Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.

Thus D' and N' commute with any polynomial in T ; ;

Thus any non-constant entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire function.

Thus, nowadays, we speak of Diophantine equations when we speak of polynomial equations to which rational or integer solutions must be found.

Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term a < sub > i </ sub > X < sup > i </ sup > is interpreted as a polynomial that has zero coefficients at all powers of X other than X < sup > i </ sup >.

Thus, it is common to speak of the quadratic polynomial

Thus an algorithm to find isolated real roots of a low-degree polynomial in one variable may bear little resemblance to an algorithm for complex roots of a " black-box " function which is not even known to be differentiable.

Thus, for example, polynomial sequences of binomial type are generated by

Thus in the common context of polynomials with rational coefficients, a polynomial is irreducible if it cannot be expressed as the product of two or more such polynomials, each of them having a lower degree than the original one.

Thus, if the permanent can be computed in polynomial time by any method, then FP = # P, which is an even stronger statement than P = NP.

Thus, it is more appropriate to speak of " the Newton form of the interpolation polynomial " rather than of " the Newton interpolation polynomial ".

Thus the Newton polynomial can be written as

Thus the function L ( x ) is a polynomial with degree at most k and where.

Thus the interpolating polynomial then is

Thus the roots of the function det (' I − A ) are the eigenvalues of A, and it is clear that this determinant is a polynomial in.

Thus, the monic polynomials form a multiplicative semigroup of the polynomial ring A.

Thus the Alexander polynomial for a cinquefoil is P ( x )

More specifically, the running time for testing whether H is a minor of G in this case is O ( n < sup > 3 </ sup >), where n is the number of vertices in G and the big O notation hides a constant that depends superexponentially on H. Thus, by applying the polynomial time algorithm for testing whether a given graph contains any of the forbidden minors, it is possible to recognize the members of any minor-closed family in polynomial time.

Thus the entire polynomial is the product of these terms, and according to vector mathematics the angles add ( or subtract, for terms in the denominator ) and lengths multiply ( or divide ).

Thus, the Weyl algebra is a quantization of the symmetric algebra, which is essentially the same as the Moyal quantization ( if for the latter one restricts to polynomial functions ), but the former is in terms of generators and relations ( considered to be differential operators ) and the latter is in terms of a deformed multiplication.

Thus eigenvalue algorithms that work by finding the roots of the characteristic polynomial can be ill-conditioned even when the problem is not.

Thus, to cite but one example, the Pax Britannica of the nineteenth century, whether with the British navy ruling the seas or with the City of London ruling world finance, was strictly national in motivation, however much other nations ( e.g., the United States ) may have incidentally benefited.

Thus, on the highroad, a troop of soldiers `` marched in their own dust and sang '', while on the footpath one man walks alone.

Thus the cocktail party would appear to be the ideal system, but there is one weakness.

Thus, circular motion is itself one of the essential characteristics of completely perfect celestial existence.

Thus in Prokofieff the Union of Soviet Socialist Republics produced one of the great composers of the Twentieth Century.

Thus it was that Barco, apprehended for mere larceny, now began to suspect that one or another of his murders had been uncovered.

Thus, a low intensity of hypothalamic-cortical discharges prevails in sleep and a high one during wakefulness, resulting in synchronous EEG potentials in the former and asynchrony in the latter condition.

Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

Thus, the writer decided to hold one experimental section of the functional preparation for marriage course in the spring semester of 1960 exclusively for persons already married -- that is, prerequisite: `` marriage ''.

Thus, when more than one distinct form leads to a particular cell in the X-region, a chain of information cells must be created to accommodate the forms, one cell in the chain for each form.

Thus, one can think of a dictionary entry as a word rather than a form.

Thus the first lines of one of Charles Wesley's hymns are as follows.

Thus, in the aggregate, the odd-lot trader is one who buys at the tops and sells at the bottoms, notwithstanding occasional individual exceptions.

Thus one line in five from The Iliad and The Odyssey is to be found somewhere else in the two poems.

Thus, when specifically permitted, the operand of a given line on the Autocoder coding sheet may be continued in the operand of from one to four additional lines which immediately follow.

Thus, if one accelerometer is properly aligned, the other is also.

Thus a man who is butting a stone wall at the office may become unusually aggressive in bed -- the one place he can still be champion.

Thus there was really an excess of eighteenth-century charm as one of these light-weight pieces followed another on Saturday night.

Thus, the planetary model of the atom was discarded in favor of one that described atomic orbital zones around the nucleus where a given electron is most likely to be observed.

Thus, according to the character Pausanias in Plato's Symposium, Aphrodite is two goddesses, one older while the other younger.

Thus arose the first monastic community, consisting of anchorites living each in his own little dwelling, united together under one superior.

Thus aesthetic judgments might be seen to be based on the senses, emotions, intellectual opinions, will, desires, culture, preferences, values, subconscious behavior, conscious decision, training, instinct, sociological institutions, or some complex combination of these, depending on exactly which theory one employs.

Thus, an analog signal is one represented by a continuous stream of data, in this case along an electrical circuit in the form of voltage, current or charge changes ( compare with digital signals below ).