Quantum harmonic oscillator with superoscillating initial datum

In this paper we study the evolution of superoscillating initial data for the quantum driven harmonic oscillator. Our main result shows that superoscillations are amplified by the harmonic potential and that the analytic solution develops a singularity in finite time. We also show that for a large class of solutions of the Schr\"odinger equation, superoscillating behavior at any given time implies superoscillating behavior at any other time.Comment: 12 page

Geometric Path Integrals. A Language for Multiscale Biology and Systems Robustness

In this paper we suggest that, under suitable conditions, supervised learning can provide the basis to formulate at the microscopic level quantitative questions on the phenotype structure of multicellular organisms. The problem of explaining the robustness of the phenotype structure is rephrased as a real geometrical problem on a fixed domain. We further suggest a generalization of path integrals that reduces the problem of deciding whether a given molecular network can generate specific phenotypes to a numerical property of a robustness function with complex output, for which we give heuristic justification. Finally, we use our formalism to interpret a pointedly quantitative developmental biology problem on the allowed number of pairs of legs in centipedes

THERMAL ALTERATION IN RECEPTOR ACTIVITY OF THE RAT FUNDAL STRIP1'2

ABSTRACT FLEISCH, JERo rs H. AND S. EHRENPREIS

Normal Positron Emission Tomography-Computerized Tomogram in a Patient with Apparent Mesenteric Panniculitis: Biopsy Is Still the Answer

Mesenteric panniculitis (also known as sclerosing mesenteritis) is a chronic inflammatory disease of the mesenteric connective tissue. It is known to have a wide spectrum of clinical and radiological presentations. In general, biopsy is recommended for diagnosis; however, a recent study proposed that a negative positron emission tomography- computerized tomography (PET-CT) scan is accurate in differentiating benign and neoplastic mesenteric processes [Br J Radiol 2006;79:37–43]. The following case report questions the accuracy of PET-CT in this setting and confirms the requirement for biopsy to rule out the presence of mesenteric lymphoma

On the problem of mass-dependence of the two-point function of the real scalar free massive field on the light cone

We investigate the generally assumed inconsistency in light cone quantum field theory that the restriction of a massive, real, scalar, free field to the nullplane $\Sigma=\{x^0+x^3=0\}$ is independent of mass \cite{LKS}, but the restriction of the two-point function depends on it (see, e.g., \cite{NakYam77, Yam97}). We resolve this inconsistency by showing that the two-point function has no canonical restriction to $\Sigma$ in the sense of distribution theory. Only the so-called tame restriction of the two-point function exists which we have introduced in \cite{Ull04sub}. Furthermore, we show that this tame restriction is indeed independent of mass. Hence the inconsistency appears only by the erroneous assumption that the two-point function would have a (canonical) restriction to $\Sigma$.Comment: 10 pages, 2 figure

Boundary Conditions for Singular Perturbations of Self-Adjoint Operators

Let A:D(A)\subseteq\H\to\H be an injective self-adjoint operator and let \tau:D(A)\to\X, X a Banach space, be a surjective linear map such that \|\tau\phi\|_\X\le c \|A\phi\|_\H. Supposing that \text{\rm Range} (\tau')\cap\H' =\{0\}, we define a family $A^\tau_\Theta$ of self-adjoint operators which are extensions of the symmetric operator $A_{|\{\tau=0\}.}$. Any $\phi$ in the operator domain $D(A^\tau_\Theta)$ is characterized by a sort of boundary conditions on its univocally defined regular component \phireg, which belongs to the completion of D(A) w.r.t. the norm \|A\phi\|_\H. These boundary conditions are written in terms of the map $\tau$, playing the role of a trace (restriction) operator, as \tau\phireg=\Theta Q_\phi, the extension parameter $\Theta$ being a self-adjoint operator from X' to X. The self-adjoint extension is then simply defined by A^\tau_\Theta\phi:=A \phireg. The case in which $A\phi=T*\phi$ is a convolution operator on LD, T a distribution with compact support, is studied in detail.Comment: Revised version. To appear in Operator Theory: Advances and Applications, vol. 13

Spherical Spectral Synthesis and Two-Radius Theorems on Damek-Ricci Spaces

We prove that spherical spectral analysis and synthesis hold in Damek-Ricci spaces and derive two-radius theorems

On the injectivity of the circular Radon transform arising in thermoacoustic tomography

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 propagation and domain dependence for the wave equation.Comment: To appear in Inverse Problem

Initial-boundary value problems for discrete evolution equations: discrete linear Schrodinger and integrable discrete nonlinear Schrodinger equations

We present a method to solve initial-boundary value problems for linear and integrable nonlinear differential-difference evolution equations. The method is the discrete version of the one developed by A. S. Fokas to solve initial-boundary value problems for linear and integrable nonlinear partial differential equations via an extension of the inverse scattering transform. The method takes advantage of the Lax pair formulation for both linear and nonlinear equations, and is based on the simultaneous spectral analysis of both parts of the Lax pair. A key role is also played by the global algebraic relation that couples all known and unknown boundary values. Even though additional technical complications arise in discrete problems compared to continuum ones, we show that a similar approach can also solve initial-boundary value problems for linear and integrable nonlinear differential-difference equations. We demonstrate the method by solving initial-boundary value problems for the discrete analogue of both the linear and the nonlinear Schrodinger equations, comparing the solution to those of the corresponding continuum problems. In the linear case we also explicitly discuss Robin-type boundary conditions not solvable by Fourier series. In the nonlinear case we also identify the linearizable boundary conditions, we discuss the elimination of the unknown boundary datum, we obtain explicitly the linear and continuum limit of the solution, and we write down the soliton solutions.Comment: 41 pages, 3 figures, to appear in Inverse Problem

Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography

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