Thus a new pattern of days began to develop, for Granny Albright did not die.

Thus it was that Barco, apprehended for mere larceny, now began to suspect that one or another of his murders had been uncovered.

Thus Portland cement began its predominant role.

Thus literary criticism became highly theoretical and some of those practicing it began referring to the theoretical dimension of their work as " critical theory "-a philosophically inspired theory of literary criticism.

Thus began a sojourn for the reformers among the Baptists within the Redstone Baptist Association ( 1815 – 1824 ).

Thus began what is now often called " The Golden Age of Hollywood ", which refers roughly to the period beginning with the introduction of sound until the late 1940s.

Thus early 1950s historian of fandom Jack Speer began to label successive generations of fans as First Fandom, Second Fandom, Third Fandom, and so forth ... all the way to Seventh Fandom and beyond.

Thus began one of the most productive periods in Telemann's life: during his tenure at Eisenach he composed a wealth of instrumental music ( sonatas and concertos ), and numerous sacred works, which included four or five complete annual cycles of church cantatas, 50 German and Italian cantatas, and some 20 serenatas.

Thus, at age 22 he began a lifelong career in government, except for a stint in journalism from 1870-78.

' Thus, gradually he began to lead to war.

Thus, competing browsers began to emulate (" cloak " or " spoof ") this string in order to also work with those sites.

Thus, when he began to develop his theory of physical education, he naturally looked to the example set by the Athenian idea of the gymnasium, a training facility that simultaneously encouraged physical and intellectual development.

Thus the division between " hardware " and " software " began with abstraction being used to deal with the complexity of computing.

Thus Puritan leaders began assuring members that if they began doing well financially in their businesses, this would be one unofficial sign they had God's approval and were among the saved – but only if they used the fruits of their labor well.

Thus began one of the most famous apprenticeships in the musical theatre, as Hammerstein designed a kind of course for Sondheim on the construction of a musical.

Thus began the Hellenistic age, a period characterized by a more absolute approach to rule, with Greek kings taking on royal trappings and setting up hereditary successions.

Thus, it was not long before fences began to appear in an arena for the competitions.

Thus the tradition of saying 150 ( or more ) Jesus Prayers every day began.

Thus, as he began recruiting supporters for an attack on slaveholders, Brown was joined by " General Tubman ", as he called her.

Thus Rome began with a fratricide, a story that was later taken to represent the city's history of internecine political strife and bloodshed.

Thus began a long wartime association between the two different personalities.

Thus began a 4-month civil war that destroyed or damaged much of the capital.

Thus began a fifteen-year correspondence between Balzac and " the object of sweetest dreams ": Ewelina Hańska.

Thus began a long-lasting dispute which eventually resulted in the signing of the Treaty of Tordesillas with Spain in 1494.

For the case of an object that is small compared with the radial distance to its axis of rotation, such as a tin can swinging from a long string or a planet orbiting in a circle around the Sun, the angular momentum can be expressed as its linear momentum,, crossed by its position from the origin, r. Thus, the angular momentum L of a particle with respect to some point of origin is

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

Thus T: X → X ** is an injective linear mapping, though it is not surjective unless X is reflexive.

Thus, although the two use a similar error metric, linear least squares is a method that treats one dimension of the data preferentially, while PCA treats all dimensions equally.

Thus if state vectors, and each solve the linear equation on ψ, then would also be a solution, in which each is a coefficient.

Thus, to the first order in, the linear equation may be solved trivially as

Thus finite-dimensional linear isometries — rotations, reflections, and their combinations — produce orthogonal matrices.

Thus, given a linear filter evaluated through, for example, reverse correlation, one can rearrange the two spatial dimensions into one dimension, thus yielding a two dimensional filter ( space, time ) which can be decomposed through SVD.

Thus it can be said that the act of differentiation is linear, or the differential operator is a linear operator.

Thus, the time travel mechanism would also have to provide instantaneous teleportation, with infinitely accurate and simultaneous adjustment of final 3D location, linear momentum, and angular momentum.

Thus, after gauge transformation, the new colors are linear combinations of the old colors.

Thus, the only non-constant irreducible polynomials over are linear polynomials.

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

for any vectors v, w and any real number c. This linear map from R < sup > n </ sup > to R is denoted df < sub > p </ sub > and called the derivative of f at p. Thus df < sub > p </ sub >( v ) = ∂< sub > v </ sub > f ( p ).

Since any vector v is a linear combination ∑ v < sup > j </ sup > e < sub > j </ sub > of its components, df is uniquely determined by df < sub > p </ sub >( e < sub > j </ sub >) for each j and each p ∈ U, which are just the partial derivatives of f on U. Thus df provides a way of encoding the partial derivatives of f. It can be decoded by noticing that the coordinates x < sup > 1 </ sup >, x < sup > 2 </ sup >,... x < sup > n </ sup > are themselves functions on U, and so define differential 1-forms dx < sup > 1 </ sup >, dx < sup > 2 </ sup >, ..., dx < sup > n </ sup >.

Thus, unitary operators are just automorphisms of Hilbert spaces, i. e., they preserve the structure ( in this case, the linear space structure, the inner product, and hence the topology ) of the space on which they act.

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.

for any p in M and smooth function f. Thus we can regard covector fields not just as sections of the cotangent bundle, but also linear mappings of vector fields into functions.

Thus, conventional, conformal, intensity-modulated, tomographic, and stereotactic radiotherapy are all produced by specially-modified linear accelerators.

Thus weights are primarily of interest for abelian Lie algebras, where they reduce to the simple notion of a generalized eigenvalue for space of commuting linear transformations.

Thus the connection ∇ defines a way of moving elements of the fibers along a curve, and this provides linear isomorphisms between the fibers at points along the curve: