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Various atoms and molecules as depicted in John Dalton’s ‘A New System of Chemical Philosophy’ (1808).
Probability theory: …”The central limit theorem (CLT) is one of the great results of mathematics.” It explains the ubiquitous occurrence of the normal distribution in nature.
The theorem states that the average of many independent and identically distributed random variables with finite variance tends towards a normal distribution irrespective of the distribution followed by the original random variables. Formally, let 
be independent random variables with mean 
and variance 
Then the sequence of random variables converges in distribution to a standard normal random variable.
Source: Wikipedia
Complex conjugate. An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.
Emergence across complex systems. Ripple patterns in a sand dune created by wind or water is an example of an emergent structure in nature.
A complex system is a system composed of interconnected parts that as a whole exhibit one or more properties (behavior among the possible properties) not obvious from the properties of the individual parts. This characteristic of every system is called emergence and is true of any system, not just complex ones.
A system’s complexity may be of one of two forms: disorganized complexity and organized complexity. In essence, disorganized complexity is a matter of a very large number of parts, and organized complexity is a matter of the subject system (quite possibly with only a limited number of parts) exhibiting emergent properties.
Examples of complex systems include ant colonies, human economiesand social structures, climate, nervous systems, cells and living things, including human beings, as well as modern energy or telecommunication infrastructures. Indeed, many systems of interest to humans are complex systems.
9,000 Buildings Burn Heating Oil From the Bottom of the Barrel. (Jan 22, 2010).Thick, black smoke from heating systems is a common sight. Switching to cleaner fuel can dramatically cut pollution.
A report from Environmental Defense Fund shows that just one percent of New York City’s buildings, those burning the dirtiest grades of heating oil, produce more pollution than all the city’s cars and trucks combined.
The pollution produced by burning No. 4 or 6 oil—some 1,000 tons of it every year—threatens the health of all New Yorkers, creating a rain of toxic soot that aggravates asthma, increases the risk of cancer, exacerbates respiratory illnesses and can cause premature death.
- Check your building with Interactive Map
- The Bottom of the Barrel: How the Dirtiest Heating Oil Pollutes Our Air and Harms Our Health advocates phasing out the dirty oil by 2020.
![Aggregate, fuzzy inference system. Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced by Lotfi A. Zadeh (1965) as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. Classical bivalent sets are in fuzzy set theory usually called ”crisp” sets. (via Mathworks)](http://24.media.tumblr.com/tumblr_kpbbn7lekh1qzvi5uo1_500.gif)
Aggregate, fuzzy inference system. Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced by Lotfi A. Zadeh (1965) as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. Classical bivalent sets are in fuzzy set theory usually called ”crisp” sets. (via Mathworks)
Airplane vortex. Turbulence in the tip vortex from an airplane wing. Studies of the critical point beyond which a system creates turbulence was important for Chaos theory, analyzed for example by the Soviet physicist Lev Landau who developed the Landau-Hopf theory of turbulence. David Ruelle and Floris Takens later predicted, against Landau, that fluid turbulence could develop through a strange attractor, a main concept of chaos theory. (chaos /limit /design dynamic systems /models)
Cessna, 182 model, wingtip vortex. The flow field around an airplane is a vector field in R3, here visualized by bubbles that follow the streamlines showing a wingtip vortex.
Sierpiński Sieve ardner (1977) and independently Watkins (Conway and Guy 1996, Krížek et al. 2001) noticed that the number of sides for constructible polygons with an odd number of sides are given by the first 32 rows of the Sierpiński sieve interpreted as binary numbers, giving 1, 3, 5, 15, 17, 51, 85, 255, … (Sloane’s A004729, Conway and Guy 1996, p. 140). In other words, every row is product of Fermat primes, with terms given by binary counting / Wolfram MathWorld.
Seventeen or Bust is a distributed computing project started in March 2002 to solve the last seventeen cases in the Sierpinski problem.
For each of those seventeen values of k, the project is searching for a prime number in the sequenceThe goal of the project is to prove that 78557 is the smallest Sierpinski number, that is, the least odd k such that k·2n+1 is composite (i.e. not prime) for all n > 0. When the project began, there were only seventeen values of k < 78557 that were still in question.
- k·21+1, k·22+1, …, k·2n+1, …
using Proth’s theorem, thereby proving that k is not a Sierpinski number. So far, the project has found prime numbers in eleven of the sequences, and is continuing to search in the remaining six. If the goal is reached, the conjectured answer 78557 to the Sierpinski problem will be proven true.
Scatter plots for sensitivity analysis bias. Sampling-based sensitivity analysis by scatterplots. Y (vertical axis) is a function of four factors. The points in the four scatterplots are always the same though sorted differently, i.e. by Z1, Z2, Z3, Z4in turn. Which factor among Z1, Z2, Z3, Z4 is most important in influencing Y? Note that the abscissa is different for each plot: (−5, +5) for Z1, (−8, +8) for Z2, (−10, +10) for Z3 and Z4. Clue: The most important factor is the one which imparts more ‘shape’ on Y.
