* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

* If the high and low signal levels have the same voltage with opposite polarity, coded signals have zero average DC voltage, thus reducing the necessary transmitting power and minimizing the amount of electromagnetic noise produced by the transmission line.

If there are to be multiple ICP-MS instruments running for long periods of time, then it will most likely be beneficial for the laboratory to install a bulk or micro bulk argon tank which will be maintained by a gas supply company, thus eliminating the need to change out tanks frequently as well as minimizing loss of argon that is left over in each used tank as well as down time for tank changeover.

If the last equality is satisfied for all t < sub > 1 </ sub >, t < sub > 2 </ sub > ∈ I, the geodesic is called a minimizing geodesic or shortest path.

For networks, they are minimizing most of their risk, but they risk losing out on most profit. If the network orders enough episodes of a show, the studio can then sell the series to other markets.

If ∇ is a metric connection, then the affine geodesics are the usual geodesics of Riemannian geometry and are the locally distance minimizing curves.

If the Black Locust is cut, split, and cured while relatively young ( within ten years ), thus minimizing beetle damage, " spitting " problems are minimal.

*: If a length-metric space ( M, d ) is complete and locally compact then any two points in M can be connected by minimizing geodesic, and any bounded closed set in M is compact.

If the law dictionary attempts to cover all the terms in the field of law, it is called a maximizing dictionary, and if it attempts to cover only a limited number of terms, it is called a minimizing dictionary.

If the Newton-Raphson search method is employed in minimizing the penalty function in beamforming, the resulting beamformer is called Newton ML beamformer.

If you can't tell, get help from your county agricultural agent or other local specialist.

If BCG is accidentally given subcutaneously, then a local abscess may form ( a BCG-oma ) that can sometimes ulcerate, and may require treatment with antibiotics.

If given subcutaneously, it may induce local infection and spread to the regional lymph nodes, causing either suppurative and nonsuppurative lymphadenitis.

If the eigenvalues are all positive, then the frequencies are all real and the stationary point is a local minimum.

If it just includes any PUBLIC identifier, it may be processed as standalone, if the XML processor knows this PUBLIC identifier in its local catalog from where it can retrieve an associated DTD entity.

If the low-order bit is 0, the remaining 7 bits of the DSAP specify an individual address, which refers to a single local service access point ( LSAP ) to which the packet should be delivered.

If the cities were treated as relatively autonomous local administrative areas, it would simplify the problems of imperial administration, which as far as Julian was concerned, should be focused on the administration of the law and defense of the empire's vast frontiers.

If set up properly, sent mail can be saved to the server also, in contrast with POP mail, where sent messages exist only in the local MUA and are not visible by other MUAs accessing the same account.

If the key signature indicates G-sharp, a local flat before a G makes it G-flat ( not G natural ), though often this type of rare accidental is expressed as a natural, followed by a flat () to make this clear.

If an unmarried couple change residence to different countries, then the local law on where the couple is last domiciled is applied to them.

If the local court is actually giving extraterritorial effect to a foreign law, it is less than sovereign and so acting in a way that is potentially unconstitutional.

If the case has been submitted to arbitration rather than a national court, say because of a forum selection clause, an arbitrator may decide not to apply local mandatory policies in the face of a choice of law by the parties if this would defeat their commercial objectives.

If the congregation does not have an installed pastor, the Presbytery appoints a minister member or elected member of the presbytery as moderator with the concurrence of the local church session.

If other products are brought inside along with pests and diseases, it would damage the ecosystem seriously and add millions of costs in the local agricultural businesses.

If the two devices are far enough apart or on separate power systems, the local ground connections at either end of the cable will have differing voltages ; this difference will reduce the noise margin of the signals.

If φ: M → N is a local diffeomorphism at x in M then dφ < sub > x </ sub >: T < sub > x </ sub > M → T < sub > φ ( x )</ sub > N is a linear isomorphism.

If the route was learned from an external peer the per-neighbor BGP process computes a LOCAL_PREFERENCE value from local policy rules and then compares the LOCAL_PREFERENCE of all routes from the neighbor.

If the universe is described by an effective local quantum field theory down to the Planck scale, then we would expect a cosmological constant of the order of.

If the object Servant exists on the same server where the method is invoked, they also mediate the local calls.

If Bell's inequalities are violated, either local realism or counterfactual definiteness must be incorrect ; but some physicists dispute that experiments have demonstrated Bell's violations, on the grounds that the sub-class of inhomogeneous Bell inequalities has not been tested or due to experimental limitations in the tests.

( If utility U = f ( x, y ), U, in the third dimension, does not have a local maximum for any x and y values.

Since a one sided maximal ideal A is not necessarily two-sided, the quotient R / A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J ( R ).

If local governments and private organizations are to carry out decentralized functions effectively, they must have an adequate level of revenues, either raised locally or transferred from the central government, as well as the authority to make decisions about expenditures.

If the people of this nation wish to deprive the states of their sovereign rights to determine what is a fair and just tax upon corporations doing a purely local business within their own state boundaries, there is a way provided by the Constitution to accomplish this purpose.

If f is a differentiable function on R ( or an open interval ) and x is a local maximum or a local minimum of f, then the derivative of f at x is zero ; points where are called critical points or stationary points ( and the value of f at x is called a critical value ).

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 the inner functions of religion are performed, the individual is a composed, ordered, motivated, and emotionally secure associate ; ;

* 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