Some examples of exponentially harmonic maps

The aim of this paper is to study some examples of exponentially harmonic maps. We study such maps firstly on flat euclidean and Minkowski spaces and secondly on Friedmann-Lema\^ itre universes. We also consider some new models of exponentially harmonic maps which are coupled with gravity which happen to be based on a generalization of the lagrangian for bosonic strings coupled with dilatonic field.Comment: 16 pages, 5 figure

Classification of unit-vector fields in convex polyhedra with tangent boundary conditions

A unit-vector field n on a convex three-dimensional polyhedron P is tangent if, on the faces of P, n is tangent to the faces. A homotopy classification of tangent unit-vector fields continuous away from the vertices of P is given. The classification is determined by certain invariants, namely edge orientations (values of n on the edges of P), kink numbers (relative winding numbers of n between edges on the faces of P), and wrapping numbers (relative degrees of n on surfaces separating the vertices of P), which are subject to certain sum rules. Another invariant, the trapped area, is expressed in terms of these. One motivation for this study comes from liquid crystal physics; tangent unit-vector fields describe the orientation of liquid crystals in certain polyhedral cells.Comment: 21 pages, 2 figure

A Mass Bound for Spherically Symmetric Black Hole Spacetimes

Requiring that the matter fields are subject to the dominant energy condition, we establish the lower bound $(4\pi)^{-1} \kappa {\cal A}$ for the total mass $M$ of a static, spherically symmetric black hole spacetime. (${\cal A}$ and $\kappa$ denote the area and the surface gravity of the horizon, respectively.) Together with the fact that the Komar integral provides a simple relation between $M - (4\pi)^{-1} \kappa A$ and the strong energy condition, this enables us to prove that the Schwarzschild metric represents the only static, spherically symmetric black hole solution of a selfgravitating matter model satisfying the dominant, but violating the strong energy condition for the timelike Killing field $K$ at every point, that is, $R(K,K) \leq 0$. Applying this result to scalar fields, we recover the fact that the only black hole configuration of the spherically symmetric Einstein-Higgs model with arbitrary non-negative potential is the Schwarzschild spacetime with constant Higgs field. In the presence of electromagnetic fields, we also derive a stronger bound for the total mass, involving the electromagnetic potentials and charges. Again, this estimate provides a simple tool to prove a ``no-hair'' theorem for matter fields violating the strong energy condition.Comment: 16 pages, LATEX, no figure

Energies of S^2-valued harmonic maps on polyhedra with tangent boundary conditions

A unit-vector field n:P \to S^2 on a convex polyhedron P \subset R^3 satisfies tangent boundary conditions if, on each face of P, n takes values tangent to that face. Tangent unit-vector fields are necessarily discontinuous at the vertices of P. We consider fields which are continuous elsewhere. We derive a lower bound E^-_P(h) for the infimum Dirichlet energy E^inf_P(h) for such tangent unit-vector fields of arbitrary homotopy type h. E^-_P(h) is expressed as a weighted sum of minimal connections, one for each sector of a natural partition of S^2 induced by P. For P a rectangular prism, we derive an upper bound for E^inf_P(h) whose ratio to the lower bound may be bounded independently of h. The problem is motivated by models of nematic liquid crystals in polyhedral geometries. Our results improve and extend several previous results.Comment: 42 pages, 2 figure

On the Evolution Equation for Magnetic Geodesics

In this paper we prove the existence of long time solutions for the parabolic equation for closed magnetic geodesics.Comment: In this paper we prove the existence of long time solutions for the parabolic equation for closed magnetic geodesic

A class of extremising sphere-valued maps with inherent maximal tori symmetries in SO(n)

In this paper we consider an energy functional depending on the norm of the gradient and seek to extremise it over an admissible class of Sobolev maps defined on an annulus and taking values on the unit sphere whilst satisfying suitable boundary conditions. We establish the existence of an infinite family of solutions with certain symmetries to the associated nonlinear Euler-Lagrange system in even dimensions and discuss the stability of such extremisers by way of examining the positivity of the second variation of the energy at these solutions

NR4A Gene Expression Is Dynamically Regulated in the Ventral Tegmental Area Dopamine Neurons and Is Related to Expression of Dopamine Neurotransmission Genes

The NR4A transcription factors NR4A1, NR4A2, and NR4A3 (also known as Nur77, Nurr1, and Nor1, respectively) share similar DNA-binding properties and have been implicated in regulation of dopamine neurotransmission genes. Our current hypothesis is that NR4A gene expression is regulated by dopamine neuron activity and that induction of NR4A genes will increase expression of dopamine neurotransmission genes. Eticlopride and γ-butyrolactone (GBL) were used in wild-type (+/+) and Nurr1-null heterozygous (+/−) mice to determine the mechanism(s) regulating Nur77 and Nurr1 expression. Laser capture microdissection and real-time PCR was used to measure Nurr1 and Nur77 mRNA levels in the ventral tegmental area (VTA). Nur77 expression was significantly elevated 1 h after both GBL (twofold) and eticlopride (fourfold). In contrast, GBL significantly decreased Nurr1 expression in both genotypes, while eticlopride significantly increased Nurr1 expression only in the +/+ mice. In a separate group of mice, haloperidol injection significantly elevated Nur77 and Nor1, but not Nurr1 mRNA in the VTA within 1 h and significantly increased tyrosine hydroxylase (TH) and dopamine transporter (DAT) mRNA expression by 4 h. These data demonstrate that the NR4A genes are dynamically regulated in dopamine neurons with maintenance of Nurr1 expression requiring dopamine neuron activity while both attenuation of dopamine autoreceptors activation and dopamine neuronal activity combining to induce Nur77 expression. Additionally, these data suggest that induction of NR4A genes could regulate TH and DAT expression and ultimately regulate dopamine neurotransmission

Calibrated Sub-Bundles in Non-Compact Manifolds of Special Holonomy

This paper is a continuation of math.DG/0408005. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of S^n by looking at the conormal bundle of appropriate submanifolds of S^n. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey-Lawson for submanifolds in R^n in their pioneering paper. We also construct calibrated submanifolds in complete metrics with special holonomy G_2 and Spin(7) discovered by Bryant and Salamon on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds in a compact manifold with special holonomy.Comment: 20 pages; for Revised Version: Minor cosmetic changes, some paragraphs rewritten for improved clarit

Is diagnosis enough to guide interventions in mental health? Using case formulation in clinical practice

While diagnosis has traditionally been viewed as an essential concept in medicine, particularly when selecting treatments, we suggest that the use of diagnosis alone may be limited, particularly within mental health. The concept of clinical case formulation advocates for collaboratively working with patients to identify idiosyncratic aspects of their presentation and select interventions on this basis. Identifying individualized contributing factors, and how these could influence the person\u27s presentation, in addition to attending to personal strengths, may allow the clinician a deeper understanding of a patient, result in a more personalized treatment approach, and potentially provide a better clinical outcome.<br /

\bbbc P^2 and \bbbc P^{1} Sigma Models in Supergravity: Bianchi type IX Instantons and Cosmologies

We find instanton/cosmological solutions with biaxial Bianchi-IX symmetry, involving non-trivial spatial dependence of the \bbbc P^{1}- and \bbbc P^{2}-sigma-models coupled to gravity. Such manifolds arise in N=1, $d=4$ supergravity with supermatter actions and hence the solutions can be embedded in supergravity. There is a natural way in which the standard coordinates of these manifolds can be mapped into the four-dimensional physical space. Due to its special symmetry, we start with \bbbc P^{2} with its corresponding scalar Ansatz; this further requires the spacetime to be $SU(2) \times U(1)$-invariant. The problem then reduces to a set of ordinary differential equations whose analytical properties and solutions are discussed. Among the solutions there is a surprising, special-family of exact solutions which owe their existence to the non-trivial topology of \bbbc P^{2} and are in 1-1 correspondence with matter-free Bianchi-IX metrics. These solutions can also be found by coupling \bbbc P^{1} to gravity. The regularity of these Euclidean solutions is discussed -- the only possibility is bolt-type regularity. The Lorentzian solutions with similar scalar Ansatz are all obtainable from the Euclidean solutions by Wick rotation