Friday 27 July 2012

Math of drugs and bodies (pharmacokinetics

__   Shahana Moideen  XB  __   


  
  
Pharmacokinetics is the process whereby substances (like food and drugs) are ingested into the body (via mouth or needles) and processed. We’ll concentrate on drugs.
 


The process of pharmacokinetics has 5 steps:
  • Liberation – the drug is released from the formulation.

  • Absorption – the drug enters the body.

  • Distribution – the drug disperses throughout the body

  • Metabolism – the drug is broken down by the body.

  • Excretion – the drug is eliminated from the body.

Of course, each drug needs to act on the body in a different way. Some drugs need to be absorbed quickly (like nitroglycerin if we are having a heart attack) and preferably eliminated quickly (otherwise toxins build up in the blood). For other drugs, we want slow absorption so we get maximum benefit and don’t lose a lot of it from excretion.
So when your doctor prescribes (say) "take 2 tablets every meal time", this is based on the desirable levels of drug concentration and known levels of distribution, metabolism and excretion in the body.

What’s the math?

When the nurse first administers the drug, the concentration of the drug in the blood stream is zero. As the drug moves around the body and is metabolized, the concentration of the drug increases.
There comes a point when the concentration no longer increases and begins to decline. This is the period when the drug is fully distributed and metabolism is taking place. As time goes on, the drug concentration gets less and less and falls below a certain effective amount. Time to take some more pills.
We can model such a situation mathematically with a differential equation  It has 2 parts – an absorption part and an elimination part. At first, absorption (increasing drug concentration) takes precedence and over time, elimination (decreasing concentration) is the most important element.
We have the following variables:
D = drug dose given
V = volume distributed in the body
C = concentration of the drug at time t
F = fraction of dose which has been absorbed (also called bioavailability)
A = absorption rate constant
E = elimination rate constant
t = time

Absorption part: This depends on the amount of the drug given, the fraction that has been absorbed and the absorption rate constant. It decreases as time goes on. The expression for absorption is given by:

A × F × D × e-At

Elimination part: The elimination dynamic is affected by the elimination constant, the volume distributed in the body and the concentration left of the drug. The expression for this part is:

E × V × C

For our model, we need to subtract the elimination part from the absorption part (since the absorption part increases the concentration of drug and the elimination part decreases it). Our differential equation is as follows:


We now substitute some typical values for our variables (without units to keep things simple. Note C is a variable, the one for which we seek an expression in t.)



Solving this differential equation (using a computer algebra system), gives the concentration at time t as:

C(t) = 533.3(e0.4t − e−0.5t)



We can see in the graph the portion where the concentration increases (up to around t = 3) and levels off. The concentration then decreases to almost zero at t = 24.

Pharmacokinetics is yet another interesting “real life” application of math.

VEDIC MATHS

Flowy XB  -
Vedic mathematics is a system of mathematics consisting of a list of 16 basic sūtras, or aphorisms. They were presented by a Hindu scholar and mathematician, Bharati Krishna Tirthaji Maharaja, during the early part of the 20th century.[1]
Tirthaji claimed that he found the sūtras after years of studying the Vedas, a set of sacred ancient Hindu texts.[2] However, Vedas do not contain any of the "Vedic mathematics" sutras.[3][4]
The calculation strategies provided by Vedic mathematics are said to be creative and useful, and can be applied in a number of ways to calculation methods in arithmetic and algebra, most notably within the education system. Some of its methods share similarities with the Trachtenberg system.
The sūtras (formulae or aphorisms)
Vedic mathematics is based on sixteen sūtras which serve as somewhat cryptic instructions for dealing with different mathematical problems. Below is a list of the sūtras, translated from Sanskrit into English:
§  "By one more than the previous one" 
§  "All from 9 and the last from 10
§  "Vertically and crosswise (multiplications)
§  "Transpose and apply"
§  "Transpose and adjust (the coefficient)"
§  "If the Samuccaya is the same (on both sides of the equation, then) that Samuccaya is (equal to) zero"
§  By the Parāvartya rule
§  "If one is in ratio, the other one is zero." 
§  "By addition and by subtraction."
§  By the completion or non-completion (of the square, the cube, the fourth power, etc.)
§  Differential calculus 
§  By the deficiency
§  Specific and general
§  The remainders by the last digit
§  "The ultimate (binomial) and twice the penultimate (binomial) (equals zero),"
§  "Only the last terms,"
§  By one less than the one before
§  The product of the sum
§  All the multipliers




Subsūtras or corollars
§  "Proportionately"
§  The remainder remains constant
§  "The first by the first and the last by the last"
§  For 7 the multiplicand is 143
§  By osculation
§  Lessen by the deficiency
§  "Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square (of the deficiency)".
§  "By one more than the previous one"
§  "Last totaling ten"
§  The sum of the products
§  "By (alternative) elimination and retention (of the highest and lowest powers)"
§  By mere observation,
§  The product of the sum is the sum of the products
§  On the flag

Tuesday 24 July 2012

IMPORTANCE OF MATHEMATICS IN EVERYDAY LIFE

Sumisha k k        
X.B        

                                    “Education should be started with mathematics. For it forms well designed brains that are able to reason right. It is even admitted that those who have studied mathematics during their childhood should be trusted, for they have acquired solid bases for arguing which become to them a sort of second nature”.
             Mathematics is around us. It is present in different forms whenever we pick up the phone, manage the money, travel to some place, play soccer, meet new friends; unintentionally in all these things mathematics is involved.

For Example

Cooking: the idea of proportion  Percentage
Medicine/Pharmacy
Bank: savings and credit
Chance to win in lottery: Probability
Area
Geometry in clothing
Geometry in house decoration
Geometry in art
Geometry in architecture
Symmetry in the nature

  • The physical sciences (chemistry, physics, engineering)
  • The life and heath sciences (biology,psychology,pharmacy,nursing,optometry)
  • The social science, including Anthropology, communications, Eco-comics, Linguists, Education, geography
  • The tech Sciences, like computer science, networking, software development
  • Business and commerce
  • Medicine
  • Actuarial science, used by insurance companies

Mathematics and applications

  • Every area of mathematics has its own unique applications to the different career options. For example
  • Algebra: computer sciences, cryptology, networking, study of symmetry in chemistry and physics
  • Calculus (differential equations): Chemistry, biology, physics engineering, the motion of water, rocket science, molecular structure, option
    price modeling in business and economics models, etc...
  • Students are encouraged to give serious attention to their future. The career world is competitive!
                      
               Most of university degree require mathematics. Students who choose not to take mathematics seriously or to ignore it in high school forfeit many future career opportunities that they could have. They essentially turn their backs on more than half the job market. The importance of mathematics for potential careers cannot be over emphasized. To get degrees in the following areas one need to have good knowledge of mathematics and statistics
                         It’s almost impossible to get through a day without using maths in some way, because our world is full of numbers to handle and problems to solve. Studying maths provides you with the tools to make sense of it all, making life that little bit easier.          

SO ,WE CONCLUDE THAT MATHEMATICS PLAY AN IMPORTANT ROLE IN OUR DAILY LIFE.

INTERESTING !!!!!!!!!

Vandana Krishnan
X A

Do you notice anything interesting in the following multiplication ?

138 X 42 = 5796

There are nine digits and all are different . Can you think of other such combinations ?

12 X 483 = 5796
18 X 297 = 5346
39 X 186 = 7246
48 X 159 = 7632
27 X198 = 5346
28 X 57 = 4396
 4  X1738 = 6952
 4  X 1963 = 7852

     ***************

An Impossible Proof
Can you prove 45 - 45 = 45 ?

Write down the figures from 9 to 1

                    987654321
subtract        123456789
                    864197532
Add up the digits .
9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45

Add up the digits
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

Add up the digits
8 + 6 + 4 + 1 + 9 + 7 + 5 + 3 + 2 = 45

Therefore 
              45 - 45 = 45
                 ************************



The Tower of Brahma

Vandana Krishnan X.A 

             
              This , so the legend has it , is a temple in Benares . They say that when the great lord Brahma created the world he put up 3 diamond sticks , mounted on a brass plate , beneath the dome that marks the center of the world . Upon one of the sticks he placed 64 gold discs .The biggest at the bottom , all decreasing in size , with the smallest at the top . The temple priests have the tasks of transposing the discs from one stick to other , using the third as an aid . They work day and night , but must transpose only one disc at a time and must not put a bigger disc on top of a smaller one . When their task is complete , the legend says , the world will disappear in a clap of thunder .
Now the total number of transpositions needed to shift the 64 disc is 2 raised to 64 minus 1 , or 18446744073709551615 . If every transpositions takes a second it will take 500000 million years to get the job done .


Origin of the Game
The Tower of Brahma, or the Tower of Hanoi, as it is sometimes called, was the invention of the French mathematician Edouard Lucas and was sold as a toy in France in 1883.
The legend of the Temple of Benares is also his invention.
This is a great game to get young children (5- or 6-year old) started and on a par with Hop Over.






Description
The original device consists of a base supporting 3 vertical pegs placed in a straight line, with a conical tower of 8 to 10 disks of the same or indeterminate color. 
The Rules
The tower of disks must be transported from the left peg to the right peg,   (a) moving only 1 disk at a time, and   (b) never placing a disk on a smaller one than itself, thus always conserving a tower, cone-like, shape. The middle peg is evidently temporarily employed in the process. The game is fiendishly difficult with this model.


Mathematics
To see the mathematics involved in the game.
Several things can be done to help the child observe and understandthe mathematical regularities of this game -(a) Place the pegs preferably in triangular rather than linear formto clearly observe the direction of the movement of each disk.(b) Color the odd-numbered disks one color and even-numbered disks another color,to clearly observe the alternation of colors at all times,disks of the same color NEVER touching each other.A detailed pedagogical approach is seen later in the "card model". 


Amazing prime numbers

  - Sneha Devan- XI . B

Here are few amazing prime numbers, these prime numbers were proved by the XVIIIth century.
31, 331, 3331 ,33331 ,333331, 3333331 ,33333331.

angel number 412

 Rules
 
Step 1  
       Select any whole number. 
Step 2  
      If it is an even number, divide by 2; if it is odd number multiply by 3 and add 1.
Step 3  
      Repeat the process mentioned in step 2 until you get the loop value 4, 2, 1 in repetition. 
Example
  Whole number is 15. 
  15 is an odd no; so (15 × 3) + 1 = 46 
  46 is an even no; so 46 / 2 = 23 
  23 is an odd no; so (23 × 3) + 1 = 70 
  70 is an even no; so 70 / 2 = 35 
  35 is an odd no; so (35 × 3) + 1 = 106 
  106 is an even no; so 106 / 2 = 53 
  53 is an odd no; so (53 × 3) + 1 = 160 
  160 is an even no; so 160 / 2 = 80 
  80 is an even no; so 80 / 2 = 40 
  40 is an even no; so 40 / 2 = 20 
  20 is an even no; so 20 / 2 = 10 
  10 is an even no; so 10 / 2 = 5 
  5 is an odd no; so (5 × 3) + 1 = 16 
  16 is an even no; so 16 / 2 = 8 
  8 is an even no; so 8 / 2 = 4 
  4 is an even no; so 4 / 2 = 2 
  2 is an even no; so 2 / 2 = 1 
  1 is an odd no; so (1 × 3) + 1 = 4 
  4 is an even no; so 4 / 2 = 2 
2 is an even no; so 2 / 2 = 1
So the loop 4..2..1 goes on and on.
Tip : The angel number 421 is the smallest prime formed by the two powers in logical order from right to left. 

million, billion, trillion ,?,?,?



Million...billion...... Trillion....!!!!!!!!
Values
Zero's
Names
100
0
One
101
1
Ten
102
2
Hundred
103
3
Thousand
104
4
Myriad
106
6
Million
109
9
Billion
1012
12
Trillion
1015
15
Quadrillion
1018
18
Quintillion
1021
21
Sextillion
1024
24
Septillion
1027
27
Octillion
1030
30
Nonillion
1033
33
Decillion
1036
36
Undecillion
1039
39
Duodecillion
1042
42
Tredecillion
1045
45
Quattuordecillion
1048
48
Quindecillion
1051
51
Sexdecillion
1054
54
Septdecillion / Septendecillion
1057
57
Octodecillion
1060
60
Nondecillion / Novemdecillion
1063
63
Vigintillion
1066
66
Unvigintillion
1069
69
Duovigintillion
1072
72
Trevigintillion
1075
75
Quattuorvigintillion
1078
78
Quinvigintillion
1081
81
Sexvigintillion
1084
84
Septenvigintillion
1087
87
Octovigintillion
1090
90
Novemvigintillionn
1093
93
Trigintillion
1096
96
Untrigintillion
1099
99
Duotrigintillion
10100
100
Googol
10102
102
Trestrigintillion
10120
120
Novemtrigintillion
10123
123
Quadragintillion
10138
138
Quinto-Quadragintillion
10153
153
Quinquagintillion
10180
180
Novemquinquagintillion
10183
183
Sexagintillion
10213
213
Septuagintillion
10240
240
Novemseptuagintillion
10243
243
Octogintillion
10261
261
Sexoctogintillion
10273
273
Nonagintillion
10300
300
Novemnonagintillion
10303
303
Centillion
10309
309
Duocentillion
10312
312
Trescentillion
10351
351
Centumsedecillion
10366
366
Primo-Vigesimo-Centillion
10402
402
Trestrigintacentillion
10603
603
Ducentillion
10624
624
Septenducentillion
10903
903
Trecentillion
102421
2421
Sexoctingentillion
103003
3003
Millillion
103000003
3000003
Milli-Millillion