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• # The Physics of Racing, Part 8: Simulating Car Dynamics with a Computer Program

Math and Physics

physicist and member of
No Bucks Racing Club

P.O. Box 662
Burbank, CA 91503

This month, we begin writing a computer program to simulate the physics of racing. Such a program is quite an ambitious one. A simple racing video game, such as "Pole Position," probably took an expert programmer several months to write. A big, realistic game like "Hard Drivin" probably took three to five people more than a year to create. The point is that the topic of writing a racing simulation is one that we will have to revisit many times in these articles, assuming your patience holds out. There are many just physics' topics still to cover too, such as springs and dampers, transients, and thermodynamics. Your author hopes you will find the computer programming topic an enjoyable sideline and is interested, as always, in your feedback.

We will use a computer programming language called Scheme. You have probably encountered BASIC, a language that is very common on personal computers. Scheme is like BASIC in that it is interactive. An interactive computer language is the right kind to use when inventing a program as you go along. Scheme is better than BASIC, however, because it is a good deal simpler and also more powerful and modern. Scheme is available for most PCs at very modest cost (MIT Press has published a book and diskette with Scheme for IBM compatibles for about \$40; I have a free version for Macintoshes). I will explain everything we need to know about Scheme as we go along. Although I assume little or no knowledge about computer programming on your part, we will ultimately learn some very advanced things.

The first thing we need to do is create a data structure that contains the mathematical state of the car at any time. This data structure is a block of computer memory. As simulated time progresses, mathematical operations performed on the data structure simulate the physics. We create a new instance of this data structure by typing the following on the computer keyboard at the Scheme prompt:

(new-race-car)This is an example of an expression. The expression includes the parentheses. When it is typed in, it is [font="Courier New"]evaluated[/font] immediately. When we say that Scheme is an interactive programming language, we mean that it evaluates expressions immediately. Later on, I show how we define this expression. It is by defining such expressions that we write our simulation program.

Everything in Scheme is an expression (that's why Scheme is simple). Every expression has a value. The value of the expression above is the new data structure itself. We need to give the new data structure a name so we can refer to it in later expressions:

(define car-161 (new-race-car))This expression illustrates two Scheme features. The first is that expressions can contain sub-expressions inside them. The inside expressions are called nested. Scheme figures out which expressions are nested by counting parentheses. It is partly by nesting expressions that we build up the complexity needed to simulate racing. The second feature is the use of the special Scheme word [font="Courier New"]define[/font]. This causes the immediately following word to become a stand-in synonym for the value just after. The technical name for such a stand-in synonym is variable. Thus, the expression [font="Courier New"]car-161[/font], wherever it appears after the [font="Courier New"]define[/font] expression, is a synonym for the data structure created by the nested expression [font="Courier New"](new-race-car)[/font].

We will have another data structure (with the same format) for [font="Courier New"]car-240[/font], another for [font="Courier New"]car-70[/font], and so on. We get to choose these names to be almost anything we like . So, we would create all the data structures for the cars in our simulation with expressions like the following:

(define car-161 (new-race-car)) (define car-240 (new-race-car)) (define car-70 (new-race-car))The state of a race car consists of several numbers describing the physics of the car. First, there is the car's position. Imagine a map of the course. Every position on the map is denoted by a pair of coordinates, and . For elevation changes, we add a height coordinate, . The position of the center of gravity of a car at any time is denoted with expressions such as the following:

(race-car-x car-161) (race-car-y car-161) (race-car-z car-161)Each of these expressions performs data retrieval on the data structure [font="Courier New"]car-161[/font]. The value of the first expression is the coordinate of the car, etc. Normally, when running the Scheme interpreter, typing an expression simply causes its value to be printed, so we would see the car position coordinates printed out as we typed. We could also store these positions in another block of computer memory for further manipulations, or we could specify various mathematical operations to be performed on them.

The next pieces of state information are the three components of the car's velocity. When the car is going in any direction on the course, we can ask "how fast is it going in the direction, ignoring its motion in the and directions?" Similarly, we want to know how fast it is going in the direction, ignoring the and directions, and so on. Decomposing an object's velocity into separate components along the principal coordinate directions is necessary for computation. The technique was originated by the French mathematician Descartes, and Newton found that the motion in each direction can be analyzed independently of the motions in the other directions at right angles to the first direction.

The velocity of our race car is retrieved via the following expressions:

(race-car-vx car-161) (race-car-vy car-161) (race-car-vz car-161)To end this month's article, we show how velocity is computed. Suppose we retrieve the position of the car at simulated time and save it in some variables, as follows:

(define x1 (race-car-x car-161)) (define y1 (race-car-y car-161)) (define z1 (race-car-z car-161))and again, at a slightly later instant of simulated time, :

(define x2 (race-car-x car-161)) (define y2 (race-car-y car-161)) (define z2 (race-car-z car-161))We have used [font="Courier New"]define[/font] to create some new variables that now have the values of the car's positions at two times. To calculate the average velocity of the car between the two times and store it in some more variables, we evaluate the following expressions:

(define vx (/ (- x2 x1) (- t2 t1))) (define vy (/ (- y2 y1) (- t2 t1))) (define vz (/ (- z2 z1) (- t2 t1)))The nesting of expressions is one level deeper than we have seen heretofore, but these expressions can be easily analyzed. Since they all have the same form, it suffices to explain just one of them. First of all, the [font="Courier New"]define[/font] operation works as before, just creating the variable [font="Courier New"]vx[/font] and assigning it the value of the following expression. This expression is

(/ (- x2 x1) (- t2 t1))In normal mathematical notation, this expression would read and in most computer languages, it would look like this:

(x2 - x1) / (t2 - t1) We can immediately see this is the velocity in the direction: a change in position divided by the corresponding change in time. The Scheme version of this expression looks a little strange, but there is a good reason for it: consistency. Scheme requires that all operations, including everyday mathematical ones, appear in the first position in a parenthesized expression, immediately after the left parenthesis. Although consistency makes mathematical expressions look strange, the payback is simplicity: all expressions have the same form. If Scheme had one notation for mathematical expressions and another notation for non-mathematical expressions, like most computer languages, it would be more complicated. Incidentally, Scheme's notation is called Polish notation. Perhaps you have been exposed to Hewlett-Packard calculators, which use reverse Polish, in in which the operator always appears in the last position. Same idea, and advantages, as Scheme, only reversed.

So, to analyze the expression completely, it is a division expression

(/ ...) whose two arguments are nested subtraction expressions

(- ...) (- ...) The whole expression has the form

(/ (- ...) (- ...))which, when the variables are filled in, is

(/ (- x2 x1) (- t2 t1))After a little practice, Scheme's style for mathematics becomes second nature and the advantages of consistent notation pay off in the long run.

Finally, we should like to store the velocity values in our data structure. We do so as follows:

(set-race-car-vx! car-161 vx) (set-race-car-vy! car-161 vy) (set-race-car-vz! car-161 vz) `The [font="Courier New"]set[/font] operations change the values in the data structure named [font="Courier New"]car-161[/font]. The exclamation point at the end of the names of these operations doesn't do anything special. It's just a Scheme idiom for operations that change data structures.

[hr]
converted by: rck@home.net
Thu Sep 29 14:10:24 PDT 1994

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# Tables in a plain text zettelkasten

Hi!

Quite often when I take notes I need tables, but so far I haven't found a satisfying method to include them in my zettelkasten system. There are several options, each with serious disadvantages:

• Option 1: Include Markdown table on a zettel directly, edit it with @ctietze's TableFlip.

• [PRO] Markdown tables resilient, no heavy reliance on any app that might seize to exist after a while for just reading them.
• [CON] Usually the Markdown tables will be too wide for a zettel, which makes it look very messy, almost unreadable.
• [CON] TableFlip is a quite limiting. You can't drag columns or rows around, sort rows alphabetically, have a line break within a cell, etc.
• Option 2: Create and edit Markdown table with TableFlip in separate file, link to the file from zettel.

• [PRO] Markdown tables resilient, no heavy reliance on any app that might seize to exist after a while for just reading them.
• [PRO] Size of table can be whatever I like.
• [CON] TableFlip is a quite limiting. You can't drag columns or rows around, sort rows alphabetically, have a line break within a cell, etc.
• Option 3: Use Excel (or other app like OmniOutliner, Numbers) to create and edit table in separate file, link to the file from zettel.

• [PRO] Most important functionality included in such apps, e.g. dragging colums and rows around. You can even include images and indent and fold rows.
• [CON] Tables can't be shared easily with people who don't have the app.
• [CON] Relies on app being available in the future, whilst it would be nice to get rid of it. In case it gets abandoned, there is the option to export table to .xlsx, which can be imported by Google Sheets. But: Formatting usually not nice, some functions might not to be supported, etc.
• Option 4: Use Google sheet and link to it from zettel.

• [PRO] Important functionality included: Rows and columns can be dragged around, hidden, links can be included, etc.
• [PRO] Can be shared easily.
• [CON] Data on Google servers, so relies on internet connection and not very protected.
• [CON] Relies on Google sheets being available in the future. Could be exported to .xlsx, Open Office or PDF if that becomes problematic.

How do you do it? Are there any tools I'm not aware of yet, e.g. other Markdown table editors with more functionality than TableFlip? And @ctietze: How is the progress in the development of TableFlip? More features coming soon?

• Option 1 (built-in iA Writer table), and Option 3 when the table stops being just a table and becomes a spreadsheet with calculations and custom formatting.

• @Splattack: Hm, the built in option in iA Writer wouldn't be enough for me as far as I can see – that's even less functionality than TableFlip has to offer...

• @Vinho Thanks for the hint on Tableflip - very nice!

• @Splattack said:
Option 1 (built-in iA Writer table), and Option 3 when the table stops being just a table and becomes a spreadsheet with calculations and custom formatting.

My position is this. I use very simple tables in Markdown in the Archive. Spreadsheets in Emacs.

But there is a good point to make that for quite some professions spreadsheets should be in the Zettelkasten from a methodological perspective.

I am a Zettler

• @Sascha said:

My position is this. I use very simple tables in Markdown in the Archive. Spreadsheets in Emacs.

But there is a good point to make that for quite some professions spreadsheets should be in the Zettelkasten from a methodological perspective.

It depends on whether you want your ZK to contain mostly ideas or also a lot of data. Of course, spreadsheets shine when storing, sorting and displaying data. You could also store ideas in a spreadsheet, but it would quickly get unwieldy if too large.

So I tend to follow that same practice as you - store my data mostly (not exclusively) outside my ZK and make reference to a relevant file name; store ideas within my ZK.

• @GeoEng51 said:

@Sascha said:

My position is this. I use very simple tables in Markdown in the Archive. Spreadsheets in Emacs.

But there is a good point to make that for quite some professions spreadsheets should be in the Zettelkasten from a methodological perspective.

It depends on whether you want your ZK to contain mostly ideas or also a lot of data. Of course, spreadsheets shine when storing, sorting and displaying data. You could also store ideas in a spreadsheet, but it would quickly get unwieldy if too large.

So I tend to follow that same practice as you - store my data mostly (not exclusively) outside my ZK and make reference to a relevant file name; store ideas within my ZK.

To me, this is not an issue of data vs ideas. For example, when describing an experiment I want to get close to the data as possible. This is: Accurately storing information of the experiment with as little interpretation as possible and covering the interpretation's conditions.

A biochemist might want to use his ZK to store data of his/her experiments.

I am a Zettler

• edited December 2020

@Sascha Didn't know that Emacs can handle tables as well – maybe I should look into it again... Don't really like it unfortunately...

The main situation in which I have needed tables in the zettelkasten so far is when I want to compare different versions of something under several aspects. Examples:

• I want to buy a new hoover and compare the different products on the market regarding price, longevity, cleaning ability, loudness, filter, dustbag volume, warranty, etc.
• I need to decide which varieties of cucumbers to grow next year and compare different varieties regarding their size, taste, price of seeds, whether the seeds are organic or not, whether the varieties are open-pollinated or hybrids, what other growers recommend them, etc.

In these situations, I would like the table editor to have the following features:

• Drag and drop columns or rows around (to move the most interesting options to the left or top and create a "ranking")
• Clickable links to image files or to where I can buy the items online
• Line breaks within cells to keep them compact (I often make longer comments in a cell).
• Option to sort rows/columns alphabetically
Post edited by Vinho on
• @Vinho

Didn't know that Emacs can handle tables as well – maybe I should look into it again... Don't really like it unfortunately...

Nobody likes Emacs. People who use it are just in an abusive relationship with this beast.

However, anything is possible in Emacs. So, any question that goes something like "Can Emacs.." the answer is: "Yes, there are 12323 possible solutions. 2 might be actionable without suffering weeks of Elisp torture."

I am a Zettler

• edited December 2020

when deciding if a table should be used in a Zettelkasten size does not matter (that's what she said). I haven't used tables in my ZK yet, because it is so fiddly to create them in plain text. When looking at tables in my old note taking (prior to ZKM) i find myself with a dilemma. Sometimes a table is both an essential reference but simultaneously a candidate for being poorly connected.

for digital Zettelkasten, how about disabling word wrap in structure notes? Do we need limited width when talking structure? I don't think so.

when thicc tables are applied on content i would try to use another method to represent it. This might not be possible, though, but i would be hesitant to allow any table to overflow the width of the body.

my first Zettel uid: 202008120915

• how about disabling word wrap in structure notes?

p.s.: this is not a rhetorical question. I accidentally made it into one but am interested what others think about it ¯(ツ)/¯

my first Zettel uid: 202008120915

• I toyed with the idea to have more horizontal screen real estate for tables etc, but it'll take a long while before I can even experiment with that because the whole text rendering thingie would have to be customized. It's on my long list of cool things

Author at Zettelkasten.de • https://christiantietze.de/

• edited December 2020

@ctietze said:
I toyed with the idea to have more horizontal screen real estate for tables etc, but it'll take a long while before I can even experiment with that because the whole text rendering thingie would have to be customized. It's on my long list of cool things

That would indeed be a cool thing, but for now I'd be quite happy with storing the table in a separate (plain text) file and being able to edit it as I want with a plain text table editor that shows the table cleanly in its whole width.
After having played around with it a bit more, the main remaining problem with plain text table editors that I have is that line breaks within a table cell seem to be impossible within the editor itself. There is the option to add a <br> tag, which makes preview tools like Marked 2 insert line breaks, but I'd like to have the line breaks in the editor itself. Is there any way to do this (and if yes, are you planning it in the foreseeable future for TableFlip)? As far as I can see, editing tables in Emacs also wouldn't solve this problem...

• I'm afraid the concept of plain text tables doesn't work with multi-line text cells in principle. There's only one kind of line break in a text file, no matter where you place it. Editors can only add conveniences to let you edit multiple lines of aligned cell contents as if it was 1 piece of text.

It's like the rowspan attribute in HTML tables. See here for an illustrative example: https://www.w3schools.com/Tags/tryit.asp?filename=tryhtml_td_rowspan

You can get that in HTML because there's syntax to group cells across line breaks, but that's not the visual format you want -- you're kinda stuck with this request, I'm afraid

Author at Zettelkasten.de • https://christiantietze.de/

• @ctietze: Hm, shit... But thanks for pointing that out!

• Instead of integrating tables in a single note, here‘s another idea (that I‘d also like to implement for my own app):

Imagine that each "table row" is simply a separate Zettel note. Within each of these notes, you can assign custom attributes/properties/metadata to the note. These custom attributes would be string-based key-value pairs, defined either in a YAML metadata block, or maybe simply as a tag. In MultiMarkdown, this could be something like [@key:value], e.g. [@maturity:draft] or [@maturity:published].

Now, imagine that your Zettelkasten app has a note list which can display multiple columns. I.e., next to the “note title” column, you’d be able to map additional columns to attribute keys. The attribute key (e.g. “maturity”) would become the column title, and the column rows would list just the attribute values (in the above example, “draft” and “published”).

With such a setup, you could:

• search your notes archive to assemble the “rows” of notes to be compared
• show/hide columns for the attributes you’re interested in
• drag+drop columns to reorder your attribute columns by priority
• click column header(s) to sort by attribute value
• change column width to have longer attribute value text display in full, or to wrap automatically.

Export to common table formats would need to be handled by a script, though. But the scripting interface could give access to these custom attributes.

With this approach, you’d benefit from “rows” being regular Zettel notes, and your Zettelkasten app being the “table editor”. And this approach could be used for all kinds of analysis.

format: plain


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# How do I find the domain of the function f(x)=5x^2+2x-1?

A polynomial function has as domain the whole set of $\mathbb{R}$, so:
$\left(- \infty , + \infty\right)$.


text:
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.

QUESTION

# Which stereoisomer of 3-hexene forms a meso compound when it reacts with Br_2 ?

trans-Hex-3-ene forms a meso compound when it reacts with bromine.

The bromination of an alkene involves the formation of an intermediate cyclic bromonium ion, followed by the anti-addition of a bromide ion to form the product.

The rigidity of the ring controls the stereochemistry of the reaction.

This means that the bromination of a trans alkene gives a meso dibromide.

For example, the bromination of trans-but-2-ene yields a meso compound.

Similarly, the bromination of hex-3-ene gives meso-2,3-dibromohexane as the product.

Attack of "Br"^- at C-3 gives (3S,4R)-3,4-dibromohexane.

Attack of "Br"^- at C-4 gives (3R,4S)-3,4-dibromohexane.

These two structures are superimposable on each other, so the product is a meso compound.


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# Other built-in calculators¶

## TIP3P¶

class ase.calculators.tip3p.TIP3P(rc=5.0, width=1.0)[source]

TIP3P potential.

rc: float

width: float

Width for cutoff function for Coulomb part.

## TIP4P¶

class ase.calculators.tip4p.TIP4P(rc=7.0, width=1.0)[source]

TIP4P potential for water.

https://doi.org/10.1063/1.445869

Requires an atoms object of OHH,OHH, … sequence Correct TIP4P charges and LJ parameters set automatically.

Virtual interaction sites implemented in the following scheme: Original atoms object has no virtual sites. When energy/forces are requested:

• virtual sites added to temporary xatoms object

• energy / forces calculated

• forces redistributed from virtual sites to actual atoms object

This means you do not get into trouble when propagating your system with MD while having to skip / account for massless virtual sites.

This also means that if using for QM/MM MD with GPAW, the EmbedTIP4P class must be used.

## Lennard-Jones¶

class ase.calculators.lj.LennardJones(**kwargs)[source]

Lennard Jones potential calculator

The fundamental definition of this potential is a pairwise energy:

u_ij = 4 epsilon ( sigma^12/r_ij^12 - sigma^6/r_ij^6 )

For convenience, we’ll use d_ij to refer to “distance vector” and r_ij to refer to “scalar distance”. So, with position vectors $$r_i$$:

r_ij = | r_j - r_i | = | d_ij |

The derivative of u_ij is:

d u_ij / d r_ij
= (-24 epsilon / r_ij) ( sigma^12/r_ij^12 - sigma^6/r_ij^6 )

We can define a pairwise force

f_ij = d u_ij / d d_ij = d u_ij / d r_ij * d_ij / r_ij

The terms in front of d_ij are often combined into a “general derivative”

du_ij = (d u_ij / d d_ij) / r_ij

The force on an atom is:

f_i = sum_(j != i) f_ij

There is some freedom of choice in assigning atomic energies, i.e. choosing a way to partition the total energy into atomic contributions.

We choose a symmetric approach:

u_i = 1/2 sum_(j != i) u_ij

The total energy of a system of atoms is then:

u = sum_i u_i = 1/2 sum_(i, j != i) u_ij

The stress can be written as ( $$(x)$$ denoting outer product):

sigma = 1/2 sum_(i, j != i) f_ij (x) d_ij = sum_i sigma_i , with atomic contributions

sigma_i  = 1/2 sum_(j != i) f_ij (x) d_ij

Implementation note:

For computational efficiency, we minimise the number of pairwise evaluations, so we iterate once over all the atoms, and use NeighbourList with bothways=False. In terms of the equations, we therefore effectively restrict the sum over $$i != j$$ to $$j > i$$, and need to manually re-add the “missing” $$j < i$$ contributions.

Another consideration is the cutoff. We have to ensure that the potential goes to zero smoothly as an atom moves across the cutoff threshold, otherwise the potential is not continuous. In cases where the cutoff is so large that u_ij is very small at the cutoff this is automatically ensured, but in general, $$u_ij(rc) != 0$$.

In order to catch this case, this implementation shifts the total energy

u'_ij = u_ij - u_ij(rc)

which ensures that it is precisely zero at the cutoff. However, this means that the energy effectively depends on the cutoff, which might lead to unexpected results!

Parameters
• sigma (float) – The potential minimum is at 2**(1/6) * sigma, default 1.0

• epsilon (float) – The potential depth, default 1.0

• rc (float, None) – Cut-off for the NeighborList is set to 3 * sigma if None. The energy is upshifted to be continuous at rc. Default None

## Morse¶

class ase.calculators.morse.MorsePotential(**kwargs)[source]

Morse potential.

Default values chosen to be similar as Lennard-Jones.

Parameters
• epsilon (float) – Absolute minimum depth, default 1.0

• r0 (float) – Minimum distance, default 1.0

• rho0 (float) – Exponential prefactor. The force constant in the potential minimum is k = 2 * epsilon * (rho0 / r0)**2, default 6.0


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Riječnik

Odaberite jednu od ključnih riječi s lijeve strane ...

Linear AlgebraSingular Value Decomposition

Vrijeme čitanja: ~60 min

In this section we will develop one of the most powerful ideas in linear algebra: the singular value decomposition. The first step on this journey is the polar decomposition.

Polar decomposition

The Gram matrix of a square matrix is a useful tool for understanding the behavior of . Let's define the matrix to be , where is the diagonalization of and is the matrix obtained by taking the square root of the diagonal entries of . Then is and satisfies

as befits the notation .

The matrix is simpler to understand than because it is symmetric and positive semidefinite, yet it transforms space very similarly to : if , then

In other words, for all , the images of under and under have equal norm. If points are the same distance from the origin, then one may be mapped to the other by a about the origin.

Therefore, for each , there is an orthogonal transformation from the range of to the range of which sends to .

It can be shown that the orthogonal transformation mapping to is the same transformation for every . Furthermore, even if the range of is not all of , we can extend this orthogonal transformation to an orthogonal transformation on by arbitrarily mapping vectors in a basis of the orthogonal complement of the range of to the orthogonal complement of the range of . In this way, we obtain the polar decomposition:

Theorem (Polar Decomposition)
For any matrix , there exists an orthogonal matrix such that

This representation is useful because it represents an arbitrary square matrix as a product of matrices whose properties are easier to understand (the orthogonal matrix because it is distance- and angle-preserving, and the positive-definite matrix because it is orthogonally diagonalizable, by the Spectral Theorem.

Exercise
Let's explore a fast method of computing a polar decomposition . This method actually works by calculating and then recovering as (since this is computationally faster than calculating the matrix square root). We call the orthogonal part of and the symmetric part of .

We set and define the iteration

Let's see why this converges to .

1. Defining and using the equation , show that Use the prior step to explain why the 's all have the same orthogonal parts and have symmetric parts converging to the identity matrix.
Hint: consider the eigendecompositions of the symmetric parts. You may assume that the sequence defined by converges to 1 regardless of the starting value .

2. Write some code to apply this algorithm to the matrix

import numpy as np
A = np.array([[1,3,4],[7,-2,5],[-3,4,11]])
A = [1 3 4; 7 -2 5; -3 4 11]

and confirm that the resulting matrices and satisfy and .

Solution. Since both conjugation and inversion reverse the order of matrix products, we get

Since is orthogonal, , as . Since is symmetric, . So this is equal to

We see that the and have the same orthogonal part, and repeating the calculation shows that all the have the same orthogonal part. As for the symmetric parts , we see that

Let's see why this averaging process converges to the identity matrix. Symmetric matrices are diagonalizable by the Spectral Theorem, so suppose diagonalizes as . Then

Thus the 's converge to the matrix , where is the diagonal matrix whose th entry is the limit obtained when you start with and repeatedly apply the function . By the fact about this iteration given in the problem statement, we conclude that is the identity matrix. Therefore, the limit of as is equal to .

For example:

import numpy as np

def polar(A,n):
R = A
for i in range(n):
R = (R + np.linalg.inv(R.T))/2
return R, np.linalg.solve(R, A)

I = np.eye(3)
A = np.array([[1, 3, 4],[7, -2, 5], [-3, 4, 11]])
R, P = polar(A,100)
R.T @ R - I, P @ P - A.T @ A
using LinearAlgebra

function polar(A,n)
R = A
for i=1:n
R = (R + inv(R'))/2
end
R, R \ A
end

A = [1 3 4; 7 -2 5; -3 4 11]
R, P = polar(A,100)
R'*R - I, P^2 - A'*A

Both of the matrices returned on the last line have entries which are within of zero.

Exercise
Show that the product of two matrices with orthonormal columns has orthonormal columns.

Solution. If and , then .

The singular value decomposition

The polar decomposition tells us that any square matrix is almost the same as some symmetric matrix, and the spectral theorem tells us that a symmetric matrix is almost the same as a simple scaling along the coordinate axes. (In both cases, the phrase "almost the same" disguises a composition with an orthogonal transformation.) We should be able to combine these ideas to conclude that any square matrix is basically the same as a simple scaling along the coordinate axes!

Let's be more precise. Suppose that is a square matrix. The polar decomposition tells us that

for some orthogonal matrix . The spectral theorem tells us that for some orthogonal matrix and a diagonal matrix with nonnegative diagonal entries.

Combining these equations, we get

Since a product of orthogonal matrices is , we can define and obtain the singular value decomposition (SVD) of :

where and are orthogonal matrices.

We can visualize the decomposition geometrically by making a figure like the one shown below, which illustrates the successive effects of each map in the composition . If we draw grid lines on the second plot (just before is applied) and propagate those grid lines to the other plots by applying the indicated maps, then we endow the domain and range of with orthogonal sets of gridlines with mapping one to the other.

We can extend the singular value decomposition to rectangular matrices (that is, matrices which are not necessarily square) by adding rows or columns of zeros to a rectangular matrix to get a square matrix, applying the SVD to that square matrix, and then trimming the resulting matrix as well as either or (depending on which dimension of is smaller). We get a decomposition of the form where is an orthogonal matrix, is an orthogonal matrix, and is a rectangular diagonal matrix.

Theorem (Singular value decomposition)
Suppose that is an matrix. Then there exist orthogonal matrices and and a rectangular diagonal matrix such that

We call the a singular value decomposition (or SVD) of . The diagonal entries of are called the singular values of .

The diagonal entries of , which are the square roots of the eigenvalues of , are called the singular values of . The columns of are called left singular vectors, and the columns of are called right singular vectors.

Looking at the bottom half of the SVD figure, we see that the singular values of are the lengths of the semi-axes of the ellipsoid in obtained as the image under of the unit ball in . Moreover, the directions of these axes are the columns of , since they are the images under of the standard basis vectors. We will see an important application of this feature of the SVD in the probability chapter when we discuss principal component analysis.

As an example of how the singular value decomposition can be used to understand the structure of a linear transformation, we introduce the Moore-Penrose pseudoinverse of an matrix . We define to be , where is the matrix obtained by inverting each nonzero element of . The pseudoinverse is a swiss-army knife for solving the linear system :

• If is square and invertible, then
• If has no solution, then is the value of which minimizes (in other words, the closest thing to a solution you can get).
• If has multiple solutions, then is the solution with minimal norm.

Exercise
Show that has SVD . Find its Moore-Penrose pseudoinverse.

Solution. Let

and let , , and be the three matrices given in the problem statement.

We need to check that are orthogonal, and that . We can verify that are orthogonal by showing that their columns are orthogonal unit vectors. Equivalently, we may compute the products and and observe that they are identity matrices. Similarly, can be verified by hand or on a computer.

The formula for the Moore-Penrose pseudoinverse is

The matrix is obtained by inverting the nonzero elements on the diagonal of , and the transposing the resulting matrix.

With a little calculation, we arrive at

SVD and linear dependence

Linear dependence is numerically fragile: if the columns of a matrix (with more rows than columns) are linearly dependent, then perturbing the entries slightly by adding tiny independent random numbers is almost certain to result in a matrix with linearly independent columns. However, intuition suggests that subverting the principles of linear algebra in this way going to solve any real-world problems that emerge from linear dependence relationships among columns of a matrix.

This intuition is accurate, and it highlights the utility of having a generalization of the idea of linear independence which can quantify how close a list of vectors is to having linear dependence relationships, rather than remaining within the confines of the binary labels "linearly dependent" or "linearly independent". The singular value decomposition provides such a tool.

Exercise
Define a matrix with 100 rows and 5 columns, and do it in such a way that two of the five columns are nearly equal to some linear combination of the other three. Calculate the singular values of the matrix, and make a conjecture about how the number of approximate linear dependencies could have been detected from the list of singular values.

import numpy as np

Solution. We see that two of the singular values are much smaller than the other three. (Remember that you have to run the cell twice to get the plot to show.)

import numpy as np
import matplotlib.pyplot as plt
A = np.random.standard_normal((100,5))
A[:,3] = A[:,2] + A[:,1] + 1e-2*np.random.standard_normal(100)
A[:,4] = A[:,1] - A[:,0] + 1e-2*np.random.standard_normal(100)
plt.bar(range(5),np.linalg.svd(A)[1])
using Plots, LinearAlgebra
A = randn(100, 5)
A[:,4] = A[:,3] + A[:,2] + 1e-2*randn(100)
A[:,5] = A[:,2] - A[:,1] + 1e-2*randn(100)
bar(1:5, svdvals(A), label = "singular values")

We conjecture that very small singular values indicates that columns would need to be removed to obtain a matrix which does not have approximate linear dependence relationships among its columns.

In fact, the idea developed in this exercise is used by the NumPy function np.linalg.matrix_rank to calculate the rank of a matrix. Because of the roundoff errors associated with representing real numbers in memory on a computer, most matrices with float entries technically have full rank. Thus np.linalg.matrix_rank computes the singular value decomposition and returns the number of of the matrix which are larger than a given threshold. The threshold is adjustable, but one common setting is times the largest entry of the matrix times the largest dimension of the matrix.

In fact, the idea developed in this exercise is used by the Julia function rank to calculate the rank of a matrix. Because of the roundoff errors associated with representing real numbers in memory on a computer, most matrices with float entries technically have full rank. Thus rank computes the singular value decomposition and returns the number of of the matrix which are larger than a given threshold. The threshold is adjustable, but one common setting is times the largest entry of the matrix times the largest dimension of the matrix.

SVD and image compression

We close this section with a computational exercise illustrating another widely applicable feature of the singular value decomposition.

Exercise

• Show that if are the columns of , are the columns of , and are the diagonal entries of , then .

• The equation is useful for compression, because terms with sufficiently small singular value factors can be dropped and the remaining vectors and singular values can be stored using less space. Suppose that is a matrix—how many entries does have, and how many entries do , , , , , have in total?

• The Python code below creates a matrix with pixel values for the image shown. How many nonzero singular values does have? Explain how you can tell just from looking at the picture.

figure: img(src="/content/linear-algebra/images/zero.svg")

import numpy as np
import matplotlib.pyplot as plt

m = 80
n = 100
a = m // 8
b = m // 4
A = np.ones((m,n))

def pixel(i,j):
if (a <= i <= b or m-b <= i <= m-a) and a <= j <= n-a:
return 0
elif (a <= j <= b or n-b <= j <= n-a) and a <= i <= m-a:
return 0
return 1

A = np.array([[pixel(i,j) for i in range(1,m+1)] for j in range(1,n+1)])

U, Σ, V = np.linalg.svd(A)

plt.matshow(A)
using LinearAlgebra, Plots
m = 80
n = 100
a = m ÷ 8
b = m ÷ 4
A = ones(m,n)

function pixel(i,j)
if (a ≤ i ≤ b || m-b ≤ i ≤ m-a) && a ≤ j ≤ n - a
0
elseif (a ≤ j ≤ b || n-b ≤ j ≤ n-a) && a ≤ i ≤ m - a
0
else
1
end
end

A = [pixel(i,j) for i=1:m,j=1:n]

U, Σ, V = svd(A)
heatmap(A)
• Now add some noise to the image:
B = A +  0.05*np.linalg.standard_normal((m,n))
B = A + 0.05 * randn(m, n)

Display this new matrix , and also find the matrix obtained by keeping only the first three terms of for this matrix . Which looks more like the original image : (i) or (ii) the three-term approximation of ?

Hint: you can achieve this computationally either by setting some singular values to 0 or by indexing the matrices , , and appropriately. Also, you will need the function np.diagonal diagm to generate a diagonal matrix from the vector of values returned by svd.

import numpy as np
import matplotlib.pyplot as plt

Solution.

• Let We need to show that We will do this by first showing that for all

Now, for all By definition of matrix-vector product, is a linear combination of the columns of with weights given by Since is diagonal, it is not hard to see that the th column of is Using definition of matrix-vector product again, we find that the th weight is the dot product of the th row of and But the th row of is by definition, and thus Therefore,

where is being treated as a matrix for all

By linearity of matrix multiplication,

and thus for all Since for all it follows that for any matrix In particular, if is the identity matrix in we have

as required.

• has entries and combined have entries.

• It can be seen from the picture that has kinds of columns: one whose components are all dark, another whose components are light in the middle, and the other whose components are dark on the outside and in the middle with strips of light in between. These columns are clearly linearly independent, and thus has rank Therefore has non-zero singular values.

We can select only the first three terms by suitably indexing the vectors, as follows:

U, Σ, V = np.linalg.svd(B)
plt.matshow(U[:,:3] * np.diag(Σ[:3]) * V.T[:3,:])
U, Σ, V = svd(B)
heatmap(U[:,1:3] * diagm(Σ[1:3]) * V'[1:3,:])
Bruno


text: I would like to obtain the node strength of each of the nodes i.e., the sum of the edge weights adjacent to the node and also the weight distribution. Cancel. Where (i,j) represent an edge originating from ith … Adjacency Matrix (AM) is a square matrix where the entry AM[i][j] shows the edge's weight from vertex i to vertex j. Sum along the row. You need to give a much better explanation of what your input matrix represents and how to generate your not_an_adjacency_matrix output from it. Graphs, The Amsterdam, Netherlands: Adjacency Matrix Let us consider a graph in which there are N vertices numbered from 0 to N-1 and E number of edges in the form (i,j). Introduction to Graph Theory, 2nd ed. An adjacency matrix is always square. We can modify the previous adjacency lists and adjacency matrices to store the weights. Incidence matrix. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. Select and move objects by mouse or move workspace. Every network can be expressed mathematically in the form of an adjacency matrix (Figure 4). The notes written after class way what I wish I said. See the example below, the Adjacency matrix for the graph shown above. the lowest distance is . Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Adjacency Matrix If a graph has n vertices, we use n x n matrix to represent the graph. Matrix has wrong format. Let's assume the n x n matrix as adj[n][n]. Our project is now open source. Knowledge-based programming for everyone. Use comma "," as separator. New York: Dover, p. 218, 1985. adj[i][j] == 1 Adjacency Matrix. matrix.reshish.com is the most convenient free online Matrix Calculator. I read an article to do this but I couldn't set up its algorithm. Unlimited random practice problems and answers with built-in Step-by-step solutions. About project and look help page. We simply use a C++/Java native 2D array of … DFS traversal of a tree using recursion. Vertex enumeration, Select the initial vertex of the shortest path, Select the end vertex of the shortest path, The number of weakly connected components is, To ask us a question or send us a comment, write us at, Multigraph does not support all algorithms, Find shortest path using Dijkstra's algorithm. The number of connected components is . Matrix should be square. Stack Exchange Network. The adjacency matrix A is the one-hop matrix. §3.1.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Calculate number of nodes in all subtrees | Using DFS. In the previous post, we introduced the concept of graphs. Depending upon the application, we use either adjacency list or adjacency matrix but most of the time people prefer using adjacency list over adjacency matrix. with the result being returned as a sparse array. graph: The graph to convert. The number of weakly connected components is . where is the number or permutations of Leave extra cells empty to enter non-square matrices. The adjacency matrix, sometimes also called the connection matrix, of a simple labeled graph is a matrix with rows and columns labeled by graph vertices, with a 1 or 0 in position according Free Matrix Adjoint calculator - find Matrix Adjoint step-by-step This website uses cookies to ensure you get the best experience. Just type matrix elements and click the button. Matrix Calculator: A beautiful, free matrix calculator from Desmos.com. For a simple graph with no self-loops, the adjacency matrix must have 0s . Adjacency Matrix A graph G = (V, E) where v= {0, 1, 2, . Save. Incidence matrix. Adjacency matrix. For a simple graph with no self-loops, the adjacency matrix must have 0s on the diagonal. 24, Jul 18. What do you think about the site? Englewood Cliffs, NJ: Prentice-Hall, pp. I'm trying to write a simple program to calculate betweenness centrality . The complexity of Adjacency Matrix representation: The adjacency matrix representation takes O(V2) amount of space while it is computed. Adjacency Matrix. An adjacency matrix provides a useful representation of a graph that can be used to compute many properties by means of simple operations on matrices. In an adjacency matrix, there is a 1 at position (i, j) if there is an edge from vertex i to vertex j.Therefore, all the outgoing edges of vertex i are in row i.A sum of row i therefore gives you the outdegree of vertex i.. For a transposed adjacency matrix, sum along the columns. Set up incidence matrix. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Adjoint Matrix Calculator The calculator will find the adjoint (adjugate, adjunct) matrix of the given square matrix, with steps shown. The time complexity to calculate the number of edges in a graph whose information in stored in form of an adjacency matrix is ..... A. O(V) B. O(E 2) C. O(E) D. O(V 2) Biology Questions answers . The components of the matrix express whether the pairs of a finite set of vertices (also called nodes) are adjacent in the graph or not. adjacency matrices of the cycle graph . Subtree of all nodes in a tree using DFS . Enter text for each vertex in separate line, Setup adjacency matrix. At any step in the algorithm, the -entry in the adjacency matrix gives the algorithm's current estimate of the shortest path from to. So your output is not an adjacency matrix. We simply use a C++/Java native 2D array of … We represent the graph by using the adjacency list instead of using the matrix. Topological Indices and Related Descriptors in QSAR and QSPR. Graph Theory. Open image in browser or Download saved image. https://mathworld.wolfram.com/AdjacencyMatrix.html, Adjacency Matrices of Manipulable Can't this be used as a way to calculate the number of hamiltonian paths? Every network can be expressed mathematically in the form of an adjacency matrix (Figure 4). Definition of an Adjacency Matrix An adjacency matrix is defined as follows: Let G be a graph with "n" vertices that are assumed to be ordered from v 1 to v n. The n x n matrix A, in which a ij = 1 if there exists a path from v i to v j (Edges are not directional). Name (email for feedback) Feedback. or not. Weisstein, Eric W. "Adjacency Matrix." Devillers, J. and A. T. Balaban (Eds.). ( answer for part 3 with adjacency list not matrix please)Please , Answer for question (3) only (use adjacency list) using this code that was done for question 2: #include #include using namespace std; struct Node {int pt, weight; Node* next; So I'm going take our matrix and I'm going to … The adjacency matrix, also called the connection matrix, is a matrix containing rows and columns which is used to represent a simple labelled graph, with 0 or 1 in the position of (V i , V j) according to the condition whether V i and V j are adjacent or not. close. It is ignored for directed graphs. I have an adjacency matrix with the non zero elements indicating the weights of the link.The weights are decimals. I tried the following code: import networkx as nx G=nx. adjacency matrix is symmetric. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). 10, Aug 17. Matrix is incorrect. There are two cases in which we can generalize the definition, depending on whether edges are directional. if there is an edge from vertex i to j, mark adj[i][j] as 1. i.e. Adjacency Matrix (AM) is a square matrix where the entry AM[i][j] shows the edge's weight from vertex i to vertex j. You can vote up the ones you like or vote down the ones you don't like, and go to the original project or source file by following the links above each example. For unweighted graphs, we can set a unit weight = 1 for all edge weights. The #1 tool for creating Demonstrations and anything technical. 1. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. On the Help page you will find tutorial video. type: Gives how to create the adjacency matrix for undirected graphs. The best way to prepare for the ARE is with practice exams. 69-73, 2000. In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph.The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.. In fact, the following is also true. Use comma "," as separator. Random graph, random matrix, adjacency matrix, Laplacian matrix, largest eigenvalue, spectral distribution, semi-circle law, free convolution. Skiena, S. "Adjacency Matrices." This Demonstration uses the Floyd–Warshall algorithm to find the shortest-path adjacency matrix and graph. Matrix representation of a graph in memory. We follow a greedy approach, wherein we prioritize the edge with the minimum weight. Adjacency Matrix In graph theory, an adjacency matrix is nothing but a square matrix utilised to describe a finite graph. Be skeptical of all statements in these notes that can be made mathematically rigorous. The illustration above shows the possible In other words, start with the n × n zero matrix , put a 1 in ( i , j ) if there is an edge whose endpoints are v i and v j . Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. And this is using matrix multiplication. Leave extra cells empty to enter non-square matrices. Gordon and Breach, pp. Send. Calculating A Path Between Vertices. 2086. Chartrand, G. Introductory © Graph Online is online project aimed at creation and easy visualization of graph and shortest path searching. If a graph has n vertices, we use n x n matrix to represent the graph. To ask us a question or send us a comment, write us at . if there is an edge from vertex i to j, mark adj[i][j] as 1. i.e. In general, the entries of this matrix can be larger than 1, i.e. Name (email for feedback) Feedback. The adjacency matrix, also called as the connection matrix, is a matrix containing rows and columns which is used to represent a simple labelled graph, with 0 or 1 in the position of (V i , V j) according to the condition whether V i and V j are adjacent or not. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. To ask us a question or send us a comment, write us at . So I'm going take our matrix and I'm going to represent it a little bit more mathematically. Let's assume the n x n matrix as adj[n][n]. Save. Contributed by: Adithya Vekatesanand Ankit Goyal (January 2014) … MA: Addison-Wesley, pp. Use A + A*A to get non-zeros in A(i, j) if you can go from node i to node j in 2 or less steps. Send. West, D. B. 1. 22, Aug 17. Observation 2: The number of k-step sequences between vertex i and vertex j in a graph with adjacency matrix M is the (i, j)entryinMk. By using this website, you agree to our Cookie Policy. But if you have an adjacency matrix representation, there's actually a really cool way to solve the two-hop neighbor problem, not for a single vertex, but in fact for all the vertices in the graph. An adjacency matrix is a compact way to represent the structure of a finite graph.If a graph has n n n vertices, its adjacency matrix is an n × n n \times n n × n matrix, where each entry represents the number of edges from one vertex to another.. Our online exams will get you test day ready! In this tutorial, we are going to see how to represent the graph using adjacency matrix. Multigraph matrix contains weight of minimum edges between vertices. Each row corresponds to a species or reaction, and each column corresponds to a species or reaction. vertex labels. The calculator will find the adjoint (adjugate, adjunct) matrix of the given square matrix, with steps shown. Please, write what kind of algorithm would you like to see on this website? Using the adjacency matrix to find eigenvector centrality. Topological Indices and Related Descriptors in QSAR and QSPR. Full report. Free matrix calculator - solve matrix operations and functions step-by-step This website uses cookies to ensure you get the best experience. For methods and operations that require complicated calculations a … Adjacency list. https://mathworld.wolfram.com/AdjacencyMatrix.html. group order is given by. Matrix is incorrect. The adjacency matrix, sometimes also called the connection matrix, of a simple labeled graph is a matrix with rows and columns labeled by graph vertices, with a 1 or 0 in position according to whether and are adjacent or not. Representing Weighted Graphs. adjMaxtrix[i][j] = 1 when there is edge between Vertex i and Vertex j, else 0. An adjacency matrix is defined as follows: Let G be a graph with "n" vertices that are assumed to be ordered from v 1 to v n. The n x n matrix A, in which a ij = 1 if there exists a path from v i to v j a ij = 0 otherwise is called an adjacency matrix. Level with maximum number of nodes using DFS in a N-ary tree. Some properties and computations in graph theory can be carried out efficiently and elegantly by using a graph's adjacency matrix. These examples are extracted from open source projects. C program to implement Adjacency Matrix of a given Graph Last Updated : 21 May, 2020 Given a undirected Graph of N vertices 1 to N and M edges in form of 2D array arr[][] whose every row consists of two numbers X and Y which denotes that there is a edge between X and Y, the task is to write C program to create Adjacency Matrix of the given Graph . Adjacency Matrix Definition. The notes written before class say what I think I should say. Show Instructions. You may check out the related API usage on the sidebar. fix matrix. n-1} can be represented using two dimensional integer array of size n x n. int adj[20][20] can be used to store a graph with 20 vertices adj[i][j] = 1, indicates presence of edge between two vertices i and j.… Read More » fix matrix. 2015 - 2021, Find the shortest path using Dijkstra's algorithm. How to calculate adjacency matrix ?. By using the matrix representation of the network we can calculate network properties such as degree, and other … This might not be very efficient especially in the case where we have a lot of vertices and only a few of those vertices are connected to each other, which translates to a very sparse adjacency matrix. In these matrices the rows and columns are assigned to the nodes in the network and the presence of an edge is symbolised by a numerical value. Adjacency Matrix: Adjacency Matrix is 2-Dimensional Array which has the size VxV, where V are the number of vertices in the graph. An 'x' means that that vertex does not exist (deleted). There are two popular data structures we use to represent graph: (i) Adjacency List and (ii) Adjacency Matrix. The adjacency matrix M G = (m i  j) of G is defined as follows: M G is an n × n matrix such that m i  j = { 1 if { v i , v j } ∈ E 0 otherwise. count and automorphism graph with adjacency matrix M is the (i, j)entryinM 2. Adjoint Matrix Calculator. The adjacency matrix of a graph can be computed in the Wolfram Language using AdjacencyMatrix[g], The Adjacency Matrix and The nth Eigenvalue Daniel A. Spielman September 5, 2012 3.1 About these notes These notes are not necessarily an accurate representation of what happened in class. An adjacency matrix allows representing a graph with a V × V matrix M = [f(i, j)] where each element f(i, j) contains the attributes of the edge (i, j). What do you think about the site? adj[i][j] == 1 . Graph was saved. The following are 30 code examples for showing how to use networkx.adjacency_matrix(). Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … this is not an adjacency matrix. Get more help from Chegg. AdjacencyMatrix returns a square matrix whose rows and columns correspond to the vertices of a graph and whose elements a ij are non-negative integers that give the numbers of (directed) edges from vertex v i to vertex v j.An adjacency matrix provides a useful representation of a graph that can be used to compute many properties by means of simple operations on matrices. Determine the degree, do an adjacency list, an adjacency matrix and an incidence matrix on the undirected graph given below: P 70 . Adjacency Matrix. The relative centrality score of vertex can be defined as: where is a set of the neighbors of and is a constant. The illustration above shows adjacency matrices for particular labelings of the claw graph, cycle graph , and complete graph . As mentioned in this article, adjacency matrix requires more memory if implemented in a program due to its requirement to store the graph information in the form of an $$N \times N$$ matrix. close. SPECTRAL OF LAPLACIAN MATRICES 2087 geometrical and topological properties can be deduced for a large class of ran-dom graph ensembles, but the spectral properties of the random graphs are still uncovered to a large extent. Create graph and find the shortest path. Learn how an adjacency matrix can be used to calculate adjacent cells within magic squares in this video tutorial by Charles Kelly. With help of this calculator you can: find the matrix determinant, the rank, raise the matrix to a power, find the sum and the multiplication of matrices, calculate the inverse matrix. The algorithm is visualized by evolving the initial directed graph to a complete digraph in which the edge weight from vertex to vertex is the weight of the shortest path from to in the initial graph. Our service already supports these features: Find the shortest path using Dijkstra's algorithm, Adjacency matrix, Incidence Matrix. An 'x' means that that vertex does not exist (deleted). The matrix A2 = A*A has a non-zero in A(i, j), if it is possible to go from node i to node j in exactly two steps. 2. Also you can create graph from adjacency matrix. If the graph is undirected (i.e. The adjacency matrix of G = (V,E) is the n ⨯ n matrix A indexed by V, whose (u, v)-entry is defined as A uv = {1 if uv ∈ E, undefined 0 if uv ∉ E. Recall that a matrix is said to be reducible if it can be transformed to the form A = [A ' B 0 A "], Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Click to workspace to add a new vertex. Explore anything with the first computational knowledge engine. All the basic matrix operations as well as methods for solving systems of simultaneous linear equations are implemented on this site. If the graph is undirected (i.e. muhammad ismat … Floyd-Warshall Algorithm on Adjacency Matrices and Directed Graphs. In time of calculation we have ignored the edges direction. Figure 1 and 2 show the adjacency matrix representation of a directed and undirected graph. Learn more about adjacency matrix Using the notation from the Wikipedia entry, this matrix contains the mutual information estimates I(X;Y) AdjacencySymmetricUncertainty: is a weighted adjacency matrix whose … Join the initiative for modernizing math education. 81-85, 1990. Logical Representation: Adjacency List Representation: Animation Speed: w: h: Reading, This reduces the overall time complexity of the process. Hints help you try the next step on your own. Use comma "," as separator. We need to calculate the minimum cost of traversing the graph given that we need to visit each node exactly once. Use comma "," as separator. . Save Graph Image. By using this website, you agree to our Cookie Policy. Possible values: upper: the upper right triangle of the matrix is used, lower: the lower left triangle of the matrix is used.both: the whole matrix is used, a symmetric matrix … On this page you can enter adjacency matrix and plot graph In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph.The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.. Your algorithm was sent to check and in success case it will be add to site. the lowest distance is . Follow this link to see it. Adjacency Matrix: Adjacency Matrix is 2-Dimensional Array which has the size VxV, where V are the number of vertices in the graph. But if you have an adjacency matrix representation, there's actually a really cool way to solve the two-hop neighbor problem, not for a single vertex, but in fact for all the vertices in the graph. Just type matrix elements and click the button. I got stuck at getting an output . Matrix Multiplication Calculator Here you can perform matrix multiplication with complex numbers online for free. Practice online or make a printable study sheet. However matrices can be not only two-dimensional, but also one-dimensional (vectors), so that you can multiply vectors, vector by matrix and vice versa. Cancel. Diameter of a tree using DFS. In the adjacency list, instead of storing the only vertex, we can store a pair of numbers one vertex and other the weight. From MathWorld--A Wolfram Web Resource. if vertex is linked to vertex , and otherwise. The matrix indicates which species and reactions are involved as reactants and products: For a given graph with vertices let be the adjacency matrix, i.e. In this post, we discuss how to store them inside the computer. Question 4 Explanation: As V entries are 0, a total of V 2-V entries are to be examined. If the graph has some edges from i to j vertices, then in the adjacency matrix at i th row and j th column it will be 1 (or some non-zero value for weighted graph), otherwise that place will hold 0. And this is using matrix multiplication. Graph(a) # a is the adjacency matrix. You can use the "update number" slider to see how the estimated distances are updated. adjMaxtrix[i][j] = 1 when there is edge between Vertex i and Vertex j, else 0. Fig. An adjacency matrix is an N-by-N matrix, where N equals the total number of species and reactions in a model. adjacency matrix code I 06, Oct 18. help. The number of weakly connected components is . In particular, the number of distinct ( answer for part 3 with adjacency list not matrix please)Please , Answer for question (3) only (use adjacency list) using this code that was done for question 2: #include #include using namespace std; struct Node {int pt, weight; Node* next; It’s easy to implement because removing and adding an edge takes only O(1) time. You need to give a much better explanation of what your input matrix represents and how to generate your not_an_adjacency_matrix output from it. Answers with built-in step-by-step solutions unit weight = 1 for all edge.. Equations are implemented on adjacency matrix calculator site code i this Demonstration uses the Floyd–Warshall algorithm find. Is the number of species and reactions in a graph G = ( V E... X ' means that that vertex does not exist ( deleted ) has the size VxV, where equals! The form of an adjacency matrix is a ( 0,1 ) -matrix with zeros on diagonal... Removing and adding an edge originating from ith … an adjacency matrix graph. Greedy approach, wherein we prioritize the edge with the non zero elements indicating the weights for Demonstrations. An adjacency matrix and graph Theory: adjacency matrix is an N-by-N,! Minimum edges between vertices | using DFS discuss how to use networkx.adjacency_matrix ( ) it will be add site...: ( i, j ) represent an edge from vertex i to j, mark adj n. Be used to calculate the number of nodes in a model be mathematically... To be examined we introduced the concept of graphs its algorithm simple to! Because removing and adding an edge from vertex i and vertex j, else 0 matrix calculator, total! ] == 1 as a way to prepare for the graph can be larger than 1, i.e )., E ) where v= { 0, 1, i.e matrix ( Figure ). Undirected graphs tool for creating Demonstrations and anything technical take our matrix and graph by mouse or workspace. Of species and reactions in a model the total number of distinct adjacency matrices and Directed graphs a way calculate. Generate your not_an_adjacency_matrix output from it E ) where v= { 0 1... Weights are decimals graph G = ( V, E ) where v= { 0, a total V! Skip the multiplication sign, so  5x  is equivalent to  5 * x  n n... # a is the one-hop matrix n matrix to represent the graph matrix Figure. Like to see how the estimated distances are updated QSAR and QSPR 1 2! J ] == 1 with built-in step-by-step solutions unit weight = 1 when there edge. Going to see on this site is computed matrix with the minimum weight with step-by-step. Total number of k-length paths in a N-ary tree indicating the weights of the link.The weights are decimals much explanation... A species or reaction. ) 30 code examples for showing how to use networkx.adjacency_matrix ( ) 1. Are is with practice exams number or permutations of vertex can be defined:. 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The next step on your own: find the shortest path searching: as V are... Number '' slider to see on this site and move objects by mouse or move.. No self-loops, the adjacency matrix, largest eigenvalue, spectral distribution, semi-circle,! Boolean matrix to save memory space ( Fig matrix to represent the.! Eds. ) V2 ) amount of space while it is computed Demonstrations and technical! - solve matrix operations as well as methods for solving systems of linear... N'T set up its algorithm or move workspace shows the possible adjacency matrices graph with adjacency matrix is. Efficiently and elegantly by using a graph with adjacency matrix species or reaction and!


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# Tag Info

17

What is the difference between clearway and stopway? The definitions linked in the question for a clearway and stopway are fairly adequate. AC 150/5300-13A Airport Design Clearway - is an area extending beyond the runway end available for completion of the takeoff operation of turbine-powered aircraft. A clearway increases the allowable aircraft operating ...

16

This is a loaded question. Every regulation in aviation seems to be written blood and the need for runway safety areas (RSA) is no exception. RSAs are defined in AC 150/5300-13A Runway Safety Area (RSA). A defined surface surrounding the runway prepared or suitable for reducing the risk of damage to aircraft in the event of an undershoot, overshoot, ...

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From the AIM, 4-3-6 (c)(2): All 14 CFR Part 139 airports report declared distances for each runway. Other airports may also report declared distances for a runway if necessary to meet runway design standards or to indicate the presence of a clearway or stopway. Where reported, declared distances for each runway end are published in the Airport/Facility ...

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No Takeoff Distance Available (TODA) can never be shorter than Takeoff Run Available (TORA). AIM 4-3-6 Takeoff Distance Available (TODA) – The takeoff run available plus the length of any remaining runway or clearway beyond the far end of the takeoff run available. I had previously downloaded the FAA runway database and determined there is no TODA ...

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No. What does the runway distance mean under "Additional Runway Information" on Jeppesen plates? What are Runway Declared Distances? How do pilots use stopway, EMAS, clearway and displaced threshold? How do runway declared distances affect my takeoff distance? Jeppesen uses its own method to calculate additional runway information. They have ...

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From USA AIP (emphasis added): The U.S. does not require or provide criteria for clearways in its design standards. It does encourage ownership and clearing of the land underlying the innermost portion of the approach out to where the approach surface is 10.5 meters above the level of the take-off surface. You may also want to check AC 150/5300-13A - ...

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From page AIRPORT-3 (page 3 overall) of this Jeppesen legend (site jump-off page here):

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Balanced field length calculations by definition are the greater of three factors. As Lnafziger noted in his answer to the linked questions above these three factors are: 115% All engine takeoff distance Accelerate-stop distance Accelerate-go distance The manufacturer will choose a $V_{1}$ speed that will all the accelerate-stop and accelerate-go ...

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Actually, the take off distance given in runways will be greater than the runway distance as it will include the distance to overcome a predetermined obstacle. Source: tc.gc.ca The balanced field length is dependent on the aircraft configuration among other things and is usually available in AFM. The airfield lengths, on the other hand are made available ...

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# Blitting With AS3: Removing a Bitmap's Background Color

Today, you will learn how to remove the background color from a sprite sheet using AS3, and blit the result to a bitmap canvas. Read on to learn more!

## Final Result Preview

Let's take a look at the final result we will be working towards:

## Step 1: Drawing the Spritesheet

So, it is time to draw your spritesheet. Open up your favourite 'pixel-art' program and create an image of 128x128 and give it a background colour of '#e731f2' which is a nice purple colour!

This is my amazing artwork:

Save your image somewhere organised and let us continue!

## Step 2: Importing the Sprite Sheet

Now, I'm call this a sprite sheet even though it is just one image. A 'sprite sheet' usually consists of more than one sprite but we can imagine we have more, right?

Anyway, if you are using Flash CS4 or higher, simply import your image via File | Import | Import to Library:

If you are using any other AS3 IDE, I have included the SWC file so you should probably skip this step. If you wish to embed your own images, I'm sure that your IDE of choice will have this feature.

## Step 3: Exporting the Sprite Sheet

We have now got our sprite sheet in the Library; we should really make it into a Class.

Right-click the image in the library and select 'Properties'. Give the image the following properties:

Hit OK. If you get a warning, just ignore it; it does not matter here.

## Step 4: Creating the Document Class

If you're not familiar with the concept of a document class, check out this Quick Tip before reading further.

Create a new 'ActionScript 3.0 Class' and give it the name 'Main'. When the file has been created, save it as 'Main.as'.

This code should be placed in our new Class:

We are not done yet, however! If you are using the Flash IDE, navigate to the 'Properties Panel' and set the 'Document Class' to 'Main'. If you are wondering what that does, it means that when your application/game is run by the Flash Player, Main.as will be the class that's linked to the SWF itself. Cool, huh?

Run the program; if you get no errors then you should be good to go!

## Step 5: Creating the Canvas

Before we do any blitting, we will first need to make a canvas to blit onto. If you are unsure of the term Blitting or would like to learn more about it, please take a look at this tutorial.

Now, we will declare a new Bitmap variable, to which we will blit (copy) the image.

After we have done this, we will add a function called Initialize() which will allow us to set everything up neatly:

Let us create the function now:

We are still not finished however, as we still have to add the imports:

Run the program; if it has a black background, it worked!

## Step 6: Initializing the SpriteSheet

Firstly, we will need to make a new variable of type SpriteSheet - which was the Class for the image we imported earlier, remember?

We shall then initialize it:

Run the program and you should see nothing; let's fix that right away!

## Step 7: Updating the Program

Now we need to add an ENTER_FRAME event. This will allow the program to update 24 times a second (24 FPS) in our case.

In the Main() function, add the following line:

Now we need to make the Update() function. Add this function after the other functions:

Don't forget the imports:

Now we are ready to do some blitting!

## Step 8: Blitting

Here comes the interesting part!

Alright, so what we want to do is:

• Clear the canvas.
• Blit the image and remove the background colour.

In the update function, type the following code:

If you run this, you will get your image on the canvas! However, this is not just what we are aiming for as we wish to remove that background colour from the image.

I shall explain some of the code above first:

• canvas.bitmapData.lock(); - This line optimizes the blitting and it is a good habit to type it most of the time!
• canvas.bitmapData.fillRect(); - This line clears the canvas by filling it with a Black colour.
• canvas.bitmapData.copyPixels(); - Not very useful in our situation but copies all the pixels from part of an image.
• canvas.bitmapData.unlock(); - This works with lock() to optimize the process.

Now you should have this on the screen...

Yes, I know, you are probably right. I think we should get rid of the purple too...

## Step 9: Removing the Colour

Finally, it's time to remove the purple colour!

What we want to do is check through every pixel; if the pixel is purple, we simply do not copy it to the canvas. To do this, we will make our own function.

Change Update() to the following:

Our new function (CustomBlitting(), which we have not written yet) takes most of the parameters that copyPixels does, along with an extra one: the colour we wish to remove.

Time to write the function. This code may look complicated if you have never done a nested for-loop before. The way this loop works is basically:

• For every row we have...
• Check every pixel in that row...
• Move to next row down...

Let me explain the getPixel and setPixel, although they should probably be self-explanatory:

• getPixel( x, y ); - This returns the colour of a pixel at the X,Y location.
• setPixel( x, y, color ); - This sets the colour of a pixel to color at the X,Y location of the canvas.

Now if you run the program, it works!

## Step 10: Challenges

I only have one challenge for you to do for this tutorial:

Accept an Array of colours as a parameter and remove any colours from the image that are in the array.

Good luck!

## Conclusion

I hope you have enjoyed this tutorial and have learnt something new today. If you'd like to show me your SWF with the completed challenges, leave a comment below!


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## Gallian- Ch.9 #49

Problem Statement: If $H\lhd G$, prove that $C(H)\lhd G$.

Proof: Recall that $C(H)$ is the set of all elements of $G$ which commute with all elements of $H$. Let $g\in G$, we wish to show that $gC(H)c^{-1}\subset C(H)$. So consider an element $s\in gC(H)g^{-1}$. Then there is some $x\in C(H)$ such that $s=gxg^{-1}$. Now, in order to show that $s\in C(H)$ we must show that $s$ commutes with any element $h\in H$.

Consider $gxg^{-1}h$. Since $H\lhd G$ it follows that there exists some $h'\in H$ such that $g^{-1}h=h'g^{-1}$.  Note that this also means that $gh'=hg$. Using these facts we may rewrite the expression in the following way:

$gxg^{-1}h=gxh'g^{-1}$

But $x$ commutes with everything from $H$ and so we may again rewrite the expression:

$=gh'xg^{-1}$

But as stated above, $gh'=hg$ and so we may rewrite one more time:

$=hgxg^{-1}$

And so we may conclude that $gC(H)g^{-1}\subset C(H)$ for every $g\in G$ and so $C(H)\lhd G$.

$\Box$

Reflection: This is one of those proofs that is not entirely difficult, but is very easy to make much more difficult than it needs to be. Thankfully I have Jeremy to straighten me out when I over think these simple problems. The key thing for me while doing this proof was to think in words, and less in symbols. Once we started talking about what it means to be in the centralizer of $H$ then it started to click.


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# Find how many different arrangements are possible?

I need some help with probability question.Can you help me with this? Below is my question, question 1(i) I get the answer, just question 1(ii)I can't get the answer.

Michael has 4 printed articles download from different online publishers. They are IEEE xplore, Elsevier, Proquest and Emerald Insight. (i) How many different arrangements are possible if he wishes to rearrange the Printed articles?

Answer: n! = 4!
= 4 x 3 x 2 x 1
= 24

(ii) Assume Proquest is one of the sub unit from Emerald Insight, how many different arrangements are possible?

Answer: ?

Can you show me the solve solutions? Thanks for your helping. Wish you have a nice day!

• "Proquest is one of the sub unit from Emerald Insight" - what does that mean? In any case the answer can't be more than $24$ so you could just list all possibilities. – David Apr 24 '19 at 8:16
• I also confuse this question. – Ason Chee Soon Apr 24 '19 at 8:19
• Maybe it means that we should consider the Proquest publications to be indistinguishable from those of Emerald Insight, in which case we want to divide by the number of permutations of the set of these indistinguishable publications... In this case, divide by $2!$. – 雨が好きな人 Apr 24 '19 at 8:28
• That mean is 4!/2! right? – Ason Chee Soon Apr 24 '19 at 8:31
• Yes. I’m not sure exactly what a ‘sub unit’ is in this case though. – 雨が好きな人 Apr 24 '19 at 8:39

As I said in my comment, I’m not sure what is meant by ‘Proquest is one of the sub unit from Emerald Insight’.

However, if it means that that Proquest is to be considered a sub-publication of Emerald Insight, then maybe we want to consider the possible arrangements where the two publications are the same.

As you said, there are $$4!$$ possible permutations of 4 articles. If we consider two of these articles to be indistinguishable, meaning that we, for example, consider the two permutations (IEEE Explore, Elsevier, Proquest, Emerald Insight) and (IEEExplore, Elsevier, Emerald Insight, Proquest) to be the same permutation, then we want $$\frac{4!}{2!}$$.

Generally, if we have $$n$$ objects and several sets of objects with the number of objects in each set given by $$a_1, a_2, \dots, a_m$$ where the objects in a given set are considered to be indistinguishable from the other objects in the set (but objects from different sets are always considered different), then the number of permutations is given by

$$\frac{n!}{a_1! a_2! \dots a_m!}$$

Intuitively, we are starting with the original permutation where all objects are different, and then dividing by the number of permutations of each set of indistinguishable objects.

In your case, we have $$n = 4$$ and we only have one set of two objects so $$a_1 = 2$$. The number of permutations is then $$\frac{4!}{2!} = 12$$.

• I also don't understanding what is meant by ‘Proquest is one of the sub unit from Emerald Insight’.Anyhow, thanks your given answer. – Ason Chee Soon Apr 26 '19 at 6:19


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Diffusion in fractal globules
Umeå University, Faculty of Science and Technology, Department of Physics.
2016 (English)Independent thesis Advanced level (professional degree), 300 HE creditsStudent thesisAlternative title
På spaning efter onormal diffusion av biomolekyler i DNA med hjälp av stokastisk simulering (Swedish)
##### Abstract [en]

Recent experiments suggest that the human genome (all of our DNA) is organised as a so-called fractal globule. The fractal globule is a knot--free dense polymer that easily folds and unfolds any genomic locus, for example a group of nearby genes. Proteins often need to locate specific target sites on the DNA, for instance to activate a gene. To understand how proteins move through the DNA polymer, we simulate diffusion of particles through a fractal globule. The fractal globule was generated on a cubic lattice as spheres connected by cylinders. With the structure in place, we simulate particle diffusion and measure how their mean squared displacement ($\langle R^2(t)\rangle$) grows as function of time $t$ for different particle radii. This quantity allows us to better understand how the three dimensional structure of DNA affects the protein's motion. From our simulations we found that $\langle R^2(t)/t\rangle$ is a decaying function when the particle is sufficiently large. This means that the particles diffuse slower than if they were free. Assuming that $\langle R^2(t) \rangle \propto t^\alpha$ for long times, we calculated the growth exponent $\alpha$ as a function of particle radius $r_p$. When $r_p$ is small compared to the average distance between two polymer segments $d$, we find that $\alpha \approx 1$. This means the polymer network does not affect the particle's motion. However, in the opposite limit $r_p\sim d$ we find that $\alpha<1$ which means that the polymer strongly slows down the particle's motion. This behaviour is indicative of sub-diffusive dynamics and has potentially far reaching consequences for target finding processes and biochemical reactions in the cell.

2016. , p. 23
##### Keyword [en]
Fractal Globule, Anomalous Diffusion, Diffusion, DNA
##### National Category
Bioinformatics (Computational Biology)
##### Identifiers
OAI: oai:DiVA.org:umu-126570DiVA, id: diva2:1034087
##### Educational program
Master of Science Programme in Engineering Physics
##### Examiners
Available from: 2016-10-14 Created: 2016-10-11 Last updated: 2018-01-14Bibliographically approved

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v. 2.33.0
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