Toward a general theory of linking invariants

Let N_1, N_2, M be smooth manifolds with dim N_1 + dim N_2 +1 = dim M$ and let phi_i, for i=1,2, be smooth mappings of N_i to M with Im phi_1 and Im phi_2 disjoint. The classical linking number lk(phi_1,phi_2) is defined only when phi_1*[N_1] = phi_2*[N_2] = 0 in H_*(M). The affine linking invariant alk is a generalization of lk to the case where phi_1*[N_1] or phi_2*[N_2] are not zero-homologous. In arXiv:math.GT/0207219 we constructed the first examples of affine linking invariants of nonzero-homologous spheres in the spherical tangent bundle of a manifold, and showed that alk is intimately related to the causality relation of wave fronts on manifolds. In this paper we develop the general theory. The invariant alk appears to be a universal Vassiliev-Goussarov invariant of order < 2. In the case where phi_1*[N_1] and phi_2*[N_2] are 0 in homology it is a splitting of the classical linking number into a collection of independent invariants. To construct alk we introduce a new pairing mu on the bordism groups of spaces of mappings of N_1 and N_2 into M, not necessarily under the restriction dim N_1 + dim N_2 +1 = dim M. For the zero-dimensional bordism groups, mu can be related to the Hatcher-Quinn invariant. In the case N_1=N_2=S^1, it is related to the Chas-Sullivan string homology super Lie bracket, and to the Goldman Lie bracket of free loops on surfaces.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper42.abs.htm

Fluid transport properties under confined conditions

This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.The problem of adequate description of transport processes of fluids in confined conditions is solved using methods of nonequilibrium statistical mechanics. The «fluid–channel wall» system is regarded as a two-phase medium, in which each phase has a particular velocity and temperature. The obtained results show that the transfer equations describing transport processes in confined spaces should contain not only the stress tensor and the heat flux vector, but also the interfacial forces responsible for the transfer of momentum and heat due to the interaction with the wall surfaces. The stress tensor and the heat flux vector fluid can be expressed in terms of the effective viscosity and thermal conductivity. However, the constitutive relations contain additive terms that correspond to the fluid–surface interactions. Thus, not only do the fluid transport coefficients in nanochannels differ from the bulk transport coefficients, but also they are not determined only by the parameters of the fluid

Using mixed data in the inverse scattering problem

Consider the fixed-$\ell$ inverse scattering problem. We show that the zeros of the regular solution of the Schr\"odinger equation, $r_{n}(E)$, which are monotonic functions of the energy, determine a unique potential when the domain of the energy is such that the $r_{n}(E)$ range from zero to infinity. This suggests that the use of the mixed data of phase-shifts $\{\delta(\ell_0,k), k \geq k_0 \} \cup \{\delta(\ell,k_0), \ell \geq \ell_0 \}$, for which the zeros of the regular solution are monotonic in both domains, and range from zero to infinity, offers the possibility of determining the potential in a unique way.Comment: 9 pages, 2 figures. Talk given at the Conference of Inverse Quantum Scattering Theory, Hungary, August 200

Barkhausen Noise in a Relaxor Ferroelectric

Barkhausen noise, including both periodic and aperiodic components, is found in and near the relaxor regime of a familiar relaxor ferroelectric, PbMg$_{1/3}$Nb$_{2/3}$O$_3$, driven by a periodic electric field. The temperature dependences of both the amplitude and spectral form show that the size of the coherent dipole moment changes shrink as the relaxor regime is entered, contrary to expectations based on some simple models.Comment: 4 pages RevTeX4, 5 figures; submitted to Phys Rev Let

A study on the effect of dislocation on the magnetic properties of nickel using magnetic NDE methods

Dislocations affect the magnetic properties of ferromagnetic materials by pinning the domain walls. The primary mechanism is interaction between the stress fields of dislocation and domain walls. Using magnetic nondestructive methods, namely the acoustic Barkhausen noise (AB), magnetic Barkhausen noise (MB), and the hysteresis curves, we have studied these interactions. The three measurements give different types of information. AB provides information about non‐180° type domain wall interaction, MB primarily provides information about 180° domain wall interaction, and the hysteresis curve about both these interactions as well as about rotation of domain walls. The paper presents results obtained on polycrystalline nickel which was first deformed and then annealed at different temperatures in order to achieve different dislocation densities. The results show that AB and hysteresis loss follow the same trend as hardness. MB results, however, change in a more complex fashion which is sensitive to grain recrystallization as well as dislocation structure. Interesting features of these results will be discussed in detail