Charge-conserving hybrid methods for the Yang-Mills equations

The Yang-Mills equations generalize Maxwell's equations to nonabelian gauge groups, and a quantity analogous to charge is locally conserved by the nonlinear time evolution. Christiansen and Winther observed that, in the nonabelian case, the Galerkin method with Lie algebra-valued finite element differential forms appears to conserve charge globally but not locally, not even in a weak sense. We introduce a new hybridization of this method, give an alternative expression for the numerical charge in terms of the hybrid variables, and show that a local, per-element charge conservation law automatically holds.Comment: New section on nonzero current, new tables and figures for numerical experiments, and expanded introduction. 27 pages, 2 figures, 1 tabl

Finite element approximation of the Levi-Civita connection and its curvature in two dimensions

We construct finite element approximations of the Levi-Civita connection and its curvature on triangulations of oriented two-dimensional manifolds. Our construction relies on the Regge finite elements, which are piecewise polynomial symmetric (0,2)-tensor fields possessing single-valued tangential-tangential components along element interfaces. When used to discretize the Riemannian metric tensor, these piecewise polynomial tensor fields do not possess enough regularity to define connections and curvature in the classical sense, but we show how to make sense of these quantities in a distributional sense. We then show that these distributional quantities converge in certain dual Sobolev norms to their smooth counterparts under refinement of the triangulation. We also discuss projections of the distributional curvature and distributional connection onto piecewise polynomial finite element spaces. We show that the relevant projection operators commute with certain linearized differential operators, yielding a commutative diagram of differential complexes.Comment: v2: Several revisions throughout, including major revisions to Section 2.4, Section 5, and the beginning of Section

rasiRNA pathway controls antisense expression of Drosophila telomeric retrotransposons in the nucleus

Telomeres in Drosophila are maintained by the specialized telomeric retrotransposons HeT-A, TART and TAHRE. Sense transcripts of telomeric retroelements were shown to be the targets of a specialized RNA-interference mechanism, a repeat-associated short interfering (rasi)RNA-mediated system. Antisense rasiRNAs play a key role in this mechanism, highlighting the importance of antisense expression in retrotransposon silencing. Previously, bidirectional transcription was reported for the telomeric element TART. Here, we show that HeT-A is also bidirectionally transcribed, and HeT-A antisense transcription in ovaries is regulated by a promoter localized within its 3′ untranslated region. A remarkable feature of noncoding HeT-A antisense transcripts is the presence of multiple introns. We demonstrate that sense and antisense HeT-A-specific rasiRNAs are present in the same tissue, indicating that transcripts of both directions may be considered as natural targets of the rasiRNA pathway. We found that the expression of antisense transcripts of telomeric elements is regulated by the RNA silencing machinery, suggesting rasiRNA-mediated interplay between sense and antisense transcripts in the cell. Finally, this regulation occurs in the nucleus since disruption of the rasiRNA pathway leads to an accumulation of TART and HeT-A transcripts in germ cell nuclei

Uncovering the Lagrangian from Observations of Trajectories

We approach the problem of automatically modeling a mechanical system from data about its dynamics, using a method motivated by variational integrators. We write the discrete Lagrangian as a quadratic polynomial with varying coefficients, and then use the discrete Euler-Lagrange equations to numerically solve for the values of these coefficients near the data points. This method correctly modeled the Lagrangian of a simple harmonic oscillator and a simple pendulum, even with significant measurement noise added to the trajectories

Yang-Mills replacement

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.Cataloged from PDF version of thesis.Includes bibliographical references (pages 87-88).We develop an analog of the harmonic replacement technique of Colding and Minicozzi in the gauge theory context. The idea behind harmonic replacement dates back to Schwarz and Perron, and the technique involves taking a function v: ... defined on a surface ... and replacing its values on a small ball B2 ... with a harmonic function u that has the same values as v on the boundary &B2 . The resulting function on ... has lower energy, and repeating this process on balls covering ..., one can obtain a global harmonic map in the limit. We develop the analogous procedure in the gauge theory context. We take a connection B on a bundle over a four-manifold X, and replace it on a small ball ... with a Yang-Mills connection A that has the same restriction to the boundary [alpha]B4 as B, and we obtain bounds on the difference ... in terms of the drop in energy. Throughout, we work with connections of the lowest possible regularity ... (X), the natural choice for this context, and so our gauge transformations are in ... (X) and therefore almost but not quite continuous, leading to more delicate arguments than are available in higher regularity.by Yakov Berchenko-Kogan.Ph. D