User:DVD206/Basic definitions and background
We will start with definitions and overview of the main mathematical objects that are involved in the inverse problems of our interest. These include the domains of definitions of the functions and operators, the boundary and spectral data and interpolation/extrapolation and restriction techniques.
Graphs and manifolds edit
Laplace equation edit
Harmonic functions edit
A harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplace's equation.
mean-value property edit
The value of a harmonic function is a weighted average of its values at the neighbor vertices.
maximum principle edit
Corollary: the maximum (and the minimum) of a harmonic functions occurs on the boundary of the graph or the manifold.
analytic continuation edit
Analytic continuation is an extension of the domain of a given analytic function.
Random walk and hitting probability edit
- We consider the following random walk of a particle on G with discrete time.
- At moment t = 0 the particle occupies a boundary vertex v of G.
- At moment t = n+1 the particle moves to a neighbor of its position at moment t = n.
Electrical networks edit
Definition: the electrical network is a directed weighted graph with a boundary such that and . The weight function defined on the edges of the graph is called conductivity.