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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. As an example, Canada's net population growth was 2.7 percent in the year 202 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Growth
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now. In more technical language, its instantaneous rate of change (that is, the derivative) of a quantity with respect to an independent variable is proportional to the quantity itself. Often the independent variable is time. 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). Exponential growth is the inverse of logarithmic growth. Not all cases of growth at an always increasing rate are instances of exponential growth. For example the function f(x) = x^3 grows at an ever increasing rate, but is much slower than growing exponentially. For example, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Decay
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda Lambda (; uppercase , lowercase ; , ''lám(b)da'') is the eleventh letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoen ...) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant: :\frac = -\lambda N(t). The solution to this equation (see #Solution_of_the_differential_equation, derivation below) is: :N(t) = N_0 e^, where is the quantity at time , is the initial quantity, that is, the quantity at time . Measuring rates of decay Mean lifetime If the decaying quantity, ''N''(''t''), is the number of discrete elements in a certain set (mathematics), se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of 72
In finance, the rule of 72, the rule of 70 and the rule of 69.3 are methods for estimating an investment's doubling time. The rule number (e.g., 72) is divided by the interest percentage per period (usually years) to obtain the approximate number of periods required for doubling. Although scientific calculators and spreadsheet programs have functions to find the accurate doubling time, the rules are useful for mental calculations and when only a basic calculator is available. These rules apply to exponential growth and are therefore used for compound interest as opposed to simple interest calculations. They can also be used for decay to obtain a halving time. The choice of number is mostly a matter of preference: 69 is more accurate for continuous compounding, while 72 works well in common interest situations and is more easily divisible. There are a number of variations to the rules that improve accuracy. For periodic compounding, the ''exact'' doubling time for an interes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Growth
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now. In more technical language, its instantaneous rate of change (that is, the derivative) of a quantity with respect to an independent variable is proportional to the quantity itself. Often the independent variable is time. 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). Exponential growth is the inverse of logarithmic growth. Not all cases of growth at an always increasing rate are instances of exponential growth. For example the function f(x) = x^3 grows at an ever increasing rate, but is much slower than growing exponentially. For example, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E-folding
In science, ''e''-folding is the time interval in which an exponentially growing quantity increases or decreases 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, theoretical physics, and cosmology. In cosmology the ''e''-folding time scale is the proper time in which the length of a patch of space or spacetime increases by the factor ''e''. In finance, the logarithmic return or continuously compounded return, also known as force of interest, is the reciprocal of the ''e''-folding time. The process of evolving to equilibrium is often characterized by a time scale called the ''e''-folding time, ''τ''. This time is used for processes which evolve exponentially toward a final state (equilibrium). In other words, if we examine an observable, ''X'', associated with a system, (temperature or density, for example) then after a time, ''τ'', the initial difference bet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albert Allen Bartlett
Albert Allen Bartlett (March 21, 1923 – September 7, 2013) was an American professor of physics at the University of Colorado at Boulder. As of July 2001, Professor Bartlett had lectured over 1,742 times since September 1969 on ''Arithmetic, Population, and Energy''. Bartlett regarded the word combination "sustainable growth" as an oxymoron, and argued that modest annual percentage population increases could lead to exponential growth. He therefore regarded human overpopulation as "The Greatest Challenge" facing humanity. Career Bartlett received a B.A. in physics at Colgate University (1944), and an M.A. (1948) and Ph.D. (1951) in physics at Harvard University. Bartlett joined the faculty at the University of Colorado at Boulder in September 1950. In 1978 he was national president of the American Association of Physics Teachers. He was a fellow of the American Physical Society and of the American Association for the Advancement of Science. In 1969 and 1970 he served two terms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Logarithm
In mathematics, the binary logarithm () is the exponentiation, power to which the number must be exponentiation, raised to obtain the value . That is, for any real number , :x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n. For example, the binary logarithm of is , the binary logarithm of is , the binary logarithm of is , and the binary logarithm of is . The binary logarithm is the logarithm to the base and is the inverse function of the power of two function. There are several alternatives to the notation for the binary logarithm; see the #Notation, Notation section below. Historically, the first application of binary logarithms was in music theory, by Leonhard Euler: the binary logarithm of a frequency ratio of two musical tones gives the number of octaves by which the tones differ. Binary logarithms can be used to calculate the length of the representation of a number in the binary numeral system, or the number of bits needed to encode a message in infor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Population Growth
Population growth is the increase in the number of people in a population or dispersed group. The World population, global population has grown from 1 billion in 1800 to 8.2 billion in 2025. Actual global human population growth amounts to around 70 million annually, or 0.85% per year. As of 2024, The United Nations projects that global population will peak in the mid-2080s at around 10.3 billion. The UN's estimates have decreased strongly in recent years due to sharp declines in global birth rates. Others have challenged many recent population projections as having underestimated population growth. The world human population has been growing since the end of the Black Death, around the year 1350. A mix of technological advancement that improved agricultural productivity and sanitation and medical advancement that reduced mortality increased population growth. In some geographies, this has slowed through the process called the demographic transition, where many nations with high ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-life
Half-life is a mathematical and scientific description of exponential or gradual decay. Half-life, half life or halflife may also refer to: Film * Half-Life (film), ''Half-Life'' (film), a 2008 independent film by Jennifer Phang * ''Half Life: A Parable for the Nuclear Age'', a 1985 Australian documentary film Literature * Half Life (Jackson novel), ''Half Life'' (Jackson novel), a 2006 novel by Shelley Jackson * Half-Life (Krach novel), ''Half-Life'' (Krach novel), a 2004 novel by Aaron Krach * Halflife (Michalowski novel), ''Halflife'' (Michalowski novel), a 2004 novel by Mark Michalowski * ''Rozpad połowiczny'' (), a 1988 award-winning dystopia novel by Edmund Wnuk-Lipiński Music *Half Life (3 album), ''Half Life'' (3 album) (2001) *Halflife (EP), ''Halflife'' (EP), an EP by Lacuna Coil and the title track *''Half-Life E.P.'', an EP by Local H * "Half Life", a song by 10 Years from ''The Autumn Effect'' * "Half Life", a song by Come from ''Near-Life Experience'' * "Ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
