Some "social technological inventions" of the 21st Century

The 21st century is the current century of the Anno Domini era or the Common Era, in accordance with the Gregorian calendar. It began on January 1, 2001, and will end on December 31, 2100. It is the first century of the 3rd millennium.
Good to know, 2015 is the 15th year in the 21st century.

Here are some of the most important social technological invention of 21st century.

Apple iPod (2001)
Portable MP3 players had been around for many years before Apple launched its version in 2001, but the iPod – together with Apple's iTunes software – was the technology that really transformed the way people listened to music. The device's large internal storage capacity meant it was no longer necessary to carry around CDs or cassette tapes, and the sleek design made it a desirable item to own.

Mozilla Firefox (2002)
Firefox was the first web browser to challenge the dominance of Microsoft's Internet Explorer, after Netscape Navigator was killed off in the 'First browser war'. It was free and open-source, so it appealed to Windows users who did not want to be locked into Microsoft's software ecosystem. However, it has since been left in the dust by Google's Chrome browser.

Skype (2003)
Skype has transformed the way people in different countries communicate. It used to cost an arm and a leg to call family or friends abroad, but Skype made it possible to speak to them – and even video chat – for free over WiFi. Initially Skype was only available as a desktop client, but over time it has launched on mobile, and many people now use it to talk loved ones and colleagues at home as well.

LinkedIn (2003)
LinkedIn is a business-oriented social networking service. It was founded in December 2002 and launched on May 5, 2003, it is mainly used for professional networking. In 2006, LinkedIn increased to 20 million members. As of March 2015, LinkedIn reports more than 364 million acquired users in more than 200 countries and territories.

Facebook (2004)
Facebook was not the first social network – it built on the success of earlier sites like MySpace and Bebo. However, it quickly overtook its predecessors thanks to its simplicity, ease of use and exclusivity, (users initially had to have a registered university email address to become a member). Today the website connects over 1.3 billion people worldwide, with users from a vast array of backgrounds, nationalities and ages.

YouTube (2005)
Created by three former PayPal employees in 2005, YouTube has grown to become the world's most popular video-sharing website. Key to its success is the ability for anyone from anywhere in the world to broadcast themselves for free. As a result, YouTube is a melting pot of news events, political messages, music clips, hilarious blunders, adverts and cat videos.

Twitter (2006)
Twitter is an online social networking service that enables users to send and read short 140-character messages called "tweets".
Registered users can read and post tweets, but unregistered users can only read them. Users access Twitter through the website interface, SMS, or mobile device app. Twitter Inc. is based in San Francisco and has more than 25 offices around the world.

Apple iPhone (2007)
Apple's iPhone was the first touchscreen smartphone to gain mass-market adoption. Part of the reason for its appeal was the fact that it could be controlled using a finger rather than requiring a stylus. The iPhone went on to spawn an entire industry, with electronics manufacturers all over the world developing their own versions of the device. Apple has now sold over 500 million iPhones globally.

Google Android (2008)
Following the ecstatic response to Apple's iPhone in 2007, other mobile phone manufacturers were desperate to get in on the action, but they needed an operating system that could compete with iOS. Android was originally created as an open source OS for cameras, but was bought by Google in 2005 and launched as a mobile OS in 2008. It is now the primary operating system for Samsung, Sony, LG and HTC phones, among others, and boasts over 80 per cent marketshare worldwide.

4G (2008)
In 2008, the International Telecommunications Union specified a set of requirements for fourth generation (4G) standards. 4G provided significantly faster mobile broadband internet access than 3G, with a much higher data capacity, supporting IP telephony, gaming services, HD mobile TV, video conferencing, and cloud computing. 4G services are now available to around 75 per cent of the UK's population.

Apple iPad (2010)
Apple did it again in 2010 with the launch of its tablet PC, the iPad. It was by no means the first device of its kind, but it succeeded in capturing the public's imagination, and kickstarted a new trend. It has remained the single most popular tablet PC ever since, but Android overtook Apple iOS to become the number one tablet operating system in terms of user numbers earlier this year, thanks to a wide variety of cheap alternatives.


Learn more: Telegraph, Wikipedia

Approximate Solution of equations by iteration: Wolfram Language Implementation of the Newton's Method

Dedicated to the memory of a Great Mathematician, a beautiful mind, John Forbes Nash.

Newton's Method

The newton method also known as the Newton-Raphson method is another important iterative method for solving equations $f\left(x\right)$, here $f$ is assumed to have a continuous derivative $f\text{'}$.

The derivation of this simple method (with great speed) can be looked up from textbooks as we are only concerned with the implementation of the method using the Wolfram Language.

Algorithm

1. Inputs: $f,x_0,\epsilon ,N$.
2. For $n=1,2\dots N-1$, compute $$x_{n+1}=x_n-\frac{f\left(x_n\right)}{f\text{'}\left(x_n\right)}$$
3. Stopping Criteria: If $\left|f\right(x_{\mathrm{n+1}}\left)\right|>\epsilon$, stop computation and Print $x_{\mathrm{n+1}}$.
4. Print $x_{\mathrm{n+1}}$.

Wolfram Language Implementation

Code 1 (newton)

This implementation defines a function "newton[fun, {var_Symbol, init_?NumericQ}, n_Integer]", where fun is the function $f\left(x\right)$ we want to find its approximate solution, var is the variable in the function $f\left(x\right)$, init is the approximation $$_x0 and n the number of iterations we want, the program terminates after $n$ iteration(s), then outputs the approximate solution $x_{n+1}$.

Code 2 (newton1)

This second implementation, slightly different, defines a function "newton1[fun, {var_Symbol, init_?NumericQ}, $\epsilon$_0.0001]" with $\epsilon$, the degree of accuracy with default value $0.0001$. The program terminates at any point where the absolute value of $\left|f\right(x_{\mathrm{n+1}}\left)\right|>\epsilon$ and returns $x_{n+1}$.

If $f\left(x\right)=x^3+x-1$, we can find the approximate solution of $f\left(x\right)$ using an initial approximation $x_0=1$.

Using the first implementation, we have

And for the second implementation,

Having any difficulty understanding the implementations? You can definitely drop a comment at the comment section. I will be glad to be at your service.

Wolfram Cloud has won me over, i want everyone to have a feel of what these implementation does. Despite not having Mathematica installed on your PC, you can test the implementation. This time, I'm going to embed the cloud deployment here, you don't have to leave this place. Its very easy. Follow me.

Wolfram Cloud Depoyment

Below is the embedded cloud deployment, very easy to use. You only have to enter the appropriate values in the text boxes and click the Submit button, anything else is automatic.



Still need a link to the cloud deployment? See it here. Will be back with Secant. Drop a comment.

References

Jain, M., Iyengar, S., & Jain, R. (2010). Numerical Method for Scientific and Engineeing Computiation (5th ed.). New Delhi: New Age International.
Kreyszig, E. (2010). Advanced Engineering Mathematics (9th ed.). USA: John Wiley and Sons Inc. Wellin, P. (2013). Programming with Mathematica (1st ed.). New York: Cambridge University Press.
Weisstein, Eric W. "Newton's Method." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NewtonsMethod.html

Approximate Solution of equations by iteration: Wolfram Language Implementation of the Bisection Method

I am going to implement four iteration methods for finding the approximate solutions of equations of the form $f\left(x\right)=0$ using the Wolfram Language, and also deploy these implementations to the cloud for the sake of those who do not have Wolfram Mathematica locally installed in their computers. The cloud deployment allows the user to test these implementations with just their web browsers.
This is the first series of this root finding article, it is based on the bisection method, while the other series will be based on the Newton method, Secant method, and the method of false position (Regula – Falsi)

Bisection Method

The bisection method is based on the repeated application of the intermediate value theorem

If the zero of a function $f\left(x\right)=0$ lies in an interval $I=(a,b)$, we bisect $I$ at the point $m=\frac{a+b}{2}$. If $f\left(a\right)f\left(m\right)<0$ replace the interval $I$ with $(a,m)$,that is $I=(a,m)$ else $f\left(m\right)f\left(b\right)<0$, then $I=(m,b)$.Thus, the new interval $I$ also contains the zero of $f\left(x\right)$. Continuing this bisection process until $\left|f\right(m\left)\right|<\epsilon$, where $\epsilon$ is the degree of accuracy, we take the last midpoint as the desired zero (approximate solution) of the function.

Algorithm

1. Inputs: Initial interval $I=(a,b)$, $\epsilon$.
2. If $f\left(a\right)f\left(b\right)>0$, "Print Approximate solution may not exist in that interval". Goto 7. Else.
3. Compute $m=\frac{a+b}{2}$ and $f\left(m\right)=0$
   
4. While $\left|f\right(m\left)\right|<\epsilon$, If $f\left(a\right)f\left(m\right)<0$, $b=m$, i.e $I=(a,m)$, else $a=m$, i.e $I=(m,b)$.
5. Goto 3. (* The loop continues until $\left|f\right(m\left)\right|<\epsilon$ *).
6. Print $m$
7. Stop

Wolfram Language Implementation

The code above is a procedural implementation of our bisection algorithm. In the code, we defined a function "bisect[fun,{var_Symbol,a_?NumericQ,b_?NumericQ},$\epsilon$_]", where fun is the equation $f\left(x\right)$ we want to find its approximate root, var is the variable in the function $f\left(x\right)$, $a$ and $b$ are the initial values of the interval where the zero of $f\left(x\right)$ lies and $\epsilon$ is the degree of accuracy that we want, with default value $0.0001$.

Given that $f\left(x\right)=e^x+x^4+x-2$, we can say that the zero of $f\left(x\right)$ lies the interval $(0,1)$ since $f\left(0\right)f\left(1\right)<0$.

Thus, we proceed to finding the approximate solution to $f\left(x\right)$ using our implementation.

Wolfram Cloud Deployment

Clicking this link, you will be presented with a web form as pictured

For $\mathrm{expr}$ , enter the function $f\left(x\right)$ you want to find its approximate root, $\mathrm{var}$ is the variable in the function $f\left(x\right)$, $a$ and $b$ where the initial values of the interval where the zero of $f\left(x\right)$ lies and $\epsilon$ is the degree of accuracy that you want, with default value $0.0001$. Then Submit.

After the form submission, comes the result

I really hope you find this interesting and important. Up next is the famous Newton method.



References

Jain, M., Iyengar, S., & Jain, R. (2010). Numerical Method for Scientific and Engineering Computation (5th ed.). New Delhi: New Age International.
Kreyszig, E. (2010). Advanced Engineering Mathematics (9th ed.). USA: John Wiley and Sons Inc.
Welin, P. (2013). Programming with Mathematica (1st ed.). New York: Cambridge University Press.

Sierpinski Triangle: An implementation with the Wolfram Language

Wow, this is my first ever blog article. I am going to make it memorable for me and my readers.
This is what I believe, all the current mind-blowing technologies has a lot to do with Mathematics, everything in the universe is Mathematical in nature, even beauty itself is Mathematical in nature. Thus, it suffice to talk about a renowned Mathematician called Waclaw Franciszek Sierpinski and the implementation of one of his work (Sierpinski triangle) using the Wolfram Language; he contributed to the beauty of Mathematics, the aspect of Mathematics called Fractals Geometry.
A Fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. If the replication is exactly the same at every scale, it is called a self-similar pattern. An example is the Sierpinski triangle. Fractals can also be nearly the same at different levels. Fractals also include the idea of a detailed pattern that repeats itself.
Waclaw Franciszek Sierpinski was a Polish mathematician. He was known for outstanding contributions to set theory, number theory, theory of functions and topology. Three well-known fractals are named after him; the Sierpinski triangle, the Sierpinski carpet and the Sierpinski curve.
The Sierpinski triangle is a fractal and attractive fixed set with the overall shape of an equilateral triangle subdivided recursively into smaller equilateral triangles.This is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction.

Without wasting much time and space (because that's what football is all about. Just kidding), let's implement the pseudo-code that constructs the Sierpinski triangle using the Wolfram Language. You will find this interesting, trust me.

Sierpinski triangle Pseudocode

The Sierpinski triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets:

1. Start with an equilateral triangle.
2. Subdivide it into four smaller congruent equilateral triangles and remove the central one.
3. Repeat step 2 with each of the remaining smaller triangles

Using the Wolfram Language

First we create the vertices of the Equilateral triangle

We can view the Equilateral triangle using the Polygon wrapper with the Graphics function

Subdividing it into four smaller congruent equilateral triangles and removing the
central one. We use the Translate and Scale function. The Translate takes a second argument, which we will denote as "vec"

Finally, we iterate the previous step 7 times using one of the most important iterative function in the Wolfram language, Nest, to get our beautiful
Sierpinski triangle.

Wow, isn't she beautiful?. I love Fractals.

Fractals are beautiful. I hope you will take out some time to study this implementation.
Happy Reading.
Cheers.