In complexity theory, computational problems that are co-NP-complete are those that are the hardest problems in co-NP, in the sense that they are the ones most likely not to be in P. If there exists a way to solve a co-NP-complete problem quickly, then that algorithm can be used to solve all co-NP problems quickly.

If the search space is fixed, the computational complexity for this class of problems is usually estimated by:

If the bath is written in terms of its set of orthogonal basis kets, that is, if it has been initially diagonalized then < font style =" vertical-align :- 35 %;"></ font > Computing the partial trace with respect to this ( computational ) basis gives:

If the sequence b < sub > n </ sub > were periodic in n with period N, then it would be a cyclic convolution of length N, and the zero-padding would be for computational convenience only.

If a majority or all of the turbulent scales are modeled, the computational cost is very low, but the tradeoff comes in the form of decreased accuracy.

If there is a unifying theme that runs through most of managerial economics, it is the attempt to optimize business decisions given the firm's objectives and given constraints imposed by scarcity, for example through the use of operations research, mathematical programming, game theory for strategic decisions, and other computational methods.

If LOTH cannot show that the mind knows that it is following the particular set of rules in question then the mind is not computational because it is not governed by computational rules.

If the field values at some distance are desired, it is likely that this distance will force the computational domain to be excessively large.

If the decisional Diffie – Hellman assumption ( DDH ) holds in, then

Between then and 2004, Diffie charted 35 cuts on the Billboard Hot Country Songs chart, including five number one singles: his debut release " Home ", " If the Devil Danced ( In Empty Pockets )", " Third Rock from the Sun ", " Pickup Man " ( his longest-lasting number one, at four weeks ) and " Bigger Than the Beatles ".

Diffie co-wrote the album's second and fourth releases, " If You Want Me To " and " New Way ( To Light Up an Old Flame )"; both peaked at number two on Billboard, and the former reached number one on the RPM country music charts in Canada.

The first three singles from the album all reached the Top Ten on the country singles charts: the title track ( which Diffie co-wrote ) and the Dennis Linde composition " John Deere Green " both peaked at number five, with the number three " Prop Me Up Beside the Jukebox ( If I Die )" in between.

Statements such as the Banach – Tarski paradox can be rephrased as conditional statements, for example, " If AC holds, the decomposition in the Banach – Tarski paradox exists.

If amber is heated under the right conditions, oil of amber is produced, and in past times this was combined carefully with nitric acid to create " artificial musk " – a resin with a peculiar musky odor.

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 this etymology is combined with the tradition reported by Geoffrey of Monmouth stating that Ambrosius Aurelianus ordered the building of Stonehenge – which is located within the parish of Amesbury ( and where Ambrosius was supposedly buried ) – and with the presence of an Iron Age hill fort also in that parish, then it may be tempting to connect Ambrosius with Amesbury.

If anything, her poems suggest the opposite – they speak of quietly experienced but intensely felt emotions, hidden from others, without any indication of being requited.

If Acorn had thought to include this small modification in the original Electron design it is likely the machine would have had a much greater impact as it would have nearly doubled the amount of motion possible in games and saved modes 0 – 3 ( including the only 16 colour mode ) from being nearly useless due to contended memory timings.

If during the 14 – 15th centuries armour seldom weighed more than 15 kg, then by the late 16th century it weighed 25 kg.

If this article / noun pair is used as the object of a verb, it ( usually ) changes to the accusative case, which entails an article shift in German – Ich sehe den Wagen.

* If the third baseman fields a ball and throws it to first, it is recorded as a 5 – 3 out.

If the score reaches 20-all, then the game continues until one side gains a two point lead ( such as 24 – 22 ), up to a maximum of 30 points ( 30 – 29 is a winning score ).

But this destruction, coupled with a protracted siege of Rain ( 9 – 16 July ), caused Prince Eugene to lament "... since the Donauwörth action I cannot admire their performances ", and later to conclude " If he has to go home without having achieved his objective, he will certainly be ruined.

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

If a " clinch " – a defensive move in which a boxer wraps his or her opponents arms and holds on to create a pause – is broken by the referee, each fighter must take a full step back before punching again ( alternatively, the referee may direct the fighters to " punch out " of the clinch ).

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.

If a cabal can be implicated in a sequence of events, there is always the hope, however tenuous, of being able to break the cabal's power – or joining it and exercising some of that power oneself.

If the corporate President is not the COO ( such as Richard Parsons of Time Warner from 1995 – 2001 ), then many division heads report directly to the CEO themselves, with the President taking on special assignments from the CEO.

If their existence past the K – T boundary can be confirmed, these hadrosaurids would be considered a Dead Clade Walking.

: " If anyone kills person – unless it be ( a punishment ) for murder or for spreading mischief in the land — it would be as if he killed all people.

* If the Venerable was not a martyr – all non-martyrs are " confessors " as they " confessed " or bore witness to their faith by how they lived their lives – it must be proven that a miracle has taken place by his or her intercession: that is, that God has shown a sign that the person is enjoying the Beatific Vision by God performing a miracle in response to the Blessed's prayers.

In the words of critic Peter Hellman, " If could recite Budweiser jingles from TV, why not give them a program that would teach the ABCs and simple number concepts?

If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

If interstellar colonization is possible, then this assumption is invalid, and the equations of population dynamics would apply instead.

If the wavepacket has a relatively large frequency spread, or if the dispersion has sharp variations ( such as due to a resonance ), or if the packet travels over very long distances, this assumption is not valid.

If the assumption of a " Proto-Human " language is accepted, its date may be set anywhere between 200, 000 years ago ( the age of Homo sapiens ) and 50, 000 years ago ( the age of behavioral modernity ).

If the Zubrin / Andrews assumption were correct then d / dt ( mv < sub > 1 </ sub >) =

If we use NC reductions, that is, reductions which can operate in polylogarithmic time on a parallel computer with a polynomial number of processors, then all P-complete problems lie outside NC and so cannot be effectively parallelized, under the unproven assumption that NC ≠ P. If we use the weaker log-space reduction, this remains true, but additionally we learn that all P-complete problems lie outside L under the weaker unproven assumption that L ≠ P. In this latter case the set P-complete may be smaller.

If instead of assuming a constant rate of usage, the assumption of a constant rate of growth of annually is made, the resource will instead last

If, however, there are some extreme events taken from thunder storms, others from snow pack melting, and others from hurricanes, then this assumption is most likely not valid.

If words and expressions that imply that women or men are inferior are constantly used, that assumption of inferiority tends to become part of our mindset ….

If one is dealing with continuous data, a typical assumption is that the continuous values associated with each class are distributed according to a Gaussian distribution.

If there were not, then the entire graph would be the union of finitely many finite sets, and thus finite, contradicting the assumption that the graph is infinite.

If that the justification of induction cannot be deductive, then it would beg the question for induction to be based on an inductive assumption about a connection.

" If they wish to assert their claim to Trans-Jordan and to avoid raising with other Powers the legal status of that area, they can only do so by proceeding upon the assumption that Trans-Jordan forms part of the area covered by the Palestine Mandate.

... it is incorrect to speak of an assumption as either true or false, since there is no way of proving it to be either ( If there were, it would no longer be an assumption ).

If such a dichotomy theorem is true, then CSPs provide one of the largest known subsets of NP which avoids NP-intermediate problems, whose existence was demonstrated by Ladner's theorem under the assumption that P ≠ NP.

If they share an assumption, belief or understanding of how the contract will be interpreted or what the legal effect will be, they are bound by that belief, assumption or understanding if:

If I ’ m persuaded that our life is predicated upon violence, that the entire structure of what we call civilization, the scaffolding that we ’ ve built up over the millennia, can collapse at any moment, then whatever I write is going to be affected by that assumption.

This most usually implicit assumption must be stated explicitly, otherwise the definition is incorrect: If water is added to a pot containing salty water, the concentration of salt decreases, although there is no chemical reaction.