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E-folding
In science, ''e''-folding is the time interval in which an exponentially growing quantity increases by a factor of ''e''; it is the base-''e'' analog of doubling time. This term is often used in many areas of science, such as in atmospheric chemistry, medicine and theoretical physics, especially when cosmic inflation is investigated. Physicists and chemists often talk about the ''e''-folding time scale that is determined by the proper time in which the length of a patch of space or spacetime increases by the factor ''e'' mentioned above. In finance, the logarithmic return or continuously compounded return, also known as force of interest, is the reciprocal of the ''e''-folding time. The term ''e''-folding time is also sometimes used similarly in the case of exponential decay, to refer to the timescale for a quantity to decrease to 1/''e'' of its previous value. The process of evolving to equilibrium is often characterized by a time scale called the ''e''-folding time, '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Growth
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing exponential decay instead. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay since the function values form a geometric progression. The formula for exponential growth of a variable at the growth rate , as time goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is x_t = x_0(1+r)^t where is the value of at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubling Time
The doubling time is the time it takes for a population to double in size/value. It is applied to population growth, inflation, resource extraction, consumption of goods, compound interest, the volume of malignant tumours, and many other things that tend to grow over time. When the relative growth rate (not the absolute growth rate) is constant, the quantity undergoes exponential growth and has a constant doubling time or period, which can be calculated directly from the growth rate. This time can be calculated by dividing the natural logarithm of 2 by the exponent of growth, or approximated by dividing 70 by the percentage growth rate (more roughly but roundly, dividing 72; see the rule of 72 for details and derivations of this formula). The doubling time is a characteristic unit (a natural unit of scale) for the exponential growth equation, and its converse for exponential decay is the half-life. For example, given Canada's net population growth of 0.9% in the year 2006, di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubling Time
The doubling time is the time it takes for a population to double in size/value. It is applied to population growth, inflation, resource extraction, consumption of goods, compound interest, the volume of malignant tumours, and many other things that tend to grow over time. When the relative growth rate (not the absolute growth rate) is constant, the quantity undergoes exponential growth and has a constant doubling time or period, which can be calculated directly from the growth rate. This time can be calculated by dividing the natural logarithm of 2 by the exponent of growth, or approximated by dividing 70 by the percentage growth rate (more roughly but roundly, dividing 72; see the rule of 72 for details and derivations of this formula). The doubling time is a characteristic unit (a natural unit of scale) for the exponential growth equation, and its converse for exponential decay is the half-life. For example, given Canada's net population growth of 0.9% in the year 2006, di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for scientific reasoning is tens of thousands of years old. The earliest written records in the history of science come from Ancient Egypt and Mesopotamia in around 3000 to 1200 BCE. Their contributions to mathematics, astronomy, and medicine entered and shaped Greek natural philosophy of classical antiquity, whereby formal attempts were made to provide explanations of events in the physical world based on natural causes. After the fall of the Western Roman Empire, knowledge of Greek conceptions of the world deteriorated in Western Europe during the early centuries (400 to 1000 CE) of the Middle Ages, but was preserved in the Muslim world during the Islamic Golden Age and later by the efforts of Byzantine Greek scholars who brought Greek ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., an integral assigns numbers to functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Derivative, differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Differential Equation
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. Ordinary differential equations (ODE) Suppose a differential equation can be written in the form :\frac f(x) = g(x)h(f(x)) which we can write more simply by letting y = f(x): :\frac=g(x)h(y). As long as ''h''(''y'') ≠0, we can rearrange terms to obtain: : = g(x) \, dx, so that the two variables ''x'' and ''y'' have been separated. ''dx'' (and ''dy'') can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of ''dx'' as a differential (infinitesimal) is somewhat advanced. Alternative notation Those who dislike Leibniz's notation may prefer to write this as :\frac \frac = g(x), but that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Differential Equation
A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written :f(x,y) \, dy = g(x,y) \, dx, where and are homogeneous functions of the same degree of and . In this case, the change of variable leads to an equation of the form :\frac = h(u) \, du, which is easy to solve by integration of the two members. Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term. History The term ''homogeneous'' was first applied to differential equations by Johann Bernoulli in section 9 of his 1726 article ''De integraionibus aequationum differentialium'' (On t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Differential Equation
In mathematics, 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) where and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable . Such an equation is an ordinary differential equation (ODE). A ''linear differential equation'' may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. 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 , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rate Constant
In chemical kinetics a reaction rate constant or reaction rate coefficient, ''k'', quantifies the rate and direction of a chemical reaction. For a reaction between reactants A and B to form product C the reaction rate is often found to have the form: r = k(T) mathrmm mathrm Here ''k''(''T'') is the reaction rate constant that depends on temperature, and and are the molar concentrations of substances A and B in moles per unit volume of solution, assuming the reaction is taking place throughout the volume of the solution. (For a reaction taking place at a boundary, one would use moles of A or B per unit area instead.) The exponents ''m'' and ''n'' are called partial orders of reaction and are ''not'' generally equal to the stoichiometric coefficients ''a'' and ''b''. Instead they depend on the reaction mechanism and can be determined experimentally. Elementary steps For an elementary step, there ''is'' a relationship between stoichiometry and rate law, as determined by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinetics (physics)
In physics and engineering, kinetics is the branch of classical mechanics that is concerned with the relationship between the motion and its causes, specifically, forces and torques. Since the mid-20th century, the term " dynamics" (or "analytical dynamics") has largely superseded "kinetics" in physics textbooks, though the term is still used in engineering. In plasma physics, kinetics refers to the study of continua in velocity space. This is usually in the context of non-thermal ( non-Maxwellian) velocity distributions, or processes that perturb thermal distributions. These " kinetic plasmas" cannot be adequately described with fluid equations. The term ''kinetics'' is also used to refer to chemical kinetics, particularly in chemical physics and physical chemistry Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm Of 2
The decimal value of the natural logarithm of 2 is approximately :\ln 2 \approx 0.693\,147\,180\,559\,945\,309\,417\,232\,121\,458. The logarithm of 2 in other bases is obtained with the formula :\log_b 2 = \frac. The common logarithm in particular is () :\log_ 2 \approx 0.301\,029\,995\,663\,981\,195. The inverse of this number is the binary logarithm of 10: : \log_2 10 =\frac \approx 3.321\,928\,095 (). By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is a transcendental number. Series representations Rising alternate factorial :\ln 2 = \sum_^\infty \frac=1-\frac12+\frac13-\frac14+\frac15-\frac16+\cdots. This is the well-known "alternating harmonic series". :\ln 2 = \frac +\frac\sum_^\infty \frac. :\ln 2 = \frac +\frac\sum_^\infty \frac. :\ln 2 = \frac +\frac\sum_^\infty \frac. :\ln 2 = \frac +\frac\sum_^\infty \frac. :\ln 2 = \frac +\frac\sum_^\infty \frac. : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
