`` Tact '', by its very derivation, implies that its possessor keeps in touch with other people, but the author of Clericis Laicos and Unam Sanctam, the wielder of the two swords, the papal sun of which the imperial moon was but a dim reflection, the peer of Caesar and vice-regent of Christ, was so high above other human beings that he had forgotten what they were like.

Forgiveness implies more than a person wanting his past sins covered by God's love.

The much stronger axiom of determinacy, or AD, implies that every set of reals is Lebesgue measurable, has the property of Baire, and has the perfect set property ( all three of these results are refuted by AC itself ).

This is to avoid the notion that a writing system that represents sounds must be either a syllabary or an alphabet, which implies that a system like Aramaic must be either a syllabary ( as argued by Gelb ) or an incomplete or deficient alphabet ( as most other writers have said ); rather, it is a different type.

He argues that this casts doubt on the notion that recessions are caused by a reallocation of resources from industrial production to consumption, since he argues that the Austrian business cycle theory implies that net investment should be below zero during recessions.

it was noted by Bohr that the existence of any sort of wave packet implies uncertainty in the wave frequency and wavelength, since a spread of frequencies is needed to create the packet itself.

Despite advances in the protection offered by ballistic armour against projectiles, as the name implies, modern ballistic body armour is much less impervious to stabbing weapons unless they are augmented with anti-knife / anti-stab armour ( usually a form of mail ).

Hence ahimsa as a binding code of conduct implies a ban on hunting, butchery, meat eating, and the use of animal products provided by violent means.

In byte-oriented systems ( i. e. most modern computers ), the term uncompressed BCD usually implies a full byte for each digit ( often including a sign ), whereas packed BCD typically encodes two decimal digits within a single byte by taking advantage of the fact that four bits are enough to represent the range 0 to 9.

To see that it is sufficient — that the parallelogram law implies that the form defined by the polarization identity is indeed a complete inner product — one verifies algebraically that this form is additive, whence it follows by induction that the form is linear over the integers and rationals.

The reverse is forward compatibility, which implies that old devices allow ( or are expected to allow ) data formats generated by new ( or future ) devices, perhaps without supporting all new features.

The Rayleigh Criterion implies that a thermoacoustic instability can be optimally controlled by having heat release oscillations 180 degrees out of phase with pressure oscillations at the same frequency.

The sculpture implies two types of cavalry, but this might be a simplification by the artist.

For instance, together with the spectral radius formula, it implies that the C *- norm is uniquely determined by the algebraic structure:

Lenin regarded colonialism as the root cause of imperialism, as imperialism was distinguished by monopoly capitalism via colonialism and as Lyal S. Sunga explains: " Vladimir Lenin advocated forcefully the principle of self-determination of peoples in his " Theses on the Socialist Revolution and the Right of Nations to Self-Determination " as an integral plank in the programme of socialist internationalism " and he quotes Lenin who contended that " The right of nations to self-determination implies exclusively the right to independence in the political sense, the right to free political separation from the oppressor nation.

Specifically, this demand for political democracy implies complete freedom to agitate for secession and for a referendum on secession by the seceding nation.

It implies the ability to defend one ’ s interests and if necessary to impose one ’ s will by any means available.

The first case was done by the Gorenstein – Walter theorem which showed that the only simple groups are isomorphic to L < sub > 2 </ sub >( q ) for q odd or A < sub > 7 </ sub >, the second and third cases were done by the Alperin – Brauer – Gorenstein theorem which implies that the only simple groups are isomorphic to L < sub > 3 </ sub >( q ) or U < sub > 3 </ sub >( q ) for q odd or M < sub > 11 </ sub >, and the last case was done by Lyons who showed that U < sub > 3 </ sub >( 4 ) is the only simple possibility.

The three classes are groups of GF ( 2 ) type ( classified mainly by Timmesfeld ), groups of " standard type " for some odd prime ( classified by the Gilman – Griess theorem and work by several others ), and groups of uniqueness type, where a result of Aschbacher implies that there are no simple groups.

It was completed in about 731, and Bede implies that he was then in his fifty-ninth year, which would give a likely birth date of about 672 – 673.

The cosmological principle implies that the metric should be homogeneous and isotropic on large scales, which uniquely singles out the Friedmann – Lemaître – Robertson – Walker metric ( FLRW metric ).

Clines ' conclusion was that the overall theme is " the partial fulfillment – which implies also the partial nonfulfillment – of the promise to or blessing of the Patriarchs.

The Paley – Wiener theorem immediately implies that if f is a nonzero distribution of compact support ( these include functions of compact support ), then its Fourier transform is never compactly supported.

The first premise is the " if – then " or conditional claim, namely that P implies Q.

The Heine – Borel theorem implies that a Euclidean n-sphere is compact.

Triton's density implies it is probably about 30 – 45 % water ice, with the remainder being rocky material.

94 – 97 ) which describes the expeditions which Antigonus sent against the Nabataeans in 312 BC is understood to throw some light upon the history of Petra, but the " petra " referred to as a natural fortress and place of refuge cannot be a proper name and the description implies that the town was not yet in existence.

The presence of these two opposing factors implies a point at which the combined blur spot is minimized ( Gibson 1975, 64 ); at that point, the < var > f </ var >- number is optimal for image sharpness, for this given depth of field – a wider aperture ( lower f-number ) causes more defocus, while a narrower aperture ( higher f-number ) causes more diffraction.

The Arzelà – Ascoli theorem implies that if is a uniformly bounded sequence of functions with bounded Lipschitz constant, then it has a convergent subsequence.

The year of their wedding is not recorded, although the historian Priscus implies it took place before the Vandal sack of Rome ( June 2 – 16, 455 ).

Wells was listed in the Domesday Book of 1086 as Welle, from the Old English wiells, which was not listed as a town, but included four manors with a population of 132, which implies a population of 500 – 600.

Ancient authorities testify to his visit to the court of Hieron ( 478 – 467 ) and this is indeed indicated by his fifth Ode ( 476 BC ), where the word xenos ( V. 11 ) implies that he had already been Hieron's guest, ( probably accompanied by his uncle ).

The " stationary " Karush – Kuhn – Tucker condition implies that the solution can be expressed as a linear combination of the training vectors

Law of Past Experience – The law of past experience implies that under some circumstances visual stimuli are categorized according to past experience.

) Note that Pearson's original paper was entitled " On Lines and Planes of Closest Fit to Systems of Points in Space " – " in space " implies physical Euclidean space where such concerns do not arise.

This implies Weierstrass ’ original statement since the polynomials on form a subalgebra of C which contains the constants and separates points.

In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of the excluded middle ( unlike in Martin-Löf type theory, where it does not ).

This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and the balance theorem implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type.

For the special case, this implies that the length of a vector is preserved as well — this is just Parseval's theorem:

In fact, Cantor's method of proof of this theorem implies the existence of an " infinity of infinities ".

Together with soundness ( whose verification is easy ), this theorem implies that a formula is logically valid if and only if it is the conclusion of a formal deduction.

This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned in a formal deduction of φ, and the soundness of the deduction system then implies φ is a logical consequence of this finite set.

Shannon's theorem also implies that no lossless compression scheme can compress all messages.

Bell's theorem implies, and it has been proven mathematically, that compatible measurements cannot show Bell-like correlations, and thus entanglement is a fundamentally non-classical phenomenon.

One might expect that by the Hahn-Banach theorem for bounded linear functionals, every bounded linear functional on C < sub > c </ sub >( X ) extends in exactly one way to a bounded linear functional on C < sub > 0 </ sub >( X ), the latter being the closure of C < sub > c </ sub >( X ) in the supremum norm, and that for this reason the first statement implies the second.

The integrability condition and Stokes ' theorem implies that the value of the line integral connecting two points is independent of the path.

If we choose the volume to be a ball of radius a around the source point, then Gauss ' divergence theorem implies that

Cox's theorem implies that any plausibility model that meets the

For instance, Bell's theorem implies that quantum mechanics cannot satisfy both local realism and counterfactual definiteness.

Noether's theorem implies that there is a conserved current associated with translations through space and time.

However, if the errors are not normally distributed, a central limit theorem often nonetheless implies that the parameter estimates will be approximately normally distributed so long as the sample is reasonably large.

In fact these uses of AC are essential: in 1950 Kelley proved that Tychonoff's theorem implies the axiom of choice.

To prove that Tychonoff's theorem in its general version implies the axiom of choice, we establish that every infinite cartesian product of non-empty sets is nonempty.

Liouville's theorem implies invariance of a kinematic measure on the unit tangent bundle.