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Bateman Equation
In nuclear physics, the Bateman equation is a mathematical model describing abundances and activities in a decay chain as a function of time, based on the decay rates and initial abundances. The model was formulated by Ernest Rutherford in 1905 and the analytical solution was provided by Harry Bateman in 1910. If, at time ''t'', there are N_i(t) atoms of isotope i that decays into isotope i+1 at the rate \lambda_i, the amounts of isotopes in the ''k''-step decay chain evolves as: : \begin \frac & =-\lambda_1 N_1(t) \\ pt\frac & =-\lambda_i N_i(t) + \lambda_N_(t) \\ pt\frac & = \lambda_N_(t) \end (this can be adapted to handle decay branches). While this can be solved explicitly for ''i'' = 2, the formulas quickly become cumbersome for longer chains. The Bateman equation is a classical master equation where the transition rates are only allowed from one species (i) to the next (i+1) but never in the reverse sense (i+1 to i is forbidden). Bateman found a general expl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nuclear Physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the atom as a whole, including its electrons. Discoveries in nuclear physics have led to nuclear technology, applications in many fields. This includes nuclear power, nuclear weapons, nuclear medicine and magnetic resonance imaging, industrial and agricultural isotopes, ion implantation in materials engineering, and radiocarbon dating in geology and archaeology. Such applications are studied in the field of nuclear engineering. Particle physics evolved out of nuclear physics and the two fields are typically taught in close association. Nuclear astrophysics, the application of nuclear physics to astrophysics, is crucial in explaining the inner workings of stars and the nucleosynthesis, origin of the chemical elements. History The history o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Methods For Ordinary Differential Equations
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution. Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved. The problem A first-order different ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nuclear History Of The United Kingdom
Nuclear may refer to: Physics Relating to the nucleus of the atom: *Nuclear engineering *Nuclear physics *Nuclear power *Nuclear reactor *Nuclear weapon *Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics * Nuclear space *Nuclear operator *Nuclear congruence *Nuclear C*-algebra Biology Relating to the nucleus of the cell: * Nuclear DNA Society *Nuclear family, a family consisting of a pair of adults and their children Music * "Nuclear" (band), group music. * "Nuclear" (Ryan Adams song), 2002 *"Nuclear", a song by Mike Oldfield from his ''Man on the Rocks'' album * ''Nu.Clear'' (EP) by South Korean girl group CLC See also *Nucleus (other) *Nucleolus *Nucleation *Nucleic acid Nucleic acids are biopolymers, macromolecules, essential to all known forms of life. They are composed of nucleotides, which are the monomers made of three components: a 5-carbon sugar, a phosphate group and a nitrogenous base. The two main ... * Nucular * * {{Disamb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pharmacokinetics
Pharmacokinetics (from Ancient Greek ''pharmakon'' "drug" and ''kinetikos'' "moving, putting in motion"; see chemical kinetics), sometimes abbreviated as PK, is a branch of pharmacology dedicated to determining the fate of substances administered to a living organism. The substances of interest include any chemical xenobiotic such as: pharmaceutical drugs, pesticides, food additives, cosmetics, etc. It attempts to analyze chemical metabolism and to discover the fate of a chemical from the moment that it is administered up to the point at which it is completely eliminated from the body. Pharmacokinetics is the study of how an organism affects a drug, whereas pharmacodynamics (PD) is the study of how the drug affects the organism. Both together influence dosing, benefit, and adverse effects, as seen in PK/PD models. Overview Pharmacokinetics describes how the body affects a specific xenobiotic/chemical after administration through the mechanisms of absorption and distributio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secular Equilibrium
In nuclear physics, secular equilibrium is a situation in which the quantity of a radioactive isotope remains constant because its production rate (e.g., due to decay of a parent isotope) is equal to its decay rate. In radioactive decay Secular equilibrium can occur in a radioactive decay chain only if the half-life Half-life (symbol ) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable at ... of the daughter radionuclide B is much shorter than the half-life of the parent radionuclide A. In such a case, the decay rate of A and hence the production rate of B is approximately constant, because the half-life of A is very long compared to the time scales considered. The quantity of radionuclide B builds up until the number of B atoms decaying per unit time becomes equal to the number being produced per unit time. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transient Equilibrium
In nuclear physics, transient equilibrium is a situation in which equilibrium is reached by a parent-daughter radioactive isotope pair where the half-life of the daughter is shorter than the half-life of the parent. Contrary to secular equilibrium, the half-life of the daughter is not negligible compared to parent's half-life. An example of this is a molybdenum-99 generator producing technetium-99 for nuclear medicine diagnostic procedures. Such a generator is sometimes called a '' cow '' because the daughter product, in this case technetium-99, is milked at regular intervals. Transient equilibrium occurs after four half-lives, on average. Activity in transient equilibrium The activity of the daughter is given by the Bateman equation: :A_d = A_P(0)\frac \times (e^-e^) \times BR + A_d(0)e^, where A_P and A_d are the activity of the parent and daughter, respectively. T_P and T_d are the half-lives (inverses of reaction rates \lambda in the above equation modulo ln(2)) of the parent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Equations In Nuclear And Particle Physics
This article summarizes equations in the theory of nuclear physics and particle physics. Definitions Equations Nuclear structure Nuclear decay Nuclear scattering theory The following apply for the nuclear reaction: :''a'' + ''b'' ↔ ''R'' → ''c'' in the centre of mass frame, where ''a'' and ''b'' are the initial species about to collide, ''c'' is the final species, and ''R'' is the resonant state. Fundamental forces These equations need to be refined such that the notation is defined as has been done for the previous sets of equations. See also *Defining equation (physical chemistry) *Defining equation (physics) * List of electromagnetism equations *List of equations in classical mechanics *List of equations in quantum mechanics *List of equations in wave theory * List of photonics equations *List of relativistic equations *Relativistic wave equations In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let be an real or complex matrix. The exponential of , denoted by or , is the matrix given by the power series e^X = \sum_^\infty \frac X^k where X^0 is defined to be the identity matrix I with the same dimensions as X. The above series always converges, so the exponential of is well-defined. If is a 1×1 matrix the matrix exponential of is a 1×1 matrix whose single element is the ordinary exponential of the single element of . Properties Elementary properties Let and be complex matrices and let and be arbitrary complex numbers. We denote the identity matrix by and the zero matrix by 0. The matrix exponential satisfies the foll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loss Of Significance
In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. For example, if there are two studs, one L_1 = 254.5\,\text long and the other L_2 = 253.5\,\text long, and they are measured with a ruler that is good only to the centimeter, then the approximations could come out to be \tilde L_1 = 255\,\text and \tilde L_2 = 253\,\text. These may be good approximations, in relative error, to the true lengths: the approximations are in error by less than 2% of the true lengths, , L_1 - \tilde L_1, /, L_1, < 2\%. However, if the ''approximate'' lengths are subtracted, the difference will be , even though the true difference between the lengths is . The difference of the approximations, |
Mathematical Model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences (such as economics, psychology, sociology, political science). The use of mathematical models to solve problems in business or military operations is a large part of the field of operations research. Mathematical models are also used in music, linguistics, and philosophy (for example, intensively in analytic philosophy). A model may help to explain a system and to study the effects of different components, and to make predictions about behavior. Elements of a mathematical model Mathematical models can take many forms, including dynamical systems, statisti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the complex frequency domain, also known as ''s''-domain, or s-plane). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms ordinary differential equations into algebraic equations and convolution into multiplication. For suitable functions ''f'', the Laplace transform is the integral \mathcal\(s) = \int_0^\infty f(t)e^ \, dt. History The Laplace transform is named after mathematician and astronomer Pierre-Simon, marquis de Laplace, who used a similar transform in his work on probability theory. Laplace wrote extensively about the use of generating functions in ''Essai philosophique sur les probabilités'' (1814), and the integral form of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
