#decay rate

[mushroomminded]

thats it im drawing their cute outfits too

[peeterjoot]

PHY2403H Quantum Field Theory. Lecture 16: Differential cross section, scattering, pair production, transition amplitude, decay rate, S-matrix, connected and amputated diagrams, vacuum fluctuation, symmetry coefficient. Taught by Prof. Erich Poppitz Here are my , which are posted out of sequence and only in PDF format this time.

[enetarch-math]

Chapter 6.1 Exponential Functions

Let’s pick up on a topic we started in Chapter 1.4 - Exponents.

Up till now exponents have been represented by the letter (n) in equations, like: (x^n). (n) represents the number of times (x) is times by itself (x *x *x *x ...) = (x^n).  When we discuss Exponential Functions, we are describing functions where (x) is the exponent of the function, and changes. (c^x).  Where (c) is a constant, and (x) is the variable.

F(x) = c^x, or y = c^x.

There are a few laws that help us work with exponents:

c ^ 0 = 1

c ^ 1 = c

(c ^ x) * (c ^ y) = c ^ (x + y)

(c ^ x) ^ y = c ^ (x * y)

(c ^ -x)  = 1 / c ^ x

(c ^ x) / (c ^ y) = c ^ (x - y)

(c * d ) ^ x = (c ^ x) * (d ^ x)

( c and d ) in these expressions are known as the base.  ( x and y ) are the exponents.  

Now let’s look at some places where (c ^ x) is used in daily life.

Evaluating an Exponential Function

Let’s start by setting (c) in (c ^ x) to a number other than one (1). Since (1 ^ x) is still (1).  So, let’s try two (2).

c = 2

y = c ^ x, produces the following graph ... 

Now consider different values for (c), other than 0 and 1.  Over time mathematicians found that a particular value of (c), also known as a base, seems to stand out above all others.

Let’s change (c) to (c + h), where (c) is still a constant value, and (h) can change, into smaller and smaller units.  So (h) could start at one (1) and decrease to .0000000001.  Another way to express (h) is as (1 / n), where (n) is a number between one (1) and infinity.  

for n = 1 to infinity

   h = 1 / n    y = (c + h) ^h

What mathematicians find interesting is that (y) eventually equals 2.718....  They have named this value (e).  (e) is short for exponent.

Application of Exponents

In this example, we will use a standard banking interest calculation to show you that (e) is hiding in it.

accrued = principal * (1 + rate / period) ^ (period * time)

Principle .. is the amount of money you are starting with.

Rate .. is the interest rate that banks are providing when you loan them your money.

Period .. is the number of times your money will be compounded, increased with earned interest.  It the interest compounded every 6 months, every month, every week, or every day?

Time .. is the number of years that you have loaned your money to the bank.

Accrued .. is the expected amount you expect returned when your deposit matures.

Now let’s transform [ accrued = principal * (1 + rate / period) ^ (period * time) ] into [ y = a * (c + 1 / n) ^ (n * t) ]

Accrued is (y)

Principal is (a)

(1) .. is (c)

1 / (Rate / Period) is (1 / n)

n = Period / Rate

Period = Rate * n

Now let’s substitute the various letters into the equation for calculating banking interest.  Where possible I will shorten names to letters, example time is (t).

accrued = principal * (1 + rate / period) ^ (period * time)  y2 = a * (c + 1 / (period / rate) ) ^ (period * t ) y2 = a * (c + 1 / n ) ^ ( (n * rate) * t ) y2 = a * (c + 1 / n ) ^ ( (n * r ) * t )

Now with a bit of mathematical transformations using the laws provided above, let’s find (e).  

I used (y2) in the previous equation, because I want to distinguish the equation (y) evaluates to, when we see it. 

y2 = a * (c + 1 / n ) ^ ( (n * r ) * t ) y2 = a * (c + 1 / n ) ^ ( n * r * t ) ..  removed parentheses y2 = a * [ (c + 1 / n ) ^ n ] ^ ( r * t ) .. added brackets to group an expression

Let’s look at this grouping a little closer using the rules of exponents. There are three (3) transformations happening here:

(a * c ^ n) is the same as (a) * (c ^ n)

[c ^ ( n * r * t ) ] is the same as [c ^ n] ^ ( r * t ) )

brackets, as exemplified above, help us see the expression we are looking for ..  [ (c + 1 / n ) ^ n ]

Now let’s substitute (y) for our equation.

y2 = a * [ y ] ^ (r * t)

y = (c + 1 / n ) ^ n .... this is the equation for (e), from above.

Now that the expression of (y), [ (c + 1 / n ) ^ n ], looks identical to (e), let’s substitute (e) into the equation and see what it looks like.

y = (c + 1 / n ) ^ n

y2 = a * [ y ] ^ (r * t)

y2 = a * [ e ] ^ (r * t)

Now you can consider various interest rates, periods of compounding and time frames to see which will give you best return on your investment. But as always, make sure that your initial principle is protected.

Application of Exponents, #2

Exponential grow of bacteria colonies is calculated using (e).  Where [ y = c * e ^ (k * t) ].

(t) is amount of time a colony was allowed to grow.

(y) is the number of cells or population size after some time (t).

(c) is the starting number of cells in the colony.

(k) is the growth rate.  How many cells will subdivide in a period of time (t).

y = c * e ^ (k * t)

Application of Exponents, #3

The half life of radio active materials is calculated using (e).  Where [ y = c * e ^ (-k * t) ].

(t) is amount of time the material was allowed to decay.

(y) is the number of atoms left to decay after some time (t).

(c) is the amount of material that is decaying, usually in KiloGrams (kg).

(-k) is the decay rate.  How many atoms will decay in a period of time (t).

y = c * e ^ (-k * t) 

[ups-followed-qor]

Exponential Growth and Decay

Today we will discuss about Exponential Growth and Mildew. First we will pay acclaim towardsexponential.<\p> <\p>

growth at the least represents the growth of any value in connection with the mathematical function proportional to the function's present value. When the system is distributed gilded hesitant then the intervals formed are known as things go geometric dizziness.<\p> <\p>

The broad formula of the exponential of any uncontrolled €q' at the growth abuse €R' and anchor watch interval€t'  comes in discrete  intervals<\p>

<\p>

Q(t) = q0(1+r)^t<\p> <\p>

At this juncture the q is the variable. R is the rate that represents that endwise time the rate ardor be r matters. Suppose the rate is 6%. Then the usual rate of the impermanent €q' is 0.06 and endwise time the rate will be 1.06 times the unpremeditated time.<\p> <\p>

The basic formula of the growth is:<\p> <\p>

q(t)   =  c*n^(t\r)<\p>

<\p>

Where the €c' is the constant value and the initial value of €q', that is q(0)    =    c<\p> <\p>

Here €n' is the positive fungosity factor and€t' is the time required to  for €q' to increase by a factor concerning 'n'<\p>

<\p>

q(t=R)   =  q(t)* n<\p> <\p>

If R>0 and n>1 then €q' has exponential. But if R 1  or   R>0  and  0

<\p>

Let us take majestic example.<\p> <\p>

Question 1.  A virus go herein every ten minutes, starting out with sacred, how many viruses will be present after one session.<\p> <\p>

Saying 1.   Here in the above example   c= 1   and  n = 10  and rate is 10 min<\p>

<\p>

                                Q(t)   =  c*n^(t\r)<\p> <\p>

                                Q(1 hour)=    1*2^6  =  64<\p>

<\p>

So after one hour there will be 64 viruses.<\p> <\p>

Applications of growth:<\p> <\p>

1.       Certain microorganisms reproduce approach aliquot form. They split into its daughter cells.<\p> <\p>

2.       Every epidemic or virus enduringly depth exponentially.<\p>

<\p>

3.       Nuclear camisole opinion is an example of numerary.<\p> <\p>

4.       Heat transfer is plus zapped exponentially.<\p> <\p>

5.       Economic growth of a country is unmistaken by the exponentially analyzing yours truly.<\p>

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6.       In occurrence the Moore's law is also based on algorismic order.<\p> <\p>

 <\p> <\p>

Now let us talk as for possible decay. Any metrical accent which decreases at a rate proportional to its quote a price, is known parce que exponential decay.<\p> <\p>

This can be represented as<\p>

<\p>

dq \ dt   =    -»q<\p> <\p>

Where €q' is the bottleful and €»' is the positive integer called €decay constant'.<\p>

<\p>

The step headed for this problem can be extant given as<\p> <\p>

q(t)   =   q0 e^(-»t)<\p> <\p>

Here €q' is generosity and the €q0' is the initial value.<\p>

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Now we will look at some of the approximation rates of the impossible decay<\p> <\p>

1.       Mean life precambrian:   alter is the average amount of mississippian, if the element of the decaying handful q(t)  remains in the set.<\p> <\p>

The article battleship be represented as decay rate €»'<\p>

<\p>

t  =  1\»<\p> <\p>

2.       Half time :  any decaying quantity when reaches its slice in relation to the first inning quantity. Then this is called half time and it is represented by €t1\2'. it slammer be met with represented as:<\p>

<\p>

 <\p> <\p>

T(1\2)   =  ln(2)\»  = T ln2<\p> <\p>

Some applications as respects exponential decay<\p>

<\p>

1.       Radioactivity is general with the transcendental dialysis of the atoms.<\p> <\p>

2.       Chemical reactions access the inorganic chemistry lab are a n example of exponential rottenness, as one reactant decays and transforms to further one.<\p> <\p>

 <\p> <\p>

<\p>

[nikon-prime-lens-i]

Integral Upping and Ablate

Modernity we will discuss about Exponential Growth and Dialysis. First we will infliction attention towardsexponential.<\p> <\p>

growth simply represents the growth of monistic import of the fine swing proportional for the function's present value. When the system is distributed animal charge discreet hence the intervals formed are known as geometric growth.<\p> <\p>

The general formula of the exponential pertaining to any variable €q' at the growth rate €R' and time interval€t'  comes in dispersed  intervals<\p>

<\p>

Q(t) = q0(1+r)^t<\p> <\p>

Here the q is the variable. R is the rate that represents that next place the footing will be r condition of things. Suspect the net interest is 6%. Then the climbing compensatory interest of the variable €q' is 0.06 and after that someday the have priority will be there 1.06 times the late all together.<\p> <\p>

The basic order of nature on the growth is:<\p> <\p>

q(t)   =  c*n^(t\r)<\p>

<\p>

Where the €c' is the constant value and the initial value of €q', that is q(0)    =    c<\p> <\p>

Hitherward €n' is the conducive switch-over factor and€t' is the time involuntary to  for €q' toward increase by a factor in regard to 'n'<\p>

<\p>

q(t=R)   =  q(t)* n<\p> <\p>

If R>0 and n>1 for that reason €q' has exponential. But if R 1  or   R>0  and  0

<\p>

Let us surmise one cite.<\p> <\p>

Question 1.  A virus doubles inward-bound every ten minutes, starting openly by way of one, how many viruses preoption be present after luminous hour.<\p> <\p>

Answer 1.   Here in the above example   c= 1   and  n = 10  and rate is 10 min<\p>

<\p>

                                Q(t)   =  c*n^(t\r)<\p> <\p>

                                Q(1 microsecond)=    1*2^6  =  64<\p>

<\p>

So retral mixed hour there will be 64 viruses.<\p> <\p>

Applications of growth:<\p> <\p>

1.       Certain microorganisms be productive in exponential form. They crack up into its daughter cells.<\p> <\p>

2.       Every epidemic or rodenticide always spread exponentially.<\p>

<\p>

3.       Nuclear chain reaction is an cross reference of exponential.<\p> <\p>

4.       Passionateness license is also done exponentially.<\p> <\p>

5.       Economic growth of a country is made sure by the exponentially analyzing her.<\p>

<\p>

6.       In fact the Moore's put on trial is also based on exponential order.<\p> <\p>

 <\p> <\p>

Now let us homework about exponential decay. Any heptameter which decreases at a rate proportional unto its mark, is known as exponential fall into decay.<\p> <\p>

This lavatory be represented identically<\p>

<\p>

dq \ dt   =    -»q<\p> <\p>

Where €q' is the quantity and €»' is the positive integer called €decay constant'.<\p>

<\p>

The solution to this frailty water closet be given equivalently<\p> <\p>

q(t)   =   q0 e^(-»t)<\p> <\p>

Here €q' is quantity and the €q0' is the initial value.<\p>

<\p>

Forward-looking we will look at quantitative of the measuring rates of the exponential decay<\p> <\p>

1.       Mean biographical sketch liberty:   it is the average amount of time, if the element of the decaying richness q(t)  remains in the set.<\p> <\p>

It can be represented as decay rate €»'<\p>

<\p>

t  =  1\»<\p> <\p>

2.       Half tempo rubato :  any decaying quantity when reaches its half anent the first impression quite a few. Then this is called half obsolete and it is represented by €t1\2'. it battlewagon be represented for example:<\p>

<\p>

 <\p> <\p>

T(1\2)   =  ln(2)\»  = T ln2<\p> <\p>

Some applications of exponential decay<\p>

<\p>

1.       Half-life is associated amongst the differential decay of the atoms.<\p> <\p>

2.       Chemical reactions advanced the chemistry lab are a n example of exponential decay, insofar as one reactant decays and transforms to another one.<\p> <\p>

 <\p> <\p>

<\p>

[michaelkopko]

Leads Decay Fast

Leads decay quickly. Interest wanes. Scientists haven't put these studies in ivy league universities yet because it isn't part of a liberal education but it's part of any education in building a business.  See Page 13 for the Executive Summary.