Articles

18.10: Movie Scripts 7-8


G.7 Determinants

Permutation Example

Lets try to get the hang of permutations. A permutation is a function which scrambles things. Suppose we had

This looks like a function $sigma$ that has values [ sigma(1) =3 , sigma(2) =2 , sigma(3) =4 , sigma(4) = 1, .]

Then we could write this as
[
egin{bmatrix}
1 & 2 & 3 & 4\
sigma(1) & sigma(2) & sigma(3) & sigma(4)
end{bmatrix}
= egin{bmatrix}
1 & 2 & 3 & 4 \
3 & 2 & 4 & 1
end{bmatrix}
]
We could write this permutation in two steps by saying that first we swap 3 and 4, and then we swap 1 and 3. The order here is important.

This is an even permutation, since the number of swaps we used is two (an even number).

Elementary Matrices

This video will explain some of the ideas behind elementary matrices. First think back to linear systems, for example (n) equations in (n) unknowns:
$$
left{
egin{array}{ccc}
a^{1}_{1} x^{1} + a^{1}_{2} x^{2}+cdots +a^{1}_{n} x^{n} &=&v^{1}
a^{2}_{1} x^{1} + a^{2}_{2} x^{2}+cdots +a^{2}_{n} x^{n} &=&v^{2}
vdots &&
a^{n}_{1} x^{1} + a^{n}_{2} x^{2}+cdots +a^{n}_{n} x^{n} &=&v^{n}, .
end{array} ight .
$$
We know it is helpful to store the above information with matrices and vectors
$$
M:=egin{pmatrix}
a^{1}_{1}&a^{1}_{2}&cdots& a^{1}_{n}
a^{2}_{1}&a^{2}_{2}&cdots& a^{2}_{n}
vdots&vdots&&vdots
a^{n}_{1}&a^{n}_{2}&cdots& a^{n}_{n}
end{pmatrix}, ,qquad
X:=egin{pmatrix}x^{1}x^{2}vdots x^{n}end{pmatrix}, ,qquad
V:=egin{pmatrix}v^{1}v^{2}vdotsv^{n}end{pmatrix}, .
$$
Here we will focus on the case the (M) is square because we are interested in its inverse (M^{-1}) (if it exists) and its determinant (whose job it will be to determine the existence of (M^{-1})).

We know at least three ways of handling this linear system problem:

  1. As an augmented matrix $$left(egin{array}{c|c}M & Vend{array} ight), .$$ Here our plan would be to perform row operations until the system looks like $$left(egin{array}{c|c}I & M^{-1}Vend{array} ight), ,$$ (assuming that (M^{-1}) exists).
  2. As a matrix equation $$MX=V, ,$$ which we would solve by finding (M^{-1}) (again, if it exists), so that $$X=M^{-1}V, .$$
  3. As a linear transformation $$L:mathbb{R}^{n}longrightarrow mathbb{R}^{n}$$ via $$mathbb{R}^{n} i X longmapsto MX in mathbb{R}^{n}, .$$ In this case we have to study the equation (L(X)=V) because (Vin mathbb{R}^{n}).

Lets focus on the first two methods. In particular we want to think about how the augmented matrix method can give information about finding (M^{-1}). In particular, how it can be used for handling determinants.

The main idea is that the row operations changed the augmented matrices, but we also know how to change a matrix (M) by multiplying it by some other matrix (E), so that (M o EM). In particular can we find ``elementary matrices'' the perform row operations?

Once we find these elementary matrices is is ( extit{very important}) to ask how they effect the determinant, but you can think about that for your own self right now.

Lets tabulate our names for the matrices that perform the various row operations:
$$left(egin{array}{r|r} Row Operation & Elementary Matrix hline R_{i}leftrightarrow R_{j} & E_{j}^{i} R_{i} o lambda R_{i} & R^{i}(lambda) R_{i} o R_{i} + lambda R_{j} & S^{i}_{j}(lambda)end{array} ight)]

To finish off the video, here is how all these elementary matrices work for a (2 imes 2) example. Lets take
$$
M=egin{pmatrix}a&bc&dend{pmatrix}, .
$$
A good thing to think about is what happens to (det M = ad-bc) under the operations below.

  1. Row swap: $$E^{1}_{2}=egin{pmatrix}0&11&0end{pmatrix}, ,qquad E^{1}_{2} M = egin{pmatrix}0&11&0end{pmatrix}egin{pmatrix}a&bc&dend{pmatrix}=egin{pmatrix}c&da&bend{pmatrix}, .$$
  2. Scalar multiplying: $$R^{1}(lambda)=egin{pmatrix}lambda&0&1end{pmatrix}, ,qquad E^{1}_{2} M = egin{pmatrix}lambda&0&1end{pmatrix}egin{pmatrix}a&bc&dend{pmatrix}=egin{pmatrix}lambda a&lambda bc&dend{pmatrix}, .$$
  3. Row sum: $$S^{1}_{2}(lambda)=egin{pmatrix}1&lambda&1end{pmatrix}, ,quad S^{1}_{2}(lambda) M = egin{pmatrix}1&lambda&1end{pmatrix}egin{pmatrix}a&bc&dend{pmatrix}=egin{pmatrix}a+lambda c&b+lambda dc&dend{pmatrix}, .$$

Elementary Determinants

This video will show you how to calculate determinants of elementary matrices. First remember that the job of an elementary row matrix is to perform row operations, so that if (E) is an elementary row matrix and (M) some given matrix, $$EM$$ is the matrix (M) with a row operation performed on it.

The next thing to remember is that the determinant of the identity is (1). Moreover, we also know what row operations do to determinants:

  1. Row swap (E^{i}_{j}): flips the sign of the determinant.
  2. Scalar multiplication (R^{i}(lambda)): multiplying a row by (lambda) multiplies the determinant by (lambda).
  3. Row addition (S^{i}_{j}(lambda)): adding some amount of one row to another does not change the determinant.

The corresponding elementary matrices are obtained by performing exactly these operations on the identity:
$$
E^{i}_{j}=egin{pmatrix}
1 & & & & & & \
& ddots & & & & &
& & 0 & & 1 & & \
& & & ddots & & &
& & 1 & & 0 & & \
& & & & & ddots&
& & & & & & 1 \
end{pmatrix}, ,
]

[
R^{i}(lambda)=
egin{pmatrix}
1 & & & & \
& ddots & & &
& & lambda & &
& & & ddots &
& & & & 1 \
end{pmatrix}
, ,]

[
S^{i}_{j}(lambda) = egin{pmatrix}
1 & & & & & & \
& ddots & & & & &
& & 1 & & lambda & &
& & & ddots & & &
& & & & 1 & & \
& & & & & ddots&
& & & & & & 1 \
end{pmatrix}
]
So to calculate their determinants, we just have to apply the above list of what happens to the determinant of a matrix under row operations to the determinant of the identity. This yields
$$
det E^{i}_{j}=-1, ,qquad
det R^{i}(lambda)=lambda, ,qquad
det S^{i}_{j}(lambda)=1, .
]

Determinants and Inverses

Lets figure out the relationship between determinants and invertibility. If we have a system of equations (Mx=b) and we have the inverse (M^{-1}) then if we multiply on both sides we get (x = M^{-1}Mx= M^{-1}b). If the inverse exists we can solve for (x) and get a solution that looks like a point.

So what could go wrong when we want solve a system of equations and get a solution that looks like a point? Something would go wrong if we didn't have enough equations for example if we were just given
[
x+y = 1
]
or maybe, to make this a square matrix (M) we could write this as
egin{align*}
x+y &= 1
0 &= 0
end{align*}
The matrix for this would be
(M =egin{bmatrix}
1 & 1\
0& 0
end{bmatrix})
and det((M) = 0). When we compute the determinant, this row of all zeros gets multiplied in every term. If instead we were given redundant equations

egin{align*}
x+y &= 1
2x+2y &= 2
end{align*}
The matrix for this would be
(M =egin{bmatrix}
1 & 1\
2& 2
end{bmatrix}) and det((M) = 0). But we know that with an elementary row operation, we could replace the second row with a row of all zeros. Somehow the determinant is able to detect that there is only one equation here. Even if we had a set of contradictory set of equations such as
egin{align*}
x+y &= 1
2x+2y &= 0,
end{align*}
where it is not possible for both of these equations to be true, the matrix (M) is still the same, and still has a determinant zero.

Lets look at a three by three example, where the third equation is the sum of the first two equations.

egin{align*}
x+y + z &= 1
y +z &= 1
x + 2y+ 2z &= 2
end{align*}

and the matrix for this is

[
M =egin{bmatrix}
1 & 1 &1\
0 & 1 & 1\
1 & 2& 2
end{bmatrix}
]

If we were trying to find the inverse to this matrix using elementary matrices
$$ left( egin{array}{ccc | ccc}
1 & 1 &1 & 1 & 0 & 0\
0 & 1 & 1 & 0 & 1 & 0 \
1 & 2 & 2 & 0 & 0 & 1
end{array} ight)
=
left( egin{array}{ccc | rrr}
1 & 1 &1 & 1 & 0 & 0\
0 & 1 & 1 & 0 & 1 & 0 \
0 & 0 & 0 & -1 & -1 & 1
end{array} ight)
$$
And we would be stuck here. The last row of all zeros cannot be converted into the bottom row of a (3 imes 3) identity matrix. this matrix has no inverse, and the row of all zeros ensures that the determinant will be zero. It can be difficult to see when one of the rows of a matrix is a linear combination of the others, and what makes the determinant a useful tool is that with this reasonably simple computation we can find out if the matrix is invertible, and if the system will have a solution of a single point or column vector.

Alternative Proof

Here we will prove more directly that} the determinant of a product of matrices is the product of their determinants. First we reference that for a matrix (M) with rows (r_{i}), if (M^{prime}) is the matrix with rows (r^{prime}_{j} = r_{j} + lambda r_{i}) for (j eq i) and (r^{prime}_{i} = r_{i}), then (det(M) = det(M^{prime})). Essentially we have (M^{prime}) as (M) multiplied by the elementary row sum matrices (S^{i}_{j}(lambda)). Hence we can create an upper-triangular matrix (U) such that (det(M) = det(U)) by first using the first row to set (m_{i}^{1} mapsto 0) for all (i > 1), then iteratively (increasing (k) by 1 each time) for fixed (k) using the (k)-th row to set (m_{i}^{k} mapsto 0) for all (i > k).

Now note that for two upper-triangular matrices (U = (u_{i}^{j})) and (U^{prime} = (u_{i}^{prime j})), by matrix multiplication we have (X = UU^{prime} = (x_{i}^{j})) is upper-triangular and (x_{i}^{i} = u_{i}^{i} u_{i}^{prime i}). Also since every permutation would contain a lower diagonal entry (which is 0) have (det(U) = prod_{i} u_{i}^{i}). Let (A) and (A^{prime}) have corresponding upper-triangular matrices (U) and (U^{prime}) respectively (i.e. (det(A) = det(U))), we note that (AA^{prime}) has a corresponding upper-triangular matrix (UU^{prime}), and hence we have
egin{align*}
det(A A^{prime}) & = det(U U^{prime}) = prod_{i} u_{i}^{i} u_{i}^{prime i}
& = left( prod_{i} u_{i}^{i} ight) left( prod_{i} u_{i}^{prime i} ight)
& = det(U) det(U^{prime}) = det(A) det(A^{prime}).
end{align*}

Practice taking Determinants

Lets practice taking determinants of (2 imes 2) and (3 imes 3) matrices.

For (2 imes 2) matrices we have a formula
[
{ m det}
egin{pmatrix}
a & b
c & d
end{pmatrix}
= ad - bc, .
]
This formula might be easier to remember if you think about this picture.


Now we can look at three by three matrices and see a few ways to compute the determinant. We have a similar pattern for (3 imes 3) matrices.
Consider the example
[
{ m det}
egin{pmatrix}
1 & 2 & 3 \
3 & 1 & 2 \
0 & 0 & 1 \
end{pmatrix}
= ( (1cdot 1cdot 1)+ (2cdot 2cdot 0) + (3cdot 3cdot 0)) - ((3cdot 1cdot 0)+ (1cdot 2cdot 0) + (3cdot 2cdot 1)) = -5
]
We can draw a picture with similar diagonals to find the terms that will be positive and the terms that will be negative.

Another way to compute the determinant of a matrix is to use this recursive formula. Here I take the coefficients of the first row and multiply them by the determinant of the minors and the cofactor. Then we can use the formula for a two by two determinant to compute the determinant of the minors

[
ext{det}
egin{pmatrix}
1 & 2 & 3 \
3 & 1 & 2 \
0 & 0 & 1 \
end{pmatrix}
= 1
egin{vmatrix}
1 & 2 \
0 &1\
end{vmatrix}
-2
egin{vmatrix}
3 & 2 \
0 & 1 \
end{vmatrix}
+ 3
egin{vmatrix}
3 & 1 \
0 & 0 \
end{vmatrix}
= 1(1-0) - 2(3-0) + 3(0-0) = -5
]
Decide which way you prefer and get good at taking determinants, you'll need to compute them in a lot of problems.

Hint for Review Problem 5

For an arbitrary (3 imes 3) matrix (A = (a^{i}_{j})), we have
[
det(A) = a^{1}_{1} a^{2}_{2} a^{3}_{3} + a^{1}_{2} a^{2}_{3} a^{3}_{1} + a^{1}_{3} a^{2}_{1} a^{3}_{2} - a^{1}_{1} a^{2}_{3} a^{3}_{2} - a^{1}_{2} a^{2}_{1} a^{3}_{3} - a^{1}_{3} a^{2}_{2} a^{3}_{1}
]
and so the complexity is (5a + 12m). Now note that in general, the complexity (c_{n}) of the expansion minors formula of an arbitrary (n imes n) matrix should be
[
c_{n} = (n-1) a + n c_{n-1} m
]
since (det(A) = sum_{i=1}^{n} (-1)^{i} a_{i}^{1} cofactor(a_{i}^{1})) and (cofactor(a_{i}^{1})) is an ((n-1) imes (n-1)) matrix. This is one way to prove part (c).

G.8 Subspaces and Spanning Sets

Linear Systems as Spanning Sets

Suppose that we were given a set of linear equations (l^{j}(x^{1}, x^{2}, dotsc, x^{n})) and we want to find out if (l^{j}(X) = v^{j}) for all (j) for some vector (V = (v^{j})). We know that we can express this as the matrix equation
[
sum_{i} l^{j}_{i} x^{i} = v^{j}
]
where (l^{j}_{i}) is the coefficient of the variable (x^{i}) in the equation (l^{j}). However, this is also stating that (V) is in the span of the vectors ({ L_{i} }_{i}) where (L_{i} = (l^{j}_{i})_{j}). For example, consider the set of equations
egin{align*}
2 x + 3 y - z & = 5
-x + 3y + z & = 1
x + y - 2 z & = 3
end{align*}
which corresponds to the matrix equation
[
egin{pmatrix}
2 & 3 & -1 \
-1 & 3 & 1 \
1 & 1 & -2
end{pmatrix} egin{pmatrix} x y z end{pmatrix} = egin{pmatrix} 5 1 3 end{pmatrix}.
]
We can thus express this problem as determining if the vector
[
V = egin{pmatrix} 5 1 3 end{pmatrix}
]
lies in the span of
[
left{ egin{pmatrix} 2 -1 1 end{pmatrix}, egin{pmatrix} 3 3 1 end{pmatrix}, egin{pmatrix} -1 1 -2 end{pmatrix} ight}.
]

Hint for Review Problem 2

For the first part, try drawing an example in (mathbb{R}^{3}):

Here we have taken the subspace (W) to be a plane through the origin and (U) to be a line through the origin. The hint now is to think about what happens when you add a vector (uin U) to a vector (win W). Does this live in the union (Ucup W)?

For the second part, we take a more theoretical approach. Lets suppose that (vin Ucap W) and (v'in Ucap W). This implies
$$
vin U quad mbox{and} quad v'in U, .
$$
So, since (U) is a subspace and all subspaces are vector spaces, we know that the linear combination
$$
alpha v+eta v'in U, .
$$
Now repeat the same logic for (W) and you will be nearly done.


18.10: Movie Scripts 7-8

NodeSource Node.js Binary Distributions

This repository contains documentation for using the NodeSource Node.js Binary Distributions via .rpm, .deb and Snap packages as well as their setup and support scripts.

If you are looking for NodeSource's low-impact Node.js performance monitoring platform, please get started here.

Please file an issue if you are experiencing a problem or would like to discuss something related to the distributions.

Pull requests are encouraged if you have changes you believe would improve the setup process or increase compatibility across Linux distributions.

Debian and Ubuntu based distributions

Available architectures:

NodeSource will continue to maintain the following architectures and may add additional ones in the future.

  • amd64 (64-bit)
  • armhf (ARM 32-bit hard-float, ARMv7 and up: arm-linux-gnueabihf)
  • arm64 (ARM 64-bit, ARMv8 and up: aarch64-linux-gnu)

Supported Ubuntu versions:

NodeSource will maintain Ubuntu distributions in active support by Canonical, including LTS and the intermediate releases.

  • Ubuntu 16.04 LTS (Xenial Xerus)
  • Ubuntu 18.04 LTS (Bionic Beaver)
  • Ubuntu 18.10 (Cosmic Cuttlefish)
  • Ubuntu 19.04 (Disco Dingo)
  • Ubuntu 19.10 (Eoan Ermine)
  • Ubuntu 20.04 LTS (Focal Fossa)
  • Ubuntu 20.10 (Groovy Gorilla)
  • Ubuntu 21.04 (Hirsute Hippo)

Supported Debian versions:

NodeSource will maintain support for stable, testing and unstable releases of Debian, due to the long release cycle a considerable number of users are running unstable and testing.

  • Debian 9 / oldstable (Stretch)
  • Debian 10 / stable (Buster)
  • Debian unstable (Sid)
  • Debian testing (Bullseye)

Supported Linux Mint versions:

  • Linux Mint 18 "Sarah" (via Ubuntu 16.04 LTS)
  • Linux Mint 18.1 "Serena" (via Ubuntu 16.04 LTS)
  • Linux Mint 18.2 "Sonya" (via Ubuntu 16.04 LTS)
  • Linux Mint 18.3 "Sylvia" (via Ubuntu 16.04 LTS)
  • Linux Mint Debian Edition (LMDE) 2 "Betsy" (via Debian 8)
  • Linux Mint 19 "Tara" (via Ubuntu 18.04 LTS)
  • Linux Mint 19.1 "Tessa" (via Ubuntu 18.04 LTS)
  • Linux Mint 19.2 "Tina" (via Ubuntu 18.04 LTS)
  • Linux Mint 19.3 "Tricia" (via Ubuntu 18.04 LTS)
  • Linux Mint 20 "Ulyana" (via Ubuntu 20.04 LTS)
  • Linux Mint 20.1 "Ulyssa" (via Ubuntu 20.04 LTS)
  • Linux Mint 20.2 "Uma" (via Ubuntu 20.04 LTS)
  • Linux Mint Debian Edition (LMDE) 3 "Cindy" (via Debian 9)
  • Linux Mint Debian Edition (LMDE) 4 "Debbie" (via Debian 10)

Supported Devuan versions:

  • Ascii / stable (via Debian 9)
  • Beowulf / testing (via Debian 10)
  • Ceres / unstable (via Debian unstable)

Supported elementary OS versions:

  • elementary OS 0.4 Loki (via Ubuntu 16.04 LTS)
  • elementary OS 5 Juno (via Ubuntu 18.04 LTS)
  • elementary OS 5.1 Hera (via Ubuntu 18.04 LTS)
  • elementary OS 6 Odin (via Ubuntu 20.04 LTS)

Supported Trisquel versions:

  • Trisquel 8 "Flidas" (via Ubuntu 16.04 LTS)
  • Trisquel 9 "Etiona" (via Ubuntu 18.04 LTS)

Supported BOSS versions:

  • BOSS 7.0 "Drishti" (via Debian 9)
  • BOSS 8.0 "Unnati" (via Debian 10)

Supported BunsenLabs versions:

Supported MX Linux versions:

  • MX-17 Horizon (via Debian 9)
  • MX-18 Continuum (via Debian 9)
  • MX-19 Patito Feo (via Debian 10)

Supported Sparky Linux versions:

  • Sparky 4.x "Tyche" (via Debian 9)
  • Sparky 5.x "Nibiru" (via Debian 10)

Supported PureOS Linux versions:

Supported Astra Linux CE versions:

Supported Ubilinux versions:

Node.js v16.x:

Node.js v14.x:

Node.js v12.x:

Node.js LTS (v14.x):

Node.js Current (v16.x):

Optional: install build tools

To compile and install native addons from npm you may also need to install build tools:

If you're not a fan of curl <url> | bash - , or are using an unsupported distribution, you can try a manual installation.

These instructions assume sudo is present, however some distributions do not include this command by default, particularly those focused on a minimal environment. In this case, you should install sudo or su to root to run the commands directly.

1. Remove the old PPA if it exists

This step is only required if you previously used Chris Lea's Node.js PPA.

2. Add the NodeSource package signing key

The key ID is 1655A0AB68576280 .

3. Add the desired NodeSource repository

4. Update package lists and install Node.js

Enterprise Linux based distributions

Available architectures:

NodeSource will continue to maintain the following architectures and may add additional ones in the future.

  • x86_64 (64-bit)
  • arm64 (ARM 64-bit, ARMv8 and up: aarch64-linux-gnu)

Supported Red Hat® Enterprise Linux® versions:

Supported CentOS versions:

  • CentOS 7 (64-bit)
  • CentOS 8 (64-bit)
  • CentOS 8 Stream (64-bit)

Supported AlmaLinux OS versions:

Supported Rocky Linux OS versions:

Supported CloudLinux versions:

Supported Fedora versions:

  • Fedora 32 (64-bit)
  • Fedora 33 (64-bit)
  • Fedora 34 (64-bit)

Supported Amazon Linux versions:

NOTE: If you are using RHEL 6 or CentOS 6, you might want to read about running Node.js on older distros.

Run on RHEL, CentOS, CloudLinux, Amazon Linux or Fedora:

Node.js v16.x

Node.js v14.x

Node.js v12.x

Node.js LTS (14.x)

Node.js Current (16.x)

Optional: install build tools

To compile and install native addons from npm you may also need to install build tools:

Snaps are containerized software packages designed to work across cloud, desktop, and IoT devices. They work natively on most popular Linux distributions and feature automatic transactional updates.

The NodeSource-managed Node.js snap contains the Node.js runtime, along with the two most widely-used package managers, npm and Yarn. They are delivered from the snapcraft store and are automatically built and pushed for each supported Node.js release line. Generally you will have a new version of Node.js automatically running on your computer the same day it is released on nodejs.org.

The Node.js snap can currently be installed on Arch Linux, Debian, Fedora, Linux Mint, Manjaro, OpenEmbedded/Yocto, OpernWrt, Solus, Ubuntu and many other distributions built on top these. NodeSource has not tested the Node.js snap on all of these distributions and feedback is welcome in this repository if you run into problems.

The snap command ships with Ubuntu, from version 16.04 and later. If you do not have it installed, follow the instructions on snapcraft to install snapd.

Snaps are delivered via "channels" for Node.js, the channel names are the major-version number of Node.js. So select a supported Node.js version and install with:

Substituting 8 for the major version you want to install. Both LTS and Current versions of Node.js are available via snapcraft.

The --classic argument is required here as Node.js needs full access to your system in order to be useful, therefore it needs snap’s "classic confinement". By default, snaps are much more restricted in their ability to access your disk and network and must request special access from you where they need it. Note that on some Linux distributions, the snap confinement mechanisms are not fully supported so --classic may not be necessary or even supported.

Once installed, the node , npm and yarn commands are available for use and will remain updated for the channel you selected.

You can use the refresh command to switch to a new channel at any time:

Once switched, snapd will update Node.js for the new channel you have selected.

Users feeling adventurous or interested in testing the latest code from the Node.js core developers can install from the "edge" channel. This has an element of risk: it is a direct pipeline from the upstream Node.js git repository to the snap store every day and previews the ongoing development work and may include breaking changes slated for the next major version of Node.js. This is only recommend for those users who are willing to participate in testing and bug reporting upstream:

Not recommended for production deployments

Due to their auto-updating nature, snaps are not necessarily appropriate for the deployment of your Node.js applications to production. NodeSource recommends a stable and integration-tested deployment pipeline for production applications such as the .deb or .rpm distributions outlined above. However, snaps are an excellent way to keep developer machines updated and allow for trivial and convenient switching between Node.js versions.

The snap man page, or Canonical’s advanced snap usage tutorial contains details of advanced snapd functionality.

To test an installation is working (and that the setup scripts are working!) use:

Q: How do I use this repo when behind a proxy?

Q: How do I pin to specific versions of Node.js?

Q: I upgraded to a new major version of Node.js using the scripts, but the old version is still being installed, what is going on?

A: You probably need to clear out your package manager's cache. Take a look at issue #191

Q: I'm trying to install Node.js on CentOS 5 / RHEL 5 and it is failing, why?

A: Due to the limitations of the compiler toolchain on EL 5 and its end of general support, we no longer support. See issue #190

Q: I'm seeing "Your distribution, identified as ".i686" or ".i386, is not currently supported, why?

A: Node.js 4.x and newer require a 64bit os for rpms. See issue #268

Q: Why have certain versions of platforms/releases stopped receiving updates to Node.js?

A: Unfortunately, newer versions of V8 require a modern compiler toolchain. On some platforms, such as ARM wheezy, that toolchain is not available. See issue #247

Q: Why is my Node.js version newer than the one of the script I’ve run?

A: Your package manager is probably installing a newer Node.js version from a different source. See issue #657

Q: What is the current status of IPv6 support?

Q: I cannot install Node.js on Debian Jessie or Ubuntu Trusty Tahr: GPG error, why?

We, unfortunately, do not have the resources necessary to support and test the plethora of Linux releases in the wild, so we rely on community members such as yourself to get support on your favorite distributions! This is a list of releases that have been requested by the community. If you are interested in contributing to this project, this would be a great place to start!

  • OpenSUSE - Issue #199
  • Scientific Linux - Issue #251
  • TANGLU Bartholomea - Issue #81
  • Korora - Issue #130
  • FreePBX - Issue #257
  • Deepin - Issue #638
  • PopOS - Issue #924
  • Kylin - Issue #1011
  • MakuluLinux - Issue #1012
  • Alma Linux - Issue #1166
Chris LeaGitHub/chrisleaTwitter/@chrislea
Rod VaggGitHub/rvaggTwitter/@rvagg
William BlankenshipGitHub/retrohackerTwitter/@retrohack3r
Harry TruongGitHub/harrytruong
Matteo BrunatiGitHub/mattbrun
Brian WhiteGitHub/mscdex
Matt LewandowskyGitHub/lewellyn
Jan-Hendrik PetersGitHub/hennr
Andris ReinmanGitHub/andris9
CarvilsiGitHub/carvilsi
Krasimir TrenchevGitHub/Ava7
Phil HelmGitHub/phelma
0xmohitGitHub/0xmohit
jdarlingGitHub/jdarling
Prayag VermaGitHub/pra85
Misha BrukmanGitHub/mbrukman
Simon LydellGitHub/lydell
Sebastian BleiGitHub/iamsebastian
Jorge Maldonado VenturaNotABug/jorgesumle
Mayank MethaGitHub/mayankmethaTwitter/@mayankmethad
Adrian EstradaGitHub/edsadrTwitter/@edsadr
Iván IguaránGitHub/igsu

Contributions are welcomed from anyone wanting to improve this project!

This material is Copyright (c) NodeSource and licensed under the MIT license. All rights not explicitly granted in the MIT license are reserved. See the included LICENSE.md file for more details.

Supported with love by the NodeSource team

This project is not affiliated with Debian, Ubuntu, Red Hat, CentOS or Fedora.
Ubuntu is a registered trademark of Canonical Ltd.
Debian is a registered trademark owned by Software in the Public Interest, Inc.
Red Hat, CentOS and Fedora are trademarks of Red Hat, Inc.
CloudLinux is a trademark of Cloud Linux, Inc


18.10: Movie Scripts 7-8

New International Version
The name of the LORD is a fortified tower the righteous run to it and are safe.

New Living Translation
The name of the LORD is a strong fortress the godly run to him and are safe.

English Standard Version
The name of the LORD is a strong tower the righteous man runs into it and is safe.

Berean Study Bible
The name of the LORD is a strong tower the righteous run to it and are safe.

King James Bible
The name of the LORD is a strong tower: the righteous runneth into it, and is safe.

New King James Version
The name of the LORD is a strong tower The righteous run to it and are safe.

New American Standard Bible
The name of the LORD is a strong tower The righteous runs into it and is safe.

NASB 1995
The name of the LORD is a strong tower The righteous runs into it and is safe.

NASB 1977
The name of the LORD is a strong tower The righteous runs into it and is safe.

Amplified Bible
The name of the LORD is a strong tower The righteous runs to it and is safe and set on high [far above evil].

Christian Standard Bible
The name of the LORD is a strong tower the righteous run to it and are protected.

Holman Christian Standard Bible
The name of Yahweh is a strong tower the righteous run to it and are protected.

American Standard Version
The name of Jehovah is a strong tower The righteous runneth into it, and is safe.

Aramaic Bible in Plain English
The name of LORD JEHOVAH is a strong tower the righteous will run to it and be strengthened.

Brenton Septuagint Translation
The name of the Lord is of great strength and the righteous running to it are exalted.

Contemporary English Version
The LORD is a mighty tower where his people can run for safety--

Douay-Rheims Bible
The name of the Lord is a strong tower: the just runneth to it, and shall be exalted.

English Revised Version
The name of the LORD is a strong tower: the righteous runneth into it, and is safe.

Good News Translation
The LORD is like a strong tower, where the righteous can go and be safe.

GOD'S WORD® Translation
The name of the LORD is a strong tower. A righteous person runs to it and is safe.

International Standard Version
The name of the LORD is a strong tower a righteous person rushes to it and is lifted up above the danger.

JPS Tanakh 1917
The name of the LORD is a strong tower: The righteous runneth into it, and is set up on high.

Literal Standard Version
The Name of YHWH [is] a tower of strength, The righteous runs into it, and is set on high.

NET Bible
The name of the LORD is like a strong tower the righteous person runs to it and is set safely on high.

New Heart English Bible
The name of the LORD is a strong tower the righteous run into it and are safe.

World English Bible
The name of Yahweh is a strong tower: the righteous run to him, and are safe.

Young's Literal Translation
A tower of strength is the name of Jehovah, Into it the righteous runneth, and is set on high.

Exodus 3:15
God also told Moses, "Say to the Israelites, 'The LORD, the God of your fathers--the God of Abraham, the God of Isaac, and the God of Jacob--has sent me to you.' This is My name forever, and this is how I am to be remembered in every generation.

2 Samuel 22:2
He said: "The LORD is my rock, my fortress, and my deliverer.

2 Samuel 22:3
My God is my rock, in whom I take refuge, my shield, and the horn of my salvation. My stronghold, my refuge, and my Savior, You save me from violence.

Psalm 18:2
The LORD is my rock, my fortress, and my deliverer. My God is my rock, in whom I take refuge, my shield, and the horn of my salvation, my stronghold.

Psalm 61:3
For You have been my refuge, a tower of strength against the enemy.

Psalm 91:2
I will say to the LORD, "You are my refuge and my fortress, my God, in whom I trust."

Psalm 144:2
He is my steadfast love and my fortress, my stronghold and my deliverer. He is my shield, in whom I take refuge, who subdues peoples under me.

The name of the LORD is a strong tower: the righteous runs into it, and is safe.

Genesis 17:1 And when Abram was ninety years old and nine, the LORD appeared to Abram, and said unto him, I am the Almighty God walk before me, and be thou perfect.

Exodus 3:13-15 And Moses said unto God, Behold, when I come unto the children of Israel, and shall say unto them, The God of your fathers hath sent me unto you and they shall say to me, What is his name? what shall I say unto them? …

Exodus 6:3 And I appeared unto Abraham, unto Isaac, and unto Jacob, by the name of God Almighty, but by my name JEHOVAH was I not known to them.

2 Samuel 22:3,51 The God of my rock in him will I trust: he is my shield, and the horn of my salvation, my high tower, and my refuge, my saviour thou savest me from violence…

Psalm 18:2 The LORD is my rock, and my fortress, and my deliverer my God, my strength, in whom I will trust my buckler, and the horn of my salvation, and my high tower.

Psalm 27:1 A Psalm of David. The LORD is my light and my salvation whom shall I fear? the LORD is the strength of my life of whom shall I be afraid?

Genesis 32:11,28,29 Deliver me, I pray thee, from the hand of my brother, from the hand of Esau: for I fear him, lest he will come and smite me, and the mother with the children…

2 Samuel 22:45-47 Strangers shall submit themselves unto me: as soon as they hear, they shall be obedient unto me…

Psalm 56:3,4 What time I am afraid, I will trust in thee…

Psalm 91:14 Because he hath set his love upon me, therefore will I deliver him: I will set him on high, because he hath known my name.

Habakkuk 3:19 The LORD God is my strength, and he will make my feet like hinds' feet, and he will make me to walk upon mine high places. To the chief singer on my stringed instruments.

Verse 10. - The Name of the Lord is a strong tower. The Name of the Lord signifies all that God is in himself - his attributes, his love, mercy, power, knowledge which allow man to regard him as a sure Refuge. "Thou hast been a Shelter for me," says the psalmist (Psalm 61:3), "and a strong Tower from the enemy." The words bring before us a picture of a capitol, or central fortress, in which, at times of danger, the surrounding population could take refuge. Into this Name we Christians are baptized and trusting in it, and doing the duties to which our profession calls, with faith and prayer, we are safe in the storms of life and the attacks of spiritual enemies. The righteous runneth into it (the tower), and is safe literally, is set on high exaltabitur , Vulgate he reaches a position where he in set above the trouble or the danger that besets him. Thus St. Peter, speaking of Christ, exclaims (Acts 4:12), "Neither is there salvation in any other for there is none other Name under heaven given among men, whereby we must be saved." "Prayer," says Tertullian ('De Orat.,' 29), "is the wall of faith, our arms and weapons against man who is always watching us. Therefore let us never go unarmed, night or day. Under the arms of prayer let us guard the standard of our Leader let us wait for the angel's trumpet, praying." Septuagint, "From the greatness of his might is the Name of the Lord and running unto it the righteous are exalted."


How to Download and Play PUBG MOBILE – Traverse on PC

Download and install BlueStacks on your PC

Complete Google sign-in to access the Play Store, or do it later

Look for PUBG MOBILE – Traverse in the search bar at the top right corner

Click to install PUBG MOBILE – Traverse from the search results

Complete Google sign-in (if you skipped step 2) to install PUBG MOBILE – Traverse

Click the PUBG MOBILE – Traverse icon on the home screen to start playing

Watch Video

Bluestacks is the groundbreaking app player that lets you enjoy the hottest Android games and apps on your computer or laptop. Take your gaming skills to the next level with a host of jaw-dropping features, like Real-time Translation, support for Macros and Scripts, and GPU Acceleration.

A single mistimed action can mean instant defeat. Don’t put your victory at risk by gaming with clunky touch controls. Advanced Keymapping lets you game with your mouse, keyboard, or gamepad for unparalleled precision and responsiveness.

Automate repetitive combos and tasks with Macros. Record your input sequence — doesn’t matter how complex! — and assign it to a key or button for instant, flawless execution.

BlueStacks makes it easy to record every second of your gameplay to standard format video that you can edit and share with friends, family, and fans!

Multiple Instances is bringing true multitasking to your favorite Android apps. Run all of your mobile games and apps at the same time on a single machine. Log in to multiple accounts for faster farming and more efficient grinding.

Get all these features and obliterate the competition when you download PUBG MOBILE – Traverse on PC with BlueStacks. BlueStacks requires a PC or Mac with at least 4 GB of RAM. Now supports simultaneous 32- and 64-bit apps.


Movies Coming Soon to DVD and Streaming

Mickey (Sebastian Stan) and Chloe (Denise Gough), two Americans in their mid-thirties.

Anne is married to a small-town minister and feels like her life and marriage have.

A young couple (Sawyer Spielberg and Malin Barr) are forced to seek shelter in the.

Nicolas Shaw is a retired U.S. special operative who becomes part of an elite 'invisible'.

In a time when monsters walk the Earth, humanity’s fight for its future sets Godzilla.

A near college graduate, Danielle, gets paid by her sugar daddy and rushes to meet.

Determined to ensure Superman’s (Henry Cavill) ultimate sacrifice was not in vain.


Ezekiel

Personal Responsibility. 1 The word of the LORD came to me: Son of man, 2 what is the meaning of this proverb you recite in the land of Israel:

but the children’s teeth are set on edge”? *

3 As I live—oracle of the Lord GOD: I swear that none of you will ever repeat this proverb in Israel. 4 For all life is mine: the life of the parent is like the life of the child, both are mine. Only the one who sins shall die!

5 If a man is just—if he does what is right, 6 if he does not eat on the mountains, * or raise his eyes to the idols of the house of Israel if he does not defile a neighbor’s wife, or have relations with a woman during her period b 7 c if he oppresses no one, gives back the pledge received for a debt, commits no robbery gives food to the hungry and clothes the naked 8 if he does not lend at interest or exact usury if he refrains from evildoing and makes a fair judgment between two opponents d 9 if he walks by my statutes and is careful to observe my ordinances, that man is just—he shall surely live—oracle of the Lord GOD. e

10 But if he begets a son who is violent and commits murder, or does any of these things, f 11 even though the father does none of them—a son who eats on the mountains, defiles the wife of his neighbor, 12 oppresses the poor and needy, commits robbery, does not give back a pledge, raises his eyes to idols, does abominable things, 13 lends at interest and exacts usury—this son certainly shall not live. Because he practiced all these abominations, he shall surely be put to death his own blood shall be on him. g

14 But, in turn, if he begets a son who sees all the sins his father commits, yet fears and does not imitate him— 15 a son who does not eat on the mountains, or raise his eyes to the idols of the house of Israel, or defile a neighbor’s wife 16 who does not oppress anyone, or exact a pledge, or commit robbery who gives his food to the hungry and clothes the naked h 17 who refrains from evildoing, accepts no interest or usury, but keeps my ordinances and walks in my statutes—this one shall not die for the sins of his father. He shall surely live! 18 Only the father, since he committed extortion and robbed his brother, and did what was not good among his people—he will die because of his sin! 19 You ask: “Why is not the son charged with the guilt of his father?” Because the son has done what is just and right and has been careful to observe all my statutes—he shall surely live! 20 i Only the one who sins shall die. The son shall not be charged with the guilt of his father, nor shall the father be charged with the guilt of his son. Justice belongs to the just, and wickedness to the wicked.

21 But if the wicked man turns away from all the sins he has committed, if he keeps all my statutes and does what is just and right, he shall surely live. He shall not die! 22 None of the crimes he has committed shall be remembered against him he shall live because of the justice he has shown. 23 j Do I find pleasure in the death of the wicked—oracle of the Lord GOD? Do I not rejoice when they turn from their evil way and live?

24 And if the just turn from justice and do evil, like all the abominations the wicked do, can they do this evil and still live? None of the justice they did shall be remembered, because they acted treacherously and committed these sins because of this, they shall die. k 25 l You say, “The LORD’s way is not fair!” * Hear now, house of Israel: Is it my way that is unfair? Are not your ways unfair? 26 When the just turn away from justice to do evil and die, on account of the evil they did they must die. 27 But if the wicked turn from the wickedness they did and do what is right and just, they save their lives 28 since they turned away from all the sins they committed, they shall live they shall not die. 29 But the house of Israel says, “The Lord’s way is not fair!” Is it my way that is not fair, house of Israel? Is it not your ways that are not fair?

30 m Therefore I will judge you, house of Israel, all of you according to your ways—oracle of the Lord GOD. Turn, turn back from all your crimes, that they may not be a cause of sin for you ever again. 31 Cast away from you all the crimes you have committed, and make for yourselves a new heart and a new spirit. Why should you die, house of Israel? n 32 o For I find no pleasure in the death of anyone who dies—oracle of the Lord GOD. Turn back and live!

* [18:2] Parents…on edge : a proverb the people quoted to complain that they were being punished for their ancestors’ sins cf. Jer 31:29.

* [18:6] Eat on the mountains : take part in meals after sacrifice at the high places.

* [18:25] The L ORD ’s way is not fair : this chapter rejects the idea that punishment is transferred from one generation to the next and emphasizes individual responsibility and accountability.


These examples are showing how to parse date in human readable form to unix timestamp in either milliseconds or seconds.

Unix time (also known as POSIX time or Epoch time) is a system for describing instants in time, defined as the number of seconds that have elapsed since 00:00:00 Coordinated Universal Time (UTC), Thursday, 1 January 1970, not counting leap seconds. It is used widely in Unix-like and many other operating systems and file formats. Because it does not handle leap seconds, it is neither a linear representation of time nor a true representation of UTC.


4. DBCA Generate scripts to create database

4.1. Run DBCA :

Open up a terminal by pressing Ctrl+Alt+T and run database configuration assistance tool dbca :

Figure-91: Run database configuration assistance tool 'dbca'

Figure-93: Select 'Create a database operation' and press 'Next'.

Figure-94: Select 'Advanced configuration' mode and press 'Next'.

Figure-95: Select 'Deployment type' and press 'Next'.

Figure-96: Enter 'Global database name', 'SID', 'PDB name' then click 'Next' button.

Figure-97: Select storage option "Use template file for database storage attributes" and click 'Next'.

/> Figure-98: Select Fast Recovery Option "Specify Fast Recovery Area" and click 'Next'

Figure-99: Choose listener to register database and click 'Next'.

Figure-100: Configure Database Vault and Label Security and click 'Next'.

Figure-101: Choose Memory option " Use Automatic Shared Memory Management (ASMM)" and configure "SGA" and "PGA" shared memory size and then hit menu 'Sizing'.

Figure-102: Specify "Processes" Number then hit menu "Character Sets".

Figure-103: Select "Unicode character" set and hit menu "Connection mode"

Figure-104: Select Dedicated server mode and hit "Sample Schemas"

Figure-105: Mark checkbox to "add sample schemas to the database" and then hit "Next".

Figure-106: Select "Configure Enterprise Manager (EM) database express " and specify port. Click 'Next'

Figure-107: Specify passwords for SYS, SYSTEM, PDBADMIN separately or select "Use the same passwords for all accounts". Click 'Next'.

Figure-108: Uncheck "Create database" checkbox to avoid creation of a database and check "Generate database creation scripts" checkbox. Click "Next" button.

/> Figure-109: Click "Save Response File " button.

Figure-110: Double-click "Documents" folder and then save a copy of dbca.rsp file by clicking "Save" button.

Save the response file dbca.rsp in

/Documents directory. This rsp file is needed later.

Figure-111: After saving "dbca.rsp" file, click "Finish" button.

Figure-112: Finish database script generation by clicking "Close" button.


18.10: Movie Scripts 7-8

Nightly automated builds available on avidemux.org.

The binaries available here are freely redistributable (cover mount CD/DVD, download site. ) BUT they must be redistributed as they are. In particular, it means you cannot alter/replace the installer to bundle avidemux with other programs (for example: browser toolbars). Doing so would invalidate your license to redistribute and you would be providing counterfeiting software.

The windows binaries are compiled on linux, the risk of viruses, trojan etc is basically zero. There is no toolbar,spyware etc . in the installer.

CHECK THE MD5SUM!, you never know and it takes 5 seconds.

Linux, source

2.7.8 Final (tar.gz), FossHub,
MD5 : 15e2389c9c526b03bd3779a6a6da9db4
see 2.6.x compilation instructions

Linux 64, Universal binaries

2.7.8 Final (appImage), FossHub
77287ae32a94d95c9540eb6f4abca568 :
(dont forget to do "chmod +x avidemux_2.7.xxxxx.appImage" !).

Ubuntu &mdash There is a PPA available, courtesy of xtradeb on the forum

Windows

If it crashes after loading a video, disable openGl in preferences.

win32 (XP)2.7.4 Final Install (32 bits), FossHub,

win64/VC++ 2.7.8 Final Install (64 bits), FossHub,
MD5: c1a43501c9f14502d7744a7b255ac15c

win64 2.6.2 Install (64 bits), SourceForge,Berlios,
MD5: af3bb48ee9c48615b866daadee930df8

win64 2.6.2 Zip (64 bits), SourceForge,Berlios,
MD5: c6c58e1c7c40df66f7511de90e0d6a14 -->

Mac OS X

2.7.8 Catalina QT5 (dmg), FossHub,
MD5 : 0950b1bf1e1b8c0e951d5eaf8f09a4ac


Info on how to build 2.6.0 bash script.
Info on how to build 2.5.6 or 2.6.0 with homebrew.


1 Double Indemnity1944

In the annals of crime fiction, one of the most legendary writers is Raymond Chandler. He created the character private eye Philip Marlowe, who appeared in Chandler&rsquos books The Big Sleep, The Long Goodbye, and The Lady in the Lake. Not only did he write some amazing novels, but he also adapted the screenplays for some of the best-known crime films, including the dark film noir Double Indemnity, which was directed by Billy Wilder and is considered a classic today.

The Easter egg is a cameo by Chandler, who was a reclusive author. At the 16-minute mark of the film, main character Walter Neff (Fred MacMurray) walks by a man sitting on a bench reading a paperback. The cameo is so minor that we have no record of anyone noticing it until January 2009, 65 years after the movie was released.

One reason no one discovered it was that cameos like this hardly ever happened back then. Also, while Chandler and Wilder admired each other for their artistic abilities, they deeply disliked each other and rarely spoke positively about one another, making it very surprising that Chandler would be featured.

Robert Grimminck is a Canadian freelance writer. You can friend him on Facebook, follow him on Twitter, or visit his website.


Watch the video: Top 10 Best Academy Award Original Screenplays (September 2021).