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Register a new userLower bound for energies of harmonic tangent unit-vector fields on convex polyhedra
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We derive a lower bound for energies of harmonic maps of convex polyhedra in $ R^3 $ to the unit sphere $S^2,$ with tangent boundary conditions on the faces. We also establish that $C^infty$ maps, satisfying tangent boundary conditions, are dense with respect to the Sobolev norm, in the space of continuous tangent maps of finite energy.
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Let O be a closed geodesic polygon in S^2. Maps from O into S^2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S^2, we compute the infimum Dirichlet energy, Ecal(H), for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H. The expression for Ecal(H) involves a topological invariant - the spelling length - associated with the (nonabelian) fundamental group of the n-times punctured two-sphere, pi_1(S^2 - {s_1,..., s_n},*). The lower bound for Ecal(H) is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representatives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes (classes containing wholly conformal and anticonformal representatives respectively), the expression for Ecal(H) reduces to a previous result involving the degrees of a set of regular values s_1,..., s_n in the target S^2 space. These degrees may be viewed as invariants associated with the abelianization of pi_1(S^2 - {s_1,..., s_n}, *). For nonconformal classes, however, Ecal(H) may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees. This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy (one-constant Oseen-Frank energy) of reflection-symmetric tangent unit-vector fields in a rectangular prism.
We work on a 4-manifold equipped with Lorentzian metric $g$ and consider a volume-preserving diffeomorphism which is the unknown quantity of our mathematical model. The diffeomorphism defines a second Lorentzian metric $h$, the pullback of $g$. Motivated by elasticity theory, we introduce a Lagrangian expressed algebraically (without differentiations) via our pair of metrics. Analysis of the resulting nonlinear field equations produces three main results. Firstly, we show that for Ricci-flat manifolds our linearised field equations are Maxwells equations in the Lorenz gauge with exact current. Secondly, for Minkowski space we construct explicit massless solutions of our nonlinear field equations; these come in two distinct types, right-handed and left-handed. Thirdly, for Minkowski space we construct explicit massive solutions of our nonlinear field equations; these contain a positive parameter which has the geometric meaning of quantum mechanical mass and a real parameter which may be interpreted as electric charge. In constructing explicit solutions of nonlinear field equations we resort to group-theoretic ideas: we identify special 4-dimensional subgroups of the Poincare group and seek diffeomorphisms compatible with their action, in a suitable sense.
We present a general construction of divergence-free knotted vector fields from complex scalar fields, whose closed field lines encode many kinds of knots and links, including torus knots, their cables, the figure-8 knot and its generalizations. As finite-energy physical fields they represent initial states for fields such as the magnetic field in a plasma, or the vorticity field in a fluid. We give a systematic procedure for calculating the vector potential, starting from complex scalar functions with knotted zero filaments, thus enabling an explicit computation of the helicity of these knotted fields. The construction can be used to generate isolated knotted flux tubes, filled by knots encoded in the lines of the vector field. Lastly we give examples of manifestly knotted vector fields with vanishing helicity. Our results provide building blocks for analytical models and simulations alike.
According to a well-know theorem by Sturm, a vibrating string is divided into exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed that one half of Sturms theorem for the strings applies to the theory of membranes: N-th eigenfunction cannot have more than N domains. He also gave an example of a eigenfunction high in the spectrum with a minimal number of nodal domains, thus excluding the existence of a non-trivial lower bound. An analogue of Sturms result for discretizations of the interval was discussed by Gantmacher and Krein. The discretization of an interval is a graph of a simple form, a chain-graph. But what can be said about more complicated graphs? It has been known since the early 90s that the nodal count for a generic eigenfunction of the Schrodinger operator on quantum trees (where each edge is identified with an interval of the real line and some matching conditions are enforced on the vertices) is exact too: zeros of the N-th eigenfunction divide the tree into exactly N subtrees. We discuss two extensions of this result in two directions. One deals with the same continuous Schrodinger operator but on general graphs (i.e. non-trees) and another deals with discrete Schrodinger operator on combinatorial graphs (both trees and non-trees). The result that we derive applies to both types of graphs: the number of nodal domains of the N-th eigenfunction is bounded below by N-L, where L is the number of links that distinguish the graph from a tree (defined as the dimension of the cycle space or the rank of the fundamental group of the graph). We also show that if it the genericity condition is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction.
We investigate the impact of operators of higher canonical dimension on the lower Higgs mass consistency bound by means of generalized Higgs-Yukawa interactions. Analogously to higher-order operators in the bare Higgs potential in an effective field theory approach, the inclusion of higher-order Yukawa interactions, e.g., $phi^3bar{psi}psi$, leads to a diminishing of the lower Higgs mass bound and thus to a shift of the scale of new physics towards larger scales by a few orders of magnitude without introducing a metastability in the effective Higgs potential. We observe that similar renormalization group mechanisms near the weak-coupling fixed point are at work in both generalizations of the microscopic action. Thus, a combination of higher-dimensional operators with generalized Higgs as well as Yukawa interactions does not lead to an additive shift of the lower mass bound, but relaxes the consistency bounds found recently only slightly. On the method side, we clarify the convergence properties of different projection and expansion schemes for the Yukawa potential used in the functional renormalization group literature so far.
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