If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

If symptoms occur, a determination of electrolytes may be helpful in determining the underlying cause.

If this goal is met by a DBMS, then the designers and builders of the specific database can concentrate on the application's aspects, and not deal with building and maintaining the underlying DBMS.

If we compress data in a manner that assumes q ( X ) is the distribution underlying some data, when, in reality, p ( X ) is the correct distribution, the Kullback – Leibler divergence is the number of average additional bits per datum necessary for compression.

If the underlying manifold is supersymmetric, the resulting theory is a supersymmetric Yang – Mills theory.

If the underlying manifold is allowed to be infinite dimensional ( for example, a Hilbert manifold ), then one arrives at the notion of an infinite-dimensional Lie group.

If LEMS is caused by an underlying cancer, treatment of the malignancy usually leads to resolution of the symptoms.

If there is an underlying cause, the condition is termed " secondary intracranial hypertension ".

If this is true, then it must be the case that the wetness of water, the hardness of ice, are not essential to the underlying substance.

If care is taken not to cut too deeply, the underlying wraps will protect the bound object from being damaged by the knife.

If g is a function from the underlying set of an object C to the underlying set of A, and if its composition with ƒ is a morphism ƒg: C → B, then g itself is a morphism.

If short-term interest rates were expected to fall in a contango market, this would narrow the spread between a futures contract and an underlying asset in good supply.

If, on the other hand, the spread between a future traded on an underlying asset and the spot price of the underlying asset was set to widen, possibly due to a rise in short-term interest rates, then an investor would be advised to sell the spread ( i. e. a calendar spread where the trader sells the near-dated instrument and simultaneously buys the future on the underlying ).

If the computational Diffie – Hellman assumption holds in the underlying cyclic group, then the encryption function is one-way.

If no underlying cause can be identified, the condition is considered idiopathic.

If the concept of belief is incoherent, then any attempt to find the underlying neural processes that support it will fail.

If dwarfism is caused by a medical disorder, the person is referred to by the underlying diagnosed disorder.

If the underlying problem is well-posed, there should be a stable algorithm for solving it.

If properly treated, typical cases of surgically correctable peritonitis ( e. g., perforated peptic ulcer, appendicitis, and diverticulitis ) have a mortality rate of about < 10 % in otherwise healthy patients, which rises to about 40 % in the elderly, and / or in those with significant underlying illness, as well as in cases that present late ( after 48 hours ).

If natural selection is the primary error correction mechanism that prevents random mutations on the Y chromosome, then fertility treatments for men with abnormal sperm ( in particular ICSI ) only defer the underlying problem to the next male generation.

If the strike is K and maturity time is T, and the buyer exercises the put at a time t, the buyer can expect to receive a payout of K-S ( t ), if the price of the underlying S ( t ) at that time is less than K. The exercise t must occur by time T ; precisely what exact times are allowed is specified by the type of put option.

If the underlying stock's market price is below the option's strike price when expiration arrives, the option owner ( buyer ) can exercise the put option, forcing the writer to buy the underlying stock at the strike price.

If tyrosine and a system generating hydrogen peroxide are added to a cell-free homogenate of the thyroid, large quantities of free mono-iodotyrosine can be formed ( Alexander, 1959 ).

If water is evaporated too quickly, the membrane dries, resistance across it increases, and eventually it will crack, creating a gas " short circuit " where hydrogen and oxygen combine directly, generating heat that will damage the fuel cell.

More precisely, if A is a finite set of generators for G then the word problem is the membership problem for the formal language of all words in A and a formal set of inverses that map to the identity under the natural map from the free monoid with involution on A to the group G. If B is another finite generating set for G, then the word problem over the generating set B is equivalent to the word problem over the generating set A.

If a simple pendulum is suspended from the cusp of an inverted cycloid, such that the " string " is constrained between the adjacent arcs of the cycloid, and the pendulum's length is equal to that of half the arc length of the cycloid ( i. e. twice the diameter of the generating circle ), the bob of the pendulum also traces a cycloid path.

If instead of simply generating random characters one restricts the generator to a meaningful vocabulary and conservatively following grammar rules, like using a context-free grammar, then a random document generated this way can even fool some humans ( at least on a cursory reading ) as shown in the experiments with SCIgen, snarXiv, and the Postmodernism Generator.

If the pH is high ( the concentration of hydrogen ions is low ), the equilibrium shifts to the left: the hydroxide ion abstracts a proton from the ammonium ion, generating ammonia.

If the resulting cavity is not large enough, it may be extended by internally generating additional gas to inject into the cavity.

If G is any group, and S is a generating subset of G, then every element of G is also of the above form ; but in general, these products will not uniquely describe an element of G.

If is a rational generating function, is also

If a < sub > n </ sub > is the probability mass function of a discrete random variable, then its ordinary generating function is called a probability-generating function.

If a < sub > n </ sub > is a Dirichlet character then its Dirichlet series generating function is called a Dirichlet L-series.

If the culture is valuable, then it holds the potential for generating sustained competitive advantages.

If boundary conditions are applied to this system ( e. g., at two physical locations in space ), then only certain values of satisfy the boundary conditions, generating corresponding discrete eigenvalues.

If computationalism is correct, and if there is no problem in generating artificial consciousness or cognition, it would establish the theoretical possibility of a simulated reality.

" Publications like The Austin Chronicle and Dallas Observer came to the conclusion that despite good acting and special effects, the movie lacked cohesion ; talking about the special effects, the Observer lamented " If only generating a soul for the film itself were so easy.

If muon-catalyzed d-t nuclear fusion were able to be realized practically, it would be a much cheaper way of generating power than conventional nuclear fission reactors because muon-catalyzed d-t nuclear fusion ( like most other types of nuclear fusion ), produces far fewer harmful ( and far less long-lived ) radioactive wastes, and hardly any greenhouse gases.

If the condition of generating a prime ideal is removed, such a set is called a projective algebraic set.

If programs are modifiable at runtime or if incremental compilation is available ( such as in C #, Forth, Frink, Groovy, JavaScript, Lisp, Lua, Perl, PHP, Python, REBOL, Ruby, Smalltalk, and Tcl ), then techniques can be used to perform metaprogramming without actually generating source code.

If an entity looks sufficiently nonhuman, its human characteristics will be noticeable, generating empathy.

If a return is not filed after the computer generated letters, such as TX11 and TX14, a non-filer officer could arbitrarily prepare a tax return for the taxpayer, normally generating a larger tax bill than what the taxpayer would expect.

If a substitute check is needed, the transmitting bank is responsible for the cost of generating and transporting it from the presentment point to the Federal Reserve Bank or other corresponding bank.

* If μ is a symmetric probability measure on Γ with support generating Γ, then convolution by μ defines an operator of norm 1 on l < sup > 2 </ sup >( Γ ) ( Kesten ).