Thursday, February 02, 2006

Critical Mass

You know, I was thinking. I am not a nuclear physicist or anything but I have learned one thing over the years and that is the idea that before there can be any kind of nuclear explosion, the radioactive material must reach critical mass first! From the way I understand it in my simplistic Pudge brain is that the radioactive material must be compressed evenly to a certain point that the energy stored in the material more energy then we could even imagine! After our small group on Sunday and the Men's group last night I think I understand the concept even better then before. Many people at NewSong have been working very hard at doing what God wants us to do in order to grow His church! God is sending us people who are hungry for his teaching and are looking for that "spiritual rudder" in their lives. I keep thinking about the scripture "if I be lifted up from the earth, I'll draw all men unto me"! We are just lifting up Jesus at NewSong and Jesus just keeps drawing people to Him! It is not going to happen, it is not about to happen, but it is right now as I type happening. People are coming to Jesus through the ministry at NewSong! I have only one question to ask, are YOU ready? We WILL have that Civic Auditorium filled before the year is out! Forget about a goal of 350 unless your are talking about by mid-summer! Plan for the harvest that God has given us! Be blessed beyond belief today.


At 8:43 AM, Blogger Russell said...


At 9:07 AM, Blogger Doug E. Pudge said...

You know LBOM, great minds think alike!!!!!!!!!!!!!!!!!! And so do ours! B4T

At 12:58 PM, Blogger Best of 1980 said...

Bring it on!!

At 8:18 PM, Blogger where's jim? said...

You know I was thinking exactly the same thing {scary huh!!}. You know I am not into time but I do mathematically understand the phenomena of exponential growth and we are just at the first knee in the curve!!!In fact just last Sunday at Cowboys I told Bobby B. we need to be looking at a bigger place, like a school with classrooms and stuff...And if you think we think BIG, just think what God must be thinking!!! Just begin to try and imagine what HE WILL DO!!!We are blessed above and beyond all measure in the name of Jesus!!!!!!!!

At 1:59 PM, Blogger where's jim? said...

The phrase exponential growth is often used in nontechnical contexts to mean merely surprisingly fast growth. In a strictly mathematical sense, though, exponential growth has a precise meaning and does not necessarily mean that growth will happen quickly. In fact, a population can grow exponentially but at a very slow absolute rate (as when money in a bank account earns a very low interest rate, for instance), and can grow surprisingly fast without growing exponentially. And some functions, such as the logistic function, approximate exponential growth over only part of their range. The "technical details" section below explains exactly what is required for a function to exhibit true exponential growth.

But the general principle behind exponential growth is that the larger a number gets, the faster it grows. Any exponentially growing number will eventually grow larger than any other number which grows at only a constant rate for the same amount of time (and will also grow larger than any function which grows only subexponentially). This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at $1 and increases by $1 each week. Although the second option, growing at a constant rate of $1/week, pays more in the short run, the first option eventually grows much larger:

Week: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Option 1: 1c, 2c, 4c, 8c, 16c, 32c, 64c, $1.28, $2.56, $5.12, $10.24, $20.48, $40.96, $81.92, $163.84, $327.68
Option 2:$1, $2, $3, $4, $5, $6, $7, $8, $9, $10, $11, $12, $13, $14, $15, $16
We can describe these cases mathematically. In the first case, the allowance at week n is 2n cents; thus, at week 15 the payout is 215 = 32768c = $327.68. All formulas of the form kn, where k is an unchanging number greater than 1 (e.g., 2), and n is the amount of time elapsed, grow exponentially. In the second case, the payout at week n is simply n + 1 dollars. The payout grows at a constant rate of $1 per week.

Anything that grows by the same percentage every year (or every month, day, hour etc.) is growing exponentially. For example, if the average number of offspring of each individual (or couple) in a population remains constant, the rate of growth is proportional to the number of individuals. Such an exponentially growing population grows three times as fast when there are six million individuals as it does when there are two million. Bank accounts with fixed-rate compound interest grow exponentially provided there are no deposits, withdrawals or service charges. Mathematically, the bank account balance for an account starting with s dollars, earning an annual interest rate r and left untouched for n years can be calculated as s(1 + r)n. So, in an account starting with $1 and earning 5% annually, the account will have after 1 year, after 10 years, and $131.50 after 100 years. Since the starting balance and rate don't change, the quantity can work as the value k in the formula kn given earlier.

Technical details
Let x be a quantity growing exponentially with respect to time t. By definition, the rate of change dx/dt obeys the differential equation:

where k > 0 is the constant of proportionality (the average number of offspring per individual in the case of the population). (See logistic function for a simple correction of this growth model where k is not constant). The solution to this equation is the exponential function -- hence the name exponential growth ('e' being a mathematical constant). The constant is determined by the initial size of the population.

In the long run, exponential growth of any kind will overtake linear growth of any kind (the basis of the Malthusian catastrophe) as well as any polynomial growth, i.e., for all α:

There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). Growth rates may also be faster than exponential. The linear and exponential models are merely simple candidates but are those of greatest occurrence in nature.

In the above differential equation, if k < 0, then the quantity experiences exponential decay.

Examples of exponential growth
Microorganisms in a culture dish will grow exponentially, at first, after the first microorganism appears (but then logistically until the available food is exhausted, when growth stops).
A virus (SARS, West Nile, smallpox) of sufficient infectivity (k > 0) will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people.
Human population, if the number of births and deaths per person per year were to remain constant (but also see logistic growth).
Many responses of living beings to stimuli, including human perception, are logarithmic responses, which are the inverse of exponential responses; the loudness and frequency of sound are perceived logarithmically, even with very faint stimulus, within the limits of perception. This is the reason that exponentially increasing the brightness of visual stimuli is perceived by humans as a smooth (linear) increase, rather than an exponential increase. This has survival value. Generally it is important for the organisms to respond to stimuli in a wide range of levels, from very low levels, to very high levels, while the accuracy of the estimation of differences at high levels of stimulus is much less important for survival.
Charging and discharging of capacitors and changes in current in inductors are also exponential growth and decay phenomena. Engineers use a rule of five time constants to estimate when a steady state has been reached.
Computer technology
Processing power of computers. See also Moore's law and technological singularity (under exponential growth, there are no such singularities).
Internet traffic growth.
Investment. The effect of compound interest over many years has a substantial effect on savings and a person's ability to retire. See also rule of 72
Atmospheric pressure decreases exponentially with increasing height above sea level, at a rate of about 12% per 1000m.
Nuclear chain reaction (the concept behind nuclear weapons). Each uranium nucleus that undergoes fission produces multiple neutrons, each of which can be absorbed by adjacent uranium atoms, causing them to fission in turn. If the probability of neutron absorption exceeds the probability of neutron escape (a function of the shape and mass of the uranium), k > 0 and so the production rate of neutrons and induced uranium fissions increases exponentially, in an uncontrolled reaction.
Newton's law of cooling where T is temperature, t is time, and, A, D, and k > 0 are constants, is an example of exponential decay.
Multi-level marketing
Exponential increases appear in each level of a starting member's downline as each subsequent member recruits more people.

Exponential stories
The surprising characteristics of exponential growth have fascinated people through the ages.

Rice on a chessboard
A courtier presented the Persian king with a beautiful, hand-made chessboard. The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third etc. The king readily agreed and asked for the rice to be brought. All went well at first, but the requirement for 2n − 1 grains on the nth square demanded over a million grains on the 21st square, more than a million million on the 41st and there simply was not enough rice in the whole world for the final squares. (From Porritt 2005)


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