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Chandrasekhar's X- And Y-function
In atmospheric radiation, Chandrasekhar's ''X''- and Y-function appears as the solutions of problems involving diffusive reflection and transmission, introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar. The Chandrasekhar's ''X''- and ''Y''-function X(\mu),\ Y(\mu) defined in the interval 0\leq\mu\leq 1, satisfies the pair of nonlinear integral equations :\begin X(\mu) &= 1+ \mu \int_0^1 \frac (\mu)X(\mu')-Y(\mu)Y(\mu')\, d\mu',\\ ptY(\mu) &= e^ + \mu \int_0^1 \frac (\mu)X(\mu')-X(\mu)Y(\mu')\, d\mu' \end where the characteristic function \Psi(\mu) is an even polynomial in \mu generally satisfying the condition :\int_0^1\Psi(\mu) \, d\mu \leq \frac, and 0, where Q is an arbitrary constant. See also *Chandrasekhar's H-function In atmospheric radiation, Chandrasekhar's ''H''-function appears as the solutions of problems involving scattering, introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar.Sparrow, Ephraim M., and Robert D. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radiation
In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes: * ''electromagnetic radiation'', such as radio waves, microwaves, infrared, visible light, ultraviolet, x-rays, and gamma radiation (γ) * ''particle radiation'', such as alpha radiation (α), beta radiation (β), proton radiation and neutron radiation (particles of non-zero rest energy) * '' acoustic radiation'', such as ultrasound, sound, and seismic waves (dependent on a physical transmission medium) * ''gravitational wave, gravitational radiation'', that takes the form of gravitational waves, or ripples in the curvature of spacetime Radiation is often categorized as either ''ionizing radiation, ionizing'' or ''non-ionizing radiation, non-ionizing'' depending on the energy of the radiated particles. Ionizing radiation carries more than 10 electron volt, eV, which is enough to ionize atoms and molecules and break ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffuse Reflection
Diffuse reflection is the reflection (physics), reflection of light or other radiation, waves or particles from a surface such that a ray (optics), ray incident on the surface is scattering, scattered at many angles rather than at just one angle as in the case of specular reflection. An ''ideal'' diffuse reflecting surface is said to exhibit Lambertian reflection, meaning that there is equal luminance when viewed from all directions lying in the half-space (geometry), half-space adjacent to the surface. A surface built from a non-absorbing powder such as plaster, or from fibers such as paper, or from a polycrystalline material such as white marble, reflects light diffusely with great efficiency. Many common materials exhibit a mixture of specular and diffuse reflection. The visibility of objects, excluding light-emitting ones, is primarily caused by diffuse reflection of light: it is diffusely-scattered light that forms the image of the object in the observer's eye. Mechanism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indian American
Indian Americans or Indo-Americans are citizens of the United States with ancestry from India. The United States Census Bureau uses the term Asian Indian to avoid confusion with Native Americans, who have also historically been referred to as "Indians" and are known as "American Indians". With a population of more than four and a half million, Indian Americans make up 1.4% of the U.S. population and are the largest group of South Asian Americans, as well as the second largest group of Asian Americans after Chinese Americans. Indian Americans are the highest-earning ethnic group in the United States.Multiple sources: * * * * * * * * * * * * * * * Terminology In the Americas, the term "Indian" had historically been used to describe indigenous people since European colonization in the 15th century. Qualifying terms such as " American Indian" and " East Indian" were and still are commonly used in order to avoid ambiguity. The U.S. government has since coined the term "Native Am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar (; ) (19 October 1910 – 21 August 1995) was an Indian-American theoretical physicist who spent his professional life in the United States. He shared the 1983 Nobel Prize for Physics with William A. Fowler for "...theoretical studies of the physical processes of importance to the structure and evolution of the stars". His mathematical treatment of stellar evolution yielded many of the current theoretical models of the later evolutionary stages of massive stars and black holes. Many concepts, institutions, and inventions, including the Chandrasekhar limit and the Chandra X-Ray Observatory, are named after him. Chandrasekhar worked on a wide variety of problems in physics during his lifetime, contributing to the contemporary understanding of stellar structure, white dwarfs, stellar dynamics, stochastic process, radiative transfer, the quantum theory of the hydrogen anion, hydrodynamic and hydromagnetic stability, turbulence, equilibrium and the stabi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optical Thickness
In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to ''transmitted'' radiant power through a material. Thus, the larger the optical depth, the smaller the amount of transmitted radiant power through the material. Spectral optical depth or spectral optical thickness is the natural logarithm of the ratio of incident to transmitted spectral radiant power through a material. Optical depth is dimensionless, and in particular is not a length, though it is a monotonically increasing function of optical path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged. In chemistry, a closely related quantity called "absorbance" or "decadic absorbance" is used instead of optical depth: the common logarithm of the ratio of incident to transmitted radiant power through a material, that is the optical depth divided by ln 10. Mathematical definitions Optical depth Op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chandrasekhar's H-function
In atmospheric radiation, Chandrasekhar's ''H''-function appears as the solutions of problems involving scattering, introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar.Sparrow, Ephraim M., and Robert D. Cess. "Radiation heat transfer." Series in Thermal and Fluids Engineering, New York: McGraw-Hill, 1978, Augmented ed. (1978). The Chandrasekhar's ''H''-function H(\mu) defined in the interval 0\leq\mu\leq 1, satisfies the following nonlinear integral equation :H(\mu) = 1+\mu H(\mu)\int_0^1 \fracH(\mu') \, d\mu' where the characteristic function \Psi(\mu) is an even polynomial in \mu satisfying the following condition :\int_0^1\Psi(\mu) \, d\mu \leq \frac. If the equality is satisfied in the above condition, it is called ''conservative case'', otherwise ''non-conservative''. Albedo is given by \omega_o= 2\Psi(\mu) = \text. An alternate form which would be more useful in calculating the ''H'' function numerically by iteration was derived by Chandrasekhar as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions. Definition by construction as an orthogonal system In this approach, the polynomials are defined as an orthogonal system with respect to the weight function w(x) = 1 over the interval 1,1/math>. That is, P_n(x) is a polynomial of degree n, such that \int_^1 P_m(x) P_n(x) \,dx = 0 \quad \text n \ne m. With the additional standardization co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integro-differential Equation
In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function. General first order linear equations The general first-order, linear (only with respect to the term involving derivative) integro-differential equation is of the form : \fracu(x) + \int_^x f(t,u(t))\,dt = g(x,u(x)), \qquad u(x_0) = u_0, \qquad x_0 \ge 0. As is typical with differential equations, obtaining a closed-form solution can often be difficult. In the relatively few cases where a solution can be found, it is often by some kind of integral transform, where the problem is first transformed into an algebraic setting. In such situations, the solution of the problem may be derived by applying the inverse transform to the solution of this algebraic equation. Example Consider the following second-order problem, : u'(x) + 2u(x) + 5\int_^u(t)\,dt = \theta(x) \qquad \text \qquad u(0)=0, where : \theta(x) = \left\{ \begin{array}{ll} 1, \qq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the List of mathematical functions contains functions that are commonly accepted as special. Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbolic c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Equations
In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ; I^1 (u), I^2(u), I^3(u), ..., I^m(u)) = 0where I^i(u) is an integral operator acting on ''u.'' Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ; D^1 (u), D^2(u), D^3(u), ..., D^m(u)) = 0where D^i(u) may be viewed as a differential operator of order ''i''. Due to this close connection between differential and integral equations, one can often convert between the two. For example, one method of solvi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scattering
Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiation) in the medium through which they pass. In conventional use, this also includes deviation of reflected radiation from the angle predicted by the law of reflection. Reflections of radiation that undergo scattering are often called ''diffuse reflections'' and unscattered reflections are called ''specular'' (mirror-like) reflections. Originally, the term was confined to light scattering (going back at least as far as Isaac Newton in the 17th century). As more "ray"-like phenomena were discovered, the idea of scattering was extended to them, so that William Herschel could refer to the scattering of "heat rays" (not then recognized as electromagnetic in nature) in 1800. John Tyndall, a pioneer in light scattering researc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
