By Biryukov O.N.

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**Additional resources for A bound for the topological entropy of homeomorphisms of a punctured two-dimensional disk**

**Example text**

To generalize Gibbs, the body is thought of as a subset B of three-dimensional space, 3 , and 2 Thermostatics 33 associated with each point are the thermodynamic variables. The physical systems to be represented are those whose distinguished thermostatic states are described by 2n thermodynamic variables divided into control variables y1 , . . , yn and state variables x 1 , . . , xn . The thermodynamic variables are, for example, components of the stress, strain and internal variable tensors, the components of flux vectors, as well as temperature and entropy.

1) x1 The force may change over the path in either magnitude or direction. In general, the work done depends on the path, as for example, when the force is friction. The configuration or shape of a body is described by indicating the position of each point in the body in a coordinate system. A system of generalized coordinates for a system is a set containing the minimum number of coordinates required to completely describe the configuration of the system. As the body goes through an imagined displacement, called a virtual displacement, the real fixed forces acting on the body do work, called the virtual work, δW , as they move through a distance during the virtual displacement.

This is the value to which the back stress tends over time. Example 4 The Fourier Law: The Fourier relation, q = −K ∇θ , is stationary. Let the gradient, ∇(1/θ ), be the control variable, and the heat flux vector, q, be the state variable. The local generalized entropy production function is ∗ (q; ∇(1/θ )) = 1 2K θ 2 q·q−q·∇ 1 . θ The Fourier relation is recovered from the zero gradient condition on (7) ∗. 4 Evolution Equations for Non-equilibrium Processes 39 Example 5 Griffith-Irwin Fracture: Classical fracture theory can be presented in terms of generalized thermodynamic functions (see Chapter 11).

### A bound for the topological entropy of homeomorphisms of a punctured two-dimensional disk by Biryukov O.N.

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