Vafa-Witten invariants for projective surfaces II: semistable case

We propose a definition of Vafa-Witten invariants counting semistable Higgs pairs on a polarised surface. We use virtual localisation applied to Mochizuki/Joyce-Song pairs. For $K_S\le0$ we expect our definition coincides with an alternative definition using weighted Euler characteristics. We prove this for deg $K_S<0$ here, and it is proved for $S$ a K3 surface in \cite{MT}. For K3 surfaces we calculate the invariants in terms of modular forms which generalise and prove conjectures of Vafa and Witten.Comment: Error found by Laarakker fixed. To appear in Pure Appl. Math. Quart. (2017), issue in honour of 60th birthday of Simon Donaldson. 37 page

Contact Lie algebras of vector fields on the plane

The paper is devoted to the complete classification of all real Lie algebras of contact vector fields on the first jet space of one-dimensional submanifolds in the plane. This completes Sophus Lie's classification of all possible Lie algebras of contact symmetries for ordinary differential equations. As a main tool we use the abstract theory of filtered and graded Lie algebras. We also describe all differential and integral invariants of new Lie algebras found in the paper and discuss the infinite-dimensional case.Comment: 20 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper1.abs.htm

Effective Temperature in a Colloidal Glass

We study the Brownian motion of particles trapped by optical tweezers inside a colloidal glass (Laponite) during the sol-gel transition. We use two methods based on passive rheology to extract the effective temperature from the fluctuations of the Brownian particles. All of them give a temperature that, within experimental errors, is equal to the heat bath temperature. Several interesting features concerning the statistical properties and the long time correlations of the particles are observed during the transition.Comment: to be published in Philosophical Magazin

Manipulability analysis of a snake robot without lateral constraint for head position control

Two dynamic manipulability criteria of a snake robot with sideways slipping are proposed with the application to head trajectory tracking control in mind. The singular posture, which is crucial in head tracking control, is characterized by the manipulability and examined for families of typical robot shapes. Differences in the singular postures from those of the robot with lateral constraints, which have not been clear in previous studies, are clarified in the analysis. In addition to the examination of local properties using the concept of manipulability, we discuss the effect of isotropic friction as a global property. It is well known that, at least empirically, a snake robot needs anisotropy in friction to move by serpentine locomotion if there are no objects for it to push around. From the point of view of integrability, we show one of the necessary conditions for uncontrollability is satisfied if the friction is isotropic

A Viscoelastic model of phase separation

We show here a general model of phase separation in isotropic condensed matter, namely, a viscoelastic model. We propose that the bulk mechanical relaxation modulus that has so far been ignored in previous theories plays an important role in viscoelastic phase separation in addition to the shear relaxation modulus. In polymer solutions, for example, attractive interactions between polymers under a poor-solvent condition likely cause the transient gellike behavior, which makes both bulk and shear modes active. Although such attractive interactions between molecules of the same component exist universally in the two-phase region of a mixture, the stress arising from attractive interactions is asymmetrically divided between the components only in dynamically asymmetric mixtures such as polymer solutions and colloidal suspensions. Thus, the interaction network between the slower components, which can store the elastic energy against its deformation through bulk and shear moduli, is formed. It is the bulk relaxation modulus associated with this interaction network that is primarily responsible for the appearance of the sponge structure peculiar to viscoelastic phase separation and the phase inversion. We demonstrate that a viscoelastic model of phase separation including this new effect is a general model that can describe all types of isotropic phase separation including solid and fluid models as its special cases without any exception, if there is no coupling with additional order parameter. The physical origin of volume shrinking behavior during viscoelastic phase separation and the universality of the resulting spongelike structure are also discussed.Comment: 14 pages, RevTex, To appear in Phys. Rev

Temperature Dependent Polarized XANES Spectra for Zn-doped LSCO system

The cuprates seem to exhibit statistics, dimensionality and phase transitions in novel ways. The nature of excitations [i.e. quasiparticle or collective], spin-charge separation, stripes [static and dynamics], inhomogeneities, psuedogap, effect of impurity dopings [e.g. Zn, Ni] and any other phenomenon in these materials must be consistently understood. Zn-doped LSCO single crystal were grown by TSFZ technique. Temperature dependent Polarized XANES [near edge local structure] spectra were measured at the BL13-B1 [Photon Factory] in the Flourescence mode from 10 K to 300 K. Since both stripes and nonmagnetic Zn impurities substituted for Cu give rise to inhomogeneous charge and spin distribution it is interesting to understand the interplay of Zn impurities and stripes. To understand these points we have used Zn-doping and some of the results obtained are as follows: The spectra show a strong dependence with respect to the polarization angle, $\theta$, as is evident at any temperature by comparing the spectra where the electric field vector is parallel with ab-plane to the one where it is parallel to the c-axis. By using the XANES [temperature] difference spectra we have determined T* [experimentally we find, T* $\approx$ 160-170 K] for this sample. The XANES difference spectra shows that the changes in XANES features are larger in the ab-plane than the c-axis, this trend is expected since zinc is doped in the ab-plane at the copper site. Our study also complements the results in literature namely that zinc doping does not affect the c-axis transport.Comment: To appear in Physica C [ISS2001 Special Issue], related talk presented at ISS2001 as PC-16, 10 pages revtex and 7 pages of figures (pdf

Strain accommodation beneath structures on Mars

A recent review of tectonic features on Mars shows that most of their subsurface structures can be confidently extended only a few kilometers deep (exceptions are rifts, in which bounding normal faults penetrate the entire brittle lithosphere, with ductile flow at deeper levels). Nevertheless, a variety of estimates of elastic lithosphere thickness and application of accepted failure criteria under likely conditions on Mars suggest a brittle lithosphere that is many tens of kilometers thick. This raises the question of how the strain (extension or shortening) accommodated by grabens and wrinkle ridges within the upper few kilometers is being accommodated at deeper levels in the lithosphere. Herein, the nonrift tectonic features present on Mars are briefly reviewed, along with their likely subsurface structures, and some inferences and implications are presented for behavior of the deeper lithosphere