Thus one can see that the Cayley – Hamilton theorem must be true.

Thus such sets of points impose one fewer condition on cubics containing them than expected, and accordingly satisfy an additional constraint, namely the Cayley – Bacharach theorem that any cubic that contains eight of the points necessarily contains the ninth.

Thus all the sets comprise a general directed graph ( ownership defines a direction ), or network construct.

Thus, it was not until two centuries had passed that in 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable.

Thus many maze solving algorithms are closely related to graph theory.

Thus, the hub and leaf nodes, and the transmission lines between them, form a graph with the topology of a star.

Thus in the graph of the supply curve, individual firms ' supply curves are added horizontally to obtain the market supply curve.

Thus in the graph of the demand curve, individuals ' demand curves are added horizontally to obtain the market demand curve.

Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing.

In one direction, the Hamiltonian path problem for graph G is equivalent to the Hamiltonian cycle problem in a graph H obtained from G by adding a new vertex and connecting it to all vertices of G. Thus, finding a Hamiltonian path cannot be significantly slower ( in the worst case, as a function of the number of vertices ) than finding a Hamiltonian cycle.

Thus, the graph forms a subgraph of the intersection graph of the subtrees.

Thus, given a graph G

Thus the graph is now ( c − 1 )- coloured.

Thus, the Petersen graph has crossing number 2.

Thus the permanent of A is equal to the sum of the weights of all perfect matchings of the graph.

Thus the permanent of a 01-matrix is equal to the number of vertex cycle covers of an unweighted directed graph.

Thus, for an implicit function to be a true ( single-valued ) function it might be necessary to use just part of the graph.

Thus, while the old " left-right " line attempted to measure politics along a one-dimensional line, Nolan's graph divided political issues into two dimensions: economic and social.

Thus one can think of the individual data points as the nodes of a graph and the kernel k defining some sort of affinity on that graph.

Thus, their theory establishes fundamental connections between graph minors and topological embeddings of graphs.

More specifically, the running time for testing whether H is a minor of G in this case is O ( n < sup > 3 </ sup >), where n is the number of vertices in G and the big O notation hides a constant that depends superexponentially on H. Thus, by applying the polynomial time algorithm for testing whether a given graph contains any of the forbidden minors, it is possible to recognize the members of any minor-closed family in polynomial time.

Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply the random graph.

Thus, abelian categories are always binormal.

Thus we obtain the global reciprocity map from the idele class group to the abelian part of the absolute Galois group of the field.

Thus, a module, like a vector space, is an additive abelian group ; a product is defined between elements of the ring and elements of the module that is distributive over both parameters and is compatible with the ring multiplication.

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, since the groups in question are abelian, K < sub > 1 </ sub >( A ) splits as the direct sum of the group of units and the special Whitehead group: K < sub > 1 </ sub >( A ) ≅ A * ⊕ SK < sub > 1 </ sub > ( A ).

Thus the representation of the center of the Heisenberg group is determined by a scale value, called the quantization value ( in physics terms, Planck's constant ), and if this goes to zero, one gets a representation of the abelian group ( in physics terms, this is the classical limit ).

Thus homotopy classes from one spectrum to another form an abelian group.

Thus, giving grades and promotions based only on an individual's performance relative to a small local group, as is common, may reduce cooperative behaviors in the group.

Thus, while altruistic persons may under some circumstances be outcompeted by less altruistic persons at the individual level, according to group selection theory the opposite may occur at the group level where groups consisting of the more altruistic persons may outcompete groups consisting of the less altruistic persons.

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, at pH between 2. 2 and 9. 4, the predominant form adopted by α-amino acids contains a negative carboxylate and a positive α-ammonium group, as shown in structure ( 2 ) on the right, so has net zero charge.

Thus this class of molecules belong to the D < sub > 2d </ sub > point group.

Thus the discovery report by the Berkeley group reads: " It is suggested that element 97 be given the name berkelium ( symbol Bk ) after the city of Berkeley in a manner similar to that used in naming its chemical homologue terbium ( atomic number 65 ) whose name was derived from the town of Ytterby, Sweden, where the rare earth minerals were first found.

Thus, the molecular formula of ethanol may be written in a paper in conformational, three-dimensional, full two-dimensional ( indicating every bond with no three-dimensional directions ), compressed two-dimensional ( CH < sub > 3 </ sub >– CH < sub > 2 </ sub >– OH ), separating the functional group from another part of the molecule ( C < sub > 2 </ sub > H < sub > 5 </ sub > OH ), or by its atomic constituents ( C < sub > 2 </ sub > H < sub > 6 </ sub > O ), according to what is discussed.

Thus the probability of two failures in a same RAID group in time proximity is much smaller ( approximately the probability squared, i. e., multiplied by itself ).

Thus the probability of two failures in a same RAID group in time proximity is much smaller ( approximately the probability squared, i. e., multiplied by itself ).

Thus, it has been argued that a voucher system would lead students who do not belong to a preferred religious or ethnic group, or those with disabilities, to become concentrated within the public school system.

Thus one has the fundamental groupoid instead of the fundamental group, and this construction is functorial.

Thus the DTFT of the s sequence is also the Fourier transform of the modulated Dirac comb function .< ref group =" note "> We may also note that:

Thus the group of left and right handed 4x4 quaternion matrcies is Sp ( 8 ) xSU ( 2 ) which does include the standard model bosons:

Thus a resin can be tailored to a particular toxic group.

Hence, those with less pigmented skin survived and had children at higher rates because their skin allowed more UV light for the production of vitamin D. Thus, the skin of those in the group that left the African continent and went far north gradually developed adaptations for relatively greater translucence compared to equatorial hues.

Thus, for example, two objects may be group isomorphic without being ring isomorphic, since the latter isomorphism selects the additional structure of the multiplicative operator.

Thus, although the composition of the group is slightly different in each instance, the large majority of men present are always close neighbors.

But this group found it increasingly difficult to prove that they were not labourers because the 1882 act defined excludables as ‘ skilled and unskilled labourers and Chinese employed in mining .’ Thus very few Chinese could enter the country under the 1882 law.

Thus Euclidean geometry corresponds to the choice of the group E ( 3 ) of distance-preserving transformations of the Euclidean space R < sup > 3 </ sup >, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group.