If the gas is inhaled, some of the radon particles may attach to the inner lining of the lung.

If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity.

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

If μ is both inner regular and locally finite, it is called a Radon measure.

If the earlier completion procedure is applied to a normed vector space, the result is a Banach space containing the original space as a dense subspace, and if it is applied to an inner product space, the result is a Hilbert space containing the original space as a dense subspace.

The parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

If one alters a Euclidean space so that its inner product becomes negative in one or more directions, then the result is a pseudo-Euclidean space.

If a charged Leyden jar is discharged by shorting the inner and outer coatings and left to sit for a few minutes, the jar will recover some of its previous charge, and a second spark can be obtained from it.

If no nucleus is present, but irregular and small dark inner spots indicating a cavity are visible, combined with concentric rings of organic substance, the pearl is likely a cultured freshwater.

If and are any complex numbers, the inner product is the standard inner product and the bar notation is used for complex conjugation then the inequality may be restated in more explicitly as

If Halley was once a long-period comet, it is likely to have originated in the Oort Cloud, a sphere of cometary bodies that has an inner edge of 20, 000 – 50, 000 AU.

If Earth's moon formed by such an impact, it is likely that other inner planets also would have been subjected to comparable impacts.

If the inner automorphism group of a perfect group G is simple, then G is called quasisimple.

If G arises as the group of units of a ring A, then an inner automorphism on G can be extended to a projectivity on the projective space over A by inversive ring geometry.

If the rainwater overflows the graduated inner cylinder, the larger outer container will catch it.

If W = V, then one can also pair the covector w * ∈ V * with the vector v ∈ V via ( w *, v ) → w *( v ), which is the duality pairing between V and its dual, sometimes called the inner product.

If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold.

If the cage is grounded, the excess charges will go to the ground instead of the outer face, so the inner face and the inner charge will cancel each other out and the rest of the cage will retain a neutral charge.

If the spacecraft's purpose is only to fly by the inner planet, then there is typically no need to slow the spacecraft.

If they fall from the utricle and become loose in the semicircular canals, they can distort the sense of movement and cause a mismatch between actual head movement and the information sent to the brain by the inner ear, causing a spinning sensation.

If any humidity subsists in the compressed air, the engine will stop due to inner icing.

If the star is in a binary system then the material will fall in through the inner Lagrangian point.

If Af denotes the space of N times continuously differentiable functions, then the space V of solutions of this differential equation is a subspace of Af.

If we are discussing differentiable complex-valued functions, then Af and V are complex vector spaces, and Af may be any complex numbers.

* If M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

If x is held fixed, then the Bessel functions are entire functions of α.

If one axiomatizes relations instead of functions, one obtains the theory of allegories.

If we take the simple molecular orbital description of the ground state and combine that function with the functions describing all possible excited states using unoccupied orbitals arising from the same set of atomic orbitals, we also reach the full configuration interaction wavefunction.

If F is clear from context then Ω < sub > F </ sub > may be denoted simply Ω, although different prefix-free universal computable functions lead to different values of Ω.

If ( 1a ) and ( 1b ) hold for a differentiable pair of functions u and v, then so do

If X is a set and M is a complete metric space, then the set B ( X, M ) of all bounded functions ƒ from X to M is a complete metric space.

If X is a topological space and M is a complete metric space, then the set C < sub > b </ sub >( X, M ) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B ( X, M ) and hence also complete.

If one identifies C with R < sup > 2 </ sup >, then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the Cauchy-Riemann equations, a set of two partial differential equations.

If a hellbender ends up in an area of slowly moving water, not enough water will pass over its skin in a given time period, making it difficult to garner enough oxygen to support necessary respiratory functions.

If we think of composition as a kind of multiplication of functions, this identity says that the inverse of a function is analogous to a multiplicative inverse.

If the Board recommends approval, and the General Conference approves the application for membership, the State must then submit its instrument of acceptance of the IAEA Statute to the United States, which functions as the depositary Government for the IAEA Statute.

* If the molecule has some symmetry, the degenerate atomic orbitals ( with the same atomic energy ) are grouped in linear combinations ( called symmetry-adapted atomic orbitals ( SO )), which belong to the representation of the symmetry group, so the wave functions that describe the group are known as symmetry-adapted linear combinations ( SALC ).

If motivation does not take place in an organization, then employees may not contribute to the other functions ( which are usually set by top-level management ).

If f and g are two multiplicative functions, one defines a new multiplicative function f * g, the Dirichlet convolution of f and g, by