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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. A small quantity is sometimes referred to as a QUANTULUM. 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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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. In mathematics education , elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree. The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano
Bernard Bolzano
in his work The Paradoxes of the Infinite
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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 . 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. 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. A vector space equipped with a scalar product is called an inner product space . The real component of a quaternion is also called its SCALAR PART
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Euclidean Vector
In mathematics , physics , and engineering , a EUCLIDEAN VECTOR (sometimes called a GEOMETRIC or SPATIAL VECTOR, 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, and denoted by A B . {displaystyle {overrightarrow {AB}}.} A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating planet rotation around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B
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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. 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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Variable (mathematics)
In elementary mathematics , a VARIABLE is an alphabetic character representing 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. The term "variable" comes from the fact that, when the argument (also called the "variable of the function") varies, then the value varies accordingly
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John Tukey
JOHN WILDER TUKEY ForMemRS (/ˈtuːki/ ; June 16, 1915 – July 26, 2000) was an American mathematician best known for development of the FFT algorithm and box plot . The Tukey range test , the Tukey lambda distribution , the Tukey test of additivity , and the Teichmüller–Tukey lemma all bear his name. CONTENTS * 1 Biography * 2 Scientific contributions * 2.1 Statistical practice * 3 Statistical terms * 4 See also * 5 Publications * 6 Notes * 7 External links BIOGRAPHYTukey was born in New Bedford, Massachusetts
New Bedford, Massachusetts
in 1915, and obtained a B.A. in 1936 and M.Sc. in 1937, in chemistry, from Brown University , before moving to Princeton University where he received a Ph.D. in mathematics . During World War II
World War II
, Tukey worked at the Fire Control Research Office and collaborated with Samuel Wilks and William Cochran
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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. The related term enumeration refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element. Counting
Counting
using tally marks at Hanakapiai Beach Counting
Counting
sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos" (2, 4, 6, 8, 10, 12, ...), or "counting by fives" (5, 10, 15, 20, 25, ...)
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Observable
In physics , an OBSERVABLE is a dynamic variable 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 . These transformation laws are automorphisms of the state space, that is bijective transformations which preserve some mathematical property
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Gérard Debreu
GéRARD DEBREU (French: ; 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
University of California, Berkeley
, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economic Sciences . CONTENTS * 1 Biography * 2 Academic career * 3 Major publications * 3.1 Books * 3.2 Book chapters * 3.3 Journal articles * 4 References * 5 External links BIOGRAPHYHis father was the business partner of his maternal grandfather in lace manufacturing, a traditional industry in Calais
Calais
. Debreu was orphaned at an early age, as his father committed suicide and his mother died of natural causes. Prior to the start of World War II
World War II
, he received his baccalauréat and went to 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) 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 . 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
Columbia University
in 1954, where he was an assistant professor in mathematical statistics and sociology
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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. The word infinitesimal comes from a 17th-century Modern Latin
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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Argument Of A Function
In mathematics , an ARGUMENT of a function is a specific input in the function, also known as an independent variable . When it is clear from the context which argument is meant, the argument is often denoted by the abbreviation arg. A mathematical function has one or more arguments in the form of independent variables designated in the function's definition, which can also contain parameters . The independent variables are mentioned in the list of arguments that the function takes, whereas the parameters are not. For example, in the logarithmic function f ( x ) = log b ( x ) {displaystyle f(x)=log _{b}(x)} , the base b {displaystyle b} is considered a parameter. A function that takes a single argument as input (such as f ( x ) = x 2 {displaystyle f(x)=x^{2}} ) is called a unary function . A function of two or more variables is considered to have a domain consisting of ordered pairs or tuples of argument values
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Calculus
CALCULUS (from Latin
Latin
calculus, literally "small pebble used for counting on an abacus ") is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations . It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves); these two branches are related to each other by the fundamental theorem of calculus . Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit . Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton
Isaac Newton
and Gottfried Leibniz . Today, calculus has widespread uses in science , engineering and economics
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Density
The DENSITY, or more precisely, the VOLUMETRIC MASS DENSITY, of a substance is its mass per unit volume . The symbol most often used for density is ρ (the lower case Greek letter rho ), although the Latin letter D can also be used. Mathematically, density is defined as mass divided by volume: = m V , {displaystyle rho ={frac {m}{V}},} where ρ is the density, m is the mass, and V is the volume. In some cases (for instance, in the United States oil and gas i