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* Soundness: If x ∉ L then for every π, V < sup > π </ sup >( x ) accepts with probability at most s ( n ).
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Soundness and for
Soundness of a deductive system is the property that any sentence that is provable in that deductive system is also true on all interpretations or structures of the semantic theory for the language upon which that theory is based.
* October 1992 – The Housing and Community Development Act of 1992 codifies within its language the Federal Housing Enterprises Financial Safety and Soundness Act of 1992 that creates the Office of Federal Housing Enterprise Oversight, and mandates HUD to set goals for lower income and underserved housing areas for the GSEs Fannie Mae and Freddie Mac
Soundness and with
* Soundness: if the statement is false, no prover, even if it doesn't follow the protocol, can convince the honest verifier that it is true, except with some small probability.
* Soundness: if the string is not in the language, no prover, however malicious, will be able to convince the verifier to accept the string with probability exceeding 1 / 3.
# Soundness: if the statement is false, no cheating prover can convince the honest verifier that it is true, except with some small probability.
Soundness and most
Soundness is among the most fundamental properties of mathematical logic.
If and x
* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.
If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.
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 F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.
If the filter shows amplitude ripple within the passband, the x dB point refers to the point where the gain is x dB below the nominal passband gain rather than x dB below the maximum gain.
* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.
* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).
If a Banach algebra has unit 1, then 1 cannot be a commutator ; i. e., for any x, y ∈ A.
If x is held fixed, then the Bessel functions are entire functions of α.
If the exponent r is even, then the inequality is valid for all real numbers x.
If x is a member of A, then it is also said that x belongs to A, or that x is in A.
If ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, x < sub > 3 </ sub >) are the Cartesian coordinates and ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, u < sub > 3 </ sub >) are the orthogonal coordinates, then
If and ∉
If ( X, Σ ) is some measurable space and A ⊂ X is a non-measurable set, i. e. if A ∉ Σ, then the indicator function 1 < sub > A </ sub >: ( X, Σ ) → R is non-measurable ( where R is equipped with the Borel algebra as usual ), since the preimage of the measurable set
If and L
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 the balance factor of P is 2, then the left subtree outweighs the right subtree of the given node, and the balance factor of the left child ( L ) must be checked.
* If the balance factor of L is + 1, a single right rotation ( with P as the root ) is needed ( Left-Left case ).
* If the balance factor of L is-1, two different rotations are needed.
If at least 50 cars had been built, sportscars like the GT40 and the Lola T70 were allowed, with a maximum of 5. 0 L. John Wyer's revised 4. 7 litre ( Bored to 4. 9 litres, and o-rings cut and installed between the deck and head to prevent head gasket failure, a common problem found with the 4. 7 engine.
If the cause of the crash is uncertain, this number is rendered as 48454C50, which stands for " HELP " in hexadecimal ASCII characters ( 48 = H, 45 = E, 4C = L, 50 = P ).
Clark L. Hull, probably the first major empirical researcher in the field, wrote If a subject after submitting to the hypnotic procedure shows no genuine increase in susceptibility to any suggestions whatever, there seems no point in calling him hypnotised ...
If the lower case L is used as the symbol, it is sometimes rendered as a cursive ℓ to help distinguish it from the capital " I ", although this usage has no official approval by any international bureau.
* Case 1: If L is a whole number, then the value will be found halfway between positions L and L + 1.
* Case 2: If L is a fraction, round to the nearest whole number.
If one considers a version of QCD with N < sub > f </ sub > flavors of massless quarks, then there is a global ( chiral ) flavor symmetry group SU < sub > L </ sub >( N < sub > f </ sub >) × SU < sub > R </ sub >( N < sub > f </ sub >) × U < sub > B </ sub >( 1 ) × U < sub > A </ sub >( 1 ).
Another more recent story involving travel to the future is Louis-Sébastien Mercier's L ' An 2440, rêve s ' il en fût jamais (" The Year 2440: A Dream If Ever There Were One "), a utopian novel in which the main character is transported to the year 2440.
If X and Y are topological vector spaces, the space L ( X, Y ) of continuous linear operators f: X → Y may carry a variety of different possible topologies.
If the array of weights consists of N weights spaced evenly over the length L
If J is a category with two objects and two parallel morphisms from object 1 to object 2 then a diagram of type J is a pair of parallel morphisms in C. The limit L of such a diagram is called an equalizer of those morphisms.
If one is given an object L of C together with a natural isomorphism Φ: Hom (–, L ) → Cone (–, F ), the object L will be a limit of F with the limiting cone given by Φ < sub > L </ sub >( id < sub > L </ sub >).
If and then
If she, Pamela, were being held responsible for his crimes, then hers must be the final act of expiation.
I showed her the shower and tub, and she said, smiling, `` If you really don't mind, I think I'll get clean in the shower, then soak for a few minutes in your tub.
If internal responsibility suggests acceptance of the socialist ideal of equality, then external responsibility implies adherence to principles of ideological supranationalism.
If Franklin was an authentic genius, then Alexander Hamilton, with his exceptional precocity, consuming energy, and high ambition, was a political prodigy.
If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.
If it is an honest feeling, then why should she not yield to it??
If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.
If a child watched its progress he whispered, `` Hay, hay, load of hay -- make a wish and turn away '', and then stared rigidly in the opposite direction until the sound of the horses' feet returned no more.
If in any one calculation Ptolemy had had to invoke 83 epicycles all at once, while Copernicus never required more than one third this number, then ( in the sense obvious to Margenau ) Ptolemaic astronomy would be simpler than Copernican.
If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.
Defoe then commented, `` If they Could Draw that young Gentleman into Their Measures They would show themselves quickly, for they are not asham'd to Say They want only a head to Make a beginning ''.
If the historian was convinced of his own correctness, then he should not allow his vision to become fogged by disturbing facts.
If man is actually the product of his environment and if science can discover the laws of human nature and the ways in which environment determines what people do, then someone -- a someone probably standing outside traditional systems of values -- can turn around and develop completely efficient means for controlling people.
If it will simply delay the debates until the qualifications are closed next spring, and then carry all the candidates on a tour of debates, it can provide a service to the state.
If they are to be commended for foresight in their planning, what then is the judgment of a town council that compounds this problem during the planning stage??
If the President of the United States really feels he won himself a `` lousy job '', then heaven help us all.
If we make it established custom that whenever butchery on the highways grows excessive, say beyond 25,000 per annum, then somebody is going to hang, it follows that the more eminent the victim, the more impressive the lesson.
If the raw population figures are crucially relevant, then it is idle to think of liberation, as idle as to suppose that Poland might liberate Russia.
If the UN troops are not mercenaries then the Hessians were not mercenaries either.
`` If they are here, then surely I have the right to be here '', Rousseau said.
If he failed to reach the riverbank in five minutes, say, then the skiffs would pull away and leave him groping in the mud.
If we had, then one of us would have had to go instead.
If our national interest lies in being able to fight and win a war rather than committing national suicide, then we must take a much more penetrating look at ballistic missiles.
If, for some reason such as economy, we are not going to develop aircraft nuclear propulsion with a sense of national urgency, then we should turn our effort to developing jet engines with a thrust-to-weight ratio of 12 or 15 to one.
If your pool is located on or near sloping ground, it may have natural drainage which is certainly more desirable than to be faced with the annual expense and labor of first pumping out the water and then scooping out all the debris.
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