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Any compact manifold has isoperimetric dimension 0.
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Any and compact
* Any finite topological space, including the empty set, is compact.
* Any space carrying the cofinite topology is compact.
* Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification.
Note that for orientable compact surfaces without boundary, the Euler characteristic equals, where is the genus of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and counts the number of handles.
Concerned with the runaway success of the Ford Mustang, Chevrolet executives realized that their compact sporty car, the Corvair, would not be able to generate the sales volume of the Mustang due to its rear-engine design, as well as declining sales, partly due to the bad publicity from Ralph Nader's book, Unsafe at Any Speed.
Any compact orientable surface and any compact surface with non-empty boundary embeds in though any closed non-orientable surface needs.
Any bounded operator L that has finite rank is a compact operator ; indeed, the class of compact operators is a natural generalisation of the class of finite-rank operators in an infinite-dimensional setting.
* Any compact operator is strictly singular, but not vice-versa.
Any action of the group by continuous affine transformations on a compact convex subset of a ( separable ) locally convex topological vector space has a fixed point.
* Any compact symplectic manifold admits a symplectic Lefschetz pencil ( Donaldson 1999 ).
( 2 ) Any sequence of continuous positive definite functions on G converging to 1 uniformly on compact subsets, converges to 1 uniformly on G.
Any compact Riemann surface of genus greater than 1 ( with its usual metric of constant curvature − 1 ) is a locally symmetric space but not a symmetric space.
* Any compact group is locally compact.
* Any discrete group is locally compact.
The canonical ring and therefore likewise the Kodaira dimension is a birational invariant: Any birational map between smooth compact complex manifolds induces an isomorphism between the respective canonical rings.
Any and manifold
# Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold.
Any real-valued differentiable function, H, on a symplectic manifold can serve as an energy function or Hamiltonian.
* Any smooth manifold is a diffeological space.
Any Calabi – Yau manifold has a finite cover that is the product of a torus and a simply-connected Calabi – Yau manifold.
Any smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology.
# Any smooth manifold of dimension n ≥ 3 admits a Riemannian metric with negative Ricci curvature.
Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system.
Any covering space of a differentiable manifold may be equipped with a ( natural ) differentiable structure that turns p ( the covering map in question ) into a local diffeomorphism – a map with constant rank n.
Any vector bundle V over a manifold may be realized as the pullback of a universal bundle over the classifying space, and the Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle ; these universal Chern classes in turn can be explicitly written down in terms of Schubert cycles.
Any smooth function on a symplectic manifold gives rise, by definition, to a Hamiltonian vector field and the set of all such form a subalgebra of the Lie Algebra of symplectic vector fields.
* Any open manifold admits a ( non-complete ) Riemannian metric of positive ( or negative ) curvature.
Any “ nice ” space such as a manifold or CW complex is semi-locally simply connected.
Any evolution of spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network.
* Any manifold with constant sectional curvature is an Einstein manifold — in particular:
Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy.
Any manifold is an n-handlebody, that is, any manifold is the union of handles.
Any handlebody decomposition of a manifold defines a CW complex decomposition of the manifold, since attaching an r-handle is the same, up to homotopy equivalence, as attaching an r-cell.
Any sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot – Carathéodory, defined as
Any symplectic manifold ( or indeed any almost symplectic manifold ) has a natural volume form.
Any and has
Any tragedy, he maintains, has six elements: plot, character, and thought ( the objects of imitation ), diction and melody ( the means of imitation ), and spectacle ( the manner of imitation ).
Any street meeting, sacred or secular, which he and his colleagues uneasily cover has as its explicit or implicit burden the cruelty and injustice of the white domination.
Any string s has at least one description, namely the program:
Any player may declare the game over at any time during his turn if either of two conditions is true: one chain has 41 or more tiles, or there is at least one chain on the board and every chain on the board has 11 or more tiles.
Any individual has one of six possible genotypes ( AA, AO, BB, BO, AB, and OO ) that produce one of four possible phenotypes: " A " ( produced by AA homozygous and AO heterozygous genotypes ), " B " ( produced by BB homozygous and BO heterozygous genotypes ), " AB " heterozygotes, and " O " homozygotes.
Any Banach algebra ( whether it has an identity element or not ) can be embedded isometrically into a unital Banach algebra so as to form a closed ideal of.
# Any collection of closed subsets of X with the finite intersection property has nonempty intersection.
Any young player who has good control will become a successful curve pitcher long before the pitcher who is endeavoring to master both curves and control at the same time.
Any use of the term " Cathar " to refer to people after the suppression of Catharism in the 14th century is a cultural or ancestral reference, and has no religious implication.
Any rational number with a denominator whose only prime factors are 2 and / or 5 may be precisely expressed as a decimal fraction and has a finite decimal expansion.
Any configuration of charges or currents has a ' dipole moment ', which describes the dipole whose field is the best approximation, at large distances, to that of the given configuration.
Any such poset has a dual poset.
:* Any total order on X has exactly one equivalence class, X itself, because x ~ y for all x and y ;
:* Any subset of the identity relation on X has equivalence classes that are the singletons of X.
# Disturbed natural forest – Any forest type above that has in its interior significant areas of disturbance by people, including clearing, felling for wood extraction, anthropogenic fires, road construction, etc.
# Disturbed natural forest – Any forest type above that has in its interior significant areas of disturbance by people, including clearing, felling for wood extraction, anthropogenic fires, road construction, etc.
Any symbolism has been added later to the three colours, although the orange comes from the House of Orange-Nassau.
Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup.
For instance, the Archbishop of Wales has criticized " atheistic fundamentalism " broadly and said " Any kind of fundamentalism, be it Biblical, atheistic or Islamic, is dangerous ".
Any attempt to define it ( X is good if it has property Y ) will simply shift the problem ( Why is Y-ness good in the first place?
Any right action has an equivalent left action, thus only left actions can be considered without any loss of generality.
Any one language has only a subset of the aspectual distinctions attested in the world's languages, and some languages ( such as Standard German ; see below ) do not have aspects.
In an inertial frame, Newton's first law ( the law of inertia ) is satisfied: Any free motion has a constant magnitude and direction.
Any conductor has inductance.
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