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

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

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

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

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

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

Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under \Gamma_0(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard-Fuchs equations of triangle subgroups of PSL(2,R) which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with \Gamma(1) is treated.Comment: 40 pages, final version, accepted by Manuscripta Mathematic

Unusual primary HIV infection with colonic ulcer complicated by hemorrhagic shock: a case report

<p>Abstract</p> <p>Introduction</p> <p>Timely diagnosis of primary HIV infection is important to prevent further transmission of HIV. Primary HIV infection may take place without symptoms or may be associated with fever, pharyngitis or headache. Sometimes, the clinical presentation includes aseptic meningitis or cutaneous lesions. Intestinal ulceration due to opportunistic pathogens (cytomegalovirus, Epstein-Barr virus, <it>Toxoplasma gondii</it>) has been described in patients with AIDS. However, although invasion of intestinal lymphoid tissue is a prominent feature of human and simian lentivirus infections, colonic ulceration has not been reported in acute HIV infection.</p> <p>Case description</p> <p>A 42-year-old Caucasian man was treated with amoxicillin-clavulanate for pharyngitis. He did not improve, and a rash developed. History taking revealed a negative HIV antibody test five months previously and unprotected sex with a male partner the month before admission. Repeated tests revealed primary HIV infection with an exceptionally high HIV-1 RNA plasma concentration (3.6 × 10⁷copies/mL) and a low CD4 count (101 cells/mm³, seven percent of total lymphocytes). While being investigated, the patient had a life-threatening hematochezia. After angiographic occlusion of a branch of the ileocaecal artery and initiation of antiretroviral therapy, the patient became rapidly asymptomatic and could be discharged. Colonoscopy revealed a bleeding colonic ulcer. We were unable to identify an etiology other than HIV for this ulcer.</p> <p>Conclusion</p> <p>This case adds to the known protean manifestation of primary HIV infection. The lack of an alternative etiology, despite extensive investigations, suggests that this ulcer was directly caused by primary HIV infection. This conclusion is supported by the well-described extensive loss of intestinal mucosal CD4⁺T cells associated with primary HIV infection, the extremely high HIV viral load observed in our patient, and the rapid improvement of the ulcer after initiation of highly active antiretroviral therapy. This case also adds to the debate on treatment for primary HIV infection, especially in the context of severe symptoms and an extremely high viral load.</p