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 Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

If D denotes the differentiation operator and P is the polynomial Af then V is the null space of the operator p (, ), because Af simply says Af.

If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.

If the quadratic polynomial is monic then the roots are quadratic integers.

If its minimal polynomial has degree, then the algebraic number is said to be of degree.

If the initial value of a polynomial ( and of its finite differences ) is calculated by some means for some value of X, the difference engine can calculate any number of nearby values, using the method generally known as the method of finite differences.

If f is a polynomial and p is big enough, then the remainder term vanishes.

* If f is an irreducible polynomial of prime degree p with rational coefficients and exactly two non-real roots, then the Galois group of f is the full symmetric group S < sub > p </ sub >.

If is a left-( respectively right -) Noetherian ring, then the polynomial ring is also a left-( respectively right -) Noetherian ring.

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.

A number a is a root of P if and only if the polynomial x − a ( of degree one in x ) divides P. It may happen that x − a divides P more than once: if ( x − a )< sup > 2 </ sup > divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that ( x − a )< sup > m </ sup > divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined ( or infinite ) multiplicity.

If, however, the set of allowed candidates is expanded to the complex numbers, every non-constant polynomial has at least one root ; this is the fundamental theorem of algebra.

If R is commutative, then one can associate to every polynomial P in R, a polynomial function f with domain and range equal to R ( more generally one can take domain and range to be the same unital associative algebra over R ).

If R is an integral domain and f and g are polynomials in R, it is said that f divides g or f is a divisor of g if there exists a polynomial q in R such that f q = g. One can show that every zero gives rise to a linear divisor, or more formally, if f is a polynomial in R and r is an element of R such that f ( r ) = 0, then the polynomial ( X − r ) divides f. The converse is also true.

If, as she walked, her steps fumbled from time to time, she chose to ignore that omen.

If the bluff failed and they ran into trouble, Brannon had told the others, they would withdraw -- and he would come after his son another time.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If the would-be joiner asks these questions he is not likely to be duped by extremists who are seeking to capitalize on the confusions and the patriotic apprehensions of Americans in a troubled time.

If the world comes to this, wouldn't it be the very time when courage and American know-how would be needed to help survivors rebuild??

If their schedules were to synchronize, there was no point in wasting time.

`` If I catch you one more time down here without stockings '' --

If there is time after the warning, the basement shielding could be improved substantially by blocking windows with bricks, dirt, books, magazines, or other heavy material.

If, in the trustee's judgment, `` reasonable market conditions '' did not prevail during any given year, he was to be allowed to petition the court for an extension of time within the ten-year period.

If the reserve components are to serve effectively in time of war, their basic organization and objectives must conform to the changing character and missions of the active forces.

If you elect to use the Standard Deduction or the Tax Table, and later find you should have itemized your deductions, you may do so by filing an amended return within the time prescribed for filing a claim for refund.

If you mail A return or tax payment, you must place it in the mails in ample time to reach the district director on or before the due date.

If the lever is pulled to clear a signal or move a switch, the tappet moves a short distance lengthwise at the same time.

If necessary to replace both halves on grill, sear cuts and allot extra time.

If the background of design is too smooth, or you wish to create a wood-grained effect, it may be added at this time with a dull tool such as the handle of a fine paintbrush.

If companies will take the time to give objective consideration to their major problems and to the questions they provoke, then a long constructive step will have been taken toward more effective marketing in next decade.

If you've travelled in Europe a time or two, it is quite certain that you've had that wanting-to-be-alone feeling or that you will get it on your next visit across the Atlantic.

`` If you are trying to get us out of the brothel, the dustbin, the kitchen sink, and the tawdry living-room, you are probably wasting your time '', Moreland told me.

If, at any time during the assignment pass, the compiler finds that there are no more index words available for assignment, the warning message `` No More Index Words Available '' will be placed in the object program listing, the table will be altered to show that index words 1 through 96 are available, and the assignment will continue as before.

If Depew had told any academic psychologist that he had a weird feeling of having lived through that identical convention session at some time in the past, he would have been informed that he was a victim of deja vue.

If baby teeth are retained too long, the incoming second teeth may be prevented from emerging at the normal time or may have to erupt in the wrong place.

If one lives near a subway or an express parkway, the solution is to have one's wines stored with a dealer and brought home a few at a time.

If this woman had delayed until after 11:20 to start her shopping, she would have had little time in which to prepare the substantial meal that was eaten at dinner in those days.

If you have taken this stroll in the morning, and you have the time and inclination, walk to the right along the crowded Corso for half a dozen blocks to visit the fine private collection of paintings -- mainly of the sixteenth and seventeenth centuries -- in the Palazzo Doria ( open Sunday, Tuesday, and Thursday, 10:00 to 1:00 ).

If they don't then for the algorithm to be effective it must provide a set of rules for extracting a square root.

If a " weak " character is followed by another " weak " character, the algorithm will look at the first neighbouring " strong " character.

If the algorithm is symmetric, the key must be known to the recipient and sender and to no one else.

If the algorithm is an asymmetric one, the enciphering key is different from, but closely related to, the deciphering key.

If one key cannot be deduced from the other, the asymmetric key algorithm has the public / private key property and one of the keys may be made public without loss of confidentiality.

If a suitably sized quantum computer capable of running Grover's algorithm reliably becomes available, it would reduce a 128-bit key down to 64-bit security, roughly a DES equivalent.

If a naive rendering algorithm is used without any filtering, high frequencies in the image function will cause ugly aliasing to be present in the final image.

If implemented using remainders of Euclidean division rather than subtractions, Euclid's algorithm computes the GCD of large numbers efficiently: it never requires more division steps than five times the number of digits ( base 10 ) of the smaller integer.

If a is smaller than b, the first step of the algorithm swaps the numbers.

If the arguments are both greater than zero then the algorithm can be written in more elementary terms as follows:

If R is a Euclidean domain in which euclidean division is given algorithmically ( as is the case for instance when R = F where F is a field, or when R is the ring of Gaussian integers ), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.

If the algorithm is unable to reduce A to triangular form, then A is not invertible.

If the true function is simple, then an " inflexible " learning algorithm with high bias and low variance will be able to learn it from a small amount of data.

If the desired output values are often incorrect ( because of human error or sensor errors ), then the learning algorithm should not attempt to find a function that exactly matches the training examples.

* Robustness: If the model, cost function and learning algorithm are selected appropriately the resulting ANN can be extremely robust.

If using efficient sorting such as mergesort or heapsort this is an O ( n log n ) average and worst-case algorithm.

The algorithm for deciding this is conceptually simple: it constructs ( the description of ) a new program t taking an argument n which ( 1 ) first executes program a on input i ( both a and i being hard-coded into the definition of t ), and ( 2 ) then returns the square of n. If a ( i ) runs forever, then t will never get to step ( 2 ), regardless of n. Then clearly, t is a function for computing squares if and only if step ( 1 ) terminates.

If we apply the run-length encoding ( RLE ) data compression algorithm to the above hypothetical scan line, we get the following:

If not, is there an algorithm to determine how many are required?

If the user wanted to see a font at another size, the Font Manager would find the closest match and apply a basic scaling algorithm.

If a particle visits every point on its trajectory then the motion is an algorithm for each point.

If two processes attempt to enter a critical section at the same time, the algorithm will allow only one process in, based on whose turn it is.