text:
## Regex Q&A¶

In [3]:
import re

### Q

can regex be used to parse through dictionaries and other objects of similar typing?

### A

re.findall() and re.sub() take a string as input. You will have to iterate over other structures and feed re the strings one at a time.

### Q

Can you search for terms by length?

### A

yes, you can specify the length of the expression match that you've defined. E.g. re.findall('[a-z]{2}',"the cat sat on the mat.") will find all strings containing lower case letters of the length 2. If you specify {1,3} as your argument, it will find strings with 1, 2, or 3 letters matching the expression (lower case a-z in this example).

In [6]:
re.findall(r'\b[a-z]{2}\b','The cat sat on the mat in the doorway.')

Out[6]:
['on', 'in']

### Q

How do I create an expression that properly uses 'or'? As in, either of two

### A

The vertical bar character (ex: 'a|b') will look for values that satisfy expressions a or b

In [8]:
re.findall('color|colour','Growing up in Ireland, I tend to write "colour" instead of "color".')

Out[8]:
['colour', 'color']
In [9]:
re.findall('colou?r','Growing up in Ireland, I tend to write "colour" instead of "color".')

Out[9]:
['colour', 'color']

### Q

How do we encrypt passwords using regex?

### A

this is not really what regex is for...

### Q

example, finding 'ar' in 'I love Richard.'

### A

re.findall("ar", ...)

### Q

Delete all words in a dictionary with a vowel?

### A

Not really a pattern. Better done with pure Python.

### Q

Does regular expressions (re.findall) work well when dealing with a document with hundreds of thousands of words (for instance a textbook). Is the runtime good?

### A

Possibly not: Python re is not very fast.

### Q

word with two letters in front of 'at', without the words with one letter in

### Q

items validates my expression if they exist in the string?

### A

In [10]:
if re.search(r'\bh[aeiou]*ck\b','what the heck?!'):
print('Got a match!')

Got a match!

### Q

how to find a list of key words? how to identify that the key words are not only being mentioned but actually being implemented?

### A

We can use a for loop to search for each keyword

### Q

Can you parse through a string and search for terms by length?

### A

Yes, you can use a for loop to go through each word until you reach the end of the string whilst isolating each term and using counts and indexing to go through each term to find the length/

### Q

Is every other letter of a string a vowel?

### A

JR: I'd be tempted to do this with pure Python. Something like this:

In [11]:
s = 'mississippi';
set(s[1::3]).issubset( set('aeiou') )

Out[11]:
True

### Q

How do you use regex on data base?

### A

Some databases have their own implementation of regular expression, it is certainly possible for SQL but specifying that here is a bit lengthy - if you mean data base like a data frame, you could define a lambda expression with the regular expression you need and apply this to a specific column of your dataframe with the map() function

### Q

when we use '.*' to find the longest string, how does python decide that it is the end of the string? Does it use the period sign? Could we customize it?

### A

In this case, the code is not explicitly searching for the end of the string, but for the last instance of the keyword, which in our case is close to the end of the string.

### Q

How do you look for a decimal number (ex: 0.5) in regex?

### A

\d.?\d

In [12]:
re.findall('\d*\.\d*','Is 3.14159 > 2.71?')

Out[12]:
['3.14159', '2.71']

### Q

with the findall example we used, how do we find the word 'at' and not part of a word

### A

In [13]:
re.findall( r'\bat\b', 'The cat spat at the mat.' )  # \b for word boundary

Out[13]:
['at']

### Q

How to find if an expression is repeated in the same word in a string?

### A

Yes, you could by choosing your expr, then creating a count for it

### Q

can we use this to create a method that find and replace words in document

### A

yes re.sub() (I think)

### Q

can you ignore certain strings/chars when looking for something? ie ignore word 'cat' when searching for 'at'.

### A

In [16]:
re.findall( '[^c]at', 'The cat spat at the mat.' )

Out[16]:
['pat', ' at', 'mat']

### Q

how to find a specific symbol (like \) in Regular Expression?

### A

by typing another '\' before your symbol

### Q

Can we use RegEx to find emojis in a text?

### A

Possibly, but not simply.

### Q

What is the difference between using this method and using split()?

### A

The difference is that the split function can be a little difficult to work with, and can lead to cluttered substrings. For example, suppose I was looping through a dictionary and extracting specific file names that began with 'Patient[X]' , where X can vary from A-Z, and ended with '.svs'. While it's true that we can split on Patient, we would then have to do more work in order to ensure that the file we are checking ends with .svs instead of something like .txt. Using this library on the other hand, makes it far more easier, effecient, and cleaner to write/read than the former. On top of this, if you needed something in the middle of a string, and if the string was huge (like 10,000 charachters), then you would not have a fun time trying to find the specific substring you want, in fact it would be a pretty slow operation as opposed to using the regex library.

JR: I'd say regex is good when you are looking to match one of a number of possibilities. Here's a good example that arose in the first names project. The task is to find all the names that are "like" Dmitri, which we interpret to mean of the form D-m-t-r-, where the blanks are vowels or sequences of vowels.

In [11]:
import glob
names = set([])
for file in glob.glob('names2019/yob*.txt'):
with open(file) as f:
for name,gender,count in [line.split(',') for line in f.read().split('\n') if len(line)>2]:
print( len(names) )
s = ' '.join(names)
import re
re.findall(r'\bD[aeiouy]*?m[aeiouy]*?t[aeiouy]*?r[aeiouy]*?\b',s)

98400

Out[11]:
['Domitri',
'Demeter',
'Dmitri',
'Demitrio',
'Dmetri',
'Dimitri',
'Demetria',
'Dmitry',
'Dametra',
'Demitra',
'Demitria',
'Dimetri',
'Demeteria',
'Damitri',
'Demitre',
'Demetrey',
'Dimitar',
'Dimitre',
'Dametria',
'Demetrie',
'Dmitriy',
'Dimitriy',
'Dimitry',
'Demitry',
'Dimitrie',
'Demetre',
'Dimitra',
'Dametri',
'Demetry',
'Demetra',
'Dametre',
'Dimitria',
'Demitri',
'Demeatra',
'Dimetra',
'Demitrie',
'Demetriu',
'Dmytro',
'Damitra',
'Demetri',
'Demetrio',
'Demetree']

### Q

Is it possible to find out error in sentence. For example find out other language in the content that cannot be read.

### A

Define a list or a dictionary contains invalid occurrences ('error') in sentences, then loop over the list or dictionaty with regex function as content to report these occurrences ('error').

JR: Regex knows nothing about human languages or meaning. It only deals with patterns in character strings.

### Q

How would you specify the number of occurrences a pattern is allowed?

### A

JR: {3}

In [17]:
re.findall( '[A-Z,a-z]{2,10}at','The cat spat on the mat.')

Out[17]:
['spat']

### Q

How to find a word of different Length containing a specific expression ?

### A

/\b\w[Yy]\w\b/g This will grab the first word containing whatever expression you are looking for. In this case, it will look for a word containing either capital or lowercase y. This grabs words of any length contianing your expression

### Q

Can we find exact capital letters in regular expressions

### A

yes, you can search upper and lower case letter, at the same time you can search with case insensitive by adding flag: re.IGNORECASE

In [22]:
re.findall(r'J.*?\b','Javier y Jesús')

Out[22]:
['Javier', 'Jesús']

### Q

How to get the word which contains some letters which is between the words?For

### A

re.findall("ar", ...)

### Q

How to find a word with specified length in a string contains the expression

### Q

How can we incorporate regular expressions into flow control and conditionals.

### A

In [23]:
if re.search(r'\bh[aeiou]*ck\b','what the heck?!'):
print('Got a match!')

Got a match!

### Q

If I have the sentence 'The flat cat sat on the mat'. How do I just extract the front of 'at'. So, in this case just the word flat. ('..at' shows all the words with a space added for 3 letter words)

### Q

suppose you find expressions with a particular string, how do you then report back only the part of the string that was not in the specific search. i.e. if were search for .*at, how do we report only the part that is not 'at'? for cat that would just be c or The c depending on where it is in the string.

### A

JR: () for marked group(s)

In [24]:
re.findall( '([A-Z,a-z]*?)at','The cat spat on the mat.')  # captured group ()

Out[24]:
['c', 'sp', 'm']

### Q

Does the re library have a way of detecting synonyms?

### A

JR: No. This requires knowledge of the natural language. Regex only concerns patterns in character strings.

### Q

How would I parse through a directory and get files that begin with 'Case_x' where x can be either A, B, or C and end with '-rendered_img.svs'.

### A

JR: Iterate over filenames and check for match with re.find( 'Case_[ABC]-rendered_img.svs' , filename )

### Q

how to delete arbitrary number of spaces between characters? for instance, "asdf gghdh dhdh. dhhd.

### A

In [25]:
re.sub( ' +', ' ', 'The  cat spat   at the      rat.' )

Out[25]:
'The cat spat at the rat.'

### Q

What if I want to search the string for thing A OR thing B? For example, if I made a todo list and the topic of the todo could be a string like "Research" OR a class title, like in your document, how could I tell it to do that?

### A

[A|B]

In [ ]:


text:
Helper functions for reading data from archives for the EQcorrscan package.

Note

These functions are tools to aid simplification of general scripts, they do not cover all use cases, however if you have a use case you want to see here, then let the authors know, or implement it yourself and contribute it back to the project.

 read_data Function to read the appropriate data from an archive for a day.
 _check_available_data Function to check what stations are available in the archive for a given day. _get_station_file Helper function to find the correct file. _parallel_checking_loop Inner loop for parallel checks.

format: plain


text:
# SpongeBob SquarePants

Nominated Article
Whoops! Maybe you were looking for Sponge, or Bob? How about Square, or Pants?
For those without comedic tastes, the self-proclaimed experts at Wikipedia think they have an article about SpongeBob SquarePants.
“Meow.”
~ Gary on SpongeBob

SpongeBob SquarePants (often referred to simply as SpongeBob, Spongebob Squarepants, Spongebob, SpongeBob Squarepants, or The Spongebob Squarepants Show, starring Spongebob Squarepants) is an animated children's television series about the everyday life of various undersea creatures which, for some reason, includes a squirrel. The show has sparked much SpongeBob-themed merchandise, including everything from boxer shorts to toilet paper. It is currently the most popular show on Nick, which isn't saying much.

## Characters

### SpongeBob SquarePants

SpongeBob SquarePants is a 23-year old anthropomorphic sea sponge with the mind capacity of a five-year-old. He currently resides in a pineapple with his snail[1], Gary. Despite being a piece of fruit, his house is not the strangest place of residence on the block.[2]

As you can see, his pants are not square.

#### Is Bob really a sponge?

Since SpongeBob does not look exactly like a sea sponge, some speculate he is really a piece of cubed cheese posing as an animate object. However, SpongeBob's relatively squishy body and wavy outline lead to the conclusion that he is really a kitchen sponge posing as a sea sponge by living at the bottom of the ocean. This would, however, pose the question of why he hasn't absorbed any sea water, given the absorbing quality of kitchen sponges. The answer is simple. It's a cartoon.

#### Are his pants really square?

Although SpongeBob's last name is SquarePants, some mathematicians claim that his pants are not really square but are more rectangular. A study of SpongeBob's pants began, and upon closer examination, it was revealed that $4s$ was not equal to $2l+2w$, with $s$ representing "side," $l$ representing "length," and $w$ representing "width." However, in a square, $l+w=2s$. Basically, Spongebob's pants do not have four equal sides, a necessary component of a square. Thus, some might speculate that his pants are a rectangle. However, this is not the case either. Since his pants contain lines on the x, y, and z axis, his pants are three-dimensional, and therefore, a rectangular prism.

Patrick's most intelligent face.

### Patrick Star

Main article: Patrick Star

Patrick Star is SpongeBob's best (and only) friend. Although he is often depicted as dimwitted, fat, lazy, ignorant, and unemployed, he is not an American. Instead, he claims to be a starfish. Yet, while starfish are shaped like stars and have multiple miniature "feet" on their underside, Patrick tends to resemble more of a lopsided triangle with a pair of limbs. Furthermore, he wears horribly ill-fitting pants and has a belly button, further enforcing the notion that he is a human with the intelligence of a starfish and possesses no other qualities of an underwater creature.

### Squidward Tentacles

Main article: Squidward Tentacles

Squidward Tentacles is SpongeBob's next-door neighbor, and, despite being named Squidward, is actually an octopus according to creator Stephen Hillenburg. He plays the clarinet at a fourth grade level and is currently the second chair in his orchestra class. However, there are only two clarinet players in the orchestra, and the other one is Used Napkin.[3] Although Squidward is often criticized for not wearing pants, nobody seems to care when Plankton frolics in the nude, and Sandy wanders around in purple undies.

### Mr. Krabs

Main articles: Mr. and Krabs

Eugene "Mr." Krabs is the father of Pearl, a sixteen-year old talking whale, and the owner of the Krusty Krab, a fast food restaurant which can somehow obtain all the necessary ingredients for a hamburger despite being underwater where no beef, wheat, or vegetables are readily avaliable. Although his last name is "Krabs," Mr. Krabs is probably a crab, not a krab (which is a type of processed seafood made from white fish that is pulverized in order to resemble crab legs). However, this still would not answer the question of how it is anatomically possible for him to have a whale as a biological daughter.

From left to right: Squidward, Mr. Krabs, Gary, SpongeBob, Patrick, Sandy, everyone else.

### Everyone else

Main article: Everyone else

Everyone else includes the rest of the major characters, such as Sandy, Plankton, and Gary, as well as other minor characters that are so unimportant that they don't even have original names, such as Tom and Fred. Everyone else has appeared in more episodes than all of the main characters combined and had several hundred cameos in the SpongeBob SquarePants Movie. Despite these achievements, a recent poll revealed that Everyone else is the least popular SpongeBob character, so unpopular, in fact, that it wasn't even included in the poll.

## Controversy and Criticism

During much of the show, the characters talk to each other in the same way that humans would: By speaking English. However, some parents accused this of being unrealistic and inaccurate. Since much of the show is set in Bikini Bottom, which is a name used by the show's creators to signify the lagoon beneath Bikini Atoll in the Marshall Islands, the characters should really be speaking in Marshallese, the official language of the Marshall Islands, not English. For example, one of SpongeBob's catchphrases is "I'm ready! I'm ready!" Realistically, he would have said, "Ña ewōr pojak! Ña ewōr pojak!" which is the correct Marshallese translation. Likewise, common phrases such as "My leg!" and "Who lives in a pineapple under the sea?" should really be "Aō ne!" and "Wōn mour i peinabōl iuṃwi-lọjet?" respectively, and Gary's classic "meow" would become "Fenōji-umfa-ñoojay-leefee-uhmhat-zhaggọt-phutta-gung."

## Merchandise

The SpongeBob Nativity Scene. Available at Target.

The producers of the show have currently made it their goal to create SpongeBob-themed versions of every single thing in existance. This trend started with SpongeBob-themed clothing, including pajamas, shirts, hats, socks, bras, thongs, gloves, turbans, saris, and parachute pants. SpongeBob versions of classic games ranging from chess and Monopoly to SpongeBob-themed black jack, Russian Roulette, and strip poker were also created. As SpongeBob's popularity grew, cereals, video games, toys, bicycles, and even college textbooks featuring SpongeBob characters were created. However, to this day, the best selling SpongeBob novelty item is the SpongeBible due to the popularity of stories such as Plankton's Ark and the crucifixion of Patrick.

## Footnotes

1. or maybe it's a cat, or a meowing snail, or a blue cat with a shell, or...
2. After all, Squidward lives in a head.
3. For more information on Used Napkin's successful music career, see his biography on Spongepedia, the free encyclopedia.

v  d  e
SpongeBob SquarePants is part of Uncyclopedia's series on Mass Media.

 Poo Lit Surprise-Winning Article This article was chosen as the Best Rewrite in the September-October 2011 Poo Lit Surprise writing competition.


text:
# Force Homework Problems Checklist

I am given a problem where I am to pick up the prism shown below by touching only the upper two faces. The coefficient of friction with each face is 0.4 and the mass of the prism is 30grams. What is the minimum force I must apply to each face in order to lift the prism?

The one thing I am stuck on are the directions of the force I apply with my fingers. I am not sure whether

a)the forces exerted by my fingers act horizontally inwards (which is slightly confusing me; theoretically, should horizontal forces not be able to counteract the downwards weight of the prism? And yet my calculations in this case give a force required of 0.61N)

b)or whether they act exactly normal to the surface (in which case i get 0.22N)

c)or whether they can act directly up the slope straight away, or at least with some upwards inclination to counteract the weight.

homework-and-exercisesnewtonian-mechanicsforcesfrictionfree-body-diagram

Alright so I think I know how to do this but I require help in calculating what acceleration would be in terms of some sort of friction coefficient.

So model a particle going down a hill. The slope is 25 degrees. The mass of the particle is 50kg and the coefficient of sliding friction between the particle and slope is 0.05. When the brake is applied, 260N acts in the opposite direction to the motion of the particle. g = 9.8ms^-2

I won't write out my entire workings are they're lengthy, but I'll give the line I am up to which is:

let $\mu$ = the coefficient of sliding friction

let $b$ = the braking force

$ma_i = (mg \cos{65}-\mu bN)\hat{\mathbf{i}} + (N-mg \sin{65})\hat{\mathbf{j}}$

So I should imagine that b is going to slow down the acceleration, so perhaps it's as simple as 0.05 x 240?

I'm not sure if it would be 240 or 2.4 though, for example - any elucidation on this?

homework-and-exercisesnewtonian-mechanicsforcesfrictionvectors


text:
${x}^{2}-4x-12$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

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MoxboasteBots5h
The given quadratic polynomial is ${x}^{2}-4x-12$.
It is factored as follows.
${x}^{2}-4x-12={x}^{2}-6x+2x-12$
=x(x-6)+2(x-6)
=(x+2)(x-6)
Not exactly what you’re looking for?
Joseph Fair
The given equation is:
${x}^{2}-4x-12$
$⇒{x}^{2}-4x=12$
Thus,
${x}^{2}-4x=12$
$⇒{x}^{2}-4x-12=0$
The product of the second degree term and the constant is $-12{x}^{2}$. Factors of $-12{x}^{2}$ that sum to -4x are -6x and 2x. Thus,
Factorizing which we get,
${x}^{2}-4x-12=0$
$⇒{x}^{2}-6x+2x-12=0$
$⇒x\left(x-6\right)+2\left(x-6\right)=0$
$⇒\left(x-6\right)\left(x+2\right)=0$
The required values of x are 6 and -2.


text:
# Suppose that f(x)= ln((9x^2)+3) (A) Find all critical values of f, compute their average.   (B) Use interval notation to indicate where f(x) is increasing. (C) Use interval notation to indicate where f(x) is decreasing.(D) Find the x-coordinates of all local maxima of x, compute their average. (E) Find the x-coordinates of all local minima of f, compute their average. (F) Use interval notation to indicate where f(x) is concave up.(G) Use interval notation to indicate where f(x) is concave down.

A) You need to solve the equation f'(x) = 0 to find critical values of the function such that:

`f'(x) = ((9x^2+3)')/(9x^2+3)`

`f'(x) = (18x)/(9x^2+3)`

You need to solve the equation f'(x) = 0, hence since `(9x^2+3)!=0` , then `18x = 0 =gt x = 0.`

Hence, the critical value...

A) You need to solve the equation f'(x) = 0 to find critical values of the function such that:

`f'(x) = ((9x^2+3)')/(9x^2+3)`

`f'(x) = (18x)/(9x^2+3)`

You need to solve the equation f'(x) = 0, hence since `(9x^2+3)!=0` , then `18x = 0 =gt x = 0.`

Hence, the critical value of the function is x=0.

B) The function is increasing over `(0,oo).`

C) The function only increases over `(0,oo).`

D) You need to remember that the x coordinate of local maximum is the critical value of function, hence x = 0.

E) Since the function has no minimum, then there is no x coordinate for local minimum.

F) Since the function increases over`(0,oo),` hence the function is concave up over `(0,oo).`

G) Since the function only increases over `(0,oo)` ,hence the function is not concave down over any interval.

Approved by eNotes Editorial Team


text:
## Classification of Magnetic Material

Subject: Physics

#### Overview

Those substances which are feebly magnetized in the direction opposite to the applied field are called diamagnetic material.This note provides us an information on classification of magnetic material.
##### Classification of Magnetic Material

There are three types of magnetic materials. They are:

1. Diamagnetic material
2. Paramagnetic material
3. Ferromagnetic material

#### Diamagnetic material

Those substances which are feebly magnetized in the direction opposite to the applied field are called diamagnetic material. Examples of diamagnetic materials are bismuth, copper, water, mercury, alcohol, argon, gold,tin, mercury, antimony etc. The magnetic moment of atoms of a diamagnetic material is zero. They acquire induced dipole moments when the material placed in an external magnetic field. These moments are in opposite in the direction to the applied field.

##### Some properties
1. The diamagnetic materials are repelled by magnets.
2. When a diamagnetic liquid in a watch glass is placed over two closely spaced pole pieces of the magnet, it is depressed at the middle while in the case of pole pieces separated by a distance, it rises at the middle. Similarly, when a diamagnetic liquid is placed in a U-tube and one of the limbs of the tube is placed between the two strong pole pieces of magnet, the liquid depressed at that limb.
3. The diamagnetic materials move from a stronger to a weaker field.
4. A diamagnetic rod, freely suspended in a magnetic field, slowly turns to set at right angle to the applied field.
5. Since magnetized is opposite in direction to an applied field, the diamagnetic materials have the small value for the intensity of magnetization, I.
6. The materials have always negative magnetic susceptibility, $\chi = (\mu_r -1)$ and accounts from -10-6 to -10-5.
7. These materials are independent of temperature.

#### Paramagnetic Material

Those materials which are weekly magnetized in the same direction of the applied magnetic field are called paramagnetic material. The examples of paramagnetic materials are aluminum, chromium, oxygen, manganese, alkali, alkaline earth metal etc.

The paramagnetic materials have permanent magnetic moments. These moments interacts weekly with each other and randomly orient in the different direction.

Some Properties

1. The paramagnetic materials are feebly attracted by magnets.
2. A paramagnetic rod, freely suspended in a magnetic field, aligns along the field.
3. The paramagnetic materials are temperature dependent and follow curve law.
4. The relative permeability is nearly unity than ranges from 1.00001 to 1.003 for common ferromagnetic materials at room temperature. So, the magnetic lines of force inside the material placed in a magnetic field are more than that outside it.
5. The susceptibility of paramagnetic substances has small positive value.

#### Ferromagnetic Material

The ferromagnetic materials are highly magnetized in a magnetic field. The examples of ferromagnetic materials are iron, nickel and cobalt, and their alloys such as alnico. Gadolinium and dysprosium are ferromagnetic at low temperature.

##### Some Properties
1. Ferromagnetic materials are highly attracted by magnets.
2. Ferromagnetic materials more from weaker to stronger field.
3. A ferromagnetic rod, freely suspended in a magnetic field, turns fast to set along the applied field.
4. The magnetic susceptibility is positive and very high and varies with applied field.
5. The relative permeability is very high in the order of 1000 to 100,000.
6. Ferromagnetic dust in a watch glass, placed over two closely spaced pole-pieces of the magnet, increases at the middle, while pole piece is separated by a distance, depresses in the middle.

#### Domain Theory of Ferro Magnetism

Each atom of ferromagnetic substance has a permanent magnetic substance; in the unmagnetised state, the atomic and molecular dipoles are arranged in random so the net magnetic moment is zero. There is a strong interaction with neighboring atoms which keeps their magnetic moment aligned parallel in small regions even in the absence of an external field. These small regions with the volume ranging between 10-12 to 10-8 m3 are called domains. When the material is placed in an external field Be, the domains tend to orient themselves parallel to field B0. As the applied field becomes stronger, the domains, having magnetic moments not aligned with the field, become very small and when the domains fully align to the applied field, the material attains magnetic saturation. On removing the field, the domain walls do not move completely into previous positions. This means material retains a magnetization in the direction of the applied field.

#### Hysteresis

• Assume a piece of magnetized iron placed in a magnetizing field H.
• When the value of H is gradually increased from zero value. The magnetic induction B in an iron also increases.
• When H is increased, B also increases till saturation point P but beyond P if H is increased B remains constant.
• When the value of H is now decreased, B does not retrace the path PO but at the lower rate and at H being zero, B is not zero but has the finite value represented by OC. The value of B at this point is called retentivity.
• Further, when H is increased in opposite direction, the value B falls to zero for the magnetizing field to be equal to OD. This value of the magnetizing field is called coercivity of material of specimen.

References

Manu Kumar Khatry, Manoj Kumar Thapa,et.al Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.

##### Things to remember

Those substances which are feebly magnetized in the direction opposite to the applied field are called diamagnetic material.

Those materials which are weekly magnetized in the same direction of the applied magnetic field are called paramagnetic material.

The ferromagnetic materials are highly magnetized in magnetic field. The examples of ferromagnetic materials are iron, nickel and cobalt, and their alloys such as alnico.

These small regions with the volume ranging between 10-12 to 10-8 mare called domains.

• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.


text:
COVID-19 cases in India are approaching 30 million while the death has surpassed 385,167 on June 18, 2021,1 with a record of 4205 deaths on Wednesday, May 14, 2021,2 a few weeks after the health minister of India declared “endgame” for the COVID-19 pandemic in the country.3 India is battling through the second wave with two mutant variants B.1.617.2 and B.1.1.7 challenging their healthcare system,3 meanwhile, World Health Organization (WHO) has shown austerity regarding the strain B.1.617.2 and on May 09, 2021, classified it as a “variant of concern”,4 which has already transmitted to 44 countries until May 13, 2021.2 India with the population of 1.4 billion, experts believe that the actual COVID-19 figures will be five to ten times more than the official figure, with cases approaching nearly 400,000 a day.5 The hospitals have nearly run out of beds, patients queuing up for 12 hours for the availability of a bed,6 and daily oxygen supply of 50,000 tonnes needed to combat hypoxic condition of COVID-19 patients in India,7 as India is reaching a “covid-hell”.3

# What is Mucormycosis and how does it transmit?

Amid the catastrophic situation of the COVID-19 pandemic in India, the re-emergence of a high mortality mucormycosis infection; a ‘black fungus’ among COVID-19 patients or recovered COVID-19 patients is posing a serious and unforeseen health threat in India. According to the reference to the experts, Kaul (2021) reports that Mucormycosis has become an epidemic.8

## Clinical forms of mucormycosis

Mucorales fungi are seen as aseptate hyphae, with some 11 genera and 27 species are known to cause human infection. Mucormycosis belongs to class Phycomycetes which is subdivided into genera Absidia, Rhizopus, and Mucor.9 Prakash and Chakrabarti (2021) mention the presence of pathogenic species including Rhizomucor, Rhizopus, Apophysomyces Lichtheimia, and, Cunninghamella in Indian soil.10 Prakash et al. (2019) discussed that the global leading species to cause mucormycosis is Rhizopus arrhizus, while, Apophysomyces variabilis, is the second most common causative agent in India, along with the presence of other agents in India including Rhizopus homothallicus, Thamnostylum lucknowense, and Mucor irregularis.11

## Transmission of the disease

Mucormycosis is caused by the fungus mucor that requires a wet surface to adhere, therefore, a humid environment is an ideal condition for it to prevail.12 It could transmit through the hospital bed linen, unsterilized oxygen pipes, and hospital ventilators.12 Mucormycosis infection targets immunocompromised patients, low levels of neutrophils, cancer patients, organ transplants, and patients who underwent stem cell transplants.13 Steroids are used for the treatment of COVID-19 infections while heavy steroid use induces immunosuppression can potentially increase chances of opportunistic infection of which is mucormycosis.13 The spores are commonly found in organic matter particularly food, soil, and plants which require humidity to grow.13 The inhaled spore forms hyphae that invade the paranasal sinuses, extending further through anatomical sites within the skull causing rhino-orbital or rhino-cerebral infection. COVID-19 patients are highly susceptible to develop opportunistic mucormycosis, due to weakened immune systems by decreased levels of CD4+T and CD8+T cells.13,14 Other factors that increase the vulnerability of COVID-19 patients include prolonging hospital stay, iatrogenic corticosteroid therapy with uncontrollable diabetes mellites, and diabetic ketoacidosis.13 and pulmonary infections.8 In India, the usual time to fumigate ICU is around 14 days, however, due to COVID-19, complete fumigation remains a challenge at large.15

# Symptoms of mucormycosis

Indian Council of Medical Research (ICMR) has released signs and symptoms of mucormycosis that may include pain and redness around the eyes and nose, cough, fever, headache, disorientation, dyspnea, hematemesis, sinusitis, toothache, facial pain and numbness, hemoptysis, blackish discharge from nose and visual disturbances including unspecified vision loss.16 ICMR has given strict instructions that any patient experiencing the mentioned symptoms must immediately seek the medical attention.

Prakash and Chakrabarti (2021) discussed the anatomical localization of mucormycosis with rhino-orbital-cerebral (ROCM) commonest area, followed by cutaneous, pulmonary, renal, gastrointestinal, disseminated, breast, ear, spine, heart, and bone.10 In a study conducted by Pagano et al. (1997) through eighteen hematology departments, fever, thoracic pain, cough, and shortness of breath were the main symptoms presented in the patient, while 89% of the patients were neutropenic at the time of onset.9 Talmi et al. (2002) found bloody nasal discharge and black nasal eschar as a sign of the infection, meanwhile, malaise was seen as the most common symptom in 89% of the cases.17

# Mucormycosis in Pre-COVID-19 and During COVID-19

In the Pre-COVID-19 era, Prakash et al. (2019) discussed hematological malignancies to be the foremost global indicator for mucormycosis infection, while, diabetes mellitus was the forerunner reason the cases in India.11 Other comorbid conditions involve skin trauma, human immunodeficiency virus, peritoneal dialysis, iron overload, deferoxamine therapy, illicit drug infusion, and voriconazole therapy.10 Pagano et al. (1997) found the lungs to be the most affected site for the infection, followed by chronic nerve compression, sinus, liver, kidney, heart, orbital space, and large intestine.9 However, currently in the mucormycosis epidemic in India, orbital space is the common site of infection followed by other organs.

# Prognosis and Outcomes

## Outcome of infection

In the study conducted by Pagano et al. (1997), the outcome of the infection resulted in thirty patients dying within 3 months of the treatment, out of which, 28 died due to mucormycosis.9 Talmi et al. (2002) discussed the orbital compromise exhibits the advancement of the fungal infection into cranial fossa leading to the option of orbital exenteration, however, this may lead to mutilation and irreversible loss of vision.17

## Prognostic factors

Pagano et al. (1997) found Phycomycetes to be resistant to triazoles and nystatin.9 Pagano et al. (1997) found the increase in neutrophils after post-chemotherapy aplasia and amphotericin B treatment was closely associated with the prognosis from the infection.9 However, oral antimycotic prophylaxis failed to prevent mucormycosis.9 Controlling the environmental exposure to neutropenic patients showed positive outcomes in restricting the development of filamentous fungal infection. Isavuconazole is a new antifungal remedy for the treatment of mucormycosis, which was unavailable in India in 2018, but recently introduced in India, therefore, it lacks research on the efficacy in long-term, however, during the limited timeframe, it has shown efficacy as compared to amphotericin B.11 Talmi et al. (2002) noted palatal involvement in the patients with poor prognosis. Talmi et al. (2002) discussed the use of granulocyte colony-stimulating factor (G-CSF) in neutropenic patients to hold value.17

## Mortality Rate

Before COVID-19 the mortality rate from mucormycosis in India was 46.7%, where, the mortality rate in North India was higher compared to South India.11 Prakash and Chakrabarti (2021) discussed that the mortality rate in the clinical presentation of mucormycosis is 64-97% in patients with gastrointestinal comorbidity, 61-77% in pulmonary, 62-79% in disseminated, 31-49% in ROCM, 23-57% in cutaneous cases.10

# Incidence and Prevalence of mucormycosis

There is a rise in mucormycosis cases globally, the rate of infection is further higher in developing countries including India.18 Prakash and Chakrabarti (2021) debated on the exact incidence of mucormycosis in India due to a deficit in a population-based study, however, with the aid of the computational-model-based method, the prevalence is 14 cases per 100,000 in India.10 Prakash and Chakrabarti (2021) in their research discussed that prevalence of mucormycosis is 70 times higher in India as compared to global.10 It was 13 cases per year from 1990 to 1999, 36 annual cases from 2000 to 2004, 50 annual cases from 2006 to 2007, and 89 annual cases from 2013 to 2015.10 One reason Prakash and Chakrabarti (2021) discussed is improvement in the diagnosis of the disease.10 In other studies, discussed by Prakash and Chakrabarti (2021) the yearly incidence is 18 cases in Southern India (Tamilnadu) from 2005 to 2015, while, 9.5 cases from 2015 to 2019.10 Talmi et al. (2002) mention that seasonal presence may contribute towards the incidence with the peak in fall and early winter.17

# Rise in the cases of mucormycosis

Mucormycosis has a mortality rate of around 50 percent12 and is costly to treat, which can further burden the health care infrastructure suffering from deficient and exhaustive resources. According to the report published in the Guardian, there were 300 cases of mucormycosis in Gujarat until May 13, 2021,19 which exponentially increased to 2,859 cases in Gujarat until May 26, 2021,20 marking over 850% increase in less than two weeks. In Gujarat 8 COVID-19 recovered patients lost their eyesight due to mucormycosis.21 similarly, 200 COVID-19 survivors are being treated for mucormycosis in Maharashtra of which 8 had died.21 58 cases from Mumbai, Pune, Delhi, Hyderabad, and Bangalore have been reported between December and February about two weeks of COVID-19 recovery.22 In Mumbai, 11 out of 40 infected individuals went through enucleation due to the extensive involvement of fungus in the orbit.22 whereas, some Indian states are at high risk of a surge in cases around 100-200 percent.15 2,770 cases for mucormycosis are recorded in Maharashtra, 768 in Andhra Pradesh, 752 in Madhya Pradesh, and 744 in Telangana, while a total of 11,717 cases are detected in India as of May 27, 202120; which is three times more than found in a decade in Europe,23 while there were only 29 cases of mucormycosis in March of this year in India.24 The cases further rose to 28,000 according to a report,25 while, cases are on the rise with 2,856 cases reported in Karnataka.26 A study by Sharma et al. (2021) conducted in Jaipur, India, among 23 COVID-19 recovered patients from August to December, the rhino-orbital fungal infection mainly starts in ethmoidal sinus then proceed to the maxillary sinus, all patients received corticosteroids for COVID-19 and the majority of 21 patients were diabetic indicating a major link with diabetes mellitus.13 According to WHO, around 8.7% population in India are diabetic, and with inevitability to use steroids for the treatment of COVID-19 makes them susceptible to contact with mucormycosis.27

## Was it the case of an emerging pathogen, rise during the COVID-19?

Mucormycosis is not a new infection, neither it has been reported recently. It had its presence but the condition remained as rare.9,10,17 However, the steroid treatment in COVID-19 patients is causing an immunosuppressed condition, together with the conditions in which mucormycosis infects humans.

# Costs, efforts, and treatment of mucormycosis

Immediate efforts are being made by Gujrat state, by an immediate supply of 5000 doses of Amphotericin b worth Indian Rupees 3.12 crore21 (approx. US$425,000) and allocating 60 beds each in two separate wards.28 The antifungal I/V treatment is the only treatment that costs around Indian Rupees 196,000 (US$2,688) per patient.22 Amphotericin b is nephrotoxic, which leads to kidney damage.8 ICMR has released advisory guidelines to inhibit the spread of mucormycosis, encouraging COVID-19 patients to keep glucose level under control, usage of sterile water, and cautious usage of steroids, while, prohibiting patients to ignore symptoms of mucormycosis, and seeking medical help as appropriate.16

# Do cases decline as the COVID-19 situation improves?

Given the current scenario, it is difficult to suggest that whether a decline in COVID-19 cases may bring improvement in the cases of mucormycosis. According to the expert prediction, India is expected to be hit by the third COVID-19 wave by October,29 and the possibility of a new mutant variant may result in a completely unknown situation. The use of steroids, along with the comorbid conditions, may still pose a serious health risk of mucormycosis among the COVID-19 recovered patients.

# Conclusions

The burden of COVID-19 in India has affected millions, which has been an emergency throughout the country during the second wave. The mucormycosis infection has been declared an epidemic in India due to the increased vulnerability of COVID-19 recovered patients with corticosteroid therapy and diabetes mellitus. Since the treatment is expensive, the burden of infection is increased on the health care system. The mortality rate of mucormycosis is much higher than of COVID-19 infection alone. Since the world has a greater population of diabetic patients in most regions and steroid therapy being used as a potential drug for COVID-19 infection, the warmer and humid regions where COVID-19 and diabetes are prevailing can be an immediate health hazard, especially in low-income and middle-income countries.

None.

## Authorship contributions

Mahnoor Sukaina is the sole author.

## Competing interests

The author completed the Unified Competing Interest form at www.icmje.org/coi_disclosure.pdf (available upon request from the corresponding author) and declare no conflicts of interest.

## Correspondence to:

Mahnoor Suakina, Karachi Medical and Dental College, D-35/2 Block-1, Clifton.

[email protected]


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# A cylinder inscribed in a cone [closed]

A cylinder inscribed in a cone with cone H = 8 and cylinder h = 4

a) Find the ratio of cone-volume / inscribed-volume. Solution:

b) Find the ratio of cone-area / inscribed-area ?

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## closed as off-topic by Goos, Michael Hoppe, Davide Giraudo, M Turgeon, egregFeb 27 at 14:22

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Goos, Michael Hoppe, Davide Giraudo, M Turgeon, egreg
If this question can be reworded to fit the rules in the help center, please edit the question.

Please tell us more about what you want help with and what you've tried so far. –  naslundx Feb 27 at 13:01
In your solution you've written $H-h = 4$, but should it not be $3$? –  naslundx Feb 27 at 13:24

## 1 Answer

If you know the radius of the base of the cone is $R$ and height $H=8$, by the pythagorean theorem the slant $S = \sqrt{R^2 + 8^2}$.

By the formulae for surface area of cylinder and cone, we have

$$A_{cone} = \pi R(R+S) = \pi R\left(R + \sqrt{R^2 + 64}\right)$$

$$A_{cylinder} = 2 \pi r(r + h) = 2 \pi \frac{R}{2}\left(\frac{R}{2} + 4\right) = \pi R \left(\frac{R}{2} + 4\right)$$

Divide the two expressions and simplify to get the ratio of the two quantities:

$$\frac{A_{cone}}{A_{cylinder}}=\frac{\pi R\left(R + \sqrt{R^2 + 64}\right)}{\pi R \left(\frac{R}{2} + 4\right)}=\frac{R + \sqrt{R^2 + 64}}{\frac{R}{2} + 4}$$

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That's why it's good to tell us more about how far you've managed yourself. I added the final step, does it make sense? –  naslundx Feb 27 at 14:11
Since we only know the heights, there is not enough information to calculate $R$. –  naslundx Feb 27 at 14:21
The expression is not constant so it cannot be simplified to an expression without the variable $R$. I would say the area ratio is dependant on the radius $R$. –  naslundx Feb 27 at 14:27
That's great, thank u :)) –  whyguy Feb 27 at 14:34


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# Talalay vs Dunlop Latex Mattresses

Types of Latex in Talalay vs Dunlop: Organic Rubber natural Latex -Synthetic Latex -Or a Combination of the Two

Organic Latex: Organic latex is made from the sap of the rubber tree which is processed (see the manufacturing process for Dunlop and Talalay latex). A Natural latex mattress made from organic latex does not contain any chemicals, synthetics, pesticides, herbicides, or other man-made products.

100% Natural Dunlop or Talalay Latex: Like organic natural latex, 100% natural is made from the sap of the rubber tree which is processed (see the process for Dunlop and Talalay latex). 100% Natural latex does not carry the associated USDA agricultural certification. The manufacturing process for organic latex and 100% natural latex is the same. The quality of the end product is identical.

Synthetic Latex: Latex that is not 100% organic is mixed with petroleum-based plasticizers like butadiene and styrene that create synthetic latex.

Blended Latex: Some companies combine organic and synthetic latex in their mattresses. Our latex mattress isn’t like this, but some combination latex mattresses also include fillers which are particles of clay (or other materials) that are mixed into the natural latex. Fillers make latex cheaper, but also less durable and stiffer.

## Latex Extraction and Sustainability In Talalay Latex vs Dunlop

The rubber for latex mattresses comes from Indonesia, Sri Lanka, Malaysia, and Thailand. Sleeping Organic extracts our rubber sap from sustainable plantations in Sri Lanka, and manufactures our mattresses in the United States. The technique for tapping latex from rubber trees eliminates a need to cut down the tree for latex extraction,

creating a sustainable relationship between sleep and sap. Natural Latex is extracted by carefully cutting the bark of rubber trees (which heals rapidly) to allow the flow of the white milk-type sap without damaging the tree. Rubber trees can yield latex for as long as 30 years.

When the trees are harvested, the land is replanted and the Rubberwood is used to create furniture. Latex is all-natural and biodegradable. In order to appreciate the value of latex, it is necessary to understand that each rubber tree weeps 15 grams of natural latex, daily.

The core of a queen-sized natural latex mattress requires the daily output of 2,500 trees that spread over 12 acres of rubber plantation. This provides a living to eight rubber tappers. Interestingly, these rubber trees account for the removal of 143 metric tons of Carbon Dioxide from the environment over one year.

## Latex Manufacturing Process For Dunlop Latex Vs Talalay

 Differences Between: Dunlop Latex Talalay Manufacturing Process has been used in latex manufacturing since 1929Thick rubber sap is whipped into a froth in a centrifugeSap is poured into a mold, the mold is covered and steam bakedNatural sediments in the mixture settle to the bottom of the mold making Dunlop latex layers slightly firmer on the bottom side Newer process that has two additional stepsThick rubber sap is whipped into a froth in a centrifugeSap is poured and the mold is sealedThe process continues in a vacuum chamber where air is extractedThe mold is flash frozen using chemicals to stabilize the materialsThe mold is then baked Consistency Dunlop latex is denser than Talalay latex which makes it a bit firmer, heavier and more durable Talalay has a more consistent cell structure, making it more soft and pillow-like Major Uses This firmer latex is typically used as a base core component Talalay is typically preferred as a top cushion layer

## Latex Allergies

Allergies are caused by the presence of proteins in a material. The protein in latex memory foam can cause an allergic reaction in some people. Some proteins in organic latex are similar to food proteins such as: bananas, avocados, chestnuts, kiwi, and tomatoes.

Allergic reactions differ in severity from one individual to another. Severity often depends on the extent of contact between latex memory foam and the human body. People often suffer from extreme allergic reactions when being exposed to balloons, gloves, condoms, and other products that come in direct contact with the body.

However, people seldom experience intense allergic reactions to organic latex mattresses due to barriers like sheets, pillows cases, and natural latex mattress covers that act as a shield between the body and the latex. One-hundred percent organic latex memory foam processes help prevent allergies by extensively washing off all of the free proteins.

We use 1.5 cubic meters of good drinkable quality water in the final wash through to keep latex foam allergies to a bare minimum. Therefore, the quality and availability of clean drinking water are essential in producing organic latex foam mattresses. In our case, the government-sponsored Board of Investment Export Processing Zone in Avissawella, Sri Lanka makes sure this quality water is available in abundance during our manufacturing process.


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Basic idea behind error recovery in recursive-descent parsers

I see that modern SW basically dominated by recursive descent parsers. These parsers try grammar production and back track to an alternative one in case of failure. Suppose BNF is

program = (declaration)* "begin" (statement)*
declaration = type_declaration | variable_declaration | function_declaration
type_declaration = "range" name "to" name | "enum" (name)*
variable_declaration = name ":" type_name
statement = ...

and suppose that user has entered user entered "range 10 to '15'" when declaration expected. The parser has failed to match the first type pattern but it also failed to match other alternatives and declarations and failed to parse the program. How do you recover? Should you track the "best match", which would be location of the "range 10 to '15'" and look for keywords met afterwards or what? Which mechanisms are used in practice, in popular parser combinators and generators like JavaCC?

• "I see that modern SW basically dominated by recursive descent parsers" -- really? My impression was that most real-world parsers in compilers used LR variants. – Raphael Jan 13 '16 at 15:12
• You are asking lots of questions. Please restrict yourself to one per post. – Raphael Jan 13 '16 at 15:13
• @Raphael Which many questions? – Valentin Tihomirov Jan 13 '16 at 15:19
• I count five. "How do you recover?", "Does my idea make sense?", plus three reference requests (which we are probably not the right site for) in the last sentence. – Raphael Jan 13 '16 at 15:20
• As you seem interested in PEG and not if CFG, arxiv.org/pdf/1405.6646.pdf seems relevant. – AProgrammer Jan 13 '16 at 19:33

Another is to add productions for typical errors in some constructs, e.g. handle a b as if it was a * b in case of expressions in a compiler for newbies.
• Can you explain what are we trying to match? Your sync token is mine keyword, as I understand. I have presented a demo grammar. Suppose that ; must follow every declaration and statement. How can you find what can follow the ; and exploit it automatically in PEG, which simply tries every alternative and says "nothing was found to match the program" if you make a slight error? – Valentin Tihomirov Jan 13 '16 at 13:02


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Absolute value

In mathematics, the absolute value (or modulus) |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3. The absolute value of a number may be thought of as its distance from zero.

Generalizations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

The graph of the absolute value function for real numbers.

## Terminology and notation

Jean-Robert Argand introduced the term "module" 'unit of measure' in French in 1806 specifically for the complex absolute value[1][2] and it was borrowed into English in 1866 as the Latin equivalent "modulus".[1] The term "absolute value" has been used in this sense since at least 1806 in French[3] and 1857 in English.[4] The notation | a | was introduced by Karl Weierstrass in 1841.[5] Other names for absolute value include "the numerical value"[1] and "the magnitude".[1]

The same notation is used with sets to denote cardinality; the meaning depends on context.

## Definition and properties

### Real numbers

For any real number a the absolute value or modulus of a is denoted by | a | (a vertical bar on each side of the quantity) and is defined as

$|a| = \begin{cases} a, & \mbox{if } a \ge 0 \\ -a, & \mbox{if } a < 0. \end{cases}$

As can be seen from the above definition, the absolute value of a is always either positive or zero, but never negative.

From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line, and more generally the absolute value of the difference of two real numbers is the distance between them. Indeed the notion of an abstract distance function in mathematics can be seen to be a generalization of the absolute value of the difference (see "Distance" below).

Since the square-root notation without sign represents the positive square root, it follows that

 $|a| = \sqrt{a^2}$ (1)

which is sometimes used as a definition of absolute value.[6]

The absolute value has the following four fundamental properties:

 $|a| \ge 0$ (2) Non-negativity $|a| = 0 \iff a = 0$ (3) Positive-definiteness $|ab| = |a||b|\,$ (4) Multiplicativeness $|a+b| \le |a| + |b|$ (5) Subadditivity

Other important properties of the absolute value include:

 $|-a| = |a|\,$ (6) Symmetry $|a - b| = 0 \iff a = b$ (7) Identity of indiscernibles (equivalent to positive-definiteness) $|a - b| \le |a - c| +|c - b|$ (8) Triangle inequality (equivalent to subadditivity) $|a/b| = |a| / |b| \mbox{ (if } b \ne 0) \,$ (9) Preservation of division (equivalent to multiplicativeness) $|a-b| \ge ||a| - |b||$ (10) (equivalent to subadditivity)

If b > 0, two other useful properties concerning inequalities are:

$|a| \le b \iff -b \le a \le b$
$|a| \ge b \iff a \le -b \mbox{ or } b \le a$

These relations may be used to solve inequalities involving absolute values. For example:

 $|x-3| \le 9$ $\iff -9 \le x-3 \le 9$ $\iff -6 \le x \le 12$

Absolute value is used to define the absolute difference, the standard metric on the real numbers.

### Complex numbers

The absolute value of a complex number z is the distance r from z to the origin. It is also seen in the picture that z and its complex conjugate z have the same absolute value.

Since the complex numbers are not ordered, the definition given above for the real absolute value cannot be directly generalized for a complex number. However the identity given in equation (1) above:

$|a| = \sqrt{a^2}$

can be seen as motivating the following definition.

For any complex number

$z = x + iy,\,$

where x and y are real numbers, the absolute value or modulus of z is denoted |z| and is defined as

$|z| = \sqrt{x^2 + y^2}.$

It follows that the absolute value of a real number x is equal to its absolute value considered as a complex number since:

$|x + i0| = \sqrt{x^2 + 0^2} = \sqrt{x^2} = |x|.$

Similar to the geometric interpretation of the absolute value for real numbers, it follows from the Pythagorean theorem that the absolute value of a complex number is the distance in the complex plane of that complex number from the origin, and more generally, that the absolute value of the difference of two complex numbers is equal to the distance between those two complex numbers.

The complex absolute value shares all the properties of the real absolute value given in (2)–(10) above. In addition, If

$z = x + i y = r (\cos \phi + i \sin \phi ) \,$

and

$\overline{z} = x - iy$

is the complex conjugate of z, then it is easily seen that

\begin{align} |z| & = r, \\ |z| & = |\overline{z}|\end{align}

and

$|z| = \sqrt{z\overline{z}},$

with the last formula being the complex analogue of equation (1) mentioned above in the real case.

The absolute square of z is defined as

$|z|^2 = z\overline{z} = x^2 + y^2.$

Since the positive reals form a subgroup of the complex numbers under multiplication, we may think of absolute value as an endomorphism of the multiplicative group of the complex numbers.

## Absolute value functions

The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0. It is monotonically decreasing on the interval (−∞, 0] and monotonically increasing on the interval [0, ∞). Since a real number and its negative have the same absolute value, it is an even function, and is hence not invertible.

Both the real and complex functions are idempotent.

It is a nonlinear function.

It is also a convex function.

### Derivatives

The derivative of the real absolute value function is given by[7]

$\frac{d|x|}{dx} = \frac{x}{|x|} = \begin{cases} -1 & x<0 \\ 1 & x>0 \end{cases}$

for x ≠ 0 (cf. signum function). The absolute value function is not differentiable at x = 0.[7] For applications in which a well-defined derivative may be needed, however, the subderivative is well defined at zero.

The complex absolute value function is continuous everywhere but complex differentiable nowhere; it violates the Cauchy-Riemann equations.

The absolute value function has no concavity at any point, the sign function is constant at all points. Therefore the second derivative of |x| with respect to x is zero everywhere except zero, where it is undefined.

### Antiderivative

The absolute value function is also integrable. Its antiderivative is

$\int|x|dx=\frac{x|x|}{2}+C,$

as evidenced by the following (using integration by parts and the fact that x2 = |x|2):

${\int|x|dx=x|x|-\int\frac{x^2}{|x|}dx= x|x|-\int|x|dx\iff 2\int|x|dx = x|x| \iff \int|x|dx =\frac{x|x|}{2}+C.}$

### Relationship to other functions

Where the absolute value function of a real number returns a value without respect to its sign, the signum function returns a number's sign without respect to its value. Therefore x = sgn(x)|x|. The real absolute value function is also related to a form of the Heaviside step function used in signal processing, defined as:

$u(x) = \begin{cases} 0, & x < 0 \\ \frac{1}{2}, & x = 0 \\ 1, & x > 0, \end{cases}$

where the value of the Heaviside function at zero is conventional. So for all nonzero points on the real number line,

$u(x) = \frac{\sgn(x) +1}{2}.\,$

## Distance

The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them.

The standard Euclidean distance between two points

$a = (a_1, a_2, \dots , a_n)$

and

$b = (b_1, b_2, \dots , b_n)$

in Euclidean n-space is defined as:

$\sqrt{(a_1-b_1)^2 + (a_2-b_2)^2 + \cdots + (a_n-b_n)^2}.$

This can be seen to be a generalization of | ab |, since if a and b are real, then by equation (1),

$|a - b| = \sqrt{(a - b)^2}.$

While if

$a = a_1 + i a_2 \,$

and

$b = b_1 + i b_2 \,$

are complex numbers, then

 $|a - b| \,$ $= |(a_1 + i a_2) - (b_1 + i b_2)|\,$ $= |(a_1 - b_1) + i(a_2 - b_2)|\,$ $= \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2}.$

The above shows that the "absolute value" distance for the real numbers or the complex numbers, agrees with the standard Euclidean distance they inherit as a result of considering them as the one and two-dimensional Euclidean spaces respectively.

The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a distance function as follows:

A real valued function d on a set X × X is called a distance function (or a metric) on X, if it satisfies the following four axioms:[8]

 $d(a, b) \ge 0$ Non-negativity $d(a, b) = 0 \iff a = b$ Identity of indiscernibles $d(a, b) = d(b, a) \,$ Symmetry $d(a, b) \le d(a, c) + d(c, b)$ Triangle inequality

## Generalizations

### Ordered rings

The definition of absolute value given for real numbers above can easily be extended to any ordered ring. That is, if a is an element of an ordered ring R, then the absolute value of a, denoted by | a |, is defined to be:

$|a| = \begin{cases} a, & \mbox{if } a \ge 0 \\ -a, & \mbox{if } a < 0, \end{cases}$

where −a is the additive inverse of a, and 0 is the additive identity element.

### Fields

The fundamental properties of the absolute value for real numbers given in (2)–(5) above, can be used to generalize the notion of absolute value to an arbitrary field, as follows.

A real-valued function v on a field F is called an absolute value (also a modulus, magnitude, value, or valuation) if it satisfies the following four axioms:

 $v(a) \ge 0$ Non-negativity $v(a) = 0 \iff a = \mathbf{0}$ Positive-definiteness $v(ab) = v(a) v(b) \,$ Multiplicativeness $v(a+b) \le v(a) + v(b)$ Subadditivity or the triangle inequality

Where 0 denotes the additive identity element of F. It follows from positive-definiteness and multiplicativeness that v(1) = 1, where 1 denotes the multiplicative identity element of F. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.

If v is an absolute value on F, then the function d on F × F, defined by d(a, b) = v(ab), is a metric and the following are equivalent:

• d satisfies the ultrametric inequality $d(x, y) \leq \max(d(x,z),d(y,z))$ for all x, y, z in F.
• $\big\{ v\Big({\textstyle \sum_{k=1}^n } \mathbf{1}\Big) : n \in \mathbb{N} \big\}$ is bounded in R.
• $v\Big({\textstyle \sum_{k=1}^n } \mathbf{1}\Big) \le 1 \text{ for every } n \in \mathbb{N}.$
• $v(a) \le 1 \Rightarrow v(1+a) \le 1 \text{ for all } a \in F.$
• $v(a + b) \le \mathrm{max}\{v(a), v(b)\} \text{ for all } a, b \in F.$

An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwise it is said to be Archimedean.[9]

### Vector spaces

Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalize the notion to an arbitrary vector space.

A real-valued function on a vector space V over a field F, represented as ||V||, is called an absolute value (or more usually a norm) if it satisfies the following axioms:

For all a in F, and v, u in V,

 $\|\mathbf{v}\| \ge 0$ Non-negativity $\|\mathbf{v}\| = 0 \iff \mathbf{v} = 0$ Positive-definiteness $\|a \mathbf{v}\| = |a| \|\mathbf{v}\|$ Positive homogeneity or positive scalability $\|\mathbf{v} + \mathbf{u}\| \le \|\mathbf{v}\| + \|\mathbf{u}\|$ Subadditivity or the triangle inequality

The norm of a vector is also called its length or magnitude.

In the case of Euclidean space Rn, the function defined by

$\|(x_1, x_2, \dots , x_n) \| = \sqrt{\sum_{i=1}^{n}(x_i)^2}$

is a norm called the Euclidean norm. When the real numbers R are considered as the one-dimensional vector space R1, the absolute value is a norm, and is the p-norm for any p. In fact the absolute value is the "only" norm on R1, in the sense that, for every norm || · || on R1, || x || = || 1 || · | x |. The complex absolute value is a special case of the norm in an inner product space. It is identical to the Euclidean norm, if the complex plane is identified with the Euclidean plane R2.

## Smooth approximation

The absolute value function does not have a derivative at x=0[citation needed]. Sometimes, a smooth approximation is required that has to be smooth at x=0. One of such approximations is given by:

$|x| \approx \frac{2x}{\pi} \mathrm{arctan}(kx)$

where k>0. The approximation improves as k increases. k=10 is good for many practical purposes.

## Notes

1. ^ a b c d Oxford English Dictionary, Draft Revision, June 2008
2. ^ Nahin, O'Connor and Robertson, and functions.Wolfram.com.; for the French sense, see Littré, 1877
3. ^ Lazare Nicolas M. Carnot, Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques pris dans l'espace, p. 105 at Google Books
4. ^ James Mill Peirce, A Text-book of Analytic Geometry at Google Books. The oldest citation in the 2nd edition of the Oxford English Dictionary is from 1907. The term "absolute value" is also used in contrast to "relative value".
5. ^ Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0-89871-420-6, p. 25
6. ^ Stewart, James B. (2001). Calculus: concepts and contexts. Australia: Brooks/Cole. ISBN 0-534-37718-1. , p. A5
7. ^ a b Bartel and Sherbert, p. 163
8. ^ These axioms are not minimal; for instance, non-negativity can be derived from the other three: 0 = d(a, a) ≤ d(a, b) + d(b, a) = 2d(a, b).
9. ^ Schechter, p 260-261.

## References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• absolute value — n. 1. the magnitude of a real number, disregarding its positive or negative sign [the absolute value of 4 or +4, written | 4| or |+4|, is 4] 2. the magnitude of a complex number; modulus …   English World dictionary

• absolute value — value of a number expressed as a positive number (i.e. 23 has an absolute value of 23) …   English contemporary dictionary

• absolute value — Math. 1. the magnitude of a quantity, irrespective of sign; the distance of a quantity from zero. The absolute value of a number is symbolized by two vertical lines, as |3| or | 3| is 3. 2. the square root of the sum of the squares of the real… …   Universalium

• Absolute Value — A business valuation method that uses discounted cash flow analysis to determine a company s financial worth. The absolute value method differs from the relative value models that examine what a company is worth compared to its competitors.… …   Investment dictionary

• absolute value — ab′solute val′ue n. 1) the magnitude of a quantity, irrespective of sign; the distance of a quantity from zero. The absolute value of a number is symbolized by two vertical lines, as |3| or |−3| is three 2) the square root of the sum of the… …   From formal English to slang

• absolute value — absoliučioji vertė statusas T sritis fizika atitikmenys: angl. absolute value vok. Absolutwert, m rus. абсолютное значение, n pranc. valeur absolue, f …   Fizikos terminų žodynas

• absolute value — skaičiaus modulis statusas T sritis informatika apibrėžtis Matematinė sąvoka, reiškianti skaičiaus absoliutinę reikšmę |x|, kuri apibrėžiama taip: Iliustraciją žr. priede. Skaičiaus modulis visuomet didesnis arba lygus nuliui. Modulį galima gauti …   Enciklopedinis kompiuterijos žodynas

• absolute value — noun Date: 1889 1. a nonnegative number equal in numerical value to a given real number 2. the positive square root of the sum of the squares of the real and imaginary parts of a complex number …   New Collegiate Dictionary

• absolute value — noun a) For a real number, its numerical value without regard to its sign; formally, 1 times the number if the number is negative, and the number unmodified if it is zero or positive. b) For a complex number, the …   Wiktionary

• absolute value — noun 1》 Mathematics the magnitude of a real number without regard to its sign. 2》 technical the actual magnitude of a numerical value or measurement, irrespective of its relation to other values …   English new terms dictionary


text:
# Remote management - Ansible as an example¶

This exercise has a long period for submission. Students who have more time in October can make this earlier, but submission are accepted until mid December.

## Exercise¶

The instructions are in a separate file and was be discussed on 05.10 lecture.

Please read and start working and send clarification questions to kari.systa (at) tuni.fi.

Give a link, and other information to access, to git-based version control of your project (e.g. cources-gitlab). The repository must include report.pdf, too.

Posting submission...


text:
# Python ANN Implementation [closed]

I recently learned about backpropagation online and tried to implement it. I am not sure I have it correct yet. I am confused and would love a second pair of eyes on this code. Please help me understand how I can improve this and if this is even correct.

Implementation:

"""
Artificial Neural Network
"""

import numpy as np

def sigmoid(x, derivative=False):
output = 1 / (1 + np.exp(-x))
if derivative:
return output * (1 - output)
return output

def tanh(x, derivative=False):
output = 2 * sigmoid(2 * x) - 1
if derivative:
return 1 - output ** 2
return output

class Layer:
def __init__(self, num_input, num_output, activation_fn=tanh):
# num_rows = num_input, num_cols = num_output
self.weights = activation_fn(np.random.rand(num_input, num_output))
self.bias = activation_fn(np.random.rand(1, num_output))
self.activation_fn = activation_fn
self.raw = None
self.activated = None
self.input_data = None

def __str__(self):
return "Weights:\n" + str(self.weights) + "\nBias:\n" + str(self.bias)

def output(self, x):
self.input_data = x
self.raw = np.dot(x, self.weights) + self.bias
self.activated = self.activation_fn(self.raw)
return self.activated

def output_der(self):
return self.activation_fn(self.activated, derivative=True)

class NeuralNetwork:
def __init__(self, layers, eta=0.1):
if len(layers) < 2:
raise Exception("Layers needs to have input and output")
self.layers = self.init_layers(layers)
self.eta = eta

def __str__(self):
s = ""
for i, l in enumerate(self.layers):
s += "Layer {}.\n{}\n\n".format(i, str(l))
s += "--------------"
return s

def init_layers(self, layers):
ann = []
i = 0
while i < len(layers) - 1:
ann.append(Layer(layers[i], layers[i + 1]))
i += 1
return ann

def train(self, dataset, times=45):
# item[0] = input
# item[1] = label
# backpropagation is an online algorithm (one example at a time)
for i in range(times):
error = 0
output = 0
for item in dataset:
output = self.forward(np.array(item[0]))
# loss_at_output_layer = -(label - output)
self.backward(output - item[1])
error += (item[1] - output) ** 2
print(error)

def hot_encode(self, output):
top = max(output)
for i in range(len(output)):
output[i] = 0 if output[i] != top else 1
return output

def test(self, dataset):
accuracy = 0
for item in dataset:
output = self.forward(np.array(item[0]))
if np.array_equal(self.hot_encode(output[0]), item[1]):
accuracy += 1
print("Accuracy: {}\n".format(accuracy / len(dataset)))

def forward(self, item):
x = item
for l in self.layers:
x = l.output(x)
return x

def backward(self, loss):
# delta_i = error * Δoutput_i
deltas = []
for l in reversed(self.layers):
deltas.append(np.multiply(loss, l.output_der()))
loss = np.dot(deltas[-1], l.weights.T)
l.weights -= self.eta * np.multiply(deltas[-1], l.weights)
l.bias -= self.eta * deltas[-1]

#---------------------------------------------------------------------#

"""
Driver Code
"""

def normalize(data):
return (2 * (data - min(data)) / (max(data) - min(data))) - 1

def get_dataset(filename):
with open(filename, "r") as f:
raw = {}
for line in f:
line_data = line.split(",")
feature = normalize(np.array(list(map(float, line_data[:-1]))))
if line_data[-1] in raw:
raw[line_data[-1]].append(feature)
else:
raw[line_data[-1]] = [feature]
hot_encoding = {}
for i, label in enumerate(raw):
hot_encoding[label] = np.zeros(len(raw))
hot_encoding[label][i] = 1
dataset = []
for label in raw:
for feature in raw[label]:
dataset.append([feature, hot_encoding[label]])
return dataset

if __name__ == "__main__":
classifier = NeuralNetwork([4, 3], eta=0.01)
print("{}\n".format(classifier))
dataset = get_dataset("./iris.data")
np.random.shuffle(dataset)
print(dataset)
split_ratio = int(2 * len(dataset) / 3)
classifier.train(dataset[:split_ratio], times=50)
classifier.test(dataset[split_ratio:])
print("{}\n".format(classifier))

I am trying to test it on the Iris dataset but it doesn't converge. I have tried the following architectures:

• 4 Input, 3 * 3 Hidden, 3 Output (Hot encoded)
• 4 '' , 3 '' , 3 ''
• 4 '' , 5 '' , 3 ''

• Please help me understand how I can improve this and if this is even correct. This site is for reviewing working code. The author should understand the code. I am not sure this is the case here. Visit our help center for more info: codereview.stackexchange.com/help/on-topic. – dfhwze Sep 8 '19 at 19:28

### Coding Style

First, I'm not a code reviewer. Your code seems to be OK though. There are some basic coding conventions in writing Python scripts, such as variable naming, commenting, docstring, and such, which I don't go through it, since I'm learning myself, and you can find it here.

### Implementation

There are a few things that hold a basic ANN not to properly converge such as:

• IRIS dataset is a pretty small dataset, to start with; which normally one uses some 70% of a dataset for training, if supervised (which is the case here), and the rest for validation.

• It is difficult for me to go through your mathematical debugging, but it might be a reason that the math might have some problems and the network doesn't converge. To make sure, you can test it step by step (Neuron by Neuron maybe, if you will) with a very simple training and testing dataset, much simpler than IRIS, to see if there might be some bugs.

• If there is no bug, the architecture of ANN is another thing that would impact the convergence. I guess you might not need three layers of hiddens, and one hidden layer with maybe 10 to 30 neurons might be just OK for IRIS. Sometimes, adding too many neurons would trap the network into mathematical local minima dilemma. You might want to make sure that the Input and Output layers have the exact correct number of neurons according to the dataset.

• There might be some related tutorials to implement ANNs from scratch, wouldn't be such a bad idea to look them up. Maybe, something with a Neuron class.

### Integration

You can also apply some already built-in modules to do that, such with KNN in this case, which I'm pretty sure you know:

from sklearn import neighbors, datasets, preprocessing
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score
from sklearn.metrics import classification_report
from sklearn.metrics import confusion_matrix

X, y = iris.data[:, :], iris.target

Xtrain, Xtest, y_train, y_test = train_test_split(X, y)
scaler = preprocessing.StandardScaler().fit(Xtrain)
Xtrain = scaler.transform(Xtrain)
Xtest = scaler.transform(Xtest)

knn = neighbors.KNeighborsClassifier(n_neighbors=4)
knn.fit(Xtrain, y_train)
y_pred = knn.predict(Xtest)

print(accuracy_score(y_test, y_pred))
print(classification_report(y_test, y_pred))
print(confusion_matrix(y_test, y_pred))

### Output

0.8947368421052632
precision    recall  f1-score   support

0       1.00      1.00      1.00        13
1       0.79      0.92      0.85        12
2       0.91      0.77      0.83        13

accuracy                           0.89        38
macro avg       0.90      0.90      0.89        38
weighted avg       0.90      0.89      0.89        38

[[13  0  0]
[ 0 11  1]
[ 0  3 10]]

### Overall

I think it is great that you are trying to implement a ANN from scratch.


text:
# Complexity of cycle-transverse matching problems

## Paper with Jing Huang and Xuding Zhu presented at IWOCA 2011

Two weeks after CanaDAM, UVic hosted the International Workshop on Combinatorial Algorithms. I presented a paper by me, Jing Huang, and Xuding Zhu which gives the first results of what would become my master’s thesis. It studies the computational complexity of the following problem: in a given graph, is there a matching which breaks all cycles of a given length?

Slides of the talk can be found here (PDF). I am grateful to NSERC for funding this research with a Alexander Graham Bell Canada Graduate Scholarship.

1. Complexity of cycle transverse matching problems. International Workshop on Combinatorial Algorithms. Ross Churchley, Jing Huang, and Xuding Zhu (2011). ↩


text:
Braverman, C.

### mathematics and statistics online

Rojas, and M. Phys , vol. Blank , Pathologies generated by round-off in dynamical systems , Physica D: Nonlinear Phenomena , vol. Blank , Small perturbations of chaotic dynamical systems , Russian Mathematical Surveys , vol. Brattka, P. CCA , pp.

Braverman , Parabolic Julia sets are polynomial time computable , Nonlinearity , vol. Braverman and M.

### Physics Faculty of St. Petersburg State University

Yampolsky , Computability of Julia sets , Moscow Math. Journ , vol.

• Scarce Natural Resources, Recycling, Innovation and Growth!
• Further Publications.
• The Birth of Head Start: Preschool Education Policies in the Kennedy and Johnson Administrations!
• Other Sites.
• Africa and the Americas: Culture, Politics, and History (Transatlantic Relations) volume 1, 2, and 3.

Brolin , Invariant sets under iteration of rational functions , Arkiv f?? Carleson and T. Gamelin , Complex Dynamics , Dellnitz and O.

1. Intelligent Robotics and Applications: 7th International Conference, ICIRA 2014, Guangzhou, China, December 17-20, 2014, Proceedings, Part II.
2. Communications in Mathematical Physics.
3. Citations per year;
4. The Problem of Pain.
5. Foundations of Applied Statistical Methods.
6. Online First!
7. Reading the Entrails an Alberta Ecohisto.
8. Ding, Q. Du, and T. Ding and A. Zhou , The projection method for computing multidimensional absolutely continuous invariant measures , Journal of Statistical Physics , vol. Hoyrup, and C.

Rojas , Randomness on Computable Probability Spaces??? Galatolo, M. Hoyrup, C. Rojas, and.

Borel-cantelli-lemma , A constructive Borel??? Cantelli lemma. Constructing orbits with required statistical properties , Theoretical Computer Science , vol.

## List of Publications

Rojas , Effective symbolic dynamics, random points, statistical behavior, complexity and entropy , Information and Computation , vol. Rojas , Dynamics and abstract computability: computing invariant measures , Disc. Rojas , Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems Proc , 7th Intl. Galatolo , I Nisoli A simple approach to rigorous approximation of invariant measures Preprint.

## Absolutely continuous spectra of quantum tree graphs with weak disorder

Hertling , Is the Mandelbrot set computable? Stable bundles and polyvector fields Hitchin, N volume 8 page Vector bundles and the icosahedron Hitchin, N Contemporary Mathematics volume page Poisson modules and generalized geometry Hitchin, N Advanced Lectures in Mathematics volume 17 page Arbeitstagung Bonn volume page A note on vanishing theorems Hitchin, N 6 March Differential and algebraic geometry, and their interaction with the equations of theoretical physics. Skip to main content.

Which Course? Which College?

You are here Home. Nigel Hitchin FRS. MA, DPhil.


text:
Procedure GMP::Linearization::RemoveDeviation(GMP, row, linNo)

# GMP::Linearization::RemoveDeviation

The procedure GMP::Linearization::RemoveDeviation removes the deviation of a linearization of a row in a generated mathematical program. That is, it deletes the extra column created (if any) when adding the linearization of the row to the generated mathematical program.

GMP::Linearization::RemoveDeviation(
GMP,    ! (input) a generated mathematical program
row,    ! (input) a scalar reference or row number
linNo   ! (input) a linearization number
)

## Arguments

GMP

An element in AllGeneratedMathematicalPrograms.

row

A scalar reference to an existing nonlinear row in the matrix or the number of that row in the range $$\{ 0 .. m-1 \}$$ where $$m$$ is the number of rows in the matrix.

linNo

An integer scalar reference to the rows and columns of the linearization.

## Return Value

The procedure returns 1 on success, or 0 otherwise.


text:
Vol. 08  No. 05 ( 2019 ), Article ID: 30571 , 12 pages
10.12677/AAM.2019.85117

The Jones Polynomials of Class of Knots

Wei Zheng, Dongyu Geng, Luying Zhao*

Liaoning Normal University, Dalian Liaoning

Received: May 9th, 2019; accepted: May 24th, 2019; published: May 31st, 2019

ABSTRACT

Knot theory is a branch of mathematical topology and the classification of knots is a very important problem in the field of knots. People solve the problem of knot by looking for knot invariants. Knot invariants are trichrome, crossover number, knot number, bridge number, knot polynomial, knot group, etc. In this paper, we study a class of special Brunnian link. The formula for calculating the Jones polynomial of this kind of chain is given. In order to realize the calculation of $B\left({c}_{1},{c}_{2},\cdots ,{c}_{n}\right)$ Jones polynomials, on the one hand, by calculating $B\left({c}_{1}\right)$ and $B\left({c}_{1},{c}_{2}\right)$ bracket polynomials, the calculating formulas of $B\left({c}_{1},{c}_{2},\cdots ,{c}_{n}\right)$ bracket polynomials are obtained. On the other hand, by dividing the given orientation into regions $B\left({c}_{1},{c}_{2},\cdots ,{c}_{n}\right)$, the enumeration method is used to find the calculation rule of the number of twists, and then the calculation formula of the number of twists $B\left({c}_{1},{c}_{2},\cdots ,{c}_{n}\right)$ is given. Combining the above two aspects, the calculation formula of the Jones polynomial $B\left({c}_{1},{c}_{2},\cdots ,{c}_{n}\right)$ is obtained. The innovation of this paper is to generalize the original definition of connected sum, to connect with a specific segment, $B\left(0\right)$ to construct $B\left(0,0\right)$, then $B\left(0,0\right)$ to construct $B\left(0,0,0\right)$, and then to find the $B\left(0,0,\cdots ,0\right)$ computational law, which greatly simplifies the computational process.

Keywords:Pretzel Knot, Jones Polynomial, Twist, Brunnian Link

Jones多项式是纽结理论中重要的不变量之一。若两个链环投影图的Jones多项式的不同，则这两个投影图对应着不同的链环，这是研究纽结和链环分类问题的重要方法。Brunnian link是一类既特殊又简单的链环，本文研究一类特殊的Brunnian链 $B\left({c}_{1},{c}_{2},\cdots ,{c}_{n}\right)$ 环，本论文给出了 $B\left({c}_{1},{c}_{2},\cdots ,{c}_{n}\right)$ 的Jones多项式的计算公式。

1. 引言

2. 预备知识

2.1. 纽结及投影图

①只有有限多个交叉点；

②每个交叉点都是二重点；

③在每个二重点处，上下两线的投影都是互相穿越交叉的则称为纽结投影图。注意，平面的选择不同随之投影图不唯一。

2.2. 链环

2.3. Reidemeister Move (R变换)

Reidemeister变换是改变纽结的正则投影图的三种方式，每一种都会改变交叉点之间的关系 [1]

RI变换：

RII变换：

RIII变换：

2.4. 纽结连通和

2.5. 尖括号多项式

1) 介绍A通道、B通道：

2.6. 拧数

2.7. Jones多项式

${S}^{3}$ 中的定向环链。因此，有Jones多项式 $V\left(L\right)$ 与之对应，且满足下列条件：

1) $V\left(L\right)$ 是合痕不变量；

2)有拆接公式： $tV\left({L}_{+}\right)+{t}^{-1}V\left({L}_{-}\right)=\left({t}^{\frac{1}{2}}-{t}^{-\frac{1}{2}}\right)V\left({L}_{0}\right)$

3) $V\left(Ο\right)=1$

2.8. 引理1

$T\left(s\right)$ 是一个排叉结如下图，当s为奇数时， $T\left(s\right)$ 是一个纽结，当s为偶数时， $T\left(s\right)$ 是一个两分支的链环，并且有 $\epsilon \left(s\right)=\frac{s}{|s|}$ 则 [4] ：

$〈T\left(s\right)〉=\left[\underset{i=1}{\overset{|s|-1}{\sum }}{A}^{-\epsilon \left(s\right)\left(2-i\right)}{\left(-{A}^{3}\right)}^{\epsilon \left(s\right)i-s}\right]-{A}^{s+2\epsilon \left(s\right)}$

2.9. 引理2

$Z>0$

$Z<0$

2.10. 引理3

${c}_{i}>0$ 时表示正的半扭转，当 ${c}_{i}<0$ 时表示负的半扭转。 $B\left({c}_{1},{c}_{2},\cdots ,{c}_{n}\right)$ 如图所示。

${c}_{i}=0\left(i=1,2,\cdots ,n\right)$ 时，此时 $B\left({c}_{1},{c}_{2},\cdots ,{c}_{n}\right)$ 变成 $B\left(0,0,\cdots ,0\right)$。这时所有的半扭转消失，如图所示 [6] 。

$n=1$ 时， $B\left({c}_{1},{c}_{2},\cdots ,{c}_{n}\right)$ 变成 $B\left({c}_{1}\right)$ ，是构成 $B\left({c}_{1},{c}_{2},\cdots ,{c}_{n}\right)$ 的单位元素，也可以理解为由 $B\left(0\right)$ 加入 $|{c}_{1}|$ 个半扭转得到的，如图所示。

3. 定理证明

$\begin{array}{c}〈B\left(0\right)〉=\left[\underset{i=1}{\overset{2}{\sum }}{A}^{i-2}{\left(-{A}^{3}\right)}^{i-3}-{A}^{5}\right]\left[\underset{i=1}{\overset{2}{\sum }}{A}^{2-i}{\left(-{A}^{3}\right)}^{3-i}-{A}^{-5}\right]\\ =\left({A}^{-7}-{A}^{-3}-{A}^{5}\right)\left({A}^{7}-{A}^{3}-{A}^{-5}\right)\end{array}$

$B\left(0\right)$ 上标记两段弧记为 $\alpha ,\beta$ ，如图所示

$〈B\left(0\right)〉=\left(1-{A}^{4}\right)〈a〉+{A}^{-2}〈b〉$

$\begin{array}{c}〈B\left(0,0,0\right)〉=\left(1-{A}^{4}\right)〈ΟΟΟ〉+{A}^{-2}〈B\left(0,0\right)〉\\ =\left(1-{A}^{4}\right){\left(-{A}^{2}-{A}^{-2}\right)}^{2}+{A}^{-2}〈B\left(0,0\right)〉\\ =\left(1-{A}^{4}\right){\left(-{A}^{2}-{A}^{-2}\right)}^{2}+{A}^{-2}\left(1-{A}^{4}\right)\left(-{A}^{2}-{A}^{-2}\right)+{\left({A}^{-2}\right)}^{2}〈B\left(0\right)〉\end{array}$

①当 $n=1$ 时如上图。

${c}_{1}$ 最右端的交叉点开始打开可得 $〈B\left(c{}_{1}\right)〉={A}^{\epsilon \left({c}_{1}\right)}〈\Lambda 〉+{A}^{-\epsilon \left({c}_{1}\right)}〈B\left({c}_{1}-\epsilon \left({c}_{1}\right)\right)〉$

$\Lambda$ 可看成如下两部分的连通和

$〈B\left({c}_{1}\right)〉={A}^{\epsilon \left({c}_{1}\right)}〈\circ \circ 〉$ $〈$ $〉$

$〈B\left({c}_{1}\right)〉={A}^{\epsilon \left({c}_{1}\right)}\left(-{A}^{2}-{A}^{-2}\right){\left(-{A}^{3}\right)}^{{c}_{1}-\epsilon \left({c}_{1}\right)}+{A}^{-\epsilon \left({c}_{1}\right)}〈B\left({c}_{1}-\epsilon \left({c}_{1}\right)\right)〉$ (1)

$〈B\left({c}_{1}-\epsilon \left({c}_{1}\right)\right)〉={A}^{\epsilon \left({c}_{1}-\epsilon \left({c}_{1}\right)\right)}\left(-{A}^{2}-{A}^{-2}\right){\left(-{A}^{3}\right)}^{\left({c}_{1}-\epsilon \left({c}_{1}\right)\right)-\epsilon \left({c}_{1}\right)}+{A}^{-\epsilon \left({c}_{1}-\epsilon \left({c}_{1}\right)\right)}〈B\left({c}_{1}-\epsilon \left({c}_{1}\right)-\epsilon \left({c}_{1}\right)\right)〉$

$〈B\left({c}_{1}-\epsilon \left({c}_{1}\right)\right)〉={A}^{\epsilon \left({c}_{1}\right)}\left(-{A}^{2}-{A}^{-2}\right){\left(-{A}^{3}\right)}^{{c}_{1}-2\epsilon \left({c}_{1}\right)}+{A}^{-\epsilon \left({c}_{1}\right)}〈B\left({c}_{1}-2\epsilon \left({c}_{1}\right)\right)〉$

$〈B\left({c}_{1}\right)〉={A}^{\epsilon \left({c}_{1}\right)}\left(-{A}^{2}-{A}^{-2}\right){\left(-{A}^{3}\right)}^{{c}_{1}-\epsilon \left({c}_{1}\right)}+{A}^{0}\left(-{A}^{2}-{A}^{-2}\right){\left(-{A}^{3}\right)}^{{c}_{1}-2\epsilon \left({c}_{1}\right)}+{A}^{-2\epsilon \left({c}_{1}\right)}〈B\left({c}_{1}-2\epsilon \left({c}_{1}\right)\right)〉$

$〈B\left({c}_{1}\right)〉=\left(-{A}^{2}-{A}^{-2}\right)\underset{i=1}{\overset{|{c}_{1}|}{\sum }}{A}^{\epsilon \left({c}_{1}\right)\left(2-i\right)}{\left(-{A}^{3}\right)}^{{c}_{1}-i\epsilon \left({c}_{1}\right)}+{A}^{-|{c}_{1}|\epsilon \left({c}_{1}\right)}〈B\left({c}_{1}-|{c}_{1}|\epsilon \left({c}_{1}\right)\right)〉$

②当 $n=2$ 时如图

${c}_{2}$ 右端的交叉点打开可得： $〈B\left({c}_{1},{c}_{2}\right)〉={A}^{\epsilon \left({c}_{2}\right)}〈L〉+{A}^{-\epsilon \left({c}_{2}\right)}〈B\left({c}_{1},{c}_{2}-\epsilon \left({c}_{2}\right)\right)〉$

L可以看成如下四部分的连通和

$\begin{array}{c}〈B\left({c}_{1},{c}_{2}\right)〉={A}^{\epsilon \left({c}_{2}\right)}\left(-{A}^{2}-{A}^{-2}\right){\left(-{A}^{3}\right)}^{{c}_{1}}\left(-{A}^{2}-{A}^{-2}\right){\left(-{A}^{3}\right)}^{{c}_{2}-\epsilon \left({c}_{2}\right)}+{A}^{-\epsilon \left({c}_{2}\right)}〈B\left({c}_{1},{c}_{2}-\epsilon \left({c}_{2}\right)\right)〉\\ ={A}^{\epsilon \left({c}_{2}\right)}{\left(-{A}^{2}-{A}^{-2}\right)}^{2}{\left(-{A}^{3}\right)}^{{c}_{1}}{\left(-{A}^{3}\right)}^{{c}_{2}-\epsilon \left({c}_{2}\right)}+{A}^{-\epsilon \left({c}_{2}\right)}〈B\left({c}_{1},{c}_{2}-\epsilon \left({c}_{2}\right)\right)〉\end{array}$

$〈B\left({c}_{1},{c}_{2}\right)〉={\left(-{A}^{2}-{A}^{-2}\right)}^{2}{\left(-{A}^{3}\right)}^{{c}_{1}}\left[\underset{i=1}{\overset{|{c}_{2}|}{\sum }}{A}^{\epsilon \left({c}_{2}\right)\left(2-i\right)}{\left(-{A}^{3}\right)}^{{c}_{2}-i\epsilon \left({c}_{2}\right)}\right]+{A}^{-{c}_{2}}〈B\left({c}_{1},0\right)〉$

${c}_{1}$ 最右面的交叉点开始有

$〈B\left({c}_{1},0\right)〉={A}^{\epsilon \left({c}_{1}\right)}〈H〉+{A}^{-\epsilon \left({c}_{1}\right)}〈B\left({c}_{1}-\epsilon \left({c}_{1}\right),0\right)〉$

$〈H〉=〈\circ \circ \circ 〉〈$ $〉$

$\begin{array}{c}〈B\left({c}_{1},0\right)〉={A}^{\epsilon \left({c}_{1}\right)}{\left(-{A}^{2}-{A}^{-2}\right)}^{2}{\left(-{A}^{3}\right)}^{{c}_{1}-\epsilon \left({c}_{1}\right)}+{A}^{-\epsilon \left({c}_{1}\right)}〈B\left({c}_{1}-\epsilon \left({c}_{1}\right),0\right)〉\\ =\left(-{A}^{2}-{A}^{-2}\right)\left[\underset{i=1}{\overset{|{c}_{1}|}{\sum }}{A}^{\epsilon \left({c}_{1}\right)\left(2-i\right)}{\left(-{A}^{3}\right)}^{{c}_{1}-i\epsilon \left({c}_{1}\right)}\right]+{A}^{-{c}_{1}}〈B\left(0,0\right)〉\end{array}$

$\begin{array}{c}〈B\left({c}_{1},{c}_{2}\right)〉={\left(-{A}^{2}-{A}^{-2}\right)}^{2}{\left(-{A}^{3}\right)}^{{c}_{1}}\left[\underset{i=1}{\overset{|{c}_{2}|}{\sum }}{A}^{\epsilon \left({c}_{2}\right)\left(2-i\right)}{\left(-{A}^{3}\right)}^{{c}_{2}-i\epsilon \left({c}_{2}\right)}\right]\\ \text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\left(-{A}^{2}-{A}^{-2}\right)}^{2}{A}^{{c}_{2}}\left[\underset{i=1}{\overset{|{c}_{1}|}{\sum }}{A}^{\epsilon \left({c}_{1}\right)\left(2-i\right)}{\left(-{A}^{3}\right)}^{{c}_{1}-i\epsilon \left({c}_{1}\right)}\right]+{A}^{-{c}_{1}-{c}_{2}}〈B\left(0,0\right)〉\end{array}$

$n=k$ 时，从 ${c}_{k}$ 最右端的交叉点开始打开可得：

$〈B\left({c}_{1},{c}_{2},\cdots ,{c}_{k}\right)〉={A}^{\epsilon \left({c}_{k}\right)}〈D〉+{A}^{-\epsilon \left({c}_{k}\right)}〈B\left({c}_{1},{c}_{2},\cdots ,{c}_{k}-\epsilon \left({c}_{k}\right)\right)〉$

$〈D〉={\left(-{A}^{2}-{A}^{-2}\right)}^{k}{\left(-{A}^{3}\right)}^{{c}_{1}+{c}_{2}+\cdots +{c}_{k-1}}{\left(-{A}^{3}\right)}^{{c}_{k}-\epsilon \left(ck\right)}$

$\begin{array}{c}〈B\left({c}_{1},{c}_{2},\cdots ,{c}_{k}\right)〉=\left[{A}^{\epsilon \left({c}_{k}\right)}{\left(-{A}^{2}-{A}^{-2}\right)}^{k}{\left(-{A}^{3}\right)}^{{c}_{1}+{c}_{2}+\cdots +{c}_{k-1}}\right]{\left(-{A}^{3}\right)}^{{c}_{k}-\epsilon \left({c}_{k}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{A}^{-\epsilon \left({c}_{k}\right)}〈B\left({c}_{1},{c}_{2},\cdots ,{c}_{k}-\epsilon \left({c}_{k}\right)\right)〉\end{array}$

$\begin{array}{l}〈B\left({c}_{1},{c}_{2},\cdots ,{c}_{k}\right)〉\\ =\left[{\left(-{A}^{2}-{A}^{-2}\right)}^{k}{\left(-{A}^{3}\right)}^{{c}_{1}+{c}_{2}+\cdots +{c}_{k-1}}\right]\underset{i=1}{\overset{|{c}_{k}|}{\sum }}{A}^{\epsilon \left({c}_{k}\right)\left(2-i\right)}{\left(-{A}^{3}\right)}^{{c}_{k}-i\epsilon \left({c}_{k}\right)}+{A}^{-{c}_{k}}〈B\left({c}_{1},{c}_{2},\cdots ,0\right)〉\end{array}$

$B\left({c}_{1},{c}_{2},\cdots ,{c}_{k-1},0\right)$ 中的 ${c}_{k-1}$ 进行一样的操作，从右面的交叉点打开得：

$〈B\left({c}_{1},{c}_{2},\cdots ,{c}_{k-1},0\right)〉={A}^{\epsilon \left({c}_{k-1}\right)}〈W〉+{A}^{-\epsilon \left({c}_{k-1}\right)}〈B\left({c}_{1},{c}_{2}\cdots ,{c}_{k-1}-\epsilon \left({c}_{k-1}\right),0\right)〉$

$\begin{array}{l}〈B\left({c}_{1},{c}_{2},\cdots ,{c}_{k-1},0\right)〉\\ =\left[{\left(-{A}^{2}-{A}^{-2}\right)}^{k}{\left(-{A}^{3}\right)}^{{c}_{1}+{c}_{2}+\cdots +{c}_{k-2}}\right]{A}^{\epsilon \left({c}_{k-1}\right)}{\left(-{A}^{3}\right)}^{{c}_{k-1}-\epsilon \left({c}_{k-1}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{A}^{-\epsilon \left({c}_{k-1}\right)}〈B\left({c}_{1},{c}_{2},\cdots ,{c}_{k-1}-\epsilon \left({c}_{k-1}\right),0\right)〉\end{array}$

$\begin{array}{l}〈B\left({c}_{1},{c}_{2},\cdots ,{c}_{k-1},0\right)〉\\ =\left[{\left(-{A}^{2}-{A}^{-2}\right)}^{k}{\left(-{A}^{3}\right)}^{{c}_{1}+{c}_{2}+\cdots +{c}_{k-2}}\right]\underset{i=1}{\overset{|{c}_{k-1}|}{\sum }}{A}^{\epsilon \left({c}_{k-1}\right)\left(2-i\right)}{\left(-{A}^{3}\right)}^{{c}_{k-1}-i\epsilon \left({c}_{k-1}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{A}^{-{c}_{k-1}}〈B\left({c}_{1},{c}_{2},\cdots ,0,0\right)〉\end{array}$

$\begin{array}{l}〈B\left({c}_{1},{c}_{2},\cdots ,{c}_{k-2},0\right)〉=\left[{\left(-{A}^{2}-{A}^{-2}\right)}^{k}{\left(-{A}^{3}\right)}^{{c}_{1}+{c}_{2}+\cdots +{c}_{k-3}}\right]\underset{i=1}{\overset{|{c}_{k-2}|}{\sum }}{A}^{\epsilon \left({c}_{k-2}\right)\left(2-i\right)}{\left(-{A}^{3}\right)}^{{c}_{k-2}-i\epsilon \left({c}_{k-2}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{A}^{-{c}_{k-2}}〈B\left({c}_{1},{c}_{2},\cdots ,0,0\right)〉\end{array}$

$\begin{array}{l}〈B\left({c}_{1},0,0,\cdots ,0\right)〉\\ ={\left(-{A}^{2}-{A}^{-2}\right)}^{k}{A}^{\epsilon \left({c}_{1}\right)}{\left(-{A}^{3}\right)}^{{c}_{1}-\epsilon \left({c}_{1}\right)}+{A}^{-\epsilon \left({c}_{1}\right)}〈B\left({c}_{1}-\epsilon \left({c}_{1}\right),0,\cdots ,0\right)〉\\ ={\left(-{A}^{2}-{A}^{-2}\right)}^{k}\underset{i=1}{\overset{|{c}_{1}|}{\sum }}{A}^{\epsilon \left({c}_{1}\right)\left(2-i\right)}{\left(-{A}^{3}\right)}^{{c}_{1}-i\epsilon \left({c}_{1}\right)}+{A}^{-{c}_{1}}〈B\left(0,0,\cdots ,0\right)〉\end{array}$

$\begin{array}{l}〈B\left({c}_{1},{c}_{2},\cdots ,{c}_{k}\right)〉\\ ={\left(-{A}^{2}-{A}^{-2}\right)}^{k}\underset{u=1}{\overset{k}{\sum }}\left[{\left(-{A}^{3}\right)}^{{c}_{1}+{c}_{2}+\cdots +{c}_{u-1}}{A}^{-{c}_{u+1}-{c}_{k+2}-\cdots -{c}_{k}}\underset{i=1}{\overset{|{c}_{u}|}{\sum }}{A}^{\epsilon \left({c}_{u}\right)\left(2-i\right)}{\left(-{A}^{3}\right)}^{{c}_{u}-i\epsilon \left({c}_{u}\right)}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{A}^{-{c}_{1}-{c}_{2}-\cdots -{c}_{k}}〈B\left(0,0,\cdots ,0\right)〉\end{array}$

$u=1$ 时，上式中的 ${\left(-{A}^{3}\right)}^{{c}_{1}+{c}_{2}+\cdots +{c}_{u-1}}{A}^{-{c}_{u+1}-{c}_{u+2}-\cdots -{c}_{k}}$ 仅剩下 ${A}^{-{c}_{2}-{c}_{3}-\cdots -{c}_{k}}$

$u=k$ 时，上式中的 ${\left(-{A}^{3}\right)}^{{c}_{1}+{c}_{2}+\cdots +{c}_{u-1}}{A}^{-{c}_{u+1}-{c}_{u+2}-\cdots -{c}_{k}}$ 仅剩下 ${\left(-{A}^{3}\right)}^{{c}_{1}+{c}_{2}+\cdots +{c}_{k-1}}$

i) 计算 ${X}_{i}\left(i=1,2,\cdots ,n\right)$ 部分的拧数

${\delta }_{i-1}=+1$${c}_{i-1}$ 为奇数、 ${\delta }_{i}=+1$ 时，如图所示，有 $\omega \left({X}_{i}\right)=0$

${\delta }_{i-1}=+1$${\delta }_{i-1}=-1$${\delta }_{i-1}=-1$${\delta }_{i-1}=+1$${\delta }_{i-1}=+1$${\delta }_{i-1}=-1$${\delta }_{i-1}=-1$、都有 $\omega \left({X}_{i}\right)=0$

ii) 计算所有 ${c}_{i}\left(i=1,2,\cdots ,n\right)$ 部分的拧数

${\delta }_{i}=+1$${c}_{i}>0$ 时，如图所示， ${c}_{i}$ 中每一个交叉点处都是+1，所以有 $\omega \left({c}_{i}\right)={c}_{i}$

$X\left(B\left({\delta }_{1},{c}_{1},{\delta }_{2},{c}_{2},\cdots ,{\delta }_{n},{c}_{n}\right)\right)={\left(-{A}^{3}\right)}^{-\underset{i=1}{\overset{n}{\sum }}{c}_{i}}〈B\left({c}_{1},{c}_{2},\cdots ,{c}_{n}\right)〉$

The Jones Polynomials of Class of Knots[J]. 应用数学进展, 2019, 08(05): 1024-1035. https://doi.org/10.12677/AAM.2019.85117

1. 1. 姜伯驹. 绳圈的数学[M]. 大连: 大连理工大学出版社, 2011: 53-69.

2. 2. 郑淑芳. 几何结构在Brunnian链环的琼斯多项式计算上的应用[D]: [硕士学位论文]. 长春: 东北师范大学, 2009.

3. 3. Adams, C.C. (2004) The Knot Book. W.H. Freeman and Company, New York.

4. 4. 戈鋆. 排叉链环的行列式[J]. 厦门大学学报, 2015, 54(6): 438-479.

5. 5. Jin, X. and Zhang, F. (2004) Zeros of the Jones Polynomials for Families of Pretzel Links. Physica, 333, 183-196. https://doi.org/10.1016/j.physa.2003.10.085

6. 6. Chan, T.H. (2000) Polynomials for a Family Brunnian Links. Journal of Knot Theory and Its Ramifications, 9, 587-609. https://doi.org/10.1142/S0218216500000335

NOTES

*通讯作者