Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals of one another, or more generally are related through Pontryagin duality.

simple: Conjugate variables

# REDIRECT Conjugate variables

# REDIRECT Conjugate variables

* Conjugate ( algebra ), used to rationalize the denominator of a fraction

* Conjugate ( graph theory ), an alternative term for a line graph

* Conjugate element ( field theory ), a generalization of complex conjugation

Conjugate harmonic functions ( and the transform between them ) are also one of the simplest examples of a Bäcklund transform ( two PDEs and a transform relating their solutions ), in this case linear ; more complex transforms are of interest in solitons and integrable systems.

The best known Krylov subspace methods are the Arnoldi, Lanczos, Conjugate gradient, GMRES ( generalized minimum residual ), BiCGSTAB ( biconjugate gradient stabilized ), QMR ( quasi minimal residual ), TFQMR ( transpose-free QMR ), and MINRES ( minimal residual ) methods.

Conjugate gradient, assuming exact arithmetic, converges in at most n steps where n is the size of the matrix of the system ( here n = 2 ).

In physics and thermodynamics, an equation of state is a relation between state variables.

Relatively simple approximate calculations, however, using the variables of equilibrium thermodynamics, are of much practical value in engineering.

The macroscopic state variables of thermodynamics have been recognized in the course of empirical work in physics and chemistry.

Classical thermodynamics can consider irreversible processes, but its account in exact terms is restricted to variables that refer only to initial and final states of thermodynamic equilibrium, or to rates of input and output that do not change with time.

The independent state variables of a small local region are only those of classical thermodynamics.

The independent state variables of a small local region include flow rates, which are not admitted as independent variables for the small local regions of local equilibrium thermodynamics.

Even within the thought-frame of classical irreversible thermodynamics, care is needed in choosing the independent variables for systems.

In some writings, it is assumed that the intensive variables of equilibrium thermodynamics are sufficient as the independent variables for the task ( such variables are considered to have no ' memory ', and do not show hysteresis ); in particular, local flow intensive variables are not admitted as independent variables ; local flows are considered as dependent on quasi-static local intensive variables.

The second law defines entropy, which may be viewed not only as a macroscopic variable of classical thermodynamics, but may also be viewed as a measure of deficiency of physical information about the microscopic details of the motion and configuration of the system, given only predictable experimental reproducibility of bulk or macroscopic behavior as specified by macroscopic variables that allow the distinction to be made between heat and work.

In chemistry, thermodynamics, and other chemical engineering, a steady state is a situation in which all state variables are constant in spite of ongoing processes that strive to change them.

Critical variables are defined, for example in thermodynamics, in terms of the values of variables at the critical point.

* Principal component analysis – transformation of a sample of correlated variables into uncorrelated variables ( called principal components ), mostly used in exploratory data analysis

In computer science, an array data structure or simply an array is a data structure consisting of a collection of elements ( values or variables ), each identified by at least one array index or key.

The problem remains NP-complete even if all expressions are written in conjunctive normal form with 3 variables per clause ( 3-CNF ), yielding the 3SAT problem.

E has two clauses ( denoted by parentheses ), four variables ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, x < sub > 3 </ sub >, x < sub > 4 </ sub >), and k = 3 ( three literals per clause ).

Once the electronic and nuclear variables are separated ( within the Born – Oppenheimer representation ), in the time-dependent approach, the wave packet corresponding to the nuclear degrees of freedom is propagated via the time evolution operator ( physics ) associated to the time-dependent Schrödinger equation ( for the full molecular Hamiltonian ).

Typically, these include all the objects referenced from anywhere in the call stack ( that is, all local variables and parameters in the functions currently being invoked ), and any global variables.

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant ( i. e., elements of their transfer function C ( s ), P ( s ), and F ( s ) do not depend on time ), the systems above can be analysed using the Laplace transform on the variables.

We may solve this by separation of variables ( moving the y terms to one side and the t terms to the other side ),

Whereas arithmetic deals with specific numbers and operators ( e. g. 3 + 2 = 5 ), algebra introduces variables, which are letters that represent non-specified numbers ( e. g. 3a + 2a = 5a ).

1 – Exponent ( power ), 2 – Coefficient, 3 – term, 4 – operator, 5 – constant, – variables

If I told you my son's age, then there would no longer be two unknowns ( variables ), and the problem becomes a linear equation with just one variable, that can be solved as described above.

To solve a linear equation with two variables ( unknowns ), requires two related equations.

In the EPR paper ( 1935 ) the authors realised that quantum mechanics was inconsistent with their assumptions, but Einstein nevertheless thought that quantum mechanics might simply be augmented by hidden variables ( i. e. variables which were, at that point, still obscure to him ), without any other change, to achieve an acceptable theory.

Critical variables affecting this acoustic impedance include: chimney length ( hole between lip-plate and head tube ), chimney diameter, and radii or curvature of the ends of the chimney and any designed restriction in the " throat " of the instrument, such as that in the Japanese Nohkan Flute.

To achieve this for a generic φ ( subject to restrictions we have already proved ), we take some one-place relation symbol F unused in φ, and two new variables y and z ..

Other methods include server side sessions, hidden variables ( when the current page contains a form ), and URL-rewriting using URI-encoded parameters, e. g., < tt >/ index. php? session_id = some_unique_session_code </ tt >.

However, each of these variables has its own stack on which it can be pushed and popped ( STASHed and RETRIEVEd, in INTERCAL terminology ), increasing the possible complexity of data structures.

However, iterative methods are often useful even for linear problems involving a large number of variables ( sometimes of the order of millions ), where direct methods would be prohibitively expensive ( and in some cases impossible ) even with the best available computing power.

It is normally written as not ( p ), where p is an atom whose variables normally have been instantiated by the time not ( p ) is invoked.

Lacking reliable data for some of the variables, we are relying on experimental data of about 20 percent internal losses for 1/4-inch long, small ( 5 - 10 M ) diameter fibers.

The user can monitor and modify the values of programs ' internal variables, and even call functions independently of the program's normal behavior.

( Since the sub-structure is usually represented in the computer by a set of integer variables with constraints, these problems can be viewed as special cases of constraint satisfaction or discrete optimization ; but they are usually formulated and solved in a more abstract setting where the internal representation is not explicitly mentioned.

It defines macroscopic variables ( such as temperature, internal energy, entropy, and pressure ) that describe average properties of material bodies and radiation, and explains how they are related and by what laws they change with time.

In 1875, in the work of Josiah Willard Gibbs, the basic thermodynamic quantities were energy potentials, such as internal energy, as dependent variables, and entropy, considered as a fundamental independent variable.

One of the methods of dealing with non-equilibrium systems is to introduce so-called ' internal variables '.

In contrast to the usual thermodynamic variables, ' internal variables ' cannot be controlled by external manipulations.

It is important that internal energy is a variable of state of the system ( see Thermodynamic state ) whereas heat and work are variables that describe processes or changes of the state of systems.

Gibbs's papers from the 1870s introduced the idea of expressing the internal energy U of a system in terms of the entropy S, in addition to the usual state variables of V ( volume ), p ( pressure ), and T ( temperature ).

Seed production in natural plant populations vary widely from year-to-year in response to weather variables, insects and diseases, and internal cycles within the plants themselves.

Let the linear, angular and internal momenta of a molecule be given by the set of r variables

Hewitt reported: "... we have found that we can do without the paraphernalia of " hairy control structure " ( such as possibility lists, non-local gotos, and assignments of values to the internal variables of other procedures in CONNIVER .)...

Shell variables, created using the set or @ statements, are internal to C shell.

Some of the shell variables are predefined and used to control various internal C shell options, e. g., what should happen if a wildcard fails to match anything.

Throughout the course of the fictional limiting quasi-static process, the internal intensive variables of the system are equal to the external intensive variables, those which describe the reactive forces exerted by the surroundings.

Compilers may also save more information in the recursion stack than is strictly necessary, such as return address, unchanging parameters, and the internal variables of the procedure.

Thus, the risk of stack overflow can be reduced by minimizing the parameters and internal variables of the recursive procedure, and / or by using an explicit stack structure.

The Gibbs free energy total differential natural variables may be derived via Legendre transforms of the internal energy.

On first glance, internal and external validity seem to contradict each other-to get an experimental design you have to control for all interfering variables.

While gaining internal validity ( excluding interfering variables by keeping them constant ) you lose ecological or external validity because you establish an artificial laboratory setting.

On the other hand with observational research you can't control for interfering variables ( low internal validity ) but you can measure in the natural ( ecological ) environment, at the place where behavior normally occurs.

Patches have a definite shape and spatial configuration, and can be described compositionally by internal variables such as number of trees, number of tree species, height of trees, or other similar measurements.