* 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 ξ ∈ gives rise to a canonical vertical vector field X < sub > ξ </ sub > by taking the derivative of the right action of the 1-parameter subgroup of H associated to ξ.

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 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 multiplication operator is a ( densely defined ) self-adjoint operator.

Any self-adjoint operator is unitarily equivalent to a multiplication operator.

* Any operator of the form

Any polynomial in D with function coefficients is also a differential operator.

If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix.

Any Fredholm operator has an index, defined as the difference between the ( finite ) dimension of the kernel of D ( solutions of Df = 0 ), and the ( finite ) dimension of the cokernel of D ( the constraints on the right-hand-side of an inhomogeneous equation like Df

Any continuity equation can be expressed in an " integral form " ( in terms of a flux integral ), which applies to any finite region, or in a " differential form " ( in terms of the divergence operator ) which applies at a point.

* Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix.

Any differential operator exhibiting this property is called a hypoelliptic operator ; thus, every elliptic operator is hypoelliptic.

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 self-adjoint operator T has a unique Borel functional calculus.

Any slice parallel to either of the axes, through the Fourier operator, is a complex exponential, i. e. the real part is a cosine wave and the imaginary part is a sine wave of the same frequency as the real part.

Any Dirac field can therefore be projected into its left-or right-handed component by the operation of the projection operator ( 1 – γ < sup > 5 </ sup >)/ 2 or ( 1 + γ < sup > 5 </ sup >)/ 2 acting on ψ.

Any normal ordered operator therefore has a vacuum expectation value of zero.

Any unitary operator on H is a partial isometry with initial and final subspaces being all of H.

Any dealer, carrier, exhibitor, handler, operator, or research facility that violates any provision of the AWA may be assessed a penalty of no more than $ 10, 000 for each violation.

Any dealer, exhibitor, carrier, handler, auction operator, or research facility may seek a review of an order within 60 days at the United States Court of Appeals.

* Any associative ring can be made into a Lie ring by defining a bracket operator.