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The index theorem
Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018
Flat structures can arise from representations of the fundamental group. Let ρ : π1M → U(k) be a representation of the fundamental group π1M. Let M¯ be the universal cover of M. We define the bundle () Wρ=M¯×ρCk.
Soil stiffness and damping
Published in W. B. Krätzig, O. T. Bruhns, H. L. Jessberger, K. Meskouris, H.-J. Niemann, G. Schmid, F. Stangenberg, A. N. Kounadis, G. I. Schuëller, Structural Dynamics, 1991
Of course, it is advisable to substitute a more fundamental group of parameters for Darcy's permeability k. This can be done through Poiseuille's idealisation of laminar flow in tubes, in conjunction with Allen Hazen's empirical correlations for sands: k=10−3d2ρgη(3)
On codimension one special Anosov endomorphisms
Published in Dynamical Systems, 2023
So to conclude the proof, it remains to consider the case of n = 4. The classifying map is a homotopy equivalence, since is a space. Then by the main theorem of section 11.5 of Freedman and Quinn [8] (this theorem shows the classification of aspherical manifolds with poly-(finite or cyclic) fundamental group), ϕ is homotopic to a homeomorphism, since all finitely generated abelian groups are polycyclic and is a space, so is aspherical. In other words, is topologically unique in -spaces.
Les vertus des défauts: The scientific works of the late Mr Maurice Kleman analysed, discussed and placed in historical context, with particular stress on dislocation, disclination and other manner of local material disbehaviour
Published in Liquid Crystals Reviews, 2022
The ‘error’ Charles Frank [83] had made in 1958 was to classify the nematic defect lines supposing that the director remained ‘in-plane’, i.e. that the order parameter manifold had been , rather than . This leads to an incorrect fundamental group and hence a different homotopic classification. The 1958 paper mention ‘topology’ on three occasions, but the use of the term is informal. There is no appeal to topology as a mathematical discipline. Frank is closing his eyes and using his ‘common sense’. Frank's common sense, of course, was uncommonly reliable – almost to the extent of genius – in general; he had a spectacular three-dimensional intuition. It half-fails him here; half, because one could use a Volterra process to define the defect line, rather than the homotopic classification.
Geometry of symplectic partially hyperbolic automorphisms on 4-torus
Published in Dynamical Systems, 2020
Note, that classification of ergodic automorphisms of the torus from a measure theory point of view is given by their entropy [21]. This follows from the Ornstein isomorphism theorem [25] and the fact that every ergodic automorphism of a torus is Bernoulli one with respect to the Lebesgue measure [21]. But we want to obtain a topological classification. The existence of a conjugating homeomorphism for two automorphisms implies the relation in the fundamental group of the torus, where H is the linear homomorphism in generated by h. Thus, matrices are similar by H. Matrix H is integer with its determinant This is not occasional due to the following Arov's theorem [4]. We formulate the consequence of the Arov's theorem