1 Lorenz Weather equations: The vertical axis is time, which goes from 0 to 5 in this plot. For each time, the third dimension in perspective show the probability distribution of the wind velocity v: the maximum of the initial bell-shape distribution corresponds to the best initial guess of what is the present state of the system. The width of the bell-shape curve quantifies the initial uncertainty of our observations: we perform an initial measurement of the wind velocity and we know that any measure has some uncertainty, here quantified by the probability that the true initial condition deviates from the best estimate corresponding to the peak. As the peak decreases in amplitude and widens this suggests increasing uncertainty in the value v.

2. “Regions of decreasing uncertainty may exist in chaotic dynamics.” Increasing the forecast horizon does not always lead to a degradation of the prediction, in contrast to standard views on chaotic dynamics.

3. English mathematician F.P Ramsey proved that compete disorder is impossible. Every large set of numbers, such as an ensemble of financial price series or points or objects, necessarily contains highly regular patterns. The relevant question is then, to figure out how many stars, numbers, or figures are required to guarantee a certain desired pattern. How probable is it to find a certain pattern in a given set? 50 of the 400 week intervals since 1910-1996 were chosen at random. The terms of the fit parameters correspond with three crashes in 1929, 1962, and 1987. “The probability that the log-periodic component results from chance is about or less than one in twenty.”

4. A crash is not the critical or singular point itself, but its triggering rate is strongly influenced by the proximity of the critical point: the closer to the critical time, the more probable is the crash.

5. The Law of One price, states that two assets should sell for the same price. If the price differs in two markets, a profitable opportunity arises to sell the asset where it is overpriced and buy where it is under-priced. “Clearly, a noise-free stock market with all information available occupied by fully rational traders of infinite analysis abilities would have a very small trade volume, if any.” A significant number of traders exhibit rational behavior. Information is incomplete and stock traders have limited abilities with respect to analyzing the available information. The market becomes rational if there are many heterogonous agents working on limited information. Perfectly rational investing won out, not because it was perfect, but because it was useful. “The machinery behind market rationality is that each investor, using the market to serve his or her own self-interest, unwittingly makes prices reflect the investor’s information and analysis.” The irrationalities should be studied concerning how they aggregate in the complex, long-lasting, repetitive, and subtle environment of the market.

6. What makes a share in a company valuable? It is earnings, which provide dividends and its potential appreciation, which gives rise to capital gains. Goldstone modes are the zero-energy infinite-wavelength mode fluctuations that attempt to restore broken symmetry. Value today is equal to its expected value tomorrow discounted by the discount factor. In the bubble component, there is not dividend! The bubble is playing the role of the Goldstone mode. “The bubble price can wander up or down and, in limit where it becomes very large in absolute value, dominate over the fundamental price, restoring the independence of the price with respect to dividend.” Goldstone modes appear spontaneous since it has no energy cost, the rational bubble can be spontaneous without any dividend. There is a competition between the increasing growth of the company and the decreasing impact of dividends further in the future due to the effect of the discount factor. The increasing growth of dividends tends to raise price. Dividends are reduced to a risk-adjusted growth rate. When the risk adjusted growth rate becomes equal or larger than the discount rate, the fundamental valuation formula becomes meaningless. The price is the sum of all future presently adjusted dividends. Speculative phases are often stopped by successive increase of the discount rate. 1929 (3.25% to 6%), 1990 in Japan ( 2.55 to 6%)

7. The Crash Hazard rate quantifies the probability that a large group of agents place sell orders simultaneously and create enough of an imbalance in the order book for market makers to be unable to absorb the other side without lowering prices substantially. Cooperative behavior results from imitation. “Our working hypothesis is that agents tend to imitate the opinions of their connections.” “A crash occurs when order wins. In stable markets buyers and sellers balance out each other, normal times are when disorder wins. When the imitation strength K gets close to a special critical value Kc, a very large group of investors share the same opinion, a may act in a coordinated way, an abrupt drop in price, infinite slope K/Kc, a crash occurs. “New demographic, technological, or economic developments prompt spontaneous innovation in financial markets and the first wave of investors and innovators become wealthy. Then imitators arrive and overdo the new techniques. In the ensuing crises, latecomers lose big before regulators and academics put out fires.”

8. Log-periodic equation: Flp(t)=A2 +B2(tc-t)^m2*[C*Cos(w*log(tc-t)/T))]. Decimal year 365 days=1.00 year, Oct 19, 1987 = 87.800 or 1 day=.00274

9. Oct 19, 1987: A2=412, B2=-165, tc=87.74, C=12, w=7.4, T=2, m2=.33

Oct 1929: A2=61 B2=-0.56 Tc=29.84 C=0.08 W=5 T=3 M2=0.63

The algorithm assumes you make a prediction of when the crash will occur. The coefficients for A2,B2,C,T,m2 will change according to a curve fitting algorithm. At best, you can approximate the potential scenerios by appending year parameters and watching for a predicted sudden change based on your assumptions about a crash in the future. If the crash does not occur than readjust your prediction and watch the profile of the curve. No one can know the future. You can only draw a power correlation that then see if it is true.
10. For a few weeks after the crash, a exponentially decaying sinusoidal function emerged. For a few weeks after the crash, a single dissipative harmonic oscillator, with a characteristic decay time of about one week equal to he period of the oscillations.

11. “The concept that emerges here is that the organization of traders in financial markets leads intrinsically to systemic instabilities, which probably result in a very robust way from the fundamental nature of human beings.” “The global behavior of the market, with its log-periodic structures that emerge as a result of the cooperative behavior of traders, is reminiscent of the process of the emergence of intelligent behavior at a macroscopic scale that individuals at the microscopic scale cannot perceive.”

12. It is estimated that the 25 companies that make up one-third of S&P500 index of market capitalization earn roughly half of their income from non-US sources. Oct 1987, to carry the bull-run, the market need to sustain corporate earnings, if not the cycle of rising prices would wither, concern over earnings may have been the straw that broke the camels back.

13. Bubble pathology: a. The bubble starts smoothly with some increasing production and sales or demand for some commodity in an otherwise relatively optimistic market. B. The attraction in investments with good potential gains leads to increasing investment with leverage coming from international investors. Price appreciation occurs. C. This in turn attracts less sophisticated investors, leveraging is further developed which leads to the demand for stock rising faster than the rate at which real money is put into the market. (earnings) D. At this stage, the behavior of the market becomes weakly coupled or practically uncoupled from real wealth. E. As price skyrockets, the number of speculative investors decreases and a period of nervousness starts until a point when the instability is revealed and the market collapses.

14.Pollution: Life expectancy, which the best overall index of the pollution level, has improved markedly as the world population has grown. Food: There is compelling reason to believe that human nutrition will continue to improve into the infinite future. Land: The amount of agricultural land has increased substantially, and is likely to continue to increase where needed. Natural Resources: Natural resources will progressively become less costly, hence less scarce, and will constitute a small proportion of our expense in the future. Energy: The long-term impact of more people is likely to speed the development of cheap energy supplies that are almost inexhaustible. Standard of Living: per capita income is likely to be higher with a growing population than with a stationary one. Human fertility: the contention that poor breed without constraint is wrong. Population growth: Even though the population for the world is increasing, the density of population on most of the world's surface will decrease.