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Fractals/Computer graphic techniques/2D/exp

=Disambiguation= Exponential mapping of the plane is:
 * is exponential transformation of the plane

Exponential mapping of the plane  is not
 * the Julia set or Mandelbrot set of the exponential function ( exponentia map), like Exponential Mandelbrot z := exp(z) + c
 * Logarithmic scale on one axis
 * Logarithmic scale on both axes: Log-log scale plot
 * polar azimuthal equidistant projection ( Exponential map in Riemannian geometry)
 * The exponential operator, anamorphosis operator which can be applied to grayscale images.

Compare exponential function by different input
 * single number ( 0D space) gives natural exponent of the number
 * number line ( 1D space). Exponential scale is not used. Logarithmic scale is used for exponential data. It gives a linear function.
 * plane ( 2D space) gives exponential mapping

number
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number x to the base b is the exponent to which b must be raised, to produce x

 * $$y = \log_b x$$

is equivalent to

 * $$x=b^y$$

if $b$ is a positive real number. (If $b$ is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.)

The real exponential function is a bijection from $$\mathbb{R}$$ to $$(0;\infty)$$. Its inverse function is the natural logarithm, denoted $$\ln, \log, \log_e;$$ because of this, some old texts refer to the exponential function as the antilogarithm.

scale
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a scale is nonlinear: the numbers 10 and 20, and 60 and 70, are not the same distance apart on a log scale. Rather, the numbers 10 and 100, and 60 and 600 are equally spaced. Thus moving a unit of distance along the scale means the number has been multiplied by 10 (or some other fixed factor). Often exponential growth curves are displayed on a log scale, otherwise they would increase too quickly to fit within a small graph. Another way to think about it is that the number of digits of the data grows at a constant rate. For example, the numbers 10, 100, 1000, and 10000 are equally spaced on a log scale, because their numbers of digits is going up by 1 each time: 2, 3, 4, and 5 digits. In this way, adding two digits multiplies the quantity measured on the log scale by a factor of 100.

Logarithmic scale
 * on 1 axis = semi-log plot
 * linear-log plot
 * log-linear plot
 * on both axes = log-log plot

Logarithmic scale is used for exponential data. It gives a linear function.

mapping

 * GRAPHICS FOR COMPLEX ANALYSIS by Douglas N. Arnold:
 * "the complex exponential maps the infinite open strip bounded by the horizontal lines through +/- pi i one-to-one onto the plane minus the negative real axis.
 * The lines of constant real part are mapped to circles, and lines of constant imaginary part to rays from the origin.
 * In the animation we view a rectangle in the strip rather than the entire strip, so the region covered is an annulus minus the negative real axis.
 * The inner boundary of the annulus is so close to the origin as to be barely visible.
 * We also make the strip a bit thinner than 2 pi, so that the annulus does not quite close up."
 * the complex exponential map takes the rectangle |Re(z)| ≤ a, |Im(z)| ≤ π onto the annulus $$e^{-a} \le |w| \le e^a$$
 * Composition of complex Mappings Viewer by Paul Falstad - the composition of selected mapping functions:  g(f(z)). One have to increase grid size to have tha same effect ( rectangle to annulus mapping)
 * The Complex Exponential Function Author:Susan Addington
 * Complex mappings Author:Juan Carlos Ponce Campuzano
 * Complex exponential function Author:Tatsuyoshi Hamada : Complex exponential function maps rectangular domain to fan shaped image ( part or full sector of punctured circle )
 * Complex exponential map by Siamak on geogebra:
 * vertical axis is transformed to the circle on the w-plane
 * horizontal axis is is transformed to the radii on the w-plane
 * Images of a square grid of size [-3,3]x[-3,3] under the (conformal) exponential map givers almost full annulus
 * The complex exponential function mapping biholomorphically a rectangle [-1,1]x[0 ,pi/2] to a quarter-annulus $$[ e^{-1}, e]x[ e^{-1}, e]$$
 * 24 Views of the Complex Exponential Function by Pacific Tech
 * mathematica question: how-do-i-visualize-expz-as-a-complex-mapping

 * virtual math museum: ConformalMaps, complex exponential function:"a Cartesian Grid is mapped “conformally” (i.e., preserving angles) to a Polar Grid: the parallels to the real axis are mapped to radial lines, and segments of length 2π that are parallel to the imaginary axis are mapped to circles around 0."

exp(x + i · y) = exp(x) · exp(i · y) = exp(x) · cos(y) + i · exp(x) · sin(y)

period
The exponential function is periodic:

exp(z + 2 π * i) = exp(z). The periodicity is also clear from the formula:

exp(x + i * y) = exp(x) * (cos(y) + i * sin(y)).

The graph of the exponential function is a two-dimensional surface ( x,y) that curves through four dimensions ( x,y,v,w).

=Name=
 * exponential mapping
 * the exponential map
 * Mercator ( it is not good name because here is only mapping from flat complex plane ( 2D Cartesian plane) to exponential plane. Here there is no mapping from sphere to flat complex plane ( Cartesian plane) )
 * Mercator map
 * Mercator projection
 * polar projection
 * log-polar mapping ( Log-polar coordinates )
 * logarithmic
 * logarithmic projection around a point c0: z-> (log(|z-c0|)
 * log(z)-Mandelbrot-Zooms = side scrolling ( side scrolling fractal zoom )

General names:
 * graphical projection
 * geometric transformation

=Description=

Features
Maps
 * Rectangle of Cartesian plane to quadratic sectors of polar coordinate,
 * x is mapped to radius, and y to angle ( radians)

 * horizontal
 * maps $$c_0$$ to infinity ( limit of the sequence = MF = the Myrberg-Feigenbaum point = the Feigenbaum Point)
 * maps important components related with above sequence to the same size
 * vertical
 * vertical coordinate are periodic of period $$2\pi$$, because they use trigonometric functions

Examples:
 * sequence of root points for periods $$p = 1*2^n$$ ( period doubling cascade) and the limit point of the sequence  is the Myrberg-Feigenbaum point $$\mathbf{MF}_{1/2} = c_0 = -1.401155...$$
 * sequence of root points for periods $$p = 1*3^n$$ and the limit point of the sequence is the Myrberg-Feigenbaum point $$\mathbf{MF} = c_0 =  -1.7864402555636388747...$$

Informal description

 * The exponential mapping transforms the entire complex plane into a strip that has unlimited length along the real axis, and a width of 2π along the imaginary axis. Exponential Map from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2022
 * " The idea is to focus on a point in or near the Mandelbrot set, and create an image where one direction is the logarithm of the distance and the other one the angle. The result is very much like the astro-ph/031571 map and theXKCD cartoon versions. This map projection is conformal, so it does not distort local angles, and objects are usually recognizable on all scales." Anders Sandberg
 * Images of the Mandelbrot set with a logarithmic projection around a point c0: z-> (log(|z-c0|), arg(z-c0)). Anders Sandberg
 * "it is a log map toward the target point (or, as some might say, a Mercator projection with the target point as South pole and complex ∞ as North pole); horizontally it is periodic and I have placed two periods side to side, whereas vertically it extends to infinity at the top and at the bottom, which corresponds to zooming infinitely far out or in, at a factor of exp(2π)≈535.5 for every size of a horizontal period. Horizontal lines (“parallels”) on the log map correspond to concentric circles around the target point, and vertical lines to radii emanating from it; and the anamorphosis preserves angles." David Madore
 * "The coordinate system is such that the angular component goes to the y-axis, and the radius goes to the x-axis of the resulting image. In addition, the x-axis (radius) is normalized with exp-function so that angles are preserved in this mapping. " Mika Seppä
 * log : "To illustrate the complexity of the boundary of the Mandelbrot set, Figure 8 renders the image of dM under the transformation log(z - c) for a certain c in the boudary of Mandelbrot set. Note the cusp on the main cardioid in the upper right; looking to the left in the figure corresponds to zooming in towards the point c. (Namely, c = -0.39054087... - 0.58678790i... the point on the boundary of the main cardioid corresponding to the golden mean Siegel disk.). Note the cusp on the main cardioid in the upper right; looking to the left in the figure corresponds to zooming in towards the point c. "
 * "Legendary side scrolling fractal zoom. 1 Month + (Interpolator+Video Editor) = Log(z). This means logarithmic projection for this location, that gives this interesting side-scrolling plane ^^)"
 * " There are no program that can render this fractal on log(Z) plane. But you can make it in Ultra Fractal or in similar software with programmable distributive. Formula is:C = exp(D), for D - is your zoomable coordinates" SeryZone X
 * just c = c0 + cexp(x + i y) and x + i y = clog(c - c0), with scaling by 2pi/width (Claude)
 * idea
 * A Map of the Universe by J. Richard Gott III, Mario Jurić, David Schlegel, Fiona Hoyle, Michael Vogeley, Max Tegmark, Neta Bahcall, Jon Brinkmann
 * XKCD cartoon versions

Formal mathematical description
Forward transformation

Here transformation is a composition of 2 transformations:
 * complex exponential function: $$ f(c) = e^c $$
 * geometric or complex translation $$t(x) = x + c_0$$

$$ c_e = t(f(c)) $$

 * $$c_e \leftarrow e^c+c_0$$

where:
 * c is a parameter point from c plane ( flat image in Cartesian coordinate = linear scale on both axes). It is a a parameter of quadratic map $$z^2 + c $$
 * $$c_e$$ is a parameter in a new transformed plane ( log-polar image )
 * $$c_0$$ is a constant parameter (for translation). It is sometimes called center, but better name seems to be: target or limit point
 * e is Euler's number, is a mathematical constant approximately equal to 2.71828,

Inverse transformation:

 * $$ c = ln(c_e - c_0)$$

where:
 * $$ln$$ is natural logarithm

exp
Complex exponential map

The complex exponential function is periodic with period $log_{b} b = 1$ and $$\exp(z+2\pi i k)=\exp z$$ holds for all $$z \in \mathbb{C}, k \in \mathbb{Z}$$.

The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. Two special cases exist:
 * when the original line is parallel to the real axis, the resulting spiral never closes in on itself;
 * when the original line is parallel to the imaginary axis, the resulting spiral is a circle of some radius.

Considering the complex exponential function as a function involving four real variables: $$v + i w = \exp(x + i y)$$ the graph of the exponential function is a two-dimensional surface curving through four dimensions.

Starting with a color-coded portion of the $$xy$$ domain, the following are depictions of the graph as variously projected into two or three dimensions.

The second image shows how the domain complex plane is mapped into the range complex plane:
 * zero is mapped to 1
 * the real $$x$$ axis is mapped to the positive real $$v$$ axis
 * the imaginary $$y$$ axis is wrapped around the unit circle at a constant angular rate
 * values with negative real parts are mapped inside the unit circle
 * values with positive real parts are mapped outside of the unit circle
 * values with a constant real part are mapped to circles centered at zero
 * values with a constant imaginary part are mapped to rays extending from zero

The third and fourth images show how the graph in the second image extends into one of the other two dimensions not shown in the second image.

The third image shows the graph extended along the real $$x$$ axis. It shows the graph is a surface of revolution about the $$x$$ axis of the graph of the real exponential function, producing a horn or funnel shape.

The fourth image shows the graph extended along the imaginary $$y$$ axis. It shows that the graph's surface for positive and negative $$y$$ values doesn't really meet along the negative real $$v$$ axis, but instead forms a spiral surface about the $$y$$ axis. Because its $$y$$ values have been extended to $log_{b} 1 = 0$, this image also better depicts the 2π periodicity in the imaginary $$y$$ value.

Log
Complex logarithm as a conformal map

Since a branch of $$\log z$$ is holomorphic, and since its derivative $$\log z$$ is never 0, it defines a conformal map.

For example, the principal branch $$w = \operatorname{Log} z$$, viewed as a mapping from $$\mathbb{C}-\mathbb{R}_{\le 0}$$ to the horizontal strip defined by $$\left| \operatorname{Im}z \right| < \pi$$, has the following properties, which are direct consequences of the formula in terms of polar form:
 * Circles in the z-plane centered at 0 are mapped to vertical segments in the w-plane connecting $$a - \pi i$$ to $$a + \pi i$$, where $$a$$ is the real log of the radius of the circle.
 * Rays emanating from 0 in the z-plane are mapped to horizontal lines in the w-plane.

Each circle and ray in the z-plane as above meet at a right angle. Their images under Log are a vertical segment and a horizontal line (respectively) in the w-plane, and these too meet at a right angle. This is an illustration of the conformal property of Log.

description for programmers
Filter Mercator from MathMap filter mercator (image in) in(xy*xy:[cos(pi/2/Y*y),1]) end

Maxima CAS src code:

(%i1) kill(all); (%i2) display2d:false; (%i3) ratprint : false; /* remove "rat :replaced " */ (%i4) c:cx +cy*%i$ (%i5) c0:c0x+c0y*%i$ (%i6) realpart(c0 + exp(c)); (%o6) %e^cx*cos(cy)+c0x (%i7) imagpart(c0 + exp(c)); (%o7) %e^cx*sin(cy)+c0y (%i8) cabs(c0 + exp(c)); (%o8) sqrt((%e^cx*sin(cy)+c0y)^2+(%e^cx*cos(cy)+c0x)^2) (%i9) carg(c0 + exp(c)); (%o9) atan2(%e^cx*sin(cy)+c0y,%e^cx*cos(cy)+c0x)

Notes
 * the mapping is periodic because there are trigonometric functions inside

(%i1) kill(all); (%i2) display2d:false; (%i3) ratprint : false; /* remove "rat :replaced " */ (%i4) ce:ceex+cey*%i; (%i5) c0:c0x+c0y*%i; (%i6) realpart(log(ce - c0)); (%o6) log((cey-c0y)^2+(ceex-c0x)^2)/2 (%i7) imagpart(log(ce - c0)); (%o7) atan2(cey-c0y,ceex-c0x)

center
$$c_0$$ is a constant parameter (for translation = offset).

It is sometimes called center, but better name seems to be:
 * target
 * limit point
 * offset or translation

The target point ( center) is projected onto -∞+0i in the exponential-mapped view. It would be intinitely far to the left and it is not seen on the image.

zoom

 * the leftmost column of the image has the minimal zoom ( the biggest plane radius )
 * the rightmost column of the image has the maximal zoom ( the smallet plane radius )

exponential grid scan
Conventions:
 * Robert Munafo generate the angle based on the horizontal pixel coordinate, and the radius from the vertical pixel coordinate ( image is in the vertical position, it is standing on the shorter dimension)
 * the normal way is to have the angle come from the vertical pixel coordinate ( image is in the horizontal position, it is lying on the longer dimension)

The grid scan with exponential coordiante mapping is different then the standard scan for flat images.

Description
 * image_width is the maximal number of pixels = width of the image in pixels ( integer number)

Here:
 * pixel coordinate j is going from 0 to image_height - 1
 * so j/pixel_height goes from 0.0 to 1.0
 * therfore angle goes from 0 to 2 pi

Claude's way is mathematically correct: the angle is usually thought of as going from -π to π, not from 0 to 2π. To fix this part, you can adjust your pixel coordinate right before computing an angle from it. Change this:

double e_angle = ((double) j) * px / r * 3.14159265359;

to this:

double e_angle = ((double) (j-h/2)) * px / r * 3.14159265359;

Notes
 * e_radius is a linear function of i
 * e_angle is a linear function of j

computing pixel coordinate
Here is example by by Robert Munafo

Compare with
 * exponential strip by Claude Heiland-Allen: The rendering of the final video can be accelerated by computing exponentially spaced rings around the zoom center, before reprojecting to a sequence of flat images. In case of problems : https://mathr.co.uk/mandelbrot/exponential_strip.svg

aspect ratio

 * For best (more conformal) results use a wide aspect ratio (9:1 works well, window size 1152x128 or 1536x170).

scale factor R
To minimize the total number of pixels that need calculating, which is proportional to

$$ K = \frac{R^2 - 1}{log R}$$

Define the efficiency of zoom by

$$\frac{2}{K}$$ then:
 * the traditional R=2 has an efficiency of only 46%,
 * R=4/3 has 74%
 * R=8/7 has 87% efficacy

where
 * video image size is W×H

=Videos=
 * Mercator - Scrolling Mandelbrot Fractal by Mathtown
 * Mappings by the exponential function by William Nesse
 * Mapping properties of the exponential function by MatheMagician
 * The function f(z)=exp(z) by Michael Robinson
 * Complex: visualizing exponential map and local inverse by James Cook
 * Mandelbrot Mercator Testing by Fractal MathPro

=Images=
 * Category:Mandelbrot sets - Mercator projection in commons

=How to read location from exponential image ?=
 * fractalforums.org : can-you-find-the-location-of-this-fractal ?
 * "An inverse mapping (using the complex logarithm function) is used to compute the location (in the strip) of needed data for any given location in the normal orthogonal coordinate space." Robert Munafo

Tips:
 * the image is not symmetric ( up and down) so imaginary part of c0 is not zero

=Implementations= Fractal programs
 * fraktaler-3
 * UltraFractal
 * Polar_Projection in fractalzoomer
 * Fractalshades - python program by Geoffroy Billotey
 * emndl: exponential strip visualisation of the Mandelbrot set, program by Claude Heiland-Allen
 * pfract - console c program by Mika Seppä
 * emndl: mathr's exponential strip mandelbrot set renderer: it is: c = c0 + cexp(x + i y) and x + i y = clog(c - c0), with scaling by 2pi/width
 * Fractalshades doc : P01-feigenbaum expmap

Kalles Fractaler and zoomasm
 * The rendering of the final video can be accelerated by computing exponentially spaced rings around the zoom center, before reprojecting to a sequence of flat images.
 * kf-2.15 supports rendering EXR keyframes in exponential map form
 * zoomasm can assemble above keyframes into a zoom video. zoomasm works from EXR, including raw iteration data, and colouring algorithms can be written in OpenGL shader source code fragments
 * kf-extras by Claude Heiland-Allen - has the exponential map (aka log polar or mercator projection) convertor

OpenCV
 * opencv 3.4 warpPolar function
 * logpolar transform using Camera by Aresh T. Saharkhiz
 * stackoverflow question: reprojecting-polar-to-cartesian-grid
 * descr
 * stackoverflow question: how-to-tune-log-polar-mapping-in-opencv
 * stackoverflow question : converting-an-image-from-cartesian-to-polar-limb-darkening

ImageMagic
 * pano2fisheye - Transforms a spherical panorama to a fisheye view = ? Reverse polar coordinates?
 * distorts depolar
 * snibgo's ImageMagick pages: Polar distortions

processing
 * forum.processing : how-to-change-an-image-from-x-y-coordinate-to-polar-coordinate

sci-kit
 * Using Polar and Log-Polar Transformations for Registration
 * skimage.transform : wrp_polar

Mathworks
 * Log-polar image sampling version 1.0.0.0 (4.33 KB) by David Young from mathworks

glsl
A typical shader looks like this ( code by Thorsten Förstemann):

Python

 * stackoverflow question: image-information-along-a-polar-coordinate-system

The images created in this way are also called:
 * Exponential Maps
 * Mercator-Mandelbrot Maps
 * Mercator projection of the Riemann number ball.

In the following program example, a logarithmic projection of the Mandelbrot set is calculated ,

The Mandelbrot set is calculated with NumPy using complex matrices. A logarithmic projection presented by David Madore and Anders Sandberg is used. This projection makes the calculation of zoom animations much easier

The logarithmic projection allows the creation of zoom animations of the Mandelbrot set to be extremely accelerated (see also the animation by Thorsten Förstemann and the coordinate analysis by Claude Heiland-Allen

When estimating the distance, the derivative $$dz_n = z_n\!'(c)$$ is also required.

The first polynomials are
 * $$z_0 = 0$$
 * $$z_1 = c$$
 * $$z_2 = c^2+c$$
 * $$z_3 = (c^2+c)^2+c$$,

the corresponding derivatives are
 * $$dz_0 = 0$$
 * $$dz_1 = 1$$
 * $$dz_2 = 2c+1$$
 * $$dz_3 = 2(c^2+c)(2c +1)+1$$.

All other polynomials and derivatives result from the iteration rule $$z_{k+1} = z_k^2+c$$ and the derivation rule.

To use the estimation formula

 * $$\delta_n = \log(|z_n|) \cdot |z_n| / |dz_n|$$

you can use the simplified sequences $$z_{k+1}=z_k^2+0$$ and $$dz_{k+1} = 2z_k dz_k+0$$ with the starting values can be considered.
 * $$z_0 = c$$
 * $$dz_0 = 1$$

The first polynomials are
 * $$z_0 = c$$
 * $$z_1 = c^2$$
 * $$z_2 = c^4$$
 * $$z_3 = c^8$$,

the corresponding derivatives are $$dz_0 = 1$$, $$ dz_1 = 2c$$, $$dz_2 = 4c^3$$ and $$dz_3 = 8c^7$$.

In general, this is how you get the polynomial $$z_n = c^{2^n}$$ and the derivative $$dz_n = 2^n c^{2^n-1}$$.

The associated Julia set is exactly the edge of the unit circle (cf. the Example on the dynamics of f(z) = z² in the article on Julia -Quantity, the short justification in van den Doel or the comprehensive analysis in Dang, Kauffman and Sandin).

The estimation formula results in

 * $$\delta_n = \log(|c|^{2^n}) \cdot |c|^{2^n} / (2^n |c|^{2^n -1}) = \ln(|c|) \cdot |c|$$

which is a good approximation for the distance to the unit circle when the point $$c$$ is close to the edge:
 * $$|c| = 1{,}1$$

gives
 * $$\delta_n = \ln(1{,}1) \cdot 1{,}1 \approx 0{,}1$$

and

 * $$|c | = 1{,}01$$ gives $$\delta_n = \ln(1{,}01) \cdot 1{,}01 \approx 0{,}01$$.

So if the real distance to the Mandelbrot set is $$\delta$$, the estimation formula gives the value

 * $$\delta_n = \log(|z_n|) \cdot |z_n| / |dz_n| \approx \ln(1+\delta) \cdot (1+\delta) \approx \delta$$.

However, it is not true that the limit $$\lim_{n \to \infty} \delta_n$$ converges to the real distance $$\delta$$; in fact, only weaker inequalities apply. You will often find a factor $$2$$ or $$0{.}5$$ in the estimation formula, depending on whether the distances are overestimated or underestimated.

=References=


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Main Text Content Parameters

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license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

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Wikijunior:Raspberry Pi/Raspberry Pi Python Individually Controllable LEDs (ws2811)

Tutorial by Andrew Oakley & Jonathan Teague - Public Domain 2020-03-14 www.cotswoldjam.org

This tutorial will cover using individually controllable strings of LEDs based on the ws2811 controller, also known as 'NeoPixels'.

The kit
In your bag you will find the following components:

 * ×3 M-F Jumper leads (pin to socket)
 * ×1 3-way 3A connector strip
 * ×1 String of 3 5V ws2811 LEDs

Putting it together
Step 1: Take the jumper lead that goes to the WHITE wire to the LEDs and plug the socket (female end) into pin 6 (GND) on the Pi.

Step 2: Take the jumper lead that goes to the RED wire to the LEDs and plug the socket (female end) into pin 2 (5V) on the Pi.

Step 3: Take the jumper lead that goes to the GREEN wire to the LEDs and plug the socket (female end) into pin 12 (GPIO 18) on the Pi.

The program
Power up your Raspberry Pi. From the desktop menu, select Programming – Python 3 (IDLE). Then use File, New File to create a new program.

Type in the following program then use File, Save to save your program as the name of your choice (don't forget to put .py on the end).

You can also find this program in the Pi Home folder

Running your program.

To run the program start a Terminal – click the icon on the top then once the terminal window opens type:

If it's worked you should see the 3 LEDs light up red, green and blue for two seconds and then turn off.

What does the program do?
The imports load the NeoPixels libraries (and time so we can insert a delay).

This tells the library we've connected our LED string to GPIO 18, there are 3 LEDs and  means only update the LEDs when we call. Lastly, we note that the colours are in the order Red, Green, Blue (RGB).

This sets the first LED to RED, the 3 values are (R, G, B) – so each of the 3 LEDs lights up a different colour.

This sets all the pixels to 0 which means off, if we don't do this the LEDs will stay on.

What next?
Now do your own thing! Make a disco sequence!

How about a traffic light sequence? Note that yellow light is a mix of red and green light, e.g: (255,255,0)

Hint: Put it in a  block so it goes on forever. And don’t forget to indent your code within the while block like this line is!

Hold down the CTRL key and press C to stop the  sequence.

You can find an example program in the Pi Home folder.

How about a sequence of lights that measures 20 seconds, the time you need to do a good job of washing your hands?

You can find an example program in the Pi Home folder.

If you stop your program and the lights are still on, you can use the  program to turn them off.

How does it work?
Each LED is connected to a ws2811 integrated circuit (IC) – it's the IC that's called a ws2811, the LED is just a three-colour LED. The Pi generates a long sequence of 0's and 1's. 24 bits for each LED (3 sets of 8-bits for each colour Red|Green|Blue), so for our string of 3 LED the Pi will generate 3 × 24 bits = 72 bits.

The first device in the string takes the fist 24 bits and uses those to set the Red|Green|Blue on its LED. It then passes the remaining 48 bits on to the second LED. That in turn takes what are now the first 24 bits from what it has received as its RGB and passes on the remaning 24 bits to the last LED.

For full details the datasheet is here: http://www.world-semi.com/DownLoadFile/129

Making it work at home
If you're not using our ready-made SD card, you need to do some setup from Raspbian Buster:

 * 1) Raspberry menu – Preferences – Raspberry Pi Configuration – Interfaces – SPI enable – I2C enable – OK
 * 2) Raspberry menu – Shutdown – Reboot

Then install the libraries:

You can download our sample code from: www.cotswoldjam.org/downloads/2020-03

Notes on using ws2811 in the real world
We’ve only given you a short string of LEDs as that’s all the Pi can reliably power. If you're using a long strip of ws2811 or ws2812 then there are two things you'll need to consider.

 * 1) ws2811 strips can be 5V or 12V, ws2812 are 5V. The Pi can only provide 5V at a low current sufficient to drive a few LEDs. To drive long strips at 5V or any at 12V you'll need to provide additional power, possibly at multiple points along the strip. If you're going to run a strip all round your bedroom ceiling 😊 then the Pi is only going to provide the control logic for the LEDs not the power.
 * 2) The control logic for ws2811 & ws2812 is specified at 5V, the Pi GPIO pins are 3.3V. For this tutorial we get away with using GPIO but to be sure it's going to work you would probably want to use a logic level shifter to turn the Pi's 3.3V into 5V.


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Pascal Programming/Examples

How Pascal can solve your problems?

= Printer driver in PCL level III= Let's say we have a printer that our operating system does not support (there is no driver). What to do ? Write your own driver in Pascal.

PCL (Printer Command Language) is a page description language (PDL) developed by Hewlett-Packard as a printer protocol and has become a de facto industry standard.

The following module allows you to print, but only black and white.

title: Pascal Programming/Examples


text:
AI Art Generation Handbook/How does AI generate art ?

The rise of artificial intelligence has led to a surge in the creation and enhancement of AI-generated images (especially in 2022, after the release of DALL-E2 ) resulting in highly detailed and imaginative artwork. This development might prompt user to ask, “How does AI produce art?"

Humans usually draw inspiration from their surroundings—such as forests, urban landscapes, and their own reflections to put their inspirations into their art.

Similarly, AI art involves the creation of artwork with the help of generative AI. This technology identifies patterns within large datasets and uses this knowledge to generate new content. To create AI art, one needs an AI art generator, like Stable Diffusion, and a concept. The AI artist inputs a detailed prompt, which the tool then interprets to generate image options based on the description provided.

Artist  Vera Molnár , a Hungrarian artist began experimenting with early programming languages to produce randomly generated artwork in 1968. Considered a pioneer in generative art, her geometric creations are included in major museum collections. She is considered as a pioneer in generative art, her geometric creations are included in major museum collections.

The core technology behind this capability is called a neural network. A neural network is a sophisticated mathematical system, or algorithm, that is designed to mimics biological neural networks in the human brain and functions to recognize patterns in extensive datasets.

The neural network have few components: [3]

(i) Input Layer: This layer receives the initial input data, such as an image, text, or numerical values.

(ii) Hidden Layer: They are intermediate layers between the input and output layers, where most of the mathematical operations (e.g matrix multiplications) occurs.

(iii) Output Layer: This layer produces the final output of the neural network, which is the generated output (e.g., an image or text).

(iv) Connections (Weights and Biases): Neurons in adjacent layers are connected by weights, which determine the strength of the connections. Biases are additional parameters that shift the activation of a neuron.

When a user prompt an AI generator to depict a dog, the neural network draws on the vast amount of information it has been trained on to create a new image. This process involves complex layers of interconnected nodes that simulate the way a human brain processes information. The role as the AI artist is to refine these generated images, guiding the AI models to produce specific scenes, such as a dog impersonate a French wearing beret, dog sitting in a bar or a dog dancing in a kitchen. These neural networks are brimming with the trained data, but it is the creativity and direction from the user that truly shapes AI-generated art.

There are 2 major parts of AI Art : Training and Inferences [4]

Training:Training is the process of teaching the neural network model to learn the patterns and relationships present in the training data.

Inference: Inference is the process of using the trained model to make predictions or generate outputs on new, previously unseen data.

As in the field of artificial intelligence, the training data is the heart of generative AI. However, as saying goes, quality over quantity. All in all, general consensus is the quality of the image aesthetic and better image description / captioning is much better than the "token" of AI model it is able to be processed. All of the AI model is trained using pair of text captions - images ,which one of the most popular method is using CLIP [5]. Here is the the known image dataset training size The picture shows how the AI Text-to-Art  oversimplifications  of process flow chart of converting from words and process it until the images is produced.

(A) The prompt text of the input in words form are being "tokenized" into tokens by the Text Encoders

(B) Then tokens are mapped to a dense vector representation (embedding vector), capturing semantic and contextual information about the token.

(C) The Diffusion Model are generating the images with the forward diffusion techniques based on the embedding vectors

(D) The dense vector representation (latent vector) represent the latent image representation essential visual features, content, and attributes of the image to be generated, compressed into a latent space representation.

(D) Image Decoders are synthesizing visual features like textures, colors, shapes, etc. based on the information encoded in latent vector. After synthetizing the images, it will upscaling the images whilst performing enhancement to increase image aesthetic.


text:
Template:DDiv

Parameters

 * The heading text: It should be placed immediately after the first cell bar, on the same line.
 * The code lines:  The entire code block should be placed in the line following the second cell bar, then apply Numbered List to it.

Example

 * Note: Code lines MUST be given numbers using the numbered lists function in toolbars. The template is made for adding numbered lines to code, and cannot be used without the  addition of numbers to stabilise the pre-formatting.
 * Refer to the example below for the layout to use.
 * Do not use parameter numbers. The first parameter is the heading and the second the code block.
 * To avoid distortion of the layout do not deviate from that below. That is; place the heading between two bars and the entire code block on the next line.
 * Select all of the lines of code and apply the Numbered List to it, from the toolbar.
 * As an alternative to numbered lists, use tags around the entire code block like > instead, and if no line numbers are needed, just miss out the line attribute, and possibly the highlight attribute.

title: Template:DDiv


text:
User:Maxmasterton

A Sixth-Form Student from Surrey, England with an interest in Mathematics

title: User:Maxmasterton


text:
Cookbook:Each

The term each is a unit of measurement equivalent to one whole unit of a designated item. It is abbreviated as ea. For example, instead of writing "2 lemons" a cook may write "2 ea. lemons" or "lemons, 2 ea." to designate two whole lemons. It is convenient to use eaches when listing multiple units next to each other, such as in "5 ea. / 1 lb / 450 g carrots," to avoid confusion by the omission of a unit.

title: Cookbook:Each


text:
User talk:2620:22:4000:1201:185A:3C8A:301B:5059

This was done by CarletonGabriel I just forgot to login
Title CarletonGabriel (discuss • contribs) 21:47, 18 November 2022 (UTC)


text:
Geometry Dash: History of Exterme Demons/Element 111 Rg

This is part of the Extreme Demon Series that I'm making. More coming soon! Anyways, let us begin with some context.

In the GW (Geometry World) community, a creator named GW Arco wanted to start a series of levels that were based on the periodic table. For example, the first level would be called "Element 1 H" and then the one after that would be called "Element 2 He", and then the one after that would be called "Element 3 Li", and so on, and so on. It didn't have a set order of creators, and anyone could join. Unfortunately, only 28 of the planned 118 levels actually created, with some being never worked on, unfinished, never uploaded, or had no set creator.

One of the creators, Foldering, or DarkX, as his Geometry Dash username was, would be making element levels 111 Rg to 118 Uuo (In 2014, scientists hadn't named elements 116-118 yet). Element 111 Rg was released on April 6, 2014 by DarkX. A few weeks before, DarkX posted a picture of the level. It showcased a flipped gravity mirror ship part with black colored blocks and a very unique and tight spike arrangement that transitioned into a regular gravity portal and a default block design, which also had some tight flying. And four days after the level was rated, RobTop (the developer of the Geometry Dash) star-rated the level Demon on April 10, 2014. The same day it was rated, players would be complaining about the difficulty of the level, with many posting pictures of the beginning of the level, showing a three-spike corner jump. There was also a picture post of a tight straight-fly part that came right after the part mentioned in the sneak-peak. Suddenly, the next day, random players started beating the level out of nowhere. It was because of a hidden path that abused the hitboxes of the spikes on the blocks, which DarkX probably used to verify the level in the first place. It spread around the GW community; players then started giving DarkX lots of backlash for making a extremely hard and adding a hidden path. Some players suggested that DarkX remove the hidden path and nerf the level. DarkX would comply, and make a nerfed version of the level and block off the hidden path. However, another secret way was discovered somewhere else in the level, which ended up giving DarkX more backlash. RobTop unrated shortly after, on April 15, 2014. DarkX would delete his Geometry Dash account. Soon after, the level would be deleted, probably by RobTop accidentally in his level deleting purge possibly around August 15, 2014.

Around April 13, 2014, a user by the name of Gw Unicorn wanted to review Element 111 Rg. What this user would do was review levels that users sent him, played it, and took images of the levels; Element 111 Rg was one of them. Unicorn talked about how it had recently been nerfed and wanted to play the level. DarkX, however, cancelled the review. However, a while before, Unicorn reviewed a level by DarkX called "Theory of Clutter". Back then, DarkX's username was DarkM, which means that his username probably was a synonym for dark matter. The level is also deleted, so it is unknown how many more levels DarkX made. The level has been restored to some extent, but most of the level has been lost.

In 2017, a Swedish player named Stormfly and a few Korean players, like Pamka, 100, etc., who remembered playing the level, decided to recreate it using pictures, videos, and memory from players. The level was mostly theoretical gameplay, and still is and was released sometime in the middle of 2020. However after that, a lot of new info came out for the level and another level that DarkX made. For one thing, a player named GW Zac began showing a lot of gameplay he remembered from 111 Rg. In Zac's level, Leap of Mistrust, a lot of gameplay in that level was taken from 111 Rg. It was then added to the recreated version of Element 111 Rg. For another, one plsyer named Geome-Try Dasher, uploaded two videos to his YouTube channel supposedly showcasing sections of Element 111 Rg. It is unknown what the level in the videos are, but images of the level secret path shows that the colors in the background align perfectly with the videos. However, the level in the videos have slopes in them, which weren't added until update 1.8, with the level being removed before then. To add more to this anomaly, the videos were recorded in March 2014, a month before the level was actually released. But the parts in the two videos were interpreted into the remake of the level. In 2022, the level was updated again, removing one of the parts that Geome-Try Dasher showcased, and improved parts.


text:
Talk:AI Art Generation Handbook

Stable Diffusion (KIV)
Currently GPU with reasonable VRAM pricing is still out of reach for AI artist hobbyist (For Stable Diffusion usage). Need to wait for the reasonable price for current gen to drop or got a windfall XD ... MrAlanKoh (discuss • contribs) 05:45, 19 November 2022 (UTC)
 * Update 1: Currently just gotten 16GB VRAM GPU and now trying to learn how to train using A1111 FDreambooth but the results seems not good. Trying bmaltais implemetnation instead. Stay tuned!
 * Update 2: Wow, ControlNet is awesome. Unable to keep up with the tools nowadays. MrAlanKoh (discuss • contribs) 02:56, 2 March 2023 (UTC)

Currently pending for SDXL to be released out MrAlanKoh (discuss • contribs) 17:55, 28 June 2023 (UTC)

title: Talk:AI Art Generation Handbook


text:
GLBasic Programming/BASIC/Variables and Data Types

Print " fun "


text:
Scratch/Lessons/Event Triggers

The "When 'Green Flag' clicked" block is an event block. When the green flag block is clicked in the top right corner of the screen, it will start/trigger any "When 'Green Flag' clicked."

title: Scratch/Lessons/Event Triggers


text:
User:Ayten.Alqahtani

= Education in Kingdom of Saudi Arabia =

Objective
This overview of the Saudi Arabian educational system covers alternatives for home schooling, special needs education, and elementary and secondary education.

The history of the Education system in Saudi Arabia
General Education

In Saudi Arabia, the educational system has significantly changed since 1932, when it was founded. During that time, only children from upper-class families had access to education. In Saudi Arabia, there are over 30,000 schools and more than 50 government and private universities. All Saudi nationals are entitled to free education. After the birth of Islam in 632 AD, Muslim states established libraries, schools, and universities that were unlike anything in the world. During this period, the Islamic world made significant contributions in fields such as physics, astronomy, philosophy, and medicine. Europe was in the Dark Ages during this time, and the Islamic world was a center for learning. Scientific methods developed by Muslim scientists and scholars during the Golden Age became the basis of modern scientific research. Saudi Arabia's education system has significantly improved over the years, and it is expected to continue growing in the future. This will allow it to become a more prominent force in the world. It is already capable of providing a wide range of education to its citizens, and there is still room for expansion. The Saudi Arabian government is also heavily involved in the country's education system, as it is the one that created and controls a large portion of the country's educational facilities. According to Al-Rashidi and Phan, education accounts for 25% of the nation's total budget.

The History of Special Education in Saudi Arabia

In Saudi Arabia, the education system had to accommodate the needs of children with various types of disabilities due to the country's growing number of students with these conditions. In 1958, male blindness was first observed, and in 1960, female learners were brought to the teachers' attention. In 1971, the system started providing assistance to those with learning disabilities. There are also resource rooms that can help children with moderate and mild impairments. The country has made significant efforts to ensure that all children have the education they need regardless of their backgrounds. The Saudi government's support for the establishment of special schools and other educational facilities for children with disabilities has been acknowledged by Aldabas. In 2015, the Ministry of Education sponsored various initiatives aimed at helping children with disabilities. Also, as the country's private schools started to emerge, the government had to accommodate students who would have a hard time following traditional methods. According to Al-Mousa, a national facility for the blind was established in 1960. It was designed to accommodate people with various kinds of disabilities. The Ministry of Education is responsible for providing accommodations for students with disabilities. It also continues to implement programs that make learning more accessible to everyone. These include segregating schools and placing children with special needs in general public schools. The implementation of special education programs in Saudi Arabia has accelerated due to the government's increasing resources. According to Alqahtani (2017), children who receive the right interventions perform 21% better than their peers. This is because the establishment of facilities for disabled children and the development of effective assistance programs are both successful. Despite the progress that has been made, there are still many issues that remain regarding the implementation of special education programs in Saudi Arabia.

Intellectual Disability in Saudi Arabia:
A Historical Perspective

The paper aims to look into the history of intellectual disabilities in Saudi Arabia. According to Aldabas, the first step in addressing these issues was in 1971 when the country's government established a special school for those with intellectual disabilities. The Saudi government has since funded and opened various programs and facilities for these individuals. In 2018, AlMedlij and Rubinstein-vila noted that education for those with intellectual disabilities is meant to fulfill the law's requirement for providing them with free and adequate education and rehabilitation services. Initially, there was only one school for those with intellectual disability in the country. Since then, the nation has greatly improved its understanding of this variety. It has become one of the most integrated countries in the region. Saudi Arabia's definition of intellectual disability is different from that of other developed countries due to the country's use of the term learning disability. According to AL-Kahtani, the condition is mainly defined by the presence of differences in an individual's functional performance. It is categorized according to the student's IQ score. According to Madhesh and Alharbi, Saudi Arabia has laws that are in line with international standards, but it is still slow to adopt the necessary approaches. In 2019, Alnhadi noted that the country's children tend to have positive attitudes toward those with intellectual disabilities. In 2011, Alquraini noted that disabled children receive various types of assistance, such as physical and occupational therapy. They are also provided with food, shelter, and money at specialized schools. The curriculum of these schools is similar to that of their peers.

Saudi Arabian and international laws governing special education

In developed countries, there are numerous laws that have been put in place to ensure that children with intellectual and developmental disabilities are treated properly. One of the most prominent laws in the US is the IDEA, which provides free education to children with disabilities. According to Alquraini, this legislation ensures that kids with special needs receive a proper education regardless of their condition. In the US, there are also laws that protect people with special needs from discrimination in their educational activities. One of these is the Rehabilitation Act, which states that people with disabilities can participate in programs and activities that are sponsored by the federal government. This law ensures that children with disabilities are able to learn in schools that are designed to accommodate them. In Saudi Arabia, there is a similar effort to follow the US' lead in developing programs and policies that accommodate people with special needs. In 2011, Alquraini noted that the country's regulations for special education programs are based on the same principles. They provide a comprehensive list of the various types of disability and the interventions that are designed to help people with these conditions.

Conclusion
Lastly, the Saudi Arabian government is committed to ensuring that children with special needs receive the necessary opportunities to learn. According to Alqahtani and Murry, in 2015, the country's education system is geared toward reducing its reliance on foreign workers. This is done through the implementation of the "Education for All" initiative, which aims to provide an education that is appropriate for all students.

title: User:Ayten.Alqahtani


text:
LMIs in Control/Click here to continue/Applications of Linear systems/Design of LMI-Based Adaptive Barrier Global Sliding Mode Fault Tolerant Control for Uncertain Systems with Faulty Actuators

LMI for adaptive barrier global sliding mode control (ABGSMC)

Uncertain systems with faulty actuators can be simplified and modeled via an LMI for adaptive barrier global sliding mode control (ABGSMC). The LMI can be derived by using a global nonlinear sliding surface in order to achieve system stability and the occurrence of sliding when actuator faults are present within the system. This LMI is important for continued research into LMI-based actuator improvements as modern systems are heavily prone to faults (including actuator faults).

The System
The following system is assumed:

$$ \begin{align} \dot{x}(t) &= A x(t) + B L u(t) + D \\ D &= \Delta \  B u(t) + b f(x) + d \\ \end{align} $$

where $$ x(t) = [x1, x2]^{\text{T}} \in R^2 $$ notes the system’s state vector, f(𝑥) is a nonlinear system function, $$ \Delta f $$ and $$ \Delta g $$ are parametric uncertainties and d represents the external disturbances. L is the post-fault coefficient which represents the actuator's effectiveness. We can assume that: $$ \begin{align} \begin{bmatrix} A \end{bmatrix} = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \end{align} $$

$$ \Delta B =  [0, \Delta B_2 ]^{\text{T}} $$

$$ B  =  [0, \Delta B_2^{\text{T}}]^{\text{T}} $$

and $$ L  =  [0,  L_2]^{\text{T}} $$

The Data
The system can then be described as:

$$ \begin{align} \dot{x_1}(t) &= A_{11} x_1(t) + A_{12} x_2(t) \\ \dot{x_2}(t) &= A_{21} x_1(t) + A_{22} x_2(t) + (B_2 L_2) u(t) + D \\ \end{align}

$$

With the state space form described above.

Assumptions
1. Pair (A, B) is totally controllable. Using this assumption for the system, the controllability of $$ (A_{11}, A_{12}) $$ can be obtained by the controllability of (A, B)

2. The parametric uncertainty and external disturbance term Dx is bounded and is less than $$ \eta_k $$

The LMI
Using the Lyapunov function, Schur Complement, the adaptive control law, and the assumption of sliding surfaces an LMI can be brought together to minimize faults via actuators

$$ \begin{align} &\text{min} \quad \eta_k < Dx \\ &\begin{bmatrix} (B_{1}W)^{T} + A+B_{1}W & B_{2} & (D_{1}W)^{T} \\ B_{2}^{T} & I & D^{T} \\ L_2W + D W & D & -L_2 \end{bmatrix} < 0 \end{align}$$

Conclusion:
As mentioned, the goal of this LMI is to mitigate the effects of external disturbances, parametric uncertainties, and actuator faults within the system. Minimizing the parametric uncertainty and disturbance term allows the system to mitigate faults within the actuators as well as determine a proper solution on how to minimize the external disturbance Dx so it falls below the parametric uncertainty.


text:
LMIs in Control/Click here to continue/Applications of Linear systems/LMI for Adaptive Barrier Global Sliding Mode

LMI for adaptive barrier global sliding mode control (ABGSMC)

Uncertain systems with faulty actuators can be simplified and modeled via an LMI for adaptive barrier global sliding mode control (ABGSMC). The LMI can be derived by using a global nonlinear sliding surface in order to achieve system stability and the occurrence of sliding when actuator faults are present within the system. This LMI is important for continued research into LMI-based actuator improvements as modern systems are heavily prone to faults (including actuator faults).

The System
The following system is assumed:

$$ \begin{align} \dot{x}(t) &= A x(t) + B L u(t) + D \\ D &= \Delta \  B u(t) + b f(x) + d \\ \end{align} $$

where $$ x(t) = [x1, x2]^{\text{T}} \in R^2 $$ notes the system’s state vector, f(𝑥) is a nonlinear system function, $$ \Delta f $$ and $$ \Delta g $$ are parametric uncertainties and d represents the external disturbances. L is the post-fault coefficient which represents the actuator's effectiveness. We can assume that: $$ \begin{align} \begin{bmatrix} A \end{bmatrix} = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \end{align} $$

$$ \Delta B =  [0, \Delta B_2 ]^{\text{T}} $$

$$ B  =  [0, \Delta B_2^{\text{T}}]^{\text{T}} $$

and $$ L  =  [0,  L_2]^{\text{T}} $$

The Data
The system can then be described as:

$$ \begin{align} \dot{x_1}(t) &= A_{11} x_1(t) + A_{12} x_2(t) \\ \dot{x_2}(t) &= A_{21} x_1(t) + A_{22} x_2(t) + (B_2 L_2) u(t) + D \\ \end{align}

$$

With the state space form described above.

Assumptions
1. Pair (A, B) is totally controllable. Using this assumption for the system, the controllability of $$ (A_{11}, A_{12}) $$ can be obtained by the controllability of (A, B)

2. The parametric uncertainty and external disturbance term Dx is bounded and is less than $$ \eta_k $$

The LMI
Using the Lyapunov function, Schur Complement, the adaptive control law, and the assumption of sliding surfaces an LMI can be brought together to minimize faults via actuators

$$ \begin{align} &\text{min} \quad \eta_k < Dx \\ &\begin{bmatrix} (B_{1}W)^{T} + A+B_{1}W & B_{2} & (D_{1}W)^{T} \\ B_{2}^{T} & I & D^{T} \\ L_2W + D W & D & -L_2 \end{bmatrix} < 0 \end{align}$$

Conclusion:
As mentioned, the goal of this LMI is to mitigate the effects of external disturbances, parametric uncertainties, and actuator faults within the system. Minimizing the parametric uncertainty and disturbance term allows the system to mitigate faults within the actuators as well as determine a proper solution on how to minimize the external disturbance Dx so it falls below the parametric uncertainty.


text: User:RequestSelector

title: User:RequestSelector


text:
User talk:159.118.167.112

--Xania   talk 04:53, 20 November 2022 (UTC)

title: User talk:159.118.167.112


text:
LMIs in Control/Click here to continue/Applications of Linear systems/Example

LMI for adaptive barrier global sliding mode control (ABGSMC)

Uncertain systems with faulty actuators can be simplified and modeled via an LMI for adaptive barrier global sliding mode control (ABGSMC). The LMI can be derived by using a global nonlinear sliding surface in order to achieve system stability and the occurrence of sliding when actuator faults are present within the system. This LMI is important for continued research into LMI-based actuator improvements as modern systems are heavily prone to faults (including actuator faults).


text:
User:Gerson Lapid Jr/sandbox

'Bold text  Developments in Canada Education System

Preparing the youths for next generation’s workforce is an important aspect of school curricula all over the world. This is emphasized in the 21st Century Skills wherein the focus of education centers on the integration of learning, literacy, and social skills into the teaching of core academic subjects worldwide. This set of skills and knowledge is viewed as necessary to prepare the young generation so they will succeed in their future career and become productive citizens of the society they live in (Explore SEL, n.d.).

Canada also viewed education in terms of its impact on the society. The country’s history shows how official policies shaped and supervised the growth of formal instructions at schools based on the changing nature of children and their respective families, the people’s behavior, and the introduction of the wage-labor economy. However, its development is reflected in various ways and distinctions depending on the experiences of people in different provinces and territories (Gaffield, 2015). Geographic Description Canada, a country in North America with a Gross Domestic Product of 1990.76 billion US dollars (World Bank, 2021), is a vast and rugged land that spans more than half the Northern Hemisphere (from north to south). It stretches almost 7,560 kilometers across six time zones (from east to west). It is the second largest country in the world, which features black-blue lakes, numerous rivers, majestic western mountains, rolling central plains, and forested eastern valleys. The Canadian Shield, a hilly region of lakes and swamps, stretches across northern Canada and has some of the oldest rocks on Earth.

Population As of November 19, 2022, Canada’s current population is 38,528,259 or about .48% of the total world population (Worldometer, 2022). In recent years, Canada has allowed the influx of immigrants from different parts of the world. The country has welcomed 492,984 immigrants in 2021/2022. While Canada’s fertility rate is 1.53 births per woman, below the population replacement rate, the population continues to grow as migration plays an increasing role in the population. Canada’s net migration rate is 6.375 per 1,000 people, the eighth-highest in the world.

Figure 1 '''Note. An overview of Canada’s education system from Canadian Information Centre for International Credentials (CICIC).'''

Canada’s Education Systems Canada is considered as one of the most educated countries in the world as it spends more on education compared to other countries (OECD, 2017). It is also home to some of the top universities in the world. However, understanding Canada’s Education System requires knowledge of the country’s 10 provinces (Alberta, British Columbia, Manitoba, New Brunswick, Newfoundland and Labrador, Nova Scotia, Ontario, Prince Edward Island, Quebec, and Saskatchewan) and three territories (Yukon, Nunavut and Northwest Territories) as some aspects of the system may vary from one province to another. This is clearly indicated in Figure 1, a diagram that shows three different pathways.

Public and private education are both available at the three main levels – primary, secondary, and post-secondary. Normally, the school year opens in September and ends in June for elementary and high school. The country’s public education is well-funded by and under the jurisdiction of the provincial or territorial governments.

Grade levels are considered part of public education system. Children in Canada attend kindergarten for a year or two at the age of four or five by choice. School then becomes mandatory as of grade one, which tends to be at the age of six years old. Schools go up to either Grade 11 or 12 (depending on the province), generally until the age of 16 years old. However, in Manitoba, New Brunswick, Northwest Territories, and Ontario, education is compulsory for people up to 18 years of age (CICIC, n.d.).

From Kindergarten (Grade 1) to Grade 8, children (aged 6 to 14) study in primary or elementary school while secondary or high school is from Grades 9 to 12 (for 14- to 18-year-olds). The province of Ontario has Grade 12+. Students in Quebec attend high school up to the age of 16 and proceed to CEGEP (or DCS) studies. CEGEP is a two-year free college education that offers either a vocational course or university preparation diploma (CICIC, n.d.).

Post-secondary schooling, meanwhile, is offered through the country’s wide network of colleges and universities.

Canada’s tertiary education has been recognized worldwide for its contributions to research. It is also known for honing a skilled workforce that ensures better labor-market results. The country boasts a high attainment in tertiary education. However, the ageing population and the rapid changing knowledge-based economy requires expansion in participation rates to ensure there is enough supply of highly skilled workforce. For this reason, the federal government encourages people from disadvantaged socio-economic groups to access tertiary education. The education system also allows flexibility of movement between institutions to students with diverse needs (Cheung, Guillemete & Mobasher-Fard, 2012).

Canada has also tapped immigration as source of skills as it believes in skills innovation development and the integration of practical industry experience within higher education programs and business, communication skills, and technical trainings (OECD, 2012). All public schools in Canada are provincially accredited, follow a standard curriculum, employ only government certified teachers and are publicly funded. The country’s education standards is high as it is supervised by the federal government. No. of Children Attending Public Education The latest statistics available on the number of enrollment in public elementary and secondary in Canada is from 2019/20. According to Statista Research (2021), a total of around 4.98 million students were enrolled in public schools in the country.

Medium of Instruction Canada has two official languages – English and French. Students are not required to be fluent in both languages to attend schools at any level. The main language of instruction in most schools across the country is English. However, in the province of Quebec, most schools use French until the end of high school.

Budget for Education The federal government provides funding for schools on reserve. At present, Canada has 518 First nations reserve schools. Provinces or territories, meanwhile, fund off-reserve public schools. Shares of public funding for private schools, on the other hand, varies from jurisdiction to jurisdiction. Some schools get full or partial funding while others get no funding at all.

In 2019, Canada spent USD$22,335 per student in tertiary education. This put Canada as no. 6 in world ranking of countries that allocated more budget in education. The country has also used 53.7% of the education budget to fund public tertiary education while the remaining 46.3% went to private education system. Overall, Canada’s public spending for primary to post-secondary education (non-tertiary) and tertiary education was 3.2% and 1.2% of the GDP, respectively, in the same year (OECD, 2022).

Other sources of education in country outside of Public Education There are about 1,700 private and independent schools that offer customized educational pathways for Grades 1-12. Postsecondary education, on the other hand, is available in both government-supported and private institutions, which offer degrees, diplomas, certificates, and attestations depending on the nature of the institution and the length of the program. However, universities are no longer the only degree-granting institutions in some provinces. A recognized postsecondary institution is a private or public institution that has been given full authority to grant degrees, diplomas, and other credentials by a public or private act of the provincial or territorial legislature or through a government-mandated quality assurance mechanism (OECD, 2017).

How is Public Education Governed? Public education is funded and overseen by provincial, territorial and local governments. Education is within provincial jurisdiction and the curriculum is overseen by the province. The departments or ministries of education are responsible for the organization, delivery, and assessment of education at the elementary and secondary levels, for technical and vocational education, and for postsecondary education (CICIC, n.d.).

An Education Superpower Canada is also considered as an education superpower given its record of having the highest proportion (55%) of working-age adults who attended tertiary education education (OECD, 2017). In an article, Coughlan (2017) emphasized that teenagers in Canada are among the best educated in the world. He also highlighted the role of migrant students in its school population.

Political and Economic Perspective Canada’s education can be viewed from a political and economic perspective. The country’s mandatory basic education can be considered as a social intervention that ensures the readiness of schools, equal opportunities even to students with diverse needs, and the promotion of social justice (Woodhead, 2016).

Social and Cultural Perspective The education system of Canada can also be studied from a social and cultural perspective as well as it recognizes the socio-cultural dimensions of education and acknowledges diversity in culture and society. The nurturance of students is an intrinsic part of the development process and Canada’s compulsory education ensures that children participate in those process from the start making it a social cultural process (Woodhead, 2016).

Conclusion The Canadian education has evolved as a result of socio-cultural and political practices, the changes in childhood and family concepts, the organization of social institutions, and the emergence of labor-market economy. Today, the country boasts of an excellent education system funded by taxes and supervised by the state. A system that is viewed as having a great impact on society and symbolizing good governance (Gaffield, 2015).

title: User:Gerson Lapid Jr/sandbox


text:
Fractals/emndl

=Programs= emndl it is the program by Claude Heiland-Allen: exponential strip visualisation of the Mandelbrot set Output options include:
 * tall thin images
 * zoom videos made from the image strips by inverting the coordinate transformation

emndl_autotune
 * attempts to find interesting points in the Mandelbrot Set that might look nice
 * is implemented in Haskell
 * currently requires a GHC built with integer-simple (not the default integer-gmp) so that the MPFR bindings work correctly

 * git repo
 * fractalforums article
 * fractalforums.org : emndl-vs-nanomb-mandelbrot-set-zooms

=instal=

prerequisites
 * for building most programs
 * gcc
 * g++
 * libqd-dev - Double-double and quad double types in C++ and F90
 * for building emndl_autotune
 * ghc
 * cabal
 * for running
 * bash
 * getopt (GNU)
 * bc
 * imagemagick             -- for image output
 * for dvd output
 * y4mscaler is a video scaler which operates on YUV4MPEG2 streams, as used by the tools in the MJPEGtools project.
 * mpeg2enc - encodes raw video into an MPEG-1/2 elementary stream using the mjpegtools library.
 * ffmpeg                  -- for mkv outpout
 * ffmpeg2theora           -- for ogv output
 * ecasound, puredata      -- for video soundtracks

For emndl:

git clone	https://code.mathr.co.uk/emndl.git cd emndl make

For emndl_autotune

cabal update cabal install --lib Vec cabal install --lib hmpfr make emndl_autotune ghc -O3 -Wall -threaded -rtsopts -fspec-constr-count=50 -package Vec -package hmpfr -package parallel -package reflection -package random -package ad --make emndl_autotune.hs

=help=

./emndl.sh --help usage options: --help -h -? show this message and exit --version      show version and exit

input options: --auto bits    find an interesting point with precision 'bits' --re real      choose coordinates --im imag --period p   --maxiters n    --size sz

render options: --quality q    horizontal size, (5 - 12)

image options: --png          generate full size PNG --jpeg         generate thumbnail JPEG

video options: --aspect w:h   output aspect (4:3, 16:9) --speed s      zoom speed (default 1) --dvd          generate PAL DVD (SD) --mkv          generate Matroska (HD) --ogv          generate Theora (LD)

other options: --quiet        less output --verbose      more output =coordinate= Coordinate

coordinate for emndl_2018-08-03T11-23-16.mkv REALCOORD=-1.2502259692930972582217787794367516384674267311920345554701091121819451616288474903375439125840028505381241666e+00 IMAGCOORD=-3.42180858457804363065272789355461346156675050268516641110743928004962993736897391706380916640147746335618768713e-01 PERIOD=396 MAXITERS=100100 SIZE=7.5089502353440369e-38

coordinates for emndl_vs_nanomb_2018-08-03 (on archive.org):

REALCOORD=2.32881715984171619523670105543682540038728554391151374122585121905032044832510538073674973239588027974526386357263542756141209509076915793136483409366732770101220857708891025180202484778852216373074628804662000573668390738172817193728347860445256777791916983537169476478249277392675920481935444775336796e-02 IMAGCOORD=8.15519594089994179859195278340598576447528880907449779065614659131971532215140192176068680822666750642391111578468988890742131839375371458052054010234250333939153906090539688498799065846221616933966400388040076657090140783996781080255541492455790725921287002240432954348425982356238830635005863927114069e-01 PERIOD=26079 MAXSUPERITERS=1000 SIZE=1.9302144566912991e-298

=run =

first example
./emndl.sh --quality 7 --png --jpeg --re -1.241733127596417466318604 --im -0.1698965841028383922879327

Result

AUTO=0 AUTOBITS=128 REAL=1 REALCOORD=-1.241733127596417466318604 IMAG=1 IMAGCOORD=-0.1698965841028383922879327 PER=0 PERIOD=1 MAXITERS=1000 SZ=0 SIZE=1 QUALITY=7 PNG=1 JPEG=1 ASPECT=16:9 SPEED=1 DVD=0 MKV=0 OGV=0 VERBOSE=1

WIDE=1 WIDTH=128 VWIDTH=128 VHEIGHT=72 E=/home/a/emndl OUT=emndl_2022-11-20T19:47:29+01:00 calculating.. Atom domain: period: 1 zmag:     1.2533 HC centre: (1.24173,0.169897)	 (relative to location) Esc R(new):1e+29 Esc R(old):1e+30 Precision: 7 --- __(-1.24173,-0.169897) i: 0	j: 0	val: (-1.24173,-0.169897) i: 0	j: 1	val: (1,0) i: 0	j: 2	val: (0,0) i: 0	j: 3	val: (0,0) i: 0	j: 4	val: (0,0) i: 0	j: 5	val: (0,0) i: 0	j: 6	val: (0,0) i: 0	j: 7	val: (0,0) i: 0	j: 8	val: (0,0) --- ... --- i: 16	j: 0	val: (0,0) i: 16	j: 1	val: (0,0) i: 16	j: 2	val: (0,0) i: 16	j: 3	val: (0,0) i: 16	j: 4	val: (0,0) i: 16	j: 5	val: (0,0) i: 16	j: 6	val: (0,0) i: 16	j: 7	val: (0,0) i: 16	j: 8	val: (0,0) --- R == 1e+30 NOTICE: precision is sufficient: 7 <= 64 finalizing..e scanline 326 / 326 equalizing.. computing depth.. DEPTH=2 downscaling.. colourizing.. making images.. making videos.. making soundtrack.. 2022-11-20T19:47:29+01:00 begin 2022-11-20T19:47:30+01:00 end

second
./emndl.sh --quality 10 --png --jpeg --dvd --ogv    --re -0.5988529658460445113700557999169873572638106766467118517477    --im -0.6627873418981974683919856287279365042423603984338094007074 AUTO=0 AUTOBITS=128 REAL=1 REALCOORD=-0.5988529658460445113700557999169873572638106766467118517477 IMAG=1 IMAGCOORD=-0.6627873418981974683919856287279365042423603984338094007074 PER=0 PERIOD=1 MAXITERS=1000 SZ=0 SIZE=1 QUALITY=10 PNG=1 JPEG=1 ASPECT=16:9 SPEED=1 DVD=1 MKV=0 OGV=1 VERBOSE=1

WIDE=1 WIDTH=1024 VWIDTH=960 VHEIGHT=540 E=/home/a/emndl OUT=emndl_2022-11-20T19:54:31+01:00 calculating.. Atom domain: period: 1 zmag:     0.893259 HC centre: (0.598853,0.662787)	 (relative to location) Esc R(new):1e+29 Esc R(old):1e+30 Precision: 9 --- __(-0.598853,-0.662787) i: 0	j: 0	val: (-0.598853,-0.662787) i: 0	j: 1	val: (1,0) i: 0	j: 2	val: (0,0) i: 0	j: 3	val: (0,0) i: 0	j: 4	val: (0,0) i: 0	j: 5	val: (0,0) i: 0	j: 6	val: (0,0) i: 0	j: 7	val: (0,0) i: 0	j: 8	val: (0,0) --- ... --- i: 16	j: 0	val: (0,0) i: 16	j: 1	val: (0,0) i: 16	j: 2	val: (0,0) i: 16	j: 3	val: (0,0) i: 16	j: 4	val: (0,0) i: 16	j: 5	val: (0,0) i: 16	j: 6	val: (0,0) i: 16	j: 7	val: (0,0) i: 16	j: 8	val: (0,0) --- R == 1e+30 NOTICE: precision is sufficient: 9 <= 64 finalizing..e scanline 2608 / 2608 equalizing.. computing depth.. DEPTH=2 downscaling.. colourizing.. making images.. making videos.. ./emndl.sh: linia 260: y4mscaler: nie znaleziono polecenia ./emndl.sh: linia 261: mpeg2enc: nie znaleziono polecenia making soundtrack.. ./emndl.sh: linia 279: ecasound: nie znaleziono polecenia ./emndl.sh: linia 282: pd: nie znaleziono polecenia ./emndl.sh: linia 292: twolame: nie znaleziono polecenia ./emndl.sh: linia 306: oggenc: nie znaleziono polecenia ./emndl.sh: linia 293: mplex: nie znaleziono polecenia ./emndl.sh: linia 307: oggz-merge: nie znaleziono polecenia 2022-11-20T19:54:31+01:00 begin 2022-11-20T19:54:48+01:00 end

third
export LD_LIBRARY_PATH=${HOME}/opt/lib export PATH=${HOME}/opt/bin:${PATH} ./emndl.sh --auto 96 --png --jpeg 12 1 0.0000e+00 0.0000e+00 7.50000000e-01 20 5 -4.4212341e-02 9.8658085e-01 2.54836585e-03 36 21 -4.08647509785e-02 9.82536914715e-01 4.88178316e-08 62 115 -4.0864860818013505432e-02 9.8253687241578015568e-01 7.76448686e-16 98 596 -4.086486081801155519398814148626e-02 9.825368724157807120516621196188e-01 1.02641550e-26 AUTO=1 AUTOBITS=96 REAL=0 REALCOORD=-4.086486081801155519398814148626e-02 IMAG=0 IMAGCOORD=9.825368724157807120516621196188e-01 PER=0 PERIOD=596 MAXITERS=1000 SZ=0 SIZE=1.3685540417099322e-26 QUALITY=8 PNG=1 JPEG=1 ASPECT=16:9 SPEED=1 DVD=0 MKV=0 OGV=0 VERBOSE=1

WIDE=1 WIDTH=256 VWIDTH=256 VHEIGHT=144 E=/home/a/emndl OUT=emndl_2022-11-23T18:42:44+01:00 calculating.. Atom domain: period: 1 zmag:     0.983386 HC centre: (0.0408649,-0.982537)	 (relative to location) Esc R(new):1e+29 Esc R(old):1e+30 Precision: 7 --- Atom domain: period: 5 zmag:     0.0821434 HC centre: (-0.0033475,0.0040441)	 (relative to location) Esc R(new):0.0129093 Esc R(old):0.129093 Precision: 15 --- Atom domain: period: 21 zmag:     0.000401242 HC centre: (1.0984e-07,4.23041e-08)	 (relative to location) Esc R(new):5.29777e-05 Esc R(old):0.000529777 Precision: 15 --- Atom domain: period: 115 zmag:     4.75912e-08 HC centre: (-1.95024e-15,-5.56367e-16)	 (relative to location) Esc R(new):7.04258e-10 Esc R(old):7.04258e-09 Precision: 16 --- Atom domain: period: 596 zmag:     5.78401e-18 HC centre: (-9.69691e-32,-9.40874e-31)	 (relative to location) Esc R(new):1.34034e-17 Esc R(old):1.34034e-16 Precision: 16 --- __(3.43351e-18,4.65465e-18) i: 0	j: 0	val: (3.43351e-18,4.65465e-18) i: 0	j: 1	val: (5.26731e+12,-3.10641e+12) i: 0	j: 2	val: (2.27498e+28,-3.95469e+28) i: 0	j: 3	val: (8.00832e+41,-2.21767e+44) i: 0	j: 4	val: (-3.90043e+59,-6.85294e+59) i: 0	j: 5	val: (-1.88333e+75,-1.10284e+75) i: 0	j: 6	val: (-4.88467e+90,-2.09717e+88) i: 0	j: 7	val: (-7.86231e+105,4.50434e+105) i: 0	j: 8	val: (-7.22294e+120,1.21745e+121) --- i: 1	j: 0	val: (0,0) i: 1	j: 1	val: (0,0) i: 1	j: 2	val: (0,0) i: 1	j: 3	val: (0,0) i: 1	j: 4	val: (0,0) i: 1	j: 5	val: (0,0) i: 1	j: 6	val: (0,0) i: 1	j: 7	val: (0,0) i: 1	j: 8	val: (0,0) --- i: 2	j: 0	val: (1.16559e+13,-2.63086e+12) i: 2	j: 1	val: (1.32558e+29,-1.18897e+29) i: 2	j: 2	val: (4.17497e+44,-1.22718e+45) i: 2	j: 3	val: (-1.13749e+60,-6.02728e+60) i: 2	j: 4	val: (-1.36998e+76,-1.61469e+76) i: 2	j: 5	val: (-5.33318e+91,-1.93651e+91) i: 2	j: 6	val: (-1.20729e+107,2.06746e+106) i: 2	j: 7	val: (-1.69961e+122,1.36808e+122) i: 2	j: 8	val: (-1.33031e+137,2.97704e+137) --- i: 3	j: 0	val: (0,0) i: 3	j: 1	val: (0,0) i: 3	j: 2	val: (0,0) i: 3	j: 3	val: (0,0) i: 3	j: 4	val: (0,0) i: 3	j: 5	val: (0,0) i: 3	j: 6	val: (0,0) i: 3	j: 7	val: (0,0) i: 3	j: 8	val: (0,0) --- i: 4	j: 0	val: (1.58875e+29,-7.00615e+28) i: 4	j: 1	val: (1.52864e+45,-2.0086e+45) i: 4	j: 2	val: (2.54531e+60,-1.7701e+61) i: 4	j: 3	val: (-2.95186e+76,-7.66605e+76) i: 4	j: 4	val: (-2.11528e+92,-1.74947e+92) i: 4	j: 5	val: (-6.9792e+107,-1.2693e+107) i: 4	j: 6	val: (-1.3956e+123,4.64333e+122) i: 4	j: 7	val: (-1.77273e+138,1.77606e+138) i: 4	j: 8	val: (-1.5096e+153,3.32683e+153) --- i: 5	j: 0	val: (0,0) i: 5	j: 1	val: (0,0) i: 5	j: 2	val: (0,0) i: 5	j: 3	val: (0,0) i: 5	j: 4	val: (0,0) i: 5	j: 5	val: (0,0) i: 5	j: 6	val: (0,0) i: 5	j: 7	val: (0,0) i: 5	j: 8	val: (0,0) --- i: 6	j: 0	val: (1.3519e+45,-9.23641e+44) i: 6	j: 1	val: (1.05008e+61,-2.06423e+61) i: 6	j: 2	val: (-5.24778e+75,-1.59172e+77) i: 6	j: 3	val: (-3.62984e+92,-6.07839e+92) i: 6	j: 4	val: (-1.97052e+108,-1.14993e+108) i: 6	j: 5	val: (-5.66073e+123,-1.94389e+122) i: 6	j: 6	val: (-1.02695e+139,4.62395e+138) i: 6	j: 7	val: (-1.34514e+154,1.31384e+154) i: 6	j: 8	val: (-1.9291e+169,2.43431e+169) --- i: 7	j: 0	val: (0,0) i: 7	j: 1	val: (0,0) i: 7	j: 2	val: (0,0) i: 7	j: 3	val: (0,0) i: 7	j: 4	val: (0,0) i: 7	j: 5	val: (0,0) i: 7	j: 6	val: (0,0) i: 7	j: 7	val: (0,0) i: 7	j: 8	val: (0,0) --- i: 8	j: 0	val: (8.03788e+60,-7.85714e+60) i: 8	j: 1	val: (4.56251e+76,-1.47389e+77) i: 8	j: 2	val: (-2.02153e+92,-1.0065e+93) i: 8	j: 3	val: (-2.81953e+108,-3.37907e+108) i: 8	j: 4	val: (-1.26107e+124,-5.16602e+123) i: 8	j: 5	val: (-3.24269e+139,1.97294e+138) i: 8	j: 6	val: (-5.84003e+154,2.49333e+154) i: 8	j: 7	val: (-1.04573e+170,6.14084e+169) i: 8	j: 8	val: (-2.51583e+185,1.6916e+185) --- i: 9	j: 0	val: (0,0) i: 9	j: 1	val: (0,0) i: 9	j: 2	val: (0,0) i: 9	j: 3	val: (0,0) i: 9	j: 4	val: (0,0) i: 9	j: 5	val: (0,0) i: 9	j: 6	val: (0,0) i: 9	j: 7	val: (0,0) i: 9	j: 8	val: (0,0) --- i: 10	j: 0	val: (3.52949e+76,-4.82147e+76) i: 10	j: 1	val: (1.11202e+92,-7.85671e+92) i: 10	j: 2	val: (-1.74272e+108,-4.78352e+108) i: 10	j: 3	val: (-1.54592e+124,-1.41788e+124) i: 10	j: 4	val: (-5.96547e+139,-1.84904e+139) i: 10	j: 5	val: (-1.46932e+155,5.06642e+153) i: 10	j: 6	val: (-3.26863e+170,5.36225e+169) i: 10	j: 7	val: (-9.21875e+185,2.22567e+185) i: 10	j: 8	val: (-2.62545e+201,1.53132e+201) --- i: 11	j: 0	val: (0,0) i: 11	j: 1	val: (0,0) i: 11	j: 2	val: (0,0) i: 11	j: 3	val: (0,0) i: 11	j: 4	val: (0,0) i: 11	j: 5	val: (0,0) i: 11	j: 6	val: (0,0) i: 11	j: 7	val: (0,0) i: 11	j: 8	val: (0,0) --- i: 12	j: 0	val: (1.17508e+92,-2.26762e+92) i: 12	j: 1	val: (-1.48869e+106,-3.27023e+108) i: 12	j: 2	val: (-9.24682e+123,-1.79275e+124) i: 12	j: 3	val: (-6.31679e+139,-4.91195e+139) i: 12	j: 4	val: (-2.23303e+155,-7.73627e+154) i: 12	j: 5	val: (-6.32356e+170,-1.31143e+170) i: 12	j: 6	val: (-2.14638e+186,-2.53377e+185) i: 12	j: 7	val: (-7.73712e+201,1.39915e+201) i: 12	j: 8	val: (-2.09628e+217,1.50196e+217) --- i: 13	j: 0	val: (0,0) i: 13	j: 1	val: (0,0) i: 13	j: 2	val: (0,0) i: 13	j: 3	val: (0,0) i: 13	j: 4	val: (0,0) i: 13	j: 5	val: (0,0) i: 13	j: 6	val: (0,0) i: 13	j: 7	val: (0,0) i: 13	j: 8	val: (0,0) --- i: 14	j: 0	val: (2.99288e+107,-8.48534e+107) i: 14	j: 1	val: (-1.37995e+123,-1.09929e+124) i: 14	j: 2	val: (-3.4462e+139,-5.61505e+139) i: 14	j: 3	val: (-1.99107e+155,-1.67848e+155) i: 14	j: 4	val: (-7.56544e+170,-4.85276e+170) i: 14	j: 5	val: (-3.22153e+186,-1.64147e+186) i: 14	j: 6	val: (-1.50281e+202,-2.57623e+201) i: 14	j: 7	val: (-5.47944e+217,1.52875e+217) i: 14	j: 8	val: (-1.31202e+233,1.2574e+233) --- i: 15	j: 0	val: (0,0) i: 15	j: 1	val: (0,0) i: 15	j: 2	val: (0,0) i: 15	j: 3	val: (0,0) i: 15	j: 4	val: (0,0) i: 15	j: 5	val: (0,0) i: 15	j: 6	val: (0,0) i: 15	j: 7	val: (0,0) i: 15	j: 8	val: (0,0) --- i: 16	j: 0	val: (5.9467e+122,-2.59355e+123) i: 16	j: 1	val: (-6.03661e+138,-3.10484e+139) i: 16	j: 2	val: (-9.10047e+154,-1.64806e+155) i: 16	j: 3	val: (-5.1965e+170,-7.04042e+170) i: 16	j: 4	val: (-2.99237e+186,-3.23487e+186) i: 16	j: 5	val: (-1.9257e+202,-1.09749e+202) i: 16	j: 6	val: (-9.57639e+217,-7.39729e+216) i: 16	j: 7	val: (-3.20512e+233,1.40031e+233) i: 16	j: 8	val: (-6.68272e+248,8.50103e+248) --- R == 1.34034e-16 NOTICE: precision is sufficient: 16 <= 64 finalizing..e scanline 3078 / 3078 equalizing.. computing depth.. DEPTH=12 downscaling.. colourizing.. making images.. making videos.. making soundtrack.. 2022-11-23T18:42:44+01:00 begin 2022-11-23T18:43:07+01:00 end

autotune
emndl_autotune +RTS -N -RTS 80 P 1 R 0.0 I 0.0 @ 1                               ▄                                                             ▄                                                              ███                                                            ▄███▄ ▄                                                     ███████████▄█▀                                                ███████████████▄                                        ▄▄▄    ▄█████████████████                                      ▄█████▄ █████████████████                          ▄▄▄▄▄▄▄▄▄▄▄█▄███████▄████████████████▀                                     ▀ ███████ █████████████████                                        ███▀▀  █████████████████▄                                               ████████████████▀                                               ▀▀█████████████                                                   ▀▀▀▀█████▀█ ▀▀                                                       ███                                                             ▀█▀                                                                ▄

P 4 R -0.1565201668337550617989867 I -1.032247108922831801671601 @ 8.2966360777899879795193e-3 ▄▄▄     █▄                                                     ████▄███████▄█                                                   ▀█████████████                                                  ███████████████                                              ▄▄███████████████▄▄                                               ██████████████████▄                                            ▀███████████████████▀                                            ███████████████████▄                                          ▄▄ ██████████████████▀                                       ▄▄████████████████████▀▀                                        ████████ ▀███████████▄▄                                         ▄██████▀    ▀▀▀█▀▀  ▀██                                         ▀▀▀▀▀█

P 31 R -0.1534817415674078270901370 I -1.030177111803473792344760 @ 2.261921892899238897e-7 -0.1534817415674078270901370 -1.030177111803473792344760 5

=exponential map=

with scaling by 2pi/width

Here are the relevant parts of the code:

=video=
 * falling into Mandelbrot (360 test) by Claude A fractal zoom with a twist. Projecting  from inside the Riemann sphere, the image expands from below and scrolls past to contract overhead. This is a first test, there are some imperfections like bad mipmap level curves.  Render time 6 hours. Coordinates:
 * Cx = -1.76915100313963348631585617219113282419006284180146489000041
 * cyu = +0.00558166050979690182753149300394271544944337923388479000001i
 * @ 1e-21 ( zoom)
 * Misaligned Butterdonut (Dinkydau) by Claude and original page Misaligned Butterdonut by DinkydauSet
 * video: 360 Mandelbrot fractal zoom to the mini-Mandelbrot-set at the center of Dinkydau's "Misaligned Butterdonut" location. Rendered with 'emndl'  using algorithms from knighty's 'nanomb2'.

=References=

title: Fractals/emndl


text:
AI Art Generation Handbook/Archive/Framing Techniques

Framing techniques is to instruct how to take photos utilizing different angles, heights and even distance to achieve certain looks.


text: User:RequestSelector/sandbox

title: User:RequestSelector/sandbox


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-Time Transient Impulse Response Bound

The System
For the single-input multi-output discrete-time LTI system with state-space realization,

\begin{align} \ x_{k+1}&=A_{d}x_{k}+B_{d}u_{k}\\ y_{k}&=C_{d}x_{k}\\ \end{align}$$

where $$A \in \mathbb{R}^{n \times n}$$, $$B_{d} \in \mathbb{R}^{n \times 1}$$ and $$C_{d} \in \mathbb{R}^{p \times n}$$ and it is assumed that $$A_{d}$$ is invertible. It is also considered that $$ Z_{k}C_{d}A^{k-1}_{d}B_{d}$$ be the unit impulse response of the system.

The Data
$$A \in \mathbb{R}^{n \times n}$$, $$B_{d} \in \mathbb{R}^{n \times 1}$$ and $$C_{d} \in \mathbb{R}^{p \times n}$$

The LMI
If the Euclidean norm of the impulse response satisfies. $$\lVert z_{k} \rVert_{2} \leq \gamma, \forall k \in \mathbb{Z}_{\geq 0}$$ and if there exist $$P \in \mathbb{S}^p$$ and $$\gamma \in \mathbb{R}_{> 0}$$,where P > 0, such that

 * $$\begin{bmatrix} P & PA_{d}^{-1}B_{d} \\ * & \gamma \end{bmatrix} \geq 0, $$

 * $$\begin{bmatrix} P & C^T_{d} \\ * & \gamma 1 \end{bmatrix} \geq 0, $$

 * $$A^T_{d}PA_{d}-P \leq 0. $$

Proof

 * The proof follows same procedure as for transient output bound for Discrete time autonomous LTI sysyems,but taking $$X_{0}=A_{d}^{-1}B_{d} $$as the initial condition, so that the unit impulse response matching the free response $$Z_{k}=C_{d}A_{d}^{k}X_{0}$$.

 * This yields the result$$X_{k}^{T}PX_{k}\leq B_{d}^{T}A_{d}^{-T}PA_{d}^{-1}B_{d}$$.

 * Using the non-strict Schur complement, the matrix inequality $$\begin{bmatrix} P & PA_{d}^-1B_{d} \\ * & \gamma \end{bmatrix} \geq 0, $$ is equivalent to the inequality $$ B_{d}^TA_{d}^-TPA_{d}^-1B_{d}\leq \gamma$$.Substituting this and  $$\begin{bmatrix} P  & C^T_{d} \\ * & \gamma 1 \end{bmatrix} \geq 0 , $$ into $$X_{k}^{T}PX_{k}\leq B_{d}^{T}A_{d}^{-T}PA_{d}^{-1}B_{d}$$.gives the desired result.

Conclusion
This LMI can be used to analyze the Discrete-Time Transient Impulse Response Bound for the given system

Implementation
This LMI can be used in a problem and can be solved using the solvers like Yalmip,sedumi,gurobi etc,.


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User:Xeverything11/sandbox

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title: User:Xeverything11/sandbox


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User:Xeverything11

Milestones

 * 20 Nov 2022 - 1st edit
 * 20 Nov 2022 - 5th edit
 * 20 Nov 2022 - 10th edit
 * 20 Nov 2022 - 20th edit
 * 22 Nov 2022 - Became autoconfirmed
 * 22 Nov 2022 - 50th edit
 * 23 Nov 2022 - 100th edit
 * 23 Nov 2022 - 200th edit
 * 23 Nov 2022 - Created 1st page and recipe: Cookbook:Apple Jelly
 * 25 Nov 2022 - 300th edit
 * 29 Nov 2022 - 400th edit
 * 29 Nov 2022 - Created 5th page: Using Chrome/Introduction
 * 30 Nov 2022 - Became a reviewer and started using SWViewer
 * 30 Nov 2022 - 500th edit
 * 5 Dec 2022 - 600th edit
 * 22 Dec 2022 - 700th edit
 * 30 Dec 2022 - Became an uploader for 7 days
 * 18 Jan 2023 - Created 10th page and 5th recipe: Cookbook:Espresso Brownies
 * 19 Jan 2023 - Became an uploader again for 7 days
 * 20 Jan 2023 - Uploaded 1st file
 * 20 Jan 2023 - 800th edit
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 * 30 Jan 2023 - Created 10th recipe: Cookbook:Biscuits III
 * 2 Feb 2023 - Created 20th page: Cookbook:Garlic Shrimp Scampi
 * 2 Feb 2023 - 1,000th edit
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 * 18 Apr 2023 - Created 30th page: Cookbook:Rosemary and Citrus Cordial
 * 21 Apr 2023 - Created 30th recipe: Cookbook:Leek, Parsley & Cheddar Mash

Pages I created

 * Using Chrome
 * Using Chrome/Introduction
 * Using Chrome/Installation
 * Using Chrome/Browsing with Tabs
 * Using Chrome/Extensions

Recipes

 * Cookbook:Apple Jelly
 * Cookbook:Banana Cream
 * Cookbook:Banana Cream II
 * Cookbook:20-Ingredient Brownies
 * Cookbook:Espresso Brownies
 * Cookbook:Two-Ingredient Brownies
 * Cookbook:Microwave Chocolate Chip Cookie
 * Cookbook:Biscuits II
 * Cookbook:Ginger Biscuits
 * Cookbook:Biscuits III
 * Cookbook:Lemon Pudding
 * Cookbook:Chocolate Chip Cookies IV
 * Cookbook:Brown Butter Toffee Chocolate Chip Cookies
 * Cookbook:Chicken Bacon Broccoli Alfredo Spaghetti
 * Cookbook:Garlic Shrimp Scampi
 * Cookbook:Spaghetti Pie
 * Cookbook:Chicken Bacon Broccoli Alfredo Spaghetti
 * Cookbook:Spinach Mushroom Pesto Spaghetti
 * Cookbook:Giant Spaghetti-Stuffed Meatball
 * Cookbook:Tomato Basil Sausage Pasta
 * Cookbook:Microwave Lemon Puddings
 * Cookbook:Coronet Cupcakes
 * Cookbook:Carrot Cake Smoothie
 * Cookbook:Slow Cooker Soda Bread with Spring Herbs
 * Cookbook:Rosemary and Citrus Cordial
 * Cookbook:Raspberry Scones
 * Cookbook:Chocolate Orange Hot Cross Bun Trifle
 * Cookbook:Eton Mess Easter Sharing Eggs
 * Cookbook:Rhubarb & Custard Pudding
 * Cookbook:Leek, Parsley & Cheddar Mash

Useful links

 * Mrs. Beeton's Book of Household Management

title: User:Xeverything11


text:
LMIs in Control/Click here to continue/Applications of Linear systems/H∞-based Synthesis of Quadrotor Guidance and Control System

System Dynamics
Consider the motion of a quadrotor along a horizontal plane alone. The control law is synthesised for roll motion(rotation wrt. X-axis) coupled with linear motion along Y-axis. The equations of motion governing the 2-D motion are given by(taken from the paper provided below):

$$ \begin{align} &\quad \quad \begin{cases} \frac{d^2x}{dt^2}=-\frac{\mu}{m_{Q}}\frac{dx}{dt}-g\frac{tan\theta}{cos\phi};\\ \frac{d^2y}{dt^2}=-\frac{\mu}{m_{Q}}\frac{dy}{dt}-gtan\phi,\\ \end{cases} \end{align}$$

$$ where $$

 * $$x$$ and $$y$$ are the displacements along X & Y axis, respectively.
 * $$\phi$$ and $$\theta$$ are the angular displacement about X & Y axis, respectively.
 * $$m_{Q}$$ and $$\mu$$ are constants that depend on the intrinsic properties of the quadrotor.

The state space representation of The $$H_\infty$$ Quadrotor Guidance is given below,

$$ \begin{align} &\quad \quad \begin{cases} \frac{dX(t)}{dt}=AX(t)+B_{u}u(t)+B_{d}(t)d(t); \\ Y=CX(t), \\ \end{cases}\end{align} $$

$$ where: $$

$$ X = [X_{1},X_{2},X_{3},X_{4},X_{5}]^T = [x,\frac{dx}{dt},y,\frac{dy}{dt},\Delta \Omega]^T,$$ (where $$\Delta \Omega$$ is the increment in motor's rotation rate)

$$A = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & -\frac{\mu}{m_{Q}} & g & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\               0 & 0 & 0 & 0 & K_{1}\\ 0 & 0 & 0 & 0 & K_{2}\\ \end{bmatrix} $$

$$ B_{u} = \begin{bmatrix} 0\\ 1\\                           K_{3} \\ 0 \\                           K_{4}\\ \end{bmatrix}$$

$$ B_{d} = \begin{bmatrix} 0\\ 0\\                           0 \\                            0 \\                            K_{3}\\ \end{bmatrix}$$

$$C = \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\                                     0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\  \end{bmatrix}$$

$$C_2 = \begin{bmatrix} I_{3x3} & 0_{3x3} \end{bmatrix} $$

$$ D_1 = 10^{-3} \times L_1, D_2 = 10^{-3} \times L_2 $$

The Optimization Problem
For the above-linearized model of the quadrotor, the linear control would be as below:

u = -K*V(t) = -K*C*X(t), $$

where V is the voltage input, $$ K \in R^{1x4}$$ and $$ C \in R^{4x5}.$$

The cost function of this controller could be defined as:

j=\int_0^\infty \! ||z(t)||^2 \, \mathrm{d}t=\int_0^\infty \! ||x^TQx+u^T Ru||^2 \, \mathrm{d}t $$ ,

where,

$$ z=

\begin{bmatrix} \sqrt{Q} & 0\\ 0 & \sqrt{R} \end{bmatrix} \begin{bmatrix} x \\ u \end{bmatrix} .$$

The LMI
$$\begin{bmatrix} PA+A^TP+Q& PB_{u}& PB_{d}&L^T \\B^T_{u}P &-R& 0& 0\\B^T_{d}P&0&-\gamma^2I&0\\L&0&0&-R \end{bmatrix}$$ < 0.

Solving the above LMI gives the value of $$\gamma$$ for X>0, $$\gamma$$ >0 and Z.

Conclusion
This LMI can be used in a problem and can be solved using the solvers like Yalmip,sedumi,gurobi etc,. This can be used to analyze the state feedback control and path tracking of a quadcopter.


text:
User talk:Adharshaeroasu

Welcome, Adharshaeroasu!

If you have any questions, you can ask in the assistance reading room or possibly contact me personally. Xeverything11 (discuss • contribs) 18:22, 20 November 2022 (UTC)

title: User talk:Adharshaeroasu


text:
User:Parismelissa

= COLUMBIAN EDUCATION SYSTEM = November 20, 2022

Contents

 * 1) Objective Overview of Chapter
 * 2) Historical Overview
 * 3) Stages in Colombian Education System
 * 4) Public Education in Colombia and the challenges to be overcome
 * 5) Colombian Education in the 21st century
 * 6) Conclusion
 * 7) References

1. At the end of the chapter, the readers should:

 * 1) understand the general history of the Colombian Education System
 * 2) develop an understand of the stages in the Colombian Education System
 * 3) become aware of the Public Education System in Colombian and its challenges
 * 4) gain an appreciation for the Colombian Education System that exists in the 21st century

2. HISTORICAL OVERVIEW OF COLOMBIA'S EDUCATION SYSTEM

The Colombian Education System dates back to its colonial years when Spanish settlers sought to teach the indigenous about Catholicism. The 16th century dawned a new day for Colombian natives when Spanish settlers officially introduced a school system with emphasis on reading, writing, language learning (Latin) and the Catholic faith ("History of Education in Colombia...", 2022).

Prior to Colombia's independence, education was seen as a privilege that was attainable mainly by the "white" male Spaniard descendants who would become professionals in the fields of law, education and medicine. Post-independence in the early 1800s ushered in a new era reform attempts were made to move the country's education system forward. Notwithstanding, steps taken to improve Education for Colombians was not without challenges. There were hurdles to overcome, one of which was the political differences that existed. This was evident in the tug of war between political parties (Hauver, 2021).

(Adapted from Hauver, 2021)

3. STAGES IN THE COLOMBIAN EDUCATION SYSTEM

(Created using draw.io)

4. PUBLIC EDUCATION IN COLOMBIA AND THE CHALLENGES FACED

Statistically, Colombia's Public Education systems can be represented as follows:

 * Only 37.2% of Colombians continue education after high school.
 * 45.4% of students dropped out of tertiary education as of 2010.
 * By the time children reach age 17, 75% drop out of the education system.
 * About 37% of students start school late and about 41% repeat at least one grade by age 15.
 * There are only 7.1 Masters of PhD degree holders per 1,000,000 inhabitants” (Mouttet, 2017).

Funding source or approach for Public Education:

Most of the funding for school education in Colombia is public funding. Statistics has shown that between the period of 2010-2017, the state financed between 73.2% and 76.1% of the total expenditure. “4.6% of annual GDP has been invested in education, but only .5% of this expenditure has gone towards rural areas. Because of this, there is a huge disparity between children in urban and rural areas” (Mouttet, 2017).

How is Public Education Governed?

In Columbia, the  Ministerio de Educación Nacional (MEN) which is located in Bogotá, is responsible for monitoring the quality of education provided. They also set the overall educational policy and school curricula. At the local or administrative level, Certified Territorial Entities (CETs), implement educational policies and oversee public and private schools within their jurisdictions. They also autonomously manage funds and teaching staff.

Admittedly, stakeholders in Education have indeed made strides in improving the Education system in Colombia. There is still much to be had as Colombia faces the challenges of low participation, 'dropouts', and drug use and trafficking.

"Even though educación media was made tuition-free at public schools in 2012 and attendance rates have been rising steadily in recent years, participation remains low by international standards. The upper-secondary net enrollment ratio—the percentage of enrollments within the relevant age group—stood at only 42.5 percent in 2018." (Carroll, 2020). Additionally, large number of students drop out of school, especially in the rural areas and among the upper secondary group. As a result, adult learners are given the opportunity to earn a diploma by sitting an equivalency exam as that of grade 11 for which they can prepare in adult education school and continuing education programs. The negative impact of drug use and trafficking on the Colombian children and Colombians in general cannot be overemphasized. Colombia is said to be the largest producer of cocaine accounting for 5% of the consumption of all drugs (Zuleta, 2022). Colombia has a history of drug production, drug trafficking, gun for drugs trading and ‘empires’ governed by drug lords/ cartels. No doubt, the drug trade has impacted various spheres of the lives of the people in Colombia with guerrilla warfare, ruthless killings and other forms of violence and crime. Education is no exception. Many children families and communities to include Guacamayal, a community located north-west of Bogota, face this issue of drug use, destroying their hope and putting their lives at risk (Drug education and prevention in Colombia, 2018). As a result, the Stand By Me Emmanuel Care Centre in Colombia, engaged parents and children in a workshop to increase their knowledge and awareness on the effects of drugs.

5. COLOMBIAN EDUCATION IN THE 21st CENTURY

Despite the many challenges, Colombia, continues to make progress it's educational system. The country, today of over 90% of its population and there are approximately 9 in 10 of primary enrollment and 8 in 10 of secondary enrollments. Internationally assessments have shown that over half of 15 years old have basic literacy skills and 35% have basic numeracy skills. There seems to be, however, gaps between genders, between rural and urban students as well as between income levels. Private institutions account for around 20% of primary and secondary schools that have been deemed to provide better education than public schools. Of note is also, the advocacy for every educational institution to establish a "Chair of Peace" in response to the degree of violence experienced by children and family and also to present a lively hope for the future (Hauver, 2021).

6. CONCLUSION

The Colombian Educational System has developed tremendously over the years. Though there are challenges, with consistency and the support and cooperation of political authorities, law enforcement agencies, educational stakeholders and communities, transformational improvements can be made.

7. REFERENCES

Carroll, L. (2020). Education in Colombia. Retrieved from: Education in Colombia (wes.org)

Colombia: Higher Education. (2022). Retrieved from: Colombia - Higher Education - Students, Universities, Institutions, and University - StateUniversity.com

Drug Education and Prevention in Colombia. (2018). Retrieved from: Drug education and prevention in Colombia - Stand By Me

Hauver, J. (2021). A Brief History of Education in Colombia. Retrieved from: A Brief History of Education in Colombia – Justin Hauver

History of Education in Colombia: Characteristics, Development – History – 2022. (2022). Retrieved from: HISTORY OF EDUCATION IN COLOMBIA: CHARACTERISTICS, DEVELOPMENT - HISTORY - 2022 (journalmural.com)

Mouttet, C. (2017). Columbia’s Education System. Retrieved from: Colombia's Education System - CHILDREN BEYOND OUR BORDERS (chbob.org)

Paris, M. (2022). Stages used in the Education system in Colombia.[jpg]. Draw.io. Concept Map.drawio - diagrams.net

Zuleta, D.L. (2022). The war on drugs: 50 years of death and failure. Retrieved from: Colombia: The war on drugs: 50 years of death and failure | International | EL PAÍS English Edition (elpais.com)

title: User:Parismelissa


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User:Soleriate

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title: User:Soleriate