Convert a square matrix to Lower and Upper Triangular Matrix

Upper and Lower triangle Matrix Using C program

In this program we will be discussing on how to convert a square matrix into corresponding upper and lower triangle matrix.

LOGIC:

If for a(ij) :

1. i=j, represents main diagonal elements
2. i>j, represents elements to the left of main diagonal elements
3. i<j, represents elements to the right of main diagonal elements

   #include<stdio.h>
   #include<conio.h>
   void main()
   {
   clrscr();
   int m,n,i,j;
   float a[100][100],cpy[100][100];
   printf("Enter number of rows ");
   scanf("%d",&m);
   printf("\n Enter elements for matrix \n");
   for(i=0;i<m;i++)
   for(j=0;j<m;j++)
   {
   scanf("%f",&a[i][j]);
   cpy[i][j]=a[i][j];
   }
   for(i=0;i<m;i++)
   for(j=0;j<m;j++)
   {
   if(i>j)
   a[i][j]=0;
   if(i<j)
   cpy[i][j]=0;
   }
   printf("Upper triangle matrix is \n");
   for(i=0;i<m;i++)
   {
   for(j=0;j<m;j++)
   printf("%f\t",a[i][j]);
   printf("\n");
   }
   printf("Lower triangle matrix is \n");
   for(i=0;i<m;i++)
   {
   for(j=0;j<m;j++)
   printf("%f\t",cpy[i][j]);
   printf("\n");
   }
   getch();
   }

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Phonons | Detailed Study

PHONONS

Phonon is a quantum of lattice vibrational mechanical energy. They are basically the ground and excited states of quantised lattice vibrations of interacting particles.

The name phonon comes from the Greek word phonē  which means voice or sound, as long wave length phonons give rise to sound.

Atoms in a crystal lattice can be considered to behave as if they are connected by tiny springs which makes mechanical waves that carry heat and sound as a result of executing Simple Harmonic Motion (SHM) by their own thermal vibrations and external forces. A packet of this wave can travel through the crystal with a definite energy and momentum, which in quantum mechanical terms can be considered as a particle (phonon).

So, we can imagine atoms vibrating in SHM as a lattice wave.

Its equation can be written as,

                                

Where:

k= Restoring force, M= Mass of atom

Potential energy =

By Schrödinger’s equation,

On solving, we get the energy Eigen values as,

The energy difference between two allowed states is  . So we define the basic packets of energyand momentumand these fundamental packets of energy exchanged are called phonons.

Phonon is a quasi- particle, that is something that is not a real particle but can behave and be considered as one. They are responsible for acoustic properties, thermal conductivity and heat capacity of solids. Besides, phonons are bosons, that is they follow Bose-Einstein statistics (Indistinguishable, but not obeying Pauli’s exclusion principle), which take the crucial role in the super conducting phenomena (low temperature) as an intermediate particle which makes electrons slightly attract each other whereas they would have normally repelled.

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Direct and Indirect Band Gap Semiconductor

DIRECT AND INDIRECT BAND GAP SEMICONDUCTORS

Semiconductors can be classified on the basis of the difference in energy level between valence band and conduction band as direct and indirect band gap semiconductors.

That is, if we plot the energy of electron (E) as a function of crystal momentum k, wave vector along the principle direction of crystal we will be able to find two types of graphs as follows.

Direct band gap semiconductor

Electrons in the conduction band occupy available states at bottom of it, whereas holes in the valence band occupy available states at the top positions where they can attain the least energy states. In the case of a direct band gap semiconductor, maximum energy and momentum of electron in valence band is same as minimum energy and momentum of electron in the conduction band which leads to the direct recombination of an electron in the with a hole in the valence band releasing the difference in energy as a photon. E_g = hv

Indirect band gap semiconductor

In the case of an indirect band gap semiconductor, maximum energy and momentum of electron in valence band is different from the minimum energy and momentum of electron in the conduction band which makes the direct recombination of electron and hole is impossible as the transition must conserve energy and momentum and the momentum of the photon is negligibly small to conserve the momentum the system. So, electron first transits to an intermediate level called recombination centre by the release of a phonon and then to the valence band by the release of infrared rays (heat).

Indirect recombination proceeds in a much slower rate than direct recombination as it involves the presence of three entities (electron, phonon and photon).

Examples of direct and indirect band gap semiconductors

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Laplace Transformation-Detailed Study

LAPLACE TRANSFORM FOR SIGNALS

Basically Laplace Transform, in mathematics, is an integral transform which takes a real variable t ( usually time ) to a complex variable s ( frequency ). It is named after its discoverer Pierre Simon Laplace ( French Mathematician & Scientist).

Laplace transform can be defined as a frequency domain approach for continuous time domain signals irrespective of the stability of the system.

Consider a function f (t) which is to be continuous and defined for all values  of t ≥0.

Then Laplace transform is

That is, Laplace transform is very much similar to the Fourier transform. But Fourier transform of a function is complex function of real variable (frequency) unlike Laplace transform which is a complex function of complex variable.

If the lower limit is 0, then the transformation is referred to as one sided or unilateral Laplace Transform while in the two sided or bilateral transform lower limit is -∞.

INVERSE LAPLACE TRANSFORM

Inverse Laplace transformation converts a frequency domain function F(s) to the time domain function  f (t).

Inverse Laplace transform is determined as follows

So, we can denote the transformation relationship between  f (t) and F[S] as

                                                            

While analysing problems Laplace transform can be applied easily if we consider it as two categories,

that is as functional and Operational transforms.

A functional Laplace transform is the Laplace transform for a specific function such as sin⁡〖θ , t,e^(-at)…  〗,

while an operational transform defines a general mathematical property of the Laplace transform.

(Note that we can attain the same result by applying direct equation also.)

FUNCTIONAL  LAPLACE  TRANSFORM

A functional transform is simply the Laplace transform of a specified function of t. On unilateral Laplace transform of the following functions the pairs obtained are:

OPERATIONAL  LAPLACE  TRANSFORM

Operational transform indicate how mathematical operations performed on either f  (t) or F[S] are converted into the other.

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HALL EFFECT

                 HALL  EFFECT

Hall effect, named after its discoverer Edwin Herbert Hall (U.S Physicist) in 1879, is a conduction phenomenon which is defined as the development of a transverse electric field in a solid conducting material when it is placed under the influence of mutually perpendicular electric and magnetic fields.

                             Hall effect is observed a the result of force due to magnetic field in a moving electric charge carrier. No matter we consider current as the movement of electrons, holes or as a combination of both ( in the case of semi conductors) a perpendicular magnetic field displaces the moving electric charge in the same direction side ways at right angles to it.

Thus, the accumulation of a specific polarity in one side results in the development of a measureable potential difference within the conductor which is indicated as positive or negative by the detector.

                               Besides solids Hall effect is observed in plasma ( ionised gases) also, but it is significantly different from what we observe in solids.

Then,

Fnet = q (E + v x B )

The net force experienced by a hole moving along y- direction is the sum of forces due to electric field and Magnetic field along y- direction

Force due to electric field     = qEy  

Force due to magnetic field  = q x y- component of v x B

                                                  = q [ - vx Bz - Bz vz]

                                                  = q [ -vx Bz] ( since vz = 0 )

               

               Fnet = Fy = q( Ey – vxBz)

In order to maintain steady state condition, Ey must balance the product vx Bz. Then, Fnet = 0. This condition occurs when the holes are shifted slightly in the negative y direction due to B. As the forces are now balanced, the lateral force becomes zero as they drift along the bar.

This establishment of Ey is known as Hall effect and the resulting voltage is known as Hall voltage Vhall = Ey w

Drift velocity           Vx= Jx  / q po

So, Ey= Jx Bz /q po= RH  Jx  Bz

Po = JxBz/qEy

     =  (Ix/wt  )Bz /  q (v/w   )                                                        

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