If the arity of the operators is fixed, the result is a syntax lacking parentheses or other brackets that can still be parsed without ambiguity.

If X and Y are topological vector spaces, the space L ( X, Y ) of continuous linear operators f: X → Y may carry a variety of different possible topologies.

If not, the amateur operators were required to stop transmitting.

: If and are normal operators and if A is a bounded linear operator such that, then.

If X is a Hilbert space and T is a normal operator, then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators ( Hermitian matrices, for example ).

If one aircraft does not follow the directive ATC issues the directive will have to be reissued and in turn increase the workload of ATM operators.

If the Governor or the other officials of Idaho accept a cent from the operators or from any other capitalist with any reference, direct or indirect, to this prosecution, they would forfeit the respect of every good citizen and I should personally feel that they had committed a real crime.

If the gain was finally set to a 50 ° PB, then 80 % power will not be applied unless the furnace is 15 ° below setpoint, so for this other application the operators will have to remember always to set the setpoint temperature 15 ° higher than actually needed.

If the Lagrangian contains combinations of field operators of high enough dimension in energy units, the counterterms required to cancel all divergences proliferate to infinite number, and, at first glance, the theory would seem to gain an infinite number of free parameters and therefore lose all predictive power, becoming scientifically worthless.

If the parameters are considered as generalized coordinates, then should be identified as the generalized force operators related to those coordinates.

If the unperturbed system is in eigenstate at time, its state at subsequent times varies only by a phase ( we are following the Schrödinger picture, where state vectors evolve in time and operators are constant ):

If i is any inclusion of compact manifolds from X to Y, they defined a ' pushforward ' operation i < sub >!</ sub > on elliptic operators of X to elliptic operators of Y that preserves the index.

If D is a differential operator with adjoint D *, then D * D and DD * are self adjoint operators whose non-zero eigenvalues have the same multiplicities.

If the operators in the family are real, then the index lies in the real K-theory of Y.

If we specialize to categories arising from partially ordered sets ( with a single morphism from to iff ), then the formalism becomes much simpler: adjoint pairs are Galois connections and monads are closure operators.

If L is the Lie algebra corresponding to the Lie group G, U ( L ) can be identified with the algebra of left-invariant differential operators ( of all orders ) on G ; with L lying inside it as the left-invariant vector fields as first-order differential operators.

If an operator is left-associative, the operators are applied in left-to-right order.

If an operator is right-associative, the operators are applied in right-to-left order.

If an operand is both preceded and followed by operators ( for example, "^ 4 ^"), and those operators have equal precedence, then the operand may be used as input to two different operations ( i. e. the two operations indicated by the two operators ).

If a large magnet undergoes a quench, the inert vapor formed by the evaporating cryogenic fluid can present a significant asphyxiation hazard to operators by displacing breathable air.

If the one- or two-digit address of an index word or electronic switch is used or is included in the operand of an XRESERVE or SRESERVE statement ( see page 99 ), the corresponding index word or electronic switch is reserved.

If is a wrapped Cauchy distribution with the parameter representing the parameters of the corresponding " unwrapped " Cauchy distribution in the variable y where, then

If measured instead per unit wavelength, using Wien's law, the peak is at 1. 06 mm corresponding to a frequency of 283 GHz.

To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If ( x, y ) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

If light consisted strictly of ordinary or classical particles, and these particles were fired in a straight line through a slit and allowed to strike a screen on the other side, we would expect to see a pattern corresponding to the size and shape of the slit.

If a polyhedron has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity.

If G = GL < sub >*</ sub >( K ), then the set of natural numbers is a proper subset of G < sub > 0 </ sub >, since for each natural number n, there is a corresponding identity matrix of dimension n. G ( m, n ) is empty unless m = n, in which case it is the set of all nxn invertible matrices.

If the flow were stopped, there would be a corresponding " stopping vortex ".

If there are four alternative symbols ( corresponding to a musical instrument that can generate four different tones, one at a time ), the first symbol may represent the bit sequence 00, the second 01, the third 10 and the fourth 11.

If countN reaches 100, reads from the corresponding channel will be disabled.

A number a is a root of P if and only if the polynomial x − a ( of degree one in x ) divides P. It may happen that x − a divides P more than once: if ( x − a )< sup > 2 </ sup > divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that ( x − a )< sup > m </ sup > divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined ( or infinite ) multiplicity.

If an atom is already in the excited state, it may be perturbed by the passage of a photon that has a frequency ν < sub > 21 </ sub > corresponding to the energy gap ΔE of the excited state to ground state transition.

If each of the player's now-separated hands beat the banker's corresponding hand then he wins the bet.

If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of the wavefunction collapsing into each of the possible eigenstates.

If X is a real-valued random variable and a is a number then the event X ≤ a is the set of outcomes whose corresponding value of X is less than or equal to a.

If a particular point on a sphere is ( arbitrarily ) designated as its north pole, then the corresponding antipodal point is called the south pole and the equator is the great circle that is equidistant to them.

# If n = 2k + 1, then the action of the unit vector u on the left ideal decomposes the space into a pair of isomorphic irreducible eigenspaces ( both denoted by Δ ), corresponding to the respective eigenvalues + 1 and − 1.

If a plane wave, confined to the xz plane, is incident on a dielectric with an angle and wavevector then a transmitted ray will be created with a corresponding angle of transmittance as shown in Fig.

If the key corresponding to a particular character was pressed at the home station, it actuated the typewheel at the distant station just as the same character moved into the printing position, in a way similar to the daisy wheel printer.

" If we placed a living organism in a box ... one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations.

If the world is reflected in a mirror which switches the left and right side of the car, the reflection of this angular velocity vector points to the right, but the actual angular velocity vector of the wheel still points to the left, corresponding to the minus sign.

If an object with comparable mass to that of the Earth were to fall towards it, then the corresponding acceleration of the Earth really would be observable.

* AAS ( Angle-Angle-Side ): If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent.

If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is equal to the length of the adjacent side multiplied by the sine of the angle, then the two triangles are congruent.

If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is greater than the length of the adjacent side multiplied by the sine of the angle ( but less than the length of the adjacent side ), then the two triangles cannot be shown to be congruent.