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If the direct-sum decomposition ( A ) is valid, how can we get hold of the projections Af associated with the decomposition??
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If and decomposition
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.
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 sugars are heated so that all water of crystallisation is driven off, then caramelization starts, with the sugar undergoing thermal decomposition with the formation of carbon, and other breakdown products producing caramel.
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 the fire was small, however, the convict would burn for some time until death from heatstroke, shock, the loss of blood and / or simply the thermal decomposition of vital body parts.
If I is a homogeneous ideal in A, then is also a graded ring, and has decomposition
If the LU decomposition is used, then the algorithm is unstable unless we use some sort of pivoting strategy.
If we are given the components of the strain tensor in an arbitrary orthonormal coordinate system, we can find the principal strains using an eigenvalue decomposition determined by solving the system of equations
If the LU decomposition exists, the LDU decomposition does too.
If Gaussian elimination produces the row echelon form without requiring any row interchanges, then P = I, so an LU decomposition exists.
If the n eigenvalues are distinct ( that is, none is equal to any of the others ), then V is invertible, implying the decomposition.
If is the singular value decomposition of, then.
If the protocol is unconditionally concealing, then Alice can unitarily transform these states into each other using the properties of the Schmidt decomposition, effectively defeating the bindingness property.
If the matrix X < sup > T </ sup > X is well-conditioned and positive definite, that is, it has full rank, the normal equations can be solved directly by using the Cholesky decomposition R < sup > T </ sup > R, where R is an upper triangular matrix, giving:
If the measure ' is complex-valued i. e. is a complex measure, its upper and lower variation cannot be defined and the Hahn – Jordan decomposition theorem can only be applied to its real and imaginary parts.
If i ≠ j, then this decomposition does not contain the trivial representation ( Otherwise, we'd have a nonzero intertwiner from V < sub > j </ sub > to V < sub > i </ sub > contradicting Schur's lemma ).
If a split-complex number z does not lie on one of the diagonals, then z has a polar decomposition.
If they settle to the bottom, phytoplankton release ammonia during their decomposition, which returns nitrogen to the waters.
If the vacuum is degenerate and we have a mixed state, the cluster decomposition property fails.
If an integer n has prime factorization, then the primary decomposition of the ideal generated by, is
If cyanogen iodide is heated enough to undergo complete decomposition, it may releases toxic fumes of nitrogen oxides, cyanide and iodide.
If a matrix A is diagonalizable, the space can be decomposed into a direct sum of eigenspaces of A, and the matrix has an eigenspace decomposition.
If and is
If the circumstances are faced frankly it is not reasonable to expect this to be true.
If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.
If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.
If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;
If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.
If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.
If it is an honest feeling, then why should she not yield to it??
If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.
If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.
If he is good, he may not be legal ; ;
If the man on the sidewalk is surprised at this question, it has served as an exclamation.
If the existent form is to be retained new factors that reinforce it must be introduced into the situation.
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 many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.
If he is a traditionalist, he is an eclectic traditionalist.
If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.
If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.
If and valid
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 the valid element indices begin at 0, the constant B is simply the address of the first element of the array.
If the exponent r is even, then the inequality is valid for all real numbers x.
If the client remains connected to a network for which this IP is valid, the server may grant the request.
If time, space, and energy are secondary features derived from a substrate below the Planck scale, then Einstein's hypothetical algebraic system might resolve the EPR paradox ( although Bell's theorem would still be valid ).
If this understanding were valid, then it would it would follow:
If this understanding were valid, then it would it would follow:
If this understanding were valid, then it would it would follow:
If this triangulated graph is colorable using four colors or less, so is the original graph since the same coloring is valid if edges are removed.
If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬ φ is not satisfiable in any structure and therefore refutable ; then ¬¬ φ is provable and then so is φ, thus Theorem 1 holds.
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 works were renewed, the copyrights would still be valid in the United States.
If we neglect the possibility of overlapping states, which is valid if the temperature is high, then the number of times we count each state is approximately N < nowiki >!</ nowiki >.
If a player cannot make a valid move after first move of 6 they must pass the die to the next player.
If the defendant asserts his right to remain silent all interrogation must immediately stop and the police may not resume the interrogation unless the police have " scrupulously honored " the defendant ’ s assertion and obtain a valid waiver before resuming the interrogation.
If the defendant does reinitiate contact, a valid waiver must be obtained before interrogation may resume.
If his reasoning is valid, he goes on to argue, either it is not very immoral to value small luxuries over saving many lives, or such affluent people are very immoral.
If one player cannot make a valid move, play passes back to the other player.
If the substitution of an expression with its value is valid only at a certain point in the execution of the program, then the expression is not referentially transparent.
If an argument form is valid, the conjunction of the premises will be logically equivalent to the conclusion and this can be clearly seen in a truth table ; it is displayed.
# If they do have a demonstrable genetic relationship, do they form a valid linguistic taxon?
If the server is case sensitive and < tt >< nowiki > http :// en. example. org / wiki / URL </ nowiki ></ tt > is correct, < tt >< nowiki > http :// en. example. org / WIKI / URL </ nowiki ></ tt > or < tt >< nowiki > http :// en. example. org / wiki / url </ nowiki ></ tt > will display an HTTP 404 error page, unless these URLs point to valid resources themselves.
If the deviations from ideality are not too strong, Raoult's law will still be valid in a narrow concentration range when approaching x = 1 for the majority phase ( the solvent ).
If the number of bits changed is even, the check bit will be valid and the error will not be detected.
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