If P is a program which outputs a string x, then P is a description of x.

If M is a Turing Machine which, on input w, outputs string x, then the concatenated string < M > w is a description of x.

If a description of s, d ( s ), is of minimal length ( i. e. it uses the fewest number of characters ), it is called a minimal description of s. Thus, the length of d ( s ) ( i. e. the number of characters in the description ) is the Kolmogorov complexity of s, written K ( s ).

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

If we take the simple molecular orbital description of the ground state and combine that function with the functions describing all possible excited states using unoccupied orbitals arising from the same set of atomic orbitals, we also reach the full configuration interaction wavefunction.

If they express much more than this, they are usually considered to be a hardware description language such as Verilog, VHDL, or any one of several specific languages designed for input to simulators.

The algorithm for deciding this is conceptually simple: it constructs ( the description of ) a new program t taking an argument n which ( 1 ) first executes program a on input i ( both a and i being hard-coded into the definition of t ), and ( 2 ) then returns the square of n. If a ( i ) runs forever, then t will never get to step ( 2 ), regardless of n. Then clearly, t is a function for computing squares if and only if step ( 1 ) terminates.

If no time range is given, use the start or end time provided in the presentation description.

If a new media stream is added to a presentation ( e. g., during a live presentation ), the whole presentation description should be sent again, rather than just the additional components, so that components can be deleted.

If the check in operation succeeds, then the version numbers of all files involved automatically increment, and the CVS-server writes a user-supplied description line, the date and the author's name to its log files.

If a job description is not available, then a systems analysis can be done to draw up a job description.

If the monks did hide the stone, they hid it well ; no other stone fitting its description has ever been found.

If one allows arbitrary temperature dependence of the prefactor, the Arrhenius description becomes overcomplete, and the inverse problem ( i. e., determining the prefactor and activation energy from experimental data ) becomes singular.

If you hear anything we've done in the last five or six years, that description is totally irrelevant ...

If dictionaries were logico-mathematical texts, then so-called circular definition would amount to infinite regress, where one of the steps involved in running the procedure is to run the procedure ; and, in the context of explanation ( as opposed to description in the form of dictionary definition ), this would be a vicious infinite regress.

If this machine U is supplied with a tape on the beginning of which is written the S. D description " of an action table of some computing machine M, then U will compute the same sequence as M ."

If a system has a probabilistic description, this description gives the probability of any configuration, and given any two different configurations, there is a state which is partly this and partly that, with positive real number coefficients, the probabilities, which say how much of each there is.

If, for example, the 43 letters long ( ignoring the spaces ) description " not divisible by any integer other than 1 and itself " were assigned to the number 43, then this would be true.

But let us have done with British-Americans and Irish-Americans and German-Americans, and so on, and all be Americans ... If a man is going to be an American at all let him be so without any qualifying adjectives ; and if he is going to be something else, let him drop the word American from his personal description.

If H < sub > f </ sub > is in this time complexity class, it means we can construct some machine K which, given some machine description and input x, decides whether the tuple (, x ) is in H < sub > f </ sub > within.

Almost all of her dialogue prior to line 32 (" If thou be he, then thou art prisoner ") is delivered direct to camera, including her incredulous description of the difference between the real Talbot, and the reports she has heard of him.

If we cannot stop warfare in our own economic system, how can we expect to abolish it internationally??

If your house is to have a forced warm air system, cooling can be a part of it.

If you plan to add cooling later to your heating system, there are things to watch for.

If the problems of combining cooling with your heating are knotty, it may be cheaper to plan on a completely separate cooling system.

If you use one of the new year-round cooling system fluids such as `` Dowguard '' be sure to check it.

If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.

If tyrosine and a system generating hydrogen peroxide are added to a cell-free homogenate of the thyroid, large quantities of free mono-iodotyrosine can be formed ( Alexander, 1959 ).

If the house is not wired adequately for electricity or if plumbing or a central heating system must be installed, check into the cost of making these improvements.

If laborers are merely commodities competing against each other in a market place like so many bags of wheat and corn ( unsupported, by the way, by any agricultural subsidy ), then they may be pardoned for reacting with complete antagonism to a system that imposes such status upon them.

If the demands of these two sovereigns upon his duty of allegiance come into conflict, those of the United States have the paramount authority in American law ; likewise, those of the foreign land have paramount authority in their legal system.

If we use a cylindrical coordinate system with basis vectors, then the gradient of and the divergence of are given by

If under the existing system he could not assemble forces quickly enough to intercept mobile Viking raiders, the obvious answer was to have a standing field force.

If administrative appeal is available, no appeal to the judicial system may be made.

If the system be entirely behind the aperture stop, then this is itself the entrance pupil ( front stop ); if entirely in front, it is the exit pupil ( back stop ).

If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their perpendicular height of incidence, i. e. their distance from the axis.

If a system consists of several particles, the total angular momentum about a point can be obtained by adding ( or integrating ) all the angular momenta of the constituent particles.

If not, the system may either be rejected or accepted on conditions previously agreed between the sponsor and the manufacturer.

If the pressure in a system remains constant ( isobaric ), a vapor at saturation temperature will begin to condense into its liquid phase as thermal energy ( heat ) is removed.

If a newer system is attempting to achieve integration with an older system which has known flaws ( or " bugs "), then the new system may be referred to as " bugwards-compatible ".

If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.

" That rich island ," he wrote on 1 December 1881, " the key to the Gulf of Mexico, is, though in the hands of Spain, a part of the American commercial system … If ever ceasing to be Spanish, Cuba must necessarily become American and not fall under any other European domination.