Fuzzy differential inclusion is tha culmination of Fuzzy concept and Differential inclusion introduced by Lotfi A. Zadeh which became popular. [1] [2]
,
f(t,x(t)] is a fuzzy valued continuous function on euclidian space which is collection of all normal, upper semi-continuous, Convex set
,Compact space, supported fuzzy subsets of .
The second order differential is
where
K is trapezoidal fuzzy number (-1,-1/2,0,1/2)
is a trianglular fuzzy number (-1,0,1) .
Fuzzy differential inclusion (FDI) has applications in
In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, Salii (1965) defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics, decision-making, and clustering, are special cases of L-relations when L is the unit interval [0, 1].
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.
In econometrics, the autoregressive conditional heteroskedasticity (ARCH) model is a statistical model for time series data that describes the variance of the current error term or innovation as a function of the actual sizes of the previous time periods' error terms; often the variance is related to the squares of the previous innovations. The ARCH model is appropriate when the error variance in a time series follows an autoregressive (AR) model; if an autoregressive moving average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model.
In differential geometry, the Einstein tensor is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field equations for gravitation that describe spacetime curvature in a manner that is consistent with conservation of energy and momentum.
A Hopfield network is a spin glass system used to model neural networks, based on Ernst Ising's work with Wilhelm Lenz on the Ising model of magnetic materials. Hopfield networks were first described with respect to recurrent neural networks by Shun'ichi Amari in 1972 and with respect to biological neural networks by William Little in 1974, and were popularised by John Hopfield in 1982. Hopfield networks serve as content-addressable ("associative") memory systems with binary threshold nodes, or with continuous variables. Hopfield networks also provide a model for understanding human memory.
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely, and allows for the quantization of the electromagnetic field by Lagrangian formulation described below.
In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary.
In condensed matter physics, Hofstadter's butterfly is a graph of the spectral properties of non-interacting two-dimensional electrons in a perpendicular magnetic field in a lattice. The fractal, self-similar nature of the spectrum was discovered in the 1976 Ph.D. work of Douglas Hofstadter and is one of the early examples of modern scientific data visualization. The name reflects the fact that, as Hofstadter wrote, "the large gaps [in the graph] form a very striking pattern somewhat resembling a butterfly."
In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0, > 0, such that
In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form
Sparse approximation theory deals with sparse solutions for systems of linear equations. Techniques for finding these solutions and exploiting them in applications have found wide use in image processing, signal processing, machine learning, medical imaging, and more.
In continuum mechanics, a compatible deformation tensor field in a body is that unique tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. Compatibility is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.
In mathematical physics, in particular electromagnetism, the Riemann–Silberstein vector or Weber vector named after Bernhard Riemann, Heinrich Martin Weber and Ludwik Silberstein, is a complex vector that combines the electric field E and the magnetic field B.
Asymptotic safety is a concept in quantum field theory which aims at finding a consistent and predictive quantum theory of the gravitational field. Its key ingredient is a nontrivial fixed point of the theory's renormalization group flow which controls the behavior of the coupling constants in the ultraviolet (UV) regime and renders physical quantities safe from divergences. Although originally proposed by Steven Weinberg to find a theory of quantum gravity, the idea of a nontrivial fixed point providing a possible UV completion can be applied also to other field theories, in particular to perturbatively nonrenormalizable ones. In this respect, it is similar to quantum triviality.
In the fields of dynamical systems and control theory, a fractional-order system is a dynamical system that can be modeled by a fractional differential equation containing derivatives of non-integer order. Such systems are said to have fractional dynamics. Derivatives and integrals of fractional orders are used to describe objects that can be characterized by power-law nonlocality, power-law long-range dependence or fractal properties. Fractional-order systems are useful in studying the anomalous behavior of dynamical systems in physics, electrochemistry, biology, viscoelasticity and chaotic systems.
Vasiliev equations are formally consistent gauge invariant nonlinear equations whose linearization over a specific vacuum solution describes free massless higher-spin fields on anti-de Sitter space. The Vasiliev equations are classical equations and no Lagrangian is known that starts from canonical two-derivative Frønsdal Lagrangian and is completed by interactions terms. There is a number of variations of Vasiliev equations that work in three, four and arbitrary number of space-time dimensions. Vasiliev's equations admit supersymmetric extensions with any number of super-symmetries and allow for Yang–Mills gaugings. Vasiliev's equations are background independent, the simplest exact solution being anti-de Sitter space. It is important to note that locality is not properly implemented and the equations give a solution of certain formal deformation procedure, which is difficult to map to field theory language. The higher-spin AdS/CFT correspondence is reviewed in Higher-spin theory article.
In measurements, the measurement obtained can suffer from two types of uncertainties. The first is the random uncertainty which is due to the noise in the process and the measurement. The second contribution is due to the systematic uncertainty which may be present in the measuring instrument. Systematic errors, if detected, can be easily compensated as they are usually constant throughout the measurement process as long as the measuring instrument and the measurement process are not changed. But it can not be accurately known while using the instrument if there is a systematic error and if there is, how much? Hence, systematic uncertainty could be considered as a contribution of a fuzzy nature.
In machine learning, diffusion models, also known as diffusion probabilistic models or score-based generative models, are a class of latent variable generative models. A diffusion model consists of three major components: the forward process, the reverse process, and the sampling procedure. The goal of diffusion models is to learn a diffusion process that generates a probability distribution for a given dataset from which we can then sample new images. They learn the latent structure of a dataset by modeling the way in which data points diffuse through their latent space.
Fuzzy differential equation are general concept of ordinary differential equation in mathematics defined as differential inclusion for non-uniform upper hemicontinuity convex set with compactness in fuzzy set.
