Stokes tomography of radio pulsar magnetospheres. I. Linear polarization

Polarimetric studies of pulsar radio emission traditionally concentrate on how the Stokes vector (I, Q, U, V) varies with pulse longitude, with special emphasis on the position angle (PA) swing of the linearly polarized component. The interpretation of the PA swing in terms of the rotating vector model is limited by the assumption of an axisymmetric magnetic field and the degeneracy of the output with respect to the orientation and magnetic geometry of the pulsar; different combinations of the latter two properties can produce similar PA swings. This paper introduces Stokes phase portraits as a supplementary diagnostic tool with which the orientation and magnetic geometry can be inferred more accurately. The Stokes phase portraits feature unique patterns in the I-Q, I-U, and Q-U planes, whose shapes depend sensitively on the magnetic geometry, inclination angle, beam and polarization patterns, and emission altitude. We construct look-up tables of Stokes phase portraits and PA swings for pure and current-modified dipole fields, filled core and hollow cone beams, and two empirical linear polarization models, L/I = \cos \theta_0 and L/I = \sin \theta_0, where \theta_0 is the colatitude of the emission point. We compare our look-up tables to the measured phase portraits of 24 pulsars in the European Pulsar Network online database. We find evidence in 60% of the objects that the radio emission region may depart significantly from low altitudes, even when the PA swing is S-shaped and/or the pulse-width-period relation is well satisfied. On the other hand, the data are explained adequately if the emission altitude exceeds ~10% of the light cylinder radius. We conclude that Stokes phase portraits should be analysed concurrently with the PA swing and pulse profiles in future when interpreting radio pulsar polarization data.Comment: 60 pages, 58 figures, submitted to MNRAS, accepted 13 Oct 201

Oval Domes: History, Geometry and Mechanics

An oval dome may be defined as a dome whose plan or profile (or both) has an oval form. The word Aoval@ comes from the latin Aovum@, egg. Then, an oval dome has an egg-shaped geometry. The first buildings with oval plans were built without a predetermined form, just trying to close an space in the most economical form. Eventually, the geometry was defined by using arcs of circle with common tangents in the points of change of curvature. Later the oval acquired a more regular form with two axis of symmetry. Therefore, an “oval” may be defined as an egg-shaped form, doubly symmetric, constructed with arcs of circle; an oval needs a minimum of four centres, but it is possible also to build polycentric ovals. The above definition corresponds with the origin and the use of oval forms in building and may be applied without problem until, say, the XVIIIth century. Since then, the teaching of conics in the elementary courses of geometry made the cultivated people to define the oval as an approximation to the ellipse, an “imperfect ellipse”: an oval was, then, a curve formed with arcs of circles which tries to approximate to the ellipse of the same axes. As we shall see, the ellipse has very rarely been used in building. Finally, in modern geometrical textbooks an oval is defined as a smooth closed convex curve, a more general definition which embraces the two previous, but which is of no particular use in the study of the employment of oval forms in building. The present paper contains the following parts: 1) an outline the origin and application of the oval in historical architecture; 2) a discussion of the spatial geometry of oval domes, i. e., the different methods employed to trace them; 3) a brief exposition of the mechanics of oval arches and domes; and 4) a final discussion of the role of Geometry in oval arch and dome design

Stokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars

The radio polarization characteristics of millisecond pulsars (MSPs) differ significantly from those of non-recycled pulsars. In particular, the position angle (PA) swings of many MSPs deviate from the S-shape predicted by the rotating vector model, even after relativistic aberration is accounted for, indicating that they have non-dipolar magnetic geometries, likely due to a history of accretion. Stokes tomography uses phase portraits of the Stokes parameters as a diagnostic tool to infer a pulsar's magnetic geometry and orientation. This paper applies Stokes tomography to MSPs, generalizing the technique to handle interpulse emission. We present an atlas of look-up tables for the Stokes phase portraits and PA swings of MSPs with current-modified dipole fields, filled core and hollow cone beams, and two empirical linear polarization models. We compare our look-up tables to data from 15 MSPs and find that the Stokes phase portraits for a current-modified dipole approximately match several MSPs whose PA swings are flat or irregular and cannot be reconciled with the standard axisymmetric rotating vector model. PSR J1939+2134 and PSR J0437$-$4715 are modelled in detail. The data from PSR J1939+2134 at 0.61\,GHz can be fitted well with a current-modified dipole at $(\alpha, i) = (22 \pm 2^\circ, 80 \pm 1^\circ)$ and emission altitude 0.4 $r_\text{LC}$. The fit is less accurate for PSR J1939+2134 at 1.414\,GHz, and for PSR J0437$-$4715 at 1.44\,GHz, indicating that these objects may have a more complicated magnetic field geometry, such as a localized surface anomaly or a polar magnetic mountain.Comment: 38 pages, 33 figures, accepted for publication by MNRA

Counting and computing regions of DD-decomposition: algebro-geometric approach

New methods for $D$-decomposition analysis are presented. They are based on topology of real algebraic varieties and computational real algebraic geometry. The estimate of number of root invariant regions for polynomial parametric families of polynomial and matrices is given. For the case of two parametric family more sharp estimate is proven. Theoretic results are supported by various numerical simulations that show higher precision of presented methods with respect to traditional ones. The presented methods are inherently global and could be applied for studying $D$-decomposition for the space of parameters as a whole instead of some prescribed regions. For symbolic computations the Maple v.14 software and its package RegularChains are used.Comment: 16 pages, 8 figure

Apparent contours of nonsingular real cubic surfaces

We give a complete deformation classification of real Zariski sextics, that is of generic apparent contours of nonsingular real cubic surfaces. As a by-product, we observe a certain "reversion" duality in the set of deformation classes of these sextics.Comment: 61 pages, 8 figures, Revised version to be published in Transactions AMS: some minor corrections, a missing lemma is include

Deformation classification of real non-singular cubic threefolds with a marked line

We prove that the space of pairs $(X,l)$ formed by a real non-singular cubic hypersurface $X\subset P^4$ with a real line $l\subset X$ has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surface $F_\mathbb{R}(X)$ formed by real lines on $X$. For another interpretation we associate with each of the 18 components a well defined real deformation class of real non-singular plane quintic curves and show that this deformation class together with the real deformation class of $X$ characterizes completely the component