The Homflypt skein module of a connected sum of 3-manifolds

If M is an oriented 3-manifold, let S(M) denote the Homflypt skein module of M. We show that S(M_1 connect sum M_2) is isomorphic to S(M_1) tensor S(M_2) modulo torsion. In fact, we show that S(M_1 connect sum M_2) is isomorphic to S(M_1) tensot S(M_2) if we are working over a certain localized ring. We show the similar result holds for relative skein modules. If M contains a separating 2-sphere, we give conditions under which certain relative skein modules of M vanish over specified localized rings.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-31.abs.htm

Mutual braiding and the band presentation of braid groups

This paper is concerned with detecting when a closed braid and its axis are 'mutually braided' in the sense of Rudolph. It deals with closed braids which are fibred links, the simplest case being closed braids which present the unknot. The geometric condition for mutual braiding refers to the existence of a close control on the way in which the whole family of fibre surfaces meet the family of discs spanning the braid axis. We show how such a braid can be presented naturally as a word in the `band generators' of the braid group discussed by Birman, Ko and Lee in their recent account of the band presentation of the braid groups. In this context we are able to convert the conditions for mutual braiding into the existence of a suitable sequence of band relations and other moves on the braid word, and derive a combinatorial method for deciding whether a braid is mutually braided.Comment: 13 pages, 10 figures. To appear in the proceedings of 'Knots in Hellas 98

On the accuracy of surface spline interpolation on the unit sphere

This paper considers a novel modification to the surface splines that have previously been used on the unit sphere. The surface splines considered are a natural analogue of surface splines in IRd and possess a unique Fourier expansion in terms of an orthonormal basis of spherical harmonics. Knowing the decay of the associated Fourier coefficients is important because they enable error estimates for spherical interpolation. In this paper we explicitly compute the Fourier coefficients of the surface splines and employ a recent theoretical result [8] to provide a useful error bound. We illuminate our theoretical findings by performing numerical experiments on the sphere and also on the hemisphere

Believing in Others

Suppose some person 'A' sets out to accomplish a difficult, long-term goal such as writing a passable Ph.D. thesis. What should you believe about whether A will succeed? The default answer is that you should believe whatever the total accessible evidence concerning A's abilities, circumstances, capacity for self-discipline, and so forth supports. But could it be that what you should believe depends in part on the relationship you have with A? We argue that it does, in the case where A is yourself. The capacity for "grit" involves a kind of epistemic resilience in the face of evidence suggesting that one might fail, and this makes it rational to respond to the relevant evidence differently when you are the agent in question. We then explore whether similar arguments extend to the case of "believing in" our significant others -- our friends, lovers, family members, colleagues, patients, and students

Maximizing Academic Success for Foster Care Students: A Trauma-Informed Approach

Children in foster care have experienced significant trauma due to the loss of primary attachment figures and the circumstances associated with that loss. Children who have suffered trauma generally present with cognitive, social, physical, and emotional vulnerabilities. These vulnerabilities are often expressed in the P–12 academic setting through difficulties with behavioral and emotional self-regulation, academic functioning, and physical ailments and illness related to chronic stress-induced compromised immune systems. This results in academic failure for half of all children in care. Training in how to respond to children who have suffered trauma is essential to ensure that children are comfortable and feel secure in the classroom so that they can access their education. To that end, a framework to support children in P–12 settings who are particularly vulnerable to academic failure due to trauma is presented

Non-axisymmetric oscillations of stratified coronal magnetic loops with elliptical cross-sections

We study non-axisymmetric oscillations of a straight magnetic tube with an elliptic cross-section and density varying along the tube. The governing equations for kink and fluting modes in the thin tube approximation are derived. We found that there are two kink modes, polarised along the large and small axes of the elliptic cross-section. We have shown that the ratio of frequencies of the first overtone and fundamental harmonic is the same for both kink modes and independent of the ratio of the ellipse axes. On the basis of this result we concluded that the estimates of the atmospheric scale height obtained using simultaneous observations of the fundamental harmonic and first overtone of the coronal loop kink oscillations are independent of the ellipticity of the loop cross-section

A constitutive model for simple shear of dense frictional suspensions

Discrete particle simulations are used to study the shear rheology of dense, stabilized, frictional particulate suspensions in a viscous liquid, toward development of a constitutive model for steady shear flows at arbitrary stress. These suspensions undergo increasingly strong continuous shear thickening (CST) as solid volume fraction $\phi$ increases above a critical volume fraction, and discontinuous shear thickening (DST) is observed for a range of $\phi$. When studied at controlled stress, the DST behavior is associated with non-monotonic flow curves of the steady-state stress as a function of shear rate. Recent studies have related shear thickening to a transition between mostly lubricated to predominantly frictional contacts with the increase in stress. In this study, the behavior is simulated over a wide range of the dimensionless parameters $(\phi,\tilde{\sigma}$, and $\mu)$, with $\tilde{\sigma} = \sigma/\sigma_0$ the dimensionless shear stress and $\mu$ the coefficient of interparticle friction: the dimensional stress is $\sigma$, and $\sigma_0 \propto F_0/ a^2$, where $F_0$ is the magnitude of repulsive force at contact and $a$ is the particle radius. The data have been used to populate the model of the lubricated-to-frictional rheology of Wyart and Cates [Phys. Rev. Lett.{\bf 112}, 098302 (2014)], which is based on the concept of two viscosity divergences or \textquotedblleft jamming\textquotedblright\ points at volume fraction $\phi_{\rm J}^0 = \phi_{\rm rcp}$ (random close packing) for the low-stress lubricated state, and at $\phi_{\rm J} (\mu) < \phi_{\rm J}^0$ for any nonzero $\mu$ in the frictional state; a generalization provides the normal stress response as well as the shear stress. A flow state map of this material is developed based on the simulation results.Comment: 12 pages, 10 figure