Analysis of Accordion DNA Stretching Revealed by The Gold Cluster Ruler

A promising new method for measuring intramolecular distances in solution uses small-angle X-ray scattering interference between gold nanocrystal labels (Mathew-Fenn et al, Science, 322, 446 (2008)). When applied to double stranded DNA, it revealed that the DNA length fluctuations are strikingly strong and correlated over at least 80 base pair steps. In other words, the DNA behaves as accordion bellows, with distant fragments stretching and shrinking concertedly. This hypothesis, however, disagrees with earlier experimental and computational observations. This Letter shows that the discrepancy can be rationalized by taking into account the cluster exclusion volume and assuming a moderate long-range repulsion between them. The long-range interaction can originate from an ion exclusion effect and cluster polarization in close proximity to the DNA surface.Comment: 9 pages, 4 figures, to appear in Phys. Rev.

Perturbations, Deformations, and Variations (and "Near-Misses") in Geometry, Physics, and Number Theory

Mathematic

Growth of Selmer Rank in Nonabelian Extensions of Number Fields

Let $p$ be an odd prime number, let E be an elliptic curve over a number field $k$, and let $F/k$ be a Galois extension of degree twice a power of p. We study the $Z_p$-corank $rk_p(E/F)$ of the $p$-power Selmer group of $E$ over $F$. We obtain lower bounds for $rk_p(E/F)$, generalizing the results in [MR], which applied to dihedral extensions. If $K$ is the (unique) quadratic extension of $k$ in $F$, if $G = Gal(F/K)$, if $G+$ is the subgroup of elements of $G$ commuting with a choice of involution of $F$ over $k$, and if $rk_p(E/K)$ is odd, then we show that (under mild hypotheses) $rkp(E/F)\ge[G:G+]$. As a very specific example of this, suppose that $A$ is an elliptic curve over $Q$ with a rational torsion point of order $p$ and without complex multiplication. If $E$ is an elliptic curve over $Q$ with good ordinary reduction at $p$ such that every prime where both $E$ and $A$ have bad reduction has odd order in $F\frac{x}{p}$ and such that the negative of the conductor of $E$ is not a square modulo $p$, then there is a positive constant $B$ depending on $A$ but not on $E$ or $n$ such that $rk_p(E/Q(A[p^n]))/geBp^{2n}$ for every $n$.Mathematic

Nearly Ordinary Galois Deformations over Arbitrary Number Fields

Let $K$ be an arbitrary number field, and let $\rho: Gal(K \bar/K) \rightarrow GL_2(E)$ be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of $\rho$. When $K$ is totally real and rho is modular, results of Hida imply that the nearly ordinary deformation space associated to rho contains a Zariski dense set of points corresponding to "automorphic" Galois representations. We conjecture that if $K$ is not totally real, then this is never the case, except in three exceptional cases, corresponding to (1) "base change", (2) "CM" forms, and (3) "Even" representations. The latter case conjecturally can only occur if the image of $\rho$ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of Leopoldt's conjecture. Second, when $K$ is an imaginary quadratic field, we prove an unconditional result that implies the existence of "many" positive dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about "$p$-adic functorality", as well as some remarks on how our methods should apply to n-dimensional representations of Gal$(Q \bar/Q)$ when $n \lt 2$.Mathematic

On Mathematics, Imagination and the Beauty of Numbers

Mathematic

The mixing of interplanetary magnetic field lines: A significant transport effect in studies of the energy spectra of impulsive flares

Using instrumentation on board the ACE spacecraft we describe short-time scale (~3 hour) variations observed in the arrival profiles of ~20 keV nucleon^(–1) to ~2 MeV nucleon^(–1) ions from impulsive solar flares. These variations occurred simultaneously across all energies and were generally not in coincidence with any local magnetic field or plasma signature. These features appear to be caused by the convection of magnetic flux tubes past the observer that are alternately filled and devoid of flare ions even though they had a common flare source at the Sun. In these particle events we therefore have a means to observe and measure the mixing of the interplanetary magnetic field due to random walk. In a survey of 25 impulsive flares observed at ACE between 1997 November and 1999 July these features had an average time scale of 3.2 hours, corresponding to a length of ~0.03 AU. The changing magnetic connection to the flare site sometimes lead to an incomplete observation of a flare at 1 AU; thus the field-line mixing is an important effect in studies of impulsive flare energy spectra

Real Time Relativity: exploration learning of special relativity

Real Time Relativity is a computer program that lets students fly at relativistic speeds though a simulated world populated with planets, clocks, and buildings. The counterintuitive and spectacular optical effects of relativity are prominent, while systematic exploration of the simulation allows the user to discover relativistic effects such as length contraction and the relativity of simultaneity. We report on the physics and technology underpinning the simulation, and our experience using it for teaching special relativity to first year university students

A Quantum Mechanical Model of the Reissner-Nordstrom Black Hole

We consider a Hamiltonian quantum theory of spherically symmetric, asymptotically flat electrovacuum spacetimes. The physical phase space of such spacetimes is spanned by the mass and the charge parameters $M$ and $Q$ of the Reissner-Nordstr\"{o}m black hole, together with the corresponding canonical momenta. In this four-dimensional phase space, we perform a canonical transformation such that the resulting configuration variables describe the dynamical properties of Reissner-Nordstr\"{o}m black holes in a natural manner. The classical Hamiltonian written in terms of these variables and their conjugate momenta is replaced by the corresponding self-adjoint Hamiltonian operator, and an eigenvalue equation for the ADM mass of the hole, from the point of view of a distant observer at rest, is obtained. Our eigenvalue equation implies that the ADM mass and the electric charge spectra of the hole are discrete, and the mass spectrum is bounded below. Moreover, the spectrum of the quantity $M^2-Q^2$ is strictly positive when an appropriate self-adjoint extension is chosen. The WKB analysis yields the result that the large eigenvalues of the quantity $\sqrt{M^2-Q^2}$ are of the form $\sqrt{2n}$, where $n$ is an integer. It turns out that this result is closely related to Bekenstein's proposal on the discrete horizon area spectrum of black holes.Comment: 37 pages, Plain TeX, no figure

Visualizing elements of Sha[3] in genus 2 jacobians

Mazur proved that any element xi of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that xi lies in the kernel of the natural homomorphism between the cohomology groups H^1(k,E) -> H^1(k,A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get explicit models of the genus 2 curves involved.Comment: 12 page