Thus a 2 × 2 matrix with determinant − 2, when applied to a region of the plane with finite area, will transform that region into one with twice the area, while reversing its orientation.

Thus for instance the determinant of a matrix with integer coefficients will be an integer, and the matrix has an inverse with integer coefficients if and only if this determinant is 1 or − 1 ( these being the only invertible elements of the integers ).

Thus, presence of high levels of peptidoglycan is the primary determinant of the characterisation of bacteria as gram-positive.

Thus the topology of the group SL ( n, C ) is the product of the topology of SU ( n ) and the topology of the group of hermitian matrices of unit determinant with positive eigenvalues.

Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant.

Thus a 2 × 2 matrix is symplectic iff it has unit determinant.

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, for Althusser's reiterating of Marxist thought, overdetermination is what " determinant contradictions ", or capital-economic incongruities ( i. e., abstract labour resulting in " isolation " -- the class struggle ), which are analogous to Freud's " potent thoughts ", apply to instances that are more really, slight, understandable.

" Thus a change in a non-price determinant of demand is reflected in a change in the x-intercept causing the curve to shift along the x axis.

Thus, if we work modulo 26 as above, the determinant must be nonzero, and must not be divisible by 2 or 13.

Thus, ancient geographic ancestry, which is highly correlated with self-identified race / ethnicity — as opposed to current residence — is the major determinant of genetic structure in the U. S. population.

Thus the cross product may be computed by writing the " symbolic determinant "

Thus every GL < sup >+</ sup >( n )- structure is reducible to an SL ( n )- structure, and GL < sup >+</ sup >( n )- structures coincide with orientations on M. More concretely, triviality of the determinant bundle is equivalent to orientability, and a line bundle is trivial if and only if it has a nowhere-vanishing section.

Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix:

Thus, the primary determinant of active layer thickness is the maximum temperature attained during the summer.

Thus, he reasoned that existence must be due to an agent cause that necessitates, imparts, gives, or adds existence to an essence.

Thus, with a bus frequency of 100 MHz, DDR SDRAM gives a maximum transfer rate of 1600 MB / s.

Thus the second experiment gives us 8 times as much precision for the estimate of a single item, and estimates all items simultaneously, with the same precision.

Thus, the conservation of electric charge, as expressed by the continuity equation, gives the result:

Thus, measuring a quantum state described by complex coefficients ( a, b ,..., h ) gives the classical probability distribution and we say that the quantum state " collapses " to a classical state as a result of making the measurement.

Thus, substituting y with a function of x gives:

f. Thus, every pair of ( mutually ) inverse elements gives rise to two idempotents, and ex

Thus, the ratio of marginal utilities gives the absolute value of the slope of the indifference curve at point.

Thus the Elohist describes Balaam constructing giving two blessings, making sacrifices on seven altars, at the high places of Baal, before each, then deciding not to seek enchantments after the third set of sacrifices, but to set his face upon the wilderness, which Balak views as a third blessing, and so Balaam then gives the three final predictions of fate.

) Thus there is a unitary freedom in the ket mixture or ensemble that gives the same density operator.

Thus, treatment of concentrated solutions of ammonium salts with strong base gives ammonia.

Thus, the innermost purpose of the creation of all rational beings is their cognition of the existence of Allah and, hence, their conscious willingness to conform their own existence to whatever they may perceive of His will and plan: and it is this twofold concept of cognition and willingness that gives the deepest meaning to what the Quran describes as " worship ".

Thus, the relative difference in velocity between a source and an observer is what gives rise to the doppler effect.

John Lemprière, in Bibliotheca Classica, notes that as the story was re-told in later versions it accumulated details from the stories of Noah and Moses: " Thus Apollodorus gives Deucalion a great chest as a means of safety ; Plutarch speaks of the pigeons by which he sought to find out whether the waters had retired ; and Lucian of the animals of every kind which he had taken with him & c ."

Thus, she carries and gives birth to a child that is she not the biological mother of.

Porro prism designs have the added benefit of folding the optical path so that the physical length of the binoculars is less than the focal length of the objective and wider spacing of the objectives gives a better sensation of depth. Thus, the size of binoculars are reduced.

Thus, finding the virus by various methods in the dog's conjunctival cells gives a definitive diagnosis.

Thus Latin facere gives Spanish hacer () ( or, in some parts of southwestern Andalusia, ).

Thus Vamana places his foot on King Mahabali's head and gives him immortality for his benevolence.

Thus Venus ' synodic period, which gives the time between two subsequent eastern ( or western ) greatest elongations, is 584 days.

Thus, profane space gives man no pattern for his behavior.

Thus, the folklorist Anatole Le Braz gives a detailed account of one ballad singer, Yann Ar Minouz, who wrote and performed songs traveling through Brittany in the late nineteenth century and selling printed versions.

Thus, although the ketose fructose is not strictly a reducing sugar, it is an alpha-hydroxy-ketone, and gives a positive test because it is converted to the aldoses glucose and mannose by the base in the reagent.

Thus they ended up on opposite sides in the conflict, as regional stability gives way to national discord and social breakdown, and the war begins to quite literally tear families apart.

Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others.

Thus congruent objects are always geometrically similar, but geometrical similarity additionally allows uniform scaling.

Thus unlike every other version of the game, Duke Nukem cannot turn ; he can only move forwards, backwards, and strafe left or right whilst scaling the graphics larger and smaller.

Thus, for example the Koch curve scales with, but the scaling holds only for values of for integer n. In addition, the Koch curve scales not only at the origin, but, in a certain sense, " everywhere ": miniature copies of itself can be found all along the curve.

Thus, just scaling up the size of an object, keeping the same material of construction ( density ), and same acceleration, would increase the thrust by the same scaling factor.

Thus, the scaling of grasping movements directed towards objects embedded within a Ponzo illusion is not subject to the size illusion.