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By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.
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By and completeness
By trading optimality, completeness, accuracy, and / or precision for speed, a heuristic can quickly produce a solution that is good enough for solving the problem at hand, as opposed to finding all exact solutions in a prohibitively long time.
By establishing output expectations, the completeness and accuracy of the specification is also a part of the targeted improvement.
By and theorem
has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.
By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.
By setting K = ker ( f ) we immediately get the first isomorphism theorem.
* By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers.
By " satisfactory " one would mean at least the equivalent of Plancherel theorem.
By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization.
By the Glivenko – Cantelli theorem, if the sample comes from distribution F ( x ), then D < sub > n </ sub > converges to 0 almost surely.
By this theorem, once a star's chemical composition and its position on the main sequence is known, so too is the star's mass and radius.
By Gödel's incompleteness theorem, Peano arithmetic is incomplete and its consistency is not internally provable.
By Rice's theorem, the 1-halting problem is undecidable.
By the corollary to the recursion theorem, there is an index such that returns.
By Liouville's theorem, Hamiltonian flows preserve the volume form on the phase space.
By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation.
By the first isomorphism theorem, the image of A under ƒ is a substructure of B isomorphic to the quotient of A by this congruence.
By the central limit theorem, this distribution approaches the normal distribution as n increases.
By analogy to the prior and posterior probability terms in Bayes ' theorem, Bayes ' rule can be seen as Bayes ' theorem in odds form.
By the extreme value theorem, a continuous function on a closed interval must attain its minimum and maximum values at least once.
By the rank-nullity theorem, a system of n vectors in k dimensions ( where all dimensions are necessary ) satisfies a ( p = n − k )- dimensional space of relations.
By Tychonoff's theorem we have that is compact since is, so the closure of in is a compactification of.
By Stone's representation theorem every Boolean ring is isomorphic to a field of sets ( treated as a ring with these operations ).
By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.
By Euler's rotation theorem, we may replace the vector with where is a 3x3 rotation matrix and is the position of the particle at some fixed point in time, say t = 0.
By the divergence theorem Gauss's law can alternatively be written in the differential form:
By and first-order
By combining the signals in various proportions, it is possible to derive any number of first-order microphones, pointing in any direction, before and after recording.
By and logic
By the time the PDP-8 / E was introduced, electronic logic had become less expensive and " one-cycle data break " became more popular.
By working through the propositions of the book the reader comes to realize that language is perfectly suited to all his needs, and that philosophy rests on a confused relation to the logic of our language.
By adding more logic in front of the adder, a single adder can be converted into much more than just an adder — an ALU.
By connecting several p-n junctions together, researchers have been able to create the basis of all logic circuits: the AND, OR, and NOT gates have all been built from semiconductor nanowire crossings.
By the late 1970s, design using programmable logic device ( PLD )' s became popular, although these designs were primarily limited to design finite state machines.
By extension, outside of classical logic, one can speak of contradictions between actions when one presumes that their motives contradict each other.
By said logic at some point the solipsist may have either evolved into a thinking being, or the solipsist created the laws that permit the solipsist's primal attributes to be explained to begin with.
By the integration of all essential sub-systems on a single chip, and the use of low-power logic, each Compute or I / O node dissipated low power ( about 17 watts, including DRAMs ).
By the 12th century, Avicennian logic had replaced Aristotelian logic as the dominant system of logic in the Islamic world.
By the late 1990s, logic synthesis tools became available.
By the same logic we can also increase the group size.
; Data and code integrity: By centralizing business logic on an individual server or on a small number of server machines, updates and upgrades to the application for all users can be guaranteed.
By the 12th century, Avicennian logic had replaced Aristotelian logic as the dominant system of logic in the Islamic world.
By this logic, the statement " I have reason to believe naturalism is valid " is self-referentially incoherent in the same manner as the sentence " One of the words of this sentence does not have the meaning that it appears to have ," or the statement " I never tell the truth.
By this logic, what makes a good political system or a good man is completely dependent upon the chance path a whole people has taken over history.
By the 1930s symbolic logic itself was adequately included within mathematics.
By this time Wotton had acquired Arabic, Syriac, and Chaldee, as well as a knowledge of logic, philosophy, mathematics, geography, chronology, and history.
By the time that the most advanced logic system was introduced for SQ, the Tate Directional Enhancement System, " quad " was already considered a failure.
" By logic and reason a north-west passage must exist, and Columbus had discovered America with far less evidence ; it was imperative for England to catch up, settle in new lands and thus challenge the Iberian powers.
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