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Quantity
Quantity is a property that can exist as a multitude or magnitude. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation. Some quantities are such by their inner nature (as number), while others are functioning as states (properties, dimensions, attributes) of things such as heavy and light, long and short, broad and narrow, small and great, or much and little. Two basic divisions of quantity, magnitude and multitude, imply the principal distinction between continuity (continuum) and discontinuity. Under the name of multitude come what is discontinuous and discrete and divisible into indivisibles, all cases of collective nouns: army, fleet, flock, government, company, party, people, chorus, crowd, mess, and number
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Counting
Counting
Counting
is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements
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Infinitesimal
In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small.[1] The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small"
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Observable
In physics, an observable is a dynamic variable[clarification needed] that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued function on the set of all possible system states. In quantum physics, it is an operator, or gauge, where the property of the system state can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value. Physically meaningful observables must also satisfy transformation laws which relate observations performed by different observers in different frames of reference
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Gérard Debreu
Gérard Debreu
Gérard Debreu
(French: [dəbʁø]; 4 July 1921 – 31 December 2004) was a French-born American economist and mathematician. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economic Sciences.[1]Contents1 Biography 2 Academic career 3 Major publications3.1 Books 3.2 Book chapters 3.3 Journal articles4 References 5 External linksBiography[edit] His father was the business partner of his maternal grandfather in lace manufacturing, a traditional industry in Calais. Debreu was orphaned at an early age, as his father committed suicide and his mother died of natural causes.[2] Prior to the start of World War II, he received his baccalauréat and went to Ambert
Ambert
to begin preparing for the entrance examination of a grande école
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R. Duncan Luce
Robert Duncan Luce (May 16, 1925 – August 11, 2012)[1] was an American mathematician and social scientist, and one of the most preeminent figures in the field of mathematical psychology. At the end of his life, he held the position of Distinguished Research Professor of Cognitive Science
Cognitive Science
at the University of California, Irvine.[2] Luce received a Bachelor of Science
Bachelor of Science
degree in Aeronautical Engineering from the Massachusetts Institute of Technology
Massachusetts Institute of Technology
in 1945, and PhD
PhD
in Mathematics
Mathematics
from the same university in 1950 under I. S. Cohen with thesis On Semigroups. He began his professorial career at Columbia University in 1954, where he was an assistant professor in mathematical statistics and sociology
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John Tukey
John Wilder Tukey ForMemRS[2] (/ˈtuːki/;[3] June 16, 1915 – July 26, 2000) was an American mathematician best known for development of the FFT algorithm and box plot.[4] The Tukey range test, the Tukey lambda distribution, the Tukey test of additivity, and the Teichmüller–Tukey lemma all bear his name.Contents1 Biography 2 Scientific contributions2.1 Statistical practice3 Statistical terms 4 See also 5 Publications 6 Notes 7 External linksBiography[edit] Tukey was born in New Bedford, Massachusetts
New Bedford, Massachusetts
in 1915, and obtained a B.A. in 1936 and M.Sc.
M.Sc.
in 1937, in chemistry, from Brown University, before moving to Princeton University
Princeton University
where he received a Ph.D.
Ph.D.
in mathematics.[5] During World War II, Tukey worked at the Fire Control Research Office and collaborated with Samuel Wilks and William Cochran
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Variable (mathematics)
In elementary mathematics, a variable is a symbol, commonly an alphabetic character, that represents a number, called the value of the variable, which is either arbitrary, not fully specified, or unknown. Making algebraic computations with variables as if they were explicit numbers allows one to solve a range of problems in a single computation. A typical example is the quadratic formula, which allows one to solve every quadratic equation by simply substituting the numeric values of the coefficients of the given equation to the variables that represent them. The concept of a variable is also fundamental in calculus. Typically, a function y = f(x) involves two variables, y and x, representing respectively the value and the argument of the function
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Set (mathematics)
In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written 2,4,6 . The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived
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Scalar (mathematics)
A scalar is an element of a field which is used to define a vector space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector.[1] In linear algebra, real numbers or other elements of a field are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector.[2][3][4] More generally, a vector space may be defined by using any field instead of real numbers, such as complex numbers. Then the scalars of that vector space will be the elements of the associated field. A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied to produce a scalar
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Euclidean Vector
In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric[1] or spatial vector,[2] or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector
Euclidean vector
is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B,[3] and denoted by A B → . displaystyle overrightarrow AB
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Tensor
In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Geometric vectors, often used in physics and engineering applications, and scalars themselves are also tensors.[1] A more sophisticated example is the Cauchy stress tensor T, which takes a direction v as input and produces the stress T(v) on the surface normal to this vector for output, thus expressing a relationship between these two vectors, shown in the figure (right). Given a reference basis of vectors, a tensor can be represented as an organized multidimensional array of numerical values. The order (also degree or rank) of a tensor is the dimensionality of the array needed to represent it, or equivalently, the number of indices needed to label a component of that array
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Argument Of A Function
In mathematics, an argument of a function is a specific input in the function, also known as an independent variable.[1] When it is clear from the context which argument is meant, the argument is often denoted by the abbreviation arg.[2] For example, the binary function f ( x , y ) = x 2 + y 2 displaystyle f(x,y)=x^ 2 +y^ 2 has two arguments, x displaystyle x and y displaystyle y , in an ordered pair ( x , y ) displaystyle (x,y) . The hypergeometric function is an example of a four-argument function. The number of arguments that a function takes is called the arity of the function
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Empirical Research
Empirical research is research using empirical evidence. It is a way of gaining knowledge by means of direct and indirect observation or experience. Empiricism
Empiricism
values such research more than other kinds. Empirical evidence (the record of one's direct observations or experiences) can be analyzed quantitatively or qualitatively. Quantifying the evidence or making sense of it in qualitative form, a researcher can answer empirical questions, which should be clearly defined and answerable with the evidence collected (usually called data). Research
Research
design varies by field and by the question being investigated
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Expression (mathematics)
In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context
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Stochastic
See also stochastic process.This article may require cleanup to meet's quality standards. No cleanup reason has been specified. Please help improve this article if you can. (September 2010) (Learn how and when to remove this template message)The word stochastic is an adjective in English that describes something that was randomly determined.[1] The word first appeared in English to describe a mathematical object called a stochastic process, but now in mathematics the terms stochastic process and random process are considered interchangeable.[2][3][4][5][6] The word, with its current definition meaning random, came from German, but it originally came from Greek στόχος (stokhos), meaning 'aim, guess'.[1] The term stochastic is used in many different fields, particularly where stochastic or random processes are used to represent systems or phenomena that seem to change in a random way
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