf evaluated at t and thus \hat{G}^{-1}(p) denotes the p'th empirical quantile, that is, the p'th quantile associated with the empirical cdf. Similarly, a one-sided lower confidence interval is computed as:
[\hat{G}^{-1}(α), \; ∞] \;\;\;\;\;\; (18)
and a one-sided upper confidence interval is computed as:
[0, \; \hat{G}^{-1}(1-α)] \;\;\;\;\;\; (19)
The function egammaAltCensored calls the R function quantile to compute the empirical quantiles used in Equations (17)-(19).
The percentile method bootstrap confidence interval is only first-order accurate (Efron and Tibshirani, 1993, pp.187-188), meaning that the probability that the confidence interval will contain the true value of μ can be off by k/√{N}, where kis some constant. Efron and Tibshirani (1993, pp.184-188) proposed a bias-corrected and accelerated interval that is second-order accurate, meaning that the probability that the confidence interval will contain the true value of μ may be off by k/N instead of k/√{N}. The two-sided bias-corrected and accelerated confidence interval is computed as:
[\hat{G}^{-1}(α_1), \; \hat{G}^{-1}(α_2)] \;\;\;\;\;\; (20)
where
α_1 = Φ[\hat{z}_0 + \frac{\hat{z}_0 + z_{α/2}}{1 - \hat{a}(z_0 + z_{α/2})}] \;\;\;\;\;\; (21)
α_2 = Φ[\hat{z}_0 + \frac{\hat{z}_0 + z_{1-α/2}}{1 - \hat{a}(z_0 + z_{1-α/2})}] \;\;\;\;\;\; (22)
\hat{z}_0 = Φ^{-1}[\hat{G}(\hat{μ})] \;\;\;\;\;\; (23)
\hat{a} = \frac{∑_{i=1}^N (\hat{μ}_{(\cdot)} - \hat{μ}_{(i)})^3}{6[∑_{i=1}^N (\hat{μ}_{(\cdot)} - \hat{μ}_{(i)})^2]^{3/2}} \;\;\;\;\;\; (24)
where the quantity \hat{μ}_{(i)} denotes the estimate of μ using all the values in \underline{x} except the i'th one, and
\hat{μ}{(\cdot)} = \frac{1}{N} ∑_{i=1}^N \hat{μ_{(i)}} \;\;\;\;\;\; (25)
A one-sided lower confidence interval is given by:
[\hat{G}^{-1}(α_1), \; ∞] \;\;\;\;\;\; (26)
and a one-sided upper confidence interval is given by:
[0, \; \hat{G}^{-1}(α_2)] \;\;\;\;\;\; (27)
where α_1 and α_2 are computed as for a two-sided confidence interval, except α/2 is replaced with α in Equations (21) and (22).
The constant \hat{z}_0 incorporates the bias correction, and the constant \hat{a} is the acceleration constant. The term “acceleration” refers to the rate of change of the standard error of the estimate of μ with respect to the true value of μ (Efron and Tibshirani, 1993, p.186). For a normal (Gaussian) distribution, the standard error of the estimate of μ does not depend on the value of μ, hence the acceleration constant is not really necessary.
When ci.method="bootstrap", the function egammaAltCensored computes both the percentile method and bias-corrected and accelerated method bootstrap confidence intervals.
### Value
a list of class "estimateCensored" containing the estimated parameters and other information. See estimateCensored.object for details.
### Note
A sample of data contains censored observations if some of the observations are reported only as being below or above some censoring level. In environmental data analysis, Type I left-censored data sets are common, with values being reported as “less than the detection limit” (e.g., Helsel, 2012). Data sets with only one censoring level are called singly censored; data sets with multiple censoring levels are called multiply or progressively censored.
Statistical methods for dealing with censored data sets have a long history in the field of survival analysis and life testing. More recently, researchers in the environmental field have proposed alternative methods of computing estimates and confidence intervals in addition to the classical ones such as maximum likelihood estimation. Helsel (2012, Chapter 6) gives an excellent review of past studies of the properties of various estimators for parameters of a normal or lognormal distribution based on censored environmental data.
In practice, it is better to use a confidence interval for the mean or a joint confidence region for the mean and standard deviation (or coefficient of variation), rather than rely on a single point-estimate of the mean. Few studies have been done to evaluate the performance of methods for constructing confidence intervals for the mean or joint confidence regions for the mean and coefficient of variation of a gamma distribution when data are subjected to single or multiple censoring. See, for example, Singh et al. (2006).
### Author(s)
Steven P. Millard (EnvStats@ProbStatInfo.com)
### References
Cohen, A.C. (1963). Progressively Censored Samples in Life Testing. Technometrics 5, 327–339
Cohen, A.C. (1991). Truncated and Censored Samples. Marcel Dekker, New York, New York, 312pp.
Cox, D.R. (1970). Analysis of Binary Data. Chapman & Hall, London. 142pp.
Efron, B. (1979). Bootstrap Methods: Another Look at the Jackknife. The Annals of Statistics 7, 1–26.
Efron, B., and R.J. Tibshirani. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York, 436pp.
Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions, Fourth Edition. John Wiley and Sons, Hoboken, NJ.
Helsel, D.R. (2012). Statistics for Censored Environmental Data Using Minitab and R, Second Edition. John Wiley \& Sons, Hoboken, New Jersey.
Johnson, N.L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York, Chapter 17.
Millard, S.P., P. Dixon, and N.K. Neerchal. (2014; in preparation). Environmental Statistics with R. CRC Press, Boca Raton, Florida.
Nelson, W. (1982). Applied Life Data Analysis. John Wiley and Sons, New York, 634pp.
Royston, P. (2007). Profile Likelihood for Estimation and Confdence Intervals. The Stata Journal 7(3), pp. 376–387.
Singh, A., R. Maichle, and S. Lee. (2006). On the Computation of a 95% Upper Confidence Limit of the Unknown Population Mean Based Upon Data Sets with Below Detection Limit Observations. EPA/600/R-06/022, March 2006. Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C.
Stryhn, H., and J. Christensen. (2003). Confidence Intervals by the Profile Likelihood Method, with Applications in Veterinary Epidemiology. Contributed paper at ISVEE X (November 2003, Chile). http://people.upei.ca/hstryhn/stryhn208.pdf.
Venzon, D.J., and S.H. Moolgavkar. (1988). A Method for Computing Profile-Likelihood-Based Confidence Intervals. Journal of the Royal Statistical Society, Series C (Applied Statistics) 37(1), pp. 87–94.
egammaCensored, GammaDist, egamma, estimateCensored.object.
### Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 # Chapter 15 of USEPA (2009) gives several examples of estimating the mean # and standard deviation of a lognormal distribution on the log-scale using # manganese concentrations (ppb) in groundwater at five background wells. # In EnvStats these data are stored in the data frame # EPA.09.Ex.15.1.manganese.df. # Here we will estimate the mean and coefficient of variation # ON THE ORIGINAL SCALE using the MLE and # assuming a gamma distribution. # First look at the data: #----------------------- EPA.09.Ex.15.1.manganese.df # Sample Well Manganese.Orig.ppb Manganese.ppb Censored #1 1 Well.1 <5 5.0 TRUE #2 2 Well.1 12.1 12.1 FALSE #3 3 Well.1 16.9 16.9 FALSE #... #23 3 Well.5 3.3 3.3 FALSE #24 4 Well.5 8.4 8.4 FALSE #25 5 Well.5 <2 2.0 TRUE longToWide(EPA.09.Ex.15.1.manganese.df, "Manganese.Orig.ppb", "Sample", "Well", paste.row.name = TRUE) # Well.1 Well.2 Well.3 Well.4 Well.5 #Sample.1 <5 <5 <5 6.3 17.9 #Sample.2 12.1 7.7 5.3 11.9 22.7 #Sample.3 16.9 53.6 12.6 10 3.3 #Sample.4 21.6 9.5 106.3 <2 8.4 #Sample.5 <2 45.9 34.5 77.2 <2 # Now estimate the mean and coefficient of variation # using the MLE, and compute a confidence interval # for the mean using the profile-likelihood method. #--------------------------------------------------- with(EPA.09.Ex.15.1.manganese.df, egammaAltCensored(Manganese.ppb, Censored, ci = TRUE)) #Results of Distribution Parameter Estimation #Based on Type I Censored Data #-------------------------------------------- # #Assumed Distribution: Gamma # #Censoring Side: left # #Censoring Level(s): 2 5 # #Estimated Parameter(s): mean = 19.664797 # cv = 1.252936 # #Estimation Method: MLE # #Data: Manganese.ppb # #Censoring Variable: Censored # #Sample Size: 25 # #Percent Censored: 24% # #Confidence Interval for: mean # #Confidence Interval Method: Profile Likelihood # #Confidence Interval Type: two-sided # #Confidence Level: 95% # #Confidence Interval: LCL = 12.25151 # UCL = 34.35332 #---------- # Compare the confidence interval for the mean # based on assuming a lognormal distribution versus # assuming a gamma distribution. with(EPA.09.Ex.15.1.manganese.df, elnormAltCensored(Manganese.ppb, Censored, ci = TRUE))$interval$limits # LCL UCL #12.37629 69.87694 with(EPA.09.Ex.15.1.manganese.df, egammaAltCensored(Manganese.ppb, Censored, ci = TRUE))$interval$limits # LCL UCL #12.25151 34.35332
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2017-04-26 02:13:47
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http://www.poissonboltzmann.org/examples/Solvation_energies/
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# Solvation Energies
Solvation energies are usually decomposed into a free energy cycle as shown in the free energy cycle below. Note that such solvation energies often performed on fixed conformations; as such, they are more correctly called “potentials of mean force”. More details on using APBS for the polar and nonpolar portions of such a cycle are given in the following sections.
### Solvation free energy cycle
Our model solvation free energy cycle is shown here
This cycle incorporates several processes into the solvation energy (step 1). Step 2 indicates charging of the solute in solution (e.g., inhomogeneous dielectric, ions present). Step 3 indicates the introduction of attractive solute-solvent dispersive interaction interactions (e.g., an integral of Weeks-Chandler-Andersen interactions over the solvent-accessible volume). Step 4 indicates the introduction of repulsive solute-solvent interaction (e.g., cavity formation). Steps 5 and 6 are basically null steps although they could be used to offset unwanted energies added in Steps 3 and 4 above. Finally, Step 6 represents the charging of the solute in a vacuum or homogeneous dielectric environment in the absence of mobile ions.
### Polar solvation
The full free energy cycle is usually decomposed into polar and nonpolar parts. The polar portion is usually represented by the charging energies in Steps 2 and 6:
$\Delta_p G = \Delta_2 G - \Delta_6 G$
Energies returned from APBS electrostatics calculations are charging free energies. Therefore, to calculate the polar contribution to the solvation free energy, we simply need to setup two calculations corresponding to Steps 2 and 6 in the free energy cycle. Note that the electrostatic charging free energies returned by APBS include self-interaction terms. These are the energies of a charge distribution interacting with itself. Such self-interaction energies are typically very large and extremely sensitive to the problem discretization (grid spacing, location, etc.). Therefore, it is very important that the two calculations in Steps 2 and 6 are performed with identical grid spacings, lengths, and centers, in order to ensure appropriate matching (or “cancellation”) of self-energy terms.
### Apolar solvation
Referring back to the solvation free energy cycle, the nonpolar solvation free energy is usually represented by the energy changes in Steps 3 through 5:
$\Delta_n G = (\Delta_3 G - \Delta_5 G) + \Delta_4 G$
where Step 4 represents the energy of creating a cavity in solution and Steps 3-5 is the energy associated with dispersive interactions between the solute and solvent. There are many possible choices for modeling this nonpolar solvation process. APBS implements a relatively general model described by Wagoner and Baker (PNAS 2006) and references therein. The implementation and invocation of this model is described in more detail in the APBS user guide. Our basic model for the cavity creation term (Step 4) is motivated by scaled particle theory and has the form
$\Delta_4 G = pV + \gamma A$
where is the solvent pressure (press keyword in the APOLAR input file section), V is the solute volume, $\gamma$ is the solvent surface tension (gamma keyword in the APOLAR input file section), and A is the solute surface area.
Our basic model for the dispersion terms (Steps 3 and 5) follow a Weeks-Chandler-Anderson framework as proposed by Levy, Zhang, and Gallicchio (J Comput Chem, 2002):
$\Delta_3 G - \Delta_5 G = \overset{-} \rho \int_\omega u^{(att)}(y)\theta(y)dy$
where $\overline{\rho}$ is the bulk solvent density (bconc keyword in the APOLAR input file section), $\Omega$ is the problem domain, $u^{\mathrm{(att)}}(y)$ is the attractive dispersion interaction between the solute and the solvent at point y with dispersive Lennard-Jones parameters specified in APBS parameter files, and $\theta(y)$ describes the solvent accessibility of point y.
The ability to independently adjust press, gamma, and bconc means that the general nonpolar solvation model presented above can be easily adapted to other popular nonpolar solvation models. For example, setting press and bconc to zero yields a typical solvent-accessible surface area model.
Unlike the Born ion, there are no good simple examples for demonstrating these types of nonpolar calculations. APBS includes several examples of calculations using the apolar model above. Interested readers should examine the alkanes apolar examples provided with APBS.
### Examples
Additional solvation energy calculations are provided below:
The Born ion
Back
Next
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2017-02-22 21:57:47
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https://ncatlab.org/nlab/show/exterior+differential+system
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# Idea
There are several different ways to think about differential systems:
• the general abstract way which we shall put forward here:
an exterior differential system is a sub-Lie-∞-algebroid $\mathfrak{a} \hookrightarrow T X$ of the tangent Lie algebroid $T X$ of a manifold that is the kernel of a morphism $p : T X \to \mathfrak{j}$ of Lie-∞-algebroids:
$\mathfrak{a} := ker(p) \hookrightarrow T X \stackrel{p}{\to} \mathfrak{j}$
• In the literature – the the references below – the term exterior differential system is instead introduced and understood in the context of dg-algebra as a dg-ideal $J \subset \Omega^\bullet(X)$ inside the deRham dg-algebra of $X$ and all concepts there are developed from this perspective.
From this the above perspective is obtained by noticing that from a dg-ideal $J$ we are naturally led to form the quotient dg-algebra $\Omega^\bullet(X)/J$ which is the cokernel of the inclusion $p^* : J \hookrightarrow \Omega^\bullet(X)$:
$J \stackrel{p^*}{\hookrightarrow} \Omega^\bullet(X) \to coker(p^*) = \Omega^\bullet(X)/J \,.$
The existing literature on exterior differential systems is actually a bit unclear about which additional assumptions on $J$ are supposed to be a crucial part of the definition. However, in most applications of interest — see the examples below — it turns out that $J$ is in fact a semifree dga (over $C^\infty(X)$).
Here we take this as indication that
• it makes good sense to understand exterior differential systems in the restricted sense where the dg-ideal $J$ is required to be a semifree dga;
• the reason that the existing literature does present the desired extra assumptions on the dg-ideal $J$ in an incoherent fashion is due to a lack of global structural insight into the role of the definition of exterior differential systems.
Because, recall that a Lie-∞-algebroid is – effectively by definition – the formal dual of a semifree $\mathbb{N}$-graded commutative dg-algebra. So precisely with that extra condition on $J$ all dg-algebras in the above may be understood as Chevalley-Eilenberg algebras of Lie-∞-algebroids and then the above cokernel sequence of dg-algebras is precisely the formal dual of the kernel sequence of Lie-∞-algebroids.
• Historically, one can trace back the basic idea of exterior differential systems to Eli Cartan’s work on partial differential equations in terms of differential forms:
for each system of partial differential equations
$\{ F^\rho(\{x^\mu\}_{\mu}, \{f^j\}_j, \{\frac{\partial f^j}{\partial x^\mu}\} ) = 0 \}_\rho$
there is a space $X$ and a dg-ideal $J \in \Omega^\bullet(X)$ such that solutions of the system of equations are given by integral manifolds of the exterior differential system determined by $J$.
The notion of an integral manifold of an exterior differential system is crucial in the theory: in terms of $J$ it is a morphism $\phi : \Sigma \to X$ of manifolds such that the pullback of the ideal vanishes, $\phi^* J = 0$.
But this says precisely that $\phi$ extends to morphism of Lie-∞-algebroids
$\phi : T \Sigma \to \mathfrak{a}$
with $CE(\mathfrak{a}) = \Omega^\bullet(X)/J$ as above. Therefore the relevance of the notion of integral manifolds in the theory we take as another indication that exterior differential systems are usefully thought of as being about Lie-∞-algebroids.
# Definition
###### Definition (exterior differential system)
An exterior differential system on a smooth manifold $X$ is a dg-ideal $J \subset \Omega^\bullet(X)$ of the deRham dg-algebra $\Omega^\bullet(X)$ of $X$.
Notice that $J$ being a dg-ideal means explicitly that
• $\forall \theta\in J \subset \Omega^\bullet(X), \omega \in \Omega^\bullet(X): \theta \wedge \omega \in J$
• the $\mathbb{N}$-grading $J = \oplus_{k \in \mathbb{N}} J_k$ on the dg-algebra $J$ is induced from that of $\Omega^\bullet(X)$ in that $J_k = J \cap \Omega^\bullet(k)$
• $\forall \theta \in J \subset \Omega^\bullet(X) : d \theta \in J$
###### Definition
An integral manifold of an exterior differential system is a submanifold $\phi : Y \hookrightarrow X$ such that the restriction of all $\theta \in J$ to $Y$ vanishes: $\that|_Y = 0$.
In other words, for an integral manifold the pullback of the ideal $J$ along the inclusion map $\phi$ vanishes: $\phi^* J = 0$.
## Common further assumptions
Often further assumptions are imposed on exterior differential systems. Here are some:
• An exterior differential system is called finitely generated if there is a finite set $\{\theta_k \in \Omega^\bullet(X)\}$ of differential forms such that $J$ is the dg-ideal generated by these, so that
$J = \{ \sum_i f_i \theta_i + \sum_j g_j d \theta_j| f_i, g_j \in C^\infty(X)\} \,.$
• Often it is assumed that $J_0 = \mathbb{R}$.
Dually in terms of Lie-∞-algebroids this assumption means that $J$ is the Chevalley-Eilenberg algebra of a Lie-∞-algebroid that is just an L-∞-algebra.
• ###### Definition (strict independence condition)
A strict independence condition on an exterior differential system $J \subset \Omega^\bullet(X)$ is an $n$-form $\omega \in \Omega^n(X)$ for some $n$ such that
• $\omega$ is decomposable into a wedge product of $n$ 1-forms mod $J^n$
• $\omega$ is pointwise not an element of $J$.
For $(J, \omega)$ am exterior differential system with strict independence condition $\omega$, an integral manifold is now more restrictively an integral manifold $\phi : \Sigma \to X$ for $J$ but now such that $\phi^* \omega$ is a volume form on $\Sigma$ (i.e. pointwise non-vanishing).
# special cases
Some special types of exterior differential systems carry their own names.
## Frobenius system
A Frobenius system is an exterior differential system $J \subset \Omega^\bullet(X)$ that is locally generated as a graded-commutative algebra from a set $\{\theta_j \in \Omega^1(U)\}_j$ of 1-forms.
Frobenius systems are in bijection with involutive subbundles of the tangent bundle of $X$, i.e. subbundles $E \hookrightarrow T X$ such that for $v,w \in \Gamma(E) \subset \Gamma(T X)$ also the Lie bracket of vector fields of $v$ and $w$ lands in $E$: $[v,w] \in \Gamma(E) \subset \Gamma(T X)$:
• given a Frobenius system the sections of $\Gamma(E)$ are defined locally to be the joint kernel of the maps $\{\theta_i : \Gamma(T U) \to \mathbb{R}\}$.
• given ab involutive subbundle $E$ the corresponding Frobenius system is the collection of 1-forms that vanishes on $E$:
$J = \{\theta \in \Omega^1(X) | \theta|_{E} = 0\}$.
Notice that the involutive subbundle may be thought of precisely as a sub-Lie algebroid
$\array{ E &&\hookrightarrow&& T X \\ & \searrow && \swarrow \\ && X }$
of the tangent Lie algebroid (i.e. as a sub Lie-∞-algebroid that happens to be an ordinary Lie algebroid). And indeed, the Chevalley-Eilenberg algebra of $E$ is the quotient $\Omega^\bullet(X)/J$ of the deRham dg-algebra by the Frobenius system:
$CE(E) = \Omega^\bullet(X)/J \,.$
### vertical tangent Lie algebroid
A special case of a Lie algebroid corresponding to a Frobenius system is the vertical tangent Lie algebroid $T_{vert} Y$ of a map $\pi : Y \to X$. This corresponds to the subbundle $ker(\pi_*) \subset T Y$ of vertical vector fields on $Y$, with respecct to $\pi$. The corresponding Frobenius system is that of horizontal differential forms
$J = \Omega^\bullet_{hor}(Y) = \{\omega \in \Omega^1(Y)| \forall v \in ker(\pi_*): \omega(v) = 0\}$
and
$CE(T_{vert} Y) = \Omega^\bullet(Y)/\Omega^\bullet_{hor}(Y)$
is the dg-algebra of vertical differential forms with respect to $Y$.
This plays a central role in the theory of Ehresmann connections and Cartan-Ehresmann ∞-connection.
## systems of partial differential equations
A system
$\{ F^\rho : \mathbb{R}^n \times \mathbb{R}^s \times \mathbb{R}^{n \cdot s} \to \mathbb{R} \}_\rho$
of partial differential equations in terms of variables $\{x^\mu\}_{\mu = 1}^n$ and functions $\{f^i\}_{i = 1}^s$ of the form
$\{ F^\rho((x^\mu), (f^i), \left(\frac{\partial f^i}{\partial x^\mu}\right)) = 0 \}$
is encoded by an exterior differential system on the 0-locus
$X := \{(x,f,p) \in \mathbb{R}^n \times \mathbb{R}^s \times \mthbb{R}^{n s} | \forall \rho : F^\rho(x,f,p) = 0 \}$
of the $\{F^\rho\}_\rho$ (assuming that this is a manifold) with the dg-ideal $J = \langle \theta_i \rangle_i$ generated by the 1-forms
$\theta^i := d f^i - \sum_{\mu=1}^n p^i_\mu d x^\mu \,.$
Namely a solution to the system of partial differential equations is precisely a section of the projection
$X \to \mathbb{R}^n$
which defined an integral manifold of the exterior differential system.
# References
The standard textbook is
• Bryant et al., Exterior differential systems
Course note are provided in
Revised on September 30, 2013 21:22:07 by Urs Schreiber (82.113.98.87)
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2017-02-20 01:48:06
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https://firas.moosvi.com/oer/physics_bank/content/public/019.Magnetism/Magnetic%20Flux/Flux%20due%20to%20a%20coil/Flux%20due%20to%20a%20coil.html
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# Flux due to a coil#
The coil whose lengthwise cross section is shown in the figure carries a current $$I$$ and has $$N$$ evenly-spaced turns distributed along the length $$\ell$$. Evaluate the magnetic flux $$\oint\vec{\mathbf{B}}\cdot d\vec{\mathbf{\ell}}$$ for each of the paths indicated.
## Part 1#
Path A:
• $$-3\mu_0 I$$
• $$-\mu_0 I$$
• zero
• $$\mu_0 I$$
• $$3\mu_0 I$$
## Part 2#
Path B:
• $$-4\mu_0 I$$
• $$-2\mu_0 I$$
• zero
• $$2\mu_0 I$$
• $$4\mu_0 I$$
## Part 3#
Path C:
• $$-7\mu_0 I$$
• $$-\mu_0 I$$
• zero
• $$\mu_0 I$$
• $$7\mu_0 I$$
## Part 4#
Path D:
• $$-2\mu_0 I$$
• $$-\mu_0 I$$
• zero
• $$\mu_0 I$$
• $$2\mu_0 I$$
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2022-09-28 10:07:18
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https://techlib.adi.com/dynamics-of-real-time-chapter-10/
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Select Page
10.1 Introduction
In this chapter we show how the real-time variable-step integration described in Chapter 9 can be combined with the multi-rate integration described in Chapter 8 to implement asynchronous real-time multi-rate simulation using either a single processor or multiple processors. Key to the successful implementation of both multi-rate integration and real-time variable-step integration is the use of accurate extrapolation formulas to convert data sequences from one frame rate to another, and to compensate for time mismatches and latencies in real-time data sequences. In both Chapters 8 and 9 we demonstrated that extrapolation based on the same algorithm used for numerical integration of the state equations can eliminate the data jitter and discontinuities so often associated with the use of more conventional extrapolation methods. In Chapter 8 we assumed that the frame-ratio N, i.e., the ratio of fast-to-slow integration frame rates in a multi-rate simulation, is always an integer. This has certainly been the choice in traditional multi-rate simulation. In this chapter we will show that multi-rate integration can be just as effective when the frame-ratio is not an integer, or indeed when the frame-ratio is not even fixed throughout a given simulation. Using the real-time variable-step predictor integration formulas introduced in Chapter 9, we will also show how processors engaged in real-time simulation can handle asynchronous interrupts without destroying the fidelity of the real-time simulation. Finally, we will demonstrate how the simulation of large complex dynamic systems can be partitioned among multiple processors, with each processor assigned to simulate identifiable physical subsystems. In this scheme each processor runs at its own frame rate, independent of the other processors, and data transfers between processors are handled with extrapolation algorithms similar to those developed in Chapters 8 and 9. Utilization of this asynchronous methodology has the potential of greatly reducing the difficulties usually associated with interconnecting many processors and hardware subsystems in a complex hardware-in-the-loop simulation.
We begin the chapter by showing how the real-time multi-rate simulation described in Chapter 8 can be modified to handle an increase in computer execution time within any given integration frame, such as might occur when an external interrupt must be processed. Later in the chapter we will show how the variable-step integration and associated extrapolation techniques can be used in multi-processor, multi-rate simulation of a combined digital-continuous system.
10.2 Accommodating a One-time Increase in Step Execution Time for a Multi-rate Simulation with Real-time Inputs and Outputs
Consider again the flight control system shown in Figure 8.2 of Chapter 8. In Section 8.9 we examined multi-rate simulation of the system using AB-2 integration, with the actuator considered to be the fast subsystem and the airframe to be the slow subsystem. We assumed that the times required to execute one integration step of the actuator and airframe were given, respectively, by and seconds. For a multi-rate frame ratio given by N = 8, Eqs. (8.1) and (8.2) yield integration step sizes for the actuator and airframe that are given, respectively, by and . For these step sizes, the resulting errors in real-time airframe and actuator outputs for an acceleration-limited unit-step input are shown in Figures 8.16 and 8.17. In this example simulation, scheduling the execution of fast and slow integration steps was based on the earliest next-half-frame occurrence. Table 8.1 shows that extrapolation of the real-time inputs is required to calculate 3 out of each 8 input data points for the actuator simulation. Also, extrapolation is needed in the calculation of 3 out of each 8 real-time actuator output data points
In this section we consider the case where, every so often, the execution time for an individual integration step may increase substantially from time to time. For example, this might occur when complex conditional code must occasionally be implemented within a particular integration step. Or perhaps it may be desirable to have the capability of servicing an external interrupt now and then in the real-time simulation. For the multi-rate simulation in Section 8.9 we chose the integration step sizes and based on fixed and known step-execution times such that the real-time simulation always runs at maximum possible speed. Under these conditions, any increase in the execution time for a single integration step will cause the overall simulation to fall permanently behind real time by the amount of the execution-time increase. In this section we will introduce a methodology which lets the real-time simulation run at maximum possible speed when step-execution times are fixed, and yet permits occasional increases in step-execution time without compromising real-time simulation accuracy.
First, let us assume that the overall time, , required to execute N actuator steps followed by 1 airframe step is measured on-line using either the computer clock or master clock which controls the real-time simulation. For our multi-rate flight-control system simulation example with frame-ratio N = 8, let us further assume that the measurement of is made just prior to executing the airframe integration formulas, i.e., just after the calculation of the airframe state-variable derivatives for the nth airframe step. Then , represents the time required to execute the airframe integration formulas for the previous (n – 1) airframe step plus the following N actuator steps plus the state-variable derivative calculations for the nth airframe step. If the integration step-size used in the airframe integration formulas is then set equal to , and the integration step-size used for the N following actuator steps is set equal to , the overall simulation will catch up with and remain synchronized with real time. By making the measurement of total execution time for N actuator steps plus 1 airframe step just prior to implementing the airframe integration formulas, we permit the airframe step size for that very same step to be adjusted to reflect any change in execution time, instead of having to wait for the next (n + 1) airframe step. Note that any increase in airframe-step execution time, such as might result from required execution of conditional code, is more likely to occur in the derivative section of the nth airframe step. If we further require that external interrupts only be allowed in the derivative section of the airframe step, it follows that the correction in airframe step-size needed to handle either contingency in a real-time environment can still be accomplished within the nth airframe step.
As a specific example, suppose we let the execution time for one particular airframe integration step be doubled, from to seconds. For all other airframe integration steps we let remain equal to its nominal value of 0.01. Also, we let the actuator integration step-execution time always remain equal to its nominal value of 0.001. Figure 10.1 shows the resulting timing diagram which, as in the earlier timing diagram of Figure 8.14, shows real time along the horizontal axis and mathematical simulation time (positive downward) along the vertical axis. The timing diagram of Figure 10.1 begins 4 actuator steps prior to the initiation of the 1st airframe step. Thus the first 4 actuator steps (j = 1, 2, 3, 4) in Figure 10.1 are
Figure 10.1. Timing diagram for real-time multi-rate simulation when the execution time for one airframe step is increased to 0.02 sec, then restored to 0.01 sec for follow-on steps.
the same as in Figure 8.14, where the 4th actuator step, representing , is completed after 0.004 seconds of real time. Then the first airframe step (n = 1) takes place, with a step-execution time of 0.02 seconds instead of the nominal value of 0.01. Thus the end of this first airframe step in Figure 10.1 occurs after 0.024 seconds of real time have elapsed. Because our measurement of , which includes the 0.01 second increase in execution time of the airframe step, is by assumption made before implementing the airframe integration calculations, we are able to increase the first airframe integration step size in Figure 10.1 from the nominal size to . In accordance with the methodology introduced in the above paragraph, Figure 10.1 shows that the step size for the next 8 actuator steps (j = 5, 6, … , 12) is increased from the nominal value of to sec. Since the measured execution time of these 8 steps plus the next airframe step (n = 2) now returns to the nominal value of 8(0.001) + 0.01 = 0.018, the size of the 2nd airframe step in Figure 10.1 is returned to , and the size of the next 8 actuator steps (j = 13, 14, …, 20) is returned to sec. In the same way, for all subsequent airframe and actuator steps, remains 0.018 for the airframe and remains 0.00225 for the actuator.
In Figure 10.1, as in the earlier timing diagram of Figure 8.14, events which occur in real time, such as real-time input samples , all lie on a straight line through the origin with slope of -1. We assume that real-time input samples continue to occur every 0.00225 seconds, i.e., with the nominal actuator step size, and that both real-time actuator and airframe outputs continue to occur every 0.018 seconds, i.e., with the nominal airframe step size. Later we will consider the case where the real-time actuator data points are required every 0.00225 seconds instead of every 0.018 seconds. The real-time input and output data points are shown in Figure 10.1 as light and dark circles, respectively, along the real-time line with slope -1. As in Figure 8.14, events which lie below this real-time line occur ahead of real time, and events which lie above the line occur behind real time. Thus the completion of the first 4 actuator steps occurs ahead of real time, as does the completion of the first airframe step. On the other hand, Figure 10.1 shows that completion of each of the next 6 actuator steps (j = 5,6, … , 10) occurs behind real time.
Table 10.1, which is similar to the earlier Table 8.1 in Chapter 8, lists the actuator and airframe step indices j and n, the integration step times, and , and the associated real time, , for the events depicted in Figure 10.1. Also shown in Table 10.1 is the input data-point time utilized at each actuator step-time , as well as the data-time used in the calculation of real-time output data points every 0.018 seconds. Finally, Table 10.1 shows the dimensionless extrapolation interval, , used to calculate each actuator real-time input data point , the dimensionless extrapolation interval, , used to calculate the actuator inputs and from the airframe simulation outputs and , and the dimensionless extrapolation interval, , used to calculate the real-time actuator output every 0.018 seconds. Comparison of Table 10.1 with the earlier Table 8.1 shows that the biggest increase in required extrapolation interval occurs in computing the real-time actuator output corresponding to (row 5, column 8). This is because when , the latest available actuator data point has the step time . This results in a required extrapolation interval of 0.009, which corresponds to 4 actuator steps. The remaining extrapolation intervals listed in Table 10.1 are not very different from those shown earlier in Table 8.1. Note, however, that the increase from 0.01 to 0.02 in execution time for the first airframe step in Figure 10.1 and Table 10.1 results in a mismatch between real-time output sample times and both actuator and airframe simulation times and , respectively, for all follow-on steps. Thus the extrapolation interval for generating real-time actuator outputs every 0.018 seconds settles down to -0.444 actuator steps (equivalent to interpolation). This is evident in the last column of Table 10.1.
Table 10.1
Integration-step Indices, Step Times and Extrapolation Intervals for Multi-rate Simulation when the Execution Time for One Airframe Step is Increased to 0.02 Sec., Then Restored to 0.01 Sec. for Subsequent Steps; Frame-ratio N = 8.
10.3. Effect of Doubled Execution Time for the 12th Airframe Integration Step
We now let the one-time increase in airframe step-execution time occur in the 12th airframe integration step rather than the 1st integration step, as in Figure 10.1 and Table 10.1. As explained in the previous section, this means that the execution time for all the actuator steps is maintained at the nominal value of 0.001 seconds, the execution time for all airframe steps except the 12th is kept at 0.01 seconds, and the real-time output sample period is held at 0.018 seconds, the nominal step size for the airframe simulation. Figures 10.2 and 10.3 show the resulting errors in airframe and actuator outputs, respectively. Also shown in Figures 10.1 and 10.2 are the real-time output errors shown earlier in Figure 8.16 for the case where the execution time for all airframe and actuator steps is fixed at 0.01 and 0.001 seconds, respectively. In both Figures 10.2 and 10.3 it is evident that the errors for the first 11 output samples (i.e., for sample-times up to 11(0.018) = 0.198 seconds) are unchanged. Then, when the airframe execution time for step 12 is doubled, the airframe and actuator real-time output errors deviate from their original values for the fixed execution-time case. Note, however, that the deviation is significantly less than the errors themselves, indicating that our methodology for handling the change in airframe step-execution time, including extrapolation for maintaining real-time outputs, has worked effectively.
Figure 10.2. Real time airframe output errors resulting from a 100 percent increase in execution time for the 12th airframe step.
In addition to considering the actuator output errors when the real-time output sample period seconds, it is also useful to examine the errors when , which is the same as the nominal actuator integration step size when the frame ratio N = 8. This has been done in
Figure 10.3. Real time actuator output errors resulting from a 100 percent increase in execution time for the 12th airframe step.
Figure 10.4, which shows the errors both with and without the increase from 0.01 to 0.02 in execution time for the 12th airframe step. Here we have reduced the length of displayed transient from 1.2 to 0.6 seconds in order to be able to distinguish better the individual error points. Note that the incremental errors associated with the much higher real-time output sample rate still remain small compared with the overall simulation errors.
Reference to row 5 in Table 10.1 shows that the maximum extrapolation interval required to generate real-time actuator outputs when occurs during the airframe step with 0.02 second execution time. The actual extrapolation interval is equal to 0.018 – 0.009 = 0.009 seconds, which is equivalent to a dimensionless extrapolation interval , as shown in row 5 of the table. On the other hand, when the real-time output sample period instead of 0.018, additional real-time outputs are required at , 0.02250 and 0.02475 seconds. The corresponding extrapolation intervals, based on seconds for the 4th actuator step, are equal to 0.01125, 0.01350 and 0.01575, respectively. For the next real-time output at seconds, row 7 in Table 10.1 shows that we can base the extrapolation on , since the next ( i.e., 5th) actuator step has been completed at . This reduces the required extrapolation interval to 0.0270 – 0.0195 = 0.0075. We conclude that the maximum extrapolation interval when computing real-time actuator outputs with occurs when and is equal to 0.01575. Note that this is considerably larger than the maximum extrapolation interval of 0.009 for the case where . In Figure 10.4, where the doubling of execution time occurs for the 12th instead of the 1st airframe integration step, the time associated with the maximum extrapolation interval of 0.01575 noted above, occurs at seconds. The corresponding extrapolation error is identified in Figure 10.4.
Figure 10.4. Real-time actuator output errors, sample-period = 0.00225, resulting from a 100 percent increase in execution time for the 12th airframe step.
It is useful to examine the local integration truncation errors when the execution time for the 12th airframe integration step is doubled. This has been done in Figure 10.5 for the airframe output and in Figure 10.6 for the actuator output . Also shown in each figure is the local truncation error when the execution time for airframe step 12 is maintained at its nominal value of 0.01 second. In both cases the truncation error has been calculated on-line using the formula in Eq. (9.7). When is doubled from 0.01 to 0.02, the 0.01 second shift in discrete airframe step-time , that occurs for is clearly evident in Figure 10.5.
Comparison of the local integration truncation errors shown in Figures 10.5 and 10.6 with the actual simulation errors in Figures 10.2 and 10.3, respectively, again demonstrates that local integration truncation errors can be used as an indicator of the effect of changes in integration step size on the overall simulation accuracy. In particular, we see in Figure 10.5 that the local truncation error associated with increasing the 12th airframe integration step size from 0.018 to 0.028 correlates directly with the increase in actual airframe output error shown in Figure 10.2. However, the airframe local truncation error increase in Figure 10.5 is confined almost entirely to
Figure 10.5. Local integration truncation error in real-time airframe output resulting from a 100 percent increase in execution time for the 12th airframe step.
Figure 10.6. Local integration truncation error in real-time actuator output resulting from a 100 percent increase in execution time for the 12th airframe step.
the 12th integration step, whereas the global error increase in Figure 10.2 dies out only after a number of follow-on integration steps. Similarly, the actuator local truncation error increase in Figure 10.6 is confined almost entirely to the 8 actuator steps following the 12th airframe step, wheras the global error increase in Figure 12 dies out more gradually.
From the flight control system example considered here, we conclude that multi-rate simulation with real-time inputs and outputs can be accomplished even when there is a significant increase in the computer execution time for any particular integration step, such as might happen when an interrupt must be handled by the simulation computer. Again, the extrapolation formulas utilized here, particularly the formula based on the same algorithm used for the numerical integration itself, permit temporary increases in the mathematical size of the multi-rate integration steps, as needed to permit the simulation to once again catch up with real time.
10.4 Multi-rate Real-time Simulation Using Multiple Processors
In all of the considerations of multi-rate integration in Chapters 8 and 9, and until now in this chapter, we have assumed that the simulation is always implemented on a single computer. However, with the emergence in recent years of high-speed, single-chip microprocessors that are quite low in cost, the use of multiple computers for complex simulations must be considered. Although the use of parallel processors to improve overall computational speed has been a popular concept for many years, a good solution to the problem of efficiently distributing a large simulation among many processors has proven to be quite elusive. In particular, it has been difficult to partition the problem in such a way that the required number of data transfers between processors is minimized. Indeed, it has been the overhead associated with these data transfers that has invariably been the stumbling block in the efficient utilization of paralleled computer resources.
In the simulation of complex dynamic systems, one obvious solution to the partitioning problem is to assign each processor in a large simulation to an identifiable physical subsystem or group of subsystems within the overall system being simulated. Assigning multi-processors in this way will in general minimize the required number of data transfers between processors. It is also compatible with exchanging individual processor simulations with actual hardware, as in real-time HITL (hardware-in-the-loop) simulations. The problem of scheduling data transfers between processors can be greatly simplified by letting each processor or system run at its own frame rate, independent of the other frame rates. This is actually what we have designated here as multi-rate simulation. When the extrapolation methods introduced in Chapters 8 and 9 are used, it turns out that the individual frame rates no longer need to be integer multiples of each other, or even commensurate. When a data block is passed from one processor to another, it is always accompanied by a time-tag that identifies the discrete time represented by the data in that block and, when available, by the time derivatives of the variables contained in the data block. This then permits the processor receiving the data to reconstruct through extrapolation the value of each variable at the time required for use in its own simulation algorithm or physical process. Transfer of data blocks between processors and/or hardware subsystems can then be accomplished in a completely asynchronous manner. Time mismatches and data transfer delays can be automatically compensated to within the accuracy inherent in the extrapolation algorithms. Figure 10.7 illustrates the multiprocessor architecture in a real-time, hardware-in-the-loop simulation, as well as a data block with time tag and typical extrapolation formula.
Bus data: each data block consists of data values, their time derivatives (when available), and a time tag, i.e., . Then, using predictor-integration extrapolation
Figure 10.7. Multi-processor architecture.
In the sections that follow, we will describe how the asynchronous methodology can be applied to the multi-rate, multi-processor simulation of complex dynamic systems, with particular emphasis on real-time simulation with real-time inputs and outputs. The variable-step integration methods introduced in Chapter 9 will permit automatic assignment of integration frame rates within a given processor, as well as the ability to handle variable frame-execution times in an ongoing real-time simulation.
10.5 Example Simulation of a Mixed-data Flight Control System
To illustrate multi-processor simulation, we consider the airframe pitch control system shown in Figure 10.8. It consists of an airframe (the slow subsystem), a control-surface actuator (the fast subsystem), and a digital autopilot. The airframe is represented with the same mathematical model and system parameters used in the previous flight control system of Section 8.3. Thus the dynamic behavior of the airframe is dominated by the 5 rad/sec undamped natural frequency, , of the short-period pitching motion. Here the airframe will be simulated with a nominal integration step size of T = 0.02 seconds. The control-surface actuator is modeled as a second-order system with undamped natural frequency, , equal to 40 rad/sec. It will be considered a fast subsystem, to be simulated with a nominal integration step size h = 0.005 seconds. The following control law is assumed for the digital controller:
(10.1)
Here represents the digital controller output at the kth frame, K is the controller gain constant, is the input pitch angle, and is the airframe output pitch angle at the kth frame, as generated by the A to D converter with sample-period . Also, is the estimated pitch rate, as obtained from a backward difference approximation, and is the effective rate constant.
Figure 10.8. Block diagram of combined continuous/digital system simulation.
For illustrative purposes we select a digital controller frame rate of 100 Hz (i.e., seconds). The zero-order extrapolator in Figure 10.8 converts the output data sequence from the digital controller to the continuous input for the control surface actuator. The actuator state equations take the form given earlier in Eq. (9.5), with X replaced by Y replaced by U(t) replaced by replaced by , and replaced by . The following parameter values are used for the simulation:
For the sample-period given by for the digital controller and the above parameter values, the flight control system response to an acceleration-limited step input is shown in Figure 10.9. Here the solution was obtained using RK-4 integration with an integration step size h = T = 0.005, which is sufficiently small to ensure negligible dynamic errors. This solution serves as a reference for calculating simulation errors in the examples that follow.
With a nominal integration step size of T = 0.02 sec for the airframe simulation and h = 0.005 sec for the actuator simulation, the ratio between actuator and airframe integration frame rates in this case is given by N = 4. Since there are two actuator integration steps per digital controller step, i.e., , there is no problem in synchronizing the output samples of the zero-order extrapolator with the integration step times for the actuator simulation. Note, however, that the zero-order extrapolator will exhibit step changes in output every seconds as the extrapolator responds to each new digital controller output . These steps become inputs to the actuator simulation, and large errors will result when the actuator is simulated using a standard predictor-integration method such as AB-2. To eliminate these errors we consider the actuator input at the beginning of the jth frame to be representative of the input halfway through the frame, i.e., . The AB-2 difference equations for the actuator simulation with step size h then become the following:
(10.2)
(10.3)
Figure 10.10. Control system response to acceleration-limited step input.
Note that the input term in Eq. (10.2) represents modified Euler integration with , whereas the remaining terms on the right side of Eq. (10.2) represent AB-2 integration. In this way, accurate integration of the fixed input over the jth frame is achieved, without the problems normally suffered by predictor methods when discontinuous step inputs occur.
We recall that the same concept of mixing modified Euler integration with AB-2 integration was used earlier in Section 8.9 in simulating the airframe, where the airframe input was integrated using the modified Euler method and AB-2 integration was used for the remaining terms on the right side of Eq. (8.5). For a fixed airframe integration step size T, the airframe difference equations are then given by
(10.4)
As in Section 8.9, the actuator output half-way through the nth airframe step is obtained using extrapolation based on the multi-rate actuator data points and their time derivatives. By analogy with Eq. (8.14), the extrapolation formula is given by
(10.5)
Here represents the nth airframe integration step size and represents the size of the n – 1 actuator step, although in our initial simulation example, both and will be constant step sizes given by T and h, respectively.
To generate the controller input data point estimate from the airframe output data points , we also use extrapolation based on second-order predictor integration. The formula is given by
(10.6)
Note that in Eq. (10.6).
We now consider the overall simulation of the flight control system in Figure 10.8 using AB-2 integration with fixed step size. The difference equations consist of (10.1) through (10.4), with Eqs. (10.5) and (10.6) used for the extrapolation required to interface the subsystem simulations with different frame rates. We will consider three cases. For the first case we set the airframe and actuator step sizes equal to their nominal values, T = 0.02 and h = 0.005, respectively, with the digital controller step size given by . For the second and third cases we let the airframe step size , i.e., T = 0.022094 and 0.017906. These step sizes correspond to frame ratios between actuator and airframe step sizes given by N = 4.41888 and N = 3.58112, respectively, compared with the nominal frame ratio of N = 4. Figure 10.11 shows the error in simulated airframe output for the all three cases. Clearly the extrapolation formulas used to compensate for the non-commensurate frame ratios are quite effective, in that they produce incremental errors which are small compared with the basic errors associated with the finite integration
Figure 10.11. Effect of airframe simulation step size on error of simulated output.
step sizes used in the simulation. Indeed, the error differences for the three cases in Figure 10.11 can be entirely explained by the second-order truncation errors associated with each different step size T, as used in the airframe simulation.
We next consider the effect of varying the actuator simulation step size h from its nominal value of 0.005 sec. In this case the ratio of the digital-controller step size, , to the actuator step size h will no longer be 2:1. Therefore, we can no longer rely on the interpretation of to represent , as in the term in Eq. (10.2), to eliminate first-order errors. The problem is illustrated in Figure 10.12, which shows the controller sample times , the controller output samples , the corresponding extrapolator output , and the actuator frame times for the case where and h = 0.0055 sec. Note that the extrapolator output sample at the time is not representative of the extrapolator output over the entire jth step when the extrapolator output jumps to a new value before the step is completed. The problem is solved by actually computing the average extrapolator output over each actuator simulation time step using the following simple formula:
(10.7)
The average extrapolator output over the jth actuator time step, is shown in Figure 10.13. This output then replaces the input in the modified AB-2 integration algorithm of Eq. (10.2). Using this procedure, we obtain the results in Figure 10.14, where the error in simulated output is shown for three different actuator simulation step sizes. For the first step size, h = 0.005, its nominal value. For the second and third step sizes, , i.e., h = 0.0060472, and 0.0039528. Note that the change in step size h has very little effect on the simulation, which indicates that the use of Eq. (10.7) to compensate for the non-integer ratio that occurs when h = 0.0060472 and 0.0039528 is indeed very effective.
Figure 10.12. Extrapolator output and effective actuator input for a non-integer frame-rate ratio.
Figure 10.13. Extrapolator output and effective actuator input based on input averaging for a non-integer frame-rate ratio.
Figure 10.14. Effect of varying the actuator step size h.
Next, with the actuator step size fixed at the nominal value h = 0.005 and the digital controller step size still set to , we investigate the effectiveness of the variable-step second-order predictor integration method when used to simulate the airframe. The AB-2 difference equations for the airframe, given earlier in Eq. (10.4), must be modified in accordance with Eq. (9.2) to accommodate the variable airframe step size . The difference equations for integrating the airframe state equations given in Eqs. (8.4) through (8.7) now become
(10.8)
(10.9)
(10.10)
Note that the modified Euler input term on the right side of Eq. (10.9) is preserved in the variable-step predictor algorithm. Eq. (10.6) is used as before to produce the controller input data estimate , as is Eq. (10.5) for the actuator output halfway through the nth frame. Eqs. (10.1), (10.2), and (10.3) are also used as shown, with the exception that in Eq. (10.2) is again replaced by the mean value, of the extrapolator output over the jth frame, as given by Eq. (10.7).
As a specific example of a variable step size, we let the airframe integration step be given by . This represents the same relative variation is step size used earlier in our second-order system simulation and illustrated in Figure 9.1. Thus the airframe step size varies between 0.016 and 0.024 seconds, with a mean value of 0.02. Since the output of the airframe simulation will now exhibit a variable frame rate, with individual data points both behind and ahead of real time, we will use extrapolation based on second-order predictor integration to generate a fixed-step real-time output. We recall that this worked well in Section 9.3 in the variable-step simulation of the simple second-order system. The real-time extrapolation is accomplished by using Eq. (10.6), but with the digital controller time replaced by the fixed-step output time . We let the step size of the real-time output data points be 0.02, the mean value of the variable step of the airframe simulation. Figure 10.15 shows the resulting airframe output errors in the variable-step case, as well as the output errors for a fixed airframe step size, (the mean of the variable-step size), and the errors when (the maximum value of the variable-step size). Note that the errors for the variable-step case are essentially the same as those for the fixed step when that step is equal to the mean variable-step size, . On the other hand, if the fixed step is set to accommodate the maximum variable step size, , the errors are significantly larger, i.e., by the ratio (0.024/0.02)2.
10.6 Multi-processor versus Single-processor Simulation
Until now we have tacitly assumed that our flight control system in Figure 10.8 is simulated using a single processor, even though each subsystem simulation employs a different frame rate (multi-rate integration). However, there may be times when it is desirable, or even necessary, to simulate each subsystem on a separate processor. For example, in hardware-in-the-loop simulation, where initially the entire simulation is accomplished using computers, it will be easier to substitute actual hardware for one of the subsystems if an individual processor has been assigned
Figure 10.15. Output error for both fixed and variable airframe simulation steps.
for simulation of that subsystem. In this case the processor, including the appropriate interfacing, is simply replaced on a one-to-one basis with the hardware. Also, when the overall system being simulated is extremely complex, a single processor may be unable to handle the simulation with acceptable accuracy in real time, in which case multiple processors will be required. The simplest way of partitioning a large problem among parallel processors is to assign each processor to given physical subsystems or groups of subsystems. In this way the required number of data-variable transfers between processors, invariably the bottleneck in parallel-processor simulation, is likely to be minimized.
To illustrate, let us assume that three processors are assigned to the simulation of the flight control system in Figure 10.8, one for the digital controller, one for the zero-order extrapolator and control-surface actuator, and one for the airframe. In this case it makes particular sense to utilize variable-step integration based on the measured execution time of each integration frame in simulating the continuous subsystems (the actuator and airframe in our example here). The extrapolation formulas presented in Section 8.4 permit each processor to reconstruct the value of its input variables at the time required for use in its own simulation. This does mean that data-block transfers from one processor to another must include a time stamp that represents the discrete time associated with the block of data. Also, in order to implement extrapolation based on the same algorithm used to compute the output data by numerical integration, the time derivatives of each variable, when available, must also be included in the data-block transfers between processors.
The asynchronous multi-processor simulation using variable-step integration, as described in the above paragraph, can result in major simplifications of the programming task associated with data transfers in a complex, multi-processor simulation. It is instructive to examine a timing chart which shows frame execution times for the individual processors in a multi-rate, multi-processor simulation. For the nominal synchronous case where the controller step size , the actuator step size h = 0.005, and the airframe step size T = 0.02 seconds, this has been done in Figure 10.16a. Here we have assumed that the mathematical step size for each subsystem simulation has been set equal to the processor time required to execute that frame plus all input/output data transfers associated with the frame.
For comparison purposes, Figure 10.16b shows the timing diagram when one processor is used for the entire simulation. In this latter case the single processor must be able to execute 1 airframe, 2 controller frames and 4 actuator frames in a total of 20 milliseconds in order for the simulation to run in real time. Also, in the single-processor program we must specify the order of execution of the different subsystem frames within each 20 millisecond total frame time. In Figure 10.16b we have chosen the frame-execution order based on the earliest next-full-frame occurrence for each subsystem. This was the first multi-rate scheduling algorithm described in Section 8.2 of Chapter 8. Thus at t = 0, the next actuator frame-end time (5 milliseconds) is the lowest, so that the actuator frame (marked Al in the figure) is the first to be executed. Starting at t = 5 milliseconds, the next frames for both the actuator (A2) and the controller (C1) end at t = 10 milliseconds. In Figure 10.16b we have assigned top priority to the controller, so that the first controller frame, Cl, is executed next. This is followed by the second and third actuator frames, A2 and A3. At the end of frame A3, the mathematical frame-end times for all three subsystems (frames A4, C2, and AF1) are equal to 20 milliseconds. Since the controller has the top priority, frame C2 is executed next, followed by A4 and finally AF1 (the lowest priority has been assigned to the airframe execution). In the synchronous case of Figure 10.16b, subsequent frame executions follow exactly the same order.
Figure 10.16. Timing diagram for one and three-processor multi-rate synchronous simulations; Tsam = 10, h = 5, T = 20 msec.
To illustrate an asynchronous case, we change the actuator step size h from 0.005 to 0.006047 seconds, with the controller and airframe step sizes retained at and T = 0.02 seconds, respectively. This is one of the cases shown earlier in Figure 10.14. The controller and airframe step sizes are now no longer commensurate with the actuator step size. The timing diagram for the three-processor simulation is shown in Figure 10.17a. Note that actuator frames no longer occur at exactly the same time as airframe or controller frames. Yet we have seen in Figure 10.14 that the extrapolation algorithms are able to compensate almost perfectly for the frame-time mismatches. In Figure 10.17b is shown the timing diagram for the single-processor simulation of the overall system. Here the order of frame executions is changed from that seen earlier in the synchronous case of Figure 10.16b. Thus frame AF1 for the airframe, instead of actuator frame A4, is executed after frame C2 for the controller. This is because the frame-end time for actuator frame A4 is now 24.189 milliseconds instead of 20 milliseconds, as in Figure 10.16b. Following frame AF1, the frame execution order in Figure 11.17b is A4, C3, A5, A6, C4, AF2, A7, … , instead of A5, C3, A6, A7, C4, A8, AF2, … , as shown earlier in the synchronous case of Figure 10.16b.
Figure 10.17. Timing diagram for one and three-processor multi-rate asynchronous simulations; Tsam = 10, h = 6.047, T = 20 msec.
In examining the multi-rate timing diagrams in Figures 10.16 and 10.17, it is important to consider Eq. (10.7), the formula used to compute , the average output of the zero-order extrapolator over the jth actuator frame. This formula requires and , the extrapolator output at times and , respectively. The determination of these outputs, in turn, requires knowledge of the digital controller output, , for the controller frames which contain the times and , respectively. For both the one and three-processor examples in Figures 10.16 and 10.17, it is readily apparent that this requirement is automatically satisfied.
The above discussion of asynchronous multi-rate simulation has been directed to the real-time case. However, it should be noted that the methodology described here works equally well in the case of non-real-time simulation. Thus variable-step multi-rate simulation for general-purpose non real-time simulation can be accomplished in a very straightforward manner using the techniques introduced in this chapter. This, in turn, has the potential of providing substantial increases in computing efficiencies when simulating complex dynamic systems.
10.7 Extension of Multi-rate Asynchronous Methodology to Real-time Interfaces
In Chapter 5 we discussed the dynamics of digital-to-analog and analog-to digital conversion. This discussion included a description of algorithms to compensate for the dynamic errors associated with zero-order DAC’s, as well as the performance improvements that can be realized by using multi-rate digital-to-analog conversion. Also described in Chapter 5 were the advantages of using multi-rate input sampling and averaging in analog-to-digital conversion. It should be noted that the asynchronous, multi-rate methodology developed here in Chapter 10 can be applied equally well to D to A and A to D interface subsystems in a real-time simulation. Each interface subsystem is simply treated as another processor in the overall multi-processor environment. Thus data-block transfers into and out of interface subsystems are accompanied by time tags for the discrete data within the data block. This then allows complete asynchronous multi-rate operations using the appropriate extrapolation formulas. Also, any time delays, fixed or variable, that occur in the data transfers are automatically compensated to within the accuracies inherent in the extrapolation formulas.
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2022-07-05 17:58:09
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http://3ecommunications.net/failed-to/failed-to-perform-redirection.html
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Home > Failed To > Failed To Perform Redirection
# Failed To Perform Redirection
Woot! this may cause by lack of the permission on the target folder, you need the right to chang permission(or simply releasecreator or userfull permission). The folder is configured to be redirected to <\\zcapad1\users\%USERNAME%>, the final expanded path was <\\zcapad1\users\hpatel>. I come to share some of the experiences and hopefully give you guys a quick workaround till MS releases a hot fix to fix this because I am positive SP3 is http://3ecommunications.net/failed-to/failed-to-delete-a-domain-from-the-database-must-be-the-owner-to-perform-this-operation.html
The folder is configured to be redirected from to <\\SERVER1\HOME$\STUDENTS\jsilva\My Pictures>. Login By creating an account, you're agreeing to our Terms of Use, Privacy Policy and to receive emails from Spiceworks. © Copyright 2006-2017 Spiceworks Inc. x 24 Private comment: Subscribers only. The new directories for the redirected folder could not be created. https://support.microsoft.com/en-us/kb/953529 Get the answer AnonymousSep 1, 2005, 7:26 AM Archived from groups: microsoft.public.win2000.group_policy (More info?)Thanks Denis. Pimiento Apr 12, 2011 Jibber It Service Provider I found that this error was caused by the initial folder not existing. Join the IT Network or Login. For example, if the previous user was set up to redirect My Documents to "\\server\share\%username%", and the settings got saved in the Default User folder, and the current user does not The folder is configured to be redirected from to <\\city-files\PDUsers\Directories\pstewart>. The folder is configured to be redirected from to <\\it055\usrdata$\nicoleta giurgiu\My Documents\My Pictures>. Application Data, MyDocuments) or creating the folders before hand.* To get the system to delete the file on second logon I had to enable Always wait for network on computer startup The new directories for the redirected folder could not be created.
Jun 19, 2012 Failed to perform redirection of folder My Documents. The following error occurred: %%5
Dec 22, 2010 Failed to perform redirection of folder My Documents. https://support.microsoft.com/en-us/kb/291087 The folder is configured to be redirected from to <>.
The folder isconfigured to be redirected to <\\server1\GENERAL\%USERNAME%\MyDocuments>. So I reverted the snapshot and logged on as local admin, and took a peek inside C:\Documents and Settings\Default User\My Documents.... SEO by vBSEO ©2011, Crawlability, Inc. Thursday, September 18, 2008 8:32 PM Reply | Quote 0 Sign in to vote Hi Guys, Like r5a and Michael_84, we too have been experiencing folder redirection issues under XP SP3, when
Folder redirection strange problem Folder Redirection Never Works Trying to setup folder redirection en shortcuts Folder Redirection move fails Error 112 - path > 255 Folder Redirection Folder Redirection... https://community.spiceworks.com/windows_event/show/260-folder-redirection-101 The shares were set up on our primary domain controller, on which active directory got FUBARed this Monday. I had similiar issues because some files couldn't be moved to the new location. The > files for the > redirected folder could not be moved to the new location.
Not a member? Check This Out The folder is configured to be redirected from <\\00-alpha\userdata\%username%\my documents> to <>. x 7 EventID.Net - Error: "Cannot create a file when that file already exists" - This issue occurs because a new folder is created when the user logs on for the Some users got all the files moved over and some didn't.The errors I've seen are SID errors, file cannot be moved over, and ect.
## Group Policy has an option to set up the Folder Redirection component as Basic, Advanced, or None.
The new directories for the redirected folder could not be created. The folder is configured to be redirected from <\\server-name\users\test user> to <\\server-name\users\year11\fbloggs>. Creating your account only takes a few minutes. The following error occurred: %%267
Jan 22, 2010 Failed to perform redirection of folder My Documents.
Files were being moved from to <\\server1\GENERAL\user1\My Documents>.The following error occurred while copying to <\\server1\GENERAL\user1\MyDocuments\desktop.ini>:The security descriptor structure is invalid.Obviously when the user first logged The folder is configured to be redirected to <\\ahs-s1\studentmydocs>, the final expanded path was <\\ahs-s1\studentmydocs>. The folder is configured to be redirected from to <\\mort-dc01\users\jwheatley>. http://3ecommunications.net/failed-to/failed-to-create-snapshots-of-replica-devices-sra-command-39-testfailoverstart-39-failed.html Friday, October 08, 2010 7:13 AM Reply | Quote 0 Sign in to vote I'm having a similar issue in windows XP SP3 with folder redirection.
In exceptional circumstances it is also possible for files that contain information about a previous user's folder redirection to be created in the "C:\Documents and Settings\Default User\Local Settings\Application Data\Microsoft\Windows\File Deployment" folder. I am still a little confused at to why I started to get this problem but,hey ho, it's working now. AnonymousSep 1, 2005, 7:26 AM Archived from groups: microsoft.public.win2000.group_policy (More info?)Thanks Lara. The following error occurred: %%267 15:57:17:343 Previous contents of folder My Documents at C:\Documents and Settings\%username%\My Documents will be deleted. 15:57:17:343 Successfully redirected folder My Documents.
The solution is to delete any files from the "C:\Documents and Settings\Default User\Local Settings\Application Data\Microsoft\Windows\File Deployment" folder. Failed to perform redirection of folder Application Data.The files for the redirected folder could not be moved to the new location.The folder is configured to be redirected to <\\SERVER\path\$\USERNAME\Application Data>. in Technical; I've been looking at a network. The following error occurred: %%1307
Sep 23, 2009 Failed to perform redirection of folder My Documents.
All clients are XP except for one Vista machine. The folder is configured to be redirected to <\\city-files\PDUsers\Directories\%username%>, the final expanded path was <\\city-files\PDUsers\Directories\jmccullough>. Can > someone please > offer me some advice on this or possibly a solution? > > Thanks, > > Kendall.Hi,Move the contents of the My Documents Manually and then in The following error occurred: %%1307
Jan 04, 2011 Failed to perform redirection of folder My Documents.
Stats Reported 7 years ago 4 Comments 4,515 Views Other sources for 107 Report Server Windows Service (MSSQLSERVER) Report Server Windows Service (SQLEXPRESS) Report Server Windows Service (MIP) MSExchange Search Indexer chkdsk can't read security descriptor stream?? The following error occurred: %%5
Sep 16, 2009 Failed to perform redirection of folder My Documents. Check the ACLs on the share and the folder.
The folder is configured to be redirected to <\\MASTER6\Users\%USERNAME%\My Documents>, the final expanded path was <\\MASTER6\Users\War room\My Documents>. The new directories for the redirected folder could not be created. The following error occurred: %%1307
Jul 06, 2009 Failed to perform redirection of folder My Documents. The folder is configured to be redirected to <\\fs01\delta\HomeShares>, the final expanded path was <\\fs01\delta\HomeShares>.
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2018-01-21 02:23:53
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https://www.ck12.org/studyguides/algebra/systems-of-linear-inequalities-study-guide.html
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# Systems of Linear Inequalities
## Big Picture
One way to solve systems of linear equalities is to graph the inequalities and see if there are any areas on the graph where the inequalities overlap. The places where the inequalities’ graphs overlap are the solutions to the system.
## Key Terms
Linear Inequality: A linear equation with the $$=$$ sign replaced with inequality signs ($$<$$,$$≤$$, $$>$$, or $$≥$$).
Absolute Value: The absolute value of a number is the distance of that number from $$0$$.
## Graphing Linear Inequaltiies
### In Two Variables
A linear equation in slope-intercept form is $$y = mx + b$$. A linear inequality is closely related to a linear equation, except the = sign is replaced with $$<$$, $$≤$$, $$>$$, or $$≥$$.
To draw a linear inequality,
• Write the equation in slope-intercept form $$y = mx + b$$ for graphing.
• Draw the line. If the inequality sign does not include an equal sign($$<$$ or $$>$$), then draw a dashed line. If the inequality sign includes an equal sign ($$≤$$ or $$≥$$), then draw a solid line.
• The line divides the plane into two halves. Shade the half plane that include the points that are part of the solution.
Examples:
Shade the half plane above the line if the inequality is greater than.
Shade the half plane below the line if the inequality is less than.
Test a point on one side of the line (not on the line) to see if the point makes the inequality true. If it does,shade that side of the line. If not, shade the other side.
### In One Variable
Linear inequalities in one variable can also be graphed on the coordinate plane. The line that gets drawn is a horizontal or vertical line. The graph looks like the solution graphed on the number line but stretched vertically.
Example: $$x > 4$$
# Systems of Linear Inequalities cont.
## Graphing Systems of Inequalities
A system of linear inequalities can be solved by graphing.
• Graph each inequality
• Find the regions of solutions that make all the inequalities true.
• Test a point in the different regions to see if the point makes the linear system of inequalities true.
### Two Linear Inequalities
The solutions of two linear inequalities are unbounded regions, which continue infinitely in at least one direction. Here are two examples:
• The orange region is the solution to $$y < 2x+6$$.
• The blue region is the solution to $$y ≥ 2x-4$$.
• The purple region is the solution to both inequalities.
Test a point on one side of the line (not on the line) to see if the point makes the inequality true. If it does,shade that side of the line. If not, shade the other side.
• The orange region is the solution to $$y > 2x+6$$.
• The blue region is the solution to $$y ≤ 2x-4$$.
• Need to shade both regions!
### More than Two Linear Inequalities
The solutions of two linear inequalities can be unbounded or bounded regions. A bounded region is a finite region with three or more sides.
Example:
$$y > 3x-4$$
$$y < -\frac{9}{4}x + 2$$
$$x ≥ 0$$
$$y ≥ 0$$
### Absolute Values
Absolute value inequalities can be re-written as a system of two inequalities.
Example: $$|x| ≥ 2$$
Rewrite as $$x ≤ -2 \text{ or } x ≥ 2$$
Example: $$|y| < 5$$
Rewrite as $$y > -5$$ and $$y < 5$$
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2021-09-27 00:49:12
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http://math.stackexchange.com/questions/194350/parabolic-branch-cut
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# Parabolic Branch Cut
The following is problem 19 from page 87 of Saff and Snider's "Fundamentals of Complex Analysis for Mathematics, Science, and Engineering,"
How would you construct a branch of $\log z$ that is analytic in the domain D consisting of all points in the plane except those lying on the half-parabola $\lbrace x+iy: x \ge 0, y = \sqrt{x}\rbrace$?
Saff and Snider have defined all logarithms to be taken to the base of $e$ unless otherwise mentioned. I understand the idea of a branch cut and its purpose in constructing a single-valued function from a multi-valued one. Saff and Snider have also defined the principal logarithm of $z$ as,
$$\text{Log}\;z = \text{Log}\;|z| + i\;Arg\; z$$
Where $\text{Arg} \; z$ lies in the half-open interval $(-\pi,\pi]$. This function has a branch cut on the nonpositive real axis. So I thought of considering $\text{Log}\;z^2.$ I figured that substituting $z^2$ for $z$ might result in a branch cut that resembled a quadratic. Unfortunately, $\text{Log}\;z^2$ just has two branch cuts. They lie on the nonnegative and nonpositive imaginary axes, respectively. I also considered using a branch cut of $\log z$ whereby the arguement is taken to be on the half-open interval $(\pi/4,9\pi/4]$. This is close to the answer, but the branch cut is still in the shape of a ray rather than a half-parabola. I'm not sure what else to try...
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I'm not sure I understand the question. What prevents you from just choosing the half-parabola to be the branch cut of $\log z$? – Robert Mastragostino Sep 11 '12 at 21:13
@RobertMastragostino: I think you're right. Actually, I think I had trouble understanding the question, and what you wrote is the correct interpretation. If you post your comment as an answer, I'll be happy to mark it as the solution. – Andrew Sep 11 '12 at 21:18
Just define the half-parabola to be the branch cut and you're done. A branch cut isn't intrinsic to a function, you choose it in whatever way you like that prevents you from circling a branch point. For example, $\log z$ has branch points at $0$ and $\infty$, so any unbounded curve that hits zero (and doesn't let you circle the origin) would separate the branches of this function.
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Actually you should say a bit more, e.g. specify the value of $\log(z)$ at some point not on the curve. – Robert Israel Sep 11 '12 at 22:33
@RobertIsrael Ah, I didn't see what it was asking. Should I update this or did you want to put that down as a separate answer? – Robert Mastragostino Sep 11 '12 at 22:37
To construct that branch of log $z$, you just define the half-parabola to be the branch cut. This would mean log $z$ = Log|$z$| + $i\theta$ where $\theta$ equals the value of arg $z$ between $\frac{\pi}{2}$ and $2\pi$ for $z$ in the second, third, or fourth quadrant, the value of arg $z$ between $0$ and $\frac{\pi}{2}$ for $z$ in the first quadrant above the half parabola, and the value of arg $z$ between $2\pi$ and $\frac{5\pi}{2}$ for z in the first quadrant below the half parabola.
Explicitly, you can solve for $\theta$ in terms of $r$ where $z=re^{i\theta}$. By the equation of the half parabola, $\frac{y}{x}=\frac{1}{y}$. Then, $\theta=\arctan(\frac{y}{x})=\arctan(\frac{1}{y})$.
$r=\sqrt{x^2+y^2}=\sqrt{x^2+x}$ , then $x^2+x-r^2=0$. By the quadratic formula, $$x=\frac{-1\pm\sqrt{1+4r^2}}{2}$$ You choose the positive square root because you want the upper half parabola so, $$y=\sqrt{\frac{\sqrt{1+4r^2}-1}{2}}$$ Therefore, the entire branch would be defined as $$\log z = \mbox{Log }|z| + i\theta , \mbox{where } \theta = \arctan \bigg(\sqrt{\frac{2}{\sqrt{1+4r^2}-1}}\bigg)$$
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2016-05-03 09:20:32
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http://www.mathcancer.org/blog/page/3/
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## Setting up gcc / OpenMP on OSX (MacPorts edition)
Note: This is the part of a series of “how-to” blog posts to help new users and developers of BioFVM and PhysiCell. This guide is for OSX users. Windows users should use this guide instead. A Linux guide is expected soon.
These instructions should get you up and running with a minimal environment for compiling 64-bit C++ projects with OpenMP (e.g., BioFVM and PhysiCell) using gcc. These instructions were tested with OSX 10.11 (El Capitan), but they should work on any reasonably recent version of OSX.
In the end result, you’ll have a compiler and key makefile capabilities. The entire toolchain is free and open source.
Of course, you can use other compilers and more sophisticated integrated desktop environments, but these instructions will get you a good baseline system with support for 64-bit binaries and OpenMP parallelization.
Note 1: OSX / Xcode appears to have gcc out of the box (you can type “gcc” in a Terminal window), but this really just maps back onto Apple’s build of clang. Alas, this will not support OpenMP for parallelization.
Note 2: This process is somewhat painful because MacPorts compiles everything from source, rather than using pre-compiled binaries. This tutorial uses Homebrew: a newer package manager that uses pre-compiled binaries to dramatically speed up the process. I highly recommend using the Homebrew version of this tutorial.
### What you’ll need:
1. XCode: This includes command line development tools. Evidently, it is required for both Macports and its competitors (e.g., Homebrew). Download the latest version in the App Store. (Search for xcode.) As of January 15, 2016, the App Store will install Version 7.2. Please note that this is a 4.41 GB download!
2. MacPorts: This is a package manager for OSX, which will let you easily download, build and install many linux utilities. You’ll particularly need it for getting gcc. Download the latest installer (MacPorts-2.3.4-10.11-ElCapitan.pkg) here. As of August 2, 2017, this will download Version 2.4.1.
3. gcc7 (from MacPorts): This will be an up-to-date 64-bit version of gcc, with support for OpenMP. As of August 2, 2017, this will download Version 7.1.1.
### Main steps:
As mentioned above, open the App Store, search for Xcode, and start the download / install. Go ahead and grab a coffee while it’s downloading and installing 4+ GB. Once it has installed, open Xcode, agree to the license, and let it install whatever components it needs.
Now, you need to get the command line tools. Go to the Xcode menu, select “Open Developer Tool”, and choose “More Developer Tools …”. This will open up a site in Safari and prompt you to log in.
Sign on with your AppleID, agree to yet more licensing terms, and then search for “command line tools” for your version of Xcode and OSX. (In my case, this is OSX 10.11 with Xcode 7.2) Click the + next to the correct version, and then the link for the dmg file. (Command_Line_Tools_OS_X_10.11_for_Xcode_7.2.dmg).
Double-click the dmg file. Double-click pkg file it contains. Click “continue”, “OK”, and “agree” as much as it takes to install. Once done, go ahead and exit the installer and close the dmg file.
#### 2) Install Macports
Double-click the MacPorts pkg file you downloaded above. OSX may complain with a message like this:
“MacPorts-2.4.1-10.11-ElCapitan.pkg” can’t be opened because it is from an unidentified developer.
If so, follow the directions here.
Leave all the default choices as they are in the installer. Click OK a bunch of times. The package scripts might take awhile.
Open a terminal window (Open Launchpad, then “Other”, then “Terminal”), and run:
sudo port -v selfupdate
to make sure that everything is up-to-date.
#### 3) Get, install, and prepare gcc
Open a terminal window (see above), and search for gcc, version 7.x or above
port search gcc7
You should see a list of packages, including gcc7.
sudo port install gcc7
You should see a list of packages, including gcc7.
This will download, build, and install any dependencies necessary for gcc7, including llvm and many, many other things. This takes even longer than the 4.4 GB download of Xcode. Go get dinner and a coffee. You may well need to let this run overnight. (On my 2012 Macbook Air, it required 16 hours to fully build gcc7 and its dependencies in a prior tutorial. We’ll discuss this point further below.)
Lastly, you need to get the exact name of your compiler. In your terminal window, type g++, and then hit tab twice to see a list. On my system, I see this:
Pauls-MBA:~ pmacklin$g++ g++ g++-mp-7 Look for the version of g++ with an “mp” in its name. In my case, it’s g++-mp-7. Double-check that you have the right one by checking its version. It should look something like this: Pauls-MBA:~ pmacklin$ g++-mp-7 --version
g++-mp-7 (MacPorts gcc7 7-20170622_0) 7.1.1 20170622
Copyright (C) 2017 Free Software Foundation, Inc.
This is free software; see the source for copying conditions. There is NO warranty; not even
for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
Notice that MacPorts shows up in the information. The correct compiler is g++-mp-7.
PhysiCell Version 1.2.2 and greater use a system variable to record your compiler version, so that you don’t need to modify the CC line in PhysiCell Makefiles. Set the PHYSICELL_CPP variable to record the compiler you just found above. For example, on the bash shell:
export PHYSICELL_CPP=g++-mp-7
echo export PHYSICELL_CPP=g++-mp-7 >> ~/.bash_profile
I wrote a sample C++ program that tests OpenMP parallelization (32 threads). If you can compile and run it, it means that everything (including make) is working! 🙂
##### Make a new directory, and enter it
Open Terminal (see above). You should be in your user profile’s root directory. Make a new subdirectory called GCC_test, and enter it.
mkdir GCC_test
cd GCC_test
##### Grab a sample parallelized program:
Download a Makefile and C++ source file, and save them to the GCC_test directory. Here are the links:
Note: The Makefiles in PhysiCell (versions > 1.2.1) can use an environment variable to specify an OpenMP-capable g++ compiler. If you have not yet done so, you should go ahead and set that now, e.g., for the bash shell:
export PHYSICELL_CPP=g++-mp-7
echo export PHYSICELL_CPP=g++-mp-7 >> ~/.bash_profile
##### Compile and run the test:
Go back to your (still open) command prompt. Compile and run the program:
make
./my_test
The output should look something like this:
Allocating 4096 MB of memory ...
Done!
Entering main loop ...
Done!
### What’s next?
Download a copy of PhysiCell and try out the included examples! Visit BioFVM at MathCancer.org.
1. PhysiCell Method Paper at bioRxiv: https://doi.org/10.1101/088773
2. PhysiCell on MathCancer: http://PhysiCell.MathCancer.org
3. PhysiCell on SourceForge: http://PhysiCell.sf.net
4. PhysiCell on github: http://github.com/MathCancer/PhysiCell
2. BioFVM on MathCancer.org: http://BioFVM.MathCancer.org
3. BioFVM on SourceForge: http://BioFVM.sf.net
4. BioFVM Method Paper in BioInformatics: http://dx.doi.org/10.1093/bioinformatics/btv730
## BioFVM: an efficient, parallelized diffusive transport solver for 3-D biological simulations
I’m very excited to announce that our 3-D diffusion solver has been accepted for publication and is now online at Bioinformatics. Click here to check out the open access preprint!
A. Ghaffarizadeh, S.H. Friedman, and P. Macklin. BioFVM: an efficient, parallelized diffusive transport solver for 3-D biological simulations. Bioinformatics, 2015.
DOI: 10.1093/bioinformatics/btv730 (free; open access)
BioFVM (stands for “Finite Volume Method for biological problems) is an open source package to solve for 3-D diffusion of several substrates with desktop workstations, single supercomputer nodes, or even laptops (for smaller problems). We built it from the ground up for biological problems, with optimizations in C++ and OpenMP to take advantage of all those cores on your CPU. The code is available at SourceForge and BioFVM.MathCancer.org.
The main idea here is to make it easier to simulate big, cool problems in 3-D multicellular biology. We’ll take care of secretion, diffusion, and uptake of things like oxygen, glucose, metabolic waste products, signaling factors, and drugs, so you can focus on the rest of your model.
### Design philosophy and main capabilities
Solving diffusion equations efficiently and accurately is hard, especially in 3D. Almost all biological simulations deal with this, many by using explicit finite differences (easy to code and accurate, but very slow!) or implicit methods like ADI (accurate and relatively fast, but difficult to code with complex linking to libraries). While real biological systems often depend upon many diffusing things (lots of signaling factors for cell-cell communication, growth substrates, drugs, etc.), most solvers only scale well to simulating two or three. We solve a system of PDEs of the following form:
$\frac{\partial \vec{\rho}}{\partial t} = \overbrace{ \vec{D} \nabla^2 \vec{\rho} }^\textrm{diffusion} – \overbrace{ \vec{\lambda} \vec{\rho} }^\textrm{decay} + \overbrace{ \vec{S} \left( \vec{\rho}^* – \vec{\rho} \right) }^{\textrm{bulk source}} – \overbrace{ \vec{U} \vec{\rho} }^{\textrm{bulk uptake}} + \overbrace{\sum_{\textrm{cells } k} 1_k(\vec{x}) \left[ \vec{S}_k \left( \vec{\rho}^*_k – \vec{\rho} \right) – \vec{U}_k \vec{\rho} \right] }^\textrm{sources and sinks by cells}$
Above, all vector-vector products are term-by-term.
#### Solving for many diffusing substrates
We set out to write a package that could simulate many diffusing substrates using algorithms that were fast but simple enough to optimize. To do this, we wrote the entire solver to work on vectors of substrates, rather than on individual PDEs. In performance testing, we found that simulating 10 diffusing things only takes about 2.6 times longer than simulating one. (In traditional codes, simulating ten things takes ten times as long as simulating one.) We tried our hardest to break the code in our testing, but we failed. We simulated all the way from 1 diffusing substrate up to 128 without any problems. Adding new substrates increases the computational cost linearly.
#### Combining simple but tailored solvers
We used an approach called operator splitting: breaking a complicated PDE into a series of simpler PDEs and ODEs, which can be solved one at a time with implicit methods. This allowed us to write a very fast diffusion/decay solver, a bulk supply/uptake solver, and a cell-based secretion/uptake solver. Each of these individual solvers was individually optimized. Theory tells us that if each individual solver is first-order accurate in time and stable, then the overall approach is first-order accurate in time and stable.
The beauty of the approach is that each solver can individually be improved over time. For example, in BioFVM 1.0.2, we doubled the performance of the cell-based secretion/uptake solver. The operator splitting approach also lets us add new terms to the “main” PDE by writing new solvers, rather than rewriting a large, monolithic solver. We will take advantage of this to add advective terms (critical for interstitial flow) in future releases.
#### Optimizing the diffusion solver for large 3-D domains
For the first main release of BioFVM, we restricted ourselves to Cartesian meshes, which allowed us to write very tailored mesh data structures and diffusion solvers. (Note: the finite volume method reduces to finite differences on Cartesian meshes with trivial Neumann boundary conditions.) We intend to work on more general Voronoi meshes in a future release. (This will be particularly helpful for sources/sinks along blood vessels.)
By using constant diffusion and decay coefficients, we were able to write very fast solvers for Cartesian meshes. We use the locally one-dimensional (LOD) method–a specialized form of operator splitting–to break the 3-D diffusion problem into a series of 1-D diffusion problems. For each (y,z) in our mesh, we have a 1-D diffusion problem along x. This yields a tridiagonal linear system which we can solve efficiently with the Thomas algorithm. Moreover, because the forward-sweep steps only depend upon the coefficient matrix (which is unchanging over time), we can pre-compute and store the results in memory for all the x-diffusion problems. In fact, the structure of the matrix allows us to pre-compute part of the back-substitution steps as well. Same for y- and z-diffusion. This gives a big speedup.
Next, we can use all those CPU cores to speed up our work. While the back-substitution steps of the Thomas algorithm can’t be easily parallelized (it’s a serial operation), we can solve many x-diffusion problems at the same time, using independent copies (instances) of the Thomas solver. So, we break up all the x-diffusion problems up across a big OpenMP loop, and repeat for y– and z-diffusion.
Lastly, we used overloaded +=, axpy and similar operations on the vector of substrates, to avoid unnecessary (and very expensive) memory allocation and copy operations wherever we could. This was a really fun code to write!
The work seems to have payed off: we have found that solving on 1 million voxel meshes (about 8 mm3 at 20 μm resolution) is easy even for laptops.
#### Simulating many cells
We tailored the solver to allow both lattice- and off-lattice cell sources and sinks. Desktop workstations should have no trouble with 1,000,000 cells secreting and uptaking a few substrates.
#### Simplifying the non-science
We worked to minimize external dependencies, because few things are more frustrating than tracking down a bunch of libraries that may not work together on your platform. The first release BioFVM only has one external dependency: pugixml (an XML parser). We didn’t link an entire linear algebra library just to get axpy and a Thomas solver–it wouldn’t have been optimized for our system anyway. We implemented what we needed of the freely available .mat file specification, rather than requiring a separate library for that. (We have used these matlab read/write routines in house for several years.)
Similarly, we stuck to a very simple mesh data structure so we wouldn’t have to maintain compatibility with general mesh libraries (which can tend to favor feature sets and generality over performance and simplicity). Rather than use general-purpose ODE solvers (with yet more library dependencies, and more work for maintaining compatibility), we wrote simple solvers tailored specifically to our equations.
The upshot of this is that you don’t have to do anything fancy to replicate results with BioFVM. Just grab a copy of the source, drop it into your project directory, include it in your project (e.g., your makefile), and you’re good to go.
### All the juicy details
The Bioinformatics paper is just 2 pages long, using the standard “Applications Note” format. It’s a fantastic format for announcing and disseminating a piece of code, and we’re grateful to be published there. But you should pop open the supplementary materials, because all the fun mathematics are there:
• The full details of the numerical algorithm, including information on our optimizations.
• Convergence tests: For several examples, we showed:
• First-order convergence in time (with respect to Δt), and stability
• Second-order convergence in space (with respect to Δx)
• Accuracy tests: For each convergence test, we looked at how small Δt has to be to ensure 5% relative accuracy at Δx = 20 μm resolution. For oxygen-like problems with cell-based sources and sinks, Δt = 0.01 min will do the trick. This is about 15 times larger than the stability-restricted time step for explicit methods.
• Performance tests:
• Computational cost (wall time to simulate a fixed problem on a fixed domain size with fixed time/spatial resolution) increases linearly with the number of substrates. 5-10 substrates are very feasible on desktop workstations.
• Computational cost increases linearly with the number of voxels
• Computational cost increases linearly in the number of cell-based source/sinks
And of course because this code is open sourced, you can dig through the implementation details all you like! (And improvements are welcome!)
### What’s next?
• As MultiCellDS (multicellular data standard) matures, we will implement read/write support for <microenvironment> data in digital snapshots.
• We have a few ideas to improve the speed of the cell-based sources and sinks. In particular, switching to a higher-order accurate solver may allow larger time step sizes, so long as the method is still stable. For the specific form of the sources/sinks, the trapezoid rule could work well here.
• I’d like to allow a spatially-varying diffusion coefficient. We could probably do this (at very great memory cost) by writing separate Thomas solvers for each strip in x, y, and z, or by giving up the pre-computation part of the optimization. I’m still mulling this one over.
• I’d also like to implement non-Cartesian meshes. The data structure isn’t a big deal, but we lose the LOD optimization and Thomas solvers. In this case, we’d either use explicit methods (very slow!), use an iterative matrix solver (trickier to parallelize nicely, except in matrix-vector multiplication operations), or start with quasi-steady problems that let us use Gauss-Seidel iterative type methods, like this old paper.
• Since advective flow (particularly interstitial flow) is so important for many problems, I’d like to add an advective solver. This will require some sort of upwinding to maintain stability.
• At some point, we’d like to port this to GPUs. However, I don’t currently have time / resources to maintain a separate CUDA or OpenCL branch. (Perhaps this will be an excuse to learn Julia on GPUs.)
Well, we hope you find BioFVM useful. If you give it a shot, I’d love to hear back from you!
Very best — Paul
## Paul Macklin profiled in New Scientist article
Paul Macklin was recently featured in a New Scientist article on multidisciplinary jobs in cancer. It profiled the non-linear path he and others took to reach a multi-disciplinary career blending biology, mathematics, and computing.
Read the article: http://jobs.newscientist.com/article/knocking-cancer-out/ (Apr. 16, 2015)
## Banner and Logo Contest : MultiCellDS Project
As the MultiCellDS (multicellular data standards) project continues to ramp up, we could use some artistic skill.
Right now, we don’t have a banner (aside from a fairly barebones placeholder using a lovely LCARS font) or a logo. While I could whip up a fancier banner and logo, I have a feeling that there is much better talent out there. So, let’s have a contest!
Here are the guidelines and suggestions:
1. The banner should use the text MultiCellDS Project. It’s up to artist (and the use) whether the “multicellular data standards” part gets written out more fully (e.g., below the main part of the banner).
2. The logo should be shorter and easy to use on other websites. I’d suggest MCDS, stylized similarly to the main banner.
3. Think of MultiCell as a prefix: MultiCellDS, MultiCellXML, MultiCellHDF, MultiCellDB. So, the “banner” version should be extensible to new directions on the project.
4. The banner and logo should be submitted in a vector graphics format, with all source.
5. It goes without saying that you can’t use clip art that you don’t have rights to. (i.e., use your own artwork or photos, or properly-attributed creative commons-licensed art.)
6. The banner and logo need to belong to the MultiCellDS project once done.
7. We may do some final tweaks and finalization on the winning design for space or other constraints. But this will be done in full consultation with the winner.
So, what are the perks for winning?
1. Permanent link to your personal research / profession page crediting you as the winner.
2. A blog/post detailing how awesome you and your banner and logo are.
3. Beer / coffee is on me next time I see you. SMB 2015 in Atlanta might be a good time to do it!
4. If we ever make t-shirts, I’ll buy yours for you. 🙂
5. You get to feel good for being awesome and helping out the project!
So, please post here, on the @MultiCellDS twitter feed, or contact me if you’re interested. Once I get a sense of interest, I’ll set a deadline for submissions and “voting” procedures.
Thanks!!
## 2015 Speaking Schedule
Here is my current speaking schedule for 2015. Please join me if you can!
Feb. 13, 2015: Seminar at the Institute for Scientific Computing Research, Lawrence Livermore National Laboratory (LLNL)
Title: Scalable 3-D Agent-Based Simulations of Cells and Tissues in Biology and Cancer [abstract]
## Paul Macklin calls for common standards in cancer modeling
At a recent NCI-organized mini-symposium on big data in cancer, Paul Macklin called for new data standards in Multicellular data in simulations, experiments, and clinical science. USC featured the talk (abstract here) and the work at news.usc.edu.
Read the article: http://news.usc.edu/59091/usc-researcher-calls-for-common-standards-in-cancer-modeling/ (Feb. 21, 2014)
## 2014 Speaking Schedule
Here is my current speaking schedule for 2014. Please join me if you can!
Feb. 16, 2014: American Association for the Advancement of Science (AAAS) Annual Meeting, Chicago
Title: Integrating Next-Generation Computational Models of Cancer Progression and Outcome [abstract]
invited by the National Cancer Institute
May 9, 2014: European Society for Medical Oncology (ESMO) 2014 IMPAKT Breast Cancer Conference, Brussels, Belgium
Title: Calibrating breast cancer simulations with patient pathology: Progress and future steps [programme]
Plenary talk
May 13, 2014: Wolfson Centre for Mathematical Biology at the University of Oxford, Oxford, UK
Title: Advances in parallelized 3-D agent-based cancer modeling and digital cell lines [abstract]
June 19, 2014: Biostatistics Seminar, University of Southern California, Los Angeles
Title: Simulating 3-D systems of 500k cells with an agent-based model, and digital cell lines [link]
Aug. 18, 2014: COMBINE (Computational Modeling in Biology Network) 2014 Symposium, University of Southern California, Los Angeles
Title: Digital cell lines and MultiCellDS: Standardizing cell phenotype data for data-driven cancer simulations[Program]
2013 public speaking schedule
I’m in the process of rolling out some updates to my website. The first thing you’ll see is a new talk / tutorial on computational modeling of biological processes, based upon my recent talk at the USC PS-OC Short Course in October 2013. I’ll make another post here when it’s ready. It will include MATLAB source code to run through the models.
In the medium term, I hope to update my list of projects to better reflect current efforts by my lab, particularly in (1) integrative modeling of cancer metastases using high-throughput in vitro experiments and sophisticated bioengineered tissues for calibration and validation, and (2) development of standardizations for cell- and tissue-scale models and experiments.
In the longer term, I hope to switch my website layout a bit to be more like the USC PSOC website. I wrote that site about a year ago, and I like the CSS and structure a lot better. 🙂
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https://wikimili.com/en/Fuchsian_group
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# Fuchsian group
Last updated
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,R) (so that it contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group (a discrete subgroup of PSL(2,C)) which is conjugate to a subgroup of PSL(2,R).
## Contents
Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In this case, the group may be called the Fuchsian group of the surface. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry. Some Escher graphics are based on them (for the disc model of hyperbolic geometry).
General Fuchsian groups were first studied by HenriPoincaré ( 1882 ), who was motivated by the paper ( Fuchs 1880 ), and therefore named them after Lazarus Fuchs.
## Fuchsian groups on the upper half-plane
Let H = {z in C : Im(z) > 0} be the upper half-plane. Then H is a model of the hyperbolic plane when endowed with the metric
${\displaystyle ds={\frac {1}{y}}{\sqrt {dx^{2}+dy^{2}}}.}$
The group PSL(2,R) acts on H by linear fractional transformations (also known as Möbius transformations):
${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}\cdot z={\frac {az+b}{cz+d}}.}$
This action is faithful, and in fact PSL(2,R) is isomorphic to the group of all orientation-preserving isometries of H.
A Fuchsian group Γ may be defined to be a subgroup of PSL(2,R), which acts discontinuously on H. That is,
An equivalent definition for Γ to be Fuchsian is that Γ be a discrete group , which means that:
• Every sequence {γn} of elements of Γ converging to the identity in the usual topology of point-wise convergence is eventually constant, i.e. there exists an integer N such that for all n > N, γn = I, where I is the identity matrix.
Although discontinuity and discreteness are equivalent in this case, this is not generally true for the case of an arbitrary group of conformal homeomorphisms acting on the full Riemann sphere (as opposed to H). Indeed, the Fuchsian group PSL(2,Z) is discrete but has accumulation points on the real number line Im z = 0: elements of PSL(2,Z) will carry z = 0 to every rational number, and the rationals Q are dense in R.
## General definition
A linear fractional transformation defined by a matrix from PSL(2,C) will preserve the Riemann sphere P1(C) = C ∪ ∞, but will send the upper-half plane H to some open disk Δ. Conjugating by such a transformation will send a discrete subgroup of PSL(2,R) to a discrete subgroup of PSL(2,C) preserving Δ.
This motivates the following definition of a Fuchsian group. Let Γ ⊂ PSL(2,C) act invariantly on a proper, open disk Δ ⊂ C ∪ ∞, that is, Γ(Δ) = Δ. Then Γ is Fuchsian if and only if any of the following three equivalent properties hold:
1. Γ is a discrete group (with respect to the standard topology on PSL(2,C)).
2. Γ acts properly discontinuously at each point z ∈ Δ.
3. The set Δ is a subset of the region of discontinuity Ω(Γ) of Γ.
That is, any one of these three can serve as a definition of a Fuchsian group, the others following as theorems. The notion of an invariant proper subset Δ is important; the so-called Picard group PSL(2,Z[i]) is discrete but does not preserve any disk in the Riemann sphere. Indeed, even the modular group PSL(2,Z), which is a Fuchsian group, does not act discontinuously on the real number line; it has accumulation points at the rational numbers. Similarly, the idea that Δ is a proper subset of the region of discontinuity is important; when it is not, the subgroup is called a Kleinian group.
It is most usual to take the invariant domain Δ to be either the open unit disk or the upper half-plane.
## Limit sets
Because of the discrete action, the orbit Γz of a point z in the upper half-plane under the action of Γ has no accumulation points in the upper half-plane. There may, however, be limit points on the real axis. Let Λ(Γ) be the limit set of Γ, that is, the set of limit points of Γz for zH. Then Λ(Γ) ⊆ R ∪ ∞. The limit set may be empty, or may contain one or two points, or may contain an infinite number. In the latter case, there are two types:
A Fuchsian group of the first type is a group for which the limit set is the closed real line R ∪ ∞. This happens if the quotient space H/Γ has finite volume, but there are Fuchsian groups of the first kind of infinite covolume.
Otherwise, a Fuchsian group is said to be of the second type. Equivalently, this is a group for which the limit set is a perfect set that is nowhere dense on R ∪ ∞. Since it is nowhere dense, this implies that any limit point is arbitrarily close to an open set that is not in the limit set. In other words, the limit set is a Cantor set.
The type of a Fuchsian group need not be the same as its type when considered as a Kleinian group: in fact, all Fuchsian groups are Kleinian groups of type 2, as their limit sets (as Kleinian groups) are proper subsets of the Riemann sphere, contained in some circle.
## Examples
An example of a Fuchsian group is the modular group, PSL(2,Z). This is the subgroup of PSL(2,R) consisting of linear fractional transformations
${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}\cdot z={\frac {az+b}{cz+d}}}$
where a, b, c, d are integers. The quotient space H/PSL(2,Z) is the moduli space of elliptic curves.
Other Fuchsian groups include the groups Γ(n) for each integer n > 0. Here Γ(n) consists of linear fractional transformations of the above form where the entries of the matrix
${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}$
are congruent to those of the identity matrix modulo n.
A co-compact example is the (ordinary, rotational) (2,3,7) triangle group, containing the Fuchsian groups of the Klein quartic and of the Macbeath surface, as well as other Hurwitz groups. More generally, any hyperbolic von Dyck group (the index 2 subgroup of a triangle group, corresponding to orientation-preserving isometries) is a Fuchsian group.
All these are Fuchsian groups of the first kind.
• All hyperbolic and parabolic cyclic subgroups of PSL(2,R) are Fuchsian.
• Any elliptic cyclic subgroup is Fuchsian if and only if it is finite.
• Every abelian Fuchsian group is cyclic.
• No Fuchsian group is isomorphic to Z × Z.
• Let Γ be a non-abelian Fuchsian group. Then the normalizer of Γ in PSL(2,R) is Fuchsian.
## Metric properties
If h is a hyperbolic element, the translation length L of its action in the upper half-plane is related to the trace of h as a 2×2 matrix by the relation
${\displaystyle |\mathrm {tr} \;h|=2\cosh {\frac {L}{2}}.}$
A similar relation holds for the systole of the corresponding Riemann surface, if the Fuchsian group is torsion-free and co-compact.
## Related Research Articles
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
In mathematics, the modular group is the projective special linear group PSL(2, Z) of 2 × 2 matrices with integer coefficients and unit determinant. The matrices A and A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic.
In mathematics, the projective special linear group PSL(2, 7), isomorphic to GL(3, 2), is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements, PSL(2, 7) is the smallest nonabelian simple group after the alternating group A5 with 60 elements, isomorphic to PSL(2, 5).
In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
In mathematics, a discrete subgroup of a topological group G is a subgroup H such that there is an open cover of G in which every open subset contains exactly one element of H; in other words, the subspace topology of H in G is the discrete topology. For example, the integers, Z, form a discrete subgroup of the reals, R, but the rational numbers, Q, do not. A discrete group is a topological group G equipped with the discrete topology.
In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, i.e. a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an asymptotic distribution law similar to the prime number theorem.
In number theory and algebraic geometry, a modular curveY(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curvesX(Γ) which are compactifications obtained by adding finitely many points to this quotient. The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field Q of rational numbers, or a cyclotomic field. The latter fact and its generalizations are of fundamental importance in number theory.
In mathematics, the Selberg trace formula, introduced by Selberg (1956), is an expression for the character of the unitary representation of G on the space L2(G/Γ) of square-integrable functions, where G is a Lie group and Γ a cofinite discrete group. The character is given by the trace of certain functions on G.
In mathematics, a Kleinian group is a discrete subgroup of PSL(2, C). The group PSL(2, C) of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic space H3, and as orientation preserving conformal maps of the open unit ball B3 in R3 to itself. Therefore, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.
In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation. The concept is named after Lazarus Fuchs.
In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space where is a discrete subgroup of PSL(2,C). Here, the subgroup , a Kleinian group, is defined so that it is isomorphic to the fundamental group of the surface N. Many authors use the terms Kleinian group and Kleinian model interchangeably, letting one stand for the other. The concept is named after Felix Klein.
In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering surface given by exactly one of the following:
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping, from M to itself, with rather clearly marked local directions of "expansion" and "contraction". Anosov systems are a special case of Axiom A systems.
In mathematics, the special linear group SL(2,R) or SL2(R) is the group of 2 × 2 real matrices with determinant one:
In mathematics, an automorphic factor is a certain type of analytic function, defined on subgroups of SL(2,R), appearing in the theory of modular forms. The general case, for general groups, is reviewed in the article 'factor of automorphy'.
In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largest possible order, 84(g − 1), of its automorphism group.
In mathematics, the trace field of a linear group is the field generated by the traces of its elements. It is mostly studied for Kleinian and Fuchsian groups, though related objects are used in the theory of lattices in Lie groups, often under the name field of definition.
Arithmetic Fuchsian groups are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. The prototypical example of an arithmetic Fuchsian group is the modular group . They, and the hyperbolic surface associated to their action on the hyperbolic plane often exhibit particularly regular behaviour among Fuchsian groups and hyperbolic surfaces.
In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs for edges. Let πα, πβ, and πγ be the interior angles at the vertices of the triangle. If any of α, β, and γ are greater than zero, then the Schwarz triangle function can be given in terms of hypergeometric functions as:
## References
• Fuchs, Lazarus (1880), "Ueber eine Klasse von Funktionen mehrerer Variablen, welche durch Umkehrung der Integrale von Lösungen der linearen Differentialgleichungen mit rationalen Coeffizienten entstehen", J. Reine Angew. Math., 89: 151–169
• Hershel M. Farkas, Irwin Kra, Theta Constants, Riemann Surfaces and the Modular Group, American Mathematical Society, Providence RI, ISBN 978-0-8218-1392-8 (See section 1.6)
• Henryk Iwaniec, Spectral Methods of Automorphic Forms, Second Edition, (2002) (Volume 53 in Graduate Studies in Mathematics ), America Mathematical Society, Providence, RI ISBN 978-0-8218-3160-1 (See Chapter 2.)
• Svetlana Katok, Fuchsian Groups (1992), University of Chicago Press, Chicago ISBN 978-0-226-42583-2
• David Mumford, Caroline Series, and David Wright, Indra's Pearls: The Vision of Felix Klein , (2002) Cambridge University Press ISBN 978-0-521-35253-6. (Provides an excellent exposition of theory and results, richly illustrated with diagrams.)
• Peter J. Nicholls, The Ergodic Theory of Discrete Groups, (1989) London Mathematical Society Lecture Note Series 143, Cambridge University Press, Cambridge ISBN 978-0-521-37674-7
• Poincaré, Henri (1882), "Théorie des groupes fuchsiens", Acta Mathematica , Springer Netherlands, 1: 1–62, doi:, ISSN 0001-5962, JFM 14.0338.01
• Vinberg, Ernest B. (2001) [1994], "F/f041890", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics , Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
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2021-09-19 08:50:39
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https://itectec.com/superuser/control-updownleftright-as-extra-keys/
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# Linux – Control-Up,Down,Left,Right as extra keys
keyboardkeyboard-layoutlinux
Is there any way of getting the arrow keys to act differently while the control key is pressed. On my system ^-Up and Up generate the same code…
From my /etc/personal-linux-console.map:
# Up
control keycode 103 = F69
# Left
control keycode 105 = F71
...
string F69 = "\033<Cu>"
string F71 = "\033<Cl>"
From my /etc/rc.local:
loadkeys -q /etc/personal-linux-console.map
I just chose these values arbitrarily. Now in the Linux console, Control-Up will send the five characters Escape-<-C-u->, and so on. You tell readline how you want to interpret that in your ~/.inputrc file.
With X, there are various places to tweak things. Some changes you can do with xmodmap. Depending on how you start X, you might be able to just save your xmodmap commands in ~/.Xmodmap or /etc/X11/Xmodmap and have them automatically loaded. Some changes will be too tricky for xmodmap, and you'll need to write XKB definition files (these reside under /usr/share/X11/xkb on my machine). These are very hairy and poorly documented. You should find what few docs and tutorials there are by Googling. In about a year we should see a new generation of XKB deployed, so I don't know how much sense it makes to invest time in learning the old format.
I don't know whether it's possible to do Control-keys with xmodmaps commands. I think it is. I used to have mine in custom XKB files (I needed the XKB files anyway, for some stuff.) Now I have the control keys configured in my X terminal (urxvt)'s config files. I use the same arbitrarily chosen sequence escape-<-C-u-> for control-up, and so on, so that I can use the settings in my .inputrc file (for readline) and for other terminal programs (mutt, elinks, and so on).
For some key redefinitions, it's also useful to write your own terminfo files so that terminfo-aware applications will be more likely to be able to handle/recognize them. I don't do that for my Control-arrow settings though. But for instance, if you have some key defined to output the string Escape-<-S-U->, you may want to use a terminfo file that declares that string as being the "Undo" key. Then in some applications you'll be able to just refer to that key as "Undo", no matter what it says on your physical keyboard.
It's complex. There's no general solution that's simpler. (Though if you were only concerned with a few keys, in a few applications, you may well be able to ignore some of the complexity.)
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2021-09-20 17:38:45
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https://zbmath.org/authors/?q=Christian+P%C3%B6tzsche
|
zbMATH — the first resource for mathematics
Pötzsche, Christian
Compute Distance To:
Author ID: potzsche.christian Published as: Poetzsche, Christian; Pötzsche, C.; Pötzsche, Ch.; Pötzsche, Christian External Links: MGP · Wikidata
Documents Indexed: 85 Publications since 2001, including 6 Books
all top 5
Co-Authors
43 single-authored 8 Rasmussen, Martin 6 Kloeden, Peter Eris 6 Siegmund, Stefan 3 Aulbach, Bernd 3 Pituk, Mihály 3 Russ, Evamaria 3 Skiba, Robert 2 Elaydi, Saber Nasr 2 Garab, Ábel 2 Heuberger, Clemens 2 Kaltenbacher, Barbara 2 Müller, Johannes 2 Rendl, Franz 1 Aarset, Christian 1 Duan, Jinqiao 1 Ey, Kristine 1 Fuchs, Thilo M. 1 Hamaya, Yoshihiro 1 Henkel, A. 1 Hense, Burkhard A. 1 Hüls, Thorsten 1 Keller, Stefan 1 Liz, Eduardo 1 Matsunaga, Hideaki 1 Palmer, Kenneth James 1 Sasu, Adina Luminiţa 1 Utz, Margarete 1 Wirth, Fabian Roger
all top 5
Serials
7 Journal of Difference Equations and Applications 5 Internationale Mathematische Nachrichten 4 Journal of Dynamics and Differential Equations 3 Journal of Mathematical Analysis and Applications 3 Journal of Differential Equations 3 Mathematische Nachrichten 3 Discrete and Continuous Dynamical Systems 3 Discrete and Continuous Dynamical Systems. Series B 2 Journal of Computational and Applied Mathematics 2 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2 Physica D 2 Communications on Pure and Applied Analysis 2 Advances in Difference Equations 2 Lecture Notes in Mathematics 2 Springer Proceedings in Mathematics & Statistics 1 Applicable Analysis 1 Computers & Mathematics with Applications 1 IMA Journal of Numerical Analysis 1 Journal of Mathematical Biology 1 Mathematical Methods in the Applied Sciences 1 Applied Mathematics and Computation 1 Integral Equations and Operator Theory 1 Numerische Mathematik 1 Proceedings of the American Mathematical Society 1 SIAM Journal on Numerical Analysis 1 Zeitschrift für Analysis und ihre Anwendungen 1 Applied Mathematics Letters 1 Dynamic Systems and Applications 1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 1 Topological Methods in Nonlinear Analysis 1 Electronic Journal of Differential Equations (EJDE) 1 Functional Differential Equations 1 Differential Equations and Dynamical Systems 1 Positivity 1 Electronic Journal of Qualitative Theory of Differential Equations 1 Nonlinear Analysis. Real World Applications 1 Dynamical Systems 1 Stochastics and Dynamics 1 SIAM Journal on Applied Dynamical Systems 1 Discrete and Continuous Dynamical Systems. Series S 1 IFIP Advances in Information and Communication Technology 1 Journal of Theoretical Biology
all top 5
Fields
55 Dynamical systems and ergodic theory (37-XX) 53 Difference and functional equations (39-XX) 32 Ordinary differential equations (34-XX) 13 Biology and other natural sciences (92-XX) 12 Numerical analysis (65-XX) 11 Operator theory (47-XX) 10 General and overarching topics; collections (00-XX) 5 Systems theory; control (93-XX) 3 Partial differential equations (35-XX) 3 Integral equations (45-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Real functions (26-XX) 1 Measure and integration (28-XX) 1 Functional analysis (46-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Probability theory and stochastic processes (60-XX) 1 Mechanics of particles and systems (70-XX) 1 Fluid mechanics (76-XX) 1 Geophysics (86-XX)
Citations contained in zbMATH
57 Publications have been cited 394 times in 274 Documents Cited by Year
A spectral characterization of exponential stability for linear time-invariant systems on time scales. Zbl 1054.34086
Pötzsche, Christian; Siegmund, Stefan; Wirth, Fabian
2003
Geometric theory of discrete nonautonomous dynamical systems. Zbl 1247.37003
Pötzsche, Christian
2010
Chain rule and invariance principle on measure chains. Zbl 1011.34045
Pötzsche, Christian
2002
Discrete-time nonautonomous dynamical systems. Zbl 1310.37011
Kloeden, P. E.; Pötzsche, C.; Rasmussen, M.
2013
Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients. Zbl 1046.34076
Pötzsche, Christian
2004
Nonautonomous bifurcation of bounded solutions. I: a Lyapunov-Schmidt approach. Zbl 1215.37017
Pötzsche, Christian
2010
Fine structure of the dichotomy spectrum. Zbl 06087704
Pötzsche, Christian
2012
Limitations of pullback attractors for processes. Zbl 1241.39004
Kloeden, Peter E.; Pötzsche, Christian; Rasmussen, Martin
2012
Topological decoupling, linearization and perturbation on inhomogeneous time scales. Zbl 1190.34120
Pötzsche, Christian
2008
Slow fiber bundles of dynamic equations on measure chains. Zbl 1198.37024
Pötzsche, Christian
2002
Exponential dichotomies for dynamic equations on measure chains. Zbl 1042.34510
Pötzsche, Christian
2001
Dichotomy spectra of triangular equations. Zbl 1364.39016
Pötzsche, Christian
2016
Taylor approximation of integral manifolds. Zbl 1106.34029
Pötzsche, Christian; Rasmussen, Martin
2006
A smoothness theorem for invariant fiber bundles. Zbl 1037.37013
Aulbach, Bernd; Pötzsche, Christian; Siegmund, Stefan
2002
Nonautonomous dynamical systems in the life sciences. Zbl 1311.37075
Kloeden, Peter E.; Pötzsche, Christian
2013
Asymptotic behavior of recursions via fixed point theory. Zbl 1138.39007
Ey, Kristine; Pötzsche, Christian
2008
$$C^m$$-smoothness of invariant fiber bundles. Zbl 1075.39014
Pötzsche, Christian; Siegmund, Stefan
2004
Bet-hedging in stochastically switching environments. Zbl 1411.92219
Müller, Johannes; Hense, Burkhard A.; Fuchs, Thilo M.; Utz, M.; Pötzsche, Christian
2013
Computation of nonautonomous invariant and inertial manifolds. Zbl 1169.65116
Pötzsche, Christian; Rasmussen, Martin
2009
Taylor approximation of invariant fiber bundles for nonautonomous difference equations. Zbl 1070.39023
Pötzsche, Christian; Rasmussen, Martin
2005
Nonautonomous continuation of bounded solutions. Zbl 1237.39005
Pötzsche, Christian
2011
Nonautonomous bifurcation of bounded solutions. II: A shovel-bifurcation pattern. Zbl 1258.37029
Pötzsche, Christian
2011
A note on the dichotomy spectrum. Zbl 1180.39030
Pötzsche, Christian
2009
Reducibility of linear dynamic equations on measure chains. Zbl 1032.39008
Aulbach, Bernd; Pötzsche, Christian
2002
PBC-based pulse stabilization of periodic orbits. Zbl 1286.39008
Liz, Eduardo; Pötzsche, Christian
2014
Computation of integral manifolds for Carathéodory differential equations. Zbl 1201.65218
Pötzsche, Christian; Rasmussen, Martin
2010
Continuity and invariance of the Sacker-Sell spectrum. Zbl 1366.39013
Pötzsche, Christian; Russ, Evamaria
2016
Qualitative analysis of a nonautonomous Beverton-Holt Ricker model. Zbl 1326.37017
Hüls, Thorsten; Pötzsche, Christian
2014
Modeling the spread of Phytophthora. Zbl 1271.92036
Henkel, A.; Müller, J.; Pötzsche, C.
2012
Bifurcations in nonautonomous dynamical systems: results and tools in discrete time. Zbl 1242.39026
Pötzsche, Christian
2011
Nonautonomous bifurcation scenarios in SIR models. Zbl 1357.37090
Kloeden, P. E.; Pötzsche, C.
2015
Bifurcations in a periodic discrete-time environment. Zbl 1278.39023
Pötzsche, Christian
2013
Dynamics of modified predator-prey models. Zbl 1202.34086
Kloeden, P. E.; Pötzsche, C.
2010
A functional-analytical approach to the asymptotics of recursions. Zbl 1180.39019
Pötzsche, Christian
2009
Invariant foliations and stability in critical cases. Zbl 1139.39034
Pötzsche, Christian
2006
Integral manifolds under explicit variable time-step discretization. Zbl 1104.34037
Keller, Stefan; Pötzsche, Christian
2006
$$\mathcal{C}^m$$-smoothness of invariant fiber bundles for dynamic equations on measure chains. Zbl 1086.37016
Pötzsche, Christian; Siegmund, Stefan
2004
Slow and fast variables in non-autonomous difference equations. Zbl 1044.39011
Pötzsche, Christian
2003
Morse decompositions for delay-difference equations. Zbl 1414.39002
Garab, Ábel; Pötzsche, Christian
2019
Order-preserving nonautonomous discrete dynamics: attractors and entire solutions. Zbl 1327.39004
Pötzsche, Christian
2015
Persistence and imperfection of nonautonomous bifurcation patterns. Zbl 1218.37035
Pötzsche, Christian
2011
Delay equations on measure chains: basics and linearized stability. Zbl 1117.39324
Pötzsche, Christian
2005
Stability of center fiber bundles for nonautonomous difference equations. Zbl 1074.37018
Pötzsche, Christian
2004
Pseudo-stable and pseudo-unstable fiber bundles for dynamic equations on measure chains. Zbl 1046.39011
Pötzsche, Christian
2003
Numerical dynamics of integrodifference equations: basics and discretization errors in a $$C^0$$-setting. Zbl 1429.65322
Pötzsche, Christian
2019
Smooth roughness of exponential dichotomies, revisited. Zbl 1334.39021
Pötzsche, Christian
2015
Book review of: K. Nipp and D. Stoffer, Invariant manifolds in discrete and continuous dynamical systems. Zbl 1335.00072
Pötzsche, C.
2014
Nonautonomous dynamical systems in the life sciences. Zbl 1282.37004
Kloeden, Peter E. (ed.); Pötzsche, Christian (ed.)
2013
Corrigendum on: “A note on the dichotomy spectrum”. Zbl 1257.39016
Pötzsche, Christian
2012
Nonautonomous bifurcation of bounded solutions: crossing curve situations. Zbl 1284.37024
Pötzsche, Christian
2012
Extended hierarchies of invariant fiber bundles for dynamic equations on measure chains. Zbl 1213.37048
Pötzsche, Christian
2010
Robustness of hyperbolic solutions under parametric perturbations. Zbl 1207.39017
Pötzsche, Christian
2009
Discrete inertial manifolds. Zbl 1146.39030
Pötzsche, Christian
2008
Attractive invariant fiber bundles. Zbl 1127.37023
Pötzsche, Christian
2007
Local approximation of invariant fiber bundles: an algorithmic approach. Zbl 1094.65128
Pötzsche, Christian; Rasmussen, Martin
2005
On periodic dynamic equations on measure chains. Zbl 1089.34035
Pötzsche, Christian
2004
Invariant manifolds with asymptotic phase for nonautonomous difference equations. Zbl 1067.39031
Aulbach, B.; Pötzsche, C.
2003
Morse decompositions for delay-difference equations. Zbl 1414.39002
Garab, Ábel; Pötzsche, Christian
2019
Numerical dynamics of integrodifference equations: basics and discretization errors in a $$C^0$$-setting. Zbl 1429.65322
Pötzsche, Christian
2019
Dichotomy spectra of triangular equations. Zbl 1364.39016
Pötzsche, Christian
2016
Continuity and invariance of the Sacker-Sell spectrum. Zbl 1366.39013
Pötzsche, Christian; Russ, Evamaria
2016
Nonautonomous bifurcation scenarios in SIR models. Zbl 1357.37090
Kloeden, P. E.; Pötzsche, C.
2015
Order-preserving nonautonomous discrete dynamics: attractors and entire solutions. Zbl 1327.39004
Pötzsche, Christian
2015
Smooth roughness of exponential dichotomies, revisited. Zbl 1334.39021
Pötzsche, Christian
2015
PBC-based pulse stabilization of periodic orbits. Zbl 1286.39008
Liz, Eduardo; Pötzsche, Christian
2014
Qualitative analysis of a nonautonomous Beverton-Holt Ricker model. Zbl 1326.37017
Hüls, Thorsten; Pötzsche, Christian
2014
Book review of: K. Nipp and D. Stoffer, Invariant manifolds in discrete and continuous dynamical systems. Zbl 1335.00072
Pötzsche, C.
2014
Discrete-time nonautonomous dynamical systems. Zbl 1310.37011
Kloeden, P. E.; Pötzsche, C.; Rasmussen, M.
2013
Nonautonomous dynamical systems in the life sciences. Zbl 1311.37075
Kloeden, Peter E.; Pötzsche, Christian
2013
Bet-hedging in stochastically switching environments. Zbl 1411.92219
Müller, Johannes; Hense, Burkhard A.; Fuchs, Thilo M.; Utz, M.; Pötzsche, Christian
2013
Bifurcations in a periodic discrete-time environment. Zbl 1278.39023
Pötzsche, Christian
2013
Nonautonomous dynamical systems in the life sciences. Zbl 1282.37004
Kloeden, Peter E. (ed.); Pötzsche, Christian (ed.)
2013
Fine structure of the dichotomy spectrum. Zbl 06087704
Pötzsche, Christian
2012
Limitations of pullback attractors for processes. Zbl 1241.39004
Kloeden, Peter E.; Pötzsche, Christian; Rasmussen, Martin
2012
Modeling the spread of Phytophthora. Zbl 1271.92036
Henkel, A.; Müller, J.; Pötzsche, C.
2012
Corrigendum on: “A note on the dichotomy spectrum”. Zbl 1257.39016
Pötzsche, Christian
2012
Nonautonomous bifurcation of bounded solutions: crossing curve situations. Zbl 1284.37024
Pötzsche, Christian
2012
Nonautonomous continuation of bounded solutions. Zbl 1237.39005
Pötzsche, Christian
2011
Nonautonomous bifurcation of bounded solutions. II: A shovel-bifurcation pattern. Zbl 1258.37029
Pötzsche, Christian
2011
Bifurcations in nonautonomous dynamical systems: results and tools in discrete time. Zbl 1242.39026
Pötzsche, Christian
2011
Persistence and imperfection of nonautonomous bifurcation patterns. Zbl 1218.37035
Pötzsche, Christian
2011
Geometric theory of discrete nonautonomous dynamical systems. Zbl 1247.37003
Pötzsche, Christian
2010
Nonautonomous bifurcation of bounded solutions. I: a Lyapunov-Schmidt approach. Zbl 1215.37017
Pötzsche, Christian
2010
Computation of integral manifolds for Carathéodory differential equations. Zbl 1201.65218
Pötzsche, Christian; Rasmussen, Martin
2010
Dynamics of modified predator-prey models. Zbl 1202.34086
Kloeden, P. E.; Pötzsche, C.
2010
Extended hierarchies of invariant fiber bundles for dynamic equations on measure chains. Zbl 1213.37048
Pötzsche, Christian
2010
Computation of nonautonomous invariant and inertial manifolds. Zbl 1169.65116
Pötzsche, Christian; Rasmussen, Martin
2009
A note on the dichotomy spectrum. Zbl 1180.39030
Pötzsche, Christian
2009
A functional-analytical approach to the asymptotics of recursions. Zbl 1180.39019
Pötzsche, Christian
2009
Robustness of hyperbolic solutions under parametric perturbations. Zbl 1207.39017
Pötzsche, Christian
2009
Topological decoupling, linearization and perturbation on inhomogeneous time scales. Zbl 1190.34120
Pötzsche, Christian
2008
Asymptotic behavior of recursions via fixed point theory. Zbl 1138.39007
Ey, Kristine; Pötzsche, Christian
2008
Discrete inertial manifolds. Zbl 1146.39030
Pötzsche, Christian
2008
Attractive invariant fiber bundles. Zbl 1127.37023
Pötzsche, Christian
2007
Taylor approximation of integral manifolds. Zbl 1106.34029
Pötzsche, Christian; Rasmussen, Martin
2006
Invariant foliations and stability in critical cases. Zbl 1139.39034
Pötzsche, Christian
2006
Integral manifolds under explicit variable time-step discretization. Zbl 1104.34037
Keller, Stefan; Pötzsche, Christian
2006
Taylor approximation of invariant fiber bundles for nonautonomous difference equations. Zbl 1070.39023
Pötzsche, Christian; Rasmussen, Martin
2005
Delay equations on measure chains: basics and linearized stability. Zbl 1117.39324
Pötzsche, Christian
2005
Local approximation of invariant fiber bundles: an algorithmic approach. Zbl 1094.65128
Pötzsche, Christian; Rasmussen, Martin
2005
Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients. Zbl 1046.34076
Pötzsche, Christian
2004
$$C^m$$-smoothness of invariant fiber bundles. Zbl 1075.39014
Pötzsche, Christian; Siegmund, Stefan
2004
$$\mathcal{C}^m$$-smoothness of invariant fiber bundles for dynamic equations on measure chains. Zbl 1086.37016
Pötzsche, Christian; Siegmund, Stefan
2004
Stability of center fiber bundles for nonautonomous difference equations. Zbl 1074.37018
Pötzsche, Christian
2004
On periodic dynamic equations on measure chains. Zbl 1089.34035
Pötzsche, Christian
2004
A spectral characterization of exponential stability for linear time-invariant systems on time scales. Zbl 1054.34086
Pötzsche, Christian; Siegmund, Stefan; Wirth, Fabian
2003
Slow and fast variables in non-autonomous difference equations. Zbl 1044.39011
Pötzsche, Christian
2003
Pseudo-stable and pseudo-unstable fiber bundles for dynamic equations on measure chains. Zbl 1046.39011
Pötzsche, Christian
2003
Invariant manifolds with asymptotic phase for nonautonomous difference equations. Zbl 1067.39031
Aulbach, B.; Pötzsche, C.
2003
Chain rule and invariance principle on measure chains. Zbl 1011.34045
Pötzsche, Christian
2002
Slow fiber bundles of dynamic equations on measure chains. Zbl 1198.37024
Pötzsche, Christian
2002
A smoothness theorem for invariant fiber bundles. Zbl 1037.37013
Aulbach, Bernd; Pötzsche, Christian; Siegmund, Stefan
2002
Reducibility of linear dynamic equations on measure chains. Zbl 1032.39008
Aulbach, Bernd; Pötzsche, Christian
2002
Exponential dichotomies for dynamic equations on measure chains. Zbl 1042.34510
Pötzsche, Christian
2001
all top 5
Cited by 368 Authors
32 Pötzsche, Christian 14 Kloeden, Peter Eris 8 Davis, John M. 8 Rasmussen, Martin 8 Xia, Yonghui 7 Caraballo Garrido, Tomás 7 Defoort, Michael 7 Djemai, Mohamed 6 Braverman, Elena 6 Gravagne, Ian A. 6 Hüls, Thorsten 6 Langa, Jose’ Antonio 6 Sasu, Adina Luminiţa 6 Sasu, Bogdan 5 DaCunha, Jeffrey J. 5 Koo, Namjip 5 Lorenz, Thomas 5 Nguyen Huu Du 5 Robledo, Gonzalo 5 Shah, Syed Omar 5 Siegmund, Stefan 5 Zada, Akbar 5 Zhang, Jimin 4 Bartosiewicz, Zbigniew 4 Castañeda, Alvaro 4 Choi, Sungkyu 4 Cui, Hongyong 4 Doan, Thai Son 4 Erbe, Lynn Harry 4 Han, Xiaoying 4 Megan, Mihail 4 Nolasco de Carvalho, Alexandre 4 Palmer, Kenneth James 4 Sacker, Robert J. 4 Taousser, Fatima Zohra 3 Babuţia, Mihai Gabriel 3 Bortolan, Matheus Cheque 3 Fan, Meng 3 Garab, Ábel 3 Hamza, Alaa E. 3 Karpuz, Başak 3 Li, Yangrong 3 Liem, Nguyen Chi 3 Liz, Eduardo 3 Obaya, Rafael 3 O’Regan, Donal 3 Peterson, Allan C. 3 Poulsen, Dylan 3 Roberts, Anthony John 3 Rodkina, Alexandra 3 Silva, César M. 3 Stehlík, Petr 3 Steyer, Andrew J. 3 Tisdell, Christopher C. 3 Van Vleck, Erik S. 3 Yang, Meihua 3 Yin, Jinyan 2 Agarwal, Ravi P. 2 Aulbach, Bernd 2 Bai, Yuzhen 2 Battelli, Flaviano 2 Bento, António J. G. 2 Bohner, Martin J. 2 Bouin, Emeric 2 Chang, Xiaoyuan 2 Chekroun, Mickaël D. 2 Cui, Yinhua 2 Do Duc Thuan 2 Girod, Alina 2 Gyori, Istvan 2 Hense, Burkhard A. 2 Hilger, Stefan 2 Horváth, László 2 Jackson, Billy J. 2 Kelly, Cónall 2 Kurbatov, Vitaliĭ Gennad’evich 2 Kurbatova, Irina Vital’evna 2 Kuttler, Christina 2 Li, Ming-Chia 2 Liu, Ping 2 Liu, Xinzhi 2 Longo, Iacopo P. 2 Lupa, Nicolae 2 Lyu, Ming-Jiea 2 Mert, Raziye 2 Müller, Johannes 2 Novo, Sylvia 2 Oganesyan, Gro R. 2 Oraby, Karima M. 2 Petropoulou, Eugenia N. 2 Pinto, Manuel 2 Piotrowska, Ewa 2 Russ, Evamaria 2 Shi, Yuming 2 Skiba, Robert 2 Slavík, Antonín 2 Smith, Hal Leslie 2 Volek, Jonáš 2 Wang, Chao 2 Wang, Yuwen ...and 268 more Authors
all top 5
Cited in 99 Serials
22 Journal of Difference Equations and Applications 19 Journal of Differential Equations 17 Journal of Mathematical Analysis and Applications 14 Discrete and Continuous Dynamical Systems. Series B 9 Journal of Dynamics and Differential Equations 9 Advances in Difference Equations 8 Applied Mathematics and Computation 7 Computers & Mathematics with Applications 7 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 7 Discrete and Continuous Dynamical Systems 6 Nonlinear Analysis. Hybrid Systems 5 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 5 Abstract and Applied Analysis 5 Qualitative Theory of Dynamical Systems 5 Communications on Pure and Applied Analysis 4 Journal of Computational and Applied Mathematics 4 Proceedings of the American Mathematical Society 4 Systems & Control Letters 4 Physica D 3 Journal of Mathematical Biology 3 Chaos, Solitons and Fractals 3 Bulletin des Sciences Mathématiques 3 Differential Equations and Dynamical Systems 2 International Journal of Control 2 Journal of the Franklin Institute 2 Journal of Statistical Physics 2 Mathematical Biosciences 2 Mathematical Methods in the Applied Sciences 2 Nonlinearity 2 Physica A 2 Bulletin of Mathematical Biology 2 Automatica 2 BIT 2 Numerische Mathematik 2 Mathematical and Computer Modelling 2 Discrete Dynamics in Nature and Society 2 Communications in Nonlinear Science and Numerical Simulation 2 Nonlinear Analysis. Real World Applications 2 Dynamical Systems 2 SIAM Journal on Applied Dynamical Systems 2 Mediterranean Journal of Mathematics 2 Journal of Biological Dynamics 2 Journal of Nonlinear Science and Applications 2 Journal of Theoretical Biology 1 Applicable Analysis 1 Archive for Rational Mechanics and Analysis 1 Communications in Mathematical Physics 1 Discrete Applied Mathematics 1 General Relativity and Gravitation 1 Inverse Problems 1 Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV) 1 Journal of Engineering Mathematics 1 Rocky Mountain Journal of Mathematics 1 Mathematics of Computation 1 Acta Mathematica Vietnamica 1 Fuzzy Sets and Systems 1 Integral Equations and Operator Theory 1 Journal of Functional Analysis 1 Mathematische Nachrichten 1 SIAM Journal on Numerical Analysis 1 Note di Matematica 1 Stochastic Analysis and Applications 1 Applied Numerical Mathematics 1 Applied Mathematics Letters 1 MCSS. Mathematics of Control, Signals, and Systems 1 Neural Computation 1 SIAM Journal on Mathematical Analysis 1 SIAM Review 1 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 1 Set-Valued Analysis 1 International Applied Mechanics 1 Combinatorics, Probability and Computing 1 Journal of Mathematical Sciences (New York) 1 Turkish Journal of Mathematics 1 Integral Transforms and Special Functions 1 Complexity 1 Nonlinear Dynamics 1 Positivity 1 Journal of Inequalities and Applications 1 Chaos 1 Communications of the Korean Mathematical Society 1 Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis 1 Computational Methods in Applied Mathematics 1 Journal of Applied Mathematics 1 Acta Mathematica Scientia. Series B. (English Edition) 1 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 1 Stochastics and Dynamics 1 Central European Journal of Mathematics 1 Journal of Function Spaces and Applications 1 Communications in Mathematical Analysis 1 Journal of Physics A: Mathematical and Theoretical 1 European Journal of Pure and Applied Mathematics 1 Discrete and Continuous Dynamical Systems. Series S 1 Kinetic and Related Models 1 Proceedings of the Estonian Academy of Sciences 1 Asian-European Journal of Mathematics 1 International Journal of Differential Equations 1 Advances in Nonlinear Analysis 1 International Journal of Analysis and Applications
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Cited in 37 Fields
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2021-02-27 09:32:31
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https://www.gradesaver.com/textbooks/math/calculus/university-calculus-early-transcendentals-3rd-edition/chapter-3-section-3-9-inverse-trigonometric-functions-exercises-page-181/8
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## University Calculus: Early Transcendentals (3rd Edition)
a) Need to calculate the value of inverse trigonometric function . $cot^{-1}(-1)=\frac{3\pi}{4}$ b) Need to calculate the value of inverse trigonometric function. $cot^{-1}({\sqrt 3})=(\frac{\pi}{6})$ c) Need to calculate the value of inverse trigonometric function. $cot^{-1}(-\frac{1}{\sqrt 3})=\frac{2\pi}{3}$
a) Need to calculate the value of inverse trigonometric function . $cot^{-1}(-1)=\frac{3\pi}{4}$ b) Need to calculate the value of inverse trigonometric function. $cot^{-1}({\sqrt 3})=(\frac{\pi}{6})$ c) Need to calculate the value of inverse trigonometric function. $cot^{-1}(-\frac{1}{\sqrt 3})=\frac{2\pi}{3}$
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2019-11-18 05:37:25
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http://gmatclub.com/forum/parallelism-157453.html?sort_by_oldest=true
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Parallelism
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06 Aug 2013, 08:52
Dear All,
I am not able to understand parallelism at all , i am always confused what i need to parallel can any one help me with basic concept
VP
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06 Aug 2013, 08:56
any-real-estate-professional-will-tell-you-that-155081.html#p1241081
or here
parallelism-confusion-in-concept-155085.html#p1241074
Hope it helps
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12 Aug 2013, 09:37
Expert's post
I believe you are talking about parallelism in lists - here is a quick explanation, hope it helps
In a list each of the items in the list must be in the same form - for instance a list of nouns or a list of verbs conjugated in the same way. You identify the list by the use of conjunctions, especially "and" and "or". when you see a conjunction you have to ask what things are being joined. It is often easiest to start with the thing that immediately follows the conjunction then read the sentence for meaning to determine which other items from the sentence are included in the list. Once you have a complete list then you must ensure that the items in the list are in the same form - if any items are not underlined then all of the other items must match that item.
For instance if the sentence says he chose sneakers and to wear shorts. there is a list that in the case of this uncorrected sentence contains sneakers and "to wear shorts" to fix this sentence both words have to be in the same form so sneakers and shorts. You could say to wear sneakers and shorts and in this case the "to wear" covers both nouns and it becomes a list of things he chose to wear, instead of a list of things he chose.
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Re: Parallelism [#permalink] 12 Aug 2013, 09:37
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2016-07-28 13:24:26
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http://www.goodmath.org/blog/2008/12/27/e-the-unnatural-natural-number-classic-repost/?replytocom=30679
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# e: the Unnatural Natural Number (classic repost)
I’m away on vacation this week, taking my kids to Disney World. Since I’m not likely to have time to write while I’m away, I’m taking the opportunity to re-run an old classic series of posts on numbers, which were first posted in the summer of 2006. These posts are mildly revised.
Anyway. Todays number is e, aka Euler’s constant, aka the natural log base. e is a very odd number, but very fundamental. It shows up constantly, in all sorts of strange places where you wouldn’t expect it.
### What is e?
e is a transcendental irrational number. It’s roughly 2.718281828459045. It’s also the base of the natural logarithm. That means that by definition, if ln(x)=y, then ey=x. Given my highly warped sense of humor, and my love of bad puns (especially bad geekpuns) , I like to call e theunnatural natural number. (It’s natural in the sense that it’s the base of the natural logarithm; but it’s not a natural number according to the usual definition of natural numbers. Hey, I warned you that it was a bad geek pun.)
But that’s not a sufficient answer. We call it the naturallogarithm. Why is that bizarre irrational number just a bit smaller than 2 3/4 natural?
Take the curve y=1/x. The area under the curve from 1 to n is the natural log of n. e is the point on the x axis where the area under the curve from 1 is equal to one:
It’s also what you get if you you add up the reciprocal of the factorials of every natural number: (1/0! + 1/1! + 1/2! + 1/3! + 1/4! + …)
It’s also what you get if you take the limit: limn → ∞ (1 + 1/n)n.
It’s also what you get if you work out this very strange looking series:
$2 + 1/(1+1/(2+2/(3+3/(4+..))))$
It’s also the base of a very strange equation: the derivative of ex is… ex.
And of course, as I mentioned in my post on i, it’s the number that makes the most amazing equation in mathematics work: e=-1.
Why does it come up so often? It’s really deeply fundamental. It’s tied to the fundamental structure of numbers. It really is a deeplynaturalnumber; it’s tied into the shape of a circle, to the basic 1/x curve. There are dozens of different ways of defining it, because it’s so deeply embedded in the structure ofeverything.
Wikipedia even points out that if you put \$1 into a bank account paying 100% interest compounded continually, at the end of the year, you’ll have exactly e dollars. (That’s not too surprising; it’s just another way of stating the integral definition of e, but it’s got a nice intuitiveness to it.)
### History
e has less history to it than the other strange numbers we’ve talked about. It’s a comparatively recent discovery.
The first reference to it indirectly by William Oughtred in the 17th century. Oughtred is the guy who invented the slide rule, which works on logarithmic principles; the moment you start looking an logarithms, you’ll start seeing e. He didn’t actually name it, or even really work out its value; but hedidwrite the first table of the values of the natural logarithm.
Not too much later, it showed up in the work of Leibniz – not too surprising, given that Liebniz was in the process of working out the basics of differential and integral calculus, and e shows up all the time in calculus. But Leibniz didn’t call it e, he called it b.
The first person to really try to calculate a value for e was Bernoulli, who was for some reason obsessed with the limit equation above, and actually calculated it out.
By the time Leibniz’s calculus was published, e was well and truly entrenched, and we haven’t been able to avoid it since.
Why the letter e? We don’t really know. It was first used by Euler, but he didn’t say why he chose that. Probably as an abbreviation for “exponential”.
### Does e have a meaning?
This is a tricky question. Does e mean anything? Or is it just an artifact – a number that’s just a result of the way that numbers work?
That’s more a question for philosophers than mathematicians. But I’m inclined to say that the number e is an artifact; but the natural logarithmis deeply meaningful. The natural logarithm is full of amazing properties – it’s the only logarithm that can be written with a closed form series; it’s got that wonderful interval property with the 1/x curve; it really is a deeply natural thing that expresses very important properties of the basic concepts of numbers. As a logarithm, some number had to be the base; it just happens that it works out to the value e. But it’s the logarithm that’s really meaningful; and you can calculate the logarithm withoutknowing the value of e.
## 36 thoughts on “e: the Unnatural Natural Number (classic repost)”
1. Jonathan Vos Post
I’m puzzled by the suggestion that “the number e is an artifact; but the natural logarithmis deeply meaningful.”
By what metaphysical argument does one have a greater epistemelogical or ontological status?
The extreme version would be to say that e doesn’t exist, except as an apparently random sequence of numerals, while natural logarithms are embedded in the physical structure of space. I know you didn’t say that, but see how silly the exaggeration becomes?
It’s akin to saying “pi is an artefact, but the ratio of circumference of diameter of circles is innate in every Keplerian orbit.”
Whoops — that brings up the question of to what extent e or pi exist, before we compare exitences. That’s the oldest known metaphysical debate in Mathematics, or at least on ontological status of mathematical objects. One one side, Platonists (Transcendent Realists) who hold that 8 and pi and e and triangles and aleph-null exist, if anything, MORE than humans do.
“O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible, — what would they answer?”
–Plato, “The Republic” [Jowell translation], Chapter 7.
On the second side, the Logicists, following Gottlieb Frege, who hold that mathematics can be known a priori, but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thus analytic, not requiring any special faculty of mathematical intuition. In this view, logic is the proper foundation
of mathematics, and all mathematical statements are necessary logical truths. Cf. Rudolf Carnap (1931).
On the third side, the Formalists, such as David Hilbert, Rudolf Carnap, Alfred Tarski and Haskell Curry, who hold that mathematical statements are equivalent to statements about the consequences of certain string manipulation rules. Some some formalists now propose
that all formal mathematical knowledge should be systematically encoded in computer-readable formats (Cf. QED project).
On the forth side, the Intuitionists whose motto is: “there are no non-experienced mathematical truths” [L.E.J. Brouwer].
Cf. Arend Heyting.
“God created the integers; all the rest is the work of Man.”
[Leopold Kronecker, tr. from German].
On the fifth side, Constructivists who hold that only mathematical objects which can be finitely and explicitly constructed in a specific sense properly belong to mathematical discourse.
On the sixth side, Fictionalists, such as Hartry Field published “Science Without Numbers” (1980), rejecting or reversing Quine’s argument on indispensability.
On the seventh side, Embodied Mind Theorists who hold that mathematical thought is a natural outgrowth of the evolved human cognitive machinery embedded in our physical universe; hence Math is not universal and does not really exist, except in human brains, which construct (not discover) mathematical objects, in efficacious ways that benefit Darwinian fitness.
Cf. “Where Mathematics Comes From” — George Lakoff and Rafael E. Núñez; Keith Devlin’s “Math Instinct.”
On the eighth side, Social Constructivists (Social Realists) such as Imre Lakatos, Thomas Tymoczko, Reuben Hersh, Philip J. Davis, and Paul Ernest, who hold that mathematics is a social/cultural construct, akin to English Common Law, or a Black Queen in Chess. Mathematical objects come from an empirical endeavor dictated by the fashions of the social group performing it, and/or by the needs of the society or power elite financing it, and are ultimately a political struggle of mathematicians seeking sex, money, or power.
Or should I classify Lakatos as a Quasi-Empiricist along with Popper and Hilary Putnam?
There are also Linguistic theorists and Aesthetic theorists of the ontology of Mathematics.
And I’m probably missing whole schools of thought.
Bottom line: we oversimplify this ancient, subtle, and
multipolar debate at peril to our scholarship, if not our right to exist and argue about “e” — which, by the way, is as barely transcedndetal as any transcendental number can be, which hardly sounds artefactual to me.
2. speedwell
I’m out of my depth here, but I did have a wild thought based on the last paragraphs of this essay: What would happen if the value of e was something else? Would something consistent result? Is it even possible?
3. speedwell
JVP, that’s all very entertaining and you’re a towering and impenetrable genius, but I think he means that “3.1416…” is the artifact, as any other representation would be, including the word “pi” or the Greek symbol we use, but the fact that the diameter and circumference of a circle have a certain relationship is the deep reality. In just the same way, we can call it “e” or “about 2.718281828459045” or “macaroni and cheese,” but the thing itself is what it is. You might have a photograph of your wife in your wallet and a painting of her in your office, but the deeper reality of your relationship with her is what makes her your wife.
4. Morgan
Speedwell, but that’s just making the point that the representation of a number, the string of digits it’s written out as, is somewhat arbitrary depending on base and writing system etc.
The value is a different thing and is not arbitrary. e is greater than two and less than three, and greater than two plus one-half but less than three, and… whatever the base.
5. Jonathan Vos Post
Morgan is correct in correcting speedwell.
I am, in fact, a rather arrogant self-centered absent-minded professor. Fortunately, I am kept in my place by my family. As to “you’re a towering and impenetrable genius” — my son is more towering, at above 6′ 3″ in height at age nineteen, and halfway through law school, having started university at age thirteen. My wife, a Physics professor, since you mention her, is also smarter than I, and that we have an even smarter son is proof at least that she is not impenetrable.
The point is that I remain puzzled by Dr. Mark Chu-Carroll’s feeling or opinion (not a claim to fact) that: “number e is an artifact; but the natural logarithm is deeply meaningful.”
Now, one might not make sense if one asked a question analogous to speedwell’s: “What would happen if the value of 2 was something else? Would something consistent result? Is it even possible?”
So the analogy hinges on whether e is intrinsic, and not contingent on any specific physical universe, as Platonists assert 2 must be intrinsic. Or, on the other hand, that the exact value (not representation in any given base or nomenclature) of e is somehow an accident.”
The ratio of circumference to diameter of a physical circle is not exactly pi, but a number contingent on the local curvature of space-time.
Philosophers of Mathematics differ as to whether there are or are not any “accidents” in Mathematics.
Many think not. Gregory J. Chaitin, however, claims that God plays dice not only in quantum mechanics, but even in the foundations of mathematics, where Chaitin discovered mathematical facts that are “true for no reason, that are true by accident.” [G. J. Chaitin, Conversations with a Mathematician: Math, Art, Science and the Limits of Reason, Springer-Verlag London, 2002, viii + 158 pages, hardcover, ISBN 1-85233-549-1].
I suspect that Chaitin’s right, but that doesn’t automatically mean that Mark is right that e is an accident. It can’t be an accident if pi is not an accident, because of the supremely beautiful Euler’s identity:
e^(i pi) + 1 = 0
6. speedwell
Ok, Morgan, I understand what you’re saying. Thanks for reducing my question to a simpler one that pointed the way to why it wasn’t a legitimate one.
JVP, you may not have appreciated my teasing (meant mostly in fun), but you often give me the impression of being somewhat bombastic. Your dismissive attitude (you’re a teacher? Really?) doesn’t do much to dispel this impression. The fact is that if I know enough to ask such a question, I know enough to understand the answer that you think you’re above having to answer. I suppose I’m to be quelled and abashed by your idle observation that the question doesn’t make sense?
7. eddie
Re: various issues but mainly speedwell’s question –
In analogy to pi, and as jvp pointed out, the numerical value of the ‘constant’ is only in euclidean (flat) space. Now, given we know this to be a special case, we resort to basic definitions, independent of local circumstances. For pi this is “the ratio of circumference to radius”, while for e it would be “where the curve is equal to it’s gradient” which also changes with local curvature.
I’m asking questions such as – “at what values of curvature are pi, e rational?” and “do they still satisfo euler’s identity?”
8. Pingu the Missing Integer
Bit of a cock-up by you maths guys, why didn’t you make pi and e both 3 when you were designing the number systems?
Tsk, tsk, is it too late for a re-design?
And no jvp, because you’re clearly a bit lacking in the brain department, I am not being serious, it’s a far more subtle point that you won’t understand, so please don’t bother with your “I’m the worst teacher of your nightmares” bleat!
9. marko
I for one are thankful to Jonathan Vos Post for writing Euler’s Identity in its real beauty, including 1 and 0:
e i π + 1 = 0
What makes it profound in my view isn’t the numerical value of π or e, nor the question if there ist any deep epistemic meaning hidden in any one of those constants; it’s the elegance in the relationships expressed in this very identity.
10. Jonathan Vos Post
“The existence of a coincidence is strong evidence for the existence of a covering theory”
“Are There Coincidences in Mathematics?”
Philip Davis
The American Mathematical Monthly, vol. 88, 1981, pp. 311-320.
11. Brian Jaress
Mark, just so you know, some of your emphasized words are coming through without spaces. For example, “We call it thenaturallogarithm” in the fourth paragraph.
Jonathan Vos Post, that was a very interesting run-down of schools of thought. I guess I’m a Formalist/Logicist who recognizes a “filtering” role for the other schools.
12. Jackson
worth noting that e shows up in analyzing how things change in a proportional manner (i.e. percentage changes); also that the Euler formula is the formula
e^(i theta) = cos(theta) + i sin(theta) when theta = pi. Maybe he hit this formula in another entry.
13. Anonymous
My favorite equation where e seems to appear mysteriously (deeper consideration makes it less obvious) is x^x. It has an absolute minima at 1/e. I actually figured out first that x^-x has an absolute maxima at 1/e back in HS calculus (too much fiddling with a graphical calc can be enlightening).
14. JoeB
Why don’t you mathematical towering intellects admit it, you’re all in that driven-by-power-sex-money group, right?
We mere lurkers, of course, are pure of heart.
15. Thony C.
The first person to really try to calculate a value for e was Bernoulli
Which one? There are something like twelve high flying mathematical Bernoullis in the first four generations. Hermann Hesse’s wife was a mathematical Bernoulli and there are still Bernoullis occupying chairs for mathematics in Switzerland.
JVP, lovely rant!
16. Chris
Some blog I’m reading (and I’m hoping it’s not this one or I’ll sound like a total idiot) has spent a fair amount of time defining power series and their derivatives, all to be able to solve a few simple differential equations.
x’ – x = 0
x” + x = 0
The first gives you exp(x), the second gives you sin(x) and cos(x). You can also get Euler’s Identity from them if you let x be complex.
The author’s point is that e comes from the first equation. There’s nothing odd about it, it’s just an identity value in some deep way like 0 and 1 are for addition and multiplication.
‘pi’ comes from the second equations. Not in relation to circles in Euclidean space, but solely as the smallest positive value where the sin(x) value crosses zero.
These values pop up in a lot of other places, of course, but you can define them entirely in terms of two simple differential equations. That’s pretty staggering when you think about it. It makes me wonder about the solutions to other simple differential equations and where those values might pop up. x” – x = 0 is still exp(x), as are all higher derivatives.
Aside: another blog was pointing out how a few simple invariants under spatial translation give you special relativity. Not just that, those invariants require that any massless particle travel at a ‘c’ defined by that transformation. We just happen to call ‘c’ ‘the speed of light’ for historical reasons, but you can argue that it really goes the other way.
17. Peter
Does e have a meaning?
This is a tricky question. Does e mean anything? Or is it just an artifact – a number that’s just a result of the way that numbers work?
That’s more a question for philosophers than mathematicians. But I’m inclined to say that the number e is an artifact; but the natural logarithm is deeply meaningful. The natural logarithm is full of amazing properties.
There can be no meaning to either the number e, or the natural logarithm. They are both artifacts of our minds that were created from a history of random chances. To look for meaning is to completely mis-understand the advances in science that has been made since Darwin’s brilliant theory of natural selection. Random chance has been responsible for the creation of every living creature, and our minds are no more relevant than that of the tiny cockroach. The great intellectuals like Denton have discovered that the universe has created itself out of nothing. Everything afterward is the product of random chance, so any mathematical system can not represent a deeper meaning. To do so would imply a deeper purpose, and we don’t want to invoke a higher intelligence like those IDiots.
18. Max
I had a month ago been pondering the nature of e myself. Rather the nature of the curve of e^x to be more precise. Even though I had been exposed to the concept in calculus it had never really struck me how weird it was. I am rather tickled to see that it will not become blase’ as I learn more.
I must also confess that I posted in part because it bothered me to think that the last response of the year to this post would be from a creationist troll.
19. Peter
Max said:
I must also confess that I posted in part because it bothered me to think that the last response of the year to this post would be from a creationist troll.
I take offense to your name calling. Could you please end the new year with an argument. I believe everything I said was easily found in many evolutionary texts.
20. Jonathan Vos Post
Sorry, Peter. I agree with Max that you’re a Creationist troll. To me, you are pathetic. My impovershed ghetto teenaged students understood Evolution by Natural Selection better than you do.
You are stomping on a straw man, and think that you are revealing profound errors in the neodarwinian synthesis. You are merely exposing the depths of you ignorance and stupidity.
“Random chance has been responsible for the creation of every living creature”
No, you poor fool. The VARIATION of individual organisms from their parents has a random component. Actually, several random components, as there are different probability distributions for different types of gene and chromosome mutations.
But NATURAL SELECTION is not random at all.
You miss the point. You show us all that you miss the point. Are you smarter than an 8th grader who has been systematically deprived of a decent education?
No. You are far, far worse off than my students. Most of my students got it. You don’t.
Try making one New Year’s resolution:
“I, Peter, promise to try reading an actual good Evolution textbook, with an open mind, and ask questions of experts instead of spitting in their faces, in the hope that maybe the brainwashing that I have received from liars and frauds may be somewhat mitigated.”
I may be wasting my time. You can’t be cured until you admit that you are sick.
My students, at least, were willing to learn. And a third to a half of them had Creationist members of their families. Can you improve your state? Can you light a candle, or will you trudge through your weary life cursing the darkness, and blogging: “behold! I’ve lit a candle that only I can see!”
21. Anonymous
Hi Mark,
Eli Maor [Loyola, Chicago], ‘”e”: The Story of a Number’ now available in paperback, includes a discussion of Napier developing logarithms and almost, but just missed devloping e.
Maor also wrote ‘To Infinity and Beyond’ [and other books on mathematics] before Buzz Lightyear appeared in Toy Story.
From PUP:
“His thesis was on an unusual subject: using mathematical methods to investigate problems in musical acoustics. This reflected his long interest in the relations between science and the arts, and in particular, music. His article, “What is There so Mathematical About Music?” received first award by the National Council of Teachers of Mathematics as the best article on teaching the applications of mathematics.”
22. Peter
JVP:
To me, you are pathetic. My impovershed ghetto teenaged students understood Evolution by Natural Selection better than you do.
Again with the insults. Don’t you know that resorting to insults is the weakest form of argument. You would know that if you were half as smart as you think you are. Insults are not persuasive. In fact, they only persuade that the insulter’s position is indefensible.
The VARIATION of individual organisms from their parents has a random component. Actually, several random components, as there are different probability distributions for different types of gene and chromosome mutations.
Well at least this is some form of argument, but this is mostly double-talk. It’s random, but it’s not really random. Is that so? It is really convenient to have it both ways isn’t it? Evolution is continuous until the evidence shows that it is not and then it is punctuated evolution. It is really convenient to be all things and change the definition when it suits you.
You do mean ask, but not question. That is your form of education. I am here to learn. Give me your best argument. I can follow the evidence no matter were it leads. Can you? I saw “Expelled, no intelligence allowed.” I had to see for myself if its claims were true. Well JVP you have shown the the caricature of the doctrinaire teacher is true. No questioning is allowed in your classes.
I may be wasting my time.
You wouldn’t be if you made a strong argument, but you resort to name calling. I however have not been wasting my time. I have learned that you have nothing to offer to support the theory of evolution.
This has been very informative JVP, but only of the vacuity of your opinions.
23. Mark C. Chu-Carroll
Peter:
While JvP can be a bit of a bombastic ass at times, I’m in complete agreement with him that you’re nothing but a troll. Your comments:
(A) Have nothing to do with the actual content of the post. They’re a non-sequitur.
(B) They’re pointless regurgitations of misrepresentations
of what scientists actually say about various topics.
And I challenge to back up your statement that what you said can easily be found in evolution textbooks. Name a single evolution textbook that says “The universe created itself out of nothing”. Name a single evolution textbook that in any way even suggests that evolution says anything about whether or not ideas have meaning.
If you want to claim that you’re not just a troll, then go ahead and cite one single evolution textbook that does either of the above. Just one.
24. Peter
Mark:
Good to hear from you. I come to this sight because, as far as I know, it has the highest caliber arguments, with the most educated people. I hope I am not detracting you from your holidays. I am on holidays too but I am starting a class in evolution next week and am doing some research here.
I couldn’t find a good definition of a troll in your meaning. If it means someone who is only interested in causing a commotion that is not me. I am here to learn. I am obviously a skeptic though.
Name a single evolution textbook that says “The universe created itself out of nothing”.
I was relying on my faulty memory when I referred to Denton above. I should have said Daniel C. Dennett, “Breaking the Spell.” He is very well known and undeniably an evolutionist. While this book is not a textbook, it is nevertheless a book about evolution. In this popular book Dennett says:
What does need its origin explained is the concrete Universe itself, and as Hume’s Philo long ago asked: Why not stop at the material world? It we have seen, does perform a version of the the ultimate bootstrapping trick; it creates itself ex nihilo, or at any rate out of something that is well-nigh indistinguishable from nothing at all. (p 244)
I trust this satisfies your requirement. You limited the requirement to textbooks, but that is not the only source of all scholarly work on evolution. Also, while Dennett earned a PhD in philosophy he was given an Honorary Doctor of Science in 2007.
As to the issue of meaning and evolution I will have to get back to you later on that with a quote. My interest lies mainly towards the philosophical area. So when I talk about meaning I am referring to an overarching framework of meaning. So I am not saying that numbers have no meaning at all. They obviously do. What I am saying is that if the universe created itself out of nothing as Dennett says, and life is the produciton of random variation, then whatever meaning we ascribe e or the natural logarithm will eventually be meaningless. While an abstract argument, I believe it is on topic
25. Jonathan Vos Post
Peter the Troll:
Mark is right that I “can be a bit of a bombastic ass at times.” Yet I don’t think you grasp why he and I agree about your illness.
As a teacher, I don’t really care what you believe, as your attempt to show it is utterly riddled with misunderstanding, fallacy, and brainwashing. As a teacher, I don’t teach my students WHAT to think. I teach them HOW to think. And that’s where you fall on your face.
You may be well intentioned, but you wildly misunderstand even what you quote.
“It [the concrete Universe] we have seen, does perform a version of the the ultimate bootstrapping trick; it creates itself ex nihilo, or at any rate out of something that is well-nigh indistinguishable from nothing at all.”
This is NOT about Biological Evolution. Nor is it, strictly, about Cosmic Evolution (same word, different meaning). It is about particular theories of the Big Bang, with which you Creationists should agree because it is consistent with one of the 3 creation myths in Genesis.
How the Cosmos began is not related to Darwin. Nor, for that matter, is Biogenesis (how life began) related to Evolution by Natural Selection.
You say Philosophy, but your rant about Evolution versus Meaning is repackaged (bad) Theology. Instead of addressing the point about whether “e” has meaning, or what numbers are, you are caught up in the era of early Medieval philosophy where Aristotle influenced Jewish and Arabic thought, which trickled down to Christian thought, and then was dumbed down recently for ignorant Americans to Intelligent Design.
“In one stunning passage of the Guide, he proclaims that he [Moses ben Maimon (1138-1204)] would be willing to discard the core religious tradition of an ex-nihilo creation if he felt that the Aristotelian argument for the eternity of the world was incontrovertible.”
The Great Eagle
Maimonides:
The Life and World of One of Civilization’s Greatest Minds
by Joel L. Kraemer
Doubleday. 640 pp. \$35.00
Reviewed by David C. Flatto
January 2009
Now, what do you think that 18th-century mathematician Leopold Kronecker meant by (as usually translated):
“God created the integers, all the rest is the work of man.”
Do you think that God created “e”, or that Man created “e”?
Show your work. Give proper citations to actual textbooks and refereed papers.
26. Peter
JVP said:
How the Cosmos began is not related to Darwin. Nor, for that matter, is Biogenesis (how life began) related to Evolution by Natural Selection.
I am afraid I am going to have to disagree with you. Dennett included a discussion on the origin of the universe in his book on the evolution of religious thought. Not biological evolution, but evolution nevertheless. So I agree with Dennett that the ORIGIN of the cosmos is related to the ORIGIN of the diverse life forms, as well as the ORIGIN of the first life.
“God created the integers, all the rest is the work of man.”
Do you think that God created “e”, or that Man created “e”?
Now that is an interesting question and why I post here. Novel ideas force me to consider and learn material I would never have considered otherwise. I think this may come closer to what Mark was talking about when he talked about philosophical meaning. I know scientist have what they call the beauty principle which says that the most beautiful solution to a problem is usually the correct approach. Some ponder the meaning of the beauty principle. Why would the most elegant solution be more often then not the correct ones. Some have conjectured that the elegance underlying the universe is the product of the mind of the creator.
BTW, this view was around a long time before ID, so don’t attribute it to the ‘creationists.’ There are religious scientists, and some very intelligent ones at that. I won’t give you the list because it is extremely long, would be incomplete, and you must be aware of that anyway.
Getting back to your question, I would have to say, being a Theist, that the (mathematical) order underlying the universe is a result of the necessary condition for its creation. So, the underlying order was created by God and the expression e was created my man to describe this order.
Regarding post #28, the word meaning was never used once in that forum, while it is central to this.
27. Jonathan Vos Post
Peter is typical of Intelligent Design trolls, in how he drags anything he wants into discussion, and falsely claims that it relates to Evolution by Natural Selection:
The Biologic Institute, Bill Dembski, and ID Research in 2008
Category: Intelligent Design
Posted on: January 2, 2009 4:30 PM, by John Lynch
So there you have it. Four very different papers with no apparent connection to the desiderata I [John Lynch] mentioned in my original post: “(a) evidence for design, (b) a method to unambiguously detect design, or (c) a theory of how the Designer did the designing”.
Even though I warned about the irrelevant Dennett quote: “This is NOT about Biological Evolution. Nor is it, strictly, about Cosmic Evolution (same word, different meaning).” we still get Peter trolling away undeterred:
“Dennett included a discussion on the origin of the universe in his book on the evolution of religious thought. Not biological evolution, but evolution nevertheless. So I agree with Dennett that the ORIGIN of the cosmos is related to the ORIGIN of the diverse life forms, as well as the ORIGIN of the first life.”
This time he is fixated on two fallacies:
(1) Any use of the word “origin” is a chance to debunk Darwin;
(2) If both subject A and subject B are discussed in the same book C, then subject A and subject B are so closely linked as to be synonymous.
As to (1) does Peter think that the (0,0) point in the Cartesian Plane has something to do with Biogenesis, or with Evolution by Natural Selection?
This is typical of ignorant people who have been sent out as the Army of God by fraudulent demogogues.
As to Cosmic Evolution, go look at the Tufts University web ages: “Cosmic Evolution: An Interdisciplinary Approach”
Link to Intro Movies: Arrow of Time and Cosmic Origins
When consciousness dawned among the ancestors of our civilization, men and women perceived two things. They noted themselves, and they noted their environment. They wondered who they were and whence they came. They longed for an understanding of the starry points of light in the nighttime sky, of the surrounding plants and animals, of the air, land, and sea. They contemplated their origin and their destiny.
Thousands of years ago, all these basic queries were treated as secondary, for the primary concern seemed well in hand: Earth was presumed to be the stable hub of the Universe. After all, the Sun, Moon, and stars all appear to revolve around our planet. It was natural to conclude, not knowing otherwise, that home and selves were special. This centrality led to a feeling of security or at least contentment–a belief that the origin, maintenance, and fate of the Universe were governed by something more than natural, something supernatural.
The ancients thought deeply and well, but not much more. Logic was paramount; empiricism less so. Their efforts nonetheless produced such notable endeavors as myth, religion, and philosophy.
Eventually, yet only a few hundred years ago, the idea of Earth’s centrality and the reliance on supernatural beings were shattered. During the Renaissance, humans began to inquire more critically about themselves and the Universe. They realized that thinking about Nature was no longer sufficient. Looking at it was also necessary. Experiments became a central part of the process of inquiry. To be effective, ideas had to be tested experimentally, either to refine them if data favored them or to reject them if they did not. The scientific method was born–probably the most powerful technique ever conceived for the advancement of factual information. Modern science had arrived.
Today, all natural scientists throughout the world employ the scientific method. Normally it works like this: First, gather some data by observing an object or event, then propose an idea to explain the data, and finally test the idea by experimenting with Nature. Those ideas that pass the tests are selected, accumulated, and conveyed, while those that don’t are discarded–a little like the evolutionary events described on this Web site. In that way, by means of a selective editing or pruning of ideas, scientists discriminate between sense and nonsense. We gain an ever-better approximation of reality. Not that science claims to reveal the truth–whatever that is–just to gain an increasingly accurate model of Nature.
Peter the Troll writes of his archaic and discredited view: “this view was around a long time before ID, so don’t attribute it to the ‘creationists.'”
Sorry, Peter’s view is a mishmash of fallacies, misunderstandings, and misrepresentations. Saying that these fallacies, misunderstandings, and misrepresentations are old does not make them true, it just makes it more pathetic when they are trotted out, as the Intelligent Desig loons insist on doing.
Peter the clueless Troll writes: “Novel ideas force me to consider and learn material I would never have considered otherwise.”
This in his response to my quote from the 1700s: “God created the integers, all the rest is the work of man.”
If that is “novel” to him, then he is confessing his ignorance.
The ignorance of Good Math and the ignorance of what the Neodarwinian Synthesis explains go hand in hand.
28. Ivan Dobrokotov
I’m interested:
Is it possible to define functions sin, cos (or e^z) and Pi-constant, without refering geometry circle, that periodicity of sin/cos will be evident? Direct algebraic represention of curve length via integration sqrt(1-x^2) is supposed to be cheating 🙂 And gluing of function pieces to make sin/cos to be defined over all reals also is seems ‘unnatural’.
sin/cos x defined as 1/0! – x^2/2! + x^4/4! – x^6/6! does not give nor evident periodicity property, nor π Pi definition.
Are there other ways to define sin/cos/Pi, without circle or tailor series? Or can be Pi/periodicity directly derived from tailor representation?
Sorry for my bad English %)
Ivan Dobrokotov.
29. Anonymous
Well, I thought about this a little more, and I think I can derive from f”(x) + f(x) == 0 periodicity of f(x).
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2021-04-11 14:53:11
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https://space.stackexchange.com/questions/30999/has-a-lander-rover-ever-examined-or-photographed-another-mission
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# Has a lander/rover ever examined or photographed another mission?
Occasionally a lander or rover (manned or unmanned) will land close enough to a previous mission to enable examination or photographs of the previous mission to be taken by the new mission. What are some examples of this?
• May be the moon, Mars, or any other body except Earth.
• Both missions must be on the surface of the body (i.e. orbital photography doesn't count).
• Must be two different missions (e.g. a rover taking a picture of its own lander doesn't count).
• This is a follow-up to Have there been long-term observations of the effects of lunar exposure to equipment?.
• I already know of Apollo 12 examining, photographing, and returning parts of Surveyor 3, so that answer is taken.
• I would appreciate refinements to this question's tags.
• Tags look good, there is always the rendezvous tag, but I'm not sure of the spirit of the tag includes a land rendezvous of a rover with another surface craft. It might apply though and I'd certainly support defining the tag that way, or making a separate surface-rendezvous tag. – uhoh Sep 29 '18 at 7:05
Within the constraints you posed, no. The premise of your first sentence is incorrect, just about all lander missions have landed very far apart.
Here's a map of US Moon missions:
The closest pair after Apollo 12/Surveyor 3 is Apollo 11/Surveyor 5, which are 25 km apart, which is too far (no line of sight, and too far to walk within the time constraints).
Similarly, Mars missions have all landed hundreds to thousands of km apart.
• Apollo 12 / Surveyor 3 is the "occasion" cited in my first sentence. Otherwise, a good answer. – DrSheldon Sep 29 '18 at 0:06
List of artifical objects on Mars and a map:
Map
As we can see, there is no landed missions close enough to each other on Mars.
List of artifical objects on the Moon and a map:
Interesting that Luna-24 landed very close to Luna-23. It's maybe the only case one lander could photograph enother. But there is no information such attempt was made.
All other lunar missions were over 10s km apart from each other.
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2020-08-04 08:19:28
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https://math.stackexchange.com/questions/2667432/show-that-if-a-topological-space-has-at-most-countable-basis-then-the-space-is
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# Show that if a topological space has at most countable basis, then the space is separable and Lindelöf
Let $(X,\mathcal{T})$ be a topological space. Show that if $(X, \mathcal{T})$ has countable base, it is separable (a) and Lindelöf (b)
My attempt:
Let $\mathcal{B}$ be a countable basis of the given topology.
(a) For every $B \in \mathcal{B}\setminus\{\emptyset\}$, choose $x_B \in B$ (here we use AC) and consider $D:= \{x_B \mid B \in \mathcal{B}\}$. It is clear that this is at most countable. Let $x \in X$ and $V \in \mathcal{V(x)}$. Then, there exists $G \in \mathcal{T}$ such that $x \in G \subseteq V$. But, there exists $\mathcal{A} \subseteq \mathcal{B}$ such that $G = \bigcup\mathcal{A}$, since $\mathcal{B}$ is a basis. Take $A \in \mathcal{A}$ with $x \in A \in \mathcal{B}$. Then $A$ is non empty, and hence $x_A \in A \cap D \subseteq V \cap D$, so $x \in \overline{D}$ and hence $D$ is dense in $X$.
(b) Let $\mathcal{G}$ be an open cover of $X$.
For every $G$ in this cover, there exists $\mathcal{A}_G \subseteq \mathcal{B}$ such that $G= \bigcup \mathcal{A_G}$. Consider the collection $\mathcal{C}:= \bigcup_{G \in \mathcal{G}}\mathcal{A}_G \subseteq \mathcal{B}$.
If $C \in \mathcal{C}$, then there is $G_C \in \mathcal{G}$ such that $C \in \mathcal{A}_{G_C}$. Thus, $C \subseteq G_C$. Consider the countable set ($\mathcal{C}$ countable because $\mathcal{B}$ countable)
$$\{G_C \mid C \in \mathcal{C}\}$$
Then:
$$X \supseteq\bigcup_{C \in \mathcal{C}}G_C \supseteq \bigcup_{C \in \mathcal{C}} C = \bigcup_{G \in \mathcal{G}} \bigcup_{H \in \mathcal{A}_G}H = \bigcup_{G \in \mathcal{G}} \bigcup\mathcal{A}_G = \bigcup_{G \in \mathcal{G}}G = \bigcup\mathcal{G} = X$$
Hence, $\{G_C \mid C \in \mathcal{C}\}$ is a countable subcover of $\mathcal{G}$, and hence $X$ is Lindelöf.
Is this correct?
• What's $\mathcal{V(x)}$? – j3M Feb 26 '18 at 13:43
• All the neighborhoods of $x \in X$. – user370967 Feb 26 '18 at 13:43
• For every $G$ you choose $\mathcal{A}_G$. You are using AC. – Henno Brandsma Feb 26 '18 at 18:57
• @HennoBrandsma Do I use AC whenever I consider the sets $G_C$ too? As I pick them and put them in a collection. Sorry for this question, I'm not formally introduced to AC yet, but this will pop up in a later course ("foundations of mathematics"). And for the rest, is my proof correct? – user370967 Feb 26 '18 at 20:08
• it’s a bit messily written but there is a good idea hidden there. – Henno Brandsma Feb 26 '18 at 20:13
The separable part I quite agree and we need countable choice there.
The other part also needs choice, I'll make my use of it explicit:
Let $\{B_n: n \in \mathbb{N}\}$ be a countable base for $X$. Let $\mathcal{O}$ be an open cover of $X$. Define for each $n \in \mathbb{N}$:
$$\mathcal{O}_n = \{O \in \mathcal{O}: B_n \subseteq O\}$$
For a given $n$ this set could be empty, finite or infinite. You cannot say beforehand. But let $N'$ be the set of $n$ for which $\mathcal{O}_n$ is non-empty, and for each $n \in N'$ we pick (yes, AC is used, and I'm not ashamed of it) some $O_n \in \mathcal{O}_n$.
Now I claim that $\{O_n: n \in N'\}$ is a countable (obvious, as $N'$ is countable as a subset of $\mathbb{N}$) subcover of $X$: Let $x \in X$. Then $x$ is covered by some $O_x \in \mathcal{O}$, and because the $B_n$ form a base, we have some $n_x \in \mathbb{N}$ such that $x \in B_{n(x)} \subseteq O_x$. We conclude that $O_x \in \mathcal{O}_{n_x}$ by definition and so as we have $O_{n_x} \in \mathcal{O}_{n_x}$ (the choice we made) and $x \in B_{n_x} \subseteq O_{n_x}$, $x$ is indeed covered by the countable subcover.
Globally it is correct.
You don't need the axiom of choice, though.
• When I pick an element out of every set $B \in \mathcal{B}$, don't I use choice? – user370967 Feb 26 '18 at 13:49
• And why is my $\mathcal{C}$ a subset of $X$? – user370967 Feb 26 '18 at 13:53
• Statement about AC is incorrect (unless you meant strictly uncountable choice). Actually, assuming every second countable space has a separable subset is equivalent to countable AC, see: math.stackexchange.com/questions/309313/… – j3M Feb 26 '18 at 13:53
• @Math_QED But you don't do that. You define $\mathcal C$ as the union of all $\mathcal{A}_G$. Where is a choice being made here? – José Carlos Santos Feb 26 '18 at 13:54
• @Math_QED Please note that Ive edited my answer. – José Carlos Santos Feb 26 '18 at 13:55
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2019-11-15 10:57:37
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https://math.stackexchange.com/questions/2409918/integral-involving-logarithm-and-dilogarithm
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# Integral involving Logarithm and Dilogarithm
I need your help in evaluating the following integral in closed form. $\displaystyle\int\limits_{0.5}^{1} \frac{\mathrm{Li}_{2}\left(x\right)\ln\left(2x - 1\right)}{x}\,\mathrm{d}x$
Since the function is singular at $x = 0.5$, we are looking for Principal Value. The integral is finite and was evaluated numerically. I expect the closed form result to contain $\,\mathrm{Li}_{3}$ and $\,\mathrm{Li}_{2}$.
Thanks
• What have you tried? Any Taylor expansions? Note that $\ln(2x-1)=\ln(1-1/2x)+\ln(x)+\ln(2)$ – Simply Beautiful Art Aug 29 '17 at 13:43
• Could you explain why, in your opinion, a closed form exists and would contain $Li_3$ and $Li_2$ ? This would be of interest to me. – Claude Leibovici Aug 29 '17 at 14:04
• Hi Claude, As a starting point, why don't you ignore my speculation" regarding the final form, and attempt to solve it like any new unsolved integral. – Hmath Aug 29 '17 at 16:52
• This is simply $-I\bigg(\dfrac12\bigg),$ where $I(a)=\displaystyle\int_a^1\frac{\text{Li}_3(x)-\text{Li}_3(a)}{x-a}~dx.$ – Lucian Sep 8 '17 at 21:23
• It resembles Frullani's integral. – Lucian Sep 8 '17 at 21:31
I will be using next integral $$\operatorname{Lv}_{n}(x,\alpha) =\int\limits_{0}^{x} \dfrac{\ln^n t}{1-\alpha t}\,\mathrm{d}t = \dfrac{n!}{\alpha}\sum\limits_{k\,=\,0}^{n}\dfrac{(-1)^k}{\left(n-k\right)!}\ln^{n-k}x\operatorname{Li}_{k+1}(\alpha x)$$ with $$x \geq 0$$, $$n \in \mathbb{N}$$ and $$\alpha$$ such that $$\alpha x < 1 \text{ or } \alpha x = 1\ \wedge\ x = 1$$ If anywhere $$\operatorname{Lv}$$ will appear I'll omit calculations of $$\operatorname{Li}$$ constants. Let \begin{align*}\mathfrak{I} &= \int\limits_{1/2}^{1}\dfrac{\ln\left(2x-1\right)}{x}\operatorname{Li}_2(x)\,\mathrm{d}x \\ &= \int\limits_{0}^{1}\dfrac{\ln x}{1+x}\operatorname{Li}_2\left(\dfrac{1+x}{2}\right)\,\mathrm{d}x \\ &= \operatorname{Li}_2\left(\dfrac{1+x}{2}\right)\Big(\ln x\ln\left(1+x\right) + \operatorname{Li}_2(-x)\Big)\Bigg\vert_{0}^{1}+\int\limits_{0}^{1}\dfrac{\ln x\ln\left(1+x\right) + \operatorname{Li}_2(-x)}{1+x}\ln \left(\dfrac{1-x}{2}\right)\,\mathrm{d}x \\ &= \operatorname{Li}_2(1)\operatorname{Li}_2(-1) + \underbrace{\int\limits_{0}^{1}\dfrac{\ln\left(1-x\right)\ln x\ln\left(1+x\right)}{1+x}\,\mathrm{d}x}_{\mathfrak{I}_1}+\underbrace{\int\limits_{0}^{1}\dfrac{\operatorname{Li}_2(-x)\ln\left(1-x\right)}{1+x}\,\mathrm{d}x}_{\mathfrak{I}_2}-\phantom{a}\\ &-\ln 2\underbrace{\int\limits_{0}^{1}\dfrac{1}{1+x}\left(\int\limits_{0}^{x}\dfrac{\ln t}{1+t}\,\mathrm{d}t\right)\,\mathrm{d}x}_{\mathfrak{I}_3} \end{align*}
## $$\mathfrak{I}_1:$$
Make substitution $$x\rightarrow \dfrac{1-x}{1+x}$$ \begin{align*} \mathfrak{I}_1 &= \int\limits_{0}^{1} \dfrac{1}{1+x}\ln\left(\dfrac{2}{1+x}\right)\ln\left(\dfrac{1-x}{1+x}\right)\ln\left(\dfrac{2x}{1+x}\right)\,\mathrm{d}x \\ &= \ln 2 \int\limits_{0}^{1} \dfrac{\ln\left(1-x\right)\ln x}{1+x}\,\mathrm{d}x - \ln 2 \int\limits_{0}^{1} \dfrac{\ln\left(1+x\right)\ln x}{1+x}\,\mathrm{d}x - \mathfrak{I}_1 + \int\limits_{0}^{1} \dfrac{\ln^2\left(1+x\right)\ln x}{1+x}\,\mathrm{d}x + \phantom{a} \\ &\,+ \color{red}{\int\limits_{0}^{1} \dfrac{1}{1+x}\ln^2\left(\dfrac{2}{1+x}\right)\ln\left(\dfrac{1-x}{1+x}\right)\,\mathrm{d}x} \\ &= \ln 2 \underbrace{\int\limits_{0}^{1} \dfrac{\ln\left(1-x\right)\ln x}{1+x}\,\mathrm{d}x}_{\mathfrak{I}_{1,1}} - \ln 2 \int\limits_{0}^{1} \dfrac{\ln\left(1+x\right)\ln x}{1+x}\,\mathrm{d}x - \mathfrak{I}_1 + \color{red}{2}\int\limits_{0}^{1} \dfrac{\ln^2\left(1+x\right)\ln x}{1+x}\,\mathrm{d}x \\ &= \dfrac{1}{2}\mathfrak{I}_{1,1}\ln 2 - \dfrac{1}{4}\ln 2 \ln^2\left(1+x\right)\ln x\Bigg\vert_{0}^{1} +\dfrac{1}{4}\ln 2 \int\limits_{0}^{1} \dfrac{\ln^2\left(1+x\right)}{x}\,\mathrm{d}x+\dfrac{1}{3}\ln^3\left(1+x\right)\ln x\Bigg\vert_{0}^{1} - \phantom{a} \\ &\ - \dfrac{1}{3}\int\limits_{0}^{1} \dfrac{\ln^3\left(1+x\right)}{x}\,\mathrm{d}x \\ &= \dfrac{1}{2}\mathfrak{I}_{1,1}\ln 2 + \dfrac{1}{4}\ln 2 \int\limits_{1}^{2} \dfrac{\ln^2 x}{x-1}\,\mathrm{d}x - \dfrac{1}{3}\int\limits_{1}^{2} \dfrac{\ln^3 x}{x-1}\,\mathrm{d}x \\ &= \dfrac{1}{2}\mathfrak{I}_{1,1}\ln 2 + \dfrac{1}{4}\ln 2 \int\limits_{1/2}^{1} \dfrac{\ln^2 x}{x\left(1-x\right)}\,\mathrm{d}x + \dfrac{1}{3}\int\limits_{1/2}^{1} \dfrac{\ln^3 x}{x\left(1-x\right)}\,\mathrm{d}x \\ &= \dfrac{1}{2}\mathfrak{I}_{1,1}\ln 2 + \dfrac{1}{4}\ln 2\left(\dfrac{1}{3}\ln^3 2 + \operatorname{Lv}_{2}(1,1)-\operatorname{Lv}_{2}\left(\dfrac{1}{2},1\right)\right) + \dfrac{1}{3}\left(-\dfrac{1}{4}\ln^4 2 + \operatorname{Lv}_{3}(1,1)-\operatorname{Lv}_{3}\left(\dfrac{1}{2},1\right)\right) \\ &= \dfrac{1}{2}\mathfrak{I}_{1,1}\ln 2 + 2\operatorname{Li}_4\left(\dfrac{1}{2}\right) + \dfrac{29}{16}\zeta(3)\ln 2 - \dfrac{1}{45}\pi^4 + \dfrac{1}{12}\ln^4 2 - \dfrac{1}{12}\pi^2\ln^2 2 \end{align*}
## $$\mathfrak{I}_{1,1}:$$
Use identity $$xy = \dfrac{1}{2}x^2 + \dfrac{1}{2}y^2 - \dfrac{1}{2}\left(x-y\right)^2$$ to have \begin{align*} \mathfrak{I}_{1,1} &= \dfrac{1}{2}\int\limits_{0}^{1} \dfrac{\ln^2 x}{1+x}\,\mathrm{d}x + \dfrac{1}{2}\int\limits_{0}^{1} \dfrac{\ln^2 \left(1-x\right)}{1+x}\,\mathrm{d}x - \dfrac{1}{2}\int\limits_{0}^{1} \dfrac{1}{1+x}\ln^2\left(\dfrac{x}{1-x}\right)\,\mathrm{d}x \\ &= \dfrac{1}{2}\operatorname{Lv}_2(1,-1)+\dfrac{1}{2}\int\limits_{0}^{1} \dfrac{\ln^2 x}{2-x}\,\mathrm{d}x-\dfrac{1}{2}\int\limits_{0}^{\infty} \dfrac{\ln^2 x}{\left(1+2x\right)\left(1+x\right)}\,\mathrm{d}x \\ &= \dfrac{1}{2}\operatorname{Lv}_2(1,-1)+\dfrac{1}{4}\operatorname{Lv}_2\left(1,\dfrac{1}{2}\right)-\dfrac{1}{2}\int\limits_{0}^{1} \dfrac{\ln^2 x}{\left(1+2x\right)\left(1+x\right)}\,\mathrm{d}x-\dfrac{1}{2}\int\limits_{0}^{1} \dfrac{\ln^2 x}{\left(2+x\right)\left(1+x\right)}\,\mathrm{d}x \\ &= \dfrac{1}{2}\operatorname{Lv}_2(1,-1)+\dfrac{1}{4}\operatorname{Lv}_2\left(1,\dfrac{1}{2}\right)-\int\limits_{0}^{1} \dfrac{\ln^2 x}{1+2x}\,\mathrm{d}x+\dfrac{1}{2}\int\limits_{0}^{1} \dfrac{\ln^2 x}{2+x}\,\mathrm{d}x \\ &= \dfrac{1}{2}\operatorname{Lv}_2(1,-1)+\dfrac{1}{4}\operatorname{Lv}_2\left(1,\dfrac{1}{2}\right)-\operatorname{Lv}_2(1,-2)+\dfrac{1}{4}\operatorname{Lv}_2\left(1,-\dfrac{1}{2}\right) \\ &= \dfrac{13}{8}\zeta(3)-\dfrac{1}{4}\pi^2\ln 2 \end{align*} So
$$\mathfrak{I}_1 = 2\operatorname{Li}_4\left(\dfrac{1}{2}\right)+\dfrac{21}{8}\zeta(3)\ln 2 - \dfrac{1}{45}\pi^4 + \dfrac{1}{12}\ln^4 2 - \dfrac{5}{24}\pi^2\ln^2 2$$
## $$\mathfrak{I}_{2}:$$
Use the same identity as for $$\mathfrak{I}_{1,1}$$
\begin{align*} -\mathfrak{I}_2 &= \int\limits_{0}^{1} \dfrac{\ln\left(1-x\right)}{1+x}\left(\int\limits_{0}^{1}\dfrac{\ln\left(1+xt\right)}{t}\,\mathrm{d}t\right)\,\mathrm{d}x \\ &= \dfrac{1}{2}\int\limits_{0}^{1} \dfrac{1}{1+x}\left(\int\limits_{0}^{1}\dfrac{\ln^2\left(1+xt\right)}{t}\,\mathrm{d}t\right)\,\mathrm{d}x + \dfrac{1}{2}\int\limits_{0}^{1}\int\limits_{0}^{1} \dfrac{1}{\left(1+x\right)t}\left(\ln^2\left(1-x\right)-\ln^2\left(\dfrac{1-x}{1+xt}\right)\right)\,\mathrm{d}t\,\mathrm{d}x \\ &= \dfrac{1}{2}\int\limits_{0}^{1} \dfrac{1}{1+x}\left(\int\limits_{0}^{x}\dfrac{\ln^2\left(1+t\right)}{t}\,\mathrm{d}t\right)\,\mathrm{d}x + \phantom{a} \\ &+ \dfrac{1}{2}\int\limits_{0}^{1} \dfrac{1}{t}\left(\int\limits_{0}^{1}\dfrac{\ln^2\left(1-x\right)}{1+x}\,\mathrm{d}x-\int\limits_{0}^{1}\dfrac{1}{1+x}\ln^2\left(\dfrac{1-x}{1+xt}\right)\,\mathrm{d}x\right)\,\mathrm{d}t \\ &= \dfrac{1}{2}\left.\ln\left(1+x\right)\int\limits_{0}^{x}\dfrac{\ln^2\left(1+t\right)}{t}\,\mathrm{d}t\right\vert_{0}^{1}-\dfrac{1}{2}\int\limits_{0}^{1}\dfrac{\ln^3\left(1+x\right)}{x}\,\mathrm{d}x + \phantom{a} \\ &+ \dfrac{1}{2}\int\limits_{0}^{1} \dfrac{1}{t}\left(\int\limits_{0}^{1}\dfrac{\ln^2 x}{2-x}\,\mathrm{d}x-\left(1+t\right)\int\limits_{0}^{1}\dfrac{\ln^2 x}{\left(1+xt\right)\left(2-x\left(1-t\right)\right)}\,\mathrm{d}x\right)\,\mathrm{d}t \\ &= \dfrac{1}{2}\ln 2\left(\dfrac{1}{3}\ln^3 2 + \operatorname{Lv}_2(1,1)-\operatorname{Lv}_2\left(\dfrac{1}{2},1\right)\right)+\dfrac{1}{2}\left(-\dfrac{1}{4}\ln^4 2+\operatorname{Lv}_3(1,1)-\operatorname{Lv}_3\left(\dfrac{1}{2},1\right)\right)+\phantom{a} \\ &+ \dfrac{1}{2}\int\limits_{0}^{1} \dfrac{1}{t}\left(\dfrac{1}{2}\operatorname{Lv}_2\left(1,\dfrac{1}{2}\right)-\int\limits_{0}^{1} \ln^2 x\left(\dfrac{t}{1+xt}+\dfrac{1-t}{2-x\left(1-t\right)}\right)\,\mathrm{d}x\right)\,\mathrm{d}t \\ &= 3\operatorname{Li}_4\left(\dfrac{1}{2}\right)+\dfrac{11}{4}\zeta(3)\ln 2 - \dfrac{1}{30}\pi^4 + \dfrac{1}{8}\ln^4 2 - \dfrac{1}{8}\pi^2\ln^2 2+\phantom{a} \\ &+ \dfrac{1}{2}\underbrace{\int\limits_{0}^{1} \dfrac{1}{t}\left(\dfrac{1}{2}\operatorname{Lv}_2\left(1,\dfrac{1}{2}\right)-t\operatorname{Lv}_2(1,-t)-\dfrac{1-t}{2}\operatorname{Lv}_2\left(1, \dfrac{1-t}{2}\right)\right)\,\mathrm{d}t}_{\mathfrak{I}_{2,1}} \end{align*} Note that $$\operatorname{Lv}_2(1,\alpha) = \dfrac{2}{\alpha}\operatorname{Li}_3(\alpha)$$ So $$\mathfrak{I}_{2,1}$$ can be simplified to \begin{align*} \mathfrak{I}_{2,1} &= \int\limits_{0}^{1} \dfrac{1}{t}\left(2\operatorname{Li}_3\left(\dfrac{1}{2}\right)+2\operatorname{Li}_3(-t)-2\operatorname{Li}_3\left(\dfrac{1-t}{2}\right)\right)\,\mathrm{d}t \\ &= 2\int\limits_{0}^{1} \dfrac{\operatorname{Li}_3(-t)}{t}\,\mathrm{d}t+2\int\limits_{0}^{1} \dfrac{1}{t}\left(\operatorname{Li}_3\left(\dfrac{1}{2}\right)-\operatorname{Li}_3\left(\dfrac{1-t}{2}\right)\right)\,\mathrm{d}t \\ &= 2\operatorname{Li}_4(-1)+2\sum\limits_{n=1}^{\infty}\dfrac{1}{n^32^n}\int\limits_{0}^{1} \dfrac{1-t^n}{1-t}\,\mathrm{d}t \\ &= 2\operatorname{Li}_4(-1)+2\sum\limits_{n=1}^{\infty}\dfrac{\mathcal{H}_n}{n^32^n} \\ &= 2\operatorname{Li}_4\left(\dfrac{1}{2}\right) - \dfrac{1}{4}\zeta(3)\ln 2 - \dfrac{1}{60}\pi^4 + \dfrac{1}{12}\ln^4 2 \end{align*} Closed form for series above you can find here. So
$$\mathfrak{I}_2 = -4\operatorname{Li}_4\left(\dfrac{1}{2}\right)-\dfrac{21}{8}\zeta(3)\ln 2 + \dfrac{1}{24}\pi^4-\dfrac{1}{6}\ln^4 2 + \dfrac{1}{8}\pi^2\ln^2 2$$
## $$\mathfrak{I}_3:$$
\begin{align*} \mathfrak{I}_3 &= \ln\left(1+x\right)\Big(\ln x\ln\left(1+x\right) + \operatorname{Li}_2(-x)\Big)\Bigg\vert_{0}^{1} - \int\limits_{0}^{1}\dfrac{\ln\left(1+x\right)\ln x}{1+x}\,\mathrm{d}x \\ &= \ln 2\operatorname{Li}_2(-1) - \dfrac{1}{2}\ln^2\left(1+x\right)\ln x\Bigg\vert_{0}^{1}+\dfrac{1}{2}\int\limits_{0}^{1} \dfrac{\ln^2\left(1+x\right)}{x}\,\mathrm{d}x \\ &= \ln 2\operatorname{Li}_2(-1) + \dfrac{1}{2}\left(\dfrac{1}{3}\ln^3 2 + \operatorname{Lv}_2(1,1) - \operatorname{Lv}_2\left(\dfrac{1}{2},1\right)\right) \\ &= \dfrac{1}{8}\zeta(3)-\dfrac{1}{12}\pi^2\ln 2 \end{align*} So
$$\mathfrak{I}_3 = \dfrac{1}{8}\zeta(3)-\dfrac{1}{12}\pi^2\ln 2$$
Collecting all parts together we have that \begin{align*} \mathfrak{I} &= \operatorname{Li}_2(1)\operatorname{Li}_2(-1)+\mathfrak{I}_1+\mathfrak{I}_2-\mathfrak{I}_3\ln 2 \\ &= -2\operatorname{Li}_4\left(\dfrac{1}{2}\right)-\dfrac{1}{8}\zeta(3)\ln 2 + \dfrac{1}{180}\pi^4 - \dfrac{1}{12}\ln^4 2 \end{align*}
• very nice solution (+1). – Ali Shadhar Mar 26 '20 at 21:19
This provisional answer is an horribly inelegant conjecture with no proof attached
$$\int\limits_{0.5}^{1}\frac{Li_2(x) \ln(2x-1)}{x} dx =-\frac{1}{2}\sum_{n=1}^{\infty}\frac{\sum_{k=0}^{n-1} \frac{\binom{n-1}{k}}{(k+1)^2}}{ \sum_{k=0}^n (k (2 k-1)) \binom{n}{k}}\tag1$$
I have checked this solution using $m=100$ approximation
$$\int_{\frac{1}{2}}^1 \frac{ \left(\sum _{k=1}^{m} \frac{x^k}{k^2}\right)\log (2 x-1)}{x} \, dx =-\frac{1}{2}\sum_{n=1}^{m}\frac{\sum_{k=0}^{n-1} \frac{\binom{n-1}{k}}{(k+1)^2}}{ \sum_{k=0}^n (k (2 k-1)) \binom{n}{k}}$$
The numerator summation arises from the number sequence given here (binomial transform of $1/(k+1)^2$: that is 1, 5/4, 29/18, 103/48, 887/300, 1517/360, etc.) and the denominator summation arises from the number sequence given here (the binomial transform of the hexagonal numbers)
Hopefully someone can make a little more sense of this than I have.
Later Edit: In working to understand user90369's answer to this question I found these identities using Mathematica
$$Li_n(\frac{2}{m})-Li_n(\frac{1}{m})=\sum _{k=1}^{\infty } \frac{1}{m^k k^{n-1}}\sum _{v=1}^k \binom{k-1}{v-1}\frac{1}{v}$$
in the case of $m=2$ $$\zeta(n)-Li_n(\frac{1}{2})=\sum _{k=1}^{\infty } \frac{1}{2^k k^{n-1}}\sum _{v=1}^k \binom{k-1}{v-1}\frac{1}{v}$$
Added Later Still: More Trivial Identities
$$Li_2(\frac{1}{2})=\sum\limits_{k=1}^\infty \frac{1}{k^3 2^k}\sum\limits_{v=1}^1 {\binom k v} \frac{1}{v}$$
$$Li_4(\frac{1}{2})=\sum\limits_{k=1}^\infty \frac{1}{k^3 2^k}\sum\limits_{v=k}^k {\binom k v} \frac{1}{v}$$
$$\int\limits_{0.5}^{1}\frac{Li_2(x) \ln(2x-1)}{x} dx=-Li_2(\frac{1}{2})-Li_4(\frac{1}{2})-\sum\limits_{k=1}^\infty \frac{1}{k^3 2^k}\sum\limits_{v=2}^{k-1} {\binom k v} \frac{1}{v}$$
Since $$\sum\limits_{k=1}^{\infty} \frac{1}{k^3 2^k}\sum\limits_{v=1}^{k} {\binom k v} \frac{1}{v}=\sum _{v=1}^{\infty} \frac{1}{v}\sum _{k=1}^{\infty} \frac{1}{k^{3} 2^k}\binom{k}{v}$$
There is a brute force pattern matching approach that I have found, using
$$S_a=\sum\limits_{k=1}^{\infty} \frac{1}{k^3 2^k}\sum\limits_{v=a}^{a} {\binom k v} \frac{1}{v}=\sum _{v=a}^a \frac{1}{v}\sum _{k=1}^{\infty} \frac{1}{k^{3} 2^k}\binom{k}{v}$$
where $$\int\limits_{0.5}^{1}\frac{Li_2(x) \ln(2x-1)}{x} dx= -\sum_{a=1}^{\infty} S_a$$
From Mathematica $$S_1=\frac{\pi ^2}{12}-\frac{\log ^2(2)}{2}=Li_2(\frac{1}{2})$$ $$S_2=1/48 \left(-\pi^2 + 12 \log(2) + 6 \log^2(2) \right)$$ $$S_3=\frac{1}{108} \left(6+\pi ^2-18 \log (2)-6 \log ^2(2)\right)$$ $$S_4=\frac{1}{192} \left(-8-\pi ^2+22 \log (2)+6 \log ^2(2)\right)$$
and so on. From $S_2$ onwards the general term as far as I have been able to determine is
$$S_a=(-1)^{k-1}C_a+(-1)^{k-1}\frac{\pi^2}{12a^2}+(-1)^{k}\frac{H_{a-1}\log(2)}{a^2}+(-1)^{k}\frac{\log^2(2)}{2a^2}$$
where $H_a$ is the Harmonic Number. I haven't been able to determine the pattern for the rational term, $C_a$ [$0$,$\frac{6}{(12\times3^2)}$,$\frac{8}{(12\times4^2)}$,$\frac{21}{2(12\times5^2)}$,$\frac{119}{10(12\times6^2)}$,$\frac{202}{15(12\times7^2)}$,$\frac{1525}{105(12\times8^2)}$,...]. Since the last three sum up to known closed forms it would be unfortunate if the rational term summation didn't do the same.
Once again I hope someone can make a little more sense of this than I have.
• Nice idea. (+1) :-D But it seems to be that the OP is not any more interested in his own question. :-( – user90369 Sep 4 '17 at 14:20
• I agree the OP seems to have lost interest. I might raise a new question in regards to determining a pattern for the rational term. – James Arathoon Sep 4 '17 at 14:57
$\displaystyle \int\limits_{0.5}^1 \frac{Li_2(x)\ln(2x-1)}{x}dx=$
$\displaystyle =\sum\limits_{k=1}^\infty \frac{1}{k^2 2^k}\sum\limits_{v=0}^{k-1} {\binom {k-1} v} \lim\limits_{h\to 0}\frac{1}{h}\left(\frac{(2x-1)^{v+h+1}}{v+h+1}-\frac{(2x-1)^{v+1}}{v+1}\right)|_{0.5}^1$
$\displaystyle =-\sum\limits_{k=1}^\infty \frac{1}{k^3 2^k}\sum\limits_{v=1}^k {\binom k v} \frac{1}{v} = -\int\limits_0^1 \frac{Li_3(\frac{x+1}{2})-Li_3(\frac{1}{2})}{x}dx$
First note:
Be $\,\displaystyle H_k(x):=x\int\limits_0^1 \frac{(xt)^k-1}{xt-1}dt=\sum\limits_{v=1}^k \frac{x^v}{v}$ . $\,$ It's $\enspace\displaystyle \sum\limits_{v=1}^k {\binom k v} \frac{1}{v}=H_k(2)-H_k(1)$ .
Second note:
We can define e.g. $\,\displaystyle Fi_n(x):=\int\limits_0^{1-x}\frac{Li_n(t+x)-Li_n(x)}{t}dt\,$ for $\,|x|\leq 1\,$ .
Then it's $\,\displaystyle \int\limits_{0.5}^1 \frac{Li_2(x)\ln(2x-1)}{x}dx=-Fi_3(\frac{1}{2})\,$ .
• I did NOT loose interest in my question! The funny thing is I am evaluating this integral because I want to evaluate an infinite sum involving binomial coefficients! You have just brought me back to binomial coefficients... Definitely not the approach I was looking for – Hmath Sep 4 '17 at 17:33
• Good to hear. I must add that you chose to hide the context behind your question. I gave a +1 to @SimplyBeautifulArt's comment "What have you tried?" If you had added context earlier (including what approach you were looking for) you could perhaps have attracted more interest and up-voting to what seems like a very interesting problem. – James Arathoon Sep 4 '17 at 18:15
• @Hmath : I agree with James Arathoon, it's indeed not possible to understand what you are looking for (beside a "closed form", which is obviously not possible). And: $H_k(x)$ is without any binomial coefficients. – user90369 Sep 4 '17 at 19:39
Start with subbing $$2x-1\to x$$ then integrate by parts we have
$$I=-\frac54\zeta(4)+\int_0^1\frac{\text{Li}_2(-x)}{1+x}\ln\left(\frac{1-x}{2}\right)\ dx+\int_0^1\frac{\ln(x)\ln(1+x)}{1+x}\ln\left(\frac{1-x}{2}\right)\ dx$$
$$=-\frac54\zeta(4)+\int_0^1\frac{\text{Li}_2(-x)\ln(1-x)}{1+x}\ dx+\int_0^1\frac{\ln(x)\ln(1+x)\ln(1-x)}{1+x}\ dx$$
$$-\ln(2)\int_0^1\frac{\ln(x)}{1+x}[\text{Li}_2(-x)+\ln(x)\ln(1+x)]\ dx$$
$$=-\frac54\zeta(4)+A+B-C$$
For $$A$$, use
$$\frac{\text{Li}_2(-x)}{1+x}=\sum_{n=1}^\infty (-1)^{n-1}H_{n-1}^{(2)}x^{n-1}\tag1$$
$$A=\sum_{n=1}^\infty (-1)^{n-1}H_{n-1}^{(2)}\int_0^1x^{n-1}\ln(1-x) \ dx=\sum_{n=1}^\infty (-1)^{n-1}H_{n-1}^{(2)}\left(-\frac{H_n}{n}\right)$$
$$=\sum_{n=1}^\infty \frac{(-1)^nH_{n}^{(2)}H_n}{n}-\sum_{n=1}^\infty \frac{(-1)^nH_n}{n^3}$$
For B, use
$$\frac{\ln(1+x)}{1+x}=\sum_{n=1}^\infty (-1)^{n}H_{n-1} x^{n-1}\tag2$$
$$B=\sum_{n=1}^\infty (-1)^{n}H_{n-1} \int_0^1x^{n-1}\ln(x)\ln(1-x)\ dx=\sum_{n=1}^\infty (-1)^{n}H_{n-1}\left(\frac{H_n}{n^2}+\frac{H_n^{(2)}}{n}-\frac{\zeta(2)}{n}\right)$$
$$=\sum_{n=1}^\infty \frac{(-1)^{n}H_{n-1}H_n}{n^2}+\sum_{n=1}^\infty \frac{(-1)^{n}H_{n-1}H_n^{(2)}}{n}-\zeta(2)\sum_{n=1}^\infty \frac{(-1)^{n}H_{n-1}}{n}$$
$$=\sum_{n=1}^\infty \frac{(-1)^{n}H_n^2}{n^2}-\sum_{n=1}^\infty \frac{(-1)^{n}H_n}{n^3}+\sum_{n=1}^\infty \frac{(-1)^{n}H_{n}H_n^{(2)}}{n}-\sum_{n=1}^\infty \frac{(-1)^{n}H_n^{(2)}}{n^2}-\zeta(2)\sum_{n=1}^\infty \frac{(-1)^{n}H_{n-1}}{n}$$
Combine the results of A and B we get
$$\small{A+B=\sum_{n=1}^\infty \frac{(-1)^{n}H_n^2}{n^2}-2\sum_{n=1}^\infty \frac{(-1)^{n}H_n}{n^3}+2\sum_{n=1}^\infty \frac{(-1)^{n}H_{n}H_n^{(2)}}{n}-\sum_{n=1}^\infty \frac{(-1)^{n}H_n^{(2)}}{n^2}-\zeta(2)\sum_{n=1}^\infty \frac{(-1)^{n}H_{n-1}}{n}}$$
The last sum can be easily obtained by integrating both sides of $$(2)$$, $$\Longrightarrow \sum_{n=1}^\infty \frac{(-1)^{n}H_{n-1}}{n}=\frac12\ln^2(2)$$
For the rest of the sums, they are already calculated:
$$\sum_{n=1}^\infty \frac{(-1)^{n}H_n}{n^3}=2\operatorname{Li_4}\left(\frac12\right)-\frac{11}4\zeta(4)+\frac74\ln2\zeta(3)-\frac12\ln^22\zeta(2)+\frac{1}{12}\ln^42\tag3$$
$$\sum_{n=1}^{\infty}\frac{(-1)^nH_n^{(2)}}{n^2}=-4\operatorname{Li}_4\left(\frac12\right)+\frac{51}{16}\zeta(4)-\frac72\ln2\zeta(3)+\ln^22\zeta(2)-\frac16\ln^42\tag4$$
$$\sum_{n=1}^{\infty}\frac{(-1)^nH_n^2}{n^2}=2\operatorname{Li}_4\left(\frac12\right)-\frac{41}{16}\zeta(4)+\frac74\ln2\zeta(3)-\frac12\ln^22\zeta(2)+\frac1{12}\ln^42\tag5$$
where $$(3)$$ can be found here and $$(4)$$ and $$(5)$$ can be found here.
Integral $$C$$ was nicely calculated by @Aknas ( check integral $$\Im_3$$ in his solution above) where he used
$$\text{Li}_2(x)+\ln(x)\ln(1+x)=\int_0^x\frac{\ln(t)}{1+t}\ dt$$
$$C= \dfrac{1}{8}\zeta(3)-\dfrac{1}{2}\ln(2)\zeta(2)$$
Combine all resuts of $$A$$, $$B$$ and $$C$$ we obtain that
$$I=-2\operatorname{Li}_4\left(\dfrac{1}{2}\right)-\dfrac{1}{8}\ln(2)\zeta(3) + \dfrac{1}{2}\zeta(4)- \dfrac{1}{12}\ln^4(2)$$
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2021-03-04 01:05:11
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http://mathoverflow.net/questions/9466/why-these-particular-numerical-factors-in-the-definition-of-gaussian-curvature/9500
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# Why these particular numerical factors in the definition of Gaussian curvature?
Wikipedia tells me that:
Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane:
$K = \lim_{r \rightarrow 0} (2 \pi r - \mbox{C}(r)) \cdot \frac{3}{\pi r^3}$
Gaussian curvature is the limiting difference between the area of a geodesic circle and a circle in the plane:
$K = \lim_{r \rightarrow 0} (\pi r^2 - \mbox{A}(r)) \cdot \frac{12}{\pi r^4}$
Can anyone explain to me why we are dividing by the factors $\frac{3}{\pi r^3}$ and $\frac{12}{\pi r^4}$ respectively? I don't understand why we are dividing by these particular factors?
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First, I guess it should say "geodesic disc" rather than "circle". At least to me, a geodesic circle is a closed geodesic loop in your surface, whereas a geodesic disc of radius r is all the points distance r from a fixed point (at least for r smaller than the injectivity radius). Note the boundary of a geodesic disc is not a geodesic.
As for the factors in those formulae, well, there's no absolute scale for Gaussian curvature. People have just agreed on the convention that the curvature of the unit sphere should be 1. (EDIT: As Greg Kuperberg points out in his answer, there are some good reasons for this convention. E.g., Gauss-Bonnet.) That then forces those factors to be what they are. It amounts to the statement that, for a small geodesic disc on the unit sphere of radius r , $$C(r) \sim 2\pi\left( r - \frac{1}{6} r^3\right),$$ and a similar formula for the area. There really is no deeper reason than that.
So, to see if the factors are right (and you should never trust what you read on the internet!) I would suggest doing exactly those calculations for the unit sphere. I've checked the first formula involving the circumference and it looks good to me. If you have problems with the calculation, leave a comment and I'll write my version down for you, but I have a feeling it's best to do these things yourself.
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Cheers. The constants seem to agree with what I have calculated. That also explains why I have seen any definitions being used with different normalising constants. Thanks again. – alext87 Dec 21 '09 at 10:44
You're welcome! – Joel Fine Dec 21 '09 at 10:51
Joel is right that it is partly just a convention to scale Gaussian curvature so that the curvature of a unit sphere is $1$. However, there are three natural motivations for this scale besides matching 1 to 1 in the case of a sphere.
First, Gauss defined his curvature as the product of the extrinsic curvatures of a surface in $\mathbb{R}^3$. So there is a coefficient of 1 in this natural formula.
Second, the unit sphere has the property that the deviation from Euclid's parallel postulate has a factor of 1. In other words, the area $A$ of a triangle with angles $\alpha, \beta, \gamma$ is $\alpha + \beta + \gamma - \pi$. In general, if you have a very small triangle with area $A$ at a point of local curvature $K$, its angle deviation is $KA$ to first order. This factor of 1 leads to a factor of $2\pi$ in the Gauss-Bonnet theorem, that the integral of Gaussian curvature is $2\pi \chi$.
Third, Gaussian curvature is the ratio of the Ricci curvature tensor to the metric, and it is also half of the scalar curvature.
In comparing these formulas, the most reasonable scales for Gaussian curvature are the standard choice, the standard choice times 2 to match scalar curvature, and the standard choice divided by $2\pi$ to match the Gauss-Bonnet theorem. The volume and area formulas are some justification for a 1/3 or a 1/12 or similar, but these are taken to be less fundamental scales.
(One irony of the discussion is that $\pi$ itself is half of the most important value in trigonometry.)
Also the volume and surface area ratios are given in Wikipedia in $n$ dimensions. It is also worth looking at the generalized Gauss-Bonnet theorem in $2n$ dimensions.
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There should not be a symbol $\pi$. Instead, there should be a symbol for what we now call $2\pi$. It would make calculations so much easier, e.g. by better illuminating when there really are strange factors of $2$. – Theo Johnson-Freyd Dec 21 '09 at 21:47
You're right, I concede there is a good choice of scale for curvature. The Gauss-Bonnet theorem is particularly convincing. I guess you can also see it infinitesimally: for a surface in R^3, the curvature is the the limit of the ratio of solid angle prescribed by the normal to surface area (which is just infinitesimal Gauss-Bonnet for a surface in R^3). I'll amend my answer now. – Joel Fine Dec 22 '09 at 12:45
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2013-12-13 21:26:14
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https://mcq.electronics-club.com/flip-flop-mcq/
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# Flip-Flop MCQ
## Flip-Flop MCQ
Flip-Flop MCQ, MCQ on Flip-Flop, Multiple Choice Questions on Flip-Flop, Objective Questions on Flip-Flop, Digital Electronics MCQ, Engineering MCQ, J-K Flip-Flop MCQ, S-R Flip-Flop MCQ, D Flip-Flop MCQ, T Flip-Flop MCQ
Logic Gate MCQ
### Objective Type Questions
Q.1. A combinational logic circuit
• must contain flip-flops
• may contain flip-flops
• does not contain flip-flops
• contains latches
Q.2. The output of a logic circuit depends upon the sequence in which the input is applied. The circuit
• is a combinational logic circuit
• is a sequential logic circuit
• may be a combinational or sequential logic circuit
• is none of the above
Answer: is a sequential logic circuit
Q.3. A sequential circuit does not use clock pulses. It is
• an asynchronous sequential circuit
• a synchronous sequential circuit
• a counter
• a shift register
Q.4. In a multi-input sequential circuit only one input is allowed to change at a time. This circuit
• consists of latches and combinational circuit
• is a clocked sequential circuit
• is a serial adder circuit
• may be a synchronous counter or a shift register
Answer: consists of latches and combinational circuit
Q.5. An asynchronous sequential circuit
• does not use clock pulses
• only one input can change at a time
• consists of combinational circuit and latches
• meets all the above conditions
Answer: meets all the above conditions
Q.6. The basic memory element in a digital circuit
• consists of a NAND gate
• consists of a NOR gate
• is a flip-flop
• is a shift register
Q.7. A flip-flop has two outputs which are
• always 0
• always 1
• always complementary
• all of the above states
Q.8. A flip-flop can be made using
• basic gates such as AND, OR and NOT
• NAND gates
• NOR gates
• any of the above
Q.9. Which of the following flip-flop is used as a latch?
• J-K flip-flop
• Master-slave flip-flop
• T flip-flop
• D flip-flop
Q.10. Which of the following flip-flop is used as a latch?
• J-K flip-flop
• Master-slave flip-flop
• S-R flip-flop
• T flip-flop
Q.11. A flip-flop can store
• one bit of data
• two bits of data
• three bits of data
• any number of bits of data
Q.12. When an inverter is placed between the inputs of an S-R flip-flop, the resulting flip-flop is a
• J-K flip-flop
• Master-slave flip-flop
• T flip-flop
• D flip-flop
Q.13. Which of the following input combinations is not allowed in an S-R flip-flop?
• S = 0, R = 0
• S = 0, R = 1
• S = 1, R = 0
• S = 1, R = 1
Answer: S = 1, R = 1
Q.14. The functional difference between an S-R flip-flop and a J-K flip-flop is that
• J-K flip-flop is faster than S-R flip-flop
• J-K flip-flop has a feedback path
• J-K flip-flop accepts both inputs 1
• J-K flip-flop does not require external clock
Answer: J-K flip-flop accepts both inputs 1
Q.15. When a flip-flop is set, its outputs will be
• $Q=0$, $\bar{Q}=0$
• $Q=1$, $\bar{Q}=0$
• $Q=0$, $\bar{Q}=1$
• $Q=1$, $\bar{Q}=1$
Answer: $Q=1$, $\bar{Q}=0$
Q.16. When a flip-flop is reset, its outputs will be
• $Q=0$, $\bar{Q}=0$
• $Q=1$, $\bar{Q}=1$
• $Q=0$, $\bar{Q}=1$
• $Q=1$, $\bar{Q}=0$
Answer: $Q=0$, $\bar{Q}=1$
Q.17. For a flip-flop with provisions of preset and clear
• while presetting, clear is disabled
• while clearing, preset is disabled
• above both are true
• preset and clear operations are performed simultaneously
Q.18. The race around condition occurs in a J-K flip-flop when
• both inputs are 0
• both inputs are 1
• the inputs are complementary
• any one of the above input combinations is present
Q.19. Master-slave configuration is used in flip-flops to
• increase its clocking rate
• reduce power dissipation
• eliminate race-around condition
• improve its reliability
Q.20. The output Qn of a J-K flip-flop is 1. It changes to 0 when a clock pulse is applied. The inputs Jn and Kn are respectively
• 0 and X
• 1 and X
• X and 1
• X and 0
Q.21. The output Qn of an S-R flip-flop is 1.1t changes to 0 when a clock pulse is applied. The inputs Sn and Rn are respectively
• X and 1
• 0 and 1
• X and 1
• 0 and X
Q.22. The output Qn of of a J-K flip-flop is 0. It changes to 1 when a clock pulse is applied. The inputs Jn and Kn are respectively.
• 1 and X
• 0 and X
• X and 0
• X and 1
Q.23. The output Qn of a J-K (or S-R) flip-flop is 0. Its output does not change when a clock pulse is applied. The inputs Jn and Kn (or Sn-Rn) are respectively
• X and 0
• X and 1
• 1 and X
• 0 and X
Q.24. The output Qn of a J-K flip-flop is 1. Its output does not change when a clock pulse is applied. The inputs Jn and Kn are respectively
• 0X
• X0
• 10
• 01
Q.25. The outputs Q and $\bar{Q}$ of a master-slave S-R flip-flop are connected to its R and S inputs respectively. Its output Q when clock pulses are applied will be
• permanently 0
• permanently 1
• fixed 0 or 1
• complementing with every clock pulse
Q.26. Flip-flops can be used to make
• latches
• bounce-elimination switches
• registers
• all of the above
Q.27. A master-slave flip-flop is triggered
• when the clock input is at HIGH logic level
• when the clock input makes a transition from LOW to HIGH
• when a pulse is applied at the clock input
• when the clock input is at LOW logic level
Answer: when a pulse is applied at the clock input
Q.28. A J-K M-S flip-flop comprises which of the following configurations?
• S-R flip-flop followed by an S-R flip-flop
• S-R flip-flop followed by a J-K flip flop
• J-K flip flop followed by a J-K flip-flop
• J-K flip-flop followed by an S-R flip-flop
Answer: J-K flip-flop followed by an S-R flip-flop
Q.29. In a J-K M-S flip-flop, race around is eliminated because of which of the following reasons?
• output of slave is fed back to the input of master
• output of master is fed back to the input of slave
• while the clock drives the master, inverted clock drives the slave
• J-K flip-flop is followed by S-R flip-flop
Answer: while the clock drives the master, inverted clock drives the slave
Q.30. The toggle mode for a J-K flip-flop is
• J = 0, K = 0
• J = 1, K = 0
• J = 0, K = 1
• J = 1, K = 1
Answer: J = 1, K = 1
Q.31. The transparent latch is
• an S-R flip-flop
• a D flip-flop
• a T flip-flop
• a flip-flop
Q.32. In a master-slave J-K flip-flop, J = K = 1. The state Qn+1 of the flip-flop after the clock pulse will be
• 0
• 1
• Qn
• $\bar{Q_{n}}$
Answer: $\bar{Q_{n}}$
Q.33. The characteristic equation of a J-K flip-flop is
• $Q_{n+1}=J\bar{Q_{n}}+\bar{K}Q_{n}$
• $Q_{n+1}=JQ_{n}+K\bar{Q_{n}}$
• $Q_{n+1}=\bar{J}Q_{n}+\bar{K}Q_{n}$
• $Q_{n+1}=\bar{J}\bar{Q_{n}}+KQ_{n}$
Answer: $Q_{n+1}=J\bar{Q_{n}}+\bar{K}Q_{n}$
Q.34. The characteristic equation of a D flip-flop is
• $Q_{n+1}=D$
• $Q_{n+1}=Q_{n}$
• $Q_{n+1}=1$
• $Q_{n+1}=\bar{Q_{n}}$
Answer: $Q_{n+1}=D$
Q.35. The characteristic equation of a T flip-flop is
• $Q_{n+1}=\bar{Q_{n}}T+Q_{n}\bar{T}$
• $Q_{n+1}=\bar{Q_{n}}\bar{T}+Q_{n}T$
• $Q_{n+1}=Q_{n}$
• $Q_{n+1}=\bar{Q_{n}}$
Answer: $Q_{n+1}=\bar{Q_{n}}T+Q_{n}\bar{T}$
Q.36. The characteristic equation of an S-R flip-flop is
• $Q_{n+1}=Q_{n}\bar{R}+S$
• $Q_{n+1}=\bar{Q_{n}}R+S$
• $Q_{n+1}=Q_{n}R+\bar{S}$
• $Q_{n+1}=Q_{n}$
Answer: $Q_{n+1}=Q_{n}\bar{R}+S$
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2022-05-26 16:44:35
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https://math.stackexchange.com/questions/2620213/what-does-the-p-harmonic-series-converge-to-when-p-1-%CE%B5
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# What does the p-harmonic series converge to when p = 1 + ε?
In infinitesimal calculus, $\epsilon$ is an infinitesimal number, that is, it is defined to be a number smaller than any real number but greater than $0$.
The p-harmonic series is:
$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^p}$
It is well known that this series diverges when $p \leq 1$ and converges when $p > 1$.
A lot of teachers like to do examples of this problem with $p$ arbitrarily close to one, but still greater, like $p = 1 + 0.001$.
What I want to know is, what happens when $p$ is infinitesimally close to $1$? In other words what does the series converge to when $p = 1 + \epsilon$?:
$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{1 + \epsilon}}$
Is this any different from just taking the limit as $p$ approaches 1?:
$\displaystyle \lim_{p \to 1^+} \sum_{n=1}^{\infty} \frac{1}{n^{p}}$
$1 + \epsilon$ is a number that is greater than $1$, whereas, the limit is getting arbitrarily close to $1$, and presumably that means it gets closer to $1$ than $1 + \epsilon$. However, I don't know what number the series could possible converge to when $p = 1 + \epsilon$.
If $\epsilon\not=0$ is infinitesimal then $\zeta(1+\epsilon)$ will be an infinite hyperreal, since the zeta function has a pole at $z=1$.
In standard mathematics there are no actual infinitesimals. But in any case: $$\sum_{n=1}^\infty \frac{1}{n^{1+\epsilon}} = \zeta(1+\epsilon) = \frac{1}{\epsilon} + \gamma + O(\epsilon) \ \text{as}\ \epsilon \to 0+$$
• You are equivocating on the meaning of the term "standard" since you are clearly not referring to it in its technical meaning (referring to Robinson's framework for analysis with infinitesimals) but rather as a generic term of approval. I object to this kind of obfuscation. -1 – Mikhail Katz Jan 25 '18 at 9:51
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2019-08-20 12:25:47
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https://www.expii.com/t/intersection-of-sets-definition-examples-4304
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Expii
# Intersection of Sets - Definition & Examples - Expii
The intersection of two or more sets consists of the elements common to all the sets. The intersection symbol ∩ means "and.".
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2023-02-08 13:35:28
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https://www.atmos-chem-phys.net/18/1835/2018/
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Journal topic
Atmos. Chem. Phys., 18, 1835–1861, 2018
https://doi.org/10.5194/acp-18-1835-2018
Atmos. Chem. Phys., 18, 1835–1861, 2018
https://doi.org/10.5194/acp-18-1835-2018
Research article 08 Feb 2018
Research article | 08 Feb 2018
# Atmospheric new particle formation at the research station Melpitz, Germany: connection with gaseous precursors and meteorological parameters
Atmospheric new particle formation at the research station Melpitz, Germany: connection with gaseous precursors and meteorological parameters
Johannes Größ1, Amar Hamed1,2,†, André Sonntag1, Gerald Spindler1, Hanna Elina Manninen3,4, Tuomo Nieminen3,2, Markku Kulmala3, Urmas Hõrrak5, Christian Plass-Dülmer6, Alfred Wiedensohler1, and Wolfram Birmili1,7 Johannes Größ et al.
• 1Leibniz Institute for Tropospheric Research, Permoserstrasse 15, 04318 Leipzig, Germany
• 2Department of Applied Physics, University of Eastern Finland, P.O. Box 1627, 70211 Kuopio, Finland
• 3Department of Physics, University of Helsinki, P.O. Box 64, 00014 Helsinki, Finland
• 4Experimental Physics Department, CERN, 1211 Geneva, Switzerland
• 5Institute of Physics, University of Tartu, Ülikooli 18, 50090 Tartu, Estonia
• 6German Meteorological Service DWD, Albin-Schwaiger-Weg 10, 82383 Hohenpeissenberg, Germany
• 7Federal Environment Agency (Umweltbundesamt), Corrensplatz 1, 14195 Berlin, Germany
• deceased
Correspondence: Wolfram Birmili (wolfram.birmili@uba.de)
Abstract
This paper revisits the atmospheric new particle formation (NPF) process in the polluted Central European troposphere, focusing on the connection with gas-phase precursors and meteorological parameters. Observations were made at the research station Melpitz (former East Germany) between 2008 and 2011 involving a neutral cluster and air ion spectrometer (NAIS). Particle formation events were classified by a new automated method based on the convolution integral of particle number concentration in the diameter interval 2–20 nm. To study the relevance of gaseous sulfuric acid as a precursor for nucleation, a proxy was derived on the basis of direct measurements during a 1-month campaign in May 2008. As a major result, the number concentration of freshly produced particles correlated significantly with the concentration of sulfur dioxide as the main precursor of sulfuric acid. The condensation sink, a factor potentially inhibiting NPF events, played a subordinate role only. The same held for experimentally determined ammonia concentrations. The analysis of meteorological parameters confirmed the absolute need for solar radiation to induce NPF events and demonstrated the presence of significant turbulence during those events. Due to its tight correlation with solar radiation, however, an independent effect of turbulence for NPF could not be established. Based on the diurnal evolution of aerosol, gas-phase, and meteorological parameters near the ground, we further conclude that the particle formation process is likely to start in elevated parts of the boundary layer rather than near ground level.
1 Introduction
Atmospheric aerosol particles have been recognised as one of the major uncertainties in predicting atmospheric radiative forcing and thus future climate (IPCC2013). As a first effect, aerosol particles influence the Earth's radiation balance by scattering and absorbing solar radiation directly . Second, aerosol particles act as cloud condensation nuclei (CCN) and thus modify the radiative properties of cloud droplets in various ways . The most influential aerosol effects are thought to be those related to changes in terrestrial temperature and precipitation patterns. Besides climate, atmospheric aerosol particles play a crucial role in the assessment of air quality and their adverse effects upon human health . Due to the complex interactions involved in the life cycle of aerosol particles, research has started with a highly integrated approach to elucidate aerosol climate effects across different temporal and spatial scales .
The nucleation of aerosol particles from gaseous precursors is one of the most important sources of atmospheric particle number. The formation of new aerosol particles in the atmosphere has been shown to occur in almost any atmospheric environment around the world . Considerable efforts have been expended to make the smallest atmospheric particles (around 1 nm in diameter) and some of their properties visible by instrumentation . The body of atmospheric and laboratory studies has clearly identified sulfuric acid as a key precursor for atmospheric particle nucleation , although the nucleation rates obtained from field and laboratory observations have been reconciled only recently . Laboratory work suggests that the acid–base interaction, such as that found between sulfuric acid and ammonia, may play a crucial role in the stabilisation of molecular clusters under conditions relevant for the troposphere .
Important open questions remain: for example, the relevance of ion-induced formation and growth or the involvement of organic molecules in the nucleation process . Several works strongly suggested looking at the atmospheric particle formation process from a micrometeorological perspective, including the role of turbulent fluctuations . These ideas have not substantiated, for example in the form of widely applicable models.
The lifetime of freshly formed ultrafine particles and thus their chance to make a further impact on the radiative balance and the budget of CCN (cloud condensation nuclei) crucially depends on their ability to grow to larger diameters. Only rapid growth by condensation can prevent the particles from being lost by coagulation with bigger particles . An assessment of the climate effects induced by atmospheric nucleation thus requires accurate descriptions of the nucleation process itself (on a molecular level) and the subsequent growth of the nucleation-mode particles into the Aitken and accumulation mode.
For computational reasons, large-scale atmospheric models generally use parameterisations of particle nucleation and growth processes . Aerosol particle growth due to the condensation of organic precursors is often treated in highly simplified form. The chemical transport model GEOS-Chem-TOMAS , for example, assumes that 10 % of monoterpene emissions will convert to secondary organic aerosol (SOA). The model then distributes this material onto the existing sectional size distribution according to either the mass in each section (thermodynamic limit) or the Fuchs-corrected surface area (kinetic limit). The work of includes even a variant in which the growth of particles by SOA condensation is highly size dependent in the nucleation-mode size range based on experimental evaluations . The rough estimate of a SOA yield and the inclusion of two alternative condensation mechanisms reflect the considerable uncertainties in current knowledge with regard to the condensational growth process. Overall, the degree to which particle nucleation is actually able to influence the budget of CCNs and thus terrestrial climate has to be considered highly uncertain .
Melpitz is an atmospheric research station in former East Germany where new particle formation events have been studied since 1996 . The frequency of new particle formation events at Melpitz tends to be high during the spring, summer, and autumn, with the fraction of NPF event days ranging between 30 and 50 % of all days in those seasons . The average particle formation and growth rates of particles in the size range of 3–11 nm have been estimated as 10 ${\mathrm{cm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$ and 4 nm h−1 in Melpitz and fall within the span of observations in the continental boundary layer . emphasised the fact that sulfuric acid alone is by far not sufficient to explain the subsequent growth of the nucleation-mode particles. suggested a connection between the observed decreasing trends in SO2 concentrations (65 %), the fraction of NPF events (45 %), and the particle formation rates (68 %) between 1996 and 2006. Conversely, the growth rates of nucleated particles increased by 22 % over that period. The delineation of these trends points to an independence of the chemical species responsible for particle nucleation and growth.
This paper revisits atmospheric new particle formation at Melpitz with a novel data set collected between 2008 and 2011. A neutral cluster and air ion spectrometer (NAIS) was used to detect aerosol particles from 2 nm in size and at a higher time resolution than previously available. For a total of 269 observation days, we examined correlations between new particle formation events and calculated proxies for gaseous precursors, ternary nucleation rates, and meteorological parameters including small-scale turbulence.
Table 1Overview of data coverage 2008–2011 encompassing four periods I–IV. The list gives the number of days for which a complete diurnal cycle of NAIS data was available. Further columns indicate the availability of additional parameters, such as the particle number size distribution from the TDMPS-APS, the H2SO4 proxy, and NH3. Also, the serial numbers of the two NAIS instruments are indicated.
2 Methods and data
## 2.1 The research station in Melpitz
Measurements of nucleation-mode particles and particle number size distributions were performed from 2008 to 2011 at the atmospheric research station in Melpitz, eastern Germany (5132 N, 1254 E; 87 $\mathrm{m}\phantom{\rule{0.125em}{0ex}}\mathrm{a}.\mathrm{s}.\mathrm{l}.$). The station is surrounded by flat grasslands, agricultural pastures, and woodlands within several tens of kilometres. No orographic obstacles or larger sources of pollution lie within the immediate vicinity of the station. The Melpitz station is a part of the observation networks WMO-GAW (World Meteorological Organization Global Atmosphere Watch), ACTRIS (Aerosols, Clouds, and Trace gases Research InfraStructure network), and GUAN (German Ultrafine Aerosol Network; Birmili et al.2016). Atmospheric particle size distributions at Melpitz have been regarded as representative of the regional atmospheric background in Central Europe . For the basic features of particle number size distributions and particle mass concentrations as a function of meteorological parameters, see and .
## 2.2 Instrumentation
Particle number size distributions were measured using three independent particle size spectrometers: a neutral cluster and air ion spectrometer (NAIS), mobility diameters 2.0–40 nm; a twin differential mobility particle size spectrometer (TDMPS), mobility diameters 3–800 nm; and an aerodynamic particle size spectrometer (APS), aerodynamic diameters 0.5–10 µm. Using these instruments, a total of four measurement periods were covered (see Table 1).
### 2.2.1 Neutral cluster and air ion spectrometer (NAIS)
The neutral cluster and air ion spectrometer (NAIS) is an extended version of the air ion spectrometer (AIS; Mirme et al.2007). The NAIS can measure the mobility distribution of ions plus the size distribution of neutral particles, while the AIS is only able to detect naturally charged ions. For the state of the art of this instrument, see . Briefly, the NAIS uses a charging-filtering section in order to measure particles that are neutrally charged in the atmosphere. The aerosol sample passes first through a charger–discharger unit. The instrument uses unipolar corona chargers for both charging and charge neutralisation. The neutraliser is also called the discharger. Charged particles are classified in the multichannel differential mobility analyser (DMA). The electric current carried by the particles is recorded by individual electrometrical amplifiers. The charged fraction of particles induced in the aerosol sample is estimated from the Fuchs theory . The corona ions generated in the unipolar charger are generally small (< 2 nm), with their exact size depending on concentration, air composition, polarity, and other factors related to particle charging . Excess corona ions are removed by electrical filters and leave an instrumental size range for aerosol particle classification between 2 and 40 nm that can be interpreted as originally atmospheric particles with confidence . The NAIS features two multichannel differential mobility analysers for detecting positively and negatively charged particles, respectively. By switching between different measurement modes, the NAIS can measure the mobility distribution of particles after positive and negative charging (“particle mode”) and also the mobility distribution of naturally charged particles and small ions (“ion mode”).
During our experiments two individual NAIS instruments were used. The instrument NAIS-4 was deployed at Melpitz between April 2008 and August 2009. Instrument NAIS-15 was deployed from June 2010 until October 2011. The NAIS-4 was calibrated in January 2008, showing an average performance compared to four other NAIS instruments . This performance could be verified in a follow-up calibration experiment in July 2009 . At Melpitz, the NAIS instruments sampled ambient air through a dedicated stainless steel pipe (diameter: 3.5 cm, length: 160 cm) at a flow rate of 60 L min−1. The sampling height was about 3.5 $\mathrm{m}\phantom{\rule{0.125em}{0ex}}\mathrm{a}.\mathrm{g}.\mathrm{l}.$ and 1 m above the roof of the measurement container. There were no obstacles in the NAIS sampling line except a metal grid that was designed to prevent insects from entering the instrument. The analyser columns of the instrument were cleaned every 4 weeks.
### 2.2.2 Twin DMPS and APS
Particle number size distributions were measured with a twin differential mobility particle size spectrometer (TDMPS). This instrument follows the design principle of but circulates sheath air in a closed loop in compliance with recommendations for atmospheric aerosol particle number size distribution measurements . Briefly, the instrument consists of two differential mobility particle analysers (Vienna type) connected to a condensation particle counter (model 3010; TSI, Shoreview, MN, USA) and an ultrafine condensation particle counter (model 3025; TSI), which encompass a total particle size range between 3 and 800 nm. A measurement cycle lasts for 10 min.
Coarse particles were measured in an aerodynamic size range between 0.8 and 10 µm using an aerodynamic particle sizer (model 3321; TSI) with the upper cut-off defined by the air inlet system. Both the TDMPS and the APS are connected to an automatic regenerating adsorption aerosol dryer , which ensures relative humidities below 30 % at all times in the aerosol sample. The sampling height of the corresponding inlets was about 5 $\mathrm{m}\phantom{\rule{0.125em}{0ex}}\mathrm{a}.\mathrm{g}.\mathrm{l}.$ and 2.5 m above the roof of the measurement container.
### 2.2.3 Merging multi-instrumental particle number size distributions
The NAIS, TDMPS, and APS number size distributions were merged as follows: from 2–10 nm, NAIS data were employed exclusively. The reason is that the current Melpitz TDMPS set-up suffers from enhanced particle losses below 10 nm because these measurements have been optimised with regard to long-term stability that involves the use of a regenerative dryer upstream of the instrument (see above). The extensive sampling system ensures low relative humidities in the sampling line at all times, but also causes non-recoverable particle losses at the lower tail of the TDMPS particle size distribution.
In the size range 10–20 nm, the NAIS and TDMPS number size distributions were cross-faded into each other using linear mixing as a function of logarithmic diameter between 10 nm (only NAIS) and 20 nm (only TDMPS). Above 20 nm, the NAIS size distributions become increasingly unreliable because the data inversion of that instrument does not take into account the multiple charges from particles bigger than 40 nm due to the limited size range of the instrument. Between 20 and 800 nm, TDMPS data were used exclusively, which exhibit their greatest reliability across this diameter range. Above 800 nm, APS data1 were used exclusively after converting the aerodynamic particle size distribution into a mobility particle size distribution using an effective particle density of 1.6 g cm−3.
### 2.2.4 Gas-phase measurements
Gaseous sulfuric acid (H2SO4) and the hydroxyl radical (OH) were measured during an intensive measurement period of EUCAARI (European integrated project on aerosol, cloud, climate, and air interactions) by chemical ionisation mass spectrometry (CIMS; Berresheim et al.2002). These measurements at Melpitz lasted from 1 to 31 May 2008. To make an estimate of H2SO4 for other periods, we calculated a proxy, which was determined on the basis of this 1-month data set. SO2 concentrations were measured by ultraviolet (UV) fluorescence using an APSA-360A gas analyser (Horiba; Kyoto, Japan). Ammonia concentrations were measured by MARGA (continuous Monitoring of AeRosol and GAses in ambient air; Metrohm Applikon B.V., Schiedam, the Netherlands).
### 2.2.5 Meteorological measurements and data
Local meteorological parameters, including temperature, pressure, relative humidity, horizontal wind speed, and wind direction, are collected at Melpitz on a routine basis. During an intensive campaign in 2010, 3-D wind speed was additionally measured on a mast of 6 m height using a sonic anemometer (model USA-1; METEK GmbH, Elmshorn). The sampling frequency of that instrument was 1 Hz. From these data, the turbulent heat flux ${w}^{\prime }{\mathit{\theta }}^{\prime }$ and the turbulent kinetic energy (TKE) were calculated for 15 min intervals. Meteorological back trajectories were determined by the Hybrid Single Particle Lagrangian Integrated Trajectory (HYSPLIT) model provided by the US NOAA Air Resources Laboratory.
Figure 1Definition of the OH proxy based on the experimental correlation with the global radiation flux during EUCAARI 2008 (see also Eq. 1).
## 2.3 Chemical mass balance model for sulfuric acid
Gaseous sulfuric acid (H2SO4) and hydroxyl radicals (OH) were only measured from 1–31 May 2008 (EUCAARI 2008). To scrutinise the relationship between H2SO4 and newly formed particles for the longer time period 2008–2011, the H2SO4 concentrations were estimated using a chemical mass balance model driven by solar radiation as a source of OH. A proxy for [H2SO4] under daytime conditions will need, in a first step, a proxy for [OH]. showed that there is a close relationship between [OH] and the UV solar flux. The latter is closely correlated with global solar irradiance . Figure 1 shows the corresponding relationship between the global radiation flux and [OH] for EUCAARI 2008 at Melpitz. On the basis of such a correlation, we devised the linear function
$\begin{array}{}\text{(1)}& \left[\mathrm{•}\mathrm{OH}\right]=A\cdot \phantom{\rule{0.25em}{0ex}}\text{Rad},\end{array}$
with Rad being global solar irradiance in W m−2 measured by a pyranometer and [OH] the hydroxyl radical concentration measured by CIMS in cm−3. The proportionality parameter A was derived by linear regression, yielding a value of 6110 ${\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{W}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{cm}}^{-\mathrm{3}}$.
In a second step, H2SO4 concentrations were estimated using a modified version of the chemical mass balance model introduced by .
$\begin{array}{}\text{(2)}& \left[{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}\right]=B\frac{\left[\mathrm{•}\mathrm{OH}\right]\left[{\mathrm{SO}}_{\mathrm{2}}\right]}{\mathrm{CS}}\left[{\mathrm{cm}}^{-\mathrm{3}}\right]\end{array}$
This mass balance assumes that OH radical attack on SO2 is the process governing the production rate of H2SO4. Here, [OH] is the hydroxyl radical concentration estimated from Eq. (1) in cm−3, [SO2] the measured sulfur dioxide concentration in cm−3B a constant related to the reaction rate of the two above-mentioned species, and CS the condensation sink in s−1. Here, CS was calculated from the particle number size distribution 3 nm–10 µm adjusted to ambient relative humidity. The hygroscopicity growth law necessary for this adjustment was derived from 1 year of hygroscopicity analyser measurements at Melpitz and is shown in Appendix A.
Figure 2Definition of the H2SO4 proxy based on the correlation of experimental and calculated values during EUCAARI 2008 (see also Eq. 2).
The term B[OH][SO2] represents the production term of H2SO4 and CS is the loss term of H2SO4 by condensation onto the pre-existing particle population. The parameter B was derived by regression analysis of measured and estimated [H2SO4] for 9 days of data during the EUCAARI 2008 campaign (Fig. 2). Linear regression analysis yielded a value of 2.79 × 10−12cm3 s−1 for B. It is worth noting that the parameter B seems to depend significantly on the observation site. notably obtained a value of 8.6 × 10−10cm3 s−1 for the boreal forest site Hyytiälä, Finland.
For reasons of consistency, this H2SO4 parameterisation was compared with proxy expressions used in previous work, particularly . Those authors performed an analysis of various linear and non-linear expressions for a H2SO4 proxy based on the same data set from Melpitz. For completeness, we reiterate these linear proxy expressions in Table B1 in Appendix B. The correlation results using these linear expressions are given in Fig. B1.
concluded that their formula (L3) provided the best fit for the Melpitz EUCAARI 2008 data set. For this work, however, we preferred Eq. (2) for two reasons. First, it simulates CS from the particle size distribution (2 nm–10 µm) after adjustment to ambient relative humidity. (Mikkonen's proxies used CS on the basis of a dry particle number size distribution.) Second, Eq. (2) is based on a mass balance calculation that is assumed to be valid at least for daytime conditions and avoids some cross-sensitivities and non-linear dependencies that lack a mechanistic explanation.
Figure 3Time series of aerosol and gas-phase parameters during four exemplary NPF events: (a) 19 June 2010, (b) 29 May 2008, (c) 7 June 2010, (d) 23 August 2008. Shown are particle number size distributions, the concentrations of sulfur dioxide SO2, the hydroxyl radical OH, sulfuric acid H2SO4, ultrafine particle number N[2;20], and the condensational sink CS.
3 Exemplary NPF events
Figure 3 presents four cases of new particle formation (NPF) events at Melpitz covering a range of different observations. Contour diagrams show the particle number size distribution (2–1000 nm), the number concentration of freshly produced particles N[2;20] (aggregated from the NAIS and TDMPS data), the condensational sink (CS), and the gas-phase concentrations of SO2, OH, and H2SO4. The four NPF events were chosen so that they represent a certain range of observations that are typical for Melpitz based on our subjective judgement. The diurnal course of solar radiation, including sunrise and sunset, can be tracked by the calculated OH concentrations.
## 3.1 Case 1: NPF and subsequent growth under clean conditions
Figure 3a shows an NPF event on 19 June 2010 when particle formation and subsequent growth up to diameters around 50 nm was clearly visible. The NPF event started around 06:00 CET in a clean Atlantic air mass, as confirmed by back trajectories. CS was constantly low throughout the day, as was [SO2]. Until 10:00 CET, the sky was cloudless, leading to OH concentrations calculated from Eq. (1) of around 4 × 106cm−3. The combination of an ideal solar radiation flux, low CS, and low [SO2] (1–2 × 1010cm−3) yielded moderate calculated concentrations of H2SO4 around 2 × 107cm−3. This case is an example in which variations in the production rate of H2SO4 correlate with the variations in [OH], while the concentrations in [SO2] remain almost constant. In the event classification to follow in Sect. 4, this event was classified as a Class I particle formation event.
## 3.2 Case 2: NPF and subsequent growth under polluted conditions
Like above, the NPF event on 29 May 2008 was marked by a pronounced particle growth up to around 70 nm (Fig. 3b). But in comparison to Case 1, significantly higher levels of both SO2 and CS prevailed. Figure 3b shows the trace of an Aitken mode (diameter around 60–100 nm) from the preceding day, which remains visible after the onset of NPF at 09:00 CET. Back trajectory analysis confirmed the presence of continental air originating from easterly directions. On this day, the high H2SO4 concentrations are caused primarily by the high level of SO2. CS was nearly constant before and during the onset of the NPF event and supposedly played a minor role in NPF and subsequent particle growth. In Sect. 4 this event is also classified as a Class I particle formation event.
## 3.3 Case 3: short-lived stationary NPF event
This case from 7 June 2010 represents a class of short-lived nucleation events, i.e. shorter than 2 h in duration (Fig. 3c). The NPF event started at 10:00 CET and was associated with a short peak in SO2. The size range from 2 to 20 nm was uniformly filled with aerosol particles and no growth was observed. Solar radiation produced [OH] levels with a maximum around 5 × 106cm−3 even later, but the reason for the cut-off of the NPF event was likely the drop in [SO2] at 12:00 CET. Back trajectory analysis suggested the advection of a clean maritime air mass from north-westerly directions. CS showed moderate values around 0.01 s−1 during daytime, but elevated values up to about 0.05 s−1 during nighttime.
## 3.4 Case 4: long-lived stationary NPF event
Like Case 3, this event from 23 August 2008 was characterised by a lack of particle growth (Fig. 3d). However, the duration of the NPF event was considerably longer than in Case 3, between 09:00 and 17:00 CET. Such observations are thought to be the result of a continuous influence by a stationary source or process. On this day, rather clean air from westerly directions prevailed with CS below 0.005 s−1 after 05:00 CET like in Case 1. Solar radiation and calculated [OH] were fluctuating due to changes in cloudiness. It might be worth noting that just before, CS dropped from its considerably higher nighttime level of 0.04 s−1 due to a change from a continentally influenced towards a maritime-influenced air mass. We are not aware of any nearby anthropogenic sources of particles which could explain this behaviour.
## 3.5 Patterns and shapes of NPF events
The case studies reveal that NPF events at Melpitz occur in a great variety of patterns and shapes. One basic reason for this variety is the stationary nature of the measurements at a single point, which is shared by many comparable observations at other fixed sites. During the measurement, air masses of more or less diverging composition blow past the measurement site. Melpitz is located in Central Europe, a region where spatial gradients in air composition are a regular feature. Only if the wind speed is low compared to the spatial inhomogeneities of the air mass, one can expect an idealistic observation of new particle formation and subsequent growth. Besides air mass changes due to advection, the atmosphere almost always involves vertical mixing during the periods of NPF events due to convection aroused by intense solar radiation. If the air aloft contains different concentrations of trace gases and/or aerosol particles, concentrations near the ground will inevitably change even during the NPF process. Surprisingly, these issues only play a marginal role in the wide body of literature on experimental NPF studies. It therefore represents a great challenge to examine and quantify the ongoing processes simply on the basis of ground-based measurements. While efforts have been made to characterise the atmosphere during NPF events vertically and spatially , such efforts will only yield a limited number of observations and usually a restricted set of parameters that can technically be measured on an airborne platform. To examine the statistical relevance of the NPF process, long-term data sets are needed, which inevitably require some categorisation or classification. The next chapter is therefore dedicated to the classification of NPF events at Melpitz by making use of the extended set of aerosol parameters available.
4 NPF event classification
## 4.1 Objectives of NPF event classification
Identifying and classifying NPF events is typically done with two intentions in mind:
1. examining the circumstances of fresh particle formation (i.e. gas-phase chemical, meteorological)
2. and evaluating the potential of NPF events to deliver total particle number concentration, CCN number concentration, and radiative forcing effects.
The main objective in this paper is to examine aspect (1), the circumstances of fresh particle formation. Our classification described in Sect. 4.2 is sensitive towards both high numbers of fresh particles (N[2;20]) and long durations of NPF events. The filter distinguishes events in which plenty of small particles occurred and/or that happened over a long duration. All events shown in Fig. 3 performed well under this method. For the purpose of examining aspect (1) we consider the filter adequate.
Aspect (2), i.e. questions regarding particle growth, CCN, and how many Aitken particles and optically active particles will be produced as a result of NPF, is another issue. While this aspect might ultimately be more relevant for climate and health implications than aspect (1), this needs a more extended analysis that would be beyond the scope of this paper. In this work we took major advantage of the NAIS instrument, which provides concentrations of particles down to 2 nm that are most suited to investigate aspect (1).
## 4.2 The convolution integral method
To examine gas-phase precursor and meteorological effects as a function of new particle formation (NPF) intensity, we developed a new method to classify the set of measured NPF events. The method is based on a convolution integral (CI) of time series of the number concentration of freshly nucleated particles (N[2;20]). The convolution integral is defined as
$\begin{array}{}\text{(3)}& \text{CI}\left(\mathit{\tau }\right)=\left(f\cdot g\right)\left(\mathit{\tau }\right)=\int f\left(t\right)g\left(\mathit{\tau }-t\right)\mathrm{d}t,\end{array}$
where f(t) is the time series of N[2;20], as averaged from a number of 27 manually selected NPF events, and g(t) the measured time series of N[2;20]; τ is a time lag between the two time series. The 27 selected NPF events featured very high peak values of N[2;20] and subsequent particle growth during a few hours. See Fig. C1 and Table C1 in Appendix C for the complete characteristics of the 27 events with respect to N[2;20]. The two events in Fig. 3a and b are representatives of this selection, with nucleation-mode particles growing to about 70 nm at midnight and eventually to about 90 nm on the next day.
The motivation behind the convolution integral (CI) method is to enable the automatic detection and classification of the NPF events. The CI function represents a simple time series in which NPF events can be detected as peaks in that series. The height of the peaks in the CI function is sensitive towards both the number concentration of new particles (N[2;20]) occurring during an event and the time duration of an event. Besides an automatic detection of the time window when NPF occurred, it is possible to objectively rank the detected NPF events according to the height of the detected peaks. The computation of the convolution integral also avoids some aspects that make the classification of NPF events problematic: (1) due to the finite width of the f(t) function, the CI function includes a smoothing of the original time series, which averages out possible experimental noise or very short-lived peak concentrations. This might help make the detection of NPF events more representative in that it captures the more significant events. (2) Any experimental data set might feature different time resolutions and limitations like data gaps. The CI method is able to even out such differences between different data sets in that it yields a standardised CI function on a regular time grid, which can be compared, for example, among different sites.
The weight function f(t) was calculated as an average of these 27 time series of N[2;20], with all time series centred around their peak value before averaging. In time, f(t) contains experimental values from 5 h prior to the maximum in N[2;20] to 10 h after. Outside this interval, f(t) was set to zero. No normalisation was made to the amplitude of N[2;20].
The 27 NPF events were selected to provide a realistic initialisation to the CI method. Of all properties of the function f(t), its width (relative to the timescale) is probably the most salient property. (The width of f(t) is visible as the red curve in Fig. C1.) Of all peaks in the original time series g(t) (N[2;20]), those peaks that have a similar width as f(t) will obtain a maximum response in the CI function in relation to their peak area. (This is a consequence of Eq. 3.) The width of f(t) is thus more important than its height because the height will come to effect in a multiplicative manner for all NPF events, while the width gains numerical relevance for such NPF events whose peak width in g(t) is the same or bigger than the width of f(t). The CI integral method will favour, in its ranking, events of such characteristics. For this reason, we selected the 27 most outstanding events (from visual inspection) with respect to both N[2;20] and also the continuous evolution of a new nucleation mode for a long duration as much as possible. We thought that these events are the ones that this analysis should ideally be looking for, although we would not aim to exclude other patterns of NPF events by default. As a matter of fact, the CI method will classify any day of observations on a continuous scale of CI ranging between values close to 0 and Max(CI). We are aware that the CI integral method might provide different results if, for instance, only very short-duration events are chosen. Such a choice would push NPF events with higher peak N[2;20] concentrations (even if only short-lived) higher in the ranking.
Figure 4Exemplary time series of the convolution integral CI from 16–30 May 2011 indicating the intensity of new particle formation.
In a second step, the time series of the CI was analysed for peak values. CI reaches a peak at the times when the peaks of f(t) and g(t) coincide. Because CI is calculated as an integral over concentration and time, higher peak values are reached when the NPF event represented by g(t) extends in time (cf. Fig. 3a, b) rather than being a short-lived event (Fig. 3c). This means that CI is not only sensitive to the absolute peak values of N[2;20] but also to the duration of the NPF event. Figure 4 illustrates a sample of the time series of CI with maximum values attained during midday, i.e. when NPF events take place.
In a third step, the peaks in CI(t) were detected and their peak values CIpeak subsequently classified according to their magnitude. Only the NPF events with peaks in CI occurring between sunrise and sunset were taken into account, i.e. those that can apparently be related to photochemical processes. (In fact, no significant nucleation was observed in Melpitz outside this period.) Figure 5 presents all peaks identified between sunrise and sunset as a function of time of day. As discussed before, the peak height is a combined measure of the attained particle number concentration N[2;20] and the event duration.
Figure 5Daily maximum of the convolution integral CI for all observation days as a function of the time of day of that maximum. Event classes were defined as Class I (red, intense new particle formation), Class II (blue, new particle formation at lower intensity), and Class III (green, NPF below significance level). See Table 2 for the exact threshold values.
Table 2Classification of NPF events according to their convolution integral peak (CImax) and two specific threshold values based on the complete NAIS-TDMPS data set.
## 4.3 Classification results
From the data cloud in Fig. 5, three event classes were defined as follows: Class I, showing CIpeak in the range 3 × 108–1.2 × 109s cm−6, Class II with CIpeak in the range 7 × 107–3 × 108s cm−6, and Class III with CIpeak below 7×107s cm−6 (see Table 2).
The motivation for the boundaries between the event classes is as follows: Class III represents the 83 NPF events of lowest intensity. As the NAIS instrument is very sensitive, it is able to detect short-lived peaks of small particles, even at very low concentration. In fact, a peak of N[2;20] can be defined for each day, no matter how low it might be. As can be seen from Fig. 5, these short and low peaks may take place any time between sunrise and sunset. We associate these very weak events with very small-scale particle bursts that do not evolve into a fully developed and spatially distributed nucleation event. In any case, this class of observations includes what most researchers would call “non-events”.
Class II represents 92 NPF events that take place at least a few hours after sunrise, i.e. when the atmospheric boundary layer has started to mix vertically. These events are usually longer-lived and reach higher concentrations in N[2;20]. The requirement of Class II events to surpass the threshold CIpeak= 7 × 107s cm−6 is clearly motivated by the shape of the data cloud in Fig. 5. Below this threshold, a daily maximum concentration of N[2;20] may take place any time between 02:00 and 20:00 CET, while the events above this threshold always exhibit a start time between sunrise and sunset, which is the case expected for photochemical NPF events.
Class I, in turn, represents the 94 most intense NPF events. These are always associated with high absolute values of N[2;20] and an event duration over several hours. Most of them, although not all, showed a clear particle growth pattern similar to that in Fig. 3a and b. The threshold in CIpeak between Class I and Class II events is somewhat arbitrary. In fact, we are facing a continuum of observations ranging from the lowest to the highest observations in NPF intensity. Guided by practical needs, we have attempted to create data subsets of similar dimension and have also tried to define a threshold above which the obvious particle growth pattern is a clear majority. This led to the threshold value of 3×108s cm−6.
Table 3Comparison of two classification schemes for new particle formation events: the CI method (Class I, Class II and Class III; see Table 2) and the University of Helsinki (UHEL) classification originally reported in .
## 4.4 Comparison with other classification methods
The introduction of a new NPF classification method requires some justification. Continuous observations of NPF events in the continental boundary layer with particle mobility spectrometers have been carried out since the mid-1990s . Continuous monitoring of air ions dates back even further to the 1980s (Hõrrak et al.2003, and references therein). Since then, there have been various attempts to classify NPF events according to their relevant features and parameters, including the following approaches.
1. The University of Helsinki classification : this elaborate method has been widely used to classify NPF events after several criteria, including the existence of a continuous trace of a nucleation mode, and whether apparent particle formation and growth rates can be derived with confidence. Somewhat problematic is the softness of some criteria, such as whether the mode concentration and diameter fluctuate strongly. Recent work has refined the nucleation-mode classification , now classifying many previously “undefined” new particle formation events.
2. Methods based on peak values in absolute particle number concentration, sometimes requiring a certain shape of the evolution of the time series of nucleation-mode particle number concentration (e.g. Birmili et al.2003).
3. Identification of new particle formation events based on the time series of multiple moments of the particle number size distribution .
Our newly developed scheme is tailored to the combined NAIS-TDMPS observations at the rural background Melpitz for the following reasons.
• The number of freshly formed particles (here N[2;20]) is, after all, the most basic and most important indicator of recent particle nucleation. Any other parameters, such as apparent particle formation rates (often estimated by ΔN∕Δt or by a time delay between precursor concentrations and N) or particle growth rates, are subject to inherent uncertainties, such as those induced through air mass changes by convection and/or advection (Sect. 3.5).
• At Melpitz, we found it hard to quantify the growth of neutral particles below 10 nm by tracking a mode in the NAIS or TDMPS size distributions. The observations indicate that if particles appear in significant numbers at the surface-based research station, they will appear across the entire interval 2–10 nm or even beyond (cf. Fig. 3a–d). When the total particle number concentration reaches its maximum, the nucleation-mode particles have very often reached the region of 20 nm in the size distribution already (Fig. 3a–d). Above that range 10–20 nm, the subsequent particle growth can usually be followed nicely using the TDMPS-based range of the size distribution (cf. Fig. 3a–b). These observations are a justification to use N[2;20] as an indicator for NPF events. The relatively wide interval N[2;20] also has the technical advantage that it produces a statistically sound signal with a low noise level.
• Our method avoids the common problem of rigorously distinguishing between NPF events and non-events. Acknowledging the true observable continuum of observations between “zero” and “top-level” concentrations, we rather introduce three classes according to different degrees of NPF intensity.
• Our method has a high degree of objectivity. (This means that it can be recorded in a way that any other researcher can reach exactly the same classification results.) This makes it similar to the approach by . Some subjectivity arises from the choice of the 27 NPF events that serve as a “calibration” of the method (Table C1) and from the threshold values for CIpeak selected to separate the events into Classes I, II, and III, although these criteria can be recorded explicitly (Table 2).
The comparison between the CI method and the University of Helsinki classification is shown in Table 3. Naturally, the two methods show a strong correlation when distinguishing between different degrees of observed particle formation. The days in UHEL class 1a coincide, for example, to 72 % with CI Class I. UHEL non-events coincide to 85 % with the analogous CI class 3. On the other hand, CI Class I splits up more evenly into UHEL classes 1a, 1b, and 2. One reason is that the UHEL scheme evaluates additional issues, such as whether the evolving nucleation mode can be clearly tracked over time (i.e unobstructed by background aerosol) or not. These are not issues in the CI method, which primarily weighs the number concentration of the observed particles and the duration of an NPF event.
Figure 6Average diurnal cycles of atmospheric parameters for the three NPF event classes: red indicates a Class I event, blue indicates a Class II event, and green indicates Class III (weak events and “non-events”). The subfigures show concentrations of (a) ultrafine particles (N[2;20]), (b) sulfur dioxide (SO2), (c) hydroxyl radicals (OH), (d) the condensational sink (CS), (e) calculated sulfuric acid (H2SO4), and (f) ammonia (NH3) using a constant value of 5 ppt, (g) relative humidity (RH), (h) temperature (T), (i) ternary nucleation rates (TNR) calculated according to and under the assumption of a constant ammonia concentration [NH3] = 5 ppt, (j) absolute humidity (AH), and (k) ozone (O3) concentrations. Data coverage: Class I (55 days), Class II (60 days), Class III (67 days). The σ values, indicated by whiskers, represent the standard error of the mean of each subpopulation. (Technically, this is calculated as $\mathit{\sigma }/\sqrt{n-\mathrm{1}}$.) The arithmetic means of the event peak times were 10:48 CET for Class I, 11:54 CET for Class II, and 11:46 CET for Class III.
Figure 7Average time series of (a) the concentrations of ultrafine particles (N[2;20]), (b) the vertical turbulent heat flux (${w}^{\prime }{\mathit{\theta }}^{\prime }$), and (c) turbulent kinetic energy (TKE) for the year 2010 and three event classes (red indicates a Class I event, blue indicates a Class II event, and green indicates Class III including weak events and “non-events”). Whiskers indicate 1 SD. Data coverage: Class I (19 days), Class II (17 days), Class III (27 days).
5 Correlations with gas-phase and meteorological parameters
## 5.1 Time evolution of NPF events
Having classified NPF events into strong, medium and weak NPF events, we now scrutinise the entire data set for correlations with gaseous precursors and meteorological parameters. Figure 6 shows the average diurnal cycles of measured atmospheric parameters that are considered relevant for the NPF process. Figure 7 adds the diurnal cycles of micro-meteorological parameters including the vertical turbulent heat flux and turbulent kinetic energy, which were collected in the year 2010. Importantly, the diurnal cycles of all parameters were moved in time prior to averaging, with the time of their peak in N[2;20] being set to t= 0. Each curve represents an arithmetic average over all days within the subsets defined in Table 2. [OH] and [H2SO4] were estimated by the proxies in Eqs. (1) and (2). The ternary nucleation rates TNR were calculated according to using the in situ measurements or estimates for T, RH, [H2SO4], and [NH3]. Because of the limited data availability of [NH3] (2010 and 2011), a sensitivity analysis for ammonia concentrations was performed separately. A corresponding Fig. D1 can be found in the Appendix. Since the inclusion of ammonia in the analysis did not alter our conclusions, we feel confident in basing the conclusions on the full observation period 2008–2011 and the constraint of using a constant ammonia concentration of 5 ppt.
### Time around sunrise (−6 h)
We start the description of the results 6 h prior to the event peak time, which is ca. 04:40 CET for Class I events, ca. 05:50 CET for Class II events, and ca. 05:10 for Class III events. This is the time before or just around sunrise on most of these days.
At this time, we see little to indication from the locally measured parameters of whether an NPF will happen or not a few hours later or which intensity the event will have: solar radiation and [⋅OH]calc are low, around 0.7–1.2×106cm−3. Ozone levels are very similar for all event classes, around 6×1011g m−3. [SO2] is the same for all event classes, just below 1.1×1010cm−3, as is RH at around 88 % on arithmetic average. [H2SO4]calc is at negligible levels, as is ternary nucleation rate (TNR). (As mentioned above, TNR was calculated according to Napari et al.2002). Also, the turbulent heat flux available for the 2010 measurements (${w}^{\prime }{\mathit{\theta }}^{\prime }$) is very similar at around 0.01–0.025 K m s−1.
The few minor indications for NPF events to come are (1) Class I events show early morning temperatures below average. (2) Class I and II events show turbulent kinetic energy TKE below average. (3) Class I and II events show a condensation sink CS above average, and this CS is declining more rapidly than on non-event days. The meteorological indications (1) and (2) point to a surface layer that is highly stratified and calm on the morning of NPF events (Classes I and II). The rapid decrease in CS can be taken as an indicator for two processes: (1) vertical mixing is more efficient on NPF days, apparently driven by solar radiation, and (2) rapidly rising temperature transfers semi-volatile particulate matter, such as ammonium nitrate, and semi-volatile organics into the gas phase. Evidence for the latter process was given by highly time resolved measurements of chemical particle composition at Melpitz .
### First indications of NPF event (−3 h)
The evolution of many parameters is already indicative of whether an NPF event will happen or not 3 h before event peak time. Most importantly, solar radiation ([OH]calc) is substantially higher on Class I and II event days compared to Class III event days. As a direct response, the near-surface temperature T is rising rapidly, and RH is decreasing. A significant increase in absolute humidity can be seen on Class I and II days 3 h before event peak time. This is interpreted as the vaporisation of the dew covering the grassland surrounding the Melpitz site.
CS decreases rapidly on the Class I and II days (Fig. 6d), while this effect is much less pronounced on Class III days. We attribute this effect to two reasons: first, semi-volatile compounds, such as ammonium nitrate, and semi-volatile organic matter present in the aerosol will partition from the particulate phase into the gas phase as ambient temperature rises. The importance of this effect has been demonstrated for the Melpitz site by mass spectrometric particulate matter measurements . Second, vertical mixing starts in the lower layers of the atmosphere under the influence of intense solar radiation. This, as an overall effect, tends to dilute aerosols present in the surface layer.
We also checked the possible influence of local sources of trace gases and particles on the diurnal cycle of CS. In the warm season, which is representative of Class I and Class II event days, local and regional emissions of primary particles are overwhelmingly made up of sources like vehicular traffic. The black carbon (BC) mass concentration, which may be regarded as representative of such emissions, exhibits a weak average diurnal cycle, changing between 0.44 µg m−3 at midnight, 0.5 µg m−3 around 07:00 CET, and 0.32 µg m−3 around 16:00 CET. We attribute this morning maximum to local anthropogenic emissions from traffic. This maximum becomes visible only after the decline in CS has started. As shown in Fig. 6d, CS changes by a factor of approximately 5 between nighttime and daytime on Class I and II event days. As BC makes up less than 10 % of total particle mass at Melpitz, we rule out the possibility that any temporal changes in local anthropogenic emissions from traffic or domestic sources can account for the observed decline in CS. We conclude that the partitioning of semi-volatile particulate matter into the gas phase and vertical mixing are the major effects reducing CS before NPF events.
A key observation is the increase in [SO2] on Class I and II event days around 3 h before event peak time. From this time, the number of newly formed particles N[2;20] increases in proportion with [SO2]. It needs to be noted that within a radius of 100 km around Melpitz, sources of SO2 are scarce. In Germany, SO2 is emitted in noticeable quantities by single point sources (power plants) and domestic heating. Point sources are, as a matter of fact, far away from Melpitz, while domestic heating is likely to be irrelevant in the warm season of concern. Our interpretation is that the morning increase in near-surface [SO2] is caused by a combination of two processes: (i) first, [SO2] depletes at night due to dry deposition onto the surface. The deposition of SO2 onto the surface was confirmed in early experiments at Melpitz by gradient measurements . This depletion of near-surface [SO2] yields the typical values of 1.1 × 1010cm−3 in the early morning hours, regardless of whether an NPF event will take place or not (Fig. 6b). (ii) Vertical mixing, starting gradually after sunrise, will cause entrainment of SO2 from greater heights where SO2 did not have the opportunity to deposit. During past field experiments, nocturnal low-level jets have been shown to advect SO2 to the Melpitz area at heights of a few hundred metres, which were entrained to the ground after the onset of convection (Beyrich1994). (Nocturnal low-level jets originate from geostrophic winds and are able to advect air over long distances above a firm temperature inversion near the ground.) Unfortunately, we did not have the means to verify this hypothesis with rigour during this experiment.
### Maximum in nucleation-mode number concentration (0 h)
Event peak time (t=0) was defined by the maximum in freshly formed particles N[2;20]. Class I events feature arithmetic mean concentrations around 1.1 × 105cm−3, and Class II events around 3.7 × 104cm−3 (Fig. 6a). Event peak time coincides with the maximum of solar radiation and [OH]calc (Fig. 6c). [SO2], [H2SO4]calc, and TNR rise from Class III to Class I events. It is worth noting that on Class I days [SO2] exhibits an additional steep rise just before event peak time, emphasising the strong connection between N[2;20] and [SO2]. This peak translates into the proxy [H2SO4]calc and TNR as well. T, RH, and [O3] do not significantly differ between Class I and II events at t=0; they show the typical features of a near-surface measurement on a cloudless day. Absolute humidity decreases on Class I and II event days towards the middle of the day, which is interpreted as mixing with relatively dry air from aloft. It is noteworthy that on Class I event days, CS increases just in time with the maximum of N[2;20] and along with a continuing rise in [SO2]. In several case studies, we observed something which we interpreted as the simultaneous entrainment of SO2 and CS (e.g. 23 August 2008 in Fig. 3d). CS correlates most strongly with the number of bigger particles, i.e. in the Aitken and accumulation mode. It is our interpretation that in these cases, CS originates from the same or similar pollution sources that emit SO2. The newly formed particles < 20 nm contribute only little to CS, at most 15 % during event peak time for event class I, and much less outside that period.
### Development after NPF event peak time (t>0h)
After event peak time, the parameters N[2;20], [OH]calc, [H2SO4]calc, and TNR decrease to their pre-event levels within a matter of a few hours. It is an interesting feature that for both Class I and II events the peak in [SO2], like the peaks in [H2SO4]calc and TNR, occurs around 1 h later than the peak in ${N}_{\left[\mathrm{2};\mathrm{20}\right].}$ This implies that the entrainment of air rich in [SO2] continues even after some other parameter has started to waive the nucleation process.
## 5.2 Micrometeorological parameters
For the third measurement period in 2010, three-dimensional (3-D) wind parameters were measured at 1 s resolution 6 m above the ground with an ultrasonic anemometer. From the 3-D wind velocities, various turbulence parameters were calculated with a time resolution of 15 min. In Fig. 7 we illustrate the parameters that proved most sensitive to the class of NPF event, the turbulent heat flux ${w}^{\prime }{\mathit{\theta }}^{\prime }$, and the turbulent kinetic energy (TKE).
A prime result is that in all cases of Class I and II events, the boundary layer was turbulently mixed. In fact, we could not see a significant difference between Class I and II days with respect to the turbulence parameters. In contrast, a rather weak flux and TKE prevailed in Class III events. The diurnal evolution of the turbulence parameters is in close correspondence with the development of solar radiation and temperature (Fig. 6).
## 5.3 Reasons for the different peak times in N[2;20]
For the event peak times shown in Table 2, the difference between Class I and II is noticeable. Class III exhibits only low peaks in N[2;20] compared to the rest so that their time of peak concentrations is subject to considerable uncertainties. Class I events take place, on average, 52 min earlier than Class II events. We observed two prime differences between those event classes: (1) temperature rises faster on the mornings of Class I events, and (2) SO2 concentrations increase faster on the mornings of Class I events. Observation (1) has implications in that air from elevated layers will be mixed down to the ground sooner on Class I days compared to Class II days. Observation (2) points to the efficient downward mixing of possible SO2 plumes that are aloft. Recent research showed the presence of SO2-enriched atmospheric plumes and layers above the Melpitz site, where particle nucleation might have taken place some time before NPF was detected on the ground . It has been suggested that certain NPF events apparently start in a layer a few hundred metres aloft to be measured near the ground only after considerable delay. Two factors might cause Class I events to occur earlier than Class II events: (a) more rapid transport of elevated layers (often SO2-enriched at Melpitz; ) where nucleation can take place before it might be observed on the ground and (b) the presence of higher SO2 concentrations requiring less time until H2SO4 concentrations pass the threshold at which nucleation can take place. These explanations are still somewhat hypothetical, and an attempt to prove them will require concurrent observations in the relevant vertical layers above the flat-terrain site Melpitz.
## 5.4 Statistical significance
Remarkable differences in observed atmospheric conditions were found between Class I, II, and III event days and are discussed from Sec. 5.1 on. To supply a statistical statement, we performed Student's t tests to check whether the parameters [OH], [SO2], [H2SO4], and CS were indeed different between these classes on a statistical level.
A Student's t test was used to decide wether the means of two populations (for example, CS on Class I and II event days) could be considered equal (null hypothesis) or different within statistical significance. Student's t distributions (Student1908) were used because they refer to the probability distribution of the mean of a normally distributed population in situations for which the sample size is small and population SD is unknown . As a significance level of the test, we chose 99 %. A test was performed for every pair of 15 min mean values of the aforementioned parameters. The tests determine the significance of the differences in mean values that can be seen in Fig. 6.
The result was that Class I and Class II are significantly different from Class III (weak events or non-events) in terms of [OH] (also called solar radiation), [SO2], and [H2SO4] for every 15 min interval of the period between 4 h prior to and past event peak time. Significant differences could even be confirmed for CS during most of that time and, in addition, for the differences between Class I and Class II regarding [OH], [SO2], and [H2SO4]. These results confirm that many of the atmospheric conditions found during Class I, Class II, and Class III events were substantially different and can thus be interpreted as influential factors for the occurrence of an NPF event of the corresponding class.
## 5.5 Examining the particle formation rate J2
To sum up the discussion of process parameters derived from the NPF events, formation rates of 2 nm particles (J2) were determined from the particle number size distributions measured by NAIS. The number concentration of particles in the size range 2–3 nm, N2–3, was integrated from the measured size distributions. Inspection of the data showed that during NPF events, the signal-to-noise ratio of the NAIS instrument at 2.06 nm is above the detection limit when averaged over 15 min intervals. J2 was calculated from the time derivative of N2–3, taking into account the coagulation losses of 2–3 nm particles onto larger particles and condensation growth out of the 2–3 nm size range as described in .
Figure E1 shows the correlation between the calculated ternary nucleation rate TNR and the measured number concentration of 2–20 nm particles N[2;20] with the calculated H2SO4 concentration. Figure E2 shows corresponding data for the particle formation rate J2. Interestingly, N[2;20] seems to correlate more strongly with H2SO4 than J2 or TNR. One reason for the lower correlation between J2 and H2SO4 could be that the calculated J2 values can be more uncertain than the directly measured N[2;20] concentrations, making the J2 vs. H2SO4 more scattered. The J2 values obtained in this study fall within the same correlation with H2SO4 as observations made at other sites during the EUCAARI 2008–2009 campaign (right graph in Fig. E2; reproduced from Kerminen et al.2010).
6 Discussion
## 6.1 Basic findings of this work
As can be seen in Fig. 6, the intensity of newly formed particles (expressed by the three different classes based on N[2;20]) correlates with [OH]calc, [H2SO4]calc, [SO2], and TNR on a diurnal scale. The most significant discrepancy between Class I–II and Class III events is made up by different levels of global radiation, manifested by [OH]calc. It can also be seen that peaks in N[2;20] and [OH]calc coincide within 30 min for event Class I and II. This simple and rather established correlation between nucleation-mode particles and solar radiation (e.g. Boy and Kulmala2002) seems to represent the most basic impact influencing NPF at Melpitz.
[H2SO4]calc turns out to be another major influential factor: the magnitudes of the daily peaks in N[2;20] and [H2SO4]calc scale in proportion across the three different classes. The effect of [H2SO4]calc can be broken down into the effects of [SO2], [OH]calc, and CS. The difference in [H2SO4]calc between Classes I–II and III is mainly made up by radiation ([OH]calc), while the difference in [H2SO4]calc between Classes I and II is primarily accounted for by different levels of [SO2]. The effect caused by differences in CS is comparatively minor; CS is slightly lower during Class II events than during Class III events, allowing for a higher steady-state [H2SO4]calc. The combination of the in situ measurements or estimates for T, RH, and [H2SO4] also yields the ternary particle nucleation rate shown in Fig. 6i. This essentially propagates the trend found for [H2SO4]calc, but does not yield significant new insights.
We obtained the following descriptions of different classes of NPF events at Melpitz.
• Class I: days with significant solar radiation and high [SO2] levels.
• Class II: days with significant solar radiation but average [SO2] levels.
• Class III (containing weak events and non-events): days with significant cloud cover.
Many other features, such as the trend towards high temperatures (T), low relative humidities (RH), and a higher ozone mixing ratio [O3], can be directly linked to solar radiation as the prime source of these meteorological and photochemical processes. It is intriguing that the diurnal cycles of T, RH, [O3], [OH]calc, ${w}^{\prime }{\mathit{\theta }}^{\prime }$, and TKE are very similar for event Classes I and II, but rather different from those in event Class III (including non-events). This suggests that the meteorological and photochemical processes on the days of Class I and II events are very similar.
## 6.2 Comparison with findings worldwide
To reinforce the findings of this work, we discuss to what extent the results depend on the observation site in Melpitz or, conversely, how they may be considered as general findings. Among the plethora of literature on the topic, we found certain key works from the following groups.
Fundamental study on the influence of solar radiation. show strong correlations between NPF events and solar radiation at the boreal forest research site in Hyytiälä (SMEAR-II) in Finland. The preferred band of solar radiation was UV-A, while the study also found an anti-correlation with water vapour. The statements about solar radiation and water vapour fully agree with this work.
Studies suggesting a critical influence of SO2 and/or H2SO4. report, for two sites in Canada and the northern US, that “SO2 and UV-B were highly correlated with particle concentration, suggesting a high association of photochemical processes with these local NPF events.” report nucleation events during the Pittsburgh Air Quality Study, concluding that “local nucleation events were usually associated with elevated SO2 concentrations”. , from the same campaign, report that sulfate appeared to be the major species involved in the early growth of nucleation-mode particles, while relevant growth due to organic species was to begin only later. report a similar strong correlation between NPF events in Atlanta, US and anthropogenic SO2 as a precursor. report, for observations in Mexico City, that “concentrations of particles with diameter greater than 10 nm increased an order of magnitude, and concentrations of sub 10 nm diameter particles increased at least 2 orders of magnitude over concentrations just before the event or on a day without nucleation. Large increases in SO2 concentrations and northerly winds also coincide with these events.”
In the Chinese megacity Beijing, located in a temperate climate and featuring high rates of anthropogenic particulate and gaseous emissions, the influence of SO2 and H2SO4 as precursors for NPF could be confirmed as well . Statistically, however, the highest nucleation-mode concentrations due to photochemical production could be found in clean air masses where CS is low . report, for a site in the South African savannah, that “the occurrence of new particle formation and growth was strongly dependent on sulfuric acid” with SO2 as a precursor and that “the contribution of sulfuric acid to the growth immediately after nucleation was significant.”
Comparative studies in Europe, usually including Melpitz data. compared observations similar to this work (NAIS measurements) at 12 observation sites across Europe. Among these sites, Melpitz exhibited the highest fraction of NPF days for the observation period (57 %). confirmed that at Melpitz, NPF events showed little sensitivity to CS, while at other background sites (Hyytiälä, Cabauw, Hohenpeissenberg, Finokalia) there was a clear trend towards lower CS on NPF event days. compared NPF event statistics and correlations for the sites Hyytiälä (Finland), Melpitz (Germany), and San Pietro Capofiume (Italy). They conclude that nucleation was found to occur frequently at all stations although “seasonal differences were observed for every station.” They conclude that in Hyytiälä the formation and growth of particles was characterised by a low pre-existing condensation sink and high biogenic VOC concentrations associated with the biological growth season, while in Melpitz and San Pietro Capofiume the high level of pollution arriving from the nearby industrial and agricultural sources plays a major role.
In summary, the correlation between NPF and solar radiation has been confirmed in a few statistically relevant studies, as has the connection of NPF events and anthropogenic SO2 plumes. On the issue of CS, the conclusions in the various works are in less agreement. In clean environments where SO2 levels are low, CS seems to be a factor unfavourable for NPF, while in areas with moderate SO2 levels, the influence of variations in CS steps back behind the dominating influence of solar radiation and SO2. In areas with extremely high levels of CS and gaseous pollutants, the occurrence of NPF events might even be limited by high CS. So far, we found no study analysing the role of ammonia on a longer statistical basis. In this respect, we consider our study a novelty.
At the research station Melpitz, NPF occurs rather frequently, with the majority of NPF events being under the influence of anthropogenic SO2 plumes as a main precursor for H2SO4 and subsequent nucleation. Among other observations, Melpitz compares best with the San Pietro Capofiume site in the Italian Po Valley and the various North American sites. NPF at Melpitz clearly behaves in a different fashion from continental background sites such as SMEAR-II in Finland, mountain sites, coastal sites, and heavily polluted locations such as Chinese megacities.
## 6.3 Where does nucleation take place?
The basic correlation of N[2;20] with [H2SO4] is interpreted as H2SO4 being a main factor responsible for the formation of new particles. There has, however, been the issue of where in the boundary layer particles would actually nucleate. If particles were formed above the ground and brought down through mixing, particles might be larger than 2–3 nm when they reach the surface. New observations have been made very recently using unmanned aircraft. Observations made by suggest that NPF events may start a few hundred metres above the surface to be measured near the ground only after considerable delay. During this delay time, it is natural to assume that particles grow to bigger diameters (up to 20 nm) before they are detected at ground level. These observations might have implications for the observed correlations between NPF parameters and gaseous precursors.
The correlation of N[2;20] with [H2SO4] being more solid than the correlation of J2 with [H2SO4] is somehow surprising because J2 refers to particles in the size range 2–3 nm that should actually be more close to the process of nucleation, which is thought to be initiated by [H2SO4]. Recent experimental work has raised one suspicion: if one assumes that particles are formed aloft and only subsequently mixed down to the ground , this would mean that many of the smallest particles have already grown into bigger sizes. Hypothetically, the actual nucleation could have terminated already when the particles from the nucleation burst reach the ground, and the particle number size distribution could well be shifted to the bigger sizes in the 2–20 nm interval. When looking at the NAIS observations from this work (Fig. 6), we almost always see new particles in a wide size range (2–10 nm or even bigger) when they first appear at the ground. This would then imply that the statistical connection between J2 and [H2SO4] is weakened because the smallest particles have already dynamically evolved at the time of measurement, while N[2;20] appears to be a better representative of the outcome of the nucleation process that happened at a previous moment upwind. This would have the consequence that ground-based measurements of the 2–3 nm particles might not necessarily be a useful indicator for nucleation processes happening at higher regions of the boundary layer and that N[2;20] might actually be a better representative of the outcome of the nucleation process. These conclusions are very tentative, since to date no comprehensive four-dimensional in situ data have been collected that would permit the establishment of the true spatial evolution of boundary layer NPF events .
7 Conclusions
This paper revisited the new particle formation process (NPF) in the Central European boundary layer at the Melpitz station using a new data set involving neutral cluster and air ion spectrometer (NAIS) data for 2008–2011. Particle formation events were classified by an automated method based on the convolution integral of particle number concentration in the diameter range 2–20 nm. In analogy to previous field studies, the intensity of solar radiation was confirmed as the main factor controlling the occurrence of NPF events. The absolute number of observed particles in the diameter range 2–20 nm, however, varied mainly in proportion with the concentration of sulfur dioxide as the presumed main precursor of sulfuric acid. This is consistent with a model picture that UV radiation is instrumental in generating OH radicals, which in turn form H2SO4 via OH radical attack on SO2. The condensation sink CS played a minor role in the NPF process in that its values were rather similar on event and non-event days. The same held for experimentally determined ammonia concentrations, a potential precursor of particle nucleation. It thus appears that at Melpitz, ammonia is always available in excess. The analysis of micrometeorological turbulence parameters demonstrated the presence of significant turbulence in the boundary layer on NPF events. Due to its close correlation with solar radiation, however, an independent effect of turbulence for NPF could not be established with certainty. An analysis of the diurnal cycles of aerosol, gas-phase, and meteorological parameters suggests that particle nucleation tends to happen aloft in the residual layer, i.e. in the remains of the mixed layer of the previous day. As a rationale we put forward the nighttime depletion of sulfur dioxide near the surface, the higher probability of particle nucleation at lower temperatures aloft, and the frequent observation of aged nucleation-mode particles (at least 10–20 nm in diameter) during observations of NPF near the ground.
Data availability
Data availability.
Particle number size distributions measured by the TDMPS are available through the German Ultrafine Aerosol Network (GUAN; Birmili et al., 2016) at hourly time resolution (doi:10.5072/guan). All other data products used in the study are available from the Leibniz Institute for Tropospheric Research (Johannes Größ and Alfred Wiedensohler) upon reasonable request.
Appendix A: Hygroscopic growth parameterisation
For the adjustment of particle number size distribution and thus CS to ambient relative humidity (Sect. 2.3), an empirical growth law based on an entire year (mid-2008–mid-2009) of hygroscopicity analyser (H-TDMA) measurements at Melpitz was used. The growth factors were measured at 90 % RH for the dry particle diameters 50, 75, 110, 165, and 265 nm. Parts of those data are illustrated in . The formula allows us to compute the hygroscopic growth factor as a function of dry particle diameter and relative humidity as follows:
$\begin{array}{}\text{(A1)}& \text{HGF}\left({D}_{\mathrm{p}},\text{RH}\right)={\left(\mathrm{1}-\frac{\text{RH}}{\mathrm{100}}\right)}^{\mathit{\gamma }\left({D}_{\mathrm{p}}\right)\frac{\text{RH}}{\mathrm{100}}},\end{array}$
with the exponent factor γ being parameterised as
$\begin{array}{}\text{(A2)}& \mathit{\gamma }\left({D}_{\mathrm{p}}\right)=\mathrm{0.20227}-\frac{\mathrm{0.1082}}{\mathrm{1}+{e}^{\frac{{D}_{\mathrm{p}}-\mathrm{118.4}}{\mathrm{21.35}}}}.\end{array}$
Appendix B: Proxy functions for sulfuric acid parameterisation
This work examined several proxy functions for the parameterisation of sulfuric acid concentration in Melpitz. Table B1 lists these linear functions, which were used in previous work . Figure B1 shows the associated data clouds based on 9 days of EUCAARI 2008 measurements.
Table B1Linear proxy functions for sulfuric acid parameterisation; reproduced from , Table 3.
Figure B1Extended version of Fig. 2 illustrating alternative expressions for the H2SO4 proxy listed in Table B1 : (a) this work, Eqs. (1) and 2, (b) L1, (c) L2, (d) L3, (e) L4, (f) L5. The data clouds refer to 9 days of measurements during the EUCAARI campaign in 2008.
Appendix C: Characteristics of the 27 manually selected NPF events
The convolution integral in Sect. 4, Eq. (3) uses a function f(t) that is characteristic of significant NPF events; f(t) is an average time series of N[2;20] based on 27 manually selected NPF events. The selected NPF events featured high peak values of N[2;20] and subsequent particle growth during a few hours, such as those shown in Fig. 3a and b. The weight function f(t) represents the average time evolution of N[2;20] during the most intense NPF events. The original time series of N[2;20] are shown in Fig. C1, with their time bases shifted so that their peaks coincide. The list of the 27 NPF events with their associated maximum N[2;20] concentrations is given in Table C1.
Figure C1Time series of N[2;20] for the 27 manually selected NPF events listed in Table C1. The red curve indicates the arithmetic average of the time series, which are shifted so that they coincide in maximum number concentration.
Table C1List of 27 manually selected NPF event days whose average diurnal profiles of N[2;20] served as a reference function f(t) in Eq. (3). This list encompasses NPF events that showed clear patterns of particle nucleation and subsequent growth in terms of particle number size distributions.
Appendix D: Alternate version of the diurnal cycles
Figure D1 provides an alternative version of Fig. 6. The data in these graphs are, however, limited to the years 2010–2011 when experimental ammonia concentrations were available.
Figure D1Alternative version of Fig. 6 limited to the years 2010–2011 when experimental ammonia concentrations were available. The graphs show average diurnal cycles of atmospheric parameters for the three NPF event classes: red indicates a Class I event, blue indicates a Class II event, and green represents a Class III event, including weak events and “non-events”. The subfigures show the concentrations of (a) ultrafine particles (N[2;20]), (b) sulfur dioxide (SO2), (c) hydroxyl radicals (OH), (d) the condensational sink (CS), (e) sulfuric acid (H2SO4), and (f) ammonia (NH3), (g) relative humidity (RH), (h) temperature (T), (i) the calculated ternary nucleation rate (TNR) according to , (j) absolute humidity (AH), and (k) ozone (O3) concentration. The σ values, indicated by whiskers, represent the standard error of the mean of each subpopulation. (Technically, this is calculated as $\mathit{\sigma }/\sqrt{n-\mathrm{1}}$.) Data coverage: Class I (33 days), Class II (35 days), Class III (40 days). The arithmetic mean event peak times were Class I (10:48 CET), Class II (11:54 CET), Class III (11:46 CET).
Additional diagrams are provided to show cross-correlations within the data set. Figure E1 provides dependencies of the calculated ternary nucleation rate (TNR) and N[2;20] from [H2SO4], while Fig. E2 gives the correlation between the particle formation rate J2 and the calculated sulfuric acid concentration [H2SO4]. The figures provide a means of consistency checking with other studies.
Figure E1(a) Correlation between the ternary nucleation rate (TNR, with NH3= 5 ppt) and ultrafine particle number concentration (N[2;20]). (b) Correlation between the particle number concentration N[2;20] and the calculated sulfuric acid concentration [H2SO4] in this work. Colours refer to event class Class I (red), Class II (blue), and Class III (green).
Figure E2Left: correlation between the particle formation rate J2 and the calculated sulfuric acid concentration [H2SO4] in this work. The diagram involves 33 cases of NPF events between 2008 and 2011. Right: the same relationship from .
Competing interests
Competing interests.
The authors declare that they have no conflict of interest.
Acknowledgements
Acknowledgements.
The measurements and data analyses for this work were performed with support from the following research and infrastructure projects: EUCAARI (European Integrated project on Aerosol, Cloud, Climate, and Air Quality Interactions), European Commission FP6 contract 34684, ACTRIS (Aerosols, Clouds, and Trace gases Research InfraStructure Network), European Commission FP7 contract 262254, and the German Federal Ministry of the Environment (BMU) grants F&E 370343200 and 371143232. U. Hõrrak acknowledges institutional research funding IUT20-11 from the Estonian Ministry of Education and Research. Credit also goes to the following TROPOS personnel: Achim Grüner (Melpitz station), Thomas Tuch (maintenance of physical aerosol measurements), Kay Weinhold (quality assurance), and Jonas Hesemann (operation of the NAIS in 2008 and 2009).
Edited by: James Allan
Reviewed by: two anonymous referees
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Riccobono, F., Schobesberger, S., Scott, C. E., Dommen, J., Ortega, I. K., Rondo, L., Almeida, J., Amorim, A., Bianchi, F., Breitenlechner, M., David, A., Downard, A., Dunne, E. M., Duplissy, J., Ehrhart, S., Flagan, R. C., Franchin, A., Hansel, A., Junninen, H., Kajos, M., Keskinen, H., Kupc, A., Kürten, A., Kvashin, A. N., Laaksonen, A., Lehtipalo, K., Makhmutov, V., Mathot, S., Nieminen, T., Onnela, A., Petäjä, T., Praplan, A. P., Santos, F. D., Schallhart, S., Seinfeld, J. H., Sipilä, M., Spracklen, D. V., Stozhkov, Y., Stratmann, F., Tomé, A., Tsagkogeorgas, G., Vaattovaara, P., Viisanen, Y., Vrtala, A., Wagner, P. E., Weingartner, E., Wex, H., Wimmer, D., Carslaw, K. S., Curtius, J., Donahue, N. M., Kirkby, J., Kulmala, M., Worsnop, D. R., and Baltensperger, U.: Oxidation products of biogenic emissions contribute to nucleation of atmospheric particles, Science, 344, 717–721, https://doi.org/10.1126/science.1243527, 2014. a
Riipinen, I., Pierce, J. R., Yli-Juuti, T., Nieminen, T., Häkkinen, S., Ehn, M., Junninen, H., Lehtipalo, K., Petäjä, T., Slowik, J., Chang, R., Shantz, N. C., Abbatt, J., Leaitch, W. R., Kerminen, V.-M., Worsnop, D. R., Pandis, S. N., Donahue, N. M., and Kulmala, M.: Organic condensation: a vital link connecting aerosol formation to cloud condensation nuclei (CCN) concentrations, Atmos. Chem. Phys., 11, 3865–3878, https://doi.org/10.5194/acp-11-3865-2011, 2011. a
Rohrer, F. and Berresheim, H.: Strong correlation between levels of tropospheric hydroxyl radicals and solar ultraviolet radiation, Nature, 442, 184–187, https://doi.org/10.1038/nature04924, 2006. a
Schobesberger, S., Franchin, A., Bianchi, F., Rondo, L., Duplissy, J., Kürten, A., Ortega, I. K., Metzger, A., Schnitzhofer, R., Almeida, J., Amorim, A., Dommen, J., Dunne, E. M., Ehn, M., Gagné, S., Ickes, L., Junninen, H., Hansel, A., Kerminen, V.-M., Kirkby, J., Kupc, A., Laaksonen, A., Lehtipalo, K., Mathot, S., Onnela, A., Petäjä, T., Riccobono, F., Santos, F. D., Sipilä, M., Tomé, A., Tsagkogeorgas, G., Viisanen, Y., Wagner, P. E., Wimmer, D., Curtius, J., Donahue, N. M., Baltensperger, U., Kulmala, M., and Worsnop, D. R.: On the composition of ammonia–sulfuric-acid ion clusters during aerosol particle formation, Atmos. Chem. Phys., 15, 55–78, https://doi.org/10.5194/acp-15-55-2015, 2015. a
Sipilä, M., Berndt, T., Petäjä, T., Brus, D., Vanhanen, J., Stratmann, F., Patokoski, J., Mauldin III, R. L., Hyvärinen, A.-P., Lihavainen, H., and Kulmala, M.: Role of sulfuric acid in atmospheric nucleation, Science, 327, 1243–1246, https://doi.org/10.1126/science.1180315, 2010. a
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Spindler, G., Mölders, N., Hanss, J., Beier, N., and Kramm, G.: Determining the dry deposition of SO2, O3, NO, and NO2 at the SANA core station Melpitz, Meteorol. Z., 5, 205–220, 1996. a
Spindler, G., Brüggemann, E., Gnauk, T., Grüner, A., Müller, K., and Herrmann, H.: A four-year size-segregated characterization study of particles PM10, PM2.5 and PM1 depending on air mass origin at Melpitz, Atmos. Environ., 44, 164–173, 2010. a
Spracklen, D. V., Carslaw, K. S., Merikanto, J., Mann, G. W., Reddington, C. L., Pickering, S., Ogren, J. A., Andrews, E., Baltensperger, U., Weingartner, E., Boy, M., Kulmala, M., Laakso, L., Lihavainen, H., Kivekäs, N., Komppula, M., Mihalopoulos, N., Kouvarakis, G., Jennings, S. G., O'Dowd, C., Birmili, W., Wiedensohler, A., Weller, R., Gras, J., Laj, P., Sellegri, K., Bonn, B., Krejci, R., Laaksonen, A., Hamed, A., Minikin, A., Harrison, R. M., Talbot, R., and Sun, J.: Explaining global surface aerosol number concentrations in terms of primary emissions and particle formation, Atmos. Chem. Phys., 10, 4775–4793, https://doi.org/10.5194/acp-10-4775-2010, 2010. a
Stanier, C. O., Khlystov, A. Y., and Pandis, S. N.: Ambient aerosol size distributions and number concentrations measured during the Pittsburgh Air Quality Study (PAQS), Atmos. Environ., 38, 3275–3284, https://doi.org/10.1016/j.atmosenv.2004.03.020, 2004. a
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Tuch, T. M., Haudek, A., Müller, T., Nowak, A., Wex, H., and Wiedensohler, A.: Design and performance of an automatic regenerating adsorption aerosol dryer for continuous operation at monitoring sites, Atmos. Meas. Tech., 2, 417–422, https://doi.org/10.5194/amt-2-417-2009, 2009. a
Vakkari, V., Laakso, H., Kulmala, M., Laaksonen, A., Mabaso, D., Molefe, M., Kgabi, N., and Laakso, L.: New particle formation events in semi-clean South African savannah, Atmos. Chem. Phys., 11, 3333–3346, https://doi.org/10.5194/acp-11-3333-2011, 2011. a
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Wehner, B., Birmili, W., Ditas, F., Wu, Z., Hu, M., Liu, X., Mao, J., Sugimoto, N., and Wiedensohler, A.: Relationships between submicrometer particulate air pollution and air mass history in Beijing, China, 2004–2006, Atmos. Chem. Phys., 8, 6155–6168, https://doi.org/10.5194/acp-8-6155-2008, 2008. a
Westervelt, D. M., Pierce, J. R., and Adams, P. J.: Analysis of feedbacks between nucleation rate, survival probability and cloud condensation nuclei formation, Atmos. Chem. Phys., 14, 5577–5597, https://doi.org/10.5194/acp-14-5577-2014, 2014. a
Wiedensohler, A., Birmili, W., Nowak, A., Sonntag, A., Weinhold, K., Merkel, M., Wehner, B., Tuch, T., Pfeifer, S., Fiebig, M., Fjäraa, A. M., Asmi, E., Sellegri, K., Depuy, R., Venzac, H., Villani, P., Laj, P., Aalto, P., Ogren, J. A., Swietlicki, E., Williams, P., Roldin, P., Quincey, P., Hüglin, C., Fierz-Schmidhauser, R., Gysel, M., Weingartner, E., Riccobono, F., Santos, S., Grüning, C., Faloon, K., Beddows, D., Harrison, R., Monahan, C., Jennings, S. G., O'Dowd, C. D., Marinoni, A., Horn, H.-G., Keck, L., Jiang, J., Scheckman, J., McMurry, P. H., Deng, Z., Zhao, C. S., Moerman, M., Henzing, B., de Leeuw, G., Löschau, G., and Bastian, S.: Mobility particle size spectrometers: harmonization of technical standards and data structure to facilitate high quality long-term observations of atmospheric particle number size distributions, Atmos. Meas. Tech., 5, 657–685, https://doi.org/10.5194/amt-5-657-2012, 2012. a
Woo, K. S., Chen, D. R., Pui, D. Y. H., and McMurry, P. H.: Measurement of Atlanta aerosol size distributions: Observations of ultrafine particle events, Aerosol Sci. Tech., 34, 75–87, https://doi.org/10.1080/02786820120056, 2001. a
Yu, F. and Turco, R. P.: The size-dependent charge fraction of sub-3-nm particles as a key diagnostic of competitive nucleation mechanisms under atmospheric conditions, Atmos. Chem. Phys., 11, 9451–9463, https://doi.org/10.5194/acp-11-9451-2011, 2011. a
Yue, D. L., Hu, M., Zhang, R. Y., Wang, Z. B., Zheng, J., Wu, Z. J., Wiedensohler, A., He, L. Y., Huang, X. F., and Zhu, T.: The roles of sulfuric acid in new particle formation and growth in the mega-city of Beijing, Atmos. Chem. Phys., 10, 4953–4960, https://doi.org/10.5194/acp-10-4953-2010, 2010. a
Zhang, Q. I., Stanier, C. O., Canagaratna, M. R., Jayne, J. T., Worsnop, D. R., Pandis, S. N., and Jimenez, J. L.: Insights into the chemistry of new particle formation and growth events in Pittsburgh based on aerosol mass spectrometry, Environ. Sci. Technol., 38, 4797–4809, https://doi.org/10.1021/es035417u, 2004. a
Zieger, P., Fierz-Schmidhauser, R., Poulain, L., Müller, T., Birmili, W., Spindler, G., Wiedensohler, A., Baltensperger, U., and Weingartner, E.: Influence of water uptake on the aerosol particle light scattering coefficients of the Central European aerosol, Tellus B, 66, 22716, https://doi.org/10.3402/tellusb.v66.22716, 2014. a
Technically, the lower limit of the APS is 500 nm in aerodynamic diameter. Converting this aerodynamic diameter into a mobility diameter yields a lower cut-off of 383 nm. Comparison with particle mobility size spectrometer data suggests, however, that the APS becomes increasingly unreliable at the lower end of its measurement range. (In principle, the APS performs best for large particles that are separated by their inertia in the instrument's counterflow. Smaller particles tend to separate only to a poor degree in the separation unit of the instrument.) Therefore, TDMPS data were used exclusively between 20 and 800 nm, with the APS data continuing the size distribution above that point.
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2020-07-14 14:43:23
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https://scicomp.stackexchange.com/tags/c/hot
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# Tag Info
### Looking for Runge-Kutta 8th order in C/C++
If you're doing celestial mechanics over long time scales, using a classical Runge-Kutta integrator will not preserve energy. In that case, using a symplectic integrator would probably be better. ...
• 29.9k
Accepted
### How to document math formulations in scientific computing codes?
I prefer using doxygen that supports C++ and LaTeX comments, both inline and as separate equations. This way, you will keep your comments, including, say, the rigorous mathematical formulation of the ...
• 8,452
Accepted
### Memory Management - Why certain initializations order are faster?
Such an effect happens because of how the data of the int** a is stored in memory (as per C/C++). This question on StackOverflow has answers with some more details (...
• 8,452
Accepted
### Numerical evaluation of the Exponential Integral Ei by rational Chebyshev approximations fails
There is a prime missing both in your code and in the expression in your question. In the original paper, the expression is: \log(x/x_0) + (x-x_0) \frac{\sum^\prime_{0\leq j\leq n}p_j T_j^*(x/6)}{\...
• 11.4k
### Use of Java or Scala in HPC?
Java has been around for almost 20 years now as a major programming language, but it hasn't caught on in scientific computing so far. I think that's a good indicator for what's going to happen in the ...
• 51.6k
Accepted
### Looking for Runge-Kutta 8th order in C/C++
Both GNU Scientific Library (GSL) (C) and Boost Odeint (C++) feature 8th order Runge-Kutta methods. Both are opensource, and under linux and mac they should be directly available from the package ...
• 2,473
Accepted
### C - OpenMP, MPI, Serial Program
I think some of your issues are more important than others and some of your emphasis is misplaced. In pursuing overhead, you are in danger of making your program unmaintainable. It is easier to write ...
• 11.4k
Accepted
### What PRNG function is this?
A little playing with the sequence of numbers generated by the C code shows that the sequence is $z_{i+1}=5z_{i}+273 \mod 2^{16}$ This is a linear congruential generator (LCG). It's easy to show ...
• 17.7k
Accepted
### C standard for computational science
In theory, as the original authors, you're free to pick and name a standard, then expect others to follow it. In practise, if you're supporting an HPC system, then your choice is likely to be ...
• 2,199
Accepted
### Suming up the series $1+1/2+...+1/n$ in C
I see several problems with your code: type error in scanf (n is an integer, not a float) i=0 and not i==0 in the for loop ...
• 196
### Numerical integration giving incorrect sign
This might be an accuracy problem in computing the second term, because of those large exponentials when $x \gg 1$. I would first work on that term: gather $e^x$ out from numerator and denominator and ...
• 9,065
Accepted
### Performance of adding eight numbers sequentially vs. in a tree
I think your analysis is basically right. Some notes. 1. Pipelining is the wrong word here; what you're looking at here is data dependency. A CPU pipeline splits an individual instruction into ...
• 11.4k
### Use of Java or Scala in HPC?
It seems unlikely to me. The Java MPI APIs haven't been worked on in years (so you're wrong about #4), and the JVM's floating-point performance is notoriously poor. Java may out perform C/C++ or ...
• 10.8k
### Use of Java or Scala in HPC?
I would argue that Java will in fact REDUCE productivity when compared with modern c++, or even with modern Fortran for the purpose of scientific computing. Writing ...
• 2,473
### Do BLAS routines compute their respective operations with minimum error?
BLAS routines do not typically use stable summation algorithms. In the case of gsl, you can look up its source code online - the source of gsl's sdot is contained ...
• 11.4k
Accepted
### Using GSL for basic operations
Of course it makes sense to use the GSL (or another library for that matter) for several reasons: Don't reinvent the wheel. The work has been done, you can spend your time on more useful things. If ...
• 6,091
### I've developed a derivative-free optimization method, looking for comments
I would like to hear comments from users that have some practical models (e.g. black-box hyperparameter optimization) which are still needed to be solved acceptably - whether this method works or not ...
• 2,116
### Looking for Runge-Kutta 8th order in C/C++
summarizing some points: If it's a long-term integration of a non-dissapative model, a symplectic integrator is what you're looking for. Otherwise, since it's an equation of motion, Runge-Kutta ...
• 11.8k
### BLAS, LAPACK or ATLAS for Matrix Multiplication in C
Yes, you want to call the BLAS routine DGEMM. The place to start for how to call it from C is to look at the documentation for DGEMM, which you can find online. Then you want to understand how to call ...
• 5,744
Accepted
### How to deal with big numbers in intermediate calculations?
I think there's a simple way to do this. You have a rational function of identical cosh/sinh terms, where every expression is a homogeneous polynomial in cosh/sinh, and the only problem is that these ...
• 11.4k
### C standard for computational science
You should definitely jump to C99, or newer(!). The C99 standard introduced the restrict keyword. Loosely speaking, with this keyword you can inform the compiler ...
• 551
Accepted
### Recommendations for ODE solvers for stiff equations
This is a huge open ended question, but I'll copy the recommendation section from the current release of DifferentialEquations.jl: Stiff Problems For stiff problems at high tolerances (>1e-2?) it ...
• 11.8k
• 399
Accepted
### C or fortran library to solve linear 2D/3D elliptic PDE
Most of the widely used finite element libraries are written in C++. If all you really care for -- and if all you will ever care for -- is solving an elliptic PDE on a rectangle, then it's probably ...
• 51.6k
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2022-08-17 17:47:44
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https://www.researchgate.net/topic/Control-Systems-Engineering
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Science topic
Control Systems Engineering - Science topic
Control engineering or control systems engineering is the engineering discipline that applies control theory to design systems with desired behaviors.
Questions related to Control Systems Engineering
• asked a question related to Control Systems Engineering
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With this question, I want to know that proper systems exist? or they are just theoretical systems?
Yes. It is causal. However, the magnitude bode plot will not roll off as frequency increases as there is exist a non-zero DC gain between the input and output.
• asked a question related to Control Systems Engineering
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I already saw some examples of GA (genetic algorithm) applications to tune PID parameters, but I (until now) don't know a way to define the bounds. The bounds are always presented in manuscripts, but they appear without great explanations. I suspect that they are obtained in empirical methods.
Could Anyone recommend me research?
Dear Italo,
In general, the bounds selection is made empirically because the "suitable range" of a PID controller is problem-dependent. The way I use to select the bounds is: 1. I tune a PID controller that produces a stable response to the closed-loop system. Then, 2. I choose a range around this "nominal" value large enough such that the GA has still some degree of freedom to search in the optimization space. Finally, 3. if the GA converges, I start decreasing/increasing this range till I got a more or less good behavior of the GA, i.e., the GA doesn't stick in a sub-optimal minimum or so.
If you want to use a more rigorous approach, I would suggest computing the set of all stabilizing PID controllers for the particular system. Then, I would establish the bounds for the GA search space to be the this computed set. In that way, you would search for the optimal controller only within those producing a stable closed-loop response.
Best,
Jorge
• asked a question related to Control Systems Engineering
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In his name is the judge
Hi everyone
In order to use controller for structure with absorber, have to connect matlab and opensees in real time :
it means in each loop in opensees data must send to matlab and then matlab do some prossess with controller like fuzzy then send back data to opensees
Relying on researching and consulting the only way is use hybrid simulation (like openfresco),
do you have any other idea or way to connect these two programs without using openfresco?
Translation results
star_border
Consult
Wish you best.
Take refuge in the right.
dear Eknara Junda thank you
i have to do optimization and design controller and for both of them i use matlab toolbox and i learned them fully, becuase of this i prefer to link matlab and opensees in real time.
• asked a question related to Control Systems Engineering
Question
How long does it take to a journal indexed in the "Emerging Sources Citation Index" get an Impact Factor? What is the future of journals indexed in Emerging Sources Citation Index?
According to Web of Science, ESCI Journal can be included in Science Citation Index (SCI), Social Science Citation Index (SSCI), or Arts and Humanities Citation Index (AHCI), if they meet "Impact Criteria".
Accordingly, journals are included in Web of Science Core Collection (SCI, SSCI, AHCI, and ESCI) if they meet 2 criteria, namely; 1) Quality 2) Impact. The "Quality criteria" comprises 24 sub-criterion, while the "Impact criterion" consist of 4 sub-criteria.
Hence, any journal captured in ESCI have already meet the quality criteria , therefore the quality criteria is the only requirement for journal to be considered in ESCI. Similarly, any journal on ESCI must wait to meet the "Impact Criteria", which can take time, and may be impossible to be predicted. This is because the Impact Criteria is evaluated according to the number of citations the journal is receiving, the performance of authors who published in the journal before, and the number of cross-references between journals in web of science, etc. Waheed Ur Rehman .
• asked a question related to Control Systems Engineering
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As IEC 61131-3 standard for PLC programming languages defines a few of them, it will be useful to learn what our colleagues apply for PLC programming.
• asked a question related to Control Systems Engineering
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Does anyone know if I can connect the attached DAQ (CASSY type) to MATLAB real time simulink for controlling purposes?
It depends on your sample frequency, which is limited for performance of DAQ. another issue should be concerned is the number analog of a processing control system. the quality of data will be bad or real-time data might be good if you collect as much data in process as possible.
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Hi
How can i correct this error?? I think it's about matrix dimensions for port e.
Error in default port dimensions function of S-function 'FeedbackLinearization/Controller'. This function does not fully set the dimensions of output port 2
I'm running a simulation based on feedback linearization control method that comes from a paper attached below.
the model is also attached.
Anyone help me, helps a poor student. (if it makes sense lol)
Hi,
file name "sfun_ abcaaaaa.c" is not available in the folder mentioned.
• asked a question related to Control Systems Engineering
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Assuming that no motion in the normal direction, how would you propose a system of hardware and interconnection as well as the control strategy that will drive the system?
Sounds analogous to an inverted pendulum. Here is one way to construct something with closely similar dynamics. Take a knee (90° angle) connector for sewage pipe, cut it nicely and mount it on a toy car such that the knee is pointing straight upwards. Place a small ball (i.e. from a roll-on bottle) on top of the knee and try to keep it there stably by moving the toy car back, forth, and sideways (hard to do manually without automated feedback control loop). I guess many feedback control schemes would keep the ball at the top of this curve subject to micromotion.
Aside: If the knee was facing down, mounted on a motor, and the ball is placed inside the 90° curve while the motor is rotated at some angular frequency, the resulting dynamics would resemble Mathieu equations whose stability analysis could be checked from Floquet theory. Incidentally, these are the same dynamics exhibited by a single ion trapped in a linear Paul trap.
• asked a question related to Control Systems Engineering
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Suppose we have a lipschitz nonlinear system with a disturbance input. The disturbance is of decaying exponential like nature with an upper bound known.
System: x_dot = f(x) + g(x)u(t) + d(t)
where, x:state; u(t): control input; d(t): Disturbance input
How can we design nonlinear observer for such system?
Disturbance is sometimes helpful while dealing with multicollinearity. Also helpful may be keeping the number of degrees of freedom minimum.
Maybe you can alernatively consider the recursive least squares algorithm (RLS). RLS is the recursive application of the well-known least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables. For the recursive least squares algorithm with forgetting factor (RLS-FF), adquired data is weighted according to its age, with increased weight given to the most recent data.
Years ago, while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS-FF algorithm to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, hence giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The power dissipated by agitation was accessed by a torque meter (pilot plant). The proposed (adaptive) control algorithm compared favourably with PID. Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):
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Hi!
I have started working on a project "floating sensor networks (FSN) for continuous water quality monitoring". For which I need simulator to measure Like, pH, Turbidity, Salinity, Temperature, DO, EC, etc. Rather than going with real time deployment of FSN for measuring water quality sensors.
Objectives of the Project:
A. Water Quality Measurement
B. Reliable Data Transform
C. Congestion Control
D. Deployment Strategy
E. Energy Harvesting
All objectives should be carried out simulation based. Kindly suggest whether this work will done via simulation design (either partially or whole)
Dear Sarang, did you find any suitable simulation software? We also need it for a small project we are working on
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Hello everyone,
I'm looking for some short courses (not online) in the field of CE (more interested in Automotive control) in Europe.
My field of interests are,
- Robotic Control Systems
- Vehicular Dynamics Control
- Motion Control
- Data-Driven Control
- Optimal Control
- System Identification
- Reinforcement Learning Control Design.
I can participate by Self-Fund but I will be happier if there will be a scholarship or something,
Yours cordially,
A. M.
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We can avoid dq transformation by using PR controller. Why don't we use PR controller in industry? Why do we prefer PI controller?
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it means the oil pump to be sometimes able to turn off during the driving cycle, and if it is possible, how efficient would it be? (more particularly in heavy vehicles like a bus).
It depends upon the size of the accumulator, and the flowrate required. If you calculate the flowrate required for the PAS pump you could work out the accumulator volume required. See link below for typical (but a little old) energy consumption info.
However, many modern vehicles are moving to electric PAS systems, so the traditional hydraulic system may not be required in future.
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I am designing a flight controller for a quadrotor.
At first I am designing a nested/cascaded controller consisting of only proportional controllers Kp . Now, if i tune the rate controller for 10 rad/s cross-over frequency, what should be the cross-over frequency for the angle, velocity and then the position loops. Also, what else do i need to know while designing a flight controller for practical implementation purposes?
Secondly, How do we implement a own flight controller such as observer based via arducopter?
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I am trying to discretize a continuous time state space model using the following code
s=tf('s');
G=1/(Iyy*(s^2))
Gs=ss(G)
Gd=c2d(Gs,0.01,'zoh');
Now, when i use this discretized model 'Discrete State-space Model' in simulink, my close loop system goes unstable. Same is happening with observer, like discretized observer is making close loop system unstable. Can someone help me here?
Your system is basicly a double integrator, so a borderline stable system. If you look at your phase value of the bode plot for the continious system (bode(G)), you will see that it has a phase of -180° over all frequencies. Since any kind of delay causes the phase of a system to drop, if you introduce an arbitrarily small amount of delay on your continious tf G, and use the command "margin(G)", you will find that your system will always have a negative phase margin. And since discretizing will always introduce delay into your system, closing the loop on it will automatically destabalize it. You can use 'tustin' as you discretizing method, which essentialy uses trapezoidal integration, a more stable integration algorithm, and simulink should probably get your discrete system stable this way.
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Optimal control and nonlinear control system.
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Consider a system with relative degree two with respect to a chosen sliding output function s. If twisting controller is applied for control, what are the expected drawbacks in control performance and steady state behaviour? You may consider that $s$ and $\dot{s}$ is available.
1. Historically twisting have been designed to substitute a discontinuous controller with Lipschitz continuous one keeping the finite-time theretically exact convergence to the sliding output for the systems with relative degree one affected by Lipschitz perturbaitons(see Levant A. IJC, 1993 https://www.tau.ac.il/~levant/slorder93.pdf).
The only adavantage of the twisting controller is that if this algorithm is applied for the systems with relative degree two it ensures theoretically exact finite-time convergence of the sliding output and its derivative to zero in finite-time.
But from the viewpoint of chattering the twiating controller is the worst one because his gains are at least twice bigger, than the upper bound of uncertainties.
Moreover, the amplitude of chattering produced by twisting controller is bigger than the amplitude of chattering produced by relay controller with a linear sliding surface see attached file, table 3
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PID controller is reportedly widely built and used in control systems engineering for industrial applications. The PID-controller has a transfer function as: (Kd*s^2+Kp*s+Ki)/s.
Therefore, it seems such a transfer function box, could be built in terms of an electronic circuit component.
Meanwhile, the order of an observer-based controller is higher than a PID-controller, as for example you see in the snap-shot from Ogata control engineering book, attached. Since Ogata is a practical engineering text-book, therefore I guess such an observer-based controller could be practically built as an electronic circuit and then embedded as a controller.
Moreover, we usually build transfer function blocks, simply in Matlab Simulink. But is it possible, practical, simple, and convenient to build such blocks as an integrated electronic element in a circuit, while they are of high order?
My main question:
There is a controller, with a transfer-function:
H_controller(s)=N(s)/D(s),
Where:
N(s)=b0*s^m+b1*s^(m-1)+b2*s^(m-2)+…+bm,
And:
D(s)=a0*s^n+a1*s^(n-1)+a2*s^(n-2)+…+an.
Moreover, m=10, n=10, or, n=11.
The digested question:
Is it possible, and practical to design an electronic circuitry for a box, with the transfer function: H_box=H_controller(s)=N(s)/D(s)?
Warning: I have no experience to build a real electronic circuitry, and my question is only the possibility and practicality of designing a circuit for such a transfer function box. Do not mind about the performance of the controller or any other stability concern with regard to that. Only the possibility and practicality of designing an electronic box which has an equivalent transfer function: H_box=H_controller(s)=N(s)/D(s).
The box is in feed-forward path.
The transfer function H(s)= N(s)/D(s)
has the form of filter characteristics.
Such general transfer characteristics has zeros and poles.
The filter which contains zeros and poles are elliptic filters.
Such filters can be implemented by using op amps+ discrete RC elements
They can be built completely integrated using switched capacitor filters
They can be also implemented by using gm-C filters.
For such filter design using active filters please follow the reference: https://www.sciencedirect.com/book/9780750675475/analog-and-digital-filter-design
Bestwishes
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I want to know that we have a real system with fractional order state space model
Dear Professor Sabatier
First, I'd like to thank you so for asking these kind of challenging questions. These kinds of questions are very important for advancing this field. As you said, our life is full of doubts. Now this question arise that: How can we trust to different integer differential models? The only sin of fractional derivative is that it has been introduced after integer derivative. Why should we accept that the first derivative (u') denotes the velocity? Why another fractional derivative (such u^{0999}) is not?
As we can check in many considerable works, fractional derivatives modify many old results in integer derivatives. As you said, you often use fractional models to build and implement very physical and useful systems (eg: battery state observers, hydrogen generator, ...). We need a comparison in study of these different natural phenomena and so a standard basic theory for the comparison which is usually integer derivative models while we can doubt it. I fee that we could improve our initial mental attitude. Again, I appreciate you.
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I'm studying digital control by myself, and I would like to know if has available on the internet the solutions manual of "Digital Control of Dynamic System 3rd edition"?
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For example, the first order Laplacian measures the difference between the local and average values of a quantity in an infinitesimal neighborhood of the point in question. So, for a flexible string, the acceleration is proportional to the Laplacian (wave equation).
For plates, the acceleration is proportional to the double Laplacian. Also, the double Laplacian occurs in the modification of the wave equation for a stiff string. Therefore, what is the double Laplacian a measure of? Thanks.
We have:
Applications of Double Laplace Transform to Boundary Value Problems DOI: 10.9790/5728-0925760
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I only have acces to historical data recorded over the course of a few years. However the data is of a closedloop proces that has already been tuned with PID. The values of P, I and D are known.
How can I determine the openloop transferfunction usign only this data?
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Why the results are not optimal?
difficult to infer all performance values? why
1- PID Tuning of Plants With Time Delay Using Root Locus
Greg Baker ,San Jose State University
2- A revision of root locus method with applications
September 2015, Journal of Process Control 34:26-34, DOI: 10.1016/j.jprocont.2015.07.007
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Is it theoretically possible, that after discretization by using Talyor Series Expansion, a non-observable nonlinear system will became an observable?
It was proved, that used continuous model of PMSM is non-observable (see attached). I want to know, if resulting discrete system is observable or not. Any comment appreciated. Thanks.
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the dc-dc converters has two transfer functions , control to output and line to output TF. which one is used when design a controller? and which one is used to test step response?
I agree with Aparna Sathya Murthy
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How can I implant ANFIS as a controller in MATLAB/SIMULINK simulation for sit to stand movement supported with functional electrical stimulation in paraplegics. I think it will be inverse dynamic model and I should use model base controller
Best regards
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I'm a beginner in control engineer, what books or websites you could recommend?
T. Bauer, J.T. Betts, W. Hallman, W.P. Huffman, K. Zondervan, Solving the optimal control problem using a nonlinear programming technique, Parts I and II, 1984.
A.E. Bryson Jr.Optimal control – 1950 to 1985
IEEE Control Systems (1996),
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Suppose a six degree of freedom simulation of an aircraft, which some aerodynamic parameters (e.g: stability derivatives), mass configuration (e.g center of mass) and etc, are randomly choose within known bounds. From the Monte Carlo sample those simulations are split in two groups: with instability (any time during the simulation) and without instability during the flight.
My question is, How could I find the more important combination of random parameters the caused the instability in flight?
I have already done a sensitivity analysis, so I have an idea how each one influence individually. What I really one to find is how the combination of parameter is causing the instability.
For more details, we have:
Monte Carlo Simulation of Real Dynamic Systems :
Best regards
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Hi every one,
here I have a problem in MATLAB, when I want to solve the following equation, relative to PI in the photo, or tau in the code, MATLAB will send me this error: Warning: Unable to find explicit solution. For options, see help.
I attached the question and the code below (in code, I rewrite pi in the photo with tau).
If you have any idea to solve this problem, analytically or numerically, I will be happy to hear it out.
NOTE:
> PI_0.1(X,t) = tau
> X = [x(t),y(t),psi(t)]^T;
** PROBLEM: Find tau in terms of X and t in which solve the mentioned equation.
Arash.
code:
______________________________________
______________________________________
clc;clear;
syms x y psi tau t
c1 = 1;c2 = 1.5;lambda = 0.1;
x_r(tau) = 0.8486*tau - 0.6949;
y_r(tau) = 5.866*sin(0.1257*tau + pi);
psi_r(tau) = 0.7958*sin(0.1257*tau - pi/2);
x_r_dot = 0.8486;
y_r_dot(tau) = 0.7374*cos(0.1257*tau + pi);
psi_r_dot(tau) = 0.1*cos(0.1257*tau - pi/2);
phrase1 = c1/2*(cos(psi)*(x - x_r) + sin(psi)*(y - y_r))*(cos(psi)*x_r_dot + sin(psi)*y_r_dot);
phrase2 = c1/2*(-sin(psi)*(x - x_r) + cos(psi)*(y - y_r))*(-sin(psi)*x_r_dot+cos(psi)*y_r_dot);
phrase3 = 0.5*(psi - psi_r)*psi_r_dot;
eq = -2*(1-lambda)^2*(phrase1 + phrase2 + phrase3) - 2*lambda^2*(t - tau)
sol = solve(eq == 0 , tau , 'IgnoreAnalyticConstraints',1)
______________________________________
______________________________________
Pass x, instead of tau, as rightly pointed out by Saeb AmirAhmadi Chomachar
syms x y psi tau t
c1 = 1;c2 = 1.5;lambda = 0.1;
x_r(tau) = 0.8486*tau - 0.6949;
y_r(tau) = 5.866*sin(0.1257*tau + pi);
psi_r(tau) = 0.7958*sin(0.1257*tau - pi/2);
x_r_dot = 0.8486;
y_r_dot(tau) = 0.7374*cos(0.1257*tau + pi);
psi_r_dot(tau) = 0.1*cos(0.1257*tau - pi/2);
phrase1 = c1/2*(cos(psi)*(x - x_r) + sin(psi)*(y - y_r))*(cos(psi)*x_r_dot + sin(psi)*y_r_dot);
phrase2 = c1/2*(-sin(psi)*(x - x_r) + cos(psi)*(y - y_r))*(-sin(psi)*x_r_dot+cos(psi)*y_r_dot);
phrase3 = 0.5*(psi - psi_r)*psi_r_dot;
eq = -2*(1-lambda)^2*(phrase1 + phrase2 + phrase3) - 2*lambda^2*(t - tau);
eqn = rewrite(eq,'log');
sol = solve(eqn == 0 , x , 'IgnoreAnalyticConstraints',1);
pretty(sol)
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I want to know the exact definition of these four tests and I am wondering which of them could work in real time?
HIL or 'hardware-in-the-loop' testing is by its very nature a resource-hungry solution to testing, requiring multi-skilled teams able to set up and configure both the execution platform and the I/Os as well as the modelling environment.
Once your model is verified (i.e., MIL in the previous step is successful), the next stage is Software-in-Loop(SIL), where you generate code only from the Controller model and replace the Controller block with this code.
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I have the nonlinear systems of Khalil, however some definitions are no so clear, is there a newer book with Matlab examples
For Video tutorials , we have:
Introduction | Nonlinear Control Systems: https://www.youtube.com/watch?v=Xgnwn0G9qoo
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Hello everybody
Despite a few TMD cost models available in the literature, I am searching for more accurate initial and lifetime cost models of translational TMDs for Life Cycle Cost Analysis (LCCA) of TMD-equipped structures.
In fact, the provided cost model affiliated with one of the companies designs and manufactures transitional TMD (such as LeMessurier CO.), which this model consists TMD initial cost (construction and installation of the TMD) and TMD damage cost (maintenance and repair losses of TMD before structural collapse)
The effectiveness of tuned mass dampers (TMDs) in reducing the seismic response of civil structures is still a debated issue. The few studies regarding TMDs on inelastic structures indicate that they would perform well under moderate earthquake loading, when the structure remains linear or weakly nonlinear, while tending to fail under severe ground shaking, when the structure experiences strong nonlinearities. TMD seismic efficiency should be therefore rationally assessed by considering to which extent moderate and severe earthquakes respectively contribute to the expected cost of damages and losses over the lifespan of the structure.
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I design feedback close loop system with a robust static feedback controller using Hinf approach where control low is u(t)=Kx(t). What is the good method to show robustness for this type of system? What plots we can make to show robustness for designed system?
Regards
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How does one usually evaluate the robustness of a controller?
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Dear community,
I am trying to build a model of a Furuta pendulum in Simulink/Simscape. Unfortunately, when I try to linearize my model with the integrated Model Linearizer, I get an unexpected result. The evaluation of the linearized system shows, that it is only poorly controllable, although a classic Furuta pendulum should be fully controllable according to literature. Therefore I assume, that there must be something wrong with my model or the way I linearized it but I can´t figure out what it is.... I´d highly appreciate any help on that, as this is bothing me for quite some time now. The model is attached to this post. Furthermore I have attached a screenshot of the linearized system.
My controllability matrix (ctrb(A,B)) then looks like this with rank = 1, which I believe can´t be right...
Controllability matrix =
1.0e+26 *
0 0.0000 0.0000 -0.0000 0.0000
0.0000 0.0000 -0.0000 0.0000 -0.0007
0 0.0000 0.0000 -0.0000 0.0000
0.0000 0.0000 -0.0000 0.0000 -0.0015
0.0000 -0.0000 0.0000 -0.0000 9.2972
Thank you and best regards, Joo
Dear Joo ...
Regards
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Hello everyone, I hope you have a good day,
As we all know, the lateral dynamic system of vehicles has two output, lateral error and heading error, and we have one input, which is steering angle, I always have one big problem:
How to Design a Controller to have zero steady-state error, when I have XY reference path?
I designed a controller to track the heading, but when the vehicle gets departed from the path, as it does not have any sense of lateral error, it will not come back to the path, it will just follow the heading with some offset.
I read a lot of papers in this area, but none of them talked about XY reference paths.
I add a photo to clear up some points, please check the attached file.
Arash
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I'm writing my thesis and I am searching for good software to draw control block diagrams!
Any suggestion?
Thank you!
Tableau Desktop is also good one for drawing the scientific illustrations.
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.
Discrete-Time Systems:
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I want to find PID parameters to regulate my system.
Manual PID tuning is done by setting the reset time to its maximum value and the rate to zero and increasing the gain until the loop oscillates at a constant amplitude. (When the response to an error correction occurs quickly a larger gain can be used.
The best method for tuning PID controller parameters is depend of your system
Regards
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I want to know all the things related and necessary which help me in control system field.
You can use control methods in everywhere. Below, there are some areas:
1. Power systems (smart grid , micro-grid , power generation , power devices (motors,...))
2. Electrical drivers for electrical motors
3. Biological systems
4. Economic systems
5. Social systems
6. Robotics
7. Aerospace and aeronautical systems
8. Automobile industry
9. Chemical Process
10. Multi agent systems
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Hello everybody
I would like to link two topics, LCCA and BIM, each of which deals with "structural and earthquake engineering" and "construction management" in civil engineering, respectively.
I am trying to investigate on LCCA of a structure equipped with Tuned Mass Dampers (TMDs) based on principles of Performance Based Earthquake Engineering (PBDE) for different seismic hazard levels (this issue is related to structural engineering). So I intend to use BIM to create a more realistic cost model including other costs are related to this topic.
I suggest that you use the 5D BIM technology, which is related to the cost of construction elements. You need to model your project using software as revit, and then link the model's elements to the cost and time schedule using software like Naviswork. This will give you visualization for the construction phases and the project life cycle.
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In order to get a better conditioned A matrix, the absolute mean of the eigenvalues of the A matrix should be one (all eigenvalues are between -1 and 1, so within the unit circle and the absolute mean is 0.4389). This could be done by scaling the time.
For the following continuous-time state-space model:
dx/dt = Ax(t) + Bu(t)
y = Cx(t)
the state-space model will look like:
dx/dtau = (1/lambda_avg)*Ax(tau/lambda_avg) + (1/lambda_avg)*Bu(tau/lambda_avg)
y = Cx(tau/lambda_avg)
with
lambda_avg, the absolute mean of the eigenvalues of the A matrix
tau, the new timescale
tau = lambda_avg*t
However, I want to scale the time of a discrete-time state-space model in order to get a better conditioned A matrix:
xi+1|k = Axi|k + Bui|k
yk = Cxk
How could I do that in the same way as for the continuous model?
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Hello everybody,
I have observed a bit confusing behavior of my system response (or may be I am missing something).
I have a transfer function in S domain converted to Z domain with a 1kHz sampling frequency at the time of conversion using matlab, When I embed this discrete version of the transfer function to my system which is also sampling on the same frequency of 1kHz. The system works the way as expected (i.e. the step response is the same as that of the s-domain analogue controller).
But if I increase the sampling frequency of my system while using the SAME discrete transfer function that i just converted from s to z domain with a SAME conversion sampling frequency of 1kHz , the step response gets further faster.
My question is that, why the discrete system gets faster response than the analogue one, despite the transfer functions of the analogue controller and the discrete controller are the same.
What I understand, the step response of any transfer function should remain the same in either case (i.e. either the function is in s-domain or in z-domain) the response should be the same ?
Does this mean the digital controllers have the ability to fast the response of the same transfer function by changing the sampling frequency of the system?
It is important, not to confuse the system sampling frequency of my u-controller at which the u-controller is collecting the samples from ADC, with the sampling frequency that I used as a parameter required to convert the s-domain transfer function to z-domain transfer function.
I thank you all for your time.
Regards,
Iftikhar Abid
The conversion from the S-domain to the Z-domain can be accomplished by using the bilinear transformation.
- A transformation for S to Z
S= (Z-1) /(Z+1)
- And frequency prewarping
w analog = tan wd/2,
where wd= 2 pi f/fs
As one sees if one changes fs , one has to change w analog as a consequence of prewarping.
Best wishes
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Would anyone help me to understand the difference between energy signals and power signals (with examples to each of them), and what is the physical intuition behind the relation between the auto-correlation function of a signal and its power spectrum density?
When the energy of a signal is finite, but power is zero, such a signal is called energy signal. for example- a pulse has a finite energy, though power is zero.
On the other hand, if the energy of a signal is infinite, but it has a finite power, it is called as power signal., for ex- sinusoid is a power signal.
The auto-correlation function is simply the convolution of the signal with itself (not flipped), where delay/lag is the parameter. Here we try to find how much a signal overlaps with itself when lag parameter is varied over time. Its Fourier transform is power spectral density, which also does the same thing in frequency domain. So, at t=0, the auto-correlation function provides the energy or power of the signal x(t). (based on whether x(t) is a power or energy signal) which is same as the power in frequency domain obtained by integrating power spectral densities for all frequencies.
so, basically, both tell the same thing, one with respect to time, other with respect to frequency.
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I want to start to work on some stepper motors bipolar with arduino mega and ramps 1.4 but I don't know how to do it. Can anyone explain how to do this?
I don't understand exactly what you mean? What is your goal? What do you want to do?
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The specifications are in term of R, L, B, J, kv, kt
The max output force of electromechanical braking system is 3500N
Hi, I initially face a same problem in my research. It solved :)
Kindly refer my publication attached.
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What is the typical pitching speed of modern MW turbine blades and do this blades pitch fast enough compared to changing wind speeds?
Significantly very fast. Standing at the control panel of a Vestas 850 kw , I saw the display was showing the instantaneous power production and pitch angle. The display was updating either every second or half second and each reading gave different readings. so a time constant for the pitch of under a second, for 40 m blades is astonishing!
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I have a brushless gimbal motor (with 12N14P windings) whose rotor shaft position is measured by means of a high-resolution absolute optical encoder. (The motor has no hall-sensors)
The amount of current fed through the three field windings are independently controlled by 16-bit DACs. (i.e. not just PWM)
What control algorithm should be used to implement precise position control of this closed-loop system, and what auto-tuning technique should be employed?
I "think" that the 12N14P winding scheme means that one full rotation of the field winding signal produces one 1/14 of a mechanical rotation. (but I could be mistaken).
It is not known what the physical angular offset relationship is between the field windings inside the motor and the zero-datum of the absolute optical encoder attached to the rotor shaft. I suspect that the auto-tuning technique will need to perform an experiment to determine this angular offset if the control algorithm is to be able to produce a field vector that leads the desired position by 90 degrees to achieve maximum torque.
From the IMU we get position and velocity after sensor funsion. The input variable of the plant seems to be the speed of the gimbal motor (since its driven by 3 sine waves), not the voltage as usual in a brushed dc motor setup. So one can not just implement a torque control for holding the position. When I investigated the Storm32BGC, it looks like the PID does influence the position and the speed at the same time: Fast movements leads to fast speed corrections, but also the deviation from the stationary position leads to a given speed.
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On what parameters the selection of continuation method depends ?
We have proposed a numerical method (denoted as MEM) for solving the
Nonlinear Dynamic problems. If you would like, you can view our published articles in this field entitled "dynamic analysis of SDOF systems using modified energy method". By the way, the presented idea is also generalized to MDOF systems in another study, which is available on my research-gate account.
Regards,
Jalili
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To design a controller for multi-degrees of freedom actuators which type of controller is better (Sliding Mode Controller or Backstepping Controller)?
There are 4 issues I have seen in this discussion:
1. Backstepping can compensate theoretically exactly only uncertainties decreasing together with the state variables. No comparison between backstepping and SMC in this sense.
2. Matched and unmatched uncertainties
For perturbed chain of integrators there are no unmatched perturbations. If you will differentiate the output till the system order all the perturbations will be matched! That is why the only reasonable to control the systems with unmatched uncertainties is a combination between backstepping and SMC, where backstepping compensates state dependent part and SMC(see Estrada et al IJRNC,27(4),2017,DOI : 10.1002/rnc.3590,Automatica 2010(11), TAC 2010(11), J. Davila TAC 2013, Ferreira et al Journal of Franklin Institute, 2014, 351(4),doi:/10.1016/j.jfranklin.2013.12.011, Automatica, September 2015, Vol. 59, 10.1016/j.automatica.2015.06.020)
3. SMC can not be smooth. Sliding mode (for definition!) is a motion on the sliding surface!
4. Chattering
4.a. There is no way to keep the sliding mode and eliminate the chattering.
4.b. It is wrong opinion that it continuous HOSM controllers (like super-twisting, Kamal et al DOI:10.1016/j.automatica.2016.02.001, Torres et al doi :10.1016/j.automatica.2017.02.035, Laghrouche et al TAC,2017) everytime produce less chattering that discontinuous ones (see https://www.researchgate.net/publication/317229996_Is_It_Reasonable_to_Substitute_Discontinuous_SMC_by_Continuous_HOSMC).
It is true for the systems with fast actuators only.
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Where can one get measurement data for ieee 34 bus radial distribution system ?
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I am making a small prototype, that consist of small compact size aerostatic bearing with active compensation. I have to control eccentricity of shaft by using active compensation. For this purpose I need a laser sensor to measure shaft eccentricity that varies from 0 to 100um. one possible suggestion is to use laser sensor of omron company but problem is that my current location is china and omron company said it will take you 5 month to receive it. Please suggest me some other possible ways so that I can finish it before time
If you want non-contact sensor, then proximity displacement sensor such as Bentley Nevada proximity sensors and Omega non-contact sensors can be used, they are cheaper than Laser sensors and have excellent accuracy:
If contact is permissible then you have wider options and LVDT sensors can be used, they have good accuracy and cheaper pices:
or search google or alibaba for LVDT sensor
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the coefficient of the plant should satisfy what kind of requirement ? and can you give some exampled?
It is generally difficult to stabilize unstable high-order non-minimum phase plants with the 'ideal' PID controller. Take Lakshmanaprabu's model as an example, which has a real pole and a real zero in the right-half plane:
Gp = (5⋅s − 1)/(16000⋅s4 + 41200⋅s3 + 2940⋅s2 − 152⋅s − 5).
It is possible to stabilize the plant with a cascading compensator
Gc = k1 − (k4⋅s2 + k3⋅s + k2)/(s3 + N3⋅s2 + N2⋅s + N1)
where
k1 = −4.4690e+04
k2 = 4.9923e+04
k3 = −2.6980e+05
k4 = 1.1096e+05
N1 = −1.1172
N2 = 6.0334
N3 = −2.4079
The closed-loop transfer function (Gc*Gp)/(1 + Gc*Gp) is a 7th-order system:
Gcl = (−2.235e+05⋅s4 + 2.793e+04⋅s3 + 4190⋅s2 − 139.7⋅s − 5.586) / (1.6e+04⋅s7 + 2674⋅s6 + 270.4⋅s5 + 17.88⋅s4 + 0.8151⋅s3 + 0.02545⋅s2 + 0.0004759⋅s + 4.249e-06),
which has two real zeros in the left-half plane and another two real zeros in the right-half plane. Cancelling the two real zeros in the left-half plane with a pre-filter
Gf = 4.249e-06/(s2 + 0.125⋅s + 0.0025)
applied to the input command outside of the feedback loop yields Gclf = Gf*Gcl
Gclf = (−5.934e-05⋅s2 + 1.483e-05⋅s − 5.934e-07) / (s7 + 0.1671⋅s6 + 0.0169⋅s5 + 0.001118⋅s4 + 5.094e-05⋅s3 + 1.59e-06⋅s2 + 2.974e-08⋅s + 2.656e-10)
Computing the DC gain of Gclf (−2234.5) and placing a reference scaling factor "1/dcgain(Gclf)" ensures the unity gain when a unit step input is applied.
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Hi,
Does someone have any experience of using PSVM for fault detection?
Thank you for the information
Dear Faeze Sdi,
Why proximal SVM are you using? You may try other algorithms for fault detection.
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Hello,
hoping that you will be in good health,
i have a 7-DOF independent wheel drive electric car model can be linear/nonlinear with longitudinal velocity, lateral velocity, yaw rate and angular rotation of the four wheels as the states of the system. the input to the system is the torque of each wheel .
i already designed a controller linear(LQR)/ nonlinear(SMC) which will turn the car with 90 degree yaw angle.
i want to find the trajectories/conditions that will achieve the control objective in minimum possible time/distance and minimum possible yaw rate, its kind of optimization "i think" can you please suggest any method or technique for doing this problem, or any starting point, i am using MATLAB/SIMULINK for my simulation ....
actually i am following the work done in this paper but they used some other tools for there work.
You can find a fully developed 14-DOF vehicle model in MATLAB with a time efficient suspension model and a reprogrammed full set MF tire model in https://arxiv.org/abs/1803.09411, in addition, this work formulated and solved the complicated optimal design and control problems of a 4-IWD electric race car on a given track in curvilinear coordinate system.
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Dear all;
1: I used SVM to fault detection, now I want to figure out effect of fault in 10 seconds time slot of 60 seconds such that I have a window of data with 10 seconds length like 0-10 seconds,0.005s-10.005 s, 0.01s -10.01s ...60s. and I want to use SVM for each 10 s window of data.
I have some data in excel file which is divided to two parts; upper section is normal data and lower section is fault data and I have 6 features. Entirely I have 24002 data. I wrote a piece of code but I'm not sure if it is correct or not and I want to know how can I correct it? 2:I would like to know how can I divide my data to train and test in for loop for each window?
clc;clear;close all;
T=2001; %length of data in 10s (Window Size)
K=(0.5*length(X))-T+1 % Number of repetitions
window=zeros(2*T,6);
for i=1:K
window=[X(i:i+T-1);X(i+12001:i+12001+T-1,:)];
%% Data Normalaization
m=length(window);
Mean_data=repmat(mean(window),m,1);
Std_data=repmat(std(window),m,1);
data_norm=(window-Mean_data)./(Std_data);
end
Dear Jiri Kovar,
I want to create motional time window in order to fault detection in each window.
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Sir,
I am unable to solve second ramp response. I need the derivation with partial fraction.
Thank you
See for paritial fraction i understand u want the values of a b or the way the roots will be arranged it is simple break roots and put s- a or s- b or whatever it be plus also then in one case put the value of one root in such a way the other root becomes zero and u will obtain value of a and b substitute back then take inverse laplace transform will obtain the transfer function in paritial fraction again the expansion of roots like repeated roots the method needs to differentiae the root once u obtaim the first root ur problem would be solved 100 percent by ogata it has detailed explanation modern control theory by ogata
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OpenDSS is a wonderful package, but it is not suitable for simulating some special features of railway traction network, it would be fantastic to have something similar related to Railway power systems. What do you think?
I totally agree with you in that
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I have designed a controller using Integrator backstepping method and that controller is based on Lyapunov Theory. I have a question How I can check stability of resulting controller?
Hi Waheed,
Since you have designed a Lyapunov-based integral backstepping controller for the 4th-order MISO system, what exactly is your difficulty in checking the stability of the control system?
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I want to know if there is any system that needs to operate on different variants (combinations of the P, I, D constants) of PID controller; like it operates on P-only controller first; then switches to PI and then may be to PD or PID alternatively or simultaneously or may be cascaded? Is there any need to have such a system? If yes where and if not why not?
thankyou all.
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I want to ascertain the level of interaction in a MIMO process and I am looking at various ways of doing that. I came across the SVD method in the book by Seborg and I would like to know how efficient it is. Is the relative gain array (RGA) method better than this ?
P.S. - I am asking the question strictly from a control system perspective.
The SVD precoding approach comes from Telatar's seminal work: http://web.mit.edu/18.325/www/telatar_capacity.pdf
He shows that it achieved the MIMO capacity. That paper is very nicely written, so I highly recommend it!
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IMC BASED FRACTIONAL ORDER PID
Anyway, you can consider derivative filter even in case of fractional PID controller in the design procedure, see e.g. But think carefully whether you really need fractional action at the controller side. It should be usually motivated by contradictory frequency domain requirements that cannot be fulfilled by classical integer order PID.
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all pole of the system are in left hand side of s -plane . But i m getting GM negative and phase margin(PM) as positive. Step response is also stable with 20 % overshoot and settling time as 6 sec. I m not able to conclude stability with tjhis results.
Dear Boris,
I hope I will be excused for writing a “long message,” (and people who are not interested can just ignore it) yet this to make sure that my messages are not misunderstood as puting “this method against the other.” Actually, I started with the claim that Nyquist theorem and Nyquist plot contain the entire information. The only problem is that, in realistic systems, where the amplitudes may go up quite a lot, even to infinite, when you need to decide what exactly occurs around 1, this could be a tough task in Nyquist plot. Here, Bode, with its logarithmic scale, is best… when it works, and, because of the two separate plots, this usually occurs in not too complex systems. Then, maybe only after some use, you learn that Nichols combines both advantages. Before Matlab (and other tools), plotting Nichols used to be a terrible task and this may explain why people might have remained reluctant to use it.
Please see my message about the Company which started with “No, we don’t use Nichols” and ended using Nichols as the main tool. So, maybe you just try to write (or ask students to write) Nichols along with Bode and just see what Nichols gives in all other than simple cases.
As you know, my main “academic” business has been (simple) adaptive control. However, as I use to tell, although I have been playing Prof. and Dr., all my years I have been mainly an engineer. In this context, first I actually refused to use adaptive control, because control practitioners do not think that, for a given specific case, one cannot find enough information to do a good fixed controller. Even when one has to fit parameters to a given specific plant, one can do it off-line. Actually, PID is so widely used because many systems are not too complicated; they could even be stable or close to stability to start with and then, you can safely fit the right parameters to give you the desired performance. If you don’t need too much, same controller can serve all machines of same type.
The problems start when you really need good performance, which usually means fast response and also very precise tracking. While doing manual tuning on one or two machines may still be not too difficult, when you produce 20, 100, or 400 machines per month, this becomes a very tough job, in particular when your customers, maybe thousands of kilometers away, may replace a motor or any other piece and the performance deteriorates. So, you may also have to keep sending your best guys all over the world.
The same complex systems and/or many not-exactly-identical systems are also the reason for my present use of adaptive control.
Same Company, which started with “we don’t use Nichols,” also responded to my idea of using automatic (adaptive) tuning with “Adaptive? Ha-Ha! You, Professor, please keep writing papers. We have real problems.” However, because the real problem was really too tough, they finally thought that they could not lose too much letting this funny Professor play for a little while.
The result? Now there is a button on each machine and all that they have to do at the end of production, or what the operator has to do when any change leads to deterioration of performance, is to push the button and wait a minute or so for the parameters to be adapted to the specific machine. Their technology, which was “dead” by 1994, is alive and kicking today and performance is something else.
This was only automatic tuning per machine, not what we understand by adaptive control, yet this was also what made me see that, when high performance is needed, on-line (simple) adaptive control is needed to maintain performance even for one given machine under uncertainty and under various and (real-time) changing operational environments. Results justify this and will still be seen.
With best regards, Itzhak
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Please read this two pictures in File, It is in Chinese I took it from a paper but the symbol and expression is worldwide and international. For a system, select sliding mode valuable as S=c1e1+c2e2......en.
What if I want to know the value of S' (dot(s)) ? In this paper, It gives the expression of S'. It derives from the error state space by letting en'=xn'-dot^n(yd)=f(x)+bu-dot^n(yd), But can I get it in an easier way?
Let see, e1=x1-yd| yd is the reference signal and x1 is the output of the system,y=x1. e2=dot(e1),e3=dot(e2)=dot(dot(e1))......So, in this way, we can get the value of S and its higher order time differentiates. I think it may work and the final value we get should be the same numerically.
What's your opioions? Thank you very much!
If you agree with me, I have another doubt.
In the simulink model of that paper, It calculated dot(s) by that expression it gave, Why didn't it use differentiator to get dot(s)? It seems it doesn't want to introduce more differentiators into its system.
Hi Dawn,
I can see where you're coming from. You are trying to generalize that
\dot{en} = \dot{xn} − yd(n) = x1(n) − yd(n) .
However,
\dot{xn} ≠ x1(n) because of
\dot{xn} = ∂(x) + β(x)u + η.
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why every passive system can be considered as stable system.
The two concepts are closely related but not completely identical. We may take a deep look at them from two perspectives.
1. The first is from the perspective of pure physics. Passivity can be considered as universal in mechanical systems, e.g., robot manipulators and satellites. Specifically, a satellite in orbit, without active control, is stable in terms of both its position and velocity and in addition the velocity of the satellite would finally decay since the system mechanical energy (kinetic and potential energy) gradually decreases due to the atmosphere drag forces (whose is role is similar to damping control). This phenomenon is due to the fact that a satellite in free motion is (output strictly) passive with respect to its output (i.e., velocity). Another common example is the bicycle and it would finally become still on a horizontal road if we do not insert energy by our legs. Passivity may also be thought to be the will of the system (free) to stabilize itself.
2. The second perspective is based on the passivity formula. Passivity is typically defined in terms of the input (u) and output (y) of a system, namely
int_0^t yTu dr >=E(t)-E(0)
where E(t) is the system storage. The interpretation is that the injected energy from the external input is not less than the variation of the system storage. "Not less than" here implies that the injected energy by the input is partitioned into two parts: 1) variation of the system storage and 2) the nonnegative second part. This intuitively means that there is some energy dissipated (by the system) and only part of the energy is converted to the system storage.
The passivity formula, however, does not naturally implies stability of the system under the external input (u) but it does imply that the free system (i.e., without the external input) is stable. The stability of the free system, yet, has a weak connection with the typical equilibrium-based Lyapunov stability since the system storage does not explicitly specify that it is defined based on certain equilibrium of the system.
If the external input is allowed to be designed, the connection between passivity and equilibrium-based Lyapunov stability can be explicitly established (in part). For instance, with a negative output feedback for the external input, output regulation (to its equilibrium zero) can typically be achieved.
In summary, we might say that passivity often implies (equilibrium-based Lyapunov) stability, but it also tells many other significant things concerning the physics/dynamics of the system.
I hope the above points would complement the existing answers and be of some help.
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Is it possible to define a 2D (or n-D) Lookup table as signal/condition generators for stimulating model variables in dSPACE Control desk NG?
Dear Arindam,
I suggest you to see attached publication in topics.
Best regards
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Electrical motor
unbalance system
There are two types of imbalances. Negative sequence and zero sequence. The negative sequence does not need a neutral connection. So you can have negative sequence imbalances in a systems without neutral and even in delta connection.
Zero sequence current can be also produced by parasitic capacitive currents is the machine is fed by a power converter operating at high switching frequency. Differential earth leaking protections are likely to trip in this case.
If there is not a fault in the winding and a converter is not used, check the power supply. If the power supply is unbalanced this will produce negative sequence currents in the machine and pulsating torques.
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Hello,
I want to control the steady state disturbances of DC DC boost converter by Luenberger observer but I am not getting a proper control scheme in any literature. So if someone could help me regarding this then it would be very helpful.
The Luenberger observer is useful to estimate the non-measurable state variables of a dynamic system. The controller is normally a LQR (linear quadratic regulator). In order to use it the system has to be observable. A DC DC boost converter is a kind of hybrid system (switched system, switching control) that need a special type of control. Therefore, hybrid observability and hybrid control are important key words. Please have a look at the following references:
1) Bemporad A., Ferrari-Trecate G., Morari M.: Observability and controllability of piecewise affine and hybrid systems. IEEE Trans A.C. 45, 1864–1876 (2000)
2) Geyer T. Papafotiou G., Morari M.: Hybrid model predictive control of the step-down DC–DC converter. IEEE Trans. Control Syst. Technol. 16(6), 1112–1124 (2008)
3) Iannelli L., Johansson K.H., Jönsson U.T., Vasca F.: Subtleties in the averaging of a class of hybrid systems with applications to power converters. Control Eng. Pract. 16, 961–975 (2008).
4) Kamri, D., Larbes, C., Observer-Based Control for DC–DC Converters. Practical Switching Control. Arabian Journal for Science and Engineering, Vol 39, No 5, pp 4089–4102, May 2014.
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Details: Consider a PID controller with pre-saturation on the output. We wish to implement integrator anti-windup, whereby once some saturation level is reached, the integrator stops integrating. I have seen two different basic policies at this point (lots of variants of these two).
* In one case the integrator accumulator is cleared, i.e. the integrator is reset and held at 0 value until we come off of saturation.
* In an alternate case, the value of the integrator is held constant, but no new input comes in, until we come off of saturation.
I guess my question is, whether there is any consensus on when it is advisable to do one versus the other.
I think it relates to the conditions that caused things to go into saturation. A large setpoint change would mean that the error history stored in the integrator were not at all related to the new setpoint. In that case, one would do well to zero out the integrator. On the other hand, going into saturation in a regulator mode would imply that the error values stored in the integrator were most likely relevant and so holding the old value constant would make more sense.
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# Peter Richtárik Randomized Dual Coordinate Ascent with Arbitrary Sampling Foundations of Computational Mathematics – Montevideo, Uruguay – December 2014.
## Presentation on theme: "Peter Richtárik Randomized Dual Coordinate Ascent with Arbitrary Sampling Foundations of Computational Mathematics – Montevideo, Uruguay – December 2014."— Presentation transcript:
1 Peter Richtárik Randomized Dual Coordinate Ascent with Arbitrary Sampling Foundations of Computational Mathematics – Montevideo, Uruguay – December 2014
2 Coauthors Zheng Qu (Edinburgh) Zheng Qu, P.R. and Tong Zhang Randomized dual coordinate ascent with arbitrary sampling arXiv: , 2014 Tong Zhang (Rutgers, Baidu)
3 Part 1 MACHINE LEARNING
4 Statistical nature of data $\phi_i(a)=\tfrac{1}{2\gamma}(a-b_i)^2$ $\Downarrow$ $\frac{1}{n}\sum_{i=1}^n \phi_i(A_i^\top w) =\frac{1}{2\gamma}\|Aw-b\|_2^2$ Data (e.g., image, text, measurements, …) Label $A_i \in \mathbb{R}^{d\times m}, \qquad y_i \in \mathbb{R}^m$
5 Prediction of labels from data $(A_i,y_i)\sim \emph{Distribution}$ $A_i^\top w \approx y_i$ Find Such that when (data, label) pair is drawn from the distribution Then Predicted labelTrue label Linear predictor
6 Measure of Success $\mathbf{E} \left[ loss(A_i^\top w, y_i)\right]$ We want the expected loss (=risk) to be small: datalabel
7 Finding a Linear Predictor via Empirical Risk Minimization $\min_{w\in \mathbb{R}^d} \frac{1}{n}\sum_{i=1}^n loss(A_i^\top w, y_i)$ $(A_1,y_1), (A_2,y_2), \dots, (A_n,y_n)\sim \emph{Distribution}$ Draw i.i.d. data (samples) from the distribution Output predictor which minimizes the empirical risk:
8 Part 2 OPTIMIZATION
9 Primal Problem d = # features (parameters) n = # samples 1 - strongly convex function (regularizer) $\min_{w \in \mathbb{R }^d}\;\; \left[ P(w) \equiv \frac{1}{n} \sum_{i=1} ^n \phi_i(A_i^ \top w) + \lambda g(w)\right]$ - smooth & convex regularization parameter
10 Assumption 1 Loss functions have Lipschitz gradient $\|\nabla \phi_i(a)-\nabla \phi_i(a')\|\;\; \leq\;\; \frac{1}{\gamma}\;\; \|a-a'\|, \quad a,a'\in \mathbb{R}^m$ Lipschitz constant
11 Assumption 2 $g(w)\geq g(w') + \left + \frac{1}{2}\|w-w'\|^2, \quad w,w'\in \mathbb{R}^d$ Regularizer is 1-strongly convex subgradient
12 Dual Problem - strongly convex $D(\alpha) \equiv - \lambda g^*\left(\frac{1}{\la mbda n}\sum_{i=1}^n A_i\alpha_i\right) - \frac{1}{n}\sum_{i= 1}^n \phi_i^*(- \alpha_i) 1 – smooth & convex \[\max_{\alpha=(\alpha_1,\dots,\alpha_n) \in \mathbb{R}^{N}=\mathbb{R}^{nm}} D(\alpha)$ $g^*(w') = \max_{w\in \mathbb{R}^d} \left\{(w')^\top w - g(w)\right\}$ $\phi_i^*(a') = \max_{a\in \mathbb{R}^m} \left\{(a')^\top a - \phi_i(a)\right\}$
13 Part 3 QUARTZ
14 Quartz
15 Fenchel Duality $P(w) - D(\alpha) \;\; = \;\; \lambda \left( g(w) + g^*\left(\bar{\alpha}\right)\right) + \frac{1}{n}\sum_{i=1}^n \phi_i(A_i^\top w) + \phi_i^*(-\alpha_i) =$ $\lambda (g(w) + g^* \left(\bar{\alpha}\right)- \left\langle w, \bar{\alpha} \right\rangle ) + \frac{1}{n}\sum_{i=1}^n \phi_i(A_i^\top w) + \phi_i^*(-\alpha_i) + \left\langle A_i^\top w, \alpha_i \right\rangle$ $\bar{\alpha} = \frac{1}{\lambda n} \sum_{i=1}^n A_i \alpha_i$ $w = \nabla g^*(\bar{\alpha})$ $\alpha_i = -\nabla \phi_i(A_i^\top w)$ Weak duality Optimality conditions
16 The Algorithm
17 Quartz: Bird’s Eye View $(\alpha^t,w^t) \qquad \Rightarrow \qquad (\alpha^{t+1},w^{t+1})$ $w^{t+1} \leftarrow (1-\theta)w^t + \theta {\color{red}\nabla g^*(\bar{\alpha}^t)}$ $\alpha_i^{t+1} \leftarrow \left(1-\frac{\theta}{{\color{blue}p_i}}\right) \alpha_i^{t} + \frac{\theta}{{\color{blue}p_i}}{\color{red} \left(-\nabla \phi_i(A_i^\top w^{t+1})\right)}$ STEP 1: PRIMAL UPDATE STEP 2: DUAL UPDATE Choose a random set ${\color{blue}S_t}$ of dual variables ${\color{blue}p_i = \mathbf{P}(i\in S_t)}$
18 The Algorithm STEP 1 STEP 2 Convex combination constant
19 Randomized Primal-Dual Methods SDCA: SS Shwartz & T Zhang, 09/2012 mSDCA M Takac, A Bijral, P R & N Srebro, 03/2013 ASDCA: SS Shwartz & T Zhang, 05/2013 AccProx-SDCA: SS Shwartz & T Zhang, 10/2013 DisDCA: T Yang, 2013 Iprox-SDCA: P Zhao & T Zhang, 01/2014 APCG: Q Lin, Z Lu & L Xiao, 07/2014 SPDC: Y Zhang & L Xiao, 09/2014 Quartz: Z Qu, P R & T Zhang, 11/2014 {\footnotesize \begin{tabular}{|c|c|c|c|c|c|c|c|c|c} \hline Algorithm & 1-nice & 1-optimal & $\tau$-nice & arbitrary &{\begin{tabular}{c}additional\\speedup\end{tabular} } & { \begin{tabular}{c}direct\\p-d\\analysis\end{tabular}} & acceleration\\ \hline SDCA & $\bullet$ & & & & & & \\ \hline mSDCA & $\bullet$ & & $\bullet$ & & $\bullet$ & & \\ \hline ASDCA & $\bullet$ & & $\bullet$ & & & & $\bullet$ \\ \hline AccProx-SDCA &$\bullet$ & & & & & &$\bullet$\\ \hline DisDCA &$\bullet$ & &$\bullet$ & & & & \\ \hline Iprox-SDCA & $\bullet$ & $\bullet$ & & & & & \\ \hline APCG &$\bullet$ & & & & & &$\bullet$ \\ \hline SPDC &$\bullet$ & $\bullet$ &$\bullet$ & & & $\bullet$ &$\bullet$ \\ \hline \bf{Quartz} &{\color{red}$\bullet$} &{\color{red}$\bullet$} &{\color{red}$\bullet$} &{\color{red}$\bullet$} &{\color{red}$\bullet$} & {\color{red}$\bullet$} & \\ \hline \end{tabular} }
20 Part 4 MAIN RESULT
21 Assumption 3 (Expected Separable Overapproximation) $\mathbf{E} \left\| \sum_{i\in \hat{S}} A_i \alpha_i\right\|^2 \;\;\leq \;\; \sum_{i=1}^n {\color{blue} p_i} {\color{red} v_i}\|\alpha_i\|^2$ ${\color{blue} p_i} = \mathbb{P}(i\in \hat{S})$ Parameters ${\color{red}v_1,\dots,\color{red}v_n}$ satisfy: inequality must hold for all $\alpha_1,\dots,\alpha_n \in \mathbb{R}^m$
22 Complexity Theorem (QRZ’14) $k\;\;\geq \;\; \max_i \left( \frac{1}{{\color{blue}p_i}} + \frac{{\color{red}v_i}}{{\color{blue}p_i} \lambda \gamma n}\right) \log \left( \frac{P(w^0)-D(\alpha^0)}{\epsilon} \right)$ $\mathbf{E}\left[P(w^t)-D(\alpha^t)\right] \leq \epsilon$
23 Part 5.A UPDATING ONE DUAL VARIABLE AT A TIME $A = [A_1,A_2,A_3,A_4,A_5] = \left(\begin{matrix} 0 & 0 & 6 & 4 & 9\\ 0 & 3 & 0 & 0 & 0\\ 0 & 0 & 3 & 0 & 1\\ 1 & 8 & 0 & 0 & 0\\ \end{matrix}\right)$
24 \begin{table} \begin{tabular}{|c|c|} \hline & \\ Optimal sampling & $\displaystyle n +\frac{\tfrac{1}{n}\sum_{i=1}^n L_i}{\lambda \gamma}$ \\ & \\ \hline & \\ Uniform sampling & $\displaystyle n +\frac{\max_i L_i}{\lambda \gamma}$ \\ & \\ \hline \end{tabular} \end{table} Complexity of Quartz specialized to serial sampling
25 Data \begin{table} \begin{tabular}{|c|c|c|c|} \hline {\bf Dataset} & \begin{tabular}{c}{\bf \# Samples}\\ $n$\end{tabular} & \begin{tabular}{c}{\bf \# features}\\ $d$ \end{tabular}& \begin{tabular}{c}{\bf density} \\ $nnz(A)/(nd)$\end{tabular}\\ \hline astro-ph & 29,882 & 99,757 & 0.08\% \\ \hline CCAT & 781,265 & 47,236 & 0.16\% \\ \hline cov1 & 522,911 & 54 & 22.22\%\\ \hline w8a & 49,749 & 300 & 3.91\% \\ \hline ijcnn1 & 49,990 & 22 & 59.09\% \\ \hline webspam & 350,000 & 254 & 33.52\% \\\hline \end{tabular} \end{table}
26 Standard primal update Experiment: Quartz vs SDCA, uniform vs optimal sampling “Aggressive” primal update Data = \texttt{cov1}, \quad $n=522,911$, \quad$\lambda = 10^{-6}$
27 Part 5.B TAU-NICE SAMPLING (STANDARD MINIBATCHING)
28 Data sparsity A normalized measure of average sparsity of the data “Fully sparse data”“Fully dense data”
29 Complexity of Quartz \begin{table} \begin{tabular}{|c|c|} \hline &\\ \begin{tabular}{c}Fully sparse data\\ (${\color{blue}\tilde{\omega}=1}$) \end{tabular} & $\displaystyle \frac{n}{{\color{red}\tau}}+\frac{\max_i L_i}{\lambda \gamma {\color{red}\tau}}$\\ & \\ \hline & \\ \begin{tabular}{c}Fully dense data \\(${\color{blue}\tilde{\omega}=n}$) \end{tabular} & $\displaystyle \frac{n}{{\color{red}\tau}}+\frac{\max_i L_i}{\lambda \gamma}$ \\ & \\ \hline & \\ \begin{tabular}{c}Any data\\ (${\color{blue}1\leq \tilde{\omega}\leq n}$) \end{tabular} & $\displaystyle \frac{n}{{\color{red}\tau}}+\frac{\left(1+\frac{({\color{blue}\tilde{\omega}}-1)({\color{red}{\color{red}\tau}}-1)}{n-1}\right)\max_i L_i}{\lambda \gamma {\color{red}\tau}}$ \\ & \\ \hline \end{tabular} \end{table}
30 Speedup $\frac{T(1)}{T({\color{red}\tau})} \geq \frac{{\color{red}\tau}}{1+\frac{\tilde{\omega}-1}{n-1}}\geq \frac{{\color{red}\tau}}{2}$ $1\leq {\color{red}\tau} \leq 2+\lambda \gamma n$ Assume the data is normalized: Then: Linear speedup up to a certain data-independent minibatch size: $T({\color{red}\tau}) \;\; = \;\; \frac{ \left(1+\frac{({\color{blue}\tilde{\omega}}-1)({\color{red}\tau}-1)}{(n-1)(1+\lambda \gamma n)}\right) }{{\color{red}\tau}} \times T({\color{red}1})$ Further data-dependent speedup, up to the extreme case: $T({\color{red}\tau}) \leq \frac{2}{{\color{red}\tau}} \times T({\color{red}1})$ ${\color{blue}\tilde{\omega}} = O(1)$ $T({\color{red}\tau}) = {\cal O} \left(\frac{T({\color{red}1})}{\color{red}\tau}\right)$
31 Speedup: sparse data
32 Speedup: denser data
33 Speedup: fully dense data
34 astro_ph: n = 29,882 density = 0.08%
35 CCAT: n = 781,265 density = 0.16%
36 \begin{tabular}{c|c|c} \hline Algorithm & Iteration complexity & $g$\\ \hline && \\ SDCA & $\displaystyle n + \frac{1}{\lambda \gamma}$ & $\tfrac{1}{2}\|\cdot\|^2$ \\ &&\\ \hline &&\\ ASDCA & $\displaystyle 4 \times \max\left\{\frac{n}{{\color{red}\tau}},\sqrt{\frac{n}{\lambda\gamma {\color{red}{\color{red}\tau}}}},\frac{1}{\lambda \gamma {\color{red}\tau}},\frac{n^{\frac{1}{3}}}{(\lambda \gamma {\color{red}\tau})^{\frac{2}{3}}}\right\}$ & $\tfrac{1}{2}\|\cdot\|^2$\\ &&\\ \hline &&\\ SPDC & $\displaystyle \frac{n}{{\color{red}\tau}}+\sqrt{\frac{n}{\lambda \gamma {\color{red}\tau}}}$ & general \\ &&\\ \hline &&\\ \bf{Quartz } & $\displaystyle \frac{n}{{\color{red}\tau}}+\left(1+\frac{(\tilde \omega -1)({\color{red}\tau}-1)}{n-1}\right)\frac{1}{\lambda \gamma {\color{red}\tau}}$ & general\\ &&\\ \hline \end{tabular} Primal-dual methods with tau-nice sampling SS-Shwartz & T Zhang ‘13 Y Zhang & L Xiao ‘14
37 \begin{tabular}{c|c|c|c|c} \hline Algorithm & $\gamma\lambda n = \Theta(\tfrac{1}{\tau})$ & $\gamma\lambda n = \Theta(1)$ & $\gamma\lambda n = \Theta(\tau)$ & $\gamma\lambda n = \Theta(\sqrt{n})$\\ \hline & $\kappa = n\tau$ & $\kappa = n$ & $\kappa = n/\tau$ & $\kappa = \sqrt{n}$\\ \hline &&&&\\ SDCA & $n \tau$ & $n$ & $n$ & $n$\\ &&&&\\ \hline &&&&\\ ASDCA & $n$ & $\displaystyle \frac{n}{\sqrt{\tau}}$ & $\displaystyle \frac{n}{\tau}$ & $\displaystyle \frac{n}{\tau} + \frac{n^{3/4}}{\sqrt{\tau}}$\\ &&&&\\ \hline &&&&\\ SPDC & $n$ & $\displaystyle \frac{n}{\sqrt{\tau}}$ & $\displaystyle \frac{n}{\tau}$ & $\displaystyle \frac{n}{\tau} + \frac{n^{3/4}}{\sqrt{\tau}}$\\ &&&&\\ \hline &&&&\\ \bf{Quartz } & $\displaystyle n + \tilde{\omega}\tau$ & $\displaystyle \frac{n}{\tau} +\tilde{\omega}$ & $\displaystyle \frac{n}{\tau}$ & $\displaystyle \frac{n}{\tau} + \frac{\tilde{\omega}}{\sqrt{n}}$\\ &&&&\\ \hline \end{tabular} For sufficiently sparse data, Quartz wins even when compared against accelerated methods Accelerated
38 Part 5.C DISTRIBUTED QUARTZ
39 Distributed Quartz: Perform the Dual Updates in a Distributed Manner Quartz STEP 2: DUAL UPDATE Data required to compute the update
40 Distribution of Data n = # dual variables Data matrix
41 Distributed sampling
42 Random set of dual variables
43 Distributed sampling & distributed coordinate descent P.R. and Martin Takáč Distributed coordinate descent for learning with big data arXiv: , 2013 Previously studied (not in the primal-dual setup): Olivier Fercoq, Zheng Qu, P.R. and Martin Takáč Fast distributed coordinate descent for minimizing non strongly convex losses 2014 IEEE Int Workshop on Machine Learning for Signal Processing, May 2014 Jakub Marecek, P.R. and Martin Takáč Fast distributed coordinate descent for minimizing partially separable functions arXiv: , June strongly convex & smooth convex & smooth
44 Complexity of distributed Quartz $\frac{n}{c\tau} + \max_i\frac{\lambda_{\max}\left( \sum_{j=1}^d \left(1+\frac{(\tau-1)(\omega_j-1)}{\max\{n/c-1,1\}}+ \left(\frac{\tau c}{n} - \frac{\tau-1}{\max\{n/c-1,1\}}\right) \frac{\omega_j'- 1}{\omega_j'}\omega_j\right) A_{ji}^\top A_{ji}\right)}{\lambda\gamma c\tau}$
45 Reallocating load: theoretical speedup n = 1,000,000 density = 100% n = 1,000,000 density = 0.01%
46 Reallocating load: experiment Data: webspam n = 350,000 density = 33.51%
47 Experiment Machine: 128 nodes of Hector Supercomputer (4096 cores) Problem: LASSO, n = 1 billion, d = 0.5 billion, 3 TB Algorithm: with c = 512 P.R. and Martin Takáč, Distributed coordinate descent method for learning with big data, arXiv: , 2013
48 LASSO: 3TB data nodes
49 Part 5.4 PRODUCT SAMPLING
50 Product sampling $X_1 := \{1,2\}, \quad X_2 := \{3,4,5\}$ Product sampling: Non-serial importance sampling ! In general: Each row (feature) has nonzeros in a single group of columns (examples) only
51 Complexity Meaning of the above notation:
52 Part 5 ESO Zheng Qu and P.R. Coordinate Descent with Arbitrary Sampling II: Expected Separable Overapproximation 2014
53 Computation of ESO parameters $\mathbf{E} \left\| \sum_{i\in \hat{S}} A_i \alpha_i\right\|^2 \;\;\leq \;\; \sum_{i=1}^n {\color{blue} p_i} {\color{red} v_i}\|\alpha_i\|^2$ $\Updownarrow$ $P \circ A^\top A \preceq Diag({\color{blue}p}\circ {\color{red}v})$ Theorem (QR’14) {For simplicity, assume that m = 1} $A = [A_1,A_2,\dots,A_n]$
54 ESO $\lambda'(M) \equiv \max_{\alpha\in \mathbb{R}^n} \frac{\alpha^\top M \alpha}{\alpha^\top Diag(M) \alpha}$ ${\color{red} v_i} = \underbrace{\min\{\lambda'(P(\hat{S})),\lambda'(A^\top A)\}}_{\equiv\beta(\hat{S},A)} \|A_i\|^2$ For any sampling, ESO holds with Theorem (QR’14) where $\mathbf{P}(|\hat{S}|=\tau)=1 \quad \Rightarrow \quad\lambda'(P(\hat{S}))=\tau$ $1 \leq \lambda'(A^\top A) \leq \underbrace{\max_j \|A_{j:}\|_0}_{\omega} \leq n$
55 ESO $\lambda'(M) \equiv \max_{\alpha\in \mathbb{R}^n} \frac{\alpha^\top M \alpha}{\alpha^\top Diag(M) \alpha}$ ${\color{red} v_i} = \underbrace{\min\{\lambda'(P(\hat{S})),\lambda'(A^\top A)\}}_{\equiv\beta(\hat{S},A)} \|A_i\|^2$ Theorem (QR’14) $\mathbf{P}(|\hat{S}|=\tau)=1 \quad \Rightarrow \quad\lambda'(P(\hat{S}))=\tau$ $1 \leq \lambda'(A^\top A) \leq \underbrace{\max_j \|A_{j:}\|_0}_{\omega} \leq n$
56 END
57 Experiment Machine: 128 nodes of Archer Supercomputer Problem: LASSO, n = 5 million, d = 50 billion, 5 TB (60,000 nnz per row of A) Algorithm: Hydra 2 with c = 256 Olivier Fercoq, Zheng Qu, P.R. and Martin Takáč, Fast distributed coordinate descent for minimizing non-strongly convex losses, IEEE Int Workshop on Machine Learning for Signal Processing, 2014
58 LASSO: 5TB data nodes
59 Part X LOSS FUNCTIONS & REGULARIZERS
60 Example 1: Quadratic Loss $\phi_i(a)=\tfrac{1}{2\gamma}(a-b_i)^2$ $\Downarrow$ $\frac{1}{n}\sum_{i=1}^n \phi_i(A_i^\top w) =\frac{1}{2\gamma}\|Aw-b\|_2^2$
61 Example 2: Logistic Loss $\phi_i(a)=\tfrac{4}{\gamma}\log\left(1+e^{-b_i a}\right)$ $\Downarrow$ $\frac{1}{n}\sum_{i=1}^n \phi_i(A_i^\top w) =\frac{4}{\gamma}\sum_{i=1}^n \log\left(1+ \exp^{-b_i A_i^\top w}\right)$
62 0-1 loss 01 1 $\phi_i(a)=\left\{\begin{array} {ll} 1 & \quad a\leq 1 \\ 0 & \quad a\geq 1 \end{array}\right.$
63 Hinge loss 01 1 $\phi_i(a)=\left\{\begin{array} {ll} 1-a & \quad a\leq 1 \\ 0 & \quad a\geq 1 \end{array}\right.$
64 Example 3: Smoothed hinge loss 01 1 $\phi_i(a)=\left\{\begin{array} {ll} 1-a-\gamma/2 & \quad a\leq 1-\gamma \\ \displaystyle\frac{(1-a)^2}{2\gamma} & \quad 1-\gamma \leq a \leq 1 \\ 0 & \quad a\geq 1 \end{array}\right.$
65 Example 4: Squared hinge loss 01 1 $\phi_i(a)=\left\{\begin{array} {ll} \displaystyle \frac{(1-a)^2}{2\gamma} & \quad a\leq 1 \\ 0 & \quad a\geq 1 \end{array}\right.$
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2016-12-06 02:31:01
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https://tex.stackexchange.com/questions/266129/flowchart-where-the-nodes-are-pspictures
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# Flowchart where the nodes are pspictures
I want to create a flowchart, in which every node is itself a pspicture (with nodes and arrows), and there are arrows between the pspictures themselves. Here is an example (which looks bad because I had to manually place the text and the arrows):
Currently, I am creating the arrows manually, but I am sure this is not the right way to do it. Here is my current code:
\documentclass[11pt,,section]{article}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
%\usepackage[a4paper]{geometry}
%\geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
\usepackage[crop=on]{auto-pst-pdf}
\usepackage{pstricks,pst-node,pst-tree}
\usepackage{graphics,graphicx}
%
\newcommand{\action}[1] {
\rput(-1.5,-.5){#1}
}
\newcommand{\goleft}[1] {
\pcline[linecolor=black]{->}(-4,-0.5)(-5,-1.5)\bput(.5){\emph{#1}}
}
\newcommand{\godown}[1] {
\pcline[linecolor=black]{->}(-1.5,-0.5)(-1.5,-1.5)\aput(.5){\emph{#1}}
}
\newcommand{\goright}[1] {
\pcline[linecolor=black]{->}(1.5,-0.5)(2.5,-1.5)\aput(.5){\emph{#1}}
}
%
\newcommand{\agent}[1]{\circlenode[linecolor=white]{#1}{\textcolor{blue}{#1}}}
\newcommand{\piece}[1]{\dianode[linecolor=white]{#1}{\textcolor{blue}{#1}}}
\newcommand{\like}[2]{\ncline[linecolor=green]{->}{#1}{#2}}
\newcommand{\likenext}[2]{\ncline[linecolor=gray,linestyle=dashed]{->}{#1}{#2}}
\newcommand{\likemaybe}[2]{\ncline[linecolor=gray,linestyle=dashed]{->}{#1}{#2}}
\newcommand{\threeagents}{
\agent{A} \agent{B} \agent{C}
\vspace{7mm}
\piece{1} \piece{2} \piece{3}
}
\newcommand{\fouragents}{
\agent{A} & \agent{B} & \agent{C} & \agent{D} \\
\piece{1} & \piece{2} & \piece{3} & \piece{4}
}
\psset{colsep=0.5cm,rowsep=1cm}
\begin{document}
\begin{psmatrix}[colsep=0,rowsep=1.7]
\makebox[5cm] {
}
&
\makebox[5cm] {
Start
\goleft{B,C,D have 3 neighbors}
\godown{2 neighbors}
\goright{1 neighbor}
}
&
\makebox[5cm] {
}
\\
\begin{psmatrix}[colsep=0cm]
\fouragents
\like{A}{1}\like{A}{2}\like{A}{3}\like{A}{4}
\like{B}{2}
\like{C}{3}
\like{D}{4}
\end{psmatrix}
\action{Done}
&
\begin{psmatrix}[colsep=0cm]
\fouragents
\like{A}{1}\like{A}{2}\like{A}{3}\like{A}{4}
\like{B}{1}
\like{C}{4}
\like{D}{4}
\end{psmatrix}
\goleft{}
\godown{}
&
\begin{psmatrix}[colsep=0cm]
\fouragents
\like{A}{1}\like{A}{2}\like{A}{3}\like{A}{4}
\like{B}{4}
\like{C}{4}
\like{D}{4}
\end{psmatrix}
\\
\begin{psmatrix}[colsep=0cm]
\fouragents
\like{A}{2}\like{A}{3}\like{A}{4}
\like{B}{1}\like{B}{2}
\like{C}{4}
\like{D}{4}
\end{psmatrix}
&
\begin{psmatrix}[colsep=0cm]
\fouragents
\like{A}{2}\like{A}{3}
\like{B}{1}\like{B}{2}
\like{C}{4}\like{C}{2}
\like{D}{4}
\end{psmatrix}
&
%%%
\end{psmatrix}
\\
\\
\end{document}
I am trying to create arrows from the "start" psmatrix to each of the psmatrix-es in the next row. But this doesn't look good.
Here is an example which shows that you can work with symbolic names for the cells:
\documentclass[11pt,,section]{article}
\usepackage[crop=on]{auto-pst-pdf}
\usepackage{pst-node}
\def\Line{\ncline[arrows=->,linecolor=green]}
\begin{document}
\begin{psmatrix}[colsep=3cm,rowsep=1.7]
& [name=start] Start & \\[1cm]
[name=Aleft]
\begin{psmatrix}[colsep=0.5cm,rowsep=0mm]
A & B & C & D \\[12mm]
1 & 2 & 3 & 4\\
\psspan{3}Done
\Line{1,1}{2,1}\Line{1,2}{2,2}\Line{1,3}{2,3}\Line{1,4}{2,4}
\Line{1,1}{2,2}\Line{1,1}{2,3}\Line{1,1}{2,4}
\end{psmatrix}
& [name=Acenter]
\begin{psmatrix}[colsep=0.5cm,rowsep=0mm]
A & B & C & D \\[12mm]
1 & 2 & 3 & 4\\
\Line{1,1}{2,1}\Line{1,2}{2,1}\Line{1,3}{2,4}\Line{1,4}{2,4}
\Line{1,1}{2,2}\Line{1,1}{2,3}\Line{1,1}{2,4}
\end{psmatrix}
& [name=Aright]
\begin{psmatrix}[colsep=0.5cm,rowsep=0mm]
A & B & C & D \\[12mm]
1 & 2 & 3 & 4\\
\Line{1,1}{2,1}\Line{1,2}{2,4}\Line{1,3}{2,4}\Line{1,4}{2,4}
\Line{1,1}{2,2}\Line{1,1}{2,3}\Line{1,1}{2,4}
\end{psmatrix}
\end{psmatrix}
\ncline[nodesepA=2mm,offsetB=-15mm]{start}{Aleft}\nbput{B,C,D have 3 neighbors}
\ncline[nodesep=2mm]{start}{Acenter}\ncput*{2 neighbors}
\ncline[nodesepA=2mm,offsetB=10mm]{start}{Aright}\naput{1 neighbor}
\end{document}
• I tried this and got two empty pages. But the problem was solved after I moved the last three "\ncline" statements above the last "\end{psmatrix}". Thanks! – Erel Segal-Halevi Sep 8 '15 at 11:38
• ah yes, I didn't ude auto-pst-pdf. The reason why it works in both ways. I always use xelatex – user2478 Sep 8 '15 at 13:00
I simplified your code. One psmatrix environment is enough. Is that like you want?
\documentclass[11pt,,section]{article}
\usepackage{pst-node,pst-tree}
\usepackage[crop=off]{auto-pst-pdf}
\usepackage{graphicx}
\newcommand{\agent}[1]{\circlenode[linecolor=white]{#1}{{#1}}}%
\newcommand{\like}[2]{\ncline[linecolor=green]{->}{#1}{#2}}
\begin{document}\color{blue}
\psset{arrows=->, arrowinset=0.25, colsep=0.75cm, nodesep=2pt}
\begin{psmatrix}
& \agent{Start}\\
\agent{A} & & \agent{E}\\
\agent{B} & [mnode=oval, linecolor=white]\agent{C}\hspace{1.5cm}\agent{D} & \agent{F}
\ncline{Start}{A}
\ncline{Start}{3,2}
\ncline{Start}{E}
\psset{linecolor=green}
\ncline{A}{B}
\ncline{F}{E}
\ncline{C}{D}
\end{psmatrix}
\end{document}
• Actually my use-case is more complicated (I tried to simplify it but probably over-simplified), because in every frame there are the same letters: A,B,C,D and more. So I cannot just send arrows from Start to different letters. – Erel Segal-Halevi Sep 7 '15 at 5:25
• I guessed so, but I'm sure it can be simplified. Could you at least post the scan of a drawing of what you try to achieve? – Bernard Sep 7 '15 at 8:08
• Perfect. I'll look at it later today. – Bernard Sep 7 '15 at 8:48
|
2019-12-11 22:25:34
|
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|
http://jnva.biemdas.com/archives/1473
|
## H.M. Srivastava, M.I. Qureshi, S.H. Malik, Some hypergeometric transformations and reduction formulas for the Gauss function and their applications involving the Clausen function
Full Text: PDF
DOI: 10.23952/jnva.5.2021.6.10
Volume 5, Issue 6, 1 December 2021, Pages 981-987
Abstract. The aim of this paper is to obtain some closed forms of hypergeometric reduction formulas for the following Gauss functions ${}_2F_1\left[\alpha,\alpha+\frac{1}{2};2\alpha-1;z\right]$ and ${_{2}F_{1}} \left[\alpha-1,\alpha-\frac{3}{2};2\alpha-1;z\right]$, and the Clausen function: ${_{3}F_{2}}\left[\gamma+1,\beta,\beta+\frac{1}{2}; \gamma,2\beta;z\right]$ by using the series rearrangement technique.
|
2021-12-04 20:13:58
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https://strawberryfields.readthedocs.io/en/latest/code/backend.html
|
# Backend API¶
Module name: strawberryfields.backends.base
This module implements the backend API. It contains the classes
as well as a few methods which apply only to the Gaussian backend.
Note
The backend API is $$\hbar$$ independent. Internally the Strawberry Fields backends use $$\hbar=2$$.
Note
Keyword arguments are denoted **kwargs, and allow additional options to be passed to the backends - these are documented where available. For more details on available keyword arguments, please consult the backends directly.
## BaseBackend¶
supports(name) Check whether the backend supports the given operating mode. begin_circuit(num_subsystems, **kwargs) Instantiate a quantum circuit. add_mode([n]) Add modes to the circuit. del_mode(modes) Delete modes from the circuit. get_modes() Return a list of the active modes for the circuit. reset([pure]) Reset the circuit so that all the modes are in the vacuum state. state([modes]) Returns the state of the quantum simulation. is_vacuum([tol]) Test whether the current circuit state is vacuum (up to given tolerance). prepare_vacuum_state(mode) Prepare the vacuum state in the specified mode. prepare_coherent_state(alpha, mode) Prepare a coherent state in the specified mode. prepare_squeezed_state(r, phi, mode) Prepare a squeezed vacuum state in the specified mode. prepare_displaced_squeezed_state(alpha, r, …) Prepare a displaced squeezed state in the specified mode. prepare_thermal_state(nbar, mode) Prepare a thermal state in the specified mode. rotation(phi, mode) Apply the phase-space rotation operation to the specified mode. displacement(alpha, mode) Apply the displacement operation to the specified mode. squeeze(z, mode) Apply the squeezing operation to the specified mode. beamsplitter(t, r, mode1, mode2) Apply the beamsplitter operation to the specified modes. loss(T, mode) Perform a loss channel operation on the specified mode. thermal_loss(T, nbar, mode) Perform a thermal loss channel operation on the specified mode. measure_homodyne(phi, mode[, shots, select]) Measure a phase space quadrature of the given mode. measure_fock(modes[, shots, select]) Measure the given modes in the Fock basis.
## Fock backends¶
Some methods are only implemented in the subclass BaseFock, which is the base class for simulators using a Fock-state representation for quantum optical circuits.
get_cutoff_dim() Returns the Hilbert space cutoff dimension used. prepare_fock_state(n, mode) Prepare a Fock state in the specified mode. prepare_ket_state(state, modes) Prepare the given ket state in the specified modes. prepare_dm_state(state, modes) Prepare the given mixed state in the specified modes. cubic_phase(gamma, mode) Apply the cubic phase operation to the specified mode. kerr_interaction(kappa, mode) Apply the Kerr interaction $$\exp{(i\kappa \hat{n}^2)}$$ to the specified mode. cross_kerr_interaction(kappa, mode1, mode2) Apply the two mode cross-Kerr interaction $$\exp{(i\kappa \hat{n}_1\hat{n}_2)}$$ to the specified modes.
## Gaussian backends¶
Likewise, some methods are only implemented in subclass BaseGaussian, which is the base class for simulators using a Gaussian symplectic representation for quantum optical circuits.
measure_heterodyne(mode[, shots, select]) Perform a heterodyne measurement on the given mode.
### Code details¶
exception strawberryfields.backends.base.NotApplicableError[source]
Exception raised by the backend when the user attempts an unsupported operation. E.g. measure_fock() on a Gaussian backend. Conceptually different from NotImplementedError (which means “not implemented, but at some point may be”).
class strawberryfields.backends.base.ModeMap(num_subsystems)[source]
Simple internal class for maintaining a map of existing modes.
reset()[source]
reset the modemap to the initial state
remap(modes)[source]
Remaps the mode list
valid(modes)[source]
checks if the mode list is valid
show()[source]
Returns the mapping
delete(modes)[source]
Deletes a mode
add(num_modes)[source]
class strawberryfields.backends.base.BaseBackend[source]
Abstract base class for backends.
short_name = 'base'
short name of the backend
Type: str
circuit_spec = None
Short name of the CircuitSpecs class used to validate Programs for this backend. None if no validation is required.
Type: str, None
supports(name)[source]
Check whether the backend supports the given operating mode.
Currently supported operating modes are:
• “gaussian”: for manipulations in the Gaussian representation using the displacements and covariance matrices
• “fock_basis”: for manipulations in the Fock representation
• “mixed_states”: for representations where the quantum state is mixed
• “batched”: allows for a multiple circuits to be simulated in parallel
Parameters: name (str) – name of the operating mode which we are checking support for True if this backend supports that operating mode. bool
begin_circuit(num_subsystems, **kwargs)[source]
Instantiate a quantum circuit.
Instantiates a representation of a quantum optical state with num_subsystems modes. The state is initialized to vacuum.
The modes in the circuit are indexed sequentially using integers, starting from zero. Once an index is assigned to a mode, it can never be re-assigned to another mode. If the mode is deleted its index becomes invalid. An operation acting on an invalid or unassigned mode index raises an IndexError exception.
Parameters: Keyword Arguments: num_subsystems (int) – number of modes in the circuit cutoff_dim (int) – Hilbert space truncation dimension (for Fock basis backends only) batch_size (int) – (optional) batch-axis dimension, enables batched operation if > 1 (for the TF backend only)
add_mode(n=1)[source]
The new modes are initialized to the vacuum state. They are assigned mode indices sequentially, starting from the first unassigned index.
Parameters: n (int) – number of modes to add indices of the newly added modes list[int]
del_mode(modes)[source]
Delete modes from the circuit.
The deleted modes are traced out. As a result the state may have to be described using a density matrix.
The indices of the deleted modes become invalid for the lifetime of the circuit object. They will never be reassigned to other modes. Deleting a mode that has already been deleted raises an IndexError exception.
Parameters: modes (Sequence[int]) – mode numbers to delete
get_modes()[source]
Return a list of the active modes for the circuit.
A mode is active if it has been created and has not been deleted.
Returns: sorted list of active (assigned, not invalid) mode indices list[int]
reset(pure=True, **kwargs)[source]
Reset the circuit so that all the modes are in the vacuum state.
After the reset the circuit is in the same state as it was after the last begin_circuit() call. It will have the original number of modes, all initialized in the vacuum state. Some circuit parameters may be changed during the reset, see the keyword args below.
Parameters: Keyword Arguments: pure (bool) – if True, initialize the circuit in a pure state representation (will use a mixed state representation if pure is False) cutoff_dim (int) – new Hilbert space truncation dimension (for Fock basis backends only)
prepare_vacuum_state(mode)[source]
Prepare the vacuum state in the specified mode.
The requested mode is traced out and replaced with the vacuum state. As a result the state may have to be described using a density matrix.
Parameters: mode (int) – which mode to prepare the vacuum state in
prepare_coherent_state(alpha, mode)[source]
Prepare a coherent state in the specified mode.
The requested mode is traced out and replaced with the coherent state $$\ket{\alpha}$$. As a result the state may have to be described using a density matrix.
Parameters: alpha (complex) – coherent state displacement parameter mode (int) – which mode to prepare the coherent state in
prepare_squeezed_state(r, phi, mode)[source]
Prepare a squeezed vacuum state in the specified mode.
The requested mode is traced out and replaced with the squeezed state $$\ket{z}$$, where $$z=re^{i\phi}$$. As a result the state may have to be described using a density matrix.
Parameters: r (float) – squeezing amplitude phi (float) – squeezing angle mode (int) – which mode to prepare the squeezed state in
prepare_displaced_squeezed_state(alpha, r, phi, mode)[source]
Prepare a displaced squeezed state in the specified mode.
The requested mode is traced out and replaced with the displaced squeezed state state $$\ket{\alpha, z}$$, where $$z=re^{i\phi}$$. As a result the state may have to be described using a density matrix.
Parameters: alpha (complex) – displacement parameter r (float) – squeezing amplitude phi (float) – squeezing angle mode (int) – which mode to prepare the squeezed state in
prepare_thermal_state(nbar, mode)[source]
Prepare a thermal state in the specified mode.
The requested mode is traced out and replaced with the thermal state $$\rho(nbar)$$. As a result the state may have to be described using a density matrix.
Parameters: nbar (float) – thermal population (mean photon number) of the mode mode (int) – which mode to prepare the thermal state in
rotation(phi, mode)[source]
Apply the phase-space rotation operation to the specified mode.
Parameters: phi (float) – rotation angle mode (int) – which mode to apply the rotation to
displacement(alpha, mode)[source]
Apply the displacement operation to the specified mode.
Parameters: alpha (complex) – displacement parameter mode (int) – which mode to apply the displacement to
squeeze(z, mode)[source]
Apply the squeezing operation to the specified mode.
Parameters: z (complex) – squeezing parameter mode (int) – which mode to apply the squeeze to
beamsplitter(t, r, mode1, mode2)[source]
Apply the beamsplitter operation to the specified modes.
It is assumed that $$|r|^2+|t|^2 = t^2+|r|^2=1$$, i.e that t is real.
Parameters: t (float) – transmitted amplitude r (complex) – reflected amplitude (with phase) mode1 (int) – first mode that beamsplitter acts on mode2 (int) – second mode that beamsplitter acts on
loss(T, mode)[source]
Perform a loss channel operation on the specified mode.
Parameters: T (float) – loss parameter, $$0\leq T\leq 1$$. mode (int) – index of mode where operation is carried out
thermal_loss(T, nbar, mode)[source]
Perform a thermal loss channel operation on the specified mode.
Parameters: T (float) – loss parameter, $$0\leq T\leq 1$$. nbar (float) – mean photon number of the environment thermal state mode (int) – index of mode where operation is carried out
measure_homodyne(phi, mode, shots=1, select=None, **kwargs)[source]
Measure a phase space quadrature of the given mode.
For the measured mode, samples the probability distribution $$f(q) = \bra{q_\phi} \rho \ket{q_\phi}$$ and returns the sampled value. Here $$\ket{q_\phi}$$ is the eigenstate of the operator
$\hat{q}_\phi = \sqrt{2/\hbar}(\cos(\phi)\hat{x} +\sin(\phi)\hat{p}) = e^{-i\phi} \hat{a} +e^{i\phi} \hat{a}^\dagger.$
Note
This method is $$\hbar$$ independent. The returned values can be converted to conventional position/momentum eigenvalues by multiplying them with $$\sqrt{\hbar/2}$$.
Updates the current state such that the measured mode is reset to the vacuum state. This is because we cannot represent exact position or momentum eigenstates in any of the backends, and experimentally the photons are destroyed in a homodyne measurement.
Parameters: phi (float) – phase angle of the quadrature to measure (x: $$\phi=0$$, p: $$\phi=\pi/2$$) mode (int) – which mode to measure shots (int) – number of measurement samples to obtain select (None or float) – If not None: desired value of the measurement result. Enables post-selection on specific measurement results instead of random sampling.
Keyword arguments can be used to pass additional parameters to the backend. Options for such arguments will be documented in the respective subclasses.
Returns: measured value float
measure_fock(modes, shots=1, select=None, **kwargs)[source]
Measure the given modes in the Fock basis.
..note::
When :code:shots == 1, updates the current system state to the conditional state of that measurement result. When :code:shots > 1, the system state is not updated.
Parameters: modes (Sequence[int]) – which modes to measure shots (int) – number of measurement samples to obtain select (None or Sequence[int]) – If not None: desired values of the measurement results. Enables post-selection on specific measurement results instead of random sampling. len(select) == len(modes) is required. measurement results tuple[int]
is_vacuum(tol=0.0, **kwargs)[source]
Test whether the current circuit state is vacuum (up to given tolerance).
Returns True iff $$|\bra{0} \rho \ket{0} -1| \le$$ tol, i.e., the fidelity of the current circuit state with the vacuum state is within the given tolerance from 1.
Parameters: tol (float) – numerical tolerance True iff current state is vacuum up to tolerance tol bool
state(modes=None, **kwargs)[source]
Returns the state of the quantum simulation.
Parameters: modes (int or Sequence[int] or None) – Specifies the modes to restrict the return state to. None returns the state containing all the modes. The returned state contains the requested modes in the given order, i.e., modes=[3,0] results in a two mode state being returned with the first mode being subsystem 3 and the second mode being subsystem 0. state description, specific child class depends on the backend BaseState
class strawberryfields.backends.base.BaseFock[source]
Abstract base class for backends capable of Fock state manipulation.
get_cutoff_dim()[source]
Returns the Hilbert space cutoff dimension used.
Returns: cutoff dimension int
prepare_fock_state(n, mode)[source]
Prepare a Fock state in the specified mode.
The requested mode is traced out and replaced with the Fock state $$\ket{n}$$. As a result the state may have to be described using a density matrix.
Parameters: n (int) – Fock state to prepare mode (int) – which mode to prepare the Fock state in
prepare_ket_state(state, modes)[source]
Prepare the given ket state in the specified modes.
The requested modes are traced out and replaced with the given ket state (in the Fock basis). As a result the state may have to be described using a density matrix.
Parameters: state (array) – Ket state in the Fock basis. The state can be given in either vector form, with one index, or tensor form, with one index per mode. For backends supporting batched mode, state can be a batch of such vectors or tensors. modes (int or Sequence[int]) – Modes to prepare the state in. If modes is not ordered this is taken into account when preparing the state, i.e., when a two mode state is prepared in modes=[3,1], then the first mode of state goes into mode 3 and the second mode goes into mode 1 of the simulator.
prepare_dm_state(state, modes)[source]
Prepare the given mixed state in the specified modes.
The requested modes are traced out and replaced with the given density matrix state (in the Fock basis). As a result the state will be described using a density matrix.
Parameters: state (array) – Density matrix in the Fock basis. The state can be given in either matrix form, with two indices, or tensor form, with two indices per mode. For backends supporting batched mode, state can be a batch of such matrices or tensors. modes (int or Sequence[int]) – which mode to prepare the state in If modes is not ordered this is take into account when preparing the state, i.e., when a two mode state is prepared in modes=[3,1], then the first mode of state goes into mode 3 and the second mode goes into mode 1 of the simulator.
cubic_phase(gamma, mode)[source]
Apply the cubic phase operation to the specified mode.
Applies the operation
$\exp\left(i \frac{\gamma}{6} (\hat{a} +\hat{a}^\dagger)^3\right)$
to the specified mode.
Note
This method is $$\hbar$$ independent. The usual definition of the cubic phase gate is $$\hbar$$ dependent:
$V(\gamma') = \exp\left(i \frac{\gamma'}{3\hbar} \hat{x}^3\right) = \exp\left(i \frac{\gamma' \sqrt{\hbar/2}}{6} (\hat{a} +\hat{a}^\dagger)^3\right).$
Hence the cubic phase gate $$V(\gamma')$$ is executed on a backend by scaling the $$\gamma'$$ parameter by $$\sqrt{\hbar/2}$$ and then passing it to this method, much in the way the $$\hbar$$ dependent X and Z gates are implemented through the $$\hbar$$ independent displacement() method.
Warning
The cubic phase gate can suffer heavily from numerical inaccuracies due to finite-dimensional cutoffs in the Fock basis. The gate implementation in Strawberry Fields is unitary, but it does not implement an exact cubic phase gate. The Kerr gate provides an alternative non-Gaussian gate.
Parameters: gamma (float) – scaled cubic phase shift, $$\gamma = \gamma' \sqrt{\hbar/2}$$ mode (int) – which mode to apply it to
kerr_interaction(kappa, mode)[source]
Apply the Kerr interaction $$\exp{(i\kappa \hat{n}^2)}$$ to the specified mode.
Parameters: kappa (float) – strength of the interaction mode (int) – which mode to apply it to
cross_kerr_interaction(kappa, mode1, mode2)[source]
Apply the two mode cross-Kerr interaction $$\exp{(i\kappa \hat{n}_1\hat{n}_2)}$$ to the specified modes.
Parameters: kappa (float) – strength of the interaction mode1 (int) – first mode that cross-Kerr interaction acts on mode2 (int) – second mode that cross-Kerr interaction acts on
state(modes=None, **kwargs)[source]
Returns the state of the quantum simulation.
Returns: state description BaseFockState
class strawberryfields.backends.base.BaseGaussian[source]
Abstract base class for backends that are only capable of Gaussian state manipulation.
measure_heterodyne(mode, shots=1, select=None)[source]
Perform a heterodyne measurement on the given mode.
Updates the current state of the circuit such that the measured mode is reset to the vacuum state.
Parameters: mode (int) – which mode to measure shots (int) – number of measurement samples to obtain select (None or complex) – If not None: desired value of the measurement result. Enables post-selection on specific measurement results instead of random sampling. measured value complex
prepare_gaussian_state(r, V, modes)[source]
Prepare a Gaussian state.
The specified modes are traced out and replaced with a Gaussian state provided via a vector of means and a covariance matrix.
Note
This method is $$\hbar$$ independent. The input arrays are the means and covariance of the $$a+a^\dagger$$ and $$-i(a-a^\dagger)$$ operators. They are obtained by dividing the xp means by $$\sqrt{\hbar/2}$$ and the xp covariance by $$\hbar/2$$.
Parameters: r (array) – vector of means in xp ordering V (array) – covariance matrix in xp ordering modes (int or Sequence[int]) – Which modes to prepare the state in. If the modes are not sorted, this is taken into account when preparing the state. I.e., when a two mode state is prepared with modes=[3,1], the first mode of the given state goes into mode 3 and the second mode goes into mode 1.
state(modes=None, **kwargs)[source]
Returns the state of the quantum simulation.
Returns: state description BaseGaussianState
|
2019-07-23 19:40:20
|
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|
https://www.mathdoubts.com/basic/number/type/real/natural/
|
# Natural numbers
A number, whose value real and represents whole is defined as a natural number.
Natural numbers is a collection of numbers which represent real and complete quantities. It is originally used in counting objects, measuring whole quantities (not fractions) and labeling anything. Due to the use of these numbers in counting, the natural numbers are also called as counting numbers. It is a primary numeric system in real numbers.
The numbers (except zero) in Hindu-Arabic Numeral System are used as numbers of Natural numbers. The first number in natural number group is
$1$
$1$
to first natural number gives second number. Add
$1$
to second number and it returns third number. Thus, the group of natural numbers is formed in real number system.
## How to use
Natural numbers are actually used for representing, the real things whose quantities are whole.
Let us understand it from two examples. Consider an apple, it is a whole apple (not a fraction). Therefore, it can be called one apple and it is written as
$1$
apple in number system. Thus, natural number
$1$
is used to represent it in numeric system.
Consider another example. Each thing is a star and it is whole. Therefore, natural numbers are used to count them. Hence, the number of stars is
$9$
. In number system, it is written as
$9$
stars.
### Representation
The set of natural numbers can be expressed in mathematics by the concept of set theory. The set of natural numbers is denoted by the letter
$N$
. The set of natural numbers of this group is written as follows.
$N=\left\{1,2,3,4,\dots \right\}$
Natural numbers is a collection of infinite numbers. Therefore, the set of natural numbers is known as infinite set.
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2018-10-17 09:15:51
|
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|
https://fiberguide.net/tech-guides/synopsis-lightscape-traps-rydberg-atoms-in-the-dark/
|
Synopsis: Lightscape Traps Rydberg Atoms in the Dark
16 Jan Synopsis: Lightscape Traps Rydberg Atoms in the Dark
A holographic technique confines excited Rydberg atoms in the central dark region of a 3D light-intensity pattern.
Neutral atoms in highly excited Rydberg states could be the next big thing in quantum computing, but only if the atoms can be held in place. Optical tweezers (laser traps) can hold ground-state atoms, but they repel Rydberg atoms, pushing them out from the bright focal point of the laser beams. Now, Daniel Barredo and colleagues at the Institute of Optics in Palaiseau, France, demonstrate a holographic method that can trap individual Rydberg atoms in 3D “lightscapes.” The team held the atoms in place with micrometer-scale precision, a requirement for quantum-information applications. Previously, 3D confinement was only achievable with millimeter precision using magnetic or electric fields.
The researchers started with a single neutral rubidium atom, which they trapped using standard optical tweezers. Deactivating the tweezers, they excited the atom to the Rydberg state. The team then immediately recaptured the atom at the center of a 3D light-intensity pattern, created by diffracting a laser beam from a spatial light modulator, where the waves interfered to form a dark spot.
Barredo and colleagues found that they could hold an excited atom for as long as the Rydberg state was maintained—about 228
$𝜇$
s at room temperature. During this time, they used microwaves to shift the atom between two Rydberg levels, a transition that the researchers say could one day be used to represent a qubit in a quantum computer. The team also demonstrated interactions between Rydberg atoms by making two atoms in adjacent traps exchange states. Such interactions are necessary to create quantum logic gates.
This research is published in Physical Review Letters.
–Marric Stephens
Marric Stephens is a freelance science writer based in Bristol, UK.
Related Articles
Optics
Synopsis: Two Lasers in One
A single semiconductor laser can produce both an amplitude-modulated and a frequency-modulated frequency comb, demonstrating a physical relationship between two types of output previously thought to be independent. Read More »
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2020-12-05 02:15:12
|
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|
https://hero.handmade.network/forums/code-discussion/t/3187-having_an_issue_with_day_5
|
6 posts
Having an issue with Day 5
Edited by skoepa on Reason: Initial post
Hello I have the exact same code as Casey does on Day 5, but it will not work for me as it gives me an unhandled exception error on line 62. I am at a complete loss as to what to do.
This is my callstack
1 2 3 4 5 > win32_handmade.exe!Win32ResizeDIBSection(int Width, int Height) Line 62 C++ win32_handmade.exe!WIN32MainWindowCallback(HWND__ * Window, unsigned int Message, unsigned int WParam, long LParam) Line 118 C++ [External Code] win32_handmade.exe!WinMain(HINSTANCE__ * hInstance, HINSTANCE__ * PrevInstance, char * CommandLine, int ShowCode) Line 184 C++ [External Code]
Mārtiņš Možeiko
2198 posts / 1 project
Having an issue with Day 5
Edited by Mārtiņš Možeiko on
In Day 5 code line 62 is a comment line in RenderWeirdGradient function. There is no way it can crash.
Please show exact code you are using, I'm guessing you either changed something too much or forgot to change something for it to work.
6 posts
Having an issue with Day 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 /* ======================================================================== $File:$ $Date:$ $Revision:$ $Creator: Casey Muratori$ $Notice: (C) Copyright 2014 by Molly Rocket, Inc. All Rights Reserved.$ ======================================================================== */ #include #include #define internal static #define local_persist static #define global_variable static typedef int8_t int8; typedef int16_t int16; typedef int32_t int32; typedef int64_t int64; typedef uint8_t uint8; typedef uint16_t uint16; typedef uint32_t uint32; typedef uint64_t uint64; struct win32_offscreen_buffer { // NOTE(casey): Pixels are alwasy 32-bits wide, Memory Order BB GG RR XX BITMAPINFO Info; void *Memory; int Width; int Height; int Pitch; }; struct win32_window_dimension { int Width; int Height; }; // TODO(casey): This is a global for now. global_variable bool GlobalRunning; global_variable win32_offscreen_buffer GlobalBackbuffer; win32_window_dimension Win32GetWindowDimension(HWND Window) { win32_window_dimension Result; RECT ClientRect; GetClientRect(Window, &ClientRect); Result.Width = ClientRect.right - ClientRect.left; Result.Height = ClientRect.bottom - ClientRect.top; return(Result); } internal void RenderWeirdGradient(win32_offscreen_buffer Buffer, int BlueOffset, int GreenOffset) { // TODO(casey): Let's see what the optimizer does uint8 *Row = (uint8 *)Buffer.Memory; for(int Y = 0; Y < Buffer.Height; ++Y) { uint32 *Pixel = (uint32 *)Row; for(int X = 0; X < Buffer.Width; ++X) { uint8 Blue = (X + BlueOffset); uint8 Green = (Y + GreenOffset); *Pixel++ = ((Green << 8) | Blue); } Row += Buffer.Pitch; } } internal void Win32ResizeDIBSection(win32_offscreen_buffer *Buffer, int Width, int Height) { // TODO(casey): Bulletproof this. // Maybe don't free first, free after, then free first if that fails. if(Buffer->Memory) { VirtualFree(Buffer->Memory, 0, MEM_RELEASE); } Buffer->Width = Width; Buffer->Height = Height; int BytesPerPixel = 4; // NOTE(casey): When the biHeight field is negative, this is the clue to // Windows to treat this bitmap as top-down, not bottom-up, meaning that // the first three bytes of the image are the color for the top left pixel // in the bitmap, not the bottom left! Buffer->Info.bmiHeader.biSize = sizeof(Buffer->Info.bmiHeader); Buffer->Info.bmiHeader.biWidth = Buffer->Width; Buffer->Info.bmiHeader.biHeight = -Buffer->Height; Buffer->Info.bmiHeader.biPlanes = 1; Buffer->Info.bmiHeader.biBitCount = 32; Buffer->Info.bmiHeader.biCompression = BI_RGB; // NOTE(casey): Thank you to Chris Hecker of Spy Party fame // for clarifying the deal with StretchDIBits and BitBlt! // No more DC for us. int BitmapMemorySize = (Buffer->Width*Buffer->Height)*BytesPerPixel; Buffer->Memory = VirtualAlloc(0, BitmapMemorySize, MEM_COMMIT, PAGE_READWRITE); Buffer->Pitch = Width*BytesPerPixel; // TODO(casey): Probably clear this to black } internal void Win32DisplayBufferInWindow(HDC DeviceContext, int WindowWidth, int WindowHeight, win32_offscreen_buffer Buffer) { // TODO(casey): Aspect ratio correction // TODO(casey): Play with stretch modes StretchDIBits(DeviceContext, /* X, Y, Width, Height, X, Y, Width, Height, */ 0, 0, WindowWidth, WindowHeight, 0, 0, Buffer.Width, Buffer.Height, Buffer.Memory, &Buffer.Info, DIB_RGB_COLORS, SRCCOPY); } LRESULT CALLBACK Win32MainWindowCallback(HWND Window, UINT Message, WPARAM WParam, LPARAM LParam) { LRESULT Result = 0; switch(Message) { case WM_CLOSE: { // TODO(casey): Handle this with a message to the user? GlobalRunning = false; } break; case WM_ACTIVATEAPP: { OutputDebugStringA("WM_ACTIVATEAPP\n"); } break; case WM_DESTROY: { // TODO(casey): Handle this as an error - recreate window? GlobalRunning = false; } break; case WM_PAINT: { PAINTSTRUCT Paint; HDC DeviceContext = BeginPaint(Window, &Paint); win32_window_dimension Dimension = Win32GetWindowDimension(Window); Win32DisplayBufferInWindow(DeviceContext, Dimension.Width, Dimension.Height, GlobalBackbuffer); EndPaint(Window, &Paint); } break; default: { // OutputDebugStringA("default\n"); Result = DefWindowProc(Window, Message, WParam, LParam); } break; } return(Result); } int CALLBACK WinMain(HINSTANCE Instance, HINSTANCE PrevInstance, LPSTR CommandLine, int ShowCode) { WNDCLASS WindowClass = {}; Win32ResizeDIBSection(&GlobalBackbuffer, 1280, 720); WindowClass.style = CS_HREDRAW|CS_VREDRAW|CS_OWNDC; WindowClass.lpfnWndProc = Win32MainWindowCallback; WindowClass.hInstance = Instance; // WindowClass.hIcon; WindowClass.lpszClassName = "HandmadeHeroWindowClass"; if(RegisterClassA(&WindowClass)) { HWND Window = CreateWindowExA( 0, WindowClass.lpszClassName, "Handmade Hero", WS_OVERLAPPEDWINDOW|WS_VISIBLE, CW_USEDEFAULT, CW_USEDEFAULT, CW_USEDEFAULT, CW_USEDEFAULT, 0, 0, Instance, 0); if(Window) { // NOTE(casey): Since we specified CS_OWNDC, we can just // get one device context and use it forever because we // are not sharing it with anyone. HDC DeviceContext = GetDC(Window); int XOffset = 0; int YOffset = 0; GlobalRunning = true; while(GlobalRunning) { MSG Message; while(PeekMessage(&Message, 0, 0, 0, PM_REMOVE)) { if(Message.message == WM_QUIT) { GlobalRunning = false; } TranslateMessage(&Message); DispatchMessageA(&Message); } RenderWeirdGradient(GlobalBackbuffer, XOffset, YOffset); win32_window_dimension Dimension = Win32GetWindowDimension(Window); Win32DisplayBufferInWindow(DeviceContext, Dimension.Width, Dimension.Height, GlobalBackbuffer); ++XOffset; YOffset += 2; } } else { // TODO(casey): Logging } } else { // TODO(casey): Logging } return(0); }
This is the full code I have.
17 posts
Hi, I'm bewwys I'm a student at the 42 codingschool . I'm glad to be here. I'm a noob so please take care of me :)
Having an issue with Day 5
Hi,
I'm wondering why in your callstack the function is define like this
Win32ResizeDIBSection(int Width, int Height)
but in your source code the function is like that
Win32ResizeDIBSection(win32_offscreen_buffer *Buffer, int Width, int Height) ?
6 posts
Having an issue with Day 5
bewwys
Hi,
I'm wondering why in your callstack the function is define like this
Win32ResizeDIBSection(int Width, int Height)
but in your source code the function is like that
Win32ResizeDIBSection(win32_offscreen_buffer *Buffer, int Width, int Height) ?
That's weird, it's like it's ignoring it or something.
Mārtiņš Možeiko
2198 posts / 1 project
Having an issue with Day 5
My suspicion is that you are running exe that is compiled from very different source code than you posted. Because line 62 is not inside Win32ResizeDIBSection function as call stack indicates. And also bewwys already mentioned - arguments to Win32ResizeDIBSection function are different from source code.
Verify where is your exe file and is it really the one you are compiling.
6 posts
Having an issue with Day 5
mmozeiko
My suspicion is that you are running exe that is compiled from very different source code than you posted. Because line 62 is not inside Win32ResizeDIBSection function as call stack indicates. And also bewwys already mentioned - arguments to Win32ResizeDIBSection function are different from source code.
Verify where is your exe file and is it really the one you are compiling.
I'm pretty sure that's the reason actually, do you know how I can fix that?
Mārtiņš Možeiko
2198 posts / 1 project
Having an issue with Day 5
Edited by Mārtiņš Možeiko on
Umm.. by debugging executable that is produced in correct place?
Basically:
1) compile to c:\whatever\file.exe
2) debug c:\whatever\file.exe (and not c:\other\location\file.exe)
Its impossible to answer without knowing what are you doing. You are asking "I want to eat oranges, but I'm eating apples. How to eat oranges?" Answer is "eat oranges, don't eat apples" :D
6 posts
Having an issue with Day 5
Edited by skoepa on
Something is just super wonky about my file, and I don't know enough to know how to fix it.
I think I messed something up when I saved it in emacs because now when I click in it it says the app cannot run on my PC.
Whenever I tried to open it in visual studio it would say attach and I don't know what to attach it to.
Edit: I'm really sorry for sounding dumb I am just so incredibly lost because I clearly screwed something up badly.
Mārtiņš Možeiko
2198 posts / 1 project
Having an issue with Day 5
Edited by Mārtiņš Možeiko on
Try without emacs just to get basics done. Open new cmd.exe, run vcvarsall.bat. Then run "cl.exe ..other..argumens.. file.cpp /Feoutput.exe" to get output.exe. And then run "devenv output.exe" and check if you can debug correctly. If you can then do small modification and try again. Repeat until you set up configuration/environment you like to use.
Whenever I tried to open it in visual studio it would say attach and I don't know what to attach it to.
What is "it"? How do you open "it"?
6 posts
Having an issue with Day 5
Edited by skoepa on
yooo that fixed it. Thank you so much!
Edit: I'm still having the attach issue, when I open the .cpp instead of start it just says to attach it to a process
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2021-09-17 23:13:39
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https://www.physicsforums.com/threads/i-just-found-a-new-way-to-calculate-force.914261/
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# B I just found a new way to calculate Force
1. May 10, 2017
### John Clement Husain
I found a new way to calculate force without getting the acceleration when you have velocity, mass, & time.
F = (m/t)v
I just find this more efficient than getting the acceleration, why not use one formula.
2. May 10, 2017
### Staff: Mentor
But as acceleration is change of velocity per time, you get $F=(m/t)v=(mv/t)=m(v/t)=ma$, which is acceleration again.
3. May 10, 2017
### Staff: Mentor
So according to your formula, force always decreases as time increases. Is this what you observe?
4. May 10, 2017
### John Clement Husain
It seems so, yes
5. May 10, 2017
### John Clement Husain
yeah, I just stated that I don't have to go to a=v/t anymore
6. May 10, 2017
### Staff: Mentor
No, it doesn't. In fact, as an object falls from a great distance the force increases as time increases, in direct opposition to your formula.
When you write a formula in physics, it is important to check if it is consistent with experiment. Yours is not.
7. May 10, 2017
### John Clement Husain
oh, I see....
also in my formula, force increases due to it's velocity. The faster the object, the greater the force?
but the formula does not contend with reality though, I shall take a note on that, thanks
8. May 10, 2017
### Ben Wilson
Also your understanding of force only works for constant accelerations.
constant a and constant m:
The force equation is F=ma, we could know the change in velocity Δv and the amount of time it took for that change to occur Δt. Then acceleration is a constant a=Δv/Δt. You could them formulate an equation F=(m/Δt)*Δv but this is trivial.
variable a and m:
Force eq. is F=dp/dt. this is the rate of change of momentum using differential calculus. You could have a changing m in a rocket, where the acceleration and velocity change as the rocket flies up, but the mass decreases as huge containers of fuel are emptied. Since p=mv, the force equation could read F=d(mv)/dt and your discovery is actually quite meaningful as you have noticed that Forces aren't limited to changes in velocity over time only.
9. May 10, 2017
### Staff: Mentor
Yes, good observation. It would mean that objects at rest would get stuck because v=0 so F=0. You could throw a ball up and have it never come down but just rest at the apex forever, and you would never be able to get out of bed in the morning.
10. May 10, 2017
### PeroK
You could have written:
$F = m\frac{dv}{dt}$
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2017-08-23 18:56:58
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https://neurips.cc/virtual/2022/poster/52908
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## Toward Efficient Robust Training against Union of $\ell_p$ Threat Models
### Gaurang Sriramanan · Maharshi Gor · Soheil Feizi
##### Hall J #803
Abstract: The overwhelming vulnerability of deep neural networks to carefully crafted perturbations known as adversarial attacks has led to the development of various training techniques to produce robust models. While the primary focus of existing approaches has been directed toward addressing the worst-case performance achieved under a single-threat model, it is imperative that safety-critical systems are robust with respect to multiple threat models simultaneously. Existing approaches that address worst-case performance under the union of such threat models ($\ell_{\infty}, \ell_2, \ell_1$) either utilize adversarial training methods that require multi-step attacks which are computationally expensive in practice, or rely upon fine-tuning of pre-trained models that are robust with respect to a single-threat model. In this work, we show that by carefully choosing the objective function used for robust training, it is possible to achieve similar, or improved worst-case performance over a union of threat models while utilizing only single-step attacks, thereby achieving a significant reduction in computational resources necessary for training. Furthermore, prior work showed that adversarial training specific to the $\ell_1$ threat model is relatively difficult, to the extent that even multi-step adversarially trained models were shown to be prone to gradient-masking. However, the proposed method—when applied on the $\ell_1$ threat model specifically—enables us to obtain the first $\ell_1$ robust model trained solely with single-step adversaries. Finally, to demonstrate the merits of our approach, we utilize a modern set of attack evaluations to better estimate the worst-case performance under the considered union of threat models.
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2023-03-31 16:11:35
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https://www.jiskha.com/questions/554597/if-80mg-of-a-radioactive-element-decays-to-10mg-in-30-minutes-the-half-life-of-this
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# Chemistry
IF 80mg of a radioactive element decays to 10mg in 30 minutes, the half-life of this element is
a.10 min
b.20 min
c.30 min
d.40 min
1. 👍
2. 👎
3. 👁
1. ln(No/N) = kt
No = 80 mg
N = 10 mg
t = 30 min
solve for k, then
k = 0.693/t1/2
Substitute k from above and solve for t1/2
10 min is correct.
1. 👍
2. 👎
2. 10 minute is correct
1. 👍
2. 👎
3. The person asking the question is asking why ? s/he did not challenge the answer 10min. But most respond just show how to get the answer and did not answer the why .. So here is my attempt.
I think first off, you need to re-affirm your understanding of the definition of half-life .
Half-life is the time taken for the element to decay to half of its original mass. so if it started off as 80gm at the start ,it reaches it first half-life when the mass is 40gm.
Since it is a MCQ multiple choice question . Let just say we find the answer by elimination . We can start from an answer we might think is most likely wrong .. For me I will try (c).
Consider (c. 30min). If this answer is true, the sample ought to reduce from 80gm to 40gm after 30min. but the question says that it is already left with 10gm after only 30 min. So (c) 30min, is quickly recognizable as a wrong answer.
if you tried to do (d) and (b) you should get similar conclusion .
Now I skipped to (a) to try.
Consider (a. 10min). if it this half-life. The sample, ought to reduce from 80gm to 40gm after 10min. And then from 40gm to 20gm in the next 10min. And then from 20gm to 10gm after another 10min. So all in all, from 80gm to 10gm in 30min. This seems consistent the original description of how it was observed to have behaved.
Note that if you have got the definition of half-life wrong, you would not recognize (a) as the correct answer.
This question is not so much a test of mathematics skill. More a test of the concept and definition.
If they really want to be confusing they can set the questions as...
"IF 80.8991234gm of a radioactive element decays to 10.12389gm in 30.1284938 minutes, the half-life of this element is ".... . But they didn't. They made the mathematics very very very simple.
1. 👍
2. 👎
4. My son with the litty explanations
1. 👍
2. 👎
## Similar Questions
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2. ### math
An element has a half-life of 30 years. If 1.0 mg of this element decays over a period of 90 years, how many mg of this element would remain? Begin amount is 1.0 elapsed time is 90y half life 30 years n=9/30 n=3 90/2^2 90/8 =
3. ### Chemistry
Gallium65, a radioactive isotope of gallium, decays by first order-kinetics. The half-life of this isotope is 15.2 min. How long would it take for 7/8 of a sample of this isotope to decay? 15.2 min 30.4 min 45.6 min 48.0 min I'm
4. ### Calculus
A certain radioactive material decays exponetially. The percent, P, of the material left after t years is given by P(t)= 100(1.2)^-t a)Determine the half life of the substance
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2. ### Precalc
Element X is a radioactive isotope such that every 86 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 730 grams, how much of the element would remain after 9 years, to the nearest whole
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1. ### Math
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2. ### Calculus-Modeling Equations
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3. ### Science
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4. ### Science
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2021-06-16 20:33:34
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http://ludovicarnold.altervista.org/teaching/optimization/gradient-descent/
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Course completion
55%
$\DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\dom}{dom} \DeclareMathOperator{\sigm}{sigm} \DeclareMathOperator{\softmax}{softmax} \DeclareMathOperator{\sign}{sign}$
A common local optimization method is the gradient descent algorithm. The gradient $\nabla f(\mathbf{x})$ has the direction of greatest increase of the function $f$ at $\mathbf{x}$.
$$\frac{\nabla f(\mathbf{x})}{\left\Vert \nabla f(\mathbf{x})\right\Vert }=\lim_{\epsilon\rightarrow0}\argmax_{\mathbf{z},\left\Vert \mathbf{z}\right\Vert \leq1}f(\mathbf{x}+\epsilon\mathbf{z})$$
The gradient can be computed using the partial derivatives w.r.t. each component of the input vector $\mathbf{x}$ :
$$\nabla f(\mathbf{x})=\left(\frac{\partial f(\mathbf{x})}{\partial x_{1}},\frac{\partial f(\mathbf{x})}{\partial x_{2}},\dots,\frac{\partial f(\mathbf{x})}{\partial x_{D}}\right)$$
In gradient descent (see the definition of the algorithm below – Gradient ascent is defined identically except for a change of sign in the update), we start at some initial guess $\mathbf{x}_{0}$ and iteratively take small steps of size $\delta\hspace{-1pt}t$ in the direction of $-\nabla f(\mathbf{x}_{k})$. In practice it is common to stop the algorithm after a predefined number of steps or when a the objective function has not decreased for some time. In the limit of infinitesimal step size, there is a guarantee that the algorithm decreases the value of $f$ at each step, and a guarantee that the algorithm converges to a local minimum if it doesn’t encounter a saddle point at which $\nabla f(\mathbf{x}_{k})=0$. However, a bigger step size allows the algorithm to move faster in the domain of $f$, possibly leading to faster convergence when it does not lead to oscillations. The figure below gives an example of gradient descent trajectory towards a local minimum.
Inputs: $f$, a function. $\delta\hspace{-1pt}t$, the step size. Outputs: $\hat{\mathbf{x}}$, approximation of a global minimum. Variables: $\mathbf{x}_{t}$, candidate solution of the algorithm at time t.
begin
repeat $K$ times:
$\mathbf{x}_{t+1}:=\mathbf{x}_{t}-\delta\hspace{-1pt}t\nabla f(\mathbf{x}_{t})$
return last position $\hat{\mathbf{x}}:=\mathbf{x}_{tmax}$.
end
Figure 3: Two possible gradient descent trajectories. In (a) the objective function is well behaved which allows the gradient to move smoothly towards the optimum. In (b) the gradient starts to oscillate as it falls into a narrow valley, thus converging more slowly.
Gradient descent often behaves poorly when the objective function has narrow valleys which cause oscillations. When confronted with such functions, a possible approach is to use $2^{\text{nd}}$ order information from the Hessian, e.g. using Newton’s method [Nocedal2006] or Hessian-Free optimization [Martens2010,Martens2011,Sutskever2011].
Surprisingly, gradient descent does not suffer from the curse of dimensionality and could in fact be considered to benefit from many dimensions. Common issues with gradient descent have to do with the gradient getting stuck in local minima and on plateaus where the derivative is zero. However, in spaces of high dimension, these issues are much less common because every dimension increases the possibility of finding a way out.
Nonetheless, the gradient descent algorithm depends on the possibility to compute the partial derivatives at each step. This is only possible when an explicit formula is available for the objective function, which is not always the case.
Next: Black-box optimization and Stochastic optimization
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2019-02-18 09:20:50
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https://linearalgebras.com/baby-rudin-chapter-1a.html
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If you find any mistakes, please make a comment! Thank you.
Solution to Principles of Mathematical Analysis Chapter 1 Part A
Chapter 1 The Real and Complex Number Systems
Exercise 1
(By ghostofgarborg) Note that $\mathbb{Q}$ is closed under the arithmetic operations of addition, subtraction, multiplication and taking multiplicative inverses.
If $r+x$ were rational, so would $(r+x) – r = x$ be, a contradiction.
If $rx$ were rational, so would $\frac 1 r rx= x$ be, a contradiction.
Exercise 2
(By ghostofgarborg) We will assume without proof that $\mathbb{Z}$ has unique prime factorizations. (This follows from $\mathbb Z$ being a primary ideal domain, and is covered in an abstract algebra course.) This implies that $p$ is a prime iff $p \mid ab \Rightarrow p \mid a$ or $p \mid b$.
Assume there are $m,n$ coprime such that $(\frac m n)^2 = 12$. This implies that $12n^2 = m^2$. Consequently, $3 \mid m^2$, and since $3$ is prime, $3 \mid m$. This means that there is a $p$ such that $m = 3p$, so that $12n^2 = 9p^2$, and consequently $4n^2 = 3p^2$. Using the primality of $3$, we can conclude that since $3 \nmid 4$, $3 \mid n^2$, and therefore $3 \mid n$. This contradicts the coprimality of $m$ and $n$. Therefore there cannot be any such $m,n$.
(By analambanomenos) If $r$ is a rational number whose square is 12, then $(r/2)^2=3$, so this is equivalent to showing there is no rational number whose square is 3. So suppose $(m/n)^2=3$ where $m,n$ are integers with no common factors. Then $m^2=3n^2$. Either $m=3p$, or $m=3p\pm 1$, and since $(3p\pm 1)^2=3(3p^2\pm 2p)+1$, $m$ must be a multiple of 3. But then we have $(3p)^2=3n^2$, or $3p^2=n^2$. Hence $n$ is also a multiple of 3, contradicting the assumption that $m$ and $n$ have no common factors.
Exercise 3
(By ghostofgarborg)
(a): Solution 1: $x \neq 0$ implies that $x$ has a multiplicative inverse. Multiply by $1/x$.
However, we can prove this without resorting to the existence of inverses, and get a solution that carries over to a larger class of rings, i.e. integral domains like $\mathbb Z$. Solution 2: Note that proposition 1.16b implies that $xy=0$ implies that $x=0$ or $y=0$. If $xy = xz$, then $x(y-z)=0$. If $x \neq 0$, then by prop. 1.16b, $(y-z) = 0$. The claim follows.
(b): Follows from (a) by letting $z = 1$.
(c): Follows from (a) by letting $z = x^{-1}$.
(d): Follows by applying (a) to
$\left( \frac 1 x \right) \frac 1 {\left( \frac 1 x \right)} = \left( \frac 1 x \right) x.$
Exercise 4
(By ghostofgarborg) Let $x \in E$. By definition of lower and upper bounds, $\alpha \leq x \leq \beta$.
Exercise 5
(By ghostofgarborg) Assume $\alpha$ is the greatest lower bound of $A$.
If $x \in (-A)$ then $-x \in A$, so $\alpha \leq -x$, and therefore $– \alpha \geq x$. This implies that $-\alpha$ is an upper bound for $(-A)$.
If $\beta < -\alpha$ then $-\beta > \alpha$, and there is an $x \in A$ such that $x < -\beta$. Then $-x \in -A$, and $-x > \beta$. This shows that $-\alpha$ is the least upper bound of $(-A)$, and we are done.
Exercise 6
(By analambanomenos)
(a): Since $np=qm$, $$((b^p)^{1/q})^{np} = ((b^p)^{1/q})^{qm}=b^{mp}.$$ Hence by Theorem 1.21, $$((b^p)^{1/q})^n=b^m.$$ A second application of Theorem 1.21 gives us $$(b^p)^{1/q}=(b^m)^{1/n}.$$ (b): Let $r=m/n$, $s=p/q$.
\begin{align*}
(b^rb^s)^{nq}=((b^m)^{1/n}(b^p)^{1/q})^{nq} &= b^{mq}b^{np} \\
(b^{r+s})^{nq}=((b^{mq+np})^{1/nq})^{nq} &= b^{mq}b^{np}
\end{align*}Hence by Theorem 1.21, $b^rb^s=b^{r+s}$.
(c): Note that the positive rational powers of $b$ are greater than 1 since the positive integral powers and roots of real numbers greater than 1 are also greater than 1. Hence if $t<r$,
$$b^t<b^tb^{r-t}=b^{t+r-t}=b^r.$$ Hence, since $b^r\in B(r)$, we have $b^r=\sup B(r)$.
(d): We need to show that $\sup B(x+y)=\sup B(x)\sup B(y)$. If $S$ and $T$ are sets of positive real numbers, it isn’t hard to show that $\sup ST = \sup S\sup T$, where $ST$ is the set of products of elements of $S$ with elements of $T$. Hence we want to show that $\sup B(x+y)=\sup(B(x)B(y))$.
Let $p,q$ be rational numbers such that $p\le x$ and $q\le y$. Then $b^pb^q=b^{p+q}\le\sup B(x+y)$, so $\sup(B(x)B(y))\le\sup B(x+y)$.
To get the reverse inequality, let $t$ be any rational number such that $t<x+y$, and let $\epsilon >0$ such that $t<x+y-\epsilon$. By Theorem 1.20(b), there are rational numbers $p\le x$, and $q\le y$ such that $x-\epsilon/2<p$ and $y-\epsilon/2<q$. Then $t<x+y-\epsilon<p+q$ so that $$b^t<b^{p+q}=b^pb^q\le\sup B(x)B(y).$$ Hence $\sup B(x+y)\le\sup(B(x)B(y))$.
Exercise 7
(By analambanomenos)
(a): Using the binomial expansion, for any positive integer $n$, $$b^n=(1+(b-1))^n=1^n+n1^{n-1}(b-1)+A$$ where $A$ is a non-negative sum of terms involving higher powers of $(b-1)$. Hence
$$b^n-1=n(b-1)+A\ge n(b-1).$$
(b): Replace $b$ in case (a) with $b^{1/n}$ to get $b-1\ge n(b^{1/n}-1)$.
(c): From case (b) we have
\begin{align*}
\frac{b-1}{t-1}\,(b^{1/n}-1) &< n(b^{1/n}-1)\le b-1 \\
b^{1/n}-1 &< t-1 \\
b^{1/n} &< t
\end{align*}(d): By case (c), if $n>(b-1)/(yb^{-w}-1)$, then $b^{1/n}<yb^{-w}$, or $b^{w+(1/n)}<y$.
(e): Applying case (c) to $t=y^{-1}b^w>1$, if $n>(b-1)/(y^{-1}b^w-1)$, then $b^{1/n}<y^{-1}b^w$, or $y<b^{w-(1/n)}$.
(f): If $b^x<y$, then from case (d) there is a sufficiently large integer $n$ such that $b^{x+(1/n)}<y$, that is, $x+(1/n)\in A$, contradicting $x=\sup A$. And if $b^x>y$, then from case (e) there is a sufficiently large integer $n$ such that $b^{x-(1/n)}>y$, so that $x-(1/n)$ is an upper bound of $A$, contradicting $x=\sup A$.
(g): Suppose there are real numbers $x_1<x_2$ such that $b^{x_1}=b^{x_2}$. Then $b^{x_1}b^{x_2-x_1}=b^{x_2}=b^{x_1}$, so that $b^{x_2-x_1}=1$. Using the definition of real powers given in exercise 6, this means there are positive integers $m,n$ such that $(b^m)^{1/n}\le 1$. However, since the positive integeral powers of numbers greater than 1 are also greater than 1, this is impossible.
Exercise 8
(By ghostofgarborg) By proposition 1.18d, an ordering $<$ that makes $\mathbb{C}$ an ordered field would have to satisfy $-1 = i^2 > 0$, contradicting $1 > 0$.
Exercise 9
(By ghostofgarborg) Let $z_i = a_i + b_i$.
Property 1.5(i): Assume $z_1 \neq z_2$. If $a_1 \neq a_2$, either $z_1 < z_2$ or $z_2 < z_1$. Otherwise $b_1 \neq b_2$, in which case $z_1 < z_2$ or $z_2 < z_1$.
Property 1.5(ii): Assume $z_1 < z_2$ and $z_2 < z_3$. Then $a_1 \leq a_2 \leq a_3$. If either of the inequalities is strict, $a_1 < a_3$, and $z_1 < z_3$. Otherwise, $b_1 \leq b_2 \leq b_3$ and all inequalities are strict, so that $z_1 < z_3$.
This proves that the order is well-defined.
The set does not have the least upper bound property. Consider the set $\mathbb R i = \{ 0 + xi : i \in \mathbb R \}$. The set is bounded above by $1$. Assume for contradiction that $\alpha = a + bi$ is a least upper bound. It is clear that we must have $a \geq 0$. If $a > 0$, then $\frac 1 2 \alpha$ is another upper bound that is strictly smaller than $\alpha$, contradicting minimality. Therefore, we must have $a=0$. Then $\alpha + i \in \mathbb R i$ is greater than $\alpha$, contradicting $\alpha$ being an upper bound.
Exercise 10
(By analambanomenos}) We have
$$(a^2-b^2) = \frac{|w|+u}{2}-\frac{|w|-u}{2} = u$$ and $$2ab = (|w|+u)^{1/2}(|w|-u)^{1/2} = (|w|^2-u^2)^{1/2} = (v^2)^{1/2} = |v|.$$ Hence $$z^2=(a^2-b^2)+2abi=u+|v|i=w$$ if $v\ge 0$, and $$(\overline{z}^2)=(a^2-b^2)-2abi=u-|v|i=w$$ if $v\le 0$. Hence every nonzero $w$ has two square roots $\pm z$ or $\pm\overline{z}$. Of course, 0 has only one square root, itself.
Baby Rudin 数学分析原理完整第一章习题解答
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2021-06-19 12:41:36
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https://www.zbmath.org/?q=an%3A1352.74022
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# zbMATH — the first resource for mathematics
A phase field model of dynamic fracture: robust field updates for the analysis of complex crack patterns. (English) Zbl 1352.74022
Summary: The numerical modeling of dynamic failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies and demands the formulation of additional branching criteria. This drawback can be overcome by a diffusive crack modeling, which is based on the introduction of a crack phase field. Following our recent works on quasi-static modeling of phase-field-type brittle fracture, we propose in this paper a computational framework for diffusive fracture for dynamic problems that allows the simulation of complex evolving crack topologies. It is based on the introduction of a local history field that contains a maximum reference energy obtained in the deformation history, which may be considered as a measure of the maximum tensile strain in the history. This local variable drives the evolution of the crack phase field. Its introduction provides a very transparent representation of the balance equation that governs the diffusive crack topology. In particular, it allows for the construction of a very robust algorithmic treatment for elastodynamic problems of diffusive fracture. Here, we extend the recently proposed operator split scheme from quasi-static to dynamic problems. In a typical time step, it successively updates the history field, the crack phase field, and finally the displacement field. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples, which show the evolution of complex crack patterns under dynamic loading.
##### MSC:
74A45 Theories of fracture and damage
Full Text:
##### References:
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On the interaction of stress waves with propagation cracks, International Journal of Fracture 26 pp 189– (1984) · doi:10.1007/BF01140627 [5] Ramulu, Mechanic of crack curving and branching-a dynamic fracture analysis, International Journal of Fracture Mechanics 27 pp 187– (1985) · doi:10.1007/BF00017967 [6] Kalthoff, Impact Loading and Dynamic Behavior of Materials pp 185– (1987) [7] Miehe, A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits, Computer Methods in Applied Mechanics and Engineering 199 pp 2765– (2010) · Zbl 1231.74022 · doi:10.1016/j.cma.2010.04.011 [8] Miehe, Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations, International Journal of Numerical Methods in Engineering 83 pp 1273– (2010) · Zbl 1202.74014 · doi:10.1002/nme.2861 [9] Griffith, The phenomena of rupture and flow in solids, Philosophical Transactions of the Royal Society London A 221 pp 163– (1921) · doi:10.1098/rsta.1921.0006 [10] Freund, Dynamic Fracture Mechanics (1990) · Zbl 0712.73072 · doi:10.1017/CBO9780511546761 [11] Francfort, Revisiting brittle fracture as an energy minimization problem, Journal of the Mechanics and Physics of Solids 46 pp 1319– (1998) · Zbl 0966.74060 · doi:10.1016/S0022-5096(98)00034-9 [12] Bourdin, The Variational Approach to Fracture (2008) · Zbl 1176.74018 · doi:10.1007/978-1-4020-6395-4 [13] Dal Maso, A model for the quasistatic growth of brittle fractures: existence and approximation results, Archive for Rational Mechanics and Analysis 162 pp 101– (2002) · Zbl 1042.74002 · doi:10.1007/s002050100187 [14] Buliga, Energy minimizing brittle crack propagation, Journal of Elasticity 52 pp 201– (1999) · Zbl 0947.74055 · doi:10.1023/A:1007545213010 [15] Mumford, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics 42 pp 577– (1989) · Zbl 0691.49036 · doi:10.1002/cpa.3160420503 [16] Ambrosio, Approximation of functionals depending on jumps by elliptic functionals via $$\Gamma$$-convergence, Communications on Pure and Applied Mathematics 43 pp 999– (1990) · Zbl 0722.49020 · doi:10.1002/cpa.3160430805 [17] Dal Maso, An Introduction to $$\Gamma$$-Convergence (1993) · doi:10.1007/978-1-4612-0327-8 [18] Braides, Approximation of Free Discontinuity Problems (1998) · Zbl 0909.49001 · doi:10.1007/BFb0097344 [19] Braides, $$\Gamma$$-Convergence for Beginners (2002) · doi:10.1093/acprof:oso/9780198507840.001.0001 [20] Bourdin, A time-discrete model for dynamic fracture based on crack regularization, International Journal of Fracture 168 pp 133– (2011) · Zbl 1283.74055 · doi:10.1007/s10704-010-9562-x [21] Hakim, Laws of crack motion and phase-field models of fracture, Journal of the Mechanics and Physics of Solids 57 pp 342– (2009) · Zbl 1421.74089 · doi:10.1016/j.jmps.2008.10.012 [22] Karma, Phase-field model of mode III dynamic fracture, Physical Review Letters 92 pp 8704.045501– (2001) [23] Eastgate, Fracture in mode I using a conserved phase-field model, Physical Review E 65 pp 036117-1-10– (2002) · doi:10.1103/PhysRevE.65.036117 [24] Capriz, Continua with Microstructure (1989) · doi:10.1007/978-1-4612-3584-2 [25] Mariano, Multifield theories in mechanics of solids, Advances in Applied Mechanics 38 pp 1– (2001) · doi:10.1016/S0065-2156(02)80102-8 [26] Frémond, Non-smooth Thermomechanics (2002) [27] Miehe, A multi-field incremental variational framework for gradient-extended standard dissipative solids, Journal of the Mechanics and Physics in Solids 59 pp 898– (2011) · Zbl 1270.74022 · doi:10.1016/j.jmps.2010.11.001 [28] Frémond, Damage, gradient of damage and principle of virtual power, International Journal of Solids and Structures 33 pp 1083– (1996) · Zbl 0910.73051 · doi:10.1016/0020-7683(95)00074-7 [29] Peerlings, Gradient enhanced damage for quasi-brittle materials, International Journal for Numerical Methods in Engineering 39 pp 3391– (1996) · Zbl 0882.73057 · doi:10.1002/(SICI)1097-0207(19961015)39:19<3391::AID-NME7>3.0.CO;2-D [30] Xu, Numerical simulations of fast crack growth in brittle solids, Journal of the Mechanics and Physics of Solids 42 pp 1397– (1994) · Zbl 0825.73579 · doi:10.1016/0022-5096(94)90003-5 [31] Camacho, Computational modelling of impact damage in brittle materials, International Journal of Solids and Structures 33 pp 2899– (1996) · Zbl 0929.74101 · doi:10.1016/0020-7683(95)00255-3 [32] Pandolfi, An efficient adaptive procedure for three-dimensional fragmentation simulations, Engineering with Computers 18 pp 148– (2002) · Zbl 01993863 · doi:10.1007/s003660200013 [33] Gürses, A computational framework of three-dimensional configurational-force-driven brittle crack propagation, Computer Methods in Applied Mechanics and Engineering 198 pp 1413– (2009) · Zbl 1227.74070 · doi:10.1016/j.cma.2008.12.028 [34] Miehe, A robust algorithm for configurational-force-driven brittle crack propagation with r-adaptive mesh alignment, International Journal for Numerical Methods in Engineering 72 pp 127– (2007) · Zbl 1194.74444 · doi:10.1002/nme.1999 [35] Belytschko, Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment, International Journal for Numerical Methods in Engineering 58 pp 1873– (2003) · Zbl 1032.74662 · doi:10.1002/nme.941 [36] Song, A method for dynamic crack and shear band propagation with phantom nodes, International Journal for Numerical Methods in Engineering 67 pp 868– (2006) · Zbl 1113.74078 · doi:10.1002/nme.1652 [37] Song, Cracking node method for dynamic fracture with finite elements, International Journal for Numerical Methods in Engineering 77 pp 360– (2009) · Zbl 1155.74415 · doi:10.1002/nme.2415 [38] Song, A comparative study on finite element methods for dynamic fracture, Computer Methods in Applied Mechanics and Engineering 42 pp 239– (2008) · Zbl 1160.74048 [39] Fagerström, Theory and numerics for finite deformation fracture modelling using strong discontinuities, International Journal for Numerical Methods in Engineering 66 pp 911– (2006) · Zbl 1110.74815 · doi:10.1002/nme.1573 [40] Armero, Numerical simulation of dynamic fracture using finite elements with embedded discontinuities, International Journal of Fracture 160 pp 119– (2009) · Zbl 1273.74422 · doi:10.1007/s10704-009-9413-9 [41] Linder, Finite elements with embedded branching, Finite Elements in Analysis and Design 45 pp 280– (2009) · doi:10.1016/j.finel.2008.10.012 [42] Radovitzky, Error estimation and adaptive meshing in strongly nonlinear dynamic problems, Computer Methods in Applied Mechanics and Engineering 172 pp 203– (1999) · Zbl 0957.74058 · doi:10.1016/S0045-7825(98)00230-8 [43] Kane, Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, International Journal for Numerical Methods in Engineering 49 pp 1295– (2000) · Zbl 0969.70004 · doi:10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W [44] Miehe, Comparison of two algorithms for the computation of fourth-order isotropic tensor functions, Computers & Structures 66 pp 37– (1998) · Zbl 0929.74128 · doi:10.1016/S0045-7949(97)00073-4 [45] Miehe, Algorithms for computation of stresses and elasticity moduli in terms of Seth-Hill’s family of generalized strain tensors, Communications in Numerical Methods in Engineering 17 pp 337– (2001) · Zbl 1049.74011 · doi:10.1002/cnm.404 [46] Gross, Fracture Mechanics (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-06-20 18:16:12
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https://www.zbmath.org/?q=an%3A0641.70014
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# zbMATH — the first resource for mathematics
Mather sets for plane Hamiltonian systems. (English) Zbl 0641.70014
Recently J. N. Mather [Topology 21, 457-467 (1982; Zbl 0506.58032)] developed a theory for area preserving monotone twist maps of an annulus, the main result of which is a generalization of the notion of an invariant curve of such maps. This theory can be applied to the Poincaré map of plane Hamiltonian system. However, the monotonicity condition is not satisfied in general, for example in the case of a periodic perturbation, the Poincaré map is the period map and the period will not be small enough in general; and thus the theory of monotone twist maps does not apply.
The purpose of this paper is to show the ideas, however, carry over to a more general case leading to a theory for plane periodic Hamiltonian differential equations. This generalization is studied under differentiability assumptions, which are not needed in Mather’s results. Given the necessary differentiability, J. Moser [Ergodic Theory and Dynamical Systems 6, 401-413 (1986; Zbl 0619.49020)] showed that an area preserving monotone twist map can be interpreted by a Hamiltonian system. The methods developed in this paper are similar to those for monotone twist maps. The author provides a reasonably self-contained exposition of Mather’s theory. The theory developed in this paper makes extensive use of the total ordering of the real line, thus it does not generalize to higher order ordinary differential equations.
Reviewer: M.Z.Nashed
##### MSC:
70H05 Hamilton’s equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
Full Text:
##### References:
[1] N. I. Akhiezer,The Calculus of Variations, Blaisdell Publ. Co., 1962. · Zbl 0119.05604 [2] S. Aubry, P. Y. Le Daeron,The discrete Fraenkel-Kontorova model and its extensions I, Physica 8D, 381-422 (1983). · Zbl 1237.37059 [3] V. Bangert,Mather Sets for Twist Mappings and Geodesics on Tori, in: Dynamics Reported1, Wiley and Teubner, 1987. [4] G. A. Bliss,Calculus of Variations (Carus Monograph1), 6. Aufl. 1971. [5] C. Caratheodory,Variationsrechnung und Partielle Differentialgleichungen erster Ordnung, Teubner 1935. · JFM 61.0547.01 [6] J. Denzler,Studium globaler Minimaler eines Variationsproblems, Diplomarbeit ETH Zürich, February 1987. [7] D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, Springer Grundlehren224 (1977). · Zbl 0361.35003 [8] M. Giaquinta,Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press; Ann. Math. Studies105. · Zbl 0516.49003 [9] O. A. Ladyzhenskaya and N. N. Ural’tseva,Linear and Quasilinear Elliptic Equations, Acad. Press (1968). [10] J. N. Mather,Existence of Quasiperiodic Orbits for Twist Homeomorphisms of the Annulus, Topology21 (1982), 457-467. · Zbl 0506.58032 · doi:10.1016/0040-9383(82)90023-4 [11] J. N. Mather,Destruction of invariant circles, submitted to Ergodic Theory and Dynamical Systems. [12] MacKay, Stark,Lectures on Orbits of Minimal Action for Area Preserving Maps; preprint, University of Warwick, 1985. [13] Ch. B. Morrey,Multiple Integrals in the Calculus of Variations, Springer Grundlehren 130, (1966). · Zbl 0142.38701 [14] J. Moser,Monotone Twist Mappings and the Calculus of Variations, Ergodic Theory and Dynamical Systems6, 401-413 (1968). · Zbl 0619.49020 [15] J. Moser,Recent Developments in the Theory of Hamiltonian Systems, SIAM Reviews28, 459-485 (1986). · Zbl 0606.58022 · doi:10.1137/1028153 [16] J. Moser,Minimal solutions of variational problems on a torus, Ann. Inst. Henri Poincaré,3, 229-272 (1986). · Zbl 0609.49029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-04-13 01:45:58
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https://www.gradesaver.com/textbooks/science/physics/college-physics-4th-edition/chapter-10-problems-page-400/58
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## College Physics (4th Edition)
The maximum kinetic energy of the body is $4.0\times 10^{-6}~J$
In general: $y(t) = A~sin(\omega~t+\phi)$ In this case: $y(t) = (4.0~cm)~sin~[~(0.70~rad/s)~t~]$ We can see that $~A = 4.0~cm~$ and $~\omega = 0.70~rad/s~$ At the equilibrium position, the upward force of the spring is equal in magnitude to the weight. We can find the mass of the body: $mg = kd$ $m = \frac{kd}{g}$ $m = \frac{(2.5~N/m)(0.040~m)}{9.80~m/s^2}$ $m = 0.0102~kg$ We can find the maximum kinetic energy: $K_m = \frac{1}{2}mv_m^2$ $K_m = \frac{1}{2}m(A~\omega)^2$ $K_m = \frac{1}{2}(0.0102~kg)~(0.040~m)^2~(0.70~rad/s)^2$ $K_m = 4.0\times 10^{-6}~J$ The maximum kinetic energy of the body is $4.0\times 10^{-6}~J$.
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2021-04-12 17:18:29
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https://brilliant.org/problems/sat-fractions-and-decimals/
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# SAT Fractions
Number Theory Level 2
Daniel ate $$\frac{1}{7}$$ of a cake. Edward ate $$\frac{1}{3}$$ of what was left. What fraction of the cake is left uneaten?
(A) $$\ \ \frac{1}{7}$$
(B) $$\ \ \frac{2}{7}$$
(C) $$\ \ \frac{11}{21}$$
(D) $$\ \ \frac{4}{7}$$
(E) $$\ \ \frac{6}{7}$$
×
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2016-10-22 16:12:50
|
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https://www.shaalaa.com/question-bank-solutions/if-difference-between-roots-equation-x-2-x-8-0-2-write-values-quadratic-equations_54302
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# If the Difference Between the Roots of the Equation X 2 + a X + 8 = 0 is 2, Write the Values of A. - Mathematics
If the difference between the roots of the equation $x^2 + ax + 8 = 0$ is 2, write the values of a.
#### Solution
Given:
$x^2 + ax + 8 = 0 .$
Let $\alpha \text { and } \beta$ are the roots of the equation.
Sum of the roots = $\alpha + \beta = \frac{- a}{1} = - a$.
Product of the roots = $\alpha\beta = \frac{8}{1} = 8$
Given:
$\alpha - \beta = 2$
$\text { Then }, \left( \alpha + \beta \right)^2 - \left( \alpha - \beta \right)^2 = 4\alpha\beta$
$\Rightarrow \left( \alpha + \beta \right)^2 - 2^2 = 4 \times 8$
$\Rightarrow \left( \alpha + \beta \right)^2 - 4 = 32$
$\Rightarrow \left( \alpha + \beta \right)^2 = 32 + 4 = 36$
$\Rightarrow \left( \alpha + \beta \right) = \pm 6$
$\alpha + \beta = - a = \pm 6$
$a = \pm 6$
Is there an error in this question or solution?
#### APPEARS IN
RD Sharma Class 11 Mathematics Textbook
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2021-04-20 20:17:11
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https://photonsquared.com/categories/
|
# Posts by Category
## Estimation of Temperature and Pressure of a Constant Volume Propane-Oxygen Mixture
Published:
A recent project required a first-order approximation to determine if an explosive gas mixture would result in a tank rupture. The following analysis done in Python and follows Coopers analysis 1 It provides a reasonable approximation, however it is sensitive to the chemical reaction hierarchy assumed. The jupyter notebook used to perform this analysis is here. Read more
## Determining Fragment and Debris Hazards
When the United States Department of Defense Explosives Safety Board (DDESB) determines fragment and debris hazards they use a 6-step process based on Technical Paper 12 (TP-12) 1. In summary, this process finds the range $R$ at which there is a probability $p$ of a person with an area $A_T=0.58\,m^2$ $(6.24\,ft^2)$ being struck by a fragment with a mass $m$ and kinetic energy $E_{CR}=58\,ft{\text -} lb\, \left( 79\,J \right)$. The process is, Read more
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2020-03-31 05:44:28
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https://pinoria.com/how-to-check-for-string-equality-with-typescript/
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### How to check for string equality with TypeScript?
Sometimes, we want to check for string equality with TypeScript.
In this article, we’ll look at how to check for string equality with TypeScript.
### How to check for string equality with TypeScript?
To check for string equality with TypeScript, we can use the `===` operator.
For instance, we write
``````if (x === y) {
} else {
}
``````
to check if `x` and `y` have the same value with `===`.
### Conclusion
To check for string equality with TypeScript, we can use the `===` operator.
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2022-11-30 09:53:05
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https://math.stackexchange.com/questions/2530182/concerning-the-derivative-definition-using-limit
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# Concerning the derivative definition using limit
If we have the following function $$\phi(x)=\frac{e^{m_2x}-e^{m_1x}}{m_2-m_1}$$ and we want to find the following limit $$\lim_{m2\to m_1}{\frac{e^{m_2x}-e^{m_1x}}{m_2-m_1}}$$ where $$m_2=m_1+h$$ then I think it must be $$\frac{d}{dm}e^{mx}|_{m=m_1}=e^{m_1x}x$$ My first question:
Can we substitute by m2 instead of m1 ? or we must substitute by m1 because m2 tends to m1 ? I mean can we write that the above limit will result in : $$\frac{d}{dm}e^{mx}|_{m=m_2}=e^{m_2x}x$$? I tried to prove that we can substitute by m2 also , I proved it but I am worried that we can not do that or that I may have proved it in a wrong way: $$\lim_{m2\to m_1}{e^{m_2x}\frac{1-e^{(m_1-m_2)x}}{-(m_1-m_2)}}$$ $$\lim_{h\to 0}{e^{m_2x}\frac{1-e^{hx}}{-h}}$$ using l'hopital rule $$\lim_{h\to 0}{e^{m_2x}\frac{e^{-hx}x}{-1}}=xe^{m_2x}$$
(this is my first question)
My second question: what about the following limit ( if we let m1 tends to m2 instead) $$\lim_{m1\to m_2}{\frac{e^{m_2x}-e^{m_1x}}{m_2-m_1}},\ \ m_2=m_1+h$$ will it result in $$\frac{d}{dm}e^{mx}|_{m=m_2}=e^{m_2x}x$$?
• I think, you are right. – Michael Rozenberg Nov 21 '17 at 3:16
• I edited the post – MCS Nov 21 '17 at 3:54
I like the following way: $$\lim_{m_2\rightarrow m_1}\frac{e^{m_2x}-e^{m_1x}}{m_2-m_1}=x\lim_{m_2\rightarrow m_1}\frac{e^{m_1x}\left(e^{(m_2-m_1)x}-1\right)}{(m_2-m_1)x}=xe^{m_1x}.$$
• I do not understand what did you do after taking the exponential common factor .. did you apply l'hopital rule considering (m2-m1)x as a single variable? if yes , I think we do not need to multiply and divide by x , we can take only m2-m1 as a single variable .. right ? ( I am just checking whether i understand ) – MCS Nov 21 '17 at 3:58
• @Sousa I used $\lim\limits_{x\rightarrow0}\frac{e^x-1}{x}=1.$ – Michael Rozenberg Nov 21 '17 at 4:01
• Note : I edited the post , I proved that we can substitute by m2 instead of m1 ..However, I feel that we can NOT substitute by m2 since m2 tends to m1 , so the proof may be wrong and I would like to know where is the fault in the proof i wrote . Thanks . – MCS Nov 21 '17 at 4:03
• I think, $m_2\rightarrow m_1$ says that $m_2$ is changed and $m_1$ is a constant. – Michael Rozenberg Nov 21 '17 at 4:07
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2019-07-17 04:29:16
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https://search.r-project.org/CRAN/refmans/eRm/html/stepwiseIt.html
|
stepwiseIt {eRm} R Documentation
Stepwise item elimination
Description
This function eliminates items stepwise according to one of the following criteria: itemfit, Wald test, Andersen's LR-test
Usage
## S3 method for class 'eRm'
stepwiseIt(object, criterion = list("itemfit"), alpha = 0.05,
verbose = TRUE, maxstep = NA)
Arguments
object Object of class eRm. criterion List with either "itemfit", "Waldtest" or "LRtest" as first element. Optionally, for the Waldtest and LRtest a second element containing the split criterion can be specified (see details). alpha Significance level. verbose If TRUE intermediate results are printed out. maxstep Maximum number of elimination steps. If NA the procedure stops when the itemset is Rasch homogeneous.
Details
If criterion = list("itemfit") the elimination stops when none of the p-values in itemfit is significant. Within each step the item with the largest chi-squared itemfit value is excluded.
If criterion = list("Waldtest") the elimination stops when none of the p-values resulting from the Wald test is significant. Within each step the item with the largest z-value in Wald test is excluded.
If criterion = list("LRtest") the elimination stops when Andersen's LR-test is not significant. Within each step the item with the largest z-value in Wald test is excluded.
Value
The function returns an object of class step containing:
X Reduced data matrix (bad items eliminated) fit Object of class eRm with the final item parameter elimination it.elim Vector contaning the names of the eliminated items res.wald Elimination results for Wald test criterion res.itemfit Elimination results for itemfit criterion res.LR Elimination results for LR-test criterion nsteps Number of elimination steps
LRtest.Rm, Waldtest.Rm, itemfit.ppar
Examples
## 2pl-data, 100 persons, 10 items
set.seed(123)
X <- sim.2pl(500, 10, 0.4)
res <- RM(X)
## elimination according to itemfit
stepwiseIt(res, criterion = list("itemfit"))
## Wald test based on mean splitting
stepwiseIt(res, criterion = list("Waldtest","mean"))
## Andersen LR-test based on random split
set.seed(123)
groupvec <- sample(1:3, 500, replace = TRUE)
stepwiseIt(res, criterion = list("LRtest",groupvec))
[Package eRm version 1.0-2 Index]
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2023-03-23 08:26:06
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|
https://www.physicsforums.com/threads/why-does-sinx-1-2i-e-ix-e-ix.102655/
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# Why does sinx=(1/2i)[e^(ix)-e^(-ix)]?
1. Dec 4, 2005
### asdf1
why does sinx=(1/2i)[e^(ix)-e^(-ix)]?
2. Dec 4, 2005
### siddharth
This can be shown from Euler's formula.
http://mathworld.wolfram.com/EulerFormula.html" [Broken]
Last edited by a moderator: May 2, 2017
3. Dec 4, 2005
### asdf1
e^ix=cosx+isinx
e^(-ix)=?
do you really add a negative sign and it becomes cos(-x)+isin(-x)?
4. Dec 4, 2005
### LeonhardEuler
Yes, and cos(-x)=cos(x), while sin(-x)=-sin(x), because they are even and odd functions, respectively.
5. Dec 4, 2005
### benorin
yes, and recall that sine is an odd function and cosine is an even function so we have
$$e^{-ix} = \cos(-x) + i\sin(-x)=\cos(x)-i\sin(x)$$
and from Euler's formula we have $e^{ix}=\cos(x)+i\sin(x)$
and so subtracting the first formula from the second gives
$$e^{ix}-e^{-ix} = 2i\sin(x)$$
hence
$$\sin(x)=\frac{1}{2i} \left( e^{ix}-e^{-ix}\right)$$
6. Dec 5, 2005
### asdf1
wow~
amazing...
thank you very much!
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2017-10-24 00:43:04
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https://forum.xwiki.org/t/what-do-we-do-with-mathjax-macro-vs-formula-macro-in-xs/10481
|
# What do we do with Mathjax macro vs Formula macro in XS?
Hi devs,
Now that we have a client-side PDF that properly exports the mathjax macro, we need to decide whether we want to promote the mathjax macro or continue promoting the formula one.
Some considerations:
• The mathjax macro was developed because it produces better rendering than the formula macro
• The formula macro is not very easy to setup since there are several renderers. Also some user is reporting problems about using the formula macro that breaks his wiki in some cases (I don’t have the details).
• With the latest macro discovery feature from Thomas/FASTEN, it’s slightly less important to have macros bundled in XS to be discovered.
• AFAICS the mathjax macro supports more feature than the formula one (including references) and is a superset of it.
I see the following options:
• Option 1: both macros at same level
• Extract the formula macro outside of XS (into xwiki-contrib) and let users decide what extension to install in their wiki.
• We’ll need to recommend one extension or if both have value, then at least do a comparison table and explain when to use one over another
• We probably also need to make the mathjax macro recommended
• Option 2: mathjax macro over formula macro
• Extract the formula macro outside of XS (into xwiki-contrib)
• Add the mathjax macro inside XS in replacement of the formula macro
• Option 3: formula macro FTW
• Deprecate the mathjax macro
• Add a new mathjax “renderer” in the formula macro and make it the default one. Note that when using mathjax you specify the font size in the content and not in the macro parameter as it’s done in the formula macro (but I guess it could be ignored in this case). Similarly mathjax doesn’t produce images so the imageType macro param would also need to be ignored.
• Optional: Also extract the formula macro outside of XS (into xwiki-contrib)
My preference would go to option 1 and recommend the mathjax macro. The rationale for not having it in XS is that I don’t think every user needs a formula macro, it’s a pretty specific case.
WDYT?
Thanks
+1 for Option 1 too
I feel Option 3 is actually pretty much Option 1 but with one more step (make the formula macro support also mathjax) so we can start with Option 1 and see later if someone wants to work on the extra step (but I don’t think it worth it for the same reason people are generally choosing between {{groovy}} and {{python}} and not between {{script language="groovy"}} and {{script language="python"}}).
+1 for option 1 also.
Thanks,
Marius
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2022-07-05 12:35:26
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https://pycbc.org/pycbc/latest/html/faithsim.html
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# Dag Generator for Doing Faithfulness Comparisons¶
## Introduction¶
This page describes how to use the faithfulness dag generator within PyCBC.
## How to generate a workflow¶
### Creating a configuration (.ini) file¶
All the choices when setting up a faithsim are contained in a single configuration file.
Below is an example.
[inspinj]
min-mass1 = 1
max-mass1 = 100
min-spin1 = 0
max-spin1 = 0
min-mass2 = 1
max-mass2 = 100
min-spin2 = 0
max-spin2 = 0
max-mtotal = 25
min-mtotal = 2
f-lower = 30
enable-spin =
aligned =
waveform = IMRPhenomB
disable-milkyway =
i-distr = uniform
l-distr = random
min-distance = 1000
d-distr = uniform
max-distance = 1000
gps-start-time = 1000000000
gps-end-time = 1000001000
time-interval = 0.
time-step = 1.
seed = 123434
m-distr = componentMass
[executables]
faithsim = /home/ahnitz/src/pc10/pycbc/bin/pycbc_faithsim
[workflow]
templates-per-job = 100
log-path = /usr1/ahnitz/
[faithsim-flatIMRC]
psd = aLIGOZeroDetHighPower
waveform1-approximant = IMRPhenomB
waveform1-start-frequency=14
waveform2-approximant = IMRPhenomC
waveform2-start-frequency=14
filter-low-frequency=15
filter-sample-rate=4096
filter-waveform-length=1024
[faithsim-flatF2]
psd = aLIGOZeroDetHighPower
waveform1-approximant = IMRPhenomB
waveform1-start-frequency=14
waveform2-approximant = TaylorF2
waveform2-start-frequency = 14
waveform2-spin-order = 5
filter-low-frequency=15
filter-sample-rate=4096
filter-waveform-length=1024
[faithsim-flatSEOBNRv1]
psd = aLIGOZeroDetHighPower
waveform1-approximant = IMRPhenomB
waveform1-start-frequency=14
waveform2-approximant = SEOBNRv1
waveform2-start-frequency = 14
filter-low-frequency=15
filter-sample-rate = 16384
filter-waveform-length=1024
[faithsim-flatEOBNRv2]
psd = aLIGOZeroDetHighPower
waveform1-approximant = IMRPhenomB
waveform1-start-frequency=14
waveform2-approximant = EOBNRv2
waveform2-start-frequency = 14
filter-low-frequency=15
filter-sample-rate = 8192
filter-waveform-length=1024
[faithsim-flatEOBNRv2HM]
psd = aLIGOZeroDetHighPower
waveform1-approximant = IMRPhenomB
waveform1-start-frequency=14
waveform2-approximant = EOBNRv2HM
waveform2-start-frequency = 14
filter-low-frequency=15
filter-sample-rate = 8192
filter-waveform-length=1024
[faithsim-flatT1]
psd = aLIGOZeroDetHighPower
waveform1-approximant = IMRPhenomB
waveform1-start-frequency=14
waveform2-approximant = TaylorT1
waveform2-start-frequency=14
filter-low-frequency=15
filter-sample-rate=4096
filter-waveform-length=1024
[faithsim-flatT2]
psd = aLIGOZeroDetHighPower
waveform1-approximant = IMRPhenomB
waveform1-start-frequency=14
waveform2-approximant = TaylorT2
waveform2-start-frequency=14
filter-low-frequency=15
filter-sample-rate=4096
filter-waveform-length=1024
[faithsim-flatT3]
psd = aLIGOZeroDetHighPower
waveform1-approximant = IMRPhenomB
waveform1-start-frequency=14
waveform2-approximant = TaylorT3
waveform2-start-frequency=14
filter-low-frequency=15
filter-sample-rate=4096
filter-waveform-length=1024
[faithsim-flatTRD]
psd = aLIGOZeroDetHighPower
waveform1-approximant = IMRPhenomB
waveform1-start-frequency=14
waveform2-approximant = PhenSpinTaylorRD
waveform2-start-frequency=14
filter-low-frequency=15
filter-sample-rate=4096
filter-waveform-length=1024
[faithsim-flatT4]
psd = aLIGOZeroDetHighPower
waveform1-approximant = IMRPhenomB
waveform1-start-frequency=14
waveform2-approximant = TaylorT4
waveform2-start-frequency=14
filter-low-frequency=15
filter-sample-rate=4096
filter-waveform-length=1024
There are four sections that must be present [inspinj], [executables], [workflow], and [faithsim-XXX].
1. inspinj
This section sets the paramaters of all of the injection waveforms. The arguments in the configuration file are fed directly to the lalapps_inspinj program to create an injection file. The same arguments are available, and the same restrictions apply. The number of injections can be set by using the gps start and end time options along with the time step. Note, however, that the waveform name is required but does not determine the actual approximants that will be compared. That is set in the [banksim] section.
2. executables
This section lists the location of the pycbc_faithsim script. Make note that the script is copied to the executables folder and that is the version that will be used.
3. workflow
This section has options that configure the workflow. The required options are ‘log-path’ and ‘templates-per-job’. The ‘log-path’ specifies the directory to store condor log files. The ‘templates-per-job’ section determines how many faithfulness calculations each job will do. The injection file is split into smaller portions to match this restriction. This option is directly proportional to the running time of each job.
Faith simulations running on LDG clusters must include the ‘accounting-group’ option in the workflow section. The value must be choosen according to the Accounting information web page.
4. faithsim-XXX Multiple sections with a a name of the form ‘faithsim-USER_STRING’ can exist. The generator will create jobs that correspond to each of these sections and each will generate an independent results file file labeled with the same USER_STRING.
These sections corresponds to the arguments sent to the banksim executable. The two approximants to compare, along with their PN order paramters (if relevant), are set here. Note that the option filter-waveform-length must be set to a value greater than the duration of the longest generated approximant.
### Generating the workflow¶
Once a configuration file as been made, create a workspace directory and place the file into it. Running the following command will generate a dag that will submit the required jobs.
pycbc_make_faithsim --conf YOUR_INI_FILE.ini
The workflow can then be submitted by running the generated shell script.
sh submit.sh
### Understanding the results¶
The main results of the faithsim are result files, one for each faithsim section in the file. These are whitespace separated ASCII files.
Some basic plots are also generated automatically and placed into the ‘plots’ folder.
The pycbc_faithsim_plots script located in the scripts folder is an example of how to read the results files.
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2020-09-20 00:25:26
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https://amathew.wordpress.com/tag/abstract-nonsense/
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This post is an exposition of the material in the paper “Homotopy is not concrete” by P. Freyd, of whose existence I learned from this MO discussion.
A category ${\mathcal{C}}$ is concrete if there is a faithful functor ${F: \mathcal{C} \rightarrow \mathbf{Sets}}$. Most of the categories one initially encounters are in fact concrete: categories of groups, rings, modules, Lie algebras, and so on, and one can think of them as consisting of “structured sets” and “morphisms respecting that structure.” Every small category is concrete, because one can take the Yoneda embedding
$\displaystyle \mathcal{C} \rightarrow \mathbf{Sets}^{\mathcal{C}^{op}}$
followed by the product functor ${\mathbf{Sets}^{\mathcal{C}^{op}}\rightarrow \mathbf{Sets}}$.
Nonetheless, not every category is concrete, and the following example shows that a very natural one is not:
Theorem 1 (Freyd) The homotopy category ${\mathcal{H}ot_*}$ of pointed spaces is not concrete.
In other words, a homotopy type is somehow too complex to be encoded simply as a set with appropriate structure.
The idea of the proof is essentially the following. In a category of structured sets, a given object can only have so many subobjects, because a set has only so many subsets. But there are categories where an object may have an enormous collection of subobjects, because the definition of a subobject is purely arrow-theoretic. So a category where objects can have lots of subobjects is probably not concrete. (more…)
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2020-07-15 06:21:40
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http://mathhelpforum.com/differential-geometry/142491-differentiable-mappings-banach-spaces-need-help.html
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# Thread: Differentiable mappings and Banach spaces - need help
1. ## Differentiable mappings and Banach spaces - need help
Please, tell me how I can prove this, seemingly, simple statement?
If $\langle{E,\|\!\cdot\!\|_1}\rangle,\,\langle{F,\|\! \cdot\!\|_2}\rangle$ - the Banach spaces, $G\subset{E}$ - an open set and mappings $G\xrightarrow{f_1}F,\,G\xrightarrow{f_2}F$ are differentiable at the point $x_0\in{G}$, then the mappings $g_1=f_1+f_2$ and $g_2=\lambda{f},\,\lambda\in\mathbb{R}$ also differentiable at the point $x_0$, and with equalities $dg_1(x_0)=df_1(x_0)+df_2(x_0),~dg_2(x_0)=\lambda{d }f_1(x_0).$
P.S. How do you define differentiability in Banach spaces?
2. Originally Posted by DeMath
Please, tell me how I can prove this, seemingly, simple statement?
If $\langle{E,\|\!\cdot\!\|_1}\rangle,\,\langle{F,\|\! \cdot\!\|_2}\rangle$ - the Banach spaces, $G\subset{E}$ - an open set and mappings $G\xrightarrow{f_1}F,\,G\xrightarrow{f_2}F$ are differentiable at the point $x_0\in{G}$, then the mappings $g_1=f_1+f_2$ and $g_2=\lambda{f},\,\lambda\in\mathbb{R}$ also differentiable at the point $x_0$, and with equalities $dg_1(x_0)=df_1(x_0)+df_2(x_0),~dg_2(x_0)=\lambda{d }f_1(x_0).$
P.S. How do you define differentiability in Banach spaces?
Probably the Fréchet derivative is what is wanted.
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2017-05-28 09:24:16
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https://unmethours.com/question/35722/does-openstudio-or-energy-plus-actually-output-loads-calculated-for-sizing-purposes/
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Question-and-Answer Resource for the Building Energy Modeling Community
Get started with the Help page
# Does OpenStudio or Energy plus actually output loads calculated for sizing purposes?
I am trying to run my energy model as a load model (rather than construct a new one in another software) and am wondering where I can find the results of the system sizing portion of the simulation. Seems like it should be pretty straightforward, but I'm having difficulty getting this information. I'm running with Ideal Air Loads enabled and simply want to know peak heating and cooling load. Originally I simply took this from peak heating/cooling demand from the annual simulation, but realized that this inaccurate as it allows for internal gains, solar gains, and thermal mass to play into the equation...things that are not desirable when sizing. I also tried summing each of the individual design loads and ventilation load, but realized that these peaks don't necessarily coincide - resulting in an overestimate in load.
Where is the system level sizing information stored and/or how can I calculate from the results?
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Sort by » oldest newest most voted
1. All EnergyPlus sizing calculations will include internal gains, solar gains, thermal mass, etc. unless the user inputs remove them. For example, all of the SizingPeriod:DesignDay objects for heating which are in the ddy files bundled with the weather files have zero sun and constant outdoor temperature for the WinterDesignDay. Also, internal gain schedules should be zero for day type "WinterDesignDay". These inputs effectively remove all of the above from the heating sizing calculations. For cooling sizing, you want all of these items to be included.
2. With ideal loads, run the design days only, and take the peak loads from there. See SimulationControl "Run Simulation for Sizing Periods". The system "coil" loads are reported as output variable "Ideal Loads Supply Air * ".
3. If you want EnergyPlus to do system-level sizing, then you'll need to add a full HVAC system with appropriate Sizing:System inputs.
more
1. Yup! And it seems that OpenStudio is smart enough to do this automatically. The default lighting, occupancy, and equipment schedules are zeroed for the winter design day and maxed for summer. The only reason my "sizing" results were not reflecting this is because I was trying to pull peak load from the annual simulation.
2. So, in OpenStudio's "Simulation Control," I see "Run Simulation For Sizing Periods." I've toggled this to "Yes" and I assume "Run Simulation For Weather File Run Periods" should be "No" (?).
( 2018-11-21 10:12:00 -0600 )edit
1. ^^
2. ^^
3. Vocabulary issue on my end, I think. I simply wanted the total (space+ventilation) load seen by the system.
( 2018-11-21 10:13:06 -0600 )edit
1. You can set both "Run Sizing Periods" and "Run Weather File Run Periods" to yes in the same simulation if you want. The time series output will have data for the sizing periods followed by the weather files periods. But many users do as you suggest, run either the sizing periods or the weather file periods.
( 2018-11-23 09:44:33 -0600 )edit
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2019-02-19 07:52:26
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https://zbmath.org/?q=an:1226.60132
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## Escape of resources in a distributed clustering process.(English)Zbl 1226.60132
Summary: In a distributed clustering algorithm introduced by E. G. Coffman jun., P.-J. Courtois, E. N. Gilbert and P. Piret [J. Appl. Probab. 28, No. 4, 737–750 (1991; Zbl 0741.60114)], each vertex of $$\mathbb Z^{d}$$ receives an initial amount of a resource, and, at each iteration, transfers all of its resource to the neighboring vertex which currently holds the maximum amount of resource. In [M. R. Hilário et al., Commun. Pure Appl. Math. 63, No. 7, 926–934 (2010; Zbl 1202.60156)], it was shown that, if the distribution of the initial quantities of resource is invariant under lattice translations, then the flow of resource at each vertex eventually stops almost surely, thus solving a problem posed in [J. van den Berg and R. W. J. Meester, Random Struct. Algorithms 2, No. 3, 335–341 (1991; Zbl 0741.05072)].
In this article, we prove the existence of translation-invariant initial distributions for which resources nevertheless escape to infinity, in the sense that the the final amount of resource at a given vertex is strictly smaller in expectation than the initial amount. This answers a question posed in [Hilário et al., loc. cit.].
### MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory 68M14 Distributed systems
### Keywords:
clustering process; random spanning tree
### Citations:
Zbl 0741.60114; Zbl 1202.60156; Zbl 0741.05072
Full Text:
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2022-05-18 07:31:01
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https://mathsmadeeasy.co.uk/maths-revision-cards/probability-higher-revision-card-answers/
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# Probability Higher Revision Card Answers
## HP1 – Probability Basics
QUESTION: Lauren, Harriet, & Amira are all playing a game which continues until someone wins. Lauren says “I have a 40% chance of winning”, whilst Harriet says “I have a $\frac{1}{4}$ chance of winning”. Amira then claims that if the Lauren & Harriet are correct, she must have a 0.35 chance of winning. Work out if Amira’s statement is true.
ANSWER: We know their probabilities must add up to 1 to make Amira’s statement true. To add these values together, we must make them all share the same format. Here, we’re going to convert them all to percentages.
Firstly, we get that
$0.35=35\%$
Then, we get
$\frac{1}{4}=1\div4=0.25=25\%$
Now, we can add the three probabilities together:
$40\%+25\%+35\%=100\%=1$
They all add to 1, so Amira’s statement is correct.
## HP2 – Tree Diagrams
QUESTION: Heloise makes two choices when getting dressed in the morning. Her first choice is whether to wear trousers (T) or shorts (S). Her second choice is whether to where a jumper (J) or no jumper (N). The probability of her wearing shorts and a jumper is 0.144.
a) Complete the tree diagram.
b) Calculate the probability of Heloise choosing to wear a jumper.
c) What is the probability the Heloise wears a jumper, given that she chose to wear trousers?
ANSWER: a) Firstly, we know she either wears a jumper or doesn’t. Therefore, to fill in the gap at the top (after she has chosen trousers), we simply subtract the probability of her wearing a jumper from 1, to get
$\text{P(N)}=1-\text{P(J)}=1-0.85=0.15$
Next, we know that the probability of her wearing shorts and a jumper is 0.144. This means that 0.144 must be the result of multiplying along the SJ branch, so in other words
$0.45\times x=0.144$
Thus, if we divide by 0.45, we get
$x=0.144\div0.45=0.32$
Then, for the final gap, we subtract 0.32 from 1 to get
$1-0.32=0.68$.
So, the completed tree diagram looks like
b) The two circumstances in which Heloise wears a jumper are: she wears trousers and a jumper, or she wears shorts and a jumper. Multiplying along the branch, we get
$\text{P(T and J)}=0.55*0.85=0.4675$
We already know the probability of her wearing shorts and a jumper: 0.144. This is an ‘or’ situation (since in either circumstance, she’s wearing a jumper), so we must add these probabilities to get
$\text{P(Jumper)}=0.144+0.4675=0.6115$
c) In this case, we know that she has chosen to wear trousers, so that means we’re limited to the options at the end of the ‘T’ branch. At that point, we can read off the probability of her wearing a jumper, and so our answer is: 0.85.
## HP3 – Venn Diagrams
QUESTION: Consider a set of numbers: $\{2, 3, 5, 6, 7, 8, 10, 11, 15\}$.Let $A$ be the set of even numbers, and $B$ be the set of prime numbers.
a) Complete the Venn diagram by writing all above numbers
in their appropriate sections.
b) State how many values are in $A\cap B$.
c) Find $P(A\cup B)$.
ANSWER: a) Firstly, let’s consider any number that are both even and prime. There is one: 2. This is the only number that will go in the section where the two circles cross over.
Then, the rest of the even numbers: 6, 8, and 10, will go in the section of the A circle that doesn’t cross over with B. Next, the rest of the prime numbers: 3, 5, 7, and 11, will go in the section of the B circle that doesn’t cross over with A.
Finally, the one number that is neither even nor prime is 15, so that goes outside the circles. The completed Venn diagram looks like the one below.
b) $A\cap B$ refers to “$A$ and $B$”. There is only one number in both $A$ and $B$, so the answer is 1.
c) $A\cup B$ refers to “$A$ or $B$”. There are 8 numbers that are contained in circle $A$ and/or circle $B$, and there are 9 numbers in total, so we get
$P(A\cup B)=\dfrac{8}{9}$
## HP4 – Averages & Spread
QUESTION: Below is some data collected on the heights, in cm, of 10 men.
$181,\,182,\,175,\,176,\,210,\,169,\,175,\,184,\,167,\,175$
a) Find the mean of these data.
b) Find the median of these data.
c) Explain why the median might be a better measure of the average in this case. (Hint: one of these values is different to the others – what difference does it make?)
ANSWER: a) We must add up all the values and divide by 10.
$\text{mean }=\dfrac{181+182+175+176+210+169+175+184+167+175}{10}=179.4\text{ cm}$
b) To find the median, we must first put the values in ascending order:
$167,\,169,\,175,\,175,\,175,\,176,\,181,\,182,\,184,\,210$
Then, if you cross off alternating biggest and smallest values, you’ll be left with two numbers: 175 and 176. Therefore, the median is 175.5cm, (the halfway point).
c) In this case, the man who is 210cm tall is significantly taller than the other men. Therefore, when we calculate the mean, the 210 value is going to make the mean much higher than otherwise, and it might not be representative of the data (try calculating the mean without 210 and see what happens). The median, however, is not affected by the value of 210, so it might be a better measure of average in this case.
## HP5 – Estimating the Mean (Grouped Frequency Tables)
QUESTION: 80 people are asked to throw a ball as far as they can, and the results are recorded. Using the data in the table below, find an estimate for the mean throwing distance in this experiment.
ANSWER: Firstly, we need to find the midpoints of each class and write them in a new column attached to the one given in the question.
Then, treat the midpoints as the actual values and find the sum of all the midpoints. To make this quicker, we multiply each midpoint by its frequency, and sum all the results. Then, since we are estimating the mean, we divide by the total number of people in the experiment: 80. Doing this, we get
$\text{estimated mean }=\dfrac{(21\times10)+(43\times30)+(12\times55)+(4\times90)}{80}= \dfrac{2,520}{80}=31.5$
## HP6 – Scatter Graphs
QUESTION: Below are some scatter graphs. State whether they show positive, negative, or no correlation. If there is correlation, state the strength of it.
ANSWER: a) We can see that these points following a straight-line pattern fairly closely, and we can see that as the $x$ value increases, so does the $y$ value. Therefore, this graph displays moderate positive correlation.
b) There appears to be no relationship followed by the points on this graph. Therefore, it displays no correlation.
c) We can see that these points following a straight-line pattern very closely, and we can see that as the $x$ value increases, the $y$ value decreases. Therefore, this graph displays strong negative correlation.
## HP7 – Cumulative Frequency
QUESTION: Gracie grows 40 sunflowers one year and records their heights in a frequency table. Complete the cumulative frequency column in the table below and draw a cumulative frequency graph of the data.
ANSWER: Obtaining cumulative frequency from a frequency table amounts to adding up the values as we go along, using the upper limit of each class as our new upper limit at each step. So, the first value is 5, then the second is $5+9=14$, then the third is $5+9+15=29$. Continuing this, the completed table looks like
Then, plotting each of these cumulative frequency values against each of the upper limits of the classes, and joining them all together with a smooth curve, we get the graph shown below.
## HP8 – Boxplots
QUESTION: From the facts given below, construct a boxplot.
The smallest value is 10,
The upper quartile is 38,
The range is 38,
The interquartile range is 12,
The median is halfway between the lower and upper quartiles.
ANSWER: We need the smallest value, largest value, lower quartile, upper quartile, and median. Given that the range is the largest value take away the smallest, if we add the range to the smallest value it will give us the largest value:
$\text{largest value }=10+38=48$
Similarly, as the interquartile range is the upper quartile take away the lower quartile, if we take away the interquartile range from the upper quartile, it will give us the lower quartile:
$\text{lower quartile }=38-12=26$
Lastly, the median is halfway between the two quartiles. So, we get
$\text{median }=\dfrac{26+38}{2}=32$
Now, we have all the information we need, and the resulting boxplot looks like
## HP9 – Histograms
QUESTION: An experiment collects data on the length of time people spend showering. Use the information in the table below to construct a histogram.
ANSWER: We need to add a third column to the table containing the frequency densities, which we calculate by dividing each frequency by its class width. So, for the first one we’d get:
$\text{frequency density }=10\div 4=2.5$
Continuing this, our table with a completed frequency density column looks like:
Now we can plot the histogram. With each bar having the width of its class interval and the height of its frequency density, our resulting histogram looks like:
QUESTION: A histogram displaying data collected on the amount of honey produced, in kg, by a selection of expert beekeepers over the course of one year is shown here. 81 beekeepers collected between 0 and 18 kg of honey. Estimate how many beekeepers collected between 40 and 68 kg of honey.
ANSWER: We need to determine what the missing frequency density scale should be. We know that frequency density is frequency divided by class width, so for the 0 to 18kg class, we get
$\text{frequency density }=81\div18=4.5$
Counting squares, we can see that the 0 to 18kg bar is 15 small squares high, therefore one square on the $y$-axis is worth
$4.5\div15=0.3$
Now, to find the estimate for the number of beekeepers between 40 and 68kg, we need to work out how many beekeepers were in the 40 to 56 class first. This bar is 25 small squares high, so its height on the $y$-axis is $25\times0.3=7.5$. The frequency is given by the area of the bar, and its width is 16, so we get
$\text{beekeepers in “40 to 56” class }=7.5\times16=120$.
Next, we need to estimate how many beekeepers collected between 56 and 68kg of honey. Looking at the histogram, we can see that 56 to 68kg is half the width of the last bar, so we will “cut” the bar in half and find the frequency of one of these new halves. (The red section below highlights how we are considering the “40 to 56” group to look)
This bar is 20 small squares high, so its height on $y$-axis is $20\times0.3=6$. As before, the frequency is given by the area of the bar, and its width is 12 (half the width of the full bar), so we get
$\text{estimate for beekeepers in “56 to 68” range }=12\times6=72$
Summing the two values together, our estimate of the total frequency of beekeepers who collected between 40 and 68kg of honey (the red section) is
$120+72=192$
## HP11 – Pie Charts
QUESTION: In one week, a bookstore sold 1,260 books. The pie chart shows information about the 3 different types of books sold. The angle for the paperback portion is 224\degree. The ratio of hardbacks to audiobooks is 3:1. Work out the number of audiobooks sold by this bookstore in one week.
ANSWER: We know that the formula for finding the angle is
$\text{angle }=\dfrac{\text{number in one category}}{\text{sum of all categories}}\times 360$
This time we know the angle (224), and the sum of all categories (1,260). So, the equation becomes
$224\degree =\dfrac{\text{paperbacks sold}}{1,260}\times360$
Divide by 360 and then multiply by 1,260 to get
$\text{paperbacks sold}=\dfrac{224}{360}\times1,260=784$
The number of paperbacks sold is 784, so the number of other books sold is $1,260-784=476$. The ratio of hardbacks:audiobooks is 3:1, so audiobooks constitute 1 part out of 4 in the ratio. Therefore, we get
$\text{audiobooks sold }=\dfrac{476}{4}=119$
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2019-05-23 17:14:25
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http://clay6.com/qa/28592/among-the-following-the-compound-that-can-be-most-readily-sulphonated-is
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Browse Questions
Among the following, the compound that can be most readily sulphonated is
(a) Benzene (b) Nitro benzene (c) Toluene (d) Chloro benzene
$-CH_3$ group that has +1 effect and due to its presence, toluene has highest electron density in the benzene ring.
For this reason Toluene can be easily sulphonated.
Hence c is the correct answer.
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2017-04-27 11:07:02
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http://mathoverflow.net/revisions/8103/list
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2 fixed terminology
Here is one set of data that will be sufficient. To get the monoidal structure you don't actually need a ("bilinear") monoidal) functor $C \to C \boxtimes C$. It is sufficient to have a bimodule category M from C to $C \boxtimes C$. You will also need a counit $C \to Vect$, and these will need to give C the structure of a (weak) comonoid in the 3-category of tensor categories, bimodule categories, intertwining functors, and natural transformations.
To compute what the induced tensor product does to two given module categories you will have to "compose" the naive tensor product with this bimodule category. This can be computed by an appropriate (homotopy) colimit of categories. It is basically a larger version of a coequalizer diagram. This is right at the category number where you will start to see interesting phenomena from the "homotopy" aspect of this colimit, which I think explains the funny behavior you're noticing with regards to bases.
Finally, you may get a braiding by having an appropriate isomorphism of bimodule categories,
$M \circ \tau \Rightarrow M$
which satisfies the obvious braiding axioms. Here $\tau$ is the usual "flip" bimodule.
1
Here is one set of data that will be sufficient. To get the monoidal structure you don't actually need a ("bilinear") functor $C \to C \boxtimes C$. It is sufficient to have a bimodule category M from C to $C \boxtimes C$. You will also need a counit $C \to Vect$, and these will need to give C the structure of a (weak) comonoid in the 3-category of tensor categories, bimodule categories, intertwining functors, and natural transformations.
To compute what the induced tensor product does to two given module categories you will have to "compose" the naive tensor product with this bimodule category. This can be computed by an appropriate (homotopy) colimit of categories. It is basically a larger version of a coequalizer diagram. This is right at the category number where you will start to see interesting phenomena from the "homotopy" aspect of this colimit, which I think explains the funny behavior you're noticing with regards to bases.
Finally, you may get a braiding by having an appropriate isomorphism of bimodule categories,
$M \circ \tau \Rightarrow M$
which satisfies the obvious braiding axioms. Here $\tau$ is the usual "flip" bimodule.
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2013-05-24 02:42:47
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https://teachingcalculus.com/2013/02/22/error-bounds/
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# Error Bounds
Whenever you approximate something you should be concerned about how good your approximation is. The error, E, of any approximation is defined to be the absolute value of the difference between the actual value and the approximation. If Tn(x) is the Taylor/Maclaurin approximation of degree n for a function f(x) then the error is $E=\left| f\left( x \right)-{{T}_{n}}\left( x \right) \right|$. This post will discuss the two most common ways of getting a handle on the size of the error: the Alternating Series error bound and the Lagrange error bound.
Both methods give you a number B that will assure you that the approximation of the function at $x={{x}_{0}}$ in the interval of convergence is within B units of the exact value. That is,
$\left( f\left( {{x}_{0}} \right)-B \right)<{{T}_{n}}\left( {{x}_{0}} \right)<\left( f\left( {{x}_{0}} \right)+B \right)$
or
${{T}_{n}}\left( {{x}_{0}} \right)\in \left( f\left( {{x}_{0}} \right)-B,\ f\left( {{x}_{0}} \right)+B \right)$.
Stop for a moment and consider what that means: $f\left( {{x}_{0}} \right)-B$ and $f\left( {{x}_{0}} \right)+B$ are the endpoints of an interval around the actual value and the approximation will lie in this interval. Ideally, B is a small (positive) number.
Alternating Series
If a series $\sum\limits_{n=1}^{\infty }{{{a}_{n}}}$ alternates signs, decreases in absolute value and $\underset{n\to \infty }{\mathop{\lim }}\,\left| {{a}_{n}} \right|=0$ then the series will converge. The terms of the partial sums of the series will jump back and forth around the value to which the series converges. That is, if one partial sums is larger than the value, the next will be smaller, and the next larger, etc. The error is the difference between any partial sum and the limiting value, but by adding an additional term the next partial sum will go past the actual value. Thus for a convergent alternating series the error is less than the absolute value of the first omitted term:
$\displaystyle E=\left| \sum\limits_{k=1}^{\infty }{{{a}_{k}}}-\sum\limits_{k=1}^{n}{{{a}_{k}}} \right|<\left| {{a}_{n+1}} \right|$.
Example: $\sin (0.2)\approx (0.2)-\frac{{{(0.2)}^{3}}}{3!}=0.1986666667$ The absolute value of the first omitted term is $\left| \frac{{{(0.2)}^{5}}}{5!} \right|=0.26666\bar{6}\times {{10}^{-6}}$. So our estimate should be between $\sin (0.2)\pm 0.266666\times {{10}^{-6}}$ (that is, between 0.1986666641 and 0.1986719975), which it is. Of course, working with more complicated series, we usually do not know what the actual value is (or we wouldn’t be approximating). So an error bound like $0.26666\bar{6}\times {{10}^{-6}}$ assures us that our estimate is correct to at least 5 decimal places.
The Lagrange Error Bound
Taylor’s Theorem: If f is a function with derivatives through order n + 1 on an interval I containing a, then, for each x in I , there exists a number c between x and a such that
$\displaystyle f\left( x \right)=\sum\limits_{k=1}^{n}{\frac{{{f}^{\left( k \right)}}\left( a \right)}{k!}{{\left( x-a \right)}^{k}}}+\frac{{{f}^{\left( n+1 \right)}}\left( c \right)}{\left( n+1 \right)!}{{\left( x-a \right)}^{n+1}}$
The number $\displaystyle R=\frac{{{f}^{\left( n+1 \right)}}\left( c \right)}{\left( n+1 \right)!}{{\left( x-a \right)}^{n+1}}$ is called the remainder.
The equation above says that if you can find the correct c the function is exactly equal to Tn(x) + R. Notice the form of the remainder is the same as the other terms, except it is evaluated at the mysterious c. The trouble is we almost never can find the c without knowing the exact value of f(x), but; if we knew that, there would be no need to approximate. However, often without knowing the exact values of c, we can still approximate the value of the remainder and thereby, know how close the polynomial Tn(x) approximates the value of f(x) for values in x in the interval, i.
Corollary – Lagrange Error Bound.
$\displaystyle \left| \frac{{{f}^{\left( n+1 \right)}}\left( c \right)}{\left( n+1 \right)!}{{\left( x-a \right)}^{n-1}} \right|\le \left( \text{max}\left| {{f}^{\left( n+1 \right)}}\left( x \right) \right| \right)\frac{{{\left| x-a \right|}^{n+1}}}{\left( n+1 \right)!}$
The number $\displaystyle \left( \text{max}\left| {{f}^{\left( n+1 \right)}}\left( x \right) \right| \right)\frac{{{\left| x-c \right|}^{n+1}}}{\left( n+1 \right)!}\ge \left| R \right|$ is called the Lagrange Error Bound. The expression $\left( \text{max}\left| {{f}^{\left( n+1 \right)}}\left( x \right) \right| \right)$ means the maximum absolute value of the (n + 1) derivative on the interval between the value of x and c. The corollary says that this number is larger than the amount we need to add (or subtract) from our estimate to make it exact. This is the bound on the error. It requires us to, in effect, substitute the maximum value of the n + 1 derivative on the interval from a to x for ${{f}^{(n+1)}}\left( x \right)$. This will give us a number equal to or larger than the remainder and hence a bound on the error.
Example: Using the same example sin(0.2) with 2 terms. The fifth derivative of $\sin (x)$ is $-\cos (x)$ so the Lagrange error bound is $\displaystyle \left| -\cos (0.2) \right|\frac{\left| {{\left( 0.2-0 \right)}^{5}} \right|}{5!}$, but if we know the cos(0.2) there are a lot easier ways to find the sine. This is a common problem, so we will pretend we don’t know cos(0.2), but whatever it is its absolute value is no more than 1. So the number $\left( 1 \right)\frac{\left| {{\left( 0.2-0 \right)}^{5}} \right|}{5!}=2.6666\bar{6}\times {{10}^{-6}}$ will be larger than the Lagrange error bound and our estimate will be correct to at least 5 decimal places.
This “trick” is fairly common. If we cannot find the number we need, we can use a value that gives us a larger number and still get a good handle on the error in our approximation.
FYI: $\displaystyle \left| -\cos (0.2) \right|\frac{\left| {{\left( 0.2-0 \right)}^{5}} \right|}{5!}\approx 2.61351\times {{10}^{-6}}$
Corrected: February 3, 2015
## 3 thoughts on “Error Bounds”
1. Jim says:
Your notation is a little confusing.
In the expression
max |f(x)(n+1)| |x-a|^(n+1)
the first occurence of “x” is bound by the max operator but the second occurence is free. You don’t indicate the scope of the max operator or the domain of the bound variable. In any event it is potentially confusing to use the same variable letter in an expression both as a free variable and a bound variable.
Then later you substitute the constant cos(.2) into both occurences of “x”. But the first occurence is bound. You can’t substitute into a bound occurence of a variable.
Like
• Jim
I made a correction to the post to make clear that $\left( \text{max}\left| {{f}^{\left( n+1 \right)}}\left( x \right) \right| \right)$ refers to the maximum of the absolute value of the (n + 1) derivative.
As for the (0.2) substitution it was just for purposes of the example. In this example the maximum value of |-cos(x)| occurs at 0.2, but it is not necessary to know this, since, as usual, we will end up substituting a larger value, namely $\cos \left( \tfrac{\pi }{2} \right)=1$.
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2020-08-05 07:31:37
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http://gmatclub.com/forum/on-the-day-of-the-performance-of-a-certain-play-each-ticket-1717.html
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Find all School-related info fast with the new School-Specific MBA Forum
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# On the day of the performance of a certain play, each ticket
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On the day of the performance of a certain play, each ticket [#permalink] 27 Jul 2003, 08:16
. On the day of the performance of a certain play, each ticket that regularly sells for less than $10.00 is sold for half price plus$0.50, and each ticket that regularly sells for $10.00 or more is sold for half price plus$1.00. On the day of the performance, a person purchases a total of y tickets, of which x regularly sell for $9.00 each and the rest regularly sell for$12.00 each. What is the amount paid, in dollars, for the y tickets ?
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X tickets for (9/2)+0.5=$5 Y-X tickets for (12/2)+1=$7
Total revenue is 5X+7(Y-X)=7Y-2X
Am I missing something?
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5 For a recent play performance, the ticket prices were $25 pe 5 31 Dec 2013, 06:21 1 On the day of the performance of a certain play, each ticket that regu 7 07 Apr 2011, 05:59 For a recent play performance, the ticket prices were$25 2 08 Sep 2009, 12:29
For a recent play performance, the ticket prices were $25 1 06 Oct 2008, 12:50 For a recent play performance, the tickets prices were$2 5 07 Mar 2008, 21:29
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2015-05-23 15:19:00
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https://epione.gitlabpages.inria.fr/flhd/federated_learning/introduction.html
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# Introduction¶
Standard machine learning approaches require to have a centralizaed dataset in order to train a model. In certain scenarios like in the biomedical field, this is not straightforward due to several reasons like:
This slows down research in healthcare and limits the generalization of certain models.
## Federated Learning¶
Federated learning (FL) is a machine learning procedure whose goal is to train a model without having data centralized. The goal of FL is to train higher quality models by having access to more data than centralized approaches, as well as to keep data securely decentralized.
### Infrastructure of a federated learning setting in healthcare¶
A common scenario of federated learning in healthcare is shown as follows:
Hospitals (a.k.a. clients) across several geographical locations hold data of interest for a researcher. These data can be “made available” for local training but, only the model is authorized to be shared with a third thrusted party (e.g. research center). Once all the models are gathered, different techniques are proposed for aggregating them as a single global model. Then, the Aggregated model can be used as purposed (e.g. training a neural network for segmentation).
### Theoretical background¶
One of the critical points in FL is knowing how to aggregate the models submitted by the clients. The main problem relies on finding the best set of parameters that define your model in function of the submissions made by the clients.
In a canonical form:
$\min_w F(w) ,\quad \textrm{where} F(w):=\sum_{k=1}^{m} p_k F_k(w)$
Where $$m$$ is the total number of clients, $$p_k>=0$$, and $$\sum_k p_k=1$$ , and $$F_k$$ is the local objective function for the $$k$$-th client. The impact (contribution) of each client to the aggregation of the global model is given by $$p_k$$.
One of the first proposed methodologies in FL for model aggregation was Federated Averaging FedAVG by (MacMahan et al, 2016), the idea behind it was to define the contribution of each client as $$p_k=\frac{n_k}{n}$$ where $$n_k$$ is the number of datapoints in the client $$k$$ and $$n$$ is the total number of observations studied.
### Challenges in federated learning¶
The main challenges in FL are associated to:
• Communication efficiency: number of iterations between clients and central location to train an optimal model.
• Data heterogeneity: how to build generalized models with heterogeneous data?
• Security: adversarial attacks and data leakage.
## References¶
1. Konečný, J., McMahan, et al. (2016). Federated learning: Strategies for improving communication efficiency. arXiv preprint arXiv:1610.05492.
2. Li, T., Sahu, et al. (2018). Federated optimization in heterogeneous networks. arXiv preprint arXiv:1812.06127.
3. Li, T., Sahu, A. K., Talwalkar, A., & Smith, V. (2020). Federated learning: Challenges, methods, and future directions. IEEE Signal Processing Magazine, 37(3), 50-60.
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2021-03-04 14:41:12
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http://mathoverflow.net/revisions/64269/list
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7 deleted 762 characters in body
If a real Banach space is all-equal, I think that it is finite dimensional and inner-product. This can be proved (at least in the finite dimensional case) using the John ellipsoid (see http://mathoverflow.net/questions/41211/easy-proof-of-the-fact-that-isotropic-spaces-are-euclidean), even though I don't like to introduce such a cumbersome structure.
If a Riemannian manifold is all-equal, then it is homogeneous and with constant curvature (how do we deal with the fundamental group?). If a Finsler manifold is all-equal, then I would expect it to be Riemannian (again using the John ellipsoid). I have no idea how to proceed without differential structure, and I really don't have any intuition of how the characterization could be true (only hope).
6 deleted 510 characters in body
If $x$, $z$ are points x$is a point and$a$, a$ and $b$, b$are lengthssuch that$a+b>d(x,z)$, then there exists$y$such that$a>d(x,y)$and$b>d(y,z)$. Which is similar to saying that the inner metric induced by the distance is not greater than the distance itself (but I can't find right now the definition of the induced inner metric). After this tweaking, it is easy to construct approximate midpoints, then midpoints (using completeness), then segments, then complete geodesics$N_b(N_a({x}))=N_{a+b}({x})$(a segment can be extended by moving the staring point to the midpoint and the midpoint to where the end). Then each pair$N$stands for open neighborhood of points can be joined by a geodesic of the expected lengthset). 5 added 326 characters in body I want to obtain a metric characterization of the classical finite dimensional spaces of Euclidean geometry. Motivation: Suppose$A$and$B$live in an$n$-dimensional Euclidean space. They are each assigned the task of constructing an equilateral triangle of side length 5. Subject$A$finds first one point$p_1$. If$n>0$, he can then find another point$p_2$at distance 5 from$p_1$. If$n>1$, he can then find$p_3$at distance 5 from$p_1$and$p_2$. Then$B$proceeds similarly. He selects a point$q_1$(probably different from$p_1$), then finds another point$q_2$and$q_3$. We wouldn't expect him to get stuck before constructing$q_3$, since$n$is the same for both subjects. I will say a space is all-equal if the possibility of completing a figure is independent of the starting points chosen. Are Euclidean spaces all-equal? Are there non Euclidean all-equal spaces? Precise statement Definition: A metric space$X$is all-equal if every time$S$and$T$are isometric subspaces of$X$(with a selected isometry$S\to T$), the isometry extends to an automorphism of$X$. Question 0: Is every Euclidean space all-equal? To state the second question we must first observe some limitations. Being connected our life intervals, we cannot visit more than one connected component of our space, and being imprecise our measurements, we cannot distinguish directly between a space and its completion. Finally, existing no natural unit of measurement, the correct category to state these questions is that of spaces in which the distance is defined up to scale. That is, it takes values in a 1-dimensional module$M$over$[0,+\infty)$, with no multiplication structure (although lengths can be tensorised to obtain areas). More definitions: A congruence space is a pair$(X,M,d)$where$X$is a set,$M$is a 1-dimensional$[0,+\infty)$module and d is a distance in$X$taking values in$M$. The morphisms from$(X,M,d)$to$(Y,N,e)$will be the subsimilarities, that is, the pairs$(f,\alpha)$where$f$is a function from$X$to$Y$and$\alpha$is a morphism from$M$to$N$such that for each$x$,$x'$in$X$we have$e(fx,fx')\leq\alpha(d(x,x'))$. Similarities are similarly defined using an$=$sign instead of$\leq$. Subspaces are defined by restriction, and the identity of$(X,M,d)$is the obvious$(id_X,id_M)$. It's easy to prove that every isomorphism is a similarity. A congruence space$X$is all-equal if every time$S$and$T$are similar subspaces, the similarity extends to an automorphism of$X$. Question 1: Is every connected complete all-equal space a finite dimensional real inner-product space? Remarks: if the connectedness hypothesis is dropped, some discrete spaces would be all-equal. If we work in the usual category of metric spaces, projective and hyperbolic spaces could be all-equal, and also Euclidean spheres, both with their inner metric and with the subspace metric. Partial results and lines of thought: If a real Banach space is all-equal, I think that it is finite dimensional and inner-product. This can be proved (at least in the finite dimensional case) using the John ellipsoid (see http://mathoverflow.net/questions/41211/easy-proof-of-the-fact-that-isotropic-spaces-are-euclidean), even though I don't like to introduce such a cumbersome structure. If a Riemannian manifold is all-equal, then it is homogeneous and with constant curvature (how do we deal with the fundamental group?). If a Finsler manifold is all-equal, then I would expect it to be Riemannian (again using the John ellipsoid). I have no idea how to proceed without differential structure, and I really don't have any intuition of how the characterization could be true (only hope). EDIT: As noted by Sergei, there are counterexamples. Generally, if$(X,d)$is an all-equal metric space, and$p>1$, then$(X,d^{\frac 1p})$could be an all-equal space. This is similar to the example in which However, I was thinking of spaces in which the distance from$x$to$y$is measured using a signal that travels form$x$to$y$at constant speed. Hence I require that the space have the following property: If$x$,$z$are points and$a$,$b$, are lengths such that$a+b>d(x,z)$, then there exists$y$such that$a>d(x,y)$and$b>d(y,z)\$.
Which is similar to saying that the inner metric induced by the distance is not greater than the distance itself (but I can't find right now the definition of the induced inner metric).
After this tweaking, it is easy to construct approximate midpoints, then midpoints (using completeness), then segments, then complete geodesics (a segment can be extended by moving the staring point to the midpoint and the midpoint to the end). Then each pair of points can be joined by a geodesic of the expected length.
Related MO questions: http://mathoverflow.net/questions/47882/characterizations-of-euclidean-space http://mathoverflow.net/questions/41211/easy-proof-of-the-fact-that-isotropic-spaces-are-euclidean http://mathoverflow.net/questions/8513/characterization-of-riemannian-metrics
4 edited body; deleted 4 characters in body
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2013-05-22 12:01:18
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http://crypto.stackexchange.com/questions?page=3&sort=votes
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# All Questions
3k views
### Why is plain-hash-then-encrypt not a secure MAC?
It seems that even in MAC-then-encrypt systems like SSL, something like HMAC is used rather than a plain hash. Why? Suppose we use some stream cipher; then why can't we use $Encrypt(m | H(m))$ as ...
1k views
### Purpose of outer key in HMAC
From what I know, the HMAC constructions has two strength: It's resistant to length extensions Since the key is consumed before the message, the attacker does not know the initial state, preventing ...
7k views
### How can SSL secure a two-way communication with only one key-pair?
As I understand it, SSL involved the use of a public-private key pair. How does this enable two-way communication? Suppose I have some server with which I wish to communicate securely. I connect to ...
6k views
### Current mathematics theory used in cryptography/coding theory
What are the mainstream techniques borrowed from algebraic geometry (or some other branch of mathematics) which are currently used in cryptography/coding theory? I've only heard about a small subset ...
2k views
### Is every output of a hash function possible?
Is every output of a hash function (e.g. SHA1, MD5, etc) guaranteed to be possible, or, conversely, are there any output values that cannot possibly be created from any input? If so, what guarantees ...
NIST SP 800-57 §5.6.1 p.62–64 specifies a correspondence between RSA modulus size $n$ and expected security strength $s$ in bits: ...
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2016-06-30 21:13:19
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http://hal.in2p3.fr/in2p3-00155193
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# The hadronic contribution to (g-2) of the muon
Abstract : The evaluation of the hadronic contribution to the muon magnetic anomaly $a_\mu$ is revisited, taking advantage of new experimental data on $e^+e^-$ annihilation into hadrons: SND and CMD-2 for the $\pi^+\pi^-$ channel, and \babar for multihadron final states. Discrepancies are observed between KLOE and CMD-2/SND data, preventing one from averaging all the $e^+e^-$ results. The long-standing disagreement between spectral functions obtained from $\tau$ decays and $e^+e^-$ annihilation is still present, and not accounted by isospin-breaking corrections, for which new estimates have been presented. The updated Standard Model value for $a_\mu$ based on $e^+e^-$ annihilation data is now reaching a precision better than experiment, and it disagrees with the direct measurement from BNL at the 3.3$\sigma$ level, while the $\tau$-based estimate is in much better agreement. The $\tau$/$e^+e^-$ discrepancy, best revealed when comparing the measured branching fraction for $\tau^- \to \pi^- \pi^0 \nu_\tau$ to its prediction from the isospin-breaking-corrected $e^+e^-$ spectral function, remains a serious problem to be understood.
Document type :
Conference papers
http://hal.in2p3.fr/in2p3-00155193
Contributor : Sabine Starita <>
Submitted on : Friday, June 15, 2007 - 5:08:57 PM
Last modification on : Thursday, January 11, 2018 - 6:14:15 AM
### Citation
M. Davier. The hadronic contribution to (g-2) of the muon. Tau06 International Workshop, Sep 2006, Pisa, Italy. pp.288-296, ⟨10.1016/j.nuclphysbps.2007.03.023⟩. ⟨in2p3-00155193⟩
Record views
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2019-04-24 15:08:58
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https://mindmatters.ai/2019/10/yes-you-can-manipulate-infinity-in-math/
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Mind Matters Natural and Artificial Intelligence News and Analysis
# Yes, You Can Manipulate Infinity—in Math
The hyperreals are bigger (and smaller) than your average number — and better!
Most of us learned the basic number systems in high school—integers (positive and negative whole numbers), fractions (ratios of numbers), real numbers (all those infinitely-continuing decimals), and maybe even complex numbers (with the evil letter “i” lurking around and causing trouble). For most everyday math, these numbers work really well.
However, some types of numbers are not well represented by the real numbers. The real numbers don’t handle numbers that are infinitely large or their inverses—numbers that are infinitely small. At most, the real numbers are extended with the positive infinity (∞ or + ∞) or negative infinity (- ∞) symbols to denote numbers too large or too small for the system to handle. However, the rules of the real numbers prevent any actual manipulation of these symbols. For instance, the relationship ∞ – ∞ is not zero, but undefined.
The hyperreal number system extends the reals so as to handle infinity precisely. Note that there are actually multiple ways to handle infinitely large numbers. But I have found one particular system to be of more practical use than other systems, which is why I think it is worth discussing. Thinking about infinities is somewhat mind-bending, but it turns out that actually manipulating infinities with the hyperreal system is incredibly easy if you are familiar with basic algebra.
For the hyperreals, a new number is introduced to represent infinity. Because the hyperreals are new, there is no universal standard for what the notation for infinity should be, so I usually adopt a lowercase Greek omega (ω). So what is ω, exactly? You should think of ω as a kind of unit but, instead of representing a physical quantity (like a foot or a mile or a kilogram), it represents a numerical quantity that is an infinity.
As we will see, there are many different infinities. So, if we can’t count to infinity (who has the time?) and there are multiple infinities, which particular one is ω? As it turns out, the particular infinity it refers to doesn’t really matter, provided that we are consistent about it. That is, we can say that ω represents some particular infinity. We don’t have to know which one, we just have to agree that it won’t change.
If you need a specific number to imagine, think of a 1 with an infinite number of zeroes following it. Once you have an infinite unit like ω, you can do a lot with it. We can multiply infinity by two and have 2 ω. We can add one to it and have ω + 1. These are all hyperreal numbers. We can even square the infinity and have ω2. What on earth is ω2? Think of it this way: If I have the number 100 (two zeroes at the end) and I multiply it by 100 (two zeroes at the end), I will get the number 10,000 (four zeroes at the end). So, if I take the number ω and multiply it by ω, I will get ω2, which you can think of as having two infinite sets of zeroes at the end.
So, like ordinary mathematical symbols, hyperreal numbers can be added, subtracted, multiplied, and divided. ω + ω is 2ω. ω x ω is ω2. ω73 is ω4. You can basically treat ω as if it were x, with the exception that it has a few special properties because it is infinite.
Real numbers are basically numbers which have no ω component. Alternatively, you can think about them as being multiplied by ω0 (since anything raised to the zero power is 1).
What if we divide by ω? What is 1/ω or ω-1? This is known as an infinitesimal, an infinitely small value. Because infinitesimals are used so often, it is useful to have a special symbol to refer to them, the lowercase Greek epsilon (ε). If I have, say, 5 + ε, I am referring to a number that is infinitely close to 5. That is, there is no real number that is between 5 and 5 + ε.
Just as we pictured ω as a 1 with an infinite number of zeroes after it, you can picture ε as a decimal looking like 0.0000001, except that there is an infinite number of zeroes between the decimal place and the 1. Likewise, if you have ε2, you can think of it as a number that has two infinite sets of zeroes between the decimal place and the 1.
So, for example, 5 + ε + 7 ε2 is a hyperreal number that is infinitely close to 5 but is slightly further out on the number line than 5 + ε (but not by much!).
How does this help us do math and solve problems? It turns out that the hyperreal number system greatly simplifies several aspects of mathematics. As a simple example, let’s say we have the following equation:
Let’s say that we wanted to find out what the value of this equation was at x = 5. What would we get? We would wind up with:
Zero divided by zero is an unknown value. So, if we can’t figure out what this is at x = 5, what if we bumped x just a little bit? What if we looked at x = 5.01? It turns out that this expression only has a problem at exactly x = 5; any other number works perfectly fine. However, with the real numbers, we would have to modify x a small, finite amount to get an answer. However, that will be inexact because we could always choose a value a little closer to 5. For instance, we could have looked at 5.0001 or 5.0000001. But, with hyperreals, we can bump x by an infinitely small amount instead. If we calculate the value at 5 + ε, the calculation will work out to:
So, the result of this is 10 + ε, which is infinitely close to the number 10. To be explicit, you can use the “standard part” function std to “round” the value to the nearest real number. In other words, std(10 + ε) = 10. Therefore, for the given fraction, we can say, for all practical purposes, when x = 5, the result is 10. This is not impossible to do with real numbers. The concept of a “limit” in the real numbers is very similar to this process. However, limits rely on a lot of proofs and additional rules that are essentially unnecessary when you use hyperreal numbers. With hyperreal numbers, you can simply calculate the values directly.
In short, hyperreal numbers are a new type of number that was developed to simplify and rethink the way that we deal with very large and very small numbers. It reduces the complexity of the task and allows us to use our well-honed high-school algebra skills to solve complex problems easily.
If you enjoyed this discussion of hyperreal numbers, you may also enjoy:
Don’t leave home without these three curves: Three mathematical curves explain a lot of what happens—and doesn’t happen—in everyday life. (Jonathan Bartlett)
and
Things exist that are unknowable. A tutorial on Chaitin’s number (Robert J. Marks)
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2022-05-28 22:49:40
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http://www.ams.org/mathscinet-getitem?mr=1813145
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MathSciNet bibliographic data MR1813145 (2002b:41011) 41A15 (41A63 65D17) Johnson, Michael J. The \$L\sb 2\$$L\sb 2$-approximation order of surface spline interpolation. Math. Comp. 70 (2001), no. 234, 719–737 (electronic). Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.
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2013-12-20 01:28:52
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https://www.recuperafranchising.it/site/random-brownian-motion-simulation-32d9b8
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Since there are so many gas molecules in the air, it will constantly bump into other molecules (roughly $$10^{14}$$ hits per second - that equals the total number of Google searches performed worldwide during 79 years!) 0000043277 00000 n If you get $$5$$ or $$6$$, roll again. By definition, $$B_{0.1} - B_0$$ is normally distributed with variance $$0.1$$, so generate one such number and let that be the value of $$B_{0.1}$$. delta : float delta determines the "speed" of the Brownian motion. 0000038859 00000 n What does it mean for something to be random and how can a surface grow randomly? This is an equation that can be solved, so we are able to predict something with certainty from a random model - this is an example of the strategy that is used in statistical mechanics. First, we want to try to model how this gas molecule moves in the simplest possible way, and you will explore one of these models in the following exercise. Central Limit Theorem, https://en.wikipedia.org/wiki/Central_limit_theorem. 0000004689 00000 n J. Matson, ‘‘Crowd Forcing: Random Movement of Bacteria Drives Gears’’, https://www.scientificamerican.com/article/brownian-motion-bacteria/. position(s)) of the Brownian motion. To see a larger example, the following is a two-dimensional random walk generated in the same way as the exercise. Imagine a gas molecule in the air: it moves around on its own until it hits another gas molecule which makes it change direction. Exercise: random motion from coin tosses and dice rolls. }); On this page, you will learn about random walks and Brownian motion. $$B_0$$ is defined to be $$0$$. 0000002208 00000 n ), but is more realistic. x�bbScc�c@ �;�f�=��OY%�H'��20�w}��� �YȺ�m���\�¬��f��ml����g�ފ_:�sNԫ&m=�-�0s�r^�������rm��1H�+c��L�s��g�+�h��땣n$�.Qs��mTP�Pe����5=b���2����)�[-i7��,Zv���Daa�U��[eN��6�������:�GR���f�5�-@��!=b�:��zy����I.g�Xeh$�ꅶ?o�W�������^gRR6��4 H@FAacc,���R��P&(���0!A0�0�J% Z�9@ �������8,b����#�YL���?�7�N0i��pĬ����+�6p��h�\����e�����M~hf��� �a��\$�Cŏ����S(�U`x�(���f���� V. 0000051392 00000 n Brownian motion is a stochastic model in which changes from one time to the next are random draws from a normal distribution with mean 0.0 and variance σ 2 × Δt. d): What way do you think would be a good way of measuring how far the random walk has gone from the origin? Now, flip a coin. 0000001260 00000 n 0000050808 00000 n $$B_{0.2} - B_{0.1}$$ is again normally distributed with variance $$0.1$$, so generate one such number and add that to $$B_{0.1}$$ to get the value of $$B_{0.2}$$. 0000041780 00000 n import random import math import numpy as np from functools import partial from bokeh.io import show, output_notebook from bokeh.layouts import row from bokeh.plotting import figure from bokeh.embed import notebook_div import plotly.plotly as py from plotly.graph _objs import * random. the commands 0000034623 00000 n 0000027959 00000 n This is a very simple model of how the gas molecule can move, but it is also close to reality! 0000016647 00000 n To learn more about this, see the references on the ‘‘central limit theorem’’ below. See the fact box below. 0000003052 00000 n 0000051247 00000 n 0000003992 00000 n If tails, mark a point one step ahead and one step below the previous one. If $$3$$, mark the one to the right, and if $$4$$, mark the one above. Continue this for a while and draw the resulting graph. 0000039997 00000 n 0000045738 00000 n Draw a coordinate system with time $$t$$ on the horizontal axis, and height $$h$$ on the vertical axis. ), but is more realistic. 0000001894 00000 n 0000023978 00000 n Mark the origin. Real gas molecules can move in all directions, not just to neighbors on a chessboard. Page generated 2017-05-18 14:49:26 EDT, by, https://www.scientificamerican.com/article/brownian-motion-bacteria/, http://www.feynmanlectures.caltech.edu/I_41.html, https://en.wikipedia.org/wiki/Central_limit_theorem, https://commons.wikimedia.org/wiki/File:Brownian_motion_large.gif, https://commons.wikimedia.org/wiki/File:Random_walk_25000.gif. Gas molecule (yellow) describing Brownian motion, Now, Einstein realized that even though the movements of all the individual gas molecules are random, there are some quantities we can measure that are not random, they are predictable and can be calculated. 0000012106 00000 n seed (10) output_notebook (hide_banner = True) In [2]: def brownian_path (N): Δt_sqrt = math. b): What is the average of $$h$$, as a function of time? The initial condition(s) (i.e. Write a program that continues this procedure! sqrt (1 / … 0000034137 00000 n 0000050488 00000 n Even though the motion is quick and jerky, the particle doesn't get very far for large times - just like for gas molecules! 0000003505 00000 n https://commons.wikimedia.org/wiki/File:Brownian_motion_large.gif. 0 0000016379 00000 n import random If you get a $$1$$, mark the square to the left of the previous square. 0000023469 00000 n \frac{\partial \rho}{\partial t} = D\frac{\partial^2 \rho}{\partial x^2}, To generate a Brownian motion, follow the following steps: we want to generate a brownian motion at times $$0, 0.1, 0.2, … , 1$$. 760 0 obj <> endobj We will use this in the next couple of pages to explain some models of randomly growing surfaces. In the beginning of the twentieth century, many physicists and mathematicians worked on trying to define and make sense of Brownian motion - even Einstein was interested in it! n : int The number of steps to take. startxref %PDF-1.4 %���� 0000046010 00000 n <<7C144B6214A5FD478B61CB26E07CCB2A>]>> 1 Simulating Brownian motion (BM) and geometric Brownian motion (GBM) ... Monte Carlo simulation can also be used to estimate other quantities of interest in nance that do not involve derivatives. a): We start with a one-dimensional motion. 0000000016 00000 n Is the average of $$h^2$$ better or worse? As $$N$$ tends to infinity, a random walk on this chessboard tends to a Brownian motion. As mentioned in the first lecture, the simplest model of Brownian motion is a random walk where the “steps” are random displacements, assumed to be IID random variables, between nearly instantaneous collisions. If $$2$$, mark the square below the previous one. 0000033741 00000 n 0000034756 00000 n randomNumber = random.gauss(0, $$s$$). 0000013071 00000 n To get started, the following is a simulation of a gas, and one particle is marked in yellow. 0000021254 00000 n To answer these questions, we will start more carefully and talk about random walks of particles. A realistic description of this is Brownian motion - it is similar to the random walk (and in fact, can be made to become equal to it. 0000039786 00000 n Simulating Brownian motion in R. This short tutorial gives some simple approaches that can be used to simulate Brownian evolution in continuous and discrete time, in the absence of and on a tree. The way the gas molecule moves will turn out to be important to studying randomly growing surfaces, so we will keep going on this track for a while! Draw a chessboard pattern around the origin, and roll a die. c): What is the average of $$h^2$$, as a function of time? Challenge question: Write a program that calculates Brownian motion at any set of times! 0000026123 00000 n 0000025874 00000 n called the diffusion equation, and where $$D$$ is the diffusion coefficient that can be calculated. 0000002585 00000 n https://commons.wikimedia.org/wiki/File:Random_walk_25000.gif. See the fact box below. 0000051538 00000 n 0000040436 00000 n Along with the Bernoulli trials process and the Poisson process, the Brownian motion process is of central importance in probability. 0000017088 00000 n 0000051617 00000 n 0000012578 00000 n The bumps therefore cancel each other out, so after a long time interval, it will barely have moved at all, even though it makes really quick jerks all the time. Feynman, ‘‘Feynman Lectures on Physics’’, http://www.feynmanlectures.caltech.edu/I_41.html. e): Next, you will draw a two-dimensional random walk. 0000039131 00000 n BROWNIAN_MOTION_SIMULATION, a MATLAB library which simulates Brownian motion in an M-dimensional region. %%EOF For example, suppose you invest in two di erent stocks, S 1(t) and S 2(t), buying N 1 shares of the rst and N 2 of the second. TeX: { equationNumbers: { autoNumber: "AMS" } } Calculate this in a table. We would therefore like to be able to describe a motion similar to the random walk above, but where the molecule can move in all directions. 0000026577 00000 n Its path describes a Brownian motion $$B_t$$ at time $$t$$. In the beginning of the twentieth century, many physicists and mathematicians worked on trying to define and make sense of Brownian motion - even Einstein was interested in it! 760 47 0000036146 00000 n Einstein's equation showed that diffusion processes, for instance seeing a drop of ink spread out in water, are caused by Brownian motion - the question we will ask for the next pages is: can Brownian motion explain also other random phenomena? and it will be just as likely to be hit from another particle on the left as it will be to be hit on the right. 0000046450 00000 n One such quantity is the density $$\rho$$ of the gas molecules. \]. 806 0 obj<>stream
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2022-05-26 11:30:46
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https://greprepclub.com/forum/if-the-average-of-p-and-4p-is-10-then-p-15670.html
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# If the average of p and 4p is 10, then p =
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If the average of p and 4p is 10, then p = [#permalink] 07 Nov 2019, 22:35
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If the average of $$p$$ and $$4p$$ is 10, then $$p =$$
(A) 1
(B) 3
(C) 4
(D) 10
(E) 18
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Re: If the average of p and 4p is 10, then p = [#permalink] 08 Nov 2019, 05:28
Expert's post
20/5=4
C
Too simple for a GRE question.
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Re: If the average of p and 4p is 10, then p = [#permalink] 08 Nov 2019, 05:28
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2019-11-12 18:59:31
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http://mathoverflow.net/questions/42706/post-correspondence-problem-variant/42721
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# post correspondence problem variant
Is there an algorithm which takes as input two lists of words $v_1,...,v_n$ and $w_1,...,w_n$ over an alphabet $X$ and decides if there is an infinite sequence $(k_i)$ where $1 \leq k_i \leq n$ for all $i$ such that $v_{k_1}v_{k_2}...=w_{k_1}w_{k_2}...$? It seems that undecidability of the original Post Correspondence problem should imply there is no such algorithm. Is there a reference that shows undecidability of this variation of Post? Thanks.
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According to this abstract, springerlink.com/content/x92705867jnu247w the Post correspondence problem is undecidable for doubly infinite words. This not quite what you want, but perhaps the method can be adapted to your problem. – John Stillwell Oct 18 '10 at 23:11
See Halava, Vesa, Harju, Tero, Karhumäki, Juhani Decidability of the binary infinite Post correspondence problem. If the alphabet consists of $\le 2$ letters, then the problem is decidable, if the number of letters is at least 7, then the problem is undecidable. The latter result is proved in Y. Matiyasevich, G. Sénizergues, Decision problems for semi-Thue systems with a few rules, in: Proceedings, 11th Annual IEEE Symposium on Logic in Computer Science, New Brunswick, NJ, 27–30 July 1996, IEEE Computer Society, Silver Spring, MD, pp. 523–531.
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2014-10-25 05:47:40
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http://zbmath.org/?q=an:0853.53069
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# zbMATH — the first resource for mathematics
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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.
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A ‘finite infinity’ version of topological censorship. (English) Zbl 0853.53069
The author gives a version of the topological censorship theorem of J. L. Friedmann, K. Schleich and D. M. Witt [Phys. Rev. Lett. 71, 1486-1489 (1993)] without the assumption of asymptotic flatness. He assumes that $\left(M,g\right)$ is a spacetime with timelike boundary $\partial M=T$, where $T$ is diffeomorphic to $ℝ×{S}^{2}$, and that for each $t$, ${{\Sigma }}_{t}:=\left\{t\right\}×{S}^{2}$ is spacelike. He further assumes that for each ${{\Sigma }}_{t}$, the null second fundamental forms corresponding to (any) inward (respectively, outward) pointing null vector field are negative (respectively, positive) definite. Each ${{\Sigma }}_{t}$ is supposed to be acausal in $M$ and the null convergence condition is assumed to hold. Under these conditions, global hyperbolicity of ${J}^{+}\left(T\right)\cap {J}^{-}\left(T\right)$ implies that this set is simply connected.
In the asymptotically flat case, there exist timelike tubes $T$ near infinity which satisfy the assumptions above and the theorem can be applied to the complement of the asymptotic region bounded by $T$.
He also shows that if the boundary of $M$ consists of several timelike tubes ${\left\{{T}_{\alpha }\right\}}_{\alpha }$ and $\left(M,g\right)$ is globally hyperbolic, then ${J}^{+}\left({T}_{a}\right)\cap {J}^{-}\left({T}_{b}\right)=\varnothing$, provided $a\ne b$. A possible interpretation of this theorem is that in globally hyperbolic spacetimes there are no wormholes connecting different asymptotic regions.
##### MSC:
53Z05 Applications of differential geometry to physics 83C75 Space-time singularities, cosmic censorship, etc.
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2014-04-24 21:35:56
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https://forum.knifetalk.net/discussion/comment/324
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## Small mill for scales
• 78
Thinking about getting a small mill for flattening and squaring off - anybody have any recommendations for something not too big?
No room (or budget) for a Bridgeport unfortunately :meh:
• 1
I'm super interested in this also. I've looked at some smaller mills, but not a lot of great reviews.
• 2
Hi, I have also been looking into getting a milling machine. I am leaning toward the Grizzly Benchtop G0761, but the $2000 tag is a bit steep (especially when you add in the cost for tooling to go along with it). If all you are going to be doing is squaring handle materials, you could probably go with one of the even smaller benchtop mills that come in around$700(ish).
• 47
How small? There are a bunch of chinese ones on Ali Express for what seem to be pretty reasonable prices. I suppose, with those, you run the risk of potentially unvetted electronic components. Maybe keep it unplugged when not in use? My buddy ended up with some no-name China export pseudo benchtop mill/drill press deal. He uses it to machine incomplete rifle lowers with pretty decent success. He's not chasing .0005 accuracy, though, so I can't speak to the little machine's repeatability.
• 78
Been thinking of getting a cheap Chinese machine but never actually seen one - wondering if quality is an issue
• 47
Just from looking, it appears that the majority of the ali express ones are the same machine with different paint/stickers. That seems to be pretty typical on Ali Express ( it is for guitars, anyhow. I've bought several off the site and there is definitely a commonality. Some large factory is making them for a bunch of retailers). The downside, in my experience, has been that you just gotta buy one to see if its shit or not. I've been fortunate with guitar purchases in that I know what I want and I know what I'm looking at. I'm able to tell pretty quickly if something is gonna be a turd. Not the case with these machines. Not for me, anyhow. I bet a machinist could look the specs and photos over to see if they're worth even considering.
• 78
Just wondering - this seems nice & small, do you guys think this would be powerful enough to work with wood (for scales)?
• 21
@Chop Knives evening Craig, ive been looking for a long while for a decent cheapish mill and found one of the major things to consider is the rear post, the cheaper mills seem to be flimsy and in turn is inaccurate due to it flexing, especially the meatier cut you make.
• 78
I know what you mean Dan. I’m hoping as most are designed for milling alu & mild steel that wood should be a breeze.
I know I need to drop a grand for something solid but I think it would be overkill - and I could do with finding something more affordable.
• 21
@Chop Knives keep your eyes on this site, they have some great machines pop up, even get in touch with them and tell them what you are after
http://fusies.co.uk/
• 78
that’s a great find!
• 1
I own one of these mills, for the money I think its great. If you are only doing jobs with fairly soft materials it will do fine. I am thinking of having a go milling fullers but not sure how it will get along with steel, but as long as I only do light passes I feel it should be ok. And having the larger table is a bonus. Also Arc euro trade are a great company to deal with in my experience.
• 47
@Chop Knives https://www.harborfreight.com/two-speed-variable-bench-mill-drill-machine-44991.html
I doubt this is available in Europe, but I got some comments from a buddy who has one of these. He mostly uses it for messing around milling receivers on rifles. He said, essentially, that he isn't bummed that he spent the money on it, but he's not real excited about it, either. He added a machinist vise to the table and it does ok for him. I would think it would be fine at just milling the sides and faces of handle material flat.
• 2
Craig, what about something like a surface grinder attachment for your grinder and some sort of vise or clamping attachment to hold scales on it? I heard this idea somewhere recently, I forget where though. Nice thing would be you might get a surface grinder attachment with the ability to flatten scales for around what you’d spend on a mill. Maybe this sort of vise would sit on the magnetic chuck? https://www.robotshop.com/en/carbide3d-low-profile-vise-nomad.html?gclid=Cj0KCQiA1sriBRD-ARIsABYdwwFlDeJgX_sPKRHr3x-2izGB-a5hu_PY_vLxnz9PTVEXVnS1xr1jkyYaAvd9EALw_wcB
Edit: looks like a small toolmakers vise and a cheap set of thin parallels might do the job, and there are cheaper options for toolmakers buses than what I linked to, though that one is nice and looks like a convenient size.
• 2
Craig have you looked at CNC routers for this application? In addition to flattening the scale material you can automate some of the profiling and shaping as well. There are quite a few options out there such as Shapeoko, XCarve, or Nextwave Automation. They’re available in a variety of sizes and price points.
I just purchased the Piranha FX from Nextwave and I looking forward to getting it set up over the next few weeks. I’ll be sure to post some info here once I’m up and running.
• 1
https://littlemachineshop.com/products/product_category.php?category=1387807683
I've owned the Fixed Column mini mill for 5 years and have loved it. We converted to cnc with the cnc fusion kit and its been a wonderful machine.
• 43
I saw a video on youtube, but I cannot remember who it was. But in any case, he used a small router table with a cheap trim router mounted in it, with a small round over bit in it. The bearing ran on the center of the knife and rounded the edges beautifully. Might be with looking into?
• 0
Little machine shop has really great mills
• 14
My mc mechanic is a machinist. Reviews and he told me not to buy the small ones. Im getting a medium sized used one when I find one. They need weight to get decent results. Oh, all them tools....
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2019-10-19 18:17:59
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http://math.stackexchange.com/questions/242079/using-divergence-theorem-on-boundary-of-surface
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# Using Divergence Theorem on boundary of surface
Use the Divergence Theorem to evaluate $$\iint_{S} \vec{F} \cdot \hat{n}\,dS,$$ where $\vec{F} = \langle 4x, 2y^2, z^2 \rangle, S$ is the boundary of the region defined by $x^2 + y^2 + z^2 \leq 4,\,\, 0 \leq z \leq 3$ and $\hat{n}$ is the unit outward normal
Attempt: The word 'boundary' is confusing me a little. So the surface is the curved part of a cylinder from $z=0$ to $z=3$. In cylindrical coordinates, $x = 2\cos\theta, y = 2\sin\theta, z = z$. Using the Div. Thm gives $\text{div} \vec{F} = 4 + 4y + 2z$, so I have $$\iiint_{E} 4+4y+2z\,dV,$$ Am I correct to write this as $$\int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{3} (4 + 8\sin\theta + 2z)\,r\,dz\,dr\,d\theta?$$ I am not sure because the question wanted the surface to be just the boundary but here I am integrating over the whole surface. Then again, I think this could be a feature of the div theorem. If we are just integrating on the boundary, then what would dV be? Many thanks
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The region described is not a cylinder. It is a sphere of radius 4 cut by the planes $z=0,3$. Spherical coordinates is most appropriate for this problem.
When you apply the divergence theorem, you establish an equality between a volume integral and a surface integral with a very particular relationship in what they integrate over. Any volume must have some boundary surface(s). The volume integral is over some region, and the corresponding surface integral in the divergence theorem is only over the boundary surfaces of that region.
If the region were a cube, the volume integral would be over the whole cube. The surface integral would be over the 6 faces of the cube.
In this case, the region is (part of) a sphere. The volume integral is over that region, and the surface integral is over two planar surfaces and part of a spherical surface.
-
Many thanks for your reply. I understand now. (Oh, and the addition of $z^2$ was a typo - it should have been $x^2 +y^2 \leq 4$) – CAF Nov 21 '12 at 16:57
One more question: In general, are surface integrals always over the boundary of some region? – CAF Nov 21 '12 at 17:00
No, surface integrals may be over "open" surfaces (think for example, a square patch of a plane). Open surfaces themselves have boundaries, however, so there is an analogue to the divergence theorem: the integral of $\nabla \times F \; dS$ on the open surface is equal to the integral of $F \cdot dr$ on the boundary curve. There is a whole broad class of these theorems--the fundamental theorem of calculus, the divergence theorem, Stokes' theorem, and so on. They all represent the same basic concept: integral of a derivative over a region equals integral of the function over the boundary. – Muphrid Nov 21 '12 at 17:05
Ok thanks. The triple integral I wrote in my first post gives me the correct answer to the Q, but to get to that integral, I had to take $y = 2\sin\theta$. I.e r takes the value 2. But then I sub that into divF and integrate over all r in the triple integral? So this is an example of what you said above right? We parametrize the surface boundary and then in the triple integral, we integrate over the whole region? – CAF Nov 21 '12 at 18:22
For the purposes of taking divergence and doing the volume integral, $y=r\sin \theta$. You can only set $r=2$ when performing the surface integration, not in the volume integration. – Muphrid Nov 21 '12 at 18:44
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2015-11-30 06:51:48
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http://kenhung.me/2017-08-23-thoughts-on-the-signal-and-the-noise.html
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Posts
Kenneth Hung | Department of Mathematics | UC Berkeley
# Thoughts on "The signal and the noise"
I came across Nate Silver’s name in two past occasions: once when I was taking Stat 215a at Berkeley, another time during the 2016 elections. With some luxury of free time when I was interning in Chicago, I took the time to finally read this book.
First the good things. Primarily due to my line of research, I have always been more fond of the frequentist approach. The book gave a very convincing argument for the Bayesian approach: as long as we do not exclude the truth, then with more data, we will be “less and less and less wrong”. While it might be odd to assign probabilities to when Newton’s Second Law does not hold, it most certainly is a valid approach for the more contemporary research, where repeated experiments show very large deviations in effect sizes; Or using his example on climate change, while climate change is irrefutable, the scale and severity of its effect might not be so agreed upon if the estimates are showing rather large variances.
The book also gave a very intuitive explanation of overfitting with the earthquake example, especially when it comes to extrapolation (vs. intrapolation). For exceptionally rare events that are rarely or never seen, an estimate for the probability of such events might be unreliable (this can go both ways), but the payoff (or penalty) can be orders of magnitude, not unlike the main focus in Talen’s book Black Swan.
I have some confusion about some of the examples used, however. In arguing that more information available can make estimation worse, the book mentioned National Journal panelists’ predictions about the 2010 midterm elections. For all 435 races, the predictions from conservatives and liberals differed by 6 seats, while for the race in a few specific states (Nevada, Illinois, Pennsylvania, Florida and Iowa) they differed by 5 out of 11. Differing by 6 out of 435 is definitely a sign of consistency in estimation compared to 5 out of 11. However, giving accurate prediction out of 11 (idealized) coin flips is definitely a harder job that doing so for 435 flips. The small number of coin flips forbids us to make full use of law of large numbers, making the prediction innately difficult. The difference in the estimates by conservatives and liberals need to be put in perspective by proper scaling (such as $\sqrt{n}$). While I agree with the sentiment, I am not sure if this example is so appropriate.
In Chapter 10, the book analyzed a poker game against a mythical opponent “the Lawyer”, and walked us through a step-by-step usage on Bayesian analysis: a prior was setup (an initial read that my fellow interns in Chicago like to talk about in poker nights) followed by Bayesian updates to our prior as the game unfolds. Each of the updates are objective as poker is a fairly mathematical game. This is a benefit of poker game with clean-cut probabilities that does not translate so well to more complicated systems, say basketball games. With a subjective prior, and Bayesian updates being subjective by nature, I am worried that it is possible that the subjectivity can add up, disallowing us to converge to the truth. However since I mostly work on more theoretical problems, I believe that other statisticians may know the tradeoffs better in practice.
Nate Silver pointed out many pitfalls and their solutions in statistical applications, some of such solutions mutually opposing, leaving room for data analysts to strike a balance. The abundance of charts and examples also made it a great summer read for me to take a break — a break from statistics by reading about statistics.
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2017-12-11 22:27:35
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https://planeta.github.io/latex/maxbibnames-for-different-bibliographies/
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# Different maximum number of authors in BibLaTeX bibliography
I have a document with multiple bibliographies, which I need to submit somewhere. Very annoyingly, the bibliography counts towards page limit, so I try to save space whenever possible. One of the places where I save space is that I limit how many authors are printed for each bibliography entry.
Later it turned out that I need to print full author list at least for the first bibliography. To minimise the space lost, I decided that I want to set the number of authors printed for the first bibliography, but not for the second. I did not find a solution on the Internet, but, fortunately, my colleague (Jan Bierbaum) is well-versed in LaTeX. He came up with the following solution.
\documentclass[a4paper]{scrbook}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[%
bibencoding=utf8,
backend=biber,
bibstyle=ieee,
citestyle=numeric-comp,
defernumbers=true,
sortcites=true,
maxcitenames=2,
mincitenames=1,
maxbibnames=3,
minbibnames=2,
]{biblatex}
\begin{filecontents}{\jobname.bib}
@inproceedings{own,
title = {Hardware Performance Variation: Comparative Study Using Lightweight Kernels},
author = {Weisbach, Hannes and Gerofi, Balazs and Kocoloski, Brian and Härtig, Hermann and Ishikawa, Yutaka},
keywords = {OWN}
}
@inproceedings{other,
title = {Accommodating Thread-Level Heterogeneity in Coupled Parallel Applications},
author = {Gutiérrez, Samuel K. and Davis, Kei and Arnold, Dorian C. and Baker, Randal S. and Robey, Robert W.
and McCormick, Patrick and Holladay, Daniel and Dahl, Jon A. and Zerr, R. Joe and Weik, Florian and Junghans, Ch
ristoph},
}
\end{filecontents}
\begin{document}
\section{Own stuff}
Own cite~\cite{own} by \citeauthor{own}
Other cite~\cite{other} by \citeauthor{other}
{
\expandafter\def\csname blx@maxbibnames\endcsname{99}%
}
\section{Other stuff}
Own cite~\cite{own} by \citeauthor{own}
Other cite~\cite{other} by \citeauthor{other}
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2021-09-23 02:12:01
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https://socratic.org/questions/how-do-you-solve-for-a-in-sqrta-9-b-3
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# How do you solve for a in sqrta + 9 = b + 3 ?
Mar 15, 2018
While this equation can't actually be solved to give an integer value for $a$, it can be rearranged, though it doesn't provide much use:
$\sqrt{a} = b + 3 - 9$
$\sqrt{a} = b - 6$
$a = \sqrt{a} \left(b - 6\right)$
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2019-10-17 12:33:41
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http://www.csam.or.kr/journal/view.html?uid=1947&&vmd=Full
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TEXT SIZE
CrossRef (0)
Fused inverse regression with multi-dimensional responses
Youyoung Choa, Hyoseon Hana, Jae Keun Yoo1,a
aDepartment of Statistics, Ewha Womans University, Korea
Correspondence to: 1Department of Statistics, Ewha Womans University, 11-1 Daehyun-Dong Seodaemun-Gu, Seoul 120-750, Korea. E-mail: peter.yoo@ewha.ac.kr
Received December 28, 2020; Revised February 3, 2021; Accepted February 5, 2021.
Abstract
A regression with multi-dimensional responses is quite common nowadays in the so-called big data era. In such regression, to relieve the curse of dimension due to high-dimension of responses, the dimension reduction of predictors is essential in analysis. Sufficient dimension reduction provides effective tools for the reduction, but there are few sufficient dimension reduction methodologies for multivariate regression. To fill this gap, we newly propose two fused slice-based inverse regression methods. The proposed approaches are robust to the numbers of clusters or slices and improve the estimation results over existing methods by fusing many kernel matrices. Numerical studies are presented and are compared with existing methods. Real data analysis confirms practical usefulness of the proposed methods.
Keywords : central subspace, fused sliced inverse regression, multivariate regression, pooled approach, sufficient dimension reduction
1. Introduction
With the recent advances in computing technology, it has become possible to perform calculations and modeling on vast amounts of data that were difficult before. With high-dimensional data modeling, the so-called curse of dimension is often faced, and it is one of main issues in such data analysis.
In regression of Y ∈ ℝu|X ∈ ℝp, sufficient dimension reduction (SDR) seeks to replace the original p-dimensional predictors X by its lower-dimensional predictor ηTX without loss of information on the conditional distribution of Y ∈ ℝu|X ∈ ℝp, where u ≥ 1, p ≥ 2 and η ∈ ℝp×d with dp. It is equivalently stated as the following independent statement:
$Y⫫X∣ηTX,$
where ⫫ stands for statistical independence.
For further usage, for p × q matrix M, we define a notation as a subspace spanned by the columns of M. Multiple η to satisfy (1.1) can exist, and then it is natural to choose the minimal one among them. The subspace spanned by the minimal one is called the central subspace. Throughout the rest of the paper, η and d will stand for an orthonormal basis matrix and the structural dimension of . The d-dimensional linearly transformed predictor ηTX is called sufficient predictors. For further insights about SDR, readers are recommended to read Yoo (2016a, b).
When the dimension of Y, u is bigger than or equal to 2, the regression is called multivariate regression. The demand of multivariate regression has rapidly grown according to advent of big data era. Repeated measures, longitudinal data, or curve or time series data often appear in big data, and the analysis of such data is difficult due to high-dimensionality of predictors. For example, the total number of regression coefficients to estimate in a classical multivariate regression of Y = (Y1, . . ., Yu)T|X = (X1, . . ., Xp)T is equal to p × u and multiply increase with adding more responses. Therefore, to avoid this complexity in the analysis, a proper dimension reduction of X is important, and SDR provides a good solution to the problem. So far, various SDR methodologies have been developed to estimate in multivariate regression. Various SDR methodologies for multivariate regression have been proposed (Lee et al., 2019; Setodji and Cook, 2004; Yin and Bura, 2006; Yoo, 2008, 2009; Yoo and Cook, 2007; Yoo et al., 2010). In Setodji and Cook (2004) and Yoo et al. (2010), an inverse regression approach, called K-means inverse regression (KIR) and K-means average variance estimation, is adopted, while the other methods combine the information from the coordinate regression of Yk|X, k = 1, . . ., u, where Yk is the kth coordinate of Y = (Y1, . . ., Yu)T.
Here, our interest is given in KIR, which is one of the widely used SDR method in multivariate regression. The key-step in KIR is to do K-means clustering Y. However, different numbers of the clusters provide different outcome by KIR, so it often causes a question regarding how many clusters must be used in KIR. So far, there is no thumb rule for it. To overcome similar issue in sliced inverse regression (SIR) (Li, 1991), Cook and Zhang (2014) propose a fused approach to combine all results from various numbers of slices, and they show that it provides robust estimation of to the number of slices and improves the estimation accuracy of . If this fusing idea is employed in KIR, we have potential advantages to have robust results to the number of clusters and to improve the estimation of like SIR. This is the main purpose of the paper. For this, we propose two fused approaches. The first one is to fuse the results based on hierarchical clustering algorithm recommended by Yoo et al. (2020), not K-means clustering algorithm in KIR. Another one is to fuse all results by the fused SIR application on the coordinate regression of Yk|X, k = 1, . . ., u.
The organization of the paper is as follows. Sliced inverse regression and hierarchical inverse regression are reviewed in Section 2. Section 3 is devoted to proposing pooled sliced inverse regression for multivariate regression and two fused approaches for multivariate regression. In Section 4, numerical studies and real data examples are presented. We summarize our work in Section 5.
2. Literature review: sliced and hierarchical inverse regressions
### 2.1. Sliced inverse regression
Understanding sliced inverse regression (SIR) (Li, 1991) is essential for methodological development, because its main methodological development is based on SIR.
Letting = cov(X), Li (1991) showed that , if E (X|ηTX) is linear in ηTX. Defining that Z = −1/2(X), the relation of is equivalent to that of according to Yoo (2016b). In practice, E(Z|Y) is restored instead of E(X|Y). Therefore, non-parametric estimation of E(Z|Y) is the primary interest in SIR. It can be done in a simple fashion by categorizing Y called slicing. Once the slicing is done, E(X|Y) can be easily replaced with sample means of X within each category. The estimation method of via −1/2E(Z|Y) is called sliced inverse regression. Its sample algorithm is as follows.
• Step 1. Slice Y to have h categories. Let Hj stand for the jth slice for j = 1, 2, . . . h.
• Step 2. Standardize the predictors X such that i = ∑̂−1/2 (Xi), i = 1, 2, . . ., n, where ∑̂ is usual moment estimator of and ∑̂−1/2∑̂−1/2 = ∑̂−1.
• Step 3. Calculate the sample means of $Z^¯k=(1/nk)∑i∈HkZ^i$ within each slice for k = 1, . . ., h, where nk stands for the size of the kth slice. Then, form a kernel matrix SIR:
$K^SIR=(n1nZ^¯1,n2nZ^¯2,…,nhnZ^¯h).$
• Step 4. Do the spectral decomposition of $M^SIR=K^SIRK^SIRT$ such that $M^SIR=∑i=1pλ^iγ^iγ^iT$, where λ̂1λ̂2 ≥ ··· ≥ λ̂p ≥ 0.
• Step 5. Let Γ̂d = (γ̂1, . . ., γ̂d) be the eigenvectors corresponding to the first d largest eigenvalues of SIR. Defining that η̂ = ∑̂−1/2Γ̂d, is the estimate of .
### 2.2. Hierarchical inverse regression
In the SIR algorithm, if following the slicing scheme for multivariate responses, it often faces the curse of dimensionality. For example, if there are five dimensional responses, the least number of slices should be 32(=25). If the number of observations in data is 50, some slices must have only one observation. Accordingly, this leads unreliable dimension reduction results. It is noted that grouping the observations based on their similarity of the response is essential in the slicing scheme. When Y is multi-dimensional, grouping by similarity can be done via clustering algorithms. Setodji and Cook (2004) and Yoo et al. (2010) successfully replace the usual slicing scheme with the K-means clustering algorithm for SIR, called K-means inverse regression (KIR), and sliced average variance estimation, respectively. In a perspective of fusing, the K-means algorithm is not be effective according to Yoo et al. (2020). The benefit of fusing mainly comes from nestness and reproducibility of slicing, but the K-means algorithm does not have the two properties. For details on nestness and reproducibility, readers refer Yoo et al. (2020).
Instead, hierarchical clustering algorithms have nestness and reproducibility, and Yoo et al. (2020) showed that the application of SIR via hierarchical clustering algorithm have advantage over the K-means clustering algorithm. So, following the guidance of Yoo et al. (2020), Ward’s hierarchical clustering algorithm will replace the usual slicing scheme.
For multivariate regression, once the responses are clustered, it replaces Step 1 in the SIR algorithm and follows the other steps in the same fashion. We call this approach hierarchical inverse regression (HIR).
3. Pooled sliced inverse regression and fused multivariate inverse regression
### 3.1. Pooled sliced inverse regression
Although clustering methods are effective and efficient alternatives to the usual slicing scheme for multivariate responses, it is inevitable for some clusters to have small sample sizes.
To overcome this issue, the following relationship between the central subspaces of Y|X and the coordinate regression of Yk|X should be noted:
$⊕k=1uSYk|X⊆SY|X,$
where is the central subspace of Yk|X and ⊕ denotes the direct sum among subspaces ( ).
This relation was firstly observed and utilized by Yoo et al. (2010), which proposed pooled sliced average variance estimation. It directly implies that combining all information on the central subspace of the coordinate regressions contains useful information on .
Following this pooling idea, we newly introduce the following pooled sliced inverse regression (pSIR). Let $MSIR(k)$ be the population kernel matrices of SIR for Yk|X. Define $MpSIR=(1/r)∑k=1rMSIR(k)$. The columns of the first d largest eigenvectors of MpSIR pre-multiplied by −1/2 span . Its sample algorithm is as follows.
• Step 1. Construct $M^SIR(k)$ for a coordinate regression of Yk|X, k = 1, . . ., u, from the usual SIR application.
• Step 2. Compute $M^pSIR=(1/r)∑k=1rM^SIR(k)$.
• Step 3. Do the spectral decomposition of pSIR such that $M^pSIR=∑i=1pλ^iγ^iγ^iT$, where λ̂1λ̂2 ≥ ··· ≥ λ̂p ≥ 0.
• Step 4. Let Γ̂d = (γ̂1, . . ., γ̂d) be the eigenvectors corresponding to the first d largest eigenvalues of pool. Defining that η̂ = ∑̂−1/2Γ̂d, is the estimate of .
### 3.2. Fused hierarchical inverse regression
Let $MHIR{g}$ indicate the kernel matrix of HIR with g clusters constructed by Ward’s hierarchical clustering algorithm. Then, the following relation is easily observed:
$Σ-12S (MHIR{g})⊆SY|X, g=2,…,h.$
The case of g = 1 is obviously ruled out, because it yields null matrix. This above relation directly indicates that
$⊕g=2hΣ-12S (MHIR{g})=Σ-12⊕g=2hS (MHIR{g})⊆SY|X.$
Based on this, we newly define $MFHIR{g}$ as
$MFHIR{g}=(MHIR{2},MHIR{3},…,MHIR{g}), g=3,…,h.$
In (3.2), the case of $MFHIR{2}$ is out of consideration, because $MFHIR{2}=MHIR{2}$. Theoretically, we can see that
$Σ-12 S (MFHIR{3})⊆Σ-12 S (MFHIR{4})⊆⋯⊆Σ-12 S (MFHIR{h})⊆SY|X.$
Therefore, $MFHIR{g}$ becomes a new kernel matrix to estimate . Further, for the exhaustive estimation of , a condition that $Σ-1/2S (MFHIR{g})=SY|X$ is forced, which is normally assumed in SDR literature. The estimation of through $MFHIR{g}$ will be called fused hierarchical inverse regression (FHIR).
The sample version $M^FHIR{g}$ is computed by replacing the population quantities with usual sample HIR kernel matrices. Fusing all information of the HIR application upto g clusters would cause potential advantages in more robust estimation results to choices of h and more accurate estimation of than KIR.
### 3.3. Fused pooled sliced inverse regression
Let $MpSIR{g}$ indicate the kernel matrix constructed by pSIR with g slices for all coordinate regressions of Yk|X. Like FHIR, the following relation is easily observed:
$Σ-/2S (MpSIR{g})⊆SY|X, g=2,…,h.$
Accordingly like (3.2), we define that
$MFpSIR{g}=(MpSIR{2},MpSIR{3},…,MpSIR{g}), g=3,⋯,h,$
and the following relation holds for the non-decreasing sequences of $MFpSIR{g}$, g = 3, . . ., h:
$Σ-12 S (MFpSIR{3})⊆Σ-12 S (MFpSIR{4})⊆⋯⊆Σ-12 S (MFpSIR{h})⊆SY|X.$
By assuming that $Σ-1/2S (MFpSIR{g})=SY|X$ for the exhaustive estimation of , the quantity $MFpSIR{g}$ becomes another kernel matrix fully informative to for multivariate regression. We call this SDR approach fused pooled sliced inverse regression (FpSIR).
The sample version $M^FpSIR{g}$ is constructed by computing $M^pSIR{g}$. Any choice of g in $M^pSIR{g}$ will provide the same asymptotic results, but their non-asymptotic behaviors can be easily affected by the choice of g. However, by fusing all the pSIR application results upto g slices for all coordinate regressions, more robust and accurate estimation of is expected than KIR.
### 3.4. Remarks on FpSIR and FHIR
For multivariate regression, one can use KIR, FpSIR and FHIR. The method FpSIR is recommended as default among the three, because FpSIR provides quite good estimation performances in various numerical studies, which are given in the next section. The two methods of KIR and FHIR require clustering application, so it cannot be implemented for some data. Also, it is known that outliers often affect clustering results, which may induce undesirable clustering results. Then, KIR and FHIR possibly produce poor estimation of . So, one fit FpSIR first, and see the results. If the dimension reduction results are not satisfactory, then it should be compared with those of FHIR and KIR.
4. Numerical studies and data analysis
### 4.1. Numerical studies
For all numerical studies, the sample sizes were 100, and each simulation model was iterated 1,000 times. To measure how the three methods of KIR, FHIR and FpSIR estimate well, absolute value |r| of the square-root of r2 from a regression of $ηiTX$ on η̂TX, i = 1, . . ., d, was computed, where η̂ stands for the sample estimate of η. Three to ten numbers of clusters or slices were considered for the three methods of KIR, FHIR and FpSIR.
The numerical studies are summarized by side-by-side boxplots of |r| for 3, 6 and 9 clusters or slices (not all reported) along with a plot of lining mean of |r|s against the number of slices, h = 3, 4, . . ., 10.
We considered the following two models, which were investigated in Setodji and Cook (2004) for KIR. In the models, all predictors Xi and random errors ɛi were independently generated from N(0, 1).
• Model 1 Each coordinate regression of Y = (Y1, . . ., Y4)T|X = (X1, . . ., X4)T is as follows.
$Y1=c11TX+c2 exp (c31TX) ɛ1;Y2=c11TX+c2 exp (c3|2-31TX|) ɛ2;Y3=c11TX+c2 exp (2c31TX) ɛ3;Y4=c11TX+c2 exp (c3|1-1TX|) ɛ4,$
where 1 is a vector all of which elements consist of 1.
• Model 2 Each coordinate regression of Y = (Y1, Y2)T|X = (X1, . . ., X10)T is as follows.
$Y1=X1(X1+X2+1)+σɛ1,Y2=X10.5+(X2+1.5)2+σɛ2.$
All coordinate regressions in Model 1 have the common linear conditional mean of c11TX. Since is spanned by the column vector 1, the structural dimension is equal to one. Depending on the choice of the value of c3, the regression has heteroscedasticity. Two cases of (1, 1, 0) and (0.1, 1, 0.1) for (c1, c2, c3) were considered. In the first case, the model is homoscedastic, while it is heteroscedastic for the second case. Through Model 1, it can be investigated how heteroscedasticity impacts the estimation of for the three methods.
Model 2 was designed to compare the estimation performances of the three methods for non-linear conditional means. Since the central subspace of Model 2 is spanned by the columns of (1, 0, 0, . . ., 0)T and (0, 1, 0, . . ., 0)T, which correspond to X1 and X2, respectively, its structural dimension is equal to two. Further, the values of σ were set to 0.5 and 1.
Numerical studies for Models 1 and 2 are summarized in Figures 16. For Model 1 with homogeneous variance, there is no notable difference among all three methods of KIR, FHIR and FpSIR, which yield very reliable estimation results. This is partially because Model 1 is just linear regression. However, according to Figure 2, the proposed FpSIR shows the best and most robust estimation results to the numbers of clusters or slices among the three. Again, KIR and FHIR provide similar estimation performances, although KIR is the worst with 3 clusters. The existence of heteroscedasticy in Model 1 can bring outliers in responses, which affect the clustering results as discussed in Section 3.3. This possibly yield undesirable clustering results of the response variables, and it induces poor estimation results of . Poor estimation of performances of KIR and FHIR in Model 1 can be partially explained by this aspect. With larger numbers of slices, FpSIR still provide good estimation of under heteroscedasticity.
For Model 2, Figures 3 and 5 show that the first sufficient predictor of X1 are well-estimated by all the three methods for σ = 0.5 and 1. However, for the second sufficient predictor X2, KIR yields very sensitive results for small numbers of clusters, while HIR is also quite robust to the numbers of clusters and FpSIR is very robust to the numbers of slices. The estimation performances of the three methods is negatively impacted by larger variability of noise ɛ. So, it can be concluded that HIR and FpSIR estimate relatively well.
This numerical studies confirm that two proposed fused methods, especially FpSIR, outperform the existing KIR in the estimation of , so we can expect potential advantages of FHIR and FpSIR over KIR for the dimension reduction of predictors in multivariate regression.
### 4.2. Minneapolis school data
For the illustration purpose, we considered a multivariate regression analyzed in Yoo (2009). The data is regarding the performance of students in n = 63 Minneapolis schools. In the data, there are four dimensional responses Y of the percents of students in a school scoring above and below average on standardized fourth and sixth grade reading comprehension tests. Among many variables the following five ones were considered as predictors: the pupil teacher ratio, and the square roots of the percentage of children receiving Aid to Families with Dependent Children, the percentage of children not living with both biological parents, the percentage of adults in the school area who completed high school, the percentage of persons in the area below the federal poverty level. The predictors were transformed to satisfy the linearity condition. The square root-scale is necessary to induce the condition required in SIR.
For this regression, KIR, FHIR and FpSIR were applied with 3, 6 and 9 clusters or slices. The estimated first and second sufficient predictors are reported in Figures 7 and 8. As seen in Figure 7, all first sufficient predictors are very close to each other regardless of the numbers of clusters or slices and methods. However, there are some differences in the second sufficient predictors. According to Figure 8, the second sufficient predictors from FpSIR with 3, 6 and 9 slices are not highly correlated. This implies that the second one would be random rather than deterministic, and this induces that it is not informative to according to Yoo (2018). So, the first sufficient predictor should be enough for the regression. On the other hand, in Figure 8, it is observed that the second sufficient predictors from KIR and FHIR with 3, 6 and 9 clusters are highly correlated to each other, so we expect that the structural dimension determination for KIR and FHIR should be, at least, two following the same rationale in Yoo (2018). To investigate this, a permutation dimension test (Yin and Bura, 2006; Yoo, 2016b) were conducted for FHIR and FpSIR, and weighted χ2 test for KIR (Setoji and Cook, 2004) starting H0:d = 0 with nominal level 5%. If H0:d = 0 is not rejected, increment d by 1 and redo the test. Then, the structural dimension d is determined as the hypothesized value in H0 that the first non-rejection occurs. The p-values for the test from KIR, FHIR and FpSIR with 3, 6 and 9 clusters or slices are summarized in Table 1. According to Table 1, as discussed, FHIR and FpSIR determine that = 1 and = 2, respectively. However, KIR determines that > 1 with 3 clusters and = 1 with 6 and 9 clusters. This is partially because of sensitiveness of KIR to the number of clusters. To decide = 1 or = 2, it is necessary for formal theoretical and numerical studies in the dimension estimation of FHIR and FpSIR, but this will be left for further research. Because FpSIR yields the best estimation results in most cases of numerical studies and Yoo (2009) suggests that = 1, we tentatively decide that = 1. So, one can investigate to find an adequate model for the multivariate regression with the first sufficient predictor, instead of the original five dimensional predictors.
5. Discussion
For multivariate regression, there are few sufficient dimension reduction methods, although multi-dimensional responses become more popular nowadays in the so-called big data era. Existing inverse regression methods are still persuasive in multivariate regression, but they are sensitive to the number of clusters or slices. A fused approach recently developed by Cook and Zhang (2014) shows clear advantage for robustness to the number of slices in slicing-based inverse regression methods. So, in this paper, two fused inverse regression methods for multivariate regression are newly proposed, which are called fused hierarchical inverse regression and fused pooled sliced inverse regression. Fused hierarchical inverse regression accumulates all kernel matrices from hierarchical inverse regression (Yoo et al., 2020) with various numbers of clusters. In fused hierarchical inverse regression, the multi-dimensional responses are clustered via Ward’s hierarchical clustering algorithm. On the other hand, the fused pooled sliced inverse regression has two step procedure. First, fused sliced inverse regression (Cook and Zhang, 2014) is implemented and fused kernel matrices are computed for each coordinate regression. Secondly, collect all kernel matrices from all coordinate regression, and the final fused kernel matrix is constructed. Different from fused hierarchical inverse regression, the clustering algorithm is not used, because the dimension of response in each coordinate regression is equal to one. So, in fused pooled sliced inverse regression, usual slicing scheme is applicable.
Numerical studies confirm that both proposed fused methods provide robustness to choice of clusters or slices and improve the estimation of the central subspace over the existing K-means inverse regression. A real data example shows their practical usefulness in multivariate regression analysis.
Theoretical asymptotics of sample kernel matrices for the dimension determination of fused hierarchical inverse regression and fused pooled sliced inverse regression should be studied and derived. Since the two proposed fused methods have similar kernel matrices, each kernel matrix is not independent. So, for theoretical development, dependency central limit theorem should be applied. This direction of research is in progress.
6. Acknowledgements
For Jae Keun Yoo, this work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Korean Ministry of Education (NRF- 2019R1F1A1050715).
Figures
Fig. 1. Model 1 with homogeneous variance.
Fig. 2. Model 1 with heterogeneous variance.
Fig. 3. Model 2 with σ = 0.5, the first sufficient predictor of X.
Fig. 4. Model 2 with σ = 0.5, the second sufficient predictor of X.
Fig. 5. Model2 with σ = 1, the first sufficient predictor of X.
Fig. 6. Model 2 with σ = 1, the second sufficient predictor of X.
Fig. 7. Scatterplot matrix of the first sufficient predictors obtained from the application of KIR, FHIR and FpSIR with 3, 6 and 9 clusters or slices: , KIR with # clusters; , FHIR with # clusters; , FpSIR with # slices.
Fig. 8. Scatterplot matrix of the second sufficient predictors obtained from the application of KIR, FHIR and FpSIR with 3, 6 and 9 clusters or slices: , KIR with # clusters; , FHIR with # clusters; , FpSIR with # slices.
TABLES
### Table 1
Dimension tests by the application of KIR, FHIR and FpSIR with 3, 6 and 9 clusters or slices: KIR#, KIR with # clusters; FHIR#, FHIR with # clusters; FpSIR#, FpSIR with # slices
KIR3KIR6KIR9FHIR3FHIR6FHIR9FpSIR3FpSIR6FpSIR9
H0 : d = 00.0000.0000.0000.0000.0000.0000.0000.0000.000
H0 : d = 10.0060.1540.0780.0010.0050.00120.1110.0800.066
H0 : d = 2NA0.9720.770NA0.7040.4290.1890.6730.741
References
1. Cook RD and Zhang X (2014). Fused estimators of the central subspace in sufficient dimension reduction. Journal of the American Statistical Association, 109, 815-827.
2. Lee K, Choi Y, Um H, and Yoo JK (2019). On fused dimension reduction in multivariate regression. Chemometrics and Intelligent Laboratory Systems, 193, 103828.
3. Li KC (1991). Sliced inverse regression for dimension reduction. Journal of the American Statistical Association, 86, 316-327.
4. Setodji CM and Cook RD (2004). K-means inverse regression. Technometrics, 46, 421-429.
5. Yin X and Bura E (2006). Moment-based dimension reduction for multivariate response regression. Journal of Statistical Planning and Inference, 136, 3675-3688.
6. Yoo C, Yoo Y, Um HY, and Yoo JK (2020). On hierarchical clustering in sufficient dimension reduction. Communications for Statistical Applications and Methods, 27, 431-443.
7. Yoo JK (2008). A Novel moment-based dimension reduction approach in multivariate regression. Computational Statistics and Data Analysis, 52, 3843-3851.
8. Yoo JK (2009). Iterative optimal sufficient dimension reduction for the conditional mean in multivariate regression. Journal of Data Science, 7, 267-276.
9. Yoo JK (2016a). Tutorial: Dimension reduction in regression with a notion of sufficiency. Communications for Statistical Applications and Methods, 23, 93-103.
10. Yoo JK (2016b). Tutorial: Methodologies for sufficient dimension reduction in regression. Communications for Statistical Applications and Methods, 23, 95-117.
11. Yoo JK (2018). Basis-adaptive selection algorithm in dr-package. The R Journal, 10, 124-132.
12. Yoo JK and Cook RD (2007). Optimal sufficient dimension reduction for the conditional mean in multivariate regression. Biometrika, 94, 231-242.
13. Yoo JK, Lee K, and Woo S (2010). On the extension of sliced average variance estimation to multi-variate regression. Statistical Methods and Applications, 19, 529-540.
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# définition - The_Limits_to_Growth
The Limits To Growth (n.prop.)
1.the title of the report commissioned to Massachusetts Institute of Technology (MIT) researchers by the Club of Rome in 1970, and published in English in 1972 and in French in 1973.
The Meadows Report describes the results of a computer-programmed stochastic algorithmic model in a finite world.
This is the first major study to highlight the ecological dangers of economic and demographic growth. These dangers stand in one expression: civilizational collapse.
## définition (complément)
voir la définition de Wikipedia
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# synonymes - The_Limits_to_Growth
The Limits To Growth (n.prop.)
voir aussi
The Limits To Growth (n.prop.)
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## dictionnaire analogique
collapsology[Dérivé]
The Limits To Growth (n. pr.)
Wikipedia
# The Limits to Growth
The Limits to Growth
The Limits to Growth first edition cover.
Jørgen Randers
William W. Behrens III
Language English
Publisher Universe Books
Publication date 1972
Pages 205
ISBN 0-87663-165-0
OCLC Number 307838
The Limits to Growth is a 1972 book about the computer modeling of unchecked economic and population growth with finite resource supplies.[1] It was commissioned by the Club of Rome and was first presented at the 3. St. Gallen Symposium. Its authors were Donella H. Meadows, Dennis L. Meadows, Jørgen Randers, and William W. Behrens III. The book used the World3 model to simulate[2] the consequence of interactions between the Earth's and human systems. The book echoes some of the concerns and predictions of Thomas Malthus in An Essay on the Principle of Population (1798).
Five variables were examined in the original model, on the assumptions that exponential growth accurately described their patterns of increase, and that the ability of technology to increase the availability of resources grows only linearly. These variables are: world population, industrialization, pollution, food production and resource depletion. The authors intended to explore the possibility of a sustainable feedback pattern that would be achieved by altering growth trends among the five variables under three scenarios. They noted that their projections for the values of the variables in each scenario were predictions "only in the most limited sense of the word," and were only indications of the system's behavioral tendencies.[3] Two of the scenarios saw "overshoot and collapse" of the global system by the mid to latter part of the 21st century, while a third scenario resulted in a "stabilized world."[4]
The most recent updated version was published on June 1, 2004 by Chelsea Green Publishing Company and Earthscan under the name Limits to Growth: The 30-Year Update. Donnella Meadows, Jørgen Randers, and Dennis Meadows have updated and expanded the original version. They had previously published Beyond the Limits in 1993 as a 20 year update on the original material.[5][6][7]
In 2008 Graham Turner at the Commonwealth Scientific and Industrial Research Organisation (CSIRO) in Australia published a paper called "A Comparison of The Limits to Growth with Thirty Years of Reality".[8][9] It examined the past thirty years of reality with the predictions made in 1972 and found that changes in industrial production, food production and pollution are all in line with the book's predictions of economic and societal collapse in the 21st century.[10] In 2010, Peet, Nørgård, and Ragnarsdóttir called the book a "pioneering report". They said that, "its approach remains useful and that its conclusions are still surprisingly valid... unfortunately the report has been largely dismissed by critics as a doomsday prophecy that has not held up to scrutiny."[11]
## Purpose
The purpose of The Limits to Growth was not to make specific predictions, but to explore how exponential growth interacts with finite resources. Because the size of resources is not known, only the general behavior can be explored. The authors state in a subsection titled The Purpose of the World Model[12]:
In this first simple world model, we are interested only in the broad behavior modes of the population-capital system. By behavior modes we mean the tendencies of the variables in the system (population or pollution, for example) to change as time progresses. A variable may increase, decrease, remain constant, oscillate, or combine several of these characteristic modes. For example, a population growing in a limited environment can approach the ultimate carrying capacity of that environment in several possible ways. It can adjust smoothly to an equilibrium below the environmental limit by means of a gradual decrease in growth rate, as shown below. It can overshoot the limit and then die back again in either a smooth or an oscillatory way, also as shown below. Or it can overshoot the limit and in the process decrease the ultimate carrying capacity by consuming some necessary nonrenewable resource, as diagrammed below. This behavior has been noted in many natural systems. For instance, deer or goats, when natural enemies are absent, often overgraze their range and cause erosion or destruction of the vegetation.
A major purpose in constructing the world model has been to determine which, if any, of these behavior modes will be most characteristic of the world system as it reaches the limits to growth. This process of determining behavior modes is "prediction" only in the most limited sense of the word. The output graphs reproduced later in this book show values for world population, capital, and other variables on a time scale that begins in the year 1900 and continues until 2100. These graphs are not exact predictions of the values of the variables at any particular year in the future. They are indications of the system's behavioral tendencies only.
The difference between the various degrees of "prediction" might be best illustrated by a simple example. If you throw a ball straight up into the air, you can predict with certainty what its general behavior will be. It will rise with decreasing velocity, then reverse direction and fall down with increasing velocity until it hits the ground. You know that it will not continue rising forever, nor begin to orbit the earth, nor loop three times before landing. It is this sort of elemental understanding of behavior modes that we are seeking with the present world model. If one wanted to predict exactly how high a thrown ball would rise or exactly where and when it would hit the ground, it would be necessary to make a detailed calculation based on precise information about the ball, the altitude, the wind, and the force of the initial throw. Similarly, if we wanted to predict the size of the earth's population in 1993 within a few percent, we would need a very much more complicated model than the one described here. We would also need information about the world system more precise and comprehensive than is currently available.
## Exponential reserve index
One key idea within the The Limits to Growth is the notion that if the rate of resource use is increasing, the amount of reserves cannot be calculated by simply taking the current known reserves and dividing by the current yearly usage, as is typically done to obtain a static index. For example, in 1972, the amount of chromium reserves was 775 million metric tons, of which 1.85 million metric tons were mined annually (see exponential growth). The static index is $775/1.85=418\text{ years}$, but the rate of chromium consumption was growing at $2.6%$ annually (Limits to Growth, pp 54–71). If instead of assuming a constant rate of usage, the assumption of a constant rate of growth of $2.6%$ annually is made, the resource will instead last
$\frac{\ln(1+0.026\times 418)}{0.026} \approx \text{95 years}$
(note that the book rounded off numbers).
In general, the formula for calculating the amount of time left for a resource with constant consumption growth is [13]:
$y = \frac{\ln((r \times s) + 1)}{r}$
where:
y = years left;
r = 0.026, the continous compounding growth rate (2.6%).
s = R/C or static reserve.
R = reserve;
C = (annual) consumption.
The authors list a number of similar exponential indices comparing current reserves to current reserves multiplied by a factor of five:
Years
Resource Consumption growth rate, annual Static index Exponential index 5 times reserves exponential index
Chromium 2.6% 420 95 154
Gold 4.1% 11 9 29
Iron 1.8% 240 93 173
Petroleum 3.9% 31 20 50
The static reserve numbers assume that the usage is constant, and the exponential reserve assumes that the growth rate is constant.
The exponential index has been interpreted as a prediction of the number of years until the world would "run out" of various resources, both by environmentalist groups calling for greater conservation and restrictions on use, and by skeptics criticizing the index when supplies failed to run out.[14][15][16][17] What The Limits to Growth actually has is the above table, which has the current reserves (that is no new sources of oil are found) for oil running out in 1992 assuming constant exponential growth. In Limits to Growth: The Thirty Year Update there are several pages explaining that new resources are found over time and that the current reserves therefore change but that ultimately resources are finite. (Earlier editions did explain this as well, but not in as much detail.) The standard model includes a resource base of double that of what they have calculated, but the book includes model runs where the assumed resources are infinite, but those model runs still result in overshoot and collapse from other factors.
## Related books
Many books about humanity’s uncertain future have appeared regularly over the years. Precursors to Limits to Growth included Harrison Brown’s The Challenge of Man’s Future (1956), Rachel Carson’s Silent Spring (1962) and Paul Ehrlich’s The Population Bomb (1968).[18]
The most notable books to be published after 1972 and up to the end of the millennium included the State of the World reports issued by the Worldwatch Institute (produced annually since 1984); the influential Our Common Future, published by the UN’s World Commission on Environment and Development (1987); Earth in the Balance, written by then-US senator Al Gore (1992); and Earth Odyssey (ISBN 978-0767900591) by journalist Mark Hertsgaard (1999), which "reported on eight years of travel all over the globe to observe the demise of Nature and the degradation of the World".[18]
Since that time, the number of similar titles published and copies sold has itself grown significantly, all documenting evidence that the world is "growing dangerously and spinning out of control".[18]
## Reception
Soon after publication prominent economists, scientists and political figures criticized the Limits to Growth. They attacked the methodology, the computer, the conclusions, the rhetoric and the people behind the project.[19] Yale economist Henry C. Wallich agreed that growth could not continue indefinitely, but that a natural end to growth was preferable to intervention. Wallich stated that technology could solve all the problems the Meadows were concerned about, but only if growth continued apace. By stopping growth too soon, Wallich warned, the world would be "consigning billions to permanent poverty".[19]
Robert M. Solow from MIT, argued that prediction in The Limit to Growth was based on a weak foundation of the data (Newsweek, March 13, 1972, page 103). Dr. Allen Kneese and Dr. Ronald Riker of Resources for the Future (RFF) stated:
"The authors load their case by letting some things grow exponentially and others not. Population, capital and pollution grow exponentially in all models, but technologies for expanding resources and controlling pollution are permitted to grow, if at all, only in discrete increments." [20]
Critics also argue that the authors of the report claimed to accept that the then-known resources of minerals and energy could, and would, grow in the future, and consumption growth rates could also decline. The theoretical expiry time for each resource would therefore need to be updated as new discoveries, technologies and trends came to light. To overcome this uncertainty, they offered an upper value for the expiry time, calculated as if the known resources were multiplied by two. Even in that case, assuming continuation of the average rate of consumption growth, virtually all major minerals and energy resources would expire within 100 years of publication (i.e., by 2070). Even if reserves were two times larger than expected, they state, ongoing growth in the consumption rate would still lead to the relatively rapid exhaustion of those reserves.[21]
In 2008 researcher Peter A. Victor wrote, that even though D.H. Meadows et al. probably underestimated price-mechanism's role in adjusting, their critics have overestimated it. He states that Limits to Growth has had a significant impact on the conception of environmental issues and notes that the models in the book were meant to be taken as predictions "only in the most limited sense of the word" as they wrote.[22]
In a 2009 article published in American Scientist titled "Revisiting the Limits to Growth After Peak Oil," Hall and Day noted that "the values predicted by the limits-to-growth model and actual data for 2008 are very close." [23] These findings are consistent with a 2010 study titled "A Comparison of the Limits of Growth with Thirty Years of Reality" which concluded: "The analysis shows that 30 years of historical data compares favorably with key features… [of the Limits to Growth] ‘standard run’ scenario, which results in collapse of the global system midway through the 21st Century." [24]
In 2011 Ugo Bardi analyzed the The Limits to Growth, its methods and historical reception and concluded that "The warnings that we received in 1972 ... are becoming increasingly more worrisome as reality seems to be following closely the curves that the ... scenario had generated." [25]
## References
1. ^ MacKenzie, Debora (10 January 2012). "Boom and doom: Revisiting prophecies of collapse". New Scientist. Retrieved 1 April 2012.
2. ^ The models were run on DYNAMO, a simulation programming language.
3. ^ Peter A. Victor (2008). Managing Without Growth, Edward Elgar Publishing, pp. 92-93, ISBN 978-1-84720-078-5
4. ^ Graham Turner (2008). "A Comparison of The Limits to Growth with Thirty Years of Reality". Page 11. Commonwealth Scientific and Industrial Research Organisation (CSIRO).
5. ^ "To Grow or not to Grow", Newsweek, March 13, 1972, pages 102–103
6. ^ Donella H. Meadows, Dennis L. Meadows, Jorgen Randers, and William W. Behrens III. (1972).
The Limits to Growth. New York: Universe Books. ISBN 0-87663-165-0
7. ^ Henry C. Wallich, "More on Growth", NewsWeek, March 13, 1972, page 86.
8. ^ Graham Turner (2008). "A Comparison of The Limits to Growth with Thirty Years of Reality". Commonwealth Scientific and Industrial Research Organisation (CSIRO).
9. ^ Graham Turner (2008). "A Comparison of The Limits to Growth with Thirty Years of Reality". Commonwealth Scientific and Industrial Research Organisation (CSIRO).
10. ^ "Prophesy of economic collapse 'coming true'", by Jeff Hecht, NewScientist, 17 November 2008
11. ^ http://www.thesolutionsjournal.com/node/569
12. ^ Meadows, D. (1974). The Limits to Growth, Second Edition Revised, Signet. ISBN 73-187907, pages 99-101
13. ^ Limits To Growth, pg 60, Derivation: $R = \int_0^y C e^{\rho t}\ dt = \frac{C}{\rho} \left(e^{\rho y} - 1\right)$ reverts to $y = \frac{\ln \left( 1 + \rho \frac{R}{C}\right)}{\rho}.$
14. ^ The Skeptical Environmentalist, p. 121
15. ^ Chapter 17: Growth and Productivity-The Long-Run Possibilities
16. ^ "Treading lightly". The Economist. 19 September 2002.
17. ^ Reason Magazine - Science and Public Policy
18. ^ a b c Alan Atkisson (2010). Believing Cassandra: How to be an Optimist in a Pessimist's World, Earthscan, pp. 17-18.
19. ^ a b Alan Atkisson (2010). Believing Cassandra: How to be an Optimist in a Pessimist's World, Earthscan, p. 13.
20. ^ Newsweek, March 13, 1972, page 103.
21. ^ http://www.clubofrome.org/docs/limits.rtf
22. ^ Peter A. Victor (2008). Managing Without Growth, Edward Elgar Publishing, pp. 92-93, ISBN 978-1-84720-078-5
23. ^ Hall, C. & Day, J. Revisiting the Limits to Growth After Peak Oil. American Scientist, 97 (2009): 230 -238.
24. ^ Turner, Graham. A Comparison of the Limits of Growth with Thirty Years of Reality. CSIRO Working Paper Series, (2010). Available at: http://www.csiro.au/files/files/plje.pdf
25. ^ Ugo Bardi. The Limits to Growth Revisited. Springer 2011 doi:10.1007/978-1-4419-9416-5 p.3
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2020-07-10 00:55:50
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https://stats.stackexchange.com/questions?sort=newest&page=2704
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# All Questions
138,960 questions
2 views
### estimate equal distribution of few points on a line
I am trying to find the best solution to estimate equal distribution of points over a line. I know I can use relative SD or similar, but I was wondering if there are more "specific" methods that can ...
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### Confidence intervals for state space models
i'm looking for on how to calculate the following ICs: smoothing, on-line filter and prediction on state space models. I'm not able to find any formula about them or any matlab command/class. Thanks....
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### Re-scaling of exponentially distributed random numbers
I am trying to generate $M$ random numbers which are exponentially distributed and whose sum adds up to $N$ (for simplicity, $N=1$). I found that the generated numbers are initially exponentially ...
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### How to deal with repeated words in training and testing set using CART?
I have a dataset where a child read a passage out loud. Each row is one word (I’m order of the passage). I have a 1/0 for whether they read the word incorrectly or correctly. I’m trying to predict ...
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### Comparer deux groupes [on hold]
J'ai deux groupe classe, un qui a reçu un cours de mathématique en utilisant la tablette et un autre ayant reçu un enseignement traditionnel. Je dois voir si la tablette a aidé les élèves à obtenir ...
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### Estimation of NegEntropy in MATLAB
I am trying to evaluate the different ICA algorithms using MATLAB. To do that, one of the measure which I use is to estimate the non-gaussianity using NegEntropy. I am trying to find a function which ...
3 views
### Measuring effects of IV (constant) on DV
I have an independent variable (a bias) that is technically a constant. I want to measure the effects of the IV on the DV. How do I do this? I'm using SPSS. Let's say, participants produced A or B ...
10 views
### StepAIC with tslm function in R
I wish to forecast a timeseries and so I use something like this : Suppose I have a data xtrain with V1,V2,V3,V4 as the attribues. Then, I will create a model as follows : form=as.formula(train ~ ...
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### Most influencer city
My Problem goes like this: Suppose the cities of world, each city has a degree of economic collaboration with other cities. I need to detect communities such that the degrees among different cities ...
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### R-Squared for a non linear curve
At the start, please forgive me if my question is too elementary. I am fitting a non-linear curve. Say a parabola. The data points I have are close to a parabola and the best output I get is a ...
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### Local Significance test for composite anomalies
Suppose I have the following 20 points: ...
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### Moderation analysis with non-significant main effects but significant interaction
I am doing a simple moderation analysis with one independent variable (IV), one moderator (M) and one dependent variable (DV). Results indicate that a regression model containing both main effects (...
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### Which statistical tool would be useful to check whether a variable has changed significantly over years
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### How to show a UMVUE exists only if $g(p)$ is a polynomial of degree at most $n$?
Let $X\sim Bin(n,p)$. The problem is to show that a UMVUE can exist for $g(p)$ only if $g(p)$ is a polynomial in $p$ of degree at most $n$. For the case when $g(p) = \frac{1}{p}$ we can show that it ...
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### BayesFactor anovaBF syntax
I want to get Bayes factors for ANOVAs that are analogous to the classical F-tests, and I just want to make sure I understand correctly how to write the syntax, especially regarding subject IDs. For ...
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### hoc test and pearson corelation
Please can you help me how to report post hoc multiple comparison? is it write to report them in that way (M=...., SD=......), i wrote them in that way. And another question how can I report a ...
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### How to transform Beta distribution of second kind into Beta distribution of first kind?
! I tried to solve it in a way but I can't complete in as I don't get any idea how to integrate the Beta kind two integrand within a finite boundary.
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### Use of adjusted R^2 when regressing using overlapping data
Context: Taking time series levels and converting to 10 day changes in levels. Then doing a linear regression on the 10 day changes over a period of time eg 500 overlapping samples, where Y is the ...
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### measurements with gaussian error
I'm reading a printed document(learning material) which says "Person A made measurements of a quantity, with Gaussian errors". To me this expression seems odd and the meaning is unclear. Is that a ...
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### confidence interval for the variance with unknown distribution
Let $X_i \sim \text{iid}(\mu, \sigma^2)$ (the question does not specify whether or not $\mu$ and $\sigma^2$ are known). I have to show the confidence interval for the variance. Since $X_i$ is not ...
16 views
26 views
### Why is random sampling a non-differentiable operation?
This answer states that we cannot back-propagate through a random node. So, in the case of VAEs, you have the reparametrisation trick, which shifts the source of randomness to another variable ...
26 views
### Commonly used example data sets
Is there a place where one can get standard example data for various statistics tools to try on? For example, if one is learning about ARIMA models, where would one get data that can be modelled well ...
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### Probability of sparse spectrum
Consider a vector $v$ such that $v \sim \mathrm{Unif}(\mathbb{S}^{d-1})$, the uniform distribution on the unit sphere in $d$ dimensions. Question: is there an upper bound on the probability that $v$ ...
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### Is this a crossover design?
I am working with collaborators who are conducting an education research study to see if there is a difference in learning outcomes between two learning experiences, one with a technology tool (let’s ...
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2019-05-25 15:12:28
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https://formulasearchengine.com/wiki/Canonical_commutation_relation
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# Canonical commutation relation
In quantum mechanics (physics), the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example,
${\displaystyle [x,p_{x}]=i\hbar }$
between the position Template:Mvar and momentum Template:Mvar in the Template:Mvar direction of a point particle in one dimension, where [x , px] = x pxpx x is the commutator of Template:Mvar and Template:Mvar, Template:Mvar is the imaginary unit, and is the reduced Planck's constant {{ safesubst:#invoke:Unsubst||\$B=h/}} . In general, position and momentum are vectors and their commutation relation between different components of position and momentum can be expressed as
${\displaystyle [r_{i},p_{j}]=i\hbar \delta _{ij}}$.
This relation is attributed to Max Born (1925),[1] who called it a "quantum condition" serving as a postulate of the theory; it was noted by E. Kennard (1927)[2] to imply the Heisenberg uncertainty principle.
## Relation to classical mechanics
By contrast, in classical physics, all observables commute and the commutator would be zero. However, an analogous relation exists, which is obtained by replacing the commutator with the Poisson bracket multiplied by i:
${\displaystyle \{x,p\}=1\,.}$
This observation led Dirac to propose that the quantum counterparts Template:Mvar, Template:Mvar of classical observables Template:Mvar, Template:Mvar satisfy
${\displaystyle [{\hat {f}},{\hat {g}}]=i\hbar {\widehat {\{f,g\}}}\,.}$
In 1946, Hip Groenewold demonstrated that a general systematic correspondence between quantum commutators and Poisson brackets could not hold consistently.[3] However, he did appreciate that such a systematic correspondence does, in fact, exist between the quantum commutator and a deformation of the Poisson bracket, the Moyal bracket, and, in general, quantum operators and classical observables and distributions in phase space. He thus finally elucidated the correspondence mechanism, Weyl quantization, that underlies an alternate equivalent mathematical approach to quantization known as deformation quantization.[3]
## Representations
The group H3(ℝ) generated by exponentiation of the Lie algebra specified by these commutation relations, [x, p] = i, is called the Heisenberg group.
According to the standard mathematical formulation of quantum mechanics, quantum observables such as x and p should be represented as self-adjoint operators on some Hilbert space. It is relatively easy to see that two operators satisfying the above canonical commutation relations cannot both be bounded—try taking the Trace of both sides of the relations and use the relation Trace(A B ) = Trace(B A ); one gets a finite number on the right and zero on the left.[4]
These canonical commutation relations can be rendered somewhat "tamer" by writing them in terms of the (bounded) unitary operators exp(i tx) and exp(i sp), which do admit finite-dimensional representations. The resulting braiding relations for these are the so-called Weyl relations
exp(i tx) exp(i sp) = exp(−iℏ s t) exp(i sp) exp(i tx).
The corresponding group commutator is then
exp(i tx) exp(i sp) exp(−i tx) exp(−i sp) = exp(−iℏ s t).
The uniqueness of the canonical commutation relations between position and momentum is then guaranteed by the Stone–von Neumann theorem.
## Generalizations
The simple formula
${\displaystyle [x,p]=i\hbar ,\,}$
valid for the quantization of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian ${\displaystyle {\mathcal {L}}}$.[5] We identify canonical coordinates (such as Template:Mvar in the example above, or a field Φ(x) in the case of quantum field theory) and canonical momenta πx (in the example above it is Template:Mvar, or more generally, some functions involving the derivatives of the canonical coordinates with respect to time):
${\displaystyle \pi _{i}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\partial {\mathcal {L}}}{\partial (\partial x_{i}/\partial t)}}.}$
This definition of the canonical momentum ensures that one of the Euler–Lagrange equations has the form
${\displaystyle {\frac {\partial }{\partial t}}\pi _{i}={\frac {\partial {\mathcal {L}}}{\partial x_{i}}}.}$
The canonical commutation relations then amount to
${\displaystyle [x_{i},\pi _{j}]=i\hbar \delta _{ij},\,}$
where δij is the Kronecker delta.
Further, it can be easily shown that
${\displaystyle [p_{i},F({\vec {x}})]=-i\hbar {\frac {\partial F({\vec {x}})}{\partial x_{i}}};\qquad [x_{i},F({\vec {p}})]=i\hbar {\frac {\partial F({\vec {p}})}{\partial p_{i}}}.}$
## Gauge invariance
Canonical quantization is applied, by definition, on canonical coordinates. However, in the presence of an electromagnetic field, the canonical momentum Template:Mvar is not gauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is
${\displaystyle p_{\textrm {kin}}=p-qA\,\!}$ (SI units) ${\displaystyle p_{\textrm {kin}}=p-{\frac {qA}{c}}\,\!}$ (cgs units),
where Template:Mvar is the particle's electric charge, Template:Mvar is the vector potential, and c is the speed of light. Although the quantity pkin is the "physical momentum", in that it is the quantity to be identified with momentum in laboratory experiments, it does not satisfy the canonical commutation relations; only the canonical momentum does that. This can be seen as follows.
The non-relativistic Hamiltonian for a quantized charged particle of mass Template:Mvar in a classical electromagnetic field is (in cgs units)
${\displaystyle H={\frac {1}{2m}}\left(p-{\frac {qA}{c}}\right)^{2}+q\phi }$
where Template:Mvar is the three-vector potential and Template:Mvar is the scalar potential. This form of the Hamiltonian, as well as the Schrödinger equation = iħ∂ψ/∂t, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation
${\displaystyle A\to A^{\prime }=A+\nabla \Lambda }$
${\displaystyle \phi \to \phi ^{\prime }=\phi -{\frac {1}{c}}{\frac {\partial \Lambda }{\partial t}}}$
${\displaystyle \psi \to \psi ^{\prime }=U\psi }$
${\displaystyle H\to H^{\prime }=UHU^{\dagger },}$
where
${\displaystyle U=\exp \left({\frac {iq\Lambda }{\hbar c}}\right)}$
and Λ=Λ(x,t) is the gauge function.
${\displaystyle L=r\times p\,\!}$
and obeys the canonical quantization relations
${\displaystyle [L_{i},L_{j}]=i\hbar {\epsilon _{ijk}}L_{k}}$
defining the Lie algebra for so(3), where ${\displaystyle \epsilon _{ijk}}$ is the Levi-Civita symbol. Under gauge transformations, the angular momentum transforms as
${\displaystyle \langle \psi \vert L\vert \psi \rangle \to \langle \psi ^{\prime }\vert L^{\prime }\vert \psi ^{\prime }\rangle =\langle \psi \vert L\vert \psi \rangle +{\frac {q}{\hbar c}}\langle \psi \vert r\times \nabla \Lambda \vert \psi \rangle \,.}$
The gauge-invariant angular momentum (or "kinetic angular momentum") is given by
${\displaystyle K=r\times \left(p-{\frac {qA}{c}}\right),}$
which has the commutation relations
${\displaystyle [K_{i},K_{j}]=i\hbar {\epsilon _{ij}}^{\,k}\left(K_{k}+{\frac {q\hbar }{c}}x_{k}\left(x\cdot B\right)\right)}$
where
${\displaystyle B=\nabla \times A}$
is the magnetic field. The inequivalence of these two formulations shows up in the Zeeman effect and the Aharonov–Bohm effect.
## Angular momentum operators
From Lx = y pzz py, etc., it follows directly from the above that
${\displaystyle [{L_{x}},{L_{y}}]=i\hbar \epsilon _{xyz}{L_{z}},}$
where ${\displaystyle \epsilon _{xyz}}$ is the Levi-Civita symbol and simply reverses the sign of the answer under pairwise interchange of the indices. An analogous relation holds for the spin operators.
All such nontrivial commutation relations for pairs of operators lead to corresponding uncertainty relations,[6] involving positive semi-definite expectation contributions by their respective commutators and anticommutators. In general, for two Hermitian operators Template:Mvar and Template:Mvar, consider expectation values in a system in the state Template:Mvar, the variances around the corresponding expectation values being , etc.
Then
${\displaystyle \Delta A\,\Delta B\geq {\frac {1}{2}}{\sqrt {\left|\left\langle \left[{A},{B}\right]\right\rangle \right|^{2}+\left|\left\langle \left\{A-\langle A\rangle ,B-\langle B\rangle \right\}\right\rangle \right|^{2}}},}$
where [A, B] ≡ A BB A is the commutator of Template:Mvar and Template:Mvar, and {A, B} ≡ A B + B A is the anticommutator.
This follows through use of the Cauchy–Schwarz inequality, since , and A B = ([A, B] + {A, B})/2 ; and similarly for the shifted operators and . (cf. Uncertainty principle derivations.)
Judicious choices for Template:Mvar and Template:Mvar yield Heisenberg's familiar uncertainty relation for Template:Mvar and Template:Mvar, as usual.
Here, for Template:Mvar and Template:Mvar,[6] in angular momentum multiplets , one has Template:LangleLx2Template:Rangle = Template:LangleLy2Template:Rangle = (Template:Ell (Template:Ell + 1) − m2) ℏ2/2 , so the above inequality yields useful constraints such as a lower bound on the Casimir invariant Template:Ell (Template:Ell + 1) ≥ m (m + 1), and hence , among others.
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2023-03-23 13:36:36
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https://math.stackexchange.com/questions/1208957/linear-independence-of-vectors-and-solutions-to-systems-of-equations
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# Linear independence of vectors and solutions to systems of equations
I am wondering whether the following argument is correct: "Let $V$ be a set of vectors in $\mathbb{R}^m$, $$V=\{v_1,v_2,…, v_n\}$$ Then if $n>m$, the vectors in $V$ are linearly dependent and $$λ_1v_1+λ_2v_2+ \ldots + λ_nv_n=0$$ has only the trivial solution." Shouldn't the second part state that the equation has infinitely many solutions including the trivial one? Thank you very much in advance for your answers.
It should say "...has non-trivial solution", because $\;\{v_1,...,v_n\}\;$ being linearly dependent means exactly that, namely: there exist scalars $\;a_i\in\Bbb R\;$ not all zero s.t.
$$\sum_{k=1}^n a_kv_k=0$$
Since the field ($\;\Bbb R\;$) is infinite, this in fact means there are infinite non-trivial choices of the above scalars .
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2020-11-26 12:04:15
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https://aquazorcarson.wordpress.com/2013/04/21/necessary-or-sufficient-conditions-for-no-arbitrage/
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## necessary or sufficient conditions for no-arbitrage
In work I have had to work with extending volatility surface interpolation and extrapolation models to fit market data. One question I kept asking myself is, what exactly is the significance of some sort of dynamics underlying the volatility movement in time? As someone from a geometric background, my first intuition for surface fitting is always to take a geometrically natural object such as a minimal surface, satisfying some obvious arbitrage free conditions such as positivity. This can be easily accomplished if one transform the surface into log space. More explicitly, we look at the set of pairs $(t_i, K_i, \log v_i)$ where $v_i$ is the implied volatility at strike $K_i$ at maturity $t_i$, and simply find a minimal surface passing through all these points, that is, ensuring that the mean curvature $H =$ trace of the curvature operator (or the sum of the principal curvatures for the 3-d hypersurface) is zero. Here using logarithm to transform the target vol values is arbitrary, any function that maps the positive axis to the entire real line would do. But surely practitioners have given thought to this highly academic approach. One immediate objection I can see is that as time moves forward, certain “continuity” behavior is not captured by this naive model: e.g., if we look at 1 yr implied vol now, versus 9 month implied vol 3 months later, the minimal surface approach might not yield any connection between the two. What’s still not clear to me is a. how seriously one should take this intuitive term structure correlation, and b. how well do existing models such as Sabr capture such correlation compared to a non-dynamical approach. This I will certainly spend time investigating in the future. But while searching for related topic, I ran into a paper by Peter Carr and D.P. Madan on equivalent conditions for no-arbitrage in a set of market quotes. Since their paper is quite short I read it pretty quickly:
Their main proposal is to discrete time and strike, and confine the set of arbitrage strategies to so-called static arbitrages. This means essentially that the strategies at any moment can only depend on the prices of assets at that moment, i.e., a Markovian control. While the equivalence between arbitrage-freeness and existence of equivalent Martingale measure holds for arbitrary adapted strategies with mild and practical regularity assumptions, the problem of identifying arbitrage is much more difficult (especially in the continuous setting) if the strategy is allowed to depend on the past history of the asset prices, even though it’s not clear to me how knowing the past in addition to the present can exploit additional arbitrages. But the main contribution of the paper, leveraging off an exotic ancient paper by Kellerer written in German, is that restricting to Markovian strategies, absence of arbitrage is equivalent to absence of three specific types of arbitrages: call spread, butterfly spread, and calendar spread arbitrages.
Knowing the absence of these three specific types of arbs, the authors argue that similar statements hold for all convex payoff instruments via standard static replication approach. This then implies via Kellerer that there is a discrete time Markovian Martingale $M_t$ with which every quoted call price can be expressed as the expected value of the standard call payoff $E_t (M_t - K)_+$. If we then price every other instrument with this Martingale, then arbitrage-freeness is guaranteed.
The significance of results in this paper is that since there is a finite (and not too large) number of quotes available in the market at any given time, checking that the three types of arbs are absent among those quotes is fairly straightforward.
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2017-07-22 02:36:11
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http://frnsys.com/ai_notes/scratch/novelty_surprise_search.html
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# Gravina, D., Liapis, A., & Yannakakis, G. N. (2016). Surprise search: Beyond objectives and novelty. In Proceedings of the Genetic and Evolutionary Computation Conference. ACM.
The problem with using only a fitness function (i.e. an explicit objective) for evolutionary algorithms: local optima.
In the context of evolutionary algorithms, deception describes when the combination of highly-fit components result in a solution further from the global optimum (towards a local optimum). Deception, in combination with sampling error and the ruggedness of a fitness landscape, are the main contributors to the difficulty of an evolutionary computation problem.
This problem can be mitigated by incorporating a novelty criteria - the dissimilarity of a new solution against existing solutions. This novelty search typically results in better solutions significantly more quickly than the conventional fitness-based approach.
The general approach involves keeping track of previous highly-novel solutions (a "novel archive"). The novelty of a solution is the average distance from either neighbors in the novel archive or in the current population. The particular distance measure is problem-dependent.
The paper proposes an additional criteria of surprise. Whereas novelty just requires that a solution be sufficiently different from existing ones, surprise also requires deviation from expectations.
One way of understanding the distinction is by viewing surprise as the time derivative of novelty; i.e. novelty is position, surprise is velocity.
To quantify the surprise of a solution, a surprise archive is maintained of the past $h$ generations and a predictive model $m$ is learned to generate expectations. A new solution is compared against the members $k$ (the level of prediction locality) groups (if $k=1$, compared to the entire population $P$, if $k=P$, compared to each individual of the population). $k$-mean sis used to form the population groups; each generation (except the first) uses the $k$ centroids from the previous generation. The expectations $p$ is a function of these, i.e. $p = m(h,k)$. Each population group is used to generate a expectation.
The surprise value $s$ of a solution $i$ is computed as the average distance to the $n$-nearest expectations:
$$s(i) = \frac{1}{n} \sum_{j=0}^n d_s (i, p_{i,j})$$
where $d_s$ is the domain-dependent measure of behavioral difference between an individual and its expectation and $p_{i,j}$ is the $j$-closest prediction point (expectation) to individual $i$.
see also: Liapis, A., Yannakakis, G. N., & Togelius, J. (2015). [Constrained novelty search: A study on game content generation](http://julian.togelius.com/Liapis2014Constrained.pdf_. Evolutionary computation, 23(1), 101-129.
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2017-11-19 04:42:43
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http://mathhelpforum.com/advanced-statistics/46216-another-goodness-fit-test-grrrr-print.html
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# Another Goodness of Fit test....GRRRR
• August 18th 2008, 09:08 PM
sebchase0625
Another Goodness of Fit test....GRRRR
I'm so confused with this goodness of fit stuff. I'm trying to solve the following problem...can anyone help??
The safety director of Honda USA took samples at random from the file of minor accidents and classified them according to the time the accident took place. Using the goodness-of-fit test and the .01 level of significance, determine whether the accidents are evenly distributed (uniform) throughout the day. Write a brief explanation of your conclusion. (accidents)
Time
Number of Accidents
8 up to 9 AM
6
9 up to 10 AM
6
10 up to 11 AM
20
11 up to 12 PM
8
1 up to 2 PM
7
2 up to 3 PM
8
3 up to 4 PM
19
4 up to 5 PM
6
• August 18th 2008, 10:20 PM
mr fantastic
Quote:
Originally Posted by sebchase0625
I'm so confused with this goodness of fit stuff. I'm trying to solve the following problem...can anyone help??
The safety director of Honda USA took samples at random from the file of minor accidents and classified them according to the time the accident took place. Using the goodness-of-fit test and the .01 level of significance, determine whether the accidents are evenly distributed (uniform) throughout the day. Write a brief explanation of your conclusion. (accidents)
Time
Number of Accidents
8 up to 9 AM
6
9 up to 10 AM
6
10 up to 11 AM
20
11 up to 12 PM
8
1 up to 2 PM
7
2 up to 3 PM
8
3 up to 4 PM
19
4 up to 5 PM
6
You can use a chi-square test for goodness of fit. The null hypothesis is
H0: $p_1 = p_2 = p_3 = p_4 = p_5 = p_6 = p_7 = p_8 = \frac{1}{8}$
where $p_i$ is the probability of an accident in the ith hourly interval for i = 1, 2, ...... 8.
The chi-square test statistic will have 8 - 1 = 7 degrees of freedom.
Now calculate the value of $X^2$ and test it for significance at the 0.01 level: $X^2 = \sum_{i=1}^{8} \frac{(n_i - n p_i)^2}{np_i}$
where n is the total number of accidents and $n_i$ is the number of accidents in the ith hourly interval (you should be familiar with this formula and where it comes from).
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2015-12-01 01:30:19
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