When a is greater than 1, the negative value of exponential function to x is very flat, and the positive value to x rises rapidly. When x equals 0, y equals 1. When a is greater than 0 and less than 1, the negative value of the exponential function to X rises rapidly and the positive value to X is very gentle. When x equals 0, y equals 1.
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In linear algebra, vector bundles are described by Euler numbers. Intersection of zero cross section and bottom space of directed vector bundle. Let ξ=(E, π, m) be an n-dimensional directed vector bundle, and m be an n-dimensional tightly connected directed (infinite) differential manifold. If the bottom space m is equal to the zero cross-section image of ξ;
Euler number is called vector bundle ξ. Let m be as above, and ξ=TM, then ξ (ξ) is called Euler characteristic of manifold m, and is denoted ξ (m). For example, χ (s...2n) = 2 (so any vector field on S^2n has zero), χ(S)=0. Euler numbers are isomorphic invariants of vector bundles. In the case of tangent bundle of manifold, the Euler characteristic number of manifold, a topological invariant widely used in algebraic topology, is obtained.