Thus, ocular albinism occurs more frequently in males as they have a single X and Y chromosome, unlike females, whose genetics are characterized by two X chromosomes.

Thus, the vector space B ( X, Y ) can be given the operator norm

Thus a point y in V is a cluster point of the net if and only if every neighborhood Y of y contains infinitely many elements of the sequence.

Thus, control unit X was either 1st or 5th, Y was either 2nd or 4th, and Z was 3rd in line.

Thus the pairs, W & Y, X & Z, are conjugates.

Thus as you move down the indifference curve you are trading consumption of units of Y for additional units of X.

Thus, the recitation of the alphabet would end in " X, Y, Z and per se and ".

Thus, peer-reviewed studies in the scientific literature document certain distinctions among the Y chromosomes of Kohanim, implying that a substantial proportion of Kohanim share some common male ancestry.

Thus, Y. Arad writes, that he had to rely, in part, on Yizkor books, which were not guaranteed to give the exact estimates of the numbers of deportees.

Thus if U < sub > p </ sub >, V < sub > p </ sub >, Y < sub > p </ sub > are three tangent vectors at p and a and b are real numbers, then

Thus let V be a vector space of dimension n over the field of scalars S, and let each of f = ( X < sub > 1 </ sub >,..., X < sub > n </ sub >) and f ' = ( Y < sub > 1 </ sub >,..., Y < sub > n </ sub >) be a basis of V .< ref group =" note "> A basis f may here profitably be viewed as a linear isomorphism from R < sup > n </ sup > to V. Regarding f as a row vector whose entries are the elements of the basis, the associated linear isomorphism is then Also, let the change of basis from f to f ′ be given by

Thus, given a tuple and the values of the attributes in X, one can determine the corresponding value of the Y attribute.

Thus the correlation coefficient is positive if X < sub > i </ sub > and Y < sub > i </ sub > tend to be simultaneously greater than, or simultaneously less than, their respective means.

Thus a metric connection is any connection Γ such that, for any two vectors X, Y ∈ T < sub > γ ( s )</ sub >

Thus the coimage of f is a quotient algebra of X much as the image of f is a subalgebra of Y ; and the bijection between them becomes an isomorphism in the algebraic sense as well ( this is the most general form of the first isomorphism theorem in algebra ).

Thus, that we could refer to an " aggregate quantity demanded " ( Y < sup > d </ sup > = C + I < sub > p </ sub > + G + NX in real or inflation-corrected terms ) at any given aggregate average price level ( such as the GDP deflator ), P.

The category Top of topological spaces has as morphisms the continuous functions, where a function f is continuous if the inverse image f < sup > − 1 </ sup >( O ) of any open set in the codomain of f is open in the domain of f. Thus any continuous function f from a space X to a space Y defines an inverse mapping f < sup > − 1 </ sup > from Ω ( Y ) to Ω ( X ).

Specifically, if X and Y are the representation spaces of two linear representations of G then a linear map f: X → Y is called an intertwiner of the representations if it commutes with the action of G. Thus an intertwiner is an equivariant map in the special case of two linear representations / actions.

Thus I ( X ; X ) ≥ I ( X ; Y ), and one can formulate the basic principle that a variable contains at least as much information about itself as any other variable can provide.

Thus mutual information can also be understood as the expectation of the Kullback-Leibler divergence of the univariate distribution p ( x ) of X from the conditional distribution p ( x | y ) of X given Y: the more different the distributions p ( x | y ) and p ( x ), the greater the information gain.

Thus, for a response Y and two variables x < sub > 1 </ sub > and x < sub > 2 </ sub > an additive model would be:

Thus, Xiahou Ba discussed the issue with Xiahou Xuan, and persuaded the latter to flee with him to Shu-Han.

Thus, Jacob ben Asher is referred to as the Ṭur or the Ba ' al ha-Ṭurim ; Joseph Caro is known as the Bet Yosef ; and Ezekiel Landau as Noda ' bi-Yehudah ; while even more frequently were authors known by contracted forms of their names, with the addition of some honorary prefix, as given above.

" Thus, Iz-ba ' al may mean " the Lord ( Ba ' al ) exists / exalts " or " where is the prince ," a name known from liturgies of the Syro-Phoenician Ba ' al cults.

Thus it is usual of Ilaro people to say " Ba mi ki Ogboni e O.

Thus we are compelled to face the urbanization of the South -- an urbanization which, despite its dramatic and overwhelming effects upon the Southern culture, has been utterly ignored by the bulk of Southern writers.

Thus, paradoxically, the beat writers resort to `` religious '' metaphors: they are in search of mana, the spiritual, the numinous, but not anything connected with formal religion.

Thus, if corporations are not to run away with us, they must become quasi-governmental institutions, subject to public control and needs.

Thus Burns's `` My love is like a red, red rose '' and Hopkins' `` The thunder-purple sea-beach, plumed purple of Thunder '' although clearly intelligible in content, hardly present ideas of the sort with which we are here concerned.

Thus there are polynomials Af such that Af.

Thus the Af are projections which correspond to some direct-sum decomposition of the space V.

Thus pencils of tangents to Q are transformed into pencils of tangents.

Thus it follows that the secants of **zg are all invariant.

Thus, although the agenda of external assistance in the economic sphere are cumulative, and many of the policies suggested for nations in the earlier stages remain relevant, the basic purpose of American economic policy during the later stages of development should be to assure that movement into a stage of self-sustaining growth is not prevented by lack of foreign exchange.

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

Thus in his friends are stranger than his sisters' strong stresses are normal for his and sisters', but in his friends are stranger than his sisters strong stresses are normal for friends and sisters.

Thus his persistence values for some stem frequencies per meaning are: stem identical in 8 languages, 100% ; ;

Thus, the white middle and lower classes are arrayed against the Negro middle and lower classes.

Thus, during any given year ( A ) if the revenues from the residential service are $7,000,000, ( B ) if the operating expenses imputed to this class of service come to $5,000,000, and ( C ) if the net investment in ( or value of ) the plant and equipment deemed devoted to this service amounts to $30,000,000, the cost analyst will report that residential service, in the aggregate, has yielded a return of $2,000,000 or 6-2/3 per cent.

Thus, in the example cited above Fromm rests his whole case on the premise that the workers are being deprived unconsciously, unknowingly, of fulfillment, and then supports this with survey data reporting conscious, experienced frustrations.

Thus, the three main categories of antisubmarine warfare operations are defense of shipping, defense of naval forces, and area defense.

Thus, the most productive areas are not necessarily the most stressed.

Thus a number of operations of the order of Af are required.

Thus, if we are to observe American folklore in the twentieth century, we will do well to establish the relationships between folklore, nationalism and imperialism at the outset.

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

" Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that " creates " output integers from an arbitrary " input " integer or integers that, in theory, can be chosen from 0 to infinity.

Thus although most species in the order are herbaceous, some no more than 15 cm high, there are a number of climbers ( e. g. some species of Asparagus ), as well as several genera forming trees ( e. g. Agave, Cordyline, Yucca, Dracaena ), some of which can exceed 10 m in height.