Revision 2418165 of "On Inverse Problems in 2D" on enwikibooks

<div style="width: 100%; text-align: center; font-size: xx-large;">'''On Inverse Problems in 2D'''</div>
</br>

<div style="width: 100%; text-align: right; font-size: large;">''Dedicated to Nicole DeLaittre''</div>


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== [[ User:Daviddaved/About the book | About the book ]] ==

[[File:Book cover.jpg|thumbnail]]

== [[ User:Daviddaved/Summary | Summary ]] ==

: 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.
== [[ User:Daviddaved/Basic definitions and background | Basic definitions and background]] ==

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.

=== [[ User:Daviddaved/Graphs and manifolds | Graphs and manifolds ]] ===

=== [[ User:Daviddaved/Harmonic functions | Harmonic functions]] ===
=== [[ User:Daviddaved/On random processes | On random processes ]] ===
=== [[ User:Daviddaved/Special matrices and determinants | Special matrices and operators ]] ===
=== [[ User:Daviddaved/Electrical networks | Electrical networks]] ===
=== [[ User:Daviddaved/The inverse problems | The inverse problems ]] ===
* [[ User:Daviddaved/Solving polynomial equation | Solving polynomial equation ]]
Rectangular directed layered grid
* [[ User:Daviddaved/Pascal traingle | Pascal triangle ]]
Rectangular grids and gluing graphs
* [[ User:Daviddaved | Monodromy operator ]]
Ordinary differential equations (ODEs)
* [[ Three-term recurrence matrices and continued fractions ]]

== [[ User:Daviddaved/Applications to classical problems | Applications to classical problems ]] ==
=== [[ User:Daviddaved/On the inverse problem of Calderon | On the inverse problem of Calderon ]] ===
=== [[ User:Daviddaved/"Can One Hear the Shape of a Drum?" | "Can One Hear the Shape of a Drum?" ]] ===
=== [[ User:Daviddaved/On inhomogeneous string of Krein | On inhomogeneous string of Krein]] ===

== [[ User:Daviddaved/Transformations of embedded graphs | Transformations of embedded graphs ]] ==
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.
<gallery>
[[File:Rectangular grid.jpg|600px|Rectangular grid]]
</gallery>
=== [[ User:Daviddaved/Y-Δ and star-mesh transforms | Y-Δ and star-mesh transforms]] ===
=== [[ User:Daviddaved/Medial graphs | Medial graphs ]] ===
=== [[ User:Daviddaved/Dual graphs and harmonic conjugates | Dual graphs and harmonic conjugates]] ===
=== [[ User:Daviddaved/Determining genus of a graph | Determining genus of a graph]] ===
=== [[ User:Daviddaved/Hamiltonian paths in graphs | Hamiltonian paths in graphs]] ===
=== [[ User:Daviddaved/The new spectral theorem | The new spectral theorem ]] ===

== [[ User:Daviddaved/The layered case | Rotation invariant layered case ]] ==
=== [[ User:Daviddaved/Fourier coordinates | Fourier coordinates ]] ===
=== [[ User:Daviddaved/Stieltjes continued fractions | Stieltjes continued fractions ]] ===

=== [[ User:Daviddaved/Blaschke products | Blaschke products]] ===

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

:<math>B(z)=\prod_k\frac{|a_k|}{a_k}(\frac{a_k-z}{1-\overline{a_k}z}).</math>

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

:<math>f:\mathbb{D}\xrightarrow[]{n\leftrightarrow 1}\mathbb{D}.</math>

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

:<math>\sum_k (1-|a_k|) <\infty.</math>

The following fact will be useful in our calculations:

:<math>B(0)=\prod_n{|a_i|}.</math>

=== [[ User:Daviddaved/Pick-Nevanlinna interpolation | Pick-Nevanlinna interpolation ]] ===
=== [[ User:Daviddaved/Cauchy matrices | Cauchy matrices ]] ===

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.

=== [[ User:Daviddaved/Solution of the inverse problem | Solution of the inverse problem ]] ===

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.

== [[ User:Daviddaved/The square root of the minus Laplacian | The square root of the minus Laplacian ]] ==

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

=== [[ User:Daviddaved/The case of the unit disc | The case of the unit disc ]] ===
=== [[ User:Daviddaved/One more graph example | One more example]] ===
=== [[ User:Daviddaved/Zolotarev problem | Zolotarev problem ]] ===

== [[ User:Daviddaved/Connections between discrtete and continuous models | Discrete and continuous models ]] ==
=== [[ User:Daviddaved/Kernel of Dirichlet-to-Neumann map |Kernel of Dirichlet-to-Neumann map]] ===
=== [[ User:Daviddaved/Riemann mapping theorem | Riemann mapping theorem]] ===
=== [[ User:Daviddaved/Hilbert transfrom | Hilbert transform ]] ===
=== [[ User:Daviddaved/Schrodinger equation | Schrodinger equation ]] ===
=== [[ User:Daviddaved/Variation diminition property | Variation diminishing property ]] ===
=== [[ User:Daviddaved/Spectral properties | Spectral properties ]] ===

== [[ User:Daviddaved/Notation | Notation ]] ==

:<math>
\mathbb{R} \mbox{ is the set of real numbers}
</math>

:<math>
\mathbb{R}^N  \mbox{ is the N-dimensional Euclidean space}
</math>

:<math>
\mathbb{C}  \mbox{ is the set of complex numbers}
</math>

:<math>
\mathbb{D}=\{z \in \mathbb{C}|, |z| \le 1\} \mbox{ is the closed unit disc}
</math>

:<math>
M \mbox{ is a 2D surface}
</math>

:<math>
\alpha, \beta, \ldots \mbox{ are analytic functions}
</math>

:<math>
\nabla \mbox{ is gradient}
</math>

:<math>
\Delta  \mbox{ is the Laplace operator}
</math>

:<math>
\Lambda \mbox{ is the Dirichlet-to-Neumann operator}
</math>

:<math>
A, B, \ldots \mbox{ are matrices}
</math>

:<math>
\lambda \mbox{ is eigenvalue}
</math>

:<math>
\rho \mbox{ is characteristic polynomial}
</math>

:<math>
P \mbox{ is permutation matrix}
</math>

:<math>
F \mbox{ is Fourier transform}
</math>

:<math>
\Gamma \mbox{ is graph}
</math>

:<math>
G \mbox{ is network}
</math>

:<math>
\gamma \mbox{ is conductivity}
</math>

:<math>
q \mbox{ is potential}
</math>

== [[ User:Daviddaved/Acknowledgements | Acknowledgements ]] ==

The author would like to thank Wiki project for the help in writing the book @ all stages of the process.

== [[User:Daviddaved/About the author | About the author ]] ==

== [[User:Daviddaved/Bibliography | Bibliography ]] ==

{{reflist}}

# 
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#

{{Subjects|Mathematics|Applied mathematics|Mathematical collections}}
{{Alphabetical|O}}
{{status|75%}}