equation that is true for all values of variables
Equation


equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variable. A conditional equation is only true for particular values of the variables. An equation is written as two expressions, connected by a equals sign ("="). The expressions on the two sides of the equals sign are called the "left-hand side" and "right-hand side" of the equation. The most common type of equation is an algebraic equation, in which
9276954
Elementary algebra


as formula_40); such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. formula_41 is true only for formula_42 and formula_43. The values of the variables which make the equation true are the solutions of the equation and can be found through equation solving. Another type of equation is an inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are: formula_44 where formula_45 represents 'greater than
9633074
GHZ experiment


"collectively" unsuitable to represent all experimental results; namely the assumption of "local hidden variables" which occur "equally in distinct trials". Since equations (1) through (4) above cannot be satisfied simultaneously when the hidden variable, λ, takes the same value in each equation, GHSZ proceed by allowing λ to take different values in each equation. They define Also, Λ is the complement of Λ. Now, equation (1) can only be true if at least one of the other three is
6694590
Equation


For example, the system has the unique solution "x" = −1, "y" = 1. An identity is an equation that is true for all possible values of the variable(s) it contains. Many identities are known in algebra and calculus. In the process of solving an equation, an identity is often used to simplify an equation making it more easily solvable. In algebra, an example of an identity is the difference of two squares: which is true for all "x" and "y".
9276965
Equation


In mathematics, an equation is a statement that asserts the equality of two expressions. 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 equality is an equation. "Solving" an equation containing variables consists of determining which values of the variables make the equality true. Variables are also called unknowns and the values of the unknowns that satisfy the equality are called solutions of the
9276953
Equation solving


space, which passes through the three points with these coordinates. The solution set of a given set of equations or inequalities is the set of all its solutions, a solution being a tuple of values, one for each unknown, that satisfies all equations or inequalities. If the solution set is empty, then there are no values "x" such that the equations or inequalities becomes true simultaneously. For example, let us examine a classic one-variable case. Using the squaring function on the integers, that is,
5425175
Equality (mathematics)


An identity is "asserted" to be true for all values of variables in a given domain. An "equation" may sometimes mean an identity, but more often it "specifies" a subset of the variable space to be the subset where the equation is true. In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties that are considered. Particularly in the case of geometry, this is where two geometric shapes are said to be equal when one may be moved to coincide
7215595
Algebraic equation


the three variables "x", "y", and "z" over the field of the elementary functions in the variable "T". As for any equation, the "solutions" of an equation are the values of the variables for which the equation is true. For univariate algebraic equations these are also called roots, even if, properly speaking, one should say "the solutions of the algebraic equation" "P"=0 "are the roots of the polynomial" "P". When solving an equation,
16565127
Elementary algebra


best-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle: This equation states that formula_39, representing the square of the length of the side that is the hypotenuse (the side opposite the right angle), is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by and . An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such
9633073
Invariant (mathematics)


there is a number to which we always arrive, regardless of the order in which we count the objects in the set. The quantity—a cardinal number—is associated with the set, and is invariant under the process of counting. An identity is an equation that remains true for all values of its variables. There are also inequalities that remain true when the values of their variables change. The distance between two points on a number line is not changed by adding the same quantity to both numbers. On the other
9741326
Linear equation


polynomial over some field, from which the coefficients are taken (the symbols used for the variables are supposed to not denote any element of the field). The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true. In the case of just one variable, there is exactly one solution (provided that formula_5. Often, the term "linear equation" refers implicitly to this particular case, in which the variable is sensibly called the "unknown". In the case
17699073
Verification-based message-passing algorithms in compressed sensing


only have one single unknown variable formula_51 common in all of those equations then the value of this common variable should be the value of the right hand side of those equations. Besides, it says that all other variables in those equations should be zero. Suppose that one of those variables formula_52 is not zero, then the right hand side of the equation which contains both formula_53 should be formula_54 (For simplicity assume that the edge weights are all 1 or zero). Besides, since we know that formula_51 is the only unique
12503911
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 . 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
12439031
Equation solving


In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equality sign. When seeking a solution, one or more free variables are designated as unknowns. A solution is an assignment of expressions to the unknown variables that makes the equality in the equation true. In other words, a solution is an expression or a collection of expressions (one for each
5425167
Diagnostic equation


In a physical (and especially geophysical) simulation context, a diagnostic equation (or diagnostic model) is an equation (or model) that links the values of these variables simultaneously, either because the equation (or model) is time-independent, or because the variables all refer to the values they have at the identical time. This is by opposition to a prognostic equation. For instance, the so-called ideal gas law ("PV" = "nRT") of classical thermodynamics relates the state variables of
21630037
Equality (mathematics)


variables, called "unknowns", for which the specified equality is true. "Equation" may also refer to an equality relation that is satisfied only for the values of the variables that one is interested in. For example, "x" + "y" = 1 is the "equation" of the unit circle. There is no standard notation that distinguishes an equation from an identity or other use of the equality relation: a reader has to guess an appropriate interpretation from the semantics of expressions and the context.
7215594
Regression analysis


between the value of the dependent variable predicted by the model, formula_67, and the true value of the dependent variable, formula_68. One method of estimation is ordinary least squares. This method obtains parameter estimates that minimize the sum of squared residuals, SSR: Minimization of this function results in a set of normal equations, a set of simultaneous linear equations in the parameters, which are solved to yield the parameter estimators, formula_70. In the case of simple regression, the formulas for the least squares estimates are formula_72 where formula_73
9775624
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 set of variables. For example, is a system of three equations in the three variables . A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than
20698258
Algebraic Riccati equation


of discrete time LQR). In infinite-horizon optimal control problems, one cares about the value of some variable of interest arbitrarily far into the future, and one must optimally choose a value of a controlled variable right now, knowing that one will also behave optimally at all times in the future. The optimal current values of the problem's control variables at any time can be found using the solution of the Riccati equation and the current observations on evolving state variables. With multiple state variables and multiple control variables, the
15880796
Identity (philosophy)


if every predicate true of "x" is true of "y" as well. Leibniz's ideas have taken root in the philosophy of mathematics, where they have influenced the development of the predicate calculus as Leibniz's law. Mathematicians sometimes distinguish identity from equality. More mundanely, an "identity" in mathematics may be an "equation" that holds true for all values of a variable. Hegel argued that things are inherently self-contradictory and that the notion of something being self-identical only made sense if it
7002884
Temporal discretization


that the time integral of a given variable is equal to a weighted average between current and future values. The integral form of the equation can be written as: where "ƒ" is a weight between 0 and 1. For any control volume this integration holds true for any discretized variable. The following equation is obtained when applied to the governing equation including full discretized diffusion, convection, and source terms. After discretizing the time derivative, function F(formula_4) remains to be evaluated. The function is now evaluated using implicit and
10745632
Boundary value problem


problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle. Boundary value problems are similar to initial value problems. A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value). A boundary
4289517
Verification-based message-passing algorithms in compressed sensing


summation equals exactly formula_62 is zero. Therefore, almost surely the value of formula_51 is zero and all other variables in these equations have value zero. There is just one scenario remained for the second part of the ECN rule as most of it has been covered in the first part. This scenario is the one that we have some equations with the same right hand side but there is two or more variable node common in all of those equations. In this case, we can say nothing about those common variable nodes; however
12503913
Uninitialized variable


In computing, an uninitialized variable is a variable that is declared but is not set to a definite known value before it is used. It will have "some" value, but not a predictable one. As such, it is a programming error and a common source of bugs in software. A common assumption made by novice programmers is that all variables are set to a known value, such as zero, when they are declared. While this is true for many languages, it is not true for all of them
11389446
Elementary mathematics


formula is: An equation is a formula of the form "A" = "B", where "A" and "B" are expressions that may contain one or several variables called unknowns, and "=" denotes the equality binary relation. Although written in the form of proposition, an equation is not a statement that is either true or false, but a problem consisting of finding the values, called solutions, that, when substituted for the unknowns, yield equal values of the expressions "A" and
20150786
BKL singularity


homogeneous models. Taking into account , equations – become The principal equations are defining the γ components; then, function ψ is found by a simple integration of –. The variable ξ runs through the values from 0 to ∞. The solution of is considered at two boundaries, ξ formula_138 1 and formula_57 1. At large ξ values, one can look for a solution that takes the form of a 1 / decomposition: whereby (equation needs condition to be true). Substituting in , one obtains in the first order where
19695823
Level set


"n" = 3, a level set is called a level surface (see also isosurface), and for higher values of "n" the level set is a level hypersurface. So a "level surface" is the set of all real-valued roots of an equation in three variables "x", "x" and "x", and a level hypersurface is the set of all real-valued roots of an equation in "n" ("n"  3) variables. A level set
10053116
Leverage (statistics)


fitted value. Note that this leverage depends on the values of the explanatory (x-) variables of all observations but not on any of the values of the dependent (y-) variables. The equation formula_7follows directly from the computation of the fitted values as formula_8. First, note that "H" is an idempotent matrix: formula_10 Also, observe that formula_11 is symmetric (i.e.: formula_12). So equating the "ii" element of "H" to that of "H" , we have and If we are
17346357
Games World of Puzzles


A second long-running contest was "Calculatrivia", where a long equation, of approximately 40 variables, was given, and each variable was associated with a clue that resulted in some numerical value. The clues were usually straightforward to interpret, but the required information would take considerable research effort to identify. When all variables were accounted for, the equation was to be worked out, and the final value mailed into "Games" along with a list of the individual values. Bygone sister publications of "Games" include
13689779
Deductive lambda calculus


is no contradiction between the two definitions. Let z be a Boolean; then we can form an equation with no solutions, To solve this equation by recursion, we introduce a new function defined by, where is an auxiliary variable to hold the recursion value. (We take it that formula_19 still returns a Boolean even if it is given a non-Boolean argument.) By an eta-reduction, we obtain, And then, Then is neither true or false, and as is a Boolean value (on any
7433926
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, 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
14917547
Tseytin transformation


this circuit be true, one final simple clause is appended: (gate8). Combining these equations results in the final instance of SAT: One possible satisfying assignment of these variables is: The values of the introduced variables are usually discarded, but they can be used to trace the logic path in the original circuit. Here, (x1,x2,x3) = (0,0,1) indeed meets the criteria for the original circuit to output true. To find a different answer, the clause (x1 ∨ x2 ∨ ) can be appended and
679149
Linear equation


each formula_245-tuple makes the equation an identity, when its components are inserted for the respective unknowns. The values of the variables with zero coefficients are taken arbitrarily from the field of coefficients. For an equation to have meaningful solutions, at least one coefficient must be non-zero. This can be formulated as If all coefficients formula_248 equal zero, then, as mentioned for one variable, the equation is either "inconsistent" (for formula_249) and there is no solution, or all The set of solutions of a linear equation
17699107
State-space representation


In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables whose values evolve through time in a way that depends on the values they have at any given time and also depends on the externally imposed values of input variables. Output variables’ values depend on the values of the state variables. The "state space" is the Euclidean space in which the variables on
7899812
State variable


possible combinations of state variable values is called the state space of the system. The equations relating the current state of a system to its most recent input and past states are called the state equations, and the equations expressing the values of the output variables in terms of the state variables and inputs are called the output equations. As shown below, the state equations and output equations for a linear time invariant system can be expressed using coefficient matrices: where "N", "L" and "M" are the dimensions
12277301
Implicational propositional calculus


by considering the case when "B" is false Φ and the case when "B" is true Φ. If the variable which is the final conclusion of a formula takes the value true, then the whole formula takes the value true regardless of the values of the other variables. Consequently if "A" is true, then Φ, Φ, Φ and Φ→(Φ→Φ) are all true. So without loss of generality, we may assume that "A" is false. Notice that Φ is a tautology if and only
6681040
Abstract differential equation


In mathematics, an abstract differential equation is a differential equation in which the unknown function and its derivatives take values in some generic abstract space (a Hilbert space, a Banach space, etc.). Equations of this kind arise e.g. in the study of partial differential equations: if to one of the variables is given a privilegiate position (e.g. time, in heat or wave equations) and all the others are put together, an ordinary "differential" equation with respect to the variable which was put in evidence is obtained
17508750
Diophantine equation


In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is such that all the unknowns take integer values). A linear Diophantine equation equates the sum of two or more monomials, each of degree 1 in one of the variables, to a constant. An exponential Diophantine equation is one in which exponents on terms can be unknowns. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly
9097145
Extraneous and missing solutions


equation, the result is the invalid equation 2 = 0. This counterintuitive result occurs because in the case where "x"=0, multiplying both sides by "x" multiplies both sides by zero, and so necessarily produces a true equation just as in the first example. In general, whenever we multiply both sides of an equation by an expression involving variables, we introduce extraneous solutions wherever that expression is equal to zero. But it is not sufficient to exclude these values, because they may have been legitimate solutions to the original
20235253
Differential-algebraic system of equations


the direction of motion to the tangent of the circle. The next derivative of this equation implies and the derivative of that last identity simplifies to formula_29 which implicitly implies the conservation of energy since after integration the constant formula_30 is the sum of kinetic and potential energy. To obtain unique derivative values for all dependent variables the last equation was three times differentiated. This gives a differentiation index of 3, which is typical for constrained mechanical systems. If initial values formula_31 and a sign for "y" are given, the other variables
2924745
Breusch–Pagan test


Ordinary least squares constrains these so that their mean is 0 and so, given the assumption that their variance does not depend on the independent variables, an estimate of this variance can be obtained from the average of the squared values of the residuals. If the assumption is not held to be true, a simple model might be that the variance is linearly related to independent variables. Such a model can be examined by regressing the squared residuals on the independent variables, using an auxiliary regression equation of the form This is the basis
10703102
Entropic value at risk


formula_3 a formula_4-algebra of subsets of formula_2 and formula_6 a probability measure on formula_3. Let formula_8 be a random variable and formula_9 be the set of all Borel measurable functions formula_10 whose moment-generating function formula_11 exists for all formula_12. The entropic value-at-risk (EVaR) of formula_13 with confidence level formula_14 is defined as follows: In finance, the random variable formula_15 in the above equation, is used to model the "losses" of a portfolio. Consider the Chernoff inequality Solving the equation formula_16 for formula_17 results in
6882603
Polynomial


the form For example, is a polynomial equation. When considering equations, the indeterminates (variables) of polynomials are also called unknowns, and the "solutions" are the possible values of the unknowns for which the equality is true (in general more than one solution may exist). A polynomial equation stands in contrast to a "polynomial identity" like , where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality. In elementary algebra, methods such
3215339
Controllability


in the systems and control literature: The state of a deterministic system, which is the set of values of all the system's state variables (those variables characterized by dynamic equations), completely describes the system at any given time. In particular, no information on the past of a system is needed to help in predicting the future, if the states at the present time are known and all current and future values of the control variables (those whose values can be chosen) are known. "Complete state controllability"
13127222
Initial condition


typically not be different values of the variable "x" at different points in time, but rather the values of "x" and its first "k" – 1 derivatives, all at some point in time such as time zero. The initial conditions do not affect the qualitative nature of the system's behavior. The characteristic equation of this dynamic equation is formula_14 whose solutions are the characteristic values formula_15 these are used in the solution equation This equation and its first "k" – 1derivatives form a system of "k
17617922
Lack-of-fit sum of squares


variable from the average value over all observations sharing its independent variable value(s). These are errors that could never be avoided by any predictive equation that assigned a predicted value for the dependent variable as a function of the value(s) of the independent variable(s). The remainder of the residual sum of squares is attributed to lack of fit of the model since it would be mathematically possible to eliminate these errors entirely. In order for the lack-of-fit sum of squares to differ from the sum of squares of residuals,
20170546
Elementary algebra


are plotted, they describe a straight line. The simplest equations to solve are linear equations that have only one variable. They contain only constant numbers and a single variable without an exponent. As an example, consider: To solve this kind of equation, the technique is add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable
9633082
Network traffic simulation


of variables. If the value of a variable changes, this represents an event, and is reflected in a change in the system’s state. As the system is dynamic, it is constantly changing, and because it is stochastic, there is an element of randomness in the system. Representation of discrete simulations is performed using state equations that contain all the variables influencing the system. Continuous simulations also contain state variables; these however change continuously with time. Continuous simulations are usually modelled using differential equations that track the state of
18434878
Consistent and inconsistent equations


In mathematics and in particular in algebra, a linear or nonlinear system of equations is consistent if there is at least one set of values for the unknowns that satisfies every equation in the system—that is, that when substituted into each of the equations makes each equation hold true as an identity. In contrast, an equation system is inconsistent if there is no set of values for the unknowns that satisfies all of the equations. If a system of equations is inconsistent, then it is possible to manipulate and combine the equations
2417202
Wald's equation


Wald's equation, the resulting value on average is Let be an infinite sequence of real-valued random variables and let be a nonnegative integer-valued random variable. Assume that: Then the random sums are integrable and If, in addition, then Remark: Usually, the name "Wald's equation" refers to this last equality. Clearly, assumption () is needed to formulate assumption () and Wald's equation. Assumption () controls the amount of dependence allowed between the sequence and the number of terms;
11514087
Narrowing of algebraic value sets


the value is not in the value set the probabilities will be zero, So, If the prior probabilities are all the same the probabilities are, A general value set is created out of the application of base value sets. The value condition rule and the probability function may be combined to give, Narrowing allows the elimination of values that do not satisfy a variable's constraints. Considered as the basis for an algorithm for solving equations, this narrowing gives a set of values consistent with the constraints on a variable. However in
15767397
Elliptic boundary value problem


differential equation relating position, tension, angle and gravity, but since the shape does not change over time, there is no time variable. Elliptic boundary value problems are a class of problems which do not involve the time variable, and instead only depend on space variables. It is not possible to discuss elliptic boundary value problems in more detail without referring to calculus in multiple variables. Unless otherwise noted, all facts presented in this article can be found in. In two dimensions, let formula_1 be the coordinates. We will
22061346
Goodman and Kruskal's lambda


In probability theory and statistics, Goodman & Kruskal's lambda (formula_1) is a measure of proportional reduction in error in cross tabulation analysis. For any sample with a nominal independent variable and dependent variable (or ones that can be treated nominally), it indicates the extent to which the modal categories and frequencies for each value of the independent variable differ from the overall modal category and frequency, i.e. for all values of the independent variable together. formula_1 can be calculated with the equation where Values for lambda range from zero (
8922587
Verification-based message-passing algorithms in compressed sensing


variable in all of these equations then there should be one equation formula_56 in which formula_51 exists and formula_52 does not exists. On the other hand, we know that the right hand side of these equations are the same; therefore, the right hand side of equation formula_56 should also be formula_60. If we remove formula_52 from this equation we should have the summation of some unknown variables to be a non-zero value formula_62. Since the non-zero elements of formula_3 are chosen randomly from a continuous distribution the probability that this
12503912
Level set


In mathematics, a level set of a real-valued function "f" of "n" real variables is a set of the form that is, a set where the function takes on a given constant value "c". When the number of variables is two, a level set is generically a curve, called a level curve, contour line, or isoline. So a level curve is the set of all real-valued solutions of an equation in two variables "x" and "x". When
10053115
Verification-based message-passing algorithms in compressed sensing


a continuous distribution. Suppose that there are formula_47 variables in that equation, if some of them in formula_48 elements are non-zero then the other formula_49 variable node value should have exactly the negative value of the summation of those formula_48 variable nodes. If the non-zero elements of the original signal are chosen from a continuous distribution then the probability of this to occur is zero. Therefore, ZCN rule works perfectly. D1CN says that if a variable node is the only unknown variable in an equation then the value of that
12503909
System of linear equations


to take any value, while the remaining variables are dependent on the values of the free variables. For example, consider the following system: The solution set to this system can be described by the following equations: Here "z" is the free variable, while "x" and "y" are dependent on "z". Any point in the solution set can be obtained by first choosing a value for "z", and then computing the corresponding values for "x" and "y". Each
20698279
Median graph


pairs of variables requiring those two variables to avoid certain combinations of values. Usually such problems are expressed in conjunctive normal form, in which each clause is expressed as a disjunction and the whole set of constraints is expressed as a conjunction of clauses, such as A solution to such an instance is an assignment of truth values to the variables that satisfies all the clauses, or equivalently that causes the conjunctive normal form expression for the instance to become true when the variable values are substituted into it. The family of all solutions has
148703
Matrix differential equation


, because our differential equations deal with two variables. The equation which involves all the pieces of information that we have previously found has the following form: Substituting the values of eigenvalues and eigenvectors yields Applying further simplification, Simplifying further and writing the equations for functions formula_61 and formula_62 separately, The above equations are, in fact, the general functions sought, but they are in their general form (with unspecified values of "A" and "B"), whilst we want to actually find their exact forms and solutions.
2798713
Cauchy–Riemann equations


z", denoted formula_36, is defined by for real "x" and "y". The Cauchy–Riemann equations can then be written as a single equation by using the Wirtinger derivative with respect to the conjugate variable. In this form, the Cauchy–Riemann equations can be interpreted as the statement that "f" is independent of the variable formula_36. As such, we can view analytic functions as true functions of "one" complex variable as opposed to complex functions of "two" real variables. A
7745461
Continuous or discrete variable


probability distributions of continuous variables can be expressed in terms of probability density functions. In continuous-time dynamics, the variable "time" is treated as continuous, and the equation describing the evolution of some variable over time is a differential equation. The instantaneous rate of change is a well-defined concept. In contrast, a discrete variable over a particular range of real values is one for which, for any value in the range that the variable is permitted to take on, there is a positive minimum distance to the
18794152
Euler equations (fluid dynamics)


, multiplying the original 1-D equation in flux-Jacobian form with P yields the characteristic equations: The original equations have been decoupled into N+2 characteristic equations each describing a simple wave, with the eigenvalues being the wave speeds. The variables "w" are called the "characteristic variables" and are a subset of the conservative variables. The solution of the initial value problem in terms of characteristic variables is finally very simple. In one spatial dimension it is: Then the solution in terms of the original conservative variables is obtained by
14033060
Linear cryptanalysis


a high bias; that is, whose probabilities of holding (over the space of all possible values of their variables) are as close as possible to 0 or 1. The second is to use these linear equations in conjunction with known plaintext-ciphertext pairs to derive key bits. For the purposes of linear cryptanalysis, a linear equation expresses the equality of two expressions which consist of binary variables combined with the exclusive-or (XOR) operation. For example, the following equation, from a hypothetical cipher, states the
14328109
EcosimPro


the following differential equation to introduce a delay to variable "x": which is equivalent to where "x" and "y" have a time dependence that will be defined in the experiment. "Tau" is datum provided given by the user; we will use a value of 0.6 seconds. This equation introduces a delay in the "x" variable with respect to "y" with value "tau". To simulate this equation we will create an EcosimPro component with the equation in it. The component to
20156702
Wald's equation


is independent of the integrable, real-valued random variable with . Define for all . Then assumptions (), (), (), and () with are satisfied, hence also () and (), and Wald's equation applies. If the distribution of is not symmetric, then () does not hold. Note that, when is not almost surely equal to the zero random variable, then () and () cannot hold simultaneously for any filtration , because cannot be independent of itself
11514092
Vector autoregression


as the following system of two equations Each variable in the model has one equation. The current (time "t") observation of each variable depends on its own lagged values as well as on the lagged values of each other variable in the VAR. A VAR with "p" lags can always be equivalently rewritten as a VAR with only one lag by appropriately redefining the dependent variable. The transformation amounts to stacking the lags of the VAR("p") variable in the new VAR(1) dependent variable and appending identities to complete
13020551
Variable (mathematics)


In elementary mathematics, a variable is a symbol, commonly a single letter, 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 for the variables that represent them
17766913
Curry's paradox


Stephen Cole Kleene and then Haskell Curry discovered with combinatory logic and Lambda calculus. The situation may be compared to defining This definition is fine as long as only positive values are allowed for the square root. In mathematics an existentially quantified variable may represent multiple values, but only one at a time. Existential quantification is the disjunction of many instances of an equation. In each equation is one value for the variable. However, in mathematics, an expression with no free variables must have one and only one value. So formula_29
20101709
Linear equation


formula_215 The motion in the formula_12-direction can be described as and similarly, the motion in the formula_13-direction can be described as These two linear equations, with variables formula_220 and formula_221, make up a "parametric form" for a linear equation with variables formula_18 that can be constructed from the two-point form with formula_208 and formula_209 as points. For formula_225 and for formula_226 For all formula_227 in the interval formula_228 the point formula_229 is on the straight line segment connecting the points for formula_207 and formula_210 This is sometimes called interpolation. For values
17699103
Lambert W function


converge), the principal branch of the Lambert function can be computed by the following integral: where the two integral expressions are equivalent due to the symmetry of the integrand. Many equations involving exponentials can be solved using the function. The general strategy is to move all instances of the unknown to one side of the equation and make it look like at which point the function provides the value of the variable in . In other words: More generally, the equation where can be transformed via the substitution of variables into giving which
9039181
Expectation–maximization algorithm


and simultaneously solving the resulting equations. In statistical models with latent variables, this is usually impossible. Instead, the result is typically a set of interlocking equations in which the solution to the parameters requires the values of the latent variables and vice versa, but substituting one set of equations into the other produces an unsolvable equation. The EM algorithm proceeds from the observation that there is a way to solve these two sets of equations numerically. One can simply pick arbitrary values for one of the two sets of unknowns, use them
12309291
Truth table


a proposition, that produces a value of "true" if its operand is false and a value of "false" if its operand is true. The truth table for NOT p (also written as ¬p, Np, Fpq, or ~p) is as follows: There are 16 possible truth functions of two binary variables: Here is an extended truth table giving definitions of all possible truth functions of two Boolean variables P and Q: where The four combinations of input values for p, q, are read by row
17512407
Klein–Gordon equation


with mass parameter formula_1 is Solutions of the equation are complex-valued functions formula_3 of the time variable formula_4 and space variables formula_5; the Laplacian formula_6 acts on the space variables only. The equation is often abbreviated as where , and is the d'Alembert operator, defined by The Klein–Gordon equation is often written in natural units: The form of the Klein–Gordon equation is derived by requiring that plane-wave solutions of the equation obey the energy–momentum relation of special relativity: Unlike the Schrödinger equation, the Klein
753548
Thermochemical equation


concept commonly known as the conservation of energy. It holds true on a physical or molecular scale. Thermochemical equations can be changed, as mentioned above, by multiplying by any numerical coefficient. All agents must be multiplied, including ΔH. Using the thermochemical equation of variables as above, one gets the following example. Assume that one needs to multiply "A" by two in order to use the thermochemical equation (as in addition, below). One must then multiply all the agents in the reaction by the same coefficient
9060744
Linear equation


(formula_91). This shows that only two of formula_92 and formula_80 can be selected independently. With the premise that at least one axis is intersected, and since both intercept values may range over the whole real number line, all parallels to both axes as well as all oblique straight lines, i.e., in fact all straight lines in the Euclidean plane, can be expressed by linear equations in two variables, and all such equations denote either oblique or axis-parallel straight lines. Therefore all equations, equivalent to one of
17699088
Vector field reconstruction


Vector field reconstruction is a method of creating a vector field from experimental or computer generated data, usually with the goal of finding a differential equation model of the system. A differential equation model is one that describes the value of dependent variables as they evolve in time or space by giving equations involving those variables and their derivatives with respect to some independent variables, usually time and/or space. An ordinary differential equation is one in which the system's dependent variables are functions of only one independent variable. Many physical, chemical, biological
15995489
Labor theory of value


commodity of variable use. Finally, the value added during the portion of the period when surplus labor is performed is called surplus value (formula_13). From the variables defined above, we find two other common expressions for the value produced during a given period: The first form of the equation expresses the value resulting from production, focusing on the costs formula_16 and the surplus value appropriated in the process of production, formula_13. The second form of the equation focuses on the value of production in terms of the values added by
17699488
Vector autoregression


Vector autoregression (VAR) is a stochastic process model used to capture the linear interdependencies among multiple time series. VAR models generalize the univariate autoregressive model (AR model) by allowing for more than one evolving variable. All variables in a VAR enter the model in the same way: each variable has an equation explaining its evolution based on its own lagged values, the lagged values of the other model variables, and an error term. VAR modeling does not require as much knowledge about the forces influencing a variable as do structural
13020546
Validity (logic)


valid and all the premises true. Model theory analyzes formulae with respect to particular classes of interpretation in suitable mathematical structures. On this reading, formula is valid if all such interpretations make it true. An inference is valid if all interpretations that validate the premises validate the conclusion. This is known as "semantic validity". In truth-preserving validity, the interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true'. In a false-preserving validity, the
7931625
Extraneous and missing solutions


. One of the basic principles of algebra is that one can multiply both sides of an equation by the same expression without changing the equation's solutions. However, strictly speaking, this is not true, in that multiplication by certain expressions may introduce new solutions that were not present before. For example, consider the following equation: If we multiply both sides by zero, we get This is true for all values of "x", so the solution set is all real numbers. But clearly not all real numbers are
20235251
Lambda lifting


subtle in the way that it works. The same value indicator is never set to true. It is only set to false if all the values cannot be matched. The value is retrieved by using "S" to build a set of the Boolean values allowed for "S". If true is a member then all the values for this parameter are equal, and the parameter may be dropped. Similarly, "def" uses set theory to query if a variable has been given a value; Let - Let
19473381
Causality


relation is unrelated to the asymmetry of any mode of implication that contraposes. Rather, a causal relation is not a relation between values of variables, but a function of one variable (the cause) on to another (the effect). So, given a system of equations, and a set of variables appearing in these equations, we can introduce an asymmetric relation among individual equations and variables that corresponds perfectly to our commonsense notion of a causal ordering. The system of equations must have certain properties, most importantly, if
16540413
Integral equation


to solve numerically requires discretizing variables and replacing integral by a quadrature rule Then we have a system with equations and variables. By solving it we get the value of the variables Integral equations are classified according to three different dichotomies, creating eight different kinds: Integral equations are important in many applications. Problems in which integral equations are encountered include radiative transfer, and the oscillation of a string, membrane, or axle. Oscillation problems may also be solved as differential equations. Both Fredholm and Volterra equations are linear integral equations, due
13367548
Tseytin transformation


true when at least some of its inputs are true, but not more than two at a time. It implements the equation . A variable is introduced for each gates' output; here each is marked in red: Notice that the output of the inverter with x as an input has two variables introduced. While this is redundant, it does not affect the equisatisfiability of the resulting equation. Now substitute each gate with its appropriate CNF sub-expression: The final output variable is gate8 so to enforce that the output of
679148
Sum of perpetuities method


It can therefore be valued using the present value of a perpetuity equation: br br br br br Substituting formula_30 for formula_19 in the equation above produces the Walter model: br And given by relationship II, formula_26 is equal to formula_34. Substituting the term, formula_34 into the above equation produces the SPM constant growth valuation model: The SPM equation requires that all variables be held constant over time which may be unreasonable in many cases. These include the assumption of constant earnings and/or dividend growth, an unchanging dividend policy, and a
6416207
Paraconsistent logic


not-true ("f") conclusion and a true ("t" or "b") hypothesis yield a not-true implication. If all the propositional variables in Γ are assigned the value "b", then Γ itself will have the value "b". If we give "X" the value "f", then So Γ→"X" will not be a tautology. Limitations: (1) There must not be constants for the truth values because that would defeat the purpose of paraconsistent logic
150966
Maxwell's equations


(apart from the initial conditions), the imperfect precision of the calculations can lead to ever-increasing violations of those laws. By introducing dummy variables characterizing these violations, the four equations become not overdetermined after all. The resulting formulation can lead to more accurate algorithms that take all four laws into account. Both identities formula_37, which reduce eight equations to six independent ones, are the true reason of overdetermination. Maxwell's equations and the Lorentz force law (along with the rest of classical electromagnetism) are extraordinarily successful at
20161190
Board puzzles with algebra of binary variables


puzzle. Below the letters in the mathematical statements are used as variables where each can take the value either 0 or 1 only. A simple example of an equation with binary variables is given below: Here there are two variables a and b but one equation. The solution is constrained by the fact that a and b can take only values 0 or 1. There is only one solution here, both a = 0, and b = 0. Another simple example is given below: The solution is straightforward: a and
2122918
Tseytin transformation


meaning that it is satisfiable if, and only if, the original input equation is satisfiable. When a satisfying assignment of variables is found, those assignments for the introduced variables can simply be discarded. A final clause is appended with a single literal: the final gate's output variable. If this literal is complemented, then the satisfaction of this clause enforces the output expression's to false; otherwise the expression is forced true. Consider the following formula formula_1 . Consider all subformulas (without variables): Introduce a new variable
679146
Implicational propositional calculus


. It is worth noting that "F" is not definable from → and arbitrary sentence variables: any formula constructed from → and propositional variables must receive the value true when all of its variables are evaluated to true. It follows as a corollary that {→} is not functionally complete. It cannot, for example, be used to define the two-place truth function that always returns "false". The following statements are considered tautologies (irreducible and intuitively true, by definition). Where in each case
6681021
Initial condition


"n"-dimensional coordinate vector), generally "nk" initial conditions are needed in order to trace the system's variables forward through time. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. A corresponding problem exists for discrete time situations.
17617916
Distributed lag


to be estimated, and "w" is called the lag weight (also to be estimated) placed on the value "i" periods previously of the explanatory variable "x". In the first equation, the dependent variable is assumed to be affected by values of the independent variable arbitrarily far in the past, so the number of lag weights is infinite and the model is called an "infinite distributed lag model". In the alternative, second, equation, there are only a finite number of lag weights,
14500980
Nomogram


a parallel coordinate system invented by d'Ocagne rather than standard Cartesian coordinates. A nomogram consists of a set of n scales, one for each variable in an equation. Knowing the values of n-1 variables, the value of the unknown variable can be found, or by fixing the values of some variables, the relationship between the unfixed ones can be studied. The result is obtained by laying a straightedge across the known values on the scales and reading the unknown value from where it crosses the scale for that variable. The virtual or
277704
Particle in a one-dimensional lattice


In the previous paragraph, the only variables not determined by the parameters of the physical system are the energy E and the crystal momentum k. If you pick a value of E, you can compute the right hand side, and then compute k by taking the formula_30 of both sides. Thus, the expression gives rise to the dispersion relation. The right hand side of the last expression above can sometimes be greater than 1 or less than -1, in which case there is no value of k that can make the equation true
10023211
System of linear equations


. The vector equation is equivalent to a matrix equation of the form where "A" is an "m"×"n" matrix, x is a column vector with "n" entries, and b is a column vector with "m" entries. The number of vectors in a basis for the span is now expressed as the "rank" of the matrix. A solution of a linear system is an assignment of values to the variables such that each of the equations is satisfied. The set of all possible solutions is called
20698265
Simultaneous equations model


uncorrelated with u's and y's are endogenous variables. Without further restrictions, the first equation is not identified because there is no excluded exogenous variable. The second equation is just identified if δ≠0, which is assumed to be true for the rest of discussion. Now we impose the cross equation restriction of δ=δ. Since the second equation is identified, we can treat δ as known for the purpose of identification. Then, the first equation becomes: y - δ z = γ y + δ z + δ
13917726
Curry (programming language)


search for solutions for existentially quantified variables. In contrast to pure logic languages, they support equation solving over nested functional expressions so that an equation like yse = is solved by instantiating ys to the list and e to the value . In Curry one can define the operation last as follows: Here, the symbol “” is used for equational constraints in order to provide a syntactic distinction from defining equations. Similarly, extra variables (i.e., variables not occurring in the left-hand side of the defining equation) are explicitly
8454701
Differential-algebraic system of equations


In mathematics, a differential-algebraic system of equations (DAEs) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. Such systems occur as the general form of (systems of) differential equations for vector–valued functions "x" in one independent variable "t", where formula_2 is a vector of dependent variables formula_3 and the system has as many equations, formula_4. They are distinct from ordinary differential equation (ODE) in that a DAE is not
2924738
Working–Hotelling procedure


resulting confidence bands are known as the Working–Hotelling–Scheffé confidence bands. Like the closely related Scheffé's method in the analysis of variance, which considers all possible contrasts, the Working–Hotelling procedure considers all possible values of the independent variables; that is, in a particular regression model, the probability that all the Working–Hotelling confidence intervals cover the true value of the mean response is the confidence coefficient. As such, when only a small subset of the possible values of the independent variable is considered, it
8179353
Inverse scattering transform


of two real variables, one space variable "x" and one time variable "t" : with formula_3 and formula_4 denoting partial derivatives with respect to "t" and "x", respectively. To solve the initial value problem for this equation where formula_5 is a known function of "x", one associates to this equation the Schrödinger eigenvalue equation where formula_7 is an unknown function of "t" and "x" and "u" is the solution of the Korteweg–de Vries equation that is unknown except
13636946