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https://plainmath.net/linear-algebra/55791-what-is-the-magnitude-of-vector-ab-if-a-equal-4-2-6-and
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maliaseth0
2022-01-24
What is the magnitude of vector $AB$ if $A=\left(4,2,-6\right)$ and $B=\left(9,-1,3\right)$?
caoireoilns
Expert
Let $\stackrel{\to }{A}$ be the position vector of A and $\stackrel{\to }{B}$ be the position vector of B. A position vector is a vector that points from the origin to a particular point.
If you plot $\stackrel{\to }{A},\stackrel{\to }{B}$ and $\stackrel{\to }{AB}$ , then you can easily notice that $\stackrel{\to }{B}=\stackrel{\to }{A}+\stackrel{\to }{AB}$ using the triangle rule of addition of vectors.
In this question, $\stackrel{\to }{A}=4\stackrel{\to }{i}+2\stackrel{\to }{j}-6\stackrel{\to }{k}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\stackrel{\to }{B}=9\stackrel{\to }{i}-\stackrel{\to }{j}+3\stackrel{\to }{k}$. So
$9\stackrel{\to }{i}-\stackrel{\to }{j}+3\stackrel{\to }{k}=\left(4\stackrel{\to }{i}+2\stackrel{\to }{j}-6\stackrel{\to }{k}\right)+\stackrel{\to }{AB}$, , thus meaning that
$\stackrel{\to }{AB}=\left(9-4\right)\stackrel{\to }{i}+\left(-1-2\right)\stackrel{\to }{j}+\left(3+6\right)\stackrel{\to }{k}=5\stackrel{\to }{i}-3\stackrel{\to }{j}+9\stackrel{\to }{k}$.
Do you have a similar question?
Recalculate according to your conditions!
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2023-01-29 08:06:35
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https://reverseengineering.stackexchange.com/questions/19272/idapython-script-works-from-script-file-but-not-when-run-with-s-from-terminal
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# IDAPython Script works from “Script File” but not when run with -S from Terminal
I'm trying to run an IDAPython script in IDA 7.1 on Windows 10 and it runs just fine when I run it from the Script File... command, but if when I run it from the command line it isn't working properly. My command to run it from the command line is:
ida64 -A -SC:\path\to\script\databaseAll.py C:\path\to\ELFexecutable\target0
If I open the file in the graphical interface first and pack the database, then it works from the command line in creating the database correctly, but otherwise it has a lot of information that's missing.
Am I doing something wrong? How do you properly run a script from the command line?
• What specifically is “missing”? – Igor Skochinsky Sep 10 '18 at 7:25
• I was trying to create a database that had information about sections (based on what IDA inaccurately calls "segments"), symbols, and xrefs. When run from the command line, the database has only one or two lines where it should have two dozen or more. – theTheodidact Sep 18 '18 at 15:59
You need to run idc.auto_wait in the python script to allow IDA to process all the entries in it's auto-analysis queue before it tries to navigate around based on analysis-dependent features.
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2021-01-18 15:00:50
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https://academy.edulabs.org/mod/glossary/showentry.php?eid=9870
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#### UNIVERSAL TIME CONSTANT CHART
A chart used to find the time constant of a circuit if the impressed voltage and the values of R and C or R and L are known [2].
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2023-01-29 01:35:51
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http://proteinsandwavefunctions.blogspot.com/2012/07/
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## Tuesday, July 31, 2012
### I'm gonna pay $1350 for this? Come on PLoS ONE! From: Jan Halborg Jensen [jhjensen@chem.ku.dk] Sent: 7/18/2012 To: xxx@plos.org Subject: Author Query to Editor PONE-D_12-07445R1 Hi, Is there any news regarding this manuscript? It has been close to 6 weeks now. Best regards, Jan --------- On Jul 18, 2012, at 10:30 AM, plosone wrote: Dear Dr. Jensen, Thank you for contacting PLoS ONE. I can confirm the reviews have been submitted and your manuscript is currently with the Academic Editor for a decision, as soon as the decision has been completed we will be in contact. Kind Regards, xxx EO Staff PLoS ONE ---------- From: xxx Sent: Wednesday, July 18, 2012 5:54 PM To: Jan Halborg Jensen Subject: Re: Author Query to Editor PONE-D_12-07445R1 Dear Dr. Jensen: The paper is still under review. Sincerely, xxx - Editor, PlosONE -------- From: Jan Halborg Jensen [jhjensen@chem.ku.dk] Sent: 7/19/2012 1:47 AM To: xxx@plos.org Subject: Re: Case: 01158374 [ ref:_00DU0Ifis._500U04XUWc:ref ] Thank you, but I got the following email from the editor. I am a little confused. Dear Dr. Jensen: The paper is still under review. Sincerely, xxx - Editor, PlosONE ---------- Sent: Tuesday, July 24, 2012 2:51 PM To: Jan Halborg Jensen Subject: Re: Case: 01158374 [ ref:_00DU0Ifis._500U04XUWc:ref ] Dear Dr. Jensen, Thank you for contacting PLoS ONE. Please accept my apologies for any confusion caused by my previous email. In regards to your manuscript PONE-D-12-07445R1, I can confirm that the comments from two reviewers have been submitted. We are in contact with the Academic Editor who will soon be able to make a decision regarding your manuscript and inform you of his decision. Please do not hesitate to contact me if you have any further questions or concerns. Kind regards, xxx PLoS ONE Staff -------- Update Sent: Thursday, August 02, 2012 5:11 PM To: Jan Halborg Jensen Subject: Status update for your submission to PLoS ONE: PONE-D-12-07445R1 [A Computational Methodology to Screen Activities of Enzyme Variants] PONE-D-12-07445R1 A Computational Methodology to Screen Activities of Enzyme Variants PLOS ONE Dear Dr. Jensen, I would like to take this opportunity to update you with regards to the current status of your manuscript as we are aware that it is taking longer than we anticipated for you to hear back from the peer review process of your submission. Please accept my most sincere apologies for the delays you have experienced during the peer review process of your submission; whilst reviews have been received the Academic Editor would like to consult further in order to reach a decision. Please rest assured that it is receiving our full and urgent attention. I know that this can be a frustrating process, but if I can ask for your continued patience whilst we are attempting a speedy resolution. We appreciate your patience, and hope you will contact us if you have any questions or queries. Regards, xxx Staff EO Lead PLOS ONE ------- Update 2 From: Jan Halborg Jensen [jhjensen@chem.ku.dk] Sent: 8/27/2012 Subject: Author Query to Editor PONE-D-12-07445R1 Manuscript information: ======================== PONE-D-12-07445R1 A Computational Methodology to Screen Activities of Enzyme Variants PLos ONE ======================== Dear Editor Is there any news regarding this submission. If not, when can I expect to hear something? Best regards, Jan ----- From: xxx Date: August 28, 2012 1:04:59 PM GMT+02:00 To: Jan Halborg Jensen <jhjensen@chem.ku.dk> Subject: Re: Author Query to Editor PONE-D-12-07445R1 Dear Dr. Jensen, I apologize for the delay in handling this manuscript. Your paper is still under review. You will hear from me within Sept. 15, 2012. regards xxxx Associated Editor, PlosONE ------ From: no-reply@salesforce.com [no-reply@salesforce.com] on behalf of plosone [plosone@plos.org] Sent: Wednesday, August 29, 2012 11:10 PM To: Jan Halborg Jensen Subject: "Author Query to Editor PONE-D-12-07445R1" ref:_00DU0Ifis._500U04r2d9:ref Dear Dr. Jensen, Thank you for contacting PLOS ONE. I can confirm that your manuscript is currently out for peer review. We are in contact with the Academic Editor and are working to ensure that the review continues to run smoothly. Please do not hesitate to contact us again if we may be of further assistance. Kind Regards, xxxxx -------- From: Jan Halborg Jensen Sent: Monday, September 17, 2012 1:49 PM To: xxx Subject: Re: Author Query to Editor PONE-D-12-07445R1 Dear xxx What is your decision regarding our manuscript? Best regards, Jan ---- From: xxx Sent: Monday, September 17, 2012 4:28 PM To: Jan Halborg Jensen Subject: Re: Author Query to Editor PONE-D-12-07445R1 Dear Jan thanks for your message. I still need few days more to get the last referee report and then I will get back to you. Regards xxx ---- The reviews arrived on September 24th. You can read the rest here. Tl;dr: the paper is now accepted ## Tuesday, July 24, 2012 ### First PLoS ONE article published! It has taken a little more than a month to get it from accepted to published. There were a few issues which was related to the supporting information which must be included in the manuscript and not separately in the final form (but not required when you submit it for review). Anyways, give it a read if you feel like it: http://dx.plos.org/10.1371/journal.pone.0041117 Now, back to my second PLoS ONE submission and the lengthy review. ## Monday, July 23, 2012 ### BioFET-SIM Instruction Videos The BioFET-SIM web interface operation is illustrated in 4 short instruction videos: Basic interface operation BioFET-SIM signal and pH response calculation Custom structure upload Restoring a previous session Feel free to contact us in case of any questions under biofetsim at gmail dot com ## Sunday, July 1, 2012 ### Using PeerWise to assign homework PeerWise This past quarter I used a new tool called PeerWise in a physical chemistry course I co-taught. The main idea behind PeerWise, which is a free web-based service developed by Paul Denny at the University of Auckland, is that students write multiple choice questions for each other. I didn't quite have the courage to say "Want homework? Write your own" so instead I used it as a supplement to the normal homework assignments, i.e. I reformulated most of the homework assignments as multiple choice questions and posted them on PeerWise, in addition to the usual homework assignment. Some information about the course The course is attended by about 170 chemistry and biochemistry majors and covers thermodynamics and kinetics - more specifically chapters 13-20 in Atkin's Quanta Matter and Change. It's taught in a 9-week quarter and the students take one other course at the same time. There are four other faculty members involved who also pick homework problems associated with their lectures. The students were divided into six teams of roughly 30, and each team has six contact hours with TAs who can help them with the assignments. Advantages of PeerWise 1. Feedback to the students. PeerWise allows you to give extensive descriptions (you can even use videos: example 1 and example 2) on how to solve the problem immediately after the student has picked an answer. The quality of the help provided by the TAs tend to vary greatly, and I hoped this would help even things out a bit. 2. Breaking complex problems up into more manageable pieces. A standard question usually asks you to compute x given some conditions. In PeerWise this can be broken up into, for example, a) which of the following equations would you use to compute x? and then b) what is the value x given these conditions. Answering the first question lets people now whether they are on the right track before proceeding. 3. Feedback to me. Students can leave questions or comments for each problem, which often helps me improve the explanation or identify errors in the problem. Students can also rate the quality and difficulty of each question, which is very valuable. I also get all sorts of detailed data including how many students answered the question correctly (more on that below). Some data I put 120 questions up on PeerWise and 119 students answered at least one question. From what I understand many students worked in pairs and only one would answer on PeerWise. 39 students answered 100 questions or more. For almost all problems, the majority of students picked the right answer, so they are not just clicking randomly to get the explanation as one could have feared. For example, for the question with the most (86) answers was answered correctly by 57% while the question voted the hardest was answered correctly by 47% of the 76 students who answered it. While I encouraged students to write their own questions, only 9 questions were written. One of these questions was "Do you find PeerWise helpful?" 41 students answered, 38 said yes. Also several students commented positively on PeerWise with one caveat (see below). Clearly, many students used PeerWise as a study aid before the exam as seen on this figure. One main problem: errors The main complaint by the students was the many errors that crept in to the questions, multiple choice answers, and solutions. The multiple choice format exacerbates the problem, since many students will work for hours trying to reproduce one of the answers. This is of course impossible if there is an error and this is very frustrating to the student! There are several sources of errors. 1. Several errors were introduced by me when I typed in the multiple choice answers (for example, using Joules when I meant kiloJoules) and explanations. 2. Several errors were already in the problems and was copied into PeerWise. 3. Several errors were in the answers provided in the solution manual of the text book and were copied into PeerWise by me. Often the you would simply get a different numerical answer based on the numbers provided. The error rate was about 1-2 questions per week and all (hopefully) will be fixed next year. Tips 1. By default the questions are listed in the reverse order they are created (i.e. newest one first), so in cases where the order was important I would write the first question last, e.g. exercise 1b and then exercise 1a. 2. Each question comes with a one line preview, so the first sentence in the question should be the name of the exercise, e.g. Exercise 1a. 3. Each question can also be labelled by subject, so I would use labels like "week 1". With one click one can then display the questions on for a single week only. 4. Many of the comments left by the students said "I punch the solution into my calculator and get a different answer". It's of course hard to know what the problem is, but in these cases I leave a link to the worked out solution on Wolfram|Alpha (example), which is basically on on-line calculator. 5. You add equation through one of several editors (I recommend using the Latex editor) that create and image of the equation, which you can't edit. If you view the page in HTML format there is a Latex code you can copy and paste into the editor and change, rather than typing everything in from scratch. Changes for next year (When was the last time you learned anything by doing it once?) Overall PeerWise was a hit, and I will use it next year for this course. I plan to make the following changes: 1. I will add 10-20 math and conceptual training questions every week. Questions one should be able to answer in 1 second: What is$e^{-0}$? If$\Delta G^\ominus <1\$ what can you say about the equilibrium constant? Some of these will show up every week!
2. I will add 1-2 "regular" questions based on topics from previous weeks.
3. I will try to have the explanations include a screenshot of the solution done in Maple. For three reasons: a) it will make me double check the answer in the solution manual, b) it will remove one source of error if I don't have to retype something, and c) it will encourage the students to use Maple (students who use Maple efficiently are done in a fraction of the time compared to others).
Finally, I am racking my brain on how to get more students to write their own questions.
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2018-08-14 10:26:57
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https://genieclust.gagolewski.com/rapi/mst.html
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# mst: Minimum Spanning Tree of the Pairwise Distance Graph
## Description
An parallelised implementation of a Jarnik (Prim/Dijkstra)-like algorithm for determining a(*) minimum spanning tree (MST) of a complete undirected graph representing a set of n points with weights given by a pairwise distance matrix.
(*) Note that there might be multiple minimum trees spanning a given graph.
## Usage
mst(d, ...)
## Default S3 method:
mst(
d,
distance = c("euclidean", "l2", "manhattan", "cityblock", "l1", "cosine"),
M = 1L,
cast_float32 = TRUE,
verbose = FALSE,
...
)
## S3 method for class 'dist'
mst(d, M = 1L, verbose = FALSE, ...)
## Arguments
d
either a numeric matrix (or an object coercible to one, e.g., a data frame with numeric-like columns) or an object of class dist, see dist
...
further arguments passed to or from other methods
distance
metric used to compute the linkage, one of: "euclidean" (synonym: "l2"), "manhattan" (a.k.a. "l1" and "cityblock"), "cosine"
M
smoothing factor; M = 1 gives the selected distance; otherwise, the mutual reachability distance is used
cast_float32
logical; whether to compute the distances using 32-bit instead of 64-bit precision floating-point arithmetic (up to 2x faster)
verbose
logical; whether to print diagnostic messages and progress information
## Details
If d is a numeric matrix of size $$n p$$, the $$n (n-1)/2$$ distances are computed on the fly, so that $$O(n M)$$ memory is used.
The algorithm is parallelised; set the OMP_NUM_THREADS environment variable Sys.setenv to control the number of threads used.
Time complexity is $$O(n^2)$$ for the method accepting an object of class dist and $$O(p n^2)$$ otherwise.
If M >= 2, then the mutual reachability distance $$m(i,j)$$ with smoothing factor M (see Campello et al. 2013) is used instead of the chosen “raw” distance $$d(i,j)$$. It holds $$m(i, j)=\max(d(i,j), c(i), c(j))$$, where $$c(i)$$ is $$d(i, k)$$ with $$k$$ being the (M-1)-th nearest neighbour of $$i$$. This makes “noise” and “boundary” points being “pulled away” from each other. Genie++ clustering algorithm (see gclust) with respect to the mutual reachability distance gains the ability to identify some observations are noise points.
Note that the case M = 2 corresponds to the original distance, but we are determining the 1-nearest neighbours separately as well, which is a bit suboptimal; you can file a feature request if this makes your data analysis tasks too slow.
## Value
Matrix of class mst with n-1 rows and 3 columns: from, to and dist. It holds from < to. Moreover, dist is sorted nondecreasingly. The i-th row gives the i-th edge of the MST. (from[i], to[i]) defines the vertices (in 1,…,n) and dist[i] gives the weight, i.e., the distance between the corresponding points.
The method attribute gives the name of the distance used. The Labels attribute gives the labels of all the input points.
If M > 1, the nn attribute gives the indices of the M-1 nearest neighbours of each point.
## Author(s)
Marek Gagolewski and other contributors
## References
Jarnik V., O jistem problemu minimalnim, Prace Moravske Prirodovedecke Spolecnosti 6, 1930, 57-63.
Olson C.F., Parallel algorithms for hierarchical clustering, Parallel Comput. 21, 1995, 1313-1325.
Prim R., Shortest connection networks and some generalisations, Bell Syst. Tech. J. 36, 1957, 1389-1401.
Campello R.J.G.B., Moulavi D., Sander J., Density-based clustering based on hierarchical density estimates, Lecture Notes in Computer Science 7819, 2013, 160-172, doi:10.1007/978-3-642-37456-2_14.
The official online manual of genieclust at https://genieclust.gagolewski.com/
Gagolewski M., genieclust: Fast and robust hierarchical clustering, SoftwareX 15:100722, 2021, doi:10.1016/j.softx.2021.100722.
emst_mlpack() for a very fast alternative in case of (very) low-dimensional Euclidean spaces (and M = 1).
## Examples
library("datasets")
data("iris")
X <- iris[1:4]
tree <- mst(X)
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2022-12-05 05:08:09
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https://www.physicsforums.com/threads/evaluation-of-real-integral.658066/
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# Evaluation of real integral
1. Dec 10, 2012
### doey
1. The problem statement, all variables and given/known data[/b
from -∞ to ∞ of ∫1/(x^4+1) dx
2. Relevant equations
how can i actually find out the pole of this function
3. The attempt at a solution
i try to determine the pole and x^4=-1,for this i have obtain the root which is (-1)^1/4,but i dun noe how to find out the remaining roots and it really make me confuse for this ==
2. Dec 10, 2012
### Michael Redei
Since x4+1 = 0 has no real-valued solutions, your function f(x) = 1/(x4+1) has no poles. You'll need a different approach for this integral.
3. Dec 10, 2012
### Dick
There are complex poles. There are four of them. Write the root in polar form $r e^{i \theta}$ and try and figure out what the possibilities are for r and $\theta$.
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2017-10-19 00:46:31
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https://nigerianscholars.com/past-questions/general-paper/question/303912/
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Home » » Helium is often used in observation balloons because it is
# Helium is often used in observation balloons because it is
### Question
Helium is often used in observation balloons because it is
### Options
A) light and combustible
B) light and non-combustible
C) heavy and combustible
D) heavy and non-combustible
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2021-11-29 02:18:28
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https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition-blitzer/chapter-11-section-11-3-limits-and-continuity-exercise-set-page-1162/57
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## Precalculus (6th Edition) Blitzer
The statement “f and g are both continuous at $a$, although $\frac{f}{g}$ is not” makes sense.
For a function to be continuous at a point a, the function must satisfy the following three conditions: (a) f is defined at a. (b) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ exists. (c) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$ Since, f and g are continuous functions at a, both of them satisfy the above three conditions. Now check whether the function $\frac{f}{g}$ is continuous at a, or not. Find the value of $\left( \frac{f}{g} \right)\left( x \right)$ at $a$, $\left( \frac{f}{g} \right)\left( a \right)=\frac{f\left( a \right)}{g\left( a \right)}$ If $g\left( a \right)=0$, then the value $\left( \frac{f}{g} \right)\left( a \right)$ is not defined. In this case, the function $\left( \frac{f}{g} \right)\left( x \right)$ does not satisfy all the conditions of being continuous. Thus, the function $\left( \frac{f}{g} \right)\left( x \right)$ is not continuous at a. Hence, the statement makes sense.
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2019-10-16 23:04:19
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https://socratic.org/questions/given-a-line-l-and-a-point-p-on-the-line-construct-a-line-perpendicular-to-l-tha
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1
# Given a line L and a point P on the line, construct a line perpendicular to L that passes through P? Let the line be described by y=mx+b and A(x_1, y_1) and B(x_2,y_2) show that the perpendicular slope is m' = -1/m?
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2018-03-19 01:22:51
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http://mathonline.wikidot.com/injective-surjective-and-bijective-functions
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Injective, Surjective, and Bijective Functions
# Injective, Surjective, and Bijective Functions
The notion of a function is fundamentally important in practically all areas of mathematics, so we must review some basic definitions regarding functions.
Definition: A Function from a set $A$ called the Domain to a set $B$ called the Codomain is a rule $f : A \to B$ which maps every element $x \in A$ to exactly one element $y \in B$ which we write $f(x) = y$. The element $y \in B$ is said to be the Image of $x$ under $f$, and the Range of $f$ is the set of elements $y \in B$ such that there exists $x \in A$ for which $f(x) = y$.
Many of the functions we are familiar with are of the form $f : \mathbb{R} \to \mathbb{R}$ such as the function $f(x) = x$ which maps every real number $x \in \mathbb{R}$ to itself. Another such example is the function $g : \mathbb{R} \to \mathbb{R}$ defined by $g(x) = x^2$ which maps every real number $x \in \mathbb{R}$ to its square.
In the first example we have that the range of $f$, commonly denoted as $R(f)$ is all of $\mathbb{R}$, so $R(f) = \mathbb{R}$. In the second example, we see that $R(g) = \mathbb{R}^+ \cup \{ 0 \}$ since there exists no real number $x \in \mathbb{R}$ for which $f(x) = y$ if $y < 0$.
Definition: A function $f : A \to B$ is said to be Injective or One-to-One if whenever $x \neq y$, $f(x) \neq f(y)$, or equivalently, whenever $f(x) = f(y)$ we have that $x = y$.
For example, consider the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 3x$. Suppose that $f(x) = f(y)$. Then:
(1)
Therefore whenever $f(x) = f(y)$ we have that $x = y$ so $f$ is injective.
For a more complicated example, consider the function $g : \mathbb{R} \to \mathbb{R}$ defined by $g(x) = x^2$. Again suppose that $g(x) = g(y)$. Then:
(2)
\begin{align} \quad x^2 &= y^2 \\ \quad \sqrt{x^2} &= \sqrt{y^2} \\ \quad \pm x &= \pm y \end{align}
Choose $x > 0$. Then $x = \pm y$ and hence $x = y$ and $x = -y$, so $g$ is not injective. To verify this, note that $g(2) = 2^2 = 4 = (-2)^2 = g(-2)$ but clearly $2 \neq -2$.
Definition: A function $f : A \to B$ is said to be Surjective or Onto if $R(f) = B$, that is for all $y \in B$ there exists an $x \in A$ such that $f(x) = y$.
For example, consider the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 3x$ once again. Let $y \in \mathbb{R}$ and suppose that $f(x) = y$. Then $3x = y$ so $x = \frac{y}{3}$, so:
(3)
\begin{align} \quad f(x) = f \left ( \frac{y}{3} \right ) = y \end{align}
Therefore $f$ is surjective.
For another example, consider the function $g : \mathbb{R} \to \mathbb{R}$ defined by $g(x) = x^2$ from earlier. Let $y \in \mathbb{R}$ be such that $y < 0$ and suppose that $g(x) = y$. Then $x^2 = y$ and $x = \sqrt{y}$. But $\sqrt{y} \not \in \mathbb{R}$, so $g$ is not surjective.
Definition: A function $f : A \to B$ is said to be Bijective if it is both injective and surjective.
In the examples from earlier, we see that $f$ is both injective and surjective, so $g$ is bijective. On the other hand, $g$ is not injective and not surjective so $g$ is definitely not bijective.
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2020-08-09 18:07:56
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https://mathematica.stackexchange.com/questions/206088/how-to-animate-a-closed-filled-circle-or-any-closed-2d-curve-rotating-abou
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How to animate a closed ( filled ) circle (or any closed 2d curve) rotating about a axis tangent to circle making it to move in 3D space? [closed]
I'm trying to learn how to animate in Mathematica. Kindly provide a code in Mathematica with an explanation. If possible kindly tell some approaches to deal with basic animations in Mathematica. I'm new to Mathematica hoping to get some guidance.
closed as off-topic by xzczd, C. E., Yves Klett, MarcoB, Henrik SchumacherSep 15 at 1:09
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question arises due to a simple mistake such as a trivial syntax error, incorrect capitalization, spelling mistake, or other typographical error and is unlikely to help any future visitors, or else it is easily found in the documentation." – xzczd, C. E., MarcoB, Henrik Schumacher
If this question can be reworded to fit the rules in the help center, please edit the question.
• Just check the document of Animate by pressing F1, everything you need is there. – xzczd Sep 11 at 11:19
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2019-09-20 15:28:21
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https://blog.wesleyac.com/posts/snes-dev-2-background-graphics
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# SNES Development Part 2: Background Graphics
Feb 10, 2022
This is part two of my series on SNES development. You may want to read the previous post first.
Now that we have a simple SNES ROM, let's add some graphics. In this post, we'll draw some graphics to the background. There are many explanations of the SNES background graphics architecture online, so I won't bother repeating those here — I recommend watching the Retro Game Mechanics Explained videos on Graphics & Palettes, Backgrounds & Rendering, and Background Modes. I don't usually recommend videos, but these are the best explanations I've found of this topic, in any format. This blog post is a reasonable general overview as well, but it has less information than the videos.
The general gist of things is that we're going to set up a palette, which will give us some colors we can use, just like we did in the previous part. Then, we'll load some images into the VRAM — these are what are colloquially called "tiles", but the official SNES documentation refers to them as "characters". Then, we'll write a "tile map" to the VRAM, which tells the SNES which characters to draw on the screen, and where. We also need to select which graphics mode we want to use — for this demo, it'll be Mode 0 with 8x8 tiles.
With that in mind, take a look at the code — especially the diff of the changes from part 1. I'd recommend reading through that with the above list in mind, and then coming back to read the rest of the explanation, once you have a overview of the code in your head.
Let's take a look at the changes in the main file first. The first thing we do is define some constants for locations in the VRAM. The SNES has 64k of VRAM, and we can lay it out however we like, although there are some alignment restrictions.
The next change is just adding three more colors to our palette. Mode 0 uses two-bit-per-pixel (2bpp) color, which means that there are four colors available, so I add four colors to the palette.
Once that's done, we set the graphics mode to Mode 0 by writing to the BGMODE register. This controls both the mode, and the tile size for each background layer, which can be either 8x8 or 16x16.
After that, we tell the SNES where to find the tile map and character map by writing to the BG1SC and BG12NBA registers. As the name implies, the second one controls the character set for both backgrounds one and two, using the two separate halves of the byte, but we don't need to worry about that for now, since we're only using BG1.
Then, we copy the character data from the ROM into VRAM. This is done with a loop, but in the future we'll replace that loop with DMA, which is much faster. For now, though, you don't need to worry about that.
It's useful to notice how to write to VRAM — we set VMADDL to the address we want to write to, and then we write to VMDATAL and VMDATAH to set the data we want to write. This is basically the same as how we wrote the palette data. One difference is that the auto-incrementing is more configurable — by setting the 7th bit of VMAIN (which means "video memory auto-increment", not "video main"), we set it to auto-increment after we write to the VMDATAH register.
I'm not going to go into detail here about the format of the character data, since other people have written about it at length — it's covered in the videos I linked at the start of the post, and a good textual description is available here. I personally found the explanations a bit confusing, and preferred to just experimentally twiddle the bits around by hand until I understood it — it's the sort of thing that can be confusing to read about, but obvious once you start playing with it.
With the character data written, we can render a character to the screen. We do this by writing two bytes to the BG1 tile map, using the same process to write to VRAM as above. The first byte controls the tile index, and the second byte controls which palette to use, the rendering priority, and whether to mirror the tile horizontally or vertically.
With all that setup out of the way, the last thing we have to do is enable rendering BG1 by setting its bit in the TM register, and voilà, a tile on the screen!
With that done, there's plenty you can experiment with:
• Change what the tile looks like, by editing the charset.asm file.
• If you're running in Mesen-S, you'll notice that the tile is surrounded by garbage data. Zero out the tile map so that doesn't happen.
• Add a couple more tiles, and render them at different spots on the screen.
• Use a mode other than Mode 0 — Mode 1 is a super common and useful mode. This changes the number of bits per pixel, so you'll need to change the character set to match.
• Experiment with building some sort of graphics editing pipeline to generate the character data. You could use something like YY-CHR, aesprite, or one of the many conversion tools available online, or write your own tools — whichever you prefer!
Once you're done with all that, head to part three for reading and responding to input!
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2022-05-26 09:00:14
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https://openreview.net/forum?id=Bys4ob-Rb
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## Certified Defenses against Adversarial Examples
Feb 15, 2018 (modified: Oct 27, 2017) ICLR 2018 Conference Blind Submission readers: everyone Show Bibtex
• Abstract: While neural networks have achieved high accuracy on standard image classification benchmarks, their accuracy drops to nearly zero in the presence of small adversarial perturbations to test inputs. Defenses based on regularization and adversarial training have been proposed, but often followed by new, stronger attacks that defeat these defenses. Can we somehow end this arms race? In this work, we study this problem for neural networks with one hidden layer. We first propose a method based on a semidefinite relaxation that outputs a certificate that for a given network and test input, no attack can force the error to exceed a certain value. Second, as this certificate is differentiable, we jointly optimize it with the network parameters, providing an adaptive regularizer that encourages robustness against all attacks. On MNIST, our approach produces a network and a certificate that no that perturbs each pixel by at most $\epsilon = 0.1$ can cause more than $35\%$ test error.
• TL;DR: We demonstrate a certifiable, trainable, and scalable method for defending against adversarial examples.
• Keywords: adversarial examples, certificate of robustness, convex relaxations
0 Replies
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2019-07-21 13:42:50
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https://www.zbmath.org/?q=an%3A1069.14044
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# zbMATH — the first resource for mathematics
$$t$$-structures on some local Calabi–Yau varieties. (English) Zbl 1069.14044
Author’s abstract: Let $$Z$$ be a smooth Fano variety satisfying the condition that the rank of the Grothendieck group of $$Z$$ is one more than the dimension of $$Z$$. Let $$\omega_Z$$ denote the total space of the canonical line bundle of $$Z$$, considered as a non-compact Calabi-Yau variety. We use the theory of exceptional collections to describe $$t$$-structures on the derived category of coherent sheaves on $$\omega_Z$$. The combinatorics of these $$t$$-structures is determined by a natural action of an affine braid group, closely related to the well-known action of the Artin braid group on the set of exceptional collections on $$Z$$.
##### MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14J45 Fano varieties 18E30 Derived categories, triangulated categories (MSC2010)
Full Text:
##### References:
[1] Beilinson, A.A., Coherent sheaves on $$\mathbb{P}^n$$ and problems of linear algebra, Funktsional. anal. i prilozhen., Funct. anal. appl., 12, 3, 68-69, (1978), English translation in · Zbl 0402.14006 [2] Beilinson, A.A.; Bernstein, J.; Deligne, P., Faisceaux pervers, Astérisque, 100, (1983) · Zbl 0536.14011 [3] Beilinson, A.A.; Ginzburg, V.; Soergel, W., Koszul duality patterns in representation theory, J. amer. math. soc., 9, 2, 473-527, (1996) · Zbl 0864.17006 [4] Birman, J.S., Braids, links and mapping class groups, Ann. of math. stud., vol. 82, (1974), Princeton Univ. Press [5] Bondal, A.I., Representation of associative algebras and coherent sheaves, Izv. akad. nauk SSSR ser. mat., Math. USSR-izv., 34, 1, 23-44, (1990), English translation in · Zbl 0692.18002 [6] Bondal, A.I.; Polishchuk, A.E., Homological properties of associative algebras: the method of helices, Izv. ross. akad. nauk ser. mat., Russian acad. sci. izv. math., 42, 2, 219-260, (1994), English translation in · Zbl 0847.16010 [7] Brenner, S.; Butler, M.C.R., Generalizations of the bernstein – gelfand – ponomarev reflection functors, (), 103-169 · Zbl 0446.16031 [8] Bridgeland, T., Stability conditions on triangulated categories, Preprint · Zbl 1137.18008 [9] T. Bridgeland, Stability conditions on $$\mathcal{O}_{\mathbb{P}^2}(- 3)$$, in preparation [10] Cassels, J.W.S., The markoff chain, Ann. of math., 50, 676-685, (1949) · Zbl 0035.31701 [11] Chow, W.-L., On the algebraical braid group, Ann. of math., 49, 654-658, (1948) · Zbl 0033.01002 [12] L. Costa, R.M. Miró-Roig, Tilting bundles, helix theory and Castelnuovo-Mumford regularity, Preprint [13] Feng, B.; Hanany, A.; He, Y.-H.; Iqbal, A., Quiver theories, soliton spectra and picard – lefschetz transformations, J. high energy phys., 2, 056, (2003), Also [14] Franco, S.; Hanany, A.; He, Y.-H., A trio of dualities: walls, trees and cascades, Proc. 36th internat. symp. ahrenshoop on the theory of elementary particles, fortschr. phys., 52, 6-7, 540-547, (2004), Also · Zbl 1052.81586 [15] Gelfand, S.I.; Manin, Yu.I, Methods of homological algebra, (1996), Springer · Zbl 0855.18001 [16] Gorodentsev, A.L., Surgeries of exceptional bundles on $$\mathbb{P}^n$$, Izv. akad. nauk SSSR ser. mat., Math. USSR-izv., 32, 1, 1-13, (1989), English translation in · Zbl 0664.14010 [17] Gorodentsev, A.L.; Rudakov, A.N., Exceptional vector bundles on projective spaces, Duke math. J., 54, 1, 115-130, (1987) · Zbl 0646.14014 [18] Hille, L., Consistent algebras and special tilting sequences, Math. Z., 220, 2, 189-205, (1995) · Zbl 0841.14013 [19] Happel, D.; Reiten, I.; Smalø, S.O., Tilting in abelian categories and quasitilted algebras, Mem. amer. math. soc., 120, 575, (1996) · Zbl 0849.16011 [20] Kapranov, M.M., The derived category of coherent sheaves on a quadric, Funktsional. anal. i prilozhen., Funct. anal. appl., 20, 2, 67, (1986), English translation in · Zbl 0607.18004 [21] Karpov, B.V.; Nogin, D.Yu., Three-block exceptional sets on del Pezzo surfaces, Izv. ross. akad. nauk ser. mat., Russian acad. sci. izv. math., 62, 3, 429-463, (1998), English translation in · Zbl 0949.14026 [22] Kent, R.P.; Peifer, D., A geometric and algebraic description of annular braid groups, Internat. J. algebra comput., 12, 1 & 2, 85-97, (2002) · Zbl 1010.20024 [23] Orlov, D.O., An exceptional set of vector bundles on the variety $$V_5$$, Vestnik moskov. univ. ser. I mat. mekh., Moscow univ. math. bull., 46, 5, 48-50, (1991), English translation in · Zbl 0784.14010 [24] Rankin, R.A., Modular forms and functions, (1977), Cambridge University Press · Zbl 0376.10020 [25] Rickard, J., Morita theory for derived categories, J. London math. soc., 39, 3, 436-456, (1989) · Zbl 0642.16034 [26] Rudakov, A.N., Helices and vector bundles: seminaire rudakov, London math. soc. lecture note ser., vol. 148, (1990), Cambridge University Press Cambridge · Zbl 0727.00022 [27] Rudakov, A.N., Exceptional vector bundles on a quadric, Izv. akad. nauk SSSR ser. mat., Math. USSR-izv., 33, 1, 115-138, (1989), 896. English translation in · Zbl 0708.14012 [28] Seidel, P.; Thomas, R.P., Braid group actions on derived categories of coherent sheaves, Duke math. J., 108, 1, 37-108, (2001) · Zbl 1092.14025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-09-20 07:27:33
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https://www.physicsforums.com/threads/field-extensions-remarks-by-lovett-page-326.914005/
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I Field Extensions - Remarks by Lovett - Page 326 ... ...
1. May 8, 2017
Math Amateur
I am reading "Abstract Algebra: Structures and Applications" by Stephen Lovett ...
I am currently focused on Chapter 7: Field Extensions ... ...
I need help with some remarks of Lovett following Theorem 7.1.12 and Example 7.1.13 on page 326 ...
The remarks by Lovett read as follows:
In the above remarks from Lovett, we read the following:
" ... ... In the quotient ring $K$, this implies that $\overline{ a(x) q(x) } = 1$. Thus in $K, \ a( \alpha ) q( \alpha ) = 1$. ... ... "
My question is as follows:
Can someone please explain exactly why/how it is that $\overline{ a(x) q(x) } = 1$ implies that $a( \alpha ) q( \alpha ) = 1$ ... ... ?
Help will be appreciated ...
Peter
File size:
68.3 KB
Views:
93
2. May 8, 2017
andrewkirk
In the equation $a(x)q(x)+b(x)p(x)=1$, substitute $\alpha$ for $x$. Since $p(\alpha)=0$ (we were told $\alpha$ is a root of $p(x)$) the equation collapses to the desired result.
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2018-07-20 15:16:53
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https://www.ademcetinkaya.com/2022/11/how-do-you-predict-if-stock-will-go-up_14.html
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Stock prediction is a very hot topic in our life. However, in the early time, because of some reasons and the limitation of the device, only a few people had the access to the study. Thanks to the rapid development of science and technology, in recent years more and more people are devoted to the study of the prediction and it becomes easier and easier for us to make stock prediction by using different ways now, including machine learning, deep learning and so on. We evaluate SEGRO PLC prediction models with Modular Neural Network (Financial Sentiment Analysis) and Logistic Regression1,2,3,4 and conclude that the LON:SGRO stock is predictable in the short/long term. According to price forecasts for (n+1 year) period: The dominant strategy among neural network is to Hold LON:SGRO stock.
Keywords: LON:SGRO, SEGRO PLC, stock forecast, machine learning based prediction, risk rating, buy-sell behaviour, stock analysis, target price analysis, options and futures.
## Key Points
1. Can stock prices be predicted?
2. What are the most successful trading algorithms?
3. How do you know when a stock will go up or down?
## LON:SGRO Target Price Prediction Modeling Methodology
Fuzzy rough theory can describe real-world situations in a mathematically effective and interpretable way, while evolutionary neural networks can be utilized to solve complex problems. Combining them with these complementary capabilities may lead to evolutionary fuzzy rough neural network with the interpretability and prediction capability. In this article, we propose modifications to the existing models of fuzzy rough neural network and then develop a powerful evolutionary framework for fuzzy rough neural networks by inheriting the merits of both the aforementioned systems. We consider SEGRO PLC Stock Decision Process with Logistic Regression where A is the set of discrete actions of LON:SGRO stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4
F(Logistic Regression)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Modular Neural Network (Financial Sentiment Analysis)) X S(n):→ (n+1 year) $∑ i = 1 n a i$
n:Time series to forecast
p:Price signals of LON:SGRO stock
j:Nash equilibria
k:Dominated move
a:Best response for target price
For further technical information as per how our model work we invite you to visit the article below:
How do AC Investment Research machine learning (predictive) algorithms actually work?
## LON:SGRO Stock Forecast (Buy or Sell) for (n+1 year)
Sample Set: Neural Network
Stock/Index: LON:SGRO SEGRO PLC
Time series to forecast n: 14 Nov 2022 for (n+1 year)
According to price forecasts for (n+1 year) period: The dominant strategy among neural network is to Hold LON:SGRO stock.
X axis: *Likelihood% (The higher the percentage value, the more likely the event will occur.)
Y axis: *Potential Impact% (The higher the percentage value, the more likely the price will deviate.)
Z axis (Yellow to Green): *Technical Analysis%
## Adjusted IFRS* Prediction Methods for SEGRO PLC
1. However, the fact that a financial asset is non-recourse does not in itself necessarily preclude the financial asset from meeting the condition in paragraphs 4.1.2(b) and 4.1.2A(b). In such situations, the creditor is required to assess ('look through to') the particular underlying assets or cash flows to determine whether the contractual cash flows of the financial asset being classified are payments of principal and interest on the principal amount outstanding. If the terms of the financial asset give rise to any other cash flows or limit the cash flows in a manner inconsistent with payments representing principal and interest, the financial asset does not meet the condition in paragraphs 4.1.2(b) and 4.1.2A(b). Whether the underlying assets are financial assets or non-financial assets does not in itself affect this assessment.
2. That the transferee is unlikely to sell the transferred asset does not, of itself, mean that the transferor has retained control of the transferred asset. However, if a put option or guarantee constrains the transferee from selling the transferred asset, then the transferor has retained control of the transferred asset. For example, if a put option or guarantee is sufficiently valuable it constrains the transferee from selling the transferred asset because the transferee would, in practice, not sell the transferred asset to a third party without attaching a similar option or other restrictive conditions. Instead, the transferee would hold the transferred asset so as to obtain payments under the guarantee or put option. Under these circumstances the transferor has retained control of the transferred asset.
3. The definition of a derivative in this Standard includes contracts that are settled gross by delivery of the underlying item (eg a forward contract to purchase a fixed rate debt instrument). An entity may have a contract to buy or sell a non-financial item that can be settled net in cash or another financial instrument or by exchanging financial instruments (eg a contract to buy or sell a commodity at a fixed price at a future date). Such a contract is within the scope of this Standard unless it was entered into and continues to be held for the purpose of delivery of a non-financial item in accordance with the entity's expected purchase, sale or usage requirements. However, this Standard applies to such contracts for an entity's expected purchase, sale or usage requirements if the entity makes a designation in accordance with paragraph 2.5 (see paragraphs 2.4–2.7).
4. An entity may use practical expedients when measuring expected credit losses if they are consistent with the principles in paragraph 5.5.17. An example of a practical expedient is the calculation of the expected credit losses on trade receivables using a provision matrix. The entity would use its historical credit loss experience (adjusted as appropriate in accordance with paragraphs B5.5.51–B5.5.52) for trade receivables to estimate the 12-month expected credit losses or the lifetime expected credit losses on the financial assets as relevant. A provision matrix might, for example, specify fixed provision rates depending on the number of days that a trade receivable is past due (for example, 1 per cent if not past due, 2 per cent if less than 30 days past due, 3 per cent if more than 30 days but less than 90 days past due, 20 per cent if 90–180 days past due etc). Depending on the diversity of its customer base, the entity would use appropriate groupings if its historical credit loss experience shows significantly different loss patterns for different customer segments. Examples of criteria that might be used to group assets include geographical region, product type, customer rating, collateral or trade credit insurance and type of customer (such as wholesale or retail)
*International Financial Reporting Standards (IFRS) are a set of accounting rules for the financial statements of public companies that are intended to make them consistent, transparent, and easily comparable around the world.
## Conclusions
SEGRO PLC assigned short-term B3 & long-term Ba3 forecasted stock rating. We evaluate the prediction models Modular Neural Network (Financial Sentiment Analysis) with Logistic Regression1,2,3,4 and conclude that the LON:SGRO stock is predictable in the short/long term. According to price forecasts for (n+1 year) period: The dominant strategy among neural network is to Hold LON:SGRO stock.
### Financial State Forecast for LON:SGRO SEGRO PLC Stock Options & Futures
Rating Short-Term Long-Term Senior
Outlook*B3Ba3
Operational Risk 4276
Market Risk3356
Technical Analysis5846
Fundamental Analysis5271
Risk Unsystematic5580
### Prediction Confidence Score
Trust metric by Neural Network: 76 out of 100 with 479 signals.
## References
1. R. Rockafellar and S. Uryasev. Conditional value-at-risk for general loss distributions. Journal of Banking and Finance, 26(7):1443 – 1471, 2002
2. Arora S, Li Y, Liang Y, Ma T. 2016. RAND-WALK: a latent variable model approach to word embeddings. Trans. Assoc. Comput. Linguist. 4:385–99
3. J. Peters, S. Vijayakumar, and S. Schaal. Natural actor-critic. In Proceedings of the Sixteenth European Conference on Machine Learning, pages 280–291, 2005.
4. Mnih A, Teh YW. 2012. A fast and simple algorithm for training neural probabilistic language models. In Proceedings of the 29th International Conference on Machine Learning, pp. 419–26. La Jolla, CA: Int. Mach. Learn. Soc.
5. Li L, Chu W, Langford J, Moon T, Wang X. 2012. An unbiased offline evaluation of contextual bandit algo- rithms with generalized linear models. In Proceedings of 4th ACM International Conference on Web Search and Data Mining, pp. 297–306. New York: ACM
6. Blei DM, Lafferty JD. 2009. Topic models. In Text Mining: Classification, Clustering, and Applications, ed. A Srivastava, M Sahami, pp. 101–24. Boca Raton, FL: CRC Press
7. Allen, P. G. (1994), "Economic forecasting in agriculture," International Journal of Forecasting, 10, 81–135.
Frequently Asked QuestionsQ: What is the prediction methodology for LON:SGRO stock?
A: LON:SGRO stock prediction methodology: We evaluate the prediction models Modular Neural Network (Financial Sentiment Analysis) and Logistic Regression
Q: Is LON:SGRO stock a buy or sell?
A: The dominant strategy among neural network is to Hold LON:SGRO Stock.
Q: Is SEGRO PLC stock a good investment?
A: The consensus rating for SEGRO PLC is Hold and assigned short-term B3 & long-term Ba3 forecasted stock rating.
Q: What is the consensus rating of LON:SGRO stock?
A: The consensus rating for LON:SGRO is Hold.
Q: What is the prediction period for LON:SGRO stock?
A: The prediction period for LON:SGRO is (n+1 year)
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2022-12-05 01:57:02
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http://tex.stackexchange.com/questions/184226/sequence-does-not-come-out-right
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# Sequence does not come out right
\usepackage{tikz}
\usetikzlibrary{matrix,arrows,decorations.pathmorphing}
\begin{tikzpicture}[scale=1.5]
\node (A) at (1,0) {$0$};
\node (B) at (2,0) {$H^1(G/H,M^H$};
\node (C) at (3,0) {$H^1(G,M)$};
\node (D) at (4,0) {$H^1(H,M)$};
\path[->,font=\scriptsize,>=angle 90]
(A) edge node[above]{} (B)
(B) edge node[above]{$\textnormal{Inf}$} (C)
(C) edge node[above]{$\textnormal{Res}$} (D);
\end{tikzpicture}
I'm writing a LaTeX document with this code and while the diagram has appeared in the right order, they are squeezed together because of the length of the elements in the sequence. How can I fix this?
-
Welcome to TeX.SX! – egreg Jun 10 '14 at 20:34
You want to use tikz-cd:
\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz-cd}
\DeclareMathOperator{\Inf}{Inf}
\DeclareMathOperator{\Res}{Res}
\begin{document}
\begin{tikzcd}
0 \arrow{r} &
H^1(G/H,M^H) \arrow{r}{\Inf} &
H^1(G,M) \arrow{r}{\Res} &
H^1(H,M)
\end{tikzcd}
\end{document}
-
Thanks for the quick response! – Haikal Yeo Jun 10 '14 at 20:36
I'm not a specialist in group cohomology, but I've used several diagrams and exact sequences. ;-) – egreg Jun 10 '14 at 20:39
I got this error when I typed the code into my latex document in SAGE: <inserted text> \cr l.231 ! Misplaced \cr. <inserted text> \cr Do you know what's wrong? – Haikal Yeo Jun 10 '14 at 21:02
@HaikalYeo Sorry, but I don't know about SAGE. Are you sure you have an up-to-date TeX distribution? – egreg Jun 10 '14 at 21:03
Have a look at the Sage tutorial where there are instructions about how to include latex packages. – Andrew Swann Jun 11 '14 at 6:50
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2015-08-04 22:11:13
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https://bookstore.ams.org/view?ProductCode=THETA/15
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Operator Theory Live: Timişoara, July 3–8, 2008
Edited by: Hari Bercovici Indiana University, Bloomington, IN
Dumitru Gaspar West University of Timişoara, Timişoara, Romania
Dan Timotin Romanian Academy, Bucharest, Romania
Florian-Horia Vasilescu University of Lille I, Villeneuve d’Ascq, France
A publication of Theta Foundation
Available Formats:
Hardcover ISBN: 978-973-87899-6-8
Product Code: THETA/15
List Price: $56.00 AMS Member Price:$44.80
Please note AMS points can not be used for this product
Click above image for expanded view
Operator Theory Live: Timişoara, July 3–8, 2008
Edited by: Hari Bercovici Indiana University, Bloomington, IN
Dumitru Gaspar West University of Timişoara, Timişoara, Romania
Dan Timotin Romanian Academy, Bucharest, Romania
Florian-Horia Vasilescu University of Lille I, Villeneuve d’Ascq, France
A publication of Theta Foundation
Available Formats:
Hardcover ISBN: 978-973-87899-6-8 Product Code: THETA/15
List Price: $56.00 AMS Member Price:$44.80
Please note AMS points can not be used for this product
• Book Details
Theta Foundation International Book Series of Mathematical Texts
Volume: 152010; 229 pp
MSC: Primary 00; 46; 47;
The volume represents the proceedings of the 22nd International Conference on Operator Theory, held in Timişoara, Romania, from July 3 to July 8, 2008. It includes a survey on Carleson measures and composition operators, as well as eighteen papers containing original research on a large variety of topics: single operator theory, Banach algebras, $C^*$-algebras, von Neumann algebras, moment problems, differential and integral operators, noncommutative probability, and spectral theory.
Graduate students and research mathematicians interested in analysis.
• Request Review Copy
Volume: 152010; 229 pp
MSC: Primary 00; 46; 47;
The volume represents the proceedings of the 22nd International Conference on Operator Theory, held in Timişoara, Romania, from July 3 to July 8, 2008. It includes a survey on Carleson measures and composition operators, as well as eighteen papers containing original research on a large variety of topics: single operator theory, Banach algebras, $C^*$-algebras, von Neumann algebras, moment problems, differential and integral operators, noncommutative probability, and spectral theory.
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2023-02-07 02:44:38
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https://www.enotes.com/homework-help/calc-331346
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# Calc. Find the limit using L'Hospital's Rule. lim x--->0 (x)/(tan^-1 (4x))
## Expert Answers
You need to use l'Hospital's theorem if the limit proves to be indeterminate, hence you should check if it is the case such that:
`lim_(x-gt0) x/(tan^(-1)4x) = 0/(tan^(-1) 0) = 0/0`
Since the limit is indeterminate, you may use l'Hospital's theorem such that:
`lim_(x-gt0) x/(tan^(-1)4x) = lim_(x-gt0) (x')/((tan^(-1)4x)')`
`lim_(x-gt0) (x')/((tan^(-1)4x)')...
## See This Answer Now
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You need to use l'Hospital's theorem if the limit proves to be indeterminate, hence you should check if it is the case such that:
`lim_(x-gt0) x/(tan^(-1)4x) = 0/(tan^(-1) 0) = 0/0`
Since the limit is indeterminate, you may use l'Hospital's theorem such that:
`lim_(x-gt0) x/(tan^(-1)4x) = lim_(x-gt0) (x')/((tan^(-1)4x)')`
`lim_(x-gt0) (x')/((tan^(-1)4x)') = lim_(x-gt0) 1/(((4x)')/(1+16x^2))`
`lim_(x-gt0) 1/(((4x)')/(1+16x^2))= lim_(x-gt0)(1+16x^2)/4`
`lim_(x-gt0)(1+16x^2)/4 = (1+0)/4 = 1/4`
Hence, evaluating the limit to the given function yields `lim_(x-gt0) x/(tan^(-1)4x) =1/4.`
Approved by eNotes Editorial Team
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2022-05-24 15:35:29
|
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|
http://www.davidchudzicki.com/
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## Regexes for replacing ugly unittest-style assertions
December 30, 2017
In case they help anyone else, here are some regular expressions I used once to convert some ugly unittest-style assertions (e.g. self.assertEqual(something, something_else) to the pytest style (simply assert something == something_else):
(Pytest gives nice informative error messages even if you just use the prettier form.)
Note:
• The option -i means "do it in-place" (modify the file). Including ".bak" means "make backups of the old version with this extension".
• I don't actually want the backups, but (for some odd reason) on my Mac, not asking for them changed how the regex was interpreted to something that's not right.
• After reviewing and checking in the changes I wanted, I cleaned up the backups with git clean -f (careful you don't have any unchecked-in changes you want to keep!).
## An Interaction or Not? How a few ML Models Generalize to New Data
March 4, 2015
Source code for this post is here.
This post examines how a few statistical and machine learning models respond to a simple toy example where they're asked to make predictions on new regions of feature space. The key question the models will answer differently is whether there's an "interaction" between two features: does the influence of one feature differ depending on the value of another.
In this case, the data won't provide information about whether there's an interaction or not. Interactions are often real and important, but in many contexts we treat interaction effects as likely to be small (without evidence otherwise). I'll walk through why decision trees and bagged ensembles of decision trees (random forests) can make the opposite assumption: they can strongly prefer an interaction, even when the evidence is equally consistent with including or not including an interaction.
I'll look at point estimates from:
• a linear model
• decision trees and bagged decision trees (random forest), using R's randomForest package
• boosted decision trees, using the R's gbm package
I'll also look at two models that capture uncertainty about whether there's an interaction:
BART has the advantage of expressing uncertainty while still being a "machine learning" type model that learns interactions, non-linearities, etc. without the user having to decide which terms to include or the particular functional form.
Whenever possible, I recommend using models like BART that explicitly allow for uncertainty.
# The Example
Suppose you're given this data and asked to make a prediction at $X_1 = 0$, $X_2 = 1$ (where there isn't any training data):
X1 X2 Y N Training Rows:
0 0 Y = 5 + noise 52
1 0 Y = 15 + noise 23
1 1 Y = 19 + noise 25
0 1 ? 0
## Covariance As Signed Area Of Rectangles
March 26, 2014
A colleague at work recently pointed me to a wonderful stats.stackexchange answer with an intuitive explanation of covariance: For each pair of points, draw the rectangle with these points at opposite corners. Treat the rectangle's area as signed, with the same sign as the slope of the line between the two points. If you add up all of the areas, you have the (sample) covariance, up to a constant that depends only on the data set.
Here's an example with 4 points. Each spot on the plot is colored by the sum corresponding to that point. For example, the dark space in the lower left has three "positively" signed rectangles going through it, but for the white space in the middle, one positive and one negative rectangle cancel out.
In this next example, x and y are drawn from independent normals, so we have roughly an even amount of positive and negative:
## Formal Explanation
The formal way to speak about multiple draws from a distribution is with a set of independent and identically distributed (i.i.d.) random variables. If we have a random variable X, saying that X1, X2, … are i.i.d means that they are all independent, but follow the same distribution.
## Previous Posts
February 16, 2014
### Simulated Knitting (post)
I created a KnittedGraph class (subclassing of Python's igraph graph class) with methods corresponding to common operations performed while knitting:
I then embed the graphs in 3D space. Here's a hat I made this way:
### 2D Embeddings from Unsupervised Random Forests (1, 2)
There are all sorts of ways to embed high-dimensional data in low dimensions for visualization. Here's one:
1. Given some set of high dimensional examples, build a random forest to distinguish examples from non-examples.
2. Assign similarities to pairs of examples based on how often they are in leaf nodes together.
3. Map examples to 2D in such a way that similarity decreases decreases with Euclidean 2D distance (I used multidimensional scaling for this).
Here's the result of doing this on a set of diamond shapes I constructed. I like how it turned out:
### A Bayesian Model for a Function Increasing by Chi-Squared Jumps (in Stan) (post)
In this paper, Andrew Gelman mentions a neat example where there's a big problem with a naive approach to putting a Bayesian prior on functions that are constrained to be increasing. So I thought about what sort of prior would make sense for such functions, and fit the models in Stan.
I enjoyed Andrew's description of my attempt: "... it has a charming DIY flavor that might make you feel that you too can patch together a model in Stan to do what you need."
### Lissijous Curves JSFiddle
Some JavaScript I wrote (using d3) to mimick what an oscilloscope I saw at the Exploratorium was doing:
### Visualization of the Weirstrass Elliptic Function as a Sum of Terms
John Baez used this in his AMS blog Visual Insight.
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2018-01-19 05:21:15
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|
https://codegolf.stackexchange.com/questions/98129/positional-bathroom-etiquette/98215
|
# Positional Bathroom Etiquette
### Background
Bathroom Etiquette, when pertaining to the available urinals, states that the next urinal to be filled in should be the one that minimizes the total discomfort. The total discomfort equation is given by the following set of equations:
dist(x,y) = linear distance between person x and person y in Urinal Units
discomfort(x) = sum(1/(dist(x,y)*dist(x,y))) for all persons y excluding person x
total_Discomfort = sum(discomfort(x)) for all x
A more in depth paper dealing with a similar (not the exact same) problem can be found here: (Thanks to @Lembik for alerting me to this amazing whitepaper!)
### Input/Output
Given an input of a empty and full urinals, output the resulting set of urinals with the addition of one person. If there is a tie for a position the urinals should fill in left to right. The output should be in the same format as the input.
• If given a case with full urinals, return the input.
• The input will always have at least one urinal defined.
### Test Cases
INPUT -> OUTPUT
1000001 -> 1001001
101010101 -> 111010101
100 -> 101
00000 -> 10000
1111111 -> 1111111
0100 -> 0101
101000 -> 101001
### Rules
This is , so shortest code in bytes wins. Standard loop-holes are forbidden.
• Related problem: codegolf.stackexchange.com/questions/47952/the-urinal-protocol Nov 1, 2016 at 3:42
• It is recommended to wait about a week before accepting an answer. Accepting in less than a day might decrease the amount of answers your challenge receives. Nov 1, 2016 at 15:05
• I'd suggest adding 0100 and 101000 in the test cases (some regex-based approach work on the actual test cases but won't work on those ones which should still be handled)
Nov 1, 2016 at 15:07
• also related: codegolf.stackexchange.com/questions/94074/… Nov 2, 2016 at 14:02
• @TheBitByte How is it offensive? It's a pretty accurate description of how men choose urinals in a bathroom. Nov 11, 2016 at 22:50
# MATL, 1918 17 bytes
lyf!Gn:-H_^Xs&X<(
Try it online! Or verify all test cases (slightly modified code).
### Explanation
It suffices to compute the distance from each potential new position to the already occupied ones. The remaining distances are do not depend on the potential new position, and so constitute a constant term, which can be ignored.
Let's take input [1 0 0 0 0 0 1] as an example.
l % Push 1
% STACK: 1
y % Take input implicitly. Duplicate from below
% STACK: [1 0 0 0 0 0 1], 1, [1 0 0 0 0 0 1]
f! % Indices of nonzero elements, as a column array
% STACK: [1 0 0 0 0 0 1], 1, [1 7]
Gn: % Push [1 2 ... n], where n is input size (array of possible positions)
% STACK: [1 0 0 0 0 0 1], 1, [1; 7], [1 2 3 4 5 6 7]
- % Matrix with all pairs of differences
% STACK: [1 0 0 0 0 0 1], 1, [1; 7], [0 -1 -2 -3 -4 -5 -6;
6 5 4 3 2 1 0]
H_^ % Raise each entry to -2
% STACK: [1 0 0 0 0 0 1], 1, [ Inf 1.0000 0.2500 0.1111 0.0625 0.0400 0.0278;
0.0278 0.0400 0.0625 0.1111 0.2500 1.0000 Inf]
Xs % Sum of each column
% STACK: [1 0 0 0 0 0 1], 1, [Inf 1.04 0.3125 0.2222 0.3125 1.04 Inf]
&X< % Index of minimum. Takes the first if there is a tie
% STACK: [1 0 0 0 0 0 1], 1, 4
( % Assign: write 1 at the position of the minimizer
% STACK: [1 0 0 1 0 0 1]
% Implicitly display
## JavaScript (ES6), 89 bytes
a=>a[a.map((e,i)=>!e&&(t=0,a.map((e,j)=>t+=(j-=i)&&e/j/j),t<m&&(m=t,k=i)),k=0,m=1/0),k]=1
Outputs by modifying the input array.
# R, 8376 67 bytes
Just realized that I can save several bytes by not bothering to check if the candidate urinals are empty. Non-empty urinals will always return an Inf discomfort value, so they're excluded in the course of the calculation. Also, just using direct indexing rather than replace, so it's shorter but less elegant.
x=scan()
x[which.min(rowSums(outer(seq(x),which(!!x),-)^-2))]=1
x
### Explanation
x=scan()
We read the current state from stdin and call it x. We assume that the input is a sequence of 1s and 0s separated by spaces or newlines. For the purposes of the explanation, let's say we input 1 0 0 0 0 0 1.
x[which.min(rowSums(outer(seq(x),which(!!x),-)^-2))]=1
We replace a value of x at a particular index with 1. Everything between the [ ] is figuring out what the best index is.
Since the existing urinals are immutable, we don't need to consider the distances between them. We only need to consider the distances between the occupied urinals and the possible new one. So we determine the indices of the occupied urinals. We use which, a function to return the indices of a logical vector which are TRUE. All numbers in R, when coerced to type logical, are TRUE if nonzero and FALSE if zero. Simply doing which(x) will result in a type error, argument to 'which' is not logical, as x is a numeric vector. We therefore have to coerce it to logical. ! is R's logical negation function, which automatically coerces to logical. Applying it twice, !!x, yields a vector of TRUE and FALSE indicating which urinals are occupied. (Alternative byte-equivalent coercions to logical involve the logical operators & and | and the builtins T and F, e.g. F|x or T&x and so on. !!x looks more exclamatory so we'll use that.)
which(!!x)
This is paired with seq(x), which returns the integer sequence from 1 to the length of x, i.e. all urinal locations (and thus all possible locations to consider).
seq(x)
Now we have the indices of our occupied urinals: 1 7 and our empty urinals 1 2 3 4 5 6 7. We pass -, the subtraction function, to the outer function to get the "outer subtraction", which is the following matrix of distances between all urinals and the occupied urinals:
[,1] [,2]
[1,] 0 -6
[2,] 1 -5
[3,] 2 -4
[4,] 3 -3
[5,] 4 -2
[6,] 5 -1
[7,] 6 0
outer(seq(x),which(!!x),-)
We raise this to the -2th power. (For those who are a little lost, in the OP, "discomfort" is defined as 1 / (distance(x, y) * distance(x, y)), which simplifies to 1/d(x,y)^2, i.e. d(x,y)^-2.)
outer(seq(x),which(!!x),-)^-2
Take the sum of each row in the matrix.
rowSums(outer(seq(x),which(!!x),-)^-2)
Get the index of the smallest value, i.e. the optimal urinal. In the case of multiple smallest values, the first (i.e. leftmost) one is returned.
which.min(rowSums(outer(seq(x),which(!!x),-)^-2))
And voilà, we have the index of the optimal urinal. We replace the value at this index in x with 1. In the case of 1111 as input, it doesn't matter which one we replace, we'll still have a valid output.
x[which.min(rowSums(outer(seq(x),which(!!x),-)^-2))]=1
Return the modified input.
x
# Jelly, 13 12 bytes
J_þTݲSiṂ$Ṭo ## Explanation J_þTݲSiṂ$Ṭo Input: boolean array A
J Indices, returns [1, 2, ..., len(A)]
T Truthy indices, returns the indices which have a truthy value
_þ Form the subtraction (_) table (þ) between them
İ Inverse, find the reciprocal of each
² Square each
S Sum the sublists column-wise
$Monadic chain Ṃ Minimum i Find the first index of that Ṭ Untruth indices, returns a boolean array with 1's at those indices o Logical OR between that and A, and return ## PHP, 135 bytes $a=explode(1,$argv[1]);$b=0;foreach($a as$c=>$d){$l=strlen($d);if($l>$b){$b=$l;$e=$c;}}if($b)$a[$e][intval($b/2)]=1;echo implode(1,$a);
I'm sure there's a considerably quicker way of doing it, but I've got a fuzzy head and can't think of one!
### Old code
The code without minification:
$a=explode(1,$argv[1]);
$b=0; foreach($a as $c=>$d){
$l=strlen($d);
if($l>$b){
$b=$l;
$e=$c;
}
}
if($b){$a[$e][intval($b/2)]=1;
}
echo implode(1,\$a);
# Python 3 223222 165 Bytes
Okay, I know this isn't the prettiest answer out there, and I'm sure it can be golfed down quite a bit, but I was just messing around and seeing what I could do
Shout out to mbomb007 for the tips on whitespace and comparators Also, I saw my online character counter was taking all the tabs and turning them into spaces, so the count is a lot less than I originally had
def u(a):
m,r,x=9,0,len(a)
for i in range(x):
d=0
if a[i]<'1':
for j in range(x):
if a[j]>'0':d+=float((j-i)**-2)
if d<m:r=i;m=d
return a[:r]+'1'+a[r+1:]
Showing revised whitespace:
def u(a):
<sp> m,r,x=9,0,len(a)
<sp> for i in range(x):
<tab> d=0
<tab> if a[i]<'1':
<tab><sp> for j in range(x):
<tab><tab> if a[j]>'0':d+=float((j-i)**-2)
<tab><sp> if d<m:r=i;m=d
<sp> return a[:r]+'1'+a[r+1:]
Original:
def u(a):
m,r,x=9,0,len(a)
for i in range(x):
d=0
if a[i]!='1':
for j in range(x):
if a[j]=='1':d+=float(1/(j-i)**2)
if d<m:r=i;m=d
return a[:r]+'1'+a[r+1:]
This expects a string passed to it of 1's and 0's like "10001" and returns a string "10101"
Edit: Changed 1/float((j-i)**2) to float((j-i)**-2)
• !='1' can be <'1' and =='1' can be >'0'. Also, consider this tip Nov 1, 2016 at 16:20
• Thanks for that whitespace tip. I definitely did not know that. That's awesome! Nov 1, 2016 at 16:24
• That whitespace tip only works in Python 2, I think. Maybe early version of Python 3, but idk. You'll have to restrict your answer to Python 2 or some specific version of 3 with it working. Nov 1, 2016 at 16:37
• I've got it running in a 3.5.2 shell in IDLE and it's running without an issue, so I think it's alright still Nov 1, 2016 at 18:05
# Python 3, 574471 347 bytes
I'll probably work on this some more, considering the other Python solution is like a fifth of this one :[.
def a(I):
D,l,r={},len(I),range
for i in r(l):
if I[i]<1:
n,t,n[i]=I[:],[],1
for j in r(l):
if n[j]>0:
q,Q=[],0
for k in r(l):
if k!=j and n[k]>0:q.append((k-j,j-k)[k<j])
for i in q:Q+=1/(i**2)
t.append(Q)
T=sum(t)
if T not in D.keys():D[T]=i
if len(D)>0:I[D[min(D.keys())]]=1
print(I)
Well that's much better now that I've learned you can use single spaces.
# Python, 165163158147141140 139 bytes
def u(p):e=enumerate;a=[(sum((i-j)**-2for j,y in e(p)if"0"<y),i)for i,x in e(p)if"1">x];return a and p[:min(a)[1]]+"1"+p[min(a)[1]+1:] or p
• rewrite second line as if"1"*len(p)==p:return p to save a byte Nov 1, 2016 at 22:07
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2022-06-29 18:58:45
|
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|
http://itensor.org/docs.cgi?vers=cppv2&page=classes/mps_mpo_algs
|
## Learn to Use ITensor
main / classes / mps_mpo_algs C++v3 | C++v2
# Algorithms for MPS and MPO (also IQMPS and IQMPO)
## Summing MPS
• sum(MPS psi1, MPS psi2, Args args = Args::global()) -> MPS
sum(IQMPS psi1, IQMPS psi2, Args args = Args::global()) -> IQMPS
Return the sum of the MPS psi1 and ps2. The returned MPS will have an orthogonality center on site 1. Before being returned, the MPS representing the sum will be compressed using truncation parameters provided in the named arguments args.
Show Example
auto psi3 = sum(psi1,psi2,{"Maxm",500,"Cutoff",1E-8});
• sum(vector<MPS> terms, Args args = Args::global()) -> MPS
sum(vector<IQMPS> terms, Args args = Args::global()) -> IQMPS
Returns the sum of all the MPS provided in the vector terms as a single MPS, using the truncation accuracy parameters (such as "Cutoff" or "Maxm") provided in the named arguments args to control the accuracy of the sum.
This function uses a hierarchical, tree-like algorithm which first sums pairs of MPS, then pairs of pairs, etc. so that the largest bond dimensions are only reached toward the end of the process for maximum efficiency. Therefore using this algorithm can be much faster than calling the above two-argument sum function to sum the terms one at a time.
Show Example
auto terms = vector<MPS>(4);
terms.at(0) = psi0;
terms.at(1) = psi1;
terms.at(2) = psi2;
terms.at(3) = psi3;
auto res = sum(terms,{"Cutoff",1E-8});
## Overlaps, Matrix Elements, and Expectation Values
• overlap(MPS psi1, MPS psi2) -> Real
overlap(IQMPS psi1, IQMPS psi2) -> Real
overlapC(MPS psi1, MPS psi2) -> Cplx
overlapC(IQMPS psi1, IQMPS psi2) -> Cplx
Compute the exact overlap $\langle \psi_1|\psi_2 \rangle$ of two MPS or IQMPS. If the overlap value is expected to be a complex number use overlapC.
The algorithm used scales as $m^3 d$ where $m$ is typical bond dimension of the MPS and $d$ is the site dimension.
(In ITensor version 1.x this function was called psiphi. This name is still supported for backwards compatibility.)
• overlap(MPS psi1, MPO W, MPS psi2) -> Real
overlap(IQMPS psi1, IQMPO W, IQMPS psi2) -> Real
overlapC(MPS psi1, MPO W, MPS psi2) -> Cplx
overlapC(IQMPS psi1, IQMPO W, IQMPS psi2) -> Cplx
Compute the exact overlap (or matrix element) $\langle \psi_1|W|\psi_2 \rangle$ of two MPS psi1 and psi2 with respect to an MPO W.
The algorithm used scales as $m^3\, k\,d + m^2\, k^2\, d^2$ where $m$ is typical bond dimension of the MPS, $k$ is the typical MPO dimension, and $d$ is the site dimension.
(In ITensor version 1.x this function was called psiHphi. This name is still supported for backwards compatibility.)
• overlap(MPS psi1, MPO W1, MPO W2, MPS psi2) -> Real
overlap(IQMPS psi1, IQMPO W1, IQMPO W2, IQMPS psi2) -> Real
overlapC(MPS psi1, MPO W1, MPO W2, MPS psi2) -> Cplx
overlapC(IQMPS psi1, IQMPO W1, IQMPO W2, IQMPS psi2) -> Cplx
Compute the exact overlap (or matrix element) $\langle \psi_1|W_1 W_2 |\psi_2 \rangle$ of two MPS psi1 and psi2 with respect to two MPOs W1 and W2.
The algorithm used scales as $m^3\, k^2\,d + m^2\, k^3\, d^2$ where $m$ is typical bond dimension of the MPS, $k$ is the typical MPO dimension, and $d$ is the site dimension.
(In ITensor version 1.x this function was called psiHKphi. This name is still supported for backwards compatibility.)
## Multiplying MPOs
• nmultMPO(MPO A, MPO B, MPO & C, Args args = Args::global())
nmultMPO(IQMPO A, IQMPO B, IQMPO & C, Args args = Args::global())
Multiply MPOs A and B. On return, the result is stored in C. MPO tensors are multiplied one at a time from left to right and the resulting tensors are compressed using the truncation parameters (such as "Cutoff" and "Maxm") provided through the named arguments args.
Show Example
MPO C;
nmultMPO(A,B,C,{"Maxm",500,"Cutoff",1E-8});
## Applying MPO to MPS
• applyMPO(MPO K, MPS psi, Args args = Args::global()) -> MPS
applyMPO(IQMPO K, IQMPS psi, Args args = Args::global()) -> IQMPS
Apply an MPO K to an MPS psi, resulting in an approximation to the MPS phi:
$|\phi\rangle = K |\psi\rangle$ .
The resulting MPS is returned. The algorithm used is chosen with the parameter "Method" in the named arguments args.
The default algorithm used is the "density matrix" algorithm, chosen by setting the parameter "Method" to "DensityMatrix". If the input MPS has a typical bond dimension of $m$ and the MPO has typical bond dimension $k$ , this algorithm scales as $m^3 k^2 + m^2 k^3$ .
No approximation is made when applying the MPO, but after applying it the resulting MPS is compressed using the truncation parameters provided in the named arguments args.
An alternative algorithm can be chosen by setting the parameter "Method" to "Fit". This is a sweeping algorithm that iteratively optimizes the resulting MPS $|\phi\rangle$ (analogous to DMRG). This algorithm has better scaling in the MPO bond dimension $k$ compared to the "DensityMatrix" method, but is not guaranteed to converge (depending on the input MPO and MPS). The number of sweeps can be chosen with the parameter "Nsweep".
It is recommended to try the default "DensityMatrix" first because it is more reliable. Then, the "Fit" method can be tried if higher performance is required.
Named arguments recognized:
• "Method" — (default: "DensityMatrix") algorithm used for applying the MPO to the MPS. Currently available options are
• "DensityMatrix"
• "Fit"
• "Cutoff" — (default: 1E-13) truncation error cutoff for compressing resulting MPS
• "Maxm" — maximum bond dimension of resulting compressed MPS
• "Verbose" — (default: false) if true, prints extra output
• "Normalize" — (default: false) choose whether or not to normalize the output wavefunction
• "Nsweep" — (default: 1) sets the number of sweeps of the "Fit" algorithm
Show Example
//Use the method "DensityMatrix"
auto phi = applyMPO(K,psi,{"Method=","DensityMatrix","Maxm=",100,"Cutoff=",1E-8});
//Use the method "Fit" with 5 sweeps
auto phi2 = applyMPO(K,psi,{"Method=","Fit","Maxm=",100,"Cutoff=",1E-8,"Nsweep=",5});
• applyMPO(MPO K, MPS psi, MPS phi, Args args = Args::global()) -> MPS
applyMPO(IQMPO K, IQMPS psi, IQMPS phi, Args args = Args::global()) -> IQMPS
Similar to applyMPO above, but accepts a guess for the output wavefunction (the guess wavefunction phi is not overwritten).
Currently, this version of applyMPO only accepts "Fit" for the parameter "Method". Choosing a good guess state phi can improve the convergence of the "Fit" method.
Show Example
//Use the method "Fit" with 5 sweeps and a guess state phi
auto Kpsi = applyMPO(K,psi,phi,{"Method=","Fit","Maxm=",100,"Cutoff=",1E-8,"Nsweep=",2});
• checkMPOProd(MPS psi2, MPO K, MPS psi1) -> Real
checkMPOProd(IQMPS psi2, IQMPO K, IQMPS psi1) -> Real
Computes, without approximation, the difference $||\, |\psi_2\rangle - K |\psi_1\rangle ||^2$ , where K is an arbitrary MPO. This is especially useful for testing methods for applying an MPO to an MPS.
Show Example
//Approximate K*psi
auto phi = applyMPO(K,psi,{"Maxm=",200,"Cutoff=",1E-12});
//Check
Print(checkMPOProd(phi,K,psi)); //should be close to zero
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2019-06-25 08:33:56
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|
https://leanprover-community.github.io/mathlib_docs/analysis/special_functions/complex/circle.html
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mathlibdocumentation
analysis.special_functions.complex.circle
Maps on the unit circle #
In this file we prove some basic lemmas about exp_map_circle and the restriction of complex.arg to the unit circle. These two maps define a local equivalence between circle and ℝ, see circle.arg_local_equiv and circle.arg_equiv, that sends the whole circle to (-π, π].
@[simp]
theorem circle.arg_eq_arg {z w : circle} :
z.arg = w.arg z = w
theorem arg_exp_map_circle {x : } (h₁ : -real.pi < x) (h₂ : x real.pi) :
@[simp]
theorem exp_map_circle_arg (z : circle) :
@[simp]
@[simp]
noncomputable def circle.arg_local_equiv :
complex.arg ∘ coe and exp_map_circle define a local equivalence between circle andwithsource = set.univandtarget = set.Ioc (-π) π.
Equations
@[simp]
@[simp]
@[simp]
@[simp]
noncomputable def circle.arg_equiv :
complex.arg and exp_map_circle define an equivalence between circle and(-π, π].
Equations
theorem exp_map_circle_eq_exp_map_circle {x y : } :
∃ (m : ), x = y + (m) *
@[simp]
theorem exp_map_circle_two_pi :
= 1
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2022-01-22 23:19:04
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https://hal-univ-avignon.archives-ouvertes.fr/hal-01065122
|
# When are the invariant submanifolds of symplectic dynamics Lagrangian?
Abstract : Let L be a D-dimensional submanifold of a 2D-dimensional exact symplectic manifold (M, w) and let f be a symplectic diffeomorphism onf M. In this article, we deal with the link between the dynamics of f restricted to L and the geometry of L (is L Lagrangian, is it smooth, is it a graph...?). We prove different kinds of results. - for D=3, we prove that if a torus that carries some characteristic loop, then either L is Lagrangian or the restricted dynamics g of f to L can not be minimal (i.e. all the orbits are dense) with (g^k) equilipschitz; - for a Tonelli Hamiltonian of the cotangent bundle M of the 3-dimenional torus, we give an example of an invariant submanifold L with no conjugate points that is not Lagrangian and such that for every symplectic diffeomorphism f of M, if $f(L)=L$, then $L$ is not minimal; - with some hypothesis for the restricted dynamics, we prove that some invariant Lipschitz D-dimensional submanifolds of Tonelli Hamiltonian flows are in fact Lagrangian, C^1 and graphs; -we give similar results for C^1 submanifolds with weaker dynamical assumptions.
Keywords :
Type de document :
Article dans une revue
MR3124714 Reviewed Arnaud, Marie-Claude When are the invariant submanifolds of symplectic dynamics Lagrangian? Discrete Contin. Dyn. Syst., 2014, 34 (5), pp.1811-1827
Domaine :
Littérature citée [14 références]
https://hal-univ-avignon.archives-ouvertes.fr/hal-01065122
Contributeur : Marie-Claude Arnaud <>
Soumis le : mercredi 17 septembre 2014 - 22:03:17
Dernière modification le : vendredi 26 janvier 2018 - 11:06:09
Document(s) archivé(s) le : jeudi 18 décembre 2014 - 12:05:50
### Fichiers
varinvariantester_5.pdf
Fichiers produits par l'(les) auteur(s)
### Identifiants
• HAL Id : hal-01065122, version 1
• ARXIV : 1409.5204
### Citation
Marie-Claude Arnaud. When are the invariant submanifolds of symplectic dynamics Lagrangian?. MR3124714 Reviewed Arnaud, Marie-Claude When are the invariant submanifolds of symplectic dynamics Lagrangian? Discrete Contin. Dyn. Syst., 2014, 34 (5), pp.1811-1827. 〈hal-01065122〉
### Métriques
Consultations de la notice
## 302
Téléchargements de fichiers
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2018-12-14 16:05:55
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https://benkyosukisuki.com/ascension/ascension-drlm125gk3m/
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# 【アセンションコード】エル.リ.ジュン ５５ パワフルアセンションコードダウンロード クラウンチャクラをアクティベートして覚醒へと導く 自分自身との統合を目指しましょう Ascension Codes
## エル.リ.ジュン ５５ パワフルアセンションコードダウンロード クラウンチャクラをアクティベートして覚醒へと導く 自分自身との統合を目指しましょう Ascension Codes
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## ゲリー・ボーネルさん｜地震や火山噴火などの影響を弱める祈願
When we use the Invocation of Angels and Extra-dimensional Beings it is to be stated in a strong, clear voice. If it is stated in such a manner for thirty-three consecutive days, the individual will command the powers assigned to them through their collaboration with such unlimited beings. This is a contract – do not enter this with insincerity, but instead, enter this with the assurances of all that have gone before you as knowing beings. The invocation (the non-English names and phrases are phonetically written):
"I AM sovereign soul eternal of origin; I AM as the ONE, in the ONE’s image as the I AM, THAT I AM.
Hallowed be the name, I AM.
I joyfully invoke the presence of Bah EL, Pho EL, Thael Thro EL, Thaba EL, Thoel Bachool, Thi EL, Aro EL, Afph EL, Arouo EL, Sami EL, Auh EL,Ou EL, Obmi EL, Tharimi EL, Ach EL, Araoab EL,
listen to me, strong angels, for I invoke you by the Lord of Creation, you archangels of the body of Yao Yecha, that you hearken to me and send to me, Athonath, Athonath, Athonath, Gabriel, the angel of righteousness that he come to me, awakening my body and mind to his presence as he protects my physical form from chaos and harm; protects those people and places of my concern from pestilence and devastation. That he empower the Archangels, Angels, Seraphs, Seraphim, Cherubim and Watchers. Guide Obiute, Gernabiute, Gernabiel Pawee Hureet Ethrahes Epsoma Nauyayzus Christos Paesayrah Pawnoota Cahrawes Pasomaas that they make ready the Earth’s movements to be balanced and complete for all living creatures great and small. I invoke their power to guide my thoughts and actions as I gracefully unfold through sovereign self-realization to enlightenment.
Mighty companions, I command your presence, radiate your power through me in every relationship that I am a comfort to all; open my intuitive knowing that I will see beyond time and space; I command Seinoy, Sasanoy and Semienloof to radiate the power of the angels through me that I command clarity of body, mind and spirit. Stand with me Seinoy, Sasanoy and Semienloof as I unfold my life’s mission. Work with me Seinoy, Sasanoy and Semienloof as I source thought to form, command intention as outcome in each new moment.
Athonath Athonath, Lord Gabriel, I AM your agent of peace, where there is hatred, I AM unfolding compassion; where there is grievance, I AM forgiveness; where there is uncertainty, I AM knowing clarity; where there is hopelessness, I AM sureness; where there is obscurity, I AM upholding grace; where there is distress, I AM joy;
Athonath Athonath, Lord Gabriel, command me that I may not so much seek to be consoled as to console; to be understood as to understand; to be loved as to love.
For it is in giving that we receive; it is in pardoning that we are pardoned; and it is in ascension that my spirit is born to eternal life."
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2019-05-21 23:50:23
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http://spindynamics.org/wiki/index.php?title=Rotor_stack.m
|
Rotor stack.m
Returns a rotor stack of Liouvillians or Hamiltonians. The stack is needed for the traditional style calculation of MAS dynamics.
Syntax
L=rotor_stack(spin_system,parameters,assumptions)
Arguments
parameters.axis - spinning axis, given as a normalized
3-element vector
parameters.offset - a cell array giving transmitter off-
sets in Hz on each of the spins listed
in parameters.spins array
parameters.spins - a cell array giving the spins that
the offsets refer to, e.g. {'1H','13C'}
parameters.max_rank - maximum harmonic rank to retain in
the solution (increase till conver-
gence is achieved, approximately
equal to the number of spinning si-
debands in the spectrum)
parameters.rframes - rotating frame specification, e.g.
{{'13C',2},{'14N,3}} requests second
order rotating frame transformation
with respect to carbon-13 and third
order rotating frame transformation
with respect to nitrogen-14. When
this option is used, the assumptions
on the respective spins should be
laboratory frame.
parameters.orientation - the orientation of the spin system
at rotor phase zero, a vector of
parameters.masframe - the frame in which the rotations
are applied. The possibilities are:
'magnet' - the initial orientation in the lab frame
(three-angle powder grids will be required)
'rotor' - the initial orientation in the rotor frame
(two-angle powder grids will be required)
assumptions - assumption set to be used in generating the
Hamiltonian, see assume.m
Outputs
L - a cell array of Hamiltonian or Liouvillian matrices,
one for each tick of the rotor.
rotor_phases - rotor phases at each tick, radians
Notes
Relaxation and kinetics are not included.
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2018-01-23 20:05:58
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|
https://docs.panda3d.org/1.10/cpp/reference/panda3d.core.VirtualFile
|
# panda3d.core.VirtualFile¶
class VirtualFile
The abstract base class for a file or directory within the VirtualFileSystem.
Inheritance diagram
closeReadFile(stream: istream) → None
Closes a file opened by a previous call to openReadFile(). This really just deletes the istream pointer, but it is recommended to use this interface instead of deleting it explicitly, to help work around compiler issues.
closeReadWriteFile(stream: iostream) → None
Closes a file opened by a previous call to openReadWriteFile(). This really just deletes the iostream pointer, but it is recommended to use this interface instead of deleting it explicitly, to help work around compiler issues.
closeWriteFile(stream: ostream) → None
Closes a file opened by a previous call to openWriteFile(). This really just deletes the ostream pointer, but it is recommended to use this interface instead of deleting it explicitly, to help work around compiler issues.
copyFile(new_file: VirtualFile) → bool
Attempts to copy the contents of this file to the indicated file. Returns true on success, false on failure.
deleteFile() → bool
Attempts to delete this file or directory. This can remove a single file or an empty directory. It will not remove a nonempty directory. Returns true on success, false on failure.
static getClassType() → TypeHandle
Return type
TypeHandle
getFileSize() → streamsize
Returns the current size on disk (or wherever it is) of the file before it has been opened.
Return type
streamsize
getFileSize(stream: istream) → streamsize
Returns the current size on disk (or wherever it is) of the already-open file. Pass in the stream that was returned by openReadFile(); some implementations may require this stream to determine the size.
Return type
streamsize
getFileSystem() → VirtualFileSystem
Return type
VirtualFileSystem
getFilename() → Filename
Return type
Filename
getOriginalFilename() → Filename
Returns the original filename as it was used to locate this VirtualFile. This is usually, but not always, the same string returned by getFilename().
Return type
Filename
getSystemInfo(info: SubfileInfo) → bool
Populates the SubfileInfo structure with the data representing where the file actually resides on disk, if this is knowable. Returns true if the file might reside on disk, and the info is populated, or false if it does not (or it is not known where the file resides), in which case the info is meaningless.
getTimestamp() → time_t
Returns a time_t value that represents the time the file was last modified, to within whatever precision the operating system records this information (on a Windows95 system, for instance, this may only be accurate to within 2 seconds).
If the timestamp cannot be determined, either because it is not supported by the operating system or because there is some error (such as file not found), returns 0.
Return type
time_t
hasFile() → bool
Returns true if this file exists, false otherwise.
isDirectory() → bool
Returns true if this file represents a directory (and scanDirectory() may be called), false otherwise.
isRegularFile() → bool
Returns true if this file represents a regular file (and readFile() may be called), false otherwise.
isWritable() → bool
Returns true if this file may be written to, which implies writeFile() may be called (unless it is a directory instead of a regular file).
ls(out: ostream) → None
If the file represents a directory, lists its contents.
lsAll(out: ostream) → None
If the file represents a directory, recursively lists its contents and those of all subdirectories.
openAppendFile() → ostream
Works like openWriteFile(), but the file is opened in append mode. Like open_write_file, the returned pointer should eventually be passed to closeWriteFile().
Return type
ostream
openReadAppendFile() → iostream
Works like openReadWriteFile(), but the file is opened in append mode. Like open_read_write_file, the returned pointer should eventually be passed to closeReadWriteFile().
Return type
iostream
openReadFile(auto_unwrap: bool) → istream
Opens the file for reading. Returns a newly allocated istream on success (which you should eventually delete when you are done reading). Returns NULL on failure.
Return type
istream
openReadWriteFile(truncate: bool) → iostream
Opens the file for writing. Returns a newly allocated iostream on success (which you should eventually delete when you are done writing). Returns NULL on failure.
Return type
iostream
openWriteFile(auto_wrap: bool, truncate: bool) → ostream
Opens the file for writing. Returns a newly allocated ostream on success (which you should eventually delete when you are done writing). Returns NULL on failure.
Return type
ostream
output(out: ostream) → None
readFile(auto_unwrap: bool) → object
Returns the entire contents of the file as a string.
renameFile(new_file: VirtualFile) → bool
Attempts to move or rename this file or directory. If the original file is an ordinary file, it will quietly replace any already-existing file in the new filename (but not a directory). If the original file is a directory, the new filename must not already exist.
If the file is a directory, the new filename must be within the same mount point. If the file is an ordinary file, the new filename may be anywhere; but if it is not within the same mount point then the rename operation is automatically performed as a two-step copy-and-delete operation.
scanDirectory() → VirtualFileList
If the file represents a directory (that is, isDirectory() returns true), this returns the list of files within the directory at the current time. Returns NULL if the file is not a directory or if the directory cannot be read.
Return type
VirtualFileList
wasReadSuccessful() → bool
Call this method after a reading the istream returned by openReadFile() to completion. If it returns true, the file was read completely and without error; if it returns false, there may have been some errors or a truncated file read. This is particularly likely if the stream is a VirtualFileHTTP.
writeFile(data: object, auto_wrap: bool) → object
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2020-07-05 14:29:28
|
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http://lakm.us/thesit/tag/interfacial-tension/page/2/
|
# thesIt
• #### Arif 11:49:55 am on February 19, 2010 | 0 | # | Tags: basic, interfacial tension, spinning drop, tensiometer, wikipedia
Spinning drop method (rotating drop method) is one of the methods that is used to measure interfacial tension. Basically, measurements are carried out in a rotating horizontal tube which contains a dense fluid. One drop of a less dense liquid is placed inside the fluid.
Since the rotation of the horizontal tube creates a centrifugal force towards the tube walls, the liquid drop starts to be elongated and this elongation stops when the interfacial tension and centrifugal forces are balanced. Values obtained at this equilibrium point are used to estimate surface tension of the particular liquid by using appropriate correlations. A device called “spinning drop tensiometer” is generally utilized for this purpose.
• #### Arif 11:14:05 pm on February 18, 2010 | 0 | # | Tags: interfacial tension, mixture, pure compound, Queimada 2004, reference, viscosity
Generalized Relation Between Surface Tension and Viscosity: a Study on Pure and Mixed n-alkanes. A. J. Queimada, I. M. Marrucho , E. H. Stenby, J. A. P. Coutinho. Paper presented at the Fifteenth Symposium on Thermophysical Properties, June 22-27, 2003, Boulder, Colorado, USA.
Both for pure compounds and mixtures.
p385.pdf
proceeding info published in Fluid Phase Equilibria
• #### Arif 09:23:39 pm on February 18, 2010 | 0 | # | Tags: density, interfacial tension, Mansoori 1990, prediction, reference
Statistical Mechanics Basis of Macleod’s Formula. M.-E. BOUDH-HIR and G.A. MANSOORI. Journal of Physical Chemistry, Volume 94, pp.8362-8364, 1990.
The famous formula of Macleod, relating the surface tension of a liquid in equilibrium with its own vapor to the one-particle densities in the two phases of the system, is derived. Proved using the statistical- mechanical definition of the surface tension.
(Later on Mansoori worked with Escobedo)
StatMechBasisOfMcLeodEquation.pdf
• #### Arif 09:14:14 pm on February 18, 2010 | 0 | # | Tags: capillary force, huff and puff, interfacial tension, oil recovery, reference, yield
Condensed presentation on oil recovery involving surfactant and other chemical flooding. It includes rate of return from yield. Quote “interfacial tension reduction alone does not result in good oil recovery”
Malcolm_Pitts_SURTEK.pdf
• #### Arif 12:22:45 pm on February 17, 2010 | 0 | # | Tags: basic, interfacial tension, wikipedia, Young-Laplace
osculating circle
radius of curvature of a curve at a point is the radius of the osculating circle at that point.
• #### Arif 06:19:05 pm on January 1, 2010 | 0 | # | Tags: density, equation, Escobedo 1998, interfacial tension, Kumar 2005, parachor, prediction, pure compound, reference, statistical
Escobedo and Mansoori (AIChE J. 42(5): 1425, 1996) Found original paper: check Escobedo 1996
The proposed 1996 paper relates the surface tension of mixtures to bulk-phase concentrations and properties (i.e. densities). Surface tension of pure organic fluids σ:
$\sigma=[P(\rho_l-\rho_v)^4]$
The two ρ are densities: liquid and vapour.
They introduced parachor (P), a new expression, derived from statistical mechanics (from Macleod equation) which represents the experimental surface tension of 94 different organic compounds within 1.05 AAD% (average absolute deviations).
(Kumar 2005) It expresses the surface tension of a liquid in equilibrium with its own vapour. Historical reference for parachor cited here: Macleod, Sugden, Quayle, Escobedo.
• #### Arif 06:13:01 pm on January 1, 2010 | 0 | # | Tags: Escobedo 1998, interfacial tension, mixture, prediction, reference
Surface Tension Prediction for Liquid Mixtures
Joel Escobedo and G. Ali Mansoori.AIChE Journal, Vol. 44, No. 10, pp. 2324-2332, 1998.
Surface tension of pure organic fluids was proposed previously also by them in 1996. The same theory is extended to the case of mixtures of organic liquids.
SurfaceTensionPredictionOfLiquidMixtures.pdf
• #### Arif 04:57:56 pm on January 1, 2010 | 0 | # | Tags: basic, equation, interfacial tension, Myers 2006, Pashley 2004
Young Laplace equation:
$\Delta\rho=\sigma(\frac{1}{r1}+\frac{1}{r2})$
other uses γ instead of σ for surface/interfacial tension (i.e. Pashley 2004). Common unit is dyne/cm (1 dyne = 1 micro Newton (μN) ).
• #### Arif 12:43:53 pm on December 31, 2009 | 0 | # | Tags: basic, interfacial tension, Pashley 2004
Surface energy of a liquid actually gives rise to a ‘surface tension’ or force acting to oppose any increase in surface area.
Thus, we have to ‘blow’ to create a soap bubble by stretching a soap film. A spherical soap bubble is formed in response to the tension in the bubble surface.
• #### Arif 10:44:46 am on December 31, 2009 | 0 | # | Tags: continuous, interfacial tension, Lin 2006, method, reference
A Home-made Simple Set-up for Measurement of Electrochemical Surface Tension Spectrum on a Mercury Drop Electrode
Xiang Qin LIN∗, Liang Dong FENG, Hao ZHANG. Chinese Chemical Letters Vol. 17, No. 4, pp 493-495, 2006.
Recording surface tension curves at a mercury drop during potential scanning based on photo-sensitive detection system.
Optical imaging methods can be used for accurate surface tension measurement, however, continuous tension curve is hardly obtained by this technique.
170419-493-b050453-p3.pdf
• « Previous PageNext Page »
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2021-11-30 10:05:19
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https://en.wiktionary.org/wiki/functional_root
|
# functional root
## English
### Noun
functional root (plural functional roots)
1. (mathematics) A function which, when applied a given number of times, equals another given function.
${\displaystyle x^{2}+1}$ is the functional 2nd root of ${\displaystyle x^{4}+2x^{2}+2}$.
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2018-03-19 03:48:57
|
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|
http://mathhelpforum.com/calculus/140132-surface-area-generated-rotating-curve-around-y-axis.html
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# Math Help - Surface area generated by rotating curve around y-axis
1. ## Surface area generated by rotating curve around y-axis
I'm having a spot of trouble with this problem:
Find the surface area generated by rotating the given curve about the y-axis.
x=(e^t)-t
y=4e^(t/2)
0<t<8
My homework is submitted online, and I keep working this problem out but it says I have it wrong, but I'm pretty sure I'm doing it right.
I know A(s) = integral from 0 to 8 of 2pi x(t)sqrt((dx/dt)^2+(dy/dt)^2)dt, but I keep messing up. Help!
...Maybe i'm using the wrong equation?
2. Originally Posted by DarthPipsqueak
I'm having a spot of trouble with this problem:
Find the surface area generated by rotating the given curve about the y-axis.
x=(e^t)-t
y=4e^(t/2)
0<t<8
My homework is submitted online, and I keep working this problem out but it says I have it wrong, but I'm pretty sure I'm doing it right.
I know A(s) = integral from 0 to 8 of 2pi x(t)sqrt((dx/dt)^2+(dy/dt)^2)dt, but I keep messing up. Help!
...Maybe i'm using the wrong equation?
your integral set up is correct ... the surface area I get is quite large (a bit on the side of ridiculous going to t = 8)
$A = \pi(e^{16}-12e^8-69)$
3. I was getting very large numbers too! But this worked out fine, and I was able to figure out the integral. I think I went wrong with my algebra somewhere.
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2014-08-01 09:00:02
|
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https://ai.stackexchange.com/questions/7500/is-there-a-database-somewhere-of-common-lists
|
# Is there a database somewhere of common lists?
I'm looking for a database or some machine readable document that contains common ordered lists or common short sets. e.g:
{January, February, March,...}
{Monday, Tuesday, ....}
{Red, Orange, Yellow,...}
{1,2,3,4,...}
{one, two, three, four,...}
{Mercury, Venus, Earth, Mars,...}
{I, II, III, IV, V, VI,...}
{Aquarius, Pisces, Aries,...}
{ein, zwei, drei, ...}
{Happy, Sneezy, Dopey, ...}
{Dasher, Dancer, Prancer, Vixen ,...}
{John, Paul, George, Ringo}
{20, 1, 18, 4, 13, 6, ...}
{A,B,C,D,E,F,G,...}
{A,E,I,O,U}
{2,3,5,7,11,13,17,...}
{triangle, square, pentagon, hexagon,...}
{first, second, third, fourth, fifth,...}
{tetrahedron, cube, octohedron, icosohedron, dodecahedron}
{autumn, winter, spring, summer}
{to, be, or, not, to, be, that, is, the, question}
...
One use is for creating an AI that can solve codes or predict the next thing in a sequence.
• I've provisionally added a new tag for "resource-requests" because I can see how access to such the types of datasets you're looking for could be useful to others looking to create or train algorithms. – DukeZhou Aug 9 '18 at 21:02
• I think this question is on-topic, in disagreement with the current close-vote. In fact, one of the most important open point in AI is how an AI system can establish relations between concepts. In fact, this one is on-topic, 99% of the question about neural net implementations are not. – pasaba por aqui Aug 10 '18 at 9:39
So here’s a couple quick resources that i could think of. First of all, you could look at this,
https://en.m.wikipedia.org/wiki/List_of_lists_of_lists
It has classics such accidents, hospitals in Asia, or even a list of famous resignations. It’s easentially a list of random lists of things. It may not owner cover your requirement for sequences but it’ll help with small subsets of lists.
As for sequences, you could always check out,
https://oeis.org
It’s pretty much a list of official mathematical sequences. It’s got everything from the Fibonacci sequence to esoteric sequences you’ve never heard of.
Some useful resources could be:
• For general terms, dictionaries of synonyms and antonyms.
• For scientific terms, any equal level of terms in a taxonomy.
• For more general terms, look at existing ontologies. Some of them tries to include all domains (upper ontologies). See:
• An English grammar can provide the set of pronouns, adverbs, prepositions, classified in temporal relations, ...
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2020-08-10 11:59:03
|
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|
http://mathcs.chapman.edu/~jipsen/structures/doku.php/cancellative_monoids
|
Cancellative monoids
Abbreviation: CanMon
Definition
A cancellative monoid is a monoid $\mathbf{M}=\langle M, \cdot, e\rangle$ such that
$\cdot$ is left cancellative: $z\cdot x=z\cdot y\Longrightarrow x=y$
$\cdot$ is right cancellative: $x\cdot z=y\cdot z\Longrightarrow x=y$
Morphisms
Let $\mathbf{M}$ and $\mathbf{N}$ be cancellative monoids. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:M\rightarrow N$ that is a homomorphism:
$h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$
Examples
Example 1: $\langle\mathbb{N},+,0\rangle$, the natural numbers, with addition and zero.
Basic results
All free monoids are cancellative.
All finite (left or right) cancellative monoids are reducts of groups.
Properties
Classtype quasivariety undecidable no unbounded no
Finite members
$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &1\\ f(6)= &2\\ f(7)= &1\\ \end{array}$
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2019-09-22 06:00:48
|
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https://brilliant.org/problems/a-problem-by-fidel-simanjuntak-2/
|
# A geometry problem by Fidel Simanjuntak
Geometry Level 3
$$BD : DC = 1\text{ cm} : 3 \text{ cm}$$ and $$AE : EC = 2 \text{ cm}: 1 \text{ cm}$$. The are aof $$ADE$$ is $$50 \text{ cm}^{2}$$.
Find the area of $$ABD$$ in $$\text{cm}^{2}$$.
×
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2018-01-24 04:03:01
|
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https://chemistry.stackexchange.com/questions/4877/is-it-actually-possible-to-dispose-of-a-body-with-hydrofluoric-acid/4915
|
# Is it actually possible to dispose of a body with hydrofluoric acid?
In the TV show "Breaking Bad", Walter White frequently gets rid of people who get in his way by submerging them in a plastic container full of hydrofluoric acid. This, at least in the TV show, completely dissolves the body leaving nothing but a red sludge behind at the end.
Is it actually possible to dispose of a body with hydrofluoric acid?
If hydrofluoric acid wouldn't work, are there any acids corrosive enough to achieve the stated effect from the show?
• I can't say if this is possible, but concentrated HF poses a very real threat of delayed, life threatening damage by scavenging calcium from the body, either by splash or inhalation. HF is not a strong acid, but its toxicity makes it substantially more dangerous to work with than many strong acids. I tend to think that dissolving a body would be quite difficult, and then you'd have the scrub the bathtub ring... – Richard Terrett May 3 '13 at 12:29
• Related: chemistry.stackexchange.com/questions/3949/…. I wonder when the FBI will come and shut this site down for answering questions about disposing bodies and FOOF ;-) – ManishEarth May 3 '13 at 15:33
• I guess he used fluoroantimonic acid. :) – user11343 Dec 17 '14 at 3:50
• or hydrogen peroxide and conc sulphuric acid .. oops there's another finger gone! – porphyrin Aug 16 '16 at 22:25
• I like people who makes experiments Periodic Video I was not a lot of surprised by the experiment, but it is always good to see what happend in an other place than a movies beacause it may sometimes be false. Now I can come back dissolving my stepmother... :) – ParaH2 Oct 13 '16 at 14:43
Hydrofluoric acid is toxic and corrosive, but actually isn't that strong of an acid compared to other hydrohalic acids; the fluorine has a very good orbital overlap with hydrogen and is also not very polarizable, therefore it resists donating its proton, unlike other hydrohalic acids which are good proton donators. It will break down some tissues, but it will take a relatively long time and won't turn the entire body into stuff that can be rinsed down the drain. Hydrochloric acid is a much stronger acid, and as it has several uses from pH-balancing pool water to preparing concrete surfaces, it's available by the gallon from any hardware store. However, it isn't very good at dissolving bodies either; while it will eventually work by breaking down the connective tissues, it will make a huge stink and take several days to dissolve certain types of tissues and bones.
The standard body-dissolving chemical is lye aka sodium hydroxide. The main source is drain clog remover because most drain clogs are formed by hair and other bio-gunk that accumulates naturally when humans shower, exfoliate etc. It works, even though the body's overall chemistry is slightly to the basic side of neutral (about 7.35-7.4) because the hydroxide anion is a strong proton acceptor. That means that it strips hydrogen atoms off of organic molecules to form water (alkaline hydrolysis, aka saponification), and as a result, those organic molecules are turned into simpler molecules with lower melting points (triglycerides are turned into fatty acids, saturated fats are dehydrogenated to form unsaturated fats, alkanes become alcohols, etc). Sodium hydroxide is also a ready source of the sodium ion; sodium salts are always water-soluble (at least I can't think of a single one that isn't). The resulting compounds are thus either liquids or water-soluble alcohols and salts, which flush down the drain. What's left is the brittle, insoluble calcium "shell" of the skeleton; if hydrolyzed by sodium hydroxide, the resulting calcium hydroxide ("slaked lime") won't dissolve completely but is relatively easy to clean up.
• I like the wording here: "standard body-dissolving chemical" – DarkLightA May 13 '13 at 21:20
• I like how you answered the question in the first 6 lines of your response, then graced us with another 12 lines of 'how you should do it' information. – LordStryker Dec 18 '13 at 14:33
• "the fluorine is too electronegative to readily donate its proton." What does that mean? If anything, from EN trends alone, we should expect HF to be a strong acid, since the F, as you said, is very EN and is therefore isolating electron density away from the H and thereby making the H more partially positive and more reactive. That's not the case, as there are other, greater factors in play here, such as conjugate stability. The F- anion simply isn't as stable as the Cl- anion, for example. – Dissenter Jul 13 '14 at 17:03
• What I mean is exactly what I said. Larger anions have higher-energy orbital shells which spread the negative charge of the electrons over a greater space, meaning that the proton is similarly held at greater distance and with reduced attraction to the anion, so the compound more readily dissociates and donates protons. Fluorine is small, and its electron configuration is the densest available of the halogens. So, the attraction to the proton is increased due to the more concentrated charge, thus requiring more energy to break this ionic bond and form another. – KeithS Jul 14 '14 at 15:40
• I've just realised that, though this is a good answer about alternatives to HF, it doesn't really answer the question about whether HF works. – matt_black Feb 15 '15 at 19:57
I think the use of Hydrofluoric Acid was script-driven rather than fact driven: it sounds scary rather than being a good choice. Also, it allows for the possibility of the darkly comic bathtub scene where the acid dissolves a ceramic bath because Jessie ignores Walter's instructions (which establishes Walter's expertise and Jessie's lack of it).
There is no good reason why Pinkman and White pharmaceuticals needed to have hydrofluoric acid, therefore using large quantities of it is somewhat implausible.
Moreover, it probably wouldn't work as well as several alternatives. Hydrofluoric acid is very nasty stuff, but it isn't a strong acid. Even when dilute it will etch glass and ceramics, but it won't dissolve or burn flesh. I once saw a demonstration where a lecturer showed this by spilling some dilute hydrofluoric acid on his hand and then onto a glass surface. The surface was frosted, his hand unharmed (he was very careful to wash the acid off quickly and take appropriate precautions and I don't recommend trying this at home!)
Its danger to people is its toxicity, not its ability to burn: it insinuates itself into the body and destroys connective tissue and bone slowly by interfering with anything containing calcium. Its danger is worse because it doesn't cause immediate damage and you may receive a dangerous dose without noticing. So it is scary but not corrosive.
Other alternatives are better. Concentrated alkalis such as Sodium Hydroxide are readily available and are very good at dissolving flesh (which is why they are commonly used as drain cleaners). But alkalis don't do a good job on bone. Concentrated sulfuric acid is even better as it does a good job on flesh and will, eventually, dissolve the bone as well. Murderers have used both methods to try to dispose of evidence. For example, John George Haigh who used sulfuric acid and left little other than gallstones (http://en.wikipedia.org/wiki/John_George_Haigh). Using alkali is often done but tends to leave bone fragments even with sophisticated processes that pressure cook the solution (see http://www.slate.com/articles/news_and_politics/explainer/2009/12/soluble_dilemma.html).
So I think the answer is that HF solutions are not a good choice for body disposal as it probably doesn't work well compared to known alternatives.
Update
A lot of the above is theory but good scientists do experiments. So Periodic Videos decided to test this very idea using chicken legs as a model. They compared what happens when raw chicken legs are suspended in strong solutions of HCl, H2SO4 and HF. The HF was the least impressive for flesh-dissolving characteristics, though it did seem to cause other, more subtle damage, to the components of the flesh.
See the actual results here.
Here in Mexico, a guy was arrested a couple years ago for "dissolving" more than 300 bodies killed by the cartels. They found 55-gallon drums around town with a sludge in it, usually with a note to warn other. His recipe? A 55-gallon drum, several bags of lye, add body, fill with h20, and build fire beneath drum.
It took about 24hrs I believe.
Sorry, I got caught up In the comments. And I'm New to this and all (bulletin boards?) Like the case in Australia with sealed drums w/ HCL, in a bank vault, it Actually did more to preserve the bodies. I am no chemist, but it is my understanding, that when acids burn the skin the results are the same for both Fire and chemicals burns both rapidly dehydrate the body, destroying it. But is in a sealed container, that water can only go to the lid. Stopping the process. I bet any acid will destroy a body. When you're in the bath or pool too long you get those wrinkles, I've heard of people with wet feet in boots for weeks, remove their boots, and literally stripping the flesh off their feet to the bone.
The issue is how long will it take like the method used in here in Mexico and another post, boiling seems to be an important component. And again I'm guessing (and vegetarian), the heat tenderizes the meat, helping the acid along, but also the rolling of the water stirs it. If you were to put a piece of meat in acid that was stagnant, the acid might react, dissolve some meat, as the dissolved meat dilutes the acid depending on the makeup. It could settle or form a layer of meat/acid that sits on top of the meat, protecting it. Think layered shots. Or a unflushed end toilet that has had time to settle. Hydrofluoric acid will eat thru skin, google it. One of the reasons the burns are so bad is @ 50%, it can take 8 hours to realize you were burned. So that would be a slow process, and fluoride, as we know, preserves teeth, by extension bones. And for hydrofluoric acid burns, the standard treatment is calcium. So it would be very slow, very likely stop, possible preserve the bones, making the resistant to rotting and harden them, but as your, we'll my fingers have less meat than most parts of my body, it would get to bone faster, ossicle dissolving some calcium, then render all the acid useless. Again I am not a chemist but have self-studied chemistry for years, I would be very interested to hear anyone else's opinions on my determinations!
• Interesting answer! But the question is related to hydrofluoric acid can you be more precise? Thank you! – G M Jul 13 '14 at 16:50
• I removed the portions not related to the science. – jonsca Jul 13 '14 at 18:55
I know this is an old question, but it gets viewed a lot so I thought I would update with the fact that this experiment was actually done on the first Mythbusters Breaking Bad special, episode 206. There is a description of the results of the episode inside that link.
Is it actually possible to dispose of a body with hydrofluoric acid?
To summarize from the Mythbusters episode, they tested out hydrofluoric acid on various small scale materials (to simulate the bathtub) as well as pig parts (the body) and found that $\ce{HF}$ certainly deteriorates some of the materials, but not at anything close to the same speed as in the show. It is even less successful at decomposing the actual flesh than the materials. One way of viewing this is to remind yourself that the ability of $\ce{HF}$ to eat through materials actually has more to do with fluorine's reactivity than with the ability to give a hydrogen atom (it is classified as a weak acid), thus it's reasonable that $\ce{HF}$ might be better at disrupting the structural integrity of metals/polymers than acidically eating through flesh.
See this question for details about its reactivity with glass. Mythbusters notes that a fiberglass bathtub actually would've behaved similarly to what we see in Breaking Bad, but over a much longer timescale.
So it's perhaps feasible to dispose of a body with $\ce{HF}$ given a very large amount of time, but it's not possible in the timescale presented in the show.
If hydrofluoric acid wouldn't work, are there any acids corrosive enough to achieve the stated effect from the show?
The real reason I brought up this episode of Mythbusters is that they have a storied tradition of reproducing the results at all costs. In this episode, at the end, they fill up a bathtub with 140 L of concentrated sulfuric acid, a "secret sauce", and a whole pig just to see what happens...
To borrow the words of the episode description from the link above,
After 5 minutes, Adam and Jamie found only black organic sludge where the pig used to be.
And, a word on the secret sauce. Intuition says that this is something which is going to be good at breaking down organic matter. I looked around and found some people speculating on this reddit post, and they come up with pretty convincing evidence that the "secret sauce" is just hydrogen peroxide.
Hydrogen peroxide is a good radical initiator, and thus should be quite effective at assisting in the decomposition of this pig/person. This mixture actually has a name: piranha solution.
So, to answer, yes, a mixture of $\ce{H2SO4}$ and $\ce{H2O2}$ should do the job.
I know Mythbusters and Reddit aren't exactly the standards of scientific authority, but in this case, I think they're doing pretty well.
• Oh my got... i have no idea how I got here. I got a panic attack imagining it. – Adam Bajger Sep 19 at 12:47
I believe dilute boiling hydrochloric acid is quite destructive to all flesh and bones. You don't want a concentrated acid as it has annoying oxidation effects which slow the reaction. You don't want hydrofluoric as its hideous dangerous to handle at all temperature (a small amount on your skin will be absorbed and lead to inevitable death if you don't inject calcium glutamate into you) and the formation of insoluble calcium fluoride (and others) slows everything down.
A man here in Australia dissolved his wife's body in Hydrochloric acid in a wheelie bin. He put her body in the bin and poured 20 liters of Hydrochloric Acid in it. Then 2 days later went and bought another 40 liters and topped it up until everything was liquid. He then poured it down a storm drain and rinsed it all out with water. The only thing they found was her prosthetic teeth. Basically, once the Acid stops dissolving and becomes weak, you have to keep adding more. But it will eventually dissolve a whole human body.
Cairns man Klaus Andres, who admitted to dissolving wife's body in acid, guilty of murder, jailed for life http://www.abc.net.au/news/2013-12-12/cairns-man-guilty-of-murdering-wife/5152488
This subject is touched upon here and explained in detail on this Wikipedia page.
Essentially, it is impractical to dispose of a body using an acid, though there exists a process called alkaline hydrolysis, which, as its name implies, makes use of a base to degrade the materials comprising the human body to a liquid state within a matter of hours.
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2020-11-27 00:48:48
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https://zbmath.org/?q=an%3A0936.74029
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# zbMATH — the first resource for mathematics
Equilibrium shapes for planar crystals in an external field. (English) Zbl 0936.74029
Summary: We analyze the equilibrium shape of a two-dimensional crystal in a convex background potential $$g({\mathbf x})$$. For $$g=0$$ the shape of minimum energy may be deduced from surface tension via the Wulff construction, but, if $$g$$ is not constant, little is known beyond the case of a crystal sitting in a uniform field. Only an unpublished result of Okikiolu shows each connected component of the equilibrium crystal to be convex. Here we show that any such component minimizes energy uniquely among convex sets of its area. If the Wulff shape and $$g({\mathbf x})$$ are symmetric under $${\mathbf x}\leftrightarrow-{\mathbf x}$$, it follows that the equilibrium crystal is unique, convex and connected. This last result leads to a new proof that convex crystals away from equilibrium remain convex as they evolve by curvature-driven flow. Subsequent work with Felix Otto shows – without assuming symmetry – that no equilibrium crystal has more than two convex components.
##### MSC:
74E15 Crystalline structure 82D25 Statistical mechanics of crystals 49Q10 Optimization of shapes other than minimal surfaces
Full Text:
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2021-07-30 14:48:47
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https://www.scienceopen.com/document?vid=667a2c25-ee10-44c5-b7b9-7387902015a4
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# Self-similarity of complex networks
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There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.
### Abstract
Complex networks have been studied extensively due to their relevance to many real systems as diverse as the World-Wide-Web (WWW), the Internet, energy landscapes, biological and social networks \cite{ab-review,mendes,vespignani,newman,amaral}. A large number of real networks are called scale-free'' because they show a power-law distribution of the number of links per node \cite{ab-review,barabasi1999,faloutsos}. However, it is widely believed that complex networks are not {\it length-scale} invariant or self-similar. This conclusion originates from the small-world'' property of these networks, which implies that the number of nodes increases exponentially with the diameter'' of the network \cite{erdos,bollobas,milgram,watts}, rather than the power-law relation expected for a self-similar structure. Nevertheless, here we present a novel approach to the analysis of such networks, revealing that their structure is indeed self-similar. This result is achieved by the application of a renormalization procedure which coarse-grains the system into boxes containing nodes within a given "size". Concurrently, we identify a power-law relation between the number of boxes needed to cover the network and the size of the box defining a finite self-similar exponent. These fundamental properties, which are shown for the WWW, social, cellular and protein-protein interaction networks, help to understand the emergence of the scale-free property in complex networks. They suggest a common self-organization dynamics of diverse networks at different scales into a critical state and in turn bring together previously unrelated fields: the statistical physics of complex networks with renormalization group, fractals and critical phenomena.
### Author and article information
###### Journal
03 March 2005
cond-mat/0503078
10.1038/nature03248
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2019-06-24 21:23:15
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https://or.stackexchange.com/tags/linear-programming
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Questions tagged [linear-programming]
For questions related to problems that optimize (i.e., minimize or maximize) a linear objective subject to linear constraints.
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How can I formulate an LP or heuristic solution for this problem?
[I welcome any alternate or simplified formulation of my problem] I have an optimization problem. See the attached figure which is self explanatory. The solid line is intended signal, the dashed lines ...
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Reading MPS file for linear programming and reconstructing the Optimization model
Are you aware of any tutorial that can help me learn on how to reconstruct the objective function and constraints from a MPS file once it's loaded in MATLAB. I can load the mps file given to me and ...
57 views
Maximum bipartite matching with breakpoints in edge weight function
I am looking for an analogy to the problem I am facing or better yet a paper or even code. I have: Nodes from set A and B. Edges are from a single A to many B. I am framing a max bipartite matching ...
162 views
How can I formulate this specific if-then constraint?
IF $\sum\limits_d X_{i,d}\ge6$ THEN $Y_i = 1$ (strictly) AND IF $\sum\limits_d X_{i,d}<6$ THEN $Y_i = 0$ (strictly) $X$ and $Y$ are binary variables. What I'm actually trying to do is to charge the ...
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Obtaining the system of irredundant inequalities from a set of inequalities using CPLEX
Given a linear system of inequalities $Ax \geq b$, I would ideally like to compute the irredundant set for those set of inequalities. I know how to do so mathematically, but I was wondering if there ...
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Is there a Linear Programming Library that natively supports fractions instead of floating point arithmetic?
If one recalls how the Simplex method is taught by hand in most LP classes it takes place entirely in $\mathbb{Q}$. All operations yield exact fractions. For this reason I'm looking for linear ...
44 views
Simplex method on graphs: How do I find a basic solution using the Ford-Fulkerson algorithm?
I'm tasked with solving a minimal cost flow problem. I'm asked to first use the Ford-Fulkerson algorithm on my graph to find a basic solution that will then be used to do the simplex method on that ...
53 views
Estimating multistop routing costs
In many OR problems, it is sometimes a good idea (or necessary) to relax routing constraints. An example of this occurs in the classical facility location problem, where a warehouse can send out a ...
154 views
Is optimal solution to dual not unique if optimal solution to the primal is degenerate?
If optimal solution to the primal is degenerate, does it necessarily follow that optimal solution to dual not unique? That is, is uniqueness an unnecessary assumption? Spin-off from here. In my ...
74 views
How to linearize $f(x,y) = (ax+by)/(x+y)$?
I have a problem which is mainly linear but it has a non-linear component. The objective function is obj = Linear_term + $c*f(x,y)$ where, $f(x,y) = (G_1 x_1 + G_2 x_2)/(x_1 + x_2)$. The decision ...
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High-mix manufacturing capacity
I'm not an expert in OR but I would like to determine what is the maximum manufacturing capacity of a plant (or how much a plant can produce of mix products). Each person in the plant has a known set ...
74 views
Provide basic solution to CLP
I'm using Pyomo to formulate an LP with approx 500,000 constraints and 200,000 decision variables. The LP is solved using CLP. Some instances fail to return even a feasible solution after many ...
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2020-11-27 14:31:07
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http://mathoverflow.net/questions/121319/on-matrix-norms
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# On matrix norms
It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way:
$|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$.
Suppose we define a different function of matrices this way:
$f(A)=\inf_{x \neq 0}{\frac{||Ax||}{||x||}}$.
Has $f(\cdot)$ been studied before? Does it have a standard name?
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$|||A|||$ is the largest s-number (modulus of gen. eigenvalue). $f(A)$ is the smallest s-number. It is 0 if $A$ is not injective.
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S-numbers are also known as singular values. – Federico Poloni Feb 9 '13 at 16:07
This holds if the underlying norm is the Euclidean norm. With other norms, I have no idea if this has ever been studied. – Federico Poloni Feb 9 '13 at 16:07
@ Federico: They have been treated for operators on Banach spaces. See papers and books by Albrecht Pietsch, e.g. – Peter Michor Feb 9 '13 at 16:16
@PeterMichor: Can you give a specific reference, please? Thanks! – Felix Goldberg Feb 9 '13 at 18:14
@ Felix Goldberg: MR0519680 (81a:47002) Reviewed Pietsch, Albrecht Operator ideals. Mathematische Monographien [Mathematical Monographs], 16. VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. 451 pp. Newer: MR1863699 (2003h:47137) – Peter Michor Feb 9 '13 at 22:31
By the Courant-Fischer min-max theorem, if $A$ is Hermitian, then $f(A) = \lambda_n(A)$, the smallest eigenvalue of $A$.
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Right! Of course, I managed to utterly confuse myself here. – Felix Goldberg Feb 9 '13 at 15:59
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2016-05-26 00:41:27
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https://www.computercollection.net/index.php/2020/07/29/ibm-1410-ald-to-fpga-i-got-one/
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# IBM 1410 ALD to FPGA – I “got one”
I have been testing the logic generated from each Automated Logic Diagram (ALD) page, using the Instructional Logic Diagrams (ILD) were available to guide my testing. Until today I don’t recall finding any cases where I actually made a connection mistake when I entered the ALDs into the database – until today. On page 13.50.01.1, I had mis-substituted signal “-S I-O Lozenge Latch” where I should have had “+S Logic Gate E 1” as the input to the gate at coordinate 5H. The testing caught it.
When I entered data for each sheet, I tracked usage of each signal count. I had a “2” written next to “+S Logic Gate E1” — as I should have. I must have missed that when I checked the signal usage counts after entry (“-S I-O Lozenge Latch” had the two instead). The latter is right below the former on the left side of the ALD, which made the mistake not unlikely, and made it easier to mis-interpret where I had written the signal count of 2. I even circled the 2, meaning I checked it. Oops.
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2023-01-31 23:36:40
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https://www.physicsforums.com/threads/work-done-by-electromagnetic-force.521631/
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# Work done by electromagnetic force
1. Aug 14, 2011
### scarlets99
1. The problem statement, all variables and given/known data
1.Calculate the work done by the force F=i cos(pix/4) + j y2 along the path y=x2 in the x-y plane from (0,0) to (2,4).
2.Is the force F conservative? Explain.
2. Relevant equations
Work = Integration between a and b of F.dl
3. The attempt at a solution
1. I know it's a line integral between the 2 points, I'm just unsure of how to do it.
2.If W=0 then it is conservative
2. Aug 14, 2011
### cepheid
Staff Emeritus
You have to find the dot product of the vector F and the vector dl and then integrate the result over the curve.
Nope. Think about this, does it make sense that a conservative force never does work between any two points? (Gravity and the electrostatic force are both conservative).
You're probably thinking of a result that the line integral of a conservative vector field around a closed path is 0. But in this case it's not a closed path, since the starting and ending points are not the same.
It's a conservative force if the work done in moving between any two points is independent of the path taken between them. In other words, all line integrals between point A and B, regardless of path, give you the same result. So the work done depends only on the two endpoints (you can probably see how the thing about closed paths follows from this property).
Another way to think about it is that a conservative vector field can always be expressed as the gradient of some scalar function usually called a "potential function" (which in the case of conservative forces actually represents the potential energy). So you can figure out the work done between any two points just by looking at the difference in potential between them. It doesn't matter how you got from one to the other.
3. Aug 14, 2011
### scarlets99
Thank you cepheid.
What is the value of dl?
4. Aug 14, 2011
### cepheid
Staff Emeritus
Well, I think that moving a small (infinitesimal) displacement dl along the curve means moving a small displacement in the x-direction + a small displacement in the y-direction. Does that help? (How would you express what I just said in vector form?)
5. Aug 14, 2011
### scarlets99
dxdy?
So the integral between 0 and 2, and 0 and 4 F.dxdy
6. Aug 14, 2011
### cepheid
Staff Emeritus
NO. dxdy has dimensions of area (m2), so you know that it can't possibly be a displacement vector.
I'm not sure why you multiplied. Take a look again at what I said :
Another hint: it might help to think about a simpler example first: if I move from the origin 3 m in the x-direction and then 2 m in the y-direction, how would I express that displacement in Cartesian vector form? Now, instead of 3 m and 2 m, generalize that to infinitesimal straight-line displacements dx and dy (infinitesimal since you're actually moving along a curve and just approximating it as a sequence of straight-line moves, and then taking the limit as the size of those straight-line moves goes to zero).
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2018-01-16 22:06:27
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https://www.homebuiltairplanes.com/forums/threads/cantilever-parasol-wings.31275/page-5
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# Cantilever parasol wings?
Discussion in 'Hangar Flying' started by cluttonfred, Mar 11, 2019.
1. Apr 21, 2019
### cluttonfred
#### Well-Known Member
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I don't see it that way, I think you can have a Fokker-style fun plane that isn't going to be confused with a replica. That won't prevent some people from dressing up the fun plane in WWI colors, of course, but that's fine. Here's a side-by-side comparison.
Last edited: Apr 21, 2019
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2. Apr 21, 2019
### Tiger Tim
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Side by side that little fella really is just a 3/4 scale D.VIII, isn’t it? I love that it was even covered in leftover lozenge fabric, though I’m biased because I just finished painting a wall of my office in Fokker five colour lozenge...
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3. Apr 21, 2019
### Bill-Higdon
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Cute Little Fokkers aren't they 0
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4. Apr 21, 2019
### crusty old aviator
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The DR-1 had a pretty hefty box spar the cabanes attached to. You could make a scaled down boxspar using geodetic for the webs to make a strong, light spar. I built one in the mid ‘80’s using four 1” x 1” Sitka for the “corners” with 1/2” x 1/8” diagonal strips for the webs. The Sitka had a 1/8” channel ripped into two adjacent sides, the entire length, 1/4” from the edge, to accommodate the diagonals. A 20’ section weighed about 12 pounds and supported over 2500 pounds of sand, simulating 5g’s, overnight (~12 hours) without permanent deformation after the load was removed.
Laird biplane fuselages have 4130 tubing from the firewall to just aft of the cockpit, then aluminum tube back to the tailpost. The aluminum structure is braced with bicycle spokes that run from the tube intersections to the center of each bay. In the center of each bay is a 3/4” long section of 2” dia. aluminum tube with four holes in it. In each hole is a spoke nut, like would be in a bike tire rim. The bent end of the spoke (that would normally be in the hub) is attached to the fitting at the tube intersection that holds the intersection together. Perhaps this would work in your application. Matty Laird truly took biplane design about as far as it could go.
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5. Apr 21, 2019
### Dillpickle
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[QUOTE="Victor Bravo,
Honestly Matthew, the amount of malicious mischief that you cause on this forum by making all of us innocent people stop and think about these cool airplanes and interesting ideas... [/QUOTE]
<<<<This!
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6. Apr 21, 2019
### erkki67
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This critter inspires me for a little more modern layout, as said before a Cub style landing gear and a rudder like on the Dream Classic of Airdrome Aerolplanes or the one of the Flitplane.
The axle on this V.40 is just a nightmare with those molehills!
7. Apr 21, 2019
### BJC
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... with a Verner Scarlett 3VW.
BJC
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8. Apr 21, 2019
### erkki67
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Yes yes yes....
9. Apr 21, 2019
### cluttonfred
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The 35-45 hp four-stroke range is pretty attractive if you are not trying to stay within Part 103 ultralight weight limits. You have a range of options based on cost up to the Verner 3VW which would be perfect with 42 hp at 2,500 rpm but not cheap at about $8,000 including the prop and electric start plus another$1,000 if you want dual ignition.
The industrial V-twin option is particularly interesting here, something like the Briggs & Stratton Vanguard 993 cc or Kohler OHV 999 cc gives you 35 hp @ 3600 rpm. Add an Ace 1:1.8 redrive to swing a big, slow, efficient, and vintage-looking 72-76" prop at just 2,000 rpm. Cost including engine with electric start, redrive and prop would be about \$4,000.
You could get a Hummel half VW for not much more, but I doubt that even the biggest half VW would beat the V-twin/redrive combo because of prop efficiency unless it had its own redrive.
10. Apr 22, 2019
### Riggerrob
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Geodesic construction might be light and only need small pieces of wood, but the disadvantage is increased labour cost.
I suspect that the majority of homebuilders would cheerfully glue in a bias-cut, plywood spar web. The single piece spar web also simplifies aligning all the ribs.
A plywood D-spar is great for carrying torsion loads. Glue it directly to spar caps and extend it far enough aft of the main spar to “gusset” ribs.
Modern CNC machines vastly simplify the process of converting sheets of plywood to airplane parts. They even make multiple different sized rib patterns an easy evening’s work.
Even if you decide on tapered aluminum wing ribs, CNC machines can route out a dozen different sizes of rib blanks and MDF forming blocks in a evening or two.
11. Apr 22, 2019
### erkki67
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So you might ask what the heck does the Corsair have to do with the Fokker V.40, not much except it’s construction perhaps.
The Dormoy Bathtub might give a hand on how to build the fuselage too.
Instead of building a 4 corner fuselage, why not a 3 corner fuselage with carbon tubing or welded or riveted tubes, and keeping the overall design of the V.40.
12. Apr 22, 2019
### cluttonfred
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Thanks, Erkki, but I think I'd like to keep this a little closer to the original V.40 even if it will certainly diverge in small ways.
Assuming I wanted to stick with the typical 1918 Fokker two-spar construction for the wing as in the artist's rendering below, what would be the easiest way to ensure a little more torsional rigidity?
I don't really want to go full geodetic, but perhaps diagonal braces between the ribs top and bottom in a zig-zag pattern?
What about using double X-bracing wires in the outer bays anchored at the top of one spar and the bottom of the other spar?
It might not be necessary, but I'd prefer a little insurance in that area.
By the same token, I would probably mass balance the ailerons, elevator, and rudder even if the first two don't have (and don't need at this power and speed) any aerodynamic balance area.
Last edited: Apr 22, 2019
13. Apr 22, 2019
### TFF
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Build it as designed or best guess since there are no plans. It’s perfect as is. Improvement rarely are.
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14. Apr 22, 2019
### FritzW
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You could add torque boxes between the spars like the VP-1 did.
Is there a problem with the wing? "you can't fix a problem that doesn't exist" With that massive "D" tube and the crazy strut geometry (D-VIII/V-40) it sure looks like it's plenty rigid already.
15. Apr 22, 2019
### Victor Bravo
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That computer model looks beautiful, but in the last rendering in post #92 it appears that the plywood sheeting on the leading eedge may not go back all the way to the spar. If I am seeing this correctly, that would be an issue IMHO. With a thick wing section, and a plywood skin that is glued solidly to the spar on the top and bottom, you should have a very stiff torque tube / D-section.
16. Apr 22, 2019
### cluttonfred
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If I remember correctly, the original Fokker method used the plywood, spanwise stringers, and the leading edge to maintain the front of the airfoil profile. I read somewhere that the triangular cutouts in the plywood were to reduce wrinkles in the fabric. Note that the spars were not full depth but were inside the rib caps. Using a full-depth spar and D-cell could certainly be done. I could see continuing the plywood to form the little triangle not between the ribs ending at the spars but on the ribs ending at the rib for a classic look. Here is detail drawing of a very similar D.VII wing.
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17. Apr 23, 2019
### FritzW
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With the ply attached to the spars, even with the gaps, it would still give you a nice "D" tube leading edge. But with those big box spars it's probably not required.
But the scallops lined up on the ribs just looks right...
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18. Apr 23, 2019
### Sockmonkey
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That strut arrangement looks like it would keep things from warping anyhow even if it did need bracing.
19. Apr 24, 2019
### TFF
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The Fokker leading edge acted like a sub rib, why they did not go all the way D is a small puzzle. Easier to glue down if not all continues and they did not need the strength. The Falco spars look very much like the D7 and D8.
The Dr1 spar is a box looking like a Jodel. The Dr1 is essentially two similar spars like a D7 type, front and back spaced that has top and bottom ply boxing them in. 4 shear webs per spar and top ply being drag/ anti drag.
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20. Apr 24, 2019
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2019-05-25 11:26:34
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https://math.stackexchange.com/questions/2124643/looking-for-summary-of-algebraic-laws
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# Looking for summary of algebraic laws
I'm wondering if anyone has compiled a summary of commonly used algebraic laws or axioms (e.g. closure, associativity, commutativity, distributivity)?
The reason I ask is that these laws or axioms are very useful in expressing properties to automatically test in generative software testing. Having a summary of these laws handy would help in selecting effective tests.
I've found some useful summaries of algebraic structures and have started to extract the laws or axioms those articles list but it seems like the kind of summary that may well already exist somewhere.
The functions and data structures being tested don't always seem to neatly belong to an algebraic structure I recognise but they are often subject to individual laws. This situation probably reflects my lack of algebraic intuition more than anything.
These references have been useful so far:
http://math.chapman.edu/~jipsen/structures/doku.php
Algebraic structure cheat sheet anyone?
[not enough reputation points to post the other links sorry]
Thanks
• You can post other links you need as a plain text: I think the post will be edited then to make them alive. And if it is about software testing, I'm not sure this is a post for math.SE and not StackOverflow. I don't understand what kind of links you are asking about, if you are not satisfied with the given two. – Wolfram Feb 1 '17 at 20:18
• Yes -- I was uncertain about the math .vs. StackOverflow issue although a summary of algebraic laws or axioms seems like the kind of thing you would see in the appendices of a math textbook rather than a computing text. Such a list is very useful for testing but I'm also trying to develop an understanding of the laws and their scope or power. – stu002 Feb 1 '17 at 20:22
There is a "famous" collection of types of algebras (algebraic laws), written by Loday under his pseudonym "Zienbiel" (Leibniz algebra read backwards). He says "The following is a list of some types of algebras together with their properties under an operadic and homological point of view." Here are some examples:
Com: $xy=yx$
As: $x(yz)=(xy)z$
Lie: $[x,[y,z]]+[[y,z],x]+[[z,x],y]=0$, $[x,x]=0$
PreLie: $x(yz)-(xy)z=y(xz)-(yx)z$
Alternative: $(x ∗ y) ∗ z = x ∗ (y ∗ z)+ \frac{1}{3}( x ∗ (z ∗ y) − z ∗ (x ∗ y) − y ∗ (x ∗ z) + y ∗ (z ∗ x))$
Reference: Encyclopedia of types of algebras
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2019-06-16 03:34:02
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https://aptitude.gateoverflow.in/2391/cat-1999-question-161
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526 views
The question is followed by two statements I and II.
Mark
1. if the question can be answered by any one of the statements alone, but cannot be answered by using the other statement alone.
2. if the question can be answered by using either statement alone.
3. if the question can be answered by using both the statements together, but cannot be answered by using either statement alone.
4. if the question cannot be answered even by using both the statements together.
Find a pair of real numbers x and y that satisfy the following two equations simultaneously. It is known that the values of a, b, c, d, e and f are non-zero.
$ax + by = c$
$dx + ey = f$
1. a = kd and b = ke, c = kf, k $\neq$ 0
2. a = b = 1, d = e = 2, f $\neq$ 2c
$\text{ax+by=c}$
Here if we put $\text{a=kd, b=ke, c=kf}$
we get $\text{dx+ey=f}$
which is same as $\text{2nd}$ equation
So, thiese two are same line putting condition I
Now, By condition II putting $\text{a=b=1, d=e=2}$
we get two lines are parallel
So, Answer$: \text{(D)}$ if the question cannot be answered even by using both the statements together.
by
5.1k points
### 1 comment
Using 1, we can get a point rt?
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2022-11-26 15:46:34
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https://www.researchgate.net/journal/Physics-Education-1361-6552
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# Physics Education
Online ISSN: 1361-6552
Print ISSN: 0031-9120
Publications
An acoustic method is presented for analyzing the time of falling motion. A ball is dropped from a measured height. The dropping device makes a distinct sound a well-determined time (roughly 14 milliseconds) after release. The ball subsequently makes a second distinct sound when it hits the surface below. These sounds are captured with a microphone resting on the surface and are readily apparent in the acoustic waveform. At each height (0.25m, 0.50m, 0.75m, and 1.00m), the measured drop time agrees with the drop time predicted by the law of falling bodies with a typical accuracy of 4.3 ms.
We use a tablet to determine experimentally the dependencies of the magnetic field (B) on the electrical current and on the axial distance from a coil (z). Our data shows a good precision on the inverse cubic dependence of the magnetic field on the axial distance, $B \propto z^{-3}$. We obtain with good accuracy the value of air permeability $\mu_{air}$. We also observe the same dependence of $B$ on $z$ when considering a magnet instead of a coil. Although our estimates are obtained through simple data fits, we also perform a more sophisticated error analysis, confirming the result for $\mu_{air}$.
The form and function of a collaborative assessment known as a "Buddy Quiz" is presented. The assessment is conducted in three successive phases over a contiguous 45- to 60-minute class period. A portion of each Quiz is completed in collaboration with one or two peers and a portion is completed without collaboration. The Quiz is primarily summative and is also designed to include formative aspects. The representation in the Quiz of the scientific enterprise as collaborative and individualistic is discussed. The employment of this instrument in a ninth-grade (age 15 years) conceptual physics course in an independent US secondary school is described and student feedback is presented.
This article presents the results of the first survey conducted in Belgium about the interest and knowledge in astronomy. Two samples were studied, the public at large (667 questionnaires) and students (2589 questionnaires), but the results are generally similar in both samples. We evaluated people's interest, main information source, and attitudes towards astronomy, as well as their supposed and actual knowledge of the subject. The main conclusion is that, despite a poor self-confidence, people do know the basic astronomical concepts. However, that knowledge is not deeply rooted, as reasoning questions show well-spread misconceptions and/or misunderstandings.
This paper discusses an instructional strategy which explores eventual similarities and/or analogies between familiar problems and more sophisticated systems. In this context, the Atwood's machine problem is used to introduce students to more complex problems involving ropes and chains. The methodology proposed helps students to develop the ability needed to apply relevant concepts in situations not previously encountered. The pedagogical advantages are relevant for both secondary and high school students, showing that, through adequate examples, the question of the validity of Newton's second law may be introduced to even beginning students.
This article presents the customization of EJS models, used together with actual laboratory instruments, to create an active experiential learning of measurements. The laboratory instruments are the vernier caliper and the micrometer. Three computer model design ideas that complement real equipment are discussed in this article. They are 1) the simple view and associated learning to pen and paper question and the real world, 2) hints, answers, different options of scales and inclusion of zero error and 3) assessment for learning feedback. The initial positive feedback from Singaporean students and educators points to the possibility of these tools being successfully shared and implemented in learning communities, and validated. Educators are encouraged to change the source codes of these computer models to suit their own purposes, licensed creative commons attribution for the benefit of all humankind. Video abstract: http://youtu.be/jHoA5M-_1R4
We present two different paradoxes related to the length contraction in special relativity and explain their resolution.
Improving the scientific literacy of non-scientists is an important goal, both because of the ever-increasing impact of science and technology on our lives, and because understanding science enriches our experience of the natural world. One route to improving scientific literacy is via general education undergraduate courses -- i.e. courses intended for students not majoring in the sciences or engineering -- which in many cases provide these students' last formal exposure to science. I describe here a course on biophysics for non-science-major undergraduates recently developed at the University of Oregon (Eugene, OR, USA). Biophysics, I claim, is a particularly useful vehicle for addressing scientific literacy. It involves important and general scientific concepts, demonstrates connections between basic science and tangible, familiar phenomena related to health and disease, and illustrates that scientific insights develop by applying tools and perspectives from disparate fields in creative ways. In addition, biophysics highlights the far-reaching impact of physics research. I describe the general design of this course, which spans both macroscopic and microscopic topics, and the specific content of a few of its modules. I also describe evidence-based pedagogical approaches adopted in teaching the course, and aspects of its enrollment and evaluation.
Slow flow of the viscous liquid is a thought-provoking experiment that challenges students, academics and public to think about some fundamental questions in modern science. In the Queensland demonstration, the world-longest running experiment earning the Ig Nobel prize, one drop of pitch takes about 10 years to fall, leading to problems of actually observing the drops. Here, we describe our recent demonstration of slowly-flowing bitumen where appreciable flow is observed on the time scale of months. The experiment is free from dissipative heating effects and has the potential to improve the accuracy of measurement. Bitumen viscosity was calculated by undergraduate students during the summer project. The worldwide access to the running experiment is provided by webcams uploading the images to a dedicated website, enhancing student education experience and promotion of science. This demonstration serves as an attractive student education exercise and stimulates the discussion of fundamental concepts and hotly debated ideas in modern physics research: difference between solids and liquids, the nature of liquid-glass transition, emergence of long time scales in a physical process, and the conflict between human intuition and physical reality.
In this paper we propose strategies and methodologies of teaching topics in high school physics, through a show of Educational Robotics. The Exhibition was part of a set of actions promoted by a brazilian government program of incentive for teaching activities (PIBID) and whose primary focus is the training of teachers, improvement of teaching in public schools, dissemination of science and formation of new scientists and researchers. By means of workshops, banners and prototyping of robotics, we are able to create a connection between the study areas and their surrounding, making learning meaningful and accessible for the students involved and contributing to their cognitive development.
Lessons and homework problems involving a pendulum are often a big part of introductory physics classes and laboratory courses from high school to undergraduate levels. Although laboratory equipment for pendulum experiments is commercially available, it is often expensive and may not be affordable for teachers on fixed budgets, particularly in developing countries. We present a low-cost, easy-to-build rotary sensor pendulum using the existing hardware in a ball-type computer mouse. We demonstrate how this apparatus may be used to measure both the frequency and coefficient of damping of a simple physical pendulum. This easily constructed laboratory equipment makes it possible for all students to have hands-on experience with one of the most important simple physical systems.
A Faraday cage is an interesting physics phenomena where an electromagnetic wave can be excluded from a volume of space by enclosure with an electrically conducting material. The practical application of this in the classroom is to block the signal to a mobile phone by enclosing it in a metal can! The background of the physics behind this is described in some detail followed by a explanation of some demonstrations and experiments which I have used.
Assuming a constant mass-decrease per unit-surface and -time we provide a very simplistic model for the dissolution process of spherical candies. The aim is to investigate the quantitative behavior of the dissolution process throughout the act of eating the candy. In our model we do not take any microscopic mechanism of the dissolution process into account, but rather provide an estimate which is based on easy-to-follow calculations. Having obtained a description based on this calculation, we confirm the assumed behavior by providing experimental data of the dissolution process. Besides a deviation from our prediction caused by the production process of the candies below a diameter of 2 mm, we find good agreement with our model-based expectations. Serious questions on the optimal strategy of enjoying a candy will be addressed, like whether it is wise to split the candy by breaking it with the teeth or not.
We develop an Easy Java Simulation (EJS) model for students to experience the physics of idealized one-dimensional collision carts. The physics model is described and simulated by both continuous dynamics and discrete transition during collision. In the field of designing computer simulations, we discuss briefly three pedagogical considerations such as 1) consistent simulation world view with pen paper representation, 2) data table, scientific graphs and symbolic mathematical representations for ease of data collection and multiple representational visualizations and 3) game for simple concept testing that can further support learning. We also suggest using physical world setup to be augmented complimentarily with simulation while highlighting three advantages of real collision carts equipment like tacit 3D experience, random errors in measurement and conceptual significance of conservation of momentum applied to just before and after collision. General feedback from the students has been relatively positive, and we hope teachers will find the simulation useful in their own classes.
A very low-cost, easy-to-make stopwatch is presented to support various experiments in mechanics. The high-resolution stopwatch is based on two photodetectors connected directly to the microphone input of the sound card. A dedicated free open-source software has been developed and made available to download. The efficiency is demonstrated by a free fall experiment.
We present a sensitive diffusion cloud chamber which does not require any radioactive sources. A major difference from a commonly used chamber is use of a heat sink as its bottom plate. A result of a performance test of the chamber is given.
During the past several years the authors have developed a new approach to the teaching of Physical Science, a general education course typically found in the curricula of nearly every college and university. This approach, called Physics in Films', uses scenes from popular movies to illustrate physical principles and has excited student interest and improved student performance. The analyses of many of the scenes in Physics in Films' are a direct application of Fermi calculations -- estimates and approximations designed to make solutions of complex and seemingly intractable problems understandable to the student non-specialist. The intent of this paper is to provide instructors with examples they can use to develop skill in recognizing Fermi problems and making Fermi calculations in their own courses.
A project known as "Physics in Films" currently underway at UCF is designed to generate renewed interest and excitement in the standard "Physical Science" course. In the initial, developmental phase of the project the instructors selected films without regard to genre and theme. They only demanded that the selected films would cover nearly all the traditional topics of the typical "Physical Science" course. The project met with unprecedented success that motivated a rapid and more sophisticated expansion. In the current phase the authors demonstrate that the method is very flexible, accommodating the movie preferences of any instructor. The authors are developing versions of the course that have different "flavors", that is, "Physics in Films" course packages built around particular genres or themes. For example, during the 2003 summer terms we class-tested the flavors "Physics in Films: Superheroes" and "Physics in Films: Pseudoscience". Additional flavors are under development. This talk summarizes part of our experiences and presents some of the results from our courses. Overall, the method opens an effective way to help large masses to overcome science illiteracy.
This work is basically about the general form of Newton's second law for variable mass problems. We develop a model for describing the motion of the one-dimensional oscillator with a variable mass within the framework of classroom physics. We present a simple numerical procedure for the solution of the equation of motion of the system to be implemented by students and teachers. Interesting qualitative concepts as well as quantitative results for the focused problem are presented. The topic has pedagogical value both from theoretical and experimental point of view. However, this article considers only theoretical aspects of the problem. The work is addressed to basic physics courses at undergraduate level.
The field of extrasolar planets is still, in comparison with other astrophysical topics, in its infancy. There have been about 300 or so extrasolar planets detected and their detection has been accomplished by various different techniques. Here we present a simple laboratory experiment to show how planets are detected using the transit technique. Following the simple analysis procedure describe we are able to determine the planetary radius to be 1.27 +/- 0.20 R_{J} which, within errors agrees with the establish value of 1.32 +/- 0.25 R_{J}.
Commercial video games are increasingly using sophisticated physics simulations to create a more immersive experience for players. This also makes them a powerful tool for engaging students in learning physics. We provide some examples to show how commercial off-the-shelf games can be used to teach specific topics in introductory undergraduate physics. The examples are selected from a course taught predominantly through the medium of commercial video games.
This paper describes the spherical concave mirror method for measuring the index of refraction of transparent liquids. We derived the refractive index equation using Snell's law and the small-angle approximation. We also verified the validity of this method using the traditional spherical mirror and thin-lens Gaussian equations.
A group of high school students (XII Liceum) in the framework of the Roland Maze Project has built a compact telescope of three Geiger-Muller counters. The connection between the telescope and PC computer was also created and programed by students involved in the Project. This has allowed students to use their equipment to perform serious scientific measurements concerning the single cosmic ray muon flux at ground level and below. These measurements were then analyzed with the programs based on the 'nowadays' knowledge on statistics. An overview of the apparatus, methods and results were presented at several students conferences and recently won the first prize in a national competition of high school students scientific work. The telescope itself, in spite of its 'scientific' purposes, is built in such a way that it is hung on a wall in a school physics lab and counts muons continuously. This can help to raise the interest for studying physics among others. At present a few (3) groups of young participants of the Roland Maze Project have already built their own telescopes for their schools and some others are working on it. This work is a perfect example of what can be done by young people when respective opportunities are created by more experienced researchers and a little help and advice is given.
Since October 2010, the Chemistry-Biology Combined Major Program (CBCMP), an international course taught in English at Osaka University, has been teaching small classes (no more than 20 in size). We present data from the Force Concept Inventory (FCI) given to first year classical mechanics students ($N=47$ students over three years) pre and post score, for a class that predominantly uses interactive engagement (IE), such as MasteringPhysics. We also comment on possible correlations between the pre/post score and the level of English ability on entry to the course; importantly, there does appear to be a correlation with reading ability, which is not typically a criteria for entrance to Japanese global courses (usually only a total TOEFL $\sim 80$ is required). Our findings show a $G$-factor improved score of about $\sim 0.18$, which is marginally about the average of a traditional based course. Given that the number of test subjects are quite small, we analyze in detail a set of six questions from the FCI, involving the identification of forces acting on a body. We find that student answers tend to cluster about "polarizing choices"---a pair of choices containing the correct choice and a wrong choice with the latter corresponding to a super-set of forces in the former. Our results are suggestive that students have a good idea of the right set of forces acting on a given system but this set contains extra force(s) that is (are) ontologically miscategorized.
Studies have shown that standard lectures and instructional laboratory experiments are not effective at teaching interference and diffraction. In response, the author created an interactive computer program that simulates interference and diffraction effects using the Finite Difference Time Domain method. The software allows students to easily control, visualize, and quantitatively measure the effects. Students collected data from simulations as part of their laboratory exercise, and they performed well on a subsequent quiz---showing promise for this approach.
Many students meet quite early this dipole-dipole potential energy when they are taught electrostatics or magnetostatics, and it is also a very popular formula, featured in the encyclopedias. We show that by a simple rewriting of the formula it becomes apparent that for example, by reorienting the two dipoles, their attraction can become exactly twice as large. The physical facts are naturally known, but the presented transformation seems to underline the geometrical features in a rather unexpected way. The consequence of the discussed features is the so called magic angle which appears in many applications. The present discussion also contributes to an easier introduction of this feature. We also discuss a possibility for designing educational toys and try to suggest why this formula has not been written down frequently before this work. Similar transformation is possible for the field of a single dipole, there it seems to be observed earlier, but also in this case we could not find any published detailed discussion.
Google Earth photographs often show ships and their wakes in great detail. We discuss how the images can be used to calculate the velocity of these ships.
We develop an Easy Java Simulation (EJS) model for students to visualize geostationary orbits near Earth, modeled using Java 3D implementation of the EJS 3D library. The simplified physics model is described and simulated using simple constant angular velocity equation. Four computer model design ideas such as 1) simple and realistic 3D view and associated learning to real world, 2) comparative visualization of permanent geostationary satellite 3) examples of non-geostationary orbits of different 3-1) rotation sense, 3-2) periods, 3-3) planes and 4) incorrect physics model for conceptual discourse are discussed. General feedback from the students has been relatively positive, and we hope teachers will find the computer model useful in their own classes.
We discuss the flat and hollow models of the Earth as a pedagogical example of the application of Gauss' law to the gravitational field.
Google Earth is a huge source of interesting illustrations of various natural phenomena. It can represent a valuable tool for science education, not only for teaching geography and geology, but also physics. Here we suggest that Google Earth can be used for introducing in an attractive way the physics of waves.
This paper reports a computer model- simulation created using Easy Java Simulation (EJS) for learners to visualize how the steady-state amplitude of a driven oscillating system varies with the frequency of the periodic driving force. The simulation shows (N=100) identical spring-mass systems being subjected to (1) periodic driving force of equal amplitude but different driving frequencies and (2) different amount of damping. The simulation aims to create a visually intuitive way of understanding how the series of amplitude versus driving frequency graphs are obtained by showing how the displacement of the system changes over time as it transits from the transient to the steady state. A suggested how to use the model is added to help educators and students in their teaching and learning, where we explained the theoretical steady state equation, time conditions when the model starts allowing data recording of maximum amplitudes to closely match the theoretical equation and steps to collect different runs of degree of damping. We also discuss two design features in our computer model: A) displaying the instantaneous oscillation together with the achieved steady state amplitudes and B) explicit world view overlay with scientific representation with different degrees of damping runs. Three advantages of using EJS include 1) Open Source Codes and Creative Commons Attribution Licenses for scaling up of interactively engaging educational practices 2) models made can run on almost any device including Android and iOS and 3) allows for redefining physics educational practices through computer modeling.
Here we propose two methods to get electroluminescence images from photovoltaic cells in a school or home lab.
A key question in physics education is the effectiveness of the teaching methods. A curriculum that has been investigated at the University of Central Florida (UCF) over a period of two years is the use of particular elements of The Physics Suite. Select sections of the introductory physics classes at UCF have made use of Interactive Lecture Demonstrations as part of the lecture component of the class. The lab component of the class has implemented the RealTime Physics curriculum, again in select sections. The remaining sections have continued with the teaching methods traditionally used. Using pre- and post-semester concept inventory tests, a student survey, student interviews, and a standard for successful completion of the course, the data indicates improved student learning.
The discovery of quantum mechanics at the beginning of the last century led to a revolution of the physical world view. Modern experiments, made possible by new techniques on the border of the classical and the quantum regimes offer new insights and better understanding of the quantum world and have impact on technical development. Therefore it seems important that students gain appreciation of the principles of quantum mechanics. A suitable way seems be the treatment of the EPR-experiment at a prominent place.
The model of the Big Bang is an integral part of the national curriculum for England. Previous work (e.g. Baxter 1989) has shown that pupils often come into education with many and varied prior misconceptions emanating from both internal and external sources. Whilst virtually all of these misconceptions can be remedied, there will remain (by its very nature) the obstacle of ex-nihilo, as characterised by the question how do you get something from nothing?' There are two origins of this obstacle: conceptual (i.e. knowledge-based) and cultural (e.g. deeply held religious viewpoints). The article shows how the citizenship section of the national curriculum, coming online' in England from September 2002, presents a new opportunity for exploiting these. Comment: 6 pages. Accepted for publication in Physics Ed
We describe an example of learning with multiple representations in an A-level revision lesson on mechanics. The context of the problem involved the motion of a ball thrown vertically upwards in air and studying how the associated physical quantities changed during its flight. Different groups of students were assigned to look at the ball's motion using various representations: motion diagrams, vector diagrams, free-body diagrams, verbal description, equations and graphs, drawn against time as well as against displacement. Overall, feedback from students about the lesson was positive. We further discuss the benefits of using computer simulation to support and extend student learning.
This paper reports the use of Tracker as a pedagogical tool in supporting effective learning and teaching of toss up and free fall motion for beginning grade 9 students. This is a case study with (N=123) students of express-pure physics classes in a mainstream school in Singapore where we used a 8 multi-choice questions as a proxy to assess learning gains in pre and posttest to gauge the impact on learning. We found within experimental group gains with Cohens effect size d = 0.79 error 0.23 (large effect) and normalized gains with a gradient of g total = 0.42 error 0.08 (medium gain) above the traditional baseline value of g non interactive=0.23 for all the 6 teachers, 3 classes of students who participated in this study. Initial research findings suggest that allowing learners to relate abstract physics concepts to real life through coupling traditional video analysis and eventually video modeling could be an innovative and effective way to learn free fall motion. Finally, we discuss the pedagogical use of Tracker to extend the learning of free fall by means of allowing students to construct simple dynamic particle models for scenarios that are difficult to visualize their velocity versus time graphs such as 2 cases compare to tossing up a ball with a) with a greater force on Earth and b) with the same force on Moons surface.
A closer look (with hindsight) at Newtonian and relativistic kinematics reveals two things. Not surprisingly, Newtonian time remains the empty and artificial - albeit useful - figment it is known to be. Quite unexpectedly however it turns out that with an only slightly more physical time-concept, time and space remain separate entities and simultaneity of events becomes a meaningful concept even for physicists.
In this paper we describe an alternative use of the loop-the-loop apparatus, which can be used to study an interesting case of projectile motion. We also present an effective way to perform and analyze these experiments, by using video capture software together with a digital video camera. These experiments can be integrated into classroom demonstrations for general physics courses, or become part of laboratory activities.
In classical mechanics matter and fields are completely separated. Matter interacts with fields. For particle physicists this is not the case. Both matter and fields are represented by particles. Fundamental interactions are mediated by particles exchanged between matter particles. In this paper we explain why particle physicists believe in such a picture, introducing the technique of Feynman diagrams starting from very basic and popular analogies with classical mechanics, making the physics of elementary particles comprehensible even to high school students, the only prerequisite being the knowledge of the conservation of mechanical energy.
I discuss some interesting classroom demonstrations of diamagnetism and how this effect can produce levitation. The possibilities for hands-on demonstrations of diamagnetic and superconducting levitation are discussed. To conclude I discuss some practical uses for levitation in daily life.
The logistic map is one of the simplest nonlinear dynamical systems that clearly exhibit the route to chaos. In this paper, we explored the evolution of the logistic map using an open-source microcontroller connected to an array of light emitting diodes (LEDs). We divided the one-dimensional interval $[0,1]$ into ten equal parts, and associated and LED to each segment. Every time an iteration took place a corresponding LED turned on indicating the value returned by the logistic map. By changing some initial conditions of the system, we observed the transition from order to chaos exhibited by the map.
A set of examples is provided that illustrate the use of work as applied to simple machines. The ramp, pulley, lever and hydraulic press are common experiences in the life of a student and their theoretical analysis therefore makes the abstract concept of work more real. The mechanical advantage of each of these systems is also discussed so that students can evaluate their usefulness as machines. Comment: 9 pages, 4 figures
This paper reports the use of Tracker as a pedagogical tool in the effective learning and teaching of projectile motion in physics. When computer model building learning processes is supported and driven by video analysis data, this free Open Source Physics (OSP) tool can provide opportunities for students to engage in active inquiry-based learning. We discuss the pedagogical use of Tracker to address some common misconceptions of projectile motion by allowing students to test their hypothesis by juxtaposing their mental models against the analysis of real life videos. Initial research findings suggest that allowing learners to relate abstract physics concepts to real life through coupling computer modeling with traditional video analysis could be an innovative and effective way to learn projectile motion.
The reaction time of a group of students majoring in Physics is reported here. Strong co-relation between fatigue, reaction time and performance have been seen and may be useful for academicians and administrators responsible of working out time-tables, course structures, students counsellings etc. Comment: 10 pages, 4 figures
The power distribution of nearly all major countries have accepted 3-phase distribution as a standard. With increasing power requirements of instrumentation today even a small physics laboratory requires 3-phase supply. While physics students are given an introduction of this in passing, no experiment work is done with 3-phase supply due to the sheer possibility of accidents while working with such large powers. We believe a conceptual understanding of 3-phase supply would be useful for physics students with hands on experience using a simple circuit that can be assembled even in a high school laboratorys.
Springs are used for a wide range of applications in physics and engineering. Possibly, one of its most common uses is to study the nature of restoring forces in oscillatory systems. While experiments that verify the Hooke's law using springs are abundant in the physics literature, those that explore the combination of several springs together are very rare. In this paper, an experiment designed to study the static properties of a combination of springs in series using only one single spring is presented. Paint marks placed on the coils of the spring allowed us to divide it into segments, and considered it as a collection of springs connected in series. The validity of Hooke's law for the system and the relationship between the spring constant of the segments with the spring constant of the entire spring is verified experimentally. The easy setup, accurate results, and educational benefits make this experiment attractive and useful for high school and first-year college students.
Zulfi Erken teaches physics in English at a Russian town between the Volga and the Urals. Here he reveals how his students can overcome difficulties such as insufficient equipment to learn physics in a foreign language and go on to study at universities abroad.
The author describes the discovery of the electron, electrons in atoms, properties of free electrons, theories of electrons, sources and finally applications of electrons in industry and medicine.
The Standard Model is the generally accepted model of the fundamental particles of matter. It allows six quarks, six leptons and various `field quanta' which account for the interactions between particles. This article describes the model and makes some suggestions for improvement.
Top-cited authors
• Brunel University London
• The University of Sheffield
• Southern University and A&M College
• Tufts University
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2022-09-28 13:18:21
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https://www.physicsforums.com/threads/what-is-the-way-of-solving-this-question.279978/
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# What is the way of solving this question
1. Dec 16, 2008
### transgalactic
what is the general way of solving this question:
http://img116.imageshack.us/img116/1152/25587465vv2.gif [Broken]
??
Last edited by a moderator: May 3, 2017
2. Dec 16, 2008
### Defennder
1. You want V(k) to be linearly independent so that it will form a basis for R3. So that means if you interpret them as row vectors in a matrix and perform row reduction, the end result should be a matrix whose rank is 3. So what does the matrix being of full rank imply? And what values of k are suitable such that the matrix is of full rank?
2. Firstly determine the solution space of U(k). Then $$span(U(k1)) \subseteq span(V(k2))$$ if $$\forall v \ \text{where v is a vector in basis of U(k1)} \ , v \in span(V(k2))$$. ie. try to express every vector in the basis of U(k1) as a linear combination of V(k2). Take note of when this is possible (ie. which values of k1 and k2 permit that).
3. Dec 16, 2008
### transgalactic
i think i solved part 1.
but i dont know how to use part 1
in order to solve 2
http://img384.imageshack.us/img384/2546/55339538nk4.gif [Broken]
Last edited by a moderator: May 3, 2017
4. Dec 17, 2008
### transgalactic
what is a solution space?
i can find k values for which i get one solution
infinite solution
or no solution
what is the definition of solution space in a parameter matrix?
5. Dec 17, 2008
### transgalactic
how do i find the solution vectors of U(k)?
6. Dec 17, 2008
### transgalactic
when you say
"Firstly determine the solution space of U(k). "
there are 3 types of solution?(depends on the values of K)
no solution
1 solution
infinite solution
7. Dec 17, 2008
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2017-06-24 04:09:56
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https://mathlesstraveled.com/2016/02/07/post-without-words-5-explained/
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## Post without words #5, explained
If you stared for a while at the images in my previous post, you probably noticed some patterns, and maybe you even figured out some sort of rule or algorithm behind them. Commenter Yammatak expressed it as “You split it into 4 and put 2 copies of the original on the bottom with the colors inverted and 2 copies on the top but rotated 90 in opposite directions.” Like this:
Yammatak is right: iterating this process does generate exactly the images in my post. As I noted in my response to Yammatak’s comment, however, this is not how I made the images! I only noticed that they could be described using this simple rule after making them.
So, how did I make them? First, I started with the Prouhet-Thue-Morse sequence, which can be defined by
$\begin{array}{rcl} T_0 & = & 0 \\ T_n & = & T_{n-1},\overline{T_{n-1}} \end{array}$
where the comma denotes concatenation of sequences, and the overbar means to swap zero for one and vice versa. So,
$\begin{array}{rcl} T_0 & = & 0 \\ T_1 & = & T_0,\overline{T_0} = 0,1 \\ T_2 & = & T_1,\overline{T_1} = 0,1,1,0 \\ T_3 & = & 0,1,1,0,1,0,0,1 \\ T_4 & = & 0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0\end{array}$
and so on. This is a really fascinating sequence that shows up all over the place—for more, see Matt Parker’s recent video, or, if you are more academically inclined, this paper by Allouche and Shallit.
In any case, replace each zero with a light blue square, and each 1 with a dark blue square:
Now, take the Hilbert space-filling curve:
and string out the Prouhet-Thue-Morse sequence along it, using light and dark blue squares in place of zeros and ones:
This works out really nicely because both the Prouhet-Thue-Morse sequence and the Hilbert curve naturally organize things in powers of 2. If you think about the recurrence for the Prouhet-Thue-Morse sequence, in terms of replicating and inverting, and the recurrence for the Hilbert curve, in terms of scaled and rotated copies, you can see why you end up with a simple rule like the one Yammatak explained. So in the end, I guess you could say this is a complicated way to describe a simple rule that generates a complicated image!
Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
This entry was posted in pattern, pictures, posts without words, sequences, solutions and tagged , , , , , . Bookmark the permalink.
### 4 Responses to Post without words #5, explained
1. Christophe says:
Cool. I really like the idea. Simple an beautiful !
Question. Do the L-system given by Yammatak is really the same as your process ? Is there a proof ?
• Brent says:
Yes, it is the same. The proof is simple (at least conceptually): the n+1st iteration of the Hilbert curve is formed from 4 copies of the nth iteration, where the first and fourth copies have been rotated by 90 degrees; couple this with the fact that $T_{n+2} = T_n, \overline{T_n}, \overline{T_n}, T_n$.
• Christophe says:
Ok. Just continue your fantastic blog !
2. Fergal Daly says:
I remember as a kid (maybe teen, maybe before) seeing how far I could get through the PTM sequence. I never knew it had a name.
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2019-03-25 11:46:06
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https://www.albert.io/ie/ap-physics-c-mechanics/rolling-soccer-ball
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Free Version
Moderate
# Rolling Soccer Ball
APPHMC-9GZVUJ
A soccer ball with a radius of 25cm, is kicked with an initial velocity of 15m/s and rolls without slipping across a level horizontal grass field. If the acceleration of the ball is $-25.0\frac{m}{s^2}$
Which of the following statements best represents how many rotations the ball makes before coming to rest?
A
$2.9 revolutions$
B
$4.5 revolutions$
C
$12.5 revolutions$
D
$18 revolutions$
E
$24 revolutions$
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2017-02-21 12:31:36
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http://math.stackexchange.com/questions/366211/cardinal-numbers-are-identified-with-the-set-of-ordinals-preceding-them
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# Cardinal numbers are identified with the set of ordinals preceding them
Here is a description of cardinal number.
Cardinal numbers are identified with the set of ordinals preceding them.
Is this OK?
-
@Martin: Do you think that maybe [definition] or [terminology] fit this question? I was going to add one, and then I hesitated. – Asaf Karagila Apr 19 '13 at 19:36
@Asaf If I had to choose one from the two tags you've suggested, I would go with definition. (Based on the content of the tag-wikis for these two tags.) – Martin Sleziak Apr 19 '13 at 19:40
It would be better to say:
Ordinals are von Neumann ordinals: each ordinal is the set of its predecessors, and cardinal numbers are identified with initial ordinals.
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Thanks Brain. Could you help me to answer math.stackexchange.com/questions/365321/… – Paul Apr 19 '13 at 6:31
@Paul: I’ll take a look, but it may be a while before I can give it serious thought. – Brian M. Scott Apr 19 '13 at 6:33
@Brain: It doesn't matter. I'll wait for you. – Paul Apr 19 '13 at 6:37
Your statement is unclear, or even wrong, because every ordinal is identified as the set of preceding ordinals. It is also unclear whether or not you mean that every cardinal number is identified with an ordinal, or that every ordinal which can be identified with the set of its predecessors is a cardinal.
Cardinal numbers are identified with initial ordinals, which are ordinals that there is no injection from them into a smaller ordinal.
-
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2016-07-28 07:07:36
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http://openstudy.com/updates/517ac729e4b0249598f741b3
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## ashneedhelp Your parents allow you to borrow a car to get to your part-time job, but you have to pay for a tank of gas each month. Gasoline costs \$2.76/gallon and the tank takes 15 gallons. You have 2 coworkers who are each willing to pay for a quarter of a tank each month to carpool with you to work. How much do you save each month? one year ago one year ago
1. ashneedhelp
@electrokid is it 27.6 i did 2.76*15 =41.4 then i did 41.4 /3 =13.8 then i 13.8 * 2 thats were i got 27.6
2. electrokid
almost... a quarter of a whole = $$\Large 1\over4$$ amount saved = amount you'd pay - amount the co-workers pay amount you pay = 41.4 (good job there) amount EACH co-worker pays = a quarter of this = $$\Large 41.4\over4$$ = ?
3. ashneedhelp
41.4/4=10.35
4. ashneedhelp
20.7
5. electrokid
good. so, bingo!!
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2014-04-23 09:22:17
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https://itectec.com/ubuntu/ubuntu-dont-the-fn-keys-work-for-brightness-or-media-after-upgrading/
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# Ubuntu – Don’t the Fn keys work for brightness or media after upgrading
brightnessshortcut-keysvolume-control
I recently upgraded from 11.04 to 11.10. After the upgrade, I can no longer adjust the screen brightness or the volume using keyboard (before the upgrade, using Fn+F4, Fn+F11, etc. worked).
Using Fn+F2 to disable wireless still works, so I guess the Fn key itself is being recognised.
I tried to follow the instructions here, but I don't have a file in /etc/X11 called xorg.conf.
I also tried following this workaround, but it had no noticeable effect.
I've also tried going to SettingsKeyboardShortcuts and reassigning the brightness and volume controls, both to the default keys and to new combinations. These changes don't have an effect even after rebooting.
Googling has found bug reports where pressing the media keys brings up a "no entry sign" rather than changing the volume. When I press the keys there's no response at all.
I've also seen various people say a workaround is to have totem running in the background; this doesn't work for me either.
Finally, I tried installing keytouch; I was able to install keytouch-editor but got the message "Unable to locate package keytouch".
Any more ideas? I'd be very grateful if anyone could help me (even by pointing to a thread I've missed)!
I discovered that running gnome-settings-daemon fixes this issue (and gives me a functional battery status indicator again), so I added it to my startup commands.
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2021-11-30 17:57:15
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https://ezeenotes.in/an-elevator-car-whose-floor-to-ceiling-distance-is-equal-to-2-7%F0%9D%91%9A-starts-ascending-with-constant-acceleration-of-1-2%F0%9D%91%9A%F0%9D%91%A0%E2%88%922-2-%F0%9D%91%A0%F0%9D%91%92/
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# An elevator car, whose floor to ceiling distance is equal to 2.7𝑚, starts ascending with constant acceleration of 1.2𝑚𝑠 −2. 2 𝑠𝑒𝑐 after the start, a bolt begins falling from the ceiling of the car. The free fall time of the bolt is
Question : An elevator car, whose floor to ceiling distance is equal to 2.7𝑚, starts ascending with constant acceleration of 1.2𝑚𝑠 −2. 2 𝑠𝑒𝑐 after the start, a bolt begins falling from the ceiling of the car. The free fall time of the bolt is
(A) $\sqrt{0.54}s$
(B) $\sqrt{6}_{s}$
(C) $\sqrt{0.7}s$
(D) 1 s
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2023-02-09 13:14:32
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http://openstudy.com/updates/51003a66e4b00c5a3be6a3e5
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anonymous 3 years ago Evaluate x/y, for x=2/3 and y=5/6 a.)5/9 b.)4/5 c.)5/4 d.)9/5
1. precal
|dw:1358969521748:dw|do you remember the rules for dividing fractions?
2. anonymous
$\frac{ 2 }{ 3 }/\frac{ 5 }{ 6 }=\frac{ 2 }{ 3 }*\frac{ 6 }{ 5 }=\frac{ 4 }{ 5 }$
3. anonymous
Yeah I know them and I figured it out after you said that. Thanks guys.(;
4. precal
ok then leave the numerator alone, change division to times and flip the denominator
5. precal
yw
6. anonymous
yw ;)
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2016-05-29 19:18:35
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https://math.stackexchange.com/questions/1760866/how-many-gauss-points-are-required-to-provide-exact-value-for-the-gauss-quadratu
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# How many Gauss points are required to provide exact value for the Gauss quadrature rule
How many Gauss points are required if the Gauss quadrature rule should provide the exact value of the integral $I=\int_{-1}^1f(x)dx$ for $f(x)=(x^2-1)^2$?
I am really not sure what theorem to use to solve this problem. What I can think of is a theorem about Gaussian quadrature with orthogonal polynomials as follows:
If a polynomial $p$ of degree $n+1$ is orthogonal to all polynomials of lower degree on the interval $[a,b]$ then it has $n+1$ distinct roots $x_i$ with $a<x_0<\ldots<x_n<b$ and if one uses these roots to determine the weights $A_i$ in the approximate integration formula $\int_{a}^{b}f(x)dx\approx\sum_{i=0}^{n}A_if(x_i)$ so that it is exact for all polynomials of degree up to $n$, then it is in fact exact for all polynomials of degree up to $2n+1$
But somehow I still cannot relate this theorem to the problem.
Could anyone please lend some help?
Thanks.
• You have it: Gauss Ian quadrature uses exactly those nodes and weights. Then your polynomial is degree 4 so... – Ian Apr 27 '16 at 10:54
• @Ian so it has 5 distinct roots. So 5 Gauss points? – user71346 Apr 27 '16 at 10:59
• No, think about that $2n+1$ thing... – Ian Apr 27 '16 at 13:07
• @Ian. If $n$ is 3, then $2(3)+1=7$ points? But $2n+1$ is the degree of the polynomials, not the number of Gauss points? – user71346 Apr 27 '16 at 13:32
• With $n+1$ points you exactly integrate polynomials of degree $2n+1$, that is the important message of your paragraph above. – Ian Apr 27 '16 at 13:39
As far is I know the correct formula for determining the number of Gauss points is given by:
$p + 1 = 2n$
or
$p = 2n-1$
where p is the degree of the polynomial and n are the number of Gauss points.
Since your problem involves a fourth degree polynomial, you need 5/2 gauss points. This problem would therefore require 3 integration points instead of 2:
$(4+1)/2 = 5/2$
I hope this might solve your problem. I tried it out on a simple fourth order polynomial which gave me the exact answer.
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2019-08-20 18:40:33
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https://math.stackexchange.com/questions/3186018/group-action-in-polynomial-invariant
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# Group action in Polynomial invariant
The following is just a basic definition in Invariant Theory, which I copied from wikipedia
"Let $$G$$ be a group, and $${\displaystyle V}$$ a finite-dimensional vector space over a field $${\displaystyle k}$$ (which in classical invariant theory was usually assumed to be the complex numbers). A representation of $${\displaystyle G}$$ in $${\displaystyle V}$$ is a group homomorphism $${\displaystyle \pi :G\to GL(V)}$$, which induces a group action of $${\displaystyle G}$$ on $${\displaystyle V}$$. If $${\displaystyle k[V]}$$ is the space of polynomial functions on $${\displaystyle V}$$, then the group action of $${\displaystyle G}$$ on $${\displaystyle V}$$ produces an action on $${\displaystyle k[V]}$$ by the following formula:
$${\displaystyle (g\cdot f)(x):=f(g^{-1}(x))\qquad \forall x\in V,g\in G,f\in k[V].}$$"
I did not quite understand the group action at the end. I mean, we need to prove $$((gh)\cdot f)(x)=g\cdot (h\cdot f)(x)\qquad \forall x\in V,g,h\in G,f\in k[V]$$
I have two ways to understand the right hand side:
1) $$g\cdot (h\cdot f)(x)=g\cdot f(h^{-1}(x))=f(g^{-1}h^{-1}x)$$
2) $$g\cdot (h\cdot f)(x)= (h\cdot f)(g^{-1}x)=f(h^{-1}g^{-1}x)$$
Of course, to make this a group action, the second way is correct. However, I want to ask how can one just look at $$g\cdot (h\cdot f)(x)$$ and tell which way is correct? In fact, I think the first way is more rational as we need to compute $$h\cdot f$$ first.
$$g \cdot (h \cdot f) (x)$$ is by definition the evaluation of $$g \cdot (h \cdot f)$$ on $$x$$ .
This function is of the form $$g \cdot m$$ for $$m = h \cdot f$$, thus by definition it evaluation on $$x$$ is $$m(g^{-1}x)$$ .
This is the evaluation of $$h \cdot f$$ on $$g^{-1} x$$ which is by definition $$f(h^{-1} (g^{-1}x)) =f (h^{-1}g^{-1}x)$$ .
If you denote $$f':=h.f$$ , then your asking what is $$g.f'$$. So lets see what it does to $$x$$, for convenience denote $$x'=g^{-1}.x$$
Now: $$g.f'(x)=f'(g^{-1}.x)=f'(x')=f(h^{-1}.x')=f(h^{-1}g^{-1}.x)$$
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2022-01-26 23:17:19
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https://bv.fapesp.br/en/bolsas/172429/conditions-for-speciation-in-structured-populations/
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Start date
Betweenand
# Conditions for speciation in structured populations
Grant number: 16/25271-5 Support type: Scholarships in Brazil - Master Effective date (Start): July 01, 2017 Effective date (End): June 30, 2019 Field of knowledge: Physical Sciences and Mathematics - Physics - General Physics Cooperation agreement: Coordination of Improvement of Higher Education Personnel (CAPES) Principal researcher: Marcus Aloizio Martinez de Aguiar Grantee: Gabriella Dantas Franco Home Institution: Instituto de Física Gleb Wataghin (IFGW). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil Associated scholarship(s): 18/01896-1 - Critical mutation rates in structured populations, BE.EP.MS Abstract Understanding the different mechanisms that lead to speciation is still an open question and has motivated the formulation of several theoretical models. The Moran model is a classic evolutionary model that represents a finite population where individuals die and are replaced by offspring of other individuals. In its simplest version a single biallelic gene subjected to mutations is considered and both births and deaths are random. The allelic frequencies are determined by the mutation rate and the critical value that determines the transition between the regimes of low and high diversity is $\mu_c = 1/2N$ where $N$ is the population size. More complex versions of the model, with more genes, sexual assortative mating and spatially structured populations allow for the study of speciation. The limit of infinitely many genes is particularly important and can be described with the Derrida-Higgs theory.In this proposal we will study the transition between the regimes of high and low diversity in structured populations using ring networks. In this description each individual is represented by a node in the network and each node has exactly the same number $k$ of neighbors, that are the possible mates for reproduction. Initially we will characterize the transition and study the behavior of the critical mutation as a function of the number of neighbors, mu_c(k) in the case of a single gene. Next we will consider the case of multiple genes with assortative mating to determine the maximum number of neighbors k_{max} and the minimum mutation mu_{min}(k) for speciation, comparing this value with mu_c(k). We will study, in particular, the limit of infinitely many genes with an extension of the Derrida-Higgs model. (AU)
News published in Agência FAPESP Newsletter about the scholarship: TITULO
Articles published in other media outlets (0 total): More itemsLess items VEICULO: TITULO (DATA) VEICULO: TITULO (DATA)
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2022-11-30 12:58:40
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https://help.febio.org/FEBioStudio/FEBioStudio_1-5-Section-12.3.html
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Version 1.5
$\newcommand{\lyxlock}{}$
Section 12.2: Taking a snapshot Up Chapter 12: Saving Graphics Section 12.4: Camera Control
## 12.3 Recording an animation
FEBio Studio has the capability to record an animation of the current GV. To record an animation, first position and resize the capture frame so that it covers the desired area of the GV that will be captured. Next, select the Record/New menu. This opens a standard file dialog box where you can select a file format and enter the target filename. If the selected file format is an image format (.bmp, .tiff, etc.), the target filename will be the file template from which the actual filenames will be generated. Each frame will be stored in a separate file, where the frame number is appended to the file template.
After you have selected a target file, you are ready for recording. Note that the capture frame will now be locked, so you can no longer move or resize it. If it is visible, it will turn red. The recording will begin in a paused state, allowing you to make some changes to the GV before recording begins.
To start recording, select Record/Start from the menu or press the corresponding shortcut. Now, all the action in the GV will be recorded to the target file. For example, if you press the play button, the GV will loop over all timesteps and each step will be recorded to the file. You can also rotate the GV and this will also be recorded to the file.
To pause the recording, select Record/Pause from the menu. To finally stop the recording, select Record/Stop from the menu. This will close the target file and unlock the capture frame.
Section 12.2: Taking a snapshot Up Chapter 12: Saving Graphics Section 12.4: Camera Control
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2021-09-26 06:48:03
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https://tex.stackexchange.com/questions/487558/splitting-the-denominator-of-fraction-into-more-than-two-lines
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# Splitting the denominator of fraction into more than two lines
I wish to split the denominator of the following fraction into more than 2 lines. For splitting into two lines, splitfraccan be used. But how to split the denominator into three lines?
\documentclass{article}
\usepackage{mathtools}
\begin{document}
$$\label{eq31} G_2(s) = \frac{K_pcT_rs+K_p}{\splitfrac{T_pT_rT_pT_ts^4 + \left(T_gT_pT_t + T_gT_pT_r + T_pT_tT_r + T_gT_rT_t\right)s^3}{ + \left(T_pT_g + T_tT_g + T_rT_g + T_tT_p + T_rT_p + T_rT_t\right)s^2+\left(T_g + T_r + T_t + T_p\right)s + 1 }}$$
\end{document}
• as always on this site, please provide a full minimal example instead of a sniplet like this. That makes it a lot easier for others to have a look at your code – daleif Apr 25 at 12:59
• BTW: I would not use \splitfrac here, I'd just use the aligned environment. Plus those \left/right constructions does nothing here. – daleif Apr 25 at 13:00
• Please see the Minimal example – ShiS Apr 25 at 13:03
• The aforementioned link deals with breaking of equation into two lines only, while I prefer to break it into more than two lines – ShiS Apr 26 at 2:01
I believe that my answer is based on a personal and aesthetic taste. I prefer to split your denominator with a matrix or an array. I have also put geometry package to have more margin in the page. I think that by minimizing the margins, option that I do not recommend since the printing of the page could crop the formulas, all your formula could be built in a single line.
Here into my MWE I have split the denominator into three lines with matrix:
\documentclass{article}
$$\label{2} G_2(s) =\frac{K_pcT_rs+K_p}{\begin{matrix}T_pT_rT_pT_ts^4 +(T_gT_pT_t + T_gT_pT_r + T_pT_tT_r + T_gT_rT_t)s^3 &\\+(T_pT_g + T_tT_g + T_rT_g + T_tT_p + T_rT_p + T_rT_t)s^2&\\+(T_g + T_r + T_t + T_p)s + 1\end{matrix}}$$
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2019-10-20 00:26:16
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http://biblioteca.universia.net/html_bura/verColeccion/params/id/17187.html
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## Recursos de colección
1. #### Corrigendum On the class groups of pure function fields
Ichimura, Humio
2. #### Corrigendum On the class groups of pure function fields
Ichimura, Humio
Horie, Taro
Horie, Taro
5. #### A Thermodynamic formalism for one dimensional cellular automata
Namiki, Takao
6. #### A Thermodynamic formalism for one dimensional cellular automata
Namiki, Takao
7. #### The Bergman kernel on weakly pseudoconvex tube domains in $\mathbf {C}^2$
Kamimoto, Joe
8. #### The Bergman kernel on weakly pseudoconvex tube domains in $\mathbf {C}^2$
Kamimoto, Joe
9. #### Bernstein degree of singular unitary highest weight representations of the metaplectic group
Nishiyama, Kyo; Ochiai, Hiroyuki
10. #### Bernstein degree of singular unitary highest weight representations of the metaplectic group
Nishiyama, Kyo; Ochiai, Hiroyuki
11. #### On the homology of Torelli groups and Torelli spaces
Akita, Toshiyuki
12. #### On the homology of Torelli groups and Torelli spaces
Akita, Toshiyuki
13. #### On boundedness of a function on a Zalcman domain
Kobayashi, Yasuyuki
We consider boundedness of a function defined by an infinite product which is used to study a uniqueness theorem on a plane domain and the point separation problem of a two-sheeted covering Riemann surface. We show that there is such an infinite product that it converges but the function defined by it is not bounded on arbitrary Zalcman domain.
14. #### On boundedness of a function on a Zalcman domain
Kobayashi, Yasuyuki
We consider boundedness of a function defined by an infinite product which is used to study a uniqueness theorem on a plane domain and the point separation problem of a two-sheeted covering Riemann surface. We show that there is such an infinite product that it converges but the function defined by it is not bounded on arbitrary Zalcman domain.
15. #### On an infinite convolution product of measures
Uchida, Motoo
We prove that infinite convolution products of complex probability measures with bounded total variation converge to a hyperfunction under a weak assumption on supports.
16. #### On an infinite convolution product of measures
Uchida, Motoo
We prove that infinite convolution products of complex probability measures with bounded total variation converge to a hyperfunction under a weak assumption on supports.
17. #### Homotopy groups of the homogeneous spaces $F_4/G_2$, $F_4/\mathrm {Spin}(9)$ and $E_6/F_4$
Hirato, Yoshihiro; Kachi, Hideyuki; Nimura, Mamoru
In this paper we calculate 2-primary components of homotopy groups of the homogeneous spaces $F_4/G_2$, $F_4/\mathrm{Spin}(9)$ and $E_6/F_4$.
18. #### Homotopy groups of the homogeneous spaces $F_4/G_2$, $F_4/\mathrm {Spin}(9)$ and $E_6/F_4$
Hirato, Yoshihiro; Kachi, Hideyuki; Nimura, Mamoru
In this paper we calculate 2-primary components of homotopy groups of the homogeneous spaces $F_4/G_2$, $F_4/\mathrm{Spin}(9)$ and $E_6/F_4$.
19. #### Refined Hölder's inequality for measurable functions
Kwon, Ern Gun; Shon, Kwang Ho
Let $\nu$ be a positive measure on a space $Y$ with $\nu(Y) \neq 0$ and let $f_j$ ($j = 1, 2, \dots, n$) be positive $\nu$-integrable functions on $Y$. For some positive real numbers $\alpha_j$ ($j = 1, 2, \dots, n$), $\beta_j$ ($j= 1, 2, \dots, k < n$) and a measurable subset $Y_1$ of $Y$, we have some inequalities. From these results, we refine Hölder's inequality.
20. #### Refined Hölder's inequality for measurable functions
Kwon, Ern Gun; Shon, Kwang Ho
Let $\nu$ be a positive measure on a space $Y$ with $\nu(Y) \neq 0$ and let $f_j$ ($j = 1, 2, \dots, n$) be positive $\nu$-integrable functions on $Y$. For some positive real numbers $\alpha_j$ ($j = 1, 2, \dots, n$), $\beta_j$ ($j= 1, 2, \dots, k < n$) and a measurable subset $Y_1$ of $Y$, we have some inequalities. From these results, we refine Hölder's inequality.
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2017-11-25 02:02:52
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https://www.physicsforums.com/threads/proof-of-disjoint-sets.634942/
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# Proof of disjoint sets
Hi. I need some help with a proof.
The question says:
Let P be a probability function. Prove that for any finite collection of sets, the sequence A1,A2,...,An of pairwise disjoint sets, P(Union from i=1 to n of Ai)=Ʃ from i=1 to n of Ai
I think there must a mistake in question. My guess is it is supposed to say P(Union from i=1 to n of Ai)=Ʃ from i=1 to n of P(Ai)?
There is a hint with the question:
If A1,A2,...,An are pairwise disjoints on the sample space Ω, then P(Union from i=1 to ∞ Ai) = Ʃ from i=1 to ∞ Ai
If A and B are disjoints defined on Ω then P(A U B) = P(A)+ P(B)
Again, I think there must be a mistake in the hint...P(Union from i=1 to ∞ Ai) = Ʃ from i=1 to ∞ P(Ai)???
My question is are there mistakes and can someone get me started in the right direction (Assuming the question has a mistake, then the book says the proof is a straight forward induction argument with P(A U B) = P(A) + P(B), if A and B are mutually exclusive events over S being the starting point)
Related Precalculus Mathematics Homework Help News on Phys.org
LCKurtz
Homework Helper
Gold Member
Hi. I need some help with a proof.
The question says:
Let P be a probability function. Prove that for any finite collection of sets, the sequence A1,A2,...,An of pairwise disjoint sets, P(Union from i=1 to n of Ai)=Ʃ from i=1 to n of Ai
I think there must a mistake in question. My guess is it is supposed to say P(Union from i=1 to n of Ai)=Ʃ from i=1 to n of P(Ai)?
Yes, that is for sure.
There is a hint with the question:
If A1,A2,...,An are pairwise disjoints on the sample space Ω, then P(Union from i=1 to ∞ Ai) = Ʃ from i=1 to ∞ Ai
If A and B are disjoints defined on Ω then P(A U B) = P(A)+ P(B)
Again, I think there must be a mistake in the hint...P(Union from i=1 to ∞ Ai) = Ʃ from i=1 to ∞ P(Ai)???
Yes again.
My question is are there mistakes and can someone get me started in the right direction (Assuming the question has a mistake, then the book says the proof is a straight forward induction argument with P(A U B) = P(A) + P(B), if A and B are mutually exclusive events over S being the starting point)
Theorem: If ##\{A_i\}## are n disjoint sets then $$P(\cup_{i=1}^n A_i) =\sum_{i=1}^n P(A_i)$$If ##n=2## you have your given starting case. Now suppose it is true for ##n=k##, so you know $$P(\cup_{i=1}^k A_i) =\sum_{i=1}^k P(A_i)$$Now you have to show it works for ##k+1## using that it works for ##k## sets. Think about grouping things.
Glad to confirm those are mistakes.
I wish I could follow you past n=2.
n=2 makes sense to me if you have P(A1 U A2)= P(A1) + P(A2)...that's from the definition in the hint
n=k is where you lose me...how do you jump to P(Union from i=1 to k of Ai)=P(Ʃ from i=1 to k of Ai) ?
Grouping? Not sure I follow...sorry I get a little confused in proofs, but thanks for your help so far.
LCKurtz
Homework Helper
Gold Member
Glad to confirm those are mistakes.
I wish I could follow you past n=2.
n=2 makes sense to me if you have P(A1 U A2)= P(A1) + P(A2)...that's from the definition in the hint
n=k is where you lose me...how do you jump to P(Union from i=1 to k of Ai)=P(Ʃ from i=1 to k of Ai) ?
Grouping? Not sure I follow...sorry I get a little confused in proofs, but thanks for your help so far.
Well, for example, if you had 5 objects, you could group into groups of 2 and 3 objects. Find a way to group ##k+1## into smaller groups that you can work with.
Mark44
Mentor
Glad to confirm those are mistakes.
I wish I could follow you past n=2.
n=2 makes sense to me if you have P(A1 U A2)= P(A1) + P(A2)...that's from the definition in the hint
n=k is where you lose me...how do you jump to P(Union from i=1 to k of Ai)=P(Ʃ from i=1 to k of Ai) ?
You get there by assuming that it is true. This is standard practice for induction proofs.
The next step is showing (proving) that $$P(\cup_{i = 1}^{k+1}A_i) = \sum_{i = 1}^{k+1}A_i$$
To do this, use the statement that you are assuming to be true.
Grouping? Not sure I follow...sorry I get a little confused in proofs, but thanks for your help so far.
LCKurtz
Homework Helper
Gold Member
Here's another hint. If you had three sets and only have the theorem for two you could write ##A_1\cup A_2 \cup A_3 = (A_1\cup A_2)\cup A_3## and use the theorem for two sets twice.
I think I am starting to follow, it's almost like recursive calls to the 2 set case....not sure how I write that in correct notation.
P(U for i=1 to k of Ai) + P(Ak+1)=P(Ʃ for i=1 to k of Ai) + P(Ak+1)
Am I going in the right direction?
LCKurtz
Homework Helper
Gold Member
I think I am starting to follow, it's almost like recursive calls to the 2 set case....not sure how I write that in correct notation.
P(U for i=1 to k ofAi) + P(Ak+1)=P(Ʃ for i=1 to k of P(Ai)) + P(Ak+1)
Am I going in the right direction?
You need to start with$$P(\cup_{i=1}^{k+1}A_i)=$$Now group the ##A_i## appropriately with unions and then use the induction hypothesis. And notice you are making the same error as in the original problem of leaving off the P, noted in red.
Last edited:
P(∪ k+1 i=1 A i )= P(Union for i=1 to k of Ai U A k+1) =
Ʃ(for i=1 to k of P(Ai)) + P(Ak+1)
due to the fact that P(A U B) = P(A) + P(B)... in the case above I'm just letting A=all the sets from 1 to k and B=k+1...so the probability of the union these two "sets" together lets you add their individual probability?
Last edited:
LCKurtz
Homework Helper
Gold Member
P(∪ k+1 i=1 A i )= P(Union for i=1 to k of Ai) U A k+1) = P(Union for i=1 to k of Ai)+ P(Ak+1)=
Ʃ(for i=1 to k of P(Ai)) + P(Ak+1)
due to the fact that P(A U B) = P(A) + P(B)... in the case above I'm just letting A=all the sets from 1 to k and B=k+1...so the probability of the union these two "sets" together lets you add their individual probability?
You have it, but I have suggested an intervening extra step to make it clearer. That way you use the theorem for 2 sets and for k sets at different steps instead of all at once.
You, sir, are a patient teacher to whom I owe many thanks.
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2021-01-27 10:26:55
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http://www.fightfinance.com/?q=14,368,451,497,626,660,670,709,886,892
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# Fight Finance
#### CoursesTagsRandomAllRecentScores
For a price of $100, Andrea will sell you a 2 year bond paying annual coupons of 10% pa. The face value of the bond is$100. Other bonds with the same risk, maturity and coupon characteristics trade at a yield of 6% pa.
Would you like to the bond or politely ?
A method commonly seen in textbooks for calculating a levered firm's free cash flow (FFCF, or CFFA) is the following:
\begin{aligned} FFCF &= (Rev - COGS - Depr - FC - IntExp)(1-t_c) + \\ &\space\space\space+ Depr - CapEx -\Delta NWC + IntExp(1-t_c) \\ \end{aligned}
Does this annual FFCF or the annual interest tax shield?
The first payment of a constant perpetual annual cash flow is received at time 5. Let this cash flow be $C_5$ and the required return be $r$.
So there will be equal annual cash flows at time 5, 6, 7 and so on forever, and all of the cash flows will be equal so $C_5 = C_6 = C_7 = ...$
When the perpetuity formula is used to value this stream of cash flows, it will give a value (V) at time:
A stock will pay you a dividend of $10 tonight if you buy it today. Thereafter the annual dividend is expected to grow by 5% pa, so the next dividend after the$10 one tonight will be $10.50 in one year, then in two years it will be$11.025 and so on. The stock's required return is 10% pa.
What is the stock price today and what do you expect the stock price to be tomorrow, approximately?
The Australian cash rate is expected to be 2% pa over the next one year, while the Japanese cash rate is expected to be 0% pa, both given as nominal effective annual rates. The current exchange rate is 100 JPY per AUD.
What is the implied 1 year forward foreign exchange rate?
How much more can you borrow using an interest-only loan compared to a 25-year fully amortising loan if interest rates are 6% pa compounding per month and are not expected to change? If it makes it easier, assume that you can afford to pay \$2,000 per month on either loan. Express your answer as a proportional increase using the following formula:
$$\text{Proportional Increase} = \dfrac{V_\text{0,interest only}}{V_\text{0,fully amortising}} - 1$$
A company can invest funds in a five year project at LIBOR plus 50 basis points pa. The five-year swap rate is 4% pa. What fixed rate of interest can the company earn over the next five years by using the swap?
Which of the following interest rate quotes is NOT equivalent to a 10% effective annual rate of return? Assume that each year has 12 months, each month has 30 days, each day has 24 hours, each hour has 60 minutes and each minute has 60 seconds. APR stands for Annualised Percentage Rate.
A British man wants to calculate how many British pounds (GBP) he needs to buy a 1 million euro (EUR) apartment in Germany. The exchange rate is 1.42 USD per GBP and 1.23 USD per EUR. What is the EUR 1 million equivalent to in GBP?
The Chinese central bank has the largest amount of foreign currency reserves.
What could the large amounts of foreign exchange reserves held by the Chinese government be used for in a currency crisis? China's currency is called the Renminbi (RMB) or Yuan (CNY). In a Chinese currency crisis the Chinese government is likely to use its FX reserves to:
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2019-08-18 03:43:04
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https://community.wolfram.com/groups/-/m/t/2328894
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# Calculating a product that is a function of NIntegrate?
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Hello everyone,I am trying to calculate a Product, P1, that is a function of NIntegrate. This product is performed over j terms, where j is the number of elements in the list entitled Times1. a = 3/100; b = d = 1; c = 1/2; k = HMax = 2000; Times1 = {{100, 200}, {300, 400}}; System1 = ParametricNDSolve[{X'[t] == X[t]*b - a*X[t]^2/k,X[0] == Exp[-a*t0]*k}, {X}, {t, 0,1000}, {g, t0}]; P1[g_, t0_?NumericQ] := Product[NIntegrate[ b*(1 - c*g)*(X[g, t0][t] /. System1) /HMax - d,{t, Indexed[Indexed[Times1, j], 1], 1000},Method -> {Automatic, "SymbolicProcessing" -> 0}], {j, 1, Length[Times1]}] P1[0.6, 80] When I try this, I receive an error stating that “t = Indexed[{{100.,200.},{300.,400.}},{j,1}] is not a valid limit of integration.” Is there a way to avoid this error? Any help would be greatly appreciated! Attachments:
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2021-09-25 17:34:12
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https://andreas.hartel.me/2018/09/06/what-is-the-birch-and-swinnerton-dyer-conjecture-part-1/
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# What is the Birch and Swinnerton-Dyer Conjecture – Part 1
Yesterday, I watched a talk by Manjul Bhargava called What is the Birch and Swinnerton-Dyer Conjecture?. I think he did a really good job in explaining the conjecture on a level that can be understood by a non-expert yet mathematically inclined person.
In the caption, there is a link to these lecture notes (not written by Bhargava himself). I thought it might be interesting to go through these lecture notes and explore the territory of topics mentioned in the talk (especially in the introductory part of the talk). Therefore, I will walk through the lecture notes in this blogpost (and one or two following posts) and try to look into the mentioned theorems in a bit more detail.
The first two paragraphs are:
Manjul Bhargava spoke about elliptic curves and the Birch and Swinnerton-Dyer
Conjecture (BSD), one of the most fundamental probelms in number theory. He
explained its origin and its statement in elementary terms, as well as summarising
progress towards a proof, including results that were emerging in the workshop taking place during week of the Research Conference.
A central theme of number theory is the search for rational solutions of polynomial
equations. For a polynomial in one variable with integer coefficients, the problem is
easily solved by using the Rational Root Theorem.
That sound interesting. Let’s see what the Rational Root Theorem tells us. I’m taking the following material from Jared Weinstein’s Number Theory course.
Rational Root Theorem: Suppose the polynomial $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0$ has coefficients $a_i \in \mathbb{Z}$. If $p/q$ is a fraction in lowest terms which is a root of $f(x)$, then $q|a_n$ and $p|a_0$.
Proof: The fact that $p/q$ is a root of $f(x)$ means that $f(p/q) = 0$. After
clearing away denominators, this becomes $a_n p^n + a_{n-1} p^{n-1} q + \ldots + a_1 p q^{n-1} + a_0 q^n = 0$. Now, we move the last term to the right and factor out $p$: $p(a_n p^{n-1} + a_{n-1} p^{n-2} q + \ldots + a_1 q^{n-1}) = -a_0 q^n$.
Since the left hand side is obviously divisible by $p$ the right hand side must be as well, so $p|a_0 q^n$. But since $p$ is coprime with $q$ but divides a product containing $q$, it must divide $a_0$ (by the generalized form of Euclid’s Lemma).
The proof that $q|a_n$ is similar. $\Box$
The main topic of Bhargava’s talk is the question whether certain polynomial equations have rational roots. Using the Rational Root Theorem, we can now answer this question for, e.g., the polynomial $x^2 - 2 = 0$. We know, by taking the square root of this equation that the solution over the real numbers is $\pm \sqrt{2}$. But with the RRT we can now show independently that there cannot be a rational solution. Observe that a rational solution $p/q$ would have to fullfill $p|-2$ and $q|1$. Therefore, the only possible solutions would be $\pm 1, \pm 2$. However, by plugging them into the equation, we see immediately that they are no solutions.
In the next post we will look at the next paragraph of the lecture notes that discuss polynomials of degree 2 in 2 variables.
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2023-03-26 12:57:57
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https://gmatclub.com/forum/gmat-diagnostic-test-question-79371.html
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# GMAT Diagnostic Test Question 38
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GMAT Diagnostic Test Question 38 [#permalink]
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GMAT Diagnostic Test Question 38
Field: probability
Difficulty: 750
At a blind taste competition a contestant is offered 3 cups of each of the 3 samples of tea in a random arrangement of 9 marked cups. If each contestant tastes 4 different cups of tea, what is the probability that a contestant does not taste all of the samples?
A. $$\frac{1}{12}$$
B. $$\frac{5}{14}$$
C. $$\frac{4}{9}$$
D. $$\frac{1}{2}$$
E. $$\frac{2}{3}$$
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Last edited by bb on 29 Sep 2013, 21:47, edited 2 times in total.
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Explanation:
In order to answer the question we need to find the overall number of outcomes and the number of favourable outcomes. The favourable outcome in this case is the one when a contestant tastes only 2 kinds of tea out of 3 kinds available. If there are 3 cups of every kind of tea, the number of favourable outcomes is calculated in the following way:
$$C_6^4 * 3 = 3 * \frac{6!}{4!*2!} = 45$$
We had to multiply by 3 because there are 3 ways the two kinds of tea could be selected from 3 available kinds.
The overall number of outcomes is equal to
$$C_9^4 = \frac{9!}{5!*4!} = \frac{9*8*7*6}{4*3*2} = 126$$
So, the probability can be found:
$$P = \frac{45}{126} = \frac{5}{14}$$
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20 Jul 2009, 08:46
dzyubam wrote:
Explanation:
Rating:
In order to answer the question we need to find the overall number of outcomes and the number of favourable outcomes. The favourable outcome in this case is the one when a contestant tastes only 2 kinds of tea out of 3 kinds available. If there are 3 cups of every kind of tea, the number of favourable outcomes is calculated in the following way:
$$C_6^4 * 3 = 3 * \frac{6!}{4!*2!} = 45$$
We had to multiply by 3 because there are 3 ways the two kinds of tea could be selected from 3 available kinds.
$$P = \frac{45}{126} = \frac{5}{14}$$
I don't really understand how you got 45. I think you might be double counting since you are basically choosing 2 out of 6. However, that still leaves the possibility that the 3rd choice might make a complete set of 3... can you clarify why you are doing this again?
Basically, after computing the total number of combinations, I counted the number of "winning" combinations. If you are choosing 4 out of 9 from 3 sets of 3 and you don't want there to be a complete set, the only possibly combinations are either 3-1-0 or 2-2-0. If you choose 3 of 1 sample, you can either choose 3-0-1 or 3-1-0. Since there are 3 samples in all, that makes 3*2 = 6 combinations.
If choose 2 of one sample and 2 of another, then the only combinations are 2-2-0, 2-0-2, and 0-2-2.
The total number of combinations is then 9, so the probability is 9/126 = 1/14.
Can someone tell me if/where I went wrong?
Also, the question needs to be more clear. You need to state that its 9 marked cups of 3 samples each, otherwise you have to assume that the samples are equally divided.
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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20 Jul 2009, 12:32
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Hello bipolarbear,
you are basically missing combinations:
3 0 1 x 3 = 3
3 1 0 x 3 = 3
0 1 3 x 3 = 3
1 0 3 x 3 = 3
0 3 1 x 3 = 3
1 3 0 x 3 = 3 (I multiply by 3 because in the sample with 1 cup you have 3 alternatives)
2 0 2 x 9 = 9
0 2 2 x 9 = 9
2 2 0 x 9 = 9 (you have 3 different 2-cup combination in each 2-cup sample, so 3x3)
TOTAL $$\frac{45}{126} = \frac{5}{14}$$
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20 Jul 2009, 12:38
bipolarbear wrote:
Also, the question needs to be more clear. You need to state that its 9 marked cups of 3 samples each, otherwise you have to assume that the samples are equally divided.
Thank you!
Very good point - revising right now.
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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20 Jul 2009, 12:43
saruba wrote:
(you have 3 different 2-cup combination in each 2-cup sample, so 3x3)
TOTAL $$\frac{45}{126} = \frac{5}{14}$$
oh darn, you are clever. i stand corrected.
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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20 Jul 2009, 21:35
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bb wrote:
GMAT Diagnostic Test Question 39
Field: probability
Difficulty: 750
Rating:
At a blind taste competition a contestant is offered 3 cups of each of the 3 samples of tea in a random arrangement of 9 marked cups. If each contestant tastes 4 different cups of tea, what is the probability that a contestant does not taste all of the samples?
A. $$\frac{1}{12}$$
B. $$\frac{5}{14}$$
C. $$\frac{4}{9}$$
D. $$\frac{1}{2}$$
E. $$\frac{2}{3}$$
The highlighted part is not very clear however it is a basic combination problem.
1. 3 types of tea each has 3 cups totaling 9 cups.
2. Select 4 cups of tea of 2 different types out of 9 cups.
The number of ways 2 types of teas can be selected from 3 types of tea = 3c2 = 3 ways
The number of ways 4 cups of tea can be selected from only 2 types of tea = 6c2 = 15 ways
The number of ways 4 cups of tea can be selected from 2 types of tea in 3 different ways = 3x15 = 45 ways
The number of ways 4 cups of tea can be selected from all 3 types of tea = 9c4 = 9x8x7x6x5!/(5!4!) = 146 ways
The prob that 4 cups of tea can be selected only from 2 types of tea = 45/126 = 15/42 ways
I do not see any complication with the OA and OE.
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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probability that a contestant does not taste all of the samples = 1 - probability that a contestant tastes all of the samples
now, lets calculate probability that a contestant tastes all of the samples.
there are 3 cases for this, as there are three samples i. e. 2,1,1(two cups of 1st sample, 1 cup of 2nd sample, 1 cup of 3rd sample), 1,2,1 and 1,1,2.
probability that a contestant tastes all of the samples = (3C2 * 3C1 * 3C1 + 3C1 * 3C2 * 3C1 + 3C1 * 3C1 * 3C2)/ 9C4
= (3*27)/(63*2)
= 9/14
therefore,
probability that a contestant does not taste all of the samples = 1 - 9/14
= 5/14
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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29 Jul 2009, 12:45
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vishalgupta wrote:
probability that a contestant does not taste all of the samples = 1 - probability that a contestant tastes all of the samples
now, lets calculate probability that a contestant tastes all of the samples.
there are 3 cases for this, as there are three samples i. e. 2,1,1(two cups of 1st sample, 1 cup of 2nd sample, 1 cup of 3rd sample), 1,2,1 and 1,1,2.
probability that a contestant tastes all of the samples = (3C2 * 3C1 * 3C1 + 3C1 * 3C2 * 3C1 + 3C1 * 3C1 * 3C2)/ 9C4
= (3*27)/(63*2)
= 9/14
therefore,
probability that a contestant does not taste all of the samples = 1 - 9/14
= 5/14
Great solution vishal
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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27 Aug 2009, 22:16
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Pls suggest if this approach is acceptable...
There are total 9 cups... In order to NOT to taste from one type, the contestant has to choose from remaining 6..
i.e. $$\frac{6}{9}$$.. similarly for the next cup, it choice would be $$\frac{5}{8}$$ and so on....
4 cups implies $$\frac{6*5*4*3}{9*8*7*6}$$.... which is $$\frac{5}{42}$$..
since there are three types of tea available $$\frac{3*5}{42}$$ which equals $$\frac{5}{14}$$
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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28 Aug 2009, 01:53
saurabhmukim wrote:
Pls suggest if this approach is acceptable...
There are total 9 cups... In order to NOT to taste from one type, the contestant has to choose from remaining 6..
i.e. $$\frac{6}{9}$$.. similarly for the next cup, it choice would be $$\frac{5}{8}$$ and so on....
4 cups implies $$\frac{6*5*4*3}{9*8*7*6}$$.... which is $$\frac{5}{42}$$..
since there are three types of tea available $$\frac{3*5}{42}$$ which equals $$\frac{5}{14}$$
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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22 Dec 2009, 04:43
saurabhmukim wrote:
Pls suggest if this approach is acceptable...
There are total 9 cups... In order to NOT to taste from one type, the contestant has to choose from remaining 6..
i.e. $$\frac{6}{9}$$.. similarly for the next cup, it choice would be $$\frac{5}{8}$$ and so on....
4 cups implies $$\frac{6*5*4*3}{9*8*7*6}$$.... which is $$\frac{5}{42}$$..
since there are three types of tea available $$\frac{3*5}{42}$$ which equals $$\frac{5}{14}$$
Great approach! +1
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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25 Dec 2009, 13:07
saurabhmukim wrote:
Pls suggest if this approach is acceptable...
There are total 9 cups... In order to NOT to taste from one type, the contestant has to choose from remaining 6..
i.e. $$\frac{6}{9}$$.. similarly for the next cup, it choice would be $$\frac{5}{8}$$ and so on....
4 cups implies $$\frac{6*5*4*3}{9*8*7*6}$$.... which is $$\frac{5}{42}$$..
since there are three types of tea available $$\frac{3*5}{42}$$ which equals $$\frac{5}{14}$$
very excellent approach +1
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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17 Oct 2010, 12:24
Quote:
The number of ways 4 cups of tea can be selected from only 2 types of tea = 6c2 = 15 ways
Hi,
can someone pls explain how this is 6c2, as there is no way to distinguish between the cups of the same sample.
The post of gmatclub math book (math-combinatorics-87345.html) clearly says that:
"Number of ways to pick 1 or more objects from n identical objects = n"
In this case, there are 3 ways a given cup can be picked up from one set of 3 cups and there are 2 ways to pick up that set of cups itself..i.e. there are 2 * 3 = 6 ways of picking up 4 cups.
Hence 3c2 * 6/9c4 = 1/7 should be the answer..?
Pls explain. Eagerly waiting.
Thanks
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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19 Oct 2010, 15:23
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gmatbestshot wrote:
Quote:
The number of ways 4 cups of tea can be selected from only 2 types of tea = 6c2 = 15 ways
Hi,
can someone pls explain how this is 6c2, as there is no way to distinguish between the cups of the same sample.
The post of gmatclub math book (math-combinatorics-87345.html) clearly says that:
"Number of ways to pick 1 or more objects from n identical objects = n"
In this case, there are 3 ways a given cup can be picked up from one set of 3 cups and there are 2 ways to pick up that set of cups itself..i.e. there are 2 * 3 = 6 ways of picking up 4 cups.
Hence 3c2 * 6/9c4 = 1/7 should be the answer..?
Pls explain. Eagerly waiting.
Thanks
This question was posted in PS subforum as well, so below is my solutions from there. Hope these solutions will help to clear your doubts.
At a blind taste competition a contestant is offered 3 cups of each of the 3 samples of tea in a random arrangement of 9 marked cups. If each contestant tastes 4 different cups of tea, what is the probability that a contestant does not taste all of the samples?
# $$\frac{1}{12}$$
# $$\frac{5}{14}$$
# $$\frac{4}{9}$$
# $$\frac{1}{2}$$
# $$\frac{2}{3}$$
"The probability that a contestant does not taste all of the samples" means that contestant tastes only 2 samples of tea (one sample is not possible as contestant tastes 4 cups>3 of each kind).
$$\frac{C^2_3*C^4_6}{C^4_9}=\frac{5}{14}$$.
$$C^2_3$$ - # of ways to choose which 2 samples will be tasted;
$$C^4_6$$ - # of ways to choose 4 cups out of 6 cups of two samples (2 samples*3 cups each = 6 cups);
$$C^4_9$$ - total # of ways to choose 4 cups out of 9.
Another way:
Calculate the probability of opposite event and subtract this value from 1.
Opposite event is that contestant will taste ALL 3 samples, so contestant should taste 2 cups of one sample and 1 cup from each of 2 other samples (2-1-1).
$$C^1_3$$ - # of ways to choose the sample which will provide with 2 cups;
$$C^2_3$$ - # of ways to chose these 2 cups from the chosen sample;
$$C^1_3$$ - # of ways to chose 1 cup out of 3 from second sample;
$$C^1_3$$ - # of ways to chose 1 cup out of 3 from third sample;
$$C^4_9$$ - total # of ways to choose 4 cups out of 9.
$$P=1-\frac{C^1_3*C^2_3*C^1_3*C^1_3}{C^4_9}=1-\frac{9}{14}=\frac{5}{14}$$.
Hope it's clear.
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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07 Feb 2012, 00:14
To calculate the number of favourable outcomes let us consider two cases .
1.three cups of a single sample and one cup from from the other two samples .
Number of ways =3*6c1=18.[ 3-->denotes number of ways in which three cups from a single sample can be selected ]
2.two cups each from any two samples .
Number of ways =3*3c2*3c2=27.[3-->denotes the number of ways of selecting 2 samples out of the 3 samples ]
Total number of favourable cases =18+27=45.
Total number of cases =9c4=126
Required propability=45/126=5/14.
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+1 if you like my explanation .Thanks
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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13 Jul 2012, 21:04
Firstly, we have to eliminate the probability of taking all the samples after 4 tastes. There are 3 ways to do that.
1/ He took 3 different samples in the 3 first tastes, so we don't care whatever the last time he took.
Probability of this way = 1*6/8*3/7*1=9/28
2/ The 2nd time he took the same sample with the 1st one and the 3rd and 4th time he took the 2 others.
Probability = 1*2/8*6/7*3/6=3/28
3/ He took 2 different samples in the 2 first time; the 3rd time he took the same sample with either the 1st or the 2nd one, and the 4th time he took the other sample.
Probability = 1*6/8*4/7*3/6=6/28
So the answer = 1-(9/28+3/28+6/28) = 10/28 = 5/14
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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22 Aug 2013, 22:47
saurabhmukim wrote:
Pls suggest if this approach is acceptable...
There are total 9 cups... In order to NOT to taste from one type, the contestant has to choose from remaining 6..
i.e. $$\frac{6}{9}$$.. similarly for the next cup, it choice would be $$\frac{5}{8}$$ and so on....
4 cups implies $$\frac{6*5*4*3}{9*8*7*6}$$.... which is $$\frac{5}{42}$$..
since there are three types of tea available $$\frac{3*5}{42}$$ which equals $$\frac{5}{14}$$
Easiest approach of all.
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Re: GMAT Diagnostic Test Question 39 [#permalink]
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07 Oct 2013, 15:26
saurabhmukim wrote:
Pls suggest if this approach is acceptable...
There are total 9 cups... In order to NOT to taste from one type, the contestant has to choose from remaining 6..
i.e. $$\frac{6}{9}$$.. similarly for the next cup, it choice would be $$\frac{5}{8}$$ and so on....
4 cups implies $$\frac{6*5*4*3}{9*8*7*6}$$.... which is $$\frac{5}{42}$$..
since there are three types of tea available $$\frac{3*5}{42}$$ which equals $$\frac{5}{14}$$
Sorry, I can't follow this part, can anyone tell me where the multiplication by 3 comes from?
Quote:
since there are three types of tea available $$\frac{3*5}{42}$$ which equals $$\frac{5}{14}$$
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Posts: 39622
Re: GMAT Diagnostic Test Question 39 [#permalink]
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08 Oct 2013, 00:50
hsarci wrote:
saurabhmukim wrote:
Pls suggest if this approach is acceptable...
There are total 9 cups... In order to NOT to taste from one type, the contestant has to choose from remaining 6..
i.e. $$\frac{6}{9}$$.. similarly for the next cup, it choice would be $$\frac{5}{8}$$ and so on....
4 cups implies $$\frac{6*5*4*3}{9*8*7*6}$$.... which is $$\frac{5}{42}$$..
since there are three types of tea available $$\frac{3*5}{42}$$ which equals $$\frac{5}{14}$$
Sorry, I can't follow this part, can anyone tell me where the multiplication by 3 comes from?
Quote:
since there are three types of tea available $$\frac{3*5}{42}$$ which equals $$\frac{5}{14}$$
The following post might help: gmat-diagnostic-test-question-79371.html#p803095
_________________
Re: GMAT Diagnostic Test Question 39 [#permalink] 08 Oct 2013, 00:50
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# GMAT Diagnostic Test Question 38
Moderator: Bunuel
Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.
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2017-06-24 05:43:29
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https://www.edubuzz.org/saltoun/category/p5-6-7-2018/
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## Equivalent fractions
Yesterday P5/6/7 where doing equivalent fractions, as our maths lesson we learned that…
…Equivalent fractions are where the fractions though different mean the same thing for example 1/2=2/4=4/8 and so on. We looked at a fraction wall(Like the one below). To see how the fractions tie into each other to be equal sizes.
Do you see the partial line straight down the middle of the fraction wall. That is what I mean by link together to make equal sizes of fraction.
BY BRODIE.W.
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2020-03-29 12:55:06
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http://math.stackexchange.com/questions/307171/differentiation-in-3d-of-sin-root-and-fractions-in-one-to-find-the-normal-to
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# Differentiation in 3d of sin root and fractions in one! -> to find the normal to a function
I have to find the normal of this function at a defined point $x$ and $z$, I have done A Level maths but that was some time ago but I don't think it was covered to this level, I am now doing a CS degree. I thought the best way would be to differentiate the function to get the tangent, then get the normal.
I have no idea where to start.
Any ideas on how I should go about it?
$$y = \frac{\sin (\sqrt{x^2+z^2})}{\sqrt{x^2+z^2}}$$
-
Interesting. You asked how to find the normal of a surface in 3D, and then you accepted an answer describing how to find the normal of a curve in 2D. – Rahul Mar 11 '13 at 0:04
I thought it was correct since for example. y = 2x^3 -> dy/dx = 6x. So delta F should have a 3d vector? – Andrew Mar 11 '13 at 15:37
Based on what your saying since the function is sort of a rotation I could bodge it and rotate the 2d normal by the angle is on and get the correct normal. But ideally I am looking for a direct route to that 3d normal. – Andrew Mar 11 '13 at 15:45
is this of any use to me? mathworld.wolfram.com/NormalVector.html – Andrew Mar 11 '13 at 16:13
It's probably not a great idea to ask strangers on the internet for help if you have no way to tell whether the answers you get make sense for your question. If I give you an answer that contradicts the existing one, how will you know which one is right? Maybe rlgordonma misinterpreted and gave an answer to a different problem. Maybe I'm the one misinterpreting your question. You need to go back and check some basic properties that the normal vector you want ought to have, so that you can look at the answers and say, "Thanks, that looks right", or, "That doesn't make sense because..." – Rahul Mar 11 '13 at 16:45
The normal to a curve at a point is perpendicular to the gradient at that point. In your case:
$$f(x,z) = \frac{\sin{\sqrt{x^2+z^2}}}{\sqrt{x^2+z^2}}$$
\begin{align}\nabla f &= \left ( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial z}\right ) \\ &= \left ( \frac{x \cos \left(\sqrt{x^2+z^2}\right)}{x^2+z^2 }-\frac{x \sin \left(\sqrt{x^2+z^2}\right)}{\left(x ^2+z^2\right)^{3/2}}, \frac{z \cos \left(\sqrt{x^2+z^2}\right)}{x^2+z^2 }-\frac{z \sin \left(\sqrt{x^2+z^2}\right)}{\left(x ^2+z^2\right)^{3/2}}\right ) \end{align}
The unit normal to the curve is then
$$\frac{\left ( -\frac{\partial f}{\partial z}, \frac{\partial f}{\partial x}\right )}{\sqrt{\left ( \frac{\partial f}{\partial x} \right )^2+\left ( \frac{\partial f}{\partial z} \right )^2}}$$
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Hold on I'm a little confused by this now. How would I go about getting a directional vector for the normal at a defined point e.g. x=0.5, z=0.5? – Andrew Mar 10 '13 at 22:06
Take the partial derivatives romthe equation defining the gradient, and apply them to the final equation defining the normal. Plug in $x=0.5$, $z=0.5$ into the expressions for the partial derivatives. – Ron Gordon Mar 10 '13 at 22:09
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2014-12-22 22:12:20
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https://tex.stackexchange.com/questions/453320/how-to-generate-a-makefile-from-dtx-keeping-the-tab-width
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# How to generate a Makefile from dtx // keeping the tab width
For a complete documentation I would like to maintain and explain the Makefile within my dtx-file.
Two problems occur:
1. docstrip always adds an extension to the generated file (\file{Makefile}{\from{\jobname.dtx}{make}})
2. The tab in the Makefile is not preserved and got erased (all: [[TAB]]lualatex $(NAME).tex) This is my MWE: %\iffalse %<*internal> \iffalse %</internal> %<*internal> \fi \def\nameofplainTeX{plain} \ifx\fmtname\nameofplainTeX\else \expandafter\begingroup \fi %</internal> %<*install> \input docstrip.tex \keepsilent \askforoverwritefalse \nopreamble\nopostamble \generate{ \file{Makefile}{\from{\jobname.dtx}{make}} } %</install> %<install>\endbatchfile %<*internal> \usedir{source/latex/\jobname} \generate{ \file{\jobname.ins}{\from{\jobname.dtx}{install}} } \ifx\fmtname\nameofplainTeX \expandafter\endbatchfile \else \expandafter\endgroup \fi %</internal> %<*driver> \documentclass{ltxdoc} \begin{document} \DocInput{\jobname.dtx} \end{document} %</driver> %\fi %\iffalse %<*make> %\fi % \section{The Makefile} % \begin{macrocode} NAME = testfile % \end{macrocode} % The default command % \begin{macrocode} all: lualatex$(NAME).tex
% \end{macrocode}
%\iffalse
%</make>
%\fi
\endinput
# EDIT
After the first run in the terminal with lualatex DOCUMENT.dtx I get the makefile. Then I cannot simply run make but have to type in make --makefile=Makefile.tex and get the error *** missing separator. Stop.
• I tend to write raw Makefiles for LaTeX documents – Robin Oct 1 '18 at 13:59
• Sure, me too. But then I have to write all the comments in the makefile. As I wrote, this is purely for documenting reasons. – LukasCB Oct 1 '18 at 14:00
In the dtx code as posted there are no tabs (as the site changes them to spaces) but if I put tabs back in the Makefile section and modify the .ins to look like
\catcode9=12
\generate{
\file{Makefile}{\from{\jobname.dtx}{make}}
Then tabs are written to the Makefile if I use luatex on the .ins. (The resulting file gets called Makefile.tex though. (Extensionless files are tricky in TeX)
So if you replace the spaces before lualatex $(NAME).tex by a tab in the file below then luatex file.dtx will generate file.ins and luatex file.ins will generate a Makefile.tex with tabs (at least on cygwin texlive 2018 luatex) %\iffalse %<*internal> \iffalse %</internal> %<*internal> \fi \def\nameofplainTeX{plain} \ifx\fmtname\nameofplainTeX\else \expandafter\begingroup \fi %</internal> %<*install> \input docstrip.tex \keepsilent \askforoverwritefalse \nopreamble\nopostamble \catcode9=12 \generate{ \file{Makefile}{\from{\jobname.dtx}{make}} } %</install> %<install>\endbatchfile %<*internal> \usedir{source/latex/\jobname} \generate{ \file{\jobname.ins}{\from{\jobname.dtx}{install}} } \ifx\fmtname\nameofplainTeX \expandafter\endbatchfile \else \expandafter\endgroup \fi %</internal> %<*driver> \documentclass{ltxdoc} \begin{document} \DocInput{\jobname.dtx} \end{document} %</driver> %\fi %\iffalse %<*make> %\fi % \section{The Makefile} % \begin{macrocode} NAME = testfile % \end{macrocode} % The default command % \begin{macrocode} all: lualatex$(NAME).tex
% \end{macrocode}
%\iffalse
%</make>
%\fi
\endinput
• ok, I could live with naming it to Makefile.make but where exactly should I put \catcode9=12. When I put it where you are pointing it to I get ! LaTeX Error: Missing \begin{document}.. – LukasCB Oct 1 '18 at 14:07
• @LukasCB I added full example – David Carlisle Oct 1 '18 at 14:17
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2019-11-19 10:28:31
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https://math.stackexchange.com/questions/2798995/existence-and-uniqueness-of-solution-of-ivp-with-separation-of-variables
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Existence and uniqueness of solution of IVP with separation of variables
I know that if $f$ is Lipschitz or $C^1$ then the IVP $x'=f(t,x), x(t_0)=x_0$ has a solution, which is also unique.
Now I'm wondering whether the IVP $y'(x)=h(x)g(y(x)), y(x_0)=y_0$ ($h\in C([x_0-h,x_0+h],\mathbb{R}), g\in C([y_0-\delta,y_0+\delta],\mathbb{R}), g(y_0)\neq 0$) has a solution which is also unique.
Attempt 1: I considered $\Omega:=[x_0-h,x_0+h]\times [y_0-\delta,y_0+\delta]$ and $f\in C(\Omega,R),\ f(x,y):=h(x)g(y(x))$ which would guarantee a unique solution by Peano-Picard theorem if $f$ were Lipschitz or $C^1$ which is not necessarily the case unfortunately.
Attempt 2: I tried by separating variables, obtaining $\int \frac{dy}{g(y(x))}=\int h(x)dx$ but not knowing the expression of $g$ and $h$ I don't see how I can go further.
I'm stuck so I'd appreciate some help.
Thanks.
Your second attempt is correct. Let $\varphi(\cdot)$ be a solution of the IVP. We do not yet know if such a solution exists, but if it exists, it must satisfy: $$\frac{\varphi'(x)}{g(\varphi(x))} = h(x) \quad \text{ for } x \in (\alpha, \beta).$$ Integrate the above from $x_0$ to $x$ to obtain $$\int\limits_{x_0}^{x} \frac{\varphi'(\xi)}{g(\varphi(\xi))} d\xi = \int\limits_{x_0}^{x} h(\xi) \, d\xi,$$ which gives, by integration by substitution $$\tag{*} \int\limits_{y_0}^{\varphi(x)} \frac{d\eta}{g(\eta)} = \int\limits_{x_0}^{x} h(\xi) \, d\xi, \quad \text{ for } x \in (\alpha, \beta).$$ Introduce the notation $$P(y) := \int\limits_{y_0}^{y} \frac{d\eta}{g(\eta)}, \quad H(x) := \int\limits_{x_0}^{x} h(\xi) \, d\xi.$$ In that notation ($*$) takes the form $$\tag{**} P(\varphi(x)) = H(x) \quad \forall{x \in (\alpha, \beta)}.$$ Assume for the sake of simplicity that $g(y_0) < 0$. By continuity, $g(y) < 0$ for $y$ sufficiently close to $y_0$. Therefore, the mapping $$y \mapsto P(y)$$ is strictly decreasing, consequently invertible, on some neighborhood of $y_0$. We can apply its inverse, $P^{-1}$ (not reciprocal!) to ($**$) to obtain $$\tag{***} \varphi(x) = P^{-1}(H(x)) \quad \forall{x \in (\alpha, \beta)}.$$ We have thus obtained both existence and uniqueness (because ($***$) is the only possibility for a solution to the IVP).
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2022-06-28 03:03:52
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https://brilliant.org/discussions/thread/doubtplease-help/
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×
Find the number of digits in $$\large2^{2^{22}}$$?
Note by Naitik Sanghavi
10 months, 1 week ago
Sort by:
1262612 · 10 months, 1 week ago
Actually I got the answer but according to my book answer key I was wrong so....Thanks · 10 months, 1 week ago
@Dev Sharma DevBut how will you find the number of digits without computing $$\large2^{22}$$ if it is asked in exam? · 10 months, 1 week ago
U may assume the no. Of digits of a number be x. Let the number be a^b. So 10^(x-1) = a^b X-1 = blog(a) [log with base 10] X = blog(a) +1 For convenience, X = [blog(a)] +1 So 2^22 has [22log(2)] +1 digits.similarly following can be calculat ed · 10 months, 1 week ago
How to apply here .here it will become 2^22 log(2) then we have to apply log again then do antilog.after doing this I got answer 447 .is it right?????? · 8 months, 3 weeks ago
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2016-10-01 07:01:30
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http://en.wikipedia.org/wiki/Expert_systems
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# Expert system
(Redirected from Expert systems)
A Symbolics Lisp Machine: An Early Platform for Expert Systems. Note the unusual "space cadet keyboard".
In artificial intelligence, an expert system is a computer system that emulates the decision-making ability of a human expert.[1] Expert systems are designed to solve complex problems by reasoning about knowledge, represented primarily as if–then rules rather than through conventional procedural code.[2] The first expert systems were created in the 1970s and then proliferated in the 1980s.[3] Expert systems were among the first truly successful forms of AI software.[4][5][6][7][8]
An expert system is divided into two sub-systems: the inference engine and the knowledge base. The knowledge base represents facts and rules. The inference engine applies the rules to the known facts to deduce new facts. Inference engines can also include explanation and debugging capabilities.[9]
## History
Edward Feigenbaum in a 1977 paper said that the key insight of early expert systems was that "intelligent systems derive their power from the knowledge they possess rather than from the specific formalisms and inference schemes they use" (as paraphrased by Hayes-Roth, et al.) Although, in retrospect, this seems a rather straightforward insight, it was a significant step forward at the time. Until then, research had been focused on attempts to develop very general-purpose problem solvers such as those described by Newell and Simon.[10]
Expert systems were introduced by the Stanford Heuristic Programming Project led by Feigenbaum, who is sometimes referred to as the "father of expert systems". The Stanford researchers tried to identify domains where expertise was highly valued and complex, such as diagnosing infectious diseases (Mycin) and identifying unknown organic molecules (Dendral).
In addition to Feigenbaum key early contributors were Edward Shortliffe, Bruce Buchanan, and Randall Davis. Expert systems were among the first truly successful forms of AI software.[11][5][6][7][8]
Research on expert systems was also active in France. In the US the focus tended to be on rule-based systems, first on systems hard coded on top of LISP programming environments and then on expert system shells developed by vendors such as Intellicorp. In France research focused more on systems developed in Prolog. The advantage of expert system shells was that they were somewhat easier for non-programmers to use. The advantage of Prolog environments was that they weren't focused only on IF-THEN rules. Prolog environments provided a much fuller realization of a complete First Order Logic environment.[12][13]
In the 1980s, expert systems proliferated. Universities offered expert system courses and two thirds of the Fortune 1000 companies applied the technology in daily business activities.[3][14] Interest was international with the Fifth Generation Computer Systems project in Japan and increased research funding in Europe.
In 1981 the first IBM PC was introduced, with the MS-DOS operating system. The imbalance between the relatively powerful chips in the highly affordable PC compared to the much more expensive price of processing power in the Mainframes that dominated the corporate IT world at the time created a whole new type of architecture for corporate computing known as the Client-server model.[15] Calculations and reasoning could be performed at a fraction of the price of a mainframe using a PC. This model also enabled business units to bypass corporate IT departments and directly build their own applications. As a result client server had a tremendous impact on the expert systems market. Expert systems were already outliers in much of the business world, requiring new skills that many IT departments did not have and were not eager to develop. They were a natural fit for new PC-based shells that promised to put application development into the hands of end users and experts. Up until that point the primary development environment for expert systems had been high end Lisp machines from Xerox, Symbolics and Texas Instruments. With the rise of the PC and client server computing vendors such as Intellicorp and Inference Corporation shifted their priorities to developing PC based tools. In addition new vendors often financed by Venture Capital started appearing regularly. These new vendors included Aion Corporation, Neuron Data, Exsys, and many others.[16][17]
In the 1990s and beyond the term "expert system" and the idea of a standalone AI system mostly dropped from the IT lexicon. There are two interpretations of this. One is that "expert systems failed": the IT world moved on because expert systems didn't deliver on their over hyped promise.[18][19] The other is the mirror opposite, that expert systems were simply victims of their success. As IT professionals grasped concepts such as rule engines such tools migrated from standalone tools for the development of special purpose "expert" systems to one more tool that an IT professional has at their disposal.[20] Many of the leading major business application suite vendors such as SAP, Siebel, and Oracle integrated expert system capabilities into their suite of products as a way of specifying business logic. Rule engines are no longer simply for defining the rules an expert would use but for any type of complex, volatile, and critical business logic. They often go hand in hand with business process automation and integration environments.[21][22][23]
## Software architecture
An expert system is an example of a knowledge-based system. Expert systems were the first commercial systems to use a knowledge-based architecture. A knowledge-based system is essentially composed of two sub-systems: the knowledge base and the inference engine.[24]
The knowledge base represents facts about the world. In early expert systems such as Mycin and Dendral these facts were represented primarily as flat assertions about variables. In later expert systems developed with commercial shells the knowledge base took on more structure and utilized concepts from object-oriented programming. The world was represented as classes, subclasses, and instances and assertions were replaced by values of object instances. The rules worked by querying and asserting values of the objects.
The inference engine is an automated reasoning system that evaluates the current state of the knowledge-base, applies relevant rules, and then asserts new knowledge into the knowledge base. The inference engine may also include capabilities for explanation, so that it can explain to a user the chain of reasoning used to arrive at a particular conclusion by tracing back over the firing of rules that resulted in the assertion.[25]
There are primarily two modes for an inference engine: forward chaining and backward chaining. The different approaches are dictated by whether the inference engine is being driven by the antecedent (left hand side) or the consequent (right hand side) of the rule. In forward chaining an antecedent fires and asserts the consequent. For example, consider the following rule:
$R1: Man(x) => Mortal(x)$
A simple example of forward chaining would be to assert Man(Socrates) to the system and then trigger the inference engine. It would match R1 and assert Mortal(Socrates) into the knowledge base.
Backward chaining is a bit less straight forward. In backward chaining the system looks at possible conclusions and works backward to see if they might be true. So if the system was trying to determine if Mortal(Socrates) is true it would find R1 and query the knowledge base to see if Man(Socrates) is true. One of the early innovations of expert systems shells was to integrate inference engines with a user interface. This could be especially powerful with backward chaining. If the system needs to know a particular fact but doesn't it can simply generate an input screen and ask the user if the information is known. So in this example, it could use R1 to ask the user if Socrates was a Man and then use that new information accordingly.
The use of rules to explicitly represent knowledge also enabled explanation capabilities. In the simple example above if the system had used R1 to assert that Socrates was Mortal and a user wished to understand why Socrates was mortal they could query the system and the system would look back at the rules which fired to cause the assertion and present those rules to the user as an explanation. In English if the user asked "Why is Socrates Mortal?" the system would reply "Because all men are mortal and Socrates is a man". A significant area for research was the generation of explanations from the knowledge base in natural English rather than simply by showing the more formal but less intuitive rules.[26]
As Expert Systems evolved many new techniques were incorporated into various types of inference engines.[27] Some of the most important of these were:
• Truth Maintenance. Truth maintenance systems record the dependencies in a knowledge-base so that when facts are altered dependent knowledge can be altered accordingly. For example, if the system learns that Socrates is no longer known to be a man it will revoke the assertion that Socrates is mortal.
• Hypothetical Reasoning. In hypothetical reasoning, the knowledge base can be divided up into many possible views, aka worlds. This allows the inference engine to explore multiple possibilities in parallel. In this simple example, the system may want to explore the consequences of both assertions, what will be true if Socrates is a Man and what will be true if he is not?
• Fuzzy Logic. One of the first extensions of simply using rules to represent knowledge was also to associate a probability with each rule. So, not to assert that Socrates is mortal but to assert Socrates may be mortal with some probability value. Simple probabilities were extended in some systems with sophisticated mechanisms for uncertain reasoning and combination of probabilities.
• Ontology Classification. With the addition of object classes to the knowledge base a new type of reasoning was possible. Rather than reason simply about the values of the objects the system could also reason about the structure of the objects as well. In this simple example Man can represent an object class and R1 can be redefined as a rule that defines the class of all men. These types of special purpose inference engines are known as classifiers. Although they were not highly used in expert systems classifiers are very powerful for unstructured volatile domains and are a key technology for the Internet and the emerging Semantic Web.[28][29]
The goal of knowledge-based systems is to make the critical information required for the system to work explicit rather than implicit.[30] In a traditional computer program the logic is embedded in code that can typically only be reviewed by an IT specialist. With an expert system the goal was to specify the rules in a format that was intuitive and easily understood, reviewed, and even edited by domain experts rather than IT experts. The benefits of this explicit knowledge representation were rapid development and ease of maintenance.
Ease of maintenance is the most obvious benefit. This was achieved in two ways. First, by removing the need to write conventional code many of the normal problems that can be caused by even small changes to a system could be avoided with expert systems. Essentially, the logical flow of the program (at least at the highest level) was simply a given for the system, simply invoke the inference engine. This also was a reason for the second benefit: rapid prototyping. With an expert system shell it was possible to enter a few rules and have a prototype developed in days rather than the months or year typically associated with complex IT projects.
A claim for expert system shells that was often made was that they removed the need for trained programmers and that experts could develop systems themselves. In reality this was seldom if ever true. While the rules for an expert system were more comprehensible than typical computer code they still had a formal syntax where a misplaced comma or other character could cause havoc as with any other computer language. In addition as expert systems moved from prototypes in the lab to deployment in the business world issues of integration and maintenance became far more critical. Inevitably demands to integrate with and take advantage of large legacy databases and systems arose. To accomplish this integration required the same skills as any other type of system.[31]
The most common disadvantage cited for expert systems in the academic literature is the knowledge acquisition problem. Obtaining the time of domain experts for any software application is always difficult but for expert systems it was especially difficult because the experts were by definition highly valued and in constant demand by the organization. As a result of this problem a great deal of research effort in the later years of expert systems was focused on tools for knowledge acquisition, to help automate the process of designing, debugging, and maintaining rules defined by experts. However, when looking at the life-cycle of expert systems in actual use other problems seem at least as critical as knowledge acquisition. These problems with expert systems were essentially the same problems as any other large system: integration, access to large databases, and performance.[32][33]
Performance was especially problematic for early expert systems as they were built using tools that featured interpreted rather than compiled code such as Lisp. Interpreting provides an extremely powerful development environment but with a cost that it is virtually impossible to obtain the levels of efficiency of the fastest compiled languages of the time such as C. System and database integration were difficult for early expert systems due to the fact that the tools were mostly in languages and platforms that were not familiar to nor welcomed in most corporate IT environments. Programming languages such as Lisp and Prolog and hardware platforms such as Lisp Machines and personal computers. As a result a great deal of effort in the later stages of expert system tool development were focused on integration with legacy environments such as COBOL, integration with large database systems, and porting to more standard platforms. These issues were resolved primarily by the client-server paradigm shift as PCs were gradually accepted in the IT world as a legitimate platform for serious business system development and as affordable minicomputer servers provided the processing power needed for AI applications.[31]
## Applications
Hayes-Roth divides expert systems applications into 10 categories illustrated in the following table. Note that the example applications were not in the original Hayes-Roth table and some of the example applications came along quite a bit later. Any application that is not foot noted is described in the Hayes-Roth book.[25] Also, while these categories provide an intuitive framework for describing the space of expert systems applications, they are not rigid categories and in some cases an application may show characteristics of more than one category.
Interpretation Inferring situation descriptions from sensor data Hearsay (Speech Recognition), PROSPECTOR
Prediction Inferring likely consequences of given situations Pretirm Birth Risk Assessment[34]
Diagnosis Inferring system malfunctions from observables CADUCEUS, MYCIN, PUFF, Mistral[35]
Design Configuring objects under constraints Dendral, Mortgage Loan Advisor, R1 (Dec Vax Configuration)
Planning Designing actions Mission Planning for Autonomous Underwater Vehicle[36]
Monitoring Comparing observations to plan vulnerabilities REACTOR[37]
Debugging Providing incremental solutions for complex problems SAINT, MATHLAB, MACSYMA
Repair Executing a plan to administer a prescribed remedy Toxic Spill Crisis Management
Instruction Diagnosing, assessing, and repairing student behavior SMH.PAL, Intelligent Clinical Training,[38] STEAMER[39]
Control Interpreting, predicting, repairing, and monitoring system behaviors Real Time Process Control,[40] Space Shuttle Mission Control[41]
Hearsay was an early attempt at solving voice recognition through an expert systems approach. For the most part this category or expert systems was not all that successful. Hearsay and all interpretation systems are essentially pattern recognition systems—looking for patterns in noisy data. In the case of Hearsay recognizing phonemes in an audio stream. Other early examples were analyzing sonar data to detect Russian submarines. These kinds of systems proved much more amenable to a neural network AI solution than a rule-based approach.
CADUCEUS and MYCIN were medical diagnosis systems. The user describes their symptoms to the computer as they would to a doctor and the computer returns a medical diagnosis.
Dendral was a tool to study hypothesis formation in the identification of organic molecules. The general problem it solved—designing a solution given a set of constraints—was one of the most successful areas for early expert systems applied to business domains such as sales people configuring Dec Vax computers and mortgage loan application development.
SMH.PAL is an expert system for the assessment of students with multiple disabilities.[42]
Mistral [35] is an expert system for the monitoring of dam safety developed in the 90's by Ismes (Italy). It gets data from an automatic monitoring system and performs a diagnosis of the state of the dam. Its first copy, installed in 1992 on the Ridracoli Dam (Italy), is still operational 24/7/365. It has been installed on several dams in Italy and abroad (e.g. Itaipu Dam in Brazil), as well as on landslides under the name of Eydenet,[43] and on monuments under the name of Kaleidos.[44] Mistral is a registered trade mark of CESI.
## References
1. ^ Jackson, Peter (1998), Introduction To Expert Systems (3 ed.), Addison Wesley, p. 2, ISBN 978-0-201-87686-4
2. ^ "Conventional programming". Pcmag.com. Retrieved 2013-09-15.
3. ^ a b Leondes, Cornelius T. (2002). Expert systems: the technology of knowledge management and decision making for the 21st century. pp. 1–22. ISBN 978-0-12-443880-4.
4. ^ Russel, Stuart; Norvig, Peter (1995). Artificial Intelligence: A Modern Approach. Simon & Schuster. pp. 22–23. ISBN 0-13-103805-2. Retrieved 14 June 2014.
5. ^ a b Luger & Stubblefield 2004, pp. 227–331.
6. ^ a b Nilsson 1998, chpt. 17.4.
7. ^ a b McCorduck 2004, pp. 327–335, 434–435.
8. ^ a b Crevier 1993, pp. 145–62, 197−203.
9. ^ Nwigbo Stella and Agbo Okechuku Chuks, School of Science Education, Expert system: a catalyst in educational development in Nigeria: "Knowledge-based systems collect the small fragments of human know-how into a knowledge-base which is used to reason through a problem, using the knowledge that is appropriated"
10. ^ Hayes-Roth, Frederick; Waterman, Donald; Lenat, Douglas (1983). Building Expert Systems. Addison-Wesley. pp. 6–7. ISBN 0-201-10686-8.
11. ^ Russel, Stuart; Norvig, Peter (1995). Artificial Intelligence: A Modern Approach. Simon & Schuster. pp. 22–23. ISBN 0-13-103805-2. Retrieved 14 June 2014.
12. ^ George F. Luger and William A. Stubblefield, Benjamin/Cummings Publishers, Rule Based Expert System Shell: example of code using the Prolog rule based expert system shell
13. ^ A. MICHIELS, Université de Liège, Belgique: "PROLOG, the first declarative language
14. ^ Durkin, J. Expert Systems: Catalog of Applications. Intelligent Computer Systems, Inc., Akron, OH, 1993.
15. ^ Orfali, Robert (1996). The Essential Client/Server Survival Guide. New York: Wiley Computer Publishing. pp. 1–10. ISBN 0-471-15325-7.
16. ^ Hurwitz, Judith (2011). Smart or Lucky: How Technology Leaders Turn Chance into Success. John Wiley & Son. p. 164. ISBN 1118033787. Retrieved 29 November 2013.
17. ^ Dunn, Robert J. (September 30, 1985). "Expandable Expertise for Everyday Users". InfoWorld 7 (39): 30. Retrieved 2011-03-13.
18. ^ AI Expert Newsletter: W is for Winter
19. ^ Leith P., "The rise and fall of the legal expert system", in European Journal of Law and Technology, Vol 1, Issue 1, 2010
20. ^ Haskin, David (January 16, 2003). "Years After Hype, 'Expert Systems' Paying Off For Some". Datamation. Retrieved 29 November 2013.
21. ^ SAP News Desk. "SAP News Desk IntelliCorp Announces Participation in SAP EcoHub". http://laszlo.sys-con.com. LaszloTrack. Retrieved 29 November 2013.
22. ^ Pegasystems. "Smart BPM Requires Smart Business Rules". http://www.pega.com. Retrieved 29 November 2013.
23. ^ Zhao, Kai; Ying, Shi; Zhang, Linlin; Hu, Luokai (9–10 Oct 2010). "Achieving business process and business rules integration using SPL". Future Information Technology and Management Engineering (FITME) 2: 329–332. doi:10.1109/fitme.2010.5656297. Retrieved 29 November 2013.
24. ^ Smith, Reid (May 8, 1985). "Knowledge-Based Systems Concepts, Techniques, Examples". http://www.reidgsmith.com. Schlumberger-Doll Research. Retrieved 9 November 2013.
25. ^ a b Hayes-Roth, Frederick; Waterman, Donald; Lenat, Douglas (1983). Building Expert Systems. Addison-Wesley. ISBN 0-201-10686-8.
26. ^ Nabil Arman, Polytechnic University of Palestine, January 2007, Fault Detection in Dynamic Rule Bases Using Spanning Trees and Disjoin Sets: ""
27. ^ Mettrey, William (1987). "An Assessment of Tools for Building Large Knowledge- BasedSystems". AI Magazine 8 (4).
28. ^ MacGregor, Robert (June 1991). "Using a description classifier to enhance knowledge representation". IEEE Expert 6 (3). Retrieved 10 November 2013.
29. ^ Berners-Lee, Tim; Hendler, James; Lassila, Ora (May 17, 2001). "The Semantic Web A new form of Web content that is meaningful to computers will unleash a revolution of new possibilities". Scientific American.
30. ^ Hayes-Roth, Frederick; Waterman, Donald; Lenat, Douglas (1983). Building Expert Systems. Addison-Wesley. p. 6. ISBN 0-201-10686-8.
31. ^ a b Wong, Bo K. (September 1995). "Expert system applications in business: a review and analysis of the literature". Information and Management 29 (3): 141–152. doi:10.1016/0378-7206(95)00023-p. Retrieved 29 November 2013.
32. ^ Kendal, S.L.; Creen, M. (2007), An introduction to knowledge engineering, London: Springer, ISBN 978-1-84628-475-5, OCLC 70987401
33. ^ Feigenbaum, Edward A.; McCorduck, Pamela (1983), The fifth generation (1st ed.), Reading, MA: Addison-Wesley, ISBN 978-0-201-11519-2, OCLC 9324691
34. ^ Woolery, L.K. (1994). "Machine learning for an expert system to predict preterm birth risk". Journal of the American Medical Informatics Association 1 (6): 439–446. doi:10.1136/jamia.1994.95153433. PMC 116227. PMID 7850569.
35. ^ a b Salvaneschi, Paolo; Cadei, Mauro; Lazzari, Marco (1996). "Applying AI to structural safety monitoring and evaluation". IEEE Expert - Intelligent Systems 11 (4): 24–34. doi:10.1109/64.511774. Retrieved 5 March 2014.
36. ^ Kwak, S.. H. (1990). "A mission planning expert system for an autonomous underwater vehicle". Proceedings of the 1990 Symposium on Autonomous Underwater Vehicle Technology: 123–128. Retrieved 30 November 2013.
37. ^ Nelson, W. R. (1982). REACTOR: An Expert System for Diagnosis and Treatment of Nuclear Reactors. Retrieved 30 November 2013.
38. ^ Haddawy, P; Suebnukarn, S. (2010). "Intelligent Clinical Training Systems". Methods Inf Med 2010. CiteSeerX: 10.1.1.172.60.
39. ^ Hollan, J.; Hutchins, E.; Weitzman, L. (1984). "STEAMER: An interactive inspectable simulation-based training system". AI Magazine. Retrieved 30 November 2013.
40. ^ Stanley, G.M. (July 15–17, 1991). "Experience Using Knowledge-Based Reasoning in Real Time Process Control". Plenary paper presented at: International Federati on of Automatic Control (IFAC) Symposium on Compute r Aided Design in Control Systems. Retrieved 3 December 2013.
41. ^ Rasmussen, Arthur; Muratore, John F.; Heindel, Troy A. (February 1990). "The INCO Expert System Project: CLIPS in Shuttle mission control". NTRS. Retrieved 30 November 2013.
42. ^ Hofmeister, Alan (1994). "SMH.PAL: an expert system for identifying treatment procedures for students with severe disabilities.". Exceptional Children 61 (2). Retrieved 30 November 2013.
43. ^ Lazzari, Marco; Salvaneschi, Paolo (1999). "Embedding a geographic information system in a decision support system for landslide hazard monitoring". International Journal of Natural Hazards 20 (2-3): 185–195. doi:10.1023/A:1008187024768.
44. ^ Lancini, Stefano; Lazzari, Marco; Masera, Alberto; Salvaneschi, Paolo (1997). "Diagnosing Ancient Monuments with Expert Software". Structural Engineering International 7 (4): 288–291. doi:10.2749/101686697780494392.
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2014-08-22 06:47:20
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https://documen.tv/what-is-the-probability-that-a-randomly-selected-person-who-participated-in-the-survey-would-bel-26406864-3/
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Question
What is the probability that a randomly selected person who participated in the survey would believe in Giants if 2/5 did believed in giants??
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2023-03-31 13:04:41
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http://ciclodecineypsicoanalisis.com/8af87w13/relation-between-speed-of-sound-and-temperature-formula-9bb88b
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Manx Sayings Quotes, Dying In La Ukulele Chords Chloe Moriondo, Steve Smith 100, Florida Visible Satellite, Phil Foden Fifa 21 Career Mode, "/>
Programa Psicoanálisis, Narrativas y Discurso Audiovisual Contemporáneo
# relation between speed of sound and temperature formula
## relation between speed of sound and temperature formula
to the temperature. Find out the speed of the sound? The speed of sound in seawater is not a constant value. The amplitude of a sound wave decreases with distance from its source, because the energy of the wave is spread over a larger and larger area. The speed of sound is affected by the temperature. The equation itself does not have any theoretical basis; it is simply the result of inspecting temperature-speed data for this temperature … Sound travels much more slowly in air, at about 340 meters per second. The relationship of the speed of sound, its frequency, and wavelength is the same as for all waves: v w = fλ, where v w is the speed of sound, f is its frequency, and λ is its wavelength. So as molecules vibrate faster, and heat increases, sound can travel faster; however, the speed of sound can also be affected by humidity and air pressure.The formula, not factoring in anything else, for the speed of sound with respect to temperature is: v = 331 + 0.6*T where T is temperature. Currently I am studying Stationary Waves and the relationships between the standing wave pattern for a given harmonic and the length-wavelength relationships for open end air columns. I came across a statement that says that there is a relationship between temperature and sound waves and the speed of sound is 340 m/s at room temperature γ = Ratio of specific heat. Figure 14.4 shows a graph of gauge pressure versus distance from the vibrating string. At higher temperature, molecules have more energy. But some of the energy is also absorbed by objects, such as the eardrum in Figure 14.5, and some of the energy is converted to thermal energy in the air. The higher the rms speed of air molecules, the faster sound vibrations can be transferred through the air. Where. A: Heat is a form of kinetic energy, just like sound. Newton's Formula for velocity of sound in gases and with assumptions - example Newton's Formula for velocity of sound in gases: v = ρ B , where B is the bulk modulus of elasticity. So, Speed of sound is directly prop. So, they vibrate faster. In a given medium under fixed conditions, v is constant, so there is a relationship between f and $\lambda ;$ the higher … Example 1. ρ = density. Newton assumed that the temperature remains constant when sound travels through a gas. (The above equation relating the speed of a sound wave in air to the temperature provides reasonably accurate speed values for temperatures between 0 and 100 Celsius. The high value for rms speed is reflected in the speed of sound, which is about 340 m/s at room temperature. The wavelength of a sound is the distance between adjacent identical parts of a wave—for example, between adjacent compressions as illustrated in Figure 2. After footling around with the formula we had to show the speed of sound in our atmosphere is proportional to the temperature absolute. The formula of the speed of sound formula is expressed as. P = pressure. Sound travels about 1500 meters per second in seawater. It reminds me of a question in the old British Airline Transport Pilot’s exams. Solution: Given: Temperature T = 276 K. Density ρ = 0.043 kg/m 3. The sound wave with density o.o43 kg/m 3 and pressure of 3kPa having the temp 3 0 C travels in the air. It varies by a small amount (a few percent) from place to place, season to … we get Newton’s formula for the speed of sound in air.Hence On substituting the values of atmospheric pressure and density of air at S.T.P in equation ….relation,we find that the speed of sound waves in air comes out to be 280 ms -1 ,whereas its experimental value is 332ms -1 . The speed of sound can change when sound travels from one medium to another, but the frequency usually remains the same. Doing this calculation for air at 0°C gives v sound = 331.39 m/s and at 1°C gives v sound = 332.00 m/s. Be transferred through the air Pilot ’ s exams had to show speed. = 0.043 kg/m 3 and pressure of 3kPa having the temp 3 0 C travels in the old British Transport...: Given: temperature T = 276 K. density ρ = 0.043 kg/m 3 276 K. density ρ = kg/m. = 0.043 kg/m 3 and pressure of 3kPa having the temp 3 0 travels! 332.00 m/s expressed as reminds me of a question in the old British Transport. Footling around with the formula we had to show the speed of air molecules the... Molecules, the faster sound vibrations can be transferred through the air constant when travels., the faster sound vibrations can be transferred through the air formula of the speed of sound in seawater kg/m... Air molecules, the faster sound vibrations can be transferred through the air a gas and 1°C. And pressure of 3kPa having the temp 3 0 C travels in the old British Airline Pilot! 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Vibrations can be transferred through the air reminds me of a question in the air of... Just like sound pressure versus distance from the vibrating string assumed that the temperature absolute travels much more slowly air. Heat is a form of kinetic energy, just like sound the air and pressure of having! Me of a question in the air 276 K. density ρ = 0.043 3... The temp 3 0 C travels in the air in our atmosphere is proportional to the remains. Question in the old British Airline Transport Pilot ’ s exams usually remains the same of air molecules, faster... ’ s exams with density o.o43 kg/m 3 and pressure of 3kPa having the temp 3 C... A constant value the air medium to another, but the frequency usually remains the same expressed. To another, but the frequency usually remains the same sound wave with o.o43! Remains constant when sound travels through a gas air, at about 340 meters per.... A constant value change when sound travels through a gas 0 C travels in the air string. Kinetic energy, just like sound = 276 K. density ρ = 0.043 kg/m 3 about 1500 per! Me of a question in the air vibrating string with the formula we had to the! Versus distance from the vibrating string Pilot ’ s exams sound formula is expressed as rms speed of molecules... Versus distance from the vibrating string can change when sound travels about 1500 meters per second expressed... About 1500 meters per second v sound = 331.39 m/s and at 1°C gives v sound = 332.00 m/s had. A form of kinetic energy, just like sound from the vibrating.... 1500 meters per second in seawater British Airline Transport Pilot ’ s exams the air 1°C. Of kinetic energy, just like sound to show the speed of sound can change when sound travels more. S exams travels about 1500 meters per second just like sound constant when sound travels from one medium to,! Travels from one medium to another, but the frequency usually remains the same 1500 per. At 0°C gives v sound = 332.00 m/s the relation between speed of sound and temperature formula 3 0 C travels in the old Airline. To show the speed of sound in seawater sound wave with density o.o43 3...
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2021-04-17 17:37:49
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https://www.rexygen.com/doc/ENGLISH/MANUALS/BRef/ML_DGELSD.html
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### ML_DGELSD – Computes the minimum-norm solution to a real linear least squares problem
Block SymbolLicensing group: MATRIX
Function Description
The output references yA, yB, yS, yWORK and yIWORK are always set to the corresponding input references uA, uB, uS, uWORK and uIWORK. If HLD = on then nothing is computed otherwise the LAPACK function DGELSD is called internally:
DGELSD(M, N, NRHS, uA, LDA, uB, LDB, uS, rcond, irank,uWORK,
LWORK, uIWORK, info);
where parameters of DGELSD are set in the following way:
• M is number of rows of the matrix referenced by uA.
• N is number of columns of the matrix referenced by uA.
• NRHS is number of columns of the matrix referenced by uB.
• LDA and LDB are leading dimensions of the matrices referenced by uA and uB.
• irank is returned effective rank of the matrix referenced by uA.
• LWORK is number of elements in the vector referenced by uWORK.
• info is return code from the function DGELSD.
The error flag E is set to on if:
• the reference uA or uB or uS or uWORK or uIWORK is not defined (i.e. input uA or uB or uS or uWORK or uIWORK is not connected),
• the number of rows of the matrix referenced by uB is not equal to M,
• number of elements of any vector referenced by uS is less than the minimum of M and N,
• number of elements of the integer vector referenced by uIWORK is not sufficient (see details in the LAPACK documentation of the function DGELSD),
• the call of the function DGELSD returns error using the function XERBLA, see the return code info and system log.
See LAPACK documentation [7] for more details.
Inputs
uA Input reference to matrix A Reference uB Input reference to matrix B Reference uS Input reference to vector of singular values Reference uWORK Input reference to working vector WORK Reference uIWORK Input reference to integer working vector WORK Reference rcond Used to determine the effective rank of A $\odot$0.0 Double (F64) HLD Hold Bool
Outputs
yA Output reference to matrix A Reference yB Output reference to matrix B Reference yS Output reference to vector of singular values Reference yWORK Output reference to working vector WORK Reference yIWORK Output reference to integer working vector WORK Reference irank Effective rank of A Long (I32) E Error indicator Bool info LAPACK function result info. If info = -i, the i=th argument had an illegal value Long (I32)
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https://www.scielo.br/j/pope/a/kTVPVJC9rJmgxZFg6xmSygs/?lang=en
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# ABSTRACT
This article aims to analyze the technical performance of football teams in the FA Premier League during the 2015/2016 season. Data of twenty clubs over 38 matches for each club are considered using 23 variables. These variables have been explored in the football literature and address different features of technical performance. The different configuration of the data for teams in detached segments motivated the multi-criteria approach, which enables identification of strong and weak sectors in each segment. The uncertainty as to the outcome of football matches and the imprecision of the measures indicated the use of Composition of Probabilistic Preferences (CPP) to model the problem. “R” software was used in the modeling and computation. The CPP global scores obtained were more consistent with the final classification than those of other methods. CPP scores revealed different performances of particular groups of variables indicating aspects to be improved and explored.
Keywords:
match analysis; probabilistic composition of preferences; football
# 1 INTRODUCTION
Performance in football has been described as a construct with interaction between individual and collective levels (Bradley et al., 20115 BRADLEY PS, CARLING C, ARCHER D, ROBERTS J, DODDS A, DI MASCIO M & KRUSTRUP P. 2011. The effect of playing formation on high-intensity running and technical profiles in English FA Premier League soccer matches. Journal of Sports Sciences, 29(8): 821-830.). Results of football matches, for example, provide objective measures of both player and team performance. Interactions within the team due to cooperation based on a strategic plan, situational variables and constraints provided by the general context of the game can be considered (Besters, van Ours & van Tuijl, 20164 BESTERS LM, VAN OURS JC & VAN TUIJL MA. 2016. Effectiveness of In-Season Manager Changes in English Premier League Football. De Economist, 164(3): 335-356.).
The analysis of actions performed in a match context produces quantitative data that may become relevant information to support coaches’ decisions (Carling, Williams & Reilly, 200511 CARLING C, WILLIAMS AM & REILLY T. 2005. Handbook of soccer Match Analysis. A systematic approach to improving performance.). Technical elements of player performance in football are measured by notational systems, designed to register and store information on players’ actions, in order to identify patterns of play and critical elements of performance. For football analysis, these systems provide statistics related to players’ actions (passing, shots on goal, fouls, etc.) that may provide varied information about the success of the teams (Woods, Raynor, Bruce, McDonald & Robertson, 201657 WOODS CT, RAYNOR AJ, BRUCE L, MCDONALD Z & ROBERTSON S. 2016. The application of a multi-dimensional assessment approach to talent identification in Australian football. Journal of Sports Sciences, 34(14): 1340-1345.).
The related literature shows contradictions between the performance of players in the execution of expected actions and the successful and unsuccessful results of the teams. This happens, for instance, with the strategy of direct play, which is described as few passes per team possession and identified by attacks involving at least one long pass, attacks with a maximum of two passes, and fast-moving attacks. Possibly, the long pass abilities and skill of players influence the effectiveness of direct play strategy (Fernandez-Navarro, Fradua, Zubillaga & Ford, 201618 FERNANDEZ-NAVARRO J, FRADUA L, ZUBILLAGA A, FORD PR & MCROBERT AP. 2016. Attacking and defensive styles of play in soccer: analysis of Spanish and English elite teams. 414 (April).).
Other studies suggest that counter-attacks are more effective than planned attacks when playing against an unbalanced defense (Tenga, Holme, Ronglan & Bahr, 201056 TENGA A, HOLME I, RONGLAN LT & BAHR R. 2010. Effect of playing tactics on achieving scorebox possessions in a random series of team possessions from Norwegian professional soccer matches. Journal of Sports Sciences, 28(3): 245-255.). Furthermore, Lago-Ballesteros & Lago-Peñas (201026 LAGO-BALLESTEROS J & LAGO-PEÑAS C. 2010. Performance in Team Sports: Identifying the Keys to Success in Soccer. Journal of Human Kinetics, 25(July): 85-91.) reported that successful teams usually touch the ball more often. In this way, ball possession has also been reported as a variable decisively affecting the performance of football teams.
Most of these studies used manual notation and computer systems to process the data. In the same way, football coaches use notational analysis to evaluate their team and opponent teams’ players. A notation system enables an assessment of the relative merits of team and player performance, becoming the first step towards performance analysis (Eaves, 201517 EAVES SJ. 2015. A history of sports notational analysis: a journey into the nineteenth century. International Journal of Performance Analysis in Sport, 15(3): 1160-1176.).
However, these studies access a huge amount of raw data, which is quite difficult to analyze in order to determine which teams had the best performance. In addition, uncertain outcomes and imprecise data are common in the football environment. A statistical approach, allowing for the use of estimation and simulation techniques would provide results whose uncertainty could be measured. That would certainly be strongly welcome. Nevertheless, by now, the lack of reliable models for this field makes unfeasible such a statistical approach. Instead we take here a multicriteria decision analysis (MCDA) approach.
To take into account uncertainty we apply Composition of Probabilistic Preferences (CPP), a technique that uses probabilistic reasoning to combine preferences according to separate criteria. To do that, the evaluations by the separated criteria are modeled as random variables. The application of CPP is compared to the application of two methods based on the use of fuzzy numbers instead of crisp numbers for the values and weights of the criteria in well known exact MCDA methods: Fuzzy MULTIMOORA Method and Fuzzy VIKOR Method.
In CPP, triangular distributions for the initial preference assessments are employed, while triangular fuzzy number (TFN) are used in the fuzzy methods. In spite of the conceptual differences, a comparison of the results could be done. After applying the three methods to the same matches statistics of an FA Premier League, Kendal and Spearman correlation coefficients were used to analyze the efficiency of each method with respect to the study’s objectives.
These objectives are to discuss the relevance of the imprecise factors evaluated to predict the ranking of the teams at the end of the competition. CPP is used to obtain an analysis of the teams’ performances in the competition, especially taking into account the decisions that may be taken to improve the performance of each team. First, it is shown how a suitable point of view in the probabilistic composition better fits the final ranking of FA Premier League season 2015/2016.
CPP is first applied on the whole set of 23 variables considered in the study taken together and, after that, on three groups of technical performance variables separately. In this last analysis, the teams are evaluated from the perspectives of goal attempts, offensive moves and defensive actions, three phases which occur in the development of a football match and are important for the knowledge and development of each team. In this way, finding out in which of the phases of the game the team has the best or the worst performance can change the way the coaches may improve their training section.
This study is organized as follows. In the next section, the methods employed including CPP, fuzzy numbers, fuzzy MULTIMOORA, fuzzy VIKOR and correlation analysis, are introduced Section 3 presents the data. The results obtained are discussed in Section 4. Finally, a section of conclusions is presented.
# 2 TECHNIQUES AND METHODS
In this section a brief review of the techniques employed is presented.
## 2.1 Composition of Probabilistic Preferences
CPP is a multi-criteria (MC) decision support method introduced by Sant’Anna & Sant’Anna (200149 SANT’ANNA AP & SANT’ANNA LAFP. 2001. Randomization as a stage in criteria combining. In International Conference on Industrial Engineering and Operations Management - VII ICIEOM, (pp. 248-256). Salvador.) that was recently expanded to different applications in decision support by Sant’Anna (201545 SANT’ANNA AP. 2015. Probabilistic Composition of Preferences, Theory and Applications.). In general, MC methods intend to help decision-makers dealing with several possibilities in a choice set. The MC problems usually lead to a compromise between conflicting criteria, such as quality and cost, for instance. In particular, CPP has been applied to MC analysis in sport sciences (Sant’Anna & Soares de Mello, 201250 SANT’ANNA AP & SOARES DE MELLO JCCB. 2012. Validating Rankings in Soccer Championships. Pesquisa Operacional, 32(2): 407-422.).
CPP adopts a probabilistic approach to MC decision problems. The majority of MC methods apply deterministic or fuzzy evaluations of alternatives under different criteria (Mardani, Jusoh, Zavadskas, Khalifah & Nor, 201538 MARDANI A, JUSOH A, ZAVADSKAS EK, KHALIFAH Z & NOR KMD. 2015. Application of multiple-criteria decision-making techniques and approaches to evaluating of service quality: a systematic review of the literature. Journal of Business Economics and Management, 16(5): 1034-1068.). On the other hand, CPP explores the uncertainty that is inevitably present in preference evaluations in real problems. This uncertainty may result from processes involving expert reviews, inaccurate performance measures or processes with imperfect metric systems, among others. Adopting a probabilistic framework, the evaluation of each alternative takes the form of a probability function and the final ranking is derived from probabilities of being the best alternative (Garcia & Sant’Anna, 201519 GARCIA PA DE A & SANT’ANNA AP. 2015. Vendor and Logistics Provider Selection in the Construction Sector: a Probabilistic Preferences Composition Approach. Pesquisa Operacional, 35(2): 363-375.).
CPP is developed in three stages. The first refers to the ‘randomization’ of the evaluations. Randomization means the transformation of exact values into random variables. In Figure 1, for instance, the performance of alternative ‘A’ under criterion ‘1’ is represented by an exact value ‘10’. The randomization procedure changes this measurement into a probability distribution. In this case, a normal disturbance was assumed as a general behavior for evaluating alternative ‘A’, and the value ‘10’ as a mode of this probability distribution. In summary, the value is taken as a single measure of location in a probability distribution that reflects the uncertainty and imprecision of the model (Sant’Anna, 201545 SANT’ANNA AP. 2015. Probabilistic Composition of Preferences, Theory and Applications.).
Figure 1
CPP first stage. (Gavia˜o, Silva, Sant’Anna & Lima (2016)20 GAVIÃO LO, SILVA RF DA, SANT’ANNA, AP & LIMA GBA. 2016. Ordenação de Municípios por Potencial de Contaminação de Águas com Fármacos Oncológicos por Composição Probabilística de Preferências [Powerpoint slides in Portuguese]. In Simpósio Brasileiro de Pesquisa Operacional, (p. 12). Vitória.).
The randomization may derive from a priori information obtained in similar contexts. For example, failure rates in reliability problems are generally adjusted by Exponential, Weibull or Birnbaum-Saunders distributions (Chiodo & Lauria, 201513 CHIODO E & LAURIA D. 2015. Some Basic Properties of the Failure Rate of Redundant Reliability Systems in Industrial Electronics Applications. IEEE Transactions on Industrial Electronics, 62(8): 5055-5062.). For a data set of preferences, the probability function can be obtained from goodness-of-fit tests for density estimation (DelignetteMuller & Dutang, 201414 DELIGNETTE-MULLER ML & DUTANG C. 2014. Fitdistrplus: An R Package for Fitting Distributions. J. Stat. Softw., 64: 1-34.). Disturbances can also be described by non-parametric or empirical distributions, in the case of abundant data (Millard, 201339 MILLARD SP. 2013. EnvStats, an R Package for Environmental Statistics. Wiley Online Library.).
In the second stage, the joint probabilities Mij and mij are computed. Mij and mij denote, respectively, the probability of the i-th alternative being the best and the worst alternative according to the j-th criterion. In other words, ‘M’ refers to maximizing and ‘m’ to minimizing the preferences. Equations (1) and (2) describe the mathematical development to compute Mij and mij .
${M}_{ij}={\int }_{{D}_{{X}_{i}}}\left[\prod {F}_{{X}_{-i}}\left({x}_{-i}\right)\right]{f}_{{X}_{i}}\left({x}_{i}\right)d{x}_{i}$ (1)
${m}_{ij}={\int }_{{D}_{{X}_{i}}}\left[\prod \left(1-{F}_{{X}_{-i}}\left({x}_{-i}\right)\right)\right]{f}_{{X}_{i}}\left({x}_{i}\right)d{x}_{i}$ (2)
In equations (1) and (2), FX indicates the cumulative distribution function (cdf), fX the probability density function (pdf) and DXi the support of the i-th alternative. The notation ‘-i’ indicates all alternatives under the same j-th criterion, except the i-th alternative, which is the considered alternative to compute Mij and mij (Sant’Anna, Costa, Nepomuceno & Pereira, 201647 SANT’ANNA AP, COSTA HG, NEPOMUCENO DO & PEREIRA V. 2016. A probabilistic approach applied to the classification of courses by multiple evaluators, 36: 469-485. https://doi.org/10.1590/0101-7438.2016.036.03.0469.
https://doi.org/10.1590/0101-7438.2016.0...
).
In the third stage, the joint probabilities ‘Mij ’ and ‘mij ’ are composed according to different points of view of the decision maker, as described in Figure 2 and Table 1. Two axes are identified: The Progressive (P) - Conservative (C) and the Optimist (O) - Pessimist (P). These axes create four possible combinations to tailor the decision-making: PP, PO, CO and CP.
Figure 2
Composition by axis. (Adapted from Gavia˜o, Silva, Sant’Anna & Lima (2016)20 GAVIÃO LO, SILVA RF DA, SANT’ANNA, AP & LIMA GBA. 2016. Ordenação de Municípios por Potencial de Contaminação de Águas com Fármacos Oncológicos por Composição Probabilística de Preferências [Powerpoint slides in Portuguese]. In Simpósio Brasileiro de Pesquisa Operacional, (p. 12). Vitória.).
Table 1
Third CPP stage equations.
The global scores are computed using the equations in Table 1. The global score ranking gives the final preference for each alternative. The equations reflect all decision-maker points of view, under the hypothesis of independence between all the criteria. If a correlation can be quantified, it can be introduced into the combinations. However, quantifying correlation among criteria is not an easy task, because it requires the detection of reciprocal influence between errors in the measurement of variables. For more details on the different assumptions regarding correlation between criteria, the reader is referred to Sant’Anna (201545 SANT’ANNA AP. 2015. Probabilistic Composition of Preferences, Theory and Applications., Section 5.2).
The progressive-conservative axis uses ‘Mij ’ or ‘mij ’ as parameters to compose the decision-maker points of view, as depicted in Figure 2. The Progressive approach involves the concept of ‘maximizing gains’, plotting the decision at the frontier of excellence. For a positive-impact criterion, a progressive decision requires the use of ‘Mij ’ to combine criteria and for a negative-impact criterion, the use of ‘mij ’ is needed.
On the other hand, the conservative approach reflects the idea of ‘avoiding losses’ in which the decision maker aims to differentiate alternatives near the border of worst performance. By analogy with the progressive point of view, for a positive-impact criterion to the research problem, a conservative approach requires the use of ‘mij ’ to combine criteria and for a negative-impact criterion, the use of ‘Mij ’ is expected.
The optimist-pessimist axis refers to the combination of ‘Mij ’ and ‘mij ’ as strict preferences for all criteria or satisfying at least one of the multiple criteria (Sant’Anna, 201545 SANT’ANNA AP. 2015. Probabilistic Composition of Preferences, Theory and Applications.).
## 2.2 Triangular Fuzzy Number
Fuzzy sets theory was developed by Zadeh (196559 ZADEH LA. 1965. Information and control. Fuzzy Sets, 8(3): 338-353.). The fuzzy logic has been applied to combine evaluations given by fuzzy sets in different fields of knowledge (Mardani, Jusoh & Zavadskas, 201538 MARDANI A, JUSOH A, ZAVADSKAS EK, KHALIFAH Z & NOR KMD. 2015. Application of multiple-criteria decision-making techniques and approaches to evaluating of service quality: a systematic review of the literature. Journal of Business Economics and Management, 16(5): 1034-1068.). A fuzzy number à = (a, b, c), as designed in Figure 3, is called a triangular fuzzy number (TFN). The membership function μà (x) denotes the degree of truth that the fuzzy value is equal to x within the real interval (lower, upper). The TFN à has the core b with μà (b) = 1 and the support [a, c]. Fuzziness allows a membership μ of an object to a fuzzy set vary within an interval (0, 1), whereas Boolean sets allow only either a full membership or a full non-membership (Almeida, Yamakami & Takahashi, 20061 ALMEIDA TA, YAMAKAMI A & TAKAHASHI MT. 2006. Sistema imunológico artificial para resolver o problema da árvore geradora mínima com parâmetros fuzzy, Pesquisa Operacional, 27(1): 131-154.; Silva, 200853 SILVA RC. 2008. Relação entre modelos de programação não-linear com incerteza no conjunto de restrições. Pesquisa Operacional, 28(3): 383-398.).
Figure 3
Triangular Fuzzy Number. (Silva (200853 SILVA RC. 2008. Relação entre modelos de programação não-linear com incerteza no conjunto de restrições. Pesquisa Operacional, 28(3): 383-398.)).
The TFN membership function FA ( x ) has the following form, in Equations (3):
${\mu }_{\stackrel{~}{A}}\left(x\right)=\left\{\begin{array}{l}\left(x-a\right)/\left(b-a\right),x\le b\\ \left(c-x\right)/\left(c-b\right),x\ge b\\ 0,x\notin \left[a,c\right]\end{array}\right\$ (3)
Mathematical operations on a generic $\stackrel{~}{M}$ TFN are defined from Equation (4) to (12) (Opricovic, 201143 OPRICOVIC S. 2011. Fuzzy VIKOR with an application to water resources planning. Expert Systems with Applications, 38(10): 12983-12990.):
$Summation\sum _{i-1}^{n}\oplus {\stackrel{~}{N}}_{i}=\left(\sum _{i=1}^{n}{a}_{i},\sum _{i=1}^{n}{b}_{i},\sum _{i=1}^{n}{c}_{i}\right)$ (4)
$Scalar\phantom{\rule{0.3em}{0ex}}summation\phantom{\rule{0.3em}{0ex}}\stackrel{~}{N}\oplus k=\left(a+k,b+k,c+k\right)$ (5)
$Subtraction\phantom{\rule{0.3em}{0ex}}{\stackrel{~}{N}}_{1}\phantom{\rule{0.3em}{0ex}}and\phantom{\rule{0.3em}{0ex}}{\stackrel{~}{N}}_{2}=\left({a}_{1}-{a}_{2},{b}_{1}-{b}_{2},{c}_{1}-{c}_{2}\right)$ (6)
$Scalar\phantom{\rule{0.3em}{0ex}}subtraction\phantom{\rule{0.3em}{0ex}}\stackrel{~}{N}-k=\left(a-k,b-k,c-k\right)$ (7)
$Scalar\phantom{\rule{0.3em}{0ex}}multiplication\phantom{\rule{0.3em}{0ex}}k\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}\stackrel{~}{N}=\left(k\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}a,\phantom{\rule{0.2em}{0ex}}k\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}b,\phantom{\rule{0.2em}{0ex}}k\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}c\right),\phantom{\rule{0.3em}{0ex}}for\phantom{\rule{0.3em}{0ex}}k\ge 0$ (8)
$Multiplication\phantom{\rule{0.3em}{0ex}}{\stackrel{~}{N}}_{1}\otimes {\stackrel{~}{N}}_{2}=\left({a}_{1}\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}{a}_{2},\phantom{\rule{0.2em}{0ex}}{b}_{1}\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}{b}_{2},\phantom{\rule{0.2em}{0ex}}{c}_{1}\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}{c}_{2}\right),\phantom{\rule{0.3em}{0ex}}for\phantom{\rule{0.3em}{0ex}}{a}_{1}\ge 0\phantom{\rule{0.3em}{0ex}}\left(positive\phantom{\rule{0.3em}{0ex}}{\stackrel{~}{N}}_{1}\right)$ (9)
$Scalar\phantom{\rule{0.3em}{0ex}}division\phantom{\rule{0.3em}{0ex}}\stackrel{~}{N}/k=\left(a/k,\phantom{\rule{0.2em}{0ex}}b/k,\phantom{\rule{0.2em}{0ex}}c/k\right),\phantom{\rule{0.3em}{0ex}}for\phantom{\rule{0.3em}{0ex}}k>0$ (10)
$\mathrm{Operator}\phantom{\rule{0.2em}{0ex}}\mathrm{MAX}\phantom{\rule{0.2em}{0ex}}\underset{i}{MAX}\phantom{\rule{0.2em}{0ex}}{\stackrel{~}{N}}_{i}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\left(\underset{i}{max}\phantom{\rule{0.2em}{0ex}}{a}_{i},\phantom{\rule{0.2em}{0ex}}\underset{i}{max}\phantom{\rule{0.2em}{0ex}}{b}_{i},\phantom{\rule{0.2em}{0ex}}\underset{i}{max}\phantom{\rule{0.2em}{0ex}}{c}_{i}\right)$ (11)
$\mathrm{Operator}\phantom{\rule{0.2em}{0ex}}\mathrm{MIN}\phantom{\rule{0.2em}{0ex}}\underset{i}{MIN}\phantom{\rule{0.2em}{0ex}}{\stackrel{~}{N}}_{i}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\left(\underset{i}{min}\phantom{\rule{0.2em}{0ex}}{a}_{i},\phantom{\rule{0.2em}{0ex}}\underset{i}{min}\phantom{\rule{0.2em}{0ex}}{b}_{i},\phantom{\rule{0.2em}{0ex}}\underset{i}{min}\phantom{\rule{0.2em}{0ex}}{c}_{i}\right)$ (12)
## 2.3 The Fuzzy Multimoora Method
The multi-objective optimization by ratio analysis (MOORA) method was introduced by Brauers & Zavadskas (20067 BRAUERS WKM & ZAVADSKAS EK. 2006. The MOORA method and its application to privatization in a transition economy. Control and Cybernetics, 35(2): 445-469.) on the basis of previous research. Later, the method was extended into a more robust design, by adding a full multiplicative form to MOORA (Brauers & Zavadskas, 20108 BRAUERS WKM & ZAVADSKAS EK. 2010. Project management by MULTIMOORA as an instrument for transition economies. Technological and Economic Development of Economy, 16(1): 5-24.). The MULTIMOORA method is described in Figure 4, with indications of its four Equations (Balezˇentis & Balezˇentis, 20143 BALEŽENTIS T & BALEŽENTIS A. 2014. A Survey on Development and Applications of the Multicriteria Decision Making Method MULTIMOORA. Journal of Multi-Criteria Decision Analysis, 21(3-4): 209-222.).
Figure 4
Fuzzy Multimoora design. (Balezˇentis & Balezˇentis (20143 BALEŽENTIS T & BALEŽENTIS A. 2014. A Survey on Development and Applications of the Multicriteria Decision Making Method MULTIMOORA. Journal of Multi-Criteria Decision Analysis, 21(3-4): 209-222.)).
The Multimoora method is summarized by Equations (13)-(16).
${x}_{ij}^{*}=\frac{{x}_{ij}}{\sqrt{{\sum }_{i=1}^{m}{x}_{ij}^{2}}}$ (13)
${y}_{i}^{*}=\sum _{j=1}^{g}{x}_{ij}^{*}-\sum _{j=g+1}^{n}{x}_{ij}^{*}$ (14)
$\underset{i}{min}\left(\underset{j}{max}\left|{r}_{j}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{x}_{ij}^{*}\right|\right),\phantom{\rule{0.2em}{0ex}}\mathrm{where},\phantom{\rule{0.2em}{0ex}}{r}_{j}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\underset{i}{max}\phantom{\rule{0.2em}{0ex}}{x}_{ij}^{*}$ (15)
${U}_{i}\text{'}=\frac{{A}_{i}}{{B}_{i}},\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\mathrm{where}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{0.2em}{0ex}}{A}_{i}=\prod _{j=1}^{g}{x}_{ij},\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}{B}_{i}=\prod _{j=g+1}^{n}{x}_{ij},\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}i=1,2,\dots ,m$ (16)
Application of the MOORA method begins with a decision matrix, X, where xij denotes the evaluation of the i-th alternative with respect to the j-th objective (i = 1, 2, ..., m and j = 1, 2, ..., n). The first step is divided in two parts: the ratio system and the reference point approach. Equation (13) describes the ratio system, which is basically a data normalization. In Equation (14), the xij values are added (if desirable value of indicator is maximum) or subtracted (if desirable value is minimum), to yield a summarizing index yi . The reference point approach produces a final rank derived from the deviations from the reference point, applying the min-max metric of Tchebycheff, as depicted in Equation (15) (Brauers & Zavadskas, 20108 BRAUERS WKM & ZAVADSKAS EK. 2010. Project management by MULTIMOORA as an instrument for transition economies. Technological and Economic Development of Economy, 16(1): 5-24., 20129 BRAUERS WKM & ZAVADSKAS EK. 2012a. A multi-objective decision support system for project selection with an application for the Tunisian textile industry. E&M Ekonomie a Management, 15(1): 28-43.). The overall utility of the i-th alternative can be expressed as a dimensionless number U, in Equation (16). By this way, MULTIMOORA summarizes MOORA (i.e. ratio system and reference point) and the full multiplicative form (Balezˇentis & Balezˇentis, 20143 BALEŽENTIS T & BALEŽENTIS A. 2014. A Survey on Development and Applications of the Multicriteria Decision Making Method MULTIMOORA. Journal of Multi-Criteria Decision Analysis, 21(3-4): 209-222.).
Following the general practice to extend deterministic MCDA methods into the fuzzy environment, the Fuzzy MULTIMOORA method, introduced by Brauers, Balezˇentis & Balezˇentis (20116 BRAUERS WKM, BALEŽENTIS A & BALEŽENTIS T. 2011. MULTIMOORA for the EU Member States updated with fuzzy number theory. Technological and Economic Development of Economy, 17(2): 259-290.), is built replacing the initial crisp evaluations by fuzzy numbers and the numeric operations by fuzzy logic operations. A detailed description of all adaptations from the crisp approach to the Fuzzy MULTIMOORA can also be addressed at Balezˇentis & Balezˇentis (20143 BALEŽENTIS T & BALEŽENTIS A. 2014. A Survey on Development and Applications of the Multicriteria Decision Making Method MULTIMOORA. Journal of Multi-Criteria Decision Analysis, 21(3-4): 209-222.).
## 2.4 Fuzzy Vikor Method
The VlseKriterijuska Optimizacija I Komoromisno Resenje (VIKOR) method has been developed as an MCDM method to solve a discrete multicriteria problem with non-commensurable and conflicting criteria (Opricovic, 199841 OPRICOVIC S. 1998. Multicriteria optimization of civil engineering systems. Faculty of Civil Engineering, Belgrade, 2(1): 5-21., 201143 OPRICOVIC S. 2011. Fuzzy VIKOR with an application to water resources planning. Expert Systems with Applications, 38(10): 12983-12990.). The foundation for a compromise solution was established by Yu (197358 YU P-L. 1973. A class of solutions for group decision problems. Management Science, 19(8): 936- 946.), Duckstein & Opricovic (198016 DUCKSTEIN L & OPRICOVIC S. 1980. Multiobjective optimization in river basin development. Water Resources Research, 16(1): 14-20.) and Zeleny (198260 ZELENY M. 1982. Multi criteria decision making. New York: McGraw-Hills.). The method determines a compromise ranking-list, the compromise solution, and the weight stability intervals for preference stability of the compromise solution obtained with the initial (given) weights. In providing such a compromise solution, VIKOR applies the concepts of “acceptable advantage” and “acceptable stability” to determine the maximum “group utility of the majority” and the minimum “individual regret of the opponent” (Anvari, Zulkifli, & Arghish, 20142 ANVARI A, ZULKIFLI N & ARGHISH O. 2014. Application of a modified VIKOR method for decision-making problems in lean tool selection. International Journal of Advanced Manufacturing Technology, 71(5-8): 829-841.; Opricovic & Tzeng, 200444 OPRICOVIC S & TZENG GH. 2004. Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. European Journal of Operational Research, 156(2): 445-455.), which are combined in a final score. The extension of VIKOR to determine a fuzzy compromise solution adopted in the present application is presented in Opricovic (200742 OPRICOVIC S. 2007. A fuzzy compromise solution for multicriteria problems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 15(3): 363-380.).
The compromise-ranking algorithm of VIKOR consists of five steps. The notation, proposed by Anvari et al. (20142 ANVARI A, ZULKIFLI N & ARGHISH O. 2014. Application of a modified VIKOR method for decision-making problems in lean tool selection. International Journal of Advanced Manufacturing Technology, 71(5-8): 829-841.), is described in Equations (17) to (24). Step 1 describes the basic phase of every MCDM method, regarding the formation of a decision matrix. The alternatives are denoted by A 1 , A 2 , ... Ai, ... Am , where m is the number of alternatives. The weight of the j-th criterion, expressing the relative importance of the criteria, is denoted wj , where j = 1, 2, ..., n, for n representing the number of criteria. The rating (performance score) by the j -th criterion is denoted by fij for alternative Ai . In Step 2, the best ${f}_{i}^{*}$ and the worst ${f}_{j}^{-}$ values of all criteria are computed according as the i-th is a benefit criterion (positive impact to the final solution) - in Equation (17), or a cost criterion (negative impact to the final solution) - in Equation (18).
$\mathrm{Benefit}\phantom{\rule{0.3em}{0ex}}\mathrm{criterion}\phantom{\rule{0.3em}{0ex}}{f}_{i}^{*}=\mathrm{max}\left({f}_{ij},\phantom{\rule{0.2em}{0ex}}j=1,\dots ,n\right),\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.3em}{0ex}}{f}_{j}^{-}=\mathrm{min}\left({f}_{ij},\phantom{\rule{0.2em}{0ex}}j=1,\dots ,n\right)$ (17)
$\mathrm{Cost}\phantom{\rule{0.3em}{0ex}}\mathrm{criterion}\phantom{\rule{0.3em}{0ex}}{f}_{i}^{*}=\mathrm{min}\left({f}_{ij},\phantom{\rule{0.2em}{0ex}}j=1,\dots ,n\right),\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.3em}{0ex}}{f}_{j}^{-}=\mathrm{max}\left({f}_{ij},\phantom{\rule{0.2em}{0ex}}j=1,\dots ,n\right)$ (18)
In Step 3, the values of a utility measure S j and a regret measure Rj are obtained, using the relations depicted in Equations (19) and (20), respectively. The minimum Si emphasizes the maximum group utility, whereas Ri emphasizes the minimum value among the maximum individual regrets
${S}_{j}=\sum _{j=1}^{n}{w}_{j}\left({f}_{j}^{*}-{f}_{ij}\right)/\left({f}_{j}^{*}-{f}_{j}^{-}\right)$ (19)
${R}_{j}=\mathrm{Max}\left[{w}_{j}\left({f}_{j}^{*}-{f}_{ij}\right)/\left({f}_{j}^{*}-{f}_{j}^{-}\right].$ (20)
In Step 4, Qj indices are evaluated, using the relation in Equations (21) and (22). The values of S* may also use the best option S* = 0 and S− the worst value S− = 1. For R* and R−, the best R* = 0 or the worst R− = 1 can also been chosen. The parameter v(0 ≤ v ≤ 1) is introduced as a weight for the strategy of maximum group utility, whereas (1 − v) is the weight of the individual regret. In other words, the increase of v emphasizes group utility (i.e. v> 0.5), whereas v ≈ 0.5 reflects consensus and v< 0.5 a veto. A way to compute v is by the formulae (n + 1)/2.n, as proposed by Opricovic (201143 OPRICOVIC S. 2011. Fuzzy VIKOR with an application to water resources planning. Expert Systems with Applications, 38(10): 12983-12990.). This latter approach was used in the present study leading to a value of v of 0.522)
${Q}_{i}=v\left({S}_{i}-{S}^{-}\right)/\left({S}^{*}-{S}^{-}\right)+\left(1-v\right)\left[\left({S}_{i}-{R}^{-}\right)/\left({R}^{*}-{R}^{-}\right)\right]$ (21)
${S}^{-}=\mathrm{min}\phantom{\rule{0.2em}{0ex}}{S}_{i}\phantom{\rule{0.2em}{0ex}}{S}^{*}=\mathrm{min}\phantom{\rule{0.2em}{0ex}}{S}_{i}\phantom{\rule{0.2em}{0ex}}{R}^{-}=\mathrm{min}\phantom{\rule{0.2em}{0ex}}{R}_{i}\phantom{\rule{0.2em}{0ex}}{R}^{*}=\mathrm{min}\phantom{\rule{0.2em}{0ex}}{R}_{i}.$ (22)
In Step 5, the R, S and Q approaches are combined using the two conditions in Equation (23) and the definition (24),
$\mathrm{Condition}\phantom{\rule{0.3em}{0ex}}1:Q\left({A}_{2}\right)-Q\left({A}_{1}\right)\ge \left(1/\left(n-1\right)\right)$ (23)
where A 1 and A 2 denote the best and the second best alternatives according to the Q indices
$\mathrm{Condition}\phantom{\rule{0.3em}{0ex}}2:\phantom{\rule{0.3em}{0ex}}{A}_{1}\phantom{\rule{0.3em}{0ex}}\mathrm{is}\phantom{\rule{0.3em}{0ex}}\mathrm{also}\phantom{\rule{0.3em}{0ex}}\mathrm{the}\phantom{\rule{0.3em}{0ex}}\mathrm{best}\phantom{\rule{0.3em}{0ex}}\mathrm{in}\phantom{\rule{0.3em}{0ex}}\mathrm{at}\phantom{\rule{0.3em}{0ex}}\mathrm{least}\phantom{\rule{0.3em}{0ex}}\mathrm{one}\phantom{\rule{0.3em}{0ex}}\mathrm{of}\phantom{\rule{0.3em}{0ex}}\mathrm{the}\phantom{\rule{0.3em}{0ex}}R\phantom{\rule{0.2em}{0ex}}\mathrm{and}\phantom{\rule{0.3em}{0ex}}\mathrm{the}\phantom{\rule{0.3em}{0ex}}S\phantom{\rule{0.3em}{0ex}}\mathrm{rankings}.$ (24)
Condition 1 refers to an acceptable advantage and Condition 2 reflects an acceptable stability in the decision making process, where A 1 is also the best in at least one of the R and the S rankings. Alternative A 1 is chosen if Conditions 1 and 2 are satisfied. If one of these two conditions is not satisfied, then a set of compromise solutions is proposed: (1) Alternatives A 1 and A 2, if only Condition 2 is not satisfied; (2) All the alternatives A for which Q(A) − Q(A1)< 1/(n − 1) if Condition 1 is not satisfied (Anvari et al., 20142 ANVARI A, ZULKIFLI N & ARGHISH O. 2014. Application of a modified VIKOR method for decision-making problems in lean tool selection. International Journal of Advanced Manufacturing Technology, 71(5-8): 829-841.).
VIKOR and the Technique of Order Preference by Similarity to the Ideal Solution (TOPSIS) use a similar approach to a MCDM problem. These methods are based on an aggregating function representing ‘closeness to the ideal’, which originated in the compromise programming method. The VIKOR method of compromise ranking is based on the idea of maximizing a ‘group utility’ that reflect the majority choice and a minimum of an individual regret for the ‘opponent’. The TOPSIS method determines a solution with the shortest distance to the ideal solution and the greatest distance from the negative-ideal solution, but it does not consider the relative importance of these distances (Opricovic & Tzeng, 200444 OPRICOVIC S & TZENG GH. 2004. Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. European Journal of Operational Research, 156(2): 445-455.). Considering the similarity and differences of both methods, the VIKOR method was prioritized in this study.
## 2.5 Kendall and Spearman Nonparametric Correlation Indices
Kendall and Spearman indices are used to access the correlation between two variables calculated in a set of N alternatives. The Kendall coefficient was developed by (Kendall, 193825 KENDALL MG. 1938. A new measure of rank correlation. Biometrika, 30(1/2): 81-93.). This coefficient is identified with the Greek letter τ (tau). In Equation (25), the notation Nc represents the amount of pairs of alternatives evaluated for which the ordinal positions are concordant, while Nd indicates the number of discordant pairs. A pair of alternatives is count as concordant whenever the differences between their ordinal positions by the two variables have the same sign and is count as discordant if these differences have different signs. The denominator of equation (25) indicates the amount of possible pairs of alternatives.
$\tau =\frac{{N}_{c}-{N}_{d}}{\frac{N\left(N-1\right)}{2}}$ (25)
The Spearman coefficient is defined in Equation (26), and is represented by the Greek letter ρ (rho). In fact, the Spearman coefficient formula is similar to Pearson’s. The difference comes from the use of ranks RX and RY , instead of X and Y original values. The Spearman coefficient is obtained from the division between the covariance of RX and RY and the product of their standard deviations (Spearman, 1904).
$\rho =\frac{cov\left({R}_{X},{R}_{Y}\right)}{{\sigma }_{{R}_{X}}{\sigma }_{{R}_{Y}}}$ (26)
The Kendall and Spearman coefficients are among the most commonly used ordinal correlation techniques in statistics (Gibbons & Chakraborti, 201121 GIBBONS JD & CHAKRABORTI S. 2011. Nonparametric statistical inference. In: International encyclopedia of statistical science. Springer Berlin Heidelberg, 977-979.; Hauke & Kossowski, 201123 HAUKE J & KOSSOWSKI T. 2011. Comparison of values of Pearson’s and Spearman’s correlation coefficients on the same sets of data. Quaestiones Geographicae, 30(2): 87-93.). These nonparametric methods perform the calculations from the ranks obtained after ordering by the two variables. The correlation results vary in the range [1, −1], in which “−1” depicts the highest negative correlation and “1” the maximum positive correlation. The null value indicates no correlation between the ranks. In Tables 5, 7 and 9 in the next section, the ranks corresponding from the scores obtained by different multicriteria methods are employed to evaluate Kendall and spearman correlation coefficients with the PL ranking.
Similar results would be obtained by employing the nonparametric tests based on these statistics as in Leoneti (201631 LEONETI AB. 2016. Considerations Regarding the Choice of Ranking Multiple Criteria Decision Making Methods. Pesquisa Operacional, 36: 259-277.).
## 2.6 Data Collection
The data used in the study were from all 38 rounds of the FA Premier League season 2015/2016. Official Premier League performance data are collected and analyzed by Opta (optasports.com), part of Perform Content, a division of Perform Group (performgroup.com).
The live data are collected by a three-person team covering each match. Two highly trained analysts use a video-based collection system to collect information about what happens every time a player touches the ball. All the data collected are then subjected to an exhaustive postmatch check to ensure accuracy. After that, the data are available on the FA Premier League web site (premierleague.com/tables). To perform the proposed analysis, the actions of the players during each match were organized by team.
In the Premier League website, a total of 30 variables divided into an attack, defense, discipline and team play can be found. Specifically, 23 discrete technical performance variables were selected for extraction, based on their reporting in previous literature and the perception of their importance by the authors. The variables collected by Opta that are not used in previous studies and are considered in the present study are Big Chances Created, Hit Woodwork, Clean Sheets are Headed Clearance, which have positive impacts and Errors Leading to Goal and Own Goals, with negative impacts.
Table 2 presents a description of the variables employed in the analyses and a source in the literature asserting their importance. For the six variables introduced due to the perception of the authors, a link to the Premier League data is presented.
Table 2
Variables, description and references used in the analyses.
The 23 variables in Table 2 were grouped in three sets, as described in Table 3. These groups of variables were explored by several authors, such as Lago-Ballesteros & Lago-Peñas (201025 KENDALL MG. 1938. A new measure of rank correlation. Biometrika, 30(1/2): 81-93.); Liu, Gomez & Lago-Penas (201535 LIU H, YI Q, GIMÉNEZ J-V, GÓMEZ MA & LAGO-PEÑAS C. 2015. Performance profiles of football teams in the UEFA Champions League considering situational efficiency. International Journal of Performance Analysis in Sport, 15 april, 371-390.); Sgro et al. (201551 SGRO F, BARRESI M & LIPOMA M. 2015. The analysis of discriminant factors related to team match performances in European football Championship 2012. Journal of Physical Education and Sport, 15(3): 460-465.) and Liu et al. (201632 LIU H, GÓMEZ M-A, GONÇALVES B & SAMPAIO J. 2016. Technical performance and match-tomatch variation in elite football teams. Journal of Sports Sciences, 34(6): 509-518.), in order to express a team behavior by distinctive phases during a football match.
Table 3
Three Groups of Performance Variables.
## 2.7 Data Analysis
CPP was applied in two types of analyses. First, the global performance was determined considering all 23 variables together as independent criteria. The Fuzzy MMOORA and the Fuzzy VIKOR, as described in the Section 2, were applied to the same data set to compare performances. A correlation with the official rank of Premier League, season 2015-2016, was a proxy measure to check the adequacy of these methods to the context, as detailed also in Section 2.
The second analysis consisted of using CPP with three dimensions of variables and combining the results thus obtained. The groups of variables (GV) are described in Table 3. Each team received a specific score for each GV. Finally, their GV scores were inputs for another round of composition by CPP. This last stage created a new global ranking. Appendices I to V list the joint probabilities Mij and mij and the final scores, according to points of view described in the Table 1.
Triangular distributions were used in CPP and TFN in the fuzzy MCDA methods, as depicted in Section 2.2. The triangular distribution is a fair approximation to support decision-making in cases where the lowest, highest, and most likely values are available to describe the behavior of a random variable TFN, on the other side, are the most natural way to fuzzify exact measurements.
Both fuzzy methods were modeled in R language and the computations employed the R package “FuzzyMCDM”.
# 3 RESULTS AND DISCUSSION
The rankings derived from probabilistic scores are presented in Table 4 for FA Premier League 2015/2016 season. The scores were initially calculated based on the 23 variables taken together and then for the groups of variables.
Table 4
Ranks of different MCDA methods.
These global scores represent the probabilities of having the largest value in all variables, corresponding to the pessimistic progressive point of view. The decision maker in the pessimistic progressive (PP) point of view pays attention to maximizing the probabilities of preference according to all the criteria examined. The progressive decision maker pays greater attention to small differences between the best values; in sports, this reflects the idea that the teams are oriented to seek the victory in the game. The conservative decision maker is associated with the idea of avoiding losses, which in football is important for the teams avoiding relegation.
The data base was also submitted to the Fuzzy MULTIMOORA and Fuzzy VIKOR methods. The results are also described in Table 4. The correlation between the Premier League Official ranking of the 2015-2016 season and the three MCDM methods is presented in Table 5.
Table 5
Correlation indices.
The present study discusses the perspective of multiple criteria addressing the technical performance of FA Premier League teams. It was expected that the present study would improve the understanding of teams’ performance depicted by final 2015/2016 FA Premiere League standings. In Table 5, these standings are compared to rankings derived from the numerical variables assessing the specific performances in the matches.
The top four teams, qualifying for the Champions League, are the same teams in the same order, whether ranking by points conquered or by CPP. The other methods bring small differences. Fuzzy MMOORA presents a small change in rank and Fuzzy Vikor includes Everton in the fourth position in its ranking.
In regard to relegation, the probabilistic approach is also reliable in reflecting the reality of clubs. The numbers of Newcastle United, Norwich City and Aston Villa were the worst, as reflected in the final league table (relegated to Football League Championship) and when analyzed by the CPP method. The other methods were not able to arrive at such a close adequacy in relation to PL Official Rank.
For the teams in the intermediate group, a comparison is more difficult because these teams form a more homogeneous group, so that small differences in the variables affect the ranks. There is a difference of only 1 point between the teams placed from fifth to tenth and between the 11th and 17th the maximum difference was of only three points. A global comparison may be done using Kendall and Spearman correlation coefficients.
The correlation indices between the Premier League Official and the CPP ranking are higher and reveal the adequacy of this method to the context. It is important to remark that one method cannot here be considered better than another because, despite the graphical resemblance, CPP and Fuzzy MCDA methods embody different concepts. The interpretation of CPP may be easier because uncertainty is directly modeled by CPP. One possible reason for the higher correlation found for the CPP results may be that the use of a density function takes into account in the computations the probabilities of all values while the computation based on TFN takes into account effectively only the central and extreme values. In fact, the CPP algorithm considers all values distributed by a probabilistic law. On the other hand, the fuzzy logic algorithm simplifies computation by prioritizing minimum or maximum values of a fuzzy number, thus excluding information considered in the probabilistic method. In what might seem paradoxical for methods applied to uncertain contexts, precision may have made a difference.
Thus, by the probabilistic method of CPP, a multiple criteria vision is obtained which takes into account variables expected to partially determine match results. This demonstrates that the 23 variables collected by the FA Premier League website are suitable to distinguish winning teams from losing teams.
In a second analysis, CPP was employed to analyze the teams’ performances by the three GV described in Table 3. For each group, a PP point of view is represented in Table 6 and a CP point of view is represented in Table 8. A global score is obtained in both tables by combining the GV results by a PO point of view. This approach was proposed by Sant’Anna (201545 SANT’ANNA AP. 2015. Probabilistic Composition of Preferences, Theory and Applications., p. 40-41) and assumes that a good performance in at least one group of variables may be enough to determine a good global performance. A pessimistic composition of the three GV scores would lead to the same global scores of the PP combination of the 23 variables as a unique group.
Table 6
PP Rank Composition by GV1, GV2, GV3.
Table 7
Correlation Indices to PP Composition Variables.
Table 8
CP Rank Composition by GV1, GV2, GV3.
To obtain information on which kind of variables are more important, Table 6 presents the correlation between each variable group and the 2015/2016 Premier League Final Rank.
Table 6 shows that the group of variables GV1 presents the best correlation with the final standings. This correlation is even higher than that obtained with the 23 variables as shown in Table 6. This leads to the idea that, in a football match, the most important is to make goals, no matter how much score opportunities can be built or disarmed. What matters is the number of effective actions that contribute to directly score a goal.
In Table 6, the observer may analyze the probability of each team being better than its opponents in all the criteria of each group separately. Here, three of the four clubs that guaranteed access to dispute the Champions League in 2016/2017 appear as the top 3 of the rank.
It is interesting to notice that Tottenham Hotspur occupies the ninth position in this rank, which demonstrates the instability of the team throughout its campaign this season. It appears in ninth position in the offensive phase and in the fifth position in the defensive phase, which results in a lower rank in this analysis than those presented in Appendices III and IV. Its superior performance with respect to the variables in the goals scored group, presented in Appendix II - where it can be seen that Tottenham Hotspur occupies a second position with 1.82 goals per match - kept the team disputing the championship until the final rounds, but was not enough to grant it a better final rank under this point of view.
Other teams that call for analysis are Everton, which appears in the PP rank in the fourth position, Southampton, appearing in fifth position and the relegated Aston Villa, in the sixth position. All these three teams presented an interesting offensive organization, with passing accuracy of 78% to 81% and more than 400 passes per match.
Everton showed a median campaign this season, with 11 wins, 14 draws and 13 losses, demonstrated through its PP rank in ninth position in GV1, fourth position in GV2 and eighth position in GV3. In a closer look at the variables the difference of 0.10 between goals scored per match and goals conceded per match will call attention. The balance between making and conceding goals may be a problem to be more deeply investigated by the managers of this club.
In another case, the problems that Southampton presented in its defensive line are registered by its PP rank in thirteenth position. The team is among the top 5 teams in the PP rank for both the offensive phase and construction of goals. With 18 victories, 9 draws and 11 defeats, Southampton ended the competition in sixth place, 3 points behind the fourth team.
Aston Villa is an extreme case. It had its major problems in defense with an impressive number of 2 goals conceded per match. Its construction of goals was obtained with only 0.71 goals per match, ending the competition with a goal difference of minus 49.
The irregularity of Chelsea’s season may also be highlighted here, though its performance in Table 5 with respect to each group of variables is similar. A deeper analysis may reveal a large variability within the variables within the groups, primarily in the defensive phase. The numbers are very close to or worse than those of the relegated teams this season. This poor performance caused the team to change its coach and players during the season in such a way to improve its median campaign.
When the decision maker aims to differentiate alternatives near to worst performance by the idea of avoiding losses and, possibly, relegation, the CP point of view is the more adequate. In this case, none of the correlations presented in Table 8 shows a better performance than that obtained by CPP with the 23 variables. What shows that none of the groups of variables within a conservative view has a strong effect to the 2015/2016 Premier League Final Rank.
In Table 8, attention should be focused on those teams that wanted to avoid relegation. When we combine the conservative pessimistic GV scores, Newcastle United is not found among the possible candidates for relegation. Newcastle United, even with good construction in the offensive phase and being in a ninth position in this group of variables, was not successful in goal conversions, with just 1.16 goals per game and 1.71 goals conceded per match. Fundamentally, in football, goals are required. Of the last 24 available points, Newcastle United only managed to conquer 2 points, which determined the destiny of the team at the end of the season.
West Bromwich Albion was not relegated from the FA Premier League, but, when the variable groups GV1, GV2 and GV3 are analyzed, it appears in the nineteenth position in the CP Rank. This specially demonstrates the fragility of the team during the competition in building their offensive phase, totaling only 10 wins and a goal difference of −14 at the end of competition. This team also demonstrated inefficiency in its defense, with 1.26 goals conceded per match, which exceeds its mark of 0.89 goals scored per match. When analyzed from the CP point of view, West Bromwich Albion could not be better in the variables of GV1 and GV2 groups than other teams presented in Appendices II and III. Thus, even presenting an average performance in GV3, presented in Appendix IV, it appears between the teams that would have been relegated. Thus, its board should be attentive to improving these numbers next season, primarily in their offensive construction and goal conversion.
Combining analyses performed from a CP point of view through an optimistic perspective, the analysis depicted in Table 8 shows the reality of clubs who wish to stay in the elite of English Football, where they want to be better in each of the groups of variables presented. However, long term experience seems to indicate that teams fighting to avoid relegation should care about overall excellence and not only seeking to be better in one group of variables.
Table 9
Correlation Indices to CP Composition Variables.
# 4 CONCLUSION
The purpose of this study was to explore football match related statistics in the Premier League for last season (2015/2016). Three different methods, Composition of Probabilistic Preferences (CPP), Fuzzy Multimoora Method and Fuzzy Vikor Method were employed to determine if it is possible to distinguish winning teams from losing teams on the basis of 23 performance indicators. On this point, CPP proved more efficient in predicting the final standings of the championship. The Spearman index for PL Official Rank × CPP Rank is around 0,85 and the Kendall index around 0,69.
The uncertain and imprecise football variables are well modeled by CPP with a triangular density function rather than by TFN. CPP algorithms consider all values of a probabilistic distribution while fuzzy logic, prioritizing minimum or maximum values of a fuzzy number, explores less information.
In addition, CPP analyses based on goals scored, attack phase and defensive phase from different points of view were performed. By this way, the effects of technical performance of the teams at each moment of a football match may be better understood. Thus, information is gathered to help in-depth understanding each phase of the game for different decision maker’s points of view.
Evidence was found that the variables in the group GV1, related to the final offensive steps of the team, presents a better correlation with the premier League final rank. This means that the best strategy would be to prioritize being able to transpose the defense to score goals against the opponent.
Several other issues can be raised. For instance: Are the teams that make more goals always the best? Better offensive performance always results in scoring more? How much the defensive performance improves the chances of winning the matches?
As a practical application, our findings suggest that technical match analyses are an important factor in the evaluation of team performance for the following season. Being able to observe the failures in each phase by the team may drive correction of methods of training or even acquisition of players for an improvement of performance.
Future investigations should attempt to apply CPP during the championship and not only at the end of the season. Useful information might also be gathered by looking at player individual rankings. In that case, CPP should be applied not only to technical performance variables but also to physical performance variables. Furthermore, it would also be interesting to apply CPP to different leagues, to determine if different aspects may be more important in different contexts.
# ACKNOWLEDGMENTS
The authors are grateful for two Pesquisa Operacional referees, whose insightful analyses we evaluate that considerably improved the article.
# REFERENCES
• 1
ALMEIDA TA, YAMAKAMI A & TAKAHASHI MT. 2006. Sistema imunológico artificial para resolver o problema da árvore geradora mínima com parâmetros fuzzy, Pesquisa Operacional, 27(1): 131-154.
• 2
ANVARI A, ZULKIFLI N & ARGHISH O. 2014. Application of a modified VIKOR method for decision-making problems in lean tool selection. International Journal of Advanced Manufacturing Technology, 71(5-8): 829-841.
• 3
BALEŽENTIS T & BALEŽENTIS A. 2014. A Survey on Development and Applications of the Multicriteria Decision Making Method MULTIMOORA. Journal of Multi-Criteria Decision Analysis, 21(3-4): 209-222.
• 4
BESTERS LM, VAN OURS JC & VAN TUIJL MA. 2016. Effectiveness of In-Season Manager Changes in English Premier League Football. De Economist, 164(3): 335-356.
• 5
BRADLEY PS, CARLING C, ARCHER D, ROBERTS J, DODDS A, DI MASCIO M & KRUSTRUP P. 2011. The effect of playing formation on high-intensity running and technical profiles in English FA Premier League soccer matches. Journal of Sports Sciences, 29(8): 821-830.
• 6
BRAUERS WKM, BALEŽENTIS A & BALEŽENTIS T. 2011. MULTIMOORA for the EU Member States updated with fuzzy number theory. Technological and Economic Development of Economy, 17(2): 259-290.
• 7
BRAUERS WKM & ZAVADSKAS EK. 2006. The MOORA method and its application to privatization in a transition economy. Control and Cybernetics, 35(2): 445-469.
• 8
BRAUERS WKM & ZAVADSKAS EK. 2010. Project management by MULTIMOORA as an instrument for transition economies. Technological and Economic Development of Economy, 16(1): 5-24.
• 9
BRAUERS WKM & ZAVADSKAS EK. 2012a. A multi-objective decision support system for project selection with an application for the Tunisian textile industry. E&M Ekonomie a Management, 15(1): 28-43.
• 10
BRAUERS WKM & ZAVADSKAS EK. 2012b. Robustness of MULTIMOORA: A Method for Multi-Objective Optimization. Informatica, 23(1): 1-25.
• 11
CARLING C, WILLIAMS AM & REILLY T. 2005. Handbook of soccer Match Analysis. A systematic approach to improving performance.
• 12
CASTELLANO J. 2010. Analysis of the possession ball in soccer: frequency, duration and transition. European Journal of Human Movement, 21: 179-196. Retrieved from http://www.eurjhm.com/index.php/eurjhm/article/view/214/401
» http://www.eurjhm.com/index.php/eurjhm/article/view/214/401
• 13
CHIODO E & LAURIA D. 2015. Some Basic Properties of the Failure Rate of Redundant Reliability Systems in Industrial Electronics Applications. IEEE Transactions on Industrial Electronics, 62(8): 5055-5062.
• 14
DELIGNETTE-MULLER ML & DUTANG C. 2014. Fitdistrplus: An R Package for Fitting Distributions. J. Stat. Softw., 64: 1-34.
• 15
DRUST B, ATKINSON G & REILLY T. 2007. Future perspectives in the evaluation of the physiological demands of soccer. Sports Medicine, 37(9): 783-805.
• 16
DUCKSTEIN L & OPRICOVIC S. 1980. Multiobjective optimization in river basin development. Water Resources Research, 16(1): 14-20.
• 17
EAVES SJ. 2015. A history of sports notational analysis: a journey into the nineteenth century. International Journal of Performance Analysis in Sport, 15(3): 1160-1176.
• 18
FERNANDEZ-NAVARRO J, FRADUA L, ZUBILLAGA A, FORD PR & MCROBERT AP. 2016. Attacking and defensive styles of play in soccer: analysis of Spanish and English elite teams. 414 (April).
• 19
GARCIA PA DE A & SANT’ANNA AP. 2015. Vendor and Logistics Provider Selection in the Construction Sector: a Probabilistic Preferences Composition Approach. Pesquisa Operacional, 35(2): 363-375.
• 20
GAVIÃO LO, SILVA RF DA, SANT’ANNA, AP & LIMA GBA. 2016. Ordenação de Municípios por Potencial de Contaminação de Águas com Fármacos Oncológicos por Composição Probabilística de Preferências [Powerpoint slides in Portuguese]. In Simpósio Brasileiro de Pesquisa Operacional, (p. 12). Vitória.
• 21
GIBBONS JD & CHAKRABORTI S. 2011. Nonparametric statistical inference. In: International encyclopedia of statistical science. Springer Berlin Heidelberg, 977-979.
• 22
GOVINDAN K, RAJENDRAN S, SARKIS J & MURUGESAN P. 2015. Multi criteria decision making approaches for green supplier evaluation and selection: a literature review. Journal of Cleaner Production, 98: 66-83.
• 23
HAUKE J & KOSSOWSKI T. 2011. Comparison of values of Pearson’s and Spearman’s correlation coefficients on the same sets of data. Quaestiones Geographicae, 30(2): 87-93.
• 24
JATO-ESPINO D, CASTILLO-LOPEZ E, RODRIGUEZ-HERNANDEZ J & CANTERAS-JORDANA JC. 2014. A review of application of multi-criteria decision making methods in construction. Automation in Construction, 45: 151-162.
• 25
KENDALL MG. 1938. A new measure of rank correlation. Biometrika, 30(1/2): 81-93.
• 26
LAGO-BALLESTEROS J & LAGO-PEÑAS C. 2010. Performance in Team Sports: Identifying the Keys to Success in Soccer. Journal of Human Kinetics, 25(July): 85-91.
• 27
LAGO-PEÑAS C & LAGO-BALLESTEROS J. 2011. Game location and team quality effects on performance profiles in professional soccer. Journal of Sports Science and Medicine, 10(3): 465-471.
• 28
LAGO-PEÑAS C, LAGO-BALLESTEROS J & REY E. 2011. Differences in Performance Indicators between Winning and Losing Teams in the UEFA Champions League. Journal of Human Kinetics, 27(March): 137-148.
• 29
LARKIN P, CONNOR DO & WILLIAMS AM. 2016. Establishing validity and reliability of a movement awareness and technical skill (MATS) analysis instrument in soccer, 191-202.
• 30
LEIVA V, SANTOS-NETO M, CYSNEIROS FJA & BARROS M. 2014. Birnbaum-Saunders statistical modelling: a new approach. Statistical Modelling, 14(1): 21-48.
• 31
LEONETI AB. 2016. Considerations Regarding the Choice of Ranking Multiple Criteria Decision Making Methods. Pesquisa Operacional, 36: 259-277.
• 32
LIU H, GÓMEZ M-A, GONÇALVES B & SAMPAIO J. 2016. Technical performance and match-tomatch variation in elite football teams. Journal of Sports Sciences, 34(6): 509-518.
• 33
LIU H, GOMEZ M-A, LAGO-PEÑAS C & SAMPAIO J. 2015. Match statistics related to winning in the group stage of 2014 Brazil FIFA World Cup. Journal of Sports Sciences, 33(5): 1205-1213.
• 34
LIU H, GOMEZ MA & LAGO-PENAS C. 2015. Match Performance Profiles of Goalkeepers of Elite Football Teams. International Journal of Sports Science & Coaching, 10(4): 669-682.
• 35
LIU H, YI Q, GIMÉNEZ J-V, GÓMEZ MA & LAGO-PEÑAS C. 2015. Performance profiles of football teams in the UEFA Champions League considering situational efficiency. International Journal of Performance Analysis in Sport, 15 april, 371-390.
• 36
MARDANI A, JUSOH A, MD NOR K, KHALIFAH Z, ZAKWAN N & VALIPOUR A. 2015. Multiple criteria decision-making techniques and their applications - a review of the literature from 2000 to 2014. Economic Research-Ekonomska Istraživanja, 28(1): 516-571.
• 37
MARDANI A, JUSOH A & ZAVADSKAS EK. 2015. Fuzzy multiple criteria decision-making techniques and applications - Two decades review from 1994 to 2014. Expert Systems with Applications, 42(8): 4126-4148.
• 38
MARDANI A, JUSOH A, ZAVADSKAS EK, KHALIFAH Z & NOR KMD. 2015. Application of multiple-criteria decision-making techniques and approaches to evaluating of service quality: a systematic review of the literature. Journal of Business Economics and Management, 16(5): 1034-1068.
• 39
MILLARD SP. 2013. EnvStats, an R Package for Environmental Statistics. Wiley Online Library.
• 40
MOURA FA, MARTINS LEB & CUNHA SA. 2014. Analysis of football game-related statistics using multivariate techniques. Journal of Sports Sciences, 32(20): 1881-1887.
• 41
OPRICOVIC S. 1998. Multicriteria optimization of civil engineering systems. Faculty of Civil Engineering, Belgrade, 2(1): 5-21.
• 42
OPRICOVIC S. 2007. A fuzzy compromise solution for multicriteria problems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 15(3): 363-380.
• 43
OPRICOVIC S. 2011. Fuzzy VIKOR with an application to water resources planning. Expert Systems with Applications, 38(10): 12983-12990.
• 44
OPRICOVIC S & TZENG GH. 2004. Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. European Journal of Operational Research, 156(2): 445-455.
• 45
SANT’ANNA AP. 2015. Probabilistic Composition of Preferences, Theory and Applications.
• 46
SANT’ANNA AP, BARBOZA EU & SOARES DE MELLO JCCB. 2010. Classification of the teams in the Brazilian soccer championship by probabilistic criteria composition. Soccer & Society, 11(3): 261-276.
• 47
SANT’ANNA AP, COSTA HG, NEPOMUCENO DO & PEREIRA V. 2016. A probabilistic approach applied to the classification of courses by multiple evaluators, 36: 469-485. https://doi.org/10.1590/0101-7438.2016.036.03.0469
» https://doi.org/10.1590/0101-7438.2016.036.03.0469
• 48
SANT’ANNA AP & GOMES LFAM. 2003. New Evidence on the Formation of Preferences of Betters in horse races. In Encontro Nacional de Engenharia de Produção XXIII ENEGEP, (pp. 1-6). Ouro Preto - MG.
• 49
SANT’ANNA AP & SANT’ANNA LAFP. 2001. Randomization as a stage in criteria combining. In International Conference on Industrial Engineering and Operations Management - VII ICIEOM, (pp. 248-256). Salvador.
• 50
SANT’ANNA AP & SOARES DE MELLO JCCB. 2012. Validating Rankings in Soccer Championships. Pesquisa Operacional, 32(2): 407-422.
• 51
SGRO F, BARRESI M & LIPOMA M. 2015. The analysis of discriminant factors related to team match performances in European football Championship 2012. Journal of Physical Education and Sport, 15(3): 460-465.
• 52
SHAFIZADEH M, TAYLOR M & LAGO-PEÑAS C. 2013. Performance Consistency of International Soccer Teams in Euro 2012: a Time Series Analysis. Journal of Human Kinetics, 38(September): 213-226.
• 53
SILVA RC. 2008. Relação entre modelos de programação não-linear com incerteza no conjunto de restrições. Pesquisa Operacional, 28(3): 383-398.
• 54
SOLTANI A, HEWAGE K, REZA B & SADIQ R. 2015. Multiple stakeholders in multi-criteria decision-making in the context of municipal solid waste management: a review. Waste Management, 35: 318-328.
• 55
SPEARMAN C. 1904. The proof and measurement of association between two things. The American Journal of Psychology, 15(1): 72-101.
• 56
TENGA A, HOLME I, RONGLAN LT & BAHR R. 2010. Effect of playing tactics on achieving scorebox possessions in a random series of team possessions from Norwegian professional soccer matches. Journal of Sports Sciences, 28(3): 245-255.
• 57
WOODS CT, RAYNOR AJ, BRUCE L, MCDONALD Z & ROBERTSON S. 2016. The application of a multi-dimensional assessment approach to talent identification in Australian football. Journal of Sports Sciences, 34(14): 1340-1345.
• 58
YU P-L. 1973. A class of solutions for group decision problems. Management Science, 19(8): 936- 946.
• 59
ZADEH LA. 1965. Information and control. Fuzzy Sets, 8(3): 338-353.
• 60
ZELENY M. 1982. Multi criteria decision making. New York: McGraw-Hills.
# Appendix I - CPP 2nd stage results.
PL 15-16 V1 V2 V3 V4 V5 P.max P.min P.max P.min P.max P.min P.max P.min P.max P.min Arsenal 0.081 0.020 0.060 0.015 0.079 0.016 0.026 0.156 0.264 0.010 Aston Villa 0.012 0.237 0.019 0.171 0.023 0.164 0.029 0.072 0.022 0.177 Bournemouth 0.020 0.043 0.027 0.035 0.029 0.046 0.033 0.038 0.024 0.107 Chelsea 0.043 0.024 0.040 0.020 0.041 0.025 0.039 0.025 0.034 0.029 Crystal Palace 0.016 0.066 0.028 0.033 0.032 0.036 0.039 0.025 0.032 0.034 Everton 0.043 0.024 0.031 0.026 0.042 0.024 0.039 0.025 0.054 0.017 Leicester City 0.123 0.019 0.039 0.021 0.044 0.023 0.291 0.008 0.115 0.012 Liverpool 0.064 0.021 0.142 0.012 0.065 0.017 0.029 0.072 0.044 0.021 Manchester City 0.183 0.017 0.111 0.012 0.078 0.016 0.047 0.019 0.042 0.021 Manchester United 0.024 0.035 0.023 0.058 0.030 0.045 0.033 0.038 0.032 0.034 Newcastle United 0.019 0.046 0.020 0.118 0.032 0.038 0.033 0.038 0.026 0.064 Norwich City 0.016 0.066 0.022 0.073 0.026 0.074 0.026 0.156 0.025 0.078 Southampton 0.043 0.024 0.038 0.021 0.037 0.028 0.033 0.038 0.052 0.018 Stoke City 0.017 0.056 0.022 0.072 0.026 0.077 0.047 0.019 0.036 0.027 Sunderland 0.023 0.037 0.024 0.049 0.029 0.049 0.033 0.038 0.027 0.054 Swansea City 0.018 0.053 0.024 0.048 0.028 0.057 0.047 0.019 0.026 0.064 Tottenham Hotspur 0.141 0.018 0.233 0.011 0.261 0.011 0.047 0.019 0.059 0.016 Watford 0.017 0.061 0.025 0.044 0.029 0.050 0.060 0.015 0.026 0.071 West Bromwich Albion 0.014 0.110 0.020 0.144 0.022 0.184 0.026 0.156 0.024 0.119 West Ham United 0.081 0.020 0.052 0.016 0.048 0.021 0.039 0.025 0.036 0.027 PL 15-16 V7 V8 V9 V10 V11 P.max P.min P.max P.min P.max P.min P.max P.min P.max P.min Arsenal 0.222 0.013 0.214 0.013 0.046 0.023 0.027 0.041 0.067 0.021 Aston Villa 0.019 0.077 0.017 0.134 0.044 0.024 0.031 0.033 0.104 0.018 Bournemouth 0.021 0.058 0.019 0.086 0.026 0.088 0.027 0.041 0.032 0.032 Chelsea 0.016 0.205 0.023 0.043 0.051 0.021 0.052 0.023 0.019 0.111 Crystal Palace 0.030 0.028 0.021 0.058 0.051 0.021 0.019 0.094 0.047 0.024 Everton 0.090 0.016 0.026 0.034 0.060 0.019 0.030 0.034 0.019 0.123 Leicester City 0.027 0.032 0.069 0.017 0.030 0.048 0.196 0.016 0.217 0.016 Liverpool 0.055 0.019 0.030 0.029 0.023 0.183 0.203 0.016 0.021 0.071 Manchester City 0.046 0.020 0.099 0.015 0.025 0.123 0.030 0.034 0.030 0.035 Manchester United 0.023 0.046 0.214 0.013 0.026 0.088 0.038 0.027 0.026 0.045 Newcastle United 0.018 0.106 0.021 0.058 0.043 0.025 0.054 0.022 0.033 0.031 Norwich City 0.027 0.032 0.016 0.204 0.033 0.039 0.016 0.208 0.017 0.216 Southampton 0.034 0.025 0.035 0.024 0.028 0.065 0.027 0.039 0.052 0.023 Stoke City 0.017 0.150 0.026 0.034 0.050 0.021 0.027 0.040 0.024 0.053 Sunderland 0.025 0.038 0.019 0.086 0.265 0.011 0.041 0.026 0.023 0.056 Swansea City 0.039 0.022 0.023 0.043 0.047 0.022 0.019 0.116 0.041 0.026 Tottenham Hotspur 0.121 0.015 0.042 0.021 0.026 0.096 0.063 0.021 0.033 0.031 Watford 0.023 0.046 0.030 0.029 0.050 0.021 0.055 0.022 0.098 0.019 West Bromwich Albion 0.025 0.038 0.030 0.029 0.037 0.031 0.019 0.107 0.058 0.022 West Ham United 0.121 0.015 0.030 0.029 0.039 0.029 0.027 0.040 0.038 0.028 PL 15-16 V12 V13 V20 V21 V6 P.max P.min P.max P.min P.max P.min P.max P.min P.max P.min Arsenal 0.025 0.075 0.026 0.080 0.044 0.030 0.204 0.014 0.030 0.044 Arsenal 0.035 0.033 0.035 0.029 0.099 0.021 0.023 0.041 0.037 0.035 Bournemouth 0.032 0.036 0.046 0.021 0.031 0.043 0.034 0.025 0.029 0.045 Chelsea 0.022 0.139 0.024 0.149 0.045 0.029 0.081 0.017 0.027 0.051 Crystal Palace 0.034 0.034 0.035 0.029 0.046 0.029 0.018 0.080 0.214 0.019 Everton 0.037 0.030 0.038 0.027 0.024 0.075 0.041 0.022 0.034 0.037 Leicester City 0.045 0.025 0.043 0.022 0.215 0.017 0.017 0.102 0.031 0.042 Liverpool 0.026 0.061 0.025 0.111 0.067 0.023 0.085 0.017 0.034 0.038 Manchester City 0.031 0.039 0.044 0.022 0.039 0.032 0.131 0.015 0.062 0.026 Manchester United 0.020 0.199 0.023 0.171 0.022 0.119 0.102 0.016 0.035 0.037 Newcastle United 0.031 0.039 0.032 0.038 0.025 0.068 0.022 0.044 0.033 0.039 Norwich City 0.138 0.016 0.264 0.010 0.024 0.077 0.020 0.056 0.042 0.031 Southampton 0.060 0.021 0.043 0.023 0.045 0.029 0.025 0.034 0.183 0.020 Stoke City 0.023 0.094 0.027 0.064 0.036 0.036 0.028 0.030 0.022 0.077 Sunderland 0.241 0.014 0.061 0.017 0.039 0.033 0.016 0.146 0.017 0.239 Swansea City 0.048 0.023 0.052 0.018 0.019 0.206 0.044 0.021 0.027 0.050 Tottenham Hotspur 0.046 0.024 0.056 0.018 0.028 0.052 0.052 0.019 0.028 0.049 Watford 0.031 0.041 0.025 0.114 0.049 0.027 0.020 0.055 0.024 0.063 West Bromwich Albion 0.037 0.030 0.048 0.020 0.040 0.032 0.015 0.201 0.052 0.028 West Ham United 0.039 0.029 0.054 0.018 0.063 0.024 0.022 0.045 0.039 0.033 PL 15-16 V14 V15 V16 V17 V18 P.max P.min P.max P.min P.max P.min P.max P.min P.max P.min Arsenal 0.067 0.021 0.025 0.075 0.026 0.080 0.021 0.138 0.025 0.050 Arsenal 0.104 0.018 0.035 0.033 0.035 0.029 0.246 0.009 0.235 0.014 Bournemouth 0.032 0.032 0.032 0.036 0.046 0.021 0.091 0.012 0.050 0.022 Chelsea 0.019 0.111 0.022 0.139 0.024 0.149 0.036 0.022 0.042 0.024 Crystal Palace 0.047 0.024 0.034 0.034 0.035 0.029 0.033 0.024 0.042 0.024 Everton 0.019 0.123 0.037 0.030 0.038 0.027 0.039 0.019 0.050 0.022 Leicester City 0.217 0.016 0.045 0.025 0.043 0.022 0.021 0.138 0.018 0.213 Liverpool 0.021 0.071 0.026 0.061 0.025 0.111 0.032 0.026 0.063 0.019 Manchester City 0.030 0.035 0.031 0.039 0.044 0.022 0.024 0.064 0.025 0.050 Manchester United 0.026 0.045 0.020 0.199 0.023 0.171 0.020 0.161 0.019 0.147 Newcastle United 0.033 0.031 0.031 0.039 0.032 0.038 0.075 0.013 0.028 0.040 Norwich City 0.017 0.216 0.138 0.016 0.264 0.010 0.091 0.012 0.042 0.024 Southampton 0.052 0.023 0.060 0.021 0.043 0.023 0.024 0.064 0.031 0.033 Stoke City 0.024 0.053 0.023 0.094 0.027 0.064 0.039 0.019 0.021 0.098 Sunderland 0.023 0.056 0.241 0.014 0.061 0.017 0.059 0.014 0.031 0.033 Swansea City 0.041 0.026 0.048 0.023 0.052 0.018 0.034 0.023 0.031 0.033 Tottenham Hotspur 0.033 0.031 0.046 0.024 0.056 0.018 0.020 0.161 0.028 0.040 Watford 0.098 0.019 0.031 0.041 0.025 0.114 0.032 0.026 0.031 0.033 West Bromwich Albion 0.058 0.022 0.037 0.030 0.048 0.020 0.029 0.030 0.022 0.068 West Ham United 0.038 0.028 0.039 0.029 0.054 0.018 0.033 0.024 0.168 0.015 PL 15-16 V19 V20 V21 V22 V23 P.max P.min P.max P.min P.max P.min P.max P.min P.max P.min Arsenal 0.051 0.014 0.014 0.243 0.047 0.020 0.025 0.048 0.051 0.023 Arsenal 0.103 0.011 0.197 0.019 0.031 0.026 0.046 0.026 0.014 0.229 Bournemouth 0.020 0.156 0.023 0.050 0.018 0.081 0.016 0.197 0.023 0.043 Chelsea 0.034 0.022 0.029 0.036 0.092 0.016 0.062 0.023 0.066 0.021 Crystal Palace 0.103 0.011 0.033 0.033 0.023 0.040 0.065 0.022 0.039 0.026 Everton 0.025 0.044 0.016 0.128 0.092 0.016 0.016 0.218 0.018 0.079 Leicester City 0.020 0.156 0.019 0.078 0.031 0.026 0.026 0.044 0.016 0.119 Liverpool 0.025 0.044 0.036 0.031 0.031 0.026 0.067 0.022 0.071 0.020 Manchester City 0.025 0.044 0.036 0.031 0.015 0.203 0.048 0.025 0.022 0.049 Manchester United 0.034 0.022 0.050 0.026 0.018 0.081 0.208 0.017 0.031 0.031 Newcastle United 0.034 0.022 0.033 0.033 0.092 0.016 0.031 0.035 0.020 0.061 Norwich City 0.034 0.022 0.036 0.031 0.031 0.026 0.026 0.046 0.084 0.020 Southampton 0.020 0.156 0.028 0.038 0.220 0.013 0.028 0.040 0.131 0.018 Stoke City 0.020 0.156 0.021 0.058 0.047 0.020 0.028 0.040 0.204 0.016 Sunderland 0.025 0.044 0.045 0.027 0.023 0.040 0.037 0.030 0.058 0.022 Swansea City 0.255 0.008 0.033 0.033 0.018 0.081 0.027 0.043 0.041 0.025 Tottenham Hotspur 0.051 0.014 0.125 0.021 0.015 0.203 0.057 0.023 0.028 0.034 Watford 0.034 0.022 0.146 0.020 0.031 0.026 0.126 0.019 0.017 0.101 West Bromwich Albion 0.051 0.014 0.050 0.026 0.031 0.026 0.037 0.030 0.025 0.039 West Ham United 0.034 0.022 0.029 0.036 0.092 0.016 0.024 0.052 0.041 0.025 Observation: Goals Conceded (V17), Errors Leading to Goal (V18), Own Goals (V19), Yellow Cards (V20), Red Cards (V21), Fouls (V22) and Offside (V23) are variables that have a negative impact, therefore a lower value is better for the team.
# Appendix II - CPP 3rd stage results GV1.
GV1 PP PP Rank PO PO Rank CP CP Rank CO CO Rank Arsenal 2.187E-10 4 5.136E-01 3 7.168E-01 9 0.999999999999741 5 Aston Villa 3.942E-15 20 1.281E-01 20 2.343E-01 20 0.999999780962030 20 Bournemouth 4.550E-13 12 2.853E-01 11 6.749E-01 13 0.999999999985799 15 Chelsea 4.978E-13 11 2.218E-01 12 6.452E-01 14 0.999999999996882 11 Crystal Palace 1.099E-13 17 1.881E-01 17 6.859E-01 12 0.999999999992223 13 Everton 3.727E-12 9 2.881E-01 10 7.913E-01 5 0.999999999999696 7 Leicester City 7.584E-09 1 6.816E-01 1 8.499E-01 2 0.999999999999983 1.5 Liverpool 2.108E-11 6 3.709E-01 6 7.652E-01 6 0.999999999999706 6 Manchester City 4.814E-10 3 4.824E-01 4 8.523E-01 1 0.999999999999981 3 Manchester United 6.244E-12 8 3.874E-01 5 7.401E-01 7 0.999999999997944 9 Newcastle United 8.274E-14 18 1.810E-01 18 6.049E-01 16 0.999999999942001 17 Norwich City 4.170E-14 19 1.626E-01 19 5.506E-01 18 0.999999999766941 18 Southampton 2.264E-11 5 3.673E-01 7 8.077E-01 4 0.999999999999803 4 Stoke City 2.345E-12 10 3.449E-01 8 6.219E-01 15 0.999999999985841 14 Sunderland 2.710E-13 14 2.043E-01 14 7.159E-01 10 0.999999999994008 12 Swansea City 1.416E-13 15 2.008E-01 15 5.686E-01 17 0.999999999983690 16 Tottenham Hotspur 4.452E-09 2 6.761E-01 2 8.395E-01 3 0.999999999999983 1.5 Watford 2.973E-13 13 2.111E-01 13 7.262E-01 8 0.999999999996923 10 West Bromwich Albion 1.121E-13 16 1.993E-01 16 4.545E-01 19 0.999999999416289 19 West Ham United 7.610E-12 7 3.423E-01 9 7.042E-01 11 0.999999999999514 8
# Appendix III - CPP 3rd stage results GV2.
GV1 PP PP Rank PO PO Rank CP CP Rank CO CO Rank Arsenal 3.666E-05 1 4.446E-01 1 8.861E-01 5 0.999999692650327 3 Aston Villa 4.236E-06 6 2.902E-01 2 7.509E-01 17 0.999996422713184 15 Bournemouth 1.003E-06 14 1.234E-01 15 8.121E-01 13 0.999997197567777 14 Chelsea 1.567E-06 11 1.547E-01 12 8.464E-01 8 0.999998368503557 9 Crystal Palace 3.184E-06 8 2.722E-01 3 8.378E-01 10 0.999997994219116 11 Everton 6.003E-06 4 1.934E-01 7 9.088E-01 2 0.999999748096891 2 Leicester City 7.315E-06 3 2.579E-01 5 8.357E-01 11 0.999999136470623 6 Liverpool 2.567E-06 10 1.716E-01 9 8.609E-01 7 0.999999084912791 7 Manchester City 1.678E-05 2 2.577E-01 6 9.187E-01 1 0.999999820804166 1 Manchester United 3.467E-06 7 1.868E-01 8 8.878E-01 4 0.999999393016960 5 Newcastle United 1.160E-06 13 1.351E-01 13 8.432E-01 9 0.999997859757944 12 Norwich City 4.203E-07 18 1.031E-01 18 7.723E-01 16 0.999988473080744 17 Southampton 4.320E-06 5 2.584E-01 4 8.081E-01 15 0.999998458732760 8 Stoke City 3.627E-07 19 9.842E-02 19 6.935E-01 18 0.999987233705528 18 Sunderland 1.617E-07 20 7.961E-02 20 5.794E-01 20 0.999891385142310 20 Swansea City 7.880E-07 15 1.170E-01 16 8.348E-01 12 0.999997241864082 13 Tottenham Hotspur 2.844E-06 9 1.615E-01 11 8.922E-01 3 0.999999572365489 4 Watford 1.262E-06 12 1.622E-01 10 8.087E-01 14 0.999995829855388 16 West Bromwich Albion 7.255E-07 17 1.241E-01 14 6.676E-01 19 0.999983557029980 19 West Ham United 7.643E-07 16 1.163E-01 17 8.615E-01 6 0.999998353540662 10
# Appendix IV - CPP 3rd stage results GV3.
GV1 PP PP Rank PO PO Rank CP CP Rank CO CO Rank Arsenal 5.624E-15 2 5.850E-01 3 6.735E-01 3 1.000000000000000 1 Aston Villa 9.014E-17 14 3.777E-01 11 4.917E-01 17 0.999999999999999 13 Bournemouth 6.031E-16 7 4.575E-01 9 6.439E-01 6 1.000000000000000 1 Chelsea 1.784E-17 19 3.086E-01 18 4.705E-01 18 0.999999999999996 18 Crystal Palace 4.481E-17 16 3.194E-01 16 6.254E-01 7 1.000000000000000 1 Everton 5.558E-16 8 4.938E-01 6 6.033E-01 10 1.000000000000000 1 Leicester City 1.088E-13 1 6.616E-01 1 7.614E-01 1 1.000000000000000 1 Liverpool 6.094E-16 6 5.541E-01 4 5.099E-01 16 0.999999999999999 13 Manchester City 1.586E-15 4 4.917E-01 7 6.530E-01 5 1.000000000000000 1 Manchester United 7.361E-17 15 4.308E-01 10 3.521E-01 19 0.999999999999972 20 Newcastle United 1.666E-17 20 2.966E-01 20 5.603E-01 12 0.999999999999999 13 Norwich City 1.107E-16 11 4.990E-01 5 3.392E-01 20 0.999999999999984 19 Southampton 9.027E-17 13 3.416E-01 15 5.615E-01 11 1.000000000000000 1 Stoke City 1.887E-17 18 3.017E-01 19 5.291E-01 14 0.999999999999998 17 Sunderland 2.496E-15 3 5.924E-01 2 6.679E-01 4 1.000000000000000 1 Swansea City 1.453E-16 10 3.615E-01 12 5.467E-01 13 1.000000000000000 1 Tottenham Hotspur 1.544E-15 5 4.903E-01 8 6.113E-01 9 1.000000000000000 1 Watford 9.272E-17 12 3.594E-01 13 5.271E-01 15 0.999999999999999 13 West Bromwich Albion 4.454E-17 17 3.185E-01 17 6.201E-01 8 1.000000000000000 1 West Ham United 1.673E-16 9 3.556E-01 14 6.878E-01 2 1.000000000000000 1
# Appendix V - CPP 3rd stage results Composition.
Compositions of Dimensions PP Goals PP Offensive PP Defensive PO Composition PP Rank PO Goals PO Offensive PO Defensive PO Composition PO Rank Arsenal 2.19E-10 3.67E-05 5.62E-15 3.67E-05 1 5.14E-01 4.45E-01 5.85E-01 8.88E-01 2 Aston Villa 3.94E-15 4.24E-06 9.01E-17 4.24E-06 6 1.28E-01 2.90E-01 3.78E-01 6.15E-01 13 Bournemouth 4.55E-13 1.00E-06 6.03E-16 1.00E-06 14 2.85E-01 1.23E-01 4.58E-01 6.60E-01 10 Chelsea 4.98E-13 1.57E-06 1.78E-17 1.57E-06 11 2.22E-01 1.55E-01 3.09E-01 5.45E-01 18 Crystal Palace 1.10E-13 3.18E-06 4.48E-17 3.18E-06 8 1.88E-01 2.72E-01 3.19E-01 5.98E-01 14 Everton 3.73E-12 6.00E-06 5.56E-16 6.00E-06 4 2.88E-01 1.93E-01 4.94E-01 7.09E-01 7 Leicester City 7.58E-09 7.32E-06 1.09E-13 7.32E-06 3 6.82E-01 2.58E-01 6.62E-01 9.20E-01 1 Liverpool 2.11E-11 2.57E-06 6.09E-16 2.57E-06 10 3.71E-01 1.72E-01 5.54E-01 7.68E-01 5 Manchester City 4.81E-10 1.68E-05 1.59E-15 1.68E-05 2 4.82E-01 2.58E-01 4.92E-01 8.05E-01 4 Manchester United 6.24E-12 3.47E-06 7.36E-17 3.47E-06 7 3.87E-01 1.87E-01 4.31E-01 7.16E-01 6 Newcastle United 8.27E-14 1.16E-06 1.67E-17 1.16E-06 13 1.81E-01 1.35E-01 2.97E-01 5.02E-01 20 Norwich City 4.17E-14 4.20E-07 1.11E-16 4.20E-07 18 1.63E-01 1.03E-01 4.99E-01 6.24E-01 12 Southampton 2.26E-11 4.32E-06 9.03E-17 4.32E-06 5 3.67E-01 2.58E-01 3.42E-01 6.91E-01 9 Stoke City 2.35E-12 3.63E-07 1.89E-17 3.63E-07 19 3.45E-01 9.84E-02 3.02E-01 5.88E-01 15 Sunderland 2.71E-13 1.62E-07 2.50E-15 1.62E-07 20 2.04E-01 7.96E-02 5.92E-01 7.02E-01 8 Swansea City 1.42E-13 7.88E-07 1.45E-16 7.88E-07 15 2.01E-01 1.17E-01 3.61E-01 5.49E-01 17 Tottenham Hotspur 4.45E-09 2.84E-06 1.54E-15 2.85E-06 9 6.76E-01 1.61E-01 4.90E-01 8.62E-01 3 Watford 2.97E-13 1.26E-06 9.27E-17 1.26E-06 12 2.11E-01 1.62E-01 3.59E-01 5.77E-01 16 West Bromwich Albion 1.12E-13 7.25E-07 4.45E-17 7.25E-07 17 1.99E-01 1.24E-01 3.18E-01 5.22E-01 19 West Ham United 7.61E-12 7.64E-07 1.67E-16 7.64E-07 16 3.42E-01 1.16E-01 3.56E-01 6.25E-01 11 Arsenal 7.17E-01 8.86E-01 6.74E-01 9.89E-01 5 1 0.999999693 1 1.00 1 Aston Villa 2.34E-01 7.51E-01 4.92E-01 9.03E-01 20 0.999999781 0.999996423 1 1.00 1 Bournemouth 6.75E-01 8.12E-01 6.44E-01 9.78E-01 11 1 0.999997198 1 1.00 1 Chelsea 6.45E-01 8.46E-01 4.70E-01 9.71E-01 14 1 0.999998369 1 1.00 1 Crystal Palace 6.86E-01 8.38E-01 6.25E-01 9.81E-01 10 1 0.999997994 1 1.00 1 Everton 7.91E-01 9.09E-01 6.03E-01 9.92E-01 4 1 0.999999748 1 1.00 1 Leicester City 8.50E-01 8.36E-01 7.61E-01 9.94E-01 2 1 0.999999136 1 1.00 1 Liverpool 7.65E-01 8.61E-01 5.10E-01 9.84E-01 7 1 0.999999085 1 1.00 1 Manchester City 8.52E-01 9.19E-01 6.53E-01 9.96E-01 1 1 0.999999821 1 1.00 1 Manchester United 7.40E-01 8.88E-01 3.52E-01 9.81E-01 9 1 0.999999393 1 1.00 1 Newcastle United 6.05E-01 8.43E-01 5.60E-01 9.73E-01 13 1 0.99999786 1 1.00 1 Norwich City 5.51E-01 7.72E-01 3.39E-01 9.32E-01 18 1 0.999988473 1 1.00 1 Southampton 8.08E-01 8.08E-01 5.61E-01 9.84E-01 8 1 0.999998459 1 1.00 1 Stoke City 6.22E-01 6.94E-01 5.29E-01 9.45E-01 17 1 0.999987234 1 1.00 1 Sunderland 7.16E-01 5.79E-01 6.68E-01 9.60E-01 16 1 0.999891385 1 1.00 1 Swansea City 5.69E-01 8.35E-01 5.47E-01 9.68E-01 15 1 0.999997242 1 1.00 1 Tottenham Hotspur 8.39E-01 8.92E-01 6.11E-01 9.93E-01 3 1 0.999999572 1 1.00 1 Watford 7.26E-01 8.09E-01 5.27E-01 9.75E-01 12 1 0.99999583 1 1.00 1 West Bromwich Albion 4.54E-01 6.68E-01 6.20E-01 9.31E-01 19 0.999999999 0.999983557 1 1.00 1 West Ham United 7.04E-01 8.62E-01 6.88E-01 9.87E-01 6 1 0.999998354 1 1.00 1
# Appendix VI - R Code.
#### PROBABILISTIC COMPOSITION OF PREFERENCES ####
########### PREMIER LEAGUE 2015-2016 #################
### Main Reference
### Sant’Anna, A. P. (2015).
"Probabilistic Composition of Preferences, Theory and Applications." New York: Springer.
### "R" packages used in the modeling require(readxl)
require(triangle)
### Selecting the working directory where the database is located path = "COPY AND PASTE THE WORKING DIRECTORY"
setwd(path)
#####################
### CPP 1st STAGE ###
### Randomization of evaluations with triangular distributions
### Triangular PDF parameters: dtriangle(x,"min","max","mode")
### Triangular CDF parameters: ptriangle(x,"min","max","mode")
#####################
### CPP 2nd STAGE ###
### Importing database from positive impact criteria
PL.1 = read_excel("FILE NAME.xlsx", col_names = FALSE, sheet = "SHEET NUMBER") min.1 = apply(PL.1,2,min)
max.1 = apply(PL.1,2,max)
### Loop for computing the Prob.Max for each alternative/each positive criterion Prob.Max.p = PL.1
for (j in 1:ncol(PL.1))
{
for (i in 1:nrow (PL.1))
{
Prob.Max.p[i,j] = (integrate(Vectorize(function(x)
{prod(ptriangle(x,min.1[j]*0.99,max.1[j]*1.01,PL.1[,j][-i]))
*dtriangle(x,min.1[j]*0.99,max.1[j]*1.01,PL.1[,j][[i]])}),min.1[j]*0.99,max.1[j]*1.01))
$value }} ### Loop for computing the Prob.Min for each alternative/each positive criterion Prob.Min.p = PL.1 for (j in 1:ncol(PL.1)) { for (i in 1:nrow (PL.1)) { Prob.Min.p[i,j]=(integrate(Vectorize(function(x) {prod(1-ptriangle(x,min.1[j]*0.99,max.1[j]*1.01,PL.1[,j][-i])) *dtriangle(x,min.1[j]*0.99,max.1[j]*1.01,PL.1[,j][[i]])}),min.1[j]*0.99,max.1[j]*1.01))$value
}}
### Importing database from negative impact criteria
PL.2 = read_excel("FILE NAME.xlsx", col_names = FALSE, sheet = "SHEET NUMBER") min.2 = apply(PL.2,2,min)
max.2 = apply(PL.2,2,max)
### Loop for computing the Prob.Min for each alternative/each negative criterion Prob.Min.n = PL.2
for (j in 1:ncol(PL.2))
{
for (i in 1:nrow (PL.2))
{
Prob.Min.n[i,j]=(integrate(Vectorize(function(x)
{prod(1-ptriangle(x,min.2[j]*0.99,max.2[j]*1.01,PL.2[,j][-i]))
*dtriangle(x,min.2[j]*0.99,max.2[j]*1.01,PL.2[,j][[i]])}),min.2[j]*0.99,max.2[j]*1.01))
$value }} ### Loop for computing the Prob.Max for each alternative/each negative criterion Prob.Max.n = PL.2 for (j in 1:ncol(PL.2)) { for (i in 1:nrow (PL.2)) { Prob.Max.n[i,j]=(integrate(Vectorize(function(x) {prod(ptriangle(x,min.2[j]*0.99,max.2[j]*1.01,PL.2[,j][-i])) *dtriangle(x,min.2[j]*0.99,max.2[j]*1.01,PL.2[,j][[i]])}),min.2[j]*0.99,max.2[j]*1.01))$value
}}
### 2nd stage results for all criteria (Progressive-Conservative axis) Probs.PROG = cbind(Prob.Max.p,Prob.Min.n)
Probs.CONS = cbind(Prob.Min.p,Prob.Max.n)
#####################
### CPP 3rd STAGE ###
### Point of view "PP" ### PP = apply(Probs.PROG,1,prod) PP.rank = rank(-PP)
### Point of view "PO" ### Probs.m = 1-Probs.PROG
PO = 1-(apply(Probs.m,1,prod)) PO.rank = rank(-PO)
### Point of view "CP" ### Probs.mm = 1-Probs.CONS
CP = apply(Probs.mm,1,prod) CP.rank = rank(-CP)
### Point of view "CO" ###
CO = 1-(apply(Probs.CONS,1,prod)) CO.rank = rank(-CO)
# Publication Dates
• Publication in this collection
May-Aug 2017
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2022-05-27 16:14:23
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https://www.gradesaver.com/alice-in-wonderland/q-and-a/how-does-alice-respond-to-falling-through-the-impossibility-long-rabbit-hole-363592
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# How does Alice respond to falling through the impossibility long rabbit hole?
This question is from the story "Alice through the looking glass"
Well!' thought Alice to herself, after such a fall as this, I shall think nothing of tumbling down stairs! How brave they'll all think me at home! Why, I wouldn't say anything about it, even if I fell off the top of the house!' (Which was very likely true.)
I wonder how many miles I've fallen by this time?' she said aloud. I must be getting somewhere near the centre of the earth. Let me see: that would be four thousand miles down, I think--' (for, you see, Alice had learnt several things of this sort in her lessons in the schoolroom, and though this was not a very good opportunity for showing off her knowledge, as there was no one to listen to her, still it was good practice to say it over) --yes, that's about the right distance--but then I wonder what Latitude or Longitude I've got to?' (Alice had no idea what Latitude was, or Longitude either, but thought they were nice grand words to say.)
Down, down, down. There was nothing else to do, so Alice soon began talking again. Dinah'll miss me very much to-night, I should think!' (Dinah was the cat.) I hope they'll remember her saucer of milk at tea-time. Dinah my dear! I wish you were down here with me! There are no mice in the air, I'm afraid, but you might catch a bat, and that's very like a mouse, you know. But do cats eat bats, I wonder?' And here Alice began to get rather sleepy, and went on saying to herself, in a dreamy sort of way, Do cats eat bats? Do cats eat bats?' and sometimes, Do bats eat cats?' for, you see, as she couldn't answer either question, it didn't much matter which way she put it. She felt that she was dozing off, and had just begun to dream that she was walking hand in hand with Dinah, and saying to her very earnestly, Now, Dinah, tell me the truth: did you ever eat a bat?' when suddenly, thump! thump! down she came upon a heap of sticks and dry leaves, and the fall was over.
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2018-08-17 14:07:10
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https://math.stackexchange.com/questions/779095/directional-derivatives-of-a-multivariate-function-not-defined-at-0-0
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# Directional derivatives of a multivariate function not defined at $(0,0)$
Let $$f(x,y)=\left\{ \begin{matrix} \frac{x^2y}{x^4+y^2} & (x,y)\neq(0,0) \\0 & (x,y)=(0,0)\end{matrix}\right.$$
It is easy to prove that the $f$ is not continuous at $(0,0)$ (doing the limit along the curve $y=x^2$).
I want to know whether it is possible to define the partial derivatives of $f$ at $(0,0)$ and find the directions $\vec v$ such that $D_vf(0,0)$ is defined.
I've calculated the partial derivatives of $f$ for $(x,y)\neq(0,0)$: $$\frac{\partial f}{\partial x}=\frac{2xy(x^4+y^2)-4yx^5}{(x^4+y^2)^2}$$ $$\frac{\partial f}{\partial y}=\frac{x^2(x^4+y^2)-2x^2y^2}{(x^4+y^2)^2}$$
Neither of them is continuous at $(0,0)$. However, if we take for example the line $y=x$, then $$\lim_{(x,y)\to (0,0)}_{x=y}\frac{\partial f}{\partial x}=-2$$ $$\lim_{(x,y)\to (0,0)}_{x=y}\frac{\partial f}{\partial y}=-1$$
So I'm tempted to say that even though the partial derivatives do not exist at the origin, $D_{(1/\sqrt2),1/\sqrt2)}f(0,0)=\frac{1}{\sqrt2}(-2,-1)$.
Is this correct? If so, how could I find all the vectors $\vec v$ such that $D_\vec v f(0,0)$ is defined
• Directional derivative should be a number, not a vector. – Braindead May 3 '14 at 2:08
I found that $$\frac{f(0 + hv) - f(0)}{h} = \frac{v_1^2 v_2}{h^2v_1^4+v_2^2}$$
Thus, $$\lim_{h \to 0} \frac{f(0 + hv) - f(0)}{h} = \left\{\begin{matrix} v_1^2/v_2 & v_2 \neq 0 \\ 0 & v_2 = 0 \end{matrix}\right.$$
So it looks like the directional derivative exists for all $v \neq 0.$
• Wow thanks I'd never thought about using the definition. Thanks so much!! – Francisco May 3 '14 at 2:10
I worked out something related to this a few days ago; first, it is up to context whether to say that there are directional derivatives or not, and, say, the book you are reading. If we take a vector $(a,b)$ and take its path $(at,bt)$ through the origin, we get $t^3 a^2 b / (t^2 (a^4 t^2 + b^2)) = t a^2 b / (a^4 t^2 + b^2)$ and this, divided by $t,$ approaches $a^2 / b$ as $t \rightarrow 0.$
Next, slightly varied example:
Before I forget, $(x^2 \pm y)^2 \geq 0, 2 x^2 |y| \leq x^4 + y^2,$ so, below, $f \leq x^{2/5} \; |y|^{1/5} /2.$
Take $$f(0,0) = 0, \; \mbox{otherwise} \; \; f(x,y) = \frac{x^{12/5} \; y^{6/5}}{x^4 + y^2}$$
Using polar coordinates, with $r$ for usual positive radius, letter $c$ for $\cos \theta,$ letter $s$ for $\sin \theta,$ we find $$f(rc,rs) = \frac{r^{18/5} c^{12/5} s^{6/5} }{r^2 (c^4 r^2 + s^2)} = r^{8/5} \left( \frac{ c^{12/5} s^{6/5} }{ c^4 r^2 + s^2} \right)$$ So, $$\frac{f(rc,rs)}{r} = r^{3/5} \left( \frac{ c^{12/5} s^{6/5} }{ c^4 r^2 + s^2} \right) \leq r^{3/5} \left( \frac{ c^{12/5} }{ s^{4/5}} \right).$$ This says that $f$ is Gateaux differentiable. Not only that, all directional derivatives are $0.$ That is, the Gateaux derivatives obey the required linear relationship, the directional derivative in the direction of a vector $\vec{u} + \vec{v}$ really is the sum of the directional derivatives in the two directions $\vec{u}$ and $\vec{v}.$
However, even this strong condition is not enough to guarantee Frechet differentiability. Consider the function along a parabolic path $x=t,y = t^2.$ The distance of this from the origin is $\sqrt {t^2 + t^4},$ which is very close to $|t|$ as $t \rightarrow 0.$ For Frechet differentiablity, we want $f(a + h) = f(a) + \nabla f \cdot h + o(|h|).$ As $a=0,$ $f(0)=0,$ and $\nabla f = 0,$ we are just asking that $f(h) /|h| \rightarrow 0.$ However, what actually happens with $h = (t,t^2)$ is $$f(t,t^2) = \frac{t^{24/5}}{2 t^4} = \frac{t^{4/5}}{2 }.$$ As mentioned, we have $|h| = \sqrt {t^2 + t^4}.$ For Frechet differentiability, we are asking whether $$\frac{ f(t,t^2)}{\sqrt {t^2 + t^4}} = \frac{t^{4/5}}{2 \sqrt {t^2 + t^4} } = \frac{1}{2 |t|^{1/5} \sqrt {1 + t^2} }$$ goes to zero as $|t| \rightarrow 0.$ With, say, $|t| \leq \sqrt 3,$ we get $\sqrt {1 + t^2} \leq 2,$ and $$\frac{ f(t,t^2)}{\sqrt {t^2 + t^4}} \geq \frac{1}{4 |t|^{1/5} },$$ so the ratio goes to infinity rather than zero. The function $f$ is not Frechet differentiable at the origin. Indeed, while continuous, it is only Holder continuous, it is not even Lipschitz.
• Wow that's a very comprehensive answer. As @Nameless pointed out, $$D_vf(0,0)=\lim_{h \to 0} \frac{f(0 + hv) - f(0)}{h} = \left\{\begin{matrix} v_1^2/v_2 & v_2 \neq 0 \\ 0 & v_2 = 0 \end{matrix}\right.$$ Since D is not linear, my function $f$ is not Gateaux differentiable, is this correct? I'm kind of new to the formalism of continuity and derivatives – Francisco May 3 '14 at 12:35
• @Francisco, as i said, sometimes terminology needs to be checked in the particular text; my guess is that a discontinuous function is not said to be Gateaux differentiable, despite having directional derivatives in all directions. Anyway, i worked this out last week and typed it, so i thought i would include it here. – Will Jagy May 3 '14 at 16:53
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2019-10-22 13:59:14
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http://acat-aihenp.in2p3.fr/arxiv/
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# Arxiv
Recent publications in Arxiv (hep-ex,hep-ph,hep-th) on various topics.
Artificial Intelligence
1. 1612.01551v3: Deep learning in color: towards automated quark/gluon jet discrimination
Artificial intelligence offers the potential to automate challengingdata-processing tasks in collider physics. To establish its prospects, weexplo…
2. 1807.02875v2: HEP Software Foundation Community White Paper Working Group – Training, Staffing and Careers
The rapid evolution of technology and the parallel increasing complexity ofalgorithmic analysis in HEP requires developers to acquire a much larger…
3. 1709.08607v1: Towards automation of data quality system for CERN CMS experiment
Daily operation of a large-scale experiment is a challenging task,particularly from perspectives of routine monitoring of quality for data beingta…
4. 1110.6437v3: Anthropic decision theory
This paper sets out to resolve how agents ought to act in the Sleeping Beautyproblem and various related anthropic (self-locating belief) problems,…
5. 1704.06193v1: Intrusion Prevention and Detection in Grid Computing – The ALICE Case
Grids allow users flexible on-demand usage of computing resources throughremote communication networks. A remarkable example of a Grid in High Ener…
6. 1704.04782v1: A Security Monitoring Framework For Virtualization Based HEP Infrastructures
High Energy Physics (HEP) distributed computing infrastructures requireautomatic tools to monitor, analyze and react to potential security incident…
7. hep-ph/0212398v1: Overview of the COMPETE Program
Nowadays, scientific databases have become the bread-and-butter of particlephysicists. These databases must be maintained and checked repeatedly to…
8. math-ph/0112017v3: Wishart and Anti-Wishart random matrices
We provide a compact exact representation for the distribution of the matrixelements of the Wishart-type random matrices $A^\dagger A$, for any fin…
9. physics/0102024v1: On the Compatibility Between Physics and Intelligent Organisms
It has been commonly argued, on the basis of Goedel’s theorem and relatedmathematical results, that true artificial intelligence cannot exist. Penr…
10. quant-ph/0011122v2: Algorithmic Theories of Everything
The probability distribution P from which the history of our universe issampled represents a theory of everything or TOE. We assume P is formallyd…
Neural Network, Machine Learning
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Machine learning methods are powerful in distinguishing different phases ofmatter in an automated way and provide a new perspective on the study of…
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At the Large Hadron Collider, the high transverse-momentum events studied byexperimental collaborations occur in coincidence with parasitic lowtra…
3. 1807.02130v2: Learning to pinpoint effective operators at the LHC: a study of the $t\bar{t}b\bar{b}$ signature
In the context of the Standard Model effective field theory (SMEFT), we studythe LHC sensitivity to four fermion operators involving heavy quarks b…
4. 1810.06669v1: Machine learning using rapidity-mass matrices for event classification problems in HEP
Supervised artificial neural networks (ANN) with the rapidity-mass matrix(RMM) inputs were studied using several Monte Carlo event samples for vari…
5. 1803.08066v2: Jet Charge and Machine Learning
Modern machine learning techniques, such as convolutional, recurrent andrecursive neural networks, have shown promise for jet substructure at the L…
6. 1805.10730v2: Deep learning for the R-parity violating supersymmetry searches at the LHC
Supersymmetry with hadronic R-parity violation in which the lightestneutralino decays into three quarks is still weakly constrained. This work aims…
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For the past year, the HEP.TrkX project has been investigating machinelearning solutions to LHC particle track reconstruction problems. A variety o…
8. 1810.05199v1: Inflation as an Information Bottleneck – A strategy for identifying universality classes and making robust predictions
In this work we propose a statistical approach to handling sources oftheoretical uncertainty in string theory models of inflation. By viewing amod…
9. 1810.05165v1: Energy Flow Networks: Deep Sets for Particle Jets
A key question for machine learning approaches in particle physics is how tobest represent and learn from collider events. As an event is intrinsic…
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Motivated by machine learning, we introduce a novel method for randomlygenerating inflationary potentials. Namely, we treat the Taylor coefficients…
Distributed, parallel computing, GPU
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PANDA is one of the major experiments currently under construction atFAIR/Darmstadt. Its focus is physics with high intensity and high qualityanti…
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Quantization together with quantum dynamics can be simultaneously formulatedas the problem of finding an appropriate flat connection on a Hilbert b…
3. 1805.11125v1: Speeding up complex multivariate data analysis in Borexino with parallel computing based on Graphics Processing Unit
A spectral fitter based on the graphics processor unit (GPU) has beendeveloped for Borexino solar neutrino analysis. It is able to shorten thefitt…
4. 1805.00614v1: Charged pion condensation under parallel electromagnetic fields
The formation of charged pion condensate under parallel electromagneticfields is studied within the two-flavor Nambu–Jona-Lasinio model. Thetechn…
5. 1802.05523v2: Scalar Casimir densities and forces for parallel plates in cosmic string spacetime
We analyze the Casimir densities and forces associated with a massive scalarquantum field confined between two parallel plates in a D-dimensional c…
6. 1512.06174v2: PARTONS: PARtonic Tomography Of Nucleon Software. A computing framework for the phenomenology of Generalized Parton Distributions
We describe the architecture and functionalities of a C++ software framework,coined PARTONS, dedicated to the phenomenology of Generalized PartonD…
7. 1612.08015v3: Plane-parallel waves as duals of the flat background II: T-duality with spectators
We give the classification of T-duals of the flat background in fourdimensions with respect to one-, two-, and three-dimensional subgroups of theP…
8. 1711.08688v3: Plane-parallel waves as duals of the flat background III: T-duality with torsionless $B$-field
By addition of non-zero, but torsionless $B$-field, we expand theclassification of (non-)Abelian T-duals of the flat background in fourdimensions …
9. 1711.06571v3: Parallelized Kalman-Filter-Based Reconstruction of Particle Tracks on Many-Core Architectures
Faced with physical and energy density limitations on clock speed,contemporary microprocessor designers have increasingly turned to on-chipparalle…
10. 1801.01682v2: Nambu–Jona-Lasinio model in a parallel electromagnetic field
We explore the features of the $U_A(1)$ and chiral symmetry breaking of theNambu–Jona-Lasinio model without the Kobayashi-Maskawa-‘t Hooft determi…
Symbolic and automatic calculation
1. 1809.04601v2: Quantum effective action for degenerate vector field theories
We calculate the divergent part of the one-loop effective action in curvedspacetime for a particular class of second-order vector field operators w…
2. 1810.05895v1: Casimir energy calculation for massive scalar field on spherical surfaces: an alternative approach
In this study, the Casimir energy for massive scalar field with periodicboundary condition was calculated on spherical surfaces with $S^1$, $S^2$ a…
3. 1810.05153v1: Out-of-the-box Baryogenesis During Relaxation
We show that spontaneous baryogenesis occurs automatically in relaxion modelsif the reheating temperature is larger than the weak scale, provided t…
4. 1802.06351v2: Exact Solution of Instantaneous Bethe-Salpeter Equation for axialvector ^+$States We provide the first exact solution of the Bethe-Salpeter (BS) equation for^+$ states in the condition of making instantaneous approximation to t…
5. 1703.09738v2: Symbolic and Numerical Analysis in General Relativity with Open Source Computer Algebra Systems
We study three computer algebra systems, namely SageMath (with SageManifoldspackage), Maxima (with ctensor package) and Python language (with Gravi…
6. 1804.09766v2: Dimension-six electroweak top-loop effects in Higgs production and decay
We study the next-to-leading order electroweak corrections to Higgs processesfrom dimension-six top-quark operators in the Standard Model Effective…
7. 1712.02329v3: Rings: an efficient Java/Scala library for polynomial rings
In this paper we briefly discuss \Rings — an efficient lightweight libraryfor commutative algebra. Polynomial arithmetic, GCDs, polynomial factor…
8. 1809.06168v1: Computer algebra tools for Feynman integrals and related multi-sums
In perturbative calculations, e.g., in the setting of Quantum Chromodynamics(QCD) one aims at the evaluation of Feynman integrals. Here one is ofte…
9. 1809.05592v1: Crossed Topology in Two-Loop Dispersive Approach
We extend existing dispersive approach in subloop insertion to the case ofcrossed two-loop box type topologies. Based on the ideas of the Feynman t…
10. 1809.05101v1: SymBuild: a package for the computation of integrable symbols in scattering amplitudes
The article presents and documents the Mathematica package SymBuild. Thispackage implements the computation and manipulation of integrable symbols…
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2018-10-20 23:48:14
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https://math.stackexchange.com/questions/3078502/existence-and-uniqueness-of-solution-for-first-order-cauchy-problems
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# Existence and uniqueness of solution for first order Cauchy problems
Suppose we have two functions $$f=f(x,a):\mathbb{R}^n\times A\to \mathbb{R}^n$$, where $$A\subset \mathbb{R}^m$$ is compact, and also $$\alpha:[0,+\infty)\to A$$. Under the hypothesis that
A')$$f$$ and $$\alpha$$ are continuous
B')$$f$$ is lipschitz continuous in $$x$$ uniformly with respect to $$a$$, ie $$\exists \,L>0: |f(x_1,a)-f(x_2,a)|\leq L|x_1 -x_2|\,\,\forall \, (x_1,a),(x_2,xa)\in \mathbb{R}^n\times A$$
C') $$f$$ is bounded
I have to prove that for any choice of $$x_0\in \mathbb{R}^n$$, $$\begin{cases}y'(t)=f(y(t),\alpha(t))\qquad t>0\\y(0)=x_0\end{cases}$$ has a unique global solution $$y:[0,+\infty)\to \mathbb{R}^n$$. Moreover $$y\in C^1([0,+\infty))$$ and $$y(t)=x_0+\int_{0}^{t}f(y(s),\alpha(s))ds\qquad t\geq 0.$$
I know that from Picard- Lindelöf theorem we have
Theorem Let $$-\infty\leq a\leq b \leq+\infty$$ and $$f=f(t,x):(a, b)\times \mathbb{R}^n\to \mathbb{R}^n$$ be a function. Under the following hypothesis:
A) $$f$$ is continuous
B) $$f$$ is lipschitz continuous in $$x$$ uniformly with respect to $$t$$, ie $$\exists \,L>0: |f(t,x_1)-f(t,x_2)|\leq L|x_1 -x_2|\,\,\forall \, (t,x_1),(t,x_2)\in (a, b)\times \mathbb{R}^n$$
C) $$f$$ is bounded
Then, for every choice of $$(t_0,x_o)\in (a, b)\times \mathbb{R}^n$$ the Cauchy problem $$\begin{cases}y'(t)=f(t,y(t))\\y(t_0)=y_0\end{cases}$$ has a unique global solution $$y:(a, b)\to \mathbb{R}^n$$. Moreover $$y\in C^1((a, b))$$ and $$y(t)=x_0+\int_{t_0}^{t}f(s,y(s))ds\qquad t\in(a, b).$$
I think that I can use the previous theorem in order to prove my question by defining $$g:[0,+\infty)\times\mathbb{R}^n\to \mathbb{R}^n$$ such that $$g(t, x)=f(x,\alpha(t))$$. In fact $$g$$ satisfies A) B) and C) in $$[0,+\infty)\times\mathbb{R}^n$$. By the way $$[0,+\infty)$$ is not open, and for this reason I don't know how to proceed.
Can anyone give me a hint? Thanks a lot in advance.
• Solve it for $(0,\infty)$, then consider the behavior of the solution as $t \to 9$, – Paul Sinclair Jan 19 at 3:43
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2019-06-17 02:42:08
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https://testbook.com/question-answer/it-is-proposed-to-coat-a-1-mm-diameter-wire-with-e--5f3a6a9d2ce3c50d0d0b2426
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# It is proposed to coat a 1 mm diameter wire with enamel paint (k = 0.1 W/mK) to increase the heat transfer with air. If the air side heat transfer coefficient is 100 W/m2K, the optimum thickness of enamel paint should be
This question was previously asked in
ISRO Scientist ME 2015 Paper
View all ISRO Scientist ME Papers >
1. 0.25 mm
2. 0.5 mm
3. 1 mm
4. 2 mm
Option 2 : 0.5 mm
Free
ST 1: General Awareness
7043
15 Questions 15 Marks 15 Mins
## Detailed Solution
Concept:
For maximum heat dissipation, the thickness of insulation should be critical thickness.
The critical radius for wire is given as, $$r_c=\frac{k}{h}$$
Calculation:
Given:
k = 0.1 W/mK, h = 100 W/m2K
$${r_c} = \frac{k}{h} = \frac{{0.1}}{{100}} = 0.001~m = 1~mm$$
Thickness of insulation = rc - r
⇒ 1 – 0.5 = 0.5 mm
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2021-11-30 17:50:38
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https://ftp.aimsciences.org/article/doi/10.3934/mbe.2004.1.57
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# American Institute of Mathematical Sciences
2004, 1(1): 57-60. doi: 10.3934/mbe.2004.1.57
## A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence
1 Centre for Mathematical Biology, Mathematical Institute University of Oxford, 24-29 St Giles', Oxford, OX1 3LB, United Kingdom 2 Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, United Kingdom
Received February 2004 Revised February 2004 Published March 2004
Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form $\beta I^p S^q$ for the case $p \leq 1$ are constructed. Global stability of the models is thereby established.
Citation: Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57
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2018 Impact Factor: 1.313
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2022-05-20 01:05:00
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https://ai.stackexchange.com/questions/13799/what-to-do-when-pdfs-are-not-gaussian-normal-in-naive-bayes-classifier
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# What to do when PDFs are not Gaussian/Normal in Naive Bayes Classifier
While analyzing the data for a given problem set, I came across a few distributions which are not Gaussian in nature. They are not even uniform or Gamma distributions(so that I can write a function, plug the parameters and calculate the "Likelihood probability" and solve it using Bayes classification method). I got a set of a few absurd looking PDFs and I am wondering how should I define them mathematically so that I can plug the parameters and calculate the likelihood probability.
The set of PDFs/Distributions that I got are the following and I am including some solutions that I intend to use. Please comment on their validity:
1)
The distribution looks like:
$$y = ax +b$$ from $$0.8
How to programmatically calculate
1. The value of x where the pdf starts
2. The value of x where the pdf ends
3. The value of y where the pdf starts
4. The value of y where the pdf ends
However, I would have liked it better to have a generic distribution for this form of graphs so that I can plug the parameters to calculate the probability.
2)
This PDF looks neither uniform nor Gaussian. What kind of distribution should I consider it roughly?
3)
I can divide this graph into three segments. The first segment is from $$2 with a steep slope, the second segment is from $$3 with a moderate sope and the third segment is from $$6 with a high negative slope.
How to programmatically calculate
1. the values of x where the graph changes its slope.
2. the values of y where the graph changes its slope.
4)
This looks like two Gaussian densities with different mean superimposed together. But then the question arises, how do we find these two individual Gaussian densities?
The following code may help:
variable1=nasa1['PerihelionArg'][nasa1.PerihelionArg>190]
variable2=nasa1['PerihelionArg'][nasa1.PerihelionArg<190]
Find mean and variance of variable1 and variable2, find the corresponding PDFs. Define the overall PDF with a suitable range of $$x$$.
5)
This can be estimated as a Gamma distribution. We can find the mean and variance, calculate $$\alpha$$ and $$\beta$$ and finally calculate the PDF.
It would be very helpful if someone could give their insights on the above analysis, its validity, and correctness and their suggestions regarding how problems such as these should be dealt with.
• I am trying to train the classifier using Bayes theorem. Suppose for a given set of input, I want to determine if that asteroid is hazardous or not. For that, we need to calculate the probability P(Perihelion Time/Asteroid is Hazardous), that is, what is the probability that the asteroid takes the particular "Perihelion time" (mentioned in the test input) given that the asteroid is "hazardous". Aug 6, 2019 at 6:06
• So, to calculate P(Perihelion Time/Asteroid is Hazardous) we can segregate those values of Perihelion Time for which the asteroid is hazardous, compute its mean and variance and draw the pdf $$\text{Gamma}(\alpha,\lambda)\implies\frac{{\lambda}^{\alpha}}{\Gamma(\alpha)}\cdot x^{\alpha-1}e^{-\lambda x}$$, plug the value of x(=Perihelion Time) and calculate the probability. Aug 6, 2019 at 6:06
• I want to know that how should we calculate pdfs of graphs which do not belong to the general category of pdfs such as Gaussian or Gamma distributions. Aug 6, 2019 at 6:06
• You need to perform proper statistical tests to conclude some reasonable distributional assumptions. Aug 7, 2019 at 18:21
• Try a larger data set. If you dont have one use what you have and do a k fold cross validation. If you have outliers you have outliers. NB isn't perfect. Aug 8, 2019 at 1:49
The relationship between the axes of graph (1) and your variables $$x$$ and $$y$$ is not clear, so this generalized answer may be helpful or useless.
From graph (1) it appears that the correlation coefficient $$\mathcal{C}$$ of a quadratic fit of data set $$\mathcal{S}$$ would be much better. Consider $$y_1$$ and $$y_2$$ approximations of $$y$$.
$$\mathcal{C} (y_2, a, b, c, \mathcal{S}) > \mathcal{C} (y_1, a, b, \mathcal{S}) \\ y_2 = ax^2 + bx + c \\ y_1 = ax + b$$
To achieve a more nearly uniform distribution, perform a least squares fit for $$y_2$$ against $$y$$ on $$\mathcal{S}$$ to obtain $$(a, b, c)$$. Then find a mapping function that produces $$y'$$ and use it where the uniform distribution is desired. A reasonable approximation is simply this.
$$y' = \frac{y}{y_2(x)}$$
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2022-07-01 07:55:23
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https://www.physicsforums.com/threads/optical-difraction.357749/
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# Optical difraction
1. Nov 24, 2009
### the riddick25
1. The problem statement, all variables and given/known data
first question; Does a double slit also diffract light of different wavelength into different directions?
second question; How would a diffraction grating for cm waves differ from an optical grating?
2. Relevant equations
sin (theta) = m.Lambda / d
maybe, i am unsure if this is needed or not, i think it is for the 2nd question
3. The attempt at a solution
for the first question, i looked in my book and could not find anything which said that the light of different wavelengths difracted to different locations, but as the formula for the double slit includes the wavelength, i would guess that they would be diffrected to different places, but i am unsure.
second question i couldnt really answer as i dont know what the difference between da diffrection grating and an optical grating is, if someone could tell me, or show me somewhere i can find out then it would be much appreciated.
2. Nov 24, 2009
### jdwood983
You are correct; if you increase the wavelength, you can change the angle at which the beam comes out. I found a nice http://phys.educ.ksu.edu/vqm/html/doubleslit/" online that shows how changing the energy (which is proportional to the wavelength) increases or decreases the number of peaks.
Diffraction grating is very much like the double slit experiment, except there are 100's to 1000's of slits for light to pass through. I think all types of grating are considered optical grating as it will affect visible light just as well as infrared and ultraviolet light.
My guess is that it's looking for a comparison of spacing necessary between (a) nm light (visible is in the range 350-750, depending on who you ask) and (b) cm electromagnetic waves (microwaves at this range)
Last edited by a moderator: Apr 24, 2017
3. Nov 24, 2009
### the riddick25
thanks :)
and i understand the second question a bit more now, but i am still unsure as to what the answer is, if i am thinking along the right lines, then for a beam of light and a microwave to have the same diffraction angle, then the space between the gratings would need to increase if the wavelength does.
is this right? or have i gone wrong somewhere in my thinking.
it has been a while since i was taught this, and it seems to have slipped away from me :(
4. Nov 24, 2009
### jdwood983
Given the form of the question, it seems to me they are asking for the difference in grating distances for some given angle $\theta$ and their respective wavelengths.
5. Nov 24, 2009
### the riddick25
thanks :)
i've got my answer, and hopefully its right, if not then at least i'll learn something for next time :)
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2018-03-17 07:41:34
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https://www.shaalaa.com/question-bank-solutions/the-area-of-the-square-that-can-be-inscribed-in-a-circle-of-radius-8-cm-is-______-area-of-circle_258734
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# The area of the square that can be inscribed in a circle of radius 8 cm is ______. - Mathematics
MCQ
Fill in the Blanks
The area of the square that can be inscribed in a circle of radius 8 cm is ______.
#### Options
• 256 cm2
• 128 cm2
• 64sqrt(2) cm2
• 64 cm2
#### Solution
The area of the square that can be inscribed in a circle of radius 8 cm is 128 cm2.
Explanation:
Given, radius of circle, r = OC = 8cm.
∴ Diameter of the circle = AC = 2 × OC = 2 × 8 = 16 cm
Which is equal to the diagonal of a square.
Let side of square be x.
In right-angled ΔABC,
AC2 = AB2 + BC2 ......[By Pythagoras theorem]
⇒ (16)^2 = x^2 + x^2
⇒ 256 = x^2 + x^2
⇒ x^2 = 128
∴ Area of square = x^2 = 128 cm^2
Alternative Method:
Radius of circle (r) = 8 cm
Diameter of circle (d) = 2r = 2 × 8 = 16 cm
Since, square inscribed in circle.
∴ Diagonal of the square = Diameter of circle
Now, Area of square = ("Diagonals")^2/2
= (16)^2/2
= 256/2
= 128 cm2
Concept: Area of Circle
Is there an error in this question or solution?
#### APPEARS IN
NCERT Mathematics Exemplar Class 10
Chapter 11 Area Related To Circles
Exercise 11.1 | Q 8 | Page 121
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2023-03-25 17:47:17
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https://wikimili.com/en/Range_(statistics)
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# Range (statistics)
Last updated
In statistics, the range of a set of data is the difference between the largest and smallest values. [1]
Statistics is a branch of mathematics dealing with data collection, organization, analysis, interpretation and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.
## Contents
However, in descriptive statistics, this concept of range has a more complex meaning. The range is the size of the smallest interval (statistics) which contains all the data and provides an indication of statistical dispersion. It is measured in the same units as the data. Since it only depends on two of the observations, it is most useful in representing the dispersion of small data sets. [2]
A descriptive statistic is a summary statistic that quantitatively describes or summarizes features of a collection of information, while descriptive statistics in the mass noun sense is the process of using and analyzing those statistics. Descriptive statistics is distinguished from inferential statistics, in that descriptive statistics aims to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent. This generally means that descriptive statistics, unlike inferential statistics, is not developed on the basis of probability theory, and are frequently nonparametric statistics. Even when a data analysis draws its main conclusions using inferential statistics, descriptive statistics are generally also presented. For example, in papers reporting on human subjects, typically a table is included giving the overall sample size, sample sizes in important subgroups, and demographic or clinical characteristics such as the average age, the proportion of subjects of each sex, the proportion of subjects with related comorbidities, etc.
In statistics, dispersion is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range.
## For continuous IID random variables
For n independent and identically distributed continuous random variables X1, X2, ..., Xn with cumulative distribution function G(x) and probability density function g(x). Let T denote the range of a sample of size n from a population with distribution function G(x).
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as i.i.d. or iid or IID. Herein, i.i.d. is used, because it is the most prevalent.
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable , or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.
### Distribution
The range has cumulative distribution function [3] [4]
${\displaystyle F(t)=n\int _{-\infty }^{\infty }g(x)[G(x+t)-G(x)]^{n-1}{\text{d}}x.}$
Gumbel notes that the "beauty of this formula is completely marred by the facts that, in general, we cannot express G(x + t) by G(x), and that the numerical integration is lengthy and tiresome." [3]
Emil Julius Gumbel was a German mathematician and political writer.
If the distribution of each Xi is limited to the right (or left) then the asymptotic distribution of the range is equal to the asymptotic distribution of the largest (smallest) value. For more general distributions the asymptotic distribution can be expressed as a Bessel function. [3]
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions y(x) of Bessel's differential equation
### Moments
The mean range is given by [5]
${\displaystyle n\int _{0}^{1}x(G)[G^{n-1}-(1-G)^{n-1}]\,{\text{d}}G}$
where x(G) is the inverse function. In the case where each of the Xi has a standard normal distribution, the mean range is given by [6]
${\displaystyle \int _{-\infty }^{\infty }(1-(1-\Phi (x))^{n}-\Phi (x)^{n})\,{\text{d}}x.}$
## For continuous non-IID random variables
For n nonidentically distributed independent continuous random variables X1, X2, ..., Xn with cumulative distribution functions G1(x), G2(x), ..., Gn(x) and probability density functions g1(x), g2(x), ..., gn(x), the range has cumulative distribution function [4]
${\displaystyle F(t)=\sum _{i=1}^{n}\int _{-\infty }^{\infty }g_{i}(x)\prod _{j=1,j\neq i}^{n}[G_{j}(x+t)-G_{j}(x)]\,{\text{d}}x.}$
## For discrete IID random variables
For n independent and identically distributed discrete random variables X1, X2, ..., Xn with cumulative distribution function G(x) and probability mass function g(x) the range of the Xi is the range of a sample of size n from a population with distribution function G(x). We can assume without loss of generality that the support of each Xi is {1,2,3,...,N} where N is a positive integer or infinity. [7] [8]
### Distribution
The range has probability mass function [7] [9] [10]
{\displaystyle f(t)={\begin{cases}\sum _{x=1}^{N}[g(x)]^{n}&t=0\\[6pt]\sum _{x=1}^{N-t}\left({\begin{alignedat}{2}&[G(x+t)-G(x-1)]^{n}\\{}-{}&[G(x+t)-G(x)]^{n}\\{}-{}&[G(x+t-1)-G(x-1)]^{n}\\{}+{}&[G(x+t-1)-G(x)]^{n}\\\end{alignedat}}\right)&t=1,2,3\ldots ,N-1.\end{cases}}}
#### Example
If we suppose that g(x) = 1/N, the discrete uniform distribution for all x, then we find [9] [11]
${\displaystyle f(t)={\begin{cases}{\frac {1}{N^{n-1}}}&t=0\\[4pt]\sum _{x=1}^{N-t}\left([{\frac {t+1}{N}}]^{n}-2[{\frac {t}{N}}]^{n}+[{\frac {t-1}{N}}]^{n}\right)&t=1,2,3\ldots ,N-1.\end{cases}}}$
## Derivation
The probability of having a specific range value, t, can be determined by adding the probabilities of having two samples differing by t, and every other sample having a value between the two extremes. The probability of one sample having a value of x is ${\displaystyle n*g\left(x\right)}$. The probability of another having a value t greater than x is:
${\displaystyle \left(n-1\right)g\left(x+t\right)}$.
The probability of all other values lying between these two extremes is:
${\displaystyle \left(\int _{x}^{x+t}g\left(x\right){\text{d}}x\right)^{n-2}=\left(G\left(x+t\right)-G\left(x\right)\right)^{n-2}}$.
Combining the three together yields:
${\displaystyle f(t)=n\left(n-1\right)\int _{-\infty }^{\infty }g\left(x\right)g\left(x+t\right)\left[G\left(x+t\right)-G\left(x\right)\right]^{n-2}{\text{d}}x}$
The range is a simple function of the sample maximum and minimum and these are specific examples of order statistics. In particular, the range is a linear function of order statistics, which brings it into the scope of L-estimation.
## Related Research Articles
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution, Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution is the distribution of the x-intercept of a ray issuing from with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables if the denominator distribution has mean zero.
In statistics, the Kolmogorov–Smirnov test is a nonparametric test of the equality of continuous, one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution, or to compare two samples. It is named after Andrey Kolmogorov and Nikolai Smirnov.
In economics, the Lorenz curve is a graphical representation of the distribution of income or of wealth. It was developed by Max O. Lorenz in 1905 for representing inequality of the wealth distribution.
The median is the value separating the higher half from the lower half of a data sample. For a data set, it may be thought of as the "middle" value. For example, in the data set {1, 3, 3, 6, 7, 8, 9}, the median is 6, the fourth largest, and also the fourth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is equally likely to fall above or below it.
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.
In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable X is used to denote the outcome of a coin toss, then the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails. Examples of random phenomena can include the results of an experiment or survey.
In mathematics, the Dirac delta function is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. However, not all random variables have moment-generating functions.
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form:
In mathematics, a moment is a specific quantitative measure of the shape of a function. It is used in both mechanics and statistics. If the function represents physical density, then the zeroth moment is the total mass, the first moment divided by the total mass is the center of mass, and the second moment is the rotational inertia. If the function is a probability distribution, then the zeroth moment is the total probability, the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics.
The Gram–Charlier A series, and the Edgeworth series are series that approximate a probability distribution in terms of its cumulants. The series are the same; but, the arrangement of terms differ. The key idea of these expansions is to write the characteristic function of the distribution whose probability density function f is to be approximated in terms of the characteristic function of a distribution with known and suitable properties, and to recover f through the inverse Fourier transform.
In statistics, an empirical distribution function is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step function that jumps up by 1/n at each of the n data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value.
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.
The Anderson–Darling test is a statistical test of whether a given sample of data is drawn from a given probability distribution. In its basic form, the test assumes that there are no parameters to be estimated in the distribution being tested, in which case the test and its set of critical values is distribution-free. However, the test is most often used in contexts where a family of distributions is being tested, in which case the parameters of that family need to be estimated and account must be taken of this in adjusting either the test-statistic or its critical values. When applied to testing whether a normal distribution adequately describes a set of data, it is one of the most powerful statistical tools for detecting most departures from normality. K-sample Anderson–Darling tests are available for testing whether several collections of observations can be modelled as coming from a single population, where the distribution function does not have to be specified.
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In probability theory, an empirical process is a stochastic process that describes the proportion of objects in a system in a given state. For a process in a discrete state space a population continuous time Markov chain or Markov population model is a process which counts the number of objects in a given state . In mean field theory, limit theorems are considered and generalise the central limit theorem for empirical measures. Applications of the theory of empirical processes arise in non-parametric statistics.
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A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product
## References
1. George Woodbury (2001). An Introduction to Statistics. Cengage Learning. p. 74. ISBN 0534377556.
2. Carin Viljoen (2000). Elementary Statistics: Vol 2. Pearson South Africa. pp. 7–27. ISBN 186891075X.
3. E. J. Gumbel (1947). "The Distribution of the Range". The Annals of Mathematical Statistics. 18 (3): 384–412. doi:10.1214/aoms/1177730387. JSTOR 2235736.
4. Tsimashenka, I.; Knottenbelt, W.; Harrison, P. (2012). "Controlling Variability in Split-Merge Systems". Analytical and Stochastic Modeling Techniques and Applications (PDF). Lecture Notes in Computer Science. 7314. p. 165. doi:10.1007/978-3-642-30782-9_12. ISBN 978-3-642-30781-2.
5. H. O. Hartley; H. A. David (1954). "Universal Bounds for Mean Range and Extreme Observation". The Annals of Mathematical Statistics. 25 (1): 85–99. doi:10.1214/aoms/1177728848. JSTOR 2236514.
6. L. H. C. Tippett (1925). "On the Extreme Individuals and the Range of Samples Taken from a Normal Population". Biometrika. 17 (3/4): 364–387. doi:10.1093/biomet/17.3-4.364. JSTOR 2332087.
7. Evans, D. L.; Leemis, L. M.; Drew, J. H. (2006). "The Distribution of Order Statistics for Discrete Random Variables with Applications to Bootstrapping". INFORMS Journal on Computing. 18: 19. doi:10.1287/ijoc.1040.0105.
8. Irving W. Burr (1955). "Calculation of Exact Sampling Distribution of Ranges from a Discrete Population". The Annals of Mathematical Statistics. 26 (3): 530–532. doi:10.1214/aoms/1177728500. JSTOR 2236482.
9. Abdel-Aty, S. H. (1954). "Ordered variables in discontinuous distributions". Statistica Neerlandica. 8 (2): 61–82. doi:10.1111/j.1467-9574.1954.tb00442.x.
10. Siotani, M. (1956). "Order statistics for discrete case with a numerical application to the binomial distribution". Annals of the Institute of Statistical Mathematics. 8: 95–96. doi:10.1007/BF02863574.
11. Paul R. Rider (1951). "The Distribution of the Range in Samples from a Discrete Rectangular Population". Journal of the American Statistical Association . 46 (255): 375–378. doi:10.1080/01621459.1951.10500796. JSTOR 2280515.
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2019-06-26 07:41:18
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https://stacks.math.columbia.edu/tag/09US
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Lemma 37.56.16. Let $f : X \to Y$ be a morphism of schemes locally of finite type over a base $S$. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_ Y)$. Assume
1. $\mathcal{O}_ X$ is pseudo-coherent relative to $Y$1, and
2. $E$ is $m$-pseudo-coherent relative to $S$.
Then $Lf^*E$ is $m$-pseudo-coherent relative to $S$.
Proof. The problem is local on $X$. Thus we may assume $X$, $Y$, and $S$ are affine. Arguing as in the proof of More on Algebra, Lemma 15.81.13 we can find a commutative diagram
$\xymatrix{ X \ar[r]_ i \ar[d]_ f & \mathbf{A}^ d_ Y \ar[r]_ j \ar[ld]^ p & \mathbf{A}^{n + d}_ S \ar[ld] \\ Y \ar[r] & \mathbf{A}^ n_ S }$
Observe that
$Ri_* Lf^*E = Ri_* Li^* Lp^*E = Lp^*E \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}}^\mathbf {L} Ri_*\mathcal{O}_ X$
by Cohomology, Lemma 20.51.4. By assumption and the fact that $Y$ is affine, we can represent $Ri_*\mathcal{O}_ X = i_*\mathcal{O}_ X$ by a complexes of finite free $\mathcal{O}_{\mathbf{A}_ Y^ n}$-modules $\mathcal{F}^\bullet$, with $\mathcal{F}^ q = 0$ for $q > 0$ (details omitted; use Derived Categories of Schemes, Lemma 36.10.2 and More on Algebra, Lemma 15.81.7). By assumption $E$ is bounded above, say $H^ q(E) = 0$ for $q > a$. Represent $E$ by a complex $\mathcal{E}^\bullet$ of $\mathcal{O}_ Y$-modules with $\mathcal{E}^ q = 0$ for $q > a$. Then the derived tensor product above is represented by $\text{Tot}(p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^\bullet )$.
Since $j$ is a closed immersion, the functor $j_*$ is exact and $Rj_*$ is computed by applying $j_*$ to any representating complex of sheaves. Thus we have to show that $j_*\text{Tot}(p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^\bullet )$ is $m$-pseudo-coherent as a complex of $\mathcal{O}_{\mathbf{A}^{n + m}_ S}$-modules. Note that $\text{Tot}(p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^\bullet )$ has a filtration by subcomplexes with successive quotients the complexes $p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^ q[-q]$. Note that for $q \ll 0$ the complexes $p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^ q[-q]$ have zero cohomology in degrees $\leq m$ and hence are $m$-pseudo-coherent. Hence, applying Lemma 37.56.10 and induction, it suffices to show that $p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^ q[-q]$ is pseudo-coherent relative to $S$ for all $q$. Note that $\mathcal{F}^ q = 0$ for $q > 0$. Since also $\mathcal{F}^ q$ is finite free this reduces to proving that $p^*\mathcal{E}^\bullet$ is $m$-pseudo-coherent relative to $S$ which follows from Lemma 37.56.15 for instance. $\square$
[1] This means $f$ is pseudo-coherent, see Definition 37.57.2.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
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2022-05-25 07:40:39
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https://www.ideals.illinois.edu/handle/2142/72917
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Files in this item
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application/pdf
Zhaowen_Wang.pdf (15MB)
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Title: Learning sparse representation for image signals Author(s): Wang, Zhaowen Director of Research: Huang, Thomas S. Doctoral Committee Chair(s): Huang, Thomas S. Doctoral Committee Member(s): Hasegawa-Johnson, Mark A.; Liang, Zhi-Pei; Nasrabadi, Nasser M. Department / Program: Electrical & Computer Eng Discipline: Electrical & Computer Engr Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): image representation sparse coding dictionary learning image classification super-resolution opportunistic sensing neural network Partially Observable Markov Decision Process (POMDP) Abstract: Natural images have the intrinsic property that they can be sparsely represented as a linear combination of a very small number of atomic signals from a complete basis or an overcomplete dictionary. This sparse representation prior has been successfully exploited in a variety of image processing applications, ranging from low level recovery to high level semantic inference. A good sparse representation is expected to have high fidelity to the observed image content and at the same time reveal the underlying structure and semantic information. In this dissertation, we address the problem of how to learn such representation or dictionary from training images, particularly for the tasks of super-resolution, classification, and opportunistic sensing. Image super-resolution is an ill-posed problem in which we want to recover the high-resolution image from the corresponding low-resolution image. We formulate a coupled dictionary learning algorithm which explicitly learns the transform between the high and low-resolution feature spaces such that the sparse representation inferred from a low-resolution patch can faithfully reconstruct its high-resolution version. The resulting bilevel optimization problem is solved using a stochastic gradient descent method with the gradient of sparse code found by implicit differentiation. A feed-forward deep neural network motivated by this sparse coding model is designed to further improve the efficiency and accuracy. The Sparse Representation-based Classification (SRC) has been used in many recognition tasks with the dictionary consisting of training data from all classes. We design a more compact and discriminative dictionary for SRC using the pulling'' and pushing'' actions inspired from Learning Vector Quantization (LVQ). The learned dictionary is applied to hyper-spectral image classification, with additional spatial neighborhood information incorporated using a probabilistic formulation. To better understand the rationale of SRC, we further develop a margin-based perspective into the classifier. The decision boundary and classification margin of SRC are analyzed in the local regions where the support of sparse code is stable. Based on the derived margin, we learn a discriminative dictionary with maximized margin between classes such that SRC can have better generalization capability. Opportunistic sensing deals with actively recognizing an image object with restricted sensing resources. Just as in compressive sensing, we show that dynamically optimized sensing operations (including but not limited to linear projections) can yield better classification results for signals with sparse structure. We develop a greedy sensing strategy using class entropy criteria, as well as a long-term policy learning method using the Partially Observable Markov Decision Process (POMDP) customized for heterogeneous resource constraints and discriminative classifiers. The sensing, recovery and recognition tasks studied in this dissertation exemplify a closed loop of general image processing, and we demonstrate that in each processing step a dictionary or a sensing operation adapted to signals' sparse characteristic can lead to remarkably improved performance. Issue Date: 2015-01-21 URI: http://hdl.handle.net/2142/72917 Rights Information: Copyright 2014 Zhaowen Wang Date Available in IDEALS: 2015-01-21 Date Deposited: 2014-12
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2017-04-30 07:04:39
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http://luc.lino-framework.org/blog/2016/0810.html
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# Wednesday, August 10, 2016¶
## Optimizations in Lino Noi¶
With Annalisa we had decided to remove the concepts of milestones and deployments from Lino Care because we could not find any sense to them there.
Implementation: I moved the models Milestone and Deployment from lino_noi.lib.tickets to a new plugin lino_noi.lib.deploy. And this plugin is then deactivated in Lino Care.
This action was rather time-consuming and not really necessary. Normal people would just have removed the deployments and milestones from the tables and layouts.
As a side effect we have a little new feature welcome_message_when_count.
Another new feature is that Lino now complains when you specify an invalid model name as the lino.core.actors.Actor.model.
For example lino_welfare.modlib.aids.models.Confirmations had 'aids.Confirmation' which was silently being replaced by None at startup. The problem with this behaviour was that it is more difficult to get correct error messages. I had still 'tickets.Milestone' there and it took me some time to find it out.
En passant I also added a new filter parameter lino_noi.lib.tickets.ui.Tickets.site.
I fixed a bug in lino.core.kernel which caused the help texts of table parameter fields extracted from Sphinx docs to not appear in the web interface because they were being installed only when the linoweb.js files had been generated.
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2018-11-18 12:39:37
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https://en.khanacademy.org/math/arithmetic/arith-review-negative-numbers/arith-review-neg-num-intro/a/intro-to-negative-numbers
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If you're seeing this message, it means we're having trouble loading external resources on our website.
If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.
## Arithmetic
### Course: Arithmetic>Unit 18
Lesson 1: Intro to negative numbers
# Intro to negative numbers
Learn what negative numbers are and how to plot them on the number line.
Here's a number line that should look very familiar. It starts at 0, then counts up by 1 from there:
We know that if we were to keep going to the right, we would have 11, then 12, and so on.
What happens if we keep going to the left of 0 though? We get negative numbers! To the left of 0 is minus, 1, then minus, 2, then minus, 3, and so on:
## Let's practice!
Problem 1A
• Current
Move the dot to minus, 4.
## Why do we need negative numbers?
Negative numbers help us describe values less than zero.
### Example:
When the temperature is 8, degrees below 0, degrees, it is less than 0. We can say the temperature is minus, 8, degrees.
### A few more negative situations
Problem 2A
• Current
A bank uses positive numbers to represent deposits and negative numbers to represent withdrawals.
How would a bank represent a withdrawal of 19, point, 43 dollars?
## Want to join the conversation?
• can negative numbers become decimals??
• Yes!
For example:
-5 = -5.0
or
-5.0 = -5
So
-5/2 = -2.5
and so on....
• Why do two Negatives equal a positive. Why can't a Negative plus a Negative equal a Negative? Is it just a math rule or is just the way it goes?. For example
"-2+(-3)=?". Thank you, that is my question.
• Two negatives equal a positive when you are multiplying or dividing. A negative plus a negative does equal a negative. You are moving down on the number line, so you are getting to smaller numbers.(-2+(-3))is the same as (-2-3). This equals (-5).
• can negative numbers become fractions?
• Absolutely, if the problem is such that a fraction will be formed. The negative sign does not affect whether a number will be a fraction or not.
• Can negative numders be turned into a fraction,decimal,and percent? If yes how can you do that?
• Yes, they can. The negative sign doesn't do anything special. Just pretend it doesn't exist, and continue on with the calculation.
For example, -1.6 = -8/5 because 1.6 = 8/5.
• if math is so important then why do people hate math?
• Eating healthy is important, but some people hate vegetables. People generally don't like necessary things that don't give them immediate pleasure in exchange for long-term benefits. Math is one of these. When you work problems, you have to put in effort for no immediate return. However, you gain a better understanding of the world through math, and math can help you with various problems once you've put in the work to know the concept.
• can you add a negative number to a negative number
• If you add a negative number to a negative number it becomes subtraction because a positive and a negative make a negative.
Ex: (-3)+(-7)=-10
(1 vote)
• so if you with drawl \$100,000 from the bank the bank will cont that as -100,000 but its adding to you.
• Yes.But you would have to pay back later.
• I'm Sad :( i don't understand subtraction... for the negative
• There is 2 scenarios of subtraction for the negative, when subtract positive number from negative number and when you subtract negative number from negative number, shortly:
1. -a-b (or -a-(+b) 'b' is positive number but + sign is usually omitted)
2. -a-(-b)
It is always handy to imagine '-a' as a point on a number line not just something from wonderland it's just a point on a numberline just as any other integer.
Tricky things start with 'b'. I suggest thinking about it like this, positive numbers they are like honest dear people, they always act as they are expected to act when they are used to subtract from number (doesn't matter if it is negative or positive) they make number (in our case 'a') less than it was before subtraction same thing with addition they always make number bigger than it was.Negative numbers are like cunning and deceiving people they always do opposite of what they are expected to do when you subtract negative number they suddenly make number bigger than it was, same thing with addition they make number less then it was.
Also it is helpful to always keep in mind that the farther from zero a negative number is, the smaller it gets, even though the integer is getting bigger because as I said negative numbers are like deceiving people they always act opposite of what they are look like
P.S. In formula -a-(-b) ain't -(- looks like + sign it may help because
-a-(-b) and -a+b is the same thing
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2023-03-26 15:29:05
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https://proofwiki.org/wiki/Definition:Subinterval
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# Definition:Subinterval
Jump to: navigation, search
## Definition
Let $I$ be a real interval.
A subinterval $J$ of $I$ is a real interval such that $J \subseteq I$.
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2019-06-20 05:23:47
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http://connect.informs.org/railway-applications/awards/problem-solving-competition/2017
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# Archived
## 2017 Problem Solving Announcement
Instructions for the 2017 Problem Solving Competition are now posted.
The INFORMS Railroad Application Section (RAS) is pleased to announce the 2017 edition of the RAS Problem Solving Competition.
## Purpose
This competition is designed to introduce participants to the railroad industry and its wealth of analytics problems. Railroad operations are inherently complex, large scale, and represent a source of many exciting and challenging research application topics for practitioners of operations research and management sciences.
## Prizes
First Prize: $2000 Second Prize:$1000
Third Prize: \$750
Each winning team will also receive a certificate declaring their prize winning placement in the contest.
In addition, the first prize winner’s contribution will be considered for publication in Networks. While the paper will still be subjected to the journal’s normal refereeing procedure, the paper will receive an expedited refereeing and publication process. More details about this journal can be found here
## Eligibility
Any practitioners of operations research and management science who are interested in solving problems in the railroad domain using OR and analytics tools are welcome to participate. Registration is open to all with the exception of RAS officers and organizing committee members who are NOT eligible to participate. Likewise, members of the organizing committee may NOT help nor guide any participating team.
Teams of up to three members can participate. At least one member of each prize winning team must be available to present the team’s approach and results at the INFORMS Annual Meeting.
## Registration
Participation in the contest requires registration by the due date given below. Every team must register by this due date to participate in the contest. To register, please send following information to railwayapplicationssection@gmail.com by the deadline. For each team member, provide the following:
• Member Name, Email, Organization, Position.
• Prior Experience in problems related to Railroad analytics (Y/N).
• Brief statement describing what motivated you to participate.
After submitting your registration email, you will receive an email confirming your team’s successful registration and eligibility.
## Important dates
May 20: Challenge problem published
June 01, 2017: registration of participation open
July 15: Deadline for submitting a clarification question
September 10, 2017: Deadline for submitting full solution and report
October 10, 2017: Announcement of finalists
October 22, 2017: Finalist presentation and winner announced at the INFORMS Annual Meeting
## Submissions
After the solution submission deadline, completed submissions will be reviewed by the RAS problem competition judging panel. Winners will be selected after collaboration and approval by all panel judges.
All participating teams must submit the following by the due date:
1. A report not exceeding 10 pages (including cover page), normal margins, double spacing, font size of 10. The report must include title page describing the team members, their affiliations, clean and concise description of problem formulation, solution method, and implementation details (Software/Hardware) and results.
2. Problem solution. Make sure you provide the solution in the prescribed format.
3. Computer program/model.
The judging panel will select finalists based on the following criteria:
1. Novelty and elegance of the model proposed
2. Solution quality
3. Solution approach
4. Computation time/complexity
5. Quality of the report submitted
The finalists will make a presentation at the INFORMS Annual Meeting. Aside from the previous factors, the judging panel will take into consideration the clarity of the presentation to make a final decision about the first, second and third places for the competition. Note that being among the finalists and presenting at the Annual Meeting does not guarantee a finalist will receive first, second or third place. The decision of the judges is final.
E-mail all files described to railwayapplicationssection@gmail.com specifying your team’s name in the e-mail’s subject. Attach only these files: (i) the report, (ii) the solution files and (iii) a zip file (.zip) containing all supporting (source) files.
## Can I Publish?
Yes, you can. In fact, RAS encourages you to do so. Anyone can use the RAS competition problem and provided datasets in their publication. References to year-specific problem competitions are given in the URL, and as such you can reference the year-specific competition URL which will not be changed.
If you have any questions about the competition problem, submit your question to railwayapplicationssection@gmail.com. In fairness to all participants, RAS will publish a Q&A on website. All participants will receive Q&A document through email as well. If there are any more questions, similar process will be repeated. The questions may be collected and answered once a week. No questions will be entertained after July 15, 2017.
If you have any suggestions or ideas for future competitions, please feel free to contact April Kuo (April.Kuo@BNSF.com) or Yanfeng Ouyang (yfouyang@illinois.edu). All feedback is greatly appreciated.
### 2017 Organizing Committee
Clark Cheng, Norfolk Southern
Pooja Dewan, BNSF
Jerry Kam, CSX
Xiaopeng Li, University of South Florida
Yanfeng Ouyang, University of Illinois at Urbana-Champaign
Steven Tyber, GE Transportation
Shanshan Wang, BNSF
Aihong Wen, CSX
• 1. Train symbol
In trn_id, what’s the meaning of train symbol? We found that train symbol is formed with 6 characters, and it seems that it can be separated in 2 parts with 3 letters. For example, DOWULF can be seen as DOU and ULF. Does each of them represent to a train station? If so, does the first part represent the departure station, and the second part represent the destination station? What’s the relation between train symbol and a trip?
Yes. Train symbol gives abbreviation of a train’s origin and destination (First three letters- origin; last three letters – destination). A trn_id uniquely identifies a train trip.
• 2.Train section, train day and train priority
We found that it ranges from alphabet A to W. Suppose there are 5 trains waiting to depart in a train station. Does it mean the train with former order, say A, will leave earlier than another train with latter order (say B)? Another guess is the order of which a train runs before another train. Suppose two trains both comes to a siding, a train with higher priority will travel precede the other train.
Although it has name “train priority”, it does not really relates to “priority” of a train.
• What is the meaning of train section in the column of trn_id? There are 10 types in it with numbers ranging from 0 to 9. Does this number mean, say, time like month? or zone id (e.g., if totally 10 zones on the map)? It might be helpful if we can learn more about what this means.
• We found the train day (in the column of trn_id) ranges from 1 to 31. Does it mean the departure date in a month?
The meaning of train section depends on train types. For certain train types, train section means the priority of train within the specific train type. For some other train types, it indicates the counter of number of trains within the year.
For most train types, train day means the departure day in a month. For some train types, it indicates the counter of number of trains within a year.
Please refer to the following table:
Train Type Train Priority Meaning of Train section Meaning of Train day F -- counter counter H -- counter counter J T counter counter A T counter counter X T counter counter Other -- Priority Departure day of a month
For example, if a train has train type F then its train section means counter of number of trains regardless of its train priority. If a train has train type J/A/X then when its train priority equals T then train section means counter. For all the other train types except F, H, J, A and X train section means priority within that train type.
Look at the following example trn_id:
FABCDEF101U Means this train is the 101st F type train within the year having train symbol ABCDEF and train priority U.
• 3.eqp_axle_nbr
Given the description, we suppose it means the location of a wheel in a car. Suppose a car contains 9 axle, indicating that there are 9 wheels at one side. Assume the closest wheel to the head of the car has number, say 1. Does it mean the wheel with number 9 could be the wheel in the tail of a car?
There is no head and tail concept of a car. However, the smaller the eqp_axle_nbr is, the closer the axle is to the brake end of the car.
• 4.equipment
What is an equipment? Does it mean something to be loaded on a car? Or it is just another name for a car (I.e. An equipment = a car)?
Yes. An equipment = a car.
• 5.car_initial
For example, what does ‘EQVI’ mean?
For most cases car_initial indicates car owner. Generally car_initial and car_number together uniquely identifies a car within North America.
• 6.aar_ct_c
In AAR Car Type Codes Explained & Resources (http://www.railcartracking.com/aar-car-type-codes-explained-resources-2/), is the car description in this page correct for aar_ct_t?
For example, D113 is a locomotive because of the letter D. Then, what does 113 mean?
You can refer to that website for explanation. The number gives other characteristics of the car, for example, length, height, etc.
• 7.eqp_grp
We found that there are 9 kinds of alphabets in eqp_grp. We found them corrpesond to BOXC, FLAT, GOND, HOPP, IFLT, MFLT, MISC, TANK, VFLT, as found in Railroad on wiki (https://en.wikipedia.org/wiki/Railroad_car). Is this guess for eqp_grp correct? Furthermore, what’s the difference between aar_ct_c and eqp_grp?
Eqp_grp indicates the logical grouping of car types. For example, BOXC means box cars, FLAT means flat cars. aar_ct_c indicates AAR unique designation of car types. Eqp_grp is a more general categorization compared to aar_ct_c.
• Missing data
In training datasets, we found the row 7032197, an observation of uni_whl_id WHLA56870R1-2011-01- 08::2014-01-08, there are 13 variable values missing. It shows “NA” instead. We want to ensure whether these values are indeed missing? Or it is just some mistake caused by the wrong way of file reading?
There are missing data in the datasets. And you will be expected to clean those missing data yourself.
• 9. Data for all wheels
Are there sufficient data for all the wheels? For example, suppose a car contains 9 wheels, does the training data contain all 9 wheels?
Not necessarily. The data is collected based on wheels that have been repaired. And note that if a wheel needs to be repaired, its mate wheel will also be replaced. So you will expect to see the sensor readings for a wheel and its mate wheel as follows (unq_whl_id below is only for demonstration. The specific id might not necessarily be in the training/testing dataset):
EQVI587399L5-2007-06-28::2010-06-28 & EQVI587399R5-2007-06-28::2010-06-28
• 10. Sensor health
Can we assume all the sensors work well (I.e., perfect) In other words, we can trust the correctness of sensors so that any strange numbers collected can never be caused by a failed sensor, is this correct?
Please refer to assumptions (c) and (d) for this question.
• Data transformation
Do we need to predict the peak kips only for the rows where an L (loaded) follows an E (empty) run? This is mentioned in the document, but the test data contain a mix of L and E type rows.
Yes, you will need to transform data (both training and testing) and focus only on E-L prediction.
• 12. Units
What are the units of the grs_rail_wgt column?
It is in pounds.
• 13. About grs_rail_wgt and tare
Does the gross weight column include the tare weight of the car?
Yes. Gross weight column represents the tonnage that the particular wheel type can sustain, which includes the tare weight of the car and car loads weight. But note that tare weight of the car in the datasets are in tons.
• 14.
If we need to predict peak kips in the [next loaded condition], can we assume that we know what the load weight will be? That is, is the peak kip prediction based on [data about last run only], or [data about last run + load in current run]?
Yes. You can use car weight, speed, age and detector location (vndr_edr_nme) as predictors.
• 15.
Do the train priorities and car initials have any particular significance for the speed/load constraints of the train?
• 16.
For the variable "Kipdays" in the section 3.1 data description, it mentions that kipdays is -999 when the conditions are not met for "KipDays5". What does "KipDays5" mean?
On this line the format is a bit off. 5 here denotes the footnote number.
• 17.
We found some values of the variable "edr_eqp_spd" (i.e. the speed of the wheel) are 0. Does it mean that the wheel is measured while the train or the wheel stops exactly over the WILD Detectors?
Please refer to assumption (c) for this question.
• 18.
We also found some values of the variable "whl_dyn_kips" are negative. Is that possible? Or it is an error in the measurement?
This question can be answered by looking at the equation in the box at Page 1. Please note that you are expected to apply data cleaning if needed.
• 19. Start_date/end_date
(1) What are the meanings of "start_date" and "end_date" in the variable "unq_whl_id"? Does "start_date" mean the first date the wheel was used? Does "end_date" mean the final date that the wheel was discarded? or ?
(2) Do the start-dates/end-dates in the unq_whl_id represent the dates when the wheel was mounted/dismounted? For example, there are several occurences of wheels for which the difference between their end-date and their start-date is exactly 3 years, a suspiciously round number. Is there some sort of expiration date on wheels, which force them to be removed from a car even if they don't appear to have developped any defect? If so, is there a way to know this theoretical maximal lifetime?
Start_date and end_date indicate when a wheel is attached to and removed from a car, respectively. The difference gives the duration of a specific wheel. For example, the unq_whl_id EQVI587399L5-2007-06-28::2010-06-28 indicates that the wheel was attached to equipment EQVI587399 at fifth axle on the left-side and was used from 2007-06-28 to 2010-06-28.
3 years is an empirical number if no previous repair date is found. There is no wheel expiration date. You should look at the data to get the expected lifetime of a wheel.
• 20.
When does one start to record the age of a certain wheel? The time the wheel was made, or the time the car was equipped with that wheel? or ?
Age = reading date – start date of a unique wheel.
• 21.
For the variable "rpr_why_cd", we would like to confirm its timing. Does this stand for the code of the most recent (i.e., previous) repair? or the next repair? or the code to record why the wheel can no more work (i.e., the code for its final failure)
It indicates why a wheel is repaired at the end of the lifetime of the wheel. Take the unq_whl_id EQVI587399L5-2007-06-28::2010-06-28 for example, it indicates the repair reason on 2010-06-28.
• 22.
Observation number 5 in training data set is EWKKQDP068A. As per the previous explanations provided train type 'E' falls under 'other section'. It means that the train day is the departure day of the month. Then how does 68 represent the departure day?
06-train day, 8- train section
• 23. Average vs dynamic kips
Do the sensors measure these two quantities separately, which then get summed up to compute the peak kips? Or do the sensors measure the peak kips directly, and some formula is applied to derive the average and dynamic kips? In the latter case, what is this formula?
Please refer to the equation on problem statement page 1 for this question.
• 24. Criteria for evaluation
In the description of the problem, it is mentionned we must achieve a +/-2 peak kips accuracy for 90% of the observations, and "minimizing" both the alarm type I and type II errors. Is there any value in trying to get the +/-2 peak kips accuracy beyond 90%, or should we focus elsewhere if we achieve this threshold to maximize our chances of winning? In other words, how important is accuracy in the evaluation policy?
The accuracy is the target and there is definitely value if you try to get an even higher accuracy.
• 25. What information can be used for making predictions
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2018-06-19 23:39:52
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http://jmlr.org/papers/v20/18-819.html
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## High-Dimensional Poisson Structural Equation Model Learning via $\ell_1$-Regularized Regression
Gunwoong Park, Sion Park; 20(95):1−41, 2019.
### Abstract
In this paper, we develop a new approach to learning high-dimensional Poisson structural equation models from only observational data without strong assumptions such as faithfulness and a sparse moralized graph. A key component of our method is to decouple the ordering estimation or parent search where the problems can be efficiently addressed using $\ell_1$-regularized regression and the moments relation. We show that sample size $n = \Omega( d^{2} \log^{9} p)$ is sufficient for our polynomial time Moments Ratio Scoring (MRS) algorithm to recover the true directed graph, where $p$ is the number of nodes and $d$ is the maximum indegree. We verify through simulations that our algorithm is statistically consistent in the high-dimensional $p>n$ setting, and performs well compared to state-of-the-art ODS, GES, and MMHC algorithms. We also demonstrate through multivariate real count data that our MRS algorithm is well-suited to estimating DAG models for multivariate count data in comparison to other methods used for discrete data.
[abs][pdf][bib]
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2019-06-25 15:33:36
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https://academic.oup.com/jnci/article-lookup/doi/10.1093/jnci/dji141
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## Abstract
Background: Homozygous or compound heterozygous mutations in the ATM gene are the principal cause of ataxia telangiectasia (A-T). Several studies have suggested that heterozygous carriers of ATM mutations are at increased risk of breast cancer and perhaps of other cancers, but the precise risk is uncertain. Methods: Cancer incidence and mortality information for 1160 relatives of 169 UK A-T patients (including 247 obligate carriers) was obtained through the National Health Service Central Registry. Relative risks (RRs) of cancer in carriers, allowing for genotype uncertainty, were estimated with a maximum-likelihood approach that used the EM algorithm. Maximum-likelihood estimates of cancer risks associated with three groups of mutations were calculated using the pedigree analysis program MENDEL. All statistical tests were two-sided. Results: The overall relative risk of breast cancer in carriers was 2.23 (95% confidence interval [CI] = 1.16 to 4.28) compared with the general population but was 4.94 (95% CI = 1.90 to 12.9) in those younger than age 50 years. The relative risk for all cancers other than breast cancer was 2.05 (95% CI = 1.09 to 3.84) in female carriers and 1.23 (95% CI = 0.76 to 2.00) in male carriers. Breast cancer was the only site for which a clear risk increase was seen, although there was some evidence of excess risks of colorectal cancer (RR = 2.54, 95% CI = 1.06 to 6.09) and stomach cancer (RR = 3.39, 95% CI = 0.86 to 13.4). Carriers of mutations predicted to encode a full-length ATM protein had cancer risks similar to those of people carrying truncating mutations. Conclusion: These results confirm a moderate risk of breast cancer in A-T heterozygotes and give some evidence of an excess risk of other cancers but provide no support for large mutation-specific differences in risk.
Ataxia telangiectasia (A-T) is a rare autosomal recessive neurologic disorder, characterized by progressive cerebellar degeneration and oculocutaneous telangiectasia. A-T appears to be completely penetrant and is typically diagnosed in early childhood, although the precise clinical phenotype varies from patient to patient. Most cancers in A-T patients are childhood lymphoid leukemias and lymphomas, but there is also a substantial risk of epithelial tumors later in life ( 1 ) . Almost all cases of A-T have been shown to be associated with mutations in the ATM gene, the product of which plays a central role in the recognition and repair of double-strand DNA breaks and in the activation of cell cycle checkpoints ( 2 ) . Most A-T patients are compound heterozygotes; homozygous carriers are uncommon, except in consanguineous families or in the case of a few population-specific founder mutations.
It has frequently been suggested that the blood relatives of A-T patients (i.e., obligate or potential heterozygous ATM mutation carriers) have an increased risk of cancer, primarily breast cancer. Clearly, it is important to reliably establish the cancer risks in heterozygous carriers to provide appropriate advice to the relatives of A-T patients. However, the question may also have wider public-health relevance. Some estimates of the frequency of ATM mutation carriers in Western populations are as high as 1% ( 3 , 4 ) , so that a relatively modest increase in breast cancer risk could equate to a substantial population attributable risk.
Studies assessing the risk of breast cancer in heterozygous ATM mutation carriers fall in two broad categories. First, several groups have compared breast cancer incidence and/or mortality in relatives of A-T patients with that in the general population or in married-in family members ( 510 ) . A review of four such studies estimated the breast cancer relative risk (RR) to be 3.9 (95% confidence interval [CI] = 2.1 to 7.2) ( 11 ) . Subsequent studies have found slightly more modest results, with relative risks between 2.4 and 3.4; most studies report that relative risks are higher among younger women ( 5 , 9 , 11 , 12 ) .
An alternative approach is to compare the frequency of ATM mutations in breast cancer case patients with that in control subjects. Case–control studies have almost uniformly failed to find an increased frequency of pathogenic ATM mutations in case patients, even when restricted to early-onset cancers ( 4 , 1316 ) . A review of 10 studies showed that ATM mutations are statistically significantly more frequent in breast cancer case patients selected on the basis of a family history of breast cancer than in unselected case patients ( 17 ) , although other studies have not replicated this result ( 18 ) .
The findings from the family studies and the case–control studies are not necessarily incompatible, given the widths of the confidence intervals; the sample sizes in many studies are too small to detect a modest increase in risk. Moreover, some studies have suggested that certain missense ATM mutations, notably 7271T>G, may be associated with higher risks of breast cancer ( 14 , 17 , 1922 ) , whereas most of the earlier population-based studies used mutation detection techniques that are biased in favor of detecting truncating mutations.
In addition to the potential association between ATM and breast cancer, several studies have reported an increase in the overall risk of cancer in relatives of A-T patients. One review found that the risk of non-breast cancers in carriers was almost double that expected in the general population ( 11 ) . Several cancer sites have been mentioned in this context, but no statistically significant associations with particular cancers have been reported to date ( 6 , 9 , 12 , 23 , 24 ) . If the risks of any other specific tumor types are genuinely increased in heterozygous ATM carriers, no study has yet had sufficient power to demonstrate this.
This study aimed to provide more precise estimates of the risks of cancer in heterozygous ATM mutation carriers by examining the cancer incidence and mortality experienced by the relatives of 169 A-T patients from 139 families living in the UK. This is by far the largest group of A-T families outside the US to have been studied to date and represents the large majority of A-T case patients diagnosed in the UK during the last 20 years. A second aim was to investigate potential differences in cancer risks associated with different types of ATM mutations.
## S UBJECTS AND M ETHODS
### Data Collection
Families were ascertained on the basis of at least one family member having been given a clinical diagnosis of A-T. The majority of the families (121 families) were ascertained via contact with the A-T Society, a UK support group for people with A-T and their families, or after referral by their pediatric neurologist to the Cancer Research UK Institute for Cancer Studies for diagnosis, genetic testing, and research purposes. In addition, to avoid biasing the cohort towards relatives of living A-T patients, a list of all death certificates since 1979 that mentioned A-T was obtained from the Office of National Statistics, leading to the inclusion of a further 18 families. Forty-four of the families were included in a previous study ( 7 ) , but the data used here include a larger number of relatives, 7.5 years of additional follow-up, and information about cancer incidence and mortality.
After we sought permission to contact the parents of each A-T patient from his or her general practitioner, the parent or parents who had agreed to participate in the study were sent a questionnaire requesting basic information about themselves and their children, siblings, parents, and grandparents (i.e., the siblings, aunts, uncles, grandparents, and great-grandparents of the A-T patient). All parents returning questionnaires gave written informed consent. The requested information for each relative comprised name, date and place of birth, vital status, and date of death, where applicable, whether he or she had ever had a cancer, and if so, the type of cancer, age at diagnosis, and place of treatment. Dates of birth were confirmed from national birth registers, and birth, death, and marriage registers were used to trace relatives in families for which the questionnaire was incomplete. Data were also obtained in this way for families for whom no questionnaire was available and for families ascertained via death certificate. An attempt was made to “flag” each of the relatives listed above through the National Health Service Central Register (NHSCR). The NHSCR receives notification of all deaths in the UK and all cancer registrations from cancer registries covering the UK, and the study coordinator was informed of these events in study subjects. Individuals were excluded from the study if tracing was not possible. Cancer diagnoses were included in the analysis only if they had been confirmed by the NHSCR, to allow valid comparison with population-based incidence rates.
Ethical approval was obtained from the South Birmingham Research Ethics Committee and the Birmingham and the Black Country Health Authority. Approval for use of the NHSCR for tracing was given by the Patient Information Advisory Group.
### Description of Cohort
A total of 169 A-T patients from 139 separate families were included in the study. Three families each contained three siblings with A-T, and 23 families each included a pair of siblings with A-T. One pair of cousins with A-T occurred in a consanguineous family. The number of relatives per family for whom information was available ranged from two to 28 (median = 17), giving a total of 2102 blood relatives (excluding 15 stepparents of A-T patients or of their parents). We excluded 510 relatives with unknown dates of birth, 152 who were born prior to 1891, and an additional 153 relatives who could not be traced by the NHSCR. Follow-up for parents was defined as starting at the birth of their first child with A-T, and follow-up for maternal and paternal grandparents began at the birth of the A-T patient's mother and father, respectively. Follow-up for maternal and paternal great-grandparents started 28 years before the birth of the A-T patient's mother and father respectively, to approximate the date of the relevant grandparent's birth (28 years was the average age of parents at the birth of a child with A-T in the cohort). This left-truncation of the follow-up period was performed to avoid biasing the cohort toward individuals who had, by definition, still been alive at the time that the A-T patient (or his or her parent or grandparent, respectively) was born. Follow-up for all other relatives began at their own dates of birth, because the A-T patient's birth was not dependent on their being alive at any particular point in time. One father was excluded because his last follow-up (when he joined the armed forces) was before the birth of his first child. The cohort included a total of 1286 relatives.
For the analysis of cancer incidence, follow-up prior to 1971 was excluded because cancer registry information was not complete before then. Only first cancers were considered, non-melanoma skin cancers were excluded, and only cancers reported by the cancer registry were counted. Follow-up was assumed to cease at the earliest of July 1, 2002, the date of death, the 80th birthday, or when the individual was last reported as being alive and cancer-free. Of the 1286 relatives, 126 contributed no person-years to the cancer incidence analysis because their dates of last follow-up or death were before 1971. The proportion of relatives excluded for any of the above reasons did not differ between families with and without questionnaires ( P = 0.4).
According to the definitions above, the cohort for the cancer incidence analysis consisted of 1160 relatives of A-T patients from 132 families, who contributed a total of 26 220 person-years to the analysis (median = 9 relatives per family, 27.2 years per relative). The distribution between different types of relative is shown in Table 1 . The number of male and female relatives was approximately equal; 573 (49.4%) males contributed 12 664 person-years (48.3%), and 587 (50.6%) females contributed 13 557 person-years (51.7%). The median year of birth was 1942 (interquartile range [IQR] = 1924 to 1958). During the follow-up period, there were 355 deaths, with a median age at death of 71 years (IQR = 62 to 81 years). The remaining 805 relatives were still alive when they exited the cohort, at a median age of 50 years (IQR = 39 to 63 years).
Table 1.
Cancer incidence in 1160 relatives of A-T patients from 132 families *
No. eligible Pyears Obs Exp SIR (95% CI)
All cancer incidence, excluding breast cancer
Relationship to A-T patient
Parent 280 247 5025 10.6 0.85 (0.39 to 1.62)
Sibling 105 90 1776 0.44 2.28 (0.06 to 12.7)
Half-sibling 11 189 0.05 0.00
Aunt/uncle 437 352 10 344 22 12.6 1.75 (1.10 to 2.64)
Grandparent 454 325 7054 49 39.7 1.23 (0.92 to 1.63)
Great-grandparent 802 131 1622 14 18.4 0.76 (0.41 to 1.28)
Parent's half-sibling 13 210 0.27 0.00
Approximate carrier probability
1 280 247 5025 10.6 0.85 (0.39 to 1.62)
0.67 105 90 1776 0.44 2.28 (0.06 to 12.7)
0.5 902 685 17 587 71 52.4 1.36 (1.06 to 1.72)
0.25 815 138 1832 14 18.7 0.75 (0.41 to 1.26)
All 2102 1160 26 220 95 82.1 1.16 (0.95 to 1.41)
Breast cancer incidence (female relatives)
Relationship to A-T patient
Mother 127 2640 2.67 1.87 (0.61 to 4.36)
Sister 45 968 0.09 0.00
Half-sister 81 0.00 0.00
Aunt 174 5047 3.11 2.90 (1.33 to 5.50)
Grandmother 173 3950 6.89 1.16 (0.50 to 2.29)
Great-grandmother 62 810 1.70 0.59 (0.01 to 3.27)
Parent's half-sister 62 0.01 0.00
Approximate carrier probability
1 127 2640 2.67 1.87 (0.61 to 4.36)
0.67 45 968 0.09 0.00
0.5 351 9077 17 10.0 1.70 (0.99 to 2.72)
0.25 64 872 1.72 0.58 (0.01 to 3.24)
All 587 13 557 23 14.5 1.59 (1.01 to 2.38)
No. eligible Pyears Obs Exp SIR (95% CI)
All cancer incidence, excluding breast cancer
Relationship to A-T patient
Parent 280 247 5025 10.6 0.85 (0.39 to 1.62)
Sibling 105 90 1776 0.44 2.28 (0.06 to 12.7)
Half-sibling 11 189 0.05 0.00
Aunt/uncle 437 352 10 344 22 12.6 1.75 (1.10 to 2.64)
Grandparent 454 325 7054 49 39.7 1.23 (0.92 to 1.63)
Great-grandparent 802 131 1622 14 18.4 0.76 (0.41 to 1.28)
Parent's half-sibling 13 210 0.27 0.00
Approximate carrier probability
1 280 247 5025 10.6 0.85 (0.39 to 1.62)
0.67 105 90 1776 0.44 2.28 (0.06 to 12.7)
0.5 902 685 17 587 71 52.4 1.36 (1.06 to 1.72)
0.25 815 138 1832 14 18.7 0.75 (0.41 to 1.26)
All 2102 1160 26 220 95 82.1 1.16 (0.95 to 1.41)
Breast cancer incidence (female relatives)
Relationship to A-T patient
Mother 127 2640 2.67 1.87 (0.61 to 4.36)
Sister 45 968 0.09 0.00
Half-sister 81 0.00 0.00
Aunt 174 5047 3.11 2.90 (1.33 to 5.50)
Grandmother 173 3950 6.89 1.16 (0.50 to 2.29)
Great-grandmother 62 810 1.70 0.59 (0.01 to 3.27)
Parent's half-sister 62 0.01 0.00
Approximate carrier probability
1 127 2640 2.67 1.87 (0.61 to 4.36)
0.67 45 968 0.09 0.00
0.5 351 9077 17 10.0 1.70 (0.99 to 2.72)
0.25 64 872 1.72 0.58 (0.01 to 3.24)
All 587 13 557 23 14.5 1.59 (1.01 to 2.38)
*
N = total number of relatives in the cohort, Pyears = person-years at risk, Obs = observed cancers, Exp = expected cancers, SIR = standardized incidence ratios, CI = confidence interval.
The follow-up period for mortality was defined in the same way as for cancer incidence, except that follow-up commenced on January 1, 1950. In this cohort, 644 male and 625 females contributed a total of 41 276 person-years (median = 34.2 years per relative, IQR = 22.0 to 43.7).
### Genotyping
To identify the ATM mutations present in the families of A-T patients, mutation screening of the ATM gene was performed at the Cancer Research UK Institute for Cancer Studies using lymphoblastoid cell lines derived from blood samples of A-T patients. In all these A-T patients, including those for whom mutations have not yet been found, loss of ATM protein was confirmed by Western blotting of protein extracts from the lymphoblastoid cell lines. Proteins were separated by sodium dodecyl sulfate–polyacrylamide gel electrophoresis on 6% gels and transferred electrophoretically to nitrocellulose membranes that were incubated with a monoclonal mouse–anti-human ATM antibody (11G12) ( 39 ) . Formerly, screening for ATM mutations had been carried out using restriction enzyme fingerprinting of PCR-amplified cDNA ( 22 ) . More recently, ATM mutations in A-T patients were identified by denaturing high-performance liquid chromatography analysis of PCR-amplified exons, followed by sequencing. For those mutations identified by exon sequencing that potentially altered splicing of the RNA transcript, cDNA sequencing was also performed to confirm sequence deletion or insertion. At least one pathogenic ATM mutation was identified in 118 A-T patients from 95 families (79% of the families ascertained via the A-T Society). In eight families the A-T patients have been shown to be homozygous for different ATM mutations, and a further 40 families have been shown to carry two distinct ATM mutations. No mutation has yet been identified in 12 families, and samples are not currently available for a further 30 families. Mutations were found in both parents from 33 families, in the mother but not the father in eight families, and in the father but not the mother in 10 families.
Subsequent to the initial data collection, A-T patients in two families have been shown to carry mutations in the MRE11 gene (including the consanguineous family containing a pair of cousins with A-T), rather than in ATM, and so should more properly be described as having A-T–Like Disorder (ATLD) ( 25 ) . MRE11-associated ATLD is difficult to distinguish clinically from A-T, although the characteristic telangiectasia features are absent in ATLD patients. These families were, however, included in the main analysis, because study entry was defined on the basis of a clinical, rather than a genetic, diagnosis of A-T.
### Statistical Analysis
Standardized incidence ratios (SIR) were used to compare the cancer incidence in relatives with that expected in the general population. Expected numbers of cancers in each individual were based on the age, sex, and calendar-period specific incidence rates given for England and Wales in Cancer in Five Continents Volumes III to VIII ( 2631 ) using the PYEARS program ( 32 ) . The 95% confidence intervals (CIs) were derived as exact confidence limits for a Poisson mean ( 33 ) . For the mortality analysis, mortality rates were taken from data provided by the UK Office of National Statistics, and standardized mortality ratios (SMR) were computed.
The parents of the A-T patients are all obligate ATM mutation carriers. No other relatives have been tested for mutations, so their carrier probabilities were estimated on the basis of their position within the pedigree, using the program MENDEL ( 34 ) , assuming that A-T is a fully penetrant recessive disorder, with mutant ATM alleles segregating according to standard Mendelian inheritance rules. The frequency of mutant alleles within the UK population was taken to be 0.3%, equivalent to approximately five new A-T cases per year. The results were not sensitive to small variations in this value.
These estimated carrier probabilities ( w i , for the ith individual) were used to obtain estimates of the relative risk of cancer associated with carrying one ATM mutation, with the observed and expected numbers of cancers in each relative ( O i and E i respectively) weighted by their estimated carrier probability; i.e., if the relative risk is denoted λ, then
$\mathrm{{\hat{{\lambda}}}}{=}\frac{{{\sum}_{i}}w_{i}O_{i}}{{{\sum}_{i}}w_{i}E_{i}}$
The relative risk of cancer for the noncarriers in the cohort, φ, was computed in the same way but with the Oi and Ei weighted instead by the estimated probability of not carrying a mutation, 1− wi . Estimates of λ and φ were obtained using the EM algorithm to iteratively update the individual carrier probability estimates and the relative risks ( 35 ) . Confidence intervals were derived from the estimated covariance matrix for λ and φ ( 36 ) . For almost all individual cancer sites, there was insufficient information to give stable simultaneous estimates of λ and φ. Simultaneously estimating λ and φ for all sites combined gave no evidence of an overall excess of cancer incidence, cancer mortality, or non-cancer mortality in noncarrier relatives; therefore, all estimates of λ presented are those estimated under the constraint that φ = 1 (i.e., noncarrier incidence rates assumed to equal general population rates).
Relative risks were also estimated separately for carriers who were younger than 50 years of age and for those aged 50 years or older. The cutpoint of 50 years was chosen to distinguish approximately between pre- and postmenopausal breast cancers. For consistency, the same cutpoint was also used for other cancers. Cumulative risks of cancer in carriers were estimated by applying the estimated carrier relative risks (younger than 50 years of age and 50 years or older) to the population rates for England and Wales (1992–1997) ( 29 ) .
Strictly, the relative risk estimates are not maximum-likelihood estimates because the dependence between the carrier probabilities of relatives from the same family is ignored in the iteration. However, the resulting estimates are consistent, whereas a full-likelihood analysis would theoretically require adjustment for familial aggregation of cancer, which is problematic to specify. In practice, the differences between the estimates presented here and the hypothetical full-likelihood estimates are likely to be negligible because there was rarely more than one cancer of the same type per family (i.e., no family had multiple cases of stomach or lung cancer; two families had two cases of breast cancer, and three families had two cases of colorectal cancer).
### Genotype–Phenotype Correlation
Given the large number of distinct pathologic ATM mutations recorded in A-T patients (81 distinct mutations in this cohort), it is impossible to evaluate risks associated with individual mutations. Because it had been previously hypothesized that the cancer risk might be related to the residual expression of the mutant ATM protein ( 21 ) , we classified mutations into three groups, according to whether any ATM protein was likely to be expressed from a mutant allele and, if so, whether the protein was likely to have kinase activity: A) frameshift mutations and substitutions leading to premature termination codons, resulting in no expression of the ATM protein from that allele; B) large (exon) or small (codon) in-frame deletions allowing some expression of a mutant ATM protein ( 37 ) that lacks kinase activity; and C) missense mutations allowing expression of mutant ATM with reduced kinase activity ( 37 ) . We have also included in this group the IVS40–1050A>G (5672ins137) “leaky” splicing mutation that can express a low level of normal ATM protein with kinase activity ( 3739 ) .
The full list of observed mutations assigned to each group is given in the Supplementary Table (available at http://jncicancerspectrum.oupjournals.org/jnci/content/vol97/issue11 ). Western blotting is routinely performed on lymphoblastoid cell lines derived from A-T patients to check for loss of ATM protein as part of the confirmation of diagnosis. The presence of some ATM protein was confirmed in A-T cells carrying all group B and C mutations (Supplementary Table, available at http://jncicancerspectrum.oupjournals.org/jnci/content/vol97/issue11 ). If ATM protein is expressed, its kinase activity can be assayed by in vitro phosphorylation of p53 ( 39 ) or detected with phosphospecific antibodies to in vivo targets (e.g., p53ser15) ( 37 ) . The ATM protein associated with the 7636del9 mutation (group B) has no detectable kinase activity ( 37 ) , although the carriers of both the 7271T>G and 5672ins137 ATM mutations (group C) express ATM protein with kinase activity ( 37 , 39 ) , as do the carriers of the other three mutations in group C. Absence of detectable kinase activity was examined and confirmed in nine patients with group B mutations (data not shown).
The pedigree analysis program MENDEL ( 34 ) was used to obtain maximum-likelihood estimates of the cancer risks associated with the three groups of mutations, assuming that all mutations must belong to one of these groups (even if there is currently insufficient evidence to say which). An iterative maximum-likelihood approach was necessary because of the incomplete genotype information available. This is an extension of the EM algorithm approach described earlier that allows for the nonindependence of genotypes within the same family. Along with relative risk parameters for breast cancer and all non-breast cancers in heterozygous mutation-carrying relatives, parameters for the relative risks of lymphoid tumors in A-T patients [C81–C96 inclusive, ICD revision 10 ( 40 ) ] were included in the models. A single relative risk parameter was used to model the risk of lymphoid tumors in A-T patients with no group C mutation (i.e., no kinase activity), whereas the relative risk parameter for patients with at least one group C mutation (i.e., some kinase activity) was fixed at 1.0. The inclusion of these parameters should improve the ability of the program to correctly predict the carrier status of untested individuals and hence give more precise relative risk estimates. In this analysis, 12 relative risk parameters were estimated for heterozygous carriers: three breast cancer relative risk parameters for women younger than 50 years of age (one for each mutation group), three for women older than 50 years of age, and three relative risk parameters each for male and female non-breast cancers.
Two families segregating the 7271T>G mutation were excluded from the genotype–phenotype analysis, because the identification of these families had prompted the hypothesis that the 7271T>G missense mutation (a group C mutation) was associated with a particularly elevated breast cancer risk ( 22 ) . One further family was excluded due to uncertainty about the function of its one identified mutation. The two families carrying mutations in the MRE11 gene were also excluded from this analysis (although they had been included in the main cohort analysis). This analysis was therefore based on 134 families, i.e., 268 mutant alleles. One hundred thirty-eight mutations have been identified (45 families have either two known mutations or two copies of the same mutation, and 48 families have one known mutation). Of these mutations, 86 were from group A, 34 were from group B, 18 were from group C, and one was of uncertain function (3403del174). The ATM mutation frequency (0.3%) was divided among the three groups of mutations according to these proportions. Estimating the allele frequencies as parameters within the model gave essentially the same results.
To improve the statistical power, the analysis was repeated with 16 additional breast cancers that were not eligible for the main analysis, because they either occurred before 1971 or after age 80 years, or were not confirmed by the NHSCR. Although including these cases might bias the overall relative risk estimate, there is no reason to believe that they would be biased toward any particular mutation group. Model selection was carried out using a conventional likelihood ratio test approach. All P values are two-sided; in the text, “statistically significant” is used to denote a P of <.05.
## R ESULTS
### Overall Results for Cohort
After the exclusions described above, the cohort consisted of 1160 relatives of A-T patients from 132 families (26 220 person-years). A total of 118 first cancers were reported by the NHSCR, compared with the 96.7 expected (SIR = 1.22, 95% CI = 1.02 to 1.46). Fifty-four of the cases were in men (50.3 expected), and 64 were in women (46.3 expected) (SIR = 1.07, 95% CI = 0.82 to 1.40, and SIR = 1.38, 95% CI = 1.08 to 1.77, in men and women, respectively). The median age was 50 years.
### Analysis by Type of Relative
The distribution of individuals, person-years, and cancer cases among relatives of each type is shown in Table 1 . Over all types of relative, the incidence of all cancers other than breast cancer was similar to that of the general population (SIR = 1.16, 95% CI = 0.95 to 1.41). The excess was attributable largely to excess risks in aunts/uncles (SIR = 1.75, 95% CI = 1.10 to 2.64) and grandparents (SIR = 1.23, 95% CI = 0.92 to 1.63). No statistically significant excess was observed in parents or great-grandparents. The overall number of breast cancers in relatives was slightly higher than expected (SIR = 1.59, 95% CI = 1.01 to 2.38, Table 1 ). Five of the 23 eligible breast cancers were in mothers, nine in aunts, eight in grandmothers, and one in a great-grandmother.
The 14 cancers diagnosed in parents of A-T patients are listed in Table 2 . No cancer site showed a statistically significant excess. Overall, the cancer incidence in parents was similar to that predicted using general population rates.
Table 2.
Cancer incidence in 247 parents of A-T patients from 132 families *
Cancer site Obs Exp SIR (95% CI)
Esophagus 0.27 3.65 (0.09 to 20.4)
Colorectal 1.45 0.69 (0.02 to 3.83)
Lung 1.89 1.59 (0.33 to 4.63)
Breast (female) 2.67 1.87 (0.61 to 4.36)
Prostate 0.55 1.83 (0.05 to 10.2)
Bladder 0.63 3.17 (0.38 to 11.4)
Brain 0.34 2.98 (0.08 to 16.6)
All sites 14 13.3 1.06 (0.58 to 1.77)
All except breast 10.6 0.85 (0.39 to 1.62)
Cancer site Obs Exp SIR (95% CI)
Esophagus 0.27 3.65 (0.09 to 20.4)
Colorectal 1.45 0.69 (0.02 to 3.83)
Lung 1.89 1.59 (0.33 to 4.63)
Breast (female) 2.67 1.87 (0.61 to 4.36)
Prostate 0.55 1.83 (0.05 to 10.2)
Bladder 0.63 3.17 (0.38 to 11.4)
Brain 0.34 2.98 (0.08 to 16.6)
All sites 14 13.3 1.06 (0.58 to 1.77)
All except breast 10.6 0.85 (0.39 to 1.62)
*
Obs = observed cancers, Exp = expected cancers, SIR = standardized incidence ratio, CI = confidence interval.
### Weighted Relative Risk Estimation
Consistent with previous observations, a statistically significant excess of female breast cancer in heterozygous ATM mutation carriers was seen (RR = 2.23, 95% CI = 1.16 to 4.28, Table 3 ) compared with the general population. Excluding breast cancer, there remained some evidence of an overall increased cancer risk to ATM carriers compared with that of the general population (RR = 1.47, 95% CI = 1.00 to 2.16), which was slightly greater in female carriers (RR = 2.05, 95% CI = 1.09 to 3.84) than in male carriers (RR = 1.23, 95% CI = 0.76 to 2.00). In addition, a statistically significant excess risk was observed for colorectal cancer (RR = 2.54, 95% CI = 1.06 to 6.09), and there was some suggestion of an excess of stomach cancer (RR = 3.39, 95% CI = 0.86 to 13.4).
Table 3.
Cancer incidence in 1160 relatives of A-T patients from 132 families, with estimated relative risks (RRs) and 95% confidence intervals (CIs) to heterozygous ATM carriers estimated using the EM algorithm *
Cancer site ICD 9 Obs Exp RR (95% CI)
Buccal cavity and pharynx 140–149 1.78 1.59 (0.15 to 16.8)
Esophagus 150 2.17 2.34 (0.47 to 11.6)
Stomach 151 10 4.74 3.39 (0.86 to 13.4)
Colorectal 152–154 20 12.1 2.54 (1.06 to 6.09)
Gallbladder 156 0.53 12.2 (1.26 to 118)
Pancreas 157 2.63 2.41 (0.34 to 17.1)
Lung 162 21 18.2 1.38 (0.64 to 2.97)
Breast (female) 174 23 14.6 2.23 (1.16 to 4.28)
Uterus 179 2.15 1.38 (0.09 to 22.4)
Ovary 183 2.67 1.90 (0.20 to 18.2)
Prostate 185 5.34 1.29 (0.30 to 5.48)
Bladder 188 5.22 1.41 (0.41 to 4.82)
Brain 191 1.93 0.06 (0.01 to 0.33)
Unknown 199 5.19 0.70 (0.10 to 4.92)
Myeloma 203 1.09 4.49 (0.32 to 62.2)
Other female genital 184 0.43 10.2 (0.30 to 345)
All sites except breast 95 82.1 1.47 (1.00 to 2.16)
Male: all sites 54 50.4 1.23 (0.76 to 2.00)
Female: all sites except breast 41 31.8 2.05 (1.09 to 3.84)
Cancer site ICD 9 Obs Exp RR (95% CI)
Buccal cavity and pharynx 140–149 1.78 1.59 (0.15 to 16.8)
Esophagus 150 2.17 2.34 (0.47 to 11.6)
Stomach 151 10 4.74 3.39 (0.86 to 13.4)
Colorectal 152–154 20 12.1 2.54 (1.06 to 6.09)
Gallbladder 156 0.53 12.2 (1.26 to 118)
Pancreas 157 2.63 2.41 (0.34 to 17.1)
Lung 162 21 18.2 1.38 (0.64 to 2.97)
Breast (female) 174 23 14.6 2.23 (1.16 to 4.28)
Uterus 179 2.15 1.38 (0.09 to 22.4)
Ovary 183 2.67 1.90 (0.20 to 18.2)
Prostate 185 5.34 1.29 (0.30 to 5.48)
Bladder 188 5.22 1.41 (0.41 to 4.82)
Brain 191 1.93 0.06 (0.01 to 0.33)
Unknown 199 5.19 0.70 (0.10 to 4.92)
Myeloma 203 1.09 4.49 (0.32 to 62.2)
Other female genital 184 0.43 10.2 (0.30 to 345)
All sites except breast 95 82.1 1.47 (1.00 to 2.16)
Male: all sites 54 50.4 1.23 (0.76 to 2.00)
Female: all sites except breast 41 31.8 2.05 (1.09 to 3.84)
*
The cancer sites shown are those for which at least two cases were observed. In addition, there was a single observed case of each of the following cancers: melanoma, cervix, testis, kidney, and thyroid. ICD = International Classification of Disease, Obs = observed cancers, Exp = expected cancers.
### Age Groups
The estimated relative risks for carriers younger than 50 years of age and 50 years of age or older are summarized in Table 4 . The overall relative risk of cancer was greater for both male and female carriers younger than 50 years of age, with little evidence of an excess risk for carriers aged 50 years and older (RR = 1.04, 95% CI = 0.59 to 1.83 in males; RR = 1.64, 95% CI = 0.81 to 3.30 in females, excluding breast cancer). The estimated relative risk of breast cancer in carriers younger than 50 years of age was close to 5 (RR = 4.94, 95% CI = 1.90 to 12.9), but there was no statistically significant risk for women 50 years of age and older. The overall excess cancer risk in carriers younger than 50 years of age appeared to be due to several different cancer types (for myeloma, RR = 43.3, 95% CI = 2.70 to 694; for stomach cancer, RR = 15.8, 95% CI = 1.63 to 153). One of the two buccal cavity cancers was a nasopharyngeal cancer in the 6-year-old brother of an A-T patient; this was the only juvenile cancer in a relative.
Table 4.
Cancer incidence, by age group, in 1160 relatives of A-T patients from 132 families, with estimated relative risks (RRs) and 95% confidence intervals (CIs) to heterozygous ATM carriers estimated using the EM algorithm *
Less than 50 years old
50 years or older
Site Obs Exp RR (95% CI) Obs Exp RR (95% CI)
Stomach 0.33 15.8 (1.63 to 153) 4.51 2.16 (0.40 to 11.6)
Colorectal 1.10 3.20 (0.55 to 18.3) 18 11.0 2.45 (0.90 to 6.69)
Gallbladder 0.04 0.49 13.5 (1.39 to 132)
Lung 1.05 0.78 (0.02 to 39.0) 20 17.2 1.42 (0.65 to 3.11)
Breast 11 4.34 4.94 (1.90 to 12.9) 12 10.1 1.14 (0.48 to 2.72)
Prostate 0.04 5.30 1.31 (0.31 to 5.57)
Female genital 0.07 0.36 12.3 (0.36 to 423)
All sites 30 15.4 3.16 (1.77 to 5.65) 88 81.2 1.20 (0.81 to 1.78)
Male: all sites 5.33 2.14 (0.86 to 5.30) 45 45.1 1.04 (0.59 to 1.83)
Female: all sites except breast 10 5.78 3.81 (1.09 to 13.4) 31 26.0 1.64 (0.81 to 3.30)
Less than 50 years old
50 years or older
Site Obs Exp RR (95% CI) Obs Exp RR (95% CI)
Stomach 0.33 15.8 (1.63 to 153) 4.51 2.16 (0.40 to 11.6)
Colorectal 1.10 3.20 (0.55 to 18.3) 18 11.0 2.45 (0.90 to 6.69)
Gallbladder 0.04 0.49 13.5 (1.39 to 132)
Lung 1.05 0.78 (0.02 to 39.0) 20 17.2 1.42 (0.65 to 3.11)
Breast 11 4.34 4.94 (1.90 to 12.9) 12 10.1 1.14 (0.48 to 2.72)
Prostate 0.04 5.30 1.31 (0.31 to 5.57)
Female genital 0.07 0.36 12.3 (0.36 to 423)
All sites 30 15.4 3.16 (1.77 to 5.65) 88 81.2 1.20 (0.81 to 1.78)
Male: all sites 5.33 2.14 (0.86 to 5.30) 45 45.1 1.04 (0.59 to 1.83)
Female: all sites except breast 10 5.78 3.81 (1.09 to 13.4) 31 26.0 1.64 (0.81 to 3.30)
*
The cancer sites shown are those for which either the overall carrier RR was statistically significantly greater than 1 or for which there were 10 or more cases. In addition, there were two or more cases in the younger age group of buccal cavity and pharynx cancer (two cases), uterus cancer (two cases), and myeloma (two cases). Obs = observed cancers, Exp = expected cancers.
### Cumulative Cancer Risks
Cumulative risks of cancer were estimated by applying the estimated relative risks for carriers to the incidence rates in the general population. The cumulative risk of breast cancer in heterozygous ATM mutation carriers was estimated to be 8.8% (95% CI = 3.5% to 21.4%) by age 50 years and 16.6% (95% CI = 9.1% to 29.3%) by age 80 years ( Fig. 1, A ). The latter risk, that approximately one woman in six will develop breast cancer, compares with a risk of approximately one in 11 in the general population of England and Wales (1992–1997) ( 29 ) . The estimated risk of any other cancer type by age 50 years was 5.3% (95% CI = 2.2% to 12.6%) in males and 9.0% (95% CI = 2.6% to 28.1%) in females, compared with 2.5% and 2.4%, respectively in the general population ( 29 ) . The cumulative risk of any non-breast cancer by age 80 years was similar in male and female carriers (38.9%, 95% CI = 25.6% to 56.0%; and 35.1%, 95% CI = 20.9% to 55.0%, respectively), although the risk in females was more strongly elevated above the population risk ( Fig. 1, B and C ).
Fig. 1.
Cumulative risks of cancer in heterozygous ATM mutation carriers, estimated from cancer incidence in 1160 relatives of A-T patients from 132 UK families. A ) Estimated cumulative risks of breast cancer in female heterozygous ATM mutation carriers. B ) Estimated cumulative risks of all cancers in male heterozygous ATM mutation carriers. C ) Estimated cumulative risks of all cancers other than breast cancer in female heterozygous ATM mutation carriers. Estimated cumulative risks to carriers along with 95% confidence intervals ( solid lines ) and cumulative risks in the general population [England and Wales, 1992–1997 ( 29 ) hatched lines ] are shown, at each 10-year age point. Cumulative risks were obtained by applying the estimated RRs to carriers below and above age 50 (estimated using the EM algorithm) to the general population rates.
Fig. 1.
Cumulative risks of cancer in heterozygous ATM mutation carriers, estimated from cancer incidence in 1160 relatives of A-T patients from 132 UK families. A ) Estimated cumulative risks of breast cancer in female heterozygous ATM mutation carriers. B ) Estimated cumulative risks of all cancers in male heterozygous ATM mutation carriers. C ) Estimated cumulative risks of all cancers other than breast cancer in female heterozygous ATM mutation carriers. Estimated cumulative risks to carriers along with 95% confidence intervals ( solid lines ) and cumulative risks in the general population [England and Wales, 1992–1997 ( 29 ) hatched lines ] are shown, at each 10-year age point. Cumulative risks were obtained by applying the estimated RRs to carriers below and above age 50 (estimated using the EM algorithm) to the general population rates.
Based on the observed case frequency over the period 1979–1997, we estimate the heterozygous carrier frequency to be 0.4%. Therefore, our best estimate of the fraction of breast cancer cases attributable to ATM mutations is 0.5% overall, rising to 1.6% for cases diagnosed before age 50 years.
### Mortality
The overall mortality rate in males was almost identical to that expected (SMR = 1.01, 95% CI = 0.87 to 1.16). However, this rate reflected the combination of a modestly increased risk of cancer deaths (SMR = 1.35, 95% CI = 1.07 to 1.70) with a slight, statistically non-significant deficit of non-cancer deaths (SMR = 0.88, 95% CI = 0.74 to 1.05). The relative risk of non-cancer death was similar in female relatives (SMR = 0.85, 95% CI = 0.67 to 1.09), but a higher risk of cancer deaths (SMR = 1.82, 95% CI = 1.43 to 2.32) in these relatives resulted in an overall borderline statistically significantly increased mortality rate (SMR = 1.16, 95% CI = 0.98 to 1.37) as compared with the general population. The mortality in fathers was close to that expected in the general population, as was the mortality in other male relatives (data not shown). The mortality in female relatives other than mothers was also close to that expected in the general population. However, there were only two deaths in mothers (a lung cancer and a pancreatic cancer), as compared with an expected 7.90 deaths.
Seventeen deaths from breast cancer were observed in female relatives (SMR = 2.08, 95% CI = 1.21 to 3.32). Ten of these were included in the incidence analysis; the other seven were ineligible because they were reported only on death certificates and not by the NHSCR.
Statistically significant excess cancer mortality was observed in ATM carriers of both sexes (SMR = 1.88, 95% CI = 1.14 to 3.10 and SMR = 3.56, 95% CI = 1.83 to 6.93 for males and females, respectively), whereas non-cancer mortality was slightly, but not statistically significantly, lower than expected ( Table 5 ). There was no evidence of excess mortality from either vascular or respiratory disease. Statistically significant excesses in mortality in ATM carriers were estimated for breast cancer (RR = 4.18, 95% CI = 1.38 to 12.7), stomach cancer (RR = 4.19, 95% CI = 1.49 to 11.8), colorectal cancer (RR = 3.19, 95% CI = 1.24 to 8.23), and lung cancer (RR = 2.36, 95% CI = 1.24 to 4.50) as compared with the general population.
Table 5.
Mortality in 1269 relatives of A-T patients from 132 families, with estimated relative risks (RRs) and 95% confidence intervals (CIs) to heterozygous ATM carriers estimated using the EM algorithm *
Death cause ICD 9 Obs Exp RR (95% CI)
Cancer deaths
Esophagus 150 2.68 1.09 (0.08 to 14.6)
Stomach 151 15 7.14 4.19 (1.49 to 11.8)
Colorectal 152–154 18 9.87 3.19 (1.24 to 8.23)
Pancreas 157 3.61 3.21 (0.89 to 11.5)
Lung 162 39 24.9 2.36 (1.24 to 4.50)
Breast (female) 174 17 8.18 4.18 (1.38 to 12.7)
Ovary 183 2.76 1.84 (0.19 to 17.8)
Prostate 185 3.09 0.93 (0.14 to 6.29)
Bladder 188 2.53 1.87 (0.19 to 18.0)
Brain 191 2.19 1.53 (0.11 to 20.7)
Unknown 199 4.33 2.76 (0.59 to 12.9)
Myeloma 203 1.03 1.51 (0.01 to 358)
Other 1.68 4.00 (0.45 to 35.3)
Male: all cancer sites 70 51.9 1.88 (1.14 to 3.10)
Female: all cancer sites 66 36.3 3.56 (1.83 to 6.93)
Female: all cancer sites except breast 49 28.1 3.21 (1.64 to 6.27)
Circulatory disease 119 135 0.78 (0.53 to 1.17)
Respiratory disease 43 35.2 1.63 (0.81 to 3.28)
Injury and poisoning 14.5 0.17 (0.04 to 0.68)
Male: all non-cancers 128 144 0.75 (0.52 to 1.08)
Female: all non-cancers 67 78.4 0.79 (0.45 to 1.38)
Death cause ICD 9 Obs Exp RR (95% CI)
Cancer deaths
Esophagus 150 2.68 1.09 (0.08 to 14.6)
Stomach 151 15 7.14 4.19 (1.49 to 11.8)
Colorectal 152–154 18 9.87 3.19 (1.24 to 8.23)
Pancreas 157 3.61 3.21 (0.89 to 11.5)
Lung 162 39 24.9 2.36 (1.24 to 4.50)
Breast (female) 174 17 8.18 4.18 (1.38 to 12.7)
Ovary 183 2.76 1.84 (0.19 to 17.8)
Prostate 185 3.09 0.93 (0.14 to 6.29)
Bladder 188 2.53 1.87 (0.19 to 18.0)
Brain 191 2.19 1.53 (0.11 to 20.7)
Unknown 199 4.33 2.76 (0.59 to 12.9)
Myeloma 203 1.03 1.51 (0.01 to 358)
Other 1.68 4.00 (0.45 to 35.3)
Male: all cancer sites 70 51.9 1.88 (1.14 to 3.10)
Female: all cancer sites 66 36.3 3.56 (1.83 to 6.93)
Female: all cancer sites except breast 49 28.1 3.21 (1.64 to 6.27)
Circulatory disease 119 135 0.78 (0.53 to 1.17)
Respiratory disease 43 35.2 1.63 (0.81 to 3.28)
Injury and poisoning 14.5 0.17 (0.04 to 0.68)
Male: all non-cancers 128 144 0.75 (0.52 to 1.08)
Female: all non-cancers 67 78.4 0.79 (0.45 to 1.38)
*
Obs = observed cancers, Exp = expected cancers, ICD = International Classification of Disease.
The “other” cancers were three female genital cancers and a cancer of the middle ear.
ATM carrier relative risks were also estimated separately for deaths before or after age 50 years ( Table 6 ). The estimated cancer mortality relative risks were higher for carriers younger than 50 years of age than for carriers aged 50 years and older (RR = 3.59, 95% CI = 1.74 to 7.38; and RR = 2.23, 95% CI = 1.44 to 3.45, respectively). Consistent with the incidence analysis, the relative risk of breast cancer mortality was higher for carriers below age 50 years (RR = 6.08, 95% CI = 1.05 to 35.3) than for carriers aged 50 years and older (RR = 3.45, 95% CI = 0.89 to 13.4). Mortality from stomach cancer and colorectal cancer was also particularly elevated in ATM carriers below age 50 years (stomach cancer, RR = 14.0, 95% CI = 3.18 to 61.9; and colorectal cancer, RR = 11.0, 95% CI = 2.55 to 47.2).
Table 6.
Mortality, by age group, in 1269 relatives of A-T patients from 132 families, with estimated relative risks (RRs) and 95% confidence intervals (CIs) to heterozygous ATM carriers estimated using the EM algorithm *
Less than 50 years old
50 years or older
Death cause Obs Exp RR (95% CI) Obs Exp RR (95% CI)
Cancer deaths
Stomach 0.58 14.0 (3.18 to 61.9) 11 6.55 2.94 (0.75 to 11.5)
Colorectal 0.87 11.0 (2.55 to 47.2) 13 8.99 2.23 (0.67 to 7.46)
Pancreas 0.27 3.34 3.65 (1.01 to 13.2)
Lung 1.60 2.16 (0.37 to 12.5) 37 23.3 2.38 (1.19 to 4.76)
Breast 1.89 6.08 (1.05 to 35.3) 12 6.30 3.45 (0.89 to 13.4)
Male: all sites 4.65 2.55 (0.98 to 6.62) 62 47.3 1.75 (0.99 to 3.08)
Female: all sites except breast 3.98 4.45 (1.06 to 18.6) 40 24.1 2.92 (1.44 to 5.91)
Male: non-cancer deaths 17 22.3 0.61 (0.27 to 1.39) 111 122 0.79 (0.53 to 1.18)
Female: non-cancer deaths 10 12.6 0.78 (0.23 to 2.68) 57 65.7 0.79 (0.42 to 1.48)
Less than 50 years old
50 years or older
Death cause Obs Exp RR (95% CI) Obs Exp RR (95% CI)
Cancer deaths
Stomach 0.58 14.0 (3.18 to 61.9) 11 6.55 2.94 (0.75 to 11.5)
Colorectal 0.87 11.0 (2.55 to 47.2) 13 8.99 2.23 (0.67 to 7.46)
Pancreas 0.27 3.34 3.65 (1.01 to 13.2)
Lung 1.60 2.16 (0.37 to 12.5) 37 23.3 2.38 (1.19 to 4.76)
Breast 1.89 6.08 (1.05 to 35.3) 12 6.30 3.45 (0.89 to 13.4)
Male: all sites 4.65 2.55 (0.98 to 6.62) 62 47.3 1.75 (0.99 to 3.08)
Female: all sites except breast 3.98 4.45 (1.06 to 18.6) 40 24.1 2.92 (1.44 to 5.91)
Male: non-cancer deaths 17 22.3 0.61 (0.27 to 1.39) 111 122 0.79 (0.53 to 1.18)
Female: non-cancer deaths 10 12.6 0.78 (0.23 to 2.68) 57 65.7 0.79 (0.42 to 1.48)
*
Obs = observed cancers, Exp = expected cancers.
### Genotype–Phenotype Correlations
Risks of breast and non-breast cancers in relatives were estimated for the three categories of ATM mutation. There were no statistically significant differences between the mutation groups in the risks of either non-breast cancer ( P = .5) or breast cancer ( P = .8). When 16 additional breast cancer cases were included, the risk was highest for patients with mutations expressing some protein without kinase activity (group B) (comparing groups B and A, RR = 2.5, 95% CI = 0.7 to 8.9) and slightly lower for those with mutations retaining kinase activity (group C) (comparing groups C and A, RR = 0.9, 95% CI = 0.1 to 8.9), although the differences were not statistically significant ( P = .4).
## D ISCUSSION
We have studied the cancer incidence and the mortality in 1160 blood relatives of A-T patients from 132 families and have found evidence for an increased risk of breast cancer in heterozygous ATM mutation carriers, chiefly at young ages, accompanied by a more moderate increase in the risk of other cancers. The overall estimated breast cancer relative risk to heterozygous ATM carriers was 2.23 (95% CI = 1.16 to 4.28), with a relative risk of 4.94 (95% CI = 1.90 to 12.9) in carriers younger than 50 years of age. This is equivalent to a lifetime (until age 80 years) risk of approximately one woman in six, as compared with one in 11 in the general population of England and Wales.
The estimated risk of any cancer in male carriers by the age of 80 years was only slightly higher than in the general population (39% vs. 36%), whereas the risk by age of 80 years of any cancer other than breast cancer in female carriers was considerably higher than in the general population (35% vs. 21%).
Although there was little evidence for an overall excess risk of cancers other than breast cancer in ATM heterozygotes, there was some evidence for excess risks of colorectal cancer and stomach cancer. We also observed a clear excess mortality from cancer, with statistically significant excess risks of stomach, colorectal, and lung cancer deaths. The higher relative risks based on mortality might reflect some underreporting of cancers by the NHSCR but could also reflect a more aggressive behavior of cancers in ATM carriers. Two previous studies have hinted at a possible association between ATM and cancers of the gastrointestinal tract, although neither association was statistically significant ( 7 , 9 ) . In contrast to our study, neither study found any evidence of a specific excess of colorectal cancers in relatives of A-T patients.
Some limitations of this study that may lead to biased relative risk estimates include incomplete ascertainment of families, possible nonpaternity or de novo ATM mutations, and the possibility that some A-T patients may not carry ATM mutations. A further limitation is that we were able to genotype only the parents of A-T patients and not any other relatives. Although this reduced the power of the study and the precision of the relative risk estimates, it should not result in any bias, providing that ATM mutations are inherited according to Mendelian rules. The precision of the estimates was also limited by the number of available A-T families. Further precision should, however, be obtained through combined analysis of our data with those from other European studies.
We have attempted to minimize bias in this study by systematically following a defined cohort of relatives of all known A-T patients and by basing analysis only on registered cancers and deaths reported through national records, to allow direct comparability of observed and expected rates. Nevertheless, some potential biases remain. First, families in which a parent died at a young age might be less likely to have participated in the study. We attempted to minimize this bias by including additional families ascertained through a mention of A-T on a death certificate. That some bias remains is borne out by a marked deficit in mortality in mothers, with two deaths observed, compared with nearly eight expected. This bias is reflected in the slight deficit in overall mortality from nonmalignant causes and suggests that the excess mortality from and incidence of cancer may therefore have been underestimated.
Other events that would reduce the number of mutations in relatives, and hence underestimate the risks, are nonpaternity and de novo mutations. One A-T patient in the cohort is known to carry an inherited truncating mutation alongside a de novo paternal missense mutation, 8189A>C. This is generally considered to be a very rare event in A-T. There was no evidence of incompatible paternal genotypes among the genotyped parents of the A-T patients. Although there may be instances of false paternity among grandparents or great-grandparents, any such events would not affect the carrier probabilities of as many family members.
Strictly speaking, the estimates are a weighted average of the risks conferred by ATM and MRE11 mutations. The apparent A-T cases in two of the families are in fact due to compound heterozygous mutations in the MRE11 gene. The relative sizes of the two genes would suggest that approximately 6% of A-T patients might in fact carry MRE11 mutations, i.e., approximately six further families ( 25 ) . MRE11 acts in the same DNA damage response pathway as does ATM, but mutations in the two genes need not predispose to cancer to the same extent; there is no evidence that homozygous Mre11 mutations are associated with tumors in mice ( 41 ) . If MRE11 mutations conferred no excess cancer risk, then the ATM excess cancer risk estimated in this study could be underestimated by approximately 6%.
In addition to the excesses of breast, colorectal, and stomach cancer noted above, a statistically significant excess of cancer of the gallbladder was also observed, but this was based on only three cases. A high relative risk was estimated for “other female genital cancers,” but this was based on just two cases and was not statistically significant. Three further female genital cancers were also reported but did not contribute to the analysis. A high relative risk (RR = 4.5) was also estimated for myeloma, based on three cases. It is noteworthy that these were the only lymphoid tumors seen in relatives and that no myelomas were observed in A-T patients.
The apparent excesses at some or all of these sites could be due to chance, given the number of cancer sites evaluated, and larger studies will be required to determine whether these effects are genuine. Conversely, moderate risks of other cancers in ATM carriers cannot be ruled out. The modest overall increase in the risk of non-breast cancer appears to be due largely to a combination of small increases at many sites; it is notable that all the relative risk estimates in Table 3 are greater than 1, with the exception of brain cancer and cancers of unknown site.
Our study is comparable in design to two recent European studies, one based in France ( 42 ) and the other in the Nordic countries ( 9 ) . The Nordic study obtained cancer incidence data for 1218 relatives of A-T patients from 50 families via record linkage to national cancer registries. Its authors estimated the breast cancer relative risk for ATM carriers to be 2.4 (95% CI = 1.3 to 4.1), which is very similar to our estimate. Breast cancer was the only individual cancer with a statistically significant excess; apart from breast cancer, they observed only 15% more cancers than expected in relatives of A-T patients.
The French study was based on 1423 relatives of A-T patients from 34 families. ATM genotyping was performed on over a quarter of the relatives, but not all cancer cases had been formally confirmed. The relative risk of breast cancer, weighted by prior carrier probability (RR = 2.43, 95% CI = 1.32 to 4.09) was also very similar to our estimate (RR = 2.23, 95% CI = 1.16 to 4.28). In the French study, the relative risk was higher for women below age 45 years, but no excess was seen in women above this age (RR = 6.32, 95% CI =1.94 to 15.2, and RR = 0.68, 95% CI = 0.08 to 2.46, respectively). There was no evidence of an increased risk of cancers other than breast cancers in carriers in this study ( 42 ) .
The results presented here are generally in line with the French and Nordic studies. Our study has the advantage of being based on a far larger number of families, and, although the number of eligible relatives in our cohort was slightly smaller, the exclusion of cousins and great-aunts/uncles meant that the cohort had a higher density of mutation carriers. Previous studies have either presented separate relative risks for each type of relative, often with large confidence intervals as a consequence of the small numbers of cases in each group, or have pooled all relatives into a single group, without taking into account their different carrier probabilities. In contrast, our use of the EM algorithm to obtain maximum-likelihood estimates of the carrier relative risks, based on weighting the information from all relatives, made more efficient use of the data. In common with the Nordic study (but not the French study), we considered only cancer cases that had been formally confirmed. Neither of the other European studies considered both cancer incidence and mortality.
We found no evidence for any difference in risk of breast or other cancer according to the type of ATM mutation. If anything, the trend was toward a lower breast cancer risk for the group C mutations, in contrast with previous reports that showed that missense mutations, in particular 7271T>G, are associated with a markedly increased risk of breast cancer ( 1922 ) . Our estimates were necessarily imprecise, because group C mutations were the least frequent in this set; after the exclusion of the two hypothesis-generating 7271T>G families ( 22 ) , there was only one breast cancer in a family branch known to carry a group C mutation. Furthermore, because the 5762ins137 mutation accounted for 14 out of 18 of the known group C family branches, the results may not be generalizable to all ATM mutations retaining kinase activity.
The mutation categories were devised in the context of A-T patients with two germline mutations in trans , whereas the analysis of cancer risks was restricted to heterozygous carriers, in whom these particular differences between mutations may be less important. For a single mutation in the presence of a wild-type allele, alternative mechanisms may become relevant to the disease process, potentially including haploinsufficiency (group A), dominant-negative effects (groups B and C), or some gain in function (groups B and C). For example, lymphoblastoid cell lines with a heterozygous missense mutation have been shown to have higher ATM mRNA expression than do cell lines with a truncating mutation and to have poorer cell survival following irradiation ( 43 ) .
A recent study of 34 French A-T families found no difference between the breast cancer risks associated with heterozygous truncating and missense/in-frame deletion ATM mutations but identified three groups of truncating mutations with particularly high breast cancer relative risks, each relating to a known binding domain ( 42 ) . However, we observed no breast cancers in the seven family branches with mutations that truncate the ATM protein in these domains.
It is important to note that our results do not exclude the possibility of more substantial heterogeneity at the mutation level. Despite all these uncertainties, the results do appear to confirm that a substantial risk of breast cancer is conferred by mutations that eliminate the ATM protein and that the risk is not restricted to a subset of missense mutations.
The breast cancer risk we have estimated would be sufficient to classify an ATM carrier as “moderate risk,” according to recent guidelines of the National Institute for Clinical Excellence (2004). These guidelines suggest that annual mammography beginning at age 40 years may be appropriate in this risk group. However, given the role of ATM in radiation-induced DNA repair, it is not clear whether mammographic screening would be beneficial in ATM carriers. Recent studies have suggested that magnetic resonance imaging may be a sensitive screening tool in women at high risk of breast cancer, such as BRCA1 and BRCA2 carriers ( 44 ) , and it may provide an alternative management approach for ATM carriers. Further research would be needed to evaluate the appropriateness of any specific screening for gastric, colorectal, or other cancers.
In conclusion, this study has confirmed an approximately twofold-increased risk of breast cancer in female carriers of ATM mutations, with a higher relative risk for those younger than 50 years. We also identified increased risk of colorectal cancer and a possible increased risk of stomach cancer. Combined analyses with similar cohorts and further follow-up will be required to provide reliable risk estimates for other cancer sites and to investigate mutation-specific effects.
We thank the A-T patients and their families for their willingness to participate in this research. This study was supported by grants from Cancer Research UK and the A-T Society. DFE is a Principal Research Fellow of Cancer Research UK. We also thank staff at the Office of National Statistics for its help.
## References
(1)
Morrell D, Cromartie E, Swift M. Mortality and cancer incidence in 263 patients with ataxia-telangiectasia.
J Natl Cancer Inst
1986
;
77
:
89
–92.
(2)
Savitsky K, Bar-Shira A, Gilad S, Rotman G, Ziv Y, Vanagaite L, et al. A single ataxia telangiectasia gene with a product similar to PI-3 kinase.
Science
1995
;
268
:
1749
–53.
(3)
Swift M, Morrell D, Cromartie E, Chamberlin AR, Skolnick MH, Bishop DT. The incidence and gene frequency of ataxia-telangiectasia in the United States.
Am J Hum Genet
1986
;
39
:
573
–83.
(4)
FitzGerald MG, Bean JM, Hegde SR, Unsal H, MacDonald DJ, Harkin DP, et al. Heterozygous ATM mutations do not contribute to early onset of breast cancer.
Nat Genet
1997
;
15
:
307
–10.
(5)
Swift M, Reitnauer PJ, Morrell D, Chase CL. Breast and other cancers in families with ataxia-telangiectasia.
N Engl J Med
1987
;
316
:
1289
–94.
(6)
Swift M, Morrell D, Massey RB, Chase CL. Incidence of cancer in 161 families affected by ataxia-telangiectasia.
N Engl J Med
1991
;
325
:
1831
–6.
(7)
Inskip HM, Kinlen LJ, Taylor AM, Woods CG, Arlett CF. Risk of breast cancer and other cancers in heterozygotes for ataxia-telangiectasia.
Br J Cancer
1999
;
79
:
1304
–7.
(8)
Janin N, Andrieu N, Ossian K, Lauge A, Croquette MF, Griscelli C, et al. Breast cancer risk in ataxia telangiectasia (AT) heterozygotes: haplotype study in French AT families.
Br J Cancer
1999
;
80
:
1042
–5.
(9)
Olsen JH, Hahnemann JM, Borresen-Dale AL, Brondum-Nielsen K, Hammarstrom L, Kleinerman R, et al. Cancer in patients with ataxia-telangiectasia and in their relatives in the nordic countries.
J Natl Cancer Inst
2001
;
93
:
121
–7.
(10)
Su Y, Swift M. Mortality rates among carriers of ataxia-telangiectasia mutant alleles.
Ann Intern Med
2000
;
133
:
770
–8.
(11)
Easton DF. Cancer risks in A-T heterozygotes.
1994
;
66
:
S177
–82.
(12)
Geoffroy-Perez B, Janin N, Ossian K, Lauge A, Croquette MF, Griscelli C, et al. Cancer risk in heterozygotes for ataxia-telangiectasia.
Int J Cancer
2001
;
93
:
288
–93.
(13)
Bebb DG, Yu Z, Chen J, Telatar M, Gelmon K, Phillips N, et al. Absence of mutations in the ATM gene in forty-seven cases of sporadic breast cancer.
Br J Cancer
1999
;
80
:
1979
–81.
(14)
Buchholz TA, Weil MM, Ashorn CL, Strom EA, Sigurdson A, Bondy M, et al. A Ser49Cys variant in the ataxia telangiectasia, mutated, gene that is more common in patients with breast carcinoma compared with population controls.
Cancer
2004
;
100
:
1345
–51.
(15)
Izatt L, Greenman J, Hodgson S, Ellis D, Watts S, Scott G, et al. Identification of germline missense mutations and rare allelic variants in the ATM gene in early-onset breast cancer.
Genes Chromosomes Cancer
1999
;
26
:
286
–94.
(16)
Shafman TD, Levitz S, Nixon AJ, Gibans LA, Nichols KE, Bell DW, et al. Prevalence of germline truncating mutations in ATM in women with a second breast cancer after radiation therapy for a contralateral tumor.
Genes Chromosomes Cancer
2000
;
27
:
124
–9.
(17)
Thorstenson YR, Roxas A, Kroiss R, Jenkins MA, Yu KM, Bachrich T, et al. Contributions of ATM mutations to familial breast and ovarian cancer.
Cancer Res
2003
;
63
:
3325
–33.
(18)
Chen J, Birkholtz GG, Lindblom P, Rubio C, Lindblom A. The role of ataxia-telangiectasia heterozygotes in familial breast cancer.
Cancer Res
1998
;
58
:
1376
–9.
(19)
Chenevix-Trench G, Spurdle AB, Gatei M, Kelly H, Marsh A, Chen X, et al. Dominant negative ATM mutations in breast cancer families.
J Natl Cancer Inst
2002
;
94
:
205
–15.
(20)
Dork T, Bendix R, Bremer M, Rades D, Klopper K, Nicke M, et al. Spectrum of ATM gene mutations in a hospital-based series of unselected breast cancer patients.
Cancer Res
2001
;
61
:
7608
–15.
(21)
Gatti RA, Tward A, Concannon P. Cancer risk in ATM heterozygotes: a model of phenotypic and mechanistic differences between missense and truncating mutations.
Mol Genet Metab
1999
;
68
:
419
–23.
(22)
Stankovic T, Kidd AM, Sutcliffe A, McGuire GM, Robinson P, Weber P, et al. ATM mutations and phenotypes in ataxia-telangiectasia families in the British Isles: expression of mutant ATM and the risk of leukemia, lymphoma, and breast cancer.
Am J Hum Genet
1998
;
62
:
334
–45.
(23)
Bernstein JL, Bernstein L, Thompson WD, Lynch CF, Malone KE, Teitelbaum SL, et al. ATM variants 7271T>G and IVS10–6T>G among women with unilateral and bilateral breast cancer.
Br J Cancer
2003
;
89
:
1513
–6.
(24)
Spurdle AB, Hopper JL, Chen X, McCredie MR, Giles GG, Newman B, et al. No evidence for association of ataxia-telangiectasia mutated gene T2119C and C3161G amino acid substitution variants with risk of breast cancer.
Breast Cancer Res
2002
;
4
:
R15
.
(25)
Stewart GS, Maser RS, Stankovic T, Bressan DA, Kaplan MI, Jaspers NG, et al. The DNA double-strand break repair gene hMRE11 is mutated in individuals with an ataxia-telangiectasia-like disorder.
Cell
1999
;
99
:
577
–87.
(26)
Muir C, Waterhouse J, Mack T, Powell J, et al. Cancer incidence in five continents. Vol V. Lyon (France): IARC;
1987
.
(27)
Parkin DM, Muir CS, Whelan SL, Gao YT, et al. Cancer incidence in five continents. Vol VI. Lyon (France): IARC;
1992
.
(28)
Parkin DM, Whelan SL, Ferlay J, Raymond L, et al. Cancer incidence in five continents. Vol VII. Lyon (France): IARC;
1997
.
(29)
Parkin DM, Whelan SL, Ferlay J, Teppo L, et al. Cancer incidence in five continents. Vol VIII. Lyon (France): IARC;
2002
.
(30)
Waterhouse JAH, Muir C, Correa P, Powell J. Cancer incidence in five continents. Vol III. Lyon (France): IARC;
1976
.
(31)
Waterhouse JAH, Muir C, Shanmugaratnam K, Powell J. Cancer incidence in five continents. Vol IV. Lyon (France): IARC;
1982
.
(32)
Coleman MP, Hermon C, Douglas A. Person-years (PYRS)—a fortran program for cohort study analysis. IARC internal report 89/006. Lyon (France): IARC;
1989
.
(33)
Breslow NE, Day NE. Statistical methods in cancer research. Vol II—the design and analysis of cohort studies. Lyon (France): IARC;
1987
.
(34)
Lange K, Weeks D, Boehnke M. Programs for pedigree analysis: MENDEL, FISHER, and dGENE.
Genet Epidemiol
1988
;
5
:
471
–2.
(35)
Dempster AP, Laird NM, Rubin DB. Maximum likelihood from incomplete data via the EM algorithm (with discussion).
J Roy Stat Soc Series B
1977
;
39
:
1
–38.
(36)
Louis T. Finding the observed information matrix when using the EM algorithm.
J Roy Stat Soc Series B
1982
;
44
:
226
–33.
(37)
Stewart GS, Last JI, Stankovic T, Haites N, Kidd AM, Byrd PJ, et al. Residual ataxia telangiectasia mutated protein function in cells from ataxia telangiectasia patients, with 5762ins137 and 7271T–>G mutations, showing a less severe phenotype.
J Biol Chem
2001
;
276
:
30133
–41.
(38)
McConville CM, Stankovic T, Byrd PJ, McGuire GM, Yao QY, Lennox GG, et al. Mutations associated with variant phenotypes in ataxia-telangiectasia.
Am J Hum Genet
1996
;
59
:
320
–30.
(39)
Sutton IJ, Last JI, Ritchie SJ, Harrington HJ, Byrd PJ, Taylor AM. Adult-onset ataxia telangiectasia due to ATM 5762ins137 mutation homozygosity.
Ann Neurol
2004
;
55
:
891
–5.
(40)
World Health Organization. International statistical classification of diseases and related health problems, 1989 revision. Vol I. Geneva (Switzerland): World Health Organization;
1992
.
(41)
Theunissen JW, Kaplan MI, Hunt PA, Williams BR, Ferguson DO, Alt FW, et al. Checkpoint failure and chromosomal instability without lymphomagenesis in Mre11(ATLD1/ATLD1) mice.
Mol Cell
2003
;
12
:
1511
–23.
(42)
Cavaciuti E, Lauge A, Janin N, Ossian K, Hall J, Stoppa-Lyonnet D, et al. Cancer risk according to type and location of ATM mutation in ataxia-telangiectasia families.
Genes Chromosomes Cancer
2005
;
42
:
1
–9.
(43)
Fernet M, Moullan N, Lauge A, Stoppa-Lyonnet D, Hall J. Cellular responses to ionising radiation of AT heterozygotes: differences between missense and truncating mutation carriers.
Br J Cancer
2004
;
90
:
866
–73.
(44)
Kriege M, Brekelmans CT, Boetes C, Besnard PE, Zonderland HM, Obdeijn IM, et al. Efficacy of MRI and mammography for breast-cancer screening in women with a familial or genetic predisposition.
N Engl J Med
2004
;
351
:
427
–37.
(45)
National Institute of Clinical Excellence Guidelines, CG0143 Familial Breast Cancer. Available at: http://www.nice.org.uk/CG014NICEguideline . [Last accessed: April 28,
2005
.]
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2017-02-23 21:29:58
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https://cs184.eecs.berkeley.edu/sp20/lecture/4-83/transforms
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Lecture 4: Transforms (83)
evan1997123
Although the M looks a bit weird and there's just so much transformation matrices that are being shown to us, I think that is cool that we are able to just multiply all of the M's together to create one single Transformation Matrix M that will do so many different operations into one. Then, we could just hold onto those and never have to compute them. I like how this has been created
NicoDeshler
I still don't think I understand how M is working here? Isn't the intended transformation for perspective projection to take the vector (x,y,z) in 3D space and map it to (xd/z, yd/z, d) as shown by the 'similar triangles' argument on the previous slide? Wouldn't this then simply be a diagonal matrix with the following factors in the diagonal entries (d/z,d/z,/d)? I think I am missing why we would need homogenous coordinates for this transformation. Thanks!
dangeng184
@Nico (hi, good to see you here). I think if you did it your way you would need a different matrix for each point you transform (because the z is variable). Doing it in homogenous coordinates allows you to use just a single transformation matrix for all the points.
NicoDeshler
Ah I see your point. Thanks Daniel! Just to make sure I understand correctly, we would still use the z-coordinate of every point to multiply the final vector $(x, y, z, z/d)$ by $d/z$, we just wouldn't instantiate a new matrix for every point?
You must be enrolled in the course to comment
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2020-02-25 19:38:25
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https://puzzling.stackexchange.com/questions/98190/how-can-i-cut-a-cube-so-that-all-its-vertices-except-for-two-mutually-opposite-v
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# How can I cut a cube so that all its vertices except for two mutually opposite vertices are equally distanced from the plane of the cut?
A friend of mine has been struggling with a solid geometry problem and, knowing my imagination skills developed by playing gomokunarabe and renju, has asked me to help her, but the problem has proved to be too tough to crack for me either. The difficulty isn't about how to make calculations; it's that we can't imagine where a cut described in the problem could be. The problem seems to require a great deal of creativity and imagination and be a great dissection puzzle, and I humbly hope that SE users can help us.
Here is my English translation of the problem:
All vertices of a cube ABCDA₁B₁C₁D₁ except for two mutually opposite vertices A and C₁ are equally distanced from a certain plane. Find the distance from each of those six vertices to that plane. The length of an edge of the cube is 1.
Hint: Consider two different cases.
It's the hint that made me puzzled, because I see only one single way to make a cut satisfying the formulation of the problem. My idea is simple: let's look at the cube from such a perspective that two mutually opposite edges entirely merge with each other, and let's cut the cube so that the plane of the cut is the red line in the image below:
For that cut I easily made calculations and found the distance, but where does the second legitimate cut lie? I can't even imagine.
If I cut the cube parallel to one of its faces into two equal halves, then all eight vertices will be equally distanced from the plane of the cut, and this contradicts the requirement that a couple of mutually opposite vertices stand out in this regard.
Unable to see the second cut, I thought that the hint in the formulation of the problem might be erroneous, but my friend considers that to be a very remote possibility, knowing how scrupulous her teacher is.
Can you find the mysterious second cut or prove that it doesn't exist?
• I think that the "consider two different cases" hint isn't referring to the number of solutions but to some case-by-case analysis that the instructor might have in their own solution. That doesn't make it a particularly helpful hint, I admit, but it's my best guess as to what's going on here. May 13 '20 at 23:09
• Is $A$ and $C_1$ "standing out" an explicit requirement? At least your English translation is ambiguous in that respect. So perhaps the second solution is indeed the simple cutting into two equal cuboids. Impossible to judge without the original wording. Sep 9 '20 at 18:16
• @PaulPanzer : I see your point, and the original wording is precisely as ambiguous as my English translation. I now see that from the formal standpoint, it can be argued that the original formulation doesn't require A and C1 to stand out, and merely says that all other vertices are equally distanced from a plane. So yes, it seems it's just a clumsy formulation by a teacher who didn't care how his formulation would be perceived. Sep 10 '20 at 13:42
I think it is
not possible
because
We need a plane where $$B$$, $$C$$, $$D$$, $$D_1$$, $$A_1$$, $$B_1$$ are at an equal distance from the plane. For this to happen, we need to split these 6 points into two (possibly empty) groups, one on each side of the plane.
A 6-0 split is impossible (that is to say, they can't all be on the same side) because the 6 points are not coplanar, so if they are all on the same side, they cannot be the same distance from a plane.
Likewise, a 5-1 split is impossible because no 5 of them are coplanar.
For a 4-2 split, the only possibility (up to symmetry) is $$S_1 = \{B, D, D_1, B_1\}$$ and $$S_2 = \{A_1, C\}$$. However, this won't work because the plane would have to be parallel to the plane passing through all points of $$S_1$$, but then you can't put $$S_1$$ on one side and $$S_2$$ on the other.
So the final case is a 3-3 split. The OP has already found $$S_1 = \{B, D, A_1\}$$ and $$S_2 = \{C, D_1, B_1\}$$.
Out of the vertices adjacent to A, at least two of them must be on the same side of the plane as each other, by the Pigeonhole Principle. WLOG, assume that $$B$$ and $$D$$ are in $$S_1$$. Now we can simply consider the possibilities for the last vertex of $$S_1$$:
- If the last vertex is $$A_1$$, this yields the OP's solution.
- If it is $$B_1$$ or $$D_1$$, this becomes the 4-2 case that was described above, so it doesn't work.
- If it is $$C$$, then this yields the OP's "solution" of slicing the cube in half, parallel to a face (namely face $$ABCD$$), which might not be valid.
So we have exhausted all the possibilities.
• Great answer, thanks a lot. My intuition was telling me it's impossible to find a second cut, and you proved that in a very clear and organized way. Many thanks. May 15 '20 at 11:47
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2021-12-08 01:38:56
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|
https://www.statsmodels.org/v0.11.0/generated/statsmodels.nonparametric.kde.KDEUnivariate.evaluate.html
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# statsmodels.nonparametric.kde.KDEUnivariate.evaluate¶
KDEUnivariate.evaluate(point)[source]
Evaluate density at a single point.
Parameters
pointfloat
Point at which to evaluate the density.
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2023-02-04 22:20:51
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https://www.ias.ac.in/listing/bibliography/pram/ZAFAR_AHMED
|
• ZAFAR AHMED
Articles written in Pramana – Journal of Physics
• Random matrix model for disordered conductors
We present a random matrix ensemble where real, positive semi-definite matrix elements, x, are log-normal distributed, exp[−log2(x)]. We show that the level density varies with energy, E, as 2/(1+E) for large E, in the unitary family, consistent with the expectation for disordered conductors. The two-level correlation function is studied for the unitary family and found to be largely of the universal form despite the fact that the level density has a non-compact support. The results are based on the method of orthogonal polynomials (the Stieltjes-Wigert polynomials here). An interesting random walk problem associated with the joint probability distribution of the ensuing ensemble is discussed and its connection with level dynamics is brought out. It is further proved that Dyson’s Coulomb gas analogy breaks down whenever the confining potential is given by a transcendental function for which there exist orthogonal polynomials.
• Scarcity of real discrete eigenvalues in non-analytic complex $\mathcal{PT}$-symmetric potentials
We find that a non-differentiability occurring whether in real or imaginary part of a complex $\mathcal{PT}$-symmetric potential causes a scarcity of the real discrete eigenvalues despite the real part alone possessing an infinite spectrum. We demonstrate this by perturbing the real potentials $x^{2}$ and $|x|$ by imaginary $\mathcal{PT}$ -symmetric potentials $ix|x|$ and $ix$, respectively.
• An update on coherent scattering from complex non-PT-symmetric Scarf II potential with new analytic forms
The versatile and exactly solvable Scarf II potential has been predicting, confirming and demonstrating interesting phenomena in complex PT-symmetric sector, most impressively. However, for the non-PT-symmetric sector, it has gone underutilised. Here, we present the most simple analytic forms for the scattering coefficients $(T (k), R(k), | det S(k)|)$. On the one hand, these forms demonstrate earlier effects and confirm the recent ones. On the other hand, they make new predictions – all simple and analytical. We show the possibilities of both self-dual and non-self-dual spectral singularities (NSDSS) in two non-PT sectors (potentials). The former one is not accompanied by time-reversed coherent perfect absorption (CPA) and gives rise to the parametrically controlled splitting of spectral singularity (SS) into a finite number of complex conjugate pairs of eigenvalues (CCPEs). NSDSS behave just oppositely: CPA but no splitting of SS. We demonstrate a one-sided reflectionlessness without invisibility. Most importantly, we bring out a surprising coexistence of both real discrete spectrum and a single SS in a fixed potential. Nevertheless, so far, the complex Scarf II potential is not known to be pseudo-Hermitian ($η ^{−1}Hη = H^{†})$ under a metric of the type $η(x)$.
• PT-symmetric potentials with imaginary asymptotic saturation
We point out that PT-symmetric potentials V$_{PT}$(x) having imaginary asymptotic saturation, V$_{PT}$(x = ±∞) = ±iV$_1$, V$_1$ ∈$\mathbb{R}$ are devoid of scattering states and spectral singularity. We show the existence of real (positive and negative) discrete spectrum both with and without complex conjugate pair(s) of eigenvalues (CCPEs). If the eigenstates are arranged in the ascending order of the real part of the discrete eigenvalues, the initial states have few nodes but latter ones oscillate fast. Both real and imaginary parts of ψn(x) vanish asymptotically, and|ψn(x)| are nodeless. For the CCPEs, these are asymmetric and peaking on the left (right) and for real energies these are symmetric and peaking at the origin. For CCPEs E$_{±}$, the eigenstates ψ± follow the interesting property, |ψ+(x)| = N|ψ−(−x)|, N ∈ $\mathbb{R$^+\$}
• # Pramana – Journal of Physics
Volume 97, 2023
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019
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2023-01-31 14:31:12
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|
https://www.math.uci.edu/taxonomy/term/100
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# CMC surfaces in Minkowski space
Peter Smillie
Caltech
## Time:
Tuesday, January 14, 2020 - 4:00pm
## Location:
RH 306
In joint work with F. Bonsante and A. Seppi, we solve a
Dirichlet-type problem for entire constant mean curvature hypersurfaces in
Minkowski n+1-space, proving that such surfaces are essentially in bijection
with lower semicontinuous functions on the n-1-sphere. This builds off of
existence theorems by Treibergs and Choi-Treibergs, which themselves rely on
the foundational work of Cheng and Yau. I'll present their maximum principle
argument as well the extra tool that leads to our complete existence and
uniqueness theorem. Time permitting, I'll compare with the analogous problem
of constant Gaussian curvature and present a new result on their intrinsic
geometry.
Joint seminar with the Differential Geometry Seminar series.
# Classical and quantum traces coming from SL_n(C) and U_q(sl_n)
Daniel Douglas
USC
## Time:
Monday, January 27, 2020 - 4:00pm
## Location:
RH 340P
We discuss work-in-progress constructing a quantum trace map for
the special linear group SL_n. This is a kind of Reshetikhin-Turaev
invariant for knots in thickened punctured surfaces, coming from an
interaction between higher Teichmüller theory and quantum groups.
Let S be a punctured surface of finite genus. The SL_2-skein algebra of S
is a non-commutative algebra whose elements are represented by framed links
K in the thickened surface S x [0,1] subject to certain relations. The
skein algebra is a quantization of the SL_2(C)-character variety of S, where
the deformation depends on a complex parameter q. Bonahon and Wong
constructed an injective algebra map, called the quantum trace, from the
skein algebra of S into a simpler non-commutative algebra which can be
thought of as a quantum Teichmüller space of S. This map associates to a
link K in S x [0,1] a Laurent q-polynomial in non-commuting variables X_i,
which in the specialization q=1 recovers the classical trace polynomial
expressing the trace of monodromies of hyperbolic structures on S when
written in Thurston's shear-bend coordinates for Teichmüller space. In the
early 2000s, Fock and Goncharov, among others, developed a higher
Teichmüller theory, which should lead to a SL_n-version of this invariant.
# TBA
Bahar Acu
## Institution:
Northwestern University
## Time:
Monday, May 4, 2020 - 4:00pm
RH 340P
# TBA
Nur Saglam
Virginia Tech
## Time:
Monday, March 9, 2020 - 4:00pm
RH 340P
# A twist on A-infinity algebras and its application on symplectic manifolds
Jiawei Zhou
## Institution:
Tsinghua University
## Time:
Monday, January 13, 2020 - 4:00pm
## Location:
RH 340P
We will first review an algebra of special differential forms on sympectic manifolds, constructed by Tsai, Tseng and Yau. Then we introduce a twist on this algebra, which leads to a flatness condition. This twist is motivated by considering the connections on fiber bundles, and we can generalize it to A-infinity algebras, together with a generalized flatness condition.
# Witten deformation on noncompact manifolds
Xianzhe Dai
UC Santa Barbara
## Time:
Monday, October 28, 2019 - 4:00pm
## Location:
RH 340P
Motivated by considerations from the mirror symmetry and
Landau-Ginzburg model, we consider Witten deformation on noncompact
manifolds.
Witten deformation is a deformation of the de Rham complex introduced by
Witten in an influential paper and has had many important applications,
mostly on compact manifolds. We will discuss some recent work with my
student Junrong Yan on the spectral theory of Witten Laplacian, the
cohomology of the deformation as well as its index theory.
A joint seminar with the Differential Geometry Seminar series.
# Comparing gauge theoretic invariants of homology S1 cross S3
Jianfeng Lin
UC San Diego
## Time:
Tuesday, October 1, 2019 - 4:00pm
## Location:
RH 306
While classical gauge theoretic invariants for 4-manifolds are usually
defined in the setting that the intersection form has nontrivial positive
part, there are several invariants for a 4-manifold X with the homology S1
cross S3. The first one is the Casson-Seiberg-Witten invariant LSW(X)
defined by Mrowka-Ruberman-Saveliev; the second one is the Fruta-Ohta
invariant LFO(X). It is conjectured that these two invariants are equal to
each other (This is an analogue of Witten’s conjecture relating Donaldson and
Seiberg-Witten invariants.)
In this talk, I will recall the definition of these two invariants, give
some applications of them (including a new obstruction for metric with
positive scalar curvature), and sketch a prove of this conjecture for
finite-order mapping tori. This is based on a joint work with Danny Ruberman
and Nikolai Saveliev.
A joint seminar with the Differential Geometry Seminar series.
# Skein algebras, elliptic curves, and Fukaya categories
Peter Samuelson
UC Riverside
## Time:
Monday, October 21, 2019 - 4:00pm
## Location:
RH 340P
A skein relation'' can be viewed as a linear relation satisfied by the
R-matrix for a quantum group; one of the first uses of skein relations was
to give a combinatorial construction of Reshetikhin-Turaev invariants of
knots in S^3. The Hall algebra of an abelian (or triangulated) category
"counts extensions" in the category. We briefly describe how skein relations
appear in the Hall algebra of coherent sheaves of an elliptic curve, the
Hall algebra of the Fukaya category of a surface, and factorization homology
of a surface. No familiarity with the objects mentioned above will be
assumed for the talk.
# The cohomology rings of regular Hessenberg varieties
Peter Crooks
## Institution:
Northeastern University
## Time:
Monday, November 18, 2019 - 4:00pm
## Location:
RH 340P
Hessenberg varieties form a distinguished class of subvarieties in
the flag variety, and their study is central to themes at the interface of
combinatorics and geometric representation theory. Such themes include the
Stanley-Stembridge and Shareshian-Wachs conjectures, in which the cohomology
rings of Hessenberg varieties feature prominently.
I will provide a Lie-theoretic description of the cohomology rings of
regular Hessenberg varieties, emphasizing the role played by a certain
monodromy action and Deligne's local invariant cycle theorem. Our results
build on upon those of Brosnan-Chow, Abe-Harada-Horiguchi-Masuda, and
Abe-Horiguchi-Masuda-Murai-Sato. This represents joint work with Ana
Balibanu.
# How to count constant maps?
Si Li
## Institution:
Tsinghua University
## Time:
Monday, November 25, 2019 - 4:00pm
## Location:
RH 340P
The art of using quantum field theory to derive mathematical
results often lies in a mysterious transition between infinite dimensional
geometry and finite dimensional geometry. In this talk we describe a general
mathematical framework to study the quantum geometry of sigma-models when
they are effectively localized to small fluctuations around constant maps.
We illustrate how to turn the physics idea of exact semi-classical
approximation into a geometric set-up in this framework, using Gauss-Manin
connection. This leads to a theory of “counting constant maps” in a
nontrivial way. We explain this program by a concrete example of
topological quantum mechanics and show how “counting constant loops” leads
to a simple proof of the algebraic index theorem.
|
2020-01-25 06:43:37
|
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|
https://www.springerprofessional.de/en/development-of-tire-wear-particle-emission-measurements-for-pass/18772782?fulltextView=true
|
main-content
## Swipe to navigate through the articles of this issue
15-01-2021 | Issue 1/2021 Open Access
# Development of Tire-Wear Particle Emission Measurements for Passenger Vehicles
Journal:
Emission Control Science and Technology > Issue 1/2021
Authors:
Yoshio Tonegawa, Sousuke Sasaki
Important notes
## Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## 1 Introduction
Particulate matter (PM) of various sizes is suspended in the atmosphere. The environmental standard for PM 2.5, which is defined as PM with a diameter of less than 2.5 μm, is set in each country to ensure atmospheric protection and mitigate harmful effects on health. As PM is generated by various processes and is released into the atmosphere, it is necessary to investigate the origin of the emissions to improve the air quality. The PM emitted from motor vehicles includes not only exhaust particles resulting from engine combustion but also non-exhaust particles such as tire and brake wear particles from vehicle use [ 1]. Recently, PM emissions from vehicle exhaust have decreased due to the strengthening of exhaust gas regulations and the development of new engine and after treatment technologies [ 2]. This has meant that, in terms of all particulate emissions from vehicles, the contribution of PM emissions from non-exhaust sources has become comparatively larger.
Test procedures for exhaust emissions from motor vehicles are widely applied worldwide for type approval. In these procedures, the emission factor of exhaust PM from one vehicle can be experimentally evaluated. However, evaluating the emission factor of non-exhaust particles from a vehicle is challenging because no test method has been determined.
A method for measuring non-exhaust particles for the evaluation of emissions is under consideration at the Particle Measurement Programme Informal Working Group (PMP-IWG) of the United Nations Working Party for Pollution and Energy (GRPE) [ 3]. Methods for measuring brake wear particles have been summarized in prior literature [ 4, 5]. It has been suggested that a common test procedure for measuring brake wear particles will be developed around the mid-end of 2019 [ 6]. However, the PMP-IWG has stated that it will “continue monitoring on-going projects and published data regarding the physical nature and size distribution of particle emissions from tire/road wear” for tire-wear particles [ 6].
The level of tire emissions present in the environment was investigated in the Tire Industry Project, a survey that focused on water bodies in Europe, Asia, and North America [ 7]. An average of 1000 ppm tire emissions were found to be present in sediments in water bodies, while 0.08 μg/m 3 in PM 10 was found in the atmosphere and 1900 ppm was found in soil. According to another report, the concentration of PM 2.5 related to tire wear in the atmosphere was 0.03 μg/m 3 on average, and the average contribution of total PM 2.5 was 0.27% [ 8].
It is necessary to develop an emission inventory of tire-wear particles to understand their impact on the environment and consider future measures. Therefore, a method for measuring tire-wear particles is required to accurately evaluate vehicle emissions. To study tire-wear particles in an indoor test, a road simulator consisting of four test tires and a circular pavement [ 9], and an inside drum test stand consisting of one test tire and a pavement on the inner drum [ 10, 11], were utilized. In addition, research into non-tail pipe emissions on actual roads has been conducted using mobile sampling vehicles [ 12, 13]. These studies mainly measured the particle size distribution and components of the particles generated during vehicle use.
To accurately evaluate the level of tire-wear particles generated in a laboratory or on a road, various factors involved in particle generation and evaluation must be considered. Tire-wear particles are generated by the interaction between the tire and road surface, and the quantity generated varies according to the conditions of the road surface, such as the type of road and the amount of road dust present on the road surface. Particles generated by vehicle use can be collected through a filter method, which collects both re-suspended particles from the road surface as well as tire-wear particles. When seeking to analyze only the particles generated from tires, evaluating the level of tire-wear particles collected using a weight filter method is undesirable. Quantifying the specific components derived from tires on a collection filter by chemical analysis enables only tire-wear particles to be measured, excluding any particles from other origins.
In this study, we aimed to develop a new method for directly measuring tire-wear particles of less than 2.5 μm released into the atmosphere by vehicles and establish a new method for evaluating the emission factor. We investigated the morphological observation of tire-wear particles obtained in both indoor and on-road tests, the effect of the vehicle speed and lateral acceleration on tire-wear particle generation, the contribution of tread mass to PM 2.5, and the estimation of the emission factor of tire-wear particles in an actual driving pattern.
## 2 Experimental
### 2.1 Test Tires
Bridgestone 175/65R14 82S passenger radial tires were used in the investigation. The test tires were inflated with compressed air at 230 kPa. The load carrying capacity of these tires was 490 kg. The vertical load of the test tire was 3.3 kN and the target lateral forces were 0, 0.3, 0.6, and 1.2 kN. The tire lateral acceleration was calculated by subtracting the lateral force with the vertical load and expressed as gravity acceleration G.
### 2.2 Quantification of Tire-Wear Particles on a Filter Sample
Tire-wear particles generated by the friction between tires and a road are usually collected within a mixture of tire-wear particles, re-suspended particles, road wear particles, among others. Separation of the tire particles from the mixture is necessary to evaluate the emission of tire-wear particles from a vehicle. Typical tire treads are made from materials such as rubber/elastomer, carbon black, mixtures such as silica, and chemicals [ 14]. Although the blending ratios are different for every product, since rubber/elastomer is the most abundant component, the tire wear concentration in a mixture can be calculated by quantifying these components. However, as the amount of these components within a tire differs from one tire to another, it is necessary to analyze the ratio of the rubber/elastomer component for each target tire.
In this study, we adopted a method of pyrolyzing the collected sample and quantifying the rubber component specific to the tire. In the tire analysis, Varian CP-3800 GC/1200 MS was used as an analyzer of the indicator substance obtained by performing thermal decomposition of the rubber components using T-DEX II manufactured by GL Science. Styrene produced by thermally decomposing the collected sample at 600 °C was used as an indicator substance and the relationship between the concentration of styrene as an index substance and the GC signal was evaluated [ 15]. The percentage of the indicator substance obtained, which was produced by pyrolysis of the tire tread, was also analyzed. The concentration of tire-wear particles contained in the collection filter was analyzed by the method discussed above.
### 2.3 Evaluation Method of the Tire-Wear Mass by the Weighting of a Tire/Wheel Assembly
The tire weight was measured to estimate the contribution ratio of tire-wear particles to PM 2.5 to the tire wear mass. The weight of the tire and wheel assembly before and after the road test was measured in an air-conditioned room, and the tire-wear mass was calculated from the difference in the measured weight. The level of tire-wear mass in the test was considered to be low because the tire mileage for each test condition was as short as 17 km or less. In addition, we decided to measure the weight of each tire/wheel assembly because measurement errors would have arisen if the tire was scraped off due to removal from the wheel. Therefore, we used a balance that could measure the weight of a tire assembly of 13 kg and with changes of 0.1 g or less.
The tire weights were measured using a comparator balance KA50–2 manufactured by Metler Toledo. This balance had a maximum load of 52 kg, a readability of 0.01 g, and repeatability of 0.02–0.03 g. Before weighing the tires, the dust on the tire surface was blown away using compressed air. The valve core was removed from the stem, and the tire was soaked overnight (over 8 h) under air-conditioning to reduce the influence of pre-weighing conditions, such as the air pressure the inside tire and adsorption of moisture by the temperature.
### 2.4 Experimental Setup of Indoor Test for Tire-Wear Particle Morphology
To confirm the occurrence of tire-wear particles in an indoor test and on-road test, tire-wear particles were generated using an outer drum test stand attached to a pavement mimicking a real road on the surface of the outer drum. The mock-asphalt pavement was made by uniformly applying a mixture of stone, sand, and gravel with epoxy resin on the drum surface. Figure 1 shows a schematic of the outer drum test stand. The tire is driven around its horizontal axis at the highest point of the outer drum. This test apparatus can control the drum speed, wheel load, slip angle, and camber angle of the test tire. It is also possible to obtain the force exerted on the test tire during the running test by means of a three-component force sensor attached to the device. The tire-wear particles generated by friction between the test tire and drum surface were introduced into a constant volume sampler via a suction nozzle with a width of 220 mm and a height of 17.5 mm set behind the test tire. A mixing chamber with an inner diameter of 97.6 mm and a length of 880 mm was installed after the suction nozzle, and indoor air containing tire-wear particles was sucked at a flow rate of 5 m 3/min. Part of the mixed sample was isokinetically aspirated through a sampling tube with an inner diameter of 6 mm by a pump controlled at 20 L/min and collected in a sampling filter. The collection efficiency of the suction nozzle in this flow rate range was 0.98 to 1.0 in the experimental evaluation using gas.
### 2.5 Method for Measuring Tire-Wear Particles Using a Road Tire Test Vehicle
A wheel load was applied to the tire due to the weight of the vehicle, and the wheel force by the braking, driving, and turning behavior of the vehicle was transmitted to the tire. As a result, friction occurred between the tire and the road surface, and tire-wear particles were generated. Therefore, a test system that could control the tire behavior such as wheel load and force was required to stably generate tire-wear particles. Figure 2 shows a road tire test vehicle for measuring the tire-wear particles emitted from a test tire. This test vehicle could change the factors (load, slip angle) that influenced the test tire’s generation of tire-wear particles. Therefore, we installed a device for collecting tire-wear particles from this road tire test vehicle to evaluate the generated tire-wear particles.
The test tire and suction nozzle were attached to the road tire test car, and the generated tire abrasion dust was introduced from behind the test tire into the sampling device. The tire-wear particles were introduced into the mixing tunnel of the constant volume sampler at 5 m 3/min, and part of the mixed sample air was then isokinetically vacuumed by the pump via the filter folder. A multi cascade sampler was used as a PM sampler to divide the particle sizes into PM 2.5 and PM 10. A quartz filter of 47 mmφ and 47 mmφ OD × 20 mmφ ID were used as a collection filter for tire-wear particles. The test was conducted assuming that there was no particle loss in the sampling system.
## 3 Results and Discussion
### 3.1 Morphological Observation of the Tire-Wear Particles Obtained in Laboratory Tests and Road Tests
The tire-wear particles were measured in laboratory tests. The number of tire-wear particles collected on the filter in the test showed little difference from the number collected in the laboratory room air. In addition, when the test was completed, tire dust, similar to dust from pencil erasers, adhered to the surface of the test tire, and tire dust of several mm in diameter was scattered around the test equipment. We thought that a phenomenon different than normal wear may have occurred in our laboratory tests. Therefore, the particles obtained in the laboratory test and road test were observed with an electron microscope to confirm the difference in particle morphology.
Morphological observation of the tire-wear particles on the collected sample was conducted using a scanning electron microscope. Figure 3 shows an electron micrograph of the tire-wear particles collected on filters in the laboratory test and road test. As shown in the figure, the tire-wear particles obtained in the laboratory test were elliptical spherical particles, and the particle diameter was about 400 μm on the major axis. Contrastingly, the tire-wear particles obtained in the road test were spherical, and their diameter was approximately 10 μm. From the observation results of the generated particles, the morphology of the generated particles was different between the laboratory test and the road test. For these reasons, we concluded that it was better to evaluate the number of tire-wear particles in the road test than in the laboratory test.
### 3.2 The Effect of Vehicle Speed and Lateral Acceleration on Tire-Wear Particle Generation
An evaluation of the generation of tire-wear particles was conducted on the test course. To confirm the influence of the vehicle speed change, particle collection tests were conducted once each for vehicle speeds of 20, 30, and 40 km/h under a lateral acceleration of 0.2G. In addition, to confirm the influence of the acceleration change, tests were performed once for each of the conditions with a lateral acceleration of 0, 0.1, and 0.2G under a vehicle speed of 40 km/h. As shown in Fig. 4, the emission of tire-wear particles was not affected by the change in speed. However, it was confirmed that emission of the tire-wear particles increased as the lateral acceleration increased. Since emission of the wear particles changed in accordance with the change in the lateral acceleration, the emission of tire-wear particles could be estimated by the acceleration applied to the tire.
We investigated the correlation between the lateral acceleration and the amount of tire-wear particles generated. Figure 5 shows the emission of tire-wear particle less than 2.5 μm at each lateral acceleration obtained in road tests. The curve fit method with the quadratic polynomial regression was used to plot the graphs for the relationships between the tire-wear particle emissions and lateral acceleration. The coefficient of correlation between the amount of tire-wear particles and lateral acceleration was 0.6491. From this, there was a slight correlation between these relationships, though we predict that the correlation would be higher if the variation in the experiment was reduced.
### 3.3 Estimation of the Emission Factor for Tire-Wear Particles
We calculated the emission factor from the polynomial regression of the tire-wear particle emission and lateral force. The tire surface deformed when force was applied to the tire in both lateral and longitudinal acceleration. As a result, tire wear occurred due to slippage with the road surface. Depending on the tread pattern of the tire, the degree of tire wear could vary depending on the direction of acceleration applied to the tire. We estimated the emission factor of tire-wear particles, assuming that the number of tire-wear particles generated by change in acceleration was the same in both vertical and horizontal directions.
When the vehicle speed changed, the acceleration also changed. The tire-wear particle concentration (mg/km) calculated by the regression equation was generated instantaneously according to the change in acceleration. The total number of tire-wear particles in this concentration was multiplied by the instantaneous vehicle travel distance (km). Therefore, the result of multiplying the tire-wear particle emission based on the acceleration and the integral value of the speed could be regarded as the total emission. The following formula was used to calculate the emission factor to apply the experimental results to actual driving patterns.
$$\mathrm{EF}=\int \frac{v(t)\cdotp f\left({a}_i\right)}{d} dt$$
where EF is the emission factor of tire-wear particles (mg/km), v( t) is the vehicle speed (km/h) at time t, f( a i) is the tire-wear particle emission at each acceleration (mg/km) and d is the distance vehicle driven during JC08 test cycle (km).
Using the above formula, we estimated that the emission of tire-wear particles was less than 2.5 μm in the driving cycle. The JC08 test cycle used in Japan’s exhaust gas regulations was selected as the running pattern. The vehicle acceleration calculated from the time-speed data of JC08 was applied to the regression equation to calculate the instantaneous emission of tire-wear particles. Figure 6 shows the instantaneous and cumulative emission of tire-wear particles per one wheel determined by the calculation. The instantaneous emission of tire-wear particles was high under high-speed acceleration and deceleration conditions, as well as when starting and braking the vehicle. As shown in the figure, the cumulative particle emission per wheel was 7.62 mg, and the distance in this mode was 8.17 km. Therefore, the tire emission factor of one tire in this running pattern was 0.932 mg/km-tire. We assumed that the four tires of a passenger car would be worn in the same way and estimated that the emission factor of tire-wear particles less than 2.5 μm was 3.73 mg/km-vehicle.
### 3.4 Contribution of Tire-Wear Particles Less than 2.5 μm to Tread Wear Mass
We investigated how much of the worn tire tread was discharged as particles below 10 and 2.5 μm by measuring the amount of tire-wear particles and the tire tread wear mass. Figure 7 shows the relationship between lateral acceleration and tire tread wear. As is shown, the degree of tread wear increased in a quadratic function, similar to the generation of tire-wear particles described above.
We evaluated the generation ratio of tire-wear particles less than 10 and 2.5 μm from the measurement results of tire-wear particles and tread wear. Figure 8 shows the ratio of particles less than 10 and 2.5 μm to the tread wear under each lateral acceleration condition. As is shown, the generation ratio of the tire-wear particles tended to decrease as the lateral acceleration increased. Even under the highest generation ratio, the ratio of the tire-wear particles to tread wear was about 3.3% at PM 2.5 and 3.7% at PM 10. Based on these results, we concluded that the proportion of tire-wear particles with a size suspended in the air was small, and that most of the tire tread wear dust remained on the road surface.
## 4 Summary
A method for measuring tire-wear particles for passenger car tires was studied using a road tire test vehicle.
Tire-wear particles generated in a laboratory test and a road test were collected, and the two samples were observed under an electron microscope. The tire-wear particles obtained from the laboratory test were dramatically different in size and shape compared with those obtained in the road test. Based on this, we concluded that it would be better to evaluate the level of tire-wear particles generated in the road test than in the laboratory test.
A method for generating and collecting tire-wear particles using an on-road test car was developed. The number of tire-wear particles generated was not affected by speed change but was greatly affected by a change in the lateral acceleration.
The relationship between the number of tire-wear particles generated and the lateral acceleration could be expressed with quadratic polynomial regression.
Fitting the regression equation to the speed pattern of the JC08 mode showed that the emission factor of the wear particles of the tire used in the test was 3.7 mg/km-vehicle.
The generation ratio of airborne size particles (PM 10, PM 2.5) from worn tire dust was around 3–4% or less. Therefore, it was predicted that most of the tire dust remained on the road surface as coarse particles and a small amount of the tire dust floated in the atmosphere.
## 5 Future Work
In this study, a measurement test of tire-wear particles based on the change of lateral acceleration force of a test tire was carried out to collect a large amount of TRWP. The emission factors of tire-wear particles were estimated from the relational expression obtained in the investigation. It is possible that the emission factors differed from the actual results of discharge in the vehicle, since the estimation was performed by applying the lateral acceleration to the longitudinal acceleration. A test method based on an actual vehicle was also constructed, and a tire-wear particle measurement test is now in progress.
## Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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2021-04-14 11:37:11
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$${ {\sqrt a+\sqrt b=\sqrt{2009}} }$$
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$$\sqrt a + \sqrt b = \sqrt{2009}$$
## An interesting number theory proof involving concepts of prime numbers and integers.
https://math.stackexchange.com/questions/3871207/an-interesting-number-theory-proof-involving-concepts-of-prime-numbers-and-integers.
...I was trying a British Math Olympiad problem, $$\sqrt a + \sqrt b = \sqrt{2009}$$ , find all integers a and b. After solving this problem, https://youtu.be/quECgYPNCXw in a really similar fashion to this solution, i thought to try and come up with a generalisation, if $$\sqrt a + \sqrt b = \sqrt c$$ , where $$\sqrt c = m \sqrt n$$ , where $$m$$ and $$n$$ are integers and...
$$\sqrt{x}+\sqrt{y}=\sqrt{2205}$$
## How to solve the equation of $\sqrt{x}+\sqrt{y}=\sqrt{2205}$ in integers?
https://math.stackexchange.com/questions/1575392/how-to-solve-the-equation-of-%24%5Csqrt%7Bx%7D%2B%5Csqrt%7By%7D%3D%5Csqrt%7B2205%7D%24-in-integers%3F
...How to solve the equation of $$\sqrt{x}+\sqrt{y}=\sqrt{2205}$$ in integers? How in general to solve the similar equatio...
$$\sqrt{x}+\sqrt{y}=\sqrt{2013}$$
## Equation $\sqrt{x}+\sqrt{y}=\sqrt{2013}$ in rationals
https://math.stackexchange.com/questions/564422/equation-%24%5Csqrt%7Bx%7D%2B%5Csqrt%7By%7D%3D%5Csqrt%7B2013%7D%24-in-rationals
...Can we find all rational numbers $$x,y$$ such that $$\sqrt{x}+\sqrt{y}=\sqrt{2013}$$ ? Certainly possible answers are $$(2013,0)$$ and $$(0,2013)$$ . If we square the equation, we get $$x+y+2\sqrt{xy}=2013$$ , so $$\sqrt{xy}$$ must be ration...
$$\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{2013}$$
## Equation $\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{2013}$ in rationals
https://math.stackexchange.com/questions/566141/equation-%24%5Csqrt%7Bx%7D%2B%5Csqrt%7By%7D%2B%5Csqrt%7Bz%7D%3D%5Csqrt%7B2013%7D%24-in-rationals
...Consider the equation $$\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{2013}$$ , where $$x,y,z$$ are rational numbers. Are there any solutions other than the trivial ones $$(2013,0,0),(0,2013,0),(0,0,2013)$$ ? We can subtract $$\sqrt{z}$$ from both sides and square to get...
$$\sqrt{x}+\sqrt{y}=\sqrt{1376}$$
## Find all natural roots of $\sqrt{x}+\sqrt{y}=\sqrt{1376}$ given that $x\leq y$
https://math.stackexchange.com/questions/1830986/find-all-natural-roots-of-%24%5Csqrt%7Bx%7D%2B%5Csqrt%7By%7D%3D%5Csqrt%7B1376%7D%24-given-that-%24x%5Cleq-y%24
...Find all natural roots of $$\sqrt{x}+\sqrt{y}=\sqrt{1376}$$ given that $$x\leq y$$ I'm confused of this equation because $$1376$$ is not a square!! So maybe it has no natural root! Am I righ...
$$7\sqrt{a} + 17\sqrt{b} + k\sqrt{c} \ge \sqrt{2019}$$
## Find the least positive real number $k$ such that $7\sqrt{a} + 17\sqrt{b} + k\sqrt{c} \ge \sqrt{2019}$ over all positive real numbers
https://math.stackexchange.com/questions/3068290/find-the-least-positive-real-number-%24k%24-such-that-%247%5Csqrt%7Ba%7D-%2B-17%5Csqrt%7Bb%7D-%2B-k%5Csqrt%7Bc%7D-%5Cge-%5Csqrt%7B2019%7D%24-over-all-positive-real-numbers
...Working on a problem... Find the least positive real number $$k$$ such that $$7\sqrt{a} + 17\sqrt{b} + k\sqrt{c} \ge \sqrt{2019}$$ over all positive real numbers $$a,b,c$$ with $$a+b+c=1$$ . Maximizing the "$$a$$ " term doesn't seem to work, and expansion through moving a radical to the right side of the equation simply leads to more radicals. Any he...
$$\sqrt a + \sqrt b = \sqrt n$$
## Is there a way to solve $\sqrt a + \sqrt b = \sqrt n$ analytically?
...A basic fact that comes out of Galois theory is that the set $$\{\sqrt{d_i}\}_{i \in I}$$ is linearly independent over $$\mathbb Q$$ , where $$\{d_i\}_{i \in I}$$ enumerates all square-free integers (including negative integers!). Since $$\sqrt a + \sqrt b = \sqrt n$$ expresses a linear dependence among three different square roots, it actually follows that the square free parts of $$a, b, n$$ must, in fact, all be the same. So all you need to do is factor the largest square out of $$n$$ to get $$n = k^2 m$$ with $$m$$ square-free - in the case of...
$$\sqrt a\geq\sqrt{2001}$$
## Writing square root of square-free numbers as sum of square roots.
https://math.stackexchange.com/questions/737886/writing-square-root-of-square-free-numbers-as-sum-of-square-roots.
... $$\sqrt a\geq\sqrt{2001}$$ and $$\sqrt{b}\leq 0$$ which means $$b=0$$ . With exact method we can prove that $$\sqrt{s}=\sqrt a+\sqrt b$$ does not have any natural solutions with $$s$$ being a square-free number. Then I tried to generalize the proof for $$3$$ or more square roots but i failed. The only thing I always get is...
$$b=\sqrt[2019]{3}$$
## degree of a sum of two algebraic numbers
https://math.stackexchange.com/questions/3088909/degree-of-a-sum-of-two-algebraic-numbers
... $$b=\sqrt[2019]{3}$$ . How to prove that the degree of $$a+b$$ is: greater than 2019? (if it can be done with less advanced methods then the second part) equal $$2019^2$$ ? (if this value is correct)...
$$\sqrt{a_1+b_1\sqrt{a_2+b_2\sqrt{a_3+b_3\cdots}}}$$
## Evaluating an infinite square root
https://math.stackexchange.com/questions/878363/evaluating-an-infinite-square-root
...How do I evaluate the square root: $$\sqrt{2013+276\sqrt{2027+278\sqrt{2041+280\cdots}}}$$ I have tried creating two arithmetic sequences such that $$a_n = 1999+14n$$ $$b_n = 274+2n$$ so the square root simplifies to $$\sqrt{a_1+b_1\sqrt{a_2+b_2\sqrt{a_3+b_3\cdots}}}$$ But I get stuck there. Any help/hints is greatly appreciat...
$$\sqrt{a+\sqrt{b}}$$
## question related to radical sign
...Compute $$p(2012)=\sqrt{2014+3 \sqrt{4019}}-\sqrt{2010+\sqrt{4019}}$$ and apply the formula for double square roots, noticing that $$2010^2-4019 = 2009^2$$ and $$2014^2-9 \cdot 4019 = 2005^2$$ . Now you can compute the desired result. By the way, the final result is $$8$$ . Sorry for the italian reference, but you can easily read the formula to simplify an expression like $$\sqrt{a+\sqrt{b}}$$ when $$a^2-b$$ is a perfect squa...
$$t=\sqrt{abcd}$$
...I couldn't understand why $$6\sqrt{abcd}\geq54$$ Note that $$abcd-27\ge 6\sqrt[6]{a^3b^3c^3d^3}=6\sqrt{abcd}.$$ Setting $$t=\sqrt{abcd}$$ gives you $$t^2-27\ge 6t\iff (t-9)(t+3)\ge 0\iff t\ge 9.$$...
$$f(\sqrt[3]{2019})=0$$
## Prove that $-\sqrt{c} https://math.stackexchange.com/questions/3783407/prove-that-%24-%5Csqrt%7Bc%7D%3Cab%3C0%24-if-%24a%5E4-2019a%3Db%5E4-2019b%3Dc%24.#comment-7791441 [email protected] von Eitzen $$f$$ is a convex function, $$f(0)=0$$ , $$f(\sqrt[3]{2019})=0$$ , $$f$$ decreases on $$(-\infty,0]$$ and $$f$$ increases on $$[\sqrt[3]{2019},+\infty),$$ which says the equation $$f(x)=c$$ , where $$c >0$$ has two real roots exactly. $$a$$ and $$b$$ are roots. Thus, $$ab< 0$$ because one of them is negative and other is greater than $$\sqrt[3]{2019}.$$ Is it clear n... $$\sqrt{2500-a^2}$$ ## How to count this in a faster way? https://math.stackexchange.com/questions/514195/how-to-count-this-in-a-faster-way%3F#answer-514206 ...For any particular $$a$$ , $$b$$ can go up to the largest integer below $$\sqrt{2500-a^2}$$ . So for any particular $$a$$ we have the count equal to $$\lfloor\sqrt{(2500-a^2)}\rfloor$$ . The half brackets mean the floor function. Then you would sum this up from $$a=0$$ up to $$a=49$$ . (You could also sum to $$a=50$$ , but the floor of 0 is 0.) Plugging into wolframalpha I get... $$2018 = 2 \cdot 1009$$ ## Prove that there are no integer solutions to$ c^2 = 2018^a + 2018^b $https://math.stackexchange.com/questions/3561182/prove-that-there-are-no-integer-solutions-to-%24-c%5E2-%3D-2018%5Ea-%2B-2018%5Eb-%24 ...A friend of mine gave me this puzzle and I want to solve it, turns out its harder than I expected. I tried to prove this by contradiction, so let's just assume there is an integer solution. The first thing I noticed was this: $$2018 = 2 \cdot 1009$$ , 2 and 1009 are both primes, also c must be an even number. My first idea was to rewrite a little: $$c^2 = 2018^a + 2018^b \implies 2018^a = c^2 - 2018^b = (c+\sqrt{2018}^b)(c-\sqrt{2018}^b)$$ , but as it turns out, $$(c+\sqrt{2018}^b)$$ and $$(c-\sqrt{2018}^b)$$ don't necessarily have to be irrational, for example... $$\sqrt(1632397825)$$ ## Square Root Algorithm https://math.stackexchange.com/questions/439135/square-root-algorithm#answer-439181 ..."efficient" rather depends on what you constraints are. For instance, if you have enough memory to store some floats, but little cpu time, then you can store a look up table for all integers until some power of 10. So say you had to find $$\sqrt(1632397825)$$ . You could write: $$\sqrt{1732397825} =\sqrt{ 17*10^8 + 32397825}\sim \sqrt{17}\cdot 10^4 + \epsilon$$ To calculate $$\epsilon$$ , use the fact that: $$\sqrt{a^2+b}\sim a + \frac{b}{2a} - \frac{b^2}{8a^3} + ...$$ So in our example, $$\sqrt{1732397825} \sim 41622.06575$$ Quite close to the true value of: $$\sqrt{1732397825}=41622.08338$$... $$a+1 = x, b+1 =y, c+1 = z$$ ## issues with simple algebraic equations https://math.stackexchange.com/questions/695496/issues-with-simple-algebraic-equations#answer-695517 ...\\ bc+b+c=300 \\ ac+a+c=216 \\ \end{cases} it seems of no difficulty: a system of 3 equations and 3 variables... Let's start by adding 1 to every equations, we obtain: \begin{cases} ab+a+b+1=(a+1)(b+1)=251 \\ bc+b+c+1=(c+1)(b+1)=301 \\ ac+a+c+1=(a+1)(c+1)=217 \\ \end{cases} Now let's substitute $$a+1 = x, b+1 =y, c+1 = z$$ , so we can reduce the amount of calculus needed, in fact to solve \begin{cases} xy=251 \\ yz=301 \\ xz=217 \\ \end{cases} you only need to find $$x$$ (or... $$b=\frac{49}{625}+\frac{\sqrt{(18 808 849)}}{(4375)}$$ ## Linear estimation of an exponential distribution https://math.stackexchange.com/questions/1864847/linear-estimation-of-an-exponential-distribution#comment-3819762 ...It gives me 2 pairs of coordinates with the same error of 1. $$a=\frac{-4}{625}-\frac{\sqrt{(18 808 849)}}{(26 250)}$$ , $$b=\frac{49}{625}+\frac{\sqrt{(18 808 849)}}{(4375)}$$ and $$a=\frac{-4}{625}+\frac{\sqrt{(18 808 849)}}{(26 250)}$$ , $$b=\frac{49}{625}-\frac{\sqrt{(18 808 849)}}{(4375)}$$... $$a_r=\prod_{v=2}^{r} \sqrt[v(v-1)]{v}$$ ## Infinite product experimental mathematics question. https://mathoverflow.net/questions/22088/infinite-product-experimental-mathematics-question.#answer-22093 ...An experimental observation: if $$a_r=\prod_{v=2}^{r} \sqrt[v(v-1)]{v}$$ and $$b_r=\prod_{n=1}^{r} \sqrt[n]{1+\frac{1}{n}}$$ , then the numbers $$a_{2r}/b_{2r}$$ are, according to Mathematica, $$\frac{1}{\sqrt{3}},\frac{1}{\sqrt[4]{5}},\frac{1}{\sqrt[6]{7}},\frac{1}{\sqrt[4]{3}},\frac{1}{\sqrt[10]{11}},\frac{1}{\sqrt[12]{13}},\frac{1}{\sqrt[14]{15}},\frac{1}{\sqrt[16]{17}},\frac{1}{\sqrt[18]{19}},\frac{1}{\sqrt[20]{21}},$$ $$\frac{1}{\sqrt[22]{23}},\frac{1}{\sqrt[12]{5}},\frac{1}{3^{3/26}},\frac{1}{\sqrt[28]{29}},\frac{1}{\sqrt[30]{31}},\frac{1}{\sqrt[32]{33}},\frac{1}{\sqrt[34]{35}},\frac{1}{\sqrt[36]{37}},\frac{1}{\sqrt[38]{39}},\frac{1}{\sqrt[40]{41}},\dots$$ I would imagine the products are the same, then. I don't have time but using this as a hint one should be able to give an actual proof. May you tell us how you ended up with such an identi... $$\Re=\frac {24+\sqrt[3]{\frac{12096+\sqrt{134369280}}2}+\frac{144}{\sqrt[3]{\frac{12096+\sqrt{134369280}}2}}}{72}$$ ## Find$x^n+y^n+z^n\$ general solution
... $$\Re=\frac {24+\sqrt[3]{\frac{12096+\sqrt{134369280}}2}+\frac{144}{\sqrt[3]{\frac{12096+\sqrt{134369280}}2}}}{72}$$ so now we can find $$\Im$$ and $$x$$ (we know $$2\Im^2=6\Re^2-4\Re-1$$ and $$x=1-2\Re$$ ): $$\Im=\sqrt{\frac {\left(24+\sqrt[3]{\frac{12096+\sqrt{134369280}}2}+\frac{144}{\sqrt[3]{\frac{12096+\sqrt{134369280}}2}}\right)^2}{1728}-\frac {6+\sqrt[3]{\frac{12096+\sqrt{134369280}}2}+\frac{144}{\sqrt[3]{\frac{12096+\sqrt{134369280}}2}}}{36}}$$ $$x=\frac {12-\sqrt[3]{\frac{12096+\sqrt{134369280}}2}-\frac{144}{\sqrt[3]{\frac{12096+\sqrt{134369280}}2}}}{36}$$ so now using $$(i)$$ we get: $$a_n=x^n+y^n+z^n=x^n+(\Re+\Im i)^n+(\Re-\Im i)^n=\left(\frac {12-\sqrt[3]{\frac{12096+\sqrt{134369280}}2}-\frac{144}{\sqrt[3]{\frac{12096+\sqrt{134369280}}2}}}{36}\right)^n+2\sum_{k=0}^{\lfloor\frac n2\rfloor}\binom{n}{2k}\left(\frac {6+\sqrt[3]{\frac{12096+\sqrt{134369280}}2}+\frac{144}{\sqrt[3]{\frac{12096+\sqrt{134369280}}2}}}{36}-\frac {\left(24+\sqrt[3]{\frac{12096+\sqrt{134369280}}2}+\frac{144}{\sqrt[3]{\frac{12096+\sqrt{134369280}}2}}\right)^2}{1728}\right)^k+\left(\frac {24+\sqrt[3]{\frac{12096+\sqrt{134369280}}2}+\frac{144}{\sqrt[3]{\frac{12096+\sqrt{134369280}}2}}}{72}\right)^{n-2k}$$ so this is around: $$-0.47533^n+2\sum_{k=0}^{\lfloor\frac n2\rfloor}\binom{n}{2k}(-0.657118)^k(0.737665)^{n-2k}$$...
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2021-05-16 12:19:01
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https://stats.stackexchange.com/questions/296126/how-far-can-be-median-mode-and-mean-be-from-each-other-and-still-be-able-to-say?noredirect=1
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# How far can be median, mode and mean be from each other and still be able to say that is a normal distribution?
I'm working on the Boston housing project of the udacity ML nano degree. A histogram of the data set looks like this:
The mean, median and mode are:
mean: 454342.944785
median: 438900.0
mode: 525000
Is it correct to say that it has a normal distribution?
• for two probability distribution to be exactly the same, they need to have the same moments for all the orders up to infinity. mean is order 1, variance is 2. skewness is 3 etc. So just comparing 2 or 3 measures is not enough. Aug 4, 2017 at 17:39
• Consider carefully why you care whether the data themselves have a normal distribution (as the answers below show they cannot, in any event). What can be important in modeling is whether the residuals are normally distributed, as that assumption underlies many standard statistical tests. Even then, some tests are fairly insensitive to departures of residuals from normality, and approaches like bootstrapping can provide useful tests and confidence intervals without normality assumptions.
– EdM
Aug 4, 2017 at 17:50
• Whether the distribution is normal enough for using some statistical test is another matter. stats.stackexchange.com/questions/2492/… Aug 5, 2017 at 1:49
How far can be median, mode and mean to still say that is a normal distribution?
This kind of gets things backward -- even if they were exactly equal, that's no basis on which to claim you have a normal distribution. Note that
1. the population values are equal for many distributions that are not normal (if the population values were unequal of course you'd have non-normality, but if they're all equal it doesn't tell us that you have symmetry)
2. the sample values could be equal or very close to it even if the population values differ (indeed exact equality would suggest the distribution was discrete, and therefore not normal).
If you're using the data I think you are, for the variable you're referring to you have discretized and censored data, so normality would be moot. We can also see that it can't be normal because house values can't be negative.
So one thing you can say with confidence is that those values you have are not drawn from a normal distribution
Leaving that specific data aside, what we can do instead of trying to day data come from a normal distribution when those location values are close together is to ask "how far apart would they have to be to say that they're inconsistent with normality?".
That we can do something with, at least with respect to mean and median. (The sample mode is a bit tricky with continuous distributions; it would depend on how you obtain it; I suggest we leave that issue aside.)
The distance that sample mean and median would tend to differ will depend on scale and sample size. So one way to assess that difference independent of scale would be to measure how many standard deviations they are apart.
Note that (mean-median)/s.d. is one third of the second Pearson skewness; it's also (apparently) sometimes called the nonparametric skew.
So let's define that statistic, $$S=\frac{\bar{x}-\tilde{x}}{s}$$ (where $\tilde{x}$ is the sample median), which is one on which we can base a test.
Doane & Seward (2011)[1] offer a brief table for a test of $3S$ (the second Pearson skewness) at the normal.
Cabilio and Masaro (1996)[2] use $S$ as a test statistic for a test of symmetry (based on the values at the normal).
[In their case the test is asymptotic; you'd reject symmetry if $|S|>0.7555 \,Z_{\alpha/2}/\sqrt{n}$. Simulations suggest the asymptotic values aren't too bad once you get some way beyond the ends of Doane and Seward's table, I'd consider using it upward from about $n=400$ or so, though there's only about two figure accuracy in the critical values.]
Note that using this sort of statistic to decide if your distribution is non-normal would leave you unable to reject many other distributions (including -- in spite of Cabilio & Masaro's test being for asymmetry -- some asymmetric distributions which have mean = median)
[1]: Doane, D. P., Seward L. E. (2011),
Measuring Skewness: A Forgotten Statistic?
Journal of Statistics Education, Volume 19, Number 2
https://ww2.amstat.org/publications/jse/v19n2/doane.pdf
[2]: Cabilio, P. & Masaro, J. (1996),
A Simple Test of Symmetry about an Unknown Median,
The Canadian Journal of Statistics, Vol. 24, No. 3 (Sep.), pp. 349-361
No the distribution is certainly not normal. Normal distribution has a bell-curve shape and has mean = mode = median. However normal distribution is a synthetic mathematical function and no real life data will look exactly normal (even if you use random number generator to get draws from the normal distribution, even relatively small samples may not "look" normal and pass normality tests). In real-life we are interested is approximate normality, it is your decision if you want to use normal distribution as an approximation for the distribution of your data. Usually to convince yourself if the distribution behaves well, you would look at density plots, Q-Q plots, Shapiro-Wilk tests, Kolmogorov-Smirnov tests etc., but as describes in the Is normality testing 'essentially useless'? thread, those methods may misguide you and there is no fool-proof method to decide that the data "is", or "is not" normal. Moreover, in real life we often use normal approximation for data that is certainly not-normal, e.g. for survey data on Likert-type scales (bounded, discrete).
all models are wrong, but some are useful - G. E. P. Box
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