2
2.0

Jun 29, 2018
06/18

by
Peter Kuchment

texts

######
eye 2

######
favorite 0

######
comment 0

This text is a somewhat reformatted (e.g., some statements that were not as such in the original paper, are given the names "Corollary" or "Theorem.") translation of the old and practically inaccessible paper: P. Kuchment, On the question of the affine-invariant points of convex bodies, (in Russian), Optimizacija No. 8(25) (1972), 48--51, 127. MR0350621. There partial solutions of some old problems of B. Gr\"unbaum concerning affine-invariant points of convex bodies...

Topics: Metric Geometry, Mathematics

Source: http://arxiv.org/abs/1602.04377

48
48

Sep 20, 2013
09/13

by
Peter Kuchment

texts

######
eye 48

######
favorite 0

######
comment 0

The purpose of this text is to set up a few basic notions concerning quantum graphs, to indicate some areas addressed in the quantum graph research, and to provide some pointers to the literature. The pointers in many cases are secondary, i.e. they refer to other surveys.

Source: http://arxiv.org/abs/0802.3442v1

41
41

Sep 23, 2013
09/13

by
Peter Kuchment

texts

######
eye 41

######
favorite 0

######
comment 0

The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and other areas. A Schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an subexponential growth integral estimate is available. A theorem on spectral gap opening for ``decorated''...

Source: http://arxiv.org/abs/math-ph/0411003v3

100
100

Jul 20, 2013
07/13

by
Peter Kuchment

texts

######
eye 100

######
favorite 0

######
comment 0

The article provides a brief survey of the mathematics of some of the newly being developed so called "hybrid" (also called "multi-physics" or "multi-wave") imaging techniques.

Source: http://arxiv.org/abs/1107.2447v1

93
93

Jul 20, 2013
07/13

by
Peter Kuchment

texts

######
eye 93

######
favorite 0

######
comment 0

The paper discusses relations between the structure of the complex Fermi surface below the spectrum of a second order periodic elliptic equation and integral representations of certain classes of its solutions. These integral representations are analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation (i.e., for generalized eigenfunctions of Laplace operator). In a previous joint work with Y. Pinchover we described all solutions...

Source: http://arxiv.org/abs/math/0604139v1

2
2.0

Jun 28, 2018
06/18

by
Peter Kuchment

texts

######
eye 2

######
favorite 0

######
comment 0

The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which influence significantly most properties of such operators. The approaches described are applicable not only to the standard model example of Schr\"odinger operator with periodic electric potential $-\Delta+V(x)$, but to a wide variety of elliptic periodic...

Topics: Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1510.00971

34
34

Jul 24, 2013
07/13

by
Peter Kuchment

texts

######
eye 34

######
favorite 0

######
comment 0

Let L be a Schroedinger operator with periodic magnetic and electric potentials, a Maxwell operator in a periodic medium, or an arbitrary self-adjoint elliptic linear partial differential operator in R^n with coefficients periodic with respect to a lattice G. Let also S be a finite part of its spectrum separated by gaps from the rest of the spectrum. We consider the old question of existence of a finite set of exponentially decaying Wannier functions such that their G-shifts span the whole...

Source: http://arxiv.org/abs/0807.1342v3

46
46

Sep 18, 2013
09/13

by
Peter Kuchment; Leonid Kunyansky

texts

######
eye 46

######
favorite 0

######
comment 0

The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic and Photoacoustic Tomography.

Source: http://arxiv.org/abs/0704.0286v2

69
69

Sep 21, 2013
09/13

by
Mark Agranovsky; Peter Kuchment

texts

######
eye 69

######
favorite 0

######
comment 0

Let f(x) belong to L^p(R^n) and R>0. The transform is considered that integrates the function f over (almost) all spheres of radius R in R^n. This operator is known to be non-injective (as one can see by taking Fourier transform). However, the counterexamples that can be easily constructed using Bessel functions of the 1st kind, only belong to L^p if p>2n/(n-1). It has been shown previously by S. Thangavelu that for p not exceeding the critical number 2n/(n-1), the transform is indeed...

Source: http://arxiv.org/abs/0905.1310v2

29
29

Sep 23, 2013
09/13

by
Peter Kuchment; Dustin Steinhauer

texts

######
eye 29

######
favorite 0

######
comment 0

Several newly developing hybrid imaging methods (e.g., those combining electrical impedance or optical imaging with acoustics) enable one to obtain some auxiliary interior information (usually some combination of the electrical conductivity and the current) about the parameters of the tissues. This information, in turn, happens to stabilize the exponentially unstable and thus low resolution optical and electrical impedance tomography. Various known instances of this effect have been studied...

Source: http://arxiv.org/abs/1110.1819v3

92
92

Sep 21, 2013
09/13

by
Peter Kuchment; Leonid Kunyansky

texts

######
eye 92

######
favorite 0

######
comment 0

Several hybrid tomographic methods utilizing ultrasound modulation have been introduced lately. Success of these methods hinges on the feasibility of focusing ultrasound waves at an arbitrary point of interest. Such a focusing, however, is difficult to achieve in practice. We thus propose a way to avoid the use of focused waves through the so called synthetic focusing, i.e. by the reconstruction of the would-be response to the focused modulation from the measurements corresponding to realistic...

Source: http://arxiv.org/abs/0901.2552v1

43
43

Sep 20, 2013
09/13

by
Peter Kuchment; Olaf Post

texts

######
eye 43

######
favorite 0

######
comment 0

An explicit derivation of dispersion relations and spectra for periodic Schr\"{o}dinger operators on carbon nano-structures (including graphen and all types of single-wall nano-tubes) is provided.

Source: http://arxiv.org/abs/math-ph/0612021v4

49
49

Sep 21, 2013
09/13

by
Peter Kuchment; Leonid Kunyansky

texts

######
eye 49

######
favorite 0

######
comment 0

We propose and test stable algorithms for the reconstruction of the internal conductivity of a biological object using acousto-electric measurements. Namely, the conventional impedance tomography scheme is supplemented by scanning the object with acoustic waves that slightly perturb the conductivity and cause the change in the electric potential measured on the boundary of the object. These perturbations of the potential are then used as the data for the reconstruction of the conductivity. The...

Source: http://arxiv.org/abs/1011.3059v3

2
2.0

Jun 30, 2018
06/18

by
David Auckly; Peter Kuchment

texts

######
eye 2

######
favorite 0

######
comment 0

Let $L$ be a periodic self-adjoint linear elliptic operator in $\R^n$ with coefficients periodic with respect to a lattice $\G$, e.g. Schr\"{o}dinger operator $(i^{-1}\partial/\partial_x-A(x))^2+V(x)$ with periodic magnetic and electric potentials $A,V$, or a Maxwell operator $\nabla\times\varepsilon (x)^{-1}\nabla\times$ in a periodic medium. Let also $S$ be a finite part of its spectrum separated by gaps from the rest of the spectrum. We address here the question of existence of a finite...

Topics: Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1704.05728

59
59

Sep 21, 2013
09/13

by
Gregory Berkolaiko; Peter Kuchment

texts

######
eye 59

######
favorite 0

######
comment 0

We study the dependence of the quantum graph Hamiltonian, its resolvent, and its spectrum on the vertex conditions and graph edge lengths. In particular, several results on the interlacing (bracketing) of the spectra of graphs with different vertex conditions are obtained and their applications are discussed.

Source: http://arxiv.org/abs/1008.0369v2

80
80

Sep 23, 2013
09/13

by
Peter Kuchment; Boris Vainberg

texts

######
eye 80

######
favorite 0

######
comment 0

The problem of absence of eigenvalues imbedded into the continuous spectrum is considered for a Schr\"{o}dinger operator with a periodic potential perturbed by a sufficiently fast decaying ``impurity'' potential. Results of this type have previously been known for the one-dimensional case only. Absence of embedded eigenvalues is shown in dimensions two and three if the corresponding Fermi surface is irreducible modulo natural symmetries. It is conjectured that all periodic potentials...

Source: http://arxiv.org/abs/math-ph/9904016v1

41
41

Sep 21, 2013
09/13

by
Mark Agranovsky; Peter Kuchment

texts

######
eye 41

######
favorite 0

######
comment 0

The paper contains a simple approach to reconstruction in Thermoacoustic and Photoacoustic Tomography. The technique works for any geometry of point detectors placement and for variable sound speed satisfying a non-trapping condition. A uniqueness of reconstruction result is also obtained.

Source: http://arxiv.org/abs/0706.0598v1

4
4.0

Jun 29, 2018
06/18

by
Peter Kuchment; Fatma Terzioglu

texts

######
eye 4

######
favorite 0

######
comment 0

In this paper, we address analytically and numerically the inversion of the integral transform (\emph{cone} or \emph{Compton} transform) that maps a function on $\mathbb{R}^3$ to its integrals over conical surfaces. It arises in a variety of imaging techniques, e.g. in astronomy, optical imaging, and homeland security imaging, especially when the so called Compton cameras are involved. Several inversion formulas are developed and implemented numerically in $3D$ (the much simpler $2D$ case was...

Topics: Data Analysis, Statistics and Probability, Physics

Source: http://arxiv.org/abs/1604.03805

42
42

Sep 20, 2013
09/13

by
Peter Kuchment; Sergei Levendorski

texts

######
eye 42

######
favorite 0

######
comment 0

The paper contains a brief description of a simplified version of A. Sobolev's proof of absolute continuity of spectra of periodic magnetic Schr\"{o}dinger operators. This approach is applicable to all periodic elliptic operators known to be of interest for math physics (including Maxwell), and in all these cases leads to the same model problem of complex analysis. The full account of this approach will be provided elsewhere.

Source: http://arxiv.org/abs/math-ph/9810002v1

2
2.0

Jun 29, 2018
06/18

by
Peter Kuchment; Fatma Terzioglu

texts

######
eye 2

######
favorite 0

######
comment 0

In this paper, we investigate the relations between the Radon and weighted divergent beam and cone transforms. Novel inversion formulas are derived for the latter two. The weighted cone transform arises, for instance, in image reconstruction from the data obtained by Compton cameras, which have promising applications in various fields, including biomedical and homeland security imaging and gamma ray astronomy. The inversion formulas are applicable for a wide variety of detector geometries in...

Topics: Functional Analysis, Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1612.06772

52
52

Sep 20, 2013
09/13

by
Gaik Ambartsoumian; Peter Kuchment

texts

######
eye 52

######
favorite 0

######
comment 0

The circular Radon transform integrates a function over the set of all spheres with a given set of centers. The problem of injectivity of this transform (as well as inversion formulas, range descriptions, etc.) arises in many fields from approximation theory to integral geometry, to inverse problems for PDEs, and recently to newly developing types of tomography. The article discusses known and provides new results that one can obtain by methods that essentially involve only the finite speed of...

Source: http://arxiv.org/abs/math-ph/0404065v3

37
37

Sep 17, 2013
09/13

by
Peter Kuchment; Yehuda Pinchover

texts

######
eye 37

######
favorite 0

######
comment 0

The paper contains integral representations for certain classes of exponentially growing solutions of second order periodic elliptic equations. These representations are the analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation. When one restricts the class of solutions further, requiring their growth to be polynomial, one arrives to Liouville type theorems, which describe the structure and dimension of the spaces of such...

Source: http://arxiv.org/abs/math/0007051v1

42
42

Sep 21, 2013
09/13

by
Gaik Ambartsoumian; Peter Kuchment

texts

######
eye 42

######
favorite 0

######
comment 0

The transform considered in the paper integrates a function supported in the unit disk on the plane over all circles centered at the boundary of this disk. Such circular Radon transform arises in several contemporary imaging techniques, as well as in other applications. As it is common for transforms of Radon type, its range has infinite co-dimension in standard function spaces. Range descriptions for such transforms are known to be very important for computed tomography, for instance when...

Source: http://arxiv.org/abs/math/0508082v1

3
3.0

Jun 30, 2018
06/18

by
Peter Kuchment; Dustin Steinhauer

texts

######
eye 3

######
favorite 0

######
comment 0

In the previous paper "Stabilizing Inverse Problems by Internal Data", the authors introduced a simple procedure that allows one to detect whether and explain why internal information arising in several novel coupled physics (hybrid) imaging modalities could turn extremely unstable techniques, such as optical tomography or electrical impedance tomography, into stable, good-resolution procedures. It was shown that in all cases of interest, the Frechet derivative of the forward mapping...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1407.0763

49
49

Sep 23, 2013
09/13

by
Peter Kuchment; Boris Vainberg

texts

######
eye 49

######
favorite 0

######
comment 0

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z^n. It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders...

Source: http://arxiv.org/abs/math-ph/0511084v1

56
56

Sep 23, 2013
09/13

by
Peter Kuchment; Sergey Lvin

texts

######
eye 56

######
favorite 0

######
comment 0

The article describes interesting nonlinear differential identities satisfied by standard exponential and trigonometric functions, which appeared as byproducts of medical imaging research. They look like some kind of non-commutative binomial formulas. A brief description of the origin of these identities is provided, as well as their direct algebraic derivation. Relations with separate analyticity theorems in several complex variables and some open problems are also mentioned.

Source: http://arxiv.org/abs/1110.6109v1

36
36

Sep 23, 2013
09/13

by
Peter Kuchment; Andrew Raich

texts

######
eye 36

######
favorite 0

######
comment 0

Precise asymptotics known for the Green's function of the Laplace operator have found their analogs for periodic elliptic operators of the second order at and below the bottom of the spectrum. Due to the band-gap structure of the spectra of such operators, the question arises whether similar results can be obtained near or at the edges of spectral gaps. As the result of this work shows, this is possible at a spectral edge in dimensions d>2.

Source: http://arxiv.org/abs/1110.0225v1

50
50

Sep 17, 2013
09/13

by
Peter Kuchment; Leonid Kunyansky

texts

######
eye 50

######
favorite 0

######
comment 0

This is the manuscript of the chapter for a planned Handbook of Mathematical Methods in Imaging that surveys the mathematical models, problems, and algorithms of the Thermoacoustic (TAT) and Photoacoustic (PAT) Tomography. TAT and PAT represent probably the most developed of the several novel ``hybrid'' methods of medical imaging. These new modalities combine different physical types of waves (electromagnetic and acoustic in case of TAT and PAT) in such a way that the resolution and contrast of...

Source: http://arxiv.org/abs/0912.2022v1

32
32

Sep 22, 2013
09/13

by
Ngoc T. Do; Peter Kuchment

texts

######
eye 32

######
favorite 0

######
comment 0

We study the dispersion relations and spectra of invariant Schr\"odinger operators on a graphyne structure (lithographite). In particular, description of different parts of the spectrum, band-gap structure, and Dirac points are provided.

Source: http://arxiv.org/abs/1302.5176v2

45
45

Sep 18, 2013
09/13

by
S. A. Fulling; Peter Kuchment

texts

######
eye 45

######
favorite 0

######
comment 0

Penrose--Lifshits mushrooms are planar domains coming in nonisometric pairs with the same geodesic length spectrum. Recently S. Zelditch raised the question whether such billiards also have the same eigenvalue spectrum for the Dirichlet Laplacian (conjecturing ``no''). Here we show that generically (in the class of smooth domains) the two members of a mushroom pair have different spectra.

Source: http://arxiv.org/abs/math/0505200v1

75
75

Sep 17, 2013
09/13

by
Peter Kuchment; Beng-Seong Ong

texts

######
eye 75

######
favorite 0

######
comment 0

The paper addresses the issue of existence and confinement of electromagnetic modes guided by linear defects in photonic crystals. Sufficient condition are provided for existence of such waves near a given spectral location. Confinement to the guide is achieved due to a photonic band gap in the bulk dielectric medium.

Source: http://arxiv.org/abs/0912.1386v1

5
5.0

Jun 28, 2018
06/18

by
Minh Kha; Peter Kuchment; Andrew Raich

texts

######
eye 5

######
favorite 0

######
comment 0

Precise asymptotics known for the Green function of the Laplacian have found their analogs for bounded below periodic elliptic operators of the second-order below and at the bottom of the spectrum. Due to the band-gap structure of the spectra of such operators, the question arises whether similar results can be obtained near or at the edges of spectral gaps. In a previous work, two of the authors considered the case of a spectral edge. The main result of this article is finding such asymptotics...

Topics: Mathematical Physics, Mathematics, Spectral Theory

Source: http://arxiv.org/abs/1508.06703

34
34

Sep 23, 2013
09/13

by
Pavel Exner; Peter Kuchment; Brian Winn

texts

######
eye 34

######
favorite 0

######
comment 0

Periodic $2$nd order ordinary differential operators on $\R$ are known to have the edges of their spectra to occur only at the spectra of periodic and antiperiodic boundary value problems. The multi-dimensional analog of this property is false, as was shown in a 2007 paper by some of the authors of this article. However, one sometimes encounters the claims that in the case of a single periodicity (i.e., with respect to the lattice $\mathbb{Z}$), the $1D$ property still holds, and spectral edges...

Source: http://arxiv.org/abs/1006.3001v1

36
36

Sep 21, 2013
09/13

by
Mark Agranovsky; David Finch; Peter Kuchment

texts

######
eye 36

######
favorite 0

######
comment 0

The paper is devoted to the range description of the Radon type transform that averages a function over all spheres centered on a given sphere. Such transforms arise naturally in thermoacoustic tomography, a novel method of medical imaging. Range descriptions have recently been obtained for such transforms, and consisted of smoothness and support conditions, moment conditions, and some additional orthogonality conditions of spectral nature. It has been noticed that in odd dimensions,...

Source: http://arxiv.org/abs/0902.4272v3

37
37

Jul 20, 2013
07/13

by
Gregory Berkolaiko; Peter Kuchment; Uzy Smilansky

texts

######
eye 37

######
favorite 0

######
comment 0

The paper addresses the the number of nodal domains for eigenfunctions of Schr\"{o}dinger operators with Dirichlet boundary conditions in bounded domains. In dimension one, the $n$th eigenfunction has $n$ nodal domains. The Courant Theorem claims that in any dimension, the number of nodal domains of the $n$th eigenfunction cannot exceed $n$. However, in dimensions higher than 1 the equality can hold for only finitely many eigenfunctions. Thus, a "nodal deficiency" arises....

Source: http://arxiv.org/abs/1107.3489v6

2
2.0

Jun 29, 2018
06/18

by
Ngoc T. Do; Peter Kuchment; Beng Ong

texts

######
eye 2

######
favorite 0

######
comment 0

In this brief paper we present some results on creating and manipulating spectral gaps for a (regular) quantum graph by inserting appropriate internal structures into its vertices. Complete proofs and extensions of the results are planned for another publication.

Topics: Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1601.04774

56
56

Jul 20, 2013
07/13

by
Xiaolei Xun; Bani Mallick; Raymond J. Carroll; Peter Kuchment

texts

######
eye 56

######
favorite 0

######
comment 0

The article addresses the problem of detecting presence and location of a small low emission source inside of an object, when the background noise dominates. This problem arises, for instance, in some homeland security applications. The goal is to reach the signal-to-noise ratio (SNR) levels on the order of $10^{-3}$. A Bayesian approach to this problem is implemented in 2D. The method allows inference not only about the existence of the source, but also about its location. We derive Bayes...

Source: http://arxiv.org/abs/1107.2980v1

46
46

Sep 23, 2013
09/13

by
Moritz Allmaras; David P. Darrow; Yulia Hristova; Guido Kanschat; Peter Kuchment

texts

######
eye 46

######
favorite 0

######
comment 0

The article addresses the possibility of robust detection of geometrically small, low emission sources on a significantly stronger background. This problem is important for homeland security. A technique of detecting such sources using Compton type cameras is developed, which is shown on numerical examples to have high sensitivity and specificity and also allows to assign confidence probabilities of the detection. 2D case is considered in detail.

Source: http://arxiv.org/abs/1012.3373v2