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

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

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

Clinical and Experimental Applications of NIR-LED Photobiomodulation

This review presents current research on the use of far-red to near-infrared (NIR) light treatment in various in vitro and in vivo models. Low-intensity light therapy, commonly referred to as “photobiomodulation,” uses light in the far-red to near-infrared region of the spectrum (630–1000 nm) and modulates numerous cellular functions. Positive effects of NIR–light-emitting diode (LED) light treatment include acceleration of wound healing, improved recovery from ischemic injury of the heart, and attenuated degeneration of injured optic nerves by improving mitochondrial energy metabolism and production. Various in vitro and in vivo models of mitochondrial dysfunction were treated with a variety of wavelengths of NIR-LED light. These studies were performed to determine the effect of NIR-LED light treatment on physiologic and pathologic processes. NIRLED light treatment stimulates the photoacceptor cytochrome c oxidase, resulting in increased energy metabolism and production. NIR-LED light treatment accelerates wound healing in ischemic rat and murine diabetic wound healing models, attenuates the retinotoxic effects of methanol-derived formic acid in rat models, and attenuates the developmental toxicity of dioxin in chicken embryos. Furthermore, NIR-LED light treatment prevents the development of oral mucositis in pediatric bone marrow transplant patients. The experimental results demonstrate that NIR-LED light treatment stimulates mitochondrial oxidative metabolism in vitro, and accelerates cell and tissue repair in vivo. NIR-LED light represents a novel, noninvasive, therapeutic intervention for the treatment of numerous diseases linked to mitochondrial dysfunction

Examining the Evidence Base for Forensic Case Formulation: An Integrative Review of Recent Research

In the past decade, forensic case formulation (FCF) has become a key activity in many forensic services. However, the evidence base for FCF remains limited. This integrative review aimed to identify and evaluate all FCF research conducted since the lack of understanding within this field was highlighted by several academics in 2011. A rigorous literature search led to the identification of 14 studies fitting the inclusion criteria. Studies were critically evaluated and synthesised to create a summary of the recent research, to identify remaining gaps in our understanding, and to create an agenda for future research

Mean Curvature Flow of Spacelike Graphs

We prove the mean curvature flow of a spacelike graph in $(\Sigma_1\times \Sigma_2, g_1-g_2)$ of a map $f:\Sigma_1\to \Sigma_2$ from a closed Riemannian manifold $(\Sigma_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(\Sigma_2,g_2)$ with bounded curvature tensor and derivatives, and with sectional curvatures satisfying $K_2\leq K_1$, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption $K_2\leq K_1$, that if $K_1>0$, or if $Ricci_1>0$ and $K_2\leq -c$, $c>0$ constant, any map $f:\Sigma_1\to \Sigma_2$ is trivially homotopic provided $f^*g_2<\rho g_1$ where $\rho=\min_{\Sigma_1}K_1/\sup_{\Sigma_2}K_2^+\geq 0$, in case $K_1>0$, and $\rho=+\infty$ in case $K_2\leq 0$. This largely extends some known results for $K_i$ constant and $\Sigma_2$ compact, obtained using the Riemannian structure of $\Sigma_1\times \Sigma_2$, and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo-Riemannian setting then in the Riemannian one.Comment: version 5: Math.Z (online first 30 July 2010). version 4: 30 pages: we replace the condition $K_1\geq 0$ by the the weaker one $Ricci_1\geq 0$. The proofs are essentially the same. We change the title to a shorter one. We add an applicatio

The classification of static vacuum space-times containing an asymptotically flat spacelike hypersurface with compact interior

We prove non-existence of static, vacuum, appropriately regular, asymptotically flat black hole space-times with degenerate (not necessarily connected) components of the event horizon. This finishes the classification of static, vacuum, asymptotically flat domains of outer communication in an appropriate class of space-times, showing that the domains of outer communication of the Schwarzschild black holes exhaust the space of appropriately regular black hole exteriors.Comment: This version includes an addendum with a corrected proof of non-existence of zeros of the Killing vector at degenerate horizons. A problem with yet another Lemma is pointed out; this problem does not arise if one assumes analyticity of the metric. An alternative solution, that does not require analyticity, has been given in arXiv:1004.0513 [gr-qc] under appropriate global condition

Dirac-harmonic maps from index theory

We prove existence results for Dirac-harmonic maps using index theoretical tools. They are mainly interesting if the source manifold has dimension 1 or 2 modulo 8. Our solutions are uncoupled in the sense that the underlying map between the source and target manifolds is a harmonic map.Comment: 26 pages, no figur

Optimal topological simplification of discrete functions on surfaces

We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2{\delta}, constructively proving the tightness of a lower bound on the number of critical points given by the stability theorem of persistent homology in dimension two for any input function. We also show that an optimal solution can be computed in linear time after persistence pairs have been computed.Comment: 27 pages, 8 figure