text:
# Io

## 简介

Input -> Process -> Output

### stdint stdout stderr

sys.stdin

sys.stdout

sys.stderr

File objects used by the interpreter for standard input, output and errors:

These streams are regular text files like those returned by the open() function. Their parameters are chosen as follows:

• The character encoding is platform-dependent. Under Windows, if the stream is interactive (that is, if its isatty() method returns True), the console codepage is used, otherwise the ANSI code page. Under other platforms, the locale encoding is used (see locale.getpreferredencoding()).

Under all platforms though, you can override this value by setting the PYTHONIOENCODING environment variable before starting Python.

• When interactive, standard streams are line-buffered. Otherwise, they are block-buffered like regular text files. You can override this value with the -u command-line option.

Note

To write or read binary data from/to the standard streams, use the underlying binary buffer object. For example, to write bytes to stdout, use sys.stdout.buffer.write(b'abc').

However, if you are writing a library (and do not control in which context its code will be executed), be aware that the standard streams may be replaced with file-like objects like io.StringIO which do not support the buffer attribute.

### text output formatting

There are ways to format your output;

• the first way is to do all the string handling yourself; using string slicing and concatenation operations you can create any layout you can imagine. The string type has some methods that perform useful operations for padding strings to a given column width; these will be discussed shortly. 比如：str.zjust(), str.center()
• The second way is to use formatted string literals, or the str.format() method.
• The string module contains a Template class which offers yet another way to substitute values into strings.

str.format 用法

print('We are the {} who say "{}!"'.format('knights', 'Ni'))
print('{1} and {0}'.format('spam', 'eggs'))
print('The story of {0}, {1}, and {other}.'.format('Bill', 'Manfred', other='Georg'))

'!a' (apply ascii()), '!s' (apply str()) and '!r' (apply repr()) can be used to convert the value before it is formatted

print('My hovercraft is full of {!r}.'.format(contents))

An optional ':' and format specifier can follow the field name. This allows greater control over how the value is formatted. Passing an integer after the ':' will cause that field to be a minimum number of characters wide.

print('The value of PI is approximately {0:.3f}.'.format(math.pi))

For a complete overview of string formatting with str.format(), see Format String Syntax.

print('The value of PI is approximately %5.3f.' % math.pi)

## File Input and Output

### mode

mode can be 'r' when the file will only be read, 'w' for only writing (an existing file with the same name will be erased), and 'a'opens the file for appending; any data written to the file is automatically added to the end. 'r+' opens the file for both reading and writing.

In text mode, the default when reading is to convert platform-specific line endings (\n on Unix, \r\n on Windows) to just \n. When writing in text mode, the default is to convert occurrences of \n back to platform-specific line endings. This behind-the-scenes modification to file data is fine for text files, but will corrupt binary data like that in JPEG or EXE files. Be very careful to use binary mode when reading and writing such files.

### with statement

It is good practice to use the with keyword when dealing with file objects. The advantage is that the file is properly closed after its suite finishes, even if an exception is raised at some point. Using with is also much shorter than writing equivalent try-finally blocks:

### api

To read a file’s contents, call f.read(size), which reads some quantity of data and returns it as a string (in text mode) or bytes object (in binary mode).size is an optional numeric argument. When size is omitted or negative, the entire contents of the file will be read and returned; it’s your problem if the file is twice as large as your machine’s memory. Otherwise, at most size bytes are read and returned. If the end of the file has been reached, f.read()will return an empty string ('').

f.readline() reads a single line from the file; a newline character (\n) is left at the end of the string, and is only omitted on the last line of the file if the file doesn’t end in a newline.

If you want to read all the lines of a file in a list you can also use list(f) or f.readlines().

f.write(string) writes the contents of string to the file, returning the number of characters written. Other types of objects need to be converted – either to a string (in text mode) or a bytes object (in binary mode) – before writing them

f.tell() returns an integer giving the file object’s current position in the file represented as number of bytes from the beginning of the file when in binary mode and an opaque number when in text mode.

To change the file object’s position, use f.seek(offset, from_what). The position is computed from adding offset to a reference point; the reference point is selected by the from_what argument. A from_what value of 0 measures from the beginning of the file, 1 uses the current file position, and 2 uses the end of the file as the reference point. from_what can be omitted and defaults to 0, using the beginning of the file as the reference point.

### binary mode

bytes data -> writer -> file

bytes data <- reader -> file

### text mode

text data -> encoder -> writer -> file

text data <- decoder <- reader <- file

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# Is there a rule of thumb for zero padding in image processing?

I see there are a lot of answers on why zero padding is necessary and how it avoids wrapping around the sides of images. However is there a rule of thumb on how much padding will be good for the image processing? I am mainly thinking from the point of view of:

a.) speed of processing the image

b.) noise

I have till now referred to the following answers:

Response will be really appreciated.

Thank you.

• you said "zero-padding against wrapping around the sides of the image"; what is the operation you do that leads to that wrapping? – Marcus Müller Mar 14 '19 at 9:36
• @MarcusMüllerI didn't get your question. Sorry. Could you please help e understand if your question was specific to my code a question in general? – Sulphur Mar 25 '19 at 19:39

See Press et al. "Numerical Recipes in C++". Chapter 13 on "Fourier and Spectral Applications", Section 13.1 with a subsection entitled "Treatment of end effects by zero padding". This is perhaps the best summary of zero-padding anywhere in the literature (but it's not referenced in either the contents or index). My edition of this book is the 2002 second edition and the page number there is 545.

• This is a gem of a book Jones. Thank you. I am reading the recommended chapter for the answer. Thank you. Much appreciated! – Sulphur Mar 25 '19 at 19:51

I do consider zero-padding as a form of an often-used combination of two other basic operations:

• "extension",
• "windowing",

for finite-length data (signals, images).

[Interlude] I don't know a generic word for that (like companding, that combines compressing and expanding). Fellow SE.DSPers, if you know of one, please share. Otherwise, I would propose "extowing". I will start with the "windowing" part.

• Windowing: While there is a huge literature in the design of 1D windows, I have neither been teached/exposed, nor seen so much references, on 2D (and nD) designs. If I focus on 2D windows per se (not for other purposes than windowing), in my bibliographic reference list (10.900+ items and counting), there is only a handful of such references, mostly ancient, like: Huang, T. S., 1970, Two-dimensional windows (quote: "Many good one-dimensional windows have been devised, however, relatively few two-dimensional windows have been investigated.") or Coulombe, S. and Dubois, E., 1996, Multidimensional windows over arbitrary lattices and their application to FIR filter design. Most per se designs I know of are:

• tensor, separable outer-products of 1D windows,
• non-separable circular extensions of 1D designs, where a centered $$W(t)=w(|t-t_0|)$$ window is converted to 2D with some norm $$l$$: $$W(x)=w(l(x-x_0))$$, with discretization and normalization side-effects,
• non-separable "1D-inspired" 2D optimization (like McClellan).

However, I have not seen a lot of them natively implemented in image processing software (apart from tensorized 1D windows and 2D discretized Gaussians).

• Extension: data extension is common practice in image processing, for different reasons. For instance, in JPEG Discrete cosine transform padding, one uses extensions to process images whose width or height are not divisible by 8. Additionally, DCT type II has beneficial symmetric features that are practically useful. A mere zero-padding can be applied, but the risk of strong artifacts at the borders is very high. Useful extensions can be strongly dependent on image applications and morphology. For instance, many sound/vibration signals are zero-mean, and can easily be zero-extended with a little tapering. Meanwhile, standard images have $$[0,255]$$ pixel values, and hence are not zero-mean. So constant (zero-order) or linear extensions are sometimes used at borders, and there exists a literature on windowing for adaptive (causal) image filtering (R.M. Mersereau ; D.E. Dudgeon, 1975, Two-dimensional digital filtering or J.H. McClellan, 1982, Multidimensional spectral estimation).

Both operations (Windowing and Extensions) are naturally combined in the design of multirate or multiscale filter banks, where parallel banks of windowed pass-band filters are designed together to allow both overlap between pixel blocks (to avoid sharp discontinuities) and perfect reconstruction (exact inversion). The Lapped Orthogonal Transform (LOT) is typical, with 50% overlap on each side. Embedded in the context of paraunitary filter filter banks, many works have derived symmetric or antisymmetric image extensions, to benefit from the inherent symmetries in the filters. The typology is often four-fold, with half-sample or whole sample symmetry, and symmetry or antisymmetry. They are sought to preserve "image" continuity or differentiability across blocks.

But let's get practical. If you have enough memory, my experience is that you are really safe, in the first instance, if you perform a 4-fold image extension (symmetric or antisymmetric depending on the data) and windowing: 50% on each edge, and a separable 1D window design, with a power-raised cosine window.

• Hi Laurent, Thaks for the detailed reply. If I understand it correctly, you tell a lot about how image continuity can be preserved and (padding) is useful for image reconstruction. However, I still did not get how it helps in speed and noise. Sorry but some rephrasing would be really helpful! – Sulphur Mar 25 '19 at 19:44
• I am not sure I understand your concerns on speed and noise. For speed, padding can help when it casts the image to a manageable size (like 8x8 blocks in JPEG). For noise, windowing can reduce symmetric wrapping artifacts. Are you digging into those directions? – Laurent Duval Mar 25 '19 at 19:53
• Umm, somewhat. For example, if I add padding (take any padding -zero or non-zero) to an image, will increasing/decreasing padding help? I can think like if we increase padding a lot that there may be too much background and hence noise but too less, will give sharper edges during processing. Definitely, we can use windowing to filter out some frequencies, but is there a rule of thumb as to how much padding would be good? Similarly, padding is basically adding more pixels in the image to be processed. How will padding affect the speed during any image processing? – Sulphur Mar 25 '19 at 19:59
• I am not sure I got all your comments. Padding can increase or decrease speed, ease or more more difficult processing depending on the image content and symmetries – Laurent Duval Jun 23 '19 at 20:22

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# $$K\backslash G/B$$ for other Cartan subgroups¶

Let us look at $$G=Sp(4,R)$$. The $$K\backslash G/B$$ elements are:

atlas> G
Value: connected split real group with Lie algebra 'sp(4,R)'
atlas> print_KGB (G)
kgbsize: 11
0:  0  [n,n]    1   2     4   5  (0,0)#0 e
1:  0  [n,n]    0   3     4   6  (1,1)#0 e
2:  0  [c,n]    2   0     *   5  (0,1)#0 e
3:  0  [c,n]    3   1     *   6  (1,0)#0 e
4:  1  [r,C]    4   9     *   *  (0,0) 1 1^e
5:  1  [C,r]    7   5     *   *  (0,0) 2 2^e
6:  1  [C,r]    8   6     *   *  (1,0) 2 2^e
7:  2  [C,n]    5   8     *  10  (0,0)#2 1x2^e
8:  2  [C,n]    6   7     *  10  (0,1)#2 1x2^e
9:  2  [n,C]    9   4    10   *  (0,0)#1 2x1^e
10:  3  [r,r]   10  10     *   *  (0,0)#3 1^2x1^e
atlas>

Recall that the first four form the fundamental fiber that go to the Cartan subgroup 0, the compact one. Elements 5 through 8 are attached to Cartan subgroup number 2, etc. The last collumn tells us that the fiber attached to the involution 2^e consists of elements 5 and 6 and the fiber corresponding to the element 1x2^e are elements 7 and 8. Here 2^e is just Cayley transform by $${\alpha}_2$$, whereas 1x2^e corresponds to conjugation by $${\alpha}_1$$ composed with the Cayley transform by $${\alpha}_2$$.

Let us recall which Cartan subgroups and Weyl groups correspond to each fiber:

atlas> set H=Cartan_class(G,0)
Variable H: CartanClass (overriding previous instance, which had type string (constant))
atlas> print_Cartan_info (H)
compact: 2, complex: 0, split: 0
canonical twisted involution: e
twisted involution orbit size: 1; fiber size: 4; strong inv: 4
imaginary root system: C2
real root system: empty
complex factor: empty
atlas>

As we know this is the Compact Cartan subgroup associated to the distinguished fiber:

atlas> print_real_Weyl (G,H)
real weyl group is W^C.((A.W_ic) x W^R), where:
W^C is trivial
A is trivial
W_ic is a Weyl group of type A1
W^R is trivial

generators for W_ic:
2,1,2
atlas>

This is a Weyl group of type A1. So, the number of KGB orbits for this Cartan is 8/4=2

Now for one of the intermediate Cartan subgroups we have:

atlas> H:=Cartan_class(G,1)
Value: Cartan class #1, occurring for 2 real forms and for 1 dual real form
atlas>
atlas> print_Cartan_info (H)
compact: 0, complex: 1, split: 0
canonical twisted involution: 2,1,2
twisted involution orbit size: 2; fiber size: 1; strong inv: 2
imaginary root system: A1
real root system: A1
complex factor: empty
atlas>

atlas> print_real_Weyl (G,H)
real weyl group is W^C.((A.W_ic) x W^R), where:
W^C is trivial
A is an elementary abelian 2-group of rank 1
W_ic is trivial
W^R is a Weyl group of type A1

generators for A
1
generators for W^R:
2,1,2
atlas>

This is a copy of $${\mathbb C}^\times$$ with Weyl group of order 4. So the number of KGB orbits is 8/4=2

Let us see what the $$W$$-orbit of one element is, say:

atlas> set x=KGB(G,4)
Variable x: KGBElt
atlas> void: for w in W do prints(cross(w,x)) od
KGB element #4
KGB element #4
KGB element #9
KGB element #9
KGB element #9
KGB element #9
KGB element #4
KGB element #4
atlas>

Starting with element 4 the order of its stabilizer has four elements. And if we list all the elements of $$W$$:

atlas> void: for (,w) in W do prints(w) od
[]
[0]
[1]
[1,0]
[0,1]
[0,1,0]
[1,0,1]
[1,0,1,0]
atlas>

We see that the elements [], [0], [1,0,1], and [1,0,1,0] all stabilize element 4. So the order of the stabilizer is 4. Similarly, for element 9.

Now for the next Cartan subgroup:

atlas> H:=Cartan_class(G,2)
Value: Cartan class #2, occurring for 1 real form and for 2 dual real forms
atlas>
atlas> print_Cartan_info (H)
compact: 1, complex: 0, split: 1
canonical twisted involution: 1,2,1
twisted involution orbit size: 2; fiber size: 2; strong inv: 4
imaginary root system: A1
real root system: A1
complex factor: empty
atlas>

This subgroup has order four. And its real Weyl group has order 2:

atlas> print_real_Weyl (G,H)
real weyl group is W^C.((A.W_ic) x W^R), where:
W^C is trivial
A is trivial
W_ic is trivial
W^R is a Weyl group of type A1

generators for W^R:
1,2,1
atlas>

Then the number of KGB orbits is 8/2=4 and we can verify also that each stabilizer is order 2:

atlas> x:=KGB(G,5)
Variable x: KGBElt
atlas>
atlas> void: for w in W do prints(cross(w,x)) od
KGB element #5
KGB element #7
KGB element #5
KGB element #8
KGB element #7
KGB element #6
KGB element #8
KGB element #6
atlas>
atlas> void: for (,w) in W do prints(w) od
[]
[0]
[1]
[1,0]
[0,1]
[0,1,0]
[1,0,1]
[1,0,1,0]
atlas>

Now for completeness, let us look at the split Cartan subgroup:

atlas> H:=Cartan_class(G,3)
Value: Cartan class #3, occurring for 1 real form and for 3 dual real forms
atlas>
atlas> print_Cartan_info (H)
compact: 0, complex: 0, split: 2
canonical twisted involution: 2,1,2,1
twisted involution orbit size: 1; fiber size: 1; strong inv: 1
imaginary root system: empty
real root system: C2
complex factor: empty
atlas>
atlas> print_real_Weyl (G,H)
real weyl group is W^C.((A.W_ic) x W^R), where:
W^C is trivial
A is trivial
W_ic is trivial
W^R is a Weyl group of type B2

generators for W^R:
1
2

A Cartan Subgroup isomorphic to $${\mathbb C}^\times \times {\mathbb C}^\times$$ and Weylgroup of type B2. So the number of KGB orbits is 8/8=1:

atlas> set x=KGB(G,10)
Variable x: KGBElt (overriding previous instance, which had type KGBElt)
atlas>  x:=KGB(G,10)
Value: KGB element #10
atlas>

atlas> void: for w in W do prints(cross(w,x)) od
KGB element #10
KGB element #10
KGB element #10
KGB element #10
KGB element #10
KGB element #10
KGB element #10
KGB element #10
atlas>

This concludes this deiscussion on $$K\backslash G/B$$ orbits. In the next chapter we will discuss the representations associated to the intermediate Cartan subgroups. The parameter includes a discrete series of a Levi factor of a parabolic subgroup. So, to some extent it reduces to the case of discrete series.

The idea is to look at the cuspidal data of an arbitrary parameter which gives a Levi factor $$M$$ and then applying what we learned about discrete series of M.


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# Water rocket

Water rocket launch

A water rocket is a type of model rocket using water as its reaction mass. The pressure vessel—the engine of the rocket—is usually a used plastic soft drink bottle. The water is forced out by a pressurized gas, typically compressed air. It is an example of Newton's third law of motion

## Operation

Simplified animation of how a water rocket works. 1) compressed air is added which creates a bubble that floats up through the water and then pressurizes the air volume in the top of the bottle. 2) The bottle is released from the pump. 3) The water is pushed out the nozzle by the compressed air. 4) The bottle moves away from the water because it follows Newton's Third Law.

The bottle is partly filled with water and sealed. The bottle is then pressurized with a gas, usually air compressed from a bicycle pump, air compressor, or cylinder up to 125 psi, but sometimes CO2 or nitrogen from a cylinder.

Water and gas are used in combination, with the gas providing a means to store potential energy, as it is compressible, and the water increasing the mass fraction and providing greater force when ejected from the rocket's nozzle. Sometimes additives are combined with the water to enhance performance in different ways. For example: salt can be added to increase the density of the reaction mass resulting in a higher specific impulse. Soap is also sometimes used to create a dense foam in the rocket which lowers the density of the expelled reaction mass but increases the duration of thrust. It is speculated that foam acts as a compressible liquid and enhances the thrust when used with De Laval nozzles.

The seal on the nozzle of the rocket is then released and rapid expulsion of water occurs at high speeds until the propellant has been used up and the air pressure inside the rocket drops to atmospheric pressure. There is a net force created on the rocket in accordance with Newton's third law. The expulsion of the water thus can cause the rocket to leap a considerable distance into the air.

In addition to aerodynamic considerations, altitude and flight duration are dependent upon the volume of water, the initial pressure, the rocket nozzle's size, and the unloaded weight of the rocket. The relationship between these factors is complex and several simulators have been written to explore these and other factors.[1][2][3]

Often the pressure vessel is built from one or more used plastic soft drink bottles, but polycarbonate fluorescent tube covers, plastic pipes, and other light-weight pressure-resistant cylindrical vessels have also been used.

Typically launch pressures vary from 75 to 150 psi (500 to 1000 kPa). The higher the pressure, the larger the stored energy.

## Predicting peak height

If aerodynamic drag and transient changes in pressure are neglected, a closed-form approximation for the peak height of a rocket fired vertically can be expressed as follows:

$h=\left({M_i \over M_R}\right)^2 \left({P_i \over \rho g}\right)$[4]

($h$ = peak height reached, $M_i$ = Initial mass of water only, $M_R$ = Rocket mass with water, $P_i$ = Initial gauge pressure inside rocket, $\rho$ = density of water, $g$ = acceleration due to gravity) Assumptions for the above equation: (1) water is incompressible, (2) flow through the nozzle is uniform, (3) velocities are rectilinear, (4) density of water is much greater than density of air, (5) no viscosity effects, (6) steady flow, (7) velocity of the free surface of water is very small compared to the velocity of the nozzle, (8) air pressure remains constant until water runs out, (9) nozzle velocity remains constant until water runs out, and (10) there are no viscous-friction effects from the nozzle (see Moody chart).

## Multi-bottle rockets and multi-stage rockets

Two multi-bottle rockets with a cat for scale.
A larger multi bottle rocket with cylindrical fins.

Multi-bottle rockets are created by joining two or more bottles in any of several different ways; bottles can be connected via their nozzles, by cutting them apart and sliding the sections over each other, or by connecting them opening to bottom, making a chain to increase volume. Increased volume leads to increased weight, but this should be offset by a commensurate increase in the duration of the thrust of the rocket. Multi-bottle rockets can be unreliable, as any failure in sealing the rocket can cause the different sections to separate. To make sure the launch goes well, pressure tests are performed beforehand, as safety is a concern. These are very good if you want to make the rocket go high however they are not very accurate and may veer off course.

Multi-stage rockets are much more complicated. They involve two or more rockets stacked on top of each other, designed to launch while in the air, much like the multi-stage rockets that are used to send payloads into space. Methods to time the launches in correct order and at the right time vary, but the crushing-sleeve method is quite popular.

## Sources of gas

Several methods for pressurizing a rocket are used including:

• A standard bicycle/car tire pump, capable of reaching at least 75 psi (520 kPa).
• Water pressure forcing all the air in an empty water hose into the rocket. Pressure is the same as the water main.
• An air compressor, like those used in workshops to power pneumatic equipment and tools. Modifying a high pressure (greater than 15 bar / 1500 kPa / 200 psi) compressor to work as a water rocket power source can be dangerous, as can using high-pressure gases from cylinders.
• Compressed gases in bottles, like carbon dioxide (CO2), air, and nitrogen gas (N2). Examples include CO2 in paintball cylinders and air in industrial and SCUBA cylinders. Care must be taken with bottled gases: as the compressed gas expands, it cools (see gas laws) and rocket components cool as well. Some materials, such as PVC and ABS, can become brittle and weak when severely cooled. Long air hoses are used to maintain a safe distance, and pressure gauges (known as manometers) and safety valves are typically utilized on launcher installations to avoid over-pressurizing rockets and having them explode before they can be launched. Highly pressurized gases such as those in diving cylinders or vessels from industrial gas suppliers should only be used by trained operators, and the gas should be delivered to the rocket via a regulator device (e.g. a SCUBA first-stage). All compressed gas containers are subject to local, state and national laws in most countries and must be safety tested periodically by a certified test centre.
• Ignition of a mixture of explosive gases above the water in the bottle; the explosion creates the pressure to launch the rocket into the air.[5]

## Nozzles

Water rocket nozzles differ from conventional combustion rocket nozzles in that they do not have a divergent section such as in a De Laval nozzle. Because water is essentially incompressible the divergent section does not contribute to efficiency and actually can make performance worse.

There are two main classes of water rocket nozzles:

• Open also sometimes referred to as "standard" or "full-bore" having an inside diameter of ~22mm which is the standard soda bottle neck opening.
• Restricted which is anything smaller than the "standard". A popular restricted nozzle has an inside diameter of 9mm and is known as a "Gardena nozzle" named after a common garden hose quick connector used to make them.

The size of the nozzle affects the thrust produced by the rocket. Larger diameter nozzles provide faster acceleration with a shorter thrust phase, while smaller nozzles provide lower acceleration with a longer thrust phase.

It can be shown that the equation for the instantaneous thrust of a nozzle is simply:[6]

$F = 2 P A_t$

where $F$ is the thrust, $P$ is the pressure and $A_t$ is area of the nozzle.

## Fins

Fins are extremely important on a water rocket. They ensure stability, and it is very likely that your water rocket flies higher with fins than without. Fins increase drag, but the stability achieved makes a much larger difference to the height your rocket will fly. A second thing that is very important is where you position the fins. It is best if they are placed near the back of the bottle where the center of mass is found. A water proof, stable, light material to make the fins would be "Coroplast". This is a cardboard like material that is almost indestructible. The only negative it has is that it is harder to glue, but with the right glue it is possible.

As the propellant level in the rocket goes down, it can be shown that the centre of mass initially moves downwards before finally moving upwards again as the propellant is depleted. This initial movement reduces stability and can cause water rockets to start tumbling end over end, greatly decreasing the maximum speed and thus the length of glide (time that the rocket is flying under its own momentum). To lower the centre of pressure and add stability, fins can be added which bring the centre of drag further back, well behind the centre of mass at all times, ensuring stability.

In the case of custom-made rockets, where the rocket nozzle is not perfectly positioned, the bent nozzle can cause the rocket to veer off the vertical axis. The rocket can be made to spin by angling the fins, which reduces off course veering.

Another simple and effective stabilizer is a straight cylindrical section from another plastic bottle. This section is placed behind the rocket nozzle with some wooden dowels or plastic tubing. The water exiting the nozzle will still be able to pass through the section, but the rocket will be stabilized.

The fins on a rocket are important for stability during flight. They should be placed near the rear of the rocket.

Aerodynamic drag acts on the fins as well as on the rocket body. Fins add to the frontal surface area on which the drag force acts (and therefore should be designed not to add too much drag). The drag forces on all frontal surfaces of the rocket can be resolved into one force acting at the center of pressure Center of pressure (fluid mechanics). This acts to oppose the forward motion, but if the rocket nose is not pointed in the direction of its motion at a given time (perhaps due to wobbling or instability), then there will be a torque, due to the resolved drag force, acting around the center of gravity. This torque will stabilize the rocket by returning its nose to the direction of travel.

Since the torque is the cross-product of the drag force magnitude and the moment arm, torque can be maximized without increasing drag force by increasing the moment arm. The larger the distance between the center of gravity and the center of pressure, the greater the moment arm on the restoring torque. Therefore, it is desirable to have the center of pressure, and therefore the fins, as far back as possible on the rocket body.

The lift force acts to push the back end of the rocket so that the nose will face the flight direction, and the drag force does the same, even though it is pointing orthogonally to the lift force.

[1]

## Landing systems

Stabilizing fins cause the rocket to fly nose-first which will give significantly higher speed, but they will also cause it to fall with a significantly higher velocity than it would if it tumbled to the ground, and this may damage the rocket or whoever or whatever it strikes upon landing.

Some water rockets have parachute or other recovery system to help prevent problems. However these systems can suffer from malfunctions. This is often taken into account when designing rockets. Rubber bumpers, Crumple zones, and safe launch practices can be utilized to minimize damage or injury caused by a falling rocket.

Another possible recovery system involves simply using the rocket's fins to slow its descent and is sometimes called backward sliding. By increasing fin size, more drag is generated. If the centre of mass is placed forward of the fins, the rocket will nose dive. In the case of super-roc or back-gliding rockets, the rocket is designed such that the relationship between centre of gravity and the centre of pressure of the empty rocket causes the fin-induced tendency of the rocket to tip nose down to be counteracted by the air resistance of the long body which would cause it to fall tail down, and resulting in the rocket falling sideways, slowly.

## Launch tubes

Some water rocket launchers use launch tubes. A launch tube fits inside the nozzle of the rocket and extends upward toward the nose. The launch tube is anchored to the ground. As the rocket begins accelerating upward, the launch tube blocks the nozzle, and very little water is ejected until the rocket leaves the launch tube. This allows almost perfectly efficient conversion of the potential energy in the compressed air to kinetic energy and gravitational potential energy of the rocket and water. The high efficiency during the initial phase of the launch is important, because rocket engines are least efficient at low speeds. A launch tube therefore significantly increases the speed and height attained by the rocket. Launch tubes are most effective when used with long rockets, which can accommodate long launch tubes.

## Safety

Water rockets employ considerable amounts of energy and can be dangerous if handled improperly or in cases of faulty construction or material failure. Certain safety procedures are observed by experienced water rocket enthusiasts:

• When a rocket is built, it is pressure tested. This is done by filling the rocket completely with water, and then pressurizing it to at least 50% greater than anticipated pressures. If the bottle ruptures, the amount of compressed air inside it (and thus the potential energy) will be very small, and the bottle will not explode.
• Using metal parts on the pressurized portion of the rocket is strongly discouraged because in the event of a rupture, they can become harmful projectiles. Metal parts can also short out power lines.
• While pressurizing and launching the rocket, bystanders are kept at a safe distance. Typically, mechanisms for releasing the rocket at a distance (with a piece of string, for example) are used. This ensures that if the rocket veers off in an unexpected direction, it is less likely to hit the operator or bystanders.
• Water rockets should only be launched in large open areas, away from structures or other people, in order to prevent damage to property and people.
• As water rockets are capable of breaking bones upon impact, they should never be fired at people, property, or animals.
• Safety goggles or a face shield are typically used.
• A typical two-litre soda bottle can generally reach the pressure of 100 psi (690 kPa) safely, but preparations must be made for the eventuality that the bottle unexpectedly ruptures.
• Glue used to put together parts of water rockets must be suitable to use on plastics, or else the glue will chemically "eat" away the bottle, which may then fail catastrophically and can harm bystanders when the rocket is launched.

## Water rocket competitions

The Oscar Swigelhoffer Trophy is an Aquajet (Water Rocket) competition held at the Annual International Rocket Week[7] in Largs, Scotland and organized by STAAR Research[8] through John Bonsor. The competition goes back to the mid-1980s, organized by the Paisley Rocketeers who have been active in amateur rocketry since the 1930s. The trophy is named after the late founder of ASTRA,[9] Oscar Swiglehoffer, who was also a personal friend and student of Hermann Oberth, one of the founding fathers of rocketry.

The competition involves team distance flying of water rockets under an agreed pressure and angle of flight. Each team consists of six rockets, which are flown in two flights. The greater distance for each rocket over the two flights is recorded, and the final team distances are collated, with the winning team having the greatest distance. The winner in 2007 was ASTRA. The competition has been regularly dominated over the last 20 years by the Paisley Rocketeers.

The United Kingdom's largest water rocket competition is currently the National Physical Laboratory's annual Water Rocket Challenge.[10] The competition was first opened to the public in 2001 and is limited to around 60 teams. It has schools and open categories, and is attended by a variety of "works" and private teams, some travelling from abroad. The rules and goals of the competition vary from year to year.

The Water Rocket Achievement World Record Association.[11] A worldwide association which administrates competitions for altitude records involving single stage and multiple stage water rockets, a flight duration competition, and speed or distance competitions for water rocket–powered cars,

The oldest and most popular water rocket competition in Germany is the Freestyle-Physics Water Rocket Competition.[12] The competition is one part of a larger part of a student physics competition, where students are tasked to construct various machines and enter them in competitive contests.

Science Olympiad also has a Water Rocket event.

## World record

Apogee photograph taken by the onboard video camera from U.S. Water Rockets' record breaking X-12 Water Rocket at an altitude of 2,068 feet (630 m).
Most water rockets launched simultaneously, by Gotta Launch

The Guinness World Record of launching most water rockets simultaneously is in hands of Gotta Launch,[13] when on 19 June 2009, they launched 213 of them at the same time, together with students of the Delft University of Technology.

The current record for greatest altitude achieved by a water and air propelled rocket[14] is 2044 feet (623 metres), held by U.S. Water Rockets[15] on 14 June 2007. This altitude was calculated by averaging two flights. The first flight achieved 2068 feet (630 meters) and the second 2020 feet (615.7 meters). The rocket also carried a video camera as payload on both flights as part of the verification required by the competition rules.

## Steam rockets

A steam rocket (or hot water rocket) is a rocket which uses steam as its propellant. Before launch, water in the sealed rocket is heated. As the rocket remains sealed, the pressure increases. This pressure is sufficient to keep the water as superheated water, rather than boiling into steam, as the boiling temperature of water increases with pressure. On launch, the pressure vessel is vented through a nozzle. The released water drops in pressure as it passes through the nozzle, allowing it to boil or 'flash' instantly into steam. The high velocity of the steam, and its expansion through the nozzle, gives rise to the usual reaction force for a rocket.

The idea of such rockets was conceived by Germany before the Second World War, with the suggested use of an alternative rocket engine for launching fighter jets. Some of the few practical steam rockets constructed have been used for drag racing and for Evel Knievel's Skycycle X-2 canyon jump.


text:
show · gl2.profinite all knowls · up · search:

A profinite group is a compact totally disconnected topological group. Equivalently, it is the inverse limit of a system of finite groups equipped with the discrete topology.

For example, if we take the finite groups $\GL(2,\Z/m\Z)$ as $m$ varies over positive integers, order them by divisibility of $m$ and consider the inverse system equipped with reduction maps $\GL(2,\Z/n\Z)\to \GL(2,\Z/m\Z)$ for all positive integers $m|n$, then the inverse limit $$\lim_{\overset{\longleftarrow}{m}} \GL(2,\Z/m\Z) \simeq \GL(2,\widehat \Z)$$ is a profinite group which is isomorphic to the group of invertible $2\times 2$ matrices over the topological ring $\widehat \Z$, which is the inverse limit of the finite rings $\Z/m\Z$ equipped with the discrete topology.

Authors:
Knowl status:
• Review status: beta
• Last edited by Andrew Sutherland on 2021-07-17 14:54:23
Referred to by:
History:


text:
#### • Class 11 Physics Demo

Explore Related Concepts

# break even point excel formula

From Wikipedia

Break-even

Break-even (or break even) is a point where any difference between plus or minus or equivalent changes side.

## In economics

In economics& business, specifically cost accounting, the break-even point (BEP) is the point at which cost or expenses and revenue are equal: there is no net loss or gain, and one has "broken even". A profit or a loss has not been made, although opportunity costs have been paid, and capital has received the risk-adjusted, expected return.

## In other fields

In nuclear fusion research, the term breakeven refers to a fusion energy gain factor equal to unity, this is also known as the Lawson criterion.

The notion can also be found in more general phenomena, such as percolation, and is rather similar to the critical threshold. In energy, the breakeven point is the point where usable energy gotten from a process exceeds the input energy.

In computer science, the (less usual) term refers to a point in the life cycle of a programming language where the language can be used to code its own compiler or interpreter. This is also called self-hosting.

In medicine, it is a postulated state when the advances of medicine permit every year an increase of one year or more of the life expectancy of the living, therefore leading to medical immortality (barring accidental death).

Question:My algebra is a little rusty. Using a random example of say phone rates what would be the break even point for two different rate plans. Plan A: $5 per month +$.05 per minute. Plan B: $.16 per minute. So that would be 5 + .05x = .16x where x = number of minutes, the break even point, right? Assuming I have the formula right what would then be the steps (please include any algebraic rules/laws) for solving for x. Thanks! Answers:Yes, that's the breakeven point. The breakeven point is the point at which neither option is better than the other (when the two plans have the same cost). 5 + .05x = .16x subtract .05x from both sides of the equal sign: 5 + .05x -.05x= .16x -.05x 5 = .11x divide both sides by .11 to get x alone by itself on one side of the equal sign. 5 = .11x 5/.11 = .11x/11 5/.11 = x 45.45 = x 45.45 minutes is the breakeven point this is 45 minutes, 27 seconds Question:a cookie store owner spends 2 dollars on each cookie and 30 dollars on additional expenses. she sells each cookie for 3 dollars. find the break even point. help? Answers:x - minimum number of cookies so she can profit 3x - 2x - 30 = 0 x=30 Question:I am using Business Plan Pro 2007 and it is giving me a negative monthly Revenue Break Even point even though all the other figures are correct. So, I may have to do it myself. Note: This is an internet site that ONLY sells advertising - however, dozens of different prices for the advertising. Important Note: This is for a startup company, so no hard figures are available, I only have projections. Thanks for your help. Answers:I'm glad to hear that you're using Business Plan Pro 2007 but sad to hear that you're having problems with it calculating your Break Even point. One thing that you might try in Business Plan Pro is to reset the default formulas in the Break Even table so that it is pulling the correct data from your Sales Forecast and Profit and Loss to calculate the break even point. You can reset the default formulas by right clicking in the Break Even table and choosing Formula Reset > Table. If you're still having problems with it calculating a negative break even point, you might have an unusual forecast like a partial year for the first year in the plan. In that case, you'll be better served by doing your own Break Even point. It's usually calculated by taking the following: Fixed Cost (1-(Unit variable Costs/Unit Price)) Where: Unit Price - The price that you charge per unit. Take into account sales discounts and special offers. For non-unit based businesses, make the per-unit revenue$1 and enter your costs as a percent of a dollar. Unit variable cost - The incremental cost of each unit of sale. If you are using a Units-Based Sales Forecast table (for manufacturing and mixed business types), you can project unit costs from the Sales Forecast table. If you are using the basic Sales Forecast table for retail, service and distribution businesses, use a percentage estimate. For example, a retail store running a 50% margin would have a per-unit cost of .5, and a per-unit revenue of 1. Fixed costs - Technically, a break-even analysis defines fixed costs as costs that would continue even if you went broke. Instead, you may want to use your regular running fixed costs, including payroll and normal expenses. This will give you a better insight on financial realities. This information is available online in the HurdleBook at: http://www.hurdlebook.com/ Here's a selection of Break Even articles: http://tinyurl.com/2bdspk (sorry, I had to make it a TinyURL, Yahoo Answers didn't like it) Lastly, we do have a free "Break Even" calculator available if you want to use it instead. You can find that on our www.Bplans.com site at: http://www.bplans.com/common/calculators/breakeven.cfm

Question:So, midterms are in 2 weeks and while trying to review for math I came across a problem in a section titled "Break-Even Point 'Use algebra to find the break-even point' " there are 4 questions in the section but I figured after I got the first one I could figure out how to do the others. I just can't even remember learning it. the problem is: E=252+3.4x I=6.7x I don't even know where to start. It would be wonderful if someone could take the time to explain how to get to the answer, not just the answer. I already have that, thanks to the answer key! Thank you so much.

Answers:The break-even point occurs when E = I, so you have 252 + 3.4x = 6.7x 252 = 3.3x x = 252/3.3 = 76.36363636 Plug the x into either equation to get the y value. I = 6.7(76.36363636) = 511.6363636 So your break-even point is (76.36363636,511.6363636)

Excel Magic Trick 744: Break Even Analysis Formulas Chart & Plotting Break Even Point On Chart :Download file: people.highline.edu See how to do Break Even Analysis Table with Formulas, Chart (Xy Scatter Chart Diagram), and Conditional Formatting with a Logical Formula (True False Formula) and Mixed Cell References. Also See a Formula and charting trick to show the approximate Break Even Point on the Chart. Accounting Break Even Analysis Excel. Managerial Accounting Cost Accounting Fixed Cost Variable Cost Accounting Contribution Margin Accounting Class Break Even Analysis XY Scatter Diagram Chart Line Chart Finance Break Even.

Perform a Break Even Analysis with Excel's Goal Seek Tool :When you are considering an investment or launching a new product you want to know at what point you will "break even." The Goal Seek tool in Excel is perfect for this scenario. In this case we want the resulting value in our formula to return "0" - our break even point. Watch how I create multiple scenaios with Goal Skke to help me to set the proper selling price and cost structure. I invite you to visit my website - www.thecompanyrocks.com - to view all of my videos and resources including "The 50 Best Tips for Excel 2007 DVD. Danny Rocks


text:
# LaTeX to WordPress

LaTeX2WP is a program that converts a LaTeX file into something that is ready to be cut and pasted into WordPress.

This way, you can write, and preview, your post in LaTeX, then run LaTeX2WP, and post into WordPress whatever comes out.

Why would you want to do that? Most users of WordPress wouldn’t.

But if you have ever found yourself typing

 <p align="center"> $latex \displaystyle \mathop{\mathbb E}_{x\sim X} f(x):= 1 \ \ \ \ (1)&fg=000000$ </p>

in order to get

$\displaystyle \mathop{\mathbb E}_{x\sim X} f(x):= 1 \ \ \ \ (1)$

and wondered whether there is a better way, maybe this is for you.

WordPress does a good job supporting simple LaTeX expressions. You type $latex 2^{2^k}$, and out comes $2^{2^k}$. There are, however, some inconveniences if you are typing something more complex:

• The editor windows is small, and it can be slow to save and produce a preview;
• Simulating LaTeX’s displayed equations is a bit painful. If one wants to have numbered equations, they have to be numbered by hand; if one wants to insert one more equation or rearrange the order of presentation, equations must be renumbered by hand, and so do all references to equations;
• If one wants to publish the notes/ expositions posted on WordPress, then one needs to convert them back to LaTeX.

There are also smaller concerns: for example some WordPress themes (including in theory‘s one) typeset LaTeX equations in gray. This can be changed by appending &fg=000000 to every formula, which is nice to automate. If you are used to having simple macros like \E for expectation, it is annoying to have to type {\mathop{\mathbb E}} each time.

With LaTeX2WP you write your whole post in LaTeX, and use LaTeX to preview it. You cannot use arbitrary macros, but any macro that can be implemented by a simple substitution can be easily added. Any set of theorem-like environments such as theorem, lemma, proposition, remark, etc. can be defined, with any combination of counters. Displayed equations are automatically numbered and centered, and all math equations are typeset with black fonts. The LaTeX commands \label, \ref and \eqref are supported, and \ref and \eqref translate to clickable links. You also maintain a LaTeX source for your post, which is ready for publication in another venue.

So far LaTeX2WP has been used at what’s new and here, at in theory. You’ll notice that the typesetting styles for theorems and section names look different. It is easy to modify the typesetting options to suit your own taste.

I would like to thank Terry Tao with his patience while using a very buggy early version and suggesting several useful improvements.

LaTeX2WP is distributed as free software with a GPL 3 license.

(More or less, this means that you can do with it whatever you like. If you create and distribute a modified version, however, you must distribute it as free software under the same GPL license and credit me for having written the original version.)

If you would like to try it out, click the download tab. If you would like to learn more about the features and limitations of LaTeX2WP, or if you are using it and have a question or a bug report, click the using LaTeX2WP tab.

If you end up using it, do let me know, by commenting on this page or by sending an email to LaTeX2WP at gmail dot com. Do include a link to your blog.

## 300 thoughts on “LaTeX to WordPress”

1. Luca,

This sounds great. Thanks for making this available. I maintain a blog for my coding theory course and the limitations of wordpress LaTeX support has been somewhat annoying. I’m looking forward to using your program to make my life much easier. Thanks again.

2. Excellent! Would you consider licensing this under GPL V2 or GPL V2+? Some people at big companies (e.g. Microsoft) can’t contribute to GPL3 software because of its terms regarding patents.

3. Cool! I’ll give it a try, definitely.

Thanks again.

Ran

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88. Using your program at studeocs. It’s very nice so far! I need to include citation information in my blog (for example, have a bibliography at the end of the post and numbered in-document links within the text). Any idea how I could automate that? I’ll be using a bib file for all my references. Thanks.

89. HI, I just started blogging and posted my very first blog with your latex to wordpress script. It is very nice, Thanks!

The only thing I noticed is that it seems the following escape sequences are not converted correctly:
1. “\{” or “\}” is rendered “{” or “}” in latex but “\{” or “\}” in latex2wp
2. “\verb|…|” is rendered “…” in latex but “\verb|…|” in latex2wp
3. “\^{}” is rendered “^” is latex but “\^{}” in latex2wp

Please let me know if there is a way to print those special symbols consistently in both latex and latex2wp 🙂

Thanks

90. To render curly brackets, put them in a math environment: $\{$ will render as {. Same for ^. Unfortunately there isn’t a simple workaround for the verbatim mode of latex

91. Thanks!

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94. Hi Luca!

I think that if you use some service like github.com to share the code, the community maybe could improve it. This script is very useful and I’ll be glad to help at least a little.

Regards.

95. I think this is among the most important info for me.
And i’m glad reading your article. But wanna remark on few general things, The site style is great, the articles is really great : D. Good job, cheers

96. Hey – I wanted to thank you for this. I’ve just started using it as well.

I maintain a mathblog at mixedmath.wordpress.com.

97. $latex \displaystyle \mathop{\mathbb E}_{x\sim X} f(x):= 1 \ \ \ \ (1)&fg=000000$

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99. Hi,
used it for my first post- works quite fine, but did have to do some editing.

100. Hello Luca,

I would like to thank you for your latex2wp program. It has helped me a lot! I have also included a link to yours in my own blog.

Thanks again.

101. Try this
$\displaystyle \li_{s}(i)=-2^{-s}\eta(s)+i\beta(s)\quad (1)$

102. $\displaystyle \li_{s}(i)=-2^{-s}\eta(s)+i\beta(s)$

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108. Awesome blog! Is your theme custom made or did you download it from somewhere?
A design like yours with a few simple tweeks would really make my
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Cheers

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114. Awesome! I was about to write an app like this. Looks like I can focus on math instead 😛

115. Thanks for this program! I started using it at amakelov.wordpress.com.

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119. Hi Luca, thank you for your wonderful LaTeX-to-WordPress program. Just wanted to let you know that I have been using it for my recent posts and have shared this page so that others can also find out about the program.

120. Thank you very much!

I wanted to construct a table and it worked great. It would be perfect if we could construct vertical lines separating the table using tabular (e.g. something like \begin{tabular}{c||cc|cc}) but it seems that the program could not parse it.

121. Hi Luca,

Thanks for the program. I played around with your code and I have implemented the align environment (not the align* one). I can send you the modified code. Since I am not really familiar with Python, I just did somethings with trial and error. I would be happy if you can check the changed code and write it in a better way and update the version, since mine looks like a mess!

If you are interested let me know.

Thanks!

122. I do consider all of the ideas you have introduced on your post.
They’re really convincing and can certainly work. Nonetheless, the posts are too quick for novices.

May just you please prolong them a little from subsequent time?
Thanks for the post.

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125. Hi. I searched for latex in wordpress and found your program. This is really great work. I have some knowledge in python too. If I have time, I may improve it. Specially, the area of using definition theoremstyle or other commands like eqnarray in latex. Anyway, than you very much for your time in this. Let me know when you try to improve it. I will help if I can. And the codes were nice and readable by the way.

126. Though I said this in mail, I think I should say it here too. This program is compatible with python version 2 only. Not with version 3(probably latter versions). Anyone who is getting errors like “float to integer version” or “s=extractbody(s)”, be sure to reinstall python 2 and it should be gone. Thanks.
BTW, you asked to include blog address too. This is mine: https://karushib.wordpress.com/2015/08/19/ease-the-pain-of-latex-writing-in-wordpress/

127. Hey Luca, thanks very much for sharing the code! I’m using a slightly modified version to run a jekyll-based website here: https://amakelov.github.io

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130. Hi and thanks for this very helpful tool! One question: Is there an easy way to change the numbering of theorems to something like “2.3.1” for Theorem 1 in Section 2, Subsection 3, instead of plain “1”? Putting _SecNumb_._SubSecNumb_ in front of _ThmNumb_ seems not to work …

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132. Is there a list somewhere of the LaTeX packages which are supported? I like to use some packages, like enumitem or tcolorbox, which maybe are not commonly supported and thus I am unsure whether I should rewrite my LaTeX documents to not use those packages or not.

133. Cool tool, thanks for making this.

I had a quick question though, you mentioned that:

If one wants to publish the notes/ expositions posted on WordPress, then one needs to convert them back to LaTeX.

Do you know a good tool for this direction? Especially for batch jobs? For example, a whole wordpress blog from back-up files. I’ve seen a few that convert straight to pdf, but it’s nice to have LaTeX in between to do some clean up. Also, prefer not to go through html scraping and html-to-TeX since it’d lose info from within the wordpress latex environments (since they just become images).

134. Hi can I indent paragraphs? I tried to set up indent, for example, \setlength{\parindent}{0.4in}. But in HTML it did not come out.

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139. Hello, Thank you for your program. I am using it in my blog

140. If you use Python 3 and see the problem
for i in range(1,(len(L)+1)/2) :
TypeError: ‘float’ object cannot be interpreted as an integer
Simply change the division operator / to the integer division operator //
There are two occurrences, one on line 140 and the other on line 153.

Explanation: Python 2’s integer division behaves like C, it rounds down. To get the same behavior in Python 3, use // instead of /.

141. You shared a great content of MATLAB. Thank you for sharing this.


text:
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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 5th 2010
• (edited Mar 5th 2010)

edited universal covering space

• moved my proof to here that it is the homotopy fiber of the morphism $X \to \Pi_1(X)$ of topological groupoids;

• then created a new section universal covering spaces in an (oo,1)-topos where I talk about an idea (the evident idea) for what one should say here.

• Also propose a definition for Whitehead tower in an (oo,1)-topos this way.

• the entry Postnikov tower in an (infinity,1)-category following JL is under way...

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeMar 5th 2010
Thanks for that. I noticed I wrote something about doing this a long time ago.
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 2nd 2017

I have added a little more point-set details to the bginning at universal covering space.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeJul 2nd 2017

I don’t mean to interrupt Urs’s train of thought (and he should feel free to ignore this while he’s hammering out details), but now returning to this topic about 7 years later, the general abstract construction of universal covering spaces in the context of cohesion seems sort of interesting to me. Part of this is the desire to get free from basepoints, and free from the path-connected assumption on the base space, and free from just groups, passing to groupoids, etc., and connect up with (so-called) “Trimble” $n$-categories…

Here is a diagram I’ve written down on my private web, an abstract construction of what I am provisionally calling the unbiased universal covering space $\widehat{X}$ with its projection $p: \widehat{X} \to X$:

$\array{ & & \Delta \Gamma X & \leftarrow & \widetilde{X} & \stackrel{\gamma}{\to} & \Delta \Pi_0(\widetilde{X}) & & \\ & \mathllap{i} \swarrow & & (pb) & \downarrow & (po) & \downarrow & & \\ X & \stackrel{eval_1}{\leftarrow} & X^I & \leftarrow & P & \to & \widehat{X} & \stackrel{p}{\to} & X\\ & & \mathllap{eval_0} \downarrow & (pb) & \downarrow & \swarrow & & & \\ & & X & \underset{i}{\leftarrow} & \Delta \Gamma X & & & & }$

I should explain the notation. We have a cohesive string of adjoints $\Pi_0 \dashv \Delta \dashv \Gamma \dashv \nabla: Set \to LocConn$ which is just what you expect: $\Gamma = \hom(1,-)$ is the underlying set functor, $\Delta$ and $\nabla$ take sets to the associated discrete and codiscrete spaces, and $\Pi_0$ is the (path-)connected components functor. (There are some point-set topology details I’m sliding over for the moment, so that we can consider the big picture.) The map $i: \Delta \Gamma X \to X$ is a counit for $\Delta \dashv \Gamma$, and the map $\gamma: \widetilde{X} \to \Delta \Pi_0 \widetilde{X}$ is a unit for $\Pi_0 \dashv \Delta$. The rest of the notation should be self-explanatory.

Ordinarily, when one constructs a universal covering space, one chooses a basepoint $x$ of the base space and looks at homotopy-rel-boundary classes of paths in $X$ whose starting point of $x$. The construction above does something similar except that it considers all possible basepoints (hence, “unbiased”). So, whereas in the ordinary construction one chooses an $x \in X$ and constructs the space of paths $P_x$ emanating from $x$ as a pullback

$\array{ P_x & \to & X^I \\ \downarrow & & \downarrow \mathrlap{eval_0} \\ 1 & \underset{x}{\to} & X }$

here we construct a space $P = \sum_{x \in X} P_x$, the coproduct over all such $P_x$, as the pullback at the bottom left. The $\widetilde{X}$ you see at top is similarly formed as a pullback; geometrically it’s the coproduct of spaces $\sum_{(x, y) \in X \times X} \widetilde{X}(x, y)$ where $\widetilde{X}(x, y)$ is the space of paths from point $x$ to point $y$. Thus $\Pi_0(\widetilde{X}(x, y))$ would be the hom-set $\Pi_1(X)(x, y)$ of the fundamental groupoid, and $\Delta \Pi_0(\widetilde{X})$ is the coproduct of all hom-sets of $\Pi_1(X)$ (seen as a discrete space). The right map $\gamma$ along the top sends a path $\alpha$ to its homotopy-rel-boundary class. Thus when we push out $\gamma$ to get $P \to \widehat{X}$, we get homotopy-rel-boundary classes of elements of $P$, which is exactly what we want for the unbiased universal covering space. (I’ll continue in the next comment.)

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeJul 2nd 2017

The composite map $X \stackrel{eval_1}{\leftarrow} X^I \leftarrow P$ factors through $P \to \widehat{X}$, and that’s where $p: \widehat{X} \to X$ comes from. (Again, all of this follows I believe from purely formal properties of cohesion.) Of course

$\array{ & & \Delta \Pi_0 \widetilde{X} = \Pi_1(X) & & \\ & \swarrow & & \searrow & \\ \Delta \Gamma X & & & & \Delta \Gamma X }$

is a monoid in the category of endospans (namely the fundamental groupoid), and the span

$\array{ & & \widehat{X} & & \\ & \swarrow & & \searrow^\mathllap{p} & \\ \Delta \Gamma X & & & & X }$

carries a canonical module structure over this monoid (I guess a right module – whatever it is you call a functor $\Pi_1(X)^{op} \to Top$). If $F: \Pi_1(X) \to Set$ is a left module, then you can form a tensor product or weighted colimit $\widehat{X} \otimes_{\Pi_1(X)} F$, an object of $Top$, in fact when you form that span composition an object of $Top/X$ and indeed a covering space over $X$. Then

$\widehat{X} \otimes_{\Pi_1(X)} -: Set^{\Pi_1(X)} \to Cov/X$

is the left adjoint in the adjoint equivalence which expresses the fundamental theorem of covering space theory, but expressed in unbiased form.

I’m not sure where to find this in the literature (some authors come close but then pull back, somehow), but the thing I find pleasant is just how formal this is, and how much you can milk on the basis of so little (cohesion plus the topological interval as data). And it hadn’t occurred to me before that the machinery of “Trimble $n$-categories”, following these developments naturally as it were, is maybe best viewed in a cohesive context.

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeJul 3rd 2017

You’d want some sort of semi-local simple-connectedness to get local triviality of your $\widetilde{X}\to X$, and for the higher stages of the Whitehead tower of $X$, local $(n-1)$-connectedness and semilocal $n$-connectedness in order to get a locally constant stack of $(n-1)$-groupoids of local sections.

There is work by Jeremy Brazas on the setup when you lose the nice local homotopical conditions, so that $\pi_1$ is naturally a semitopological group (multiplication is separately continuous in each variable) as a quotient of the based loop space, and you can also put a group topology on it, and get the nice construction of universal “covering spaces” for that framework (fibres are no longer discrete, but one has good lifting properties).

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeJul 3rd 2017

Interesting! Fills me with questions.

1. Can you then extract a space that’s equivalent to the usual universal covering spaces, instead of a coproduct of a bunch of copies of it, by taking some kind of quotient of $\widehat{X}$?
2. What do you get if you do it all homotopically, i.e. use ʃ instead of $\Pi_0$ and a homotopy pushout instead of a set-pushout?
3. What if instead of the pushout, we considered $P$ as living over $X\times \flat X$ (i.e. $X\times \Delta\Gamma X$) and take its fiberwise ʃ or $\Pi_0$?
• CommentRowNumber8.
• CommentAuthorDavidRoberts
• CommentTimeJul 3rd 2017

@Mike Todd has explained this construction before, but not using formal cohesion. Essentially any two copies of a universal cover of a component $X_\alpha \subset X$ corresponding to basepoints $x_1,s_2$ get identified as there is an arrow between them in $\Pi_1(X)$. A homotopy pushout would remember which arrow one is using to identify these two covering spaces.

Note that Ronnie Brown would be happy to only go halfway and choose a basepoint in each path component of $X$, but this does away with that by using every single point, then gluing them all back together. It’s really very nice!

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeJul 3rd 2017

But the pushout all happens over the bottom copy of $\Delta\Gamma X$, so it seems to me that it can’t identify things in one component of $P$ with another?

• CommentRowNumber10.
• CommentAuthorDavidRoberts
• CommentTimeJul 3rd 2017

That map $\widehat{X}\to \Delta\Gamma X$ looks wrong to me. Where does it come from?

• CommentRowNumber11.
• CommentAuthorDavidRoberts
• CommentTimeJul 3rd 2017
• (edited Jul 3rd 2017)

(duplicate)

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJul 3rd 2017
• (edited Jul 3rd 2017)

Mike, the version in $\infty$-cohesion is what we have in the $n$Lab entry since rev 6 in 2010: the universal covering space is the homotopy fiber of

$X \longrightarrow \Delta \Pi_1(X) = \tau_1 ʃ X$

over any point (if $X$ is connected).

(As one runs through the truncation degree $\tau_n$ here, one obtains the whole Whitehead tower. This is something that David Roberts and myself made notes about years ago at Whitehead tower in an (infinity,1)-topos. Looking back at it, it could do with some polishing.)

If we don’t like to choose a point, we could instead take the homotopy fiber product with $\flat X = \Delta \Gamma \to \Delta \Pi_1 X$ (the points-to-pieces transform). This would give the $\Gamma X$-many-copies-version of the covering space which Todd is considering above.

In the entry instead the perspective is taken that one takes the fiber product along a choice of section $\pi_0(X) = \tau_0(\Delta \Pi_1 X) \to \Delta \Pi_1 X$ of the 0-truncation. This is not as bad as it sounds, since this section is essentially unique.

But I am inclined to view this as the pullback along something like the “smallest 1-epimorphism out of a 0-truncated object into $\Delta \Pi_1 X$.” I wish I had a solid general definition of what this should be, because something similar happens in the definition of ordinary differential cohomology:

There we form the homotopy fiber product of a canonical map $\mathbf{B}^n \mathbb{Z} \to \mathbf{B}^n \mathbb{Z}$ along something like the “smallest 1-epimorphism-when-evaluated-on-manifolds out of a 0-truncated object”, namely the morphism $\Omega^n_{cl} \longrightarrow \mathbf{B}^n \mathbb{R}$ (from closed differental $n$-forms to the coefficients for degree $n$ real cohomology, which is an epimorphism when evaluated on manifolds (or anything smooth with a partition of unity) by the de Rham theorem).

So it seems a common theme that we want to take homotopy fibers of morphisms into some $\infty$-stack by choosing a smallest 0-truncated object that hits all the connected components of the thing we are pullback back over, at least after evaluating everything on some prescribed class of objects. I keep wanting to have a good general formalization of this rough idea. It might be the answer to some other open problems that I have.

• CommentRowNumber13.
• CommentAuthorDavidRoberts
• CommentTimeJul 3rd 2017

This is not as bad as it sounds, since this section is essentially unique.

its mere existence is unique, but it can have many automorphisms!

• CommentRowNumber14.
• CommentAuthorTodd_Trimble
• CommentTimeJul 3rd 2017

Guys, thanks very much for your interest and questions! I’m going to be very busy today with other things, but I’m looking forward to getting back to this soon, hopefully by sometime tomorrow. I can tell I’m going to learn a great deal from this, and I’m sure to have questions in return.

I did want to address David’s question in #10 about $\Delta \Gamma X \leftarrow \widehat{X}$, which is induced from the map $P \to \Delta \Gamma X$. Set-theoretically it’s supposed to take an equivalence class of paths $[\alpha]$ to $\alpha(0)$. I think it’s well-defined because two paths $\alpha, \beta \in P$ map to the same $[\alpha] = [\beta]$ in $\widehat{X}$ iff when we regard $\alpha, \beta$ as belonging to $\widetilde{X}$ (remember that set-theoretically $\widetilde{X} \to P$ is an identity because it’s a pullback of the set-theoretic identity map $\Delta \Gamma X \to X$ on the left), we have $\gamma(\alpha) = \gamma(\beta)$: that only happens if $\alpha(0) = \beta(0)$ (call it $x$) and $\alpha(1) = \beta(1)$ (call it $y$) and there’s a path between $\alpha, \beta$ in $\widetilde{X}(x, y)$ (homotopy-rel-boundary). Anyway, we do have $\alpha(0) = \beta(0)$, so $[\alpha] \mapsto \alpha(0)$ is well-defined.

• CommentRowNumber15.
• CommentAuthorDavidRoberts
• CommentTimeJul 3rd 2017

Hmm, so you do have a disjoint union of simply-connected covering spaces, one for each point of the space? To my mind, a universal covering space of a disconnected space should induce an isomorphism on $\Pi_0$. But perhaps that’s just experience with pointed covering spaces talking…

• CommentRowNumber16.
• CommentAuthorMike Shulman
• CommentTimeJul 3rd 2017

Thanks Urs! Do you think that can be made to coincide formally with Todd’s definition?

This is not as bad as it sounds, since this section is essentially unique.

its mere existence is unique

… assuming the axiom of choice in the discrete world (right?). (-: Otherwise it might not exist at all. In fact, if sets cover fails, there may not be any epimorphism to an object out of a 0-truncated one.

• CommentRowNumber17.
• CommentAuthorDavidRoberts
• CommentTimeJul 3rd 2017

@Mike

assuming the axiom of choice in the discrete world (right?)

of course, and that’s a reason to use the cohesive treatment over the version that assumes one has a section of $X\to \Delta\Pi_0(X)$!

• CommentRowNumber18.
• CommentAuthorTodd_Trimble
• CommentTimeJul 3rd 2017

David: yes. And you raise a good point. I’d like to mull over that further…

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeJul 3rd 2017
• (edited Jul 3rd 2017)

I have written out now the abstract argument that the definition in cohesive homotopy theory comes out right, this is now the first part of the section (now) titled Universal covering space – In cohesive homotopy theory.

The previous argument that used to be in the entry (now the second part of that section, not notationally harmonized yet), argues that the homotopy fiber of the morphism from $X$ to its fundamental groupoid, both regarded as topological stacks, canonically yields the classical construction via homotopy classes of paths, the way Todd is discussing above. For this to be a special case of this abstract construction one needs a result, namely that the shape modality is represented by a “path $\infty$-groupoid”.

This is true in “Euclidean topological infinity-groupoids” at least, due to a result that Dmiti Pavlov has announced long ago, following a similar result by Bunke-Nikolaus-Völkl. I am recording this now at shape via cohesive path ∞-groupoid.

One would like this to be true also over the $\infty$-cohesive site of locally contractible topological spaces (or at least over the 1-cohesive site of locally simply connected topological spaces), but I don’t know if it is. Somebody should think about this!

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeJul 3rd 2017
• (edited Jul 3rd 2017)

Regarding choice: Right, I am tacitly assuming choice in the base topos.

From the discussion of the sharp modality $\sharp$ as the Aufhebung of the initial opposition $\emptyset \dashv \ast$ (here), with its minimal solution necessarily being Boolean (this remark) I grew fond of the idea that it is quite natural to assume the base of a cohesive theory to be classical. I might go as far as feeling that this neatly captures the whole dichotomy between classical and constructive logics: we do want both, but the former precisely only in the base of discrete types, while the latter in the full theory of cohesive types.

In any case, if we don’t allow ourselves to choose single basepoints at all, then I don’t see how to get rid of the the spurious copies of the universal covering space.

• CommentRowNumber21.
• CommentAuthorMike Shulman
• CommentTimeJul 3rd 2017

I would be somewhat doubtful that local contractibility is enough, but I could be wrong. My hope would instead be that the identification can be proved internally from the real-cohesion axiom.

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeJul 4th 2017

Do you know if the internal real cohesion satisfies the property that shape preserves pullbacks over discrete types?

• CommentRowNumber23.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 9th 2021

I see Peter Scholze returned to an MO question on universal covers from a condensed mathematics perspective here. Presumably cohesive homotopy types can do a similar job.

• CommentRowNumber24.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 9th 2021
• (edited Feb 9th 2021)

Oh, maybe David R.’s new answer is about working in a cohesive setting.

Felix and David Jaz Myers have approached this in cohesive type theory, here.

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeDec 25th 2021

• William S. Massey, Chapter 5, Section 10 of: Algebraic Topology: An Introduction, Harcourt Brace & World 1967, reprinted in: Graduate Texts in Mathematics, Springer 1977 (ISBN:978-0-387-90271-5)

for an original reference (maybe the construction via equivalence classes of based paths originates with the discussion on p. 175 here?)

and pointer to

for generalization to equivariant homotopy theory.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeDec 26th 2021
• (edited Dec 26th 2021)

prodded by the exchange at “fundamental theorem of covering spaces” (here)

have added pointer to definition of universal covering spaces

in cohesive homotopy theory

and explicitly so in cohesive homotopy type theory:


text:
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Get Full Access to University Physics - 13 Edition - Chapter 18 - Problem 46e
Get Full Access to University Physics - 13 Edition - Chapter 18 - Problem 46e

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# For a gas of nitrogen molecules(N2) what must the

ISBN: 9780321675460 31

## Solution for problem 46E Chapter 18

University Physics | 13th Edition

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University Physics | 13th Edition

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Problem 46E

For a gas of nitrogen molecules(N2) what must the temperature be if 94.7% of all the molecules have speeds less than Use Table 18.2. The molar mass of N? i2?28.0 g/mol.

Step-by-Step Solution:

Solution 46E Step 1: Relation between root mean square speed and temperature is, v = 3kT/M rms Where, k - Boltzmann constant T - temperature of the gas M - molar mass

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