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Indeterminate
Indeterminate may refer to: In mathematics * Indeterminate (variable), a symbol that is treated as a variable * Indeterminate system, a system of simultaneous equations that has more than one solution * Indeterminate equation, an equation that has more than one solution * Indeterminate form, an algebraic expression with certain limiting behaviour in mathematical analysis Other * Indeterminate growth, a term in biology and especially botany * Indeterminacy (philosophy), describing the shortcomings of definition in philosophy * Indeterminacy (music), music for which the composition or performance is determined by chance * Statically indeterminate In statics and structural mechanics, a structure is statically indeterminate when the static equilibrium equations force and moment equilibrium conditions are insufficient for determining the internal forces and reactions on that structure. Mat ..., in statics, describing a structure for which the static equilibrium equations are insuf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indeterminate Form
In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. More specifically, an indeterminate form is a mathematical expression involving at most two of 0~, 1 or \infty, obtained by applying the algebraic limit theorem in the process of attempting to determine a limit, which fails to restrict that limit to one specific value or infinity, and thus does not determine the limit being sought. A limit confirmed to be infinity is not indeterminate since it has been determined to have a specific value (infinity). The term was originally introduced by Cauchy's student Moigno in the middle of the 19th century. There are seven indeterminate forms which are typically cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indeterminate (variable)
In mathematics, particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else except itself. It may be used as a placeholder in objects such as polynomials and formal power series. In particular: * It does not designate a constant or a parameter of the problem. * It is not an unknown that could be solved for. * It is not a variable designating a function argument, or a variable being summed or integrated over. * It is not any type of bound variable. * It is just a symbol used in an entirely formal way. When used as placeholders, a common operation is to substitute mathematical expressions (of an appropriate type) for the indeterminates. By a common abuse of language, mathematical texts may not clearly distinguish indeterminates from ordinary variables. Polynomials A polynomial in an indeterminate X is an expression of the form a_0 + a_1X + a_2X^2 + \ldots + a_nX^n, where the ''a_i'' are called the coeffici ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indeterminate Equation
In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution. For example, the equation ax + by =c is a simple indeterminate equation, as is x^2=1. Indeterminate equations cannot be solved uniquely. In fact, in some cases it might even have infinitely many solutions. Some of the prominent examples of indeterminate equations include: Univariate polynomial equation: :a_nx^n+a_x^+\dots +a_2x^2+a_1x+a_0 = 0, which has multiple solutions for the variable x in the complex plane—unless it can be rewritten in the form a_n(x-b)^n = 0. Non-degenerate conic equation: :Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0, where at least one of the given parameters A, B, and C is non-zero, and x and y are real variables. Pell's equation: :\ x^2 - Py^2 = 1, where P is a given integer that is not a square number, and in which the variables x and y are required to be integers. The equation of Pythagorean triples: :x^2+y^2=z^2, in which the vari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indeterminate System
In mathematics, particularly in algebra, an indeterminate system is a system of simultaneous equations (e.g., linear equations) which has more than one solution (sometimes infinitely many solutions). In the case of a linear system, the system may be said to be underspecified, in which case the presence of more than one solution would imply an infinite number of solutions (since the system would be describable in terms of at least one free variable), but that property does not extend to nonlinear systems (e.g., the system with the equation x^2=1 ). An indeterminate system by definition is consistent, in the sense of having at least one solution. For a system of linear equations, the number of equations in an indeterminate system could be the same as the number of unknowns, less than the number of unknowns (an underdetermined system), or greater than the number of unknowns (an overdetermined system). Conversely, any of those three cases may or may not be indeterminate. Examples Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indeterminacy (music)
Indeterminacy is a composing approach in which some aspects of a musical work are left open to chance or to the interpreter's free choice. John Cage, a pioneer of indeterminacy, defined it as "the ability of a piece to be performed in substantially different ways". The earliest significant use of music indeterminacy features is found in many of the compositions of American composer Charles Ives in the early 20th century. Henry Cowell adopted Ives's ideas during the 1930s, in works allowing players to arrange the fragments of music in a number of different possible sequences. Beginning in the early 1950s, the term came to refer to the (mostly American) movement which grew up around Cage. This group included the other members of the New York School. In Europe, following the introduction of the expression "aleatory music" by Werner Meyer-Eppler, the French composer Pierre Boulez was largely responsible for popularizing the term. Definition Describing indeterminacy, composer John C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indeterminate Growth
In biology and botany, indeterminate growth is growth that is not terminated in contrast to determinate growth that stops once a genetically pre-determined structure has completely formed. Thus, a plant that grows and produces flowers and fruit until killed by frost or some other external factor is called indeterminate. For example, the term is applied to tomato varieties that grow in a rather gangly fashion, producing fruit throughout the growing season, and in contrast to a determinate tomato plant, which grows in a more bushy shape and is most productive for a single, larger harvest, then either tapers off with minimal new growth or fruit, or dies. Inflorescences In reference to an inflorescence (a shoot specialised for bearing flowers, and bearing no leaves other than bracts), an indeterminate type (such as a raceme) is one in which the first flowers to develop and open are from the buds at the base, followed progressively by buds nearer to the growing tip. The growth of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statically Indeterminate
In statics and structural mechanics, a structure is statically indeterminate when the static equilibrium equations force and moment equilibrium conditions are insufficient for determining the internal forces and reactions on that structure. Mathematics Based on Newton's laws of motion, the equilibrium equations available for a two-dimensional body are: : \sum \mathbf F = 0 : the vectorial sum of the forces acting on the body equals zero. This translates to: :: \sum \mathbf H = 0 : the sum of the horizontal components of the forces equals zero; :: \sum \mathbf V = 0 : the sum of the vertical components of forces equals zero; : \sum \mathbf M = 0 : the sum of the moments (about an arbitrary point) of all forces equals zero. In the beam construction on the right, the four unknown reactions are , , , and . The equilibrium equations are: : \begin \sum \mathbf V = 0 \quad & \implies \quad \mathbf V_A - \mathbf F_v + \mathbf V_B + \mathbf V_C = 0 \\ \sum \mathbf H = 0 \quad & \im ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indeterminacy (philosophy)
Indeterminacy, in philosophy, can refer both to common scientific and mathematical concepts of uncertainty and their implications and to another kind of indeterminacy deriving from the nature of definition or meaning. It is related to deconstructionism and to Nietzsche's criticism of the Kantian noumenon. Indeterminacy in philosophy Introduction The problem of indeterminacy arises when one observes the eventual circularity of virtually every possible definition. It is easy to find loops of definition in any dictionary, because this seems to be the only way that certain concepts, and generally very important ones such as that of existence, can be defined in the English language. A definition is a collection of other words, and in any finite dictionary if one continues to follow the trail of words in search of the precise meaning of any given term, one will inevitably encounter this linguistic indeterminacy. Philosophers and scientists generally try to eliminate indeterminate ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
