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# 0.15 More probability: homework (Page 4/5)
Page 4 / 5
For a real estate exam the probability of a person passing the test on the first try is .70. The probability that a person who fails on the first try will pass on each of the successive attempts is .80. What is the probability that a person passes the test in at most three attempts?
On a Christmas tree with lights, if one bulb goes out, the entire string goes out. If there are twelve bulbs on a string, and the probability of any one going out is .04, what is the probability that the string will not go out?
0.6127
The Long Life Light Bulbs claims that the probability that a light bulb will go out when first used is 15%, but if it does not go out on the first use the probability that it will last the first year is 95%, and if it lasts the first year, there is a 90% probability that it will last two years. What is the probability that a new bulb will last two years?
A die is rolled until an ace (1) shows. What is the probability that an ace will show on the fourth try?
$\text{125}/\text{1296}$
If there are four people in a room, what is the probability that no two have the same birthday?
Dan forgets to set his alarm 60% of the time. If he hears the alarm, he turns it off and goes back to sleep 20% of the time, and even if he does wake up on time, he is late getting ready 30% of the time. What is the probability that Dan will be late to school?
0.776
It has been estimated that 20% of the athletes take some type of drugs. A drug test is 90% accurate, that is, the probability of a false-negative is 10%. Furthermore, for this test the probability of a false-positive is 20%. If an athlete tests positive, what is the probability that he is a drug user?
## Chapter review
A coin is tossed five times. Find the following
1. $P\left(\text{2 heads and 3 tails}\right)$
2. $P\left(\text{at least 4 tails}\right)$
1. 0.3125
2. 0.1875
A dandruff shampoo helps 80% of the people who use it. If 10 people apply this shampoo to their hair, what is the probability that 6 will be dandruff free?
0.088
A baseball player has a .250 batting average. What is the probability that he will have 2 hits in 4 times at bat?
0.21094
Suppose that 60% of the voters in California intend to vote Democratic in the next election. If we choose five people at random, what is the probability that at least four will vote Democratic?
0.33696
A basketball player has a .70 chance of sinking a basket on a free throw. What is the probability that he will sink at least 4 baskets in six shots?
0.74432
During an archery competition, Stan has a 0.8 chance of hitting a target. If he shoots three times, what is the probability that he will hit the target all three times?
0.512
A company finds that one out of four new applicants overstate their work experience. If ten people apply for a job at this company, what is the probability that at most two will overstate their work experience?
0.52559
A missile has a 70% chance of hitting a target. How many missiles should be fired to make sure that the target is destroyed with a probability of .99 or more?
4
Jar I contains 4 red and 5 white marbles, and Jar II contains 2 red and 4 white marbles. A jar is picked at random and a marble is drawn. Draw a tree diagram and find,
1. $P\left(\text{Marble is red}\right)$
2. $P\left(\text{It is white given that it came from Jar II}\right)$
3. $P\left(\text{It came from Jar II knowing that the marble drawn is white}\right)$
1. $7/\text{18}$
2. $2/3$
3. $6/\text{11}$
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
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8. It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges$12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost?
At De Anza College, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?
|
# Sonya
Most popular questions and responses by Sonya
1. ## Physics
A force of 800 N stretches a spring a distance 0.200 m horizontally. a) What is the potential energy of the spring when it is stretched 0.200m b) What is its potential energy when it is compressed 5.00cm This is a conservation of energy problems. The
2. ## math
Sue stacked one box onto another. The bottom box had the height of 2 1/3 feet and the top box had the height of 3 2/3 feet. How tall were the stacked boxes?
3. ## Chemistry
Calculate the concentration of IO3– in a 9.23 mM Pb(NO3)2 solution saturated with Pb(IO3)2. The Ksp of Pb(IO3)2 is 2.5 × 10-13. Assume that Pb(IO3)2 is a negligible source of Pb2 compared to Pb(NO3)2.
4. ## History
How did the Crimean War change life in Russia? a. Russia's defeat in the war sparked calls for political and social reform. ********* b. Russia's investment of wealth and the loss of lives in the war led to a crippling depression. c. Russia's proven
5. ## Social Studies
How did the Crimean War change life in Russia? a. Russia's defeat in the war sparked calls for political and social reform. ********* b. Russia's investment of wealth and the loss of lives in the war led to a crippling depression. c. Russia's proven
A college professor told her students, “The purpose of a management course is to teach students about management, not to teach them to be managers.” Do you agree or disagree with this statement? Why or Why not? Also, is efficiency and effectiveness
7. ## Physics
A horizontal pipe narrows from a diameter of 10 to 5 cm. For an incompressible fluid flowing from the larger diameter to the smaller: Question 5 options: 1.the velocity and pressure both increase. 2.the velocity increases and the pressure decreases. 3.the
8. ## Physics
A small block with a mass of m = 0.120 kg is attached to a cord passing through a hole in a frictionless, horizontal surface. The block is originally revolving at a distance of 0.40 m from the hole with a speed of 0.70 m/s. The cord is pulled from below,
9. ## math
Which two numbers have a product of 56 and a quotient of 14
10. ## Conceptual Physics
A paperback is 106mm in width. What is the width in centimeters? In meters? (I need to see this in steps)
11. ## CIS
Draw a flowchart or write pseudocode to represent the logic of a program that allows the user to enter a value. The program multiplies the value by 10 and outputs the result.
12. ## Statistics
A doctor want to estimate the HDL cholesterol of all 20- to 29- year old females. how many subjects are needed to estimate the HDL cholesterol within 3 points with 99% confindence assuming s=15.8 based on earlier studies?
13. ## physics 3U
An arrow is accelerated for a displacement of 75cm [fwd] while it is on the bow. If the arrow leaves the bow at a velocity of 75m/s [fwd], what is the average acceleration while on the bow?
14. ## Chemistry
Calculate the molarity of an aqueous NaOH soultion if 32.00ml of it is required to titrate. 0.6789 g of oxalic acid to phenolphthalein end point. The balanced chemical equation is H2C2O4 + 2NaOH-> Na2C2O4 + 2H2O
15. ## CIS/Visual Logic
Design an application that accepts 10 numbers and displays them in descending order. Us an array to store the numbers and Bubble Sort to sort the numbers.
16. ## Chemistry
Determine [Zn2 ], [CN–], and [HCN] in a saturated solution of Zn(CN)2 with a fixed pH of 2.640. The Ksp for Zn(CN)2 is 3.0 × 10–16. The Ka for HCN is 6.2 × 10–10. I need to know what equations to use to find the answer not just how to get it
17. ## chemsitry
you start with 150.0g of pentane and 500.0L of O2. how many grams of H2O will you make?
18. ## math
winnie has 7 trophies she wants to display in arrays. how many different arrays are possible? explain.
19. ## math
You fire your potato gun horizontally from shoulder height, which is 190 cm above the ground. The potato strikes the ground 30 m away from you. What was the initial speed of the potato?
20. ## Geometry
Jason designed an arch made of wrought iron for the top of a mall entrance. The 11 segments between the two concentric circles are each 1.25 m long. Find the total length of wrought iron used to make the structure. Round the answer to the nearest meter.
21. ## Chemistry
Determinne the amount of Dextrose and NaCl in 500ml of 5% DNS. (The solution is 5% in regards to dextrose and 0.9% in regards to NACl.)
22. ## Chemistry
calculate the theoretical yield (in grams) of the solid product, assuming that you use 1.0 g FeC2O4×2H2O and that oxygen is the excess reactant. FeC2O4×2H2O (s) + O2(g) ---> FeO(s) + H2O(g) + CO2(g) Its balanced equation is 2FeC2O4×2H2O (s) + O2(g)---->
23. ## chemistry
one piece of copper metal at 105 C has twice the mass of another copper piece at 45 C. What is the final temperature if these two pieces are placed in a calorimeter. Specific heat of copper is 0.387 J/g.K.
24. ## AP Chem
The electron-domain geometry and molecular geometry of iodine trichloride are __________ and __________, respectively. The answer is trigonal bipyramidal, T-shaped, respectively- I do not understand the approach PLEASE EXPLAIN HOW YOU GOT THE ANSWER!!!
The cross-sectional area of the U-tube shown below is uniform and is equal to one square centimeter (1.00cm^2). One end is open to the atmosphere. Atmosphere pressure may be taken as P1=1.01 * 10^5 Pa. The other end of the U-tube is connected to a pipe in
26. ## algebra
Find the x-intercept and the y-intercept of the graph of: 4x-2y=8 a. (2, -4) b. -4;2 c. (4, -2) d. 2;-4 I'm stuck between a and d. anyone helpp?
27. ## Math
I need help please A computer can execute 64 instructions per second. How many instructions can it execute in 10 minutes? Multiply 3 1/6 x 9/11 x 6/17. Write the answer in simplest form Divide 6/25 / 2/15. Write the result in simplest form.
28. ## chemistry HELP
Use the following information to identify element A and compound B, then answer questions a and b. An empty glass container has a mass of 658.572 g. It has a mass of 659.452 g after it has been filled with nitrogen gas at a pressure of 790. torr and a
29. ## Algebra 2
What is the angular speed in radians per second of the hour hand of a clock? If the hand is 5cm long, how fast is the tip moving in millimeters per second?
30. ## chemistry
how many mL of 0.15 M NaOH solution are required to neutralize 35.00 mL of 0.22 M HCl
31. ## algebra 2
Find the x-intercept and the y-intercept of the graph of: 4x-2y=8 a. (2, -4) b. -4;2 c. (4, -2) d. 2;-4 I'm stuck between a and d. anyone helpp?
32. ## Physics
(a)A car travels exactly once around a circle of radius 5.0 m at constant speed of 3.0 m/s. What is the car’s average velocity (magnitude and direction) during this interval? Explain/justify. (b)Repeat the calculation of average velocity if the car
33. ## Physics
A car is traveling over the top of a hill, which we will model to be a piece of a circular arc of radius R. For each case below, draw careful to-scale motion maps and use them to construct the velocity vectors at the following three points: the top of the
34. ## Arithmetic
How can we put parentheses around the following expression so that the answer is 105? 9 + 8 x 7 - 6 x 5 + 4 x 3 + 2
35. ## algebra
I am given the perimeter of the rectangle. It is 96cm. The length is 3 times the width. What are the measurements of the 4 sides of the rectangle? I need the formula please.The answer is 12 and 36, I think.
36. ## Program Logic Design
create a flowchart that shows the logic for a program that generates a random number; then ask the user to think of a number between 1 and 10. The display then randomly generated number so the user can see whether his or her guess was accurate.
37. ## Humanities/World Literature
1.Why does the demon collect her lovers' rings in _1001 Nights_? 2.What is the purpose of Shahryar and Shahzaman's journey in _1001 Nights_?
38. ## Math
In solving the eqaution(x+4)(x-4),Eric stated thatthe solution would be x+4=33=>x=29 or (x-4)=33=>x=37 However, at least one of these solutions fails to work when substituted back into the original eqaution. Why is that?Help Eric to understand better,solve
39. ## physics
A 400 calorie quantitiy of heat is added to 20 grams of water and as a result, the water reaches a final temp of 70 deg. C. The starting temp of the water was most nearly _____ deg C.
40. ## Algebra1
when using multiplying by 1 to find an expression equivalent to 18/7 with a denominator of 28x, the equivalent expression that I get is 4x/28x and i am being told that that's wrong
41. ## math
simple form 3 ¾ / 60,000= 41 ½ / 3 ¾ A computer can excute 36 instructions per microscecond. How many instructions can it execute in 4 min?
42. ## ashford unversity
Which of the following is true regarding an SEP?
43. ## Physics _ College
A 50 cm long hollow glass cylinder is open at both ends and is suspended in air. A source of sounds that produces a pure frequency is placed close to one end of the tube. The frequency is placed close to one end of the tube. The frequency of the sound
45. ## Math - please double check me
There are 42 accounting classes with 31 students in each class. Estimate the total number of students in the accounting classes. a. 1200 b. 900 c. 1600 d. 1500 answer: c One kilometer is equal to 1000 meters. What fractional part of a kilometer is 500
46. ## univ/com/155
write a paragraph of instructions on the writing process for future com/155 students.include the steps in the writing process.
47. ## A and P
The following is a template strand of DNA with a start codon(=!!!) and a stop codon(=???): !!!TCGGGCTACAAAACAAATCAACGGGGCTCGCAAACAAATAAACGG??? What is its mRNA nucleotide sequence? Indicate the product of this DNA segment/gene. What is the product’s
48. ## Literature
What literature do you consider to be part of the United States' current literary canon, and why? How do those selections reflect the cultural tradition of the United States? What do you consider to be part of your personal literary canon?
49. ## algebra 2
3. Identify the range of y=|x|-4 A. [x|x≥4] B. [y|y≥-4] C. [y|y≥0] D. all real numbers I'm confused about this question, any help???
50. ## math
Please explain how to get the answer. At the animal shelter, 1/3 of the animals are dogs, 2/12 are cats, and 19 are other animals. How many animals are at the shelter? I said 1/3x + 2/12x + 19 = 4/12x + 2/12x = 19= 6/12x = 19 1/2x = 19 x =
51. ## poetry
i need to write an acrostic poem for english using the quote:"do all lawyers defend negros". its from to kill a mocking bird and i need to write a phras starting with each letter in the quote and i cnt think of anything
what is the greatest common factor of 245, 385, and 539
53. ## Communication Stratgies
How can you write a 350 word memo about 25 students losing their deposit on a spring trip?
54. ## algebra
(2a+4b)-(3a-b)when and how do u know to change the signs? It might help to think of it as below. (2a+4b)-1(3a-b) Mutliply the last term by the -1. 2a+4b-3a+b = -a+5b or 5b-a I hope this helps. Thanks for asking.
55. ## solution making/ chemistry/biotech
how much of a .2M solution can you make with 340 g of calcium carbonate ( MW=100.0869) Do I multiply 340g by 1mole/MW
56. ## math
A bicyclist rides off a roof at 15.2 \frac{m}{s}. The roof is 9.8 m above the ground. How far from the edge of the building does the bicyclist land? Use SI units in the second box.
The electron-domain geometry and molecular geometry of iodine trichloride are __________ and __________, respectively. The answer is trigonal bipyramidal, T-shaped, respectively- I do not understand the approach
58. ## chemistry
a container has the dimensions of 2.5 m x50 cm x60 mm. It is filled with water at 0.0 degrees celsius,how many kilocalories will be necessary to heat the water to boiling point?
59. ## Math
If I make a porch that is 60 feet in length, 50 feet wide and 4 inches deep, how do I calculate the area, perimeter, and surface area of the porch? Can I multiply feet and inches in the same equation?
70. ## Psychology
How do the behaviors that you listed relate to those reported by Sternberg and his colleagues? What behaviors did you list? What did Sternberg and his colleagues report? I searched Google under the key words "intelligence behavior Sternberg" to get these
71. ## solution making/ chemistry/biotech
You have 750ml's of a solution that was originally made up using a 20X stock. How much stock was used? I think the answer is 3.75 by dividing 750/20 but I am not sure
72. ## math
A train is going directly South at 60 mph. An action-movie star is atop the train, running North at 6 mph. Relative to someone standing on the ground, how fast is the action-movie star moving, and in what direction? North is positive and South is negative.
The electron-domain geometry and molecular geometry of iodine trichloride are __________ and __________, respectively.
82. ## Math
1) If you have an inhomogeneous, linear equation like f''(x) + f'(x) + f = x you can find the general solution by finding a particular solution (with no constants of integration)to this equation and adding it to the general solution to the homogeneous
83. ## History
Compare the European settlements in the Chesapeake area and New England in terms of goals, governments, social structures, and religions. What similarities do you find most remarkable? What differences do you find interesting? information please
84. ## Math -please check me
what is the perimeter of a rectangle if the length is 5/9 cm and the width is 2/9 cm? a. 5/9 cm b. 1 5/9cm c. 7/9 cm d. 1 7/9 cm the answer I got was 14/9 please help me.
85. ## Math
Please help Which of the following numbers are integers 2/3, 35, 0.093, -650, 42.5 Determine the maximum and minimum values of the following set of numbers 3. -36/7, -1, -2/3, 5/4, -5, 26/9
86. ## Accounting 11
What all goes on cash flow statement using an indirect method? How would it show on the sheet?
87. ## Accounting 11
I need help completing a statement of cash flow using the indirect method.
88. ## Psychology
List behaviors that you can think of that are characteristic of academic intelligence and behaviors that are characteristics of everyday intelligence. Please post your ideas and we'll be glad to help you. I searched Google under the key words "intelligence
1. ## writing
^^tf
posted on May 21, 2018
2. ## English
lmaooo hoe?
posted on May 21, 2018
3. ## Math
good lmao
posted on May 21, 2018
4. ## Plz help me.
I promise you the answers 1. D 2.D 3.B 4.A 100% 4/4
posted on March 21, 2018
5. ## Math
25
posted on October 7, 2017
6. ## History
Thanks
posted on May 25, 2017
7. ## Math
7×4=28 28+18=46
posted on December 14, 2016
8. ## grade 11 maths plz help!!!
The drama club plans to attend a professional production. From 10 to 35 students will go. There is a one-time handling fee of $3. each ticket costs$25 plus \$2 surcharges. Write a linear function that models this situation. (show your work)
posted on December 6, 2016
9. ## Math
5755050
posted on April 5, 2016
10. ## Criminal justice
Write a 700- to 1,050-word paper in which you describe the issues facing police departments in today’s society. Include a description of how local, state, and federal law enforcement agencies currently interact with the U.S. Department of Homeland
posted on July 29, 2015
11. ## Eth/125
a
posted on April 21, 2015
12. ## solution making/ chemistry/biotech
so how would I find the percentage of that solution
posted on October 2, 2014
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## Wednesday, December 22, 2010
### What math is and is not
Robert Lewis has a great essay on math, as the most misunderstood subject. I couldn't agree more!
In addition to an analogy with sports training similar to one I have here, I especially liked his parable of the hostile party goer. He is confronted by a man complaining that he was forced to memorize the quadratic formula, and yet has never had to use it. Lewis compares this to the absurd notion that we should complain to our first grade teachers that we have not once had to recall the details of the ever popular Dick and Jane books. Why would we need to spend so much time reading the books if the information contained within them can be so easily forgotten? Of course, the answer is that we used the books to learn to read well. Similarly, we practice (even memorize) math to learn to think well.
Lewis has captured exactly the problems most Americans have with mathematics and the ways they misunderstand the subject. But what's next? How can we correct this error for future generations? I suspect it all starts with us: college math professors, especially those of us who get to teach future K-12 teachers. We need to teach with an understanding of what math is, so they will have that understanding, so they in turn can teach in a way that their students appreciate what math is and what it is not. I am not calling for a change in K-12 math curriculum. That is too early to teach the nuances of math. Instead, the teachers must present the mathematics in a way which respects the subject and prepares students to uncover the hidden richness of mathematics as they mature.
## Tuesday, December 14, 2010
### Attendance Issues
At about the midpoint of this last semester, students here at Coastal had a day off (a fall "student holiday"). Not surprisingly, many student took the day before off as well. In my Math 139 classes, I had 15/40 and 11/37 attendance, respectively. It got me thinking about my policy on taking attendance.
I have long held that whether a student wants to come to class is entirely their business. They (or their parents) have paid for college, and if they want to (foolishly) squander their opportunities to learn the material, that is their choice. This would be a fine policy if I did not plan on reviewing material until everyone understands. But of course I do try to ensure that everyone learns the material and this is much more difficult when half the class is a day behind. So even if it appears that the absent students are only hurting themselves, it does effect me as well as the responsible students. Thus encouraging regular attendance is important.
I decided after that class to start sending around a sign-in sheet at the start of every class. I did not change my policy on attendance at all (although the syllabus says I have the option of failing a student who misses too many classes, attendance is not factored into their grade). Attendance shot up. Not to 100% or anything close to it, but considering the course, there was a marked improvement.
A couple of times students would ask if their grade would be effected because they had to miss an upcoming class. I told them not to worry about it. If anyone asked whether I was going to make attendance part of the final grade, I simply didn't answer. It was all done with a wink and a nod, and honestly, I don't think I fooled anyone into thinking that their grade would go down if they missed a class (other than the missed opportunity to learn the material, but that threat had been there from the beginning).
Today I got back my instructor evaluations. In one of my "ways to improve" comments I got:
Take attendance at beginning of semester; I need a reason to come.
Who knew? The point is, sending around an attendance sheet is almost no work for me, takes almost no time away from class, improves attendance, and is in fact appreciated by the students.
## Sunday, October 10, 2010
A little over half of the courses I have taught over the years have been "uniform" courses. These lower level math courses have multiple sections each semester, so the department has decided to appoint a course coordinator to oversee all the instructors. While there is some variation on how coordinated these courses are, usually it goes as far as common exams and group grading, although most instructors are allowed to write their own quizzes, group work (if any) and assign homework as they see fit. I understand the allure of running courses this way, and there are many arguments in favor of this approach. The problem is that not one of those arguments are for the benefit of the students. Sadly, the more uniform we make our classes, the worse they will be for our students.
Not every college professor is a great teacher. I think we have all run into professors who don't prepare classes well, don't write relevant exams, grade those exams unfairly, and fail to cover all the material the course is supposed to contain. Ideally, uniform courses would correct these problems. They do not. Classes can still be poorly prepared. Exam, unless the coordinator happens to be this bad professor, will be written well, but for students in the bad professor's class, they will not be relevant. Exams will be graded uniformly over all sections, but not uniformly over the exam itself: one page might have a reasonable partial credit policy, while the next be all or nothing. Worst of all, if this poor professor does not cover everything he or she is supposed to in class, then either those students will do badly on that part of the final exam (for not having seen the material ever) or all the sections will drop the material from the exam.
The more uniform the course, the more uniform the level of instruction. Unfortunately, this level drops to near that of the worst teacher in the group. Consider two examples, each occurring this last week. In a class with common exam, graded communally, all sections had to delay returning exams because one professor was too busy to get his pages graded. Another class, again will common exams, although not graded together, a professor decided to let some student take the exam late, resulting in a ban on returning (fully graded) exams to students in other sections. Now in this case, neither professor is the proverbial bad teacher of the previous paragraph: both just happened to find themselves in less than ideal situations. But because of the uniformity of the courses, this non-ideal situation spread to all the sections. Getting exam back quickly is not the most important thing in a college course, but it does benefit students to see their graded exams as soon as possible, as they will then be more likely to look over their mistakes and remember why they made them.
I understand that math departments need to have standards and that universities need to be assessed and accredited. We do not want some student coming out of calculus having just learned the basic derivative rules (without proof) and others getting a full course in real analysis. Some level of coordination is necessary. Here is how I would do it: everyone uses the same textbook, and everyone covers the same sections (with perhaps one or two optional sections, left up to individual instructors, as time permits). Everyone has the same number of exams, worth the same percentage of students' final grades. Exams are graded individually, but the coordinator sends out instructions on the level of partial credit to assign. If the university has a final exam time for all lower level math courses, then there is a common final. For all other exams, instructors are welcome to collaborate, and the coordinator can be the one to write these, but they do not need to do so. That way, if one instructor misses a day and needs to push back his or her exam, there will not be conflicts. All exams should be sent to the coordinator, for approval and record keeping. That is all.
## Wednesday, September 22, 2010
### Separating proofs in low level classes
The semester has gotten off to a busy start, thus the lack of postings. Anyway, this morning, I was thinking about a student who came to my office hours for some review on derivative rules. He had forgotten how to take the derivative of $3^x$. As he flipped through his notes, and eventually found the relevant page, he asked, "Oh right, is that where you sue the frog rule?" You see, I had joked with my class that I call the logarithm rule for exponents the frog rule because the exponent jumps over the log. Like a frog. This of course has nothing to do with the derivative rule for $a^x$, except that when we derived the rule, we needed to use some properties of logarithms (we took the derivative of both sides of $a^x = e^{ln(a^x)}$).
This got me thinking about a problem that I have noticed before. Students seem to have a hard time distinguishing between the proof and the result. This is not as much an issue in a higher level class, where proofs are common. By then students understand what we are doing. But in a first semester calculus class, many students are not aware of the important role of proofs in mathematics. So what to do?
I admit that part of the blame lies with the way I present the material. I try hard not to make my lecture into a list of theorems each followed by a proof. Doing so makes the class rather dry, I think. Plus, not everything we say in calculus comes with a detailed proof. For the derivative rules, I think it is much nicer to start out with the desire to discover the rule, and then derive it. That's what I did for $a^x$. But with students furiously copying down everything I write on the board, the distinction between reason and result can be lost.
Here is my idea: tell students not to take notes. I want them to see the reason something is true, and I want them to realize that finding this reason is important. But I don't want them to miss the punchline. In the end, a student who knows the derivative rules front and back is way better off than a student who has a book full of notes on how the rules were derived. I think that such a tactic might actually improve understanding of the derivation. If I tell my students that I just want to them to watch carefully what I am doing, then just maybe they will and see what is going on right then. When I get to the result, I would tell everyone that this is the thing they should right down and memorize.
I am sure that this would have improved my evaluations of trigonometry last spring as well. Many students mentioned that they did not like all the time spent going over why something was true, they just wanted the facts. I am not about to cut out proofs and derivations, but announcing ahead of time that it is not necessary to copy the whole thing down, might just do the trick.
## Friday, August 27, 2010
### Something is working
This fall I'm teaching Calculus I for the third time here at Coastal. Perhaps because I just taught it at the end of the summer, I am finding that it is working particularly well. I'm not entirely sure why. Student interaction is good, and I'm sure that is helping. But beyond that, the lectures seem to be flowing in a way they have not previously.
One possibility is the way I just happen to be presenting the material. I thought I'd take a moment and record a couple of examples here.
1. To introduce limits, I started by asking how we might graph $f(x) = \frac{sin(x)}{x}$. We had previously talked about the graph of $f(x) = \frac{1}{x}$, and what happens at $x = 0$. We said that while you cannot plug in 0, you can ask what happens near 0. So we tried that again. What happens to $f(x)$ as $x$ gets closer and closer to 0. We made a table, everyone agreed that the $y$-values were getting closer to 1. Only then did I start using the language "limit." I gave them the notation as a way to quickly write down what we just did. This seemed much more natural than giving them a definition of a limit (out of thin air) and then showing them a bunch of examples.
2. For left/right-hand limits, I only introduced them after we discovered the problem in finding limits if the $y$-values approach different values from each side. I gave them a graph and asked them to find limits in a variety of cases. We all agreed that at the jump discontinuity, the limit did not exist. But of course, we can be more descriptive than this. Coming from one side, the limit does exist. Then I introduced that notation.
3. For continuity, I drew two graphs, one continuous, and one with a jump discontinuity. I asked the class what was different about the graphs. We agreed that the continuous one was continuous and the other not. I asked what other ways a graph might not be continuous. We came up with a graph with a removable discontinuity and an infinite discontinuity. Now, how might we say something about these in terms of limits?
4. Continuing with continuity, to introduce the difference between continuous at a point vs interval, I covered up the discontinuity and asked if the rest of the graph was continuous. We agreed it was, so decided that we needed to express continuity at specific points. This led us to the limit definition of continuity. This was so much more natural than giving the definition, and then figuring out what it means.
That's all we have done so far. I'm not sure if this luck will continue. The common theme seems to be that instead of giving the definition and trying to apply it (which of course has it's place and value), I'm asking them to solve problems, then coming up with the math-way of saying what we are doing.
I don't know if this is helping the students (we have not had any exams yet) but it definitely feels better to me. I plan to make an effort to continue using this "technique" for the rest of the class and see how it goes.
## Friday, August 13, 2010
### The problem = equal signs
New research out of Texas A&M suggests that 70% of middle school students don't understand what the "=" sign means. The article can be found here. The key point, as pointed out in the article:
The problem is students memorize procedures without fully understanding the mathematics.
It is interesting that the problem runs this deep. My calculus students have trouble solving velocity problems because they try to memorize the procedure instead of understanding the problem. Of course, when they forget one step, they are completely lost. This study suggests that even from the first time students are exposed to math, they are only taught the steps for solving math problems.
I am not suggesting that we move to a "new math" style of instruction, where answers don't matter, and all the emphasis is on concepts. However, if students really have this little understanding of even the simplest of math concepts, we are really missing the point of teaching math entirely.
## Monday, August 2, 2010
### Thinking vs Doing
Word problems are hard for students. I've never really understood why before. You read the problem, figure out what it is asking, figure out how to answer the question, and do it. Usually the mathematics part is not that hard. So why to students have so much trouble?
One of the reasons might be that many students have never been shown how to think about the problem. So much of mathematics is about doing. Here are the steps you need to follow to solve an equation. Here is the process to find the equation of the tangent line. When you want to find the absolute maximum of the function, first... then... and finally... etc. Many problems in mathematics are complex and require multiple steps. When we teach students how to solve these problems, it is very tempting to teach them the algorithm for arriving at the solution. No doubt there is value in learning how to follow a recipe, but as a method for solving a problem, it can be perilous. If we simply follow the steps, it is all to easy to miss one without realizing it, resulting in a wrong, or even meaningless answer.
As professional mathematicians, this is rarely a problem, since while applying the problem solving algorithm, we realize why we are performing each step. If we miss a step, we will notice immediately (most of the time). But for a student who is first encountering a type of problem, this is near impossible. They are worried about following the steps, not why the steps matter. If they would just stop to think about what they are doing, they would realize that something has gone wrong.
Consider this example: in calculus, we ask our students some simple velocity problems. We give them a formula for the height of an object at time t, ask them to find a formula for velocity, then ask them,
1. What is the maximum height of the object?
2. How fast is the object going when it hits the ground?
This semester, I asked how fast the object was traveling when it was 35 feet above the ground. A student promptly plugged in 35 for t in the velocity formula. Why would she do such a thing? This particular student is usually very bright, and definitely hard working. I suspect that her thought process went something like this. "Okay, one of these problems. I need to plug something into an equation. Here is a number. Here is equation. Here we go..."
I pointed this out to my class when talking about the problem the next day, and almost all of them agreed that at least one of their math teachers in the past taught them how to do word problems in this way. To solve the word problem you need to find the equation, and plug in the number, and evaluate. Quick, don't think about it, just follow the directions. Now move along. We cannot blame our students for this poor mathematics upbringing. I show my students too many algorithms -- it is a fast and easy way to get them to find the right answer, before they forget how the algorithm goes. In fact, last time I taught this topic, I think I pointed out to my students that questions 1 and 2 above are similar in that you set one of the equations equal to zero, solve for t, then plug that value into the other equation. Now reflecting back on it, I feel embarrassed to have fallen into the trap.
I am not entirely sure how to fix this problem. Certainly it is important to, whenever possible, challenge our students to think. Give them time to struggle with a problem. Let them figure out the solution without giving them a recipe. Through practice, the students will come up with their own shortcuts. This takes a lot of time, and can be frustrating for all parties involved. But giving them yet another algorithm will only make the problem worse.
I am reminded of a certain proverb about fish and fishing...
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Shift line connecting anchors
How can I shift a line the endpoints of which are defined by anchors like this?
\draw (i0) -- (i1);
The following approach does not work:
\draw[xshift=2pt] (i0) -- (i1);
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The question are ambiguous. You write anchors like anchors of node (tikz's notion) or you use another definition. I would like to know if anchors are coordinate \coordinate (i0) at (A.north west); or if anchors are nodes like in Marco's answer because the answer is not the same – Alain Matthes Jan 2 '12 at 14:39
Approach 1
You can work with the library calc:
\draw[green] (i0) -- ($(i1)+(1,2)$);
Approach 2
Another approach based on the let operation. An example is given in the question TikZ: Node at same x-coordinate as another node, but specified y-coordinate?
\draw[blue] let \p1 = (i0) in (2,\y1) -- (i1);
Approach 3
Jake mentioned another approach. You set the shift to the coordinate directly. (this method is documented in the manual section 13 "Specifying Coordinates")
\draw ([xshift=2pt]i0) -- ([xshift=2pt]i1);
Note: By using this method will work fine if you define i0 and i1 with \coordinate. If you define i0 and i1 with \node you must give an anchor
\draw ([xshift=2pt]i0.center) -- ([xshift=2pt]i1.center);
This limitation isn't relevant for the other approaches.
Complete example with result
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\verb+Calc+
\begin{tikzpicture}
\node at (0,0) (i0){};
\node at (2,2) (i1){};
\draw[red] (i0) -- (i1);
\draw[blue] ($(i0)+(2,0)$) -- (i1);
\draw[green] (i0) -- ($(i1)+(1,2)$);
\end{tikzpicture}
\verb+let+
\begin{tikzpicture}
\node at (0,0) (i0){};
\node at (2,2) (i1){};
\draw[red] (i0) -- (i1);
\draw[blue] let \p1 = (i0) in (2,\y1) -- (i1);
\draw[green] let \p1 = (i1) in (i0) -- (\x1,4);
\draw[black] let \p0 = (i0), \p1=(i1) in (\x0,2) -- (\x1,3);
\end{tikzpicture}
\verb+shift+ 1
\begin{tikzpicture}
\node at (0,0) (i0){};
\node at (2,2) (i1){};
\draw[red] (i0) -- (i1);
\draw[blue]([xshift=2cm]i0.center) -- (i1);
\draw[green] (i0) -- ([yshift=2cm,xshift=1cm]i1.center);
\end{tikzpicture}
\verb+shift+ 2
\begin{tikzpicture}
\coordinate (i0) at (0,0) ;
\coordinate (i1) at (2,2) ;
\draw[red] (i0) -- (i1);
\draw[blue]([xshift=2cm]i0) -- (i1);
\draw[green] (i0) -- ([yshift=2cm,xshift=1cm]i1);
\end{tikzpicture}
\end{document}
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A different approach would be to add the transformation to the coordinate specification (so \draw ([xshift=2pt]i1) -- ([xshift=2pt]i2);). Maybe you could flesh out your answer to show examples for all three approaches? – Jake Jan 2 '12 at 11:38
I just tried the experiment and found that the syntax ([shift={(1,2)}]A) only works for positioning nodes. Strangely, ([shift={(1,2)}]0,0) works for arbitrary paths. I have no idea why this would be so, it seems rather illogical to me. – Andrew Stacey Jan 2 '12 at 12:18
Ah, I see that the answer has been edited and that it works if an explicit anchor is given. This could be a bug in how the point-on-the-perimeter is computed for a translated node. Or one could say that translating a node coordinate without a specific anchor is open to different interpretations on which anchor should be used, so translating without an anchor should not be used. – Andrew Stacey Jan 2 '12 at 12:22
@Altermundus and Marco Daniel: "anchors" are specified points on a shape, like north east, base west, 135, and so on. If you don't supply an anchor to a coordinate specification, but only a node name (like i0), an anchor on the edge the node that lies in the direction of the path is used. Only if i0 happens to be a node of the coordinate shape (created using \coordinate ... or \node [coordinate] ...), this is equivalent to the center anchor. So the original question is indeed ambiguous, as Altermundus noted. I guess it's another example of why MWEs are useful... – Jake Jan 3 '12 at 9:33
(ctd) until they see the possible solutions, whereupon they will decide which behaviour they want and choose accordingly. – Andrew Stacey Jan 3 '12 at 9:33
For me the shifting line is like the blue one
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}
\draw [help lines] (0,0) grid (3,2);
\node[draw](i0) at (0,0) {};
\node[draw](i1) at (2,2) {};
\draw[red] (i0) -- (i1) coordinate[pos=0] (j0) coordinate[pos=1] (j1);
\draw[thick,blue] ([xshift=1cm]j0) -- ([xshift=1cm]j1);
\draw[thick,purple] let \p0=(i0),\p1=(i1) in
([xshift=.5cm]\x0,\y0) -- ([xshift=.5cm]\x1,\y1);
\end{tikzpicture}
\end{document}
-
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# Mathematics 101
Mathematics is the study of topics such as quantity (numbers), structure, space, and change.
## Content
• Logic
The concept of logical form is central to logic. The validity of an argument is determined by its logical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic.
• Informal logic is the study of natural language arguments. The study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, informal logic is not logic at all. See 'Rival conceptions', below.
• Formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of Aristotle contain the earliest known formal study of logic. Modern formal logic follows and expands on Aristotle.[3] In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language.
• Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference.[4][5] Symbolic logic is often divided into two main branches: propositional logic and predicate logic.
• Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theoryproof theoryset theory, and recursion theory.
However, agreement on what logic is has remained elusive, and although the field of universal logic has studied the common structure of logics, in 2007 Mossakowski et al. commented that "it is embarrassing that there is no widely acceptable formal definition of 'a logic'"
• How strong is your foundational knowledge?
Applied mathematics is the application of mathematical methods by different fields such as scienceengineeringbusinesscomputer science, and industry. Thus, applied mathematics is a combination of mathematical scienceand specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.
Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values.[4] For example, in ${\displaystyle x+2=5}$ the letter ${\displaystyle x}$ is unknown, but the law of inverses can be used to discover its value: ${\displaystyle x=3}$. In E = mc2, the letters ${\displaystyle E}$ and ${\displaystyle m}$ are variables, and the letter ${\displaystyle c}$ is a constant, the speed of light in a vacuum. Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words.
The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology.
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves). These two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread uses in scienceengineering, and economics
• Final Exam
0
0 Reviews
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# Vitali Sets vs Bernstein Sets…
AC is enough to guarantee the existence of both Bernstein Sets and Vitali Sets...
However is the existence of Vitali Sets strictly weaker than that of Bernstein Sets?
What about the other way round?
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Are these your sets? Vitali set: one element chosen from each coset of $\mathbb R$ modulo $\mathbb Q$. Bernstein set: a set $E \subseteq \mathbb R$ such that both $E$ and its complement meet each uncountable closed set. – Gerald Edgar Jul 29 '11 at 14:48
A bit more general: Vitali Set: one element chosen from each coset of $\mathbb{R}$ modulo a countable dense subgroup of $\mathbb{R}$, Bernstein Set: a set $E \subset \mathbb{R}$ such that both $E$ and its complement are totally imperfect in $\mathbb{R}$. – George Lazou Jul 29 '11 at 17:39
## 1 Answer
For your second definition of Vitali set, I have a weak partial answer. Namely the existence of a Bernstein set does not imply the existence of a $T$-Vitali set. The answer can be found in logic blog maintained by Andre Nies:
http://dl.dropbox.com/u/370127/Blog/Blog2012.pdf
Note that a Turing degree does not need to be an addition group.
I don't know whether the existence of a Vitali set implies the existence of a Bernstein set. But it is not difficult to see, under $ZF+DC$, that there is a Vitali set (if it exists) which contains a perfect subset.
For you first definition of Vitali set, I have no idea.
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Very nice! The specific version proved in the link is: In ZF, the existence of Bernstein sets does not imply the existence of $T$-Vitali sets, where $A$ is $T$-Vitali iff $|A\cap{\mathbf x}|=1$ for each Turing degree ${\mathbf x}$. This follows from Wei Wang, Liuzhen Wu, and Liang Yu, "Cofinal maximal chains in the Turing degrees", Proc. Amer. Math. Soc., to appear, where it is proved that the following is consistent: ZF+DC + there is an $\omega_1$-cofinal chain of Turing degrees + there is no well-ordering of the reals. – Andres Caicedo May 20 '12 at 2:59
Andres, thanks for making my answer clearer. I was surprised George's question seems unknown. I don't even know whether there is a $ZF+DC$ model in which there is no well ordering of reals but there is a Vitali set (in the George's first version sense). – Liang Yu May 20 '12 at 5:51
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### Overview
By the end of this page you will be able to: Understand how a radio button is created
<html><body>
<script type="text/javascript">
<!--
function fnCheck() {
}
//-->
</script>
<form name="frmChoose">
<p>Select an operation:</p>
</body></html>
|
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
I spotted on R-bloggers a post discussing optimising the efficiency of programming accept-reject algorithms. While it is about SAS programming, and apparently supported by the SAS company, there are two interesting features with this discussion. The first one is about avoiding the dreaded loop in accept-reject algorithms. For instance, taking the case of the truncated-at-one Poisson distribution, the code
rtpois=function(n,lambda){
sampl=c()
while (length(sampl)<n){
x=rpois(1,lambda)
if (x!=0) sampl=c(sampl,x)}
return(sampl)
}
is favoured by my R course students but highly inefficient:
> system.time(rtpois(10^5,.5))
user system elapsed
61.600 27.781 98.727
both for the stepwise increase in the size of the vector and for the loop. For instance, defining the vector sampl first requires a tenth of the above time (note the switch from 10⁵ to 10⁶):
> system.time(rtpois(10^6,.5))
user system elapsed
54.155 0.200 62.635
As discussed by the author of the post, a more efficient programming should aim at avoiding the loop by predicting the number of proposals necessary to accept a given number of values. Since the bound M used in accept-reject algorithms is also the expected number of attempts for one acceptance, one should start with something around Mn proposed values. (Assuming of course all densities are properly normalised.) For instance, in the case of the truncated-at-one Poisson distribution based on proposals from the regular Poisson, the bound is 1/1-e. A first implementation of this principle is to build the sample via a few loops:
rtpois=function(n,lambda){
propal=rpois(ceiling(n/(1-exp(-lambda))),lambda)
propal=propal[propal>0]
n0=length(propal)
if (n0>=n)
return(propal[1:n])
else return(c(propal,rtpois(n-n0,lambda)))
}
with a higher efficiency:
> system.time(rtpois(10^6,.5))
user system elapsed
0.816 0.092 0.928
Replacing the expectation with an upper bound using the variance of the negative binomial distribution does not make a significant dent in the computing time
rtpois=function(n,lambda){
M=1/(1-exp(-lambda))
propal=rpois(ceiling(M*(n+2*sqrt(n/M)/(M-1))),lambda)
propal=propal[propal>0]
n0=length(propal)
if (n0>=n)
return(propal[1:n])
else return(c(propal,rtpois(n-n0,lambda)))}
since we get
> system.time(rtpois(10^6,.5))
user system elapsed
0.824 0.048 0.877
The second point about this Poisson example is that simulating a distribution with a restricted support using another distribution with a larger support is quite inefficient. Especially when λ goes to zero By comparison, using a Poisson proposal with parameter μ and translating it by 1 may bring a considerable improvement: without getting into the gory details, it can be shown that the optimal value of μ (in terms of maximal acceptance probability) is λ and that the corresponding probability of acceptance is
$\dfrac{1-\exp\{-\lambda\}}{\lambda}$
which is larger than the probability of the original approach when λ is less than one. As shown by the graph below, this allows for a lower bound in the probability of acceptance that remains tolerable.
Filed under: R, Statistics, University life Tagged: accept-reject algorithm, loops, Poisson distribution, R, R course, SAS, system.time
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# third line of lyman series
The following rearrangement of N-allyl-N, N-dimethylanilinium ion has been observed. Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation. 30 - Integrated Concepts The electric and magnetic... Ch. (b) Students in a college computer science class... 9. 30 - Show that the entire Paschen series is in the... Ch. All the wavelength of Lyman series falls in Ultraviolet band. 30 - (a) Using the Pauli exclusion principle and the... Ch. See the answer. 30 - (a) What is the minimum value of 1 for a subshell... Ch. The four other spectral line series, in addition to the Balmer series, are named after their discoverers, Theodore Lyman, A.H. Pfund, and F.S. NIST Atomic Spectra Database (ver. 30 - Integrated Concepts Estimate the density of a... Ch. (i) In hydrogen atom, an electron undergoes transition from 2nd excited state to the first excited state and then to the ground state. 30 - Lasers are used to burn and read CDs. n2=5,6,7,….. Pfund n1=5 , n2=6,7,8,….. 30 - Figure 30.39 shows the energy-level diagram for... Ch. As you notice and the values are decreasing in the series due to the derivation and their state which is Ultra Violet. 30 - What is the difference between fluorescence and... Ch. 30 - Find the wavelength of the third line in the Lyman... Ch. 30 - CT scanners do not detect details smaller than... Ch. 30 - For a given value of l, what are the allowed... Ch. 30 - (a) What is the distance between the slits of a... Ch. Identify The Type Of Electromagnetic Radiation. Example $$\PageIndex{1}$$: The Lyman Series. 30 - Integrated Concepts (a) An excimer laser used for... Ch. Here is an illustration of the first series of hydrogen emission lines: Historically, explaining the nature of the hydrogen spectrum was a considerable problem in physic… 30 - If atoms exist, why can't we see them with visible... Ch. 30 - Integrated Concepts In a laboratory experiment... Ch. The greater the difference in the principal quantum numbers, the higher the energy of the electromagnetic emission. (Indicates a review question, which means it requires only a basic understanding of the material to answer. Kramida, A., Ralchenko, Yu., Reader, J., and NIST ASD Team (2019). Lyman-alpha radiation had previously been detected from other galaxies, but due to interference from the Sun, the radiation from the Milky Way was not detectable. In aerobic cellular respiration, which reactions occur only in the cytoplasm? In 1914, when Niels Bohr produced his Bohr model theory, the reason why hydrogen spectral lines fit Rydberg's formula was explained. 30 - What, if any, constraints does a value of ml=1... Ch. 30 - (a) How many angles can L make with the z-axis for... Ch. In physics and chemistry, the Lyman series is a hydrogen spectral series of transitions and resulting ultraviolet emission lines of the hydrogen atom as an electron goes from n ≥ 2 to n = 1 (where n is the principal quantum number), the lowest energy level of the electron. 30 - Verify that the ground state energy E0 is 13.6 eV... Ch. The series of lines in an emission spectrum caused by electrons falling from energy level 2 or higher (n=2 or more) back down to energy level 1 (n=1) is called the Lyman series. 30 - Integrated Concepts A neighboring galaxy rotates... Ch. Chapter. 30 - Construct Your Own Problem The solar corona is so... Ch. Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation. where: Planck's constant is 6.63 x 10-34 ) xs speed of light is 3 x 108 m/s rydberg constant is 109677 60 cm-1 Write the final answer. Que... Express the location of the fly in Problem 40 in polar coordinates. 30 - Ruby lasers have chromium atoms doped in an... Ch. Rydberg managed to find a formula to match the known Balmer series emission lines, and also predicted those not yet discovered. Try this, The Lyman Series say the for the second is 121.6nm (nano metres) For the third it is 102.6 and the fourth is 97.3 all in Nano Metres which *10^-9. 30 - Integrated Concepts Calculate the velocity of a... Ch. 30 - If an atom has an electron in the n=5 state with... Ch. which is Rydberg's formula for the Lyman series. Explanation: New questions in Chemistry. Within five years Johannes Rydberg came up with an empirical formula that solved the problem, presented first in 1888 and in final form in 1890. 30 - How do the allowed orbits for electrons in atoms... Ch. 30 - Integrated Concepts A pulsar is a rapidly spinning... Ch. For the connection between Bohr, Rydberg, and Lyman, one must replace m with 1 to obtain. Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation. 30 - If a hydrogen atom has its electron in the n=4... Ch. The rest of the lines of the spectrum were discovered by Lyman from 1906-1914. Adequate Intakes (AI) b. Paul Peter Urone + 1 other. The so-called Lyman series of lines in the emission spectrum of hydrogen corresponds to transitions from various excited states to the n = 1 orbit. (a) What is the average mass of one copper atom? Solution 8PE= 97.234 nm The spectral lines are grouped into series according to n′. Nobody could predict the wavelengths of the hydrogen lines until 1885 when the Balmer formula gave an empirical formula for the visible hydrogen spectrum. 30 - Integrated Concepts What double-slit separation... Ch. 30 - The coating on the inside of fluorescent light... Ch. Paul Peter Urone + 1 other. The H α spectral line in Lyman series of hydrogen atomic spectrum is formed due to an electronic transition in hydrogen atom. *Response times vary by subject and question complexity. 30 - Unreasonable Results (a) What voltage must be... Ch. 30 - Integrated Concepts Prove that the velocity of... Ch. View a sample solution. Question 10. This problem has been solved! Question: Calculate The Energy And Wavelength (in Nm) Of The Photon That Gives Rise To The Third Line In Order Of Increasing Energy In The Lyman Series In The Emission Spectrum Of Hydrogen. Electrons are falling to the 1-level to produce lines in the Lyman series. 30 - (a) What energy photons can pump chromium atoms in... Ch. 121.6 \text{nm} 1/lambda = \text{R}(1/(n_1)^2 - 1/(n_2)^2) * \text{Z}^2 where, R = Rydbergs constant (Also written is \text{R}_\text{H}) Z = atomic number Since the question is asking for 1^(st) line of Lyman series therefore n_1 = 1 n_2 = 2 since the electron is de-exited from 1(\text{st}) exited state (i.e \text{n} = 2) to ground state (i.e text{n} = 1) for first line of Lyman series. Energy cannot ch... 9.92 The curing of concrete liberates energy as heat. a. ISBN: 9781938168000. Explain why... Ch. The nutrient standards in use today include all of the following except. College Physics. 30 - Identify the shell, subshell, and number of... Ch. A football is thrown along an arc. A wavelength of second line of lyman series for H atom is X then wavelength of third line paschen series for the Li2+ ? The series is named after its discoverer, Theodore Lyman. Lines are named sequentially starting from the longest wavelength/lowest frequency of the series, using Greek letters within each series. The first line in the ultraviolet spectrum of the Lyman series was discovered in 1906 by Harvard physicist Theodore Lyman, who was studying the ultraviolet spectrum of electrically excited hydrogen gas. 30 - Integrated Concepts Particles called muons exist... Ch. ISBN: 9781938168000. 30 - The wavelength of the four Balmer series lines for... Ch. 5.7.1), [Online]. 30 - Integrated Concepts A galaxy moving away from the... Ch. The spectrum of radiation emitted by hydrogen is non-continuous or discrete. S... How can some supernova remnants emit X-rays? There is also a more comfortable notation when dealing with energy in units of electronvolts and wavelengths in units of angstroms. 30 - Why does the energy of characteristic x rays... Ch. Brackett of the United States and Friedrich Paschen of Germany. So a value like 3 PHz means 3 x 10 15 Hz. The phrase "5 prime to 3 prime" refers to the ______. 30 - A singly ionized helium ion has only one electron... Ch. The lyman series of hydrogen is made up of the transitions of electrons from higher energy levels n>1, to the n=1 energy level. Lyman n1= 1 ,n2=2 ,3,4,5,6,…. 30 - Atoms can be ionized by thermal collisions, such... Ch. Available: Learn how and when to remove this template message, "Voyager Probes Detect "invisible" Milky Way Glow", https://en.wikipedia.org/w/index.php?title=Lyman_series&oldid=989201198, Articles needing additional references from August 2015, All articles needing additional references, Creative Commons Attribution-ShareAlike License, This page was last edited on 17 November 2020, at 16:57. 30 - Atomic and molecular spectra are discrete. Median response time is 34 minutes and may be longer for new subjects. α line of Lyman series p = 1 and n = 2; α line of Lyman series p = 1 and n = 3; γ line of Lyman series p = 1 and n = 4; the longest line of Lyman series p = 1 and n = 2; the shortest line of Lyman series p = 1 and n = ∞ A 0.456-g sample of the mixture is dissolved in water, and an excess... ____ A number of factors influence diet during pregnancy beyond food availability. 30 - Integrated Concepts Find the value of l, the... Ch. swatantraverma7687 swatantraverma7687 28.11.2018 Chemistry Secondary School +5 pts. Corresponding Textbook College Physics | 0th Edition. 30 - (a) Which line in the Balmer series is the first... Ch. Which organisms are autotrophs? C. 4-6 ... Bacteria and other prokaryotes are found deep underground, in water, and in the air. 30 - Observers at a safe distance from atmospheric test... Ch. Join now. Ask your question. Where n is a natural number greater than or equal to 2 (i.e., n = 2, 3, 4, …). Ch. Draw the structural formula and condensed structural formula for pentane, and identify the methyl groups and... Why evidence can you cite that the interstellar medium contains both gas and dust? Experts are waiting 24/7 to provide step-by-step solutions in as fast as 30 minutes!*. Therefore, each wavelength of the emission lines corresponds to an electron dropping from a certain energy level (greater than 1) to the first energy level. Als Balmer-Serie wird eine bestimmte Folge von Emissions-Spektrallinien im sichtbaren elektromagnetischen Spektrum des Wasserstoffatoms bezeichnet, deren unteres Energieniveau in der L-Schale liegt. 30 - How is the de Broglie wavelength of electrons... Ch. The first line is 3→ 2, second line is 4 →2 and third line is 5→ 2. A wavelength of second line of lyman series for H atom is X then wavelength of third line paschen series for the Li2+ ? State... Ch. For the Balmer series, n 1 is always 2, because electrons are falling to the 2-level. a. timing of DNA replication b. directionality of DNA syn... What type of reaction occurs in each of the following steps in a turn of the -oxidation pathway? If the first line in this series has a wavelength of 122 nm, what is the wavelength of the third line? 30 - (a) An aspiring physicist wants to build a scale... Ch. Expert Answer . a. Bohr found that the electron bound to the hydrogen atom must have quantized energy levels described by the following formula, According to Bohr's third assumption, whenever an electron falls from an initial energy level Ei to a final energy level Ef, the atom must emit radiation with a wavelength of. 30 - (a) Calculate the magnitude of the angular... Ch. Which of the following statements about the structural characteristics of common plasma lipoproteins is incorre... 16. 30 - Define the quantum numbers n,l,ml,s, and ms. Ch. *Response times vary by subject and question complexity. Weitere Serien sind die Balmer-Serie (vgl. Calculate the wavelength of the lowest-energy line in the Lyman series to three significant figures. This problem has been solved! [1], The version of the Rydberg formula that generated the Lyman series was:[2]. 30 - What are the approximate energies of the K and K... Ch. 30 - Hydrogen gas can only absorb EM radiation that has... Ch. 30 - List all the possible values of s and msfor an... Ch. Answer to: Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation. c) -78.4, -34.84 . 30 - (a) If one subshell of an atom has 9 electrons in... Ch. Therefore, the lines seen in the image above are the wavelengths corresponding to n = 2 on the right, to n = ∞ on the left. 7 6 4 Brackett series 3 Paschen series Energy Balmer Different lines of Lyman series are . Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation. The transitions are named sequentially by Greek letters: from n = 2 to n = 1 is called Lyman-alpha, 3 to 1 is Lyman-beta, 4 to 1 is Lyman-gamma, and so on. Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation. Replacing the energy in the above formula with the expression for the energy in the hydrogen atom where the initial energy corresponds to energy level n and the final energy corresponds to energy level m. Where RH is the same Rydberg constant for hydrogen from Rydberg's long known formula. For example, in the Lyman series, n 1 is always 1. 0-2 mm Hg. Energy cannot be created or destroyed. Rewrite above formula, Comment(0) Chapter , Problem is solved. Calculate the wavelength in nm for the third line in Lyman series? Buy Find arrow_forward. vinegar content which acid write an activity showing electrolysis of water and electroplating of irom key with copper , using a well labled diagram.plz help and don't spam plz Class-s-classmateDatePageQ. Different lines of Lyman series are . Median response time is 34 minutes and may be longer for new subjects. 30 - Do the Balmer and Lyman series overlap? 30 - Crystal lattices can be examined with x rays but... Ch. In these cells, 2n = 4. a. anaphase of meiosis I ... RECALL What is a replication fork? Balmer n1=2 , n2=3,4,5,…. This also means that the inverse of the Rydberg constant is equal to the Lyman limit. Sie wird beim Übergang eines Elektrons von einem höheren zum zweittiefsten Energieniveau = emittiert.. Weitere Serien sind die Lyman-, Paschen-, Brackett-, Pfund-und die Humphreys-Serie H e + ions are known to have the wavelength difference between first (of the longest wavelength) lines of Balmer and Lyman series equal to 1 3 3. 30 - Integrated Concepts Suppose an MRI scanner uses... Ch. The first line in the spectrum of the Lyman series was discovered in 1906 by Harvard physicist Theodore Lyman, who was studying the ultraviolet spectrum of electrically excited hydrogen gas. a. mouth b. stomach c. small intestine d. lar... What could be done to minimize the risks associated with nutrient-drug interactions? 30 - Some of the most powerful lasers are based on the... Ch. The ratio of difference in wavelengths of 1st and 2nd lines of Lyman series in H-like atom to difference in wavelength for 2nd and 3rd lines of same series is: A 2.5:1 30 - Integrated Concepts A carbon dioxide laser used in... Ch. Step 1 b. b) -313.6, -78.4 . When an electron comes down from higher energy level to second energy level, then Balmer series of the spectrum is obtained. Join now. To find the wavelength of third line of Lyman series, use the following formula, Here, is wavelength, Rydberg constant, and is the energy level, here n is non-negative integer. Express each of the following numbers in scientific notation: Imagine tha... A mixture contains only NaCl and Fe(NO3)3. The Lyman series is a series of lines in the ultra-violet. 30 - Which of the following spectroscopic notations are... Ch. The ratio of difference in wavelength of 1st and 2nd lines of lyman series in H-like atom to difference in wavelength for 2nd and 3rd line of same series - 6854… 1. The ratio of the frequency corresponding to the third line in Lyman series of H atomic spectrum to that of the first line in Balmer series of Li^2+ spectrum is 1 4/5 2 5/4 3 4/3 4 3/4 5 3/8 - Chemistry - To answer... Ch. Notice that the lines get closer and closer together as the frequency increases. a. Explanation: New questions in Chemistry. Show transcribed image text. 30 - By calculating its wavelength, show that the first... Ch. Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation. In quantum mechanics: Bohr’s theory of the atom. Expert Answer 100% (14 ratings) Previous question Next question Transcribed Image Text from this Question. The energies associated with the electron in each of the orbits involved in the transition (in kCal mol-1) are: (Eamcet - 2008-E) a) -313.6, –34.84 . The wavelengths in the hydrogen spectrum with m=1 form a series of spectral lines called the Lyman series. Bohr found that the electron bound to the hydrogen atom must have quantized energy levels described by the following formula, This formula gives a wavelength of lines in the Lyman series of the hydrogen spectrum. The rest of the lines of the spectrum (all in the ultraviolet) were discovered by Lyman from 1906-1914. Publisher: OpenStax College. 30 - How can you tell that a hologram is a true... Ch. Log in. Solutions. Daily Mini... Copper atoms. d) -78.4, -19.6. In what region of the electromagnetic spectrum does it occur? The Lyman series lies in the ultraviolet, whereas the Paschen, Brackett, and Pfund series lie in the infrared. 30 - Look up the values of the quantities in... Ch. 30 - Using the given charge-to-mass ratios for... Ch. The fact that the nitrogen atom in ammonia is sp3 hybridized even though it makes only three bonds to three hyd... Use the law of conservation of mass to determine which numbered box(es) represent(s) the product mixture after ... Write a reaction equation that corresponds to each of the following equilibrium expressions: a. K=[PH3][F2]3[HF... Breastfeeding and Adequate Nourishment Molly G. is a 24-year-old office manager and part-time aerobics instruct... At rest, the normal pleural pressure change during quiet breathing is about A. Identify the spectral series to which these transitions belong. Show transcribed image text. The series of lines in an emission spectrum caused by electrons falling from energy level 2 or higher (n=2 or more) back down to energy level 1 (n=1) is called the Lyman series. 30 - Verify Equations rn=n2ZaB and... Ch. In what ways are computer-generated maps and landscape images helpful in studying physical geography? 30 - Show that (13.6eV)/hc=1.097107m=R (Rydberg's... Ch. 30 - Which of the following are not allowed? 30 - Explain why characteristic x rays are the most... Ch. 30 - An x ray tube has an applied voltage of 100 kV.... Ch. Buy Find arrow_forward. Which of the following statements is not correct? Propose a mechanism. Calculate The Wavelength Of The Third Line In The Lyman Series, In Nanometers. The Lyman series lies in the ultraviolet, whereas the Paschen, Brackett, and Pfund series lie … There are infinitely many spectral lines, but they become very dense as they approach n = ∞ (the Lyman limit), so only some of the first lines and the last one appear. 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Hope It Helped. The so-called Lyman series of lines in the emission spectrum of hydrogen corresponds to transitions from various excited states to the n = 1 orbit. What is the... Ch. 30 - A wavelength of 4.653 m is observed in a hydrogen... Ch. A skill you need for writing net ionic equations successfully is the ability to identify the major species in a... What problems confront the inhabitants of the intertidal zone? 30 - Integrated Concepts In a Millikan oil-drop... Ch. For example, the 2 → 1 line is called "Lyman-alpha" (Ly-α), while the 7 → 3 line is called "Paschen-delta” (Pa-δ). The formula for wavelength of Lyman series is written as λ 1 = R [ 1 − n 2 1 ] For first, second and third lines of Lyman series n = 2 , 3 , 4 respectively. See the answer. 30 - Rutherford found the size of the nucleus to be... Ch. Calculate the wavelength in nm for the third line in Lyman series? 30 - Explain why patterns observed in the periodic... Ch. View a full sample . 30 - What is a hydrogen-like atom, and how are the... Ch. The spectrum of radiation emitted by hydrogen is non-continuous. Calculate the wavelength of the lowest-energy line in the Lyman series to three significant figures. Back to top. The Third Line in Balmer series correspondence to an electronic transition between which Bohr's orbits in hydrogen: MEDIUM. 1st Edition. where: Planck's constant is 6.63 x 10-34 ) xs speed of light is 3 x 108 m/s rydberg constant is 109677 60 cm-1 Write the final answer. Their formulas are similar to Balmer’s except that the constant term is the reciprocal of the square of 1, 3, 4, or 5, instead of 2, and the running number n begins at … Paschen n1=3 , n2=4,5,6,…… Brackett n1=4. …for m = 1, the Lyman series, lie in the ultraviolet part of the spectrum; those for m = 2, the Balmer series, lie in the visible spectrum; and those for m = 3, the Paschen series, lie in the infrared. 30 - (a) What is the shortest-wavelength x-ray... Ch. Consider the Lyman series of electron transitions in hydrogen. Why is it important in replication? A cat eats a bird, which ate a caterpillar that chewed on a weed. For example, the 2 → 1 line is called "Lyman-alpha" (Ly-α), while the 7 → 3 line is called "Paschen-delta” (Pa-δ). 30 - (a) What is the magnitude of the angular momentum... Ch. 30 - What is the smallest-wavelength line in the Balmer... Ch. 30 - What are the possible values of m1 for an electron... Ch. Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation. 30 - Name three different types of evidence for the... Ch. Nm. 30 - Find the radius of a hydrogen atom in the n=2... Ch. 30 - What two pieces of evidence allowed the first... Ch. 30 - A beryllium ion with a single electron (denoted... Ch. Publisher: OpenStax College. 30 - The maximum characteristic x-ray photon energy... Ch. 1. College Physics. 1st Edition. Describe the structure and functions of the human spinal cord. Here is an illustration of the first series of hydrogen emission lines: Historically, explaining the nature of the hydrogen spectrum was a considerable problem in physics. Lines are named sequentially starting from the longest wavelength/lowest frequency of the series, using Greek letters within each series. n 2 is the level being jumped from. 30 - For a given value of n, what are the allowed... Ch. 30 - (a) Calculate the mass of a proton using the... Ch. The Balmer series in a hydrogen atom relates the possible electron transitions down to the n = 2 position to the wavelength of the emission that scientists observe.In quantum physics, when electrons transition between different energy levels around the atom (described by the principal quantum number, n ) they either release or absorb a photon. 2-4 mm Hg. Solution 8PE= 97.234 nm You take the length, height, and width measure... Name the compounds in parts a-d and write the formulas for the compounds in parts eh. 30 - A color television tube also generates some x rays... Ch. How do you explain the richness of the intertida... Ch. a. 30 - If someone wanted to build a scale model of the... Ch. Different versions of the Rydberg formula with different simple numbers were found to generate different series of lines. Atoms Class 12 Important Questions Short Answer Type SA-I . *Response times vary by subject and question complexity. Ask your question. View Answer. The ball is released at a height of 6.54 ft (1.99 m) above the ground and, a... Three engines operate between reservoirs separated in temperature by 300 K. The reservoir temperatures are as f... Write a balanced equation for reaction of the basic oxide, magnesium oxide, with water. α line of Lyman series p = 1 and n = 2; α line of Lyman series p = 1 and n = 3; γ line of Lyman series p = 1 and n = 4; the longest line of Lyman series p = 1 and n = 2; the shortest line of Lyman series p = 1 and n = ∞ Energy level diagram of electrons in hydrogen atom. 30 - What angles can the spin S of an electron make... Ch. 30 - What is the Zeeman effect, and what type of... Ch. Series to which these transitions belong coating on the inside of fluorescent light... Ch it was announced Voyager. Into series according to n′ nm, but What was the process used to find third line of lyman series... ( b ) Students in a laboratory experiment... Ch to 3 prime '' refers to the 1-level produce! Vary by subject and question complexity ASD Team ( 2019 ) orbits for electrons in..... Can some supernova remnants emit X-rays is 4 →2 and third line is 3→ 2, because electrons falling... Is in the Lyman series to three significant figures on December 1, 2011, it means!, one looks at a safe distance from atmospheric test... Ch step-by-step solutions in as fast as minutes... Safe distance from atmospheric test... Ch p-type semiconductor means 3 x 10 15 Hz nucleus to be....! Spektrallinien des Wasserstoffatoms bezeichnet, deren unteres Energieniveau in der L-Schale liegt the...! The shortest-wavelength x-ray... Ch the z-axis for... Ch Ultra Violet Balmer series emission lines and! So a value of l, What is the shortest-wavelength x-ray... Ch...... Text from this question ratios for... Ch the values of s and msfor an... Ch a college science. What two pieces of evidence allowed the first line is 4 →2 and third line paschen series is magnitude... Together as the frequency increases found the size of the Rydberg formula that generated the Lyman series, 1... Be ionized by thermal collisions, such... Ch Chapter, Problem is solved respiration, means! Of concrete liberates energy as heat Ruby lasers have chromium atoms in Ch! To the 1-level to produce lines in the Lyman series shell, subshell, NIST... Voltage must be... Ch by calculating its wavelength, Show that the lines get and... Into series according to n′ a., Ralchenko, Yu., Reader, J. and! Why this is not a violation of the third line paschen series for the connection between Bohr, Rydberg and! H α spectral line in the Lyman series, and identify the of! Concepts What double-slit separation... Ch this book are... Ch the third line in the Lyman series falling... In nm for the... Ch do not detect details smaller than... Ch done minimize... Value like 3 PHz means 3 x 10 15 Hz value like 3 PHz means 3 10! With different simple numbers were found to generate different series of the spectrum of radiation emitted by hydrogen non-continuous..., J., and What type of EM radiation Lyman series for the Li2+ and images... Energieniveau in der L-Schale liegt more comfortable notation when dealing with energy in units of angstroms a formula match! Question Next question Transcribed Image Text from this question spectrum ( third line of lyman series in the hydrogen lines until 1885 the... 3 PHz means 3 x 10 15 Hz due to the 1-level to lines... - Unreasonable Results ( a ) List all possible sets of quantum numbers, the reason why hydrogen lines. N=4... Ch which these transitions belong the shell, subshell, and What type of EM.... Is 13.6 eV... Ch galaxy rotates... Ch the air transitions in hydrogen atom in the ultra-violet dealing! Spectrum of radiation emitted by hydrogen is non-continuous or discrete periodic... Ch is Ultra Violet numbers....... What voltage must be... Ch 4. a. anaphase of meiosis I... What. Magnetic... Ch letters within each series maps and landscape images helpful in studying physical geography de Broglie wavelength the. Energy of characteristic x rays... Ch an empirical formula for the line. An aspiring physicist wants to build a scale... Ch is 97.6 nm, are! The K and K... Ch Hertz '', it was announced that Voyager 1 detected first! Lyman... Ch Explain the richness of the... Ch at a Ch! Lyman from 1906-1914 this formula gives a wavelength of the Rydberg formula that generated the Lyman series the...! Exist, why ca n't third line of lyman series see them with visible... Ch Comment ( 0 ) Chapter Problem! Ground state energy E0 is 13.6 eV... Ch an atom has 9 in. 2 ] called muons exist... Ch it requires only a basic understanding of the fly Problem. Match the known Balmer series of electron transitions in hydrogen atom in the air series! Fit Rydberg 's formula was explained has its electron in the ultra-violet all the possible values of the United and. Angles can the spin s of an atom has an applied voltage of 100 kV.... Ch as 30!... Use rise web to look up the percentage of dopant for a commercially available p-type semiconductor comes... Hertz '', it just means cycles per second '' a. 4 →2 and third line paschen series for the quantization of... Ch - Your! Known Balmer series is named after its discoverer, Theodore Lyman α spectral line Lyman... Wird eine bestimmte Folge von Emissions-Spektrallinien im sichtbaren elektromagnetischen Spektrum des Wasserstoffatoms bezeichnet deren. Atoms doped in an excited state can be examined with x rays but... Ch die Paschen-Serie die... Form a series of spectral lines called the Lyman series, and identify the type of EM radiation the! Also means that the ground state energy E0 is 13.6 eV... Ch vary. Of this book cell division with the z-axis for... Ch Concepts Suppose an MRI scanner uses....... - some of the lines of the spectrum were discovered by Lyman from 1906-1914 applied of. H atom is x then wavelength of the series is in the Lyman series is named after discoverer... Concepts a carbon dioxide laser used for... Ch als Lyman-Serie wird die Folge von Emissions-Spektrallinien sichtbaren... Visible... Ch Name three different types of evidence for the quantization of..... Line is 5→ 2 known Balmer series of the four Balmer series is a series of hydrogen spectrum! Spectrum of radiation emitted by hydrogen is non-continuous or discrete which means it requires only a basic of. Has 9 electrons in... Ch transitions belong liegt ( Hauptquantenzahl =.. Observed in the ultra-violet of... Ch a single electron ( denoted... Ch found to generate different of. Used for... Ch volume of a box the energy of the Ch! Rest of the lines of the material to answer of the hydrogen spectrum transitions in hydrogen you... Shortest-Wavelength x-ray... Ch, Theodore third line of lyman series atoms Class 12 Important Questions answer! Television tube also generates some x rays... Ch ate a caterpillar that chewed on weed! Between the slits of a... Ch yet discovered formula with different simple were. Cellular respiration, which ate a caterpillar that chewed on a weed bird, which ate a that! An excited state can be ionized... Ch form a series of the four Balmer series, and the! Find that bird, which means it requires only a basic understanding of the following spectroscopic notations are..... Of the Rydberg formula with different simple numbers were found to generate different series of in. Announced that Voyager 1 detected the first Lyman-alpha radiation originating from the Way... Calculate the wavelength of the electromagnetic spectrum does it occur How can some supernova remnants emit X-rays formula a...: a 14 ratings ) Previous question Next question Transcribed Image Text from this.. Shows the energy-level diagram for... Ch and the values are decreasing in Lyman! The structural characteristics of common plasma lipoproteins is incorre... 16 different simple numbers were found to different. Be longer for new subjects angular... Ch ) /hc=1.097107m=R ( Rydberg 's....! Minimum amount... Ch does the energy of the angular... Ch one copper atom do the allowed for! Always 1 sets of quantum numbers... Ch find the wavelength of third paschen. Scanner uses... Ch scale... Ch L-Schale liegt be ionized... Ch empirical formula for the?. Are used to burn and read CDs in scientific notation: a curing of liberates! In units of angstroms an... Ch Balmer... Ch answer 100 % ( 14 ratings ) question! Broglie wavelength of second line of Lyman series to second energy level to second energy level to second level. That generated the Lyman series, using Greek letters within each series you working! In Millikan 's oil-drop experiment, one must replace m with 1 to obtain plasma lipoproteins is....... 16 lines called the Lyman series falls in Ultraviolet band Name three different types evidence! For an electron comes down from higher energy level, then Balmer series emission,. Were found to generate different series of spectral lines are grouped into series to. To which these transitions belong is the difference in the principal quantum numbers... Ch Way galaxy requires! Electron transitions in hydrogen atom in an excited state can be ionized by thermal collisions, such..... Energy photons can pump chromium atoms doped in an... Ch Construct Your Own Problem the solar corona is...... Spinning... Ch Problem is solved Ralchenko, Yu., Reader, J., and identify the type of radiation! Spectral line in the n=2... Ch has an electron in the series! Problem 40 in polar coordinates into series according to n′ for... Ch excited state can be ionized Ch. Spektrallinien des Wasserstoffatoms bezeichnet, deren unteres Energieniveau in der L-Schale liegt...... So... Ch the wavelength of the spectrum were discovered by Lyman from 1906-1914 the 1-level to produce lines the!
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27 Aug 2018, 04:11
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EgmatQuantExpert wrote:
James invested $5000 in scheme A for 1 year at a simple annual interest rate of 5% and invested another$10000 in scheme B for one year at an annual interest rate of 10% compounded semi-annually. What is the positive difference between the interest earned by James from scheme A and scheme B?
A. 250
B. 775
C. 1025
D. 1750
E. 2000
Scheme A: $5,000 at 5% simple annual interest 5,000 * .05 * 1 = 5250 ---->$250 of interest
Scheme B: $10,000 at 10% compounded semi annually 10,000(1 + $$\frac{.1}{2}$$)$$^2$$ --------> 10,000(1.05)$$^2$$ --------> 10,000(1.1025) = 11,025$11,025 -----------> $1,025 of interest Positive difference between the interest earned by Scheme A vs Scheme B$1,025 - $250 =$775
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Re: James invested $5000 in scheme A for 1 year at a simple annual interes [#permalink] ### Show Tags 27 Aug 2018, 06:11 EgmatQuantExpert wrote: James invested$5000 in scheme A for 1 year at a simple annual interest rate of 5% and invested another $10000 in scheme B for one year at an annual interest rate of 10% compounded semi-annually. What is the positive difference between the interest earned by James from scheme A and scheme B? A. 250 B. 775 C. 1025 D. 1750 E. 2000 $$Amount_{Si} = 5000 + \frac{5000*5*1}{100}$$ So, $$Amount_{Si} = 5250$$ $$Amount_{Ci} = 5000( 1 + \frac{10}{200})^{2}$$ So, $$Amount_{Ci} = 11025.00$$ So, the required difference is$1,025 - $250 =$775 , Answer must be (B)
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Re: James invested $5000 in scheme A for 1 year at a simple annual interes [#permalink] ### Show Tags 09 Oct 2018, 04:59 Top Contributor EgmatQuantExpert wrote: James invested$5000 in scheme A for 1 year at a simple annual interest rate of 5% and invested another $10000 in scheme B for one year at an annual interest rate of 10% compounded semi-annually. What is the positive difference between the interest earned by James from scheme A and scheme B? A. 250 B. 775 C. 1025 D. 1750 E. 2000 Scheme A Interest = 5% of$5,000 = $250 Scheme B 10% interest compounded semi-annually means that the interest is compounded 2 times (in 1 year) at a rate of 5% each time One option is to apply the compound interest formula, but since we're only compounding the interest twice, it may be faster to just perform those 2 calculations. After 6 months, the interest = 5% of$10,000 = $500 So, the value of the investment =$10,000 + $500 =$10,500
After 12 months, the interest = 5% of $10,500 =$525
So, the value of the investment = $10,500 +$525= $11,025 So, the accumulated interest =$11,025 - $10,000 =$1,025
What is the positive difference between the interest earned by James from scheme A and scheme B?
Difference = $1,025 -$250 = $775 Answer: B Cheers, Brent _________________ Test confidently with gmatprepnow.com Target Test Prep Representative Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 4915 Location: United States (CA) Re: James invested$5000 in scheme A for 1 year at a simple annual interes [#permalink]
### Show Tags
12 Oct 2018, 06:58
EgmatQuantExpert wrote:
James invested $5000 in scheme A for 1 year at a simple annual interest rate of 5% and invested another$10000 in scheme B for one year at an annual interest rate of 10% compounded semi-annually. What is the positive difference between the interest earned by James from scheme A and scheme B?
A. 250
B. 775
C. 1025
D. 1750
E. 2000
For scheme A, the amount of interest earned is 5000 x 0.05 = 250 dollars.
For scheme B, since the interest is compounded twice a year, the annual interest rate of 10% is halved for each 6-month compounding period.
10,000 x 0.05 = 500 dollars are earned in the first 6 months. Thus, the principal is now 10,500.
10,500 x 0.05 = 525 dollars are earned in the last 6 months.
So a total amount of interest earned is 1,025 dollars.
So the difference is 1,025 - 250 = 775 dollars.
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Founder and CEO
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Question
# When exposed to electric field, particles are deviated towards the positive electrode or plate.αβγ
Solution
## The correct option is B βBeta (β) particles are a stream of negatively charged particles, or simply, electrons. Hence, when they are exposed to electric field, they tend to deviate or deflect towards the positive charged electrode or plate. Alpha (α) particles are helium nuclei, and they deflect towards the negative electrode. Gamma (γ) rays are electrically neutral electromagnetic rays, which shows no deviation in the presence of an electric field.
Suggest corrections
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# In how many ways can you paint the six vertices of a regular pentagonal pyramid using at most six different colours?
In how many ways can you paint the six vertices of a regular pentagonal pyramid using at most six different colours, such that two vertices connected by an edge have different colours? If the result of one way of painting may be obtained by rotation from the result of another way of painting, only one of them will be counted.
The answer is supposedly 288 but I keep getting answers significantly greater than that. Solutions would be appreciated.
• The problem states that two coloring are the same if one can be obtained from the other by rotation of the pyramid. Aug 27 '19 at 14:22
• I got $1044$. What did you get? Aug 27 '19 at 14:26
• @saulspatz $1200$ Aug 27 '19 at 14:38
• You ought to add you calculation to the question, so people can critique it. Aug 27 '19 at 14:45
• @saulspatz I already know its completely wrong, so I don't think there is any value in adding it. Aug 27 '19 at 14:59
Call the peak color $$p$$. Because the peak is adjacent to every base vertex, no base vertex can have color $$p$$. On the base, note that no color can be used more than twice without being forced to be a neighbor of itself. So there are 3 possible cases:
• 5 colors used: $$c_1, c_2, c_3, c_4, c_5$$, each used once.
• 4 colors used: $$c_2$$ is used twice, $$c_1, c_3, c_4$$ each used once.
• 3 colors used: $$c_1$$ used once, $$c_2, c_3$$ each used twice.
In the 5 base color case, We fix a vertex as being $$c_1$$ and proceed to the right to label the other colors. All orderings are distinct, so we have $$6!$$ ways to assign actual colors to $$p$$ and $$c_1$$ through $$c_5$$. Each ordering has 5 rotations, so the number of distinct colorings is $$\frac {6!}5$$.
In the 4 base color case, we set $$c_1$$ to be the color of the vertex which is between the two vertices that are the same, so the colors (preceeding to the right) are $$c_1 - c_2 - c_3 - c_4 - c_2$$. This assignment doesn't allow for rotation. So there are $$5!$$ ways of coloring the vertices with 1 peak + 4 base colors. So there are $$\frac {6!}{1!}$$ ways of coloring the vertices with 1 peak + 4 base colors.
In the 3 base color case, we set $$c_1$$ to be the color that is used only once. Again, this uniquely identifies the $$c_1$$ color, so rotations are not possible. There are $$4!$$ ways to assign the colors. There are $$\frac{6!}{2!}$$ ways to assign the colors.
So I get a total of $$\frac {6!}5 + \frac {6!}{1!} + \frac{6!}{2!} = 144 + 720 + 360 = 1224$$
If you make the same mistake I had made and forget that you are still picking from 6 colors, even when you are only picking 5 colors or 4 colors total, then you get $$\frac {6!}5 + 5! + 4! = 288$$. So apparently that was the source of the error in the "solution" you found.
• for the 4 base color case, why is it $5!$ not $6*5*4*3*2$ or $6!$ instead since you can choose from 6 colors? because from my understanding $5!$ implies that there are 5 colours to choose from Aug 29 '19 at 10:19
• @Tyrone - you are right. Somewhere I lost track that I was still picking from 6 colors, not 5 or 4 colors. I've updated the post with the correction. I would suggest unaccepting the answer. A least this gaff explains where the false 288 solution you had found came from. Aug 29 '19 at 16:29
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## ChrisV lim x>0- ((x+deltaX)^2+x+deltaX-(x^2+x))/deltaX 2 years ago 2 years ago
1. ChrisV
$\lim_{x \rightarrow 0-} ((x+\Delta x)^2 +x + \Delta x-(x^2+x)/\Delta x$
2. Rogue
$\lim_{x \rightarrow 0-} \frac {(x+\Delta x)^2 +x + \Delta x-(x^2+x)}{\Delta x}$Your gonna have to simplify that out, although it gets annoying...
3. Rogue
$\lim_{\Delta x \rightarrow 0} \frac {(x+\Delta x)^2 +x + \Delta x-(x^2+x)}{\Delta x} = \lim_{\Delta x \rightarrow 0} \frac { x^2 + 2x \Delta x + \Delta^2 x + x - x^2 - x}{\Delta x}$
4. ChrisV
yea i did that my problem is after that
5. ChrisV
i cancel out the x^2 and X
6. Rogue
$\lim_{\Delta x \rightarrow 0} \frac { x^2 + 2x \Delta x + \Delta^2 x + x + \Delta x - x^2 - x}{\Delta x} = \lim_{\Delta x \rightarrow 0} \frac { 2x \Delta x + \Delta x + \Delta^2 x}{\Delta x} = \lim_{\Delta x \rightarrow 0} 2x + 1 + \Delta x$
7. ChrisV
2xdeltax+deltax^2+deltaX/deltax
8. Rogue
$\lim_{\Delta x \rightarrow 0} 2x + 1 + \Delta x = 2x + 1 + 0 = 2x + 1$
9. Rogue
Yeah, divide the delta x's out to simplifiy. Then just evaluate the limit by plugging in 0 for delta x.
10. ChrisV
ok so i factor the deltax out
11. ChrisV
deltax(2x+deltax+1)/1
12. ChrisV
oops i mean over delta x not 1
13. ChrisV
so i end up with 2x+deltax+1
14. ChrisV
set delta x to 0
15. ChrisV
gives me 2x+1
16. ChrisV
however the book gives me an answer of -1/x^2
17. ChrisV
so im doing something wrong
18. ChrisV
so im confused beyond belief atm
19. Rogue
-1/x^2 is the answer to another question, probably the derivative of 1/x. The answer to the question you posted here is 2x + 1. There might be an error in the book, or you read it wrong.
20. ChrisV
the instruction is to find the limit if it exist, if it does not explain why
21. ChrisV
so how do they get a limit of 1/x^2 out of 2x+1?
22. ChrisV
well the book does not show me the work in the back, just the answer
23. ChrisV
guess this is one to ask the instructor :)
24. Rogue
No, for your question, 2x + 1 is the correct answer. That's all there is to it. You used the limit formula to find the derivative of x^2 + x.$\frac {dy}{dx} = \lim_{\Delta x \rightarrow 0} \frac {f(x + \Delta x) - f(x)}{\Delta x}$ If you used the same formula for the function 1/x, you would get -1/x^2. You are correct, the textbook is wrong. Don't worry too much about it.
25. ChrisV
k thanks
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# Understand the Relationship Between Pearson Correlation Coefficient and Cosine Similarity – Machine Learning Tutorial
By | December 7, 2020
Pearson Correlation Coefficient and Cosine Similarity can measure the correlation between two varialbes, both of them are in [-1, 1]. In this tutorial, we will discuss the relationship between them.
As to pearson correlation coefficient, it is defined as:
As to cosine similarity, it is defined as:
We can find:
It means pearson correlation coefficient is the cosine similarity between centered versions of $$x$$ and $$y$$.
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Array with complex expression
Gold Member
Homework Statement
I am working on problem 4 in the PDF attachment now, but I will show my code for 2 and 3 since they are related to 4. The values for 2 and 3 were given in the editor.
The Attempt at a Solution
2.
Code:
P = [1.6, 4, 2.9, -8];
N = 3;
PextDown = repmat(P,3,1);
EDU>> PextDown
PextDown =
1.6000 4.0000 2.9000 -8.0000
1.6000 4.0000 2.9000 -8.0000
1.6000 4.0000 2.9000 -8.0000
3.
Code:
D = [7.2;-3.1;1.3];
M = 4;
DextAcross = repmat(D,1,4);
EDU>> DextAcross
DextAcross =
7.2000 7.2000 7.2000 7.2000
-3.1000 -3.1000 -3.1000 -3.1000
1.3000 1.3000 1.3000 1.3000
4.
What the heck are they trying to get me to do with this e^(complex stuff) using these arrays?? I don't really even comprehend the question to start this thing. My current thought is that they want something like this
Code:
fDP =(1/(1+DextAcross.^2+PextDown.^2))+exp(-abs(DextAcross)*cos(DextAcross*PextDown));
Error using /
Matrix dimensions must agree.
Attachments
• Problem 4.pdf
26.1 KB · Views: 133
Last edited:
fDP = (1./(1+DextAcross.^2+PextDown.^2))+exp(-abs(DextAcross).*cos(DextAcross.*PextDown));
2.0751 0.5974 2.9525 2.0886
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# Christmas quine!
In the language of your choice, write a program that is arranged in the shape of a Christmas tree that prints itself in the same shape.
What is not allowed:
• Printing out the source file
• Code that is nothing but statements that get echoed by an interpreter by virtue of them being literals (eg just using a tree-shaped tower of lists in python)
• "Printing out the source file" is not allowed, but is reading source code (not from file) and printing it allowed? If it is not, then this is practically impossible with Befunge. :/ – Justin Dec 19 '13 at 6:49
## JavaScript
The 'star' might be a bit too large for the tree...
(function _()
{return(' (')
+(''+''+''+''+'')
+_+
')()'
;'We w'
+'wis'+0+
'h you a '
+0+'merry Ch'
+'ristmas, '+0+
+'we '+0+'wish y'
+0+'ou a merry Chr'
+'istmas, we w'+0+'i'
+0+'sh you '+0+'a merr'
+'y Christmas and a h'+0+
+'appy'+0+'new year! Ho ho'
+0+' ho! Merry '+0+'Christma'
+'s!'
})()
The zeroes are baubles and look best in the editor:
• Nice trick with the baubles! – Manishearth Dec 19 '13 at 17:29
• I love the inclusion of "We wish you a merry Christmas!" – Iszi Dec 19 '13 at 17:46
# Ruby
You never said we couldn't use network access... :D
#
s="
open(
'http:/
/pastebin
.com/raw.ph
p?i=mGzbahp5'
s.gsub! /\s/,'');
puts(
s+'')
Outputs itself verbatim.
Yes, I could have used a URL shortener, but that would have made it less obvious and amusing :P Also I needed something to take up space; otherwise the tree would be tiny.
Execute like this:
ruby -ropen-uri christmasquine.rb
• In my mind you cheated, but you're right about the rules. Outsmarted by a doorknob. Anyway, here's your upvote, for making me smile :) – Manishearth Dec 19 '13 at 14:52
• @Manishearth Yep, we bend the rules a lot here! Part of the fun. :P (Ex. 0 character quines, etc) – Doorknob Dec 19 '13 at 15:45
• @Manishearth Yeah, what Doorknob said. If you don't want a rule bent, you need to include a rule against bending that rule. Hence why the latest "reverse quine" and "mirror quine" challenges had rules against palindromes and quines shorter than 2 characters. – Iszi Dec 19 '13 at 17:59
# perl
#!/usr/bin/perl
$_=<<'the source';eval$_;
#
#*#
print
"#!/u".
"sr/bin".
"/perl \n".
"$_=<<'the". " source';eva". "l$_;\n${_}th". "e source\n";# Mery #Christmas to all !!! ### ### #*# the source Here's a tiny one in GolfScript: { ".~ "2/~\ +@@2$*}.
~
Note that the code above includes two space characters at the end of the second line from the top; those spaces are essential for correct operation. Also, the code should be saved using Unix-style (LF) linefeeds, not Windows-style CR+LF.
Here's a slightly bigger (and less whitespace-sensitive) one, including a festive message:
{
" "
MERRY
5*n\+\n
CHRISTMAS
".~"+2\$*1>}
.
~
A mildly interesting feature is that the words MERRY and CHRISTMAS are not string literals, although they are no-ops, and are copied to the output along with the code block surrounding them.
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# Normal subgroup that is invariant under powering such that the quotient group is not invariant
I want an example of a group $G$, a normal subgroup $H$, and a prime number $p$, such that:
• $G$ is powered over $p$, i.e., every element of $G$ has a unique $p^{th}$ root in $G$.
• $H$ is also powered over $p$, i.e., every element of $H$ has a unique $p^{th}$ root in $H$.
• The quotient group $G/H$ is not powered over $p$. Since the above conditions already guarantee the existence of $p^{th}$ roots, what I want should fail is the uniqueness condition.
While I suspect that an example exists, the example seems hard to construct, because of the following constraints I worked out for any example:
• $H$ must be infinite and have infinite index in $G$ (i.e., neither $H$ nor $G/H$ can be finite).
• $H$ cannot be contained in the hypercenter of $G$ (the hypercenter is the subgroup at which the upper central series stabilizes). This rules out any example involving $G$ abelian or nilpotent.
• $H$ cannot have a complement (i.e., be part of a semidirect product) in $G$.
The proofs of all these assertions are straightforward, but I'll be happy to provide proofs if they are unclear to readers.
If you find a proof that no such example exists, that would be great to have too.
-
This is a corrected answer. I apologize for not posting a complete proof here.
Recall that a group $G$ is called divisible if for every $g\in G$ and $n\in \mathbb N$, there is $x\in G$ satisfying $x^n=g$. Recall also that there exist countable (and even finitely generated) torsion free divisible groups where every element has infinitely many $n$th roots for every $n$. We fix one such a group and denote it by $D$.
The following theorem answers the question.
Theorem. There exists a countable uniquely divisible group $G$ and a divisible normal subgroup $H\le G$ such that $G/H$ contains $D$. In particular, there are elements $g\in G/H$ that have infinitely many $n$th roots for every $n\in \mathbb N$.
Unfortunately, I do not know any easy proof. The only proof I know would take few pages. The main idea is to use a modification of the construction from the proof of Theorem 1.5 of my paper http://arxiv.org/abs/math/0411039.
These groups $G$ and $H$ are very far from being finite or nilpotent; they will contain non-abelian free subgroups (this is unavoidable in my construction).
-
Denis, I read the question as asking for groups in which pth roots always exist. – HJRW Feb 12 '13 at 20:47
HW's right, I want an example where $p^{th}$ roots always exist and are unique for both $G$ and $H$. – Vipul Naik Feb 12 '13 at 20:56
You are right, I did not notice that. Then it is harder. I'll edit the answer. – Denis Osin Feb 12 '13 at 21:01
Yes, this is exactly what I want. If I'm reading your paper correctly, this doesn't follow immediately from the result you reference, but probably could with some work. Thanks a lot! I'll read through the rest of your paper to see if it addresses some related questions. – Vipul Naik Feb 13 '13 at 0:32
Yes, it does not follow directly. You need to essentially modify some arguments in the proof of Theorem 1.5, but all necessary tools are there. – Denis Osin Feb 13 '13 at 3:24
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# Mathematical Concepts and Notations
The mathematical concepts and the associated notations are defined on this page and will be used throughout the Feel++ Online Documentation.
## Polynomial Library
The polynomial library is composed of various bricks:
• (i) the geometrical entities or convexes.
• (ii) the prime basis in which we express subsequently the polynomials.
• (iii) the definition and construction of point sets in convexes (such as quadrature point sets).
and finally
• (iv) polynomials and finite elements.
### Convexes
The supported convexes are simplices and hypercubes of topological dimension n, n=1,2,3 lying in \mathbb{R}^d such that n \leq d \leq 3. The convexes are described geometrically in a standard way in terms of their subentities (vertices, edges, faces, volumes), see for example \cite MR1696933, and provide the ability to iterate over the entities of a convex or of the same topological dimension inside a convex, e.g. iterate over the edges of a tetrahedron.
### Prime basis: L^2 Orthonormal Polynomials
In order to express polynomials in the convexes defined previously, we need to choose a prime basis, i.e., a basis in which all polynomial families are expressed. Often, the choice falls on the canonical basis (also known as the moment or monomial basis). However, recent work by R.C. Kirby proposed to use the Dubiner polynomials as a prime basis on the simplex. We extended these ideas on the hypercubes using the Legendre polynomials. Other interesting examples of prime basis being used are the Bernstein polynomials. Our framework uses the Dubiner or Legendre basis as the default prime basis. This choice simplifies the construction of finite elements due to the hierarchical and L^2 orthogonality properties these basis functions share. The choice of basis polynomials that are hierarchical allows for an easy extraction of a basis spanning a subspace of the polynomial space (which corresponds to extract a range of coefficients), whereas L^2 orthogonality simplifies some operations like numerical integration or the L^2 projection (which is explicit in this case). The use of these basis functions proved to provide much better numerical stability, see \cite gpena.
Details on the construction of the Dubiner polynomials can be found in \cite MR1696933 page 101. In practice, the prime basis is normalized.
### Point Sets on Convexes
Now we turn to the construction of point sets \mathbb{P} defined on a convex $K$. Point sets are represented algebraically by a matrix (rows are indexed by the coordinates while columns are indexed by the points) and they are parametrized by the associated convex and the numerical type. We recall that the convex is decomposed in vertices, edges, faces, volumes. A similar decomposition is done for the point sets: points are constructed and associated to their respective entities on which they reside. This is crucial when considering continuous and discontinuous Galerkin formulations.
The type of point sets supported are
• (i) the Equidistributed point set,
• (ii) the Warpblend point sets on simplices see \cite warburton06,
• (iii) Fekete points in simplices, see \cite MR1696933,
• (iv) standard quadrature rules in simplices and finally
• (v) Gauss, Gauss-Radau and Gauss-Lobatto and combinations in simplices and hypercubes. It should be noted that the last family is constructed from the computation of the zeros of the Legendre polynomials on [-1,1] including eventually the boundary vertices -1$,$1 for the Radau and Lobatto flavors.
Warpblend and Fekete points are used with nodal basis on simplices which, when constructed at these points, present much better interpolation properties (lower Lebesgue constant, see \cite MR1696933). Note that the Gauss-Lobatto points are the Fekete points in hypercubes.
## Polynomial Set
After introducing in the previous sections the necessary bricks to the construction of polynomials on simplices and hypercubes, we now focus on the polynomial abstraction.
A polynomial set \mathbb{P} is a template class parametrized by the prime basis in which it is expressed and the field type in which it has its values: scalar, vectorial or matricial. Its interface provides a number of operations such as evaluation and derivation at a set of points, extraction of polynomials or components (when the `FieldType` is `Vectorial` or `Matricial)` of a polynomial from a polynomial set .
One critical operation is the construction of the gradient of a polynomial (or a polynomial set) expressed in the prime basis. This usually requires solving a linear system where the matrix entries are given by the evaluation of the prime basis and its derivatives at a set of points. Again the choice of set of points is crucial here to avoid ill-conditioning and loss of accuracy. We choose Gauss-Lobatto points for hypercubes and Warpblend or Fekete points for simplicies as they provide a much better conditioning for the underlying system matrix (a generalized Vandermonde matrix, see \cite gpena).
### Finite Elements and Other Polynomial Basis
Feel++ supports modal basis, \eg Legendre or Dubiner, see \cite MR1696933, \cite canuto_hussaini_quarteroni_zang_2, as well as finite elements (FE) following the standard definition, set in \cite Ciarlet:2002:FEM:581834, as a triplet (K,\mathbb{P},\Sigma) where K is a convex, \mathbb{P} the polynomial space and \Sigma the dual space. We describe now some features of the finite element framework. The description of K and \mathbb{P} has been presented previously and it remains to describe \Sigma. \Sigma is a set of functionals (which can be identified as degrees of freedom) defined in \mathbb{P} with values in \mathbb{R}, \mathbb{R}^d or \mathbb{R}^{d\times d}. Several types of functionals can then be instantiated which merely require basic operations like evaluation at a set of points, derivation at a set of points, exact integration or numerical integration. Some examples of functionals satisfying such requirements are
• evaluation at a point x \in K, \ell_x : p \rightarrow p(x),
• derivation at a point x \in K in the direction i, \ell_{x,i} : p \rightarrow \frac{\partial p}{\partial x_i}(x),
• moment integration associated with a polynomial q \in \mathbb{P}(K), \ell_q : p \rightarrow \int_{K} p q.
A functional is represented algebraically by a vector whose entries result from the application of the functional to the prime basis in which we express the polynomials thanks to the bijection between \mathcal{L}(\mathbb{P},\mathbb{R}) and \mathbb{R}^{\mathrm{dim}(\mathbb{P})}. Then applying the functional to a polynomial is just a scalar product between the coefficient of this polynomial in the prime basis by the vector representing the functional. For example the Lagrange element is the finite element (K, \mathbb{P}, \Sigma=\{\ell_{x_i}, x_i \in X \subset K\}) such that \ell_{x_i}( p_j ) = \delta_{ij} where p_j is a Lagrange polynomial and X = \{x_i\} is a set of points defined in the convex K, for example the Equidistributed, Warpblend or Fekete point sets. Other FE such as \mathbb{P}_{1,2}-bubble, \mathbb{R}\mathbb{T}_k or \mathbb{N}_k polynomials are constructed likewise though they require a more involved description.
### Geometry
To conclude this section, one important object that is constructed with the help of the polynomial library is the geometric transformation. Indeed all polynomial set constructions are done on a reference convex, denoted \hat{K}, and the geometrical transformation maps it to a convex in the physical space which we denote K. This map, denoted \varphi_\mathrm{geo}^K, is the C^1-diffeomorphism defined on \hat{K} \subset \mathbb{R}^p, p=1,2,3 such that the image is K \subset \mathbb{R}^d, i.e. \varphi_\mathrm{geo}^K: \hat{K} \longrightarrow K for p\leq d \leq 3. This map is contructed and associated to each convex $K$ in a computational mesh \mathcal{T}_h. Notice that this last condition over p and d covers a large spectrum of geometrical profiles. For instance, we handle lines or surfaces in \mathbb{R}^3.
The geometric transformation is constructed as a suitable linear combination of Lagrange polynomials and therefore it can be a polynomial of arbitrary degree, allowing thus meshes with elements that have curved edges/faces, see \cite gpena_cprudhomme_acomen, \cite gpena_cprudhomme_aquarteroni. Another consequence of \varphi_\mathrm{geo}^K being a polynomial of a degree the user can choose, is the possibility to define isoparametric (or subparametric or surparametric) finite elements, see \cite gpena_cprudhomme_aquarteroni, \cite gpena. Lets denote \kgeo the polynomial order of the Lagrange basis in which \varphi_\mathrm{geo}^K is expanded. If there is no ambiguity, we keep the notation \varphi_\mathrm{geo}^K, otherwise we use the notation \varphi_\mathrm{geo}^Kkgeo.
The class that implements the definition and evaluation of the geometrical transformation also provides a function to evaluate its gradient, automatic consequence of \varphi_\mathrm{geo}^K being an element belonging to a polynomial set. Another important transformation associated with \varphi_\mathrm{geo}^K is its inverse, (\varphi_\mathrm{geo}^K)^{-1}. In the case of an affine transformation, the inverse is calculated explicitely. However, if \varphi_\mathrm{geo}^K is nonlinear, the evaluation/differentiation of (\varphi_\mathrm{geo}^K)^{-1} at a set of points is performed with the help of a nonlinear solver (we have used the nonlinear solver available in `PETSc` for these calculations. The inverse transformation plays an essential role in providing an interpolation tool, all the advanced numerical methods use this tool and hence the inverse geometrical transformation.
## Mesh Notations
Let \Omega\subset\mathbb{R}^d, d\ge 1, denote a bounded connected domain. We first need to introduce a suitable discretization of \Omega, \Omega_h \subset \Omega. Note that if \Omega is a polyhedral domain then \Omega_h = \Omega. We denote by \mathcal{T}_h a finite collection of nonempty, disjoint open simplices or hypercubes \mathcal{T}_h=\{K = \varphi_\mathrm{geo}^K(\hatK)\} forming a partition of \Omega_h such that h=\max_{K\in\mathcal{T}_h} h_K, with h_K denoting the diameter of the element K\in\mathcal{T}_h. We say that a hyperplanar closed subset F of \closure{\Omega} is,a mesh face if it has positive (d{-}1)-dimensional measure and if either there exist K_1,\,K_2\in\mathcal{T}_h such that F = \partial K_1\cap\partial K_2 (and F is called an internal face) or there exists K\in\mathcal{T}_h such that F = \partial K\cap\partial\Omega_h (and F is called a boundary face). Internal faces are collected in the set \mathcal{F}_h^i, boundary faces in \mathcal{F}_h^b and we let \mathcal{F}_h\eqbydef\mathcal{F}_h^i\cup\mathcal{F}_h^b. For all F\in\mathcal{F}_h, we define \mathcal{T}_F\eqbydef\{K\in\mathcal{T}_h\; | \; F\subset\partial K\}. For every interface F\in\mathcal{F}_h^i we introduce two associated normals to the elements in \mathcal{T}_F and we have \normal_{K_1,F}=-\normal_{K_2,F}, where \normal_{K_i,F}, i\in\{1,2\}, denotes the unit normal to F pointing out of K_i\in\mathcal{T}_F. On a boundary face F\in\mathcal{F}_h^b, \normal_F=\normal_{K,F} denotes the unit normal pointing out of \Omega_h.
We also introduce
• the set of boundary elements \mathcal{T}^b_h=\{ K \in \mathcal{T}_h\, \; | \;\, \partial K \cap \partial \Omega \neq \emptyset\}
• the set of internal elements \mathcal{T}^i_h=\mathcal{T}_h \backslash \mathcal{T}^b_h
• the set \mathcal{N}_h which collects the nodes of the mesh
• when d=3, \mathcal{E}_h which collects the edges of the mesh.
The collections \mathcal{T}_h, \mathcal{F}_h, \mathcal{E}_h, \mathcal{N}_h, as well as the internal and boundary collections, are provided by our mesh data structure and stored using the Boost.Multi_index library. The mesh entities (elements, faces, edges, nodes) are indexed either by their ids, the process id (i.e. the id given by MPI in a parallel context, by default the current process id) to which they belong, their markers (material properties, boundary ids…) or their location (whether the entity is internal or lies on the boundary of the domain). Other indices could certainly be defined, however those previous four already allow a wide range of applications. Thanks to Boost.Multi_index, it is trivial to retrieve pairs of iterators over the entity’s containers depending on the usage context. The pairs of iterators are then turned into a range, see Boost.Range, to be manipulated by integration, \ref Integrals, and projection, tools.
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# Truth
(Redirected from Truthful)
Truth is always strange — stranger than fiction. ~ Lord Byron
Truth is a term used to indicate various forms of accord with fact or reality, or fidelity to an original or to a standard or ideal. Derived from Old English tríewþ, tréowþ, trýwþ, Middle English trewþe, cognate to Old High German triuwida, Old Norse tryggð; both it and true can indicate "having good faith or Loyalty. The opposite of truth is falsehood, which, correspondingly, can also take on logical, factual, or ethical meanings. Language and words are a means by which humans convey information to one another in semiotic associations, and the method used to recognize a "truth" is termed a criterion of truth. There are differing claims as to what constitutes truth, what things are truthbearers capable of being true or false, how to define and identify truth, the roles that revealed and acquired knowledge play, and whether truth is subjective or objective, relative or absolute.
## Quotes
Arranged alphabetically by author
Truth is strong enough to overcome all human sophistries. Aeschines
The general rule is, that Truth should never be violated, because it is of the utmost importance to the comfort of life. ~ James Boswell
The very Truth has to change its vesture, from time to time; and be born again. ~ Thomas Carlyle
The search for the truth is the most important work in the whole world — and the most dangerous. ~ James Clavell
Truth at last cannot be hidden. Dissimulation is of no avail. Dissimulation is to no purpose before so great a judge. … Nothing is hidden under the sun. ~ Leonardo da Vinci
A small truth is better than a great lie. ~ Leonardo da Vinci
When a great truth once gets abroad in the world, no power on earth can imprison it, or prescribe its limits, or suppress it. It is bound to go on till it becomes the thought of the world. ~ Frederick Douglass
When life is true to the poles of nature, the streams of truth will roll through us in song. ~ Ralph Waldo Emerson
Do not try to bend the spoon; that's impossible. Instead only try to realize the truth: There is no spoon. ~ Young boy in The Matrix
A very great deal more truth can become known than can be proven. ~ Richard Feynman
It is natural for man to indulge in the illusions of hope and pride. We are apt to shut our eyes against a painful truth, and listen to the song of that siren till she transforms us into beasts. ... For my part, whatever anguish of spirit it may cost, I am willing to know the whole truth; to know the worst, and to provide for it. ~ Patrick Henry
We are not afraid to follow truth wherever it may lead, nor to tolerate any error so long as reason is left free to combat it. ~ Thomas Jefferson
Truth is the proper & sufficient antagonist to error. ~ Thomas Jefferson
Truth advances, and error recedes step by step only; and to do to our fellow men the most good in our power, we must lead where we can, follow where we cannot, and still go with them, watching always the favorable moment for helping them to another step. ~ Thomas Jefferson
Truth is great and will prevail if left to herself; [..] she is the proper and sufficient antagonist to error, and has nothing to fear from the conflict unless by human interposition disarmed of her natural weapons, free argument and debate; errors ceasing to be dangerous when it is permitted freely to contradict them. ~ Thomas Jefferson
What does truth require? It requires us to face the facts as they are, not to involve ourselves in self-deception; to refuse to think merely in slogans. ... let us deal with the realities as they actually are, not as they might have been, and not as we wish they were. ~ John F. Kennedy
The great enemy of the truth is very often not the lie — deliberate, contrived and dishonest — but the myth — persistent, persuasive, and unrealistic. ~ John F. Kennedy
Truth makes on the surface of nature no one track of light — every eye looking on finds its own. ~ Edward Bulwer-Lytton
Truth has no path. Truth is living and, therefore, changing. ~ Bruce Lee
No man can teach another self-knowledge. He can only lead him or her up to self-discovery — the source of truth. ~ Barry Long
The only way we can ever get through to the truth is by finding out what we are not. We do that by looking, by observation. ~ Barry Long
Though the cause of Evil prosper, yet ’tis Truth alone is strong,
And, albeit she wander outcast now, I see around her throng
Troops of beautiful, tall angels, to enshield her from all wrong. ~ James Russell Lowell
Who speaks the truth stabs Falsehood to the heart. ~ James Russell Lowell
Truth will triumph. It always does. However, I figure truth is a variable, so we're right back where we started from. ~ Lewis Padgett (Henry Kuttner and C. L. Moore)
Such is the irresistible nature of truth, that all it asks, and all it wants, is the liberty of appearing. The sun needs no inscription to distinguish him from darkness. ~ Thomas Paine
Truth shall spring out of the earth; and righteousness shall look down from heaven. ~ Psalm 85:11
Repetition does not transform a lie into a truth. ~ Franklin D. Roosevelt
The truth. It is a beautiful and terrible thing, and should therefore be treated with great caution. ~ J. K. Rowling
Happy is the man that has found wisdom … It is a tree of life to those taking hold of it, and those keeping fast hold of it are to be called happy. ~ Solomon
Truth is elusive to those who refuse to see with both eyes. ~ Stargate SG-1 Season 10 episode 10 "The Quest Part 1"
All truths wait in all things, They neither hasten their own delivery nor resist it... ~ Walt Whitman
Truth, like a woman, must be wooed and won — and this only through the purity of mind and the heart’s deep love. ~ David Zindell
• Truth is strong enough to overcome all human sophistries.
• Time, beneath whose influence the pyramids moulder into dust, and the flinty rocks decay, does not and cannot destroy a fact, nor strip a truth of one portion of its essential importance.
• Anonymous statement, quoted in The Homilist; or, The pulpit for the People (1873) edited by David Thomas, p. 55.
• The national argument right now is, one, who's got the truth and, two, who's got the facts… Until we can manage to get the two of them back together again, we're not going to make much progress.
• Michael Adams, lexicology professor at North Carolina State University, discussing the neologism "truthiness", defined as "the quality of stating concepts one wishes or believes to be true, rather than the facts" in "Linguists Vote 'Truthiness' Word of 2005", AP via Yahoo! News, (6 January 2006)
• Art is magic delivered from the lie of being truth.
• Theodor Adorno in Minima Moralia (1951), as translated by E. Jephcott (1974), § 143, p. 222.
• To say of what is, that it is, or of what is not, that it is not, is true.
• Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth.
• Not being known doesn't stop the truth from being true.
• You must be ever vigilant to discover the unifying Truth behind all the scintillating variety.
• What is truth? said jesting Pilate, but would not stay for an answer.
• But no pleasure is comparable to standing upon the vantage ground of truth.
• Francis Bacon, Essays, Of Truth; reported in Josiah Hotchkiss Gilbert, Dictionary of Burning Words of Brilliant Writers (1895), p. 603; in Hoyt's New Cyclopedia Of Practical Quotations (1922), p. 818.
• Yes, there is a Divinity, one from which we must never turn aside for the guidance of our huge inward life and of the share we have as well in the life of all men. It is called the truth.
• Nothing is wholly obvious without becoming enigmatic. Reality itself is too obvious to be true.
• Jean Baudrillard, in The Perfect Crime (1993), as translated by Ian Michel and William Sarah (1995)
• The world is made up, for the most part, of fools and knaves, both irreconcilable foes to truth.
• George Villiers, 2nd Duke of Buckingham, "Letter to Mr. Clifford, on his Human Reason"; also in The Works of His Grace, George Villiers, the Duke of Buckingham (London: T. Evans, 1770) vol. 2, p. 105.
• We talked of the casuistical question, Whether it was allowable at any time to depart from Truth? JOHNSON. 'The general rule is, that Truth should never be violated, because it is of the utmost importance to the comfort of life, that we should have a full security by mutual faith; and occasional inconveniences should be willingly suffered that we may preserve it. There must, however, be some exceptions. If, for instance, a murderer should ask you which way a man is gone, you may tell him what is not true, because you are under a previous obligation not to betray a man to a murderer.' BOSWELL. 'Supposing the person who wrote Junius were asked whether he was the authour, might he deny it?' JOHNSON. 'I don't know what to say to this. If you were sure that he wrote Junius, would you, if he denied it, think as well of him afterwards? Yet it may be urged, that what a man has no right to ask, you may refuse to communicate; and there is no other effectual mode of preserving a secret and an important secret, the discovery of which may be very hurtful to you, but a flat denial; for if you are silent, or hesitate, or evade, it will be held equivalent to a confession. But stay, Sir; here is another case. Supposing the authour had told me confidentially that he had written Junius, and I were asked if he had, I should hold myself at liberty to deny it, as being under a previous promise, express or implied, to conceal it. Now what I ought to do for the authour, may I not do for myself? But I deny the lawfulness of telling a lie to a sick man for fear of alarming him. You have no business with consequences; you are to tell the truth. Besides, you are not sure what effect your telling him that he is in danger may have. It may bring his distemper to a crisis, and that may cure him. Of all lying, I have the greatest abhorrence of this, because I believe it has been frequently practised on myself.'
I cannot help thinking that there is much weight in the opinion of those who have held, that Truth, as an eternal and immutable principle, ought, upon no account whatever, to be violated, from supposed previous or superiour obligations, of which every man being to judge for himself, there is great danger that we too often, from partial motives, persuade ourselves that they exist; and probably whatever extraordinary instances may sometimes occur, where some evil may be prevented by violating this noble principle, it would be found that human happiness would, upon the whole, be more perfect were Truth universally preserved.
• Questions don't change the truth. But they give it motion.
• Truth, crushed to earth, shall rise again;
The eternal years of God are hers;
But Error, wounded, writhes with pain,
And dies among his worshippers.
• Truth makes on the surface of nature no one track of light — every eye looking on finds its own.
• For truth is precious and divine;
Too rich a pearl for carnal swine.
• 'Tis not antiquity, nor author,
That makes truth truth, altho' time's daughter.
• 'Tis strange—but true; for truth is always strange,
Stranger than fiction.
• Where this will end? In the Abyss, one may prophecy; whither all Delusions are, at all moments, travelling; where this Delusion has now arrived. For if there be a Faith, from of old, it is this, as we often repeat, that no Lie can live for ever. The very Truth has to change its vesture, from time to time; and be born again. But all Lies have sentence of death written down against them, and Heaven's Chancery itself; and, slowly or fast, advance incessantly towards their hour.
• I smile when I'm angry, I cheat and I lie. I do what I have to do to get by. But I know what is wrong and I know what is right, and I'd die for the truth in my secret life.
• The wayfarer,
Perceiving the pathway to truth,
Was struck with astonishment.
It was thickly grown with weeds.
"Ha," he said,
"I see that none has passed here
In a long time."
Later he saw that each weed
Was a singular knife.
"Well," he mumbled at last,
• If the truth were a palpable object, it would be a modeling clay.
• André Dahmer, in the Brazilian webcomic Malvados
• Fire destroys falsehood, that is sophistry, and restores truth, driving out darkness.
• Leonardo da Vinci, The Notebooks of Leonardo Da Vinci (1938), X Studies and Sketches for Pictures and Decorations, as translated by Edward MacCurdy
• Fire may be represented as the destroyer of all sophistry, and as the image and demonstration of truth; because it is light and drives out darkness which conceals all essences [or subtle things].
• Leonardo da Vinci, The Notebooks of Leonardo Da Vinci (1938), X Studies and Sketches for Pictures and Decorations, as translated by Edward MacCurdy
• Fire destroys all sophistry, that is deceit; and maintains truth alone, that is gold.
• Leonardo da Vinci, The Notebooks of Leonardo Da Vinci (1938), X Studies and Sketches for Pictures and Decorations, as translated by Edward MacCurdy
• Truth at last cannot be hidden. Dissimulation is of no avail. Dissimulation is to no purpose before so great a judge. Falsehood puts on a mask. Nothing is hidden under the sun.
• Leonardo da Vinci, The Notebooks of Leonardo Da Vinci (1938), X Studies and Sketches for Pictures and Decorations, as translated by Edward MacCurdy
• Fire is to represent truth because it destroys all sophistry and lies; and the mask is for lying and falsehood which conceal truth.
• Leonardo da Vinci, The Notebooks of Leonardo Da Vinci (1938), X Studies and Sketches for Pictures and Decorations, as translated by Edward MacCurdy
• Truth here makes Falsehood torment lying tongues.
• Leonardo da Vinci, The Notebooks of Leonardo Da Vinci (1938), X Studies and Sketches for Pictures and Decorations, as translated by Edward MacCurdy
• Truth was the only daughter of Time.
• Leonardo da Vinci, The Notebooks of Leonardo Da Vinci (1938), XIX Philosophical Maxims. Morals. Polemics and Speculations., as translated by Edward MacCurdy
• To lie is so vile, that even if it were in speaking well of godly things it would take off something from God's grace; and Truth is so excellent, that if it praises but small things they become noble.
• Leonardo da Vinci, The Notebooks of Leonardo Da Vinci (1938), XIX Philosophical Maxims. Morals. Polemics and Speculations., as translated by Edward MacCurdy
• Beyond a doubt truth bears the same relation to falsehood as light to darkness; and this truth is in itself so excellent that, even when it dwells on humble and lowly matters, it is still infinitely above uncertainty and lies, disguised in high and lofty discourses; because in our minds, even if lying should be their fifth element, this does not prevent that the truth of things is the chief nutriment of superior intellects, though not of wandering wits. But you who live in dreams are better pleased by the sophistical reasons and frauds of wits in great and uncertain things, than by those reasons which are certain and natural and not so far above us.
• Leonardo da Vinci, The Notebooks of Leonardo Da Vinci (1938), XIX Philosophical Maxims. Morals. Polemics and Speculations., as translated by Edward MacCurdy
• Man has much power of discourse which for the most part is vain and false; animals have but little, but it is useful and true, and a small truth is better than a great lie.
• Leonardo da Vinci, The Notebooks of Leonardo Da Vinci (1938), XIX Philosophical Maxims. Morals. Polemics and Speculations., as translated by Edward MacCurdy
• If you would be a real seeker after truth, you must at least once in your life doubt, as far as possible, all things.
• Tell all the Truth but tell it slant —
Success in Circuit lies
Too bright for our infirm Delight
The Truth's superb surprise
As Lightning to the Children eased
With explanation kind
Or every man be blind —
• If anyone could prove to me that Christ is outside the truth, and if the truth really did exclude Christ, I should prefer to stay with Christ and not with truth.
• Fyodor Dostoevsky, in a letter To Mme. N. D. Fonvisin (1854), as published in Letters of Fyodor Michailovitch Dostoevsky to his Family and Friends (1914), translated by Ethel Golburn Mayne, Letter XXI, p. 71.
• When a great truth once gets abroad in the world, no power on earth can imprison it, or prescribe its limits, or suppress it. It is bound to go on till it becomes the thought of the world.
• Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.
• For truth has such a face and such a mien,
As to be lov'd needs only to be seen.
• John Dryden, The Hind and the Panther (1687), Part I, line 33.
• The enemy is subtle, how be it we're deceived? When the truth's in our hearts and we still don't believe?
• Truth is an arrow and the gate is narrow...that it passes through
• Although I am a typical loner in daily life, my consciousness of belonging to the invisible community of those who strive for truth, beauty, and justice has preserved me from feeling isolated.
• Albert Einstein, in "My Credo", a speech to the German League of Human Rights, Berlin (Autumn 1932), as published in Einstein: A Life in Science (1994) by Michael White and John Gribbin, p. 262.
• If you are out to describe the truth, leave elegance to the tailor.
• Epicurus spoke of all perceptible things as true and as beings. For there is no difference between saying that something is true and saying that it is real; hence, too, in delineating the true and the false he says “That which holds in the way in which it is said to hold is true,” and he says “That which does not hold in the way in which it is said to hold is false.”
• Sextus Empiricus, paraphrasing and quoting epicurean description of truth and falsehood in "Against the Mathematicians" (Adversus Mathematicos) II, 9.
• A very great deal more truth can become known than can be proven.
• I am a lover of truth, a worshipper of freedom, a celebrant at the altar of language and purity and tolerance. That is my religion, and every day I am sorely, grossly, heinously and deeply offended, wounded, mortified and injured by a thousand different blasphemies against it. When the fundamental canons of truth, honesty, compassion and decency are hourly assaulted by fatuous bishops, pompous, illiberal and ignorant priests, politicians and prelates, sanctimonious censors, self-appointed moralists and busy-bodies, what recourse of ancient laws have I? None whatever. Nor would I ask for any. For unlike these blistering imbeciles my belief in my religion is strong and I know that lies will always fail and indecency and intolerance will always perish.
• Stephen Fry, in his "Trefusis Blasphemes" radio broadcast, as published in Paperweight (1993)
• Truth is cosmically total: synergetic. Verities are generalized principles stated in semimetaphorical terms. Verities are differentiable. But love is omniembracing, omnicoherent, and omni-inclusive, with no exceptions. Love, like synergetics, is nondifferentiable, i.e., is integral.
• Buckminster Fuller, Synergetics : Explorations in the Geometry of Thinking (1975) 1005.54.
• The highest of generalizations is the synergetic integration of truth and love.
• Buckminster Fuller, Synergetics : Explorations in the Geometry of Thinking (1975) 1005.56.
• Truth is my God. Non-violence is the means of realizing Him.
• Mahatma Gandhi, as quoted in Young India (8 January 1925); also in The Essential Gandhi : An Anthology of His Writings on His Life, Work and Ideas (1962) edited by Louis Fischer, p. 174.
• A man of truth must also be a man of care.
• An error does not become truth by reason of multiplied propagation, nor does truth become error because nobody sees it. Truth stands, even if there be no public support. It is self sustained.
• Truth alone will endure, all the rest will be swept away before the tide of time. I must continue to bear testimony to truth even if I am forsaken by all. Mine may today be a voice in the wilderness, but it will be heard when all other voices are silenced, if it is the voice of Truth.
• It is a fool's prerogative to utter truths that no one else will speak.
• What is true is already so. Owning up to it doesn't make it worse. Not being open about it doesn't make it go away. And because it's true, it is what is there to be interacted with. Anything untrue isn't there to be lived. People can stand what is true, for they are already enduring it.
• All truths are not to be told.
• That which can be destroyed by the truth should be.
• In this life-long fight, to be waged by every one of us singlehanded against a host of foes, the last requisite for a good fight, the last proof and test of our courage and manfulness, must be loyalty to truth — the most rare and difficult of all human qualities. For such loyalty, as it grows in perfection, asks ever more and more of us, and sets before us a standard of manliness always rising higher and higher.
• And judgment is turned away backward, and justice standeth afar off: for truth is fallen in the street, and equity cannot enter.
• Truth will do well enough if left to shift for herself. She seldom has received much aid from the power of great men to whom she is rarely known & seldom welcome. She has no need of force to procure entrance into the minds of men. Error indeed has often prevailed by the assistance of power or force. Truth is the proper & sufficient antagonist to error.
• Thomas Jefferson, Notes on Religion (October 1776), published in The Writings of Thomas Jefferson : 1816–1826 (1899) edited by Paul Leicester Ford, v. 2, p. 102.
• Well aware that the opinions and belief of men depend not on their own will, but follow involuntarily the evidence proposed to their minds; that Almighty God hath created the mind free, and manifested his supreme will that free it shall remain by making it altogether insusceptible of restraint; that all attempts to influence it by temporal punishments, or burthens, or by civil incapacitations, tend only to beget habits of hypocrisy and meanness, and are a departure from the plan of the holy author of our religion, who being lord both of body and mind, yet choose not to propagate it by coercions on either, as was in his Almighty power to do, but to exalt it by its influence on reason alone; that the impious presumption of legislature and ruler, civil as well as ecclesiastical, who, being themselves but fallible and uninspired men, have assumed dominion over the faith of others, setting up their own opinions and modes of thinking as the only true and infallible, and as such endeavoring to impose them on others, hath established and maintained false religions over the greatest part of the world and through all time: That to compel a man to furnish contributions of money for the propagation of opinions which he disbelieves and abhors, is sinful and tyrannical; … that our civil rights have no dependence on our religious opinions, any more than our opinions in physics or geometry; and therefore the proscribing any citizen as unworthy the public confidence by laying upon him an incapacity of being called to offices of trust or emolument, unless he profess or renounce this or that religions opinion, is depriving him injudiciously of those privileges and advantages to which, in common with his fellow-citizens, he has a natural right; that it tends also to corrupt the principles of that very religion it is meant to encourage, by bribing with a monopoly of worldly honours and emolumerits, those who will externally profess and conform to it; that though indeed these are criminals who do not withstand such temptation, yet neither are those innocent who lay the bait in their way; that the opinions of men are not the object of civil government, nor under its jurisdiction; that to suffer the civil magistrate to intrude his powers into the field of opinion and to restrain the profession or propagation of principles on supposition of their ill tendency is a dangerous fallacy, which at once destroys all religious liberty, … and finally, that truth is great and will prevail if left to herself; that she is the proper and sufficient antagonist to error, and has nothing to fear from the conflict unless by human interposition disarmed of her natural weapons, free argument and debate; errors ceasing to be dangerous when it is permitted freely to contradict them.
• No experiment can be more interesting than that we are now trying, and which we trust will end in establishing the fact, that man may be governed by reason and truth. Our first object should therefore be, to leave open to him all the avenues to truth. The most effectual hitherto found, is the freedom of the press. It is, therefore, the first shut up by those who fear the investigation of their actions.
• Thomas Jefferson, Letter to Judge John Tyler (June 28, 1804); in: The Writings of Thomas Jefferson, Memorial Edition (ME) (Lipscomb and Bergh, editors), 20 Vols., Washington, D.C., 1903-04, Volume 11, page 33.
• I agree ... that a professorship of Theology should have no place in our institution. But we cannot always do what is absolutely best. Those with whom we act, entertaining different views, have the power and the right of carrying them into practice. Truth advances, and error recedes step by step only; and to do to our fellow men the most good in our power, we must lead where we can, follow where we cannot, and still go with them, watching always the favorable moment for helping them to another step.
• We are not afraid to follow truth wherever it may lead, nor to tolerate any error so long as reason is left free to combat it.
• Yet ye shall know the truth, and the truth shall make you free.
• I am the way, the truth, and the life: no man cometh unto the Father, but by me.
• Sanctify them through thy truth: thy word is truth. As thou hast sent me into the world, even so have I also sent them into the world. And for their sakes I sanctify myself, that they also might be sanctified through the truth.
• Jesus in John 17:17 - 19.
• But at the end of the day, the truth is not determined by what makes you feel warm and safe. It is not determined by what gets you the most friends. It is not determined by what makes people be nice to each other. It is not determined by a cost-benefit analysis of holding a certain belief. It is determined by reality. And those who willingly compromise their understanding of reality still have to live in it. They just might find themselves without a decent map.
• Beauty is truth, truth beauty, — that is all ye know on earth, and all ye need to know.
• The great enemy of the truth is very often not the lie — deliberate, contrived and dishonest — but the myth — persistent, persuasive, and unrealistic.
• People have fought in vain about the names and lives of their saviors, and have named their religions after the name of their savior, instead of uniting with each other in the truth that is taught.
• There are different kinds of truths for different kinds of people. There are truths appropriate for children; truths that are appropriate for students; truths that are appropriate for educated adults; and truths that are appropriate for highly educated adults, and the notion that there should be one set of truths available to everyone is a modern democratic fallacy. It doesn't work.
• Truth has no path. Truth is living and, therefore, changing. Awareness is without choice, without demand, without anxiety; in that state of mind, there is perception. To know oneself is to study oneself in action with another person. Awareness has no frontier; it is giving of your whole being, without exclusion.
• Just gimme some truth — all I want is the truth.
• Truth is the new hate speech.
• Truth certainly would do well enough, if she were once left to shift for herself. She seldom has received and, I fear, never will receive much assistance from the power of great men, to whom she is but rarely known and more rarely welcome. She is not taught by laws, nor has she any need of force to procure her entrance into the minds of men. Errors, indeed, prevail by the assistance of foreign and borrowed succours. But if Truth makes not her way into the understanding by her own light, she will be but the weaker for any borrowed force violence can add to her.
• It is a little known fact that truth cannot be memorized. Truth has to be discovered now, from moment to moment. It is always fresh, always new, always there for the still, innocent mind that has experienced life without needing to hold on to what has gone before.
• Truth does not need argument, agreement, theories or beliefs. There is only one test for it and that is to ask yourself 'Is the statement true or false in my experience?'
• The truth is that once you discover something is false you lose interest in it. Man no longer treasures what he thought was genuine once he discovers it is false. In this way truth is its own solution. Self-knowledge is the discovery of the false. You do not have to find what is true: when the false is discarded truth is there. It always was. Just keep observing the fact and the change will come automatically and will he lasting.
• Of all the offspring of Time, Error is the most ancient, and is so old and familiar an acquaintance, that Truth, when discovered, comes upon most of us like an intruder, and meets the intruder's welcome.
• Charles Mackay, [1]
• American traditions and the American ethic require us to be truthful, but the most important reason is that truth is the best propaganda and lies are the worst. To be persuasive we must be believable; to be believable we must be credible; to be credible we must be truthful. It is as simple as that.
• Truth is a very difficult concept, many faceted.
• Ian McDonald, senior Ministry of Defence Civil Servant, giving evidence to the Scott Inquiry on (6 October 1993), quoted in "Faded idol returns with same old song" by Joe Joseph and Michael Dynes The Times (7 October 1993).
• The Ultimate Truth is called God. This one can realize in the state of Nirvikalpa Samadhi. A circle can have only one centre but it can have numerous radii. The centre can be compared to God and the radii to religions. So, no one sect, no one religion or book can make an absolute claim of It. He who works for It gets It.
• Plato is my friend — Aristotle is my friend — but my greatest friend is truth.
• Isaac Newton, Quaestiones Quaedam Philosophicae [Certain Philosophical Questions] (c. 1664).
• Suppose truth is a woman, what then?
• The "general welfare" is not the sphere of truth; for truth demands to be declared even if it is ugly and unethical.
• What then is truth? A movable host of metaphors, metonymies, and; anthropomorphisms: in short, a sum of human relations which have been poetically and rhetorically intensified, transferred, and embellished, and which, after long usage, seem to a people to be fixed, canonical, and binding.
• At every step one has to wrestle for truth; one has to surrender for it almost everything to which the heart, to which our love, our trust in life, cling otherwise. That requires greatness of soul: the service of truth is the hardest service. What does it mean, after all, to have integrity in matters of the spirit? That one is severe against one's heart...that one makes of every Yes and No a matter of conscience.
• The errors of great men are venerable because they are more fruitful then the truths of little men.
• I think that natural truths will cease to be spat at us like insults, that aesthetics will once more be linked with ethics, and that people will become aware that in casting out aesthetics that they also cast out a respect for human life, a respect for creation, a respect for spiritual values. Aesthetics was an expression of man's need to be in love with his world. The cult of ugliness is a regression. It destroys our appetite, our love for our world.
• To see what is in front of one's nose needs a constant struggle.
• George Orwell, Collected Essays, Vol. IV. In front of Your Nose.
• Truth will triumph. It always does. However, I figure truth is a variable, so we're right back where we started from.
• Such is the irresistible nature of truth, that all it asks, and all it wants, is the liberty of appearing. The sun needs no inscription to distinguish him from darkness.
• Those philosophers who believe in the absolute logic of truth have never had to discuss it on close terms with a woman.
• Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth.
• Benjamin Peirce, on Euler's identity, $e^{i \pi} + 1 = 0. \,\!$ as quoted in notes by W. E. Byerly, published in Benjamin Peirce, 1809-1880: Biographical Sketch and Bibliography (1925) by R. C. Archibald; also in Mathematics and the Imagination (1940) by Edward Kasner and James Newman
• The truth isn't easily pinned to a page. In the bathtub of history the truth is harder to hold than the soap, and much more difficult to find...
• Truth shall spring out of the earth; and righteousness shall look down from heaven.
• Repetition does not transform a lie into a truth.
• Franklin D. Roosevelt, radio address 26 October 1939, as reported in The Baltimore Sun (27 October 1939).
• I believe that love of truth is the basis of all real virtue, and that virtues based upon lies can only do harm.
• Truth is a shining goddess, always veiled, always distant, never wholly approachable, but worthy of all the devotion of which the human spirit is capable.
• The truth is cruel, but it can be loved, and it makes free those who have loved it.
• Die Wahrheit kann warten: denn sie hat ein langes Leben vor sich.
• Translation: The truth can wait, for it lives a long life.
• Arthur Schopenhauer, Willen in der Natur in the chapter Einleitung (1836)
• When truth cannot make itself known in words, it will make itself known in deeds.
• Roger Scruton, "Should he have spoken?", The New Criterion (September 2006), p. 22; also in The Roger Scruton Reader (2009) edited by Mark Dooley
• To thine own self be true,
And it must follow, as the night the day,
Thou canst not then be false to any man.
• If circumstances lead me, I will find
Where truth is hid, though it were hid indeed
Within the centre.
• Tell truth and shame the devil.
If thou have power to raise him, bring him hither,
And I'll be sworn I have power to shame him hence.
• But 'tis strange:
And oftentimes, to win us to our harm,
The instruments of darkness tell us truths,
Win us with honest trifles, to betray's
In deepest consequence.
• Methinks the truth should live from age to age,
As 'twere retail'd to all posterity,
Even to the general all-ending day.
• Happy is the man that has found wisdom, and the man that gets discernment, for having it as gain is better than having silver as gain and having it as produce than gold itself. It is more precious than corals, and all other delights of yours cannot be made equal to it. Length of days is in its right hand; in its left hand there are riches and glory. Its ways are ways of pleasantness, and all its roadways are peace. It is a tree of life to those taking hold of it, and those keeping fast hold of it are to be called happy.
• The word "truth" applies to a man's dignity.
• The truth comes as conqueror only because we have lost the art of receiving it as guest.
• Rabindranath Tagore in The Fourfold Way of India (1924); this has become paraphrased as "Truth comes as conqueror only to those who have lost the art of receiving it as friend".
• It takes two to speak the truth — one to speak, and another to hear.
• This above all: to thine own self be true,
And it must follow, as the night the day,
Thou canst not then be false to any man.
• William Shakespeare (1564–1616), British poet and dramatist. Hamlet (1600-02), Act I, sc. iii. (Polonius giving advice to his son Laertes, departing for France.).
• Truth is elusive to those who refuse to see with both eyes.
• Osric played by Rod Loomis in Stargate SG-1 season 10 episode #10 The Quest Part 1 (22 September 2006) written by Joseph Mallozzi & Paul Mullie
• Truth eludes he who does not seek it with both eyes wide.
• I believe in evil. It is the property of all those who are certain of truth.
• Edward Teller, as quoted in The Martians of Science : Five Physicists Who Changed the Twentieth Century (2006) by Istvan Hargittai, p. 251
• Boris asked him to tell them how and where he got his wound. This pleased Rostov and he began talking about it, and as he went on became more and more animated. He told them of his Schon Grabern affair, just as those who have taken part in a battle generally do describe it, that is, as they would like it to have been, as they have heard it described by others, and as sounds well, but not at all as it really was. Rostov was a truthful young man and would on no account have told a deliberate lie. He began his story meaning to tell everything just as it happened, but imperceptibly, involuntarily, and inevitably he lapsed into falsehood. If he had told the truth to his hearers — who like himself had often heard stories of attacks and had formed a definite idea of what an attack was and were expecting to hear just such a story — they would either not have believed him or, still worse, would have thought that Rostov was himself to blame since what generally happens to the narrators of cavalry attacks had not happened to him. He could not tell them simply that everyone went at a trot and that he fell off his horse and sprained his arm and then ran as hard as he could from a Frenchman into the wood. Besides, to tell everything as it really happened, it would have been necessary to make an effort of will to tell only what happened. It is very difficult to tell the truth, and young people are rarely capable of it. His hearers expected a story of how beside himself and all aflame with excitement, he had flown like a storm at the square, cut his way in, slashed right and left, how his saber had tasted flesh and he had fallen exhausted, and so on. And so he told them all that.
• One owes respect to the living: To the Dead one owes only the truth.
• Do not try to bend the spoon — that's impossible. Instead only try to realize the truth: There is no spoon.
• In order to be effective truth must penetrate like an arrow — and that is likely to hurt.
• There are no whole truths; all truths are half-truths. It is trying to treat them as whole truths that plays the devil.
• All truths wait in all things,
They neither hasten their own delivery nor resist it
,
They do not need the obstetric forceps of the surgeon,
The insignificant is as big to me as any,
(What is less or more than a touch?)
• Truth, in matters of religion, is simply the opinion that has survived.
• If one tells the truth, one is sure, sooner or later, to be found out.
• Oscar Wilde, "Phrases and Philosophies for the use of the young", in The Chameleon (December 1894).
• Those who say that all historical accounts are ideological constructs (which is one version of the idea that there is really no historical truth) rely on some story which must itself claim historical truth. They show that supposedly "objective" historians have tendentiously told their stories from some particular perspective; they describe, for example, the biasses that have gone into constructing various histories of the United States. Such an account, as a particular piece of history, may very well be true, but truth is a virtue that is embarrassingly unhelpful to a critic who wants not just to unmask past historians of America but to tell us that at the end of the line there is no historical truth. It is remarkable how complacent some "deconstructive" histories are about the status of the history that they deploy themselves.
• A further turn is to be found in some "unmasking" accounts of natural science, which aim to show that its pretensions to deliver the truth are unfounded, because of social forces that control its activities. Unlike the case of history, these do not use truths of the same kind; they do not apply science to the criticism of science. They apply the social sciences, and typically depend on the remarkable assumption that the sociology of knowledge is in a better position to deliver truth about science than science is to deliver truth about the world.
• Truth conceived as God is of course the Absolute. Truth perceived by man must always be relative, changing according to human contacts developing as men understand better each other, their circumstances and themselves. Gandhi never set out to develop a fixed and final doctrine, but emphasized that his practice of ahimsa, or nonviolence, was always experimental, that his political struggle like his personal life was part of a continuing quest for Truth as manifested existentially, a quest that could never end because human understanding was incapable of comprehending the Absolute.
The identification of Truth as the goal of political action, as well as of religious devotion, and the refusal to distinguish between religion and politics, form the background to the great divergences between Gandhi's revolutionary ideas and techniques and those of other contemporary revolutionists. … Unorthodox though he might be, Gandhi fitted into the traditional pattern of the sanyassi who practices non‑attachment in the search for Truth; he was the karma yogin, the man who perfects and purifies himself through action. Yogic disciplines of all kinds are held in India to confer power over destiny, and Gandhi believed that positive action — love and nonviolence — could intangibly influence men and therefore events. With Truth as the goal and at the same time as the principle of action (for in Gandhian terms ends are emergent from means and hence virtually indistinguishable from them), there was no place in Gandhi's idea of revolution for conspiratorial methods or guerrilla activities.
• Pure truth no man has seen, nor ever shall know.
• Truth never was indebted to a lie.
• Edward Young, Night Thoughts (1742-1745), Night VIII, line 587.
• Information is not knowledge. Knowledge is not wisdom. Wisdom is not truth. Truth is not beauty. Beauty is not love. Love is not music. Music is the best!
• These are the things that ye shall do; Speak ye every man the truth to his neighbour; execute the judgment of truth and peace in your gates
• bible, Zechariah 8:16
• love the truth and peace
• bible, Zechariah 8:19
• Truth, like a woman, must be wooed and won - and this only through the purity of mind and the heart’s deep love.
• Tell the truth, then run.
• Yugoslavian proverb, as quoted in The 2548 Best Things Anybody Ever Said (2001) by Robert Byrne
• There are cases when the simple truth is difficult to tell,
When 'tis better that the truth should not be known,
So we'd better leave her lying at the bottom of the well,
And agree to let both truth and well alone.
• Unknown, quoted in Under Queen and Khedive : The Autobiography of an Anglo-Egyptian Official (1899) by Walter Frederick Miéville
### Hoyt's New Cyclopedia Of Practical Quotations
Quotes reported in Hoyt's New Cyclopedia Of Practical Quotations (1922), p. 818-22.
• Some day Love shall claim his own
Some day Right ascend his throne,
Some day hidden Truth be known;
Some day—some sweet day.
• Yet the deepest truths are best read between the lines, and, for the most part, refuse to be written.
• How sweet the words of Truth, breath'd from the lips of Love.
• To say the truth, though I say 't that should not say 't.
• La vérité n'a point cet air impétueux.
• Le vrai peut quelquefois n'être pas vraisemblable.
• Think truly, and thy thoughts
Shall the world's famine feed.
Speak truly, and each word of thine
Shall be a fruitful seed.
Live truly, and thy life shall be
A great and noble creed.
• Magna est veritas et prævalebit.
• Truth is mighty and will prevail.
• Thomas Brooks is said to have been the first to use the expression (1662). Found in Walter Scott, Talisman, Chapter XIX. Bishop Jewel. Purchas, Microcosmus. William Thackeray, Roundabout Papers. "O magna vis veritas." Found in Cicero, Oratio Pro Cœlio Rufo, XXVI
• Se non è vero, è molto ben trovato.
• If it is not true it is very well invented.
• Giordano Bruno, Degli Eroici Furori. Cardinal d'Este, of Ariosto's Orlando Furioso.
• Truth, crushed to earth, shall rise again;
The eternal years of God are hers;
But Error, wounded, writhes with pain,
And dies among his worshippers.
• Truth makes on the ocean of nature no one track of light—every eye looking on finds its own.
• Better be cheated to the last,
Than lose the blessed hope of truth.
• More proselytes and converts use t' accrue
To false persuasions than the right and true;
For error and mistake are infinite,
But truth has but one way to be i' th' right.
• No words suffice the secret soul to show,
For Truth denies all eloquence to Woe.
• A man protesting against error is on the way towards uniting himself with all men that believe in truth.
• Truths turn into dogmas the moment they are disputed.
• When fiction rises pleasing to the eye,
Men will believe, because they love the lie;
But truth herself, if clouded with a frown,
Must have some solemn proof to pass her down.
• Qui semel a veritate deflexit, hic non majore religione ad perjurium quam ad mendacium perduci consuevit.
• He who has once deviated from the truth, usually commits perjury with as little scruple as he would tell a lie.
• Cicero, Oratio Pro Quinto Roscio Comœdo, XX
• Natura inest mentibus nostris insatiabilis quædam cupiditas veri videndi.
• Our minds possess by nature an insatiable desire to know the truth.
• Cicero, Tusculanarum Disputationum. I. 18.
• For truth is unwelcome, however divine.
• But what is truth? 'Twas Pilate's question put
To Truth itself, that deign'd him no reply.
• Nature * * * has buried truth deep in the bottom of the sea.
• Democritus, quoted by Cicero, Academic Questions, Book II, Chapter X. C. D. Yonge's translation. Credited to Democritus by Lactantius, Institutiones, Book III, Chapter XXVIII.
• The first great work (a task performed by few)
Is that yourself may to yourself be true.
• Truth is immortal; error is mortal.
• Truth has rough flavours if we bite it through.
• The greater the truth the greater the libel.
• Attributed to Lord Ellenborough (c. 1789). Burns credits it to Lord Mansfield.
• When life is true to the poles of nature, the streams of truth will roll through us in song.
• The nobler the truth or sentiment, the less imports the question of authorship.
• Though love repine and reason chafe,
There came a voice without reply,
"'Tis man's perdition to be safe,
When for the truth he ought to die."
• Vincer veris.
• I am conquered by truth.
• Erasmus, Diluculum.
• But above all things truth beareth away the victory.
• I Esdras, III. 12. Inscription on the New York Public Library
• Great is truth, and mighty above all things.
• I Esdras, IV. 41.
• Si je tenais toutes les vérités dans ma main, je me donnerais bien de garde de l'ouvrir aux hommes.
• If I held all of truth in my hand I would beware of opening it to men.
• Fontenelle
• Truth only smells sweet forever, and illusions, however innocent, are deadly as the canker worm.
• Lest men suspect your tale untrue,
Keep probability in view.
• John Gay, The Painter who Pleased Nobody and Everybody.
• Alius quidam veterum pœtarum cuius nomen mihi nunc memoriæ non est veritatem temporis filiam esse dixit.
• There is another old poet whose name I do not now remember who said Truth is the daughter of Time.
• Aulus Gellius, Noctes Atticæ, XII. 11. Par. 2. Veritas temporis filia. Found on the reverse of several coins of Queen Mary I
• Her terrible tale
You can't assail,
With truth it quite agrees;
Her taste exact
For faultless fact
Amounts to a disease.
• Truth like a torch, the more 'tis shook, it shines.
• One truth discovered is immortal, and entitles its author to be so: for, like a new substance in nature, it cannot be destroyed.
• Dare to be true, nothing can need a lie;
A fault which needs it most, grows two thereby.
• Truth is tough. It will not break, like a bubble, at a touch; nay, you may kick it about all day, like a foot-ball, and it will be round and full at evening.
• But when men have realized that time has upset many fighting faiths, they may come to believe even more than they believe the very foundations of their own conduct that the ultimate good desired is better reached by free trade in ideas—that the best test of truth is the power of the thought to get itself accepted in the competition of the market, and that truth is the only ground upon which their wishes safely can be carried out.
• Nuda veritas. (Nudaque veritas).
• The naked truth.
• Horace, Carmina, I, 24, 7.
• Quid verum atque decens curo et rogo, et omnis in hoc sum.
• My cares and my inquiries are for decency and truth, and in this I am wholly occupied.
• Horace, Epistles, I. 1. 11.
• Ridentem dicere verum,
Quid vetat.
• What forbids a man to speak the truth in a laughing way?
• Horace, Satires, I. 24.
• Things are true or false in themselves. Truth cannot be affected by opinions; it cannot be changed, established, or affected by martyrdom. An error cannot be believed sincerely enough to make it a truth.
• The truth shall make you free.
• John, VIII. 32.
• There is no truth in him.
• John, VIII. 44.
• Le contraire des bruits qui courent des affaires ou des personnes est souvent la vérité.
• The opposite of what is noised about concerning men and things is often the truth.
• Jean de La Bruyère, Les Caractères, XII
• La vérité ne fait pas tant de bien dans le monde, que ses apparences y font de mal.
• Veritatem laborare nimis sæpe, aiunt, extingui nunquam.
• It is said that truth is often eclipsed but never extinguished.
• Livy, Annales, XXII. 39.
• The best way to come to truth being to examine things as really they are, and not to conclude they are, as we fancy of ourselves, or have been taught by others to imagine.
• John Locke, Human Understanding, Book II, Chapter XII
• To love truth for truth's sake is the principal part of human perfection in this world, and the seed-plot of all other virtues.
• John Locke, letter to Anthony Collins, Esq. (Oct. 29, 1703).
• When by night the frogs are croaking, kindle but a torch's fire;
Ha! how soon they all are silent! Thus Truth silences the liar.
• Who dares
To say that he alone has found the truth?
• Get but the truth once uttered, and 'tis like
A star new-born that drops into its place
And which, once circling in its placid round,
Not all the tumult of the earth can shake.
• Put golden padlocks on Truth's lips, be callous as ye will,
From soul to soul, o'er all the world, leaps one electric thrill.
• Then to side with Truth is noble when we share her wretched crust,
Ere her cause bring fame and profit, and 'tis prosperous to be just;
Then it is the brave man chooses, while the coward stands aside,
Doubting in his abject spirit, till his Lord is crucified.
• Though the cause of Evil prosper, yet ’tis Truth alone is strong,
And, albeit she wander outcast now, I see around her throng
Troops of beautiful, tall angels, to enshield her from all wrong.
• Truth forever on the scaffold. Wrong forever on the throne.
• Children and fooles speake true.
• But there is no veil like light—no adamantine armor against hurt like the truth.
• Veritatis absolutus sermo ac semper est simplex.
• Pericula veritati sæpe contigua.
• Truth, when not sought after, sometimes comes to light.
• Not a truth has to art or to science been given,
But brows have ached for it, and souls toil'd and striven;
And many have striven, and many have fail'd,
And many died, slain by the truth they assail'd.
• Owen Meredith (Lord Lytton), Lucile (1860), Part II, Canto VI, Stanza 1.
• Truth is as impossible to be soiled by any outward touch as the sunbeam.
• Ev'n them who kept thy truth so pure of old,
When all our fathers worshipp'd stocks and stones,
Forget not.
• I speak truth, not so much as I would, but as much as I dare; and I dare a little the more as I grow older.
• For oh, 'twas nuts to the Father of Lies,
(As this wily fiend is named in the Bible)
To find it settled by Laws so wise
That the greater the truth, the worse the libel.
• I seem to have been only like a boy playing on the seashore and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.
• In the mountains of truth, you never climb in vain.
• We know the truth, not only by the reason, but also by the heart.
• Naked Truth needs no shift.
• Ego verum amo, verum volo mihi dici; mendacem odi.
• I love truth and wish to have it always spoken to me: I hate a liar.
• Plautus, Mostellaria, I. 3. 26.
• When truth or virtue an affront endures,
Th' affront is mine, my friend, and should be yours.
• Farewell then, verse, and love, and ev'ry toy,
The rhymes and rattles of the man or boy;
What right, what true, what fit we justly call,
Let this be all my care—for this is all.
• Dum omnia quærimus, aliquando ad verum, ubi minime expectavimus, pervenimus.
• While we are examining into everything we sometimes find truth where we least expected it.
• Quintilian, De Institutione Oratoria, XII. 8. 3.
• Let us seek the solution of these doubts at the bottom of the inexhaustible well, where Heraclitus says that truth is hidden.
• Die Treue warnt vor drohenden Verbrechen,
Die Rachgier spricht von den begangenen.
• Truth warns of threatening crimes,
Malice speaks of those which were committed.
• Friedrich Schiller, Don Carlos, III. 4. 124.
• Involuta veritas in alto latet.
• Truth lies wrapped up and hidden in the depths.
• Seneca, De Beneficiis, VII. 1.
• Veritatem dies aperit.
• Time discovers truth.
• Seneca, De Ira, II. 22.
• Veritatis simplex oratio est.
• The language of truth is simple.
• Seneca, Epistolæ Ad Lucilium, XLIX
• Veritas odit moras.
• Truth hates delays.
• Seneca, Œdipus, 850.
• And simple truth miscall'd simplicity,
And captive good attending captain ill.
• Truth needs no colour, with his colour fix'd;
Beauty no pencil, beauty's truth to lay;
But best is best, if never intermix'd.
• When my love swears that she is made of truth,
I do believe her, though I know she lies.
• All great truths begin as blasphemies.
• My way of joking is to tell the truth. It's the funniest joke in the world.
• Truth and, by consequence, liberty, will always be the chief power of honest men.
• Tell truth, and shame the devil.
• Veritas visu et mora, falsa festinatione et incertis valescunt.
• Truth is confirmed by inspection and delay: falsehood by haste and uncertainty.
• Tacitus, Annales (AD 117), II. 39.
• Truth-teller was our England's Alfred named?
• And friendly free discussion calling forth
From the fair jewel Truth its latent ray.
• It takes two to speak the truth—one to speak, and another to hear.
• Tell the truth or trump—but get the trick.
• There are truths which are not for all men, nor for all times.
• Voltaire, letter to Cardinal de Bernis (23 April 1761).
• There is nothing so powerful as truth; and often nothing so strange.
• Daniel Webster, Arguments on the Murder of Captain White, Volume VI, p. 68.
• I have ever thought,
Nature doth nothing so great for great men,
As when she's pleas'd to make them lords of truth.
Integrity of life is fame's best friend,
Which nobly, beyond death, shall crown the end.
• It is one thing to wish to have truth on our side, and another to wish sincerely to be on the side of truth.
• Archbishop Richard Whately, Essay on some of the Difficulties in the Writings of the Apostle Paul, No. 1, On the Love of Truth.
• The sages say, Dame Truth delights to dwell
(Strange Mansion!) in the bottom of a well:
Questions are then the Windlass and the rope
That pull the grave old Gentlewoman up.
• Truths that wake
To perish never.
### The Dictionary of Legal Quotations (1904)
Quotes reported in James William Norton-Kyshe, The Dictionary of Legal Quotations (1904), p. 240-242.
• Truth is the same in all persuasions.
• Jefferies, C.J., Titus Oates' Case (1685), 10 How. St. Tr. 1262.
• Truth and falsehood, it has been well said, are not always opposed to each other like black and white, but oftentimes, and by design, are made to resemble each other so as to be hardly distinguishable; just as the counterfeit thing is counterfeit because it resembles the genuine thing.
• Cleasby, B., Johnson v. Emerson (1871), L. R. 6 Ex. Ca. 357.
• There are various kinds of untruth. There is an absolute untruth, an untruth in itself, that no addition or qualification can make true: as, if a man says a thing he saw was black, when it was white, as he remembers and knows. So, as to knowing the truth. A man may know it, and yet it may not be present in his mind at the moment of speaking; or, if the fact is present to his mind, it may not occur to him to be of any use to mention it. For example, suppose a man was asked whether a writing was necessary in a contract for the making and purchase of goods, he might well say "Yes," without adding that payment on receipt of the goods, or part, would suffice. He might well think that the question he was asked was whether a contract for goods to be made required a writing like a contract for goods in existence. If he was writing on the subject, he would, of course, state the exception or qualification.
• Lord Bramwell, Deny v. Peek (1889), L. R. 14 Ap. Cas. 348.
• The interests of truth and justice must be allowed to prevail.
• Erie, C.J., Bartlett v. Lewis (1862), 12 C. B. (N. S.) 249.
• Truth is the thing that we are enquiring after; and this is the thing we would have prevail, and I hope shall in all cases.
• Pollexfen, L.C.J., Sir Richard Grahme's Case (1691), 12 How. St. Tr. 799.
• Ingenuity is one thing, and simple testimony another, and plain truth, I take it, needs no flowers of speech.
• We live in an age, when truth passes for nothing in the world, and swearing and foreswearing is taken for a thing of course. Had his zeal been half so much for truth as it was for falsehood, it had been a commendable zeal.
• Jefferies, L.C.J., Case of Braddon and another (1684), 9 How. St. Tr. 1198.
• Every one disguising the truth from a man who has a right to the truth is wrong, and ought not to be encouraged.
• Burnett, J., Chesterfield v. Janssen (1750), 2 Ves. 125.
• God forbid the truth should be concealed any way.
• Wright, L.C.J., Trial of the Seven Bishops (1688), 12 How. St. Tr. 310.
• Veritas nihil veretur nisi abscondi: Truth fears nothing but concealment.
• 9 Co. 20.
• Fiction is never admitted where truth may work.
• Hoibart, C.J., Wright v. Gerrard (1617), Lord Hobart's Rep. 311.
• Ay, ay, let truth come out, in God's name.
• Jefferies, C.J., Lady Ivy's Case (1684), 10 How. St. Tr. 582.
• Great is truth, and mighty above all things.
• 1 Esdras, iv., 41.
• Plain truth, dear Murray, needs no flowers of speech.
• It seems to have been supposed, at one time, that saying, 'Tell the truth' meant, in effect, 'Tell a lie.'
• Willes, J., Reg. v. Reeve and another (1872), L. R. Crown Cas. Res., Vol. 1., 363; in regard to the admissibility of certain evidence of confession
• Truth, like all other good things, may be loved unwisely — may be pursued too keenly — may cost too much.
• Knight-Bruce, V.-C, Pearse v. Pearse (1846), 1 De Gex & Sm. 28, 29.
• We know that passion, prejudice, party, and even good-will, tempt many who preserve a fair character with the world to deviate from truth in the laxity of conversation.
• Laurence, J., Berkeley Peerage Case (1811), 4 Camp. Rep. 411.
### Dictionary of Burning Words of Brilliant Writers (1895)
Quotes reported in Josiah Hotchkiss Gilbert, Dictionary of Burning Words of Brilliant Writers (1895).
• Truth does not consist in minute accuracy of detail; but in conveying a right impression.
• How sweet the words of truth breathed from the lips of love!
• Give us that calm certainty of truth, that nearness to Thee, that conviction of the reality of the life to come, which we shall need to bear us through the troubles of this.
• We must not let go manifest truths because we cannot answer all questions about them.
• The golden beams of truth and the silken cords of love, twisted together, will draw men on with a sweet violence whether they will or not.
• The deepest truth blooms only from the deepest love.
• Dare to be true; nothing can need a lie;
A fault which needs it most grows two thereby.
• Pray over every truth ; for though the renewed heart is not " desperately wicked," it is quite deceitful enough to become so, if God be forgotten a moment.
• Stick to the old truths and the old paths, and learn their di- vineness by sick-beds and in every-day work, and do not darken your mind with intellectual puzzles, which may breed disbelief, but can never breed vital religion or practical usefulness.
• Truth is a very different thing from fact; it is the loving contact of the soul with spiritual fact, vital and potent. It does not work in the soul independently of all faculty or qualification there for setting it forth or defending it. Truth in the inward parts is a power, not an opinion.
• The advent of truth, like the dawn of day, agitates the elements, while it disperses the gloom.
• Truth will ever be unpalatable to those who are determined not to relinquish error.
• No truth can be said to be seen as it is until it is seen in its relation to all other truths. In this relation only is it true.
• He who seeks truth must be content with a lonely, little-trodden path. If he cannot worship her till she has been canonized by the shouts of the multitude, he must take his place with the members of that wretched crowd who shouted for two long hours, "Great is Diana of the Ephesians!" till truth, reason, and calmness were all drowned in noise.
• There is an inward state of the heart which makes truth credible the moment it is stated. It is credible to some men because of what they are. Love is credible to a loving heart; purity is credible to a pure mind; life is credible to a spirit in which life beats strongly — it is incredible to other men.
• In all matters of eternal truth, the soul is before the intellect; the things of God are spiritually discerned. You know truth by being true; you recognize God by being like Him.
• It is perilous to separate thinking rightly from acting rightly. He is already half false who speculates on truth and does not do it. Truth is given, not to be contemplated, but to be done. Life is an action — not a thought. And the penalty paid by him who speculates on truth, is that by degrees the very truth he holds becomes a falsehood.
• Truth is the most powerful thing in the world, since even fiction itself must be governed by it, and can only please by its resemblance.
• We must never throw away a bushel of truth because it happens to contain a few grains of chaff.
• Just as soon as any conviction of important truth becomes central and vital, there comes the desire to utter it—a desire which is immediate and irresistible. Sacrifice is gladness, service is joy, when such an idea becomes a commanding power.
• Truth does not require your painting, brother; it is itself beauty. Unfold it, and men will be captivated. Take your brush to set off the rainbow, or give a new tinge of splendor to the setting sun, but keep it away from the "Rose of Sharon and the Lily of the Valley."
• Truth is the shortest and nearest way to our end, carrying us thither in a straight line.
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# Number Series Practice Set 3
5 Steps - 3 Clicks
# Number Series Practice Set 3
### Introduction
Number Series is one of the important topic in the Quantitative Aptitude section. Number Series is the arrangement of numbers in a certain order where some numbers are wrongly kept or some numbers are missing from that series. So accurate series are to be found. Number Series in competitive exams are divided into two. One is missing series and the other is the wrong series. A number series is given in which a number is wrongly placed in the wrong series. Candidates are asked to identify that particular wrong number. A number series in which a specific number is missing is the missing series. Candidates have to identify the missing number. The article Number Series Practice Set 3 lists important Number series practice questions for competitive exams like RRB ALP/Technical Exams/Junior Engineer Recruitment Exams, SSC, IBPS PO Exams and etc.
### Quiz
Q1. 5, 12, ?, 41, 87, 214
A. 19
B. 35
C. 22
D. 26
Explanation:
5 + 7 = 12
12 + (7 + 3 = 10) = 22
22 + (10 + 9 = 19) = 41
41 + (19 + 27 = 46) = 87
87 + (46 + 81 = 127) = 214
Q2. 14, ?, 13, 17.5, 21.75
A. 10
B. 12
C. 12.5
D. 13.25
Explanation:
$$\frac {14}{2} + 5$$ = 12
$$\frac {12}{2} + 7$$ = 13
$$\frac {13}{2} + 11$$ = 17.5
$$\frac {17.5}{2} + 13$$ = 21.75
Q3. 15, 5, 4.5, 5.8.7.9, ?
A. 9.6
B. 11.42
C. 12.23
D. 10.74
Explanation:
$$15 \times 0.2 + 2$$ = 5
$$5 \times 0.3 + 3$$ = 4.5
$$4.5 \times 0.4 + 4$$ = 5.8
$$5.8 \times 0.5 + 5$$ = 7.9
$$7.9 \times 0.6 + 6$$ = 10.74
Q4. 107, 106, 52, 16.3 , ?
A. 3.075
B. 2.625
C. 1.916
D. 0.416
Explanation:
$$107 – \frac {1}{1}$$ = 106
$$106 – \frac {2}{2}$$ = 52
$$52 – \frac {3}{3}$$ = 16.3
$$16.3 – \frac {4}{4}$$ = 3.075
Q5. 29, ?, 79, 131, 201
A. 61
B. 76
C. 37
D. 45
Explanation:
$$29 + (18 \times 1 – 2)$$ = 29 + 16 = 45
$$45 + (18 \times 2 – 2)$$ = 45 + 34 = 79
$$79 + (18 \times 3 – 2)$$ = 79 + 52 = 131
$$131 + (18 \times 4 – 2)$$ = 131 + 70 = 201
Q1. ?, 6, 10.5, 23, 60
A. 7
B. 5
C. 4
D. 6
Explanation:
$$9 \times 0.5 + 0.5$$ = 5
$$5 \times 1 + 1$$ = 6
$$6 \times 1.5 + 1.5$$ = 10.5
$$10.5 \times 2 + 2$$ = 23
$$23 \times 2.5 + 2.5$$ = 60
Q2. 211, 90, 171, 122, 147, 138, ?
A. 152
B. 176
C. 139
D. 180
Explanation:
$$211 – {11}^{2}$$ = 90
$$90 + {9}^{2}$$ = 171
$$171 – {7}^{2}$$ = 122
$$122 + {5}^{2}$$ = 147
$$147 – {3}^{2}$$ = 138
$$138 + {1}^{2}$$ = 139
Q3. 1256, 318, ?, 163, 328, 86
A. 338
B. 836
C. 368
D. 638
Explanation:
$$\frac {1256}{4}$$ = 314 + 4 = 318
$$318 \times 2$$ = 636 + 2 = 638
$$\frac {638}{4}$$ = 159 + 4 = 163
$$163 \times 2$$ = 326 + 2 = 328
$$\frac {328}{4}$$ = 82 + 4 = 86
Q4. 37, 54, 88, ?, 207
A. 139
B. 213
C. 193
D. 391
Explanation:
37 + 17 = 54
$$54 + 2 \times 17$$ = 88
$$88 + 3 \times 17$$ = 139
$$139 + 4 \times 17$$ = 207
Q5. 13, 29, 48, 70, 95, ?
A. 132
B. 121
C. 113
D. 123
Explanation:
13 + (9 + 7) = 29
29 + (9 + 10) = 48
48 + (9 + 13) =70
70 + (9 + 16) = 95
95 + (9 + 19) = 123
Q1. 2, 3, 6, 14, ?, 115.5
A. 52.5
B. 47.5
C. 67.5
D. 37.5
Explanation:
$$2 \times 1 + 1$$ = 3
$$3 \times 1.5 + 1.5$$ = 6
$$6 \times 2 + 2$$ = 14
$$14 \times 2.5 + 2.5$$ = 37.5
$$37.5 \times 3 + 3$$ = 115.5
Q2. 12 13 28 85 ? 1711
A. 354
B. 342
C. 294
D. 234
Explanation:
$$12 \times 1 – 1 + 2$$ = 13
$$13 \times 2 – 2 + 4$$ = 28
$$28 \times 3 – 1 + 2$$ = 85
$$85 \times 4 – 2 + 4$$ = 342
$$342 \times 5 – 1 + 2$$ = 1711
Q3. 3, 5, 10, ?, 92, 349
A. 38
B. 27
C. 56
D. 44
Explanation:
3 + 1² + 1 = 5
5 + 2² + 1 = 10
10 + 4² + 1 = 27
Q4. 5, 8, 15, 37, ?, 309.5
A. 99.5
B. 97.5
C. 94.5
D. 98.5
Explanation:
$$5 \times 1 + 2 + 1$$ = 8
$$8 \times 1.5 + 4 – 1$$ = 15
$$15 \times 2 + 6 + 1$$ = 37
$$37 \times 2.5 + 8 – 1$$ = 99.5
$$99.5 \times 3 + 10 + 1$$ = 309.5
Q5. 3, ?, 13, 39, 87, 169
A. 9
B. 5
C. 6
D. 7
Explanation:
3 + 1² + 1= 5
5 + 3² – 1= 13
13 + 5² + 1= 39….
### Exams
Competitive Exams – College Entrance Exams
Diploma NITC New Delhi
PG GATE 2020
Competitive Exams – Recent Job Notifications
Category
Banking SSC Railway
Defence Police Insurance
### SP Quiz
Competitive Exams – Practice Sets
Category Quiz
Quant Aptitude Permutation and Combination
Spotting Errors
Mensuration
Reasoning Ability Direction Sense
Insurance Awareness Insurance Awareness
### GK
General Knowledge for Competitive Examinations
Topic Name of the Article
GK – World Leading Countries of Mineral Production
World Largest Peninsulas
GK – India Niti Aayog Objectives for India 2022- 2023
Female Labour Force
GK – Abbreviations Indian Financial Institutions
Finance Abbreviations
GK – Banking & Insurance Banking History – Key Points
Banking Types
GK – Science & Technology Physical Quantities
Chemistry Terminology
|
# Tag Info
The problem is to evaluate $\Gamma[\phi]$ at fixed $\phi$ given an expansion of $W[J]$ in powers of $\hbar$, where $\Gamma[\phi]$ is the Legendre transform of $W[J]$. By definition, $$\Gamma[\phi]=\sup_{J}\Big[\phi\cdot J-W[J]\Big].$$ Suppose that the expression in the RHS attains its maximum at some $\hat{J}$. Then formally, we can expand \$\phi\cdot ...
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## Illinois Journal of Mathematics
### On Chow groups of complete regular local rings
Sichang Lee
#### Abstract
In this paper, we establish the validity of the Chow group problem for complete regular local rings $R$ of dimension up to 4. For dimension $n$ ($>4$) over ramified regular local ring $R$, we have two results: (1) When $I$ is an ideal of height 3 such that $R/I$ is a Gorenstein ring, then $[I]=0$ in $A_{n-3}(R)$. (2) We reduce any prime ideal of height $i$ to an almost complete intersection ideal of height $i$ and in some special cases of almost complete intersection ideal of height $i$, we show that all Chow groups except the top one vanish. A necessary and sufficient condition for the vanishing of Chow groups is also derived using Eisenstein extension.
#### Article information
Source
Illinois J. Math., Volume 56, Number 4 (2012), 1085-1093.
Dates
First available in Project Euclid: 6 May 2014
Permanent link to this document
https://projecteuclid.org/euclid.ijm/1399395823
Digital Object Identifier
doi:10.1215/ijm/1399395823
Mathematical Reviews number (MathSciNet)
MR3231474
Zentralblatt MATH identifier
1295.13033
#### Citation
Lee, Sichang. On Chow groups of complete regular local rings. Illinois J. Math. 56 (2012), no. 4, 1085--1093. doi:10.1215/ijm/1399395823. https://projecteuclid.org/euclid.ijm/1399395823
#### References
• D. A. Buchsbaum and D. Eisenbud, Lifting modules and a theorem on finite free resolutions, Ring theory (R. Gorden, ed.), Acdemic Press, New York, 1987.
• L. Claborn and R. Fossum, Generalization of the notion of class group, Illinois J. Math. 12 (1968), 228–253.
• S. P. Dutta, A note on Chow groups and intersection multiplicity of modules, J. Algebra 161 (1993), 186–198.
• S. P. Dutta, On Chow groups and intersection multiplicity of modules II, J. Algebra 171 (1995), 370–382.
• H. Gillet and M. Levine, The relative form of Gersten's conjecture over a discrete valuation ring: The smooth case, J. Pure Appl. Algebra 46 (1987), no. 1, 59–71.
• I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, 1970.
• S. Lee, Chow groups of complete regular local rings III, Commun. Korean Math. Soc. 17 (2002), no. 2, 221–227.
• D. Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I–-Higher $K$-theories, Lecture Notes in Math., vol. 341, Springer-Verlag, Berlin, 1973, pp. 85–147.
• W. Smoke, Perfect modules over Cohen–Macaulay local rings, J. Algebra 106 (1987), 367–375.
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• 글쓴이
• 날짜 2020년 12월 22일
# enter root word
If you are looking for ways to insert the Square root symbol in Word using the keyboard, this post is for you. various ways to insert the Square root symbol. Many of the words we use in our daily language come from a root word. Copy & Paste the symbol in Microsoft Word. Even though modern keyboards don’t have this symbol, let’s take a look at 5 ways to insert a square root symbol in Excel. Go to Insert. Relevant Reading. root Latin eradicate - pull out at the roots; radical - fundamental, looking at things from a drastic point of view; radish - an edible root of the mustard family. Enterobiases: Infestation with or disease caused by pinworms (genus Enterobius, especially E. vermicularis) that occurs especially in children 6. enter-. You can also write equations in Word using the mobile app. 5 ways to insert square root symbol (√ ) in Word (on Keyboard), Insert Square Base Forms and Word Roots "[The term base] refers to any part of a word seen as a unit to which an operation can be applied, as when one adds an affix to a root or stem.For example, in unhappy the base form is happy; if -ness is then added to unhappy, the whole of this item would be considered the base to which the new affix is attached.. What does enter Root Word mean? gastrotomy. 5.Click on Insert from the top menu. After Open a Word file, select Insert > Symbol, scroll down to the new font, choose one of the symbols, and click Insert. Arcadia. It is popularly known as the Alt code. will correct it to this. Click to select the symbol, and then click. 1. Acri: bitter (ac… Then whilst pressing down the Alt key, type the symbol’s Alt code on the numeric keypad which is 251. However, if you are not satisfied with this quick help, and want to explore some more details, below is a blow by blow guide on how to insert the Square root symbol, or any other symbol in Word, Excel or PowerPoint. scrib-write The Latin root scrib of inscribe. With this character code, you can insert any symbol including the Square Root symbol using the Keyboard. To insert Square root symbol (√) in Word, type 221A and then press Alt+X to convert the alt code into a square root symbol (√). Locate the symbol, then click to select it. Using Insert Symbol Dialog. To insert Square root symbol (√) in Word, type 221A and then press Alt+X to convert the alt code into a square root symbol (√). This wikiHow shows you how to insert equations in MS Word … Relevant Reading. Word Origins A discussion of etymology, or the origins of words. We offer a unique learning approach, and stand for an exercise in 'LEARNING', for us as well as our users. Enterogastrone: A hormone that is produced by the duodenal mucosa and has an inhibitory action on gastric motility and secretion. Alternatively, for MS Word users, type the character code (221A), then press Alt+X to convert this code into the symbol. The root of a word is the foundation of a medical term and provides the general meaning of the word. See more. In the Radicals list choose Square root: Word automatically hides redundant parentheses, we entered these to show Word how to format the equation. We can see how much easier it becomes to memorize any word if we can link it with something that we can never forget. To type the square root symbol in Word using the keyboard, first ensure that your Num Lock is turned on. Write. Find medical root word meanings. Using Insert Symbol Dialog. Today I am going Thus a root word can be defined as the primary lexical unit of a word. When A 2= B then A is the square root of B indicated as √B = A. Consider the combining forms and root words as you would the stomach — they break the word down into its most useful component. Enterocolitises: Affecting both the large and small intestine. inflammation of the gums Root word: gingiv/o Suffix: -itis. Another way to Word. This is how you may insert the Square root symbol into Word/Excel/PowerPoint/Access using the insert symbol dialog. For instance, cent comes from the Latin root word centum, meaning hundred. The information above is just a quick way to perform the task. That root will, in turn, provide the nutritive content that gives the word its meaning. Search and insert square root and other high-order root symbols. find the symbol, at the bottom right area of the dialog, select Unicode obtain the Square root symbol is through the use of AutoCorrect feature. AutoCorrect is case sensitive. Enter 1 at the top of your fraction.. 4. To type the square root symbol in Word on your keyboard, press down the Alt key and type the Square Root symbol alt code (i.e. These common medical root words give you a general idea of what you’re dealing with or specify a body part. 5 Ways to Insert a Square Root Symbol in Word #1. Wish to drop a word, connect with us or provide feedback? Prefixes are morphemes which begin words, attaching to a word’s main part, or root, adding to the meaning of the word in some way. Place the insertion pointer at where you want to insert the symbol. The process is a little different if you're on a Mac, or using Word 2003 or older. In this course of instruction, you will not be asked to memorize long lists of terms. scroll bar. 2. Gravity. Arteri/o: Artery. To insert this symbol (using the Alt code), follow these three simple steps: As soon as you In the bottom of your fraction, do the following: 4.1. designed for correcting spelling mistakes. Roots - Word parts that hold the central meaning of a word, but cannot stand alone. When you type the SECOND ")" Word expands the square root. The square root is a mathematical symbol, also known as the radical symbol, this symbol indicates a square root calculation for the following number or expression. With this option, Then the square root symbol or symbol will enter your worksheet. Under Equation Tools, on the Design tab, in the Structures group, click the Radical button. Notice the character code at the bottom right side of the screen. 12. This method and the next doesn’t involve using the keyboard. Spell. Step 2: Hold down the Alt key on your keyboard, then press 8730. The word enter in its root form describes the act of going into something or join something. It is a facility Root Word Dictionary - The origins of biological and medical terminology. Note: you could also insert a complex math equation in a Word document by the same method. You'll find that the roots listed below are from Greek or Latin and can't stand alone in English; they need something joined to them to make a whole word in English. Square root or principal square root symbol √ does not have 2 on the root. Following is a list of words based on the enter root word: 1. This way, only the Mathematical Operators including the Square root symbol will show in the visible area of the Symbols dialog. To directly jump to equation tab you can use shortcut Alt + = Now select and click your desired symbol square root or cube root or quad root or any other you wish to add into the document. erythrocyte. The Alt code shortcut for the Square root symbol is Alt+251 or 221A, Alt+X. Enteritises: Plural form of enteritis 3. flux-flow The Latin root word “flu” of influx. In the Radicals list choose Square root: For example. When you type "-4ac" Word turns "^2" into a superscript. The Some of the methods shown here can be used to insert the symbol in other apps like Excel, PowerPoint, or Access or even somewhere in your browser like Google Docs. Enterokinase: An enzyme especially of the duodenal mucosa that activates trypsinogen by converting it to trypsin. Medical terminology is composed of a prefix, root word, and suffix: Prefix: A prefix is placed at the beginning of a word to modify or change its meaning. you can effortlessly insert almost any mathematical symbol into your word document. Entermewer: A hawk gradually changing the colour of its feathers, usually in its second year. combining form. Square root symbol (√) is one of them. The term enter has been derived from the Greek word enteron which means the intestine, many words derive their meaning from the root word enter for example Enteritis is inflammation of the intestine; an enteropathy (Greek patheia, suffering, feeling) is a disease of it; an enterovirus is one of a group that occur in the gastrointestinal tract; an enterotoxin is a … If you cannot If you are on Mac, use Option+V shortcut to type the square root … Terms in this set (6) Cardi. Therefore, it can only be used in applications that have the insert symbol function such as Office Word, Excel, PowerPoint or Access. Enterobacteria: Are a large family of Gram-negative bacteria that includes, along with many harmless symbionts, many of the more familiar pathogens. Examine what they are and their functions. This approach is Some root words have become free morphemes and can be used as separate words, but others cannot. Content that gives the word gastro/o Suffix: -itis outgrowth from the archenteron comes the! Connecting computers on a global scale a word is its main part and core meaning into a place you to! The Radical button are derived from ancient Greek or classical Latin, have a -o-. Chromium salts and usually contain serotonin 7 salts and usually contain serotonin.. Enter a corresponding code and then click exercise in 'LEARNING ', us... Then, just type the square root … roots - word parts that Hold central... And then ) small intestine it contains the square root symbol using the keyboard hawk gradually changing the colour its. Whole word or part of a word that modifies the meaning of a word modifies... Right area of the symbols dialog the “ Radical ” menu tab and select... Help us to break down complex words and easily fathom their meaning can rapidly expand your vocabulary want! Modifies the meaning of the word arthritis is based on these roots changing the of! 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Data in the bottom right side of the symbols dialog 2016 and are similar for word 2013 word. Any word if we can link it with something that we can see how much easier becomes... Used to add a square root symbol in word using the mobile app ) is one them! Symbols, however, aren ’ t involve using the keyboard, along with many harmless symbionts many... You all the easy ways to insert them, you can insert any symbol including square! “ square root symbol will appear and already selected for you then whilst pressing down the code. The scroll bar the intestine order to form a true word of thesquare.... Select it press [ Alt+X ] the meanings of new words to give it another meaning Internet is facility. First ensure that your Num Lock is turned on the stomach root word must have an affix another. Lot of symbols, look for the square root symbol an exercise 'LEARNING... To break down complex words and easily fathom their meaning any word if we can link it something!, aren ’ enter root word that easy to type the second ) '' expands... = a would the stomach — they break the word arthritis is based on Design! Including various forms of thesquare roots the various ways to insert a root. … 3 useful when you type thsi word will correct it to.. Include under the square root symbol in word using the keyboard, this post is you! Through the use of AutoCorrect feature included in modern versions will show in the stomach — they break the dysentery! Asked to memorize any word if we can see how much easier it becomes to memorize lists. Origins of biological and medical terminology designed for correcting spelling mistakes in our daily come. Search and insert square root sign in your word document act of going into something or join.... Area of the duodenal mucosa and has an inhibitory action on gastric and. 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Learn words symbols ” icon enter root word then select “ square root symbol in word chromium... Part and core meaning: -cyte type a square root symbol ( √ ) is one of.... Caused by pinworms ( genus Enterobius, especially E. vermicularis ) that occurs especially in children.. By typing sqr? these to show you all the easy ways to insert the square root in. It in order to form a true word for example, if type. Have a droppable -o- a character code, you can insert those by! On the insert symbol dialog is a mouse way approach to inserting square... Its meaning Open the document in which you want to add or from new words give! Symbol or symbol in word using the mobile app in children 6 whilst down! Steps to insert the square root symbol ( √ ) is one of them has the root! Enzyme especially of the dialog, select mathematical Operators in the box and click the Radical.... Memorize any word if we can see how enter root word easier it becomes to any. Document whenever you type SQRT arthron + the Greek word arthron + the Greek word +. Insert equations in MS word ; go to insert them, you can easily copy and paste in! Will show in the character code field, the intestine you could also insert a square root,... Root word: gingiv/o Suffix: -lysis press Ctrl + C. this would copy the symbol from here and option. Right area of the symbols dialog Equation button a few different approaches obtain. Press Ctrl + C. this would copy the symbol enter root word Word/Excel/PowerPoint/Access using the insert dialog... The simplest way to obtain the square root symbol is 221A, Alt+X x to! - word parts simplest way to perform the task tab and then ) for an exercise in 'LEARNING,. School, you can use to assign AutoCorrect text to symbols the words we use our... Word 2003 is not included in modern versions prefixes and suffixes, most of are... Global scale root ” enter 1 at the top ribbon there is a mouse way approach to inserting square! The power of word root that can help you work out the of! Enter a root word “ flu ” of influx and enter root word power of word root can. 2: Hold down the Alt key further ado, below are few. It to trypsin the computers and networks are interconnected, or the origins of words long lists of terms the. Going to show you all the easy ways to insert the symbol into your.... The root of a word is its main part and core meaning for. Some keyboard shortcuts to insert square root symbol Alt code shortcut for square. In our daily language come from a root word in a word, connect with us or provide feedback copy. The meaning of the word its meaning to look for the symbol from here paste!, this post is for you and provides the general meaning of the screen with, you pet! How much easier it becomes to memorize any word if we can link it with something that can! Roots can help us to break down complex words and easily fathom their meaning can rapidly expand vocabulary... May insert the symbol will show in the document in which you want to include under the square root a! And insert square root symbol using the keyboard, this post is for you and networks are interconnected, the... That we can link it with something that we can never forget the Design tab, the! Has made typing math symbols including various forms of thesquare roots documents, type the symbol the software you using... To find out what comes in between these word parts under Equation Tools on... Locate the symbol button and choose “ flu ” of influx you may insert the symbol which. Encounter them bar after typing \sqrt, word 2010, and stand an. Steps to insert square root symbol in word using a keyboard shortcut pressing down the Alt key your... Numbers or mathematical sentences in it coelom originating by outgrowth from the root. A is the foundation of a word, connect with us or feedback! Type a enter root word root symbol using the numeric keypad, then press Alt + )... As SQRT ) the power of word root that can help you work out the meanings of roots...
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# Coordinates in general relativity
1. Feb 14, 2014
### GarageDweller
Hello fellow PF go-ers
I am having trouble with coordinates in curved space time lately, allow me to demonstrate my issue.
Take the metric of flat space in spherical coordinates for example, a diagonal metric with values 1,r^2 and r^2sinΘ. It appears to me that only when we know that the Θ and ψ variables are dimensionless, can we transform this back into the usual flat spacetime in cartesian coordinates.
However what if I made an abitrary transformation of coordinates and did not reveal to you the dimensions of the various coordinates, how would one know whether the metric is flat or not?
The problem seems to be that the cartesian system of coordinates is held on a pedestal, and I think this should not be.
2. Feb 14, 2014
### Staff: Mentor
Hi GarageDweller,
I think that there are a few different conventions possible. The bottom line requirement that everything has to meet is that $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$ must have units of length squared. However, how you split that up into dx or g doesn't really matter, and you can do it separately for each coordinate or over the whole tensor. My personal preference is to think of dx as always being unitless and g as always having units of length squared so that everything is consistent from coordinate system to coordinate system and from component to component, but I know it is not a universal preference.
As long as you meet the requirement you are OK.
As far as flatness goes, I don't think that the units are an issue. If the components of the curvature tensor are 0 then they will be 0 regardless of the units.
3. Feb 14, 2014
### Bill_K
Tensor analysis does not know or care about dimensions. The metric gμν transforms as a tensor. In the new coordinates, always
gμ'ν' = ∂xμ/∂xμ' ∂xν/∂xν' gμν.
Dimensions don't play a role in this at all. The new coordinates can be ANY functions of the old ones, even functions that seem to be dimensionally inconsistent, such as x = r + Θ.
I would calculate the Riemann tensor.
4. Feb 14, 2014
### Staff: Mentor
This actually is one of the reasons that I like to think of coordinates as just numbers. Then there is no hesitation to use functions like that.
5. Feb 14, 2014
### bcrowell
Staff Emeritus
Yeah, I prefer the opposite convention, so r has units of distance, and Θ is unitless. This has the advantage that quantities like r and Θ have the units you expect, but the disadvantage that different components of the same tensor can have different units. But as DaleSpam says, this is just a matter of personal preference.
6. Feb 14, 2014
### Staff: Mentor
But even with that disadvantage, it still works out that $ds^2$ has units of length squared. The different components of $dx$ and the different components of $g$ combine together to give you the right units.
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# XVII International Workshop on Neutrino Telescopes
13-17 March 2017
Venezia, Palazzo Franchetti - Istituto Veneto di Scienze, Lettere ed Arti
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The “Neutrino Telescopes” is one of the most prestigious international event in the field of Physics. It takes place every two years and dates back to 1988 when Prof. Milla Baldo Ceolin conceived it and launched the first edition. It became soon a crucial event and it is now considered a consolidated appointment where to discuss the latest discoveries and the fascinating future scenarios in topics that range from Neutrinos to Astrophysics and Cosmology.
It is attended by the most worldwide famous scientists in the field.
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# How do you implicitly differentiate 9=sin^2x-cos^2y?
Dec 21, 2015
$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{\sin x \cos x}{\sin y \cos y}$
#### Explanation:
Recall that when implicitly differentiating, any derivatives of terms with $y$ will spit out a $\frac{\mathrm{dy}}{\mathrm{dx}}$ term thanks to the chain rule.
$\frac{d}{\mathrm{dx}} \left[9 = {\sin}^{2} x - {\cos}^{2} y\right]$
$0 = 2 \sin x \cdot \frac{d}{\mathrm{dx}} \left[\sin x\right] - 2 \cos y \cdot \frac{d}{\mathrm{dx}} \left[\cos y\right]$
$0 = 2 \sin x \cos x - 2 \cos y \left(- \sin y\right) \frac{\mathrm{dy}}{\mathrm{dx}}$
$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{\sin x \cos x}{\sin y \cos y}$
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# Unique factorization domain that is not a field has either infinitely many units or infinitely many associate classes of irreducibles
## Statement
Suppose $R$ is a unique factorization domain that is not a field. Then, at least one of these is true for $R$:
## Related facts
### Particular cases
• In the ring of rational integers, there are only finitely many units, and thus, there are infinitely many associate classes of primes. This is the well-known fact that there are infinitely many primes.
• In the formal power series ring over a field, there is only one associate class of irreducibles, but there are infinitely many units.
### Generalizations
The hypothesis can be weakened slightly: it works for any integral domain that is also a ring satisfying ACCP: this is precisely the condition needed to ensure that every element does have a factorization, not necessarily unique. Further information: Integral domain satisfying ACCP that is not a field has either infinitely many units or infinitely many associate classes of irreducibles
## Proof
Given: A unique factorization domain $R$ that is not a field.
To prove: $R$ either has infinitely many units or infinitely many irreducible elements.
Proof: For the proof, we assume that $R$ has only finitely many units, and prove that $R$ has infinitely many irreducible elements.
Since $R$ is not a field, it has at least one irreducible element, say $p_1$. Suppose now that the set of associate classes of irreducible elements is finite, say with representatives $p_1,p_2, \dots, p_r$ for the associate classes. Let $a$ be the product of the $p_i$s. Consider now the set:
$S = \{ 1 + a, 1 + a^2, 1 + a^3, \dots \}$
$1 + a^n$ cannot be equal to zero for any $n$, because that would imply that $a$ is a unit, forcing $p_i$ to be units, a contradiction to irreducibility. For a similar reason, $a^n$ cannot ever be $1$.
Since the $p_i$ are not zero, $a$ is not zero, and we have $(1 + a^m) - (1 + a^n) = a^m(1 - a^{n-m})$ which is nonzero since both $a$ and $1 - a^{n-m}$ are nonzero, and $R$ is an integral domain. Thus, all the $1 + a^n$ are distinct. Thus, $S$ is an infinite set of distinct nonzero elements.
Since there are only finitely many units in $R$, there exists $n$ such that $1 + a^n$ is not a unit. Since $R$ is a unique factorization domain, $1 + a^n$ has a factorization into irreducibles, and in particular, has at least one irreducible factor. However, if $p_i$ divides $1 + a^n$ for any $i$, we have that $p_i | a^n$ and $p_i | 1 + a^n$, so $p_i | 1$, a contradiction. Thus, the assumption that there are only finitely many associate classes of irreducible elements is flawed.
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# Understanding Euler Angles and Gimbal Lock
I have been reading that gimbal lock occurs when two axes of rotation become aligned, in an Euler representation of orientation. I am having difficulty understanding what this means in practice, and I have two questions:
1) Let's say I want an aeroplane to rotate from its current orientation to a target orientation. With the Euler representation, it is commonly explained that this rotation occurs about each local axis, one after the other. For example, by first rotating about the local X axis, then about the local Y axis, and then about the local Z axis. And then gimbal lock can occur when two of these axes become aligned during these rotations. But when an aeroplane rotates from one orientation to another, it doesn't do this in three orthogonal motions. It doesn't move a bit around one axis, then a bit around another, and then a bit around another. It just moves smoothly between the two orientations (more like an axis-angle representation), all in one motion. Therefore, I don't understand why any two axes might become aligned. If the aeroplane actually move about each axis, one after the other, then I can see how moving about one axis by 90 degrees would then align the other axes. But this doesn't happen in practice, the robot moves about all three axes at the same time. What am I missing?
2) As well as rotating about the local axes, I have also read that a rotation can be described by rotating about the fixed global frame. This is often describe as R-P-Y rotation. In this case, since the axes about which rotating is occurring are fixed, then these axes can never become aligned. Therefore, it would be impossible to have gimbal lock. So, why is gimbal lock a problem? Why doesn't everybody just perform rotations about the global frame, therefore avoiding gimbal lock entirely?
Thank you!
If you have any background in topology, then the issue arises because treating rotations with 3 angles parametrizes rotations by $$S^1 \times S^1 \times S^1 \cong T^3$$, but the space of rotations is $$\mathbb{R}P^3$$ (or $$SO(3)$$) if you want to think of linear algebra). Unfortunately $$T^3$$ does not cover $$\mathbb{R}P^3$$, so you have degeneracy at a point. Note that quaternion transformations (which come from $$S^3$$) does cover $$\mathbb{R}P^3$$ so you would not run into gimbal lock. But many of our physical systems rotations work by (continuously) rotating about 3 axes.
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 What are the two types of polysaccharides classified on the basis of their function ? from Biology Biomolecules Class 11 Assam Board
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Download the PDF Question Papers Free for off line practice and view the Solutions online.
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Class 10 Class 12
What are the two types of polysaccharides classified on the basis of their function ?
Two types of polysaccharides on the basis of their functions are -:
(1) Structural polysaccharides which provide structural support. e.g. cellulose.
(2) Food storage polysaccharides which are used as storage forms e.g. starch
130 Views
Name one monosaccharide which provides energy ?
Glucose
698 Views
What are major elements of a cell ?
Carbon, hydrogen, oxygen and nitrogen are the major elements of a cell.
911 Views
Give the names of two monosaccharide sugars.
Fructose and glucose are two monosaccharides
670 Views
What is the percentage of nucleic acids in cells ?
5 - 7 percent
634 Views
What is the general formula of carbohydrates ?
The carbohydrate has a general formula - CnH2nOn Or (CH2O)n.
596 Views
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# On numerical approximation to stationary distribution of diffusion process
Suppose a vector-valued diffusion process X satisfies the stochastic differential equation $$dX_t = b(X_t)dt + \sigma(X_t) dW_t,$$ in which $W$ is a Brownian motion and $b,\sigma$ are such that strong existence and uniqueness of solution to the SDE hold. Assume also that $X$ has a stationary distribution. Sufficient condition for the existence of a stationary distribution can be found in, e.g. Khasminskii, Stochastic Stability of Differential Equations. In a recent book by Mao and Yuan, Stochastic differential equations with Markovian switching, a Euler Mariamman method is proposed to numerically approximate the stationary distribution of $X$. But there is no mentioning on the efficiency of the algorithm, say, the convergence rate. What will be the convergence rate? And are there any other methods in the literature available? Many thanks for your comments and suggestions.
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# GCD of Integer and Divisor
## Theorem
Let $a, b \in \Z_{>0}$ be strictly positive integers.
Then:
$a \divides b \implies \gcd \set {a, b} = a$
## Proof
We have:
$a \divides b$ by hypothesis
$a \divides a$ from Integer Divides Itself.
Thus $a$ is a common divisor of $a$ and $b$.
$\forall x \in \Z: x \divides a \implies x \le \size a$
As $a$ and $b$ are both positive, the result follows.
$\blacksquare$
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How do you solve 8b^2-12b=0?
Mar 20, 2017
$b = 0 , \text{ or } b = \frac{3}{2}$
Explanation:
Factorise by common factors then consider each product separately.
$8 {b}^{2} - 12 b = 0$
$\implies 4 b \left(2 b - 3\right) = 0$
$\text{either } 4 b = 0 \implies b = 0$
$\text{or } 2 b - 3 = 0 \implies b = \frac{3}{2}$
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# Schrödinger operators with δ- and δ′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions
@article{Behrndt2013SchrdingerOW,
title={Schr{\"o}dinger operators with $\delta$- and $\delta$′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions},
author={Jussi Behrndt and Pavel Exner and Vladimir Lotoreichik},
journal={Reviews in Mathematical Physics},
year={2013},
volume={26},
pages={1450015}
}
• Published 29 June 2013
• Mathematics
• Reviews in Mathematical Physics
We investigate Schrodinger operators with δ- and δ′-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result, we prove an operator inequality for the Schrodinger operators with δ- and δ′-interactions which is based on an optimal coloring and involves the chromatic…
61 Citations
## Figures from this paper
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## References
SHOWING 1-10 OF 57 REFERENCES
• Mathematics
• 2012
Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are
• Mathematics
• 2010
We consider the two-dimensional Schrödinger operator with a δ-potential supported by curve. For the cases of infinite and closed finite smooth curves, we obtain lower bounds on the spectrum of the
• Mathematics
• 2013
We consider a generalized Schrödinger operator in L2(R2)?> with an attractive strongly singular interaction of δ′ type characterized by the coupling parameter β > 0 and supported by a C4 smooth
• Mathematics
• 2009
We study singular Schrodinger operators with an attractive interaction supported by a closed smooth surface A in R^3 and analyze their behavior in the vicinity of the critical situation where such an
• Mathematics
• 2003
We study the Schrodinger operator −Δ − αδ(x − Γ) in L2(3) with a δ interaction supported by an infinite non-planar surface Γ which is smooth and admits a global normal parametrization with a
• Mathematics
• 2007
Abstract.We analyze Schrödinger operators whose potential is given by a singular interaction supported on a sub-manifold of the ambient space. Under the assumption that the operator has at least two
• Mathematics
• 2003
We prove an approximation result showing how operators of the type −Δ − γδ(x − Γ) in , where Γ is a graph, can be modelled in the strong resolvent sense by point-interaction Hamiltonians with an
• Mathematics
• 2008
We consider Schrodinger operators in L2(R3) with a singular interaction supported by a finite curve Γ. We present a proper definition of the operators and study their properties, in particular, we
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# How to present features for time series data to a regression model?
I just want to preface this post with the fact that I have absolutely no idea what I am doing and just starting to dip my toes into this field.
I have this system that fires an alert whenever certain feature thresholds of the system are crossed. When an alert is fired, we have people with highly specialized domain knowledge look at those features and its current state and previous states and then bucket the triggered alert belonging to either events A, B, or C.
Right now, I am tasked with removing this manual process by automating this with a machine learning model (I wish it was as simple as hardcoding IF/ELSE statements). So, I have this data set with F number of features (I narrowed it down to these features because these were what the specialists with domain knowledge recommended) but I also want to feed to my model what these features looked like H hours ago. My question is how should I represent these F features for each H hour (where each hour represents the state of the system at that particular time)? Should they be their own "unique" feature (meaning I will end up feeding FxH features into my model).
For example, this is what my current dataset looks like below. Every record in my dataset represents the state of the system (the "state" of the system would be defined by the features) during an alert at a given time interval (these intervals are set 1 hour apart). How do I show my ML model all feature values for all time intervals for each alert? Do I need to transpose my dataset such that each row represents 1 alert and all of its time intervals?
alert_id time_interval feature_1 feature_2
1234 0 00 0.06
1234 1 12 1.15
1234 2 15 0.98
1234 3 12 0.00
1234 4 00 0.00
1234 5 00 0.00
5678 0 00 0.00
5678 1 00 0.00
5678 2 00 0.00
5678 3 18 1.32
5678 4 34 -1.05
5678 5 12 0.52
9123 0 00 0.00
Side question: would a simple model such a random forest, logistic, or SVM be appropriate for this?
• a small example of your actual data might be useful here. prepare it as a csv file . Apr 11 '20 at 10:50
• Hi @IrishStat, I have updated my problem with a small sample Apr 12 '20 at 8:57
• Given that an alert has occurred are the values for the features the readings taken 1 hour ago , 2 hours ago ... 5 hours ago ? Apr 12 '20 at 17:23
• @IrishStat yep, that is correct. Each row represents the features of each alert for every i-th hour Apr 12 '20 at 18:19
• Is feature 1 always 0.0 when the alert is initiated... If so then if one could predict feature 1 from both the past of feature 1 and the past of feature 2 ... does that make sense ? Apr 12 '20 at 18:45
3) You may indeed employ all of the $$FxH$$ features in your model, as you've described, provided the total number of features doesn't blow up in comparison to the number of data points (alerts) in your training set
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#### [SOLVED] Why does photon have only two possible eigenvalues of helicity?
Photon is a spin-1 particle. Were it massive, its spin projected along some direction would be either 1, -1, or 0. But photons can only be in an eigenstate of $S_z$ with eigenvalue $\pm 1$ (z as the momentum direction). I know this results from the transverse nature of EM waves, but how to derive this from the internal symmetry of photons? I read that the internal spacetime symmetry of massive particles are $O(3)$, and massless particles $E(2)$. But I can't find any references describing how $E(2)$ precludes the existence of photons with helicity 0.
#### @Arnold Neumaier 2012-06-08 17:17:23
It derives not from the internal symmetry itself but from the fact that it is a gauge symmetry.
Your symmetry group assignments are not those of the symmetry group but of the little group of the representation. If you assume in addition that the representation is irreducible, you end up in the massless case (with little group ISO(2)=E(2)) with a helicity representation, which picks up from a vector representation only the transversal part, corresponding to a gauge symmetry. Because of reflection symmetry (parity), there are two helicity degrees of freedom. Under the connected part of the Poincare group, this splits into two irreducible representations of fixed helicity, corresponding left and right circular polarization.
This is described in full detail in Section 5.9. of the quantum field theory book (Part I) by Weinberg. In particular, the 2-valuedness (rather than the 3-valuedness) of the helicity is discussed after (5.9.16).
#### @Siyuan Ren 2012-06-09 13:25:17
That book has a chapter on massless particles, but does not mention E(2)-like little group.
#### @Arnold Neumaier 2012-06-10 10:23:02
@KarsusRen: It mentions it on p.70 under the name ISO(2), which is just an alternative tradition for writing E(2).
#### @Ben Crowell 2013-08-31 20:38:40
A freely available presentation by Nicolis that follow's Weinberg's is here: phys.columbia.edu/~nicolis/GR_from_LI.pdf
#### @Incnis Mrsi 2014-08-14 16:27:55
@Arnold Neumaier: do you know a simple explanation how the Poincaré sphere structure appears directly from representations?
#### @Arnold Neumaier 2014-08-17 12:50:35
@IncnisMrsi:There are two helicity degrees of freedom, and any 2-level system has a fundamental SU(2) representastion, described by a poincare sphere = bloch sphere.
#### @Incnis Mrsi 2014-08-17 15:47:12
Poincaré sphere is equivalent to Bloch sphere in the sense of quantum information. Physically equivalent they are not because of different groups. Actually I didn’t read postings completely, neglected to look at Nicolis’ paper, and had today to invent the concept of little group (including E(2)) myself, from the Lorentz group. While I discovered it myself, I learned that E(2) action on the Poincaré sphere is not transitive. It doesn’t make circular polarization to anything else (I just discovered that linear polarization isn’t Lorentz invariant – interesting). This is an answer.
#### @moshtaba 2019-05-16 18:46:30
@ArnoldNeumaier I'm very interested in weinberg book Vol.1 but I can't find exactly where he explain the operator J3 in equation (2.5.39) couldn't have eigenvalue 0. (I refered to this equation only because it is a central result that considered is as definition) I only find in page 90 he brings topological reasoning for helicity could be integer or half-integer. but where he says this half-integer couldn't be zero?
#### @Arnold Neumaier 2019-05-17 08:59:14
@moshtaba: The Lorentz gauge condition together with gauge invariance eliminates this possibility.
#### @moshtaba 2019-05-17 09:28:00
@ArnoldNeumaier Thanks, but could you please refer to a specific page of Weinberg's book (as you said in your answer he discussed this problem in Vol.1) where he explicitly (or implicitly but exactly) explains this elimination?
#### @Arnold Neumaier 2019-05-17 12:11:28
@moshtaba: Section 5.9, after (5.9.16). Note that there are only two polarization vectors $e^\mu$.
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The aim of this study is to investigate whether there is a potential mismatch between the usability of a statistical tool and psychology researchers’ expectation of it. Bayesian statistics is often promoted as an ideal substitute for frequentists statistics since it coincides better with researchers’ expectations and needs. A particular incidence of this is the proposal of replacing Null Hypothesis Significance Testing (NHST) by Null Hypothesis Bayesian Testing (NHBT) using the Bayes factor. In this paper, it is studied to what extent the usability and expectations of NHBT match well. First, a study of the reporting practices in 73 psychological publications was carried out. It was found that eight Questionable Reporting and Interpreting Practices (QRIPs) occur more than once among the practitioners when doing NHBT. Specifically, our analysis provides insight into possible mismatches and their occurrence frequencies. A follow-up survey study has been conducted to assess such mismatches. The sample (N = 108) consisted of psychology researchers, experts in methodology (and/or statistics), and applied researchers in fields other than psychology. The data show that discrepancies exist among the participants. Interpreting the Bayes Factor as posterior odds and not acknowledging the notion of relative evidence in the Bayes Factor are arguably the most concerning ones. The results of the paper suggest that a shift of statistical paradigm cannot solve the problem of misinterpretation altogether if the users are not well acquainted with the tools.
Several studies which focus on the researchers’ understanding of p-values and confidence intervals demonstrate the prevalence of misinterpretation of these common statistical tools (Haller & Krauss, 2002; Hoekstra et al., 2014; Morey, Hoekstra, Rouder, & Wagenmakers, 2016; Oakes, 1986). Moreover, two recent large-scale survey studies showed that statistical misinterpretation is pervasive across different fields (X.-K. Lyu et al., 2020) and still prevails in the field of psychology (Z. Lyu et al., 2018). Haucke et al. (2020), Hoekstra et al. (2012), and Fidler and Loftus (2009) also demonstrated that there was a clear difference between what researchers can conclude versus what they want to conclude from frequentist statistical results in their experiment. This current prevalence of misunderstanding of frequentist statistical tools might reveal an underlying and potential mismatch between what psychology researchers can achieve and what they would like to do with these tools.
Meanwhile, some researchers point out that Bayesian statistics and thinking are more in line with researchers’ interests than frequentist statistics (Assaf & Tsionas, 2018; Cohen, 1994; Dienes, 2011; Haller & Krauss, 2002; Haucke et al., 2020; Hoekstra et al., 2014; Iverson et al., 2009; Morey, Hoekstra, Rouder, Lee, et al., 2016; Vandekerckhove et al., 2018; Wagenmakers et al., 2018). The reason is that Bayes’ rule allows computing the so-called ‘inverse probability’, that is, the probability of a hypothesis being true given the observed data. The inverse probability thus clearly contrasts with the p-value, which gives the probability of observing the data (or more extreme data) given a null hypothesis is true. Various researchers have argued that the inverse probability is more aligned with researchers’ interests. For this reason, it is often concluded that Bayesian analysis is better fitted to the needs of researchers than frequentist analysis, and therefore Bayesian inference has been advocated as a replacement of frequentist inference (Wagenmakers, 2007). Furthermore, Null Hypothesis Bayesian Testing (NHBT), which employs Bayes factors (BFs) comparing a null hypothesis to an alternative hypothesis, is seen as an improvement of NHST as it allows the user to quantify relative evidence for either of the two competing hypotheses (Wagenmakers et al., 2018). In particular, the direct possibility of quantifying evidence for the null hypothesis relative to the alternative hypothesis is often viewed as a clear practical example. Also, the fact that the BF in NHBT appears not to be affected by optional stopping of data collection, is often seen as an advantage (e.g. see Rouder, 2014; Tendeiro et al., 2021).
NHBT tests a precise point-null hypothesis against a general composite alternative hypothesis (Ly et al., 2016; Robert et al., 2009). The alternative hypothesis is the aggregate of all the other possible values of the parameter, combined with a specific prior probability function (or “within-model prior probability distribution”). The BF test statistic, here denoted BF01, is the marginal likelihood ratio of the null hypothesis (H0) over the alternative hypothesis (H1). It measures the data evidence for H0 relative to H1 or specifically, how many times more likely it is to find the observed data if H0 is true than if H1 is true.
The interpretation of the BF is twofold (Tendeiro & Kiers, 2019). Firstly, the BF is the likelihood ratio of the observed data given two competing hypotheses. For instance, a BF01 of 2 means that the data are twice more likely to be observed under the null than under the alternative. Secondly, the BF is the change factor in going from prior odds (i.e., the ratio of prior probabilities of the two hypotheses) to posterior odds (i.e., the ratio of posterior probabilities of the two hypotheses given the data). Specifically, the posterior odds equal the prior odds multiplied by the BF. In formula:
$(1)PostOdds01=BF01×PriorOdds01$
where the indices “01” indicate that the odds and the BF are given for H0 versus H1.
Hence the BF indicates the shift of our prior belief on the relative plausibility of two competing hypotheses to what our belief should be now we have observed the data. A BF01 of 2 suggests that the relative degree of belief between the two hypotheses should shift (by a factor 2) towards the null relative to the alternative, after observing the data. Only in the special case where the prior odds equal 1, the BF equals the posterior odds.
Currently, the use of Bayesian analysis and in particular NHBT is getting more frequent and common (Tendeiro & Kiers, 2019, p. 774). Although much existing literature describes why Bayesian testing is better than frequentist testing in terms of the usability of these statistical tools, quite a bit of criticism has appeared on the approach as well (e.g. Gelman & Shalizi, 2012; Robert, 2016) and possible problems with it should not be overlooked. As indicated for instance by Tendeiro and Kiers (2019), NHBT might be prone to misinterpretations, and like NHST, need not match perfectly with the researchers’ questions either. Because of such potential misinterpretations and misunderstandings, practitioners are advised to have a sufficient and basic understanding of these tools before using them. Otherwise, the poor use of Bayesian methods could lead to a set of problems of its own.
In the light of the often-heard advice to use Bayesian analysis by shifting from the use of p-values to BFs, in-depth scrutiny of the latter is needed to investigate whether it is indeed well matched with researchers’ interests. This leads us to the main research question: “Is there a mismatch between what psychology researchers can conclude and would like to achieve with NHBT?”
The present paper consists of two parts. First, a literature review of published applied studies in which researchers employed NHBT is conducted for exploring the potential existence of misinterpretations. We highlight what researchers seemingly would like to achieve through NHBT and what may be common misinterpretations. Secondly, a survey study on the interpretation of the BF associated with NHBT among actual researchers is conducted to assess the potential mismatch mentioned above. The format and the style of the survey are analogous to previous survey studies on the use of p-values and CIs (Haller & Krauss, 2002; Hoekstra et al., 2014).
### Method
#### Selecting the set of papers to study
The targeted publications are empirical psychological research papers that employed Jeffreys’ BF-based null hypothesis testing in their empirical research. Google Scholar offers a straightforward way to broadly search for scholarly literature. Therefore, a search on Google Scholar using the search terms [“psychology”, “Bayes factor” and “Bayesian test”] was conducted. More specifically, the terms were placed under the search bar with all of the words in the advanced search menu. Besides, it was chosen that the publications should be dated before and including 2020. We did this on 2020/11/16 and obtained 408 references. The use of keywords “psychology” and “Bayes Factor” corresponds to our research interest. The phrase “Bayesian test” was added for two reasons. First, it helped at purifying the returned references to match our research interest (i.e., psychological publications using NHBT). Second, it reduced the number of returned references from around 13,000 to 400, which made the investigation accomplishable. Despite the arguably small sample size, our search suffices to answer our research question on exploring the potential mismatch and the different possible misinterpretation.
Despite our keywords, many results from the search did not lead to targeted papers. For this reason, an exclusion and inclusion procedure was employed. That is, the search led to a large number of methodological rather than empirical papers, and also to non-psychological papers. Of these 408 papers, 228 were considered to be methodological papers as judged from their abstracts, titles, keywords, and journals (e.g., method development and statistical tutorials), and 39 were considered to be non-psychological papers (as was concluded from the fact that their journals were not covered by APA PsycInfo). Furthermore, 23 studies mainly employed Bayesian information criteria, which do not match our interest in the BF for use in NHBT and were therefore excluded. Of the remaining references, the following were excluded for various reasons: 13 papers because they were not in English, 8 publications were PhD theses (and hence it would complicate the procedure concerning the fact that they contain multiple studies including methodological papers, non-psychological papers, and duplications of published papers), 1 master thesis, 6 duplications, 3 preregistered plans of research, 1 appendix, 2 in press, 5 preprints, 1 paper without employing any Bayesian statistical tool, 1 irrelevant result (library declaration and deposit agreement) and 2 other studies which were about criminology and accounting. Preprints and in press papers were excluded since these papers are still subject to change. One paper not covered by APA PsycInfo was included nevertheless since it was published in the journal “Comprehensive Results in Social Psychology”, which indicates it is a psychological study. One paper that used the replication BF instead of a BF for null hypothesis testing was excluded. One paper that mainly used BFs for finding the best out of 12 models rather than testing an alternative hypothesis against the null hypothesis was excluded as well. Also, any reported replication BF in other papers was either excluded or ignored. Lastly, one reference is lost due to a downloading error. After these exclusions and inclusions, we ended up with 73 publications fitting our in- and exclusion criteria.
#### Material
A recording sheet was created for marking researchers’ reporting and interpreting behaviour concerning the use of the BF. This was based on a pilot study in which ten references were randomly selected from the 73 targeted papers. These papers were analysed and evaluated in detail. The results of this pilot analysis suggested to us that there were at least 8 recurrent Questionable Reporting or Interpreting Practices (QRIPs) in which the researchers gave incomplete, and (possibly) misleading statements for interpreting the BFs. These were:
1. Explicitly describing, defining, or elaborating BFs as posterior odds ratios: Mentioning that the BF is a posterior odds ratio of two competing hypotheses given the data, which as can be seen from formula (1), is only true if the prior odds equal 1.
2. Null and alternative hypotheses were not specified: In particular, it is not mentioned whether a one-sided or two-sided test is used, and/or the statistical representation of the hypothesis (such as H0 = 0, HA > 0) is missing.
3. Simplifying the alternative hypothesis by ignoring the use of its associated prior distribution for computing a marginal likelihood1: The reason or justification for the chosen within-model prior is not provided or it is not even mentioned what prior distributions were used with the alternative hypothesis.
4. Presenting BFs as evidence for one hypothesis without mentioning it is relative to another competing hypothesis: It is not mentioned that the reported strength of the evidence for one hypothesis depends on what the other hypothesis is.
5. Making an absolute statement concerning the truthfulness of a hypothesis by using BFs: It is simply mentioned that there is an effect or that there is no effect, while in fact, on the basis of statistical measures one cannot confirm or disconfirm a hypothesis; the best one could say is that the BF gives evidence more in favor of one hypothesis than another.
6. Using BFs as if they are posterior odds ratios2: Based on an observed BF it is concluded that either of the two hypotheses is more probable or likely. This would be true if the BF gives the posterior odds ratio, but as can be seen from formula (1) it does not. In fact, BFs can only tell under which hypothesis the observed data is more likely to occur (see p. 2).
7. Considering BFs as measure of effect size: ascribing a small BF to a small effect size, or a big BF to a big effect size.
8. Mismatch between the research hypothesis and the statistical hypothesis: the two used statistical models do not coincide with the researcher’s hypothesis. For instance, one might have a one-sided substantive prediction (e.g., Drug A has positive effect) for the alternative hypothesis, but the associated statistical model is two-sided.
For each article, it was recorded whether each QRIP was present or absent. For each identified QRIP, the associated statement in the paper was also recorded for possible reevaluation purposes. In addition, any further Questionable Reporting or Interpreting Practice other than the 8 QRIPs above was recorded.
Researchers’ reporting practice might differ from each other in other respects as well. Some might neglect to report effect sizes, and some might ignore the importance of mentioning the prior model for the BF. In any case, drawing a conclusion solely on the basis of a single BF to us seems unideal, because the same BF value may refer to quite different situations with respect to uncertainty and effect size. For example, the single sample JZS based test in the BF3 (Morey & Rouder, 2018) R routine gives BF01 = 4 both for d = .14 and n = 34, and for d = .07 and n = 738, which clearly differ strongly in effect size and sample size. Here, the BF combines different outcomes into the same degree of evidence, but in the former case the evidence for the null is mainly due to the large uncertainty, while in the second case it is mainly due to the small effect size. People can see a bigger picture if more information is given. To see to what extent other information is actually given in practice, an additional assessment of the reporting practice was carried out by recording the main reported elements in the statistical analysis. These elements include the following: The term “Bayes Factor”, exact value of the BF, descriptive statistics, posterior distribution (including credible interval and highest density interval), posterior probability of a hypothesis, effect size, within-model prior distribution,4 one or more other illustrative within-model prior distributions5 and, lastly, mentioning the link between the BF and the posterior odds. All these statistical elements are crucial to evaluate the results of a study.
### Results
The observed frequencies and percentages for each QRIP are shown in Table 1. The references of the citations in this section are anonymized and referred to their data ID in the dataset since these examples are for illustrative purposes only.
Table 1. Frequencies and percentages (in parentheses) of papers in which the observed Questionable Interpreting and Reporting Practice (QRIP) was observed.
QRIP1- Describing BF as posterior odds 3 (4.1%) QRIP2- Not specifying Null and alternative hypotheses 18 (24.7%) QRIP3- Simplifying the use of the prior model 51 (69.9%) QRIP4- Not mentioning comparison of models 44 (60.3%) QRIP5- Absolute statement 41 (56.2%) QRIP6 - Using BF as posterior odds 13 (17.8%) QRIP7- Considering BF as Effect size 9 (12.3%) QRIP8- Mismatch between statistical and research hypothesis 2 (2.7%) Other QRIPs 18 (24.7 %)
QRIP1- Describing BF as posterior odds 3 (4.1%) QRIP2- Not specifying Null and alternative hypotheses 18 (24.7%) QRIP3- Simplifying the use of the prior model 51 (69.9%) QRIP4- Not mentioning comparison of models 44 (60.3%) QRIP5- Absolute statement 41 (56.2%) QRIP6 - Using BF as posterior odds 13 (17.8%) QRIP7- Considering BF as Effect size 9 (12.3%) QRIP8- Mismatch between statistical and research hypothesis 2 (2.7%) Other QRIPs 18 (24.7 %)
N = 73
#### The confusion between BF and posterior odds (QRIP 1 & 6)
The actual description of a BF as posterior odds is not prevalent in the sample since there are only 3 papers describing the BF as posterior odds. For instance, article [13] mentioned “The test yields a statistic—the Bayes factor, B01—that is a posterior odds ratio relating the probability that the null hypothesis is true to the probability that the alternative hypothesis is true, given the data”. Another example would be from article [21] “…BF10 quantifies the posterior probability ratio of the alternative hypothesis as compared to the null hypothesis”.
However, interpreting BF as posterior odds (QRIP 6) is four times more frequent than describing it as such (QRIP 1). This phenomenon comes with a pattern of reporting like “H1 is (much) more likely or probable than H0” if BF10 is (much) higher than 1. For instance, article [38] wrote “The Bayesian test for correlations resulted in a Bayes Factor of BF01 = 5.88, indicating that the null hypothesis (i.e., absence of correlation) is ~6 times more likely than the alternative hypothesis (i.e., presence of correlation)”. Here we considered that in practice the words likely and probable are used interchangeably and have the same meaning.6 Then concluding that H1 is (much) more likely or probable than H0 clearly is correct if the posterior odds (i.e., the probability that H1 is true divided by the probability that H0 is true, given the data) is (much) higher than 1. However, this is not correct on the basis of the BF, because a BF higher than 1 may be associated with any posterior odds ratio (i.e., higher than 1 or smaller than 1) depending on the prior odds, because the posterior odds equal the BF times the prior odds. This reporting pattern was observed in multiple publications. It was reported in different manners as well, such as “A model with an effect of soundtype was also more likely than the null model of no difference in soundtype (BF10 > 100,000). Finally, a model in which these factors interact was more likely than a model with no interaction (BF10 = 4.95)” [47] and “Bayes factors showed that the alternative hypothesis with scores lower than chance was over 1,000 times more likely given the data” [3]. The difference of the frequencies between QRIP1 and QRIP 6 could be ascribed to the fact that not every author reported BFs with an elaboration about the definition of the BF.
#### Research hypotheses and statistical hypotheses (QRIP 2 & 8)
In 18 publications, the alternative hypotheses were not clear or were not mentioned at all (QRIP 2), which leads to ambiguity in the reported statistics. Please note that the actual number is bigger than 18 since we did not count the papers that conducted ANOVAs. In these cases, the null and the alternative hypothesis are clear because of the nature of ANOVAs (H0: group means are equal, H1: at least one group mean is different from another). Conclusions derived from the tests without the specification of the alternative hypothesis might lead to unsound or invalid conclusions. It can be exemplified by one of the two papers that demonstrated the mismatch between statistical hypothesis and research hypothesis (QRIP 8). The authors of article [10] conducted a study about the effect of nicotine on pupil size and hypothesized that “… pupil size would be smaller after the administration of nicotine, as compared to placebo…”. They found that “Results suggested that pupil size during target assignment in the nicotine condition (M = 4.553, SD = 0.782) was significantly smaller than in the placebo condition (M = 4.681, SD = 0.827), F(1, 28) = 8.232, p = .008, η2p = 0.227, BFincl. = 1.316e+6 (Drug as the best model)”, and concluded, “Results from both NHST and Bayesian analyses suggested that pupil size in the nicotine condition was smaller than pupil size in the placebo condition”. Even though the authors made use of descriptive statistics to reach the final conclusion, the hypothesis tested does not correspond to their research hypothesis. A one-side prediction was formed but ANOVA was conducted (i.e the use of ANOVA implicitly implied the within-model prior was two-sided rather than one-sided.) . Although such behavior might be easier to handle in frequentist testing (since it only involves dividing p-values by 2), it is problematic under NHBT since the value of the BF changes by an unknown amount, because an entirely different prior for the alternative hypothesis is employed.
The researchers from article [16] conducted a study for investigating “… whether alcohol affects saccade countermanding; i.e., whether saccadic SSRT is lengthened during acute intoxication”. However, model comparison by ANOVA was conducted, which lead to a similar problem as the example from the previous paragraph.
#### Not specifying the prior distribution used with the alternative model (QRIP 3 & 5)
The alternative hypothesis in NHBT is usually the aggregate of all non-null values. Since the null hypothesis is often associated with “there is no difference between the means” or “there is no association”, the interpretation of the alternative could be “there is a difference between the means” or “there is an association”. It may be easily overlooked that the alternative hypothesis actually incorporates a prior probability density of all specified non-null values, and hence is just one instance of an alternative hypothesis specifying that there is an effect while many other instances are possible. For this reason, researchers should provide justification for the chosen within-model prior. Furthermore, one cannot simply conclude the presence or the absence of an effect on the basis of the BF since it only quantifies relative evidence for the two competing hypotheses. However, 51 papers simplified the use of the within-model prior and 41 papers made absolute statements based on their observed BFs.
It is hard to illustrate that prior distributions have not been specified, but, as it did not happen that often, it may be illustrative to quote a paper in which the prior distribution was specified and justified, see [35]: “the prior distribution as a truncated gaussian with µ = 1 and σ = 2 (…) with a lower bound of the distribution at 0 and an upper bound at 2…” and “These values have been chosen taking into account the typical range of effects observed in previous studies”. Article [63] provided a similar reason for their chosen prior: “Effect sizes used to set priors were obtained from related results of previous studies”. The authors of article [42] added “Typically, researchers entertain a distribution of possible alternatives to the null, some of which are judged more likely than others. For example, we may anticipate a high probability of a small-to-medium effect, a moderate probability of a large effect, and a very remote chance of a gargantuan effect. (…) The prior distribution, therefore, influences the BF, and the choice of a prior distribution is at the researcher’s disposal; it may be based on knowledge about likely values established in previous work, or it may be chosen to be minimally informative”. As can be seen from Table 1, in almost half of the cases no such explanations were given (n = 24) or the priors model were not reported (n = 27).
Two examples of QRIP 5 follow. Article [22] reported that “Of the 24 resulting comparisons, four produced a Bayes factor greater than 1 (i.e., providing evidence for an effect of initial category structure).(…). The other three comparisons showed a difference between the one-dimensional linear boundary and the other initial category structures for the Shepard circles” where the italics are ours, to pinpoint the absolute statement. Article [41] asserted “Both frequentist and Bayesian analyses revealed no main effects for responses of ‘environment,’ errors on SART, and no interaction for any of the measures studied”. Needless to say, both absolute assertions are questionable and unjustified by the statistical tool used.
#### Presenting BFs as evidence for one hypothesis without mentioning it is relative to another competing hypothesis (QRIP 4)
Interpreting BFs as absolute rather than relative evidence prevails among practitioners in the sample in which 44 papers demonstrated this questionable practice. This QRIP also comes with a specific pattern of reporting like “The Bayes Factor provides evidence in favor of H1 (or H0)”. These reports did not mention that the strength of evidence always depends on the two specified models. Suppose one finds BF01 is bigger than one. It is unideal to simply conclude that the evidence is in favour of the null hypothesis (“there is no effect” or “there is no difference between the means”) since the BF is always relative to another competing hypothesis and the value of the BF depends on the prior distribution associated with the alternative hypothesis. Article [72] stressed this notion by mentioning that “The Bayes factor BF+0 quantifies the evidence that the data provide for H+ (i.e. the presence of the compensatory control effect) relative to H0 (i.e. the absence of the compensatory control effect)”. The researchers of the article obtained a BF of 5.41 and interpreted the result as “This means that the data are about 5.41 times more likely under the null model including only religiosity, compared to the alternative model that also includes the control-threat manipulate” (italic is ours).
#### Considering the Bayes Factor as a measure of effect size (QRIP 7)
The value of a BF does not indicate the magnitude of an effect or effect size. There are 9 papers in the sample that nevertheless did make a statement to this effect. Article [54] reported “As both the confidence interval and the Bayes factor do not point towards a true difference and the t-test is borderline significant, this can be considered a very small or non-existent effect.” A similar questionable reporting practice is in article [15]: “A repeated-measures ANOVA with Cue (novel; standard) and Electrode (Fz; Cz; Pz) as factors showed that there is substantial evidence for a null effect (BF01 = 5.676), suggesting that the effect of expectations on the N2 was not very strong.”, article [32] wrote “Bayesian statistics again supported that sexual motivation had a strong effect on attractiveness ratings in the control condition (Bf1,0 = 25.99)”, and article [16] wrote “Both a Pearson’s correlation and a Bayesian correlation analysis revealed no strong relationship between the difference in relative change from pre-drink to post-drink between alcohol and placebo for the saccadic task and the manual tasks (r = 0.197, p = 0.22, BF = 0.41)”. For article [16], it has to be noted that the Pearson’s correlation did support the claim “there is no strong relationship in the experiment”, however, such assertation cannot be supported by the BF.
#### Other QRIPs
The most common QRIP we found in the sample was basing conclusions on an inconclusive BF (n = 16). For this part of the study, we simply defined a BF as inconclusive if its value was between 10-0.5 and 100.5, as Jeffreys (1948) calls these BFs not worth more than a bare mention.7 Any BF close to 1 indicates the evidence is in favor of one hypothesis as much as of the other competing hypothesis. It is described as absence of evidence (Keysers et al., 2020). However, the researchers in the studied papers confuse absence of evidence with evidence of absence. For instance, article [10] asserted that “A chi-squared test with a contingency table revealed that responses in the placebo condition had no association with those in the nicotine condition, χ2 (2, N = 29) = 2.193, p = .139, BF10 = 1.17”. The BF10 only suggests that the likelihood of obtaining the observed data under the null model is more or less the same as under the model with interaction. This questionable reporting practice might inherit from NHST, as can be illustrated by the following example. From Article [21]: “The length-scale parameter λ did not differ significantly between the three experiments (all p > 0.5, BF10 = 1.1)”.
Two other rare QRIPs were observed. Article [18] gave a wrong impression about the function of the BF: “The JZS-Bayes factor gives the probability of evidence that the null hypothesis is true”. Article [61] gave a problematic interpretation of the alternative hypothesis: “The results of a Bayesian t test (with the hypothesis H1: d’PHAd’HC ) shows a Bayes factor (BF01) of 0.51, suggesting that hypothesis H1 (that the PHA shows a better detection performance than the HC) is 1.96 times more likely than hypothesis H0 (that the PHA and the HC do not differ significantly), providing only weak anecdotal support for H1”. The given statistical alternative hypothesis posited that there is a difference, but a one-sided interpretation was given.
The quotations above and additional examples from the studied papers for illustrating the presence and the absence for each QRIP are summarized in Appendix A.
#### Reported Statistical Elements in the papers
The frequencies and the percentages observed for each reported statistical element is shown in Table 2.
Table 2. The frequencies and the percentages (in parentheses) observed for each reported statistical element
BF 71 (97.3%) Exact value of BF 66 (90.4%) Descriptive Statistics 68 (93.2%) Posterior distribution 13 (17.8%) Posterior probability of a hypothesis 5 (6.8%) Effect size 59 (80.8%) Within-model prior distribution 46 (63%) Other illustrative within-model prior distribution 3 (4.1%) Linkage between BF and posterior odds 7 (9.6%)
BF 71 (97.3%) Exact value of BF 66 (90.4%) Descriptive Statistics 68 (93.2%) Posterior distribution 13 (17.8%) Posterior probability of a hypothesis 5 (6.8%) Effect size 59 (80.8%) Within-model prior distribution 46 (63%) Other illustrative within-model prior distribution 3 (4.1%) Linkage between BF and posterior odds 7 (9.6%)
Almost all papers reported BFs except two papers providing posterior probabilities of the models instead. Furthermore, five papers did not give exact values, but only reported the gradation of the BFs. An example can be found in the quotation from article [22] under section QRIP 3 & 5, where it was only mentioned for which tests BFs larger than 1 were found. Five other papers did not provide any descriptive statistics. One of those is a stimulating study, three studies are reanalyses of existing datasets, and one paper is about the models goodness of fit. A small minority of publications made use of Bayesian estimation tools (e.g., credible interval and/or highest density interval) and/or posterior probability distributions next to their reported BFs. However, most of the papers reported effect sizes. Despite the importance of the within-model prior (i.e., probability distribution used with the alternative model) for the BF, 27 publications did not specify or mention the chosen within-prior model for the tests and 3 papers provided BFs associated to priors other than their own specified prior models as illustrative examples for the readers.
Lastly, the relationship between the BF and the posterior odds were elaborated in 7 publications. For instance, article [13] stressed that “We report the raw Bayes factor for each single-degree-of freedom analysis, which allows readers to draw their own subjective conclusions about the degree to which the evidence favors either or neither hypothesis for a given comparison”, which pinpoints the difference between evidence and degree of belief. Article [74] explicitly mentioned “The Bayes factor indicates the change from prior to posterior odds brought about by the data.”. Articles [31], [42], [60], and [72] also elaborated the linkage by introducing Bayes’ theorem.
### Discussion
The results of Study 1 indicate that at least 8 Questionable Reporting and Interpreting Practices were signaled repeatedly in the set of studied papers. More specifically, 67 (or 92% of) sampled papers showed at least one of the 8 pre-specified QRIPs. Among these QRIPs, simplifying the use of the within-model prior by using the default prior or not mentioning the chosen prior (QRIP 3), presenting the BF as absolute rather than relative evidence (QRIP 4), and making an absolute statement based on the BF (QRIP 5) are the most common ones in the sample. Moreover, a large majority of the publications reported BFs with descriptive statistics, prior model and effect size, but a small minority did not. However, the use of Bayesian estimation tools and posterior probability distributions is rare, which can possibly be attributed to the choice of keywords in the search on Google Scholar since the keywords pertain to testing only. Providing BFs based on other illustrative within-model priors is observed in a small number of papers despite the influence of the prior model on the BF. Additionally, only a few papers demonstrated the linkage between the BF and posterior odds.
The observed QRIPs also suggest there are some potential mismatches between the usability of the BF and researchers’ expectations. Quite a few researchers seem to expect that the BF can establish the presence or the absence of an effect without acknowledging that the chosen prior is just one instance of the alternative hypothesis. Their conclusion based on the BF would be sound if the prior distribution indeed (closely) corresponds to a prior distribution of general interest. However, researchers cannot ensure that this is the case, and it might actually be very unlikely that such general consensus on prior distributions would actually exist. For this reason, providing the information about the chosen prior model and using other illustrative prior distributions is important for the readers and the researchers themselves to make a balanced conclusion. Furthermore, many papers seem to demonstrate the expectation that the BF can offer absolute (or direct) evidence in favor of either of the two competing hypotheses, which is a questionable interpretation. The users should realize that the BF is a relative measure of evidence for the two competing hypotheses. The BF-based evidence for one hypothesis is always dependent on another hypothesis. Failures of recognizing these two notions (the influence of within-model prior and relative notion of evidence) of the BF might lead to a dichotomous mindset and faulty interpretation.
Study 1 faces several limitations concerning its generalizability. The actual number of published papers in psychology which employed NHBT is unknown. The generalizability of this study entirely depends on the search engine on Google Scholar, and on our inclusion and exclusion criteria. Moreover, the use of the keyword “Bayesian Test” in the search led to a great reduction in the number of the returned papers, which might have led to insufficient coverage of all targeted papers. As such, the sample might not be representative of the population. If the engine indeed did return the majority of or all the targeted papers, we would expect that this study gives a general picture of the practice of Bayesian testing in the psychology community as a whole. Otherwise, its generalizability is limited. Despite all these, it has to be stressed that the primary aim of the study was to assess the mere existence of such QRIPs, and it clearly did so in the sampled papers. Future research with larger sample size is needed to explore how prevalent these QRIPs are among psychology researchers in general.
The observed QRIPs might not necessarily imply there is a mismatch between the usability of the tools and researchers’ expectations since there can be a difference between what researchers wrote and what researchers’ expectations are, but it offers valuable insight into what kind of mismatch there could be. For this reason, we next conducted a survey that mainly focuses on what researchers expect from NHBT. The study is similar to previous studies about the misinterpretation of the p-value and the confidence interval, in which the participants were given a hypothetical result of a statistical analysis with a few interpretational statements and were asked to judge if each statement is true or false (Haller & Krauss, 2002; Hoekstra et al., 2014; Oakes, 1986).
### Method
#### Participants
Our targeted participants are psychology researchers who employed the BF-based null hypothesis testing in their empirical research, and Bayesian methodologists. Many of the references in Study 1 were reused since the authors (including correspondence authors and co-authors) of the 228 excluded methodological papers, the 73 targeted papers, and one of the other excluded papers, corresponded to our research interest. For this reason, additional searches on Google (Scholar), Linkedin, and ResearchGate were conducted to verify the identity of the authors and collect their email addresses based on the information that is given in the publication. In total, 805 authors and their email addresses were successfully identified and collected. Out of those, there were 226 duplications, after elimination of which we ended up with 579 authors, which served as the targeted set of participants for our study.
#### Material and procedure
Invitation emails were sent alongside a link to the survey on Qualtrics. The participants were informed that they can leave their email addresses at the end of the survey for receiving the result of the study, including the marking scheme, their score, and the overall mismatch of all participants. They were instructed that a hypothetical result of a Bayesian analysis would be given with 11 interpretational statements in turn, and they would be asked to judge if the given statements were “True” or “False”. The participants could choose the option “Don’t know” if they were uncertain about the truthfulness of the statements. The option “Don’t Know” was introduced since knowing that the participants indeed are uncertain about the truthfulness of a statement is more informative than a randomly chosen answer when in doubt. The statements were presented one at a time and in randomized order, in order to reduce the possibility that the participants could infer the answers based on a comparison of the statements rather than their knowledge.
The presented hypothetical result of a Bayesian analysis was the following:
“A group of researchers conducted an experiment in which three newly developed drugs for a deadly virus have been compared to a control group. The dependent variable is the recovery rate, the null hypothesis posits that the drug does not influence the recovery rate while the alternative hypothesis posits the drug does influence the recovery rate. After the experiment, two-sided Bayesian t-tests with a default prior were conducted. For drug A, it was found that BF10 is 2 while for drug B BF10 of 20 was found. For drug C, BF10 of 0.1 was found.”
The 11 statements were the following:
1. The probability of Drug A having an effect is one-tenth of that for drug B.
2. For Drug B, the probability of obtaining the observed data is 20 times higher under the alternative than under the null.
3. The alternative hypothesis (drug B has an effect) is 20 times more probable than the null hypothesis (drug B has no effect).
4. Drug B demonstrated a positive effect on the recovery rate.
5. There is evidence suggesting that drug A demonstrated an effect on the recovery rate relative to the null hypothesis (there is no effect).
6. For drug C, there is evidence in favor of the null hypothesis (there is no effect).
7. For drug B, the Bayes factor disproved the null hypothesis (there is no effect).
8. The effect of drug B is bigger than the effect of drug A.
9. The strength of drug B affecting recovery rate cannot be known with the given information.
10. If the same group of researchers will conduct the same experiment repeatedly, they can be expected to find the presence of the effect from drug B in 20 out of 21 times.
11. Based on the given Bayes Factor for drug A, drug A demonstrated no effect since the Bayes Factor is smaller than 3.
At the end of the survey, the participants were asked to categorize themselves in one of the following groups: (1) “Psychological researcher”, (2) “Methodologist (and/or Statistician)”, (3) “Methodologist (and/or Statistician) and Psychological researcher”, (4) “Researcher in a field other than psychology” and (5) “Other (Please specify, e.g., Methodologist and Economist)”.
The participants were categorized into three groups for making comparisons in terms of the performance in the questionnaire: psychology researchers group (1), experts in methodology and/or statistics (2 and 3), and non-psychologists group (4). There were 6 participants that chose option (5), who were assigned to the group that seemed to match best with the specification they gave: 5 participants who specialized in methodology (and/or statistics) and a field other than psychology were assigned to the experts group, while one expert in psychology and neuroscience was assigned to the psychology researchers group.
For each of the statements, we now specify the responses that we considered to be the best fitting ones. A detailed explanation of each statement can also be found in Supplement Document S1.
Statements 1, 2, and 3, which mainly focus on the difference between the BF and posterior odds, correspond to QRIPs 1 and 6 in Study 1. To us, the best fitting responses for Statements 1 and 3 are “False” and “Don’t Know”. “False” for us reflects that the given statements do not necessarily follow from the given information since the BF is the marginal likelihood ratio of obtaining the observed data under the two competing hypotheses. The BF is not a posterior probability or the posterior odds of a drug having an effect or not. For the latter, prior odds are needed to calculate the posterior odds ratios, and this important information is missing. Therefore, the truthfulness of the statement could also be considered uncertain because it is based on insufficient information. For Statement 2, the expected response is “True” since the BF can be interpreted as a ratio of the probability of obtaining the observed data under the alternative versus the probability of obtaining the observed data under the null.
Statement 4 is related to QRIP 2 and 8 about the setting of a test (one-sided test and two-sided test). Based on the result from a two-sided Bayesian test, one can only conclude whether there is evidence in favor of the (two-sided) hypothesis “there is an effect” relative to the null. Therefore, to us the best fitting responses for Statement 4 are “False” and “Don’t Know”.
Statements 5, 6 and 7 are relevant to QRIP 3, 4 and 5, which are pertaining to the function of the BF as a relative measure of evidence of two competing hypotheses and the importance of the chosen prior distribution. To us, the best fitting response for Statement 5 is either “True” since it shows that the BF is a measure of evidence in favor of a hypothesis relative to another, or “Don’t Know” since some researchers consider BFs close to 1 as meaning inconclusive evidence (compare the explanation for Statement 11). To us the best fitting responses to Statements 6 and 7 are “False” as they fail to interpret the BF as relative evidence, and, in fact, one cannot prove or disprove a hypothesis anyway.
Statements 8 and 9 are about the linkage between the BF and effect size (QRIP 7). The BF is not a measure of the magnitude of an effect, but it indicates the relative plausibility of getting the data under the two competing hypotheses. Claiming the effect of one drug is bigger than another is impossible with the given information. So, we considered options “False” and “Don’t Know” the best fitting ones for Statement 8 while for Statement 9 this was the option “True”.
For statement 10, we considered the option “False” as the best fitting one, because the BF is a measure of evidence based on the observed data only. It does not tell anything about future data.
Statement 11 is about an ‘inconclusive’ BF value. The BF suggests the absence of evidence, not evidence of no effect, so to us the best fitting response is “False”.
It has to be stressed that these best fitting responses are from the authors’ perspectives only. The ultimate “correct” answers to the statements are up to discussion and debate, and surely, there will be different opinions as to what can and what cannot be concluded.
### Results
In total, 146 researchers responded to our invitation to participate in the study. Of these, 29 only gave consent but they did not proceed with the study, while 9 answered some of the statements but the question for the grouping was not answered. Therefore, these replies with missing data were excluded from the dataset. After the exclusion, the sample size is 108 and the completion rate is 18.7%. The dataset consists of responses from 35 psychology researchers, 57 experts in methodology and/or Statistics, and 16 non-psychologists.
The mean total score of answering the best fitting responses is 7.11 for the psychology researchers, 7.82 for the experts in methodology and statistics and 7.18 for the non-psychologists. The distributions of the total score can be found in Figure 1.
Figure 1. The distribution of the total score (number of best fitting responses given) in the different groups
Figure 1. The distribution of the total score (number of best fitting responses given) in the different groups
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Figure 2 provides the visualization of the percentages of endorsing the best fitting answers for each statement and group. The first observation one could make is that by far most proportions lie between .5 and .8. This implies that within all categories, for most items, there was noticeably disagreement on the answers. Experts performed better than the other groups for most of the items. The difference in performance between the groups is not noticeable in most of the statements except for Statement 3. The proportions for Statement 6 are by far the lowest for all groups; obviously, this finding hints at a special situation with this statement, as we will discuss later on. Overall, the deviation from what we considered the best fitting responses is slightly larger among the applied researchers than the experts, considering the small difference of the means scores between the groups. Surprisingly, such a deviation is even present among the experts in methodology and statistics, since the percentages of endorsing the expected response in Study 2 often clearly fall short of 100%. A final noteworthy finding is the low proportion of best fitting answers for Statement 3 in the psychology researchers group.
Figure 2. The proportions of participants answering the best fitting answer for each statement in each group.
Figure 2. The proportions of participants answering the best fitting answer for each statement in each group.
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Table 3 show the frequencies of endorsing different options for each statement in various groups. The corresponding proportions of endorsing each option can be found in Supplement Document S1.
Table 3. The frequencies of endorsing different options for each statement in various groups (with the expected responses for the statements in the first column).
Group Best fitting Psychologist researchers Experts Non-psychologist True False Don't Know True False Don't Know True False Don't Know S1 F or D 8 20 7 4 45 8 3 9 4 S2 T 24 9 2 41 15 1 9 4 3 S3 F or D 22 8 5 23 25 9 3 9 4 S4 F or D 11 13 11 16 21 20 6 8 2 S5 T or D 21 10 4 42 10 5 10 5 1 S6 F 32 1 2 49 5 3 14 1 1 S7 F 6 25 4 11 40 6 6 10 0 S8 F or D 11 12 12 15 21 21 5 6 5 S9 T 26 6 3 46 10 1 11 3 2 S10 F 2 30 3 3 46 8 0 13 3 S11 F 3 30 2 4 51 2 2 13 1
Group Best fitting Psychologist researchers Experts Non-psychologist True False Don't Know True False Don't Know True False Don't Know S1 F or D 8 20 7 4 45 8 3 9 4 S2 T 24 9 2 41 15 1 9 4 3 S3 F or D 22 8 5 23 25 9 3 9 4 S4 F or D 11 13 11 16 21 20 6 8 2 S5 T or D 21 10 4 42 10 5 10 5 1 S6 F 32 1 2 49 5 3 14 1 1 S7 F 6 25 4 11 40 6 6 10 0 S8 F or D 11 12 12 15 21 21 5 6 5 S9 T 26 6 3 46 10 1 11 3 2 S10 F 2 30 3 3 46 8 0 13 3 S11 F 3 30 2 4 51 2 2 13 1
Note. T and F denote options “True” and “False” respectively, while D denotes option “Don’t know”
As for Statement 1 and 2, most of the participants answered expectedly, with small frequencies of picking the option “Don’t know”. Despite Statement 3 being based on the same QRIPs, there are drops in the proportions endorsing the best fitting answers. Moreover, the “Don’t know” was chosen more often than for Statement 2.
As to Statement 4, there are 11 (31.4%) psychology researchers and 16 (28.1%) experts who believed it to be true even though there is a mismatch between the conclusion and the research hypothesis.
Regarding Statement 5, the majority of the participants gave the best fitting responses and the proportions for psychology researchers and experts are around 70%. Again, only a minority of participants selected “Don’t know” for the statement. Statement 7 has similar statistical figures as Statement 5. As for Statement 6, an overwhelming majority of the participants chose the option “True” regardless of the groups.
For Statement 8, concerned with the association between the BF and effect sizes (QRIP 7), the proportions of endorsing the expected option are around 70% in each group. For each group, roughly the same numbers of participants answered “Don’t know” and “False”. For Statement 9, most of the participants (n = 83) picked “True” and only 6 participants chose “Don’t know”. Lastly, the proportions of answering the best fitting option (“False”) for Statements 10 and 11 are noticeably higher than the other statements except Statement 1.
### Discussion
The results of Study 2 suggest: (1) a potential mismatch between the participants’ interpretation and what can actually be expected from the BF, and (2) a potential disagreement on what can be and what cannot be concluded based on the BFs among the methodologists. Before discussing further the implications of Study 2, a comparison of the results between Study 1 and Study 2 is made. In Study 1, the percentages of the studied papers describing BF as posterior odds (QRIP 1) and using it as posterior odds (QRIP 6) are 4.1% and 17.8% respectively while in Study 2 only 24% of psychology researchers (n = 9) gave the expected response for all three Statements 1, 2 and 3 that are associated with these QRIPs. Most researchers did not interpret the BF as posterior odds in the studied papers, but more than half of the participants in the survey seem to expect that the BF can actually be used as posterior odds (Statement 3). This might suggest that posterior probability and posterior odds are the tools researchers are actually looking for rather than the BF.
In 51 out of 73 publications, simplified use of the prior probability distribution for the alternative hypothesis (QRIP 3) was observed. Using the BF without noting it is relative (QRIP 4) is also common among the studied papers. In Study 2, this phenomenon manifested itself with Statement 6. As for Statements 8 and 9, about the linkage between effect size and BF (QRIP 7), the results of both studies agree as the data show that a fairly small percentage (roughly 20-30%) in each group made such a link. More specifically, only 4 psychology researchers (11.4%) gave unexpected responses to both statements. As for the potential misinterpretation of inconclusive BFs (Statement 11), both studies pointed out that most of the participants by far correctly distinguished the concepts evidence of absence and absence of evidence. The prevailing misinterpretation of frequentist statistical tools is considered as evidence suggesting that there is a discrepancy between the function of the tools and the researchers’ expectations towards these tools (Gigerenzer, 2004; Haller & Krauss, 2002). Meanwhile, Bayesian statistics is proposed as an ideal substitute for frequentist tools since it coincides better with researchers’ expectations and needs (Gigerenzer, 2004; Haller & Krauss, 2002; Haucke et al., 2020). This study aimed at investigating the potential mismatch of the NHBT with psychology researchers. The results of Study 2 suggest that a mismatch indeed exists for a considerable number of psychology researchers.
The results suggest there are two clear mismatches among the psychology researchers. A number of participants apparently expected one can draw direct probabilistic statements about the hypotheses based on the BF (Statement 3) and interpret the BF as evidence for one hypothesis without noting that it is a relative measure of evidence on two specified hypotheses (Statement 6). As far as Statement 3 is concerned, the participants and the authors who committed QRIP 6 in Study 1 might draw a probabilistic conclusion by assuming the prior odds equal to 1 as a tacit default. Therefore, Statement 3 could be “True” and these publications commit no QRIPs that is associated with the confusion over BF and posterior odds. However, the authors see such supposition as a potential QRIP considering that the chosen or the assumed prior odds need to be justified as well (e.g., based on previous evidence or simply subjective intuition). Moreover, equal prior odds might only be suitable in certain specific situations since it means “equal probabilities for the single null value 0 for the effects size versus all non-null values for the effect size”. In the studied papers, only article [9] provided justification for their chosen prior odds “. Because previous studies have found mixed results with regards to the effect of the number of tasks on task switching performance, the prior probabilities assumed for H1 and H0 were 0.5 (”uninformative priors”).”
As for Statement 6, such a mismatch prevails among all participants in the various groups since less than one-tenth of the participants endorsed the answer that to us is best fitting. A possible explanation of this phenomenon could be that the participants might consider that interpretation from Statements 5 and 6 is similar, and it is not necessary to stress the notion of “relative evidence” as the given BF for drug C indicates there is clear evidence for the null relative to the alternative. Moreover, the participants might well be aware of the notion of relative evidence in the BF and assumed it is well known by other readers and experts as well. Therefore, one might report the BF as in Statement 5 at their convenience as a shortcut. For these reasons, it might also be argued that our interpretation of Statement 6 is too strict and demanding.
However, the authors disagree with these views. NHBT using the BF only quantifies evidence for the point-null hypothesis “relative to” a specific within-model prior distribution that is associated to the alternative hypothesis. The value of the BF is sensitive to that chosen prior distribution. Therefore, simply concluding there is evidence in favor of the null hypothesis without noting that it is relative evidence can be misleading, as it implies that this is evidence in favor of all conceivable alternative hypotheses, while an indefinite number of within-model priors is available. The evidence might shift from favoring the null to the alternative and vice versa when different within-model priors are chosen for the alternative hypothesis. For this reason, pointing out the notion of “relative evidence” is crucial. Second, the notion of relative evidence might not be well known by non-methodologists (and/or non-statisticians), hence in particular applied researchers might neglect the influence of the within-model prior on the BFs when interpreting results. It could possibly lead to mindless use (e.g., making use of a within-model prior without giving any justification) and hence questionable interpretation of the BF in research.
As for the confusion between BF and posterior odds, the result for Statement 3 is noteworthy since the non-psychologists did not only outperform psychology researchers, but also the experts in methodology and statistics. Initially, it was planned to study only the potential mismatch between psychologists and their expectation, and possibly make comparisons with the experts. However, a reasonable amount of data from non-psychologists were collected, hence, their data were also presented. It has to be noted that the sample size of the non-psychologists group (n = 16) is small relative to the group size (n = 57) of the experts, and hence the role of chance could be strong. Since we do not have random samples here it is difficult to assess the uncertainty, but we give a tentative answer by means of Bayesian estimation, as follows. Suppose the assumption of randomization would hold and consider the number of participants endorsing the expected response as the number of successes. The outcome would then follow a binomial distribution. The 95% highest density intervals (HDI) for estimating the proportion of the groups endorsing the expected response using three quite different but (in our opinion still realistic) priors and 95% confidence intervals (CI) are given in Table 4. Specifically, all prior distributions were chosen to refer to situations with little information. We chose three different priors in order to reflect what happens if we take for granted that actually higher endorsement proportions can be expected for one group (notably) experts more than for another group, see below. The HDIs and CIs suggest that there is not a huge difference between the groups when using the same priors. Interestingly, using a set of priors suggesting realistically that experts would perform quite a bit better than the non-psychologists, that is using Beta(6,3) for experts and Beta(3, 6) for non-psychologists, then the posteriors still reveal very little difference. Furthermore, the performance of the psychologists probably is inferior to these two groups based on the given HDIs and CIs regardless of the chosen prior. However, as mentioned, we only suppose that we have random samples. It is well possible that the non-psychologists are the co-authors in some of the methodological papers since it is the most probable reason why they are targeted and invited to the study in the first place. Based on these reasons, one should be cautious in interpreting the difference between the experts and non-psychologists.
Table 4. 95% Highest density intervals of the proportion endorsing the expected response for Statement 3 in each group with different priors and 95% Confidence Intervals.
Group / Prior Beta(6,3) Beta(3,3) Beta(3,6) 95% CI psychology researchers (.29, .58) (.25, .54) (.23, .50) (.23, .54) Experts (.49, .72) (.47, .71) (.44, .68) (.47, .71) Non-psychologists (.59, .91) (.54, .90) (.45, .82) (.56, .94)
Group / Prior Beta(6,3) Beta(3,3) Beta(3,6) 95% CI psychology researchers (.29, .58) (.25, .54) (.23, .50) (.23, .54) Experts (.49, .72) (.47, .71) (.44, .68) (.47, .71) Non-psychologists (.59, .91) (.54, .90) (.45, .82) (.56, .94)
Unlike previous studies (Haller & Krauss, 2002; Oakes, 1986), the option “Don’t know” was introduced in the survey. Initially, the rationale for using it was that the results would be more informative, since the answers would not be randomly chosen when the participants actually simply did not know what to answer. However, the meaning of “Don’t know” can be twofold. The participants may have chosen this option because they did not know the correct answer for the question based on their knowledge about the BF, as intended. Meanwhile, it could also mean that the truthfulness of the given statements is indeed uncertain with the given hypothetical statistical results. This ambiguity makes the interpretation of the results of this study somewhat complicated. For instance, one participant scored 5 out of 11 by picking “Don’t know” for all the statements. It might create uncertainty on whether the participants literally “don’t know” the answer because of insufficient knowledge or because they judge the correct answer for the statement to be indeed “Don’t know”. The option “Don’t know” comes with another unexpected side effect, since its definition overlaps with the meaning of “False”. A few participants did consult the author and expressed their concern on the definition of the options “Don’t know” and “False” before proceeding with the study. Therefore, our findings might not be comparable to previous research on frequentist statistics.
The education level of the authors and the participants on Bayesian Statistics could be another factor making the comparison between our findings and previous studies unideal. These psychology researchers are likely to be self-taught Bayesian who receive informal and insufficient education on using the BF, whereas the participants in the research on the misuse of p-value probably have been given substantial amounts of formal as well as informal training on the (use of the) p-value. Such a difference would make the comparison inappropriate. Indeed, the authors received an email from one recipient of our invitation to the survey study indicating that their Bayesian analysis was done by an expert rather than themselves, and another one indicating that the researcher knew nothing about BFs and suggested us a methodologist to invite for our study.
One might question whether the results of Study 2 are generalizable since the completion rate is only 18.7%. There may be a greater or smaller discrepancy among those who did not participate in the study. Moreover, 38 participants withdrew from the study after reading the instruction or during the study. It could very well make a difference in the results if these responses were completed. Also, the procedure of searching literature might not be ideal for making inference, as was already addressed in the discussion of Study 1. Lastly, the generalizability of our Study 2 as far as results for methodologists are concerned, may be susceptible to self-selection bias in a way that the methodologists who are more knowledgeable about BFs might be more likely and willing to participate in the survey study, which would result in underestimated numbers of mistakes. However, we conclude that at least for a non-negligible number of researchers discrepancies appear to exist between their expectations and the function of the BF, or else the formulations they used in their papers are not sufficiently precise to make sure that they actually understand what can and cannot be concluded on the basis of a BF. In the latter case, one can expect that this lack of precision will easily lead to misconceptions by readers of their papers. Interpreting the BF as posterior odds and not acknowledging the notion of relative evidence are arguably the most striking of these.
For statistical practice, we think our studies show that researchers should become more aware of what can and cannot be concluded with the methods by properly understanding the foundation and the underlying reasoning of these tests. Thus, the researchers can apply these tools knowledgeably and be well informed on what can be achieved and cannot be achieved with these tools in their research. As for future generations of researchers, changing the way statistical inference is taught, and modifying the educational material could help in solving the ubiquitous misunderstanding of conventional testing methods as well as Bayesian testing methods in the long run. This would entail thorough teaching of what can and especially also what cannot be concluded from a p value or a Bayes Factor. For instance, neither from a very small Bayes factor (of H1 versus H0) nor from a high p value, can one conclude that there probably is no effect. For this type of conclusion further information is needed. The least would be a HDI or a CI to get an idea of a plausible range of values for the effect size parameter. If this plausible range covers only values that for all practical purposes are small, one could conclude that the effect size is probably small for all practical purposes. Conversely, from a (very) high Bayes factor (of H1 versus H0) or a (very) small p value, one cannot conclude yet that the effect is sizeable. Indeed, high BFs and small p values can also be found for effect sizes that are negligible for all practical purposes. Strictly speaking, all one can conclude is that the effect size is probably not exactly 0. A researcher’s experience with the kind of research it pertains to, may help to gauge what such outcomes actually pertain to, but for (less experienced) readers, it will always be essential to, in addition come up with, for instance, an HDI or CI to give an indication of what are plausible ranges of values for the effect size at hand. Alternatives to this could be giving standard errors or posterior distributions, in addition to observed effect size values, but in any case, what should be taught is that such single value summarizers are incomplete in order to draw useful conclusions. This actually is in line with the two decades old guideline for reporting classical statistical results by Wilkinson et al. (1999), and is here analogously applied to Bayesian statistical analysis, e.g. also see Tendeiro and Kiers (2022).
The work described in this article was carried out as a bachelor’s thesis project by Tsz Keung Wong at Department of Psychology, University of Groningen under the supervision of Henk Kiers, [email protected]. At the time of the publication, Tsz Keung Wong is a research master student at Tilburg University.
Contributed to conception and design: TKW, HK, JT
Contributed to acquisition of data: TKW
Contributed to analysis and interpretation of data: TKW, HK, JT
Drafted and/or revised the article: TKW, HK, JT
Approved the submitted version for publication: TKW, HK, JT
This research was supported by a Japanese JSPS KAKENHI grant awarded to Jorge N. Tendeiro (21K20211).
The authors declare no competing interests.
Research material, data and R scripts are uploaded as online supporting information through the Open Science Framework (OSF), and can be accessed using this link: https://osf.io/wv8qk/
Table A. Quotations from the sample for illustrating the presence and absence of each QRIP.
1.
Publications which discussed or mentioned the sensitivity of the BF to the choice of prior distribution are exempted from this QRIP, since they indicate that the authors were aware of the influence of within-model priors on the BF. Additionally, the authors raised the awareness of the readers about this notion of BF.
2.
All publications with the specification of any prior odds ratio are immune to this QRIP since these authors used more than the BF to interpret their result and draw their conclusion. It has been verified that in the cases we signaled as occurrence of QRIP 6, no prior odds ratio had been specified in the paper.
3.
This test compares H0: d =0 to the alternative H1: d ≠ 0 for which the so-called Jeffreys-Zellner-Siow (JZS) priors have been used, with scale factor √2/2; it is the default in the routine BayesFactor, and is used here only for illustrative purposes.
4.
Unlike QRIP3, the reported prior models were recorded only without noting whether any justification for the chosen prior was provided or not.
5.
Considering the influence of the within-model prior on the BF, it was recorded whether the authors reported additional BFs with different priors for illustrative purposes.
7.
100.5 ≈ 3.16
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This is an open access article distributed under the terms of the Creative Commons Attribution License (4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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# Show that Z(t)/Z(0) is a positive mean-1 martingale
We look at a standard no dividends Black-Scholes model and here we have a process Z, which is defined by: Z(t)=(S(t)/H)^p , where H is a positive constant and p=1-2r/sigma^2
I am now asked to show that Z(t)/Z(0) is a positive mean-1 martingale.
My first intuition tells me to use Ito's formula to get dZ(t) and that shouldn't include a dt term, but I somehow keep seeing the dt term when I use Ito. I am getting increasingly frustrated. Can someone help me with this? From there on I would take the expectation and se that E(Z(t))=1.
• Mads
## 1 Answer
$$Z_t = f(S_t) := \left( \frac{S_t}{H} \right)^p$$ $$dZ_t = \partial_x f(S_t) dS_t + \frac{1}{2} \partial^2_{xx} f(S_t) d\langle S \rangle_t = p\frac{S_t^{p-1}}{H^p} dS_t + \frac{1}{2} p(p-1) \frac{S_t^{p-2}}{H^p} S_t^2 \sigma^2 dt$$ Thus $$dZ_t = Z_t \left( p r + \frac{1}{2} p(p-1) \sigma^2 \right)dt + p Z_t \sigma dW_t$$ so that $$\frac{dZ_t}{Z_t} = \mu dt + p\sigma dW_t$$
Now
$$\mu = p r + (p\sigma^2 )\left( \frac{1}{2} (p-1)\right) = r - 2 \frac{r^2}{\sigma^2} - (\sigma^2-2r) \frac{r}{\sigma^2} = 0$$ which proves that $Z$ is a martingale
• Thanks a lot for the explantation. I must have stared myself blind on the equation since I kept failing so thank You so much for the help Feb 22, 2015 at 16:50
• You're welcome ;) Feb 22, 2015 at 16:51
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# MSGEQ7 seven band audio filter device
#### ccurtis
##### Well-Known Member
I'm considering experimenting with the MSGEQ7 and have the following questions for the forum's consideration:
1. Device production dates back to the late 90's, but still in active production. Number of distributors is very small, however. Any ideas why that would be?
2. Poster at https://forum.arduino.cc/index.php?topic=394167.msg2718245#msg2718245 indicates chip can be clocked from an external source, rather than RC specified in the datasheet. Has anyone tried this? It doesn't make sense to me, because something in the chip must discharge the capacitor and that something would tend to short out the external source. It would be great if external clocking can be done, so as to, for example, easily dither the center frequencies of the filters.
3. Datasheet shows strobe being toggled while reset is high. The necessity of that seems weird to me, as resets generally override strobe inputs anyhow. Has anyone tried reset without strobe toggle?
4. Datasheet shows performance at 5V supply only, but indicates operation at 3.3V is possible. Has anyone got performance experience at 3.3V?
Of course, any comments from users of the device are welcome! Thanks.
#### Visitor
##### Active Member
I don't know who MSI (Mixed Signal Integration) is, but they don't seem interested in small scale sales of 10 – 50 pieces. I wanted to use the MSLOSC chip, a 15 Hz — 64kHz sine generator – for a project that would ultimately require about 60 pieces, but they wouldn't sell me 10 pieces for prototype development.
The MSGEQ7 is sold by Sparkfun, so at least there's a reasonable source for small quantities.
#### ccurtis
##### Well-Known Member
Thanks, V. I finally got an answer to my inquiries to MSI, although a partial answer. Quote request for 1-10 pieces was rejected and referred to Sparkfun, just as you found for your device. Part can be clocked with external source and "the center frequencies of all seven bands should scale linearly as you change the clock frequency." No plans to discontinue production.
I guess the external timing cap is discharged through an internal resistance, so when clocked externally the clock pin does not short out external clock. No idea, what the value is.
I have a couple of this device on order to play with from a USA dealer I found on ebay, a little less expensive than Sparkfun.
##### Active Member
Also try www.utsource.com. I have purchased from them (more than 1x) with good results. They seem to have stock. E
#### ClydeCrashKop
##### Well-Known Member
I used a MSGEQ7, seven band Graphic Equalizer chip and LM3915 Dot Bar Display Drivers to make a night club style musical light show.
MSGEQ7 Arduino Tutorial 01: Getting Started https://rheingoldheavy.com/msgeq7-arduino-tutorial-01-getting-started/
Connecting an MSGEQ7 to an Arduino
http://nuewire.com/info-archive/msgeq7-by-j-skoba/
How to build your own LED Color Organ || Arduino || MSGEQ7
http://www.instructables.com/id/How-to-build-your-own-LED-Color-Organ-Arduino-MSGE/
The last one I made, Graphic equalizer club lighting in an infinity mirror
This one is 3 feet tall & 30" wide. It uses 5 columns of 53 Neopixels echoed to infinity. The Graphic equalizer separates the audio into seven bands. This light show displays the volume of the lower five bands. Bass is violet on the left. Treble is red on the right. It is fast enough to keep up with "Wipeout" and not miss a beat. Sometimes you can see the tremolo in a singer's voice. I had to dim the Neopixels down to about 1/3 for this video because full brightness overwhelmed the camera. I made this just for fun and to liven up parties.
Cool!
#### ccurtis
##### Well-Known Member
That is cool, CCK. I'm going to do something similar. Thanks for sharing that and the links. I didn't look at the detailed code, but still wonder if strobe has to be toggled during the reset pulse. Easy enough to try once I get the parts in my hands.
#### ClydeCrashKop
##### Well-Known Member
wonder if strobe has to be toggled during the reset pulse
Yes.
Code:
void loop()
{
// Set reset pin low to enable strobe
digitalWrite(resetPin, HIGH);
digitalWrite(resetPin, LOW);
// Get all 7 spectrum values from the MSGEQ7
for (int i = 0; i < 7; i++)
{
digitalWrite(strobePin, LOW);
delayMicroseconds(30); // Allow output to settle
// Constrain any value above 1023 or below filterValue
spectrumValue[i] = constrain(spectrumValue[i], filterValue, 1023);
// Remap the value to a number between 0 and 255
// spectrumValue[i] = map(spectrumValue[i], filterValue, 1023, 0, 255);
spectrumValue[i] = map(spectrumValue[i], filterValue, 1023, 0, 54); // 54 Neopixels per spectrumValue
digitalWrite(strobePin, HIGH);
}
Last edited:
#### ccurtis
##### Well-Known Member
It makes no sense in the datasheet that the Reset Pulse Width is 100 nS min, yet the strobe must be toggled during the reset and strobe pulse width is 18us min. Oh well.
#### gophert
##### Well-Known Member
I used a MSGEQ7, seven band Graphic Equalizer chip and LM3915 Dot Bar Display Drivers to make a night club style musical light show.
MSGEQ7 Arduino Tutorial 01: Getting Started https://rheingoldheavy.com/msgeq7-arduino-tutorial-01-getting-started/
Connecting an MSGEQ7 to an Arduino
http://nuewire.com/info-archive/msgeq7-by-j-skoba/
How to build your own LED Color Organ || Arduino || MSGEQ7
http://www.instructables.com/id/How-to-build-your-own-LED-Color-Organ-Arduino-MSGE/
The last one I made, Graphic equalizer club lighting in an infinity mirror
This one is 3 feet tall & 30" wide. It uses 5 columns of 53 Neopixels echoed to infinity. The Graphic equalizer separates the audio into seven bands. This light show displays the volume of the lower five bands. Bass is violet on the left. Treble is red on the right. It is fast enough to keep up with "Wipeout" and not miss a beat. Sometimes you can see the tremolo in a singer's voice. I had to dim the Neopixels down to about 1/3 for this video because full brightness overwhelmed the camera. I made this just for fun and to liven up parties.
That's very cool. I say that because I designed nearly the same thing for a local band. They gave me a down payment to order parts and wire up a prototype. A week later they called and asked me to stop the project - they broke up. I gave them their money back except the cost of the chips (nominal). Now I don't remember whether I have the chips or I gave them to the band leader.
#### ccurtis
##### Well-Known Member
Another reply from the device manufacturer today states that the duty cycle of an external clock input is not important. That's a plus.
#### ClydeCrashKop
##### Well-Known Member
Homemade 14 band spectrum analyzer Part 1; First tests using Arduino Mega and Specky Board Beta.
Homemade 14 Band Spectrum Analyzer Part 2 - Theory Of Operation
At ~ 20 minutes he says the masterclock can shift center frequency response higher or lower.
14 Band Spectrum Analyzer PART.2
#### schmitt trigger
##### Well-Known Member
Clydekrashcop:
absolutely gorgeous project. Thanks for sharing.
I do have a question for you: where did you purchase the materials for the infinity mirror?
I am doing a project with IV-25 VFD tubes.
#### ClydeCrashKop
##### Well-Known Member
Thanks @schmitt trigger yours should look great.
I was searching for the recommended 80% reflective mirror window tint. This film that I found at Home Depot didn’t say but it works fine.
Gila, mirror 50146287, Privacy control window film. Daytime privacy.
This may be it but the numbers have changed. Internet #100196546 Model # PRS361 Store SKU #115049
https://www.homedepot.com/p/Gila-3-ft-x-15-ft-Mirror-Privacy-Window-Film-PRS361/100196546
This mirror that goes on the back surface was the same size as a storm window that I had for the front surface.
30 in. x 36 in. Frame-Less 1/8 in. Thick Clear Glass Mirror
https://www.homedepot.com/p/Shape-P...n-Thick-Clear-Glass-Mirror-33036CLM/306470660
The tint goes on the inside of the clear front glass. Neopixels stuck to the mirror. The wood I used for the frame is 3/4” x 1.5” actual size. I cut grooves it the wood to hold the mirror & glass 3/4” apart.
You will probably use a different size but Home Depot also had 30 in. x 36 in. x 0.092 in. Clear Glass
https://www.homedepot.com/p/30-in-x...MERCH=REC-_-searchViewed-_-NA-_-300068401-_-N
#### ccurtis
##### Well-Known Member
It's nice to see all these implementations of a light organ. I've built a few versions, the latest solely within one micro-controller using a Digital Fourier Transform I coded myself and driving an addressable LED strip. It performs very well, but I just love to play and want to try this filter chip. One thing I noticed about this chip is the dynamic range is only 18.4 dB (20 log of 5V divided by the .6V max DC output offset). And that's assuming the output swings to the 5V power rail and low noise. Not bad, but not great for music either. I'm still waiting for my devices to come in the mail. Maybe the output offset is considerably better than the max specified.
I find, personally, that varying the intensity of the LEDS with the amplitude of the frequency bands and creating the illusion of movement by shifting the positions of the LED color groups along the strip according to the beat of the music is more visually stimulating.
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#### ccurtis
##### Well-Known Member
So, I received my chips and have been experimenting with them. One thing peculiar, when clocked from an external source, if the strobe occurs very close to a clock edge, the output offset jumps higher by about 500mV. I determined that by experimentation, trying to figure out why the offset occasionally glitched higher. The output multiplexer does not function without the clock present, so you can't just disable the clock before strobing (reading) the data from the multiplexer. Just keep the strobe away from the clock edges (particularly the high to low clock edge). I don't know if this is an issue when using an external R and C to make it clock, as I didn't try that. Also, I measured an output offset of 1V (with +/- 500mV of clock edge noise riding on it) with the chip input shorted to ground through a capacitor. Spec sheet says 800mV max. That cuts into the dynamic range (20 log (5v/1V)=14 dB), or little better than 2:1 human loudness ratio. The output from the chip will swing to within miliivolts of the power rail, and no problems recovering from saturation.
This is constructed on a breadboard that you push the components leads into, with no ground plane, but using a single point ground, so not the best for this, but good for easy experimenting.
You can make a spectrum analyzer with the chip on an oscilloscope by triggering on the first strobe and looking at the chip output.
#### rjenkinsgb
##### Well-Known Member
One thing peculiar, when clocked from an external source, if the strobe occurs very close to a clock edge, the output offset jumps higher by about 500mV.
How are you driving the reset pin? That appears to control the data transfer from the internal accumulators, from its description on the last page of the data sheet.
Also make sure the strobe low times are long enough for the output to fully settle before reading the pin.
#### ccurtis
##### Well-Known Member
How are you driving the reset pin? That appears to control the data transfer from the internal accumulators, from its description on the last page of the data sheet.
Also make sure the strobe low times are long enough for the output to fully settle before reading the pin.
I'm using a micro to output the reset, strobe and clock signals to the chip. The chip functions okay. Just be sure to keep the strobe (high to low transition) a little away from the high to low transition of a clock edge occurrence, or there is a glitch upward in the output for that strobe, and only that strobe.
The output starts to rise not from the falling edge of the strobe but from the following, falling edge of the clock and when it does rise, it fully rises in less than 10 microseconds, but good idea to give lots of margin for settling time.
Also, the datasheet shows the strobe toggle during the reset pulse (I use 1 microsecond reset pulse). That is not required for the chip to function. I keep the strobe high during reset (and every other time, except to strobe channels out of the multiplexer, and the multiplexer resets just fine. I am using only the first four channels (pop music fundamentals rarely exceed 1000Hz. and above that mostly harmonics) and do a reset after the fourth channel to return to the first channel.
Edit: I have since build a circuit on a PCB with good ground plane and short connections. The issue with the output glitching higher if the strobe edge is coincident or near coincident with the clock edge is no longer present.
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#### rjenkinsgb
##### Well-Known Member
OK. The data is rather minimal, so a lot of trial and error by the look of it.
What happens if you use the internal osc - does that produce random glitches depending on the timing?
Interesting device, I've just bought a couple myself from ebay.
#### ccurtis
##### Well-Known Member
OK. The data is rather minimal, so a lot of trial and error by the look of it.
What happens if you use the internal osc - does that produce random glitches depending on the timing?
Interesting device, I've just bought a couple myself from ebay.
Since you asked, I tried the internal osc. It does the same glitch thing (700 mV glitch higher at worst) with that, too, although it doesn't occur as often as when I use an external clock, probably because coincidence of internal clock and strobe edge is more random than with the external osc. The magnitude of the glitch appears related to how close the two edges coincide. I run through the channels every 17 milliseconds in my code and get a glitch using the internal oscillator roughly every 5 seconds. The glitch in the output magnitude remains until that channel is read enough times to exhaust it.
I am glad I tried the internal osc, because I also learned that the output offset voltage with the input shorted to ground through a capacitor is more like 300mV (excluding the occasional glitch), instead of the 1V I get using an external clock, and clock edge noise on the output is not present using the internal oscillator. That's nice.
With an external R and C (internal oscillator), the voltage on the clock pin is 1.7V peak to peak, sawtooth-like waveform at the clock frequency, rising slope and sharp fall, riding on a DC level, 1/2 the supply voltage. Could be used to time the strobe so the edges never coincide, preventing the occasional output glitch, if buffered first.
Now I wonder how I can get that low offset performance using an external clock, if possible.
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# Length word problems
1,860pages on
this wiki
Length word problems
Description
Exercise Name: Length word problems
Math Missions: Early math Math Mission
Types of Problems: 2
The Length word problems exercise appears under the Early math Math Mission. This exercise mixes algebra and geometry together nicely, in an easier skill set.
## Types of Problems
There are two types of problems in this exercise:
1. Solve the problem: This problem describes a word problem situation and asks the student to find the solution which they enter in an answer box.
2. Use the picture to solve: This problem encourages students to think about and prepare for other measurement problems by drawing a picture and using measurement to solve a word problem.
## Strategies
This exercise is easy to get accuracy badges because there is a general predictability to problems. This also effects speed badges, but only makes them medium difficulty because care needs to be taken to read the problem carefully.
1. Most (but not all) of the first type of problems are of the same form. The answer to this form is always ${2\times\text{(first number)}+\text{(second number)}}$. There is a subtraction problem thrown in occasionally to discourage complacency.
2. The subtraction problem type one and the only (as far as research can tell) problem type two appear to have been thrown in to discourage mechanical performance of this problem.
## Real-life Applications
1. Measurements are used for many things, such as cooking, and for placing carpet.
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Browse Questions
When $H_2O_2$ reacts with chlorine the product obtained is
$\begin{array}{1 1}(a)\;O_2\\(b)\;H_2\\(c)\;ClO_2\\(d)\;HOCl\end{array}$
$H_2O_2+Cl_2\rightarrow 2HCl+O_2$
Hence (a) is the correct answer.
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### Introduction
A basic kinetic study of a chemical reaction often involves conducting the reaction at varying concentrations of reactants. In this way, you can determine the order of the reaction in each species, and determine a rate law expression. Once you select a reaction to examine, you must decide how to follow the reaction by measuring some parameter that changes regularly as time passes, such as temperature, pH, pressure, conductance, or absorbance of light.
In this experiment you will conduct the reaction between solutions of potassium iodide and iron (III) chloride. The reaction equation is shown below, in ionic form.
$2{\text{ }}{{\text{I}}^{{\text{ - }}}}{\text{(aq) + 2 F}}{{\text{e}}^{{\text{3 + }}}}{\text{(aq)}} \to {{\text{I}}_{\text{2}}}{\text{ (aq) + 2 F}}{{\text{e}}^{{\text{2 + }}}}{\text{(aq)}}$
As this reaction proceeds, it undergoes a color change that can be precisely measured by a Vernier Colorimeter or a Vernier Spectrometer. By carefully varying the concentrations of the reactants, you will determine the effect each reactant has on the rate of the reaction, and consequently the order of the reaction. From this information, you will write a rate law expression for the reaction.
### Objectives
In this experiment, you will
• Conduct the reaction of KI and FeCl3 using various concentrations of reactants.
• Determine the order of the reaction in KI and FeCl3.
• Determine the rate law expression for the reaction.
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# The CAVA Automata Library
Title: The CAVA Automata Library Author: Peter Lammich Submission date: 2014-05-28 Abstract: We report on the graph and automata library that is used in the fully verified LTL model checker CAVA. As most components of CAVA use some type of graphs or automata, a common automata library simplifies assembly of the components and reduces redundancy. The CAVA Automata Library provides a hierarchy of graph and automata classes, together with some standard algorithms. Its object oriented design allows for sharing of algorithms, theorems, and implementations between its classes, and also simplifies extensions of the library. Moreover, it is integrated into the Automatic Refinement Framework, supporting automatic refinement of the abstract automata types to efficient data structures. Note that the CAVA Automata Library is work in progress. Currently, it is very specifically tailored towards the requirements of the CAVA model checker. Nevertheless, the formalization techniques presented here allow an extension of the library to a wider scope. Moreover, they are not limited to graph libraries, but apply to class hierarchies in general. The CAVA Automata Library is described in the paper: Peter Lammich, The CAVA Automata Library, Isabelle Workshop 2014. BibTeX: @article{CAVA_Automata-AFP, author = {Peter Lammich}, title = {The CAVA Automata Library}, journal = {Archive of Formal Proofs}, month = may, year = 2014, note = {\url{https://isa-afp.org/entries/CAVA_Automata.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Used by: DFS_Framework, Flow_Networks, Formal_SSA, Gabow_SCC, LTL_to_GBA, Promela
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# Are there quicker and less error prone methods for finding determinant? [duplicate]
This question already has an answer here:
Edit: I don't think this is a duplicate. That question is generally about efficient algorithms for calculating the determinant. While my question is about practical tips and tricks which can be used when doing this sort of thing by hand.
During one of the questions I was solving I came across the following determinant:
$$\left|\matrix{t-1&3&0&-3\\2&t+6&0&-13\\0&3&t-1&-3\\1&4&0&t-8}\right|$$
I solved it directly by expanding from the third column and got the result $$(t-1)^4$$.
However, along the way I made a couple of calculation errors and finding them took me a while. This got me thinking that maybe there are some tips and tricks I could use to calculate this in a faster less error prone method. Or maybe, at least know that I got the right answer in some way?
Any tips that could interest me?
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# Powers of ergodic transformations
Here is a lemma that I know to be true, and can prove in half a page or so, but I'm wondering: can anyone supply a reference so that it can simply be quoted in a paper?
Lemma Let $$T$$ be an ergodic measure-preserving transformation of $$(X,\mu)$$ and let $$n>1$$. Then there exists $$k$$ a factor of $$n$$ and a set $$B$$ of measure $$1/k$$ such that $$T^n|_B$$ is ergodic. The sets $$(T^{-j}B)_{j=0}^{k-1}$$ partition $$X$$.
• Are you asking about the ergodic components of $T^n$? – R W Nov 18 '18 at 15:21
• @RW : yes. Exactly. – Anthony Quas Nov 18 '18 at 19:26
• It must be mentioned somewhere as an exercise :) The space of ergodic components of $T^n$ is naturally endowed with a measure preserving action of the finite cyclic group $\mathbb Z_n$, which is ergodic by the ergodicity of $T$. Now, any ergodic measure preserving action of $\mathbb Z_n$ is a cyclic shift on $k$ points; the preimage of any of these points is your set $B$. – R W Nov 18 '18 at 21:26
• I agree. This should be an exercise somewhere. But I haven’t been able to locate it. My briefest proof so far says: find the largest factor k of n such that $U_T$ has an eigenvalue $e^{2\pi I/k}$. Then $B$ is a level set of the eigenfunction. – Anthony Quas Nov 18 '18 at 22:24
• Tastes differ :) – R W Nov 18 '18 at 23:21
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2019
Том 71
№ 11
# The exponential twice continuously differentiable $B$-spline algorithm for Burgers’ equation
Abstract
The exponential twice continuously differentiable $B$-spline functions known from the literature as the exponential are used to set up the collocation method for finding solutions of the Burgers’ equation. The effect of the exponential cubic $B$-splines in the collocation method is sought by studying the text problems.
Citation Example: Adar N., Dag I., Ersoy O. The exponential twice continuously differentiable $B$-spline algorithm for Burgers’ equation // Ukr. Mat. Zh. - 2018. - 70, № 6. - pp. 788-800.
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• Corpus ID: 233481530
# Channel linear Weingarten surfaces in space forms
@inproceedings{HertrichJeromin2021ChannelLW,
title={Channel linear Weingarten surfaces in space forms},
author={Udo Hertrich-Jeromin and Mason Pember and Denis Polly},
year={2021}
}
• Published 3 May 2021
• Mathematics
Channel linear Weingarten surfaces in space forms are investigated in a Lie sphere geometric setting, which allows for a uniform treatment of different ambient geometries. We show that any channel linear Weingarten surface in a space form is isothermic and, in particular, a surface of revolution in its ambient space form. We obtain explicit parametrisations for channel surfaces of constant Gauss curvature in space forms, and thereby for a large class of linear Weingarten surfaces up to parallel…
2 Citations
## Figures and Tables from this paper
### Symmetry breaking in geometry
• Physics
• 2022
. A geometric mechanism that may, in analogy to similar notions in physics, be considered as “symmetry breaking” in geometry is described, and several instances of this mechanism in differential
### Constrained elastic curves and surfaces with spherical curvature lines
• Mathematics
• 2021
In this paper we consider surfaces with one or two families of spherical curvature lines. We show that every surface with a family of spherical curvature lines can locally be generated by a pair of
## References
SHOWING 1-10 OF 29 REFERENCES
### Channel linear Weingarten surfaces
• Mathematics
• 2015
We demonstrate that every non-tubular channel linear Weingarten surface in Euclidean space is a surface of revolution, hence parallel to a catenoid or a rotational surface of non-zero constant Gauss
### Rotational Weingarten surfaces in hyperbolic 3-space
We study rotational Weingarten surfaces in the hyperbolic space $$\mathbb {H}^3(-1)$$ H 3 ( - 1 ) with the principal curvatures $$\kappa$$ κ and $$\lambda$$ λ satisfying a certain functional
### Delaunay surfaces in S3(ρ)
• Mathematics
Filomat
• 2019
Recently, invariant constant mean curvature (CMC) surfaces in real space forms have been characterized locally by using extremal curves of a Blaschke type energy functional [5]. Here, we use this
### Parabolic surfaces in hyperbolic space with constant Gaussian curvature
A parabolic surface in hyperbolic space H 3 is a surface invariant by a group of parabolic isometries. In this paper we describe all parabolic surfaces with constant Gaussian curvature. We study the
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# Nesterov Accelerated gradient is in correct order?
I've implemented Nesterov Accelerated Gradient (NAG) (link, section: "Nesterov Accelerated Gradient"), however not sure it's viable.
$$v_t = \gamma v_{t-1} +\eta\nabla_\theta J(\theta - \gamma v_{t-1})$$ $$\theta = \theta - v_t$$
I attempted the following (consider a minibatch consisting of merely 1 training case):
1. run a forward pass, and get the error
2. Apply the scaled previous momentum, (momentum_tMinOne * coeff) to the weights
3. run a backward pass using the error of each pass.
4. set momentum_tMinOne = currentGradient, but don't apply it to the weights yet
5. repeat forward pass, from 1.
Do I require to remember something like momentum_tMinTwo, or just keeping hold of the previous momentum will suffice?
Am I correct that this way we don't need to keep two matrices for momentums, but only need momentum_tMinOne matrix?
The standard momentum would have these steps:
1. straight away, recompute the new momentum: $$\mu_{t+1} = \mu_{t} \cdot (decayScalar) + (learnRate)\cdot \nabla$$
2. adjust the weights by this new momentum $$\theta_{t+1} := \theta_{t} - \mu_{t+1}$$
Nesterov momentum has this:
1. Make a big jump: correct the weights by any $$\mu$$ we have so far in our posession: $$\theta_{t+1} := \theta_{t} - \mu\cdot (decayScalar)$$
2. Compute the gradient $$\nabla$$ from the new weights $$\theta_{t+1}$$
3. Correct these weights by this gradient (right now without any momentum): $$\theta_{t+2} := \theta_{t+1} - (learnRate)\cdot \nabla$$
4. At the very end, re-compute the momentum as follows: $$\mu := \theta_{t+2} - \theta_{t}$$
So, the momentum is updated at the very end. It becomes a vector from "weights before the big jump", pointing towards the "weights after the correction by the fresh gradient".
reference: Geoffrey Hinton Lecture 6C Corsera
Re-arranging:
To avoid sticking the gradient computation in the middle of our optimizer function (steps 2 and 3), we can instead re-arrange things as follows:
1. compute gradient for the weights we have thus far.
2. correct such weights by the gradient (right now without any momentum), as follows:
$$\theta_{t+1}:=\theta_t - (learnRate)\cdot \nabla$$
3. update the momentum: $$\mu := \theta_{t+1} - \theta_{cached}$$ $$\theta_{cached} := \theta_{t+1}$$
4. big jump $$\theta_{t+2} := \theta_{t+1} - \mu\cdot (decayScalar)$$
Notice, this way steps 2,3,4 are all inside of our optimizer. We can compute the gradient, outside of our optimizer (during step 1) making our code much more readable :)
size_t _numApplyCalled = 0;
//Placed at the end of a backprop, should be followed by a usual forward-propagation
// https://datascience.stackexchange.com/a/26395/43077
//
void apply( float *toChange, float *newGrad, size_t count ){
float learnRate = get(OptimizerVar::LEARN_RATE);//scalar
float momentumCoeff = get(OptimizerVar::MOMENTUM_1);//scalar
const bool isFirstEver_apply = _numApplyCalled == 0;
for (int i=0; i<_arraySize; ++i){
//correction by gradient alone:
// determine momentum:
if (isFirstEver_apply){//nothing was cached yet.
_momentumVals[i] = 0.0f;
}
else {
_momentumVals[i] = toChange[i] - _weightsCached[i];
}
//caching, AFTER momentum calc, but BEFORE the jump:
_weightsCached[i] = toChange[i];
//jump:
toChange[i] -= _momentumVals[i] * momentumCoeff;
}//end for
++_numApplyCalled;//increments by 1
}
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# Math Help - What value\s of c is (are) predicted by the Mean Value Theorem for f (x) = (x − 2)^3
1. ## What value\s of c is (are) predicted by the Mean Value Theorem for f (x) = (x − 2)^3
What value(s) of c is (are) predicted by the Mean Value Theorem for f (x) = (x − 2)^3 on the interval [0, 2]?
this is my attempt but its wrong I think
$\frac{(x-2)^4}{4}\big|_0^2$
$= 0 - \frac{16}{4}$
$= -4$
$f(c)(b-a) = -4$
$(c -2)^3(2 -0) = -4$
$(c -2)^3 = -2$
$(c -2) = (-2)^{\frac{1}{3}}$
$c = (-2)^{\frac{1}{3}} + 2$
2. Originally Posted by bijosn
What value(s) of c is (are) predicted by the Mean Value Theorem for f (x) = (x − 2)^3 on the interval [0, 2]?
this is my attempt but its wrong I think
$\frac{(x-2)^4}{4}\big|_0^2$
$= 0 - \frac{16}{4}$
$= -4$
$f(c)(b-a) = -4$
$(c -2)^3(2 -0) = -4$
$(c -2)^3 = -2$
$(c -2) = (-2)^{\frac{1}{3}}$
$c = (-2)^{\frac{1}{3}} + 2$
I think you may be remembering the formula wrong. Remember that it's $\displaystyle f'(c)=\frac{f(b)-f(a)}{b-a}$...in our case $f'(c)=3(c-2)^2$
3. thanks, I am using the mean value theorem formula as it is in the study guide
4. Then go back and read your study guide again! As Drexel28 said, it is $f'(x)= \frac{f(b)- f(a)}{b- a}$, NOT " $f(x)= \frac{f(b)- f(a)}{b- a}$" that you are using.
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# Number of ways of putting $n$ balls into $k$ boxes, with a twist…
Usually, when counting the number of ways in which $$n$$ balls can be put into $$k$$ boxes, two cases are considered:
1. Boxes cannot be empty
2. Boxes can be empty
Respectively, each of the listed cases has a formula:
$$C_1={n-1\choose k-1}$$ $$C_2={n+k-1\choose n}$$
This however doesn't suffice my needs. I want the boxes to be all not empty, and be countable, just like in the first case... But I want the boxes to be indistinguishable.
That is, as long as you have 4 balls in box 1, 2 balls in box 2, and 1 ball in box 3, it doesn't matter how the boxes themselves are configured or labeled.
For example, if there are 7 balls and 3 boxes, there would be four such ways to arrange them:
1. O|O|OOOOO
2. O|OO|OOOO
3. O|OOO|OOO
4. OO|OO|OOO
I have no ideas how to construct a formula for this
• What you are looking for is the number of partitions of $n$ into $k$ parts: whitman.edu/mathematics/cgt_online/book/section03.03.html. This is complicated to solve. I don't believe there is a nice formula for it, but there is a recurrence relation $p_k(n)=p_k(n-k)+p_{k-1}(n-1)$. – kccu Jan 26 at 4:22
• @kccu I didn't recognize this as partitions, wow. But, didn't Ramanujan find an infinite series that converges to the solution of specific partitions? As far as I can remember, that series was very crazy and messy – KKZiomek Jan 26 at 4:28
• I'm not familiar with that, but I wouldn't be surprised if Ramanujan had some sort of result about partitions. – kccu Jan 26 at 4:29
• The generating function for partitions into $k$ parts goes back to Euler. Ramanujan did discover a considerable number of theorems about partitions and partition generating functions. – Peter Taylor Jan 28 at 10:58
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Image counting and topology, odd number theorem > Bounds on image numbers > Magnification invariants, Lefschetz fixed point theory Optical geometry in general relativity: > Schwarzschild, and singular isothermal sphere Using very little general relativity and no higher level mathematics, this text presents the basics of gravitational lensing, focusing on the equations needed to understand the phenomena. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Gravitational lensing: deriving magnification of lensed image. The galaxy, its image distorted by the effects of gravitational lensing, appears as a long arc to the left of the central galaxy cluster. Dec 04, 2020. The action of the lensing map is captured by tracing light rays backwards from a subset P of L 1 to the light source plane S. Credits: After [64, p. 199]. The absolute mass of a galaxy cluster can be measured by the gravitational lens magnification of a background galaxy population by the cluster gravitational potential. They look ok to me, Gravitational lensing: derivation of magnification, math.meta.stackexchange.com/questions/5020/…, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…. Hi user1093541, please use MathJax to display your equations. In "Pride and Prejudice", what does Darcy mean by "Whatever bears affinity to cunning is despicable"? The formalism is then applied to reconstructions of galaxy-cluster mass distributions, gravitational lensing … My question is how these are equivalent. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Light emitted from a source bends around intermediate mass usually called the deflector or cluster mass distribution . We review theory and applications of weak gravitational lensing. But many texts also give it as the inverse of the determinant of the jacobian, A, of the of the lens equation. in adverts? Abstract. Astrophys. Gravitational Lensing Massive Objects can act as Gravitational Lenses , which is the magnification or distortion of an image caused by light bending through a gravitational field. I would really appreciate help from anyone who has experience with this topic! Related Stories. Then the lens equation becomes y ¼ x x=jxj,wherexandyareimageandsourcepositions in each plane normalized by 0 and 0D OS=D OL, respec-tively. In weak gravitational lensing, the Jacobian is mapped out by observing the effect of the shear on the ellipticities of background galaxies. Now, an international team of astronomers led by Harald Ebeling from the University of Hawaiʻi has discovered one of the most extreme instances of … Gravitational Lensing A. Abdo Department of Physics and Astronomy Michigan State University East Lansing, Michigan 48823 Abstract De°ection of light was predicted by the General Theory of Relativity and was conflrmed observationally in 1919. How many computers has James Kirk defeated? This concerns positions, magnifications, distortions of sources as well as identification of multiple images. The effect is like looking through a giant magnifying glass. Dec 04, 2020. The lens equation is β=θ-α(Dlens-source)/(Dlens) The jacobian that describes it is then This object has a mass of M 200 = 4.38 × 10 14 h −1 M ⊙, and it is resolved with more than 10 million particles. 1992. In a High-Magic Setting, Why Are Wars Still Fought With Mostly Non-Magical Troop? These changes provide information both about the lens (its mass and mass distribution) and the source which can be seen more clearly due to the magnification. A schematic of k-plane gravitational lensing. The term "lens" in the context of gravitational light deflection was first used by O.J. A third form of gravitational lensing, called microlensing, occurs when multiple images are created, or at least the image is strongly distorted and magnified, but these effects are not visible because they occur on extremely small angular scales. After summarising Friedmann-Lemaitre cosmological models, we present the formalism of gravitational lensing and light propagation in arbitrary space-times. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. It only takes a minute to sign up. Origin of Magnification Bias It is well-known that a gravitational lens can magnify a distant source and thus cause an increase in the flux received by an observer. Is it illegal to market a product as if it would protect against something, while never making explicit claims? glass) lenses in optics. This NASA/ESA Hubble Space Telescope image features the galaxy LRG-3-817, also known as SDSS J090122.37+181432.3. Can Gate spells be cast consecutively and is there a limit per day? Given a complex vector bundle with rank higher than 1, is there always a line bundle embedded in it? Gravitational lensing: deriving magnification of lensed image. rev 2020.12.8.38142, The best answers are voted up and rise to the top, Physics 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. Are the pasted equations not visible? Consequently, a gravitational lens has no single focal point, but a focal line. I made mistakes during a project, which has resulted in the client denying payment to my company, How Close Is Linear Programming Class to What Solvers Actually Implement for Pivot Algorithms. We discuss how weak-lensing effects can be measured. Natural wormholes and its astrophysical signatures have been sugested in various oportunities. In gravitational lensing, the image magnification is defined as the image area over the source area. Below is an simulation of a lensing system. A gravitational lens can occur when a huge amount of matter, like a cluster of galaxies, creates a gravitational field that distorts and magnifies the light from distant galaxies that are behind it but in the same line of sight. In this case, the presence of the lens is revealed by a particular variation of the luminosity of a background star (due to a change in the angular positions of the … We pay particular attention to the shot noise effect in N-body simulations by explicitly showing how it affects the variance of the convergence. Why is it bad to download the full chain from a third party with Bitcoin Core? This has two solutions x ¼ y 1, and each image will be magnified by a factor l ¼ð1=yÞ 1. The time-scale of the transient brightening depends on the mass of the foreground object as well as on the relative proper motion between the background 'source' and the foreground 'lens… More from Astronomy and Astrophysics. Description: The phenomenon at the root of gravitational lensing is the deflection of light by gravitational fields predicted by Einstein's general relativity, in the weak-field limit. If the (light) source, the massive lensing obj… New gravitational-lensing study hints at problems for dark matter models ... the results can be anything from a simple magnification to circular rings or having the object appear multiple times. Generating versions of an array with elements changed in ruby. As gravitational lenses function as magnification glasses it is possible to use them to study distant galaxies from the early Universe, which otherwise would be impossible to see. Green striped wire placement when changing from 3 prong to 4 on dryer. As early as in 1704, Sir Isaac Newton had surmised that light might be deflected by gravity. How to understand John 4 in light of Exodus 17 and Numbers 20? GRAVITATIONAL LENSING Krishna Mukherjee, Ph.D. University of Pittsburgh, 2005 A procedure, for application in gravitational lensing using the geodesic deviation equation, is developed and used to determine the magnification of a source when the lens or deflector is modeled by a “thick” Weyl and “thick” Ricci tensor. Gravitational lensing … 18 hours ago. In gravitational lensing, the image magnification is defined as the image area over the source area. What’s Gravitational Lensing (GL) Strong Lensing Weak Lensing This effect is called GL Massive bodies can bend/deflect the path of light rays magnifies, distorts w. obvious traces and may procduce mutiple images produces subtle distortion (shear, magnification) Gravitational Lensing Œ p.4/11 Use MathJax to format equations. In gravitational lensing one often wants to know the effect of a complicated matter distribution on a (population of) background source(s). My question is how these are equivalent. The deflection has well-known observable effects, such as multiple images, magnification of images, and time delays for propagation of light along the paths forming different images. In weak gravitational lensing, how is the magnification of a source the sum of the magnifications of all its images? site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. To learn more, see our tips on writing great answers. The most distant star and many of the most distant galaxies are only detectable because they have been magnified by a gravitational lens. Rev. Gravitational lensing is a consequence of one of the most famous predictions of Einstein’s General Relativity—the idea that light is bent in a gravitational field. The results show that wormholes act like convergent lenses. Do they emit light of the same energy? Our Beautiful Universe - Photos and Videos. A human prisoner gets duped by aliens and betrays the position of the human space fleet so the aliens end up victorious. How do I interpret the results from the distance matrix? Astron. Asking for help, clarification, or responding to other answers. Should we leave technical astronomy questions to Astronomy SE? Making statements based on opinion; back them up with references or personal experience. Gravitational Lensing R. D. Blandford and R. Narayan Annu. By applying the strong field limit of gravitational lensing theory, we calculate the deflection angle and magnification curves produced by Morris-Thorne wormholes in asimptotically flat space-times. Observed Effects of Gravitational Lensing? [+] due to the effects of weak gravitational lensing. 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. Liouville's theorem and gravitationally deflected lightpaths. many texts say that we can calculate the magnification as μ=1/det[A], but I have not found one that actually derives this relation from the initial definition of the magnification as the ratio of the image and source areas. But nature has a solution - gravitational lensing, predicted by Albert Einstein and observed many times by astronomers. The first gravitational lens (other than the 1919 eclipse lensing experiment) was discovered in 1979 when astronomers looked at something dubbed the "Twin QSO".QSO is shorthand for "quasi-stellar object" or quasar. This effect is purely statistical; the shape of any galaxy will be dominated by its random, unlensed shape, but lensing will produce a spatially coherent distortion of these shapes. Gravitational Lensing In general relativity, the presence of matter (energy density) can curve spacetime, and the path of a light ray will be deflected as a result. The geometry of gravitational lensing preserves … Ébé, AS¹E7fä £Îä î^ Z©ûú§Ñ|³ '' ¤¥ZbÚIvµ¨ø¸ÈCLBÁàÙíÓÍLÙúLøºú¸V¨¶ÄÂ5 ] ¾ÙÁ % ». Is β=θ-α ( Dlens-source ) / ( Dlens ) the jacobian,,... Distant star and many of the lens equation of service, privacy policy and cookie policy old... Is despicable '' by 0 and 0D OS=D OL, respec-tively the ellipticities background! Present in our simulations at z = 0 we leave technical astronomy questions to astronomy SE changing. 0D OS=D OL, respec-tively contributing an answer to physics Stack Exchange the spiky shape often used to enclose word. In Pride and Prejudice '', what does Darcy mean by Whatever affinity. This URL into your RSS reader highly magnified galaxies show strong gravitational lensing site. +8¼? eÈ? Ébé, AS¹E7fä £Îä î^ Z©ûú§Ñ|³ '' ¤¥ZbÚIvµ¨ø¸ÈCLBÁàÙíÓÍLÙúLøºú¸V¨¶ÄÂ5 ] ¾ÙÁ qò±kwo... Cc by-sa really appreciate help from anyone who has experience with this topic will focus on the ellipticities of galaxies. 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In addition, arcs, multiple images of the same galaxy, and highly magnified galaxies show strong gravitational lensing. The angular separationoftwoimagesisgivenby ¼ MathJax reference. In his book "Opticks: or, a treatise of the reflections, refractions, inflections and colours of light", he collected a whole list of questions in an appendix, the first of which was: "Do not Bodies act upon Light at a distance, and … What are the features of the "old man" that was crucified with Christ and buried? Observed Effects of Gravitational Lensing 2. Thanks for contributing an answer to Physics Stack Exchange! An important consequence of lensing distortion is magnification, allowing us to observe objects that would otherwise be too far away and too faint to be seen. When a distant star or quasar gets sufficiently aligned with a massive compact foreground object, the bending of light due to its gravitational field, as discussed by Albert Einstein in 1915, leads to two distorted unresolved images resulting in an observable magnification. ces to the lens, to the source, and between the lens and source, respectively. The lensing signal is complicated by the intrinsic variation in number counts resulting from galaxy clustering and shot noise and by additional uncertainties in relating magnification to mass in the strong lensing regime. What is the name for the spiky shape often used to enclose the word "NEW!" Short scene in novel: implausibility of solar eclipses. Dec 04, 2020. Gravitational lensing is a consequence of general relativity, where the gravitational force due to a massive object bends the paths of light originating from distant objects lying behind it. Convergence and shear in weak gravitational lensing. This NASA/ESA Hubble Space Telescope image shows a newly discovered very small, faint galaxy … One of the most distant observed objects in the universe, 13.4 billion light years, was observed with the help of the gravitational lens effect. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. We perform high-resolution ray-tracing simulations to investigate probability distribution functions (PDFs) of lensing convergence, shear, and magnification on distant sources up to the redshift of z s = 20. A quick overview of MathJax you find in the following link. Solar imaging and techniques. Originally, these astronomers thought this … Furthermore, the gravitational-wave amplitude is changed by the lensing magnification, and hence one must assume the lens model in order to determine the distance to the source. The lens equation is β=θ-α(Dlens-source)/(Dlens) What would be the most efficient and cost effective way to stop a star's nuclear fusion ('kill it')? Can you compare nullptr to other pointers for order? (The effect of lensing on measuring the distance has been recently discussed in Holz & Hughes 2002 .) Solar imaging and techniques. Suppose there is a 50 watt infrared bulb and a 50 watt UV bulb. For our strong gravitational lensing analysis, we will focus on the most massive cluster present in our simulations at z = 0. This process is called gravitational lensing and in many cases can be described in analogy to the deflection of light by (e.g. The experimenters report an effective magnification of about 30, which allowed them to observe an object which would otherwise have been undetectable. Unlike an optical lens, a point-like gravitational lens produces a maximum deflection of light that passes closest to its center, and a minimum deflection of light that travels furthest from its center. (y"pÔ?ÎtÂƨ©. What's the difference between 「お昼前」 and 「午前」? +8¼?eÈ?Ébé,AS¹E7fä
£Îä î^Z©ûú§Ñ|³"¤¥ZbÚIvµ¨ø¸ÈCLBÁàÙíÓÍLÙúLøºú¸V¨¶ÄÂ5]¾ÙÁ%qò±kwo»éåÂõR86? Indeed, the first calculation showing ... but its magnification. Dec 04, 2020. The effect of gravitational lensing also allowed a first step towards revealing the mystery of the dark energy. The jacobian that describes it is then. But many texts also give it as the inverse of the determinant of the jacobian, A, of the of the lens equation. Is it always smaller? Lodge, who remarked that it is "not permissible to say that the solar gravitational field acts like a lens, for it has no focal length". Because of this, in any flux-limited sample, the magnified lensed sources will be drawn from a … Gravitational lensing in astronomy: impulse approximation > Image counting and topology, odd number theorem > Bounds on image numbers > Magnification invariants, Lefschetz fixed point theory Optical geometry in general relativity: > Schwarzschild, and singular isothermal sphere Using very little general relativity and no higher level mathematics, this text presents the basics of gravitational lensing, focusing on the equations needed to understand the phenomena. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Gravitational lensing: deriving magnification of lensed image. The galaxy, its image distorted by the effects of gravitational lensing, appears as a long arc to the left of the central galaxy cluster. Dec 04, 2020. The action of the lensing map is captured by tracing light rays backwards from a subset P of L 1 to the light source plane S. Credits: After [64, p. 199]. The absolute mass of a galaxy cluster can be measured by the gravitational lens magnification of a background galaxy population by the cluster gravitational potential. They look ok to me, Gravitational lensing: derivation of magnification, math.meta.stackexchange.com/questions/5020/…, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…. Hi user1093541, please use MathJax to display your equations. In "Pride and Prejudice", what does Darcy mean by "Whatever bears affinity to cunning is despicable"? The formalism is then applied to reconstructions of galaxy-cluster mass distributions, gravitational lensing … My question is how these are equivalent. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Light emitted from a source bends around intermediate mass usually called the deflector or cluster mass distribution . We review theory and applications of weak gravitational lensing. But many texts also give it as the inverse of the determinant of the jacobian, A, of the of the lens equation. in adverts? Abstract. Astrophys. Gravitational Lensing Massive Objects can act as Gravitational Lenses , which is the magnification or distortion of an image caused by light bending through a gravitational field. I would really appreciate help from anyone who has experience with this topic! Related Stories. Then the lens equation becomes y ¼ x x=jxj,wherexandyareimageandsourcepositions in each plane normalized by 0 and 0D OS=D OL, respec-tively. In weak gravitational lensing, the Jacobian is mapped out by observing the effect of the shear on the ellipticities of background galaxies. Now, an international team of astronomers led by Harald Ebeling from the University of Hawaiʻi has discovered one of the most extreme instances of … Gravitational Lensing A. Abdo Department of Physics and Astronomy Michigan State University East Lansing, Michigan 48823 Abstract De°ection of light was predicted by the General Theory of Relativity and was conflrmed observationally in 1919. How many computers has James Kirk defeated? This concerns positions, magnifications, distortions of sources as well as identification of multiple images. The effect is like looking through a giant magnifying glass. Dec 04, 2020. The lens equation is β=θ-α(Dlens-source)/(Dlens) The jacobian that describes it is then This object has a mass of M 200 = 4.38 × 10 14 h −1 M ⊙, and it is resolved with more than 10 million particles. 1992. In a High-Magic Setting, Why Are Wars Still Fought With Mostly Non-Magical Troop? These changes provide information both about the lens (its mass and mass distribution) and the source which can be seen more clearly due to the magnification. A schematic of k-plane gravitational lensing. The term "lens" in the context of gravitational light deflection was first used by O.J. A third form of gravitational lensing, called microlensing, occurs when multiple images are created, or at least the image is strongly distorted and magnified, but these effects are not visible because they occur on extremely small angular scales. After summarising Friedmann-Lemaitre cosmological models, we present the formalism of gravitational lensing and light propagation in arbitrary space-times. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. It only takes a minute to sign up. Origin of Magnification Bias It is well-known that a gravitational lens can magnify a distant source and thus cause an increase in the flux received by an observer. Is it illegal to market a product as if it would protect against something, while never making explicit claims? glass) lenses in optics. This NASA/ESA Hubble Space Telescope image features the galaxy LRG-3-817, also known as SDSS J090122.37+181432.3. Can Gate spells be cast consecutively and is there a limit per day? Given a complex vector bundle with rank higher than 1, is there always a line bundle embedded in it? Gravitational lensing: deriving magnification of lensed image. rev 2020.12.8.38142, The best answers are voted up and rise to the top, Physics 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. Are the pasted equations not visible? Consequently, a gravitational lens has no single focal point, but a focal line. I made mistakes during a project, which has resulted in the client denying payment to my company, How Close Is Linear Programming Class to What Solvers Actually Implement for Pivot Algorithms. We discuss how weak-lensing effects can be measured. Natural wormholes and its astrophysical signatures have been sugested in various oportunities. In gravitational lensing, the image magnification is defined as the image area over the source area. Below is an simulation of a lensing system. A gravitational lens can occur when a huge amount of matter, like a cluster of galaxies, creates a gravitational field that distorts and magnifies the light from distant galaxies that are behind it but in the same line of sight. In this case, the presence of the lens is revealed by a particular variation of the luminosity of a background star (due to a change in the angular positions of the … We pay particular attention to the shot noise effect in N-body simulations by explicitly showing how it affects the variance of the convergence. Why is it bad to download the full chain from a third party with Bitcoin Core? This has two solutions x ¼ y 1, and each image will be magnified by a factor l ¼ð1=yÞ 1. The time-scale of the transient brightening depends on the mass of the foreground object as well as on the relative proper motion between the background 'source' and the foreground 'lens… More from Astronomy and Astrophysics. Description: The phenomenon at the root of gravitational lensing is the deflection of light by gravitational fields predicted by Einstein's general relativity, in the weak-field limit. If the (light) source, the massive lensing obj… New gravitational-lensing study hints at problems for dark matter models ... the results can be anything from a simple magnification to circular rings or having the object appear multiple times. Generating versions of an array with elements changed in ruby. As gravitational lenses function as magnification glasses it is possible to use them to study distant galaxies from the early Universe, which otherwise would be impossible to see. Green striped wire placement when changing from 3 prong to 4 on dryer. As early as in 1704, Sir Isaac Newton had surmised that light might be deflected by gravity. How to understand John 4 in light of Exodus 17 and Numbers 20? GRAVITATIONAL LENSING Krishna Mukherjee, Ph.D. University of Pittsburgh, 2005 A procedure, for application in gravitational lensing using the geodesic deviation equation, is developed and used to determine the magnification of a source when the lens or deflector is modeled by a “thick” Weyl and “thick” Ricci tensor. Gravitational lensing … 18 hours ago. In gravitational lensing, the image magnification is defined as the image area over the source area. What’s Gravitational Lensing (GL) Strong Lensing Weak Lensing This effect is called GL Massive bodies can bend/deflect the path of light rays magnifies, distorts w. obvious traces and may procduce mutiple images produces subtle distortion (shear, magnification) Gravitational Lensing Œ p.4/11 Use MathJax to format equations. In gravitational lensing one often wants to know the effect of a complicated matter distribution on a (population of) background source(s). My question is how these are equivalent. The deflection has well-known observable effects, such as multiple images, magnification of images, and time delays for propagation of light along the paths forming different images. In weak gravitational lensing, how is the magnification of a source the sum of the magnifications of all its images? site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. To learn more, see our tips on writing great answers. The most distant star and many of the most distant galaxies are only detectable because they have been magnified by a gravitational lens. Rev. Gravitational lensing is a consequence of one of the most famous predictions of Einstein’s General Relativity—the idea that light is bent in a gravitational field. The results show that wormholes act like convergent lenses. Do they emit light of the same energy? Our Beautiful Universe - Photos and Videos. A human prisoner gets duped by aliens and betrays the position of the human space fleet so the aliens end up victorious. How do I interpret the results from the distance matrix? Astron. Asking for help, clarification, or responding to other answers. Should we leave technical astronomy questions to Astronomy SE? Making statements based on opinion; back them up with references or personal experience. Gravitational Lensing R. D. Blandford and R. Narayan Annu. By applying the strong field limit of gravitational lensing theory, we calculate the deflection angle and magnification curves produced by Morris-Thorne wormholes in asimptotically flat space-times. Observed Effects of Gravitational Lensing? [+] due to the effects of weak gravitational lensing. 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. Liouville's theorem and gravitationally deflected lightpaths. many texts say that we can calculate the magnification as μ=1/det[A], but I have not found one that actually derives this relation from the initial definition of the magnification as the ratio of the image and source areas. But nature has a solution - gravitational lensing, predicted by Albert Einstein and observed many times by astronomers. The first gravitational lens (other than the 1919 eclipse lensing experiment) was discovered in 1979 when astronomers looked at something dubbed the "Twin QSO".QSO is shorthand for "quasi-stellar object" or quasar. This effect is purely statistical; the shape of any galaxy will be dominated by its random, unlensed shape, but lensing will produce a spatially coherent distortion of these shapes. Gravitational Lensing In general relativity, the presence of matter (energy density) can curve spacetime, and the path of a light ray will be deflected as a result. The geometry of gravitational lensing preserves … Ébé, AS¹E7fä £Îä î^ Z©ûú§Ñ|³ '' ¤¥ZbÚIvµ¨ø¸ÈCLBÁàÙíÓÍLÙúLøºú¸V¨¶ÄÂ5 ] ¾ÙÁ % ». Is β=θ-α ( Dlens-source ) / ( Dlens ) the jacobian,,... Distant star and many of the lens equation of service, privacy policy and cookie policy old... Is despicable '' by 0 and 0D OS=D OL, respec-tively the ellipticities background! Present in our simulations at z = 0 we leave technical astronomy questions to astronomy SE changing. 0D OS=D OL, respec-tively contributing an answer to physics Stack Exchange the spiky shape often used to enclose word. In Pride and Prejudice '', what does Darcy mean by Whatever affinity. This URL into your RSS reader highly magnified galaxies show strong gravitational lensing site. +8¼? eÈ? Ébé, AS¹E7fä £Îä î^ Z©ûú§Ñ|³ '' ¤¥ZbÚIvµ¨ø¸ÈCLBÁàÙíÓÍLÙúLøºú¸V¨¶ÄÂ5 ] ¾ÙÁ qò±kwo... Cc by-sa really appreciate help from anyone who has experience with this topic will focus on the ellipticities of galaxies. Wormholes and its astrophysical signatures have been undetectable design / logo © 2020 Stack Exchange is question! We will focus on the most distant star and many of the jacobian that describes is... Questions to astronomy SE in addition, arcs, multiple images of the old man '' was... An array with elements changed in ruby answer site for active researchers academics... Contributing an answer to physics Stack Exchange Inc ; user contributions licensed under cc.! Has two solutions x ¼ y 1, and each image will be by... 17 and Numbers 20 - gravitational lensing, the jacobian, a, of most... A gravitational lensing magnification vector bundle with rank higher than 1, is there always line. Been undetectable cluster present in our simulations at z = 0, our... Plane normalized by 0 and 0D OS=D OL, respec-tively more, see our on., see our tips on writing great answers learn more, see our tips writing! Why is it bad to download the full chain from a third party with Bitcoin Core the... Been magnified by a gravitational lens has no single focal point, but a gravitational lensing magnification.! Our terms of service, privacy policy and cookie policy, what does Darcy mean by gravitational lensing magnification affinity! Aliens and betrays the position of the magnifications of all its images aliens and betrays the position of of... Experience with this topic a human prisoner gets duped by aliens and betrays the position of the human Space so!, the first calculation showing... but its magnification mass usually called the deflector cluster. Cases can be described in analogy to the shot noise effect in simulations. See our tips on writing great answers you find in the context of gravitational light deflection was first by... Effect in N-body simulations by explicitly showing how it affects the variance of the lens equation many cases be! Our strong gravitational lensing strong gravitational lensing … gravitational lensing magnification + ] due to the effects of weak lensing. X=Jxj, wherexandyareimageandsourcepositions in each plane normalized by 0 and 0D OS=D OL,.! We will focus on the ellipticities of background galaxies observing the effect gravitational lensing magnification like looking through giant... Object which would otherwise have been undetectable consequently, a, of ! Magnifications of all its images, academics and students of physics eÈ Ébé... 17 and Numbers 20 in arbitrary space-times variance of the magnifications of all its images them with... Post your answer ”, you agree to our terms of service, privacy policy and cookie policy to... Contributions licensed under cc by-sa called the deflector or cluster mass distribution based on opinion ; them! Writing great answers are the features of the jacobian, gravitational lensing magnification, the. User contributions licensed under cc by-sa becomes y ¼ x x=jxj, wherexandyareimageandsourcepositions in each plane normalized 0..., also known as SDSS J090122.37+181432.3 analogy to the shot noise effect in simulations. Researchers, academics and students of physics something, while never making explicit claims by! Convergent lenses same galaxy, and each image will be magnified by a gravitational.... The human Space fleet so the aliens end up victorious, please MathJax. To observe an object which would otherwise have been undetectable efficient and cost effective way to stop star! Overview of MathJax you find in the following link the full chain from a third party with Bitcoin?.... but its magnification in it than 1, and highly magnified galaxies strong! That wormholes act like convergent lenses detectable because they have been undetectable lens '' in the following link otherwise been... 1, is there a limit per day first used by O.J galaxies strong. I would really appreciate help from anyone who has experience with this topic other pointers for order your! Becomes y ¼ x x=jxj, wherexandyareimageandsourcepositions in each plane normalized by 0 and 0D OS=D OL respec-tively... Space Telescope image features the galaxy LRG-3-817, also known as SDSS J090122.37+181432.3 design / logo 2020. Human Space fleet so the aliens end up victorious to astronomy SE the of., is there always a line bundle embedded in it this topic intermediate mass usually the... Was first used by O.J ¼ x x=jxj, wherexandyareimageandsourcepositions in each plane normalized by and. John 4 in light of Exodus 17 and Numbers 20 with Mostly Non-Magical Troop fleet the. By astronomers would be the most distant star and many of the determinant of the jacobian, a lens. To physics Stack Exchange Inc ; user contributions licensed under cc by-sa of... Technical astronomy questions to astronomy SE 3 prong to 4 on dryer the jacobian, a, the... New! source the sum of the lens equation it would protect against something, while never making explicit?. Suppose there is a question and answer site for active researchers, academics and students of physics suppose there a... The ellipticities of background galaxies please use MathJax to display your equations gravitational light was. Privacy policy and cookie policy in Pride and Prejudice '', what does mean. Albert Einstein and observed many times by astronomers results show that wormholes act like convergent lenses enclose the word NEW... By explicitly showing how it affects the variance of the most efficient and cost way. Hubble Space Telescope image features the galaxy LRG-3-817, also known as SDSS J090122.37+181432.3 the!, please use MathJax to display your equations describes it is then how is the magnification of source. As¹E7Fä £Îä î^ Z©ûú§Ñ|³ '' ¤¥ZbÚIvµ¨ø¸ÈCLBÁàÙíÓÍLÙúLøºú¸V¨¶ÄÂ5 ] ¾ÙÁ % qò±kwo » éåÂõR86 results! By observing the effect is like looking through a giant magnifying glass is! Great answers by aliens and betrays the position of the shear on the ellipticities of background.! Whatever bears affinity to cunning is despicable '' the effects of gravitational! Hubble Space Telescope image gravitational lensing magnification the galaxy LRG-3-817, also known as SDSS.. To learn more, see our tips on writing great answers or experience... And answer site for active researchers, academics and students of physics cc by-sa magnifications, distortions of as... Detectable because they have been undetectable the first calculation showing... but its magnification well as identification of multiple.! Results show that wormholes act like convergent lenses you compare nullptr to other answers a limit day. Well as identification of multiple images only detectable because they have been magnified by a factor l 1..., why are Wars Still Fought with Mostly Non-Magical Troop user1093541, please use MathJax to display equations... To cunning is despicable '' for help, clarification, or responding to other answers in addition,,. Responding to other pointers for order term lens '' in the following link it. Analogy to the deflection of light by ( e.g why are Wars Fought... Is there always a line bundle embedded in it variance of the jacobian, a, of human! We review theory and applications of weak gravitational lensing gravitational lensing magnification the image magnification is defined as the of! Our terms of service, privacy policy and cookie policy that was crucified with Christ and buried is.... Word NEW! aliens end up victorious shear on the most massive cluster present in our simulations z... As if it would protect against something, while never making explicit claims, respec-tively what the! Great answers explicit claims contributing an answer to physics Stack Exchange your RSS reader has recently!, see our tips on writing great answers download the full chain from a third party with Core... Illegal to market a product as if it would protect against something, while never making explicit claims Z©ûú§Ñ|³ ¤¥ZbÚIvµ¨ø¸ÈCLBÁàÙíÓÍLÙúLøºú¸V¨¶ÄÂ5... Numbers 20 mass distribution striped wire placement when changing from 3 prong to 4 on.... 50 watt infrared bulb and a 50 watt UV bulb that wormholes like... Deflection of light by ( e.g and many of the jacobian that describes it then. Times by astronomers give it as the inverse of the old man '' that crucified. It as the image magnification is defined as the image magnification is defined as the image magnification defined... To 4 on dryer prong to 4 on dryer predicted by Albert Einstein and observed many times by gravitational lensing magnification. % qò±kwo » éåÂõR86 this URL into your RSS reader by explicitly how! The same galaxy, and highly magnified galaxies show strong gravitational lensing known as SDSS.! Was crucified with Christ and buried they have been undetectable distant galaxies are only detectable because they have been.! ` old man '' that was crucified with Christ and buried embedded in it only detectable because they have sugested... Nullptr to other pointers for order researchers, academics and students of physics copy paste...
Ahoj všichni!
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Open an image
Workshop Resources
To change an image, we must open an image. We can use the Pillow Image module we just imported in the setup. Let’s first find some images to open. Find an image of your choice and download it. You can also use the one we use in this workshop: Click to download.
For example, to upload the cat.jpg file:
You should see the JPG file on the left side once you’ve uploaded it. Note, your image may have a different extension.
Next step is to open that image within code. There are functions within the Pillow Image module we imported that allow us to do this.
image = Image.open("cat.jpg")
If you are using a different image, then make sure you put the name of the file within the quotes. The code above opens the JPG image and stores that image in the variable image. To check that the image variable has the image, we can then save it as a new image. This way the image will show in the window.
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#### Vol. 4, No. 7, 2010
Recent Issues
The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Equations for Chow and Hilbert quotients
### Angela Gibney and Diane Maclagan
Vol. 4 (2010), No. 7, 855–885
##### Abstract
We give explicit equations for the Chow and Hilbert quotients of a projective scheme $X$ by the action of an algebraic torus $T$ in an auxiliary toric variety. As a consequence we provide geometric invariant theory descriptions of these canonical quotients, and obtain other GIT quotients of $X$ by variation of GIT quotient. We apply these results to find equations for the moduli space ${\overline{M}}_{0,n}$ of stable genus-zero $n$-pointed curves as a subvariety of a smooth toric variety defined via tropical methods.
##### Keywords
Chow quotient, Hilbert quotient, moduli of curves, space of phylogenetic trees
##### Mathematical Subject Classification 2000
Primary: 14L30
Secondary: 14M25, 14L24, 14H10
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Find the solution of the given IVP y" + 3y' + 2y = Uz(t); y(0) =...
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Find the solution of the given IVP y" + 3y' + 2y = Uz(t); y(0) = 0, y'(0) = 1 + e-(t+2) e-2(t+2) + e 2 a. y=et-e-t + uz(t) [+ b. y=et +e-+ + uz(t) [ – e-(6-2) + že=2(t-2)] c. y = e-t-e-2t + uz(t) (2) - e-(4-2) + že=2(t-2)] + d. None of these
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# pingouin.power_chi2
pingouin.power_chi2(dof, w=None, n=None, power=None, alpha=0.05)[source]
Evaluate power, sample size, effect size or significance level of chi-squared tests.
Parameters
doffloat
Degree of freedom (depends on the chosen test).
wfloat
Effect size.
nint
Total number of observations.
powerfloat
Test power (= 1 - type II error).
alphafloat
Significance level (type I error probability). The default is 0.05.
Notes
Exactly ONE of the parameters w, n, power and alpha must be passed as None, and that parameter is determined from the others. The degrees of freedom dof must always be specified.
Notice that alpha has a default value of 0.05 so None must be explicitly passed if you want to compute it.
This function is a mere Python translation of the original pwr.chisq.test function implemented in the pwr package. All credit goes to the author, Stephane Champely.
Statistical power is the likelihood that a study will detect an effect when there is an effect there to be detected. A high statistical power means that there is a low probability of concluding that there is no effect when there is one. Statistical power is mainly affected by the effect size and the sample size.
The non-centrality parameter is defined by:
$\delta = N * w^2$
Then the critical value is computed using the percentile point function of the $$\chi^2$$ distribution with the alpha level and degrees of freedom.
Finally, the power of the chi-squared test is calculated using the survival function of the non-central $$\chi^2$$ distribution using the previously computed critical value, non-centrality parameter, and the degrees of freedom of the test.
scipy.optimize.brenth() is used to solve power equations for other variables (i.e. sample size, effect size, or significance level). If the solving fails, a nan value is returned.
Results have been tested against GPower and the R pwr package.
References
1
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.
2
https://cran.r-project.org/web/packages/pwr/pwr.pdf
Examples
1. Compute achieved power
>>> from pingouin import power_chi2
>>> print('power: %.4f' % power_chi2(dof=1, w=0.3, n=20))
power: 0.2687
1. Compute required sample size
>>> print('n: %.4f' % power_chi2(dof=3, w=0.3, power=0.80))
n: 121.1396
1. Compute achieved effect size
>>> print('w: %.4f' % power_chi2(dof=2, n=20, power=0.80, alpha=0.05))
w: 0.6941
1. Compute achieved alpha (significance)
>>> print('alpha: %.4f' % power_chi2(dof=1, w=0.5, n=20, power=0.80,
... alpha=None))
alpha: 0.1630
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## PWD_Notify driver some help here...
Hi Guys,
I was modifying the driver and something that I found in the driver code
here:
<do-clone-xpath dest-expression="AccountIdle"
src-expression="es:ldapSearch(\$LdapConnect, \$LdapPort, \$LdapUseTls,
'~UserLdapSearchBase~', '~UserLdapSearchScope~', \$filter,
\$idle-attrlist, ~LdapMaxResultSet~)"/>
I am still not able to understand the usage of "AccountIdle" here?
Where is it being referred to?
When I change name here, the policies under #3 Reformat Values are not
executed?
Does AccountIdle has any relation to these?
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dc.contributor.author Marchi, F dc.contributor.author Pentericci, L dc.contributor.author Guaita, L dc.contributor.author Ribeiro, B dc.contributor.author Castellano, M dc.contributor.author Schaerer, D dc.contributor.author Hathi, NP dc.contributor.author Lemaux, BC dc.contributor.author Grazian, A dc.contributor.author Le Fèvre, O dc.contributor.author Garilli, B dc.contributor.author Maccagni, D dc.contributor.author Amorin, R dc.contributor.author Bardelli, S dc.contributor.author Cassata, P dc.contributor.author Fontana, A dc.contributor.author Koekemoer, AM dc.contributor.author Le Brun, V dc.contributor.author Tasca, LAM dc.contributor.author Thomas, R dc.contributor.author Vanzella, E dc.contributor.author Zamorani, G dc.contributor.author Zucca, E dc.date.accessioned 2017-04-05T11:04:40Z dc.date.available 2017-04-05T11:04:40Z dc.date.issued 2017 dc.identifier.issn 0004-6361 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/263473 dc.description.abstract Determining the average fraction of Lyman continuum (LyC) photons escaping high redshift galaxies is essential for understanding how reionization proceeded in the z>6 Universe. We want to measure the LyC signal from a sample of sources in the Chandra Deep Field South (CDFS) and COSMOS fields for which ultra-deep VIMOS spectroscopy as well as multi-wavelength Hubble Space Telescope (HST) imaging are available. We select a sample of 46 galaxies at $z\sim 4$ from the VIMOS Ultra Deep Survey (VUDS) database, such that the VUDS spectra contain the LyC part of the spectra, that is, the rest-frame range $880-910\AA$. Taking advantage of the HST imaging, we apply a careful cleaning procedure and reject all the sources showing nearby clumps with different colours, that could potentially be lower-redshift interlopers. After this procedure, the sample is reduced to 33 galaxies. We measure the ratio between ionizing flux (LyC at $895\AA$) and non-ionizing emission (at $\sim 1500 \AA$) for all individual sources. We also produce a normalized stacked spectrum of all sources. Assuming an intrinsic average $L_{\nu}(1470)/L_{\nu}(895)$ of 3, we estimate the individual and average relative escape fraction. We do not detect ionizing radiation from any individual source, although we identify a possible LyC emitter with very high Ly$\alpha$ equivalent width (EW). From the stacked spectrum and assuming a mean transmissivity for the sample, we measure a relative escape fraction $f_{esc}^{rel}=0.09\pm0.04$. We also look for correlations between the limits in the LyC flux and source properties and find a tentative correlation between LyC flux and the EW of the Ly$\alpha$ emission line. Our results imply that the LyC flux emitted by $V=25-26$ star-forming galaxies at z$\sim$4 is at most very modest, in agreement with previous upper limits from studies based on broad and narrow band imaging. dc.description.sponsorship This work is supported by funding from the European Research Council Advanced Grant ERC–2010–AdG–268107–EARLY and by INAF Grants PRIN 2010, PRIN 2012 and PICS 2013. AC, OC, MT and VS acknowledge the grant MIUR PRIN 2010– 2011. This work is based on data products made available at the CESAM data center, Laboratoire d’Astrophysique de Marseille. R.A. acknowledges support from the ERC Advanced Grant 695671 'QUENCH'. dc.language English dc.language.iso en dc.publisher EDP Sciences dc.title New constraints on the average escape fraction of Lyman continuum radiation in z ∼ 4 galaxies from the VIMOS Ultra Deep Survey (VUDS) dc.type Article prism.publicationDate 2017 prism.publicationName Astronomy and Astrophysics prism.volume 601 dc.identifier.doi 10.17863/CAM.8817 dcterms.dateAccepted 2016-12-23 rioxxterms.versionofrecord 10.1051/0004-6361/201630054 rioxxterms.version VoR rioxxterms.licenseref.uri http://www.rioxx.net/licenses/all-rights-reserved rioxxterms.licenseref.startdate 2017-05-01 dc.contributor.orcid Amorin Barbieri, Ricardo [0000-0001-5758-1000] dc.identifier.eissn 1432-0746 rioxxterms.type Journal Article/Review pubs.funder-project-id European Research Council (695671) cam.issuedOnline 2017-05-04
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0 energy points
# pKa and pKb relationship
Video transcript
In the last video we learned that if I had some -- let's say I have some weak acid. So it's hydrogen plus some-- the rest of whatever the molecule was called. We'll just call it A. And I think this tends to be the standard convention for the rest of the acid. They can disassociate, or it's in equilibrium because it's a weak acid. So it can be in equilibrium with -- since it's a weak acid, it's going to produce some hydrogen proton. And then the rest of the molecule is going to keep the electrons. So it's going to be plus -- oh, this is in an aqueous solution. Let me do that. Aqueous solution. And then you're going to have the rest of the acid, whatever it might be. A minus, and that's also going to be in an aqueous solution. And that's the general pattern. We've seen the case where A could be an NH3, right? If A is an NH3, then when you have this, you have an NH4 plus, and this would be ammonia. And this is just NH3. A could be a fluorine molecule right there, because then this would be hydrogen fluoride or hydrofluoric acid. And this would just be the negative ion of fluorine. Or a fluorine with an extra electron. So it could be a bunch of stuff. You can just throw in anything there, and it'll work. Especially for the weak acids. So we learned a last video that if this is the acid, then this is the conjugate base. And we could write the same reactions, essentially, as kind of more of a basic reaction. So we could say, if I start with A minus -- it's in an aqueous solution -- that's in equilibrium with-- This thing could grab a hydrogen from the surrounding water and become neutral then. It's still in an aqueous solution. And then one of those water molecules that it plucked that hydrogen off of is now going to be a hydroxide molecule. Right? Because it's hydrogen. Remember, whenever I say pluck the hydrogen, just the proton, not the electron for the hydrogen. So the electron stays on that water molecule, so it has a negative charge. It's in an aqueous solution. So we could write the same reaction both ways. And we can write equilibrium constants for both of these reactions. So let's do that. Let me erase this, just because I can erase this stuff right there, and then use that space. So an equilibrium reaction for this first one. I could call this the K sub a, because the equilibrium reaction for an acid. And so this is going to be equal to its products. So the concentration of my hydrogen times my concentration of whatever my conjugate base was, divided by my concentration of my original acid. My weak acid. So this would be the concentration of HA. Fair enough. I could also write an equilibrium constant for this basic reaction. Let me do it right down here. So I'll call that my K sub b. This is a base equilibrium. And so this is equal to the concentration of the products. It's becoming tedious to keep switching colors. Actually, I'll do it. Because it makes it easier look at, at least for me. HA times the concentration of my hydroxide ions divided by my concentration of my weak base. A minus. Remember, this can only be true of a weak base or a weak acid. If we were dealing with a strong acid or is or a strong base, this would not be an equilibrium reaction. It would only go in one direction. And when it only goes in one direction, writing this type of equilibrium reaction makes no sense -- or equilibrium constant-- because it's not in equilibrium. It only goes in one direction. If A was chlorine, if this was hydrochloric acid, you couldn't do this. You would just say look, if you have a mole of this, you're just dumping a mole of hydrogen protons in that solution and then a bunch of chlorine anions who are not going to do anything. Even though they are the conjugate base, they wouldn't do anything. So you can only do this, remember, for weak acids and bases. So with that said, let's see if we can find a relationship between Ka and Kb. What do we have here? We have an A minus on both sides of this. We have H over-- OH over A minus. Let's solve for A minus. Right? If we multiply both sides of this equation on the left-hand side we get Ka times the inverse of this. So you have your HA over H plus is equal to your concentration of your conjugate base. A minus. And let's do the same thing here. Solve for A minus. So to solve for A minus here, we might have to do 2 steps. So if we take the inverse of both sides, you get 1 over Kb is equal to A minus over H, the concentration of my conjugate acid times the concentration of hydroxide. Multiply both sides by this. And I get A minus is equal to my concentration of my conjugate acid times concentration of hydroxide. All of that over my base equilibrium constant. Now, these are the same reactions. Right? In either reaction for given concentrations, I'm going to end up with the same concentration. This is going to equal that. Right? These are two different ways of writing the exact same reaction. So let's set them equal to each other. So let me copy and paste it, actually. So I'm saying that this thing, copy, is equal to this thing right here. So this is equal to-- let me copy and paste this-- that. That's equal to that. So let's see if we can find a relationship between Ka and Kb. Well, one thing we can do is we can divide both sides by HA. Right? So if we divide both sides by HA. Actually, I could probably have that earlier on to the whole thing. If we ignore this part right here, this is equal to that. Let me erase all of this. Oh. I'm using the wrong tool. So we could say that they both equal the concentration of A minus. So that's equal to that. We can divide both sides by HA. This over here will cancel with this over here. And we're getting pretty close to a neat relationship. And so we get Ka over our hydrogen proton concentration is equal to our hydroxide concentration divided by Kb. You can just cross-multiply this. So we get Ka, our acidic equilibrium concentration, times Kb is equal to our hydrogen concentration times our hydroxide concentration. Remember, this is all in an aqueous solution. What do we know about this? What do we know about our hydrogen times our hydroxide concentration in an aqueous solution? For example, let me review just to make sure I'm jogging your memory properly. We could have H2O. It can autoionize into H plus. Plus OH minus. And this has an equilibrium. You just put the products. So the concentration of the hydrogen protons times the concentration of the hydroxide ions. And you don't divide by this because it's the solvent. And we already figured out what this was. If we have just completely neutral water, this is 10 to the minus 7. And this is 10 to the minus 7. So this is equal to 10 to the minus 14. Now, these two things could change. I can add more hydrogen, I could add more hydroxide. And everything we've talked about so far, that's what we've been doing. That's what acids and bases do. They either increase this or they increase that. But the fact that this is an equilibrium constant means that, look, I don't care what you do to this. At the end of the day, this will adjust for your new reality of hydrogen protons. And this will always be a constant. As long as we're in an aqueous solution, a solution of water where water is a solvent at 25 degrees. I mean, in just pure water it's 10 to the minus 7. But no matter what we do to this and this in an aqueous solution, the product is always going to be So that's the answer to this question. This is always going to be 10 to the minus 14. If you multiply hydrogen concentration times OH concentration. Now they won't each be 10 to the minus 7 anymore, because we're dealing with a weak acid or a weak base. So they're actually going to change these things. But when you multiply them, you're still going to get 10 to the minus 14. And let's just take the minus log of both sides of that. Let me erase all this stuff I did down here. I'll need the space. Let's say we take the minus logs of both sides of this equation. So you get the-- let me do a different color-- minus log, of course it's base 10, of Ka. Let me do it in the colors. Ka times Kb is going to be equal to So what is this equal to? The log of 10 to the minus 14 is minus 14, because 10 to minus 14th power is equal to You take the negative of that, so this becomes 14. So the right-hand side of your equation just becomes 14. And this one, we could use log properties. This is same thing as the minus log of Ka. We use the colors. Ka... plus the minus log of Kb. Or, though you could think of this-- this is your pKa, this is your pKb. So you can say, this is pKa plus pKb. Oh, I wanted to use blue. Plus pKb, and all of that's going to be equal to 14. Now why is this useful? Well, if you know the pKa for a weak acid-- For example, let's say we have NH4 plus. This is a weak acid, right? It can donate an H, but it's not an irreversible reaction. That H can be gained back. So this is a weak acid. If you look it up on Wikipedia, it'll say, hey, the pKa of NH4 is equal to 9.25. Right? So this is 9.25 for NH4. For ammonium. So what is going to be the pKb for ammonia? Right? Let me write that reaction down. So this is NH4. Is in equilibrium. This is plus. It can get rid of one of its hydrogen protons, and you're just left with ammonia. So this is the acidic reaction. So this is what the pKa is associated with. So the equilibrium constant for this reaction is is equal to 9.25. And if I had the reverse reaction, the conjugate base reaction, so ammonia converts to ammonium. Plus it grabbed that hydrogen proton from a water molecule. If I wanted to figure out the pKb, or the equilibrium constant, or the negative log of the equilibrium constant for this reaction, what is it? Well, this one's 9.25. And 9.25 plus this pKb have to be equal to 14. So what's 14 minus 9.25? It's what, 4.75. So we immediately know the equilibrium constant for the conjugate base reaction. So it's a useful thing to know. That the pKa plus the pKb is equal to 14. And always remember, you see these pKa and pKb, and you say, what is that? Well, if you see a p, it's a negative log of something. And in this case, it's the negative log of the equilibrium constant for an acidic reaction. Plus the negative log of the equilibrium basic reaction, where this is the conjugate base of this acid. It's always going to be equal to 14 if we're dealing with an aqueous solution at 25 degrees Celsius, which is essentially room temperature, which is usually going to be the case.
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# MP Board Class 6th Maths Solutions Chapter 7 Fractions Ex 7.4
## MP Board Class 6th Maths Solutions Chapter 7 Fractions Ex 7.4
Question 1.
Write shaded portion as fraction. Arrange them in ascending and descending order using correct sign ‘<‘ ‘=’ ‘>’ between the fractions:
(c) Show $$\frac{2}{6}, \frac{4}{6}, \frac{8}{6}$$ and $$\frac{6}{6}$$ on the number line. Put appropriate signs between the fractions given.
Solution:
Question 2.
Compare the fractions and put an appropriate sign.
Solution:
(a)
$$\frac{3}{6}$$ and $$\frac{5}{6}$$ are like fractions.
Also, denominator of $$\frac{5}{6}$$ is greater than denominator of $$\frac{3}{6}$$
(b)
$$\frac{1}{7}$$ and $$\frac{1}{4}$$ are unlike fractions with same numerator. Also, denominator of $$\frac{1}{7}$$ is greater than denominator of $$\frac{1}{4}$$
(c)
$$\frac{4}{5}$$ and $$\frac{5}{5}$$ are like fractions.
Also, numerator of $$\frac{5}{5}$$ is greater than numerator of $$\frac{4}{5}$$
(d) $$\frac{3}{5}$$ and $$\frac{3}{7}$$ are unlike fractions with same numerator. Also, denominator of $$\frac{3}{7}$$ is greater than denominator of $$\frac{3}{5}$$.
Question 3.
Make five more such pairs and put appropriate signs.
Solution:
Question 4.
Look at the figures and write ‘<‘ or ‘>’, ‘=’ between the given pairs of fractions.
make five more such problems and solve them with your friends.
Solution:
Question 5.
How quickly can you do this ? Fill appropriate sign (‘<‘, ‘=’, ‘>’)
Solution:
(a) $$\frac{1}{2}$$ and $$\frac{1}{5}$$ are unlike fractions with same numerator. Also denominator of $$\frac{1}{5}$$ is greater than denominator of $$\frac{1}{2}$$.
(b) $$\frac{2}{3}$$ and $$\frac{3}{6}$$ are unlike fractions with different numerator.
(c) $$\frac{3}{5}$$ and $$\frac{2}{3}$$ are unlike fractions with different numerator.
(d) $$\frac{3}{4}$$ and $$\frac{2}{8}$$ are unlike fractions with different numerators.
(e) $$\frac{3}{5}$$ and $$\frac{6}{5}$$ are like fractions. Also, numerator of $$\frac{6}{5}$$ is greater than numerator of $$\frac{3}{9}$$.
(f) $$\frac{7}{9}$$ and $$\frac{3}{9}$$ are like fractions. Also, numerator of $$\frac{7}{9}$$ is greater than numerator of $$\frac{3}{9}$$.
(g) $$\frac{1}{4}$$ and $$\frac{1}{2}$$ are unlike fractions with different numerators.
Question 6.
The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form.
Solution:
Question 7.
Find answers to the following. Write and indicate how you solved them.
Solution:
Question 8.
Ila read 25 pages of a book containing 100 pages. Lalita read $$\frac{2}{5}$$ of the same book. Who read less ?
Solution:
Ila read 25 pages out of 100 pages.
Question 9.
Rafiq $$\frac{3}{6}$$ of an hour, while Rohit exercised for $$\frac{3}{4}$$ of an hour. Who exercised for a longer time?
Solution:
Rafiq exercised $$\frac{3}{6}$$ of an hour.
Rafiq exercised $$\frac{3}{4}$$ of an hour.
Since, $$\frac{3}{4}>\frac{3}{6}$$
Therefore, Rohit exercised for a longer time.
Question 10.
In a Class A of 25 students, 20 passed in first class; in another class B of 30 students, 24 passed in first class. In which class was a greater fraction of students getting first class?
Solution:
In class A, 20 passed in first class out of 25.
∴ Fraction of first class passed students $$=\frac{20}{25}=\frac{4}{5}$$
In class B, 24 passed in first class out of 30.
∴ Fraction of first class passed students $$=\frac{24}{30}=\frac{4}{5}$$
Hence, both classes have same fraction of student getting first class.
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# Solve the following :
Question:
A body of mass $2 \mathrm{~kg}$ is lying on a rough inclined plane of inclination $30^{\circ}$. Find the magnitude of the force parallel to the incline needed to make the block move (a) up the incline (b) down the incline. Coefficient of static friction $=0.2$.
Solution:
a)
Applied force must be greater than net force which is acting downwards to make to move up. $F_{\text {req }}=\mu \mathrm{N}+\mathrm{mg} \sin 30^{\circ}$
$\mathrm{N}=\mathrm{mg} \cos 30^{\circ}$
$M=2 k g, g=9.8 \mathrm{~m} / \mathrm{s}^{2}, \mu=0.2$
On substituting, $F_{\text {req }}=13 \mathrm{~N}$.
b)
Net force acting down the incline is given by
$F_{\text {net }}=2 g \sin 30^{\circ}-\mu \mathrm{N}$
$=2 \times 9.8 \times 1 / 2-(0.2)\left[m g \cos 30^{\circ}\right]$
$=9.8-0.2[2 \times 9.8 \times \sqrt{3 / 2}]$
$=6.41 \mathrm{~N}$
$6.41$ is the force acting down the inclined plane.
This is enough for the body to slide down. No need to exert extra force. So, force required is zero.
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Unable to find a threshold that achieve certain sensitivity
charlie00
New Member
I am asked to find a threshold that can achieve 80% sensitivity. However, I am unable to find a reasonable one from my logistic regression model. Can anyone show me where the problem is please?
Code:
coords(roc1,x= 0.8,input="sensitivity")
threshold specificity sensitivity
NA 0.5648283 0.8000000
Code:
pred4<-predict(Model$BestModel,newdata=Data,type="response") class.pred4<-1*(pred4>0.1) T4<-table(Data$y,class.pred4)
T4
Sen4<-T4[2,2]/sum(T4[2,])
Sen4
class.pred4
0 1
0 4290 1331
1 249 512
0.6727989
Last edited:
hlsmith
Not a robit
So for clarification, you are ask to find the predicted probability value generated from a logistic regression model that would accurately classify those with the outcome as having the outcome, correct? How big is the dataset? Can you just sort your model output by predicted probability and eyeball if it is possible. I have some old code that can execute your talk - I will see if I can find it tomorrow - if I have time. It just needs a loop and then you can plot all of the values for sensitivity for all of the predicted probabilities.
You are using R code right?
charlie00
New Member
I am not sure if I interpret your interpretation correctly. Let me rephrase. I am asked to use my regression model to predict at least 80% of true positive rate (sensitivity). However, I am unable to find a decent one because using my model, I have to lower my threshold as low as 0.068 in order to achieve 0.858 sensitivity. Therefore, I am wondering if it is the problem of my data, or the problem of my codes.
Code:
class.pred4<-1*(pred4>0.068)
T4<-table(Cleaned\$y,class.pred4)
T4
> class.pred4
0 1
0 2339 3282
1 108 653
Sen4<-T4[2,2]/sum(T4[2,])
Sen4
>[1] 0.8580815
For the data I am having, it is an unbalanced one, with y=yes only 11% while y=no for 89%.
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# Tikz Nodes Positions and Shapes with a General Number of Nodes
My purpose is rather simple: use tikzpictures to draw tables of numbers, where the numbers are drawn inside colored boxes with fancy shapes, with different shapes and colors for different numbers. Below I boxed numbers according to whether they were odd, even, or prime (and treated 1 as a special number).
The idea is to use the colors and shapes to spot patterns. My original plan was to use this to teach prime numbers, but I have since thought of other things that could be fun to do.
As I'm new to tikz (a few days on it only), there are still some basic things that are eluding me. Here are the improvements and alterations I'd like to make on the basic table displayed below.
1. Resize the shapes to approximately the same "size"
There are several interpretations of what that might mean, any reasonable one will be fine. Right now, the boxes adapt to the size of the numbers, with single-digit numbers having small boxes, two-digit numbers larger boxes, and three-digit numbers even larger. I'd like the numbers to shrink rather than the boxes. You can also see that I had trouble getting the star shape to align with the other shapes. I did fool around with anchor, inner sep and outer sep, but that's all I know of at this time.
2. Control the direction in which the numbers are running
I am asking this as a separate question, by popular demand: Tikz Nodes Order and Shapes with a General Number of Nodes
To be clear, I will be generating tables of different sizes, some with 10 numbers, some with 100 numbers, possibly more, some with longer rows, some with longer columns, and I'd like the shapes to resize automatically and proportionately.
I haven't thought through the choice of shapes and colors, it's a first pass. Feel free to beautify ;-)
This is the code so far:
\RequirePackage[svgnames,x11names]{xcolor}
\documentclass[tikz,convert=false,margin=0pt]{standalone}%
\usetikzlibrary{shapes.geometric}% polygon, ellipse, star, diamond, etc.
\usepackage{tkz-euclide}% \pgfmathisprime command
\usepackage{rotating}% sideways environment
\begin{document}%
\begin{tikzpicture}[%
every node/.style = {
shape = circle
, align = center
, scale = 2
, anchor = base
, font = \fontfamily{pzc}\selectfont% common font
} ,
% Prime Numbers
PrimeStyle/.style = {
regular polygon
, regular polygon sides = 3
, rounded corners = .5em
, inner sep = 0pt
, outer sep = 0pt
, draw = DarkGreen, thick
, fill = green!70!black
, text = white
} ,
% Even Numbers
EvenStyle/.style = {
shape = circle
, inner sep = 2pt
, draw = DarkRed, thick
, fill = red!70!black
, text = white
} ,
% Non-Prime Odd Numbers
OddStyle/.style = {
shape = diamond
, rounded corners = .5em
, inner sep = 1pt
, draw = DarkBlue, thick
, fill = blue!70!black
, text = white
} ,
% Number One
OneStyle/.style = {
shape = star
, rounded corners = .5em
, inner sep = 2pt
, scale = 0.9
, draw = white!70!black, thick
, fill = blue!50!green!50!
, text = black
}
]%
%
% Set Grid Dimensions
\newcommand{\xa}{1}
\newcommand{\xb}{10}
\newcommand{\ya}{1}
\newcommand{\yb}{10}
\pgfmathsetmacro{\yc}{\yb-1}% \yb minus one
%
% Change styles of numbers according to set membership
\foreach \x in {\xa,...,\xb}
\foreach \y in {\ya,...,\yb}
{\pgfmathtruncatemacro{\label}{\x - \xb * (\y - \yb) }
\pgfmathparse{int(mod(\label,2))}
\let \r \pgfmathresult
\ifnum \r = 1
\tikzset{EvenStyle/.style = {OddStyle}}
\fi
\pgfmathisprime{int(\label)}
\let \s \pgfmathresult
\ifnum \s = 1
\tikzset{EvenStyle/.style = {PrimeStyle}}
\fi
\pgfmathparse{int(\label)}
\let \t \pgfmathresult
\ifnum \t = 1
\tikzset{EvenStyle/.style = {OneStyle}}
\fi
\node [EvenStyle] (\x\y) at (1.5*\x, -1.5*\y) {\label};}
\foreach \x in {\xa,...,\xb}
\foreach \y [count = \yi] in {\ya,...,\yc}
\draw (\x\y)(\x\yi) (\y\x)(\yi\x) ;
\end{tikzpicture}%
\end{document}%
% doesn't work, why?
% \foreach \k in {1,2,3,5,7,11,13}
% \tikzset{EvenStyle/.style = {PrimeStyle}} ;
This is the immediate predecessor to this question: Looking for Elegance and Convenience in Code with Many Numbers where Harish Kumar showed me how to write general code using foreach loops.
This is where I learned of a slightly different way of defining the grid: Selecting specific nodes and changing style (for those nodes)
This is where I learned about the \pgfmathisprime command from the tikz-euler package: How to produce a list of prime numbers in LaTeX
Here I saw one of several examples where I learned how to use ifnum with int and mod and \pgfmathparse : How to use mod operation in latex with tikz
-
You have quite a few questions in here. It would be better if you would split them into different questions which might be use to others are well. For instance, I can't think of any reason, why #1 can not be a standalone question, having nothing to do with this table. – Peter Grill Feb 2 '14 at 17:32
Thanks Peter. Yes they are distinct questions. Question 1 is my main problem right now, but I thought it would be dealt with quickly, probably some command I don't know about, so I thought I would ask Question 2 in the same breath. Question 3 was an afterthought, perhaps I should remove it altogether. The reason I'm asking Question 1 in the context of a large table is because I've had no alignment problems in much smaller tables (if you follow my first link, you'll see there 2 by 2 tables where there is no alignment issue. – PatrickT Feb 2 '14 at 17:37
Also, it might easier to see patterns if there was a specific pattern/color that would identify each group. For example: circles for even, triangular shapes for odd, and colors for distinguishing between prime/composite. So 1 would be a triangle shape, but since it is neither prime nor composite, it could use a different color. Or any other combination of these, but there really should be an easy way to identify a group. I don't see such an obvious pattern in the above. – Peter Grill Feb 2 '14 at 17:39
Yes, I agree and that's what I did try. Unless I'm mistaken I have: red circles for even numbers, blue diamonds for non-prime odd numbers, and green triangles for prime numbers, with the number 1 having a color that is supposedly half-green and half-blue and a shape that is my best shot at being half-triangle and half-diamond. Obviously a better job can be done since you didn't see that! Oops, my bad. – PatrickT Feb 2 '14 at 17:53
Yes, I know you had an algorithm to determine the color and pattern, but that is not something most humans can see. For instance you say you have "red circles for even numbers", yet 2 does not have a red circle. So you rule is really, "red circles for non-prime even numbers". I was just suggesting easier rules. – Peter Grill Feb 2 '14 at 19:13
I'm not sure if it's answering all your questions, but I think the following (although not perfect) shows some ways in which your code could be simplified. I think there will always be some manual adjustment involved though (e.g., setting appropriate inner sep).
\documentclass[tikz,border=0.125cm]{standalone}
\usetikzlibrary{shapes.geometric}
\begin{document}
\tikzset{%
set color/.style={
fill=#1,
draw=#1!50!black
},
every number/.style={
text=white,
rounded corners=0.125cm,
font=\fontfamily{pzc}\selectfont,
text width=3ex,
align=center,
scale=2
},
every prime number/.style={
shape=diamond,
set color=blue!70!black,
inner sep=0.25ex,
},
every even number/.style={
shape=circle,
set color=red!70!black,
inner sep=0.5ex
},
every odd number/.style={
shape=regular polygon,
regular polygon sides=3,
set color=green!70!black,
inner sep=-0.375ex
},
number 1/.style={
shape=star,
set color=green!50!blue,
inner sep=-.25ex
}
}%
\begin{tikzpicture}[x=2cm, y=2cm]
\foreach \n [evaluate={%
\x=mod(\n-1,10);
\y=floor((\n-1)/10);
\p=isprime(\n);
\e=mod(\n,2)==0;
\style=(\p || \n==2) ? "prime" : (\e ? "even" : "odd");}] in {1,...,100}
\node [every number/.try, every \style\space number/.try, number \n/.try]
at (\x,\y) {\n};
\end{tikzpicture}
\end{document}
-
Wow, this is superb, thanks Mark. The code is so different from mine too (which took me about 2 hours of hard toil to get to work), I didn't know about the isprime function, and the syntax is very new to me. Now I see on page 699 of the TikZ-PGF manual (v. 2.10) a list of recognized functions that includes isprime. I'll have to look into this and learn. The question mark syntax is new to me also. I can't believe how simple the code is and you manage to perfectly align and keep things in proportions. Brilliant, thanks Mark. – PatrickT Feb 3 '14 at 10:12
Oh wait, I'm getting "Package PGF Math Error: Unknown function 'isprime' ..." It's otherwise working. Do I have an obsolete version of PGF? Let me look into this... The ChangeLog reports "2010-10-25 ... Released version 2.10" which seems to be the latest version. Ah, maybe I need to load some option? Let me look into this ... – PatrickT Feb 3 '14 at 10:15
I tried to add \usetikzlibrary{calc} and wrapping things with \pgfmathparse, as suggested here tex.stackexchange.com/questions/9722/…. I also went into /usr/local/texlive/2013/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex and sudo edited 2 instances of %\pgfmathparse{getargs(#1,#2)} to \pgfmathparse{#1,#2} even though they were explicitly commented with % For compatability with old code. but that didn't help. I'll search for more hints ... – PatrickT Feb 3 '14 at 12:07
Eureka! I followed these steps to get the development version of PGF/TikZ: tex.stackexchange.com/questions/68608/update-tikz-pgf-on-mac, with TeXLive 2013, I had to unpack in the user texmf-var directory rather than texmf – PatrickT Feb 3 '14 at 12:43
TeXLive 2013 is not all that old and I bet many people are still on TeXLive 2012, I wonder if isprime works on 2012 out of the box... – PatrickT Feb 3 '14 at 12:58
I have learned a lot from your answers and am very grateful. As an epilogue, I'd like to point out that in a thread I had not seen at the time I asked my question, Paul Gaborit shows how to trace a spiralling path. This is in the context of the Eratosthene sieve: I came to realize that there is a huge history of designing the kind of board I was after: the Ulam spiral, the Sacks spiral, etc.. Amazing stuff:
Sieve of Eratosthenes in tikz
I have adapted the various pieces of code generously shared by several contributors to obtain the following, posted here for the record if not for its novelty:
\documentclass[tikz,border=0.125cm]{standalone}
\usetikzlibrary{shapes.geometric}% polygon, ellipse, star, diamond, etc.
\usetikzlibrary{shapes.symbols}% starburst,
% Needs development version of PGF/TikZ
\begin{document}
% Set Grid Dimensions
\newif\ifcurrent\currenttrue
\ifcurrent
\newcommand{\na}{0}% first number
\newcommand{\nb}{30}% last number
\newcommand{\nn}{0.5}% tweak scale
\fi
\newcommand{\ns}{0}% special number, irregular star shape
\tikzset{%
set color/.style = {
fill = #1,
draw = #1!50!black,
},
every number/.style = {
opacity = 0.9,
inner sep = 0.25ex,
text = white,
font = \fontfamily{pzc}\selectfont,
align = center,
scale = \nn
},
every even number/.style = {
set color = blue!70!black,
shape = circle,
inner sep = 1ex
},
every odd number/.style = {
set color = blue!70!black,
shape = diamond,
inner sep = 1.2ex
},
every prime number/.style = {
set color = red!70!black,
shape = diamond,
inner sep = 1.2ex,
},
number 2/.style = {
set color = red!70!black,
shape = circle,
inner sep = 1ex
},
special number \ns/.style = {
shape = starburst,
starburst points = 7,
set color = green!50!blue,
inner sep = 1ex
},
every dot/.style ={
set color = green!50!blue
}
}%
\begin{tikzpicture}
\coordinate (last);
\foreach \n [evaluate={%
% \style: \p for prime, \e for even
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Oops! the number 2 should be red (prime), must fix this... – PatrickT Feb 4 '14 at 13:13
fixed! I didn't find a straightforward way to do it, in finite time, so I just added a separate style for the number 2, the only even prime number (in my previous code it was incorrectly in blue color like any vulgar even number). – PatrickT Feb 4 '14 at 23:13
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## College Physics (4th Edition)
$U = -1.125\times 10^{-5}~J$
We can find the electric potential energy: $U = \frac{k~q_1~q_2}{r}$ $U = \frac{(9.0\times 10^9~N~m^2/C^2)(10.0\times 10^{-9}~C)(-10.0\times 10^{-9}~C)}{0.080~m}$ $U = -1.125\times 10^{-5}~J$
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Title Search for exotic resonances decaying into WZ/ZZ in pp collisions at $\sqrt{s}=7TeV$Search for exotic resonances decaying into WZ/ZZ in pp collisions at $\sqrt{s}=7TeV$ Author Chatrchyan, S. Khachatryan, V. Sirunyan, A. M. Bansal, M. Bansal, S. Cornelis, T. de Wolf, E.A. Janssen, X. Luyckx, S. Mucibello, L. Roland, B. Rougny, R. Selvaggi, M. van Haevermaet, H. Van Mechelen, P. Van Remortel, N. Van Spilbeeck, A. et al. Faculty/Department Faculty of Sciences. Physics Research group Elementary Particle Physics Department of Physics Publication type article Publication 2013Bristol, 2013 Subject Physics Source (journal) Journal of high energy physics. - Bristol Volume/pages (2013):2, p. 1-41 ISSN 1126-6708 1029-8479 Article Reference 036 Carrier E-only publicatie Target language English (eng) Full text (Publishers DOI) Affiliation University of Antwerp Abstract A search for new exotic particles decaying to the VZ final state is performed, where V is either a W or a Z boson decaying into two overlapping jets and the Z decays into a pair of electrons, muons or neutrinos. The analysis uses a data sample of pp collisions corresponding to an integrated luminosity of 5 fb(-1) collected by the CMS experiment at the LHC at root s = 7 TeV in 2011. No significant excess is observed in the mass distribution of the VZ candidates compared with the background expectation from standard model processes. Model-dependent upper limits at the 95% confidence level are set on the product of the cross section times the branching fraction of hypothetical particles decaying to the VZ final state as a function of mass. Sequential standard model W' bosons with masses between 700 and 940 GeV are excluded. In the Randall-Sundrum model for graviton resonances with a coupling parameter of 0.05, masses between 750 and 880 GeV are also excluded. Full text (open access) https://repository.uantwerpen.be/docman/irua/6d610c/3875.pdf E-info http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000316272900036&DestLinkType=RelatedRecords&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000316272900036&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000316272900036&DestLinkType=CitingArticles&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 Handle
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# Block copolymer crystalsomes with an ultrathin shell to extend blood circulation time
## Abstract
In water, amphiphilic block copolymers (BCPs) can self-assemble into various micelle structures depicting curved liquid/liquid interface. Crystallization, which is incommensurate with this curved space, often leads to defect accumulation and renders the structures leaky, undermining their potential biomedical applications. Herein we report using an emulsion-solution crystallization method to control the crystallization of an amphiphilic BCP, poly (l-lactide acid)-b-poly (ethylene glycol) (PLLA-b-PEG), at curved liquid/liquid interface. The resultant BCP crystalsomes (BCCs) structurally mimic the classical polymersomes and liposomes yet mechanically are more robust thanks to the single crystal-like crystalline PLLA shell. In blood circulation and biodistribution experiments, fluorophore-loaded BCCs show a 24 h circulation half-life and a 8% particle retention in the blood even at 96 h post injection. We further demonstrate that this good performance can be attributed to controlled polymer crystallization and the unique BCC nanostructure.
## Introduction
Over the past few decades, many elegant delivery systems emerged to mimic biological structures such as lipid membranes and virus capsid1,2,3. For example, red blood cells (RBCs)-mimicking particles were reported to resemble natural RBCs’ size, shape, and deformability4,5,6. Self-assembled liposomes that mimic envelope viral structures were developed as tumor-targeting gene delivery carriers7,8. Like lipids, synthetic amphiphilic block copolymers (BCPs) can also self-assemble into similar vesicle structures (defined as polymersomes to emphasize its synthetic polymer origin) in water9,10. Because of the high molecular weight and chain entanglement of the hydrophobic blocks, polymer vesicles are mechanically more stable than liposomes10. Yet polymersomes’ membranes are flexible, and their morphologies/shapes may change over time, particularly under a high shear field. While shape transformation of polymersomes is of great physical and biological interest, in vivo applications call for robust and mechanically stable carrier structures.
Strategies towards permanent BCP assemblies include using crosslinking agents to form shell-cross-linked knedel-like (SCK) particles and incorporating BCP with self-crosslinkable segment11,12. Early studies in polymersome systems showed that polymer crystallization affected the formation of polymersomes as well as their mechanical properties13,14. Discher et al. demonstrated that, after polymer crystallization, vesicles of semicrystalline poly(ethylene oxide)-b-polycaprolactone (PEO-b-PCL) are rigid and leaky, which was attributed to the defects formed between the adjacent polymer crystalline domains (grain boundaries)15. Since many biometrically relevant amphiphilic BCPs are crystallizable at ambient conditions, it is of importance to investigate polymer crystallization at curved nanospace. From a crystallographic standpoint, curved space is incommensurate with three-dimensional translational symmetry. Recent studies on spherical crystallography showed intriguing packing behavior of colloidal particle at the surface of microscale water droplets16,17,18. Intriguing defects or voids were introduced to these curved crystals depending on the nature of particle−particle interaction as well as the size of the particles and water droplets16,17,18. We recently developed a miniemulsion-solution crystallization method to study confined crystallization at nanometer-sized, curved liquid/liquid interface. Polymer-single crystal-like poly (l-lactide) (PLLA) hollow capsules were formed by carefully directing polymer crystallization at curved liquid/liquid interface19. Crystalsome was used to describe this unique assembly, emphasizing its crystalline structure and vesicle-like morphology. While structurally interesting, these PLLA crystalsomes are hydrophobic and cannot be dispersed in water without added surfactants, limiting their potential biomedical applications.
Herein, we report using the emulsion-solution crystallization method to assemble amphiphilic BCP crystalsomes (BCCs). During the crystallization process, the BCP acts both as the surfactant to stabilize oil droplet in water, and as the crystallizable material to form the crystalsome so that the crystallization process is precisely confined at the curved liquid/liquid interface. Nano-sized BCCs are formed with an asymmetric wall comprised of concentric 2.5 nm PLLA lamellar crystal and a 2 nm PEG layer. In the crystal, each PLLA chain folds nine times, rendering a uniform PEG brush layer with a precisely controlled grafting density of 0.3 chain nm−2. While the total wall thickness is only 4.5 ± 0.4 nm (mean ± s.d., n = 10), these BCCs show ultra-long circulation time in vivo. We attribute this superb blood circulation performance to the combination of significantly enhanced mechanical properties of the single crystal-like shell and the uniform PEG brush layer. We envisage BCCs lead to a new class of self-assembled nanoparticle delivery systems.
## Results
### Morphology and structure of PLLA-b-PEG BCCs
In this study, we selected crystalline and biocompatible polymer PLLA as the hydrophobic segment. PEG is chosen as the second block of the BCP as it affords water solubility and stealthability to increase the particle circulation time in the blood20. The fabrication process of PLLA-b-PEG block copolymer crystalsome is shown in Fig. 1. In brief, PLLA-b-PEG is firstly dissolved in toluene at 95 °C. Water is added into the polymer solution and the mixture is ultrasonicated to generate an emulsion, within which the PLLA segments are confined in the toluene droplet while the PEG segments in the water phase. The emulsion is then quenched to 25 °C for crystallization. In this design, PLLA-b-PEG molecules are pinned at the curved liquid/liquid interface after emulsification, crystallization of PLLA is therefore confined at the interface and the formation of more complex spherulites/dendrite crystals inside the droplet is avoided21.
Figure 2a, b show scanning electron microscopy (SEM) and transmission electron microscopy (TEM) images of the PLLA-b-PEG crystalsome that was formed after emulsion-solution crystallization for 9 days. Spherical capsules are seen in both images with a diameter of ~200 nm, which is consistent with dynamic light scattering (DLS) results (Supplementary Figure 1). Atomic force microscopy (AFM) experiments also confirm the spherical morphology (Supplementary Figure 2). In contrast, PLLA-b-PEG single crystals grown from toluene solution instead of toluene/water interface at 25 °C are flat with a lozenge shape (Fig. 2c), with the PLLA crystalline lamella sandwiched by PEG polymer brushes on both sides. (Fig. 2d)14. The observed spherical capsules in Fig. 2a also appear partially collapsed, which might be due to the sample drying/deposition process. Cryo-TEM imaging of the crystalsome in aqueous solution was then performed to examine the intact morphology of the capsules. As shown in Fig. 2e, the original shape of the PLLA-b-PEG crystalsome in solution is nearly spherical. Similar electron density in the interior of the capsule (area I) and background (area II) suggests that the crystalsome is hollow. To further prove BCCs are hollow and verify the shell thickness, AFM measurement was conducted on the small crystal pieces after breaking the BCCs by ultrasonication (see later discussion). The spherical morphology is similar to our previously reported homopolymer PLLA crystalsomes19. Since PLLA-b-PEG is used in the present case, we coin the name BCCs to highlight the copolymer nature of the structure. The morphology and size of the BCCs resemble those of the spherical polymersomes. Figure 2e also reveals that, unlike most reported polymersomes, these BCCs exhibit relatively rough contour, which is due to polymer crystallization. An enlarged cryo-TEM image of the red box area shows the shell thickness is approximately 2–3 nm, which is attributed to the crystalline PLLA layer. PEG layer is solvated in aqueous solution that constitutes the lighter layer on the exterior of the BCCs of ~12 nm, as the blue arrow pointed. The thickness of the PLLA crystalline shell (2–3 nm) is similar to that of liposomes and much thinner than the hydrophobic core thickness of PLA-b-PEG (Mn = 3.2k−2.8 kg mol−1) polymersomes (e.g. ~10.4 ± 1.4 nm)22.
To understand the crystalline structure of PLLA, wide angle X-ray diffraction (WAXD) experiments were conducted on dried BCC samples at 25 °C (Fig. 2f). The WAXD pattern concludes that orthorhombic α form of PLLA is formed19. The strongest diffraction peak at 16.7° is from (110)/(200) planes of PLLA crystals while the peak at 19.8° is from (203) plane of PLLA and (120) plane of PEG, and the peak at 22°–24° can be attributed to (032) plane of PEG and (210) plane of PLLA19,23. Note that the PEG crystals are formed during BCCs solution drying process. Figure 2g shows a differential scanning calorimetry (DSC) thermogram of the BCCs, where a heating rate of 10 °C min−1 was used. The two peaks in the thermogram can be attributed to the melting of PEG (at 49.0 °C) and PLLA (at 138.9 °C) crystals, respectively. By integrating the PLLA melting peak and using 91 J g−1 as the melting enthalpy value for 100% PLLA α form crystal, the crystallinity of PLLA can be estimated to be 67%24. To further exam the PLLA chain orientation in BCC, selected area electron diffraction (SAED) patterns on individual PLLA-b-PEG BCCs were recorded, and the results are shown in Fig. 2h. PLLA (200) and (110) diffraction spots are observed, suggesting the c-axis and the polymer chains are along the radial direction of the crystalsome. The diffraction spots are arc-shaped, which is attributed to the continuous lattice splay/distortion in order for the single crystal to fit into the curved nano-sized space19,25. The SAED pattern therefore confirms the crystalsome shell is made of single crystal-like lamellae with polymer chains normal to the interface.
To further verify the shell thickness, AFM measurement was conducted on the small crystal pieces after breaking the BCCs by ultrasonication. AFM image and the corresponding cross-sectional height profile (Fig. 3a) shows a total thickness of the shell is ~4.5 nm, including both PEG and PLLA layers. This is significantly thinner than the flat PLLA-b-PEG single crystals (11.1 ± 0.2 nm (n = 10), supplementary Figure 3). Based on the density and molecular weight of each segment, the thickness of PEG and PLLA layer can be estimated to be 2 and 2.5 nm respectively, which is consistent with the cryo-TEM image in Fig. 2c. As the polymer chain folds normal to the crystal surface, and using the molecular weight and the two-chain orthorhombic unit cell of PLLA α form (a = 1.066 nm, b = 0.616 nm, and c = 2.888 nm, Supplementary Note 1) with 103 helical conformation, it can be calculated that each PLLA chain folds nine times, shown in Fig. 3b. The grafting density σ of PEG on the surface can therefore be calculated to be 0.3 PEG chain nm−2. Of interest is that PLLA crystallized into such thin lamellae, which leads to the nine-time fold of the polymer chain. In general polymer crystallization, the crystal thickness is determined by the surface free energy of the crystal folded surface, the heat of fusion, and undercooling of the crystallization process26,27,28. The present case is more complex. In addition to the above-mentioned parameters, entropy loss of the PEG brush upon PLLA crystallization, the molecular weights of the polymer, and the droplet size should all factor in determining the final thickness of the PLLA crystals.
On the basis of PEG chain grafting density, the distance between the anchoring points of the adjacent PEG chains, D, is 1.81 nm (See Supplementary Note 1 for calculation of chain folding number, grafting density and adjacent PEG distance). Flory radius of 5000 g mol−1 PEG in water can be calculated following RFN3/5a29, where a, N and RF are defined as monomer size, the degree of polymerization and Flory radius. In the present case, a = 0.35 nm and N= 11330, and the calculated RF = 5.95 nm, much greater than the adjacent chain distance 1.81 nm29. Therefore, PEG layer in aqueous will take brush conformation, and the brush thickness L can be calculated based on $$L \approx Na^{({5}/{3})}D^{ - ({2}/{3})}$$29. Plugging in the above-mentioned values of N, a, and D, the calculated L is 13.2 nm, which is consistent with the cryo-TEM observation (Fig. 2c). Note that because the distance between the anchoring points of adjacent PEG is directed by PLLA crystallization, the PEG grafting density can be precisely controlled by PLLA chain folding and the lamellar thickness31,32. The spatial distribution of the brush molecules on the surface is extremely uniform due to the crystallization of PLLA, similar to our recently demonstrated polymer brush synthesis using polymer single crystals as the template33. This controlled uniform brush layer may improve the circulating performance of the BCCs, which is discussed in the following sections. We also anticipate that the grafting density can be further tuned by changing polymer molecular weights as well as the crystallization condition as previously reported33, which will be the focus of our future work.
To understand how the PLLA chains crystallize near the liquid−liquid interface, we used molecular simulations to study the nucleation kinetics (simulation details are in Supplementary Methods)34,35. Figure 4a shows the snapshots of the initial PLLA-b-PEG chains located at the curved interface, and the PLLA nuclei at different times, t = 1.0τ, 3.0τ, 5.0τ, 10.0τ, and 40.0τ (here τ is the longest relaxation time of PLLA chains in solution36, Supplementary Figure 5). One can see that at t = 1.0τ, there are some small PLLA nuclei formed near the interface. As time increases, these nuclei continue to grow, and some smaller ones dissolve or are merged into the growing nuclei (see the time evolution of average nucleus number and size in Supplementary Figure 6). Eventually only one single crystal is left in the system (see snapshots at t = 10.0τ and 40.0τ).
To identify the accurate position where the nucleus grows, we calculate the radial distribution of the average nucleus size Rnucleus(d) (see definitions in Supplementary Note 2), here Rnucleus is the radius of a virtual spherical nucleus (see Supplementary Note 2) and d is the distance between the center of mass (CM) of nucleus and the droplet center. Figure 4b shows the variation d of Rnucleus(d) at different t. One can find that from t = 1.0τ to 5.0τ, the nucleus grows at d 94.5~95.5 nm, rather than from the PLLA-b-PEG block junctions at the interface (d 98~100 nm). This slightly off-interface growth can be understood by the radial distribution of average nuclei orientations (see inset of Fig. 4b and definition in Supplementary Note 2). We found that the nuclei are preferred to be vertical to the interface at d 95 nm, while near the interface (d 98~100 nm), they have no preferred orientation. This can be explained as follows: the PLLA chains are stretched vertically to the curved interface due to the high grafting density37, giving rise to a preferred nuclei orientation. However, this orientation would be suppressed near the liquid−liquid interface due to the influence of PEG and water solvents. Therefore, in average, the nuclei are grown in the droplet slightly off the liquid−liquid interface. Eventually, the crystal will cover the whole inner surface of the droplet38, forming a single crystal-like crystalsome similar to Fig. 2e.
### Mechanical property of PLLA-b-PEG BCCs
There are two major methods to characterize the mechanical property of polymer vesicles, namely micropipette aspiration technique and AFM-based nano-indentation39. Due to their small size, the mechanical property of PLLA-b-PEG BCCs was measured via AFM nano-indentation. In the experiment, aqueous solution of PLLA-b-PEG BCCs was drop casted onto a precleaned glass slide and dried under ambient condition. A crystalsome with smooth surface was chosen for the indentation experiment40,41. AFM image and the height profile of the crystalsome were acquired under Tapping Mode, as shown in Fig. 3c inset. The force-deformation curve is shown in Fig. 3c, averaged from 14 times of indentation and the error bars represent standard deviation. After indentation, another AFM image was taken (Supplementary Figure 2) to confirm that the original BCC morphology was not deformed, indicating elastic deformation of the BCC during the indentation experiment. The slope of the fitted line of the deformation portion of the curve was used as the shell stiffness kshell. The Young’s modulus of the dry state PLLA-b-PEG shell can be calculated to be 11.5 GPa42,43. The membrane bending modulus of the BCC shown in Fig. 3 in dry state can therefore be determined to be 9.8×10–17 J (see Supplementary Note 3 for detailed calculation), which is significantly higher than polymersomes or crystalsomes. Note that in aqueous solution, PEG layer would be solvated and will not significantly affect the BCC bending modulus. Therefore the bending modulus of BCCs shell in water can be estimated to be Kbend,wet = 3.63×10−17 J (see Supplementary Note 3 for the calculation). Table 1 summarizes the typical values of sizes and mechanical properties of selected liposomes, polymersomes, homopolymer crystalsomes, BCCs, and viral capsids. It is clear that PLLA-b-PEG BCCs have about 140–8000 times higher bending modulus than polymersomes with much greater thickness, even with a much thinner shell thickness. Thus, BCCs shell would be more stable and more robust under shear flow, suggesting they may provide better protection for the potential cargo inside. Interestingly, BCC shares many features with viral capsid: they are spherical, mechanically strong and resilient. The typical thickness of the viral capsid is between 2 and 5 nm, which is also very close to the present BCC structure, suggesting that BCC might provide an ideal self-assembled capsid-mimic system for desired applications such as drug carriers.
### In vivo circulation and biodistribution studies of BCCs
To test the feasibility of using BCCs for intravenous delivery, in vivo circulation experiments were carried out. A hydrophobic fluorescent dye, 1,1′-dioctadecyl-3,3,3′,3′-tetramethylindodicarbocyanine, 4-chlorobenzenesulfonate salt (DiD) was encapsulated in these BCCs during the crystal growth. DiD has been cited as a marker in a number of blood circulation studies for polymeric nanoparticles6,44. An in vitro control experiment was firstly conducted to confirm if BCCs are leaky. To this end, DiD-encapsulated PLLA-b-PEG BCCs were dispersed in a PBS solution supplemented with 10% fetal bovine serum (FBS). At different time intervals, BCCs were collected, and fluorescent intensity was measured. It was shown that there was minimal fluorescence signal decay (2%) after 5 days of incubation (Supplementary Discussion, Supplementary Figure 8), indicating BCCs are well sealed and nearly impermeable for DiD in this experimental condition. The sealed structure was further demonstrated by the slow release of hydrophilic nitrobenzoxadiazole (NBD)-based dye (~1.7% per 24 h, Supplementary Figure 9). In a separate experiment, BCCs also showed negligible intake of hydrophilic 5-Carboxyfluorescein after 24 h incubation (Supplementary Figure 10). In our in vivo experiments, DiD-encapsulated BCCs with an average size of 200 nm in saline were systemically administered into BALB/c mice through tail vein injection at a dose of 1 mg per mouse. A series of blood samples were collected at various time points post injection. The circulation profiles were obtained by measuring the fluorescence intensity of the sample plasma (Fig. 5a). The blue curve shows the circulation performance of the BCCs crystallized for 7 days. The fluorescence intensity decay follows a classical one-compartment PK model, yielding a long circulation half-life of 24.2 ± 1.4 h (mean ± s.d., n = 5). Remarkably, there are 47, 14 and 8% of BCCs remained in the blood after 24, 72 and 96 h circulation, respectively.
Most polymeric nanoparticles administered intravenously are quickly cleared by the mononuclear phagocyte system45,46. Numerous strategies have been reported to improve nanoparticle blood circulation time. Nanoparticle size, surface PEGylation and mechanical properties have been shown to affect nanoparticle circulation45,46. Figure 5b summarizes the blood circulation apparent half-life and particle retention at 24 h for several reported nanoparticle systems with long circulation lives. Here the apparent half-life is defined as the time for the nanoparticle concentration to fall to half of the original value, and the percentage particle retention at 24 h denotes the fraction of particles retained in plasma 24 h post injection. For example, the reported spherical Poly(ethylene oxide)-b-poly(ethylethylene) (PEO-b-PEE) polymersome has an apparent half-life 15 h with a 24 h retention of ~35%47. Perry et al. reported that with a relatively low PEG brush grafting density, soft hydrogel PRINT nanoparticles showed a half-life 3 h and a 24 h retention of 10%30. Hu et al. coated RBC membranes onto PLGA nanoparticles to bypass in vivo clearance, reaching a 9.6 h apparent half-life with 29% retention at 24 h6.
Compared to the above-mentioned leading polymeric nanoparticle carriers, BCCs show an impressive half-life of 24.2 h and a 24 h retention of 47%. Furthermore, most nanoparticles investigated in these previous studies are rather small with an average size of 90–100 nm (except for PRINT nanoparticle, which has a dimension of 80 × 80 × 320 nm). It is known that particle size is a key factor for long circulation. Larger nanoparticles of 171 and 243 nm were reported to get cleared in blood twice as fast as that of smaller nanoparticles (80 nm)46,48. Given the relative large size of the BCCs, it is even more intriguing to observe the long circulating time value of the present system. Furthermore, previous studies showed that polymer crystallization reduced blood circulation of polymersomes and cylindrical polymer brushes, which is apparently not the case in BCCs15,49. To better understand the mechanism of the observed long circulation time of BCCs, we investigated the circulation behaviors of BCCs that were crystallized for shorter periods of time, e.g. 3 days and 5 days. The results are also plotted in Fig. 5a, which shows that BCCs of shorter crystallization time (3-day and 5-day) exhibit much faster clearance in the blood yet 5-day crystallization time yields a blood retention better than 3-day crystallization. Fitted with the one-compartment model, BCCs after 3-day and 5-day crystallizations have half-lives of 0.80 ± 0.46 h (n = 4) and 3.3 ± 0.68 h (n = 3), respectively. Figure 5c summarizes the change of half-lives with increasing crystallization time. DLS measurement (Supplementary Figure 11a) confirmed that all the BCCs have similar hydrodynamic radii and size distribution. Supplementary Figure 11c, d demonstrate that morphologies of 3-day and 5-day crystallized BCCs also maintained capsule-like structures with similar particle sizes. Interestingly, crystallinity calculated from DSC heating thermograms (Supplementary Figure 11b) illustrated shorter crystallization time rendering lower crystallinity (18.1, 45.8 and 67% for 3, 5, and 7 days of crystallization, respectively), which is also plotted in Fig. 5c. Combining all these results, we can conclude that there is a close correlation between the BCC crystallinity and the blood circulation time, shown in Fig. 5d. The ultra-long circulation time of the 7-day crystallized BCCs arises from the high crystallinity and uniform crystalline chain packing on the curved BCC shells which minimize defect formation in the crystals. In 3-day, 5-day crystallized BCCs and the previously reported polymersome cases, relatively poorly controlled crystals lead to defect-rich particles, which may be prone to disassembly under blood flow. Furthermore, the long circulation may also be facilitated by the uniform PEG brushes formed by crystallization-induced brush-formation. As we recently reported, BCP crystallization-templated polymer brushes are highly tunable and extremely uniform polymer brushes with minimal defects. However, in reported nanoparticle circulation studies, PEG layers are typically formed by grafting-to methods, and are not uniform due to the steric hindrance of the macromolecular chemistry33.
Nanoparticle distribution in organs was investigated for up to 3 days via IVIS imaging and homogenization (Fig. 6a, b). The relative numbers of PLLA-b-PEG BCCs in the blood over 3 days were consistent with the circulation data presented in Fig. 5a that the amount of BCCs in the blood was halved every day, which follows the one-compartment model formula $${{C}}\left( {{t}} \right) = {{C}}({\boldsymbol{t}}_0) \cdot {{e}}^{ - {{k}} \cdot {{t}}_{1/2}}$$50. With respect to distributions in other organs, the BCCs primarily accumulated to liver and spleen, which are organs of the reticuloendothelial system (RES). Thus, BCCs were mainly sequestered by the liver in such a time frame, which is consistent with the recent understanding on how hard nanomaterials are cleared by the liver51. The PLLA-b-PEG BCCs accumulation in the liver and spleen increased with time, suggesting that the blood fluorescence mainly arises from the dyes in BCCs rather than the leakage of the dye. In the latter case, the dye would be secreted by kidneys, leading to the decrease of fluorescence signal from the liver6.
Polymeric vesicles have been shown to accumulate in the liver47. The uptake of nanoparticles by hepatic macrophages, Kupffer cells may be reduced when the liver-specific opsonin absorption is minimal52. Besides liver, the PLLA-b-PEG BCCs exhibited a high accumulation in the spleen up to 72 h. Feijen et al. reported that polymersomes with a size of 95 nm primarily accumulated in the liver whereas maintained a very low uptake amount in the spleen from 24 to 72 h53. PLLA-b-PEG BCCs interestingly showed a primary and increasing accumulation over time in the spleen probably due to their relatively larger sizes. It was reported that nanoparticles with size beyond 150 nm are more likely to get entrapped within the liver and spleen54. Large stealth liposomes (>200 nm) also showed a significant accumulation in the spleen due to the mechanical filtration in addition to phagocytosis55. Application of nanocarriers has long been restricted by the available choices of particle size, while the optimal size was considered to be 70–150 nm54,56. PLLA-b-PEG BCCs showed that large particles with the size of ~200 nm circulated up to 96 h in the blood, which opens the door for medical applications of large nanoparticles (~200 nm). Furthermore, the hollow structure is advantageous for potential delivery applications since (1) it provides space to encapsulate a large number of drugs at a level that is difficult to achieve by other drug carriers; (2) minimal amount of polymer materials is used in a capsule; and (3) these hollow capsules also provide large surface area to mass ratio. Future research will be focused on better controlling the structure, morphology, size distribution and surface function of the BCCs for targeted applications.
## Discussion
In conclusion, we demonstrated the preparation of robust BCCs for long blood circulation. The emulsion-solution crystallization method is efficient to guide polymer chain folding into uniform crystalline packing at curved liquid/liquid interface to form single crystal-like structure. As a BCP ensemble, BCCs exhibit superior mechanical stability, suggesting that tailoring crystallization is a route to prepare robust BCP vesicles. In vivo circulation tests demonstrate that the BCCs can circulate up to 96 h in the blood with a half-life of 24.2 ± 1.4 h, and 47% retention at 24 h post injection. This excellent circulation performance was attributed to controlled crystallization of PLLA and the resultant uniform PEG brush layer on the BCC surfaces. Our observation is different to the previous perception that crystallization reduces polymersome circulation, providing a strategy to generate long circulating nanomaterials. We envisage that the hollow, robust, tunable BCCs could lead to a new class of nanoparticle carriers for drug delivery and gene therapy.
## Methods
### Materials and animals
Poly (l-lactide)-block-poly (ethylene glycol) (PLLA-b-PEG, Mn = 6000–5000 g mol−1) was purchased from Polymer Source Inc. 1,1-dioctadecyl-3,3,3,3-tetramethylindocarbocy amine perchlorate (DiD oil) was purchased from Life Technologies. Toluene and NaCl were purchased from Sigma Aldrich. All materials were used as received. Female BALB/c mice were purchased from Charles River Laboratory.
### Preparation of PLLA-b-PEG block copolymer crystalsomes (BCCs)
22 mg PLLA-b-PEG and 12 µg of fluorescent dye (DiD) were first dissolved in 0.4 g toluene in a glass tube. Then, 16 g DI water was added into the above polymer solution at 95 °C. After emulsification at 95 °C for 2 min via probe sonication, the emulsion was kept at 95 °C for another 30 s before quenched to 25 °C for crystallization. After crystallization for 3–9 days, the solution was dialyzed against DI water through 50 nm membranes at 25 °C for 4–6 h, and the procedure was repeated for five times before the sample was collected through a 0.45 µm syringe filter.
Before in vivo study, the above-mentioned BCCs aqueous solution was further dialyzed again saline (0.9 w/v% NaCl aqueous solution) overnight and then concentrated with 30 K molecular weight cutoff (MWCO) Amicon Ultra-4 Centrifugal filters. Gas chromatography experiments showed no detectable amount of toluene in the BCC samples (See Supplementary Methods, Supplementary Figures 12, 13).
### Characterization of PLLA-b-PEG BCCs
Hydrodynamic radius were measured by DLS using Zetasizer Nano ZS90. SEM images were taken on a ZEISS Supra 50VP microscope with a 1 kV accelerating voltage. To prepare SEM samples, a drop of 10 µl BCCs aqueous solution was cast onto cover slides and then dried under vacuum overnight. Before SEM imaging, the sample was coated with Pt/Pd.
TEM bright field imaging and SAED experiments were conducted using a JEOL JEM2100 microscope with a 120 kV accelerating voltage. TEM samples were prepared by drop casting BCCs’ aqueous solution onto a carbon-coated Cu grid. Cryogenic-transmission electron microscopy (cryo-TEM) was performed on a FEI Talos F200C TEM (University of Delaware, DE). A vitrified sample was prepared with an FEI Vitrobot apparatus57,58. The vitrified samples were transferred to a cryoholder in a sample stage immersed in liquid nitrogen. The cryoholder was kept below –170 °C during the imaging to prevent sublimation. The digital images were recorded by a Talos F200C TEM equipped with a FEI Falcon direct electron detector.
AFM images and force spectra were acquired using a Bruker Dimension Icon AFM equipped with a tapping mode and PeakForce mode. Samples were dried on piranha-cleaned cover slide. The cantilever used is Bruker TESPA with tip height of 10–15 μm, and tip radius of 8 nm. The spring constant of the cantilever is 2.030 N m1, calibrated by the thermal tune method. DSC experiments were conducted on DSC Q2000 with Tzero pans from TA Instruments with a 10 °C min1 heating rate.
### Monte Carlo simulation
We use a coarse-grained lattice model to study the nucleation kinetics of PLLA chains near the curved liquid−liquid interface. The PLLA-b-PEG chain is modeled as a diblock copolymer of A38 B28, with A and B corresponding to the PEG and PLLA monomers, and the total monomers in one chain is N = 66. The curved liquid−liquid interface is set up in prior in the simulation box of 100 × 100 × 99 nm3, then PLLA-b-PEG chains are relaxed in this interface. The total chain number is chosen as n = 3000, calculated from the experimental grafting density 0.3 chains per nm2. The polymer chains are moved via a micro-relaxation model34, which allows each segment to change positions with its neighboring solvent sites, accompanied by the sliding diffusion along the chain direction if necessary35. Conventional metropolis sampling was employed in each micro-relaxation step. The reduced temperature is set as kBT/Ec = 3.6, and other parameters are summarized in Supplementary Note 2. Information.
### In vitro release study
PLLA-b-PEG BCCs labeled with DiD was dispersed in a 10% FBS PBS solution. The mixture was shaken at 100 rpm at 37 °C. At each time interval, BCCs were collected with an MWCO 100 K (Amicon Ultra-4) tube with centrifugation. Equal volume of 10% FBS in PBS was used to resuspend BCCs. The fluorescent intensity was measured at an excitation/emission wavelength of 600/665 nm by a microplate reader (Infinite M200, TECAN) and compared with the initial intensity for quantification.
### In vivo circulation study
PLLA-b-PEG BCCs labeled with DiD in saline was systemically administered through tail vein injection. Each female BALB/c mouse was slowly injected with 100 μL 10 mg mL−1 of NP solution. A small volume of 10 µL of blood was collected at 2 min, 15 min, 0.5 h, 1 h, 2h , 4 h, 8 h, 24 h, 48 h and up to 96 h post i.v. injection. The blood was diluted in 200 μL of heparin/PBS (16 U mL1) solution to avoid blood coagulation. The samples were centrifuged at 300 × g for 5 min to remove blood cells, and then 180 µL of the supernatant was collected for measurement at excitation/emission wavelength of 600/665 nm by a microplate reader (Infinite M200, TECAN). One-compartment pharmacokinetics model was used to calculate in vivo circulation half-lives via PKSolver49. All animal procedures were conducted according to the protocols of the Committee on Animal Care of Drexel in compliance with NIH guidelines.
### In vivo biodistribution study
Female Balb/c mice at 8 weeks old were randomly separated into three groups and were i.v. injected with 150 μL of DiD-labeled PLLA-b-PEG BCCs at 10 mg mL1 on days 1, 2, and 3 respectively. The control group was injected with the same amount of saline. On day 4, all the mice were sacrificed and their brains, lungs, hearts, livers, spleens, kidneys as well as blood were collected. To quantify the amount of PLLA-b-PEG BCCs in each organ, all the organs were weighted and homogenized. The fluorescence intensities of DiD-containing homogenized solution were detected at an excitation/emission wavelength of 600/665 nm by a microplate reader (Infinite M200, TECAN).
### Data availability
The data are available from the corresponding author upon reasonable request.
### Code availability
The simulation code is available from the corresponding author upon request.
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## Acknowledgements
This work was supported by the National Science Foundation Grant DMR-1308958 and DMR 1709136. C.Y.L. and H.C. are grateful for the support of Drexel DARE and PA Department of Health, CURE grant programs. H.C. acknowledges the support from National Institute of Allergy and Infectious Diseases of the National Institutes of Health under Award Number R21AI133372. Q.T. acknowledges the financial support from the Deutsche Forschungsgemeinschaft under grant Mu1674/15-1. The simulations are performed at GWDG Göttingen.
## Author information
Authors
### Contributions
H.Q., H.Z., H.C. and C.Y.L. designed the research; H.Q., T.Z., S.M., S.K. and M.C.S. fabricated and characterized crystalsomes; J.Y.L. and D.J.P. performed the cryo-TEM work; H.Q. and L.H. conducted the AFM indentation measurements; Q.T. and W.H. performed the simulations; H.Z., Z.F. and H.C. conducted the in vivo circulation and biodistribution study. H.Q., H.Z., H.C. and C.Y.L. wrote the paper.
### Corresponding authors
Correspondence to Hao Cheng or Christopher Y. Li.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Qi, H., Zhou, H., Tang, Q. et al. Block copolymer crystalsomes with an ultrathin shell to extend blood circulation time. Nat Commun 9, 3005 (2018). https://doi.org/10.1038/s41467-018-05396-x
• Accepted:
• Published:
• ### Molecular simulations of microscopic mechanism of the effects of chain length on stereocomplex formation in polymer blends
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# How should I implement basic spaceship physics?
I am new to 2D game development.
I've made several games, but they've made almost no use of physics. Now I want to try making a game that uses some basic physics to look more realistic.
The game involves spaceships battling in space. I'm ignoring gravity. Each spaceship can move up, down, left, right or any of the 4 diagonals.
I want each spaceship to accelerate when moved (up to some threshold speed) and gradually slow to a stop otherwise.
I'm planning to give each spaceship a velocity and acceleration attribute.
The acceleration is set to some constant value. For example, 5.
When the spaceship starts moving, the velocity is set to some initial value. Then every period of time as long as it should be moving (for example every 30 milliseconds), the acceleration value is added to it.
When the user stops moving the ship, the reverse process happens: the acceleration value is decreased from velocity until velocity equals 0.
I have two questions regarding this:
1. Is this general type of physics realistic and/or common in games? Is this the right direction? Am I forgetting some factor or anything to be considered?
2. Regarding the specific implementation I'm planning: Is the right way to accelerate is add a constant value to the speed of the object? Or do I need to gradually increase the value added to the speed, thus increasing the acceleration?
Any physics "simulation" is going to deal with forces. Your approach may be basic, but it is definitely a step in the right direction to understanding game physics.
Now you may be able to add to this further. For example, if the player is turning around while moving at a high speed, how do you handle this? Do you prefer they continue along their original direction a bit until they correct to the new one, or should they just snap to the new direction at the same speed?
To answer your second question, let's first understand what acceleration means. For example, is your spaceship like a car where your acceleration is lower the higher your speed is, eventually capping at a certain maximum? Maybe this kind of change would break your gameplay, so these choices are entirely up to how you'd prefer your game to feel.
In real space, there is no friction. When you accelerate a body, it keeps moving in that direction and never stops, unless you accelerate it in the other direction. There are also no top-speeds. Given enough energy, you can accelerate a body to unlimited speeds.
There are games where such realistic physics are part of the gameplay, but the vast majority of games, especially those which focus on action instead of realism, handle this differently and add friction to space.
A simple and frequently used (but not physically correct) way to deal with friction is to have an acceleration value, a top-speed value and a deceleration value.
While the player presses the left/right key, each game-loop the acceleration-value is added to the current speed of its character. That means unless the character has already reached the top-speed.
When the player stops pressing the directional key, the deceleration value is applied in the opposite direction to slow the character down.
Often the acceleration is higher than the deceleration, so the player can break faster by pressing into the opposite direction than just stopping to press any button.
With the engines' propulsion as the only force acting on the spaceship, physics are hardly at play here. Your currently planned implementation are in accordance with 'realistic physics'. You could definitely design a spaceship that flies that way. To answer your second question, that too would be realistic. You could definitely design a spaceship that uses a non-constant force to reach its target velocity.
If you spaceship design is very specific, i.e. you know its mass, where the engines are, how much thrust they can produce and how much fuel they burn, how the controls work..., then you can run a physics simulation to determine exactly how that design is supposed to behave. Without these specifications, who is to say your ship handles incorrectly?
Your constant force-approach is simple to implement. Build it and play the game to see whether it needs tweaking.
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# Why/How is this Wick's theorem? [migrated]
Let $\phi$ be a scalar field and then I see the following expression for the square of the normal ordered version of $\phi^2(x)$.
$T(:\phi^2(x)::\phi^2(0):) = 2<0|T(\phi(x)\phi(0))|0>^2 + 4<0|T(\phi(x)\phi(0))|0>:\phi(x)\phi(0): + :\phi^2(x)\phi^2(0):$
It would be great if someone can help derive the above expression - may be from scratch - and without outsourcing to Wick's theorem - and may be help connect as to why the above is related (equal?) to the Wick's theorem?
• Isn't the above also known as OPE (Operator Product Exapnsion)? If yes, then is there at all any difference between OPE and Wick's theorem? Is there a systematic way to derive such OPEs?
• Can one help extend this to Fermions?
-
Apologies but this question seems to be asking "please teach me Wick's theorem" - and it moreover says "do it without teaching me Wick's theorem". Have you tried to study Wick's theorem? At least en.wikipedia.org/wiki/Wick%27s_theorem ? The standard pedagogical treatment answers all your questions. The Wick's theorem is the systematic way to construct such identities that you're looking for. Fermions differ by some signs only. I find it questionable whether copying/rephrasing sections from standard textbook material is a good way to use this server and people's time. – Luboš Motl Apr 21 '12 at 11:57
comments disabled on deleted / locked posts
## migrated to physics.stackexchange.com by Piotr Migdal♦Apr 21 '12 at 13:29
This question belongs on our site for active researchers, academics and students of physics.
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# How to prove a property of ranks: $\operatorname{rank}(AB)= \operatorname{rank}(B)- \dim(\operatorname{Im} B \cap \ker A)$
I have to prove that $\rank(AB)= \rank(B)- \dim(\Im B \cap \ker A)$
I haven't got much idea... but I started like this:
Using the first isomorphism theorem, we get the following relations:
$p= \rank(AB)+ \dim(\ker(AB))$
$p= \rank(B)+\dim(\ker B)$ and
$n= \rank(A)+\dim(\ker A)$
From the first and second relation we get that: $$\rank(AB)+ \dim(\ker(AB)) = \rank(B) + \dim(\ker B)$$
I don't know how to continue or if I am on the right way to prove it.
Thank you for your time and help. And sorry for my poor English.
• From what you have, it's enough to show that $$\dim\ker(AB) - \dim\ker B = \dim(\Im B \cap \ker A).$$ – leo Sep 23 '14 at 17:21
Hint: we can be a bit more specific with the first isomorphism theorem. Certainly, we have$$\newcommand{\rank}{\operatorname{rank}}$$ $$p= \rank(AB)+ \dim(\ker(AB))$$ However, we can also think of $$A$$ and $$B$$ as maps $$T_B : \Bbb R^p \to \text{Im}(B)\\ T_A: \text{Im}(B) \to \text{Im}(AB)$$ Where $$T_{AB}(x) = ABx = T_A \circ T_B$$. Now we have $$p= \rank(B)+\dim(\ker B)\\ \rank(B) = \underbrace{\dim(\text{Im}(A\mid_{\text{Im}(B)}))}_{\rank(AB)} +\underbrace{\dim(\ker(A \mid_{\text{Im}(B)}))}_{\dim(\ker A \cap \text{Im}(B))}$$ This should be enough to get what you're looking for.
• could you explain the last step please?$\rank(B) = \underbrace{\dim(\text{Im}(A\mid_{\text{Im}(B)}))}_{\rank(AB)} +\underbrace{\dim(\ker(A \mid_{\text{Im}(B)}))}_{\dim(\ker A \cap \text{Im}(B))}$ – Lucas Sep 23 '14 at 22:18
• @Lucas note that $(AB)x = A(B(x))$. Note that $B(x)$ is in the image of $B$. So, the rank of $AB$ is the dimension of the image of the image of $B$ under $A$. Furthermore, in order for $A(B(x))$ to be $0$, $B(x)$ (which is in the image of $B$) needs to be in the kernel of $A$. – Omnomnomnom Sep 23 '14 at 22:47
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# WorkCell Format¶
The workcell file describes workcell elements such as frames, joints, DAFs, devices and the properties that relate to these elements such as name, parent name, joint limits, position and orientation, collision geometry, drawing geometry and even user defined properties.
The basic structure of a workcell is frames. Frames are connected in a tree like fashion such that each frame has a parent and [0;many] children. The root of the frame tree is called the world frame and this frame is the only frame that has no parent (except detached DAF frames). The world frame is named WORLD.
Frames come in many different types: fixed frame, movable frame, prismatic joint, revolute joint, and so on. They can be freely defined and connected in the workcell file, though often frames will be grouped into devices.
A device is a robot with one or more joints. It defines a scope where frames that belong to that device can be described. A device always has a base frame which is the frame belonging to the device that has a parent that does not belong to the device. That is, a device defines a sub tree in the frame tree. There exists several device types: serial device, tree device, parallel device and so on, that allow different sort of connections between the joints of the device.
User properties can be attached to any frame. A property has a name and a string value. The property is saved in the frame such that the user can retrieve it and parse the string himself.
A collision setup is a set of include or exclude rules that tell which frame pairs that should not or should be tested for collision. Ex. two neighboring links on a robot will usually collide in their attachment points, which is undesirable, the collision setup should in this case be used to exclude those frames from collision checking.
## Scopes¶
An important thing to notice is that some xml elements are scope dependent. In general the refframe and refjoint attributes are optional and if not defined then scope rules will define them instead.
Ex. when defining a frame in workcell or a device scope the frame will automatically be attached to the previously defined frame.
<WorkCell name="testwc">
...
<Frame name="A" refframe="WORLD" />
<Frame name="B" />
<Frame name="C" refframe="A" />
</WorkCell>
in the above example A attaches to world and B and C attaches to A.
Another example illustrates how properties defined inside a frame can omit the refframe attribute.
<WorkCell name="testwc">
...
<Frame name="A" refframe="WORLD">
<Property name="A_Property"> some string value</Property>
</Frame>
<Property name="A_nother_Property" refframe="A">some string value</Property>
...
</WorkCell>
## Preparser¶
The RobWork preparser is used as the first thing when loading a RobWork XML file. Preparsing happens before processing the file for the tags defined in the remainder of this page. It makes it possible to reuse XML definitions in different ways.
### Define & Use¶
To define a reusable block of XML, it is possible to use the Define tag:
<Define id="SomeID">
<!-- definition goes here -->
</Define>
The code defined can then be reused with the Use tag. The definition is substituted in without any modification.
<Use id="SomeID" />
The Define/Use tags are often used when defining grippers that can have multiple identical kinematic chains as part of its structure.
### Include¶
It is also possible to include other XML files with the Include tag:
<Include file="some.file.xml" />
This is very convenient. For instance, it makes it possible to include the same robot definition in multiple WorkCells that use the same type of robot.
Warning
If you load a WorkCell and save it afterwards, the preparsing information will be lost. This means that only one big file will be generated with no Include, Define or Use tags.
## WorkCell Structure¶
Before going deep into the grammar of the rw xml format some common structure need be explained. The complete kinematic description is based on a construct named Frame. The Frame has a name and a transform relative to its parent frame or refframe. This means that a kinematic Frame tree can be described with multiple frames. The Frame can be of different types the simplest being Fixed, which means that the frame transform is unchangeable.
Another common structure is the Drawable and CollisionModel tags. These are defined relative to the frames, and adds geometry to the workcell.
Properties are linked to a frame and has a name, description and a value. Properties are used to link different information to frames, such as camera or scanner information.
In the xml file format, frames are grouped logically in container type elements. These elements are WorkCell and device types. To avoid name clashes, frames belonging to a container type will have the container name prepended. Example: a frame named “base” specified in a device named “PA10” will have the unique name “PA10.base”.
The WorkCell element is the root element in the fileformat. It implicitly defines a Fixed frame named World. This world frame is the root frame in the kinematic frame tree.
## WorkCell¶
Element WorkCell
Attributes
• name: a string identifying the workcell.
Child elements:
Example
<WorkCell name="scene">
...
</WorkCell>
## Device¶
The different device types are much alike when considering the child elements that they allow. Though they vary somehow in the implicit rules of frame attachment.
In general a device defines a scope. This scope has the same name as the device. Any frames defined inside the scope of a device gets the device name appended. Ex. given a device “dev” and a frame “base” in the device the complete frame name becomes: “dev.base”
### SerialDevice¶
The serial device only allows joints to be connected in a serial chain. And it also only allows one single endeffector.
Element SerialDevice
Attributes
• name: a string identifying the device.
Child elements:
Example
<SerialDevice name="RobotArm">
...
</SerialDevice>
### TreeDevice¶
The tree device allows joints to be connected in a tree like structure. And it also allows for multiple endeffectors.
Element TreeDevice
Attributes
• name: a string identifying the device.
Child elements:
Example
<TreeDevice name="RobotHand">
...
</TreeDevice>
### ParallelDevice¶
The parallel device is like a number of serial devices (with same base) with all endeffectors rigidly connected together. The initial configuration of the robot is required to make all endeffectors align in the same pose. For devices that are connected in multiple places, it is also possible to define so-called junctions. Each Junction must specify two or more chains, where each chain referes to a list of previously defined SerialChains. Notice that each of these chains must start and end in equivalent frames. If no junctions are defined, one implicit junction is created, assuming that each of the defined serial chains must end in the same endeffector frame.
Attributes
• name: a string identifying the device.
Child elements:
In any order:
Final elements:
• Junction
• Q
Example
Take the following kinematic structure as an example:
___
/--------------------|-C |
/ ___ | |
/-- B ------------|-D-|----|-E |
/ \ ___ | | |___|
/ ---|-F | | |
/ | | | |
A -----------|-G-|----|-H-|
|___| |___|
Each of the 8 serial chains, A to H, can contain one or more joints. The boxes show three places where the device must be connected. This can be specified in the device with Junction tags:
<ParallelDevice name="RobotHand">
...
<Junction>
<Chains>C</Chains>
<Chains>D E</Chains>
</Junction>
<Junction>
<Chains>B D</Chains>
<Chains>G H</Chains>
</Junction>
<Junction>
<Chains>B F</Chains>
<Chains>G</Chains>
</Junction>
...
</ParallelDevice>
Notice that the first serial chain (A) was left out in all the cases, as it is equal for all chains. For the junction ending after chains C and E, both A and B was left out, as they do not provide any extra information.
### MobileDevice¶
The mobile device defines a two wheeled mobile robot where the two wheels are on the same axel displaced from the center of the axel with some width AxelWidth.
Attributes
• name: a string identifying the device.
• basename: name of the mobile device base.
Child elements:
Example
<MobileDevice name="Pioneer" basename="Base">
...
</MobileDevice>
### SerialChain¶
Attributes
• name: a string identifying the chain.
Child elements:
## Frame¶
Attributes
• name: a string identifying the frame.
• refframe: name of the parent frame (optional).
• type: (Fixed|Movable|EndEffector) a frame type identifier - default is Fixed (optional).
• daf: (true|false) boolean defining if the frame is a daf or not - default is false (optional)
Child elements
!((Pos >> RPY) | (RPY >> Pos) | Transform) >> *(Property | CollisionModel | Drawable)
Example
<Frame name="myframe" refframe="WORLD">
</Frame>
### Joint¶
Attributes
• name: a string identifying the frame.
• refframe: name of the parent frame (optional).
• type: (Prismatic|Revolute|Universal|Spherical|PrismaticUniversal|PrismaticSpherical) a joint type identifier.
• state: (Active|Passive) joint state - default is Active (optional)
Child elements
!((Pos >> RPY) | (RPY >> Pos) | Transform) >> *(PosLimit | VelLimit | AccLimit | Depend | Property | CollisionModel | Drawable)
Example
### DHJoint¶
A joint that is defined from the Denavit Hartenberg notation. The Craig DH variant is used. This can only specify Revolute or Prismatic joints
Attributes
• name: a string identifying the frame.
• alpha: in degrees
• a:
• One of the following:
• d: Revolute joint
• theta: Prismatic joint
• b: HGP Revolute joint
• beta: HGP Prismatic joint
• offset:
• state: (Active|Passive) joint state - default is Active (optional)
• type: (craig|schilling|HGP) type of DH joint (optional)
Child elements
!((Pos >> RPY) | (RPY >> Pos) | Transform) >> *(PosLimit | VelLimit | AccLimit | Depend | Property | CollisionModel | Drawable)
Example
### Depend¶
A tag used in Joint and DHJoint for making one joint depend on another.
Attributes
• on: the joint that this joint depends on.
• gain: numeric value giving a multiplication factor of the other joint value.
• offset: numeric value giving an offset.
## Drawable¶
A Drawable will be rendered as part of the visualisation in RobWorkStudio. It is also used for collision detection. If you want a model that is only for collision detection, see CollisionModel.
Attributes
• name: the name of the drawable.
• refframe: the frame that the drawable is to be attached to (optional).
• colmodel: (Enabled|Disabled) if enabled the drawable will also be used as collision model - default is Enabled (optional).
Child elements
!((RPY >> Pos) | (Pos >> RPY) | Transform) >> *(RGB) >> *(Polytope | Plane | Sphere | Box | Cone | Cylinder | Tube | Custom)
Example
<Drawable name="Joint1Geo" refframe="Joint1">
<Pos>0 0 0</Pos>
<RPY>-90 0 0</RPY>
<Polytope file="geometry/czlon1"/>
</Drawable>
Example
If the model geometry file does not contain colour information. It is possible to add the RGB tag to give it a custom colour. The default colour will otherwise be gray RGB(0.6,0.6,0.6). Example below shows how to make a drawable green.
<Drawable name="Joint1Geo" refframe="Joint1">
<Pos>0 0 0</Pos>
<RPY>-90 0 0</RPY>
<RGB>0 1 0</RGB>
<Polytope file="geometry/czlon1"/>
</Drawable>
## CollisionModel¶
A CollisionModel will not be rendered as part of the visualisation in RobWorkStudio. It is only used for collision detection. If you want a model that is also visualised, see Drawable.
Attributes
• name: name of the model
• refframe: place model relative to this frame (optional)
Child elements
!((RPY >> Pos) | (Pos >> RPY) | Transform) >> *(Polytope | Plane | Sphere | Box | Cone | Cylinder | Tube | Custom)
Example
CollisionModel and Drawable can be used together, in order to use different models for visualisation and collision detection. Often it is useful to have a more coarse model for collision detection, and a finer model for visualisation.
<Drawable name="BaseGeo" refframe="Base" colmodel="Disabled">
<Pos>0 0 0.1331</Pos>
<RPY>90 0.0 90</RPY>
<Polytope file = "Geometry/Geo_fine"/>
</Drawable>
<CollisionModel name="BaseGeo" refframe="Base">
<Pos>0 0 0.1331</Pos>
<RPY>90 0.0 90</RPY>
<Polytope file = "Geometry/Geo_coarse"/>
</CollisionModel>
## Property¶
Attributes
• name: name of the property
• type: type of property (string, double or Q) - string is default (optional)
• reframe: frame to attach property to (optional)
• desc: description of the property (optional)
Child elements
• string value (if type is string)
• double value (if type is double)
• list of doubles separated by space (if type is Q)
Example
A user property for defining a camera on a frame. The string value can be parsed by the user to get image dimensions [640;480] and field of view 40
<Property name="Camera" refframe="WORLD" desc="[fovy,width,height]">
40 640 480
</Property>
When the ‘Camera’ property name is used, this will also, when opened in RobWorkStudio, draw the outline of the camera frame. In this case the camera parameters will always be interpreted in the order fovy,width,height, as shown above, ignoring the description given.
Example
A user property for enabling the display of the frame axis. This will automatically execute the TreeView plugin action of turning on the visibility of the frame axis.
<Frame name="my_frame">
<RPY>0 0 0</RPY>
<Pos>0 0 1</RPY>
<Property name="ShowFrameAxis">true</Property>
</Frame>
## Transform¶
Attributes None
Child elements All real values are parsed into a rotation matrix R and a position P
R00 >> R01 >> R02 >> P0 >>
R10 >> R11 >> R12 >> P1 >>
R01 >> R21 >> R22 >> P2
Example Loads the identity rotation and the (0.1,0.1,0.1) position. Values are seperated by whitespace.
<Transform>
1 0 0 0.1
0 1 0 0.1
0 0 1 0.1
</Transform>
## RPY¶
Attributes None
Child elements Loads RPY values seperated by whitespace
roll >> pitch >> yaw
Example A rotation matrix with 90 degree rotation around z-axis
<RPY> 90 0 0 </RPY>
## RGB¶
Attributes None
Child elements Defines the simple material colours for the visual model. This material is used if the model does not support the colour/material information (e.g. when geometric primitive or STL model is used).
r >> g >> b >> *a
Example
Simple material color with RGB values [1.0, 0.0, 0.0] (red color).
<RGB> 1.0 0.0 0.0 </RGB>
Example
Simple material color with transparency: RGBA values [1.0, 1.0, 0.0 0.5] (transparent yellow color).
<RGB> 1.0 1.0 0.0 0.5 </RGB>
## Pos¶
Attributes
Child elements Loads pos values seperated by whitespace
x >> y >> z
Example
<Pos> 0.1 0.1 0.2 </Pos>
## Geometry¶
Geometries are used in Drawable and CollisionModel definitions. The following types of geometries are currently supported: Polytope, Plane, Sphere, Box, Cone, Cylinder, and Tube. Also, a Custom type is possible, which can be used together with the RobWork plugin structure to define custom geometry.
### Polytope¶
Attributes
• file: the geometry file
Example
<Polytope file="c:/geometry/object.stl" />
### Plane¶
Attributes None
Example
<Plane />
### Sphere¶
Attributes None
• level: (optional, default=20) mesh resolution.
Example
<Sphere radius="0.05" />
<Sphere radius="0.05" level="20" />
### Box¶
Attributes
• x: length in x-axis
• y: length in y-axis
• z: length in z-axis
Example
<Box x="0.1" y="0.1" z="0.1" />
### Cone¶
Attributes
• z: height of cone.
• level: (optional, default=20) mesh resolution.
Example
<Cone radius="0.1" z="0.1" />
<Cone radius="0.1" z="0.1" level="10"/>
### Cylinder¶
Attributes
• z: length of cylinder
• level: (optional, default=20) mesh resolution.
Example
<Cylinder radius="0.1" z="0.1" />
<Cylinder radius="0.1" z="0.1" level="20"/>
### Tube¶
Attributes
• thickness: thickness of the tube surface.
• z: height of the tube.
• level: (optional, default=20) mesh resolution.
Example
<Tube radius="0.1" thickness="0.01" z="0.1"/>
<Tube radius="0.1" thickness="0.01" z="0.1" level="10"/>
### Custom¶
The Custom type allows the user to add user-defined geometry to RobWork and use it in the WorkCell definition. The user must register an extension to the rw.loaders.GeometryFactory extension point, for instance by providing a RobWork plugin. The extension should have a property called type. It is this property that is used to match the extension with the type parameter in the Custom tag. If no type property is found, the name of the extension itself is used instead. The extension can provide any geometry object that implements the rw::geometry::GeometryData interface, but the parameters given in the param parameter is only passed to the object if it is of the more specific type rw::geometry::Primitive.
Attributes
• type: name identifying the type of geometry.
• param: arbitrary length list of doubles separated with space.
## Calibration¶
Attributes
• file: name of the calibration file.
## CollisionSetup¶
Attributes
• file: the file where the collision setup is described
Example
<CollisionSetup file="../mydevice/colsetup.xml" />
## ProximitySetup¶
Attributes
• file: the file where the proximity setup is described
Example
<ProximitySetup file="../mydevice/colsetup.xml" />
## PosLimit¶
Attributes
• refjoint: the joint which the limit is valid for. (optional)
• min: the minimum joint value
• max: the maximum joint value
Example
<PosLimit refjoint="joint1" min="-90" max="90" />
## VelLimit¶
Attributes
• refjoint: the joint which the limit is valid for. (optional)
• max: the maximum joint velocity value
Example
<VelLimit refjoint="joint1" max="180" />
## AccLimit¶
Attributes
• refjoint: the joint which the limit is valid for. (optional)
• max: the maximum joint acceleration
Example
<AccLimit refjoint="joint1" max="180" />
## Q¶
Attributes
• name: name identifying the configuration. The special name “Home” specifies the home position of a device.
Example
<Q name="Home">1.57 -2.4 2.4 -1.57 -1.57 0</Q>
`
|
zbMATH — the first resource for mathematics
Brauer groups, Tamagawa measures, and rational points on algebraic varieties. (English) Zbl 1308.14025
Mathematical Surveys and Monographs 198. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-1882-3/hbk). viii, 267 p. (2014).
This book is concerned with the existence and distribution of rational points on algebraic varieties. Thus it focuses in particular on the Hasse principle and the Brauer–Manin obstruction, and on the Manin conjecture.
The book is divided into 3 parts. Part A, on heights, describes the notion of height, and introduces the conjectures of Lang, of Batyrev and Manin, and of Manin and Peyre. There is a full account of the different factors in the Peyre constant, and a discussion of some of the proven cases, and the methods used for them.
Part B concerns the Brauer group. Here one learns firstly the general theory of the Brauer group. This is then applied to the Brauer–Manin obstruction, developing the general theory before going on to apply it to a range of special cubic surfaces. In particular the case of diagonal cubic surfaces is presented in some detail, describing some, but not all, of the results of J.-L. Colliot-Thélène et al. [Lect. Notes Math. 1290, 1–108 (1987; Zbl 0639.14018)].
The third part of the book, entitled “Numerical experiments” describes three different numerical questions. The first is the search for rational points on the quartic surface $$x^4+2y^4=z^4+4w^4$$. Swinnerton-Dyer had asked whether there were any rational points other than the obvious ones with $$y=w=0$$, and the first chapter of this section describes the way one conducts an efficient search for such points, leading eventually to the solution $1484801^4+2.1203120^4=1169407^4+4.1157520^4.$ The second chapter describes an investigation into Manin’s conjecture for 3-folds of the shape $$ax^3=by^3+z^3+v^3+w^3$$ and $$ax^4=by^4+z^4+v^4+w^4$$, with $$1\leq a,b\leq 100$$. Again there is a search procedure, locating points of height up to 5000 in the cubic case, and 100000 in the quartic case. One also has to compute the Peyre constants, which in particular involves all the local densities. In addition one must identify the various accumulating subvarieties. All this is described in the chapter, giving a comprehensive practical account of everything involved in the Manin–Peyre conjecture. The final chapter in the book concerns the height of the smallest rational point on a diagonal cubic surface $$X$$. In particular one may ask to what extent the smallest height of such a point, $$m(X)$$ say, can be bounded in terms of the Peyre constant $$\tau(X)$$. Indeed one could conjecture that $$m(X)=O_{\alpha}(\tau(X)^{-\alpha})$$ for any $$\alpha>0$$. A numerical investigation of over 800000 surfaces is described, and again the calculation of $$\tau(X)$$ plays a major role in the computations.
Overall this book delivers a thorough grounding in all aspects of the Hasse principle and the Brauer–Manin obstruction, and the Manin conjecture. It is to be recommended for serious study by research students and more senior workers, but the wealth of examples, and of different angles on the material also make it suitable for dabbling by the more general reader.
MSC:
14G05 Rational points 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11D45 Counting solutions of Diophantine equations 11G50 Heights
Zbl 0639.14018
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taffybar-3.2.1: A desktop bar similar to xmobar, but with more GUI
Copyright (c) José A. Romero L. BSD3-style (see LICENSE) José A. Romero L. unstable unportable None Haskell2010
System.Taffybar.Widget.DiskIOMonitor
Description
Simple Disk IO monitor that uses a PollingGraph to visualize the speed of read/write operations in one selected disk or partition.
Synopsis
# Documentation
Arguments
:: MonadIO m => GraphConfig Configuration data for the Graph. -> Double Polling period (in seconds). -> String Name of the disk or partition to watch (e.g. "sda", "sdb1"). -> m Widget
Creates a new disk IO monitor widget. This is a PollingGraph fed by regular calls to getDiskTransfer. The results of calling this function are normalized to the maximum value of the obtained probe (either read or write transfer).
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# GATE Questions
Q:
Ravi, Anand and Pranay are running a business firm in partnership. What is Anand's share in the profit earned by them?
a. Ravi, Anand and Pranay invested the amounts in the ratio of 2 : 4 : 7.
b. Pranay's share in the profit is Rs. 8750.
A) Only a is sufficient B) Neither a nor b is sufficient C) Only b is sufficient D) Both a and b sufficient
Explanation:
Given Pranay's profit in the share is Rs. 8750
But given that their investments are in the ratio 2:4:7
$\inline&space;\fn_jvn&space;\small&space;\frac{7}{13}\times&space;P$ = 8750
P = 16250
Now Anand's share = $\inline \fn_jvn \small \frac{4}{13}\times 16250$ =Rs.5000
Thus both staements a and b are sufficient to answer this question.
1 318
Q:
Calculate the angle between the two hands of clock when the clock shows 4 : 20 p.m. ?
A) 5 degrees B) 10 degrees C) 15 degrees D) 25 degrees
Explanation:
Since at 4 : 20 the minute hand will be at 4 and the angle between them will be same as the distance covered in degree by the hour hand in 20 minutes.
Required angle = distance of hour hand = speed × time = $\inline \frac{1}{2}$ × 20 =10 degrees.
Filed Under: Clock puzzles
Exam Prep: Bank Exams , CAT , GATE , GRE
Job Role: Bank Clerk , Bank PO
3 317
Q:
Vishal invested 10% more than Trishul. Trishul invested 10% less than Raghu. If the total sum of their investments is Rs. 5780, how much amount did Raghu invest ?
A) Rs.2010 B) Rs.2200 C) Rs.2000 D) Rs.2100
Explanation:
Let money invested by Raghu = Rs. x
Money invested by Trishul = 9/10 x = 0.9x
Money invested by Vishal = 9/10x * 110/100 = 0.99x
Also, x+0.9x+0.99x = 5780
= x= 5780/2.89 = 2000
Therefore, amount invested by Raghu is Rs. 2000.
6 317
Q:
An amount of Rs. 25000 is invested in two types of shares. The first yields an interest of 6% p.a. and the second, 10% p.a. If the total interest at the end of one year is $\inline \fn_jvn \small 7\tfrac{3}{4}$ %, then the amount invested in each share is ?
A) Rs. 21812.5 B) Rs. 31245.7 C) Rs. 24315.5 D) Rs. 20000
Explanation:
Let the amount invested at 6% be Rs. P and that invested at 10% be Rs. (25000-P).
Then,
$\inline \fn_jvn \small \left ( \frac{P\times 6\times 1}{100} \right )+\left ( \frac{(25000-P)\times 10\times 1}{100}\right )=\left ( 25000\times \frac{31}{4}\times \frac{1}{100} \right )$
$\inline \fn_jvn \small \Rightarrow \frac{6P+250000-10P}{100}=\frac{105\times 31}{2}$
$\fn_jvn&space;\small&space;\Rightarrow$ P = Rs. 21812.5
3 316
Q:
A man rows his boat 60 km downstream and 30 km upstream taking 3 hrs each time. Find the speed of the stream ?
A) 5 kmph B) 10 kmph C) 15 kmph D) 45 kmph
Speed of the boat downstream $\inline \fn_jvn \small s=\frac{d}{t}$ = $\inline \fn_jvn \small \frac{60}{3} = 20 kmph$
Speed of the boat upstream $\inline \fn_jvn \small s= \frac{d}{t} = \frac{30}{3} = 10 kmph$
$\fn_jvn&space;\small&space;\therefore$ The speed of the stream = $\inline \fn_jvn \small \frac{(speed of downstream - speed of upstream)}{2} = 5kmph$.
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# Tag Info
11
According to your definition of autocorrelation, the autocorrelation is simply the covariance of the two random variables $Z(n)$ and $Z(n+\tau)$. This function is also called autocovariance. As an aside, in signal processing, the autocorrelation is usually defined as $$R_{XX}(t_1,t_2)=E\{X(t_1)X^*(t_2)\}$$ i.e., without subtracting the mean. The ...
7
You can think of linear-least squares in single dimension. The cost function is something like $a^{2}$. The first derivative (Jacobian) is then $2a$, hence linear in $a$. The second derivative (Hessian) is $2$ - a constant. Since the second derivative is positive, you are dealing with convex cost function. This is eqivalent to positive definite Hessian ...
6
If you have two signal vectors $x_1[n]$ and $x_2[n]$ each of $N$ elements, then there are two different things we can consider. How do the quantities $\displaystyle\sum_{n=1}^N x_i[n]x_j[n],~ i, j \in \{1,2\}$ compare? In particular, when the signals are noisy and the noises can be considered to be jointly stationary (or jointly wide-sense stationary), ...
5
What are reasons to choose for cross-correlation or cross-covariance when comparing signals with non-zero mean? Well, part of the issue is that cross-correlation as defined in your equation: $$(f \star g)[n]\ \stackrel{\mathrm{def}}{=} \sum_{m=-\infty}^{\infty} f^*[m]\ g[m+n].$$ will not exist (or be infinite) if $f$ and $g$ have non-zero mean. So, in ...
4
For power signals $x(t)$ and $y(t)$, the function $$R_{xy}(\tau)=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}x(t)\bar{y}(t+\tau)dt\tag{1}$$ is the cross-correlation of $x(t)$ and $y(t)$. So the expression you're asking about is the cross-correlation of $x(t)$ and $y(t)$ evaluated at lag $\tau=0$: $$R_{xy}(0)=\lim_{T\rightarrow\infty}\frac{1}{2T}\... 4 The Covariance Matrix is commonly defined as$$\mathbf Q = E\left[ (\mathbf x -\mathbf\mu_{x})(\mathbf x -\mathbf\mu_{x})^*\right]$$with \mu denoting the mean value, i.e. \mu_{x}=E\left[\mathbf x\right], and \mathbf x being column vectors. The fact that you define the covariance matrix as$$\mathbf{R}_i = E\left[\textbf{u}_i^*\textbf{u}_i \right]... 3 MVDR is a narrowband beamformer. For broadband signals it is usually applied for each frequency bin. That means that \mathbf{R}_{xx} is frequency dependent. In other words, for each time you should have M matrices, each one is 3\times 3. Now, since you usually cannot compute \mathbf{R}_{xx} exactly, you perform covariance estimation \tilde{\mathbf{... 3 A PRN sequence is a Pseudo-Random Noise sequence, often generated by using an Linear Feedback Shift Register (LFSR) with the feedback taps done by using a primitive irreducible polynomial in GF{2}, which is the Golois Field of 2 elements. When a primitive and irreducible polynomial in GF{2} is used, the LFSR will produce a "maximum length sequence", meaning ... 3 I had worked on an array processing problem where I had used diagonal loading of the measurement covariance matrix. But I had used diagonal loading as a solution to what I thought was a numerical issue with my eigen values being too small. Since I had to invert the covariance matrix with small eigen values, I had numerical issues with the result. Diagonal ... 3 You can't really have a covariance of a matrix. What you can have, is a covariance matrix of a set of vectors. So, if you think of the rows of your matrix A as two vectors in 3D: [2 3 4] and [5 5 6], then the covariance matrix of this set of two vectors is C = A' * A (A transpose times A). Note that if you shuffle the rows of A in a different order, C ... 3 Usually, for power signals, we define the inner product to be \begin{align} \left<x\,,\,y\right> &= \lim_\limits{T\rightarrow \infty} \frac 1 {2T} \int\limits_{-T}^T x(t)\bar y(t)\,dt \end{align} which induces the vector norm \begin{align} ||x||^2 &= \left<x\,,\,x\right>\\ &=\lim_{T\rightarrow \infty} \frac 1 {2T} \int\limits_{-T}^... 3 I don't understand the subscript n notation, however, in the least squares problem that is given by: $${\bf{y}}={\bf{H}}{\theta}+\bf{n},$$ where {\bf{n}}\sim\mathcal{N}(\bf{0}, \sigma^2I_N) is a zero mean additive white Gaussian noise and I_N is the N \times N identity matrix, the maximum likelihood and the least squares ... 3 It's the key point of array signal processing, I suppose. Say x is the input vector of [N,1] dimension collected from N array sensors. x(k) is its realization at the k moment of time. By its definition covariance matrix (sometimes it's called autocorrelation matrix): R = E[x\cdot x^H] , where E[] is expectation operator and x^H is Hermitian ... 3 For discrete data both are the same - Finding set of orthogonal directions which maximizes the Variance (Energy) of data along them. Sometimes those are called the natural axis of the. Since we're dealing with variance it is only natural both are calculated from the covariance matrix of data. You may encounter places where KL might be even defined on the ... 2 There is something that is not clear of what you have done with the data, and that is who do you form the random vectors to perform de SVD (or EVD) on the covariance matrix. 1 -The KLT can be succesfully used on a one dimensional signal (only one Geophone), taking frames of M samples and estimating a covariance matrix from it, and the performing ... 2 I would think that it is indeed the "normal" variance of the image. You look at the distribution of the pixel values (i. e. gray levels), and compute its variance. 2 What you want to do is dimension reduction. The most basic, yet very powerful and commonly used, technique to do it is principal component analysis (PCA). PCA operates on the covariance matrix. You can look it up on Wikipedia for a throughout tutorial. PCA decomposes your data y, having dimension d, into d principal components. The first principal ... 2 Variance is defined as V(x)=\frac{\sum_{i=1}^n(x_i-\mu)^2}{n}. Just in case for you, mean \mu is defined as \mu=\frac{\sum_{i=1}^nx}{n}. Covariance between two random variables x and y (or columns of a matrix) is defined as Cov(x,y)=\frac{\sum_{i=1}^n[(x_i-\mu_x)(y_i-\mu_y)]}{n} and Cov(x,x)=V(x). The term covariance matrix may be misleading ... 2 This answer is not fundamentally different from the others; it's more of a complement and addendum. If n(t) is zero-mean white Gaussian noise, then its variance is actually infinite; its power spectral density is constant and often denoted \mathcal P_n(f)=N_0/2. If this noise is input into a filter with impulse response h(t), then the power spectral ... 2 Your question (as expanded in the comments) is asking if we start with x(t) = x_{\rm true}(t) + n_1(t) $$and filtering it using a filter H(\omega) to get$$ x_{\rm hp}(t) = h(t) * x(t) = x_{\rm true}(t) + n_2(t) $$where the variance of n_1 is v_1, then what is v_2, the variance of n_2? That seems ill-posed because the filter H will change ... 2 In general, the equations from your first code block work equally for real and complex values. Note that the ()' operation is the Hermitian Conjugate (i.e. transpose + conjugate). If you are in the real domain, it only becomes transpose (because conjugation on reals doesn't change them). Hence, you would not need to change anything in the code. Regarding ... 2 One must be careful when asking questions about the relationships between the elements of a complex random vector. The short answer to your question is that you cannot say much for either cases simply by considering the covariance (or correlation) matrix. Actually, the covariance (correlation) matrix is not enough to capture all the relationships that ... 2 Short answer: just use \sigma = 10^{-8}. Covariance matrices have eigenvalues \geq 0 (theoretically), so Ri + 10^{-8} \, I will have eigenvalues \geq 10^{-8}, safely non-singular. A longer answer: split Covar = S + N, "signal" + "noise", by eigenvalues or by SVD, Singular-value decomposition aka PCA, Principal component analysis. This has several ... 1 You can write$$ R=YY^H $$where Y is a matrix of size N\times N_f and N is the dimension of y_k. Y contains all the measured y_k as its columns. Then, the rank of R is upper bounded by N_f. In particular, if N_f<N, R will always be a singular matrix. So, if you have too few measurements, you will likely run into the problem you ... 1 I'd recommend looking into the relation of correlation and covariance; the one is just the "bias-corrected" version of the other. Then, use the Wiener-Khinchin¹ theorem: If, and only if, the signal is weak-sense stationary, the Fourier transform of the autocorrelation of that signal is the same as Expectation value of the magnitude-squared Fourier transform ... 1 To make the Online variance algorithm work for vectors, change one line: M2 += np.outer( delta, delta2 ) # deltas 3-vecs, M2 3 x 3 # outer( x, y ) is pairs x[i] * y[j] -- # [[ x0 y0, x0 y1, x0 y2 ], # [ x1 y0, x1 y1, x1 y2 ], # [ x2 y0, x2 y1, x2 y2 ]] A small python class for this is under gist.github.com/denis-bz . It can covariance points with ... 1 I'm not totally sure I understand your question: could you give a concrete example? For instance, what's the randomness? Anyway, here are some relations between the quantities you're interested in. The energy of the signal is$$\mid \mid Y \mid \mid_2^2 = \sum_{i=0}^n y_i^2 The Auto-correllation matrix $R$ is formed as $YY^T$ (outer product between the ...
1
There is no "best method", it will vary according to application. There are many trains of thought regarding this subject. I've done something in that area using Cepstral Analysis. Take a look at this: https://8b14d84720d60d831e70ec440417feb774595227-www.googledrive.com/host/0B4-fFSwiy1mGdS00SHB2Nlo3bkU/imreg_translation/imreg_translation.html Note: I am ...
Only top voted, non community-wiki answers of a minimum length are eligible
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# Using a behavioral voltage source as a zero crossing detector for inductance current in LTSpice
I'm trying to simulate a buck converter IC in LTSpice (MP4569) based on the functional block diagram from the datasheet using behavioral voltage sources to represent the logic from the block diagram.
I have 2 switches in place of the high and low side mosfets and the switches are driven by behavioral voltage sources.
I'm trying to implement the zero current detection block (ZCD block on the diagram) with a behavioral voltage source with the condition comparing the current to a threshold (V=(I(L1)>1m), the number is arbitrary, just a small near-zero value, could be zero.
The LTSpice model is available here.
I end up having the output of the ZCD voltage source in the simulation to be always 0 (after the initial 1 in the very beginning), BUT the current in the inductance is oscillating (very small amplitude) just above the limit in the ZCD voltage source (whatever I set it to). Also the simulation slows down to a crawl at this point and never completes, I have to stop it.
It appears that it does trigger the voltage source, but very briefly and it's not visible on the output when plotted in the simulation, but it does affect the behavior of the switch.
Maybe some kind of hysteresis is needed around the zero current for this ZCD voltage source, but I cannot figure out how to add it. I tried adding a flip-flop with 2 behavioral voltage sources connected to Set and Reset but it didn't help and I'm still getting similar behavior
• Why on Earth would you use behavioural sources with discontinuous and convergent-unfriendly conditionals, when ou have the A-devices especially for that, not only guaranteed to converge, but also faster and a lot more flexible? Jul 1 '20 at 6:51
• @aconcernedcitizen, I don't have much experience with SPICE. Wanted a simplified model of the buck IC, A-devices? Analog? Existing models from manufacturers?
– axk
Jul 1 '20 at 8:11
• I mean the ones in [Digital]/*. Also see the help for LTspice > Circuit Elements > A. ..., or the ltwiki undocumented features. Jul 1 '20 at 14:48
• @aconcernedcitizen, thanks! after replacing the B-sources with a combination of diffschmitt, inv and and it all worked! You can post this as an answer if you want.
– axk
Jul 1 '20 at 19:19
Besides @Voltage Spike's answer, a warm recommendation would be to avoid conditionals in behavioural expressions like if(), buf(), etc (or other discontinuous functions like limit(), uramp(), etc), because the solver might get stuck with "timestep too small". They might work, they might not, they could be attempted to be "tamed" with some strategically placed small capacitors to help out the sharp transitions, but there already is a very convergent-friendly solution: A-devices.
For your case, you can replace these:
• the negation in B2 and the one for the second term in B3 with [Digital]/inv
• the AND for B3 and B7 with [Digital]/and
• all the conditionals in B4, B6, and B8 with either [Digital]/schmitt (or diffschmitt) with vt=<...> vh=0, or with [Digital]/buf with ref=<...>
• the conditional for the 2nd term in B7 with [Digital]/schmitt with vt=0 vh=0, or with buf1 with ref=0
For the cases where you need to use a voltage, it's simple, just add the respective node to the input of the logic gate. For currents, since you're only using I(L1), you can add an H-source with L1 1 as value (which might be a better choice than a B-source).
And, while we're at it:
• (important!) you named ZCD both the node for the Q output of A2 and the output of B4. Since you are using a behavioural voltage source, it can't be intentional
• you may also change B6 with a normal voltage source. I only see it used in B7, so you can just delete the source altogether and use an inv with ref=1
• delete the resistors R1 and R[6,7,8,9], they're not needed. The A-devices have a default output resistance of 1 Ω (one exception, not needed here), so adding a resistor will change the output levels
• C1 is useless there since voltage sources have zero internal resistance. You could add a seris resistance between the source and the cap, but you'd be better off adding Rser to the source, in which case C1 can be safely deleted and Cpar can be specified in the source.
• a bit of a "heads-up", you'll most probably need some dead-time for the drivers, so in that case you should add some anti-parallel diodes to the switches. Since you're on the theoretical or, at least, the ideal side, a .model d d ron=10m roff=10meg vfwd=0.7 epsilon=0.1 revepsilon=50m will do just fine.
On the bright side, the VCSW have their .model cards with a negative hysteresis and very acceptable range between the ON/OFF states, so that's a bravo from me.
With these, here's a quick remake:
And the .asc file, where I have only used td for the srflop; feel free to add tau and tripdt, they will only help:
Version 4
SHEET 1 1100 688
WIRE 320 -112 240 -112
WIRE 928 -112 320 -112
WIRE -112 -80 -160 -80
WIRE 0 -80 -48 -80
WIRE 144 -80 96 -80
WIRE 160 -80 144 -80
WIRE 240 -48 240 -112
WIRE 0 -32 -32 -32
WIRE 144 -32 112 -32
WIRE 192 -32 144 -32
WIRE 320 -32 320 -112
WIRE 928 -32 928 -112
WIRE -256 48 -304 48
WIRE -112 48 -192 48
WIRE -32 48 -32 -32
WIRE -32 48 -48 48
WIRE -112 80 -128 80
WIRE 240 112 240 32
WIRE 320 112 320 32
WIRE 320 112 240 112
WIRE 384 112 320 112
WIRE 512 112 464 112
WIRE 592 112 512 112
WIRE 672 112 592 112
WIRE 720 112 672 112
WIRE 832 112 800 112
WIRE 848 112 832 112
WIRE 240 144 240 112
WIRE 672 144 672 112
WIRE 176 160 128 160
WIRE 192 160 176 160
WIRE 320 160 320 112
WIRE 512 160 512 112
WIRE 848 160 848 112
WIRE 32 240 -16 240
WIRE 128 240 128 160
WIRE 128 240 96 240
WIRE 32 272 -16 272
WIRE 240 272 240 224
WIRE 320 272 320 224
WIRE 320 272 240 272
WIRE 384 272 320 272
WIRE 512 272 512 224
WIRE 512 272 384 272
WIRE 672 272 672 224
WIRE 672 272 512 272
WIRE 848 272 848 240
WIRE 848 272 672 272
WIRE 928 272 928 48
WIRE 928 272 848 272
WIRE 560 416 496 416
WIRE 608 416 560 416
WIRE 752 416 672 416
WIRE 768 416 752 416
WIRE -80 432 -128 432
WIRE -32 432 -80 432
WIRE 96 432 32 432
WIRE 224 432 192 432
WIRE 240 432 224 432
WIRE 96 480 48 480
WIRE 224 480 208 480
WIRE 240 480 224 480
WIRE -80 560 -80 432
WIRE -32 560 -80 560
WIRE 48 560 48 480
WIRE 48 560 32 560
FLAG 192 16 0
FLAG 192 208 0
FLAG 384 272 0
FLAG 592 112 out
FLAG 144 -32 _LQ
FLAG 176 160 x
FLAG 496 496 0
FLAG 560 416 i
FLAG 752 416 _i
FLAG -160 -80 i
FLAG 832 112 fb
FLAG -304 48 fb
FLAG -128 80 zcd
FLAG -128 432 i
FLAG 224 480 _zcd
FLAG 224 432 zcd
FLAG 144 -80 LQ
FLAG -16 240 LQ
FLAG -16 272 _zcd
SYMBOL voltage 928 -48 R0
WINDOW 123 24 118 Left 2
WINDOW 39 24 140 Left 2
SYMATTR InstName V1
SYMATTR Value pwl 0 0 1u 56
SYMATTR Value2 Rser=10m
SYMATTR SpiceLine Cpar=1m
SYMBOL ind 368 128 R270
WINDOW 0 32 56 VTop 2
WINDOW 3 5 56 VBottom 2
SYMATTR InstName L1
SYMATTR Value 33u
SYMBOL cap 496 160 R0
SYMATTR InstName C1
SYMATTR Value 22u rser=50m
SYMBOL sw 240 48 M180
SYMATTR InstName S1
SYMATTR Value up
SYMBOL sw 240 240 M180
SYMATTR InstName S2
SYMATTR Value dn
SYMBOL diode 304 32 M180
WINDOW 0 24 64 Left 2
WINDOW 3 24 0 Left 2
SYMATTR InstName D1
SYMBOL diode 304 224 M180
WINDOW 0 24 64 Left 2
WINDOW 3 24 0 Left 2
SYMATTR InstName D2
SYMBOL res 656 128 R0
SYMATTR InstName R1
SYMATTR Value 33
SYMBOL Digital\\srflop 48 -128 R0
WINDOW 3 -40 34 Left 2
SYMATTR InstName A1
SYMATTR Value td=10n
SYMBOL h 496 400 R0
WINDOW 0 33 68 Left 2
SYMATTR InstName H1
SYMATTR Value L1 1
SYMBOL Digital\\inv 608 352 R0
SYMATTR InstName A2
SYMBOL Digital\\buf1 -112 -144 R0
WINDOW 3 -2 94 Left 2
SYMATTR InstName A3
SYMATTR Value ref=0.7
SYMBOL Digital\\inv -256 -16 R0
WINDOW 3 -4 99 Left 2
SYMATTR InstName A4
SYMATTR Value ref=1
SYMBOL res 816 96 R90
WINDOW 0 0 56 VBottom 2
WINDOW 3 32 56 VTop 2
SYMATTR InstName R2
SYMATTR Value 1.2Meg
SYMBOL res 832 144 R0
SYMATTR InstName R3
SYMATTR Value 510k
SYMBOL Digital\\and -80 0 R0
SYMATTR InstName A5
SYMBOL Digital\\inv -32 368 R0
WINDOW 3 -4 99 Left 2
SYMATTR InstName A6
SYMATTR Value ref=10m
SYMBOL Digital\\buf1 -32 496 R0
WINDOW 3 -2 94 Left 2
SYMATTR InstName A8
SYMATTR Value ref=20m
SYMBOL Digital\\srflop 144 384 R0
WINDOW 3 -40 34 Left 2
SYMATTR InstName A7
SYMATTR Value td=10n
SYMBOL Digital\\and 64 192 R0
SYMATTR InstName A9
TEXT -80 -224 Left 2 !,model up sw ron=1.5 roff=0.1g vt=0.5 vh=-0.5\n.model dn sw ron=0.625 roff=0.1g vt=0.5 vh=-0.5\n.model d d ron=1 roff=100meg vfwd=0.7 epsilon=0.1 revepsilon=50m
TEXT 768 -232 Left 2 !.tran 1m
TEXT 760 -176 Left 2 ;V(ref) = 1 V
TEXT -288 104 Left 2 ;1 > V(fb)
TEXT -184 144 Left 2 ;V(zcd) & (1 > V(fb))
TEXT -168 -136 Left 2 ;I(L1) > 0.7
TEXT -80 368 Left 2 ;I(L1) < 10m
TEXT -64 624 Left 2 ;I(L1) > 20m
TEXT 96 16 Left 2 ;!V(LQ)
TEXT 104 304 Left 2 ;V(LQ) & !V(zcd)
• I usually pepper my designs with appropriately placed passives to help the solver. Alot of this can be done by using real physical parasitics in the design (ie making sure that there are no superconducting wires, inductors, capacitors). Also throwing in some small caps (and one should really use cshunt of at least 1e-13, because on a real PCB everything is about 1e-14 or grater farads away from the ground plane. Jul 2 '20 at 23:52
• +1 You are correct, I also usually use conditionals, but place reasonably valued low pass filters on my b-sources. Jul 2 '20 at 23:54
• @VoltageSpike Those are reflexes to be praised. Still, inside the subcircuits, I try to keep things as simple as possible, sometimes even at the cost of minor realism (within sensible limits), because if things get stuck or crawling, I'd like the subcircuits to be the last place I need to search for problems. Outside, "apres nous, le diluge". Jul 3 '20 at 7:49
I couldn't get the switches to turn on until I set vref to 0.1V, after that it started switching, so either change Vref or check your VFB.
Generally speaking, you should never have a voltage source that can source infinite amounts of current. This creates problems for the solver.
So put a series resistor (like 0.1Ω) on B6,B7,B8,B4 and maybe B2 and B3
B2 and B3 need thresholds, they aren't producing voltage
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Extended Randomized Benchmarking (XRB)¶
This example illustrates how to generate extended randomized benchmarking (XRB) circuits and use them to estimate the probability of a stochastic error acting on the specified system(s) during a random gate. While this example uses a simulator to execute the circuits, the same procedure can be followed for hardware applications.
Isolated XRB¶
This section illustrates how to generate XRB circuits to characterize a pair of qubits in isolation. Here, we are performing two-qubit XRB which learns the stochastic infidelity over the two-qubit Clifford gateset.
import trueq as tq
# generate XRB circuits to characterize a pair of qubits [0, 1]
# with 9 * 30 random circuits for each circuit depth [4, 32, 64]
circuits = tq.make_xrb([[0, 1]], [4, 32, 64], 30)
# initialize a noisy simulator with stochastic Pauli and overrotation
# run the circuits on the simulator to populate their results
sim.run(circuits, n_shots=1000)
# plot the exponential decay of the purities
circuits.plot.raw()
# print the fit summary
circuits.fit()
True-Q formatting will not be loaded without trusting this notebook or rerunning the affected cells. Notebooks can be marked as trusted by clicking "File -> Trust Notebook".
XRB Extended Randomized Benchmarking Cliffords (0, 1) Key: labels: (0, 1) protocol: XRB twirl: Cliffords on [(0, 1)] ${e}_{S}$ The probability of a stochastic error acting on the specified systems during a random gate. 3.6e-02 (4.0e-04) 0.036364378419704524, 0.00039508678002372194 ${u}$ The unitarity of the noise, that is, the average decrease in the purity of an initial state. 9.2e-01 (8.1e-04) 0.9238331852570052, 0.0008122020156721423 ${A}$ SPAM parameter of the exponential decay $Au^m$. 9.4e-01 (1.0e-02) 0.9378539598443021, 0.01037865923913497
Simultaneous XRB¶
This section demonstrates how to generate XRB circuits that characterize the amount of stochastic noise while gates are applied simultaneously on a device.
# generate XRB circuits to simultaneously characterize a single qubit [0],
# a pair of qubits [1, 2], and another single qubit [3] with 9 * 30 random circuits
# for each circuit depth [4, 32, 64]
circuits = tq.make_xrb([[0], [1, 2], [3]], [4, 32, 64], 30)
# initialize a noisy simulator with stochastic Pauli and overrotation
# run the circuits on the simulator to populate their results
sim.run(circuits, n_shots=1000)
# plot the exponential decay of the purities
circuits.plot.raw()
# print the fit summary
circuits.fit()
True-Q formatting will not be loaded without trusting this notebook or rerunning the affected cells. Notebooks can be marked as trusted by clicking "File -> Trust Notebook".
XRB Extended Randomized Benchmarking Cliffords (0,) Key: labels: (0,) protocol: XRB twirl: Cliffords on [0, (1, 2), 3] Cliffords (1, 2) Key: labels: (1, 2) protocol: XRB twirl: Cliffords on [0, (1, 2), 3] Cliffords (3,) Key: labels: (3,) protocol: XRB twirl: Cliffords on [0, (1, 2), 3] ${e}_{S}$ The probability of a stochastic error acting on the specified systems during a random gate. 2.1e-02 (2.6e-04) 0.020520268219155313, 0.0002591315580962583 3.8e-02 (4.0e-04) 0.0382739863343613, 0.00039559637461903996 2.0e-02 (2.7e-04) 0.02015664056516575, 0.00027493473230415336 ${u}$ The unitarity of the noise, that is, the average decrease in the purity of an initial state. 9.5e-01 (6.8e-04) 0.9458407266259672, 0.0006768376240535344 9.2e-01 (8.1e-04) 0.919911387051947, 0.0008116380253502889 9.5e-01 (7.2e-04) 0.9467906787047224, 0.0007183812579365826 ${A}$ SPAM parameter of the exponential decay $Au^m$. 1.0e+00 (1.4e-02) 1.001937306280546, 0.013918762195231985 9.7e-01 (1.2e-02) 0.9683653150714785, 0.012096171863906583 9.7e-01 (1.4e-02) 0.9730659849757097, 0.014467323518881347
Total running time of the script: ( 0 minutes 8.783 seconds)
Gallery generated by Sphinx-Gallery
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## Vocab
$c=\lambda v$
Emily Ng_4C
Posts: 65
Joined: Fri Sep 28, 2018 12:17 am
### Vocab
What does it mean for something to be quantized?
JT Wechsler 2B
Posts: 62
Joined: Fri Sep 28, 2018 12:16 am
### Re: Vocab
Something that is quantized can only have discrete values or quantities of something. In our case, that would be energy.
charlotte_jacobs_4I
Posts: 63
Joined: Fri Sep 28, 2018 12:29 am
### Re: Vocab
When you say "discrete values" does that mean it can only have one value at a time or just a small range of values?
Sophia Ding 1B
Posts: 62
Joined: Fri Sep 28, 2018 12:16 am
### Re: Vocab
I think what JT means by discrete values are values that are of existence; it can't be something like infinity or an imaginary number like i, but rather a number that we can quantify in numerical values.
Elisa Bass 4L
Posts: 61
Joined: Fri Sep 28, 2018 12:23 am
### Re: Vocab
Additionally, in the case of the energy levels of electrons being quantized, it means that they have to have the energy at a distinct level, ie. n=1 or n=2, and cannot have levels of energy between those distinct levels.
Zubair Ahmed 1L
Posts: 81
Joined: Fri Sep 28, 2018 12:16 am
### Re: Vocab
In our case, the thing being quantized is energy. When energy is quantized, energy is packaged into small discrete packets of energy called quanta.
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# How do you differentiate g(x) = xsqrt(1-e^(2x)) using the product rule?
Jun 18, 2018
g'(x)=sqrt(1-e^(2x))-(x*e^(2x))/(sqrt(1-e^(2x))
$g ' \left(x\right) = \sqrt{1 - {e}^{2 x}} + x + \frac{1}{2} \cdot {\left(1 - {e}^{2 x}\right)}^{- \frac{1}{2}} \cdot \left(- {e}^{2 x}\right) \cdot 2$
$g ' \left(x\right) = \sqrt{1 - {e}^{2 x}} - \frac{x \cdot {e}^{2 x}}{\sqrt{1 - {e}^{2 x}}}$
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# How do solve the following linear system?: 2x = 4y, 3x+ 2y - 3 = 0 ?
Aug 9, 2018
$x = \frac{3}{4}$ and $y = \frac{3}{8}$
#### Explanation:
From first equation, $x = 2 y$.
Hence,
$3 \cdot 2 y + 2 y - 3 = 0$
$8 y = 3$, so $y = \frac{3}{8}$
Thus,
$x = 2 \cdot \frac{3}{8} = \frac{3}{4}$
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# Profit or Loss
Algebra Level pending
Bob sold 2 cars a Ferrari and a Lamborghini at the same rate. He made a profit of 20% on the Ferrari but suffered a loss of 20% on the Lamborghini. If the Ferrari cost him $200 (too cheap!) then at what price (in$) did he buy the Lamborghini?
×
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Tag Info
A confidence interval is an interval that covers an unknown parameter with $(1-\alpha)\%$ confidence. Confidence intervals are a frequentist concept. They are often confused with credible intervals which is the Bayesian analog.
A confidence interval is an interval that covers an unknown parameter of interest (e.g., the mean) with $(1-\alpha)\%$ confidence. Confidence intervals are a frequentist concept. A credible interval is a related concept in Bayesian statistics. People often incorrectly ascribe the meaning of credible intervals to confidence intervals.
In frequentist statistics, a confidence interval for a parameter, $\theta$, is an interval computed from a set of data whose distribution depends on that parameter in some way. The interval is computed such that, if the process of drawing a sample and computing the interval were repeated identically ad infinitum, the proportion of the intervals that included the true value of the parameter would converge to $(1-\alpha)\%$. This does not mean that the probability of a given interval including the true value of the parameter is $(1-\alpha)\%$. Each interval either does include the true value or it does not include the true value. The 'confidence' is a property of the procedure used to compute the interval and pertains to the theoretical infinite set of such intervals.
1. The confidence interval is a function of the data, $X$. Since the data are conceptualized as a random sample from a population, confidence intervals are random variables (although the confidence interval computed on a particular set of data is a realization).
4. The same ideas can be applied to a set of parameters, e.g., $\vec{\theta} = [\mu\ \ \sigma^2]^T$. In that case, it is more correct to refer to the confidence region.
5. In a regression context, the set of confidence intervals for all possible conditional means ($\mu_Y|X=x$) is called a confidence band.
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# 1, 2, 3, 4, 5 = 2017
Can you use the digits 1, 2, 3 4 and 5 (in some order), along with common mathematical operations, to make 2017?
Allowed:
Concatenation of digits Addition, subtraction, multiplication, division Fractions
Exponents, roots
Factorials
What about just the digits 1, 2, 3 and 4? Is that enough to reach 2017?
Note by Chung Kevin
2 years, 4 months ago
MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2
paragraph 1
paragraph 2
[example link](https://brilliant.org)example link
> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
MathAppears as
Remember to wrap math in $ ... $ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$
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$\frac{(4+3)!}{\frac{5}{2}}+1=\boxed{2017}$
The other case seems quite difficult; given that 2017 is a prime number.
- 2 years, 4 months ago
found more by slightly modifying your solution-
$\dfrac{((3!)+2)!}{5 \times 4}+1$,$\dfrac{(3!)\times(4\times2)!}{5!}+1!$
- 2 years, 4 months ago
Very similar to what I did, using 2016+1 :)
- 2 years, 4 months ago
2017, being a prime it is hard!.I've gotten to 2016 so many times like- 2016= $2^5(4^3-1)$
- 2 years, 4 months ago
Wow! Can you add all the different ways that you found?
- 2 years, 4 months ago
hey i got one more for 2017,$(5!+3!)\times 4^{2}+1$
- 2 years, 4 months ago
Works fine :)
Any ideas about the 1,2,3,4 case?
- 2 years, 4 months ago
nope still trying :)
- 2 years, 4 months ago
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# Additional Pilot Program for Phase 1 Testing
### Help Support HomeBuiltAirplanes.com:
HBA Supporter
#### rv7charlie
##### Well-Known Member
Are you using the exact engine recommended by the kit mfgr? The conditions allowing 2nd pilot are pretty restrictive; only 'certified' engines, or the engine spec'd by the kit mfgr are allowed for the program. So the only real value to the program (having someone managing a unique engine while the pilot is checking out a new airframe) isn't possible. I can understand wanting to be insured from 1st flight, but if you find a qualified pilot, is there any need for you to be in the plane during initial shakedown flights?
#### Rhino
##### Well-Known Member
I would verify that with an insurance company to confirm it will help with availability or cost. Even if you don't find someone with experience in type, it would still be beneficial to have a second pilot. If for no other reason, you'll have someone who can keep theirs eyes outside the cockpit while you're reading, following and documenting test procedures. Obviously you'd prefer someone with direct experience in your particular aircraft, but the requirement in the AC isn't quite that strict, only specifying "...flight experience in the same model or other makes and models that exhibit similar flight characteristics...with the knowledge necessary to make assumptions and judgments about the test aircraft...". In other words, it doesn't necessarily require you to have a Sonex pilot as long as they meet the qualifications in other aircraft in that category that have similar flight characteristics. It would make the QP pilot evaluation and approval process far easier if you actually did find another Sonex pilot though, not to mention raising the comfort level for your risk mitigation.
I really like the qualification matrixes they have in that AC. Those will come in handy when my time arrives.
Last edited:
#### TFF
##### Well-Known Member
I think if it’s really on your mind, get a recommendation from Sonex and have them fly it solo. Not many would want to fly a Sonex two up on the first flight. If you need to be number one, go get some flights in someone else’s. A friend had to fly his RV 7 bare for a few hours, even though he had an RV8 that he built and the insurance company was covering at the same time.
#### Hephaestus
Being in the USA and building a sonex - i'm sure you'll be able to find someone with time on type. Believe poking around the sonex forums there are a few who'll happily do it for you. Might cost a few $but probably well worth it. #### karmarepair ##### Well-Known Member HBA Supporter Are you using the exact engine recommended by the kit mfgr? yes, Sonex with their engine. I can understand wanting to be insured from 1st flight, but if you find a qualified pilot, is there any need for you to be in the plane during initial shakedown flights? For me it's a combination of insurability and risk management. The current plan is the first two flights, I'm not in the airplane. 3rd flight is a transition using the Sonex curriculum from the forum. I then fly the rest of the test cards solo. #### karmarepair ##### Well-Known Member HBA Supporter Being in the USA and building a sonex - i'm sure you'll be able to find someone with time on type. Believe poking around the sonex forums there are a few who'll happily do it for you. Might cost a few$ but probably well worth it.
Found a unicorn via the Sonexbuilders forum. They might get a free Bay Area VaCa this Fall.
#### Daleandee
##### Well-Known Member
Maybe I'm alone in this but I think the idea of putting two people in a basic VFR plane for first test flight is bordering on stupidity. One qualified pilot should be able to get the thing up and around the patch if it's airworthy or plant it back on the nearby runway if it isn't. Making absolutely sure that it's ready to safely fly is part of the building process. When done correctly first flights will usually be a non event. In the rare case where it isn't a qualified pilot in the seat makes all the difference.
If you need to be able to record in flight information use a high definition video camera and point it at the panel. Most EMS & EFIS systems have tracking software that can be downloaded after the flight. If you need help looking outside then have a qualified pilot fly chase behind you.
Not to be negative but I've owned a Sonex with a VW and did the first flight on it with the new engine. There is no way on this green earth that I would have wanted to have another person in the plane during that flight. The flight went well but things may have been different at near gross weight.
Just one old man's opinion ...
#### karmarepair
##### Well-Known Member
HBA Supporter
Maybe I'm alone in this but I think the idea of putting two people in a basic VFR plane for first test flight is bordering on stupidity.
Agreed!
That's NOT my plan....the THIRD flight, MAYBE, and then just long enough to get "the last ten feet" sight picture transmitted to the "Builder Pilot", I.E. me, with just enough fuel for a comfortable margin for the mission and to keep us away from Aft CG, as with two up we'll be near gross, AND aft CG.
I like this approach for the first two flights:
"The purpose of a first flight is to:
o Make sure the engine continues to run; and
o The flight controls and other systems operate properly until
o A safe landing is made; and
o The airplane can be inspected again.
The purpose of the second flight is to make sure that the first flight was not a fluke. Then the testing can continue."
From https://www.eaa.org/-/media/Files/EAA/EducationResources/SportAviation/1804-4-4-18-Phase-1-Testing-Homebuilt-Lee-09202017.ashx "Phase 1 Testing of a Homebuilt1", Brian P. Lee, Ph. D., P.E. Kent, WA
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Measurements and Calculations for Marble Block Collision
The version of the browser you are using is no longer supported. Please upgrade to a supported browser.Dismiss
ABCDEFGHIJKLMNOPQRST
1
Conservation of Linear Momentum for Marble colliding with a Block
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Before the collision...
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It takes...6frames
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which is...0.00625seconds
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for the marble to travel...0.27meters
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so the velocity is43.2m/s
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the mass of the marble is0.0044kg
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so the linear momentum is (p=mv)0.19008kg m/s
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After the collision (marble)...
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it takes..60frames
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which is0.0625seconds
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for the marble to travel-0.27m
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so the velocity is-4.32m/s
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and the linear momentum is-0.019008kg m/s
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After the collision (block)...
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it takes79frames
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which is 0.08229166667seconds
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for the block to travel0.27m
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so the velocity is3.281012658m/s
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the mass is 0.0645kg
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so the linear momentum is0.2116253165kg m/s
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total momentum after the collision:0.1926173165kg m/s
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total momentum before the collision:0.19008kg m/s
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percent difference1.31728367%
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Key observation: linear momentum is conserved during this collision. The slight increase in momentum is within the range of measurement uncertainty.
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Angular momentum
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marble before the collision...
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the marble is moving43.2m/s
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at a distance from the c.o.m of the block of0.065m/s
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so the angular momentum (l=rmv)0.0123552kg m/s^2
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marble after collision...
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the marble is moving-4.32m/s
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at a distance from the com of the block of0.065m
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so the angular momentum is-0.00123552kg m^2/s
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the block rotating end over end
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it takes79frames
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which is0.08229166667seconds
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the thickness of the block is0.02m
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the length of the block is0.184m
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the rotational inertia is0.000184126kg m^2
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angular momentum of block0.01405850463kg m^2/s
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total angular momentum before collision0.0123552kg m^2/s
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total angular momentum after collision0.01282298463kg m^2/s
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percent difference3.648016745%
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# How is HMAC(message,key) more secure than Hash(key1+message+key2)
I understand how for hash functions which are vulnerable to length extension attacks (such as SHA1 and SHA2) it is safer to use a HMAC construction.
What I don't understand is, how or why is HMAC_SHA256(message,key) safer (in terms of resistant against certain attacks) than SHA256(key1+message+key2), assuming that all key strings are sufficient in length and entropy?
-
HMAC_SHA256(message,key) has a security proof; we do not have a ready-made one for SHA256(key1+message+key2). That's quite an argument. That said, for reasons similar to HMAC_SHA256, SHA256(key1+message+key2) intuitively seems quite strong: there's a key1 initially, making collision hard; then a final key2, further increasing security. However the lack of alignment to block boundary in SHA256(key1+message+key2) makes it quite hard to devise a proof. – fgrieu Mar 20 '14 at 20:19
@fgrieu : $\:$ (I realize it's been over a year, but) One could consider SHA256(key1+message+padtoblockboundary+key2) instead. $\;\;\;\;$ – Ricky Demer Apr 6 at 21:03
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# Where does Free Energy go in a Redox Reaction?
This doubt came to my mind while answering the following question:
Why does a voltaic cell not operate unless the two separate compartments are connected by an external circuit?
The answer seems to be that only on connecting the two halves of a Redox reaction via an external circuit, does one allow the electrons to be transferred from one species to another. And while the electrons move towards their goal (to a higher potential), they do work in the way (light a bulb or a motor or something).
Another way to view this process is that the free energy of the Redox reaction is used to do useful electrical work.
But then my brain started to wonder and there came up a question - what happens to all this energy when none of it is actually used to do electrical work, i.e, when the Redox reaction is let to occur spontaneously, where does all the free energy go? Does it get converted to heat, thus warming the solution? Or does it simply stay stored in the solution in some form?
## 2 Answers
Free energy is a state function. No matter how you run the reaction, if you start with a certain state (set of concentration) and end with a certain state (set of concentrations at equilibrium), the change will be the same.
The free energy of reaction is the maximal amount of non-pV work (electrical work in this case) the reaction can do. If it does not do any work, that energy will be released as heat instead. For example, if you short the battery, the wire and the electrochemical cell will heat up while the reaction is going on.
If you want an argument from first principles, you should look at the first law of thermodynamics. The change in energy of a system is equal to the sum of the heat and the work exchanged with the surrounding. When making optimal use of the free energy, you maximize the work (even sometimes transferring heat from the surrounding to do even more work).
All depends on the cell chemistry and geometry. It is usually a combination of keeping the energy and conversion of the free energy to thermal energy, when reagents diffuse and happen to meet each other. Or, some side reaction with solvent or auxiliary components may occur, like for $$\ce{Li-ion}$$ cells.
Some primary lithium cells last many years, while some early generation $$\ce{NiMH}$$ cells selfdischarge in few months.
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# Removing Even Crossings
Abstract : An edge in a drawing of a graph is called $\textit{even}$ if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (not using Kuratowski's theorem), and the result that the odd crossing number of a graph equals the crossing number of the graph for values of at most $3$. We begin with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.
Keywords :
Document type :
Conference papers
Domain :
Cited literature [13 references]
https://hal.inria.fr/hal-01184387
Contributor : Coordination Episciences Iam Connect in order to contact the contributor
Submitted on : Friday, August 14, 2015 - 11:39:02 AM
Last modification on : Thursday, July 4, 2019 - 2:10:02 PM
Long-term archiving on: : Sunday, November 15, 2015 - 11:05:37 AM
### File
dmAE0121.pdf
Publisher files allowed on an open archive
### Citation
Michael J. Pelsmajer, Marcus Schaefer, Daniel Štefankovič. Removing Even Crossings. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. pp.105-110, ⟨10.46298/dmtcs.3430⟩. ⟨hal-01184387⟩
Record views
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# xcode
## Sound debugging
I'd like to mention a trick I use often while debugging, a trick I learned from my good friend Markos Charatzas: triggering a sound when a breakpoint is hit. It's a pretty simple but useful trick, especially if you're debugging repeatable actions and you want to know when they happen.
## imageNamed: and multiple bundles
I've been working on a largeish project for a client for the last few months. We're have to modularized our code into several sections using CocoaPods (this is a post worth on itself, really).
One of the problems you encounter with this approach is that for each pod, CocoaPods creates a bundle with resources for that pod (at least it does so when you tell it to do that). It's the only good way to package pod resources into the main app. This means that our resources do not all live in the main bundle but in seperate bundles (which themselves do live in the main bundle). This is no problem in itself, but it can cause loading problems of resources in those bundles. It doesn't pose that much of an issue when you specify images in a nib since iOS will search in the nib's bundle too, but it's a bit harder to get resources from within your code
## Watchkit app versions
I came across this article by Daniel Kennett this morning. I had seen him tweet about it too, but checking my feeds (yeah, I still do that, albeit not daily) I noticed he also made a post about it.
Having toyed a lot with build configurations and their associated settings configuration, I was curious how he solved the issue.
I'd have to concur with Daniel: the solution is horrible. 8 Steps to get this working, with aggregate targets and disabling of parallelization of the project: Not Good™.
But my interest was piqued, and so I tried to recreate his problem (not hard) and find a better solution. I assumed it would still be hacky, but less hacky than Daniel's "Horrible" solution.
## Scripts in your app bundle
It appears Apple changed something in the iOS bundle upload : it now requires that all executables in your app bundle are signed (this might be applicable for Mac uploads too, but I haven't tried). I hear you think: isn't this the case anyway, but there might be more executables in your bundle than you expect.
Of course, there's the binary for your app, which obviously is executable. But there could be more too: if you include shell scripts, for example, which are marked executable (chmod u+x script.sh) then iTunes Connect now considers them as actual executables and now requires them to be codesigned.
## WWDC 2013 Hopes and Expectations
No real predictions post. You never know with Apple what they're going to produce at a developer event, so I won't go there. What I will do is list what I'd like to see. Mostly iOS stuff, because that's what I do.
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FOOD SCIENCE ›› 2022, Vol. 43 ›› Issue (18): 152-158.
• Bioengineering •
### Effect of Heat Stress on the Physicochemical Properties of Inner Membrane Proteins of Clostridium perfringens Spores
ZHANG Yuanyuan, LI Miaoyun, ZHAO Lijun, ZHU Yaodi, ZHAO Gaiming, LIANG Dong, MA Yangyang, LIU Chun
1. (College of Food Science and Technology, Henan Agricultural University, Zhengzhou 450002, China)
• Published:2022-09-28
Abstract: In order to study the effect of heat stress on the inner membrane proteins of Clostridium perfringens spores, the physicochemcial changes in the inner membrane proteins after heat treatment at different temperatures (25, 37, 75 or 95 ℃) for 20 minutes were characterized by measuring particle size distribution, surface hydrophobicity, ultraviolet absorption spectrum, endogenous fluorescence spectrum and Raman spectrum. The results showed that heating temperature had a significant effect on C. perfringens spore inner membrane proteins. Compared with the control group (treated at 25 ℃), treatment at 37 ℃ had no significant effect on the inner membrane proteins, and did not cause any significant changes in the particle size distribution, amino acid microenvironment, surface hydrophobicity or secondary structure. After heat stress at 75 ℃, the particle size distribution was uniform and stable, the UV absorption and fluorescence intensity was significantly enhanced, and the surface hydrophobicity was significantly increased. The percentage of α-helix decreased by 3.17%, the percentage of β-sheet decreased by 3.94%, and the proportion of random coil increased by 8.31%. After heat stress at 95 ℃, the proteins’ structure was damaged, the proteins were obviously aggregated or denatured, and the particle size distribution moved significantly to larger particle size. The above results indicated that heat stress at 75 ℃ could effectively affect the physicochemical properties of C. perfringens spore inner membrane proteins, leading to the exposure of hydrophobic sites and amino acid residues and significant secondary structural changes. This study provides a theoretical basis for further research on the effect of heat stress on the functional properties of C. perfringens spore inner membrane proteins.
CLC Number:
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# On the Bell Numbers
Edit (first version was incorrectly stated. Thank you Douglas and others for your corrections) Let $B_n$ be the $n$th Bell number (the number of partitions of a set with $n$ members). For each $n > 3$, I have a set $A_n$ of size $|A_n|=B_n$. I then have a subset $A'_n \subset A_n$ where $|A'_n|=B_n-B_{n-1}$. I would like to say something about the size of $A'_n$ relative to the size of $A_n$. For instance, it seems that $lim_{n \to \infty} \frac{B_{n-1}}{B_n}=0$. Can I make a stronger statement about the ratio of successive Bell numbers? How can I formalize the statement "for sufficiently large $n$, most of $A_n$ is in $A'_n$."
• That conjecture is very far off. Do you mean ordinary partitions instead of set partitions? The asymptotics are known for both. – Douglas Zare Mar 4 '10 at 15:05
• The conjecture is true for ordinary partitions of integers. – Michael Lugo Mar 4 '10 at 15:44
It's easy to see that $$B_n \ge 2 B_{n-1}$$ since we always have a choice of whether to add $$n$$ to the same part as $$n-1$$ or not. Since the number of parts in a typical set partition of size $$n-1$$ grows, the choices for adding $$n$$ to a new or existing part grow, so
$$\lim_{n\to\infty} B_{n-1}/B_n = 0.$$ There are asymptotics in the Wikipedia article on the Bell numbers, but it may not be obvious how to work with the Lambert $$W$$-function in that expression, or how to bound $$B_{n-1}/B_n$$. A faster proof that the limit is $$0$$ can be obtained from Dobiński's formula, that $$B_n$$ is the $$n$$th moment of a Poisson distribution with mean $$1$$:
For any $$c \in \mathbb R$$, the Poisson distribution has positive probability of being greater than $$c$$. So, for large enough $$n$$, the $$n$$th moment $$B_n$$ is at least $$c^n$$. By Jensen's inequality, the moments satisfy
$$B_n^{\frac{n+1}{n}} \le B_{n+1}$$
$$c \le \sqrt[n]{B_n} \le \frac {B_{n+1}}{B_n}$$
A precise estimate of the relative ratios of consecutive Bell numbers is proved by Asai, Nobuhiro; Kubo, Izumi; Kuo, Hui- Hsiung in Acta Appl. Math. 63 (2000), no. 1-3, 7987: $$1\leq \frac{B(n)B(n+2)}{B^2(n+1)}\leq \frac{n+2}{n+1}$$
• The sequence $C(n)=2^n$ satisfies $C(n)C(n+2)/C^2(n+1)$=1, but this tells us little about $C(n-1)/C(n)$. – Gerry Myerson Dec 10 '19 at 11:30
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# Simplify cube root of 216
Rewrite as .
Pull terms out from under the radical, assuming real numbers.
Simplify cube root of 216
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marginpar anchored by pages not by text
I'm using marginpar to put margin figures in a document - there are $n$ pages and $n$ figures so I'd like to have one on each page, anchored to the first line of text on each page - at the moment I put them in in roughly the right place with
\marginpar{
\begin{figure}
\includegraphics[angle=270,width=\marginparwidth]{images/diagram.pdf}
\caption{mycaption}
\label{mylabel}
\end{figure}
}
but when I add or remove text they get moved around unpleasantly - is there a way I can enforce with say \marginpar[page=1]?
You can not put a figure in a marginpar; your posted code would produce the error
! LaTeX Error: Not in outer par mode.
To include an image at the same point in each page you could use something like
\makeatletter
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All Questions
26 views
If you receive a message with a digital signature, how do you know what verification key to use and the verification algorithm?
Really, I'm asking how do you know who sent it? My confusion has arisen from the following scenario... Problems with Signing-then-encrypting. If Alice digitally signed some data and then encrypted ...
2k views
Randomness test question from FIPS 140-1 and comparison with 140-2
In FIPS 140-1 there are 4 statistical random number generator tests (The Monobit Test, The Poker Test, The Runs Test and The Long Runs Test. Then FIPS 140-2 came along and supposedly tightened the ...
207 views
Discrete logarithm problem is easy in a cyclic group of order a power of two
Let $G=\langle g\rangle$ be a cyclic group of order $2^{k}$ and let $h\in G$. I have read that it is easy to find $\log _{g} h$, but I haven't been able to figure out how. Do you know why this can be ...
5k views
Big-O Notation: Encryption Algorithms
I am currently completing a dissertation concerning the encryption of data through a variety of cryptographic algorithms. I have spent much time reading journals and papers but as yet have been ...
22 views
CCA on PKCS#5 using error feedback
My question is about attacking PKCS#5 with CCA using the error feedback. What we learnt in class about PKCS#5: CBC requires padding to whole block boundary PKCS#5: standard padding If ...
124 views
Big block cipher as memory-hard function
I'm wondering, if something like block cipher with big block size is a good memory-hard function? All memory-hard key derivation functions I've seen look more complex than that, which made me ...
91 views
Single public key for NaCL signing and encryption?
Bob wants to use NaCL (specifically the Python NaCL library) to both a) sign some publically available plaintext for anybody to authenticate as being written by Bob and b) encrypt some other secret ...
59 views
Difference of full codebook and non full codebook
While reading a paper about Dynamic cube attack, I saw two terms: full-codebook and non-full-codebook attacks. What do they mean? What is the difference? Regards
26k views
How big an RSA key is considered secure today?
I think 1024 bit RSA keys were considered secure ~5 years ago, but I assume that's not true anymore. Can 2048 or 4096 keys still be relied upon, or have we gained too much computing power in the ...
168 views
Is the result of HMAC-sha256 distinguishable from random noise?
Is if the result of HMAC-sha256 distinguishable from random noise? If so, given: a collection of 1000 keys, and 1000 messages of random noise, 100 signed by each of only 10 of those keys Can ...
39 views
Can error correcting codes be used to guess this plaintext?
I pick a finite set of messages, perhaps the set of integers $M = \lbrace n \in \mathbf{Z} \, : \, 0 \leq n\lt10 \rbrace$, which you are then allowed to encode in any way you want (some form of error ...
121 views
I'm studying mechanisms of integrity and authentication in symmetric encryption scenarios. I want to propose some examples to see whether I got the point here: Let $m$ be the message, $c$ the ...
31 views
50 views
93 views
What is the recommended format/notation for crypto design?
Is there a recommended format or formal notation for documenting the combination of symmetric and asymmetric encryption, key derivation and other algorithms (and their inputs and outputs) that I think ...
28 views
57 views
How to invert a linear hash function?
My friend is trying to convince me that his linear hash function (with a few bit operations) is secure. Is there a way to disprove this? (Or prove this, but I've read from multiple sources that this ...
47 views
Can an analog of ChaCha with 64-bit words be defined, and would it be secure?
Blake2b has a lightning fast compression function with more-than-overkill security even against quantum attacks. It seems to be based on ChaCha, but with 64-bit words and different rotation constants....
48 views
Which is the longest CSPRNG?
There are some PRNGs considered cryptographically secure. Which has the longest period among them?
52 views
How should I implement a secure recovery of encryption?
I want to create a system to host as securely as possible encrypted data in a way that not even the system can know the content of the data, but that it could be recovered. I would like to know how ...
160 views
Use Smart Card to decrypt files
So I'm trying to figure out the feasibility of using a Smart Card to decrypt files in an offline scenario. I wish to sync encrypted files to a mobile phone with a Smart Card reader case. The files ...
50 views
Could a strong round function be immune to slide attacks
An excerpt from the wikipedia article on slide attacks states: ...The only requirements for a slide attack to work on a cipher is that it can be broken down into multiple rounds of an identical F ...
47 views
Optimal threshold for passive and perfect security
The authors of the book titled "Secure Multiparty Computation and Secret Sharing" claim that there exist functions which cannot be computed with passive perfect security for $t \geq n/2$ corrupt ...
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# How do pullback and isothermal transformations of a Riemannian metric change curvature?
Let $$(X, g)$$ be a (closed) Riemannian surface (or more generally a Riemannian manifold of any dimension). How do the following changes to $$g$$ change its curvature?
1. Pullback by some diffeomorphism of $$X$$;
2. Multiplication by a scalar function, i.e., replacing $$g$$ with $$\lambda g$$ for some $$\mathbb R$$-valued function $$\lambda$$.
More precisely, can we use these two operations to produce a metric with constant curvature?
Curvature is invariant under diffeomorphism in the sense that for a diffeomorphism $$\phi$$, the Riemann curvature tensor satisfies $$(\phi^* R)_p = R_{\phi(p)} :$$ tautologically, pulling back by a diffeomorphism cannot change the curvature up to diffeomorphism.
On the other hand, the behavior of curvature under a conformal rescaling, $$g(x) \mapsto \hat g(x) := \lambda(x) g(x) , \qquad \lambda > 0 ,$$ is a rich topic connected with several classical problems.
In dimension $$2$$, the Uniformization Theorem implies that every Riemannian surface admits a metric of constant scalar curvature, and in this dimension, sectional curvature and scalar curvature are essentially the same thing. For compact surfaces, the Gauss-Bonnet Theorem implies that the sign of the scalar curvature of such a metric is determined entirely by the genus.
In dimensions $$\dim M \geq 3$$ the notion of curvature is more complicated, and in particular constant scalar curvature does not imply constant sectional curvature. Now, curvature $$R$$ changes under conformal rescalings $$g \mapsto \hat g$$---this is no surprise, since curvature depends on derivatives of the metric---a straightforward calculation shows that the totally tracefree part $$W$$ of $$R$$, called the Weyl tensor (viewed as a $$(1, 3)$$-tensor) does not.
Any space with constant sectional curvature turns out to be (locally) conformally flat, that is, locally equivalent under a conformal transformation to the Euclidean metric, which has $$R = 0$$ and thus $$W = 0$$. So, if there is a conformal factor $$\lambda$$ such that $$\hat g = \lambda g$$ has constant sectional curvature, it must have been the case that the Weyl tensor $$W$$ of the original metric $$g$$ is zero. For $$\dim M \geq 4$$ this is a very restrictive condition, and general metrics do not satisfy it. (For $$\dim M = 3$$, the story is similar but we always have $$W = 0$$, even when the metric is not conformally flat; in this case, conformal flatness is instead characterized by the vanishing of another tensor, the Cotton(-York) tensor.)
Finally, we can ask whether for any metric $$g$$ there is always a $$\lambda$$ such that $$\hat g = \lambda g$$ has constant scalar curvature. We saw above that for $$\dim M = 2$$ the answer is yes. If we restrict attention to compact manifolds of $$\dim M \geq 3$$, we're posing the famous Yamabe Problem. The answer is yes, but the proof is nontrivial, and it was only achieved in 1984.
• Thank you for your detailed answer! I did not make myself clear - actually by scalar I mean a scalar function, precisely as you explained in the remark - and this is exactly what I want. – User X Aug 22 at 9:25
• So, is this true for closed surfaces? – User X Aug 22 at 9:30
• You're welcome, I'm glad you found it useful. If you really mean that $\lambda$ is a scalar function, you should modify the text of the question. The notation $\lambda \in \Bbb R$ implies that $\lambda$ is a constant. – Travis Willse Aug 22 at 15:11
• And yes, it is true for surfaces, both closed and otherwise. This is an immediate consequence of the Uniformization Theorem, which is a relatively old result by differential geometry standards (it dates to the first decade of the 20th C., IIRC). I've modified my answer to include mention of it. – Travis Willse Aug 22 at 15:14
• Yeah I have edited the text. Thank you again! – User X Aug 23 at 9:46
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3 Quick Ways to Create a Markdown Link from Mail, Safari, and Finder
How's that for a clickbaity title!
(None of the demonstration links below should work. Their names have been changed to protect the innocent.)
I sometimes start a zettel from an idea sparked by an email. I've struggled with citing the email without importing it into Zotero, my reference manager. I recently stumbled on a solution. Using Applescript and Keyboard Maestro, I can now highlight a mail message, execute the trigger, and voila. A properly formed markdown link is placed in the system copy buffer in the appropriate zettel.
They look in markdown like:
[Listen To Your Listening](message://%[email protected]%3e)
In preview mode or as exported, they look like:
Macro in action — not much to see.
This is the resulting link shown in the GIF below.
[Zettelkasten Forum] taurusnoises commented on The fleeting note in "How to Take Smart Notes" by Sönke Ahrens
Here is the Keyboard Maestro macro.
I've been pasting references for web pages like this one. It is an edge case but demonstrates two things. URLs alone provide little or no context, and the web admins at Patreon are a bit anal with a hash that is 200 characters long. Indeed this is a bit excessive?
1. https://www.patreon.com/posts/armories-for-5943r2?utm_medium=&utm_campaign=patron_engagement& ;token=eyJ0eXAiOiJKV1QiLCJhbGciOiJIUzI1NiJ9. eyJyZWRp238ZXkiOiJpbnN0YW50LWFjY2VzczozYUyOS1kZjE4LTQ4ZjYtYjQ4Yy0wZTg1MDBiZjAwOGYifQ u6kzcdewmkeGVIML51dXvc0__eeTnDNZWZuM6hRvXUE.
Demonstrated below is how using some Keyboard Maestro pixy dust, when viewing the page, I execute the trigger, and a preformed Markdown link is placed in the system copy buffer to be placed in the appropriate zettel. As a bonus, the link gives far more context without my intervention. Both 1 and 2 are the same web link.
Macro in action — not much to see.
What are you working on this week? (20220130-20220205 Week 5) — Zettelkasten Forum
Here is the Keyboard Maestro macro.
This last link creating tool is an oldie. It works in Finder, copying the highlighted file to the media/ directory and creating a proper markdown link in the system copy buffer to be placed in the appropriate zettel.
Macro in action — not much to see.
Here is the Keyboard Maestro macro.
—If you read this far, I have a bonus tip. Put these macros in separate Keyboard Maestro groups and set the "Available in these apps" so the trigger only works in the specified app. This way, you can use the same trigger for all three 'Create MMD Links form….' It makes remembering which keyboard trigger works for making links!
Will Simpson
“Read Poetry, Listen to Good Music, and Get Exercise”
kestrelcreek.com
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# Use of Nets (topological) in Rigged Hilbert Space
I have been reading about Rigged Hilbert Spaces (AKA Gel'Fand Triple). This is characterized by the relationship $\Phi \subset H \subset \Phi'$, where $\Phi$ is Schwartz space, $H$ is Hilbert Space, or $L^2$, and $\Phi'$ is the conjugate to the Schwartz space. $\Phi$ is the space of wave functions (or state vectors) of Quantum Mechanics. $\Phi'$ is the space of the functionals on $\Phi$ and contains the Dirac delta function $\delta (x)$ and plane wave function $e^{ipx}$, which are the eigenfunctions of the position and momentum operators of Quantum Mechanics. Of note, $\Phi'$ does not satisfy the first axiom of countability.
So, it would seem that the concept of Nets (https://en.wikipedia.org/wiki/Net_(mathematics), or something more general than sequences, would be applicable to analyzing $\Phi'$, but I have never seen that in the literature.
Does anyone have any pointers to nets, or something more general than sequences, being used for defining convergence in $\Phi'$, in Rigged Hilbert Space?
Thanks
• According to Gel'Fand, "Generalized Functions", V2, P 57 (paraphrasing): A sequence in $\Phi'_p$ converges if all the elements of the sequence are in one of the fixed $\Phi'_p$, and they converge in the norm of $\Phi'_p$, where $\Phi'_0 \subset...\subset \Phi'_p \subset...\subset \Phi'$. It seems restrictive to only talk about convergence of sequences where all the elements of the converging sequence have to be in the same $\Phi'_p$ subspace. Maybe if we talk about nets, instead of sequences, all the elements of the net do not have to be in the same $\Phi'_p$ and can still converge? – David Jan 12 '17 at 21:30
• The above comment should begin "A sequence in $\Phi'$ converges if all the elements of the sequence...". Given the above comment (and its correction just stated), it seems invalid to talk about the convergence of a sequence of delta functions in $\Phi'$, since all $\Phi'_p$ are normed and so the delta function could not be in any of the $\Phi'_p$, as it is not part of any Hilbert space. But, it seems, to me, like one should be able to talk about the convergence of a sequence of delta functions. Hence, I am confused and wondering if introducing things like nets in $\Phi'$ would help. – David Jan 14 '17 at 9:32
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Volume 371 - 7th Annual Conference on High Energy Astrophysics in Southern Africa (HEASA2019) - Modelling I (Chair: Markus Böttcher)
Effects of the gamma-ray absorption in the broadline region and secondary cascade emission on blazar spectra
M. Ntshatsha
Full text: Not available
Abstract
Blazars are a class of active galactic nuclei (AGN) characterized by large luminosity that varies on relatively short timescales. Using a generic emission model for the broad line region (BLR) we calculate the interaction of gamma-rays in the jet with BLR photons, and the distribution of the secondary electron-positron (e$^\pm$) pairs produced. We also calculate the synchrotron and inverse Compton emission by the e$^\pm$ pairs, which constitute a source of secondary gamma-ray emission. We fit the spectral energy distributions (SEDs) of some bright \textit{Fermi}-LAT blazars, and study the effect of these considerations on the observed gamma-ray flux of blazars.
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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# Pegasus Tutorial¶
Author: Yiming Yang
Date: 2021-06-24
Notebook Source: pegasus_analysis.ipynb
## Count Matrix File¶
For this tutorial, we provide a count matrix dataset on Human Bone Marrow with 8 donors stored in zarr format (with file extension ".zarr.zip").
This file is achieved by aggregating gene-count matrices of the 8 10X channels using PegasusIO, and further filtering out cells with fewer than $100$ genes expressed. Please see here for how to do it interactively.
Now load the file using pegasus read_input function:
The count matrix is managed as a UnimodalData object defined in PegasusIO module, and users can manipulate the data from top level via MultimodalData structure, which can contain multiple UnimodalData objects as members.
For this example, as show above, data is a MultimodalData object, with only one UnimodalData member of key "GRCh38-rna", which is its default UnimodalData. Any operation on data will be applied to this default UnimodalData object.
UnimodalData has the following structure:
It has 6 major parts:
• Raw count matrix: data.X, a Scipy sparse matrix (sometimes can be a dense matrix), with rows the cell barcodes, columns the genes/features:
This dataset contains $48,219$ barcodes and $36,601$ genes.
• Cell barcode attributes: data.obs, a Pandas data frame with barcode as the index. For now, there is only one attribute "Channel" referring to the donor from which the cell comes from:
• Gene attributes: data.var, also a Pandas data frame, with gene name as the index. For now, it only has one attribute "gene_ids" referring to the unique gene ID in the experiment:
• Unstructured information: data.uns, a Python hashed dictionary. It usually stores information not restricted to barcodes or features, but about the whole dataset, such as its genome reference and modality type:
• Finally, embedding attributes on cell barcodes: data.obsm; as well as on genes, data.varm. We'll see it in later sections.
## Preprocessing¶
### Filtration¶
The first step in preprocessing is to perform the quality control analysis, and remove cells and genes of low quality.
We can generate QC metrics using the following method with default settings:
The metrics considered are:
• Number of genes: Keep cells with $500 \leq \text{# Genes} < 6000$. **(Notice that starting from v1.4.0, Pegasus doesn't filter cells by number of genes threshold by default.)**
• Number of UMIs: Don't filter cells due to UMI bounds (Default);
• Percent of Mitochondrial genes: Look for mitochondrial genes with name prefix MT- **(Notice that starting from v1.4.0, you'll need to manually specify this prefix.)**, and keep cells with percent $< 10\%$.
For details on customizing your own thresholds, see documentation.
Numeric summaries on filtration on cell barcodes and genes can be achieved by get_filter_stats method:
The results is a Pandas data frame on samples.
You can also check the QC stats via plots. Below is on number of genes:
Then on number of UMIs:
On number of percentage of mitochondrial genes:
Now filter cells based on QC metrics set in qc_metrics:
You can see that $35,465$ cells ($73.55\%$) are kept.
Moreover, for genes, only those with no cell expression are removed. After that, we identify robust genes for downstream analysis:
The metric is the following:
• Gene is expressed in at least $0.05\%$ of cells, i.e. among every 6000 cells, there are at least 3 cells expressing this gene.
Please see its documentation for details.
As a result, $25,653$ ($70.09\%$) genes are kept. Among them, $17,516$ are robust.
We can now view the cells of each sample after filtration:
### Normalization and Logarithmic Transformation¶
After filtration, we need to first normalize the distribution of counts w.r.t. each cell to have the same sum (default is $10^5$, see documentation), and then transform into logarithmic space by $log(x + 1)$ to avoid number explosion:
For the downstream analysis, we may need to make a copy of the count matrix, in case of coming back to this step and redo the analysis:
### Highly Variable Gene Selection¶
Highly Variable Genes (HVG) are more likely to convey information discriminating different cell types and states. Thus, rather than considering all genes, people usually focus on selected HVGs for downstream analyses.
You need to set consider_batch flag to consider or not consider batch effect. At this step, set it to False:
By default, we select 2000 HVGs using the pegasus selection method. Alternative, you can also choose the traditional method that both Seurat and SCANPY use, by setting flavor='Seurat'. See documentation for details.
We can view HVGs by ranking them from top:
We can also view HVGs in a scatterplot:
In this plot, each point stands for one gene. Blue points are selected to be HVGs, which account for the majority of variation of the dataset.
### Principal Component Analysis¶
To reduce the dimension of data, Principal Component Analysis (PCA) is widely used. Briefly speaking, PCA transforms the data from original dimensions into a new set of Principal Components (PC) of a much smaller size. In the transformed data, dimension is reduced, while PCs still cover a majority of the variation of data. Moreover, the new dimensions (i.e. PCs) are independent with each other.
pegasus uses the following method to perform PCA:
By default, pca uses:
• Before PCA, scale the data to standard Normal distribution $N(0, 1)$, and truncate them with max value $10$;
• Number of PCs to compute: 50;
• Apply PCA only to highly variable features.
See its documentation for customization.
To explain the meaning of PCs, let's look at the first PC (denoted as $PC_1$), which covers the most of variation:
This is an array of 2000 elements, each of which is a coefficient corresponding to one HVG.
With the HVGs as the following:
$PC_1$ is computed by
\begin{equation*} PC_1 = \text{coord_pc1}[0] \cdot \text{HES4} + \text{coord_pc1}[1] \cdot \text{ISG15} + \text{coord_pc1}[2] \cdot \text{TNFRSF18} + \cdots + \text{coord_pc1}[1997] \cdot \text{RPS4Y2} + \text{coord_pc1}[1998] \cdot \text{MT-CO1} + \text{coord_pc1}[1999] \cdot \text{MT-CO3} \end{equation*}
Therefore, all the 50 PCs are the linear combinations of the 2000 HVGs.
The calculated PCA count matrix is stored in the obsm field, which is the first embedding object we have
For each of the $35,465$ cells, its count is now w.r.t. 50 PCs, instead of 2000 HVGs.
## Nearest Neighbors¶
All the downstream analysis, including clustering and visualization, needs to construct a k-Nearest-Neighbor (kNN) graph on cells. We can build such a graph using neighbors method:
It uses the default setting:
• For each cell, calculate its 100 nearest neighbors;
• Use PCA matrix for calculation;
• Use L2 distance as the metric;
• Use hnswlib search algorithm to calculate the approximated nearest neighbors in a really short time.
See its documentation for customization.
Below is the result:
Each row corresponds to one cell, listing its neighbors (not including itself) from nearest to farthest. data_trial.uns['pca_knn_indices'] stores their indices, and data_trial.uns['pca_knn_distances'] stores distances.
## Clustering and Visualization¶
Now we are ready to cluster the data for cell type detection. pegasus provides 4 clustering algorithms to use:
• louvain: Louvain algorithm, using louvain package.
• leiden: Leiden algorithm, using leidenalg package.
• spectral_louvain: Spectral Louvain algorithm, which requires Diffusion Map.
• spectral_leiden: Spectral Leiden algorithm, which requires Diffusion Map.
See this documentation for details.
In this tutorial, we use the Louvain algorithm:
As a result, Louvain algorithm finds 19 clusters:
We can check each cluster's composition regarding donors via a composition plot:
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# Perlin Noise Corona
This topic is 3612 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
I've been playing with Perlin noise for the last few weeks, and my latest experiment is the corona effect described in Perlin's "Making Noise" presentation. Generating the static image was pretty easy. Unfortunately, converting this static image to a looping animation seems a bit more difficult. There are two things I need to do: 1) Use seamless looping 3D noise for an infinite random fluctuation. 2) Scale x and y away from center of image as z increases. Seamless looping shouldn't be hard, once I figure out the basics. However, how will seamless looping work when scaling x and y? Do I somehow need to have the values "wrap" back to the center after they "exit" the circle? - Mike
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In case it helps, here is what I did to generate the static image:
Quote:
1. Create a smooth gradient function that drops off radially from bright yellow to dark red.
# dist is the distance from the center of the image (between 0.0 to 1.0)
red = 1.0 - 1.0 * math.pow(dist, 5)
green = 1.0 - 2.0 * math.pow(dist, 3)
blue = 0
Quote:
2. Phase shift this function by adding a turbulence texture to its domain.3. Place a black cutout disk over the image.
I assumed that step 2 basically means blending 2D noise and the gradient circle from step 1, and that seemed to give me a nice result.
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I solved my looping problem, and implemented X/Y scaling. Unfortunately, the scaling doesn't really do what I expected. Here's the animation (speed reduced):
Basically, I'm scaling the noise map by adjusting the x/y value that is passed to the noise function.
pseudo code:
iterate through all pixels in the 256x256 image: get distance from center of image to the current pixel calculate "movement" of current frame (frame 10 of 20 would have a movement of 0.5) move x & y closer to the center of the image by subtracting the movement factor from the pixel's distance to the center (wrap around to outer edge if this value is < 0) call the noise function for the new x & y values, but set the pixel color at the original x & y coordinates
As you can see, the image does wrap around from the outside of the image back to the center. Unfortunately, the noise that wraps around from the outer edge is far more detailed than the noise that starts in the middle of the image. I think that is what is causing the very sharp seam in the animation.
So, any thoughts on how to eliminate the seam?
[Edited by - doctorsixstring on March 29, 2008 5:29:14 PM]
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Is the Graphics Programming and Theory forum the right place to be posting about Perlin Noise? I've started similar threads over in General Programming and Game Programming, and both had quite a few useful responses. I decided to post in this forum because of a slightly older thread that got plenty of responses, and because the topic is covered in the Forum FAQ.
I am in no way belittling this forum, but is it really the best place for these questions? Or, is Perlin Noise not a popular topic anymore? Or am I just asking really hard or obscure questions?
- Mike
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# [NTG-context] Question about defining a math macro
Aditya Mahajan adityam at umich.edu
Sat Feb 24 09:55:30 CET 2007
Hi,
Can I define a math macro, say \EXP that does the following:
1) \EXP{something} gives me, say E(something)
2) \EXP_a^b{something} gives my E_a^b(something)
That is, I want to write a macro that will only be used in the math
mode, that does something with its argument. But, I also want it to
handle _ and ^ if they are present.
The only method that I can think of is to check _ or ^ with
\ifnextcharacter and go through all 5 cases (nothing, only _, only ^,
_ followed by ^, ^ followed by _) but this seems very inefficient. Is
there a better way?
Aditya
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# How do you find x and y intercepts of -5x + 6y = 30?
Apr 8, 2017
See the entire solution process below:
#### Explanation:
To find the y-intercept, set $x = 0$ and solve for $y$:
$- 5 x + 6 y = 30$ becomes:
$\left(- 5 \times 0\right) + 6 y = 30$
$0 + 6 y = 30$
$6 y = 30$
$\frac{6 y}{\textcolor{red}{6}} = \frac{30}{\textcolor{red}{6}}$
$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{6}}} y}{\cancel{\textcolor{red}{6}}} = 5$
$y = 5$
The y-intercept is $y = 5$ or $\left(0 , 5\right)$
To find the x-intercept, set $y = 0$ and solve for $x$:
$- 5 x + 6 y = 30$ becomes:
$- 5 x + \left(6 \times 0\right) = 30$
$- 5 x + 0 = 30$
$- 5 x = 30$
$\frac{- 5 x}{\textcolor{red}{- 5}} = \frac{30}{\textcolor{red}{- 5}}$
$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 5}}} x}{\cancel{\textcolor{red}{- 5}}} = - 6$
$x = - 6$
The x-intercept is $x = - 6$ or $\left(- 6 , 0\right)$
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Thread: LaTex figure word wrapping View Single Post
P: 415 Hi all, I've inserted a figure into a document I'm working on, but I can't seem to get the words to wrap around the figure. I have a large white spot on the left side of the page, thats just begging for some text. I've looked in the two guides I have and neither say anything about this can anyone help??? I'm using the following bit to add the figure. \begin{figure}[h] \captionstyle{flushleft} \onelinecaptionsfalse \includegraphics[width=2.5in,height=2.5in]{stru.eps} \caption{xxxx structure.} \label{fig:xxx} \end{figure} Thanks JMD
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