Tag: math

Saul Griffith, "Climate Change Recalculated"

According to Saul's calculations, in order to reach a goal of 450 ppm of atmospheric carbon dioxide in time to limit a global rise in temperature to 2 degrees C., we'll have to reduce the amount of burned fossil fuels to 3 terawatts (of power). That means we'll need to replace 11.5 terawatts with new clean sources.

From Stewart Brand's Long Now Blog:

That would mean the following. (Here I'm drawing on notes and extrapolations I've written up previously from discussion with Griffith):

"Two terawatts of photovoltaic would require installing 100 square meters of 15-percent-efficient solar cells every second, second after second, for the next 25 years. (That's about 1,200 square miles of solar cells a year, times 25 equals 30,000 square miles of photovoltaic cells.) Two terawatts of solar thermal? If it's 30 percent efficient all told, we'll need 50 square meters of highly reflective mirrors every second. (Some 600 square miles a year, times 25.) Half a terawatt of biofuels? Something like one Olympic swimming pools of genetically engineered algae, installed every second. (About 15,250 square miles a year, times 25.) Two terawatts of wind? That's a 300-foot-diameter wind turbine every 5 minutes. (Install 105,000 turbines a year in good wind locations, times 25.) Two terawatts of geothermal? Build 3 100-megawatt steam turbines every day-1,095 a year, times 25. Three terawatts of new nuclear? That's a 3-reactor, 3-gigawatt plant every week-52 a year, times 25."

All of it! Please listen to Saul's Long Now Lecture to hear how, despite the odds, he remains an optimist.

Calculate and compare your own power consumption at WattzOn.com.

Check out the very cool 100 mpg, charge-overnight-from-a-wall-socket, ready-for-the-carpool-lane, 3-wheeled (California only) Aptera.
(As mentioned in Saul's talk.)

See also: Google PowerMeter is currently being tested by employees and is not yet available to the public.

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Six Degrees of Separation/Connection may be in trouble. Why can't 6/degrees find Osama bin laden?

Six Degrees has problems

ScienceFriday.com's version

Chances are you've heard of the 'small world' idea of six degrees of separation. But is it correct?

The idea traces back to an experiment begun in 1967 by Stanley Milgram, in which he tried to trace how many acquaintances it would take to pass a letter between two randomly selected people. The result that entered the public consciousness was that in general it took six steps or fewer to bridge the gap between any two people. But is that result accurate? Judith Kleinfeld,

http://www.sciencefriday.com/program/archives/200801256

tags, 6 degrees, Kevin Bacon Effect, six degrees, Milgram,

Haber-Bosch process has often been called the most important invention of the 20th century

this ties in with the earlier post on Peak Phosphorus

from Juergen Schmidhuber's site

Since age 15 or so Prof. Jürgen Schmidhuber's main scientific ambition has been to build an optimal scientist, then retire. In 2028 they will force him to retire anyway. By then he shall be able to buy hardware providing more raw computing power than his brain

Their Haber-Bosch process has often been called the most important invention of the 20th century (e.g., V. Smil, Nature, July 29 1999, p 415) as it "detonated the population explosion," driving the world's population from 1.6 billion in 1900 to 6 billion in 2000.

Haber-Bosch process:

Under high temperatures and very high pressures, hydrogen and nitrogen (from thin air) are combined to produce ammonia.

Nearly one century after its invention, the process is still applied all over the world to produce 500 million tons of artificial fertilizer per year. 1% of the world's energy supply is used for it (Science 297(1654), Sep 2002); it still sustains roughly 40% of the population (M. D. Fryzuk, Nature 427, p 498, 5 Feb 2004).

http://www.idsia.ch/~juergen/worldpopgrowth1.jpg

http://www.idsia.ch/~juergen/

tags needed

Why Aren't There Any Millionaire's in Russia? I thought everyone got 1000 shares of RussPetrol?!

Moscow's suburb for billionaires
By Rupert Wingfield-Hayes
BBC News, Moscow

Most people in Britain are now familiar with the scruffy, boyish and invariably unshaven features of Roman Abramovich, owner of Chelsea football club, and Russia's most famous billionaire.

This week we learned that Mr Abramovich is one of a growing list of hyper-rich Russians.
According to Forbes magazine Russia now has 60 billionaires.

A mile down the road, firmly back in Russia, I went to see Mrs Rima. The 75-year-old showed me around the one-room shack she built with her own hands.

She survives on a pension of £60 a month.

I asked her what she thinks of the rich people who live behind the high green walls.

"They're all thieves," she said. "All that money is stolen from the people."

It's a view millions of Russians would agree with. Fifteen years ago everything in Russia was owned by the state. Today a quarter of Russia's economy is owned by 36 men.

From Our Own Correspondent was broadcast on Saturday, 21 April, 2007 at 1130 GMT on BBC Radio 4. Please check the programme schedules for World Service transmission times.

Three Ways to Listen From BBC.

Mysteries of Percents

Suppose inflation was 11% last year, and 11% this year. Is that a total inflation of 22% for the two years?

Consider a sample of purchases costing $100. After the first year, 11% inflation means you will pay 11% more for the same items.

11% of $100 is $11. Therefore, you will pay $111 by the end of the year for those same items.

In the second year, another 11% inflation assaults your pocketbook. By the end of the second year, you will pay 11% more on $111.

11% of $111 is $12.21. Therefore, by the end of the second year, you will pay $111 + $12.21 = $123.21 for the same goods.

So after two years, you are paying an additional $23.21, or 23.21% more for those same items, which is higher than 22%.

This is because percent increase (or percent decrease) is exponential, and not linear. Exponential growth is what grows your savings account over the years, and what allows inflation to eat it away.

Note that the longer the time period, the bigger the difference between the exponential and the linear value.

For instance, suppose there was 11% inflation for 10 years. What would the price of your $100 basket of goods be?

Using the exponential equation, we have

$100(1 + 0.11)^10 = $283.94

That's a 283% increase!

Can you figure?: If you lost 5% of your weight one year, and then lost another 5% the next year, would you have lost a total of 10% of your original weight?

There are many confusions related to percents because of this principle of exponential growth. For instance, suppose you had investments totaling $1000 and during a market downturn, you lost 30% of that investment.

30% of $1000 is $300, so your investments are now worth $1000 – $300 = $700.

Now, suppose the market goes back up 30%. Are you back to your original investment value of $1000?

No! 30% of $700 is $210. Adding $700 + $210 = $910. You are still down $90.

$300 is approximately 43% of $700. So to increase your $700 back up to $1000, there would have to be a 43% increase in the market!

Can you figure?: If you lost 20% in the stock market, what percent would the market need to increase to recoup your loss?

Such is the mysterious way of percents.

A Pretty Good Shot

In the September 4, 2006 issue of Space News, Thomas Christie, the former chief weapons tester at the Pentagon, was quoted as saying the Ground Based Mid-course Defense System "likely would have less than a 20% chance of shooting down an incoming missile from North Korea". When asked his take on the system's effectiveness, the president of the Missile Defense Advocacy Alliance, Mr. Riki Ellison, responded "that could be from firing just one missile, and, there are about 9 or 10 interceptors that could take multiple shots at the incoming target, thereby increasing the chances of a hit."

If one interceptor has a 20% chance of hitting a target, what exactly are the chances of a successful defense against an incoming missile with 10 interceptors?

Basic probability theory can help answer that question. Let's start simple. Suppose an antiballistic missile (ABM) interceptor has exactly a 20% chance of hitting an incoming missile. That means it has an 80% chance of missing. Not good.

Suppose two interceptors are shot, and for simplicity, let's assume that each interceptor  launched  is an independent event (the action of one interceptor does not effect the other). In this case, there are several possible situations that can occur. The first ABM could hit the target, and the second could miss. Likewise, the first ABM could miss and the second one could hit. Or, they both could hit the target, or, both miss it.

Let's abbreviate these four possibilities {HM, MH, HH, MM} using H for a hit and M for a miss. This list of all the possible outcomes for two interceptors is called the sample space. An element of the sample space is called an event. Now assuming each interceptor is independent, and one interceptor does not effect the other, the probability of any compound event of two interceptors can be computed by multiplying the probability of each of the single events.

For example, if both ABMs hit the target, we can multiply the probability of each one hitting to get the probability of the compound event of them both hitting:

P(HH) = P(H)P(H)

                  = (0.20)(0.20)

                  = 0.04

We have a 4% chance of BOTH missiles hitting the target.

The probability that both miss the target is computed as

P(MM) = P(M)P(M)

                    = (0.80)(0.80)

                    = 0.64

giving a 64% chance of both ABM's missing the target.

Now, what if one hits, and the other misses:

P(HM) = P(H)P(M)

                  = (0.20)(0.80)

                  = 0.16

or a 16% chance of one hitting and one missing. It's the same computation and the same result for the first ABM missing and the second one hitting:

P(MH) = (0.80)(0.20)

                   = 0.16

Now it is a basic tenent of probability that the sum of all probabilities of events in the sample space must add to 1. We can confirm this by noting that

                  0.04 + 0.64 + 0.16 + 0.16 = 1

Let's now ask what is the probability that AT LEAST one hit will occur? There are three possibilites in the sample space where at least one hit can occur: HM or MH or HH. To find the probability of at least one hit occuring, we add the probabilities for each of these compound events.   (In these simple cases, OR measn ADD in probability.)

P(at least one hit) = P( HM or MH or HH)

                                                        = 0.16 + 0.16 + 0.04

                                                        = 0.36

So there is a 36% chance that at least one of the two ABMs will it the target missile.     Since all probabilities in a sample space must add to 1, and the probability that both interceptors miss is P(MM) = 0.64,  we have that:

                  P(at least one hit) + P(all miss) = 1

                                                                               0.36 + 0.64 = 1

Subtracting the P(all miss) from both sides of the equation, we have

                  P(at least one hit) = 1 – P(all miss)

Since there is only  ONE WAY  that any number of interceptors can  ALL MISS  the target, this simple equation gives an easier way to compute the probability of at least one hit for any number of ABMs.

With three ABMs you have a sample set of eight different possible outcomes.

                  {HMM, MHM, MMH, HHM, HMH, MHH, HHH, MMM}

To computer the probabiltiy that at least one of the three interceptors will hit the target, we have  

                  P(at least one hit) = 1 – P(no hit at all)

or

        P(HMM or MHM or MMH or HHM or HMH or MHH or HHH) = 1 – P(MMM)

It's easy to compute the probability that all three interceptors miss:

                    P(MMM) = (0.80)(0.80)(0.80)

                                                = 0.512

                                                = 51.2%

Then, substituting this into our equation, we get the probability that at least one of the interceptors will hit as

                    P(at least one hit) = 1 – P(all miss)

                                                                              = 1 – P(MMM)

                                                                              = 1 – 0.512

                                                                               = 0.488

                                                                               = 48.8%

At least with three ABMs we are getting closer to a 50% chance of at least one of them taking out the target.

So what if we had 10 interceptors.   The number of total outcomes increases to 2^10.   What is the probability that at least one of the ten interceptors will hit the target?

                  P(at least one hit) = 1 – P(all miss)

                                                                            = 1 – P(MMMMMMMMMM)

                                                                              = 1 – (0.8)^10

                                                                               = 1 – 0.107374

                                                                              = 0.892626

                                                                                                           > 89%

It's close to a 90% chance that at least one of the ten antiballistic missiles will hit an incoming missile, assuming independent events.

Now, inreality, the events are not independent. In other words, if the first missile missed, one assumes there would be some information gleaned that would add to the accuracy of the second one. However, enemy efforts to disguise the incoming missile containing the warhead with a number of decoys further complicate the computation, as well as the probability of a hit.

Nevertheless, Mr. Ellison is correct; 10 interceptors increase the chances of a successful hit. As we have shown, the probability increases from 20% to 89%. He also states that "while the system's capability might not be 100%, I think it would have a pretty good shot at intercepting the North Korean missile." Given the price tag in the trillions through 2015 for Ballistic Missile Defense, it had better be "a pretty good shot".

 

Preparing Math Students for a World of Collapse

I have been a math teacher since 1991 when I taught my first algebra class at Philadelphia Community College. I had just received my Bachelors degree in Physics. Bolstered by my girlfriend Val's seemingly cushy part-time employment as a math instructor, and the fact that the math department was in quick need of an algebra instructor, I interviewed with the math chairman, and convinced him that I would be perfect for the job.

I was right. As it turned out, I was good at it. Not that I didn't have my problems, I had many. However, I seemed to have a rapport with a class of math students that allowed me to teach in a relaxed atmosphere and keep everyone engaged. I have always enjoyed teaching and felt lucky to have had the kind of job that continually allows me to learn as much as my students.

But I work hard at teaching. More than most, I think, but it may be that every teacher thinks that. I know there are alot of people who look at teachers and say, "Boy, what a job, summers off." These folks think that it's an easy task to walk in a room, stand up in front of 45 young people, and keep them directly engaged in mathematics for 90 minutes three to five days a week. Plus, they don't think about all the lost nights and weekends a teacher spends preparing exercies and grading papers.

Preparing and psyching up for the experience of teaching is not only exciting, but scary too. As a musician, I've played in countless bands, and I gained some experience on stage. I can liken the first day of a new class to the same kind of butterflies that one can get before Saturday nite's performance.

To keep my anxiety at a minimum, I have always prepared intensely for all my classes, producing notes, websites, hand-outs, transparencies, examples, and finding news stories relevant to our topics. I have learned how to create websites, and use the Internet (significant as I am no adolescent.) and create online classes. I have learned how to create a syllabus of information, pace it through the semester, and determine whether or not it's being understood.

But the biggest thing I've learned from teaching is patience. Early on, I would start to lose it when students would not understand what I was s-o-c-a-r-e-f-u-l-l-y-t-e-l-l-i-n-g-t-h-e-m. I didn't know why they asked the same thing over and over again.

I had to learn that each student's mind will absorb the patterns of math differently, and that I have to frame each concept so that it may be understood as quickly as each mind can possibly get it. It's somewhat like using the right phrase that instantly communicates the message, and the student grasps the pattern all at once in a simple and clear understanding. Finding those key words, and saying them in the right tone is the Teacher's Holy Grail for which there is no resolution. Each semester, the new class of students enlarges the catalogue of Magick Words, and we all climb the Mountain of Math together.

My subject is greeted with much noise and moaning. "Aaaawrrhhh, math…." eyes rolling, head twisting, soul writing in agony at the thought of math class. "Why do we have to learn this?" or "What good will this ever be?" they whine.

I have answered that question with a plethora of responses:

It's a workout for your left head muscle.

You never know what you might be doing in the future.

It's history.

It's cool.

Someday, somebody could pay you to tell them what's in these books.

And while I still believe that math and science are an important part of every educated mind, I've had problems being motivated to teach the topics given in the basic curriculum.

After learning about Peak Oil, and then, economic imbalances, topped off with ecological collapse, what good did learning to solve equations with rational expressions do? How does spending two weeks of precious classtime on factoring prepare student to think critically?

I admit, I'd had reservations about college math curricula even before I'd learned about the impending slide of civilization. Any criticism or suggestion of restructure would be met by the math department with " Well they have to learn this."

"But, why?", I'd ask.

"Because they're learning critical thinking." was always the last response.

Nowhere have I ever found evidence of that claim.

So what then? What should my motivation to teach algebra, and the students' learning it, be?

Personally, I find the subject fascinating. The manipulation of tiny symbolic squiggles representing the unknown quantities of the universe dancing about a page is akin to a beautiful work of art or music. And I have always tried to communicate my own fascination and love of this subject, but many young college students just don't see it that way.

But more importantly, how will learning math help students navigate the challenges they are most certainly going to face as they live their lives in the coming years?

The easiest answer is to use mathematics to help students understand what is happening in the first place. In a class of Pre-algebra or general Algebra, the math topics are rudimentary, but amenable to using energy and population data for percent problems and linear equations. This kind of data is perfect for descriptive statistics analysis as well.

In a liberal arts math course, one can go even further. Looking at exponentials, compound interest, and annuity equations leads directly to the finite resource equation and finding the exponential reserve, which gives the amount of time left for a finite resource that is being used at an increasing percent annually. This allows analysis of gas, coal, oil, and even domestic wellwater timeframes.

In this way, by using actual data and mathematical analysis, communicating to students a realistic picture of the world outside the classroom is neither political, or, makes the teacher sound like a nut case.

Beyond that, re-learning all the skills lost by our cheap oil-cheap imports society will be a difficult task. If some Peak Oil theorists are right, we're going to have to learn to do many forgotten tasks ourselves. How much power can we get out of a nearby stream? How would we build a hydro-electric system?

Indeed, mathematics in a post-crash world will be used in carpentry, agriculture, domestic item production, civic engineering, food storage, the list is endless. McLuhan wrote we are returning to a cultural oral bias, this time with our eyes wide open. Perhaps we will have to re-live the entire history of mathematics from it's first applications to commerce and agriculture millenia ago in order to succeed in preserving our evolving culture.

In any case, mathematics curriculum must adapt to a post-oil reality, or the institutions that push it will be relegated to the dust bin. If only schools and universities just listened to their students asking "Why math?", and responded honestly, they would be much more successful in graduating productive students with higher quantitative and critical thinking skills. And we would all be better off as a society at large.