Any quadratic polynomial over the complex numbers ( polynomials of the form where,, and ∈ ) can be factored into an expression with the form using the quadratic formula.

for each p ∈ U. Any differential 1-form arises this way, and by using (*) it follows that any differential 1-form α on U may be expressed in coordinates as

Any element y ∈ Y := Ran ( S − I ) has a distance larger than or equal to 1 from u ( otherwise y < sub > i </ sub > = x < sub > i + 1 </ sub >-x < sub > i </ sub > would be positive and bounded away from zero, whence x < sub > i </ sub > could not be bounded ).

* Any Hilbert space H is a Hilbert manifold with a single global chart given by the identity function on H. Moreover, since H is a vector space, the tangent space T < sub > p </ sub > H to H at any point p ∈ H is canonically isomorphic to H itself, and so has a natural inner product, the " same " as the one on H. Thus, H can be given the structure of a Riemannian manifold with metric

* Any expression formed using any combination of the basic arithmetic operations and extraction of nth roots gives an algebraic number.

Any accelerating electric charge, and therefore any changing electric current, gives rise to an electromagnetic wave that propagates at very high speed outside the surface of the conductor.

Any second-countable space is separable: if is a countable base, choosing any gives a countable dense subset.

* Properly with a generic referent: " Any cow gives milk.

Any meromorphic function f gives rise to a divisor denoted ( f ) defined as

* Any Lie group gives a Lie groupoid with one object, and conversely.

* Any foliation gives a Lie groupoid.

* Any principal bundle with structure group G gives a groupoid, namely over M, where G acts on the pairs componentwise.

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 point x gives rise to a continuous function p < sub > x </ sub > from the one element topological space 1 ( all subsets of which are open ) to the space X by defining p < sub > x </ sub >( 1 ) = x. Conversely, any function from 1 to X clearly determines one point: the element that it " points " to.

Any prime number p gives rise to an ideal pO < sub > K </ sub > in the ring of integers O < sub > K </ sub > of a quadratic field K.

Any n-dimensional formal group law gives an n dimensional Lie algebra over the ring R, defined in terms of the quadratic part F < sub > 2 </ sub > of the formal group law.

Any surface that intersects the wire has current I passing through it so Ampère's law gives the correct magnetic field.

Any ring homomorphism A → B gives a map K < sub > 0 </ sub >( A ) → K < sub > 0 </ sub >( B ) by mapping ( the class of ) a projective A-module M to M ⊗< sub > A </ sub > B, making K < sub > 0 </ sub > a covariant functor.

:( 1 ) Any person who manufactures, sells or hires or offers for sale or hire, or exposes or has in his possession for the purpose of sale or hire or lends or gives to any other person —

Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy.

Any skew semistandard tableau of shape / with positive integer entries gives rise to a sequence of partitions ( or Young diagrams ), by starting with, and taking for the partition places further in the sequence the one whose diagram is obtained from that of by adding all the boxes that contain a value ≤ in ; this partition eventually becomes equal to.

Any attempt to drastically increase this range ( by increasing the angular extent of the true radius lobe nose region ) soon gives rise to problems with excessive rates of valve acceleration etc.

Any nontrivial line arrangement on RP < sup > 2 </ sup > defines a graph in which each face is bounded by at least three edges, and each edge bounds two faces ; so, double counting gives the additional inequality F ≤ 2E / 3.

Let now v and w be two vertices in G. Any spanning tree contains precisely one simple path between v and w. Taking this path in the uniform spanning tree gives a random simple path.

Any symmetric space gives an involutory quandle, where is the result of ' reflecting through '.

Any topological space X gives rise to a category where the objects are the points of X and the morphisms are the homotopy classes of paths.

In his letter, Polding gives Wardell a completely free hand in the design, saying " Any plan, any style, anything that is beautiful and grand, to the extent of our power.

Any premetric gives rise to a preclosure operator cl as follows:

Any place called Altenberg may have given rise to Altenberg as a family name, such as:

Any significant rise in plasma osmolality is detected by the hypothalamus, which communicates directly with the posterior pituitary gland.

Any knowledge of the early years of human civilization – the development of agriculture, cult practices of folk religion, the rise of the first cities – must come from archaeology.

Any compound that causes a rise in acetylcholine concentration can potentially overcome BZ-induced inhibition and restore normal functioning ; even the nerve agent VX has been shown to be effective when given under carefully controlled conditions.

Any object that has a mass that is less than the mass of an equal volume of air will rise in air-in other words, any object less dense than air will rise.

Any object that has a mass that is less than the mass of an equal volume of air will rise in air-in other words, any object less dense than air will rise.

Any rise between used with any going ( tread ) between

Any purchase of souvenirs will help the local communities and enabling them to rise out of poverty.

Any natural scene or color photograph can be optically and physiologically dissected into three primary colors, red, green and blue, roughly equal amounts of which give rise to the perception of white, and different proportions of which give rise to the visual sensations of all other colors.

Any number of Fellows may rise to degrees below the ninetieth, but there are only ever three Fellows in the ranks 90 to 99.

1 ) Any non-diagonalizable intersection form gives rise to a four-dimensional topological manifold with no differentiable structure ( so cannot be smoothed ).

Any convex polyhedron can be distorted into a canonical form, in which a midsphere ( or intersphere ) exists tangent to every edge, such that the average position of these points is the center of the sphere, and this form is unique up to congruences.

Any ellipse can be obtained by rotation and translation of a canonical ellipse with the proper semi-diameters.

Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows.

Any particular Boolean function can be represented by one and only one full disjunctive normal form, one of the two canonical forms.

Any two meromorphic 1-forms will yield linearly equivalent divisors, so the canonical divisor is uniquely determined up to linear equivalence ( hence " the " canonical divisor ).

Any canonical transformation involving a type-2 generating function G < sub > 2 </ sub >( q, P, t ) leads to the relations

Any sheaf has a canonical embedding into the flasque sheaf of all possibly discontinuous sections of the étalé space, and by repeating this we can find a canonical flasque resolution for any sheaf.

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 base change of a finite morphism is finite, i. e. if is another ( arbitrary ) morphism, then the canonical morphism is finite.