|
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] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematics), function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equation, ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with stochastic differential equations, ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stiff Equation
In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution. When integrating a differential equation numerically, one would expect the requisite step size to be relatively small in a region where the solution curve displays much variation and to be relatively large where the solution curve straightens out to approach a line with slope nearly zero. For some problems this is not the case. In order for a numerical method to give a reliable solution to the differential system sometimes the step size is required to be at an unacceptably small level in a region where the solution curve is very smooth. The phenomenon is known as ''stiffness''. In some cases there may b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Medical Imaging
Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to reveal internal structures hidden by the skin and bones, as well as to diagnose and treat disease. Medical imaging also establishes a database of normal anatomy and physiology to make it possible to identify abnormalities. Although imaging of removed organ (anatomy), organs and Tissue (biology), tissues can be performed for medical reasons, such procedures are usually considered part of pathology instead of medical imaging. Measurement and recording techniques that are not primarily designed to produce images, such as electroencephalography (EEG), magnetoencephalography (MEG), electrocardiography (ECG), and others, represent other technologies that produce data susceptible to representation as a parameter graph versus time or maps that contain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cybernetics
Cybernetics is the transdisciplinary study of circular causal processes such as feedback and recursion, where the effects of a system's actions (its outputs) return as inputs to that system, influencing subsequent action. It is concerned with general principles that are relevant across multiple contexts, including in engineering, ecological, economic, biological, cognitive and social systems and also in practical activities such as designing, learning, and managing. Cybernetics' transdisciplinary character has meant that it intersects with a number of other fields, leading to it having both wide influence and diverse interpretations. The field is named after an example of circular causal feedback—that of steering a ship (the ancient Greek κυβερνήτης (''kybernḗtēs'') refers to the person who steers a ship). In steering a ship, the position of the rudder is adjusted in continual response to the effect it is observed as having, forming a feedback loop throu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atmospheric Dispersion Modeling
Atmospheric dispersion modeling is the mathematical simulation of how air pollutants disperse in the ambient atmosphere. It is performed with computer programs that include algorithms to solve the mathematical equations that govern the pollutant dispersion. The dispersion models are used to estimate the downwind ambient concentration of air pollutants or toxins emitted from sources such as industrial plants, vehicular traffic or accidental chemical releases. They can also be used to predict future concentrations under specific scenarios (i.e. changes in emission sources). Therefore, they are the dominant type of model used in air quality policy making. They are most useful for pollutants that are dispersed over large distances and that may react in the atmosphere. For pollutants that have a very high spatio-temporal variability (i.e. have very steep distance to source decay such as black carbon) and for epidemiological studies statistical land-use regression models are also u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Differential Inclusion
Fuzzy differential inclusion is the extension of differential inclusion to fuzzy sets introduced by Lotfi A. Zadeh. x'(t) \in f(t , x(t))\alpha with x(0) \in _0\alpha Suppose f(t,x(t)) is a fuzzy valued continuous function on Euclidean space. Then it is the collection of all normal, upper semi-continuous, convex, compactly supported fuzzy subsets of \mathbb^n. Second order differential The second order differential is x''(t) \in x \alpha where k \in \alpha, K is trapezoidal fuzzy number (-1,-1/2,0,1/2), and x_0 is a trianglular fuzzy number (-1,0,1). Applications Fuzzy differential inclusion (FDI) has applications in * Cybernetics * Artificial intelligence, Neural network, * Medical imaging * Robotics * Atmospheric dispersion modeling * Weather forecasting * Cyclone * Pattern recognition * Population biology The term population biology has been used with different meanings. In 1971, Edward O. Wilson ''et al''. used the term in the sense of applying m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Concept
A fuzzy concept is an idea of which the boundaries of application can vary considerably according to context or conditions, instead of being fixed once and for all. This means the idea is somewhat vague or imprecise. Yet it is not unclear or meaningless. It has a definite meaning, which can be made more exact only through further elaboration and specification — including a closer definition of the context in which the concept is used. The colloquial meaning of a "fuzzy concept" is that of an imprecise idea which is "somewhat vague" for any kind of reason, or which is "approximately true" in a situation. The inverse of a "fuzzy concept" is a "crisp concept" (i.e. a precise concept). Fuzzy concepts are often used to navigate imprecision in the real world, when exact information is not available, but where an indication is sufficient to be helpful. Although the linguist George Philip Lakoff already defined the semantics of a fuzzy concept in 1973, inspired by an unpublished 197 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siconos
SICONOS is an open source scientific software primarily targeted at modeling and simulating non-smooth dynamical systems (NSDS): * Mechanical systems (Rigid body or solid) with Unilateral contact and Coulomb friction as we find in Non-smooth mechanics, Contact dynamics or Granular material. * Switched Electrical Circuit such as Power converter, Rectifier, Phase-locked loop (PLL) or Analog-to-digital converter * Sliding mode control systems Other applications are found in Systems and Control (hybrid systems, differential inclusions, optimal control with state constraints), Optimization ( Complementarity problem and Variational inequality) Biology Gene regulatory network, Fluid Mechanics and Computer graphics, etc. Components The software is based on 3 main components * Siconos/Numerics (C API). Collection of low-level algorithms for solving basic Algebra and optimization problems arising in the simulation of nonsmooth dynamical systems ** Linear complementarity problem (LCP) ** ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
INRIA
The National Institute for Research in Digital Science and Technology (Inria) () is a French national research institution focusing on computer science and applied mathematics. It was created under the name French Institute for Research in Computer Science and Automation (IRIA) () in 1967 at Rocquencourt near Paris, part of Plan Calcul. Its first site was the historical premises of SHAPE (central command of NATO military forces), which is still used as Inria's main headquarters. In 1980, IRIA became INRIA. Since 2011, it has been styled ''Inria''. Inria is a Public Scientific and Technical Research Establishment (EPST) under the double supervision of the French Ministry of National Education, Advanced Instruction and Research and the Ministry of Economy, Finance and Industry. Administrative status Inria has nine research centers distributed across France (in Bordeaux, Grenoble- Inovallée, Lille, Lyon, Nancy, Paris- Rocquencourt, Rennes, Saclay, and Sophia Antipolis) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Impact (mechanics)
In mechanics, an impact is when two bodies Collision, collide. During this collision, both bodies decelerate. The deceleration causes a high force or Shock (mechanics), shock, applied over a short time period. A high force, over a short duration, usually causes more damage to both bodies than a lower force applied over a proportionally longer duration. At normal speeds, during a perfectly inelastic collision, an object struck by a projectile will Deformation (engineering), deform, and this deformation will absorb most or all of the force of the collision. Viewed from a conservation of energy perspective, the kinetic energy of the projectile is changed into heat and sound energy, as a result of the deformations and vibrations induced in the struck object. However, these deformations and vibrations cannot occur instantaneously. A high-velocity collision (an impact) does not provide sufficient time for these deformations and vibrations to occur. Thus, the struck material behaves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dry Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of the processes involved is called tribology, and has a history of more than 2000 years. Friction can have dramatic consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Another important consequence of many types of friction can be wear, which may lead to performance degradation or damage to components. It is known that frictional energy losses account for about 20% of the total energy expenditure of the world. As briefly discussed later, there are many different contributors to the retarding force in friction, ranging from asperity deformation to the generation of charges and changes in local structure. When two bodies in contact move relative to each other, due to these various ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
