Laplace transform has been used to solve Initial Value Problems. This is a topic covered in a standard differential equation course at most universities. In this post I will describe a strange situation: the solution of 2nd order ODE obtained by using Laplace transform appears to violate one of the two initial conditions. Then a [ ]
Let be a neighborhood of , be a local diffeomorphism fixing . Then is said to be stable if every open neighborhood of contains an invariant neighborhood of . It is said to be unstable if there is an open neighborhood of such that . It is said to be mixed if it is neither [ ]
Let be a homeomorphism on a compact space . Given a ceiling function , we consider the mapping torus and the suspension flow on , which is just the flow on , respecting the equivalence relation . Let be an exact symplectic manifold, be a Hamiltonian diffeomorphism. Then for any primitive 1-form , that is, [ ]
Let be a set, and be a -algebra on . A function is called a (signed) measure if it satisfies and -additive. Note that is automatic whenever there exists with finite measure. Now suppose is a topological space. There is a natural -algebra, the Borel -algebra on . Then a measure is called Borel if [ ]
Let be the universal cover of the annulus , be a diffeomorphism on preserving the 2-form , be a lift of . Suppose is monotone twist. That is, given , the function is strictly increasing. Let be the set of points such that . It follows that we can reinterpret the function as . In [ ]
Newton s laws of motion are three physical laws that, together, laid the foundation for classical mechanics. The second law provides a differential equation for the motion: given an initial position and an initial velocity, one can find the position at any given time . After Newton, one might wonder why the nature behaves this way. [ ]
Let be an ellipse with , be the billiard map on the phase space . Note that is a monotone twist map. The rotation interval of is . Moreover, for any , , there is a unique invariant curve of rotation number . The case is special: it consists of a periodic orbit of period [ ]
Let be the stand sphere with the induced Riemannian metric, be the unit tangent bundle of . There is a natural map from to . That is, let , set . It is easy to see that . Moreover, is bijective and hence a homeomorphism. Let , be the equator, which is a simple closed [ ]
Consider a smooth one-parameter family of diffeomorphisms on a manifold . It is a flow if and for every , , . Set . This generates a vector field . More generally, one consider a time-dependent vector field. That is, , where may change in time. This in turn genertes a one-parameter family (non-autonomous) of [ ]
Consider a monic polynomial with integer coefficients: , .The complex roots of such polynomials are called algebraic integers. For example, integers and the roots of integers are algebraic integers. Note that the Galois conjugates of an algebraic integer are also algebraic integers. Consider a square matrix with positive integer entries. By Perron-Frobenius Theorem, there is [ ]