** This follows by noting f < sub > ω </ sub >( n ) > 2 ↑< sup > n-1 </ sup > n > 3 ↑< sup > n-2 </ sup > 3 + 2, and hence f < sub > ω </ sub >( g < sub > k </ sub > + 2 ) > g < sub > k + 1 </ sub > + 2.

It follows from Jensen's covering theorem that the existence of 0 < sup >#</ sup > is equivalent to ω < sub > ω </ sub > being a regular cardinal in the constructible universe L.

It follows from ZF + axiom of determinacy that ω < sub > 1 </ sub > is measurable, and that every subset of ω < sub > 1 </ sub > contains or is disjoint from a closed and unbounded subset.

Since ω and g are nondegenerate, each induces a bundle isomorphism TM → T * M, where the first map, denoted φ < sub > ω </ sub >, is given by the interior product φ < sub > ω </ sub >( u ) = < sub > u </ sub > ω = ω ( u, •) and the other, denoted φ < sub > g </ sub >, is given by the analogous operation for g. With this understood, the three structures ( g, ω, J ) form a compatible triple when each structure can be specified by the two others as follows:

In astrodynamics the argument of periapsis ω can be calculated as follows:

Actually an apparently stronger result follows: If there is a measurable cardinal, then every game in the first ω < sup > 2 </ sup > levels of the difference hierarchy over is determined.

The Riemannian metric g and associated ( 1, 1 ) form ω are related by the almost complex structure J as follows

This uniquely determines the Lie derivative and it follows that for vector fields the Lie derivative is the commutator

The system is based on a " 666-sided " dice, the designers ' name for three six-sided dice interpreted as follows: the first and second dice read as the tens and units, are used to determine success or failure, while the third die determines the degree of success.

It follows that the sign of M ( x ) determines the nature of the medium, with positive M ( x ) corresponding to medium A as described above and negative M ( x ) corresponding to medium B.

* If the economy follows adaptive expectations, future inflation is partly determined by past experience with inflation, since experience determines expected inflation and this is a major determinant of realized inflation.

The NTSB probable cause statement is as follows: " The National Transportation Safety Board determines that the probable cause of the accident was the flightcrew's failure to use the taxi checklist to ensure the flaps and slats were extended for takeoff.

After an investigation The National Transportation Safety Board determines the probable cause ( s ) of this accident as follows: The pilot ’ s failure to select the proper fuel tank for takeoff, which resulted in a loss of engine power.

Supporters of the existence of qualia argue that, since we can imagine this happening without contradiction, it follows that we are imagining a change in a property that determines the way things look to us, but that has no physical basis.

Mykros determines to discover the truth and follows the Maylin, Renis, into the Borad ’ s power chamber.

The National Transportation Safety Board determines the probable cause ( s ) of this accident as follows:

The Birth episode follows Astro's adventures and other characters he interacts with, and ends with robots being destroyed by a device called Death Mask, after it determines that the robots are too dangerous to be left alive, leaving Sharaku able to conquer the Earth.

The talker volume distribution follows a log-normal curve and the medium-power talker is uniquely determined by the average talker volume.

This follows from the rationalist position that man is born to think and that language use is a uniquely human trait impossible in other species.

This means that the restriction of ρ < sub > n </ sub > to elements of degree at most n is a bijective linear map, and ρ < sub > n </ sub >( e < sub > k </ sub >( X < sub > 1 </ sub >,…, X < sub > n + 1 </ sub >)) = e < sub > k </ sub >( X < sub > 1 </ sub >,…, X < sub > n </ sub >) for all k ≤ n. The inverse of this restriction can be extended uniquely to a ring homomorphism φ < sub > n </ sub > from R < sup > S < sub > n </ sub ></ sup > to R < sup > S < sub > n + 1 </ sub ></ sup >, as follows for instance from the fundamental theorem of symmetric polynomials.

* In words: If all the parameters A, B, C, ..., W of an internal formula F have standard values then holds for all xs as soon as it holds for all standard xs — from which it follows that all uniquely defined concepts or objects within classical mathematics are standard.

As the rational canonical form is uniquely determined by the unique invariant factors associated to A, and these invariant factors are independent of basis, it follows that two square matrices A and B are similar if and only if they have the same rational canonical form.

The strongest conclusion that validly follows from the proposed definition is that if any conceivable individual actually is uniquely both divine and existent, then God actually exists.

The Cantor – Bendixson theorem states that closed sets of a Polish space X have the perfect set property in a particularly strong form ; any closed set C may be written uniquely as the disjoint union of a perfect set P and a countable set S. Thus it follows that every closed subset of a Polish space has the perfect set property.

Hence it follows that the involution within the bundle must be a perspective De Jonquieres involution of order Af and the invariant locus must have a multiple line of multiplicity either Af or Af.

The Klein bottle can be seen as a fiber bundle over S < sup > 1 </ sup > as follows: one takes the square ( modulo the edge identifying equivalence relation ) from above to be E, the total space, while the base space B is given by the unit interval in y, modulo 1 ~ 0.

The Monthly Baen Bundles installments are as follows: first includes roughly a half of every book in the bundle, second includes roughly three quarters, and the third, coinciding with the print release, includes the full text.

In 1938, Franco adopted a variant of the Coat of Arms reinstating some elements originally used by the House of Trastámara such as Saint John's eagle and the yoke and bundle, as follows: Quarterly, 1 and 4. quarterly Castile and León, 2 and 3. per pale Aragon and Navarra, enté en point of Granada.

for all smooth functions f on M and all smooth sections σ of E. It follows that the difference ∇< sub > 1 </ sub > − ∇< sub > 2 </ sub > is induced by a one-form on M with values in the endomorphism bundle End ( E ) = EE *:

It follows that this composite descends to a first order differential operator D from sections of V = P ×< sub > H </ sub > V to sections of the bundle.

As in the nonsupersymmetric case, we have a Spin ( 3, 1 ) principal bundle over M. We have an R < sup > 4 | 4 </ sup > vector bundle T over M. The fiber of T transforms under the local Lorentz group as follows ; the four real bosonic dimensions transform as a vector and the four real fermionic dimensions transform as a Majorana spinor.

It then follows that f < sup >*</ sup > E is a fiber bundle over B ′ with fiber F. The bundle f < sup >*</ sup > E is called the pullback of E by f or the bundle induced by f. The map is then a bundle morphism covering f.

alt = A map of Australia with the cane toad's distribution highlighted: The area follows the northeastern coast of Australia, ranging from the Northern Territory through to the north end of New South Wales.

From these two axioms, it follows that for every g in G, the function which maps x in X to g · x is a bijective map from X to X ( its inverse being the function which maps x to g < sup >− 1 </ sup >· x ).

We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows.

It therefore follows from the pigeonhole principle that it is not possible to map every sequence of length N or less to a unique sequence of length N − 1 or less.

This map is characterized by a differential equation as follows:

The inverse map of Φ can be described as follows.

The technical statement is as follows: if M is a given m-dimensional Riemannian manifold ( analytic or of class C < sup > k </ sup >, 3 ≤ k ≤ ∞), then there exists a number n ( with n ≤ m ( 3m + 11 )/ 2 if M is a compact manifold, or n ≤ m ( m + 1 )( 3m + 11 )/ 2 if M is a non-compact manifold ) and an injective map ƒ: M → R < sup > n </ sup > ( also analytic or of class C < sup > k </ sup >) such that for every point p of M, the derivative dƒ < sub > p </ sub > is a linear map from the tangent space T < sub > p </ sub > M to R < sup > n </ sup > which is compatible with the given inner product on T < sub > p </ sub > M and the standard dot product of R < sup > n </ sup > in the following sense:

The proof follows by the fact that if f were indeed a map onto P ( S )), then we could find r in S, such that f (

In the case of abelian groups or modules over some ring, the map d can be constructed as follows.

Juni follows a map given to him by Demetra to a lava-filled gorge.

The Val Tevere is marked on the map by Highway A1, the Autostrada del Sole, which enters it in the vicinity of Monte Rotondo north of Rome and follows the course of the river to the point where the latter flows from Lago di Cabara in the vicinity of Baschi and then goes up the valley of the Fiume Paglia.

A map depicting the route of the Ice Age Trail, including the trail's official logoThe trail roughly follows the location of the terminal moraine from the last Ice Age.

The tangent space to S < sup > 2 </ sup > at a point m is naturally identified with the vector sub-space of ℝ < sup > 3 </ sup > consisting of all vectors orthogonal to m. It follows that a vector field Y on S < sup > 2 </ sup > can be seen as a map Y: S < sup > 2 </ sup > → ℝ < sup > 3 </ sup >, which satisfies

Some authors omit the assumption of surjectivity, for if X is connected and C is nonempty then surjectivity of the covering map actually follows from the other axioms.

Surjectivity of for follows from the fact that for all such n, the n-sphere S < sup > n </ sup > is simply connected and hence every continuous map from S < sup > n </ sup > to X can be lifted to C.

An equivalent, but purely algebraic description of linear differential operators is as follows: an R-linear map P is a kth-order linear differential operator, if for any k + 1 smooth functions we have

It follows the form of the mathematical set-builder notation ( set comprehension ) as distinct from the use of map and filter functions.

Fourth World follows the First World, Second World, and Third World hierarchy of nation-state status ; however, unlike the former categories, Fourth World is not spatially bounded, and is usually used to refer to populations whose size and shape does not map onto citizenship in a specific nation-state.

It follows that the full pre-image ƒ < sup >− 1 </ sup >( q ) in M of a regular value q ∈ N under a differentiable map ƒ: M → N is either empty or is a differentiable manifold of dimension dim M − dim N, possibly disconnected.

A squarish map that largely follows Haff and Martin appears in The Dictionary of Imaginary Places.

His reasoning was that the degree of the Gauss map must be preserved in such " turning "— in particular it follows that there is no such turning of S < sup > 1 </ sup > in R < sup > 2 </ sup >.

That this map is a homomorphism follows from the fact that the multiplication of unit complex numbers corresponds to addition of angles: