Painleve versus Fuchs

The sigma form of the Painlev{\'e} VI equation contains four arbitrary parameters and generically the solutions can be said to be genuinely ``nonlinear'' because they do not satisfy linear differential equations of finite order. However, when there are certain restrictions on the four parameters there exist one parameter families of solutions which do satisfy (Fuchsian) differential equations of finite order. We here study this phenomena of Fuchsian solutions to the Painlev{\'e} equation with a focus on the particular PVI equation which is satisfied by the diagonal correlation function C(N,N) of the Ising model. We obtain Fuchsian equations of order $N+1$ for C(N,N) and show that the equation for C(N,N) is equivalent to the $N^{th}$ symmetric power of the equation for the elliptic integral $E$. We show that these Fuchsian equations correspond to rational algebraic curves with an additional Riccati structure and we show that the Malmquist Hamiltonian $p,q$ variables are rational functions in complete elliptic integrals. Fuchsian equations for off diagonal correlations $C(N,M)$ are given which extend our considerations to discrete generalizations of Painlev{\'e}.Comment: 18 pages, Dedicated to the centenary of the publication of the Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard Fuchs in 190

Tetramethylenedisulfotetramine alters Ca²⁺ dynamics in cultured hippocampal neurons: mitigation by NMDA receptor blockade and GABA(A) receptor-positive modulation.

Tetramethylenedisulfotetramine (TETS) is a potent convulsant that is considered a chemical threat agent. We characterized TETS as an activator of spontaneous Ca²⁺ oscillations and electrical burst discharges in mouse hippocampal neuronal cultures at 13-17 days in vitro using FLIPR Fluo-4 fluorescence measurements and extracellular microelectrode array recording. Acute exposure to TETS (≥ 2 µM) reversibly altered the pattern of spontaneous neuronal discharges, producing clustered burst firing and an overall increase in discharge frequency. TETS also dramatically affected Ca²⁺ dynamics causing an immediate but transient elevation of neuronal intracellular Ca²⁺ followed by decreased frequency of Ca²⁺ oscillations but greater peak amplitude. The effect on Ca²⁺ dynamics was similar to that elicited by picrotoxin and bicuculline, supporting the view that TETS acts by inhibiting type A gamma-aminobutyric acid (GABA(A)) receptor function. The effect of TETS on Ca²⁺ dynamics requires activation of N-methyl-D-aspartic acid (NMDA) receptors, because the changes induced by TETS were prevented by MK-801 block of NMDA receptors, but not nifedipine block of L-type Ca²⁺ channels. Pretreatment with the GABA(A) receptor-positive modulators diazepam and allopregnanolone partially mitigated TETS-induced changes in Ca²⁺ dynamics. Moreover, low, minimally effective concentrations of diazepam (0.1 µM) and allopregnanolone (0.1 µM), when administered together, were highly effective in suppressing TETS-induced alterations in Ca²⁺ dynamics, suggesting that the combination of positive modulators of synaptic and extrasynaptic GABA(A) receptors may have therapeutic potential. These rapid throughput in vitro assays may assist in the identification of single agents or combinations that have utility in the treatment of TETS intoxication

Quantifying evolutionary constraints on B cell affinity maturation

The antibody repertoire of each individual is continuously updated by the evolutionary process of B cell receptor mutation and selection. It has recently become possible to gain detailed information concerning this process through high-throughput sequencing. Here, we develop modern statistical molecular evolution methods for the analysis of B cell sequence data, and then apply them to a very deep short-read data set of B cell receptors. We find that the substitution process is conserved across individuals but varies significantly across gene segments. We investigate selection on B cell receptors using a novel method that side-steps the difficulties encountered by previous work in differentiating between selection and motif-driven mutation; this is done through stochastic mapping and empirical Bayes estimators that compare the evolution of in-frame and out-of-frame rearrangements. We use this new method to derive a per-residue map of selection, which provides a more nuanced view of the constraints on framework and variable regions.Comment: Previously entitled "Substitution and site-specific selection driving B cell affinity maturation is consistent across individuals

Functional Forms for the Squeeze and the Time-Displacement Operators

Using Baker-Campbell-Hausdorff relations, the squeeze and harmonic-oscillator time-displacement operators are given in the form $\exp[\delta I] \exp[\alpha (x^2)]\exp[\beta(x\partial)] \exp[\gamma (\partial)^2]$, where $\alpha$, $\beta$, $\gamma$, and $\delta$ are explicitly determined. Applications are discussed.Comment: 10 pages, LaTe

Zero--Temperature Quantum Phase Transition of a Two--Dimensional Ising Spin--Glass

We study the quantum transition at $T=0$ in the spin-$\frac12$ Ising spin--glass in a transverse field in two dimensions. The world line path integral representation of this model corresponds to an effective classical system in (2+1) dimensions, which we study by Monte Carlo simulations. Values of the critical exponents are estimated by a finite-size scaling analysis. We find that the dynamical exponent, $z$, and the correlation length exponent, $\nu$, are given by $z = 1.5 \pm 0.05$ and $\nu = 1.0 \pm 0.1$. Both the linear and non-linear susceptibility are found to diverge at the critical point.Comment: RevTeX 10 pages + 4 figures (appended as uuencoded, compressed tar-file), THP21-9

Critical Behavior and Griffiths-McCoy Singularities in the Two-Dimensional Random Quantum Ising Ferromagnet

We study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one-dimension. At the critical point, the dynamical exponent is infinite and the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point there are Griffiths-McCoy singularities, characterized by a single, continuously varying exponent, z', which diverges at the critical point, as in one-dimension. Consequently, the zero temperature susceptibility diverges for a RANGE of parameters about the transition.Comment: 4 pages RevTeX, 3 eps-figures include

Boundary bound states and boundary bootstrap in the sine-Gordon model with Dirichlet boundary conditions.

We present a complete study of boundary bound states and related boundary S-matrices for the sine-Gordon model with Dirichlet boundary conditions. Our approach is based partly on the bootstrap procedure, and partly on the explicit solution of the inhomogeneous XXZ model with boundary magnetic field and of the boundary Thirring model. We identify boundary bound states with new ``boundary strings'' in the Bethe ansatz. The boundary energy is also computed.Comment: 25 pages, harvmac macros Report USC-95-001

Mean Field Renormalization Group for the Boundary Magnetization of Strip Clusters

We analyze in some detail a recently proposed transfer matrix mean field approximation which yields the exact critical point for several two dimensional nearest neighbor Ising models. For the square lattice model we show explicitly that this approximation yields not only the exact critical point, but also the exact boundary magnetization of a semi--infinite Ising model, independent of the size of the strips used. Then we develop a new mean field renormalization group strategy based on this approximation and make connections with finite size scaling. Applying our strategy to the quadratic Ising and three--state Potts models we obtain results for the critical exponents which are in excellent agreement with the exact ones. In this way we also clarify some advantages and limitations of the mean field renormalization group approach.Comment: 16 pages (plain TeX) + 8 figures (PostScript, appended), POLFIS-TH.XX/9

Analyticity and Integrabiity in the Chiral Potts Model

We study the perturbation theory for the general non-integrable chiral Potts model depending on two chiral angles and a strength parameter and show how the analyticity of the ground state energy and correlation functions dramatically increases when the angles and the strength parameter satisfy the integrability condition. We further specialize to the superintegrable case and verify that a sum rule is obeyed.Comment: 31 pages in harvmac including 9 tables, several misprints eliminate

Structure analysis of the Ga-stabilized GaAs(001)-c(8x2) surface at high temperatures

Structure of the Ga-stabilized GaAs(001)-c(8x2) surface has been studied using rocking-curve analysis of reflection high-energy electron diffraction (RHEED). The c(8x2) structure emerges at temperatures higher than 600C, but is unstable with respect to the change to the (2x6)/(3x6) structure at lower temperatures. Our RHEED rocking-curve analysis at high temperatures revealed that the c(8x2) surface has the structure which is basically the same as that recently proposed by Kumpf et al. [Phys. Rev. Lett. 86, 3586 (2001)]. We found that the surface atomic configurations are locally fluctuated at high temperatures without disturbing the c(8x2) periodicity.Comment: 14 pages, 4 figures, 1 tabl