Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Abstractly, in trying to prove a proposition P, one assumes that it is false, and that therefore there is at least one counterexample.

Abstractly programmable toolpath guidance began with mechanical solutions, such as in musical box cams and Jacquard looms.

Abstractly, 1 / 8-schisma tuning may be considered the analog, among schismatic tunings, of 1 / 4-comma meantone among meantone tunings, as it also has pure interval ratios of 2: 1 and 5: 4, though with much more accurate interval ratios of 3: 2 and 6: 5 ( less than a quarter of a cent off from just intonation ) than its meantone counterpart.

Abstractly, this circle is a cyclic group of order twelve, and may be identified with the residue classes modulo twelve.

Abstractly, an array reference is simply a procedure of two arguments: an array and a subscript vector, which could be expressed as ; but many languages provide special syntax like ; similarly an array element update is abstractly something like, but many languages provide syntax like.

Abstractly, it can be defined as the integer triples ( a, b, c ), associated to 3 < sup > a </ sup > 5 < sup > b </ sup > 7 < sup > c </ sup >, where the distance measure is not the usual Euclidean distance but rather the Euclidean distance deriving from the vector space norm

Abstractly we can say that this is a projective line in the space of all conics, on which we take

Morgan hesitated, thinking that if this was a trick, it was a good one.

There's only one way they can get out now and that's through the Gap -- if we ride hard we can take them ''.

As I dug in behind one of the bales we were using as protection, I grudgingly found myself agreeing with Oso's logic, especially when I imagined what would have happened to Missy if Old Knife's large party of screeching warriors had overrun our company.

But she was caught in it, and she faced the terrible possibility that, if it were a dream, it was one from which she might never awaken.

Greg climbed into the cockpit feeling as if he had never been in one before.

The clearly identifiable enemy continued on as if no one else were around.

The spokes were tight again, the iron tires gripped onto the wheels as if of one piece.

While his comrades cocked the trap, that one behaved as if it was some dull maneuver.

What if one or more of them turn irrational or suddenly, coolly, decide to clobber the Russians??

I asked Wisman what would happen if he broke out the go codes and tried to start transmitting one.

The Negro faces as much, if not more, difficulty in fitting himself into an urban economy as he did in an agrarian one.

I was having lunch not long ago ( apologies to N. V. Peale ) with three distinguished historians ( one specializing in the European Middle Ages, one in American history, and one in the Far East ), and I asked them if they could name instances where the general mores had been radically changed with `` deliberate speed, majestic instancy '' ( Francis Thompson's words for the Hound Of Heaven's Pursuit ) by judicial fiat.

Operating as a one man police force in fact if not in name, he is at once more independent and more dedicated than the police themselves.

Even if people do, in a not far distant future, begin to read one another's minds, there will still be the question of whether what you find in another man's mind is especially worth reading -- worth more, that is, than what you can read in good books.

Even if men eventually find themselves able to look through walls and around corners, one may question whether this will help them to live better lives.

At one time it seemed as if the Soviet Union had done us a favor by providing a striking example of how not to behave towards other peoples and other nations.

but both groups were so closely knit that despite individual differences the family life in both cases was remarkably similar in atmosphere if not entirely in content -- the one being definitely Jewish and the other vaguely Christian.

At the same time, I am aware that my recoil could be interpreted by readers of the tea leaves at the bottom of my psyche as an incestuous sign, since theirs is a science of paradox: if one hates, they say it is because one loves ; ;

if one bullies, they say it is because one is afraid ; ;

if one shuns, they say it is because one desires ; ;

As I see it, if war starts and we survive the initial attack enough to be able to fight back, the nuclear weapons we now have -- at least the bombs -- can inflict all the demage that is necessary.

All will do a good job if sharp, but the twist drills don't cut quite as smoothly as the others, since they do not have the outlining spurs that sever the fibers before actual boring starts.

and independent silhouettes are apt to coincide with the recognizable contours of the subject from which a picture starts ( if it does start from a subject ).

Watterson opposed the structure publishers imposed on Sunday newspaper cartoons: the standard cartoon starts with a large, wide rectangle featuring the cartoon's logo or a throwaway panel tangential to the main area so that newspapers pressed for space can remove the top third of the cartoon if they wish ; the rest of the strip is presented in a series of rectangles of different widths.

because if one starts with the bra

The board starts off with a portion of the map left blank: when players expand into the blank region, terrain hexes are drawn at random from a supply and placed in the empty space, and, if a land hex is " discovered ", a number token may be assigned.

He refers to British director Christopher Nolan ’ s The Dark Knight and Inception as British rather than as American films, and yet " when a movie which looks quintessentially ‘ British ’, such as The King's Speech, achieves equivalent success, everyone suddenly starts writing articles about the state of our national cinema as if it somehow exists in isolation.

For example, Tropaeolum speciosum, while beautiful, can be considered a pest if it seeds and starts to grow where it is not wanted.

I just say that if a man dances effeminately he dances badly — just as if a woman comes out on stage and starts to sing bass.

An image starts with a fixed-length Image Descriptor, which may specify the presence and size of a Local Color Table ( which follows next if present ).

It would be a dangerous illusion to think that, if war once starts, it will come to an early end even if a success on any one of the several fronts on which it will be engaged should have been secured " Ribbentrop for his part told Hitler that Chamberlain's letter was just a bluff, and urged his master to call it.

While data reduction ( compression, be it lossy or lossless ) is a main goal of transform coding, it also allows other goals: one may represent data more accurately for the original amount of space – for example, in principle, if one starts with an analog or high-resolution digital master, an MP3 file of a given size should provide a better representation than a raw uncompressed audio in WAV or AIFF file of the same size.

Likewise, if the clock starts at 12: 00 ( noon ) and 21 hours elapse, then the time will be 9: 00 the next day, rather than 33: 00.

For example, if one starts with Euler's totient function, and repeatedly applies the transformation process, one obtains:

The game starts with any number of players and a number of chairs one fewer than the number of players ; the chairs are arranged in a circle ( or other closed figure if space is constrained ; a double line is sometimes used ) facing outward, with the people standing in a circle just outside of that.

Then, we can show that if the game starts with n spots, it will end in no more than 3n − 1 moves and no fewer than 2n moves.

Before the wedding starts, Steve takes Fran for a walk and asks her if they ’ re doing the right thing.

* The number of Xs is always either equal to or exactly 1 more than the number of Os ( if X starts ).

* In Ruby a variable is automatically considered immutable if its identifier starts with a capital letter.

Similarly, if your length measurement on your micrometer " starts at one " instead of " starting at zero ", then you must remember to subtract that one from your total value to yield the true length of the object by itself.

He buys his dream car ( a 1970 Pontiac Firebird ) and starts working out after he overhears Angela tell Jane that she would find him sexually attractive if he improved his physique.

It will automatically tighten and become stiff if the bird starts to lose hold of the branch on which it is perching.

In this protocol, before sending the data, the station senses the channel and if the channel is idle it starts transmitting the data.