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Logistic Function
A logistic function or logistic curve is a common S-shaped curve ( sigmoid curve) with the equation f(x) = \frac where The logistic function has domain the real numbers, the limit as x \to -\infty is 0, and the limit as x \to +\infty is L. The exponential function with negated argument (e^ ) is used to define the standard logistic function, depicted at right, where L=1, k=1, x_0=0, which has the equation f(x) = \frac and is sometimes simply called the sigmoid. It is also sometimes called the expit, being the inverse function of the logit. The logistic function finds applications in a range of fields, including biology (especially ecology), biomathematics, chemistry, demography, economics, geoscience, mathematical psychology, probability, sociology, political science, linguistics, statistics, and artificial neural networks. There are various generalizations, depending on the field. History The logistic function was introduced in a series of three papers by Pier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigmoid Function
A sigmoid function is any mathematical function whose graph of a function, graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the logistic function, which is defined by the formula :\sigma(x) = \frac = \frac = 1 - \sigma(-x). Other sigmoid functions are given in the #Examples, Examples section. In some fields, most notably in the context of artificial neural networks, the term "sigmoid function" is used as a synonym for "logistic function". Special cases of the sigmoid function include the Gompertz curve (used in modeling systems that saturate at large values of ''x'') and the ogee curve (used in the spillway of some dams). Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (''y'' axis) in the range 0 to 1. Another commonly used range is from −1 to 1. A wide variety of sigmoid functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Political Science
Political science is the scientific study of politics. It is a social science dealing with systems of governance and Power (social and political), power, and the analysis of political activities, political philosophy, political thought, political behavior, and associated constitutions and laws. Specialists in the field are political scientists. History Origin Political science is a social science dealing with systems of governance and power, and the analysis of political activities, political institutions, political thought and behavior, and associated constitutions and laws. As a social science, contemporary political science started to take shape in the latter half of the 19th century and began to separate itself from political philosophy and history. Into the late 19th century, it was still uncommon for political science to be considered a distinct field from history. The term "political science" was not always distinguished from political philosophy, and the modern dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Curve
In mathematics, logarithmic growth describes a phenomenon whose size or cost can be described as a logarithm function of some input. e.g. ''y'' = ''C'' log (''x''). Any logarithm base can be used, since one can be converted to another by multiplying by a fixed constant.. Logarithmic growth is the inverse of exponential growth and is very slow. A familiar example of logarithmic growth is a number, ''N'', in positional notation, which grows as log''b'' (''N''), where ''b'' is the base of the number system used, e.g. 10 for decimal arithmetic. In more advanced mathematics, the partial sums of the harmonic series :1+\frac+\frac+\frac+\frac+\cdots grow logarithmically. In the design of computer algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...s, logari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Growth
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate direct proportion, 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 (mathematics)#Of change, rate of change (that is, the derivative) of a quantity with respect to an independent variable is proportionality (mathematics), proportional to the quantity itself. Often the independent variable is time. Described as a Function (mathematics), function, a quantity undergoing exponential growth is an Exponentiation#Power functions, 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 Inverse function, the inverse of logarithmic growth. Not all cases of growth at an always increasing rate are instances of exponenti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Growth
In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term '' affine function'' is often used. * In linear algebra, mathematical analysis, and functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ..., a linear function is a linear map. As a polynomial function In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero). When the function is of only one variable, it is of the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PRIMUS (journal)
''PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies'' is a peer-reviewed academic journal covering the teaching of undergraduate mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ..., established in 1991. The journal has been published by Taylor & Francis since March 2007. It is abstracted and indexed in Cambridge Scientific Abstracts, MathEduc, PsycINFO, and '' Zentralblatt MATH''. PRIMUS is an affiliated journal of the Mathematical Association of America, so all MAA members have access to PRIMUS. Editorial Team PRIMUS was started by founding editor-in-chief Brian Winkel in 1991 to address the lack of venues for tertiary mathematics educators to share their pedagogical work. In 2011, Jo Ellis-Monaghan became the second editor-in-chief ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adolphe Quetelet
Lambert Adolphe Jacques Quetelet FRSF or FRSE (; 22 February 1796 – 17 February 1874) was a Belgian- French astronomer, mathematician, statistician and sociologist who founded and directed the Brussels Observatory and was influential in introducing statistical methods to the social sciences. His name is sometimes spelled with an accent as ''Quételet''. He also founded the science of anthropometry and developed the body mass index (BMI) scale, originally called the Quetelet Index. His work on measuring human characteristic to determine the ideal ''l'homme moyen'' ("the average man"), played a key role in the origins of eugenics. Biography Adolphe was born in Ghent (which, at the time was a part of the new French Republic). He was the son of François-Augustin-Jacques-Henri Quetelet, a Frenchman and Anne Françoise Vandervelde, a Flemish woman. His father was born at Ham, Picardy, and being of a somewhat adventurous spirit, he crossed the English Channel and ... [...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] |
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] |
Pierre François Verhulst
Pierre François Verhulst (28 October 1804, in Brussels – 15 February 1849, in Brussels) was a Belgian mathematician and a doctor in number theory from the University of Ghent in 1825. He is best known for the logistic growth model. Logistic equation Verhulst developed the logistic function in a series of three papers between 1838 and 1847, based on research on modeling population growth that he conducted in the mid 1830s, under the guidance of Adolphe Quetelet; see for details. Verhulst published in the equation: : \frac = rN - \alpha N^2 where ''N''(''t'') represents number of individuals at time ''t'', ''r'' the intrinsic growth rate, and ''\alpha'' is the density-dependent crowding effect (also known as intraspecific competition). In this equation, the population equilibrium (sometimes referred to as the carrying capacity, ''K''), N^*, is : N^* = \frac . In he named the solution the logistic curve. Later, Raymond Pearl and Lowell Reed popularized the equatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalizations
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
