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, if what is at issue is whether `` All S is P '', it is indifferent whether `` Some S is not P '' or `` No S is P '', since in either case the judgment in question is false.

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, a single " rule ," like mapping every real number x to x < sup > 2 </ sup >, can lead to distinct functions and, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals.

Thus, only Division I wins ( whether FBS, FCS, or FCS non scholarship ) are considered for playoff selection.

Thus, texts like the Annals will often state whether the calendar they use ( the calendar of Lu ) is in phase with the Royal calendar ( used by the Zhou kings ).

Thus, whether " dog " is a species or a genus depends on context.

Thus John asserts Jesus ' innate superiority over all divine messengers, whether angels or prophets.

Thus, the debate of when, where, why and whether to use horseshoes is a hot topic today.

Thus, whether S2, S3, both, or neither is the same ship as S1 is a matter of convention and what purposes we have for considering things to be the same or different.

Thus Kant defines idealism as " the assertion that we can never be certain whether all of our putative outer experience is not mere imagining ".

Thus the way they voted was kept secret because the jurists would hold their disk by the axle by thumb and forefinger, thus hiding whether its axle was hollow or solid.

Thus Legendre's contribution lay in introducing a convenient notation that recorded quadratic residuosity of a mod p. For the sake of comparison, Gauss used the notation, according to whether a is a residue or a non-residue modulo p.

Thus in determining whether to shut down a firm should compare total revenue to total variable costs ( VC ) rather than total costs ( FC + VC ).

Thus architects, archaeologists and design historians sometimes use the word to describe the accumulated iterations of a design or a site, whether in literal layers of archaeological remains, or by the figurative accumulation and reinforcement of design ideas over time.

Thus, philosophy of law addresses such diverse topics as theories of contract law, theories of criminal punishment, theories of tort liability, and the question whether judicial review is justified.

Thus, many scholars will call a body of stories " mythology ", leaving open the question of whether the stories are true or false.

" Thus without holding a proposition up against the real world, we cannot tell whether the proposition is true or false.

Expected to make use of devices with which listeners could press a button to indicate whether they liked or disliked a particular piece of music, Adorno bristled with distaste and astonishment: “ I reflected that culture was simply the condition that precluded a mentality that tried to measure it .” Thus Adorno suggested using individual interviews to determine listener reactions and, only three months after meeting Lasarzfeld, completed a 160-page memorandum on the Project ’ s topic, “ Music in Radio .” Adorno was primarily interested in how the musical material was affected by its distribution through the medium of radio and thought it imperative to understand how music was affected by its becoming part of daily life.

Thus, the answer naturally arises whether the Gaius of 3 John is the same man mention in any of the others contexts.

Thus Tantra, especially in its nondual forms, rejected the renunciant values of Patañjalian yoga, offering instead a vision of the whole of reality as the self-expression of a single, free and blissful Divine Consciousness under whatever name, whether Śiva or Buddha-nature.

Thus for finite n, the expression " n-th element " of a well-ordered set requires context to know whether this counts from zero or one.

Thus, the legal definition of a narcotic is whether or not it is listed on the Schedules of the Convention.

Thus there is disagreement among botanists as to whether the genera Chionodoxa and Scilla are sufficiently different to be kept separate.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match.

Thus, the absence of the ignorable coordinate q < sub > k </ sub > from the Lagrangian implies that the Lagrangian is unaffected by changes or transformations of q < sub > k </ sub >; the Lagrangian is invariant, and is said to exhibit a symmetry under such transformations.

Thus violations in T symmetry are often referred to as CP violations.

Thus, there is a chiral symmetry of these fermions, e. g., SU ( N < sub > f </ sub >)< sub > L </ sub > ⊗ SU ( N < sub > f </ sub >)< sub > R </ sub >, if they all transform according the same complex representation of G < sub > TC </ sub >.

Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for their respective symmetry groups.

Thus one cannot cancel the anomaly by some UV completion of the theory — an anomalous symmetry is simply not a symmetry of the theory, even though classically it looks as if it is.

Thus the Kajkavian vowel system has a symmetry between front and back closed vocalic phonemes: */ ẹ / (< */ ě /, */ ь /) and */ ọ / (< */ ǫ /, ).

Thus, they can be thought of as the excitations of the field in the broken symmetry directions in group space — and are massless if the spontaneously broken symmetry is not also broken explicitly.

Thus there is approximate reflection symmetry.

Thus, and shifts become and shifts with heteroatoms, while preserving symmetry considerations.

Thus, life emerged when the universe was extremely young, to within Planck Time of the Big Bang, and there were intelligent civilizations that witnessed, and indeed consciously influenced, major events in the early universe such as inflationary expansion, symmetry breaking, and various phase changes.

Thus steric effects are often contrasted and complemented by electronic effects implying the influence of effects such as induction, conjunction, orbital symmetry, electrostatic interactions, and spin state.

Thus homothety is added, self-similarity is considered a symmetry.

Thus s < sub > p </ sub > is a geodesic symmetry and, since p was arbitrary, M is a Riemannian symmetric space.

Thus two bears are facing each other in perfect symmetry (" confronted "), forming the shape of a heart.

Thus, by enforcing the symmetry of the molecule, the degeneracy lifting effect caused by inter-state couplings is prevented.

Thus 5-fold rotational symmetry cannot be eliminated by an argument missing either of those assumptions.

Thus, for example, wallpaper and crystals cannot have 8-fold rotational symmetry.

Thus the actual symmetry group contains elements of the form gag < sup >− 1 </ sup >= b, which is a conjugate of a.

) Thus, the symmetry group of the sphere with proper rotations is.