Decomposition theorem

Since the deformation tensor F is nonsingular, it may be decomposed uniquely into a proper orthogonal tensor R and a positive-definite symmetric tensor U by the polar decomposition theorem... 

Theorem 6.8 (Decomposition theorem for a semismall morphism [10, Theorem 1.7]). Let 7T Z y be a semismall projective morphism. As,sume that Z is nonsingular. Then... 

This result is deduced from the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber [7], which has been proved via the Gadic intersection cohomology of varieties over fields of positive characteristic. Another proof was given by M. Saito s mixed Hodge modules [75], which is also very deep. 

Theorem 6.8 (Decomposition theorem for a semismall morphism [10, Theorem 1.7]). 

The decomposition Theorem 7.1.1 (of (r, elementary polycycles) is the main reason why we prefer the property to be elementary to kernel-elementary. Another reason is that if an (r, g)g(m-polycycle is elementary, then its universal cover is also elementary. However, the notion of kernel elementariness will be useful in the classification of infinite elementary ( 3,4, 5, 3)- and ( 2, 3, 5)-polycycles. 

We also need to say a few words regarding the spectrum, a, of a general unbounded (closable) operator H, defined on a complete separable Hilbert space. We will make use of the Lebesque decomposition theorem that divides the spectrum into three parts (with respect to the spectral measure on a) the pure point part erP, the absolutely continuous part erAc, and the singularly... 

This is actually our second decomposition theorem for abelian groups in (6.8) we decomposed finite abelian group schemes into connected and etale factors. Moreover, that result is of the same type, since by (8.5) we see it is equivalent to a decomposition of the dual into unipotent and multiplicative parts. As this suggests, the theorem in fact holds for all abelian affine group schemes. To introduce the version of duality needed for this extension, we first prove separately a result of some interest in itself. 

Comparison of non-definite fimctions such as d , computed as a difference of two positive definite elements, can be done by means of the concept of a Signed Measure, applying the so called Hahn Decomposition Theorem [11]. 

Noetherian rings, and the decomposition theorem of ideals in these rings. 

Recall that by the noetherian decomposition theorem, if A C k [Xi,..., Xn is an ideal such that A = y/A, then A can be written in exactly one way as an intersection of a finite set of prime ideals, none of which contains any other. And a prime ideal is not the intersection of any two strictly bigger ideals. Therefore ... 

We will describe the PCA method following the treatment of Ressler et al. (2000) but with the above notation. PCA can be derived from the singular-value decomposition theorem from linear algebra, which says that any rectangular matrix can be decomposed as follows... 

Another approach to the analysis of Jones and Mueller-Jones matrix exploits the polar decomposition theorem [18]. This approach was first suggested in [19] and was explored in [20,21]. The polar decomposition of a Jones matrix J can be represented as ... 

As experimental evidence has shown that the standard Folgar-Tucker model predicts a faster transient orientation evolution than that observed experimentally. Tucker et al. (2007), Wang et al. (2008) and Phelps and Tucker (2009) have proposed a new evolution equation, i.e., the so-called reduced-strain closure (RSC) model, to slow down the fiber orientation kinetics. Their approach is based on the spectral decomposition theorem. The theorem states that if T is a symmetric second-order tensor, then there is a basis e, i — 1, 2, 3 consisting entirely of eigenvectors of T and the corresponding eigenvalues Aj, i — 1, 2, 3 forming the entire spectrum of T, thus T can be represented by T = A,e,e,. 

The notion of poset fibrations satisfies the following universality property. Theorem 11.9. (Decomposition theorem)... 

The decomposition theorem 11.9, is often used as a rationale to construct an acyclic matching on a poset P in several steps first map P to some other poset Q, then construct acyclic matchings on the fibers of this map. By the observation above, these acyclic matchings will patch together to form an acyclic matching for the whole poset. See Figure 11.5 for an example. For future reference, we summarize this observation in the next theorem. 

Proof. The role of the base space here is played by the poset Q, and the fiber maps gq are given by the acyclic matchings on the subposets ip q). The decomposition theorem tells us that there exists a poset map from P to the total space of the corresponding poset fibration, and that the fibers of this map are the same as the fibers of the fiber maps Qq. Since the latter are given by acyclic matchings, we conclude that we have a poset map from P with small fibers that corresponds precisely to the patching of acyclic matchings on the subposets 95 (g), for q Q. ... 

Our main innovation in Section 11.1 is the equivalent reformulation of acyclic matchings in terms of poset maps with small fibers, as well as the introduction of the universal object connected to each acyclic matching. The patchwork theorem 11.10 is a standard tool, used previously by several authors. We think that the terminology of poset fibrations together with the decomposition theorem 11.9 give the patchworking particular clarity. 

Tomescu I, Balaban AT (1989) Decomposition Theorems for Calculating the Number of Kekule Structures in Coronoids Fused via Perinaphthyl Units. Match 24 289... 

In the following we are going to introduce the generalized Onsager constitutive theory for a non-linear system of constitutive (9) satisfying the (18) reciprocity relations, the (13) equilibrium conditions and the (10) second law of thermodynamics. In the following we shall present that the Edelen s decomposition theorem [4] is valid in every class of the thermodynamic forces, which are two times continuously differentiable with respect to fluxes. 

This is the Edelen s decomposition theorem. Its essence is that every force has a definite separate part X (j(,rj) which satisfies the reciprocity relations (19) and a certain other separate part Vy jir k)r which identically satisfies the relation (20). We refer to this second type of force as non dissipative force. The flux dissipation potential i uniquely... 

From the Edelen s decomposition theorem follows that the existence of the scalar potential ii k) the associated thermodynamic forces... 

Substituting the (14) quasi-linear Onsager constitutive equations for the second equation of the Edelen s decomposition theorem (21), then we get for the flux potential... 

The well known matrix decomposition theorem states that a positive square matrix P can be decomposed as... 

We can define other deformation tensors, also, in terms of the deformation gradient tensor F. According to the polar decomposition theorem of the second-order tensor (see Appendix 2A), the deformation gradient tensor F, which is an asymmetric tensor and is assumed to be nonsingular (i.e., det F 0), can be expressed as a product of a positive symmetric tensor with an orthogonal tensor (Jaunzemis 1967) ... 

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