|
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 starting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and 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 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, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space (controls)
In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables whose values evolve over time in a way that depends on the values they have at any given time and on the externally imposed values of input variables. Output variables’ values depend on the values of the state variables. The "state space" is the Euclidean space in which the variables on the axes are the state variables. The state of the system can be represented as a ''state vector'' within that space. To abstract from the number of inputs, outputs and states, these variables are expressed as vectors. If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form. The state-space method is characterized by significant algebraization of general syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimal Control
Optimal control theory is a branch of mathematical optimization 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 calc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set-valued Analysis
A set-valued function (or correspondence) 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 herein and in many others references in mathematical analysis, a multivalued function is a set-valued function that has a further continuity property, namely that the choice of an element in the set f(x) defines a corresponding element in each set f(y) for close to , and thus defines locally an ordinary function. Examples The argmax of a function is in general, multivalued. For example, \operatorname_ \cos(x) = \. Set-valued analysis Set-valued analysis is the study of sets in the spirit of mathematical analysis and general topology. Instead of considering collections of only points, set-valued analysis co ... [...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, interpreting the processes of evolution to show that life is only the result of natural processes by "''pure chance''". The basic tenet of this book is that systems in nature with molecular biology, such as enzymatic biofeedback loops, can be explained without having to invoke final causality. Teleonomic In this book, Monod adopted the term teleonomic to permit recognition of purpose in biology without appealing to a final cause. Inspiration 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." Awards The first U.S. edition (New York: Vintage, 1971), translated by Austryn Wainhouse, won the National Book Award in category Translation. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacques Monod
Jacques Lucien Monod (February 9, 1910 – May 31, 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 proteins, such as the ones encoded within the ''lac'' (lactose) operon, is prevented when a repressor, encoded by a regulatory gene, binds to its operator, a specific site in the DNA sequence that is close to the genes encoding the proteins. (It is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polysemous
Polysemy ( or ; ) is the capacity for a sign (e.g. a symbol, a morpheme, a word, or a 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 d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Human Brain
The human brain is the central organ of the human nervous system, and with the spinal cord makes up the central nervous system. The brain consists of the cerebrum, the brainstem and the cerebellum. It controls most of the activities of the body, processing, integrating, and coordinating the information it receives from the sense organs, and making decisions as to the instructions sent to the rest of the body. The brain is contained in, and protected by, the skull bones of the head. 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 neuronal layers, while the allocortex has three or four. Each hemisphere is conventionally divided into four lobes – the frontal, temporal, parietal, and occipital lo ... [...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 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, 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 Biology, 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 biology, molecular and cell biology, cellular studies of individual neurons to neuroimaging, imaging ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autonomous Agency Theory
Autonomous agency theory (AAT) is a viable system theory (VST) which models autonomous social complex adaptive systems. It can be used to model the relationship between an agency and its environment(s), and these may include other interactive agencies. The nature of that interaction is determined by both the agency's external and internal attributes and constraints. Internal attributes may include immanent dynamic "self" processes that drive agency change. History Stafford Beer coined the term ''viable systems'' in the 1950s, and developed it within his management cybernetics theories. He designed his viable system model as a diagnostic tool for organisational pathologies (conditions of social ill-health). This model involves a system concerned with operations and their direct management, and a meta-system that "observes" the system and controls it. Beer's work refers to Maturana's concept of autopoiesis, which explains why living systems actually live. However, Beer did not mak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viable System Theory
Viable system theory (VST) concerns cybernetic processes in relation to the development/evolution of dynamical systems. They are considered to be living systems in the sense that they are complex and adaptive, can learn, and are capable of maintaining an autonomous existence, at least within the confines of their constraints. These attributes involve the maintenance of internal stability through adaptation to changing environments. One can distinguish between two strands such theory: formal systems and principally non-formal system. Formal viable system theory is normally referred to as viability theory, and provides a mathematical approach to explore the dynamics of complex systems set within the context of control theory. In contrast, principally non-formal viable system theory is concerned with descriptive approaches to the study of viability through the processes of control and communication, though these theories may have mathematical descriptions associated with them. Hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hélène Frankowska
Hélène Frankowska, or Halina Frankowska is a Polish and French mathematician known for her research in control theory and set-valued analysis. She is a director of research at the Centre national de la recherche scientifique, and works in the Institut de Mathématiques de Jussieu of Pierre and Marie Curie University. Education Frankowska completed her undergraduate studies in 1979 at the University of Warsaw, in the Department of Mathematics, Informatics and Mechanics. After a year of studies at the International School for Advanced Studies in Trieste, Italy, she completed her doctorate of the third cycle in 1983 and her state doctorate in mathematics in 1984 at Paris Dauphine University. Her dissertation, jointly supervised by Czesław Olech Czesław Olech (22 May 1931 – 1 July 2015) was a Polish mathematician. He was a representative of the Kraków school of mathematics, especially the differential equations school of Tadeusz Ważewski. Education and career In 1954 he c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
