Pareto Dominance and Mean Variance. Cardinality. Fractal Geometry. Fractals and Chaos Theory. The picture above is an example of a Julia set.
The picture is a graph of the results of entering a series of numbers into the VERY SIMPLE function f(x) = x2 + c. For this particular graph, the value of c was -0.74 + 0.11i. There are several interesting things about this picture: 1. Laplace operator. The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics, where the operator gives a constant multiple of the mass density when it is applied to a given gravitational potential.
Solutions of the equation ∆f = 0, now called Laplace's equation, are the so-called harmonic functions, and represent the possible gravitational fields in free space. The Laplacian occurs in differential equations that describe many physical phenomena, such as electric and gravitational potentials, the diffusion equation for heat and fluid flow, wave propagation, and quantum mechanics. The Laplacian represents the flux density of the gradient flow of a function. Ordinal number. Representation of the ordinal numbers up to ωω.
Each turn of the spiral represents one power of ω In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals. General Systems Theory. © 1993, David S.
Walonick, Ph.D. General systems theory was originally proposed by biologist Ludwig von Bertalanffy in 1928. Stochastic process. Stock market fluctuations have been modeled by stochastic processes.
In probability theory, a stochastic process /stoʊˈkæstɪk/, or sometimes random process (widely used) is a collection of random variables; this is often used to represent the evolution of some random value, or system, over time. This is the probabilistic counterpart to a deterministic process (or deterministic system). Riemannian manifold. In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real smooth manifold M equipped with an inner product on the tangent space at each point that varies smoothly from point to point in the sense that if X and Y are vector fields on M, then.
Barc. Nonlinear Geoscience (Fractals) Let's Generate Fractals A simple artificial self-similar form that has been closely studied for a long time is Koch curve (named after Swedish mathematician Helge von Koch).
To generate a Koch curve and others with similar properties is simple; all we need is a construction rule which will then be applied repeatedly (iteratively). The rule is as follows: Start with a straight line segment of unit length, divide it up into three equal segments of length 1/2, next substitute the middle segment with a V-shaped indentation whose sides are also 1/3. Fractal Markets.