Thus, the difference of any two solutions to the equation Ax = b lies in the null space of A.

Thus a possible diagnostic outcome would be that all tosses yield the same outcome, and the probability of 5 of a kind is 6 % under the null hypothesis.

Thus the test procedure can be defined ( that is the critical values can be defined ) for testing the null hypothesis H < sub > T </ sub > exactly as if the null hypothesis of interest was the reduced version H < sub > 0 </ sub >.

Thus if a baserunner running from third base crosses home plate with two out before a batter is put out on a ground out or before a fly ball is caught after two outs, then the run that would otherwise score is null ; likewise, if a conventional double play is made with one out on a ground ball involving forces at other bases, the run that would otherwise score before the third out is registered is void.

Thus he protested against the practice of declaring null and void all oaths and promises which may be made during the coming year, and against the refusal to grant an honorable burial to excommunicated persons and their connections

Thus, there may be a linear nature of photoreceptors, one inhibiting its following neighbor when moving in the null direction, but arriving too late at the adjacent cell when traveling in the preferred direction.

Thus, nonsecretive adenomas may be either null cell adenomas or a more specific adenoma that, however, remains nonsecretive.

Thus, level 0 is associated with the value of null, level 1 is associated with I < sub > base </ sub >

Thus we may consider the allele for the O blood type as a null allele.

Thus the null hypothesis is rejected if ( where is the upper tail critical value for the distribution ).

Thus, every null electrovacuum has one quadruple eigenvalue, namely zero.

Thus the " null distribution " of the test statistic, i. e. its probability distribution assuming the null hypothesis is true, is stochastically smaller than the Kolmogorov – Smirnov distribution.

Thus Shiva's greatest advance was in its failure, a not entirely obvious example of the null result.

Thus, if public pressure sets the effective limit to the price that the industry may charge, this pressure is itself a function of the wage rate.

Thus, in `` The Story Of Ruth '' we have Ruth, Naomi, and Boaz and sets that are meticulously authentic.

Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.

Thus all the sets comprise a general directed graph ( ownership defines a direction ), or network construct.

Thus intuitionists absolutely disallow the blanket assertion: " For all propositions P concerning infinite sets D: P or ~ P " ( Kleene 1952: 48 ).

Thus, 10 % of possible sets differ in one feature, 30 % in two features, 40 % in three features, and 20 % in all four features.

( Thus, the nowhere dense sets need not form a sigma-ideal.

Thus, sets of complementary wavelets are useful in wavelet based compression / decompression algorithms where it is desirable to recover the original information with minimal loss.

Thus, what the axiom is really saying is that two sets are equal if and only if they have precisely the same members.

Thus the Third Age of Middle-earth begins, with Isildur's fateful decision to keep the Ring of Power, which sets in motions the events that will lead to the War of the Ring.

Thus the axioms of ZFC refer only to sets, not to urelements ( elements of sets which are not themselves sets ) or classes ( collections of mathematical objects defined by a property shared by their members ).

Thus an oracle machine with a noncomputable oracle will be able to compute sets that are not computable without an oracle.

Thus, the supplier sets a lower price for that consumer because the student or senior has a more elastic Price elasticity of demand ( see the discussion of Price elasticity of demand as it applies to revenues from the first degree price discrimination, above ).

Thus the relatively small amount of network color programming, combined with the high cost of color television sets, meant that as late as 1964 only 3. 1 percent of television households in the U. S. had a color set.

Thus, the cardinality of the Cartesian product of no sets is 1.

Thus crystal sets produce rather weak sound and must be listened to with sensitive earphones, and can only pick up stations within a limited range.

Thus, unlike copy editing, proofreading's defining procedure is to work directly with two sets of information at the same time.

Thus a sheaf can often be defined by giving its values on the open sets of a basis, and verifying the sheaf axioms relative to the basis.

Thus, one could instead of sequences also consider sets.

Thus there might be several such sets.

Thus, players are more vulnerable than in other action games, and the hostility of the environment sets up a narrative where the odds are weighed decisively against the avatar.

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 the member of an industrial union comes to regard his officers as business agents who may proceed without interference or recall ; ;

Thus the student of literature may sometimes find it helpful to classify a poem or an essay as being in idea or in ideal content or subject matter typical or atypical of its period.

Thus, as a development program is being launched, commitments and obligations must be entered into in a given year which may exceed by twofold or threefold the expenditures to be made in that year.

Thus, a child's Skeletal Age `` dots '' may be classified as `` advanced '' when they appear above the middle curve, `` moderate '' when they appear immediately above or below the middle curve, and `` delayed '' when they appear below the lower curve.

Thus he may be referring to some concrete thing, or incident, in his immediate environment by some symbolic-sounding, hyperbolic reference to transcendental events on the global scene.

Thus in Mary wrote an account of the trip first strong stress on Mary marks Mary as the first in a series of people who wrote accounts of the trip, strong stress on wrote marks the writing as the first of a series of actions of Mary's concerned with an account of her trip ( about which she may later have made speeches, for example ), and strong stress on trip makes the trip the first of a series of subjects about which Mary wrote accounts.

Thus while his theory or technique may not be oversubscribed, it is commonplace for bullish and bearish positions to become temporarily over-subscribed.

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 creativity may run all the way from making a cake, building a chicken coop, or producing a book, to founding a business, creating a League of Nations or, developing a mature character.

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 he will be in a position to disabuse the Soviet leader of any notions he may have about grave Allied disunity.

Thus, while it remains possible that the Babylonians and/or the Pythagoreans may perhaps have had the magic square of three before the Chinese did, more definite evidence will have to turn up from the Middle East or the Classical World before China can lose her claim to the earliest known magic square by more than a thousand years.

Thus, giving grades and promotions based only on an individual's performance relative to a small local group, as is common, may reduce cooperative behaviors in the group.

Thus, while altruistic persons may under some circumstances be outcompeted by less altruistic persons at the individual level, according to group selection theory the opposite may occur at the group level where groups consisting of the more altruistic persons may outcompete groups consisting of the less altruistic persons.

Thus, although some things may be certain, they have little to do with Dasein's sense of care and existential anxiety, e. g., in the face of death.

Thus, a court officer taking possession of goods under a court order may use force if reasonably necessary.

Thus, a gene for antibiotic resistance that evolves via natural selection may be shared.

This latter construal is sometimes expressed by saying " there is no fact of the matter as to whether or not P ." Thus, we may speak of anti-realism with respect to other minds, the past, the future, universals, mathematical entities ( such as natural numbers ), moral categories, the material world, or even thought.

Thus it may be roughly translated as " protector of man ".

Thus, three eclipse calculations may be evidence for the suggestion that Romulus reigned from 746 BC to 709 BC, and Rome was founded during 745 BC.

Thus if the instrument depends on the pressure or suction effect alone, and this pressure or suction is measured against the air pressure in an ordinary room, in which the doors and windows are carefully closed and a newspaper is then burnt up the chimney, an effect may be produced equal to a wind of 10 mi / h ( 16 km / h ); and the opening of a window in rough weather, or the opening of a door, may entirely alter the registration.