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Viability Theory
Viability theory is an area of mathematics that studies the evolution of dynamical systems under constraints on the system state. It was developed to formalize problems arising in the study of various natural and social phenomena, and has close ties to the theories of optimal control and set-valued analysis. Motivation Many systems, organizations, and networks arising in biology and the social sciences do not evolve in a deterministic way, nor even in a stochastic way. Rather they evolve with a Darwinian flavor, driven by random fluctuations but yet constrained to remain "viable" by their environment. Viability theory started in 1976 by translating mathematically the title of the book Chance and Necessity by Jacques Monod to the differential inclusion x '(t) \in F (x (t)) for chance and x (t) \in K for necessity. The differential inclusion is a type of “evolutionary engine” (called an evolutionary system associating with any initial state x a subset of evolutions startin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
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, such as in a parametric curve. 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space (controls)
In control engineering and system identification, a state-space representation is a mathematical model of a physical system that uses state variables to track how inputs shape system behavior over time through first-order differential equations or difference equations. These state variables change based on their current values and inputs, while outputs depend on the states and sometimes the inputs too. The state space (also called time-domain approach and equivalent to phase space in certain dynamical systems) is a geometric space where the axes are these state variables, and the system’s state is represented by a state vector. For linear, time-invariant, and finite-dimensional systems, the equations can be written in matrix form, offering a compact alternative to the frequency domain’s Laplace transforms for multiple-input and multiple-output (MIMO) systems. Unlike the frequency domain approach, it works for systems beyond just linear ones with zero initial condi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimal Control
Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the Moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory. Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set-valued Analysis
A set-valued function, also called a correspondence or set-valued relation, is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set. Set-valued functions are used in a variety of mathematical fields, including optimization, control theory and game theory. Set-valued functions are also known as multivalued functions in some references, but this article and the article Multivalued function follow the authors who make a distinction. Distinction from multivalued functions Although other authors may distinguish them differently (or not at all), Wriggers and Panatiotopoulos (2014) distinguish multivalued functions from set-valued functions (which they called ''set-valued relations'') by the fact that multivalued functions only take multiple values at finitely (or denumerably) many points, and otherwise behave like a function. Geometrically, this means that the graph of a multivalued function is necessarily a line of zero a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chance And Necessity
''Chance and Necessity: Essay on the Natural Philosophy of Modern Biology'' (French: ''Le Hasard et la Nécessité: Essai sur la philosophie naturelle de la biologie moderne'') is a 1970 book by Nobel Prize winner Jacques Monod. Aimed at a general audience, the book describes the basic characteristics of life, reviews findings of modern biochemistry and molecular biology, and argues that life arose by blind chance guided by natural selection, could not have been predicted, and does not have a higher purpose. The author concludes that traditional philosophical or religious systems of value, explanation and meaning are untenable, and advocates for an "ethic of knowledge": committing to objectivity in the pursuit of knowledge. According to the introduction, the book's title was inspired by a line attributed to Democritus, "Everything existing in the universe is the fruit of chance and necessity." Summary Monod starts the preface of the book by saying that biology is both margin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacques Monod
Jacques Lucien Monod (; 9 February 1910 – 31 May 1976) was a French biochemist who won the Nobel Prize in Physiology or Medicine in 1965, sharing it with François Jacob and André Lwoff "for their discoveries concerning genetic control of enzyme and virus synthesis".'' Chance and Necessity: An Essay on the Natural Philosophy of Modern Biology'' by Jacques Monod, New York, Alfred A. Knopf, 1971, Monod and Jacob became famous for their work on the '' E. coli'' ''lac'' operon, which encodes proteins necessary for the transport and breakdown of the sugar lactose (lac). From their own work and the work of others, they came up with a model for how the levels of some proteins in a cell are controlled. In their model, the manufacture of a set of related proteins, such as the ones encoded within the ''lac'' (lactose) operon, is prevented when a repressor protein, encoded by a regulatory gene, binds to its operator, a specific site in the DNA sequence that is close to the genes en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Inclusion
In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form :\frac(t)\in F(t,x(t)), where ''F'' is a multivalued map, i.e. ''F''(''t'', ''x'') is a ''set'' rather than a single point in \R^d. Differential inclusions arise in many situations including differential variational inequalities, projected dynamical systems, Moreau's sweeping process, linear and nonlinear complementarity dynamical systems, discontinuous ordinary differential equations, switching dynamical systems, and fuzzy set arithmetic. For example, the basic rule for Coulomb friction is that the friction force has magnitude ''μN'' in the direction opposite to the direction of slip, where ''N'' is the normal force and ''μ'' is a constant (the friction coefficient). However, if the slip is zero, the friction force can be ''any'' force in the correct plane with magnitude smaller than or equal to ''μN''. Thus, writing the friction force as a function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polysemous
Polysemy ( or ; ) is the capacity for a sign (e.g. a symbol, morpheme, word, or phrase) to have multiple related meanings. For example, a word can have several word senses. Polysemy is distinct from '' monosemy'', where a word has a single meaning. Polysemy is distinct from homonymy—or homophony—which is an accidental similarity between two or more words (such as ''bear'' the animal, and the verb ''bear''); whereas homonymy is a mere linguistic coincidence, polysemy is not. In discerning whether a given set of meanings represent polysemy or homonymy, it is often necessary to look at the history of the word to see whether the two meanings are historically related. Dictionary writers often list polysemes (words or phrases with different, but related, senses) in the same entry (that is, under the same headword) and enter homonyms as separate headwords (usually with a numbering convention such as ''¹bear'' and ''²bear''). Polysemes A polyseme is a word or phrase with diffe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Human Brain
The human brain is the central organ (anatomy), organ of the nervous system, and with the spinal cord, comprises the central nervous system. It consists of the cerebrum, the brainstem and the cerebellum. The brain controls most of the activities of the human body, body, processing, integrating, and coordinating the information it receives from the sensory nervous system. The brain integrates sensory information and coordinates instructions sent to the rest of the body. The cerebrum, the largest part of the human brain, consists of two cerebral hemispheres. Each hemisphere has an inner core composed of white matter, and an outer surface – the cerebral cortex – composed of grey matter. The cortex has an outer layer, the neocortex, and an inner allocortex. The neocortex is made up of six Cerebral cortex#Layers of neocortex, neuronal layers, while the allocortex has three or four. Each hemisphere is divided into four lobes of the brain, lobes – the frontal lobe, frontal, pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Social Dynamics
Social dynamics (or sociodynamics) is the study of the behavior of groups and of the interactions of individual group members, aiming to understand the emergence of complex social behaviors among microorganisms, plants and animals, including humans. It is related to sociobiology but also draws from physics and complex system sciences. In the last century, sociodynamics was viewed as part of psychology, as shown in the work: "Sociodynamics: an integrative theorem of power, authority, interfluence and love". In the 1990s, social dynamics began being viewed as a separate scientific discipline 'By whom?''/sup>. An important paper in this respect is: "The Laws of Sociodynamics". Then, starting in the 2000s, sociodynamics took off as a discipline of its own, many papers were released in the field in this decade. Overview The field of social dynamics brings together ideas from economics, sociology, social psychology, and other disciplines, and is a sub-field of complex adaptive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neuroscience
Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions, and its disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, developmental biology, cytology, psychology, physics, computer science, chemistry, medicine, statistics, and mathematical modeling to understand the fundamental and emergent properties of neurons, glia and neural circuits. The understanding of the biological basis of learning, memory, behavior, perception, and consciousness has been described by Eric Kandel as the "epic challenge" of the biological sciences. The scope of neuroscience has broadened over time to include different approaches used to study the nervous system at different scales. The techniques used by neuroscientists have expanded enormously, from molecular and cellular studies of individual neurons to imaging of sensory, motor and cognitive tasks in the brain. Hist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty Set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called ''non-empty''. In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). Notation Common notations for the empty set include "", "\emptyset", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø () in the Danish orthography, Danish and Norwegian orthography, Norwegian a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
