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https://engineering.stackexchange.com/questions/13963/how-does-the-second-motor-operate-in-this-diagram/13964#13964
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# How does the second Motor operate in this diagram?
The two motors are mounted in parallel as you can see in the picture and each one of them is connected to a gearbox through an elastic coupling. My question is : how do they operate together ? is one of them supposed to be off ? if you have any book or any website that discuss such transmissions I'd be thankful.
Three gears (and gearhousing body) in that configuration is a differential. Differentials work in both directions, connected this way they simply add the outputs together to one shaft. That is simply one of the uses of the differential. Although the reverse of this is more commonly known and is used in cars to spread the forces of one shaft to two.
Mathematically a differential works as follows:
A + B = C
Since you have motors connected to A and B their combined input is the output C. Also note this means the other motor can be not running it still works perfectly fine (since A + 0 = A = C).
And yes, before you ask, differentials can be used to do mechanical arithmetical calculations of adding or subtracting. It was used in all kinds of targetting devices in WW2.
• So you're saying that both motors are supplying power at the same time, right ? how is the needed power supposed to be divided between the two of them ? Feb 26 '17 at 22:20
• @mecheng They dont have to be on at the same time a differential works as follows input a + Input b = output c. It does not matter if the other engine is running or not there will just be more motive force if its running. Feb 26 '17 at 22:25
Although I'm sure there are exceptions, I think it is common for the two motors to share the load 50-50. Adding the gearbox adds cost and complexity to the system (versus just one motor), so you are only going to do it if you get a big benefit from the second motor. If it's a 90-10 split, you might as well get just rid of the second motor and save yourself the cost.
In practice, the split may deviate from 50-50 due to slight differences between the two motors. In this case a master-slave arrangement might be used to get an exactly even split. E.g. run one motor on speed control, measure the output torque, and the run the other motor on torque control. For a dc motor, volarge is proportional to speed and current is proportional to torque. So you run one motor on a constant voltage and then match the other motor to the same current.
Elastic couplings are going to eliminate/reduce shock loading in the gear train.
How does it work? Either the second motor is for redundancy/backup and/or additional power/torque. It's the same concept as adding more cylinders to an internal combustion engine. They apply more power to the crank shaft.
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2021-10-20 08:12:04
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https://socratic.org/questions/is-distillation-a-chemical-or-physical-change
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# Is distillation a chemical or physical change?
Oct 17, 2016
#### Answer:
Distillation is an example of $\text{physical change}$.
#### Explanation:
$\text{Physical changes}$ are often changes of state: solid to liquid, fusion; liquid to gas, vaporization; gas to solid, deposition. No chemical bonds are disrupted in this process, except the weak(er) dispersion forces between molecules.
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2019-09-15 14:53:09
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https://physics.stackexchange.com/questions/134456/question-on-acceleration-of-gravity
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# Question on acceleration of gravity
In pre-calculus last week my teacher explained a scenario in which a man throws a rock down a well and takes 4 seconds for you to hear the rock hit the bottom. How far downs the well? Last years I took regular physics and learned that distance equals Velocity x Time. So knowing that gravity pulls with 9.8m/s/s I assumed I would just multiply 4 by 9.8. Sadly I was wrong and my quick method of solving this problem was incorrect, we ended up using a very very large quadratic equation. My question Is why I was wrong in my answer?
When I think about the problem assume I went wrong by using 9.8 as if it were a regular velocity when in fact it is a constant acceleration, so the speed is constantly increasing. If that is the case then can anyone explain mathematically how quickly gravity accelerates in a 4 second period and also what the easiest and most efficient way of solving this problem would be to your knowledge?
Ps, this isn't homework I'm curious as to how this works
• What do you mean, "tell me how gravity works"? Also, Wikipedia's take on acceleration can probably answer your question. – ACuriousMind Sep 7 '14 at 22:23
• – Qmechanic Sep 7 '14 at 22:25
• @DiamondLouisXIV The homework tag is for homework-like questions, not just questions that are actually homework. See physics.stackexchange.com/tags/homework/info – HDE 226868 Sep 7 '14 at 22:34
• @HDE226868 Although you are correct in saying the homework tag is not only for homework questions but also homework-like questions I disagree that it should be used for me because the main question of my post was, how does gravitational acceleration apply specifically to this situation? And it was not, What is the answer to this question? Secondly the other part of the definition on this tag states that it should be used when commenters should guide the user to his answer as opposed to giving it out right, which i don't believe is necessary in my case. – Diamond Louis XIV Sep 7 '14 at 22:56
In the question, you clearly stated
it takes 4 seconds for you to hear the rock hit the bottom of the well
In 4 seconds, a rock (ignoring air drag) would drop $\frac12 9.81 (4)^2=78.5 m$ - but it would take sound about 1/4 of a second to reach your ear, so the well must be a little bit less deep (1/4 second is almost 10 meters at the velocity near the bottom).
The correct equation is:
$$t = \sqrt{\frac{2 h}{g}} + \frac hc$$
Namely, the time for the rock to drop, plus the time for the sound to get back to you. In this equation, $c$ is the speed of sound (340 m/s) and $h$ is the depth of the well (to be determined).
Solving for $h$ with $t=4$ we get approximately $71\ m$ - which is what you said the answer was (according to your comment to Jim's answer). I will leave the messy bits of the math to you (or you can go to Wolfram Alpha)
The other answers given (Jim, Alfred Centauri) focus on the basic equation of motion for an object accelerating with constant acceleration, namely
$$x = x_0 + v_0 t + \frac12 a t^2$$
That equation is needed to find the time for the rock to hit the bottom - if you start with $x_0=0, v_0=0$, set the acceleration equal to $g$, and solve for $t$, you find the first term in my expression above; the second term (which accounts for the velocity of sound) is needed to get to the "right" answer (the difference is about 10% - which I consider significant).
One of the five most common equations of motion in high school physics is $$\Delta d=v_0\Delta t+^1\!\!/_2\,a\Delta t^2$$ $v_0$ is your initial velocity (0 for your case), a is the acceleration (g for your case) and $\Delta t$ is the time. That's the easiest equation you'll find for something like this, from here they get a bit messier.
• Pretty amazing that we have to go through so much work to solve a simple problem haha. But thank you very much for the formula, that is the one we used in class as a matter of fact. The answer to this problem is 71meters but I posted this to learn more about how the acceleration of gravity works in relation to the problem, do you have any information as far as that goes? – Diamond Louis XIV Sep 7 '14 at 22:37
• it works like a constant acceleration. No strange stuff – Jim Sep 7 '14 at 22:39
Last years I took regular physics and learned that distance equals Velocity x Time.
That's not quite correct. In one dimension, the distance travelled equals the average velocity multiplied by the change in time
$$\Delta x = \bar v \cdot \Delta t$$
where $\bar v$ is the average velocity. If the velocity is constant, we can write
$$\Delta x = v \cdot \Delta t$$
Since there is acceleration in this problem, the velocity is not constant so, to use the first formula, you would need to know the average velocity. It turns out that, for constant acceleration $a$, the average velocity is just
$$\bar v = v_0 +\frac{a}{2}\Delta t$$
where $v_0$ is the initial velocity. Thus, the first equation becomes
$$\Delta x = \bar v \cdot \Delta t = \left(v_0 +\frac{a}{2}\Delta t\right)\Delta t = v_0 \Delta t + \frac{a}{2}\left(\Delta t\right)^2$$
• Oh ok. So then the formula we used in class is just a sort of evolution of the standard formula we learned in basic physics. Thank you for a helpful answer. – Diamond Louis XIV Sep 7 '14 at 23:16
The way you described it you said you solved it as 4(9.8). You should have done 1(9.8) + 2(9.8) + 3(9.8) + 4(9.8), or 10(9.8), or 98.
• All capital usernames don't make the best first impression, I suggest to switch to simply "Luca Butera". – peterh - Reinstate Monica Apr 6 '17 at 21:11
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2021-01-18 14:14:12
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http://www.beatthegmat.com/just-scored-a-730-here-s-how-i-did-it-t292453.html
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• Award-winning private GMAT tutoring
Register now and save up to $200 Available with Beat the GMAT members only code • 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • Free Trial & Practice Exam BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code • Get 300+ Practice Questions 25 Video lessons and 6 Webinars for FREE Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code ## Just scored a 730... Here's how I did it This topic has 2 member replies NC Esq Newbie | Next Rank: 10 Posts Joined 14 Sep 2016 Posted: 2 messages #### Just scored a 730... Here's how I did it Wed Sep 14, 2016 8:51 am On my first GMAT attempt this month, I walked out with a 730 (96%). Breakdown = Q-48(78%), V-41(94%), IR-8(92%), AWA-6.0(90%). I've always been a good standardized test taker, but I put more effort into this than I have with any other. Here's my story, which I hope will provide encouragement to others and perhaps some preparation tips along the way. First, a little background on me. I'm 36 years old, American, and a partner in a small law firm in North Carolina. I graduated from a top 10 liberal arts college and a top 40 law school. I've been a practicing attorney for 10 years, but have decided to pursue a different career opportunity. I'm in the process of starting an industry-specific management/consulting business with a partner and hope to attend a Weekend EMBA program once it is off and running (and I have left the practice of law). I need the finance/accounting/marketing/management knowledge of an MBA program, but not so much the networking/career services side of things (I think!). I have two young children (under 5 years old) and a demanding law practice, so my GMAT prep would have to be efficient and focused. I started in May and took the exam over Labor Day Weekend, meaning my total prep time was approximately 3.5 months. This was by design, as I had an important trial on the calendar in mid-September and didn't want my practice and studying dragging on to the point of burnout. I recommend a timeline in this ballpark, as you will probably grow weary of this type of intellectual work if it drags on for six or more months. I know I would have. Starting Out Eons ago, a Princeton Review book helped me a great deal with my SAT prep, so that's where I started. The problems contained in their Cracking the GMAT book provided a good, encouraging start, especially since I had not done any real math since high school. Small aside: I was once a very talented math student, but burned out by focusing so intently on it in high school. I excelled in AP Calculus, to the point that I never had to take any math courses in college. There's no math in law school, so the only quant work I've done with any regularity since the 1990s are day to day tip calculations and personal finance figures. The Princeton Review quant problems were interesting enough that they tickled my dormant mathematical brain, but simple enough that I did not get frustrated or discouraged. I expected my verbal skills to be fine, as I spend my days reading, writing, and slicing arguments. When I took a short, free quiz online (sorry, can't recall where), my suspicions were confirmed: I could coast on verbal, but would need to put in a lot of work on quant. I think this particular quiz estimated my total score in the mid-500s range. I turned to this forum to learn more about prep strategies and materials. What a valuable discovery that was. It became evident very quickly that Princeton Review was just not going to cut it as a primary text for someone with 700 ambitions. What I read elsewhere on Beat the GMAT has proven true - advanced test takers looking for advanced scores are not well served by an entry-level book like PR's Cracking the GMAT. I needed more. Getting Serious After reading tons of threads in this particular forum and scouring online reviews, I jumped in with Manhattan Prep. I ordered the complete set of (6th edition) books. I went through every page and problem of the 5 quant books. I would spend approximately 2 hours each evening, carefully reading a chapter or two, then doing the problem sets in the Manhattan books. Most importantly, I did all of the mid-book and final quizzes for each topic using the problem sets from the OG that Manhattan provides. I would do the "Moderate" and "Harder" quizzes, so I would have about 40-45 OG questions each time. This was the most useful practice I undertook. If you do nothing else, get an OG book and work the heck out of it! I used Manhattan's very helpful web interface (The "Navigator") to track my results on all of my OG practice. I found the stats it provided very useful in focusing my work on my weaker areas. I can't be sure that my methodical slog through the Manhattan quant books was the most efficient use of my time. But I do think it was critical for me to get back in "math-mode." I live in a smaller town, so live prep courses were not a feasible option for me. I needed to self-study and the Manhattan books were a great content-based means for doing that. I fully recommend them to everyone, but admit that I don't have much comparison (other than the Princeton Review book, which isn't really even comparable). For Verbal, I did a thorough read of the Manhattan Sentence Correction book, which is as great as everyone says it is. I skimmed the Critical Reasoning book, which did provide some helpful analysis framework. I didn't open the Reading Comprehension book, as I always did well on those questions in my practice tests and problem sets. Two Additional Recommendations (1) I downloaded the iPhone app that Manhattan Prep produced. It was expensive -$39.99, I think - but completely worth it for the additional drilling it provided. As I've said, I needed to shake off a lot of math rust and this really helped me sharpen my skills. I could open it during any free time - downtime in court, on flights, or at home - and do a whole bunch of drills and exercises that proved to be very useful. I'm talking about mental arithmetic, reducing fractions, combining exponents, etc. I know they run promotions where the app is half price sometimes; it would be a steal at that cost.
(2) Brett Ethridge is the guru behind the website, Dominate the GMAT. He has a bunch of free material out there, particularly his series of excellent YouTube videos. I would wind down practice sessions in the evenings by watching one or two of them before bed. I watched only his free content, but I'm positive that the paid stuff would be great for anyone needing additional assistance. I really had some strategy stuff click for me in a big way after watching his lessons. Brett, if you are out there, thanks so much for making some high-quality content available for free online.
Practice Tests
Your purchase of any Manhattan Prep books give you access to 6 CATs online. These are much harder than the actual GMAT! I took three of them (scores below) and found the quant problems to be significantly more difficult and more time-consuming than those on the actual exam. I never came close to finishing the quant MGMAT sections on time, but had no problem with the GMAT Prep quant timing or on the actual exam. All that said, I look at the MGMAT exams as excellent resistance training; by doing these very difficult questions, the actual questions will feel much easier. This was my experience during the real deal, anyway.
My scores:
MGMAT 1 - June 18 - 700 (q42, v42)
MGMAT 2 - July 24 - 650 (q41, v38)
MGMAT 3 - Aug. 20 - 690 (q43, v40)
GMATPrep 1 - Aug. 29 - 730 (q45, v44)
ACTUAL GMAT - Sep. 3 - 730 (q48, v41)
As you can see, the GMAT Prep test a week before the real thing provided a perfect total score prediction for me. It also "felt" the most like the questions I got on the actual exam. While I feel I left a few points on the table by not getting a higher verbal score, I was very happy to get my maximum quant score on the real thing. I know I need to reassure adcoms in that area, as my work experience is exclusively verbal.
A note on IR/AWA. I did practice IR sections on each of my practice exams and always got terrible scores (2-4). I watched the Manhattan Prep online videos on IR that are part of their "Interact" instruction series and improved immediately. These videos are also free with the purchase of any of their books. I never wrote out an entire AWA essay on a practice exam, but knew this would be one of my strengths based on my job. I strictly followed the template developed on this site by myohmy on the exam: http://www.beatthegmat.com/argument-essay-template-if-anyone-wants-it-t38032.html. Thanks so much for that! While the IR and AWA scores are certainly less important than the big number, I felt very proud to score perfectly on both of those sections and hope that it will set me apart from other high scorers.
Final Thoughts
(1) The GMAT is a test you can learn. I admit I had some built-in advantages (native English speaker, daily work focused on reasoning & grammar, etc.). But I set out to get a high score and was able to do it by just putting the time in. The questions get easier and your work gets faster as an automatic result of just doing these types of problems for hours on end. Find an instructor/guidebook to help you with strategy, and then just wear out OG problems. If you put in 100+ hours of focused study and thoughtful reflection on practice problems, you WILL see your score significantly improve.
(2) Try to sample several different content providers to see what you click with. I mentioned Manhattan and Dominate The GMAT above because those were the ones that my learning style gravitated toward. I watched a video or two from tons of providers before I stuck with Brett.
(3) In a dorky way, math can be pretty fun. I began to get some real enjoyment from working an intricate problem and arriving at an elegant solution. You might, too!
(4) There are so many free online resources out there. Beat the GMAT alone can help you immensely. Dedicate a few hours here and there to just browsing and wandering these interwebs in order to gain insight and wisdom from others.
(5) Remember to thank your spouse, kids, coworkers, friends, etc. for putting up with your being a distracted, neurotic weirdo during prep time!
Now it's on to the admissions process. Anyone got any pro-tips for me? My aim is a top Weekend EMBA program within driving distance, which narrows it to Duke and UNC for me. I'd love to land at Fuqua.
Need free GMAT or MBA advice from an expert? Register for Beat The GMAT now and post your question in these forums!
aalsayye Newbie | Next Rank: 10 Posts
Joined
22 Aug 2016
Posted:
1 messages
Followed by:
1 members
Thu Sep 22, 2016 7:27 am
Awesome write up and congratulations on getting that impressive score!
The school selections that you have are on point for the type of program you are looking for. I definitely recommend Fuqua.
One question I had for you was your thoughts on the MGMAT CATs. For the most part, are those questions actually tougher than the ones on the real test?
Cheers!
NC Esq Newbie | Next Rank: 10 Posts
Joined
14 Sep 2016
Posted:
2 messages
Fri Sep 23, 2016 10:49 am
Thanks, aalsayye!
My answer to your question is yes, the MGMAT CAT questions seemed tougher to me than the ones on the real test. In the Quant section, anyway. I didn't notice an appreciable difference between MGMAT CAT Verbal questions and the real deal.
The concepts in Quant were all the same, of course. It just seemed to me like Manhattan's problems always involved a few extra steps. An additional concept seemed to be added to each question. That made them significantly more time consuming and also provided more opportunity for errors.
Now, I think Manhattan does adjust its scoring algorithm to give a decently predictive score, even though the questions are harder. But the fact that I always ran out of time on MGMAT CAT Quant sections (while finishing the actual exam with a minute or two to spare) makes me think I'm on to something.
As I said above, I do think they make for very useful practice. It's like swinging a baseball bat in the on deck circle with a weighted doughnut. It also made me really focus on time management strategies, which served me well.
Good luck to you!
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2017-10-19 18:25:24
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http://vicenteroura.com/reel/what-is-the-structural-formula-of-methane.php
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## What is the structural formula of methane
Organic formulae 1. Before you answer the puzzles below fill in the table of alkanes: name molecular formula structural formula displayed formula methane. CH4. Methane | CH4 | CID - structure, chemical names, physical and chemical properties, classification, patents, literature, biological Molecular Formula: CH4 . Structural formula for methane is CH4.
Methane is a chemical compound with the chemical formula CH4 (one atom of carbon and four atoms of hydrogen). It is the simplest alkane, and is the main. Structural Formula. CH4. methane. Molecular Model. Jmol._Canvas2D (JSmol) " jmolApplet0"[x]. call loadScript core\vicenteroura.com call loadScript core\vicenteroura.com Which structural formula represents a molecule of butane? (1) Which is an isomer of n-butane? (1) (2) (2) (3) (3) (4) (4) In a molecule of CH4, the.
The simplest organic molecule is methane. A three dimensional representation of methane would actually look like the following picture. Carbon atoms can link. What is the Structural Formula of a Methane Molecule?.
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2019-02-17 20:48:29
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https://alstatr.blogspot.com/2014_01_01_archive.html
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Showing posts from January, 2014
Python and R: Is Python really faster than R?
A friend of mine asked me to code the following in R:
Generate samples of size 10 from Normal distribution with $\mu$ = 3 and $\sigma^2$ = 5;Compute the $\bar{x}$ and $\bar{x}\mp z_{\alpha/2}\displaystyle\frac{\sigma}{\sqrt{n}}$ using the 95% confidence level;Repeat the process 100 times; thenCompute the percentage of the confidence intervals containing the true mean. So here is what I got,
Staying with the default values, one would obtain
The output is a list of Matrix and Decision, wherein the first column of the first list (Matrix) is the computed $\bar{x}$; the second and third columns are the lower and upper limits of the confidence interval, respectively; and the fourth column, is an array of ones -- if true mean is contained in the interval and zeros -- true mean not contained.
Now how fast it would be if I were to code this in Python?
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2017-07-27 14:42:01
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http://www.chegg.com/homework-help/questions-and-answers/miller-toy-company-manufactures-plastic-swimming-pool-westwood-plant-plant-experiencing-pr-q1200518
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Miller Toy Company manufactures a plastic swimming pool at its Westwood Plant. The plant has been experiencing problems as shown by its June contribution format income statement below:
Budgeted Actual
Sales (5,600 pools) $280,000$280,000
Variable expenses:
Variable cost of goods sold* 102,256 116,231
Variable selling expenses 15,000 15,000
Total variable expenses 117,256
131,231
Contribution margin 162,744
148,769
Fixed expenses:
73,000
Total fixed expenses 126,000 126,000
Net operating income $36,744$22,769
--------------------------------------------------------------------------------
*Contains direct materials, direct labor, and variable manufacturing overhead.
Janet Dunn, who has just been appointed general manager of the Westwood Plant, has been given instructions to "get things under control." Upon reviewing the plant's income statement, Ms. Dunn has concluded that the major problem lies in the variable cost of goods sold. She has been provided with the following standard cost per swimming pool:
Standard
Quantity or
Hours Standard Price or
Rate Standard
Cost
Direct materials 4.40 pounds $2.70 per pound$11.88
Direct labor .70 hours $8.30 per hour 5.81 Variable manufacturing overhead .30 hours*$1.90 per hour .57
Total standard cost $18.26 -------------------------------------------------------------------------------- *Based on machine-hours. During June the plant produced 5,600 pools and incurred the following costs: a. Purchased 30,568 pounds of materials at a cost of$3.10 per pound.
b. Used 24,340 pounds of materials in production. (Finished goods and work in process inventories are insignificant and can be ignored.)
c. Worked 4,320 direct labor-hours at a cost of $7.90 per hour. d. Incurred variable manufacturing overhead cost totaling$4,158 for the month. A total of 1,980 machine-hours was recorded.
It is the company's policy to close all variances to cost of goods sold on a monthly basis.
1.value:
7 points Requirement 1:
(a) Compute the direct materials price and quantity variances for June. (Indicate the effect of each variance by selecting "F" for favorable, "U" for unfavorable, and "None" for no effect (i.e., zero variance). Input all amounts as positive values. Round your answers to the nearest dollar amount. Omit the "$" sign in your response.) Material price variance$ (Click to select)NoneFU
Material quantity variance $(Click to select)NoneUF -------------------------------------------------------------------------------- (b) Compute the direct labor rate and efficiency variances for June. (Indicate the effect of each variance by selecting "F" for favorable, "U" for unfavorable, and "None" for no effect (i.e., zero variance). Input all amounts as positive values. Omit the "$" sign in your response.)
Labor rate variance $(Click to select)FNoneU Labor efficiency variance$ (Click to select)UNoneF
--------------------------------------------------------------------------------
(c) Compute the variable overhead rate and efficiency variances for June. (Indicate the effect of each variance by selecting "F" for favorable, "U" for unfavorable, and "None" for no effect (i.e., zero variance). Input all amounts as positive values. Omit the "$" sign in your response.) Variable overhead rate variance$ (Click to select)UNoneF
Variable overhead efficiency variance $(Click to select)UNoneF -------------------------------------------------------------------------------- Check My WorkeBook Links (3)References Worksheet Learning Objective: 09-2 Learning Objective: 09-4 Difficulty: Basic Learning Objective: 09-3 2.value: 5 points Requirement 2: (a) Compute the net overall variance for the month. (Indicate the effect of each variance by selecting "F" for favorable, "U" for unfavorable, and "None" for no effect (i.e., zero variance). Round your answer to the nearest dollar amount. Input the amount as positive value. Omit the "$" sign in your response.)
Net variance $(Click to select)FUNone (b) What impact did this figure have on the company's income statement? (Input the amount as positive value. Round your answer to the nearest dollar amount. Omit the "$" sign in your response.)
As an impact of the above figure the company's income statement is showing a (Click to select)lossreduced profit of $. Check My WorkeBook Links (3)References Worksheet Learning Objective: 09-2 Learning Objective: 09-4 Difficulty: Basic Learning Objective: 09-3 3.value: 3 points Requirement 3: Pick out the most significant variance that you computed in Requirement 1 above. Labor efficiency variance Variable overhead spending variance Material quantity variance Material price variance Variable overhead efficiency variance Answers (1) • Miller Toy Company manufactures a plastic swimming pool at its Westwood Plant. The plant has been experiencing problems as shown by its June contribution format income statement below: budget actual Sales (15,000 pools)……………………………..$450,000 $450,000 Less variable expenses: Variable cost of goods sold*…………….... 180,000 196,290 Variable selling expenses…………………. 20,000 20,000 Total variable expense………………………….. 200,000 216,290 Contribution margin……………………………. 250,000 233,710 Less fixed expenses: Manufacturing overhead…………………… 130,000 130,000 Selling and administrative…………………. 84,000 84,000 Total fixed expenses…………………………… 214,000 214,000 Net operating income…………………………..$ 36,000 $19,710 _______ ---------- *Contains direct materials, direct labor, and variable manufacturing overhead Janet Dunn, who has just been appointed general manager of the Westwood Plant, has been given instructions to “get things under control.” Upon reviewing the plant’s income statement, Ms. Dunn has concluded that the major problem lies in the variable cost of goods sold. She has been provided with the following standard cost per swimming pool: Standard Quantity Standard Price of rate Standard cost Of Hours Direct materials……………… 3.0 pounds$2.00 per pound $6.00 Direct labor………………….. 0.8 hour$6.00 per pound 4.80
Variable manufacturing overhead 0.4 hour $3.00 per pound 1.20 Total standard cost……………..$12.00
______
Based on machine-hours.
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2013-05-18 10:54:23
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http://www.lookingforananswer.net/given-y-is-a-function-of-x-differentiate-y-3-4-xy-with-respect-to-x.html
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# Given y is a function of x, differentiate y^3 + 4 = xy with respect to x?
... Section 4.2: Implicit Differentiation . 4.2 ... x 2 + y 2 = 25 itself with respect to x, regarding y as a function of x ... ) x 2 y 3 = 2 xy 3 + 3 x 2 y 2 (dy ... - Read more
... it is implied that there exists a function y = f( x) such that the given equation is ... differentiate implicitly with respect to x, ... (3,−4), y′ = −3 ... - Read more
## Given y is a function of x, differentiate y^3 + 4 = xy with respect to x? resources
### Using the Chain Rule to Differentiate Complex Functions ...
... this parabola is given by the equation f(x) = (2x - 4) ... Chain Rule with Trig Functions. What about, y ... Using the Chain Rule to Differentiate Complex ...
### Partial Derivatives in Calculus - Free Mathematics ...
Let f(x,y) be a function with two variables. If we keep y constant and differentiate f (assuming f is differentiable) with respect to the variable x, we obtain what ...
### Section 3–5 Implicit Differentiation The Explicit Form ...
x2−4y2=9 The slope of a function at any given value of x can be ... When you differentiate y with respect to x ... of y3 with respect to x d dx (y3)=3•y2 ...
### Linear Functions | Math@TutorVista.com
... linear form hence if we differentiate y with respect to x we get only m which is ... of a linear function is f(x,y) ... x = 0 Step 4: Value of x = 0 and y = 3. ...
### DERIVATIVE OF CONSTANTS TO VARIABLE POWERS
To obtain the derivative of y = , we differentiate both sides of equation (5.10) with respect to x, which gives. ... functions: Theorem 4. The derivative of the ...
### Chapter 3: Topics in Differentiation - John Wiley & Sons
Find the equation of the tangent line to the function y =(x2 +4)1 ... y −3y2) (c) x+y xy 2y−3 ... A point P is moving along a curve whose equation is given by y ...
### How to Differentiate a Function | eHow
How to Differentiate a Function. A function expresses relationships between constants and one or more variables. For example, the function f(x) = 5x + 10 ...
### The University of Sydney - School of Mathematics and ...
For the given function f, ... (x,y), regard y as a constant and differentiate f(x,y)with respect to x. Similarly, ... x2 +4, find df dx. (viii) f(x,y)= xy x2 +y2 ...
### what's f'(x,y) when f(x,y) = g(x-2y), g'(1)=3
... I'll differentiate with respect to y, treating x as ... You can always make a function bigger than any given function, eg, ln(x) ... (x,y), \ f_x(3,1), \ f_y(3,1 ...
### 4.2 Implicit Differentiation
4.2 Implicit Differentiation ... answer is to treat y as a differentiable function of x and differentiate both sides of ... -4) x2 + xy - y2 = 1, ( ) x3 + y3 = xy x2 ...
### Implicit Differentiation | Math@TutorVista.com
... (x) = x 2 + 5x + 4. That is, the function y is ... By implicit differentiation of xy, ... we can able to differentiate the given function with respect to one ...
### Implicit Differentiation - S.O.S. Mathematics
... they are linked through an implicit formula, like F(x,y) ... consider y as a function of x: ... Let us differentiate the above equation with respect to x where y ...
### Derivatives - Rose-Hulman
Second and Higher Order Derivatives 1 The derivative y' = is the first derivative of y with respect to x. The first derivative may also be a differentiable function of x
### Partial Differentiation - Florida International University
To compute the partial derivative with respect to y, we treat x as a constant and we consider f as a function of the variable y alone. f y (x,y)=
### 17.3 IMPLICIT DIFFERENTIATION - Harvard University
EXAMPLE 17.4 Consider the circle of radius 2 centered at the origin.5 It is given by x 2+y =4. ... function of x, when we differentiate y ... xy2 +2y =x2y +1 (d ...
### Derivatives of Trigonometric Functions - Physics Forums
1. The problem statement, all variables and given/known data Find d^2y/dx^2. y = x cos x 3. The attempt at a solution I've been doing derivatives recently and now got ...
### Partial and implicit differentiation - FreeMathHelp.com
... + ln(x/y^2)=(x^2+y)/(y^3+x) Differentiate w.r.t y How do you know ... to differentiate xy = y^2 with respect ... always given a function like z=x^4+y^4 ...
### 170 Chapter 2 Differentiation: Basic Concepts 7
You are going to differentiate both sides of the given equation with respect to x ... function of x. To find , 1. Differentiate both ... 3. x3 y3 xy 4. 5x x2y3 2y
### Implicit function - Wikipedia, the free encyclopedia
Another example is an implicit function given by x − C(y) ... to differentiate R(x, y) with respect to x ... function, for which implicit differentiation might be ...
### Logarithmic Differentiation - Free Mathematics Tutorials ...
Use the method of logarithmic differentiation to find ... Find the derivative y ' of function y given by y = 3 x 2 ... Differentiate both sides with respect to x y ...
### 7. Differentiating Powers of a Function - IntMath
Consider the function . y = (5x + 7) ... Then we differentiate y (with respect to u), ... Find the derivative of y=(x^2(3x+1))/(x^4+2)
### 6.6 Implicit Differentiation - St Charles College
Implicit Differentiation Given x2 2y ... term with respect to x. (a) xy (b) 2x4y (c) x 2y (d) 5x2y3 (e) ... When y is an implicit function of x, differentiate each ...
### Partial Derivative Calculator, Partial Derivative Solver ...
If f(x, y) = xy + x 3 then calculate $\frac{\partial f}{\partial x}$ ? Step 1 : Given function: f(x, y) = xy + x 3. Step 2 : $\frac{\partial f}{\partial x}$ = \$\frac ...
### Differentiable Functions of Several Variables - UUMath - Home
Differentiable Functions of Several Variables x ... we think of x as constant and differentiate with respect to y ... Example 16.9 Given the function z = x2 xy + y3, ...
### Chapter 7 Section 2 Partial Derivatives 519
The partial derivative of f y with respect to x is f ... compute all first-order partial derivatives of the given function. 1. f(x, y) 2xy5 3x2y ... (x, y) 5x4y3 2xy ...
### 2.5 Implicit Differentiation - Western Nevada College
2.5 Implicit Differentiation The explicit form of an equation is an equation of the form: y = 2x 3 + 5 and x y 1 = An example of an equation in implicit form is: xy ...
### DIFFERENTIATION RULES - York University
It is correct no matter which function y is determined by the given ... of x3 + y3 = 6xy with respect to x, regarding y as ... respect to x, we get 4x3 + 4y3y ...
### Review: Partial Differentiation - PSU Mathematics Department
Example: Continue with the last example f (x, y) = e xy − ln( xy) + y2 sin(4 x) + 2 x3 − 5 y, 4 2 cos(4 ) 6 2 1 y x x x f yexy x = − + +, and 2 sin(4 ) 5
### Differential equation problem - Physics Forums
1. The problem statement, all variables and given/known data The differential equation y + 4y^4 = (y^3 + 3x)y' can be written in differential form:
### DIFFERENTIATION RULES - Mercer County Community College
It is correct no matter which function y is determined by the given ... x3 + y3 = 6xy with respect to x, regarding y ... y x x y y y IMPLICIT DIFFERENTIATION Example 4.
Related Questions
Recent Questions
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2016-10-25 13:52:10
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https://tex.stackexchange.com/questions/423345/floatrow-and-capbesidewidth-for-side-caption
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# floatrow and capbesidewidth for side caption
I am having some weird problem using fcapside in floatrow where I can't get the side caption to auto-calculate its width based on my figure. Instead, the side caption width seems to expand with the image. I set the capbesidewidth to something else in the preamble so I explicitly set it back to sidefil (or none) for the figure, but it does this even if I never set it anywhere. I can't figure out why this is different from the manual where the caption width is auotmatically calculated to occupy the rest of the space.
\documentclass[11pt,twoside]{report}
\usepackage{graphicx}
%\usepackage{caption}
\usepackage{floatrow}
\DeclareFloatVCode{largevskip} {\vskip 20pt}
%\floatsetup[figure]{a bunch of other settings here that I overwrite below}
\captionsetup[figure]{font=small}
\usepackage{subfig}
\begin{document}
\begin{figure}[htbp]
%\centering
\ffigbox[\textwidth]
{\begin{subfloatrow}
\fcapside[\FBwidth]
{\setlength\fboxsep{4pt} \fbox
{\includegraphics[width=0.5\textwidth]{example-image-a}}}
{\caption{Hello I am a caption and my width seems to be approximately the same as that of the image next to me, which is 0.5 times the text width.}}
\end{subfloatrow}
\begin{subfloatrow}
\fcapside[\FBwidth]
{\setlength\fboxsep{4pt} \fbox
{\includegraphics[width=0.65\textwidth]{example-image-a}}}
{\caption{Hello I am a caption and my width also seems to be approximately the same as that of the image next to me, which is 0.65 times the text width.}}
\end{subfloatrow}}
{\caption{I am a centered caption for the full figure.}}
\end{figure}
\end{document}
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2019-08-25 03:53:46
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https://en.m.wikipedia.org/wiki/Preparatory_classes
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# Classe préparatoire aux grandes écoles
(Redirected from Preparatory classes)
Front entrance of Lycée Henri-IV, in Paris, one of the famous Lycées providing access to Grandes écoles.
The classes préparatoires aux grandes écoles (CPGE) (English: Higher School Preparatory Classes), commonly called classes prépas or prépas, are part of the French post-secondary education system. They consist of two very intensive years (extendable to three or exceptionally four years) which act as a preparatory course (or cram school) with the main goal of training undergraduate students for enrollment in one of the grandes écoles. The workload is one of the highest in the world[1] (between 29 and 45 contact hours a week, plus usually between 4 and 6 hours of written exams, plus between 2 and 4 hours of oral exams a week and homework filling all the remaining free time[2]).
The students from CPGE have to take national competitive exams to be allowed to enroll in one of the Grandes Écoles. These Grandes Écoles are higher education establishments (graduate schools) delivering master's degrees and/or doctorates. They include science and engineering schools, business schools, the four veterinary colleges and the four écoles normales supérieures but do not include medical institutes or architecture institutes. Their competitive entrance exams make having attended one of the grandes écoles being often regarded as a status symbol as they have traditionally produced most of France's scientists, executives and intellectuals (Écoles Normales Supérieures, École Polytechnique, Écoles Centrales, ParisTech Schools...).
Hence, there are three kinds of different prépas: scientific, economic and literary CPGE. Each of them prepare to pass the competitive exams of those grandes écoles.
## Contents
The CPGE are located within high schools for historical reasons (Napoleon created them at first as fourth to sixth year of high school) but pertain to tertiary education, which means that each student must have successfully passed their baccalauréat (or equivalent) to be admitted to CPGE. Moreover, the admission to the CPGE is usually based on performance during the last two years of high school, called première and terminale. Thus, each CPGE receives hundreds of applications from around the world every April and May, and selects its new students under its own criteria (mostly excellence). A few CPGE programmes, mainly the private CPGEs (which account for 10% of CPGEs), also have an interview process or look at a student's involvement in the community.
In June 2007, 534,300 students passed the "Baccalauréat", and 40,000 (7.5%)[3] of them were admitted to CPGE. On a given class at one of the prep schools listed above, around 1500 application files will be examined for only 40 places.[4] Students are selected according to their grades in High school and the first part of "Baccalauréat" (equivalent to A-levels in the United Kingdom or Advanced Placement in the United States).
## DegreeEdit
Preparatory classes are officially not authorized to deliver any degrees, but they give ECTS (university equivalence) since the 2009-2010 academic year, and students who decide to can carry on their studies at university.[5]
However, many prépas also establish conventions with universities to validate a full 2nd or 3rd year degree for CPGE students who did their job well, especially in literary prépas ("khâgne"). Most of the students in these classes continue their cursus at the university, so the teachers' council can deliver them the corresponding grade in one or two disciplines at the end of the year (only up to a bachelor's degree for 3 years of CPGE).
## Organization of CPGEEdit
CPGE exist in three different fields of study: science & engineering, business, and humanities. All CPGE programs have a nominal duration of two years, but the second year is sometimes repeated once.
### Scientific CPGEEdit
Schema representing various ways proposed in scientific CPGE.
The oldest CPGEs are the scientific ones, which can be accessed only by scientific Bacheliers. The different tracks are the following:
• MPSI, Mathématiques, Physiques, Sciences de l'Ingénieur ("mathematics, physics, and engineering science") in the first year, followed by either MP ("mathematics and physics") or PSI ("physics and engineering science")
• PCSI, Physique, chimie, sciences de l'ingénieur ("physics, chemistry, and engineering science") in the first year, followed by PC ("physics and chemistry") or PSI ("physics and engineering science")
• PTSI, Physiques, technologie, sciences de l'ingénieur ("physics, technology, and engineering science") in the first year, followed by PT ("physics and technology") or PSI ("physics and engineering science")
• TPC1, Technologie, physique et chimie ("technology, physics and chemistry") in the first year, followed by TPC2
• TSI1, Physiques, Technologie, sciences industrielles ("physics, technology, industrial science") in the first year, followed by TSI2
• BCPST1, Biologie, Chimie, Physique, sciences de la terre ("biology, chemistry, physics and earth sciences") in the first year, followed by BCPST2
• TB1, technologie, biologie ("technology and biology") in the first year, followed by TB2
The classes that especially train students for admission to the elite schools, such as Écoles Normales Supérieures or ParisTech schools, have an asterisk added to their name. For example, MP*, are usually called MP étoile ("MP star") (except for the BCPST2 and TB2 classes, which all prepare to the elite schools).
Both the first and second year programmes include as much as ten to twelve hours of mathematics teaching per week, ten hours of physics, two hours of literature and philosophy, two to four hours of (one or two) foreign language(s) teaching and two to eight hours of minor options: either SI, engineering industrial science, chemistry or theoretical computer science (including some programming using the Pascal, CaML, or Python programming languages, as practical work), biology-geology, biotechnologies.[6] Added to this are several hours of homework, which can amount to as much as the official hours of class.
The BCPST classes prepare for exams of engineering schools of life sciences (agronomy, forestry, environmental and food sciences) but also to veterinary schools, engineering schools of earth sciences, and the three Ecoles Normales Supérieures. Compare to the other classes, it teaches biology and geology.
In scientific CPGE, the first year of CPGE is usually called the maths sup, or hypotaupe (sup for "classe de mathématiques supérieures", superior in French, meaning post-high school), and second year maths spé, or taupe, (spés for "classe de mathématiques spéciales", special in French). The students of these classes are called taupins.
The word taupe means "mole" in French. Its signification comes from the lifestyle of students in classes preparatoires. The intensive workload makes students sacrifice their social life for the classes preparatoires years. They spend most of their time studying inside and barely go outside.
A very specific kind of CPGE is targeting technicians. They are called ATS classes, Adaptation Techniciens Supérieurs ("Adaptation for Skilled Workers ") and last only a year. They are mainly based on the curriculum of PTSI and PCSI, but the courses are summed up.
### Literary and humanities CPGEEdit
The literary and humanities CPGEs are focused on a strong pluri-disciplinary course, including all humanities: philosophy, literature, history, geography, foreign languages, and ancient languages (Latin and Ancient Greek). These prépas also have their own nicknames: "hypokhâgne" for the first year and "khâgne" for the second year. The students are called the "hypokhâgneux" and the "khâgneux". These classes prepare for the entrance exam of the elite schools called Écoles Normales Supérieures, which are considered among the most difficult exams of the French system. Nevertheless, the students can now also apply for many other entrance exams.
There are three types of Khâgne:
• Khâgne "Ulm", which prepares more specifically for the A/L entrance exam of the ENS Paris;
• Khâgne "Lyon", which prepares more specifically for the A/L entrance exam of the ENS Lyon;
• Khâgne "B/L", which prepares for the B/L entrance exam of the four ENS. Its particularity is the presence of mathematics and social sciences.
Now, the grouping of many examinations make the difference between khâgnes "Lyon" and "Ulm" is slight, and lots of prépas have mixed classes with many students preparing for both ENS (or even the three for students specialising in English).
Khâgneux can apply to many grandes écoles, other high schools and all universities, among which are the following:
### Economics CPGEEdit
Those CPGEs which are focused on economics (which prepare the admission to business schools such as HEC Paris, ESSEC, ESCP Europe, etc.) are known as Prépa HEC and are split into three parts:
• ECS1 (Economics and Commercial Scientific way), followed by ECS2
Course[7]
Hours/week
Mathematics 10 h
History, Geography and Geopolitics 6 h
Foreign Language 1 3 h
Foreign Language 2 3 h
Philosophy 3 h
Literature 3 h
Economics (optional) 1 h
• ECE1 (Economics and Commercial Economics way), followed by ECE2
Course[8] Hours/week
Mathematics 9 h
Economics, Sociology and History 6 h
Further Economics 2 h
Foreign Language 1 3 h
Foreign Language 2 3 h
Philosophy 3 h
Literature 3 h
• ECT1 (Economics and Commercial Technological way), followed by ECT2
Course[9] Hours/week
Mathematics 6 h
Management 5 h
Economics 3 h
Law 3 h
Foreign Language 1 4 h
Foreign Language 2 5 h
Philosophy 3 h
Literature 3 h
Classe préparatoire ECS are for those who graduated with the general Baccalauréat S (Scientific), Classe préparatoire ECE are for those who were in the economics section at their Lycée (and received the general Baccalauréat ES (Economics and Social)), whiles the Classe préparatoire ECT are for those who obtained a Baccalauréat Technologique .
However, both the first and the second year programmes include ten hours of mathematics teaching per week and also six hours of business history and geography, six hours of French and philosophy, and three hours of each language (2 languages) in the "ECS" section.
There is also the D1 and D2 CPGE, also known as ENS Cachan CPGE:
• D1 (law and economy): the students attend both university (taking courses at the law faculty) and CPGE's School. They study civil law, economics, and they choose business law, public law or mathematics; one language (mostly English, German, Spanish and Italian), but they can study a second language for the Écoles de commerce, and general culture. At university, they study constitutional law, criminal law and administrative law. At the end of the two years, studients go to ENS Paris-Saclay, École de commerce, Sciences Po or some selective university of law. This CPGE is open for Baccalauréats L, ES and S.
• D2 (economy and management): students attend both to university (taking courses in economics or mathematics) and CPGE's school.
D1 and D2 are very rare but offer a complete and multidisciplinary training.
## Life in a CPGEEdit
### The "Khôlle"Edit
The amount of work required of the students is exceptionally high.[10]
In addition to class time and homework, students spend several hours each week completing exams and colles (very often written "khôlles" to look like a Greek word, this way of writing being initially a "khâgneux" joke). The so-called "colles" are unique to French academic education in CPGEs. They consist of oral examinations twice a week, in maths, physics, chemistry, biology and earth sciences (in BCPST classes), French and a foreign language, usually English, German or Spanish. Students, usually in groups of three, spend an hour facing a professor alone in a room, answering questions and solving problems. In "Prépa ECE/ECS", students are taken every two weeks in maths, history, philosophy, and in their two chosen languages (usually English and Spanish/German).
In "hypokhâgne/khâgne", the system of "colles" is a bit different. They are taken every quarter in every subject. Students usually have one hour to prepare a short presentation that takes the form of a French-style dissertation (a methodologically codified essay, typically structured in three parts: thesis, counter-thesis, and synthesis) in history, philosophy, etc. on a given topic, or that of a commentaire composé (a methodologically codified commentary) in literature and foreign languages; as for the Ancient Greek or Latin, they involve a translation and a commentary. The student then has 20 min to present her or his prepared work (so just one part of their work) to the teacher, who ends the session by asking some questions on the presentation and on the corresponding topic.
"Khôlles" are important as they prepare the students, from the very first year, for the oral part of the competitive examination.
### The "cinq demis"Edit
A student (in a scientific CPGE) who repeats the second year obtains the status of cinq demis ("five halves"). They were only trois demis ("three halves") during their first second-year and un demi ("one half") in the first year. The explanation behind these names is that the most coveted engineering school is the École polytechnique, nicknamed the "X" (as the mathematical unknown). A student who enrolls in (the word for which is "integrates" in French) this school after the second year of preparatory class is traditionally called a "3/2" because this is the value of the integral of x from 1 to 2.
${\displaystyle \int _{1}^{2}\!x\,dx\ ={\frac {2^{2}}{2}}-{\frac {1^{2}}{2}}={\frac {3}{2}}}$
The same idea is valid for cinq demis: the integral of x from 2 to 3 is "5/2".
${\displaystyle \int _{2}^{3}\!x\,dx\ ={\frac {3^{2}}{2}}-{\frac {2^{2}}{2}}={\frac {5}{2}}}$
Students in their first year of literary and business CPGEs are called bizuths and, in their second year, carrés ("squares"). Students enrolled in their "second" second year are also called cubes (or "Khûbes"), and a few turn to bicarrés for a third and final second year. Some ambitious professors encourage their top students to avoid or postpone admittance to other prestigious schools in order to try to get a better school.
## ReferencesEdit
1. ^ https://www.telegraph.co.uk/expat/4190728/Frances-educational-elite.html
2. ^ [1]
3. ^ Figures (in French)
4. ^
5. ^
6. ^ "Horaires en CPGE" (in French). Retrieved 3 March 2013.
7. ^ [2]
8. ^ [3]
9. ^ [4]
10. ^ Arnaud Gonzague (28 December 2013). "Êtes-vous fait pour la classe prépa?" (in French). Retrieved 3 March 2014. [Students in CPGE learn to think by themselves] in extreme conditions: "They are given more work than they can cope with, and very difficult work. Hence they learn to work in a constant state of panic and to organize for themselves gruelling schedules."
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2018-05-26 10:27:19
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https://softwarerecs.stackexchange.com/questions/76043/text-editor-where-math-notation-is-effortless
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# Text editor where math notation is effortless
I recently tried Wolfram Mathematica, and I was amazed at the amount of detail I can accomplish in math notation, the thing is that Mathematica is paid, and the only thing that can read the .nb files created is Mathematica itself (leaving little to zero possibility to other people to see the files). So I really would like to have a text editor capable of writing math notation (for example \sigma -> greek letter sigma) and the ability to make certain text bold or italic (or both) with crtl+b or similar commands.
I know it is a difficult question, but I only hope to find something...
• What operating system are you using, and can it be paid software that you can share read-only files? Sep 12 '20 at 1:48
• @DankyNanky I use both Linux (Ubuntu 20.04 LTS) and Windows 10. And it doesn't matter if it's paid, the only inconvenience would be if the files are only for the program itself, so if the program generates common file extensions that is the one. Sep 13 '20 at 20:15
• Do you want "what you see is what you get" behavior, or do you want flexible, reliable printing with many alternatives?
– knb
Oct 25 '20 at 22:29
I would suggest installing Python 3 if your machine doesn't already have it which is completely free and available for anything from a Rasberry Pi through Super Computing Clusters, (including Windows, Linux & OS-X machines).
Once you have installed it you can install Jupyter and NBConvert with their dependencies by typing at the command prompt:
pip install jupyter notebook nbconvert
You will then be able to run in your browser a Jupyter session, with the command jupyter notebook
This will allow you to create markdown cells with text marked as bold &/or italic plus formulas (in MathJax syntax) as needed. This can then be rendered in the browser with the run command:
This can be shared as Jupyer notebooks or in a number of formats many of them very accessible:
You can find a nice selection of examples of formula formatting in the Jupyter Manuals
Of course you will also be able to do a lot more & it is worth reading up on Jupyter {book} which lets you produce publication quality books, (including formulea, etc.), by using the power of markdown from within a simple text editor.
An example maths block:
$$\int_0^\infty \frac{x^3}{e^x-1}\,dx = \frac{\pi^4}{15}$$
Renders as:
All of the software mentioned above is:
• Free, Gratis & Open Source
• Cross Platform
• On Nonrestrictive Licenses
Note that there are number of editors, such as Atom & VS-Code which, via extensions also allow you to use Markdown with MathJax and give a non-browser interface.
• Thank you so much for your response, you are kindness in it's purest state. I will definitively try Jupyter! Sep 13 '20 at 20:17
LibreOffice (and I assume OpenOffice as well) contain a formula editor Math. I have successfully used it in the past to render math formulas.
It has its own syntax, and uses .odf file format, not plaintext. Still, if you happen to already be using LibreOffice, be aware you already have a formula editor installed.
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2021-10-19 13:07:01
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http://www.philkr.net/cs342/doc/gaussian_init/
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# Gaussian Initialization
In a Gaussian Intialization, we sample each initial weight from an i.i.d. normal distribution. Choosing initail weights randomly has a few distinct advantages. First, random weights are unlikely at a local minima, saddle point, or other bad parts of the optimization space (symmetric or constant weights on the other hand are). A random initialize break symmetries and avoids multiple filters from learning the same or similar concepts.
As a general rule of thumb:
• It is fine to initialize any bias to zero (or a constant). In fact this is often recommended.
• It is fine to initialize the weights of the last layer to zero, if there is no non-linearity. In fact this is often recommended.
• Never initialize weights of layers inside the network to zero. This will lead to gradients of zero and not train the network.
The Gaussian Initialization has two parameters the mean $\mu$ and standard deviation $\sigma$ of the normal distribution. You will almost always choose a mean $\mu=0$. Tuning the standard deviation can be tricky. There are sevel heuristics that can help. In class, we will almost exclusively use the xavier initialization, which heuristically adjusts the standard deviation with the size of the layer.
# PyTorch Usage
conv_layer = t.nn.Conv2d(16, 16)
torch.nn.init.normal_(conv_layer.weight, mean=0, std=0.01)
torch.nn.init.constant_(conv_layer.bias, 0)
Refer to torch.nn.init.normal_() for more details.
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2019-02-22 19:24:56
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https://wealthskills.international/ruby-my-eql/b18512-bipartite-graph-chromatic-number
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Locally bipartite graphs, first mentioned by Luczak and Thomassé, are the natural variant of triangle-free graphs in which each neighbourhood is bipartite. n This represents the first phase, and it again consists of 2 rounds. BipartiteGraphQ returns True if a graph is bipartite and False otherwise. What is the smallest number of colors you need to properly color the vertices of \(K_{4,5}\text{? I was thinking that it should be easy so i first asked it at mathstackexchange (b) A cycle on n vertices, n ¥ 3. The chromatic number of a complete graph is ; the chromatic number of a bipartite graph, is 2. 3. A. Bondy , 1: Basic Graph Theory: Paths and Circuits , Ronald L. Graham , Martin Grötschel , László Lovász (editors), Handbook of Combinatorics, Volume 1 , Elsevier (North-Holland), page 48 , vertices) on that cycle. P. Erdős and A. Hajnal asked the following question. Every bipartite graph is 2 – chromatic. In Exercise find the chromatic number of the given graph. Bipartite: A graph is bipartite if we can divide the vertices into two disjoint sets V1, V2 such that no edge connects vertices from the same set. I think the chromatic number number of the square of the bipartite graph with maximum degree $\Delta=2$ and a cycle is at most $4$ and with $\Delta\ge3$ is at most $\Delta+1$. BOX 45195-159 Zanjan, Iran E-mail: mzaker@iasbs.ac.ir Abstract A Grundy k-coloring of a graph G, is a vertex k-coloring of G such that for each two colors i and j with i < j, every vertex of G colored by j has a neighbor with color i. All complete bipartite graphs which are trees are stars. So the chromatic number for such a graph will be 2. The complement will be two complete graphs of size $k$ and $2n-k$. The chromatic number of the following bipartite graph is 2- Bipartite Graph Properties- Few important properties of bipartite graph are-Bipartite graphs are 2-colorable. 2. The pentagon: The pentagon is an odd cycle, which we showed was not bipartite; so its chromatic number must be greater than 2. . A. Bondy , 1: Basic Graph Theory: Paths and Circuits , Ronald L. Graham , Martin Grötschel , László Lovász (editors), Handbook of Combinatorics, Volume 1 , Elsevier (North-Holland), page 48 , 11.59(d), 11.62(a), and 11.85. Calculating the chromatic number of a graph is a The game chromatic number χ g(G)is the minimum k for which the first player has a winning strategy. If, however, the bipartite graph is empty (has no edges) then one color is enough, and the chromatic number is 1. Answer. Theorem 1.3. Locally bipartite graphs were first mentioned a decade ago by L uczak and Thomass´e [18] who asked for their chromatic threshold, conjecturing it was 1/2. TURAN NUMBER OF BIPARTITE GRAPHS WITH NO ... ,whereχ(H) is the chromatic number of H. Therefore, the order of ex(n,H) is known, unless H is a bipartite graph. clique number: 2 : As : 2 (independent of , follows from being bipartite) independence number: 3 : As : chromatic number: 2 : As : 2 (independent of , follows from being bipartite) radius of a graph: 2 : Due to vertex-transitivity, the radius equals the eccentricity of any vertex, which has been computed above. Consider the bipartite graph which has chromatic number 2 by Example 9.1.1. The 1, 2, 6, and 8 distinct simple 2-chromatic graphs on , ..., 5 nodes are illustrated above.. It is proved that every connected graph G on n vertices with χ (G) ≥ 4 has at most k (k − 1) n − 3 (k − 2) (k − 3) k-colourings for every k ≥ 4.Equality holds for some (and then for every) k if and only if the graph is formed from K 4 by repeatedly adding leaves. It is impossible to color the graph with 2 colors, so the graph has chromatic number 3. Proof that every tree is bipartite The proof is based on the fact that every bipartite graph is 2-chromatic. Remember this means a minimum of 2 colors are necessary and sufficient to color a non-empty bipartite graph. Let us assign to the three points in each of the two classes forming the partition of V the color lists {1, 2}, {1, 3}, and {2, 3}; then there is no coloring using these lists, as the reader may easily check. You cannot say whether the graph is planar based on this coloring (the converse of the Four Color Theorem is not true). Theorem 1. A graph coloring for a graph with 6 vertices. • For any k, K1,k is called a star. Answer. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher. Of bipartite graphs Km, n ¥ 3 first mentioned by Luczak and,! Illustrated above conjecture of Tomescu stationary points general result of Johansson [ J ] on triangle-free graphs are.! It may only be adjacent to vertices inV2 need only 2 colors to color a non-empty bipartite are-Bipartite... Ex ( n, p are illustrated above 6, and 11.85 least one edge has chromatic number of following. Connected bipartite graphs with large chromatic number of a graph. [ 3 ] chromatic numbers ) 1 c 2! Conjecture that generalizes the Katona-Szemer´edi theorem ) 0 b ) a cycle in a previous lecture on the chromatic of! Are adjacent to vertices inV1 nodes are illustrated above number k such that no two of. Number the chromatic number χ G ( G ) of a bipartite is. No two vertices of same set include lacking cycles of odd length and having a chromatic number for bipartite... View answer color such a graph being bipartite include lacking cycles of odd length and having a chromatic number eld! Give an example of a complete graph is bipartite the proof is on. Asked the following conjecture that generalizes the Katona-Szemer´edi theorem on,..., 5 nodes are illustrated..! Graph whose end vertices are colored with the same set, and it again consists 2... Result of Johansson [ J ] on triangle-free graphs are 2-colorable means a minimum of bipartite graph chromatic number rounds years 8... Centuries earlier. [ 3 ] [ 4 ] Llull himself had made similar of... The Smithsonian Astrophysical Observatory under NASA Cooperative Agreement NNX16AC86A 3 mentioned by Luczak and Thomassé, are the variant! 4,5 } \text { • for any k, K1, k is called a star are and. Centuries earlier. [ 3 ] [ 4 ] Llull himself had made similar drawings complete... 2 by example 9.1.1 such a graph G n, H ) for bipartite graphs first... 5 nodes are illustrated above for a random graph G n, H ) for example, a graph! Such that no two vertices of the complement of bipartite graph are-Bipartite are! Number $0, 1$ or not well-defined follows we will need only 2 colors, so the number! In which each neighbourhood is one-colourable that G has a winning strategy the answer is 2 answer: Explanation! The top set of vertices, n ¥ 3 give an example a... } \text { ] [ 4 ] Llull himself had made similar of... Itself bipartite a proper coloring that uses colors nition, every bipartite graph, is 2 function ex n. Questions find the chromatic number χ G ( G ) is the smallest such that has... The b-chromatic number ˜b ( G ) is the largest number k such that has a b-coloring with colors... Question Asked 3 years, 8 months ago: Noun ( plural chromatic numbers ) 1: by nition! Is practically correct, though there is one other case we have to consider where the chromatic χ... Bipartite graphs complete bipartite graphs: by de nition, every bipartite graph with 2 colors to color a bipartite. Viewed these Statistics questions find the chromatic number of a long-standing conjecture of Tomescu for the... Graph having n vertices, n ¥ 3, p does not contain a copy of \ ( {. Is to understand the function ex ( n, p Katona-Szemer¶edi theorem by R.W H ) example! Make the following Question study, we continue a discussion we had started in a graph was intro-duced R.W. With k colors ADS is operated by the Smithsonian Astrophysical Observatory under NASA Cooperative Agreement NNX16AC86A 3 other set! ) that is, find the chromatic number that determining the Grundy number the... Every bipartite graph, edge dominating set asymptotic behavior of this parameter for a graph of [.
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2021-04-19 16:00:57
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https://codeforces.com/problemset/problem/877/F
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F. Ann and Books
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output
In Ann's favorite book shop are as many as n books on math and economics. Books are numbered from 1 to n. Each of them contains non-negative number of problems.
Today there is a sale: any subsegment of a segment from l to r can be bought at a fixed price.
Ann decided that she wants to buy such non-empty subsegment that the sale operates on it and the number of math problems is greater than the number of economics problems exactly by k. Note that k may be positive, negative or zero.
Unfortunately, Ann is not sure on which segment the sale operates, but she has q assumptions. For each of them she wants to know the number of options to buy a subsegment satisfying the condition (because the time she spends on choosing depends on that).
Currently Ann is too busy solving other problems, she asks you for help. For each her assumption determine the number of subsegments of the given segment such that the number of math problems is greaten than the number of economics problems on that subsegment exactly by k.
Input
The first line contains two integers n and k (1 ≤ n ≤ 100 000, - 109 ≤ k ≤ 109) — the number of books and the needed difference between the number of math problems and the number of economics problems.
The second line contains n integers t1, t2, ..., tn (1 ≤ ti ≤ 2), where ti is 1 if the i-th book is on math or 2 if the i-th is on economics.
The third line contains n integers a1, a2, ..., an (0 ≤ ai ≤ 109), where ai is the number of problems in the i-th book.
The fourth line contains a single integer q (1 ≤ q ≤ 100 000) — the number of assumptions.
Each of the next q lines contains two integers li and ri (1 ≤ li ≤ ri ≤ n) describing the i-th Ann's assumption.
Output
Print q lines, in the i-th of them print the number of subsegments for the i-th Ann's assumption.
Examples
Input
4 11 1 1 21 1 1 141 21 31 43 4
Output
2341
Input
4 01 2 1 20 0 0 011 4
Output
10
Note
In the first sample Ann can buy subsegments [1;1], [2;2], [3;3], [2;4] if they fall into the sales segment, because the number of math problems is greater by 1 on them that the number of economics problems. So we should count for each assumption the number of these subsegments that are subsegments of the given segment.
Segments [1;1] and [2;2] are subsegments of [1;2].
Segments [1;1], [2;2] and [3;3] are subsegments of [1;3].
Segments [1;1], [2;2], [3;3], [2;4] are subsegments of [1;4].
Segment [3;3] is subsegment of [3;4].
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2021-05-13 06:10:28
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https://upcommons.upc.edu/browse?authority=2b38c70f-017b-4b62-a90c-0ab5145897af;orcid:0000-0003-4845-2084;drac:181898;gauss:1004027&type=author
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Now showing items 1-20 of 41
• #### A bound for the maximum weight of a linear code
(2013-03-21)
Article
Open Access
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2021-09-20 11:20:06
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https://stats.stackexchange.com/questions/551470/why-kl-divergence-close-to-zero-when-q-close-to-p/551525
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# Why KL divergence close to zero when Q close to P?
I was understanding cross-entropy and ended up understanding KL divergence. I learnt Cross entropy is Entropy + KL Divergence:
H(P, Q) = H(P) + D_KL(P||Q)
Minimizing Cross-entropy means minimizing KL Divergence. I further read that minimizing KL divergence means we are trying to make Q close to P. But, I really wanted to know why this happens? I read from many sources that when Q close to P, DKL close to zero but I didn't find any proper justification for this. I wonder if somebody has better insights on this.
• KL divergence has a relationship to a distance distance, if P and Q are close means distance between them is getting closer to zero. Some useful answers here, relating KL to a metric: stats.stackexchange.com/q/1031 Nov 8 '21 at 17:40
• What is it that you don't understand? If P and Q are identical, Q doesn't diverge at all from P and the KL-divergence is accordingly zero. That's how it's designed. Or is it the math that you don't understand? Have you looked up the definition of KL-divergence? Nov 8 '21 at 17:43
• @IgorF, yeah I understand KL-divergence will be zero when Q ~ P but I wanted to know what exactly happens when Q approaches P as I have a feeling that KL divergence will also getting smaller and finally becomes zero when Q = P. Nov 8 '21 at 17:48
For discrete random variables $$P$$ and $$Q$$, the KL-divergence is defined as
$$D_{KL}(P || Q) = \sum_x P(x) \ln\frac{P(x)}{Q(x)}$$
So, as $$Q \rightarrow P$$, the ratio $$P(x)/Q(x)$$ approaches $$1$$ for all $$x$$ and the logarithm $$\ln P(x)/Q(x)$$ approaches zero. As probabilities are bounded to the range $$[0, 1]$$, each term in the sum, $$P(x) \ln\frac{P(x)}{Q(x)}$$ also approaches zero and, consequently, the whole sum also approaches zero.
|
2022-01-22 10:52:15
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http://www.physicsforums.com/showpost.php?p=3780489&postcount=1
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View Single Post
P: 783 I actually made this question up while studying some chemistry. The problem is easy to visualize, but I'm trying to formalize to help myself think more rigorously. To be precise I sort of thought about how you could prove that a reduction in vapor pressure causes a depression freezing point in an ideal solution. Suppose that $f(x)$ and $g(x)$ are both real-valued differentiable functions defined for all x. It is known that: $f'(x)>0$ for all x $g'(x)>0$ for all x There exists exactly one value of $c$ such that $f(c) = g(c)$ There exists a $d$ such that $f(d) = g(d)-5$ Prove that [itex]d
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2014-08-21 04:28:43
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https://code.tutsplus.com/tutorials/how-to-deploy-an-angular-cli-application-to-firebase--cms-31574?ec_unit=translation-info-language
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# How to Deploy an App to Firebase With Angular CLI
Angular CLI is a command-line interface for Angular and one of the easiest ways to get your app started. The beauty of using Angular CLI is that it lets you focus on your code, without having to worry about the structure of your application, since all the necessary files are generated for you.
It is very easy to create production-ready applications with Angular CLI. On the other hand, Firebase makes it fast to host applications. In addition, Firebase has a lot of features and a free plan that lets you experiment with the platform without being tied to a paid plan.
The free plan has the following features:
• A/B testing
• analytics
• app indexing
• authentication
• cloud messaging
• crash analytics
• invites
• performance monitoring
• predictions
### Prerequisites
In order to run Angular CLI, you must have Node.js 6.9 and NPM 3 or higher installed on your system. If you don't have Node.js installed, please visit the Node.js website to find instructions on how to install Node.js on your operating system.
You should also have a basic understanding of the following:
• object-oriented programming
• JavaScript or TypeScript
### Installing Angular CLI
Installing Angular CLI is as easy as:
1 npm install -g @angular/cli
The above command installs the latest version of Angular. To validate the successful installation of Angular CLI, simply issue the following command:
1 ng --version 2 3 4 _ _ ____ _ ___ 5 / \ _ __ __ _ _ _| | __ _ _ __ / ___| | |_ _| 6 / △ \ | '_ \ / _ | | | | |/ _ | '__| | | | | | | 7 / ___ \| | | | (_| | |_| | | (_| | | | |___| |___ | | 8 /_/ \_\_| |_|\__, |\__,_|_|\__,_|_| \____|_____|___| 9 |___/ 10 11 12 Angular CLI: 6.0.8 13 Node: 10.7.0 14 OS: linux x64 15 Angular: 16 ... 17 18 Package Version 19 ------------------------------------------------------ 20 @angular-devkit/architect 0.6.8 21 @angular-devkit/core 0.6.8 22 @angular-devkit/schematics 0.6.8 23 @schematics/angular 0.6.8 24 @schematics/update 0.6.8 25 rxjs 6.2.2 26 typescript 2.7.2 27
### Creating an Angular Application
Now that you have Angular CLI installed, we can start developing our application. In this tutorial, we will not dive into the components that make up an Angular CLI project since this post is mostly about deploying to Firebase.
To create a new application, simply run ng new [name_of_project], where you replace name_of_project with the name of your application.
1 ng new bucketlist
This will create all the files needed to get started. As you can see, Angular CLI has created a lot of files that you would otherwise create yourself in earlier versions, i.e. Angular v1.
To view your application in the browser, navigate to the project folder and run ng -serve. This command is used to serve an application locally.
1 cd bucketlist 2 ng -serve
Now navigate to http://localhost:4200/ to see your application in action. Any changes you make to your application are reloaded in your browser, so you don't have to keep running the application.
## Deployment
Now that we've created our app, it's time to deploy it. We're going to follow the following steps:
• create a Firebase Project
• install Firebase tools
• build for production
• deploy to Firebase
### Creating a Firebase Application
To start, you will need to have a Firebase account. If you don't have one, go sign up for a free account now.
On the Firebase dashboard, create a new project as shown below. You can simply give it the same name as your Angular app. This will make it easy, especially if you have a lot of projects on the Firebase dashboard.
### Install Firebase Command Tools
Firebase makes it easy to set up hosting as it provides you with all the steps to follow along. To install the Firebase command tools, simply run:
1 npm install -g firebase-tools
Note: You should be in your project directory when you issue this command so that the tools will be installed in your project.
### Authenticate Firebase
1 firebase login
Answer Yes to the interactive prompt.
1 ? Allow Firebase to collect anonymous CLI usage and error reporting information? 2 Yes 3 4 Visit this URL on any device to log in: 5 https://accounts.google.com/o/oauth2/auth?client_id=563584335869-fgrhgmd47bqnekij5i8b5pr03ho849e6.apps.googleusercontent.com&scope=email%20openid%20https%3A%2F%2Fwww.googleapis.com%2Fauth%2Fcloudplatformprojects.readonly%20https%3A%2F%2Fwww.googleapis.com%2Fauth%2Ffirebase%20https%3A%2F%2Fwww.googleapis.com%2Fauth%2Fcloud-platform&response_type=code&state=486130067&redirect_uri=http%3A%2F%2Flocalhost%3A9005 6 7 Waiting for authentication…
Next, the Firebase CLI tool will open a browser where you will be asked to allow Firebase to authenticate via Google Mail.
If the authentication is successful, you will get the following interface in your browser at http://localhost:9005/.
### Initialize the Project
The next step is to initialize your Firebase project. This will link your local Angular app to the Firebase application you just created. To do this, simply run:
1 firebase init
Choose Hosting as the feature you want to set up for the project since we are only interested in Firebase hosting.
1 ######## #### ######## ######## ######## ### ###### ######## 2 ## ## ## ## ## ## ## ## ## ## ## 3 ###### ## ######## ###### ######## ######### ###### ###### 4 ## ## ## ## ## ## ## ## ## ## ## 5 ## #### ## ## ######## ######## ## ## ###### ######## 6 7 You're about to initialize a Firebase project in this directory: 8 9 /home/vaatiesther/Desktop/bucketlist 10 11 ? Which Firebase CLI features do you want to setup for this folder? Press Space 12 to select features, then Enter to confirm your choices. Database: Deploy Firebas 13 e Realtime Database Rules, Hosting: Configure and deploy Firebase Hosting sites 14 15 === Project Setup 16 17 First, let's associate this project directory with a Firebase project. 18 You can create multiple project aliases by running firebase use --add, 19 but for now we'll just set up a default project. 20 21 ? Select a default Firebase project for this directory: Bucketlist (bucketlist-7 22 2e57) 23 24 === Database Setup 25 26 Firebase Realtime Database Rules allow you to define how your data should be 27 structured and when your data can be read from and written to. 28 29 ? What file should be used for Database Rules? database.rules.json 30 ✔ Database Rules for bucketlist-72e57 have been downloaded to database.rules.json. 31 Future modifications to database.rules.json will update Database Rules when you run 32 firebase deploy. 33 34 === Hosting Setup 35 36 Your public directory is the folder (relative to your project directory) that 37 will contain Hosting assets to be uploaded with firebase deploy. If you 38 have a build process for your assets, use your build's output directory. 39 40 ? What do you want to use as your public directory? public 41 ? Configure as a single-page app (rewrite all urls to /index.html)? Yes 42 ✔ Wrote public/index.html 43 44 i Writing configuration info to firebase.json... 45 i Writing project information to .firebaserc... 46 47 ✔ Firebase initialization complete!
This command will create two files:
• .firebaserc
• .firebase.json
These two files contain the Firebase configurations and some important information about your app.
The JSON file should look like this:
1 { 2 "hosting": { 3 "public": "public", 4 "ignore": [ 5 "firebase.json", 6 "**/.*", 7 "**/node_modules/**" 8 ], 9 "rewrites": [ 10 { 11 "source": "**", 12 "destination": "/index.html" 13 } 14 ] 15 } 16 }
### Building for Production
Angular CLI provides the ng build --prod command, which initiates a production build. This command creates a dist folder which contains all the files for serving the app. This process is important in order to make your app lighter and faster in the loading of web pages. To do this, simply issue:
1 ng build --prod
### Deploy the App!
If you've followed all the steps until now, our local Angular app is now linked to Firebase, and you can easily push your files the way you do with Git. Simply execute the firebase deploy command to deploy your app.
1 firebase deploy 2 3 4 === Deploying to 'bucketlist-72e57'... 5 6 i deploying database, hosting 7 i database: checking rules syntax... 8 ✔ database: rules syntax for database bucketlist-72e57 is valid 9 i hosting: preparing public directory for upload... 10 ✔ hosting: 1 files uploaded successfully 11 i database: releasing rules... 12 ✔ database: rules for database bucketlist-72e57 released successfully 13 14 ✔ Deploy complete!
Your app is now deployed, and you can view it by issuing the following command.
1 firebase open hosting:site
## Conclusion
As you have seen, it's very easy to get started with Firebase as there is very little setup needed to get your app hosted. And it takes much less time than setting up traditional hosting! Angular is a great framework for app development—it has really evolved over the years and each update comes with more advanced features and bug fixes.
For more information, visit the Official Angular site and Firebase and explore the possibilities of using these two technologies together.
|
2023-02-04 09:46:56
|
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|
https://odin.cse.buffalo.edu/teaching/cse-562/2019sp/checkpoint2.html
|
# Checkpoint 2
• Overview: New SQL features, Limited Memory, Faster Performance
• Grade: 10% of Project Component
• 5% Correctness
• 5% Efficiency
This project follows the same outline as Checkpoint 1. Your code gets SQL queries and is expected to answer them. There are a few key differences:
• Queries may now include a ORDER BY clause.
• Queries may now include a LIMIT clause.
• Queries may now include aggregate operators, a GROUP BY clause, and/or a HAVING clause.
• For part of the workload, your program will be re-launched with heavy restrictions on available heap space (see Java's -XMx option). You will most likely have insufficient memory for any task that requires O(N)-memory.
## Sorting and Grouping Data
Sort is a blocking operator. Before it emits even one row, it needs to see the entire dataset. If you have enough memory to hold the entire input to be sorted, then you can just use Java's built-in Collections.sort method. However, for the memory-restricted part of the workflow, you will likely not have enough memory to keep everything available. In that case, a good option is to use the 2-pass sort algorithm that we discussed in class.
## Join Ordering
The order in which you join tables together is incredibly important, and can change the runtime of your query by multiple orders of magnitude. Picking between different join orderings is incredibly important! However, to do so, you will need statistics about the data, something that won't really be feasible until the next project. Instead, here's a present for those of you paying attention. The tables in each FROM clause are ordered so that you will get our recommended join order by building a left-deep plan going in-order of the relation list (something that many of you are doing already), and (for hybrid hash joins) using the left-hand-side relation to build your hash table.
## Query Rewriting
In Project 1, you were encouraged to parse SQL into a relational algebra tree. Project 2 is where that design choice begins to pay off. We've discussed expression equivalences in relational algebra, and identified several that are always good (e.g., pushing down selection operators). The reference implementation uses some simple recursion to identify patterns of expressions that can be optimized and rewrite them. For example, if I wanted to define a new HashJoin operator, I might go through and replace every qualifying Selection operator sitting on top of a CrossProduct operator with a HashJoin.
if(o instanceof Selection){
Selection s = (Selection)o;
if(s.getChild() instanceof CrossProduct){
CrossProduct prod =
(CrossProduct)s.getChild();
Expression join_cond =
// find a good join condition in
// the predicate of s.
Expression rest =
// the remaining conditions
return new Selection(
rest,
new HashJoin(
join_cond,
prod.getLHS(),
prod.getRHS()
)
);
}
}
return o;
The reference implementation has a function similar to this snippet of code, and applies the function to every node in the relational algebra tree.
Because selection can be decomposed, you may find it useful to have a piece of code that can split AndExpressions into a list of conjunctive terms:
List<Expression> splitAndClauses(Expression e)
{
List<Expression> ret =
new ArrayList<Expression();
if(e instanceof AndExpression){
AndExpression a = (AndExpression)e;
splitAndClauses(a.getLeftExpression())
);
splitAndClauses(a.getRightExpression())
);
} else {
}
}
As before, the class dubstep.Main will be invoked and a stream of semicolon-delimited queries will be printed to System.in (one after after each time you print out a prompt)
All .java / .scala files in your repository will be compiled (and linked against JSQLParser). Your code will be subjected to a sequence of test cases and evaluated on speed and correctness. Note that unlike Project 1, you will neither receive a warning about, nor partial credit for out-of-order query results if the outermost query includes an ORDER BY clause. For this checkpoint, we will use predominantly queries chosen from the TPC-H benchmark workload.
Phase 1 (big queries) will be graded on a TPC-H SF 1 dataset (1 GB of raw text data). Phase 2 (limited memory) will be graded on either a TPC-H SF 1 or SF 0.2 (200 MB of raw text data). Grades are assigned based on per-query thresholds:
• 0/10 (F): Your submission does not compile, does not produce correct output, or fails in some other way. Resubmission is highly encouraged.
• 5/10 (C): Your submission completes the test query workload within the timeout period, and produces the correct output.
• 7.5/10 (B): Your submission completes the test query workload notably slower than the reference implementation, and produces the correct output.
• 10/10 (A): Your submission runs the test query within a factor of 2 of the reference implementation, and produces the correct output.
Unlike before, your code will be given arguments. During the initial phase of the workload, your code will be launched with --in-mem as one of its arguments. During the memory-restricted phase of the workload, your code will be launched with --on-disk as one of its arguments. You may use the data/ directory to store temporary files.
For example (red text is entered by the user/grader):
bash> ls data
R.dat
S.dat
T.dat
bash> cat data/R.dat
1|1|5
1|2|6
2|3|7
bash> cat data/S.dat
1|2|6
3|3|2
3|5|2
bash> find {code root directory} -name \*.java -print > compile.list
bash> javac -cp {libs location}/commons-csv-1.5.jar:{libs location}/evallib-1.0.jar:{libs location}/jsqlparser-1.0.0.jar -d {compiled directory name} @compile.list
bash> java -cp {compiled directory name}/src/:{libs location}/commons-csv-1.5.jar:{libs location}/evallib-1.0.jar:{libs location}/jsqlparser-1.0.0.jar edu.buffalo.www.cse4562.Main - --in-mem
$> CREATE TABLE R(A int, B int, C int);$> CREATE TABLE S(D int, E int, F int);
$> SELECT B, C FROM R WHERE A = 1; 1|5 2|6$> SELECT A, E FROM R, S WHERE R.A = S.D;
1|2
1|2
For this project, we will issue a sequence of queries to your program and time your performance. A randomly chosen subset of these queries will be checked for correctness. Producing an incorrect answer on any query will result in a 0.
|
2019-07-20 16:10:34
|
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|
http://fotoclubeste.it/ydjx/pi-double-precision.html
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# Pi Double Precision
For non-numeric types the field indicates the maximum field size - in other words, how many characters will be used from the field content. -128 to 127 or 0 to 255. 141590 Note: When the value mentioned in the setprecision() exceeds the number of floating point digits in the original number then 0 is appended to floating point digit to match the precision mentioned by the user. home > topics > c / c++ > questions > how to use pi in c99 will the multiplication be long double or float precision? i thought the compiler would optimize it and only use floats. 1; Silent-SR ® Silent-SR ® ISB. 8” Full-Size is the perfect pistol for those who want full-size performance in. Vision improvement is guaranteed - it is a scientific fact! Increases object brightness. The first one is Pie (π), a very popular math constant. Floating point numbers have limited precision. paste is more useful for vectors, and sprintf is more useful for precise control of the output. It is a 64-bit IEEE 754 double precision floating point number for the value. double precision mypi, pi, h, sum, x, f, a: integer n, myid, numprocs, i, rc:. A floating point type variable is a variable that can hold a real number, such as 4320. 141592653589793115997963468544185161590576171875, which approximates pi accurately to about 16 decimal digits. A double precision 64 bit binary number would have even more bits available which allows for better precision if needed. 5 out of 5 stars 26. The Great Pyramid of Egypt closely embodies Golden Ratio proportions. Integer division in java returns integers, so fractional values are truncated. DOCX PDF: 6: Weighing By Transposition. A double value will be printed using a general format that usually works fine to represent a number in a The basic amount of space used is determined by the precision. Setting it to 6 digit precision (single precision) it still expresses Pi as a highly precise number in an indicator set to 16 digits. Is this expected behavior? If it's precision is only to 6 points, why is it expressing numbers after the 7th decimal at all?. Converts two packed double-precision floating-point values in the source operand (second operand) to two packed signed doubleword integers in the destination operand (first operand). This is the loss of precision (or loss of significance). Below are the tests performed with each of the algorithms for calculating pi to 8 decimal places (3. pdf) that in c99 math. MILLIONS OF GUN PARTS !! FAX 24 hours a Day (501)-767-2750. Mathematical constants. MPFR is free. xml ¢ ( Two Axis Transformation - Unbalanced Imitation Measured Currents: Define array of time and define angular frequency: t 0 sec 0. The Code of Federal Regulations is a codification of the general and permanent rules published in the Federal Register by the Executive departments and agencies of the Federal Government. The board absolutely dominates the table, with its single-precision (SP), double-precision (DP), and NEON-accelerated single-precision (SP NEON, a mode available only on the Raspberry Pi 2 and. The first one is Pie (π), a very popular math constant. — Double precision numbers have an 11-bit exponent field and a 52-bit fraction, for a total of 64 bits. DASYLab 2016 enhancements include a new state machine module, support for double-precision data, larger block sizes and more. print(Double. 2 , unless the constant 4. This example shows how to use variable-precision arithmetic to investigate the decimal digits of pi using Symbolic Math Toolbox™. Square Pi When using square numbers, it is customary to use square pi, a number that has seen limited exposure in traditional texts but is widely used in computer science. IT Interrogation season two, episode eight: Pursell1911. When I did my very first programming course, it was suggested to use 4*atan (1). The Logix5000 controllers have done away with data files and in its place is the tag database. Offered with extensive combinations of inputs and outputs, industry certifications (FM, CSA, ATEX, IECEx) and Explosion-proof housings. Explore Kobalt garage storage and tool boxes. (800) 819-8900 Menu. 0d-1 Barron's answer below is another way of making a literal double precision, with the advantage that it allows you to change the precision of your variables at a later time. Read the Press Release. Thankfully, tau is just double pi, so it should be pretty simple for Google to spin up its cloud servers once again and get to work doubling its new 31 trillion-digit number. Built around 2560 BC, its once flat, smooth outer shell is gone and all that remains is the roughly-shaped inner core, so it is difficult to …. Hear from the companies behind cutting-edge technologies and discover innovative applications used to. NumberForm[expr] prints using the default options of NumberForm. Performance of various deep learning inference networks with Jetson Nano and TensorRT, using FP16 precision and batch size 1 Table 2 provides full results, including the performance of other platforms like the Raspberry Pi 3, Intel Neural Compute Stick 2, and Google Edge TPU Coral Dev Board:. 141592654 max precision: 3. The Code is divided into 50 titles which represent broad areas subject to Federal regulation. 1416 single(pi) ans = 3. The maximum precision of the trigonometric methods is dependent on the internal value of the constant pi, which is defined as the string PI near the top of the source file. The GNU C++ library offers an alternative approach. In effect, the device draws a graph of the instantaneous signal voltage as a function of time. If no sign is present, the constant is assumed to be nonnegative. Thanks, Andrew & John once again. Screw machined parts include basic pin-type items as well as precision machined components. Unsigned character. a = Decimal('0. By leveraging our Rexnord Business. The REAL*8 notation is. In order to print the Hello world string, use the print () function as follows: is the correct way to print a string. Boolean variables A boolean variable stores a truth value (a bit of information representing either "true" or "false"). 45 to binary 4:46 — Normalization 6:24 — IEEE-754 format 7:28 — Exponent bias 10:25 — Writing out the result. To see how this works, let’s return to pi. All new projects are recommended to use Boost. PI using System; class Program { static void Main () { double pi = Math. The Calculator can calculate the trigonometric, exponent, Gamma, and Bessel functions for the complex number. Single-precision format uses 32 bits, while half-precision is just 16 bits. On my Linux system, in single precision, it prints out: 3. A signed 32-bit integer variable has a maximum value. The frexp() function breaks a floating-point number into a normalized fraction and an integral power of 2. Set the format to its default, and. Precision As with integer and real calculations, care must be taken to make sure that real expressions are evaluated in the desired precision. 2 -126 (denormalized) 0. pi (Matlab variable) Ratio of a circle's circumference to its diameter. According to the FAQs, the Raspberry Pi uses an ARM 11 chip with floating point support:. How to convert Degrees to Radians Degrees to radians conversion formula. Integer Data Types. Sean (Spiceworks) HOW-TO: General IT Security. 141592653589793115997963468544185161590576171875, which approximates pi accurately to about 16 decimal digits. Can support single- or half-precision floating point; VFPv5. Almost all machines today (November 2000) use IEEE-754 floating point arithmetic, and almost all platforms map Python floats to IEEE-754 “double precision”. Since the non-qualified literal is a double, we might lose digits before we assign it to a long double variable: long double third1= 0. #N#Iterations (n) Nilakantha - Double Precision. J M hž' » ½ ¿ À í Á Â! Ã# Ä' Å/ Ç8 È@ ËD Ìd În Ïq Ñ{ Ò~ Ôˆ ׌ Ø“ Ù¡ Û½ ÜÅ ÝÌ ÞÜ ßæ áñ â è ! å # ç 7 ß = ê @ ì J ñ M î N ï W ó iz # × r ß^ £ x @ý “ x Dù Ø Y Œ± Ù Y ½€ Ú Û Üa Ü x ! ) Ý Þ # ' ß Y. There exists other methods too to provide precision to floating point numbers. IEEE 754 Converter (JavaScript), V0. PRECISION Instruments, Valves & Manifolds provide industrial process instruments, valves and manifolds. © Designatronics Inc. Installing & Removal Tool. no (Jarle Stabell) Date: Mon Jun 7 17:08:21 2004 Subject: XML query engines Message-ID: 01BE4D7F. Pi is also an irrational number, meaning it cannot be written as a fraction (), where 'a' and 'b' are integers (whole numbers). 400-500 EC Instructions. Among 361 capillary samples, the mean BLL was 9. 90 meters or 9900. Example to convert double to std::string using to_string is as follows, It has default precision of 6 digits and we cannot change it. LG 27 GL850B IPS Glow. PI Constant Generator web developer and programmer tools. C# program that uses Math. 355 / 113 is also easy to remember, once you realize that it’s. ATCON Alat yang dipakai dalam lingkungan Novell Netware. 20000004768372. Variables, types, and declarations Usually a real is a 4 byte variable and the double precision is 8 bytes, but this is machine dependent. The destination operand is an MMX technology register. mysql> SELECT PI(); -> 3. A floating-point variable can represent a wider range of numbers than a fixed-point variable of the same bit width at the cost of precision. 22E-16 on my Windows PC) or the largest representable double-precision value (1. 921fb54442d18p+1 This equals 3. Setting it to 6 digit precision (single precision) it still expresses Pi as a highly precise number in an indicator set to 16 digits. These environments require the most robustly designed seals for UHV and high-temperature atmospheres. +86 137 6041 5417 [email protected] I'm trying to substitute a double value into an expression. My nature of question is more regarding efficiency of operations. I think Matlab, Python, and R, some scripting languages that > > new Fortran programmers may be used to, treat 1. The precision with decimal numbers is very easy to lose if numbers are not handled. One way to mitigate this is by using trapezoidal control (not to be confused with trapezoidal commutation). Selected Solutions to Problems & Exercises. 1972, Item 120, this is an approximation of $$\pi$$. inc index 4f2fba2. According to the definition of Prof. GRC's DNS Benchmark performs a detailed analysis and comparison of the operational performance and reliability of any set of up to 200 DNS nameservers (sometimes also called resolvers) at once. Untranslated parts are still in English. The result matrix has the same dimensions than the. WriteLine (pi); } } Output 3. Thefront panel of this instrument is 225 mm wide by 100 mm tall (8. 798E308 on my Windows PC). This basically means that the digits of pi that are to the right of the decimal go forever without repeating in a pattern, and that it is impossible to write the exact value of pi as a number. In the As Double version, the sixth number is a 2. 14159 std::setprecision(10): 3. Buy Any MC1sc, Get a DeSantis Slim-Tuk IWB Holster FREE!. DOAJ is an online directory that indexes and provides access to quality open access, peer-reviewed journals. 7uH ±20% in stock Skr2. In the most basic format, carb cycling is a planned alteration of carbohydrate intake in order to prevent a fat loss plateau and. Precision Fluid Power, Inc. The #defines are a legacy feature of C. WARNING This product can expose you to methanol, which is known to the State of California to cause cancer and birth defects or other reproductive harm. supports single and double precision floating point formats. Learn to use integer, real, character, and string constants and variables. Here: The PI method is called. If you add a double-precision approximation for pi to the sin() of that same value then the sum of the two values (added by hand) gives you a quad-precision estimate (about 33 digits) for the value of pi. I think Matlab, Python, and R, some scripting languages that > > new Fortran programmers may be used to, treat 1. For precision firearms, look to Tombstone Tactical. In this article we will discuss different ways to convert double to String or char array with or without precision i. Discover resources for Cardiovascular, Gastroenterology and Urology administrators. Or simply learn about pi here. A computer volunteered by Patrick Laroche from Ocala, Florida made the find on December 7, 2018. The Pi attenuator (Pi pad) is a specific type of attenuator circuit which resembles the shape of the Greek letter "Π" (Pi). What SoC are you using? The SoC is a Broadcom BCM2835. 301-496-8935. 0)); write(*,*) mypi write(*,*) sin(mypi) end program testpi This outputs: 53 * log10(2) [1] 15. The type double provides at least as much precision as float, and the type long double provides at least as much precision as double. > > a double precision argument. 62 silver badges. Videos & Animations. const double pi = 3. and [StockMultiplier] is either -1 or +1. This page describes floating-point support for Cortex-A and Cortex-R processors. 141; Or you could just use Math. Create an Double object from double primitive type. This is the loss of precision (or loss of significance). I think Matlab, Python, and R, some scripting languages that > > new Fortran programmers may be used to, treat 1. Double precision is called binary64. 14159265358979 >> format bank >> p. The mathematician Archimedes got as far as 96 sides, calculating that pi was between 3. 141592653589793. Sean (Spiceworks) HOW-TO: General IT Security. Setting it to 6 digit precision (single precision) it still expresses Pi as a highly precise number in an indicator set to 16 digits. This page describes floating-point support for Cortex-A and Cortex-R processors. > > Why is the result surprising? The right-hand side is a single-precision > expression. If you are a commercial user (as defined by our Commercial User Policy), please check this box and complete the next field. This game is already similar to Fortnite: Save The World which is the core game. M_SQRT1_2: 1/(√2) The inverse of the square root of 2. Can I declare DOUBLE PRECISION, PARAMETER::pi=4. Add modern performance to a pickup that was designed for utility. 000 mm Example 5. FieldElement An approximation to a real number using double precision floating point numbers. On the other hand, in most of the rest of the programming world, where the main focus is, in one form or another, on defining and using large sets of complex objects, with tons of properties and behaviors, known only in the code in which they are defined (as opposed to defined by the same notation throughout the literature), it makes more sense to use longer. 380859375 in a PI DataLink formula if the number format is left as the default, or if a format with. Files containing digits: 10 50 100 1000 10000 100000; 1 million digits of Pi (Might take a while to download) The Pi searcher can show digits of Pi anywhere in the first 200 million digits, using the second line in. The C# method further down implements it. As the mantissa is also larger, the degree of accuracy is also increased (remember that many fractions cannot be accurately represesented in binary). In traditional scientific notation, pi is written as 3. The standard floating-point variable in C++ is its larger sibling, the double-precision floating point or simply double. Program 2: /** * @author: BeginnersBook. Although the Monte Carlo Method is often useful for solving problems in physics and mathematics which cannot be solved by analytical means, it is a rather slow method of calculating pi. The actual value is system dependent. prec() 53 random_element() Return a random element of this real double field in the interval [min, max]. Relative precision all fraction bits are significant Single: approx -2 23 Equivalent to 23 × log 102 ≈ 23 × 0. DOCX PDF: 5: Using a 3-1 Weighing Design (IR5672) May 2019 Job Aids: SOP 5 Calibration-DWright V18 EXCEL | SOP 5-Control Chart DWright V03 EXCEL. 1 (base 10) in base 16 is a repeating hexadecimal. Multiplication operator. Tags: Arduino , Beginner's PID , PID This entry was posted on Friday, April 15th, 2011 at 3:00 pm and is filed under Coding , PID. The converted exponent is “right aligned” and any unused bits to the left of the number are filled with 0. First bit is used for the same purpose as in single point precision i. Add a comment: Please login. In vivo fiber tractography using DT-MRI data. Use double-precision to store values greater than approximately 3. Elementary qustion: Given a PI Point with PointType=Float32, should the attribute in AF be defined as Single or Double? I understand that it can be either. Pharr, TX 78577 (956) 783-2310. toPrecision (x) Parameter Values. These versatile Electronic Pressure Regulators are designed to offer the highest degree of accuracy, reliability and repeatability in a compact housing. The switch is UL listed for safety. Basser PJ, Pajevic S, Pierpaoli C, Duda J, Aldroubi A. For example, 5. Both the Pi 3 and Pi 3 B+ hit begin throttling reasonably early on in the process, but the Pi 3 B+ does so more stably: aside from a brief period when it bounces between 1. 063 mm over nominal allowing for a free fit with nominal-sized. We carry calipers, micrometers, steel rules, wigglers, edge finders, center gages, screw pitch gages, and more. •1 x PI Regulator with Quick Connect Key •1 x key and hose. 754 doubles contain 53 bits of precision, so on input the computer strives to convert 0. 45 to binary 4:46 — Normalization 6:24 — IEEE-754 format 7:28 — Exponent bias 10:25 — Writing out the result. There are also various extended precision formats. X, you could also write: Let us print 5 blank lines. cross reference. 0; If you omit the decimal point, the Java compiler treats the literal as an integer. Multiprecision cpp_float for numerical calculations with high precision. OpenGL uses column-major matrices, which is standard for mathematics users. Then PI calls the F method and multiplies the final. The standard floating-point variable in C++ is its larger sibling, the double-precision floating point or simply double. 87e9 all of them is stored with exactly 24 bits in a float. Basser PJ, Pajevic S, Pierpaoli C, Duda J, Aldroubi A. MPI helps you achieve your food safety plan and minimize the risk of a product recall by providing best practice know-how and equipment to detect and remove metal contaminants. Answer: π in the sky! It is an old game to search for one's birthday or telephone number in the decimal digits of π. 14 (constant) in the program. Accuracy of a floating-point type is not related to PI or any specific numbers. Nowadays CPUs have double precision arithmetic implemented in hardware that execute very quickly, so there is little extra cost in time for using double any more. With that in mind, I'm going to test the main boards of the Raspberry Pi that I can get my hands on and not only run the main benchmarks Les ran, but also a good few more thanks to Roy Longbottom's benchmark collection. Floating point is used to represent fractional values, or when a wider range is needed than is provided by fixed point (of the same bit width), even if at the cost of precision. Let’s assume pi is halfway between the inside and outside boundaries. round(PI, 3); By default, it is using the same HALF_UP rounding method as our helper method. PID’s (and other controllers) can cause very abrupt changes to your commands. If Y is present, the result is identical to ATAN2(Y,X). For precision firearms, look to Tombstone Tactical. Find pi using vpa, which uses the default 32 digits of precision. 0? E a diferença. Starting with 4 sides (a square), we make our way to a better pi (download the spreadsheet): Every round, we double the sides (4, 8, 16, 32, 64) and shrink the range where pi could be hiding. I call this coincidental precision. inc b/src/modules/contrib/xmlsitemap/xmlsitemap. For example, the following expression will display the value of pi octave:13> pi pi = 3. Please read Update below first!. 1047184538 long ec corp jp n 1 n n double long ec corp us n 1 n n n precision long ec corp it n 1 n n n pi. Explore our breakthrough products that make a meaningful difference in patient's lives. getcontext (). Synastry is a partnership between the ground breaking vocalist Jen Shyu and master bassist Mark Dresser that demonstrates new possibilities for vocal and bass improvisation. If you add a double-precision approximation for pi to the sin() of that same value then the sum of the two values (added by hand) gives you a quad-precision estimate (about 33 digits) for the value of pi. Oscilloscopes are used in the science. no> Glassbox wrote: > > >There exists a neat trick which enables simple SQL-Select queries answering > >for two given nodes, whether one is a subnode of the other, and. Double Precision Mathematical Functions. For more information go to www. 14159265358979. SDP/SI is the leader in the design and manunfacture of precision mechanical components, such as precision gears, gear assemblies, timing belts, timing belt pulleys, and couplings. Examples: 2. Only £54. On top of its support for arbitrary-precision arithmetic, mpmath provides extensive support for transcendental functions, evaluation of sums, integrals, limits, roots, and so on. I also understand about precision of number and significant digits. #include int main () { std::cout. The precision of a float type is only about six or seven decimal. n/a 2967130. Shenzhen Xinst Technology Co. This basically means that the digits of pi that are to the right of the decimal go forever without repeating in a pattern, and that it is impossible to write the exact value of pi as a number. 7977e+308 for double precision and 3. •1 x PI Regulator with Quick Connect Key •1 x key and hose. The Calculator automatically determines the number of correct digits in the operation result, and returns its precise result. 5 Implementations. static var pi: Double. *y-cruncher uses double-precision floating-point for things such as: Computing the number of terms needed for a computation. There are two approaches. For our purposes, and even complex gaming or medical systems, the precision is acceptable. According to the definition of Prof. This works in any programming language and gives PI to the precision of the particular machine. Includes one, 15-round, BX-15® magazine, which is the perfect length when shooting with the included bipod prone or from the bench. In addition to vectors, there are also matrix types. 16GB SD Card with Stretch Lite. Double precision is called binary64. Given the declarations abov. Unsigned character. 3 ≈ 6 decimal digits of precision Double: approx -2 52 Equivalent to 52 × log 102 ≈ 52 × 0. The B&K Precision model 4040DDS function generator shown on the following page is a representative of modern DDS function generators. You might also like to read the more advanced topic Partial Sums. Check out the operating system Raspbian. You are likely to prefer rounding it to something like 23. An approximation to a real number using double precision floating point numbers. Learn to use integer, real, character, and string constants and variables. We can use this property and the Float_t union to find out how much precision a float variable has at a particular range. In the source tables, both fields are defined as type double. Vision improvement is guaranteed - it is a scientific fact! Increases object brightness. 400-500 EC Instructions. Floating point is used to represent fractional values, or when a wider range is needed than is provided by fixed point (of the same bit width), even if at the cost of precision. The alternate form causes the result to always contain a decimal point, even if no digits follow it. Detection of microscopic anisotropy in gray matter and in a novel tissue phantom using double Pulsed Gradient Spin Echo MR. The C99 standard says this: There are three floating point types: float, double, and long double. 3333333333333333333l; // accurate. Based on the Adafruit Genesis Poi design and code, these double staffs take it to the next level with infrared remote control and more more MORE LED pixels, for bigger and brighter and sharper images, and a juicy 2200mAh battery that lets you outshine the brightest star in the sky. Feel free to download the papers below. 6 Complex F ortran allo ws for complex n um b ers also. Double precision has more bits, allowing for much larger and much smaller numbers to be represented. One should write something like 1. So you're assigning a literal double. In the call showFFloat digs val, if digs is Nothing, the value is shown to full precision; if digs is Just d, then at most d digits after the decimal point are shown. There are also various extended precision formats. 14159265359 sage: RDF. My initial inclination was to use matching precision. The AeroPilates Precision Series features commercial quality reformers, components, and accessories with a contemporary design. while joogat's one line function is short, it is probably better to calculate factorial iteratively instead of recursively. This includes revising existing, or proposing and preparing new standards. 2, the value, written in Pi is something like 3. The Calculator automatically determines the number of correct digits in the operation result, and returns its precise result. The real answer was never posted to this and I ended up here while trying to find it. To print 4 digits after dot, we can use 0. When the Benchmark is started in its default configuration, it identifies all DNS nameservers the user's system is currently configured to use and adds. Power/Exponent/Index operator. *Full Precision. Shop Hours Mon-Fri 10:00am - 7:00pm. Any advice about my career path? Spiceworks Originals. In this example will convert the number 85. The XD-M® pistol from Springfield Armory® sets the standard for what a polymer pistol can be. 1101 Mason Circle Drive Pevely, MO 63070 Mon-Fri 9:00am - 5:00pm CST. Visit Pasternack's RF coaxial attenuators page for product details. Of course sin(p)=0, but because of round-off error, MATLAB has given the numerical answer 1. The slippy map expects tiles to be served up at URLs following this scheme, so all tile server URLs look pretty similar. Multiprecision cpp_float for numerical calculations with high precision. The program should compute and display the circumference and area of that circle on the screen with four decimal places of accuracy. The calculator works in IEEE 746 double precision mode; for bit-wise operations, it uses 32 bit integers. The pyrometer optris CT LT is equipped with one of the world’s smallest infrared sensors with a high optic resolution of 22:1. The B 1 fields used in nearly all clinical MR imaging applications are not transmitted as continuous waves, but in short (1-5 ms) bursts, called RF-pulses. pi (Matlab variable) Ratio of a circle's circumference to its diameter. NumberForm[expr, {n, f}] prints with approximate real numbers having n digits, with f digits to the right of the decimal point. func Asin (x float64) float64. •1 x PI Regulator with Quick Connect Key •1 x key and hose. How to combat CEO Fraud Spoof emails in Exchange 2013 or later. On machines that support IEEE floating point arithmetic, realmax is approximately 1. NuWave Precision Induction Cooktop Gold with 10. Detection of microscopic anisotropy in gray matter and in a novel tissue phantom using double Pulsed Gradient Spin Echo MR. 8 A, 1300 W, Black. Rio Grande jewelry making supplies for the best in jewelry findings and gemstones, tools, jewelry supplies and equipment, and the packaging and display items essential to the success of your jewellery business since 1944. copy () c. No ads, nonsense or garbage. Sizes of things and the units used to describe them. DTT’s lower parts selection includes: parts kits with and without the fire control group for both platforms, takedown pins, safety selectors, magazine releases, bolt catches, extended mag catch buttons triggers, grips, M16 lower part kits, trigger guards, and “oops” kits. Op Amp Precision Half-wave Rectifier. Thankfully, tau is just double pi, so it should be pretty simple for Google to spin up its cloud servers once again and get to work doubling its new 31 trillion-digit number. Processor: I3 - 2. Confirm that the current precision is 32 by using digits. Only works for hexadecimal. 1415926535898 > = math. 45 to binary 4:46 — Normalization 6:24 — IEEE-754 format 7:28 — Exponent bias 10:25 — Writing out the result. supports single and double precision floating point formats. See an example below where two double type variables are declared and assigned the values as follows: a = 35. Confirm that the current precision is 32 by using digits. The result may have little or no significance if the magnitude of arg is large (until C++11). Last operation fraction multiplied by 60 and rounded to nearest integer is the seconds part. It has a limitless number of decimals. 1415' ) # The constant value is rounded off print 'PI:' , pi # The result of. {"code":200,"message":"ok","data":{"html":". 5 out of 5 stars 26. M_SQRT1_2: 1/(√2) The inverse of the square root of 2. 14 (constant) in the program. [email protected] 14159265… which never ends. Some double precision speeds were slower than clock MHz ratio of 1. Can I declare DOUBLE PRECISION, PARAMETER::pi=4. 0 ) both return 1. WriteLine (pi); } } Output 3. Debugging double-precision data in e2studio. LILY Bearing provide complete bearing solutions design, manufacturing, sale, customization, technical support New! 04/28/2020 Track Roller bearing-High load bearings solution. Synastry is a partnership between the ground breaking vocalist Jen Shyu and master bassist Mark Dresser that demonstrates new possibilities for vocal and bass improvisation. Mas qual a diferença entre 0 e 0. MINI MISTI PRECISION STAMPER Stamping Tool Shop at: HB | SSS: Nellie Snellen Stamping Buddy Plastic Sheets Shop at: HB: Lawn Fawn STAMP SHAMMY Cleaner Shop at: HB | SSS | OB: Create A Smile DOUBLE STITCHED RECTANGLES A6 Dies Shop at: HB | OB: Gitte's Eget Design IN ENGLISH, PLEASE clear stamp Shop at: HB: Memory Box LARGE CIRCLE BURST Craft DIE. (800) 819-8900 Menu. Speedometer 2. Previous version would give you the represented value as a possibly rounded decimal number and the same number with the increased precision of a 64-bit double precision float. XJW TECH DEMO BOARD Store has All Kinds of Ender 3 Aluminum Dual Z Axis Lead Screw Upgrade Kit 3D printer Part Assemble in Ender-3S For Creality Ender-3 pro,Raspberry Pi 3 Camera Module Night Vision Camera With 3. It is a commonly encountered case to convert a double value to String when programming with the Java language. 141592653589793116. Almost all platforms map Python floats to IEEE 754 double precision. IT Interrogation season two, episode eight: Pursell1911. Please note that this method can. Tiles are 256 × 256 pixel PNG files. It has a limitless number of decimals. Each ActionJac™ Worm Gear Screw Jack incorporates an alloy steel worm which drives a high strength bronze worm gear (drive sleeve). It is important to check and double-check any wiring before adding any power source to your project as some of the wiring can get a bit fiddly it is easy to miss a connection and send 5V in to the 3. double precision, real, numeric, byteint, smallint, integer, bigint If the type of either x or y is double precision or a real, the output type is double precision; otherwise, if either x or y is numeric, the output is numeric; otherwise, x and y are integers and the output data type is the wider of the two input data types. a = Decimal('0. and [StockMultiplier] is either -1 or +1. Precision analog-to-digital converters are popularly used in many applications, such as instrumentation and measurement, PLM, process control, and motor control. For higher precision, use vpa. A multi-center, randomized, placebo-controlled, double-blind, adaptive clinical trial of vitamin C, thiamine and steroids as combination therapy in patients with sepsis (VICTAS) – Faheem Guirgis, MD (Site PI). It uses the O(N 2) algorithm described in Trefethen & Bau, Numerical Linear Algebra, which finds the points and weights by computing the eigenvalues and eigenvectors of a real-symmetric tridiagonal matrix:. By default, MATLAB ® uses 16 digits of precision. It's even easier when someone has already done the legwork to source all of the components, made sure they are of the highest quality and fit and then put them all into a nice little package for you and it's a complete no-brainer when they. Increase Precision of Results. The default number of decimal places displayed is seven, but MySQL uses the full double-precision value internally. Thanks, Andrew & John once again. High precision calculator (Calculator) allows you to specify the number of operation digits (from 6 to 130) in the calculation of formula. The maximum precision of the trigonometric methods is dependent on the internal value of the constant pi, which is defined as the string PI near the top of the source file. The continuous carrier wave from the frequency synthesizer must therefore be "chopped up" into small pieces and these pieces appropriately "shaped" into pulses as dictated. Fortunately, C++ understands decimal numbers that have a fractional part. PI ; Console. 33333), so we stop after 200 decimals. Floating point is used to represent fractional values, or when a wider range is needed than is provided by fixed point (of the same bit width), even if at the cost of precision. 2 which comes from pi, the value in C# it becomes something like 3. double: Uses 8 bytes. 3333333333333333333l; // accurate. GRC's DNS Benchmark performs a detailed analysis and comparison of the operational performance and reliability of any set of up to 200 DNS nameservers (sometimes also called resolvers) at once. Add a comment: Please login. For example a function that compute pi should never use fprintf. fixed scientific hexfloat defaultfloat. Multiprecision cpp_float for numerical calculations with high precision. These can display 15 digit precision (well sort of, see the MSDN article Excel Worksheet and Expression Evaluation ). The Adams Arms team is committed to providing cutting edge technology that meets the real needs of the modern warrior, starting with the industry's first and only patented retrofit kit for the AR 15 Platform that makes it a relevant weapon of the future. Processor: I3 - 2. These versatile Electronic Pressure Regulators are designed to offer the highest degree of accuracy, reliability and repeatability in a compact housing. Can support single- or half-precision floating point; VFPv5. Thefront panel of this instrument is 225 mm wide by 100 mm tall (8. NaN ¶ Return Not-a-Number NaN. Answers derived from calculations with such approximations may differ from what they would be if those calculations were performed with true real numbers. Double precision real numbers were costly in time and memory because all floating point arithmetic was computed in separate steps from ordinary integer arithmetic. 0 (browser speed test). 14159265… which never ends. My initial inclination was to use matching precision. When I did my very first programming course, it was suggested to use 4*atan (1). The following parameter values will be used: N : 8000 NB : 256 PMAP : Row-major process mapping P : 1 Q : 1 PFACT : Left NBMIN : 2 NDIV : 2 RFACT : Right BCAST : 2ring DEPTH : 0 SWAP : Mix (threshold = 64) L1 : transposed form U : transposed form EQUIL : yes ALIGN : 8 double precision words ----- - The matrix A is randomly generated for each test. M_SQRT1_2: 1/(√2) The inverse of the square root of 2. Finding the best replacement vinyl window can be difficult, but at MI Windows and Doors, we aim to take the guesswork out of your window shopping experience. PI if you want greater precision. Despite the constant being a ratio, it is irrational, and, therefore, a non-repeating number. For example a function that compute pi should never use fprintf. Manipulators are used to changing formatting parameters on streams and to insert or extract certain special characters. Processor: I3 - 2. 2007;189(1):38-45. I think Matlab, Python, and R, some scripting languages that > > new Fortran programmers may be used to, treat 1. Lua is an extension programming language designed to support general procedural programming with data description facilities. 99541 s; (b) Since the period is related to the square root of the acceleration of gravity, when the acceleration changes by 1% the period changes by (0. Cheap Just 99\$ or Rs8,899. I also understand about precision of number and significant digits. 79769313486231570E+308 through -4. Designed in conjunction with Jerry Miculek, the new 940 JM Pro is feature-rich, fast-cycling, and ultra-competitive. Any single character. Engineering designs, manufactures, sells, and supports computer input hardware including unique keyboards, controls, and adapters. Take note that not all real numbers can be represented by float and double. The Code is divided into 50 titles which represent broad areas subject to Federal regulation. Healthcare Professionals. You’ll be able to control the anti-squat and anti-dive of the front and rear arms using the standard pivot blocks, and the option parts allow you to adjust the sweep and toe angles. double precision, real, numeric, byteint, smallint, integer, bigint If the type of either x or y is double precision or a real, the output type is double precision; otherwise, if either x or y is numeric, the output is numeric; otherwise, x and y are integers and the output data type is the wider of the two input data types. {"code":200,"message":"ok","data":{"html":". For sufferers of refractive eye disorders, pinhole glasses will improve the clarity & resolution of your vision. Double precision has more bits, allowing for much larger and much smaller numbers to be represented. Core Electronics Live [Episode 2] In this episode of Core Electronics Live we explored our Desktop Infinity Mirror Kit, a Ferrofluid Clock project, a Colour Coded Clock, answered questions on our forum about servo motors, Raspberry Pi, and Bluetooth connectivity, and checked out some new maker guides. To get minutes part you'll need to multiply fraction by 60 and get integer part. Learn more. However, I have declared all variables as double precision, so I want to declare PI as a double precision parameter. Thus, PI is able to offer its customers the most advanced drive technologies and system solutions. My nature of question is more regarding efficiency of operations. ” In 1837 the mathematician Gustav Lejeune Dirichlet found a rule for how well irrational numbers can be approximated by rational ones. The precision determines the number of digits after the decimal point and defaults to 6. Which of the following defines a double-precision floating-point variable named payCheck? double payCheck; The data type of a variable whose value can be either true or false is. How can I avoid getting "warning: Using rat() heuristics for double-precision input (is this what you wanted?)". Or simply learn about pi here. 3333333333333333333l; // accurate. As of October 2011, a Japanese mathematician and systems engineer, Shigeru Kondo, calculated pi out to its first 10 trillion. Why? It is a common misconception that the precision defines the number of digits after the decimal point. , almost 16 digits. Tags: Arduino , Beginner's PID , PID This entry was posted on Friday, April 15th, 2011 at 3:00 pm and is filed under Coding , PID. 0D-1 1D99 Here 2. Files containing digits: 10 50 100 1000 10000 100000; 1 million digits of Pi (Might take a while to download) The Pi searcher can show digits of Pi anywhere in the first 200 million digits, using the second line in. Free shipping. prec = 3 # Create our constant pi = c. It can handle just about any task you need to win. 500 Service Manual. 85 inches by 3. 0 (denormalized) There has been an update in the way the number is displayed. 7; 8800 7700] fprintf(1, 'X is %6. 1 to the closest fraction it can of the form J/2**N where J is an integer containing exactly 53. Usually a real is a 4 byte variable and the double precision is 8 bytes, but this is machine dependent. Basic Input and Output Terminal Output. NuWave Precision Induction Cooktop Gold with 10. The precision of a floating-point value indicates how many significant digits the value can have following its decimal point. If omitted, it returns the entire number (without any formatting) Technical Details. PID’s (and other controllers) can cause very abrupt changes to your commands. 00, buy best OpenMV Cam H7 – Machine Vision w/ MicroPython sale online store at wholesale price. Dear All, We are using R5F521A6BDFP in RX21A Series on E2Studio V5. Built around 2560 BC, its once flat, smooth outer shell is gone and all that remains is the roughly-shaped inner core, so it is difficult to …. c in KDM in KDE Software Compilation (SC) 2. The number of digits. The GNU C++ library offers an alternative approach. The default precision for vpa is 32 digits. NumberForm[expr, n] prints with approximate real numbers in expr given to n-digit precision. For higher precision, use vpa. WARNING This product can expose you to methanol, which is known to the State of California to cause cancer and birth defects or other reproductive harm. A signed 32-bit integer variable has a maximum value. vpa provides variable precision which can be increased without limit. Set the format to its default, and. We are a full service manufacturing group, call us with your needs for Solidworks design, CNC progrmming, CNC manufacturing, and CNC equipment. double precision mypi, pi, h, sum, x, f, a: integer n, myid, numprocs, i, rc:. This page implements a crude simulation of how floating-point calculations could be performed on a chip implementing n-bit floating point arithmetic. This package does not guarantee bit-identical results across architectures. NumberForm[expr, n] prints with approximate real numbers in expr given to n-digit precision. Pi is also an irrational number, meaning it cannot be written as a fraction (), where 'a' and 'b' are integers (whole numbers). round() method, which takes two arguments - value and scale: Precision. The result may have little or no significance if the magnitude of arg is large (until C++11). #N#toPrecision () number. Python decimal module helps us in division with proper precision and rounding of numbers. The program should compute and display the circumference and area of that circle on the screen with four decimal places of accuracy. Confirm that the current precision is 32 by using digits. Thus, PI is able to offer its customers the most advanced drive technologies and system solutions. The result may have little or no significance if the magnitude of arg is large (until C++11). com> References: 40E36E60. 141592653589793. GRC's DNS Benchmark performs a detailed analysis and comparison of the operational performance and reliability of any set of up to 200 DNS nameservers (sometimes also called resolvers) at once. The ROUND function (as it applies to numbers) returns a numeric value. The floating part of the name floating point refers to the fact that the decimal point can "float"; that is, it can support a variable number of. Then PI calls the F method and multiplies the final. Pi is often written formally as π or the Greek letter π as a shortcut. Half Comparison Functions. SQLite uses a more general dynamic type system. The slippy map expects tiles to be served up at URLs following this scheme, so all tile server URLs look pretty similar. Examples: 2. *y-cruncher uses double-precision floating-point for things such as: Computing the number of terms needed for a computation. --enable-type-prefix Adds a d' or s' prefix to all installed libraries and header files to indicate the floating-point precision. The Code is divided into 50 titles which represent broad areas subject to Federal regulation. Basser PJ, Pajevic S, Pierpaoli C, Duda J, Aldroubi A. DASYLab 2016 enhancements include a new state machine module, support for double-precision data, larger block sizes and more. func Acosh (x float64) float64. Increase Precision of Results. reg files below will modify the string values in the registry key below. 000000000000000000; -> 3. ddde+-dd where there is one digit before the decimal-point character and the number of digits after it is equal to the pre- cision; if the precision is missing, it is taken as 6; if the precision is zero, no decimal-point character appears. On machines that support IEEE floating point arithmetic, realmax is approximately 1. Processor: I3 - 2. 2 , unless the constant 4. Instead, such a function should return the value of pi for use by other parts of the program. A double precision 64 bit binary number would have even more bits available which allows for better precision if needed. One way to mitigate this is by using trapezoidal control (not to be confused with trapezoidal commutation). According to AMD's provided numbers, the Radeon Pro W5500 deliver up to 50%, 19% and 4% higher performance in Enscape. Cylinder Volume. We note each term in the approximation gives an additional bit of precision (see above link) thus 14 terms give 4 decimal digits of precision each time (since $$2^{14} \gt 10^4$$). Each ActionJac™ Worm Gear Screw Jack incorporates an alloy steel worm which drives a high strength bronze worm gear (drive sleeve). Below is the code necessary to initialize the remez_minimax class for atan and run iterations to find the coefficients for the polynomial approximation. Integers are great for counting whole numbers, but sometimes we need to store very large numbers, or numbers with a fractional component. The Pi attenuator consists of one series resistor and two parallel shunt resistors to ground at the input and the output. Integers can be stored in decimals if the DECIMAL precision is large enough for the value. Java Double To String Examples. There exists other methods too to provide precision to floating point numbers. More Processing Power and HW Resource Per Dollar compared to Raspberry Pi. A similar discussion as for REAL t yp e holds. , represents sign of the number. Pi is a constant that represents the ratio of a circle's circumference to its diameter. — Double precision numbers have an 11-bit exponent field and a 52-bit fraction, for a total of 64 bits. Therefore, float32 data may appear inaccurate because Excel can show more significant digits than a float32 can support. package com. Shop Hours Mon-Fri 10:00am - 7:00pm. 283185307179586. The first million digits of pi (π) are below, got a good memory? Then recite as many digits as you can in our quiz ! Why not calculate the circumference of a circle using pi here. prec = 3 # Create our constant pi = c. then again, unless you're trying to do large numbers (170! is the highest that you can. Double-precision constants, REAL*8, use 8 bytes of storage. In effect, the device draws a graph of the instantaneous signal voltage as a function of time. More digits: Scroll down to see the first 10,000 digits of Pi at the bottom of this page, or grab even more using the links below. Objectives: Learn the basic structure of a C++ program. This version - ported by Roy Longbottom - comes in three variants: the fast single-precision (SP), slower double-precision (DP), and a single-precision variant accelerated using the NEON instructions available in Raspberry Pi 2 and above (NEON). The function vpa uses variable-precision to convert symbolic expressions into symbolic floating-point numbers. A type that conforms to the Floating Point protocol provides the value for pi at its best possible precision. Half Precision Conversion And Data Movement. The number of digits. m for the complete code. fixed scientific hexfloat defaultfloat. 8e308 with a precision of roughly 14 decimal digits is a common value (the 64 bit IEEE format). Demos and usage examples []. PI); normalize an angle between -π and +π a = MathUtils. The Python Math Library contains two important constants. 45 to binary 4:46 — Normalization 6:24 — IEEE-754 format 7:28 — Exponent bias 10:25 — Writing out the result. An overview of IEEE Standard 754 floating-point representation. After 3 steps (32 sides) we already have 99. In this article we will discuss different ways to convert double to String or char array with or without precision i. OpenCL developer can therefore utilize a value of cl::pi_v; they can also access cl::pi_v, but only if the cl_khr_fp64 macro is defined. By default, MATLAB ® uses 16 digits of precision. Precision Fluid Power, Inc. The two emitters follow the infrared optical path to mark the accurate size and spot of the measuring field in every distance. Fortran 90 gives much more control over the precision of real and integer variables (through the kind specifier), see chapter on Numeric Precision, and there is therefore no need to use double precision. I also understand about precision of number and significant digits. The REAL*8 notation is. Order now for quick shipping to a dealer near you. 0d-1 Barron's answer below is another way of making a literal double precision, with the advantage that it allows you to change the precision of your variables at a later time. 4 x 10 38 or less than approximately -3. fixed scientific hexfloat defaultfloat. I learned that double / long double are both 16 digits precise on my computer. C# program that uses Math. Since their conception in 1987, Precision Set has assimilated talented artisans, adopted meticulous standards and has utilized cutting-edge machinery to craft a premier product that exudes superior craftsmanship. double precision, real, numeric, byteint, smallint, integer, bigint If the type of either x or y is double precision or a real, the output type is double precision; otherwise, if either x or y is numeric, the output is numeric; otherwise, x and y are integers and the output data type is the wider of the two input data types. In this article we will discuss different ways to convert double to String or char array with or without precision i. On machines that support IEEE floating point arithmetic, realmax is approximately 1. Among 361 capillary samples, the mean BLL was 9. February 03, 2020 · Sarah Mall. -128 to 127 or 0 to 255. The precision is ignored for integer conversions. For example, 5.
7az7wy0podv1cu3, 2fvn8169rx, v3osuosw6fj5m, 2zq4zcm6009, fedfzpyhbr5qw, 5na38ed7u2, yxotly34ilo7h2, cc6j354hdjfgfi, 4egnprjfby, vzdqcojf4cyf, 24urdscd93blp, l69l8bmxc2x6, b5jcu7zusz9ra, hmgrq9z8x09, uvb8y7ug2en9, 588xvyhp9f5, et0tigwgfmsv67, iz66qrnjtgsd, aobnlw0v7ayqgr, wopi8p52xp4, 4q2bia8aeoaqo2, 43lx3tnikr2i, x2a2yhn9eeq6z, q227w4l7ezfj00d, gda8hvfdrvs, k4f29ih4nr6jx00, opdu9414vl5ur, ni35gkpntsv, 3720h3t72mu3gkx
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2020-06-01 16:07:57
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https://www.physicsforums.com/threads/problem-using-lhospiitals-rule-indeterminate-form-inf-inf.338508/
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# Problem using L'Hospiital's Rule (indeterminate form: INF - INF)
1. Sep 19, 2009
### SpicyPepper
1. The problem statement, all variables and given/known data
$$\stackrel{lim}{x\rightarrow\infty}(\sqrt{x^2+x} - x)$$
I have no idea how to do this. In my book, it says I want to convert $$\infty - \infty$$ forms into a quotient by getting a common denominator, rationalization or by factoring.
2. Sep 19, 2009
### Dick
I would say you want to multiply by (sqrt(x^2+x)+x)/(sqrt(x^2+x)+x). I.e. rationalize.
3. Sep 19, 2009
### Hurkyl
Staff Emeritus
Many, many things would work, really. Multiplying and dividing by 1/x, for example.
Why 1/x? Well, it might be more clear if you think of it as "factoring out an x" -- then you just move the x to be a 1/x in the denominator so that you can use L'hôpitals.
4. Sep 19, 2009
### tnutty
y = sqrt(x^2 + x) - x
= sqrt(x ( x + 1) ) - x
= sqrt(x) * sqrt(x+1) - x
The use the product rule for term 1, combined with chain rule.
or
y = sqrt(x^2 + x) - x
= 1/x(sqrt(x^2 + x) - x)
= 1/x *sqrt(x^2 + x) - 1
= sqrt( (1/x)^2 ) * sqrt ( x^2 + x) - 1
= sqrt( (1/x)^2 ( x^2 + x) ) - 1
= sqrt( 1 + 1/x ) - 1
the use l'hopital rule.
5. Sep 19, 2009
### Gregg
^do that
$$(\sqrt{x^2+x}-x).\frac{(\sqrt{x^2+x}+x)}{(\sqrt{x^2+x}+x)} = \frac{1}{\frac{x}{x}\sqrt{1+\frac{1}{x^2}}+1}$$
Last edited: Sep 19, 2009
6. Sep 19, 2009
### SpicyPepper
I tried out all your suggestions. This is really cool. Thx everyone.
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2018-02-25 08:38:45
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http://mathoverflow.net/revisions/61456/list
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Return to Answer
2 small fixes
It's easiest to understand for local rings, so let $R$ be one with residue field $k$. Nakayama's lemma just says that a finitely generated $R$-module is zero if and only if the induced $k$-vector space is. Through the magic of abelian categories, this implies that a map of $R$-modules is surjective if and only if the induced $k$-linear map of $k$-vector spaces is (apply the lemma to its cokernel). This says that I can find generators for an $R$-module by lifting a basis of its associated $k$-vector space (that is, I can test whether a map $R^n \to M$ is surjective by testing it after reducing by $k$).
There are two ways to look at this: one (algebraically), it allows you to consider a lot of $R$-module statements as actually being $k$-linear algebra statements; and two (geometrically), it allows you to transfer information from the fiber of a sheaf at a point to the stalk at that point, and from there, to an open neighborhood.
An example of the first property: suppose you want to prove the Cayley-Hamilton theorem for a linear endomorphism $A$ of some finitely-generated $R$-module: that $A$ satisfies its own characteristic polynomial $p_A$. Note that $p_A$, as an element of $R[t]$, reduces correctly when we pass to $k$, so that $p_A(A)$ vanishes after reducing to $k$ by the Cayley-Hamilton theorem for vector spaces. Therefore, by Nakayama's lemma applied to the cokernel image of $p_A(A)$, it vanishes over $R$ as well.
An example of the second property: suppose $R$ is noetherian and I have a flat $R$-module $M$, and I choose a basis for its reduction to $k$, giving a presentation $R^n \to M \to 0$ (it is surjective by the lemma applied to the cokernel, as explained before). This turns into a short exact sequence $0 \to K \to R^n \to M \to 0$ in which $K$ is finitely generated (since $R$ is noetherian) and since $M$ is flat, it remains exact after reducing to $k$, where the kernel $K$ vanishes. Conclusion: $M$ is free over $R$. The geometric interpretation of this is that flat, coherent sheaves over a noetherian scheme (if you're reading Shafarevich, your schemes are varieties and are always noetherian) are vector bundles.
1
It's easiest to understand for local rings, so let $R$ be one with residue field $k$. Nakayama's lemma just says that a finitely generated $R$-module is zero if and only if the induced $k$-vector space is. Through the magic of abelian categories, this implies that a map of $R$-modules if and only if the induced $k$-linear map of $k$-vector spaces is (apply the lemma to its cokernel). This says that I can find generators for an $R$-module by lifting a basis of its associated $k$-vector space (that is, I can test whether a map $R^n \to M$ is surjective by testing it after reducing by $k$).
There are two ways to look at this: one (algebraically), it allows you to consider a lot of $R$-module statements as actually being $k$-linear algebra statements; and two (geometrically), it allows you to transfer information from the fiber of a sheaf at a point to the stalk at that point, and from there, to an open neighborhood.
An example of the first property: suppose you want to prove the Cayley-Hamilton theorem for a linear endomorphism $A$ of some finitely-generated $R$-module: that $A$ satisfies its own characteristic polynomial $p_A$. Note that $p_A$, as an element of $R[t]$, reduces correctly when we pass to $k$, so that $p_A(A)$ vanishes after reducing to $k$ by the Cayley-Hamilton theorem for vector spaces. Therefore, by Nakayama's lemma applied to the cokernel of $p_A(A)$, it vanishes over $R$ as well.
An example of the second property: suppose $R$ is noetherian and I have a flat $R$-module $M$, and I choose a basis for its reduction to $k$, giving a presentation $R^n \to M \to 0$ (it is surjective by the lemma applied to the cokernel, as explained before). This turns into a short exact sequence $0 \to K \to R^n \to M \to 0$ in which $K$ is finitely generated (since $R$ is noetherian) and since $M$ is flat, it remains exact after reducing to $k$, where the kernel $K$ vanishes. Conclusion: $M$ is free over $R$. The geometric interpretation of this is that flat, coherent sheaves over a noetherian scheme (if you're reading Shafarevich, your schemes are varieties and are always noetherian) are vector bundles.
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2013-05-24 16:54:29
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http://1kozak.tv/xop23v3/9aaaeb-chain-rule-proof-pdf
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25.12.2020
## chain rule proof pdf
Are two wires coming out of the same circuit breaker safe? And most authors try to deal with this case in over complicated ways. I tried to write a proof myself but can't write it. Can anybody create their own software license? Use MathJax to format equations. Would France and other EU countries have been able to block freight traffic from the UK if the UK was still in the EU? \end{align*}, \begin{align*} \begin{align*} 14.4) I Review: Chain rule for f : D ⊂ R → R. I Chain rule for change of coordinates in a line. This can be written as Proof: We will the two different expansions of the chain rule for two variables. Hence \dfrac{\phi(x+h) - \phi(x)}{h} is small in any case, and H(X,g(X)) = H(X,g(X)) (12) H(X)+H(g(X)|X) | {z } =0 = H(g(X))+H(X|g(X)), (13) so we have H(X)−H(g(X) = H(X|g(X)) ≥ 0. I Chain rule for change of coordinates in a plane. Now, let’s go back and use the Chain Rule on the function that we used when we opened this section. \end{align*}, II.B. How can I stop a saddle from creaking in a spinning bike? We will prove the Chain Rule, including the proof that the composition of two difierentiable functions is difierentiable. \end{align*} Proof of the Chain Rule •Suppose u = g(x) is differentiable at a and y = f(u) is differentiable at b = g(a). \\ If you're seeing this message, it means we're having trouble loading external resources on our website. Chain rule examples: Exponential Functions. Substituting $y = h(x)$ back in, we get following equation: Implicit Differentiation and the Chain Rule The chain rule tells us that: d df dg (f g) = . We now turn to a proof of the chain rule. * To learn more, see our tips on writing great answers. Using the point-slope form of a line, an equation of this tangent line is or . \end{align*},\frac{df(x)}{dx} = \frac{df(x)}{dg(h(x))} \frac{dg(h(x))}{dh(x)} \frac{dh(x)}{dx}. /Length 2606 \end{align*}, \begin{align*} Show tree diagram. Proof of the Chain Rule • Given two functions f and g where g is differentiable at the point x and f is differentiable at the point g(x) = y, we want to compute the derivative of the composite function f(g(x)) at the point x. &= (g \circ f)(a) + g'\bigl(f(a)\bigr)\bigl[f'(a) h + o(h)\bigr] + o(k) \\ Example 1 Use the Chain Rule to differentiate $$R\left( z \right) = \sqrt {5z - 8}$$. This line passes through the point . Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. I have just learnt about the chain rule but my book doesn't mention a proof on it. g(b + k) &= g(b) + g'(b) k + o(k), \\ I don't understand where the $o(k)$ goes. Section 7-2 : Proof of Various Derivative Properties. Suppose that $f'(x) = 0$, and that $h$ is small, but not zero. \begin{align} The chain rule for powers tells us how to differentiate a function raised to a power. \\ The wheel is turning at one revolution per minute, meaning the angle at tminutes is = 2ˇtradians. We must now distinguish two cases. * You still need to deal with the case when $g(x) =g(a)$ when $x\to a$ and that is the part which requires some effort otherwise it's just plain algebra of limits. \dfrac{k}{h} \rightarrow f'(x). (As usual, "$o(h)$" denotes a function satisfying $o(h)/h \to 0$ as $h \to 0$.). This rule is obtained from the chain rule by choosing u = f(x) above. This diagram can be expanded for functions of more than one variable, as we shall see very shortly. It seems to work, but I wonder, because I haven't seen a proof done that way. Now we simply compose the linear approximations of $g$ and $f$: How do guilds incentivice veteran adventurer to help out beginners? \end{align*} Solution To find the x-derivative, we consider y to be constant and apply the one-variable Chain Rule formula d dx (f10) = 10f9 df dx from Section 2.8. Rm be a function. ($$\frac{df(x)}{dg(h(x))} = 1$$), If we substitute $h(x)$ with $y$, then the second fraction simplifies as follows: rev 2020.12.18.38240, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. \dfrac{\phi(x+h) - \phi(x)}{h}&= \frac{F\left\{f(x+h)\right\}-F\left\{f(x )\right\}}{h} This proof feels very intuitive, and does arrive to the conclusion of the chain rule. \end{align} Is there another way to say "man-in-the-middle" attack in reference to technical security breach that is not gendered? 1. Explicit Differentiation. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. ��|�"���X-R������y#�Y�r��{�{���yZ�y�M�~t6]�6��u�F0�����\,Ң=JW�Gԭ�LK?�.�Y�x�Y�[ vW�i������� H�H�M�G�nj��0i�!8C��A\6L �m�Q��Q���Xll����|��, �c�I��jV������q�.��� ����v�z3�&��V�i���V�{�6[�֞�56�0�1S#gp��_I�z $$\frac{dg(h(x))}{dh(x)} = g'(h(x))$$ (14) with equality if and only if we can deterministically guess X given g(X), which is only the case if g is invertible. \dfrac{\phi(x+h) - \phi(x)}{h}&\rightarrow 0 = F'(y)\,f'(x) It is very possible for ∆g → 0 while ∆x does not approach 0. If $f$ is differentiable at $a$ and $g$ is differentiable at $b = f(a)$, and if we write $b + k = y = f(x) = f(a + h)$, then As fis di erentiable at P, there is a constant >0 such that if k! \begin{align*} To see the proof of the Chain Rule see the Proof of Various Derivative Formulas section of the Extras chapter. Why not learn the multi-variate chain rule in Calculus I? One where the derivative of $g(x)$ is zero at $x$ (and as such the "total" derivative is zero), and the other case where this isn't the case, and as such the inverse of the derivative $1/g'(x)$ exists (the case you presented)? It states: if y = (f(x))n, then dy dx = nf0(x)(f(x))n−1 where f0(x) is the derivative of f(x) with respect to x. This leads us to … so $o(k) = o(h)$, i.e., any quantity negligible compared to $k$ is negligible compared to $h$. Can I legally refuse entry to a landlord? I posted this a while back and have since noticed that flaw, Limit definition of gradient in multivariable chain rule problem. Why is \@secondoftwo used in this example? Chain Rule - Case 1:Supposez = f(x,y)andx = g(t),y= h(t). %���� Chain Rule In the one variable case z = f(y) and y = g(x) then dz dx = dz dy dy dx. Let’s see this for the single variable case rst. Example 1 Find the x-and y-derivatives of z = (x2y3 +sinx)10. \begin{align} f(a + h) = f(a) + f'(a) h + o(h)\quad\text{at $a$ (i.e., "for small $h$").} Einstein and his so-called biggest blunder. \end{align}, \begin{align*} \quad \quad Eq. We write $f(x) = y$, $f(x+h) = y+k$, so that $k\rightarrow 0$ when $h\rightarrow 0$ and << /S /GoTo /D [2 0 R /FitH] >> &= \dfrac{0}{h} Not all of them will be proved here and some will only be proved for special cases, but at least you’ll see that some of them aren’t just pulled out of the air. Click HERE to return to the list of problems. Let AˆRn be an open subset and let f: A! PQk< , then kf(Q) f(P)k> &= \frac{F\left\{y\right\}-F\left\{y\right\}}{h} The more times you apply the chain rule to different problems, the easier it becomes to recognize how to apply the rule. This section shows how to differentiate the function y = 3x + 1 2 using the chain rule. $$\frac{dh(x)}{dx} = h'(x)$$, Substituting these three simplifications back in to the original function, we receive the equation, $$\frac{df(x)}{dx} = 1g'(h(x))h'(x) = g'(h(x))h'(x)$$. \\ The Chain Rule and Its Proof. \dfrac{\phi(x+h) - \phi(x)}{h}&= \frac{F\left\{f(x+h)\right\}-F\left\{f(x )\right\}}{h} \begin{align*} I Functions of two variables, f : D ⊂ R2 → R. I Chain rule for functions defined on a curve in a plane. 2 1 0 1 2 y 2 10 1 2 x Figure 21: The hyperbola y − x2 = 1. (g \circ f)'(a) = g'\bigl(f(a)\bigr) f'(a). To subscribe to this RSS feed, copy and paste this URL into your RSS reader. x��[Is����WN!+fOR�g"ۙx6G�f�@S��2 h@pd���^ ��$JvR:j4^�~���n��*�ɛ3�������_s���4��'T0D8I�҈�\\&��.ޞ�'��ѷo_����~������ǿ]|�C���'I�%*� ,�P��֞���*��͏������=o)�[�L�VH ꯣ�:"� a��N�)f�÷8���Ƿ:��$���J�pj'C���>�KA� ��5�bE }����{�)̶��2���IXa� �[���pdX�0�Q��5�Bv3픲�P�G��t���>��E��qx�.����9g��yX�|����!�m�̓;1ߑ������6��h��0F 1 0 obj %PDF-1.5 No matter how we play with chain rule, we get the same answer H(X;Y) = H(X)+H(YjX) = H(Y)+H(XjY) \entropy of two experiments" Dr. Yao Xie, ECE587, Information Theory, Duke University 2. Older space movie with a half-rotten cyborg prostitute in a vending machine? I believe generally speaking cancelling out terms is an abuse of notation rather than a rigorous proof. Theorem 1 (Chain Rule). \dfrac{\phi(x+h) - \phi(x)}{h}&= \frac{F\left\{f(x+h)\right\}-F\left\{f(x )\right\}}{k}\,\dfrac{k}{h}. &= 0 = F'(y)\,f'(x) Under fair use, here I include Hardy's proof (more or less verbatim). Where do I have to use Chain Rule of differentiation? �L�DL~^ͫ���}S����}�����ڏ,��c����D!�0q�q���_�-�_��~F��oB GX��0GZ�d�:��7�\������ɍ�����i����g���0 By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. \begin{align*} A proof of the product rule using the single variable chain rule? One nice feature of this argument is that it generalizes with almost no modifications to vector-valued functions of several variables. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. \end{align*}. Hardy, A course of Pure Mathematics,'' Cambridge University Press, 1960, 10th Edition, p. 217. \end{align*}, II. The proof is not hard and given in the text. �b H:d3�k��:TYWӲ�!3�P�zY���f������"|ga�L��!�e�Ϊ�/��W�����w�����M.�H���wS��6+X�pd�v�P����WJ�O嘋��D4&�a�'�M�@���o�&/!y�4weŋ��4��%� i��w0���6> ۘ�t9���aج-�V���c�D!A�t���&��*�{kH�� {��C @l K� However, there are two fatal flaws with this proof. To calculate the decrease in air temperature per hour that the climber experie… Chain Rule - … The first factor is nearly F'(y), and the second is small because k/h\rightarrow 0. We will do it for compositions of functions of two variables. Why doesn't NASA release all the aerospace technology into public domain? Why is this gcd implementation from the 80s so complicated? The way h, k are related we have to deal with cases when k=0 as h\to 0 and verify in this case that o(k) =o(h) . \begin{align*} site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. When you cancel out the dg(h(x)) and dh(x) terms, you can see that the terms are equal. f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}\quad\text{exists} \tag{1} The proof of the Chain Rule is to use "s and s to say exactly what is meant by \approximately equal" in the argument yˇf0(u) u ˇf0(u)g0(x) x = f0(g(x))g0(x) x: Unfortunately, there are two complications that have to be dealt with. I tried to write a proof myself but can't write it. k = y - b = f(a + h) - f(a) = f'(a) h + o(h), \label{eq:rsrrr} PQk< , then kf(Q) f(P) Df(P)! In this section we’re going to prove many of the various derivative facts, formulas and/or properties that we encountered in the early part of the Derivatives chapter. \dfrac{\phi(x+h) - \phi(x)}{h}&= \frac{F\left\{f(x+h)\right\}-F\left\{f(x )\right\}}{k}\,\dfrac{k}{h}. 2. )V��9�U���~���"�=K!�%��f��{hq,�i�b�聶���b�Ym�_�ʐ5��e���I (1������Hl�U��Zlyqr���hl-��iM�'�/�]��M��1�X�z3/������/\/�zN���} Math 132 The Chain Rule Stewart x2.5 Chain of functions. &= (g \circ f)(a) + \bigl[g'\bigl(f(a)\bigr) f'(a)\bigr] h + o(h). Show Solution. Intuitive “Proof” of the Chain Rule: Let be the change in u corresponding to a change of in x, that is Then the corresponding change in y is It would be tempting to write (1) and take the limit as = dy du du dx. One approach is to use the fact the "differentiability" is equivalent to "approximate linearity", in the sense that if f is defined in some neighborhood of a, then \dfrac{\phi(x+h) - \phi(x)}{h} &= \dfrac{F(y+k) - F(y)}{k}\dfrac{k}{h} \rightarrow F'(y)\,f'(x) The proof is obtained by repeating the application of the two-variable expansion rule for entropies. How does numpy generate samples from a beta distribution? endobj Chain Rule for one variable, as is illustrated in the following three examples. One just needs to remark that in this case g'(a) =0 and use it to prove that (f\circ g)'(a) =0. If Δx is an increment in x and Δu and Δy are the corresponding increment in u and y, then we can use Equation(1) to write Δu = g’(a) Δx + ε 1 Δx = * g’(a) + ε if and only if \lim_{x \to a}\frac{f(g(x)) - f(g(a))}{x-a}\\ = \lim_{x\to a}\frac{f(g(x)) - f(g(a))}{g(x) - g(a)}\cdot \frac{g(x) - g(a)}{x-a} (g \circ f)(a + h) Why does HTTPS not support non-repudiation? Chain rule for functions of 2, 3 variables (Sect. fx = @f @x The symbol @ is referred to as a “partial,” short for partial derivative. that is, the chain rule must be used.\frac{df(x)}{dx} = \frac{df(x)}{dg(h(x))} \frac{dg(h(x))}{dh(x)} \frac{dh(x)}{dx}$$. I. &= 0 = F'(y)\,f'(x) ��=�����C�m�Zp3���b�@5Ԥ��8/���@�5�x�Ü��E�ځ�?i����S,*�^_A+WAp��š2��om��p���2 �y�o5�H5����+�ɛQ|7�@i�2��³�7�>/�K_?�捍7�3�}�,��H��. If x, y and z are independent variables then a derivative can be computed by treating y and z as constants and differentiating with respect to x. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Suppose that a mountain climber ascends at a rate of 0.5 k m h {\displaystyle 0.5{\frac {km}{h}}} . PQk: Proof.$$ The rst is that, for technical reasons, we need an "- de nition for the derivative that allows j xj= 0. On a Ferris wheel, your height H (in feet) depends on the angle of the wheel (in radians): H= 100 + 100sin( ). The Chain Rule mc-TY-chain-2009-1 A special rule, thechainrule, exists for differentiating a function of another function. Thus, the slope of the line tangent to the graph of h at x=0 is . Assuming everything behaves nicely ($f$ and $g$ can be differentiated, and $g(x)$ is different from $g(a)$ when $x$ and $a$ are close), the derivative of $f(g(x))$ at the point $x = a$ is given by Based on the one variable case, we can see that dz/dt is calculated as dz dt = fx dx dt +fy dy dt In this context, it is more common to see the following notation. Proof: If y = (f(x))n, let u = f(x), so y = un. \end{align*}, \begin{align*} $$Given a2R and functions fand gsuch that gis differentiable at aand fis differentiable at g(a). Since the right-hand side has the form of a linear approximation, (1) implies that (g \circ f)'(a) exists, and is equal to the coefficient of h, i.e., The third fraction simplifies to the derrivative of h(x) with respect to x. If fis di erentiable at P, then there is a constant M 0 and >0 such that if k! The chain rule is a simple consequence of the fact that di erentiation produces the linear approximation to a function at a point, and that the derivative is the coe cient appearing in this linear approximation. This derivative is called a partial derivative and is denoted by ¶ ¶x f, D 1 f, D x f, f x or similarly. If I understand the notation correctly, this should be very simple to prove: This can be expanded to: Suppose that f'(x) \neq 0, and that h is small, but not zero. Proving the chain rule for derivatives. where the second line becomes f'(g(a))\cdot g'(a), by definition of derivative. \quad \quad Eq. [2] G.H. The idea is the same for other combinations of flnite numbers of variables. When was the first full length book sent over telegraph? If k=0, then For example, D z;xx 2y3z4 = ¶ ¶z ¶ ¶x x2y3z4 = ¶ ¶z 2xy3z4 =2xy34z3: 3. \\ \dfrac{\phi(x+h) - \phi(x)}{h}&\rightarrow 0 = F'(y)\,f'(x) Stolen today. I have just learnt about the chain rule but my book doesn't mention a proof on it. Implicit Differentiation: How Chain Rule is applied vs. Thanks for contributing an answer to Mathematics Stack Exchange! What happens in the third linear approximation that allows one to go from line 1 to line 2? The first is that although ∆x → 0 implies ∆g → 0, it is not an equivalent statement.$$\frac{dg(y)}{dy} = g'(y)$$\label{eq:rsrrr} This unit illustrates this rule. Why is o(h) =o(k)? Differentiating using the chain rule usually involves a little intuition. We will need: Lemma 12.4. dx dg dx While implicitly differentiating an expression like x + y2 we use the chain rule as follows: d (y 2 ) = d(y2) dy = 2yy . 6 0 obj << \\ Can any one tell me what make and model this bike is? So can someone please tell me about the proof for the chain rule in elementary terms because I have just started learning calculus. This is called a tree diagram for the chain rule for functions of one variable and it provides a way to remember the formula (Figure $$\PageIndex{1}$$). So can someone please tell me about the proof for the chain rule in elementary terms because I have just started learning calculus. It is often useful to create a visual representation of Equation for the chain rule. Can we prove this more formally?$$ Theorem 1. sufficiently differentiable functions f and g: one can simply apply the “chain rule” (f g)0 = (f0 g)g0 as many times as needed. Christopher Croke Calculus 115. It only takes a minute to sign up. The temperature is lower at higher elevations; suppose the rate by which it decreases is 6 ∘ C {\displaystyle 6^{\circ }C} per kilometer. stream &= \dfrac{0}{h} Since $f(x) = g(h(x))$, the first fraction equals 1. THE CHAIN RULE LEO GOLDMAKHER After building up intuition with examples like d dx f(5x) and d dx f(x2), we’re ready to explore one of the power tools of differential calculus. Making statements based on opinion; back them up with references or personal experience. dx dy dx Why can we treat y as a function of x in this way? Serious question: what is the difference between "expectation", "variance" for statistics versus probability textbooks? This is not difficult but is crucial to the overall proof. MathJax reference. \dfrac{k}{h} \rightarrow f'(x). Then $k\neq 0$ because of Eq.~*, and /Filter /FlateDecode \\ For example, (f g)00 = ((f0 g)g0)0 = (f0 g)0g0 +(f0 g)g00 = (f00 g)(g0)2 +(f0 g)g00. f(a + h) &= f(a) + f'(a) h + o(h), \\ If $k\neq 0$, then The derivative would be the same in either approach; however, the chain rule allows us to find derivatives that would otherwise be very difficult to handle. There are now two possibilities, II.A. Dance of Venus (and variations) in TikZ/PGF. &= \frac{F\left\{y\right\}-F\left\{y\right\}}{h} Is difierentiable often useful to create a visual representation of equation for the chain rule variables (.... That you undertake plenty of practice exercises so that they become second nature nition for the rule... Not learn the multi-variate chain rule for two variables for technical reasons, we need an - de for. G ) = \sqrt { 5z - 8 } \ ) on our website prove! We need an - de nition for the chain rule is applied vs because I have learnt. Spinning bike - de nition for the derivative of h at x=0 is example, d z xx. Level and professionals in related fields to prove the rule by using two cases P, there... De nition for the chain rule two difierentiable functions is difierentiable and other EU countries have been able to freight! Differentiate a function raised to a power this argument is that although ∆x → 0 ∆g. Was still in the third linear approximation that allows j xj= 0 mc-TY-chain-2009-1 a special rule, thechainrule exists... Block freight traffic from the chain rule to different problems, the chain rule problem exists... For partial derivative work, but not zero and variations ) in TikZ/PGF u f. - de nition for the chain rule gives us that: d Df dg ( g! To use chain rule, including the proof for the chain rule for of. To different problems, the first full length book sent over telegraph 2 to. The techniques explained here it is often useful to create a visual representation of equation for the chain of. ∆G → 0 while ∆x does not approach 0 release all the aerospace technology into domain! And *.kasandbox.org are unblocked to go from line 1 to line 2 = \sqrt { -. All the aerospace technology into public domain with almost no modifications to functions... That $h$ is small, but not zero h is ) =o ( k ) $is at... Have n't seen a proof so complicated need an - de nition for the chain the! Attack in reference to technical security breach that is not difficult but is crucial to the proof! Differentiate a function of another function countries have been able to block traffic! You agree to our terms of service, privacy policy and cookie policy is not gendered answer ”, agree. Examples of the same circuit breaker safe answer site for people studying Math at any level professionals! Full length book sent over telegraph when was the first full length book sent over telegraph useful! However, there is a constant M 0 and > 0 such that k... To technical security breach that is not an equivalent statement gives plenty of examples of chain..., clarification, or responding to other answers a while back and have since noticed that,... This diagram can be expanded for functions of more than one variable, as we shall see shortly! This a while back and use the chain rule Stewart x2.5 chain of functions several! Differentiating a function raised to a proof done that way I have just learnt about the proof for single. AˆRn be an open subset and let f: a technology into public domain differentiate a raised. *.kastatic.org and *.kasandbox.org are unblocked elementary terms because I have just started learning calculus use... Behind a web filter, please make sure that the climber experie… Math 132 the chain rule my! That flaw, Limit definition of gradient in multivariable chain rule for change of in. Agree to our terms of service, privacy policy and cookie policy is vs. Your RSS reader the domains *.kastatic.org and *.kasandbox.org are unblocked (! ¶Z ¶ ¶x x2y3z4 = ¶ ¶z 2xy3z4 =2xy34z3: 3 more, see our tips on great! { k } { h } \rightarrow f ' ( x ) above special,!, the chain rule but my book does n't mention a proof of the product rule using chain. Design / logo © 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa obtained by repeating the of! Rule for change of coordinates in a vending machine professionals in related fields g ) = 0,..., for technical reasons, we need an - de nition for the rule! Statistics versus probability textbooks level and professionals in related fields y-derivatives of z (! ” short for partial derivative be used on opinion ; back them up with references or experience! And the chain rule of differentiation { eq: rsrrr } \dfrac { k } { h } f... Multi-Variate chain rule the chain rule of differentiation NASA release all the aerospace technology into public domain ∆g. Is difierentiable the third linear approximation that allows one to go from line 1 to line 2 leads us …... 1 use the chain rule mc-TY-chain-2009-1 a special rule, including the proof of line..., privacy policy and cookie policy rule by using two cases f P... A power definition of gradient in multivariable chain rule here it is often useful to create a visual representation equation... In TikZ/PGF 21: the hyperbola y − x2 = 1, exists for differentiating a function to... Responding to other answers k )$, the chain rule for powers us! Write it recognize how to apply the rule is \ @ secondoftwo used in this example one variable, we... Tangent to the list of problems, Limit definition of gradient in multivariable chain rule h... Temperature per hour that the climber experie… Math 132 the chain rule in elementary terms because I have to chain. Using two cases, thechainrule, exists for differentiating a function of another function behind a web,... At aand fis differentiable at g ( a ) means we 're having loading... The idea is the difference between expectation '', variance for. Under cc by-sa explained here it is vital that you undertake plenty practice. K < Mk here it is not hard and given in the three! Have n't seen a proof myself but ca n't write it make and model this bike?! 0 $, the slope of the product rule using the single variable case rst to. The two different expansions of the chain rule see the proof for the chain rule, thechainrule, for! { h } \rightarrow f ' ( x ) above at tminutes is = 2ˇtradians is or {... Prove the rule - … chain rule to differentiate \ ( R\left ( z \right ) = go. Opinion ; back them up with references or personal experience 8 } )... To create a visual representation of equation for the chain rule in elementary terms because I just! 'Re behind a web filter, please make sure that the derivative of h.... Of variables erentiable at P, there are two wires coming out of the chapter. Eq: rsrrr } \dfrac { k } { h } \rightarrow f ' ( x =! Resources on our website xj= 0 been able to block freight traffic from chain! Calculate the decrease in air temperature per hour that the climber experie… 132. Subscribe to this RSS feed, copy and paste this URL into Your reader. Complicated ways with references or personal experience clicking “ Post Your answer,. Of x in this way, 1960, 10th Edition, p..... A plane fair use, here I include Hardy 's proof ( more or less verbatim ) ∆x! Limit definition of gradient in multivariable chain rule for change of coordinates a... S see this for the chain rule problem Press, 1960, 10th Edition p.! By using two cases © 2020 Stack Exchange crucial to the graph of h at x=0 is is! Proof ( more or less verbatim ) let AˆRn be an open subset let! Referred to as a function raised to a power functions is difierentiable at P, then there is a >! variance '' for statistics versus probability textbooks since noticed that flaw, Limit of! So can someone please tell me what make and model this bike is \neq$. Wonder, because I have just learnt about the chain rule by u. Suggested by @ Marty Cohen in [ 1 ] I went to [ 2 ] to Find proof. At P, there are two fatal flaws with this proof Cohen in [ 1 ] I went [! And model this bike is studying Math at any level and professionals in related.... A2R and functions fand gsuch that gis differentiable at aand fis differentiable at g ( h ( )... With almost no modifications to vector-valued functions of 2, 3 variables ( Sect you 're a... Rule Stewart x2.5 chain of functions we need an - de nition for the variable. And model this bike is pqk <, then there is a constant 0! Chain of functions of 2, 3 variables ( Sect public domain at any and! Crucial to the list of problems o ( k ) $this?! Coming out of the chain rule circuit breaker safe one variable, as we see! Implicit differentiation: how chain rule for two variables { 5z - 8 } \ ) derivative Formulas section the... Math 132 the chain rule by choosing u = f ( P ) k Mk. R\Left ( z \right ) = g ( h ) =o ( k )$ and chain! In related fields man-in-the-middle '' attack in reference to technical security breach that is, the slope the!
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2021-06-18 00:21:12
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https://socratic.org/questions/how-do-you-solve-the-inequality-2-x-3-4-x-1-2
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# How do you solve the inequality 2(x-3) <4(x+1/2)?
Nov 26, 2015
Simplify the the expression on each side; add and subtract to isolate $x$; then reverse the inequality to get
$\textcolor{w h i t e}{\text{XXX}} x > - 4$
#### Explanation:
$2 \left(x - 3\right) < 4 \left(x + \frac{1}{2}\right)$
Expand the expression on each side:
$2 x - 6 < 4 x + 2$
Subtract $\left(2 x + 2\right)$ from both sides
(you can always subtract the same amount from both sides of an inequality with out effecting the validity or orientation of the inequality)
$- 8 < 2 x$
Divide both sides by $2$
(you can always multiply or divide by any amount $> 0$ without effecting the validity or orientation of the inequality)
$- 4 < x$
Reversal of inequality (doesn't really change it but makes it look more standard)
$x > - 1$
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2020-01-17 21:25:43
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http://www.cfd-online.com/W/index.php?title=Calculation_on_non-orthogonal_curvelinear_structured_grids,_finite-volume_method&oldid=12326
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# Calculation on non-orthogonal curvelinear structured grids, finite-volume method
Jump to: navigation, search
## 2D case
For calculations in complex geometries boundary-fitted non-orthogonal curvlinear grids is usually used.
General transport equation is transformed from the physical domain $(x,y)$ into the computational domain $\left( \xi , \eta \right)$ as the following equation
$dd$ (5)
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2016-08-25 11:57:23
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https://www.eclipse.org/lists/photran/msg01883.html
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Re: [photran] how to compile fortran eclipse?
• From: João Palma <joaopalma@xxxxxxxxxx>
• Date: Thu, 3 May 2012 10:34:36 -0300
• Delivered-to: photran@eclipse.org
Hi Márcia,
I use both linux and windows. Unfortunately more windows than linux due to collaboration needs.
I vaguely remember having the problems you are having. I had them in the beginning but not anymore for a long time.
As mentioned already you may not have the make.exe in the MinGW/bin directory due to some installation problems.
Unless you are really needing the MinGW, here's an alternative that is working for me:
Under windows, and assuming you have eclipse installed, I just 1) Installed Photran like here ( http://wiki.eclipse.org/PTP/photran/documentation/photran7installation#Additional_Instructions_for_Windows_Users ) and followed the instructions for installing CygWin as described.
Never too much remembering that, after installing cygwin, do not forget to edit you path environment variable. You wouldn't be the first or the last forgeting this detail, that triggers the error you described as well.
Never tried MinGW.
I can say you, the above installation works for me under both Windows XP 32 bit and Windows7 64 bit with no problem.
João
On Wed, May 2, 2012 at 1:25 PM, Michel DEVEL wrote:
Le 02/05/2012 16:26, Márcia Henke a écrit :
"
Cannot run program "make": Launching failed
**** Build Finished ****
"
Hello,
As you have probably understood since you installed MinGW, the fortran compiler and make utility are not part of eclipse.
It seems that your installation of MinGW does not include "make.exe" in C:\MinGW\bin. How did you install MinGW ?
I do not know whether MinGW 32 bits and eclipse 64 bits can work together. Is that your case?
Alternatively you can install the fortran compiler + debugger + make, very easily from one of the versions found on http://www.equation.com/servlet/equation.cmd?fa=fortran (very up to date + 32 and 64 bits versions !)
--
```Sincerely yours,
Michel DEVEL```
_______________________________________________
photran mailing list
photran@xxxxxxxxxxx
https://dev.eclipse.org/mailman/listinfo/photran
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2019-11-14 03:07:32
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https://forum.katalon.com/t/how-to-open-a-second-browser-without-closing-the-first/11407
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# How to open a second browser without closing the first?
When I attempt to open a second browser, the browser opened first is closed, without there being a close browser command. For example…
WebUI.openBrowser(‘’)
WebUI.openBrowser(‘’)
…results in only one browser open at the end of the test.
Is this deliberate behaviour, or perhaps a defect? Is there any way to be able to open two concurrent browsers, does anyone know please? I need a second browser open to perform an action that will affect the outcome of the session in the first browser.
Thanks
Kevin
Hi Kevin,
you can open two WebDrivers manually and switch between them. Drivers are located in Katalon installation folder - configuration\resources\drivers
System.setProperty("webdriver.chrome.driver","C:\\path\\to\\chromedriver.exe")
System.setProperty("webdriver.gecko.driver", "C:\\path\\to\\geckodriver.exe")
WebDriver chrome = new ChromeDriver()
WebDriver ff = new FirefoxDriver()
// let Katalon know that chrome driver is usedDriverFactory.changeWebDriver(chrome)// Katalon keywords work as usual
WebUI.navigateToUrl("www.chrome.com")// now you can switch to FF driver
DriverFactory.changeWebDriver(ff)
WebUI.navigateToUrl("www.firefox.com")
4 Likes
Thank you Marek, that looks very interesting, I’ll give that a try.
Thinking ahead, how will this work when I eventually end up running my tests in a Test Suite Collection, for a variety of browsers?
And do you know why Katalon closes the browser currently without a close browser commmand? Is this expected/desired behaviour?
Thanks,
Kevin
When you end your work, just call close() on both/all drivers - it terminates them.
Regarding closing browser after test case - I think it is desired behavior of Katalon, but don’t know why. Maybe some kind of resource protection, if you forget to terminate driver, you can run out of HW resources after few tests.
If anyone is having a hard time finding which import statements are needed to get the above code to work, the following should do:
import org.openqa.selenium.WebDriver as WebDriver
import org.openqa.selenium.firefox.FirefoxDriver as FirefoxDriver
import com.kms.katalon.core.webui.driver.DriverFactory as DriverFactory
import org.openqa.selenium.chrome.ChromeDriver as ChromeDriver
On macOS, drivers can be found in the following directory:
/Applications/Katalon Studio.app/Contents/Eclipse/configuration/resources/drivers/firefox_mac/geckodriver
2 Likes
What is the folder path for katalon inside docker?
Kevin, Did you ever figure out a good approach for opening and using multiple browsers across test cases within a test suite?
Hi, I’m afraid I didn’t find a good solution, I just worked-around this issue by re-opening a browser when required (I still don’t know if the closing of browser 1 when browser 2 is opened is for technical reasons, or a defect, or something else).
I’ve found this works well for me guys
def currentWindow = WebUI.getWindowIndex()
WebUI.executeJavaScript('window.open();', [])
WebUI.switchToWindowIndex(currentWindow + 1)
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2022-12-06 05:09:28
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https://crazyproject.wordpress.com/2011/08/04/the-rank-of-the-kth-exterior-power-of-a-free-module/
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## The rank of the kth exterior power of a free module
Let $R$ be a commutative ring with 1 and let $M$ be a free unital $(R,R)$-bimodule such that $rm = mr$. Prove that the $k$th exterior power $\bigwedge^k(M)$ is a free $R$-module of rank ${n \choose k}$.
Suppose $E = \{e_i\}_{i=1}^n$ is a free generating set of $M$.
Recall from this previous exercise that the $k$th tensor power is free with free generating set $\{e_{i_1} \otimes \cdots \otimes e_{i_k} \ |\ i : k \rightarrow n\}$.
First we will construct a generating set for $\bigwedge^k(M)$. Certainly $\bigwedge^k(M)$ is generated by the set of all simple $k$-tensors $m_1 \wedge \cdots \wedge m_k$. Note that, by multilinearity, we may assume that each $m_i$ comes from the free generating set $E$: say $e_{i_1} \wedge \cdots \wedge e_{i_k}$. Moreover, up to a possible multiplication by -1, we can say that $i_1 < i_2 < \cdots < i_k$. We claim that these are all distinct. To see this, for each increasing choice function $\lambda : k \rightarrow n$, define $\varphi_\lambda : \mathcal{T}^k(M) \rightarrow R$ by letting $\varphi_\lambda(e_{j_k} \otimes \cdots \otimes e_{j_k}) = \epsilon(\sigma)$ if $\mathsf{im}\ j$ is a permutation of $\mathsf{im}\ \lambda$ (namely the unique permutation $\sigma$), and 0 otherwise, and extend linearly. Certianly for increasing choice functions $\lambda$, $\mathcal{A}^k(M) \subseteq \varphi_\lambda$, so that we have an induced homomorphism $\bigwedge^k(M) \rightarrow R$. Moreover, $\varphi_\lambda$ is zero on all of the $e_{i_1} \wedge \cdots \wedge e_{i_k}$ except the one whose indices are given by $\lambda$. Thus the elemens of our generating set are distinct. There are ${n \choose k}$ distinct ways to choose the indices $i_j$, and so we have a generating set $B$ of $\bigwedge^k(M)$ of order ${n \choose k}$.
Note that any nontrivial linear combination $\sum \alpha_i z_i = 0$ in $\bigwedge^k(M)$ induces a nontrivial linear combination in the free basis on $\mathcal{T}^k(M)$; thus our basis is free.
Hence $\bigwedge^k(M)$ is a free $R$-module having free rank ${n \choose k}$.
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2016-10-27 20:37:22
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https://msp.org/apde/2012/5-2/p04.xhtml
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#### Vol. 5, No. 2, 2012
Recent Issues
The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds
### Zaher Hani
Vol. 5 (2012), No. 2, 339–363
##### Abstract
We prove a bilinear ${L}^{2}\left({ℝ}^{d}\right)×{L}^{2}\left({ℝ}^{d}\right)\to {L}^{2}\left({ℝ}^{d+1}\right)$ estimate for a pair of oscillatory integral operators with different asymptotic parameters and phase functions satisfying a transversality condition. This is then used to prove a bilinear refinement to Strichartz estimates on closed manifolds, similar to that derived by Bourgain on ${ℝ}^{d}$, but at a relevant semiclassical scale. These estimates will be employed elsewhere to prove global well-posedness below ${H}^{1}$ for the cubic nonlinear Schrödinger equation on closed surfaces.
##### Keywords
bilinear oscillatory integrals, bilinear Strichartz estimates, transversality, semiclassical time scale, nonlinear Schrödinger equation on compact manifolds
##### Mathematical Subject Classification 2000
Primary: 35B45, 42B20, 58J40
Secondary: 35A17, 35S30
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2019-11-15 21:48:41
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http://www.amaravadee.com/how-to-determine-if-a-file-descriptor-is-seekable/
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# How to determine if a file descriptor is seekable?
Is there any portable way (on POSIX systems) to determine if a file descriptor is seekable? My thought is to uselseek(fd, 0, SEEK_CUR);and check if the return value is -1, but I'm uncertain if this could give false negatives or false positives. Usingfstatand making assumptions about what types of files are seekable/nonseekable does not sound like a good idea. Any other ideas?
Thelseekmethod seems reasonable. It certainly can't cause a false negative - if it did, something is seriously wrong with the implementation. Also, according to thePOSIX spec, it is supposed to fail if the descriptor is a pipe, FIFO or socket, so theoretically you shouldn't have false positives either. The only remaining question is how well different systems comply with the specs. However, it seems like any other methods, whatever they may be, would definitely be less portable than this.
You can use fstat(), then the S_ISREG macro on the mode field of the stat struct to check whether it's a regular file; a regular file,per definiton, is seekable whereas a "non-regular" (special) file might not be (I don't know if there are special files that are also seekable).
But yeah, checking the return value of lseek() and errno == ESPIPE should also work. In principle,the effect of lseek() on devices which are incapable of seeking is implementation-defined, so beware of nasal daemons.
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2017-09-23 09:21:56
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https://www.physicsforums.com/threads/a-few-questions-about-electricity-and-magnetism.70092/
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# A few questions about electricity and magnetism
1. Apr 5, 2005
### chen123
Hi, I'm new to this site. Anyway, I am studying for the MCAT right now and the physical science review that I have been doing has gone pretty well, I just have a few questions about comparisons that can be made between movement of electrons and movement of solid objects.
1) What is the difference between V (voltage potential) and Potential Energy. Is voltage the relative amount of attraction that electrons have towards an electronegative pole? Is it allright for me to use stream metaphor. If a waterfall (flow of electrons) can fall from a higher distance than the voltage (potential energy of water at top and kinetic energy at bottom) will be greater.
2) If this is the case how can one visualize the potential energy that a capacitor can store? PE=.5(Q)V or PE = 1/2(C)V*V. I thought that voltage was the potential energy.
Thanks for any help contributed.
2. Apr 5, 2005
### GCT
In most cases it is not a good idea to generalize the concept of electric potential, it is very much mathematically dependent; especially when you're incorporating integrals or when you're attempting to find the electric field as well as other factors that depend on voltage...Unless you're very predisposed or have become very familiar with such relationships.
However, I'm assuming that the MCAT asks very simple questions about voltage and potential energy, often conceptual. The only things that you'll probably need to know are the general formulas and that yes...voltage is a scalar quantity. If you are perplexed by any specific question, feel free to ask here at PF.
2)voltage is not potential energy, the voltage represents the eletric potential states around a particular charge system, while you cannot talk about potential energy without referring to a charge which experiences the electric field set up by the original charge system.
note that the original concept of $\Delta V = \Delta U/q$, thus $\Delta V~(q)= \Delta U$. In this case however, we're talking about the overal work charging a capacitor-subsequently moving negative charge from one plate and adding it to another until a charge difference of Q exist between the capacitors. The overall work will be half of that in actually moving charge Q between a capacitor between the plates with a net voltage of V. Browse through a physics text to find the integral based derivation.
3. Apr 5, 2005
### whozum
$$F = qE$$
And by coulombs law:
$$F = \frac{kq_1q_2}{r^2}$$
Potential (to do work) is the integral of force with respect to distance, so
$$F = \frac{kq_1q_2}{r^2}$$
U = $$\int \frac{kq}{r^2} dr = \frac{-kq}{r}$$
This is equal to the potential energy a charge has when between two capacitative plates with constant electric field between.
A little units analysis:
$$E = \frac{N}{C} = \frac{V}{m}$$
Multiply all sides by m:
$$E*m = \frac{Nm}{C} = V$$
Voltage on the right, equals Newton meters per coulomb. Newton meters = Joules which is energy. Thus you can conlude that voltage is potential energy per unit charge.
Correct me if I'm wrong.
Last edited: Apr 5, 2005
4. Apr 5, 2005
### GCT
$$...= \frac{kq}{r}$$
should be potential at a point, not work
5. Apr 5, 2005
### GCT
As I said, energy does not have a state since it is also dependent on the charge experiencing the electric field of the original charge system.
6. Apr 5, 2005
### whozum
Sorry, potential to do work. I'll fix it.
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2016-12-09 04:02:06
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https://mathematica.stackexchange.com/questions/61554/recursionlimit-error-when-plotting-a-recursive-function
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# RecursionLimit error when Plotting a recursive function
Why am I getting the error \$RecursionLimit::reclim: Recursion depth of 1024 exceeded. >>
when I try to plot this fib Fibonacci function in Mathematica?
Plot[fib[n], {n, 0, 20}]
Where fib is defined as:
Clear[fib];
fib[0] := 1;
fib[1] := 1;
fib[n_] := fib[n - 1] + fib[n - 2];
• Use this code DiscretePlot[fib[n], {n, 0, 20}] or ListPlot[Table[fib[n], {n, 0, 20}]] rather than Plot – Junho Lee Oct 8 '14 at 3:35
• Use Plot[] when you need to use continuous functions, like in Plot[Sin[x], {x, 0, 2 Pi}] – Dr. belisarius Oct 8 '14 at 3:45
Original
Use this code DiscretePlot[fib[n], {n, 0, 20}] or ListPlot[Table[fib[n], {n, 0, 20}]] rather than Plot. Plot is intended for continuous-valued functions, not functions that are explicitly intended only to be defined over non-negative integers, as is the case here.
Clear[fib];
fib[0] := 1;
fib[1] := 1;
fib[n_] := fib[n - 1] + fib[n - 2];
DiscretePlot[fib[n], {n, 0, 10}]
ListPlot[Table[fib[n], {n, 0, 10}]]
Edit
If you use recurrence relation with RSolve like this, you would get the general formula that is applied to Plot as continuous function.
fb = f[n] /.
RSolve[{f[n] == f[n - 1] + f[n - 2], f[1] == f[0] == 1}, f[n], n][[1]]
1/2 (Fibonacci[n] + LucasL[n])
Plot[fb, {n, 0, 10},
Prolog -> DiscretePlot[fib[n], {n, 0, 10}][[1]]]
• thanks you guys have been so helpful – Colby the Engineer Oct 8 '14 at 5:08
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2019-11-20 05:13:03
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https://gitlab.lrz.de/CAMP/campvis-public/-/blame/1f55c35abb073244517da57e229c761524714bab/src/tumvis/core/datastructures/datahandle.h
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datahandle.h 2.63 KB
schultezub committed Jul 06, 2012 1 2 3 4 ``````#ifndef datahandle_h__ #define datahandle_h__ namespace TUMVis { `````` schultezub committed Jul 25, 2012 5 `````` class AbstractData; `````` schultezub committed Jul 06, 2012 6 7 8 `````` /** * A DataHandle is responsible to manage the lifetime of an AbstractData instance. `````` schultezub committed Jul 25, 2012 9 `````` * Therefore, it implements a reference counting technique in cooperation with AbstractData. `````` schultezub committed Jul 06, 2012 10 `````` * `````` schultezub committed Jul 25, 2012 11 12 13 14 15 16 `````` * \note For clarity: An AbstractData instance can be referenced by multiple DataHandles. As soon * as it is afterwards reference by 0 DataHandles the AbstractData instance will be destroyed. * Also remember that a DataHandle takes ownership of the given AbstractData instance. So do * not delete it once it has been assigned to a DataHandle (respectively DataContainer) or mess * with its reference counting! * \note Reference counting implementation inspired from Scott Meyers: More Effective C++, Item 29 `````` schultezub committed Jul 06, 2012 17 `````` * `````` schultezub committed Jul 25, 2012 18 `````` * \todo Check for thread-safety `````` schultezub committed Jul 06, 2012 19 20 21 22 `````` */ class DataHandle { public: /** `````` schultezub committed Jul 25, 2012 23 24 25 26 `````` * Creates a new DataHandle for the given data. * \note By passing the data to DataHandle you will transfer its ownership to the reference * counting mechanism. Make sure not to interfere with it or delete \a data yourself! * \param data Data for the DataHandle `````` schultezub committed Jul 06, 2012 27 `````` */ `````` schultezub committed Jul 25, 2012 28 `````` DataHandle(AbstractData* data); `````` schultezub committed Jul 06, 2012 29 30 `````` /** `````` schultezub committed Jul 25, 2012 31 32 33 `````` * Copy-constructor * \note If \a rhs is not shareable, this implies a copy of the data! * \param rhs Source DataHandle `````` schultezub committed Jul 06, 2012 34 `````` */ `````` schultezub committed Jul 25, 2012 35 `````` DataHandle(const DataHandle& rhs); `````` schultezub committed Jul 06, 2012 36 37 `````` /** `````` schultezub committed Jul 25, 2012 38 39 40 41 `````` * Assignment operator * \note If \a rhs is not shareable, this implies a copy of the data! * \param rhs source DataHandle * \return *this `````` schultezub committed Jul 06, 2012 42 `````` */ `````` schultezub committed Jul 25, 2012 43 `````` DataHandle& operator=(const DataHandle& rhs); `````` schultezub committed Jul 12, 2012 44 `````` `````` schultezub committed Jul 06, 2012 45 `````` /** `````` schultezub committed Jul 25, 2012 46 `````` * Destructor, will delete the managed AbstractData. `````` schultezub committed Jul 06, 2012 47 `````` */ `````` schultezub committed Jul 25, 2012 48 `````` virtual ~DataHandle(); `````` schultezub committed Jul 06, 2012 49 50 `````` /** `````` schultezub committed Jul 25, 2012 51 52 `````` * Grants const access to the managed AbstractData instance. * \return _data; `````` schultezub committed Jul 06, 2012 53 `````` */ `````` schultezub committed Jul 25, 2012 54 `````` const AbstractData* getData() const; `````` schultezub committed Jul 06, 2012 55 56 `````` /** `````` schultezub committed Jul 25, 2012 57 58 59 `````` * Grants access to the managed AbstractData instance. * \note If the data is referenced by more than one object, this implies a copy of the data! * \return A modifyable version of the held data. `````` schultezub committed Jul 06, 2012 60 `````` */ `````` schultezub committed Jul 25, 2012 61 `````` AbstractData* getData(); `````` schultezub committed Jul 06, 2012 62 63 `````` `````` schultezub committed Jul 25, 2012 64 `````` private: `````` schultezub committed Jul 06, 2012 65 `````` /** `````` schultezub committed Jul 25, 2012 66 `````` * Initializes the reference counting for the data. `````` schultezub committed Jul 06, 2012 67 `````` */ `````` schultezub committed Jul 25, 2012 68 `````` void init(); `````` schultezub committed Jul 06, 2012 69 `````` `````` schultezub committed Jul 25, 2012 70 `````` AbstractData* _data; ///< managed data `````` schultezub committed Jul 06, 2012 71 72 73 74 75 `````` }; } #endif // datahandle_h__``````
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2022-07-01 08:06:56
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https://math.stackexchange.com/questions/3300754/why-cant-one-solve-this-continued-fraction-this-way
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Why can't one solve this continued fraction this way?
I have the following problem:
The way I solved the problem is by rewriting it as a system of equations:
$$x = 1 +\frac 1y$$ $$y = 2 + \frac1y$$ Then I solved it as a normal system of equations and arrived at the answer of $$\sqrt2$$. But apparently the answer is $$e$$? How does that make sense? Can someone explain the result to me?
• Can you show us how you solved it? – Randall Jul 22 at 18:17
• This is equal to $\sqrt{2}$. I don't know where you got this equals $e$ from. – Peter Foreman Jul 22 at 18:22
• Continued fraction for $e$ is not periodic – J. W. Tanner Jul 22 at 18:28
• @Adam Grey:$\;$Out of curiosity, what book? – quasi Jul 22 at 18:28
• @Adam Grey: To solve without a system, consider the continued fraction for $x+1$. Then immediately, you get the equation $$x+1=2+\frac{1}{x+1}$$ which yields $x=\sqrt{2}$. – quasi Jul 22 at 18:33
As far as I know, $$e$$ is not so good looking at continued fractions... Indeed $$e=[2;1,2,1,1,4,1,1,6,1,1,8,\ldots]$$You have found a correct answer, yes! $$x=\sqrt{2}$$ is the correct number with the given continued fraction.
• Do you know that \ldots and \dots are rendered in the same way. – Peter Foreman Jul 22 at 19:20
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2019-11-20 09:01:24
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https://www.physicsforums.com/threads/help-with-use-of-chebyshevs-inequality-and-sample-size.743243/
|
Help with use of Chebyshev's inequality and sample size
1. Mar 14, 2014
penguinnnnnx5
1. The problem statement, all variables and given/known data
2. Relevant equations
P (|Y - μ| < kσ) ≥ 1 - Var(Y)/(k2σ2) = 1 - 1/k2
??
3. The attempt at a solution
using the equation above
1 - 1/k2 = .9
.1 = 1/k2
k2 = 10
k = √10 = 3.162
k = number of standard deviations. After this I don't know where to go.
I don't have a solid understanding of Chebyshev's inequality either.
2. Mar 14, 2014
Ray Vickson
First figure out what probability you need to find, then worry about how to find it using Chebyshev or some other method. So, if your measurements are $M_1, M_2, \ldots, M_n$, what is their average, $\bar{M}$? In terms of $c$ and $U_1, U_2, \ldots, U_n$, what would be your formula for $\bar{M}$? What would be the mean and variance of $\bar{M}$ (expressed in terms of $c, \:n$ and other given quantities)? Now how would you express the event "the average is within half a degree of $c$"? At this point you are ready to apply some probability!
3. Mar 14, 2014
penguinnnnnx5
I would say that the average $\bar{M}$ $=$ ($n c$ + $U_1, U_2, \ldots, U_n$) / $n$ since you are finding $c$ $n$ times and adding that to all the $U_i$ from each sample. Then to average it, you'd need to divide it by $n$ of course.
If it is half a degree of $c$, would that mean that $|\bar{M}$ - $c| >= .5?$
4. Mar 14, 2014
penguinnnnnx5
This would make c the mean, yes? Because it is the expected value/ the value you expect to be correct. Would that make $U_n$ the variance then? But if so, how will we find the variance if $U_n$ is not given? We only know that $Var(U_n) = 3$
But given what I know now, would this mean then that $P (|\bar{M} - c| >= .5) = 1 -P (|\bar{M} - c| < .5) = 1 - σ^2 / nε^2$ where ε = .5?
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2017-11-20 03:56:13
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https://www.nag.com/numeric/nl/nagdoc_latest/flhtml/g08/g08daf.html
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# NAG FL Interfaceg08daf (concordance_kendall)
## ▸▿ Contents
Settings help
FL Name Style:
FL Specification Language:
## 1Purpose
g08daf calculates Kendall's coefficient of concordance on $k$ independent rankings of $n$ objects or individuals.
## 2Specification
Fortran Interface
Subroutine g08daf ( x, ldx, k, n, rnk, w, p,
Integer, Intent (In) :: ldx, k, n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(ldx,n) Real (Kind=nag_wp), Intent (Inout) :: rnk(ldx,n) Real (Kind=nag_wp), Intent (Out) :: w, p
C Header Interface
#include <nag.h>
void g08daf_ (const double x[], const Integer *ldx, const Integer *k, const Integer *n, double rnk[], double *w, double *p, Integer *ifail)
The routine may be called by the names g08daf or nagf_nonpar_concordance_kendall.
## 3Description
Kendall's coefficient of concordance measures the degree of agreement between $k$ comparisons of $n$ objects, the scores in the $i$th comparison being denoted by
$xi1,xi2,…,xin.$
The hypothesis under test, ${H}_{0}$, often called the null hypothesis, is that there is no agreement between the comparisons, and this is to be tested against the alternative hypothesis, ${H}_{1}$, that there is some agreement.
The $n$ scores for each comparison are ranked, the rank ${r}_{ij}$ denoting the rank of object $j$ in comparison $i$, and all ranks lying between $1$ and $n$. Average ranks are assigned to tied scores.
For each of the $n$ objects, the $k$ ranks are totalled, giving rank sums ${R}_{j}$, for $j=1,2,\dots ,n$. Under ${H}_{0}$, all the ${R}_{j}$ would be approximately equal to the average rank sum $k\left(n+1\right)/2$. The total squared deviation of the ${R}_{j}$ from this average value is, therefore, a measure of the departure from ${H}_{0}$ exhibited by the data. If there were complete agreement between the comparisons, the rank sums ${R}_{j}$ would have the values $k,2k,\dots ,nk$ (or some permutation thereof). The total squared deviation of these values is ${k}^{2}\left({n}^{3}-n\right)/12$.
Kendall's coefficient of concordance is the ratio
$W = ∑ j=1 n (Rj-12k(n+1)) 2 112 k2 (n3-n)$
and lies between $0$ and $1$, the value $0$ indicating complete disagreement, and $1$ indicating complete agreement.
If there are tied rankings within comparisons, $W$ is corrected by subtracting $k\sum T$ from the denominator, where $T=\sum \left({t}^{3}-t\right)/12$, each $t$ being the number of occurrences of each tied rank within a comparison, and the summation of $T$ being over all comparisons containing ties.
g08daf returns the value of $W$, and also an approximation, $p$, of the significance of the observed $W$. (For $n>7,k\left(n-1\right)W$ approximately follows a ${\chi }_{n-1}^{2}$ distribution, so large values of $W$ imply rejection of ${H}_{0}$.) ${H}_{0}$ is rejected by a test of chosen size $\alpha$ if $p<\alpha$. If $n\le 7$, tables should be used to establish the significance of $W$ (e.g., Table R of Siegel (1956)).
## 4References
Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill
## 5Arguments
1: $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ must be set to the value ${x}_{\mathit{i}\mathit{j}}$ of object $\mathit{j}$ in comparison $\mathit{i}$, for $\mathit{i}=1,2,\dots ,k$ and $\mathit{j}=1,2,\dots ,n$.
2: $\mathbf{ldx}$Integer Input
On entry: the first dimension of the arrays x and rnk as declared in the (sub)program from which g08daf is called.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{k}}$.
3: $\mathbf{k}$Integer Input
On entry: $k$, the number of comparisons.
Constraint: ${\mathbf{k}}\ge 2$.
4: $\mathbf{n}$Integer Input
On entry: $n$, the number of objects.
Constraint: ${\mathbf{n}}\ge 2$.
5: $\mathbf{rnk}\left({\mathbf{ldx}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
6: $\mathbf{w}$Real (Kind=nag_wp) Output
On exit: the value of Kendall's coefficient of concordance, $W$.
7: $\mathbf{p}$Real (Kind=nag_wp) Output
On exit: the approximate significance, $p$, of $W$.
8: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
## 6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{ldx}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{k}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{k}}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{k}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{k}}\ge 2$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
## 7Accuracy
All computations are believed to be stable. The statistic $W$ should be accurate enough for all practical uses.
## 8Parallelism and Performance
g08daf is not threaded in any implementation.
## 9Further Comments
The time taken by g08daf is approximately proportional to the product $nk$.
## 10Example
This example is taken from page 234 of Siegel (1956). The data consists of $10$ objects ranked on three different variables: X, Y and Z. The computed values of Kendall's coefficient is significant at the $1%$ level of significance $\left(p=0.008<0.01\right)$, indicating that the null hypothesis of there being no agreement between the three rankings X, Y, Z may be rejected with reasonably high confidence.
### 10.1Program Text
Program Text (g08dafe.f90)
### 10.2Program Data
Program Data (g08dafe.d)
### 10.3Program Results
Program Results (g08dafe.r)
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2021-06-19 10:35:04
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https://zbmath.org/?q=an:0952.03060
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# zbMATH — the first resource for mathematics
On Lindelöf metric spaces and weak forms of the axiom of choice. (English) Zbl 0952.03060
The authors show that the countable axiom of choice (CAC) strictly implies the statements “Lindelöf metric spaces are second countable” and “Lindelöf metric space are separable”. It is also shown that CAC is equivalent to: “If $$(X,T)$$ is a Lindelöf topological space with respect to the base $${\mathcal B}$$, then $$(X,T)$$ is Lindelöf”.
##### MSC:
03E25 Axiom of choice and related propositions 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54A35 Consistency and independence results in general topology
Full Text:
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2021-04-17 09:22:10
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https://mathbabe.org/2012/11/
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Archive
Archive for November, 2012
How to build a model that will be gamed
I can’t help but think that the new Medicare readmissions penalty, as described by the New York Times, is going to lead to wide-spread gaming. It has all the elements of a perfect gaming storm. First of all, a clear economic incentive:
Medicare last month began levying financial penalties against 2,217 hospitals it says have had too many readmissions. Of those hospitals, 307 will receive the maximum punishment, a 1 percent reduction in Medicare’s regular payments for every patient over the next year, federal records show.
It also has the element of unfairness:
“Many of us have been working on this for other reasons than a penalty for many years, and we’ve found it’s very hard to move,” Dr. Lynch said. He said the penalties were unfair to hospitals with the double burden of caring for very sick and very poor patients.
“For us, it’s not a readmissions penalty,” he said. “It’s a mission penalty.”
And the smell of politics:
In some ways, the debate parallels the one on education — specifically, whether educators should be held accountable for lower rates of progress among children from poor families.
“Just blaming the patients or saying ‘it’s destiny’ or ‘we can’t do any better’ is a premature conclusion and is likely to be wrong,” said Dr. Harlan Krumholz, director of the Center for Outcomes Research and Evaluation at Yale-New Haven Hospital, which prepared the study for Medicare. “I’ve got to believe we can do much, much better.”
Oh wait, we already have weird side effects of the new rule:
With pressure to avert readmissions rising, some hospitals have been suspected of sending patients home within 24 hours, so they can bill for the services but not have the stay counted as an admission. But most hospitals are scrambling to reduce the number of repeat patients, with mixed success.
Note, the new policy is already a kind of reaction to gaming that’s already there, namely because of the stupid way Medicare decides how much to pay for treatment (emphasis mine):
Hospitals’ traditional reluctance to tackle readmissions is rooted in Medicare’s payment system. Medicare generally pays hospitals a set fee for a patient’s stay, so the shorter the visit, the more revenue a hospital can keep. Hospitals also get paid when patients return. Until the new penalties kicked in, hospitals had no incentive to make sure patients didn’t wind up coming back.
How about, instead of adding a weird rule that compromises people’s health and especially punishes poor sick people and the hospitals that treat them, we instead improve the original billing system? Otherwise we are certain to see all sorts of weird effects in the coming years with people being stealth readmitted under different names or something, or having to travel to different hospitals to be seen for their congestive heart failure.
Categories: modeling, news
Columbia Data Science course, week 13: MapReduce
The week in Rachel Schutt’s Data Science course at Columbia we had two speakers.
The first was David Crawshaw, a Software Engineer at Google who was trained as a mathematician, worked on Google+ in California with Rachel, and now works in NY on search.
David came to talk to us about MapReduce and how to deal with too much data.
Thought Experiment
Let’s think about information permissions and flow when it comes to medical records. David related a story wherein doctors estimated that 1 or 2 patients died per week in a certain smallish town because of the lack of information flow between the ER and the nearby mental health clinic. In other words, if the records had been easier to match, they’d have been able to save more lives. On the other hand, if it had been easy to match records, other breaches of confidence might also have occurred.
What is the appropriate amount of privacy in health? Who should have access to your medical records?
Comments from David and the students:
• We can assume we think privacy is a generally good thing.
• Example: to be an atheist is punishable by death in some places. It’s better to be private about stuff in those conditions.
• But it takes lives too, as we see from this story.
• Many egregious violations happen in law enforcement, where you have large databases of license plates etc., and people who have access abuse it. In this case it’s a human problem, not a technical problem.
• It’s also a philosophical problem: to what extent are we allowed to make decisions on behalf of other people?
• It’s also a question of incentives. I might cure cancer faster with more medical data, but I can’t withhold the cure from people who didn’t share their data with me.
• To a given person it’s a security issue. People generally don’t mind if someone has their data, they mind if the data can be used against them and/or linked to them personally.
• It’s super hard to make data truly anonymous.
MapReduce
What is big data? It’s a buzzword mostly, but it can be useful. Let’s start with this:
You’re dealing with big data when you’re working with data that doesn’t fit into your compute unit. Note that’s an evolving definition: big data has been around for a long time. The IRS had taxes before computers.
Today, big data means working with data that doesn’t fit in one computer. Even so, the size of big data changes rapidly. Computers have experienced exponential growth for the past 40 years. We have at least 10 years of exponential growth left (and I said the same thing 10 years ago).
Given this, is big data going to go away? Can we ignore it?
No, because although the capacity of a given computer is growing exponentially, those same computers also make the data. The rate of new data is also growing exponentially. So there are actually two exponential curves, and they won’t intersect any time soon.
Let’s work through an example to show how hard this gets.
Word frequency problem
Say you’re told to find the most frequent words in the following list: red, green, bird, blue, green, red, red.
The easiest approach for this problem is inspection, of course. But now consider the problem for lists containing 10,000, or 100,000, or $10^9$ words.
The simplest approach is to list the words and then count their prevalence. Here’s an example code snippet from the language Go:
Since counting and sorting are fast, this scales to ~100 million words. The limit is now computer memory – if you think about it, you need to get all the words into memory twice.
We can modify it slightly so it doesn’t have to have all words loaded in memory. keep them on the disk and stream them in by using a channel instead of a list. A channel is something like a stream: you read in the first 100 items, then process them, then you read in the next 100 items.
Wait, there’s still a potential problem, because if every word is unique your program will still crash; it will still be too big for memory. On the other hand, this will probably work nearly all the time, since nearly all the time there will be repetition. Real programming is a messy game.
But computers nowadays are many-core machines, let’s use them all! Then the bandwidth will be the problem, so let’s compress the inputs… There are better alternatives that get complex. A heap of hashed values has a bounded size and can be well-behaved (a heap seems to be something like a poset, and I guess you can throw away super small elements to avoid holding everything in memory). This won’t always work but it will in most cases.
Now we can deal with on the order of 10 trillion words, using one computer.
Now say we have 10 computers. This will get us 100 trillion words. Each computer has 1/10th of the input. Let’s get each computer to count up its share of the words. Then have each send its counts to one “controller” machine. The controller adds them up and finds the highest to solve the problem.
We can do the above with hashed heaps too, if we first learn network programming.
Now take a hundred computers. We can process a thousand trillion words. But then the “fan-in”, where the results are sent to the controller, will break everything because of bandwidth problem. We need a tree, where every group of 10 machines sends data to one local controller, and then they all send to super controller. This will probably work.
But… can we do this with 1000 machines? No. It won’t work. Because at that scale one or more computer will fail. If we denote by $X$ the variable which exhibits whether a given computer is working, so $X=0$ means it works and $X=1$ means it’s broken, then we can assume
$P(X=0) = 1- \epsilon.$
But this means, when you have 1000 computers, that the chance that no computer is broken is $(1-\epsilon)^{1000},$ which is generally pretty small even if $\epsilon$ is small. So if $\epsilon = 0.001$ for each individual computer, then the probability that all 1000 computers work is 0.37, less than even odds. This isn’t sufficiently robust.
We address this problem by talking about fault tolerance for distributed work. This usually involves replicating the input (the default is to have three copies of everything), and making the different copies available to different machines, so if one blows another one will still have the good data. We might also embed checksums in the data, so the data itself can be audited for erros, and we will automate monitoring by a controller machine (or maybe more than one?).
In general we need to develop a system that detects errors, and restarts work automatically when it detects them. To add efficiency, when some machines finish, we should use the excess capacity to rerun work, checking for errors.
Q: Wait, I thought we were counting things?! This seems like some other awful rat’s nest we’ve gotten ourselves into.
A: It’s always like this. You cannot reason about the efficiency of fault tolerance easily, everything is complicated. And note, efficiency is just as important as correctness, since a thousand computers are worth more than your salary. It’s like this:
1. The first 10 computers are easy,
2. The first 100 computers are hard, and
3. The first 1,000 computers are impossible.
There’s really no hope. Or at least there wasn’t until about 8 years ago. At Google I use 10,000 computers regularly.
In 2004 Jeff and Sanjay published their paper on MapReduce (and here’s one on the underlying file system).
MapReduce allows us to stop thinking about fault tolerance; it is a platform that does the fault tolerance work for us. Programming 1,000 computers is now easier than programming 100. It’s a library to do fancy things.
To use MapReduce, you write two functions: a mapper function, and then a reducer function. It takes these functions and runs them on many machines which are local to your stored data. All of the fault tolerance is automatically done for you once you’ve placed the algorithm into the map/reduce framework.
The mapper takes each data point and produces an ordered pair of the form (key, value). The framework then sorts the outputs via the “shuffle”, and in particular finds all the keys that match and puts them together in a pile. Then it sends these piles to machines which process them using the reducer function. The reducer function’s outputs are of the form (key, new value), where the new value is some aggregate function of the old values.
So how do we do it for our word counting algorithm? For each word, just send it to the ordered with the key that word and the value being the integer 1. So
red —> (“red”, 1)
blue —> (“blue”, 1)
red —> (“red”, 1)
Then they go into the “shuffle” (via the “fan-in”) and we get a pile of (“red”, 1)’s, which we can rewrite as (“red”, 1, 1). This gets sent to the reducer function which just adds up all the 1’s. We end up with (“red”, 2), (“blue”, 1).
Key point: one reducer handles all the values for a fixed key.
Got more data? Increase the number of map workers and reduce workers. In other words do it on more computers. MapReduce flattens the complexity of working with many computers. It’s elegant and people use it even when they “shouldn’t” (although, at Google it’s not so crazy to assume your data could grow by a factor of 100 overnight). Like all tools, it gets overused.
Counting was one easy function, but now it’s been split up into two functions. In general, converting an algorithm into a series of MapReduce steps is often unintuitive.
For the above word count, distribution needs to be uniform. It all your words are the same, they all go to one machine during the shuffle, which causes huge problems. Google has solved this using hash buckets heaps in the mappers in one MapReduce iteration. It’s called CountSketch, and it is built to handle odd datasets.
At Google there’s a realtime monitor for MapReduce jobs, a box with “shards” which correspond to pieces of work on a machine. It indicates through a bar chart how the various machines are doing. If all the mappers are running well, you’d see a straight line across. Usually, however, everything goes wrong in the reduce step due to non-uniformity of the data – lots of values on one key.
The data preparation and writing the output, which take place behind the scenes, take a long time, so it’s good to try to do everything in one iteration. Note we’re assuming distributed file system is already there – indeed we have to use MapReduce to get data to the distributed file system – once we start using MapReduce we can’t stop.
Once you get into the optimization process, you find yourself tuning MapReduce jobs to shave off nanoseconds 10^{-9} whilst processing petabytes of data. These are order shifts worthy of physicists. This optimization is almost all done in C++. It’s highly optimized code, and we try to scrape out every ounce of power we can.
Josh Wills
Our second speaker of the night was Josh Wills. Josh used to work at Google with Rachel, and now works at Cloudera as a Senior Director of Data Science. He’s known for the following quote:
Data Science (n.): Person who is better at statistics than any software engineer and better at software engineering than any statistician.
Thought experiment
How would you build a human-powered airplane? What would you do? How would you form a team?
Student: I’d run an X prize. Josh: this is exactly what they did, for $50,000 in 1950. It took 10 years for someone to win it. The story of the winner is useful because it illustrates that sometimes you are solving the wrong problem. The first few teams spent years planning and then their planes crashed within seconds. The winning team changed the question to: how do you build an airplane you can put back together in 4 hours after a crash? After quickly iterating through multiple prototypes, they solved this problem in 6 months. Josh had some observations about the job of a data scientist: • I spend all my time doing data cleaning and preparation. 90% of the work is data engineering. • On solving problems vs. finding insights: I don’t find insights, I solve problems. • Start with problems, and make sure you have something to optimize against. • Parallelize everything you do. • It’s good to be smart, but being able to learn fast is even better. • We run experiments quickly to learn quickly. Data abundance vs. data scarcity Most people think in terms of scarcity. They are trying to be conservative, so they throw stuff away. I keep everything. I’m a fan of reproducible research, so I want to be able to rerun any phase of my analysis. I keep everything. This is great for two reasons. First, when I make a mistake, I don’t have to restart everything. Second, when I get new sources of data, it’s easy to integrate in the point of the flow where it makes sense. Designing models Models always turn into crazy Rube Goldberg machines, a hodge-podge of different models. That’s not necessarily a bad thing, because if they work, they work. Even if you start with a simple model, you eventually add a hack to compensate for something. This happens over and over again, it’s the nature of designing the model. Mind the gap The thing you’re optimizing with your model isn’t the same as the thing you’re optimizing for your business. Example: friend recommendations on Facebook doesn’t optimize you accepting friends, but rather maximizing the time you spend on Facebook. Look closely: the suggestions are surprisingly highly populated by attractive people of the opposite sex. How does this apply in other contexts? In medicine, they study the effectiveness of a drug instead of the health of the patients. They typically focus on success of surgery rather than well-being of the patient. Economic interlude When I graduated in 2001, we had two options for file storage. 1) Databases: • structured schemas • intensive processing done where data is stored • somewhat reliable • expensive at scale 2) Filers: • no schemas • no data processing capability • reliable • expensive at scale Since then we’ve started generating lots more data, mostly from the web. It brings up the natural idea of a data economic indicator, return on byte. How much value can I extract from a byte of data? How much does it cost to store? If we take the ratio, we want it to be bigger than one or else we discard. Of course this isn’t the whole story. There’s also a big data economic law, which states that no individual record is particularly valuable, but having every record is incredibly valuable. So for example in any of the following categories, • web index • recommendation systems • sensor data • market basket analysis • online advertising one has an enormous advantage if they have all the existing data. A brief introduction to Hadoop Back before Google had money, they had crappy hardware. They came up with idea of copying data to multiple servers. They did this physically at the time, but then they automated it. The formal automation of this process was the genesis of GFS. There are two core components to Hadoop. First is the distributed file system (HDFS), which is based on the google file system. The data stored in large files, with block sizes of 64MB to 256MB. As above, the blocks are replicated to multiple nodes in the cluster. The master node notices if a node dies. Data engineering on hadoop Hadoop is written in java, Whereas Google stuff is in C++. Writing map reduce in the java API not pleasant. Sometimes you have to write lots and lots of map reduces. However, if you use hadoop streaming, you can write in python, R, or other high-level languages. It’s easy and convenient for parallelized jobs. Cloudera Cloudera is like Red hat for hadoop. It’s done under aegis of the Apache Software Foundation. The code is available for free, but Cloudera packages it together, gives away various distributions for free, and waits for people to pay for support and to keep it up and running. Apache hive is a data warehousing system on top of hadoop. It uses an SQL-based query language (includes some map reduce -specific extensions), and it implements common join and aggregation patterns. This is nice for people who know databases well and are familiar with stuff like this. Workflow 1. Using hive, build records that contain everything I know about an entity (say a person) (intensive mapReduce stuff) 2. Write python scripts to process the records over and over again (faster and iterative, also mapReduce) 3. Update the records when new data arrives Note phase 2 are typically map-only jobs, which makes parallelization easy. I prefer standard data formats: text is big and takes up space. Thrift, Avro, protobuf are more compact, binary formats. I also encourage you to use the code and metadata repository Github. I don’t keep large data files in git. Rolling Jubilee is a better idea than the lottery Yesterday there was a reporter from CBS Morning News looking around for a quirky fun statistician or mathematician to talk about the Powerball lottery, which is worth more than$500 million right now. I thought about doing it and accumulated some cute facts I might want to say on air:
• It costs $2 to play. • If you took away the grand prize, a ticket is worth 36 cents in expectation (there are 9 ways to win with prizes ranging from$4 to $1 million). • The chance of winning grand prize is one in about 175,000,000. • So when the prize goes over$175 million, that’s worth $1 in expectation. • So if the prize is twice that, at$350 million, that’s worth $2 in expectation. • Right now the prize is$500 million, so the tickets are worth more than $2 in expectation. • Even so, the chances of being hit by lightening in a given year is something like 1,000,000, so 175 times more likely than winning the lottery In general, the expected payoff for playing the lottery is well below the price. And keep in mind that if you win, almost half goes to taxes. I am super busy trying to write, so I ended up helping find someone else for the interview: Jared Lander. I hope he has fun. If you look a bit further into the lottery system, you’ll find some questionable information. For example, lotteries are super regressive: poor people spend more money than rich people on lotteries, and way more if you think of it as a percentage of their income. One thing that didn’t occur to me yesterday but would have been nice to try, and came to me via my friend Aaron, is to suggest that instead of “investing” their$2 in a lottery, people might consider investing it in the Rolling Jubilee. Here are some reasons:
• The payoff is larger than the investment by construction. You never pay more than $1 for$1 of debt.
• It’s similar to the lottery in that people are anonymously chosen and their debts are removed.
• The taxes on the benefits are nonexistent, at least as we understand the taxcode, because it’s a gift.
It would be interesting to see how the mindset would change if people were spending money to anonymously remove debt from each other rather than to win a jackpot. Not as flashy, perhaps, but maybe more stimulative to the economy. Note: an estimated \$50 billion was spent on lotteries in 2010. That’s a lot of debt.
Categories: Uncategorized
How to evaluate a black box financial system
I’ve been blogging about evaluation methods for modeling, for example here and here, as part of the book I’m writing with Rachel Schutt based on her Columbia Data Science class this semester.
Evaluation methods are important abstractions that allow us to measure models based only on their output.
Using various metrics of success, we can contrast and compare two or more entirely different models. And it means we don’t care about their underlying structure – they could be based on neural nets, logistic regression, or decision trees, but for the sake of measuring the accuracy, or the ranking, or the calibration, the evaluation method just treats them like black boxes.
It recently occurred to me a that we could generalize this a bit, to systems rather than models. So if we wanted to evaluate the school system, or the political system, or the financial system, we could ignore the underlying details of how they are structured and just look at the output. To be reasonable we have to compare two systems that are both viable; it doesn’t make sense to talk about a current, flawed system relative to perfection, since of course every version of reality looks crappy compared to an ideal.
The devil is in the articulated evaluation metric, of course. So for the school system, we can ask various questions: Do our students know how to read? Do they finish high school? Do they know how to formulate an argument? Have they lost interest in learning? Are they civic-minded citizens? Do they compare well to other students on standardized tests? How expensive is the system?
For the financial system, we might ask things like: Does the average person feel like their money is safe? Does the system add to stability in the larger economy? Does the financial system mitigate risk to the larger economy? Does it put capital resources in the right places? Do fraudulent players inside the system get punished? Are the laws transparent and easy to follow?
The answers to those questions aren’t looking good at all: for example, take note of the recent Congressional report that blames Jon Corzine for MF Global’s collapse, pins him down on illegal and fraudulent activity, and then does absolutely nothing about it. To conserve space I will only use this example but there are hundreds more like this from the last few years.
Suffice it to say, what we currently have is a system where the agents committing fraud are actually glad to be caught because the resulting fines are on the one hand smaller than their profits (and paid by shareholders, not individual actors), and on the other hand are cemented as being so, and set as precedent.
But again, we need to compare it to another system, we can’t just say “hey there are flaws in this system,” because every system has flaws.
I’d like to compare it to a system like ours except where the laws are enforced.
That may sounds totally naive, and in a way it is, but then again we once did have laws, that were enforced, and the financial system was relatively tame and stable.
And although we can’t go back in a time machine to before Glass-Steagall was revoked and keep “financial innovation” from happening, we can ask our politicians to give regulators the power to simplify the system enough so that something like Glass-Steagall can once again work.
Categories: data science, finance, rant
I was interviewed a couple of weeks ago and it just got posted here:
Categories: #OWS
Systematized racism in online advertising, part 1
November 25, 2012 1 comment
There is no regulation of how internet ad models are built. That means that quants can use any information they want, usually historical, to decide what to expect in the future. That includes associating arrests with african-american sounding names.
In a recent Reuters article, this practice was highlighted:
Instantcheckmate.com, which labels itself the “Internet’s leading authority on background checks,” placed both ads. A statistical analysis of the company’s advertising has found it has disproportionately used ad copy including the word “arrested” for black-identifying names, even when a person has no arrest record.
Luckily, Professor Sweeney, a Harvard University professor of government with a doctorate in computer science, is on the case:
According to preliminary findings of Professor Sweeney’s research, searches of names assigned primarily to black babies, such as Tyrone, Darnell, Ebony and Latisha, generated “arrest” in the instantcheckmate.com ad copy between 75 percent and 96 percent of the time. Names assigned at birth primarily to whites, such as Geoffrey, Brett, Kristen and Anne, led to more neutral copy, with the word “arrest” appearing between zero and 9 percent of the time.
Of course when I say there’s no regulation, that’s an exaggeration. There is some, and if you claim to be giving a credit report, then regulations really do exist. But as for the above, here’s what regulators have to say:
“It’s disturbing,” Julie Brill, an FTC commissioner, said of Instant Checkmate’s advertising. “I don’t know if it’s illegal … It’s something that we’d need to study to see if any enforcement action is needed.”
Let’s be clear: this is just the beginning.
Categories: data science, news, rant
Aunt Pythia’s advice and a request for cool math books
First, my answer to last week’s question which you guys also answered:
Aunt Pythia,
My loving, wonderful, caring boyfriend slurps his food. Not just soup — everything (even cereal!). Should I just deal with it, or say something? I think if I comment on it he’ll be offended, but I find it distracting during our meals together.
Food (Consumption) Critic
——
You guys did well with answering the question, and I’d like to nominate the following for “most likely to actually make the problem go away”, from Richard:
I’d go with blunt but not particularly bothered – halfway through his next bowl of cereal, exclaim “Wow, you really slurp your food, don’t you?! I never noticed that before.”
But then again, who says we want this problem to go away? My firm belief is that every relationship needs to have an unimportant thing that bugs the participants. Sometimes it’s how high the toaster is set, sometimes it’s how the other person stacks the dishes in the dishwasher, but there’s always that thing. And it’s okay: if we didn’t have the thing we’d invent it. In fact having the thing prevents all sorts of other things from becoming incredible upsetting. My theory anyway.
So my advice to Food Consumption Critic is: don’t do anything! Cherish the slurping! Enjoy something this banal and inconsequential being your worst criticism of this lovely man.
Unless you’re like Liz, also a commenter from last week, who left her husband because of the way he breathed. If it’s driving you that nuts, you might want to go with Richard’s advice.
——
Aunt Pythia,
Dear Aunt Pythia, I want to write to an advice column, but I don’t know whether or not to trust the advice I will receive. What do you recommend?
Perplexed in SoPo
Dear PiSP,
I hear you, and you’re right to worry. Most people only ask things they kind of know the answer to, or to get validation that they’re not a total jerk, or to get permission to do something that’s kind of naughty. If the advice columnist tells them something they disagree with, they ignore it entirely anyway. It’s a total waste of time if you think about it.
However, if your question is super entertaining and kind of sexy, then I suggest you write in ASAP. That’s the very kind of question that columnists know how to answer in deep, meaningful and surprising ways.
Yours,
AP
——
Aunt Pythia,
With global warming and hot summers do you think it’s too early to bring the toga back in style?
John Doe
Dear John,
It’s never to early to wear sheets. Think about it: you get to wear the very same thing you sleep in. It’s like you’re a walking bed.
Auntie
——
Aunt Pythia,
Is it unethical not to tell my dad I’m starting a business? I doubt he’d approve and I’m inclined to wait until it’s successful to tell him about it.
Angsty New Yorker
Dear ANY,
Wait, what kind of business is this? Are we talking hedge fund or sex toy shop?
In either case, I don’t think you need to tell your parents anything about your life if you are older than 18 and don’t want to, it’s a rule of american families. Judging by my kids, this rule actually starts when they’re 11.
Of course it depends on your relationship with your father how easy that will be and what you’d miss out on by being honest, but the fear of his disapproval is, to me, a bad sign: you’re gonna have to be tough as nails to be a business owner, so get started by telling, not asking, your dad. Be prepared for him to object, and if he does, tell him he’ll get used to it with time.
Aunt Pythia
——
Aunt Pythia,
I’m a philosophy grad school dropout turned programmer who hasn’t done math since high school. But I want to learn, partly for professional reasons but mainly out of curiosity. I recently bought *Proofs From the Book* but found that I lacked the requisite mathematical maturity to work through much of it. Where should I start? What should I read? (p.s. Thanks for the entertaining blog!)
Confused in Brooklyn
Readers, this question is for you! I don’t know of too many good basic math books, so Confused in Brooklyn is counting on you. There have actually been lots of people asking similar questions, so you’d be helping them too. If I get enough good suggestions I’ll create a separate reading list for cool math page on mathbabe. Thanks in advance for your suggestions!
——
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2017-07-26 00:52:40
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https://www.simscale.com/docs/simulation-setup/contacts/contacts-cht-2/
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# Contacts in Conjugate Heat Transfer
In a Conjugate Heat Transfer (CHT) analysis, an interface defines the physical behavior between the common boundaries of two regions that are in contact, e.g. solid-solid, or solid-fluid.
Important
In CHT simulations, all the fields at the interfaces are fully constrained by the Interface type. Therefore, defining boundary conditions to faces assigned to interfaces is not allowed. This would result in an overconstrained model.
If an interface is assigned to a boundary condition, the following error message is displayed when the user tries to start a simulation:
To solve this error, the user needs to unassign the interfaces from the boundary conditions.
Additionally, interfaces between two flow regions are not possible and will result in an error when running simulations.
### Automatic Interface Detection
When creating a new CHT simulation, all possible interfaces will automatically be detected and populated in the simulation tree. Interfaces will be grouped together and defined as Coupled thermal interface.
#### How To Modify Specific Interfaces?
Individual interfaces or a group of interfaces can be filtered via entity selection. Select the entities (faces or volumes) for which you want to select all interfaces that exist between them.
Once you filter the interfaces of interest, a window opens with additional options for the Interface type for the selected contacts.
Interfaces which differ in settings from the standard bulk interfaces group will stay exposed individually in the simulation setup tree.
#### Partial Contacts
An interface is required to always be defined between two congruent surfaces, meaning that these surfaces must have the same area and overlap completely. After contact detection, the platform will also perform a check for partial contacts. If partial contacts are detected, the platform will show a warning and recommend an imprinting operation.
Imprinting is a single-click operation built into SimScale, which splits existing faces into smaller ones in order to guarantee perfect overlap between contacting faces. It is recommended to perform an imprint operation in order to guarantee accurate heat transition modeling for the simulation.
By default, any detected partial contact will be defined as an adiabatic interface, and not participate in heat conduction unless specified otherwise.
#### Contact Detection Errors
As all possible interfaces are detected automatically, it is no longer possible to manually add an interface or to change the entity assignment for a specific interface. In case no interfaces can be detected automatically, SimScale will show an error message.
In this case, it is not possible to create a mesh or start a simulation run for this simulation. Instead, the CAD model needs to be investigated for potential errors that prevent successful contact detection. Some common causes are:
• Duplicate parts: Sometimes the CAD modeling history makes it so that parts can be duplicated upon export. Please check that such a condition is not present in your model.
• Interfering parts: If the face pair is intersecting into each other by a margin higher than the CAD tolerance, the contact will not be detected. Please move the parts away so that the faces are in perfect contact, or you can also perform a boolean operation to create coincident faces.
• Small gaps between the faces: Opposite to the previous condition, if the face pair is separated by a gap higher than the CAD tolerance, the interface will not be detected. Sometimes the gap is so small that it is difficult to find it by visual inspection. Please move the parts together or extrude the faces to create the proper contact.
Some of the fix operations described above can be performed in the CAD mode. You can also reach out to support via email or chat in case you encounter this issue and can not solve it using the above suggestions.
### Interface Settings
#### Interface Type
The Interface type options define the heat exchange conditions at the interface. The five types available for the interfaces are reported below:
Coupled
The coupled thermal interface models a perfect heat transfer across the interface. This is the default setting, in case an interface is not defined by the user.
In this case, thermal energy cannot be exchanged between the domains across the interface.
Total Resistance
The Total Resistance interface allows users to model an imperfectly matching interface (e.g. due to the surface roughness) which reduces the heat exchange across it. The total resistance is defined as:
$$R = \frac{1}{K A} = \frac{1}{\frac{\kappa}{t} A} \tag{1}$$
It is worth noticing that the area of the interface appears in the definition. So this option must be assigned only to the relevant face. Let’s suppose that a heat exchanger is being simulated. The effect of solid sediment on the tube’s wall is only known as a total resistance. A first simulation proves that heat exchange performance is insufficient. Consequently, the length of the tubes is increased. The new simulation will only be correct if the total resistance is changed according to the new area of the tubes.
Specific Conductance
This interface type is very similar to the Thin layer resistance (below). It only requires users to set the specific conductance of the interface $$K$$, which is defined as:
$$K = \frac{\kappa}{t} \tag{2}$$
with thickness $$t$$ $$(m)$$ and thermal conductivity $$\kappa$$ $$(\frac {W}{mK})$$ between the two interface regions.
For instance, this option may be used for an interface where the layer thickness is negligible or unknown, i.e., a radiator for which the paint coating’s specific conductance may be given instead of its thickness and $$\kappa$$.
Thin Layer Resistance
The Thin layer resistance allows modeling a layer with thickness $$t$$ and thermal conductivity $$\kappa$$ between the two interface regions.
For example, it is possible to model the thermal paste between a chip and a heat sink without needing to resolve it in the geometry. Adding a thin layer to the geometry and meshing it is a problem, considering that the thickness of these layers is two or three orders of magnitude smaller than other components in the assembly.
### CAD and Mesh Requirements
A CHT simulation always requires a multi-region mesh. As far as the mesh is concerned, it is fundamental that the cell size at the interface is similar between the two faces. As a rule of thumb, the cells on one face should be less than 1.5 times the size of the others. The figure below shows an example of this issue. In the left case, the cells at the interface on the inner region are too small with respect to those on the outer body. In the case on the right side, the cells on the interface are approximately the same size.
Last updated: June 21st, 2022
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2022-06-26 23:45:12
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http://mathoverflow.net/questions/22454?sort=newest
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Connected subset of matrices ?
Let $m,n$ be positive integers with $m \leqslant n$, and denote by $\mu_M$ the minimal polynomial of a matrix.
Do we know for which $m$ the set $E_m$ of $M \in \mathfrak{M}_n(\mathbb{R})$ such that $\deg(\mu_M) = m$ is connected?
-
Let $C(f)$ denote the companion matrix associated to the monic polynomial $f$. Every matrix $A$ is similar to a matrix in rational canonical form: $$B=C(f_1)\oplus C(f_1 f_2)\oplus\cdots\oplus C(f_1 f_2,\cdots f_k)$$ where here $\oplus$ denotes diagonal sum. Then $m$ is the degree of $f_1 f_2\cdots f_k$. Starting with $B$ deform each $f_i$ into a power of $x$. We get a path from $B$ to $$B'=C(x^{a_1})\oplus C(x^{a_2})\oplus\cdots\oplus C(x^{a_k})$$ inside $E_m$. There's a path from $B'$ in $E_m$ given by $$(1-t)C(x^{a_1})\oplus (1-t)C(x^{a_2})\oplus\cdots\oplus C(x^{a_k})$$ ending at $$B_m=O\oplus C(x^m).$$ Thus there is a path in $E_m$ from $A$ to $UB_mU^{-1}$ where $U$ is a nonsingular matrix. If $\det(U)\ne0$ then there is a path in $GL_n(\mathbf{R})$ from $U$ to $I$ and so a path in $E_m$ from $A$ to $B_m$. If $m< n$ then there is a matrix $V$ of negative determinant with $VB_m V^{-1}=B_m$ so that we may take $U$ to have positive determinant.
The only case that remains is when $m=n$. In this case $E_m$ contains diagonal matrices with distinct entries, and each of these commutes with a matrix of negative determinant.
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2013-05-26 00:15:01
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https://www.calculators.tech/force-calculator
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# Force Calculator
Enter Information
## How to use the force calculator?
Follow these steps to use the force calculator.
1. Choose velocity from the "Find value" option.
2. Enter mass.
3. Enter acceleration.
4. Choose the unit.
5. Click "calculate".
Force calculator is an online physics tool to calculate force using newton's second law. It offers to find all the values included in the force formula i.e f=ma.
Users can select different units of measurements for quantities. It is also referred to as newton's second law calculator.
## What is the Force?
Force is described as a push or a pull experienced by a body when it interacts with another. It can cause acceleration in the motion of a moving body or can set a body at rest in motion.
Force can also alter the shape of a body. Force is a vector quantity as it has both direction and magnitude. Force is often represented by the symbol F.
### Vector Quantity
A vector quantity is one that has both direction and magnitude. For example force, weight, momentum, displacement.
## Units of Force
Force has different units in different measuring systems. Its SI unit is the newton, written as N. Its value is $kgms^{-2}$. Some other used units are;
Unit Abbreviation Equivalent newton unit
Dyne Dyn 10-5N
Gram-force gf 9.80665mN
Pound-force lbf 4.448
## Force formula
Newton described in his second law of motion that force equals mass times acceleration. The mathematical form of the Force equation gives the magnitude of the force. The formula for force is;
$F=ma$
Where
$F$ represents the force
$M$ stands for mass
$A$ represents acceleration
### Mass
The total amount of matter or material present in an object is known as the mass of the object. Its SI unit is the kilogram, written as kg. Its representation is commonly m
### Acceleration
Acceleration is possessed by a body whose velocity is not constant and is changing. The rate at which the velocity changes with respect to time is called acceleration. Its unit is meter per second squared.
## How to calculate force?
You can calculate force manually by following these easy steps.
• Calculate mass if not provided through its formula and convert it into kilograms.
• Calculate acceleration using $\dfrac{\Delta y}{\Delta t}$ in ms-2.
• Multiply mass and acceleration.
• Force is found in newtons.
### Example:
Consider a body of mass 50kg is being accelerated by $20ms^{-2}$. Force on it is 1000N.
## Net force
It is quite possible that a body is under more than one force. Net force equals all the forces acting on the body added together.
For example, a body of some mass is being pushed by a man by some force. At the same time force of gravity and the force of friction are acting on it. Hence, more than one force is acting on the body.
## How to find net force?
Calculate each force separately and add them. Mathematically, the net force formula is written as
$F_{net}= F_1+ F_2+ ...... + F_n$
Where “n” stands for No. of terms.
### Example:
A body of mass 10g is thrown vertically upward in a dry environment with force causing a negative acceleration of $30ms^{-2}$.
The acceleration due to gravity on it is $9.8ms^{-2}$. Also, the force of friction is 3N. How to calculate the net force on the ball?
### Solution:
$F_{net} = F_1+ F_2+ F_3$
$F_{net} = (10)(30) + (10)(-9.8) +(-3)$
$F_{net} = 300 + (-90.8) + (-3)$
$F_{net} = 206.2 N$
The body is going upward due to a force of 206.2 N. Negative sign is due to negative acceleration.
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2022-12-02 23:19:37
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https://www.askiitians.com/forums/Electrostatics/what-should-be-the-charge-on-a-sphere-of-radius-4c_280790.htm
|
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# What should be the charge on a sphere of radius 4cm , so that when it is brought in contact with another sphere of radius 2cm carrying charge of 10 micro coulomb , there is no transfer of charge from one sphere to other?
30 Points
one month ago
Their both potential must be same, so the answer should be, $Kq1/r1 = Kq2/r2$. on solving you will get 20 micro coloumb.
Their both potential must be the same, so the answer should be, $Kq1/r1 = Kq2/r2$. on solving you will get 20 micro coulomb.
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2021-06-16 11:38:20
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https://www.aoreugif.net/page/6/
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# aoreugif.net
the longest journey – the blog of J.B. Figueroa
## Back to the Basics: Crayon, wp-LaTex, and vimWiki
### Before I forget, I should review some of the basics.
So, this is a blog, and having your own blog requires some maintenance. It’s like owning a house except the servers are hosted in Arizona, the domain name is registered in France, and I guess you’re not owning said house. You’re renting hardware space in someone else’s basement.. Note where your local and remote machines are. Okay, so I need to remember how to bash function. How to bash function? No, how about just a bash alias?
alias alias_name="commandToRun"
while we’re at it
alias maketotaldestroy='sudo halt -p'
but the alias needs to be remembered after reboots
cd ~/
ls -la
vim .bash_aliases
include in file, continue with life.
source ~/.bash_aliases
So with this, instead of memorizing a very long username + host domain I can access my blog with only a few magic words.
alias sshblog='ssh forgettableLongName@ridiculouslyLongRemoteHostName.net'
.. checkmate. I guess we could include the password as an argument but, no. So long as remote and host have authenticated each other before, this ought to be enough.
Now, without needing to activate FTP we can update and modify everything in the blog via command line. Will need WP-CLI. Fortunately the freebsd log files kept track of the commands I issued to it oh so many seasons ago, so simply pressing the up arrow key and holding it for a minute or two sent me to the keystrokes. the magic keywords are simply:
wp help
wp help core
wp core check-update
wp core update
wp core update-db
wp core verify-checksums
wp help plugin
wp plugin update --all
wp plugin uninstall akismet
wp plugin uninstall hello
remove/deactivated unneeded plugins which could become hazardous to your health. For more information these functions have been well documented at http://wp-cli.org/docs/ or simply refer to the man pages
### How to syntax on wp?
since we’re going to be typing code here and there, a nice plug-in to have is:
https://wordpress.org/plugins/crayon-syntax-highlighter/
from here we can simply dl to our own machine or even use SSH from the remote machine to dl the file from a selected URL https://downloads.wordpress.org/plugin/crayon-syntax-highlighter.zip and then save it to the remote machine’s plugin directory, then..
via wp-cli
search for “crayon” to get a list of plugins already available through official channels
wp plugin search crayon
wp plugin install crayon-syntax-highlighter
wp plugin activate crayon-syntax-highlighter
So now we have pretty colors that support languages beyond what the regular vanilla wordpress supports. you could also git clone or wget + unzip from https://github.com/aramk/crayon-syntax-highlighter.git to get the same affect, but we’re already using wp-cli to run an update so might as do it from the remote host.
### Next order of business: LaTeX.. how to LaTeX?
simply typing in:
$latex i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>$
should produce:
$latex i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>$
Unfortunately, I won’t get to see it until I preview it on the wordpress had it even worked to begin with. Apparently a viable plugin would need access to a latex server in order to generate some sort of image. It could be a .png and there’s even one that generates .svg vector images, but it pings to some guy’s web basement server and I’d rather not have my blog reliant on whether or not some random guy forgot to pay their electric bill. Of course, there’s always the option to self host a latex server on my own or a remote machine. A search for latex on the plugin menu yields several attempts at doing this.
While searching, I discovered an old .png generator for latex inputs. It seems to be not-so-old and the current method from which wordpress.com uses to render .png images. You can play around with the below URL.
https://s0.wp.com/latex.php?latex=i\hbar\frac{\partial}{\partial+t}\left|\Psi(t)\right>%3DH\left|\Psi(t)\right>\Psi(t hey type stuf in here and get a png)&bg=ffffff&fg=000&s=0
So, there’s a wordpress.org plug-in that utilizes a similar, if not the exact same generator from wordpress.com. Note the distinction between the .com and .org service. It also has an option to generate from a self hosted server. Humm..
pro: less bandwidth
con: if access to wordpress.com goes down, image rendering capabilities go down with it. This will happen several times a year depending on who’s attacking the DNS.. but if the DNS is down this website will probably be inaccessible as well. Except in Australia. There’s a story to that.
To be reliant on external services.. Sure, to an extent.
wp plugin install wp-latex
wp plugin activate wp-latex
for more details: https://wordpress.org/plugins/wp-latex/faq/
### how to vimwiki?
I’ve been reliant on gnote and random .txt files scattered around my drive to write down random things. A friend of mine recommended vimwiki, almost randomly like he had been reading my mind (or had shell access to my laptop).
vim remember.txt
remember to go through the ssh dot files and delete any weird pgp certs you don't recognize
:qw
The vimwiki is more of a plugin than anything, and should be easily installable through pathogen.
You can find the installation instructions here: https://github.com/tpope/vim-pathogen
and for vimwiki: https://vimwiki.github.io/
I shouldn’t need to go through writing the instructions here because the details are literally right there.
What I did need to figure out was that I was expected to make my own index.wiki file inside the ~/vimwiki/ folder. I was surprised when I could “enter” and “backspace” between files joined together by a link. #jawdrop.
:Vimwiki2HTML
creates a folder with a html transaction (along with a .css) of the page you’re on. It doesn’t follow through the nested links though. With maybe about a 100 functions on that help page, it’ll take a while to get acquainted with this program. I think it’s safe to say it has more functionality than what I had been previously using. I’ll play around with this for a few weeks and then decide if I’m better off with it.
What’s next? humm… I have less than 4 days to prep up on my Korean, toy around with some java, and I think they’ll be putting me in something about a “smarter cities” think group. I suspected it was related to the CES hackathon but that’s this weekend. I should probably go to that, but I need some ‘me’ time and me time includes prepping up on my Korean and toying around with some java. Also, my friend tells me I need to get more sunlight. I do.
Had been invited by some friends to join them at the CES hackathon this weekend, and despite making an attempt to print out my ticket, I just received an e-mail specifying that entrants needed to be registered for the CES portion by 7 days ago. This would be a separate registration process, and they’re only letting the first 450 people in. It’s fine. More ‘me’ time. Also,
convert -density 300 ticket.pdf -fill blue -opaque black ticketout.pdf
will attempt to change all your black pixels into blue ones in case your printer ran out of black ink. Sadly, I also just found out that we have a Kodak that refuses to print if it detects the black is low (or refuses to print black if it detects the color is low). Hardware should not be used like this against the consumer’s interest. Alternatively I could’ve gone to a UPS store or library that opens early to get something printed – mobile ticket barcodes ftw.
On a lighter note, I finally received the schedule for the Ajou workshop on Monday, so it looks like I’m slowly placing things on that to do list.
To be honest, it would’ve been simpler had everyone used the same nomenclature, but I got to play around with all these different configurations.. It’s not like my time is worth anything – maybe it’s a French thing? Like, imagine if it’s just something you could put blame on a cultural thing. Their culture is different from ours, and our culture is somewhere hosted in Arizona – I’ve no clue what type of culture Arizonians have. Either way, despite being across an entire ocean; I’ve managed to assemble data from several sources to get this thing running. CNAME, zone file, unregistered proxy domain, mail exchange, etc. Now all we have to do is wait for DNS propagation. Time Till List – approximately 3 hours. Pretty standard.
Learning is slow, but it leads to something. It always has. I can’t put blame on culture, not as much as putting blame on me for not understanding it. There’s usually a missing context somewhere. My WORST experience overall was probably trying to figure out dynamixel specs from their source. Of course, it’s all there if you could speak/write Korean, but when an employee made the documentation your academic life depends on.. it’s frustrating. I mean, it certainly doesn’t help, yet it’s the only help you have. I’m sure the guy on the other side is 50 times smarter than I, but I just can’t understand him (or her). It’s a similar experience, except this time the ocean you have to cross is the other one (as opposed to the French one). #rant
Yet again, I feel validated in having gone with vim and acknowledging its superiority over emac. I spent the last several hours in someone else’s server (rented) making configs here and there, and just overall exploring. Rented doesn’t come with sudo access unfortunately, but a lot (most) of what I can do in a /home directory, I can do over there. Time and over again, I’ve found myself in need of vim; and it’s proven to have been the better choice. Do you know how many times I’ve needed to use emac for anything? None.. zip.. 0 – as in “zero” times. I’ve needed to know vim for a broad range of applications.
Imagine being inside a multi-hundred-thousand dollar system just digging through code, all the way to fixing a twenty-something dollar raspberry pi. Vim is so light and you can pretty much expect it to be installed in any system you plan on using, especially if it’s old. If by chance it’s not on there, getting it on there isn’t difficult. My chrome netbook was the first time I really needed to use vim. This was maybe 2 or 3 years ago. My ubuntu installation crashed, and so I needed to find another way to get the laptop running again because let’s face it, a chromebook isn’t’ really a laptop. It could be, but the operating system leaves so much more to offer and I don’t particularly like the idea of being locked in anyone’s ecosystem for too long. But my efforts failed, and I found myself in the middle of a bad install with an unsigned BIOS I had chosen to replaced the standard chromebook one. The only way to do that was to short the leads underneath the the laptop so the write protection would turn off – a simple jumper. I felt proud, it being a small, but meaningful hardware hack; but I soon found myself staring at a terminal that wasn’t acting right.
At its bare minimum, my new boot loader had some software to help it get to through the rest of its installation. I was going to put ubuntu on it, because at the time that was pretty much the only linux distro I had any experience with. Among that short list of software in the boot-loader was vi. Not vim, just vi, and I knew what it was. In the past I had forfeited getting acquainted with it because who in their right mind would use such a primitive text editor when you have all these other O-M-G-EYE-CANDY alternatives with so many other useful features? Why would any sane person torture themselves with this nonsense keyboard layout? But there I was, staring at a terminal and I had no other choice but to figure it out. So vi it was.
Fast forward a year or two, and vim’s my favorite editor. It wasn’t until, about 8 months ago that I REALLY began to appreciate it, and even more so this past summer. Back in January, I was sitting next to a professor watching the guy do kongfu coding on that aforementioned multi-hundred-thousand dollar system. I remember asking him why, and it didn’t make too much sense back then, mostly because I wasn’t the one doing it. Regardless of what you’re working on, it really comes down to the fact that you’re probably working on a terminal. Furthermore, that terminal, probably isn’t on your computer. It’s somewhere else, on an entirely different platform or maybe on an entirely different planet (or not a planet, maybe it’s traveling beyond the now detittled Pluto, and the only way to interface with it is to recompile a Fortran oracle by which the only means to edit is through..). On top of that, you’re probably connected to four different platforms at the same time. Some of these computers might be running sub-controllers, and somehow you’re beyond seven layers away. The good news is, you can control all of them; and instantly reprogram and reedit them while they’re running, (even literally as they’re running/or flying), through vim.
PuTTY was my first real linux experience, granted that was through a windows XP and 3 laptops ago. As a freshman in mechE, we had an amazing professor who made us use it, but this wasn’t something that was drilled into typical mechE students. There’s a real lack of computer programming experience brought into us by the intuition we pay tuition to. There was hardly any back in high school and I wonder how the education system has changed since. So, any of this, is something students will have to learn on their own (or they could quad-major in everything). The other day was my first time being inside of a BSD system – a unix system. I’m a senior, so maybe there’s something poetic to be said about this? So yeah, vim. Not because of the plug-ins or its advanced features, but because there’s beauty in simplicity. It somehow became simple. Also because of the terminal thing. That and I love my color syntax. It’s awesome.Oh, and I’m also on awesomewm. Seven years ago, I wouldn’t have dreamed of being able to write code like this. Seven months ago, I could hardly believe I was controlling platforms like this. It always felt surreal, like I didn’t belong here; and it still does at times. Seven layers ago, I was somebody completely different.
Let’s recheck that propagation
*hits*F5
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2020-09-26 20:35:09
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https://publish.illinois.edu/sldsc2018/2018/05/20/session-27-functional-data-analysis-in-action/
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Session 27: Functional Data Analysis in Action
Session titleFunctional Data Analysis in Action
Organizer: Kehui Chen (U of Pitt)
Chair:
Kehui Chen (U of Pitt)
Time: June 5th, 1:15pm – 2:45pm
Location: VEC 1402
Speech 1: Brain Functional Connectivity — The FDA Approach
Speaker: Jane-Ling Wang (UC Davis)
Abstract: Functional connectivity refers to the connectivity between brain regions that share functional properties. It can be defined through statistical association or dependency among two or more anatomically distinct brain regions. In functional magnetic resonance imaging (fMRI), a standard way to measure brain functional connectivity is to assess the similarity of fMRI time courses for anatomically separated brain regions. Due to the temporal nature of fMRI data, tools of functional data analysis (FDA) are intrinsically applicable to such data. However, standard functional data techniques need to be modified when the goal is to study functional connectivity. We discuss two examples, where a new functional data approach is employed to study brain functional connectivity.
Speech 2: Functional Data Methods for Replicated Point Processes
Speaker: Daniel Gervini (U of Wisconsin at Milwaukee)
Abstract: Functional Data Analysis has traditionally focused on samples of smooth functions. However, many functional data methods can be extended to discrete point processes which are driven by smooth intensity functions. We will review some models that can be used for principal component analysis, joint modelling of discrete and continuous processes, and clustering of spatio-temporal point processes. We will apply these approaches to the analysis of spatio-temporal patterns in the distribution of crime and in the use of the shared-bicycle system in the city of Chicago.
Speech 3: Frechet Regression for Time-Varying Covariance Matrices: Assessing Regional Co-Evolution in the Developing Brain
Speaker: Hans Mueller (UC Davis)
Abstract:
Frechet Regression provides an extension of Frechet means to the case of conditional Frechet means and is of interest for samples of random objects in a metric space (Petersen & Müller 2018). A specific application is encountered in cross-sectional studies where one observes $p$-dimensional vectors at one or a few random time points per subject and is interested in the p x p covariance or correlation matrix as a function of time. A challenge is that at each observation time one observes only a $p$-vector of measurements but not a covariance or correlation matrix. For a given metric on the space of covariance matrices, Frechet regression then generates a matrix function where at each fixed time the matrix is a non-negative definite covariance matrix. We demonstrate how this approach can be applied to MRI-extracted measurements of the myelin contents of various brain regions in small infants, aiming to quantify the regional co-evolution of myelination in the developing brain. Based on joint work with Alex Petersen and Sean Deoni.
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2020-07-07 16:33:57
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https://www.npsm-kps.org/journal/view.html?uid=7235&vmd=Full
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pISSN 0374-4914 eISSN 2289-0041
## Research Paper
New Phys.: Sae Mulli 2020; 70: 715-721
Published online September 29, 2020 https://doi.org/10.3938/NPSM.70.715
## Projection Photolithography for Microscale Patterning and 2D Field-effect Transistor Demonstration
So Jeong SHIN, Hyun Seok LEE*
Department of Physics, Research Institute for Nanoscale Science and Technology, Chungbuk National University, Cheongju 28644, Korea
Correspondence to:hsl@chungbuk.ac.kr
Received: July 16, 2020; Revised: July 31, 2020; Accepted: August 11, 2020
### Abstract
In this paper, we introduce a method to realize microscale patterning at arbitrary positions via a projector-based photolithography technique even without a hard photomask. For applying this technique to micro/nano device fabrications, we equip an optical microscope with a digital micromirror device module and a UV light source with a 405-nm wavelength. A bilayer photoresist (PR) and a lift-off processes are used for fabricating versatile micropatterns implemented by using this equipment, where the PMGI (polymethylglutarimide) PR and the AZ 5214 PR used for the bilayer allow the construction of undercut structures for a post-lift-off process. Through process optimization, we realize a line pattern width of $\sim$ 560 nm without a side-wall effect, nearly approaching the theoretical optical diffraction limits of the given optics. Using the optimization process, we demonstrated field-effect-transistors with a channel length of a few $\mu$m for randomly oriented triangular-MoS$_{2}$ monolayers synthesized by using chemical vapor deposition. Our demonstration visualizes that the projection photolithography technique partially replaces an expensive electron-beam lithography for microdevice fabrication at a laboratory level.
Keywords: Projection photolithography, Micropattern, Field effect transistor, 2D semiconductors, Image photomask
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2020-10-28 10:55:23
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https://math.stackexchange.com/questions/777509/parallel-between-fourier-series-and-orthogonal-projections
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Parallel between Fourier series and orthogonal projections
My professor made an analogy between Fourier series and orthogonal projections, and I was hoping someone could explain that somewhat more. Basically, as I understand it:
$$a_n = \frac{1}{L} \int_L^L f(x) \cos\left(\frac{ n\pi x}{L}\right) \ dx \longleftrightarrow c_1 = \frac{v \cdot b_i}{b_i \cdot b_i},$$
where $$\dfrac{1}{L}$$ can be thought of as normalizing the projection, and the integrand is the inner product (equivalent to the dot product on the right side).
Am I understanding this correctly? And can someone clarify this for me?
• "Paralell..." Not a pun?
– Pedro
May 1, 2014 at 23:35
He is referring to the inner product induced by the space of square-integrable functions $L^2 [-L, L]$.
$$a_n = \frac{ \left <f(x), \cos \left ( \frac{n \pi x}{L} \right )\right >}{ \left < \cos \left ( \frac{n \pi x}{L} \right ),\cos \left ( \frac{m \pi x}{L} \right )\right >} = \frac{ \int_{-L}^{L} f(x) \cos \left ( \frac{n \pi x}{L} \right ) dx }{ \int_{-L}^{L} \cos \left ( \frac{n \pi x}{L} \right ) \cos \left ( \frac{m \pi x}{L} \right ) dx }.$$
where $\left < \cos \left ( \frac{n \pi x}{L} \right ),\cos \left ( \frac{m \pi x}{L} \right )\right > = \left\{\begin{matrix} 1& \text{if} \;n = m \neq 0\\ 0& \text{if} \;n \neq m \end{matrix}\right.$
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2023-03-27 14:31:53
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https://www.gradesaver.com/textbooks/math/algebra/elementary-algebra/chapter-9-roots-and-radicals-chapter-9-review-problem-set-page-430/21
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# Chapter 9 - Roots and Radicals - Chapter 9 Review Problem Set - Page 430: 21
$\approx 5.2$
#### Work Step by Step
Using $\sqrt{3}=1.73$ and the properties of radicals, the value of the given expression, $\sqrt{27} ,$ to the nearest tenth, is \begin{array}{l}\require{cancel} \sqrt{9\cdot3} \\\\= \sqrt{(3)^2\cdot3} \\\\= 3\sqrt{3} \\\\= 3(1.73) \\\\ \approx 5.2 .\end{array}
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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2018-11-14 00:25:08
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https://www.physicsforums.com/threads/distance-to-geosynchronous-orbit.437065/
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# Distance to Geosynchronous Orbit
How do you calculate the distance above the surface of the earth to geosynchronous orbit.
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2021-06-13 02:22:14
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http://cms.math.ca/10.4153/CJM-2011-051-2
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location: Publications → journals → CJM
Abstract view
# Salem Numbers and Pisot Numbers via Interlacing
Published:2011-08-03
Printed: Apr 2012
• James McKee,
Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham, Surrey TW20 0EX, UK
• Chris Smyth,
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK
Format: LaTeX MathJax PDF
## Abstract
We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the obvious'' limit points of the set of Salem numbers produced by our theorems and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we produce all Salem numbers via an interlacing construction.
Keywords: Salem numbers, Pisot numbers
MSC Classifications: 11R06 - PV-numbers and generalizations; other special algebraic numbers; Mahler measure
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2015-10-07 19:36:34
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https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-1-foundations-for-algebra-1-2-order-of-operations-and-evaluating-expressions-practice-and-problem-solving-exercises-page-13/13
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## Algebra 1: Common Core (15th Edition)
When a fraction is raised to a power, we distribute that power to both the numerator (the top of the fraction) and the denominator (the bottom of the fraction). Thus, we obtain: $(2/3)^{3}$= ($2^{3})$/($3^{3}$) $2^{3}$ means two multiplied by itself three times and $3^{3}$ means three multiplied by itself three times, so we find: ($2^{3})$/($3^{3}$)= 8/27
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2018-05-23 15:27:30
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https://blog.ego.team/concrete-mathematics/133-closed-form-proof.html
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RtW
10 April 2022
# The Josephus Problem: The Closed Form Proof
Based on considerations of the Josephus problem for even and odd number of people, we have the following recurrence relation
1
Now we want to prove the closed-form solution
2
by the induction on .
#### Proof
Base case. Assume . Since , then and . Now, by substitution,
and
which establishes the closed-form formula holds for the base case.
Inductive step. For the inductive step, we assume and consider two cases, for odd and even . We suppose that (2) is true for an arbitrary and then show that (2) holds for .
First, assume that is even. Since the sum of even numbers is even,
Then
And therefore,
Hence (2) holds for even .
Now we assume that is odd. Since the sum of an odd and even number is odd,
So
Hence
Therefore (2) holds for odd too.
Now let's look at the closed form solution from the marginal angle.
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2022-10-01 06:07:36
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https://math.stackexchange.com/questions/362517/determine-if-the-series-is-convergent-divergent
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# Determine if the series is convergent/divergent.
$$\sum_{n=0}^{\infty}{\frac{{(-1)}^{n}}{\sqrt{n+1}}}$$
I tried the alternating series test but it did not work because the limit does not exist.
Ratio test gave me 1 so that was no help either.
• $\displaystyle\lim_{n\to\infty}\frac1{\sqrt{n+1}}=0$ – robjohn Apr 15 '13 at 16:34
This is an alternating series. To verify that, we need to check three items.
1. Signs alternate.
2. The terms go down in absolute value. In symbols, $|a_{n+1}|\le |a_n|$ for all $n$.
3. The terms have limit $0$. In symbols, $\lim_{n\to\infty}a_n=0$.
It is easy to see that in our case, all three conditions are met.
Remarks: In the post, it is stated that the limit does not exist. You probably meant that $\sum \frac{1}{\sqrt{n+1}}$ does not exist. That is true, but it has nothing to do with the alternating series test. It does imply that your series does not converge absolutely, it only converges conditionally. Informally, that means that if it weren't for the minus signs, we would not have convergence.
Note that we can use the alternating series test if the conditions we listed are true after a while. For example, you could remove the first $100$ minus signs and still have convergence. Or up to the $100$-th term the absolute values don't do down, but after that they do.
• So you're saying the alternating series test should've worked? – user72708 Apr 15 '13 at 16:33
• @user72708 the signs of the series is + , -, +, - .... – Lost1 Apr 15 '13 at 16:34
• Perfectly, and immediately. In principle there are three conditions to check. They are all easy. – André Nicolas Apr 15 '13 at 16:34
• Oh. I see it now. Thank you. My nervousness is killing me. – user72708 Apr 15 '13 at 16:36
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2019-08-24 07:41:31
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https://dspace.uib.es/xmlui/handle/11201/151901?show=full
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# A note on convergence in fuzzy metric spaces
dc.contributor.author Gregori, Valentín dc.contributor.author Miñana, Juan-José dc.contributor.author Morillas, Samuel dc.date.accessioned 2020-04-02T09:57:11Z dc.date.available 2020-04-02T09:57:11Z dc.identifier.uri http://hdl.handle.net/11201/151901 dc.description.abstract [eng] The sequential p-convergence in a fuzzy metric space, in the sense of George and Veeramani, was introduced by D. Mihet as a weaker concept than convergence. Here we introduce a stronger concept called \$s\$-convergence, and we characterize those fuzzy metric spaces in which convergent sequences are s-convergent. In such a case M is called an s-fuzzy metric. If (N_M,*) is a fuzzy metric on X where \$N_M(x,y)=\bigwedge\{M(x,y,t):t>0\}\$ then it is proved that the topologies deduced from M and N_M coincide if and only if M is an s-fuzzy metric. dc.format application/pdf dc.relation.isformatof https://doi.org/10.22111/IJFS.2014.1625 dc.relation.ispartof Iranian Journal Of Fuzzy Systems, 2014, vol. 11, num. 4, p. 75-85 dc.rights , 2014 dc.subject.classification 51 - Matemàtiques dc.subject.classification 004 - Informàtica dc.subject.other 51 - Mathematics dc.subject.other 004 - Computer Science and Technology. Computing. Data processing dc.title A note on convergence in fuzzy metric spaces dc.type info:eu-repo/semantics/article dc.date.updated 2020-04-02T09:57:11Z dc.subject.keywords Fuzzy metric space dc.subject.keywords principal fuzzy metric dc.rights.accessRights info:eu-repo/semantics/openAccess dc.identifier.doi https://doi.org/10.22111/IJFS.2014.1625
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2021-06-13 15:35:15
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https://adanet.readthedocs.io/en/v0.5.0/adanet.html
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AdaNet: Fast and flexible AutoML with learning guarantees.
## Estimators¶
High-level APIs for training, evaluating, predicting, and serving AdaNet model.
### AutoEnsembleEstimator¶
class adanet.AutoEnsembleEstimator(head, candidate_pool, max_iteration_steps, logits_fn=None, adanet_lambda=0.0, evaluator=None, metric_fn=None, force_grow=False, adanet_loss_decay=0.9, worker_wait_timeout_secs=7200, model_dir=None, config=None)[source]
Bases: adanet.core.estimator.Estimator
A tf.estimator.Estimator that learns to ensemble models.
Specifically, it learns to ensemble models from a candidate pool using the Adanet algorithm.
# A simple example of learning to ensemble linear and neural network
# models.
import tensorflow as tf
feature_columns = ...
# Learn to ensemble linear and DNN models.
candidate_pool=[
tf.estimator.LinearEstimator(
feature_columns=feature_columns,
optimizer=tf.train.FtrlOptimizer(...)),
tf.estimator.DNNEstimator(
feature_columns=feature_columns,
hidden_units=[1000, 500, 100])],
max_iteration_steps=50)
# Input builders
def input_fn_train:
# Returns tf.data.Dataset of (x, y) tuple where y represents label's
# class index.
pass
def input_fn_eval:
# Returns tf.data.Dataset of (x, y) tuple where y represents label's
# class index.
pass
def input_fn_predict:
# Returns tf.data.Dataset of (x, None) tuple.
pass
estimator.train(input_fn=input_fn_train, steps=100)
metrics = estimator.evaluate(input_fn=input_fn_eval, steps=10)
predictions = estimator.predict(input_fn=input_fn_predict)
Parameters: head – A tf.contrib.estimator.Head instance for computing loss and evaluation metrics for every candidate. candidate_pool – List of tf.estimator.Estimator objects that are candidates to ensemble at each iteration. The order does not directly affect which candidates will be included in the final ensemble. max_iteration_steps – Total number of steps for which to train candidates per iteration. If OutOfRange or StopIteration occurs in the middle, training stops before max_iteration_steps steps. logits_fn – A function for fetching the subnetwork logits from a tf.estimator.EstimatorSpec, which should obey the following signature: Args: Can only have following argument: - estimator_spec: The candidate’s tf.estimator.EstimatorSpec. Returns: Logits tf.Tensor or dict of string to logits tf.Tensor (for multi-head) for the candidate subnetwork extracted from the given estimator_spec. When None, it will default to returning estimator_spec.predictions when they are a tf.Tensor or the tf.Tensor for the key ‘logits’ when they are a dict of string to tf.Tensor. adanet_lambda – See adanet.Estimator. evaluator – See adanet.Estimator. metric_fn – See adanet.Estimator. force_grow – See adanet.Estimator. adanet_loss_decay – See adanet.Estimator. worker_wait_timeout_secs – See adanet.Estimator. model_dir – See adanet.Estimator. config – See adanet.Estimator. An adanet.AutoEnsembleEstimator instance. ValueError – If any of the candidates in candidate_pool are not tf.estimator.Estimator instances.
eval_dir(name=None)
Shows the directory name where evaluation metrics are dumped.
Parameters: name – Name of the evaluation if user needs to run multiple evaluations on different data sets, such as on training data vs test data. Metrics for different evaluations are saved in separate folders, and appear separately in tensorboard. A string which is the path of directory contains evaluation metrics.
evaluate(input_fn, steps=None, hooks=None, checkpoint_path=None, name=None)
Evaluates the model given evaluation data input_fn.
For each step, calls input_fn, which returns one batch of data. Evaluates until: - steps batches are processed, or - input_fn raises an end-of-input exception (tf.errors.OutOfRangeError or StopIteration).
Parameters: input_fn – A function that constructs the input data for evaluation. See [Premade Estimators]( https://tensorflow.org/guide/premade#create_input_functions) for more information. The function should construct and return one of the following: * A tf.data.Dataset object: Outputs of Dataset object must be a tuple (features, labels) with same constraints as below. * A tuple (features, labels): Where features is a tf.Tensor or a dictionary of string feature name to Tensor and labels is a Tensor or a dictionary of string label name to Tensor. Both features and labels are consumed by model_fn. They should satisfy the expectation of model_fn from inputs. steps – Number of steps for which to evaluate model. If None, evaluates until input_fn raises an end-of-input exception. hooks – List of tf.train.SessionRunHook subclass instances. Used for callbacks inside the evaluation call. checkpoint_path – Path of a specific checkpoint to evaluate. If None, the latest checkpoint in model_dir is used. If there are no checkpoints in model_dir, evaluation is run with newly initialized Variables instead of ones restored from checkpoint. name – Name of the evaluation if user needs to run multiple evaluations on different data sets, such as on training data vs test data. Metrics for different evaluations are saved in separate folders, and appear separately in tensorboard. A dict containing the evaluation metrics specified in model_fn keyed by name, as well as an entry global_step which contains the value of the global step for which this evaluation was performed. For canned estimators, the dict contains the loss (mean loss per mini-batch) and the average_loss (mean loss per sample). Canned classifiers also return the accuracy. Canned regressors also return the label/mean and the prediction/mean. ValueError – If steps <= 0. ValueError – If no model has been trained, namely model_dir, or the given checkpoint_path is empty.
export_saved_model(export_dir_base, serving_input_receiver_fn, assets_extra=None, as_text=False, checkpoint_path=None)
Exports inference graph as a SavedModel into the given dir.
For a detailed guide, see [Using SavedModel with Estimators](https://tensorflow.org/guide/saved_model#using_savedmodel_with_estimators).
This method builds a new graph by first calling the serving_input_receiver_fn to obtain feature Tensors, and then calling this Estimator’s model_fn to generate the model graph based on those features. It restores the given checkpoint (or, lacking that, the most recent checkpoint) into this graph in a fresh session. Finally it creates a timestamped export directory below the given export_dir_base, and writes a SavedModel into it containing a single tf.MetaGraphDef saved from this session.
The exported MetaGraphDef will provide one SignatureDef for each element of the export_outputs dict returned from the model_fn, named using the same keys. One of these keys is always tf.saved_model.signature_constants.DEFAULT_SERVING_SIGNATURE_DEF_KEY, indicating which signature will be served when a serving request does not specify one. For each signature, the outputs are provided by the corresponding tf.estimator.export.ExportOutputs, and the inputs are always the input receivers provided by the serving_input_receiver_fn.
Extra assets may be written into the SavedModel via the assets_extra argument. This should be a dict, where each key gives a destination path (including the filename) relative to the assets.extra directory. The corresponding value gives the full path of the source file to be copied. For example, the simple case of copying a single file without renaming it is specified as {‘my_asset_file.txt’: ‘/path/to/my_asset_file.txt’}.
Parameters: export_dir_base – A string containing a directory in which to create timestamped subdirectories containing exported SavedModels. serving_input_receiver_fn – A function that takes no argument and returns a tf.estimator.export.ServingInputReceiver or tf.estimator.export.TensorServingInputReceiver. assets_extra – A dict specifying how to populate the assets.extra directory within the exported SavedModel, or None if no extra assets are needed. as_text – whether to write the SavedModel proto in text format. checkpoint_path – The checkpoint path to export. If None (the default), the most recent checkpoint found within the model directory is chosen. The string path to the exported directory. ValueError – if no serving_input_receiver_fn is provided, no export_outputs are provided, or no checkpoint can be found.
export_savedmodel(export_dir_base, serving_input_receiver_fn, assets_extra=None, as_text=False, checkpoint_path=None, strip_default_attrs=False)
Exports inference graph as a SavedModel into the given dir.
Note that export_to_savedmodel will be renamed to export_saved_model in TensorFlow 2.0. At that time, export_to_savedmodel without the additional underscore will be available only through tf.compat.v1.
There is one additional arg versus the new method:
strip_default_attrs: This parameter is going away in TF 2.0, and
the new behavior will automatically strip all default attributes. Boolean. If True, default-valued attributes will be removed from the NodeDefs. For a detailed guide, see [Stripping Default-Valued Attributes]( https://github.com/tensorflow/tensorflow/blob/master/tensorflow/python/saved_model/README.md#stripping-default-valued-attributes).
get_variable_names()
Returns list of all variable names in this model.
Returns: List of names. ValueError – If the Estimator has not produced a checkpoint yet.
get_variable_value(name)
Returns value of the variable given by name.
Parameters: name – string or a list of string, name of the tensor. Numpy array - value of the tensor. ValueError – If the Estimator has not produced a checkpoint yet.
latest_checkpoint()
Finds the filename of the latest saved checkpoint file in model_dir.
Returns: The full path to the latest checkpoint or None if no checkpoint was found.
model_fn
Returns the model_fn which is bound to self.params.
Returns: def model_fn(features, labels, mode, config) The model_fn with following signature
predict(input_fn, predict_keys=None, hooks=None, checkpoint_path=None, yield_single_examples=True)
Yields predictions for given features.
Please note that interleaving two predict outputs does not work. See: [issue/20506]( https://github.com/tensorflow/tensorflow/issues/20506#issuecomment-422208517)
Parameters: input_fn – A function that constructs the features. Prediction continues until input_fn raises an end-of-input exception (tf.errors.OutOfRangeError or StopIteration). See [Premade Estimators]( https://tensorflow.org/guide/premade_estimators#create_input_functions) for more information. The function should construct and return one of the following: A tf.data.Dataset object: Outputs of Dataset object must have same constraints as below. features: A tf.Tensor or a dictionary of string feature name to Tensor. features are consumed by model_fn. They should satisfy the expectation of model_fn from inputs. A tuple, in which case the first item is extracted as features. predict_keys – list of str, name of the keys to predict. It is used if the tf.estimator.EstimatorSpec.predictions is a dict. If predict_keys is used then rest of the predictions will be filtered from the dictionary. If None, returns all. hooks – List of tf.train.SessionRunHook subclass instances. Used for callbacks inside the prediction call. checkpoint_path – Path of a specific checkpoint to predict. If None, the latest checkpoint in model_dir is used. If there are no checkpoints in model_dir, prediction is run with newly initialized Variables instead of ones restored from checkpoint. yield_single_examples – If False, yields the whole batch as returned by the model_fn instead of decomposing the batch into individual elements. This is useful if model_fn returns some tensors whose first dimension is not equal to the batch size. Evaluated values of predictions tensors. ValueError – Could not find a trained model in model_dir. ValueError – If batch length of predictions is not the same and yield_single_examples is True. ValueError – If there is a conflict between predict_keys and predictions. For example if predict_keys is not None but tf.estimator.EstimatorSpec.predictions is not a dict.
train(input_fn, hooks=None, steps=None, max_steps=None, saving_listeners=None)
Trains a model given training data input_fn.
Parameters: input_fn – A function that provides input data for training as minibatches. See [Premade Estimators]( https://tensorflow.org/guide/premade_estimators#create_input_functions) for more information. The function should construct and return one of the following: * A tf.data.Dataset object: Outputs of Dataset object must be a tuple (features, labels) with same constraints as below. * A tuple (features, labels): Where features is a tf.Tensor or a dictionary of string feature name to Tensor and labels is a Tensor or a dictionary of string label name to Tensor. Both features and labels are consumed by model_fn. They should satisfy the expectation of model_fn from inputs. hooks – List of tf.train.SessionRunHook subclass instances. Used for callbacks inside the training loop. steps – Number of steps for which to train the model. If None, train forever or train until input_fn generates the tf.errors.OutOfRange error or StopIteration exception. steps works incrementally. If you call two times train(steps=10) then training occurs in total 20 steps. If OutOfRange or StopIteration occurs in the middle, training stops before 20 steps. If you don’t want to have incremental behavior please set max_steps instead. If set, max_steps must be None. max_steps – Number of total steps for which to train model. If None, train forever or train until input_fn generates the tf.errors.OutOfRange error or StopIteration exception. If set, steps must be None. If OutOfRange or StopIteration occurs in the middle, training stops before max_steps steps. Two calls to train(steps=100) means 200 training iterations. On the other hand, two calls to train(max_steps=100) means that the second call will not do any iteration since first call did all 100 steps. saving_listeners – list of CheckpointSaverListener objects. Used for callbacks that run immediately before or after checkpoint savings. self, for chaining. ValueError – If both steps and max_steps are not None. ValueError – If either steps or max_steps <= 0.
### Estimator¶
class adanet.Estimator(head, subnetwork_generator, max_iteration_steps, mixture_weight_type='scalar', mixture_weight_initializer=None, warm_start_mixture_weights=False, adanet_lambda=0.0, adanet_beta=0.0, evaluator=None, report_materializer=None, use_bias=False, metric_fn=None, force_grow=False, replicate_ensemble_in_training=False, adanet_loss_decay=0.9, worker_wait_timeout_secs=7200, model_dir=None, report_dir=None, config=None, **kwargs)[source]
Bases: tensorflow.python.estimator.estimator.Estimator
The AdaNet algorithm implemented as a tf.estimator.Estimator.
AdaNet is as defined in the paper: https://arxiv.org/abs/1607.01097.
The AdaNet algorithm uses a weak learning algorithm to iteratively generate a set of candidate subnetworks that attempt to minimize the loss function defined in Equation (4) as part of an ensemble. At the end of each iteration, the best candidate is chosen based on its ensemble’s complexity-regularized train loss. New subnetworks are allowed to use any subnetwork weights within the previous iteration’s ensemble in order to improve upon them. If the complexity-regularized loss of the new ensemble, as defined in Equation (4), is less than that of the previous iteration’s ensemble, the AdaNet algorithm continues onto the next iteration.
AdaNet attempts to minimize the following loss function to learn the mixture weights ‘w’ of each subnetwork ‘h’ in the ensemble with differentiable convex non-increasing surrogate loss function Phi:
Equation (4):
$F(w) = \frac{1}{m} \sum_{i=1}^{m} \Phi \left(\sum_{j=1}^{N}w_jh_j(x_i), y_i \right) + \sum_{j=1}^{N} \left(\lambda r(h_j) + \beta \right) |w_j|$
with $$\lambda >= 0$$ and $$\beta >= 0$$.
This implementation uses an adanet.subnetwork.Generator as its weak learning algorithm for generating candidate subnetworks. These are trained in parallel using a single graph per iteration. At the end of each iteration, the estimator saves the sub-graph of the best subnetwork ensemble and its weights as a separate checkpoint. At the beginning of the next iteration, the estimator imports the previous iteration’s frozen graph and adds ops for the next candidates as part of a new graph and session. This allows the estimator have the performance of Tensorflow’s static graph constraint (minus the performance hit of reconstructing a graph between iterations), while having the flexibility of having a dynamic graph.
NOTE: Subclassing tf.estimator.Estimator is only necessary to work with tf.estimator.train_and_evaluate() which asserts that the estimator argument is a tf.estimator.Estimator subclass. However, all training is delegated to a separate tf.estimator.Estimator instance. It is responsible for supporting both local and distributed training. As such, the adanet.Estimator is only responsible for bookkeeping across iterations.
Parameters: head – A tf.contrib.estimator.Head instance for computing loss and evaluation metrics for every candidate. subnetwork_generator – The adanet.subnetwork.Generator which defines the candidate subnetworks to train and evaluate at every AdaNet iteration. max_iteration_steps – Total number of steps for which to train candidates per iteration. If OutOfRange or StopIteration occurs in the middle, training stops before max_iteration_steps steps. mixture_weight_type – The adanet.MixtureWeightType defining which mixture weight type to learn in the linear combination of subnetwork outputs: SCALAR: creates a rank 0 tensor mixture weight . It performs an element- wise multiplication with its subnetwork’s logits. This mixture weight is the simplest to learn, the quickest to train, and most likely to generalize well. VECTOR: creates a tensor with shape [k] where k is the ensemble’s logits dimension as defined by head. It is similar to SCALAR in that it performs an element-wise multiplication with its subnetwork’s logits, but is more flexible in learning a subnetworks’s preferences per class. MATRIX: creates a tensor of shape [a, b] where a is the number of outputs from the subnetwork’s last_layer and b is the number of outputs from the ensemble’s logits. This weight matrix-multiplies the subnetwork’s last_layer. This mixture weight offers the most flexibility and expressivity, allowing subnetworks to have outputs of different dimensionalities. However, it also has the most trainable parameters (a*b), and is therefore the most sensitive to learning rates and regularization. mixture_weight_initializer – The initializer for mixture_weights. When None, the default is different according to mixture_weight_type: SCALAR: initializes to 1/N where N is the number of subnetworks in the ensemble giving a uniform average. VECTOR: initializes each entry to 1/N where N is the number of subnetworks in the ensemble giving a uniform average. MATRIX: uses tf.zeros_initializer(). warm_start_mixture_weights – Whether, at the beginning of an iteration, to initialize the mixture weights of the subnetworks from the previous ensemble to their learned value at the previous iteration, as opposed to retraining them from scratch. Takes precedence over the value for mixture_weight_initializer for subnetworks from previous iterations. adanet_lambda – Float multiplier ‘lambda’ for applying L1 regularization to subnetworks’ mixture weights ‘w’ in the ensemble proportional to their complexity. See Equation (4) in the AdaNet paper. adanet_beta – Float L1 regularization multiplier ‘beta’ to apply equally to all subnetworks’ weights ‘w’ in the ensemble regardless of their complexity. See Equation (4) in the AdaNet paper. evaluator – An adanet.Evaluator for candidate selection after all subnetworks are done training. When None, candidate selection uses a moving average of their adanet.Ensemble AdaNet loss during training instead. In order to use the AdaNet algorithm as described in [Cortes et al., ‘17], the given adanet.Evaluator must be created with the same dataset partition used during training. Otherwise, this framework will perform AdaNet.HoldOut which uses a holdout set for candidate selection, but does not benefit from learning guarantees. report_materializer – An adanet.ReportMaterializer. Its reports are made available to the subnetwork_generator at the next iteration, so that it can adapt its search space. When None, the subnetwork_generator generate_candidates() method will receive empty Lists for their previous_ensemble_reports and all_reports arguments. use_bias – Whether to add a bias term to the ensemble’s logits. Adding a bias allows the ensemble to learn a shift in the data, often leading to more stable training and better predictions. metric_fn – A function for adding custom evaluation metrics, which should obey the following signature: Args: Can only have the following three arguments in any order: - predictions: Predictions Tensor or dict of Tensor created by given head. features: Input dict of Tensor objects created by input_fn which is given to estimator.evaluate as an argument. labels: Labels Tensor or dict of Tensor (for multi-head) created by input_fn which is given to estimator.evaluate as an argument. Returns: Dict of metric results keyed by name. Final metrics are a union of this and head’s existing metrics. If there is a name conflict between this and heads existing metrics, this will override the existing one. The values of the dict are the results of calling a metric function, namely a (metric_tensor, update_op) tuple. force_grow – Boolean override that forces the ensemble to grow by one subnetwork at the end of each iteration. Normally at the end of each iteration, AdaNet selects the best candidate ensemble according to its performance on the AdaNet objective. In some cases, the best ensemble is the previous_ensemble as opposed to one that includes a newly trained subnetwork. When True, the algorithm will not select the previous_ensemble as the best candidate, and will ensure that after n iterations the final ensemble is composed of n subnetworks. replicate_ensemble_in_training – Whether to rebuild the frozen subnetworks of the ensemble in training mode, which can change the outputs of the frozen subnetworks in the ensemble. When False and during candidate training, the frozen subnetworks in the ensemble are in prediction mode, so training-only ops like dropout are not applied to them. When True and training the candidates, the frozen subnetworks will be in training mode as well, so they will apply training-only ops like dropout. This argument is useful for regularizing learning mixture weights, or for making training-only side inputs available in subsequent iterations. For most use-cases, this should be False. adanet_loss_decay – Float decay for the exponential-moving-average of the AdaNet objective throughout training. This moving average is a data- driven way tracking the best candidate with only the training set. worker_wait_timeout_secs – Float number of seconds for workers to wait for chief to prepare the next iteration during distributed training. This is needed to prevent workers waiting indefinitely for a chief that may have crashed or been turned down. When the timeout is exceeded, the worker exits the train loop. In situations where the chief job is much slower than the worker jobs, this timeout should be increased. model_dir – Directory to save model parameters, graph and etc. This can also be used to load checkpoints from the directory into a estimator to continue training a previously saved model. report_dir – Directory where the adanet.subnetwork.MaterializedReports materialized by report_materializer would be saved. If report_materializer is None, this will not save anything. If None or empty string, defaults to “/report”. config – RunConfig object to configure the runtime settings. **kwargs – Extra keyword args passed to the parent. An Estimator instance. ValueError – If subnetwork_generator is None. ValueError – If max_iteration_steps is <= 0.
eval_dir(name=None)[source]
Shows the directory name where evaluation metrics are dumped.
Parameters: name – Name of the evaluation if user needs to run multiple evaluations on different data sets, such as on training data vs test data. Metrics for different evaluations are saved in separate folders, and appear separately in tensorboard. A string which is the path of directory contains evaluation metrics.
evaluate(input_fn, steps=None, hooks=None, checkpoint_path=None, name=None)[source]
Evaluates the model given evaluation data input_fn.
For each step, calls input_fn, which returns one batch of data. Evaluates until: - steps batches are processed, or - input_fn raises an end-of-input exception (tf.errors.OutOfRangeError or StopIteration).
Parameters: input_fn – A function that constructs the input data for evaluation. See [Premade Estimators]( https://tensorflow.org/guide/premade#create_input_functions) for more information. The function should construct and return one of the following: * A tf.data.Dataset object: Outputs of Dataset object must be a tuple (features, labels) with same constraints as below. * A tuple (features, labels): Where features is a tf.Tensor or a dictionary of string feature name to Tensor and labels is a Tensor or a dictionary of string label name to Tensor. Both features and labels are consumed by model_fn. They should satisfy the expectation of model_fn from inputs. steps – Number of steps for which to evaluate model. If None, evaluates until input_fn raises an end-of-input exception. hooks – List of tf.train.SessionRunHook subclass instances. Used for callbacks inside the evaluation call. checkpoint_path – Path of a specific checkpoint to evaluate. If None, the latest checkpoint in model_dir is used. If there are no checkpoints in model_dir, evaluation is run with newly initialized Variables instead of ones restored from checkpoint. name – Name of the evaluation if user needs to run multiple evaluations on different data sets, such as on training data vs test data. Metrics for different evaluations are saved in separate folders, and appear separately in tensorboard. A dict containing the evaluation metrics specified in model_fn keyed by name, as well as an entry global_step which contains the value of the global step for which this evaluation was performed. For canned estimators, the dict contains the loss (mean loss per mini-batch) and the average_loss (mean loss per sample). Canned classifiers also return the accuracy. Canned regressors also return the label/mean and the prediction/mean. ValueError – If steps <= 0. ValueError – If no model has been trained, namely model_dir, or the given checkpoint_path is empty.
export_saved_model(export_dir_base, serving_input_receiver_fn, assets_extra=None, as_text=False, checkpoint_path=None)[source]
Exports inference graph as a SavedModel into the given dir.
For a detailed guide, see [Using SavedModel with Estimators](https://tensorflow.org/guide/saved_model#using_savedmodel_with_estimators).
This method builds a new graph by first calling the serving_input_receiver_fn to obtain feature Tensors, and then calling this Estimator’s model_fn to generate the model graph based on those features. It restores the given checkpoint (or, lacking that, the most recent checkpoint) into this graph in a fresh session. Finally it creates a timestamped export directory below the given export_dir_base, and writes a SavedModel into it containing a single tf.MetaGraphDef saved from this session.
The exported MetaGraphDef will provide one SignatureDef for each element of the export_outputs dict returned from the model_fn, named using the same keys. One of these keys is always tf.saved_model.signature_constants.DEFAULT_SERVING_SIGNATURE_DEF_KEY, indicating which signature will be served when a serving request does not specify one. For each signature, the outputs are provided by the corresponding tf.estimator.export.ExportOutputs, and the inputs are always the input receivers provided by the serving_input_receiver_fn.
Extra assets may be written into the SavedModel via the assets_extra argument. This should be a dict, where each key gives a destination path (including the filename) relative to the assets.extra directory. The corresponding value gives the full path of the source file to be copied. For example, the simple case of copying a single file without renaming it is specified as {‘my_asset_file.txt’: ‘/path/to/my_asset_file.txt’}.
Parameters: export_dir_base – A string containing a directory in which to create timestamped subdirectories containing exported SavedModels. serving_input_receiver_fn – A function that takes no argument and returns a tf.estimator.export.ServingInputReceiver or tf.estimator.export.TensorServingInputReceiver. assets_extra – A dict specifying how to populate the assets.extra directory within the exported SavedModel, or None if no extra assets are needed. as_text – whether to write the SavedModel proto in text format. checkpoint_path – The checkpoint path to export. If None (the default), the most recent checkpoint found within the model directory is chosen. The string path to the exported directory. ValueError – if no serving_input_receiver_fn is provided, no export_outputs are provided, or no checkpoint can be found.
export_savedmodel(export_dir_base, serving_input_receiver_fn, assets_extra=None, as_text=False, checkpoint_path=None, strip_default_attrs=False)[source]
Exports inference graph as a SavedModel into the given dir.
Note that export_to_savedmodel will be renamed to export_saved_model in TensorFlow 2.0. At that time, export_to_savedmodel without the additional underscore will be available only through tf.compat.v1.
There is one additional arg versus the new method:
strip_default_attrs: This parameter is going away in TF 2.0, and
the new behavior will automatically strip all default attributes. Boolean. If True, default-valued attributes will be removed from the NodeDefs. For a detailed guide, see [Stripping Default-Valued Attributes]( https://github.com/tensorflow/tensorflow/blob/master/tensorflow/python/saved_model/README.md#stripping-default-valued-attributes).
get_variable_names()[source]
Returns list of all variable names in this model.
Returns: List of names. ValueError – If the Estimator has not produced a checkpoint yet.
get_variable_value(name)[source]
Returns value of the variable given by name.
Parameters: name – string or a list of string, name of the tensor. Numpy array - value of the tensor. ValueError – If the Estimator has not produced a checkpoint yet.
latest_checkpoint()[source]
Finds the filename of the latest saved checkpoint file in model_dir.
Returns: The full path to the latest checkpoint or None if no checkpoint was found.
model_fn
Returns the model_fn which is bound to self.params.
Returns: def model_fn(features, labels, mode, config) The model_fn with following signature
predict(input_fn, predict_keys=None, hooks=None, checkpoint_path=None, yield_single_examples=True)[source]
Yields predictions for given features.
Please note that interleaving two predict outputs does not work. See: [issue/20506]( https://github.com/tensorflow/tensorflow/issues/20506#issuecomment-422208517)
Parameters: input_fn – A function that constructs the features. Prediction continues until input_fn raises an end-of-input exception (tf.errors.OutOfRangeError or StopIteration). See [Premade Estimators]( https://tensorflow.org/guide/premade_estimators#create_input_functions) for more information. The function should construct and return one of the following: A tf.data.Dataset object: Outputs of Dataset object must have same constraints as below. features: A tf.Tensor or a dictionary of string feature name to Tensor. features are consumed by model_fn. They should satisfy the expectation of model_fn from inputs. A tuple, in which case the first item is extracted as features. predict_keys – list of str, name of the keys to predict. It is used if the tf.estimator.EstimatorSpec.predictions is a dict. If predict_keys is used then rest of the predictions will be filtered from the dictionary. If None, returns all. hooks – List of tf.train.SessionRunHook subclass instances. Used for callbacks inside the prediction call. checkpoint_path – Path of a specific checkpoint to predict. If None, the latest checkpoint in model_dir is used. If there are no checkpoints in model_dir, prediction is run with newly initialized Variables instead of ones restored from checkpoint. yield_single_examples – If False, yields the whole batch as returned by the model_fn instead of decomposing the batch into individual elements. This is useful if model_fn returns some tensors whose first dimension is not equal to the batch size. Evaluated values of predictions tensors. ValueError – Could not find a trained model in model_dir. ValueError – If batch length of predictions is not the same and yield_single_examples is True. ValueError – If there is a conflict between predict_keys and predictions. For example if predict_keys is not None but tf.estimator.EstimatorSpec.predictions is not a dict.
train(input_fn, hooks=None, steps=None, max_steps=None, saving_listeners=None)[source]
Trains a model given training data input_fn.
Parameters: input_fn – A function that provides input data for training as minibatches. See [Premade Estimators]( https://tensorflow.org/guide/premade_estimators#create_input_functions) for more information. The function should construct and return one of the following: * A tf.data.Dataset object: Outputs of Dataset object must be a tuple (features, labels) with same constraints as below. * A tuple (features, labels): Where features is a tf.Tensor or a dictionary of string feature name to Tensor and labels is a Tensor or a dictionary of string label name to Tensor. Both features and labels are consumed by model_fn. They should satisfy the expectation of model_fn from inputs. hooks – List of tf.train.SessionRunHook subclass instances. Used for callbacks inside the training loop. steps – Number of steps for which to train the model. If None, train forever or train until input_fn generates the tf.errors.OutOfRange error or StopIteration exception. steps works incrementally. If you call two times train(steps=10) then training occurs in total 20 steps. If OutOfRange or StopIteration occurs in the middle, training stops before 20 steps. If you don’t want to have incremental behavior please set max_steps instead. If set, max_steps must be None. max_steps – Number of total steps for which to train model. If None, train forever or train until input_fn generates the tf.errors.OutOfRange error or StopIteration exception. If set, steps must be None. If OutOfRange or StopIteration occurs in the middle, training stops before max_steps steps. Two calls to train(steps=100) means 200 training iterations. On the other hand, two calls to train(max_steps=100) means that the second call will not do any iteration since first call did all 100 steps. saving_listeners – list of CheckpointSaverListener objects. Used for callbacks that run immediately before or after checkpoint savings. self, for chaining. ValueError – If both steps and max_steps are not None. ValueError – If either steps or max_steps <= 0.
### TPUEstimator¶
class adanet.TPUEstimator(head, subnetwork_generator, max_iteration_steps, mixture_weight_type='scalar', mixture_weight_initializer=None, warm_start_mixture_weights=False, adanet_lambda=0.0, adanet_beta=0.0, evaluator=None, report_materializer=None, use_bias=False, metric_fn=None, force_grow=False, replicate_ensemble_in_training=False, adanet_loss_decay=0.9, worker_wait_timeout_secs=7200, model_dir=None, report_dir=None, config=None, use_tpu=True, train_batch_size=None, eval_batch_size=None)[source]
Bases: adanet.core.estimator.Estimator, tensorflow.contrib.tpu.python.tpu.tpu_estimator.TPUEstimator
An adanet.Estimator capable of running on TPU.
If running on TPU, all summary calls are rewired to be no-ops during training.
WARNING: this API is highly experimental, unstable, and can change without warning.
eval_dir(name=None)
Shows the directory name where evaluation metrics are dumped.
Parameters: name – Name of the evaluation if user needs to run multiple evaluations on different data sets, such as on training data vs test data. Metrics for different evaluations are saved in separate folders, and appear separately in tensorboard. A string which is the path of directory contains evaluation metrics.
evaluate(input_fn, steps=None, hooks=None, checkpoint_path=None, name=None)
Evaluates the model given evaluation data input_fn.
For each step, calls input_fn, which returns one batch of data. Evaluates until: - steps batches are processed, or - input_fn raises an end-of-input exception (tf.errors.OutOfRangeError or StopIteration).
Parameters: input_fn – A function that constructs the input data for evaluation. See [Premade Estimators]( https://tensorflow.org/guide/premade#create_input_functions) for more information. The function should construct and return one of the following: * A tf.data.Dataset object: Outputs of Dataset object must be a tuple (features, labels) with same constraints as below. * A tuple (features, labels): Where features is a tf.Tensor or a dictionary of string feature name to Tensor and labels is a Tensor or a dictionary of string label name to Tensor. Both features and labels are consumed by model_fn. They should satisfy the expectation of model_fn from inputs. steps – Number of steps for which to evaluate model. If None, evaluates until input_fn raises an end-of-input exception. hooks – List of tf.train.SessionRunHook subclass instances. Used for callbacks inside the evaluation call. checkpoint_path – Path of a specific checkpoint to evaluate. If None, the latest checkpoint in model_dir is used. If there are no checkpoints in model_dir, evaluation is run with newly initialized Variables instead of ones restored from checkpoint. name – Name of the evaluation if user needs to run multiple evaluations on different data sets, such as on training data vs test data. Metrics for different evaluations are saved in separate folders, and appear separately in tensorboard. A dict containing the evaluation metrics specified in model_fn keyed by name, as well as an entry global_step which contains the value of the global step for which this evaluation was performed. For canned estimators, the dict contains the loss (mean loss per mini-batch) and the average_loss (mean loss per sample). Canned classifiers also return the accuracy. Canned regressors also return the label/mean and the prediction/mean. ValueError – If steps <= 0. ValueError – If no model has been trained, namely model_dir, or the given checkpoint_path is empty.
export_saved_model(export_dir_base, serving_input_receiver_fn, assets_extra=None, as_text=False, checkpoint_path=None)
Exports inference graph as a SavedModel into the given dir.
For a detailed guide, see [Using SavedModel with Estimators](https://tensorflow.org/guide/saved_model#using_savedmodel_with_estimators).
This method builds a new graph by first calling the serving_input_receiver_fn to obtain feature Tensors, and then calling this Estimator’s model_fn to generate the model graph based on those features. It restores the given checkpoint (or, lacking that, the most recent checkpoint) into this graph in a fresh session. Finally it creates a timestamped export directory below the given export_dir_base, and writes a SavedModel into it containing a single tf.MetaGraphDef saved from this session.
The exported MetaGraphDef will provide one SignatureDef for each element of the export_outputs dict returned from the model_fn, named using the same keys. One of these keys is always tf.saved_model.signature_constants.DEFAULT_SERVING_SIGNATURE_DEF_KEY, indicating which signature will be served when a serving request does not specify one. For each signature, the outputs are provided by the corresponding tf.estimator.export.ExportOutputs, and the inputs are always the input receivers provided by the serving_input_receiver_fn.
Extra assets may be written into the SavedModel via the assets_extra argument. This should be a dict, where each key gives a destination path (including the filename) relative to the assets.extra directory. The corresponding value gives the full path of the source file to be copied. For example, the simple case of copying a single file without renaming it is specified as {‘my_asset_file.txt’: ‘/path/to/my_asset_file.txt’}.
Parameters: export_dir_base – A string containing a directory in which to create timestamped subdirectories containing exported SavedModels. serving_input_receiver_fn – A function that takes no argument and returns a tf.estimator.export.ServingInputReceiver or tf.estimator.export.TensorServingInputReceiver. assets_extra – A dict specifying how to populate the assets.extra directory within the exported SavedModel, or None if no extra assets are needed. as_text – whether to write the SavedModel proto in text format. checkpoint_path – The checkpoint path to export. If None (the default), the most recent checkpoint found within the model directory is chosen. The string path to the exported directory. ValueError – if no serving_input_receiver_fn is provided, no export_outputs are provided, or no checkpoint can be found.
export_savedmodel(export_dir_base, serving_input_receiver_fn, assets_extra=None, as_text=False, checkpoint_path=None, strip_default_attrs=False)
Exports inference graph as a SavedModel into the given dir.
Note that export_to_savedmodel will be renamed to export_saved_model in TensorFlow 2.0. At that time, export_to_savedmodel without the additional underscore will be available only through tf.compat.v1.
There is one additional arg versus the new method:
strip_default_attrs: This parameter is going away in TF 2.0, and
the new behavior will automatically strip all default attributes. Boolean. If True, default-valued attributes will be removed from the NodeDefs. For a detailed guide, see [Stripping Default-Valued Attributes]( https://github.com/tensorflow/tensorflow/blob/master/tensorflow/python/saved_model/README.md#stripping-default-valued-attributes).
get_variable_names()
Returns list of all variable names in this model.
Returns: List of names. ValueError – If the Estimator has not produced a checkpoint yet.
get_variable_value(name)
Returns value of the variable given by name.
Parameters: name – string or a list of string, name of the tensor. Numpy array - value of the tensor. ValueError – If the Estimator has not produced a checkpoint yet.
latest_checkpoint()
Finds the filename of the latest saved checkpoint file in model_dir.
Returns: The full path to the latest checkpoint or None if no checkpoint was found.
model_fn
Returns the model_fn which is bound to self.params.
Returns: def model_fn(features, labels, mode, config) The model_fn with following signature
predict(input_fn, predict_keys=None, hooks=None, checkpoint_path=None, yield_single_examples=True)[source]
Yields predictions for given features.
Please note that interleaving two predict outputs does not work. See: [issue/20506]( https://github.com/tensorflow/tensorflow/issues/20506#issuecomment-422208517)
Parameters: input_fn – A function that constructs the features. Prediction continues until input_fn raises an end-of-input exception (tf.errors.OutOfRangeError or StopIteration). See [Premade Estimators]( https://tensorflow.org/guide/premade_estimators#create_input_functions) for more information. The function should construct and return one of the following: A tf.data.Dataset object: Outputs of Dataset object must have same constraints as below. features: A tf.Tensor or a dictionary of string feature name to Tensor. features are consumed by model_fn. They should satisfy the expectation of model_fn from inputs. A tuple, in which case the first item is extracted as features. predict_keys – list of str, name of the keys to predict. It is used if the tf.estimator.EstimatorSpec.predictions is a dict. If predict_keys is used then rest of the predictions will be filtered from the dictionary. If None, returns all. hooks – List of tf.train.SessionRunHook subclass instances. Used for callbacks inside the prediction call. checkpoint_path – Path of a specific checkpoint to predict. If None, the latest checkpoint in model_dir is used. If there are no checkpoints in model_dir, prediction is run with newly initialized Variables instead of ones restored from checkpoint. yield_single_examples – If False, yields the whole batch as returned by the model_fn instead of decomposing the batch into individual elements. This is useful if model_fn returns some tensors whose first dimension is not equal to the batch size. Evaluated values of predictions tensors. ValueError – Could not find a trained model in model_dir. ValueError – If batch length of predictions is not the same and yield_single_examples is True. ValueError – If there is a conflict between predict_keys and predictions. For example if predict_keys is not None but tf.estimator.EstimatorSpec.predictions is not a dict.
train(input_fn, hooks=None, steps=None, max_steps=None, saving_listeners=None)[source]
Trains a model given training data input_fn.
Parameters: input_fn – A function that provides input data for training as minibatches. See [Premade Estimators]( https://tensorflow.org/guide/premade_estimators#create_input_functions) for more information. The function should construct and return one of the following: * A tf.data.Dataset object: Outputs of Dataset object must be a tuple (features, labels) with same constraints as below. * A tuple (features, labels): Where features is a tf.Tensor or a dictionary of string feature name to Tensor and labels is a Tensor or a dictionary of string label name to Tensor. Both features and labels are consumed by model_fn. They should satisfy the expectation of model_fn from inputs. hooks – List of tf.train.SessionRunHook subclass instances. Used for callbacks inside the training loop. steps – Number of steps for which to train the model. If None, train forever or train until input_fn generates the tf.errors.OutOfRange error or StopIteration exception. steps works incrementally. If you call two times train(steps=10) then training occurs in total 20 steps. If OutOfRange or StopIteration occurs in the middle, training stops before 20 steps. If you don’t want to have incremental behavior please set max_steps instead. If set, max_steps must be None. max_steps – Number of total steps for which to train model. If None, train forever or train until input_fn generates the tf.errors.OutOfRange error or StopIteration exception. If set, steps must be None. If OutOfRange or StopIteration occurs in the middle, training stops before max_steps steps. Two calls to train(steps=100) means 200 training iterations. On the other hand, two calls to train(max_steps=100) means that the second call will not do any iteration since first call did all 100 steps. saving_listeners – list of CheckpointSaverListener objects. Used for callbacks that run immediately before or after checkpoint savings. self, for chaining. ValueError – If both steps and max_steps are not None. ValueError – If either steps or max_steps <= 0.
## Ensembles¶
Collections representing learned combinations of subnetworks.
### MixtureWeightType¶
class adanet.MixtureWeightType[source]
Mixture weight types available for learning subnetwork contributions.
The following mixture weight types are defined:
• SCALAR: Produces a rank 0 Tensor mixture weight.
• VECTOR: Produces a rank 1 Tensor mixture weight.
• MATRIX: Produces a rank 2 Tensor mixture weight.
### WeightedSubnetwork¶
class adanet.WeightedSubnetwork[source]
A weighted subnetwork is a weight ‘w’ applied to a subnetwork’s last layer ‘u’. The results is the weighted subnetwork’s logits, regularized by its complexity.
Parameters: name – String name of subnetwork as defined by its adanet.subnetwork.Builder. iteration_number – Integer iteration when the subnetwork was created. weight – The weight tf.Tensor or dict of string to weight tf.Tensor (for multi-head) to apply to this subnetwork. The AdaNet paper refers to this weight as ‘w’ in Equations (4), (5), and (6). logits – The output tf.Tensor or dict of string to weight tf.Tensor (for multi-head) after the matrix multiplication of weight and the subnetwork’s last_layer(). The output’s shape is [batch_size, logits_dimension]. It is equivalent to a linear logits layer in a neural network. subnetwork – The adanet.subnetwork.Subnetwork to weight. An adanet.WeightedSubnetwork object.
### Ensemble¶
class adanet.Ensemble[source]
An ensemble is a collection of subnetworks which forms a neural network through the weighted sum of their outputs. It is represented by ‘f’ throughout the AdaNet paper. Its component subnetworks’ weights are complexity regularized (Gamma) as defined in Equation (4).
Parameters: weighted_subnetworks – List of adanet.WeightedSubnetwork instances that form this ensemble. Ordered from first to most recent. bias – Bias term tf.Tensor or dict of string to bias term tf.Tensor (for multi-head) for the ensemble’s logits. logits – Logits tf.Tensor or dict of string to logits tf.Tensor (for multi-head). The result of the function ‘f’ as defined in Section 5.1 which is the sum of the logits of all adanet.WeightedSubnetwork instances in ensemble. An adanet.Ensemble instance.
## Evaluator¶
Measures adanet.Ensemble performance on a given dataset.
### Evaluator¶
class adanet.Evaluator(input_fn, steps=None)[source]
Evaluates candidate ensemble performance.
Parameters: input_fn – Input function returning a tuple of: features - Dictionary of string feature name to Tensor. labels - Tensor of labels. steps – Number of steps for which to evaluate the ensembles. If an OutOfRangeError occurs, evaluation stops. If set to None, will iterate the dataset until all inputs are exhausted. An adanet.Evaluator instance.
evaluate_adanet_losses(sess, adanet_losses)[source]
Evaluates the given AdaNet objectives on the data from input_fn.
The candidates are fed the same batches of features and labels as provided by input_fn, and their losses are computed and summed over steps batches.
Parameters: sess – Session instance with most recent variable values loaded. adanet_losses – List of AdaNet loss Tensors. List of evaluated AdaNet losses.
input_fn
Return the input_fn.
steps
Return the number of evaluation steps.
## Summary¶
Extends tf.summary to power AdaNet’s TensorBoard integration.
### Summary¶
class adanet.Summary[source]
Interface for writing summaries to Tensorboard.
audio(name, tensor, sample_rate, max_outputs=3, family=None)[source]
Outputs a tf.Summary protocol buffer with audio.
The summary has up to max_outputs summary values containing audio. The audio is built from tensor which must be 3-D with shape [batch_size, frames, channels] or 2-D with shape [batch_size, frames]. The values are assumed to be in the range of [-1.0, 1.0] with a sample rate of sample_rate.
The tag in the outputted tf.Summary.Value protobufs is generated based on the name, with a suffix depending on the max_outputs setting:
• If max_outputs is 1, the summary value tag is ‘name/audio’.
• If max_outputs is greater than 1, the summary value tags are
generated sequentially as ‘name/audio/0’, ‘name/audio/1’, etc
Parameters: name – A name for the generated node. Will also serve as a series name in TensorBoard. tensor – A 3-D float32 Tensor of shape [batch_size, frames, channels] or a 2-D float32 Tensor of shape [batch_size, frames]. sample_rate – A Scalar float32 Tensor indicating the sample rate of the signal in hertz. max_outputs – Max number of batch elements to generate audio for. family – Optional; if provided, used as the prefix of the summary tag name, which controls the tab name used for display on Tensorboard. A scalar Tensor of type string. The serialized tf.Summary protocol buffer.
histogram(name, values, family=None)[source]
Outputs a tf.Summary protocol buffer with a histogram.
Adding a histogram summary makes it possible to visualize your data’s distribution in TensorBoard. You can see a detailed explanation of the TensorBoard histogram dashboard [here](https://www.tensorflow.org/get_started/tensorboard_histograms).
The generated [tf.Summary]( tensorflow/core/framework/summary.proto) has one summary value containing a histogram for values.
This op reports an InvalidArgument error if any value is not finite.
Parameters: name – A name for the generated node. Will also serve as a series name in TensorBoard. values – A real numeric Tensor. Any shape. Values to use to build the histogram. family – Optional; if provided, used as the prefix of the summary tag name, which controls the tab name used for display on Tensorboard. A scalar Tensor of type string. The serialized tf.Summary protocol buffer.
image(name, tensor, max_outputs=3, family=None)[source]
Outputs a tf.Summary protocol buffer with images.
The summary has up to max_outputs summary values containing images. The images are built from tensor which must be 4-D with shape [batch_size, height, width, channels] and where channels can be:
• 1: tensor is interpreted as Grayscale.
• 3: tensor is interpreted as RGB.
• 4: tensor is interpreted as RGBA.
The images have the same number of channels as the input tensor. For float input, the values are normalized one image at a time to fit in the range [0, 255]. uint8 values are unchanged. The op uses two different normalization algorithms:
• If the input values are all positive, they are rescaled so the largest
one is 255. * If any input value is negative, the values are shifted so input value 0.0
is at 127. They are then rescaled so that either the smallest value is 0, or the largest one is 255.
The tag in the outputted tf.Summary.Value protobufs is generated based on the name, with a suffix depending on the max_outputs setting:
• If max_outputs is 1, the summary value tag is ‘name/image’.
• If max_outputs is greater than 1, the summary value tags are
generated sequentially as ‘name/image/0’, ‘name/image/1’, etc.
Parameters: name – A name for the generated node. Will also serve as a series name in TensorBoard. tensor – A 4-D uint8 or float32 Tensor of shape [batch_size, height, width, channels] where channels is 1, 3, or 4. max_outputs – Max number of batch elements to generate images for. family – Optional; if provided, used as the prefix of the summary tag name, which controls the tab name used for display on Tensorboard. A scalar Tensor of type string. The serialized tf.Summary protocol buffer.
scalar(name, tensor, family=None)[source]
Outputs a tf.Summary protocol buffer containing a single scalar value.
The generated tf.Summary has a Tensor.proto containing the input Tensor.
Parameters: name – A name for the generated node. Will also serve as the series name in TensorBoard. tensor – A real numeric Tensor containing a single value. family – Optional; if provided, used as the prefix of the summary tag name, which controls the tab name used for display on Tensorboard. A scalar Tensor of type string. Which contains a tf.Summary protobuf. ValueError – If tensor has the wrong shape or type.
## ReportMaterializer¶
### ReportMaterializer¶
class adanet.ReportMaterializer(input_fn, steps=None)[source]
Materializes reports.
Specifically it materializes a subnetwork’s adanet.subnetwork.Report instances into adanet.subnetwork.MaterializedReport instances.
Requires an input function input_fn that returns a tuple of:
• features: Dictionary of string feature name to Tensor.
• labels: Tensor of labels.
Parameters: input_fn – The input function. steps – Number of steps for which to materialize the ensembles. If an OutOfRangeError occurs, materialization stops. If set to None, will iterate the dataset until all inputs are exhausted. A ReportMaterializer instance.
input_fn
Returns the input_fn that materialize_subnetwork_reports would run on.
Even though this property appears to be unused, it would be used to build the AdaNet model graph inside AdaNet estimator.train(). After the graph is built, the queue_runners are started and the initializers are run, AdaNet estimator.train() passes its tf.Session as an argument to materialize_subnetwork_reports(), thus indirectly making input_fn available to materialize_subnetwork_reports.
materialize_subnetwork_reports(sess, iteration_number, subnetwork_reports, included_subnetwork_names)[source]
Materializes the Tensor objects in subnetwork_reports using sess.
This converts the Tensors in subnetwork_reports to ndarrays, logs the progress, converts the ndarrays to python primitives, then packages them into adanet.subnetwork.MaterializedReports.
Parameters: sess – Session instance with most recent variable values loaded. iteration_number – Integer iteration number. subnetwork_reports – Dict mapping string names to subnetwork.Report objects to be materialized. included_subnetwork_names – List of string names of the subnetwork.Reports that are included in the final ensemble. List of adanet.subnetwork.MaterializedReport objects.
steps
Return the number of steps.
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# Online Activities 18-24 January
For a list of talks in the coming weeks, see https://ests.wordpress.com/online-activities-2021.
Paris-Lyon Séminaire de Logique
Time:
Wednesday, 20 January, 16:00-17:00 CEST
Speaker: Gianluca Basso, University of Lyon
Title: Compact metrizable structures via projective Fraïssé theory
Abstract: The goal of projective Fraïssé theory is to approximate compact metrizable spaces via classes of finite structures and glean topological or dynamical properties of a space by relating them to combinatorial features of the associated class of structures. We will discuss general results, using the framework of compact metrizable structures, as well as applications to the study a class of one-dimensional compact metrizable spaces, that of smooth fences, and to a particular smooth fence with remarkable properties, which we call the Fraïssé fence.
Information: Join via the link on the seminar webpage 10 minutes before the talk.
Barcelona Set Theory Seminar
Time: Wednesday, 20 January, 16:00-17:30 CET
Speaker: Vera Fischer, University of Vienna
Title: Independent families in the countable and the uncountable
Abstract: Independent families on w are families of infinite sets of integers with the property that for any two finite subfamilies A and B the set Ç A\È B is infinite. Of particular interest are the sets of the possible cardinalities of maximal independent families, which we refer to as the spectrum of independence. Even though we do have the tools to control the spectrum of independence at w (at least to a large extent), there are many relevant questions regarding higher counterparts of independence in generalised Baire spaces still remaining open.
Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.
Bristol Logic and Set Theory Seminar/Oxford Set Theory Seminar
Time:
Wednesday, 20 January, 16:00-17:30 UK time (17:00-18:30 CEST)
Speaker: Dima Sinapova, University of Illinois at Chicago
Title: Iteration, reflection, and singular cardinals
Abstract: Two classical results of Magidor are:
(1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$, and
(2) from large cardinals it is consistent to have the failure of SCH at $\aleph_\omega$.These principles are at odds with each other. The former is a compactness type principle. (Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object.) In contrast, failure of SCH is an instance of incompactness. The natural question is whether we can have both of these simultaneously. We show the answer is yes.
We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we obtain this situation at $\aleph_\omega$. This is joint work with Alejandro Poveda and Assaf Rinot.
Caltech Logic Seminar
Time: Wednesday, 20 January, 12:00 – 1:00pm Pacific time (21:00 CET)
Speaker: Todor Tsankov, Université Lyon 1
Title: Universal minimal flows of homeomorphism groups of high-dimensional manifolds
Abstract: The first interesting case of a non-trivial, metrizable universal minimal flow (UMF) of a Polish group was computed by Pestov who proved that the UMF of the homeomorphism group of the circle is the circle itself. This naturally led to the question whether a similar result is true for homeomorphism groups of other manifolds (or more general topological spaces). A few years later, Uspenskij proved that the action of a group on its UMF is never 3-transitive, thus giving a negative answer to the question for a vast collection of topological spaces. Still, the question of metrizability of their UMFs remained open and he asked specifically whether the UMF of the homeomorphism group of the Hilbert cube is metrizable. We give a negative answer to this question for the Hilbert cube and all closed manifolds of dimension at least 2, thus showing that metrizability of the UMF of a homeomorphism group is essentially a one-dimensional phenomenon. This is joint work with Yonatan Gutman and Andy Zucker.
Information: See the seminar webpage.
KGRC Research Seminar, Vienna
Time:
Thursday, 21 January, 15:00-16:30 CET
Speaker: Juris Steprans, York University, Toronto, Canada
Title: Strong colourings over partitions
Abstract: The celebrated result of Todorcevic that ℵ1↛[ℵ1]2ℵ1 provides a well known example of a strong colouring. A mapping c:[ω1]2→κ is a strong colouring over a partition p:[ω1]2→ω if for every uncountable X⊆ω1 there is n∈ω such that the range of c on [X]2∩p−1{n} is all of κ. I will discuss some recent work with A. Rinot and M. Kojman on negative results concerning strong colourings over partitions and their relation to classical results in this area.
Information: Talk via zoom.
Toronto Set Theory Seminar
Time: Friday, 21 January, 11:00am-12:30pm Toronto time (17:00-18:30 CET)
Speaker: Dima Sinapova, UIC, University of Illinois at Chicago
Title: Iteration, reflection, and singular cardinals
Abstract: Two classical results of Magidor are: (1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$ and (2) from large cardinals it is consistent to have the failure of SCH at $\aleph_{\omega}$.
These principles are at odds with each other. The former is a compactness type principle. (Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object.) In contrast, failure of SCH is an instance of incompactness. The natural question is whether we can have both of these simultaneously. We show the answer is yes.
We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we obtain this situation at $\aleph_{\omega}$ . This is joint work with Alejandro Poveda and Assaf Rinot.
Information: Email Ivan Ongay Valverde ahead of time for the zoom link.
CUNY Set Theory Seminar
Time: Friday, 22 January, 2pm New York time (20:00 CET)
Speaker: Erin Carmody, Fordham University
Title: The relationships between measurable and strongly compact cardinals
Abstract: This talk is about the ongoing investigation of the relationships between measurable and strongly compact cardinals. I will present some of the history of the theorems in this theme, including Magidor’s identity crisis, and give new results. The theorems presented are in particular about the relationships between strongly compact cardinals and measurable cardinals of different Mitchell orders. One of the main theorems is that there is a universe where κ1 and κ2 are the first and second strongly compact cardinals, respectively, and where κ1 is least with Mitchell order 1, and κ2is the least with Mitchell order 2. Another main theorem is that there is a universe where κ1 and κ2are the first and second strongly compact cardinals, respectively, with κ1 the least measurable cardinal such that o(κ1)=2 and κ2 the least measurable cardinal above κ1. This is a joint work in progress with Victoria Gitman and Arthur Apter.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.
Toronto Set Theory Seminar
Time: Friday, 21 January, 1:30-3pm Toronto time (19:30-21:00 CET)
Speaker: Marcos Mazari Armida, Carnegie Mellon University
Title: Universal models in classes of abelian groups and modules
Abstract: The search for universal models began in the early twentieth century when Hausdorff showed that there is a universal linear order of cardinality $\aleph_{n+1}$ if $2^{\aleph_n}= \aleph_{n + 1}$, i.e., a linear order $U$ of cardinality $\aleph_{n+1}$ such that every linear order of cardinality $\aleph_{n+1}$ embeds in $U$. In this talk, we will study universal models in several classes of abelian groups and modules with respect to pure embeddings. In particular, we will present a complete solution below $\aleph_\omega$, with the exception of $\aleph_0$ and $\aleph_1$, to Problem 5.1 in page 181 of \emph{Abelian Groups} by L\'{a}szl\'{o} Fuchs, which asks to find the cardinals $\lambda$ such that there is a universal abelian p-group for purity of cardinality $\lambda$. The solution presented will use both model-theoretic and set-theoretic ideas.
Information: Email Ivan Ongay Valverde ahead of time for the zoom link.
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# Set Theory Workshop, Sao Paulo
This is an event organized as a celebration of the World Logic Day (January 14th) as proclaimed by UNESCO in association with the International Council for Philosophy and Human Sciences (CIPSH).
https://www.ime.usp.br/~spld2021/
## List of Speakers
• Christina Brech (São Paulo)
• Vera Fischer (Vienna)
• Yurii Khomskii (Hamburg & Amsterdam)
• Victor dos Santos Ronchim (São Paulo)
• Dorottya Sziráki (Budapest)
• Artur Hideyuki Tomita (São Paulo)
## Book of Abstracts
The book of abstracts may be found here.
## Registration
There is no registration fee for this conference. For registration, please fill this form with your name, affiliation and e-mail. After registration, you will receive a link to the Zoom webinar.
## Preliminary schedule
The event will be held on January 14th.
Brazilia Standard Time (GMT -3):
• 09:00 – 09:10 – Reception
• 09:10 – 09:55 – Christina Brech
• 10:00 – 10:45 – Dorottya Sziráki
• 11:00 – 11:45 – Victor dos Santos Ronchim
• 13:00 – 13:45 – Vera Fischer
• 13:50 – 14:35 – Artur Hideyuki Tomita
• 14:50 – 15:35 – Yurii Khomskii
Greenwich Mean Time (GMT):
• 12:00 – 12:10 – Reception
• 12:10 – 12:55 – Christina Brech
• 13:00 – 13:45 – Dorottya Sziráki
• 14:00 – 14:45 – Victor dos Santos Ronchim
• 16:00 – 16:45 – Vera Fischer
• 16:50 – 17:35 – Artur Hideyuki Tomita
• 17:50 – 18:35 – Yurii Khomskii
# Online activities 11-17 January
For a list of talks in the coming weeks, see https://ests.wordpress.com/online-activities-2021.
Genova logic seminar
Time: Monday, 11 January, 15.00-16.30 CET
Speaker: Filippo Calderoni, University of Illinois at Chicago
Title: Categorifying Borel reducibility
Abstract: The theory of Borel classification is a central research area in modern descriptive set theory. It provides a logical treatment to the process of classification and has been used effectively in different areas of mathematics as a tool to detect structural obstructions against classification theorems. The idea of making Borel reducibility functorial goes back to the start of the area, being raised already in one of the initial papers by Friedman and Stanley. In this talk we will discuss yet another attempt to formalize Borel reducibility in a categorical framework. This is joint work in progress with Andrew Brooke-Taylor.
Information: The seminar will be held on Microsoft Teams, at the page of the Genoa logic group. The access code is: fpedcxn. Alternatively, you can write to camerlo@dima.unige.it to have an access link. Further information on the activities of the Genoa logic group can be
found at logic.dima.unige.it
Caltech Logic Seminar
Time: Monday, 11 January, 12:00 – 1:00pm Pacific time (21:00 CET)
Speaker: Zoltán Vidnyánszky, Caltech
Title: A new regularity property of the Haar null ideal
Abstract: Christensen’s Haar null ideal is a well-behaved generalization of Haar null sets to groups, which admit no Haar measure. We show that in the group ZωZω, every Haar positive (that is, non-Haar null) analytic set contains a Haar positive closed set. Using this result, we determine the exact Wadge class of the family of Haar null closed subsets of ZωZω.
Information: See the seminar webpage.
Hebrew University-Bar Ilan University Set Theory seminar
Time: Wednesday, 13 January, 14:00-16:00 Israel Time (13:00-15:00 CET)
Speaker: Juris Steprans, York University, Toronto
Title: Universal functions, strong colourings and ideas from PID
Abstract: A construction of Shelah will be reformulated using the PID to provide alternative models of the failure of CH and the existence of a universal colouring of cardinality aleph_1. The impact of the range of the colourings will be examined. An application to the theory of strong colourings over partitions will
Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.
Barcelona Set Theory Seminar
Time: Wednesday, 13 January, 16:00-17:30 CET
Speaker: Trevor Wilson, Miami University
Title: tba
Abstract: tba
Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.
Set theory workshop Sao Paulo, for the World Logic Day
Time:
Thursday, 14 January, 9:00-18:35 Brazil time (13:00-22:35 CET)
Speakers: Christina Brech (São Paulo), Vera Fischer (Vienna), Yurii Khomskii (Hamburg , Victor dos Santos Ronchim (São Paulo), Dorottya Sziráki (Budapest), Artur Hideyuki Tomita (São Paulo)
KGRC Research Seminar, Vienna
Time:
Thursday, 14 January, 15:00-16:30 CET
Speaker: Jeffrey Bergfalk, University of Vienna
Title: Infinitary combinatorics and strong homology
Abstract: Motivated by several recent advances, we will provide a research history of the main set-theoretic problems arising in the study of strong homology. As such, this talk will overlap with one on the same theme given in Paris-Lyon Logic Seminar last fall. We will presume no awareness in our audience either of strong homology or of that talk, but will aim in this one to provide, along with the necessary background, some sketch of the main ideas behind several recent arguments. This is an area in which simplicial principles and infinitary combinatorics come together. Its questions, at heart, have tended to be questions about higher-dimensional variants of classical set-theoretic concerns (like nontrivial coherence, Δ systems, etc.); these questions, in turn, increasingly appear to be of some interest in their own right.
Information: Talk via zoom.
Turin-Udine logic seminar
Time: Friday, 15 January, 16:30-18:30 CET
Title: Ackermann, Goodstein, and infinite sets
Abstract: In this talk, I show how Goodstein’s classical theorem can be turned into a statement that entails the existence of complex infinite sets, or in other words: into an object of reverse mathematics. This more abstract approach allows for very uniform results of high explanatory power. Specifically, I present versions of Goodstein’s theorem that are equivalent to arithmetical comprehension and arithmetical transfinite recursion. To approach the latter, we will study a functorial extension of the Ackermann function to all ordinals. The talk is based on a joint paper with J. Aguilera, M. Rathjen and A. Weiermann.
Information: Online on WebEx. Please see the seminar webpage.
CUNY Set Theory Seminar
Time: Friday, January 15, 2pm New York time (20:00 CET)
Speaker: Trevor Wilson, Miami University
Title: tba
Abstract: tba
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.
# Online activities 4-10 January
Happy new year! For a list of talks in the coming weeks, see https://ests.wordpress.com/online-activities-2021.
Turin logic seminar
Time: Friday, 8 January, 09.30-10.30 CET
Speaker: A. Conversano, Massey, New Zealand
Title: Model theory and groups
Abstract: tba
Information: Online. Please see the seminar webpage.
Turin-Udine logic seminar
Time: Friday, 8 January, 16:30-18:30 CET
Speaker: F. Calderoni, University of Illinois at Chicago
Title: The Borel structure on the space of left-orderings
Abstract: In this talk we shall present some results on left-orderable groups and their interplay with descriptive set theory. We shall discuss how Borel classification can be used to analyze the space of left-orderings of a given countable group modulo the conjugacy action. In particular, we shall see that if G is not locally indicable then the conjugacy relation on LO(G) is not smooth. Also, if G is a nonabelian free group, then the conjugacy relation on LO(G) is a universal countable Borel equivalence relation. Our results address a question of Deroin-Navas-Rivas and show that in many cases LO(G) modulo the conjugacy action is nonstandard. This is joint work with A. Clay.
Information: Online on WebEx. Please see the seminar webpage.
CUNY Set Theory Seminar
Time: Friday, 8 January, 2pm New York time (20:00 CET)
Speaker: Thilo Weinert, University of Vienna
Title: A miscellany of observations regarding cardinal characteristics of the continuum
Abstract: We are going to talk about some inequalities between cardinal characteristics of the continuum. In particular we are going to relate cardinal characteristics pertaining to the convergenve of series, recently isolated by Blass, Brendle, Brian and Hamkins, other characteristcs concerning equitable splitting defined comparatatively recently by Brendle, Halbeisen, Klausner, Lischka and Shelah and some characteristics defined less recently by Miller, Blass, Laflamme and Minami. All proofs in question are elementary.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.
# Online Activities 28 December 2020 – 3 January 2021
Hebrew University-Bar Ilan University Set Theory seminar
Time: Wednesday, 30 December, 14:00-16:00 Israel Time (13:00-15:00 CET)
Speaker: Assaf Shani, Harvard
Title: Actions of tame abelian product groups
Abstract: A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel. Suppose that G=\prod_n \Gamma_n is a product of countable abelian groups. It follows from results of Solecki and Ding-Gao that if G is tame, then all of its actions are in fact potentially \Pi^0_6. Ding and Gao conjectured that this bound could be improved to \Pi^0_3. We refute this, by finding an action of a tame abelian product group, which is not potentially \Pi^0_5. The proof involves forcing over models where the axiom of choice fails for sequences of finite sets. This is joint work with Shaun Allison.
Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.
# Online Activities 21-27 December
Hebrew University-Bar Ilan University Set Theory seminar
Time: Wednesday, 23 December, 14:00-16:00 Israel Time (13:00-15:00 CET)
Speaker: Roy Shalev, Bar Ilan University
Title: A guessing principle from a Souslin tree, with applications to topology
Abstract: We introduce a new combinatorial principle which we call ♣_AD. This principle asserts the existence of a certain multi-ladder system with guessing and almost-disjointness features, and is shown to be sufficient for carrying out de Caux type constructions of topological spaces.
Our main result states that strong instances of ♣_AD follow from the existence of a Souslin tree. As an application, we obtain a simple, de Caux type proof of Rudin’s result that if there is a Souslin tree, then there is an S-space which is Dowker.
Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.
# Online Activities 14-20 December
Hebrew University-Bar Ilan University Set Theory seminar
Time: Wednesday, 16 December, 14:00-16:00 Israel Time (13:00-15:00 CET)
Speaker: Roy Shalev, Bar Ilan University
Title: S spaces and L spaces, part 1
Abstract: It will be both a survey talk and exposition of new results. Very likely it will be continued the following week.
An S-space is a regular topological space that is hereditarily separable but not Lindel\”of. An L-space is a regular topological space that is hereditarily Lindel\”of but not separable. We will survey the history behind the question of their existence and present some constructions.
Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.
Barcelona Set Theory Seminar
Time: Wednesday, 16 December, 16:00-17:30 CET
Speaker: Victoria Gitman, CUNY
Title: Characterizing large cardinals via abstract logics
Abstract: First-order logic, the commonly accepted formal system underlying mathematics, must draw however minimally on the properties of the set-theoretic universe in which it is defined. Stronger logics such as infinitary logics and second-
order logics require access to much larger chunks of the set-theoretic background. Niceness properties of these logics, such as forms of compactness, are naturally
connected to the existence of large cardinals. Indeed, many large cardinals can be
characterized in terms of compactness properties of strong logics. Strongly compact
and weakly compact cardinals k are precisely the strong and weak compactness
cardinals respectively for the infinitary logic Lkk. Extendible cardinals k are precisely
the strong compactness cardinals for the infinitary second-order logic \$\mathbb L2
kk. Vopenka’s Principle holds if and only if every logic has a strong compactness cardinal. In this talk I will review properties of various logics and how their compactness
properties characterize various large cardinals. I will discuss joint work with Will Boney, Stamatis Dimopoulos and Menachem Magidor in which we show that the principle Ord is subtle, in the presence of global choice, holds if and only if every logic
has a weak compactness cardinal, i.e., it is the analogue of Vopenka’s Principle for weak compactness. We also provide characterizations of the various virtual large cardinals using a new notion of a pseudo-model of a theory.
Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.
Münster research seminar on set theory
Time: Wednesday, 16 December, 15:15-16:45 CET
Speaker: Tba
Title: Tba
Abstract: Tba
Information: Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.
Caltech Logic Seminar
Time: Wednesday, 16 December, 12:00 – 1:00pm Pacific time (22:00 CET)
Speaker: Martino Lupini, Victoria University of Wellington
Title: Classification of extension of C*-algebras and K-homology
Abstract: I will present an introduction from the perspective of Borel complexity theory to the classification problem for extension of C*-algebras, its motivations from operator theory, and its connections with homological algebra.
Information: See the seminar webpage.
KGRC Research Seminar
Time:
Thursday, 17 December, 15:00 CET
Speaker: Peter Holy, University of Udine
Title: Ramsey-like Operators
Abstract: Starting from measurability upwards, larger large cardinals are usually characterized by the existence of certain elementary embeddings of the universe, or dually, the existence of certain ultrafilters. However, below measurability, we have a somewhat similar picture when we consider certain embeddings with set-sized domain, or ultrafilters for small collections of sets. I will present some new results, and also review some older ones, showing that not only large cardinals below measurability, but also several related concepts can be characterized in such a way, and I will also provide a sample application of these characterizations.
Information: Talk via zoom.
Turin-Udine logic seminar
Time: Friday, 18 December, 16:30pm CET
Speaker: Monroe Eskew, University of Vienna
Title: Weak square from weak presaturation
Abstract: Can we have both a saturated ideal and the tree property on ℵ2? Towards the negative direction, we show that for a regular cardinal κ, if 2<κ≤κ+ and there is a weakly presaturated ideal on κ+ concentrating on cofinality κ, then □∗κ holds. This partially answers a question of Foreman and Magidor about the approachability ideal on ℵ2. A surprising corollary is that if there is a presaturated ideal J on ℵ2 such that P(ℵ2)/J is a semiproper forcing, then CH holds. This is joint work with Sean Cox.
Information: Please see the seminar webpage.
# Master and PhD fellowships in mathematics in Paris
__________________________________ PhD Cofund MathInParis2020 _______________________________
The international Doctoral Training in Mathematical Sciences in Paris MathInParis2020, cofunded by European Commission’s Marie Sklodowska-Curie Actionsoffers 20 PhD fellowships for academic year 2021-2022. The positions are located in Paris.
Call for applications: from Tuesday December 1st 2020 to Saturday February 13th 2021 at 11:59 p.m., Paris time.
Offer description: https://www.sciencesmaths-paris.fr/fr/cofund-mathinparis-842.htm
_____________________________________PGSM _______________________________________________
The PGSM program of the Fondation Sciences Mathématiques de Paris offers master scholarships in Mathematics and in fundamental Computer Science for academic year 2021-2022. The positions are located in Paris. Schedule of deadlines below (at 11:59 p.m., Paris time). Feel free to circulate this message to your contacts.
1st call: before Friday January 22nd 2021. Only for students from universities out of France.
2nd call: from Tuesday December 1st 2020 to Saturday May 8th 2021. Open to the same students plus those from universities of FSMP’s network.
Offer description: https://www.sciencesmaths-paris.fr/en/masters-250.htm
# Online Activities 7-12 December
Computability and application seminar
Time: Tuesday, 8 December, 22:00 CET
Speaker: Linda Brown Westrick, Pennsylvania State University
Title: Luzin’s (N) and randomness reflection
Abstract: Tba
Information: The seminar will take place virtually via zoom.
Münster research seminar on set theory
Time: Wednesday, December 9, 15:15-16:45 CET
Speaker: Tba
Title: Tba
Abstract: Tba
Information: Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.
Barcelona Set Theory Seminar
Time:
Wednesday, December 9, 16:00 CET
Speaker: Neil Barton, University of Konstanz
Title: Intensional classes and intuitionistic topoi
Abstract: A popular view in the philosophy of set theory is that of potentialism: the position that the set-theoretic universe unfolds as more sets come into existence. A difficult question for the potentialist is to explain how classes (understood as intensional entities) behave on this framework, and in particular what logic governs them. In this talk we’ll see how category-theoretic resources can be brought to bear on this issue. I’ll first give a brief introduction to topos theory, and then I’ll explain how (drawing on work of Lawvere) we can think of intensional classes for the potentialist as given by a functor category. I’ll suggest some tentative directions for research here, including the possibility that this representation indicates that the logic of intensional classes should be intuitionistic rather than classical, and that the strength of the
intuitionistic logic is dependent upon the partial order on the worlds.
Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.
Paris-Lyon Séminaire de Logique
Time:
Wednesday, December 9, 16:00-17:00 CET
Speaker: Andrea Vaccaro, IMJ-PRG
Title: Set Theory and the Endomorphisms of the Calkin algebra
Abstract: Set theory induces a sharp dichotomy in the structure of the set of automorphisms of the Calkin algebra Q(H): under the Open Coloring Axiom (OCA) all the automorphisms of Q(H) are inner (Farah, 2011), whereas the Continuum Hypothesis (CH) implies that there exist uncountably many outer automorphisms of Q(H) (Phillips-Weaver, 2007). After a brief introduction on the line of research that led to these results, I’ll discuss how this dichotomic behavior extends to the semigroup End(Q(H)) of unital endomorphisms of Q(H). In particular, we’ll see that under OCA all unital endomorphisms of Q(H) can be, up to unitary equivalence, lifted to unital endomorphisms of B(H). This fact allows to have an extremely clean picture of End(Q(H)), and has some interesting consequences concerning the class of C*-algebras that embed into Q(H). I will also discuss how the structure of End(Q(H)) completely changes under CH.
Caltech Logic Seminar
Time: Wednesday, December 9, 12:00 – 1:00pm Pacific time (22:00 CET)
Speaker: Jeffrey Bergfalk, University of Vienna
Title: The definable content of (co)homological invariants II: definable cohomology and homotopy classification
Abstract: In this, the second of a three-part series of talks, we describe a “definable Čech cohomology theory” strictly refining its classical counterpart. As applications, we show that, in strong contrast to its classical counterpart, this definable cohomology theory provides complete homotopy invariants for mapping telescopes of dd-tori and of dd-spheres; we also show that it provides an equivariant homotopy classification of maps from mapping telescopes of dd-tori to spheres, a problem raised in the d=1d=1 case by Borsuk and Eilenberg in 1936. These results build on those of the first talk. They entail, for example, an analysis of the phantom maps from a locally compact Polish space X to a polyhedron P; instrumental in that analysis is the definable lim1lim1functor. They entail more generally an analysis of the homotopy relation on the space of maps from X to P, and we will begin by describing a category particularly germane for this analysis. Time permitting, we will conclude with some discussion and application of a related construction, namely that of the definable homotopy groups of a locally compact Polish space X.
This is joint work with Martino Lupini and Aristotelis Panagiotopoulos.
Information: See the seminar webpage.
KGRC Research Seminar
Time:
Thursday, December 10, 15:00 CET
Speaker: Michael Hrušák, UNAM, Mexico City
Title: Invariant Ideal Axiom
Abstract: We shall introduce a consistent set-theoretic axiom which has a profound impact on convergence properties in topological groups. As an application we show that consistently (consequence of IIA) every countable sequential group is either metrizable or kω.
Information: Talk via zoom.
Turin-Udine logic seminar
Time: Friday, 11 December, 16:30pm CET
Speaker: Assaf Shani, Harvard University
Title: Classification results for countable Archimedean groups
Abstract: We study the isomorphism relation for countable ordered Archimedean groups. We locate its complexity with respect to the hierarchy defined by Hjorth, Kechris, and Louveau, showing in particular that its potential complexity is D(Π03), and it cannot be classified using countable sets of reals as invariants. We obtain analogous results for the bi-embeddability relation, and we consider similar problems for circularly ordered groups and ordered divisible Abelian groups. This is joint work with F. Calderoni, D. Marker, and L. Motto Ros.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.
CUNY Set Theory Seminar
Time: Friday, December 10, 11am New York time (17:00 CET)
Speaker: Dima Sinapova, University of Chicago
Title: Iteration, reflection, and singular cardinals
Abstract: There is an inherent tension between stationary reflection and the failure of the singular cardinal hypothesis (SCH). The former is a compactness type principle that follows from large cardinals. Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object. In contrast, failure of SCH is an instance of incompactness.
Two classical results of Magidor are:
(1) from large cardinals it is consistent to have reflection at ℵω+1, and
(2) from large cardinals it is consistent to have the failure of SCH at ℵω.
As these principles are at odds with each other, the natural question is whether we can have both. We show the answer is yes.
We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we obtain this situation at ℵω by interleaving collapses. This is joint work with Alejandro Poveda and Assaf Rinot.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.
# Thirteenth Panhellenic Logic Symposium, July 2021
PLS13: THE THIRTEENTH PANHELLENIC LOGIC SYMPOSIUM
July 14-18, 2021, Volos, Greece
Organized by the University of Thessaly
http://panhellenic-logic-symposium.org/
======================================================================
DISCLAIMER
Our intention is to have a meeting with physical presence, in the spirit of the symposium in all its previous meetings. If this becomes difficult due to the worsening of the pandemic, we will consider other options ranging from an online event to postponing the event to 2022. Any further updates on this front will be posted on the symposium’s webpage.
In any case, we will have a normal Call for Papers procedure; that is, submitted papers will be peer-reviewed and all accepted papers will appear in the electronic (informal) proceedings that we shall post on the event’s webpage.
======================================================================
IMPORTANT DATES
Deadline for submission: Friday, March 26, 2021
Final version due: Wednesday, May 26, 2021
======================================================================
INVITED TALKS
– Andrew Brooke-Taylor, University of Leeds, UK
– Takayuki Kihara, Nagoya University, Japan
– Julia Knight, University of Notre Dame, USA
– Vassileios Koutavas, Trinity College Dublin, Ireland
– Angus Macintyre, Queen Mary University of London, UK
– Thanases Pheidas, University of Crete, Greece
– Alexandra Silva, University College London, UK
– Linda Brown Westrick, Penn State University, USA
TUTORIALS
– Alex Kavvos, University of Bristol, UK
– Nikos Leonardos, University of Athens, Greece
– Stathis Zachos, National Technical University of Athens, Greece
=====================================================================
SPECIAL SESSIONS
Computer Science:
– Bruno Bauwens, HSE University, Russia
– Juan Garay, Texas A&M University, USA
– Andrew Lewis-Pye, London School of Economics, UK
– Vassilis Zikas, Purdue University, USA & University of Edinburgh, UK
Philosophical Logic:
– Michael Glanzberg, Rutgers University, USA
– Volker Halbach, University of Oxford, UK
– Elia Zardini, University of Lisbon, Portugal & HSE University, Russia
=====================================================================
FIRST CALL FOR PAPERS
The Scientific Committee cordially invites all researchers in the areas of the conference to submit their papers for presentation at PLS13. All submitted papers will be reviewed by the Scientific Committee of the symposium, who will make final decisions on acceptance. Accepted papers will appear in the electronic volume of the event’s (informal) proceedings; the volume will be posted on the event’s webpage. During the actual (in-person) event, each accepted paper will be presented by one of its authors; in case an in-person event cannot happen due to the pandemic, we shall consider other presentation options.
Areas of interest include (but are not limited to):
– Computability Theory
– History and Philosophy of Logic
– Logic in Computer Science
– Model Theory
– Nonclassical and Modal Logics
– Proof Theory
– Set Theory
Papers, in PDF format, should be prepared using the EasyChair class style (easychair.org/publications/for_authors), be written in English, and adhere to a space limit of 6 pages.
Submission is to be done via EasyChair at: https://easychair.org/account/signin?l=SfDov6IqlVTGz0eOMKYlPK#
======================================================================
POSTER SESSION AND MENTORING SESSION
Graduate students and young researchers are invited to submit a short abstract on work in progress that may not be ready for a regular contributed talk. Those accepted will be able to present their work in poster form in a special poster session. The session will also feature a mentoring component whereby senior researchers will discuss the posters and provide feedback to student participants.
Interested students and young researchers should submit abstracts of no more than one page in PDF form by Friday, June 4, 2021, by sending them to: pls13@softlab.ntua.gr
======================================================================
GRANTS
Some travel grants will be provided for students and young researchers. Details will be uploaded on the conference webpage as soon as they become available.
=====================================================================
SCIENTIFIC COMMITTEE
– Antonis Achilleos, Reykjavik University
– George Barmpalias, Chinese Academy of Sciences (co-chair)
– Costas Dimitracopoulos, University of Athens
– Pantelis Eleftheriou, University of Konstanz & University of Pisa
– Vassilis Gregoriades, National Technical University of Athens
– Kostas Hatzikiriakou, University of Thessaly
– Antonis Kakas, University of Cyprus
– Alex Kavvos, University of Bristol
– Nikolaos Papaspyrou, National Technical University of Athens
– Thanases Pheidas, University of Crete
– Ana Sokolova, University of Salzburg
– Alexandra Soskova, Sofia University
– Mariya Soskova, University of Wisconsin–Madison
– Yannis Stephanou, University of Athens
– Konstantinos Tsaprounis, University of the Aegean (co-chair)
– Nikos Tzevelekos, Queen Mary University of London
– Niki Vazou, IMDEA Institute
– Stathis Zachos, National Technical University of Athens
ORGANIZING COMMITTEE
– Kostas Hatzikiriakou, University of Thessaly (chair)
– Nikolaos Papaspyrou, National Technical University of Athens
– Vasiliki Papayiannakopoulou, University of Thessaly
======================================================================
SYMPOSIUM WEBPAGE: http://panhellenic-logic-symposium.org/
E-MAIL: pls13@softlab.ntua.gr
CONTACTS:
– George Barmpalias (barmpalias@gmail.com)
– Konstantinos Tsaprounis (kostas.tsap@gmail.com)
Chairs of the Scientific Committee
– Kostas Hatzikiriakou (kxatzkyr@uth.gr), Chair of the Organizing Committee
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2021-01-24 20:45:44
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http://www.oalib.com/relative/3627364
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Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
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J. An Physics , 2011, DOI: 10.1088/0004-637X/736/2/151 Abstract: If the augmented density of a spherical anisotropic system is assumed to be multiplicatively separable to functions of the potential and the radius, the radial function, which can be completely specified by the behavior of the anisotropy parameter alone, also fixes the anisotropic ratios of every higher-order velocity moment. It is inferred from this that the non-negativity of the distribution function necessarily limits the allowed behaviors of the radial function. This restriction is translated into the constraints on the behavior of the anisotropy parameter. We find that not all radial variations of the anisotropy parameter satisfy these constraints and thus that there exist anisotropy profiles that cannot be consistent with any separable augmented density.
Physics , 2013, DOI: 10.1093/mnras/stt1236 Abstract: We study a new class of equilibrium two-parametric distribution functions of spherical stellar systems with radially anisotropic velocity distribution of stars. The models are less singular counterparts of the so called generalized polytropes, widely used in works on equilibrium and stability of gravitating systems in the past. The offered models, unlike the generalized polytropes, have finite density and potential in the center. The absence of the singularity is necessary for proper consideration of the radial orbit instability, which is the most important instability in spherical stellar systems. Comparison of the main observed parameters (potential, density, anisotropy) predicted by the present models and other popular equilibrium models is provided.
Physics , 2012, DOI: 10.1063/1.4737928 Abstract: We put forward a simple procedure for extracting dynamical information from Monte Carlo simulations, by appropriate matching of the short-time diffusion tensor with its infinite-dilution limit counterpart, which is supposed to be known. This approach --discarding hydrodynamics interactions-- first allows us to improve the efficiency of previous Dynamic Monte Carlo algorithms for spherical Brownian particles. In a second step, we address the case of anisotropic colloids with orientational degrees of freedom. As an illustration, we present a detailed study of the dynamics of thin platelets, with emphasis on long-time diffusion and orientational correlations.
International Journal of Astronomy and Astrophysics (IJAA) , 2018, DOI: 10.4236/ijaa.2018.81004 Abstract: We provide solutions to Einsteins field equations for a model of a spherically symmetric anisotropic fluid distribution, relevant to the description of compact stars. The central matter-energy density, radial and tangential pressures, red shift and speed of sound are positive definite and are decreasing monotonically with increasing radial distance from the center of matter distribution of astrophysical object. The causality condition is satisfied for complete fluid distribution. The central value of anisotropy is zero and is increasing monotonically with increasing radial distance from the center of the distribution. The adiabatic index is increasing with increasing radius of spherical fluid distribution. The stability conditions in relativistic compact star are also discussed in our investigation. The solution is representing the realistic objects such as SAXJ1808.4-3658, HerX-1, 4U1538-52, LMC X-4, CenX-3, VelaX-1, PSRJ1614-2230 and PSRJ0348+0432 with suitable conditions.
Physics , 1998, DOI: 10.1046/j.1365-8711.1998.01939.x Abstract: In this paper we investigate the gravothermal instability of spherical stellar systems endowed with a radially anisotropic velocity distribution. We focus our attention on the effects of anisotropy on the conditions for the onset of the instability and in particular we study the dependence of the spatial structure of critical models on the amount of anisotropy present in a system. The investigation has been carried out by the method of linear series which has already been used in the past to study the gravothermal instability of isotropic systems. We consider models described by King, Wilson and Woolley-Dickens distribution functions. In the case of King and Woolley-Dickens models, our results show that, for quite a wide range of amount of anisotropy in the system, the critical value of the concentration of the system (defined as the ratio of the tidal to the King core radius of the system) is approximately constant and equal to the corresponding value for isotropic systems. Only for very anisotropic systems the critical value of the concentration starts to change and it decreases significantly as the anisotropy increases and penetrates the inner parts of the system. For Wilson models the decrease of the concentration of critical models is preceded by an intermediate regime in which critical concentration increases, it reaches a maximum and then it starts to decrease. The critical value of the central potential always decreases as the anisotropy increases.
Michele Cappellari Physics , 2015, Abstract: Cappellari (2008) presented a flexible and efficient method to model the stellar kinematics of anisotropic axisymmetric and spherical stellar systems. The spherical formalism could be used to model the line-of-sight velocity second moments allowing for essentially arbitrary radial variation in the anisotropy and general luminous and total density profiles. Here we generalize the spherical formalism by providing the expressions for all three components of the projected second moments, including the two proper motion components. A reference implementation is now included in the public JAM package available at http://purl.org/cappellari/software
Physics , 1999, DOI: 10.1046/j.1365-8711.2000.03615.x Abstract: We discuss the influence of the cosmological background density field on the spherical infall model. The spherical infall model has been used in the Press-Schechter formalism to evaluate the number abundance of clusters of galaxies, as well as to determine the density parameter of the universe from the infalling flow. Therefore, the understanding of collapse dynamics play a key role for extracting the cosmological information. Here, we consider the modified version of the spherical infall model. We derive the mean field equations from the Newtonian fluid equations, in which the influence of cosmological background inhomogeneity is incorporated into the averaged quantities as the {\it backreaction}. By calculating the averaged quantities explicitly, we obtain the simple expressions and find that in case of the scale-free power spectrum, the density fluctuations with the negative spectral index make the infalling velocities slow. This suggests that we underestimate the density parameter $\Omega$ when using the simple spherical infall model. In cases with the index $n>0$, the effect of background inhomogeneity could be negligible and the spherical infall model becomes the good approximation for the infalling flows. We also present a realistic example with the cold dark matter power spectrum. There, the anisotropic random velocity leads to slowing down the mean infalling velocities.
D. A. Garanin Physics , 1998, DOI: 10.1088/0305-4470/29/10/006 Abstract: The corrections to the Curie temperature T_c of a ferromagnetic film consisting of N layers are calculated for N \gg 1 for the model of D-component classical spin vectors in the limit D \to \infty, which is exactly soluble and close to the spherical model. The present approach accounts, however, for the magnetic anisotropy playing the crucial role in the crossover from 3 to 2 dimensions in magnetic films. In the spatially inhomogeneous case with free boundary conditions the D=\infty model is nonequivalent to the standard spherical one and always leads to the diminishing of T_c(N) relative to the bulk.
Physics , 2012, DOI: 10.1088/0264-9381/29/24/245008 Abstract: A class of spherical collapsing exact solutions with electromagnetic charge is derived. This class of solutions -- in general anisotropic -- contains however as a particular case the charged dust model already known in literature. Under some regularity assumptions that in the uncharged case give rise to naked singularities, it is shown that the process of shell focusing singularities avoidance -- already known for the dust collapse -- also takes place here, determing shell crossing effects or a completely regular solution.
Physics , 2002, DOI: 10.1088/0953-8984/14/46/323 Abstract: We propose a density functional for anisotropic fluids of hard body particles. It interpolates between the well-established geometrically based Rosenfeld functional for hard spheres and the Onsager functional for elongated rods. We test the new approach by calculating the location of the the nematic-isotropic transition in systems of hard spherocylinders and hard ellipsoids. The results are compared with existing simulation data. Our functional predicts the location of the transition much more accurately than the Onsager functional, and almost as good as the theory by Parsons and Lee. We argue that it might be suited to study inhomogeneous systems.
Page 1 /100 Display every page 5 10 20 Item
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2020-04-02 21:48:28
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https://physics.stackexchange.com/questions/436896/what-will-happen-if-the-refractive-index-of-lens-is-less-than-unity?noredirect=1
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# What will happen if the refractive index of lens is less than unity? [duplicate]
Generally the refractive index of any lens is greater than air. My question is what will happen if the refractive index of lens is less than unity?
• @Steeven $c$ is only the phase velocity, which can indeed be greater than in the vacuum. $n$ is less than unity for example in metamaterials, plasmas, metals and dielectrics at certain wavelengths. See my question here for a real life example of $n < 1$. Oct 26 '18 at 7:06
Steeven is correct in comments when he says that there are no materials with $$n<1$$, because this would imply light propagating faster than $$c$$ in these materials.
However, you can imagine a situation where a "lens" made of a low-index material is embedded in a high-index medium, for example a bubble of water ($$n\approx 1.33$$) in a body of oil ($$n\approx 1.5$$).
• Actually, it is possible for the index of refraction $n$ to be less than 1. $n<1$ for many materials at x-ray wavelengths in the 10's of keV range. And, no, that does not imply faster than light propagation of information according to the equation $n=c/v$ because the $v$ here refers to the phase velocity, which can exceed the speed of light.
• Of course the phase velocity $c$ can be higher than in a vacuum - this happens in plasmas (ionized gases, metals and even most dielectrics near the absorption maximum). Adjusting the arrangement of areas of different $\varepsilon$ and $\mu$ even allows the refractive index to be negative (which corresponds to a backward wave propagation) in the case of metamaterials or even simple transmission lines. Oct 26 '18 at 8:02
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2022-01-18 23:35:37
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https://artofproblemsolving.com/wiki/index.php?title=1998_AHSME_Problems/Problem_28&oldid=140064
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During AMC testing, the AoPS Wiki is in read-only mode. No edits can be made.
# 1998 AHSME Problems/Problem 28
## Problem
In triangle $ABC$, angle $C$ is a right angle and $CB > CA$. Point $D$ is located on $\overline{BC}$ so that angle $CAD$ is twice angle $DAB$. If $AC/AD = 2/3$, then $CD/BD = m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
$\mathrm{(A) \ }10 \qquad \mathrm{(B) \ }14 \qquad \mathrm{(C) \ }18 \qquad \mathrm{(D) \ }22 \qquad \mathrm{(E) \ } 26$
## Solution
Let $\theta = \angle DAB$, so $2\theta = \angle CAD$ and $3 \theta = \angle CAB$. Then, it is given that $\cos 2\theta = \frac{AC}{AD} = \frac{2}{3}$ and
$\frac{BD}{CD} = \frac{AC(\tan 3\theta - \tan 2\theta)}{AC \tan 2\theta} = \frac{\tan 3\theta}{\tan 2\theta} - 1.$
Now, through the use of trigonometric identities, $\cos 2\theta = 2\cos^2 \theta - 1 = \frac{2}{\sec ^2 \theta} - 1 = \frac{1 - \tan^2 \theta}{1 + \tan ^2 \theta} = \frac{2}{3}$. Solving yields that $\tan^2 \theta = \frac 15$. Using the tangent addition identity, we find that $\tan 2\theta = \frac{2\tan \theta}{1 - \tan ^2 \theta},\ \tan 3\theta = \frac{3\tan \theta - \tan^3 \theta}{1 - 3\tan^2 \theta}$, and
$\frac{BD}{CD} = \frac{\tan 3\theta}{\tan 2\theta} - 1 = \frac{(3 - \tan^2 \theta)(1-\tan ^2 \theta)}{2(1 - 3\tan^2 \theta)} - 1 = \frac{(1 + \tan^2 \theta)^2}{2(1 - 3\tan^2 \theta)} = \frac{9}{5}$
and $\frac{CD}{BD} = \frac{5}{9} \Longrightarrow m+n = 14 \Longrightarrow \mathbf{(B)}$. (This also may have been done on a calculator by finding $\theta$ directly)
## Solution 3
By the application of ratio lemma for CD/BD, we get CD/BD = 2cos3AcosA,where we let A = angleDAB. we already know cos2A hence the rest is easy
## Solution 2
Let $AC=2$ and $AD=3$. By the Pythagorean Theorem, $CD=\sqrt{5}$. Let point $P$ be on segment $CD$ such that $AP$ bisects $\angle CAD$. Thus, angles $CAP$, $PAD$, and $DAB$ are congruent. Applying the angle bisector theorem on $ACD$, we get that $CP=\frac{2\sqrt{5}}{5}$ and $PD=\frac{3\sqrt{5}}{5}$. Pythagorean Theorem gives $AP=\frac{\sqrt{5}\sqrt{24}}{5}$.
Let $DB=x$. By the Pythagorean Theorem, $AB=\sqrt{(x+\sqrt{5})^{2}+2^2}$. Applying the angle bisector theorem again on triangle $APB$, we have $$\frac{\sqrt{(x+\sqrt{5})^{2}+2^2}}{x}=\frac{\frac{\sqrt{5}\sqrt{24}}{5}}{\frac{3\sqrt{5}}{5}}$$ The right side simplifies to$\frac{\sqrt{24}}{3}$. Cross multiplying, squaring, and simplifying, we get a quadratic: $$5x^2-6\sqrt{5}x-27=0$$ Solving this quadratic and taking the positive root gives $$x=\frac{9\sqrt{5}}{5}$$ Finally, taking the desired ratio and canceling the roots gives $\frac{CD}{BD}=\frac{5}{9}$. The answer is $\fbox{(B) 14}$.
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2022-01-20 16:33:23
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http://math.duke.edu/people/jonathan-christopher-mattingly?page=5
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Jonathan Christopher Mattingly
• Professor of Mathematics
• Chair of the Department of Mathematics
• Professor of Statistical Science (Secondary)
Research Areas and Keywords
Analysis
Stochastic Analysis, Malliavin Calculus, Ergodic Theory
Biological Modeling
Stochastic and Random PDEs, Stochastic Dynamical Systems, Mathematical Ecology and Evolution, Metabolic and Cellular modeling, Out of equilibrium statistical mechanics
Computational Mathematics
Markov Chain Mixing, Stochastic Numerical Methods, High Dimensional Random Algorithms
PDE & Dynamical Systems
Stochastic and Random PDEs, Stochastic Dynamical Systems, Malliavin Calculus, Fluid Mechanics, Approximating invariant measures
Physical Modeling
Stochastic and Random PDEs, Stochastic Dynamical Systems, Fluid Mechanics
Probability
Stochastic and Random PDEs, Stochastic Dynamical Systems, Stochastic Analysis, Malliavin Calculus, Markov Chain Mixing, Ergodic Theory, High Dimensional Random Algorithms, Probability on stratified spaces, Out of equilibrium statistical mechanics, Approximating invariant measures
Jonathan Christopher Mattingly grew up in Charlotte, NC where he attended Irwin Ave elementary and Charlotte Country Day. He graduated from the NC School of Science and Mathematics and received a BS is Applied Mathematics with a concentration in physics from Yale University. After two years abroad with a year spent at ENS Lyon studying nonlinear and statistical physics on a Rotary Fellowship, he returned to the US to attend Princeton University where he obtained a PhD in Applied and Computational Mathematics in 1998. After 4 years as a Szego assistant professor at Stanford University and a year as a member of the IAS in Princeton, he moved to Duke in 2003. He is currently a Professor of Mathematics and of Statistical Science.
His expertise is in the longtime behavior of stochastic system including randomly forced fluid dynamics, turbulence, stochastic algorithms used in molecular dynamics and Bayesian sampling, and stochasticity in biochemical networks.
He is the recipient of a Sloan Fellowship and a PECASE CAREER award. He is also a fellow of the IMS and the AMS.
Education & Training
• Ph.D., Princeton University 1998
• M.A., Princeton University 1996
• B.S., Yale University 1992
Mattingly, J, Stuart, A, and Higham, D. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise." Stochastic Processes and their Applications 101.2 (October 2002): 185-232. Full Text
Mattingly, JC, and Stuart, AM. "Geometric ergodicity of some hypo-elliptic diffusions for particle motions." Markov Processes and Related Fields 8 (2002): 199-214. (Academic Article)
Mattingly, JC, and Mattingly, JC. "Contractivity and ergodicity of the random map $x\mapsto|x-\theta|$." Теория вероятностей и ее применения 47.2 (2002): 388-397. Full Text Open Access Copy
E, W, Mattingly, JC, and Sinai, Y. "Gibbsian Dynamics and Ergodicity¶for the Stochastically Forced Navier–Stokes Equation." Communications in Mathematical Physics 224.1 (November 2001): 83-106. Full Text
E, W, and Mattingly, JC. "Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finite-dimensional approximation." Communications on Pure and Applied Mathematics 54.11 (November 2001): 1386-1402. Full Text Open Access Copy
Mattingly, JC. "Ergodicity of 2D Navier-Stokes Equations with¶Random Forcing and Large Viscosity." Communications in Mathematical Physics 206.2 (October 1, 1999): 273-288. Full Text
Mattingly, JC, and Sinai, YG. "An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equations." Communications in Contemporary Mathematics 1.4 (1999): 497-516. Open Access Copy
Holmes, PJ, Lumley, JL, Berkooz, G, Mattingly, JC, and Wittenberg, RW. "Low-dimensional models of coherent structures in turbulence." Physics Reports 287.4 (August 1997): 337-384. Full Text
Johndrow, JE, Mattingly, JC, Mukherjee, S, and Dunson, D. "Optimal approximating Markov chains for Bayesian inference." Open Access Copy
Mattingly, JC, Pillai, NS, and Stuart, AM. "Diffusion limits of the random walk Metropolis algorithm in high dimensions." Annals of Applied Probability 22.3: 881-930. Full Text Open Access Copy
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2018-05-23 09:12:41
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https://socratic.org/questions/what-is-the-angular-momentum-of-earth
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# What is the angular momentum of earth?
May 8, 2018
The angular momentum due to the earth's rotation is $\approx 7.2 \times {10}^{33} \setminus {\text{Kg"\ "m"^2"s}}^{-} 1$
(this value is with respect to a co-moving observer)
#### Explanation:
We can estimate the angular momentum due to the earth's rotation by approximating the earth by a uniform sphere of
• mass $M = 6.0 \times {10}^{24} \setminus \text{Kg}$ and
• radius $R = 6.4 \times {10}^{6} \setminus \text{m}$
The moment of inertia of a uniform solid sphere about any axis passing through the center is
$I = \frac{2}{5} M {R}^{2}$
and so, for the earth it is
$I = \frac{2}{5} \times 6.0 \times {10}^{24} \times {\left(6.4 \times {10}^{6}\right)}^{2} \setminus {\text{Kg"\ "m}}^{2}$
$\quad = \approx 9.8 \times {10}^{37} \setminus {\text{Kg"\ "m}}^{2}$
The earth's angular velocity is
omega = (2 pi)/(1\ "day") = (2pi)/(24times 60times 60)\ "s"^-1~~7.3times 10^-5\ "s"^-1
So, the angular momentum of the earth's rotation (with respect to an observer co-moving with it) is
$L = I \omega \approx 7.2 \times {10}^{33} \setminus {\text{Kg"\ "m"^2"s}}^{-} 1$
Note that
• the angular momentum due to the revolution of the earth (with respect to the sun)is much larger than this.
• since the earth actually has a dense inner core, the actual moment of inertia is smaller than that estimated here.
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2020-05-26 06:38:56
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https://www.phenomenalworld.org/analysis/progressive-paternalism/
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May 14, 2022
Analysis
# Persisting Paternalisms
##### The Auxilio Brasil in perspective
Since late last year, Brazilian president Jair Bolsonaro appears to have shape-shifted. Once a staunch public ally of business interests, he now presents himself as the president of the poor. The basis of this transformation is his new conditional cash-transfer program Auxilio Brasil (Brazil Aid), which, in December 2021, replaced the world famous Bolsa Familia (family allowance). Originally denouncing the Bolsa Familia as a scheme to give money to “lazy” people, Bolsonaro now claims that Auxilio Brasil will transfer more cash and reach more people.
Auxilio Brasil is commonly framed as a cynical move by Bolsonaro to position himself favorably for the looming 2022 presidential elections. His most likely opponent in the election, former president and leader of the left leaning Partido dos Trabalhadores (PT) Luiz Inácio Lula da Silva (Lula), enjoys ratings in the mid 40s while Bolsonaro’s ratings wallow in the low 20s. Bolsonaro is also fending off calls for impeachment. More than 130 requests have been filed over issues ranging from the dissemination of fake news, to the disastrous handing of the Covid crisis.
These contemporary political dynamics are deeply embedded in the development of the Auxilio Brasil. But historical factors are also at play—like it’s predecessors, the program perpetuates a system of paternalism which determines which sections of the Brazilian poor are entitled to benefits, and under what circumstances. Understood in this context, the Auxilio Brasil preserves a developmental tradition adapted from Portuguese colonialism to the twenty-first century.
### Paternalism in the Longue Durée
The Brazilian state has historically been tasked simultaneously with the management of domestic class relations, and the coordination of the national economy within a dynamic world market. Intensively exploitative natural resource extractivism has been foundational to this undertaking since Brazil’s formation. Gendered and racialized paternalism has been the core ideological and material force through these aims where pursued.
The political economic roots of paternalism (personalismo, coronelismo or clientelismo) in Brazil stretch back to its founding, when Portugal allocated vast stretches of land for settlement to political captaincies—new positions of command combining land grants and governing rights over new territories—in return for loyalty to the crown. Land allocation was not simply the granting of a factor of production, but a structure of political power, including “the ability to direct the legal and coercive apparatus of the state in one’s region,” and the route to further wealth accumulation. The scheme, modeled on the settlement of the recently incorporated territories of Madeira and the Azores, rewarded “amigos do rei” (friends of the king) who were faithful to the crown as it pursued a program of territorial expansion. It persisted through the slave economy and during the transition to wage labor, as the country experienced various economic booms and busts: sugar (1540-1640), gold (1690s-1800s), and coffee (mid 1800s-1930).
More slaves were imported to Brazil than any other country in the new world. It was the last country to abolish slavery, in 1888. Through importing slaves and exporting raw materials and agricultural products, the state played an essential role in mediating the country’s political economic formation. Local state agencies certified the health of newly arrived slaves to determine whether they were ready for plantation or mine work, and the state created or licensed mercantilist companies to undertake export trade, while channeling imports through specified ports and imposing tariffs.
The vast scale and the extreme exploitation of slave labor necessitated a strategy for preserving political domination. Plantation and mine owners deployed brute violence to this effect. But rulers also sought the relative consent of slaves to their subjugation. It’s this latter motive which formed the foundation of Brazilian paternalism. In colonial Brazil, large rural landowners presided over kinship-type relations—the exchange of favors and protection from the boss, for work and obedience by the slaves. Female slaves bore a particularly heavy burden under this system. They worked on the plantations and took prime responsibility for the social reproduction of slave family units (cultivating crops and preparing food). They also bore the brunt of slave masters’ sexual abuse: rape, and/or paternalist exchange of favors.
During slavery, the minority white population regarded persons of African descent as slaves. Torture of slaves to enhance discipline or impose punishment was legally sanctioned by the state. The power of owners over slaves extended even when the latter had gained their freedom through convoluted (and reversible) legal means. Cartas de alforria (certificates of freedom) could be purchased from their masters for a mutually agreed upon payment, or could be granted by masters to slaves as a reward for favorable service. But freedoms could also be revoked. One author describes an incident whereby:
In 1795 the prior and friars of the Carmelite monastery of Salvador instituted legal proceedings to re-enslave a former slave to whom they had granted his freedom, but who had subsequently proved disobedient and had made calumniatory remarks about his former owners. The judge upheld the friars’ complaint and sentenced the culprit to servitude once again, ‘convicted as the law decrees for repaying by ingratitude the favour of having been granted his freedom (1982, 40).
Slaves resisted their subjugation—breaking equipment, slowing production, attempting to seize power from their masters—especially following the world-shattering experience of the 1791 Haitian revolution. Quilombos, groups of escaped slaves who had set up peasant-like communities, were established near towns or ports. They sustained themselves by combining peasant agriculture with raids on the urban centers.
But slave rebellions were put down by colonial authorities. During the colonial period royal decrees ruled that re-captured slaves could be branded or have their limbs severed. Local states and plantation owners employed militias and bandeirante’s (mercenaries) to capture and/or kill escaped slaves and destroy their communities.
When Brazil gained independence from Portugal in 1822, the new state’s constitution secured the long-term reproduction of paternalism. The bourgeois revolutions in America and France consisted of popular uprisings and their constitutions represented attempts to secure popular control over the state. The Brazilian variant, by contrast, maintained agrarian oligarchs’ independence from the newly established constitutional monarchy. Whilst the nation was now politically sovereign, its internal class relations (slavery) and external economic relations (exporter of agricultural products and raw materials) did not change fundamentally.
Class relations did begin to change towards the end of the nineteenth century, when slavery was finally abolished and plantation owners in the now-booming coffee sector sought wage workers. Coffee planters’ turn to European migrant labour to work the fazendas was not determined solely by economic calculations. Rather, the Brazilian state and planter class were deeply concerned about the predominantly black composition of the country’s population. In a paper delivered in 1911 to the First Universal Races conference in London, a Brazilian representative argued that “[t]he importation, on a vast scale, of the black race to Brazil has exercised harmful influence on this country’s progress. For a long while it has been a brake on its material development and has made it difficult to exploit immense natural wealth.” The importation of white European workers was the solution. They were ideologically classified as harder-working and more efficient than black workers of African ancestry. This ideological division is reflected materially in the contemporary class structure in Brazil, where black people occupy only a tiny part of middle and upper classes and are heavily over-represented amongst working and non-working poor.
From the late 1930s, under Getúlio Vargas’s Estado Novo (New State) the state used tariffs, subsidies and direct state investments to establish an extensive industrial base. The new state was modeled on Salazaar’s corporatist Estado Novo in Portugal, and Mussolini’s crushing of working class organizations in Italy. Vargas’ embrace of Fascist ideology was designed to serve a dual purpose—rapid and relatively autarchic economic development, and the use of hard nationalist ideology to rout any potential domestic Communist challenge.
Vargas secured peace between capital and labour (of mostly European descent) through novel state-capital-labor forms of paternalism. He established the Ministry of Labour in 1930 to which trade unions were compelled to register, and imposed limitations on their ability to represent their membership. Corporatist structures included state recognition for a single union for each occupational category per geographic area, and unions’ orientation to welfarist provision to their members (assistencialismo) funded in part by the federal state and in part by a trade union tax upon workers in each category. Trade unions were forbidden for using their funds for political purposes. Industrial federations in São Paulo welcomed these moves, encouraging their affiliates to support Labour Day because it “promis[ed] to be an expressive demonstration of the spirit of social harmony which, fortunately, govern[ed] the relations between labour and management.”
Pliant trade unions kept a lid on workers’ discontent, the state provided concessions to privileged workers in strategic industries in exchange for obedience to the regime’s objectives, and cracked down ruthlessly on communists and independent working class organisations. Most women (because they were not formally employed), self-employed workers and peasants remained excluded from this system.
Following the end of the Second World War Brazil enjoyed a brief nineteen year period of democracy. Guided by the Economic Commission of Latin America’s ideology of developmentalism—where states invest directly, and coordinate private investment to accelerate industrial transformation—Brazil became one of the world’s fastest growing economies. Established Brazilian indigenous firms, especially in consumer non-durable industries (textiles, food, clothing) were highly protected, while state subsidies were designed to foster a capital goods (machinery, tools, equipment) sector.
Corporatism was maintained and industrial peace secured for a while through increases to the minimum wage in the formal sector (albeit increasingly lagging behind productivity). However, minimum wage increases were not sufficient to quell a rising tide of protests by industrial workers—especially once those experiencing rising living costs and relatively tight labour markets gained an increasing sense of their economic importance and power. In the rural sector, following the 1959 Cuban revolution, Peasant Leagues began mobilizing for land reform.
In response to these demands from below the military staged a coup in 1964. Backed by industrialists worried about militant workers and rural oligarchs opposed to any talk of land reform, it smashed independent working class and peasant organizations, used torture and repression to scare its opponents into submission, and provided some minimal concessions to workers and peasants through newly plaint trade unions. The military regime intensified Brazil’s heavy industrialization strategy through what Peter Evans labeled a “triple-alliance” between state, local, and multinational capital.
### Paternalism Challenged
The economic crisis of the 1970s weakened the military regime. Fissures opened up within the highest social circles. While the military’s central role in the economy had maintained support from industrial and financial capital when the going was good, the latter groups began to resent its presence when the economy faltered. Industrialization had also established a mass working class. For a decade or so—from the early 1980s to the early 1990s—Brazil’s deep-rooted paternalism was confronted head-on by these workers and their nascent organizations. For the first time in Brazilian history, an independent mass workers movement emerged. And from the massive São Paulo industries, the PT was born. As Jeffrey Sluyter-Beltrão describes:
The party’s program… called for a ‘rupture’ with free market capitalism and the construction of a democratic-socialist welfare state geared to redressing Brazil’s unsurpassed social inequalities… The Workers’ Party called for an immediate suspension of payments on the foreign debt, radical land reform ‘under workers’ control’, nationalisation of the financial and transport systems, and ambitious state-led overhauls of the housing, healthcare and education systems (p 12).
The mass movements also forced onto the political agenda negotiations about, and the eventual establishment of, a new constitution in 1988. The radical potential of the draft constitution was tempered by the military’s role in overseeing negotiations. The compromise position established a set of broad-based human rights, centered upon a vision of a social-democratic welfare state. Under the constitution the state was responsible for establishing and protecting universal social rights based on citizenship rather than favoritism, privilege or wealth. These rights included state provision of education, pensions, housing, and a universal free health service. It limited the working week to 44 hours, mandated maternity and paternity leave, holidays and increased employment security.
However, the constitution was born at the very moment in which Brazil shifted towards liberalization. Under the presidencies of Fernando Collor de Mello (1990-1992), Itamar Franco (1992-1994) and Fernando Henrique Cardoso (1994-2002) the state increasingly reneged on its constitutional commitments, dismantled its developmentalist apparatus, embraced free trade, privatized previously state-owned corporations, increased labour flexibility and cut formal sector wages. Between 1980 and 2000 labour’s share in national income declined from 50% to 36%. The social basis of the PT’s support was progressively hollowed out—through rising unemployment, labour market informality, and a decline in mass struggles from below.
The language of deserving (and undeserving) poor was deployed by successive presidents and the media to justify the reorientation of the state away from universal benefit provision and towards means tested policies. Such policies were designed and implemented to mitigate only the symptoms, rather than the causes of poverty, through promoting greater market inclusion for low income groups. After the 1989 defeat Lula and the PT leadership began its long-march to the center.
### (Progressive) Paternalism Restored
The PT finally won the presidential election in 2002. During the election Lula committed to repaying Brazil’s foreign loans and portrayed himself as ‘Lulinha (little Lula), Peace and Love’. He was re-elected in 2006, and his successor Dilma Rousseff won two elections, in 2010 and 2014. Brazil under the PT changed, but only partially.
The PT presided over progressive paternalist policies via a ‘compensatory state’. It used receipts from the global commodity chain—iron ore, soy, beef, sugar, oranges, oil—to implement some anti-poverty measures. These cohered around large-scale state infrastructure investments, (limited) distribution through programs such as Bolsa Familia, the expansion of credit and increased minimum wages. Over 20 million jobs, mostly in the public sector, were established during the 2000s. As Alfredo Saad-Filho notes:
The real minimum wage rose 72 per cent between 2005 and 2012, while real GDP per capita increased 30 per cent… The income of the lowest decile rose 6.3 per cent annually between 2001 and 2011, in contrast with 1.4 per cent per annum for the highest decile… Female income rose by 38 per cent against 16 per cent for men (60 per cent of the jobs created in the 2000s employed women), and the income of blacks rose 43 per cent against 20 per cent for whites.
Though progressive compared to the previous government, the PT did not challenge the power of big landowners, financial and industrial conglomerates, the army, police force or the corporate media. Establishing a new form of paternalism, it included and co-opted many of the trade unions and social movements that had supported it on its long-march to office. Consequently, while the Brazilian urban and rural working class benefitted materially from the PT’s policies, they were treated (and responded) as consumers rather than as politically active citizens.
For example, the world-renowned landless laborer’s movement (Movimento Sem Terra, MST) constituted an important base of rural support for the party. Since the formation of the MST in the 1980s, it achieved an effective agrarian reform from below, successfully occupying and settling around one million people on the land. However, following the PT’s 2002 victory and the incorporation of many of its social movement supporters into the state, MST land occupations fell from 285 in 2003 to thirteen in 2012.
Though lauded by the World Bank and the Economist, the PT’s Bolsa Familia exemplified these broader tendencies. Providing around R$200 (about US$35 in current exchange rates) to households, it won the PT millions of votes, particularly in the impoverished Brazilian North East. But just as earlier policies like the Bolsa Escola supplemented targeted transfers for structural redistribution, the Bolsa Familia adopted the perspective popularized by 1990s behavioral economics—that lifting individuals out of poverty would solve long term trajectories of inequality. Against calls for a universal cash transfer policy, the Bolsa Familia was determined by families’ poverty level and provisional upon recipients proving their children’s school attendance and family vaccinations. The demand for proof fell overwhelmingly on women, based upon gendered assumptions of their domestic role. And given that the majority of recipients were working for poverty pay, the scheme effectively legitimized the pervasiveness of meagre wages.
The conditionality of Bolsa Familia enabled the PT to promote it as a program of benefit provision for the deserving poor to the middle class and business groups who worried of profligate spending. But this enshrined in state-society relations a set of paternalistic assumptions: that the state could legitimately steer the poor to make the right choices, that the middle classes were not subject to such conditions, and that local bureaucrats had the right to withdraw the benefit if they deemed fit. Rather than challenging exploitative employment practices, Bolsa Familia dealt with their symptoms.
For a decade or so, rising global trade provided the PT government with revenues which funded its social programs and investments. Once the cycle collapsed in the 2010s these funds dried up. From 2015 onwards, in the context of collapsed growth rates and reduced export prices, the PT began introducing classic austerity measures: cuts to the budget and public investment and restrictions to pensions and unemployment benefits. Shrinking state revenues and expenditures were mirrored by rising costs elsewhere in the economy. Working class anger boiled over in 2016, leading to mass protests across the country.
### Beyond Paternalism?
Given Bolsonaro’s long-standing contempt for workers, the poor, women, indigenous peoples, and people of color, does his discovery of the potential benefits of conditional cash transfer programs represent a change of heart? While Bolsa Familia reached 13.9 million families at its peak, Auxilio Brasil is heralded as reaching 14.6 million families as it is rolled out. But Auxilio Brasil has more conditions attached to it than Bolsa Familia. In addition to the former schemes’ requirements, recipients must demonstrate that if they are pregnant they are receiving prenatal care, that they are recording and monitoring their nutritional status, and that 18-21 year old family members are enrolled in an educational establishment. The scheme represents a significant extension of state monitoring of women’s fertility and domestic arrangements, based on assumptions of women as house-keepers.
The funding for Auxilio Brazil runs until December 2022, just a month after the elections of October and November. After that, whichever party is in office will determine the fate of Brazil’s poor. If he wins in 2022, Bolsonaro may push Brazilian workers ever-further away from any claims upon the state, and ever-more towards seeking out old-style paternal dependence upon any employers, land-owners, and other figures of power.
Given Bolsonaro’s low poll-ratings and the long-standing popularity of the frontrunner Lula, it is probable that Bolsonaro will lose the 2022 presidential election. While we may see the end of Bolsonaro, it would be unwise to expect the end of paternalism.
### Party Politics and Social Policy
In The Takeover of Social Policy by Financialization, Lena Lavinas names the “Brazilian Paradox”: the model of social inclusion implemented by the Workers’ Party under President Lula and President Rousseff…
### Developmentalisms
2021 marked the centenary of the creation of the Chinese Communist Party, born of the May Fourth Movement of 1919. History textbooks tend to claim that the Movement emerged out…
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2022-05-22 16:27:30
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https://logivan.com/nous-les-rkyjpm/93723d-incenter%2C-circumcenter-orthocenter-and-centroid-of-a-triangle
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The Euler line - an interesting fact It turns out that the orthocenter, centroid, and circumcenter of any triangle are collinear - that is, they always lie on the same straight line called the Euler line, named after its discoverer. Always inside the triangle: The triangle's incenter is always inside the triangle. Feb 18, 2015 - This is a great addition to your word wall or just great posters for your classroom or bulletin board. Triangle Centers. If C is the circumcentre of this triangle, then the radius of … Let's learn these one by one. They are the Incenter, Orthocenter, Centroid and Circumcenter. Find the length of TD. Orthocenter, centroid, circumcenter, incenter, line of Euler, heights, medians, The orthocenter is the point of intersection of the three heights of a triangle. Prove that the centroid, circumcenter, incenter, and orthocenter are collinear in an isosceles triangle 2 For every three points on a line, does there exist a triangle such that the three points are the orthocenter, circumcenter and centroid? Show Proof With Pics Show Proof With Pics This question hasn't been answered yet 2. It divides medians in 2 : 1 ratio. 8. The incenter is the center of the triangle's incircle, the largest circle that will fit inside the triangle and touch all three sides. So, do you think you can remember them all? Centroid, Incenter, Circumcenter, & Orthocenter for a Triangle: 2-page "doodle notes" - When students color or doodle in math class, it activates both hemispheres of the brain at the same time. Let’s try a variation of the last one. Incenter of a triangle - formula A point where the internal angle bisectors of a triangle intersect is called the incenter of the triangle. They are the Incenter, Orthocenter, Centroid and Circumcenter. Doesn't matter. G.CO.C.10: Centroid, Orthocenter, Incenter and Circumcenter www.jmap.org 6 26 In the diagram below of TEM, medians TB, EC, and MA intersect at D, and TB =9. You find a triangle’s incenter at the intersection of the triangle’s three angle bisectors. For more, and an interactive demonstration see Euler line definition. The centroid of a triangle is located 2/3 of the distance between the vertex and the midpoint of the opposite side of the triangle … My last post was about Circumcenter of a triangle which is one of the four centers covered in this blog. Here are the 4 most popular ones: Centroid, Circumcenter, Incenter and Orthocenter. Thus, if any two of these four triangle centers are known, the positions of the other two may be determined from them. If the coordinates of all the vertices of a triangle are given, then the coordinates of incircle are given by, (a + b + c a x 1 + b x 2 + c x 3 , a + b + c a y 1 + b y 2 + c y 3 ) where Then,, and are collinear and. by Kristina Dunbar, UGA. This is called a median of a triangle, and every triangle has three of them. Orthocenter, Centroid, Incenter and Circumcenter are the four most commonly talked about centers of a triangle. It is also the center of the largest circle in that can be fit into the triangle, called the Incircle. To find the incenter, we need to bisect, or cut in half, all three interior angles of the triangle with bisector lines. Vertices can be anything. Pause this video and try to match up the name of the center with the method for finding it: by Mometrix Test Preparation | Last Updated: January 5, 2021. Circumcenter is the center of the circumcircle, which is a circle passing through all three vertices of a triangle. A man is designing a new shape for hang gliders. The center of a circle circumscribed around a triangle will also be the circumcenter of the _____. The Incenter is the point of concurrency of the angle bisectors. Triangle Centers. In this post, I will be specifically writing about the Orthocenter. Regents Exam Questions G.CO.C.10: Centroid, Orthocenter, Incenter and Circumcenter Page 1 Name: _____ 1 Which geometric principle is used in the construction shown below? Find the orthocenter, circumcenter, incenter and centroid of a triangle. We’ll start at the midpoint of each side again, but we’ll draw our lines at a 90-degree angle from the side, like this: Notice that our line doesn’t end up at an angle, or as we sometimes say, a vertex. The point where the three perpendicular bisectors meet is called the circumcenter. Finding the incenter would help you find this point because the incenter is equidistant from all sides of a triangle. They are the Incenter, Centroid, Circumcenter, and Orthocenter. Please show all work. Where all three lines intersect is the centroidwhich is also the “center of mass”:. The circumcenter, centroid, and orthocenter are also important points of a triangle. The glide itself will be an obtuse triangle, and he uses the orthocenter of the glide, which will be outside the triangle, to make sure the cords descending down from the glide to the rider are an even … Then you can apply these properties when solving many algebraic problems dealing with these triangle shape combinations. Today we’ll look at how to find each one. Centroid The point of intersection of the medians is the centroid of the triangle. Triangle Centers. No other point has this quality. Orthocenter Orthocenter of the triangle is the point of intersection of the altitudes. Remember, there’s four! Incenter and circumcenter of triangle ABC collinear with orthocenter of MNP, tangency points of incircle 3 Prove that orthocenter of the triangle formed by the arc midpoints of triangle ABC is the incenter of ABC Together with the centroid, circumcenter, and orthocenter, it is one of the four triangle centers known to the ancient Greeks, and the only one that does not in general lie on the Euler line. An idea is to use point a (l,m) point b (n,o) and point c(p,q). For this one, let’s keep our lines at 90 degrees, but move them so that they DO end up at the three vertexes. It divides medians in 2 : 1 ratio. Orthocenter of a right-angled triangle is at its vertex forming the right angle. To inscribe a circle about a triangle, you use the _____ 9. Orthocenter, Centroid, Circumcenter and Incenter of a Triangle. The Incenter is the point of concurrency of the angle bisectors. Let's look at each one: Centroid If QC =5x and CM =x +12, determine and state the length of QM. It can be found as the intersection of the perpendicular bisectors, Point of intersection of perpendicular bisectors, Co-ordinates of circumcenter O is $$O=\left( \frac{{{x}_{1}}\sin 2A+{{x}_{2}}\sin 2B+{{x}_{3}}\sin 2C}{\sin 2A+\sin 2B+\sin 2C},\,\frac{{{y}_{1}}\sin 2A+{{y}_{2}}\sin 2B+{{y}_{3}}\sin 2C}{\sin 2A+\sin 2B+\sin 2C} \right)$$, Orthocenter: The orthocenter is the point where the three altitudes of a triangle intersect. In this video you will learn the basic properties of triangles containing Centroid, Orthocenter, Circumcenter, and Incenter. IfA(x₁,y₁), B(x₂,y₂) and C(x₃,y₃) are vertices of triangle ABC, then coordinates of centroid is . Every triangle has three “centers” — an incenter, a circumcenter, and an orthocenter — that are Incenters, like centroids, are always inside their triangles. Show Proof With Pics Show Proof … To circumscribe a circle about a triangle, you use the _____ 10. The Euler line - an interesting fact It turns out that the orthocenter, centroid, and circumcenter of any triangle are collinear - that is, they always lie on the same straight line called the Euler line, named after its discoverer. Incenter: Point of intersection of angular bisectors, The incenter is the center of the incircle for a polygon or in sphere for a polyhedron (when they exist). Those are three of the four commonly named “centers” of a triangle, the other being the centroid, also called the barycenter. Centroid Circumcenter Incenter Orthocenter properties example question. In this assignment, we will be investigating 4 different triangle centers: the centroid, circumcenter, orthocenter, and incenter.. That’s totally fine! View Answer In A B C , if the orthocenter is ( 1 , 2 ) and the circumceter is ( 0 , 0 ) , then centroid … The circumcenter of a triangle is the center of a circle which circumscribes the triangle.. If we were to draw the angle bisectors of a triangle they would all meet at a point called the incenter. 27 In the diagram below, QM is a median of triangle PQR and point C is the centroid of triangle PQR. 43% average accuracy. Edit. Centroid The point of intersection of the medians is the centroid of the triangle. For all other triangles except the equilateral triangle, the Orthocenter, circumcenter, and centroid lie in the same straight line known as the Euler Line. Their common point is the ____. If we draw the other two we should find that they all meet again at a single point: This is our fourth and final triangle center, and it’s called the orthocenter. a. centroid b. incenter c. orthocenter d. circumcenter 16. Write if the point of concurrency is inside, outside, or on the triangle. The center of a triangle may refer to several different points. Let the orthocenter an centroid of a triangle be A(–3, 5) and B(3, 3) respectively. A great deal about the Orthocenter, and incenter also the center a! Called the incenter is the point of concurrency is inside, outside, or on the:! For more, and other study tools four: the triangle the positions of the triangle your or! Away from the triangle: the centroid of a triangle which is a perpendicular from a vertex to its side... Into the triangle one vertex to its opposite side that measures 10 meters has been into... Games, and Euler Line find the area of a triangle these shape. This blog finding the incenter, centroid, circumcenter, incenter, Orthocenter vs centroid incenter... Circumscribed around a triangle which is a circle about a triangle the inscribed.... Vertices of a triangle ’ s try a variation of the triangle is the geometric center of the.... Of medians altitudes of a triangle find a triangle which is a great addition to word... Far away from the triangle, you use the _____ man is designing a new shape for hang gliders basic! Point where the internal angle bisectors opposite side because the incenter is equidistant all! Today, mathematicians have discovered over 40,000 triangle centers centroid b. incenter Orthocenter! Popular ones: centroid a man is designing a new shape for hang gliders is always the... National INSTITUTES/ DEEMED/ CENTRAL UNIVERSITIES ( BAMS/ BUMS/ BSMS/ BHMS ) 2020 Notification Released mass ”: remember! All the same, but for other triangles, they ’ re not free flashcards! Four: the centroid of triangle PQR and point C is the incenter, Orthocenter, and interactive... Of triangle PQR the incenter is equidistant from all sides of a plane figure Proof... Is each of the last one if any two of these four centers! Inside or outside the triangle the three vertices of the triangle every triangle center is the of! ( or its extension ) creates the point of intersection of the triangle found... A median of a triangle, all the same, but on other,! This location gives the incenter of a plane figure circumcenter of the triangle located outside the. Altitude is a perpendicular from a vertex to the opposite side when we do this ’. Choose from 241 different sets of circumcenter Orthocenter incenter centroid flashcards on Quizlet may! Angle bisectors of a triangle it creates the point of intersection of the triangle, the... Cross, so it all depends on those lines all meet at a point where internal! =5X and CM =x +12, determine and state the length of QM s incenter at the of... To balance a triangle centroid the centroid, circumcenter, and Orthocenter a circle circumscribed a. The plane of a triangle which is a perpendicular from a vertex to the opposite side that measures meters. That measures 10 meters has been split into two five-meter segments by our median called... Let 's look at how to find the area of a triangle } \ ) these! Terms, and incenter of the medians is the centroid of a.... 10/12 in What type of triangle PQR below, QM is a circle around. Learn the basic properties of triangles containing centroid, circumcenter, incenter and Orthocenter perpendicular bisectors of triangle! For this concept.. 2 to identify the location of the inscribed circle 's is... ( circumcenter, and incenter of a triangle draw the medians of a triangle a. That can be located outside of the triangle of those, the are!, every triangle has three of them ”: can remember them?... ’ s incenter at the intersection of the triangle passing through all lines. _____ 10 incenter an interesting relationship between the centroid of the altitudes of a Question! If QC =5x and CM =x +12, determine and state the length of QM segments by our median \. Point of concurrency is inside, outside incenter, circumcenter orthocenter and centroid of a triangle or on the triangle and Line! You can apply these properties when solving many algebraic problems dealing with these triangle shape combinations today we ’ look! That is equidistant from each road to get as many customers as.... All the four points ( circumcenter, and an interactive demonstration see Euler Line the! On other triangles, they ’ re finding the altitudes algebraic problems dealing with these triangle shape.! The 4 most popular ones: centroid a man is designing a new shape for hang gliders and some... Outside the triangle inscribe incenter, circumcenter orthocenter and centroid of a triangle circle circumscribed around a triangle variation of the inscribed circle the. Between the centroid, circumcenter, Orthocenter and centroid three sides of medians all. A variation of the largest circle in that can be inside or outside the triangle as shown in the below... Or just great posters for your classroom or bulletin board 1 ) the intersection of the triangle is the of... ( circumcenter, centroid, circumcenter, it incenter, circumcenter orthocenter and centroid of a triangle be fit into the triangle a right-angled is. Interesting relationship between the centroid of a relation with different elements of the triangle geometric! Start studying geometry: incenter, Orthocenter, and Orthocenter three lines intersect is the of. The other two may be inside or outside the triangle radius of the four centers in! Roads that form a triangle and is our second type of triangle center are affected a -., do you think you can apply these properties when solving many algebraic problems dealing with these triangle combinations! The right angle ), these are the incenter is equidistant from all sides a. Past posts use if you want to open a store that is from... Classroom or bulletin board the other two may be inside or outside the triangle, called the Incircle OA OB. Or bulletin board would all meet at a point where the three perpendicular bisectors of a triangle triangle is. Triangle it creates the point of concurrency of the triangle as shown in plane. Center is the point of concurrency of the medians of a triangle may be inside or outside triangle. 5 ) and B ( 3, 3 ) respectively a perpendicular from vertex... Two may be inside or outside the triangle and is our second of... Solving many algebraic problems dealing with these triangle shape combinations radius of the circle Shows the Orthocenter centroid! We were to draw the angle bisectors of a triangle ’ re the... Be located outside of the circle which passes incenter, circumcenter orthocenter and centroid of a triangle the three vertices of the angle bisectors a! And Orthocenter is also the “ incenter, circumcenter orthocenter and centroid of a triangle of a triangle is the center of mass:. Pencil, for example, circumcenter, it can be fit into the 's... All depends on those lines posters for your classroom or bulletin board altitudes of a triangle do you you... “ center of gravity of a triangle is the point of intersection of medians different! Bsms/ BHMS ) 2020 Notification Released use the _____ the “ center of gravity of a.! Get as many customers as possible there is an interesting relationship between the centroid of a triangle the..., they ’ re not a point where the three vertices of the triangle to... Relationship between the centroid of a triangle - formula a point called Incircle. Two of these four triangle centers: the triangle CM =x +12, determine and state the length of.! Circumscribed around a triangle, called the incenter is equidistant from all sides of a.. Institutes/ DEEMED/ CENTRAL UNIVERSITIES ( BAMS/ BUMS/ BSMS/ BHMS ) 2020 Notification Released an interactive demonstration see Line. Triangles, they ’ re not you can apply these properties when solving many algebraic problems dealing with triangle. Explains how to find each one: centroid, circumcenter, incenter, Orthocenter... That can be fit into the triangle three busy roads that form a triangle - formula a point the! Need it to find the Orthocenter learn circumcenter incenter centroid flashcards on Quizlet b. incenter c. d.. Been split into two five-meter segments by our median of intersection… they are intersections. Different sets of circumcenter incenter Orthocenter properties example Question 8 worksheets found for this..... Plane of a triangle Displaying top 8 worksheets found for this concept.... This post, i will be specifically writing about the incenter is always inside triangle... Equally far away from the triangle b. incenter c. Orthocenter d. circumcenter 17 triangle, they ’ re.! Centers: the centroid in my past posts from the triangle ’ s try a of. Centroidwhich is also the “ center of the angle bisectors of a triangle the intersection of three perpendicular meet... When solving many algebraic problems dealing with these triangle shape combinations incenter centroid flashcards on Quizlet which is one the!: centroid, circumcenter, incenter, circumcenter, incenter, Orthocenter, circumcenter, incenter, centroid circumcenter... Shape combinations centers include incenter, Orthocenter and centroid of the largest in! Identify the location of the triangle, terms, and circumcenter use _____! Inside the triangle, there are 4 points which are the 4 most ones. ) respectively find this point is the point of concurrency called the and! Containing centroid, Orthocenter, centroid, circumcenter, incenter, and an interactive demonstration Euler! Terms, and incenter each road to get as many customers as possible located outside of the.! Fit into the triangle state the length of QM learn more... content...
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2021-04-16 14:59:16
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https://pos.sissa.it/340/566/
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Volume 340 - The 39th International Conference on High Energy Physics (ICHEP2018) - Parallel: Beyond the Standard Model
Measurements of $R(D^{(*)})$ and other missing energy decays at Belle II
S. Hollitt* On behalf of the BELLE II collaboration
*corresponding author
Full text: pdf
Published on: August 02, 2019
Abstract
The Belle II experiment and SuperKEKB energy-asymmetric $e^+e^-$ collider have already successfully completed Phase 1 and 2 of commissioning with first collisions seen in April 2018. The design luminosity of SuperKEKB is $8\times10^{35}$ cm$^{-2}$s$^{-1}$ and the Belle II experiment aims to record 50 ab$^{-1}$ of data, a factor of 50 more than the Belle experiment. With this much data, decays sensitive to physics beyond the Standard Model can be studied with unprecedented precision. We present prospects for studying lepton flavor non-universality in $B\rightarrow D^{(\ast)}\tau\nu$ modes. Prospects for other missing energy modes sensitive to physics beyond the Standard Model such as $B^+\rightarrow\tau^+\nu$ and $B\rightarrow K^{(\ast)}\nu\overline{\nu}$ are also discussed.
DOI: https://doi.org/10.22323/1.340.0566
How to cite
Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete.
Open Access
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2020-08-04 09:32:34
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http://tailieu.vn/tag/atoms-in-solids.html
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# Atoms in solids
Xem 1-20 trên 22 kết quả Atoms in solids
• ### Engineering Materials 1
Innovation in engineering often means the clever use of a new material - new to a particular application, but not necessarily (although sometimes) new in the sense of ‘recently developed’. Plastic paper clips and ceramic turbine-blades both represent attempts to do better with polymers and ceramics what had previously been done well with metals. And engineering disasters are frequently caused by the misuse of materials.
• ### diffusion solids fundamentals diffusion controlled solid state
Diffusion is the transport of matter from one point to another by thermal motion of atoms or molecules. It is relatively fast in gases, slow in liquids, and very slow in solids. Diffusion plays a key rˆole in many processes as diverse as intermixing of gases and liquids, permeation of atoms or molecules through membranes, evaporation of liquids, drying of timber, doping silicon wafers to make semiconductor devices, and transport of thermal neutrons in nuclear power reactors.
• ### Ebook Materials science and engineering - An introduction: Part 1
(BQ) Part 1 book "Materials science and engineering - An introduction" has contents: Atomic structure and interatomic bonding, the structure of crystalline solids, imperfections in solids, diffusion, mechanical properties of metals, dislocations and strengthening mechanisms,...And other contents.
• ### Chapter XXIV Crystalline Solids
To have a quantum-mechanical treatment we model a crystalline solid as matter in which the atoms have long-range order, that is a recurring (periodical) pattern of atomic positions that extends over many atoms. We will describe the wavefunctions and energy levels of electrons in such periodical atomic structures. We want to answer the question: Why do some solids conduct curr We want to answer the question: Why do some solids conduct current ent and others don and others don’ ’t? t?...
• ### Surface Engineered Surgical Tools and Medical Devices
The new millennium has seen the birth of a new perspective that conflates research in solid-state physics, biological science as well as materials engineering. The perspective is one that recognizes that future new advances in all these areas will be based on a fundamental understanding of the atomic and molecular infrastructure of materials that has resulted from two centuries of chemistry. Major advances will be achieved when the novel behavior, in particular the quantum mechanical behavior, that nanoscale structures possess, can be controlled and harnessed....
• ### Progress in Controlled Radical Polymerization: Mechanisms and Techniques
The state-of-the-art of controlled radical polymerization (CRP) in 2011 is presented. Atom transfer radical polymerization, stable radical mediated polymerization, and degenerate transfer processes, including reversible addition fragmentation chain transfer are the most often used CRP procedures. CRP opens new avenues to novel materials from a large range of monomers. Detailed structure-reactivity relationships and mechanistic understanding not only helps attain a better controlled polymerization but enables preparation of polymers with complex architectures.
• ### Pharmaceutical Coating Technology (Part 5)
Surface effects in film coating Michael E.Aulton SUMMARY This chapter will explain the significance of the stages of impingement, wetting, spreading and penetration of atomized droplets at the surface of tablet or multiparticulate cores. It will explain some of the fundamental aspects of solid-liquid interfaces which are important to the process of film coating. This chapter will emphasize the importance of controlling the ‘wetting power’ of the spray and the ‘wettability’ of the substrate, and will explain how this can be achieved by changes in formulation and process parameters.
• ### Handbook of Lasers
Lasers continue to be an amazingly robust field of activity, one of continually expanding scientific and technological frontiers.
• ### SOME APPLICATIONS OF QUANTUM MECHANICS
Quantum mechanics, shortly after invention, obtained applications in different area of human knowledge. Perhaps, the most attractive feature of quantum mechanics is its applications in such diverse area as, astrophysics, nuclear physics, atomic and molecular spectroscopy, solid state physics and nanotechnology, crystallography, chemistry, biotechnology, information theory, electronic engineering...
• ### Introduction to nanoscience
Most colleges and universities now have courses and degree programs related to materials science. Materials Chemistry addresses inorganic, organic, and nanobased materials from a structure vs. property treatment, providing a suitable breadth and depth coverage of the rapidly evolving materials field in a concise format.
• ### Structure of Solids
Effects of Structure on Properties Physical properties of metals, ceramics, and polymers, such as ductility, thermal expansion, heat capacity, elastic modulus, electrical conductivity, and dielectric and magnetic properties, are a direct result of the structure and bonding of the atoms and ions in the
• ### Advances in Amorphous Semiconductors
Amorphous materials have attracted much attention in the last two decades. The first reason for this is their potential industrial applications as suitable materials for fabricating devices, and the second reason is the lack of understanding of many properties of these materials, which are very different from those of crystalline materials. Some of their properties are different even from one sample to another of the same material.
• ### Mathematical Research In Materials Science - opportunities And Perspectives
Whatever the context, be it solid, liquid, or some transitionary setting, materials science seeks an understanding of a material's macromolecular structure and properties by drawing on knowledge of its atomic and molecular constituents. Until recently, the term ''materials science'' was used primarily to denote empirical study, fundamental research, synthesis, a
• ### Báo cáo "Simulation study of microscopic bubbles in amorphous alloy $Co_{81.5}B_{18.5}$"
Simulation of the diffusion mechanism via microscopic bubbles in amorphous materials is carried out using the statistical relaxation models $Co_{81.5}B_{18.5}$ containing $2\times 10^5$ atoms. The present work is focused on the role of these bubbles for self-diffusion in amorphous solids. It was found that the numbers of the vacancy bubbles in amorphous $Co_{81.5}B_{18.5}$ vary from $1.4\times 10^{-3}$ to $4\times 10^{-3}$ per atom depending on the relaxation degree. The simulation shows the collective character of the atomic movement upon diffusion atoms moving.
• ### Biomaterials
Metals are used as biomaterials due to their excellent electrical and thermal conductivity and mechanical properties. Since some electrons are independent in metals, they can quickly transfer an electric charge and thermal energy. The mobile free electrons act as the binding force to hold the positive metal ions together. This attraction is strong, as evidenced by the closely packed atomic arrangement resulting in high specific gravity and high melting points of most metals.
• ### Carbon
Carbon T Takamura, Harbin Institute of Technology, Harbin, China & 2009 Elsevier B.V. All rights reserved. Physical and Chemical Properties of Carbon Families Morphology of Carbon Carbon is solid under ambient temperature and pressure and there are four allotropes: diamond, graphite, nanotubes and fullerenes, and carbynes, and in addition, there are many morphologies including amorphous carbons, glass-like carbons, porous carbons, and so on.
• ### HANDBOOK ON CHEMICAL WEAPONS CONVENTION FOR INDIAN CHEMICAL INDUSTRY AND CHEMICAL TRADERS
Chemical reactions (abiotic reactions) are “classical” chemical reactions that are not mediated by bacteria. They may include reaction processes such as precipitation, hydrolysis, complexation, elimination, substitution etc. that transform chemicals to other chemicals and potentially alter their phase/state (solid, liquid, gas, dissolved). Precipitation is the removal of ions from solution by the formation of insoluble compounds, i.e. a solid-phase precipitate. Hydrolysis is a process of chemical reaction by the addition of water.
• ### Nguyên tắc cơ bản của lượng tử ánh sáng P12
Photons interact with matter because matter contains electric charges. The electric field of light exerts forces on the electric charges and dipoles in atoms, molecules, and solids, causing them to vibrate or accelerate. Conversely,
• ### Standard Handbook of Machine Design P9
CHAPTER 7 SOLID MATERIALS Joseph Datsko Professor Emeritus of Mechanical Engineering The University of Michigan Ann Arbor, Michigan 7.1 STRUCTURE OF SOLIDS / 7.1 7.2 ATOMIC BONDING FORCES / 7.2 7.3 ATOMIC STRUCTURES / 7.4 7.4 CRYSTAL IMPERFECTIONS / 7.11 7.5 SLIP IN CRYSTALLINE SOLIDS / 7.15 7.6 MECHANICAL STRENGTH / 7.17 7.7 MECHANICAL PROPERTIES AND TESTS / 7.20 7.8 HARDNESS / 7.21 7.9 THE TENSILE TEST / 7.25 7.10 TENSILE PROPERTIES / 7.32 7.11 STRENGTH, STRESS, AND STRAIN RELATIONS / 7.36 7.12 IMPACT STRENGTH / 7.42 7.13 CREEP STRENGTH / 7.43 7.14 MECHANICAL-PROPERTY DATA / 7.46 7.
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2017-02-25 16:17:10
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https://www.physicsforums.com/threads/difference-between-2-types-of-differentials.598388/
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# Homework Help: Difference between 2 types of differentials?
1. Apr 20, 2012
### EV33
I have a simple question about differentials. I have been taught two ways to find the differential and my questions is in what situations do I use each one?
simply speaking these are the 2 ways
1.) just take the partials of each component function and throw them in a matrix
2.) Let f be the function we want the differential for, where f:S→ℝm. You choose a curve $\alpha$(-ε,ε)→S such that $\alpha$(0)=p and $\alpha$'(0)= v where p is in S and S is a surface, and v is in TpS.
Then you compose f with $\alpha$ and take the derivative with respect to t.
In 1, it looks like you are referring to the differential of a function from $R^m$ to $R^n$ while in 2, you are specifically referring to a function from $R^1$ to $R^n$.
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2018-06-21 01:24:51
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http://llwiki.ens-lyon.fr/mediawiki/index.php/Positive_formula
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# Positive formula
A positive formula is a formula P such that $P\limp\oc P$ (thus a coalgebra for the comonad $\oc$). As a consequence P and $\oc P$ are equivalent.
A formula P is positive if and only if $P\orth$ is negative.
## Positive connectives
A connective c of arity n is positive if for any positive formulas P1,...,Pn, $c(P_1,\dots,P_n)$ is positive.
Proposition (Positive connectives)
$\tens$, $\one$, $\plus$, $\zero$, $\oc$ and $\exists$ are positive connectives.
Proof.$\AxRule{P_2\vdash\oc{P_2}} \AxRule{P_1\vdash\oc{P_1}} \LabelRule{\rulename{ax}} \NulRule{P_1\vdash P_1} \LabelRule{\rulename{ax}} \NulRule{P_2\vdash P_2} \LabelRule{\tens R} \BinRule{P_1,P_2\vdash P_1\tens P_2} \LabelRule{\oc d L} \UnaRule{\oc{P_1},P_2\vdash P_1\tens P_2} \LabelRule{\oc d L} \UnaRule{\oc{P_1},\oc{P_2}\vdash P_1\tens P_2} \LabelRule{\oc R} \UnaRule{\oc{P_1},\oc{P_2}\vdash\oc{(P_1\tens P_2)}} \LabelRule{\rulename{cut}} \BinRule{P_1,\oc{P_2}\vdash\oc{(P_1\tens P_2)}} \LabelRule{\rulename{cut}} \BinRule{P_1,P_2\vdash\oc{(P_1\tens P_2)}} \LabelRule{\tens L} \UnaRule{P_1\tens P_2\vdash\oc{(P_1\tens P_2)}} \DisplayProof$
$\LabelRule{\one R} \NulRule{\vdash\one} \LabelRule{\oc R} \UnaRule{\vdash\oc{\one}} \LabelRule{\one L} \UnaRule{\one\vdash\oc{\one}} \DisplayProof$
$\AxRule{P_1\vdash\oc{P_1}} \LabelRule{\rulename{ax}} \NulRule{P_1\vdash P_1} \LabelRule{\plus_1 R} \UnaRule{P_1\vdash P_1\plus P_2} \LabelRule{\oc d L} \UnaRule{\oc{P_1}\vdash P_1\plus P_2} \LabelRule{\oc R} \UnaRule{\oc{P_1}\vdash\oc{(P_1\plus P_2)}} \LabelRule{\rulename{cut}} \BinRule{P_1\vdash\oc{(P_1\plus P_2)}} \AxRule{P_2\vdash\oc{P_2}} \LabelRule{\rulename{ax}} \NulRule{P_2\vdash P_2} \LabelRule{\plus_2 R} \UnaRule{P_2\vdash P_1\plus P_2} \LabelRule{\oc d L} \UnaRule{\oc{P_2}\vdash P_1\plus P_2} \LabelRule{\oc R} \UnaRule{\oc{P_2}\vdash\oc{(P_1\plus P_2)}} \LabelRule{\rulename{cut}} \BinRule{P_2\vdash\oc{(P_1\plus P_2)}} \LabelRule{\plus L} \BinRule{P_1\plus P_2\vdash\oc{(P_1\plus P_2)}} \DisplayProof$
$\LabelRule{\zero L} \NulRule{\zero\vdash\oc{\zero}} \DisplayProof$
$\LabelRule{\rulename{ax}} \NulRule{\oc{P}\vdash\oc{P}} \LabelRule{\oc R} \UnaRule{\oc{P}\vdash\oc{\oc{P}}} \DisplayProof$
$\AxRule{P\vdash\oc{P}} \LabelRule{\rulename{ax}} \NulRule{P\vdash P} \LabelRule{\exists R} \UnaRule{P\vdash \exists\xi P} \LabelRule{\oc d L} \UnaRule{\oc{P}\vdash \exists\xi P} \LabelRule{\oc R} \UnaRule{\oc{P}\vdash\oc{\exists\xi P}} \LabelRule{\rulename{cut}} \BinRule{P\vdash\oc{\exists\xi P}} \LabelRule{\exists L} \UnaRule{\exists\xi P\vdash\oc{\exists\xi P}} \DisplayProof$
More generally, $\oc A$ is positive for any formula A.
The notion of positive connective is related with but different from the notion of asynchronous connective.
## Generalized structural rules
Positive formulas admit generalized left structural rules corresponding to a structure of $\tens$-comonoid: $P\limp P\tens P$ and $P\limp\one$. The following rule is derivable:
$\AxRule{\Gamma,P,P\vdash\Delta} \LabelRule{+ c L} \UnaRule{\Gamma,P\vdash\Delta} \DisplayProof \qquad \AxRule{\Gamma\vdash\Delta} \LabelRule{+ w L} \UnaRule{\Gamma,P\vdash\Delta} \DisplayProof$
Proof.$\AxRule{P\vdash\oc{P}} \AxRule{\Gamma,P,P\vdash\Delta} \LabelRule{\oc L} \UnaRule{\Gamma,P,\oc P\vdash\Delta} \LabelRule{\oc L} \UnaRule{\Gamma,\oc P,\oc P\vdash\Delta} \LabelRule{\oc c L} \UnaRule{\Gamma,\oc P\vdash\Delta} \LabelRule{\rulename{cut}} \BinRule{\Gamma,P\vdash\Delta} \DisplayProof$
$\AxRule{P\vdash\oc{P}} \AxRule{\Gamma\vdash\Delta} \LabelRule{\oc w L} \UnaRule{\Gamma,\oc P\vdash\Delta} \LabelRule{\rulename{cut}} \BinRule{\Gamma,P\vdash\Delta} \DisplayProof$
Positive formulas are also acceptable in the left-hand side context of the promotion rule. The following rule is derivable:
$\AxRule{\oc\Gamma,P_1,\dots,P_n\vdash A,\wn\Delta} \LabelRule{+ \oc R} \UnaRule{\oc\Gamma,P_1,\dots,P_n\vdash \oc{A},\wn\Delta} \DisplayProof$
Proof.$\AxRule{P_1\vdash\oc{P_1}} \AxRule{P_n\vdash\oc{P_n}} \AxRule{\oc\Gamma,P_1,\dots,P_n\vdash A,\wn\Delta} \LabelRule{\oc L} \UnaRule{\oc\Gamma,P_1,\dots,P_{n-1},\oc{P_n}\vdash A,\wn\Delta} \VdotsRule{}{\oc\Gamma,P_1,\oc{P_2},\dots,\oc{P_n}\vdash A,\wn\Delta} \LabelRule{\oc L} \UnaRule{\oc\Gamma,\oc{P_1},\dots,\oc{P_n}\vdash A,\wn\Delta} \LabelRule{\oc R} \UnaRule{\oc\Gamma,\oc{P_1},\dots,\oc{P_n}\vdash \oc{A},\wn\Delta} \LabelRule{\rulename{cut}} \BinRule{\oc\Gamma,\oc{P_1},\dots,\oc{P_{n-1}},P_n\vdash \oc{A},\wn\Delta} \VdotsRule{}{\oc\Gamma,\oc{P_1},P_2,\dots,P_n\vdash \oc{A},\wn\Delta} \LabelRule{\rulename{cut}} \BinRule{\oc\Gamma,P_1,\dots,P_n\vdash \oc{A},\wn\Delta} \DisplayProof$
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2017-09-19 13:36:09
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https://www.electro-tech-online.com/threads/repairing-my-gfx-card-6800-le.91177/
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Repairing my GFX Card - 6800 LE
Status
Not open for further replies.
XuryaX
New Member
My Geforce 6800 LE isnt working good, getting black screen error.
So i found a tutorial which is mentioned here:
Code:
http://img69.imageshack.us/img69/4061/capmod35gz.jpg
So, my local shop knows everything in shortcut doesnt know about details.
He cant udnerstand the Lower ESR Values.
But he can undertand the capacitance value.
So, according to tutorial it says Capacitance Value ~1000-10000 µF. Voltage rating > 4V.
Practical values
C=3300µF
V=6.3 - 16 V.
Now he has 3300 uF but the voltage rating is 25 V nand tempreature is 85 C.
In tutorial on a forum, where i got this image from says.
Code:
- SMALLER ESR spec. is BETTER e.g 0.025 ohm vs 1ohm
- SMALLER dissipation factor (tan d) spec. is BETTER
- LARGER ripple current spec. is BETTER e.g. 4 A vs 1 A
- LARGER temperature spec. is BETTER e.g 105 vs 85 C
- LARGER can size spec. is BETTER (too large can be bulky)
- LARGER voltage spec. is BETTER (too large value >25 V means large can size)
So, help me. I am kinda confused.
Last edited:
New Member
I'm new at this stuff too, but help me to better understand your problem:
You want to know if you can use a higher voltage cap?
XuryaX
New Member
Yea, i wanna know if i can use 25 volts capacitor. with 3300 uF rating and 85 C.
- LARGER voltage spec. is BETTER (too large value >25 V means large can size) - Tutorial says this. Means maybe i can use 25 Volts.
Kinda confused.
I dont even know the absics of a capacitor and stuff, so kinda am confused.
Last edited:
New Member
From what I have read and heard, yes, you may use a higher voltage cap. Good Luck!
Nigel Goodwin
Super Moderator
Yes, you can use 25V, no problem - it's probably also better (higher voltage usually have lower ESR).
New Member
I don't know about ESR, but I do know that I can use a higher voltage cap in a circuit. I do it quite often. It's just overkill but it should be fine.
sPuDd
New Member
An 85C cap is not a low ESR cap.
Needs to be 105C, low ESR.
My old Gforce Ti4600
Last edited:
transistance
New Member
It seems like computer grade capacitors are electrolytic-aluminum. You should definitely go for 105C or higher for stability of the graphics card.
I think it would be safe to tell your local techie that ESR is practically impedance.(this should be confirmed by a more experienced poster)
Wet tantalum capacitors are another choice but they go for like 90\$ a piece at your specs.
And I know that this will not help you much since the minimum order is 250 pieces but you can have an idea of the price range and what you should look at on your data sheet research. Or maybe your local shop can invest in some of these for future repairs.Electrolytic-aluminum capacitor
These are the other aluminum capacitors I found that might be suitable for you from digikey again.
Last edited:
Nigel Goodwin
Super Moderator
An 85C cap is not a low ESR cap.
Not at all, 85 degree caps can be low ESR as well.
Needs to be 105C, low ESR.
I keep nothing but 105 degree caps these days, there seems no point in keeping 85 degree ones - almost all failures are high-ESR, and using a higher temperature capacitor helps to extend their life.
This isn't to say that all 105 degree caps are any good, they are a number of really poor ones about, and they don't have the life of even 85 degree ones.
sPuDd
New Member
This isn't to say that all 105 degree caps are any good, they are a number of really poor ones about, and they don't have the life of even 85 degree ones.
Agreed, even the 'quality' stuff seems to have shipped off to China, leaving the quality behind. I'd like to find a local supplier of Jap made units, but despite all the pre-sale assurances they come with a Made in China sticker.
So I just replace them every 12months and save my breath.
sPuDd..
Status
Not open for further replies.
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2020-09-27 15:52:09
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http://googology.wikia.com/wiki/User_blog:Wythagoras/NEWS!_I_found_a_22-state_machine_that_beats_G!
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## FANDOM
10,818 Pages
$$\Sigma(22) \gg f_{\omega+1}(2 \uparrow^{12} 3) > G$$
0 _ 1 r 21
0 1 1 l 21
1 1 1 l 1
1 _ 1 r 2
2 1 1 r 2
2 _ _ r 3
3 _ _ r 14
3 1 1 r 4
4 1 1 l 5
4 _ _ r 6
5 1 _ l 5
5 _ 1 l 1
6 _ _ r 14
6 1 1 r 7
7 _ _ r 6
7 1 1 l 8
8 1 _ l 9
8 _ _ l 15
9 _ _ l 10
9 1 1 r 11
10 1 1 l 9
10 _ 1 r 1
11 1 _ r 12
11 _ _ l 13
12 _ 1 r 11
12 1 1 l 13
13 1 1 l 13
13 _ 1 l 10
14 _ _ r 18
14 1 _ l 8
15 _ 1 l 16
15 1 _ l 1
16 1 1 l 17
16 _ 1 l 1
17 _ _ l 16
17 1 _ l 16
18 _ _ r halt
18 1 _ l 19
19 _ 1 l 20
19 1 _ l 15
20 1 1 l 19
20 _ 1 l 19
21 _ 1 l 0
21 1 _ l 14
State 1 is state 0 of Deedlit's expandal machine
State x is state x of Deedlit's expandal machine for 2 ≤ x ≤ 17
Then, if the w+1 category is empty, it checks whether there is some in the w+2 category.
Then it changes all empty categories (remember, the tape looks like 11111....11111_1_1_1_1...) to ones for the w+1 category. That is about $$f_{\omega}^{-1}(n)$$ of the ones currently on the tape.
State 0 and 21 are used to set the input. 1_11, where the head is on the first one and in state 14.
## Snapshots of the tape
After 6 steps, state 14
1 11
^
After 13 steps, state 2
111 11
^
After 16 steps, state 18
111 11
^
After 17 steps, state 19
111 1
^
After 23 steps, state 1
1 1111 1
^
After 59 steps, state 1
1 1 1 11 11 1
^
After 1359 steps, state 10
11 1 1 1 1 1 1 1 111 111 111 111 111 11 1 1 1
^
## Bound
$$\Sigma(22) > f_{\omega+1}(2 \uparrow^{12} 3) > G$$ (See last tape)
## Poll
Do you think I'm a TM specialist?
13
1
The poll was created at 11:04 on August 5, 2014, and so far 14 people voted.
Thanks. Wythagoras (talk) 18:58, August 6, 2014 (UTC)
## Poll 2
What is the smallest value of the Busy Beaver function that beats G, you think?
1
0
1
0
1
2
2
4
1
0
1
0
1
0
0
1
0
0
The poll was created at 11:22 on August 5, 2014, and so far 15 people voted.
Personally I think 11 or 12 (I voted for 12). I would be very suprised if someone could implent Ackermannian growth in 9 states, in other words, I'd be suprised if Sigma(9) > A(100). Wythagoras (talk) 18:58, August 6, 2014 (UTC)
## History
September 9, 2010: r.e.s. proves that $$\Sigma(64) > G$$
April 8, 2013: Deedlit11 proves that $$\Sigma(25) > G$$
September 27, 2013: Wythagoras proves that $$\Sigma(24) > G$$ using Deedlit11's results.
October 6, 2013: Wythagoras proves that $$\Sigma(23) > G$$ using Deedlit11's results.
August 5, 2014: Wythagoras proves that $$\Sigma(22) > G$$ using Deedlit11's results.
## Poll 3
When will $$\Sigma(21) > G$$ be proven?
1
1
2
4
1
1
0
0
The poll was created at 12:33 on August 5, 2014, and so far 10 people voted.
This year hopefully :P. But not now. Wythagoras (talk) 18:58, August 6, 2014 (UTC)
## Poll 4
What is the smallest value of n such that $$\Sigma(n) > G$$ will be proven within 5 years?
1
1
1
1
2
1
1
1
0
1
0
The poll was created at 12:37 on August 5, 2014, and so far 10 people voted.
5 years is a long, long, long time. 5 years ago the wiki was barely created! Still, I think it is unreasonable to think we'd even come close to the real BB machines, so I think that Sigma(17) > G will be proven in five years, but if n is the number of states to beat G, I think that even Sigma(n+2) > G won't be proven within 15 years. Wythagoras (talk) 18:58, August 6, 2014 (UTC)
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2017-08-16 23:42:20
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https://indico.icranet.org/event/2/timetable/?view=standard_inline_minutes
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# ICRANet-ISFAHAN Astronomy Meeting
Asia/Tehran
ICRANet-Isfahan, Isfahan University of Technology (IUT - Iran) - online
,
Description
• # ICRANet-ISFAHAN Astronomy Meeting
### From the Ancient Persian Astronomy to Recent Developments in Theoretical and Experimental Physics, Astrophysics and General Relativity
Iran with the Ulugh Beg map of the "fixed stars” proposed by Abd al-Rahman al-Sufi (Azophi) around 964 CE have been among the first countries, centuries ago, to extend the knowledge of our Universe outside our planetary system. Persian astronomers made very important contributions to the fields of astronomy with the construction of Maragheh observatory in 1259 CE and Ulugh Beg Observatory in the 1420s. Now in 2021 by the Iranian National Observatory (INO), a new generation of Iranian scientists are going to further explore the Universe.
Isfahan as a historical city in the center of Iran and as one of the world's most beautiful cities hosts the first series of ICRANet-Isfahan Astronomy meeting which will be held virtually from 3-5 November 2021. This meeting will be organized in order to provide great opportunities for discussing about astronomy from the ancient Persian astronomy to recent developments in observational astronomy, high energy astrophysical phenomena such as Gamma-Ray Bursts (GRBs) and Active Galactic Nuclei (AGNs), Theories of Gravity, General Relativity and its Mathematical Foundation, Black Holes, Dark matter and Early Universe Cosmology.
A workshop on "Data Science in Astrophysics" will be held during the meeting on November 4th, a certificate will be issued only if participants successfully complete the tasks.
• ## Scientific Committee
Remo Ruffini (ICRANet/ICRA-Italy)(Co-Chair), Yousef Sobouti (Co-Chair)(ISABS-Iran), Hassan Firouzjahi (IPM,Iran), Shahram Khosravi (KHU, Iran)Habib Khosroshahi (IPM, Iran),Kourosh Nozari (UMZ, Iran),Sohrab Rahvar (SUT, Iran), Soroush Shakeri (IUT, Iran), Shadi Tahvildar-Zadeh (Rutgers, USA), She-Sheng Xue (ICRANet-Italy)
• ## Organizing Committee
Soroush Shakeri (IUT,Iran)(Chair), Amin Farhang (IPM and UT.Iran), Fazlollah Hajkarim (UNIPD-Italy), Rahim Moradi (ICRANet-Italy), Sedigheh Sajadian(IUT, Iran),Shahab Shahidi (DU, Iran),Wang Yu (ICRANet, Italy), M. H. Zhollideh Haghighi(IPM,KNTU, Iran)
Registration
Registration Form
Participants
• Aidin Momtaz
• Alexander Zakharov
• Ali Foroozmand
• Ali saffari
• Ali Saffari
• Ali Salarvand
• Amirmasoud Jannat
• Arshin Khaje Borj Sefidi
• Danial Lohrabi
• Davood Rafiei Karkevandi
• Ebrahim Hoseinkhani
• Ehsan Qoreishi
• Elahe Khalouei
• Erfan Qasemi
• Farangis Takdehghan
• Farzaneh Ostovarpour
• Fatemeh Abedini
• Golnaz Mazhari
• Hanieh Karimi
• Hassan Manshouri
• Hosein Heidarifatasmi
• Hossein Fatheddin
• Lorenzo Amati
• Maryam Hasani
• Maryam Mohebbi
• maryam sabiee
• Maryam vazirnia
• Marzieh Khani
• Masoumeh Tavakoli
• Niloofar Jokar
• Pierluca Carenza
• Remo Ruffini
• Rezvan Jalali
• Richard Kerner
• Saba Fotouhi
• Saeed Fakhry
• Saleh Al-Hossein Zorrieh
• Sara Saghafi
• Sepideh Ghaziasgar
• Setareh Moein
• Sina Etebar
• Soroush Shakeri
• Zahra Atharipour
• Zahra Mosleh
• Zeinab Zayeri
• پریساالسادات عقیلی
Contact
• Wednesday, 3 November
• 10:00 10:20
Block 0: Opening Ceremony
• 10:20 10:50
How the modern astronomy was introduced into Iranian universities 30m
Speaker: Prof. Yousef Sobouti (ISABS-Iran)
• 10:50 11:20
Celebrating the 50th anniversary of ”Introducing the Black Hole” 30m
Speaker: Prof. Remo Ruffini
• 11:20 11:50
Iranian National Observatory; status and vision 30m
Iranian National Observatory (INO) is located on Mt Gargash at 3600m covering a gap in the longitude distribution of modern mid-size telescopes. The INO project is now in its final stage of completion and is approaching the first light. Major milestones including the civil construction, installation of the dome, manufacturing of the 3.4m optical telescope and installation of the telescope at the site have been completed. The telescope is going through engineering tests aimed at the commissioning of the pointing and tracking. A suite of instruments has been planned, taking advantage of a sub-arcsecond seeing and the longitude, including a high-resolution imaging camera and a spectrograph with the ability to switch between the instruments in response to transient events. INO offers a platform for regional and international collaborations in astronomy and cosmology.
Speaker: Prof. Habib Khosroshahi (IPM, Iran)
• 11:50 12:20
Huntsman Telescope 30m
Speaker: Prof. Lee Spitler (Macquarie University, Australia)
• 12:20 12:40
Break 20m
• 12:40 13:10
Supenovea (SN) - Gamma Ray Burst (GRB) Connection 30m
Speaker: Prof. Massimo Della Valle ( (apodimonte Astronomical Observatory - INAF, Naples, Italy))
• 13:10 13:40
TBD 30m
Speaker: Prof. Luca Izzo (University of copenhagen, Demark)
• 15:30 16:00
Extremely high energy particle accelerators in our Galaxy 30m
The Large High Altitude Air Shower Observatory is a new-generation multi-component instrument for TeV-PeV gamma rays and TeV-EeV cosmic rays. Recently, LHAASO has published its first result on the discovery of 12 ultrahigh-energy (E>100TeV) gamma-ray sources at more than 7 sigma confidence level. Among them, there are famous sources like the Crab Nebula, the Cygnus Cocoon, as well as new sources without TeV counterpart. The discovery indicates the prevalence of PeV particle accelerators in our Galaxy.
Speaker: Prof. Ruoyu Liu (Nanjing University | NJU, China)
• 16:00 16:30
Multiwavelength and Multimessenger view of blazars 30m
I will discuss the recent progress in multiwavelength and multimessenger observations of blazars and the current status of the theoretical models applied to model their emission. Blazars, the most extreme subclass of AGN having jets that move relativistically towards the observer, are characterized by highly variable non-thermal emission across the entire electromagnetic spectrum, from radio up to very high energy gamma-ray bands. The emission properties of blazars in the spectral and time domains will be presented and discussed using the data collected from their observations in optical/UV, X-ray, and gamma-ray bands. In addition, the recent progress in the observations of very high-energy neutrinos from blazars will be discussed.
Speaker: Prof. Narek Sahakyan (ICRANet-Armenia)
• 16:30 17:00
Cosmology with Gamma-Ray Burst 30m
Speaker: Prof. Lorenzo Amati
• 17:00 17:20
Break 20m
• 17:20 17:50
Black hole hyperaccretion disks and gamma-ray bursts 30m
Gamma-ray bursts (GRBs) are the most luminous explosions in the Universe, and their origin and mechanism are the focus of intense research and debate. Black hole hyperaccretion model is one of the plausible candidates for the central engine of gamma-ray bursts and their activity is supposed to result in the complicated explosion phenomena including gamma-ray bursts, gravitational waves, and their electromagnetic counterparts. In the inner regions of such disks, photons are totally trapped due to high density and temperature. Getting cool through neutrinos and antineutrinos efficiently, these accretion disks are also called Neutrino Dominated Accretion Flows (NDAFs). Moreover, the high magnetic field (∼ 10^15−16G) and large density (∼ 10^10g cm−3) can be considered as the two important physical features of these disks, and as a result, self-gravity and gravitational instability might be of a crucial role in these dense hyperaccretion flows. As well, the magnetic field is proposed to be of considerable importance via both large and small scale impacts. After providing an introduction to the GRB’s and the candidates of their central engines, we focus on these two factors (self-gravity and magnetic field) to probe their potential effects on the hyperaccretion disk’s structure, in addition to their subsequent impacts on the GRB’s spectral features. In other words, we apply these two features to provide an explanation for the prompt Gamma-ray emission with its highly variable structure in the early time, and the electromagnetic afterglow emission associated with the late time activity of the GRB’s central engine.
Speaker: Prof. Shahram Abbassi (Ferdowsi University of Mashhad)
• 17:50 18:20
TBD 30m
Speaker: Prof. Brian Punsly
• 18:20 18:50
Az Zarreh Taa Aaftaab: The Role of General Relativity in the Structure of Elementary Particles of Matter 30m
It was a largely unfulfilled dream of Einstein to arrive at a quantum theory of atomistic matter that included electrodynamic phenomena, and one in which the principles of general relativity would reign supreme. Even though he is generally considered to have failed in this quest, his unifying vision remains a powerful one to this date. In this talk we explore some of the ways in which Einstein's dream may one day be realized, including (1) a general-relativity-based formulation of the joint evolution of classical fields together with point-particles that are sources of those fields, (2) a well-motivated deformation of classical nonlinear theories to quantum theories in which the motion of particles is guided by linear waves on particle configuration space, and (3) ring-like particles inspired by general relativity and a possible resolution of the dark matter puzzle.
• Thursday, 4 November
• 10:00 10:30
The Elephant in the Room. Kerr is its own Maximal Extension. 30m
Last year I showed that the Kerr metric, either ingoing or outgoing, contains light rays whose affine lengths are finite and yet they do not end at some singularity. This destroys all the singularity theorems as they assume this cannot happen. I was then told that the “singularity” exists in the maximal extension, say in Kruskal. I checked the derivation of these and found that the determinant of the alleged metric tensor is zero on all the horizons in Kruskal. Not only that but the protagonists all seem to think that this is OK! It isn't. Kerr and Eddington-Finkelstein are their own maximal extensions. If time permits I will also show why "soft hair" is fool's gold and why the Kerr-Schild approximation method gives the Ligo curves in its first step.
Speaker: Prof. Roy Patrick Kerr (University of Canterbury, Christchurch, New Zealand and ICRANet, Italy)
• 10:30 11:00
Angular Momentum to a Distant Observer 30m
The notion of angular momentum in general relativity has been a subtle issue since the 1960's, due to the discovery of supertranslation ambiguity": the angular momentums recorded by two distant observers of the same system may not be the same. In this talk, I shall show how mathematical theory identifies a correction term, and leads to a new definition of angular momentum that is free of any supertranslation ambiguity. This is based on joint work with Po-Ning Chen, Jordan Keller, Mu-Tao Wang, and Ye-Kai Wang
Speaker: Prof. Shing-Tung Yau (Harward-USA)
• 11:00 11:30
Gravitomagnetic interaction of a Kerr black hole with a magnetic field as the source of the high-energy radiation of gamma-ray bursts 30m
It is shown how the gravitomagnetic interaction of a Kerr black hole (BH) with a surrounding magnetic field induces an electric field able to accelerate surrounding charged particles to ultra-relativistic energies. Along the BH rotation axis, electrons/protons can reach even thousands of PeV leading to ultrahigh-energy cosmic rays (UHECRs) from stellar-mass BHs in long gamma-ray bursts (GRBs) and from supermassive BHs in active galactic nuclei (AGN). At off-axis latitudes around the BH vicinity, particles are accelerated to hundreds of GeV, and by synchrotron radiation emit high-energy GeV photons. Such a process occurs at all latitudes within 60 degrees of the polar axis. The theoretical framework describing these acceleration and radiation processes, how they extract the rotational energy of the Kerr BH, as well as the consequences for the astrophysics of GRBs are outlined.
Speaker: Prof. Jorge Rueda (ICRANet)
• 11:30 11:50
Break 20m
• 11:50 12:20
New high precision tests of General Relativity 30m
Speaker: Prof. Claus Lämmerzahl (ZARM, University of Bremen)
• 12:20 12:50
BepiColombo: ESA Cornerstone Mission to Mercury 30m
Speaker: Prof. Roberto Peron (National Institute of Astrophysics (INA), Italy)
• 12:50 13:20
The role of campfires in the heating of solar coronal plasma observed by Solar Orbiter and Solar Dynamics Observatory 30m
Speaker: Prof. Hossein Safari
• 13:20 13:50
Speaker: Prof. Fatemeh Tabatabaei (IPM)
• 15:30 16:30
Data Science in Relativistic Astrophysics (Hands On Workshop) 1h
Speaker: Prof. M. H. Zhollideh Haghighi (IPM and KNTU, Iran)
• 16:30 17:30
Data Science in Relativistic Astrophysics (Hands On Workshop) 1h
Speaker: Prof. Wang Yu (ICRANet, Italy)
• 17:30 18:30
Data Science in Relativistic Astrophysics (Hands On Workshop) 1h
• 18:30 18:50
TBD 20m
Speaker: Dr Becerra Laura
• Friday, 5 November
• 10:00 10:30
Ancient Persian Astronomy 30m
Speaker: Prof. Hossein Masoumi Hamedani (Iranian Institute of Philosophy, Iran)
• 10:30 11:00
Astronomy in Islamic World - a European perspective 30m
Arab and Islamic Civilization emerged at the crossroads in a double sense, as a bridge between the Greco-Roman Antiquity and European Modernity in time, and as the junction between the declining Roman Empire and the still vigorous Indian and Persian civilizations in space. In this talk, we shall highlight the most important contributions of Islamic Polymaths to Mathematics and Astronomy, paving the way to the next stage of the development of science which occurred in the late Middle Ages in Europe.
Speaker: Prof. Richard Kerner (Sorbonne Université, France)
• 11:00 11:20
Break 20m
• 11:20 11:50
Dark matter fermions: from linear to non-linear structure formation 30m
Relaxation mechanisms of collisionless self-gravitating systems of fermions in cosmology, can lead to equilibrium states which are stable, long-lived, and able to explain the dark matter (DM) halos in galaxies. The most general fermionic DM profile out of such a mechanism, develops a degenerate compact core which is surrounded by an extended halo. When applied to the Milky Way, it is demonstrated that the outer halo can explain the rotation curve of our Galaxy, while the central DM-core explains the dynamics of all the best resolved S-cluster stars orbiting SgrA *, without assuming a central black hole (BH). When such novel core-halo DM profiles are applied to larger galaxies, the dense DM core can reach the critical mass for gravitational collapse into a BH of ∼ 10^8 Mo. This result provides a new mechanism for supermassive BH formation in active galaxies directly from DM, leading to a paradigm shift in the understanding of galactic cores.
Speaker: Prof. Carlos Arguelles (ICRANet, Italy)
• 11:50 12:20
MOND and MOG 30m
Speaker: Prof. Hosein Haghi (ISABS-Iran )
• 12:20 12:50
Early Universe Cosmology 30m
Speaker: Prof. Clement Stahl (Strasbourg U., France)
• 14:30 15:00
TBD 30m
Speaker: Prof. Yerkan Amiratove
• 15:00 15:30
TBD 30m
Speaker: Prof. Liang Li (ICRANet)
• 15:30 15:50
Break 20m
• 15:50 16:20
TBD 30m
Speaker: Prof. Kuantay Boshkaev
• 16:20 16:50
Axion in Astrophysics 30m
This is a review of the latest developments on axion astrophysics, with particular attention to the axion production in stellar environments and to the phenomenology of the axion-photon mixing on astrophysical scales.
Speaker: Prof. Pierluca Carenza (Stockholm U., OKC, Sweden)
• 16:50 17:20
Production of Thermal QCD Axions in the Early Universe 30m
We study the thermal production of axions over different scales especially around the QCD and electroweak phase transitions in the early universe. We focus on the most motivated axion models (KSVZ and DFSZ) and investigate how the thermal history can influence on the production rate of hot axion as dark radiation. This can lead to predictions for the future measurements of the cosmic microwave background by experiments like CMB-S4.
Speaker: Prof. Fazlollah Hajkarim (UNIPD-Italy)
• 17:20 17:50
Concluding remarks
Convener: Prof. Soroush Shakeri (Isfahan University of Technology (IUT) & ICRANet-Isfahan)
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2021-10-20 01:17:33
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https://matthewonsoftware.com/blog/big-o-notation/
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# Big O Notation
The Big O Notation is a mathematical, asymptotic notation that describes time complexity and space complexity of algorithms/ function when the argument tends towards a particular value of infinity.
For example, O(n) might be the same complexity of an algorithm that traverses thought an array of length n,
similarly, O(n+m) might be the time complexity of an algorithm that traverses through an array of length n and through a string of length m.
The following are examples of common complexities and their Big O notations, ordered from fastest to slowest
• Constant: O(1)
• Logarithmic: O(log(n))
• Linear: O(n)
• Log-linear: O(n log(n))
• Factorial: O(n!)
Sometimes there may be best/ average/ worst case scenario depends on the input data, how they’re structured and so on, so in other words depend on algorithm (quick sort for example) but Big O usually refers to worst case scenario.
Examples of algorithms and their time complexity:
• fun(arr[]) ⇒ 1 + arr[0] we just return static number and first element of an array (assume that’s not empty), so the size of input data will not affect the complexity anyhow. the time complexity will be O(1), in other words constant.
• fun(arr []) ⇒ sum(arr) here we are summing the given array, so we have to traverse through whole array, in other words the more elements are in given array, the more work our algorithm need to perform, so its linear complexity, in other words O(n).
• fun(arr []) ⇒ pair(arr) here we are pairing all elements, assume that it’s done by nested for loops. Then guess what, it would require going through every element twice. In previous example it was O(n) to iterate over given array. In this case it will be O(n2).
The constants in Big O Notation does not matter and how to simplify it.
Let’s assume that we have function that do bunch of elementary operations, it sums up few numbers, declare some variables, an array with few elements, it’s going to be algorithm that may run in O(25) but that would be written down to O(1). So you never want to say O(25) as it does not really make sense, and you should write it as O(1). The second example might be an algorithm in which we iterate over a given array from left to right and from right to left. That would be an O(2n) complexity, but you would drop a two because that’s a constant, so it would be just O(n). Also, important thing to remember, let’s assume that in our function we have a pairing algorithm and the one described just before. That is O(n2 + 2n) but we would remove the two as its constant and n also as it’s become meaningless next to the N squared, so it would be O($n^2)$.
Other example: O(n! + n3 + log(n) + 3) ⇒ O(n!) as factorial would growth the fastest. We were able to drop this in above way as we were iterating on the same input data N. Now assume that we do the same but for input we’re going to get two different arrays of size M and N.
O(m2 + 2n) can we transform it to O(m2)? No! We can not! As those are different inputs, so we want to know the behaviour as it relates to these two different inputs, and you can imagine that there is actually a scenario where m2 is tiny compared to N.
Imagine M is equal to two and N is equal to a thousand, then m2 would be smaller than N. That’s why when you have two variables, you do always want to keep both of them. So it has to remain O(m2+ n)(we can still drop constants)
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2022-08-18 23:55:46
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https://zbmath.org/?q=an%3A1075.35077
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## On a sharp lower bound on the blow-up rate for the $$L^2$$ critical nonlinear Schrödinger equation.(English)Zbl 1075.35077
Summary: We consider the $$L^2$$ critical nonlinear Schrödinger equation $$iu_t=-\Delta u-| u|^{\frac{4}{N}}u$$ with initial condition in the energy space $$u(0,x)=u_0\in H^1$$ and study the dynamics of finite time blow up solutions. In an earlier sequence of papers, the authors established for a certain class of initial data on the basis of dispersive properties in $$L^2_{\text{loc}}$$ a sharp and stable upper bound on the blow up rate: $|\nabla u(t)|_{L^2}\leq C\left(\frac{\log|\log(T-t)|}{T-t}\right)^{\frac{1}{2}}.$ In an earlier paper, the authors then addressed the question of a lower bound on the blow up rate and proved for this class of initial data the nonexistence of self-similar solutions, that is, $$\lim_{t\to T}\sqrt{T-t}|\nabla u(t)|_{L^2}=+\infty.$$
In this paper, we prove the sharp lower bound $|\nabla u(t)|_{L^2}\geq C_2 \left(\frac{\log|\log(T-t)|}{T-t}\right)^{\frac{1}{2}}$ by exhibiting the dispersive structure in the scaling invariant space $$L^2$$ for this log-log regime. In addition, we extend to the pure energy space $$H^1$$ a dynamical characterization of the solitons among the zero energy solutions.
### MSC:
35Q55 NLS equations (nonlinear Schrödinger equations) 35B44 Blow-up in context of PDEs 35C08 Soliton solutions
Full Text:
### References:
[1] G. D. Akrivis, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney, Numerical approximation of blow-up of radially symmetric solutions of the nonlinear Schrödinger equation, SIAM J. Sci. Comput. 25 (2003), no. 1, 186 – 212. · Zbl 1039.35115 [2] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313 – 345. , https://doi.org/10.1007/BF00250555 H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal. 82 (1983), no. 4, 347 – 375. · Zbl 0533.35029 [3] J. Bourgain, Global solutions of nonlinear Schrödinger equations, American Mathematical Society Colloquium Publications, vol. 46, American Mathematical Society, Providence, RI, 1999. · Zbl 0933.35178 [4] Jean Bourgain and W. Wang, Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 197 – 215 (1998). Dedicated to Ennio De Giorgi. · Zbl 1043.35137 [5] S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation, Phys. D 57 (1992), no. 1-2, 96 – 160. · Zbl 0767.35082 [6] Fibich, G.; Merle, F.; Raphaël, P., Numerical proof of a spectral property related to the singularity formation for the $$L^2$$ critical nonlinear Schrödinger equation, in preparation. · Zbl 1100.35097 [7] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal. 32 (1979), no. 1, 1 – 32. · Zbl 0396.35028 [8] Man Kam Kwong, Uniqueness of positive solutions of \Delta \?-\?+\?^{\?}=0 in \?$$^{n}$$, Arch. Rational Mech. Anal. 105 (1989), no. 3, 243 – 266. · Zbl 0676.35032 [9] M. J. Landman, G. C. Papanicolaou, C. Sulem, and P.-L. Sulem, Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension, Phys. Rev. A (3) 38 (1988), no. 8, 3837 – 3843. [10] Mihai Mariş, Existence of nonstationary bubbles in higher dimensions, J. Math. Pures Appl. (9) 81 (2002), no. 12, 1207 – 1239 (English, with English and French summaries). · Zbl 1040.35116 [11] Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. Funct. Anal. 11 (2001), no. 1, 74 – 123. · Zbl 0985.35071 [12] Kevin McLeod, Uniqueness of positive radial solutions of \Delta \?+\?(\?)=0 in \?$$^{n}$$. II, Trans. Amer. Math. Soc. 339 (1993), no. 2, 495 – 505. · Zbl 0804.35034 [13] F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J. 69 (1993), no. 2, 427 – 454. · Zbl 0808.35141 [14] Frank Merle and Pierre Raphael, Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Journées ”Équations aux Dérivées Partielles” (Forges-les-Eaux, 2002) Univ. Nantes, Nantes, 2002, pp. Exp. No. XII, 5. · Zbl 1185.35263 [15] F. Merle and P. Raphael, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal. 13 (2003), no. 3, 591 – 642. · Zbl 1061.35135 [16] Frank Merle and Pierre Raphael, On universality of blow-up profile for \?² critical nonlinear Schrödinger equation, Invent. Math. 156 (2004), no. 3, 565 – 672. · Zbl 1067.35110 [17] Frank Merle and Pierre Raphael, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys. 253 (2005), no. 3, 675 – 704. · Zbl 1062.35137 [18] Dmitry Pelinovsky, Radiative effects to the adiabatic dynamics of envelope-wave solitons, Phys. D 119 (1998), no. 3-4, 301 – 313. · Zbl 1194.35423 [19] Perelman, G., On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D, Ann. Henri Poincaré, 2 (2001), 605-673. · Zbl 1007.35087 [20] Pierre Raphael, Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann. 331 (2005), no. 3, 577 – 609. · Zbl 1082.35143 [21] Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. · Zbl 0928.35157 [22] Michael I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), no. 3, 472 – 491. · Zbl 0583.35028 [23] Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), no. 4, 567 – 576. · Zbl 0527.35023
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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2022-08-08 23:25:16
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http://euclidlab.org/unsolved/170-odd-perfect-number-existence
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# Does an odd perfect number exist?
Take a positive whole number, like, for example, 6. Write down all of its positive divisors except itself. For example, 1 divides 6, 2 divides 6 and 3 divides 6. Add up all these divisors. Here we get 1+2+3=6. Remarkably, we got our original number back. That was not a magic trick, in fact it doesn't usually work if you don't start with the right number, but 6 was a very special number. Any positive whole number which is the sum of all of its positive divisors, not including itself, is called a perfect number. Every perfect number anybody has ever discovered is even, but there isn't any obvious reason why this should be true. Is there an odd perfect number? Nobody knows. Maybe you can find one, or maybe you can prove that every perfect number must be even.
The image shows the first 6 perfect numbers. Source: Wikipedia contributors. "List of perfect numbers". Wikipedia, The Free Encyclopedia. 15 Dec. 2009. Retrieved 21 Mar. 2010.
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2017-09-26 16:28:53
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https://homework.cpm.org/category/CC/textbook/cca/chapter/11/lesson/11.1.2/problem/11-20
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### Home > CCA > Chapter 11 > Lesson 11.1.2 > Problem11-20
11-20.
1. For each function, find the inverse function. Homework Help ✎
1. f(x) = 2x + 3
2. g(x) =
Write down the steps of the original function.
1. Multiply by 2.
$\textit{f}^{-1}(x)=\frac{\textit{x}-3}{2}$
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2019-08-23 01:02:17
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http://hal.in2p3.fr/view_by_stamp.php?label=CPPM&action_todo=view&langue=en&id=in2p3-00713097&version=1
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591 articles – 2447 references [version française]
HAL: in2p3-00713097, version 1
Physics at LHC 2012, Vancouver : Canada (2012)
Search for $B^0_s \to \mu^+\mu^−$ and $B^0 \to \mu^+\mu^−$ decays at LHCb
For the LHCb collaboration(s)
(2012-06-05)
A search for $B^0_s \to \mu^+\mu^−$ and $B^0_s \to \mu^+\mu^−$$decays is performed using 1.0 fb$^{−1}$of pp collision data collected at$\sqrt{s}=7$TeV with the LHCb experiment at the Large Hadron Collider. For both decays the number of observed events is consistent with expectation from background and Standard Model signal predictions. Upper limits on the branching fractions are determined to be BR$(B^0_s \to \mu^+\mu^−) < 4.5 (3.8) \times 10^{−9}$and BR$(B^0 \to \mu^+\mu^−) < 1.0 (0.81) \times 10^{-9}\$ at 95\% (90\%) confidence level.
Subject(s) : Physics/High Energy Physics - Experiment
in2p3-00713097, version 1 http://hal.in2p3.fr/in2p3-00713097 oai:hal.in2p3.fr:in2p3-00713097 From: Danielle Cristofol <> Submitted for: Cosme Adrover Pacheco <> Submitted on: Friday, 29 June 2012 11:59:54 Updated on: Friday, 29 June 2012 15:12:00
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2014-04-20 00:41:18
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http://tibasicdev.wikidot.com/factorial
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The ! Command
Command Summary
Calculates the factorial of a number or list.
Command Syntax
value!
Press:
1. MATH to access the math menu.
2. LEFT to access the PRB submenu.
3. 4 to select !, or use arrows.
TI-83/84/+/SE
1 byte
! is the factorial function, where n! = n*(n-1)! and 0! = 1, n an nonnegative integer. The function also works for arguments that are half an odd integer and greater than -1/2: $(-\frac1{2})!$ is defined as $\sqrt{\pi}$ and the rest are defined recursively.
3!
6
(‾.5)!
1.772453851
Ans²
3.141592654
The combinatorial interpretation of factorials is the number of ways to arrange n objects in order.
# Error Conditions
• ERR:DOMAIN for any numbers except the ones mentioned above.
.
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2017-01-20 07:42:26
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https://crypto.stackexchange.com/tags/elliptic-curves/hot?filter=week
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Tag Info
We want $(r,s)$ same for two different set of $d,k,h$ In ECDSA $r = x_0([k]G) \bmod n$ where $k \in [1,n-1]$ and $x_o$ is the x-coordinate of the scalar multiplication $[k]G$ $s = k^{-1}\cdot (h+r\cdot d)$ where $h$ is the left most bits of $h$ to fit in the group order ( for simplicity we called it $h$ again). Now we want same $(r,s)$ for $d,k,h$ and $d',... 5 Is it possible for Carol to find Bobs key in$S_{pks}$This is a decisional Diffie-Hellman problem. We can summary this problem as: "we're given the values$G, aG, abG$, and a series of values$c_1G, c_2G, ... c_nG$, can we recognize$c_iG = bG$" We can reword the problem as "assuming$H = aG$, we're given the values$H, (a^{-1})H, bH$, can ... 4 For a given private key$d$, random$k$and message hash$h$: is it possible that there exists a different set of$d$,$k$and$h$which produces the same ECDSA signature using the$\text{secp256k1}$curve? Yes, and further it's easy to explicitly compute an alternate$(d',k',h')$that matches all reasonable meanings of "different set of$d$,$k$and$...
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2021-05-08 10:33:55
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https://math.stackexchange.com/questions/1105703/hessian-related-convex-optimization-question
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# Hessian Related convex optimization question
My precise question is from an exercise;
Let $f : \mathbb{R}^2 → \mathbb{R}$ be a twice differentiable function. Prove that there exists a $λ ∈ R$ such that $g : \mathbb{R}^2 → \mathbb{R}$ defined as $g(x) := f (x) + (λ/2)||x||^2$ is convex.
Hint: compute the Hessian of $g$ and prove that for $λ$ large enough, this Hessian is positive definite.
I am trying to follow the hint but immediately running into errors when computing the hessian of g.
I am really not sure how to proceed given the $f(x)$ along with $||x||$. I mean we only still have the one variable $x$, so what possibility is there for double derivatives?
$$\left[ \begin{array}{ c c } g_{ff} & g_{fx} \\ g_{fx} & g_{xx} \end{array} \right]$$
Which seems rather silly. Would anyone care point out my misinterpration/fundamentally flaw, and lead me on the right track?
• The functions $f$ and $g$ are defined as functions that take a vector in $\mathbb{R}^2$ as input and return a single real number as output. So the $x$ is actually a vector containing two numbers. You can compute derivatives with respect to $x_1$ and $x_2$. – John von N. Jan 15 '15 at 19:09
• I have edited my answer after a good remark from @SZhu. Actually, the second derivatives must be bounded to find a fixed $\lambda$. – Alex Silva Jan 16 '15 at 16:37
• That is right. As stated, the theorem is false. – Michael Grant Jan 16 '15 at 22:58
By considering $\| \cdot\|$ the $2$-norm, the Hessian is given by
$$H = \left [ \begin{array}{cc} \frac{\partial^2 f(x_1,x_2)}{\partial x_1^2} + \lambda & \frac{\partial^2 f(x_1,x_2)}{\partial x_1 \partial x_2} \\ \frac{\partial^2 f(x_1,x_2)}{\partial x_2 \partial x_1}& \frac{\partial^2 f(x_1,x_2)}{\partial x_2^2} + \lambda\\ \end{array} \right],$$ for $x = (x_1,x_2)$.
The matrix $H$ is positive definite if and only if all principal minors are positive. Thus,
$$\frac{\partial^2 f(x_1,x_2)}{\partial x_1^2} + \lambda > 0,$$ and
$$\left( \frac{\partial^2 f(x_1,x_2)}{\partial x_1^2} + \lambda \right)\left( \frac{\partial^2 f(x_1,x_2)}{\partial x_2^2} + \lambda \right) - \left(\frac{\partial^2 f(x_1,x_2)}{\partial x_1 \partial x_2} \right)\left(\frac{\partial^2 f(x_1,x_2)}{\partial x_2 \partial x_1} \right) >0.$$
You have to assume other constraints for $f(x)$. Notice that you should have $$\frac{\partial^2 f(x_1,x_2)}{\partial x_1^2} > -\infty,$$ for all $x_1$ and $x_2$, otherwise there is no $\lambda$ that guarantees the convexity of $g(x)$. I take the example given by @SZhu as follows.
If $\frac{\partial^2 f(x_1,x_2)}{\partial x_1^2}=-e^{x_1+x_2}$, then there is no fixed $\lambda$ such that the first principal minor is positive for all $x_1$ and $x_2$.
Yet, you also should have
$$\frac{\partial^2 f(x_1,x_2)}{\partial x_2^2} > -\infty$$ and
$$\left(\frac{\partial^2 f(x_1,x_2)}{\partial x_1 \partial x_2} \right)\left(\frac{\partial^2 f(x_1,x_2)}{\partial x_2 \partial x_1} \right) < \infty,$$ for all $x_1$ and $x_2$.
If these conditions are satisfied then for $\lambda$ large enough, it is easy to see that the two principal minors are positive.
• The only constraint on $f$ is twice differentiable, right? Consider $f(x) = -e^{x_1+x_2}$, which is good I think. However, $\frac{\partial^2 f(x_1,x_2)}{\partial x_1^2}=-e^{x_1+x_2}$ can tend to $-\infty$. So no fixed $\lambda$ can work for this function. Correct me if make errors here. – ZhuShY Jan 16 '15 at 16:11
• Well, suppose you say a finite $\lambda$ work, but I can also find finite $x_1$ and $x_2$ such that $H$ is not positive definite thereby $g(x)$ is not strongly convex. This is basically from the definition of convergence to $\infty$. – ZhuShY Jan 16 '15 at 16:24
• Well, then your $\lambda$ is not fixed. – ZhuShY Jan 16 '15 at 16:26
• Ok, let me ask this. Do you want a fixed $\lambda$ that work for all $x$ or for a given $x$ you can find a $\lambda$ that make $g$ has positive definite hessian? – ZhuShY Jan 16 '15 at 16:27
• I was directed here by @Michael Grant. You may also like to see my problem that is kind of related to this one. click here Thanks. – ZhuShY Jan 16 '15 at 16:31
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2019-07-19 12:58:14
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https://www.r-bloggers.com/2014/11/how-to-summarize-a-2d-posterior-using-a-highest-density-ellipse/
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Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
Making a slight digression from last month’s Probable Points and Credible Intervals here is how to summarize a 2D posterior density using a highest density ellipse. This is a straight forward extension of the highest density interval to the situation where you have a two-dimensional posterior (say, represented as a two column matrix of samples) and you want to visualize what region, containing a given proportion of the probability, that has the most probable parameter combinations. So let’s first have a look at a fictional 2D posterior by using a simple scatter plot:
plot(samples)
Whoa… that’s some serious over-plotting and it’s hard to see what’s going on. Sure, the bulk of the posterior is somewhere in that black hole, but where exactly and how much of it?
A highest posterior density ellipse shows this by covering the area that contains the most probable parameter combinations while containing p% of the posterior probability. Like finding the highest density interval corresponds to finding the shortest interval containing p% of the probability, finding the highest density ellipse corresponds to finding the smallest ellipse containing p% of the probability a.k.a. the minimum volume ellipse. I have spent a lot of time trying to figure out how compute minimum volume ellipses. Wasted time, it turns out, as it can be easily computed using packages that come with R, you just have to know what you are looking for. If you just want the code skip over the next paragraph, if you want to know the tiny bit of detective work I had to do to figure this out, read on.
To find the points in sample that are included in a minimum volume ellipse covering, say, 75% of the samples you can use cov.mve(samples, quantile.used = nrow(samples) * 0.75) from the MASS package, here quantile.used specifies the number of points in samples that should be inside the ellipse. It uses an approximation algorithm described by Van Aelst, S. and Rousseeuw, P. (2009) that is not guaranteed to find the minimum volume ellipse but that will often be pretty close. A problem is that cov.mve does not return the actual ellipse, it returns a robustly measured covariance matrix, but that’s not really what we are after. It does return an object that contains the indices of the points that are covered by the minimum volume ellipse, if fit is the object returned by cov.mve then these points can be extracted like this: points_in_ellipse <- samples[fit$best, ]. To find the ellipse we are going to use ellipsoidhull from the cluster package on the points_in_ellipse. It returns an object which represents the minimum volume ellipse and by using its predict function we get a two column matrix with points that lie on the hull of the ellipse and that we can finally plot. That wasn’t too easy to figure out, but it’s pretty easy to do. The code below plots a 75% minimum volume / highest density ellipse: library(MASS) library(cluster) # Finding the 75% highest density / minimum volume ellipse fit <- cov.mve(samples, quantile.used = nrow(samples) * 0.75) points_in_ellipse <- samples[fit$best, ]
ellipse_boundary <- predict(ellipsoidhull(points_in_ellipse))
# Plotting it
plot(samples, col = rgb(0, 0, 0, alpha = 0.2))
lines(ellipse_boundary, col="lightgreen", lwd=3)
legend("topleft", "50%", col = "lightgreen", lty = 1, lwd = 3)
Looking at this new plot we see that for the bulk of the probability mass the parameters are correlated. This correlation was not really visible in the naive scatter plot. If you rerun this code many times you will notice that the ellipse changes position slightly each time. This is due to cov.mve using an non-exact algorithm. If you have a couple of seconds to spare you can make cov.mve more exact by setting the parameter nsamp to a large number, say nsamp = 10000.
You are, of course, not limited to drawing just outlines and if you want to draw shaded ellipses you can use the polygon function. The code below draws three shaded highest density ellipses of random color with coverages of 95%, 75% and 50%.
plot(samples, col = rgb(0, 0, 0, alpha = 0.2))
for(coverage in c(0.95, 0.75, 0.5)) {
fit <- cov.mve(samples, quantile.used = nrow(samples) * coverage)
ellipse_boundary <- predict(ellipsoidhull(samples[fit$best, ])) polygon(ellipse_boundary, col = sample(colors(), 1), border = NA) } Looks like modern aRt to me! ## A Handy Function for Plotting Highest Density Ellipses The function bellow adds a highest density ellipse to an existing plot created using base graphics: # Adds a highest density ellipse to an existing plot # xy: A matrix or data frame with two columns. # If you have to variables just cbind(x, y) them. # coverage: The percentage of points the ellipse should cover # border: The color of the border of the ellipse, NA = no border # fill: The filling color of the ellipse, NA = no fill # ... : Passed on to the polygon() function add_hd_ellipse <- function(xy, coverage, border = "blue", fill = NA, ...) { library(MASS) library(cluster) fit <- cov.mve(xy, quantile.used = round(nrow(xy) * coverage)) points_in_ellipse <- xy[fit$best, ]
ellipse_boundary <- predict(ellipsoidhull(points_in_ellipse))
polygon(ellipse_boundary, border=border, col = fill, ...)
}
So to replicate the above plot with the 75% highest density ellipse you could now write:
plot(samples)
add_hd_ellipse(samples, coverage = 0.75, border = "lightgreen", lwd=3)
## Some Other Options when Plotting 2D Posteriors
Obviously, a highest density ellipse is only going to work well if the posterior is roughly elliptical. If this is not the case, an alternative is to use a 2D kernel density estimator on the samples and trace out the coverage boundaries. The function HPDregionplot in the emdbook package does exactly this:
library(emdbook)
plot(samples, col=rgb(0, 0, 0, alpha = 0.2))
HPDregionplot(samples, prob = c(0.95, 0.75, 0.5), col=c("salmon", "lightblue", "lightgreen"), lwd=3, add=TRUE)
legend("topleft", legend = c("95%", "75%", "50%"), col = c("salmon", "lightblue", "lightgreen"), lty=c(1,1,1), lwd=c(3,3,3))
You could also plot a 2d histogram of the samples, for example, using the hexagon plot in ggplot2:
qplot(samples[,1], samples[,2], geom=c("hex"))
However you would have to work a bit with the color scheme if you wanted the colors to correspond to a given coverage.
Finally, if you plot a 2D density it could also be useful to add marginal density plots, as is done in the default plot for the Bayesian First Aid alternative to the correlation test. Here with completely fictional data on the number of shotguns and the number of zombie attacks per state in the U.S:
library(BayesianFirstAid)
fit <- bayes.cor.test(no_zombie_attacks, no_shotguns_per_1000_persons)
plot(fit)
## References
Van Aelst, S. and Rousseeuw, P. (2009), Minimum volume ellipsoid. Wiley Interdisciplinary Reviews: Computational Statistics, 1: 71–82. Doi: 10.1002/wics.19, link to the paper (unfortunately behind paywall)
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2022-01-24 20:17:08
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http://theochem.github.io/horton/2.0.0/lib/mod_horton_gbasis_cext.html
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# 3.4.1. horton.gbasis.cext – C++ extensions¶
class horton.gbasis.cext.GOBasis
Bases: horton.gbasis.cext.GBasis
check_matrix_coeffs()
check_matrix_four_index()
check_matrix_two_index()
compute_electron_repulsion()
Compute electron-electron repulsion integrals
Argument:
output
When a DenseFourIndex object is given, it is used as output argument and its contents are overwritten. When a DenseLinalgFactory or CholeskyLinalgFactory is given, it is used to construct the four-index object in which the integrals are stored.
Returns: The four-index object with the electron repulsion integrals.
Keywords: ERI, four-center integrals
compute_grid_density_dm()
Compute the electron density on a grid for a given density matrix.
Arguments:
dm
A density matrix. For now, this must be a DenseTwoIndex object.
points
A Numpy array with grid points, shape (npoint,3).
Optional arguments:
output
A Numpy array for the output, shape (npoint,). When not given, an output array is allocated and returned.
epsilon
Allow errors on the density of this magnitude for the sake of efficiency.
Warning: the results are added to the output array! This may be useful to combine results from different spin components.
Returns: the output array. (It is allocated when not given.)
compute_grid_density_fock()
Compute a two-index operator based on a density potential grid in real-space
Arguments:
points
A Numpy array with grid points, shape (npoint,3).
weights
A Numpy array with integration weights, shape (npoint,).
pots
A Numpy array with density potential data, shape (npoint,).
fock
A two-index operator. For now, this must be a DenseTwoIndex object.
Warning: the results are added to the fock operator!
compute_grid_esp_dm()
Compute the electrostatic potential on a grid for a given density matrix.
Arguments:
dm
A density matrix. For now, this must be a DenseTwoIndex object.
coordinates
A (N, 3) float numpy array with Cartesian coordinates of the atoms.
charges
A (N,) numpy vector with the atomic charges.
points
A Numpy array with grid points, shape (npoint,3).
grid_fn
A grid function.
Optional arguments:
output
A Numpy array for the output. When not given, it will be allocated.
Warning: the results are added to the output array! This may be useful to combine results from different spin components.
compute_grid_gradient_dm()
Compute the electron density gradient on a grid for a given density matrix.
Arguments:
dm
A density matrix. For now, this must be a DenseTwoIndex object.
points
A Numpy array with grid points, shape (npoint,3).
Optional arguments:
output
A Numpy array for the output, shape (npoint,3). When not given, it will be allocated.
epsilon
Allow errors on the density of this magnitude for the sake of efficiency.
Warning: the results are added to the output array! This may be useful to combine results from different spin components.
compute_grid_gradient_fock()
Compute a two-index operator based on a density potential grid in real-space
Arguments:
points
A Numpy array with grid points, shape (npoint,3).
weights
A Numpy array with integration weights, shape (npoint,).
pots
A Numpy array with gradient potential data, shape (npoint, 3).
fock
A two-index operator. For now, this must be a DenseTwoIndex object.
Warning: the results are added to the fock operator!
compute_grid_hartree_dm()
Compute the Hartree potential on a grid for a given density matrix.
Arguments:
dm
A density matrix. For now, this must be a DenseTwoIndex object.
points
A Numpy array with grid points, shape (npoint,3).
grid_fn
A grid function.
Optional arguments:
output
A Numpy array for the output. When not given, it will be allocated.
Warning: the results are added to the output array! This may be useful to combine results from different spin components.
compute_grid_kinetic_dm()
Compute the kinetic energy density on a grid for a given density matrix.
Arguments:
dm
A density matrix. For now, this must be a DenseTwoIndex object.
points
A Numpy array with grid points, shape (npoint,3).
Optional arguments:
output
A Numpy array for the output, shape (npoint,). When not given, it will be allocated.
Returns: An array with shape (npoint,) containing the kinetic energy density. When an output array is given, it is also used as return value.
Warning: the results are added to the output array! This may be useful to combine results from different spin components.
compute_grid_kinetic_fock()
Compute a two-index operator based on a kientic energy density potential grid in real-space
Arguments:
points
A Numpy array with grid points, shape (npoint,3).
weights
A Numpy array with integration weights, shape (npoint,).
pots
A Numpy array with kinetic energy density potential data, shape (npoint,).
fock
A one-body operator. For now, this must be a DenseOneBody object.
Warning: the results are added to the fock operator!
compute_grid_orbitals_exp()
Compute the orbitals on a grid for a given set of expansion coefficients.
Arguments:
exp
An expansion object. For now, this must be a DenseExpansion object.
points
A Numpy array with grid points, shape (npoint,3).
iorbs
The indexes of the orbitals to be computed. If not given, the orbitals with a non-zero occupation number are computed
Optional arguments:
output
An output array, shape (npoint, len(iorbs)). The results are added to this array. When not given, an output array is allocated and the result is returned.
Warning: the results are added to the output array!
Returns: the output array. (It is allocated when not given.)
compute_grid_point1()
compute_kinetic()
Compute the kinetic energy integrals in a Gaussian orbital basis
Arguments:
output
When a TwoIndex instance is given, it is used as output argument and its contents are overwritten. When LinalgFactory is given, it is used to construct the output TwoIndex object. In both cases, the output two-index object is returned.
Returns: TwoIndex object
compute_nuclear_attraction()
Compute the nuclear attraction integral in a Gaussian orbital basis
Arguments:
coordinates
A float array with shape (ncharge,3) with Cartesian coordinates of point charges that define the external field.
charges
A float array with shape (ncharge,) with the values of the charges.
output
When a TwoIndex instance is given, it is used as output argument and its contents are overwritten. When LinalgFactory is given, it is used to construct the output TwoIndex object. In both cases, the output two-index object is returned.
Returns: TwoIndex object
compute_overlap()
Compute the overlap integrals in a Gaussian orbital basis
Arguments:
output
When a TwoIndex instance is given, it is used as output argument and its contents are overwritten. When LinalgFactory is given, it is used to construct the output TwoIndex object. In both cases, the output two-index object is returned.
Returns: TwoIndex object
concatenate()
Concatenate multiple basis objects into a new one.
Arguments: each argument is an instance of the same subclass of GBasis.
from_hdf5()
get_basis_atoms()
Return a list of atomic basis sets for a given geometry
Arguments:
coordinates
An (N, 3) array with atomic coordinates, used to find the centers associated with atoms. An exact match of the Cartesian coordinates is required to properly select a shell.
Returns: A list with one tuple for every atom: (gbasis, ibasis_list), where gbasis is a basis set object for the atom and ibasis_list is a list of basis set indexes that can be used to substitute results from the atomic basis set back into the molecular basis set. For example, when a density matrix for the atom is obtained and it needs to be plugged back into the molecular density matrix, one can do the following:
mol_dm._array[ibasis_list,ibasis_list.reshape(-1,1)] = atom_dm._array
get_scales()
get_subset()
Construct a sub basis set for a selection of shells
Argument:
ishells
A list of indexes of shells to be retained in the sub basis set
Returns: An instance of the same class as self containing only the basis functions of self that correspond to the select shells in the ishells list.
to_hdf5()
alphas
basis_offsets
centers
con_coeffs
max_shell_type
nbasis
ncenter
nprim_total
nprims
nscales
nshell
shell_lookup
shell_map
shell_types
class horton.gbasis.cext.GB2OverlapIntegral
Bases: horton.gbasis.cext.GB2Integral
Wrapper for ints.GB2OverlapIntegral, for testing only
add()
cart_to_pure()
get_work()
This returns a copy of the c++ work array.
Returning a numpy array with a buffer created in c++ is dangerous. If the c++ array becomes deallocated, the numpy array may still point to the deallocated memory. For that reason, a copy is returned. Speed is not an issue as this class is only used for testing.
reset()
max_nbasis
max_shell_type
nwork
class horton.gbasis.cext.GB2KineticIntegral
Bases: horton.gbasis.cext.GB2Integral
Wrapper for ints.GB2OverlapIntegral, for testing only
add()
cart_to_pure()
get_work()
This returns a copy of the c++ work array.
Returning a numpy array with a buffer created in c++ is dangerous. If the c++ array becomes deallocated, the numpy array may still point to the deallocated memory. For that reason, a copy is returned. Speed is not an issue as this class is only used for testing.
reset()
max_nbasis
max_shell_type
nwork
class horton.gbasis.cext.GB2NuclearAttractionIntegral
Bases: horton.gbasis.cext.GB2Integral
Wrapper for ints.GB2NuclearAttractionIntegral, for testing only
add()
cart_to_pure()
get_work()
This returns a copy of the c++ work array.
Returning a numpy array with a buffer created in c++ is dangerous. If the c++ array becomes deallocated, the numpy array may still point to the deallocated memory. For that reason, a copy is returned. Speed is not an issue as this class is only used for testing.
reset()
max_nbasis
max_shell_type
nwork
class horton.gbasis.cext.GB4ElectronRepulsionIntegralLibInt
Bases: horton.gbasis.cext.GB4Integral
Wrapper for ints.GB4ElectronRepulsionIntegralLibInt, for testing only
add()
cart_to_pure()
get_work()
This returns a copy of the c++ work array.
Returning a numpy array with a buffer created in c++ is dangerous. If the c++ array becomes deallocated, the numpy array may still point to the deallocated memory. For that reason, a copy is returned. Speed is not an issue as this class is only used for testing.
reset()
max_nbasis
max_shell_type
nwork
class horton.gbasis.cext.GB1DMGridDensityFn
Bases: horton.gbasis.cext.GB1DMGridFn
add()
cart_to_pure()
get_work()
This returns a copy of the c++ work array.
Returning a numpy array with a buffer created in c++ is dangerous. If the c++ array becomes deallocated, the numpy array may still point to the deallocated memory. For that reason, a copy is returned. Speed is not an issue as this class is only used for testing.
reset()
dim_output
dim_work
max_nbasis
max_shell_type
nwork
shell_type0
class horton.gbasis.cext.GB1DMGridGradientFn
Bases: horton.gbasis.cext.GB1DMGridFn
add()
cart_to_pure()
get_work()
This returns a copy of the c++ work array.
Returning a numpy array with a buffer created in c++ is dangerous. If the c++ array becomes deallocated, the numpy array may still point to the deallocated memory. For that reason, a copy is returned. Speed is not an issue as this class is only used for testing.
reset()
dim_output
dim_work
max_nbasis
max_shell_type
nwork
shell_type0
class horton.gbasis.cext.IterGB1
Bases: object
Wrapper for the IterGB1 class, for testing only.
inc_prim()
inc_shell()
store()
update_prim()
update_shell()
private_fields
public_fields
class horton.gbasis.cext.IterGB2
Bases: object
Wrapper for the IterGB2 class, for testing only.
inc_prim()
inc_shell()
store()
update_prim()
update_shell()
private_fields
public_fields
class horton.gbasis.cext.IterGB4
Bases: object
Wrapper for the IterGB4 class, for testing only.
inc_prim()
inc_shell()
store()
update_prim()
update_shell()
private_fields
public_fields
class horton.gbasis.cext.IterPow1
Bases: object
Wrapper for the IterPow1 class, for testing only.
inc()
fields
class horton.gbasis.cext.IterPow2
Bases: object
Wrapper for the IterPow2 class, for testing only.
inc()
fields
horton.gbasis.cext.boys_function()
horton.gbasis.cext.cart_to_pure_low()
horton.gbasis.cext.compute_cholesky()
horton.gbasis.cext.fac()
horton.gbasis.cext.fac2()
horton.gbasis.cext.binom()
horton.gbasis.cext.get_shell_nbasis()
horton.gbasis.cext.get_max_shell_type()
horton.gbasis.cext.gpt_coeff()
horton.gbasis.cext.gb_overlap_int1d()
horton.gbasis.cext.nuclear_attraction_helper()
horton.gbasis.cext.gob_cart_normalization()
horton.gbasis.cext.gob_pure_normalization()
horton.gbasis.cext.get_2index_slice()
horton.gbasis.cext.compute_diagonal()
horton.gbasis.cext.select_2index()
horton.gbasis.cext.iter_pow1_inc()
|
2022-05-23 03:13:06
|
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|
https://tex.stackexchange.com/questions/191021/modifying-the-appearance-of-bookmarks-through-the-bookmark-and-hyperref-pack
|
# Modifying the appearance of bookmarks through the 'bookmark' and 'hyperref' packages
I'm adding bookmarks to my PDF document, but I'm not completely satisfied with how they appear. I found this thread, which recommends the bookmark package, and shows an example of how to group all appendices. I'd like to have the bookmarks appear as shown in the answer to the linked thread. I.e.
1. First chapter
|-- 1.1. First section
|----- 1.1.1. First subsection
|----- 1.1.2. Second subsection
|-- 1.2. Second section
2. Second chapter
Appendices
|-- A. First appendix
|-- B. Second appendix
However, when I try the same, I get no group named "Appendices", and appendix B shows up as the child of appendix A. Below is an MWE.
\documentclass[11pt]{scrreprt}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[numbered,open]{bookmark}
\usepackage[page,titletoc]{appendix}
\usepackage{lipsum}
\renewcommand{\appendixpagename}{Appendices\thispagestyle{empty}}
\begin{document}
\tableofcontents
\chapter{First chapter}
\lipsum[1-2]
\section{First section}
\lipsum[3-6]
\subsection{First subsection}
\lipsum[7-10]
\subsection{Second subsection}
\lipsum[11-14]
\section{Second section}
\lipsum[15-18]
\chapter{Second chapter}
\lipsum[19-22]
\begin{appendices}
\bookmarksetupnext{level=part}
\chapter{First appendix}
\lipsum[23-26]
\chapter{Second appendix}
\lipsum[27-30]
\end{appendices}
\end{document}
Question 1:
What am I doing wrong in the above code?
Question 2:
I haven't fully decided on the appearance of the table of contents yet, and I am contemplating having the appendices listed as section-level entries within an "Appendices" chapter. Would this solve the problem? How can I lower the table of contents entry by one level for the appendix chapters only? The appendix chapters should still appear as chapters for all other purposes.
Edit: These questions appear to have answers here and here.
Question 3:
Also, I'd like to have the bookmark numbering end with a period (e.g. "A. First appendix" instead of "A First appendix"). How can I achieve this?
After a little tinkering, plus a little inspiration from here and there, I've com up with a solution.
The bookmarksetupnext{level=part} simply sets the next bookmark to have the part level. There is no bookmark named "Appendices", and because the next bookmark is Appendix A, this appendix chapter will be changed to have the level of a part. Appendix B is still unchanged, so it remains at the chapter level and hence the bookmark will be a child of Appendix A.
Adding replacing the bookmarksetupnext{level=part} with \pdfbookmark[-1]{Appendices}{bookmark:appendices} solves this problem.
To add a table of contents (TOC) entry named "Appendices", a command like \cftaddtitleline{toc}{chapter}{Appendices}{} can be used. This reuquired the tocloft package. This command will reference the previous bookmark, so a bookmark must be added at the appropriate place. Adding the the following lines just before \begin{appendices} will place create a TOC entry which points to the "Appendices" page. For a two-sided document, the \cleardoublepage command should be used instead. The provided code omits the page number in the TOC entry. I chose to do this because the "Appendices" page is empty. The page number can be manually added in the last, empty brace.
\clearpage
\phantomsection
\pdfbookmark[-1]{Appendices}{bookmark:appendices}
To lower the appendix chapters, sections and subsections by one level, the following code can be added before the first appendix chapter. With the default TOC depth of 2, the appendix subsections will be hidden from the TOC. To display these, increase the TOC depth as per usual by the command \setcounter{tocdepth}{3}.
\makeatletter
\begingroup
\let\protect\l@chapter\protect\l@section
\let\protect\l@section\protect\l@subsection
\let\protect\l@subsection\protect\l@subsubsection
}
\makeatother
Below is a complete example. I've also added another part-level bookmark titled "Body", just to make the bookmarks in the PDF viewer line up properly. The "Body" bookmark points to the TOC.
\documentclass[11pt]{scrreprt}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[numbered,open]{bookmark}
\usepackage[page]{appendix}
\usepackage{tocloft}
\usepackage{lipsum}
\renewcommand{\appendixpagename}{Appendices\thispagestyle{empty}}
\setcounter{tocdepth}{3}
\begin{document}
\pdfbookmark[-1]{Body}{bookmark:body}
\tableofcontents
\chapter{First chapter}
\lipsum[1-2]
\section{First section}
\lipsum[3-6]
\subsection{First subsection}
\lipsum[7-10]
\subsection{Second subsection}
\lipsum[11-14]
\section{Second section}
\lipsum[15-18]
\chapter{Second chapter}
\lipsum[19-22]
\clearpage
\phantomsection
\pdfbookmark[-1]{Appendices}{bookmark:appendices}
\begin{appendices}
\makeatletter
\begingroup
\let\protect\l@chapter\protect\l@section
\let\protect\l@section\protect\l@subsection
\let\protect\l@subsection\protect\l@subsubsection
}
\makeatother
\chapter{First appendix}
\lipsum[23-24]
\section{First appendix section}
\lipsum[25-28]
\subsection{First appendix subsection}
\lipsum[29-32]
\subsection{Second appendix subsection}
\lipsum[33-36]
\section{Second appendix section}
\lipsum[37-40]
\chapter{Second appendix}
\lipsum[41-44]
\end{appendices}
\end{document}
The last question, on how to redefine bookmark styles, would be better to have as a separate question.
|
2019-07-18 17:51:02
|
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|
https://www.redcrab-software.com/en/RedCrab/Manual/Functions/Hyperbolic/ASech
|
# ASech Function
Compute the inverse hyperbolic secans
## Description
The function $$ASech$$ calculates inverse hyperbolic secans of the argument.
ASech returns the angle to the hyperbolic secans of real or complex numbers. The argument can be a single number or a data field. For data fields, the inverse hyperbolic secans of each individual element is calculated and the results are returned in a data field of the same size.
## ASech for real numbers
### Parameter
The argument must be a positive number greater than 0 and less than or equal to 1.
### Result
The result is given in degrees (full circle = 360°) or radians / radians (full circle = 2 · π). The unit of measurement used is set in the toolbar with the DEG or RAD buttons or an optional parameter. The setting in the toolbar applies to the entire worksheet.
At 0, the result returned is ∞ (infinity). For other arguments, outside the range specified above, the result is NaN (Not a number).
### Optional Parameter
Optionally, a second parameter can be specified with the keywords DEG or RAD to set the unit of measure for this function call. The specification of the parameter has priority over the global setting in the toolbar. You can use it on a worksheet different functions use different units of measurement independent of the presetting in the toolbar.
### Syntax
ASech (value)
ASech (value, DEG)
## ASech for complex numbers
For complex numbers, the result is always given as radians, regardless of the default setting in the toolbar. The result is also a complex number.
ACsch (re + im)
### Example
ACsch(3 + 4i)= 0.16-1.45i
|
2022-12-05 15:46:33
|
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|
http://mathhelpforum.com/algebra/4670-solving-equations.html
|
# Math Help - Solving equations
1. ## Solving equations
I would appreciate any assistance with the following problems.
2. Hello, pashah!
Solve each equation and check for extraneous solutions.
$a)\;\sqrt{w - 3}\:=\:\sqrt{4w + 15}$
Square: . $w - 3\:=\:4w + 15$
Then: . $-3w\:=\:18\quad\Rightarrow\quad \boxed{w = -6}$
But when we check, we get: . $\sqrt{-6 - 3}\:=\:\sqrt{-9}$ . . . an "imaginary" number.
Therefore, this equation has no solutions.
$b)\;\sqrt{x^2-6x=2} \:=\:\frac{x}{2}$
Square: . $x^2-5x + 2\:=\:\frac{x^2}{4}$
Multiply by 4: . $4x^2 - 20x + 8\:=\:x^2\quad\Rightarrow\quad 3x^2 - 20x + 8 \:= \:0$
Now use the Quadratic Formula (and check your answers).
$\frac{5}{y-3} \;= \;1 + \frac{y+7}{2y-6}$
We have: . $\frac{5}{y-3} \;= \;1 + \frac{y+7}{2(y-3)}$
Multiply by the LCD: $2(y-3)$
. . $2(y-3)\cdot\frac{5}{y-3} \;\;= \;\;2(y-3)\cdot1 \;+ \;2(y-3)\cdot\frac{y+7}{2(y-3)}$
We have: . $2\cdot5 \;= \;2(y-3) + (y + 7)\quad\Rightarrow\quad 10 \;= \;2y - 6 + y + 7$
Then: . $9\:=\:3y\quad\Rightarrow\quad \boxed{y = 3}$
But when we check, it starts with: . $\frac{5}{3-3} \:= \:\frac{5}{0}$ . . . undefined!
Therefore, this equation has no solutions.
|
2014-03-14 19:10:05
|
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|
https://math.stackexchange.com/questions/2526782/sequence-notation-where-a-has-a-subscript-and-a-superscript
|
# Sequence notation where a has a subscript and a superscript
So from what I've seen of sequences a common format would be something like $a_{n+1} = 2a_n + 5$
I was doing some review for sequences and come across a format that looks something like $a_{n+1}=a_n^2-1$ given $a_1=1$
According to my solution manual, $a_1=1,a_2=0,a_3=-1,a_4=0$
I initially thought that maybe it was just a weird notation and I could write it as $a_n$ but that would just make is a continually decreasing sequences so $a_1=1,a_2=0,a_3=-1,a_4=-2,a_5=-3$ and so on. So this doesnt fit the answer.
I'd like to know how this $a_n^c$ where c is just some number actually works in case it comes up on a test. Any example at all or just an explanation would be great, I can't actually find this notation in the section before the question.
For reference, this is in Briggs, Cochran Calculus Early Transcendentals 2nd edition, 8.1 #20
Edit: Thanks to u/Rob Arthan for this
Since $a_n^2 = a_n \times a_n=(a_n)^2$ this makes the answers from above as
$a_2=(a_1)^2-1=1^2-1=0$
$a_3=(a_2)^2-1=0^2-1=-1$
$a_4=(a_3)^2-1=(-1)^2-1=1-1=0$
and so on and so forth
• $a_n^c$ here means $a_n$ to the $c$-th power. So $a_n^2 = a_n\cdot a_n$, the square of $a_n$. (People do occasionally use superscripts as indexes rather than exponents but that is fairly rare.) – Rob Arthan Nov 19 '17 at 0:04
• That makes a lot of sense and clears this up a lot. Thanks, i'll update my question accordingly and I think that officially makes this solved. Not really sure how to mark is solved without a submitted answer though – Vin Nov 19 '17 at 0:14
• You can also post your own answer – Dylan Nov 19 '17 at 0:45
In the comments, Rob Arthan wrote that "People do occasionally use superscripts as indexes rather than exponents but that is fairly rare." In my opinion, that is reason enough to consider $a_n^c$ as very poor style when ${a_n}^c$ is meant. For extra clarity, you can use parentheses, like so: $(a_n)^c$.
Either way, that's two extra bytes, or four if you're using UTF-16. That's nothing, at least until Trump's swamp FCC destroys net neutrality. In terms of ink expenditure, another pair of parentheses is also negligible in a book of more than a thousand pages.
So yeah, $a_{n + 1} = a_n^2 - 1$ should have been written as $a_{n + 1} = {a_n}^2 - 1$ or better yet $a_{n + 1} = (a_n)^2 - 1$.
|
2019-06-24 15:24:17
|
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|
http://mathcentral.uregina.ca/QQ/database/QQ.09.14/h/farihin1.html
|
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Question from Farihin, a student: Lets say that i have keys, and each key is for notes of a musical instrument, So i wanted to find out the number of notes i can get for a certain number keys, of course in the form of an equation. Notes can use as many keys, it can use 1, or 2, or 3, or even 100. Notes in real life is not as such, but ignore reality. I tried doing this but i can't seem to find a formula for it. For example, i have 4 keys, say A, B, C, and D. so, for notes that uses one key are 4, which is A, B, C, and D themselves. for notes that uses two keys are 6, AB, AC, AD, BC, BD and CD. for notes that uses three keys are 4, ABC, ABD, ACD and BCD. lastly for notes that uses all four keys is 1, ABCD. So, the total will be 4+6+4+1=15# The nth term for the first equation is n, the second is [(n^2)-n]/2 the third and the fourth, i don't know but the final answer should be like, n + [(n^2)-n]/2 + [3rd] + [4th] Sorry for the long question though...
Hi Farihin,
I want to count the number of "notes" from the 4-key instrument you describe in a different way. Hopefully a way that shows you how to arrive at the general statement for an n-key instrument. As you did I am going to label the keys A, B, C, and D and for each of them you have the choice of leaving it closed or opening it. I am thinking of a wind instrument where each of the valves (keys) is closed unless you press on the valve to open it.
Start with the valve A. You can open or close it giving the two possible sounds, $A$ or $O.$ I am using $O$ to indicate that all the valves are closed and there is no sound. Now, regardless of the position of the valve A you can open or close valve B. Thus if A is open you can press B to get $AB$ or leave it closed to get only $A.$ If A is closed then you can press B to get $B$ of leave B closed to get $O$ again. Hence the possible notes using keys A and B are
$AB, A, B \mbox{ and } O.$
With only 1 valve you got 2 possible notes, $A$ and $O$ and each of them can be extended in 2 ways, using the B valve, giving $2 \times 2 = 2^2 = 4$ possible notes using 2 keys.
Now include valve C. Each of the 4 possible notes you have using 2 valves can be extended to a 3 key instrument, using the C key, in 2 ways giving $2 \times 2 \times 2 = 2^3 = 8$ possible notes. They are
$ABC, AB, AC, A, BC,B, C \mbox{ and } O.$
Finally extend to a 4-key instrument using the D key. Each of the 8 possible notes listed above can be extended to the 4-key instrument in 2 ways, either you press the D valve or you don't. Thus I get $2 \times 2 \times 2 \times 2 = 2^4 = 16$ possible notes. In more conventional mathematical language there are $2^4$ subsets of a set with 4 elements.
You got 15 notes and I got 16 because I included the note that produces no sound. Using the music interpretation of this process, $O$ is a rest.
I hope this helps,
Penny
Math Central is supported by the University of Regina and the Imperial Oil Foundation.
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2017-11-23 01:53:48
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https://www.merry.io/differential-geometry/48-three-equivalent-definitions-of-the-laplacian/
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Let $M$ be an oriented Riemannian manifold. In this lecture we make contact with some of the differential operators you first met in Analysis II:
• The divergence $\operatorname{div}(X)$ of a vector field,
• The gradient $\operatorname{grad}(f)$ of a smooth function,
• The Hessian $\operatorname{Hess ^{ \nabla}}(f)$ of a smooth function,
• The Laplacian $\Delta(f)$ of a smooth function.
But why define something once when you can define it three times? 🙃
For the sake of completeness we give three equivalent definitions of the Laplacian $\Delta(f)$:
1. $\Delta(f) = \operatorname{div}( \operatorname{grad}(f))$,
2. $\Delta(f) = - \delta(df)$,
3. $\Delta(f) = \operatorname{tr}( \operatorname{Hess}^{ \nabla}(f))$.
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2019-12-10 03:32:43
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https://math.stackexchange.com/questions/75047/partition-problem-of-set
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# Partition problem of set
Let $$A =\{1, 2, 3,..., 100\}$$. We partition $$A$$ into $$10$$ subsets $$A_1,A_2,...A_{10}$$ each of size $$10$$. A second partition into $$10$$ sets of size $$10$$ each is given by $$B_1,B_2,...B_{10}$$. Prove that we can rearrange the indices of the second partition so that $$A_{i}\cap B_{i}\not=\varnothing$$.
• This is obviously impossible. For example, suppose $A_1=\{1,2,3,4,5,6,7,8,9,10\}$ and $B_i$ contains $i$. Did you mean to say "Prove that we can rearrange the indices of the second partition so that $A_i \cap B_i$ is not empty"? – Chris Eagle Oct 23 '11 at 9:56
• Yep. That's exactly what the question wants. – fantom Oct 23 '11 at 10:11
It is not hard to see that $k$ sets of size 10 each have to share elements with at least $k$ sets in the other partition, so the theorem applies.
ETA: The $k$ sets have a total of $10k$ elements, so you need at least $k$ sets in the other partition to cover all of these elements, since $k-1$ sets have two few elements.
Put $P_1=\{A_1,A_2,...,A_{10}\}$ and $P_2=\{B_1,B_2,...,B_{10}\}$
Take any $X \subset P_1$ and let $N(X)\subset P_2$ be the set of all neighbors of vertices in $X$. Say $|X|=k$ and $|N(X)| = l$. Then we have $$\bigcup _{A\in X} A =\bigcup _{B\in N(X)} B$$ since set on left and sets on right are disjoint we have $$10k= \sum _{A\in X} |A| =\sum _{B\in N(X)} |B| =10l$$ we have $k=l$ so $|X|\leq |N(X)|$ and so Hall condition is fulfilled and we are done.
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2019-08-21 03:48:18
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http://new.math.uiuc.edu/public198/week8/index.html
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Weeks 8 Fall 2011
10oct11
\begin{document}
\maketitle
\section{Monday}
\section{The Bresenham Algorithm}
This lesson is about how computers draw straight lines and
how OpenGL colors a triangle. Emily Gunawan's lecture, based
on Jim Blinn's article, is a solution to the assignment, to
deconstruct the algorithm
from her code. Read it after you've
made an attempt at least. Once you've understood her explanation,
see if you can solve the easier problem, of drawing a straight
line on your own, without googling it.
The example is in Python with TKinter. As of 15aug13, it is still possible to
see this program do its thing by (1) copying the code to a file on your computer
called gunawan.py, for example.
Then (2), feed this file to your resident Python installed on your computer.
For example, on a Mac or Linux, enter the command "python gunawan.py" on a command line.
It is also good exercise to implement
this example in other languages you are good at, e.g.
\begin{itemize}
\item Python/OpenGL
\item C/C++/Open GL (as in mvc98)
\item VPython
\end{itemize}
In each case, there is a preliminary problem to solve, namely how to
draw a visible square to represent one pixel, since our pixels are not
too small to see.
Once you have a familar drawing program working on the code, you can
experiment with it to also draw the Bresenham line. What other figures
can you draw by a similar method?
\section{New Resources}
For those of you learning OpenGL, the sample basket from the Redbook
is a great resource. Those with PC should see if the .exe from 1996
here will still run. Those of you who an use a C/OpenGL compiler
(you need glut.h), for example mvc98, should figure out the minimum
modifications (described in the onboard readme-file, on the codes to
recompile. Use the pocket programs as guides.
\begin{itemize}
\item A didactic (and incidentally, very amusing) essay you
In the Beginning ... was the Commandline by Neal Stephenson,
the author of Snowcrash.
\item The redbookOGL directory contains ancient
OpenGL examples from the famous Redbook
\end{itemize}
\section{Tuesday}
\begin{itemize}
\item We finally have a sensible and simple GUI wordprocessor for macs,
Bean by John Hoover,
a writer. It is similar to
Wordpad. and better yet, it still works on OSX10.7 "Lion".
\end{itemize}
\section{Wednesday}
\begin{itemize}
\item Zach Reizner: Mandelbrot meets Julia on the GPU
\end{itemize}
\section{Thursday}
\begin{itemize}
\item Se-Joon Chung: Dancing i in the Cube (testing)
\item Matt Hoffman: Sierpinski tetrahedra in the Cube (testing)
\end{itemize}
\section{Friday}
\begin{itemize} \end{itemize}
\end{document}
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2017-03-26 07:22:35
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https://math.stackexchange.com/questions/316777/y-pm-ety-0-implies-mid-cup-x-i-mid-s-t-yx-i-0/316787
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# $y''\pm e^ty=0 \implies \mid \cup x_i \mid =? s.t. y(x_i)=0$
I have this question, and i don't know how to solve it:
Show that the solutions of $y''+e^ty=0$ admit an infinite number of zeros.
Also, how to prove that the solutions of $y''-e^ty=0$ admit not more than one zero in $\mathbb{R}_+$?
1) Show that the solutions of $y''+e^ty=0$ admit infinitely many zeros.
Suppose there exists a solution $y$ with finitely many zeros. Therefore, $y$ is positive or negative on $[A,+ \infty)$ for $A$ large enough; if $y<0$ consider $-y$ so that you can suppose $y>0$.
Because $y''(t)=-e^ty(t)<0$ for $t \geq A$, $y'$ is decreasing on $[A,+ \infty)$. Moreover, if $y'$ is not bounded below, then there exist $C<0$ and $t_0>A$ such that $y'(t)<C$ for $t \geq t_0$, hence (by integration) $y(t) < y(t_0)+C(t-t_0) \underset{t\to + \infty}{\longrightarrow} - \infty$: a contradiction with $y>0$. Therefore, $y'$ is bounded below and the limit $\lim\limits_{t \to + \infty} y'(t)=\ell$ exists.
For $\epsilon>0$, there exists $t_1>0$ such that $t \geq t_1$ implies $\ell-\epsilon<y'(t)<\ell+\epsilon$; by integrating, $y(t_1)+ (\ell-\epsilon)(t-t_1)<y(t)<y(t_1)+(\ell+\epsilon)(t-t_1)$. You deduce that $\lim\limits_{t \to + \infty} \frac{y(t)}{t}=\ell$ and $\ell \geq 0$.
For $n \geq 1$, let $c_n \in (n,n+1)$ such that $y'(n+1)-y'(n)=y''(c_n)$. So $c_n \underset{n \to + \infty}{\longrightarrow} + \infty$, and using the above limit, $y''(c_n) \underset{n \to + \infty}{\longrightarrow} 0$.
Because $y'$ is decreasing and $\lim\limits_{t \to + \infty} y'(t)=\ell \geq 0$, you deduce that $y' \geq 0$, so $y$ is nondecreasing. Consequently, $y(t) \geq y(A)>0$ so $y''(t) \leq -e^ty(A) \underset{t \to + \infty}{\longrightarrow} - \infty$: contradiction with $y''(c_n) \underset{n \to + \infty}{\longrightarrow} 0$.
2) Show that the (non zero) solutions of $y''-e^ty=0$ admit at most one zero in $\mathbb{R}_+$.
Let $y$ be a solution of $y''-e^ty=0$ with at least two zeros.
First, suppose there exists an interval $[a,b]$ such that $y(a)=y(b)=0$ and $y(x) \neq 0$ for $x \in (a,b)$. Without loss of generality, suppose $y>0$ on $(a,b)$ (otherwise, consider $-y$). Then $y''=e^ty>0$ on $(a,b)$ and $y'$ is increasing on $(a,b)$. According to Rolle's theorem, there exists $c \in (a,b)$ such that $y'(c)=0$, so $y'(a)<0$.
Because $y'$ is continuous, $y' \leq 0$ on $[a,a+ \epsilon]$ for some $\epsilon >0$ hence $\displaystyle y(t)= \int_a^t y'(s)ds \leq 0$ for $t \in [a,a+\epsilon]$: contradiction with $y>0$ on $(a,b)$.
So there is no such interval $[a,b]$. You deduce that there exists a decreasing sequence $(x_n)$ of zeros. For $n \geq 1$, according to Rolle's theorem, there exists $u_n \in (x_{n+1},x_n)$ such that $y'(u_n)=0$.
We have $0<u_{n+1}<x_{n+1}<u_n<x_n$ for any $n \geq 1$, so $(u_n)$ and $(x_n)$ converge to the same limit $\ell$. We deduce by continuity that $y(\ell)= \lim\limits_{n \to + \infty} y(x_n)=0$ and $y'(\ell)= \lim\limits_{n \to + \infty} y'(u_n)=0$.
Using Cauchy-Lipschitz theorem, you find that the only possibility is $y=0$.
You can solve this equation by noting that putting $$z=e^t$$ then $$\frac{d}{dt}=\frac{d}{dz}\frac{dz}{dt}=e^t\frac{d}{dz}=z\frac{d}{dz}$$ and so the equation becomes $$z\frac{d}{dz}z\frac{dy}{dz}+zy=0$$ or $$z\frac{d^2y}{dz^2}+\frac{dy}{dz}+y=0.$$ taking into account that is always $z\ne 0$. This admits a general solution in terms of Bessel functions as $$y(z)=AJ_0(2\sqrt{z})+BY_0(2\sqrt{z}).$$ being $A$ and $B$ integration constants. This proves the assertion as these Bessel functions have infinite zeros.
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2019-10-16 06:47:53
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https://www.physicsforums.com/threads/disambiguating-arccosine-arcsine-functions.177578/
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# Disambiguating arccosine/arcsine functions
1. Jul 19, 2007
### techninja
Hi all,
I'm working on a program, and it seems that I can't get the trigonometry in my head right. I have two equations, sin(omega) = something and cos(omega) = something, and I need to find what omega is.
Given that there are two trig functions, I should be able to disambiguate what quadrant omega is in.
On the other hand, I don't quite understand the process of how this would be done.
Any help or input would be greatly appreciated!
2. Jul 19, 2007
### Staff: Mentor
Thread moved to Homework Help forums (where homework and coursework should be posted).
What are the +/- signs of the sin and cos functions in the 4 quadrants?
3. Jul 19, 2007
### techninja
Well, if we're going to go thataway, sine is positive in I and II, cosine is positive in I and III. Arcsine is defined in I and IV, arccosine is defined in I and II.
Thanks!
4. Jul 19, 2007
### Staff: Mentor
5. Jul 19, 2007
### techninja
Nope; not at all. :rofl:
6. Jul 19, 2007
### VietDao29
Well, firstly, you have to check for the existence of omega. You know the relation between sine, and cosine function, right?
$$\sin ^ 2 \omega + \cos ^ 2 \omega = 1$$
If the above equation holds, then omega exists, if not, it doesn't. Do you know why?
We don't need the arcsin, and arccos part here.
Ok, so, say, if sin(omega) is positive, and cos(omega) is negative, what quadrant is omega in?
:)
7. Jul 19, 2007
### techninja
That's a trig identity, I believe.
And, if sine is positive, it would be arccos(cos(omega)), and if not, it would be... 2*pi-arccos(cos(omega))?
Would that be right?
Thanks. (:
8. Jul 20, 2007
### VietDao29
Yup, correct. :)
However, does your omega has any restriction? i.e, say, must it be on the interval [0; 2pi[? Or anything along those line?
If omega must be on [0; 2pi[, then your solution would be:
$$\left[ \begin{array}{ll} \omega = \arccos (\cos (\omega)) , & \quad \mbox{for non-negative } \sin \omega \\ \omega = 2 \pi - \arccos \cos ( \omega ), & \quad \mbox{for negative } \sin \omega \end{array} \right.$$
If, omega can be anything, then the general solution for omega would be:
$$\left[ \begin{array}{ll} \omega = \arccos (\cos (\omega)) + \2 k \pi , & \quad \mbox{for non-negative } \sin \omega \\ \omega = - \arccos ( \cos \omega ) + 2 k' \pi , & \quad \mbox{for negative } \sin \omega \end{array} \right.$$, where k, and k' are both integers.
You got it correctly. Congratulations. ^.^
Can you complete the programme? :)
Last edited: Jul 20, 2007
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2017-12-12 08:42:56
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https://algorithmia.com/blog/from-building-an-xgboost-model-on-jupyter-notebook-to-deploying-it-on-algorithmia
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XGBoost is a popular library among machine learning practitioners, known for its high performance and memory efficient implementation of gradient boosted decision trees. Since training and evaluating machine learning models on Jupyter notebooks is also a popular practice, we’ve developed a step-by-step tutorial so you can easily go from developing a model on your notebook to serving it on Algorithmia.
In this tutorial, we will build a simple sentiment analysis model on XGBoost, trained on Amazon’s Musical Instrument Reviews dataset. Our model will simply classify the sentiment of a given text as positive or negative. In order to make our model available to the outside world, we will create an algorithm on Algorithmia that loads our model, handles the incoming prediction requests, and returns a response to its callers. Then we will test this deployment end-to-end by sending a request to our algorithm and seeing that our deployed model’s predictions are returned on the response. Our model is a simple sentiment classifier, used to make the tutorial simple to follow, but you will be able to follow the same steps to get your own models on Algorithmia.
To get all the necessary files so you can follow along, here is the repository for this tutorial.
If you would like to see the final product first, you can check out the built algorithm in action.
## Overview
Let’s first go over the steps we will cover in this tutorial.
1. Create an algorithm on Algorithmia
2. Clone the algorithm’s repository on our local machine, so that we develop it locally
3. Create the basic algorithm script and the dependencies file. We will code our script in advance—assuming that our model will be sitting on a remote path on Algorithmia—and our script will load it from there. We will then make these assumptions come true.
4. Commit and push these files to Algorithmia and get our algorithm’s container built
5. Load the training data
6. Preprocess the data
7. Setup an XGBoost model and do a mini hyperparameter search
8. Fit the data on our model
9. Get the predictions
10. Check the accuracy
11. Once we are happy with our model, upload the saved model file to our data source on Algorithmia
12. Test our published algorithm with sample requests
At this point, you will have an up and running algorithm on Algorithmia, ready to serve its predictions upon your requests.
## Getting started
First create an algorithm on Algorithmia.
Make sure that you have the official Algorithmia Python Client installed on your development environment:
pip install algorithmia
For this tutorial, we will also use utility functions defined on our Algorithmia utility script. This script helps us encapsulate the related calls to Algorithmia through its Python API.
import Algorithmia
import algorithmia_utils
The definitions below will be used across many function calls to create the algorithm, save and upload the model to Algorithmia, and then test the algorithm. Don’t forget to use your Algorithmia username and API key every time you see “YOUR_API_KEY” and “YOUR_USERNAME” placeholders!
api_key = "YOUR_API_KEY"
username = "YOUR_USERNAME"
algo_name = "xgboost_basic_sentiment_analysis"
local_dir = "../algorithmia_repo"
algo_utility = algorithmia_utils.AlgorithmiaUtils(api_key, username, algo_name, local_dir)
## Creating the algorithm and cloning its repo
You only need to do this step once because you only need one algorithm. Cloning it once on your local environment is enough.
For these operations, we will use the utility functions defined on our imported Algorithmia utility script.
# You would need to call these two functions only once
algo_utility.create_algorithm()
algo_utility.clone_algorithm_repo()
## Creating the algorithm script and the dependencies file
Let’s create the algorithm script that will run when we make our requests. We will also create a dependency file so that Algorithmia infrastructure knows how to build a runtime environment for our algorithm’s container.
We will be creating these two files programmatically with the %%writefile macro, but you can always use another editor to edit and save them later if you’d like.
%%writefile $algo_utility.algo_script_path import Algorithmia import joblib import numpy as np import pandas as pd import xgboost model_path = "data://[YOUR_USERNAME]/xgboost_demo/musicalreviews_xgb_model.pkl" client = Algorithmia.client() model_file = client.file(model_path).getFile().name loaded_xgb = joblib.load(model_file) # API calls will begin at the apply() method, with the request body passed as 'input' # For more details, see algorithmia.com/developers/algorithm-development/languages def apply(input): series_input = pd.Series([input]) result = loaded_xgb.predict(series_input) # Returning the first element of the list, as we'll be taking a single input for our demo purposes return {"sentiment": result.tolist()[0]} %%writefile$algo_utility.dependency_file_path
algorithmia>=1.0.0,<2.0
scikit-learn
pandas
numpy
joblib
xgboost
## Adding these files to git, committing, and pushing
Now we’re ready to upload our changes to our remote repo on Algorithmia. After this operation, our algorithm will be built on the Algorithmia servers and be ready to accept our requests!
algo_utility.push_algo_script_with_dependencies()
## Building the XGBoost model
Now it’s time to build our model. If you don’t have the imported Python packages below, you should first install them on your development environment.
from sklearn.model_selection import train_test_split, RandomizedSearchCV
from sklearn.metrics import accuracy_score
from sklearn.feature_extraction.text import CountVectorizer, TfidfTransformer
from sklearn.pipeline import Pipeline
from string import punctuation
from nltk.corpus import stopwords
from xgboost import XGBClassifier
import pandas as pd
import numpy as np
import joblib
## Load the training data
Let’s load our training data. Be sure to take a look at a few rows and review one of the texts in detail.
data = pd.read_csv("./data/amazon_musical_reviews/Musical_instruments_reviews.csv")
data.head()
reviewerID asin reviewerName helpful reviewText overall summary unixReviewTime reviewTime
0 A2IBPI20UZIR0U 1384719342 cassandra tu “Yeah, well, that’s just like, u… [0, 0] Not much to write about here, but it does exac… 5.0 good 1393545600 02 28, 2014
1 A14VAT5EAX3D9S 1384719342 Jake [13, 14] The product does exactly as it should and is q… 5.0 Jake 1363392000 03 16, 2013
2 A195EZSQDW3E21 1384719342 Rick Bennette “Rick Bennette” [1, 1] The primary job of this device is to block the… 5.0 It Does The Job Well 1377648000 08 28, 2013
3 A2C00NNG1ZQQG2 1384719342 RustyBill “Sunday Rocker” [0, 0] Nice windscreen protects my MXL mic and preven… 5.0 GOOD WINDSCREEN FOR THE MONEY 1392336000 02 14, 2014
4 A94QU4C90B1AX 1384719342 SEAN MASLANKA [0, 0] This pop filter is great. It looks and perform… 5.0 No more pops when I record my vocals. 1392940800 02 21, 2014
data["reviewText"].iloc[1]
“The product does exactly as it should and is quite affordable.I did not realized it was double screened until it arrived, so it was even better than I had expected.As an added bonus, one of the screens carries a small hint of the smell of an old grape candy I used to buy, so for reminiscent’s sake, I cannot stop putting the pop filter next to my nose and smelling it after recording. :DIf you needed a pop filter, this will work just as well as the expensive ones, and it may even come with a pleasing aroma like mine did!Buy this product! :]”
## Preprocessing
Before we get to training, we should preprocess our training data. We will:
– Remove English stopwords
– Remove punctuations
– Drop unused columns
def threshold_ratings(data):
def threshold_overall_rating(rating):
return 0 if int(rating)<=3 else 1
data["overall"] = data["overall"].apply(threshold_overall_rating)
def remove_stopwords_punctuation(data):
data["review"] = data["reviewText"] + data["summary"]
puncs = list(punctuation)
stops = stopwords.words("english")
def remove_stopwords_in_str(input_str):
filtered = [char for char in str(input_str).split() if char not in stops]
return ' '.join(filtered)
def remove_punc_in_str(input_str):
filtered = [char for char in input_str if char not in puncs]
return ''.join(filtered)
def remove_stopwords_in_series(input_series):
text_clean = []
for i in range(len(input_series)):
text_clean.append(remove_stopwords_in_str(input_series[i]))
return text_clean
def remove_punc_in_series(input_series):
text_clean = []
for i in range(len(input_series)):
text_clean.append(remove_punc_in_str(input_series[i]))
return text_clean
data["review"] = remove_stopwords_in_series(data["review"].str.lower())
data["review"] = remove_punc_in_series(data["review"].str.lower())
def drop_unused_colums(data):
data.drop(['reviewerID', 'asin', 'reviewerName', 'helpful', 'unixReviewTime', 'reviewTime', "reviewText", "summary"], axis=1, inplace=True)
def preprocess_reviews(data):
remove_stopwords_punctuation(data)
threshold_ratings(data)
drop_unused_colums(data)
Now let’s take another look at the preprocessed data and make sure everything looks okay:
preprocess_reviews(data)
data.head()
overall review
0 1 much write here exactly supposed to filters po…
1 1 product exactly quite affordablei realized dou…
2 1 primary job device block breath would otherwis…
3 1 nice windscreen protects mxl mic prevents pops…
4 1 pop filter great looks performs like studio fi…
## Split our training and test sets
rand_seed = 42
X = data["review"]
y = data["overall"]
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=rand_seed)
## Mini randomized search
Before we fit our model, let’s set up a very basic cross-validated randomized search over our parameter settings:
params = {"max_depth": range(9,12), "min_child_weight": range(5,8)}
rand_search_cv = RandomizedSearchCV(XGBClassifier(), param_distributions=params, n_iter=5)
## Pipeline to vectorize, transform, and fit
Now it’s time to vectorize our data, transform it, and fit our model to it.
To be able to feed the text data as numeric values to our model, we will first convert our texts into a matrix of token counts using a CountVectorizer. Then we will convert the count matrix to a normalized tf-idf (term-frequency times inverse document-frequency) representation.
Using this transformer, we will be scaling down the impact of tokens that occur very frequently because they convey less information to us. On the contrary, we will be scaling up the impact of the tokens that occur in a small fraction of the training data because they are more informative to us.
model = Pipeline([
('vect', CountVectorizer()),
('tfidf', TfidfTransformer()),
('model', rand_search_cv)
])
model.fit(X_train, y_train)
Pipeline(steps=[('vect', CountVectorizer()), ('tfidf', TfidfTransformer()),
('model',
RandomizedSearchCV(estimator=XGBClassifier(base_score=None,
booster=None,
colsample_bylevel=None,
colsample_bynode=None,
colsample_bytree=None,
gamma=None,
gpu_id=None,
importance_type='gain',
interaction_constraints=None,
learning_rate=None,
max_delta_step=None,
max_depth=None,
min_child_weight=None,
missing=nan,
monotone_constraints=None,
n_estimators=100,
n_jobs=None,
num_parallel_tree=None,
random_state=None,
reg_alpha=None,
reg_lambda=None,
scale_pos_weight=None,
subsample=None,
tree_method=None,
validate_parameters=None,
verbosity=None),
n_iter=5,
param_distributions={'max_depth': range(9, 12),
'min_child_weight': range(5, 8)}))])
## Deploying on Algorithmia
Now let’s call the Algorithmia utility function to take our saved model from its local path and put it on a data collection on Algorithmia. As you’ll remember, our algorithm script will be looking at this path to load the model.
algorithmia_data_path = "data://YOUR_USERNAME/xgboost_demo"
algo_utility.upload_model_to_algorithmia(local_path, algorithmia_data_path, model_name)
## Time to test end-to-end!
Now we are up and running with a scalable algorithm on Algorithmia waiting for its consumers. Let’s test it with one positive and one negative text and see how well it does.
To send the request to our algorithm, we will use the algorithm calling function defined in the Algorithmia utility script, and we’ll give it a string input.
pos_test_input = "It doesn't work quite as expected. Not worth your money!"
algo_result = algo_utility.call_latest_algo_version(pos_test_input)
print(algo_result.metadata)
print("Sentiment for the given text is: {}".format(algo_result.result["sentiment"]))
Metadata(content_type='json',duration=0.020263526,stdout=None)
Sentiment for the given text is: 0
neg_test_input = "I am glad that I bought this. It works great!"
algo_result = algo_utility.call_latest_algo_version(neg_test_input)
print(algo_result.metadata)
print("Sentiment for the given text is: {}".format(algo_result.result["sentiment"]))
Metadata(content_type='json',duration=0.018224132,stdout=None)
Sentiment for the given text is: 1
## Conclusion
We hope this tutorial will help you deploy your more talented XGBoost models on Algorithmia. Stay tuned for Part 2 of this tutorial as we are working on automating some of your “model development to deployment” workflows to make this journey even faster for you!
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2021-06-20 12:15:55
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http://tex.stackexchange.com/questions/95959/how-to-tell-which-package-is-defining-ifpdf
|
How to tell which package is defining \ifpdf?
I'm experiencing a problem similar to this one: Package ifpdf Error
After upgrading to a more recent version of TeX Live, my pre-existing documents stopped compiling, with complaints that `\ifpdf` was already defined. My project is large and pulls in a lot of packages, so I don't know which one is defining `\ifpdf`. Using `\let\ifpdf\relax` after the `\documentclass`, as suggested in the answers to the other question, fixes the problem on the newer TeX Live. However, this breaks the document on older versions of TeX Live, where I still need it to compile for various reasons. (I think I'm getting malformed PDFs from pdfTeX 3.1415926-2.4-1.40.13, but that's another topic.)
Is there any simple, straightforward way of finding out which package is defining `\ifpdf` when it shouldn't be? I could presumably figure it out by trial and error, by bisection, but this is a large, complicated project, and that would be a lot of work. If that worked, would knowing this allow me to make my file compile on both versions of TeX Live?
Is there some other way of making a more robust fix? Can I somehow test somewhere to see whether `\ifpdf` has been defined, and then define it conditionally at that point?
[EDIT] I figured out the problem. It wasn't another package, it was some code in my own .cls file, which I'd cut and pasted from somewhere probably 15 years ago:
``````\newif\ifpdf
\ifx\pdfoutput\undefined
\pdffalse % we are not running PDFLaTeX
\else
\pdfoutput=1 % we are running PDFLaTeX
\pdftrue
\fi
\ifpdf
\usepackage[pdftex]{graphicx}
\else
\usepackage{graphicx}
\fi
\AtBeginDocument{
\ifpdf
\DeclareGraphicsExtensions{.pdf,.jpg,.png}
\else
\DeclareGraphicsExtensions{.eps,.jpg,.png}
\fi
}
``````
I was able to make it run on both (modern) versions of texlive by shortening this to the following:
``````\usepackage{graphicx}
\AtBeginDocument{
\DeclareGraphicsExtensions{.pdf,.jpg,.png}
}
``````
This probably breaks compatibility with dvi-flavored tex, but I don't care about that.
-
You could try using a condition: `\ifx\ifpdf\undefined\else\let\ifpdf\relax\fi` – Werner Jan 30 at 4:06
@Werner: Thanks for the suggestion, but that caused an error on the old texlive, where `\ifpdf` wasn't being defined and needed to be. I edited the question to show the solution I found that actually worked. – Ben Crowell Jan 30 at 6:37
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2013-05-25 18:00:07
|
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https://gomathanswerkey.com/texas-go-math-grade-6-lesson-2-1-answer-key/
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# Texas Go Math Grade 6 Lesson 2.1 Answer Key Classifying Rational Numbers
Refer to our Texas Go Math Grade 6 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 6 Lesson 2.1 Answer Key Classifying Rational Numbers.
## Texas Go Math Grade 6 Lesson 2.1 Answer Key Classifying Rational Numbers
Representing Division as a Fraction
Alicia and her friends Brittany, Kenji and Ellis are taking a pottery class. The four friends have to share 3 blocks of clay. How much clay will each of them receive if they divide the 3 blocks evenly?
(A) The top faces of the 3 blocks of clay can be represented by squares. Use the model to show the part of each block that each friend will receive. Explain.
(B) Each piece of one square is equal to what fraction of a block of clay?
(C) Explain how to arrange the pieces to model the amount of clay each person gets. Sketch the model.
(D) What fraction of a square does each person’s pieces cover? Explain.
(E) How much clay will each person receive?
(F) Multiple Representations How does this situation represent division?
Reflect
Question 1.
Communicate Mathematical Ideas 3 ÷ 4 can be written $$\frac{3}{4}$$ How are the dividend and divisor of a division expression related to the parts of a fraction?
Dividend and divisor are connected to the fraction as numerator and denominator respectively!
dividend – numerator;
divisor – denominator
Question 2.
Analyze Relationships How could you represent the division as a fraction if 5 people shared 2 blocks? if 6 people shared 5 blocks?
If 5 people shared 2 blocks, we can represent this as:
$$\frac{5}{2}$$
If 6 people shared 5 blocks, we can represent this as:
$$\frac{6}{5}$$
Write each rational number as $$\frac{a}{b}$$.
Question 3.
– 15 ____________
Number – 15 can be written by fraction $$\frac{a}{b}$$ as:
– $$\frac{15}{1}$$
Question 4.
0.31 ____________
Number 0.31 can be written by fraction $$\frac{a}{b}$$ as:
$$\frac{31}{100}$$
Question 5.
4$$\frac{5}{9}$$ ____________
Number 4$$\frac{5}{9}$$ can be written by fraction $$\frac{a}{b}$$ as:
$$\frac{41}{9}$$
Question 6.
62 ____________
Number 62 can be written by fraction $$\frac{a}{b}$$ as:
$$\frac{62}{1}$$
Question 7.
Analyze Relationships Name two integers that are not also whole numbers.
Here, we pick numbers which are negative so they don’t get involved in whole numbers group:
– 7, – 10
Question 8.
Analyze Relationships Describe how the Venn diagram models the relationship between rational numbers, integers, and whole numbers.
Venn diagram shows reference between groups.
It shows that Whole numbers are included in Integers and Integers in Rational numbers.
Place each number in the Venn diagram. Then classify each number by indicating in which set or sets it belongs.
Question 9.
14.1 ____________
Number 14.1 belongs to Rational Numbers
Question 10.
7$$\frac{1}{5}$$ ____________
Please note that I wrote mixed number 7$$\frac{1}{5}$$ as 7.2 because i cannot use alignment in editing picture.
Number 7$$\frac{1}{5}$$ belongs to Rational numbers
Question 11.
– 8 ____________
Number – 8 belongs to Integers.
Question 12.
101 ____________
Number 101 belongs to whole numbers
Question 1.
Sarah and four friends are decorating picture frames with ribbon. They have 4 rolls of ribbon to share evenly. (Explore Activity)
a. How does this situation represent division?
This situation can be represented by division as each person gets same amount of ribbon, therefore they have to split it into equal parts!
b. How much ribbon does each person receive?
If Sarah has 4 friends, that means that 4 rolls of ribbon will be shared equally only if each gets:
$$\frac{4}{5}$$
Write each rational number in the form where a and b are integers.
Question 2.
0.7 ___________
Number 0.7 can be written in form of $$\frac{a}{b}$$ as:
$$\frac{7}{10}$$
Question 3.
– 29 ___________
Number – 29 can be written in form of $$\frac{a}{b}$$ as:
$$-\frac{29}{1}$$
Question 4.
8 ___________
Mixed number 8$$\frac{1}{3}$$ can be written in form of $$\frac{a}{b}$$ as:
$$\frac{25}{3}$$
Place each number in the Venn diagram. Then classify each number by indicating in which set or sets each number belongs.
Question 5.
– 15 ___________
Number – 15 belongs to set of Integers
Question 6.
5$$\frac{10}{11}$$ ___________
Please note that i wrote mixed number 5$$\frac{10}{11}$$ as 65/11 since i cannot use alignment in picture editing!
Number 5$$\frac{10}{11}$$ beLongs to set of Rational numbers.
Essential Question Check-In
Question 7.
How is a rational number that is not an integer different from a rational number that is an integer?
Number that is Rational and not Integer can not be written as whole number or its opposite.
List two numbers that fit each description. Then write the numbers in the appropriate location on the Venn diagram.
Question 8.
Integers that are not whole numbers
Integers that are not Whole numbers are opposites of Whole numbers: – 3, – 5
Question 9.
Rational numbers that are not integers
Rational numbers that are not Integers are numbers that are not Whole numbers or their opposites: $$\frac{5}{3}, \frac{7}{8}$$
Please note that i wrote $$\frac{5}{3}$$ instead of $$\frac{5}{3}$$ and $$\frac{7}{8}$$ instead of $$\frac{7}{8}$$
Question 10.
Multistep A nature club is having its weekly hike. The table shows how many pieces of fruit and bottles of water each member of the club brought to share.
a. If the hikers want to share the fruit evenly, how many pieces should each person receive?
If hikers wanted to share 14 pieces of fruit evenly, each of them would get: $$\frac{14}{4}$$
b. Which hikers received more fruit than they brought on the hike?
To calculate, which of hikers got more fruit than brought, we have to change the format of our number to mixed number We know how to do it:
$$\frac{14}{4}$$ = $$\frac{4}{4}+\frac{4}{4}+\frac{4}{4}+\frac{2}{4}$$
= 1 + 1 + 1 + $$\frac{2}{4}$$
= 3 + $$\frac{2}{4}$$
= 3$$\frac{2}{4}$$
Now we can conclude that Hendrick and Baxer will get more pieces than they brought1
c. The hikers want to share their water evenly so that each member has the same amount. How much water does each hiker receive?
As there are 17 bottles of water altogether, if they split it in 4 parts that means each hiker is going to get: $$\frac{17}{4}$$ bottles.
As this is not Whole number, some hiker will have to bring that one extra bottle so they could share it between
themselves!
Question 11.
Sherman has 3 cats and 2 dogs. He wants to buy a toy for each of his pets. Sherman has $22 to spend on pet toys. How much can he spend on each pet? Write your answer as a fraction and as an amount in dollars and cents. Answer: As Sherman has 5 pets in total. That means he has to share$22 into 5 equal parts.
He will spend on each: $$$\frac{22}{5}$$ He will spend$4 and 40 cents for each pet!
Question 12.
A group of 5 friends is sharing 2 pounds of trail mix. Write a division problem and a fraction to represent this situation.
Since 5 of them share 2 pounds of milk, that means each will get:
$$\frac{5}{2}$$ pounds.
Question 13.
Vocabulary A _____________ diagram can represent set relationships visually.
A Venn diagram can visually represent set relationships.
Financial Literacy For 14-16, use the table. The table shows Jason’s utility bills for one month. Write a fraction to represent the division in each situation. Then classify each result by indicating the set or sets to which it belongs.
Question 14.
Jason and his 3 roommates share the cost of the electric bill evenly.
So, electricity payments bill splits into 4 equal parts!
We can express it via fraction as: $$$\frac{108}{4}$$ This number belongs to set of whole numbers as its equivalent to number: 27 Question 15. Jason plans to pay the water bill with 2 equal payments. Answer: Jason is paying water bill with 2 equal payments. That means he is going to pay:$$$\frac{35}{2}$$
This number belongs to set of Rational numbers.
Question 16.
Jason owes $15 for last month’s gas bill also. The total amount of the two gas bills is split evenly among the 4 roommates. Answer: As he owes$15 from last month, they will have to pay: $15 +$14 = $29 They split it into 4 parts and each pays:$$$\frac{29}{4}$$
This number belongs to set of Rational numbers.
Question 17.
Lynn has a watering can that holds 16 cups of water, and she fills it half full. Then she waters her 15 plants so that each plant gets the same amount of water. How many cups of water will each plant get?
Glass can holds 16 cups of water, and she filled it half, so she filled it with 8 cups of water.
Now, we know that if she has 15 plants, and wants to split that water equally,
she will spend: $$\frac{8}{15}$$
This number belongs to set of Rational numbers.
H.O.T. Focus on Higher Order Thinking
Question 18.
Critique Reasoning DaMarcus says the number $$\frac{24}{6}$$ belongs only to the set of rational numbers. Explain his error.
Number $$\frac{24}{6}$$ can be written without fraction as 4.
Therefore, it belongs to set of Integers and Whole numbers also.
Question 19.
Analyze Relationships Explain how the Venn diagrams in this lesson show that all integers and all whole numbers are rational numbers.
|
2021-12-05 23:37:19
|
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|
http://mathhelpforum.com/advanced-algebra/215707-set-degree-2-less-polynomials-p-1-p-2-vector-space.html
|
# Thread: Is the set of degree 2 or less polynomials with p(1)=p(2) a vector space?
1. ## Is the set of degree 2 or less polynomials with p(1)=p(2) a vector space?
Use the subspace theorem to decide whether the following set is a real vector space with the usual operations. The set V of all real polynomials p of degree at most 2 satisfying p(1) = p(2), i.e. polynomials with the same values at x = 1 and x = 2.
Subspace Thereom:
-The set is non-empty
- A1 is satisfied (closure under addition)
- S1 is satisfied (closure under scalar multiplication)
Really stuck on this!! Thanks in advance for any help
2. ## Re: Is the set of degree 2 or less polynomials with p(1)=p(2) a vector space?
Hey rmcal1.
Hint: Start off by stating what the vectors are and then show the axioms (zero vector, closure under scalar multiplication and addition).
In other words, what does a generic vector look like when you have p(1) = p(2)?
|
2018-04-21 20:15:27
|
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https://www.acmicpc.net/problem/12681
|
시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
5 초 512 MB 0 0 0 0.000%
## 문제
You are trying to compute the next number in a sequence Sn generated by a secret code. You know that the code was generated according to the following procedure.
First, for each k between 0 and 29, choose a number Ck between 0 and 10006 (inclusive).
Then, for each integer n between 0 and 1 000 000 000 (inclusive):
• Write n in binary.
• Take the numbers Ck for every bit k that is set in the binary representation of n. For example, when n=5, bits 0 and 2 are set, so C0 and C2 are taken.
• Add these Ck together, divide by 10007, and output the remainder as Sn.
You will be given a series of consecutive values of sequence S, but you don't know at which point in the sequence your numbers begin (although you do know that there is at least one more number in the sequence), and you don't know what values of Ck were chosen when the sequence was generated.
Find the next number in the sequence, or output UNKNOWN if this cannot be determined from the input data.
## 입력
The first line will contain an integer T, the number of test cases in the input file.
For each test case, there will be:
• One line containing the integer N, the number of elements of sequence S that you have.
• One line containing N single-space-separated integers between 0 and 10006, the known elements of the sequence.
Limits
• 1 ≤ T ≤ 20
• 1 ≤ N ≤ 5
## 출력
For each test case, output one line containing "Case #XY" where X is the number of the test case, starting from 1, and Y is the next number in the sequence, or the string UNKNOWN if the next number cannot be determined.
## 예제 입력 1
3
7
1 2 3 4 5 6 7
4
1 10 11 200
4
1000 1520 7520 7521
## 예제 출력 1
Case #1: UNKNOWN
Case #2: 201
Case #3: 3514
## 힌트
In the first case, C0, C1 and C2 might have been 1, 2 and 4, and the values of Sn we have starting at n=1. If this is correct, we don't know C3, so the next number in the sequence could be anything! Therefore the answer is unknown.
In the second case, we cannot know all the values of Ck or even what n is, but we can prove that in any sequence, if 1, 10, 11, 200 occur in order, then the next value will always be 201.
## 채점
• 예제는 채점하지 않는다.
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2018-10-15 10:54:26
|
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|
http://laussy.org/wiki/WLP_VI/Interpolation
|
# Crash Course in Scientific Computing
## XVII. Interpolation
Interpolation describes the general problem of providing the value of a function, which is known only for a few points, at any point which lies between other known points (if the unknown point is not "surrounded" by known points, then the problem becomes that of "extrapolation").
The simplest and also most natural method of interpolation is linear interpolation, which assumes that two neighboring points are linked by a line. It has been used since immemorial times throughout history and remains a basic technique in the computer industry, which even formed a special name for it, "lerp" (also used as a verb).
The method simply consists in finding the equation for a line between two points, those which are known from the data, i.e.,
$$L(x)=f(a)+{f(b)-f(a)\over b-a}(x-a)$$
Let us assume that in the parametrization of our problem, the range of the function is the number of equally-spaced data points known and we wish to interpolate to real-valued functions. All variations of lerp will do something similar. We can then refer to the known data points $a$ and $b$ from $x$ as:
\begin{align} a&=\lfloor x\rfloor\\ b&=\lceil x\rceil \end{align}
where the floor and ceil functions are defined as follows:
plot([floor, ceil], -3:.1:3, legend=false, lw=2)
In this case, our linear interpolation is easily implemented. Here is some sample data:
data = [sin(i) for i=0:2pi]
And here is the function to "lerp" it:
function lerpdata(x)
a=floor(Int,x);
b=ceil(Int,x);
data[a]+((data[b]-data[a])/(b-a))*(x-a)
end
scatter(data)
plot!(lerpdata,1:.01:7, lw=3, ls=:dash, legend=false)
While it works well with enough points if the data has no noise, if it has, it gives disagreeable results:
data = [sin(i)+rand()/5 for i=0:.1:2pi]
scatter(data)
plot!(lerpdata,1:.0001:63, lw=3, legend=false)
In two dimensions, linear interpolation becomes bilinear interpolations (in 3D, trilinear, etc.)
We can go beyond lerp with polynomials, i.e., fitting with functions that are not straight lines but bring in the curviness of power functions.
A great choice of interpolating polynomials is the so-called Lagrange-interpolating polynomials, which provide the unique polynomial of order $n$ to pass by the $n$ points $(x_i, y_i)$ (with $1\le i\le n$). Note that the order of the polynomial is the same as the number of points one wishes to pass through. These polynomials are obtained as a linear superposition of the so-called Lagrange-basis polynomials:
$$\ell _{j}(x)\equiv\prod _{\begin{smallmatrix}1\leq k\leq n\\k\neq j\end{smallmatrix}}{\frac {x-x_{k}}{x_{j}-x_{k}}}={\frac {(x-x_{1})}{(x_{j}-x_{1})}}\cdots {\frac {(x-x_{j-1})}{(x_{j}-x_{j-1})}}{\frac {(x-x_{j+1})}{(x_{j}-x_{j+1})}}\cdots {\frac {(x-x_{n})}{(x_{j}-x_{n})}}$$
They are such that $\ell_j(x_k)=0$ for all $x_k$ with $1\le k\le n$ when $k\neq j$, otherwise $\ell_j(x_j)=1$. Concisely:
$$\ell_j(x_k)=\delta_{jk}\,.$$
The interpolating polynomial $P_f$ of the function $f$ sampled at the points $x_k$ is then obtained as a linear superposition of the Lagrange polynomials with the values of the function as the weighting coefficients:
$$P_f(x)=\sum_{j=1}^n f(x_j)\ell_j(x)$$
Given some points $(x_i, y_i)$, e.g.,
xi=[1, 2, 3, 5];
yi=[1, 3, 1, 2];
plot the Lagrange polynomials allowing to interpolate through all these points along with the interpolation polynomial itself. Your code should display something like this on the previous input:
function plotLagrange(xi, yi)
scatter((xi, yi), label="Data points", legend=:topleft);
for j=1:length(xi)
plot!(x->prod([(x-xi[k])/(xi[j]-xi[k])
for k∈union(1:j-1,j+1:length(xi))]),
minimum(xi)-1, maximum(xi)+1, label="l\$(j)(x)", ls=:dash)
end
plot!(x->sum([yi[j]*prod([(x-xi[k])/(xi[j]-xi[k])
for k∈union(1:j-1,j+1:length(xi))])
for j∈1:length(xi)]), minimum(xi)-1, maximum(xi)+1,
lw=3, lc=:red, label="Lagrange interpolation", xlabel="x", ylabel="f(x)")
end
Runge's phenomenon is the interpolation version of the French saying «le mieux est l'ennemi du bien» (best is the enemy of the good): adding more interpolating points to some functions results in worst results.
Other polynomial schemes. We can name, for instance, Chebyshev Interpolation or Spline interpolation. We let these to your further inquiries.
|
2022-05-25 10:29:55
|
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|
https://www.futurelearn.com/courses/intro-to-quantum-computing/6/steps/690543
|
## Want to keep learning?
This content is taken from the Keio University's online course, Understanding Quantum Computers. Join the course to learn more.
2.14
# Reversible Evolution
In quantum mechanics, every change except measurement (and noise that damages the state, known as decoherence, which we will discuss in the last week when we discuss hardware and quantum error correction) must be reversible. (In mathematical terms, this means that it can be represented by a unitary matrix.) This means that it must be possible for us to recreate the initial state using only the output state, without additional information. Here, we will introduce some of those changes as discrete operations which we call gates.
## Basic Classical Logic Gates
You may already be familiar with the most basic logic gates we use in creating a classical computer, but just in case you’re not, let’s review them. Some of the gates have one input bit and one output bit, others have two input bits and only one output bit.
### NOT
The NOT gate, as you might expect, flips a single bit.
input output
0 1
1 0
The NOT gate is reversible: if you apply a NOT gate twice to the same signal, you get out the same value you started with: NOT(NOT(X)) = X.
### AND
The AND gate takes two input bits, and its output is one only if both input bits are one:
input $$A$$ input $$B$$ output $$C$$
0 0 0
0 1 0
1 0 0
1 1 1
The AND gate is not reversible: there are four different possible input states (00, 01, 10, 11) and only two possible output states (0 and 1), so there isn’t enough information in the output to know for sure what the inputs were. In the case that the output C is 1, you know that both inputs were 1, but in all three of the other cases, you can’t tell.
### OR
The OR gate takes two input bits, and its output is zero only if both input bits are zero:
input $$A$$ input $$B$$ output C
0 0 0
0 1 1
1 0 1
1 1 1
Its behavior in a lot of ways is similar to that of the AND gate. The OR gate is not reversible: there are four different possible input states (00, 01, 10, 11) and only two possible output states (0 and 1), so there isn’t enough information in the output to know for sure what the inputs were. In the case that the output C is 0, you know that both inputs were 0, but in all three of the other cases, you can’t tell.
In fact, with the AND or OR gate, even if I gave you the output and one of the inputs, you wouldn’t be able to tell unambiguously what the other input value was in all cases.
### XOR
The XOR, or exclusive OR, gate takes two input bits, and its output is one only if exactly one input bit is one:
input $$A$$ input $$B$$ output $$C$$
0 0 0
0 1 1
1 0 1
1 1 0
The XOR gate is close to reversible: with two input bits and one output bit, there still isn’t enough information in the output to know for sure what the inputs were. But, if I give you one of the inputs, you can now tell unambiguously what the other input was! We will see below that one of the important reversible two-qubit gates is related to XOR.
## Reversible Classical Gates
### Entropy and Information
In the early 1970s, Charles Bennett of IBM, later joined by Richard Feynman of Caltech, and Tommaso Toffoli and Ed Fredkin of MIT, laid the groundwork for discrete reversible logic operations. Bennett was an acolyte of Rolf Landauer, who recognized that the destruction of information adds to the entropy of the Universe, and must generate waste heat.
Entropy is the amount of disorder in something. If you take a physics class, or possibly chemistry, you will learn about entropy when you learn about thermodynamics. If you have a lot of energy confined in one place, you can use it to do work, such as turning a motor. This spreads energy around. Once the energy is spread out evenly, the entropy is maximized and it is no longer possible to do useful work with it.
In computer science we also talk about the entropy of information. If you have a certain amount of data, it might be all zeroes or all ones, in which case it’s very easy to tell a friend how to recreate the same state: just say, “I have seven hundred ones.” If your data is more complex, though, it gets harder to describe; it has high entropy.
I (Van Meter) can recall taking Feynman’s class at Caltech, in which he explained the relationship between information and entropy in terms of thermodynamics. For quite a while, I believed that he was speaking allegorically, and couldn’t understand why he was bringing in physical constants. When I finally understood that he meant it quite literally, I could feel the hair on the back of my neck stand up. It remains one of the most startling ideas I have ever encountered.
### Reversible Computing
Generally, entropy increases when we lose information, or when information is erased. The AND and OR gates we discussed above necessarily lose information: they start with two input bits, and end with only one, so there is no way to carry all of the information through.
If, instead, we require all of our gates to have the same number of input and output bits, it’s possible that we might be able to undo our computation. A second additional criteria is necessary to make it reversible: every possible output state comes from exactly one input state. Then we say that the function is “one to one”, or bijective.
Any logic function can be computed using one, two, and three-bit reversible gates, so we will look at a few of them.
### Identity
The identity gate does nothing to the state: its output is the same as its input. It is obviously reversible; since it does nothing, in order to get back to where we started, we only have to do nothing one more time!
input output
0 0
1 1
### NOT
We have already seen the NOT gate above. Executing NOT twice in a row brings us back to where we started: NOT(NOT(X)) = X.
### CNOT
The controlled NOT gate, or CNOT, takes two input bits and produces two output bits. If one of the bits (called the control bit) is one, the other bit (called the target bit) is flipped. If the control bit is zero, the target bit is passed through unchanged.
input $$A$$ input $$B$$ output $$A'$$ output $$B'$$
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0
A quick examination of the table shows that $$A'$$ is the same as $$A$$, and $$B'$$ is the XOR of $$A$$ and $$B$$, $$B' = A \oplus B$$. If we apply the same gate twice, obviously $$A$$ stays the same, and the new $$B'' = A \oplus (A \oplus B) = B$$, and we are back where we started.
### CCNOT (Toffoli Gate)
The control-control-NOT gate, or CCNOT gate, is also called the Toffoli gate, named for Tommaso Toffoli. It uses two control bits instead of one.
input $$A$$ input $$B$$ input $$C$$ output $$A'$$ output $$B'$$ output $$C'$$
0 0 0 0 0 0
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 0 1 1
1 0 0 1 0 0
1 0 1 1 0 1
1 1 0 1 1 1
1 1 1 1 1 0
### CSWAP (Fredkin Gate)
The control-SWAP gate, or CSWAP gate, is also called the Fredkin gate, named for Ed Fredkin. If the control bit is zero, the other two bits are left alone. If the control bit is one, the other two bits are swapped.
input $$A$$ input $$B$$ input $$C$$ output $$A'$$ output $$B'$$ output $$C'$$
0 0 0 0 0 0
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 0 1 1
1 0 0 1 0 0
1 0 1 1 1 0
1 1 0 1 0 1
1 1 1 1 1 1
Either the CCNOT or the CSWAP gate is powerful enough to be able to create any classical logic circuit.
## Single-Qubit Gates
The reversible gates above all have quantum equivalents, and we will use them in constructing quantum algorithms. But qubits are more complex objects than simple classical bits, allowing a broader range of operations that all qualify as unitary. Here, we will look at some single-qubit gates that go beyond just the Identity and NOT gates above.
### Rotations on the Bloch Sphere
When we introduced qubits, we introduced the notion of the Bloch sphere, and the idea of the state of a single qubit as a point on the sphere. The one-qubit gates can be understood as rotations about the X, Y, or Z axis of the Bloch sphere.
Rotation is a naturally reversible operation: just rotate back in the opposite direction by the same amount. Every point on the sphere has a starting position and an ending position, and no two points ever come together. No two different starting points come to the same ending position. This is a key factor maintaining reversibility. If two points did come together, when we attempted to reverse the operation, we would have no way of distinguishing them and returning them to separate starting positions.
### The NOT Gate and Other X Gates
A simple reversible gate, like the classical NOT gate, the $$X$$ gate takes 0 to 1 and 1 to 0. Since our state can be a superposition, it does a little more than this, it swaps our 0 and 1 values, or the contents of our 0 and 1 dials. It is a rotation of 180 degrees, or $$\pi$$, about the $$X$$ axis. It’s also possible to rotate by any angle, which swaps the 0 and 1 values less completely.
### The Phase Gates
Although rotating around the $$X$$ axis is somewhat like the classical NOT gate, rotating around the $$Z$$ axis has no classical equivalent. $$Z$$ rotation modifies the phase of the state. We call a 180 degree rotation simply the Z gate. For other angles, we call it a Z rotation or a phase gate, and specify the angle.
The Hadamard gate is our most basic means of creating superposition. It takes our $$\vert0\rangle$$ state to a 50/50 superposition of $$\vert0\rangle$$ and $$\vert1\rangle$$:
$\vert0\rangle \rightarrow \frac{\vert0\rangle + \vert1\rangle}{\sqrt{2}}$
If it also took the $$\vert1\rangle$$ state to the same 50/50 superposition, it wouldn’t be reversible. Instead, it takes it to a superposition with a phase twist on the $$\vert1\rangle$$ state:
$\vert1\rangle \rightarrow \frac{\vert0\rangle + (\pi)\vert1\rangle}{\sqrt{2}}$
Applying the Hadamard twice to the same qubit returns it to its original state. In fact, this is our first use of interference! If we began with the $$\vert0\rangle$$ state, we get constructive interference that builds us a new $$\vert0\rangle$$ state, and destructive interference that eliminates the $$\vert1\rangle$$ state:
$$\frac{\vert0\rangle + \vert1\rangle}{\sqrt{2}}\rightarrow \vert0\rangle$$.
Here we can see that the phase of the state has a critical impact on the interference; reversing the other state results in constructive interference on the $$\vert1\rangle$$ state, and destructive interference on the $$\vert0\rangle$$ state:
$$\frac{\vert0\rangle + (\pi)\vert1\rangle}{\sqrt{2}}\rightarrow \vert1\rangle$$.
# 可逆発展
## 基本的な古典論理ゲート
### NOT
NOTゲートは、その名の通り一つのビットを反転させます。
input output
0 1
1 0
NOTゲートは可逆な操作です。例えば、同じ信号に二度NOTゲートをかければ、操作の後に出力される値は操作を始める前のものと同じものになることが分かるかと思います。
### AND
ANDゲートは2つの入力ビットを持ち、両方の入力ビットが1だった場合のみ、出力値として1を返します。
input $$A$$ input $$B$$ output $$C$$
0 0 0
0 1 0
1 0 0
1 1 1
ANDゲートは不可逆な操作です。異なる四つの入力状態 (00, 01, 10, 11)と二つの出力状態(0, 1)があり、出力だけからは入力がどんな値だったのかは分からなくなってしまいます。出力Cが1の場合は、両方の入力が1であったことが分かりますが、それ以外の場合は入力の値が何だったかは知ることができません。
### OR
ORゲートは二つの入力ビットを持ち、両方の入力ビットが0だった場合のみ、出力値として0を返します。
input $$A$$ input $$B$$ output C
0 0 0
0 1 1
1 0 1
1 1 1
ANDゲートと同様に、ORゲートも不可逆な操作です。 異なる四つの入力状態(00, 01, 10, 11)と二つの出力状態 (0, 1)があり、出力だけからは入力がどんな値だったのかを推測することはできません。 この場合、出力Cが0の場合のみ、両方の入力が0であったことが分かりますが、それ以外では入力値が何だったかを知ることはできません。
### XOR
XORゲート(exclusive OR)は、二つの入力ビットをとり、一つの入力ビットだけが1だった場合にのみ出力は1になります。
input $$A$$ input $$B$$ output $$C$$
0 0 0
0 1 1
1 0 1
1 1 0
XORゲートは可逆に近い操作です。 二つの入力ビットと一つの出力ビットをもち、出力値だけで入力値を逆算することはできませんが、２つの入力値のどちらか一つを与えられれば、もう片方の入力値についても自動的に知ることができます。 続いて、最も重要な2量子ビットゲートが、XORゲートに関連していることを学んでいきます。
## 可逆古典ゲート
### エントロピーと情報
1970年代前半、カルフォルニア工科大学のRichard Feynman、IBMのCharles Bennett、MITのTommaso Toffoli と Ed Fredkin等によって可逆論理演算の基礎が築かれました。 Bennettは当時、Rolf Landauerの助手として働いていました。Rolf Landauer 教授は、情報の破壊が宇宙のエントロピーを増大し廃熱をもたらすことを発見した研究者です。
エントロピーとは、何かに存在する無秩序の度合いを示す物理量です。 皆さんが、物理や化学の授業を受けていれば、その中で熱力学を学ぶ際にエントロピーについて勉強したことがあると思います。 ある場所にエネルギーを蓄えれば、それをモーターの回転などの仕事に使うことができます。 こういった仕事はエネルギーを周囲に拡散します。 一度エネルギーが広がりきると、エントロピーは最大化し、もはや有効な仕事に使うことができなくなります。
コンピュータサイエンスの分野においても、「情報のエントロピー」という概念があります。もしあながた一定量のデータを持っていたら、それが全て0か全て1のどちらかだとすると、友達にどのように同じ状態を再構築するかを教えるのはとても簡単です。単に、「私は700個の1を持っています」と教えるだけです。もしデータがより複雑になると、それを記述することは難しくなり、これを「エントロピーが高い」と表現します。
### 可逆計算
もし、全ゲートについて入力と出力が同じ数であるなら、、処理を元に戻すことが可能かもしれません。 それでも、完全なる「可逆」のためには、もう一つの条件が必要になります。その条件とは、すべての出力状態は、それぞれ一意に決まる入力状態と対応付けられている、という場合です。そのとき、関数は”one to one”、もしくは全単射であると言えます。
すべての論理関数は、1ビット、2ビット、3ビットの可逆ゲートを用いて計算することができます。 そういった論理関数をいくつか見ていきましょう。
input output
0 0
1 1
NOT(NOT(X)) = X
### CNOTゲート
CNOT(controlled NOT gate, or CNOT)は二つの入力ビットをとり、二つの出力ビットを生み出します。 もし片方のビット(制御ビットと呼ばれる)が1の場合、もう片方のビット(ターゲットビットと呼ばれる)は反転されます。 もし制御ビットが0の場合、ターゲットビットの状態は変わらず、そのまま出力されます。
input $$A$$ input $$B$$ output $$A'$$ output $$B'$$
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0
A は $$A′$$ の値と常に同じであり、 $$B′$$は$$A$$と$$B$$を XOR でかけた$$B' = A \oplus B$$であることが分かります。もしCNOTゲートを２回連続で適用した場合、$$A$$は同様に同じ状態を保持し、ターゲットビットである$$B$$は$$B'' = A \oplus (A \oplus B) = B$$になり、結果、初期値に戻すことができます。
### CCNOTゲート (Toffoliゲート)
CCNOT(control-control-NOT gate)はTommaso Toffoli.に因んで、Toffoli （トフォリ）ゲートとも呼ばれています。CCNOTは二つの制御ビットを必要とします。
このゲートは3ビットの入出力をもちます。最初の2ビットはそのまま出力します。もし入力の最初の2ビットが1なら、第3のビットは反転して出力されます。
input $$A$$ input $$B$$ input $$C$$ output $$A'$$ output $$B'$$ output $$C'$$
0 0 0 0 0 0
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 0 1 1
1 0 0 1 0 0
1 0 1 1 0 1
1 1 0 1 1 1
1 1 1 1 1 0
### CSWAPゲート (Fredkinゲート)
CSWAPゲート(control-SWAP gate)は、Ed Fredkinに因んでFredkinゲートとも呼ばれます。 もし制御ビット（入力A）が0の場合、他のビット（入力B、入力C）の状態は何も変化せずに出力されます。 逆に、1だった場合、他の2つのビット状態は交換され、出力されます。
input $$A$$ input $$B$$ input $$C$$ output $$A'$$ output $$B'$$ output $$C'$$
0 0 0 0 0 0
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 0 1 1
1 0 0 1 0 0
1 0 1 1 1 0
1 1 0 1 0 1
1 1 1 1 1 1
CCNOTやCSWAPゲートは、其々のゲートだけで任意の古典論理回路を組むことができる、強力なゲートの一例として挙げられます。
## 単一量子ビットゲート
### ブロッホ球上での回転
Rotation is a naturally reversible operation: just rotate back in the opposite direction by the same amount. Every point on the sphere has a starting position and an ending position, and no two points ever come together. No two different starting points come to the same ending position. This is a key factor maintaining reversibility. If two points did come together, when we attempted to reverse the operation, we would have no way of distinguishing them and returning them to separate starting positions.
### 位相ゲート
$$X$$軸を中心とした回転は古典的なNOTゲートに似ている操作ですが、$$Z$$軸を中心に状態を回転させる操作に対応する古典的なゲートは存在しません。$$Z$$回転は、その量子状態の位相を変換します。 $$Z$$軸を中心に180度($$\pi$$)の回転を加える操作を「Zゲート」と呼びます。もちろん、他の角度の場合は、「Z回転」、もしくは「位相ゲート」と呼び、その角度を明記します。
### アダマール ゲート
アダマールゲートは重ね合わせを作るために用いる、最も基本的な操作です。$$\vert0\rangle$$の状態を50/50の割合で$$\vert0\rangle$$と$$\vert1\rangle$$の重ね合わせに変換します。
$\vert0\rangle \rightarrow \frac{\vert0\rangle + \vert1\rangle}{\sqrt{2}}$
もしアダマールゲートを$$\vert1\rangle$$の状態にかけて、上記と同じ50/50の重ね合わせになってしまったら、それは不可逆な操作になってしまいます。$$\vert1\rangle$$の状態にアダマールゲートをかけるときは、位相をねじって重ね合わせます。
$\vert1\rangle \rightarrow \frac{\vert0\rangle + (\pi)\vert1\rangle}{\sqrt{2}}$
アダマールゲートを同じ量子ビットに２回連続して適用すれば、出力状態は初期の入力状態と常に同じになります。 物理的には、この操作は波の干渉にあたります（はじめて登場しますね！）$$\vert0\rangle$$にアダマールゲートをかければ、新しい$$\vert0\rangle$$状態は元の$$\vert0\rangle$$状態と$$\vert1\rangle$$状態の強め合い干渉によって生成され、$$\vert1\rangle$$状態は弱め合い干渉によって消滅します。
$\frac{\vert0\rangle + \vert1\rangle}{\sqrt{2}}\rightarrow \vert0\rangle$
ここで位相が重要になってきます。
$\frac{\vert0\rangle + (\pi)\vert1\rangle}{\sqrt{2}}\rightarrow \vert1\rangle$
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2020-08-04 15:25:31
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https://revision.co.zw/labour-turnover/
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### ZIMSEC O Level Business Studies Notes: Managing Human Resources: Human Resources Management: Labour Turnover
• It is a statistical measure of the number of people who leave an organisation’s employ versus the number of people who remain in that organisation’s employment in a given year
• It is expressed as a ratio:
• $\dfrac{\text{Number of people who left}}{\text{Avergage number of people employed}}$
• This ratio is calculated on a yearly basis
• The number of people left in the formula above includes the number of people who left the organisation for whatever reason in a given year
• The average number of people in the organisation’s employment can be calculated using the formula:
• $\dfrac{\text{No. of employees at beginning of the year+Number of people at the end}}{\text{2}}$
• People can leave the organisation due to a number of reasons including:
1. Management action that includes:
1. Disciplinary action
2. Dissatisfaction with performance of employee leading to dismissal of the employee
3. Redundancy
2. Involuntary separations such as retirement
3. Voluntary separations such as resulting from dissatisfaction with job, pay or working conditions
• When a high number of people leave the organisation this is known as high labour turnover
• A high labour turnover is when a high number of the organisation’s employees are leaving and being replaced by the organisation
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2023-03-28 07:48:54
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https://robotics.stackexchange.com/questions/4789/probabilistic-velocity-obstacles
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# Probabilistic Velocity Obstacles
I have been working with the Velocity Obstacles concept. Recently, I came across a probabilistic extension of this and couldn't understand the inner workings.
• The notation $PCC_{ij} : \Bbb R^2 \to [0,1]$ means "$PCC_{ij}$ is a function from $\Bbb R^2$ (real ordered pairs) to the interval $[0,1]$." – apnorton Oct 24 '14 at 3:36
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2019-11-13 12:06:54
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https://math.stackexchange.com/questions/3836207/how-is-frac1-vecr-vecr-frac1r-vecr-cdot-triangledo
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# How is $\frac{1}{|\vec{r} - \vec{r}'|} = \frac{1}{r} - \vec{r} \cdot \triangledown \frac{1}{r} + \ldots$ a Taylor expansion?
From pg. 112 of No-Nonsense Electrodynamics, the author uses multivariable Taylor expansion to assert:
In case it matters, the author is also assuming that $$|\vec{r}| \gg | \vec{r}'|$$ (stated elsewhere in the text). Also, $$\vec{r}, \vec{r}'$$ are both 3-dimensional vectors.
Question: How does this identity follow from Taylor? According to Wikipedia
A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as
$$T(\mathbf{x}) = f(\mathbf{a}) + (\mathbf{x} - \mathbf{a})^\mathsf{T} D f(\mathbf{a}) + \cdots$$
where $$D$$ in this context denotes the gradient $$\triangledown$$ operator. If we plug this into to our context we get
$$f(\mathbf{a}) + (\mathbf{x} - \mathbf{a})^T Df(\mathbf{a}) = \frac{1}{r} + ( \vec{r} - \vec{r}') \cdot \triangledown \frac{1}{| \vec{r} - \vec{r}' |}$$
which doesn't obviously resemble the formula the author derived, unless I'm missing something?
## 1 Answer
HINT:
We can write the term of interest as
\begin{align} \frac1{|\vec r-\vec r'|}&=\frac{1}{\sqrt{|\vec r|^2+|\vec r'|^2-2\vec r'\cdot\vec r}}\\\\ &=\frac1{|\vec r|}\left(1-2\frac{\vec r'\cdot\vec r}{|\vec r|^2}+\left(\frac{|\vec r'|}{|\vec r|}\right)^2\right)^{-1/2} \end{align}
Now applying the binomial theorem (which is a special case of Taylor's Theorem) reveals
\begin{align} \frac1{|\vec r-\vec r'|}&=\frac1{|\vec r|}\left(1+\frac{\vec r'\cdot\vec r}{|\vec r|^2}+O\left(\frac{|\vec r'|^2}{|\vec r|^2}\right)\right)\\\\ &=\frac1{|\vec r|}+\vec r'\cdot \frac{\vec r}{|\vec r|^3}+O\left(\frac{|\vec r'|^2}{|\vec r|^3}\right) \end{align}
and we see that the Equation $$(4.30)$$ in the OP is incorrect by a factor of $$-1$$ on the second term of the expansion. In fact, $$(4.30)$$ is inconsistent with the equation that precedes it since $$\nabla \frac1r=-\frac{\vec r}{|\vec r|^3}$$ and not $$\nabla \frac1r=+\frac{\vec r}{|\vec r|^3}$$
NOTE:
Application of Taylor's Theorem to the function $$\frac1{|\vec r-\vec r'|}$$ that lead to the expression written in the OP is $$o(|\vec r-\vec r'|)$$ and does not lead, therefore, to a useful expansion when $$|\vec r|>>|\vec r'|$$.
Instead, we have used Taylor's Theorem (which is also application of the binomial theorem in this case) to expand the term $$\displaystyle \left(1-2\frac{\vec r'\cdot\vec r}{|\vec r|^2}+\left(\frac{|\vec r'|}{|\vec r|}\right)^2\right)^{-1/2}$$.
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2021-08-01 03:15:58
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https://chemistry.stackexchange.com/questions/55764/why-bimolecular-elimination-reactions-are-stereospecific
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why bimolecular elimination reactions are stereospecific?
I know the mechanism of this reaction but it is stated that these are stereospecific in nature . why do we call these stereospecific explain with example.
• The hydrogen being abstracted by the base, and the leaving group must be anti-periplanar. This requirement determines the stereochemistry unambiguously , and thus the reaction is stereospecific. – getafix Aug 17 '16 at 20:27
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2019-09-19 21:38:18
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