A general property of finite groups implies that a finite subset of a symmetric group is again a group if and only if it is closed under the group operation.

To these rules we must add a further one for closed loops that implies an integration on momenta, since these internal (" virtual ") particles are not constrained to any specific energy-momentum-even that usually required by special relativity ( see this article for details ).

More generally, a Dedekind cut is a partition of a totally ordered set into two non-empty parts ( A and B ), such that A is closed downwards ( meaning that for all a in A, x ≤ a implies that x is in A as well ) and B is closed upwards, and A contains no greatest element.

* Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.

Indeed, this is essentially an algebraic translation of these geometric terms: the coordinate ring of any affine variety is, by definition, a finitely generated k-algebra, so Noetherian ; moreover curve means dimension one and nonsingular implies ( and, in dimension one, is equivalent to ) normal, which by definition means integrally closed.

The Menger sponge is a closed set ; since it is also bounded, the Heine – Borel theorem implies that it is compact.

The geometrization conjecture implies that a closed 3-manifold is hyperbolic if and only if it is irreducible, atoroidal, and has infinite fundamental group.

For closed surfaces, this classification is consistent with the Gauss – Bonnet theorem, which implies that for a closed surface with constant curvature, the sign of that curvature must match the sign of the Euler characteristic.

Additionally, if M is a closed symplectic manifold, then the 2nd de Rham cohomology group H < sup > 2 </ sup >( M ) is nontrivial ; this implies, for example, that the only n-sphere that admits a symplectic form is the 2-sphere.

Tsen's theorem implies that the Brauer group of a function field K in one variable over an algebraically closed field vanishes.

When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is of particular interest: A simple module over k-algebra is said to be absolutely simple if its endomorphism ring is isomorphic to k. This is in general stronger than being irreducible over the field k, and implies the module is irreducible even over the algebraic closure of k.

Homology and cohomology groups are defined to be zero for negative degrees, so Poincaré duality in particular implies that the homology and cohomology groups of orientable closed n-manifolds are zero for degrees bigger than n.

Success in reaching the other stone with their eyes closed implies that the pilgrim will find love, or true love.

Haboush's theorem implies that if G is a reductive algebraic group acting regularly on an affine algebraic variety, then disjoint closed invariant sets X and Y can be separated by an invariant function f ( this means that f is 0 on X and 1 on Y ).

Although the term campus radio implies full-power AM or FM transmission over the air, many radio stations experiment with low-power broadcasting, closed circuit or carrier current systems, often to on-campus listeners only.

" The organization formally adopts agnostic philosopher Paul Draper's definition of metaphysical naturalism as " the hypothesis that the natural world is a closed system, which means that nothing that is not a part of the natural world affects it .... implies that there are no supernatural entities, or at least none that actually exercises its power to affect the natural world.

1 and gcd ( b, n ) = 1 implies gcd ( ab, n ) = 1 the set of classes relatively prime to n is closed under multiplication.

The existence of a taut foliation implies various useful properties about a closed 3-manifold.

For example, affine space is a Zariski-open subset of projective space, and since any closed affine subset can be expressed as an intersection of the projective completion and the affine space embedded in the projective space, this implies that any affine variety is quasiprojective.

Discreteness implies points in B < sup > 3 </ sup > have finite stabilizers, and discrete orbits under the group G. But the orbit Gp of a point p will typically accumulate on the boundary of the closed ball.

Thus, boundedness of T < sup >∗</ sup > on its domain does not imply boundedness of T. On the other hand, if T < sup >∗</ sup > is defined on the whole space then T is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space .< ref > Proof: being closed, the everywhere defined T < sup >∗</ sup > is bounded, which implies boundedness of T < sup >∗∗</ sup >, the latter being the closure of T. See also for the case of everywhere defined T .</ ref > If the domain of T < sup >∗</ sup > is dense, then it has its adjoint T < sup >∗∗</ sup >.

An operator T is said to be self-adjoint if T < sup >∗</ sup > = T. Note that, when T is self-adjoint, the existence of the adjoint implies that T is dense and since is necessarily closed, is closed.

Because string theory implies that the Universe has more dimensions than we expect — 26 for bosonic string theories and 10 for superstring theories — we have to find a reason why the extra dimensions are not apparent.

The required cancellation implies that the spacetime dimensionality must be equal to the critical dimension which is either 26 in the case of bosonic string theory or 10 in the case of superstring theory.

This implies that the IIA string theory can identify with the IIB string theory: any object which can be described with the IIA theory has an equivalent, although seemingly different, description in terms of the IIB theory.

In the case of string field theory, this implies that a classical configuration, usually called the string field, is given by an element of the free string Fock space.

A magnetic monopole implies a mathematical singularity in the vector potential, which can be expressed as a Dirac string of infinitesimal diameter that contains the equivalent of all of the 4πg flux from a monopole " charge " g. The Dirac string starts from, and terminates on, a magnetic monopole.

The definition of the Myhill – Nerode relation implies that the transition function is well-defined: no matter which representative string x is chosen for state X, the same transition function value will result.

A state of this automaton is accepting if the corresponding equivalence class contains a string in L ; in this case, again, the definition of the relation implies that all strings in the same equivalence class must also belong to L, for otherwise the empty string would be a distinguishing string for some pairs of strings in the class.

This implies that if our machine accepts x, it must also accept the string of ' a's followed by ' b's, which is not in the language of strings containing an equal number of ' a's and ' b's.

The fact that T-duality interchanges the usual Neumann boundary conditions with Dirichlet boundary conditions was discovered independently by Horava and by Dai, Leigh, and Polchinski in 1989 ; this result implies that such boundary conditions must necessarily appear in regions of the moduli space of any open string theory.

The Dirac string acts as the solenoid in the Aharonov-Bohm effect, and the requirement that the position of the Dirac string should not be observable implies the Dirac quantization rule: the product of a magnetic charge and an electric charge must always be an integer multiple of.

This ' relict stress ' theory implies that the ringing rock boulders act much like a guitar string.

That implies that Shann was on the enemy side.

In I knew you when you were a child, and you were pretty then dominant stress on then implies that the young woman spoken to is still pretty.

In contrast to this Stoic-patristic view, Utopia implies that the nature of man is such that to rely on individual conscience to supply the deficiencies of municipal law is to embark on the bottomless sea of human sinfulness in a sieve.

The assumption of unit-treatment addivity implies that, for every treatment, the th treatment have exactly the same effect on every experiment unit.

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.

He shows how a satisfying argument for the validity of experience can be based neither on demonstration ( since " it implies no contradiction that the course of nature may change ") nor experience ( since that would be a circular argument ).

As Ásatrú implies a focus on polytheistic belief in the Æsir usage of the term in Scandinavia has declined somewhat.

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.

" Love and Theft is, as the title implies, a kind of homage ," writes Kot, " never more so than on ' High Water ( for Charley Patton ),' in which Dylan draws a sweeping portrait of the South's racial history, with the unsung blues singer as a symbol of the region's cultural richness and ingrained social cruelties.

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 ).

The epilogue, in which Judah is assigned a leadership role twice, implies pro-Judah political leanings on the part of the author.

Cosmology as a science originated with the Copernican principle, which implies that celestial bodies obey identical physical laws to those on Earth, and Newtonian mechanics, which first allowed us to understand those laws.

" But this can't be accurate because it implies the cat is actually both dead and alive until the box is opened to check on it.

As that history implies, CCL was written for the Macintosh, but Clozure CL now runs on Mac OS X, FreeBSD, Linux, Solaris and Windows.

where ε denotes the Levi-Civita symbol, the metric tensor is used to lower the index on F, and the Einstein summation convention implies that repeated indices are summed over.

is in C. In other words, every point on the line segment connecting x and y is in C. This implies that a convex set in a real or complex topological vector space is path-connected, thus connected.

Supporters consider cheerleading, as a whole, a sport, citing the heavy use of athletic talents while critics do not see it as deserving of that status since sport implies a competition among squads and not all squads compete, along with subjectivity of competitions where-as with gymnastics, diving, and figure skating-scores are assessed based on human judgment and not an objective goal or measurement of time.

In British English, it is generally accepted that collective nouns can take either singular or plural verb forms depending on the context and the metonymic shift that it implies.

Thus, for example, controversies in physics are limited to subject-areas where experiments cannot be carried out yet, whereas Benford's Law implies that controversy is inherent to politics, where communities must frequently decide on courses of action based on insufficient information.

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.

For pan-and polytheistic faiths this usually implies the direct action of one god or another on the course of human events.

Data structures ( fields, records, files and objects ) optimized to deal with very large amounts of data stored on a permanent data storage device ( which implies relatively slow access compared to volatile main memory ).