text:
How to use TikZ nodes within tabular and within other nodes, drawing paths between them?

I'm trying to use TikZ to reimplement a diagram I did using PSTricks, and I think I must have the wrong mental model. Here's how I'm tackling the problem:

• The overall diagram layout is made with a tabular environment.

• The typical cell within the tabular layout needs to be a TikZ node, so I can draw curves that point to it.

• Sometimes a cell contains a bullet, which needs to be the origin of a curve drawn to the edge of another cell.

• Some truly exciting cells stack text on top of another little diagram which itself contains two bullets, each of which originates an arrow that is drawn to another node.

Here's how I've tried to solve the problem:

1. Each cell within the tabular has its own \begin{tikzpicture}...\end{tikzpicture, with the remember picture option.

2. Each cell is a node that I can point to.

3. If the contents of a cell is just text, then it is just text.

4. If the contents of a cell is more complicated, I put another \begin{tikzpicture}...\end{tikzpicture} in the contents of the node.

5. Each curve, more or less, is in its own \begin{tikzpicture}...\end{tikzpicture}.

I'm thrashing around getting this to work, mostly by throwing overlay in more or less at random. But I don't have a mental model. What I've learned from searching TeX.se and elsewhere is "don't do that." What should I be doing?

What should I read in order to develop a suitable mental model for specifying diagrams within TeX within diagrams within TeX? And so on?

• One option is to make each cell content a \tikzmarknode, which requires the tikzmark library. Then the picture in which you draw the connections needs to have the keys remember picture,overlay. (You may use collcell to automatize that.) Another way is to use a TikZ matrix of nodes, which requires the matrix library, to do the whole table in one tikzpicture.
– user194703
Dec 18, 2019 at 1:51
• I second making the whole diagram a single tikzpicture and using a matrix to control the layout. Dec 18, 2019 at 6:45

Here is an example using tikzmark. The picture in which you draw the connections needs to have the keys remember picture,overlay.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{tikzmark}
\tikzset{bullet/.style={circle,inner sep=0.2ex,fill}}
\begin{document}
\begin{table}[htb]
% note that the table borders are just for illustration
% this works with bookmarks, too
\centering
\begin{tabular}{|l|c|l|}
\hline
\tikzmarknode{bla}{bla} & \tikzmarknode{bla2}{bla} & \tikzmarknode{blub}{blub}\\
\hline
\tikzmarknode[bullet]{b1}{} &
\begin{tikzpicture}[remember picture,baseline={(0,0)}]
\draw[thick,blue] plot[variable=\x,domain=0:3] (\x,{sin(\x*120)});
\path (0.5,1.2) node[bullet] (b3){} (1.5,1.2) node[bullet] (b4){}
(0,1.4) (0,-1.2);
\end{tikzpicture}
& \tikzmarknode[bullet]{b2}{} \\
\hline
\end{tabular}
\begin{tikzpicture}[overlay,remember picture]
\draw[red,-stealth] (bla) to[bend left] (bla2);
\draw[red,-stealth] (bla2) to[bend left] (blub);
\draw[red,-stealth] (bla) to[bend left] (b1);
\draw[red,-stealth] (bla) to[bend left] (b3);
\draw[red,-stealth] (blub) to[bend left] (b2);
\draw[red,-stealth] (blub) to[bend left] (b4);
\end{tikzpicture}
\caption{A table with \texttt{\textbackslash tikzmarknode}s,
\texttt{tikzpicture}s and Ti\emph{k}Z annotations.}
\end{table}

\end{document}


text:
# parameter distribution and predictive distribution

This question is from Bishop's pattern recognition's chapter 3 about Bayesian linear regression. Chapter 3 starts with MLE and regularization to solve the overfitting problem.

But MLE and regularization are not enough to solve the overfitting problem, hence the book comes up with Bayesian linear regression methods and I find it confusing.

they define the ${\bf w}$ as a random variable $$p({\bf w}) = N({\bf w}|{\bf m}_0, {\bf S}_0) \qquad{(3.48)}$$ I think this ${\bf w } \$is only to represent the belief of the prior(subjective) then they just defined it as mean ${\bf m_0}$ and ${\bf S_0}\$ as variance.

$${\bf m}_N = {\bf S}_N(S_0^{-1}{\bf m}_0+\beta\Phi^T{\bf t}) \qquad{(3.50)}$$

$${\bf S}_N^{-1} = {\bf S}_0^{-1}+\beta\Phi^T\Phi \qquad{(3.51)}$$

and finally they come up with log likelihood function

$$\ln p({\bf w}|{\bf t}) = -\frac{\beta}{2}\sum_{n=1}^{N}\{t_n-{\bf w}^T\phi({\bf x}_n)\}^2 - \frac{\alpha}{2}{\bf w}^T{\bf w}+const \qquad{(3.55)}$$

I'm not really sure with the ${\bf t} \$ terms in (3.55) equation . I think this ${\bf t} \$ represent the target from some sample data.(if it's right is this sample data also selected as randomly?)

after they find the log likelihood function(3.55). they are doing "predictive distribution".

$$p(t|{\bf t}, \alpha, \beta) = \int p(t|{\bf w}, \beta)p({\bf w}|{\bf t}, \alpha, \beta)d{\bf w} \qquad{(3.57)}$$

also confused about what is t and ${\bf t} \$ represent in the equation. and also what they are predicting?

this (3.57) equation mean and variance is founded as

$$p(t\;|\;{\bf t}, \alpha, \beta) = N(t\;|\;{\bf m}_N^T\phi({\bf x}), \sigma_N^2({\bf x})) \qquad{(3.58)}$$

$$\sigma_N^2({\bf x}) = \frac{1}{\beta} + \phi({\bf x})^T S_N\phi({\bf x}) \qquad{(3.59)}$$

this variance's first term represent the original noise's variance and they said this second term represent the uncertainty of the ${\bf w }\$ how they know this part represent the uncertainty of the ${\bf w }\$

and they said $( \sigma\_{N+1}^2({\bf x}) \le \sigma\_N^2({\bf x}) )$ and when N is bigger it becomes narrower(I don't understand how second term is going narrower when N is increase)

this picture shows when N grows. I guess this red line just represent the random variable and it changes as N. but what does green line represent for?

I'm trying to clarify the each terms. thank you

format: plain


text:
# An equation relating Time complexity, Space complexity, and entropy of output

Is there an equation that relates minimum time complexity, minimum space complexity, and entropy of the output of a function? It seems to me that there should be a relatively intuitive relationship between the three, something like $$E=f(T+S)$$ where $$E$$ is shannon entropy of the output, $$T$$ is time, $$S$$ is space, and f is a linear function or something like that. The main argument being that any function can be written as a hash map that takes its input to its output. This hashmap could be compressed based on its entropy.

I have a hunch that the only way to decrease the space an algorithm uses more than that is to increase the time complexity, as if the hashmap is moved onto a time axis instead of a space one. Is this crazy? Has there been any research done on this subject?

Edit for clarification: What I mean by "entropy of the output of a function" is this. Assume a function $$g$$ takes as input a bit string $$x$$ and outputs a message $$y$$. Now $$g$$ can thought of a source who's $$x$$th output is $$y$$. By "entropy of the output" I mean the shannon entropy of $$g$$ when thought of as a source.

Again, the argument is that $$g$$ can be implemented using a hash map in exponential space but in $$O(1)$$ time. However, I have a hunch that it seems reasonable that this space complexity can be reduced based on the entropy of $$g$$ (i.e. the entropy of the hashmap) and still maintain $$O(1)$$ time, essentially by compressing the hash map.

Beyond this reduction I also have a hunch that that you can decrease the space further, but in doing so you have to increase time proportionally. The idea being that you can't actually compress the information in the hash map any more than you already have (even when you compress using other measures of entropy like kolmogorov complexity), you can just "move" it along the time axis.

• What do you mean by entropy of the output of a function? – Martin Berger Feb 9 '19 at 10:27
• @MartinBerger Edited for clarification – William Oliver Feb 9 '19 at 14:22
• Shannon entropy requires a random source. Where's the randomness in your setup? – Aryeh Feb 9 '19 at 19:21
• @Aryeh Assume that when we think of $g$ as a source it actually is a random source. The probability of each symbol being given by the limit of the sample probability of each symbol as the length of the input goes to infinity. For the sake of this question, you can assume each event is independent. – William Oliver Feb 9 '19 at 20:05

(Too long for a comment.)

There are two aspects to your question. There is the idea of a time-"space" tradeoff, and the idea of entropy as a measure or bound for how hard this tradeoff must be for a problem. Both aspects have some issues relating to how theorists usually study these problems. Here is why and some suggestions.

Regarding entropy: This is an interesting idea but either wrong or hard to formalize for several reasons. Notice that a program that just spits out a long sequence of independent bits (whether it produces them or reads them from an input) has very high entropy but is also very simple. Its running time and space requirements are minimal. And in general there are deterministic algorithms with 0 entropy that solve very difficult problems on both time and space axis. Finally, many problems of theoretical interest are decision problems where the output is just one bit.

So entropy seems unrelated at least from usual perspectives, and you have to think it through more or else switch to a notion like Kolmogorov complexity (e.g. compressibility of the truth-table of a decision problem or something).

Regarding time-space tradeoffs: It's hard to reconcile your ideas with what we traditionally call space-time tradeoffs, which refer to time and memory usage of an algorithm starting from scratch (unlike your hashtables etc). It is studied but very, very hard. You are asking to go even farther and relate this tradeoff to some other notion of complexity of the problem, so we are probably even farther from those results (maybe not in some specific domain?).

However, I do think your intuition can be captured by complexity classes involving "advice". For instance, P/poly are problems solved by polynomial time algorithms with access to polynomial-length advice strings. You seem to be observing that all problems are in O(n)/EXP: trivial linear-time algorithm, or constant time in word RAM model, that has access to a lookup table for all answers. Whereas the problem might not be in P/poly or might not be in Exp/O(1) or so on. We trade off advice space for running time. I don't know of work on this -- I'm not an expert so maybe it exists, but it sounds hard. A general theorem relating this tradeoff to a measure of complexity of the problem would be very cool, but even harder.

• I think this answers my question about whether there is existing research on the subject, but the criticisms you've come up with have crossed my mind. The key thing is that the "a program that just spits out a long sequence of independent bits" only has high entropy if the input has high entropy, so the entropy is actually the entropy of the input, not the entropy of the output. An encryption algorithm, for example, is pretty useless unless the method of key generation has high entropy. – William Oliver Feb 10 '19 at 20:57
• Furthermore, a decision problem outputs just one bit for one input, yes, but this doesn't mean that the hashtable mentioned doesn't have high entropy. This will be as many bits (computer bits, not bits as in shannon bits) as there are possible inputs. – William Oliver Feb 10 '19 at 20:58


text:
# Cabibbo Theory

Hi,

I'm trying to cover the basics of Cabibbo theory, yet the materials Ive been presented with give a very jumbled description, and I'd just like to ask here to obtain some clarity...

Is it fair to say that the Cabibbo angle is a means of quantifying the different coupling strengths between different generations of quarks?

Or is this too much a simplification or just plain ol wrong?

Regards

Luke


text:
# Fatigue analysis type in structural design

Fatigue analysis type serves to determine normal and shear stress range between two load cases. The stresses correspond to nominal stresses and must be further evaluated using code design methods. It is assumed to be used for the design of high-cycle fatigue details, where no yielding is expected.
$$The fatigue analysis type does not provide any final resistance or number of cycles the detail can take. It just provides inputs to further calculations according to codes.Always, at least two load cases must be set. The first load case is the reference one. It is assumed as, e.g., a self-weight of the structure and can contain zero loads. The other load cases are simulating fatigue actions. The nominal normal and shear stress provided by IDEA StatiCa is the stress range between the fatigue action, e.g. LE2, and the reference load case.For example, the shear stress at a certain location is 50 MPa in the Reference load case and 180 MPa in LE2. The shown nominal shear stress at this location is:$\tau = 180-50=130\, \textrm{MPa}$Note that there should be no yielding of plates due to fatigue actions, otherwise the stress ranges are distorted. The stresses are available for: Bolts Welds Plates BoltsAt bolts, the stresses are determined simply by dividing the force by the corresponding area: $$\sigma = F_t / A_s$$ $$\tau = V / A$$ where: $$F_t$$ – tensile force in the bolt $$A_s$$ – tensile stress area of the bolt $$V$$ – shear force in bolt; if there are multiple shear planes, the highest shear force is used $$A$$ – area of the bolt resisting in shear; tensile stress area if threads are intercepted by the shear plane and gross cross-sectional area otherwise WeldsWelds in CBFEM consist of the weld element with multipoint constraints connecting the plates. The stress distribution in the weld is disrupted by the constraints and, therefore, the stresses are taken from a section located at 1.5 times the leg size from the weld toe. Three sections are created for a double-sided fillet weld. Two sections are in the same detail category, and only the more stressed one is shown. The maximum normal stress and the corresponding shear stress at the same location, as well as the maximum shear stress and the corresponding normal stress at the same location, are shown. PlatesThe stress in plates may be visualized by creating a user-defined section by a Workplane manufacturing operation. In the figure below, two workplaces were created to see the stresses around bolt holes. The maximum normal stress and the corresponding shear stress at the same location, as well as the maximum shear stress and the corresponding normal stress at the same location, are shown.$$


text:
# Integration

Intergrate ¡ì sec^3(x) dx

= ¡ì sec x d tan x
= sec x tan x - ¡ì tan x d sec x
= sec x tan x - ¡ì sec x tan^2(x) dx
= sec x tan x + ¡ì sec x dx - ¡ì sec^3(x) dx
= sec x tan x + ln |sec x + tan x| - ¡ì sec^3(x) dx
=¡ì sec^3(x) dx = (1/2)(sec x tan x + ln |sec x + tan x|) + C1

¡ì [3x sin x/cos^4(x)] dx
= -3 ¡ì [x/cos^4(x)] d cos x
= ¡ì x d sec^3(x)
= x sec^3(x) - ¡ì sec^3(x) dx
= x sec^3(x) - (1/2) sec x tan x - (1/2) ln |sec x + tan x| + C2

I'm not sure if your integration is correct or not, not all of your symbols converted to ASCII. I plugged sec3(x) into a piece of software and got an answer that looks slightly different from yours, but I'm not positive. If you still need help with this post a new question so it's easy to find.

1. s=integral
let i=S(sec^3x)dx
i=S(sec^3x)=S(sec^2x.secx).dx
=tanxsecx-S(tan^2x.secx)dx=
tanxsecx-S(sec^2x-1)secx.dx=
tanxsecx-S(sec^3x)dx+S(secx)dx
2i=tanxsecx+ln|secx+tanx|+x
i=(tanxsecx+ln|secx+tanx|+c)(0.5)

posted by sherif
2. -cos(2x)upon2

posted by G.K

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text:
Simple Git Guide

This post is based primarily on a great git-tutorial git-scm.com.

1. Types of files

The files in your working directory may be of different states:

• tracked files are files that were in your last snapshot (they are already preserved in your git repository),
• untracked files are all the others, i.e. that are files that were not in your last commit and not in your staging area (not ready to commit).

Starting a new git repository

The post briefly describes how to create git repository.

Schedule R Code for Windows

This post  explains how to set your code to work monthly/weekly/daily which is very useful for data monitoring, scraping or automatic reports. In Mac and Linux, cron is usually used for this purpose. Windows users can work with Task Schedular, using the command line or special R features.

Principal Component Analysis in R

The full information on the theory of principal component analysis may be found here. This article is about practice in R. It covers main steps in data preprocessing, compares R results with theoretical calculations, shows how to analyze principal components and use it for dimensionality reduction. The last section is devoted to modelling using principal components and comparing it to LDA.

Principal Component Analysis

Principal component analysis (PCA) is an unsupervised method of generating components from a large set of variables available if a data set which represent a combinations of features that capture as much information in the data as possible. Or in a Wikipedia way:

A statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components.

Eigenvectors and eigenvalues

Spectral theorem:

If A is symmetric, then A is orthogonally diagonalizable and has only real eigenvalues. In other words, there exist real numbers λ_1, …, λ_n (the eigenvalues) and orthogonal, non-zero vectors v_1, …, v_n (the eigenvectors) such that for each i = 1,2,…,n:

$A v_i = \lambda_i v_i$.Read More »

Matrices in NumPy

NumPy: Matrices

import numpy as np
a = np.array([[1, 2, 3], [2, 5, 6], [6, 7, 4]])

b = np.eye(5)
c = np.ones((7, 5))
d = np.zeros((7, 5)) # tuple as argument!

v = np.arange(0, 24, 2) # start, stop, step as arguments
d = v.reshape((3, 4)) # reshape tells dim of matrix

Indexing:

print (d[2, 1]) # print an element
print (d[[1, 0], [2, 3]]) # print two elements
print(d[1,:]) # print a row
print([d[:,3]) # print a column

Multiplication:

r1 = np.dot(a, b)
r2 = a.dot(b)

r = a * b # is a multiplication by coordinates

Matrix operations

# Transposing:
b = np.transpose(a)
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text:
#### Maximum entropy models capture melodic styles

##### Jason Sakellariou, Francesca Tria, Vittorio Loreto, François Pachet

We introduce a Maximum Entropy model able to capture the statistics of melodies in music. The model can be used to generate new melodies that emulate the style of the musical corpus which was used to train it. Instead of using the $n-$body interactions of $(n-1)-$order Markov models, traditionally used in automatic music generation, we use a $k-$nearest neighbour model with pairwise interactions only. In that way, we keep the number of parameters low and avoid over-fitting problems typical of Markov models. We show that long-range musical phrases don't need to be explicitly enforced using high-order Markov interactions, but can instead emerge from multiple, competing, pairwise interactions. We validate our Maximum Entropy model by contrasting how much the generated sequences capture the style of the original corpus without plagiarizing it. To this end we use a data-compression approach to discriminate the levels of borrowing and innovation featured by the artificial sequences. The results show that our modelling scheme outperforms both fixed-order and variable-order Markov models. This shows that, despite being based only on pairwise interactions, this Maximum Entropy scheme opens the possibility to generate musically sensible alterations of the original phrases, providing a way to generate innovation.

arrow_drop_up


text:
### Блог пользователя MikeMirzayanov

Автор MikeMirzayanov, 10 месяцев назад, ,

Hello!

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The XTX Markets Global Forecasting Challenge is an online competition for aspiring quantitative professionals, where contestants are tasked to develop a predictive model based on training data provided by XTX.

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The competition is open to all data-minded contestants from 1st of July to 30th of September, 2019.

APPLY HERE→

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• +134

 » 9 месяцев назад, # |   -65 Sorry I'm not interested.
•  » » 9 месяцев назад, # ^ | ← Rev. 2 →   -26 Sorry I'm not interested. Me too!
 » 9 месяцев назад, # |   -34 Great Contest from HEADQUARTERS MikeMirzayanov
 » 9 месяцев назад, # |   +1 I don't know why but that page is not responding. It just freezed with title "redirecting...". Has anyone encountered this problem?
•  » » 9 месяцев назад, # ^ |   +20 Oooops... Now everything is fine xd.


text:
# Boaler (2002) FLM

The Development of Disciplinary Relationships: Knowledge, Practice and Identity in Mathematics Classrooms

## Abstract

The article does not have an abstract.

• The relationships between knowledge and practice
• Relationships between knowledge, practice and identity
• Developing relationships with the discipline of mathematics
• Discussion and conclusion

## Summary

Boaler wrote this article as a reflection of how she'd come to see the learning experience and "relationships between teaching and learning, between knowledge and practice, and between learning and believing" (p. 42). Boaler's experience in classrooms coincides with the "cognitive revolution" (Schoenfeld, 1999; Resnick, 1991) that superseded behaviorist views of learning that claimed knowledge was stable, individual, and transferrable. Mathematical learning was seen to rely on opportunities to acquire knowledge that was taught and practiced to reinforce behaviors. Constructivists believed that knowledge was not acquired by students, but that students actively organized and reorganized ideas as part of their own constructions (Lerman, 1996). While different, both of these perspectives see knowledge as characteristic of the individual and something that can be taken from place to place.

Learning can also be seen from a situated perspective where knowledge is distributed among people, activities, and their environment (Lave, 1988; Greeno and MMAP, 1998; Boaler, 2000; Cobb, 2000). Situated learning reflects how people use and co-construct knowledge in ways that depend on their environment. Therefore, while cognitive processes may still be important (Greeno, 1997), they are seen as occurring as part of a broader context in which they occur (Greeno and MMAP, 1998) and knowledge is no longer viewed as exclusive to the individual.

In her research of two contrasting high schools, "Amber Hill" and "Phoenix Park," Boaler (1997) investigated the relationship between teaching, student beliefs, and student understanding. She showed that in Amber Hill, where students spent much of their time working on textbook exercises, students came away with an ability to do textbook-style problems but struggled in more open or applied problem-solving situations. In contrast, students at Phoenix Hill spent most of their time in project-based approaches to mathematics. Those students outperformed the Amber Hill students on the national examination, despite similar scores prior to the curriculum intervention, and despite the exam looking mostly like textbook exercises. A cognitive interpretation of these results would be that the Phoenix Park students simply learned more. Boaler, however, acknowledged that the Amber Hill students learned a great deal but struggled to apply that knowledge beyond the practice of completing prescriptive textbook exercises. Therefore, the practices students engage in as they do mathematics are important — not because it necessarily affects the amount learned, but because the practices shape the students' knowledge itself (Dowling, 1996). Boaler concluded that for students to use and apply their mathematics knowledge flexibly, they would have to engage in mathematical practices that engage them in similarly flexible learning experiences.

Along with this more situated view of mathematical learning, Boaler included student identity as a dimension along with knowledge and practice. In a study of AP Calculus students (Boaler & Greeno, 2000), students in four of the six high schools in the study engaged in a more traditional, procedure-focused mathematics similar to students at Amber Hill. In the other two high schools, there was a greater focus on student discussion and collaboration around problem solving. Boaler related the traditional approach to the idea of "received knowing" (Belencky, Clinchy, Goldberger, & Tarule, 1986) where knowledge is presented to students in a way that gives mathematical authority to the teacher and the textbook (Ball, 1993), making the students "received knowers." Students in these classes, while generally successful, reported not liking mathematics and feeling too constrained in practices that "left no room for their own interpretation or agency" (p. 44). Many of these students wished to not take mathematics in the future, even though the pedagogical practices of their mathematics experience were unrelated to the nature of mathematics itself (Burton, 1999a, 1999b). Some students in these classes reported being satisfied and motivated, apparently pleased to be a "received knower":

Jerry, Lemon School: I always like subjects where there is a definite right or wrong answer. That's why I'm not a very inclined or good English student. Because I don't really think about how or why something is the way it is. I just like math because it is or isn't. (p. 44)

In the discussion-oriented classes, students formed a relationship with mathematics that did not seem to conflict with their identities as creative problem-solvers:

Veena, Orange school: Sometimes you sit there and go 'It's fun!' I'm a very verbal person and I'll just ask a question and even if I sound like a total idiot and it's a stupid question I'm just not seeing it, but usually it clicks for me pretty easily and then I can go on and work on it. But at first sometimes you just sit there and ask — 'What is she teaching us?' 'What am I learning?' But then it clicks, there's this certain point when it just connects and you see the connection and you get it. (p. 44)

Student identity, which includes the relationship students had with mathematics, was an important factor in Boaler's study even though student achievement in the two types of schools was similar. In the discussion-oriented classes students described their participation in mathematics in ways consistent with how Wenger (1998) described learning as a process of "becoming":

Because learning transforms who we are and what we can do, it is an experience of identity. It is not just an accumulation of skills and information, but a process of becoming — to become a certain person or, conversely, to avoid becoming a certain person. Even the learning that we do entirely by ourselves eventually contributes to making us into a specific kind of person. We accumulate skills and information, not in the abstract as ends in themselves, but in the service of an identity. (p. 215)

With this Boaler had reflexively related knowledge and practice, and seen how practice influenced identity, but had yet to connect knowledge to identity. For this she engaged in new research involving about 600 students in three high schools. (See note.) As students engaged in either traditional or reform-based courses, student identity and agency played an important role. In reform classrooms, students developed agency as they proposed "theories" and critiqued each other's ideas in ways that did not center mathematical authority with the teacher or the textbook. Boaler acknowledged that such agency raised concerns (Rosen, 2000) that students will spend much of their time and effort unproductively and are left with a "fuzzy" mathematics (Becker & Jacob, 2000). But in her study, Boaler found that the nature of the agency bore similarities to what Pickering (1995) observed in the work of professional mathematicians. Pickering saw two main types of agency at work when professional mathematicians made mathematical advances: a human agency to propose new approaches and ideas, as well as a disciplinary agency that seemed to prescribe certain standard methods or approaches. In Boaler's reform-oriented classrooms, students could more regularly engage in both types of agency, whereas traditional classrooms were dominated by disciplinary agency. This "dance of agency" in the reform classrooms was evident as students described proposing new approaches, testing those approaches, and evaluating their success as they found ways forward.

While other researchers have described the importance of student beliefs and dispositions (Schoenfeld, 1992; McLeod, 1992), Boaler saw a connection between knowledge and identity she described as a "disciplinary relationship" where knowledge is not so much mastered as it is appropriated. Herrenkohl & Wertsch (1999) described the appropriation of knowledge as a student's need to not just develop skills, but to develop a disposition to use them. Boaler believes the students at Phoenix Park and in other reform classrooms had developed a "positive, active relationship with mathematics" (p. 46) that made them able to apply more knowledge in different contexts. In the case of professional mathematicians, Boaler believes that a mathematician without some particular knowledge will make progress on a problem because she would know a set of practices for problem-solving and have a productive disposition towards doing so. This combination of practices, identity, and agency will put her in a position to find the resources needed to fill the gaps in knowledge.

Boaler admits that the relationships between knowledge, practice, and identity need to be more well-defined, and more needs to be known about if or how learners appropriate particular knowledge. Also, more needs to be known about how advantages for learners vary in the "dance of agency" and how factors like gender, race, and class intersect with students' relationships with mathematics.

Note: This is almost certainly what became Boaler's well-known Railside study, described in Boaler & Staples (2008). --Raymond Johnson (talk) 03:52, 21 November 2013 (UTC)

## Also

APA
Boaler, J. (2002). The development of disciplinary relationships: Knowledge, practice and identity in mathematics classrooms. For the Learning of Mathematics, 22(1), 42–47.
BibTeX
@article{Boaler2002,
author = {Boaler, Jo},
journal = {For the Learning of Mathematics},
number = {1},
pages = {42--47},
title = {{The development of disciplinary relationships: Knowledge, practice and identity in mathematics classrooms}},
url = {http://www.jstor.org/stable/40248383},
volume = {22},
year = {2002}
}


text:
# How can I find minimum and maximum of a region?

I have a regionplot like the following:

m1 = 0.547;
m1p = 0.958;
m2 = 0.137;
r1 = Sqrt[m1^2 + (Sqrt[w1^2 - m2^2] + Sqrt[w2^2 - m2^2])^2] // Expand;
r2 = Sqrt[m1^2 + (Sqrt[w1^2 - m2^2] - Sqrt[w2^2 - m2^2])^2] // Expand;
RegionPlot[
m1p - w1 - w2 < Re[r1] &&
m1p - w1 - w2 > Re[r2], {w1, .1, .25}, {w2, .1, .25},
BoundaryStyle -> Blue, FrameLabel -> {"w1", "w2"}]

How can I find minimum and maximum of w1 and w2?

-
Thanks for registering. Besides upvoting correct answers you can accept one answer to any of your questions which you find the best (by clicking a tickmark under the vote counter). Otherwise you should explain more exactly what you are expecting. For more information see mathematica.stackexchange.com/faq –  Artes Jan 17 '13 at 9:51

If you're happy with your plot you can just recycle it :

rp = RegionPlot[m1p - w1 - w2 < Re[r1] && m1p - w1 - w2 > Re[r2], {w1, .1, .25}, {w2, .1, .25},
BoundaryStyle -> Blue, FrameLabel -> {"w1", "w2"}];

minw1 = Min@rp[[1, 1, All, 1]];
maxw1 = Max@rp[[1, 1, All, 1]];
minw2 = Min@rp[[1, 1, All, 2]];
maxw2 = Max@rp[[1, 1, All, 2]];

Show[rp,
Graphics[{Red, Line[{{minw1, 0}, {minw1, 0.5}}]}],
Graphics[{Red, Line[{{maxw1, 0}, {maxw1, 0.5}}]}],
Graphics[{Red, Line[{{0, minw2}, {0.5, minw2}}]}],
Graphics[{Red, Line[{{0, maxw2}, {0.5, maxw2}}]}]]

-

You can take an initial point in your plot region and feed it to functions like FindMinimum to search for a local minimum.

For example the max value of w2 (upper boundary of the region):

FindMinimum[{-w2,
m1p - Re[r1] < w1 + w2 < m1p - Re[r2]},
{{w1, 0.1424}, {w2, 0.2399}}]

{-0.244612, {w1 -> 0.142883, w2 -> 0.244612}}

-
I tried NMinimize and didn't work too well. –  b.gatessucks Jan 16 '13 at 10:43
@b.gatessucks I tried it too with no luck. For this kind of problems, I tend more and more to a local extremum searcher now. –  Silvia Jan 16 '13 at 10:45


text:
# Prove that $f:[0,\infty)\to\mathbb{R}$ where $f(x) := {1\over x}\cos({1\over x}),x>0$ ,does $f$ has the intermediate value property on $[0,\infty)$?

Prove that $f:[0,\infty)\to\mathbb{R}$ where $f(x) := {1\over x}\cos({1\over x}),x>0$ does $f$ has the intermediate value property on $[0,\infty)$?

Attempts: In $\mathbb{R}$, if $f$ is continuous then it has intermediate value property and as we know ${1\over x}\cos({1\over x})$ are continuous on $\mathbb{R}$ so it has the intermediate value property but it doesn't seem to be that straight forward.

-
You write $f : [0, \infty) \to \mathbb{R}$ but you only define $f(x)$ for $x > 0$. Is $f$ defined at $0$? –  Michael Albanese Feb 2 '13 at 8:39
@MichaelAlbanese well, the question didn't mention, but actually any value is not a matter –  Mathematics Feb 2 '13 at 8:45
Near zero, $f$ takes on arbitrarily large positive values, immediately after or before taking on arbitrarily "large negative" values, meaning negative with large absolute value. Use the intermediate value theorem to get your result. –  coffeemath Feb 2 '13 at 8:48
@Mathematics: That is true, but if you don't know how to solve this problem, it isn't clear how you know that it doesn't matter what $f(0)$ is. On the other hand, it does matter that $f(0)$ is something. Just so we're all talking about the same thing, could you please choose some value for $f(0)$? –  Jonas Meyer Feb 2 '13 at 8:52
@JonasMeyer I guess choosing $f(0)=0$ would still work. –  Mathematics Feb 2 '13 at 8:56

The function $f\colon [0,+\infty)\to \mathbb R$ $$f(x) = \begin{cases} \frac {1}{x} \cos \frac 1 x & \text{if x>0}\\\\ 0 & \text {if x=0} \end{cases}$$ is not continuous but has the intermediate value property. In fact given two points $a,b \in [0,\infty)$ with $a<b$ you have two possibilities:

1. $a>0$. In this case notice that the function is continuous on $[a,b]$ hence it has the property
2. $a=0$. In this case you should notice that it is possible to find $\epsilon<b$ such that $f(\epsilon)=0$ (see where $\cos(1/x)=0$). Since $f$ is continuous on $[\epsilon,b]$ you can apply the intermediate value theorem there.
-

This answer is motivated by a question in the comments from the OP @Mathematics.

Note that $\cos(2\pi n)=1$ and that $\cos(2\pi n + \pi) = -1.$ It follows that the function $f(x)=(1/x) \cos(1/x)$ (with $x>0$, we don't need a definition of $f(0)$ for this approach) has the following particular values:

$$f(1/(2 \pi n))=2 \pi n,$$ $$f(1/(2 \pi n + \pi))=-(2 \pi n + \pi).$$ So given any real number $c$ we can choose $n$ large enough so that $c$ lies in the open interval $I_n=(-(2\pi n + \pi), 2 \pi n).$ By the intermediate value theorem $f(x)=c$ for some $c \in I_n.$ So we can say that $f(x)$ takes on every real number value near (but no need for at) $0$. And looking at the above sequence of intervals $I_n$ we can in fact say that $f(x)$ takes on every real number infinitely often.

-
If $f(0)$ isn't defined, then the domain of the function is $(0,\infty)$, and it is continuous there. Every continuous function on an interval has the intermediate value property. Every choice of definition for $f(0)$ will work, which perhaps is what is meant. For all $b>0$, $f((0,b))=\mathbb R$, so there is no problem hitting everything between $f(0)$ and $f(b)$, whatever $f(0)$ may be. –  Jonas Meyer Feb 2 '13 at 20:50


text:
##### Page tree

If you have any questions concerning Compliance Checker, contact cchecker-support@chemaxon.com.

## KNOWN ISSUES

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Contents of Japanese hits' (Category#: 7261-7268) What is controlled field cannot be correctly saved in PDF and SDF reports without installing an appropriate Japanese fontset to the /usr/lib/jvm/.../jre/lib/fonts/ folder and adding the report.font.name=<fontname> to the application.properties.
(E.g. the kochi-gothic-subst.ttf fontset, that can be downloaded from https://osdn.net/projects/efont/releases/ - but you need to clarify it's licensing with the issuer.)
This will be fixed after licensing issues of an appropriate font set will cleared.

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We suggest to use the Chrome browser but Firefox is working as well.
These incompatibility issues with Microsoft's browsers are investigated and will be fixed ASAP.

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The previously customized application.properties file should be saved to a backup before upgrading with the Mac and Windows installers, because these overwrite this configuration file in its default location.
After the upgrade the application.properties file (in the default ../cc-config/config folder) should be replaced with the saved config file.

#### Single check

• ##### 2D-Clean issue

Sometimes when NAME or SMILES is entered into the input box - and the 2D-Clean is slow - then the displayed structure is not 2D-Cleaned. This distorted structure image will appear even in the reports.
Clicking into the structure image the Marvin JS popup will already display the structure 2D-Cleaned. Or you can 2D-Clean it in Marvin JS by clicking the 2D-Clean button, or make any changes. When the structure image is acceptable for you then click OK in Marvin JS and Check on the Single check page.

#### File check

• ##### Cannot read molecule file

If the file contains an invalid structure record in the first 6 lines of the file then the File check fails with the following error:

In this case the file - that contains single-line format structures - still can be checked with the Extended search function, and the invalid structure will probably get passed - as a "Not found" text.

A more correct workaround can be to rename the file to CSV extension and still use the File check function. This will send the invalid structure to Error.

If the first 6 lines of the file do not contain invalid structure records then this file import error does not occur!

#### Report generation

• ##### Finished status

Sometimes the web interface does not show the real status, but usually this can be solved by a Page refresh or reloading the page with Ctrl+F5.

A new report was generated from an earlier batch check and Finished was displayed in the Generated reports section of the batch check Summary dropdown when the report generation was finished.

After a Page refresh the Download button and the Generated file name is displayed correctly.

#### Startup

• ##### Slow startup on Fedora based systems

On Fedora based systems (Fedora / RedHat / CentOS / etc) Java uses /dev/random to generate the seed of SecureRandom that is used to generate Session IDs. This causes a very slow first session generation on Compliance Checker services. You can change this value according to Oracle's documentation to have a faster but less safer SecureRandom seed.


text:
Browse Questions

# The dimension of a physical quantity can be written as $[M^aL^bT^c]$, where a, b, and c are constants. The dimension of which of the following physical quantities is correctly paired with the values of given constants?

Can you answer this question?


text:
# Sharp singular Adams inequalities in high order Sobolev spaces

@article{Lam2011SharpSA,
title={Sharp singular Adams inequalities in high order Sobolev spaces},
author={Nguyen Huong Lam and Guozhen Lu},
journal={arXiv: Analysis of PDEs},
year={2011}
}
• Published 29 December 2011
• Mathematics
• arXiv: Analysis of PDEs
In this paper, we prove a version of weighted inequalities of exponential type for fractional integrals with sharp constants in any domain of finite measure in $\mathbb{R}^{n}$. Using this we prove a sharp singular Adams inequality in high order Sobolev spaces in bounded domain at critical case. Then we prove sharp singular Adams inequalities for high order derivatives on unbounded domains. Our results extend the singular Moser-Trudinger inequalities of first order in \cite{Ad2, R, LR, AdY} to…
Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4
• Mathematics
• 2018
Abstract In this paper, we establish a sharp concentration-compactness principle associated with the singular Adams inequality on the second-order Sobolev spaces in ℝ 4 {\mathbb{R}^{4}} . We also
Sharp Affine and Improved Moser–Trudinger–Adams Type Inequalities on Unbounded Domains in the Spirit of Lions
• Mathematics
• 2017
The purpose of this paper is threefold. First, we prove sharp singular affine Moser–Trudinger inequalities on both bounded and unbounded domains in $${\mathbb {R}}^{n}$$Rn. In particular, we will
The sharp Adams type inequalities in the hyperbolic spaces under the Lorentz-Sobolev norms.
Let $2\leq m < n$ and $q \in (1,\infty)$, we denote by $W^mL^{\frac nm,q}(\mathbb H^n)$ the Lorentz-Sobolev space of order $m$ in the hyperbolic space $\mathbb H^n$. In this paper, we establish the
Sharp weighted Trudinger–Moser–Adams inequalities on the whole space and the existence of their extremals
• Mathematics
Calculus of Variations and Partial Differential Equations
• 2019
Though there have been extensive works on the existence of maximizers for sharp first order Trudinger–Moser inequalities, much less is known for that of the maximizers for higher order Adams’
Sharp Singular Trudinger-Moser-Adams Type Inequalities with Exact Growth
• Mathematics
• 2015
The main purpose of this paper is two fold. On the one hand, we review some recent progress on best constants for various sharp Moser-Trudinger and Adams inequalities in Euclidean spaces
Equivalence of critical and subcritical sharp Trudinger-Moser-Adams inequalities
• Mathematics
• 2015
Sharp Trudinger-Moser inequalities on the first order Sobolev spaces and their analogous Adams inequalities on high order Sobolev spaces play an important role in geometric analysis, partial
The main purpose of our paper is to prove sharp Adams-type inequalities in unbounded domains of R for the Sobolev space W n m (R) for any positive integer m less than n. Our results complement those

## References

SHOWING 1-10 OF 75 REFERENCES
An improved Hardy-Sobolev inequality and its application
• Mathematics
• 2001
For\Omega \subset $IR^n$,n\geq 2, a bounded domain, and for 1 < p < n, we improve the Hardy-Sobolev inequality, by adding a term with a singular weight of the type \frac{1}{log(1/|x|)}$^2$ . We show
EXISTENCE OF NONTRIVIAL SOLUTIONS TO POLYHARMONIC EQUATIONS WITH SUBCRITICAL AND CRITICAL EXPONENTIAL GROWTH
• Mathematics
• 2012
The main purpose of this paper is to establish the existence of nontrivial solutions to semilinear polyharmonic equations with exponential growth at the subcritical or critical level. This growth


text:
Electric fields problem [closed]

Why is this the case..

I would have thought it would have followed the E field?

-
Just remember: "electrons swim upstream" –  Alfred Centauri Jun 19 '13 at 20:34
after which, they are inevitably all eaten by bears –  Jim Jun 19 '13 at 20:37
@Jim I should flag that comment for being funny :P On topic though: take a look at the Lorentz force wikipedia page and keep in mind that the charge $q = -e$ of an electron is negative. Edit: I see that I completely missed the fact that there was already a good answer to your question. You might want to accept that answer if it cleared up your confusion. ;) –  Wouter Jun 19 '13 at 20:54


text:
Simple Math

I ran across a puzzling math problem today. Subtracting a whole number from a fractional number does not give the correct answer. 10.4 - 10 = 0.4 BUT in SNAP! the answer is 0.4lotsofzeros36.

I tried subtracting 10.0 instead, but get the same value.

I use Chrome on a PC.

Computers do not hold decimal numbers totally accurately so when you do this sort of maths - the inac curacies become apparent.

A solution I thought of was to multiply both operands by the same power of 10 to make the integers, doing the addition or subtraction, then dividing the result by the same power of 10. I haven't tested it yet, but I think it will work because I haven't seen the problem with addition and subtraction of integers. Now I have: https://snap.berkeley.edu/snap/snap.html#present:Username=snapenilk&ProjectName=alt%20add%20and%20sub

Oh, I'm happy -- I get to teach someone something!

You'll see the same problem when you do the division. The problem is that computers have only a finite memory. In the case of integers, that finiteness manifests itself in a limit to the size of integers you can represent. The arithmetic built into computer hardware has a relatively small limit, less than 100 factorial, but even with software arithmetic (as in our bignum library), there are only a finite number of atoms in the universe.

But with real numbers, or even just rational numbers, the problem is much worse, because those numbers are dense: there are infinitely many of them between any two of them. So, for example, there are infinitely many rational numbers between 0 and 1. There are infinitely many between 0.000000001 and 0.000000002. And so on. But there are only finitely many representations available in a computer. So, no matter what, almost all rational numbers can't be represented exactly in a computer. If you use the computer's hardware representation for real numbers, called floating point, then the only rational numbers that can be represented exactly are the ones whose denominator is a power of two.

For real numbers in general, that's the end of the story. But for rational numbers there's a better representation available: You represent the number as two integers, the numerator and the denominator. (If you want a unique representation, you require that the fraction be in lowest terms, i.e., that the numerator and denominator have no common factor larger than 1. And that the denominator be positive.) Then every rational number can be represented exactly, up to the limit on representability of integers, namely, the numerator and denominator can't be too big. The same library that gives us extra-large integers (the feature is called "infinite precision" but of course it's not really infinite) also gives us exact rationals. So the problem the OP posed is

What's more, if you use exact rationals inside the computer, you can give an exact decimal representation (0.4) externally if you generate it as a character string, not as a number. (Of course if the fraction is 1/3 then you can't generate an exact decimal representation no matter how you go about it.)

So, since there are only finitely many floating point representations, each of them actually represents a range of numbers. The floating point representation that is displayed as 0.40000000000000036 represents a range of values that includes 4/10. But there's only one number that it represents exactly, and that's 0.40000000000000036.

We could look at the decimal representation of a value, treat it as a character string, and look for long substrings of zeros (or nines, if the value exactly represented in floating point is a little less than the value we want), and manipulate the digits as text to get a rounded version. (That's sort of what variable watchers do, since if you put 0.40000000000000036 in a variable, its watcher shows 0.4. But I think it's just rounding to the nearest some-small-number-of digits, rather than checking the characters in the string.)

So far we are only talking about ugly representations seen by users, but not actual program bugs. Those come in when you do equality comparisons:

Depending on what your program is going to do based on the result of the comparison, this could result in horrible misbehavior.

They make you learn all this stuff when you major in computer science, but honestly, floating point computation is one of the two things (the other is security) that I think you should never try to program yourself, but instead hire a specialist. In cases where your program really matters, I mean, of course; knock yourself out experimenting with these ideas.

Good Golly, Brian, you do go in for completeness!

Much of this can be kind of ignored by truncation (specification of number of digits to maintain; not the same as rounding) but I did not see a trunc function. Is there something in Snap! that can be set to specify precision? I did not recognize this is Settings, and I haven't found a function/operation that purports to do this.

I went down this rabbit hole when I was trying to separate a value into an integer and it's decimal part. I don't have a project, but was just doodling around.

I just tested my solution and it works.

Huh. Right you are. I don't understand why, though... I'm going to have to think harder about this.

In the pulldown menu of the block, you'll find floor and ceiling.

We don't have a precision setting. I was taught that that's a wrong approach, that every so often you'll get really strange results if you just truncate. Although I suppose we could keep full precision internally and just truncate for display purposes, as the variable watchers do.

But why don't you want round? If the value you have is 3.99999999999997, taking the floor is the wrong thing.

I think the really right thing is to examine the digits of the numeral and recognize the case of a bunch of consecutive zeros or nines, and only then round. To take a somewhat artificial example,

should really show all five significant digits. And even

benefits pedagogically from showing a lot of digits.

But why does @snapenilk's algorithm work? What that tells me is that 0.40000000000000036 isn't the closest representable value to 0.4. So why doesn't

give 0.4?

Oh, wait, when you subtract 10 from 10.4, the normalization process left-shifts the answer so that the most significant bit is 1. Whereas @snapenilk's version doesn't result in a denormalized intermediate form. So I guess I have to look at the internal floating point actual bits to understand what it's doing... Ugh...

@bh do you know the more common name for one trio-trigintillion?
[poll type=regular results=on_vote public=true chartType=bar]

• nothing
[/poll]

Nope, I've never heard of a trigintillion so have no hope of working it out.

it's about 30-illion if a million was 1-illion and a billion was 2-illion

First of all, Google is a search engine, not a number.

Secondly, googol has

one hundred zeros (one hundred one decimal places), which is called:
Ten duotrigintillion.
https://keelyhill.github.io/tuppers-formula/

you are correct!

for snapenilk only

haha! snapenilk has already seen this

Now that I saw it, you can delete it so that nobody else can see it.

now look

The circle is complete now! What started me down this hole was playing with the floor and ceiling functions.

I've implemented my own training-wheels version of floating point in an old project when it mattered, but the REAL take-away here is to beware of using equality in tests. GT or LT might be better.


text:
+0

# Using absolute value to determine distance: help!

0
49
3

So I started off by subtracting 1422 with 1016, then 1016 with 1202 and so forth... but I end up with the wrong answer? I don't understand what I'm doing wrong...

The right answer is 1743 miles and I got 1753 miles

Guest Apr 17, 2018
edited by Guest  Apr 17, 2018

#1
+19207
+2

Using absolute value to determine distance

$$\begin{array}{|rcll|} \hline && (1422-1016)-(1016-1202)+(1202-918)+(918-496)+(496-51) \\ &=& 406 + 186 + 284 +422 +445 \\ &=& 1743 \\ \hline \end{array}$$

heureka  Apr 17, 2018
Sort:

#1
+19207
+2

Using absolute value to determine distance

$$\begin{array}{|rcll|} \hline && (1422-1016)-(1016-1202)+(1202-918)+(918-496)+(496-51) \\ &=& 406 + 186 + 284 +422 +445 \\ &=& 1743 \\ \hline \end{array}$$

heureka  Apr 17, 2018
#2
+1

Sorry, I don't understand why the distance between Delta Junction and Haines Junction is subtracted with the distance between the two destinations travelled on Thursday?

Guest Apr 17, 2018
#3
+19207
+2

Sorry, I don't understand why the distance between Delta Junction and Haines Junction is subtracted with the distance between the two destinations travelled on Thursday?

Because 1016 -1202 = -186  is negative. Using the absolute value is -(-186) = + 186

heureka  Apr 17, 2018

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text:
reaction_graphs.pl 6.33 KB
 MARTINEZ Thierry committed Oct 19, 2015 1 2 3 :- module( reaction_graphs, [  MARTINEZ Thierry committed Mar 18, 2016 4  % Commands  MARTINEZ Thierry committed Oct 19, 2015 5  reaction_graph/0,  MARTINEZ Thierry committed Oct 20, 2015 6  import_reactions_from_graph/0,  MARTINEZ Thierry committed Mar 18, 2016 7 8 9  draw_reactions/0, % Public API reaction_graph/1  MARTINEZ Thierry committed Oct 19, 2015 10 11 12  ] ).  MARTINEZ Thierry committed Oct 20, 2015 13   MARTINEZ Thierry committed Oct 19, 2015 14 15 :- devdoc('\\section{Commands}').  FAGES Francois committed Aug 08, 2016 16 17 18 19 20 :- devcom('FF: It would be nice to have a notion of input/output and draw graphs accordingly, with inputs on top and outputs bottom in all (reaction or influence) graphs. The input species could be those present at initial_state. The output species could be those not reactant (nor catalyst) of any reaction, nor source of any nfluence. Alternatively we could have input/output annotations on molecular species. It could be useful also for some analyses.').  MARTINEZ Thierry committed Oct 20, 2015 21   MARTINEZ Thierry committed Oct 19, 2015 22 23 reaction_graph :- biocham_command,  MARTINEZ Thierry committed Oct 20, 2015 24  doc('Builds the reaction graph of the current model.'),  MARTINEZ Thierry committed Oct 23, 2015 25  delete_items([kind: graph, key: reaction_graph]),  MARTINEZ Thierry committed Oct 20, 2015 26 27  new_graph, set_graph_name(reaction_graph),  MARTINEZ Thierry committed Mar 18, 2016 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  get_current_graph(GraphId), reaction_graph(GraphId). import_reactions_from_graph :- biocham_command, doc(' Updates the set of reactions of the current model with the current graph. '), make_bipartite_graph, make_reactions. draw_reactions :- biocham_command, doc(' Draws the reaction graph of the current model. Equivalent to \\texttt{reaction_graph. draw_graph.} \\begin{example} '), biocham_silent(clear_model), biocham(load(library:examples/mapk/mapk)), biocham(draw_reactions), doc(' \\end{example} '), reaction_graph, draw_graph. :- devdoc('\\section{Public API}'). reaction_graph(GraphId) :-  MARTINEZ Thierry committed Oct 20, 2015 62 63 64 65 66 67 68  set_counter(reaction, 0), \+ ( item([kind: reaction, item: Item]), reaction(Item, Kinetics, Reactants, Products, Reversible), \+ ( count(reaction, ReactionCount), format(atom(ReactionCounter), 'reaction~d', [ReactionCount]),  MARTINEZ Thierry committed Mar 18, 2016 69  transition(GraphId, [ReactionCounter]),  MARTINEZ Thierry committed Oct 20, 2015 70  (  MARTINEZ Thierry committed Oct 24, 2015 71  Reversible = true  MARTINEZ Thierry committed Oct 20, 2015 72  ->  MARTINEZ Thierry committed Mar 24, 2016 73  set_attribute(GraphId, [ReactionCounter], reversible = Reversible)  MARTINEZ Thierry committed Oct 20, 2015 74 75 76 77 78 79 80 81  ; true ), ( Kinetics = 'MA'(1) -> true ;  MARTINEZ Thierry committed Mar 24, 2016 82  set_attribute(GraphId, [ReactionCounter], kinetics = Kinetics)  MARTINEZ Thierry committed Oct 20, 2015 83 84 85  ), \+ ( (  MARTINEZ Thierry committed Oct 24, 2015 86  member(Stoichiometry * Object, Reactants),  MARTINEZ Thierry committed Oct 20, 2015 87 88 89  From = Object, To = ReactionCounter ;  MARTINEZ Thierry committed Oct 24, 2015 90  member(Stoichiometry * Object, Products),  MARTINEZ Thierry committed Oct 20, 2015 91 92 93 94  From = ReactionCounter, To = Object ), \+ (  MARTINEZ Thierry committed Mar 18, 2016 95 96  place(GraphId, Object), add_edge(GraphId, From -> To, EdgeId),  MARTINEZ Thierry committed Oct 20, 2015 97  (  MARTINEZ Thierry committed Mar 24, 2016 98  get_attribute(EdgeId, stoichiometry = OldStoichiometry)  MARTINEZ Thierry committed Oct 20, 2015 99  ->  MARTINEZ Thierry committed Oct 24, 2015 100  NewStoichiometry is OldStoichiometry + Stoichiometry  MARTINEZ Thierry committed Oct 20, 2015 101  ;  MARTINEZ Thierry committed Oct 24, 2015 102  NewStoichiometry is Stoichiometry  MARTINEZ Thierry committed Oct 20, 2015 103 104  ), (  MARTINEZ Thierry committed Oct 24, 2015 105  NewStoichiometry = 1  MARTINEZ Thierry committed Oct 20, 2015 106 107  -> catch(  MARTINEZ Thierry committed Mar 18, 2016 108  delete_attribute(EdgeId, stoichiometry),  MARTINEZ Thierry committed Oct 20, 2015 109 110 111 112  error(unknown_item), true ) ;  MARTINEZ Thierry committed Mar 24, 2016 113  set_attribute(EdgeId, stoichiometry = NewStoichiometry)  MARTINEZ Thierry committed Oct 20, 2015 114 115 116 117 118 119 120  ) ) ) ) ).  MARTINEZ Thierry committed Feb 16, 2016 121 122 123 :- devdoc('\\section{Private predicates}').  MARTINEZ Thierry committed Oct 20, 2015 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 :- dynamic(vertex_transition/1). :- dynamic(vertex_place/1). :- dynamic(fixpoint/0). make_bipartite_graph :- get_current_graph(GraphId), retractall(vertex_transition(_)), retractall(vertex_place(_)), retractall(vertex_unknown(_)), \+ ( item([parent: GraphId, kind: vertex, id: VertexId, item: Vertex]),  MARTINEZ Thierry committed Mar 24, 2016 140  get_attribute(VertexId, kind = Kind),  MARTINEZ Thierry committed Oct 20, 2015 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  \+ ( ( Kind = place -> assertz(vertex_place(Vertex)) ; Kind = transition -> assertz(vertex_transition(Vertex)) ; true ) ) ), \+ \+ ( repeat, assertz(fixpoint), \+ ( item([parent: GraphId, kind: edge, item: (A -> B)]), \+ ( reaction_object(A, B), object_reaction(A, B), reaction_object(B, A), object_reaction(B, A) ) ), fixpoint ), check_bipartite_graph. check_bipartite_graph :- get_current_graph(GraphId), findall( AmbiguousVertex, ( vertex_transition(AmbiguousVertex), vertex_place(AmbiguousVertex) ), AmbiguousVertices ), ( AmbiguousVertices = [] -> true ; throw(error(ambiguous_vertices(AmbiguousVertices))) ), findall( AmbiguousEdge, ( item([parent: GraphId, kind: edge, item: AmbiguousEdge]), AmbiguousEdge = (From -> _To), \+ vertex_transition(From), \+ vertex_place(From) ), AmbiguousEdges ), ( AmbiguousEdges = [] -> true ; throw(error(ambiguous_edges(AmbiguousEdges))) ). reaction_object(A, B) :- ( vertex_transition(A), \+ vertex_place(B) -> assertz(vertex_place(B)), retractall(fixpoint) ; true ). object_reaction(A, B) :- ( vertex_place(A), \+ vertex_transition(B) -> assertz(vertex_transition(B)), retractall(fixpoint) ; true ). make_reactions :- delete_items([kind: reaction]), get_current_graph(GraphId), \+ ( vertex_transition(Reaction), \+ (  MARTINEZ Thierry committed Mar 18, 2016 238 239  get_kinetics(GraphId, Reaction, Kinetics), get_reversible(GraphId, Reaction, Reversible),  MARTINEZ Thierry committed Oct 20, 2015 240  findall(  MARTINEZ Thierry committed Oct 24, 2015 241  Stoichiometry * Reactant,  MARTINEZ Thierry committed Oct 20, 2015 242 243 244 245 246 247 248 249  ( vertex_place(Reactant), item([ parent: GraphId, kind: edge, item: (Reactant -> Reaction), id: EdgeId ]),  MARTINEZ Thierry committed Oct 24, 2015 250  get_stoichiometry(EdgeId, Stoichiometry)  MARTINEZ Thierry committed Oct 20, 2015 251 252 253 254  ), Left ), findall(  MARTINEZ Thierry committed Oct 24, 2015 255  Stoichiometry * Product,  MARTINEZ Thierry committed Oct 20, 2015 256 257 258 259 260 261 262 263  ( vertex_place(Product), item([ parent: GraphId, kind: edge, item: (Reaction -> Product), id: EdgeId ]),  MARTINEZ Thierry committed Oct 24, 2015 264  get_stoichiometry(EdgeId, Stoichiometry)  MARTINEZ Thierry committed Oct 20, 2015 265 266 267 268 269 270 271 272  ), Right ), add_reaction(Kinetics, Left, Right, Reversible) ) ).  MARTINEZ Thierry committed Mar 18, 2016 273 get_kinetics(GraphId, Vertex, Kinetics) :-  MARTINEZ Thierry committed Oct 20, 2015 274  (  MARTINEZ Thierry committed Mar 24, 2016 275  get_attribute(GraphId, Vertex, kinetics = Kinetics)  MARTINEZ Thierry committed Oct 20, 2015 276 277 278 279 280 281 282  -> true ; Kinetics = 'MA'(1) ).  MARTINEZ Thierry committed Mar 18, 2016 283 get_reversible(GraphId, Vertex, Reversible) :-  MARTINEZ Thierry committed Oct 20, 2015 284  (  MARTINEZ Thierry committed Mar 24, 2016 285  get_attribute(GraphId, Vertex, reversible = Reversible)  MARTINEZ Thierry committed Oct 20, 2015 286 287 288  -> ( (  MARTINEZ Thierry committed Oct 24, 2015 289  Reversible = true  MARTINEZ Thierry committed Oct 20, 2015 290  ;  MARTINEZ Thierry committed Oct 24, 2015 291  Reversible = false  MARTINEZ Thierry committed Oct 20, 2015 292 293 294 295  ) -> true ;  MARTINEZ Thierry committed Oct 24, 2015 296  throw(error(reversible_attribute_boolean))  MARTINEZ Thierry committed Oct 20, 2015 297 298  ) ;  MARTINEZ Thierry committed Oct 24, 2015 299  Reversible = false  MARTINEZ Thierry committed Oct 20, 2015 300 301 302  ).  MARTINEZ Thierry committed Mar 18, 2016 303 get_stoichiometry(EdgeId, Stoichiometry) :-  MARTINEZ Thierry committed Oct 20, 2015 304  (  MARTINEZ Thierry committed Mar 24, 2016 305  get_attribute(EdgeId, stoichiometry = Stoichiometry)  MARTINEZ Thierry committed Oct 20, 2015 306 307 308  -> true ;  MARTINEZ Thierry committed Oct 24, 2015 309  Stoichiometry = 1  MARTINEZ Thierry committed Oct 20, 2015 310  ).


text:
# Tevatron constraints on models of the Higgs boson with exotic spin and parity using decays to bottom-antibottom quark pairs

Aaltonen, T A; Amerio, S; Amidei, D E; et al; Canelli, F; Kilminster, B; CDF Collaboration; D0 Collaboration (2015). Tevatron constraints on models of the Higgs boson with exotic spin and parity using decays to bottom-antibottom quark pairs. Physical Review Letters, 114:151802.

## Abstract

Combined constraints from the CDF and D0 Collaborations on models of the Higgs boson with exotic spin $J$ and parity $P$ are presented and compared with results obtained assuming the standard model value $J^P=0^+$. Both collaborations analyzed approximately 10~fb$^{-1}$ of proton-antiproton collisions with a center-of-mass energy of 1.96 TeV collected at the Fermilab Tevatron. Two models predicting exotic Higgs bosons with $J^P=0^-$ and $J^P=2^+$ are tested. The kinematic properties of exotic Higgs boson production in association with a vector boson differ from those predicted for the standard model Higgs boson. Upper limits at the 95% credibility level on the production rates of the exotic Higgs bosons, expressed as fractions of the standard model Higgs boson production rate, are set at 0.36 for both the $J^P=0^-$ hypothesis and the $J^P=2^+$ hypothesis. If the production rate times the branching ratio to a bottom-antibottom pair is the same as that predicted for the standard model Higgs boson, then the exotic bosons are excluded with significances of 5.0 standard deviations and 4.9 standard deviations for the $J^P=0^-$ and $J^P=2^+$ hypotheses, respectively.

## Abstract

Combined constraints from the CDF and D0 Collaborations on models of the Higgs boson with exotic spin $J$ and parity $P$ are presented and compared with results obtained assuming the standard model value $J^P=0^+$. Both collaborations analyzed approximately 10~fb$^{-1}$ of proton-antiproton collisions with a center-of-mass energy of 1.96 TeV collected at the Fermilab Tevatron. Two models predicting exotic Higgs bosons with $J^P=0^-$ and $J^P=2^+$ are tested. The kinematic properties of exotic Higgs boson production in association with a vector boson differ from those predicted for the standard model Higgs boson. Upper limits at the 95% credibility level on the production rates of the exotic Higgs bosons, expressed as fractions of the standard model Higgs boson production rate, are set at 0.36 for both the $J^P=0^-$ hypothesis and the $J^P=2^+$ hypothesis. If the production rate times the branching ratio to a bottom-antibottom pair is the same as that predicted for the standard model Higgs boson, then the exotic bosons are excluded with significances of 5.0 standard deviations and 4.9 standard deviations for the $J^P=0^-$ and $J^P=2^+$ hypotheses, respectively.

## Statistics

### Citations

2 citations in Web of Science®
8 citations in Scopus®

### Altmetrics

Detailed statistics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Physics Institute 530 Physics English 3 February 2015 12 Feb 2016 16:46 28 Apr 2017 03:22 American Physical Society 0031-9007 https://doi.org/10.1103/PhysRevLett.114.151802 arXiv:1502.00967v2


text:
## Calculus: Early Transcendentals 8th Edition

$\lim _{n\rightarrow \infty }\dfrac {2+n}{1-2n}=\lim _{n\rightarrow \infty }\dfrac {\dfrac {2}{n}+1}{\dfrac {1}{n}-2}=\dfrac {1}{-2}\neq 0 \Rightarrow \sum ^{\infty }_{n=1}\dfrac {2+n}{1-2n}$ Thus, it diverges.


text:
# Multi-species occupancy with covariates

In the previous unit we worked with a data set with capture histories but no information on habitat. Here we want to show how habitat covariates can be included in the model.

## Birds on islands

Greg Irving and colleagues studied resident forest bird species on islands in the Chiew Larn hydropower reservoir in southern Thailand and the near-by mainland. Details are here. Thanks to Greg Irving for providing the data files.

They collected data for a number of habitat variables, but a preliminary analysis showed that the area of the island and distance from the mainland were enough to get good predictions of occupancy; adding further covariates to the model did not improve performance.

You can download and look at the data with the following code:

CL <- read.csv("http://bcss.org.my/data/ChiewLarn_birds.csv",
comment="#")
str(CL)
# 'data.frame':   24 obs. of  83 variables:
# $siteID: chr "IS05" "IS06" "IS07" "IS39" ... #$ area  : num  9.763 49.785 1.499 0.835 0.827 ...
# $MLdist: num 725 250 150 425 67 575 325 320 1000 25 ... #$ PinTailed_Parrotfinch : int  3 3 1 1 3 3 3 3 3 3 ...
# $BlackNaped_Monarch : int 3 3 0 0 1 3 2 3 1 2 ... # [...] #$ RedBearded_BeeEater   : int  0 0 0 0 0 0 0 0 0 0 ...

Ymat <- as.matrix(CL[, -(1:3)])
rownames(Ymat) <- CL$siteID table(Ymat) # 0 1 2 3 # 1370 249 137 164 colSums(Ymat) # PinTailed_Parrotfinch BlackNaped_Monarch # 68 52 # Everetts_Whiteeye Wreathed_Hornbill # 52 49 # [...] # Oriental_Magpie_Robin RedBearded_BeeEater # 1 1 nOcc <- 3 # Each site visited 3 times ( nSobs <- ncol(Ymat) ) # Number of species observed # 80 ( nSites <- nrow(Ymat) ) # Number of sites # 24 # Prepare predictors logArea <- log(CL$area)
exp(mean(logArea))  # 10.13575 ha
logAreaS <- wiqid::standardize(logArea)
mean(CL$MLdist) # 285.2918 m MLdistS <- wiqid::standardize(CL$MLdist)

We have data for 24 sites, 23 islands and the mainland, and 80 species. As with the previous example, the number of detections per species ranges widely, with a maximum of 68, and 6 species with just 1 detection each.

The predictors we’ll use are log(area) and distance from mainland. We standardise both to have mean = 0 and SD = 1.

The probability of occupancy differs among sites, depending on the covariates, just as for the willow tits example, but now replicated for each species $k$: $${\rm logit}(\psi_{ik}) = \beta_{0,k} + \beta_{area,k} logArea_i + \beta_{dist,k} MLdist_i$$ where the coefficients, $\beta$, differ among species.

#### Species level priors

We model the intercept and each of the coefficients as random variables drawn from a normal distribution with mean and SD to be estimated:$$\beta_{x,k} \sim {\rm Normal}(\mu_{x}, \sigma_{x})$$

In the previous unit we included correlation between $\psi$ and $p$; similarly here we include correlation between $\beta_0$ and $p$.

#### Community level hyperpriors

The intercept, $\beta_{0,k}$, is the probability on the logit scale of occupancy of species $k$ at a site with area = 10.14 ha, 285 m from the mainland, and $\mu_0$ is the mean of the intercepts for the community. We thus use a uniform Beta(1,1) prior for the probability and convert to the logit scale:$$\overline{\beta_0} \sim {\rm Beta}(1,1)$$ $$\mu_0 = {\rm logit}(\overline{\beta_0})$$and we use a uniform prior for the SD as usual: $$\sigma_0 \sim {\rm Uniform}(0, 5)$$

We are working on the logit scale with standardised predictors, so a value for a coefficient outside $\pm 5$ would indicate a problem. We use Uniform(-5, 5) priors for $\mu$ and check the output to see if the posterior is constrained by this:$$\mu_{area} \sim {\rm Uniform}(-5,5)$$ and we use a uniform prior for the SD also$$\sigma_{area} \sim {\rm Uniform}(0,5)$$We use similar hyperpriors for distance.

Here’s the JAGS code for the model:

# MSOM model with covariates
# File name "MSOM_cov.jags"

model{
for(k in 1:nSobs){  # Loop through species
# Likelihood
for(i in 1:nSites) {
# Ecological model
logit(psi[i, k]) <- b0[k] + bArea[k] * logArea[i] +
bDist[k] * MLdist[i]
z[i, k] ~ dbern(psi[i, k])
# Observation model
y[i, k] ~ dbin(p[k] * z[i, k], nOcc)
}

# Priors (species level)
b0[k] ~ dnorm(mu.b0, tau.b0)
mu.eta[k] <- mu.lp + rho * sd.lp/sd.b0 *
(b0[k] - mu.b0)
lp[k] ~ dnorm(mu.eta[k], tau.eta)
p[k] <- ilogit(lp[k])

bArea[k] ~ dnorm(mu.area, tau.area)
bDist[k] ~ dnorm(mu.dist, tau.dist)

}

# Hyperpriors (community level)
b0.mean ~ dbeta(1, 1)
mu.b0 <- logit(b0.mean)
sd.b0 ~ dunif(0, 5)
tau.b0 <- 1/sd.b0^2

mu.betalpsi1 ~ dnorm(0,0.1)
sd.area ~ dunif(0, 5)
tau.area <- 1/sd.area^2
mu.dist ~ dnorm(0,0.1)
sd.dist ~ dunif(0,5)
tau.dist <- 1/sd.dist^2

p.mean ~ dbeta(1, 1)
mu.lp <- logit(p.mean)
sd.lp ~ dunif(0, 5)
tau.lp <- 1/sd.lp^2

rho ~ dunif(-1, 1)
tau.eta <- tau.lp/(1 - rho^2)
}

## Running the analysis

Here’s the code to prepare the data, run the analysis and check the output. Mixing tends to be poor for the SDs of the coefficients, so to get good MCEs we increase the number of iterations. We are monitoring a lot of nodes, so we do some brutal thinning to keep the size of the output object reasonable. We also tell jags not to calculate statistics for psi and z, which have 1920 nodes each, just return the MCMC chains.

# Pack up everything for JAGS
z <- (Ymat > 0)*1
z[z == 0] <- NA
jagsData <- list(y = Ymat, nSobs = nSobs, nSites = nSites,
nOcc = nOcc, z = z, logArea = logAreaS, MLdist = MLdistS)
str(jagsData)
# List of 5
# $y : int [1:24, 1:80] 3 3 1 1 3 3 3 3 3 3 ... #$ nSobs : int 80
# $nSites: int 24 #$ nOcc  : num 3
# $z : num [1:24, 1:80] 1 1 1 1 1 1 1 1 1 1 ... #$ logArea: num [1:24] -0.0205 0.8702 -1.045 -1.3652 -1.3702 ...
# $MLdist : num [1:24] 1.543 -0.124 -0.475 0.49 -0.766 ... wanted <- c("b0.mean", "mu.b0", "sd.b0", "mu.area", "sd.area", "mu.dist", "sd.dist", "p.mean", "sd.lp", "mu.lp", "b0", "bArea", "bDist", "p", "psi", "z") # takes 10 mins library(jagsUI) out <- jags(jagsData, NULL, wanted, "MSOM_cov.jags", n.chains=3, n.adapt=1000, n.iter=60000, n.thin=30, codaOnly=c("psi", "z"), parallel=TRUE, DIC=FALSE) mco <- mcmcOutput(out) mcoKey <- mco[1:10] diagPlot(mcoKey) summary(mcoKey) # mean sd median l95 u95 Rhat MCEpc # b0.mean 0.233 0.055 0.229 0.133 0.342 1.000 1.506 # mu.b0 -1.216 0.309 -1.217 -1.829 -0.624 1.000 1.487 # sd.b0 2.521 0.312 2.497 1.906 3.098 0.999 1.881 # mu.area 2.587 0.271 2.570 2.085 3.132 1.000 2.393 # sd.area 0.865 0.332 0.874 0.145 1.462 0.999 3.582 # mu.dist -0.642 0.132 -0.636 -0.900 -0.392 0.994 1.782 # sd.dist 0.494 0.207 0.499 0.028 0.840 1.008 3.001 # p.mean 0.470 0.030 0.471 0.411 0.526 1.001 1.383 # sd.lp 0.766 0.128 0.751 0.528 1.022 1.002 1.883 # mu.lp -0.120 0.119 -0.117 -0.360 0.104 1.001 1.383 The Rhats and MCEs are good, and the diagnostic plots are ok. Posterior distributions are not constrained by our uniform priors. ## Plotting$\psi$and$p$#### Intercepts For the basic model we plotted the posterior distributions of$\psi$for each species. With this model, that differs among sites, so we’ll plot the intercept,$\psi_0$, the probability of occupancy for a site with area = 10.14 ha, 285 m from the mainland. # Convert intercepts to the probability scale psi0 <- plogis(out$sims.list$b0) psi0.mean <- colMeans(psi0) psi0.CRI <- apply(psi0, 2, quantile, probs=c(0.025, 0.975)) op <- par(mfrow=1:2) plot(psi0.mean, 1:nSobs, xlim=0:1, pch=16, xlab="psi0", ylab="Species") segments(psi0.CRI[1,], 1:nSobs, psi0.CRI[2,], 1:nSobs) abline(v=plogis(out$mean$mu.b0), col='red') abline(v=plogis(out$mean$mu.b0 + c(2, -2)*out$mean$sd.b0), col='red', lty=3) plot(out$mean$p, 1:nSobs, xlim=0:1, pch=16, xlab="p", ylab="Species") segments(out$q2.5$p, 1:nSobs,out$q97.5$p, 1:nSobs) abline(v=plogis(out$mean$mu.lp), col='red') abline(v=plogis(out$mean$mu.lp + c(2, -2)*out$mean$sd.lp), col='red', lty=3) par(op) The species with most detections are at the bottom of the plot and have high$\psi_0$and high$p$, as we’d expect. The top 6 species have only 1 detection each: the estimates of$p$for these are relatively high, around 0.4, and the small number of detections seems to be due to very low occupancy rates. #### Effect of covariates We use the posterior means for the coefficients for each species to predict its occupancy probability,$\psi$, for a range of values of area and distance from the mainland. Then we plot each of these. # Values for the x axis for Area range(logArea) logxx <- seq(-0.28, 8.62,, 101) xxA <- exp(logxx) # Standardise logxx with the mean and SD of logArea logxxS <- wiqid::standardize2match(logxx, logArea) # Get the predictions for each value of xxA predA <- matrix(NA, 101, nSobs) for(i in 1:101) predA[i,] <- plogis(out$mean$b0 + out$mean$bArea * logxxS[i]) matplot(xxA, predA, type='l', log='x', lty=1, ylab="Occupancy", xlab="Area (ha)") # Do the same for distance to mainland range(CL$MLdist)
xxD <- seq(0, 1000,, 101)
xxDS <- wiqid::standardize2match(xxD, CL$MLdist) predD <- matrix(NA, 101, nSobs) for(i in 1:101) predD[i,] <- plogis(out$mean$b0 + out$mean$bDist * xxDS[i]) matplot(xxD, predD, type='l', lty=1, ylab="Occupancy", xlab="Distance from mainland (m)") For all species the probability of occupancy increases with the area of the island, and for all but one (Cream-vented Bulbul) it decreases with distance from the mainland. ## Working with the$z$matrix The$z$matrix – a sites x species matrix with 1 for presence and 0 for absence – is the key to a lot of community analysis. The JAGS output above includes z. We extract and check it with this code: z <- out$sims.list$z str(z) # num [1:6000, 1:24, 1:80] 1 1 1 1 1 1 1 1 1 1 ... range(z) # [1] 0 1 #### How many species at each site? The first question we might ask is “How many of these 80 species occur at each site?” The answer is going to be higher than the number observed since detection is imperfect. For each iteration of the MCMC chain we sum across rows in the$z$matrix, so get an MCMC chain for richness at each site, and we can then plot the posterior distribution for each. We’ll add the number of species observed at each site. siteN <- apply(z, 1:2, sum) siteNobs <- rowSums(Ymat > 0) windows(5,5) op <- par(mfrow=c(3,3), mar=c(5,1,4,1), ask=TRUE) for(i in 1:nSites) { plot(table(siteN[, i]), frame=FALSE, yaxt='n', ylab="", xlab="Number of species", main=CL$siteID[i])
abline(v=siteNobs[i]-0.1, col='grey', lwd=2, lend='butt')
}
par(op)

The code above plots all 24 sites, but we’ll only show a selection here:

It may be more interesting to plot species richness against the site covariates.

op <- par(mfrow=1:2)
# Richness vs distance from mainland
plot(CL$MLdist, colMeans(siteN), pch=16, las=1, ylab="Species richness", xlab="Distance from mainland (m)") cri <- apply(siteN, 2, quantile, probs=c(0.025, 0.975)) segments(CL$MLdist, cri[1,], CL$MLdist, cri[2,]) # Similarity vs area plot(CL$area, colMeans(siteN), pch=16, las=1, log="x",
ylab="Species Richness", xlab="Area (ha)")
cri <- apply(siteN, 2, quantile, probs=c(0.025, 0.975))
segments(CL$area, cri[1,], CL$area, cri[2,])
par(op)

#### How many sites per species?

We could ask the question the other way around: “At how many sites does each species occur?” We sum the columns in the $z$ matrix to get this:

spsN <- apply(z, c(1,3), sum)
spsNobs <- colSums(Ymat > 0)

The plotting code is similar to that for site-wise species richness. Again we show a selection only.

The first species, Pin-tailed Parrotfinch, had the highest total detections and was detected at all 24 sites. The last 2 species, Oriental Magpie Robin and Red-bearded Bee-eater were each detected once, but $\psi_0$ for the Magpie Robin has a wide CrI, so it could (just!) plausibly be present everywhere, while for the Bee-eater it’s more precisely estimated.

#### Which islands are most similar to the mainland?

The $z$ matrix is the starting point for estimating site similarity using the Jaccard or Sørensen indices for example. Any pair of sites can be compared, but the obvious question here is to compare each of the islands to the mainland. We’ll use the Jaccard index.

The code below calculates the indices and plots the results against the covariates.

# Function to calculate Jaccard index
Jaccard <- function(z1, z2)
sum(z1 & z2) / sum(z1 | z2)

ref <- 24  # reference site (mainland)
niter <- dim(z)[1]
jaccSites <- matrix(NA, niter, nSites)
for(i in 1:niter) {
tmp <- z[i,,]
jaccSites[i, ] <- apply(tmp, 1, Jaccard, z2=tmp[ref,])
}

op <- par(mfrow=1:2)
# Similarity vs distance from mainland
plot(CL$MLdist, colMeans(jaccSites), pch=16, ylim=0:1, las=1, ylab="Jaccard index", xlab="Distance from mainland (m)") cri <- apply(jaccSites, 2, HDInterval::hdi) segments(CL$MLdist, cri[1,], CL$MLdist, cri[2,]) # Similarity vs area plot(CL$area, colMeans(jaccSites), pch=16, ylim=0:1, las=1, log="x",
ylab="Jaccard index", xlab="Area (ha)")
cri <- apply(jaccSites, 2, HDInterval::hdi)
segments(CL$area, cri[1,], CL$area, cri[2,])
title(main="Similarity of islands to mainland",
outer=TRUE, line=-3)
par(op)

Clearly, proximity to the mainland does not lead to similarity, but large islands are more similar to the mainland in their species composition than small islands.

#### Which species have similar distributions?

And of course we can ask a similar question about similarity of species: “Which species are most similar in the sites they occupy?” This time we’ll do a full species x species matrix and plot the results as a dendrogram.

Base R has a dist function, but it only has a limited number of distance/similarity options; the wiqid::distShell function allows us to plug in our Jaccard function, but it expects a species x sites matrix, so we need to transpose our matrix. We’ll use the posterior means of the Jaccard indices for the next step.

library(wiqid)
simMCMC <- array(NA_real_, c(niter, nSobs, nSobs))
dimnames(simMCMC) <- list(NULL, names(spsNobs), names(spsNobs))
for(i in 1:niter)
simMCMC[i,,] <- as.matrix(distShell(t(z[i,,]), Jaccard))
# Takes 2.5 mins to do 6000
meanSim <- apply(simMCMC, 2:3, mean)
str(meanSim)
# num [1:80, 1:80] 0 0.877 1 0.894 0.888 ...
# - attr(*, "dimnames")=List of 2
#  ..$: chr [1:80] "PinTailed_Parrotfinch" ... # ..$ : chr [1:80] "PinTailed_Parrotfinch" ...

For the dendrogram we need the distances (or dissimilarities) between species, and they need to be in an object of class dist. With 80 species, the dendrogram is too big to display on laptop screens, so we provide code to create a PDF file.

JaccDistSps <- as.dist(1 - meanSim)
par("mar" = c(4,2,1, 5), cex = 0.5)
plot(as.dendrogram(hclust(JaccDistSps), hang=-1), horiz=TRUE)

pdf("Jaccard species plot.pdf", 7, 10, paper="a4")
par("mar" = c(4,2,1,15), cex=0.5)
plot(as.dendrogram(hclust(JaccDistSps), hang=-1), horiz=TRUE)
dev.off()

## What’s next?

There are more than 80 species of forest birds in the Chiew Larn study area; some were spotted there outside the formal surveys; we know they are there but have no data to estimate occupancy or probability of detection.

Hierarchical modelling allows us to get better estimates for species with only 1 detection. Can it help to estimate parameters for species with 0 detections?

In the next unit we’ll try this out and see if we can estimate the number of species in the community which were not detected, and hence the total species richness.


text:
# zbMATH — the first resource for mathematics

Why Gaussian quadrature in the complex plane? (English) Zbl 0977.41013
Numer. Algorithms 26, No. 3, 251-280 (2001); addendum ibid. 27, No. 2, 215-217 (2001).
Authors’ abstract: This paper synthesizes formally orthogonal polynomials, Gaussian quadrature in the complex plane and the bi-conjugate gradient method together with an application. Classical Gaussian quadrature approximates an integral over (a region of) the real line. We present an extension of Gaussian quadrature over an arc in the complex plane, which we call complex Gaussian quadrature. Since there has not been any particular interest in the numerical evaluation of integrals over the long history of complex function theory, complex Gaussian quadrature is in need of motivation. Gaussian quadrature in the complex plane yields approximations of certain sums connected with the bi-conjugate gradient method. The scattering amplitude $$c^TA^{-1 }b$$ is an example where $$A$$ is a discretization of a differential integral operator corresponding to the scattering problem and $$b$$ and $$c$$ are given vectors. The usual method to estimate this is to use $$c^Tx^{(k)}$$. A result of Warnick is that this is identically equal to the complex Gaussian quadrature estimate of $$1/\lambda$$. Complex Gaussian quadrature thereby replaces this particular inner product in the estimate of the scattering amplitude.

##### MSC:
 41A55 Approximate quadratures 65D32 Numerical quadrature and cubature formulas 65F10 Iterative numerical methods for linear systems
Full Text:


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# Message protocol to probabilistically infer missing object from Union of two subsets of a larger set

This was a challenge problem I read some time ago and just remembered it:

Say you have two people, $A$ and $B$, collect objects distinctly labeled $1,...,n$. They will each separately collect sets $S_A$ and $S_B$ such that $|S_A \cup S_B| = n-k$ where $k\leq \sqrt n$. There is no guarantee in what order any of these will be collected in by each person. The goal is for $A$ to send $B$ a short message of bits so that $B$ can infer (at least) ONE of the $k$ objects missing from the union of their sets, i.e., $$k \in \{1,\ldots,n\} \setminus (S_A \cup S_B).$$ The protocol or composition of the message is agreed upon before hand, but since neither of them know the value of $k$ a priori the message can't be based on that. Before entering the maze, both parties have access to shared randomness or a set of random bits they know and can be used to communicate later (i.e., an adversarial maze generator does not have access to this randomness). (I am not sure if this detail changes the problem.)

We know $k \geq 1$ and the sets $S_A$ and $S_B$ are disjoint.

$A$ can of course base his answer on $S_A$ but can't transmit all of $S_A$. For example I came up with: if $A$ sends the sum of the elements of $S_A$ and the size of $S_A$, then $B$ can always infer if only one element is missing ($k=1$) but will fail otherwise. This takes $O(\log n)$ bits.

But how can we design a protocol to achieve the following:

$A$ sends a message with $O(\log^3 n)$ bits such that $B$ with probability $1-\frac{1}{n^c}$ can infer (at least) ONE missing object for any $1 \leq k \leq \sqrt n$ (and with remaining probability fails), where $c$ is some number given ahead of time (i.e., it is not under the control of or chosen by $A$ but the constant factor of $O(\log^3 n)$ can depend on it).

To clarify, the goal is to design a message protocol that will achieve the desired result for any values of $n$ and $c$. So the program itself can't pick the value of $c$.

• This question seems to be going completely unanswered! I guess it has stumped a lot of people! I guess that is why they call it a challenge problem... :) – Four_FUN Nov 18 '14 at 7:49
• Patience, young Padawan. – Raphael Nov 19 '14 at 17:38
• To be clear, we don't know anything about "fairness" here? In particular, $S_A = \emptyset$ is a distinct possibility? By $\log^3 n$ we mean $(\log n)^3$ here? – Raphael Nov 19 '14 at 18:04
• Both of those statements are indeed correct. I am very patient. I just like to make observational jokes! :) – Four_FUN Nov 21 '14 at 2:06


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# Tag Info

55

You are indeed correct that the point where the coin hits the train must momentarily stop because of the change of direction. However, this does not mean that the train will stop! It will slightly decrease its speed due to the kinetic energy of the coin, but this will be unmeasurably small. What will happen is that the point of contact will stop (kinda, see ...

41

What does change the velocity of an object? Force of course. (Acceleration is the change of velocity). The question is how much time does it take to make that change? A long time? Infinitely small time? Momentarily? Well, it depends on the acceleration itself! If you accelerate in your car, it'd take a significant amount of time to go from $0$ to $100\ \... 15 I think this is a great question. I'm not 100% sure this is a complete correct answer, but I'm at least well over 50% sure. Here are the paths of the penny (in green) and the train (in red): First observation: The penny might never have velocity zero; at time$T$it can jump discontinuously from a positive to a negative velocity. Second and more ... 9 It looks like there's a hole in your reasoning. If I understand correctly, you're saying this: At some point, the penny's velocity must be zero. At some point, the penny's velocity must be equal to the train's velocity. Therefore, at some point, the train's velocity must be zero. Both 1 and 2 are true (assuming that the collision is exactly head-on). But 3 ... 7 Two objects in contact have the same velocity No, not always. Take two balls and make one roll faster than the other and make them collide. They do collide with different velocities! But, in your case, if the coin hits the train and exerts a force capable of accelerating it in the opposite direction of its movement until its velocity becomes zero then yeah,... 5 For an object rotating about an axis, every point on the object has the same angular velocity. The tangential velocity of any point is proportional to its distance from the axis of rotation, i.e.$\mathbf{v_{\perp}} = \boldsymbol{\omega} \times \mathbf{r}.$So it depends what you mean by "the velocity of the string." 4 We are interested whether a given force $${\bf F}~=~{\bf F}({\bf r},{\bf v},{\bf a},t) \tag{1}$$ has a velocity-dependent potential $$U~=~U({\bf r},{\bf v},t),\tag{2}$$ which by definition means that $${\bf F}~\stackrel{?}{=}~\frac{d}{dt} \frac{\partial U}{\partial {\bf v}} - \frac{\partial U}{\partial {\bf r}}. \tag{3}$$ If we define the potential ... 4 For different points of the string to have same velocities, their direction of movement and speeds would to be the same as well. If the string is taut then all points are moving in the same direction at any given time, but it's easy to show that the speeds are not the same: If you have your 2 meter long string, the endpoint will move the total distance of 2·... 4 Constant velocity and speed have no meaning unless you specify the frame of reference with respect to which it is measured or observed. (The only exception is the speed of light). If you see an object “moving”, then it is moving with respect to your frame of reference. If you see it as “still” it is still with respect to your frame of reference. But in ... 4 This is actually in interesting question, and definitely not worth all of the down votes in my opinion. You're actually asking a different question than you realize. It's not like people were sitting around saying "We need a suitable definition for velocity. What should it be?" This doesn't make sense. Without a definition how would anyone know to even ... 4 It depends. In principle it could, in any realistic situation it is not even close. As you said, it comes down to the fundamentals. When you throw a coin at a train, the train will exert some force on the coin via electrostatic repulsion. On the atomic level this is due to the electrons in the coin getting too close to the electrons in the train and ... 3 The Maxwell-Boltzmann graph represents just this.$\hskip2in$Source: (wewwchemistry.com) It can represent different temperatures too. You see a probability density (simplified here to # of molecules) against speed or kinetic energy. The area underneath gives the proportion of molecules at a speed range. With altitude changes, the pressure would change. ... 3 Your example of pushing a car at a constant velocity is doing work, because you are applying force on the car. The difference is that in that case, there is a friction force that also does work, in direct opposition to the work provided by pushing. The work done on the car by you is counteracted by the work done on the car due to friction, and thus net ... 3 The four-velocity is the tangent four-vector of a timelike world line, that is$U^\mu = dx^\mu / d\tau$, and this definition applies to any coordinate system. The difference of the coordinates$dx^\mu$is a vector, i.e. an invariant, and together with the proper time$\tau$, an invariant by definition, produces the four-velocity$U^\mu$, which is an ... 2 You cannot tell the difference. In fact many would say there is no difference. If you're really in a universe where there's nothing except the object like you say, there would be no difference between just sitting still and moving at a constant speed. There are no stars to see whizzing by, no atmosphere to make you feel the wind, nothing to make the light so ... 2 Yes, integrating with respect to one$v_k$will give a right-hand side that depends on every$v_l$with$l\neq k$. But notice we have one equation for every possible value of$k$, and in the end, the left hand side of the equation is always the same, and so should be the right hand side. Thus, we conclude that the function at the right hand side cannot ... 2 The principle that two objects in contact have the same velocity is only true if they are in contact for a period of time. In the simplest modeling of the situation, the coin and the train are only in contact for an instant, where an impulse is transferred to the coin to send it clattering out of the way. The rule that two objects in contact have the same ... 1 Model the train as a tiny mass (the part that the coin is going to hit) connected to a great mass (the rest of the train) by a spring (representing the elasticity of the metal that’s hit). In this model, the tiny mass will indeed equalize its velocity with that of the coin, at some point being at rest relative to the rails, before elasticity causes the coin ... 1 This may be easier to understand assuming that the train and the coin interact during some short and measurable time via forces, caused by elastic (spring-like) deformation of the metal surfaces involved. In other words, imagine there is a tiny short spring between the coin and the train. Both train and coin are subject to the same interaction force so ... 1 Depending on the 'type' of velocity. • Angular velocity will be constant for every particle / unit length of the string i.e. 5/2 revolutions s-1 • Linear velocity will change according to the distance from the point of rotation V= r. w, where V is the linear velocity, r is the length of the string from the point of rotation and w is the angular velocity 1 To simply put, Kinetic energy can be calculated by the basic process of computing the work (W) that is done by a force (F). If the body has a mass of m that was pushed for a distance of d on a surface with a force that’s parallel to it.$W=F.d=m.a.d$The acceleration in this equation can be substituted by the initial$(v_i)$and final$(v_f)\$ velocity and ...

Only top voted, non community-wiki answers of a minimum length are eligible


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# What is the formula for cobalt (III) chlorate?

$H C l {O}_{3} \left(a q\right) r i g h t \le f t h a r p \infty n s {H}^{+} \left(a q\right) + C l {O}_{3}^{-} \left(a q\right)$
$C o {\left(C l {O}_{3}\right)}_{3}$


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# How do you write the partial fraction decomposition of the rational expression  (x-1)/(x^3 +x)?

Oct 29, 2016

The answer is $= \frac{x + 1}{{x}^{2} + 1} - \frac{1}{x}$

#### Explanation:

Let's factorise the denominator ${x}^{3} + x = x \left({x}^{2} + 1\right)$
so the partial fraction decomposition is
$\frac{x - 1}{{x}^{3} + x} = \frac{x - 1}{x \left({x}^{2} + 1\right)} = \frac{A x + B}{{x}^{2} + 1} + \frac{C}{x}$
$= \frac{x \left(A x + B\right) + C \left({x}^{2} + 1\right)}{\left(x\right) \left({x}^{2} + 1\right)}$
So we have $x - 1 = x \left(A x + B\right) + C \left({x}^{2} + 1\right)$
let $x = 0$$\implies$$- 1 = C$$\implies$$C = - 1$
Coefficient of x, $1 = B$$\implies$$B = 1$
coefficients of x^2, $0 = A + C$ $\implies$$A = 1$

And finally we have
$\frac{x - 1}{{x}^{3} + x} = \frac{x + 1}{{x}^{2} + 1} - \frac{1}{x}$