** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

** For example, consider the map ( which is the " realification " of the complex square function ) where U

For example, if u < sub > 1 </ sub > is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u < sub > 1 </ sub >, with α ∈ R, is an invariant curve of the map.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.

For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.

For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category.

For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, which only share a point that also belongs to Arizona and Colorado, are not.

For instance, consider a simplified map:

For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.

For example, the exponential map of SL < sub > 2 </ sub >( R ) is not surjective.

For example, the width of roads and small streams are exaggerated when they are too narrow to be shown on the map at true scale ; that is, on a printed map they would be narrower than could be perceived by the naked eye.

For example, this map of Europe has been distorted to show population distribution, while the rough shape of the continent is still discernable.

For example, cartographers may simply omit military installations or remove features solely in order to enhance the clarity of the map.

For example, a road map may not show railroads, smaller waterways or other prominent non-road objects, and even if it does, it may show them less clearly ( e. g. dashed or dotted lines / outlines ) than the main roads.

For the purposes of this article, foot orienteering serves as a point of departure for discussion of all other variations, but basically any sport that involves racing against a clock and requires navigation using a map is a type of orienteering.

For the location with respect to French Polynesia, see the inset of: Image: French Polynesia map. jpg.

For example, a linear map can be represented by a matrix, a 2-dimensional array, and therefore is a 2nd-order tensor.

For instance, any isometry ( distance-preserving map ) between metric spaces is uniformly continuous.

For example, an 1893 pleasure map of the Thames shows 8 " charging stations for electric launches " between Kew ( Strand-on-the-Green ) and Reading ( Caversham ).

Theorem ( Modified Schwarz inequality for 2-positive maps ) For a 2-positive map Φ between C *- algebras, for all a, b in its domain,

For example, a subway map intended for riders may represent a subway station with a dot ; the dot doesn't resemble the actual station at all but gives the viewer information without unnecessary visual clutter.

For systems such as the logistic map, the bifurcation diagram shows a range of parameter values for which apparently the only cycle has period 3.

For a more mathematical discussion, see the articles on nonlinearity, dynamical systems, bifurcation theory, and chaos theory.

The Hénon map does not have a strange attractor for all values of the parameters a and b. For example, by keeping b fixed at 0. 3 the bifurcation diagram shows that for a

For example, the period one orbit only appears for values of around 4 and is never found again in the bifurcation diagram.

For three straight years, Tom Horn patrolled the southern Wyoming pastures, and how many men he killed after Lewis and Powell ( if he killed Lewis and Powell ) will never be known.

For pride's sake, I will not say that the coy and leering vade mecum of those verses insinuated itself into my soul.

William Wimsatt and Cleanth Brooks, it seems to me, have a penetrating insight into the way in which this control is effected: `` For if we say poetry is to talk of beauty and love ( and yet not aim at exciting erotic emotion or even an emotion of Platonic esteem ) and if it is to talk of anger and murder ( and yet not aim at arousing anger and indignation ) -- then it may be that the poetic way of dealing with these emotions will not be any kind of intensification, compounding, or magnification, or any direct assault upon the affections at all.

For them a box will be lodgment enough.

For that reason any democratic reform and effort to bring genuine representative government to the Dominican Republic will need the greatest sympathy and help.

`` For if ye forgive men their trespasses, your heavenly Father will also forgive you: but if ye forgive not men their trespasses, neither will your Father forgive -- your trespasses ''.

For the policy officer will know that action can almost never be secret and that in general the effectiveness of policy will be conditioned by the readiness of the country to sustain it.

For example, it probably will be necessary for the Corps to have authority to pay medical expenses of volunteers.

For me, these will belong more completely to their surroundings if they are conceived in this early stage, though I freely admit that I do not hesitate to add or eliminate figures on the full sheet when it serves my final purpose.

For the most part, this discussion will be confined to results obtained since the introduction of the reference standard.

For any such square the middle corner of these will be called the vertex of the square and the corner not on the curve will be called the diagonal point of the square.

For the vertex at Af, these values will be denoted by Af.

For winning Larson will receive a $100 U.S. Savings Bond from the Junior Achievement national organization.

For most of them, it will be their first experience in membership training, since this is a recent development in many churches.

For the army of compulsive eaters -- from the nibblers and the gobblers to the downright gluttons -- reducing is a war with the will that is rarely won.

" Swift extends the metaphor to get in a few jibes at England ’ s mistreatment of Ireland, noting that " For this kind of commodity will not bear exportation, and flesh being of too tender a consistence, to admit a long continuance in salt, although perhaps I could name a country, which would be glad to eat up our whole nation without it.

For example, if an author is paid a modest advance of $ 2000. 00, and their royalty rate is 10 % of a book priced at $ 20. 00-that is, $ 2. 00 per book-the book will need to sell 1000 copies before any further payment will be made.

For Tarrou, plague is the destructive impulse within every person, the will and the capacity to do harm, and it is everyone's duty to be on guard against this tendency within themselves, lest they infect someone else with it.

For an example of its use, analysis of the concentration of elements is important in managing a nuclear reactor, so nuclear scientists will analyze neutron activation to develop discrete measurements within vast samples.

For reception, a local oscillator will typically restore the suppressed carrier so the signal can be demodulated with a product detector.

For the living know that they will die ; But the dead know nothing, And they have no more reward, For the memory of them is forgotten.