A Mathematical Gift III: The Interplay Between Topology, by Kenji Ueno, Koji Shiga, Shigeyuki Morita

By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This publication will deliver the sweetness and enjoyable of arithmetic to the study room. It bargains severe arithmetic in a full of life, reader-friendly variety. integrated are workouts and lots of figures illustrating the most innovations.
The first bankruptcy provides the geometry and topology of surfaces. between different subject matters, the authors speak about the Poincaré-Hopf theorem on serious issues of vector fields on surfaces and the Gauss-Bonnet theorem at the relation among curvature and topology (the Euler characteristic). the second one bankruptcy addresses a number of elements of the concept that of measurement, together with the Peano curve and the Poincaré technique. additionally addressed is the constitution of 3-dimensional manifolds. particularly, it truly is proved that the 3-dimensional sphere is the union of 2 doughnuts.
This is the 1st of 3 volumes originating from a chain of lectures given by way of the authors at Kyoto college (Japan).

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Extra resources for A Mathematical Gift III: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 23)

Sample text

If the dimension of V is 2, show the induced action on the slope m of a line is m= a c b d • m = (c + dm)/(a + bm). Hint: consider the induced action on y/x. 4. 1 Euclidean action on smooth curves in the plane. 4 Induced action on derivatives: the prolonged action Suppose there is an action of the group G in the plane with coordinates (x, y). If we take a curve in the plane given by y = f (x), so that we consider y to be a function of x, then there is an induced action on the derivatives yx , yxx and so forth, called the prolonged action.

45) α j vj . 2. 47) j vh · uαK = α φK,j αj . 45), for a prolonged action is vj = ξji i,α,K ∂ ∂ ∂ α + φ,jα α + φK,j . 14, x= ax + b , cx + d y = 6c(cx + d) + (cx + d)2 y, ad − bc = 1. Take local coordinates near the identity to be (a, b, c) so that e = (1, 0, 0). 6. Hint: (α, β, γ ) = (α 1 , α 2 , α 3 ). 10 to the prolonged action is the first step of Sophus Lie’s algorithm for calculating the symmetry group of a differential equation. This algorithm is discussed in detail in textbooks, for example Bluman and Cole (1974), Ovsiannikov (1982), Bluman and Kumei (1989), Stephani (1989), Olver (1993), Hydon (2000) and Cantwell (2002), and we refer the interested reader to these.

What to do: integrating the system dx/dt = 2x with initial condition x = x at t = 0 yields x = exp(2t)x. Show this is a reparametrisation of the scaling transformation that satisfies the one parameter group property. What not to do: integrating the system dx/dλ = 2x with initial condition x = x at λ = 1 yields x = exp(2(λ − 1))x. Show this is not a group action of (R+ , ·). 24 in practice. These infinitesimals arose in a study of non-classical reductions of the equation ut = uxx + f (u), for f (u) a cubic (Clarkson and Mansfield, 1993).

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