Thus T is not diagonalizable.

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 the multiplicity of Af for a given T must be an even number.

Thus for clocks on or near the geoid, T < sub > eph </ sub > ( within 2 milliseconds ), but not so closely TCB, can be used as approximations to Terrestrial Time, and via the standard ephemerides T < sub > eph </ sub > is in widespread use.

Thus this code number of a proof of contradition of T must be a non-standard number.

This identifies the tangent space T < sub > e </ sub > at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur Thus the Lie bracket on is given explicitly by = < sub > e </ sub >.

Thus for a theory " T ", " G " is true, but not provable in " T ".

Thus, T ' ai chi ch ' uan in spite of its roots in martial arts has become similar in scope to Qigong, the purely meditative practice based on notions of circulation of qi.

Thus, one would expect to find a factor that roughly corresponds to " left " and " right ", as this is the dominant framing for politics in our society, but the basis of Eysenck's " tough / tender-minded " thesis ( the second, " T "- factor ) may well represent nothing beyond an abstract mathematical construct.

Thus, in DNA, the purines adenine ( A ) and guanine ( G ) pair up with the pyrimidines thymine ( T ) and cytosine ( C ), respectively.

where D denotes the continuum derivative operator, mapping f to its derivative f. The expansion is valid when both sides act on analytic functions, for sufficiently small h. Thus, T < sub > h </ sub >= e < sup > D </ sup >, and formally inverting the exponential yields

Its change is equal to the amount of reversible work done on, or obtainable from, a system at constant T. Thus its appellation " work content ", and the designation A from Arbeit, the German word for work.

Thus the identifier " Model T " identifies the class ( model ) of automobiles that Ford's Model T comprises ; whereas the unique identifier " Model T Serial Number 159, 862 " identifies one specific member of that class — that is, one particular Model T car, owned by one specific person.

This question has two fundamental physical units: time t and length, and three dimensional variables: distance D, time taken T, and velocity V. Thus there is 3 − 2 = 1 dimensionless quantity.

There are 3 fundamental physical units in this equation: time t, mass m, and length l, and 4 dimensional variables, T, M, L, and g. Thus we need only 4 − 3 = 1 dimensionless parameter, denoted π, and the model can be re-expressed as

Thus T is uncountable: it cannot be placed in one-to-one correspondence with the set of natural numbers.

Thus the Fourier coefficients are just the values of the Fourier transform sampled on a grid of width 1 / T.

Thus, the definition of T < sub > 3 </ sub > may cite T < sub > 0 </ sub >, T < sub > 1 </ sub >, or T < sub > 2½ </ sub > instead of T < sub > 2 </ sub > ( Hausdorffness ); all are equivalent in the context of regular spaces.

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 there is a natural bijection from the set of all possible equivalence relations on X and the set of all partitions of X.

Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are three equivalent ways of specifying the same thing.

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

Thus, by definition, one mole of pure < sup > 12 </ sup > C has a mass of exactly 12 g. It also follows from the definition that X moles of any substance will contain the same number of molecules as X moles of any other substance.

Thus the Corpus Juris Canonici ( C. ii, X, De cleric.

Thus, it is exactly the unique minimal convex set containing X.

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, a non-mutant MECP2 gene is necessary for a Rett's-affected embryo to survive in most cases, and the embryo, male or female, must have another X chromosome.

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

Thus, the DCT-I corresponds to the boundary conditions: x < sub > n </ sub > is even around n = 0 and even around n = N-1 ; similarly for X < sub > k </ sub >.

Thus different countries will create their own unique X. 500 services.

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

Thus, as δn → 0 and by rearranging, we have:

Thus a covering map p: C → X defines a conjugacy class of subgroups of π < sub > 1 </ sub >( X, x ) and one can show that equivalent covers of X define the same conjugacy class of subgroups of π < sub > 1 </ sub >( X, x ).

Given two tensor bundles E → M and F → M, a map A: Γ ( E ) → Γ ( F ) from the space of sections of E to sections of F can be considered itself as a tensor section of if and only if it satisfies A ( fs ,...) = fA ( s ,...) in each argument, where f is a smooth function on M. Thus a tensor is not only a linear map on the vector space of sections, but a C < sup >∞</ sup >( M )- linear map on the module of sections.

The notion is closely related to the idea of a multiplicative character in group theory, which is a homomorphism χ from a group G to the multiplicative group of a field F. Thus χ: G → F < sup >×</ sup > satisfies χ ( e ) = 1 ( where e is the identity element of G ) and

: Thus 3 → 4

Thus P is a smooth manifold with a smooth map π: P → M which looks locally like the trivial bundle M × H → M. The frame bundle of M is a principal GL ( n )- bundle, while if M is a Riemannian manifold, then the orthonormal frame bundle is a principal O ( n )- bundle.

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 the compound p → q is false if and only if both p is true and q is false.

Thus → is a function from pairs of truth values of the components p, q to truth values of the compound p → q, whose truth value is entirely a function of the truth values of the components.

Thus the atomic number and the number of neutrons each decrease by 2 ( Z: 92 → 90, N: 146 → 144 ), so that the mass number decreases by 4 ( A = 238 → 234 ); the result is an atom of thorium-234 and an alpha particle ():

An arbitration scheme ( or constitution or voting scheme ) is a way of producing a ranking R < sub > soc </ sub > for the whole society from any profile P. Thus an arbitration scheme is itself a function P → R < sub > soc </ sub >.

Thus for instance if T is an operator, applying the squaring function s → s < sup > 2 </ sup > to T yields the operator T < sup > 2 </ sup >.

Thus M is called the underlying space, or the body, of M. The quotient map O < sub > M </ sub > → O < sub > M </ sub >/ I corresponds to an injective map M → M ; thus M is a submanifold of M.

The two different extensions of γ on each ball now becomes a map S³ → G. However, the homotopy group π < sub > 3 </ sub >( G ) = Z for any compact, connected simple Lie group G. Thus we have

Thus a cartesian section consists of a choice of one object x < sub > S </ sub > in F < sub > S </ sub > for each object S in E, and for each morphism f: T → S a choice of an inverse image m < sub > f </ sub >: x < sub > T </ sub > → x < sub > S </ sub >.