Saturday, February 14, 2015

The Algebraic and parametric equation of conic section and its Application in Linear Algebra and Singular Value Decomposition and Image or Data compression.



Today i will start from the definition of circle .The circle is the locus of a point P(x,y) which moves in such a way that it's distance from fixed point called center is constant that constant is called the radius of circle Using this definition we can derive equation of circle  using Distance formula from coordinate Geometry.

                   d=sqrt[(X2-X1)^2 + (Y2-Y1)^2]
A circle is a simple shape in Euclidean geometry. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius.
A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.
A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter, using calculus of variations.

Cartesian coordinates

n an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that

\left(x - a \right)^2 + \left( y - b \right)^2=r^2.  
 Is the equation of a circle whose origin is shifted by X=a and Y=b from O(0,0)
This equation, known as the Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x − a and y − b. If the circle is centred at the origin (0, 0), then the equation simplifies to
x^2 + y^2 = r^2.\!\  
The equation can be written in parametric form using the trigonometric functions sine and cosine as
x = a+r\,\cos t,\,
y = b+r\,\sin t\,
where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the x-axis.
An alternative parametrisation of the circle is:
x = a + r \frac{2t}{1+t^2}.\,
y = b + r \frac{1-t^2}{1+t^2}\,
In this parametrisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the line passing through the centre parallel to the x-axis .

Parametric equation

In mathematics, parametric equations of a curve express the coordinates of the points of the curve as functions of a variable, called a parameter. For example,
\begin{align} x&=\cos t\\ y&=\sin t \end{align}
are parametric equations for the unit circle, where t is the parameter. Together, these equations are called a parametric representation of the curve.
A common example occurs in kinematics, where the trajectory of a point is usually represented by a parametric equation with time as the parameter.

Ellipse

It is the locus of a point which moves in such a way that the sum of it's distance from two focus is a constant which is equal to 2a.

An ellipse in canonical position (center at origin, major axis along the X-axis) with semi-axes a and b can be represented parametrically as
x(t)=a\,\cos t
y(t)=b\,\sin t. 
The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is      \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1. This can be explained as follows:
If we let
{x} = {a}\cos\theta.
{y} = {b}\sin\theta.
Then plotting x and y values for all angles of θ between 0 and 2π results in an ellipse (e.g. at θ = 0, x = a, y = 0 and at θ = π/2, y = b, x = 0).
Squaring both equations gives:

{x}^2 = {a}^2\cos^2\theta.





{y}^2 = {b}^2\sin^2\theta.
Dividing these two equations by a2 and b2 respectively gives:
\frac{{x}^2}{{a}^2} = \cos^2\theta.\frac{{y}^2}{{b}^2} = \sin^2\theta.


  • Adding these two equations together gives:
    • \frac{{x}^2}{{a}^2} + \frac{{y}^2}{{b}^2} = \cos^2\theta + \sin^2\theta.



    \frac{{x}^2}{{a}^2} + \frac{{y}^2}{{b}^2} = 1.


    This means any noncircular ellipse is a compressed (or stretched) circle. If a circle is treated like an ellipse, then the area of the ellipse would be proportional to the length of either axis (i.e. doubling the length of an axis in a circular ellipse would create an ellipse with double the area of the original circle).
    An ellipse in general position can be expressed as

    x(t)=X_c + a\,\cos t\,\cos \varphi - b\,\sin t\,\sin\varphi
    y(t)=Y_c + a\,\cos t\,\sin \varphi + b\,\sin t\,\cos\varphi
    as the parameter t varies from 0 to 2π. Here (X_c,Y_c) is the center of the ellipse, and \varphi is the angle between the X-axis and the major axis of the ellipse

      
    Focus
    The distance from the center C to either focus is f = ae, which can be expressed in terms of the major and minor radii:

    f = \sqrt{a^2-b^2}.
    The sum of the distances from any point P = P(x,y) on the ellipse to those two foci is constant and equal to the major axis

    PF_1 + PF_2 = \sqrt{(x+f)^2+y^2} + \sqrt{(x-f)^2+y^2} = 2a.

    Eccentricity

    The eccentricity of the ellipse (commonly denoted as either e or \varepsilon) is
    e=\varepsilon=\sqrt{\frac{a^2-b^2}{a^2}} =\sqrt{1-\left(\frac{b}{a}\right)^2} =f/a
     

     

     




     
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    Friday, February 13, 2015

    Balancing of Redox Reaction in terms of Oxidation number in Neutral and basic medium,Gibb's free energy change in Redox reaction,Standard Reduction potential and Application of redox reaction in Biochemistry.


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    Tuesday, February 10, 2015

    Hiyperconjugation and its stabalizing effect and it's corrosponding Resonance Structure for stability.

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    Hyperconjugation is a general stabilising interaction. It involves delocalisation of σ electrons of C—H bond of an alkyl group directly attached to an atom of unsaturated system or to an atom with an unshared p orbital. The σ electrons of C—H bond of the alkyl group enter into partial conjugation with the attached unsaturated system or with the unshared p orbital. Hyperconjugation is a permanent effect.
    To understand hyperconjugation effect, let us take an example of  (ethyl cation) in which the positively charged carbon atom has an empty p orbital. One of the C-H bonds
    of the methyl group can align in the plane of this empty p orbital and the electrons constituting the C-H bond in plane with this p orbital can then be delocalised into the
    empty p orbital    .
    This type of overlap stabilises the carbocation because electron density from the adjacent σ bond helps in dispersing the
    positive charge.
    Hyperconjugation is also possible in alkenes and alkylarenes. Delocalisation of electrons by hyperconjugation in the case of alkene as shown in the video.
    There are various ways of looking at the hyperconjugative effect. One of the way is to regard C—H bond as possessing partial ionic character due to resonance.
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    Resonance and stability of benzene,Mechanism of electrophilic substitution reactions and Hyperconjugation,Carbocation,RNA,ATP and Mathematics Behind them.

    Today i will start from the forces at molecular level and their stabilizing effect responsible for the existence of a particular compound or structure and also how resonance mathematically it is the oscillation of molecule between its all possible combination possible,at molecular level resonance is mainly found in compound having double bond.so what are the forces mainly responsible for stabilizing effects at molecular level and this altering physical property like boiling point,polarity etc.
    These interaction includes
    1)Covalent interaction  2) Resonance Interaction leading to dispersal of charge 3)Hyperconjugative interaction 3)ionic interaction 4)Hydrogen bond interaction 3)van der wall's Force interaction 4)London Dispersion interaction 5)Electromeric Effect +E Effect and -E effect. etc

    Let's start from various domain of bio molecular Science and molecular science where they play vital role.
    a)RNA (Ribo nucleic Acid) :RNA is similar to DNA, but with two major chemical differences. First, RNA molecules contain ribose sugars in which the number 2 carbon is bonded to a hydroxyl group. In DNA, this hydroxyl group is replaced by a hydrogen atom. Second, RNA molecules utilize uracil in place of thymine. Uracil has the same structure as thymine, except that one of its carbons lacks a methyl (—CH3) group. Transcribing the DNA message into a chemically different
    molecule such as RNA allows the cell to tell which is the original information storage molecule and which is the transcript. DNA molecules are always double-stranded (except
    for a few single-stranded DNA viruses)while the RNA molecules transcribed from DNA are typically single-stranded.  
    The figure of  DNA and RNA can develop some  motivation here for their stability in terms of fundamental forces of interaction.
    In DNA it is hydrogen bond as discussed in the previous article as depicted here
      Although there is no chemical reason why RNA cannot form double helices as DNA does, cells do not possess the enzymes necessary to assemble double strands of RNA, as they do for DNA. Using two different molecules, one single-stranded and the other double-stranded, separates the role of DNA in storing hereditary information from the role of RNA in using this information to specify protein structure.
    But my question is which bio molecular interaction is responsible for the stability of single strand ,why doesn't it decompose down?
    Let's analyses it in terms of figure 
    DNA versus RNA. DNA forms a double helix, uses deoxyribose as the sugar in its sugar-phosphate backbone, and utilizes thymine among its nitrogenous bases. RNA, on the other hand, is usually single-stranded, uses ribose as the sugar in its sugar-phosphate backbone, and utilizes uracil in place of thymine.

    Let's come to another instance where these molecular force will stabilize ATP molecule by storing Energy in bond.

     
    ATP. Adenosine triphosphate (ATP) contains adenine, a fivecarbon sugar, and three phosphate groups. This molecule serves to transfer energy rather than store genetic information.
    What molecular force is responsible for stability of ATP. In addition to serving as subunits of DNA and RNA, nucleotide bases play other critical roles in the life of a cell.
    For example, adenine is a key component of the molecule adenosine triphosphate the energy currency of the cell. It also occurs in the molecules nicotinamide adenine dinucleotide (NAD+) and flavin adenine dinucleotide (FAD+), which carry electrons whose energy is used to make ATP. A nucleic acid is a long chain of five-carbon sugars with
    an organic base protruding from each sugar. DNA is a
    double-stranded helix that stores hereditary information as a specific sequence of nucleotide bases. RNA is a single-stranded molecule that transcribes this information to direct protein synthesis.

    Benzene:A completely different phenomenon will come to define the stability of benzene and mathematically it is equivalent to oscillation ie benzene switching on and off between to energy equivalent term and hence the average energy of benzene decreases on an average ie it's bind dissociation enthalpy is in between double and single bond and bond length also between double and single bond.
    and it's resonance can be shown as

    its Equivalence is as following

    We can say that the three pi bond is executing oscillation so the six pi electron is present everywhere the hexagonal ring.


      Its other energetically same structure is as following
    The sum up of the above two resonance structure is equivalent to the following hybrid.



     The electron free electron density or pi electron density is increasing above and below the hexagonal benzene where each carbon is SP2 hybridized and thus trigonal planar and the unhybrid P orbital oriented perpendicular above and below the hexagonal plane total six having six electron  in this orthogonal p orbital forming pi bond leads to electron density above and below the plane.
    I will come to different interactions at molecular level which will lead to stability of molecular structure or system and try to correlate them using mathematics till then wait.

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