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# Blender 2.78 Manual for offline use - how - where?

Blender Help says I can download the Blender 2.78 Manual for offline use - how - where?

• blender (which I use, love and support since two decades...) has a terrible history of downloadable manuals... worst blender aspect ever, imho. when it was on mediawiki I started a project to regularly create a pdf version from it... then they switched to a version based on "reStructuredText " and although every project I've seen using this kind of system (see readthedocs.org) has html, pdf, epub export at least... blender never had one. You could try to download manual sources and build it locally: see docs.blender.org/manual/en/dev/about/contribute/index.html – m.ardito Feb 9 '17 at 22:40

There is a link for the online documents here: https://docs.blender.org/manual/en/dev . I'm not sure why it doesn't work, but there is a link to download a zip file, but the actually link tag isn't working in the wiki. For now, you can download directly, I guess: https://docs.blender.org/manual/en/dev/blender_manual.zip

The make.bat file in the blender documentation subversion repository includes a "make epub" option that will use their tooling to build a local epub file locally. It's reasonably easy to do this same thing on Mac or Linux box if that's your platform.

A step-by-step of what I did was:

• svn checkout https://svn.blender.org/svnroot/bf-manual/trunk/blender_docs
• cd blender_docs
• virtualenv .venv_blender
• source .venv_blender/bin/activate
• pip install -r requirements.txt

and I added this to the local Makefile:

epub: .FORCE .SPHINXBUILD_EXISTS
# './' (input), './epub/' (output)
QUICKY_CHAPTERS=$(QUICKY_CHAPTERS) \$(SPHINXBUILD) -b epub $(SPHINXOPTS) ./manual "$(BUILDDIR)/epub"

Then make epub generates an epub file into build/epub directory.

## protected by Community♦Sep 17 at 0:02

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# Insertion Sort Implementation Using For And While Loop

### Introduction

I'm fairly new to programming (and Java) and I've actually never implemented Insertion Sort.

I know that there's pseudo-code defined in the wikipedia article (linked above), but the goal was to implement the algorithm without looking at the pseudo-code and only using the wiki article's graphic for reference.

### Concerns

• After I completed my implementation, I looked at a couple implementations on CodeReview.

### Implementation

public class InsertionSorter {
public static int[] sort(int[] numbers) {
int[] sorted = numbers.clone();
for (int i = 1; i < sorted.length; i++) {
int j = i - 1;
int numberToInsert = sorted[i];
while (j >= 0 && numberToInsert < sorted[j]) {
sorted[j + 1] = sorted[j];
sorted[j] = numberToInsert;
j--;
}
}
return sorted;
}
}

### Thoughts on best practices

Method signature should be void and should simply return the modified original array. Again, is this behavior better / more-expected? My argument against this is that maybe it could lead to some unexpected behavior if the user does not expected the original array to be modified? Or unexpected behavior down the road if the user forgets that the returned array has been modified. That being said, with proper documentation, this could be avoided.

This is a variant of the Single Responsibility Principle. Your method is supposed to sort something. Should it also copy something? Or only sort? Note that there is no way to sort in place with your version (except to copy back over the original array). But with a sort-in-place version, you can easily duplicate your method's functionality:

int[] sorted = unsorted.clone();
InPlaceSorter.sort(sorted);

Now we have unsorted, which is the original array, and sorted, which is a new sorted version of the original array.

Documentation and the fact that saying

int[] sorted = InPlaceSorter.sort(unsorted);

will produce a compiler error should be enough to make clear that the array is being modified.

By contrast, calling

InsertionSorter.sort(unsorted);

without assigning the result will silently not do anything useful. So I'd find your version riskier, as it doesn't warn when someone tries to do the wrong thing.

If you do stick to your original version, consider changing the name to make it clearer. E.g. generateSortedCopyOf. If I read that, I expect it to return a copy without modifying the original. And I wouldn't try calling it to do a sort-in-place.

One uses double for-loops vs. a for-loop and a while-loop. Is the double for-loop more readable?

Personally, I don't have a strong feeling about this one way or the other. All for loops can be written as while loops and sometimes I find it more convenient.

      int j = i - 1;
int numberToInsert = sorted[i];
while (j >= 0 && numberToInsert < sorted[j]) {
sorted[j + 1] = sorted[j];
sorted[j] = numberToInsert;
j--;
}

But this does translate nicely to a for loop.

      int numberToInsert = sorted[i];
for (int j = i - 1; j >= 0 && numberToInsert < sorted[j]; j--) {
sorted[j + 1] = sorted[j];
sorted[j] = numberToInsert;
}

No gymnastics required to fit it into the for pattern. I think that I'd vote for this solution as simpler in this case. But see ahead.

### Optimizations

This solution does more assignments than needed. Consider

      int j = i - 1;
int numberToInsert = sorted[i];
for (; j >= 0 && numberToInsert < sorted[j]; j--) {
sorted[j + 1] = sorted[j];
}
j++;

sorted[j] = numberToInsert;

Now it's not nearly as obvious that this should be a for loop.

      int j = i - 1;
int numberToInsert = sorted[i];
while (j >= 0 && numberToInsert < sorted[j]) {
sorted[j + 1] = sorted[j];
j--;
}
j++;

sorted[j] = numberToInsert;

Either way though, this could save you a bunch of assignments at the expense of always doing an assignment (we can check that i != j if we want to avoid that useless assignment). But looking at this, we have another option:

      int j = i - 1;
int numberToInsert = sorted[i];
while (j >= 0 && numberToInsert < sorted[j]) {
j--;
}
j++;

if (j != i) {
System.arraycopy(sorted, j, sorted, j + 1, i - j);
sorted[j] = numberToInsert;
}

Now we allow the Java compiler to do any optimizations that System.arraycopy allows. At worst this should be the same as the original.

And we might do even better if we take advantage of the sorted portion of the array.

      int numberToInsert = sorted[i];
int j = Math.abs(1 + Arrays.binarySearch(sorted, 0, i - 1, numberToInsert));

if (j != i) {
System.arraycopy(sorted, j, sorted, j + 1, i - j);
sorted[j] = numberToInsert;
}

Of course, one of the benefits of an insertion sort is its simplicity. This replaces two things we originally did in a single loop with two system calls. That may not be faster in some cases. In particular, consider the case of an already sorted list. The original algorithm runs in $O(n)$ time in that case, as the inner loop simply doesn't iterate. This version would run in $O(n \log n)$ because binarySearch is a logarithmic operation.

Note: adding one to the binarySearch result and taking the absolute value always gives the insertion point. This is because a positive result is the current location of the element, and we want to insert after the current location in case of duplicates (technically we want to insert after the last duplicate, but I didn't add that logic); and a negative result is one less than the insertion point.

Another issue is that the original version might parallelize better. One core can do comparisons while the other core writes values. Or maybe the compiler optimizes binarySearch and arraycopy better. You'd have to test to see, and you might get different results on different platforms.

format: plain


text:
# Testing small values

What is the smallest positive integer $n$ such that $\gcd( 2n-33, 5n + 73 ) \neq 1?$

×


text:
# Questions Asked onMarch 18, 2013

1. ## ....

hans uses a wheelchair. although he has a ramp to get in his house he finds it difficult to get his wheelchair up the ramp. which modification would make it easier for hans to get in his house? a. increase the length of the ram b. increase friction between

2. ## ....

in the digestive tract, mouth is to bolus as a. stomach is to chyme b. gallbladder is to insulin c. small intestine is t carbohydrates d. esophagus is to peristalsis

3. ## Cultural Anthropology

In a band society, who would most likely be responsible for punishing a murderer

4. ## science

in the digestive tract, mouth is bolus to a. stomach is to chyme b. gallbladder is to insulin c. small intestine is to carbohydrate d. esophagus is to peristalsis

5. ## Chemistry

Calculate the pH at the equivalence point for the titration of 0.130M methalamine (CH3NH2) with 0.13M Hcl. The Kb of methyl amine is 5.0x10-4.

6. ## Chemistry (reposting)

In which of the following solution would CaC2O4 have the highest molar solubility? Please help. A) 0.01 M NaCl B) 0.01 M H2C2O4 C) 0.01 M Na2C2O4 D) 0.01 M NaHC2O4 E) 0.01 M HCl I think it is E because it is a strong acid and will completely dissociate

7. ## Quantum Physics

a) What is QFT6 of 1/√2(|0>+|3>)? b) What is QFT6 of 1/√2(|1>+|4>)? c) What is QFT6 of 1/√3(|0>+|2>+|4>)? d) What is QFT6 of 1/√3(|1>+|3>+|5>)? Answer like: a) |0> = ; |1> = ; |2> = ; |3> = ; |4> = ; |5> = ...

8. ## science

a man wishes to swin across a river 600 m wide. If can swim at the rate of 4 km/hr in still water and the river flows at 2 km/hr. Then in what direction must he swim to reach a point exactly opposite to the starting point bnd when will he reach it ?

9. ## statistics

The time to fly between New York City and Chicago is uniformly distributed with a minimum of 120 minutes and a maximum of 150 minutes. What is the standard deviation?

10. ## Chemistry

Calculate the molar concentration of uncomplexed Zn2+ (aq) in a solution that contains 0.22 mol of Zn(NH3)4 2+ per liter and 0.3109 M NH3 at equilibrium. Kf for Zn(NH3)4 2+ is 2.9 X 10^9 I started this way... (0.3109 + 4)2/0.22-x)= 2.9 X 10^9

11. ## maths-surface area

A cone 7dm high and 6 dm in diameter is placed on the ground. Sand is poured over it until conical heap 21 dm high and 33 dm circumference at the bottom is formed. Find how many m^3 of sand is there?

12. ## physics

A spherical bowling ball with mass m = 4.1 kg and radius R = 0.116 m is thrown down the lane with an initial speed of v = 8.9 m/s. The coefficient of kinetic friction between the sliding ball and the ground is ¦Ì = 0.34. Once the ball begins to roll

13. ## math

a right triangle has an acute angle(theta) suck that cot(theta)=15. find cos(theta)

14. ## alg 1

the length of a rectangular garden is 8 feet longer than the width. The garden is surrounded by a 4-foot sidewalk. the sidewalk has an area of 320 square feet. find the dimensions of the garden. please help

15. ## help?

find the altitude of an equilateral triangle if a side is 6 mm long.

16. ## English

The team ______ won the game is my favorite. which word correctly completes the sentence above? a. which b.whom c. who d. that

17. ## stats.

The GMAT test is required for admission to most graduate programs in business. In a recent year, the GMAT test scores were normally distributed with a mean of 550 and standard deviation of 100. A. Find the first quartile for the distribution of GMAT

18. ## Chemistry

A reaction mixture of 4.0 mL of 0.002 M SCN- and 5.0 mL of 0.002 M Fe3+ is diluted to 10.0 mL with 0.1 M HNO3 to form the blood-red FeNCS2+ complex. The equilibrium molar concentration of the FeNCS2+ determined from a standardization curve is 1.5x10^-4

19. ## math

A community college has 2,000 students and 50 instructors. Next year the enrollment will be 6,000. How many new instructors should be hired if the college wants to keep the same studentto instructor ratio?

20. ## Algebra

You roll a pair of number cubes and then flip a coin. Is this an example of independent or dependent events? A.independent B.dependent This is A...right?

21. ## math

a species of orchid is being studied. The population 'p' at time 't' years after the study started is assumed to be p = 2800ae^0.2t/1 + ae^0.2t where 'a' is a constant. Given tnat there were 300 orchids when the study started. a. show that a =0.12 b. use

22. ## English

hans uses a wheelchair. although he has a ramp to get in his house, he finds it difficult to get his wheelchair chair up the ramp. which modification would make it easier for hans to get in his house? a. increase the length of the ramp b. increase the

23. ## math

How do I do this? Without a calculator, determine two angles between 0 degrees and 360 degrees that have a cosine of negative square root 3 over 2.

24. ## math/ratios

There are 77 students in the student council. The ratio of girls to boys is 7:4. How many girls are in the student council?

26. ## math

A bookshop had 60 boxes of pencils. Each box contained 85 pencils. Half of the pencils were packed into packets of 25 pencils and the rest were packed into packets of 20. How many packets of pencils were there altogether? My answer is 229 am i right? How

27. ## maths-surface area

Cylindrical groove is made in a cylindrical piece of wood of length 17.5cm. The groove is filled with graphite and the cylindrical graphite block is of same length as that of wooden piece. If the diameter of the graphite cylinder is 3mm and outer diameter

28. ## algebra

an object is launched vertically in the air at 29.4 meters per second from 10 meter tall platform . find its maximum height in seconds and in meters ?

29. ## Chemistry

Solutions of sodium thiosulfate are used to dissolve unexposed AgBr (Ksp = 5.0 10-13) in the developing process for black-and-white film. What mass of AgBr can dissolve in 1.40 L of 0.425 M Na2S2O3? Ag+ reacts with S2O32- to form a complex ion. Ag+(aq) + 2

30. ## Chemistry

How many chlorine atoms are contained in a 1.00 g sample of iron(III) chloride, FeCl3? Use the masses to two decimal places from the chart in the front of your text. Use the format 6.02E23 for 6.02 x 1023, or 6.02E-23 for 6.02 x 10-23,use two decimal

whatever you can do or dream you can, begin it. Boldness has genius, power and magic in it !" describe in 25 word what the author is saying.

32. ## Stats.

The average age of a vehicle registered in the United States is 8 years, or 96 months. Assume the standard deviation is 16 months. If a random sample of a group of 48 vehicles is selected find the probability that the mean of the group’s age is between

33. ## physics

A thin spherical shell of mass M = 2.00 kg is released from rest at the top of an incline of height H=1.31 m and rolls without slipping to the bottom. The ramp is at an angle of θ = 24.1o to the horizontal. Calculate the speed of the sphere's CM at the

34. ## Cultural Anthrolpogy

Anthropological research has demonstrated that supernatural beings and forces exist in what percentage of the world's cultures? A. 50 percent B. 80 percent C. 95 percent D. 100 percent My answer: C

35. ## Ýndustrial Engineering

The figure shows the paths of a golf ball your friend drops from the window of her apartment and of the rock you throw from the ground at the same instant. The rock and the ball collide at x = 41.00 m, y = 17.02 m, and t = 2.94 s. If the ball was dropped

36. ## Accounting 202

Tse does not want to give Towry this budget without making constructive suggestions for steps Towry could take to improve expected performance. Write a memo to Towry outlining your suggestions.

37. ## geometry

find the perimeter of the rectangle if the area is 6m+3m

38. ## Physics

A 10 g bullet is fired horizontally into a 1 kg block of wood hanging from a 1 m long string attached to the ceiling, getting stuck in the block. Due to the impact, the block then swings on the string to a maximum angle of 60 degrees with respect to the

78. ## chemistry

Rank the following 3 compounds in terms of increasing boiling point: CCl4, CH4, CH2Cl2 - Rank the following 3 compounds in terms of increasing boiling point: CF4, CH4, CH2F2 - Water, H2O, is a liquid at room temperature. Hydrogen selenide, H2Se, is a

79. ## Geometry

ABC is a right angled triangle with ∠ABC=90∘ and side lengths AB=24 and BC=7. A semicircle is inscribed in ABC, such that the diameter is on AC and it is tangent to AB and BC. If the radius of the semicircle is an improper fraction of the form a/b,

80. ## Combinations Maths

25 independent, fair coins are tossed in a row. What is the expected number of consecutive HH pairs? If 6 coin tosses in a row give HHTHHH, the number of consecutive HH pairs is 3.

81. ## Chemistry

I could use some help please. Ammmonia, NH3, can be made by reacting Nitrogen with Hydrogen. N2+3H2=2NH3 How many moles of NH3 can be made if 7.5 moles of H2 react eith enough N2? Thank-you in advance for any help

82. ## Chemistry

Question 2 Marks: 2 How many grams of oxygen gas must react to give 174.3 grams of ZnO according to the balanced equation. 2 Zn(s) + O2(aq) --> 2 ZnO(s)

83. ## math

The total capacity of 2 water bottles, A and B, is 680 ml. The ratio of bottle A's capacity to bottle B's capacity is 4:6. How much larger is bottle B's capacity than bottle A's? My answer is 136, am i right?

84. ## Chemistry

How many grams of silver nitrate must react to give 67.5 grams of Ag according to the balanced equation. Cu(s) + 2 AgNO3(aq) --> Cu(NO3)2(aq) + 2 Ag(s)

85. ## World History (Ms. Sue)

Why was the immediate impact of the emancipation proclamation limited? My answer: The impact was limited because most of the slaves that Lincoln intended to liberate lived in areas distant from the Union troops that could enforce his proclamation?

86. ## Chemistry

How many grams of zinc must react to give 13.0 grams of ZnO according to the balanced equation. 2 Zn(s) + O2(aq) --> 2 ZnO(s)

87. ## math

The time is 5:15. Is the hour hand closer to the 5 or the 6? Why?

88. ## Chemistry

Question 9 Marks: 2 In the reaction: Zn + 2 HCl ZnCl2 + H2 An excess of Zn is reacted with 300.0 g of HCl. How many grams of H2 will be produced?

89. ## Math

Frank has 3 nickels, 5 dimes, and 2 quarters. What is the range, mean, median, and mode of the values of Frank's coins?

90. ## Chemistry

why is it important that the equations be balanced when performing mass-mass stoichiometry problems?

91. ## Mathematic

Haw many meters are in 2.7 kilometers

92. ## Chemistry

What is the mass of iron (g) produced from 227.4 of iron(III) oxide ?

93. ## POLITICAL SCIENCE

describe the European origins of American conservative and liberal ideologies. Then, explain how these two American ideologies have evolved from their European origins, and the key features of each American ideology today. Do “Tea Party” beliefs neatly

94. ## Mathematics

For what values of m and n Does the expression ( mx^4 + nx^3 + 14x^2 + 9x + 2) divided by ( x^2 + 3x + 1 ) leaves a remainder , (-11x - 4 ) .. ? I divided it , nut the whole m and n remainder thingy made me so confused !!!

95. ## Rhts

If 13sinx+5=0 and x[0;270],determine witout the use of a calculator,the value of sin2x

96. ## LAN

1. System A consists of a single ring with 100 stations, one per repeater. System B consists of four 25 stations rings linked by a bridge. If the probability of a link failure is , a repeater failure is , and a bridge failure is , derive an expression for

97. ## algebra 2

how do you find all real zeros of the polynomial function. y=4x+15x -8x-3

98. ## stats.

The mean number of bumped airline passengers per day is 1.35 and the standard deviation is 0.25. For a random selection of a group of 40 days, what is the probability that the mean of bumped passengers for the 40 days will be between 1.25 and 1.50?

99. ## English

1. I am good at computers. 2. I am good with computers. (Are both the same in meaning?) 3. I am good at using computers. 4. I am good with using computers. (Are both the same and grammatical?)

100. ## Propability

Briefly summarize the experimental design for deciding whether a proposed treatment should go on for further study. Note where independence came into play. If you were more conservative and reject the treatment only if the probability that it is 30%

101. ## US History

I have 3 History question that I am completely stuck on. Please help? Thanks Which answer best describes an important outcome of the Gibbons v. Ogden court case? The case set definitions for navigation and placed state control over all coastal and river

102. ## physics

A SUPPOSED TREASURE OF OLD SILVER JEWELRY IS UNCONVERED. a.02M^3 BOX OF THE "OLD JEWELRY" HAS A MASS OF 177KG.IS IT R EALLY OLD SILVER? IF NOT WHAT IS IT?

103. ## geometry

Consider the region in Quadrant 1 totally bounded by the 4 lines: x = 3, x = 9, y = 0, and y = mx (where m is positive). Determine the value of c such that the vertical line x = c bisects the area of that totally bounded region. Needless to say, your first

104. ## Physics

A bungee jumper (m = 66.00 kg) tied to a 39.00 m cord, leaps off a 69.00 m tall bridge. He falls to 5.00 m above the water before the bungee cord pulls him back up. What size impulse is exerted on the bungee jumper while the cord stretches?

105. ## World History (Ms. Sue)

Why did black soldiers often face greater hardships that white soldiers? My answer: They face greater hardships because of the color of their skin. They were often given menial jobs, lower pay, and worked under white officers. They were organized in

106. ## math

There are 77 students in the syudent council. The ratio of girls to boys is 7:4. How many girls are in the student council?

107. ## asap math

The measure of a triangle the measure of d is 35° and the measure of cab is 35° what is acd

108. ## English

1. He worked hard to be a best baseball player. 2. He worked hard to be the best baseball player. (which one is right? Do we have to use 'a' or 'the'? Do they have a different meaning?)

109. ## Physical Science

The transfer of heat that takes place because of the density difference in fluids is

110. ## language arts Ms. Sue please

4. "Brer" is dialect for briar. brother. Mister. brawny. 5. What is the main reason to ask questions as you read a story? to describe the characters to summarize the plot to identify the author’s purpose to aid in your comprehension My answers: 4.

111. ## math

A basket contains 15 red numbered balls and 10 white numbered balls. A sample of five balls is selected. How many samples contain three red balls and two white balls? Explain your answer.

112. ## Algebra

3 real numbers that form a geometric progression have sum equal to 175 and product equal to 17576. What is the sum of the largest and smallest numbers?

113. ## Chemistry

At a given Temperature, the elementary reaction AB in the forward direction is first order in A with a rate constant of 3.90*10^-2s^-1. The reverse reaction is first order in B and the rate constant is 8.20*10^-2s^-1. What is the value at equillibrium for

explain All the parallelograms are rhombus but all rhombus are not a parallelograms

115. ## Algebra

Consider all quadruples of real numbers (x_1,x_2,x_3,x_4), such that x_1\leq x_2 \leq x_3 \leq x_4 and each of the four numbers plus the product of the other three equals 130. If S is the sum of all possible values of x_4, what is S, rounded to the nearest

116. ## Math

Do you think regular or irregular polygons are used more frequently in the design of objects and buildings in the world???

117. ## geometry

Consider the region in Quadrant 1 totally bounded by the 4 lines: x = 3, x = 9, y = 0, and y = mx (where m is positive). Determine the value of c such that the vertical line x = c bisects the area of that totally bounded region. Needless to say, your first

118. ## logirythms

1/2LOG(x-1)-3LOG(X+4)

119. ## Geometry

A two-player game is played with two piles of stones, with sizes m,n. On a player's turn, that player can remove any number of stones from one pile, or the same number of stones from each pile. A player loses when they are unable to take a stone. If 1 \leq

120. ## PHYSICS

A bottle containing 4 kgof water at a temperature of 15°Cis placed in a refrigerator where the temperature is kept at 2°C.How much heat is transferred from the water to cool it to 2°C?

121. ## probability math

bill is on the school archery team. the target has a center bull's-eye and two rings around the bull's-eye. the table gives the probabilities of each outcome. what is the probability that bill will get the next arrow in the inner or outer ring?express your

122. ## math

A box consists of white,yellow,green and black balls, in the ratio 2,3,1,6.Total number of balls are 432.How many yellow balls in the box

123. ## Algebra

a and b are positive numbers that satisfy the equation \frac {1}{a} - \frac {1}{b} = \frac {1}{a+b} . Determine the value of \frac {a^6}{b^6} + \frac {b^6} {a^6} .

124. ## Math

The volume of a cylinder is given by formula V= pi(3.14) r^2h where r is base radius and h is height a) the height of a cylinder of radius 5cm and volume 500. Cm^3 B) radius of base of a cylinder of volume 300 cm^3 and height 10 cm Our teacher didn't

125. ## geometry

Carlos want to show a rectangle in his drawing. What dotes he have to do to the drawing below to make the rectangle a closed shape?

126. ## Economics

Protecting intellectual property rights can result in new inventions that help the economy to grow. True False

127. ## Algebra

3 real numbers that form a geometric progression have sum equal to 175 and product equal to 17576. What is the sum of the largest and smallest numbers?

128. ## physics

Show that the energy-momentum tensors for electromagnetism and for scalar field theory satisfy the dominant energy condition, and thus also the weak, null, and null dominant conditions. Show that they also satisfy w>-1

129. ## probability math

bill is on the school archery team. the target has a center bull's-eye and two rings around the bull's-eye. the table gives the probabilities of each outcome. what is the probability that bill will get the next arrow in the inner or outer ring?express your

130. ## urgent pls education

Assuming in your class,you have noticed that a particular child who is an adolescent has suddenly become withdrawn,discuss how you are going to help such a child???

131. ## science help?

(a) One such reaction fuses two nuclei of carbon-12 ( 6 12C ) to form an atom of neon ( 10 20Ne ) with the emission of an α-particle. Write an equation for this process, indicating the atomic number and mass number of each of the products in the usual

132. ## Trigonometry

using csc = 3.75, solve for sides: a= 3 b= ? c= ? and the remaining five trigonometric functions: sin= (1/3.75) cos= ? tan= ? sec= ? cot= ? *b is not 1; and c is not 3.75

133. ## math/ratios

There are 77 students in the student council. The ratio of girls to boys is 7:4. How many girls are in the student council?

134. ## Math

Teo rectangles with dimensions 8cm by 10cm and 9cm by 12cm partly cover each other(see the diagram). the grey area 37cm2. what is the area of the shape marked by dots? How can I upload diagram?

135. ## geography

1. Bolivia is pretty isolated--why? 2. Columbia has an important resource--what is it, where is it? 3. Brazil has many resources--where are they? Access? 4. Why is Brazilia a forward capital? 5. What is Cuba's access? Why does the US have a trade embargo

136. ## algebra 1

write a quadratic functions for graph that has an axis of symmetry of x= -3/2. summarize your steps.

137. ## World History (Ms. Sue)

How did the Emancipation Proclamation change the role of African Americans in the war? My answer: The Emancipation Proclamation delared that African-American men would be able to "be received into the armed services of the United States," which also lead

138. ## Chem

What is the specific heat in J/g Celsius for a metal sample with a mass 95.6 g which absorbs 841 J of energy when it's temperature increases from 30.0 degrees Celsius to 98.0 degrees Celsius?

144. ## calculus

A publisher prints and sells both hardcover and paperback copies of the same book. Two machines M and N are needed to manufacture these books. To produce one hardcover copy,machine M works 1/6 hours and N works 1/12 hours. For a paperback copy M and N work

145. ## chemistry

define elements, compound and molecules.

146. ## physics

A 7·cm tall candle is placed 125·cm from a lens with a focal length of +25·cm. Note: enter the absolute value of your answers for all distances and heights ... use the signs to determine the type and orientation of the images. a) How far from the lens

147. ## calculus

A publisher prints and sells both hardcover and paperback copies of the same book. Two machines M and N are needed to manufacture these books. To produce one hardcover copy,machine M works 1/6 hours and N works 1/12 hours. For a paperback copy M and N work

148. ## math

A publisher prints and sells both hardcover and paperback copies of the same book. Two machines M and N are needed to manufacture these books. To produce one hardcover copy,machine M works 1/6 hours and N works 1/12 hours. For a paperback copy M and N work

149. ## ENGLISH, HELP?

WHAT WOULD A GOOD ESSAY BE FOR A PROBLEM IN THE WORLD? LIKE POLLUTION, OR BANKRUPTCY? ANY OTHER IDEA'S?

150. ## Algebra

Find the value of p such that (–6, 3) is the midpoint of the points (-9, -1) and (–3, p).

157. ## biomechanics

A Softball is Thrown at an angle of 45 degrees with the horizontal. It leaves the hand 2m above the ground. Its Initial Velocity is 20 m/s . Neglecting all external resistance, how far would it travel ? The point at which it hits the ground is at the same

158. ## math

Why is 1/8 greater than 1/10 but less than 1/3?

159. ## word puzzle

Foreign object - fellow has one

160. ## Math

What does it mean when it says: Which state has the largest population growth

161. ## College Math - Probability

Briefly summarize the experimental design for deciding whether a proposed treatment should go on for further study. Note where independence came into play. If you were more conservative and reject the treatment only if the probability that it is 30%

162. ## physics

A Softball is Thrown at an angle of 45 degrees with the horizontal. It leaves the hand 2m above the ground. Its Initial Velocity is 20 m/s . Neglecting all external resistance, how far would it travel ? The point at which it hits the ground is at the same

163. ## math

find the volume of a square pyramid. Round to the nearest tenth(slant height=20 base=13)

164. ## Math

Need help on the question below. Which number belongs to the set of rational numbers but does not belong to the set of whole numbers? a. -1 b. 0 c. 1 d. 2 Thanks...

165. ## math

Marty ate 1/4 of his pizza and Luis ate 1/3 of his pizz.marty ate more pizza than luis.how is that possible?

166. ## Physics

What is the gibbs free energy for the reaction a)temperature b)800 degrees celsius c)comment on whether the reactions are spontaneous or not (CHEMICAL REACTION) deltaH = -100kJ delta S= -200 J/K

167. ## math

Greater than40 less than70divisible evenly 15 digits add up to6

168. ## math

How do you solve: imgur[dot]com/PlcCL8s [I had to upload cause its impossible to write proper notation here.

169. ## math

2log_3(9) solve

170. ## Chem

264.6 g of water is in a Styrofoam calorimeter of negligible heat capacity. The initial T of the water is 16.9oC. After 270.9 g of an unknown compound at 71.3oC is added, the equilibrium T is 33.2oC. What is the specific heat of the unknown compound in

171. ## math

log_3 (3)^(2x)

178. ## math

what does |-7| mean?

179. ## Math

18 play baseball, 17 play basketball, 30 play neither or both. I'm trying to make a Venn diagram but don't understand how. Please help!

180. ## math

find the coefficient of x^29 in the expansion (1+2x)^12 .(1+x)^18

181. ## Physics

A light beam containing red and violet wavelengths is incident on a slab of quartz at an angle of incidence of 81.0°. The index of refraction of quartz is 1.455 at 660 nm (red light), and its index of refraction is 1.468 at 410 nm (violet light). Find the

182. ## pre algerba

For the past 4 weeks, you have saved $5 per week. You know have$65 in your savings account. Write an equation that represents your balance A after t weeks.

183. ## algebra 1

write a quadratic functions for graph that has an axis of symmetry of x= -3/2. summarize your steps.

184. ## statistics

a. A single student (n=1) b. For a mean of 50 student (n=50) c. For a mean of 500 students (n=500)

185. ## Chemistry

A 20.0mL 0.3M HCl solution is titrated with 0.4M NaOH. What is the exact pH of the solution before titration? And what is the pH at the equivalence point?(are there any straightforward formulas to use on this kind of problem?)

186. ## college physics

if a girl pulls on a 15 kg wagon with a constant horizontal force of 35N.Her friend pushes on the wagon in the same direction with a constant horizontal force of 10 N. and there is a friction force of 15 N. I need to draw a diagram of this problem and

187. ## Chemistry

How many grams ammonium chloride and how many milliliters 3.0 M sodium hydroxide should be added to 200 ml of water and diluted to 500 ml to prepare a buffer of 9.50 with a salt concentration of 0.1 M? Ka=9.24

188. ## geometry

The angles in triangle ABC satisfy 6sin∠A=3√3sin∠B=2√2sin∠C. If sin2∠A=a/b, where a and b are coprime positive integers, what is the value of a+b?

189. ## algebra

an object is launched vertically in the air at 29.4 meters per second from 10 meter tall platform . find its maximum height in seconds and in meters ?

190. ## government

How to do informational paragtaph on structure of our local goverment (mississippi)

191. ## Chemistry

A 20.0mL 0.3M HCl solution is titrated with 0.4M NaOH. What is the exact pH of the solution before titration? And what is the pH at the equivalence point?(are there any straightforward formulas to use on this kind of problem?)

192. ## physics

a disc of radius 5.70 cm rotates about its axis and a point 1.90 cm from the center of the disc moves 34.5 cm in 12.2 s .calculate the angular velocity of the disc.

193. ## Math

5 books and 9 magazines cost $150.50 altogether. 9 books and 5 magazines cost$175.70. How much does 1 book and 1 magazine cost?

What is arithmetic and geometric sequences?

195. ## english

Write the pronoun in these sentences. Then write the underlined words,that they replace. " Let us think of a way to clean up litter," said Matt and Lynn. ( the underlined word is Matt and Lynn)

196. ## college physics

If a girl drops a coin into a wishing well. After exactly 5 seconds she hears it splash into the water. How fast was the coin going after 1 second? After 2 seconds? How fast was the coin going when it hit the water? How far down is the water? Could you

197. ## Math

Precalc (1+i)^3 and the cube root of 8i

198. ## math

There are 3 pitchers of juice. Each student will be served ^ of the juice in a pitcher. How many students will get juice? l a. Draw lines on the pitchers to show how many students these pitchers will serve. b. Write a number model to describe the problem.

199. ## Chemistry

What is the pH of a saturated zinc hydroxide solution?

200. ## Algebra 1

Can you please help me with this question? I put parenthesis to make it look less confusing. (10)/(x)-(10)/(x-3)=(-3)

223. ## Physics

A golf ball with a mass of 50.3 g can be blasted from rest to a speed of 72.1 m/s during impact with a club head. Assume that the impact lasts only about 0.910 ms. Calculate the change in momentum of the ball. Calculate the average force applied.

224. ## math

the letters g and g stand for numbers. if g + 9=h + 9, what is true about g and h?

225. ## Physics

A physics teacher (mass = 84.5 kg) is jogging through the woods and runs straight into a large oak tree at 3.5 m/s. Rebound speed is measured at 2.3 m/s in the opposite direction. If the time of contact with the tree is 45 milliseconds, what is the

226. ## Math

If someone is renting a truck that has a 40 gallon tank and it gets 10 miles per gallon and they are traveling 252 miles, what will they pay for gas for the trip?

227. ## physics

A ball whirls around in a vertical circle at the end of a string. The other end of the string is fixed at the center of the circle. Assume that the total energy of the ball-Earth system remains constant. (a) What is the tension in the string at the bottom?

228. ## Chemistry

What volume of a 6.0 M NaOH solution would you need to prepare 250.0 milliliters of a 3.0 M NaOH solution?

229. ## Math

1. What two numbers can be multiplied to equal -60 and added to be -11? 2. What two numbers can be multiplied to equal -36 and added to be -16? 3. What two numbers can be multiplied to equal 10 and added to equal -7? Thanks

230. ## Stats

1. The question asks to choose a data point in our graph (we chose 476,44) of median home prices (x) and percentage of students who scored Advanced on math MCAS (y). We have to find the residual. I'm confused because the equation I was given is e=y-yhat I

231. ## DES

A rock is being pushed by a stream with 30 N of force. The rock has a static friction of 40 N and a sliding friction of 20 N. Does the rock move?

232. ## College Mathematics

Counting Principles and Permutation. A social security number consists of 9 digits. How many different social security numbers are possible if repetition of digits is permitted

233. ## Grammar

By, who were telephones improved? How can I know if this is right with who or should it be whom?

234. ## Life orientation

In 10-15 lines critically discuss 5 ways in which the human or environmental problem impact on the community

235. ## jiskha

Lol, I have a quick question...If you don't mind. Do you hink jiskha will have thier own app one day? It's so tiring, because you need wifi right lol just curious.

236. ## Physics

Suppose that each of the three masses in the figure below has a mass of 5.50 kg, a radius of 0.0700 m, and are separated by a distance of 10.5 m. If the balls are released from rest, what speed will they have when they collide at the center of the

237. ## English

Should I use a comma or colon after as in this sentence? The sentence is not complete that I posted here I have learned many things in this class I will list a few such as different writing styles, write descriptive paragraphs,

238. ## precalc

sine curve in the form of y = a sin(pi/2x) that passes through the point (2.5,-9.546)

239. ## Math HELP!! :(

a contractor's designing one side of an office building 48 feet in length the windows are each 5 feet long and will be placed Six feet apart and 5 feet from the end of the building wall how many windows can a contractor design

240. ## math

Hi, This is the third time I posted this because it wasn't answered by anyone. PLEASE help. I think the first part is correct but I am not sure if the V=lw10 is correct. I got 10 for the number of cubes. Here is the question There is a picture with stacked

307. ## stats

Use this z-score formula for this problem: z = (x - mean)/(sd/√n) x = 1.25, 1.50 mean = 1.35 sd = 0.25 n = 40 Calculate two z-scores, then use a z-table to determine probability between the two scores. I hope this will help get you started. •stats. -

308. ## Physics

A bungee jumper (m = 66.00 kg) tied to a 39.00 m cord, leaps off a 69.00 m tall bridge. He falls to 5.00 m above the water before the bungee cord pulls him back up. What size impulse is exerted on the bungee jumper while the cord stretches?

309. ## math

Two angles between 0 degrees and 360 degrees have a tangent undefined. What are the angles? Why is the tangent undefined for each of those?

315. ## Physics

A charge of -3.66C is fixed at the center of a compass. Two additional charges are fixed on the circle of the compass (radius = 0.0978 m). The charges on the circle are -4.04C at the position due north and +6.20C at the position due east. What is (a) the

316. ## algebra

1.You flip a coin 30 times and get heads 11 times, so the chance of getting heads is 11/30. Is this an example of theoretical or experimental probability? theoretical experimental * 2.You have made 3 out of 10 of your last free throws. How many free throws

317. ## Biology

Where is the clitoris situated? What is transpiration? What is the function of Bertholin's Glands? What is the chemical composition of melanin? Where are the testicles situated in the human body?

318. ## math 140

A football team gained 12 yards on one play, lost 40 yards on another, and gained 34 yards on the last play of the first half. They had already gained 345 already during the half. What was the total yardage gain for the first half?

319. ## math

. Find the roots of the polynomial equation. x2 + x undefined 72 = 0 (Points : 1) 9 and 8 6 and 6 2 and 16 9 and 8

320. ## Quantum Physics

Let |\psi\rangle=\frac{1-i}{2}|0\rangle-\frac{1+i}{2}|1\rangle and |\phi\rangle=\frac{2+i}{3}|0\rangle-\frac{2i}{3}|1\rangle. What is \langle \psi\,|\,\phi\rangle? Please tell the ans...

321. ## Language Arts

what kind of behavior were greek myths meant to encourage? i think it is mainly to discourage selfishness and greed.

322. ## chemistry

Construct the correct structure for AsClF42-. Please construct the molecule such that the dipole is oriented vertically up or vertically down. - Name the type of hybrid orbitals the central atom forms. - Name the molecular geometry of the compound. - State

323. ## algebra

What is the probability that a month picked at random will have 31 days? A.5/12 B.4/12 C.7/12 D.8/12 I think it is C...?

324. ## Life orientation

describe the course of Ap chdmistry

325. ## Physics

Betty weighs 449 N and she is sitting on a playground swing seat that hangs 0.55 m above the ground. Tom pulls the swing back and releases it when the seat is 0.76 m above the ground.The acceleration of gravity is 9.8 m/s^2. How fast is Betty moving when

326. ## Algebra

If you rolled a number cube 100 times, how many times would you expect to get an even number? A.25 B.75 C.50 D.38 I think it is C...?

327. ## simple queston

what is -430 in scientific notation?

328. ## chemistry

Which of the following combinations of atomic orbitals can combine to form sigma bonds? s-s ,s-p, p-p ,hybrid orbital - hybrid orbital ,hybrid orbital - s ,hybrid orbital - p

329. ## Algebra

1. Simplify the expression: (x^3*y^5) (-y^7) 2. Write the expression in radical form: (125x)^1/3

330. ## math

The crater is circular and has a radius of about 0.4 mile. About how many square miles does this crater cover. Use 3.14 for pi answer my 1.256square am i correct

331. ## chemistry

Carbon reacts with chlorine and hydrogen to form the compound CH3Cl. (a) Construct the structure of CH3Cl. What is the dominant form of secondary bonding for CH3Cl? Dipole-dipole interactions, London dispersion forces, Covalent bonding, Ionic bonding

332. ## physics

A(n) 96.1 g ball is dropped from a height of59.1 cm above a spring of negligible mass.The ball compresses the spring to a maximum displacement of 4.75403 cm.The acceleration of gravity is 9.8m/s^2. Calculate the spring force constant k. Answer in units of

333. ## chemistry

a 225.0 g sample of an unknown, silver, pure metal was heated to 95.5 degrees celsius then put into 150.50 g of water at 22.5 degrees celsius. the water was heated by the hot metal to a temperature of 31.4 degrees celsius. what is the specific heat? what

334. ## Algebra

We define n♡ recursively as follows. 1♡=1; n♡=((n−1)♡)⋅n+1 Find the largest n

335. ## physics

In an arcade game a 0.099 kg disk is shot across a frictionless horizontal surface by compressing it against a spring and releasing it.If the spring has a spring constant of 244 N/m and is compressed from its equilibrium position by 4 cm, find the speed

336. ## Math

Is the opposite of 0 still 0 or does 0 have no opposite? Thanks...

337. ## Algebra

Let S(N) denote the digit sum of the integer N. Let M denote the maximum value of N/S(N), where N is a 3-digit number. How many 3-digit numbers N satisfy N/S(N)=M? The digit sum of an integer is the sum of all its digits. For example, the digit sum of

338. ## Number theroy

What is the 50th smallest positive integer that can be written as the sum of distinct non-negative integer powers of 3?

339. ## AP Chemistry

What mass in grams of sodium hydroxide is produced if 20.0 g of sodium metal reacts with excess water according to the chemical equation? 2Na(s)+2H20(l)=2NaOH(aq)+Hs(g) How many moles of Ag can be produced if 350. g of Cu are reacted with excess AgNO3

340. ## Geometry

BC is a triangle with ∠BAC=60∘,AB=5 and AC=25. D is a point on the internal angle bisector of ∠BAC such that BD=DC. What is AD^2? It is not stated that D lies on BC. This assumption is not necessarily true.

341. ## Science

how many kw is an airflow of 30m³/s @130°C

342. ## Combinations Maths

For how many positive integers n are there exactly ⌊n/2⌋ or ⌈n/2⌉ primes less than or equal to n?

343. ## English

Can someone check of this email is grammatically correct? Thank you. How are things? 1) I’ve been on holiday in England and I’ve been attending an English summer course for two weeks. 2) I’ve met an interesting girl from London, Mary, at a party

344. ## Science

Need help on the question below... Which is the warmest climate zone? A. Arctic Zone B. Temperate Zone C. Tropical Zone D. Intermediate Zone Thanks :)

345. ## English

1. He seldom tatoos roses on his arms. 2. He sometimes woos when he meets a pretty lady at a meeting. 3. He seldom tatooes roses on his arms. 4. He sometimes wooes when he meets a pretty lady at a meeting (Which ones are grammatical?) 5. He tatooed a

346. ## aig math

A bookshop had 60 boxes of pencils. Each box contained 85 pencils. Half of the pencils were packed into packets of 25 pencils and the rest were packed into packets of 20. How many packets of pencils were there altogether? My answer is 229 am i right? How

The strongest recorded earthquake in the world took place in Chile on May 22 1960. It had a magnitude of 9.5 on the Richter scale. The strongest quake actually recorded in Canada was magnitude 8.1 on August 22 1949 and occurred off the Queen Charlotte

348. ## physics

An aluminum wing on a passenger jet is 38 mlong when its temperature is 23°C.At what temperature would the wing be 2 cm (0.02 m) shorter?

349. ## chemistry

A student extracts 6.3 g of water when performing the following combustion: C2H8 + 4O2 --> 2 CO2 + 4H2O If the theoretical yield is 8.6, what is the percent yield?

350. ## Math

Principal 450.00 x interest rate 9.25 x compounded monthly x 4 years

351. ## math

What is the rule for this input/output table Input 2 output 10 Input 6 output 26 Input 8 output 34 Input 10 output 42

352. ## Statistic

In a jar there 12 white balls 8 red balls 4 greens all with the same size if a person selects one ball at random find the probability it is a red ball.

353. ## organic chemistry

when i add methyl salicylate to aqueous sodium hydroxide, immediately a white solid forms... that is sodium salicylate is it?? Since it hasn't been heating under reflux yet.

354. ## proprotions

why did the envelope lose the race with the cardboard box?

355. ## math

2log[base3]9 = ?

How do you find an answer to a MAD (mean absolute deviation) chart?

357. ## World History (Ms. Sue)

What was Lincoln's reasons for not emancipating slaves when the war began? My answer: Lincoln did not believe he had the power under the Constitution to abolish slavery where it already existed. Nor did he want to anger the four slave states that remained

358. ## urgent maths pls

A goat is tied at point A with a rope of length(L). How long can L be so that the goat can feed on exactly half of the field if the field is a circle.

359. ## Chemistry

Question 2 Marks: 2 How many grams of oxygen gas must react to give 174.3 grams of ZnO according to the balanced equation. 2 Zn(s) + O2(aq) --> 2 ZnO(s)

360. ## algebra II

i just placed $1500 in an account which earns 8% per year compunded quartely. how much money will be in the account in 20 years? how long will it tke for this account ot have$7000 in it?

361. ## Chemistry

Question 2 Marks: 2 How many grams of oxygen gas must react to give 174.3 grams of ZnO according to the balanced equation. 2 Zn(s) + O2(aq) --> 2 ZnO(s)

362. ## physics

A 1.03 kg mass is attached to a spring of force constant 10.6 N/cm and placed on a frictionless surface. By how much will the spring stretch if the mass moves along a circular path of radius 0.525 m at a rate of 2.20 revolutions per second?


text:
# Miscellaneous¶

Miscellaneous stuff that doesn’t really fit anywhere else.

diofant.utilities.misc.debug(*args)[source]

Print *args if DIOFANT_DEBUG is True, else do nothing.

diofant.utilities.misc.filldedent(s, w=70)[source]

Strips leading and trailing empty lines from a copy of $$s$$, then dedents, fills and returns it.

Empty line stripping serves to deal with docstrings like this one that start with a newline after the initial triple quote, inserting an empty line at the beginning of the string.


text:
Ben Green and I have just uploaded our paper “The quantitative behaviour of polynomial orbits on nilmanifolds” to the arXiv (and shortly to be submitted to a journal, once a companion paper is finished). This paper grew out of our efforts to prove the Möbius and Nilsequences conjecture MN(s) from our earlier paper, which has applications to counting various linear patterns in primes (Dickson’s conjecture). These efforts were successful – as the companion paper will reveal – but it turned out that in order to establish this number-theoretic conjecture, we had to first establish a purely dynamical quantitative result about polynomial sequences in nilmanifolds, very much in the spirit of the celebrated theorems of Marina Ratner on unipotent flows; I plan to discuss her theorems in more detail in a followup post to this one.In this post I will not discuss the number-theoretic applications or the connections with Ratner’s theorem, and instead describe our result from a slightly different viewpoint, starting from some very simple examples and gradually moving to the general situation considered in our paper.

To begin with, consider a infinite linear sequence $(n \alpha + \beta)_{n \in {\Bbb N}}$ in the unit circle ${\Bbb R}/{\Bbb Z}$, where $\alpha, \beta \in {\Bbb R}/{\Bbb Z}$. (One can think of this sequence as the orbit of $\beta$ under the action of the shift operator $T: x \mapsto x +\alpha$ on the unit circle.) This sequence can do one of two things:

1. If $\alpha$ is rational, then the sequence $(n \alpha + \beta)_{n \in {\Bbb N}}$ is periodic and thus only takes on finitely many values.
2. If $\alpha$ is irrational, then the sequence $(n \alpha + \beta)_{n \in {\Bbb N}}$ is dense in ${\Bbb R}/{\Bbb Z}$. In fact, it is not just dense, it is equidistributed, or equivalently that

$\displaystyle\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N F( n \alpha + \beta ) = \int_{{\Bbb R}/{\Bbb Z}} F$

for all continuous functions $F: {\Bbb R}/{\Bbb Z} \to {\Bbb C}$. This statement is known as the equidistribution theorem.

We thus see that infinite linear sequences exhibit a sharp dichotomy in behaviour between periodicity and equidistribution; intermediate scenarios, such as concentration on a fractal set (such as a Cantor set), do not occur with linear sequences. This dichotomy between structure and randomness is in stark contrast to exponential sequences such as $( 2^n \alpha)_{n \in {\Bbb N}}$, which can exhibit an extremely wide spectrum of behaviours. For instance, the question of whether $(10^n \pi)_{n \in {\Bbb N}}$ is equidistributed mod 1 is an old unsolved problem, equivalent to asking whether $\pi$ is normal base 10.

Intermediate between linear sequences and exponential sequences are polynomial sequences $(P(n))_{n \in {\Bbb N}}$, where P is a polynomial with coefficients in ${\Bbb R}/{\Bbb Z}$. A famous theorem of Weyl asserts that infinite polynomial sequences enjoy the same dichotomy as their linear counterparts, namely that they are either periodic (which occurs when all non-constant coefficients are rational) or equidistributed (which occurs when at least one non-constant coefficient is irrational). Thus for instance the fractional parts $\{ \sqrt{2}n^2\}$ of $\sqrt{2} n^2$ are equidistributed modulo 1. This theorem is proven by Fourier analysis combined with non-trivial bounds on Weyl sums.

For our applications, we are interested in strengthening these results in two directions. Firstly, we wish to generalise from polynomial sequences in the circle ${\Bbb R}/{\Bbb Z}$ to polynomial sequences $(g(n)\Gamma)_{n \in {\Bbb N}}$ in other homogeneous spaces, in particular nilmanifolds. Secondly, we need quantitative equidistribution results for finite orbits $(g(n)\Gamma)_{1 \leq n \leq N}$ rather than qualitative equidistribution for infinite orbits $(g(n)\Gamma)_{n \in {\Bbb N}}$.

Before we extend to nilmanifolds, let us briefly review what happens for higher-dimensional torii ${\Bbb R}^m/{\Bbb Z}^m$. From the theory of Weyl sums, one can show that an infinite polynomial sequence $(P(n) \hbox{ mod } {\Bbb Z}^m)_{n \in {\Bbb N}}$ in a torus is either equidistributed, or is contained in a finite union of proper subtorii (cf. Kronecker’s theorem in the case when P is linear). Iterating this, we get a Ratner-type theorem for the torus, namely that every infinite polynomial sequence in a torus is equidistributed within a finite union of subtorii.

It turns out that a similar Ratner-type result holds for nilmanifolds, and is due to Leibman (with some earlier results in this direction by Leon Green, by Parry, and by Shah). Recall that a nilmanifold is a quotient space $G/\Gamma$, where G is a nilpotent Lie group (which for simplicity we shall take to be connected and simply connected, although these restrictions can be removed with a bit of effort), and $\Gamma$ is a discrete cocompact subgroup. A good example is the Heisenberg nilmanifold

$\displaystyle G/\Gamma := \begin{pmatrix} 1 & {\Bbb R} & {\Bbb R} \\ 0 & 1 & {\Bbb R} \\ 0 & 0 & 1 \end{pmatrix} /\begin{pmatrix} 1 & {\Bbb Z} & {\Bbb Z} \\ 0 & 1 & {\Bbb Z} \\ 0 & 0 & 1 \end{pmatrix}.$

There is a well-defined notion of a polynomial sequence in G – a sequence $g(n)$ which becomes trivial after finitely many applications of the differentiation operator $\partial$, defined as $\partial g(n) := g(n+1) g(n)^{-1}$. Leibman showed that an infinite polynomial sequence $(g(n)\Gamma)_{n \in {\Bbb N}}$ is either equidistributed in the nilmanifold $G/\Gamma$, or is else contained in a finite union of proper subnilmanifolds; iterating this, one can conclude that every infinite polynomial sequence is equidistributed in a finite union of nilmanifolds. One can also phrase this as a factorisation theorem: every polynomial sequence is the product of a constant sequence, a sequence equidistributed in a nilmanifold, and a periodic (and rational) sequence.

A typical application of Leibman’s theorem would be the assertion that for any real numbers $\alpha, \beta$, the sequence $( \lfloor \alpha n \rfloor \beta n )_{n \in {\Bbb N}}$ is either periodic, or equidistributed modulo 1, where $\lfloor x \rfloor$ is the integer part of x (except in some rare but explicitly describable cases, when $\alpha, \beta$ lie in a quadratic extension of ${\Bbb Q}$, in which one has a different distribution; see comments). I do not know of a way to prove this result which does not basically require one to prove a large part of Leibman’s theorem.

Leibman first proves his theorem in the linear case by heavy reliance on the ergodic theorem, an induction on the step of the nilmanifold, and some Fourier analysis (or representation theory) to handle the “vertical” behaviour of the nilmanifold, using some arguments of Parry. He then passes to the polynomial case by using a lifting trick of Furstenberg to linearise the sequence.

Extensions of these results beyond nilmanifolds, and in particular to finite volume homogeneous spaces, are certainly of interest (see for instance this article of Margulis) but are considerably more difficult, as one loses the ability to work by induction from the abelian case.

Now we turn to quantitative versions of the above statements, in which we look at the equidistribution properties of a finite polynomial sequence $(g(n) \Gamma)_{1 \leq n \leq N}$. One can define the notion of a finite sequence being equidistributed up to some error $\delta$, which roughly means that irregularities in the sequence can only be detected at scales $O(\delta)$ or below. Our first main result is that a finite polynomial sequence is either equidistributed up to error $\delta$, or is else concentrated within $O(\delta)$ of $O(\delta^{-O(1)})$ proper subnilmanifolds, whose “slopes” have height $O(\delta^{-O(1)})$. One can iterate this and conclude a Ratner-type theorem, namely that every finite polynomial sequence is within $O(\delta)$ of being equidistributed on a polynomial number of proper subnilmanifolds, whose slopes also have polynomially bounded heights. These polynomial bounds turn out to be important for our number theoretic applications, which will be discussed in a subsequent post. There is also a factorisation theorem, which asserts that every polynomial sequence can be expressed as the product of a slowly varying sequence, a sequence which is equidistributed in a nilmanifold, and a periodic rational sequence, with quantitative polynomial bounds on all of these assertions.

To illustrate our result with a concrete instance, we can assert that for any real numbers $\alpha, \beta$, any error tolerance $\delta > 0$, and any large integer, one can subdivide $\{1,\ldots,N\}$ into arithmetic progressions, each of density $\gg \delta^{O(1)}$, such that $\lfloor \alpha n \rfloor \beta n$ is either within $\delta$ of a constant, or is $\delta$-equidistributed, or is the push-forward of a $\delta$-equidistributed sequence by a quadratic polynomial with coefficients $O(\delta^{O(1)})$ (this latter case is technical, and rather rare, having to do with special subnilmanifolds of the Heisenberg nilmanifold).

One of the key difficulties in working in the quantitative setting (especially when one is insisting on polynomial bounds everywhere, and when N is fixed in advance) is that one can no longer use the ergodic theorem (unless one has quantitative control on the ergodicity, such as spectral gaps, which appear to be unavailable in this setting). Because of this, we were eventually forced to find a different approach than Leibman’s to these problems, and ended up with one based primarily on Fourier analysis in the vertical direction combined with the van der Corput inequality (very much in the spirit of the Weyl’s theory of equidistribution and exponential sums). Another new feature of the quantitative setting is the presence of error terms: a sequence may not lie exactly in a subnilmanifold, but instead deviates from it by a small error. This error can be eliminated, but often at the cost of increasing the “degree” (or other measures of “complexity”) of the polynomial sequence $g(n)$. This makes it far from obvious that iterative arguments guaranteed to terminate; it also seems to prevent one from using the Furstenberg lifting trick to linearise a sequence in the middle of an iteration argument, and so we were forced to work directly with polynomial sequences. For similar reasons, we had to perform a fair amount of algebraic computation to set up a good notion of the “complexity” of a polynomial sequence, which decreased with every stage of the main iteration argument. (In the end, this complexity is modeled by three parameters: the step s of the nilmanifold, the “commutator degree” D of the sequence g(n), and the “nonlinearity dimension” of the coefficient spaces for g(n), thus leading to a triple induction on these parameters in order to conclude the argument.)

[Update, Sep 26: inaccuracies with the $\alpha n \lfloor \beta n \rfloor$ example fixed; thanks to Ben Green for the corrections.]


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## Chemistry (7th Edition)

a)$MnO_{4}^{-}(aq)+3e^{-} \rightarrow MnO_{2}(s)$ , $IO_{3}^{-}(aq) \rightarrow IO_{4}^{-}(aq)+2e^-$ b)$NO_{3}^{-}(aq)+e^- \rightarrow NO_{2}(g)$ , $SO_{2}(aq) \rightarrow SO_{4}^{-}(aq)+3e^-$
a)$Mn$ is reduced from 7+ to 4+ and $I$ is oxidized from 5+ to 7+ b)$N$ is reduced from 5+ to 4+ and $S$ is oxidized from 4+ to 7+


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# Quotes and Stories from “Turing 50”

::

The ACM recently hosted a celebration of 50 years of the A.M. Turing award. These are some notes and thoughts from the event, including how Fred Brooks once rented a bus, Don Knuth’s outrageous implementation of batch processing, and Judea Pearl’s theory of homo sapiens.

Conventions / Disclaimers:

• The blockquotes below are paraphrased, may be incorrect, and may be incorrectly attributed. Make sure to watch the ACM’s live stream before quoting anything here!!!

• Section-breaks are labeled as “panel”, “talk”, “question”, etc.

• This is intentionally “bad writing” in the Peter Lee sense (see below) — primarily “what I saw”, very little about “what I thought and felt”. A summary in my own words just wouldn’t do justice to the panelists.

• The “Augmented Reality” session was my favorite.

### Opening Remarks

#### Alan Turing is with us today

At the start of the event, the emcee unveiled a bronze bust of Alan Turing. This statue was on display at center stage during the whole event.

It’s a good sculpture and it’s good we remember Alan Turing, but I’m sad that the ACM would encourage this kind of idol-worship. Let’s not forget Turing’s excellent teachers and colleagues!

### talk: Impact of Turing Recipients’ Work

##### Barbara Liskov

the first awards recognized achievements in the standard fields of theory, AI, and systems

hostile environment around the first awards, trepidation about future awards

with Unix, Ritchie and Thompson got the design right

Niklaus Wirth: “If I understood how important Pcode was, I would have spent more time designing it”

#### thoughts

What is “systems” — does that even have a definition? And Unix is definitely NOT an example of a “right design”; rather it’s a landmark of worse is better design.

### panel: Advances in Deep Neural Networks

#### Stuart Russell

I work in all areas of AI except for deep learning

#### Judea Pearl

I am a foreigner in this field … left because human beings are not good at handling information … people are very good with causal inference, not with statistical inference … deep learning is statistical

there is a very old existence proof, homo sapiens took over the planet … I believe because they had an internal model of their environment … a drawing of a lion with wings is evidence of this model, you have to have such a model before you can experiment with it and imagine … snakes have superb optics, result of a long evolution process … very specific but they cannot build eyeglasses … humans have an internal model, can build a market based on promises and build large communities based on promises

I see four levels … second level is predicting events, if I do X then what? … third level is counterfactual, if I did things differently then how would the outcome change … very hard to advance between levels, are we working to help machine learning ‘level up’?

data science is about the relation between data and reality … data alone is not data science

##### Michael Jordan

today we can’t think without holding a piece of metal

machine learning is part of computer science rather than AI … AI is about how to make human … machine learning is about allocating resources … matrices are not all of human intelligence … neural nets are part of a wider toolbox … too much hype in NLP its just syntax

huge gap between syntax and semantics … chat bots are just syntax, don’t learn … faking intelligence with neural nets, so well that you can build a company …

if I say ‘a GLEEB walked across the airport’ then true intelligence can make a lot of educated guesses about a ‘GLEEB’ without any other context

##### Fei-Fei Li

I disagree, ML is part of AI … understanding intelligence and making intelligent methods for solving AI problems

to quote Churchhill ‘its not beginning of end, not end, not beginning of end, probably end of beginning’

AI cannot yet find our keys

quote: ‘todays AI is making a perfect chess move while the world is on fire’ … ignores context

##### Stuart Russell

Turing … a program is a mathematical object … math community did not recognize this

lots of grad student descent … tuning to get performance … deep learning is neglecting the problem of exponential data … deep learning is just circuits, circuits lack expressive power … a human can process data from CERN but a neural net cannot, need to know physics

probabilistic programming, somewhat under the radar, maybe on the right track … 10-line program running/generating/updating a large network of possibilities … more composable and flexible

##### Ilya Sutskever

why I like deep learning … philosophically satisfying … the hypothesis class is a circuit … powerful hypothesis class not too many parameters … can actually find circuits … ‘violates all theory’ … really amazing … humans can see and hear pretty fast, even though our neurons are pretty slow, perhaps because we do a massively parallel process that doesn’t take many steps … works well enough to be useful

models e.g. for vision are very hard to understand … fight fire with fire … incomprehensible solution to incomprehensible problem

##### Raquel Urtasun

the breakthrough in neural nets is not algorithms … it is tricks, hardware, and grad students

with neural nets we forget about modeling, uncertainty, and prior knowledge … perception is a canonical example

#### question: boundaries

##### Judea Pearl:

glad to see people in deep learning understand its limitations … is there a clearer definition of the boundaries? Are you worried about bridging the levels factual/inferential/counterfactural?

##### Michael Jordan

the big problem is decision making under uncertainty

##### Fei-Fei Li

cognition is a hard problem

##### Judea Pearl

do you have a clear idea of the boundaries?

##### Michael Jordan

neural nets use back-propagation … its non-modular, sad fact … performance and explainability is the tradeoff … then again people are non-modular

##### Stuart Russell

AlphaGo is not deep learning … basically an improved version of the machines Arthur Samuel made in the late 1950s … the interesting code is in C++ … rules of go, next moves, searching future states … depends on transitive closure

##### Judea Pearl

can AlphaGo take advice from a human?

##### Michael Jordan

not currently, but that would be a new policy to add to the toolbox … just as neural nets are one tool within AlphaGo

##### Raquel Urtasun

no reason to ask if deep learning is going to solve all problems

#### question: education?

##### Judea Pearl

indeed, what DO you teach in your neural networks classes?

##### Fei-Fei Li

… chain rule, Taylor expansion

##### Judea Pearl

teaching is communicating truths … what is true about neural nets? what are some things that will definitely not happen?

##### Stuart Russell

Peter Norvig and I have a problem with our AI book … chapter on vision, chapter on speech, will probably post just point to the neural nets chapter … we don’t really understand! … really selling students short

##### Fei-Fei Li

in labs we talk about what we cannot do … we all have open problems

Stuart I hope you have a very good author for the chapters. There are so many open problems to communicate to students!

##### Michael Jordan

CS cirriculum needs more statistics, inferential thinking … revise the whole cirriculum bottom-up to weave this in

#### question: could a neural net fix my phone without breaking it?

##### Judea Pearl

right! big problem that neural nets have no internal model to manipulate

#### question: generalizability?

##### Ilya Sutskever

special-purpose vs. general purpose solution depends on the problem … most things we give special-purpose solutions … I guess if you wanted to automate a mathematician that would need to be general

##### Stuart Russell

always argue with your self … try to break what you’ve built … there’s a system that plays video games just using the pixels on screen as hints … it’s very good at mazes; if a newborn baby learned to play maze games in 2 hours that would be amazing! … does the system scale? absolutely not

#### thoughts

When Michael Jordan said “people are non-modular”, I think he means that people are able to break abstraction barriers when needed.

### panel: Restoring Personal Privacy without Compromising National Security

##### Joan Feigenbaum

… wikileaks … russian hackers … social emergency …

##### Whitfield Diffie

everything I say today is copyleft

its a misunderstanding to talk about a conflict between security and privacy … two aspects … problem goes back to feudalism … the right to build a castle was granted by the king … on one hand a castle improves national security … on the other hand a castle can be used to attack the king … technology is upsetting the basic notion of private vs. public security … governments cannot protect citizens and cannot protect themselves … extremely difficult to prove that a small process is secure

exceptional access makes it more complex

##### Paul Syverson

major concern are national security threats and ability of authorities to confound threats … analogy to printing press … proclimation of 1635 that only state messengers can carry letters … 1663 treatise by the national censor, no printing house can have a back door … the general topic is very old … title of this session isn’t very good, the real dilemma is investigation vs privacy

##### Bryan Ford

code is law for better or worse, tech is not a tool like a watch … tech can monitor us and decide when it works … tech is government, not obedient tools … the mind is a warrant-proof space … 5th amendment rights should extend to wearables

cannot divorce the security/privacy issues from the current political context … the serious vulnerabilities are not in math … they are in users and implementors

#### question: back doors

##### Joan Feigenbaum

perhaps we should explain what a back door is

##### Bryan Ford

agency keeps a master key in escrow

non-lawyers can and should take a stand on basic issues

there are legitimate warrant-proof spaces … electronic extensions of the mind need to be recognized as warrant-proof spaces

the set of authorities with backdoor access should change as I travel between countries … but this will lead to a global race to the bottom

##### Whitfield Diffie

germany has a law against sex tourism (committed by German citizens visiting other countries) … neither government will be willing to lose backdoor access

technical reasons against backdoors … (1) ‘weak crypto’ was implemented, nobody turned it off, is now breakable by anyone in 2015 … (2) Juniper used non-default crypto parameters, someone (inside?) changed the parameters … (3) attackers exploit back doors

##### Paul Syverson

quote ‘you can put a man on the moon, surely you can put a man on the sun’

##### Whitfield Diffie

trouble is getting him back safely

##### Bryan Ford

I think back doors are okay, but not for personal devices … need public lab and transparent processes, need separation of powers … prosecutors are getting cases thrown out because courts do not accept their backdoors … there is a place for transparent back door tools

politicians are rarely technical people

##### Bryan Ford

tech is not a set of policy-neutral tools, need to address gap of understanding

#### question: ???

##### Whitfield Diffie

we don’t know how to build good crypto programs … opponents are debugging our programs with different goals … we’re trying for-all-paths safety (universal) … they’re trying exists-bad-path (existential)

##### Bryan Ford

cybersecurity market is a lemon market

##### Joan Feigenbaum

question from audience ‘I am an advisor to a company working with nuclear energy, they are terrified of being attacked, how should I advise them?’

##### Whitfield Diffie

a network like that is probably separated enough to be safe … the problem is being safe AND connected to the web

##### Bryan Ford

because the internet of things

#### question: what should the ACM do?

maybe we need increased regulation, the ACM could help bring experts together

#### question: what is true security

##### Paul Syverson

it’s all the same thing … gets labeled differently … just trying to control which bits can go where and who gets to read them

security is the absense of being violated

##### Joan Feigenbaum

problem of our community, have strict standards, may be unrealistic … maybe a lot more tolerance in practice than our model accepts

##### Paul Syverson

security and privacy are environmental problems

#### question: can we stop the needle-in-haystack search for vulnerabilities?

##### Paul Syverson

need to build in security from the start

##### Bryan Ford

need rule of law, transparency, separation of powers

stop delaying, instead of spending $on fixing problems, we should invest in solving the fundamental issues ### panel: Preserving our Past for the Future Note: I was volunteering during this session; quotes are sparse ##### Mahadev Satyanarayanan the running system is the total documentation … there are too many details for prose to capture ##### ?? running old code has a danger of running old bugs ##### ??: what not to save? … it’s very hard to tell in advance ##### Mahadev Satyanarayanan: there is no absolute censor in a world with caching ##### Brewster Kahle asking UNESCO to solve the problem is unrealistic … need to empower the fanatics, given them tools to preserve data #### thoughts I totally agree with the “empower the fanatics” sentiment. Today, because of “volunteer librarians”, I think we’re doing pretty well about preserving the past. Suppose I found an old PowerPoint file. I’m sure I could find a way to read it with help from the internet — either by searching Google, pirating an old version of PowerPoint, or asking online forums. So personally I’m not worried about losing data we have currently; I’m more worried about the future, the internet becoming “less chaotic”. The panel raised a good question about how to preserve research and encourage reproducibility. A .pdf or .tex document is not enough; a virtual machine is okay. Really I think we need a stronger cultural emphasis on literate programming and a mature library like TeX to help authors store and share their work. The Gamma seems on the right track. I was surprised that the panel did not discuss search, version control, and the ACM’s open-access policy. ### panel: Moore’s Law is Really Dead: What’s Next? Note: I was volunteering during this session ##### Butler Lampson there’s plenty of room at the top … with Moore’s Law we got improvements at the bottom of the software stack, everything above got to benefit and it was easy to integrate the changes … there’s lots of opportunities to trim fat in the middle/top of the software stack … these improvements will be harder to integrate, but there’s lots of opportunities ##### Margaret Martonosi By the way, don’t believe the brochure that says I’m at Google. My affiliation is Princeton, Google and I are just friends. ##### Butler Lampson important to distinguish approximate vs. precise software … precise software has a specification and the customer cares about that specification … approximate software doesn’t have a hard spec, just needs to approximately work … the web is approximate, it doesn’t work and it doesn’t need to! … windows is precise, definitely has a spec and users definitely care #### thoughts The recording of this panel should be good; it was very lively, very practical. And the first audience question (by David Patterson) was “an A+ question”. The panel reminded me of a talk by Yale Patt about “the end” of the Von Neumann architecture. His opinion is future computers will be Von Neumann machines that rely on “accelerators” like a GPU — computer organization is not going to change, but will expand to have a bigger toolbox. So sure, Moore’s Law is dead, but there are many opportunities to make computers faster at places other than the bottom of the software stack. ### panel: Challenges in Ethics and Computing Note: I was volunteering during this session ##### Raj Reddy there are more slaves in the world currently than there were in the US during the civil war … here is one way technology could help, by giving everone a device to record their location … if someone’s time and location is constant, they may be held against their will ##### ?? do you believe every problem has a technological solution? ##### Noel Sharkey yes the training set may be biased against people similar to me, but I want you to consider my case as an individual ##### ?? a very nice Washington Post article ##### ?? whether to encrypt the back hall ##### Raj Reddy we can sit here and wring our hands, but nothing will come of it unless it is written in the US constitution #### thoughts I did not enjoy this panel. This is an ACM event, not a United Nations event. An ACM-sponsored panel about social and political problems should look for constructive ways that computer science can address these problems. Raj Reddy tried to give constructive solutions, but the panel seemed more interested in complaining about how hopeless things are. The comment by Noel Sharkey about “consider me as an individual” was something I hadn’t thought about. Instead of worrying about biased datasets, let’s use technology to collect data on an individual instead of abstracting a person by their race, age, or neighborhood. ### talk: Computer Science as a Major Body of Accumulated Knowledge ##### Donald Knuth don’t applaud me, just read my books at the time, computer science was AI, numerical analysis, and programming languages a colleague said ‘I will believe that computer science is a science when it has 1000 deep theorems’ … I am not sure what a deep theorem is but I think its different from what’s proven by deep learning great privilege that we can invent the problems we work on … imagination … physicists can only guess the size of the sun I’ve always believed computer science and math are two parallel subjects … sometimes you hear people wondering if one subsumes the other when I won the Turing Award, the prize money was about$1,000,000 less than it is today … I did get a nice Tiffany bowl that my wife and I use to serve strawberries … strawberries actually taste better …

very fortunate in this field … I’m completely worthless as an economic advisor … it’s a game I’ve been able to take advantage of

#### question: how could you offer to pay for TeX bug reports?

##### Donald Knuth

well there were many, many bugs … I stopped doubling at 32768 … brought people out of nowhere … next time I check the bug reports will be 2021 … someone is queueing the bugs reports … I believe strongly in batch rather than swap-in/swap-out … last time I checked reports was 2014 so 2021 will be next

TeX was a literate program, and it helped that I wrote ‘The Errors of TeX’ about the first N bugs

#### question: do you think computers will become good composers of music? do you see a role for computer-assisted proving?

##### Donald Knuth

Yes in part, assisted is the key word … I have a program running now that I hope will help me prove a theorem

#### question: favorite algorithm?

##### Donald Knuth

Tarjan’s strong components … short deep useful

#### question: thoughts on AI, computers taking over?

##### Donald Knuth

I get scared when I see Stuart Russell making assumptions based on people acting rationally … then you look at election results

#### question: if you could start over and do things differently, what would you change?

##### Donald Knuth

I would use decimal internally in TeX instead of binary

#### question: how to record history?

##### Donald Knuth

a video ‘Lets not dumb down the history of CS’ … used to be history of algorithms … trouble with funding … the history is nothing that a non-CS person could not understand … the whole field of history changed from internal to external … historians need to be external to get published in journals … no CS department supports a historian … recently read a dissertation about the ALGOL 60 copmiler … very careful, describes data structures and organization … this kind of thing is what deserves to be called history

#### question: teachers

##### Donald Knuth

hardest thing for me is choosing between two hypotheses (1) could teach this to anyone (2) only 2% of the world is geeks … suppose the second is true then you can’t talk about how to teach if the teacher is not in the 2% …

the newest issue of CACM has a fun typo, ‘deep earning’

### panel: Quantum Computing: Far Away? Around the Corner? Or Maybe Both at the Same Time?

##### John Martinis

goal to have a 45–50 qbit machine … 1 error per 1000 operations … to test, run sample algorithm, chart output vs. a classical supercomputer … got to be a supercomputer to finish the computation in time

##### Andrew Yao

I’m a believer … one suggested benchmark is to factor 1000-digit numbers … impossible to attain … need to expore new possibilities, take physics attitute

CS did well attracting attention to quantum … science should be more open … share results between physics chemistry CS … don’t just stick to your specialized conferences

CS departments reception to quantum is less than satisfactory … 15 years ago, maybe 4 or 5 universities … now, maybe 7 or 8 .. China doing much better in this regard

##### Jay Gambetta

not useful to make analogy to anything classical … universal fault tolerance? or computation in the presence of error … either would be excellent, still a long way off

IBM put quantum on the cloud … picked an instruction set that tries to abstract away … have been 19 published papers on the behavior of this quantum hardware

##### Dorit Aharonov

two paths … finding algorithms, besides Shor’s algorithm … make quantum computer to realize the algorithms … finding algorithms is very difficult … information-processing point-of-view

error correction still small scale … can we use entanglement between probes to improve accuracy?

##### Umesh Vazirani

_different goals … maybe you want perfect Qbits for a perfect Hilbert space … reality is a noisy space … short run, how to compute with noise … how to correct errors …

##### Jay Gambetta:

_those 2 paths are the same to me … we want larger devices with fidelity

lets build hardware see where goes … exciting prospect, computer scientists will explore what they can do with these erroneous qbits … that’s why IBM has the instruction set open to the community

#### question: why isn’t adding 10 qbits only 10x harder?

##### John Martinis:

building infrastructure to scale … not just grad student code … we’re all good coders using standard industry practices for coding

##### Jay Gambetta

fidelity is hard to achieve

#### question: both IBM and Google use superconducting storage?

##### John Martinis

superconducting scales … ion traps harder to scale, but we still watch, keep eye on data

#### question: education

##### John Martinis

I like talking to engineering colleges … physics and engineering need to work together

#### question: is quantum going to change programing languages?

##### Jay Gambetta

yes very different to handle errors … current challenge is building an abstraction over the superconducting hardware

##### John Martinis

hoping to first expose hardware, then get a model, eventually a language

#### question: what would Feynman do?

experiments!

##### Dorit Aharonov

yes he’d tell us to keep playing, and play with us

### panel: Augmented Reality: From Gaming to Cognitive Aids and Beyond

Peter Lee starts off wearing a headset.

##### Ivan Sutherland:

I can tell you how VR started. Bell Helicopter company wanted to land at night … put an infrared camera on the landing site and a display in the cockpit … to test they used the roof of their building … one day an observer in a Bell office is watching, though the camera, two men playing catch on the roof … one player threw the ball at the camera and the observer ducked … he had identified his position with the camera … my observation was that you didn’t need a camera, could substitute a computer … the rest is history

##### Yvonne Rogers

my goal is to augment people … Englebart very inspiring … ok 2 stories … (1) a student of mine wanted to help picky eaters … computer vision for when they tried to hide peas under the plate … projected colors onto the peas, called them ‘disco peas’, kids were distracted enough to get over their aversion … children and parents got involved, new social interaction … (2) opera makeup for schoolchildren, virtually getting into character … teenage boys in the classes got to try makeup for the first time … singers found it useful for rehearsals

##### Peter Lee

I feel socially awkward wearing this headset, but I have some of my slides here … making a wearable headset requires huge effort … research prototypes can be uncomfortable … a product needs to be perfect and its very hard to do perfect … one goal, give Lowe’s VR to demo a virtual kitchen … Case Western anatomy class used virtual cadaver, great collective experience

##### Fred Brooks

two stories … (1) Henry Fuchs 1998, working with breast surgeon, try augmented reality to improve the precision of biopsy probe insertion … 2 years to a working prototype, hard to track surgeon’s eyes, display where probe is, where ultrasound is, provide low latency … one day trying on live patient, worked 100% perfect probe right on the mark, jubilation … then the doctor had to tell the patient ‘yes it is really cancer’ … (2) a challenge, augmented reality EMT training … real teams, virtual patient, virtual surround … track real tools, 8 eyes, 8 images, team needs to interact

#### question: what are current uses of augmented reality?

##### Ivan Sutherland

the pilot of a jumbo jet typically has 1 hour flight experience before he flies for the first time, but extensive training in a flight simulator

the best AR

##### Ivan Sutherland

once I was in a flight simulator with the chief pilot … and he turned to me and asked ‘have you ever experienced a slow roll in a 747?’ … a slow roll is a twisting motion, a very benign maneuver, constant one-G pressure the plane doesn’t know its upside down … ‘here we go’ and suddenly the world inverted … I remarked that it was certainly impressive, but didn’t you treat the simulator as a real experience, and never attempted anything you would not do in reality? … ‘that is true, but I am the chief pilot’

##### Peter Lee

construction, architecture

##### Fred Brooks

where’s the ‘augmented’?

##### Peter Lee

whether augmented or virtual

##### Fred Brooks

yes we did my kitchen that way, made my wife sick when she tried it

surgical

##### Fred Brooks

still sounds virtual

##### Yvonne Rogers

displays on a car, superimposed directions on the tarmac … one of the users took a vacation and had to use the old GPS technology … found it very difficult to go back

#### question: AR tools for developers?

##### Blair MacIntyre

can developers write apps for the Microsoft Hololens?

##### Peter Lee

we belive in experience, anything we can do to foster experiences is good

##### Fred Brooks

faking things … subtle and important … I remember using a flight simulator, navigating the runway, and I turned my head to see if my wing was going to clip a plane … turned and there was nothing there … emotional shock to leave the simulation, I had been flying for 1 hour

##### Ivan Sutherland

pilot training is an early adopter because the cost of real planes is so high, impossible to train for emergency situations

the ultimate goal, you can sit in a virtual chair … and if the chair has handcuffs you cannot get up … a virtual bullet is lethal … probably impossible because bits don’t weigh anything … you know Ben Franklin invented augmented reality, eyeglasses … the desire outweighs cost … I cannot see the audience here, maybe it would be good if I had a headset! but Peter took his off

##### Peter Lee

because of my slides I couldn’t see the audience, but then without the headset I couldn’t see them either

#### question: any challenges to suggest to the audience?

##### Peter Lee

if we had holographic transport, we wouldn’t need planes!

##### Yvonne Rogers

maybe, but you need to give attendees a physical presence … smell, touch

##### Ivan Sutherland

what makes us willing to work together? I had a collaboration with three people … all in different locations .. communicated with a phone … worked perfectly, because we had worked in the same location first and knew one another so well … how to get to that point, where a simulation could be a useful tool … another good observation by Fred Brooks, given a domain X ask how good does the simulation need to be for X … Licklider told me, you’d need damn good fiction to land someone on the moon, the simulation would need to provide every detail … for flight simulation the user’s imagination can fill some gaps, a pilot can recognize an aircraft carrier from a rougher picture

##### Fred Brooks

at IBM I once hired buses to bring the Poughkeepsie secretaries to the main office … the secretaries at the two offices only knew one another from the phone … this one lunch did so much good … only \$75 to rent a bus

##### Peter Lee

how important is it to shake hands, to bump into a table?

##### Yvonne Rogers

for this conference, I think the live stream is getting a better experience because the cameras zoom in on us, the panelists … the audience in the back cannot see us, only a picture of us on the monitors

##### Peter Lee

_one excellent video game, starts in the dark, you hear a voice … turn around and there’s a character sitting on a chair … if you rearrange your furniture he finds a new seat …

##### Yvonne Rogers

games are a great example … Pokemon Go … Apple jusr released an app toolkit … need to get those in schools, in the hands of kids who can build with them

#### question: Ivan, about your ‘ultimate display’ paper, what has since surprised or frustrated you?

##### Ivan Sutherland

I wasn’t surprised because I never had any expectations … of course sticks are not real … no assumptions so no strong feelings

#### question: people already distracted by cell phones, how to manage all this input?

##### Yvonne Rogers

good question, how much data you can present to people … and then the problem with google glass, your companions don’t know what you are looking at … at least with snapchat glasses, you can trust the device is simpler

##### Peter Lee

good writing defines reality, bad writing reports it … with the printing press, quickly went from 30,000 books to over 13,000,000 … novels evolved shortly after, a new form of expression

#### question: Peter, how long do your people wear the hololens?

##### Peter Lee

hard to say … but often longer than the battery lasts

##### Fred Brooks

how long does it last?

##### Peter Lee

depends what you’re doing, 3 hours

#### question: nausea

##### Peter Lee

I get nauseous in our minecraft VR … but there’s a pop-out feature where you keep playing, but the game world is in a TV set instead of around you … can pop back in when you’re feeling better

##### Yvonne Rogers

we’ve seen about 20% of the population gets nauseous

##### audience member

Dana Boyd conducted an experiment, found the nausea was worse for wemon

##### Yvonne Rogers

oculus makes me feel sick, but the hololens has never given me trouble

##### Peter Lee

have models to predict head motion, to keep the VR world steadier

##### Blair MacIntyre

I remember reading papers that measured framerate … would be interesting to revisit

##### Fred Brooks

framerate not important, its the latency that gets you … one colleague of mine, we call her ‘the canary’ because she’s so sensitive, in fact …

##### Peter Lee

talking about nausea is part of the problem, people experience it more … every time I talk about it in public my co-workers tell me to stop!

another cool application, there’s a hololens app to give blind people a tour of the Redmond office … you say a building and it takes you there

##### Fred Brooks

one challenge, the relative brightness of the real and virtual worlds

#### question: any last remarks

##### Ivan Sutherland

_I hoped from the beginning that AR would be a teaching tool … I learned that F = MA not from a book but from a large flywheel in the school’s basement … very substantial inertia … the greatest value for AR would be to show people things in a way that makes the underlying meaning clear … what color should the hydrogen atoms in a benzene ring be? the color will be fiction, but the quality of learning will depend on that fiction … challenge for content makers … what is the haptic experience of feeling bits?

### Small Group Session

After the last panel, I got to attend a small group session with other students, Dick Karp, and Don Knuth. It doesn’t feel right to summarize or quote from the session, but there’s one thing I want to write about.

During the group session, I said something that I now regret. There was a brief silence as the group changed subjects, and someone suggested that we do a round of introductions. I objected, this will take so long, but in fact the introductions were a very good idea.

Normally, I don’t like introductions because they focus on names, backgrounds, and credentials. I don’t care about any of these when I’m meeting someone! Rather, I prefer to just talk and by-the-way learn about the other person(s). There’s an anaology to double-blind reviewing — the focus should be content and not credentials.

These introductions were successful for two reasons. First, they gave everyone in the room a turn to speak, and this seemed to help people join the actual discussion sooner. That was strange to me. I always feel a little nervous the first time I speak up in front of a group, but if I really feel like speaking then I can always get over this little barrier. I guess it’s not right to assume the nervousness is “little” for everyone. Second, the introductions format was “say your name and a funny fact”. This prompt by itself led to some nice conversation topics:

• Could a computer program decide whether a statement was funny or not funny?
• What kind of humor works in a classroom? In a textbook?
• Would this kind of introduction be acceptable in another era or culture, for instance Victorian England?

“Nice” in the sense that everyone could contribute, which was a real challenge. Even the question “does anyone have a favorite algorithm?” didn’t have much success fostering discussion.

Related: a useful greeting at the event was “what SIG are you?”. The answer was a good hint about what level of abstraction you two could best communicate at.


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# the energy of scattering solution of cubic Klein-Gordon equation in $n=3$?

Let $$n=3$$ and $$u$$ be the solution to Klein-Gordon equation $$$$\begin{cases}\ddot{u}-\Delta u +u=u^3 \\ u(0)=u_0, \partial_t u(0)=u_1, \end{cases}$$$$ where $$(u_0,u_1) \in H^1 \times L^2$$. If we assume that $$u$$ exists globally and scatters to a solution $$v$$ of a free Klein-Gordon equation (the nonlinear term=0) as $$t \to \infty$$ with initial data $$(v_0,v_1)$$, then the energy $$E(u,\dot{u})$$ would equal to the energy of free Klein-Gordon $$E(v,\dot{v})$$?

where the energies are defined as below: $$$$E(u,\dot{u})(t)=\int_{\mathbb{R}^3} \frac{1}{2} \left( |u|^2+|\nabla u|^2 +|\dot{u}|^2 \right) -\frac{1}{4}|u|^4 dx, \\ E(v,\dot{v})(t)= \int_{\mathbb{R}^3} \frac{1}{2} \left( |v|^2+|\nabla v|^2 +|\dot{v}|^2 \right) dx.$$$$ And $$u$$ scatters to $$v$$ means that $$\lVert u(t)-v(t) \rVert_{H^1}+\lVert \dot u(t)-\dot v(t) \rVert_{L^2} \to 0$$ as $$t\to \infty$$.

With some classical results of conservation of energy, we can see that $$E(u,\dot{u})(t)$$ doesn't depend on time, and the same as $$v$$. But my question is: does $$E(u,\dot{ u})$$ equal to $$E(v,\dot{v})$$? And how can I check it precisely?

• In principle, the equation can be solved exactly and you can answer this question. See arxiv.org/abs/1504.02299.
– Jon
Mar 5, 2020 at 13:10
• By Sobolev inequality if $u - v \to 0$ in $H^1$ we also have $u - v \in L^4$. This means that we expect $$E(v,\dot{v})(t) - \frac14 \int_{\mathbb{R}^3} |v|^4 ~dx - E(u,\dot{u})(t) \to 0$$ So it remains to show that the $L^4$ norm of a free solution decays to 0 as $t\to \infty$. This should follow by Strichartz (I don't remember the admissible exponents for KG off the top of my head, but $L^4$ in space should be admissible). Mar 5, 2020 at 14:46
• @WillieWong thank you very much! But with Strichartz estimates, which is $\lVert u \rVert_{L_t^3 L_x^6(I \times \mathbb{R}^3)} \lesssim \lVert (u_0,u_1) \rVert_{H^1 \times L^2} + \lVert F\rVert_{L_t^1L_x^2(I \times \mathbb{R}^3)}$, where $F=u^3$ is the nonlinear term, I still cannot figure out why $\lVert u(t) \rVert_{L_x^4} \to 0$?
– Tao
Mar 5, 2020 at 15:34
• You are using the wrong estimate. I'll post an answer. Mar 5, 2020 at 15:51

Step 1: assuming scattering, there exists a solution $$v$$ to the linear Klein-Gordon equation such that $$u-v \to 0$$ in $$H^1(\mathbb{R}^3)$$ as $$t \to \infty$$. By Sobolev embedding this means that for any $$p\in [2,6]$$ you also have $$u-v \to 0$$ in $$L^p(\mathbb{R}^3)$$.

In particular, this means that $$E(u,\dot{u})(t) - E(v,\dot{v})(t) + \frac14 \int_{\mathbb{R}^3} |v(t)|^4 ~\mathrm{d}x \to 0$$ as $$t \to \infty$$. Since the first two quantity is constant. It suffices to show that $$\liminf \|v\|_{L^4_x} \to 0$$

Step 2: Since $$v$$ solves the linear Klein-Gordon equation, by the granddaddy of Strichartz estimates, you see that in three dimensions (see final page of the linked article), for any $$q\in [10/3,\infty)$$ the space-time integral estimate

$$\|v\|_{L^q_{t,x}} \lesssim E(v, \dot{v}).$$

In particular you can apply this to $$q = 4 > 10/3$$ and have that $$\|v(t)\|_{L^4_x}^4$$ is integrable in time, and hence has vanishing liminf.

• Get it! Thanks for your help.
– Tao
Mar 5, 2020 at 16:36