If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

If each node additionally records the size of its subtree ( including itself and its descendants ), then the nodes can be retrieved by index in O ( log n ) time as well.

If there is refraction at a collective spherical surface, or through a thin positive lens, O ' 2 will lie in front of O ' 1 so long as the angle u2 is greater than u1 ( under correction ); and conversely with a dispersive surface or lenses ( over correction ).

If the angle u1 is very small, O ' 1 is the Gaussian image ; and O ' 1 O ' 2 is termed the longitudinal aberration, and O ' 1R the lateral aberration of the pencils with aperture u2.

If the pencil with the angle u2 is that of the maximum aberration of all the pencils transmitted, then in a plane perpendicular to the axis at O ' 1 there is a circular disk of confusion of radius O ' 1R, and in a parallel plane at O ' 2 another one of radius O ' 2R2 ; between these two is situated the disk of least confusion.

If rays issuing from O ( fig.

If the object point O is infinitely distant, u1 and u2 are to be replaced by h1 and h2, the perpendicular heights of incidence ; the sine condition then becomes sin u ' 1 / h1 = sin u ' 2 / h2.

If evolutionary processes are blind to the difference between function F being performed by conscious organism O and non-conscious organism O *, it is unclear what adaptive advantage consciousness could provide.

If a diatomic molecule consists of two atoms of the same element, such as H < sub > 2 </ sub > and O < sub > 2 </ sub >, then it is said to be homonuclear, but otherwise it is heteronuclear.

If we insert vertices in random order, it turns out ( by a somewhat intricate proof ) that each insertion will flip, on average, only O ( 1 ) triangles – although sometimes it will flip many more.

If a skip list is used, the insertion time is brought down to O ( log n ), and swaps are not needed because the skip list is implemented on a linked list structure.

If the items are stored in a linked list, then the list can be sorted with O ( 1 ) additional space.

If a large, b-bit number is the product of two primes that are roughly the same size, then no algorithm has been published that can factor in polynomial time, i. e., that can factor it in time O ( b < sup > k </ sup >) for some constant k. There are published algorithms that are faster than O (( 1 + ε )< sup > b </ sup >) for all positive ε, i. e., sub-exponential.

If the table size n is large enough, linear search will be faster than binary search, whose cost is O ( log n ).

Anderson is best remembered as a film maker for his " Mick Travis trilogy ", all of which star Malcolm McDowell as the title character: If .... ( 1968 ), a satire on public schools ; O Lucky Man!

If the carbonyl is at position 1 ( that is, n or m is zero ), the molecule begins with an formyl group H ( C = O )-, and is technically an aldehyde.

In the biography, Paul Allen wrote, regarding a suggestion in Cosmopolitan that his plays were becoming autobiographical: " If we take that to mean that his plays tell his own life story, he still hasn't started.

If the rearcourt attacker plays a dropshot, his partner will move into the forecourt to threaten the net reply.

* If the offence fails to make ten yards in three plays, the defence takes over on downs.

The song If You Wanna Be Happy by Jimmy Soul plays in the background.

If the modem plays a melody consisting of 1000 tones per second, the symbol rate is 1000 symbols / second, or baud.

If new Shakespearean plays were being written especially for presentation at the Blackfriars ' theatre after 1608, they could not have been written by Edward de Vere.

If no range is specified, the stream is played from the beginning and plays to the end, or, if the stream is paused, it is resumed at the point it was paused.

# Opposite corner: If the opponent is in the corner, the player plays the opposite corner.

* If X plays center opening move, O should pay attention and not allow a fork.

** If O plays a corner or side-middle first, X is guaranteed to win:

*** If O plays a side-middle, X takes the only corner that O's blocking won't make 2 in a row.

* If a player can do an approach and a withdrawal at the same time, he has to choose which one he plays.

If White plays 1.

" If the next player also plays a seven, he announces " have a very nice day " and the player after that draws two penalty cards.

If many of these plays have been judged less compelling than his earlier work, his later work, in which satiric fury is tempered and broadened, also includes three of his acknowledged masterpieces.

If the closest boules from each team are an equal distance from the jack, then the team that played last plays again.

* If, after the trump card is turned up, and before the dealer accepts or refuses, or, if there is no proposal, before he plays, it be discovered that he has dealt himself more than five cards, the non-dealer may claim a fresh deal.

* If, after the trump card is turned up, and before the dealer accepts or refuses, or, if there is no proposal, before he plays, it be discovered that the dealer has fewer than five cards, the non-dealer may permit the dealer to complete his hand from the stock, or may claim a fresh deal.

If the king be the card first played by the dealer, he may announce it at any time before he plays again.

If a contestant plays a false hidden immunity idol, the host would toss it into the fire pit.

* If a contestant plays the hidden immunity idol, any votes cast for that contestant will not count, and the person with the next largest number of votes will be eliminated.

If White plays a regular opening and Black responds in an unconventional way, the opening is not categorized A00.

If South ( declarer ) is on lead he can lead to the queen ; that is, he leads a small spade and, if West plays low, plays the queen from dummy.

-- If there are outside windows in the basement corner where you build a shelter, they should be shielded as shown in the Appendix, page 29.

If the vertex is at Af, and if the interior of C is on the left as one moves in the direction of increasing t, then every such corner can be found from the curve obtained by rotating C clockwise through 90-degrees about the vertex.

If you want a picture get to the corner of Adams and Clark just as fast as you can.

If one set is played, the pieces usually go into the opponent's starting corner, and the number of pieces per side is often increased to 15 ( instead of the usual 10 ).

If two sets are played, the pieces can either go into the opponent's starting corners, or one of the players ' two sets can go into an opposite empty corner.

If a is a central idempotent, then the corner ring is a ring with multiplicative identity a.

If a decomposition exists with each c < sub > i </ sub > a centrally primitive idempotent, then R is a direct sum of the corner rings c < sub > i </ sub > Rc < sub > i </ sub >, each of which is ring irreducible.

( If it is the first move of the game, playing on a corner gives " O " more opportunities to make a mistake and may therefore be the better choice ; however, it makes no difference between perfect players.

* If X does not play center opening move ( playing a corner is the best opening move ), take center, and then a side middle.

** If O takes center ( best move for him ), X should take the corner opposite the original, and proceed as detailed above.

*** If corner, X simply takes any of the other 2 corners, and then the last, a fork.

If binding around a not fully convex, or square-edged object, arrange the knot so the overhand knot portion is stretched across a convex portion, or a corner, with the riding turn squarely on top of it.

If there is more than one reflective surface in the antenna, as in a corner reflector antenna, each surface forms its own image of the antenna elements.

" If you had Jack Gould in your corner, you could not believe what it meant ," said Cooney decades later.

If the corner refuses to open, move to an adjacent corner and keep flexing.

If you spot a fire or emergency you should call 911 or run to the nearest street corner and pull the lever down.