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Multistability
In a dynamical system, multistability is the property of having multiple stable equilibrium points in the vector space spanned by the states in the system. By mathematical necessity, there must also be unstable equilibrium points between the stable points. Points that are stable in some dimensions and unstable in others are termed unstable, as is the case with the first three Lagrangian points. Bistability Bistability is the special case with two stable equilibrium points. It is the simplest form of multistability, and can occur in systems with only one state variable, as it only takes a one-dimensional space to separate two points. Initial instability Near an unstable equilibrium, any system will be sensitive to noise, initial conditions and system parameters, which can cause it to develop in one of multiple divergent directions. In economics and social sciences, path dependence gives rise to divergent directions of development. Some path dependent processes are adequately ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multistability
In a dynamical system, multistability is the property of having multiple stable equilibrium points in the vector space spanned by the states in the system. By mathematical necessity, there must also be unstable equilibrium points between the stable points. Points that are stable in some dimensions and unstable in others are termed unstable, as is the case with the first three Lagrangian points. Bistability Bistability is the special case with two stable equilibrium points. It is the simplest form of multistability, and can occur in systems with only one state variable, as it only takes a one-dimensional space to separate two points. Initial instability Near an unstable equilibrium, any system will be sensitive to noise, initial conditions and system parameters, which can cause it to develop in one of multiple divergent directions. In economics and social sciences, path dependence gives rise to divergent directions of development. Some path dependent processes are adequately ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a Set (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, a dynamical system has a State ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stability Theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance. In dynamical systems, an orbit is called ''Lyapunov stable'' if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well-studied problem involvi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian Points
In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of the restricted three-body problem in which two bodies are far more massive than the third. Normally, the two massive bodies exert an unbalanced gravitational force at a point, altering the orbit of whatever is at that point. At the Lagrange points, the gravitational forces of the two large bodies and the centrifugal force balance each other. This can make Lagrange points an excellent location for satellites, as few orbit corrections are needed to maintain the desired orbit. Small objects placed in orbit at Lagrange points are in equilibrium in at least two directions relative to the center of mass of the large bodies. For any combination of two orbital bodies there are five Lagrange points, L1 to L5, all in the orbital plane of the two lar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Market Share
Market share is the percentage of the total revenue or sales in a market that a company's business makes up. For example, if there are 50,000 units sold per year in a given industry, a company whose sales were 5,000 of those units would have a 10percent share in that market. "Marketers need to be able to translate and incorporate sales targets into market share because this will demonstrate whether forecasts are to be attained by growing with the market or by capturing share from competitors. The latter will almost always be more difficult to achieve. Market share is closely monitored for signs of change in the competitive landscape, and it frequently drives strategic or tactical action."Farris, Paul W.; Neil T. Bendle; Phillip E. Pfeifer; David J. Reibstein (2010). ''Marketing Metrics: The Definitive Guide to Measuring Marketing Performance.'' Upper Saddle River, New Jersey: Pearson Education, Inc. . The Marketing Accountability Standards Board (MASB) endorses the definitions, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monopoly
A monopoly (from Greek language, Greek el, μόνος, mónos, single, alone, label=none and el, πωλεῖν, pōleîn, to sell, label=none), as described by Irving Fisher, is a market with the "absence of competition", creating a situation where a specific person or company, enterprise is the only supplier of a particular thing. This contrasts with a monopsony which relates to a single entity's control of a Market (economics), market to purchase a good or service, and with oligopoly and duopoly which consists of a few sellers dominating a market. Monopolies are thus characterized by a lack of economic Competition (economics), competition to produce the good (economics), good or Service (economics), service, a lack of viable substitute goods, and the possibility of a high monopoly price well above the seller's marginal cost that leads to a high monopoly profit. The verb ''monopolise'' or ''monopolize'' refers to the ''process'' by which a company gains the ability to raise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multistable Perception
Multistable perception (or bistable perception) is a perceptual phenomenon in which an observer experiences an unpredictable sequence of spontaneous subjective changes. While usually associated with visual perception (a form of optical illusion), multistable perception can also be experienced with auditory and olfactory percepts. Classification Perceptual multistability can be evoked by visual patterns that are too ambiguous for the human visual system to definitively and uniquely interpret. Familiar examples include the Necker cube, Schroeder staircase, structure from motion, monocular rivalry, and binocular rivalry, but many more visually ambiguous patterns are known. Because most of these images lead to an alternation between two mutually exclusive perceptual states, they are sometimes also referred to as bistable perception. Auditory and olfactory examples can occur when there are conflicting, and so rival, inputs into the two ears or two nostrils. Characterization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necker Cube
The Necker cube is an optical illusion that was first published as a Rhomboid in 1832 by Swiss crystallographer Louis Albert Necker. It is a simple wire-frame, two dimensional drawing of a cube with no visual cues as to its orientation, so it can be interpreted to have either the lower-left or the upper-right square as its front side. Ambiguity The Necker cube is an ambiguous drawing. Each part of the picture is ambiguous by itself, yet the human visual system picks an interpretation of each part that makes the whole consistent. The Necker cube is sometimes used to test computer models of the human visual system to see whether they can arrive at consistent interpretations of the image the same way humans do. Humans do not usually see an inconsistent interpretation of the cube. A cube whose edges cross in an inconsistent way is an example of an impossible object, specifically an impossible cube (compare Penrose triangle). With the cube on the left, most people see the lower ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monocular Rivalry
Monocular rivalry is a phenomenon of human visual perception that occurs when two different images are optically superimposed. During prolonged viewing, one image becomes clearer than the other for a few moments, then the other image becomes clearer than the first for a few moments. These alternations in clarity continue at random for as long as one looks. Occasionally one image will become exclusively visible and the other image invisible. In the demonstration, one image is a green grating and the other is a red grating. During prolonged inspection, the viewer can see the green grating as clearer than the red grating for a few moments, then the reverse. Occasionally the green grating will be all that is visible and occasionally the red grating will be all that is visible. Occasionally, at transitions, one will briefly see irregular composites of the two gratings (such as the red and green gratings superimposed but with one or two bars of the green grating invisible). Monocular ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binocular Rivalry
Binocular rivalry is a phenomenon of visual perception in which perception alternates between different images presented to each eye. When one image is presented to one eye and a very different image is presented to the other (also known as dichoptic presentation), instead of the two images being seen superimposed, one image is seen for a few moments, then the other, then the first, and so on, randomly for as long as one cares to look. For example, if a set of vertical lines is presented to one eye, and a set of horizontal lines to the same region of the retina of the other, sometimes the vertical lines are seen with no trace of the horizontal lines, and sometimes the horizontal lines are seen with no trace of the vertical lines. At transitions, brief, unstable composites of the two images may be seen. For example, the vertical lines may appear one at a time to obscure the horizontal lines from the left or from the right, like a traveling wave, switching slowly one image for the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
