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Simultaneous Equation
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a: * System of linear equations, * System of nonlinear equations, * System of bilinear equations, * System of polynomial equations, * System of differential equations, or a * System of difference equations See also * Simultaneous equations model, a statistical model in the form of simultaneous linear equations * Elementary algebra Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variable (mathematics ..., for elementary methods {{set index article Equations Broad-concept articles de:Gleichung#Gleichungssysteme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set S is called finite if there exists a bijection for some natural number n (natural numbers are defined as sets in Zermelo-Fraenkel set theory). The number n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equation
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in French an ''équation'' is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. The " =" symbol, which appears in every equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A ''Solution (mathematics), solution'' to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the Tuple, ordered triple (x,y,z)=(1,-2,-2), since it makes all three equations valid. Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A Nonlinear system, system of non-linear equations can often be Approximation, approximated by a linear system (see linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Nonlinear Equations
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Bilinear Equations
In mathematics, a system of bilinear equations is a special sort of system of polynomial equations, where each equation equates a bilinear form with a constant (possibly zero). More precisely, given two sets of variables represented as coordinate vectors and ''y'', then each equation of the system can be written y^TA_ix=g_i, where, is an integer whose value ranges from 1 to the number of equations, each A_i is a matrix, and each g_i is a real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re .... Systems of bilinear equations arise in many subjects including engineering, biology, and statistics. See also * Systems of linear equations References * Charles R. Johnson, Joshua A. Link 'Solution theory for complete bilinear systems of equations' - http://onlinelibrary.wiley.co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Polynomial Equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations where the are polynomials in several variables, say , over some Field (mathematics), field . A ''solution'' of a polynomial system is a set of values for the s which belong to some algebraically closed field extension of , and make all equations true. When is the field of rational numbers, is generally assumed to be the field of complex numbers, because each solution belongs to a field extension of , which is isomorphic to a subfield of the complex numbers. This article is about the methods for solving, that is, finding all solutions or describing them. As these methods are designed for being implemented in a computer, emphasis is given on fields in which computation (including equality testing) is easy and efficient, that is the field of rational numbers and finite fields. Searching for solutions that belong to a specific set is a problem which is generally much more d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Differential Equations
In mathematics, a system of differential equations is a finite set of differential equations. Such a system can be either linear or non-linear. Also, such a system can be either a system of ordinary differential equations or a system of partial differential equations. Linear systems of differential equations A first-order linear system of ODEs is a system in which every equation is first order and depends on the unknown functions linearly. Here we consider systems with an equal number of unknown functions and equations. These may be written as \frac = a_(t) x_1 + \ldots + a_(t)x_n + g_(t), \qquad j=1,\ldots,n where n is a positive integer, and a_(t),g_(t) are arbitrary functions of the independent variable t. A first-order linear system of ODEs may be written in matrix form: \frac \begin x_1 \\ x_2 \\ \vdots \\ x_n \end = \begin a_ & \ldots & a_ \\ a_ & \ldots & a_ \\ \vdots & \ldots & \vdots \\ a_ & & a_ \end \begin x_1 \\ x_2 \\ \vdots \\ x_n \end + \begin g_1 \\ g_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Difference Equation
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. The order of the equation is the maximum time gap between any two indicated values of the variable vector. For example, :\mathbf x_t = \mathbf_ + \mathbf_ is an example of a second-order matrix difference equation, in which is an vector of variables and and are matrices. This equation is homogeneous because there is no vector constant term added to the end of the equation. The same equation might also be written as :\mathbf x_ = \mathbf_ + \mathbf_ or as :\mathbf x_n = \mathbf_ + \mathbf_ The most commonly encountered matrix difference equations are first-order. Nonhomogeneous first-order case and the steady state An example of a nonhomogeneous first-order matrix difference equation is :\mathbf x_t = \mathbf_ + \mathbf with additive cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simultaneous Equations Model
Simultaneous equations models are a type of statistical model in which the dependent variables are functions of other dependent variables, rather than just independent variables. This means some of the explanatory variables are jointly determined with the dependent variable, which in economics usually is the consequence of some underlying equilibrium mechanism. Take the typical supply and demand model: whilst typically one would determine the quantity supplied and demanded to be a function of the price set by the market, it is also possible for the reverse to be true, where producers observe the quantity that consumers demand ''and then'' set the price. Simultaneity poses challenges for the estimation of the statistical parameters of interest, because the Gauss–Markov assumption of strict exogeneity of the regressors is violated. And while it would be natural to estimate all simultaneous equations at once, this often leads to a computationally costly non-linear optimization p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Algebra
Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variable (mathematics), variables (quantities without fixed values). This use of variables entails use of algebraic notation and an understanding of the general rules of the Operation (mathematics), operations introduced in arithmetic: addition, subtraction, multiplication, division, etc. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real number, real and complex numbers. It is typically taught to secondary school students and at introductory college level in the United States, and builds on their understanding of arithmetic. The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
