User:Daviddaved/On 2D Inverse Problems
Contents
 1 About the book
 2 Summary
 3 Basic definitions and background
 4 Applications to classical problems
 5 Transformations of embedded graphs
 6 The layered case
 7 The square root of the minus Laplacian
 8 Connections between discrete and continuous models
 9 Notation
 10 Acknowledgements
 11 About the author
 12 Bibliography
About the book [edit]
Summary [edit]
 The main object of study of this book is the relationship between local and global properties of twodimensional 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, twodimensional 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 selfcontained, and is intended for the use by a wide audience of people interested in the subject.
Basic definitions and background[edit]
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]
Harmonic functions[edit]
On random processes [edit]
Special matrices and determinants [edit]
Electrical networks[edit]
The inverse problems [edit]
 === Solving polynomial equation ===
Rectangular directed layered grid
 === Pascal triangle ===
Rectangular grids and gluing graphs
 === Monodromy operator ===
Ordinary differential equations (ODEs)
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 Krein[edit]
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 YΔ or starmesh transforms.

Rectangular grid.jpg
Rectangular grid
YΔ and starmesh transforms[edit]
Medial graphs [edit]
Dual graphs and harmonic conjugates[edit]
Determining genus of a graph[edit]
Hamilton paths in graphs[edit]
The new spectral theorem [edit]
The layered case [edit]
Fourier coordinates [edit]
Stieltjes continued fractions [edit]
Blaschke products[edit]
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 nto1 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:
PickNevanlinna interpolation [edit]
Cauchy matrices [edit]
 The Cayley transform provides the link between the Stieltjes continued fractions and Blaschke products and the PickNevanlinna interpolation problem at the unit disc and the halfspace.
Solution of the inverse problem [edit]
Rotation invariant layered networks
A. Elementary symmetric functions and permutations B. Continued fractions and interlacing properties of zeros of polynomials C. Waveparticle duality and identities involving integrals of paths in a graph and its Laplacian eigenvalues D. Square root and finitedifferences
Given the DirichlettoNeumann 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
The case of the unit disc [edit]
Zolotarev problem [edit]
One more graph example[edit]
Connections between discrete and continuous models [edit]
Kernel of DirichlettoNeumann map[edit]
Riemann mapping theorem[edit]
Hilbert transform [edit]
Schrodinger equation [edit]
Variation diminishing property [edit]
Spectral properties [edit]
Notation [edit]
Acknowledgements [edit]
The author would like to thank Wiki project for the help in all stages of writing the book.
About the author [edit]
Bibliography [edit]
 Astala, K. and P¨aiv¨arinta, L. "Calder´on’s inverse conductivity problem in the plane", http://annals.math.princeton.edu/wpcontent/uploads/annalsv163n1p05.pdf
 Biesel, O. D., Ingerman D. V., Morrow J. A. and Shore W. T. "Layered Networks, the Discrete Laplacian, and a Continued Fraction Identity", http://www.math.washington.edu/~reu/papers/2008/william/layered.pdf
 Borcea, L. , Druskin, V. and Knizhnerman, L. "On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids", Communications on Pure and Applied Mathematics, Vol. 000, 0001–0048 (2000)
 Borcea, L., Druskin, V., Vasquez, G. F. and Mamonov, A. V. "Resistor network approaches to electrical impedance tomography", http://arxiv.org/abs/1107.0343
 Cannon, J. W. "The combinatorial Riemann mapping theorem", http://www.springerlink.com/content/9w0608p039151254/
 Curtis, E. B. and Morrow, J. A. "Inverse Problems for Electrical Networks", http://books.google.com/books/about/Inverse_Problems_for_Electrical_Networks.html?id=gLnohh95zFIC
 Curtis, E. B., Ingerman, D. V. and Morrow, J. A. "Circular Planar Graphs and Resistor Networks", http://www.math.washington.edu/~morrow/papers/cim.pdf
 De Verdière, Yves Colin, "Reseaux electrique planaires I", http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.224.133
 Doyle, P. G. and Snell, L. J. "Random walks and electric networks", http://www.cse.buffalo.edu/~hungngo/classes/2005/Expanders/papers/general/randomWalk.pdf
 Edelman, A. and Strang, G. "Pascal matrices", http://web.mit.edu/18.06/www/pascalwork.pdf
 Fomin, S. "Looperased walks and total positivity", http://arxiv.org/pdf/math.CO/0004083.pdf
 Gantmacher, F. R. and Krein, M. G. "Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems", Revised Edition, http://www.ams.org/bookstore?fn=20&arg1=diffequ&ikey=CHEL345H
 Ingerman, D. V. "The Square of the DirichlettoNeumann map equals minus Laplacian", http://arxiv.org/ftp/arxiv/papers/0806/0806.0653.pdf
 Kac, M. "Can one hear the shape of a drum?", American Mathematical Monthly 73 (4, part 2): 1–23, (1966), doi:10.2307/2313748
 Karlin, S. "Total positivity", http://books.google.com/books/about/Total_Positivity.html?id=yIarAAAAIAAJ
 Kohn, R. and Vogelius, M. "Determining conductivity by boundary measurements", Comm. Pure Appl. Math., 37(1984), 289–298, http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160370302/abstract
 Marshall, D. E. "An elementary proof of the PickNevanlinna interpolation theorem", http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.mmj/1029001307&page=record
 Nachman, A. I. "Global uniqueness for a twodimensional inverse boundary value problem", http://www.jstor.org/discover/10.2307/2118653?uid=3739960&uid=2&uid=4&uid=3739256&sid=21101022816693
 Petrushev, P. P. and Popov, A. V. "Rational approximation of real functions", 1987, http://books.google.com/books/about/Rational_Approximation_of_Real_Functions.html?id=0up9c_uo2xQC
 Pinkus, A. "Totally positive matrices", http://www.amazon.com/TotallyPositiveMatricesCambridgeMathematics/dp/0521194083
 Sylvester, J. and Uhlmann, G. "A global uniqueness theorem for an inverse boundary value problem", Ann. of Math., 125 (1987), 153–169.
 Uhlmann, G. "Electrical impedance tomography and Calder´on’s problem", http://www.math.washington.edu/~gunther/publications/Papers/calderoniprevised.pdf