**On Inverse Problems in 2D**

*Dedicated to Nicole DeLaittre*

## About the book Edit

## Summary Edit

- The main object of study of this book is the relationship between local and global properties of two-dimensional manifolds (surfaces) and embedded graphs. The dimension of the unknown parameter fits the dimension of the data of the measurements in several important instances of the inverse problems. Also, two-dimensional setting has an additional structure, due to the duality between harmonic functions on embedded graphs and manifolds and the connection to special matrices. The context of the inverse problems provides a unified point of view on the work of many great mathematicians. Some of the problems simplify significantly in the graph theoretical setting, but their solutions nevertheless convey the main ideas of the solutions for their continuous analogs. These are some of the main motivations for writing this book. Even though there are references to many mathematical areas in this book, it is practically self-contained, and is intended for the use by a wide audience of people interested in the subject.

## Basic definitions and backgroundEdit

This chapter gives the definitions and the 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

### Harmonic functionsEdit

### On random processes Edit

### Special matrices and operators Edit

### Electrical networksEdit

### The inverse problems Edit

Rectangular directed layered grid

Rectangular grids and gluing graphs

Ordinary differential equations (ODEs)

- Three-term recurrence matrices and continued fractions

## Applications to classical problems Edit

### On the inverse problem of Calderon Edit

### "Can One Hear the Shape of a Drum?" Edit

### On inhomogeneous string of KreinEdit

## Transformations of embedded graphs Edit

The rules for replacing conductors in series or parallel connection by a single electrically equivalent conductor follow from the equivalence of the Y-Δ or star-mesh transforms.

### Y-Δ and star-mesh transformsEdit

### Medial graphs Edit

### Dual graphs and harmonic conjugatesEdit

### Determining genus of a graphEdit

### Hamiltonian paths in graphsEdit

### The new spectral theorem Edit

## Rotation invariant layered case Edit

### Fourier coordinates Edit

### Stieltjes continued fractions Edit

### Blaschke productsEdit

Let *a_i* be a set of *n* points in the complex unit disc. The corresponding **Blaschke product** is defined as

If the set of points is finite, the function defines the *n*-to-*1* map of the unit disc onto itself,

If the set of points is infinite, the product converges and defines an automorphism of the complex unit disc, given the Blaschke condition

The following fact will be useful in our calculations:

### Pick-Nevanlinna interpolation Edit

### Cauchy matrices Edit

The Cayley transform provides the link between the Stieltjes continued fractions and Blaschke products and the Pick-Nevanlinna interpolation problem for the unit disc and the half-space.

### Solution of the inverse problem Edit

A. Elementary symmetric functions and permutations B. Continued fractions and interlacing properties of zeros of polynomials C. Wave-particle duality and identities involving path integrals and Laplacian eigenvalues of a graph D. Square root and finite-differences

Given the Dirichlet-to-Neumann map of a layered network, find the eigenvalues and the interpolate, calculate the Blaschke product and continued fraction. That gives the conductivities of the layeres.

## The square root of the minus Laplacian Edit

- We will now consider an important special case of the inverse problem