* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.

Instead, in order to solve such a puzzle, the solver must find a solution that satisfies the given conditions.

A fundamental solution of Laplace's equation satisfies

A Green's function is a fundamental solution that also satisfies a suitable condition on the boundary S of a volume V. For instance, may satisfy

* A solution is found that satisfies minimum criteria

A typical problem for Laplace's equation is to find a solution that satisfies arbitrary values on the boundary of a domain.

" In this usage the satisficing solution satisfies some criteria and sacrifices others.

The purpose of testing is to achieve organizational acceptance that the solution satisfies the recovery requirements.

The solved, but simplified problem is then " perturbed " to make the conditions that the perturbed solution actually satisfies closer to the real problem, such as including the gravitational attraction of a third body ( the Sun ).

In computer science, brute-force search or exhaustive search, also known as generate and test, is a trivial but very general problem-solving technique that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problem's statement.

As long as the total force is nonzero, this equation has a unique solution, and it satisfies the torque requirement.

Aside from insights that such constants of motion give into the nature of a system, they are a useful calculational tool ; for example, an approximate solution can be corrected by finding the nearest state that satisfies the suitable conservation laws.

A solution is an evaluation that satisfies all constraints.

A solution to a system of equations is a particular specification of the values of all variables that simultaneously satisfies all of the equations.

There is one and only one solution u ( x ) which satisfies

# The boolean satisfiability problem, in which a candidate solution is a truth assignment, and the target is to maximize the number of clauses satisfied by the assignment ; in this case, the final solution is of use only if it satisfies all clauses

# The nurse scheduling problem where a solution is an assignment of nurses to shifts which satisfies all established constraints

Instead, to solve such a puzzle, the solver must find a solution that satisfies the given conditions.

Tabu search can be used to find a satisficing solution for the traveling salesman problem ( that is, a solution that satisfies an adequacy criterion, rather than the absolutely optimal solution ).

Actually, is the only solution of the functional equation that is monotone on and satisfies.

Assuming that all the particles start from the origin at the initial time t = 0, the diffusion equation has the solution

When Newton's method is used to find the solution for the maximum likelihood estimate, the middle 24 % order statistics can be used as an initial solution for x < sub > 0 </ sub >.

In the problems of finding the root of an equation ( or a solution of a system of equations ), an iterative method uses an initial guess to generate successive approximations to a solution.

* the adiabatic quantum computer ( computation decomposed into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contains the solution ),

In the solution, c < sub > 1 </ sub > and c < sub > 2 </ sub > are two constants determined by the initial conditions, and the origin is set to be the equilibrium position.

The NF operator can also be applied on an initial solution obtained by NN algorithm for further improvement in an elitist model, where better solutions are only accepted.

The equation alone does not specify a solution ; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the value and velocity of the wave.

In the theory of differential equations, Lipschitz continuity is the central condition of the Picard – Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem.

This formula implies that the solution at ( t, x ) depends only upon the data on the segment of the initial line that is cut out by the characteristic curves

The idea of using an initial property is to set up the problem in terms of some auxiliary category E, and then identify that what we want is to find an initial object of E. This has an advantage that the optimization — the sense that we are finding the most efficient solution — means something rigorous and is recognisable, rather like the attainment of a supremum.

Therefore, the assertion that an object R → R * is initial in E, that is, that there is a morphism from it to any other element of E, means that the ring R * is a most efficient solution to our problem.

The particle motion is described by a solution to in dependence on initial condition, with the phase of.

A solution was proposed by Andreas Hinz, and is based on the observation that in a shortest sequence of moves, the largest disk that needs to be moved ( obviously one may ignore all of the largest disks that will occupy the same peg in both the initial and final configurations ) will move either exactly once or exactly twice.

where the ψ ( r ) cancels, so solving this equation for implies the solution of the time-dependent equation with this initial condition is:

The differential equation is a general description of the application and may be adjusted appropriately for a specific situation, the solution describes exactly how the system will behave for all times after the initial conditions, and according to the boundary conditions.

The solution r ( or p ) to the equation of motion, combined with the initial values, describes of the system for all times after t = 0.

* Algorithms for finding the solution to a NLLSQ problem require initial values for the parameters, LLSQ does not.

It has a unique solution, given an initial position and an initial velocity.

The solution proposed was that once the initial residents had moved in their families would be given priority for new housing when it became available.

Variations in the character of the solution or in the conditions of deposition may cause a corresponding variation in the successive layers, so that bands of chalcedony often alternate with layers of crystalline quartz.

This means that they were at first fragmental rocks like limestone, shale and sandstone and have never been in a molten condition nor entirely in solution, but the high temperature and pressure conditions of metamorphism have acted on them by erasing their original structures and inducing recrystallization in the solid state.

These range from simplified forms of the first-principles equations that are easier or faster to solve, to approximations limiting the size of the system ( for example, periodic boundary conditions ), to fundamental approximations to the underlying equations that are required to achieve any solution to them at all.

The solution sought is then separated from the remaining six based on physical conditions.

For example, if an electric field is placed across a solution of Na < sup >+</ sup > and Cl < sup >−</ sup > ( and conditions are right ) the sodium ions move towards the negative electrode ( cathode ), while the chloride ions move towards the positive electrode ( anode ).

It is also useful for determining the tertiary structure and quaternary structure of purified proteins, especially since it can be carried out under native solution conditions.

If < math > c ^ 2 < 4km </ math > there are two complex conjugate roots a ± ib, and the solution ( with the above boundary conditions ) will look like this:

From 1950 to 1955, Simon studied mathematical economics and during this time, together with David Hawkins, discovered and proved the Hawkins – Simon theorem on the “ conditions for the existence of positive solution vectors for input-output matrices.

FARC-EP remains open to a negotiated solution to the nation's conflict through dialogue with a flexible government that agrees to certain conditions, such as the demilitarization of certain areas, cessation of paramilitary and government violence against rural peasants, social reforms to reduce poverty and inequality, and the release of all jailed ( and extradited ) FARC-EP rebels.

The solution is not to force migrant workers into ghettos, but to urge companies to improve living conditions for workers – and not to accommodate large numbers of workers in inadequate space, and to improve the standard of living for them.

Stoichiometry calculations can predict how elements and components diluted in a standard solution react in experimental conditions.

In aqueous solution, tungstate gives the heteropoly acids and polyoxometalate anions under neutral and acidic conditions.

Titration, by definition, is the determination of rank or concentration of a solution with respect to water with a pH of 7 ( the pH of pure H < sub > 2 </ sub > O under standard conditions ).

For example, proteins and larger RNA molecules cannot be crystallized if their tertiary structure has been unfolded ; therefore, the range of crystallization conditions is restricted to solution conditions in which such molecules remain folded.

The solution conditions that favor the first step ( nucleation ) are not always the same conditions that favor the second step ( subsequent growth ).

The crystallographer's goal is to identify solution conditions that favor the development of a single, large crystal, since larger crystals offer improved resolution of the molecule.