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# Thread: Partially Ordered Set, example from Wiki.

1. ## Partially Ordered Set, example from Wiki.

Hi,

I was reading in the Naive Set Theory book by Halmos, and encountered partially ordered sets.
It seems like a lot of people understand these things immediately. I am not one of those

I did a bit of searching and read through the wiki on posets. Here's an example they give:

Set of natural numbers equipped with the relation of divisibility.
$A=\{1,2,3,4,5,10,12,15,20,30,60\}$ is the set of all positive divisors of 60.

Then this relation (divisibility) is reflexive, transitive and antisymmetric.

(1)Reflexive: $aRa,\;for\;all\;a\in A$
So, $(a,a)\in R$

As I understand it, the ordered pair $(a,a)$ is in a partially ordered subset of $A$.
Now, I guess I can "create" these ordered pairs by taking the Cartesian Product, $AxA$.
This gives me among other things the diagonal terms $(1,1),(2,2),(3,3),\cdots,(60,60)$, all in all 11 diagonal terms.

I can then go ahead and create a subset of A that is reflexive:
perhaps, $\{(1,2),(1,1),(2,2)\}$
or, $\{(5,3),(5,5),(3,3)\}$

There are then $n^2-n$ off-diagonal elements, so there are $2^{n^2-n}$ subsets of the off-diagonal pairs.
I can combine any of these subsets with the diagonal to get a reflexive relation on A.
All in all, $2^{n^2-n}$ reflexive relations..

Is this a good way to think of the reflexive property, or am I way off?

Thanks!

2. Originally Posted by Mollier
Hi,

I was reading in the Naive Set Theory book by Halmos, and encountered partially ordered sets.
It seems like a lot of people understand these things immediately. I am not one of those

I did a bit of searching and read through the wiki on posets. Here's an example they give:

Set of natural numbers equipped with the relation of divisibility.
$A=\{1,2,3,4,5,10,12,15,20,30,60\}$ is the set of all positive divisors of 60.

Then this relation (divisibility) is reflexive, transitive and antisymmetric.

(1)Reflexive: $aRa,\;for\;all\;a\in A$
So, $(a,a)\in R$

As I understand it, the ordered pair $(a,a)$ is in a partially ordered subset of $A$.
No. All subsets of $A$ consist of members of a, not ordered pairs! Saying that divisibility is reflexive has nothing directly to do with ordered subsets of A anyway. Divisibility is reflexive because a= 1*a so a divides itself.

Now, I guess I can "create" these ordered pairs by taking the Cartesian Product, $AxA$.
This gives me among other things the diagonal terms $(1,1),(2,2),(3,3),\cdots,(60,60)$, all in all 11 diagonal terms.

I can then go ahead and create a subset of A that is reflexive:
perhaps, $\{(1,2),(1,1),(2,2)\}$
or, $\{(5,3),(5,5),(3,3)\}$
I really have no idea what you are doing. What is your objective?

There are then $n^2-n$ off-diagonal elements, so there are $2^{n^2-n}$ subsets of the off-diagonal pairs.
I can combine any of these subsets with the diagonal to get a reflexive relation on A.
All in all, $2^{n^2-n}$ reflexive relations..

Is this a good way to think of the reflexive property, or am I way off?

Thanks!
Yes, you certainly can but I don't know why you are doing it. Why do you want to know how many "reflexive relations" there are on a set? That doesn't appear to have anything to do with partial orders.

3. Hi,
the reason I am writing all this rubbish is to try and understand what reflexive,transitive and antisymmetric really mean.
In my last post, I just focused on reflexive, and of course go it all wrong.

So, a reflexive relation is a relation on a set for which every element is related to itself.
Can I say that $\leq$ is a reflexive relation on the set $A=\{1,2,3\}$, because $1=1,2=2,3=3$?

$<$ then is not a reflexive relation on set $A=\{1,2,3\}$..

You mentioned that divisibility is reflexive because $a=1*a$, I think that makes sense.

I've read somewhere that $aRa$ for all $a\in A$ can be seen as $(a,a)\in R$ What is that all about?
I actually read it here on the forums:
http://www.mathhelpforum.com/math-he...ial-order.html

Thanks

4. Hi,
the reason I am writing all this rubbish is to try and understand what reflexive,transitive and antisymmetric really mean.
In my last post, I just focused on reflexive, and of course go it all wrong.

So, a reflexive relation is a relation on a set for which every element is related to itself.
Can I say that $\leq$ is a reflexive relation on the set $A=\{1,2,3\}$, because $1=1,2=2,3=3$?

$<$ then is not a reflexive relation on set $A=\{1,2,3\}$..

You mentioned that divisibility is reflexive because $a=1*a$, I think that makes sense.

Thanks

5. the reason I am writing all this rubbish is to try and understand what reflexive,transitive and antisymmetric really mean.
In my last post, I just focused on reflexive, and of course go it all wrong.
One has to understand first what a relation is. A relation $R$ on a set $A$ is a function of two arguments that returns true or false: for all $a_1\in A$, $a_2\in A$ we have $R(a_1,a_2)\in\{\text{true},\text{false}\}$. (And, of course, one first has to understand what a function of several arguments is.) The fact that $R(a_1,a_2)=\text{true}$ is conventionally written $a_1Ra_2$, e.g., $1\le 3$.

Now, there are many ways to specify a function, or even to define what a function is. Some options are: it's a law that puts a single result into correspondence with each argument; it's an algorithm that computes the result from an argument; it's a set of pairs $F$ with one condition: $\langle x,y_1\rangle\in F$ and $\langle x,y_2\rangle\in F$ imply $y_1=y_2$ for all $x, y_1, y_2$. It makes sense to specify the details only when one really needs them, e.g., in a proof. In set theory, the last option above is taken as a definition, though it is only one "implementation" of an abstract concept of a function.

A relation viewed as a function of two arguments, would be considered a set of nested pairs, e.g., $\langle\langle a_1,a_2\rangle,\text{true}rangle$. One prefers a simpler implementation, where one removes pairs where the last element is $\text{false}$ and keeps only the first element $\langle a_1,a_2\rangle$ of other pairs. There is nothing special about this implementation, except that it is probably the simplest way to implement a relation. An important point is that it should be easily possible to convert one implementation into another.

Therefore, the notation $a_1Ra_2$, where $R$ is an "abstract" relation, becomes $\langle a_1,a_2\rangle\in R$ where $R$ is implemented as a set of pairs.

Also, it is important to keep in mind the type of every object one deals with. For example, a set of numbers consists of numbers, which are themselves not sets. An partial order is a ordered pair $\langle A, R\rangle$ where $A$ is a set and $R$ is a relation on $R$ satisfying some axioms. I.e., here $R$ can be viewed consisting of ordered pairs of elements of $A$. Of course, one becomes a little sloppy soon and, e.g., talks about a subset of a partial order $\langle A, R\rangle$ meaning a subset $A'$ of $A$ with the restriction of $R$ on $A'$. Literally, one can't take a subset of an ordered pair $\langle A, R\rangle$ unless one considers a specific implementation of ordered pairs. Such implicit coercions are normal, but one has to be able to identify them when needed.

6. Emarkov: Thanks for the reply.

Let me rephrase my question and not attempt any answer myself

How can I tell that the set of natural numbers equipped with the relation of divisibility is a partially ordered set?

Thanks.

7. Originally Posted by Mollier

Let me rephrase my question and not attempt any answer myself

How can I tell that the set of natural numbers equipped with the relation of divisibility is a partially ordered set?

Thanks.
Why not just verify the axioms?

Clearly $n\mid n$. If $n\mid n'$ then $n'=nz$ for some $z\in\mathbb{Z}$ and similarly we have that $n=nz'$. So then we have that $z=\frac{1}{z'}$ but since $z\in\mathbb{Z}$ it follows that $z'=\pm 1$. But, since we are only dealing with natural numbers $z'=1\implies z=1\implies n=n'$ so it has anti-symmetry. If $n\mid m,m\mid \ell$ then $\ell=zm=zz'n\implies n\mid \ell$ so it has transitivity.

8. Just rewriting a bit here:

If $n|n'$ and $n'|n$ then $n=n'$.

$n|n' \Rightarrow n=zn'$

$n'|n \Rightarrow n'=zn$

$n'=z^2n' \Rightarrow z^2=1 \Rightarrow z= \pm 1$

Since only natural numbers: $z=1$

Hence, $n=n'$

Thanks mate!

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Compact and matrix operators on the space $(| A_{f}^{\theta})|_{k}$

F.Gökçe, G. C. Hazar Güleç

In this study, we introduce a new space $\left\vert A_{f}^{\theta }\right\vert _{k}$ \ by using factorable matrix and investigate its certain topological and algebraic structures where $\theta$ is a positive sequence. Also, we characterize some matrix operators on this space and determine their norms and the Hausdorff measure of noncompactness. In the particular case, we get some well known results.

Tbilisi Mathematical Journal, Vol. 12(4) (2019), pp. 1-13


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The Kinetic Theory of Gases Introduction (Exercises)

Conceptual Questions

2.1 Molecular Model of an Ideal Gas

1. Two $$\displaystyle H_2$$ molecules can react with one (O_2\) molecule to produce two $$\displaystyle H_2O$$ molecules. How many moles of hydrogen molecules are needed to react with one mole of oxygen molecules?

2. Under what circumstances would you expect a gas to behave significantly differently than predicted by the ideal gas law?

3. A constant-volume gas thermometer contains a fixed amount of gas. What property of the gas is measured to indicate its temperature?

4. Inflate a balloon at room temperature. Leave the inflated balloon in the refrigerator overnight. What happens to the balloon, and why?

5. In the last chapter, free convection was explained as the result of buoyant forces on hot fluids. Explain the upward motion of air in flames based on the ideal gas law.

2.2 Pressure, Temperature, and RMS Speed

6. How is momentum related to the pressure exerted by a gas? Explain on the molecular level, considering the behavior of molecules.

7. If one kind of molecule has double the radius of another and eight times the mass, how do their mean free paths under the same conditions compare? How do their mean free times compare?

8. What is the average velocity of the air molecules in the room where you are right now?

9. Why do the atmospheres of Jupiter, Saturn, Uranus, and Neptune, which are much more massive and farther from the Sun than Earth is, contain large amounts of hydrogen and helium?

10. Statistical mechanics says that in a gas maintained at a constant temperature through thermal contact with a bigger system (a “reservoir”) at that temperature, the fluctuations in internal energy are typically a fraction $$\displaystyle 1/\sqrt{N}$$ of the internal energy. As a fraction of the total internal energy of a mole of gas, how big are the fluctuations in the internal energy? Are we justified in ignoring them?

11. Which is more dangerous, a closet where tanks of nitrogen are stored, or one where tanks of carbon dioxide are stored?

2.3 Heat Capacity and Equipartition of Energy

12. Experimentally it appears that many polyatomic molecules’ vibrational degrees of freedom can contribute to some extent to their energy at room temperature. Would you expect that fact to increase or decrease their heat capacity from the value R? Explain.

13. One might think that the internal energy of diatomic gases is given by $$\displaystyle E_{int}=5RT/2$$.. Do diatomic gases near room temperature have more or less internal energy than that? Hint: Their internal energy includes the total energy added in raising the temperature from the boiling point (very low) to room temperature.

14. You mix 5 moles of $$\displaystyle H_2$$ at 300 K with 5 moles of He at 360 K in a perfectly insulated calorimeter. Is the final temperature higher or lower than 330 K?

2.4 Distribution of Molecular Speeds

15. One cylinder contains helium gas and another contains krypton gas at the same temperature. Mark each of these statements true, false, or impossible to determine from the given information.

(a) The rms speeds of atoms in the two gases are the same.

(b) The average kinetic energies of atoms in the two gases are the same.

(c) The internal energies of 1 mole of gas in each cylinder are the same.

(d) The pressures in the two cylinders are the same.

16. Repeat the previous question if one gas is still helium but the other is changed to fluorine, $$\displaystyle F_2$$.

17. An ideal gas is at a temperature of 300 K. To double the average speed of its molecules, what does the temperature need to be changed to?

Problems

2.1 Molecular Model of an Ideal Gas

18. The gauge pressure in your car tires is $$\displaystyle 2.50×10^5N/m^2$$ a temperature of 35.0°C when you drive it onto a ship in Los Angeles to be sent to Alaska. What is their gauge pressure on a night in Alaska when their temperature has dropped to −40.0°C? Assume the tires have not gained or lost any air.

19. Suppose a gas-filled incandescent light bulb is manufactured so that the gas inside the bulb is at atmospheric pressure when the bulb has a temperature of 20.0°C.

(a) Find the gauge pressure inside such a bulb when it is hot, assuming its average temperature is 60.0°C (an approximation) and neglecting any change in volume due to thermal expansion or gas leaks.

(b) The actual final pressure for the light bulb will be less than calculated in part (a) because the glass bulb will expand. Is this effect significant?

20. People buying food in sealed bags at high elevations often notice that the bags are puffed up because the air inside has expanded. A bag of pretzels was packed at a pressure of 1.00 atm and a temperature of 22.0°C.When opened at a summer picnic in Santa Fe, New Mexico, at a temperature of 32.0°C, the volume of the air in the bag is 1.38 times its original volume. What is the pressure of the air?

21. How many moles are there in

(a) 0.0500 g of $$\displaystyle N_2$$ gas (M=28.0g/mol)?

(b) 10.0 g of $$\displaystyle CO_2$$ gas (M=44.0g/mol)?

(c) How many molecules are present in each case?

22. A cubic container of volume 2.00 L holds 0.500 mol of nitrogen gas at a temperature of 25.0°C. What is the net force due to the nitrogen on one wall of the container? Compare that force to the sample’s weight.

23. Calculate the number of moles in the 2.00-L volume of air in the lungs of the average person. Note that the air is at 37.0°C (body temperature) and that the total volume in the lungs is several times the amount inhaled in a typical breath as given in Example 2.2.

24. An airplane passenger has $$\displaystyle 100cm^3$$ of air in his stomach just before the plane takes off from a sea-level airport. What volume will the air have at cruising altitude if cabin pressure drops to $$\displaystyle 7.50×10^4N/m^2$$?

25. A company advertises that it delivers helium at a gauge pressure of $$\displaystyle 1.72×10^7Pa$$ in a cylinder of volume 43.8 L. How many balloons can be inflated to a volume of 4.00 L with that amount of helium? Assume the pressure inside the balloons is 1.01×105Pa1.01×105Pa and the temperature in the cylinder and the balloons is 25.0°C25.0°C.

26. According to http://hyperphysics.phy-astr.gsu.edu.../venusenv.html, the atmosphere of Venus is approximately $$\displaystyle 96.5%CO_2$$ and $$\displaystyle 3.5%N_2$$ by volume. On the surface, where the temperature is about 750 K and the pressure is about 90 atm, what is the density of the atmosphere?

27. An expensive vacuum system can achieve a pressure as low as $$\displaystyle 1.00×10^{−7}N/m^2$$ at 20.0°C. How many molecules are there in a cubic centimeter at this pressure and temperature?

28. The number density N/V of gas molecules at a certain location in the space above our planet is about $$\displaystyle 1.00×10^{11}m^{−3}$$, and the pressure is $$\displaystyle 2.75×10^{−10N}/m^2$$ in this space. What is the temperature there?

29. A bicycle tire contains 2.00 L of gas at an absolute pressure of $$\displaystyle 7.00×10^5N/m^2$$ and a temperature of 18.0°C. What will its pressure be if you let out an amount of air that has a volume of $$\displaystyle 100cm^3$$ at atmospheric pressure? Assume tire temperature and volume remain constant.

30. In a common demonstration, a bottle is heated and stoppered with a hard-boiled egg that’s a little bigger than the bottle’s neck. When the bottle is cooled, the pressure difference between inside and outside forces the egg into the bottle. Suppose the bottle has a volume of 0.500 L and the temperature inside it is raised to 80.0°C while the pressure remains constant at 1.00 atm because the bottle is open.

(a) How many moles of air are inside?

(b) Now the egg is put in place, sealing the bottle. What is the gauge pressure inside after the air cools back to the ambient temperature of 25°C but before the egg is forced into the bottle?

31. A high-pressure gas cylinder contains 50.0 L of toxic gas at a pressure of $$\displaystyle 1.40×10^7N/m^2$$ and a temperature of 25.0°C. The cylinder is cooled to dry ice temperature (−78.5°C) to reduce the leak rate and pressure so that it can be safely repaired.

(a) What is the final pressure in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change?

(b) What is the final pressure if one-tenth of the gas escapes? (c) To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (assuming the gas does not change phase and that there is no leakage during cooling)? (d) Does cooling the tank as in part

(c) appear to be a practical solution?

32. Find the number of moles in 2.00 L of gas at 35.0°Cand under $$\displaystyle 7.41×10^7N/m^2$$ of pressure.

33. Calculate the depth to which Avogadro’s number of table tennis balls would cover Earth. Each ball has a diameter of 3.75 cm. Assume the space between balls adds an extra 25.0% to their volume and assume they are not crushed by their own weight.

34. a) What is the gauge pressure in a 25.0°C car tire containing 3.60 mol of gas in a 30.0-L volume?

(b) What will its gauge pressure be if you add 1.00 L of gas originally at atmospheric pressure and 25.0°C ? Assume the temperature remains at 25.0°C and the volume remains constant.

2.2 Pressure, Temperature, and RMS Speed

In the problems in this section, assume all gases are ideal.

35. A person hits a tennis ball with a mass of 0.058 kg against a wall. The average component of the ball’s velocity perpendicular to the wall is 11 m/s, and the ball hits the wall every 2.1 s on average, rebounding with the opposite perpendicular velocity component.

(a) What is the average force exerted on the wall?

(b) If the part of the wall the person hits has an area of $$\displaystyle 3.0m^2$$, what is the average pressure on that area?

36. A person is in a closed room (a racquetball court) with $$\displaystyle V=453m^3$$ hitting a ball (m=42.0g)(m=42.0g) around at random without any pauses. The average kinetic energy of the ball is 2.30 J.

(a) What is the average value of $$\displaystyle v^2_x$$? Does it matter which direction you take to be x?

(b) Applying the methods of this chapter, find the average pressure on the walls?

(c) Aside from the presence of only one “molecule” in this problem, what is the main assumption in Pressure, Temperature, and RMS Speed that does not apply here?

37. Five bicyclists are riding at the following speeds: 5.4 m/s, 5.7 m/s, 5.8 m/s, 6.0 m/s, and 6.5 m/s. (a) What is their average speed? (b) What is their rms speed?

38. Some incandescent light bulbs are filled with argon gas. What is $$\displaystyle v_{rms}$$ for argon atoms near the filament, assuming their temperature is 2500 K?

39. Typical molecular speeds ($$\displaystyle v_{rms}$$) are large, even at low temperatures. What is $$\displaystyle v_{rms}$$ for helium atoms at 5.00 K, less than one degree above helium’s liquefaction temperature?

40. What is the average kinetic energy in joules of hydrogen atoms on the 5500°C surface of the Sun?

(b) What is the average kinetic energy of helium atoms in a region of the solar corona where the temperature is $$\displaystyle 6.00×10^5K$$ ?

41. What is the ratio of the average translational kinetic energy of a nitrogen molecule at a temperature of 300 K to the gravitational potential energy of a nitrogen-molecule−Earth system at the ceiling of a 3-m-tall room with respect to the same system with the molecule at the floor?

42. What is the total translational kinetic energy of the air molecules in a room of volume $$\displaystyle 23m^3$$ if the pressure is $$\displaystyle 9.5×10^4Pa$$ (the room is at fairly high elevation) and the temperature is 21°C? Is any item of data unnecessary for the solution?

43. The product of the pressure and volume of a sample of hydrogen gas at 0.00°C is 80.0 J.

(a) How many moles of hydrogen are present?

(b) What is the average translational kinetic energy of the hydrogen molecules?

(c) What is the value of the product of pressure and volume at 200°C?

44. What is the gauge pressure inside a tank of $$\displaystyle 4.86×10^4mol$$ of compressed nitrogen with a volume of $$\displaystyle 6.56m^3$$ if the rms speed is 514 m/s?

45. If the rms speed of oxygen molecules inside a refrigerator of volume $$\displaystyle 22.0ft.^3$$ is 465 m/s, what is the partial pressure of the oxygen? There are 5.71 moles of oxygen in the refrigerator, and the molar mass of oxygen is 32.0 g/mol.

46. The escape velocity of any object from Earth is 11.1 km/s. At what temperature would oxygen molecules (molar mass is equal to 32.0 g/mol) have root-mean-square velocity $$\displaystyle v_{rms}$$ equal to Earth’s escape velocity of 11.1 km/s?

47. The escape velocity from the Moon is much smaller than that from the Earth, only 2.38 km/s. At what temperature would hydrogen molecules (molar mass is equal to 2.016 g/mol) have a root-mean-square velocity vrmsvrms equal to the Moon’s escape velocity?

48. Nuclear fusion, the energy source of the Sun, hydrogen bombs, and fusion reactors, occurs much more readily when the average kinetic energy of the atoms is high—that is, at high temperatures. Suppose you want the atoms in your fusion experiment to have average kinetic energies of $$\displaystyle 6.40×10^{−14}J$$. What temperature is needed?

49. Suppose that the typical speed ($$\displaystyle v_{rms}$$) of carbon dioxide molecules (molar mass is 44.0 g/mol) in a flame is found to be 1350 m/s. What temperature does this indicate?

50. (a) Hydrogen molecules (molar mass is equal to 2.016 g/mol) have vrmsvrms equal to 193 m/s. What is the temperature? (b) Much of the gas near the Sun is atomic hydrogen (H rather than $$\displaystyle H_2$$). Its temperature would have to be $$\displaystyle 1.5×10^7K$$ for the rms speed vrmsvrms to equal the escape velocity from the Sun. What is that velocity?

51. There are two important isotopes of uranium, $$\displaystyle ^{235}U$$ and $$\displaystyle ^{238}U$$; these isotopes are nearly identical chemically but have different atomic masses. Only $$\displaystyle ^{235}U$$ is very useful in nuclear reactors. Separating the isotopes is called uranium enrichment (and is often in the news as of this writing, because of concerns that some countries are enriching uranium with the goal of making nuclear weapons.) One of the techniques for enrichment, gas diffusion, is based on the different molecular speeds of uranium hexafluoride gas, $$\displaystyle UF_6$$.

(a) The molar masses of $$\displaystyle ^{235}U$$ and $$\displaystyle ^{238}UF_6$$ are 349.0 g/mol and 352.0 g/mol, respectively. What is the ratio of their typical speeds vrmsvrms?

(b) At what temperature would their typical speeds differ by 1.00 m/s?

(c) Do your answers in this problem imply that this technique may be difficult?

52. The partial pressure of carbon dioxide in the lungs is about 470 Pa when the total pressure in the lungs is 1.0 atm. What percentage of the air molecules in the lungs is carbon dioxide? Compare your result to the percentage of carbon dioxide in the atmosphere, about 0.033%.

53. Dry air consists of approximately 78%nitrogen,21%oxygen,and 1%argon by mole, with trace amounts of other gases. A tank of compressed dry air has a volume of 1.76 cubic feet at a gauge pressure of 2200 pounds per square inch and a temperature of 293 K. How much oxygen does it contain in moles?

54. (a) Using data from the previous problem, find the mass of nitrogen, oxygen, and argon in 1 mol of dry air. The molar mass of $$\displaystyle N_2$$ is 28.0 g/mol, that of $$\displaystyle O_2$$ is 32.0 g/mol, and that of argon is 39.9 g/mol.

(b) Dry air is mixed with pentane ($$\displaystyle C_5H_{12}$$, molar mass 72.2 g/mol), an important constituent of gasoline, in an air-fuel ratio of 15:1 by mass (roughly typical for car engines). Find the partial pressure of pentane in this mixture at an overall pressure of 1.00 atm.

55. (a) Given that air is 21% oxygen, find the minimum atmospheric pressure that gives a relatively safe partial pressure of oxygen of 0.16 atm.

(b) What is the minimum pressure that gives a partial pressure of oxygen above the quickly fatal level of 0.06 atm?

(c) The air pressure at the summit of Mount Everest (8848 m) is 0.334 atm. Why have a few people climbed it without oxygen, while some who have tried, even though they had trained at high elevation, had to turn back?

56. (a) If the partial pressure of water vapor is 8.05 torr, what is the dew point? (760torr=1atm=101,325Pa)

(b) On a warm day when the air temperature is 35°C and the dew point is 25°C, what are the partial pressure of the water in the air and the relative humidity?

2.3 Heat Capacity and Equipartition of Energy

57. To give a helium atom nonzero angular momentum requires about 21.2 eV of energy (that is, 21.2 eV is the difference between the energies of the lowest-energy or ground state and the lowest-energy state with angular momentum). The electron-volt or eV is defined as $$\displaystyle 1.60×10^{−19}J.$$ Find the temperature T where this amount of energy equals $$\displaystyle k_BT/2$$. Does this explain why we can ignore the rotational energy of helium for most purposes? (The results for other monatomic gases, and for diatomic gases rotating around the axis connecting the two atoms, have comparable orders of magnitude.)

58. (a) How much heat must be added to raise the temperature of 1.5 mol of air from 25.0°C to 33.0°C at constant volume? Assume air is completely diatomic.

(b) Repeat the problem for the same number of moles of xenon, Xe.

59. A sealed, rigid container of 0.560 mol of an unknown ideal gas at a temperature of 30.0°C is cooled to −40.0°C. In the process, 980 J of heat are removed from the gas. Is the gas monatomic, diatomic, or polyatomic?

60. A sample of neon gas (Ne, molar mass M=20.2g/mol) at a temperature of 13.0°C is put into a steel container of mass 47.2 g that’s at a temperature of −40.0°C. The final temperature is −28.0°C. (No heat is exchanged with the surroundings, and you can neglect any change in the volume of the container.) What is the mass of the sample of neon?

61. A steel container of mass 135 g contains 24.0 g of ammonia, $$\displaystyle NH_3$$, which has a molar mass of 17.0 g/mol. The container and gas are in equilibrium at 12.0°C. How much heat has to be removed to reach a temperature of −20.0°C? Ignore the change in volume of the steel.

62. A sealed room has a volume of $$\displaystyle 24m^3$$. It’s filled with air, which may be assumed to be diatomic, at a temperature of 24°C and a pressure of 9.83×104Pa.9.83×104Pa. A 1.00-kg block of ice at its melting point is placed in the room. Assume the walls of the room transfer no heat. What is the equilibrium temperature?

63. Heliox, a mixture of helium and oxygen, is sometimes given to hospital patients who have trouble breathing, because the low mass of helium makes it easier to breathe than air. Suppose helium at 25°C is mixed with oxygen at 35°C to make a mixture that is 70% helium by mole. What is the final temperature? Ignore any heat flow to or from the surroundings, and assume the final volume is the sum of the initial volumes.

64. Professional divers sometimes use heliox, consisting of 79% helium and 21% oxygen by mole. Suppose a perfectly rigid scuba tank with a volume of 11 L contains heliox at an absolute pressure of $$\displaystyle 2.1×10^7Pa$$ at a temperature of 31°C

(a) How many moles of helium and how many moles of oxygen are in the tank?

(b) The diver goes down to a point where the sea temperature is 27°C while using a negligible amount of the mixture. As the gas in the tank reaches this new temperature, how much heat is removed from it?

65. In car racing, one advantage of mixing liquid nitrous oxide ($$\displaystyle N_2O$$) with air is that the boiling of the “nitrous” absorbs latent heat of vaporization and thus cools the air and ultimately the fuel-air mixture, allowing more fuel-air mixture to go into each cylinder. As a very rough look at this process, suppose 1.0 mol of nitrous oxide gas at its boiling point, −88°C, is mixed with 4.0 mol of air (assumed diatomic) at 30°C30°C. What is the final temperature of the mixture? Use the measured heat capacity of $$\displaystyle N_2O$$ at 25°C, which is 30.4J/mol°C. (The primary advantage of nitrous oxide is that it consists of 1/3 oxygen, which is more than air contains, so it supplies more oxygen to burn the fuel. Another advantage is that its decomposition into nitrogen and oxygen releases energy in the cylinder.)

2.4 Distribution of Molecular Speeds

66. In a sample of hydrogen sulfide (M=34.1g/mol) at a temperature of $$\displaystyle 3.00×10^2K$$, estimate the ratio of the number of molecules that have speeds very close to $$\displaystyle v_{rms}$$ to the number that have speeds very close to $$\displaystyle 2v_{rms}$$.

67. Using the approximation $$\displaystyle ∫^{v_1+Δv}_{v_1}f(v)dv≈f(v1)Δv$$ for small Δv, estimate the fraction of nitrogen molecules at a temperature of $$\displaystyle 3.00×10^2K$$ that have speeds between 290 m/s and 291 m/s.

68. Using the method of the preceding problem, estimate the fraction of nitric oxide (NO) molecules at a temperature of 250 K that have energies between $$\displaystyle 3.45×10^{−21}J$$ and $$\displaystyle 3.50×10^{−21}J$$.

69. By counting squares in the following figure, estimate the fraction of argon atoms at T=300K that have speeds between 600 m/s and 800 m/s. The curve is correctly normalized. The value of a square is its length as measured on the x-axis times its height as measured on the y-axis, with the units given on those axes.

70. Using a numerical integration method such as Simpson’s rule, find the fraction of molecules in a sample of oxygen gas at a temperature of 250 K that have speeds between 100 m/s and 150 m/s. The molar mass of oxygen ($$\displaystyle O_2$$) is 32.0 g/mol. A precision to two significant digits is enough.

71. Find (a) the most probable speed,

(b) the average speed, and

(c) the rms speed for nitrogen molecules at 295 K.

72. Repeat the preceding problem for nitrogen molecules at 2950 K.

73. At what temperature is the average speed of carbon dioxide molecules (M=44.0g/mol) 510 m/s?

74. The most probable speed for molecules of a gas at 296 K is 263 m/s. What is the molar mass of the gas? (You might like to figure out what the gas is likely to be.)

75. a) At what temperature do oxygen molecules have the same average speed as helium atoms (M=4.00g/mol) have at 300 K?

b) What is the answer to the same question about most probable speeds?

c) What is the answer to the same question about rms speeds?

76. In the deep space between galaxies, the density of molecules (which are mostly single atoms) can be as low as $$\displaystyle 10^6atoms/m^3$$, and the temperature is a frigid 2.7 K. What is the pressure?

(b) What volume (in $$\displaystyle m^3$$) is occupied by 1 mol of gas?

(c) If this volume is a cube, what is the length of its sides in kilometers?

77. (a) Find the density in SI units of air at a pressure of 1.00 atm and a temperature of 20°C20°C, assuming that air is $$\displaystyle 78%N_2,21%O_2,$$ and $$\displaystyle 1%Ar$$

(b) Find the density of the atmosphere on Venus, assuming that it’s $$\displaystyle 96%CO_2$$ and $$\displaystyle 4%N_2$$, with a temperature of 737 K and a pressure of 92.0 atm.

78. The air inside a hot-air balloon has a temperature of 370 K and a pressure of 101.3 kPa, the same as that of the air outside. Using the composition of air as $$\displaystyle 78%N_2,21%O_2,$$ and $$\displaystyle 1%Ar$$, find the density of the air inside the balloon.

79. When an air bubble rises from the bottom to the top of a freshwater lake, its volume increases by 80%. If the temperatures at the bottom and the top of the lake are 4.0 and 10 °C, respectively, how deep is the lake?

80. (a) Use the ideal gas equation to estimate the temperature at which 1.00 kg of steam (molar mass M=18.0g/mol) at a pressure of $$\displaystyle 1.50×10^6Pa$$ occupies a volume of $$\displaystyle 0.220m^3$$.

(b) The van der Waals constants for water are $$\displaystyle a=0.5537Pa⋅m^6/mol^2$$ and $$\displaystyle b=3.049×10^{−5}m^3/mol$$. Use the Van der Waals equation of state to estimate the temperature under the same conditions.

(c) The actual temperature is 779 K. Which estimate is better?

81. One process for decaffeinating coffee uses carbon dioxide (M=44.0g/mol) at a molar density of about $$\displaystyle 14,600mol/m^3$$ and a temperature of about 60°C.

(a) Is $$\displaystyle CO_2$$ a solid, liquid, gas, or supercritical fluid under those conditions?

(b) The van der Waals constants for carbon dioxide are $$\displaystyle a=0.3658Pa⋅m^6/mol^2$$ and $$\displaystyle b=4.286×10^{−5}m^3/mol$$. Using the van der Waals equation, estimate the pressure of $$\displaystyle CO_2$$ at that temperature and density.

82. On a winter day when the air temperature is 0°C, the relative humidity is 50%. Outside air comes inside and is heated to a room temperature of 20°C. What is the relative humidity of the air inside the room. (Does this problem show why inside air is so dry in winter?)

83. On a warm day when the air temperature is 30°C, a metal can is slowly cooled by adding bits of ice to liquid water in it. Condensation first appears when the can reaches 15°C. What is the relative humidity of the air?

84. (a) People often think of humid air as “heavy.” Compare the densities of air with 0% relative humidity and 100%100% relative humidity when both are at 1 atm and 30°C. Assume that the dry air is an ideal gas composed of molecules with a molar mass of 29.0 g/mol and the moist air is the same gas mixed with water vapor.

(b) As discussed in the chapter on the applications of Newton’s laws, the air resistance felt by projectiles such as baseballs and golf balls is approximately $$\displaystyle F_D=CρAv^2/2$$, where ρ is the mass density of the air, A is the cross-sectional area of the projectile, and C is the projectile’s drag coefficient. For a fixed air pressure, describe qualitatively how the range of a projectile changes with the relative humidity.

(c) When a thunderstorm is coming, usually the humidity is high and the air pressure is low. Do those conditions give an advantage or disadvantage to home-run hitters?

85. The mean free path for helium at a certain temperature and pressure is $$\displaystyle 2.10×10^{−7}m$$. The radius of a helium atom can be taken as $$\displaystyle 1.10×10^{−11}m$$. What is the measure of the density of helium under those conditions

(a) in molecules per cubic meter and

(b) in moles per cubic meter?

86. The mean free path for methane at a temperature of 269 K and a pressure of $$\displaystyle 1.11×10^5Pa$$ is $$\displaystyle 4.81×10^{−8}m$$. Find the effective radius r of the methane molecule.

87. In the chapter on fluid mechanics, Bernoulli’s equation for the flow of incompressible fluids was explained in terms of changes affecting a small volume dV of fluid. Such volumes are a fundamental idea in the study of the flow of compressible fluids such as gases as well. For the equations of hydrodynamics to apply, the mean free path must be much less than the linear size of such a volume, $$\displaystyle a≈dV^{1/3}$$. For air in the stratosphere at a temperature of 220 K and a pressure of 5.8 kPa, how big should a be for it to be 100 times the mean free path? Take the effective radius of air molecules to be $$\displaystyle 1.88×10^{−11}m$$, which is roughly correct for $$\displaystyle N_2$$.

88. Find the total number of collisions between molecules in 1.00 s in 1.00 L of nitrogen gas at standard temperature and pressure (0°C, 1.00 atm). Use $$\displaystyle 1.88×10^{−10}m$$ as the effective radius of a nitrogen molecule. (The number of collisions per second is the reciprocal of the collision time.) Keep in mind that each collision involves two molecules, so if one molecule collides once in a certain period of time, the collision of the molecule it hit cannot be counted.

89. (a) Estimate the specific heat capacity of sodium from the Law of Dulong and Petit. The molar mass of sodium is 23.0 g/mol.

(b) What is the percent error of your estimate from the known value, 1230J/kg⋅°C?

90. A sealed, perfectly insulated container contains 0.630 mol of air at 20.0°C and an iron stirring bar of mass 40.0 g. The stirring bar is magnetically driven to a kinetic energy of 50.0 J and allowed to slow down by air resistance. What is the equilibrium temperature?

91. Find the ratio $$\displaystyle f(v_p)/f(v_{rms})$$ for hydrogen gas (M=2.02g/mol) at a temperature of 77.0 K.

92. Unreasonable results. (a) Find the temperature of 0.360 kg of water, modeled as an ideal gas, at a pressure of $$\displaystyle 1.01×10^5Pa$$ if it has a volume of $$\displaystyle 0.615m^3$$.

93. Unreasonable results. (a) Find the average speed of hydrogen sulfide, $$\displaystyle H_2S$$, molecules at a temperature of 250 K. Its molar mass is 31.4 g/mol

(b) The result isn’t very unreasonable, but why is it less reliable than those for, say, neon or nitrogen?

Challenge Problems

94. An airtight dispenser for drinking water is 25cm×10cm in horizontal dimensions and 20 cm tall. It has a tap of negligible volume that opens at the level of the bottom of the dispenser. Initially, it contains water to a level 3.0 cm from the top and air at the ambient pressure, 1.00 atm, from there to the top. When the tap is opened, water will flow out until the gauge pressure at the bottom of the dispenser, and thus at the opening of the tap, is 0. What volume of water flows out? Assume the temperature is constant, the dispenser is perfectly rigid, and the water has a constant density of $$\displaystyle 1000kg/m^3$$.

95. Eight bumper cars, each with a mass of 322 kg, are running in a room 21.0 m long and 13.0 m wide. They have no drivers, so they just bounce around on their own. The rms speed of the cars is 2.50 m/s. Repeating the arguments of Pressure, Temperature, and RMS Speed, find the average force per unit length (analogous to pressure) that the cars exert on the walls.

96. Verify that $$\displaystyle v_p=\sqrt{\frac{2k_BT}{m}}$$.

97. Verify the normalization equation $$\displaystyle ∫^∞_0f(v)dv=1$$. In doing the integral, first make the substitution $$\displaystyle u=\sqrt{\frac{m}{2k_BT}}v=\frac{v}{v_p}$$.This “scaling” transformation gives you all features of the answer except for the integral, which is a dimensionless numerical factor. You’ll need the formula $$\displaystyle ∫^∞_0x^2e^{−x^2}dx=\frac{\sqrt{π}}{4}$$ find the numerical factor and verify the normalization.

98. Verify that $$\displaystyle \bar{v}=\sqrt{\frac{8}{π}\frac{k_BT}{m}}.$$. Make the same scaling transformation as in the preceding problem.

99. Verify that $$\displaystyle v_{rms}=\sqrt{\bar{v^2}=\sqrt{\frac{3k_BT}{m}}$$.


text:
# qml¶

This is the top level module from which all basic functions and classes of PennyLane can be directly imported.

## Functions¶

 Prints the information for pennylane installation. adjoint(fn[, lazy]) Create the adjoint of an Operator or a function that applies the adjoint of the provided function. adjoint_metric_tensor(circuit[, device, hybrid]) Implements the adjoint method outlined in Jones to compute the metric tensor. apply(op[, context]) Apply an instantiated operator or measurement to a queuing context. apply_controlled_Q(fn, wires, target_wire, …) Provides the circuit to apply a controlled version of the $$\mathcal{Q}$$ unitary defined in this paper. batch_input(tape[, argnum]) Transform a QNode to support an initial batch dimension for gate inputs. batch_params(tape[, all_operations]) Transform a QNode to support an initial batch dimension for operation parameters. batch_partial(qnode[, all_operations, …]) Create a batched partial callable object from the QNode specified. broadcast(unitary, wires, pattern[, …]) Applies a unitary multiple times to a specific pattern of wires. commutation_dag(circuit) Construct the pairwise-commutation DAG (directed acyclic graph) representation of a quantum circuit. compile(tape[, pipeline, basis_set, …]) Compile a circuit by applying a series of transforms to a quantum function. cond(condition, true_fn[, false_fn]) Condition a quantum operation on the results of mid-circuit qubit measurements. ctrl(fn, control[, control_values]) Create a method that applies a controlled version of the provided method. cut_circuit(tape[, auto_cutter, …]) Cut up a quantum circuit into smaller circuit fragments. cut_circuit_mc(tape[, …]) Cut up a circuit containing sample measurements into smaller fragments using a Monte Carlo method. Quantum function transform that substitutes operations conditioned on measurement outcomes to controlled operations. density_matrix(wires) Quantum density matrix in the computational basis. device(name[, wires]) Load a Device and return the instance. dot(x, y) Lazily perform the dot product between arrays, tensors, and QNodeCollection. draw(qnode[, wire_order, show_all_wires, …]) Create a function that draws the given qnode. draw_mpl(qnode[, wire_order, …]) Draw a qnode with matplotlib eigvals(op[, k, which]) The eigenvalues of one or more operations. equal(op1, op2[, check_interface, …]) Function for determining operator equality. execute(tapes, device, gradient_fn[, …]) Execute a batch of tapes on a device in an autodifferentiable-compatible manner. expval(op) Expectation value of the supplied observable. from_pyquil(pyquil_program) Loads pyQuil Program objects by using the converter in the PennyLane-Forest plugin. from_qasm(quantum_circuit) Loads quantum circuits from a QASM string using the converter in the PennyLane-Qiskit plugin. from_qasm_file(qasm_filename) Loads quantum circuits from a QASM file using the converter in the PennyLane-Qiskit plugin. from_qiskit(quantum_circuit) Loads Qiskit QuantumCircuit objects by using the converter in the PennyLane-Qiskit plugin. from_quil(quil) Loads quantum circuits from a Quil string using the converter in the PennyLane-Forest plugin. from_quil_file(quil_filename) Loads quantum circuits from a Quil file using the converter in the PennyLane-Forest plugin. generator(op[, format]) Returns the generator of an operation. import_operator(qubit_observable[, format, …]) Convert an external operator to a PennyLane operator. is_commuting(operation1, operation2) Check if two operations are commuting using a lookup table. jacobian(func[, argnum]) Returns the Jacobian as a callable function of vector-valued (functions of) QNodes. layer(template, depth, *args, **kwargs) Repeatedly applies a unitary a given number of times. load(quantum_circuit_object, format) Load external quantum assembly and quantum circuits from supported frameworks into PennyLane templates. map(template, observables, device[, …]) Map a quantum template over a list of observables to create a QNodeCollection. matrix(op, *[, wire_order]) The matrix representation of an operation or quantum circuit. measure(wires) Perform a mid-circuit measurement in the computational basis on the supplied qubit. measurement_grouping(tape, obs_list, coeffs_list) Returns a list of measurement optimized tapes, and a classical processing function, for evaluating the expectation value of a provided Hamiltonian. metric_tensor(tape[, approx, …]) Returns a function that computes the metric tensor of a given QNode or quantum tape. mutual_info(wires0, wires1[, log_base]) Mutual information between the subsystems prior to measurement: pattern_matching(circuit_dag, pattern_dag) Function that applies the pattern matching algorithm and returns the list of maximal matches. pattern_matching_optimization(tape, …[, …]) Quantum function transform to optimize a circuit given a list of patterns (templates). paulix_ops(generators, num_qubits) Generate the single qubit Pauli-X operators $$\sigma^{x}_{i}$$ for each symmetry $$\tau_j$$, such that it anti-commutes with $$\tau_j$$ and commutes with all others symmetries $$\tau_{k\neq j}$$. probs([wires, op]) Probability of each computational basis state. qfunc_transform(tape_transform) Given a function which defines a tape transform, convert the function into one that applies the tape transform to quantum functions (qfuncs). qnode(func, device[, interface, …]) Represents a quantum node in the hybrid computational graph. quantum_monte_carlo(fn, wires, target_wire, …) Provides the circuit to perform the quantum Monte Carlo estimation algorithm. Scan installed PennyLane plugins to refresh the device list. sample([op, wires, counts]) Sample from the supplied observable, with the number of shots determined from the dev.shots attribute of the corresponding device, returning raw samples (counts=False) or the number of counts for each sample (counts=True). simplify(operation) Simplify the (controlled) rotation operations Rot, U2, U3, and CRot into one of RX, CRX, RY, CRY, RZ, CZ, Hadamard and controlled-Hadamard where possible. snapshots(qnode) Create a function that retrieves snapshot results from a QNode. specs(qnode[, max_expansion, expansion_strategy]) Resource information about a quantum circuit. Quantum state in the computational basis. Lazily sum the constituent QNodes of a QNodeCollection. Compute the generators $$\{\tau_1, \ldots, \tau_k\}$$ for a Hamiltonian over the binary field $$\mathbb{Z}_2$$. taper(h, generators, paulixops, paulix_sector) Transform a Hamiltonian with a Clifford operator and taper qubits. var(op) Variance of the supplied observable. Returns the PennyLane version number. vn_entropy(wires[, log_base]) Von Neumann entropy of the system prior to measurement.

## Classes¶

 AdagradOptimizer([stepsize, eps]) Gradient-descent optimizer with past-gradient-dependent learning rate in each dimension. AdamOptimizer([stepsize, beta1, beta2, eps]) Gradient-descent optimizer with adaptive learning rate, first and second moment. AllSinglesDoubles(weights, wires, hf_state) Builds a quantum circuit to prepare correlated states of molecules by applying all SingleExcitation and DoubleExcitation operations to the initial Hartree-Fock state. AmplitudeDamping(gamma, wires[, do_queue, id]) Single-qubit amplitude damping error channel. AmplitudeEmbedding(features, wires[, …]) Encodes $$2^n$$ features into the amplitude vector of $$n$$ qubits. AngleEmbedding(features, wires[, rotation, …]) Encodes $$N$$ features into the rotation angles of $$n$$ qubits, where $$N \leq n$$. ApproxTimeEvolution(hamiltonian, time, n[, …]) Applies the Trotterized time-evolution operator for an arbitrary Hamiltonian, expressed in terms of Pauli gates. ArbitraryStatePreparation(weights, wires[, …]) Implements an arbitrary state preparation on the specified wires. ArbitraryUnitary(weights, wires[, do_queue, id]) Implements an arbitrary unitary on the specified wires. Barrier(wires) The Barrier operator, used to separate the compilation process into blocks or as a visual tool. BasicEntanglerLayers(weights[, wires, …]) Layers consisting of one-parameter single-qubit rotations on each qubit, followed by a closed chain or ring of CNOT gates. BasisEmbedding(features, wires[, do_queue, id]) Encodes $$n$$ binary features into a basis state of $$n$$ qubits. BasisState(n, wires) Prepares a single computational basis state. BasisStatePreparation(basis_state, wires[, …]) Prepares a basis state on the given wires using a sequence of Pauli-X gates. Beamsplitter(theta, phi, wires[, do_queue, id]) Beamsplitter interaction. BitFlip(p, wires[, do_queue, id]) Single-qubit bit flip (Pauli $$X$$) error channel. Wrapper for simple callables with boolean output that can be manipulated and combined with bit-wise operators. CNOT(wires) The controlled-NOT operator CPhase alias of pennylane.ops.qubit.parametric_ops.ControlledPhaseShift CRX(phi, wires[, do_queue, id]) The controlled-RX operator CRY(phi, wires[, do_queue, id]) The controlled-RY operator CRZ(phi, wires[, do_queue, id]) The controlled-RZ operator CRot(phi, theta, omega, wires[, do_queue, id]) The controlled-Rot operator CSWAP(wires) The controlled-swap operator CVNeuralNetLayers(theta_1, phi_1, varphi_1, …) A sequence of layers of a continuous-variable quantum neural network, as specified in Killoran et al. (2019). CY(wires) The controlled-Y operator CZ(wires) The controlled-Z operator CatState(a, phi, p, wires[, do_queue, id]) Prepares a cat state. CircuitGraph(ops, obs, wires[, par_info, …]) Represents a quantum circuit as a directed acyclic graph. CoherentState(a, phi, wires[, do_queue, id]) Prepares a coherent state. CommutingEvolution(hamiltonian, time[, …]) Applies the time-evolution operator for a Hamiltonian expressed as a linear combination of mutually commuting Pauli words. Configuration(name) Configuration class. ControlledAddition(s, wires[, do_queue, id]) Controlled addition operation. ControlledPhase(s, wires[, do_queue, id]) Controlled phase operation. ControlledPhaseShift(phi, wires[, do_queue, id]) A qubit controlled phase shift. ControlledQubitUnitary(U, control_wires, …) Apply an arbitrary fixed unitary to wires with control from the control_wires. CrossKerr(kappa, wires[, do_queue, id]) Cross-Kerr interaction. CubicPhase(gamma, wires[, do_queue, id]) Cubic phase shift. DepolarizingChannel(p, wires[, do_queue, id]) Single-qubit symmetrically depolarizing error channel. Device([wires, shots, analytic]) Abstract base class for PennyLane devices. DeviceError Exception raised by a Device when it encounters an illegal operation in the quantum circuit. DiagonalQubitUnitary(D, wires) Apply an arbitrary fixed diagonal unitary matrix. DisplacedSqueezedState(a, phi_a, r, phi_r, wires) Prepares a displaced squeezed vacuum state. Displacement(a, phi, wires[, do_queue, id]) Phase space displacement. DisplacementEmbedding(features, wires[, …]) Encodes $$N$$ features into the displacement amplitudes $$r$$ or phases $$\phi$$ of $$M$$ modes, where $$N\leq M$$. DoubleExcitation(phi, wires[, do_queue, id]) Double excitation rotation. DoubleExcitationMinus(*params[, wires, …]) Double excitation rotation with negative phase-shift outside the rotation subspace. DoubleExcitationPlus(phi, wires[, do_queue, id]) Double excitation rotation with positive phase-shift outside the rotation subspace. ECR(wires) An echoed RZX(pi/2) gate. ExpvalCost(ansatz, hamiltonian, device[, …]) Create a cost function that gives the expectation value of an input Hamiltonian. FermionicDoubleExcitation(weight[, wires1, …]) Circuit to exponentiate the tensor product of Pauli matrices representing the double-excitation operator entering the Unitary Coupled-Cluster Singles and Doubles (UCCSD) ansatz. FermionicSingleExcitation(weight[, wires, …]) Circuit to exponentiate the tensor product of Pauli matrices representing the single-excitation operator entering the Unitary Coupled-Cluster Singles and Doubles (UCCSD) ansatz. FockDensityMatrix(state, wires[, do_queue, id]) Prepare subsystems using the given density matrix in the Fock basis. FockState(n, wires[, do_queue, id]) Prepares a single Fock state. FockStateProjector(n, wires[, do_queue, id]) The number state observable $$\ket{n}\bra{n}$$. FockStateVector(state, wires[, do_queue, id]) Prepare subsystems using the given ket vector in the Fock basis. GateFabric(weights, wires, init_state[, …]) Implements a local, expressive, and quantum-number-preserving ansatz proposed by Anselmetti et al. (2021). GaussianState(V, r, wires[, do_queue, id]) Prepare subsystems in a given Gaussian state. GeneralizedAmplitudeDamping(gamma, p, wires) Single-qubit generalized amplitude damping error channel. GradientDescentOptimizer([stepsize]) Basic gradient-descent optimizer. GroverOperator([wires, work_wires, do_queue, id]) Performs the Grover Diffusion Operator. Hadamard(wires) The Hadamard operator Hamiltonian(coeffs, observables[, simplify, …]) Operator representing a Hamiltonian. Hermitian(A, wires[, do_queue, id]) An arbitrary Hermitian observable. HilbertSchmidt(*params, v_function, v_wires, …) Create a Hilbert-Schmidt template that can be used to compute the Hilbert-Schmidt Test (HST). IQPEmbedding(features, wires[, n_repeats, …]) Encodes $$n$$ features into $$n$$ qubits using diagonal gates of an IQP circuit. ISWAP(wires) The i-swap operator Identity(wires) The identity observable $$\I$$. Interferometer(theta, phi, varphi, wires[, …]) General linear interferometer, an array of beamsplitters and phase shifters. InterferometerUnitary(U, wires[, do_queue, id]) A linear interferometer transforming the bosonic operators according to the unitary matrix $$U$$. IsingXX(phi, wires[, do_queue, id]) Ising XX coupling gate IsingXY(phi, wires[, do_queue, id]) Ising (XX + YY) coupling gate IsingYY(phi, wires[, do_queue, id]) Ising YY coupling gate IsingZZ(phi, wires[, do_queue, id]) Ising ZZ coupling gate Kerr(kappa, wires[, do_queue, id]) Kerr interaction. LieAlgebraOptimizer(circuit[, stepsize, …]) Lie algebra optimizer. LocalHilbertSchmidt(*params, v_function, …) Create a Local Hilbert-Schmidt template that can be used to compute the Local Hilbert-Schmidt Test (LHST). MERA(wires, n_block_wires, block, n_params_block) The MERA template broadcasts an input circuit across many wires following the architecture of a multi-scale entanglement renormalization ansatz tensor network. MPS(wires, n_block_wires, block, n_params_block) The MPS template broadcasts an input circuit across many wires following the architecture of a Matrix Product State tensor network. MomentumOptimizer([stepsize, momentum]) Gradient-descent optimizer with momentum. MottonenStatePreparation(state_vector, wires) Prepares an arbitrary state on the given wires using a decomposition into gates developed by Möttönen et al. (2004). MultiControlledX(control_wires, wires, …) Apply a Pauli X gate controlled on an arbitrary computational basis state. MultiRZ(theta[, wires, do_queue, id]) Arbitrary multi Z rotation. NesterovMomentumOptimizer([stepsize, momentum]) Gradient-descent optimizer with Nesterov momentum. NumberOperator(wires) The photon number observable $$\langle \hat{n}\rangle$$. OrbitalRotation(phi, wires[, do_queue, id]) Spin-adapted spatial orbital rotation. P(wires) The momentum quadrature observable $$\hat{p}$$. ParticleConservingU1(weights, wires[, …]) Implements the heuristic VQE ansatz for quantum chemistry simulations using the particle-conserving gate $$U_{1,\mathrm{ex}}$$ proposed by Barkoutsos et al. in arXiv:1805.04340. ParticleConservingU2(weights, wires[, …]) Implements the heuristic VQE ansatz for Quantum Chemistry simulations using the particle-conserving entangler $$U_\mathrm{ent}(\vec{\theta}, \vec{\phi})$$ proposed in arXiv:1805.04340. PauliError(operators, p[, wires, do_queue, id]) Pauli operator error channel for an arbitrary number of qubits. PauliRot(theta, pauli_word[, wires, …]) Arbitrary Pauli word rotation. PauliX(wires) The Pauli X operator PauliY(wires) The Pauli Y operator PauliZ(wires) The Pauli Z operator Permute(permutation, wires[, do_queue, id]) Applies a permutation to a set of wires. PhaseDamping(gamma, wires[, do_queue, id]) Single-qubit phase damping error channel. PhaseFlip(p, wires[, do_queue, id]) Single-qubit bit flip (Pauli $$Z$$) error channel. PhaseShift(phi, wires[, do_queue, id]) Arbitrary single qubit local phase shift PolyXP(q, wires[, do_queue, id]) An arbitrary second-order polynomial observable. Projector(basis_state, wires[, do_queue, id]) Observable corresponding to the computational basis state projector $$P=\ket{i}\bra{i}$$. QAOAEmbedding(features, weights, wires[, …]) Encodes $$N$$ features into $$n>N$$ qubits, using a layered, trainable quantum circuit that is inspired by the QAOA ansatz. QFT(wires) Apply a quantum Fourier transform (QFT). QNGOptimizer([stepsize, approx, lam]) Optimizer with adaptive learning rate, via calculation of the diagonal or block-diagonal approximation to the Fubini-Study metric tensor. QNode(func, device[, interface, …]) Represents a quantum node in the hybrid computational graph. QNodeCollection([qnodes]) Represents a sequence of independent QNodes that all share the same signature. QuadOperator(phi, wires[, do_queue, id]) The generalized quadrature observable $$\x_\phi = \x cos\phi+\p\sin\phi$$. QuadraticPhase(s, wires[, do_queue, id]) Quadratic phase shift. QuantumFunctionError Exception raised when an illegal operation is defined in a quantum function. QuantumMonteCarlo(probs, func, target_wires, …) Performs the quantum Monte Carlo estimation algorithm. QuantumPhaseEstimation(unitary, …[, …]) Performs the quantum phase estimation circuit. QubitCarry(wires) Apply the QubitCarry operation to four input wires. QubitChannel(K_list[, wires, do_queue, id]) Apply an arbitrary fixed quantum channel. QubitDensityMatrix(state, wires) Prepare subsystems using the given density matrix. QubitDevice([wires, shots, r_dtype, …]) Abstract base class for PennyLane qubit devices. QubitStateVector(state, wires) Prepare subsystems using the given ket vector in the computational basis. QubitSum(wires) Apply a QubitSum operation on three input wires. QubitUnitary(U, wires) Apply an arbitrary fixed unitary matrix. Abstract base class for classes that exposes a queue for objects. QutritUnitary(*params, wires[, do_queue]) Apply an arbitrary, fixed unitary matrix. RMSPropOptimizer([stepsize, decay, eps]) Root mean squared propagation optimizer. RX(phi, wires[, do_queue, id]) The single qubit X rotation RY(phi, wires[, do_queue, id]) The single qubit Y rotation RZ(phi, wires[, do_queue, id]) The single qubit Z rotation RandomLayers(weights, wires[, ratio_imprim, …]) Layers of randomly chosen single qubit rotations and 2-qubit entangling gates, acting on randomly chosen qubits. ResetError(p0, p1, wires[, do_queue, id]) Single-qubit Reset error channel. Rot(phi, theta, omega, wires[, do_queue, id]) Arbitrary single qubit rotation Rotation(phi, wires[, do_queue, id]) Phase space rotation. RotoselectOptimizer([possible_generators]) Rotoselect gradient-free optimizer. RotosolveOptimizer([substep_optimizer, …]) Rotosolve gradient-free optimizer. S(wires) The single-qubit phase gate SISWAP(wires) The square root of i-swap operator. SPSAOptimizer([maxiter, alpha, gamma, c, A, a]) The Simultaneous Perturbation Stochastic Approximation method (SPSA) is a stochastic approximation algorithm for optimizing cost functions whose evaluation may involve noise. SQISW alias of pennylane.ops.qubit.non_parametric_ops.SISWAP SWAP(wires) The swap operator SX(wires) The single-qubit Square-Root X operator. ShotAdaptiveOptimizer(min_shots[, …]) Optimizer where the shot rate is adaptively calculated using the variances of the parameter-shift gradient. SimplifiedTwoDesign(initial_layer_weights, …) Layers consisting of a simplified 2-design architecture of Pauli-Y rotations and controlled-Z entanglers proposed in Cerezo et al. (2021). SingleExcitation(phi, wires[, do_queue, id]) Single excitation rotation. SingleExcitationMinus(phi, wires[, do_queue, id]) Single excitation rotation with negative phase-shift outside the rotation subspace. SingleExcitationPlus(phi, wires[, do_queue, id]) Single excitation rotation with positive phase-shift outside the rotation subspace. Snapshot([tag, do_queue]) The Snapshot operation saves the internal simulator state at specific execution steps of a quantum function. SparseHamiltonian(H[, wires, do_queue, id]) A Hamiltonian represented directly as a sparse matrix in Compressed Sparse Row (CSR) format. SqueezedState(r, phi, wires[, do_queue, id]) Prepares a squeezed vacuum state. Squeezing(r, phi, wires[, do_queue, id]) Phase space squeezing. SqueezingEmbedding(features, wires[, …]) Encodes $$N$$ features into the squeezing amplitudes $$r \geq 0$$ or phases $$\phi \in [0, 2\pi)$$ of $$M$$ modes, where $$N\leq M$$. StronglyEntanglingLayers(weights, wires[, …]) Layers consisting of single qubit rotations and entanglers, inspired by the circuit-centric classifier design arXiv:1804.00633. T(wires) The single-qubit T gate TTN(wires, n_block_wires, block, n_params_block) The TTN template broadcasts an input circuit across many wires following the architecture of a tree tensor network. TensorN([wires]) The tensor product of the NumberOperator acting on different wires. ThermalRelaxationError(pe, t1, t2, tq, wires) Thermal relaxation error channel. ThermalState(nbar, wires[, do_queue, id]) Prepares a thermal state. Toffoli(wires) Toffoli (controlled-controlled-X) gate. Tracker([dev, callback, persistent]) This class stores information about device executions and allows users to interact with that data upon individual executions and batches, even within parameter-shift gradients and optimization steps. TwoModeSqueezing(r, phi, wires[, do_queue, id]) Phase space two-mode squeezing. U1(phi, wires[, do_queue, id]) U1 gate. U2(phi, delta, wires[, do_queue, id]) U2 gate. U3(theta, phi, delta, wires[, do_queue, id]) Arbitrary single qubit unitary. UCCSD(weights, wires[, s_wires, d_wires, …]) Implements the Unitary Coupled-Cluster Singles and Doubles (UCCSD) ansatz. VQECost(*args, **kwargs) Create a cost function that gives the expectation value of an input Hamiltonian. WireCut(wires) The wire cut operation, used to manually mark locations for wire cuts. X(wires) The position quadrature observable $$\hat{x}$$. batch_transform(*args, **kwargs) Class for registering a tape transform that takes a tape, and outputs a batch of tapes to be independently executed on a quantum device. grad(fun[, argnum]) Returns the gradient as a callable function of (functions of) QNodes. kUpCCGSD(weights, wires[, k, delta_sz, …]) Implements the k-Unitary Pair Coupled-Cluster Generalized Singles and Doubles (k-UpCCGSD) ansatz. op_transform(*args, **kwargs) Convert a function that applies to operators into a functional transform. single_tape_transform(transform_fn) For registering a tape transform that takes a tape and outputs a single new tape.


text:
# Summary

In this chapter we learned about input polling and how it is implemented in XNA using structures representing input state and static GetState() methods. We saw the three primary forms of input we use in the MonoGame framework - the keyboard, the mouse, and the gamepad.

We also saw how a variety of game controllers (i.e. RockBand gear, steering wheels, flight sticks, etc.) are mapped to the standard gamepad; how its state struct is actually composed of several sub-structs; and how to turn on and off the vibration motors.

Finally, we discussed how we can use two copies of a state struct - one from the prior frame and one from the current frame - to determine if a button was just pressed or released.


text:
# Can't use Input Generators with Binary install?

When I download and install the binaries from OpenChemistry, it doesn’t display any input generators besides GAMESS in the Quantum tab.

On Windows, it installed to C:\Program Files\Avogadro2 and this contains a bin and share folder. The bin folder contains the Avogadro2 executable and a number of .dll files. The share folder just contains a doc folder. I haven’t been able to find the inputGenerators in any of these folders. I have also looked to see if there was a particular folder where I could add the files manually, but I can’t seem to find one that Avogadro will read.

On Mac, I found the input generators at /Applications/Avogadro2.app/Contents/lib/avogadro2/scripts/inputGenerators, yet the program can’t seem to find them. I tried to copy them to /usr/local/lib/avogadro2/scripts/inputGenerators, but it still couldn’t find them there.

## Environment Information

Operating system and version: Mac OS 10.15.6 & Windows 10.0.19041

1 Like

Thanks for the heads-up about the 1.91 builds - we clearly need to make a new release ASAP.

I don’t know about Windows. I’d give you a build from GitHub, but the Windows compile seems broken at the moment. I’ll revise this when I get that working again.

On the Mac, the input generators live in different places. Apple actively discourages developers from using /usr/local. I’d suggest creating a symlink from wherever you have them to \${HOME}/Library/Application Support/OpenChemistry/Avogadro/inputGenerators

I will try to find an hour or so to code up the “please install this plugin” feature today or tomorrow - I appreciate the concern.

For the Mac, should that directory exist already with the binary install of Avogadro2? I checked Library/Application Support for both the system and my user account and neither had an OpenChemistry folder. Manually creating the folders and placing the various .py files there didn’t seem to help Avogadro2 to find them.

No, there won’t be an OpenChemistry or an Avogadro directory unless they’re created. Since there are a variety of plugin types, you need something like this:

• Library
• Application Support
• Open Chemistry


text:
# Why are there exactly 2$^\omega$ perfect subsets of the real numbers?

How can you proof that there are $2^{\omega}$ perfect subsets of the real numbers?

• If $\omega$ is the cardinality of the integers, then this is not the case. Or by $\omega$, do you mean the cardinality of the reals? If so, then how do you define $2^\omega$ anyway? – Omnomnomnom Aug 21 '13 at 19:21
• @Omnomnomnom: $\omega$ is the cardinality of the integers, and the assertion is correct: there are $2^\omega$ perfect subsets of $\Bbb R$. – Brian M. Scott Aug 21 '13 at 19:28
• 2$^\omega$ is defined as the cardinality of the reals (as is \omega the cardinality of the natural numbers) – ABC Aug 21 '13 at 19:29
• @BrianM.Scott I misread the "perfect subsets" part of the question, and then I said something stupid. Thank you for the correction. – Omnomnomnom Aug 21 '13 at 19:47
• @Omnomnomnom: I wondered if that’s what had happened. You’re welcome. – Brian M. Scott Aug 21 '13 at 19:49

Let $C$ be the middle-thirds Cantor set, and for each $x\in\Bbb R$ let $C+x=\{y+x:y\in C\}$ be the translate of $C$ by $x$. Clearly each $C+x$ is perfect, and $C+x\ne C+y$ if $x,y\in\Bbb R$ with $x\ne y$, so $\{C+x:x\in\Bbb R\}$ is a family of $2^\omega$ perfect subsets of $\Bbb R$. This shows that there are at least $2^\omega$ perfect subsets of $\Bbb R$.

By definition a perfect set is closed, so to finish the argument it suffices to show that $\Bbb R$ has only $2^\omega$ closed subsets. Since there is an obvious bijection between the closed sets in $\Bbb R$ and the open sets, we need only show that there are just $2^\omega$ open sets in $\Bbb R$. $\Bbb R$ has a countable base $\mathscr{B}$ (e.g., the family of open intervals with rational endpoints). For each open set $U\subseteq\Bbb R$ there is a $\mathscr{B}_U\subseteq\mathscr{B}$ such that $U=\bigcup\mathscr{B}_U$. There are only $\omega^\omega=2^\omega$ subsets of $\mathscr{B}$, so there are at most $2^\omega$ open sets in $\Bbb R$ and hence at most $2^\omega$ perfect sets.

Thus, there are exactly $2^\omega$ perfect subsets of $\Bbb R$.

There are at least $2^{\aleph_0}$ perfect sets, namely those of the form $[0,a]$, for $a$ a positive real. So it is enough to show that there are no more (by the Bernstein-Schroeder theorem).

But a perfect set is closed, and a closed set is the complement of an open set, so it is enough to check that there are at most $2^{\aleph_0}$ open sets.

Now, an open set is a countable union of open intervals with rational endpoints. There are countably many such intervals, so at most $\aleph_0^{\aleph_0}=2^{\aleph_0}$ open sets.

A very similar idea, coupled with something like transfinite induction, shows that in fact there are $2^{\aleph_0}$ Borel sets, see for example here.

Moreover: The Borel $\sigma$-algebra admits a natural stratification (or hierarchy) as $\bigcup_{0<\alpha<\omega_1}\mathbf\Sigma^0_\alpha$. Here, $\omega_1$ is the smallest uncountable ordinal, and:

• $\mathbf\Sigma^0_1$ is the collection of open sets. Their complements (the closed sets) form the $\mathbf\Pi^0_1$ sets.
• $\mathbf\Sigma^0_\alpha$ ($\alpha>1$) is the collection of countable unions of the form $\bigcup_{n=1}^\infty A_n$, where each $A_n$ belongs to some $\mathbf\Pi^0_{\beta_n}$ with $\beta_n<\alpha$. The complements of these sets form the $\mathbf\Pi^0_\alpha$ sets.

One can check that $\mathbf\Sigma^0_\alpha\subsetneq\mathbf\Sigma^0_\beta$ whenever $\alpha<\beta<\omega_1$. And a more refined argument than the one above shows that, for each $0<\alpha<\omega_1$, $|\mathbf\Sigma^0_\alpha\setminus\bigcup_{\beta<\alpha}\mathbf\Sigma^0_\beta|=2^{\aleph_0}$, and the same happens replacing each $\mathbf\Sigma$ with $\mathbf\Pi$. That is, it is not just that there are continuum many perfect sets: We also have that each level of the Borel hierarchy adds continuum many new sets.

(For a deep extension of this result in the choiceless context of determinacy, see here.)


text:
# How do I draw a high-performanced scatter plot

I am new to DirectX and currently I'm trying to use it for data visualization, to be specific, a 3D scatter plot. There are very limited resources on the internet, and I have read part of Frank D Luna's book as suggested by many. I studied some samples provided by Microsoft and finally have a grasp on how the loops work inside programs.

However, since I need to manipulate a huge amount of data and require performance as much as possible, I run into something that I hardly find reference with current material.

1. Buffer Usage I expect the points I plot to be constant after they are created, but the creation of points happens all the time, should I set the D3D11_USAGE to Default or Dynamic, and which method to call when I need to update the buffer? I have noticed How to use UpdateSubresource and Map/Unmap?, and the presentation https://developer.nvidia.com/sites/default/files/akamai/gamedev/files/gdc12/Efficient_Buffer_Management_McDonald.pdf by NVIDIA. But what should I do if I do need frequent updates of buffer, while no update for each data in the buffer needed?
2. I noticed in the example "Simple Direct3D game" by Microsoft, it seem to rewrite the entire buffer each time it is updated.

        auto objects = m_game->RenderObjects();
for (auto object = objects.begin(); object != objects.end(); object++)
{
(*object)->Render(d3dContext, m_constantBufferChangesEveryPrim.Get());
}

Is it really so? And How should I only add new vertices into the buffer every time updating it?

3. Since the program is expected to run very fast, the buffer is expected to be updated much faster than once per frame. Can I write a method to update the buffer and add a new point into the buffer every time I call it? If I could, can I put it in another thread or is there any better method to realize it?

4. In the example the main loop of the game seems to be in GameMain::StartRenderLoop(), is this loop running once per frame? And what determines the frequency of the loop?

I might have many misconception on frame and rendering, so thank you very much for the patience and answers.

This is called 'dynamic' submission in which you generate vertices on-the-fly and draw them. The more efficient approach is to use 'static' submission where you upload the vertex data to the video card and draw it many times, but most real-world graphics programs are a mix of both.

If you are doing 2D rendering, you should look at Direct2D if that's an option for you.

How to: Use dynamic resources

You could also use the PrimitiveBatch class in the DirectX Tool Kit

• Thanks for your answer. The static submission you mentioned, should it be submitted every frame or for every primitive, and when I draw it, can I only add elements to the current buffer rather than redraw the whole thing? – 王凯越 Kaiyue Wang Oct 19 '16 at 5:23
• Static submission means that data doesn't change at all. For example in a game, the characters and buildings. What you are describing is better suited to dynamic submission. – Chuck Walbourn Oct 19 '16 at 16:16
• Yeah, that's what I thought it would be. But still, is there a way to avoid discarding the previous data in the buffer during dynamic update? And how do I control the update rate? – 王凯越 Kaiyue Wang Oct 20 '16 at 15:25


text:
### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 2013

 FINAL EXAMINATION April 1998 TIME: 3 HOURS

CALCULATORS PERMITTED

 MARKS (20) 1. (a) Evaluate , where R is the region in between the curves y = x and y = x4, for . (b) Show that , where Da is the disc of radius a > 0, centred at the origin. (c) Evaluate where E is the region in bounded by the spheres of radii a and b (b > a > 0), centred at the origin. (10) 2. (a) Show that the integral satisfies , where S is the unit square . (b) Compute an approximate value for I by partitioning S by the lines and then using double sum with the integrand evaluated at the mid-points of the sub-regions defined by the partition. (20) 3. Consider the vector field defined on . (a) Compute div and curl. (b) Find a potential function for . (c) Evaluate the line integral where and is any smooth curve joining A to B. (d) Compute the flux of across the ellipsoid a2x2 + b2y2 + c2z2 = 1. (e) Show that where (10) 4. Use the change of coordinates in defined by , to show that where T is the triangular region in the (x,y) plane whose vertices are and (0,1). (15) 5. In this problem you may use the standard formulas for area of a disc and surface area of a sphere: and , respectively, where R is the radius. (a) Evaluate where is the circle of radius 3 centred at the origin, oriented clockwise. (b) Compute the value of where is the sphere x2 + y2 + z2 = a2, oriented by its outward normal and , where e is a constant and is the unit normal. Express e in terms of Q and a. (c) Suppose that two vector fields and , defined and smooth on , are related by curl . Show that the flux of J through a surface is equal to the circulation of around , where is the boundary of . [Assume that and are smooth and compatibly oriented.] (10) 6. (a) Determine which of the following sequences converges and obtain the limit in the convergent cases. (i) (ii) (b) Determine whether each of the following series is absolutely convergent , conditionally convergent or divergent. (i) (ii) (iii) (15) 7. (a) Find either the general (n ) term, or the terms of degree , in the Taylor series expansion about x = 0 (Maclaurin series) for (i) ; (ii) and determine the interval of convergence in each case. (b) Set x = 1/3 in (a) (ii) and compute the approximation to given by the sum of the first three non-zero terms of the Taylor series. Compute the error, if the exact value is . (c) Compute an approximation to by using the first three non-zero terms in the Taylor expansion for f(t) about t = 0. Use this result to approximate F(1/2). (100)


text:
## Algebraic & Geometric Topology

### A family of pseudo-Anosov braids with small dilatation

#### Abstract

This paper describes a family of pseudo-Anosov braids with small dilatation. The smallest dilatations occurring for braids with 3,4 and 5 strands appear in this family. A pseudo-Anosov braid with $2g+1$ strands determines a hyperelliptic mapping class with the same dilatation on a genus–$g$ surface. Penner showed that logarithms of least dilatations of pseudo-Anosov maps on a genus–$g$ surface grow asymptotically with the genus like $1∕g$, and gave explicit examples of mapping classes with dilatations bounded above by $log11∕g$. Bauer later improved this bound to $log6∕g$. The braids in this paper give rise to mapping classes with dilatations bounded above by $log(2+3)∕g$. They show that least dilatations for hyperelliptic mapping classes have the same asymptotic behavior as for general mapping classes on genus–$g$ surfaces.

#### Article information

Source
Algebr. Geom. Topol., Volume 6, Number 2 (2006), 699-738.

Dates
Received: 23 July 2005
Revised: 13 April 2006
Accepted: 26 April 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796542

Digital Object Identifier
doi:10.2140/agt.2006.6.699

Mathematical Reviews number (MathSciNet)
MR2240913

Zentralblatt MATH identifier
1126.37014

#### Citation

Hironaka, Eriko; Kin, Eiko. A family of pseudo-Anosov braids with small dilatation. Algebr. Geom. Topol. 6 (2006), no. 2, 699--738. doi:10.2140/agt.2006.6.699. https://projecteuclid.org/euclid.agt/1513796542

#### References

• M Bauer, 1094556
• M Bestvina, M Handel, Train-tracks for surface homeomorphisms, Topology 34 (1995) 109–140
• J S Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J. (1974)
• D W Boyd, Small Salem numbers, Duke Math. J. 44 (1977) 315–328
• D W Boyd, Speculations concerning the range of Mahler's measure, Canad. Math. Bull. 24 (1981) 453–469
• P Boyland, Rotation sets and monotone periodic orbits for annulus homeomorphisms, Comment. Math. Helv. 67 (1992) 203–213
• P Brinkmann, A note on pseudo-Anosov maps with small growth rate, Experiment. Math. 13 (2004) 49–53
• A J Casson, S A Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts 9, Cambridge University Press, Cambridge (1988)
• A de Carvalho, T Hall, Braid forcing and star-shaped train tracks, Topology 43 (2004) 247–287
• A Fathi, F Laudenbach, V Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66, Société Mathématique de France, Paris (1979)
• T Hall, The creation of horseshoes, Nonlinearity 7 (1994) 861–924
• J-Y Ham, W Song, The minimum dilatation of pseudo-Anosov 5–braids, preprint (2005)
• E Hironaka, Salem–Boyd sequences and Hopf plumbing, Osaka J. Math. 43 (2006)
• T Kanenobu, Module d'Alexander des nœ uds fibrés et polynôme de Hosokawa des lacements fibrés, Math. Sem. Notes Kobe Univ. 9 (1981) 75–84
• A Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. (1980) 137–173
• A Kawauchi, A survey of knot theory, Birkhäuser Verlag, Basel (1996)
• E Kin, The forcing partial order on a family of braids forced by pseudo-Anosov 3–braids, in preparation
• K H Ko, J E Los, W T Song, Entropies of braids, J. Knot Theory Ramifications 11 (2002) 647–666
• D H Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. $(2)$ 34 (1933) 461–479
• C J Leininger, On groups generated by two positive multi-twists: Teichmüller curves and Lehmer's number, Geom. Topol. 8 (2004) 1301–1359
• J Los, On the forcing relation for surface homeomorphisms, Inst. Hautes Études Sci. Publ. Math. (1997) 5–61
• T Matsuoka, Braids of periodic points and $2$–dimensional analogue of Shorkovskii's ordering, from: “Dynamical systems and Nonlinear Oscillations”, (G Ikegami, editor), World Scientific Press (1986) 58–72
• C T McMullen, Polynomial invariants for fibered 3-manifolds and Teichmüller geodesics for foliations, Ann. Sci. École Norm. Sup. $(4)$ 33 (2000) 519–560
• H Minakawa, Examples of pseudo-Anosov homeomorphisms with small dilatations, preprint
• R C Penner, 1068128
• D Rolfsen, Knots and links, Publish or Perish, Berkeley, Calif. (1976)
• E Rykken, Expanding factors for pseudo-Anosov homeomorphisms, Michigan Math. J. 46 (1999) 281–296
• R Salem, A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan, Duke Math. J. 11 (1944) 103–108
• P Walters, An introduction to ergodic theory, Graduate Texts in Mathematics 79, Springer, New York (1982)
• A Y Zhirov, On the minimum dilation of pseudo-Anosov diffeomorphisms of a double torus, Uspekhi Mat. Nauk 50 (1995) 197–198


text:
Does connectedness follow?

(It is an other question than my previous question, despite it starts with the same discussion about connectors.)

This question arose in my research of generalized connectedness (see this draft article for the overall idea):

Let $U$ is a set, $r$ is a binary relation on $\mathcal{P} U$. I call $r$ a connector.

Informal note: The relation $r$ of two sets $A$ and $B$ represents that $A$ and $B$ are in some sense "near" or "touch". For example $r$ may be a proximity.

I will define the set $\mathrm{CC} (r)$ of connected subsets of $U$ by the formula

$\mathrm{CC} (r) = \lbrace A \in \mathcal{P} U | \forall X, Y \in \mathcal{P} A \setminus \lbrace \emptyset \rbrace : (X \cup Y = A \wedge X \cap Y = \emptyset \Rightarrow X r Y) \rbrace .$

As I mentioned above, $r$ may be a proximity and in this case $\mathrm{CC}(r)$ is proximal connectedness, that is a set a $A$ is connected iff every partition of the set a $A$ consists of two near sets.

As an other important example $ArB$ may mean that the topological closure (given some topological space) of the set $A$ in the subspace generated by the set $A\cup B$ intersects $B$ or the closure of $B$ intersect $A$. This is equivalent to the classic definition of connectedness of a set on topological space, because it happens if and only if $A$ and $B$ are not both open-closed on $A\cup B$.

There are other examples of connectedness following this scheme: graph connectedness, digraph strong connectedness, uniform connectedness, etc. (see my draft article)

I will call a link a ternary relation $\tau\in U\times U\times\mathcal{P}U$. (The link $\tau(a,b,A)$ may mean for example that there are some path from a point $a$ to a point $b$ through a set $A$.) I will call a link $\tau$ increasing when $A\subseteq B\implies \tau_A\subseteq\tau_B$.

I will define connector $Q(\tau)$ by the formula $\forall X,Y\in\mathcal{P}U:(X Q(\tau) Y\iff \exists x\in X,y\in Y:\tau(x,y,X\cup Y))$.

Let $\tau$ is an increasing link.

Prove or give a counter example: $A\in\mathrm{CC}(Q(\tau)) \implies \forall x,y\in A:\tau(x,y,A)$.

Does it hold if we remove the requirement to be increasing?

-
You should probably tell us what $\tau_A$ means. – Mariano Suárez-Alvarez Jul 23 2010 at 21:23
I think that $\tau_A(a,b) =\tau (a,b,A)$, for what I understand. Isn't it right? – dan232 Aug 9 2011 at 23:57
@dan232: You are right. – porton Aug 10 2011 at 11:06


text:
# Other Modular Mathematical Objects

Models of this type are also automatically listed in: abstract modular, abstract, geometric, mathematical object, modular, multi-sheet, other mathematical object

Mathematical objects not grouped elsewhere, folded as modular origami.

This page lists models of a single type. You might be interested in folding instructions instead.

## Möbius Strip V (CBU)

A Möbius strip consisting of a single Conveyor Belt Unit (CBU) twisted and locked with itself by both ends. This is the cleanest representation of a Möbius s...

## Trefoil Knot

A trefoil knot from a single Cross Lap Unit.

## Möbius Strip IV (CLU)

The Möbius band (aka Möbius strip) is an interesting mathematical object, a single-sided surface. This origami version is folded using my Cross Lap Unit (CLU...

## Möbius Strip III (BBU)

A single-sided surface, the Möbius Band is one of the more interesting mathematical objects that can be reproduced in origami.

## Hamiltonia Cycle of the Cube

A Hamiltonian cycle is a closed path on a polyhedron which visits each vertex exactly once. This model represents such a path for a cube. It can also be used...

## Ring

This ring can also be worn as a headband. It uses a non-standard way of connecting the modules. Any even number of modules can be connected this way, though ...

## Klein Bottle

This model was quite difficult to design, as the two sides of surfaces made with PHiZZ modules differ a lot (due to the presence of “bumps” where units join)...

## Möbius Strip I (Trimodule)

This is one of the rather few modular origami designs which require an odd number of modules. Compare also with another similar model.

## Möbius Strip II (Trimodule)

This is one of the rather few modular origami designs which use an odd number of units. Compare also with another similar model.


text:
## Class/Course - Class XI

### Subject - Math

#### Total Number of Question/s - 3865

Just Exam provide question bank for Class XI standard. Currently number of question's are 3865. We provide this data in all format (word, excel, pdf, sql, latex form with images) to institutes for conducting online test/ examinations. Here we are providing some demo contents. Interested person may contact us at info@justexam.in

• 1. Sets - Quiz

1. If n(A) = 4, n(B) = 3, n(A x B x C) = 24, then n(C) =
a) 288
b) 1
c) 12
d) 17
e) 2

2. For any two sets A and B if A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X , then
a) A - B = A ∩ B
b) A = B
c) B - A = A ∩ B
d) None of these

• 2. Relations and Functions - Quiz

• 3. Trigonometric Functions - Quiz

• 4. Principle of Mathematical Induction - Quiz

• 5. Complex Numbers and Quadratic Equations - Quiz

1. If P is the point in the argand diagram corresponding to the complex number $\sqrt{3}$ + i and if OPQ is an isosceles right angled triangle, right angled at O, then Q represents the complex number
a) $\sqrt{3}$ - i or 1 - i$\sqrt{3}$
b) -1 ± i$\sqrt{3}$
c) -1 + i$\sqrt{3}$ or 1 - i$\sqrt{3}$
d) 1 ± i$\sqrt{3}$

2. If z = x + iy and $arg\left (\frac{z - 2}{z + 2} \right )$ = $\frac{\pi}{6}$, then locus of z is
a) A straight line
b) A circle
c) A parabola
d) An ellipse

• 6. Linear Inequalities - Quiz

• 7. Permutations and Combinations - Quiz

1. The value of 20P2 is
a) 20
b) 19
c) 380
d) None of these

2. 20 persons are invited for a party. In how many different ways can they and the host be seated at a circular table. If the two particular persons are to be seated on either side of the host
a) 20!
b) 2 . 18!
c) 18!0
d) None of these

• 8. Binomial Theorem - Quiz

1. In the expansion of (x + a)n, the sum of odd terms is P and sum of even terms is Q, then the value of (P2 - Q2) will be
a) (x2 + a2)n
b) (x2 - a2)n
c) (x - a)2n
d) (x + a)2n

2. The term independent of x in $\left [\sqrt{\frac{x}{3}} + \frac{\sqrt{3}}{x^{2}} \right ]^{10}$ is
a) 2/3
b) 5/3
c) 4/3
d) None of these

• 9. Sequences and Series - Quiz

• 10. Straight Lines - Quiz

1. The equation of the perpendicular bisector of the line segment joining A(-2,3) and B(6,-5) is
a) x - y = -1
b) x - y = 3
c) x + y = 3
d) x + y = 1
e) x + y = -1

2. The equation of the line which bisects the obtuse angle between the lines x - 2y + 4 = 0 and 4x - 3y + 2 = 0, is
a) $\left (4 - \sqrt{5}\right )x - \left (3 -2 \sqrt{5} \right )y + \left (2 - 4\sqrt{5} \right )$ = 0
b) $\left (4 + \sqrt{5}\right )x - \left (3 +2 \sqrt{5} \right )y + \left (2 + 4\sqrt{5} \right )$ = 0
c) $\left (4 + \sqrt{5}\right )x + \left (3 +2 \sqrt{5} \right )y + \left (2 + 4\sqrt{5} \right )$ = 0
d) None of these

• 11. Conic Sections - Quiz

1. The locus of the point of intersection of lines (x + y)t = a and x - y = at, where t is the parameter , is
a) A circle
b) An ellipse
c) A rectangular hyperbola
d) None of these

2. Three normals are drawn to the parabola y2 = x through point (a,0) . Then
a) a = 1/2
b) a = 1/4
c) a = 1/2
d) None of these

• 12. Introduction to Three Dimensional Geometry - Quiz

• 13. Limits and Derivatives - Quiz

• 14. Mathematical Reasoning - Quiz

1. An AND gate is the Boolean function defined by
a) f(x1,x2) = x1.x2, x1x2 Îµ {0,1}
b) f(x1,x2) = x1 + x2, x1x2 Îµ {0,1}
c) f(x1,x2) = x1, x1x2 Îµ {0,1}
d) f(x1,x2) = x2, x1x2 Îµ {0,1}

2. Which of the following is not a proposition
a) $\sqrt{3}$ is a prime
b) $\sqrt{2}$ is irrational
c) Mathematics is interesting
d) 5 is an even integer

• 15. Statistics - Quiz

1. A force $\sqrt{5}$ unit act along the line $\frac{x-3}{2}$ = $\frac{y-4}{-1}$, the moment of the force about point (4,1) along z-axis is
a) 0
b) 5$\sqrt{5}$
c) -$\sqrt{5}$
d) 5

2. A bowler throws a bumper with a speed of 25 m/sec. The moment the ball touches the ground , it losses its energy by 1.5 kgm. If the weight of the ball is 225gm, the speed of the ball at which it hits the bat is
a) 2.22 m/sec
b) 22.2 m/sec
c) 4.44m/sec
d) 44.4 n/sec

• 16. Probability - Quiz

• 17. Quadratic Equation and Inequation - Quiz

1. If the equation anxn + an-1xn-1 + ….. + a1x = 0, a1 ≠ 0, n ≥ 2, has a positive root x = α , then the equation nanxn-1 + (n-1)an-1xn-2 + ..... + a1 = - has a positive root, which is
a) Greater than or equal to α
b) Equal to α
c) Greater than α
d) Smaller than α

2. The condition that x3 - px2 + qx - r = 0 may have two of its roots equal to each other but opposite in sign is
a) r = pq
b) r = 2p3 + pq
c) r = p2q
d) None of these

• 18. Progression - Quiz

1. The sum of the series 5.05 + 1.212 + 0.29088 + …. ∞ is
a) 6.93378
b) 6.87342
c) 6.74384
d) 6.64474

2. A boy goes to school from his home at a speed of x km/hour and comes back at a speed of y km/hour, then the average speed is given by
a) A.M.
b) G.M.
c) H.M.
d) None of these

• 19. Exponential and Logarithmic Series - Quiz

1. If x,y,x are three consecutive positive integers, then $\frac{1}{2}log_{e}x + \frac{1}{2}log_{e}z + \frac{1}{2xz + 1} + \frac{1}{3}\left (\frac{1}{2xz + 1} \right )^{3} + ....$ =
a) logex
b) logey
c) logez
d) None of these

2. If S = $\frac{1}{1.2} - \frac{1}{2.3} + \frac{1}{3.4} - \frac{1}{4.5} + .... + \infty$, then eS =
a) $log_{e}\left (\frac{4}{e} \right )$
b) $\frac{4}{e}$
c) $log_{e}\left (\frac{e}{4} \right )$
d) $\frac{e}{4}$

• 20. Trigonometric Ratio, Function and Identities - Quiz

1. If x + y + z = 1800, then cos2x + cos2y - cos2z is equal to
a) 4sinx.siny.sinz
b) 1 - 4sinx.siny.cosz
c) 4sinx.siny.sinz-1
d) cosA.cosB.cosC

2. $sin^{4}\frac{\pi}{8} + sin^{4}\frac{3\pi}{8} + sin^{4}\frac{5\pi}{8} + sin^{4}\frac{7\pi}{8}$ = =
a) $\frac{1}{2}$
b) $\frac{1}{4}$
c) $\frac{3}{2}$
d) $\frac{3}{4}$

• 21. Trigonometric Equation and Inequation - Quiz

1. If a, b,c are the sides of a triangle ABC, then which of the following inequalities is not true
a) 8abc ≤ (a + b)(b + c)(c + a)
b) 2bc ≤ a3 + b3 + c3
c) 6abc ≤ bc(b + c) + ca(c + a) + ab(a + b)
d) abc ≤ (a + b - c)(b + c - a)(c + a - b)

2. In ΔABC, $\frac{1}{a}cos^{2}\frac{A}{2} + \frac{1}{b}cos^{2}\frac{B}{2} + \frac{1}{c}cos^{2}\frac{C}{2}$ =
a) s
b) $\frac{s}{abc}$
c) $\frac{s^{2}}{abc}$
d) $\frac{s^{3}}{abc}$

• 22. Hyperbolic Function - Quiz

1. sin2(ix) + cosh2x is equal to
a) 1
b) -1
c) 2cosh2x
d) cosh2x

2. cosech-1 equals
a) $log\left (\frac{1 + \sqrt{1 + x^{2}}}{x} \right )$
b) $log\left (\frac{1 + \sqrt{1 - x^{2}}}{x} \right )$
c) $log\left (\frac{1 - \sqrt{1 - x^{2}}}{x} \right )$
d) $log\left (\frac{1 - \sqrt{1 + x^{2}}}{x} \right )$

• 23. Rectangular Cartesian Coordinate - Quiz

1. Two vertices of a triangle are (4,-3) and (-2,5) . If the orthocentre of the triangle is at (1,2), then the third vertex is
a) (-33,-26)
b) (33,26)
c) (26,33)
d) None of these

2. The following points A(2a,4a), B(2a,6a) and C(2a, $\sqrt{3}a$,5a), (a>0) are the vertices of
a) An acute angled triangle
b) A right angles triangle
c) An isosceles triangle
d) None of these

• 24. Circle and System of Circle - Quiz

1. Area of the circle in which a cord of length $\sqrt{2}$ makes an angle $\frac{\pi}{2}$ at the centre is
a) $\frac{\pi}{2}$
b) 2π
c) π
d) $\frac{\pi}{4}$

2. The equation of the circle which passes through (1,0) and (0-,1) and has its radius as small as possible , is
a) x2 + y2 - 2x - 2y +1 = 0
b) x2 + y2 - x - y = 0
c) 2x2 + 2y2 - 3x - 3y + 1 = 0
d) x2 + y2 - 3x - 3y + 2 = 0

• 25. Pair of Straight Line - Quiz

1. Let a and b be non-zero real numbers. Then, the equation (ax2 + by2 + c)(x2 - 5y + 6y2) = 0 represents
a) Four straight lines, when c = 0 and a,b are of the same sign
b) Two straight lines and a circle, when a = b and c is of sign opposite to that of a
c) Two straight line and a hyperbola, when a and b are of the same sign and c is of sign opposite to that of a
d) A circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a

2. If in general quadratic equation f(x,y) = 0, Δ = 0 and h2 = ab, then the equation represents
a) Two parallel then the equation represents
b) Two perpendicular straight lines
c) Two coincident lines
d) None of these